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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/38722-pdf.pdf b/38722-pdf.pdf Binary files differnew file mode 100644 index 0000000..c083292 --- /dev/null +++ b/38722-pdf.pdf diff --git a/38722-pdf.zip b/38722-pdf.zip Binary files differnew file mode 100644 index 0000000..2bb29af --- /dev/null +++ b/38722-pdf.zip diff --git a/38722-t.zip b/38722-t.zip Binary files differnew file mode 100644 index 0000000..48b8d40 --- /dev/null +++ b/38722-t.zip diff --git a/38722-t/38722-t.tex b/38722-t/38722-t.tex new file mode 100644 index 0000000..0149a5e --- /dev/null +++ b/38722-t/38722-t.tex @@ -0,0 +1,15606 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% The Project Gutenberg EBook of The Tides and Kindred Phenomena in the % +% Solar System, by Sir George Howard Darwin % +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.net % +% % +% % +% Title: The Tides and Kindred Phenomena in the Solar System % +% The Substance of Lectures Delivered in 1897 at the Lowell % +% Institute, Boston, Massachusetts % +% % +% Author: Sir George Howard Darwin % +% % +% Release Date: January 31, 2012 [EBook #38722] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK TIDES AND KINDRED PHENOMENA *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{38722} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% %% +%% multicol: Multicolumn environment for index. Required. %% +%% makeidx: Indexing. Required. %% +%% %% +%% footmisc: Extended footnote capabilities. Required. %% +%% %% +%% indentfirst: Indent first word of each sectional unit. Required. %% +%% icomma: Make the comma a decimal separator in math. Required. %% +%% %% +%% calc: Length calculations. Required. %% +%% %% +%% array: Enhanced arrays. Required. %% +%% %% +%% yfonts: Gothic font on title page. Optional. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% caption: Caption customization. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% Changes are noted in this file in multiple ways. %% +%% 1. \DPnote{} for in-line `placeholder' notes. %% +%% 2. \DPtypo{}{} for typographical corrections, showing original %% +%% and replacement text side-by-side. %% +%% 3. \DPchg{}{} for stylistic changes made for consistency. %% +%% 4. [** TN: Note]s for lengthier or stylistic comments. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behavior may be controlled by boolean flags. %% +%% %% +%% ForPrinting (false by default): %% +%% Compile a print-optimized PDF file. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.net + + +Title: The Tides and Kindred Phenomena in the Solar System + The Substance of Lectures Delivered in 1897 at the Lowell + Institute, Boston, Massachusetts + +Author: Sir George Howard Darwin + +Release Date: January 31, 2012 [EBook #38722] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK TIDES AND KINDRED PHENOMENA *** +\end{PGtext} +\end{minipage} +\end{center} +\clearpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Bryan Ness and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/Canadian Libraries) +\end{PGtext} +\end{minipage} +\vfill +\TranscribersNote{\TransNoteText} +\end{center} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\cleardoublepage +\pagenumbering{roman} +\PageSep{iii} +\begin{center} +\textbf{\Huge THE TIDES} +\vfil +\textbf{\large AND KINDRED PHENOMENA IN THE \\[4pt] +SOLAR SYSTEM} +\vfil +{\footnotesize THE SUBSTANCE OF LECTURES DELIVERED \\[4pt] +IN 1897 AT THE LOWELL INSTITUTE, \\[4pt] +BOSTON, MASSACHUSETTS} +\vfil +BY +\vfil +\textbf{\Large GEORGE HOWARD DARWIN} +\vfil +\scriptsize PLUMIAN PROFESSOR AND FELLOW OF TRINITY COLLEGE IN THE \\[4pt] +UNIVERSITY OF CAMBRIDGE +\vfil\vfil +\Graphic[png]{1.25in}{riverside} +\vfil\vfil\vfil +\footnotesize BOSTON AND NEW YORK \\ +\normalsize HOUGHTON, MIFFLIN AND COMPANY \\ +\textgoth{The Riverside Press, Cambridge} \\ +1899 +\end{center} +\PageSep{iv} +\clearpage +\null\vfill +\begin{center} +\footnotesize +COPYRIGHT, 1898, BY GEORGE HOWARD DARWIN \\ +ALL RIGHTS RESERVED +\end{center} +\vfill +\PageSep{v} + + +\Preface + +\First{In} 1897 I delivered a course of lectures on +the Tides at the Lowell Institute in Boston, +Massachusetts, and this book contains the substance +of what I then said. The personal form +of address appropriate to a lecture is, I think, +apt to be rather tiresome in a book, and I have +therefore taken pains to eliminate all traces of +the lecture from what I have written. + +A mathematical argument is, after all, only +organized common sense, and it is well that men +of science should not always expound their work +to the few behind a veil of technical language, +but should from time to time explain to a larger +public the reasoning which lies behind their +mathematical notation. To a man unversed in +popular exposition it needs a great effort to shell +away the apparatus of investigation and the +technical mode of speech from the thing behind +it, and I owe a debt of gratitude to Mr.~Lowell, +trustee of the Institute, for having afforded me +the occasion for making that effort. +\PageSep{vi} + +It is not unlikely that the first remark of +many who see my title will be that so small a +subject as the Tides cannot demand a whole volume; +but, in fact, the subject branches out in +so many directions that the difficulty has been to +attain to the requisite compression of my matter. +Many popular works on astronomy devote a few +pages to the Tides, but, as far as I know, none +of these books contain explanations of the practical +methods of observing and predicting the +Tides, or give any details as to the degree of +success attained by tidal predictions. If these +matters are of interest, I invite my readers not +to confine their reading to this preface. The +later chapters of this book are devoted to the +consideration of several branches of speculative +Astronomy, with which the theory of the Tides +has an intimate relationship. The problems involved +in the origin and history of the solar +and of other celestial systems have little bearing +upon our life on the earth, yet these questions +can hardly fail to be of interest to all those +whose minds are in any degree permeated by +the scientific spirit. + +I think that there are many who would like to +understand the Tides, and will make the attempt +to do so provided the exposition be sufficiently +\PageSep{vii} +simple and clear; it is to such readers I address +this volume. It is for them to say how far I +have succeeded in rendering these intricate subjects +interesting and intelligible, but if I have +failed it has not been for lack of pains. + +The figures and diagrams have, for the most +part, been made by Mr.~Edwin Wilson of Cambridge, +but I have to acknowledge the courtesy +of the proprietors of \Title{Harper's}, the \Title{Century}, +and the \Title{Atlantic Monthly} magazines, in supplying +me with some important illustrations. + +A considerable portion of \Ref{Chapter}{III}.\ on the +``Bore'' is to appear as an article in the \Title{Century +Magazine} for October, 1898, and the reproductions +of Captain Moore's photographs of the +``Bore'' in the Tsien-Tang-Kiang have been +prepared for that article. The \Title{Century} has also +kindly furnished the block of Dr.~Isaac Roberts's +remarkable photograph of the great nebula in +the constellation of Andromeda; it originally +appeared in an article on Meteorites in the number +for October,~1890. The greater portion of +the text and the whole of the illustrations of +\Ref{Chapter}{XX}.\ were originally published in \Title{Harper's +Magazine} for June,~1889. Lastly, portions +of Chapters \Ref{}{XV}.~and~\Ref{}{XVI}.\ appeared in +the \Title{Atlantic Monthly} for April, 1898, published +\PageSep{viii} +by Messrs.\ Houghton, Mifflin~\&~Co., who also +make themselves responsible for the publication +of the American edition of this book. + +In conclusion, I wish to take this opportunity +of thanking my American audience for the cordiality +of their reception, and my many friends +across the Atlantic for their abundant hospitality +and kindness. + +\Signature{G. H. DARWIN.} +{\textsc{Cambridge}, \textit{August}, 1898.} +\PageSep{ix} + + +\Contents + +\ToCChap{I} +{TIDES AND METHODS OF OBSERVATION} + +\ToCSect{Definition of tide}{\PgNos{1}{3}} +\ToCSect{Oceanic tides}{\PgNos[,]{4}{5}} +\ToCSect{Methods of observation}{\PgNo{6}} +\ToCSect{Tide-gauge}{\PgNos{7}{12}} +\ToCSect{Tide-curve}{\PgNo{12}} +\ToCSect{Site for tide-gauge}{\PgNo{13}} +\ToCSect{Irregularities in tide-curve}{\PgNos[,]{14}{15}} +\ToCSect{Authorities}{\PgNo{16}} + + +\ToCChap{II} +{SEICHES IN LAKES} + +\ToCSect{Meaning of seiche}{\PgNo{17}} +\ToCSect{Uses of scientific apparatus}{\PgNo{18}} +\ToCSect{Forel's plemyrameter}{\PgNos[,]{19}{20}} +\ToCSect{Records of the level of the lake}{\PgNo{20}} +\ToCSect{Interpretation of record}{\PgNos{21}{23}} +\ToCSect{Limnimeter}{\PgNo{24}} +\ToCSect{Mode of oscillation in seiches}{\PgNos{25}{28}} +\ToCSect{Wave motion in deep and in shallow water}{\PgNos{29}{32}} +\ToCSect{Composition of waves}{\PgNos{32}{36}} +\ToCSect{Periods of seiches}{\PgNos[,]{37}{38}} +\ToCSect{Causes of seiches}{\PgNos[,]{39}{40}} +\ToCSect{Vibrations due to wind and to steamers}{\PgNos{41}{47}} +\ToCSect{Aerial waves and their action on lakes and on the sea}{\PgNos{48}{53}} +\ToCSect{Authorities}{\PgNos[,]{53}{54}} +\PageSep{x} + + +\ToCChap{III} +{TIDES IN RIVERS---TIDE MILLS} + +\ToCSect{Definition of ebb and flow}{\PgNo{56}} +\ToCSect{Tidal currents in rivers}{\PgNos{56}{58}} +\ToCSect{Progressive change of wave in shallow water}{\PgNos[,]{58}{59}} +\ToCSect{The bore}{\PgNo{59}} +\ToCSect{Captain Moore's survey of the Tsien-Tang-Kiang}{\PgNos{60}{64}} +\ToCSect{Diagram of water-levels during the flow}{\PgNos[,]{64}{65}} +\ToCSect{Chinese superstition}{\PgNos{68}{71}} +\ToCSect{Pictures of the bore}{\PgNo{69}} +\ToCSect{Other cases of bores}{\PgNo{71}} +\ToCSect{Causes of the bore}{\PgNo{72}} +\ToCSect{Tidal energy}{\PgNos[,]{73}{74}} +\ToCSect{Tide mills}{\PgNos[,]{74}{75}} +\ToCSect{Authorities}{\PgNo{75}} + + +\ToCChap{IV} +{HISTORICAL SKETCH} + +\ToCSect{Theories of the Chinese}{\PgNos[,]{76}{77}} +\ToCSect{Theories of the Arabs}{\PgNos{77}{79}} +\ToCSect{Theories of the Norsemen}{\PgNos[,]{79}{80}} +\ToCSect{Writings of Posidonius and Strabo}{\PgNos{80}{84}} +\ToCSect{Seleucus the Babylonian on the diurnal inequality}{\PgNos[,]{84}{85}} +\ToCSect{Galileo and Kepler}{\PgNo{85}} +\ToCSect{Newton and his successors}{\PgNos{86}{88}} +\ToCSect{Empirical method of tidal prediction}{\PgNos{88}{90}} +\ToCSect{Authorities}{\PgNo{90}} + + +\ToCChap{V} +{TIDE-GENERATING FORCE} + +\ToCSect{Inertia and centrifugal force}{\PgNos{91}{93}} +\ToCSect{Orbital motion of earth and moon}{\PgNos{93}{95}} +\ToCSect{Tide-generating force}{\PgNos{96}{100}} +\ToCSect{Law of its dependence on the moon's distance}{\PgNos{101}{103}} +\ToCSect{Earth's rotation}{\PgNos[,]{103}{104}} +\PageSep{xi} +\ToCSect{Second explanation of tide-generating force}{\PgNos[,]{104}{105}} +\ToCSect{Horizontal tide-generating force}{\PgNos[,]{105}{106}} +\ToCSect{Successive changes of force in the course of a day}{\PgNos[,]{107}{108}} +\ToCSect{Authorities}{\PgNo{108}} + + +\ToCChap{VI} +{DEFLECTION OF THE VERTICAL} + +\ToCSect{Deflection of a pendulum by horizontal tidal force}{\PgNos{109}{111}} +\ToCSect{Path pursued by a pendulum under tidal force}{\PgNos{111}{113}} +\ToCSect{Object of measuring the deflection of a pendulum}{\PgNos{113}{115}} +\ToCSect{Attempt to measure deflection by bifilar pendulum}{\PgNos{115}{125}} +\ToCSect{Microsisms}{\PgNos{125}{127}} +\ToCSect{The microphone as a seismological instrument}{\PgNos{127}{130}} +\ToCSect{Paschwitz's work with the horizontal pendulum}{\PgNos[,]{131}{132}} +\ToCSect{Supposed measurement of the lunar deflection of gravity}{\PgNo{132}} +\ToCSect{Authorities}{\PgNo{133}} + + +\ToCChap{VII} +{THE ELASTIC DISTORTION OF THE EARTH'S SURFACE BY +VARYING LOADS} + +\ToCSect{Distortion of an elastic surface by superincumbent load}{\PgNos{134}{137}} +\ToCSect{Application of the theory to the earth}{\PgNos[,]{137}{138}} +\ToCSect{Effects of tidal load}{\PgNos{138}{140}} +\ToCSect{Probable deflections at various distances from the coast}{\PgNos{140}{143}} +\ToCSect{Deflections observed by M.~d'Abbadie and by Dr.~Paschwitz}{\PgNos{143}{145}} +\ToCSect{Effects of atmospheric pressure on the earth's surface}{\PgNos{145}{147}} +\ToCSect{Authorities}{\PgNo{148}} + + +\ToCChap{VIII} +{EQUILIBRIUM THEORY OF TIDES} + +\ToCSect{Explanation of the figure of equilibrium}{\PgNos{149}{151}} +\ToCSect{Map of equilibrium tide}{\PgNos{151}{153}} +\ToCSect{Tides according to the equilibrium theory}{\PgNos{153}{156}} +\ToCSect{Solar tidal force compared with lunar}{\PgNos{156}{158}} +\PageSep{xii} +\ToCSect{Composition of lunar and solar tides}{\PgNos[,]{158}{159}} +\ToCSect{Points of disagreement between theory and fact}{\PgNos{159}{162}} +\ToCSect{Authorities}{\PgNo{162}} + + +\ToCChap{IX} +{DYNAMICAL THEORY OF THE TIDE WAVE} + +\ToCSect{Free and forced waves in an equatorial canal}{\PgNos{163}{165}} +\ToCSect{Critical depth of canal}{\PgNos{165}{167}} +\ToCSect{General principle as to free and forced oscillations}{\PgNos{167}{174}} +\ToCSect{Inverted and direct oscillation}{\PgNos[,]{172}{173}} +\ToCSect{Canal in latitude~$60°$}{\PgNos[,]{174}{175}} +\ToCSect{Tides where the planet is partitioned into canals}{\PgNo{175}} +\ToCSect{Removal of partitions; vortical motion of the water}{\PgNos[,]{176}{177}} +\ToCSect{Critical latitude where the rise and fall vanish}{\PgNos[,]{177}{178}} +\ToCSect{Diurnal inequality}{\PgNos{178}{180}} +\ToCSect{Authorities}{\PgNo{181}} + + +\ToCChap{X} +{TIDES IN LAKES---COTIDAL CHART} + +\ToCSect{The tide in a lake}{\PgNos{182}{185}} +\ToCSect{The Mediterranean Sea}{\PgNos[,]{185}{186}} +\ToCSect{Derived tide of the Atlantic}{\PgNos{186}{188}} +\ToCSect{Cotidal chart}{\PgNos{188}{192}} +\ToCSect{Authorities}{\PgNo{192}} + + +\ToCChap{XI} +{HARMONIC ANALYSIS OF THE TIDE} + +\ToCSect{Tide in actual oceans due to single equatorial satellite}{\PgNos{193}{196}} +\ToCSect{Substitution of ideal satellites for the moon}{\PgNos{197}{199}} +\ToCSect{Partial tide due to each ideal satellite}{\PgNos{199}{201}} +\ToCSect{Three groups of partial tides}{\PgNo{201}} +\ToCSect{Semidiurnal group}{\PgNos{201}{204}} +\ToCSect{Diurnal group}{\PgNos{204}{206}} +\ToCSect{Meteorological tides}{\PgNos[,]{206}{207}} +\ToCSect{Shallow water tides}{\PgNos{207}{210}} +\ToCSect{Authorities}{\PgNo{210}} +\PageSep{xiii} + + +\ToCChap{XII} +{REDUCTION OF TIDAL OBSERVATIONS} + +\ToCSect{Method of singling out a single partial tide}{\PgNos{211}{214}} +\ToCSect{Variety of plans adopted}{\PgNos{214}{217}} +\ToCSect{Tidal abacus}{\PgNos{217}{220}} +\ToCSect{Authorities}{\PgNo{220}} + + +\ToCChap{XIII} +{TIDE TABLES} + +\ToCSect{Definition of special and general tables}{\PgNo{221}} +\ToCSect{Reference to moon's transit}{\PgNos[,]{222}{223}} +\ToCSect{Examples at Portsmouth and at Aden}{\PgNos{223}{228}} +\ToCSect{General inadequacy of tidal information}{\PgNos[,]{229}{230}} +\ToCSect{Method of calculating tide tables}{\PgNos{230}{233}} +\ToCSect{Tide-predicting machine}{\PgNos{233}{241}} +\ToCSect{Authorities}{\PgNo{241}} + + +\ToCChap{XIV} +{THE DEGREE OF ACCURACY OF TIDAL PREDICTION} + +\ToCSect{Effects of wind and barometric pressure}{\PgNos[,]{242}{243}} +\ToCSect{Errors at Portsmouth}{\PgNos[,]{243}{244}} +\ToCSect{Errors at Aden}{\PgNos{245}{249}} +\ToCSect{Authorities}{\PgNo{250}} + + +\ToCChap{XV} +{CHANDLER'S NUTATION---THE RIGIDITY OF THE EARTH} + +\ToCSect{Nutation of the earth and variation of latitude}{\PgNos{251}{254}} +\ToCSect{Elasticity of the earth}{\PgNos[,]{254}{255}} +\ToCSect{Tide due to variation of latitude}{\PgNos[,]{255}{256}} +\ToCSect{Rigidity of the earth}{\PgNos{256}{260}} +\ToCSect{Transmission of earthquake shocks}{\PgNos[,]{261}{262}} +\ToCSect{Authorities}{\PgNos[,]{262}{263}} +\PageSep{xiv} + + +\ToCChap{XVI} +{TIDAL FRICTION} + +\ToCSect{Friction retards the tide}{\PgNos{264}{267}} +\ToCSect{Retardation of planetary rotation}{\PgNos{267}{269}} +\ToCSect{Reaction on the satellite}{\PgNos{269}{272}} +\ToCSect{Ancient eclipses of the sun}{\PgNos[,]{272}{273}} +\ToCSect{Law of variation of tidal friction with moon's distance}{\PgNos{273}{275}} +\ToCSect{Transformations of the month and of the day}{\PgNos{275}{280}} +\ToCSect{Initial and final conditions of motion}{\PgNos[,]{280}{281}} +\ToCSect{Genesis of the moon}{\PgNos{281}{285}} +\ToCSect{Minimum time requisite for the evolution}{\PgNos[,]{285}{286}} +\ToCSect{Rotation of the moon}{\PgNos[,]{286}{287}} +\ToCSect{The month ultimately to be shorter than the day}{\PgNos{287}{289}} + + +\ToCChap{XVII} +{TIDAL FRICTION (\textit{continued})} + +\ToCSect{Discovery of the Martian satellites}{\PgNos{290}{298}} +\ToCSect{Rotation of Mercury, of Venus, and of the Jovian satellites}{\PgNos[,]{298}{299}} +\ToCSect{Adaptation of the earth's figure to changed rotation}{\PgNos{299}{302}} +\ToCSect{Ellipticity of the internal strata of the earth}{\PgNos{302}{304}} +\ToCSect{Geological evidence}{\PgNos{304}{306}} +\ToCSect{Distortion of a plastic planet and trend of continents}{\PgNos{306}{308}} +\ToCSect{Obliquity of the ecliptic}{\PgNos{308}{312}} +\ToCSect{Eccentricity of lunar orbit}{\PgNos[,]{312}{313}} +\ToCSect{Eccentricity of the orbits of double stars}{\PgNo{313}} +\ToCSect{Plane of the lunar orbit}{\PgNos[,]{313}{314}} +\ToCSect{Short summary}{\PgNos[,]{314}{315}} +\ToCSect{Authorities}{\PgNo{315}} + + +\ToCChap{XVIII} +{THE FIGURES OF EQUILIBRIUM OF A ROTATING MASS OF +LIQUID} + +\ToCSect{Plateau's experiment}{\PgNos{316}{319}} +\ToCSect{Stability of a celestial sphere of liquid}{\PgNos{319}{321}} +\ToCSect{The two ellipsoids of Maclaurin and that of Jacobi}{\PgNos{321}{323}} +\PageSep{xv} +\ToCSect{Transitions with change of rotation}{\PgNos[,]{323}{324}} +\ToCSect{Coalescence of Jacobi's with Maclaurin's ellipsoid}{\PgNos{324}{326}} +\ToCSect{Poincaré's law of stability and coalescence}{\PgNos[,]{326}{327}} +\ToCSect{Poincaré's pear-shaped figure}{\PgNos[,]{327}{328}} +\ToCSect{Hour-glass figure of equilibrium}{\PgNos{328}{332}} +\ToCSect{Figures of planets}{\PgNos[,]{332}{333}} +\ToCSect{Authorities}{\PgNo{333}} + + +\ToCChap{XIX} +{THE EVOLUTION OF CELESTIAL SYSTEMS} + +\ToCSect{The Nebular Hypothesis}{\PgNos{334}{338}} +\ToCSect{Nebula in Andromeda}{\PgNos[,]{338}{339}} +\ToCSect{Distribution of satellites in the solar system}{\PgNos{339}{341}} +\ToCSect{Genesis of celestial bodies by fission}{\PgNo{342}} +\ToCSect{Dr.~See's speculations as to systems of double stars}{\PgNos{342}{344}} +\ToCSect{Diversity of celestial bodies}{\PgNos{344}{346}} +\ToCSect{Authorities}{\PgNo{346}} + + +\ToCChap{XX} +{SATURN'S RINGS} + +\ToCSect{Description}{\PgNos[,]{347}{348}} +\ToCSect{Discovery of Saturn's rings}{\PgNos{348}{352}} +\ToCSect{Diagram of the rings}{\PgNos{353}{356}} +\ToCSect{Roche's investigation}{\PgNos[,]{356}{357}} +\ToCSect{Roche's limit}{\PgNos{358}{360}} +\ToCSect{The limit for the several planets}{\PgNos[,]{360}{361}} +\ToCSect{Meteoric constitution of Saturn's rings}{\PgNos[,]{361}{362}} +\ToCSect{Maxwell's investigations}{\PgNos{362}{367}} +\ToCSect{Spectroscopic examination of the rings}{\PgNos{367}{369}} +\ToCSect{Authorities}{\PgNo{369}} +\PageSep{xvi} +%[Blank Page] +\PageSep{xvii} + + +% [** TN: Plates (Figs. 23, 33, and 40) listed separately in the original; +% omitted separate "FULL-PAGE" and "IN TEXT" headings, folded plate +% references into main illustration list.] +% LIST OF ILLUSTRATIONS +\Illustrations + +\LoFFig{1}{Well for Tide-Gauge}{7} +\LoFFig{2}{Pipe of Tide-Gauge}{9} +\LoFFig{3}{Indian Tide-Gauge}{10} +\LoFFig{4}{Légé's Tide-Gauge}{11} +\LoFFig{5}{Bombay Tide-Curve from Noon, April 22, to Noon, April 30, 1884}{14} +\LoFFig{6}{Sites for a Tide-Gauge}{15} +\LoFFig{7}{Plemyrameter}{20} +\LoFFig{8, 9}{Records of Seiches at Évian}{23} +\LoFFig{10}{Map of Lake of Geneva}{26} +\LoFFig{11}{Wave in Deep Water}{30} +\LoFFig{12}{Wave in Shallow Water}{31} +\LoFFig{13}{Simple Wave}{33} +\LoFFig{}{Composition of Two Equal and Opposite Waves}{34} +\LoFFig{14}{Vibrations due to Steamers}{45} +\LoFFig{15}{Progressive Change of a Wave in Shallow Water}{59} +\LoFFig{16}{Chart of the Estuary of the Tsien-Tang-Kiang}{61} +\LoFFig{17}{Bore-Shelter on the Tsien-Tang-Kiang}{64} +\LoFFig{18}{Diagram of the Flow of the Tide on the Tsien-Tang-Kiang}{66} +\LoFFig{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{69} +\LoFFig{20}{Earth and Moon}{93} +\PageSep{xviii} +\LoFFig{21}{Revolution of a Body without Rotation}{98} +\LoFFig{22}{Tide-generating Force}{100} +\LoFFig{23}{Horizontal Tide-generating Force}{106} +\LoFFig{24}{Deflection of a Pendulum; the Moon and Observer on the Equator}{111} +\LoFFig{25}{Deflection of a Pendulum; the Moon in N. Declination~$15°$, +the Observer in N. Latitude~$30°$}{112} +\LoFFig{26}{Bifilar Pendulum}{115} +\LoFFig{27}{Form of Dimple in an Elastic Surface}{135} +\LoFFig{28}{Distortion of Land and Sea-Bed by Tidal Load}{139} +\LoFFig{29}{Chart of Equilibrium Tides}{152} +\LoFFig{30}{Forced Oscillations of a Pendulum}{171} +\LoFFig{31}{The Tide in a Lake}{184} +\LoFFig{32}{Chart of Cotidal Lines}{190} +\LoFFig{33}{Tidal Abacus}{218} +\LoFFig{34}{Curves of Intervals and Heights at Portsmouth and at Aden}{227} +\LoFFig{35}{Diagram of Tide-predicting Instrument}{235} +\LoFFig{36}{Frictionally Retarded Tide}{266} +\LoFFig{37}{Maclaurin's and Jacobi's Ellipsoids of Equilibrium}{323} +\LoFFig{38}{Figures of Equilibrium}{325} +\LoFFig{39}{Hour-glass Figure of Equilibrium}{329} +\LoFFig{40}{Nebula in Andromeda}{339} +\LoFFig{41}{The Planet Saturn}{349} +\LoFFig{42}{Diagram of Saturn and his Rings}{353} +\LoFFig{43}{Roche's Figure of a Satellite when elongated to the utmost}{357} + +\MainMatter +\PageSep{1} +\index{Barometric pressure|see{Atmospheric pressure}}% +\index{Bifilar|see{Pendulum}}% +\index{D'Abbadie|see{Abbadie}}% +\index{Horizontal tide-generating force|see{also Pendulum}}% +\index{Rebeur|see{Paschwitz}}% +\index{Thomson, Sir W.|see{Kelvin}}% +\index{Tidal problem|see{Laplace, Harmonic Analysis, etc.}}% +\index{Tide|see{also other headings; \eg\ for tide-generating force \textit{see} Force}}% +\index{Vertical|see{Deflection}}% +% [** TN: Text printed by \Chapter macro] +% THE TIDES + + +\Chapter{I} +{Tides and Methods of Observation} + +\First{The} great wave caused by an earthquake is +\index{Tide!definition|(}% +often described in the newspapers as a tidal +wave, and the same name is not unfrequently +applied to such a short series of enormous waves +as is occasionally encountered by a ship in the +open sea. We must of course use our language +in the manner which is most convenient, but as +in this connection the adjective ``tidal'' implies +simply greatness and uncommonness, the use of +the term in such a sense cannot be regarded as +appropriate. + +The word ``tidal'' should, I think, only be +used when we are referring to regular and persistent +alternations of rise and fall of sea-level. +Even in this case the term may perhaps be used +in too wide a sense, for in many places there is a +regular alternation of the wind, which blows in-shore +during the day and out during the night +with approximate regularity, and such breezes +\PageSep{2} +alternately raise and depress the sea-level, and +thus produce a sort of tide. Then in the Tropics +there is a regularly alternating, though small, +periodicity in the pressure of the atmosphere, +which is betrayed by an oscillation in the height +of the barometer. Now the ocean will respond +to the atmospheric pressure, so that the sea-level +will fall with a rising barometer, and rise +with a falling barometer. Thus a regularly +periodic rise and fall of the sea-level must result +from this cause also. Again, the melting of the +snows in great mountain ranges, and the annual +variability in rainfall and evaporation, produce +approximately periodic changes of level in the +estuaries of rivers, and although the period of +these changes is very long, extending as they do +over the whole year, yet from their periodicity +they partake of the tidal character. + +These changes of water level are not, however, +tides in the proper sense of the term, and a true +tide can only be adequately defined by reference +to the causes which produce it. A tide, in fact, +means a rising and falling of the water of the +ocean caused by the attractions of the sun and +moon. + +Although true tides are due to astronomical +causes, yet the effects of regularly periodic winds, +variation of atmospheric pressure, and rainfall +are so closely interlaced with the true tide that +in actual observation of the sea it is necessary to +\PageSep{3} +consider them both together. It is accordingly +practically convenient to speak of any regular +alternation of sea-level, due to the wind and to +the other influences to which I have referred, as +a Meteorological Tide. The addition of the adjective +``meteorological'' justifies the use of the +term ``tide'' in this connection. + +We live at the bottom of an immense sea of +air, and if the attractions of the sun and moon +affect the ocean, they must also affect the air. +This effect will be shown by a regular rise and +fall in the height of the barometer. Although +such an effect is undoubtedly very small, yet it +is measurable. The daily heating of the air by +the sun, and its cooling at night, produce marked +alternations in the atmospheric pressure, and this +effect may by analogy be called an atmospheric +meteorological tide. + +The attractions of the moon and sun must certainly +act not only on the sea, but also on the +solid earth; and, since the earth is not perfectly +rigid or stiff, they must produce an alternating +change in its shape. Even if the earth is now +so stiff that the changes in its shape escape +detection through their minuteness, yet such +changes of shape must exist. There is much +evidence to show that in the early stages of their +histories the planets consisted largely or entirely +of molten rock, which must have yielded to tidal +influences. I shall, then, extend the term ``tide'' +\index{Tide!definition|)}% +\PageSep{4} +so as to include such alternating deformations of +a solid and elastic, or of a molten and plastic, +globe. These corporeal tides will be found to +lead us on to some far-reaching astronomical +speculations. The tide, in the sense which I +\index{Tide!general description|(}% +have attributed to the term, covers a wide field +of inquiry, and forms the subject of the present +volume. + +I now turn to the simplest and best known +form of tidal phenomena. When we are at the +seashore, or on an estuary, we see that the water +rises and falls nearly twice a day. To be more +exact, the average interval from one high water +to the next is twelve hours twenty-five minutes, +and so high water falls later, according to the +clock, by twice twenty-five minutes, or by fifty +minutes, on each successive day. Thus if high +water falls to-day at noon, it will occur to-morrow +at ten minutes to one. Before proceeding, it +may be well to remark that I use high water and +low water as technical terms. In common parlance +the level of water may be called high or +low, according as whether it is higher or lower +than usual. But when the level varies periodically, +there are certain moments when it is highest +and lowest, and these will be referred to as +the times of high and low water, or of high and +of low tide. In the same way I shall speak of +the heights at high and low water, as denoting +the water-level at the moments in question. +\PageSep{5} + +The most elementary observations would show +that the time of high water has an intimate relationship +to the moon's position. The moon, in +fact, passes the meridian on the average fifty +minutes later on each succeeding day, so that if +high water occurs so many hours after the moon +is due south on any day, it will occur on any +other day about the same number of hours after +the moon was south. This rule is far from being +exact, for it would be found that the interval +from the moon's passage to high water differs +considerably according to the age of the moon. +I shall not, however, attempt to explain at present +how this rough rule as to the time of high +water must be qualified, so as to convert it into +an accurate statement. + +But it is not only the hour of high water which +changes from day to day, for the height to which +the water rises varies so conspicuously that the +fact could not escape the notice of even the most +casual observer. It would have been necessary +to consult a clock to discover the law by which +the hour of high water changes from day to day; +but at the seashore it would be impossible to +avoid noticing that some rocks or shoals which +are continuously covered by the sea at one part +of a fortnight are laid bare at others. It is, in +fact, about full and new moon that the range +from low to high water is greatest, and at the +moon's first or third quarter that the range is +\PageSep{6} +least. The greater tides are called ``springs,'' +and the smaller ``neaps.'' + +The currents produced in the sea by tides are +\index{Gauge, tide!description of|(}% +often very complicated where the open sea is +broken by islands and headlands, and the knowledge +of tidal currents at each place is only to be +gained by the practical experience of the pilot. +Indeed, in the language of sailors, the word +``tide'' is not unfrequently used as meaning +tidal current, without reference to rise and fall. +These currents are often of great violence, and +vary from hour to hour as the water rises and +falls, so that the pilot requires to know how the +water stands in-shore in order to avail himself of +his practical knowledge of how the currents will +make in each place. A tide table is then of +\index{Tide!general description|)}% +much use, even at places where the access to a +harbor is not obstructed by a bar or shoal. It +is, of course, still more important for ships to +have a correct forecast of the tides where the +entrance to the harbor is shallow. + +I have now sketched in rough outline some +of the peculiarities of the tides, and it will have +become clear that the subject is a complicated +one, not to be unraveled without regular observation. +\index{Observation!methods of tidal|(}% +I shall, therefore, explain how tides are +observed scientifically, and how the facts are collected +upon which the scientific treatment of the +tides is based. + +The rise and fall of the sea may, of course, be +\PageSep{7} +roughly estimated by observing the height of the +water on posts or at jetties, which jut out into +moderately deep water. But as the sea is continually +disturbed by waves, observations of this +kind are not susceptible of accuracy, and for +\Figure{1}{Well for Tide-Gauge}{png} +scientific purposes more elaborate apparatus is +required. The exact height of the water can +only be observed in a place to which the sea has +a moderately free access, but where the channel +is so narrow as to prevent the waves from sensibly +disturbing the level of the water. This result +\PageSep{8} +is obtained in a considerable variety of ways, +but one of them may be described as typical of +all. + +A well (\fig{1}) about two feet in diameter is +dug to a depth of several feet below the lowest +tide and in the neighborhood of deep water. The +well is lined with iron, and a two-inch iron pipe +runs into the well very near its bottom, and passes +down the shore to the low-water line. Here it is +joined to a flexible pipe running out into deep +water, and ending with a large rose pierced with +many holes, like that of a watering can. The +rose (\fig{2}) is anchored to the bottom of the +sea, and is suspended by means of a buoy, so as +to be clear of the bottom. The tidal water can +thus enter pretty freely into the well, but the +passage is so narrow that the wave motion is not +transmitted into the well. Inside the well there +floats a water-tight copper cylinder, weighted at +the bottom so that it floats upright, and counter-poised +so that it only just keeps its top clear of +the water. To the top of the float there is fastened +a copper tape or wire, which runs up to +the top of the well and there passes round a +wheel. Thus as the water rises and falls this +wheel turns backwards and forwards. + +It is hardly necessary to describe in detail the +simple mechanism by which the turning of this +wheel causes a pencil to move backwards and forwards +in a straight line. The mechanism is, +\PageSep{9} +however, such that the pencil moves horizontally +backwards and forwards by exactly the same +amount as the water rises or falls in the well; or, +\Figure{2}{Pipe of Tide-Gauge}{png} +if the rise and fall of the tide is considerable, +the pencil only moves by half as much, or one +third, or even one tenth as much as the water. +At each place a scale of reduction is so chosen +as to bring the range of motion of the pencil +within convenient limits. We thus have a pencil +which will draw the rise and fall of the tide +on the desired scale. + +It remains to show how the times of the rise +and fall are indicated. The end of the pencil +touches a sheet of paper which is wrapped round +a drum about five feet long and twenty-four +\PageSep{10} +\index{Curve, tide!irregularities in|(}% +inches in circumference. If the drum were kept +still the pencil would simply draw a straight line +to and fro along the length of the drum as the +water rises and falls. But the drum is kept +turning by clockwork, so that it makes exactly +one revolution in twenty-four hours. Since the +drum is twenty-four inches round, each inch of +circumference corresponds to one hour. If the +water were at rest the pencil would simply draw +a circle round the paper, and the beginning and +ending of the line would join, whilst if the drum +remained still and the water moved, the pencil +\Figure{3}{Indian Tide-Gauge}{jpg} +would draw a straight line along the length of +the cylinder; but when both drum and water +are in motion, the pencil draws a curve on the +cylinder from which the height of water may be +read off at any time in each day and night. At +the end of twenty-four hours the pencil has returned +to the same part of the paper from which +\PageSep{11} +\Figure[0.7]{4}{Légé's Tide-Gauge}{jpg} +\index{Gauge, tide!description of|)}% +\PageSep{12} +it started, and it might be thought that there +would be risk of confusion between the tides of +to-day and those of yesterday. But since to-day +the tides happen about three quarters of an hour +later than yesterday, it is found that the lines +keep clear of one another, and, in fact, it is +usual to allow the drum to run for a fortnight +before changing the paper, and when the old +sheet is unwrapped from the drum, there is +written on it a tidal record for a fortnight. + +The instrument which I have described is +called a ``tide-gauge,'' and the paper a ``tide-curve.'' +As I have already said, tide-gauges +may differ in many details, but this description +will serve as typical of all. Another form of +tide-gauge is shown in \fig{4}; here a continuous +sheet of paper is placed over the drum, so that +there is no crossing of the curves, as in the first +example. Yet another form, designed by Lord +Kelvin, is shown on p.~170 of vol.~iii.\ of his +``Popular Lectures.'' + +The actual record for a week is exhibited in +\index{Curve, tide!at Bombay}% +\fig{5}, on a reduced scale. This tide-curve was +drawn at Bombay by a tide-gauge of the pattern +first described. When the paper was wrapped +on the drum, the right edge was joined to the +left, and now that it is unwrapped the curve +must be followed out of the paper on the left +and into it again on the right. The figure +shows that spring tide occurred on April~26, +\PageSep{13} +1884; the preceding neap tide was on the~18th, +and is not shown. It may be noticed that the +law of the tide is conspicuously different from +that which holds good on the coast of England, +for the two successive high or low waters which +occur on any day have very different heights. +Thus, for example, on April~26 low water occurred +at 5.50~\PM, and the water fell to $5$~ft.\ +$2$~in., whereas the next low water, occurring at +5.45~\AM\ of the~27th, fell to $1$~ft.\ $3$~in., the +heights being in both cases measured from a +certain datum. When we come to consider the +theory of the tides the nature of this irregularity +will be examined. + +The position near the seashore to be chosen +for the erection of the tide-gauge is a matter of +much importance. The choice of a site is generally +limited by nature, for it should be near +the open sea, should be sheltered from heavy +weather, and deep water must be close at hand +even at low tide. + +In the sketch map shown in \fig{6} a site such +as~$A$ is a good one when the prevailing wind +blows in the direction of the arrow. A position +such as~$B$, although well sheltered from heavy +seas, is not so good, because it is found that +tide-curves drawn at~$B$ would be much zigzagged. +These zigzags appear in the Bombay curves, +although at Bombay they are usually very +smooth ones. +\PageSep{14} + +These irregularities in the tide-curve are not +due to tides, and as the object of the observation +\index{Observation!methods of tidal|)}% +is to determine the nature of the tides it is +\Figure{5}{Bombay Tide-Curve from Noon, April~22, +to Noon, April~30, 1884}{jpg} +desirable to choose a site for the gauge where +\index{Gauge, tide!site for}% +the zigzags shall not be troublesome; but it is +not always easy to foresee the places which will +furnish smooth tide-curves. + +Most of us have probably at some time or +other made a scratch on the sand by the seashore, +\PageSep{15} +and watched the water rise over it. We +generally make our mark on the sand at the +furthest point, where the wash of a rather large +wave has brought up the water. For perhaps +five or ten minutes no wave brings the water up +as far as the mark, and one begins to think that +it was really an extraordinarily large wave which +was marked, although it did not seem so at the +time. Then a wave brings up the water far over +the mark, and immediately all the waves submerge +it. This little observation simply points +\Figure{6}{Sites for a Tide-Gauge}{png} +to the fact that the tide is apt to rise by jerks, +and it is this irregularity of rise and fall which +marks the notches in the tide-curves to which I +have drawn attention. + +Now in scientific matters it is well to follow +up the clues afforded by such apparently insignificant +facts as this. An interesting light is +indeed thrown on the origin of these notches on +tide-curves by an investigation, not very directly +\PageSep{16} +connected with tidal observation, on which I shall +make a digression in the following chapter. +\index{Curve, tide!irregularities in|)}% + +\begin{Authorities} +Baird's \Title{Manual for Tidal Observations} (Taylor \& Francis, +\index{Baird, \Title{Manual for Tidal Observation}}% +1886). Price 7\textit{s.}~6\textit{d.} Figs.\ \figref{1},~\figref{2},~\figref{3},~\figref{6} are reproduced from this +work. + +The second form of tide-gauge shown in \fig{4} is made by +Messrs.\ Légé, and is reproduced from a woodcut kindly provided +by them. + +Sir William Thomson's (Lord Kelvin's) \Title{Popular Lectures and +Addresses}, vol.~iii. (Macmillan,~1891). +\end{Authorities} +\PageSep{17} + + +\Chapter{II} +{Seiches in Lakes} + +\First{It} has been known for nearly three centuries +\index{Lakes!seiches in|(}% +\index{Seiches!definition}% +that the water of the Lake of Geneva is apt to +\index{Geneva!seiches in lake|(}% +rise and fall by a few inches, sometimes irregularly +and sometimes with more or less regularity; +and the same sort of oscillation has been observed +in other Swiss lakes. These quasi-tides, +called seiches, were until recently supposed only +to occur in stormy weather, but it is now known +that small seiches are of almost daily occurrence.\footnote + {The word ``seiche'' is a purely local one. It has been alleged + to be derived from ``sèche,'' but I can see no reason for + associating dryness with the phenomenon.} + +Observations were made by Vaucher in the +\index{Vaucher, record of a great seiche at Geneva}% +last century on the oscillations of the Lake of +Geneva, and he gave an account of a celebrated +seiche in the year~1600, when the water oscillated +through three or four feet; but hardly any +systematic observation had been undertaken when +Professor Forel, of Lausanne, attacked the subject, +\index{Forel!on seiches|(}% +and it is his very interesting observations +which I propose to describe. + +Doctor Forel is not a mathematician, but is +\PageSep{18} +rather a naturalist of the old school, who notes any +interesting fact and then proceeds carefully to +investigate its origin. His papers have a special +charm in that he allows one to see all the workings +of his mind, and tells of each difficulty as it +arose and how he met it. To those who like to +read of such work, almost in the form of a narrative, +I can strongly recommend these papers, +which afford an admirable example of research +thoroughly carried out with simple appliances. + +People are nowadays too apt to think that +science can only be carried to perfection with +elaborate appliances, and yet it is the fact that +many of the finest experiments have been made +with cardboard, cork, and sealing-wax. The principal +reason for elaborate appliances in the laboratories +of universities is that a teacher could not +deal with a large number of students if he had +to show each of them how to make and set up +his apparatus, and a student would not be able +to go through a large field of study if he had to +spend days in preparation. Great laboratories +have, indeed, a rather serious defect, in that they +tend to make all but the very best students helpless, +and thus to dwarf their powers of resource +and inventiveness. The mass of scientific work +is undoubtedly enormously increased by these +institutions, but the number of really great investigators +seems to remain almost unaffected by +them. But I must not convey the impression +\PageSep{19} +that, in my opinion, great laboratories are not +useful. It is obvious, indeed, that without them +science could not be taught to large numbers of +students, and, besides, there are many investigations +in which every possible refinement of apparatus +is necessary. But I do say that the +number of great investigators is but little increased +by laboratories, and that those who are +interested in science, but yet have not access to +laboratories, should not give up their study in +despair. + +Doctor Forel's object was, in the first instance, +to note the variations of the level of the lake, after +obliterating the small ripple of the waves on the +surface. The instrument used in his earlier investigations +was both simple and delicate. Its principle +was founded on casual observation at the port +of Morges, where there happens to be a breakwater, +pierced by a large ingress for ships and a small +one for rowing boats. He accidentally noticed +that at the small passage there was always a current +setting either inwards or outwards, and it +occurred to him that such a current would form +a very sensitive index of the rise and fall of the +water in the lake. He therefore devised an instrument, +illustrated in \fig{7}, and called by him +a plemyrameter, for noting currents of even the +\index{Plemyrameter, observation of seiches with|(}% +most sluggish character. Near the shore he +made a small tank, and he connected it with the +lake by means of an india-rubber siphon pipe of +\PageSep{20} +small bore. Where the pipe crossed the edge +of the tank he inserted a horizontal glass tube +of seven millimetres diameter, and in that tube +he put a float of cork, weighted with lead so that +it should be of the same density as water. At +the ends of the glass tube there were stops, so +that the float could not pass out of it. When +the lake was higher than the tank, the water ran +\Figure{7}{Plemyrameter}{png} +through the siphon pipe from lake to tank, and +the float remained jammed in the glass tube +against the stop on the side towards the tank; +and when the lake fell lower than the tank, the +float traveled slowly to the other end and remained +there. The siphon pipe being small, the +only sign of the waves in the lake was that the +float moved with slight jerks, instead of uniformly. +Another consequence of the smallness +of the tube was that the amount of water which +could be delivered into the tank or drawn out of +it in one or two hours was so small that it might +\PageSep{21} +practically be neglected, so that the water level +in the tank might be considered as invariable. + +This apparatus enabled Forel to note the rise +and fall of the water, and he did not at first +attempt to measure the height of rise and fall, +as it was the periodicity in which he was principally +interested. + +In order to understand the record of observations, +\index{Seiches!records of}% +it must be remembered that when the +float is towards the lake, the water in the tank +stands at the higher level, and when the float +is towards the tank the lake is the higher. In +the diagrams, of which \fig{8} is an example, the +straight line is divided into a scale of hours and +minutes. The zigzag line gives the record, and +the lower portions represent that the water of +the lake was below the tank, and the upper line +that it was above the tank. The fact that the +float only moved slowly across from end to end +of the glass tube, is indicated by the slope of +the lines, which join the lower and upper portions +of the zigzags. Then on reading \fig{8} we +see that from $2$~hrs.\ $1$~min.\ to $2$~hrs.\ $4$~mins.\ the +water was high and the float was jammed against +the tank end of the tube, because there was a +current from the lake to the tank. The float +then slowly left the tank end and traveled +across, so that at $2$~hrs.\ $5$~mins.\ the water was +low in the lake. It continued, save for transient +changes of level, to be low until $2$~hrs.\ $30$~mins., +\PageSep{22} +when it rose again. Further explanation seems +unnecessary, as it should now be easy to read +this diagram, and that shown in~\fig{9}. +\index{Plemyrameter, observation of seiches with|)}% + +The sharp pinnacles indicate alternations of +level so transient that the float had not time to +travel across from one end of the glass tube +to the other, before the current was reversed. +These pinnacles may be disregarded for the +present, since we are only considering seiches of +considerable period. + +These two diagrams are samples of hundreds +which were obtained at various points on the +shores of Geneva, and of other lakes in Switzerland. +In order to render intelligible the method +by which Forel analyzed and interpreted these +records, I must consider \fig{8} more closely. In +this case it will be noticed that the record shows +a long high water separated from a long low +water by two pinnacles with flat tops. These +pieces at the ends have an interesting significance. +When the water of the lake is simply +oscillating with a period of about an hour we +have a trace of the form shown in~\fig{9}. But +when there exists concurrently with this another +oscillation, of much smaller range and of short +period, the form of the trace will be changed. +When the water is high in consequence of the +large and slow oscillation, the level of the lake +cannot be reduced below that of the tank by +the small short oscillation, and the water merely +\PageSep{23} +stands a little higher or a little lower, but always +remains above the level of the tank, so that the +trace continues on the higher level. But when, +in course of the changes of the large oscillation, +the water has sunk to near the mean level of the +%[** TN: Figures 8 and 9 combined; cross-refs handled by preamble code] +\Figure{8}{Records of Seiches at Évian}{png} +lake, the short oscillation will become manifest, +and so it is only at the ends of the long flat +pieces that we shall find evidence of the quick +oscillation. + +Thus, in these two figures there was in one +case only one sort of wave, and in the other +there were two simultaneous waves. These +records are amongst the simplest of those obtained +by Forel, and yet even here the oscillations +of the water were sufficiently complicated. +It needed, indeed, the careful analysis of many +records to disentangle the several waves and to +determine their periods. + +After having studied seiches with a plemyrameter +for some time, Forel used another form of +\PageSep{24} +apparatus, by which he could observe the amplitude +of the waves as well as their period. His +apparatus was, in fact, a very delicate tide-gauge, +which he called a limnimeter. The only +\index{Limnimeter, a form of tide gauge}% +difference between this instrument and the one +already described as a tide-gauge is that the +drum turned much more rapidly, so that five +feet of paper passed over the drum in twenty-four +hours, and that the paper was comparatively +narrow, the range of the oscillation being +small. The curve was usually drawn on the full +scale, but it could be quickly reduced to half +scale when large seiches were under observation. + +It would be impossible in a book of this kind +\index{Lakes!mode of rocking in seiches}% +to follow Forel in the long analysis by which he +interpreted his curves. He speaks thus of the +complication of simultaneous waves: ``All these +oscillations are embroidered one on the other +and interlace their changes of level. There is +here matter to disturb the calmest mind. I +must have a very stout faith in the truth of my +hypothesis to persist in maintaining that, in the +midst of all these waves which cross and mingle, +there is, nevertheless, a recognizable rhythm. +This is, however, what I shall try to prove.''\footnote + {\Title{Deuxième Étude}, p.~544.} +The hypothesis to which he here refers, and +triumphantly proves, is that seiches consist of a +rocking of the whole water of the lake about +fixed lines, just as by tilting a trough the water +\PageSep{25} +in it may be set swinging, so that the level at +the middle remains unchanged, while at the two +ends the water rises and falls alternately. + +In another paper he remarks: ``If you will +\index{Lakes!mode of rocking in seiches}% +follow and study with me these movements you +will find a great charm in the investigation. +When I see the water rising and falling on the +shore at the end of my garden I have not before +me a simple wave which disturbs the water of +the bay of Morges, but I am observing the manifestation +of a far more important phenomenon. +It is the whole water of the lake which is rocking. +It is a gigantic impulse which moves the +whole liquid mass of Leman throughout its +length, breadth, and depth\dots. It is probable +that the same thing would be observed in far +larger basins of water, and I feel bound to +recognize in the phenomenon of seiches the +\index{Seiches!longitudinal and transverse|(}% +grandest oscillatory movement which man can +study on the face of our globe.''\footnote + {\Title{Les Seiches, Vagues d'Oscillation}, p.~11.} + +It will now be well to consider the map of +Geneva in~\fig{10}. Although the lake somewhat +resembles the arc of a circle, the curvature of +its shores will make so little difference in the +nature of the swinging of the water that we +may, in the first instance, consider it as practically +straight. + +Forel's analysis of seiches led him to conclude +that the oscillations were of two kinds, the longitudinal +\PageSep{26} +and the transverse. In the longitudinal +seiche the water rocks about a line drawn across +the lake nearly through Morges, and the water +at the east end of the lake rises when that at +the west falls, and vice versa. The line about +which the water rocks is called a node, so that in +this case there is one node at the middle of the +lake. This sort of seiche is therefore called a +uninodal longitudinal seiche. The period of the +\Figure{10}{Map of Lake of Geneva}{png} +oscillation is the time between two successive +high waters at any place, and it was found to be +seventy-three minutes, but the range of rise and +fall was very variable. There are also longitudinal +seiches in which there are two nodes, +dividing the lake into three parts, of which the +central one is twice as long as the extreme parts; +such an oscillation is called a binodal longitudinal +seiche. In this mode the water at the middle +\PageSep{27} +of the lake is high when that at the two ends +is low, and vice versa; the period is thirty-five +minutes. + +Other seiches of various periods were observed, +\index{Seiches!periods of}% +some of which were no doubt multinodal. Thus +in a trinodal seiche, the nodes divide the lake +into four parts, of which the two central ones +are each twice as long as the extreme parts. If +there are any number of nodes, their positions +are such that the central portion of the lake is +divided into equal lengths, and the terminal +parts are each of half the length of the central +part or parts. This condition is necessary in +order that the ends of the lake may fall at places +where there is no horizontal current. In all such +modes of oscillation the places where the horizontal +current is evanescent are called loops, and +these are always halfway between the nodes, +where there is no rise and fall. + +A trinodal seiche should have a period of +about twenty-four minutes, and a quadrinodal +seiche should oscillate in about eighteen minutes. +The periods of these quicker seiches would, no +doubt, be affected by the irregularity in the form +and depth of the lake, and it is worthy of notice +that Forel observed at Morges seiches with +periods of about twenty minutes and thirty minutes, +which he conjectured to be multinodal. + +The second group of seiches were transverse, +\index{Seiches!longitudinal and transverse|)}% +being observable at Morges and Évian. It was +\PageSep{28} +clear that these oscillations, of which the period +was about ten minutes, were transversal, because +at the moment when the water was highest at +Morges it was lowest at Évian, and vice versa. +As in the case of the longitudinal seiches, the +principal oscillation of this class was uninodal, +but the node was, of course, now longitudinal to +the lake. The irregularity in the width and +depth of the lake must lead to great diversity of +period in the transverse seiches appropriate to +the various parts of the lake. The transverse +seiches at one part of the lake must also be +transmitted elsewhere, and must confuse the +seiches appropriate to other parts. Accordingly +there is abundant reason to expect oscillations of +such complexity as to elude complete explanation. + +The great difficulty of applying deductive +reasoning to the oscillations of a sheet of water +of irregular outline and depth led Forel to construct +a model of the lake. By studying the +waves in his model he was able to recognize +many of the oscillations occurring in the real +lake, and so obtained an experimental confirmation +of his theories, although the periods of +oscillation in the model of course differed enormously +from those observed in actuality. + +The theory of seiches cannot be considered as +demonstrated, unless we can show that the water +of such a basin as that of Geneva is capable of +\index{Geneva!seiches in lake|)}% +\index{Geneva!model of lake}% +\PageSep{29} +swinging at the rates observed. I must, therefore, +now explain how it may be proved that the +periods of the actual oscillations agree with the +facts of the case. + +As a preliminary let us consider the nature of +wave motion. There are two very distinct cases +of the undulatory motion of water, which nevertheless +graduate into one another. The distinction +lies in the depth of the water compared with +the length of the wave, measured from crest to +crest, in the direction of wave propagation. The +wave-length may be used as a measuring rod, +and if the depth of the water is a small fraction +of the wave-length, it must be considered shallow, +but if its depth is a multiple of the wave-length, +it will be deep. The two extremes of +course graduate into one another. + +In a wave in deep water the motion dies out +pretty rapidly as we go below the surface, so that +when we have gone down half a wave-length +below the surface, the motion is very small. In +shallow water, on the other hand, the motion extends +quite to the bottom, and in water which is +neither deep nor shallow, the condition of affairs +is intermediate. The two figures, \figref{11}~and~\figref{12}, +show the nature of the movement in the two +classes of waves. In both cases the dotted lines +\index{Waves!in deep and shallow water}% +show the position of the water when at rest, and +the full lines show the shapes assumed by the +rectangular blocks marked out by the dotted +\PageSep{30} +lines, when wave motion is disturbing the water. +It will be observed that in the deep water, as +shown in \fig{11}, the rectangular blocks change +their shape, rise and fall, and move to and fro. +Taking the topmost row of rectangles, each block +of water passes successively in time through all +the forms and positions shown by the top row +of quasi-parallelograms. So also the successive +changes of the second row of blocks are indicated +by the second strip, and the third and the fourth +indicate the same. The changes in the bottom +\Figure[0.7]{11}{Wave in Deep Water}{png} +row are relatively very small both as to shape +and as to displacement, so that it did not seem +worth while to extend the figure to a greater +depth. + +Turning now to the wave in shallow water in~\fig{12}, +we see that each of the blocks is simply +displaced sideways and gets thinner or more +\PageSep{31} +squat as the wave passes along. Now, I say that +we may roughly classify the water as being deep +with respect to wave motion when its depth is +more than half a wave-length, and as being shallow +when it is less. Thus the same water may +be shallow for long waves and deep for short +\index{Waves!speed of}% +ones. For example, the sea is very shallow for +\Figure[0.7]{12}{Wave in Shallow Water}{png} +the great wave of the oceanic tide, but it is very +deep even for the largest waves of other kinds. +Deepness and shallowness are thus merely relative +to wave-length. + +The rate at which a wave moves can be exactly +calculated from mathematical formulæ, +from which it appears that in the deep sea a +wave $63$~metres in length travels at $36$~kilometres +per hour, or, in British measure, a wave of +$68$~yards in length travels $22\frac{1}{2}$~miles an hour. +Then, the rule for other waves is that the speed +varies as the square root of the wave-length, so +that a wave $16$~metres long---that is, one quarter +of $63$~metres---travels at $18$~kilometres an +hour, which is half of $36$~kilometres an hour. +Or if its length were $7$~metres, or one ninth as +\PageSep{32} +long, it would travel at $12$~kilometres an hour, +or one third as quick. + +Although the speed of waves in deep water +depends on wave-length, yet in shallow water the +speed is identical for waves of all lengths, and +depends only on the depth of the water. In +water $10$~metres deep, the calculated velocity of +a wave is $36$~kilometres an hour; or if the water +were $2\frac{1}{2}$~metres deep (quarter of $10$~metres), it +would travel $18$~kilometres (half of $36$~kilometres) +an hour; the law of variation being that +the speed of the wave varies as the square root +of the depth. For water that is neither deep nor +shallow, the rate of wave propagation depends +both on depth and on wave-length, according to +a law which is somewhat complicated. + +In the case of seiches, the waves are very long +compared with the depth, so that the water is to +be considered as shallow; and here we know +that the speed of propagation of the wave depends +only on depth. The average depth of the +Lake of Geneva may be taken as about $150$~metres, +and it follows that the speed of a long wave +in the lake is about $120$~kilometres an hour. + +In order to apply this conclusion to the study +of seiches, we have to consider what is meant by +the composition of two waves. If I take the +series of numbers +\[ +\text{\&c.}\quad +100\quad 71\quad 0\quad -71\quad -100\quad -71\quad 0\quad 71\quad 100\quad +\text{\&c.} +\] +and plot out, at equal distances, a figure of +\PageSep{33} +\index{Waves!composition of|(}% +heights proportional to these numbers, setting +off the positive numbers above and the negative +numbers below a horizontal line, I get the simple +wave line shown in~\fig{13}. Now, if this +wave is traveling to the right, the same series of +numbers will represent the wave at a later time, +when they are all displaced towards the right, as +in the dotted line. + +Now turn to the following schedule of numbers, +and consider those which are written in the +top row of each successive group of three rows. +The columns represent equidistant spaces, and +the rows equidistant times. The first set of +numbers, $-100$,~$-71$,~$0$,~\&c., are those which +were plotted out as a wave in~\fig{13}; in the top +\Figure[0.7]{13}{Simple Wave}{png} +row of the second group they are the same, but +moved one space to the right, so that they represent +the movement of the wave to the right in +one interval of time. In the top row of each +successive group the numbers are the same, but +always displaced one more space to the right; +they thus represent the successive positions of a +\PageSep{34} +\begin{figure}[hp!] +\centering +\ifthenelse{\boolean{ForPrinting}}{% + \makebox[0pt][c]{\includegraphics[height=0.9\textheight]{./images/fig13b.pdf}} +}{ + \includegraphics[width=0.75\textwidth]{./images/fig13b.pdf} +} +\caption{Composition of Two Equal and Opposite Waves} +\label{fig:13b} +\iffalse +-100 -71 0 71 100 71 0 -71 -100 + +-100 -71 0 71 100 71 0 -71 -100 +[*--] +-200 -142 0 142 200 142 0 -142 -200 +[*--] + -71 -100 -71 0 71 100 71 0 -71 + + -71 0 71 100 71 0 -71 -100 -71 +[*--] +-142 -100 0 100 142 100 0 -100 -142 +[*--] + 0 -71 -100 -71 0 71 100 71 0 + + 0 71 100 71 0 -71 -100 -71 0 +[*--] + 0 0 0 0 0 0 0 0 0 +[*--] + 71 0 -71 -100 -71 0 71 100 71 + + 71 100 71 0 -71 -100 -71 0 71 +[*--] + 142 100 0 -100 -142 -100 0 100 142 +[*--] + 100 71 0 -71 -100 -71 0 71 100 + + 100 71 0 -71 -100 -71 0 71 100 +[*--] + 200 142 0 -142 -200 -142 0 142 200 +[*--] + 71 100 71 0 -71 -100 -71 0 71 + + 71 0 -71 -100 -71 0 71 100 71 +[*--] + 142 100 0 -100 -142 -100 0 100 142 +[*--] + 0 71 100 71 0 -71 -100 -71 0 + + 0 -71 -100 -71 0 71 100 71 0 +[*--] + 0 0 0 0 0 0 0 0 0 +[*--] + -71 0 71 100 71 0 -71 -100 -71 + + -71 -100 -71 0 71 100 71 0 -71 +[*--] +-142 -100 0 100 142 100 0 -100 -142 +[*--] +-100 \DPtypo{71}{-71} 0 71 100 71 0 -71 -100 + +-100 -71 0 71 100 71 0 -71 -100 +[*--] +-200 -142 0 142 200 142 0 -142 -200 +[*--] +\fi +\end{figure} +\PageSep{35} +wave moving to the right. The table ends in +the same way as it begins, so that in eight of +these intervals of time the wave has advanced +through a space equal to its own length. + +If we were to invert these upper figures, so +that the numbers on the right are exchanged +with those on the left, we should have a series of +numbers representing a wave traveling to the +left. Such numbers are shown in the second +row in each group. + +When these two waves coëxist, the numbers +must be compounded together by addition, and +then the result is the series of numbers written +in the third rows. These numbers represent the +resultant of a wave traveling to the right, and of +an equal wave traveling simultaneously to the +left. + +It may be well to repeat that the first row of +each group represents a wave moving to the +right, the second row represents a wave moving +to the left, and the third represents the resultant +of the two. Now let us consider the nature +of this resultant motion; the third and the +seventh columns of figures are always zero, and +therefore at these two places the water neither +rises nor falls,---they are, in fact, nodes. If the +schedule were extended indefinitely both ways, +exactly halfway between any pairs of nodes +there would be a loop, or line across which there +is no horizontal motion. In the schedule, as it +\PageSep{36} +stands, the first, fifth, and ninth columns are +loops. + +At the extreme right and at the extreme left +the resultant numbers are the same, and represent +a rise of the water from $-200$ to~$+200$, +and a subsequent fall to~$-200$ again. If these +nine columns represent the length of the lake, +the motion is that which was described as binodal, +for there are two nodes dividing the lake +into three parts, there is a loop at each end, and +when the water is high in the middle it is low at +the ends, and vice versa. It follows that two +equal waves, each as long as the lake, traveling +in opposite directions, when compounded together +give the motion which is described as the +binodal longitudinal seiche. + +Now let us suppose that only five columns of +the table represent the length of the lake. The +resultant numbers, which again terminate at +each end with a loop, are:--- +\begin{align*} +-200&& -142&& 0&& 142&& 200& \displaybreak[0] \\ +-142&& -100&& 0&& 100&& 142& \displaybreak[0] \\ + 0&& 0&& 0&& 0&& 0& \displaybreak[0] \\ + 142&& 100&& 0&& -100&& -142& \displaybreak[0] \\ + 200&& 142&& 0&& -142&& -200& \displaybreak[0] \\ + 142&& 100&& 0&& -100&& -142& \displaybreak[0] \\ + 0&& 0&& 0&& 0&& 0& \displaybreak[0] \\ +-142&& -100&& 0&& 100&& 142& \displaybreak[0] \\ +-200&& -142&& 0&& 142&& 200& +\end{align*} + +Since the middle column consists of zero +throughout, the water neither rises nor falls +\PageSep{37} +there, and there is a node at the middle. Again, +since the numbers at one end are just the same +as those at the other, but reversed as to positive +and negative, when the water is high at one end +it is low at the other. The motion is, in fact, a +simple rocking about the central line, and is that +described as the uninodal longitudinal seiche. + +The motion is here again the resultant of two +equal waves moving in opposite directions, and +the period of the oscillation is equal to the time +which either simple wave takes to travel through +its own length. But the length of the wave is +now twice that of the lake. Hence it follows +that the period of the rocking motion is the +time occupied by a wave in traveling twice the +length of the lake. We have already seen that +in shallow water the rate at which a wave moves +is independent of its length and depends only +on the depth of the water, and that in water of +the same depth as the Lake of Geneva the wave +travels $120$~kilometres an hour. The Lake of +Geneva is $70$~kilometres long, so that the two +waves, whose composition produces a simple rocking +\index{Waves!composition of|)}% +of the water, must each of them have a +length of $140$~kilometres. Hence it follows that +the period of a simple rocking motion, with one +node in the middle of the Lake of Geneva, will +be almost exactly $\frac{140}{120}$~of an hour, or $70$~minutes. +Forel, in fact, found the period to be $73$~minutes. +He expresses this result by saying that +\PageSep{38} +a uninodal longitudinal seiche in the Lake of +Geneva has a period of $73$~minutes. His observations +also showed him that the period of a +binodal seiche was $35$~minutes. It follows from +the previous discussion that when there are two +nodes the period of the oscillation should be +half as long as when there is one node. Hence, +we should expect that the period would be +about $36$~or $37$~minutes, and the discrepancy +between these two results may be due to the +fact that the formula by which we calculate the +period of a binodal seiche would require some +correction, because the depth of the lake is not +so very small compared with the length of these +shorter waves. + +It is proper to remark that the agreement +between the theoretical and observed periods is +suspiciously exact. The lake differs much in +depth in different parts, and it is not quite certain +what is the proper method of computing +the average depth for the determination of the +period of a seiche. It is pretty clear, in fact, +that the extreme closeness of the agreement is +accidentally due to the assumption of a round +number of metres as the average depth of the +lake. The concordance between theory and observation +must not, however, be depreciated too +much, for it is certain that the facts of the case +agree well with what is known of the depth of +the lake. +\index{Forel!on seiches|)}% +\PageSep{39} + +The height of the waves called ``seiches'' is +\index{Seiches!causes of}% +\index{Storms a cause of seiches}% +very various. I have mentioned an historical +seiche which had a range of as much as four +feet, and Forel was able by his delicate instruments +still to detect them when they were only a +millimetre or a twenty-fifth of an inch in height. +It is obvious, therefore, that whatever be the +cause of seiches, that cause must vary widely +in intensity. According to Forel, seiches arise +from several causes. It is clear that anything +which heaps up the water at one end of the +lake, and then ceases to act, must tend to produce +an oscillation of the whole. Now, a rise +of water level at one end or at one side of the +lake may be produced in various ways. Some, +and perhaps many, seiches are due to the tilting +of the whole lake bed by minute earthquakes. +\index{Earthquakes!a cause of seiches}% +Modern investigations seem to show that this is +a more fertile cause than Forel was disposed to +allow, and it would therefore be interesting to +see the investigation of seiches repeated with the +aid of delicate instruments for the study of +earthquakes, some of which will be described in +\Ref{Chapter}{VI}. I suspect that seiches would be +observed at times when the surface of the earth +is much disturbed. + +The wind is doubtless another cause of seiches. +\index{Wind!a cause of seiches}% +When it blows along the lake for many hours in +one direction, it produces a superficial current, +and heaps up the water at the end towards +\PageSep{40} +\index{Atmospheric pressure!cause of seiches}% +which it is blowing. If such a wind ceases +somewhat suddenly, a seiche will certainly be +started, and will continue for hours until it dies +out from the effects of the friction of the water +on the lake bottom. Again, the height of the +barometer will often differ slightly at different +parts of the lake, and the water will respond, just +as does the mercury, to variations of atmospheric +pressure. About a foot of rise of water should +correspond to an inch of difference in the height +of barometer. The barometric pressure cannot +be quite uniform all over the Lake of Geneva, +and although the differences must always be +exceedingly small, yet it is impossible to doubt +that this cause, combined probably with wind, +will produce many seiches. I shall return later +\index{Seiches!causes of}% +to the consideration of an interesting speculation +as to the effects of barometric pressure on +the oscillation of lakes and of the sea. Lastly, +Forel was of opinion that sudden squalls or local +storms were the most frequent causes of seiches. +\index{Storms a cause of seiches}% +I think that he much overestimated the efficiency +of this cause, because his theory of the path of +the wind in sudden and local storms is one that +would hardly be acceptable to most meteorologists. + +Although, then, it is possible to indicate causes +competent to produce seiches, yet we cannot as +yet point out the particular cause for any individual +seiche. The complication of causes is so +\PageSep{41} +\index{Lakes!vibrations|(}% +\index{Vibration of lakes|(}% +great that this degree of uncertainty will probably +never be entirely removed. + +But I have not yet referred to the point which +justifies this long digression on seiches in a book +on the tides. The subject was introduced by +the irregularities in the line traced by the tide-gauge +at Bombay, which indicated that there +are oscillations of the water with periods ranging +from two minutes to a quarter of an hour or +somewhat longer. Now these zigzags are not +found in the sea alone, for Forel observed on +the lake oscillations of short period, which resembled +seiches in all but the fact of their more +rapid alternations. Some of these waves are +perhaps multinodal seiches, but it seems that +they are usually too local to be true seiches +affecting the whole body of the lake at one time. +Forel calls these shorter oscillations ``vibrations,'' +thus distinguishing them from proper seiches. +A complete theory of the so-called vibrations +has not yet been formulated, although, as I shall +show below, a theory is now under trial which +serves to explain, at least in part, the origin of +vibrations. + +Forel observed with his limnimeter or tide-gauge +that when there is much wind, especially +\index{Wind!vibrations of lakes due to}% +from certain quarters, vibrations arise which are +quite distinct from the ordinary visible wave +motion. The period of the visible waves on the +\PageSep{42} +Lake of Geneva is from $4$~to $5$~seconds,\footnote + {I observed when it was blowing half a gale on Ullswater, in + Cumberland, that the waves had a period of about a second.} +whereas +vibrations have periods ranging from $45$~seconds +to $4$~minutes. Thus there is a clear line separating +waves from vibrations. Forel was unable +to determine what proportion of the area of the +lake is disturbed by vibrations at any one time, +and although their velocity was not directly observed, +there can be no doubt that these waves +are propagated at a rate which corresponds to +their length and to the depth of the water. I +have little doubt but that the inequalities which +produce notches in a tide-curve have the same +origin as vibrations on lakes. + +It is difficult to understand how a wind, whose +\index{Wind!vibrations of lakes due to}% +only visible effect is short waves, can be responsible +for raising waves of a length as great as a +thousand yards or a mile, and yet we are driven +to believe that this is the case. But Forel also +found that steamers produce vibrations exactly +like those due to wind. The resemblance was +indeed so exact that vibrations due to wind +could only be studied at night, when it was +known that no steamers were traveling on the +lake, and, further, the vibrations due to steamers +could only be studied when there was no wind. + +His observations on the steamer vibrations are +amongst the most curious of all his results. +When a boat arrives at the pier at Morges, the +\PageSep{43} +water rises slowly by about $5$~to $8$~millimetres, +and then falls in about $20$~to $30$~seconds. The +amount and the rapidity of the rise and fall +vary with the tonnage of the boat and with the +rate of her approach. After the boat has passed, +the trace of the limnimeter shows irregularities +with sharp points, the variations of height ranging +from about two to five millimetres, with a +period of about two minutes. These vibrations +continue to be visible during two to three hours +after the boat has passed. As these boats travel +at a speed of $20$~kilometres an hour, the vibrations +persist for a long time after any renewal +of them by the boat has ceased. These vibrations +are called by Forel ``the subsequent steamer +vibrations.'' + +That the agitation of the water should continue +for more than two hours is very remarkable, +and shows the delicacy of the method of +observation. But it seems yet more strange +that, when a boat is approaching Morges, the +vibrations should be visible during $25$~minutes +before she reaches the pier. These he calls +``antecedent steamer vibrations.'' They are +more rapid than the subsequent ones, having a +period of a minute to a minute and a quarter. +Their height is sometimes two millimetres (a +twelfth of an inch), but they are easily detected +when less than one millimetre in height. It +appears that these antecedent vibrations are first +\PageSep{44} +noticeable when the steamer rounds the mole of +Ouchy, when she is still at a distance of $10$~kilometres. +As far as one can judge from the speed +at which waves are transmitted in the Lake of +Geneva, the antecedent vibrations, which are +noticed $25$~minutes before the arrival of the +boat, must have been generated when she was at +a distance of $12$~kilometres from Morges. \Fig{14} +gives an admirable tracing of these steamer +vibrations.\footnote + {From \Title{Les Seiches, Vagues d'Oscillation fixe des Lacs}, 1876.} + +In this figure the line~$\Seg{a}{a'}$ was traced between +two and three o'clock in the morning, and shows +scarcely any sign of perturbation. Between +three and eight o'clock in the morning no observations +were taken, but the record begins again +at eight o'clock. The portion marked~$\Seg{b}{b'}$ shows +weak vibrations, probably due to steamers passing +along the coast of Savoy. The antecedent +vibrations, produced by a steamer approaching +Morges, began about the time of its departure +from Ouchy, and are shown at~$\Seg{c}{c'}$. The point~$d$ +shows the arrival of this boat at Morges, and +$d'$~shows the effect of another boat coming from +Geneva. The portion marked~$\Seg[e]{e}{e}$ shows the +subsequent steamer vibrations, which were very +clear during more than two hours after the boats +had passed. + +Dr.~Forel was aware that similar vibrations occur +in the sea, for he says: ``What are these +\index{Sea!vibrations of}% +\PageSep{45} +oscillations with periods of +$5$,~$10$, $20$, or $100$~minutes, +which are sometimes irregular? +Are they analogous to +our seiches? Not if we define +seiches as uninodal oscillations, +for it is clear that +if, in a closed basin of $70$~kilometres +in length, uninodal +seiches have a period of +$73$~minutes, in the far greater +basin of the Mediterranean, +or of the ocean, a uninodal +wave of oscillation must have +a much longer period. They +resemble much more closely +what I have called vibrations, +and, provisionally, I +shall call them by the name +of `vibrations of the sea.' I +\index{Sea!vibrations of}% +venture to invite men of science +who live on the seacoast +to follow this study. +It presents a fine subject for +research, either in the interpretation +of the phenomenon +or in the establishment of +the relations between these +movements and meteorological +conditions.''\footnote + {\Title{Seiches et Vibrations des Lacs et de la Mer}, 1879, p.~5.} +%[** TN: Figure wrapped in the original] +\TallFig{14}{Vibrations due to Steamers}{png} +\PageSep{46} + +These vibrations are obviously due to the wind +or to steamers, but it is a matter of no little surprise +that such insignificant causes should produce +even very small waves of half a mile to a +mile in length. + +The manner in which this is brought about is +undoubtedly obscure, yet it is possible to obtain +some sort of insight into the way in which these +long waves arise. When a stone falls into calm +water waves of all sorts of lengths are instantaneously +generated, and the same is true of +any other isolated disturbance. Out of all these +waves the very long ones and the very short +ones are very small in height. Theoretically, +waves of infinitely great and of infinitely small +lengths, yet in both cases of infinitely small +heights, are generated at the instant of the impulse, +but the waves of enormous length and +those of very small length are of no practical +importance, and we need only consider the moderate +waves. For the shorter of these the water +is virtually deep, and so they will each travel +outwards at a pace dependent on length, the +longer ones outstripping the shorter ones. But +for the longer waves the water will be shallow, +and they will all travel together. Thus the general +effect at a distance is the arrival of a long +wave first, followed by an agitated rippling. +The point which we have to note is that an isolated +disturbance will generate long waves and +\PageSep{47} +that they will run ahead of the small ones. It +is important also to observe that the friction of +the water annuls the oscillation in the shorter +waves more rapidly than it does that of the +longer ones, and therefore the long waves are +more persistent. Now we may look at the disturbance +due to a steamer or to the wind as consisting +of a succession of isolated disturbances, +each of which will create long waves outstripping +the shorter ones. These considerations afford a +sort of explanation of what is observed, but I do +not understand how it is that the separation of +the long from the short waves is so complete, nor +what governs the length of the waves, nor have +I made any attempt to evaluate the greater rapidity +of decrease of short waves than long ones.\footnote + {See, however, S.~S. Hough, \Title{Proc.\ Lond.\ Math.\ Soc.}, xxviii.\ +\index{Hough, S. S.!frictional extinction of waves}% + p.~276.} +It must then be left to future investigators to +elucidate these points. + +\TB + +The subject of seiches and vibrations clearly +affords an interesting field for further research. +The seiches of Lake George in New South Wales +have been observed by Mr.~Russell, the government +\index{Russell, observation of seiches in New South Wales}% +astronomer at Sydney; but until last year +they do not seem to have been much studied on +any lakes outside of Switzerland. The great +lakes of North America are no doubt agitated by +seiches on a much larger scale than those on the +\PageSep{48} +comparatively small basin of Geneva. This idea +appears to have struck Mr.~Napier Denison of +\index{Denison, F. Napier, vibrations and seiches on lakes|(}% +Toronto, and he has been so fortunate as to enlist +the interest of Mr.~Bell Dawson, the chief of +\index{Dawson coöperates in investigation of seiches}% +the Canadian Tidal Survey, and of Mr.~Stupart, +\index{Stupart coöperates in investigation of seiches}% +the director of the Meteorological Department. +Mr.~Denison's attention has been, in the first instance, +principally directed towards those notches +in tide-curves which have afforded the occasion +for the present discussion of this subject. He +has made an interesting suggestion as to the +origin of these oscillations, which I will now +explain. + +The wind generally consists of a rather shallow +\index{Waves!in atmosphere|(}% +current, so that when it is calm at the earth's +surface there is often a strong wind at the top +of a neighboring mountain; or the wind aloft +may blow from a different quarter from that below. +If we ascend a mountain or go up in a +balloon, the temperature of the air falls on the +average by a certain definite number of degrees +per thousand feet. But the normal rate of fall +of temperature is generally interrupted on passing +into an upper current, which blows from a +different direction. This abrupt change of temperature +corresponds with a sudden change of +density, so that the upper layer of air must be +regarded as a fluid of different density from that +of the lower air, over which it slides. + +Now Helmholtz has pointed out that one layer +\index{Atmospheric waves, Helmholtz on|(}% +\index{Helmholtz!on atmospheric waves|(}% +\PageSep{49} +of fluid cannot slide over another, without generating +waves at the surface of separation. We +are familiar with this fact in the case of sea-waves +generated by wind. A mackerel sky +\index{Mackerel sky, evidence of air-waves}% +proves also the applicability to currents of air of +Helmholtz's observation. In this case the moisture +of the air is condensed into clouds at the +crests of the air waves, and reabsorbed in the +hollows, so that the clouds are arranged in a visible +ripple-mark. A mackerel sky is not seen in +stormy weather, for it affords proof of the existence +of an upper layer of air sliding with only +moderate velocity over a lower layer. The distance +from crest to crest must be considerable +as measured in yards, yet we must regard the +mackerel sky as a mere ripple formed by a slow +relative velocity of the two layers. If this is so, +it becomes of interest to consider what wave-lengths +may be expected to arise when the upper +current is moving over the lower with a speed of +perhaps a hundred miles an hour. The problem +is not directly soluble, for even in the case of +sea-waves it is impossible to predict the wave-lengths. +We do know, however, that the duration +of the wind and the size of the basin are +material circumstances, and that in gales in the +open ocean the waves attain a very definite magnitude. + +Although the problem involved is not a soluble +one, yet Helmholtz has used the analogy of +\PageSep{50} +oceanic waves for an approximate determination +of the sizes of the atmospheric ones. His +method is a very fertile one in many complex +physical investigations, where an exact solution +is not attainable. The method may be best illustrated +by one or two simple cases. + +It is easy for the mathematician to prove that +the period of a swing of a simple pendulum must +vary as the square root of its length. The proof +does not depend on the complete solution of the +problem, so that even if it were insoluble he +would still be sure of the correctness of his conclusion. +If, then, a given pendulum is observed +to swing in a certain period, it is certain that a +similar pendulum of four times the length will +take twice as long to perform its oscillation. In +the same way, the engine power required for a +ship is determinable from experiments on the +resistance suffered by a small model when towed +through the water. The correct conclusion is +discovered in this case, although it is altogether +impossible to discover the resistance of a ship +by \textit{à ~priori} reasoning. + +The wave motion at the surface separating +two fluids of different densities presents another +problem of the same kind, and if the result is +known in one case, it can be confidently predicted +in another. Now oceanic waves generated +\index{Waves!in atmosphere|)}% +by wind afford the known case, and Helmholtz +has thence determined by analogy the +\PageSep{51} +lengths of the atmospheric waves which must +exist aloft. By making plausible suppositions +as to the densities of the two layers of air and +as to their relative velocity, he has shown that +sea-waves of ten yards in length will correspond +with air-waves of as much as twenty miles. A +wave of this length would cover the whole sky, +and might have a period of half an hour. It is +clear then that mackerel sky will disappear in +stormy weather, because we are too near to the +crests and furrows to observe the orderly arrangement +of the clouds. + +Although the waves are too long to be seen as +such, yet the unsteadiness of the barometer in a +gale of wind affords evidence of the correctness +of this theory. In fact, when the crest of denser +air is over the place of observation the barometer +rises, and it falls as the hollow passes. The +waves in the continuous trace of the barometer +have some tendency to regularity, and have +periods of from ten minutes to half an hour. +The analogy seems to be pretty close with the +confused and turbulent sea often seen in a gale +of wind in the open ocean.\footnote + {A gust of wind will cause the barometer to vary, without a + corresponding change in the density of the air. It is not therefore + safe to interpret the oscillations of the barometer as being + due entirely to true changes of pressure. If, however, the intermittent + squalls in a gale are connected with the waves aloft, + the waviness of the barometric trace would still afford signals + of the passage of crests and hollows above.} +\index{Atmospheric waves, Helmholtz on|)}% +\index{Helmholtz!on atmospheric waves|)}% +\PageSep{52} + +Mr.~Denison's application of this theory consists +in supposing that the vibrations of the sea +and of lakes are the response of the water to +variations in the atmospheric pressure. The sea, +being squeezed down by the greater pressure, +should fall as the barometer rises, and conversely +should rise as the barometer falls. He is engaged +in a systematic comparison of the simultaneous +excursions of the water and of the barometer +on Lake Huron. Thus far the evidence +seems decidedly favorable to the theory. He +concludes that when the water is least disturbed, +so also is the barometric trace; and that when +the undulations of the lake become large and +rapid, the atmospheric waves recorded by the +barometer have the same character. There is +also a considerable degree of correspondence +between the periods of the two oscillations. The +smaller undulations of the water correspond with +the shorter air-waves, and are magnified as they +run into narrower and shallower places, so as to +make conspicuous ``vibrations.'' + +It is interesting to note that the vibrations of +the water have a tendency to appear before those +in the barometer, so that they seem to give a +warning of approaching change of weather. It +is thus not impossible that we here have the +foreshadowing of a new form of meteorological +instrument, which may be of service in the forecasting +of the weather. +\PageSep{53} + +I must, however, emphasize that these conclusions +are preliminary and tentative, and that +much observation will be needed before they can +be established as definite truths. Whatever +may be the outcome, the investigation appears +promising, and it is certainly already interesting. +\index{Denison, F. Napier, vibrations and seiches on lakes|)}% +\index{Lakes!vibrations|)}% +\index{Vibration of lakes|)}% + +\begin{Authorities} +Papers by Dr.~Forel on Seiches. +\index{Forel!list of papers}% + +\Journal ``Bibliothèque Universelle, Archives des Sciences physiques +et naturelles,'' Geneva:--- + +\Paper{\Title{Formule des Seiches}, 1876.} + +\Paper{\Title{Limnimètre Enregistreur}, 1876.} + +\Paper{\Title{Essai monographique}, 1877.} + +\Paper{\Title{Causes des Seiches}, \DPchg{Sept.~15}{15~Sept.}, 1878.} + +\Paper{\Title{Limnographe}, 15~Déc., 1878.} + +\Paper{\Title{Seiche du \emph{20}~Février, \emph{1879}}, 15~Avril, 1879.} + +\Paper{\Title{Seiches dicrotes}, 15~Jan., 1880.} + +\Paper{\Title{Formules des Seiches}, 15~Sept., 1885.} + +\Journal ``Bulletin de la Soc.\ Vaudoise des Sciences naturelles:''--- + +\Paper{\Title{Première Étude}, 1873.} + +\Paper{\Title{Deuxième Étude}, 1875.} + +\Paper{\Title{Limnimétrie du Lac Léman}. I\iere~Série. Bull.~xiv.\ 1877. +II\ieme~Série. Bull.~xv. III\ieme~Série. Bull.~xv. 1879.} + +\Journal ``Actes de la Soc.\ helv.\ Andermatt:''--- + +\Paper{\Title{Les Seiches, Vagues d'Oscillation}, 1875.} + +\Journal ``Association Française pour l'avancement,'' etc.:--- + +\Paper{\Title{Seiches et Vibrations}, Congrès de Montpelier, 1879.} + +\Journal ``Annales de Chimie et de Physique:''--- + +\Paper{\Title{Les Seiches, Vagues d'Oscillation}, 1876.} + +\Paper{\Title{Un Limnimètre Enregistreur}, 1876.} + +\TB + +Helmholtz, Sitzungsberichte der Preuss.\ Akad.\ der Wissenschaft, +July~25, 1889; transl.\ by Abbe in \Title{Smithsonian Reports}. +\PageSep{54} + +F.~Napier Denison:--- + +\Paper{\Title{Secondary Undulations~\dots\ found in Tide-Gauges.} ``Proc.\ +Canadian Institute,'' Jan.~16, 1897.} + +\Paper{\Title{The Great Lakes as a Sensitive Barometer.} ``Proc.\ Canadian +Institute,'' Feb.~6, 1897.} +\index{Lakes!seiches in|)}% + +\Paper{Same title, but different paper, ``Canadian Engineer,'' Oct.\ +and Nov., 1897.} +\end{Authorities} +\index{Forel!list of papers}% +\PageSep{55} + + +%[** TN: Footnote mark handled by \Chapter logic] +\Chapter{III} +{Tides in Rivers---Tide Mills} + +\footnotetext{The account of the bore in this chapter appeared as an + article in the \Title{Century Magazine} for August,~1898. The illustrations + then used are now reproduced, through the courtesy of + the proprietors.} + +\First{Since} most important towns are situated on +\index{Rivers!tide wave in|(}% +rivers or on estuaries, a large proportion of our +tidal observations relates to such sites. I shall +therefore now consider the curious, and at times +very striking phenomena which attend the rise +and fall of the tide in rivers. + +The sea resembles a large pond in which the +water rises and falls with the oceanic tide, and a +river is a canal which leads into it. The rhythmical +rise and fall of the sea generate waves +which would travel up the river, whatever were +the cause of the oscillation of the sea. Accordingly, +a tide wave in a river owes its origin +directly to the tide in the sea, which is itself +produced by the tidal attractions of the sun and +moon. + +We have seen in \Ref{Chapter}{II}.\ that long waves +progress in shallow water at a speed which depends +only on the depth of the water, and that +\PageSep{56} +waves are to be considered as long when their +length is at least twice the depth of the water. +Now the tide wave in a river is many hundreds +of times as long as the depth, and it must therefore +progress at a speed dependent only on the +depth. That speed is very slow compared with +the motion of the great tide wave in the open +ocean. + +The terms ``ebb'' and ``flow'' are applied to +\index{Ebb and flow defined}% +\index{Flow and ebb defined}% +tidal currents. The current ebbs when the +water is receding from the land seaward, and +flows when it is approaching the shore. On the +open seacoast the water ebbs as the water-level +falls, and it flows as the water rises. Thus at +high and low tide the water is neither flowing +landward nor ebbing seaward, and we say that +it is slack or dead. In this case ebb and flow +are simultaneous with rise and fall, and it is not +uncommon to hear the two terms used synonymously; +but we shall see that this usage is incorrect. + +I begin by considering the tidal currents in a +river of uniform depth, so sluggish in its own +proper current that it may be considered as a +stagnant canal, and the only currents to be considered +are tidal currents. At any point on the +river bank there is a certain mean height of +water, such that the water rises as much above +that level at high water as it falls below it at +low water. The law of tidal current is, then, +\index{Currents, tidal, in rivers}% +\PageSep{57} +very simple. Whenever the water stands above +the mean level the current is up-stream and progresses +along with the tide wave; and whenever +it stands below mean level the current is down-stream +and progresses in the direction contrary +to the tide wave. Since the current is up-stream +when the water is higher than the mean, and +down-stream when it is lower, it is obvious that +when it stands exactly at mean level the current +is neither up nor down, and the water is slack +or dead. Also, at the moment of high water +the current is most rapid up-stream, and at low +water it is most rapid down-stream. Hence the +tidal current ``flows'' for a long time after high +water has passed and when the water-level is +falling, and ``ebbs'' for a long time after low +water and when the water-level is rising. + +The law of tidal currents in a uniform canal +communicating with the sea is thus very different +from that which holds on an open seacoast, +where slack water occurs at high and at low +water, instead of at mean water. But rivers +gradually broaden and become deeper as they +approach the coast, and therefore the tidal currents +in actual estuaries must be intermediate +between the two cases of the open seacoast and +the uniform canal. + +A river has also to deliver a large quantity of +water into the sea in the course of a single tidal +oscillation, and its own proper current is superposed +\PageSep{58} +on the tidal currents. Hence in actual +rivers the resultant current continues to flow up +stream after high water is reached, with falling +water-level, but ceases flowing before mean water-level +is reached, and the resultant current ebbs +down-stream after low water, and continues to +ebb with the rising tide until mean water is +reached, and usually for some time afterward. +The downward stream, in fact, lasts longer than +the upward one. The moments at which the +currents change will differ in each river according +to the depth, the rise and fall of the tide at +the mouth, and the amount of water delivered +by the river. An obvious consequence of this +is that in rivers the tide rises quicker than it +falls, so that a shorter time elapses between low +water and high water than between high water +and low water. + +The tide wave in a river has another peculiarity +of which I have not yet spoken. The complete +theory of waves would be too technical for a book +of this sort, and I must ask the reader to accept +as a fact that a wave cannot progress along a +river without changing its shape. The change +is such that the front slope of the wave gradually +gets steeper, and the rear slope becomes more +gradual. This is illustrated in~\fig{15}, which +shows the progress of a train of waves in shallow +water as calculated theoretically. If the +steepening of the advancing slope of a wave +\PageSep{59} +were carried to an extreme, the wave would present +the form of a wall of water; but the mere +advance of a wave into shallow water would by +itself never suffice to produce so great a change +of form without the concurrence of the natural +\Figure{15}{Progressive Change of a Train +of Waves in Shallow Water}{png} +stream of the river. The downward current in +the river has, in fact, a very important influence +in heading the sea-water back, and this coöperates +with the natural change in the shape of a +wave as it runs into shallow water, so as to exaggerate +the steepness of the advancing slope of +the wave. + +There are in the estuaries of many rivers +\index{Rivers!tide wave in|)}% +broad flats of mud or sand which are nearly dry +at low water, and in such situations the tide not +unfrequently rises with such great rapidity that +the wave assumes the form of a wall of water. +This sort of tide wave is called a ``bore,'' and in +\index{Bore!definition}% +French \textit{mascaret}. Notwithstanding the striking +nature of the phenomenon, very little has been +published on the subject, and I know of only one +series of systematic observations of the bore. +As the account to which I refer is contained in +the official publications of the English Admiralty, +it has probably come under the notice of only a +\PageSep{60} +small circle of readers. But the experiences of +the men engaged in making these observations +were so striking that an account of them should +prove of interest to the general public. I have, +moreover, through the kindness of Admiral Sir +William Wharton and of Captain Moore, the +\index{Moore, Captain!survey of Tsien-Tang-Kiang|(}% +advantage of supplementing verbal description +by photographs. + +The estuary on which the observations were +made is that of the Tsien-Tang-Kiang, a considerable +\index{Tsien-Tang-Kiang, the bore in|(}% +river which flows into the China Sea about +sixty miles south of the great Yang-Tse-Kiang. +At most places the bore occurs only intermittently, +but in this case it travels up the river at +every tide. The bore may be observed within +seventy miles of Shanghai, and within an easy +walk of the great city of Hangchow; and yet +\index{Hangchow, the bore at|(}% +nothing more than a mere mention of it is to be +found in any previous publication. + +In 1888 Captain Moore, R.~N., in command +of Her Majesty's surveying ship Rambler, +thought that it was desirable to make a thorough +survey of the river and estuary. He returned +to the same station in~1892; and the account +which I give of his survey is derived from reports +drawn up after his two visits. The annexed +sketch-map shows the estuary of the +Tsien-Tang, and the few places to which I shall +have occasion to refer are marked thereon. + +On the morning of September~19, 1888, the +\PageSep{61} +Rambler was moored near an island, named +after the ship, to the southwest of Chapu Bay; +and on the~20th the two steam cutters Pandora +and Gulnare, towing the sailing cutter +\Figure{16}{Chart of the Estuary of the Tsien-Tang-Kiang}{png} +Brunswick, left the ship with instruments for +observing and a week's provisions. + +Captain Moore had no reason to suspect that +the tidal currents would prove dangerous out +in the estuary, and he proposed to go up the +estuary about thirty miles to Haining, and then +follow the next succeeding bore up-stream to +Hangchow. Running up-stream with the flood, +all went well until about~11.30, when they were +about fifteen miles southwest by west of Kanpu. +The leading boat, the Pandora, here grounded, +and anchored quickly, but swung round violently +as far as the keel would let her. The other +boats, being unable to stop, came up rapidly; +and the Gulnare, casting off the Brunswick, +\PageSep{62} +struck the Pandora, and then drove on to and +over the bank, and anchored. The boats soon +floated in the rising flood, and although the engines +of the steam cutters were kept going +full speed, all three boats dragged their anchors +in an eleven-knot stream. When the flood +slackened, the three boats pursued their course +to the mouth of the river, where they arrived +about 4~\PM. The ebb was, however, so violent +that they were unable to anchor near one another. +Their positions were chosen by the advice of +some junkmen, who told Captain Moore, very +erroneously as it turned out, that they would be +safe from the night bore. + +The night was calm, and at~11.29 the murmur +of the bore was heard to the eastward; it could +be seen at~11.55, and passed with a roar at~12.20, +well over toward the opposite bank, as predicted +by the Chinese. The danger was now supposed +to be past; but at~1~\AM\ a current of extreme +violence caught the Pandora, and she had +much difficulty to avoid shipwreck. In the +morning it was found that her rudder-post and +propeller-guard were broken, and the Brunswick +and Gulnare were nowhere to be seen. +They had, in fact, been in considerable danger, +and had dragged their anchors three miles up +the river. At 12.20~\AM\ they had been struck +by a violent rush of water in a succession of big +ripples. In a few moments they were afloat in +\PageSep{63} +an eight-knot current; in ten minutes the water +rose nine feet, and the boats began to drag their +anchors, although the engines of the Gulnare +were kept going full speed. After the boats had +dragged for three miles, the rush subsided, and +when the anchor was hove up the pea and the +greater part of the chain were as bright as polished +silver. + +This account shows that all the boats were in +imminent danger, and that great skill was needed +to save them. After this experience and warning, +the survey was continued almost entirely +from the shore. + +The junks which navigate the river are well +aware of the dangers to which the English boats +were exposed, and they have an ingenious method +of avoiding them. At various places on the +bank of the river there are shelter platforms, of +which I show an illustration in~\fig{17}. Immediately +after the passing of the bore the +\index{Bore!bore-shelter}% +junks run up-stream with the after-rush and +make for one of these shelters, where they allow +themselves to be left stranded on the raised +platform shown in the picture. At the end of +this platform there is a sort of round tower +jutting out into the stream. The object of this +is to deflect the main wave of the bore so as to +protect the junks from danger. After the passage +of the bore, the water rises on the platform +very rapidly, but the junks are just able to float +\PageSep{64} +in safety. Captain Moore gives a graphic account +of the spectacle afforded by the junks as +they go up-stream, and describes how on one +occasion he saw no less than thirty junks swept +\Figure{17}{Bore-Shelter on the Tsien-Tang-Kiang}{jpg} +up in the after-rush, at a rate of ten knots, past +the town of Haining toward Hangchow, with all +sail set but with their bows in every direction. + +Measurements of the water-level were made +in the course of the survey, and the results, in +the form of a diagram, \fig{18}, exhibit the nature +of the bore with admirable clearness. The +observations of water-level were taken simultaneously +at three places, viz., Volcano Island +in the estuary, Rambler Island near the mouth +of the river, and Haining, twenty-six miles up +the river. In the figure, the distance between +\PageSep{65} +the lines marked Rambler and Volcano represents +fifty-one miles, and that between Rambler +and Haining twenty-six miles. The vertical +scales show the height of water, measured in +feet, above and below the mean level of the +water at these three points. The lines joining +these vertical scales, marked with the hours of +the clock, show the height of the water simultaneously. +The hour of~8.30 is indicated by +the lowest line it shows that the water was +one foot below mean level at Volcano Island, +twelve feet below at Rambler Island, and eight +feet below at Haining. Thus the water sloped +down from Haining to Rambler, and from Volcano +to Rambler; the water was running up the +estuary toward Rambler Island, and down the +estuary to the same point. At 9~and at~9.30 +there was no great change, but the water had +risen two or three feet at Volcano Island and at +Rambler Island. By ten~o'clock the water was +rising rapidly at Rambler Island, so that there +was a nearly uniform slope up the river from +Volcano Island to Haining. The rise at Rambler +Island then continued to be very rapid, +while the water at Haining remained almost +stationary. This state of affairs went on until +midnight, by which time the water had risen +twenty-one feet at Rambler Island, and about +six feet at Volcano Island, but had not yet risen +at all at Haining. No doubt through the whole +\PageSep{66} +of this time the water was running down the +river from Haining towards its mouth. It is +clear that this was a state of strain which could +not continue long, for there was over twenty +feet of difference of level between Rambler +Island, outside, and Haining, in the river. Almost +exactly at midnight the strain broke down +and the bore started somewhere between Rambler +\index{Bore!diagram of rise in Tsien-Tang}% +Island and Kanpu, and rushed up the river +in a wall of water twelve feet high. This result +is indicated in the figure by the presence of two +lines marked ``midnight.'' After the bore had +\Figure{18}{Diagram of the Flow of the Tide on +the Tsien-Tang-Kiang}{jpg} +passed there was an after-rush that carried the +water up eight feet more. It was on this that +the junks were swept up the stream, as already +described. At~1.30 the after-rush was over, +\PageSep{67} +but the water was still somewhat higher at +Rambler Island than at Haining, and a gentle +current continued to set up-stream. The water +then began to fall at Rambler Island, while it +continued to rise at Haining up to three o'clock. +At this point the ebb of the tide sets in. I do +not reproduce the figure which exhibits the fall +of the water in the ebbing tide, for it may suffice +to say that there is no bore down-stream, +\index{Bore!pictures}% +although there is at one time a very violent +current. + +In 1892 Captain Moore succeeded, with considerable +\index{Moore, Captain!illustrations of bore}% +difficulty, in obtaining photographs of +the bore as it passed Haining. They tell more +of the violence of the wave than could be conveyed +by any amount of description. The photographs, +reproduced in~\fig{19}, do not, however, +show that the broken water in the rear of the +crest is often disturbed by a secondary roller, or +miniature wave, which leaps up, from time to +time, as if struck by some unseen force, and disappears +in a cloud of spray. These breakers +were sometimes twenty to thirty feet above the +level of the river in front of the bore. + +The upper of these pictures is from a photograph, +taken at a height of twenty-seven feet +above the river, as the bore passed Haining on +October~10, 1892. The height of this bore was +eleven feet. The lower pictures, also taken at +Haining, represent the passage of the bore on +\PageSep{68} +October~9, 1892. The first of these photographs +was taken at 1.29~\PM, and the second +represents the view only one minute later. + +The Chinese regard the bore with superstitious +\index{Bore!Chinese superstition|(}% +\index{Chinese!superstition as to bore|(}% +reverence, and their explanation, which I quote +from Captain Moore's report, is as follows: +``Many hundred years ago there was a certain +general who had obtained many victories over +the enemies of the Emperor, and who, being +constantly successful and deservedly popular +among his countrymen, excited the jealousy of +his sovereign, who had for some time observed +with secret wrath his growing influence. The +Emperor accordingly caused him to be assassinated +and thrown into the Tsien-Tang-Kiang, +where his spirit conceived the idea of revenging +itself by bringing the tide in from the ocean in +such force as to overwhelm the city of Hangchow, +then the magnificent capital of the empire. +As my interpreter, who has been for some years +in America, put it, `his sowl felt a sort of ugly-like +arter the many battles he had got for the +Emperor.' The spirit so far succeeded as to +flood a large portion of the country, when the +Emperor, becoming alarmed at the distress and +loss of property occasioned, endeavored to enter +into a sort of compact with it by burning paper +and offering food upon the sea-wall. This, however, +did not have the desired effect, as the high +tide came in as before; and it was at last determined +\PageSep{69} +\ifthenelse{\boolean{ForPrinting}}{% + \TallFig[0.7]{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{jpg} +}{% + \Figure[0.75]{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{jpg} +} +\index{Wharton, Sir W. J., illustration of bore}% +\PageSep{70} +to erect a pagoda at the spot where the +worst breach in the embankment had been made. +Hence the origin of the Bhota Pagoda. A +pagoda induces the good \textit{fungshui}, or spirit. +After it was built the flood tide, though it still +continued to come in the shape of a bore, did +not flood the country as before.'' + +We ``foreign devils'' may take the liberty of +suspecting that the repairs to the embankment +had also some share in this beneficial result. + +This story is remarkable in that it refers to +the reign of an Emperor whose historical existence +is undoubted. It thus differs from many +of the mythical stories which have been invented +by primitive peoples to explain great natural +phenomena. There is good reason to suppose, +in fact, that this bore had no existence some centuries +ago; for Marco Polo, in the thirteenth +\index{Marco Polo, resident of Hangchow}% +century, stayed about a year and a half at +Hangchow, and gives so faithful and minute +\index{Hangchow, the bore at|)}% +an account of that great town that it is almost +impossible to believe that he would have omitted +to notice a fact so striking. But the Emperor +referred to in the Chinese legend reigned some +centuries before the days of Marco Polo, so that +we have reason to believe that the bore is intermittent. +\index{Bore!Chinese superstition|)}% +I have also learned from Captain +Moore himself that at the time of the great +\index{Moore, Captain!survey of Tsien-Tang-Kiang|)}% +Taiping rebellion, the suppression of which was +principally due to ``Chinese'' Gordon, the intensity +\index{Chinese!superstition as to bore|)}% +\index{Tsien-Tang-Kiang, the bore in|)}% +\PageSep{71} +of the bore was far less than it is to-day. +\index{Bore!rivers where found}% +This shows that the bore is liable to great variability, +according as the silting of the estuary +changes. + +The people at Haining still continue to pay +religious reverence to the bore, and on one of +the days when Captain Moore was making observations +some five or six thousand people assembled +on the river-wall to propitiate the god of +the waters by throwing in offerings. This was +the occasion of one of the highest bores at spring +tide, and the rebound of the bore from the sea-wall, +and the sudden heaping up of the waters +as the flood conformed to the narrow mouth of +the river, here barely a mile in width at low +water, was a magnificent spectacle. A series of +breakers were formed on the back of the advancing +flood, which for over five minutes were not +less than twenty-five feet above the level of the +river in front of the bore. On this occasion +Captain Moore made a rough estimate that a +million and three quarters of tons of water passed +the point of observation in one minute. + +The bore of which I have given an account is +perhaps the largest known; but relatively small +ones are to be observed on the Severn and Wye +\index{Severn, bore in the}% +\index{Wye, bore in the}% +in England, on the Seine in France, on the Petitcodiac +\index{Petitcodiac, bore in the}% +\index{Seine, bore in the}% +in Canada, on the Hugli in India, and +\index{Hugli, bore on the}% +doubtless in many other places. In general, +however, it is only at spring tides and with certain +\PageSep{72} +winds that the phenomenon is at all striking. +In September,~1897, I was on the banks of the +Severn at spring tide; but there was no proper +bore, and only a succession of waves up-stream, +\index{Bore!causes}% +and a rapid rise of water-level. + +I have shown, at the beginning of this chapter, +that the heading back of the sea water by +the natural current of a river, and the progressive +change of shape of a wave in shallow water combine +to produce a rapid rise of the tide in rivers. +But the explanation of the bore, as resulting +from these causes, is incomplete, because it leaves +their relative importance indeterminate, and +serves rather to explain a rapid rise than an absolutely +sudden one. I think that it would be +impossible, from the mere inspection of an estuary, +to say whether there would be a bore there; +we could only say that the situation looked +promising or the reverse. + +The capriciousness of the appearance of the +bore proves in fact that it depends on a very nice +balance between conflicting forces, and the irregularity +in the depth and form of an estuary renders +the exact calculation of the form of the +rising tide an impossibility. It would be easy +to imitate the bore experimentally on a small +scale; but, as in many other physical problems, +we must rest satisfied with a general comprehension +of the causes which produce the observed +result. +\PageSep{73} + +The manner in which the Chinese avail themselves +of the after-rush for ascending the river +affords an illustration of the utilization by mankind +of tidal energy. In going up-stream, a +\index{Energy, tidal, utilization of}% +barge, say of one hundred tons, may rise some +twenty or thirty feet. There has, then, been +done upon that barge a work of from two to +three thousand foot-tons. Whence does this +energy come? Now, I say that it comes from +the rotation of the earth; for we are making the +tide do the work for us, and thus resisting the +tidal movement. But resistance to the tide has +the effect of diminishing the rate at which the +earth is spinning round. Hence it is the earth's +rotation which carries the barge up the river, and +we are retarding the earth's rotation and making +the day infinitesimally longer by using the tide +in this way. This resistance is of an analogous +character to that due to tidal friction, the consideration +of which I must defer to a future +chapter, as my present object is to consider the +uses which may be made of tidal energy. + +It has been supposed by many that when the +coal supply of the world has been exhausted we +shall fall back on the tides to do our work. But +a little consideration will show that although this +source of energy is boundless, there are other far +more accessible funds on which to draw. + +I saw some years ago a suggestion that the +rise and fall of old hulks on the tide would afford +\PageSep{74} +\index{Energy, tidal, utilization of}% +serviceable power. If we picture to ourselves the +immense weight of a large ship, we may be deluded +for a moment into agreement with this +project, but numerical calculation soon shows its +futility. The tide takes about six hours to rise +from low water to high water, and the same +period to fall again. Let us suppose that the +water rises ten feet, and that a hulk of $10,000$ +tons displacement is floating on it; then it is +easy to show that only twenty horse-power will +be developed by its rise and fall. We should +then require ten such hulks to develop as much +work as would be given by a steam engine of +very moderate size, and the expense of the installation +would be far better bestowed on water-wheels +in rivers or on wind-mills. I am glad to +\index{Mills worked by the tide}% +say that the projector of this scheme gave it up +when its relative insignificance was pointed out +to him. It is the only instance of which I ever +heard where an inventor was deterred by the impracticability +of his plan. + +We may, then, fairly conclude that, with existing +mechanical appliances, the attempt to utilize +the tide on an open coast is futile. But +where a large area of tidal water can be easily +trapped at high water, its fall may be made to +work mill-wheels or turbines with advantage. +The expense of building long jetties to catch the +water is prohibitive, and therefore tide mills are +only practicable where there exists an easily +\PageSep{75} +adaptable configuration of shoals in an estuary. +There are, no doubt, many such mills in the +\index{Mills worked by the tide}% +world, but the only one which I happen to have +seen is at Bembridge, in the Isle of Wight. At +this place embankments formed on the natural +shoals are furnished with lock-gates, and inclose +many acres of tidal water. The gates open automatically +with the rising tide, and the incipient +outward current at the turn of the tide closes +the gates again, so that the water is trapped. +The water then works a mill wheel of moderate +size. When we reflect on the intermittence of +work from low water to high water and the great +inequality of work with springs and neaps, it +may be doubted whether this mill is worth the +expense of retaining the embankments and lock-gates. + +We see then that, notwithstanding the boundless +energy of the tide, rivers and wind and fuel +are likely for all time to be incomparably more +important for the use of mankind. + +\begin{Authorities} +On waves in rivers see Airy's article on \Title{Tides and Waves} in +\index{Airy, Sir G. B.!tides in rivers}% +\index{Rivers!Airy on tide in}% +the ``Encyclopædia Metropolitana.'' Some of his results will +also be found in the article \Title{Tides} in the ``Encyclopædia Britannica.'' + +Commander Moore, R.~N., \Title{Report on the Bore of the Tsien-Tang-Kiang.} +Sold by Potter, Poultry, London,~1888. + +\Title{Further Report},~\&c., by the same author and publisher,~1893. +\end{Authorities} +\PageSep{76} + + +\Chapter{IV} +{Historical Sketch} + +\First{I Cannot} claim to have made extensive investigations +\index{History!of tidal theories|(}% +as to the ideas of mankind at different +periods on the subject of the tides, but I propose +in the present chapter to tell what I have +been able to discover. + +No doubt many mythologies contain stories +explanatory of the obvious connection between +the moon and the tide. But explanations, professing +at least to be scientific, would have been +brought forward at periods much later than +those when the mythological stories originated, +and I shall only speak of the former. + +I have to thank my colleagues at Cambridge +for the translations from the Chinese, Arabic, +\index{Chinese!theories of tide}% +Icelandic, and classical literatures of such passages +as they were able to discover. + +I learn from Professor Giles that Chinese +\index{Giles on Chinese theories of the tide}% +writers have suggested two causes for the tides: +first, that water is the blood of the earth, and +that the tides are the beating of its pulse; and +secondly, that the tides are caused by the earth +breathing. Ko~Hung, a writer of the fourth +century of our era, gives a somewhat obscure +\PageSep{77} +\index{Chinese!theories of tide}% +\index{Giles on Chinese theories of the tide}% +explanation of spring and neap tides. He says +that every month the sky moves eastward and +then westward, and hence the tides are greater +and smaller alternately. Summer tides are said +to be higher than winter tides, because in summer +the sun is in the south and the sky is $15,000$~li +($5,000$~miles) further off, and therefore in +summer the female or negative principle in nature +is weak, and the male or positive principle +strong. + +In China the diurnal inequality is such that +in summer the tide rises higher in the daytime +than in the night, whilst the converse is true +in winter. I suggest that this fact affords the +justification for the statement that the summer +tides are great. + +\TB + +Mr.~E.~G. Browne has translated for me the +\index{Arabian theories of tide|(}% +\index{Browne, E. G., Arabian theories of tide|(}% +following passage from the ``Wonders of Creation'' +of Zakariyy\bar{a} ibn Muhammad ibn Mahm\bar{u}d +al Qazv\bar{i}n\bar{i}, who died in \AD~1283.\footnote + {Wüstenfeld's edition, pp.~103,~104.} + +``Section treating of certain wonderful conditions +of the sea. + +``Know that at different periods of the four +seasons, and on the first and last days of the +months, and at certain hours of the night and +day, the seas have certain conditions as to the +rising of their waters and the flow and agitation +thereof. +\PageSep{78} + +``As to the rising of the waters, it is supposed +that when the sun acts on them it rarefies them, +and they expand and seek a space ampler than +that wherein they were before, and the one part +repels the other in the five directions eastwards, +westwards, southwards, northwards, and upwards, +and there arise at the same time various winds +on the shores of the sea. This is what is said +as to the cause of the rising of the waters. + +``As for the flow of certain seas at the time +of the rising of the moon, it is supposed that at +the bottom of such seas there are solid rocks +and hard stones, and that when the moon rises +over the surface of such a sea, its penetrating +rays reach these rocks and stones which are at +the bottom, and are then reflected back thence; +and the waters are heated and rarefied and seek +an ampler space and roll in waves towards the +seashore~\dots\ and so it continues as long as +the moon shines in mid-heaven. But when she +begins to decline, the boiling of the waters +ceases, and the particles cool and become dense +and return to their state of rest, and the currents +run according to their wont. This goes +on until the moon reaches the western horizon, +when the flow begins again, as it did when the +moon was in the eastern horizon. And this +flow continues until the moon is at the middle +of the sky below the horizon, when it ceases. +Then when the moon comes upward, the flow +\PageSep{79} +begins again until she reaches the eastern horizon. +This is the account of the flow and ebb +of the sea. + +``The agitation of the sea resembles the agitation +of the humours in men's bodies, for verily +as thou seest in the case of a sanguine or bilious +man,~\&c., the humours stirring in his body, and +then subsiding little by little; so likewise the +sea has matters which rise from time to time as +they gain strength, whereby it is thrown into +violent commotion which subsides little by little. +And this the Prophet (on whom be the blessings +of God and his peace) hath expressed in a poetical +manner, when he says: `Verily the Angel, +who is set over the seas, places his foot in the +sea and thence comes the flow; then he raises it +and thence comes the ebb.'\,'' +\index{Arabian theories of tide|)}% +\index{Browne, E. G., Arabian theories of tide|)}% + +\TB + +Mr.~Magnússon has kindly searched the old +\index{Icelandic theory of tides}% +\index{Magnússon on Icelandic theories of tides}% +Icelandic literature for references to the tides. +In the Rimbegla he finds this passage:--- + +``Beda the priest says that the tides follow +the moon, and that they ebb through her blowing +on them, but wax in consequence of her +movement.'' + +And again:--- + +``(At new moon) the moon stands in the way +of the sun and prevents him from drying up the +sea; she also drops down her own moisture. +For both these reasons, at every new moon, the +\PageSep{80} +\index{Icelandic theory of tides}% +\index{Magnússon on Icelandic theories of tides}% +ocean swells and makes those tides which we call +spring tides. But when the moon gets past the +sun, he throws down some of his heat upon +the sea, and diminishes thereby the fluidity of +the water. In this way the tides of the sea +are diminished.'' + +In another passage the author writes:--- + +``But when the moon is opposite to the sun, +the sun heats the ocean greatly, and as nothing +impedes that warmth, the ocean boils and the +sea flood is more impetuous than before---just +as one may see water rise in a kettle when it +boils violently. This we call spring tide.'' + +There seems to be a considerable inconsistency +in explaining one spring tide by the interception +of the sun's heat by the moon, and the next one +by the excess of that heat. + +But it is not necessary to search ancient literature +for grotesque theories of the tides. In +1722 E.~Barlow, gentleman, in ``An Exact Survey +of the Tide,''\footnote + {``The Second Edition, with Curious Maps.'' (London: John + Hooke, 1722.)} +attributes it to the pressure +of the moon on the atmosphere. And theories +not less absurd have been promulgated during +the last twenty years. + +\TB + +The Greeks and Romans, living on the shores +of the Mediterranean, had not much occasion to +learn about the tide, and the passages in classical +\PageSep{81} +\index{Greek!theory and description of tides|(}% +\index{Roman description of tides|(}% +literature which treat of this matter are but +few. But where the subject is touched on we +see clearly their great intellectual superiority over +those other peoples, whose ideas have just been +quoted. + +The only author who treats of the tide in any +\index{Strabo on tides|(}% +detail is Posidonius, and we have to rely for our +\index{Posidonius on tides|(}% +knowledge of his work entirely on quotations +from him by Strabo.\footnote + {My attention was drawn to Strabo by a passage in Sir W. + Thomson's (Lord Kelvin's) Popular Lectures, \Title{The Tides}, vol.~ii. + I have to thank Mr.~Duff for the translations which follow from + Strabo and Posidonius. The work consulted was Bake's \Title{Posidonius} + (Leiden,~1810), but Mr.~Duff tells me that the text is very + corrupt in some places, and he has therefore also consulted a + more recent text.} + +Posidonius says that Aristotle attributed the +\index{Aristotle on tides}% +flow and ebb of the sea at Cadiz to the mountainous +formation of the coast, but he very justly +pronounces this to be nonsense, particularly as +the coast of Spain is flat and sandy. He himself +attributes the tides to the moon's influence, and +the accuracy of his observations is proved by the +following interesting passage from Strabo:\footnotemark--- +\footnotetext{Teubner's \Title{Strabo},~i.\ p.~236.} + +``Posidonius says that the movement of the +ocean observes a regular series like a heavenly +body, there being a daily, monthly, and yearly +movement according to the influence of the +moon. For when the moon is above the (eastern) +horizon by the distance of one sign of the +zodiac (\ie~$30°$) the sea begins to flow, and encroaches +\PageSep{82} +visibly on the land until the moon +reaches the meridian. When she has passed the +meridian, the sea in turn ebbs gradually, until +the moon is above the western horizon by the +distance of one sign of the zodiac. The sea then +remains motionless while the moon is actually +setting, and still more so (\textit{sic}) so long as the +moon is moving beneath the earth as far as a +sign of the zodiac beneath the horizon. Then +the sea again advances until the moon has +reached the meridian below the earth; and retreats +while the moon is moving towards the east, +until she is the distance of a sign of the zodiac +below the horizon; it remains at rest until the +moon is the same distance above the horizon, and +then begins to flow again. Such is the daily +movement of the tides, according to Posidonius. + +``As to their monthly movement, he says that +the ebbs are greatest at the conjunctions [of +the sun and moon], and then grow less until the +time of half moon, and increase again until the +time of full moon, and grow less again until +the moon has waned to half. Then the increase +of the tide follows until the conjunction. But +the increases last longer and come quicker [this +phrase is very obscure]. + +``The yearly movements of the tides he says +he learned from the people of Cadiz. They told +him that the ebb and flow alike were greatest at +the summer solstice. He guesses for himself +\PageSep{83} +that the tides grow less from the solstice to the +equinox, and then increase between the equinox +and the winter solstice, and then grow less until +the spring equinox, and greater until the summer +solstice.'' + +This is an excellent account of the tides at +Cadiz, but I doubt whether there is any foundation +\index{Polibius on tides at Cadiz}% +for that part which was derived from hearsay. +Lord Kelvin remarks, however, that it is interesting +to note that inequalities extending over +the year should have been recognized. + +Strabo also says that there was a spring near +Cadiz in which the water rose and fell, and that +this was believed by the inhabitants, and by +Polybius, to be due to the influence of the ocean +tide, but Posidonius was not of this opinion. +Strabo says:--- + +``Posidonius denies this explanation. He says +there are two wells in the precinct of Hercules at +Cadiz, and a third in the city. Of the two former +the smaller runs dry while people are drawing +water from it, and when they stop drawing water +it fills again; the larger continues to supply +water all day, but, like all other wells, it falls +during the day but is replenished at night, when +the drawing of water has ceased. But since the +ebb tide often coincides with the replenishing of +the well, therefore, says Posidonius, the idle story +of the tidal influence has been believed by the +inhabitants.'' +\PageSep{84} + +Since the wells follow the sun, whilst the tide +follows the moon, the criticism of Posidonius is +a very just one. But Strabo blames him for +distrusting the Cadizians in a simple matter of +everyday experience, whilst accepting their evidence +as to an annual inequality in the tides. + +There is another very interesting passage in +Strabo, the meaning of which was obviously unknown +to the Dutch commentator Bake---and +indeed must necessarily have been unintelligible +to him at the time when he wrote, on account of +the then prevailing ignorance of tidal phenomena +in remoter parts of the world. Strabo +writes:--- + +``Anyhow Posidonius says that Seleucus of +\index{Diurnal inequality!observed by Seleucus}% +\index{Posidonius on tides|)}% +\index{Seleucus, observation of tides of Indian Ocean}% +the Red Sea [also called the Babylonian] declares +that there is a certain irregularity and regularity +in these phenomena [the tides], according +to the different positions [of the moon] in the +zodiac. While the moon is in the equinoctial +signs, the phenomena are regular; but while she +is in the signs of the solstices, there is irregularity +both in the height and speed of the tides, +and in the other signs there is regularity or the +reverse in proportion to their nearness to the solstices +or to the equinoxes.'' + +Now let us consider the meaning of this. +When the moon is in the equinoxes she is on +the equator, and when she is in the solstices she +is at her maximum distances to the north or +\PageSep{85} +south of the equator---or, as astronomers say, in +her greatest north or south declination. Hence +Seleucus means that, when the moon is on the +\index{Seleucus, observation of tides of Indian Ocean}% +equator, the tides follow one another, with two +equal high and low waters a day; but when she +is distant from the equator, the regular sequence +is interrupted. In other words, the diurnal +inequality (which I shall explain in a later chapter) +vanishes when the moon is on the equator, +and is at its maximum when the declination is +greatest. This is quite correct, and since the +diurnal inequality is almost evanescent in the +\index{Diurnal inequality!observed by Seleucus}% +Atlantic, whilst it is very great in the Indian +Ocean, especially about Aden, it is clear that +Seleucus had watched the sea there, just as we +should expect him to do from his place of origin. + +\TB + +Many centuries elapsed after the classical +period before any scientific thought was bestowed +on the tides. Kepler recognized the +\index{Kepler!ideas concerning tides}% +tendency of the water on the earth to move +towards the sun and the moon, but he was unable +to submit his theory to calculation. Galileo +\index{Galileo!blames Kepler for his tidal theory}% +expresses his regret that so acute a man as +Kepler should have produced a theory, which +appeared to him to reintroduce the occult qualities +of the ancient philosophers. His own explanation +referred the phenomenon to the rotation +of the earth, and he considered that it afforded +a principal proof of the Copernican system. +\index{Greek!theory and description of tides|)}% +\index{Roman description of tides|)}% +\index{Strabo on tides|)}% +\PageSep{86} +\index{Kepler!ideas concerning tides}% + +The theory of tide-generating force which will +be set forth in \Ref{Chapter}{V}.\ is due to Newton, +\index{Newton!founder of tidal theory}% +who expounded it in his ``Principia'' in~1687. +His theory affords the firm basis on which all +subsequent work has been laid. + +In 1738 the Academy of Sciences of Paris +offered the theory of the tides as the subject for +a prize. The authors of four essays received +prizes, viz., Daniel Bernoulli, Euler, Maclaurin, +\index{Bernoulli, Daniel, essay on tides}% +\index{Euler, essay on tides}% +\index{Maclaurin!essay on tides}% +and Cavalleri. The first three adopted, not only +\index{Cavalleri, essay on tides}% +the theory of gravitation, but also Newton's +theory to its fullest extent. A considerable +portion of Bernoulli's work is incorporated in +the account of the theory of the tides which I +shall give later. The essays of Euler and Maclaurin +contained remarkable advances in mathematical +knowledge, but did not add greatly to +the theory of the tides. The Jesuit priest +Cavalleri adopted the theory of vortices to explain +the tides, and it is not worth while to +follow him in his erroneous and obsolete speculations. + +Nothing of importance was added to our +knowledge until the great French mathematician +Laplace took up the subject in~1774. It was he +\index{Laplace!theory of tides|(}% +who for the first time fully recognized the difficulty +of the problem, and showed that the earth's +rotation is an essential feature in the conditions. +The actual treatment of the tidal problem is in +effect due to Laplace, although the mode of +\PageSep{87} +presentment of the theory has come to differ +considerably from his. + +Subsequently to Laplace, the most important +workers in this field have been Sir John Lubbock +\index{Lubbock, Sir J., senior, on tides}% +senior, Whewell, Airy, and Lord Kelvin. +\index{Kelvin, Lord!initiates harmonic analysis}% +\index{Whewell!on tides}% +\index{Whewell!empirical construction of tide tables|(}% +The work of Lubbock and Whewell is chiefly +remarkable for the coördination and analysis of +enormous masses of data at various ports, and +the construction of trustworthy tide tables. +Airy contributed an important review of the +whole tidal theory. He also studied profoundly +the theory of waves in canals, and considered +the effects of frictional resistances on the progress +of tidal and other waves. + +Lord Kelvin initiated a new and powerful +method of considering tidal oscillations. His +method possesses a close analogy with that already +used in discussing the irregularities in the +motions of the moon and planets. His merit +consists in the clear conception that the plan of +procedure which has been so successful in the +one case would be applicable to the other. The +difference between the laws of the moon's motion +and those of tidal oscillations is, however, +so great that there is scarcely any superficial +resemblance between the two methods. This +so-called ``harmonic analysis'' of the tides is +\index{Harmonic analysis!initiated by Lord Kelvin}% +daily growing in favor in the eyes of men of +science, and is likely to supersede all the older +methods. I shall explain it in a future chapter. +\PageSep{88} + +Amongst all the grand work which has been +bestowed on this difficult subject, Newton stands +out first, and next to him we must rank Laplace. +However original any future contribution to the +science of tides may be, it would seem as though +it must perforce be based on the work of these +two. The exposition which I shall give hereafter +of the theory of oceanic tides is based on +the work of Newton, Bernoulli, Laplace, and +\index{Bernoulli, Daniel, essay on tides}% +\index{Laplace!theory of tides|)}% +Kelvin, in proportions of which it would be +difficult to assign the relative importance. + +\TB + +The connection between the moon and the +tide is so obvious that long before the formulation +of a satisfactory theory fairly accurate predictions +of the tides were made and published. +On this head Whewell\footnote + {\Title{History of the Inductive Sciences}, 1837, vol.~ii.\ p.~248 \textit{et~seq.}} +\index{History!of tidal theories|)}% +has the following interesting +passage:--- + +``The course which analogy would have recommended +for the cultivation of our knowledge of +tides would have been to ascertain by an analysis +of long series of observations, the effects of +changes in the time of transit, parallax, and +declination of the moon, and thus to obtain the +laws of phenomena; and then to proceed to +investigate the laws of causation. + +``Though this was not the course followed by +mathematical theorists, it was really pursued by +those who practically calculated tide tables; and +\PageSep{89} +the application of knowledge to the useful purposes +of life, being thus separated from the +promotion of the theory, was naturally treated +as a gainful property, and preserved by secrecy. +\dots~Liverpool, London, and other places, had +their tide tables, constructed by undivulged +methods, which methods, in some instances at +least, were handed down from father to son for +several generations as a family possession; and +the publication of new tables accompanied by a +statement of the mode of calculation was resented +as an infringement of the rights of property. + +``The mode in which these secret methods +were invented was that which we have pointed +out,---the analysis of a considerable series of +observations. Probably the best example of this +was afforded by the Liverpool tide tables. These +were deduced by a clergyman named Holden, +from observations made at that port by a harbor +master of the name of Hutchinson, who was +led, by a love of such pursuits, to observe the +tides for above twenty years, day and night. +Holden's tables, founded on four years of these +observations, were remarkably accurate. + +``At length men of science began to perceive +that such calculations were part of their business; +and that they were called upon, as the +guardians of the established theory of the universe, +to compare it in the greatest possible +\PageSep{90} +detail with the facts. Mr.~Lubbock was the +first mathematician who undertook the extensive +labors which such a conviction suggested. Finding +that regular tide observations had been made +at the London docks from~1795, he took nineteen +years of these (purposely selecting the +length of the cycle of the motions of the lunar +orbit), and caused them (in~1831) to be analyzed +by Mr.~Dessiou, an expert calculator. He thus +obtained tables for the effect of the moon's +declination, parallax, and hour of transit, on the +tides; and was enabled to produce tide tables +founded upon the data thus obtained. Some +mistakes in these as first published (mistakes unimportant +as to the theoretical value of the work) +served to show the jealousy of the practical tide +table calculators, by the acrimony with which the +oversights were dwelt upon; but in a very few +years the tables thus produced by an open and scientific +process were more exact than those which +resulted from any of the secrets; and thus practice +was brought into its proper subordination to +theory.'' +\index{Whewell!empirical construction of tide tables|)}% + +\begin{Authorities} +The history from Galileo to Laplace is to be found in the +\Title{Mécanique Céleste} of Laplace, book~xiii, chapter~i. + +The other authorities are quoted in the text or in footnotes. +\end{Authorities} +\PageSep{91} + + +\Chapter{V} +{Tide-generating Force} + +\First{It} would need mathematical reasoning to fully +\index{Centripetal and centrifugal forces|(}% +explain how the attractions of the sun and moon +give rise to tide-generating forces. But as this +\index{Forces!centripetal and centrifugal|(}% +book is not intended for the mathematician, I +must endeavor to dispense with technical language. + +A body in motion will move in a straight line, +unless it is deflected from its straight path by +some external force, and the resistance to the +deflection is said to be due to inertia. The motion +of the body then is equivalent in its effect +to a force which opposes the deflection due to +the external force, and in many cases it is permissible +to abstract our attention from the motion +of the system and to regard it as at rest, if +at the same time we introduce the proper ideal +forces, due to inertia, so that they shall balance +the action of the real external forces. + +If I tie a string to a stone and whirl it round, +the string is thrown into a state of tension. The +natural tendency of the stone, at each instant, is +to move onward in a straight line, but it is continuously +deflected from its straight path by the +\PageSep{92} +tension of the string. In this case the ideal +force, due to inertia, whereby the stone resists +its continuous deflection, is called centrifugal +force. This force is in reality only a substitute +for the motion, but if we withdraw our attention +from the motion, it may be regarded as a reality. + +The centrifugal force is transmitted to my +hand through the string, and I thus experience +an outward or centrifugal tendency. But the +stone itself is continually pulled inward by the +string, and the force is called centripetal. When +a string is under tension, as in this experiment, +it is subject to equal and opposite forces, so that +the tension implies the existence of a pair of +forces, one towards and the other away from the +centre of rotation. The force is to be regarded +as away from the centre when we consider the +sensation of the whirler, and as towards the centre +when we consider the thing whirled. A similar +double view occurs in commerce, where a +transaction which stands on the credit side in the +books of one merchant appears on the debit side +in the books of the other. + +This simple experiment exemplifies the mechanism +by which the moon is kept revolving round +the earth. There is not of course any visible +connection between the two bodies, but an invisible +bond is provided by the attraction of gravity, +which replaces the string which unites the +stone to the hand. The moon, then, whirls +\PageSep{93} +\index{Forces!tide-generating|(}% +\index{Orbit!of moon and earth|(}% +round the earth at just such a rate and at just +\index{Earth and moon!diagram}% +such a distance, that her resistance to circular +motion, called centrifugal force, is counterbalanced +by the centripetal tendency of gravity. If +\index{Centripetal and centrifugal forces|)}% +she were nearer to us the attraction of gravity +would be greater, and she would have to go +round the earth faster, so as to make enough +centrifugal force to counterbalance the greater +\Figure[0.8]{20}{Earth and Moon}{png} +gravity. The converse would be true, and the +moon would go round slower, if she were further +from us. + +The moon and the earth go round the sun in +companionship once in a year, but this annual +motion does not affect the interaction between +them, and we may put aside the orbital motion +of the earth, and suppose the moon and earth to +\index{Moon and earth!diagram}% +be the only pair of bodies in existence. When +the principle involved in a purely lunar tide is +grasped, the action of the sun in producing a +\index{Forces!centripetal and centrifugal|)}% +\PageSep{94} +solar tide will become obvious. But the analogy +of the string and stone is imperfect in one +respect where the distinction is important; the +moon, in fact, does not revolve exactly about +the earth, but about the centre of gravity of +the earth and moon. The earth is eighty times +as heavy as the moon, and so this centre of gravity +is not very far from the earth's centre. The +upper part of \fig{20} is intended to represent a +planet and its satellite; the lower part shows +the earth and the moon in their true proportions. +The upper figure is more convenient for +our present argument, and the planet and satellite +may be described as the earth and the moon, +notwithstanding the exaggeration of their relative +proportions. The point~$G$ is the centre of +gravity of the two, and the axis about which +they revolve passes through~$G$. This point is +sufficiently near to the centre of the earth to +permit us, for many purposes, to speak of the +moon as revolving round the earth. But in the +present case we must be more accurate and must +regard the moon and earth as revolving round~$G$, +their centre of gravity. The moon and earth +are on opposite sides of this point, and describe +circles round it. The distance of the moon's +centre from~$G$ is $237,000$~miles, whilst that of +the earth's centre is only $\DPchg{3000}{3,000}$~miles in the opposite +direction. The $\DPchg{3000}{3,000}$~and $237,000$~miles +together make up the $240,000$~miles which separate +the centres of the two bodies. +\PageSep{95} + +A system may now be devised so as to resemble +the earth and moon more closely than that +of the string and stone with which I began. If +a large stone and a small one are attached to one +another by a light and stiff rod, the system can +be balanced horizontally about a point in the rod +called the centre of gravity~$G$. The two weights +may then be set whirling about a pivot at~$G$, so +that the rod shall always be horizontal. In consequence +of the rotation the rod is brought into +a state of stress, just as was the string in the +first example, and the centripetal stress in the +rod exactly counterbalances the centrifugal force. +The big and the little stones now correspond to +the earth and the moon, and the stress in the rod +plays the same part as the invisible bond of +gravity between the earth and the moon. Fixing +our attention on the smaller stone or moon +at the end of the longer arm of the rod, we see +that the total centrifugal force acting on the +moon, as it revolves round the centre of gravity, +is equal and opposite to the attraction of the +earth on the moon. On considering the short +arm of the rod between the pivot and the big +stone, we see also that the centrifugal force acting +on the earth is equal and opposite to the +attraction of the moon on it. In this experiment +as well as in the former one, we consider +the total of centrifugal force and of attraction, +but every particle of both the celestial bodies is +\index{Orbit!of moon and earth|)}% +\PageSep{96} +\index{Davis, method of presenting tide-generating force}% +acted on by these forces, and accordingly a +closer analysis is necessary. + +It will now simplify matters if we make a supposition +which departs from actuality, introducing +the true conditions at a later stage in the +argument. + +The earth's centre describes a circle about the +centre of gravity~$G$, with a radius of $\DPchg{3000}{3,000}$~miles, +and the period of the revolution is of course one +month. Now whilst this motion of revolution +of the earth's centre continues, let it be supposed +that the diurnal rotation is annulled. As this +is a mode of revolution which differs from that +of a wheel, it is well to explain exactly what is +meant by the annulment of the diurnal rotation. +This is illustrated in~\fig{21}, which shows the +successive positions assumed by an arrow in revolution +without rotation. The shaft of the arrow +always remains parallel to the same direction in +space, and therefore it does not rotate, although +the whole arrow revolves. It is obvious that every +particle of the arrow describes a circle of the +same radius, but that the circles described by +them are not concentric. The circles described +by the point and by the base of the arrow are +shown in the figure, and their centres are separated +by a distance equal to the length of the +arrow. Now the centrifugal force on a revolving +particle acts along the radius of the circle described, +and in this case the radii of the circles +\PageSep{97} +described by any two particles in the arrow are +always parallel. The parallelism of the centrifugal +forces at the two ends of the arrow is +indicated in the figure. Then again, the centrifugal +force must everywhere be equal as well +as parallel, because its intensity depends both on +the radius and on the speed of revolution, and +these are the same for every part. It follows +that if a body revolves without rotation, every +part of it is subject to equal and parallel centrifugal +forces. The same must therefore be +true of the earth when deprived of diurnal rotation. +Accordingly every particle of the idealized +non-rotating earth is continuously subject to +equal and parallel centrifugal forces, in consequence +of the revolution of the earth's centre +in its monthly orbit with a radius of $\DPchg{3000}{3,000}$~miles.\footnote + {I owe the suggestion of this method of presenting the origin + of tide-generating force to Professor Davis of Harvard +\index{Davis, method of presenting tide-generating force}% + University.} + +We have seen that the total of centrifugal +force acting on the whole earth must be just +such as to balance the total of the centripetal +forces due to the moon's attraction. If, then, +the attractional forces, acting on every particle +of the earth, were also equal and parallel, there +would be a perfect balance throughout. We +shall see, however, that although there is a perfect +balance on the whole, there is not that uniformity +\PageSep{98} +which would render the balance perfect +at every particle. + +As far as concerns the totality of the attraction +the analogy is complete between the larger +stone, revolving at the end of the shorter arm +of the rod, and the earth revolving in its small +\Figure[0.7]{21}{Revolution of a Body without Rotation}{png} +orbit round~$G$. But a difference arises when we +compare the distribution of the tension of the +rod with that of the lunar attraction; for the +rod only pulls at the stone at the point where it +is attached to it, whereas the moon attracts every +particle of the earth. She does not, however, +attract every particle with equal force, for she +pulls the nearer parts more strongly than the +further, as is obvious from the nature of the law +of gravitation. The earth's centre is distant +sixty times its radius from the moon, so that the +nearest and furthest parts are distant fifty-nine +\PageSep{99} +and sixty-one radii respectively. Hence the attractions +at the nearest and furthest parts differ +only a little from the average, namely, that at +the centre; but it is just these small differences +which are important in this matter. + +Since on the whole the attractions and the centrifugal +forces are equal and opposite, and since +the centrifugal forces acting on the non-rotating +earth are equal and parallel at every part, and +since the attraction at the earth's centre is the +average attraction, it follows that where the attraction +is stronger than the average it overbalances +the centrifugal force, and where it is weaker +it is overbalanced thereby. + +The result of the contest between the two sets +of forces is illustrated in~\fig{22}. The circle +represents a section of the earth, and the moon +is a long way off in the direction~$M$. + +Since the moon revolves round the earth, +whilst the earth is still deprived of rotation, the +figure only shows the state of affairs at a definite +instant of time. The face which the earth exhibits +to the moon is always changing, and the +moon returns to the same side of the earth only +at the end of the month. Hence the section of +the earth shown in this figure always passes +through the moon, while it is continually shifting +with respect to the solid earth. The arrows in +the figure show by their directions and lengths +the magnitudes and directions of the overbalance +\PageSep{100} +in the contest between centrifugal and centripetal +tendencies. The point~$V$ in the figure is +the middle of the hemisphere, which at the moment +portrayed faces full towards the moon. It +\Figure[0.7]{22}{Tide-Generating Force}{png} +is the middle of the round disk which the man in +the moon looks at. The middle of the face invisible +to the man in the moon is at~$I$. The +point of the earth which is only fifty-nine earth's +radii from the moon is at~$V$. Here attraction +overbalances centrifugal force, and this is indicated +by an arrow pointing towards the moon. +The point distant sixty-one earth's radii from +the moon is at~$I$, and attraction is here overbalanced, +as indicated by the arrow pointing away +from the moon. + +I shall have to refer hereafter to the intensities +\PageSep{101} +of these forces, and will therefore here pause +to make some numerical calculations. + +The moon is distant from the earth's centre +sixty times the earth's radius, and the attraction +of gravity varies inversely as the square of the +distance. Hence we may take $\frac{1}{60^{2}}$ or $\frac{1}{3,600}$ as a +measure of the intensity of the moon's attraction +at the earth's centre. The particle which occupies +the centre of the earth is also that particle +which is at the average distance of all the particles +constituting the earth's mass. Hence $\frac{1}{60^{2}}$ or +$\frac{1}{3,600}$ may be taken as a measure of the average +attraction of the moon on every particle of the +earth. + +Now the point~$V$ is only distant fifty-nine +earth's radii from the moon, and therefore, on +the same scale, the moon attraction is measured +by $\frac{1}{59^{2}}$ or~$\frac{1}{3,481}$. + +The attraction therefore at~$V$ exceeds the average +by $\frac{1}{59^{2}} - \frac{1}{60^{2}}$, or $\frac{1}{3,481} - \frac{1}{3,600}$. It will be well to +express these results in decimals; now $\frac{1}{3,481}$ is +$.000,287,27$, and $\frac{1}{3,600}$ is $.000,277,78$, so that the +difference is~$.000,009,49$. It is important to +notice that $\frac{2}{60^{3}}$ or $\frac{2}{216,000}$ is equal to~$.000,009,26$; +so that the difference is nearly equal to~$\frac{2}{60^{3}}$. + +{\Loosen Again, the point~$I$ is distant sixty-one earth's +radii from the moon, and the moon's attraction +there is to be measured by $\frac{1}{61^{2}}$ or $\frac{1}{3,721}$. The attraction +at~$I$ therefore falls below the average by +$\frac{1}{60^{2}} - \frac{1}{61^{2}}$, or $\frac{1}{3,600} - \frac{1}{3,721}$; that is, by~$.000,277,78 - .000,268,75$, +\PageSep{102} +which is equal to~$.000,009,03$. +This again does not differ much from~$\frac{2}{60^{3}}$.} + +These calculations show that the excess of the +actual attraction at~$V$ above the average attraction +is nearly equal to the excess of the average +above the actual attraction at~$I$. These two +excesses only differ from one another by $5$~per +cent.\ of either, and they are both approximately +equal to~$\frac{2}{60^{3}}$ on the adopted scale of measurement. + +The use of any particular scale of measurement +is not material to this argument, and we +should always find that the two excesses are +nearly equal to one another. And further, if +the moon were distant from the earth by any +other number of earth's radii, we should find +that the two excesses are each nearly equal to $2$ +divided by the cube of that number.\footnote + {\Loosen This argument is very easily stated in algebraic notation. + If $x$~be the number of earth's radii at which the moon is + placed, the points $V$~and~$I$ are respectively distant $x - 1$ and + $x + 1$ radii. Now $(x - 1)^{2}$~is nearly equal to~$x^{2} - 2x$ or to + $x^{2}(1 - \frac{2}{x})$, and therefore $\frac{1}{(x - 1)^{2}}$~is nearly equal to~$\frac{1}{x^{2}(1 - \frac{2}{x})}$, which is + nearly equal to~$\frac{1}{x^{2}}(1 + \frac{2}{x})$. Hence $\frac{1}{(x - 1)^{2}} - \frac{1}{x^{2}}$~is nearly equal to~$\frac{2}{x^{3}}$. + By a similar argument $(x + 1)^{2}$~is nearly equal to~$x^{2}(1 + \frac{2}{x})$, + and $\frac{1}{(x + 1)^{2}}$~is nearly equal to~$\frac{1}{x^{2}}(1 - \frac{2}{x})$; so that $\frac{1}{x^{2}} - \frac{1}{(x + 1)^{2}}$~is nearly + equal to~$\frac{2}{x^{3}}$.} + +We conclude then that the two overbalances +at $V$~and~$I$, which will be called tide-generating +forces, are nearly equal to one another, and vary +\PageSep{103} +inversely as the cube of the distance of the moon +from the earth. + +The fact of the approximate equality of the +overbalance or excess on the two sides of the +earth is noted in the figure by two arrows at $V$ +and $I$ of equal lengths. The argument would +be a little more complicated, if I were to attempt +to follow the mathematician in his examination +of the whole surface of the earth, and to trace +from point to point how the balance between +the opposing forces turns. The reader must +accept the results of such an analysis as shown +in \fig{22} by the directions and lengths of the +arrows. + +We have already seen that the forces at $V$ and~$I$, +the middles of the faces of the earth which +are visible and invisible to the man in the moon, +are directed away from the earth's centre. The +edges of the earth's disk as seen from the moon +are at $D$ and~$D$, and here the arrows point inwards +to the earth's centre and are half as long +as those at $V$ and~$I$. At intermediate points, +they are intermediate both in size and direction. + +The only point in which the system considered +differs from actuality is that the earth has +been deprived of rotation. But this restriction +may be removed, for, when the earth rotates +once in $24$~hours, no difference is made in the +forces which I have been trying to explain, +\PageSep{104} +although of course the force of gravity and the +shape of the planet are affected by the rotation. +This figure is called a diagram of tide-generating +forces, because the tides of the ocean are due to +the action of this system of forces. + +The explanation of tide-generating force is +the very kernel of our subject, and, at the risk +of being tedious, I shall look at it from a slightly +different point of view. If every particle of the +earth and of the ocean were acted on by equal +and parallel forces, the whole system would +move together and the ocean would not be displaced +relatively to the earth; we should say +that the ocean was at rest. If the forces were +not quite equal and not quite parallel, there +would be a slight residual effect tending to make +the ocean move relatively to the solid earth. In +other words, any defect from equality and parallelism +in the forces would cause the ocean to +move on the earth's surface. + +The forces which constitute the departure +from equality and parallelism are called ``tide-generating +forces,'' and it is this system which +is indicated by the arrows in~\fig{22}. Tide-generating +force is, in fact, that force which, +superposed on the average force, makes the actual +force. The average direction of the forces +which act on the earth, as due to the moon's +attraction, is along the line joining the earth's +centre to the moon's centre, and its average +\PageSep{105} +intensity is equal to the force at the earth's +centre. + +Now at~$V$ the actual force is straight towards~$M$, +in the same direction as the average, but of +greater intensity. Hence we find an arrow +directed towards~$M$, the moon. At~$I$, the actual +force is again in the same direction as, but of +less intensity than, the average, and the arrow is +directed away from~$M$, the moon. At~$D$, the +actual force is almost exactly of the same intensity +as the average, but it is not parallel thereto, +and we must insert an inward force as shown by +the arrow, so that when this is compounded with +the average force we may get a total force in +the right direction. + +Now let us consider how these forces tend to +affect an ocean lying on the surface of the earth. +The moon is directly over the head of an inhabitant +of the earth, that is to say in his zenith, +when he is at~$V$; she is right under his feet in +the nadir when he is at~$I$; and she is in the +observer's horizon, either rising or setting, when +he is anywhere on the circle~$D$. When the +inhabitant is at~$V$ or at~$I$ he finds that the tide-generating +force is towards the zenith; when he +is anywhere on the circle~$D$ he finds it towards +the nadir. At other places he finds it directed +towards or away from some point in the sky, +except along two circles halfway between $V$ and~$D$, +or between $I$ and~$D$, where the tide-generating +\PageSep{106} +force is level along the earth's surface, and may +be called horizontal. + +A vertical force cannot make things move +sideways, and so the sea will not be moved horizontally +by it. The vertical part of the tide-generating +force is not sufficiently great to +overcome gravity, but will have the effect of +making the water appear lighter or heavier. It +will not, however, be effective in moving the +water, since the water must remain in contact +with the earth. We want, then, to omit the +vertical part of the force and leave behind only +the horizontal part, by which I mean a force +which, to an observer on the earth's surface, is +not directed either upwards or downwards, but +along the level to any point of the compass. + +If there be a force acting at any point of the +earth's surface, and directed upwards or downwards +away from or towards some point in the +sky other than the zenith, it may be decomposed +into two forces, one vertically upwards or downwards, +and another along the horizontal surface. +Now as concerns the making of the tides, +no attention need be paid to that part which +is directed straight up or down, and the only +important part is that along the surface,---the +horizontal portion. + +Taking then the diagram of tide-generating +forces in~\fig{22}, and obliterating the upward +and downward portions of the force, we are left +%% Plate 1 +\Figure[0.9]{23}{Horizontal Tide-Generating Force}{jpg} +%% Facing page +%[Blank Page] +\PageSep{107} +\index{Horizontal tide-generating force}% +with a system of forces which may be represented +by the arrows in the perspective picture of horizontal +tide-generating force shown in~\fig{23}. + +If we imagine an observer to wander over the +earth, $V$~is the place at which the moon is vertically +over his head, and the circle~$D$, shown by +the boundary of the shadow, passes through all +the places at which the moon is in the horizon, +just rising or setting. Then there is no horizontal +force where the moon is over his head or under +his feet, or where the moon is in his horizon +either rising or setting, but everywhere else there +is a force directed along the surface of the earth +in the direction of the point at which the moon +is straight overhead or underfoot. + +Now suppose $P$ to be the north pole of the +earth, and that the circle $A_{1}$,~$A_{2}$, $A_{3}$, $A_{4}$,~$A_{5}$ is a +parallel of latitude---say the latitude of London. +Then if we watch our observer from external +space, he first puts in an appearance on the picture +at~$A_{1}$, and is gradually carried along to~$A_{2}$ +by the earth's rotation, and so onwards. Just before +he comes to~$A_{2}$, the moon is due south of him, +and the tide-generating force is also south, but +not very large. It then increases, so that nearly +three hours later, when he has arrived at~$A_{3}$, it +is considerably greater. It then wanes, and +when he is at~$A_{4}$ the moon is setting and the +force is nil. After the moon has set, the force +is directed towards the moon's antipodes, and it +\PageSep{108} +is greatest about three hours after moonset, and +vanishes when the moon, still being invisible, is +on the meridian. + +It must be obvious from this discussion that +the lunar horizontal tide-generating force will +differ, both as to direction and magnitude, according +to the position of the observer on the +earth and of the moon in the heavens, and that +it can only be adequately stated by means of +mathematical formulæ. I shall in the following +chapter consider the general nature of the +changes which the forces undergo at any point +\index{Forces!tide-generating|)}% +on the earth's surface. + +But before passing on to that matter it should +be remarked that if the earth and sun had been +the only pair of bodies in existence the whole of +the argument would have applied equally well. +Hence it follows that there is also a solar tide-generating +force, which in actuality coëxists +with the lunar force. I shall hereafter show +how the relative importance of these two influences +is to be determined. + +\begin{Authorities} +Any mathematical work on the theory of the tides; for example, +Thomson and Tait's \Title{Natural Philosophy}, Lamb's \Title{Hydrodynamics}, +Bassett's \Title{Hydrodynamics}, article \Title{Tides}, ``Encycl.\ Britan.,'' +Laplace's \Title{Mécanique Céleste},~\&c. +\end{Authorities} +\PageSep{109} + + +\Chapter{VI} +{Deflection of the Vertical} + +\First{The} intensity of tide-generating force is to be +\index{Deflection of the vertical|(}% +estimated by comparison with some standard, and +it is natural to take as that standard the force of +gravity at the earth's surface. Gravity acts in a +vertical direction, whilst that portion of the tidal +force which is actually efficient in disturbing the +ocean is horizontal. Now the comparison between +a small horizontal force and gravity is +easily effected by means of a pendulum. For if +the horizontal force acts on a suspended weight, +the pendulum so formed will be deflected from +the vertical, and the amount of deflection will +measure the force in comparison with gravity. +A sufficiently sensitive spirit level would similarly +show the effect of a horizontal force by the +displacement of the bubble. When dealing with +tidal forces the displacements of either the pendulum +\index{Forces!numerical estimate|(}% +\index{Forces!deflection of vertical by|(}% +or the level must be exceedingly minute, +but, if measurable, they will show themselves as +a change in the apparent direction of gravity. +Accordingly a disturbance of this kind is often +described as a deflection of the vertical. + +The maximum horizontal force due to the +\PageSep{110} +moon may be shown by a calculation, which involves +the mass and distance of the moon, to +have an intensity of $\frac{1}{11,660,000}$ of gravity.\footnote + {It does not occur to me that there is any very elementary + method of computing the maximum horizontal tidal force, but it + is easy to calculate the vertical force at the points $V$~or~$I$ in~\fig{22}. + + {\Loosen The moon weighs $\frac{1}{80}$~of the earth, and has a radius $\frac{1}{4}$~as large. + Hence lunar gravity on the moon's surface is~$\frac{1}{80} × 4^{2}$, or $\frac{1}{5}$~of + terrestrial gravity at the earth's surface. The earth's radius is + $4,000$~miles and the moon's distance from the earth's centre + $240,000$~miles. Hence her distance from the nearer side of the + earth is $236,000$~miles. Therefore lunar gravity at the earth's + centre is $\frac{1}{5} × \frac{1}{240^{2}}$~of terrestrial gravity, and lunar gravity at the + point~$V$ is $\frac{1}{5} × \frac{1}{236^{2}}$~of the same. Therefore the tidal force at~$V$ + is $\frac{1}{5} × \frac{1}{236^{2}} - \frac{1}{5} × \frac{1}{240^{2}}$~of terrestrial gravity. On multiplying the + squares of~$236$ and of~$240$ by~$5$, we find that this difference is + $\frac{1}{278,480} - \frac{1}{288,000}$. If these fractions are reduced to decimals + and the subtraction is performed, we find that the force at~$V$ + is $.000,000,118,44$~of terrestrial gravity. When this decimal is + written as a fraction, we find the result to be $\frac{1}{8,450,000}$~of + gravity.} + + Now it is the fact, although I do not see how to prove it in an + equally elementary manner, that the maximum horizontal tide-generating + force has an intensity equal to $\frac{3}{4}$~of the vertical force + at $V$~or~$I$. To find $\frac{3}{4}$~of the above fraction we must augment the + denominator by one third part. Hence the maximum horizontal + force is $\frac{1}{11,260,000}$~of gravity. This number does not agree exactly + with that given in the text; the discrepancy is due to the + fact that round numbers have been used to express the sizes and + distance apart of the earth and the moon, and their relative + masses.} +Such a +force must deflect the bob of a pendulum by the +same fraction of the length of the cord by which +it is suspended. If therefore the string were $10$~metres +or $33$~feet in length, the maximum deflection +of the weight would be $\frac{1}{11,660,000}$~of $10$~metres, +\PageSep{111} +or $\frac{1}{1,166}$~of a millimetre. In English measure this +is $\frac{1}{29,000}$~of an inch. But the tidal force is reversed +in direction about every six hours, so that the +pendulum will depart from its mean direction by +\index{Pendulum!curves traced by, under tidal force}% +as much in the opposite direction. Hence the +\Figure[0.8]{24}{Deflection of a Pendulum; the Moon and +Observer on the Equator}{png} +excursion to and fro of the pendulum under the +lunar influence will be $\frac{1}{14,500}$~of an inch. With a +pendulum one metre, or $3$~ft.\ $3$~in.\ in length, +the range of motion of the pendulum bob is +$\frac{1}{145,000}$~of an inch. For any pendulum of manageable +length this displacement is so small, that it +seems hopeless to attempt to measure it by direct +observation. Nevertheless the mass and distance +of the moon and the intensity of gravity being +known with a considerable degree of accuracy, it +is easy to calculate the deflection of the vertical +at any time. +\index{Forces!numerical estimate|)}% + +The curves which are traced out by a pendulum +present an infinite variety of forms, corresponding +\PageSep{112} +to various positions of the observer on +the earth and of the moon in the heavens. Two +illustrations of these curves must suffice. \Fig{24} +shows the case when the moon is on the celestial +equator and the observer on the terrestrial +equator. The path is here a simple ellipse, +which is traversed twice over in the lunar day by +the pendulum. The hours of the lunar day at +\index{Pendulum!curves traced by, under tidal force}% +which the bob occupies successive positions are +marked on the curve. + +If the larger ellipse be taken to show the displacement +of a pendulum when the sun and +\Figure[0.8]{25}{Deflection of a Pendulum; the Moon in N. +Declination~$15°$, the Observer in N. Latitude~$30°$}{png} +moon coöperate at spring tide, the smaller one +will show its path at the time of neap tide. + +In \fig{25} the observer is supposed to be in +latitude~$30°$, whilst the moon stands $15°$~N. of +the equator; in this figure no account is taken +\PageSep{113} +of the sun's force. Here also the hours are +marked at the successive positions of the pendulum, +which traverses this more complex curve +only once in the lunar day. These curves are +somewhat idealized, for they are drawn on the +hypothesis that the moon does not shift her +position in the heavens. If this fact were taken +into account, we should find that the curve +would not end exactly where it began, and that +the character of the curve would change slowly +from day to day. + +But even after the application of a correction +for the gradual shift of the moon in the heavens, +the curves would still be far simpler than in actuality, +because the sun's influence has been left +out of account. It has been remarked in the +last chapter that the sun produces a tide-generating +force, and it must therefore produce a +deflection of the vertical. Although the solar +deflection is considerably less than the lunar, yet +it would serve to complicate the curve to a great +degree, and it must be obvious then that when +the full conditions of actuality are introduced +the path of the pendulum will be so complicated, +that mathematical formulæ are necessary for +complete representation. + +Although the direct observation of the tidal +deflection of the vertical would be impossible +even by aid of a powerful microscope, yet several +attempts have been made by more or less +\PageSep{114} +indirect methods. I have just pointed out that +the path of a pendulum, although drawn on an +ultra-microscopic scale, can be computed with a +high degree of accuracy. It may then occur to +the reader that it is foolish to take a great deal +of trouble to measure a displacement which is +scarcely measurable, and which is already known +with fair accuracy. To this it might be answered +that it would be interesting to watch the direct +gravitational effects of the moon on the earth's +surface. But such an interest does not afford +the principal grounds for thinking that this +attempted measurement is worth making. If the +solid earth were to yield to the lunar attraction +with the freedom of a perfect fluid, its surface +would always be perpendicular to the direction +of gravity at each instant of time. Accordingly +a pendulum would then always hang perpendicularly +to the average surface of the earth, and so +there would be no displacement of the pendulum +with reference to the earth's surface. If, then, +the solid earth yields partially to the lunar attraction, +the displacements of a pendulum must be +of smaller extent relatively to the earth than if +the solid earth were absolutely rigid. I must +therefore correct my statement as to our knowledge +of the path pursued by a pendulum, and +say that it is known if the earth is perfectly +unyielding. The accurate observation of the +movement of a pendulum under the influence of +\PageSep{115} +the moon, and the comparison of the observed +oscillation, with that computed on the supposition +that the earth is perfectly stiff, would afford +the means of determining to +what extent the solid earth is +yielding to tidal forces. Such +a result would be very interesting +as giving a measure of the +stiffness of the earth as a whole. + +I must pass over the various +\index{Cambridge, experiments with bifilar pendulum at|(}% +\index{Darwin, G. H.!bifilar pendulum|(}% +\index{Darwin, Horace, bifilar pendulum|(}% +\index{Deflection of the vertical!experiments to measure|(}% +\index{Pendulum!bifilar|(}% +earlier attempts to measure the +lunar attraction, and will only +explain the plan, although it +was abortive, used in~1879 by +my brother Horace and myself. + +Our object was to measure +the ultra-microscopic displacements +of a pendulum with reference +to the ground on which it +stood. The principle of the apparatus +used for this purpose is +due to Lord Kelvin; it is very +simple, although the practical +application of it was not easy. + +%[** TN: Figure wrapped in the original] +\Figure[0.2]{26}{Bifilar Pendulum}{png} + +\Fig{26} shows diagrammatically, and not drawn +to scale, a pendulum~$\Seg{A}{B}$ hanging by two wires. +At the foot of the pendulum there is a support~$C$ +attached to the stand of the pendulum; $D$~is a +small mirror suspended by two silk fibres, one +being attached to the bottom of the pendulum~$B$ +\PageSep{116} +and the other to the support~$C$. When the +two fibres are brought very close together, any +movement of the pendulum perpendicular to the +plane of the mirror causes the mirror to turn +through a considerable angle. The two silk +fibres diverge from one another, but if two vertical +lines passing through the two points of suspension +are $\frac{1}{1,000}$~of an inch apart, then when the +pendulum moves one of these points through a +millionth of an inch, whilst the other attached to~$C$ +remains at rest, the mirror will turn through +an angle of more than three minutes of arc. +A lamp is placed opposite to the mirror, and +the image of the lamp formed by reflection in +the mirror is observed. A slight rotation of the +mirror corresponds to an almost infinitesimal +motion of the pendulum, and even excessively +small movements of the mirror are easily detected +by means of the reflected image of the light. + +In our earlier experiments the pendulum was +hung on a solid stone gallows; and yet, when +the apparatus was made fairly sensitive, the image +of the light danced and wandered incessantly. +Indeed, the instability was so great that +the reflected image wandered all across the room. +We found subsequently that this instability was +due both to changes of temperature in the stone +gallows, and to currents in the air surrounding +the pendulum. + +To tell of all the difficulties encountered +\PageSep{117} +might be as tedious as the difficulties themselves, +so I shall merely describe the apparatus in its +ultimate form. The pendulum was suspended, +as shown in~\fig{26}, by two wires; the two wires +being in an east and west plane, the pendulum +could only swing north and south. It was hung +inside a copper tube, just so wide that the solid +copper cylinder, forming the pendulum bob, did +not touch the sides of the tube. A spike projected +from the base of the pendulum bob +through a hole in the bottom of the tube. The +mirror was hung in a little box, with a plate-glass +front, which was fastened to the bottom of the +copper tube. The only communication between +the tube and the mirror-box was by the hole +through which the spike of the pendulum projected, +but the tube and mirror-box together +formed a water-tight vessel, which was filled with +a mixture of spirits of wine and boiled water. +The object of the fluid was to steady the +mirror and the pendulum, while allowing its +slower movements to take place. The water was +boiled to get rid of air in it, and the spirits of +wine was added to increase the resistance of the +fluid, for it is a remarkable fact that a mixture +of spirits and water has considerably more viscosity +or stickiness than either pure spirits or +pure water. + +The copper tube, with the pendulum and mirror-box, +was supported on three legs resting on +\PageSep{118} +a block of stone weighing a ton, and this stood +on the native gravel in a north room in the laboratory +at Cambridge. The whole instrument +was immersed in a water-jacket, which was furnished +with a window near the bottom, so that +the little mirror could be seen from outside. A +water ditch also surrounded the stone pedestal, +and the water jacketing of the whole instrument +made the changes of temperature very slow. + +A gas jet, only turned up at the moment of +observation, furnished the light to be observed +by reflection in the little mirror. The gas +burner could be made to travel to and fro along +a scale in front of the instrument. In the preliminary +description I have spoken of the motion +of the image of a fixed light, but it clearly +amounts to the same thing if we measure the +motion of the light, keeping the point of observation +fixed. In our instrument the image of +the movable gas jet was observed by a fixed telescope +placed outside of the room. A bright +light was unfortunately necessary, because there +was a very great loss of light in the passages to +and fro through two pieces of plate glass and a +considerable thickness of water. + +Arrangements were made by which, without +entering the room, the gas jet could be turned +up and down, and could be made to move to and +fro in the room in an east and west direction, +until its image was observed in the telescope. +\PageSep{119} +There were also adjustments by which the two +silk fibres from which the mirror hung could be +brought closer together or further apart, thus +making the instrument more or less sensitive. +There was also an arrangement by which the image +of the light could be brought into the field +of view, when it had wandered away beyond the +limits allowed for by the traverse of the gas jet. + +When the instrument was in adjustment, an +observation consisted of moving the gas jet until +its image was in the centre of the field of +view of the telescope; a reading of the scale, by +another telescope, determined the position of the +gas jet to within about a twentieth of an inch. + +The whole of these arrangements were arrived +at only after laborious trials, but all the precautions +were shown by experience to be necessary, +and were possibly even insufficient to guard the +instrument from the effects of changes of temperature. +I shall not explain the manner in which +we were able to translate the displacements of +the gas jet into displacements of the pendulum. +It was not very satisfactory, and only gave approximate +results. A subsequent form of an +instrument of this kind, designed by my brother, +has been much improved in this respect. It was +he also who designed all the mechanical appliances +in the experiment of which I am speaking. + +It may be well to reiterate that the pendulum +was only free to move north and south, and that +\PageSep{120} +our object was to find how much it swung. The +east and west motion of a pendulum is equally +interesting, but as we could not observe both +displacements at the same time, we confined our +attention in the first instance to the northerly +and southerly movements. + +When properly adjusted the apparatus was so +sensitive that, if the bob of the pendulum moved +through $\frac{1}{40,000}$~of a millimetre, that is, a millionth +part of an inch, we could certainly detect the +movement, for it corresponded to a twentieth +of an inch in our scale of position of the gas +jet. When the pendulum bob moved through +this amount, the wires of the pendulum turned +through one two-hundredth of a second of arc; +this is the angle subtended by one inch at $770$~miles +distance. I do not say that we could actually +measure with this degree of refinement, but +we could detect a change of that amount. In +view of the instability of the pendulum, which +still continued to some extent, it may be hard to +gain credence for the statement that such a small +deflection was a reality, so I will explain how we +were sure of our correctness. + +In setting up the apparatus, work had to be +conducted inside the room, and some preliminary +observations of the reflected image of a stationary +gas jet were made without the use of the telescope. +The scale on which the reflected spot +of light fell was laid on the ground at about +\PageSep{121} +seven feet from the instrument; in order to +watch it I knelt on the pavement behind the +scale, and leant over it. I was one day watching +on the scale the spot of light which revealed +the motion of the pendulum, and, being tired +with kneeling, supported part of my weight on +my hands a few inches in front of the scale. +The place where my hands rested was on the +bare earth, from which a paving stone had been +removed. I was surprised to find quite a large +change in the reading. It seemed at first incredible +that my change of position was the cause, +but after several trials I found that light pressure +with one hand was quite sufficient to produce +an effect. It must be remembered that this was +not simply a small pressure delivered on the bare +earth at, say, seven feet distance, but it was the +difference of effect produced by the same pressure +at seven feet and six feet; for, of course, +the change only consisted in the distribution of +the weight of a small portion of my body. + +It is not very easy to catch the telescopic image +of a spot of light reflected from a mirror of +the size of a shilling. Accordingly, in setting +up our apparatus, we availed ourselves of this result, +for we found that the readiest way of bringing +the reflected image into the telescopic field +of view was for one of us to move slowly about +the room, until the image of the light was +brought, by the warping of the soil due to his +\PageSep{122} +weight, into the field of view of the telescope. +He then placed a heavy weight on the floor +where he had been standing; this of course +drove the image out of the field of view, but +after he had left the room the image of the flame +was found to be in the field. + +We ultimately found, even when no special +pains had been taken to render the instrument +sensitive, that if one of us was in the room, and +stood at about sixteen feet south of the instrument +with his feet about a foot apart, and slowly +shifted his weight from one foot to the other, a +distinct change was produced in the image of the +gas flame, and of course in the position of the +little mirror, from which the image was derived +by reflection. It may be well to consider for +a moment the meaning of this result. If one +presses with a finger on a flat slab of jelly, a sort +of dimple is produced, and if a pin were sticking +upright in the jelly near the dimple, it would tilt +slightly towards the finger. Now this is like +what we were observing, for the jelly represents +the soil, and the tilt of the pin corresponds to +that of the pendulum. But the scale of the displacement +is very different, for our pendulum +stood on a block of stone weighing nearly a ton, +which rested on the native gravel at two feet below +the level of the floor, and the slabs of the +floor were removed from all round the pendulum. +The dimple produced by a weight of $140$~lbs.\ on +\PageSep{123} +\index{Distortion of soil!by weight}% +\index{Elastic distortion!of soil by weight}% +the stone paved floor must have been pretty +small, and the slope of the sides of that dimple +\index{Dimple!in soil, due to weight}% +at sixteen feet must have been excessively slight; +but we were here virtually observing the change +of slope at the instrument, when the centre of +the dimple was moved from a distance of fifteen +feet to sixteen feet. + +It might perhaps be thought that all observation +would be rendered impossible by the street +traffic and by the ordinary work of the laboratory. +But such disturbances only make tremors +of very short period, and the spirits and water +damped out quick oscillations so thoroughly, that +no difference could be detected in the behavior +of the pendulum during the day and during the +night. Indeed, we found that a man could stand +close to the instrument and hit the tub and pedestal +smart blows with a stick, without producing +any sensible effect. But it was not quite easy to +try this experiment, because there was a considerable +disturbance on our first entering the room; +and when this had subsided small movements of +the body produced a sensible deflection, by slight +changes in the distribution of the experimenter's +weight. + +It is clear that we had here an instrument of +amply sufficient delicacy to observe the lunar +tide-generating force, and yet we completely +failed to do so. The pendulum was, in fact, +always vacillating and changing its position by +\PageSep{124} +many times the amount of the lunar effect which +we sought to measure. + +An example will explain how this was: A series +of frequent readings were taken from July +21st to~25th, 1881, with the pendulum arranged +to swing north and south. We found that there +was a distinct diurnal period, with a maximum at +noon, when the pendulum bob stood furthest +northward. The path of the pendulum was interrupted +by many minor zigzags, and it would +sometimes reverse its motion for an hour together. +But the diurnal oscillation was superposed on a +gradual drift of the pendulum, for the mean +diurnal position traveled slowly southward. Indeed, +in these four days the image disappeared +from the scale three times over, and was brought +back into the field of view three times by the +appliance for that purpose. On the night between +the 24th and~25th the pendulum took an +abrupt turn northward, and the scale reading +was found, on the morning of the~25th, nearly +at the opposite end of the scale from that towards +which it had been creeping for four days +previously. + +Notwithstanding all our precautions the pendulum +was never at rest, and the image of the +flame was always trembling and dancing, or waving +slowly to and fro. In fact, every reading of +our scale had to be taken as the mean of the +excursions to right and left. Sometimes for two +\PageSep{125} +or three days together the dance of the image +would be very pronounced, and during other +days it would be remarkably quiescent. + +The origin of these tremors and slower movements +\index{Earthquakes!microsisms and earth tremors|(}% +\index{Italian investigations in seismology|(}% +\index{Tremors, earth}% +is still to some extent uncertain. Quite +recent investigations by Professor Milne seem to +\index{Milne on seismology}% +show that part of them are produced by currents +in the fluid surrounding the pendulum, that +others are due to changes in the soil of a very +local character, and others again to changes +affecting a considerable tract of soil. But when +all possible allowance is made for these perturbations, +it remains certain that a large proportion +of these mysterious movements are due to minute +earthquakes. + +Some part of the displacements of our pendulum +was undoubtedly due to the action of the +moon, but it was so small a fraction of the whole, +that we were completely foiled in our endeavor +to measure it.\footnote + {Since the date of our experiment the bifilar pendulum has + been perfected by my brother, and it is now giving continuous + photographic records at several observatories. It is now made + to be far less sensitive than in our original experiment, and no + attempt is made to detect the direct effect of the moon.} +\index{Cambridge, experiments with bifilar pendulum at|)}% +\index{Darwin, G. H.!bifilar pendulum|)}% +\index{Darwin, Horace, bifilar pendulum|)}% +\index{Deflection of the vertical!experiments to measure|)}% +\index{Pendulum!bifilar|)}% + +The minute earthquakes of which I have +\index{Microsisms, minute earthquakes|(}% +spoken are called by Italian observers microsisms, +and this name has been very generally +adopted. The literature on the subject of seismology +is now very extensive, and it would be +out of place to attempt to summarize here the +\PageSep{126} +conclusions which have been drawn from observation. +I may, however, permit myself to add a +few words to indicate the general lines of the research, +which is being carried on in many parts +of the world. + +Italy is a volcanic country, and the Italians +have been the pioneers in seismology. Their +observations have been made by means of pendulums +of various lengths, and with instruments +of other forms, adapted for detecting vertical +movements of the soil. The conclusions at +which Father Bertelli arrived twenty years ago +\index{Bertelli on Italian seismology}% +may be summarized as follows:--- + +The oscillation of the pendulum is generally +\index{Pendulum!as seismological instrument}% +parallel to valleys or chains of mountains in the +neighborhood. The oscillations are independent +of local tremors, velocity and direction of wind, +rain, change of temperature, and atmospheric +electricity. + +Pendulums of different lengths betray the +movements of the soil in different manners, according +to the agreement or disagreement of +their natural periods of oscillation with the period +of the terrestrial vibrations. + +The disturbances are not strictly simultaneous +in the different towns of Italy, but succeed one +another at short intervals. + +After earthquakes the ``tromometric'' or microseismic\DPnote{** [sic]} +\index{Tromometer, a seismological instrument}% +movements are especially apt to be in +a vertical direction. They are always so when +\PageSep{127} +\index{Bertelli on Italian seismology}% +the earthquake is local, but the vertical movements +are sometimes absent when the shock +occurs elsewhere. Sometimes there is no movement +at all, even when the shock occurs quite +close at hand. + +The positions of the sun and moon appear to +have some influence on the movements of the +pendulum, but the disturbances are especially +\index{Pendulum!as seismological instrument}% +frequent when the barometer is low. + +The curves of ``the monthly means of the +tromometric movement'' exhibit the same forms +in the various towns of Italy, even those which +are distant from one another. + +The maximum of disturbance occurs near the +winter solstice and the minimum near the summer +solstice. + +At Florence a period of earthquakes is presaged +\index{Earthquakes!microsisms and earth tremors|)}% +\index{Tromometer, a seismological instrument}% +by the magnitude and frequency of oscillatory +movements in a vertical direction. These +movements are observable at intervals and during +several hours after each shock. + +Some very curious observations on microsisms +\index{Microsisms, minute earthquakes|)}% +have also been made in Italy with the microphone, +by which very slight movements of the +soil are rendered audible. + +Cavaliere de Rossi, of Rome, has established a +``geodynamic'' observatory in a cave $700$~metres +above the sea at Rocca di Papa, on the external +slope of an extinct volcano. + +At this place, remote from all carriages and +\PageSep{128} +\index{Rossi on Italian seismology|(}% +roads, he placed his microphone at a depth of $20$~metres +\index{Microphone as a seismological instrument|(}% +below the ground. It was protected +against insects by woolen wrappings. Carpet +was spread on the floor of the cave to deaden +the noise from particles of stone which might +possibly fall. Having established his microphone, +he waited till night, and then heard noises which +he says revealed ``natural telluric phenomena.'' +The sounds which he heard he describes as +``roarings, explosions occurring isolated or in +volleys, and metallic or bell-like sounds'' (\textit{fremiti, +scopii isolati o di moschetteria, e suoni-metallici +o di campana}). They all occurred +mixed indiscriminately, and rose to maxima at +irregular intervals. By artificial means he was +able to cause noises which he calls ``rumbling (?) +or crackling'' (\textit{rullo o crepito}). The roaring +(\textit{fremito}) was the only noise which he could reproduce +artificially, and then only for a moment. +It was done by rubbing together the conducting +wires, ``in the same manner as the rocks must +rub against one another when there is an earthquake.'' + +A mine having been exploded in a quarry at +some distance, the tremors in the earth were +audible in the microphone for some seconds +subsequently. + +There was some degree of coincidence between +the agitation of the pendulum-seismograph and +the noises heard with the microphone. +\PageSep{129} + +At a time when Vesuvius became active, +Rocca di Papa was agitated by microsisms, and +the shocks were found to be accompanied by the +very same microphonic noises as before. The +noises sometimes became ``intolerably loud;'' +especially on one occasion in the middle of the +night, half an hour before a sensible earthquake. +The agitation of the microphone corresponded +exactly with the activity of Vesuvius. + +Rossi then transported his microphone to +Palmieri's Vesuvian observatory, and worked in +conjunction with him. He there found that +each class of shock had its corresponding noise. +The sussultorial shocks, in which I conceive the +movement of the ground is vertically up and +down, gave the volleys of musketry (\textit{i~colpi di +moschetteria}), and the undulatory shocks gave +the roarings (\textit{i~fremiti}). The two classes of +noises were sometimes mixed up together. + +Rossi makes the following remarks: ``On +Vesuvius I was put in the way of discovering +that the simple fall and rise in the ticking which +occurs with the microphone [\textit{battito del orologio +unito al microfono}] (a phenomenon observed +by all, and remaining inexplicable to all) is a +consequence of the vibration of the ground.'' +This passage alone might perhaps lead one to +suppose that clockwork was included in the circuit; +but that this was not the case, and that +``ticking'' is merely a mode of representing a +\PageSep{130} +\index{Paschwitz, von Rebeur!on horizontal pendulum|(}% +\index{Pendulum!horizontal|(}% +natural noise is proved by the fact that he subsequently +says that he considers the ticking to +be ``a telluric phenomenon.'' + +Rossi then took the microphone to the Solfatara +\index{Rossi on Italian seismology|)}% +of Pozzuoli, and here, although no sensible +tremors were felt, the noises were so loud as +to be heard simultaneously by all the people in +the room. The ticking was quite masked by +other natural noises. The noises at the Solfatara +were imitated by placing the microphone +on the lid of a vessel of boiling water. Other +seismic noises were then imitated by placing the +microphone on a marble slab, and scratching +\index{Microphone as a seismological instrument|)}% +and tapping the under surface of it. + +The observations on Vesuvius led him to the +conclusion that the earthquake oscillations have +sometimes fixed ``nodes,'' for there were places +on the mountain where no effects were observed. +There were also places where the movement was +intensified, and hence it may be concluded that +the centre of disturbance may sometimes be very +distant, even when the observed agitation is +considerable. + +At the present time perhaps the most distinguished +investigator in seismology is Professor +Milne, formerly of the Imperial College of Engineering +\index{Milne on seismology}% +at Tokyo. His residence in Japan gave +\index{Japan, frequency of earthquakes}% +him peculiar opportunities of studying earthquakes, +for there is, in that country, at least one +earthquake per diem of sufficient intensity to +\index{Italian investigations in seismology|)}% +\PageSep{131} +\index{Japan, frequency of earthquakes}% +affect a seismometer. The instrument of which +he now makes most use is called a horizontal +pendulum. The principle involved in it is old, +but it was first rendered practicable by von +Rebeur-Paschwitz, whose early death deprived +the world of a skillful and enthusiastic investigator. + +The work of Paschwitz touches more closely +on our present subject than that of Milne, because +he made a gallant attempt to measure the +moon's tide-generating force, and almost persuaded +himself that he had done so. + +The horizontal pendulum is like a door in its +mode of suspension. If a doorpost be absolutely +vertical, the door will clearly rest in any +position, but if the post be even infinitesimally +tilted the door naturally rests in one definite +position. A very small shift of the doorpost is +betrayed by a considerable change in the position +of the door. In the pendulum the door is +replaced by a horizontal boom, and the hinges +by steel points resting in agate cups, but the +principle is the same. + +The movement of the boom is detected and +registered photographically by the image of a +light reflected from certain mirrors. Paschwitz +made systematic observations with his pendulum +at Wilhelmshaven, Potsdam, Strassburg, and +Orotava. He almost convinced himself at one +time that he could detect, amidst the wanderings +\PageSep{132} +of the curves of record, a periodicity corresponding +to the direct effect of the moon's action. +But a more searching analysis of his results left +the matter in doubt. Since his death the observations +at Strassburg have been continued by +M.~Ehlert. His results show an excellent consistency +\index{Ehlert, observation with horizontal pendulum}% +with those of Paschwitz, and are therefore +\index{Paschwitz, von Rebeur!on horizontal pendulum|)}% +confirmatory of the earlier opinion of the +latter. I am myself disposed to think that the +detection of the lunar attraction is a reality, but +the effect is so minute that it cannot yet be +relied on to furnish a trustworthy measurement +of the amount of the yielding of the solid earth +to tidal forces. + +It might be supposed that doubt could hardly +arise as to whether or not the direct effect of +the moon's attraction had been detected. But +I shall show in the next chapter that at many +places the tidal forces must exercise in an indirect +manner an effect on the motion of a pendulum +\index{Pendulum!horizontal|)}% +much greater than the direct effect. + +It was the consideration of this indirect effect, +and of other concomitants, which led us to +abandon our attempted measurement, and to +conclude that all endeavors in that direction +were doomed to remain for ever fruitless. I can +but hope that a falsification of our forecast by +M.~Ehlert and by others may be confirmed. +\PageSep{133} + +\begin{Authorities} +G.~H. Darwin and Horace Darwin, ``Reports to the British +Association for the Advancement of Science:''--- + +\Title{Measurement of the Lunar Disturbance of Gravity.} York +meeting, 1881, pp.~93--126. + +\Title{Second Report on the same}, with appendix. Southampton +meeting, 1882, pp.~95--119. + +E.~von Rebeur-Paschwitz, \Title{Das Horizontalpendel}. + +``Nova Acta Leop.\ Carol.\ Akad.,'' 1892, vol.~lx.\ no.~1, p.~213; +also ``Brit.\ Assoc.\ Reports,''~1893. + +E.~von Rebeur-Paschwitz, \Title{Ueber Horizontalpendel-Beobach\-tungen +in Wilhelmshaven, Potsdam und Puerto Orotava auf Tenerifa}. + +``Astron.\ Nachrichten,'' vol.~cxxx.\ pp.~194--215. + +R.~Ehlert, \Title{Horizontalpendel-Beobachtungen}. + +``Beiträge zur Geophysik,'' vol.~iii.\ Part~I., 1896. + +C.~Davison, \Title{History of the Horizontal and Bifilar Pendulums}. +\index{Davison, history of bifilar and horizontal pendulums}% + +``Appendix to Brit.\ Assoc.\ Report on Earth Tremors.'' Ipswich +meeting, 1895, pp.~184--192. + +``British Association Reports of Committees.'' + +\Title{On Earth Tremors}, 1891--95 (the first being purely formal). + +\Title{On Seismological Investigation}, 1896. + +The literature on Seismology is very extensive, and would +\index{Seismology}% +need a considerable index; the reader may refer to \Title{Earthquakes} +and to \Title{Seismology} by John Milne. Both works form volumes in +the International Scientific Series, published by Kegan Paul, +Trench, Trübner \&~Co. +\end{Authorities} +\index{Deflection of the vertical|)}% +\index{Forces!deflection of vertical by|)}% +\PageSep{134} + + +\Chapter[Distortion of the Earth's Surface]{VII} +{The Elastic Distortion of the Earth's Surface +by Varying Loads} + +\First{When} the tide rises and falls on the seacoast, +\index{Darwin, G. H.!distortion of earth's surface by varying loads|(}% +\index{Deflection of the vertical!due to tide|(}% +\index{Distortion of soil!by varying loads|(}% +\index{Elastic distortion!of earth by varying loads|(}% +many millions of tons of water are brought alternately +nearer and further from the land. Accordingly +a pendulum suspended within a hundred +miles or so of a seacoast should respond to the +attraction of the sea water, swinging towards the +sea at high water, and away from it at low water. +Since the rise and fall has a lunar periodicity the +pendulum should swing in the same period, even +if the direct attraction of the moon did not affect +it. But, as I shall now show, the problem is +further confused by another effect of the varying +tidal load. + +We saw in \Ref{Chapter}{VI}.\ how a weight resting +on the floor in the neighborhood of our pendulum +produced a dimple by which the massive +stone pedestal of our instrument was tilted over. +Now as low tide changes to high tide the position +of an enormous mass of water is varied with +respect to the land. Accordingly the whole +coast line must rock to and fro with the varying +tide. We must now consider the nature of the +\PageSep{135} +distortion of the soil produced in this way. The +mathematical investigation of the form of the +dimple in a horizontal slab of jelly or other elastic +\index{Dimple!form of, in elastic slab}% +material, due to pressure at a single point, +shows that the slope at any place varies inversely +as the square of the distance from the centre. +That is to say, if starting from any point we +proceed to half our original distance, we shall +find four times as great a slope, and at one third +\Figure[0.7]{27}{Form of Dimple in an Elastic Surface}{png} +of the original distance the slope will be augmented +ninefold. + +The theoretical form of dimple produced by +pressure at a single mathematical point is shown +in~\fig{27}. The slope is exaggerated so as to +render it visible, and since the figure is drawn on +the supposition that the pressure is delivered at +a mathematical point, the centre of the dimple +is infinitely deep. If the pressure be delivered +by a blunt point, the slope at a little distance +\PageSep{136} +will be as shown, but the centre will not be infinitely +deep. If therefore we pay no attention to +the very centre, this figure serves to illustrate +the state of the case. When the dimple is produced +by the pressure of a weight, that weight, +being endowed with gravitation, attracts any +other body with a force varying inversely as the +square of the distance. It follows, therefore, +that the slope of the dimple is everywhere exactly +\index{Slope of soil!due to elastic distortion}% +proportional to the gravitational attraction +\index{Attraction!of weight resting on elastic slab proportional to slope}% +of the weight. Since this is true of a single +weight, it is true of a group of weights, each +producing its own dimple by pressure and its own +attraction, strictly proportional to one another. +Thus the whole surface is deformed by the superposition +of dimples, and the total attraction is +the sum of all the partial attractions. + +Let us then imagine a very thick horizontal +slab of glass supporting any weights at any parts +of its surface. The originally flat surface of the +slab will be distorted into shallow valleys and +low hills, and it is clear that the direct attraction +of the weights will everywhere be exactly proportional +to the slopes of the hillsides; also the +direction of the greatest slope at each place must +agree with the direction of the attraction. The +direct attraction of the weights will deflect a +pendulum from the vertical, and the deflection +must be exactly proportional to the slope produced +by the pressure of the weights. It may +\PageSep{137} +be proved that if the slab is made of a very stiff +glass the angular deflection of the pendulum +under the influence of attraction will be one fifth +\index{Attraction!of weight resting on elastic slab proportional to slope}% +of the slope of the hillside; if the glass were +of the most yielding kind, the fraction would be +one eighth. The fraction depends on the degree +of elasticity of the material, and the stiffer it is +the larger the fraction. + +The observation of a pendulum consists in +noting its change of position with reference to +the surface of the soil; hence the slope of the +soil, and the direct attraction of the weight +which causes that slope, will be absolutely fused +together, and will be indistinguishable from one +another. + +Now, this conclusion may be applied to the +tidal load, and we learn that, if rocks are of the +same degree of stiffness as glass of medium +quality, the direct attraction of the tidal load +produces one sixth of the apparent deflection of +a pendulum produced by the tilting of the soil. + +If any one shall observe a pendulum, within +say a hundred miles of the seacoast, and shall +detect a lunar periodicity in its motion, he can +only conclude that what he observes is partly +due to the depression and tilting of the soil, +partly to attraction of the sea water, and partly +to the direct attraction of the moon. Calculation +indicates that, with the known average elasticity +of rock, the tilting of the soil is likely to +\PageSep{138} +\index{Elastic distortion!calculation and illustration|(}% +\index{Slope of soil!calculation and illustration of|(}% +be far greater than the other two effects combined. +Hence, if the direct attraction of the +moon is ever to be measured, it will first be +necessary to estimate and to allow for other important +oscillations with lunar periodicity. The +difficulty thus introduced into this problem is so +serious that it has not yet been successfully met. +It may perhaps some day be possible to distinguish +the direct effects of the moon's tidal attraction +from the indirect effects, but I am not +very hopeful of success in this respect. It was +pointed out in \Ref{Chapter}{VI}.\ that there is some +reason to think that a lunar periodicity in the +swing of a pendulum has been already detected, +and if this opinion is correct, the larger part of +the deflection was probably due to these indirect +effects. + +The calculation of the actual tilting of the +coast line by the rising tide would be excessively +complex even if accurate estimates were obtainable +of the elasticity of the rock and of the tidal +load. It is, however, possible to formulate a +soluble problem of ideal simplicity, which will +afford us some idea of the magnitude of the +results occurring in nature. + +In the first place, we may safely suppose the +earth to be flat, because the effect of the tidal +load is quite superficial, and the curvature of the +earth is not likely to make much difference in +the result. In the second place, it greatly simplifies +\PageSep{139} +the calculation to suppose the ocean to +consist of an indefinite number of broad canals, +separated from one another by broad strips of +land of equal breadth. Lastly, we shall suppose +that each strip of sea rocks about its middle line, +so that the water oscillates as in a seiche of the +Lake of Geneva; thus, when it is high water +on the right-hand coast of a strip of sea, it is +low water on the left-hand coast, and vice versa. +We have then to determine the change of shape +of the ocean-bed and of the land, as the tide +rises and falls. The problem as thus stated is +\Figure{28}{Distortion of Land and Sea-bed by Tidal Load}{png} +vastly simpler than in actuality, yet it will suffice +to give interesting indications of what must +occur in nature. + +The figure~\figref{28} shows the calculated result, the +slopes being of course enormously exaggerated. +The straight line represents the level surface of +land and sea before the tidal oscillation begins, +the shaded part being the land and the dotted +part the sea. Then the curved line shows the +form of the land and of the sea-bed, when it is +low water at the right of the strip of land and high +\PageSep{140} +water at the left. The figure would be reversed +when the high water interchanges position +with the low water. Thus both land and sea +rock about their middle lines, but the figure +shows that the strip of land remains nearly flat +although not horizontal, whilst the sea-bed becomes +somewhat curved. + +It will be noticed that there is a sharp nick at +the coast line. This arises from the fact that +deep water was assumed to extend quite up to +the shore line; if, however, the sea were given +a shelving shore, as in nature, the sharp nick +would disappear, although the form of the distorted +rocks would remain practically unchanged +elsewhere. + +Thus far the results have been of a general +character, and we have made no assumptions as +to the degree of stiffness of the rock, or as to +the breadths of the oceans and continents. Let +us make hypotheses which are more or less +plausible. At many places on the seashore the +tide ranges through twenty or thirty feet, but +these great tides only represent the augmentation +of the tide-wave as it runs into shallow +water, and it would not be fair to suppose our +tide to be nearly so great. In order to be moderate, +I will suppose the tide to have a range of +$160$~centimetres, or, in round numbers, about $5$~feet. +Then, at the high-water side of the sea, +the water is raised by $80$~centimetres, and at the +\index{Elastic distortion!calculation and illustration|)}% +\index{Slope of soil!calculation and illustration of|)}% +\PageSep{141} +low-water side it is depressed by the same +amount. The breadth of the Atlantic is about +$4,000$~or $5,000$~miles. I take then, the breadth of +the oceans and of the continents as $3,900$~miles, +or $6,280$~kilometres. Lastly, as rocks are usually +stiffer than glass, I take the rock bed to +be twice as stiff as the most yielding glass, and +quarter as stiff again as the stiffest glass; this +assumption as to the elasticity of rock makes the +attraction at any place one quarter of the slope. +For a medium glass we found the fraction to be +about one sixth. These are all the data required +for determining the slope. + +It is of course necessary to have a unit of +measurement for the slope of the surface. Now +a second of arc is the name for the angular +magnitude of an inch seen at $3\frac{1}{4}$~miles, and accordingly +a hundredth of a second of arc, usually +written~$0''.01$, is the angular magnitude of +an inch seen at $325$~miles; the angles will then +be measured in hundredths of seconds. + +Before the tide rises, the land and sea-bed +are supposed to be perfectly flat and horizontal. +Then at high tides the slopes on the land are as +follows:--- +\[ +\begin{array}{r@{\ }lc} +\multicolumn{2}{c}{\ColHead{Distance from high-water mark}} & +\multicolumn{1}{c}{\ColHead[2in]{Slope of the land measured in hundredths of seconds of arc}} \\ + 10 &\text{metres} & 10 \\ +100 &\text{metres} & \Z8 \\ + 1 &\text{kilometre} & \Z6 \\ + 10 &\text{kilometres} & \Z4 \\ + 20 &\text{kilometres} & \Z3\rlap{$\frac{1}{2}$} \\ +100 &\text{kilometres} & \Z2 +\end{array} +\] +\PageSep{142} +The slope is here expressed in hundredths of a +second of arc, so that at $100$~kilometres from the +coast, where the slope is~$2$, the change of plane +amounts to the angle subtended by one inch at +$162$~miles. + +When high water changes to low water, the +slopes are just reversed, hence the range of +change of slope is represented by the doubles of +these angles. If the change of slope is observed +by some form of pendulum, allowance must be +made for the direct attraction of the sea, and it +appears that with the supposed degree of stiffness +of rock these angles of slope must be augmented +in the proportion of $5$~to~$4$. Thus, we +double the angles to allow of change from high +to low water, and augment the numbers as $5$ is +to~$4$, to allow for the direct attraction of the sea. +Finally we find results which may be arranged +in the following tabular form:--- +\[ +\begin{array}{r@{\ }lc} +\multicolumn{2}{c}{\ColHead{Distance from high-water mark}} & +\multicolumn{1}{c}{\ColHead[1.75in]{Apparent range of deflection of the vertical}} \\ + 10 &\text{metres} & 0''.25\Z \\ +100 &\text{metres} & 0''.20\Z \\ + 1 &\text{kilometre} & 0''.15\Z \\ + 10 &\text{kilometres} & 0''.10\Z \\ + 20 &\text{kilometres} & 0''.084 \\ +100 &\text{kilometres} & 0''.050 +\end{array} +\] +At the centre of the continent, $1,950$~miles from +the coast, the range will be~$0''.012$. + +If all the assumed data be varied, the ranges +of the slopes are easily calculable, but these +\PageSep{143} +results may be taken as fairly representative, although +perhaps somewhat underestimated. Lord +Kelvin has made an entirely independent estimate +\index{Kelvin, Lord!calculation of tidal attraction}% +of the probable deflection of a pendulum +by the direct attraction of the sea at high tide. +\index{Attraction!of tide calculated}% +He supposes the tide to have a range of $10$~feet +from low water to high water, and he then estimates +the attraction of a slab of water $10$~feet +thick, $50$~miles broad perpendicular to the coast, +and $100$~miles long parallel to the coast, on a +plummet $100$~yards from low-water mark and +opposite the middle of the $100$~miles. This +would, he thinks, very roughly represent the +state of things at St.~Alban's Head, in England. +He finds the attraction such as to deflect the +plumb-line, as high water changes to low water, +by a twentieth of a second of arc. The general +law as to the proportionality of slope to +attraction shows that, with our supposed degree +of stiffness of rock, the apparent deflection of a +plumb-line, due to the depression of the coast +and the attraction of the sea as high water +changes to low water, will then be a quarter of a +second of arc. Postulating a smaller tide, but +spread over a wider area, I found the result +would be a fifth of a second; thus the two results +present a satisfactory agreement. + +This speculative investigation receives confirmation +from observation. The late M.~d'Abbadie +\index{Abbadie, tidal deflection of vertical}% +established an observatory at his château of +\index{Deflection of the vertical!due to tide|)}% +\PageSep{144} +\index{Abbadie, tidal deflection of vertical}% +Abbadia, close to the Spanish frontier and within +a quarter of a mile of the Bay of Biscay. Here +he constructed a special form of instrument for +detecting small changes in the direction of gravity. +Without going into details, it may suffice +to state that he compared a fixed mark with its +image formed by reflection from a pool of mercury. +He took $359$~special observations at the +times of high and low tide in order to see, as he +says, whether the water exercised an attraction +on the pool of mercury, for it had not occurred +to him that the larger effect would probably +arise from the bending of the rock. He found +that in $243$~cases the pool of mercury was tilted +towards the sea at high water or away from it at +low water; in $59$~cases there was no apparent +effect, and in the remaining $57$~cases the action +was inverted. The observations were repeated +later by his assistant in the case of $71$~successive +high waters\footnote + {Presumably the observation at one high water was defective.} +and $73$~low waters, and he also +found that in about two thirds of the observations +the sea seemed to exercise its expected +influence. We may, I think, feel confident that +on the occasions where no effect or a reversal +was perceived, it was annulled or reversed by a +warping of the soil, such as is observed with +seismometers. + +Dr.~von Rebeur-Paschwitz also noted deflections +\index{Paschwitz, von Rebeur!tidal deflection of vertical at Wilhelmshaven}% +due to the tide at Wilhelmshaven in Germany. +\PageSep{145} +\index{Atmospheric pressure!distortion of soil by}% +\index{Elastic distortion!by atmospheric pressure|(}% +\index{Pressure of atmosphere, elastic distortion of soil by}% +The deflection was indeed of unexpected +magnitude at this place, and this may probably +be due to the peaty nature of the soil, which +renders it far more yielding than if the observatory +were built on rock. + +This investigation has another interesting application, +for the solid earth has to bear another +varying load besides that of the tide. The +atmosphere rests on the earth and exercises a +variable pressure, as shown by the varying +height of the barometer. The variation of +pressure is much more considerable than one +would be inclined to suspect off-hand. The +height of the barometer ranges through nearly +two inches, or say five centimetres; this means +that each square yard of soil supports a weight +greater by $1,260$~lbs.\ when the barometer is very +high, than when it is very low. If we picture +to ourselves a field loaded with half a ton to +each square yard, we may realize how enormous +is the difference of pressure in the two cases. + +In order to obtain some estimate of the effects +of the changing pressure, I will assume, as before, +that the rocks are a quarter as stiff again +as the stiffest glass. On a thick slab of this +material let us imagine a train of parallel waves +of air, such that at the crests of the waves the +barometer is $5$~centimetres higher than at the +hollow. Our knowledge of the march of barometric +gradients on the earth's surface makes it +\PageSep{146} +plausible to assume that it is $1,500$~miles from +the line of highest to that of lowest pressure. +Calculation then shows that the slab is distorted +into parallel ridges and valleys, and that the +tops of the ridges are $9$~centimetres, or $3\frac{1}{2}$~inches, +higher than the hollows. Although the actual +distribution of barometric pressures is not of this +simple character, yet this calculation shows, with +a high degree of probability, that when the +barometer is very high we are at least $3$~inches +nearer the earth's centre than when it is very +low. + +The consideration of the effects of atmospheric +\index{Atmospheric pressure!distortion of soil by}% +\index{Level of sea affected by atmospheric pressure}% +\index{Pressure of atmosphere, elastic distortion of soil by}% +pressure leads also to other curious conclusions. +I have remarked before that the sea must respond +\index{Sea!level affected by atmospheric pressure}% +to barometric pressure, being depressed +by high and elevated by low pressure. Since a +column of water $68$~centimetres ($2$~ft.\ $3$~in.)\ in +height weighs the same as a column, with the +same cross section, of mercury, and $5$~centimetres +in height, the sea should be depressed by $68$~centimetres +under the very high barometer as compared +with the very low barometer. But the +height of the water can only be determined with +reference to the land, and we have seen that the +land must be depressed by $9$~centimetres. Hence +the sea would be apparently depressed by only $59$~centimetres. + +It is probable that, in reality, the larger barometric +inequalities do not linger quite long +\PageSep{147} +enough over particular areas to permit the sea to +attain everywhere its due slope, and therefore the +full difference of water level can only be attained +occasionally. On the other hand the elastic compression +\index{Elastic distortion!by atmospheric pressure|)}% +of the ground must take place without +sensible delay. Thus it seems probable that this +compression must exercise a very sensible effect +in modifying the apparent depression or elevation +of the sea under high and low barometer. + +If delicate observations are made with some +form of pendulum, the air waves and the consequent +distortions of the soil should have a sensible +effect on the instrument. In the ideal case +which I have described above, it appears that +the maximum apparent deflection of the plumb-line +would be $\frac{1}{90}$~of a second of arc; this would +be augmented to $\frac{1}{70}$~of a second by the addition +of the true deflection, produced by the attraction +of the air. Lastly, since the slope and attraction +would be absolutely reversed when the air wave +assumed a different position with respect to the +observer, it is clear that the range of apparent +oscillation of the pendulum might amount to +$\frac{1}{35}$~of a second of arc. + +This oscillation is actually greater than that +due to the direct tidal force of the moon acting +on a pendulum suspended on an ideally unyielding +earth. Accordingly we have yet another +reason why the direct measurement of the tidal +force presents a problem of the extremest difficulty. +\PageSep{148} + +\begin{Authorities} +G.~H. Darwin, \Title{Appendix to the Second Report on Lunar Disturbance +of Gravity}. ``Brit.\ Assoc.\ Reports.'' Southampton,~1882. + +Reprint of the same in the ``Philosophical Magazine.'' + +d'Abbadie, \Title{Recherches sur la verticale}. ``Ann.\ de~la Soc.\ Scient.\ +de~Bruxelles,'' 1881. + +von Rebeur-Paschwitz, \Title{Das Horizontalpendel}. ``Nova Acta +K.~Leop.\ Car.\ Akad.,'' Band~60, No.~1, 1892. +\end{Authorities} +\index{Darwin, G. H.!distortion of earth's surface by varying loads|)}% +\index{Distortion of soil!by varying loads|)}% +\index{Elastic distortion!of earth by varying loads|)}% +\PageSep{149} + + +\Chapter{VIII} +{Equilibrium Theory of Tides} + +\First{It} is clearly necessary to proceed step by step +\index{Equilibrium theory of tides|(}% +towards the actual conditions of the tidal problem, +and I shall begin by supposing that the +oceans cover the whole earth, leaving no dry +land. It has been shown in \Ref{Chapter}{V}.\ that the +tidal force is the resultant of opposing centrifugal +and centripetal forces. The motion of the +system is therefore one of its most essential features. +We may however imagine a supernatural +being, who carries the moon round the earth and +makes the earth rotate at the actual relative +speeds, but with indefinite slowness as regards +absolute time. This supernatural being is further +to have the power of maintaining the tidal forces +at exactly their present intensities, and with their +actual relationship as regards the positions of +the moon and earth. Everything, in fact, is to +remain as in reality, except time, which is to be +indefinitely protracted. The question to be considered +is as to the manner in which the tidal +forces will cause the ocean to move on the slowly +revolving earth. + +It appears from \fig{23} that the horizontal +\PageSep{150} +tidal force acts at right angles to the circle, where +the moon is in the horizon, just rising or just +setting, towards those two points, $V$~and~$I$, where +the moon is overhead in the zenith, or underfoot +in the nadir. The force will clearly generate +currents in the water away from the circle of +moonrise and moonset, and towards $V$~and~$I$. +The currents will continue to flow until the water +level is just so much raised above the primitive +surface at $V$~and~$I$, and depressed along the circle, +that the tendency to flow downhill towards +the circle is equal to the tendency to flow uphill +under the action of the tide-generating force. +When the currents have ceased to flow, the figure +of the ocean has become elongated, or egg-shaped +with the two ends alike, and the longer +axis of the egg is pointed at the moon. When +this condition is attained the system is at rest or +in equilibrium, and the technical name for the +egg-like form is a ``prolate ellipsoid of revolution''---``prolate'' +because it is elongated, and +``of revolution'' because it is symmetrical with +respect to the line pointing at the moon. Accordingly +the mathematician says that the figure +of equilibrium under tide-generating force is a +prolate ellipsoid of revolution, with the major +axis directed to the moon. + +It has been supposed that the earth rotates and +that the moon revolves, but with such extreme +slowness that the ocean currents have time +\PageSep{151} +\index{Figure of equilibrium!of ocean under tidal forces|(}% +\index{Forces!figure of equilibrium under tidal|(}% +enough to bring the surface to its form of equilibrium, +\index{Equilibrium theory of tides!chart and law of tide|(}% +at each moment of time. If the time be +sufficiently protracted, this is a possible condition +of affairs. It is true that with the earth spinning +at its actual rate, and with the moon revolving +as in nature, the form of equilibrium can +never be attained by the ocean; nevertheless it +is very important to master the equilibrium +theory. + +\Fig{29} represents the world in two hemispheres, +as in an ordinary atlas, with parallels +of latitude drawn at $15°$~apart. At the moment +represented, the moon is supposed to be in the +zenith at $15°$~of north latitude, in the middle of +the right-hand hemisphere. The diametrically +opposite point is of course at $15°$~of south latitude, +in the middle of the other hemisphere. +These are the two points $V$~and~$I$ of figs.\ \figref{22}~and~\figref{23}, +towards which the water is drawn, so that the +vertices of the ellipsoid are at these two spots. +A scale of measurement must be adopted for +estimating the elevation of the water above, and +its depression below the original undisturbed surface +of the globe. It will be convenient to measure +the elevation at these two spots by the +number~$2$. A series of circles are drawn round +these points, but one of them is, of necessity, +presented as partly in one hemisphere and partly +in the other. In the map they are not quite concentric +with the two spots, but on the actual +\PageSep{152} +\TallFig{29}{Chart of Equilibrium Tide}{png} +\PageSep{153} +\index{Semidiurnal tide!in equilibrium theory|(}% +globe they would be so. These circles show +where, on the adopted scale of measurement, the +elevation of height is successively $1\frac{1}{2}$,~$1$,~$\frac{1}{2}$. The +fourth circle, marked in chain dot, shows where +there is no elevation or depression above the original +surface. The next succeeding and dotted +circle shows where there is a depression of~$\frac{1}{2}$, and +the last dotted line is the circle of lowest water +where the depression is~$1$; it is the circle~$\Seg{D}{D}$ of~\fig{22}, +and the circle of the shadow in~\fig{23}. + +The elevation above the original spherical surface +at the vertices or highest points is just twice +as great as the greatest depression. But the +greatest elevation only occurs at two points, +whereas the greatest depression is found all along +a circle round the globe. The horizontal tide-generating +force is everywhere at right angles to +these circles, and the present figure is in effect a +reproduction, in the form of a map, of the perspective +picture in~\fig{23}. + +Now as the earth turns from west to east, let +us imagine a man standing on an island in the +otherwise boundless sea, and let us consider what +he will observe. Although the earth is supposed +to be revolving very slowly, we may still call the +twenty-fourth part of the time of its rotation an +hour. The man will be carried by the earth's +rotation along some one of the parallels of latitude. +If, for example, his post of observation is +in latitude $30°$~N., he will pass along the second +\index{Equilibrium theory of tides!chart and law of tide|)}% +\index{Figure of equilibrium!of ocean under tidal forces|)}% +\index{Forces!figure of equilibrium under tidal|)}% +\PageSep{154} +parallel to the north of the equator. This parallel +cuts several of the circles which indicate the +elevation and depression of the water, and therefore +he will during his progress pass places where +the water is shallower and deeper alternately, and +he would say that the water was rising and falling +rhythmically. Let us watch his progress +across the two hemispheres, starting from the +extreme left. Shortly after coming into view he +is on the dotted circle of lowest water, and he +says it is low tide. As he proceeds the water +rises, slowly at first and more rapidly later, until +he is in the middle of the hemisphere; he arrives +there six hours later than when we first began to +watch him. It will have taken him about $5\frac{1}{2}$~hours +to pass from low water to high water. At +low water he was depressed by~$1$ below the original +level, and at high water he is raised by~$\frac{1}{2}$ +above that level, so that the range from low +water to high water is represented by~$1\frac{1}{2}$. After +the passage across the middle of the hemisphere, +the water level falls, and after about $5\frac{1}{2}$~hours +more the water is again lowest, and the depression +is measured by~$1$ on the adopted scale. +Soon after this he passes out of this hemisphere +into the other one, and the water rises again +until he is in the middle of that hemisphere. +But this time he passes much nearer to the vertex +of highest water than was the case in the other +hemisphere, so that the water now rises to a +\PageSep{155} +height represented by about~$1\frac{4}{5}$. In this half of +his daily course the range of tide is from $1$~below +to $1\frac{4}{5}$~above, and is therefore~$2\frac{4}{5}$, whereas before +it was only~$1\frac{1}{2}$. The fact that the range of two +successive tides is not the same is of great importance +in tidal theory; it is called the diurnal +inequality of the tide. + +It will have been noticed that in the left hemisphere +the range of fall below the original spherical +surface is greater than the range of rise +above it; whereas in the right hemisphere the +rise is greater than the fall. Mean water mark +is such that the tide falls on the average as much +below it as it rises above it, but in this case the +rise and fall have been measured from the originally +undisturbed surface. In fact the mean +level of the water, in the course of a day, is not +identical with the originally undisturbed surface, +although the two levels do not differ much from +one another. + +The reader may trace an imaginary observer +in his daily progress along any other parallel of +latitude, and will find a similar series of oscillations +in the ocean; each latitude will, however, +present its own peculiarities. Then again the +moon moves in the heavens. In \fig{29} she has +been supposed to be $15°$~north of the equator, +but she might have been yet further northward, +or on the equator, or to the south of it. Her +extreme range is in fact $28°$~north or south of +\PageSep{156} +\index{Forces!those of sun and moon compared|(}% +\index{Lunar!tide-generating force compared with solar|(}% +\index{Moon and earth!tide-generating force compared with sun's|(}% +\index{Solar!tide-generating force compared with lunar|(}% +the equator. To represent each such case a new +map would be required, which would, however, +only differ from this one by the amount of displacement +of the central spots from the equator. + +It is obvious that the two hemispheres in \fig{29} +are exactly alike, save that they are inverted +with respect to north and south; the right hemisphere +is in fact the same as the left upside down. +It is this inversion which causes the two successive +tides to be unlike one another, or, in other +words, gives rise to the diurnal inequality. But +\index{Diurnal inequality!according to equilibrium theory}% +there is one case where inversion makes no difference; +this is when the central spot is on the +equator in the left hemisphere, for its inversion +then makes the right hemisphere an exact reproduction +of the left one. In this case therefore +the two successive tides are exactly alike, and +there is no diurnal inequality. Hence the diurnal +inequality vanishes when the moon is on the +equator. + +Our figure exhibits another important point, +for it shows that the tide has the greater range +in that hemisphere where the observer passes +nearest to one of the two central spots. That is +to say, the higher tide occurs in that half of the +daily circuit in which the moon passes nearest to +the zenith or to the nadir of the observer. + +Thus far I have supposed the moon to exist +alone, but the sun also acts on the ocean according +\index{Sun!tide-generating force of, compared with that of moon|(}% +to similar laws, although with less intensity. +\index{Semidiurnal tide!in equilibrium theory|)}% +\PageSep{157} +We must now consider how the relative strengths +of the actions of the two bodies are to be determined. +It was indicated in \Ref{Chapter}{V}.\ that +tide-generating force varies inversely as the cube +of the distance from the earth of the tide-generating +body. The force of gravity varies inversely +as the square of the distance, so that, as +we change the distance of the attracting body, +tidal force varies with much greater rapidity than +does the direct gravitational attraction. Thus if +the moon stood at half her present distance from +the earth, her tide-generating force would be $8$~times +as great, whereas her direct attraction would +only be multiplied $4$~times. It is also obvious +that if the moon were twice as heavy as in reality, +her tide-generating force would be doubled; +and if she were half as heavy it would be halved. +Hence we conclude that tide-generating force +varies directly as the mass of the tide-generating +body, and inversely as the cube of the distance. + +The application of this law enables us to compare +the sun's tidal force with that of the moon. +The sun is $25,500,000$~times as heavy as the +moon, so that, on the score of mass, the solar +tidal force should be $25\frac{1}{2}$~million times greater +than that of the moon. But the sun is $389$~times +as distant as the moon. And since the +cube of~$389$ is about $59$~millions, the solar tidal +force should be $59$~million times weaker than +that of the moon, on the score of distance. +\PageSep{158} + +We have, then, a force which is $25\frac{1}{2}$~million +times stronger on account of the sun's greater +weight, and $59$~million times weaker on account +of his greater distance; it follows that the sun's +tide-generating force is $25\frac{1}{2}$-$59$ths, or a little +less than half of that of the moon. + +We conclude then that if the sun acted alone +on the water, the degree of elongation or distortion +of the ocean, when in equilibrium, would +be a little less than half of that due to the moon +alone. When both bodies act together, the distortion +of the surface due to the sun is superposed +on that due to the moon, and a terrestrial +observer perceives only the total or sum of the +two effects. + +When the sun and moon are together on the +same side of the earth, or when they are diametrically +opposite, the two distortions conspire +together, and the total tide will be half as great +again as that due to the moon alone, because +the solar tide is added to the lunar tide. And +when the sun and moon are at right angles to +\index{Sun!tide-generating force of, compared with that of moon|)}% +one another, the two distortions are at right +angles, and the low water of the solar tide conspires +with the high water of the lunar tide. +The composite tide has then a range only half as +great as that due to the moon alone, because the +solar tide, which has a range of about half that +\index{Solar!tide-generating force compared with lunar|)}% +of the lunar tide, is deducted from the lunar +\index{Lunar!tide-generating force compared with solar|)}% +tide. Since one and a half is three times a half, +\index{Forces!those of sun and moon compared|)}% +\index{Moon and earth!tide-generating force compared with sun's|)}% +\PageSep{159} +\index{Spring and neap tides!in equilibrium theory}% +it follows that when the moon and sun act together +the range of tide is three times as great +as when they act adversely. The two bodies +are together at change of moon and opposite at +full moon. In both of these positions their +actions conspire; hence at the change and the +full of moon the tides are at their largest, and +are called spring tides. When the two bodies +are at right angles to one another, it is half +moon, either waxing or waning, the tides have +their smallest range, and are called neap tides. +\index{Neap and spring tides!in equilibrium theory}% + +The observed facts agree pretty closely with +this theory in several respects, for spring tide +occurs about the full and change of moon, neap +tide occurs at the half moon, and the range at +springs is usually about three times as great as +that at neaps. Moreover, the diurnal inequality +conforms to the theory in vanishing when the +moon is on the equator, and rising to a maximum +when the moon is furthest north or south. The +amount of the diurnal inequality does not, however, +agree with theory, and in many places the +tide which should be the greater is actually the +less. + +The theory which I have sketched is called +the Equilibrium Theory of the Tides, because +it supposes that at each moment the ocean is +in that position of rest or equilibrium which it +would attain if indefinite time were allowed. +The general agreement with the real phenomena +\PageSep{160} +\index{Equilibrium theory of tides!defects of}% +proves the theory to have much truth about it, +but a detailed comparison with actuality shows +that it is terribly at fault. The lunar and solar +tidal ellipsoids were found to have their long +axes pointing straight towards the tide-generating +bodies, and, therefore, at the time when the +moon and sun pull together, it ought to be high +water just when they are due south. In other +words, at full and change of moon, it ought +to be high water exactly at noon and at midnight. +\index{High water!under moon in equilibrium theory}% +Now observation at spring tides shows +that at most places this is utterly contradictory +to fact. + +It is a matter of rough observation that the +tides follow the moon's course, so that high +water always occurs about the same number of +hours after the moon is due south. This rule +has no pretension to accuracy, but it is better +than no rule at all. Now at change and full of +the moon, the moon crosses the meridian at the +same hour of the clock as the sun, for at change +of moon they are together, and at full moon +they are twelve hours apart. Hence the hour +of the clock at which high water occurs at +change and full of moon is in effect a statement +of the number of hours which elapse after the +moon's passage of the meridian up to high +water. This clock time affords a rough rule for +the time of high water at any other phase of the +moon; if, for example, it is high water at eight +\PageSep{161} +o'clock at full and change, approximately eight +hours will always elapse after the moon's passage +until high water occurs. Mariners call the clock +time of high water at change and full of moon +\index{Establishment of port!definition}% +\index{Establishment of port!zero in equilibrium theory}% +``the establishment of the port,'' because it +establishes a rough rule of the tide at all other +times. + +According to the equilibrium theory, high +water falls at noon and midnight at full and +change of moon, or in the language of the mariner +the establishment of all ports should be +zero. But observation shows that the establishment +at actual ports has all sorts of values, and +that in the Pacific Ocean (where the tidal forces +have free scope) it is at least much nearer to six +hours than to zero. High water cannot be more +than six hours before or after noon or midnight +on the day of full or change of moon, because if +it occurs more than six hours after one noon, it +is less than six hours before the following midnight; +hence the establishment of any port +cannot possibly be more than six hours before or +after. Accordingly, the equilibrium theory is +nearly as much wrong as possible, in respect to +the time of high water. In fact, in many places +it is nearly low water at the time that the equilibrium +theory predicts high water. + +It would seem then as if the tidal action of +the moon was actually to repel the water instead +of attracting it, and we are driven to ask whether +\PageSep{162} +\index{Establishment of port!definition}% +this result can possibly be consistent with the +theory of universal gravitation. + +The existence of continental barriers across +the oceans must obviously exercise great influence +on the tides, but this fact can hardly be +responsible for a reversal of the previsions of the +equilibrium theory. It was Newton who showed +that a depression of the ocean under the moon +is entirely consistent with the theory of gravitation. +In the following chapter I shall explain +Newton's theory, and show how it explains the +discrepancy which we have found between the +equilibrium theory and actuality. + +\begin{Authorities} +An exposition of the equilibrium theory will be found in any +mathematical work on the subject, or in the article \Title{Tides} in the +``Encyclopædia Britannica.'' +\end{Authorities} +\index{Equilibrium theory of tides|)}% +\PageSep{163} + + +\Chapter{IX} +{Dynamical Theory of the Tide Wave} + +\First{The} most serious difficulties in the complete +\index{Dynamical theory of tide-wave|(}% +tidal problem do not arise in a certain special +case which was considered by Newton. His supposition +was that the sea is confined to a canal +\index{Canal!critical depth|(}% +circling the equator, and that the moon and sun +move exactly in the equator. + +An earthquake or any other gigantic impulse +may be supposed to generate a great wave in this +equatorial canal. The rate of progress of such +a wave is dependent on the depth of the canal +only, according to the laws sketched in \Ref{Chapter}{II}., +and the earth's rotation and the moon's attraction +make no sensible difference in its speed +of transmission. If, for example, the canal were +$5$~kilometres ($3$~miles) in depth, such a great +wave would travel $796$~kilometres ($500$~miles) +per~hour. If the canal were shallower the speed +would be less than this; if deeper, greater. +Now there is one special depth which will be +found to have a peculiar importance in the theory +of the tide, namely, where the canal is $13\frac{3}{4}$~miles +deep. In this case the wave travels $1,042$~miles +an hour, so that it would complete the +\PageSep{164} +$25,000$~miles round the earth in exactly $24$~hours. +It is important to note that if the depth of the +equatorial canal be less than $13\frac{3}{4}$~miles, a wave +takes more than a day to complete the circuit of +the earth, and if the depth be greater the circuit +is performed in less than a day. + +The great wave, produced by an earthquake or +other impulse, is called a ``free wave,'' because +\index{Free wave, explanation and contrast with forced}% +\index{Waves!forced and free}% +when once produced it travels free from the action +of external forces, and would persist forever, +were it not for the friction to which water is +necessarily subject. But the leading characteristic +of the tide wave is that it is generated and +kept in action by continuous forces, which act +on the fluid throughout all time. Such a wave +is called a ``forced wave,'' because it is due to +\index{Forced wave, explanation and contrast with free wave}% +the continuous action of external forces. The +rate at which the tide wave moves is moreover +dependent only on the rate at which the tidal +forces travel over the earth, and not in any degree +on the depth of the canal. It is true that +the depth of the canal exercises an influence on +the height of the wave generated by the tidal +forces, but the wave itself must always complete +the circuit of the earth in a day, because the +earth turns round in that period. + +We must now contrast the progress of any +long ``free wave'' in the equatorial canal with +that of the ``forced'' tide wave. I may premise +that it will here be slightly more convenient to +\PageSep{165} +consider the solar instead of the lunar tide. The +lunar wave is due to a stronger tide-generating +force, and since the earth takes $24$~hours $50$~minutes +to turn round with respect to the moon, that +is the time which the lunar tide wave takes to +complete the circuit of the earth; but these differences +are not material to the present argument. +The earth turns with respect to the sun +in exactly one day, or as we may more conveniently +say, the sun completes the circuit of the +earth in that time. Therefore the solar tidal +influence travels over the surface of the earth +at the rate of $1,042$~miles an hour. Now this is +exactly the pace at which a ``free wave'' travels +in a canal of a depth of $13\frac{3}{4}$~miles; accordingly +\index{Canal!theory of tide wave in|(}% +in such a canal any long free wave just keeps +pace with the sun. + +We have seen in \Ref{Chapter}{V}.\ that the solar +tide-generating force \emph{tends} to make a wave crest, +at those points of the earth's circumference where +it is noon and midnight. At each moment of +time the sun is generating a new wave, and after +it is generated that wave travels onwards as a +free wave. If therefore the canal has a depth +\index{Canal!critical depth|)}% +of $13\frac{3}{4}$~miles, each new wave, generated at each +moment of time, keeps pace with the sun, and +the summation of them all must build up two +enormous wave crests at opposite sides of the +earth. + +If the velocity of a free wave were absolutely +\PageSep{166} +the same whatever were its height, the crests of +the two tide waves would become infinite in +height. As a fact the rate of progress of a wave +is somewhat influenced by its height, and therefore, +when the waves get very big, they will +cease to keep pace exactly with the sun, and +the cause for continuous exaggeration of their +heights will cease to exist. We may, however, +express this conclusion by saying that, when the +canal is $13\frac{3}{4}$~miles deep, the height of the tide +wave becomes mathematically infinite. This does +not mean that mathematicians assert that the +wave would really become infinite, but only that +the simple method of treatment which supposes +the wave velocity to depend only on the depth +of water becomes inadequate. If the ocean was +really confined to an equatorial canal, of this exact +depth, the tides would be of very great +height, and the theory would be even more complex +than it is. It is, however, hardly necessary +to consider this special case in further detail. + +We conclude then that for the depth of $13\frac{3}{4}$~miles, +the wave becomes infinite in height, in +the qualified sense of infinity which I have described. +We may feel sure that the existence +of the quasi-infinite tide betokens that the behavior +of the water in a canal shallower than +$13\frac{3}{4}$~miles differs widely from that in a deeper +one. It is therefore necessary to examine into +the essential point in which the two cases differ +\PageSep{167} +from one another. In the shallower canal a free +wave covers less than $25,000$~miles a day, and +thus any wave generated by the sun would tend +to be left behind by him. On the other hand, +in the deeper canal a free wave would outstrip +the sun, and a wave generated by the sun tends +to run on in advance of him. But these are +only tendencies, for in both the shallower and +the deeper canal the actual tide wave exactly +keeps pace with the sun. + +It would be troublesome to find out what +would happen if we had the water in the canal +at rest, and were suddenly to start the sun to +work at it; and it is fortunately not necessary +to attempt to do so. It is, however, certain that +for a long time the motion would be confused, +but that the friction of the water would finally +produce order out of chaos, and that ultimately +there would be a pair of antipodal tide crests +traveling at the same pace as the sun. Our +task, then, is to discover what that final state of +motion may be, without endeavoring to unravel +the preliminary chaos. + +Let us take a concrete case, and suppose our +canal to be $3$~miles deep, in which we have seen +that a free wave will travel $500$~miles an hour. +Suppose, then, we start a long free wave in the +equatorial canal of $3$~miles deep, with two crests +\index{Canal!theory of tide wave in|)}% +$12,500$~miles apart, and therefore antipodal to +one another. The period of a wave is the time +\PageSep{168} +between the passage of two successive crests +past any fixed point. In this case the crests +are antipodal to one another, and therefore the +wave length is $12,500$~miles, and the wave +travels $500$~miles an hour, so that the period of a +free wave is $25$~hours. But the tide wave keeps +pace with the sun, traveling $1,042$~miles an +hour, and there are two antipodal crests, $12,500$~miles +apart; hence, the time between the passage +of successive tide crests is $12$~hours. + +In this case a free wave would have a period +of $25$~hours, and the tide wave, resulting from +the action of solar tide-generating force, has a +period of $12$~hours. The contrast then lies between +the free wave, with a period of $25$~hours, +and the forced wave, with a period of $12$~hours. + +For any other depth of ocean the free wave +will have another period depending on the depth, +but the period of the forced wave is always $12$~hours, +because it depends on the sun. If the +ocean be shallower than $3$~miles, the free period +will be greater than $25$~hours, and, if deeper, +less than $25$~hours. But if the ocean be deepened +to $13\frac{3}{4}$~miles, the free wave travels at the +same pace as the forced wave, and therefore the +two periods are coincident. For depths greater +than $13\frac{3}{4}$~miles, the period of the free wave is +less than that of the forced wave; and the +converse is true for canals less than $13\frac{3}{4}$~miles in +depth. +\PageSep{169} +\index{Forced oscillation!principle of}% +\index{Free oscillation contrasted with forced}% +\index{Principle of forced oscillations}% + +Now let us generalize this conception; we +have a system which if disturbed and left to +itself will oscillate in a certain period, called the +free period. Periodic disturbing forces act on +this system and the period of the disturbance +is independent of the oscillating system itself. +The period of the disturbing forces is called the +forced period. How will such a system swing, +when disturbed with this forced periodicity? + +A weight tied to the end of a string affords +an example of a very simple system capable of +oscillation, and the period of its free swing depends +on the length of the string only. I will +suppose the string to be $3$~feet, $3$~inches, or one +metre in length, so that the period of the swing +from right to left, or from left to right is one +second.\footnote + {A pendulum of one metre in length is commonly called a + seconds-pendulum, although its complete period is two seconds.} +If, holding the string, I move my +hand horizontally to and fro through a small +distance with a regular periodicity, I set the +pendulum a-swinging. The period of the movement +of my hand is the forced period, and the +free period is two seconds, being the time occupied +by a metre-long pendulum in moving from +right to left and back again to right. If I time +the to and fro motion of my hand so that its +period from right to left, or from left to right, +is exactly one second, the excursions of the pendulum +bob grow greater and greater without limit, +\PageSep{170} +\index{Forced oscillation!principle of}% +\index{Free oscillation contrasted with forced}% +\index{Principle of forced oscillations}% +because the successive impulses are stored up in +the pendulum, which swings further and further +with each successive impulse. This case is +exactly analogous with the quasi-infinite tides +which would arise in a canal $13\frac{3}{4}$~miles deep, and +here also this case is critical, separating two +modes of oscillation of the pendulum of different +characters. + +Now when the hand occupies more than one +second in moving from right to left, the forced +period is greater than the free period of the +pendulum; and when the system is swinging +steadily, it will be observed that the excursion +of the hand agrees in direction with the excursion +of the pendulum, so that when the hand is +furthest to the right so is also the pendulum, +and vice versa. When the period of the force +is greater than the free period of the system, at +the time when the force tends to make the pendulum +move to the right, it is furthest to the +right. The excursions of the pendulum agree +in direction with that of the hand. + +Next, when the hand occupies less than one +second to move from right to left or from left +to right, the forced period is less than the free +period, and it will be found that when the hand +is furthest to the right the pendulum is furthest +to the left. The excursions of the pendulum +are opposite in direction from those of the hand. +These two cases are illustrated by~\fig{30}, which +\PageSep{171} +\index{High water!position in shallow and deep canals in dynamical theory}% +will, perhaps, render my meaning more obvious. +We may sum up this discussion by saying that +in the case of a slowly varying disturbing force, +the oscillation and the force are consentaneous, +but that with a quickly oscillating force, the +oscillation is exactly inverted with respect to the +force. + +Now, this simple case illustrates a general +dynamical principle, namely, that if a system +\Figure{30}{Forced Oscillations of a Pendulum}{png} +capable of oscillating with a certain period is +acted on by a periodic force, when the period of +the force is greater than the natural free period +of the system, the oscillations of the system +agree with the oscillations of the force; but if +the period of the force is less than the natural +free period of the system the oscillations are +inverted with reference to the force. + +This principle may be applied to the case of +the tides in the canal. When the canal is more +than $13\frac{3}{4}$~miles deep, the period of the sun's disturbing +force is $12$~hours and is greater than the +\PageSep{172} +\index{Newton!theory of tide in equatorial canal}% +natural free period of the oscillation, because a +free wave would go more than half round the +earth in $12$~hours. We conclude, then, that when +the tide-generating forces are trying to make it +high water, it will be high water. It has been +\index{High water!position in shallow and deep canals in dynamical theory}% +shown that these forces are tending to make high +water immediately under the sun and at its antipodes, +and there accordingly will the high water +be. In this case the tide is said to be direct. + +But when the canal is less than $13\frac{3}{4}$~miles +deep, the sun's disturbing force has, as before, a +period of $12$~hours, but the period of the free +wave is more than $12$~hours, because a free wave +would take more than $12$~hours to get half round +the earth. Thus the general principle shows +that where the forces are trying to make high +water, there will be low water, and vice versa. +Here, then, there will be low water under the sun +and at its antipodes, and such a tide is said to +be inverted, because the oscillation is the exact +inversion of what would be naturally expected. + +All the oceans on the earth are very much +shallower than fourteen miles, and so, at least +near the equator, the tides ought to be inverted. +The conclusion of the equilibrium theory will +therefore be the exact opposite of the truth, near +the equator. + +This argument as to the solar tide requires +but little alteration to make it applicable to the +lunar tide. In fact the only material difference +\PageSep{173} +\index{Waves!of tide in equatorial canal}% +in the conditions is that the period of the lunar +tide is $12$~hours $25$~minutes, instead of $12$~hours, +and so the critical depth of an equatorial canal, +\index{Equatorial canal, tide wave in}% +which would allow the lunar tide to become +quasi-infinite, is a little less than that for the +solar tide. This depth for the lunar tide is in +fact nearly $13$~miles.\footnote + {It is worthy of remark that if the canal had a depth of between + $13\frac{3}{4}$ and $13$~miles, the solar tides would be inverted, and + the lunar tides would be direct. We should then, at the equator, + have spring tide at half moon, when our actual neaps occur; + and neap tide at full and change, when our actual springs occur. + The tides would also be of enormous height, because the depth + is nearly such as to make both tides quasi-infinite. If the depth + of the canal were very nearly $13\frac{3}{4}$~miles the solar tide might be + greater than the lunar. But these exceptional cases have only + a theoretical interest.} + +This discussion should have made it clear that +any tidal theory, worthy of the name, must take +account of motion, and it explains why the prediction +of the equilibrium theory is so wide from +the truth. Notwithstanding, however, this condemnation +of the equilibrium theory, it is of the +utmost service in the discussion of the tides, +because by far the most convenient and complete +way of specifying the forces which act on the +ocean at each instant is to determine the figure +which the ocean would assume, if the forces had +abundant time to act. + +\TB + +When the sea is confined to an equatorial +canal, the tidal problem is much simpler than +\PageSep{174} +when the ocean covers the whole planet, and +this is much simpler than when the sea is interrupted +by continents. Then again, we have +thus far supposed the sun and moon to be always +exactly over the equator, whereas they actually +range a long way both to the north and to the +south of the equator; and so here also the true +problem is more complicated than the one under +consideration. Let us next consider a case, still +far simpler than actuality, and suppose that +whilst the moon or sun still always move over +the equator, the ocean is confined to several +canals which run round the globe, following parallels +of latitude. +\index{Latitude!tidal wave in canal in high|(}% + +The circumference of a canal in latitude~$60°$ +\index{Canal!canal in high latitude|(}% +is only $12,500$~miles, instead of~$25,000$. If a +free wave were generated in such a canal with +two crests at opposite sides of the globe, the distance +from crest to crest would be $6,250$~miles. +Now if an equatorial canal and one in latitude~$60°$ +have equal depths, a free wave will travel at +the same rate along each; and if in each canal +there be a wave with two antipodal crests, the +time occupied by the wave in latitude~$60°$ in +traveling through a space equal to its length will +be only half of the similar period for the equatorial +waves. The period of a free wave in latitude~$60°$ +\index{Waves!in canal in high latitude|(}% +is therefore half what it is at the +equator, for a pair of canals of equal depths. +But there is only one sun, and it takes $12$~hours +\PageSep{175} +to go half round the planet, and therefore for +both canals the forced tide wave has a period of +$12$~hours. If, for example, both canals were +$8$~miles deep, in the equatorial canal the +\index{Canal!tides in ocean partitioned into canals}% +period of the free wave would be greater than +$12$~hours, whilst in the canal at $60°$~of latitude +it would be less than $12$~hours. It follows then +from the general principle as to forced and free +oscillations, that whilst the tide in the equatorial +canal would be inverted, that in latitude~$60°$ +would be direct. Therefore, whilst it would be +low water under the moon at the equator, it +would be high water under the moon in latitude~$60°$. +Somewhere, between latitude~$60°$ and the +equator, there must be a place at which the free +period in a canal $8$~miles deep is the same as +the forced period, and in a canal at this latitude +the tide would be infinite in height, in the modified +sense explained earlier. It follows therefore +that there is for any given depth of canal, less +than $14$~miles, a critical latitude, at which the +tide tends to become infinite in height. + +We conclude, that if the whole planet were +divided up into canals each partitioned off from +its neighbor, and if the canals were shallower +than $14$~miles, we should have inverted tides in +the equatorial region, and direct tides in the +polar regions, and, in one of the canals in some +middle latitude, very great tides the nature of +which cannot be specified exactly. +\PageSep{176} + +The supposed partitions between neighboring +canals have introduced a limitation which must +be removed, if we are to approach actuality, but +I am unable by general reasoning to do more +than indicate what will be the effect of the removal +of the partitions. It is clear that when +the sea swells up to form the high water, the +water comes not only from the east and the west +of the place of high water, but also from the +north and south. The earth, as it rotates, carries +with it the ocean; the equatorial water is +carried over a space of $25,000$~miles in $24$~hours, +whereas the water in latitude~$60°$ is carried over +only $12,500$~miles in the same time. When, in +the northern hemisphere, water moves from north +to south it passes from a place where the surface +of the earth is moving slower, to where it is +moving quicker. Then, as the water goes to the +south, it carries with it only the velocity adapted +to the northern latitude, and so it gets left behind +by the earth. Since the earth spins from +west to east, a southerly current acquires a westward +trend. Conversely, when water is carried +northward of its proper latitude, it leaves the +\index{Latitude!tidal wave in canal in high|)}% +earth behind and is carried eastward. Hence +the water cannot oscillate northward and southward, +without at the same time oscillating eastward +and westward. Since in an ocean not +partitioned into canals, the water must necessarily +move not only east and west but also north +\index{Canal!canal in high latitude|)}% +\index{Waves!in canal in high latitude|)}% +\PageSep{177} +\index{Earth and moon!rotation of, effects on tides}% +\index{Rotation!of earth involved in tidal problem}% +\index{Vortical motion in oceanic tides}% +and south, it follows that tidal movements in the +ocean must result in eddies or vortices. The +\index{Eddies, tidal oscillation involves}% +eddying motion of the water must exist everywhere, +but it would be impossible, without mathematical +reasoning, to explain how all the eddies +fit into one another in time and place. It must +suffice for the present discussion for the reader +to know that the full mathematical treatment of +the problem shows this general conclusion to be +correct. + +The very difficult mathematical problem of +the tides of an ocean covering the globe to a +uniform depth was first successfully attacked by +Laplace. He showed that whilst the tides of a +\index{Laplace!theory of tides|(}% +shallow ocean are inverted at the equator, as +proved by Newton, that they are direct towards +the pole. We have just arrived at the same +conclusion by considering the tide wave in a +canal in latitude~$60°$. But our reasoning indicated +that somewhere in between higher latitudes +and the equator, the tide would be of an undefined +character, with an enormous range of rise +and fall. The complete solution of the problem +shows, however, that this indication of the +canal theory is wrong, and that the tidal variation +of level absolutely vanishes in some latitude +intermediate between the equator and the pole. +The conclusion of the mathematician is that +there is a certain circle of latitude, whose position +depends on the depth of the sea, where +there is neither rise nor fall of tide. +\PageSep{178} +\index{Vortical motion in oceanic tides}% + +At this circle the water flows northward and +southward, and to and fro between east and +west, but in such a way as never to raise or depress +the level of the sea. It is not true to say +that there is no tide at this circle, for there are +tidal currents without rise and fall. When the +ocean was supposed to be cut into canals, we +thereby obliterated the northerly and southerly +currents, and it is exactly these currents which +prevent the tides becoming very great, as we +were then led to suppose they would be. + +It may seem strange that, whereas the first +rough solution of the problem indicates an oscillation +of infinite magnitude at a certain parallel +of latitude, the more accurate treatment of the +case should show that there is no oscillation of +level at all. Yet to the mathematician such a +result is not a cause of surprise. But whether +strange or not, it should be clear that if at the +equator it is low water under the moon, and if +near the pole it is high water under the moon, +there must in some intermediate latitude be a +place where the water is neither high nor low, +that is to say, where there is neither rise nor fall.\footnote + {The mathematician knows that a quantity may change sign, + either by passing through infinity or through zero. Where a + change from positive to negative undoubtedly takes place, and + where a passage through infinity can have no physical meaning, + the change must take place by passage through zero.} + +\TB + +Now let us take one more step towards actuality, +and suppose the earth's equator to be +\PageSep{179} +oblique to the orbits of the moon and sun, so +that they may sometimes stand to the north and +sometimes to the south of the equator. We +have seen that in this case the equilibrium theory +indicates that the two successive tides on any +one day have unequal ranges. The mathematical +solution of the problem shows that this conclusion +is correct. It appears also that if the +ocean is deeper at the poles than at the equator, +that tide is the greater which is asserted to be +so by the equilibrium theory. If, however, the +ocean is shallower at the poles than at the equator, +it is found that the high water which the +equilibrium theory would make the larger is actually +the smaller and vice versa. + +If the ocean is of the same depth everywhere, +we have a case intermediate between the two, +where it is shallower at the poles, and where it is +deeper at the poles. Now in one of these cases +it appears that the higher high water occurs +where in the other we find the lower high water +to occur; and so, when the depth is uniform, +the higher high water and the lower high water +must attain the same heights. We thus arrive +at the remarkable conclusion that, in an ocean +of uniform depth, the diurnal inequality of the +\index{Diurnal inequality!in Laplace's solution}% +tide is evanescent. There are, however, diurnal +inequalities in the tidal currents, which are so +adjusted as not to produce a rise or fall. This +result was first arrived at by the great mathematician +Laplace. +\PageSep{180} + +According to the equilibrium theory, when the +moon stands some distance north of the equator, +the inequality between the successive tides on +the coasts of Europe should be very great, but +the difference is actually so small as to escape +ordinary observation. In the days of Laplace, +the knowledge of the tides in other parts of the +world was very imperfect, and it was naturally +thought that the European tides were fairly +representative of the whole world. When, then, +it was discovered that there would be no diurnal +inequality in an ocean of uniform depth covering +the whole globe, it was thought that a fair explanation +had been found for the absence of +that inequality in Europe. But since the days +of Laplace much has been learnt about the tides +\index{Laplace!theory of tides|)}% +in the Pacific and Indian oceans, and we now +know that a large diurnal inequality is almost +\index{Diurnal inequality!in Atlantic, Pacific, and Indian Oceans}% +universal, so that the tides of the North Atlantic +are exceptional in their simplicity. In fact, the +evanescence of the diurnal inequality is not much +closer to the truth than the large inequality +predicted by the equilibrium theory; and both +theories must be abandoned as satisfactory explanations +of the true condition of affairs. But +notwithstanding their deficiencies both these +theories are of importance in teaching us how +the tides are to be predicted. In the next chapter +I shall show how a further approximation to +the truth is attainable. +\PageSep{181} + +\begin{Authorities} +The canal theory in its elementary form is treated in many +works on Hydrodynamics, and in \Title{Tides}, ``Encyclopædia Britannica.'' + +An elaborate treatment of the subject is contained in Airy's +\index{Airy, Sir G. B.!attack on Laplace}% +\Title{Tides and Waves}, ``Encyclopædia Metropolitana.'' Airy there +attacks Laplace for his treatment of the wider tidal problem, +but his strictures are now universally regarded as unsound. + +Laplace's theory is contained in the \Title{Mécanique Céleste}, but it +is better studied in more recent works. + +A full presentment of this theory is contained in Professor +Horace Lamb's \Title{Hydrodynamics}, Camb.\ Univ.\ Press, 1895, chapter~viii. +\index{Lamb, H., presentation of Laplace's theory}% + +Important papers, extending Laplace's work, by Mr.~S.~S. +Hough, are contained in the \Title{Philosophical Transactions of the +\index{Hough, S. S.!dynamical solution of tidal problem}% +Royal Society}, A.~1897, pp.~201--258, and A.~1898, pp.~139--185. +\end{Authorities} +\index{Dynamical theory of tide-wave|)}% +\PageSep{182} + + +\Chapter{X} +{Tides in Lakes---Cotidal Chart} + +\First{If} the conditions of the tidal problem are to +\index{Lakes!tides in|(}% +agree with reality, an ocean must be considered +which is interrupted by continental barriers of +land. The case of a sea or lake entirely surrounded +by land affords the simplest and most +complete limitation to the continuity of the +water. I shall therefore begin by considering +the tides in a lake. + +The oscillations of a pendulum under the tidal +attraction of the moon were considered in \Ref{Chapter}{VI}., +and we there saw that the pendulum +would swing to and fro, although the scale of +displacement would be too minute for actual +observation. Now a pendulum always hangs +perpendicularly to the surface of water, and +must therefore be regarded as a sort of level. +As it sways to and fro under the changing action +of the tidal force, so also must the surface +of water. If the water in question is a lake, the +rocking of the level of the lake is a true tide. + +A lake of say a hundred miles in length is +very small compared with the size of the earth, +and its waters must respond almost instantaneously +\PageSep{183} +to the changes in the tidal force. Such +a lake is not large enough to introduce, to a +perceptible extent, those complications which +make the complete theory of oceanic tides so +difficult. The equilibrium theory is here actually +true, because the currents due to the changes in +the tidal force have not many yards to run before +equilibrium is established, and the lake may +be regarded as a level which responds almost +instantaneously to the tidal deflections of gravity. +The open ocean is a great level also, but sufficient +time is not allowed it to respond to the changes +in the direction of gravity, before that direction +has itself changed. + +It was stated in \Ref{Chapter}{V}.\ that the maximum +horizontal force due to the moon has an intensity +equal to $\frac{1}{11,664,000}$~part of gravity, and that +therefore a pendulum $10$~metres long is deflected +through $\frac{1}{11,664,000}$~of $10$~metres, or through $\frac{1}{1,166}$~of +a millimetre. Now suppose our lake, $200$~kilometres +in length, runs east and west, and that +our pendulum is hung up at the middle of the +lake, $100$~kilometres from either end. In \fig{31} +let $\Seg{C}{D}$ represent the level of the lake as +undisturbed, and $\Seg{A}{B}$~an exaggerated pendulum. +When the tide-generating force displaces the +pendulum to~$\Seg{A}{B'}$, the surface of the lake must +assume the position~$\Seg{C'}{D'}$. Now $\Seg{A}{B}$~being $10$~metres, +$\Seg{B}{B'}$~may range as far as $\frac{1}{1,166}$~of a millimetre; +and it is obvious that $\Seg{C}{C'}$~must bear the +\PageSep{184} +same relation to~$\Seg{C}{B}$ that $\Seg{B}{B'}$ does to~$\Seg{A}{B}$. +Hence $\Seg{C}{C'}$ at its greatest may be $\frac{1}{11,664,000}$~of half +the length of the lake. The lake is supposed +to be twice $100$~kilometres in length, and $100$~kilometres +is $10$~million centimetres; thus $\Seg{C}{C'}$~is +$\frac{1}{1.1664}$~centimetre, or $\frac{9}{10}$~of a centimetre. When +the pendulum is deflected in the other direction +the lake rocks the other way, and $C'$~is just as +much above~$C$ as it was below it before. It +follows from this that the lunar tide at the ends +of a lake, $200$~kilometres or $120$~miles in length, +has a range of $1\frac{3}{4}$~centimetres or $\frac{2}{3}$~of an inch. +The solar tidal force is a little less than half as +strong as that due to the moon, and when the +two forces conspire together at the times of +spring tide, we should find a tide with a range +of $2\frac{1}{2}$~centimetres. +\Figure{31}{The Tide in a Lake}{png} + +If the same rule were to apply to a lake $2,000$~kilometres +or $1,200$~miles in length, the range +of lunar tide would be about $17$~centimetres or +$7$~inches, and the addition of solar tides would +bring the range up to $25$~centimetres or $10$~inches. +\PageSep{185} +I dare say that, for a lake of such a +size, this rule would not be very largely in error. +But as we make the lake longer, the currents +set up by the tidal forces have not sufficient time +to produce their full effects before the intensity +and direction of the tidal forces change. Besides +this, if the lake were broad from north to south, +the earth's rotation would have an appreciable +effect, so that the water which flows from the +north to the south would be deflected westward, +and that which flows from south to north would +tend to flow eastward. The curvature of the +earth's surface must also begin to affect the +motion. For these reasons, such a simple rule +would then no longer suffice for calculating the +tide. + +Mathematicians have not yet succeeded in +solving the tidal problem for a lake of large +dimensions, and so it is impossible to describe +the mode of oscillation. It may, however, be asserted +that the shape, dimensions, and depth of +the lake, and the latitudes of its boundaries will +affect the result. The tides on the northern and +southern shores will be different, and there will +be nodal lines, along which there will be no rise +and fall of the water. +\index{Lakes!tides in|)}% + +The Straits of Gibraltar are so narrow, that +\index{Mediterranean Sea, tides in}% +the amount of water which can flow through +them in the six hours which elapse between +high and low water in the Atlantic is inconsiderable. +\PageSep{186} +\index{Waves!propagated northward in Atlantic|(}% +Hence the Mediterranean Sea is virtually +\index{Mediterranean Sea, tides in}% +a closed lake. The tides of this sea are +much complicated by the constriction formed +by the Sicilian and Tunisian promontories. Its +tides probably more nearly resemble those of two +lakes than of a single sheet of water. The tides +of the Mediterranean are, in most places, so inconspicuous +that it is usually, but incorrectly, +described as a tideless sea. Every visitor to +Venice must, however, have seen, or may we say +smelt, the tides, which at springs have a range of +some four feet. The considerable range of tide +at Venice appears to indicate that the Adriatic +\index{Adriatic, tide in}% +acts as a resonator for the tidal oscillation, in the +same way that a hollow vessel, tuned to a particular +note, picks out and resonates loudly when +that note is sounded. + +We see, then, that whilst the tides of a small +lake are calculable by the equilibrium theory, +those of a large one, such as the Mediterranean, +remain intractable. It is clear, then, that the +tides of the ocean must present a problem yet +more complex than those of a large lake. + +In the Pacific and Southern oceans the tidal +\index{Pacific Ocean, tide in, affects Atlantic}% +forces have almost uninterrupted sway, but the promontories +of Africa and of South America must +profoundly affect the progress of the tide wave +from east to west. The Atlantic Ocean forms a +\index{Atlantic, tide in|(}% +great bay in this vaster tract of water. If this +inlet were closed by a barrier from the Cape of +\PageSep{187} +\index{Pacific Ocean, tide in, affects Atlantic}% +Good Hope to Cape Horn, it would form a lake +large enough for the generation of much larger +tides than those of the Mediterranean Sea, although +probably much smaller than those which +we actually observe on our coasts. Let us now +suppose that the tides proper to the Atlantic are +non-existent, and let us remove the barrier between +the two capes. Then the great tide wave +sweeps across the Southern ocean from east to +west, and, on reaching the tract between Africa +and South America, generates a wave which +travels northward up the Atlantic inlet. This +secondary wave travels ``freely,'' at a rate dependent +only on the depth of the ocean. The +energy of the wave motion is concentrated, where +the channel narrows between North Africa and +Brazil, and the height of the wave must be augmented +in that region. Then the energy is +weakened by spreading, where the sea broadens +again, and it is again reconcentrated by the projection +of the North American coast line towards +Europe. Hence, even in this case, ideally simplified +as it is by the omission of the direct action +of the moon and sun, the range of tide would +differ at every portion of the coasts on each side +of the Atlantic. + +The time of high water at any place must also +depend on the varying depth of the ocean, for it +is governed by the time occupied by the ``free +wave'' in traveling from the southern region to +\PageSep{188} +\index{Cotidal chart}% +the north. But in the south, between the two +capes of Africa and South America, the tidal +oscillation is constrained to keep regular time +with the moon, and so it will keep the same +rhythm at every place to the northward, at whatever +variable pace the wave may move. The +time of high water will of course differ at every +point, being later as we go northward. The +wave may indeed occupy so long on its journey, +that one high water may have only just arrived +at the northern coast of Africa, when another is +rounding the Cape of Good Hope. + +Under the true conditions of the case, this +``free'' wave, generated in and propagated from +the southern ocean, is fused with the true +``forced'' tide wave generated in the Atlantic itself. +\index{Atlantic, tide in|)}% +It may be conjectured that on the coast of +Europe the latter is of less importance than the +\index{Europe, tides on coasts of}% +former. It is interesting to reflect that our tides +to-day depend even more on what occurred yesterday +or the day before in the Southern Pacific +and Indian oceans, than on the direct action of +the moon to-day. But the relative importance +of the two causes must remain a matter of conjecture, +for the problem is one of insoluble complexity. + +Some sixty years ago Whewell, and after him +\index{Whewell!on cotidal charts}% +Airy, drew charts illustrative of what has just +\index{Airy, Sir G. B.!cotidal chart}% +been described. A map showing the march +of the tide wave is reproduced from Airy's +\index{Waves!propagated northward in Atlantic|)}% +\PageSep{189} +\index{Establishment of port!shown in cotidal chart}% +``Tides and Waves,'' in~\fig{32}. It claims to +show, from the observed times of high water at +the various parts of the earth, how the tide wave +travels over the oceans. Whewell and Airy were +\index{Whewell!on cotidal charts}% +well aware that their map could only be regarded +as the roughest approximation to reality. Much +has been learnt since their days, and the then +incomplete state of knowledge hardly permitted +them to fully realize how very rough was their +approximation to the truth. No more recent attempt +has been made to construct such a map, +and we must rest satisfied with this one. Even +if its lines may in places depart pretty widely +from the truth, it presents features of much interest. +I do not reproduce the Pacific Ocean, +because it is left almost blank, from deficiency +of data. Thus, in that part of the world where +the tides are most normal, and where the knowledge +of them would possess the greatest scientific +interest, we are compelled to admit an almost +total ignorance. + +The lines on the map, \fig{32}, give the Greenwich +times of high water at full and change of +moon. They thus purport to represent the successive +positions of the crest of the tide wave. +For example, at noon and midnight (XII~o'clock), +at full and change of moon, the crest +of the tide wave runs from North Australia to +Sumatra, thence to Ceylon, whence it bends back +to the Island of Bourbon, and, passing some hundreds +\PageSep{190} +\TallFig[0.875]{32}{Chart of Cotidal Lines}{jpg} +\PageSep{191} +\index{Cotidal chart!for diurnal tide hitherto undetermined}% +of miles south of the Cape of Good Hope, +trends away towards the Antarctic Ocean. At +the same moment the previous tide crest has +traveled up the Atlantic, and is found running +across from Newfoundland to the Canary +Islands. A yet earlier crest has reached the +north of Norway. At this moment it is low +water from Brazil to the Gold Coast, and again +at Great Britain. + +The successive lines then exhibit the progress +of the wave from hour to hour, and we see how +the wave is propagated into the Atlantic. The +crowding together of lines in places is the graphical +representation of the retardation of the +wave, as it runs into shallower water. + +But even if this chart were perfectly trustworthy, +it would only tell us of the progress of +the ordinary semidiurnal wave, which produces +high water twice a day. We have, however, seen +reason to believe that two successive tides should +not rise to equal heights, and this figure does +not even profess to give any suggestion as to +how this inequality is propagated. In other +words, it is impossible to say whether two successive +tides of unequal heights tend to become +more or less unequal, as they run into any of +the great oceanic inlets. Thus the map affords +no indication of the law of the propagation of +the diurnal inequality. +\index{Diurnal inequality!not shown in cotidal chart}% + +This sketch of the difficulties in the solution +\PageSep{192} +\index{Cotidal chart!for diurnal tide hitherto undetermined}% +of the full tidal problem might well lead to despair +of the possibility of tidal prediction on our +coasts. I shall, however, show in the next chapter +how such prediction is possible. + +\begin{Authorities} +For cotidal charts see Whewell, \Title{Phil.\ Trans.\ Roy.\ Soc.}\ 1833, +or Airy's \Title{Tides and Waves}, ``Encyclopædia Metropolitana.'' +\index{Airy, Sir G. B.!\Title{Tides and Waves}}% +\end{Authorities} +\PageSep{193} + + +\Chapter{XI} +{Harmonic Analysis of the Tide} + +\First{It} is not probable that it will ever be possible +\index{Analysis, harmonic, of tide|(}% +\index{Harmonic analysis!account of|(}% +\index{Moon and earth!tide due to ideal, moving in equator}% +to determine the nature of the oceanic oscillation +as a whole with any accuracy. It is true that +we have already some knowledge of the general +march of the tide wave, and we shall doubtless +learn more in the future, but this can never suffice +for accurate prediction of the tide at any +place. + +Although the equilibrium theory is totally +false as regards its prediction of the time of passage +and of the height of the tide wave, yet it +furnishes the stepping-stone leading towards the +truth, because it is in effect a compendious statement +of the infinite variety of the tidal force in +time and place. + +I will begin my explanation of the practical +method of tidal prediction by obliterating the +sun, and by supposing that the moon revolves in +an equatorial circle round the earth. In this +case the equilibrium theory indicates that each +tide exactly resembles its predecessors and its +successors for all time, and that the successive +and simultaneous passages of the moon and of +\PageSep{194} +\index{Moon and earth!tide due to ideal, moving in equator}% +the wave crests across any place follow one +another at intervals of $12$~hours $25$~minutes. It +would always be exactly high water under or +opposite to the moon, and the height of high +water would be exactly determinate. In actual +oceans, even although only subject to the action +of such a single satellite, the motion of the water +would be so complex that it would be impossible +to predict the exact height or time of high or +of low water. But since the tidal forces operate +in a stereotyped fashion day after day, there will +be none of that variability which actually occurs +on the real earth under the actions of the real +sun and moon, and we may positively assert that +whatever the water does to-day it will do to-morrow. +Thus, if at a given place it is high water +at a definite number of hours after the equatorial +moon has crossed the meridian to-day, it will be +so to-morrow at the same number of hours after +the moon's passage, and the water will rise and +fall every day to the same height above and below +the mean sea level. If then we wanted to +know how the tide would rise and fall in a given +harbor, we need only watch the motion of the +sea at that place, for however the water may +move elsewhere its motion will always produce +the same result at the port of observation. +Thus, apart from the effects of wind, we should +only have to note the tide on any one day +to be able to predict it for all time. For by a +\PageSep{195} +\index{Satellites!tide due to single equatorial}% +single day of observation it would be easy to +note how many hours after the moon's passage +high water occurs, and how many feet it rises +and falls with reference to some fixed mark on +the shore. The delay after the moon's passage +and the amount of rise and fall would differ geographically, +but at each place there would be two +definite numbers giving the height of the tide +and the interval after the moon's passage until +high water. These two numbers are called the +tidal constants for the port; they would virtually +\index{Constants, tidal, explained}% +contain tidal predictions for all time. + +Now if the moon were obliterated, leaving the +sun alone, and if he also always moved over +the equator, a similar rule would hold good, +but exactly $12$~hours would elapse from one +high water to the next, instead of $12$~hours $25$~minutes +as in the case of the moon's isolated +action. Thus two other tidal constants, expressive +of height and interval, would virtually contain +tidal prediction for the solar tide for all +time. + +Theory here gives us some power of foreseeing +the relative importance of the purely lunar +and of the purely solar tide. The two waves +due to the sun alone or to the moon alone have +the same character, but the solar waves follow +one another a little quicker than the lunar waves, +and the sun's force is a little less than half the +moon's force. The close similarity between the +\PageSep{196} +\index{Satellites!tide due to single equatorial}% +actions of the sun and moon makes it safe to conclude +that the delay of the isolated solar wave +after the passage of the sun would not differ +much from the delay of the isolated lunar wave +after the passage of the moon, and that the +height of the solar wave would be about half of +that of the lunar wave. But theory can only be +trusted far enough to predict a rough proportionality +of the heights of the two tide waves to their +respective generating forces, and the approximate +equality of the intervals of retardation; but the +height and retardation of the solar wave could +not be accurately foretold from observation of +the lunar wave. + +When the sun and moon coëxist, but still +move in equatorial circles, the two waves, which +we have considered separately, are combined. +The four tidal constants, two for the moon and +two for the sun, would contain the prediction of +the height of water for all time, for it is easy at +any future moment of time to discover the two +intervals of time since the moon and since the +sun have crossed the meridian of the place of +observation; we should then calculate the height +of the water above some mark on the shore on +the supposition that the moon exists alone, and, +again, on the supposition that the sun exists +alone, and adding the two results together, should +obtain the required height of the water at the +moment in question. +\PageSep{197} + +But the real moon and sun do not move in +equatorial circles, but in planes which are oblique +to the earth's equator, and they are therefore +sometimes to the north and sometimes to the +south of the equator; they are also sometimes +nearer and sometimes further from the earth on +account of the eccentricity of the orbits in which +they move. Now the mathematician treats this +complication in the following way: he first considers +the moon alone and replaces it by a number +of satellites of various masses, which move +in various planes. It is a matter of indifference +that such a system of satellites could not maintain +the orbits assigned to them if they were allowed +to go free, but a mysterious being may be +postulated who compels the satellites to move in +the assigned orbits. One, and this is the largest +of these ideal satellites, has nearly the same mass +as the real moon and moves in a circle over the +equator; it is in fact the simple isolated moon +whose action I first considered. Another small +satellite stands still amongst the stars; others +move in such orbits that they are always vertically +overhead in latitude~$45°$; others repel instead +of attracting; and others move backwards +amongst the stars. Now all these satellites are +so arranged as to their masses and their orbits, +that the sum of their tidal forces is exactly the +same as those due to the real moon moving in +her actual orbit. +\PageSep{198} +\index{Interval from moon's transit to high water!in case of ideal satellite}% + +So far the problem seems to be complicated +rather than simplified, for we have to consider a +dozen moons instead of one. The simplification, +however, arises from the fact that each satellite +either moves uniformly in an orbit parallel to the +equator, or else stands still amongst the stars. +It follows that each of the ideal satellites creates +a tide in the ocean which is of a simple character, +and repeats itself day after day in the same +way as the tide due to an isolated equatorial +moon. If all but one of these ideal satellites +were obliterated the observation of the tide for +a single day would enable us to predict the tide +for all time; because it would only be necessary +to note the time of high water after the ideal +satellite had crossed the meridian, and the height +\index{Height of tide!due to ideal satellite}% +of the high water, and then these two data would +virtually contain a tidal prediction for that tide +at the place of observation for all future time. +The interval and height are together a pair of +``tidal constants'' for the particular satellite in +question, and refer only to the particular place +at which the observation is made. + +In actuality all the ideal satellites coëxist, and +the determination of the pair of tidal constants +appropriate to any one of them has to be made +by a complex method of analysis, of which I shall +say more hereafter. For the present it will suffice +to know that if we could at will annul all +the ideal satellites except one, and observe its +\PageSep{199} +\index{Moon and earth!ideal satellites replacing actual}% +\index{Partial tides in harmonic method}% +tide even for a single day, its pair of constants +could be easily determined. It would then only +be necessary to choose in succession all the satellites +\index{Satellites!ideal replacing sun and moon in harmonic analysis}% +as subjects of observation, and the materials +for a lunar tide table for all time would be obtained. + +The motion of the sun round the earth is analogous +to that of the moon, and so the sun has +also to be replaced by a similar series of ideal +suns, and the partial tide due to each of them +has to be found. Finally at any harbor some +twenty pairs of numbers, corresponding to twenty +ideal moons and suns, give the materials for tidal +prediction for all time. Theoretically an infinite +number of ideal bodies is necessary for an absolutely +perfect representation of the tides, but +after we have taken some twenty of them, the +remainder are found to be excessively small in +mass, and therefore the tides raised by them are +so minute that they may be safely omitted. This +method of separating the tide wave into a number +of partial constituents is called ``harmonic +analysis.'' It was first suggested, and put into +practice as a practical treatment of the tidal +problem, by Sir William Thomson, now Lord +Kelvin, and it is in extensive use. +\index{Kelvin, Lord!initiates harmonic analysis}% + +In this method the aggregate tide wave is considered +as the sum of a number of simple waves +following one another at exactly equal intervals +of time, and always presenting a constant rise +\PageSep{200} +\index{Moon and earth!ideal satellites replacing actual}% +\index{Prediction of tide!due to ideal satellite}% +and fall at the place of observation. When the +time of high water and the height of any one of +these constituent waves is known on any one +day, we can predict, with certainty, the height +of the water, as due to it alone, at any future +time however distant. The period of time which +elapses between the passage of one crest and of +the next is absolutely exact, for it is derived from +a study of the motions of the moon or sun, and +is determined to within a thousandth of a second. +The instant at which any one of the satellites +\index{Satellites!ideal replacing sun and moon in harmonic analysis}% +passes the meridian of the place is also +known with absolute accuracy, but the interval +after the passage of the satellite up to the high +water of any one of these constituent waves, and +the height to which the water will rise are only +derivable from observation at each port. + +Since there are about twenty coëxistent waves +of sensible magnitude, a long series of observations +is requisite for disentangling any particular +wave from among the rest. The series must +also be so long that the disturbing influence of +the wind, both on height and time, may be eliminated +by the taking of averages. It may be +well to reiterate that each harbor has to be considered +by itself, and that a separate set of tidal +constants has to be found for each place. If it +is only required to predict the tides with moderate +accuracy some eight partial waves suffice, but +if high accuracy is to be attained, we have to +\PageSep{201} +consider a number of the smaller ones, bringing +the total up to $20$ or~$25$. + +When the observed tidal motions of the sea +have been analyzed into partial tide waves, they +are found to fall naturally into three groups, +which correspond with the dissections of the sun +\index{Sun!ideal, replacing real sun in harmonic analysis}% +and moon into the ideal satellites. In the first +and most important group the crests follow one +another at intervals of somewhere about $12$~hours; +these are called the semidiurnal tides. +In the second group, the waves of which are in +most places of somewhat less height than those +of the semidiurnal group, the crests follow one +another at intervals of somewhere about $24$~hours, +and they are called diurnal. The tides +of the third group have a very slow periodicity, +for their periods are a fortnight, a month, half +a year, and a year; they are commonly of very +small height, and have scarcely any practical +importance; I shall therefore make no further +reference to them. + +Let us now consider the semidiurnal group. +The most important of these is called ``the principal +lunar semidiurnal tide.'' It is the tide +\index{Lunar!tide, principal}% +\index{Semidiurnal tide!in harmonic method|(}% +raised by an ideal satellite, which moves in a circle +round the earth's equator. I began my explanation +of this method by a somewhat detailed +consideration of this wave. In this case, the +wave crests follow one another at intervals of +$12$~hours $25$~minutes $14\frac{1}{6}$~seconds. The average +\PageSep{202} +interval of time between the successive visible +transits of the moon over the meridian of the +place of observation is $24$~hours $50$~minutes $28\frac{1}{3}$~seconds; +and as the invisible transit corresponds +to a tide as well as the visible one, the interval +between the successive high waters is the time +between the successive transits, of which only +each alternate one is visible. + +The tide next in importance is ``the principal +solar semidiurnal tide.'' This tide bears the +\index{Solar!principal tide}% +same relationship to the real sun that the principal +lunar semidiurnal tide bears to the real moon. +The crests follow one another at intervals of +exactly $12$~hours, which is the time from noon to +midnight and of midnight to noon. The height +of this partial wave is, at most places, a little less +than half of that of the principal lunar tide. + +The interval between successive lunar tides is +$25\frac{1}{4}$~minutes longer than that between successive +solar tides, and as there are two tides a day, the +lunar tide falls behind the solar tide by $50\frac{1}{2}$~minutes +a day. If we imagine the two tides to start +together with simultaneous high waters, then in +about $7$~days the lunar tide will have fallen about +$6$~hours behind the solar tide, because $7$~times +$50\frac{1}{2}$~minutes is $5$~hours $54$~minutes. The period +from high water to low water of the principal +solar semidiurnal tide is $6$~hours, being half the +time between successive high waters. Accordingly, +when the lunar tide has fallen $6$~hours +\PageSep{203} +\index{Spring and neap tides!represented by principal lunar and solar tides}% +behind the solar tide, the low water of the solar +tide falls in with the high water of the lunar +tide. It may facilitate the comprehension of +this matter to take a numerical example; suppose +then that the lunar tide rises $4$~feet above +and falls by the same amount below the mean +level of the sea, and that the solar tide rises and +falls $2$~feet above and below the same level; +then if the two partial waves be started with their +high waters simultaneous, the joint wave will at +first rise and fall by $6$~feet. But after $7$~days it +is low solar tide when it is high lunar tide, and +so the solar tide is subtracted from the lunar +tide, and the compound wave has a height of +$4$~feet less $2$~feet, that is to say, of $2$~feet. +After nearly another $7$~days, or more exactly +after $14\frac{1}{2}$~days from the start, the lunar tide has +lost another $6$~hours, so that it has fallen back +$12$~hours in all, and the two high waters agree +together again, and the joint wave has again a +rise and fall of $6$~feet. When the two high +waters conspire it is called spring tide, and when +the low water of the solar tide conspires with the +high water of the lunar tide, it is called neap +tide. It thus appears that the principal lunar +and principal solar semidiurnal tides together +represent the most prominent feature of the tidal +oscillation. + +The next in importance of the semidiurnal +waves is called the ``lunar elliptic tide,'' and here +\PageSep{204} +\index{Neap and spring tides!represented by principal lunar and solar tides}% +the crests follow one another at intervals of $12$~hours +$39$~minutes $30$~seconds. Now the interval +between the successive principal lunar tides was +\index{Lunar!elliptic tide}% +$12$~hours $25$~minutes $14$~seconds; hence, this +new tide falls behind the principal lunar tide by +$14\frac{1}{2}$~minutes in each half day. If this tide starts +so that its high water agrees with that of the +principal lunar tide, then after $13\frac{3}{4}$~days from +the start, its hollow falls in with the crest of the +former, and in $27\frac{1}{2}$~days from the start the two +crests agree again. + +The moon moves round the earth in an ellipse, +and if to-day it is nearest to the earth, in $13\frac{3}{4}$~days +it will be furthest, and in $27\frac{1}{2}$~days it will +be nearest again. The moon must clearly exercise +a stronger tidal force and create higher +tides when she is near than when she is far; +hence every $27\frac{1}{2}$~days the tides must be larger, +and halfway between they must be smaller. +But the tide under consideration conspires with +the principal lunar tide every $27\frac{1}{2}$~days, and, +accordingly, the joint wave is larger every $27\frac{1}{2}$~days +and smaller in between. Thus this lunar +elliptic tide represents the principal effect of the +\index{Elliptic tide, lunar}% +elliptic motion of the moon round the earth. +There are other semidiurnal waves besides the +three which I have mentioned, but it would +hardly be in place to consider them further +here. +\index{Semidiurnal tide!in harmonic method|)}% + +Now turning to the waves of the second kind, +\PageSep{205} +\index{Diurnal inequality!in harmonic method}% +which are diurnal in character, we find three, all +of great importance. In one of them the high +waters succeed one another at intervals of $25$~hours +$49$~minutes $9\frac{1}{2}$~seconds, and of the second +and third, one has a period of about $4$~minutes +less than $24$~hours and the other of about $4$~minutes +greater than the $24$~hours. It would +hardly be possible to show by general reasoning +how these three waves arise from the attraction +of three ideal satellites, and how these satellites +together are a substitute for the actions of the +true moon and sun. It must, however, be obvious +that the oscillation resulting from three coëxistent +waves will be very complicated. + +All the semidiurnal tides result from waves of +essentially similar character, although some follow +one another a little more rapidly than others, +and some are higher and some are lower. An +accurate cotidal map, illustrating the progress of +any one of these semidiurnal waves over the +ocean, would certainly tell all that we care to +know about the progress of all the other waves +of the group. + +Again, all the diurnal tides arise from waves +of the same character, but they are quite diverse +in origin from the semidiurnal waves, and have +only one high water a day instead of two. A +complete knowledge of the behavior of semidiurnal +waves would afford but little insight into +the behavior of the diurnal waves. At some +\PageSep{206} +time in the future the endeavor ought to be +made to draw a diurnal cotidal chart distinct +from the semidiurnal one, but our knowledge is +not yet sufficiently advanced to make the construction +of such a chart feasible. + +\TB + +All the waves of which I have spoken thus +far are generated by the attractions of the sun +and moon and are therefore called astronomical +tides, but the sea level is also affected by other +oscillations arising from other causes. + +Most of the places, at which a knowledge of +the tides is practically important, are situated in +estuaries and in rivers. Now rain is more prevalent +\index{Rivers!annual meteorological tide in}% +at one season than at another, and mountain +snow melts in summer; hence rivers and +estuaries are subject to seasonal variability of +level. In many estuaries this kind of inequality +may amount to one or two feet, and such a considerable +change cannot be disregarded in tidal +prediction. It is represented by inequalities with +periods of a year and of half a year, which are +called the annual and semiannual meteorological +\index{Annual and semi-annual tides}% +\index{Meteorological!tides}% +tides. + +Then again, at many places, especially in the +Tropics, there is a regular alternation of day and +night breezes, the effect of which is to heap up +% [** TN: "inland", "off-shore" on line breaks in the original; sole instances] +the water in-shore as long as the wind blows in-land, +\index{Wind!a cause of meteorological tides}% +and to lower it when the wind blows off-shore. +Hence there results a diurnal inequality +\PageSep{207} +\index{Estuary, annual meteorological tide in}% +of sea-level, which is taken into account in tidal +prediction by means of a ``solar diurnal meteorological +\index{Meteorological!tides}% +tide.'' Although these inequalities depend +entirely on meteorological influences and +have no astronomical counterpart, yet it is necessary +to take them into account in tidal prediction. + +\TB + +But besides their direct astronomical action, +the sun and moon exercise an influence on the +sea in a way of which I have not yet spoken. +We have seen how waves gradually change their +shape as they progress in a shallow river, so that +the crests become sharper and the hollows flatter, +while the advancing slope becomes steeper and +the receding one less steep. An extreme exaggeration +of this sort of change of shape was +found in the bore. Now it is an absolute rule, +in the harmonic analysis of the tide, that the +partial waves shall be of the simplest character, +and shall have a certain standard law of slope +on each side of their crests. If then any wave +ceases to present this standard simple form, it is +necessary to conceive of it as compound, and to +build it up out of several simple waves. By the +composition of a simple wave with other simple +waves of a half, a third, a quarter of the wave +length, a resultant wave can be built up which +shall assume any desired form. For a given +compound wave, there is no alternative of choice, +\PageSep{208} +for it can only be built up in one way. The +analogy with musical notes is here complete, for +a musical note of any quality is built up from +a fundamental, together with its octave and +twelfth, which are called overtones. So also the +distorted tide wave in a river is regarded as consisting +of simple fundamental tide, with over-tides +of half and third length. The periods of +these over-tides are also one half and one third +of that of the fundamental wave. + +Out in the open ocean, the principal lunar +semidiurnal tide is a simple wave, but when it +runs into shallow water at the coast line, and +still more so in an estuary, it changes its shape. +\index{Estuary, annual meteorological tide in}% +The low water lasts longer than the high water, +and the time which elapses from low water to +high water is usually shorter than that from +high water to low water. The wave is in fact +no longer simple, and this is taken into account +by considering it to consist of a fundamental +lunar semidiurnal wave with a period of $12$~hours +$50$~minutes, of the first over-tide or octave +with a period of $6$~hours $25$~minutes, of the second +over-tide or twelfth with a period of $4$~hours +$17$~minutes, and of the third over-tide or +double octave with a period of $3$~hours $13$~minutes. +In estuaries, the first over-tide of the +lunar semidiurnal tide is often of great importance, +and even the second is considerable; the +third is usually very small, and the fourth and +\PageSep{209} +higher over-tides are imperceptible. In the same +way over-tides must be introduced, to represent +the change of form of the principal solar semidiurnal +tide. But it is not usually found necessary +to consider them in the cases of the less +important partial tides. The octave, the twelfth, +and the upper octave may be legitimately described +as tides, because they are due to the +attractions of the moon and of the sun, although +they arise indirectly through the distorting influence +of the shallowness of the water. + +\TB + +I have said above that about twenty different +simple waves afford a good representation of the +tides at any port. Out of these twenty waves, +some represent the seasonal change of level in +the water due to unequal rainfall and evaporation +at different times of the year, and others +represent the change of shape of the wave due +to shallowing of the water. Deducting these +quasi-tides, we are left with about twelve to +represent the true astronomical tide. It is not +possible to give an exact estimate of the number +of partial tides necessary to insure a good representation +of the aggregate tide wave, because +the characteristics of the motion are so different +at various places that partial waves, important +at one place, are insignificant at others. For +example, at an oceanic island the tides may be +more accurately represented as the sum of a +\PageSep{210} +dozen simple waves than by two dozen in a tidal +river. + +The method of analyzing a tide into its constituent +parts, of which I have now given an +account, is not the only method by which the +tides may be treated, but as it is the most recent +and the best way, I shall not consider the older +methods in detail. The nature of the procedure +adopted formerly will, however, be indicated in +\Ref{Chapter}{XIII}. + +\begin{Authorities}[Authority] +G.~H. Darwin, \Title{Harmonic Analysis of Tidal Observations}: +\index{Analysis, harmonic, of tide|)}% +\index{Darwin, G. H.!harmonic analysis}% +\index{Harmonic analysis!account of|)}% +``Report to British Association.'' Southport,~1883. + +An outline of the method is also contained in \Title{Tides}, ``Encyclopædia +Britannica.'' +\end{Authorities} +\PageSep{211} + + +\Chapter{XII} +{Reduction of Tidal Observations} + +\First{I have} now to explain the process by which +\index{Reduction of tidal observations|(}% +the several partial tides may be disentangled +from one another. + +The tide gauge furnishes a complete tidal record, +so that measurement of the tide curve gives +the height of the water at every instant of time +during the whole period of observation. The +\index{Observation!reduction of tidal|(}% +record may be supposed to begin at noon of a +given day, say of the first of January. The +longitude of the port of observation is of course +known, and the Nautical Almanack gives the +positions of the sun and moon on the day and +at the hour in question, with perfect accuracy. +The real moon has now to be replaced by a +series of ideal satellites, and the rules for the +substitution are absolutely precise. Accordingly, +the position in the heavens of each of +the ideal satellites is known at the moment of +time at which the observations begin. The +same is true of the ideal suns which replace the +actual sun. + +I shall now refer to only a single one of the +ideal moons or suns, for, \textit{mutatis mutandis}, +\PageSep{212} +what is true of one is true of all. It is easy to +calculate at what hour of the clock, measured in +the time of the place of observation, the satellite +in question will be due south. If the ideal +satellite under consideration were the one which +generates the principal lunar semidiurnal tide, it +would be due south very nearly when the real +moon is south, and the ideal sun which generates +the principal solar tide is south exactly at noon. +But there is no such obvious celestial phenomenon +associated with the transit of any other of +the satellites, although it is easy to calculate the +time of the southing of each of them. We have +then to discover how many hours elapse after +the passage of the particular satellite up to the +high water of its tide wave. The height of +the wave crest above, and the depression of the +wave hollow below the mean water mark must +also be determined. When this problem has +been solved for all the ideal satellites and suns, +the tides are said to be reduced, and the reduction +furnishes the materials for a tide table for +the place of observation. + +The difficulty of finding the time of passage +and the height of the wave due to any one of +the satellites arises from the fact that all the +waves really coëxist, and are not separately +manifest. The nature of the disentanglement +may be most easily explained from a special +case, say for example that of the principal lunar +\PageSep{213} +semidiurnal tide, of which the crests follow one +another at intervals of $12$~hours $25$~minutes $14\frac{1}{6}$~seconds. + +Since the waves follow one another at intervals +of approximately, but not exactly, a half-day, it +is convenient to manipulate the time scale so as +\index{Time!lunar}% +to make them exactly semidiurnal. Accordingly +we describe $24$~hours $50$~minutes $28\frac{1}{3}$~seconds as +a lunar day, so that there are exactly two waves +\index{Lunar!time}% +following one another in the lunar day. + +The tide curve furnishes the height of the +\index{Curve, tide!partitioned into lunar time}% +water at every moment of time, but the time +having been registered by the clock of the tide +gauge is partitioned into ordinary days and +hours. It may, however, be partitioned at intervals +of $24$~hours $50$~minutes $28\frac{1}{3}$~seconds, and +into the twenty-fourth parts of that period, and +it will then be divided into lunar days and hours. +On each lunar day the tide for which we are +searching presents itself in the same way, so +that it is always high and low water at the same +hour of the lunar clock, with exactly two high +waters and two low waters in the lunar day. + +Now the other simple tides are governed by +other scales of time, so that in a long succession +of days their high waters and low waters occur +at every hour of the lunar clock. If then we +find the average curve of rise and fall of the +water, when the time is divided into lunar days +and hours, and if we use for the average a long +\PageSep{214} +succession of days, all the other tide waves will +disappear, and we shall be left with only the +lunar semidiurnal tide, purified from all the +others which really coëxist with it. + +The numerical process of averaging thus leads +to the obliteration of all but one of the ideal +satellites, and this is the foundation of the +method of harmonic analysis. The average +lunar tide curve may be looked on as the outcome +of a single day of observation, when all +but the selected satellite have been obliterated. +The height of the average wave, and the interval +after lunar noon up to high water, are the +two tidal constants for the lunar semidiurnal +tide, and they enable us to foretell that tide for +all future time. + +If the tide curve were partitioned into other +days and hours of appropriate lengths, it would +be possible by a similar process of averaging to +single out another of the constituent tide waves, +and to determine its two tidal constants, which +contain the elements of prediction with respect +to it. By continued repetition of operations of +this kind, all the constituents of practical importance +can be determined, and recorded numerically +by means of their pairs of tidal constants. + +The possibility of the disentanglement has +now been demonstrated, but the work of carrying +out these numerical operations would be +\PageSep{215} +\index{Schedule for reducing tidal observations}% +fearfully laborious. The tide curve would have +to be partitioned into about a dozen kinds of +days of various lengths, and the process would +entail measurements at each of the $24$~hours of +each sort of day throughout the whole series. +There are about nine thousand hours in a year, +and it would need about a hundred thousand +measurements of the curve to evaluate twelve +different partial tides; each set of measured +heights would then have to be treated separately +to find the several sorts of averages. Work of +this kind has usually to be done by paid computers, +and the magnitude of the operation +would make it financially prohibitive. It is, +however, fortunately possible to devise abridged +methods, which bring the work within manageable +limits. + +In order to minimize the number of measurements, +the tide curve is only measured at each +of the $24$~exact hours of ordinary time, the +height at noon being numbered $0$~hr., and that +at midnight $12$~hrs., and so on up to $24$~hrs. +After obtaining a set of $24$~measurements for +each day, the original tide curve is of no further +use. The number of measurements involved is +still large, but not prohibitive. It would be +somewhat too technical, in a book of this kind, +to explain in detail how the measured heights of +the water at the exact hours of ordinary time +may be made to give, with fair approximation, +\PageSep{216} +\index{Schedule for reducing tidal observations}% +the heights at the exact hours of other time +scales. It may, however, be well to explain that +this approximate method is based on the fact, +that each exact hour of any one of the special +time scales must of necessity fall within half an +hour of one of the exact hours of ordinary time. +The height of the water at the nearest ordinary +hour is then accepted as giving the height at the +exact hour of the special time. The results, as +computed in this way, are subjected to a certain +small correction, which renders the convention +accurate enough for all practical purposes. + +A schedule, serviceable for all time and for +all places, is prepared which shows the hour of +ordinary time lying nearest to each successive +hour of any one of the special times. The successive +$24$~hourly heights, as measured on the tide +curve, are entered in this schedule, and when +the entry is completed the heights are found to +be arranged in columns, which follow the special +time scale with a sufficiently good approximation +to accuracy. A different form of schedule is +required for each partial tide, and the entry of +the numbers therein is still enormously laborious, +although far less so than the re-partitions and +re-measurements of the tide curve would be. + +The operation of sorting the numbers into +schedules has been carried out in various ways. +In the work of the Indian Survey, the numbers +\index{Indian Survey!method of reducing tidal observations}% +have been re-copied over and over again. In +\PageSep{217} +\index{Abacus for reducing tidal observations|(}% +\index{Darwin, G. H.!tidal abacus|(}% +\index{Indian Survey!method of reducing tidal observations}% +the office of the United States Coast Survey use +\index{United States Coast Survey!method of reducing tidal observations}% +is made of certain card templates pierced with +holes. These templates are laid upon the tabulation +of the measurements of the tide curve, +and the numbers themselves are visible through +the holes. On the surface of the template lines +are drawn from hole to hole, and these lines +indicate the same grouping of the numbers as +that given by the Indian schedules. Dr.~Börgen, +\index{Borgen@Börgen, method of reducing tidal observations}% +of the Imperial German Marine Observatory +\index{German method of reducing tidal observations}% +at Wilhelmshaven, has used sheets of tracing +paper to attain the same end. The Indian procedure +is unnecessarily laborious, and the American +and German plans appear to have some +disadvantage in the fact that the numbers to be +added together lie diagonally across the page. +I am assured by some professional computers +that diagonal addition is easy to perform correctly; +nevertheless this appeared to me to be +so serious a drawback, that I devised another +plan by which the numbers should be brought +into vertical columns, without the necessity of +re-copying them. In my plan each day is treated +as a unit and is shifted appropriately. It might +be thought that the results of the grouping +would be considerably less accurate than in the +former methods, but in fact there is found to be +no appreciable loss of accuracy. + +I have $74$~narrow writing-tablets of xylonite, +divided by lines into $24$~compartments; the +\PageSep{218} +tablets are furnished with spikes on the under +side, so that they can be fixed temporarily in any +position on an ordinary drawing-board. The +compartments on each strip are provided for the +entry of the $24$~tidal measurements appertaining +to each day. Each strip is stamped at its end +with a number specifying the number of the day +to which it is appropriated. + +The arrangement of these little tablets, so that +the numbers written on them may fall into columns, +is indicated by a sheet of paper marked +with a sort of staircase, which shows where each +tablet is to be set down, with its spikes piercing +the guide sheet. When the strips are in place, +as shown in~\fig{33}, the numbers fall into $48$~columns, +numbered $0$,~$1$,~\dots~$23$, $0$,~$1$,~\dots~$23$ +twice over. The guide sheet shown in the figure~\figref{33} +is the one appropriate for the lunar semidiurnal +tide for the fourth set of $74$~days of +a year of observation. The upper half of the +tablets are in position, but the lower ones are +left unmounted, so as the better to show the +staircase of marks. + +Then I say that the average of all the $74$~numbers +standing under the two~$0$'s combined +will give the average height of water at $0$~hr.\ +of lunar time, and the average of the numbers +under~$1$, that at $1$~hr.\ of lunar time, and so forth. +Thus, after the strips are pegged out, the computer +has only to add the numbers in columns in +%% Plate 2 +\TallFig{33}{Tidal Abacus}{jpg} +%% Facing page +%[Blank Page] +\PageSep{219} +order to find the averages. There are other +sheets of paper marked for such other rearrangements +of the strips that each new setting gives +one of the required results; thus a single writing +of the numbers serves for the whole computation. +It is usual to treat a whole year of +observations at one time, but the board being +adapted for taking only $74$~successive days, five +series of writings are required for $370$~days, +which is just over a year. The number~$74$ was +chosen for simultaneous treatment, because $74$~days +is almost exactly five semilunations, and +accordingly there will always be five spring tides +on record at once. + +In order to guard the computer against the +use of the wrong paper with any set of strips, +the guide sheets for the first set of $74$~days are +red; for the second they are yellow; for the +third green; for the fourth blue; for the fifth +violet, the colors being those of the rainbow. + +The preparation of these papers entailed a +great deal of calculation in the first instance, but +the tidal computer has merely to peg out the +tablets in their right places, verifying that the +numbers stamped on the ends of the strips agree +with the numbers on the paper. The addition +of the long columns of figures is certainly laborious, +but it is a necessary incident of every +method of reducing tidal observations. + +The result of all the methods is that for each +\PageSep{220} +partial tide we have a set of $24$~numbers, which +represent the oscillations of the sea due to the +isolated action of one of the ideal satellites, during +the period embraced between two successive +passages of that satellite to the south of the +place of observation. The examination of each +partial tide wave gives its height, and the interval +of time which elapses after its satellite has +passed the meridian until it is high water for +that particular tide. The height and interval +are the tidal constants for that particular tide, at +the port of observation. +\index{Observation!reduction of tidal|)}% + +The results of this ``reduction of the observations'' +are contained in some fifteen or twenty +pairs of tidal constants, and these numbers contain +a complete record of the behavior of the sea +at the place in question. + +\begin{Authorities} +G.~H. Darwin, \Title{Harmonic Analysis, \&c.}: ``Report to British Association,'' +1883. + +G.~H. Darwin, \Title{On an apparatus for facilitating the reduction of +\index{Darwin, G. H.!tidal abacus|)}% +tidal observations}: ``Proceedings of the Royal Society,'' vol.~lii.\ +1892. +\end{Authorities} +\index{Abacus for reducing tidal observations|)}% +\index{Reduction of tidal observations|)}% +\PageSep{221} + + +\Chapter{XIII} +{Tide Tables} + +\First{A tide table} professes to tell, at a given +\index{Tables, tide|(}% +place and on a given day, the time of high and +low water, together with the height of the rise +and the depth of the fall of the water, with +reference to some standard mark on the shore. +A perfect tide table would tell the height of the +water at every moment of the day, but such a +table would be so bulky that it is usual to predict +only the high and low waters. + +There are two kinds of tide table, namely, +those which give the heights and times of high +and low water for each successive day of each +year, and those which predict the high and low +water only by reference to some conspicuous +celestial phenomenon. Both sorts of tide table +refer only to the particular harbor for which they +are prepared. + +The first kind contains definite forecasts for +each day, and may be called a special tide table. +Such a table is expensive to calculate, and must +be published a full year beforehand. Special +tide tables are published by all civilized countries +for their most important harbors. I believe that +\PageSep{222} +\index{Indian Survey!tide tables}% +\index{United States Coast Survey!tide tables of}% +the most extensive publications are those of the +Indian Government for the Indian Ocean, and +of the United States Government for the coasts +of North America. The Indian tables contain +\index{America, North, tide tables for}% +predictions for about thirty-seven ports. + +The second kind of table, where the tide is +given by reference to a celestial phenomenon, +may be described as a general one. It is here +necessary to refer to the Nautical Almanack for +the time of occurrence of the celestial phenomenon, +and a little simple calculation must then be +made to obtain the prediction. The phenomenon +to which the tide is usually referred is the passage +of the moon across the meridian of the place of +observation, and the table states that high and +low water will occur so many hours after the +moon's passage, and that the water will stand at +such and such a height. + +The moon, at her change, is close to the sun +and crosses the meridian at noon; she would +then be visible but for the sun's brightness, and +if she did not turn her dark side towards us. +She again crosses the meridian invisibly at midnight. +At full moon she is on the meridian, +visibly at midnight, and invisibly at noon. At +waxing half moon she is visibly on the meridian +at six at night, and at waning half moon at six +in the morning. The hour of the clock at which +the moon passes the meridian is therefore in effect +a statement of her phase. Accordingly the +\PageSep{223} +relative position of the sun and moon is directly +involved in a statement of the tide as corresponding +to a definite hour of the moon's passage. A +table founded on the time of the moon's passage +must therefore involve the principal lunar and +solar semidiurnal tides. + +At places where successive tides differ but little +from one another, a simple table of this kind +suffices for rough predictions. The curves marked +Portsmouth in \fig{34} show graphically the interval +after the moon's passage, and the height +of high water at that port, for all the hours of +the moon's passage. We have seen in \Ref{Chapter}{X}.\ +that the tide in the North Atlantic is principally +due to a wave propagated from the Southern +Ocean. Since this wave takes a considerable +time to travel from the Cape of Good Hope to +England, the tide here depends, in great measure, +on that generated in the south at a considerable +time earlier. It has therefore been found better +to refer the high water to a transit of the moon +which occurred before the immediately preceding +one. The reader will observe that it is noted on +the upper figure that $28$~hours have been subtracted +from the Portsmouth intervals; that is +to say, the intervals on the vertical scale marked +$6$,~$7$,~$8$ hours are, for Portsmouth, to be interpreted +as meaning $34$,~$35$,~$36$ hours. These are +the hours which elapse after any transit of the +moon up to high water. The horizontal scale is +\PageSep{224} +\index{Moon and earth!tidal prediction by reference to transit|(}% +one of the times of moon's transit and of phases +of the moon; the vertical scale in the lower figure +is one of feet, and it shows the height to +which the water will rise measured from a certain +mark ashore. These Portsmouth curves do not +extend beyond 12~o'clock of moon's transit; this +is because there is hardly any diurnal inequality, +\index{Diurnal inequality!complicates prediction}% +and it is not necessary to differentiate the hours +as either diurnal or nocturnal, the statement being +equally true of either day or night. Thus +if the Portsmouth curves had been extended onward +from $12$~hours to $24$~hours of the clock time +of the moon's passage, the second halves of the +curves would have been merely the duplicates of +the first halves.\footnote + {Before the introduction of the harmonic analysis of the tides + described in preceding chapters, tidal observations were ``reduced'' + by the construction of such figures as these, directly from + the tidal observations. Every high water was tabulated as appertaining + to a particular phase of the moon, both as to its height + and as to the interval between the moon's transit and the occurrence + of high water. The average of a long series of observations + may be represented in the form of curves by such figures + as these.} + +But the time of the moon's passage leaves her +angular distance from the equator and her linear +distance from the earth indeterminate; and since +the variability of both of these has its influence +on the tide, corrections are needed which add +something to or subtract something from the +tabular values of the interval and height, as dependent +solely on the time of the moon's passage. +\PageSep{225} +\index{Diurnal inequality!complicates prediction}% +\index{Interval from moon's transit to high water!at Portsmouth and at Aden}% +The sun also moves in a plane which is oblique +to the equator, and so similar allowances must be +made for the distance of the sun from the equator, +and for the variability in his distance from the +earth. In order to attain accuracy with a tide +table of this sort, eight or ten corrections are +needed, and the use of the table becomes complicated. + +It is, however, possible by increasing the number +of such figures or tables to introduce into +them many of the corrections referred to; and +the use of a general tide table then becomes comparatively +simple. The sun occupies a definite +position with reference to the equator, and stands +at a definite distance from the earth on each day +of the year; also the moon's path amongst the +stars does not differ very much from the sun's. +Accordingly a tide table which states the interval +after the moon's passage to high or low water +and the height of the water on a given day of +\index{Height of tide!at Portsmouth and at Aden}% +the year will directly involve the principal inequalities +in the tides. As the sun moves slowly +amongst the stars, a table applicable to a given +day of the year is nearly correct for a short time +before and after that date. If, then, a tide table, +stating the time and height of the water by reference +to the moon's passage, be computed for +say every ten days of the year, it will be very +nearly correct for five days before and for five +days after the date for which it is calculated. +\PageSep{226} + +The curves marked Aden, March and June, in +\index{Prediction of tide!example at Aden|(}% +\fig{34}, show the intervals and heights of tide, +on the 15th of those months at that port, for all +the hours of the moon's passage. The curves are +to be read in the same way as those for Portsmouth, +but it is here necessary to distinguish the +hours of the day from those of the night, and +accordingly the clock times of moon's transit are +numbered from $0$~hr.\ at noon up to $24$~hrs.\ at +the next noon. The curves for March differ so +much from those for June, that the corrections +would be very large, if the tides were treated at +Aden by a single pair of average curves as at +Portsmouth. + +The law of the tides, as here shown graphically, +may also be stated numerically, and the +use of such a table is easy. The process will be +best explained by an example, which happens to +be retrospective instead of prophetic. It will involve +that part of the complete table (or series of +curves) for Aden which applies to the 15th of +March of any year. Let it be required then to +find the time and height of high water on March~17, +1889. The Nautical Almanack for that year +shows that on that day the moon passed the meridian +of Aden at eleven minutes past noon of +Aden time, or in astronomical language at $0$~hr.\ +$11$~mins. Now the table, or the figure of intervals, +shows that if the moon had passed at $0$~hr., +or exactly at noon, the interval would have been +\PageSep{227} +$8$~hrs.\ $9$~mins., and that if she had passed at $0$~hr.\ +$20$~mins., or 12.20~\PM\ of the day, the interval +would have been $7$~hrs.\ $59$~mins. But on +March~17th the moon actually crossed at $0$~hr.\ +\Figure{34}{Curves of Intervals and Heights at Portsmouth +and at Aden}{png} +$11$~mins., very nearly halfway between noon and +$20$~mins.\ past noon. Hence the interval was +halfway between $8$~hrs.\ $9$~mins.\ and $7$~hrs.\ $59$~mins., +so that it was $8$~hrs.\ $4$~mins. Accordingly +it was high water $8$~hrs.\ $4$~mins.\ after the moon +\PageSep{228} +crossed the meridian. But the moon crossed at +$0$~hr.\ $11$~mins., therefore the high water occurred +at 8.15~\PM. + +Again the table of heights, or the figure, shows +that on March~15th, if the moon crossed at $0$~hr.\ +$0$~min.\ the high water would be $6.86$~ft.\ above +a certain mark ashore, and if she crossed at $0$~hr.\ +$20$~mins.\ the height would be $6.92$~ft. But on +March~17th the moon crossed halfway between +$0$~hr.\ $0$~min.\ and $0$~hr.\ $20$~mins., and therefore +the height was halfway between $6.86$~ft.\ and +$6.92$~ft., that is to say, it was $6.89$~ft., or $6$~ft.\ +$11$~in. We therefore conclude that on March~17, +1889, the sea at high water rose to $6$~ft.\ +$11$~in., at 8.15~\PM. I have no information as +to the actual height and time of high water on +that day, but from the known accuracy of other +predictions at Aden we may be sure that this +agrees pretty nearly with actuality. The predictions +derived from this table are markedly improved +when a correction, either additive or subtractive, +is applied, to allow for the elliptic motion +of the moon round the earth. On this particular +occasion the moon stood rather nearer the earth +than the average, and therefore the correction to +the height is additive; the correction to the time +also happens to be additive, although it could +not be foreseen by general reasoning that this +would be the case. The corrections for March~17, +1889, are found to add about $2$~mins.\ to the +\PageSep{229} +time, bringing it to 8.17~\PM, and nearly two +inches to the height, bringing it to $7$~ft.\ $1$~in. + +This sort of elaborate general tide table has +been, as yet, but little used. It is expensive to +calculate, in the first instance, and it would occupy +two or three pages of a book. The expense +is, however, incurred once for all, and the table +is available for all time, provided that the tidal +observations on which it is based have been good. +A sea captain arriving off his port of destination +would not take five minutes to calculate the two +or three tides he might require to know, and the +information would often be of the greatest value +to him. + +As things stand at present, a ship sailing to +most Chinese, Pacific, or Australian ports is only +furnished with a statement, often subject to considerable +error, that the high water will occur at +so many hours after the moon's passage and will +rise so many feet. The average rise at springs and +neaps is generally stated, but the law of the variability +according to the phases of the moon is wanting. +But this is not the most serious defect in the +information, for it is frequently noted that the +tide is ``affected by diurnal inequality,'' and this +note is really a warning to the navigator that he +cannot foretell the time of high water within two +or three hours of time, or the height within several +feet. + +Tables of the kind I have described would +\PageSep{230} +\index{Prediction of tide!method of computing|(}% +banish this extreme vagueness, but they are more +likely to be of service at ports of second-rate importance +than at the great centres of trade, because +at the latter it is worth while to compute +full special tide tables for each year. +\index{Tables, tide!method of calculating|(}% + +It is unnecessary to comment on the use of +tables containing predictions for definite days, +since it merely entails reference to a book, as to +a railway time table. Such special tables are undoubtedly +the most convenient, but the number +of ports which can ever be deemed worthy of the +great expense incidental to their preparation +must always be very limited. + +\TB + +We must now consider the manner in which +tide tables are calculated. It is supposed that +careful observations have been made, and that +the tidal constants, which state the laws governing +the several partial tides, have been accurately +determined by harmonic analysis. The analysis +of tidal observations consists in the dissection of +the aggregate tide wave into its constituent partial +waves, and prediction involves the recomposition +or synthesis of those waves. In the synthetic +\index{Synthesis of partial tides for prediction|(}% +process care must be taken that the partial +waves shall be recompounded in their proper +relative positions, which are determined by the +places of the moon and sun at the moment of +time chosen for the commencement of prediction. + +The synthesis of partial waves may be best +\index{Moon and earth!tidal prediction by reference to transit|)}% +\index{Prediction of tide!example at Aden|)}% +\PageSep{231} +arranged in two stages. It has been shown in +\Ref{Chapter}{XI}.\ that the partial waves fall naturally +into three groups, of which the third is practically +insignificant. The first and second are the +semidiurnal and diurnal groups. The first process +is to unite each group into a single wave. + +We will first consider the semidiurnal group. +Let us now, for the moment, banish the tides +from our minds, and imagine that there are two +trains of waves traveling simultaneously along a +straight canal. If either train existed by itself +every wave would be exactly like all its brethren, +both in height, length, and period. Now suppose +that the lengths and periods of the waves +of the two coëxistent trains do not differ much +from one another, although their heights may +differ widely. Then the resultant must be a single +train of waves of lengths and periods intermediate +between those of the constituent waves, +but in one part of the canal the waves will be +high, where the two sets of crests fall in the +same place, whilst in another they will be low, +where the hollow of the smaller constituent wave +falls in with the crest of the larger. If only one +part of the canal were visible to us, a train of +waves would pass before us, whose heights would +gradually vary, whilst their periods would change +but little. + +In the same way two of the semidiurnal tide +waves, when united by the addition of their separate +\PageSep{232} +displacements from the mean level, form a +single wave of variable height, with a period still +semidiurnal, although slightly variable. But +there is nothing in this process which limits the +synthesis to two waves, and we may add a third +and a fourth, finally obtaining a single semidiurnal +wave, the height of which varies according +to a very complex law. + +A similar synthesis is then applied to the second +group of waves, so that we have a single +variable wave of approximately diurnal period. +The final step consists in the union of the single +semidiurnal wave with the single diurnal one into +a resultant wave. When the diurnal wave is +large, the resultant is found to undergo very +great variability both in period and height. The +principal variations in the relative positions of +the partial tide waves are determined by the +phases of the moon and by the time of year, and +there is, corresponding to each arrangement of +the partial waves, a definite form for the single +resultant wave. The task of forming a general +tide table therefore consists in the determination +of all the possible periods and heights of the resultant +wave and the tabulation of the heights +and intervals after the moon's passage of its high +and low waters. + +I supposed formerly that the captain would +himself calculate the tide he required from the +general tide table, but such calculation may be +\PageSep{233} +\index{Machine, tide-predicting}% +\index{Predicting machine for tides|(}% +done beforehand for every day of a specified +year, and the result will be a special tide table. +There are about $\DPchg{1400}{1,400}$ high and low waters in +a year, so that the task is very laborious, and +has to be repeated each year. + +\TB + +It is, however, possible to compute a special +tide table by a different and far less laborious +method. In this plan an ingenious mechanical +device replaces the labor of the computer. The +first suggestion for instrumental prediction of +tides was made, I think, by Sir William Thomson, +now Lord Kelvin, in~1872. Mr.~Edward +\index{Kelvin, Lord!tide predicting machine}% +Roberts bore an important part in the practical +\index{Roberts, E., the tide-predicting machine}% +realization of such a machine, and a tide predicter +was constructed by Messrs.\ Légé for the +\index{Lege@Légé, constructor of tide-predicting machine}% +Indian Government under his supervision. This +is, as yet, the only complete instrument in existence. +But others are said to be now in course +of construction for the Government of the +United States and for that of France. The +Indian machine cost so much and works so well, +that it is a pity it should not be used to the full +extent of its capacity. The Indian Government +has, of course, the first claim on it, but the use +of it is allowed to others on the payment of a +small fee. I believe that, pending the construction +of their own machine, the French authorities +are obtaining the curves for certain tidal +predictions from the instrument in London. +\index{Prediction of tide!method of computing|)}% +\index{Synthesis of partial tides for prediction|)}% +\PageSep{234} + +Although the principle involved in the tide +predicter is simple, yet the practical realization +of it is so complex that a picture of the whole +machine would convey no idea of how it works. +I shall therefore only illustrate it diagrammatically, +in~\fig{35}, without any pretension to scale +or proportion. The reader must at first imagine +that there are only two pulleys, namely, $A$~and~$B$, +so that the cord passes from the fixed end~$F$ +under~$A$ and over~$B$, and so onward to the pencil. +The pulley~$B$ is fixed, and the pulley~$A$ can slide +vertically up and down in a slot, which is not +shown in the diagram. If $A$~moves vertically +through any distance, the pencil must clearly +move through double that distance, so that +when $A$~is highest the pencil is lowest, and vice +versa. + +The pencil touches a uniformly revolving +drum, covered with paper; thus if the pulley~$A$ +executes a simple vertical oscillation, the pencil +draws a simple wave on the drum. Now the +pulley is mounted on an inverted T-shaped +frame, and a pin, fixed in a crank~$C$, engages in +the slit in the horizontal arm of the T-piece. +When the crank~$C$ revolves, the pulley~$A$ executes +a simple vertical oscillation with a range depending +on the throw of the crank.\footnote + {I now notice that the throw of the crank~$C$ is too small to + have allowed the pencil to draw so large a wave as that shown + on the drum. But as this is a mere diagram, I have not thought + it worth while to redraw the whole.} +The position +\PageSep{235} +of the pin is susceptible of adjustment on the +crank, so that its throw and the range of oscillation +of the pulley can be set to any required +\Figure[0.7]{35}{Diagram of Tide-predicting Instrument}{png} +length---of course within definite limits determined +by the size of the apparatus. + +The drum is connected to the crank~$C$ by a +train of wheels, so that as the crank rotates the +drum also turns at some definitely proportional +rate. If, for example, the crank revolves twice +for one turn of the drum, the pencil will draw a +simple wave, with exactly two crests in one circumference +of the drum. If one revolution of +the drum represents a day, the graphical time +scale is $24$~hours to the circumference of the +\PageSep{236} +drum. If the throw of the crank be one inch, +the pulley will oscillate with a total range of two +inches, and the pencil with a total range of four +inches. Then taking two inches lengthwise on +the drum to represent a foot of water, the curve +drawn by the pencil might be taken to represent +the principal solar semidiurnal tide, rising one +foot above and falling one foot below the mean +sea level. + +I will now show how the machine is to be +adjusted so as to give predictions. We will +suppose that it is known that, at noon of the +first day for which prediction is required, the +solar tide will stand at $1$~ft.\ $9$~in.\ above mean +sea level and that the water will be rising. Then, +the semi-range of this tide being one foot, the +pin is adjusted in the crank at one inch from +the centre, so as to make the pencil rock through +a total range of $4$~inches, representing $2$~feet. +The drum is now turned so as to bring the noon-line +of its circumference under the pencil, and +the crank is turned so that the pencil shall be +$3\frac{1}{2}$~inches (representing $1$~ft.\ $9$~in.\ of water) +below the middle of the drum, and so that when +the machine starts, the pencil will begin to descend. +The curve being drawn upside-down, +the pencil is set below the middle line because +the water is to be above mean level, and it must +begin to descend because the water is to ascend. +The train of wheels connecting the crank and +\PageSep{237} +drum is then thrown into gear, and the machine +is started; it will then draw the solar tide curve, +on the scale of $2$~inches to the foot, for all +time. + +If the train of wheels connecting the crank to +the drum were to make the drum revolve once +whilst the crank revolves $1.93227$~times, the +curve would represent a lunar semidiurnal tide. +The reason of this is that $1.93227$~is the ratio +of $24$~hours to $12$~h.\ $25$~m.\ $14$~s., that is to say, +of a day to a lunar half day. We suppose the +circumference of the drum still to represent an +ordinary day of $24$~hours, and therefore the +curve drawn by the pencil will have lunar semidiurnal +periodicity. In order that these curves +may give predictions of the future march of that +tide, the throw of the crank must be set to give the +correct range and its angular position must give +the proper height at the moment of time chosen +for beginning. When these adjustments are +made the curve will represent that tide for all +time. + +We have now shown that, by means of appropriate +trains of wheels, the machine can be made +to predict either the solar or the lunar tide; but +we have to explain the arrangement for combining +them. If, still supposing there to be +only the two pulleys $A$,~$B$, the end~$F$ of the cord +were moved up or down, its motion would be +transmitted to the pencil, whether the crank~$C$ +\PageSep{238} +and pulley~$A$ were in motion, or at rest; but if +they were in motion, the pencil would add the +motion of the end of the cord to that of the +pulley. If then there be added another fixed +pulley~$B'$, and another movable pulley~$A'$, driven +by a crank and T-piece (not shown in the diagram), +the pencil will add together the movements +of the two pulleys $A$~and~$A'$. There must +now be two trains of wheels, one connecting $A$ +with the drum and the other for~$A'$. If a single +revolution of the drum causes the crank~$C$ to +turn twice, whilst it makes the crank of~$A'$ rotate +$1.93227$~times, the curve drawn will represent +the union of the principal solar and lunar semidiurnal +tides. The trains of wheels requisite for +transmitting motion from the drum to the two +cranks in the proper proportions are complicated, +but it is obviously only a matter of calculation +to determine the numbers of the teeth in the +several wheels in the trains. It is true that rigorous +accuracy is not attainable, but the mechanism +is made so nearly exact that the error in the +sum of the two tides would be barely sensible +even after $\DPchg{3000}{3,000}$~revolutions of the drum. It is +of course necessary to set the two cranks with +their proper throws and at their proper angles +so as to draw a curve which shall, from the noon +of a given day, correspond to the tide at a given +place. + +It must now be clear that we may add as +\PageSep{239} +many more movable pulleys as we like. When +the motion of each pulley is governed by an +appropriate train of wheels, the movement of +the pencil, in as far as it is determined by that +pulley, corresponds to the tide due to one of our +ideal satellites. The resultant curve drawn on +the drum is then the synthesis of all the partial +tides, and corresponds with the motion of the +sea. + +The instrument of the Indian Government +unites twenty-four partial tides. In order to +trace a tide curve, the throws of all the cranks +are set so as to correspond with the known +heights of the partial tides, and each crank is set +at the proper angle to correspond with the moment +of time chosen for the beginning of the tide +table. It is not very difficult to set the cranks +and pins correctly, although close attention is of +course necessary. The apparatus is then driven +by the fall of a weight, and the paper is fed +automatically on to the drum and coiled off on +to a second drum, with the tide curve drawn on +it. It is only necessary to see that the paper +runs on and off smoothly, and to write the date +from time to time on the paper as it passes, in +order to save future trouble in the identification +of the days. It takes about four hours to run +off the tides for a year. + +The Indian Government sends home annually +the latest revision of the tidal constants for +\PageSep{240} +thirty-seven ports in the Indian Ocean. Mr.~Roberts +sets the machine for each port, so as to +correspond with noon of a future 1st~of January, +and then lets it run off a complete tide +curve for a whole year. The curve is subsequently +measured for the time and height of +each high and low water, and the printed tables +are sold at the moderate price of four rupees. +The publication is made sufficiently long beforehand +to render the tables available for future +voyages. These tide tables are certainly amongst +the most admirable in the world. + +\TB + +It is characteristic of England that the machine +is not, as I believe, used for any of the +home ports, and only for a few of the colonies. +The neglect of the English authorities is not, +however, so unreasonable as it might appear to +be. The tides at English ports are remarkably +simple, because the diurnal inequality is practically +absent. The applicability of the older +methods of prediction, by means of such curves +as that for Portsmouth in~\fig{34}, is accordingly +easy, and the various corrections are well determined. +The arithmetical processes are therefore +not very complicated, and ordinary computers +are capable of preparing the tables with but +little skilled supervision. Still it is to be regretted +that this beautiful instrument should not +be more used for the home and colonial ports. +\PageSep{241} + +The excellent tide tables of the Government +of the United States have hitherto been prepared +by the aid of a machine of quite a different +character, the invention of the late Professor +Ferrel. This apparatus virtually carries out +\index{Ferrel, tide-predicting instrument}% +\index{Predicting machine for tides!Ferrel's}% +that process of compounding all the waves together +into a single one, which I have described +as being done by a computer for the formation +of a general tide table. It only registers, however, +the time and height of the maxima and +minima---the high and low waters. I do not +think it necessary to describe its principle in +detail, because it will shortly be superseded by a +machine like, but not identical with, that of the +Indian Government. + +\begin{Authorities} +G.~H. Darwin, \Title{On Tidal Prediction}. ``Philosophical Transactions +of the Royal Society,'' A.~1891, pp.~159--229. + +In the example of the use of a general tide table at Aden, +given in this chapter, the datum from which the height is measured +is $0.37$~ft.\ higher than that used in the Indian Tide Tables; +\index{Tables, tide|)}% +\index{Tables, tide!method of calculating|)}% +accordingly $4\frac{1}{2}$~inches must be added to the height, in order to +bring it into accordance with the official table. + +Sir William Thomson, \Title{Tidal Instruments}, and the subsequent +discussion. ``Institute of Civil Engineers,'' vol.~lxv. + +William Ferrel, \Title{Description of a Maxima and Minima Tide-predicting +\index{Machine, tide-predicting}% +Machine}. ``United States Coast Survey,'' 1883. +\end{Authorities} +\index{Predicting machine for tides|)}% +\PageSep{242} + + +\Chapter[Accuracy of Tidal Prediction]{XIV} +{The Degree of Accuracy of Tidal Prediction} + +\First{The} success of tidal predictions varies much +\index{Atmospheric pressure!influence on tidal prediction}% +\index{Prediction of tide!errors in|(}% +according to the place of observation. They are +not unfrequently considerably in error in our +latitude, and throughout those regions called by +sailors ``the roaring forties.'' The utmost that +can be expected of a tide table is that it shall +be correct in calm weather and with a steady +barometer. But such conditions are practically +non-existent, and in the North Atlantic the great +variability in the meteorological elements renders +tidal prediction somewhat uncertain. + +The sea generally stands higher when the +barometer is low, and lower when the barometer +is high, an inch of mercury corresponding to +rather more than a foot of water. The pressure +of the air on the sea in fact depresses it in those +places where the barometer is high, and allows it +to rise where the opposite condition prevails. + +Then again a landward wind usually raises the +\index{Wind!perturbation of, in tidal prediction}% +sea level, and in estuaries the rise is sometimes +very great. There is a known instance when the +Thames at London was raised by five feet in a +strong gale. Even on the open coast the effect +\PageSep{243} +\index{Atmospheric pressure!influence on tidal prediction}% +of wind is sometimes great. A disastrous example +\index{Wind!perturbation of, in tidal prediction}% +of this was afforded on the east coast of England +in the autumn of~1897, when the conjunction +of a gale with springtide caused the sea to +do an enormous amount of damage, by breaking +embankments and flooding low-lying land. + +But sometimes the wind has no apparent effect, +and we must then suppose that it had been blowing +previously elsewhere in such a way as to depress +the water at the point at which we watch it. +The gale might then only restore the water to its +normal level, and the two effects might mask one +another. The length of time during which the +wind has lasted is clearly an important factor, +because the currents generated by the wind must +be more effective in raising or depressing the sea +level the longer they have lasted. + +It does not then seem possible to formulate +any certain system of allowance for barometric +pressure and wind. There are, at each harbor, +certain rules of probability, the application of +which will generally lead to improvement in the +prediction; but occasionally such empirical corrections +will be found to augment the error. + +But notwithstanding these perturbations, good +tide tables are usually of surprising accuracy +even in northern latitudes; this may be seen +from the following table showing the results of +comparisons between prediction and actuality at +Portsmouth. The importance of the errors in +\index{Errors in tidal prediction|(}% +\PageSep{244} +height depends of course on the range of tide; +it is therefore well to note that the average ranges +of tide at springs and neaps are $13$~ft.\ $9$~in.\ and +$7$~ft.\ $9$~in.\ respectively. +\begin{table}[hbtp!] +\caption{Table of Errors in the Prediction of High Water at +Portsmouth in the Months of January, May, And +\index{Portsmouth, table of errors in tidal predictions}% +September, 1897.} +\[ +\begin{array}{|c|c||c|c|} +\hline +\multicolumn{2}{|c||}{\ColHead{Time}} & +\multicolumn{2}{c|}{\ColHead{Height}} \\ +\hline +\ColHead[1.2in]{Magnitude of error} & +\ColHead[0.5in]{Number of cases} & +\ColHead[1.2in]{Magnitude of error} & +\ColHead[0.5in]{Number of cases} \\ +\hline +&& \TEntry{Inches} & \\ +\Z0\mm\text{ to }\Z5\mm& 69 & \Z0\text{ to }\Z6 & 89 \\ +\Z6\mm\text{ to }10\mm & 50 & \Z7\text{ to }12 & 58 \\ + 11\mm\text{ to }15\mm & 25 & 13\text{ to }18 & 24 \\ + 16\mm\text{ to }20\mm & 10 & 19\text{ to }24 & \Z6 \\ + 21\mm\text{ to }25\mm & 11 & \Dash & \Dash \\ + 26\mm\text{ to }30\mm & \Z7 & \Dash & \Dash \\ + 31\mm\text{ to }35\mm & \Z4 & \Dash & \Dash \\ + 52\mm & \Z1 & \Dash & \Dash \\ +\hline +\Strut\Dash & \llap{$1$}77&\Dash & \llap{$1$}77 \\ +\hline +\end{array} +\] +\end{table} +\begin{table}[hbtp!] +\caption{Errors in Height for the Year 1892, +Excepting Part of July} +\[ +\begin{array}{|c|c|} +\hline +\ColHead{Magnitude of error} & \ColHead{Number of cases} \\ +\hline +\TEntry{Inches} & \\ +\Z0\text{ to }\Z6& 381 \\ +\Z7\text{ to }12 & 228 \\ + 13\text{ to }18 & \Z52 \\ + 19\text{ to }24 & \Z\Z8 \\ + 31 & \Z\Z1 \\ +\hline +\Strut\Dash & 670 \\ +\hline +\end{array} +\] +\end{table} +\PageSep{245} + +\begin{Remark} +\NB---The comparison seems to indicate that these predictions +might be much improved, because the predicted height is +nearly always above the observed height, and because the diurnal +inequality has not been taken into account sufficiently, if at +all. +\end{Remark} + +In tropical regions the weather is very uniform, +and in many places the ``meteorological +tides'' produced by the regularly periodic variations +of wind and barometric pressure are taken +into account in tidal predictions. + +The apparent irregularity of the tides at Aden +is so great, that an officer of the Royal Engineers +has told me that, when he was stationed there +many years ago, it was commonly believed that +the strange inequalities of water level were due +to the wind at distant places. We now know +that the tide at Aden is in fact marvelously +regular, although the rule according to which it +proceeds is very complex. In almost every month +in the year there are a few successive days when +there is only one high water and one low water +in the $24$~hours; and the water often remains +almost stagnant for three or four hours at a +time. This apparent irregularity is due to the +diurnal inequality, which is very great at Aden, +whereas on the coasts of Europe it is insignificant. + +I happen to have a comparison with actuality +of a few predictions of high water at Aden, +where the maximum range of the tide is about +$8$~ft.\ $6$~in. They embrace the periods from March~10 +\index{Errors in tidal prediction|)}% +\PageSep{246} +\index{Aden, errors of tidal prediction at}% +\index{Tables, tide!amount of error in}% +to April~9, and again from November~12 to +December~12, 1884. In these two periods there +were $118$~high waters, but through an accident +to the tide gauge one high water was not registered. +On one occasion, when the regular semidiurnal +sequence of the tide would lead us to +expect high water, there occurred one of those +periods of stagnation to which I have referred. +Thus we are left with $116$~cases of comparison +between the predicted and actual high waters. + +The results are exhibited in the following +table:--- +\[ +\begin{array}{|c|c||c|c|} +\hline +\multicolumn{2}{|c||}{\ColHead{Time}} & +\multicolumn{2}{c|}{\ColHead{Height}} \\ +\hline +\ColHead{Magnitude of errors} & +\ColHead[0.5in]{Number of high waters} & +\ColHead{Magnitude of errors} & +\ColHead[0.5in]{Number of high waters} \\ +\hline +&& \TEntry{Inches} & \\ +\Z0\mm\PadTxt{ and }{to}\Z5\mm & 35 & 0 & 15 \\ +\Z5\mm\PadTxt{ and }{to}10\mm & 32 & 1 & 48 \\ + 10\mm\PadTxt{ and }{to}15\mm & 19 & 2 & 28 \\ + 15\mm\PadTxt{ and }{to}20\mm & 19 & 3 & 14 \\ + 20\mm\PadTxt{ and }{to}25\mm & \Z5 & 4 & 11 \\ + 26\mm\text{ and }28\mm & \Z2 & +\multicolumn{1}{c|}{\TEntry{No high water}} & \Z1 \\ + 33\mm\text{ and }36\mm & \Z2 & \Dash & \Dash \\ + 56\mm\text{ and }57\mm & \Z2 & \Dash & \Dash \\ +\multicolumn{1}{|c|}{\TEntry{No high water}} & \Z1 & \Dash & \Dash \\ +\hline +\Strut & 117 && 117 \\ +\hline +\end{array} +\] + +It would be natural to think that when the +prediction is erroneous by as much as $57$~minutes, +it is a very bad one; but I shall show that +\PageSep{247} +\index{Tables, tide!amount of error in}% +this would be to do injustice to the table. On +several of the occasions comprised in this list +the water was very nearly stagnant. Now if the +water only rises about a foot from low to high +water in the course of four or five hours, it is +almost impossible to say with accuracy when it +was highest, and two observers might differ in +their estimate by half an hour or even by an +hour. + +In the table of comparison there are $11$~cases +in which the error of time is equal to or greater +than twenty minutes, and I have examined these +cases in order to see whether the water was then +nearly stagnant. A measure of the degree of +stagnation is afforded by the amount of the rise +from low water to high water, or of the fall from +high water to low water. The following table +gives a classification of the errors of time according +to the rise or fall:--- +\begin{table}[hp!] +\caption{Analysis of Errors in Time.} +\centering +\begin{tabular}{|c|c|} +\hline +\ColHead[1.5in]{Ranges from low water to high water} & +\ColHead{Errors of time} \\ +\hline +\Strut +Nil & \Dash \\ +$6$ in.\ to $8$ in. & $22$, $26$, $28$, $56$, $57$~minutes \\ +$13$ in. & $36$~minutes \\ +$17$ in. & $22$ \Ditto{minutes} \\ +$19$ in. & $33$ \Ditto{minutes} \\ +$2$ ft.\ $10$ in. & $22$ \Ditto{minutes} \\ +$3$ ft.\ $\Z9$ in. & $23$ \Ditto{minutes} \\ +$3$ ft.\ $11$ in. & $20$ \Ditto{minutes} \\ +\hline +\end{tabular} +\end{table} +\PageSep{248} + +There are then only three cases when the rise +of water was considerable, and in the greatest of +them it was only $3$~ft.\ $11$~in. + +If we deduct all the tides in which the range +between low and high water was equal to or less +than $19$~inches, we are left with $108$~predictions, +and in these cases the greatest error in time is +$23$~mins. In $86$~cases the error is equal to or less +than a quarter of an hour. This leaves $22$~cases +where the error was greater than $15$~mins.\ made +up as follows: $18$~cases with error greater than +$15$~mins.\ and less than $20$~mins.\ and $3$~cases with +errors of $20$~mins., $22$~mins., $23$~mins. Thus in +$106$ out of~$108$ predictions the error of time was +equal to or less than $20$~minutes. + +Two independent measurements of a tide +curve, for the determination of the time of high +water, lead to results which frequently differ by +five minutes, and sometimes by ten minutes. It +may therefore be claimed that these predictions +have a very high order of accuracy as regards +time. + +Turning now to the heights, out of $116$~predictions +the error in the predicted height was +equal to or less than $2$~inches in $91$~cases, it +amounted to $3$~inches in $14$~cases, and in the +remaining $11$~cases it was $4$~inches. It thus appears +that, as regards the height of the tide also, +the predictions are of great accuracy. This +short series of comparisons affords a not unduly +\PageSep{249} +favorable example of the remarkable success attainable, +where tidal observation and prediction +have been thoroughly carried out at a place +subject to only slight meteorological disturbance. + +If our theory of tides were incorrect, so that +we imagined that there was a partial tide wave +of a certain period, whereas in fact such a wave +has no true counterpart in physical causation, +the reduction of a year of tidal observation would +undoubtedly assign some definite small height, +and some definite retardation of the high water +after the passage of the corresponding, but +erroneous, satellite. But when a second series +of observations is reduced, the two tidal constants +would show no relationship to their previous +evaluations. If then reductions carried +out year after year assign, as they do, fairly +consistent values to the tidal constants, we may +feel confident that true physical causation is involved, +even when the heights of some of the +constituent tide waves do not exceed an inch +or two. + +Prediction must inevitably fail, unless we have +lighted on the true causes of the phenomena; +success is therefore a guarantee of the truth of +the theory. When we consider that the incessant +variability of the tidal forces, the complex +outlines of our coasts, the depth of the sea and +the earth's rotation are all involved, we should +\PageSep{250} +regard good tidal prediction as one of the +greatest triumphs of the theory of universal +gravitation. + +\begin{Authorities} +The Portsmouth comparisons were given to the author by the +Hydrographer of the Admiralty, Admiral Sir W.~J. Wharton. + +G.~H. Darwin, \Title{On Tidal Prediction}. ``Philosophical Transactions +of the Royal Society,'' A.~1891. +\end{Authorities} +\index{Prediction of tide!errors in|)}% +\PageSep{251} + + +\Chapter[Rigidity of the Earth]{XV} +{Chandler's Nutation---The Rigidity of the +Earth} + +\First{In} the present chapter I have to explain the +\index{Nutation!Chandler's|(}% +\index{Variation of latitude|(}% +origin of a tide of an entirely different character +from any of those considered hitherto. It may +fairly be described as a true tide, although it is +not due to the attraction of either the sun or +the moon. + +We have all spun a top, and have seen it, as +boys say, go to sleep. At first it nods a little, +but gradually it settles down to perfect steadiness. +Now the earth may be likened to a top, +and it also may either have a nutational or nodding +motion, or it may spin steadily; it is only +by observation that we can decide whether it is +nodding or sound asleep. + +The equator must now be defined as a plane +through the earth's centre at right angles to the +axis of rotation, and not as a plane fixed with +reference to the solid earth. The latitude of +\index{Latitude!periodic variations of|(}% +any place is the angle\footnote + {This angle is technically called the geocentric latitude; the + distinction between true and geocentric latitude is immaterial in + the present discussion.} +between the equator and +\PageSep{252} +a line drawn from the centre of the earth to the +place of observation. Now when the earth +nutates, the axis of rotation shifts, and its +extremity describes a small circle round the spot +which is usually described as the pole. The +equator, being perpendicular to the axis of rotation, +of course shifts also, and therefore the +latitude of a place fixed on the solid earth varies. +During the whole course of the nutation, the +earth's axis of rotation is always directed towards +the same point in the heavens, and therefore the +angle between the celestial pole and the vertical +or plumb-line at the place of observation must +oscillate about some mean value; the period of +the oscillation is that of the earth's nutation. +This movement is called a ``free'' nutation, +because it is independent of the action of external +forces. + +There are, besides, other nutations resulting +from the attractions of the moon and sun on the +protuberant matter at the equator, and from the +same cause there is a slow shift in space of the +earth's axis, called the precession. These movements +are said to be ``forced,'' because they are +due to external forces. The measurements of +the forced nutations and of the precession afford +the means of determining the period of the free +nutation, if it should exist. It has thus been +concluded that if there is any variation in the +latitude, it should be periodic in $305$~days; but +\PageSep{253} +only observation can decide whether there is +such a variation of latitude or not. + +Until recently astronomers were so convinced +of the sufficiency of this reasoning, that, when +they made systematic examination of the latitudes +of many observatories, they always searched +for an inequality with a period of $305$~days. +Some thought that they had detected it, but +when the observations extended over long periods, +it always seemed to vanish, as though what +they had observed were due to the inevitable +errors of observation. At length it occurred +to Mr.~Chandler to examine the observations +\index{Chandler, free nutation of earth, and variation of latitude|(}% +of latitude without any prepossession as to the +period of the inequality. By the treatment of +enormous masses of observation, he came to the +conclusion that there is really such an inequality, +but that the period is $427$~days instead of $305$~days. +He also found other inequalities in the +motion of the axis of rotation, of somewhat +obscure origin, and of which I have no occasion +to say more.\footnote + {They are perhaps due to the unequal melting of polar + ice and unequal rainfall in successive years. These irregular + variations in the latitude are such that some astronomers are + still skeptical as to the reality of Chandler's nutation, and think + that it will perhaps be found to lose its regularly rhythmical + character in the future.} + +The question then arises as to how the theory +can be so amended as to justify the extension of +the period of nutation. It was, I believe, Newcomb, +\PageSep{254} +\index{Newcomb, S., theoretical explanation of Chandler's nutation}% +of the United States Naval Observatory, +who first suggested that the explanation is to be +sought in the fact that the axis of rotation is an +axis of centrifugal repulsion, and that when it +shifts, the distribution of centrifugal force is +changed with reference to the solid earth, so +that the earth is put into a state of stress, to +which it must yield like any other elastic body. +The strain or yielding consequent on this stress +must be such as to produce a slight variability +in the position of the equatorial protuberance +with reference to places fixed on the earth. +Now the period of $305$~days was computed on +the hypothesis that the position of the equatorial +protuberance is absolutely invariable, but +periodic variations of the earth's figure would +operate so as to lengthen the period of the free +nutation, to an extent dependent on the average +elasticity of the whole earth. +\index{Elasticity of earth}% + +Mr.~Chandler's investigation demanded the +utmost patience and skill in marshaling large +masses of the most refined astronomical observations. +His conclusions are not only of the +greatest importance to astronomy, but they also +give an indication of the amount by which the +solid earth is capable of yielding to external +forces. It would seem that the average stiffness +of the whole earth must be such that it yields a +little less than if it were made of steel.\footnote + {Mr.~S. S. Hough, p.~338 of the paper referred to in the list +\index{Hough, S. S.!rigidity of earth}% + of authorities at the end of the chapter.} +But +\PageSep{255} +\index{Elasticity of earth}% +the amount by which the surface yields remains +unknown, because we are unable to say what +proportion of the aggregate change is superficial +and what is deep-seated. It is, however, certain +that the movements are excessively small, because +the circle described by the extremity of +the earth's axis of rotation, about the point on +the earth which we call the pole, has a radius of +only fifteen feet. + +It is easily intelligible that as the axis of +rotation shifts in the earth, the oceans will tend +to swash about, and that a sort of tide will be +generated. If the displacement of the axis were +considerable, whole continents would be drowned +by a gigantic wave, but the movement is so +small that the swaying of the ocean is very +feeble. Two investigators have endeavored to +detect an oceanic tide with a period of $427$~days; +they are Dr.~Bakhuyzen of Leyden and +\index{Bakhuyzen on tide due to variation of latitude}% +Mr.~Christie of the United States Coast Survey. +\index{Christie, A. S., tide due to variation of latitude}% +The former considered observations of sea-level +on the coasts of Holland, the latter those on the +coasts of the United States; and they both conclude +that the sea-level undergoes a minute +variability with a period of about $430$~days. A +similar investigation is now being prosecuted by +the Tidal Survey of India, and as the Indian +tidal observations are amongst the best in the +world, we may hope for the detection of this +minute tide in the Indian Ocean also. +\PageSep{256} +\index{Earth and moon!rigidity of|(}% +\index{Rigidity of earth|(}% + +The inequality in water level is so slight and +extends over so long a period that its measurement +cannot yet be accepted as certain. The +mean level of the sea is subject to slight irregular +variations, which are probably due to unequal +rainfall and unequal melting of polar ice in +successive years. But whatever be the origin of +these irregularities they exceed in magnitude the +one to be measured. The arithmetical processes, +employed to eliminate the ordinary tides and the +irregular variability, will always leave behind +some residual quantities, and therefore the examination +of a tidal record will always apparently +yield an inequality of any arbitrary period whatever. +It is only when several independent determinations +yield fairly consistent values of the +magnitude of the rise and fall and of the moment +of high water, that we can feel confidence +in the result. Now although the reductions of +Bakhuyzen and Christie are fairly consistent +\index{Bakhuyzen on tide due to variation of latitude}% +\index{Christie, A. S., tide due to variation of latitude}% +with one another, and with the time and height +suggested by Chandler's nutation, yet it is by no +means impossible that accident may have led to +this agreement. The whole calculation must +therefore be repeated for several places and at +several times, before confidence can be attained +in the detection of this latitudinal tide. + +\TB + +The prolongation of the period of Chandler's +nutation from $305$ to $427$~days seems to indicate +\index{Latitude!periodic variations of|)}% +\index{Nutation!Chandler's|)}% +\index{Variation of latitude|)}% +\PageSep{257} +that our planet yields to external forces, and we +naturally desire to learn more on so interesting +a subject. Up to fifty years ago it was generally +held that the earth was a globe of molten +matter covered by a thin crust. The ejection of +lava from volcanoes and the great increase of +temperature in mines seemed to present evidence +in favor of this belief. But the geologists and +physicists of that time seemed not to have perceived +that the inference might be false, if great +pressure is capable of imparting rigidity to matter +at a very high temperature, because the interior +of the earth might then be solid although +very hot. Now it has been proved experimentally +that rock expands in melting, and a physical +corollary from this is that when rock is under +great pressure a higher temperature is needed to +melt it than when the pressure is removed. The +pressure inside the earth much exceeds any that +can be produced in the laboratory, and it is uncertain +up to what degree of increase of pressure +the law of the rise of the temperature of +melting would hold good; but there can be no +doubt that, in so far as experiments in the laboratory +can be deemed applicable to the conditions +prevailing in the interior of the earth, they +tend to show that the matter there is not improbably +solid. + +But Lord Kelvin reinforces this argument +\index{Kelvin, Lord!rigidity of earth|(}% +from another point of view. Rock in the solid +\index{Chandler, free nutation of earth, and variation of latitude|)}% +\PageSep{258} +condition is undoubtedly heavier than when it is +molten. Now the solidified crust on the surface +of a molten planet must have been fractured +many times during the history of the planet, +and the fragments would sink through the liquid, +and thus build up a solid nucleus. It will +be observed that this argument does not repose +on the rise in the melting temperature of rock +through pressure, although it is undoubtedly +reinforced thereby. + +Hopkins was, I think, the first to adduce arguments +\index{Hopkins on rigidity of earth}% +of weight in favor of the earth's solidity. +He examined the laws of the precession and +nutation of a rigid shell inclosing liquid, and +found that the motion of such a system would +differ to a marked degree from that of the earth. +From this he concluded that the interior of the +earth was not liquid. + +Lord Kelvin has pointed out that although +Hopkins's investigation is by no means complete, +yet as he was the first to show that the +motion of the earth as a whole affords indications +of the condition of the interior, an important +share in the discovery of the solidity of the +earth should be assigned to him. Lord Kelvin +then resumed Hopkins's work, and showed that +if the liquid interior of the planet were inclosed +in an unyielding crust, a very slight departure +from perfect sphericity in the shell would render +the motion of the system almost identical with +\PageSep{259} +\index{Height of tide!reduced by elastic yielding of earth}% +that of a globe solid from centre to surface, +although this would not be the case with the +more rapid nutations. A yet more important +deficiency in Hopkins's investigation is that he +\index{Hopkins on rigidity of earth}% +did not consider that, unless the crust were more +rigid than the stiffest steel, it would yield to the +surging of the imprisoned liquid as freely as +india-rubber; and, besides, that if the crust +yielded freely, the precession and nutations of +the whole mass would hardly be distinguishable +from those of a solid globe. Hopkins's argument, +as thus amended by Lord Kelvin, leads +to one of two alternatives: either the globe is +solid throughout, or else the crust yields with +nearly the same freedom to external forces as +though it were liquid. + +We have now to show that the latter hypothesis +is negatived by other considerations. The +oceanic tides, as we perceive them, consist in a +motion of the water relatively to the land. Now +if the solid earth were to yield to the tidal forces +with the same freedom as the superjacent sea, +the cause for the relative movement of the sea +would disappear. And if the solid yielded to +some extent, the apparent oceanic tide would be +proportionately diminished. The very existence +of tides in the sea, therefore, proves at least that +the land does not yield with perfect freedom. + +Lord Kelvin has shown that the oceanic tides, +on a globe of the same rigidity as that of glass, +\PageSep{260} +would only have an apparent range of two fifths +of those on a perfectly rigid globe; whilst, if +the rigidity was equal to that of steel, the fraction +of diminution would be two thirds. I have +myself extended his argument to the hypothesis +that the earth may be composed of a viscous +material, which yields slowly under the application +of continuous forces, and also to the hypothesis +of a material which shares the properties +of viscosity and rigidity, and have been led to +\index{Rigidity of earth|)}% +analogous conclusions. + +The difficulty of the problem of oceanic tides +is so great that we cannot say how high the tides +would be if the earth were absolutely rigid, but +Lord Kelvin is of opinion that they certainly +\index{Kelvin, Lord!rigidity of earth|)}% +would not be twice as great as they are, and he +concludes that the earth possesses a greater average +stiffness than that of glass, although perhaps +not greater than that of steel. It is proper to +add that the validity of this argument depends +principally on the observed height of an inequality +of sea level with a period of a fortnight. This +is one of the partial tides of the third kind, which +I described in \Ref{Chapter}{XI}.\ as practically unimportant, +and did not discuss in detail. The value +of this inequality in the present argument is due +to the fact that it is possible to form a much +closer estimate of its magnitude on a rigid earth +\index{Earth and moon!rigidity of|)}% +than in the case of the semidiurnal and diurnal +tides. +\PageSep{261} +\index{Darwin, G. H.!rigidity of earth}% +\index{Earthquakes!shock perceptible at great distance}% + +It may ultimately be possible to derive further +indications concerning the physical condition of +the inside of the earth from the science of seismology. +The tremor of an earthquake has frequently +been observed instrumentally at an enormous +distance from its origin; as, for example, +when the shock of a Japanese earthquake is +perceived in England. + +The vibrations which are transmitted through +the earth are of two kinds. The first sort of wave +is one in which the matter through which it passes +is alternately compressed and dilated; it may be +described as a wave of compression. In the +second sort the shape of each minute portion of +the solid is distorted, but the volume remains +unchanged, and it may be called a wave of distortion. +These two vibrations travel at different +speeds, and the compressional wave outpaces +the distortional one. Now the first sign of a +distant earthquake is that the instrumental record +shows a succession of minute tremors. +These are supposed to be due to waves of compression, +and they are succeeded by a much +more strongly marked disturbance, which, however, +lasts only a short time. This second phase +in the instrumental record is supposed to be due +to the wave of distortion. + +If the natures of these two disturbances are +correctly ascribed to their respective sources, it +is certain that the matter through which the vibration +\PageSep{262} +\index{Darwin, G. H.!rigidity of earth}% +has passed was solid. For, although a +compressional wave might be transmitted without +much loss of intensity, from a solid to a +liquid and back again to a solid, as would have +to be the case if the interior of the earth is molten, +yet this cannot be true of the distortional +wave. It has been supposed that vibrations due +to earthquakes pass in a straight line through +the earth; if then this could be proved, we +should know with certainty that the earth is +solid, at least far down towards its centre. + +Although there are still some---principally +amongst the geologists---who believe in the existence +of liquid matter immediately under the +solid crust of the earth,\footnote + {See the Rev.~Osmond Fisher's \Title{Physics of the Earth's Crust}.} +\index{Fisher, Osmond, on molten interior of earth}% +yet the arguments which +I have sketched appear to most men of science +conclusive against such belief. + +\begin{Authorities} +Mr.~S.~C. Chandler's investigations are published in the ``Astronomical +Journal,'' vol.~11 and following volumes. A summary +is contained in ``Science,'' May~3, 1895. + +R.~S. Woodward, \Title{Mechanical Interpretation of the Variations of +\index{Woodward on variation of latitude}% +Latitude}, ``Ast.\ Journ.'' vol.~15, May,~1895. + +Simon Newcomb, \Title{On the Dynamics of the Earth's Rotation}, +``Monthly Notices of the R.~Astron.\ Soc.,'' vol.~52 (1892), +p.~336. + +S.~S. Hough, \Title{The Rotation of an Elastic Spheroid}, ``Philosoph.\ +\index{Hough, S. S.!Chandler's nutation}% +Trans.\ of the Royal Society,'' A.~1896, p.~319. He indicates a +slight oversight on the part of Newcomb. + +H.~G. van~de Sande Bakhuyzen, \Title{Ueber die Aenderung der Polhoehe}, +``Astron.\ Nachrichten,'' No.~3261. +\PageSep{263} + +A.~S. Christie, \Title{The Latitude-variation Tide}, ``Phil.\ Soc.\ of +Washington, Bulletin,'' vol.~12 (1895), p.~103. + +Lord Kelvin, in Thomson and Tait's ``Natural Philosophy,'' +on the Rigidity of the Earth; and ``Popular Lectures,'' vol.~3. + +G.~H. Darwin, \Title{Bodily Tides of Viscous and Semi-elastic Spheroids,~\&c.}, +``Philosoph.\ Trans.\ of the Royal Society,'' Part.~I. +1879. +\end{Authorities} +\PageSep{264} + + +%[** TN: Footnote mark handled by \Chapter logic] +\Chapter{XVI} +{Tidal Friction} +\footnotetext{A considerable portion of this and of the succeeding chapter + appeared as an article in \Title{The Atlantic Monthly} for April,~1898.} + +\First{The} fact that the earth, the moon, and the +\index{Friction of tides|(}% +planets are all nearly spherical proves that in +early times they were molten and plastic, and +assumed their present round shape under the +influence of gravitation. When the material of +which any planet is formed was semi-liquid +through heat, its satellites, or at any rate the +sun, must have produced tidal oscillations in the +molten rock, just as the sun and moon now produce +the tides in our oceans. + +Molten rock and molten iron are rather sticky +or viscous substances, and any movement which +agitates them must be subject to much friction. +Even water, which is a very good lubricant, is +not entirely free from friction, and so our present +oceanic tides must be influenced by fluid +friction, although to a far less extent than the +molten solid just referred to. Now, all moving +systems which are subject to friction gradually +come to rest. A train will run a long way when +the steam is turned off, but it stops at last, and +\PageSep{265} +a fly-wheel will continue to spin for only a limited +time. This general law renders it certain that +the friction of the tide, whether it consists in the +swaying of molten lava or of an ocean, must be +retarding the rotation of the planet, or at any +rate retarding the motion of the system in some +way. + +It is the friction upon its bearings which brings +a fly-wheel to rest; but as the earth has no bearings, +it is not easy to see how the friction of the +tidal wave, whether corporeal or oceanic, can +tend to stop its rate of rotation. The result +must clearly be brought about, in some way, by +the interaction between the moon and the earth. +Action and reaction must be equal and opposite, +and if we are correct in supposing that the friction +of the tides is retarding the earth's rotation, +there must be a reaction upon the moon which +must tend to hurry her onwards. To give a +homely illustration of the effects of reaction, I +may recall to mind how a man riding a high +bicycle, on applying the brake too suddenly, was +thrown over the handles. The desired action +was to stop the front wheel, but this could not +be done without the reaction on the rider, which +sometimes led to unpleasant consequences. + +The general conclusion as to the action and +reaction due to tidal friction is of so vague a +character that it is desirable to consider in detail +how they operate. +\PageSep{266} + +The circle in \fig{36} is supposed to represent +the undisturbed shape of the planet, which rotates +in the direction of the curved arrow. A portion +of the orbit of the satellite is indicated by part +\Figure{36}{Frictionally retarded Tide}{png} +of a circle, and the direction of its motion is +shown by an arrow. I will first suppose that the +water lying on the planet, or the molten rock of +which it is formed, is a perfect lubricant devoid +of friction, and that at the moment represented +in the figure the satellite is at~$M'$. The fluid will +then be distorted by the tidal force until it assumes +the egg-like shape marked by the ellipse, +projecting on both sides beyond the circle. It +will, however, be well to observe that if this figure +represents an ocean, it must be a very deep +one, far deeper than those which actually exist +on the earth; for we have seen that it is only in +deep oceans that the high water stands underneath +and opposite to the moon; whereas in +shallow water it is low water where we should +\PageSep{267} +naturally expect high water. Accepting the hypothesis +that the high tide is opposite to the +moon, and supposing that the liquid is devoid of +friction, the long axis of the egg is always directed +straight towards the satellite~$M'$, and the +liquid maintains a continuous rhythmical movement, +so that as the planet rotates and the satellite +revolves, it always maintains the same shape +and attitude towards the satellite. + +But when, as in reality, the liquid is subject to +friction, it gets belated in its rhythmical rise and +fall, and the protuberance is carried onward by +the rotation of the planet beyond its proper +place. In order to make the same figure serve +for this condition, I set the satellite backward to~$M$; +for this amounts to just the same thing, and +is less confusing than redrawing the protuberance +in its more advanced position. The planet +then constantly maintains this shape and attitude +with regard to the satellite, and the interaction +between the two will be the same as though the +planet were solid, but continually altering its +shape. + +We have now to examine what effects must +follow from the attraction of the satellite on an +egg-shaped planet, when the two constantly +maintain the same attitude relatively to each +other. It will make the matter somewhat easier +of comprehension if we replace the tidal protuberances +by two particles of equal masses, one at~$P$, +\PageSep{268} +\index{Earth and moon!rotation retarded by tidal friction}% +\index{Retardation of earth's rotation}% +and the other at~$P'$. If the masses of these +particles be properly chosen, so as to represent +the amount of matter in the protuberances, the +proposed change will make no material difference +in the action. + +The gravitational attraction of the satellite is +greater on bodies which are near than on those +which are far, and accordingly it attracts the +particle~$P$ more strongly than the particle~$P'$. It +is obvious from the figure that the attraction on~$P$ +must tend to stop the planet's rotation, whilst +\index{Rotation!retarded by tidal friction}% +that on~$P'$ must tend to accelerate it. If a man +pushes equally on the two pedals of a bicycle, +the crank has no tendency to turn, and besides +there are dead points in the revolution where +pushing and pulling have no effect. So also in +the astronomical problem, if the two attractions +were exactly equal, or if the protuberances were +at a dead point, there would be no resultant effect +on the rotation of the planet. But it is +obvious that here the retarding pull is stronger +than the accelerating one, and that the set of the +protuberances is such that we have passed the +dead point. It follows from this that the primary +effect of fluid friction is to throw the tidal +protuberance forward, and the secondary effect +is to retard the planet's rotation. + +It has been already remarked that this figure is +drawn so as to apply only to the case of corporeal +tides or to those of a very deep ocean. If +\PageSep{269} +\index{Moon and earth!retardation of motion by tidal friction}% +the ocean were shallow and frictionless, it would +be low water under and opposite to the satellite. +If then the effect of friction were still to throw +the protuberances forward, the rotation of the +planet would be accelerated instead of retarded. +But in fact the effect of fluid friction in a shallow +ocean is to throw the protuberances backward, +and a similar figure, drawn to illustrate such a +displacement of the tide, would at once make it +clear that here also tidal friction will lead to the +retardation of the planet's rotation. Henceforth +then I shall confine myself to the case illustrated +by~\fig{36}. + +Action and reaction are equal and opposite, +and if the satellite pulls at the protuberances, +they pull in return on the satellite. The figure +shows that the attraction of the protuberance~$P$ +tends in some measure to hurry the satellite onward +in its orbit, whilst that of~$P'$ tends to retard +it. But the attraction of~$P$ is stronger than that +of~$P'$, and therefore the resultant of the two is a +force tending to carry the satellite forward in the +direction of the arrow. + +If a stone be whirled at the end of an elastic +string, a retarding force, such as the friction of +the air, will cause the string to shorten, and an +accelerating force will make it lengthen. In the +same way the satellite, being as it were tied to +the planet by the attraction of gravitation, when +subjected to an onward force, recedes from the +\PageSep{270} +\index{Moon and earth!retardation of motion by tidal friction}% +planet, and moves in a spiral curve at ever increasing +distances. The time occupied by the +satellite in making a circuit round the planet is +prolonged, and this lengthening of the periodic +time is not merely due to the lengthening of the +arc described by it, but also to an actual retardation +of its velocity. It appears paradoxical that +the effect of an accelerating force should be a +retardation, but a consideration of the mode in +which the force operates will remove the paradox. +The effect of the tangential accelerating +force on the satellite is to make it describe an +increasing spiral curve. Now if the reader will +draw an exaggerated figure to illustrate part of +such a spiral orbit, he will perceive that the central +force, acting directly towards the planet, +must operate in some measure to retard the velocity +of the satellite. The central force is very +great compared with the tangential force due to +the tidal friction, and therefore a very small +fraction of the central force may be greater than +the tangential force. Although in a very slowly +increasing spiral the fraction of the central force +productive of retardation is very small, yet it is +found to be greater than the tangential accelerating +force, and thus the resultant effect is a +retardation of the satellite's velocity. + +The converse case where a retarding force results +in increase of velocity will perhaps be more +intelligible, as being more familiar. A meteorite, +\PageSep{271} +rushing through the earth's atmosphere, moves +faster and faster, because it gains more speed +from the attraction of gravity than it loses by the +friction of the air. + +Now let us apply these ideas to the case of the +earth and the moon. A man standing on the +planet, as it rotates, is carried past places where +the fluid is deeper and shallower alternately; at +the deep places he says that it is high tide, and +at the shallow places that it is low tide. In \fig{36} +it is high tide when the observer is carried +past~$P$. Now it was pointed out that when there +is no fluid friction we must put the moon at~$M'$, +but when there is friction she must be at~$M$. +Accordingly, if there is no friction it is high tide +when the moon is over the observer's head, but +when there is friction the moon has passed his +zenith before he reaches high tide. Hence he +would remark that fluid friction retards the time +of high tide. + +A day is the name for the time in which the +earth rotates once, and a month for the time in +which the moon revolves once. Then since tidal +friction retards the earth's rotation and the +moon's revolution, we may state that both the +day and the month are being lengthened, and +that these results follow from the retardation of +the time of high tide. + +It must also be noted that the spiral in which +the moon moves is an increasing one, so that her +\PageSep{272} +\index{Assyrian records of eclipses}% +distance from the earth also increases. These +are absolutely certain and inevitable results of +the mechanical interaction of the two bodies. + +At the present time the rates of increase of +the day and month are excessively small, so that +it has not been found possible to determine them +with any approach to accuracy. It may be well +to notice in passing that if the rate of either increase +of element were determinable, that of the +other would be deducible by calculation. + +The extreme slowness of the changes within +historical times is established by the early records +in Greek and Assyrian history of eclipses of the +\index{Eclipses, ancient, and earth's rotation}% +\index{Greek!records of ancient eclipses}% +sun, which occurred on certain days and in certain +places. Notwithstanding the changes in the +calendar, it is possible to identify the day according +to our modern reckoning, and the identification +of the place presents no difficulty. +Astronomy affords the means of calculating the +exact time and place of the occurrence of an +eclipse even three thousand years ago, on the +supposition that the earth spun at the same rate +then as now, and that the complex laws governing +the moon's motion are unchanged. + +The particular eclipse referred to in history is +known, but any considerable change in the +earth's rotation and in the moon's position would +have shifted the position of visibility on the +earth from the situation to which modern computation +would assign it. Most astronomical +\PageSep{273} +observations would be worthless if the exact time +of the occurrence were uncertain, but in the +case of eclipses the place of observation affords +\index{Eclipses, ancient, and earth's rotation}% +just that element of precision which is otherwise +wanting. As, then, the situations of the ancient +eclipses agree fairly well with modern computations, +we are sure that there has been no great +change within the last three thousand years, +either in the earth's rotation or in the moon's +motion. There is, however, a small outstanding +discrepancy which indicates that there has been +some change. But the exact amount of change +involves elements of uncertainty, because our +knowledge of the laws of the moon's motion is +not yet quite accurate enough for the absolutely +perfect calculation of eclipses which occurred +many centuries ago. In this way, it is known +that within historical times the retardation of the +earth's rotation and the recession of the moon +have been at any rate very slow. + +It does not, however, follow from this that +the changes have always been equally slow; indeed, +it may be shown that the efficiency of tidal +friction increases with great rapidity as we bring +the tide-generating satellite nearer to the planet. + +It has been shown in \Ref{Chapter}{V}.\ that the intensity +of tide-generating force varies as the inverse +cube of the distance between the moon and +the earth, so that if the moon's distance were +reduced successively to $\frac{1}{2}$,~$\frac{1}{3}$,~$\frac{1}{4}$, of its original distance, +\PageSep{274} +the force and the tide generated by it +would be multiplied $8$,~$27$,~$64$ times. But the +efficiency of tidal friction increases far more rapidly +than this, because not only is the tide itself +augmented, but also the attraction of the moon. +In order to see how these two factors will coöperate, +let us begin by supposing that the +height of the tide remains unaffected by the approach +or retrogression of the moon. Then the +same line of argument, which led to the conclusion +that tide-generating force varies inversely as +the cube of the distance, shows that the action +of the moon on protuberances of definite magnitude +must also vary inversely as the cube of the +distance. But the height of the tide is not in +fact a fixed quantity, but varies inversely as the +cube of the distance, so that when account is +taken both of the augmentation of the tide and +of the increased attraction of the moon, it follows +that the tidal retardation of the earth's rotation +must vary as the inverse sixth power of +the distance. Now since the sixth power of~$2$ is~$64$, +the lunar tidal friction, with the moon at +half her present distance, would be $64$~times as +efficient as at present. Similarly, if her distance +were diminished to a third and a quarter of what +it is, the tidal friction would act with $729$ and +$4,096$~times its present strength. Thus, although +the action may be insensibly slow now, it must +have gone on with much greater rapidity when +the moon was nearer to us. +\PageSep{275} + +There are many problems in which it would +be very difficult to follow the changes according +to the times of their occurrence, but where it is +possible to banish time from consideration, and +to trace the changes themselves, in due order, +without reference to time. In the sphere of +common life, we know the succession of stations +which a train must pass between London and +Edinburgh, although we may have no time-table. +This is the case with our astronomical +problem; for although we have no time-table, +yet the sequence of the changes in the system +can be traced accurately. + +Let us then banish time, and look forward to +the ultimate outcome of the tidal interaction of +the moon and earth. The day and the month +\index{Day, change in length of, under tidal friction}% +\index{Month, change in, under tidal friction|(}% +are lengthening at relative rates which are calculable, +although the absolute rates in time are +unknown. It will suffice for a general comprehension +of the problem to know that the present +rate of increase of the day is much more rapid +than that of the month, and that this will hold +good in the future. Thus, the number of rotations +of the earth in the interval comprised in +one revolution of the moon diminishes; or, in +other words, the number of days in the month +diminishes, although the month itself is longer +than at present. For example, when the day +shall be equal in length to two of our actual +days, the month may be as long as thirty-seven +\PageSep{276} +of our days, and then the earth will spin round +only about eighteen times in the month. + +This gradual change in the day and month +\index{Day, change in length of, under tidal friction}% +proceeds continuously until the duration of a +rotation of the earth is prolonged to fifty-five of +our present days. At the same time the month, +or the time of revolution of the moon round the +earth, will also occupy fifty-five of our days. +Since the month here means the period of the +return of the moon to the same place among the +stars, and since the day is to be estimated in +the same way, the moon must then always face +the same part of the earth's surface, and the +two bodies must move as though they were +united by a bar. The outcome of the lunar +tidal friction will therefore be that the moon +and the earth go round as though locked together, +in a period of fifty-five of our present +days, with the day and the month identical in +length. + +Now looking backward in time, we find the +day and the month shortening, but the day +changing more rapidly than the month. The +earth was therefore able to complete more revolutions +in the month, although that month was +itself shorter than it is now. We get back in +fact to a time when there were $29$~rotations of +the earth in a month instead of~$27\frac{1}{3}$, as at present. +This epoch is a sort of crisis in the history +of the moon and the earth, for it may be proved +\PageSep{277} +that there never could have been more than $29$~days +in the month. Earlier than this epoch, the +days were fewer than~$29$, and later fewer also. +Although measured in years, this epoch in the +earth's history must be very remote, yet when we +contemplate the whole series of changes it must +be considered as a comparatively recent event. +In a sense, indeed, we may be said to have passed +recently through the middle stage of our history. + +Now, pursuing the series of changes further +back than the epoch when there was the maximum +number of days in the month, we find the +earth still rotating faster and faster, and the +moon drawing nearer and nearer to the earth, +and revolving in shorter and shorter periods. +But a change has now supervened, so that the +rate at which the month is shortening is more +rapid than the rate of change in the day. Consequently, +the moon now gains, as it were, on +the earth, which cannot get round so frequently +in the month as it did before. In other words, +the number of days in the month declines from +the maximum of~$29$, and is finally reduced to +one. When there is only one day in the month, +\index{Month, change in, under tidal friction|)}% +the earth and the moon go round at the same +rate, so that the moon always looks at the same +side of the earth, and so far as concerns the +motion they might be fastened together by a +rigid bar. + +This is the same conclusion at which we arrived +\PageSep{278} +with respect to the remote future. But +the two cases differ widely; for whereas in the +future the period of the common rotation will +be $55$~of our present days, in the past we find +the two bodies going round each other in between +three and five of our present hours. A +satellite revolving round the earth in so short a +period must almost touch the earth's surface. +The system is therefore traced until the moon +nearly touches the earth, and the two go round +each other like a single solid body in about three +to five hours. + +The series of changes has been traced forward +and backward from the present time, but it will +make the whole process more intelligible, and +the opportunity will be afforded for certain further +considerations, if I sketch the history again +\index{History!of earth and moon|(}% +in the form of a continuous narrative. + +Let us imagine a planet attended by a satellite +which revolves so as nearly to touch its surface, +and continuously to face the same side of the +planet's surface. If now, for some reason, the +satellite's month comes to differ very slightly +from the planet's day, the satellite will no longer +continuously face the same side of the planet, +but will pass over every part of the planet's +equator in turn. This is the condition necessary +for the generation of tidal oscillations in the +planet, and as the molten lava, of which we +suppose it to be formed, is a sticky or viscous +\PageSep{279} +fluid, the tidal oscillations must be subject to +friction. Tidal friction will then begin to do its +work, but the result will be very different according +as the satellite revolves a little faster or +a little slower than the planet. If it revolves a +little faster, so that the month is shorter than +the day, we have a condition not contemplated +in~\fig{36}; it is easy to see, however, that as +the satellite is always leaving the planet behind +it, the apex of the trial protuberance must be +directed to a point behind the satellite in its +orbit. In this case the rotation of the planet +must be \DPtypo{acclerated}{accelerated} by the tidal friction, and the +satellite will be drawn inward towards the planet, +into which it must ultimately fall. In the application +of this theory to the earth and moon, it +is obvious that the very existence of the moon +negatives the hypothesis that the initial month +was even infinitesimally shorter than the day. +We must then suppose that the moon revolved +a little more slowly than the earth rotated. In +this case the tidal friction would retard the +earth's rotation, and force the moon to recede +from the earth, and so perform her orbit more +slowly. Accordingly, the primitive day and the +primitive month lengthen, but the month increases +much more rapidly than the day, so that +the number of days in a month increases. This +proceeds until that number reaches a maximum, +which in the case of our planet is about~$29$. +\PageSep{280} +\index{Instability!nature of dynamical, and initial of moon's motion|(}% +\index{Stability!nature of dynamical}% + +After the epoch of the maximum number of +days in the month, the rate of change in the +length of the day becomes less rapid than that +in the length of the month; and although both +periods increase, the number of days in the +month begins to diminish. The series of +changes then proceeds until the two periods +come again to an identity, when we have the +earth and the moon as they were at the beginning, +revolving in the same period, with the +moon always facing the same side of the earth. +But in her final condition the moon will be a +long way off the earth instead of being quite +close to it. + +Although the initial and final states resemble +each other, yet they differ in one respect which +is of much importance, for in the initial condition +the motion is unstable, whilst finally it is +stable. The meaning of this is, that if the +moon were even infinitesimally disturbed from +the initial mode of motion, she would necessarily +either fall into the planet, or recede therefrom, +and it would be impossible for her to continue +to move in that neighborhood. She is unstable +in the same sense in which an egg when balanced +on its point is unstable; the smallest mote +of dust will upset it, and practically it cannot +stay in that position. But the final condition +resembles the case of the egg lying on its side, +which only rocks a little when we disturb it. +\PageSep{281} +\index{Stability!nature of dynamical}% +So if the moon were slightly disturbed from her +final condition, she would continue to describe +very nearly the same path round the earth, and +would not assume some entirely new form of +orbit. + +It is by methods of rigorous argument that +the moon is traced back to the initial unstable +condition when she revolved close to the earth. +But the argument here breaks down, and calculation +is incompetent to tell us what occurred +before, and how she attained that unstable mode +of motion. If we were to find a pendulum +swinging in a room, where we knew that it had +been undisturbed for a long time, we might, by +observing its velocity and allowing for the resistance +of the air, conclude that at some previous +moment it had just been upside down, but +calculation could never tell us how it had +reached that position. We should of course +feel confident that some one had started it. +Now a similar hiatus must occur in the history +of the moon, but it is not so easy to supply the +missing episode. It is indeed only possible to +speculate as to the preceding history. + +But there is some basis for our speculation; +for I say that if a planet, such as the earth, +made each rotation in three hours, it would very +nearly fly to pieces. The attraction of gravity +would be barely strong enough to hold it together, +just as the cohesive strength of iron is +\PageSep{282} +\index{Forced oscillation!due to solar tide, possibly related to birth of moon|(}% +\index{Moon and earth!origin of}% +insufficient to hold a fly-wheel together if it is +spun too fast. There is, of course, an important +distinction between the case of the ruptured +fly-wheel and the supposed break-up of the +earth; for when a fly-wheel breaks, the pieces +are hurled apart as soon as the force of cohesion +fails, whereas when a planet breaks up through +too rapid rotation, gravity must continue to +hold the pieces together after they have ceased +to form parts of a single body. + +Hence we have grounds for conjecturing that +the moon is composed of fragments of the primitive +planet which we now call the earth, which +detached themselves when the planet spun very +swiftly, and afterwards became consolidated. It +surpasses the power of mathematical calculation +to trace the details of the process of this rupture +and subsequent consolidation, but we can hardly +doubt that the system would pass through a +period of turbulence, before order was reëstablished +in the formation of a satellite. + +I have said above that rapid rotation was probably +the cause of the birth of the moon, but it +may perhaps not have been brought about by +this cause alone. There are certain considerations +which make it difficult to ascertain the +initial common period of revolution of the moon +and the earth with accuracy; it may lie between +three and five hours. Now I think that such +a speed might not quite suffice to cause the +\index{Instability!nature of dynamical, and initial of moon's motion|)}% +\PageSep{283} +\index{Moon and earth!origin of}% +primitive planet to break up. In \Ref{Chapter}{XVIII}.\ +we shall consider in greater detail the conditions +under which a rotating mass of liquid would +rupture, but for the present it may suffice to say +that, where the rotating body is heterogeneous in +density, like the earth, the exact determination +of the limiting speed of rotation is not possible. +Is there, then, any other cause which might coöperate +with rapid rotation in producing rupture? +I think there is such a cause, and, although +we are here dealing with guesswork, I +will hazard the suggestion. + +The primitive planet, before the birth of the +moon, was rotating rapidly with reference to the +sun, and it must therefore have been agitated by +solar tides. In \Ref{Chapter}{IX}.\ it was pointed out +that there is a general dynamical law which enables +us to foresee the magnitude of the oscillations +of a system under the action of external +forces. That law depended on the natural or +free period of the oscillation of the system when +disturbed and left to itself, free from the intervention +of external forces. We saw that the +more nearly the periodic forces were timed to +agree with the free period, the greater was the +amplitude of the oscillations of the system. Now +it is easy to calculate the natural or free period +of the oscillation of a homogeneous liquid globe +of the same density as the earth, namely, five +and a half times as heavy as water; the period +\PageSep{284} +\index{Sun!possible influence of, in assisting birth of moon}% +is found to be $1$~hour $34$~minutes. The heterogeneity +of the earth introduces a complication of +which we cannot take account, but it seems likely +that the period would be from $1\frac{1}{2}$ to $2$~hours. +The period of the solar semidiurnal tide is half a +\index{Solar!possible effect of tide in assisting birth of moon}% +day, and if the day were from $3$ to $4$ of our present +hours the forced period of the tide would +be in close agreement with the free period of +oscillation. + +May we not then conjecture that as the rotation +of the primitive earth was gradually reduced +by solar tidal friction, the period of the solar tide +was brought into closer and closer agreement +with the free period, and that consequently the +solar tide increased more and more in height? +In this case the oscillation might at length become +so violent that, in coöperation with the +rapid rotation, it shook the planet to pieces, and +that huge fragments were detached which ultimately +became our moon. + +There is nothing to tell us whether this theory +affords the true explanation of the birth of the +moon, and I say that it is only a wild speculation, +incapable of verification. + +But the truth or falsity of this speculation +does not militate against the acceptance of the +general theory of tidal friction, which, standing +on the firm basis of mechanical necessity, throws +much light on the history of the earth and the +moon, and correlates the lengths of our present +day and month. +\index{Forced oscillation!due to solar tide, possibly related to birth of moon|)}% +\PageSep{285} +\index{Sun!possible influence of, in assisting birth of moon}% + +I have said above that the sequence of events +has been stated without reference to the scale of +time. It is, however, of the utmost importance +\index{Time!requisite for evolution of moon}% +to gain some idea of the time requisite for all the +changes in the system. If millions of millions +of years were necessary, the theory would have +to be rejected, because it is known from other +lines of argument that there is not an unlimited +bank of time on which to draw. The uncertainty +as to the duration of the solar system is +\index{Solar!possible effect of tide in assisting birth of moon}% +wide, yet we are sure that it has not existed for +an almost infinite past. + +Now, although the actual time scale is indeterminate, +it is possible to find the minimum time +adequate for the transformation of the moon's +orbit from its supposed initial condition to its +present shape. It may be proved, in fact, that +if tidal friction always operated under the conditions +most favorable for producing rapid change, +the sequence of events from the beginning until +to-day would have occupied a period of between +$50$ and $60$~millions of years. The actual period, +of course, must have been much greater. Various +lines of argument as to the age of the solar +system have led to results which differ widely +among themselves, yet I cannot think that the +applicability of the theory is negatived by the +magnitude of the period demanded. It may be +that science will have to reject the theory in its +full extent, but it seems unlikely that the ultimate +\PageSep{286} +\index{Moon and earth!rotation annulled by tidal friction and present libration}% +verdict will be adverse to the preponderating +influence of the tide in the evolution of our +planet. + +\TB + +If this history be true of the earth and moon, +\index{History!of earth and moon|)}% +it should throw light on many peculiarities of the +solar system. In the first place, a corresponding +series of changes must have taken place in the +moon herself. Once on a time the moon must +have been molten, and the great extinct volcanoes +revealed by the telescope are evidences of +her primitive heat. The molten mass must have +been semi-fluid, and the earth must have raised +in it enormous tides of molten lava. Doubtless +the moon once rotated rapidly on her axis, and +the frictional resistance to her tides must have +impeded her rotation. This cause must have +\index{Rotation!of moon annulled by tidal friction}% +added to the moon's recession from the earth, +but as the moon's mass is only an eightieth part +of that of the earth, the effect on the moon's +orbit must have been small. The only point to +which we need now pay attention is that the +rate of her rotation was reduced. She rotated +then more and more slowly until the tide solidified, +and thenceforward and to the present day +she has shown the same face to the earth. Kant +\index{Kant!rotation of moon}% +and Laplace in the last century, and Helmholtz +\index{Helmholtz!on rotation of the moon}% +\index{Laplace!on rotation of moon}% +in recent times, have adduced this as the explanation +of the fact that the moon always shows +us the same face. Our theory, then, receives a +\PageSep{287} +striking confirmation from the moon; for, having +ceased to rotate relatively to us, she has actually +advanced to that condition which may be +foreseen as the fate of the earth. + +The earth tide in the moon is now solidified +so that the moon's equator is not quite circular, +and the longer axis is directed towards the earth. +Laplace has considered the action of the earth +\index{Laplace!on rotation of moon}% +on this solidified tide, and has shown that the +moon must rock a little as she moves round the +earth. In consequence of this rocking motion or +libration of the moon, and also of the fact that +her orbit is elliptic, we are able to see just a little +more than half of the moon's surface. + +\TB + +Thus far I have referred in only one passage +to the influence of solar tides, but these are of +considerable importance, being large enough to +cause the conspicuous phenomena of spring and +neap tides. Now, whilst the moon is retarding +the earth's rotation, the sun is doing so also. +But these solar tides react only on the earth's +motion round the sun, leaving the moon's motion +round the earth unaffected. It might perhaps +be expected that parallel changes in the +earth's orbit would have proceeded step by step, +and that the earth might be traced to an origin +close to the sun. The earth's mass is less than $\frac{1}{300,000}$~part +of the sun's, and the reactive effect on the +earth's orbit round the sun is altogether negligible. +\PageSep{288} +It is improbable, in fact, that the year is, +from this cause at any rate, longer by more than +a few seconds than it was at the very birth of +the solar system. + +Although the solar tides cannot have had any +perceptible influence upon the earth's movement +in its orbit, they will have affected the rotation +of the earth to a considerable extent. Let us +imagine ourselves transported to the indefinite +future, when the moon's orbital period and the +earth's diurnal period shall both be prolonged to +$55$~of our present days. The lunar tide in the +earth will then be unchanging, just as the earth +tide in the moon is now fixed; but the earth will +be rotating with reference to the sun, and, if +there are still oceans on the earth, her rotation +will be subject to retardation in consequence of +the solar tidal friction. The day will then become +longer than the month, whilst the moon +will at first continue to revolve round the earth +in $55$~days. Lunar tides will now be again generated, +but as the motion of the earth will be +very slow relatively to the moon, the oscillations +will also be very slow, and subject to little friction. +But that friction will act in opposition to +the solar tides, and the earth's rotation will to +some slight extent be assisted by the moon. +The moon herself will slowly approach the earth, +moving with a shorter period, and must ultimately +fall back into the earth. We know that +\PageSep{289} +there are neither oceans nor atmosphere on the +moon, but if there were such, the moon would +have been subject to solar tidal friction, and +would now be rotating slower than she revolves. + +%[** TN: Not hyperlinking chapter reference] +\begin{Authorities} +See the end of Chapter~XVII. +\end{Authorities} +\PageSep{290} + + +\Chapter[Tidal Friction]{XVII} +{Tidal Friction (Continued)} + +\First{It} has been shown in the last chapter that the +prolongation of the day and of the month under +the influence of tidal friction takes place in such +a manner that the month will ultimately become +longer than the day. Until recent times no case +had been observed in the solar system in which +a satellite revolved more rapidly than its planet +rotated, and this might have been plausibly adduced +as a reason for rejecting the actual efficiency +of solar tidal friction in the process of +celestial evolution. At length however, in~1877, +Professor Asaph Hall discovered in the system +\index{Hall, Asaph, discovery of Martian satellites|(}% +of the planet Mars a case of the kind of motion +\index{Mars!discovery of satellites|(}% +which we foresee as the future fate of the moon +and earth, for he found that the planet was attended +by two satellites, the nearer of which has +\index{Satellites!discovery of those of Mars|(}% +a month shorter than the planet's day. He gives +an interesting account of what had been conjectured, +partly in jest and partly in earnest, as to +the existence of satellites attending that planet. +This foreshadowing of future discoveries is so +curious that I quote the following passage from +Professor Hall's paper. He writes:--- +\PageSep{291} + +``Since the discovery of the satellites of Mars, +the remarkable statements of Dean Swift and +Voltaire concerning the satellites of this planet, +and the arguments of Dr.~Thomas Dick and +others for the existence of such bodies, have attracted +so much attention, that a brief account +of the writings on this subject may be interesting. + +``The following letter of Kepler was written +\index{Kepler!argument respecting Martian satellites}% +to one of his friends soon after the discovery by +Galileo in~1610 of the four satellites of Jupiter, +\index{Galileo!discovery of Jupiter's satellites}% +and when doubts had been expressed as to the +reality of this discovery. The news of the discovery +was communicated to him by his friend +Wachenfels; and Kepler says:--- + +``\,`Such a fit of wonder seized me at a report +which seemed to be so very absurd, and I was +thrown into such agitation at seeing an old dispute +between us decided in this way, that between +his joy, my coloring, and the laughter of +both, confounded as we were by such a novelty, +we were hardly capable, he of speaking, or I of +listening. On our parting, I immediately began +to think how there could be any addition to the +number of the planets without overturning my +``Cosmographic Mystery,'' according to which +Euclid's five regular solids do not allow more +than six planets round the sun\dots. I am so +far from disbelieving the existence of the four +circumjovial planets, that I long for a telescope, +to anticipate you, if possible, in discovering \emph{two} +\PageSep{292} +\index{Kepler!argument respecting Martian satellites}% +round Mars, as the proportion seems to require, +\emph{six} or \emph{eight} round Saturn, and perhaps \emph{one} each +round Mercury and Venus.' + +``Dean Swift's statement concerning the satellites +\index{Swift, satire on mathematicians|(}% +of Mars is in his famous satire, `The +Travels of Mr.~Lemuel Gulliver.' After describing +\index{Gulliver@\Title{Gulliver's Travels}, satire on mathematics|(}% +his arrival in Laputa, and the devotion +of the Laputians to mathematics and music, +Gulliver says:--- + +``\,`The knowledge I had in mathematics gave +me great assistance in acquiring their phraseology, +which depended much upon that science, +and music; and in the latter I was not unskilled. +Their ideas were perpetually conversant in lines +and figures. If they would, for example, praise +the beauty of a woman, or of any other animal, +they describe it by rhombs, circles, parallelograms, +ellipses, and other geometrical terms, or +by words of art drawn from music, needless here +to repeat\dots. And although they are dexterous +enough upon a piece of paper, in the management +of the rule, the pencil, and the divider, +yet in the common actions and the behavior of +life, I have not seen a more clumsy, awkward, +and unhandy people, nor so slow and perplexed +in their conceptions upon all subjects, except +those of mathematics and music. They are very +bad reasoners, and vehemently given to opposition, +unless when they happen to be of the right +opinion, which is seldom their case\dots. These +\PageSep{293} +people are under continual disquietudes, never +enjoying a minute's peace of mind; and their +disturbances proceed from causes which very +little affect the rest of mortals. Their apprehensions +arise from several changes they dread +in the celestial bodies. For instance, that the +earth, by the continual approaches of the sun +towards it, must, in the course of time, be absorbed, +or swallowed up. That the face of the +sun will, by degrees, be encrusted with its own +effluvia, and give no more light to the world. +That the earth very narrowly escaped a brush +from the tail of the last comet, which would +have infallibly reduced it to ashes; and that the +next, which they have calculated for one-and-thirty +years hence, will probably destroy us. +For if, in its perihelion, it should approach +within a certain degree of the sun (as by their +calculations they have reason to dread,) it will +receive a degree of heat ten thousand times +more intense than that of red-hot glowing iron; +and, in its absence from the sun, carry a blazing +tail ten hundred thousand and fourteen miles +long; through which, if the earth should pass +at the distance of one hundred thousand miles +from the nucleus, or main body of the comet, it +must, in its passage, be set on fire, and reduced +to ashes. That the sun, daily spending its rays, +without any nutriment to supply them, will at +last be wholly consumed and annihilated; which +\PageSep{294} +must be attended with the destruction of this +earth, and of all the planets that receive their +light from it. + +``\,`They are so perpetually alarmed with the +apprehension of these, and the like impending +dangers, that they can neither sleep quietly in +their beds, nor have any relish for the common +pleasures and amusements of life. When they +meet an acquaintance in the morning, the first +question is about the sun's health, how he looked +at his setting and rising, and what hopes they had +to avoid the stroke of the approaching comet\dots. +They spend the greatest part of their lives +in observing the celestial bodies, which they do +by the assistance of glasses, far excelling ours in +goodness. For although their largest telescopes +do not exceed three feet, they magnify much +more than those of a hundred with us, and show +the stars with greater clearness. This advantage +has enabled them to extend their discoveries +much further than our astronomers in Europe; +for they have made a catalogue of ten thousand +fixed stars, whereas the largest of ours do not +contain above one-third of that number\dots. +They have likewise discovered two lesser stars, +or satellites, which revolve about Mars; whereof +the innermost is distant from the centre of the +primary planet exactly three of his diameters, +and the outermost, five; the former revolves in +the space of ten hours, and the latter in twenty-one +\PageSep{295} +and a half; so that the squares of their +periodical times are very near in the same proportion +with the cubes of their distance from +the centre of Mars; which evidently shows them +to be governed by the same law of gravitation +that influences the other heavenly bodies.' + +``The reference which Voltaire makes to the +\index{Voltaire, satire on mathematicians, and Martian satellites}% +moons of Mars is in his `Micromegas, Histoire +Philosophique.' Micromegas was an inhabitant +of Sirius, who, having written a book which a +suspicious old man thought smelt of heresy, left +Sirius and visited our solar system. Voltaire +says:--- + +``\,`Mais revenons à nos voyageurs. En sortant +de Jupiter, ils traversèrent un espace d'environ +cent millions de lieues, et ils côtoyèrent +la planète de Mars, qui, comme on sait, est cinq +fois plus petite que noire petit globe; ils virent +deux lunes qui servent à cette planète, et qui ont +échappé aux regards de nos astronomes. Je sais +bien que le père \emph{Castel} écrira, et même plaisamment, +\index{Castel, Father, ridiculed by Voltaire}% +contre l'existence de ces deux lunes; mais +je m'en rapporte à ceux qui raisonnent par analogie. +Ces bons philosophes-là savent combien il +serait difficile que Mars, qui est si loin du soleil, +se passât à moins de deux lunes.' + +``The argument by analogy for the existence +of a satellite of Mars was revived by writers like +Dr.~Thomas Dick, Dr.~Lardner, and others. In +\index{Dick, argument as to Martian satellites}% +\index{Lardner, possibility of Martian satellites}% +addition to what may be called the analogies of +\index{Gulliver@\Title{Gulliver's Travels}, satire on mathematics|)}% +\index{Swift, satire on mathematicians|)}% +\PageSep{296} +astronomy, these writers appear to rest on the +idea that a beneficent Creator would not place +a planet so far from the sun as Mars without +giving it a satellite. This kind of argument has +passed into some of our handbooks of astronomy, +and is stated as follows by Mr.~Chambers +\index{Chambers on possible existence of Martian satellites}% +in his excellent book on `Descriptive Astronomy,' +2d~edition, p.~89, published in~1867:--- + +``\,`As far as we know, Mars possesses no satellite, +though analogy does not forbid, but rather, +on the contrary, infers the existence of one; and +its never having been seen, in this case at least, +proves nothing. The second satellite of Jupiter +is only $\frac{1}{43}$~of the diameter of the primary, and +a satellite $\frac{1}{43}$~of the diameter of Mars would +be less than $100$~miles in diameter, and therefore +of a size barely within the reach of our largest +telescopes, allowing nothing for its possibly close +proximity to the planet. The fact that one of +the satellites of Saturn was only discovered a +few years ago renders the discovery of a satellite +of Mars by no means so great an improbability +as might be imagined.' + +``Swift seems to have had a hearty contempt +for mathematicians and astronomers, which he +has expressed in his description of the inhabitants +of Laputa. Voltaire shared this contempt, +\index{Voltaire, satire on mathematicians, and Martian satellites}% +and delighted in making fun of the philosophers +whom Frederick the Great collected at Berlin. +The `père Castel' may have been le~père Louis +\index{Castel, Father, ridiculed by Voltaire}% +\PageSep{297} +Castel, who published books on physics and +mathematics at Paris in 1743 and~1758. The +probable origin of these speculations about the +moons of Mars was, I think, Kepler's analogies. +Astronomers failing to verify these, an opportunity +was afforded to satirists like Swift and +Voltaire to ridicule such arguments.''\footnote + {\Title{Observations and Orbits of the Satellites of Mars}, by Asaph + Hall. Washington, Government Printing Office, 1878.} + +As I have already said, these prognostications +were at length verified by Professor Asaph Hall +in the discovery of two satellites, which he named +Phobos and Deimos---Fear and Panic, the dogs +\index{Deimos, a satellite of Mars}% +\index{Phobos, a satellite of Mars}% +of war. The period of Deimos is about $30$~hours, +and that of Phobos somewhat less than $8$~hours, +whilst the Martian day is of nearly the same +length as our own. The month of the inner +minute satellite is thus less than a third of the +planet's day; it rises to the Martians in the west, +and passes through all its phases in a few hours; +sometimes it must even rise twice in a single +Martian night. As we here find an illustration +of the condition foreseen for the earth and moon, +it seems legitimate to suppose that solar tidal +friction has retarded the planet's rotation until it +has become slower than the revolution of one of +the satellites. It would seem as if the ultimate +fate of Phobos will be absorption in the planet. + +Several of the satellites of Jupiter and of Saturn +present faint inequalities of coloring, and +\PageSep{298} +\index{Jupiter!satellites constantly face planet}% +\index{Saturn!satellites always face the planet}% +telescopic examination has led astronomers to believe +that they always present the same face to +their planets. The theory of tidal friction would +\index{Planets!rotation of some, annulled by tidal friction}% +certainly lead us to expect that these enormous +planets should work out the same result for their +relatively small satellites that the earth has produced +\index{Satellites!discovery of those of Mars|)}% +\index{Satellites!rotation of those of Jupiter and Saturn annulled}% +in the moon. + +The proximity of the planets Mercury and +\index{Mercury, rotation of}% +Venus to the sun should obviously render solar +\index{Venus, rotation of}% +tidal friction far more effective than with us. +The determination of the periods of rotation of +\index{Rotation!of Mercury, Venus, and satellites of Jupiter and Saturn annulled by tidal friction}% +these planets thus becomes a matter of much interest. +But the markings on their disks are so +obscure that the rates of their rotations have remained +under discussion for many years. Until +recently the prevailing opinion was that in both +cases the day was of nearly the same length as +ours; but a few years ago Schiaparelli of Milan, +\index{Schiaparelli on rotation of Venus and Mercury}% +an observer endowed with extraordinary acuteness +of vision, announced as the result of his observations +that both Mercury and Venus rotate +only once in their respective years, and that +each of them constantly presents the same face +to the sun. These conclusions have recently been +confirmed by Mr.~Percival Lowell from observations +\index{Lowell, P., on rotations of Venus and Mercury}% +made in Arizona. Although on reading +the papers of these astronomers it is not easy +to see how they can be mistaken, yet it should +be noted that others have failed to detect the +markings on the planet's disks, although they +\index{Hall, Asaph, discovery of Martian satellites|)}% +\index{Mars!discovery of satellites|)}% +\PageSep{299} +\index{Lowell, P., on rotations of Venus and Mercury}% +apparently enjoyed equal advantages for observation.\footnote + {Dr.~See, a member of the staff of the Flagstaff Observatory, + Arizona, tells me that he has occasionally looked at these planets + through the telescope, although he took no part in the systematic + observation. In his opinion it would be impossible for any one + at Flagstaff to doubt the reality of the markings. There are, + however, many astronomers of eminence who suspend their + judgment, and await confirmation by other observers at other + stations.} + +If, as I am disposed to do, we accept these observations +as sound, we find that evidence favorable +to the theory of tidal friction is furnished +by the planets Mercury and Venus, and by the +\index{Mercury, rotation of}% +\index{Venus, rotation of}% +satellites of the earth, Jupiter and Saturn, whilst +\index{Earth and moon!figure of}% +\index{Earth and moon!adjustment of figure to suit change of rotation|(}% +the Martian system is yet more striking as an +instance of an advanced stage in evolution. + +\TB + +It is well known that the figure of the earth +is flattened by the diurnal rotation, so that the +polar axis is shorter than any equatorial diameter. +At the present time the excess of the equatorial +radius over the polar radius is $\frac{1}{290}$~part of +either of them. Now in tracing the history of +the earth and moon, we found that the earth's +rotation had been retarded, so that the day is +now longer than it was. If then the solid earth +has always been absolutely unyielding, and if an +ocean formerly covered the planet to a uniform +depth, the sea must have gradually retreated +towards the poles, leaving the dry land exposed +at the equator. If on the other hand the solid +\PageSep{300} +\index{Geological evidence of earth's plasticity}% +\index{Plasticity of earth under change of rotation|(}% +earth had formerly its present shape, there must +then have been polar continents and a deep equatorial +sea. + +But any considerable change in the speed of +the earth's rotation would, through the action of +gravity, bring enormous forces to bear on the +solid earth. These forces are such as would, if +they acted on a plastic material, tend to restore +the planet's figure to the form appropriate to its +changed rotation. It has been shown experimentally +by M.~Tresca and others that even very +\index{Tresca on flow of solids}% +rigid and elastic substances lose their rigidity +and their elasticity, and become plastic under the +action of sufficiently great forces. It appears to +me, therefore, legitimate to hold to the belief in +the temporary rigidity of the earth's mass, as explained +in \Ref{Chapter}{XV}., whilst contending that +under a change of rotational velocity the earth +may have become plastic, and so have maintained +a figure adapted to its speed. Geological observation +shows that rocks have been freely twisted +and bent near the earth's surface, and it is impossible +to doubt that under altered rotation the +deeper portions of the earth would have been +subjected to very great stress. I conjecture that +the internal layers might adapt themselves by +continuous flow, whilst the superficial portion +might yield impulsively. Earthquakes are probably +due to unequal shrinkage of the planet in +cooling, and each shock would tend to bring the +\PageSep{301} +strata into their position of rest; thus the earth's +surface would avail itself of the opportunity afforded +by earthquakes of acquiring its proper +shape. The deposit in the sea of sediment, derived +from the denudation of continents, affords +another means of adjustment of the figure of the +planet. I believe then that the earth has always +maintained a shape nearly appropriate to its rotation. +The existence of the continents proves +that the adjustment has not been perfect, and we +shall see reason to believe that there has been +also a similar absence of complete adjustment in +the interior. + +But the opinion here maintained is not shared +by the most eminent of living authorities, Lord +Kelvin; for he holds that the fact that the average +\index{Kelvin, Lord!denies adjustment of earth's figure to changed rotation}% +figure of the earth corresponds with the +actual length of the day proves that the planet +was consolidated at a time when the rotation was +but little more rapid than it is now. The difference +between us is, however, only one of degree, +for he considers that the power of adjustment is +slight, whilst I hold that it would be sufficient +to bring about a considerable change of shape +within the period comprised in geological history. + +If the adjustment of the planet's figure were +perfect, the continents would sink below the +ocean, which would then be of uniform depth. +But there is no superficial sign, other than the +dry land, of absence of adaptation to the present +\PageSep{302} +\index{Moon and earth!inequality in motion indicates internal density of earth}% +rotation---unless indeed the deep polar sea discovered +by Nansen be such. Yet, as I have +hinted above, some tokens still exist in the earth +\index{Earth and moon!internal density}% +of the shorter day of the past. The detection of +this evidence depends however on arguments of +so technical a character that I cannot hope in +such a work as this to do more than indicate the +nature of the proof. + +The earth is denser towards the centre than +outside, and the layers of equal density are concentric. +\index{Density!of earth, law of internal}% +If then the materials were perfectly +plastic throughout, not only the surface, but +also each of these layers would be flattened to a +definite extent, which depends on the rate of rotation +and on the law governing the internal +density of the earth. Although the rate at +which the earth gets denser is unknown, yet it is +possible to assign limits to the density at various +depths. Thus it can be proved that at any internal +point the density must lie between two +values which depend on the position of the point +in question. So also, the degree of flattening at +any internal point is found to lie between two +extreme limits, provided that all the internal layers +are arranged as they would be if the whole +mass were plastic. + +Now variations in the law of internal density +and in the internal flattening would betray themselves +to our observation in several ways. In +the first place, gravity on the earth's surface +\index{Earth and moon!adjustment of figure to suit change of rotation|)}% +\index{Gravity, variation according to latitude}% +\index{Plasticity of earth under change of rotation|)}% +\PageSep{303} +\index{Meteorological!conditions dependent on earth's rotation}% +\index{Moon and earth!inequality in motion indicates internal density of earth}% +\index{Nutation!value of, indicates internal density of earth}% +would be changed. The force of gravity at the +\index{Gravity, variation according to latitude}% +poles is greater than at the equator, and the law +of its variation according to latitude is known. +In the second place the amount of the flattening +of the earth's surface would be altered, and the +present figure of the earth is known with considerable +exactness. Thirdly the figure and law of +density of the earth govern a certain irregularity +or inequality in the moon's motion, which has +been carefully evaluated by astronomers. Lastly +the precessional and nutational motion of the +earth is determined by the same causes, and these +motions also are accurately known. These four +facts of observation---gravity, the ellipticity of +\index{Ellipticity of earth's strata in excess for present rotation}% +the earth, the lunar inequality, and the precessional +and nutational motion of the earth---are +so intimately intertwined that one of them cannot +be touched without affecting the others. + +Now Édouard Roche, a French mathematician, +\index{Roche, E.!ellipticity of internal strata of earth}% +has shown that if the earth is perfectly plastic, +so that each layer is exactly of the proper shape +for the existing rotation, it is not possible to adjust +the unknown law of internal density so as +\index{Precession, value of, indicates internal density of earth}% +to make the values of all these elements accord +with observation. If the density be assumed +such as to fit one of the data, it will produce a +disagreement with observation in others. If, +however, the hypothesis be abandoned that the +internal strata all have the proper shapes, and if +it be granted that they are a little more flattened +\PageSep{304} +\index{Ellipticity of earth's strata in excess for present rotation}% +\index{Geological evidence of earth's plasticity!as to retardation of earth's rotation|(}% +than is due to the present rate of rotation, the +data are harmonized together; and this is just +what would be expected according to the theory +of tidal friction. But it would not be right to +attach great weight to this argument, for the +absence of harmony is so minute that it might +be plausibly explained by errors in the numerical +data of observation. I notice, however, that the +most competent judges of this intricate subject +are disposed to regard the discrepancy as a +reality. + +\DPchg{}{\TB} + +We have seen in the preceding chapter that +the length of day has changed but little within +historical times. But the period comprised in +written history is almost as nothing compared +with the whole geological history of the earth. +We ought then to consider whether geology furnishes +any evidence bearing on the theory of +tidal friction. The meteorological conditions on +the earth are dependent to a considerable extent +on the diurnal rotation of the planet, and therefore +those conditions must have differed in the +past. Our storms are of the nature of aerial eddies, +and they derive their rotation from that of +the earth. Accordingly storms were probably +more intense when the earth spun more rapidly. +The trunks of trees should be stronger than they +are now to withstand more violent storms. But +I cannot learn that there is any direct geological +evidence on this head, for deciduous trees with +\PageSep{305} +\index{Ripple mark in sand preserved in geological strata}% +stiff trunks seem to have been a modern product +of geological time, whilst the earlier trees more +nearly resembled bamboos, which yield to the +wind instead of standing up to it. It seems possible +that trees and plants would not be exterminated, +even if they suffered far more wreckage +than they do now. If trees with stiff trunks +could only withstand the struggle for existence +when storms became moderate in intensity, their +absence from earlier geological formations would +be directly due to the greater rapidity of the +earth's rotation in those times. + +According to our theory the tides on the seacoast +must certainly have had a much wider +range, and river floods must probably have been +more severe. The question then arises whether +these agencies should have produced sedimentary +deposits of coarser grain than at present. Although +I am no geologist, I venture to express a +doubt whether it is possible to tell, within very +wide limits, the speed of the current or the range +of the tide that has brought down and distributed +any sedimentary deposit. I doubt whether any +geologist would assert that floods might not have +been twice or thrice as frequent, or that the tide +might not have had a very much greater range +than at present. + +In some geological strata ripple-marks have +been preserved which exactly resemble modern +ones. This has, I believe, been adduced as an +\PageSep{306} +argument against the existence of tides of great +range. Ripples are, however, never produced +by a violent scour of water, but only by gentle +currents or by moderate waves. The turn of +the tide must be gentle to whatever height it +rises, and so the formation of ripple-mark should +have no relationship to the range of tide. + +It appears then that whilst geology affords no +direct confirmation of the theory, yet it does not +present any evidence inconsistent with it. Increased +activity in the factors of change is important +to geologists, since it renders intelligible +a diminution in the time occupied by the history +of the earth; and thus brings the views of the +\index{Earth and moon!probably once molten}% +geologist and of the physicist into better harmony. + +Although in this discussion I have maintained +the possibility that a considerable portion of the +changes due to tidal friction may have occurred +within geological history, yet it seems to me +probable that the greater part must be referred +back to pre-geological times, when the planet +was partially or entirely molten. + +\TB + +The action of the moon and sun on a plastic +and viscous planet would have an effect of which +some remains may perhaps still be traceable. +The relative positions of the moon and of the +frictionally retarded tide were illustrated in the +last chapter by~\fig{36}. That figure shows that +\index{Geological evidence of earth's plasticity!as to retardation of earth's rotation|)}% +\PageSep{307} +the earth's rotation is retarded by forces acting +\index{Earth and moon!distortion under primeval tidal friction}% +on the tidal protuberances in a direction adverse +to the planet's rotation. As the plastic substance, +of which we now suppose the planet to +be formed, rises and falls rhythmically with the +tide, the protuberant portions are continually +subject to this retarding force. Meanwhile the +internal portions are urged onward by the +inertia due to their velocity. Accordingly there +must be a slow motion of the more superficial +portions with reference to the interior. From +the same causes, under present conditions, the +whole ocean must have a slow westerly drift, although +it has not been detected by observation. + +Returning however to our plastic planet, the +equatorial portion is subjected to greater force +than the polar regions, and if meridians were +painted on its surface, as on a map, they would +gradually become distorted. In the equatorial +belt the original meridional lines would still run +north and south, but in the northern hemisphere +they would trend towards the northeast, and in +the southern hemisphere towards the southeast. +This distortion of the surface would cause the +surface to wrinkle, and the wrinkles should be +warped in the directions just ascribed to the +meridional lines. If the material yielded very +easily I imagine that the wrinkles would be +small, but if it were so stiff as only to yield with +difficulty they might be large. +\PageSep{308} + +There can be no doubt as to the correctness +\index{History!of earth and moon|(}% +of this conclusion as to a stiff yet viscous planet, +but the application of these ideas to the earth is +hazardous and highly speculative. We do, however, +observe that the continents, in fact, run +\index{Continents, trend of, possibly due to primeval tidal friction}% +roughly north and south. It may appear fanciful +to note, also, that the northeastern coast of +America, the northern coast of China, and the +southern extremity of South America have the +proper theoretical trends. But the northwestern +coast of America follows a line directly adverse +to the theory, and the other features of the globe +are by no means sufficiently regular to inspire +much confidence in the justice of the conjecture.\footnote + {See, also, W. Prinz, \Title{Torsion apparente des planètes}, ``Annuaire + de l'Obs.~R. de~Bruxelles,'' 1891.} + +\TB + +We must now revert to the astronomical aspects +of our problem. It is natural to inquire +whether the theory of tidal friction is competent +to explain any peculiarities of the motion of the +moon and earth other than those already considered. +It has been supposed thus far that the +moon moves over the earth's equator in a circular +orbit, and that the equator coincides with the +plane in which the earth moves in its orbit. But +the moon actually moves in a plane different +from that in which the earth revolves round the +sun, her orbit is not circular but elliptic, and the +\PageSep{309} +earth's equator is oblique to the orbit. We must +consider, then, how tidal friction will affect these +three factors. + +Let us begin by considering the obliquity of +the equator to the ecliptic, which produces the +seasonal changes of winter and summer. The +problem involved in the disturbance of the motion +of a rotating body by any external force is +too complex for treatment by general reasoning, +and I shall not attempt to explain in detail the +interaction of the moon and earth in this respect. + +The attractions of the moon and sun on the +equatorial protuberance of the earth causes the +earth's axis to move slowly and continuously +with reference to the fixed stars. At present, +the axis points to the pole-star, but $13,000$~years +hence the present pole-star will be $47°$~distant +from the pole, and in another $13,000$~years it +will again be the pole-star. Throughout this +precessional movement the obliquity of the equator +to the ecliptic remains constant, so that winter +and summer remain as at present. There is +also, superposed on the precession, the nutational +or nodding motion of the pole to which I referred +in \Ref{Chapter}{XV}. In the absence of tidal +friction the attractions of the moon and sun on +the tidal protuberance would slightly augment +the precession due to the solid equatorial protuberance, +and would add certain very minute +nutations of the earth's axis; the amount of +\PageSep{310} +these tidal effects, is, however, quite insignificant. +But under the influence of tidal friction, +the matter assumes a different aspect, for the +earth's axis will not return at the end of each +nutation to exactly the same position it would +have had in the absence of friction, and there is +a minute residual effect which always tends in +the same direction. A motion of the pole may +be insignificant when it is perfectly periodic, but +it becomes important in a very long period of +time when the path described is not absolutely +reëntrant. Now this is the case with regard to +the motion of the earth's axis under the influence +of frictionally retarded tides, for it is found +to be subject to a gradual drift in one direction. + +In tracing the history of the earth and moon +backwards in time we found the day and month +growing shorter, but at such relative speeds that +the number of days in the month diminished until +the day and month became equal. This conclusion +remains correct when the earth is oblique +to its orbit, but the effect on the obliquity is +\index{Ecliptic, obliquity of, due to tidal friction|(}% +\index{Obliquity of ecliptic, effects of tidal friction on|(}% +found to depend in a remarkable manner upon +the number of days in the month. At present +and for a long time in the past the obliquity +is increasing, so that it was smaller long ago. +But on going back to the time when the day +was six and the month twelve of our present +hours we find that the tendency for the obliquity +to increase vanishes. In other words, if +\PageSep{311} +there are more than two days in a month the +obliquity will increase, if less than two it will +diminish. + +Whatever may be the number of days in the +month, the rate of increase or diminution of +obliquity varies as the obliquity which exists at +the moment under consideration. If, then, a +planet be spinning about an axis absolutely perpendicular +to the plane of its satellite's orbit, the +obliquity remains invariable. But if we impart +infinitesimal obliquity to a planet whose day is +less than half a month, that infinitesimal obliquity +will increase; whilst, if the day is more +than half a month, the infinitesimal obliquity +will diminish. Accordingly, the motion of a +planet spinning upright is stable, if there are +less than two days in a month, and unstable if +there are more than two. + +It is not legitimate to ascribe the whole of +the present obliquity of~$23\frac{1}{2}°$ to the influence of +tidal friction, because it appears that when there +were only two days in the month, the obliquity +was still as much as~$11°$. It is, moreover, impossible +to explain the considerable obliquity of the +other planets to their orbits by this cause. It +must, therefore, be granted that there was some +unknown cause which started the planets in rotation +about axes oblique to their orbits. It remains, +however, certain that a planet, rotating primitively +without obliquity, would gradually become +\PageSep{312} +inclined to its orbit, although probably not to so +great an extent as we find in the case of the +earth. + +The next subject to be considered is the fact +that the moon's orbit is not circular but eccentric. +Here, again, it is found that if the tides +were not subject to friction, there would be no +sensible effect on the shape of the moon's path, +but tidal friction produces a reaction on the +moon tending to change the degree of eccentricity. +In this case, it is possible to indicate by +general reasoning the manner in which this reaction +operates. We have seen that tidal reaction +tends to increase the moon's distance from the +earth. Now, when the moon is nearest, in perigee, +the reaction is stronger than when she is +furthest, in apogee. The effect of the forces in +perigee is such that the moon's distance at the +next succeeding apogee is greater than it was at +the next preceding apogee; so, also, the effect +of the forces in apogee is an increase in the perigeal +distance. But the perigeal effect is stronger +than the apogeal, and, therefore, the apogeal distances +increase more rapidly than the perigeal +ones. It follows, therefore, that, whilst the orbit +as a whole expands, it becomes at the same time +more eccentric. +\index{Ecliptic, obliquity of, due to tidal friction|)}% +\index{Obliquity of ecliptic, effects of tidal friction on|)}% + +The lunar orbit is then becoming more eccentric, +and numerical calculation shows that in +very early times it must have been nearly circular. +\PageSep{313} +\index{Eccentricity of orbit!due to tidal friction}% +\index{Moon and earth!eccentricity of orbit increased by tidal friction}% +\index{Saint@St.\ Vénant on flow of solids}% +\index{See, T. J. J.!eccentricity of orbits of double stars}% +But mathematical analysis indicates that in +this case, as with the obliquity, the rate of +increase depends in a remarkable manner upon +the number of days in the month. I find in +fact that if eighteen days are less than eleven +months the eccentricity will increase, but in the +converse case it will diminish; in other words +the critical stage at which the eccentricity is +stationary is when $1\frac{7}{11}$~days is equal to the +month. It follows from this that the circular +orbit of the satellite is dynamically stable or +\index{Orbit!of double stars, very eccentric}% +unstable according as $1\frac{7}{11}$~days is less or greater +than the month. + +The effect of tidal friction on the eccentricity +has been made the basis of extensive astronomical +speculations by Dr.~See. I shall revert to +this subject in \Ref{Chapter}{XIX}., and will here +merely remark that systems of double stars are +\index{Stars!double, eccentricity of orbits}% +found to revolve about one another in orbits of +great eccentricity, and that Dr.~See supposes +that the eccentricity has arisen from the tidal +action of each star on the other. + +The last effect of tidal friction to which I +have to refer is that on the plane of the moon's +orbit. The lunar orbit is inclined to that of the +earth round the sun at an angle of~$5°$, and the +problem to be solved is as to the nature of the +effect of tidal friction on that inclination. The +nature of the relation of the moon's orbit to the +ecliptic is however so complex that it appears +\index{History!of earth and moon|)}% +\PageSep{314} +\index{Eccentricity of orbit!due to tidal friction}% +\index{Moon and earth!eccentricity of orbit increased by tidal friction}% +hopeless to explain the effects of tidal action +without the use of mathematical language, and +I must frankly give up the attempt. I may, +however, state that when the moon was near the +earth she must have moved nearly in the plane +of the earth's equator, but that the motion gradually +changed so that she has ultimately come to +move nearly in the plane of the ecliptic. These +two extreme cases are easily intelligible, but the +transition from one case to the other is very +complicated. It may suffice for this general +account of the subject to know that the effects +of tidal friction are quite consistent with the +present condition of the moon's motion, and +with the rest of the history which has been +traced. + +This discussion of the effects of tidal friction +may be summed up thus:--- + +If a planet consisted partly or wholly of molten +lava or of other fluid, and rotated rapidly about +an axis perpendicular to the plane of its orbit, +and if that planet was attended by a single satellite, +revolving with its month a little longer than +the planet's day, then a system would necessarily +be developed which would have a strong resemblance +to that of the earth and moon. + +A theory reposing on \textit{veræ causæ} which brings +into quantitative correlation the lengths of the +present day and month, the obliquity of the +ecliptic, the eccentricity and the inclination of +\PageSep{315} +the lunar orbit, should have strong claims to +acceptance. + +\begin{Authorities} +G.~H. Darwin. A series of papers in the ``Phil.\ Trans.\ Roy.\ +\index{Darwin, G. H.!papers on tidal friction}% +Soc.'' pt.~i.\ 1879, pt.~ii.\ 1879, pt.~ii.\ 1880, pt.~ii.\ 1881, pt.~i.\ 1882, +and abstracts (containing general reasoning) in the corresponding +Proceedings; also ``Proc.\ Roy.\ Soc.''\ vol.~29, 1879, p.~168 (in +part republished in Thomson and Tait's \Title{Natural Philosophy}), +and vol.~30, 1880, p.~255. + +Lord Kelvin, \Title{On Geological Time}, ``Popular Lectures and +\index{Kelvin, Lord!on geological time}% +Addresses,'' vol.~iii. Macmillan, 1894. + +Roche. The investigations of Roche and of others are given +in Tisserand's \Title{Mécanique Céleste}, vol.~ii. Gauthier-Villars, 1891. +\index{Tisserand, Roche's investigations as to earth's figure}% + +Tresca and St.~Vénant, \Title{Sur l'écoulement des Corps Solides}, +``Mémoires des Savants Étrangers,'' Académie des Sciences de +Paris, vols.\ 18~and~20. + +Schiaparelli, \Title{Considerazioni sul moto rotatorio del pianeta +\index{Schiaparelli on rotation of Venus and Mercury}% +Venere}. Five notes in the ``Rendiconti del R.~Istituto Lombardo,'' +vol.~23, and \Title{Sulla rotazione di Mercurio}, ``Ast.\ Nach.,'' +No.~2944. An abstract is given in ``Report of Council of R. +Ast.\ Soc.,'' Feb.~1891. + +Lowell, Mercury, ``Ast.\ Nach.,'' No.~3417. \Title{Mercury and Determination +\index{Lowell, P., on rotations of Venus and Mercury}% +of Rotation Period~\dots\ of Venus}, ``Monthly Notices +R. Ast.\ Soc.,'' vol.~57, 1897, p.~148. \Title{Further proof},~\&c., \textit{ibid}.\ +p.~402. + +Douglass, \Title{Jupiter's third Satellite}, ``Ast.\ Nach.,'' No.~3432. +\index{Douglass, rotation of Jupiter's satellites}% +\Title{Rotation des IV~Jupitersmondes}, ``Ast.\ Nach.,'' No.~3427, confirming +Engelmann, \Title{Ueber~\dots\ Jupiterstrabanten}, Leipzig, 1871. + +Barnard, \Title{The third and fourth Satellites of Jupiter}, ``Ast.\ +\index{Barnard, rotation of Jupiter's satellites}% +Nach.,'' No.~3453. +\end{Authorities} +\index{Friction of tides|)}% +\PageSep{316} + + +\Chapter[Figures of Equilibrium]{XVIII} +{The Figures of Equilibrium of a Rotating +Mass of Liquid} + +\First{The} theory of the tides involves the determination +\index{Equilibrium, figures of, of rotating liquid|(}% +\index{Figure of equilibrium!of rotating liquid|(}% +\index{Rotating liquid, figures of equilibrium|(}% +of the form assumed by the ocean under +the attraction of a distant body, and it now +remains to discuss the figure which a rotating +mass of liquid may assume when it is removed +from all external influences. The forces which +act upon the liquid are the mutual gravitation +of its particles, and the centrifugal force due to +its rotation. If the mass be of the appropriate +shape, these two opposing forces will balance +one another, and the shape will be permanent. +The problem in hand is, then, to determine +what shapes of this kind are possible. + +In 1842 a distinguished Belgian physicist, M.~Plateau,\footnote + {He is justly celebrated not only for his discoveries, but also + for his splendid perseverance in continuing his researches after + he had become totally blind.} +devised an experiment which affords +\index{Capillarity of liquids, and Plateau's experiment|(}% +\index{Plateau, experiment on figure of rotating globule|(}% +a beautiful illustration of the present subject. +The experiment needs very nice adjustment in +several respects, but I refer the reader to +Plateau's paper for an account of the necessary +\PageSep{317} +\index{Surface tension of liquids}% +precautions. Alcohol and water may be so +mixed as to have the same density as olive oil. +If the adjustment of density is sufficiently exact, +a mass of oil will float in the mixture, in the +form of a spherical globule, without any tendency +to rise or fall. The oil is thus virtually +relieved from the effect of gravity. A straight +wire, carrying a small circular disk at right +angles to itself, is then introduced from the top +of the vessel. When the disk reaches the +globule, the oil automatically congregates itself +round the disk in a spherical form, symmetrical +with the wire. + +The disk is then rotated slowly and uniformly, +and carries with it the oil, but leaves the surrounding +mixture at rest. The globule is then +seen to become flattened like an orange, and as +the rotation quickens it dimples at the centre, +and finally detaches itself from the disk in the +form of a perfect ring. This latter form is only +transient; for the oil usually closes in again +round the disk, or sometimes, with slightly different +manipulation, the ring may break into +drops which revolve round the centre, rotating +round their axes as they go. + +The force which holds a drop of water, or +this globule of oil, together is called ``surface +tension'' or ``capillarity.'' It is due to a certain +molecular attraction, quite distinct from +that of gravitation, and it produces the same +\PageSep{318} +effect as if the surface of the liquid were enclosed +in an elastic skin. There is of course no +actual skin, and yet when the liquid is stirred +the superficial particles attract their temporary +neighbors so as to restore the superficial elasticity, +continuously and immediately. The intensity +of surface tension depends on the nature +\index{Surface tension of liquids}% +of the material with which the liquid is in contact; +thus there is a definite degree of tension +in the skin of olive oil in contact with spirits +and water. + +A globule at rest necessarily assumes the form +of a sphere under the action of surface tension, +but when it rotates it is distorted by centrifugal +force. The polar regions become less curved, +and the equatorial region becomes more curved, +until the excess of the retaining power at the +equator over that at the poles is sufficient to +restrain the centrifugal force. Accordingly the +struggle between surface tension and centrifugal +force results in the assumption by the globule +of an orange-like shape, or, with greater speed +of rotation, of the other figures of equilibrium. +\index{Capillarity of liquids, and Plateau's experiment|)}% + +In very nearly the same way a large mass of +gravitating and rotating liquid will naturally +assume certain definite forms. The simplest +case of the kind is when the fluid is at rest in +space, without any rotation. Then mutual gravitation +is the only force which acts on the system. +The water will obviously crowd together +\PageSep{319} +into the smallest possible space, so that every +particle may get as near to the centre as its +neighbors will let it. I suppose the water to be +incompressible, so that the central portion, although +pressed by that which lies outside of it, +does not become more dense; and so the water +does not weigh more per cubic foot near the +centre than towards the outside. Since there +is no upwards and downwards, or right and +left about the system, it must be symmetrical in +every direction; and the only figure which possesses +this quality of universal symmetry is the +sphere. A sphere is then said to be a figure of +equilibrium of a mass of fluid at rest. + +If such a sphere of water were to be slightly +deformed, and then released, it would oscillate +to and fro, but would always maintain a nearly +spherical shape. The speed of the oscillation +depends on the nature of the deformation impressed +upon it. If the water were flattened to +the shape of an orange and released, it would +spring back towards the spherical form, but +would overshoot the mark, and pass on to a +lemon shape, as much elongated as the orange +was flattened. It would then return to the +orange shape, and so on backwards and forwards, +passing through the spherical form at +each oscillation. This is the simplest kind of +oscillation which the system can undergo, but +there is an infinite number of other modes of +\index{Plateau, experiment on figure of rotating globule|)}% +\PageSep{320} +any degree of complexity. The mathematician +can easily prove that a liquid globe, of the same +density as the earth, would take an hour and a +half to pass from the orange shape to the lemon +shape, and back to the orange shape. At present, +the exact period of the oscillation is not +the important point, but it is to be noted that if +the body be set oscillating in any way whatever, +it will continue to oscillate and will always remain +nearly spherical. We say then that the +sphere is a stable form of equilibrium of a mass +of fluid. The distinction between stability and +instability has been already illustrated in \Ref{Chapter}{XVI}.\ +by the cases of an egg lying on its +side and balanced on its end, and there is a +similar distinction between stable and unstable +modes of motion. + +Let us now suppose the mass of water to rotate +slowly, all in one piece as if it were solid. +We may by analogy with the earth describe the +axis of rotation as polar, and the central plane, +at right angles to the axis, as equatorial. The +equatorial region tends to move outwards in consequence +of the centrifugal force of the rotation, +and this tendency is resisted by gravitation which +tends to draw the water together towards the +centre. As the rotation is supposed to be very +slow, centrifugal force is weak, and its effects are +small thus the globe is very slightly flattened at +the poles, like an orange or like the earth itself. +\PageSep{321} +Such a body resembles the sphere in its behavior +when disturbed; it will oscillate, and its average +figure in the course of its swing is the orange +shape. It is therefore stable. + +But it has been discovered that the liquid may +also assume two other alternative forms. One +of these is extremely flattened and resembles a +flat cheese with rounded edges. As the disk of +liquid is very wide, the centrifugal force at the +equator is very great, although the rotation is +very slow. In the case of the orange-shaped figure, +the slower the rotation the less is the equatorial +centrifugal force, because it diminishes +both with diminution of radius and fall of speed. +But in the cheese shape the equatorial centrifugal +force gains more by the increase of equatorial +radius than it loses by diminution of rotation. +Therefore the slower the rotation the broader the +disk, and, if the rotation were infinitely slow, the +liquid would be an infinitely thin, flat, circular +disk. + +The cheese-like form differs in an important +respect from the orange-like form. If it were +slightly disturbed, it would break up, probably +into a number of detached pieces. The nature +of the break-up would depend on the disturbance +from which it started, but it is impossible to trace +the details of the rupture in any case. We say +then that the cheese shape is an unstable figure +of equilibrium of a rotating mass of liquid. +\PageSep{322} +\index{Stability!of figures of equilibrium}% + +The third form is strikingly different from +either of the preceding ones. We must now imagine +the liquid to be shaped like a long cigar, +and to be rotating about a central axis perpendicular +to its length. Here again the ends of +the cigar are so distant from the axis of rotation +that the centrifugal force is great, and with infinitely +slow rotation the figure becomes infinitely +long and thin. Now this form resembles the +cheese in being unstable. It is remarkable that +these three forms are independent of the scale on +which they are constructed, for they are perfectly +similar whether they contain a few pounds of +water or millions of tons.\footnote + {It is supposed that they are more than a fraction of an inch + across, otherwise surface tension would be called into play.} +If the period of rotation +and the density of the liquid are given, +the shapes are absolutely determinable. + +The first of the three figures resembles the +earth and may be called the planetary figure, and +\index{Planetary figure of equilibrium of rotating liquid}% +I may continue to refer to the other two as the +cheese shape and the cigar shape. The planetary +and cheese shape are sometimes called the spheroids +of Maclaurin, after their discoverer, and +\index{Maclaurin!figure of equilibrium of rotating liquid|(}% +the cigar shape is generally named after Jacobi, +\index{Jacobi, figure of equilibrium of rotating liquid|(}% +the great German mathematician. For slow rotations +the planetary form is stable, and the +cheese and cigar are unstable. There are probably +other possible forms of equilibrium, such as +a ring, or several rings, or two detached masses +\PageSep{323} +revolving about one another like a planet and +satellite, but for the present I only consider these +three forms. + +Now imagine three equal masses of liquid, infinitely +distant from one another, and each rotating +\Figure[0.85]{37}{}{png} +at the same slow speed, and let one of them +have the planetary shape, the second the cheese +shape, and the third the cigar shape. When the +rotations are simultaneously and equally augmented, +we find the planetary form becoming +flatter, the cheese form shrinking in diameter +and thickening, and the cigar form shortening +and becoming fatter. There is as yet no change +in the stability, the first remaining stable and +\index{Stability!of figures of equilibrium}% +the second and third unstable. The three figures +are illustrated in~\fig{37}, but the cigar shape +is hardly recognizable by that name, since it has +already become quite short and its girth is +considerable. +\PageSep{324} + +Now it has been proved that as the cigar shape +shortens, its tendency to break up becomes less +marked, or in other words its degree of instability +diminishes. At a certain stage, not as yet +exactly determined, but which probably occurs +when the cigar is about twice as long as broad, +the instability disappears and the cigar form just +becomes stable. I shall have to return to the +consideration of this phase later. The condition +of the three figures is now as follows: The planetary +form of Maclaurin has become much flattened, +but is still stable; the cigar form of Jacobi +has become short and thick, and is just stable; +and the cheese form of Maclaurin is still unstable, +\index{Maclaurin!figure of equilibrium of rotating liquid|)}% +but its diameter has shrunk so much that the +figure might be better described as a very flat +orange. + +On further augmenting the rotation the form +of Jacobi still shrinks in length and increases in +girth, until its length becomes equal to its +greater breadth. Throughout the transformation +the axis of rotation has always remained the +shortest of the three, so that when the length +becomes equal to the shorter equatorial diameter, +the shape is not spherical, but resembles that of +a much flattened orange. In fact, at this stage +Jacobi's figure of equilibrium has degenerated to +\index{Jacobi, figure of equilibrium of rotating liquid|)}% +identity with the planetary shape. One of the +upper ovals in \fig{38} represents the section of +the form in which the planetary figure and the +\PageSep{325} +cigar figure coalesce, the former by continuous +flattening, the latter by continuous shortening. +The other upper figure represents the form to +which the cheese-like figure of Maclaurin has +\Figure[0.85]{38}{}{png} +\index{Poincaré!figure of rotating liquid}% +been reduced; it will be observed that it presents +some resemblance to the coalescent form. + +When the rotation is further augmented, there +is no longer the possibility of an elongated Jacobian +figure, and there remain only the two +spheroids of Maclaurin. But an important change +has now supervened, for both these are now unstable, +and indeed no stable form consisting of a +single mass of liquid has yet been discovered. + +Still quickening the rotation, the two remaining +forms, both unstable, grow in resemblance to +one another, until at length they become identical +in shape. This limiting form of Maclaurin's +spheroids is shown in the lower part of~\fig{38}. +If the liquid were water, it must rotate in $2$~hours +\PageSep{326} +$25$~minutes to attain this figure, but it would be +unstable. + +A figure for yet more rapid rotation has not +been determined, but it seems probable that +dimples would be formed on the axis, that the +dimples would deepen until they met, and that +the shape would then be annular. The actual +existence of such figures in Plateau's experiment +is confirmatory of this conjecture. + +We must now revert to the consideration of +the cigar-shaped figure of Jacobi, at the stage +when it has just become stable. The whole of +this argument depends on the fact that any figure +of equilibrium is a member of a continuous +series of figures of the same class, which gradually +transforms itself as the rotation varies. Now +M.~Poincaré has proved that, when we follow a +\index{Poincaré!law of interchange of stability}% +given series of figures and find a change from instability +to stability, we are, as it were, served with +a notice that there exists another series of figures +coalescent with the first at that stage. We have already +seen an example of this law, for the planetary +figure of Maclaurin changed from stability +to instability at the moment of its coalescence +with the figure of Jacobi. Now I said that when +the cigar form of Jacobi was very long it was +unstable, but that when its length had shrunk to +about twice its breadth it became stable; hence +we have notice that at the moment of change +another series of forms was coalescent with the +\PageSep{327} +cigar. It follows also from Poincaré's investigation +\index{Poincaré!law of interchange of stability}% +\index{Poincaré!figure of rotating liquid}% +that the other series of forms must have +been stable before the coalescence. + +Let us imagine then a mass of liquid in the +form of Jacobi's cigar-shaped body rotating at +the speed which just admits of stability, and let +us pursue the series of changes backwards by +making it rotate a little slower. We know that +this retardation of rotation lengthens Jacobi's +figure, and induces instability, but Poincaré has +not only proved the existence and stability of the +other series, but has shown that the shape is +something like a pear. + +Poincaré's figure is represented approximately +in~\fig{38}, but the mathematical difficulty of the +problem has been too great to admit of an absolutely +exact drawing. The further development +of the pear shape is unknown, when the rotation +slackens still more. There can, however, be +hardly any doubt that the pear becomes more +constricted in the waist, and begins to resemble +an hour-glass; that the neck of the hour-glass +becomes thinner, and that ultimately the body +separates into two parts. It is of course likewise +unknown up to what stage in these changes +Poincaré's figure retains its stability. + +I have myself attacked this problem from an +entirely different point of view, and my conclusions +throw an interesting light on the subject, +although they are very imperfect in comparison +\PageSep{328} +\index{Darwin, G. H.!hour-glass figure of rotating liquid|(}% +with Poincaré's masterly work. To understand +this new point of view, we must consider a new +series of figures, namely that of a liquid planet +attended by a liquid satellite. The two bodies +are supposed to move in a circle round one another, +and each is also to revolve on its axis at +such a speed as always to exhibit the same face +to its neighbor. Such a system, although divided +into two parts, may be described as a figure of +equilibrium. If the earth were to turn round +once in twenty-seven days, it would always show +to the moon the same side, and the moon actually +does present the same side to us. In this +case the earth and the moon would form such a +system as that I am describing. Both the planet +and the satellite are slightly flattened by their +rotations, and each of them exercises a tidal influence +on the other, whereby they are elongated +towards the other. + +The system then consists of a liquid planet +and liquid satellite revolving round one another, +so as always to exhibit the same face to one another, +and each tidally distorting the other. It +is certain that if the two bodies are sufficiently +far apart the system is a stable one, for if any +slight disturbance be given, the whole system will +not break up. But little is known as yet as to +the limiting proximity of the planet and satellite, +which will insure stability. + +Now if the rotations and revolutions of the +\PageSep{329} +bodies be accelerated, the two masses must be +brought nearer together in order that the greater +attraction may counterbalance the centrifugal +force. But as the two are brought nearer the +tide-generating force increases in intensity with +great rapidity, and accordingly the tidal elongation +of the two bodies is much augmented. + +A time will at length come when the ends of +the two bodies will just touch, and we then have +a form shaped like an hour-glass with a very +\Figure[0.7]{39}{Hour-glass Figure of Equilibrium}{png} +\PageLabel[pg]{329}% [** TN: Used by reference on p. 356 of the original] +thin neck. The form is clearly Poincaré's figure, +at an advanced stage of its evolution. + +The figure~\figref{39} shows the form of one possible +\PageSep{330} +figure of this class; it arises from the coalescence +of two equal masses of liquid, and the +shape shown was determined by calculation. +But there are any number of different sorts of +hour-glass shapes, according to the relative sizes +of the planet and satellite which coalesce; and +in order to form a continuous series with Poincaré's +pear, it would be necessary to start with +a planet and satellite of some definitely proportionate +sizes. Unfortunately I do not know +what the proportion may be. There are, however, +certain indications which may ultimately +lead to a complete knowledge of the series of +figures from Jacobi's cigar shape down to the +planet and satellite. It may be shown---and I +shall have in \Ref{Chapter}{XX}.\ to consider the point +more in detail---that if our liquid satellite had +only, say, a thousandth of the mass of the planet, +and if the two bodies were brought nearer one +another, at a certain calculable distance the tidal +action of the big planet on the very small satellite +would become so intense that it would tear +it to pieces. Accordingly the contact and coalescence +of a very small satellite with a large +planet is impossible. It is, however, certain that +a large enough satellite---say of half the mass +of the planet---could be brought up to contact +with the planet, without the tidal action of the +planet on the satellite becoming too intense to +admit of the existence of the latter. There +\PageSep{331} +must then be some mass of the satellite, which +will just allow the two to touch at the same +moment that the tidal action of the larger on +the smaller body is on the point of disrupting +it. Now I suspect, although I do not know, +that the series of figures which we should find in +this case is in fact Poincaré's series. This discussion +shows that the subject still affords an +interesting field for future mathematicians. + +These investigations as to the form of rotating +masses of liquid are of a very abstract character, +and seem at first sight remote from practical +conclusions, yet they have some very interesting +applications. + +The planetary body of Maclaurin is flattened +at the poles like the actual planets, and the +degree of its flattening is exactly appropriate to +the rapidity of its rotation. Although the planets +are, at least in large part, composed of solid +matter, yet that matter is now, or was once, +sufficiently plastic to permit it to yield to the +enormous forces called into play by rotation and +gravitation. Hence it follows that the theory +of Maclaurin's figure is the foundation of that +of the figures of planets, and of the variation of +gravity at the various parts of their surfaces. +In the liquid considered hitherto, every particle +attracted every other particle, the fluid was +equally dense throughout, and the figure assumed +was the resultant of the battle between +\PageSep{332} +\index{Figure of planets and their density}% +\index{Gravity, variation according to latitude}% +\index{Saturn!law of density and figure}% +the centrifugal force and gravitation. At every +part of the liquid the resultant attraction was +directed nearly, but not quite, towards the +centre of the shape. But if the attraction had +everywhere been directed exactly to the centre, +the degree of flattening would have been +diminished. We may see that this must be so, +because if the rotation were annulled, the mass +would be exactly spherical, and if the rotation +were not annulled, yet the forces would be such +as to make the fluid pack closer, and so assume +a more nearly spherical form than when the +forces were not absolutely directed to the centre. +It may be shown in fact that the flattening is +$2\frac{1}{2}$~times greater in the case of Maclaurin's +body than it is when the seat of gravitation is +exactly central. + +In the case of actual planets the denser matter +\index{Planets!figures and internal densities}% +must lie in the centre and the less dense outside. +If the central matter were enormously +denser than superficial rock, the attraction would +be directed towards the centre. There are then +two extreme cases in which the degree of flattening +can be determined,---one in which the density +\index{Density!of planets determinable from their figures}% +of the planet is the same all through, giving +Maclaurin's figure; the other when the density +is enormously greater at the centre. The flattening +in the former is $2\frac{1}{2}$~times as great as in +the latter. The actual condition of a real planet +must lie between these two extremes. The +\index{Darwin, G. H.!hour-glass figure of rotating liquid|)}% +\PageSep{333} +\index{Figure of planets and their density}% +knowledge of the rate of rotation of a planet +and of the degree of its flattening furnishes us +with some insight into the law of its internal +density. If it is very much less flat than Maclaurin's +\index{Density!of planets determinable from their figures}% +figure, we conclude that it is very dense +in its central portion. In this way it is known +with certainty that the central portions of the +planets Jupiter and Saturn are much denser, +\index{Jupiter!figure and law of internal density}% +\index{Planets!figures and internal densities}% +compared with their superficial portions, than is +the case with the earth. + +I do not propose to pursue this subject into +the consideration of the law of the variation of +gravity on the surface of a planet; but enough +has been said to show that these abstract investigations +have most important practical applications. + +\begin{Authorities} +Plateau, ``Mémoires de l'Académie Royale de~Belgique,'' +vol.~xvi. 1843. + +Thomson and Tait's \Title{Natural Philosophy} or other works on +hydrodynamics give an account of figures of equilibrium. + +Poincaré, \Title{Sur l'équilibre d'une masse fluide animée d'un mouvement +de rotation}, ``Acta Mathematica,'' vol.~7, 1885. + +An easier and different presentation of the subject is contained +in an inaugural dissertation by Schwarzschild (Annals of Munich +\index{Schwarzschild!exposition of Poincaré's theory}% +Observatory, vol.~iii. 1896). He considers that Poincaré's +proof of the stability of his figure is not absolutely conclusive. + +G.~H. Darwin, \Title{Figures of Equilibrium of Rotating Masses of +\index{Darwin, G. H.!Jacobi's ellipsoid}% +Fluid}, ``Transactions of Royal Society,'' vol.~178, 1887. + +G.~H. Darwin, \Title{Jacobi's Figure of Equilibrium},~\&c., ``Proceedings +\index{Equilibrium, figures of, of rotating liquid|)}% +\index{Figure of equilibrium!of rotating liquid|)}% +Roy.\ Soc.,'' vol.~41, 1886, p.~319. + +S.~Krüger, \Title{Ellipsoidale Evenwichtsvormen},~\&c., Leeuwen, Leiden, +\index{Krüger, figures of equilibrium of liquid}% +1896; \Title{Sur l'ellipsoïde de Jacobi}, ``Nieuw Archief voor Wiskunde,'' +2d~series, 3d~part, 1898. The author shows that G.~H. +Darwin had been forestalled in much of his work on Jacobi's +figure, and he corrects certain mistakes. +\end{Authorities} +\index{Rotating liquid, figures of equilibrium|)}% +\PageSep{334} + + +\Chapter{XIX} +{The Evolution of Celestial Systems} + +\First{Men} will always aspire to peer into the remote +\index{Evolution of celestial systems|(}% +\index{Solar!system, nebular hypothesis as to origin of|(}% +past to the utmost of their power, and the fact +that their success or failure cannot appreciably +influence their life on the earth will never deter +them from such endeavors. From this point +of view the investigations explained in the last +chapter acquire much interest, since they form +the basis of the theories of cosmogony which +seem most probable by the light of our present +knowledge. + +We have seen that an annular figure of equilibrium +\index{Kant!nebular hypothesis|(}% +\index{Nebular hypothesis|(}% +actually exists in Plateau's experiment, +and it is almost certainly a possible form amongst +celestial bodies. Plateau's ring has however +only a transient existence, and tends to break up +into globules, spinning on their axes and revolving +round the centre. In this result we saw a +close analogy with the origin of the planets, and +regarded his experiment as confirmatory of the +Nebular Hypothesis, of which I shall now give a +short account.\footnote + {My knowledge of the history of the Nebular Hypothesis is + entirely derived from an interesting paper by Mr.~G.~F. Becker, +\index{Becker, G. F., on Nebular Hypothesis}% + on ``Kant as a Natural Philosopher,'' \Title{American Journal of Science}, + vol.~v. Feb.~1898.} +\PageSep{335} + +The first germs of this theory are to be found +in Descartes' ``Principles of Philosophy,'' published +\index{Descartes, vortical theory of cosmogony}% +in~1644. According to him the sun and +planets were represented by eddies or vortices in +a primitive chaos of matter, which afterwards +formed the centres for the accretion of matter. +As the theory of universal gravitation was propounded +for the first time half a century later +than the date of Descartes' book, it does not +seem worth while to follow his speculations +further. Swedenborg formulated another vortical +cosmogony in~1734, and Thomas Wright of +\index{Wright, Thomas, on a theory of cosmogony}% +Durham published in 1750 a book of preternatural +dullness on the same subject. It might not +have been worth while to mention Wright, but +that Kant acknowledges his obligation to him. + +The Nebular Hypothesis has been commonly +associated with the name of Laplace, and he undoubtedly +\index{Laplace!nebular hypothesis|(}% +avoided certain errors into which his +precursors had fallen. I shall therefore explain +Laplace's theory, and afterwards show how he +was, in most respects, really forestalled by the +great German philosopher Kant. + +Laplace supposed that the matter now forming +the solar system once existed in the form of a +lens-shaped nebula of highly rarefied gas, that it +rotated slowly about an axis perpendicular to the +present orbits of the planets, and that the nebula +extended beyond the present orbit of the furthest +planet. The gas was at first expanded by heat, +\PageSep{336} +and as the surface cooled the central portion +condensed and its temperature rose. The speed +of rotation increased in consequence of the contraction, +according to a well known law of mechanics +called ``the conservation of moment of +momentum;''\footnote + {Kant fell into error through ignorance of the generality of + this law, for he imagined that rotation could be generated from + rest.} +the edges of the lenticular mass +of gas then ceased to be continuous with the +more central portion, and a ring of matter was +detached, in much the same way as in Plateau's +experiment. Further cooling led to further contraction +and consequently to increased rotation, +until a second ring was shed, and so on successively. +The rings then ruptured and aggregated +themselves into planets whilst the central nucleus +formed the sun. + +Virtually the same theory had been propounded +by Kant many years previously, but I am not +aware that there is any reason to suppose that +Laplace had ever read Kant's works. In a paper, +to which I have referred above, Mr.~G.~F. +Becker makes the following excellent summary +\index{Becker, G. F., on Nebular Hypothesis|(}% +of the relative merits of Kant and Laplace; he +writes:--- + +``Kant seems to have anticipated Laplace almost +completely in the more essential portions +of the nebular hypothesis. The great Frenchman +was a child when Kant's theory was issued, +\PageSep{337} +and the `Système du Monde,' which closes with +the nebular hypothesis, did not appear until +1796. Laplace, like Kant, infers unity of origin +for the members of the solar system from the +similarity of their movements, the small obliquity +and small eccentricity of the orbits of either +planets or satellites.\footnote + {``The retrograde satellites of Uranus were discovered by + Herschel in~1787, but Laplace in his hypothesis does not refer to + them.''} +Only a fluid extending +throughout the solar system could have produced +such a result. He is led to conclude that the +atmosphere of the sun, in virtue of excessive +heat, originally extended beyond the solar system +and gradually shrank to its present limits. This +nebula was endowed with moment of momentum +which Kant tried to develop by collisions. Planets +formed from zones of vapor, which on breaking +agglomerated\dots. The main points of +comparison between Kant and Laplace seem to +be these. Kant begins with a cold, stationary +nebula which, however, becomes hot by compression +and at its first regenesis would be in a state +of rotation. It is with a hot, rotating nebula +that Laplace starts, without any attempt to account +for the heat. Kant supposes annular +zones of freely revolving nebulous matter to +gather together by attraction during condensation +of the nebula. Laplace supposes rings left +\index{Laplace!nebular hypothesis|)}% +behind by the cooling of the nebula to agglomerate +\PageSep{338} +in the same way as Kant had done. While +both appeal to the rings of Saturn as an example +of the hypothesis, neither explains satisfactorily +why the planetary rings are not as stable +as those of Saturn. Both assert that the positive +rotation of the planets is a necessary +consequence of agglomeration, but neither is +sufficiently explicit. The genesis of satellites is +for each of them a repetition on a small scale of +the formation of the system\dots. While Laplace +assigns no cause for the heat which he ascribes +to his nebula, Lord Kelvin goes further +back and supposes a cold nebula consisting of +separate atoms or of meteoric stones, initially +possessed of a resultant moment of momentum +equal or superior to that of the solar system. +Collision at the centre will reduce them to a +vapor which then expanding far beyond Neptune's +orbit will give a nebula such as Laplace +postulates.\footnote + {\Title{Popular Lectures}, vol.~i.\ p.~421.} +Thus Kelvin goes back to the same +initial condition as Kant, excepting that Kant +endeavored (of course vainly) to develop a moment +of momentum for his system from collisions.''\footnote + {Becker, \Title{Amer.\ Journ.\ Science}, vol.~v. 1898, pp.~107,~108.} +\index{Becker, G. F., on Nebular Hypothesis|)}% + +There is good reason for believing that the +Nebular Hypothesis presents a true statement in +outline of the origin of the solar system, and of +the planetary subsystems, because photographs +%% Plate 3 +\TallFig[0.85]{40}{Nebula in Andromeda}{jpg} +%% Facing page +%[Blank Page] +\PageSep{339} +of nebulæ have been taken recently in which we +can almost see the process in action. \Fig{40} is +a reproduction of a remarkable photograph by +Dr.~Isaac Roberts of the great nebula in the constellation +\index{Roberts, I., photograph of nebula in Andromeda}% +of Andromeda. In it we may see the +\index{Andromeda, nebula in}% +\index{Nebula in Andromeda}% +lenticular nebula with its central condensation, +the annulation of the outer portions, and even +the condensations in the rings which will doubtless +at some time form planets. This system is +built on a colossal scale, compared with which +our solar system is utterly insignificant. Other +\index{Solar!system, nebular hypothesis as to origin of|)}% +\index{Solar!system, distribution of satellites in|(}% +nebulæ show the same thing, and although they +are less striking we derive from them good +grounds for accepting this theory of evolution +as substantially true. + +\TB + +I explained in \Ref{Chapter}{XVI}.\ how the theory +of tidal friction showed that the moon took her +origin very near to the present surface of the +earth. But it was also pointed out that the same +theory cannot be invoked to explain an origin +for the planets at a point close to the sun. They +must in fact have always moved at nearly their +present distances. In the same way the dimensions +of the orbits of the satellites of Mars, Jupiter, +\index{Satellites!distribution of, in solar system|(}% +Saturn, and Neptune cannot have been +largely augmented, whatever other effects tidal +friction may have had. We must therefore still +rely on the Nebular Hypothesis for the explanation +\index{Nebular hypothesis|)}% +of the main features of the system as a +whole. +\index{Kant!nebular hypothesis|)}% +\PageSep{340} + +It may, at first sight, appear illogical to maintain +that an action, predominant in its influence +on our satellite, should have been insignificant +in regulating the orbits of all the other bodies +of the system. But this is not so, for whilst the +earth is only $80$~times as heavy as the moon, Saturn +weighs about $4,600$~times as much as its +satellite Titan, which is by far the largest satellite +in the solar system; and all the other satellites +are almost infinitesimal in comparison with their +primaries. Since, then, the relationship of the +moon to the earth is unique, it may be fairly contended +that a factor of evolution, which has been +predominant in our own history, has been relatively +insignificant elsewhere. + +There is indeed a reason explanatory of this +singularity in the moon and earth; it lies in the +fact that the earth is nearer to the sun than any +other planet attended by a satellite. To explain +the bearing of this fact on the origin of satellites +and on their sizes, I must now show how tidal +friction has probably operated as a perturbing +influence in the sequence of events, which would +be normal according to the Nebular Hypothesis. + +We have seen that rings should be shed from +the central nucleus, when the contraction of the +nebula has induced a certain degree of augmentation +of rotation. Now if the rotation were +retarded by some external cause, the genesis of +a ring would be retarded, or might be entirely +prevented. +\PageSep{341} + +The friction of the solar tides in a planetary +nebula furnishes such an external cause, and accordingly +the rotation of a planetary nebula near +to the sun might be so much retarded that a ring +would never be detached from it, and no satellite +would be generated. From this point of view +it is noteworthy that Mercury and Venus have +no satellites; that Mars has two, Jupiter five, +and that all the exterior planets have several +satellites. I suggest then that the solar tidal +friction of the terrestrial nebula was sufficient to +retard the birth of a satellite, but not to prevent +it, and that the planetary mass had contracted +to nearly the present dimensions of the earth +and had partially condensed into the solid and +liquid forms, before the rotation had augmented +sufficiently to permit the birth of a satellite. +When satellites arise under conditions which are +widely different, it is reasonable to suppose that +their masses will also differ much. Hence we can +understand how it has come about that the relationship +between the moon and the earth is so +unlike that between other satellites and their +planets. In \Ref{Chapter}{XVII}.\ I showed that there +are reasons for believing that solar tidal friction +\index{Solar!system, distribution of satellites in|)}% +has really been an efficient cause of change, and +this makes it legitimate to invoke its aid in explaining +the birth and distribution of satellites. +\index{Satellites!distribution of, in solar system|)}% + +\TB + +In speaking of the origin of the moon I have +\PageSep{342} +\index{Eccentricity of orbit!theory of, in case of double stars}% +\index{See, T. J. J.!theory of evolution of double stars|(}% +been careful not to imply that the matter of +which she is formed was necessarily first arranged +in the form of a ring. Indeed, the genesis of +the hour-glass figure of equilibrium from Jacobi's +form and its fission into two parts indicate the +possibility of an entirely different sequence of +events. It may perhaps be conjectured that the +moon was detached from the primitive earth in +this way, possibly with the help of tidal oscillations +due to the solar action. Even if this suggestion +is only a guess, it is interesting to make +such speculations, when they have some basis of +reason. + +In recent years astronomers have been trying, +principally by aid of the spectroscope, to determine +the orbits of pairs of double stars around +\index{Stars!theory of evolution|(}% +one another. It has been observed that, in the +majority of these systems, the masses of the two +component stars do not differ from one another +extremely; and Dr.~See, who has specially devoted +himself to this research, has drawn attention +to the great contrast between these systems +and that of the sun, attended by a retinue of +infinitesimal planets. He maintains, with justice, +that the paths of evolution pursued in the two +cases have probably also been strikingly different. + +It is hardly credible that two stars should +have gained their present companionship by an +accidental approach from infinite space. They +cannot always have moved as they do now, and +\PageSep{343} +so we are driven to reflect on the changes which +might supervene in such a system under the +action of known forces. + +The only efficient interaction between a pair +of celestial bodies, which is known hitherto, is +a tidal one, and the friction of the oscillations +introduces a cause of change in the system. +Tidal friction tends to increase the eccentricity +of the orbit in which two bodies revolve about +one another, and its efficiency is much increased +when the pair are not very unequal in mass and +when each is perturbed by the tides due to the +other. The fact that the orbits of the majority +of the known pairs are very eccentric affords a +reason for accepting the tidal explanation. The +only adverse reason, that I know of, is that the +eccentricities are frequently so great that we +may perhaps be putting too severe a strain on +the supposed cause. + +But the principal effect of tidal friction is the +repulsion of the two bodies from one another, +so that when their history is traced backwards +we ultimately find them close together. If then +this cause has been as potent as Dr.~See believes +it to have been, the two components of a binary +system must once have been close together. +From this stage it is but a step to picture to +ourselves the rupture of a nebula, in the form +of an hour-glass, into two detached masses. + +The theory embraces all the facts of the case, +\PageSep{344} +and as such is worthy of at least a provisional +acceptance. But we must not disguise from +ourselves that out of the thousands, and perhaps +millions of double stars which may be visible +from the earth, we only as yet know the orbits +and masses of a dozen. + +Many years ago Sir John Herschel drew a +\index{Herschel, observations of twin nebulæ}% +number of twin nebulæ as they appear through +a powerful telescope. The drawings probably +possess the highest degree of accuracy attainable +by this method of delineation, and the shapes +present evidence confirmatory of the theory of +the fission of nebulæ adopted by Dr.~See. But +since Herschel's time it has been discovered that +many details, to which our eyes must remain forever +blind, are revealed by celestial photography. +The photographic film is, in fact, sensitive to +those ``actinic'' rays which we may call invisible +light, and many nebulæ are now found to be +hardly recognizable, when photographs of them +are compared with drawings. A conspicuous +example of this is furnished by the great nebula +in Andromeda, illustrated above in~\fig{40}. + +Photographs, however, do not always aid interpretation, +for there are some which serve only +to increase the chaos visible with the telescope. +We may suspect, indeed, that the complete system +of a nebula often contains masses of cold +and photographically invisible gas, and in such +cases it would seem that the true nature of the +whole will always be concealed from us. +\PageSep{345} + +Another group of strange celestial objects is +that of the spiral nebulæ, whose forms irresistibly +\index{Nebulae@Nebulæ, description of various}% +suggest violent whirlpools of incandescent +gas. Although in all probability the motion of +the gas is very rapid, yet no change of form has +been detected. We are here reminded of a +rapid stream rushing past a post, where the form +of the surface remains constant whilst the water +itself is in rapid movement; and it seems reasonable +to suppose that in these nebulæ it is +only the lines of the flow of the gas which are +visible. Again, there are other cases in which +the telescopic view may be almost deceptive in +its physical suggestions. Thus the Dumb-Bell +\index{Dumb-bell nebula, description of photograph of}% +nebula (27~Messier Vulpeculæ), as seen telescopically, +might be taken as a good illustration of a +nebula almost ready to split into two stars. If +this were so, the rotation would be about an +axis at right angles to the length of the nebula. +But a photograph of this object shows that the +system really consists of a luminous globe surrounded +by a thick and less luminous ring, and +that the opacity of the sides of the ring takes a +bite, as it were, out of each side of the disk, and +so gives it the apparent form of a dumb-bell. +In this case the rotation must be about an axis +at right angles to the ring, and therefore along +the length of the dumb-bell. It is proper to +add that Dr.~See is well aware of this, and does +not refer to this nebula as a case of incipient +fission. +\PageSep{346} + +I have made these remarks in order to show +that every theory of stellar evolution must be +full of difficulty and uncertainty. According to +our present knowledge Dr.~See's theory appears +to have much in its favor, but we must await its +confirmation or refutation from the results of +future researches with the photographic plate, +the spectroscope, and the telescope. + +\begin{Authorities} +Mr.~G.~F. Becker (\Title{Amer.\ Jour.\ Science}, vol.~v. 1898, art.~xv.)\ +gives the following references to Kant's work: \Title{Sämmtliche +Werke}, ed.~Hartenstein, 1868 (Tidal Friction and the Aging of +the Earth), vol.~i.\ pp.~179--206; (Nebular Hypothesis), vol.~i.\ +pp.~207--345. + +Laplace, \Title{Système du Monde}, last appendix; the tidal retardation +of the moon's rotation is only mentioned in the later +editions. + +T.~J.~J. See, \Title{Die Entwickelung der Doppelstern-systeme}, ``Inaugural +Dissertation,'' 1892. Schade, Berlin. + +T.~J.~J. See, \Title{Evolution of the Stellar Systems}, vol.~i.\ 1896. +Nichols Press, Lynn, Massachusetts. Also a popular article, +\Title{The Atlantic Monthly}, October, 1897. + +G.~H. Darwin, \Title{Tidal Friction~\dots\ and Evolution}, ``Phil.\ Trans.\ +\index{Darwin, G. H.!evolution of satellites}% +Roy.\ Soc.,'' part~ii.\ 1881, p.~525. +\end{Authorities} +\index{Evolution of celestial systems|)}% +\index{See, T. J. J.!theory of evolution of double stars|)}% +\index{Stars!theory of evolution|)}% +\PageSep{347} + + +\Chapter[Saturn's Rings]{XX} +{Saturn's Rings\protect\footnotemark} + +\footnotetext{Part of this chapter appeared as an article in \Title{Harper's + Magazine} for June,~1889.} + +\First{To} the naked eye Saturn appears as a brilliant +\index{Saturn!description and picture|(}% +star, which shines, without twinkling, with a +yellowish light. It is always to be found very +nearly in the ecliptic, moving slowly amongst +the fixed stars at the rate of only thirteen degrees +per annum. It is the second largest +planet of the solar system, being only exceeded +in size by the giant Jupiter. It weighs $91$~times +as much as our earth, but, being as light as cork, +occupies $690$~times the volume, and is nine times +as great in circumference. Notwithstanding its +great size it rotates around its axis far more +rapidly than does the earth, its day being only +$10\frac{1}{2}$~of our hours. It is ten times as far from +the sun as we are, and its year, or time of revolution +round the sun, is equal to thirty of our +years. It was deemed by the early astronomers +to be the planet furthest from the sun, but that +was before the discovery by Herschel, at the +end of the last century, of the further planet +Uranus, and that of the still more distant Neptune +by Adams and Leverrier in the year~1846. +\PageSep{348} + +The telescope has shown that Saturn is attended +by a retinue of satellites almost as numerous +as, and closely analogous to, the planets +circling round the sun. These moons are eight +in number, are of the most various sizes, the +largest as great as the planet Mars, and the +smallest very small, and are equally diverse in +respect of their distances from the planet. But +besides its eight moons Saturn has another attendant +absolutely unique in the heavens; it is +girdled with a flat ring, which, like the planet +itself, is only rendered visible to us by the +illumination of sunlight. \Fig{41}, to which +further reference is made below, shows the general +appearance of the planet and of its ring. +The theory of the physical constitution of that +ring forms the subject of the present chapter. + +A system so rich in details, so diversified and +so extraordinary, would afford, and doubtless +has afforded, the subject for many descriptive +essays; but description is not my present object. + +The existence of the ring of Saturn seems +now a very commonplace piece of knowledge, +and yet it is not $300$~years since the moons of +Jupiter and Saturn were first detected, and since +suspicion was first aroused that there was something +altogether peculiar about the Saturnian +system. These discoveries, indeed, depended +entirely on the invention of the telescope. It +may assist the reader to realize how necessary +\PageSep{349} +%[** TN: Oriented vertically in the original] +\Figure{41}{The Planet Saturn}{jpg} +\PageSep{350} +the aid of that instrument was when I say that +Saturn, when at his nearest to us, is the same in +size as a sixpenny piece held up at a distance of +$210$~yards. + +It was the celebrated Galileo who first invented +\index{Galileo!Saturn's ring}% +a combination of lenses such as is still +used in our present opera-glasses, for the purpose +of magnifying distant objects. + +In July of~1610 he began to examine Saturn +with his telescope. His most powerful instrument +only magnified $32$~times, and although +such an enlargement should have amply sufficed +to enable him to make out the ring, yet he persuaded +himself that what he saw was a large +bright disk, with two smaller ones touching it, +one on each side. His lenses were doubtless +imperfect, but the principal cause of his error +must have been the extreme improbability of the +existence of a ring girdling the planet. He +wrote an account of what he had seen to the +Grand Duke of Tuscany, Giuliano de'~Medici, +and to others; he also published to the world an +anagram which, when the letters were properly +arranged, read as follows: ``Altissimum planetam +tergeminum observavi'' (I have seen the +furthest planet as triple), for it must be remembered +that Saturn was then the furthest known +planet. + +In 1612 Galileo again examined Saturn, and +was utterly perplexed and discouraged to find +\PageSep{351} +his triple star replaced by a single disk. He +writes, ``Is it possible that some mocking demon +has deceived me?'' And here it may be well to +remark that there are several positions in which +Saturn's rings vanish from sight, or so nearly +vanish as to be only visible with the most powerful +modern telescopes. When the plane of the +ring passes through the sun, only its very thin +edge is illuminated; this was the case in~1612, +when Galileo lost it; secondly, if the plane of +the ring passes through the earth, we have only +a very thin edge to look at; and thirdly, when +the sun and the earth are on opposite sides of +the ring, the face of the ring which is presented +to us is in shadow, and therefore invisible. + +Some time afterwards Galileo's perplexity was +increased by seeing that the planet had then a +pair of arms, but he never succeeded in unraveling +the mystery, and blindness closed his career +as an astronomer in~1626. + +About thirty years after this, the great Dutch +astronomer Huyghens, having invented a new +\index{Huyghens, discovery of Saturn's ring}% +sort of telescope (on the principle of our present +powerful refractors), began to examine the planet +and saw that it was furnished with two loops or +handles. Soon after the ring disappeared; but +when, in~1659, it came into view again, he at +last recognized its true character, and announced +that the planet was attended by a broad, flat +ring. +\PageSep{352} + +A few years later it was perceived that there +were two rings, concentric with one another. +The division, which may be easily seen in drawings +of the planet, is still named after Cassini, +\index{Cassini, discovery of division in Saturn's rings}% +one of its discoverers. Subsequent observers +have detected other less marked divisions. + +Nearly two centuries later, namely, in~1850, +Bond in America and Dawes in England, independently +\index{Bond, discovery of inner ring of Saturn}% +\index{Dawes, discovery of inner ring of Saturn}% +and within a fortnight of the same +time, observed that inside of the well-known +bright rings there is another very faint dark +ring, which is so transparent that the edge of +the planet is visible through it. There is some +reason to believe that this ring has really become +more conspicuous within the last $200$~years, +so that it would not be right to attribute the +lateness of its detection entirely to the imperfection +of earlier observations. + +It was already discovered in the last century +that the ring is not quite of the same thickness +at all points of its circumference, that it is not +strictly concentric with the planet, and that it +revolves round its centre. Herschel, with his +magnificent reflecting telescope, detected little +beads on the outer ring, and by watching these +he concluded that the ring completes its revolution +in $10\frac{1}{2}$~hours. + +This sketch of the discovery and observation +of Saturn's rings has been necessarily very incomplete, +but we have perhaps already occupied +too much space with it. +\PageSep{353} + +\Fig{41} exhibits the appearance of Saturn and +his ring. The drawing is by Bond of Harvard +University, and is considered an excellent one. + +It is usual to represent the planets as they are +seen through an astronomical telescope, that is +\Figure{42}{Diagram of Saturn and his Rings}{png} +to say, reversed. Thus in \fig{41} the south +pole of the planet is at the top of the plate, and +unless the telescope were being driven by clockwork, +the planet would appear to move across +the field of view from right to left. + +The plane of the ring is coincident with the +equator of the planet, and both ring and equator +are inclined to the plane of the planet's orbit at +an angle of $27$~degrees. + +A whole essay might be devoted to the discussion +of this and of other pictures, but we must +confine ourselves to drawing attention to the +well-marked split, called Cassini's division, and +\PageSep{354} +to the faint internal ring, through which the +edge of the planet is visible. + +The scale on which the whole system is constructed +is best seen in a diagram of concentric +circles, showing the limits of the planet's body +and of the successive rings. Such a diagram, +with explanatory notes, is given in~\fig{42}. + +An explanation of the outermost circle, called +\emph{Roche's limit}, will be given later. The following +are the dimensions of the system:--- +\begin{center} +\begin{tabular}{l>{\qquad}r} +Equatorial diameter of planet & $73,000$ miles \\ +Interior diameter of dark ring & $93,000$ \Ditto{miles} \\ +Interior diameter of bright rings & $111,000$ \Ditto{miles} \\ +Exterior diameter of bright rings & $169,000$ \Ditto{miles} +\end{tabular} +\end{center} +We may also remark that the radius of the +limit of the rings is $2.38$~times the mean radius +of the planet, whilst Roche's limit is $2.44$~such +radii. The greatest thickness of the ring is uncertain, +but it seems probable that it does not +exceed $200$~or $300$~miles. + +The pictorial interest, as we may call it, of all +this wonderful combination is obvious, but our +curiosity is further stimulated when we reflect on +the difficulty of reconciling the existence of this +strange satellite with what we know of our own +planet and of other celestial bodies. + +It may be admitted that no disturbance to our +ordinary way of life would take place if Saturn's +\index{Saturn!description and picture|)}% +rings were annihilated, but, as Clerk-Maxwell +has remarked, ``from a purely scientific point of +\PageSep{355} +view, they become the most remarkable bodies in +the heavens, except, perhaps, those still less \emph{useful} +bodies---the spiral nebulæ. When we have +actually seen that great arch swung over the +equator of the planet without any visible connection, +we cannot bring our minds to rest. We +cannot simply admit that such is the case, and +describe it as one of the observed facts of nature, +not admitting or requiring explanation. We +must either explain its motion on the principles +of mechanics, or admit that, in Saturnian realms, +there can be motion regulated by laws which we +are unable to explain.'' + +I must now revert to the subject of \Ref{Chapter}{XVIII}.\ +and show how the investigations, there +explained, bear on the system of the planet. We +then imagined a liquid satellite revolving in a +circular orbit about a liquid planet, and supposed +that each of these two masses moved so as always +to present the same face to the other. It was +pointed out that each body must be somewhat +flattened by its rotation round an axis at right +angles to the plane of the orbit, and that the +tidal attraction of each must deform the other. +In the application of this theory to the system of +Saturn it is not necessary to consider further the +tidal action of the satellite on the planet, and we +must concentrate our attention on the action of +the planet on the satellite. We have found reason +to suppose that the earth once raised enormous +\PageSep{356} +\index{Saturn!theory of ring|(}% +tides in the moon, when her body was +molten, and any planet must act in the same way +on its satellite. When, as we now suppose, the +satellite moves so as always to present the same +face to the planet, the tide is fixed and degenerates +into a permanent distortion of the equator +of the satellite into an elliptic shape. If the +satellite is very small compared with its planet, +and if it is gradually brought closer and closer +to the planet, the tide-generating force, which +varies inversely as the cube of the distance, increases +with great rapidity, and we shall find the +satellite to assume a more and more elongated +shape. When the satellite is not excessively +small, the two bodies may be brought together +until they actually touch, and form the hour-glass +figure exhibited in \fig{39}, \PageRef{p.}{329}. + +The general question of the limiting proximity +of a liquid planet and satellite which just insures +stability is as yet unsolved. But it has been +proved that there is one case in which instability +sets in. Édouard Roche has shown that this approach +\index{Roche, E.!theory of limit and Saturn's ring|(}% +up to contact is not possible when the +satellite is very small, for at a certain distance +the tidal distortion of a small satellite becomes +so extreme that it can no longer subsist as a +single mass of fluid. He also calculated the +form of the satellite when it is elongated as much +as possible. \Fig{43} represents the satellite in +its limiting form. We must suppose the planet +\PageSep{357} +about which it revolves to be a large globe, with +its centre lying on the prolongation of the longest +axis of the egg-like body in the direction +of~$E$. As it revolves, the longest axis of the satellite +always points straight towards its planet. +The egg, though not strictly circular in girth, is +\Figure[0.7]{43}{Roche's Figure of a Satellite when elongated +to the utmost}{png} +very nearly so. Thus another section at right +angles to this one would be of nearly the same +shape. One diameter of the girth is in fact only +longer than the other by a seventeenth part. +The shortest of the three axes of the slightly flattened +egg is at right angles to the plane of the +orbit in which the satellite revolves. The longest +axis of the body is nearly twice as long as +either of the two shorter ones; for if we take +the longest as~$\DPchg{1000}{1,000}$, the other two would be $496$ +and~$469$. \Fig{43} represents a section through +the two axes equal respectively to~$\DPchg{1000}{1,000}$ and to~$469$, +so that we are here supposed to be looking +at the satellite's orbit edgewise. +\PageSep{358} + +But, as I have said, Roche determined not +\index{Earth and moon!Roche's limit for}% +only the shape of the satellite when thus elongated +to the utmost possible extent, but also in +its nearness to the planet, and he proved that if +the planet and satellite be formed of matter of +the same density, the centre of such a satellite +must be at a distance from the planet's centre of +$2\frac{11}{25}$~of the planet's radius. This distance of $2\frac{11}{25}$ +or $2.44$~of a planet's radius I call Roche's limit +for that planet. The meaning of this is that inside +of a circle drawn around a planet at a distance +so proportionate to its radius no small +satellite can circulate; the reason being that if +a lump of matter were started to revolve about +the planet inside of that circle, it would be torn +to pieces under the action of the forces we have +been considering. It is true that if the lump of +matter were so small as to be more properly described +as a stone than as a satellite, then the +cohesive force of stone might be strong enough +to resist the disruptive force. But the size for +which cohesion is sufficient to hold a mass of +matter together is small compared with the +smallest satellite. + +I have said that Roche's limit as evaluated at +$2.44$~radii is dependent on the assumption of +equal densities in the satellite and planet. If +the planet be denser than the satellite, Roche's +limit is a larger multiple of the planet's radius, +and if it be less dense the multiple is smaller. +\PageSep{359} +But the variation of distance is not great for +considerable variations in the relative densities +of the two bodies, the law being that the~$2.44$ +must be multiplied by the cube root of the ratio +of the density of the planet to that of the satellite. +If for example the planet be on the average +of its whole volume twice as dense as the +satellite, the limit is only augmented from $2.44$ +to $3$~times the planet's radius; and if it be half +as dense, the $2.44$ is depressed to~$1.94$. Thus +the variation of density of the planet from a +half to twice that of the planet---that is to +say, the multiplication of the smaller density by +four---only changes Roche's limit from $2$ to $3$~radii. +It follows from this that, within pretty +wide limits of variation of relative densities, +Roche's limit changes but little. + +The only relative density of planet and satellite +that we know with accuracy is that of the +earth and moon. Now the earth is more dense +than the moon in the proportion of $8$~to~$5$; hence +Roche's limit for the earth is the cube root of~$\frac{8}{5}$ +multiplied by~$2.44$, that is to say, it is $2.86$~times +the earth's radius. It follows that if the moon +were to revolve at a distance of less than $2.86$~radii, +or $11,000$~miles, she would be torn to pieces +by the earth's tidal force. + +If this result be compared with the conclusions +drawn from the theory of tidal friction, it follows +that at the earliest stage to which the moon was +\PageSep{360} +\index{Saturn!Roche's limit for}% +traced, she could not have existed in her present +form, but the matter which is now consolidated +in the form of a satellite must then have been a +mere swarm of loose fragments. Such fragments, +if concentrated in one part of the orbit, would +be nearly as efficient in generating tides in the +planet as though they were agglomerated in the +form of a satellite. Accordingly the action of +tidal friction does not necessitate the agglomeration +of the satellite. The origin and earliest history +of the moon must always remain highly +speculative, and it seems fruitless to formulate +exact theories on the subject.\footnote + {Mr.~Nolan has criticised the theory of tidal friction from +\index{Nolan, criticism of tidal theory of moon's origin}% + this point of view (\Title{Genesis of the Moon}, Melbourne, 1885; also + \Title{Nature}, Feb.~18 and July~29, 1886).} + +When we apply this reasoning to the other +planets, exact data are wanting. The planet +Mars resembles the earth in so many respects +\index{Mars!Roche's limit}% +that it is reasonable to suppose that there is much +the same relationship between the densities of +the planet and satellites as with us. As with the +case of the earth and moon, this would bring +Roche's limit to $2.86$~times the planet's radius. +The satellite Phobos, however, revolves at a +distance of $2.75$~radii of Mars; hence we are +bound to suppose that the density of Phobos is +a very little more nearly equal to that of Mars +than in the case of the moon and earth; if +it were not so, Phobos would be disrupted by +\PageSep{361} +tidal action. How interesting it will be if future +generations shall cease to see the satellite Phobos, +for they will then conclude that Phobos has been +drawn within the charmed circle, and has been +broken to pieces. + +In considering the planets Jupiter and Saturn, +\index{Jupiter!Roche's limit for}% +we are deprived of the indications which are useful +in the case of Mars. The satellites are probably +solid, and these planets are known to have +a low mean density. Hence it is probable that +Roche's limit is a somewhat smaller multiple than +$2.44$~of the radii of Jupiter and Saturn. The +only satellite which is in danger is the innermost +and recently discovered satellite of Jupiter, which +revolves at $2.6$~times the planet's mean radius, +for with the same ratio of densities as obtains +here the satellite would be broken up. This confirms +the conclusion that the mean density of +Jupiter is at least not greater than that of the +satellite. + +We are also ignorant of the relative densities +of Saturn and its satellites, and so in the figure +Roche's limit is placed at $2.44$~times the planet's +radius, corresponding to equal densities. But +the density of the planet is very small, and therefore +the limit is almost certainly slightly nearer +to the planet than is shown. + +This system affords the only known instance +where matter is clearly visible circulating round +an attractive centre at a distance certainly less +\PageSep{362} +than the theoretical limit, and the belief seems +justified that Saturn's rings consist of dust and +fragments. + +Although Roche himself dismissed this matter +in one or two sentences, he saw the full bearing +of his remarks, and to do him justice we should +date from~1848 the proof that Saturn's rings +consist of meteoric stones. + +The theoretical limit lies just outside the limit +of the rings, but we may suspect that the relative +densities of the planet and satellite are such that +the limit should be displaced to a distance just +inside of the outer edge of the ring, because any +solid satellite would almost necessarily have a +mean density greater than that of the planet. + +Although Roche's paper was published about +fifty years ago, it has only recently been mentioned +in text-books and general treatises. Indeed, +it has been stated that Bond was the first +in modern times to suggest the meteoric constitution +of the rings. His suggestion, based on +telescopic evidence, was however made in~1851. + +\TB + +And now to explain how a Cambridge mathematician +to whom reference was made above, in +ignorance of Roche's work of nine years before, +\index{Roche, E.!theory of limit and Saturn's ring|)}% +arrived at the same conclusion. In~1857, Clerk-Maxwell, +one of the most brilliant men of science +who have taught in the University of Cambridge, +and whose early death we still deplore, attacked +\PageSep{363} +\index{Instability!of Saturn's ring}% +the problem of Saturn's rings in a celebrated +essay, which gained for him what is called the +Adams prize. Laplace had early in the century +considered the theory that the ring is solid, and +Maxwell first took up the question of the motion +\index{Maxwell on Saturn's ring|(}% +of such a solid ring at the point where it had +been left. He determined what amount of +weighting at one point of a solid uniform ring is +necessary to insure its steady motion round the +planet. He found that there must be a mass +attached to the circumference of the ring weighing +$4\frac{1}{2}$~times as much as the ring itself. In fact, +the system becomes a satellite with a light ring +attached to it. + +``As there is no appearance,'' he says, ``about +the rings justifying a belief in so great an irregularity, +the theory of the solidity of the rings +becomes very improbable. When we come to +consider the additional difficulty of the tendency +of the fluid or loose parts of the ring to accumulate +at the thicker parts, and thus to destroy that +nice adjustment of the load on which the stability +depends, we have another powerful argument +against solidity. And when we consider the immense +size of the rings and their comparative +thinness, the absurdity of treating them as rigid +bodies becomes self-evident. An iron ring of +such a size would be not only plastic, but semi-fluid, +under the forces which it would experience, +and we have no reason to believe these rings to +\PageSep{364} +\index{Instability!of Saturn's ring}% +be artificially strengthened with any material +unknown on this earth.'' + +The hypothesis of solidity being condemned, +Maxwell proceeds to suppose that the ring is +composed of a number of equal small satellites. +This is a step towards the hypothesis of an indefinite +number of meteorites of all sizes. The +consideration of the motion of these equal satellites +affords a problem of immense difficulty, for +each satellite is attracted by all the others and +by the planet, and they are all in motion. + +If they were arranged in a circle round the +planet at equal distances, they might continue to +revolve round the planet, provided that each +satellite remained in its place with mathematical +exactness. Let us consider that the proper place +of each satellite is at the ends of the spokes of +a revolving wheel, and then let us suppose that +none of them is exactly in its place, some being +a little too far advanced, some a little behind, +some too near and some too far from the centre +of the wheel---that is to say, from the planet---then +we want to know whether they will swing +to and fro in the neighborhood of their places, +or will get further and further from their places, +and whether the ring will end in confusion. + +Maxwell treated this problem with consummate +skill, and showed that if the satellites were +not too large, confusion would not ensue, but +each satellite would oscillate about its proper +place. +\PageSep{365} +\index{Stability!of Saturn's ring}% + +At any moment there are places where the +satellites are crowded and others where they are +spaced out, and he showed that the places of +crowding and of spacing out will travel round +the ring at a different speed from that with +which the ring as a whole revolves. In other +words, waves of condensation and of rarefaction +are propagated round the ring as it rotates. + +He constructed a model, now in the laboratory +at Cambridge, to exhibit these movements; it is +pretty to observe the changes of the shape of the +ring and of the crowding of the model satellites +as they revolve. + +I cannot sum up the general conclusions at +which Maxwell arrived better than by quoting +his own words. + +In the summary of his paper he says:--- + +``If the satellites are unequal, the propagation +of waves will no longer be regular, but the disturbances +of the ring will in this, as in the +former case, produce only waves, and not growing +confusion. Supposing the ring to consist, +not of a single row of large-satellites, but of a +cloud of evenly distributed unconnected particles, +we found that such a cloud must have a +very small density in order to be permanent, and +that this is inconsistent with its outer and inner +parts moving with the same angular velocity. +Supposing the ring to be fluid and continuous, +we found that it will necessarily be broken up +into small portions. +\PageSep{366} +\index{Stability!of Saturn's ring}% + +``We conclude, therefore, that the rings must +consist of disconnected particles; these may be +either solid or liquid, but they must be independent. +The entire system of rings must therefore +consist either of a series of many concentric +rings, each moving with its own velocity, and +having its own system of waves, or else of a confused +multitude of revolving particles, not arranged +in rings, and continually coming into +collision with each other. + +``Taking the first case, we found that in an +indefinite number of possible cases the mutual +perturbation of two rings, stable in themselves, +might mount up in time to a destructive magnitude, +and that such cases must continually occur +in an extensive system like that of Saturn, the +only retarding cause being the possible irregularity +of the rings. + +``The result of long-continued disturbance +was found to be the spreading out of the rings +in breadth, the outer rings pressing outward, +while the inner rings press inward. + +``The final result, therefore, of the mechanical +theory is, that the only system of rings which +can exist is one composed of an indefinite number +of unconnected particles, revolving round the +planet with different velocities according to their +respective distances. These particles may be +arranged in a series of narrow rings, or they may +move through each other irregularly. In the +\PageSep{367} +\index{Keeler, spectroscopic examination of Saturn's ring|(}% +first case the destruction of the system will be +very slow, in the second case it will be more +rapid, but there may be a tendency towards an +arrangement in narrow rings, which may retard +the process. + +``We are not able to ascertain by observation +the constitution of the two outer divisions of the +system of rings, but the inner ring is certainly +transparent, for the limb (\ie~edge) of Saturn +has been observed through it. It is also certain, +that though the space occupied by the ring is +transparent, it is not through the material particles +of it that Saturn was seen, for his limb was +observed without distortion; which shows that +there was no refraction, and therefore that the +rays did not pass through a medium at all, but +between the solid or liquid particles of which the +ring is composed. Here then we have an optical +argument in favor of the theory of independent +particles as the material of the rings. The +two outer rings may be of the same nature, but +not so exceedingly rare that a ray of light can +pass through their whole thickness without encountering +one of the particles.'' +\index{Maxwell on Saturn's ring|)}% + +\TB + +The last link in the chain of evidence has been +furnished by recent observations made in America. +If it can be proved that every part of the +apparently solid ring moves round the planet's +centre at a different rate, and that the speed at +\PageSep{368} +\index{Meteoric constitution of Saturn's ring}% +\index{Spectroscopic proof of rotation of Saturn's ring}% +each part is appropriate at its distance from the +centre, the conclusion is inevitable that the ring +consists of scattered fragments. + +Every one must have noticed that when a +train passes at full speed with the whistle blowing, +there is an abrupt fall in the pitch of the +note. This change of note is only apparent to +the stationary listener, and is caused by the +crowding together of the waves of sound as the +train approaches, and by their spacing out as it +recedes. The same thing is true of light-waves, +and if we could imagine a colored light to pass +us at an almost inconceivable velocity it would +change in tint as it passed.\footnote + {This statement is strictly correct only of monochromatic + light. I might, in the subsequent argument, have introduced + the limitation that the moving body shall emit only monochromatic + light. The qualification would, however, only complicate + the statement, and thus render the displacement of the lines of + the spectrum less easily intelligible.} +Now there are certain +lines in the spectrum of sunlight, and the +shifting of their positions affords an excessively +delicate measure of a change which, when magnified +enormously, would produce a change of +tint. For example, the sun is a rotating body, +and when we look at its disk one edge is approaching +us and the other is receding. The +two edges are infinitesimally of different colors, +and the change of tint is measurable by the displacement +of the lines I have mentioned. In +the same way Saturn's ring is illuminated by +sunlight, and if different portions are moving at +\PageSep{369} +\index{Spectroscopic proof of rotation of Saturn's ring}% +different velocities, those portions are infinitesimally +of different colors. Now Professor Keeler, +the present director of the Lick Observatory, has +actually observed the reflected sunlight from the +several parts of Saturn's ring, and he finds that +the lines in the spectrum of the several parts +are differently displaced. From measurement of +these displacements he has concluded that every +part of the ring moves at the same pace as if it +were an independent satellite. The proof of the +meteoric constitution of the ring is therefore +\index{Meteoric constitution of Saturn's ring}% +complete. + +It would be hard to find in science a more +beautiful instance of arguments of the most +diverse natures concentrating themselves on a +definite and final conclusion. + +\begin{Authorities} +Édouard Roche, \Title{La figure d'une masse fluide soumise à l'attraction +\index{Roche, E.!stability of ellipsoid of}% +d'un point éloigné}, ``Mém.\ Acad.\ de~Montpelier,'' vol.~i.\ +(Sciences), 1847--50. + +Maxwell, \Title{Stability of Saturn's Rings}, Macmillan, 1859. + +Keeler, \Title{Spectroscopic Proof of the Meteoric Constitution of +Saturn's Rings}, ``Astrophysical Journal,'' May, 1895; see also +\index{Keeler, spectroscopic examination of Saturn's ring|)}% +\index{Saturn!theory of ring|)}% +the same for June, 1895. + +Schwarzschild, \Title{Die Poincarésche Theorie des Gleichgewichts}, +\index{Schwarzschild!stability of Roche's ellipsoid}% +``Annals of Munich Observatory,'' vol.~iii.\ 1896. He considers +the stability of Roche's ellipsoid. +\end{Authorities} +\PageSep{370} +%[Blank Page] +\PageSep{371} +\BackMatter +\printindex +\iffalse +INDEX + +Abacus for reducing tidal observations#abacus, 217-220. + +Abbadie, tidal deflection of vertical#Abbadie, 143, 144. + +Aden, errors of tidal prediction at#Aden, 246. + +Adriatic, tide in#Adriatic, 186. + +Airy, Sir G. B.#Airy, + tides in rivers, 75; + attack on Laplace, 181; + cotidal chart, 188; + \Title{Tides and Waves}, 192. + +America, North, tide tables for#America, 222. + +Analysis, harmonic, of tide#analysis, 193-210. + +Andromeda, nebula in#Andromeda, 339. + +Annual and semi-annual tides#annual, 206. + +Arabian theories of tide#Arab, 77-79. + +Aristotle on tides#Aristotle, 81. + +Assyrian records of eclipses, 272. + +Atlantic, tide in#Atlantic, 186-188. + +Atmospheric pressure, + cause of seiches, 40; + distortion of soil by, 145, 146; + influence on tidal prediction, 242, 243. + +Atmospheric waves, Helmholtz on#Helmholtz, 48-51. + +Attraction, + of weight resting on elastic slab proportional to slope, 136, 137; + of tide calculated, 143. + +Baird, \Title{Manual for Tidal Observation}#Baird, 16. + +Bakhuyzen on tide due to variation of latitude#Bakhuyzen, 255, 256. + +Barnard, rotation of Jupiter's satellites#Barnard, 315. + +Barometric pressure. |see{Atmospheric pressure}. 0 + +Becker, G. F., on Nebular Hypothesis#Becker, 334, 336-338. + +Bernoulli, Daniel, essay on tides#Bernoulli, 86, 88. + +Bertelli on Italian seismology#Bertelli, 126, 127. + +Bifilar. |see{Pendulum}. 0 + +Borgen@Börgen, method of reducing tidal observations#Börgen, 217. + +Bond, discovery of inner ring of Saturn#Bond, 352. + +Bore, + definition, 59; + bore-shelter, 63; + diagram of rise in Tsien-Tang, 66; + pictures, 67; + rivers where found, 71; + causes, 72; + Chinese superstition, 68-70. + +Browne, E. G., Arabian theories of tide#Browne, 77-79. + +Cambridge, experiments with bifilar pendulum at#bifilar, 115-125. + +Canal, + theory of tide wave in, 165-167; + critical depth, 163-165; + tides in ocean partitioned into canals, 175; + canal in high latitude, 174-176. + +Capillarity of liquids, and Plateau's experiment#Plateau, 316-318. + +Cassini, discovery of division in Saturn's rings#Cassini, 352. + +Castel, Father, ridiculed by Voltaire#Castel, 295, 296. + +Cavalleri, essay on tides#Cavalleri, 86. + +Centripetal and centrifugal forces#Centripetal, 91-93. + +Chambers on possible existence of Martian satellites#Chambers, 296. +\PageSep{372} + +Chandler, free nutation of earth, and variation of latitude#Chandler, 253-257. + +Chinese + superstition as to bore, 68-70; + theories of tide, 76, 77. + +Christie, A. S., tide due to variation of latitude#Christie, 255, 256. + +Constants, tidal, explained#constants, 195. + +Continents, trend of, possibly due to primeval tidal friction#continents, 308. + +Cotidal chart, 188; + for diurnal tide hitherto undetermined, 191, 192. + +Currents, tidal, in rivers#tidal current, 56. + +Curve, tide#tide curve, + irregularities in, 10-16; + at Bombay, 12; + partitioned into lunar time, 213. + +D'Abbadie. |see{Abbadie}. 0 + +Darwin, G. H.#Darwin, + bifilar pendulum, 115-125; + harmonic analysis, 210; + tidal abacus, 217-220; + distortion of earth's surface by varying loads, 134-148; + rigidity of earth, 261, 262; + papers on tidal friction, 315; + hour-glass figure of rotating liquid, 328-332; + Jacobi's ellipsoid, 333; + evolution of satellites, 346. + +Darwin, Horace, bifilar pendulum#Horace, 115-125. + +Davis, method of presenting tide-generating force#Davis, 96, 97. + +Davison, history of bifilar and horizontal pendulums#Davison, 133. + +Dawes, discovery of inner ring of Saturn#Dawes, 352. + +Dawson coöperates in investigation of seiches#Dawson, 48. + +Day, change in length of, under tidal friction#day, 275, 276. + +Deflection of the vertical, 109-133; + experiments to measure, 115-125; + due to tide, 134-143. + +Deimos, a satellite of Mars#Deimos, 297. + +Denison, F. Napier, vibrations and seiches on lakes#Denison, 48-53. + +Density + of earth, law of internal, 302; + of planets determinable from their figures, 332, 333. + +Descartes, vortical theory of cosmogony#Descartes, 335. + +Dick, argument as to Martian satellites#Dick, 295. + +Dimple, + in soil, due to weight, 123; + form of, in elastic slab, 135. + +Distortion of soil + by weight, 123; + by varying loads, 134-148. + +Diurnal inequality + observed by Seleucus, 84, 85; + according to equilibrium theory, 156; + in Laplace's solution, 179; + in Atlantic, Pacific, and Indian Oceans, 180; + not shown in cotidal chart, 191; + in harmonic method, 205; + complicates prediction, 224, 225. + +Douglass, rotation of Jupiter's satellites#Douglass, 315. + +Dumb-bell nebula, description of photograph of#dumb-bell, 345. + +Dynamical theory of tide-wave, 163-181. + +Earth and moon#Earth, + diagram, 93; + rotation of, effects on tides, 177; + rigidity of, 256-260; + rotation retarded by tidal friction, 268; + figure of, 299; + adjustment of figure to suit change of rotation, 299-302; + internal density, 302; + probably once molten, 306; + distortion under primeval tidal friction, 307; + Roche's limit for, 358. + +Earthquakes, + a cause of seiches, 39; + microsisms and earth tremors, 125-127; + shock perceptible at great distance, 261. + +Ebb and flow defined#ebb, 56. + +Eccentricity of orbit + due to tidal friction, 313, 314; + theory of, in case of double stars, 342. +\PageSep{373} + +Eclipses, ancient, and earth's rotation#eclipses, 272, 273. + +Ecliptic, obliquity of, due to tidal friction#ecliptic, 308-312. + +Eddies, tidal oscillation involves#eddies, 177. + +Ehlert, observation with horizontal pendulum#Ehlert, 132. + +Elastic distortion#elastic + of soil by weight, 123; + of earth by varying loads, 134-148; + calculation and illustration, 138-140; + by atmospheric pressure, 145-147. + +Elasticity of earth#elasticity, 254, 255. + +Elliptic tide, lunar#elliptic tide, 204. + +Ellipticity of earth's strata in excess for present rotation#ellipticity, 303, 304. + +Energy, tidal, utilization of#tidal energy, 73, 74. + +Equatorial canal, tide wave in#canal, 173. + +Equilibrium, figures of, of rotating liquid#equilibrium, 316-333. + +Equilibrium theory of tides#equilibrium, 149-162; + chart and law of tide, 151-153; + defects of, 160. + +Errors in tidal prediction#errors, 243-245. + +Establishment of port, + definition, 161, 162; + zero in equilibrium theory, 161; + shown in cotidal chart, 189. + +Estuary, annual meteorological tide in#estuary, 207, 208. + +Euler, essay on tides#Euler, 86. + +Europe, tides on coasts of#Europe, 188. + +Evolution of celestial systems, 334-346. + +Ferrel, tide-predicting instrument#Ferrel, 241. + +Figure of equilibrium + of ocean under tidal forces, 151-153; + of rotating liquid, 316-333. + +Figure of planets and their density, 332, 333. + +Fisher, Osmond, on molten interior of earth#Fisher, 262. + +Flow and ebb defined#ebb, 56. + +Forced oscillation, + principle of, 169, 170; + due to solar tide, possibly related to birth of moon, 282-284. + +Forced wave, explanation and contrast with free wave#forced wave, 164. + +Forces, + centripetal and centrifugal, 91-93; + tide-generating, 93-108; + numerical estimate, 109-111; + deflection of vertical by, 109-133; + figure of equilibrium under tidal, 151-153; + those of sun and moon compared, 156-158. + +Forel + on seiches, 17-38; + list of papers, 53, 54. + +Free oscillation contrasted with forced#free oscillation, 169, 170. + +Free wave, explanation and contrast with forced#free wave, 164. + +Friction of tides, 264-315. + +Galileo, + blames Kepler for his tidal theory, 85; + discovery of Jupiter's satellites, 291; + Saturn's ring, 350. + +Gauge, tide#tide gauge, + description of, 6-11; + site for, 14. + +Geneva, + seiches in lake, 17-28; + model of lake, 28. + +Geological evidence of earth's plasticity#plasticity, 300; + as to retardation of earth's rotation, 304-306. + +German method of reducing tidal observations#German, 217. + +Giles on Chinese theories of the tide#Giles, 76, 77. + +Gravity, variation according to latitude#gravity, 302, 303, 332. + +Greek + theory and description of tides, 81-85; + records of ancient eclipses, 272. + +Gulliver@\Title{Gulliver's Travels}, satire on mathematics#Gulliver, 292-295. + +Hall, Asaph, discovery of Martian satellites#Hall, 290-298. + +Hangchow, the bore at#Hangchow, 60-70. +\PageSep{374} + +Harmonic analysis + initiated by Lord Kelvin, 87; + account of, 193-210. + +Height of tide#height + due to ideal satellite, 198; + at Portsmouth and at Aden, 225; + reduced by elastic yielding of earth, 259. + +Helmholtz + on atmospheric waves, 48-51; + on rotation of the moon, 286. + +Herschel, observations of twin nebulæ#Herschel, 344. + +High water + under moon in equilibrium theory, 160; + position in shallow and deep canals in dynamical theory, 171, 172. + +History + of tidal theories, 76-88; + of earth and moon, 278-286, 308-313. + +Hopkins on rigidity of earth#Hopkins, 258, 259. + +Horizontal tide-generating force, 107. + +Horizontal tide-generating force |see{also Pendulum}. 0 + +Hough, S. S.#Hough, + frictional extinction of waves, 47; + dynamical solution of tidal problem, 181; + rigidity of earth, 254; + Chandler's nutation, 262. + +Hugli, bore on the#Hugli, 71. + +Huyghens, discovery of Saturn's ring#Huyghens, 351. + +Icelandic theory of tides, 79, 80. + +Indian Survey, + method of reducing tidal observations, 216, 217; + tide tables, 222. + +Instability, + nature of dynamical, and initial of moon's motion, 280-282; + of Saturn's ring, 363, 364. + +Interval from moon's transit to high water + in case of ideal satellite, 198; + at Portsmouth and at Aden, 225. + +Italian investigations in seismology, 125-130. + +Jacobi, figure of equilibrium of rotating liquid#Jacobi, 322-324. + +Japan, frequency of earthquakes#Japan, 130, 131. + +Jupiter, + satellites constantly face planet, 298; + figure and law of internal density, 333; + Roche's limit for, 361. + +Kant, + rotation of moon, 286; + nebular hypothesis, 334-339. + +Keeler, spectroscopic examination of Saturn's ring#Keeler, 367-369. + +Kelvin, Lord#Kelvin, + initiates harmonic analysis, 87, 199; + calculation of tidal attraction, 143; + tide predicting machine, 233; + rigidity of earth, 257-260; + denies adjustment of earth's figure to changed rotation, 301; + on geological time, 315. + +Kepler, + ideas concerning tides, 85, 86; + argument respecting Martian satellites, 291, 292. + +Krüger, figures of equilibrium of liquid#Krüger, 333. + +Lakes, + seiches in, 17-54; + mode of rocking in seiches, 24, 25; + vibrations, 41-53; + tides in, 182-185. + +Lamb, H., presentation of Laplace's theory#Lamb, 181. + +Laplace, + theory of tides, 86-88, 177-180; + on rotation of moon, 286, 287; + nebular hypothesis, 335-337. + +Lardner, possibility of Martian satellites#Lardner, 295. + +Latitude, + tidal wave in canal in high, 174-176; + periodic variations of, 251-256. + +Lege@Légé, constructor of tide-predicting machine#Légé, 233. + +Level of sea affected by atmospheric pressure#atmospheric pressure, 146. + +Limnimeter, a form of tide gauge#limnimeter, 24. + +Lowell, P., on rotations of Venus and Mercury#Lowell, 298, 299, 315. + +Low water. |see{High water}. +\PageSep{375} + +Lubbock, Sir J., senior, on tides#Lubbock, 87. + +Lunar + tide-generating force compared with solar, 156-158; + tide, principal, 201; + elliptic tide, 204; + time, 213. + +Machine, tide-predicting#tide-predicting, 233, 241. + +Mackerel sky, evidence of air-waves#mackerel, 49. + +Maclaurin, + essay on tides, 86; + figure of equilibrium of rotating liquid, 322-324. + +Magnússon on Icelandic theories of tides#Magnússon, 79, 80. + +Marco Polo, resident of Hangchow#Marco, 70. + +Mars, + discovery of satellites, 290-298; + Roche's limit, 360. + +Maxwell on Saturn's ring#Maxwell, 363-367. + +Mediterranean Sea, tides in#Mediterranean, 185, 186. + +Mercury, rotation of#Mercury, 298, 299. + +Meteoric constitution of Saturn's ring#meteoric, 368, 369. + +Meteorological + tides, 206, 207; + conditions dependent on earth's rotation, 303. + +Microphone as a seismological instrument#microphone, 128-130. + +Microsisms, minute earthquakes#microsisms, 125-127. + +Mills worked by the tide#mills, 74, 75. + +Milne on seismology#Milne, 125, 130. + +Month, change in, under tidal friction#month, 275-277. + +Moon and earth, + diagram, 93; + tide-generating force compared with sun's, 156-158; + tide due to ideal, moving in equator, 193, 194; + ideal satellites replacing actual, 199, 200; + tidal prediction by reference to transit, 224-230; + retardation of motion by tidal friction, 269, 270; + origin of, 282, 283; + rotation annulled by tidal friction and present libration, 286; + inequality in motion indicates internal density of earth, 302, 303; + eccentricity of orbit increased by tidal friction, 313, 314. + +Moore, Captain#Moore, + illustrations of bore, 67; + survey of Tsien-Tang-Kiang, 60-70. + +Neap and spring tides#neap + in equilibrium theory, 159; + represented by principal lunar and solar tides, 204. + +Nebula in Andromeda#Andromeda, 339. + +Nebulae@Nebulæ, description of various#Nebulæ, 345. + +Nebular hypothesis, 334-339. + +Newcomb, S., theoretical explanation of Chandler's nutation#Newcomb, 254. + +Newton, + founder of tidal theory, 86; + theory of tide in equatorial canal, 172. + +Nolan, criticism of tidal theory of moon's origin#Nolan, 360. + +Nutation, + value of, indicates internal density of earth, 303; + Chandler's, 251-256. + +Obliquity of ecliptic, effects of tidal friction on#obliquity, 310-312. + +Observation, + methods of tidal, 6-14; + reduction of tidal, 211-220. + +Orbit + of moon and earth, 93-95; + of double stars, very eccentric, 313. + +Pacific Ocean, tide in, affects Atlantic#Pacific, 186, 187. + +Partial tides in harmonic method#partial tides, 199. + +Paschwitz, von Rebeur#Paschwitz, + on horizontal pendulum, 130-132; + tidal deflection of vertical at Wilhelmshaven, 144. + +Pendulum, + curves traced by, under tidal force, 111, 112; + bifilar, 115-125; + as seismological instrument, 126, 127; + horizontal, 130-132. +\PageSep{376} + +Petitcodiac, bore in the#Petitcodiac, 71. + +Phobos, a satellite of Mars#Phobos, 297. + +Planetary figure of equilibrium of rotating liquid#planetary figure, 322. + +Planets, + rotation of some, annulled by tidal friction, 298; + figures and internal densities, 332, 333. + +Plasticity of earth under change of rotation#plasticity, 300-302. + +Plateau, experiment on figure of rotating globule#Plateau, 316-319. + +Plemyrameter, observation of seiches with#plemyrameter, 19-22. + +Poincaré, + law of interchange of stability, 326, 327; + figure of rotating liquid, 325, 327. + +Polibius on tides at Cadiz#Polibus, 83. + +Portsmouth, table of errors in tidal predictions#Portsmouth, 244. + +Posidonius on tides#Posidonius, 81-84. + +Precession, value of, indicates internal density of earth#internal density, 303. + +Predicting machine for tides, 233-241; + Ferrel's, 241. + +Prediction of tide, + due to ideal satellite, 200; + example at Aden, 226-230; + method of computing, 230-233; + errors in, 242-250. + +Pressure of atmosphere, elastic distortion of soil by#distortion, 145, 146. + +Principle of forced oscillations#forced oscillations, 169, 170. + +Rebeur. |see{Paschwitz}. 0 + +Reduction of tidal observations, 211-220. + +Retardation of earth's rotation, 268. + +Rigidity of earth#rigidity, 256-260. + +Ripple mark in sand preserved in geological strata#ripple mark, 305. + +Rivers, + tide wave in, 55-59; + Airy on tide in, 75; + annual meteorological tide in, 206. + +Roberts, E., the tide-predicting machine#Roberts, 233. + +Roberts, I., photograph of nebula in Andromeda#Roberts, 339. + +Roche, E.#Roche, + ellipticity of internal strata of earth, 303; + theory of limit and Saturn's ring, 356-362; + stability of ellipsoid of, 369. + +Roman description of tides#Roman, 81-85. + +Rossi on Italian seismology#Rossi, 128-130. + +Rotating liquid, figures of equilibrium#rotating liquid, 316-333. + +Rotation + of earth involved in tidal problem, 177; + retarded by tidal friction, 268; + of moon annulled by tidal friction, 286; + of Mercury, Venus, and satellites of Jupiter and Saturn annulled by tidal friction, 298. + +Russell, observation of seiches in New South Wales#Russell, 47. + +Saint@St.\ Vénant on flow of solids#Vénant, 313. + +Satellites, + tide due to single equatorial, 195, 196; + ideal replacing sun and moon in harmonic analysis, 199, 200; + discovery of those of Mars, 290-298; + rotation of those of Jupiter and Saturn annulled, 298; + distribution of, in solar system, 339-341. + +Saturn, + satellites always face the planet, 298; + law of density and figure, 332; + description and picture, 347-354; + theory of ring, 356-369; + Roche's limit for, 360. + +Schedule for reducing tidal observations, 215, 216. + +Schiaparelli on rotation of Venus and Mercury#Schiaparelli, 298, 315. + +Schwarzschild, + exposition of Poincaré's theory, 333; + stability of Roche's ellipsoid, 369. + +Sea, + vibrations of, 44, 45; + level affected by atmospheric pressure, 146. +\PageSep{377} + +See, T. J. J., + eccentricity of orbits of double stars, 313; + theory of evolution of double stars, 342-346. + +Seiches, + definition, 17; + records of, 21; + longitudinal and transverse, 25-27; + periods of, 27; + causes of, 39, 40. + +Seine, bore in the#Seine, 71. + +Seismology, 133. + +Seleucus, observation of tides of Indian Ocean#Seleucus, 84, 85. + +Semidiurnal tide + in equilibrium theory, 153-156; + in harmonic method, 201-204. + +Severn, bore in the#Severn, 71. + +Slope of soil + due to elastic distortion, 136; + calculation and illustration of, 138-140. + +Solar + tide-generating force compared with lunar, 156-158; + principal tide, 202; + possible effect of tide in assisting birth of moon, 284, 285; + system, nebular hypothesis as to origin of, 334-339; + system, distribution of satellites in, 339-341. + +Spectroscopic proof of rotation of Saturn's ring#spectroscop, 368, 369. + +Spring and neap tides + in equilibrium theory, 159; + represented by principal lunar and solar tides, 203. + +Stability, + nature of dynamical, 280, 281; + of figures of equilibrium, 322, 323; + of Saturn's ring, 365, 366. + +Stars, + double, eccentricity of orbits, 313; + theory of evolution, 342-346. + +Storms a cause of seiches#storms, 39, 40. + +Strabo on tides#Strabo, 81-85. + +Stupart coöperates in investigation of seiches#Stupart, 48. + +Sun, + tide-generating force of, compared with that of moon, 156-158; + ideal, replacing real sun in harmonic analysis, 201; + possible influence of, in assisting birth of moon, 284, 285. + +Surface tension of liquids#surface tension, 317, 318. + +Swift, satire on mathematicians#Swift, 292-295. + +Synthesis of partial tides for prediction#synthesis, 230-233. + +Tables, tide#tide tables, 221-241; + method of calculating, 230-241; + amount of error in, 246, 247. + +Thomson, Sir W. |see{Kelvin}. 0 + +Tidal problem. |see{Laplace, Harmonic Analysis, etc.} 0 + +Tide, + definition, 1-3; + general description, 4-6. + +Tide, |see{also other headings; \eg\ for tide-generating force, |see{Force}}. 0 + +Time, + lunar, 213; + requisite for evolution of moon, 285. + +Tisserand, Roche's investigations as to earth's figure#Tisserand, 315. + +Tremors, earth#tremors, 125. + +Tresca on flow of solids#Tresca, 300. + +Tromometer, a seismological instrument#tromometer, 126, 127. + +Tsien-Tang-Kiang, the bore in#Tsien, 60-70. + +United States Coast Survey, + method of reducing tidal observations, 217; + tide tables of, 222. + +Variation of latitude, 251-256. + +Vaucher, record of a great seiche at Geneva#Vaucher, 17. + +Venus, rotation of#Venus, 298, 299. + +Vertical. |see{Deflection}. 0 + +Vibration of lakes, 41-53. + +Voltaire, satire on mathematicians, and Martian satellites#Voltaire, 295, 296. + +Vortical motion in oceanic tides#vorticial, 177, 178. +\PageSep{378} + +Waves + in deep and shallow water, 29; + speed of, 31; + composition of, 33-37; + in atmosphere, 48-50; + forced and free, 164; + of tide in equatorial canal, 173; + in canal in high latitude, 174-176; + propagated northward in Atlantic, 186-188. + +Wharton, Sir W. J., illustration of bore#Wharton, 69. + +Whewell + on tides, 87; + empirical construction of tide tables, 87-90; + on cotidal charts, 188, 189. + +Wind, + a cause of seiches, 39; + vibrations of lakes due to, 41, 42; + a cause of meteorological tides, 206; + perturbation of, in tidal prediction, 242, 243. + +Woodward on variation of latitude#Woodward, 262. + +Wright, Thomas, on a theory of cosmogony#Wright, 335. + +Wye, bore in the#Wye, 71. +\fi +\PageSep{379} +\clearpage +\null\vfill +\begin{center} +\textgoth{The Riverside Press} + +\footnotesize +CAMBRIDGE, MASSACHUSETTS, U. S. A. \\ +ELECTROTYPED AND PRINTED BY \\ +H. O. 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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.net % +% % +% % +% Title: The Tides and Kindred Phenomena in the Solar System % +% The Substance of Lectures Delivered in 1897 at the Lowell % +% Institute, Boston, Massachusetts % +% % +% Author: Sir George Howard Darwin % +% % +% Release Date: January 31, 2012 [EBook #38722] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK TIDES AND KINDRED PHENOMENA *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{38722} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.net + + +Title: The Tides and Kindred Phenomena in the Solar System + The Substance of Lectures Delivered in 1897 at the Lowell + Institute, Boston, Massachusetts + +Author: Sir George Howard Darwin + +Release Date: January 31, 2012 [EBook #38722] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK TIDES AND KINDRED PHENOMENA *** +\end{PGtext} +\end{minipage} +\end{center} +\clearpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Bryan Ness and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/Canadian Libraries) +\end{PGtext} +\end{minipage} +\vfill +\TranscribersNote{\TransNoteText} +\end{center} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\cleardoublepage +\pagenumbering{roman} +\PageSep{iii} +\begin{center} +\textbf{\Huge THE TIDES} +\vfil +\textbf{\large AND KINDRED PHENOMENA IN THE \\[4pt] +SOLAR SYSTEM} +\vfil +{\footnotesize THE SUBSTANCE OF LECTURES DELIVERED \\[4pt] +IN 1897 AT THE LOWELL INSTITUTE, \\[4pt] +BOSTON, MASSACHUSETTS} +\vfil +BY +\vfil +\textbf{\Large GEORGE HOWARD DARWIN} +\vfil +\scriptsize PLUMIAN PROFESSOR AND FELLOW OF TRINITY COLLEGE IN THE \\[4pt] +UNIVERSITY OF CAMBRIDGE +\vfil\vfil +\Graphic[png]{1.25in}{riverside} +\vfil\vfil\vfil +\footnotesize BOSTON AND NEW YORK \\ +\normalsize HOUGHTON, MIFFLIN AND COMPANY \\ +\textgoth{The Riverside Press, Cambridge} \\ +1899 +\end{center} +\PageSep{iv} +\clearpage +\null\vfill +\begin{center} +\footnotesize +COPYRIGHT, 1898, BY GEORGE HOWARD DARWIN \\ +ALL RIGHTS RESERVED +\end{center} +\vfill +\PageSep{v} + + +\Preface + +\First{In} 1897 I delivered a course of lectures on +the Tides at the Lowell Institute in Boston, +Massachusetts, and this book contains the substance +of what I then said. The personal form +of address appropriate to a lecture is, I think, +apt to be rather tiresome in a book, and I have +therefore taken pains to eliminate all traces of +the lecture from what I have written. + +A mathematical argument is, after all, only +organized common sense, and it is well that men +of science should not always expound their work +to the few behind a veil of technical language, +but should from time to time explain to a larger +public the reasoning which lies behind their +mathematical notation. To a man unversed in +popular exposition it needs a great effort to shell +away the apparatus of investigation and the +technical mode of speech from the thing behind +it, and I owe a debt of gratitude to Mr.~Lowell, +trustee of the Institute, for having afforded me +the occasion for making that effort. +\PageSep{vi} + +It is not unlikely that the first remark of +many who see my title will be that so small a +subject as the Tides cannot demand a whole volume; +but, in fact, the subject branches out in +so many directions that the difficulty has been to +attain to the requisite compression of my matter. +Many popular works on astronomy devote a few +pages to the Tides, but, as far as I know, none +of these books contain explanations of the practical +methods of observing and predicting the +Tides, or give any details as to the degree of +success attained by tidal predictions. If these +matters are of interest, I invite my readers not +to confine their reading to this preface. The +later chapters of this book are devoted to the +consideration of several branches of speculative +Astronomy, with which the theory of the Tides +has an intimate relationship. The problems involved +in the origin and history of the solar +and of other celestial systems have little bearing +upon our life on the earth, yet these questions +can hardly fail to be of interest to all those +whose minds are in any degree permeated by +the scientific spirit. + +I think that there are many who would like to +understand the Tides, and will make the attempt +to do so provided the exposition be sufficiently +\PageSep{vii} +simple and clear; it is to such readers I address +this volume. It is for them to say how far I +have succeeded in rendering these intricate subjects +interesting and intelligible, but if I have +failed it has not been for lack of pains. + +The figures and diagrams have, for the most +part, been made by Mr.~Edwin Wilson of Cambridge, +but I have to acknowledge the courtesy +of the proprietors of \Title{Harper's}, the \Title{Century}, +and the \Title{Atlantic Monthly} magazines, in supplying +me with some important illustrations. + +A considerable portion of \Ref{Chapter}{III}.\ on the +``Bore'' is to appear as an article in the \Title{Century +Magazine} for October, 1898, and the reproductions +of Captain Moore's photographs of the +``Bore'' in the Tsien-Tang-Kiang have been +prepared for that article. The \Title{Century} has also +kindly furnished the block of Dr.~Isaac Roberts's +remarkable photograph of the great nebula in +the constellation of Andromeda; it originally +appeared in an article on Meteorites in the number +for October,~1890. The greater portion of +the text and the whole of the illustrations of +\Ref{Chapter}{XX}.\ were originally published in \Title{Harper's +Magazine} for June,~1889. Lastly, portions +of Chapters \Ref{}{XV}.~and~\Ref{}{XVI}.\ appeared in +the \Title{Atlantic Monthly} for April, 1898, published +\PageSep{viii} +by Messrs.\ Houghton, Mifflin~\&~Co., who also +make themselves responsible for the publication +of the American edition of this book. + +In conclusion, I wish to take this opportunity +of thanking my American audience for the cordiality +of their reception, and my many friends +across the Atlantic for their abundant hospitality +and kindness. + +\Signature{G. H. DARWIN.} +{\textsc{Cambridge}, \textit{August}, 1898.} +\PageSep{ix} + + +\Contents + +\ToCChap{I} +{TIDES AND METHODS OF OBSERVATION} + +\ToCSect{Definition of tide}{\PgNos{1}{3}} +\ToCSect{Oceanic tides}{\PgNos[,]{4}{5}} +\ToCSect{Methods of observation}{\PgNo{6}} +\ToCSect{Tide-gauge}{\PgNos{7}{12}} +\ToCSect{Tide-curve}{\PgNo{12}} +\ToCSect{Site for tide-gauge}{\PgNo{13}} +\ToCSect{Irregularities in tide-curve}{\PgNos[,]{14}{15}} +\ToCSect{Authorities}{\PgNo{16}} + + +\ToCChap{II} +{SEICHES IN LAKES} + +\ToCSect{Meaning of seiche}{\PgNo{17}} +\ToCSect{Uses of scientific apparatus}{\PgNo{18}} +\ToCSect{Forel's plemyrameter}{\PgNos[,]{19}{20}} +\ToCSect{Records of the level of the lake}{\PgNo{20}} +\ToCSect{Interpretation of record}{\PgNos{21}{23}} +\ToCSect{Limnimeter}{\PgNo{24}} +\ToCSect{Mode of oscillation in seiches}{\PgNos{25}{28}} +\ToCSect{Wave motion in deep and in shallow water}{\PgNos{29}{32}} +\ToCSect{Composition of waves}{\PgNos{32}{36}} +\ToCSect{Periods of seiches}{\PgNos[,]{37}{38}} +\ToCSect{Causes of seiches}{\PgNos[,]{39}{40}} +\ToCSect{Vibrations due to wind and to steamers}{\PgNos{41}{47}} +\ToCSect{Aerial waves and their action on lakes and on the sea}{\PgNos{48}{53}} +\ToCSect{Authorities}{\PgNos[,]{53}{54}} +\PageSep{x} + + +\ToCChap{III} +{TIDES IN RIVERS---TIDE MILLS} + +\ToCSect{Definition of ebb and flow}{\PgNo{56}} +\ToCSect{Tidal currents in rivers}{\PgNos{56}{58}} +\ToCSect{Progressive change of wave in shallow water}{\PgNos[,]{58}{59}} +\ToCSect{The bore}{\PgNo{59}} +\ToCSect{Captain Moore's survey of the Tsien-Tang-Kiang}{\PgNos{60}{64}} +\ToCSect{Diagram of water-levels during the flow}{\PgNos[,]{64}{65}} +\ToCSect{Chinese superstition}{\PgNos{68}{71}} +\ToCSect{Pictures of the bore}{\PgNo{69}} +\ToCSect{Other cases of bores}{\PgNo{71}} +\ToCSect{Causes of the bore}{\PgNo{72}} +\ToCSect{Tidal energy}{\PgNos[,]{73}{74}} +\ToCSect{Tide mills}{\PgNos[,]{74}{75}} +\ToCSect{Authorities}{\PgNo{75}} + + +\ToCChap{IV} +{HISTORICAL SKETCH} + +\ToCSect{Theories of the Chinese}{\PgNos[,]{76}{77}} +\ToCSect{Theories of the Arabs}{\PgNos{77}{79}} +\ToCSect{Theories of the Norsemen}{\PgNos[,]{79}{80}} +\ToCSect{Writings of Posidonius and Strabo}{\PgNos{80}{84}} +\ToCSect{Seleucus the Babylonian on the diurnal inequality}{\PgNos[,]{84}{85}} +\ToCSect{Galileo and Kepler}{\PgNo{85}} +\ToCSect{Newton and his successors}{\PgNos{86}{88}} +\ToCSect{Empirical method of tidal prediction}{\PgNos{88}{90}} +\ToCSect{Authorities}{\PgNo{90}} + + +\ToCChap{V} +{TIDE-GENERATING FORCE} + +\ToCSect{Inertia and centrifugal force}{\PgNos{91}{93}} +\ToCSect{Orbital motion of earth and moon}{\PgNos{93}{95}} +\ToCSect{Tide-generating force}{\PgNos{96}{100}} +\ToCSect{Law of its dependence on the moon's distance}{\PgNos{101}{103}} +\ToCSect{Earth's rotation}{\PgNos[,]{103}{104}} +\PageSep{xi} +\ToCSect{Second explanation of tide-generating force}{\PgNos[,]{104}{105}} +\ToCSect{Horizontal tide-generating force}{\PgNos[,]{105}{106}} +\ToCSect{Successive changes of force in the course of a day}{\PgNos[,]{107}{108}} +\ToCSect{Authorities}{\PgNo{108}} + + +\ToCChap{VI} +{DEFLECTION OF THE VERTICAL} + +\ToCSect{Deflection of a pendulum by horizontal tidal force}{\PgNos{109}{111}} +\ToCSect{Path pursued by a pendulum under tidal force}{\PgNos{111}{113}} +\ToCSect{Object of measuring the deflection of a pendulum}{\PgNos{113}{115}} +\ToCSect{Attempt to measure deflection by bifilar pendulum}{\PgNos{115}{125}} +\ToCSect{Microsisms}{\PgNos{125}{127}} +\ToCSect{The microphone as a seismological instrument}{\PgNos{127}{130}} +\ToCSect{Paschwitz's work with the horizontal pendulum}{\PgNos[,]{131}{132}} +\ToCSect{Supposed measurement of the lunar deflection of gravity}{\PgNo{132}} +\ToCSect{Authorities}{\PgNo{133}} + + +\ToCChap{VII} +{THE ELASTIC DISTORTION OF THE EARTH'S SURFACE BY +VARYING LOADS} + +\ToCSect{Distortion of an elastic surface by superincumbent load}{\PgNos{134}{137}} +\ToCSect{Application of the theory to the earth}{\PgNos[,]{137}{138}} +\ToCSect{Effects of tidal load}{\PgNos{138}{140}} +\ToCSect{Probable deflections at various distances from the coast}{\PgNos{140}{143}} +\ToCSect{Deflections observed by M.~d'Abbadie and by Dr.~Paschwitz}{\PgNos{143}{145}} +\ToCSect{Effects of atmospheric pressure on the earth's surface}{\PgNos{145}{147}} +\ToCSect{Authorities}{\PgNo{148}} + + +\ToCChap{VIII} +{EQUILIBRIUM THEORY OF TIDES} + +\ToCSect{Explanation of the figure of equilibrium}{\PgNos{149}{151}} +\ToCSect{Map of equilibrium tide}{\PgNos{151}{153}} +\ToCSect{Tides according to the equilibrium theory}{\PgNos{153}{156}} +\ToCSect{Solar tidal force compared with lunar}{\PgNos{156}{158}} +\PageSep{xii} +\ToCSect{Composition of lunar and solar tides}{\PgNos[,]{158}{159}} +\ToCSect{Points of disagreement between theory and fact}{\PgNos{159}{162}} +\ToCSect{Authorities}{\PgNo{162}} + + +\ToCChap{IX} +{DYNAMICAL THEORY OF THE TIDE WAVE} + +\ToCSect{Free and forced waves in an equatorial canal}{\PgNos{163}{165}} +\ToCSect{Critical depth of canal}{\PgNos{165}{167}} +\ToCSect{General principle as to free and forced oscillations}{\PgNos{167}{174}} +\ToCSect{Inverted and direct oscillation}{\PgNos[,]{172}{173}} +\ToCSect{Canal in latitude~$60°$}{\PgNos[,]{174}{175}} +\ToCSect{Tides where the planet is partitioned into canals}{\PgNo{175}} +\ToCSect{Removal of partitions; vortical motion of the water}{\PgNos[,]{176}{177}} +\ToCSect{Critical latitude where the rise and fall vanish}{\PgNos[,]{177}{178}} +\ToCSect{Diurnal inequality}{\PgNos{178}{180}} +\ToCSect{Authorities}{\PgNo{181}} + + +\ToCChap{X} +{TIDES IN LAKES---COTIDAL CHART} + +\ToCSect{The tide in a lake}{\PgNos{182}{185}} +\ToCSect{The Mediterranean Sea}{\PgNos[,]{185}{186}} +\ToCSect{Derived tide of the Atlantic}{\PgNos{186}{188}} +\ToCSect{Cotidal chart}{\PgNos{188}{192}} +\ToCSect{Authorities}{\PgNo{192}} + + +\ToCChap{XI} +{HARMONIC ANALYSIS OF THE TIDE} + +\ToCSect{Tide in actual oceans due to single equatorial satellite}{\PgNos{193}{196}} +\ToCSect{Substitution of ideal satellites for the moon}{\PgNos{197}{199}} +\ToCSect{Partial tide due to each ideal satellite}{\PgNos{199}{201}} +\ToCSect{Three groups of partial tides}{\PgNo{201}} +\ToCSect{Semidiurnal group}{\PgNos{201}{204}} +\ToCSect{Diurnal group}{\PgNos{204}{206}} +\ToCSect{Meteorological tides}{\PgNos[,]{206}{207}} +\ToCSect{Shallow water tides}{\PgNos{207}{210}} +\ToCSect{Authorities}{\PgNo{210}} +\PageSep{xiii} + + +\ToCChap{XII} +{REDUCTION OF TIDAL OBSERVATIONS} + +\ToCSect{Method of singling out a single partial tide}{\PgNos{211}{214}} +\ToCSect{Variety of plans adopted}{\PgNos{214}{217}} +\ToCSect{Tidal abacus}{\PgNos{217}{220}} +\ToCSect{Authorities}{\PgNo{220}} + + +\ToCChap{XIII} +{TIDE TABLES} + +\ToCSect{Definition of special and general tables}{\PgNo{221}} +\ToCSect{Reference to moon's transit}{\PgNos[,]{222}{223}} +\ToCSect{Examples at Portsmouth and at Aden}{\PgNos{223}{228}} +\ToCSect{General inadequacy of tidal information}{\PgNos[,]{229}{230}} +\ToCSect{Method of calculating tide tables}{\PgNos{230}{233}} +\ToCSect{Tide-predicting machine}{\PgNos{233}{241}} +\ToCSect{Authorities}{\PgNo{241}} + + +\ToCChap{XIV} +{THE DEGREE OF ACCURACY OF TIDAL PREDICTION} + +\ToCSect{Effects of wind and barometric pressure}{\PgNos[,]{242}{243}} +\ToCSect{Errors at Portsmouth}{\PgNos[,]{243}{244}} +\ToCSect{Errors at Aden}{\PgNos{245}{249}} +\ToCSect{Authorities}{\PgNo{250}} + + +\ToCChap{XV} +{CHANDLER'S NUTATION---THE RIGIDITY OF THE EARTH} + +\ToCSect{Nutation of the earth and variation of latitude}{\PgNos{251}{254}} +\ToCSect{Elasticity of the earth}{\PgNos[,]{254}{255}} +\ToCSect{Tide due to variation of latitude}{\PgNos[,]{255}{256}} +\ToCSect{Rigidity of the earth}{\PgNos{256}{260}} +\ToCSect{Transmission of earthquake shocks}{\PgNos[,]{261}{262}} +\ToCSect{Authorities}{\PgNos[,]{262}{263}} +\PageSep{xiv} + + +\ToCChap{XVI} +{TIDAL FRICTION} + +\ToCSect{Friction retards the tide}{\PgNos{264}{267}} +\ToCSect{Retardation of planetary rotation}{\PgNos{267}{269}} +\ToCSect{Reaction on the satellite}{\PgNos{269}{272}} +\ToCSect{Ancient eclipses of the sun}{\PgNos[,]{272}{273}} +\ToCSect{Law of variation of tidal friction with moon's distance}{\PgNos{273}{275}} +\ToCSect{Transformations of the month and of the day}{\PgNos{275}{280}} +\ToCSect{Initial and final conditions of motion}{\PgNos[,]{280}{281}} +\ToCSect{Genesis of the moon}{\PgNos{281}{285}} +\ToCSect{Minimum time requisite for the evolution}{\PgNos[,]{285}{286}} +\ToCSect{Rotation of the moon}{\PgNos[,]{286}{287}} +\ToCSect{The month ultimately to be shorter than the day}{\PgNos{287}{289}} + + +\ToCChap{XVII} +{TIDAL FRICTION (\textit{continued})} + +\ToCSect{Discovery of the Martian satellites}{\PgNos{290}{298}} +\ToCSect{Rotation of Mercury, of Venus, and of the Jovian satellites}{\PgNos[,]{298}{299}} +\ToCSect{Adaptation of the earth's figure to changed rotation}{\PgNos{299}{302}} +\ToCSect{Ellipticity of the internal strata of the earth}{\PgNos{302}{304}} +\ToCSect{Geological evidence}{\PgNos{304}{306}} +\ToCSect{Distortion of a plastic planet and trend of continents}{\PgNos{306}{308}} +\ToCSect{Obliquity of the ecliptic}{\PgNos{308}{312}} +\ToCSect{Eccentricity of lunar orbit}{\PgNos[,]{312}{313}} +\ToCSect{Eccentricity of the orbits of double stars}{\PgNo{313}} +\ToCSect{Plane of the lunar orbit}{\PgNos[,]{313}{314}} +\ToCSect{Short summary}{\PgNos[,]{314}{315}} +\ToCSect{Authorities}{\PgNo{315}} + + +\ToCChap{XVIII} +{THE FIGURES OF EQUILIBRIUM OF A ROTATING MASS OF +LIQUID} + +\ToCSect{Plateau's experiment}{\PgNos{316}{319}} +\ToCSect{Stability of a celestial sphere of liquid}{\PgNos{319}{321}} +\ToCSect{The two ellipsoids of Maclaurin and that of Jacobi}{\PgNos{321}{323}} +\PageSep{xv} +\ToCSect{Transitions with change of rotation}{\PgNos[,]{323}{324}} +\ToCSect{Coalescence of Jacobi's with Maclaurin's ellipsoid}{\PgNos{324}{326}} +\ToCSect{Poincaré's law of stability and coalescence}{\PgNos[,]{326}{327}} +\ToCSect{Poincaré's pear-shaped figure}{\PgNos[,]{327}{328}} +\ToCSect{Hour-glass figure of equilibrium}{\PgNos{328}{332}} +\ToCSect{Figures of planets}{\PgNos[,]{332}{333}} +\ToCSect{Authorities}{\PgNo{333}} + + +\ToCChap{XIX} +{THE EVOLUTION OF CELESTIAL SYSTEMS} + +\ToCSect{The Nebular Hypothesis}{\PgNos{334}{338}} +\ToCSect{Nebula in Andromeda}{\PgNos[,]{338}{339}} +\ToCSect{Distribution of satellites in the solar system}{\PgNos{339}{341}} +\ToCSect{Genesis of celestial bodies by fission}{\PgNo{342}} +\ToCSect{Dr.~See's speculations as to systems of double stars}{\PgNos{342}{344}} +\ToCSect{Diversity of celestial bodies}{\PgNos{344}{346}} +\ToCSect{Authorities}{\PgNo{346}} + + +\ToCChap{XX} +{SATURN'S RINGS} + +\ToCSect{Description}{\PgNos[,]{347}{348}} +\ToCSect{Discovery of Saturn's rings}{\PgNos{348}{352}} +\ToCSect{Diagram of the rings}{\PgNos{353}{356}} +\ToCSect{Roche's investigation}{\PgNos[,]{356}{357}} +\ToCSect{Roche's limit}{\PgNos{358}{360}} +\ToCSect{The limit for the several planets}{\PgNos[,]{360}{361}} +\ToCSect{Meteoric constitution of Saturn's rings}{\PgNos[,]{361}{362}} +\ToCSect{Maxwell's investigations}{\PgNos{362}{367}} +\ToCSect{Spectroscopic examination of the rings}{\PgNos{367}{369}} +\ToCSect{Authorities}{\PgNo{369}} +\PageSep{xvi} +%[Blank Page] +\PageSep{xvii} + + +% [** TN: Plates (Figs. 23, 33, and 40) listed separately in the original; +% omitted separate "FULL-PAGE" and "IN TEXT" headings, folded plate +% references into main illustration list.] +% LIST OF ILLUSTRATIONS +\Illustrations + +\LoFFig{1}{Well for Tide-Gauge}{7} +\LoFFig{2}{Pipe of Tide-Gauge}{9} +\LoFFig{3}{Indian Tide-Gauge}{10} +\LoFFig{4}{Légé's Tide-Gauge}{11} +\LoFFig{5}{Bombay Tide-Curve from Noon, April 22, to Noon, April 30, 1884}{14} +\LoFFig{6}{Sites for a Tide-Gauge}{15} +\LoFFig{7}{Plemyrameter}{20} +\LoFFig{8, 9}{Records of Seiches at Évian}{23} +\LoFFig{10}{Map of Lake of Geneva}{26} +\LoFFig{11}{Wave in Deep Water}{30} +\LoFFig{12}{Wave in Shallow Water}{31} +\LoFFig{13}{Simple Wave}{33} +\LoFFig{}{Composition of Two Equal and Opposite Waves}{34} +\LoFFig{14}{Vibrations due to Steamers}{45} +\LoFFig{15}{Progressive Change of a Wave in Shallow Water}{59} +\LoFFig{16}{Chart of the Estuary of the Tsien-Tang-Kiang}{61} +\LoFFig{17}{Bore-Shelter on the Tsien-Tang-Kiang}{64} +\LoFFig{18}{Diagram of the Flow of the Tide on the Tsien-Tang-Kiang}{66} +\LoFFig{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{69} +\LoFFig{20}{Earth and Moon}{93} +\PageSep{xviii} +\LoFFig{21}{Revolution of a Body without Rotation}{98} +\LoFFig{22}{Tide-generating Force}{100} +\LoFFig{23}{Horizontal Tide-generating Force}{106} +\LoFFig{24}{Deflection of a Pendulum; the Moon and Observer on the Equator}{111} +\LoFFig{25}{Deflection of a Pendulum; the Moon in N. Declination~$15°$, +the Observer in N. Latitude~$30°$}{112} +\LoFFig{26}{Bifilar Pendulum}{115} +\LoFFig{27}{Form of Dimple in an Elastic Surface}{135} +\LoFFig{28}{Distortion of Land and Sea-Bed by Tidal Load}{139} +\LoFFig{29}{Chart of Equilibrium Tides}{152} +\LoFFig{30}{Forced Oscillations of a Pendulum}{171} +\LoFFig{31}{The Tide in a Lake}{184} +\LoFFig{32}{Chart of Cotidal Lines}{190} +\LoFFig{33}{Tidal Abacus}{218} +\LoFFig{34}{Curves of Intervals and Heights at Portsmouth and at Aden}{227} +\LoFFig{35}{Diagram of Tide-predicting Instrument}{235} +\LoFFig{36}{Frictionally Retarded Tide}{266} +\LoFFig{37}{Maclaurin's and Jacobi's Ellipsoids of Equilibrium}{323} +\LoFFig{38}{Figures of Equilibrium}{325} +\LoFFig{39}{Hour-glass Figure of Equilibrium}{329} +\LoFFig{40}{Nebula in Andromeda}{339} +\LoFFig{41}{The Planet Saturn}{349} +\LoFFig{42}{Diagram of Saturn and his Rings}{353} +\LoFFig{43}{Roche's Figure of a Satellite when elongated to the utmost}{357} + +\MainMatter +\PageSep{1} +\index{Barometric pressure|see{Atmospheric pressure}}% +\index{Bifilar|see{Pendulum}}% +\index{D'Abbadie|see{Abbadie}}% +\index{Horizontal tide-generating force|see{also Pendulum}}% +\index{Rebeur|see{Paschwitz}}% +\index{Thomson, Sir W.|see{Kelvin}}% +\index{Tidal problem|see{Laplace, Harmonic Analysis, etc.}}% +\index{Tide|see{also other headings; \eg\ for tide-generating force \textit{see} Force}}% +\index{Vertical|see{Deflection}}% +% [** TN: Text printed by \Chapter macro] +% THE TIDES + + +\Chapter{I} +{Tides and Methods of Observation} + +\First{The} great wave caused by an earthquake is +\index{Tide!definition|(}% +often described in the newspapers as a tidal +wave, and the same name is not unfrequently +applied to such a short series of enormous waves +as is occasionally encountered by a ship in the +open sea. We must of course use our language +in the manner which is most convenient, but as +in this connection the adjective ``tidal'' implies +simply greatness and uncommonness, the use of +the term in such a sense cannot be regarded as +appropriate. + +The word ``tidal'' should, I think, only be +used when we are referring to regular and persistent +alternations of rise and fall of sea-level. +Even in this case the term may perhaps be used +in too wide a sense, for in many places there is a +regular alternation of the wind, which blows in-shore +during the day and out during the night +with approximate regularity, and such breezes +\PageSep{2} +alternately raise and depress the sea-level, and +thus produce a sort of tide. Then in the Tropics +there is a regularly alternating, though small, +periodicity in the pressure of the atmosphere, +which is betrayed by an oscillation in the height +of the barometer. Now the ocean will respond +to the atmospheric pressure, so that the sea-level +will fall with a rising barometer, and rise +with a falling barometer. Thus a regularly +periodic rise and fall of the sea-level must result +from this cause also. Again, the melting of the +snows in great mountain ranges, and the annual +variability in rainfall and evaporation, produce +approximately periodic changes of level in the +estuaries of rivers, and although the period of +these changes is very long, extending as they do +over the whole year, yet from their periodicity +they partake of the tidal character. + +These changes of water level are not, however, +tides in the proper sense of the term, and a true +tide can only be adequately defined by reference +to the causes which produce it. A tide, in fact, +means a rising and falling of the water of the +ocean caused by the attractions of the sun and +moon. + +Although true tides are due to astronomical +causes, yet the effects of regularly periodic winds, +variation of atmospheric pressure, and rainfall +are so closely interlaced with the true tide that +in actual observation of the sea it is necessary to +\PageSep{3} +consider them both together. It is accordingly +practically convenient to speak of any regular +alternation of sea-level, due to the wind and to +the other influences to which I have referred, as +a Meteorological Tide. The addition of the adjective +``meteorological'' justifies the use of the +term ``tide'' in this connection. + +We live at the bottom of an immense sea of +air, and if the attractions of the sun and moon +affect the ocean, they must also affect the air. +This effect will be shown by a regular rise and +fall in the height of the barometer. Although +such an effect is undoubtedly very small, yet it +is measurable. The daily heating of the air by +the sun, and its cooling at night, produce marked +alternations in the atmospheric pressure, and this +effect may by analogy be called an atmospheric +meteorological tide. + +The attractions of the moon and sun must certainly +act not only on the sea, but also on the +solid earth; and, since the earth is not perfectly +rigid or stiff, they must produce an alternating +change in its shape. Even if the earth is now +so stiff that the changes in its shape escape +detection through their minuteness, yet such +changes of shape must exist. There is much +evidence to show that in the early stages of their +histories the planets consisted largely or entirely +of molten rock, which must have yielded to tidal +influences. I shall, then, extend the term ``tide'' +\index{Tide!definition|)}% +\PageSep{4} +so as to include such alternating deformations of +a solid and elastic, or of a molten and plastic, +globe. These corporeal tides will be found to +lead us on to some far-reaching astronomical +speculations. The tide, in the sense which I +\index{Tide!general description|(}% +have attributed to the term, covers a wide field +of inquiry, and forms the subject of the present +volume. + +I now turn to the simplest and best known +form of tidal phenomena. When we are at the +seashore, or on an estuary, we see that the water +rises and falls nearly twice a day. To be more +exact, the average interval from one high water +to the next is twelve hours twenty-five minutes, +and so high water falls later, according to the +clock, by twice twenty-five minutes, or by fifty +minutes, on each successive day. Thus if high +water falls to-day at noon, it will occur to-morrow +at ten minutes to one. Before proceeding, it +may be well to remark that I use high water and +low water as technical terms. In common parlance +the level of water may be called high or +low, according as whether it is higher or lower +than usual. But when the level varies periodically, +there are certain moments when it is highest +and lowest, and these will be referred to as +the times of high and low water, or of high and +of low tide. In the same way I shall speak of +the heights at high and low water, as denoting +the water-level at the moments in question. +\PageSep{5} + +The most elementary observations would show +that the time of high water has an intimate relationship +to the moon's position. The moon, in +fact, passes the meridian on the average fifty +minutes later on each succeeding day, so that if +high water occurs so many hours after the moon +is due south on any day, it will occur on any +other day about the same number of hours after +the moon was south. This rule is far from being +exact, for it would be found that the interval +from the moon's passage to high water differs +considerably according to the age of the moon. +I shall not, however, attempt to explain at present +how this rough rule as to the time of high +water must be qualified, so as to convert it into +an accurate statement. + +But it is not only the hour of high water which +changes from day to day, for the height to which +the water rises varies so conspicuously that the +fact could not escape the notice of even the most +casual observer. It would have been necessary +to consult a clock to discover the law by which +the hour of high water changes from day to day; +but at the seashore it would be impossible to +avoid noticing that some rocks or shoals which +are continuously covered by the sea at one part +of a fortnight are laid bare at others. It is, in +fact, about full and new moon that the range +from low to high water is greatest, and at the +moon's first or third quarter that the range is +\PageSep{6} +least. The greater tides are called ``springs,'' +and the smaller ``neaps.'' + +The currents produced in the sea by tides are +\index{Gauge, tide!description of|(}% +often very complicated where the open sea is +broken by islands and headlands, and the knowledge +of tidal currents at each place is only to be +gained by the practical experience of the pilot. +Indeed, in the language of sailors, the word +``tide'' is not unfrequently used as meaning +tidal current, without reference to rise and fall. +These currents are often of great violence, and +vary from hour to hour as the water rises and +falls, so that the pilot requires to know how the +water stands in-shore in order to avail himself of +his practical knowledge of how the currents will +make in each place. A tide table is then of +\index{Tide!general description|)}% +much use, even at places where the access to a +harbor is not obstructed by a bar or shoal. It +is, of course, still more important for ships to +have a correct forecast of the tides where the +entrance to the harbor is shallow. + +I have now sketched in rough outline some +of the peculiarities of the tides, and it will have +become clear that the subject is a complicated +one, not to be unraveled without regular observation. +\index{Observation!methods of tidal|(}% +I shall, therefore, explain how tides are +observed scientifically, and how the facts are collected +upon which the scientific treatment of the +tides is based. + +The rise and fall of the sea may, of course, be +\PageSep{7} +roughly estimated by observing the height of the +water on posts or at jetties, which jut out into +moderately deep water. But as the sea is continually +disturbed by waves, observations of this +kind are not susceptible of accuracy, and for +\Figure{1}{Well for Tide-Gauge}{png} +scientific purposes more elaborate apparatus is +required. The exact height of the water can +only be observed in a place to which the sea has +a moderately free access, but where the channel +is so narrow as to prevent the waves from sensibly +disturbing the level of the water. This result +\PageSep{8} +is obtained in a considerable variety of ways, +but one of them may be described as typical of +all. + +A well (\fig{1}) about two feet in diameter is +dug to a depth of several feet below the lowest +tide and in the neighborhood of deep water. The +well is lined with iron, and a two-inch iron pipe +runs into the well very near its bottom, and passes +down the shore to the low-water line. Here it is +joined to a flexible pipe running out into deep +water, and ending with a large rose pierced with +many holes, like that of a watering can. The +rose (\fig{2}) is anchored to the bottom of the +sea, and is suspended by means of a buoy, so as +to be clear of the bottom. The tidal water can +thus enter pretty freely into the well, but the +passage is so narrow that the wave motion is not +transmitted into the well. Inside the well there +floats a water-tight copper cylinder, weighted at +the bottom so that it floats upright, and counter-poised +so that it only just keeps its top clear of +the water. To the top of the float there is fastened +a copper tape or wire, which runs up to +the top of the well and there passes round a +wheel. Thus as the water rises and falls this +wheel turns backwards and forwards. + +It is hardly necessary to describe in detail the +simple mechanism by which the turning of this +wheel causes a pencil to move backwards and forwards +in a straight line. The mechanism is, +\PageSep{9} +however, such that the pencil moves horizontally +backwards and forwards by exactly the same +amount as the water rises or falls in the well; or, +\Figure{2}{Pipe of Tide-Gauge}{png} +if the rise and fall of the tide is considerable, +the pencil only moves by half as much, or one +third, or even one tenth as much as the water. +At each place a scale of reduction is so chosen +as to bring the range of motion of the pencil +within convenient limits. We thus have a pencil +which will draw the rise and fall of the tide +on the desired scale. + +It remains to show how the times of the rise +and fall are indicated. The end of the pencil +touches a sheet of paper which is wrapped round +a drum about five feet long and twenty-four +\PageSep{10} +\index{Curve, tide!irregularities in|(}% +inches in circumference. If the drum were kept +still the pencil would simply draw a straight line +to and fro along the length of the drum as the +water rises and falls. But the drum is kept +turning by clockwork, so that it makes exactly +one revolution in twenty-four hours. Since the +drum is twenty-four inches round, each inch of +circumference corresponds to one hour. If the +water were at rest the pencil would simply draw +a circle round the paper, and the beginning and +ending of the line would join, whilst if the drum +remained still and the water moved, the pencil +\Figure{3}{Indian Tide-Gauge}{jpg} +would draw a straight line along the length of +the cylinder; but when both drum and water +are in motion, the pencil draws a curve on the +cylinder from which the height of water may be +read off at any time in each day and night. At +the end of twenty-four hours the pencil has returned +to the same part of the paper from which +\PageSep{11} +\Figure[0.7]{4}{Légé's Tide-Gauge}{jpg} +\index{Gauge, tide!description of|)}% +\PageSep{12} +it started, and it might be thought that there +would be risk of confusion between the tides of +to-day and those of yesterday. But since to-day +the tides happen about three quarters of an hour +later than yesterday, it is found that the lines +keep clear of one another, and, in fact, it is +usual to allow the drum to run for a fortnight +before changing the paper, and when the old +sheet is unwrapped from the drum, there is +written on it a tidal record for a fortnight. + +The instrument which I have described is +called a ``tide-gauge,'' and the paper a ``tide-curve.'' +As I have already said, tide-gauges +may differ in many details, but this description +will serve as typical of all. Another form of +tide-gauge is shown in \fig{4}; here a continuous +sheet of paper is placed over the drum, so that +there is no crossing of the curves, as in the first +example. Yet another form, designed by Lord +Kelvin, is shown on p.~170 of vol.~iii.\ of his +``Popular Lectures.'' + +The actual record for a week is exhibited in +\index{Curve, tide!at Bombay}% +\fig{5}, on a reduced scale. This tide-curve was +drawn at Bombay by a tide-gauge of the pattern +first described. When the paper was wrapped +on the drum, the right edge was joined to the +left, and now that it is unwrapped the curve +must be followed out of the paper on the left +and into it again on the right. The figure +shows that spring tide occurred on April~26, +\PageSep{13} +1884; the preceding neap tide was on the~18th, +and is not shown. It may be noticed that the +law of the tide is conspicuously different from +that which holds good on the coast of England, +for the two successive high or low waters which +occur on any day have very different heights. +Thus, for example, on April~26 low water occurred +at 5.50~\PM, and the water fell to $5$~ft.\ +$2$~in., whereas the next low water, occurring at +5.45~\AM\ of the~27th, fell to $1$~ft.\ $3$~in., the +heights being in both cases measured from a +certain datum. When we come to consider the +theory of the tides the nature of this irregularity +will be examined. + +The position near the seashore to be chosen +for the erection of the tide-gauge is a matter of +much importance. The choice of a site is generally +limited by nature, for it should be near +the open sea, should be sheltered from heavy +weather, and deep water must be close at hand +even at low tide. + +In the sketch map shown in \fig{6} a site such +as~$A$ is a good one when the prevailing wind +blows in the direction of the arrow. A position +such as~$B$, although well sheltered from heavy +seas, is not so good, because it is found that +tide-curves drawn at~$B$ would be much zigzagged. +These zigzags appear in the Bombay curves, +although at Bombay they are usually very +smooth ones. +\PageSep{14} + +These irregularities in the tide-curve are not +due to tides, and as the object of the observation +\index{Observation!methods of tidal|)}% +is to determine the nature of the tides it is +\Figure{5}{Bombay Tide-Curve from Noon, April~22, +to Noon, April~30, 1884}{jpg} +desirable to choose a site for the gauge where +\index{Gauge, tide!site for}% +the zigzags shall not be troublesome; but it is +not always easy to foresee the places which will +furnish smooth tide-curves. + +Most of us have probably at some time or +other made a scratch on the sand by the seashore, +\PageSep{15} +and watched the water rise over it. We +generally make our mark on the sand at the +furthest point, where the wash of a rather large +wave has brought up the water. For perhaps +five or ten minutes no wave brings the water up +as far as the mark, and one begins to think that +it was really an extraordinarily large wave which +was marked, although it did not seem so at the +time. Then a wave brings up the water far over +the mark, and immediately all the waves submerge +it. This little observation simply points +\Figure{6}{Sites for a Tide-Gauge}{png} +to the fact that the tide is apt to rise by jerks, +and it is this irregularity of rise and fall which +marks the notches in the tide-curves to which I +have drawn attention. + +Now in scientific matters it is well to follow +up the clues afforded by such apparently insignificant +facts as this. An interesting light is +indeed thrown on the origin of these notches on +tide-curves by an investigation, not very directly +\PageSep{16} +connected with tidal observation, on which I shall +make a digression in the following chapter. +\index{Curve, tide!irregularities in|)}% + +\begin{Authorities} +Baird's \Title{Manual for Tidal Observations} (Taylor \& Francis, +\index{Baird, \Title{Manual for Tidal Observation}}% +1886). Price 7\textit{s.}~6\textit{d.} Figs.\ \figref{1},~\figref{2},~\figref{3},~\figref{6} are reproduced from this +work. + +The second form of tide-gauge shown in \fig{4} is made by +Messrs.\ Légé, and is reproduced from a woodcut kindly provided +by them. + +Sir William Thomson's (Lord Kelvin's) \Title{Popular Lectures and +Addresses}, vol.~iii. (Macmillan,~1891). +\end{Authorities} +\PageSep{17} + + +\Chapter{II} +{Seiches in Lakes} + +\First{It} has been known for nearly three centuries +\index{Lakes!seiches in|(}% +\index{Seiches!definition}% +that the water of the Lake of Geneva is apt to +\index{Geneva!seiches in lake|(}% +rise and fall by a few inches, sometimes irregularly +and sometimes with more or less regularity; +and the same sort of oscillation has been observed +in other Swiss lakes. These quasi-tides, +called seiches, were until recently supposed only +to occur in stormy weather, but it is now known +that small seiches are of almost daily occurrence.\footnote + {The word ``seiche'' is a purely local one. It has been alleged + to be derived from ``sèche,'' but I can see no reason for + associating dryness with the phenomenon.} + +Observations were made by Vaucher in the +\index{Vaucher, record of a great seiche at Geneva}% +last century on the oscillations of the Lake of +Geneva, and he gave an account of a celebrated +seiche in the year~1600, when the water oscillated +through three or four feet; but hardly any +systematic observation had been undertaken when +Professor Forel, of Lausanne, attacked the subject, +\index{Forel!on seiches|(}% +and it is his very interesting observations +which I propose to describe. + +Doctor Forel is not a mathematician, but is +\PageSep{18} +rather a naturalist of the old school, who notes any +interesting fact and then proceeds carefully to +investigate its origin. His papers have a special +charm in that he allows one to see all the workings +of his mind, and tells of each difficulty as it +arose and how he met it. To those who like to +read of such work, almost in the form of a narrative, +I can strongly recommend these papers, +which afford an admirable example of research +thoroughly carried out with simple appliances. + +People are nowadays too apt to think that +science can only be carried to perfection with +elaborate appliances, and yet it is the fact that +many of the finest experiments have been made +with cardboard, cork, and sealing-wax. The principal +reason for elaborate appliances in the laboratories +of universities is that a teacher could not +deal with a large number of students if he had +to show each of them how to make and set up +his apparatus, and a student would not be able +to go through a large field of study if he had to +spend days in preparation. Great laboratories +have, indeed, a rather serious defect, in that they +tend to make all but the very best students helpless, +and thus to dwarf their powers of resource +and inventiveness. The mass of scientific work +is undoubtedly enormously increased by these +institutions, but the number of really great investigators +seems to remain almost unaffected by +them. But I must not convey the impression +\PageSep{19} +that, in my opinion, great laboratories are not +useful. It is obvious, indeed, that without them +science could not be taught to large numbers of +students, and, besides, there are many investigations +in which every possible refinement of apparatus +is necessary. But I do say that the +number of great investigators is but little increased +by laboratories, and that those who are +interested in science, but yet have not access to +laboratories, should not give up their study in +despair. + +Doctor Forel's object was, in the first instance, +to note the variations of the level of the lake, after +obliterating the small ripple of the waves on the +surface. The instrument used in his earlier investigations +was both simple and delicate. Its principle +was founded on casual observation at the port +of Morges, where there happens to be a breakwater, +pierced by a large ingress for ships and a small +one for rowing boats. He accidentally noticed +that at the small passage there was always a current +setting either inwards or outwards, and it +occurred to him that such a current would form +a very sensitive index of the rise and fall of the +water in the lake. He therefore devised an instrument, +illustrated in \fig{7}, and called by him +a plemyrameter, for noting currents of even the +\index{Plemyrameter, observation of seiches with|(}% +most sluggish character. Near the shore he +made a small tank, and he connected it with the +lake by means of an india-rubber siphon pipe of +\PageSep{20} +small bore. Where the pipe crossed the edge +of the tank he inserted a horizontal glass tube +of seven millimetres diameter, and in that tube +he put a float of cork, weighted with lead so that +it should be of the same density as water. At +the ends of the glass tube there were stops, so +that the float could not pass out of it. When +the lake was higher than the tank, the water ran +\Figure{7}{Plemyrameter}{png} +through the siphon pipe from lake to tank, and +the float remained jammed in the glass tube +against the stop on the side towards the tank; +and when the lake fell lower than the tank, the +float traveled slowly to the other end and remained +there. The siphon pipe being small, the +only sign of the waves in the lake was that the +float moved with slight jerks, instead of uniformly. +Another consequence of the smallness +of the tube was that the amount of water which +could be delivered into the tank or drawn out of +it in one or two hours was so small that it might +\PageSep{21} +practically be neglected, so that the water level +in the tank might be considered as invariable. + +This apparatus enabled Forel to note the rise +and fall of the water, and he did not at first +attempt to measure the height of rise and fall, +as it was the periodicity in which he was principally +interested. + +In order to understand the record of observations, +\index{Seiches!records of}% +it must be remembered that when the +float is towards the lake, the water in the tank +stands at the higher level, and when the float +is towards the tank the lake is the higher. In +the diagrams, of which \fig{8} is an example, the +straight line is divided into a scale of hours and +minutes. The zigzag line gives the record, and +the lower portions represent that the water of +the lake was below the tank, and the upper line +that it was above the tank. The fact that the +float only moved slowly across from end to end +of the glass tube, is indicated by the slope of +the lines, which join the lower and upper portions +of the zigzags. Then on reading \fig{8} we +see that from $2$~hrs.\ $1$~min.\ to $2$~hrs.\ $4$~mins.\ the +water was high and the float was jammed against +the tank end of the tube, because there was a +current from the lake to the tank. The float +then slowly left the tank end and traveled +across, so that at $2$~hrs.\ $5$~mins.\ the water was +low in the lake. It continued, save for transient +changes of level, to be low until $2$~hrs.\ $30$~mins., +\PageSep{22} +when it rose again. Further explanation seems +unnecessary, as it should now be easy to read +this diagram, and that shown in~\fig{9}. +\index{Plemyrameter, observation of seiches with|)}% + +The sharp pinnacles indicate alternations of +level so transient that the float had not time to +travel across from one end of the glass tube +to the other, before the current was reversed. +These pinnacles may be disregarded for the +present, since we are only considering seiches of +considerable period. + +These two diagrams are samples of hundreds +which were obtained at various points on the +shores of Geneva, and of other lakes in Switzerland. +In order to render intelligible the method +by which Forel analyzed and interpreted these +records, I must consider \fig{8} more closely. In +this case it will be noticed that the record shows +a long high water separated from a long low +water by two pinnacles with flat tops. These +pieces at the ends have an interesting significance. +When the water of the lake is simply +oscillating with a period of about an hour we +have a trace of the form shown in~\fig{9}. But +when there exists concurrently with this another +oscillation, of much smaller range and of short +period, the form of the trace will be changed. +When the water is high in consequence of the +large and slow oscillation, the level of the lake +cannot be reduced below that of the tank by +the small short oscillation, and the water merely +\PageSep{23} +stands a little higher or a little lower, but always +remains above the level of the tank, so that the +trace continues on the higher level. But when, +in course of the changes of the large oscillation, +the water has sunk to near the mean level of the +%[** TN: Figures 8 and 9 combined; cross-refs handled by preamble code] +\Figure{8}{Records of Seiches at Évian}{png} +lake, the short oscillation will become manifest, +and so it is only at the ends of the long flat +pieces that we shall find evidence of the quick +oscillation. + +Thus, in these two figures there was in one +case only one sort of wave, and in the other +there were two simultaneous waves. These +records are amongst the simplest of those obtained +by Forel, and yet even here the oscillations +of the water were sufficiently complicated. +It needed, indeed, the careful analysis of many +records to disentangle the several waves and to +determine their periods. + +After having studied seiches with a plemyrameter +for some time, Forel used another form of +\PageSep{24} +apparatus, by which he could observe the amplitude +of the waves as well as their period. His +apparatus was, in fact, a very delicate tide-gauge, +which he called a limnimeter. The only +\index{Limnimeter, a form of tide gauge}% +difference between this instrument and the one +already described as a tide-gauge is that the +drum turned much more rapidly, so that five +feet of paper passed over the drum in twenty-four +hours, and that the paper was comparatively +narrow, the range of the oscillation being +small. The curve was usually drawn on the full +scale, but it could be quickly reduced to half +scale when large seiches were under observation. + +It would be impossible in a book of this kind +\index{Lakes!mode of rocking in seiches}% +to follow Forel in the long analysis by which he +interpreted his curves. He speaks thus of the +complication of simultaneous waves: ``All these +oscillations are embroidered one on the other +and interlace their changes of level. There is +here matter to disturb the calmest mind. I +must have a very stout faith in the truth of my +hypothesis to persist in maintaining that, in the +midst of all these waves which cross and mingle, +there is, nevertheless, a recognizable rhythm. +This is, however, what I shall try to prove.''\footnote + {\Title{Deuxième Étude}, p.~544.} +The hypothesis to which he here refers, and +triumphantly proves, is that seiches consist of a +rocking of the whole water of the lake about +fixed lines, just as by tilting a trough the water +\PageSep{25} +in it may be set swinging, so that the level at +the middle remains unchanged, while at the two +ends the water rises and falls alternately. + +In another paper he remarks: ``If you will +\index{Lakes!mode of rocking in seiches}% +follow and study with me these movements you +will find a great charm in the investigation. +When I see the water rising and falling on the +shore at the end of my garden I have not before +me a simple wave which disturbs the water of +the bay of Morges, but I am observing the manifestation +of a far more important phenomenon. +It is the whole water of the lake which is rocking. +It is a gigantic impulse which moves the +whole liquid mass of Leman throughout its +length, breadth, and depth\dots. It is probable +that the same thing would be observed in far +larger basins of water, and I feel bound to +recognize in the phenomenon of seiches the +\index{Seiches!longitudinal and transverse|(}% +grandest oscillatory movement which man can +study on the face of our globe.''\footnote + {\Title{Les Seiches, Vagues d'Oscillation}, p.~11.} + +It will now be well to consider the map of +Geneva in~\fig{10}. Although the lake somewhat +resembles the arc of a circle, the curvature of +its shores will make so little difference in the +nature of the swinging of the water that we +may, in the first instance, consider it as practically +straight. + +Forel's analysis of seiches led him to conclude +that the oscillations were of two kinds, the longitudinal +\PageSep{26} +and the transverse. In the longitudinal +seiche the water rocks about a line drawn across +the lake nearly through Morges, and the water +at the east end of the lake rises when that at +the west falls, and vice versa. The line about +which the water rocks is called a node, so that in +this case there is one node at the middle of the +lake. This sort of seiche is therefore called a +uninodal longitudinal seiche. The period of the +\Figure{10}{Map of Lake of Geneva}{png} +oscillation is the time between two successive +high waters at any place, and it was found to be +seventy-three minutes, but the range of rise and +fall was very variable. There are also longitudinal +seiches in which there are two nodes, +dividing the lake into three parts, of which the +central one is twice as long as the extreme parts; +such an oscillation is called a binodal longitudinal +seiche. In this mode the water at the middle +\PageSep{27} +of the lake is high when that at the two ends +is low, and vice versa; the period is thirty-five +minutes. + +Other seiches of various periods were observed, +\index{Seiches!periods of}% +some of which were no doubt multinodal. Thus +in a trinodal seiche, the nodes divide the lake +into four parts, of which the two central ones +are each twice as long as the extreme parts. If +there are any number of nodes, their positions +are such that the central portion of the lake is +divided into equal lengths, and the terminal +parts are each of half the length of the central +part or parts. This condition is necessary in +order that the ends of the lake may fall at places +where there is no horizontal current. In all such +modes of oscillation the places where the horizontal +current is evanescent are called loops, and +these are always halfway between the nodes, +where there is no rise and fall. + +A trinodal seiche should have a period of +about twenty-four minutes, and a quadrinodal +seiche should oscillate in about eighteen minutes. +The periods of these quicker seiches would, no +doubt, be affected by the irregularity in the form +and depth of the lake, and it is worthy of notice +that Forel observed at Morges seiches with +periods of about twenty minutes and thirty minutes, +which he conjectured to be multinodal. + +The second group of seiches were transverse, +\index{Seiches!longitudinal and transverse|)}% +being observable at Morges and Évian. It was +\PageSep{28} +clear that these oscillations, of which the period +was about ten minutes, were transversal, because +at the moment when the water was highest at +Morges it was lowest at Évian, and vice versa. +As in the case of the longitudinal seiches, the +principal oscillation of this class was uninodal, +but the node was, of course, now longitudinal to +the lake. The irregularity in the width and +depth of the lake must lead to great diversity of +period in the transverse seiches appropriate to +the various parts of the lake. The transverse +seiches at one part of the lake must also be +transmitted elsewhere, and must confuse the +seiches appropriate to other parts. Accordingly +there is abundant reason to expect oscillations of +such complexity as to elude complete explanation. + +The great difficulty of applying deductive +reasoning to the oscillations of a sheet of water +of irregular outline and depth led Forel to construct +a model of the lake. By studying the +waves in his model he was able to recognize +many of the oscillations occurring in the real +lake, and so obtained an experimental confirmation +of his theories, although the periods of +oscillation in the model of course differed enormously +from those observed in actuality. + +The theory of seiches cannot be considered as +demonstrated, unless we can show that the water +of such a basin as that of Geneva is capable of +\index{Geneva!seiches in lake|)}% +\index{Geneva!model of lake}% +\PageSep{29} +swinging at the rates observed. I must, therefore, +now explain how it may be proved that the +periods of the actual oscillations agree with the +facts of the case. + +As a preliminary let us consider the nature of +wave motion. There are two very distinct cases +of the undulatory motion of water, which nevertheless +graduate into one another. The distinction +lies in the depth of the water compared with +the length of the wave, measured from crest to +crest, in the direction of wave propagation. The +wave-length may be used as a measuring rod, +and if the depth of the water is a small fraction +of the wave-length, it must be considered shallow, +but if its depth is a multiple of the wave-length, +it will be deep. The two extremes of +course graduate into one another. + +In a wave in deep water the motion dies out +pretty rapidly as we go below the surface, so that +when we have gone down half a wave-length +below the surface, the motion is very small. In +shallow water, on the other hand, the motion extends +quite to the bottom, and in water which is +neither deep nor shallow, the condition of affairs +is intermediate. The two figures, \figref{11}~and~\figref{12}, +show the nature of the movement in the two +classes of waves. In both cases the dotted lines +\index{Waves!in deep and shallow water}% +show the position of the water when at rest, and +the full lines show the shapes assumed by the +rectangular blocks marked out by the dotted +\PageSep{30} +lines, when wave motion is disturbing the water. +It will be observed that in the deep water, as +shown in \fig{11}, the rectangular blocks change +their shape, rise and fall, and move to and fro. +Taking the topmost row of rectangles, each block +of water passes successively in time through all +the forms and positions shown by the top row +of quasi-parallelograms. So also the successive +changes of the second row of blocks are indicated +by the second strip, and the third and the fourth +indicate the same. The changes in the bottom +\Figure[0.7]{11}{Wave in Deep Water}{png} +row are relatively very small both as to shape +and as to displacement, so that it did not seem +worth while to extend the figure to a greater +depth. + +Turning now to the wave in shallow water in~\fig{12}, +we see that each of the blocks is simply +displaced sideways and gets thinner or more +\PageSep{31} +squat as the wave passes along. Now, I say that +we may roughly classify the water as being deep +with respect to wave motion when its depth is +more than half a wave-length, and as being shallow +when it is less. Thus the same water may +be shallow for long waves and deep for short +\index{Waves!speed of}% +ones. For example, the sea is very shallow for +\Figure[0.7]{12}{Wave in Shallow Water}{png} +the great wave of the oceanic tide, but it is very +deep even for the largest waves of other kinds. +Deepness and shallowness are thus merely relative +to wave-length. + +The rate at which a wave moves can be exactly +calculated from mathematical formulæ, +from which it appears that in the deep sea a +wave $63$~metres in length travels at $36$~kilometres +per hour, or, in British measure, a wave of +$68$~yards in length travels $22\frac{1}{2}$~miles an hour. +Then, the rule for other waves is that the speed +varies as the square root of the wave-length, so +that a wave $16$~metres long---that is, one quarter +of $63$~metres---travels at $18$~kilometres an +hour, which is half of $36$~kilometres an hour. +Or if its length were $7$~metres, or one ninth as +\PageSep{32} +long, it would travel at $12$~kilometres an hour, +or one third as quick. + +Although the speed of waves in deep water +depends on wave-length, yet in shallow water the +speed is identical for waves of all lengths, and +depends only on the depth of the water. In +water $10$~metres deep, the calculated velocity of +a wave is $36$~kilometres an hour; or if the water +were $2\frac{1}{2}$~metres deep (quarter of $10$~metres), it +would travel $18$~kilometres (half of $36$~kilometres) +an hour; the law of variation being that +the speed of the wave varies as the square root +of the depth. For water that is neither deep nor +shallow, the rate of wave propagation depends +both on depth and on wave-length, according to +a law which is somewhat complicated. + +In the case of seiches, the waves are very long +compared with the depth, so that the water is to +be considered as shallow; and here we know +that the speed of propagation of the wave depends +only on depth. The average depth of the +Lake of Geneva may be taken as about $150$~metres, +and it follows that the speed of a long wave +in the lake is about $120$~kilometres an hour. + +In order to apply this conclusion to the study +of seiches, we have to consider what is meant by +the composition of two waves. If I take the +series of numbers +\[ +\text{\&c.}\quad +100\quad 71\quad 0\quad -71\quad -100\quad -71\quad 0\quad 71\quad 100\quad +\text{\&c.} +\] +and plot out, at equal distances, a figure of +\PageSep{33} +\index{Waves!composition of|(}% +heights proportional to these numbers, setting +off the positive numbers above and the negative +numbers below a horizontal line, I get the simple +wave line shown in~\fig{13}. Now, if this +wave is traveling to the right, the same series of +numbers will represent the wave at a later time, +when they are all displaced towards the right, as +in the dotted line. + +Now turn to the following schedule of numbers, +and consider those which are written in the +top row of each successive group of three rows. +The columns represent equidistant spaces, and +the rows equidistant times. The first set of +numbers, $-100$,~$-71$,~$0$,~\&c., are those which +were plotted out as a wave in~\fig{13}; in the top +\Figure[0.7]{13}{Simple Wave}{png} +row of the second group they are the same, but +moved one space to the right, so that they represent +the movement of the wave to the right in +one interval of time. In the top row of each +successive group the numbers are the same, but +always displaced one more space to the right; +they thus represent the successive positions of a +\PageSep{34} +\begin{figure}[hp!] +\centering +\ifthenelse{\boolean{ForPrinting}}{% + \makebox[0pt][c]{\includegraphics[height=0.9\textheight]{./images/fig13b.pdf}} +}{ + \includegraphics[width=0.75\textwidth]{./images/fig13b.pdf} +} +\caption{Composition of Two Equal and Opposite Waves} +\label{fig:13b} +\iffalse +-100 -71 0 71 100 71 0 -71 -100 + +-100 -71 0 71 100 71 0 -71 -100 +[*--] +-200 -142 0 142 200 142 0 -142 -200 +[*--] + -71 -100 -71 0 71 100 71 0 -71 + + -71 0 71 100 71 0 -71 -100 -71 +[*--] +-142 -100 0 100 142 100 0 -100 -142 +[*--] + 0 -71 -100 -71 0 71 100 71 0 + + 0 71 100 71 0 -71 -100 -71 0 +[*--] + 0 0 0 0 0 0 0 0 0 +[*--] + 71 0 -71 -100 -71 0 71 100 71 + + 71 100 71 0 -71 -100 -71 0 71 +[*--] + 142 100 0 -100 -142 -100 0 100 142 +[*--] + 100 71 0 -71 -100 -71 0 71 100 + + 100 71 0 -71 -100 -71 0 71 100 +[*--] + 200 142 0 -142 -200 -142 0 142 200 +[*--] + 71 100 71 0 -71 -100 -71 0 71 + + 71 0 -71 -100 -71 0 71 100 71 +[*--] + 142 100 0 -100 -142 -100 0 100 142 +[*--] + 0 71 100 71 0 -71 -100 -71 0 + + 0 -71 -100 -71 0 71 100 71 0 +[*--] + 0 0 0 0 0 0 0 0 0 +[*--] + -71 0 71 100 71 0 -71 -100 -71 + + -71 -100 -71 0 71 100 71 0 -71 +[*--] +-142 -100 0 100 142 100 0 -100 -142 +[*--] +-100 \DPtypo{71}{-71} 0 71 100 71 0 -71 -100 + +-100 -71 0 71 100 71 0 -71 -100 +[*--] +-200 -142 0 142 200 142 0 -142 -200 +[*--] +\fi +\end{figure} +\PageSep{35} +wave moving to the right. The table ends in +the same way as it begins, so that in eight of +these intervals of time the wave has advanced +through a space equal to its own length. + +If we were to invert these upper figures, so +that the numbers on the right are exchanged +with those on the left, we should have a series of +numbers representing a wave traveling to the +left. Such numbers are shown in the second +row in each group. + +When these two waves coëxist, the numbers +must be compounded together by addition, and +then the result is the series of numbers written +in the third rows. These numbers represent the +resultant of a wave traveling to the right, and of +an equal wave traveling simultaneously to the +left. + +It may be well to repeat that the first row of +each group represents a wave moving to the +right, the second row represents a wave moving +to the left, and the third represents the resultant +of the two. Now let us consider the nature +of this resultant motion; the third and the +seventh columns of figures are always zero, and +therefore at these two places the water neither +rises nor falls,---they are, in fact, nodes. If the +schedule were extended indefinitely both ways, +exactly halfway between any pairs of nodes +there would be a loop, or line across which there +is no horizontal motion. In the schedule, as it +\PageSep{36} +stands, the first, fifth, and ninth columns are +loops. + +At the extreme right and at the extreme left +the resultant numbers are the same, and represent +a rise of the water from $-200$ to~$+200$, +and a subsequent fall to~$-200$ again. If these +nine columns represent the length of the lake, +the motion is that which was described as binodal, +for there are two nodes dividing the lake +into three parts, there is a loop at each end, and +when the water is high in the middle it is low at +the ends, and vice versa. It follows that two +equal waves, each as long as the lake, traveling +in opposite directions, when compounded together +give the motion which is described as the +binodal longitudinal seiche. + +Now let us suppose that only five columns of +the table represent the length of the lake. The +resultant numbers, which again terminate at +each end with a loop, are:--- +\begin{align*} +-200&& -142&& 0&& 142&& 200& \displaybreak[0] \\ +-142&& -100&& 0&& 100&& 142& \displaybreak[0] \\ + 0&& 0&& 0&& 0&& 0& \displaybreak[0] \\ + 142&& 100&& 0&& -100&& -142& \displaybreak[0] \\ + 200&& 142&& 0&& -142&& -200& \displaybreak[0] \\ + 142&& 100&& 0&& -100&& -142& \displaybreak[0] \\ + 0&& 0&& 0&& 0&& 0& \displaybreak[0] \\ +-142&& -100&& 0&& 100&& 142& \displaybreak[0] \\ +-200&& -142&& 0&& 142&& 200& +\end{align*} + +Since the middle column consists of zero +throughout, the water neither rises nor falls +\PageSep{37} +there, and there is a node at the middle. Again, +since the numbers at one end are just the same +as those at the other, but reversed as to positive +and negative, when the water is high at one end +it is low at the other. The motion is, in fact, a +simple rocking about the central line, and is that +described as the uninodal longitudinal seiche. + +The motion is here again the resultant of two +equal waves moving in opposite directions, and +the period of the oscillation is equal to the time +which either simple wave takes to travel through +its own length. But the length of the wave is +now twice that of the lake. Hence it follows +that the period of the rocking motion is the +time occupied by a wave in traveling twice the +length of the lake. We have already seen that +in shallow water the rate at which a wave moves +is independent of its length and depends only +on the depth of the water, and that in water of +the same depth as the Lake of Geneva the wave +travels $120$~kilometres an hour. The Lake of +Geneva is $70$~kilometres long, so that the two +waves, whose composition produces a simple rocking +\index{Waves!composition of|)}% +of the water, must each of them have a +length of $140$~kilometres. Hence it follows that +the period of a simple rocking motion, with one +node in the middle of the Lake of Geneva, will +be almost exactly $\frac{140}{120}$~of an hour, or $70$~minutes. +Forel, in fact, found the period to be $73$~minutes. +He expresses this result by saying that +\PageSep{38} +a uninodal longitudinal seiche in the Lake of +Geneva has a period of $73$~minutes. His observations +also showed him that the period of a +binodal seiche was $35$~minutes. It follows from +the previous discussion that when there are two +nodes the period of the oscillation should be +half as long as when there is one node. Hence, +we should expect that the period would be +about $36$~or $37$~minutes, and the discrepancy +between these two results may be due to the +fact that the formula by which we calculate the +period of a binodal seiche would require some +correction, because the depth of the lake is not +so very small compared with the length of these +shorter waves. + +It is proper to remark that the agreement +between the theoretical and observed periods is +suspiciously exact. The lake differs much in +depth in different parts, and it is not quite certain +what is the proper method of computing +the average depth for the determination of the +period of a seiche. It is pretty clear, in fact, +that the extreme closeness of the agreement is +accidentally due to the assumption of a round +number of metres as the average depth of the +lake. The concordance between theory and observation +must not, however, be depreciated too +much, for it is certain that the facts of the case +agree well with what is known of the depth of +the lake. +\index{Forel!on seiches|)}% +\PageSep{39} + +The height of the waves called ``seiches'' is +\index{Seiches!causes of}% +\index{Storms a cause of seiches}% +very various. I have mentioned an historical +seiche which had a range of as much as four +feet, and Forel was able by his delicate instruments +still to detect them when they were only a +millimetre or a twenty-fifth of an inch in height. +It is obvious, therefore, that whatever be the +cause of seiches, that cause must vary widely +in intensity. According to Forel, seiches arise +from several causes. It is clear that anything +which heaps up the water at one end of the +lake, and then ceases to act, must tend to produce +an oscillation of the whole. Now, a rise +of water level at one end or at one side of the +lake may be produced in various ways. Some, +and perhaps many, seiches are due to the tilting +of the whole lake bed by minute earthquakes. +\index{Earthquakes!a cause of seiches}% +Modern investigations seem to show that this is +a more fertile cause than Forel was disposed to +allow, and it would therefore be interesting to +see the investigation of seiches repeated with the +aid of delicate instruments for the study of +earthquakes, some of which will be described in +\Ref{Chapter}{VI}. I suspect that seiches would be +observed at times when the surface of the earth +is much disturbed. + +The wind is doubtless another cause of seiches. +\index{Wind!a cause of seiches}% +When it blows along the lake for many hours in +one direction, it produces a superficial current, +and heaps up the water at the end towards +\PageSep{40} +\index{Atmospheric pressure!cause of seiches}% +which it is blowing. If such a wind ceases +somewhat suddenly, a seiche will certainly be +started, and will continue for hours until it dies +out from the effects of the friction of the water +on the lake bottom. Again, the height of the +barometer will often differ slightly at different +parts of the lake, and the water will respond, just +as does the mercury, to variations of atmospheric +pressure. About a foot of rise of water should +correspond to an inch of difference in the height +of barometer. The barometric pressure cannot +be quite uniform all over the Lake of Geneva, +and although the differences must always be +exceedingly small, yet it is impossible to doubt +that this cause, combined probably with wind, +will produce many seiches. I shall return later +\index{Seiches!causes of}% +to the consideration of an interesting speculation +as to the effects of barometric pressure on +the oscillation of lakes and of the sea. Lastly, +Forel was of opinion that sudden squalls or local +storms were the most frequent causes of seiches. +\index{Storms a cause of seiches}% +I think that he much overestimated the efficiency +of this cause, because his theory of the path of +the wind in sudden and local storms is one that +would hardly be acceptable to most meteorologists. + +Although, then, it is possible to indicate causes +competent to produce seiches, yet we cannot as +yet point out the particular cause for any individual +seiche. The complication of causes is so +\PageSep{41} +\index{Lakes!vibrations|(}% +\index{Vibration of lakes|(}% +great that this degree of uncertainty will probably +never be entirely removed. + +But I have not yet referred to the point which +justifies this long digression on seiches in a book +on the tides. The subject was introduced by +the irregularities in the line traced by the tide-gauge +at Bombay, which indicated that there +are oscillations of the water with periods ranging +from two minutes to a quarter of an hour or +somewhat longer. Now these zigzags are not +found in the sea alone, for Forel observed on +the lake oscillations of short period, which resembled +seiches in all but the fact of their more +rapid alternations. Some of these waves are +perhaps multinodal seiches, but it seems that +they are usually too local to be true seiches +affecting the whole body of the lake at one time. +Forel calls these shorter oscillations ``vibrations,'' +thus distinguishing them from proper seiches. +A complete theory of the so-called vibrations +has not yet been formulated, although, as I shall +show below, a theory is now under trial which +serves to explain, at least in part, the origin of +vibrations. + +Forel observed with his limnimeter or tide-gauge +that when there is much wind, especially +\index{Wind!vibrations of lakes due to}% +from certain quarters, vibrations arise which are +quite distinct from the ordinary visible wave +motion. The period of the visible waves on the +\PageSep{42} +Lake of Geneva is from $4$~to $5$~seconds,\footnote + {I observed when it was blowing half a gale on Ullswater, in + Cumberland, that the waves had a period of about a second.} +whereas +vibrations have periods ranging from $45$~seconds +to $4$~minutes. Thus there is a clear line separating +waves from vibrations. Forel was unable +to determine what proportion of the area of the +lake is disturbed by vibrations at any one time, +and although their velocity was not directly observed, +there can be no doubt that these waves +are propagated at a rate which corresponds to +their length and to the depth of the water. I +have little doubt but that the inequalities which +produce notches in a tide-curve have the same +origin as vibrations on lakes. + +It is difficult to understand how a wind, whose +\index{Wind!vibrations of lakes due to}% +only visible effect is short waves, can be responsible +for raising waves of a length as great as a +thousand yards or a mile, and yet we are driven +to believe that this is the case. But Forel also +found that steamers produce vibrations exactly +like those due to wind. The resemblance was +indeed so exact that vibrations due to wind +could only be studied at night, when it was +known that no steamers were traveling on the +lake, and, further, the vibrations due to steamers +could only be studied when there was no wind. + +His observations on the steamer vibrations are +amongst the most curious of all his results. +When a boat arrives at the pier at Morges, the +\PageSep{43} +water rises slowly by about $5$~to $8$~millimetres, +and then falls in about $20$~to $30$~seconds. The +amount and the rapidity of the rise and fall +vary with the tonnage of the boat and with the +rate of her approach. After the boat has passed, +the trace of the limnimeter shows irregularities +with sharp points, the variations of height ranging +from about two to five millimetres, with a +period of about two minutes. These vibrations +continue to be visible during two to three hours +after the boat has passed. As these boats travel +at a speed of $20$~kilometres an hour, the vibrations +persist for a long time after any renewal +of them by the boat has ceased. These vibrations +are called by Forel ``the subsequent steamer +vibrations.'' + +That the agitation of the water should continue +for more than two hours is very remarkable, +and shows the delicacy of the method of +observation. But it seems yet more strange +that, when a boat is approaching Morges, the +vibrations should be visible during $25$~minutes +before she reaches the pier. These he calls +``antecedent steamer vibrations.'' They are +more rapid than the subsequent ones, having a +period of a minute to a minute and a quarter. +Their height is sometimes two millimetres (a +twelfth of an inch), but they are easily detected +when less than one millimetre in height. It +appears that these antecedent vibrations are first +\PageSep{44} +noticeable when the steamer rounds the mole of +Ouchy, when she is still at a distance of $10$~kilometres. +As far as one can judge from the speed +at which waves are transmitted in the Lake of +Geneva, the antecedent vibrations, which are +noticed $25$~minutes before the arrival of the +boat, must have been generated when she was at +a distance of $12$~kilometres from Morges. \Fig{14} +gives an admirable tracing of these steamer +vibrations.\footnote + {From \Title{Les Seiches, Vagues d'Oscillation fixe des Lacs}, 1876.} + +In this figure the line~$\Seg{a}{a'}$ was traced between +two and three o'clock in the morning, and shows +scarcely any sign of perturbation. Between +three and eight o'clock in the morning no observations +were taken, but the record begins again +at eight o'clock. The portion marked~$\Seg{b}{b'}$ shows +weak vibrations, probably due to steamers passing +along the coast of Savoy. The antecedent +vibrations, produced by a steamer approaching +Morges, began about the time of its departure +from Ouchy, and are shown at~$\Seg{c}{c'}$. The point~$d$ +shows the arrival of this boat at Morges, and +$d'$~shows the effect of another boat coming from +Geneva. The portion marked~$\Seg[e]{e}{e}$ shows the +subsequent steamer vibrations, which were very +clear during more than two hours after the boats +had passed. + +Dr.~Forel was aware that similar vibrations occur +in the sea, for he says: ``What are these +\index{Sea!vibrations of}% +\PageSep{45} +oscillations with periods of +$5$,~$10$, $20$, or $100$~minutes, +which are sometimes irregular? +Are they analogous to +our seiches? Not if we define +seiches as uninodal oscillations, +for it is clear that +if, in a closed basin of $70$~kilometres +in length, uninodal +seiches have a period of +$73$~minutes, in the far greater +basin of the Mediterranean, +or of the ocean, a uninodal +wave of oscillation must have +a much longer period. They +resemble much more closely +what I have called vibrations, +and, provisionally, I +shall call them by the name +of `vibrations of the sea.' I +\index{Sea!vibrations of}% +venture to invite men of science +who live on the seacoast +to follow this study. +It presents a fine subject for +research, either in the interpretation +of the phenomenon +or in the establishment of +the relations between these +movements and meteorological +conditions.''\footnote + {\Title{Seiches et Vibrations des Lacs et de la Mer}, 1879, p.~5.} +%[** TN: Figure wrapped in the original] +\TallFig{14}{Vibrations due to Steamers}{png} +\PageSep{46} + +These vibrations are obviously due to the wind +or to steamers, but it is a matter of no little surprise +that such insignificant causes should produce +even very small waves of half a mile to a +mile in length. + +The manner in which this is brought about is +undoubtedly obscure, yet it is possible to obtain +some sort of insight into the way in which these +long waves arise. When a stone falls into calm +water waves of all sorts of lengths are instantaneously +generated, and the same is true of +any other isolated disturbance. Out of all these +waves the very long ones and the very short +ones are very small in height. Theoretically, +waves of infinitely great and of infinitely small +lengths, yet in both cases of infinitely small +heights, are generated at the instant of the impulse, +but the waves of enormous length and +those of very small length are of no practical +importance, and we need only consider the moderate +waves. For the shorter of these the water +is virtually deep, and so they will each travel +outwards at a pace dependent on length, the +longer ones outstripping the shorter ones. But +for the longer waves the water will be shallow, +and they will all travel together. Thus the general +effect at a distance is the arrival of a long +wave first, followed by an agitated rippling. +The point which we have to note is that an isolated +disturbance will generate long waves and +\PageSep{47} +that they will run ahead of the small ones. It +is important also to observe that the friction of +the water annuls the oscillation in the shorter +waves more rapidly than it does that of the +longer ones, and therefore the long waves are +more persistent. Now we may look at the disturbance +due to a steamer or to the wind as consisting +of a succession of isolated disturbances, +each of which will create long waves outstripping +the shorter ones. These considerations afford a +sort of explanation of what is observed, but I do +not understand how it is that the separation of +the long from the short waves is so complete, nor +what governs the length of the waves, nor have +I made any attempt to evaluate the greater rapidity +of decrease of short waves than long ones.\footnote + {See, however, S.~S. Hough, \Title{Proc.\ Lond.\ Math.\ Soc.}, xxviii.\ +\index{Hough, S. S.!frictional extinction of waves}% + p.~276.} +It must then be left to future investigators to +elucidate these points. + +\TB + +The subject of seiches and vibrations clearly +affords an interesting field for further research. +The seiches of Lake George in New South Wales +have been observed by Mr.~Russell, the government +\index{Russell, observation of seiches in New South Wales}% +astronomer at Sydney; but until last year +they do not seem to have been much studied on +any lakes outside of Switzerland. The great +lakes of North America are no doubt agitated by +seiches on a much larger scale than those on the +\PageSep{48} +comparatively small basin of Geneva. This idea +appears to have struck Mr.~Napier Denison of +\index{Denison, F. Napier, vibrations and seiches on lakes|(}% +Toronto, and he has been so fortunate as to enlist +the interest of Mr.~Bell Dawson, the chief of +\index{Dawson coöperates in investigation of seiches}% +the Canadian Tidal Survey, and of Mr.~Stupart, +\index{Stupart coöperates in investigation of seiches}% +the director of the Meteorological Department. +Mr.~Denison's attention has been, in the first instance, +principally directed towards those notches +in tide-curves which have afforded the occasion +for the present discussion of this subject. He +has made an interesting suggestion as to the +origin of these oscillations, which I will now +explain. + +The wind generally consists of a rather shallow +\index{Waves!in atmosphere|(}% +current, so that when it is calm at the earth's +surface there is often a strong wind at the top +of a neighboring mountain; or the wind aloft +may blow from a different quarter from that below. +If we ascend a mountain or go up in a +balloon, the temperature of the air falls on the +average by a certain definite number of degrees +per thousand feet. But the normal rate of fall +of temperature is generally interrupted on passing +into an upper current, which blows from a +different direction. This abrupt change of temperature +corresponds with a sudden change of +density, so that the upper layer of air must be +regarded as a fluid of different density from that +of the lower air, over which it slides. + +Now Helmholtz has pointed out that one layer +\index{Atmospheric waves, Helmholtz on|(}% +\index{Helmholtz!on atmospheric waves|(}% +\PageSep{49} +of fluid cannot slide over another, without generating +waves at the surface of separation. We +are familiar with this fact in the case of sea-waves +generated by wind. A mackerel sky +\index{Mackerel sky, evidence of air-waves}% +proves also the applicability to currents of air of +Helmholtz's observation. In this case the moisture +of the air is condensed into clouds at the +crests of the air waves, and reabsorbed in the +hollows, so that the clouds are arranged in a visible +ripple-mark. A mackerel sky is not seen in +stormy weather, for it affords proof of the existence +of an upper layer of air sliding with only +moderate velocity over a lower layer. The distance +from crest to crest must be considerable +as measured in yards, yet we must regard the +mackerel sky as a mere ripple formed by a slow +relative velocity of the two layers. If this is so, +it becomes of interest to consider what wave-lengths +may be expected to arise when the upper +current is moving over the lower with a speed of +perhaps a hundred miles an hour. The problem +is not directly soluble, for even in the case of +sea-waves it is impossible to predict the wave-lengths. +We do know, however, that the duration +of the wind and the size of the basin are +material circumstances, and that in gales in the +open ocean the waves attain a very definite magnitude. + +Although the problem involved is not a soluble +one, yet Helmholtz has used the analogy of +\PageSep{50} +oceanic waves for an approximate determination +of the sizes of the atmospheric ones. His +method is a very fertile one in many complex +physical investigations, where an exact solution +is not attainable. The method may be best illustrated +by one or two simple cases. + +It is easy for the mathematician to prove that +the period of a swing of a simple pendulum must +vary as the square root of its length. The proof +does not depend on the complete solution of the +problem, so that even if it were insoluble he +would still be sure of the correctness of his conclusion. +If, then, a given pendulum is observed +to swing in a certain period, it is certain that a +similar pendulum of four times the length will +take twice as long to perform its oscillation. In +the same way, the engine power required for a +ship is determinable from experiments on the +resistance suffered by a small model when towed +through the water. The correct conclusion is +discovered in this case, although it is altogether +impossible to discover the resistance of a ship +by \textit{à~priori} reasoning. + +The wave motion at the surface separating +two fluids of different densities presents another +problem of the same kind, and if the result is +known in one case, it can be confidently predicted +in another. Now oceanic waves generated +\index{Waves!in atmosphere|)}% +by wind afford the known case, and Helmholtz +has thence determined by analogy the +\PageSep{51} +lengths of the atmospheric waves which must +exist aloft. By making plausible suppositions +as to the densities of the two layers of air and +as to their relative velocity, he has shown that +sea-waves of ten yards in length will correspond +with air-waves of as much as twenty miles. A +wave of this length would cover the whole sky, +and might have a period of half an hour. It is +clear then that mackerel sky will disappear in +stormy weather, because we are too near to the +crests and furrows to observe the orderly arrangement +of the clouds. + +Although the waves are too long to be seen as +such, yet the unsteadiness of the barometer in a +gale of wind affords evidence of the correctness +of this theory. In fact, when the crest of denser +air is over the place of observation the barometer +rises, and it falls as the hollow passes. The +waves in the continuous trace of the barometer +have some tendency to regularity, and have +periods of from ten minutes to half an hour. +The analogy seems to be pretty close with the +confused and turbulent sea often seen in a gale +of wind in the open ocean.\footnote + {A gust of wind will cause the barometer to vary, without a + corresponding change in the density of the air. It is not therefore + safe to interpret the oscillations of the barometer as being + due entirely to true changes of pressure. If, however, the intermittent + squalls in a gale are connected with the waves aloft, + the waviness of the barometric trace would still afford signals + of the passage of crests and hollows above.} +\index{Atmospheric waves, Helmholtz on|)}% +\index{Helmholtz!on atmospheric waves|)}% +\PageSep{52} + +Mr.~Denison's application of this theory consists +in supposing that the vibrations of the sea +and of lakes are the response of the water to +variations in the atmospheric pressure. The sea, +being squeezed down by the greater pressure, +should fall as the barometer rises, and conversely +should rise as the barometer falls. He is engaged +in a systematic comparison of the simultaneous +excursions of the water and of the barometer +on Lake Huron. Thus far the evidence +seems decidedly favorable to the theory. He +concludes that when the water is least disturbed, +so also is the barometric trace; and that when +the undulations of the lake become large and +rapid, the atmospheric waves recorded by the +barometer have the same character. There is +also a considerable degree of correspondence +between the periods of the two oscillations. The +smaller undulations of the water correspond with +the shorter air-waves, and are magnified as they +run into narrower and shallower places, so as to +make conspicuous ``vibrations.'' + +It is interesting to note that the vibrations of +the water have a tendency to appear before those +in the barometer, so that they seem to give a +warning of approaching change of weather. It +is thus not impossible that we here have the +foreshadowing of a new form of meteorological +instrument, which may be of service in the forecasting +of the weather. +\PageSep{53} + +I must, however, emphasize that these conclusions +are preliminary and tentative, and that +much observation will be needed before they can +be established as definite truths. Whatever +may be the outcome, the investigation appears +promising, and it is certainly already interesting. +\index{Denison, F. Napier, vibrations and seiches on lakes|)}% +\index{Lakes!vibrations|)}% +\index{Vibration of lakes|)}% + +\begin{Authorities} +Papers by Dr.~Forel on Seiches. +\index{Forel!list of papers}% + +\Journal ``Bibliothèque Universelle, Archives des Sciences physiques +et naturelles,'' Geneva:--- + +\Paper{\Title{Formule des Seiches}, 1876.} + +\Paper{\Title{Limnimètre Enregistreur}, 1876.} + +\Paper{\Title{Essai monographique}, 1877.} + +\Paper{\Title{Causes des Seiches}, \DPchg{Sept.~15}{15~Sept.}, 1878.} + +\Paper{\Title{Limnographe}, 15~Déc., 1878.} + +\Paper{\Title{Seiche du \emph{20}~Février, \emph{1879}}, 15~Avril, 1879.} + +\Paper{\Title{Seiches dicrotes}, 15~Jan., 1880.} + +\Paper{\Title{Formules des Seiches}, 15~Sept., 1885.} + +\Journal ``Bulletin de la Soc.\ Vaudoise des Sciences naturelles:''--- + +\Paper{\Title{Première Étude}, 1873.} + +\Paper{\Title{Deuxième Étude}, 1875.} + +\Paper{\Title{Limnimétrie du Lac Léman}. I\iere~Série. Bull.~xiv.\ 1877. +II\ieme~Série. Bull.~xv. III\ieme~Série. Bull.~xv. 1879.} + +\Journal ``Actes de la Soc.\ helv.\ Andermatt:''--- + +\Paper{\Title{Les Seiches, Vagues d'Oscillation}, 1875.} + +\Journal ``Association Française pour l'avancement,'' etc.:--- + +\Paper{\Title{Seiches et Vibrations}, Congrès de Montpelier, 1879.} + +\Journal ``Annales de Chimie et de Physique:''--- + +\Paper{\Title{Les Seiches, Vagues d'Oscillation}, 1876.} + +\Paper{\Title{Un Limnimètre Enregistreur}, 1876.} + +\TB + +Helmholtz, Sitzungsberichte der Preuss.\ Akad.\ der Wissenschaft, +July~25, 1889; transl.\ by Abbe in \Title{Smithsonian Reports}. +\PageSep{54} + +F.~Napier Denison:--- + +\Paper{\Title{Secondary Undulations~\dots\ found in Tide-Gauges.} ``Proc.\ +Canadian Institute,'' Jan.~16, 1897.} + +\Paper{\Title{The Great Lakes as a Sensitive Barometer.} ``Proc.\ Canadian +Institute,'' Feb.~6, 1897.} +\index{Lakes!seiches in|)}% + +\Paper{Same title, but different paper, ``Canadian Engineer,'' Oct.\ +and Nov., 1897.} +\end{Authorities} +\index{Forel!list of papers}% +\PageSep{55} + + +%[** TN: Footnote mark handled by \Chapter logic] +\Chapter{III} +{Tides in Rivers---Tide Mills} + +\footnotetext{The account of the bore in this chapter appeared as an + article in the \Title{Century Magazine} for August,~1898. The illustrations + then used are now reproduced, through the courtesy of + the proprietors.} + +\First{Since} most important towns are situated on +\index{Rivers!tide wave in|(}% +rivers or on estuaries, a large proportion of our +tidal observations relates to such sites. I shall +therefore now consider the curious, and at times +very striking phenomena which attend the rise +and fall of the tide in rivers. + +The sea resembles a large pond in which the +water rises and falls with the oceanic tide, and a +river is a canal which leads into it. The rhythmical +rise and fall of the sea generate waves +which would travel up the river, whatever were +the cause of the oscillation of the sea. Accordingly, +a tide wave in a river owes its origin +directly to the tide in the sea, which is itself +produced by the tidal attractions of the sun and +moon. + +We have seen in \Ref{Chapter}{II}.\ that long waves +progress in shallow water at a speed which depends +only on the depth of the water, and that +\PageSep{56} +waves are to be considered as long when their +length is at least twice the depth of the water. +Now the tide wave in a river is many hundreds +of times as long as the depth, and it must therefore +progress at a speed dependent only on the +depth. That speed is very slow compared with +the motion of the great tide wave in the open +ocean. + +The terms ``ebb'' and ``flow'' are applied to +\index{Ebb and flow defined}% +\index{Flow and ebb defined}% +tidal currents. The current ebbs when the +water is receding from the land seaward, and +flows when it is approaching the shore. On the +open seacoast the water ebbs as the water-level +falls, and it flows as the water rises. Thus at +high and low tide the water is neither flowing +landward nor ebbing seaward, and we say that +it is slack or dead. In this case ebb and flow +are simultaneous with rise and fall, and it is not +uncommon to hear the two terms used synonymously; +but we shall see that this usage is incorrect. + +I begin by considering the tidal currents in a +river of uniform depth, so sluggish in its own +proper current that it may be considered as a +stagnant canal, and the only currents to be considered +are tidal currents. At any point on the +river bank there is a certain mean height of +water, such that the water rises as much above +that level at high water as it falls below it at +low water. The law of tidal current is, then, +\index{Currents, tidal, in rivers}% +\PageSep{57} +very simple. Whenever the water stands above +the mean level the current is up-stream and progresses +along with the tide wave; and whenever +it stands below mean level the current is down-stream +and progresses in the direction contrary +to the tide wave. Since the current is up-stream +when the water is higher than the mean, and +down-stream when it is lower, it is obvious that +when it stands exactly at mean level the current +is neither up nor down, and the water is slack +or dead. Also, at the moment of high water +the current is most rapid up-stream, and at low +water it is most rapid down-stream. Hence the +tidal current ``flows'' for a long time after high +water has passed and when the water-level is +falling, and ``ebbs'' for a long time after low +water and when the water-level is rising. + +The law of tidal currents in a uniform canal +communicating with the sea is thus very different +from that which holds on an open seacoast, +where slack water occurs at high and at low +water, instead of at mean water. But rivers +gradually broaden and become deeper as they +approach the coast, and therefore the tidal currents +in actual estuaries must be intermediate +between the two cases of the open seacoast and +the uniform canal. + +A river has also to deliver a large quantity of +water into the sea in the course of a single tidal +oscillation, and its own proper current is superposed +\PageSep{58} +on the tidal currents. Hence in actual +rivers the resultant current continues to flow up +stream after high water is reached, with falling +water-level, but ceases flowing before mean water-level +is reached, and the resultant current ebbs +down-stream after low water, and continues to +ebb with the rising tide until mean water is +reached, and usually for some time afterward. +The downward stream, in fact, lasts longer than +the upward one. The moments at which the +currents change will differ in each river according +to the depth, the rise and fall of the tide at +the mouth, and the amount of water delivered +by the river. An obvious consequence of this +is that in rivers the tide rises quicker than it +falls, so that a shorter time elapses between low +water and high water than between high water +and low water. + +The tide wave in a river has another peculiarity +of which I have not yet spoken. The complete +theory of waves would be too technical for a book +of this sort, and I must ask the reader to accept +as a fact that a wave cannot progress along a +river without changing its shape. The change +is such that the front slope of the wave gradually +gets steeper, and the rear slope becomes more +gradual. This is illustrated in~\fig{15}, which +shows the progress of a train of waves in shallow +water as calculated theoretically. If the +steepening of the advancing slope of a wave +\PageSep{59} +were carried to an extreme, the wave would present +the form of a wall of water; but the mere +advance of a wave into shallow water would by +itself never suffice to produce so great a change +of form without the concurrence of the natural +\Figure{15}{Progressive Change of a Train +of Waves in Shallow Water}{png} +stream of the river. The downward current in +the river has, in fact, a very important influence +in heading the sea-water back, and this coöperates +with the natural change in the shape of a +wave as it runs into shallow water, so as to exaggerate +the steepness of the advancing slope of +the wave. + +There are in the estuaries of many rivers +\index{Rivers!tide wave in|)}% +broad flats of mud or sand which are nearly dry +at low water, and in such situations the tide not +unfrequently rises with such great rapidity that +the wave assumes the form of a wall of water. +This sort of tide wave is called a ``bore,'' and in +\index{Bore!definition}% +French \textit{mascaret}. Notwithstanding the striking +nature of the phenomenon, very little has been +published on the subject, and I know of only one +series of systematic observations of the bore. +As the account to which I refer is contained in +the official publications of the English Admiralty, +it has probably come under the notice of only a +\PageSep{60} +small circle of readers. But the experiences of +the men engaged in making these observations +were so striking that an account of them should +prove of interest to the general public. I have, +moreover, through the kindness of Admiral Sir +William Wharton and of Captain Moore, the +\index{Moore, Captain!survey of Tsien-Tang-Kiang|(}% +advantage of supplementing verbal description +by photographs. + +The estuary on which the observations were +made is that of the Tsien-Tang-Kiang, a considerable +\index{Tsien-Tang-Kiang, the bore in|(}% +river which flows into the China Sea about +sixty miles south of the great Yang-Tse-Kiang. +At most places the bore occurs only intermittently, +but in this case it travels up the river at +every tide. The bore may be observed within +seventy miles of Shanghai, and within an easy +walk of the great city of Hangchow; and yet +\index{Hangchow, the bore at|(}% +nothing more than a mere mention of it is to be +found in any previous publication. + +In 1888 Captain Moore, R.~N., in command +of Her Majesty's surveying ship Rambler, +thought that it was desirable to make a thorough +survey of the river and estuary. He returned +to the same station in~1892; and the account +which I give of his survey is derived from reports +drawn up after his two visits. The annexed +sketch-map shows the estuary of the +Tsien-Tang, and the few places to which I shall +have occasion to refer are marked thereon. + +On the morning of September~19, 1888, the +\PageSep{61} +Rambler was moored near an island, named +after the ship, to the southwest of Chapu Bay; +and on the~20th the two steam cutters Pandora +and Gulnare, towing the sailing cutter +\Figure{16}{Chart of the Estuary of the Tsien-Tang-Kiang}{png} +Brunswick, left the ship with instruments for +observing and a week's provisions. + +Captain Moore had no reason to suspect that +the tidal currents would prove dangerous out +in the estuary, and he proposed to go up the +estuary about thirty miles to Haining, and then +follow the next succeeding bore up-stream to +Hangchow. Running up-stream with the flood, +all went well until about~11.30, when they were +about fifteen miles southwest by west of Kanpu. +The leading boat, the Pandora, here grounded, +and anchored quickly, but swung round violently +as far as the keel would let her. The other +boats, being unable to stop, came up rapidly; +and the Gulnare, casting off the Brunswick, +\PageSep{62} +struck the Pandora, and then drove on to and +over the bank, and anchored. The boats soon +floated in the rising flood, and although the engines +of the steam cutters were kept going +full speed, all three boats dragged their anchors +in an eleven-knot stream. When the flood +slackened, the three boats pursued their course +to the mouth of the river, where they arrived +about 4~\PM. The ebb was, however, so violent +that they were unable to anchor near one another. +Their positions were chosen by the advice of +some junkmen, who told Captain Moore, very +erroneously as it turned out, that they would be +safe from the night bore. + +The night was calm, and at~11.29 the murmur +of the bore was heard to the eastward; it could +be seen at~11.55, and passed with a roar at~12.20, +well over toward the opposite bank, as predicted +by the Chinese. The danger was now supposed +to be past; but at~1~\AM\ a current of extreme +violence caught the Pandora, and she had +much difficulty to avoid shipwreck. In the +morning it was found that her rudder-post and +propeller-guard were broken, and the Brunswick +and Gulnare were nowhere to be seen. +They had, in fact, been in considerable danger, +and had dragged their anchors three miles up +the river. At 12.20~\AM\ they had been struck +by a violent rush of water in a succession of big +ripples. In a few moments they were afloat in +\PageSep{63} +an eight-knot current; in ten minutes the water +rose nine feet, and the boats began to drag their +anchors, although the engines of the Gulnare +were kept going full speed. After the boats had +dragged for three miles, the rush subsided, and +when the anchor was hove up the pea and the +greater part of the chain were as bright as polished +silver. + +This account shows that all the boats were in +imminent danger, and that great skill was needed +to save them. After this experience and warning, +the survey was continued almost entirely +from the shore. + +The junks which navigate the river are well +aware of the dangers to which the English boats +were exposed, and they have an ingenious method +of avoiding them. At various places on the +bank of the river there are shelter platforms, of +which I show an illustration in~\fig{17}. Immediately +after the passing of the bore the +\index{Bore!bore-shelter}% +junks run up-stream with the after-rush and +make for one of these shelters, where they allow +themselves to be left stranded on the raised +platform shown in the picture. At the end of +this platform there is a sort of round tower +jutting out into the stream. The object of this +is to deflect the main wave of the bore so as to +protect the junks from danger. After the passage +of the bore, the water rises on the platform +very rapidly, but the junks are just able to float +\PageSep{64} +in safety. Captain Moore gives a graphic account +of the spectacle afforded by the junks as +they go up-stream, and describes how on one +occasion he saw no less than thirty junks swept +\Figure{17}{Bore-Shelter on the Tsien-Tang-Kiang}{jpg} +up in the after-rush, at a rate of ten knots, past +the town of Haining toward Hangchow, with all +sail set but with their bows in every direction. + +Measurements of the water-level were made +in the course of the survey, and the results, in +the form of a diagram, \fig{18}, exhibit the nature +of the bore with admirable clearness. The +observations of water-level were taken simultaneously +at three places, viz., Volcano Island +in the estuary, Rambler Island near the mouth +of the river, and Haining, twenty-six miles up +the river. In the figure, the distance between +\PageSep{65} +the lines marked Rambler and Volcano represents +fifty-one miles, and that between Rambler +and Haining twenty-six miles. The vertical +scales show the height of water, measured in +feet, above and below the mean level of the +water at these three points. The lines joining +these vertical scales, marked with the hours of +the clock, show the height of the water simultaneously. +The hour of~8.30 is indicated by +the lowest line it shows that the water was +one foot below mean level at Volcano Island, +twelve feet below at Rambler Island, and eight +feet below at Haining. Thus the water sloped +down from Haining to Rambler, and from Volcano +to Rambler; the water was running up the +estuary toward Rambler Island, and down the +estuary to the same point. At 9~and at~9.30 +there was no great change, but the water had +risen two or three feet at Volcano Island and at +Rambler Island. By ten~o'clock the water was +rising rapidly at Rambler Island, so that there +was a nearly uniform slope up the river from +Volcano Island to Haining. The rise at Rambler +Island then continued to be very rapid, +while the water at Haining remained almost +stationary. This state of affairs went on until +midnight, by which time the water had risen +twenty-one feet at Rambler Island, and about +six feet at Volcano Island, but had not yet risen +at all at Haining. No doubt through the whole +\PageSep{66} +of this time the water was running down the +river from Haining towards its mouth. It is +clear that this was a state of strain which could +not continue long, for there was over twenty +feet of difference of level between Rambler +Island, outside, and Haining, in the river. Almost +exactly at midnight the strain broke down +and the bore started somewhere between Rambler +\index{Bore!diagram of rise in Tsien-Tang}% +Island and Kanpu, and rushed up the river +in a wall of water twelve feet high. This result +is indicated in the figure by the presence of two +lines marked ``midnight.'' After the bore had +\Figure{18}{Diagram of the Flow of the Tide on +the Tsien-Tang-Kiang}{jpg} +passed there was an after-rush that carried the +water up eight feet more. It was on this that +the junks were swept up the stream, as already +described. At~1.30 the after-rush was over, +\PageSep{67} +but the water was still somewhat higher at +Rambler Island than at Haining, and a gentle +current continued to set up-stream. The water +then began to fall at Rambler Island, while it +continued to rise at Haining up to three o'clock. +At this point the ebb of the tide sets in. I do +not reproduce the figure which exhibits the fall +of the water in the ebbing tide, for it may suffice +to say that there is no bore down-stream, +\index{Bore!pictures}% +although there is at one time a very violent +current. + +In 1892 Captain Moore succeeded, with considerable +\index{Moore, Captain!illustrations of bore}% +difficulty, in obtaining photographs of +the bore as it passed Haining. They tell more +of the violence of the wave than could be conveyed +by any amount of description. The photographs, +reproduced in~\fig{19}, do not, however, +show that the broken water in the rear of the +crest is often disturbed by a secondary roller, or +miniature wave, which leaps up, from time to +time, as if struck by some unseen force, and disappears +in a cloud of spray. These breakers +were sometimes twenty to thirty feet above the +level of the river in front of the bore. + +The upper of these pictures is from a photograph, +taken at a height of twenty-seven feet +above the river, as the bore passed Haining on +October~10, 1892. The height of this bore was +eleven feet. The lower pictures, also taken at +Haining, represent the passage of the bore on +\PageSep{68} +October~9, 1892. The first of these photographs +was taken at 1.29~\PM, and the second +represents the view only one minute later. + +The Chinese regard the bore with superstitious +\index{Bore!Chinese superstition|(}% +\index{Chinese!superstition as to bore|(}% +reverence, and their explanation, which I quote +from Captain Moore's report, is as follows: +``Many hundred years ago there was a certain +general who had obtained many victories over +the enemies of the Emperor, and who, being +constantly successful and deservedly popular +among his countrymen, excited the jealousy of +his sovereign, who had for some time observed +with secret wrath his growing influence. The +Emperor accordingly caused him to be assassinated +and thrown into the Tsien-Tang-Kiang, +where his spirit conceived the idea of revenging +itself by bringing the tide in from the ocean in +such force as to overwhelm the city of Hangchow, +then the magnificent capital of the empire. +As my interpreter, who has been for some years +in America, put it, `his sowl felt a sort of ugly-like +arter the many battles he had got for the +Emperor.' The spirit so far succeeded as to +flood a large portion of the country, when the +Emperor, becoming alarmed at the distress and +loss of property occasioned, endeavored to enter +into a sort of compact with it by burning paper +and offering food upon the sea-wall. This, however, +did not have the desired effect, as the high +tide came in as before; and it was at last determined +\PageSep{69} +\ifthenelse{\boolean{ForPrinting}}{% + \TallFig[0.7]{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{jpg} +}{% + \Figure[0.75]{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{jpg} +} +\index{Wharton, Sir W. J., illustration of bore}% +\PageSep{70} +to erect a pagoda at the spot where the +worst breach in the embankment had been made. +Hence the origin of the Bhota Pagoda. A +pagoda induces the good \textit{fungshui}, or spirit. +After it was built the flood tide, though it still +continued to come in the shape of a bore, did +not flood the country as before.'' + +We ``foreign devils'' may take the liberty of +suspecting that the repairs to the embankment +had also some share in this beneficial result. + +This story is remarkable in that it refers to +the reign of an Emperor whose historical existence +is undoubted. It thus differs from many +of the mythical stories which have been invented +by primitive peoples to explain great natural +phenomena. There is good reason to suppose, +in fact, that this bore had no existence some centuries +ago; for Marco Polo, in the thirteenth +\index{Marco Polo, resident of Hangchow}% +century, stayed about a year and a half at +Hangchow, and gives so faithful and minute +\index{Hangchow, the bore at|)}% +an account of that great town that it is almost +impossible to believe that he would have omitted +to notice a fact so striking. But the Emperor +referred to in the Chinese legend reigned some +centuries before the days of Marco Polo, so that +we have reason to believe that the bore is intermittent. +\index{Bore!Chinese superstition|)}% +I have also learned from Captain +Moore himself that at the time of the great +\index{Moore, Captain!survey of Tsien-Tang-Kiang|)}% +Taiping rebellion, the suppression of which was +principally due to ``Chinese'' Gordon, the intensity +\index{Chinese!superstition as to bore|)}% +\index{Tsien-Tang-Kiang, the bore in|)}% +\PageSep{71} +of the bore was far less than it is to-day. +\index{Bore!rivers where found}% +This shows that the bore is liable to great variability, +according as the silting of the estuary +changes. + +The people at Haining still continue to pay +religious reverence to the bore, and on one of +the days when Captain Moore was making observations +some five or six thousand people assembled +on the river-wall to propitiate the god of +the waters by throwing in offerings. This was +the occasion of one of the highest bores at spring +tide, and the rebound of the bore from the sea-wall, +and the sudden heaping up of the waters +as the flood conformed to the narrow mouth of +the river, here barely a mile in width at low +water, was a magnificent spectacle. A series of +breakers were formed on the back of the advancing +flood, which for over five minutes were not +less than twenty-five feet above the level of the +river in front of the bore. On this occasion +Captain Moore made a rough estimate that a +million and three quarters of tons of water passed +the point of observation in one minute. + +The bore of which I have given an account is +perhaps the largest known; but relatively small +ones are to be observed on the Severn and Wye +\index{Severn, bore in the}% +\index{Wye, bore in the}% +in England, on the Seine in France, on the Petitcodiac +\index{Petitcodiac, bore in the}% +\index{Seine, bore in the}% +in Canada, on the Hugli in India, and +\index{Hugli, bore on the}% +doubtless in many other places. In general, +however, it is only at spring tides and with certain +\PageSep{72} +winds that the phenomenon is at all striking. +In September,~1897, I was on the banks of the +Severn at spring tide; but there was no proper +bore, and only a succession of waves up-stream, +\index{Bore!causes}% +and a rapid rise of water-level. + +I have shown, at the beginning of this chapter, +that the heading back of the sea water by +the natural current of a river, and the progressive +change of shape of a wave in shallow water combine +to produce a rapid rise of the tide in rivers. +But the explanation of the bore, as resulting +from these causes, is incomplete, because it leaves +their relative importance indeterminate, and +serves rather to explain a rapid rise than an absolutely +sudden one. I think that it would be +impossible, from the mere inspection of an estuary, +to say whether there would be a bore there; +we could only say that the situation looked +promising or the reverse. + +The capriciousness of the appearance of the +bore proves in fact that it depends on a very nice +balance between conflicting forces, and the irregularity +in the depth and form of an estuary renders +the exact calculation of the form of the +rising tide an impossibility. It would be easy +to imitate the bore experimentally on a small +scale; but, as in many other physical problems, +we must rest satisfied with a general comprehension +of the causes which produce the observed +result. +\PageSep{73} + +The manner in which the Chinese avail themselves +of the after-rush for ascending the river +affords an illustration of the utilization by mankind +of tidal energy. In going up-stream, a +\index{Energy, tidal, utilization of}% +barge, say of one hundred tons, may rise some +twenty or thirty feet. There has, then, been +done upon that barge a work of from two to +three thousand foot-tons. Whence does this +energy come? Now, I say that it comes from +the rotation of the earth; for we are making the +tide do the work for us, and thus resisting the +tidal movement. But resistance to the tide has +the effect of diminishing the rate at which the +earth is spinning round. Hence it is the earth's +rotation which carries the barge up the river, and +we are retarding the earth's rotation and making +the day infinitesimally longer by using the tide +in this way. This resistance is of an analogous +character to that due to tidal friction, the consideration +of which I must defer to a future +chapter, as my present object is to consider the +uses which may be made of tidal energy. + +It has been supposed by many that when the +coal supply of the world has been exhausted we +shall fall back on the tides to do our work. But +a little consideration will show that although this +source of energy is boundless, there are other far +more accessible funds on which to draw. + +I saw some years ago a suggestion that the +rise and fall of old hulks on the tide would afford +\PageSep{74} +\index{Energy, tidal, utilization of}% +serviceable power. If we picture to ourselves the +immense weight of a large ship, we may be deluded +for a moment into agreement with this +project, but numerical calculation soon shows its +futility. The tide takes about six hours to rise +from low water to high water, and the same +period to fall again. Let us suppose that the +water rises ten feet, and that a hulk of $10,000$ +tons displacement is floating on it; then it is +easy to show that only twenty horse-power will +be developed by its rise and fall. We should +then require ten such hulks to develop as much +work as would be given by a steam engine of +very moderate size, and the expense of the installation +would be far better bestowed on water-wheels +in rivers or on wind-mills. I am glad to +\index{Mills worked by the tide}% +say that the projector of this scheme gave it up +when its relative insignificance was pointed out +to him. It is the only instance of which I ever +heard where an inventor was deterred by the impracticability +of his plan. + +We may, then, fairly conclude that, with existing +mechanical appliances, the attempt to utilize +the tide on an open coast is futile. But +where a large area of tidal water can be easily +trapped at high water, its fall may be made to +work mill-wheels or turbines with advantage. +The expense of building long jetties to catch the +water is prohibitive, and therefore tide mills are +only practicable where there exists an easily +\PageSep{75} +adaptable configuration of shoals in an estuary. +There are, no doubt, many such mills in the +\index{Mills worked by the tide}% +world, but the only one which I happen to have +seen is at Bembridge, in the Isle of Wight. At +this place embankments formed on the natural +shoals are furnished with lock-gates, and inclose +many acres of tidal water. The gates open automatically +with the rising tide, and the incipient +outward current at the turn of the tide closes +the gates again, so that the water is trapped. +The water then works a mill wheel of moderate +size. When we reflect on the intermittence of +work from low water to high water and the great +inequality of work with springs and neaps, it +may be doubted whether this mill is worth the +expense of retaining the embankments and lock-gates. + +We see then that, notwithstanding the boundless +energy of the tide, rivers and wind and fuel +are likely for all time to be incomparably more +important for the use of mankind. + +\begin{Authorities} +On waves in rivers see Airy's article on \Title{Tides and Waves} in +\index{Airy, Sir G. B.!tides in rivers}% +\index{Rivers!Airy on tide in}% +the ``Encyclopædia Metropolitana.'' Some of his results will +also be found in the article \Title{Tides} in the ``Encyclopædia Britannica.'' + +Commander Moore, R.~N., \Title{Report on the Bore of the Tsien-Tang-Kiang.} +Sold by Potter, Poultry, London,~1888. + +\Title{Further Report},~\&c., by the same author and publisher,~1893. +\end{Authorities} +\PageSep{76} + + +\Chapter{IV} +{Historical Sketch} + +\First{I Cannot} claim to have made extensive investigations +\index{History!of tidal theories|(}% +as to the ideas of mankind at different +periods on the subject of the tides, but I propose +in the present chapter to tell what I have +been able to discover. + +No doubt many mythologies contain stories +explanatory of the obvious connection between +the moon and the tide. But explanations, professing +at least to be scientific, would have been +brought forward at periods much later than +those when the mythological stories originated, +and I shall only speak of the former. + +I have to thank my colleagues at Cambridge +for the translations from the Chinese, Arabic, +\index{Chinese!theories of tide}% +Icelandic, and classical literatures of such passages +as they were able to discover. + +I learn from Professor Giles that Chinese +\index{Giles on Chinese theories of the tide}% +writers have suggested two causes for the tides: +first, that water is the blood of the earth, and +that the tides are the beating of its pulse; and +secondly, that the tides are caused by the earth +breathing. Ko~Hung, a writer of the fourth +century of our era, gives a somewhat obscure +\PageSep{77} +\index{Chinese!theories of tide}% +\index{Giles on Chinese theories of the tide}% +explanation of spring and neap tides. He says +that every month the sky moves eastward and +then westward, and hence the tides are greater +and smaller alternately. Summer tides are said +to be higher than winter tides, because in summer +the sun is in the south and the sky is $15,000$~li +($5,000$~miles) further off, and therefore in +summer the female or negative principle in nature +is weak, and the male or positive principle +strong. + +In China the diurnal inequality is such that +in summer the tide rises higher in the daytime +than in the night, whilst the converse is true +in winter. I suggest that this fact affords the +justification for the statement that the summer +tides are great. + +\TB + +Mr.~E.~G. Browne has translated for me the +\index{Arabian theories of tide|(}% +\index{Browne, E. G., Arabian theories of tide|(}% +following passage from the ``Wonders of Creation'' +of Zakariyy\bar{a} ibn Muhammad ibn Mahm\bar{u}d +al Qazv\bar{i}n\bar{i}, who died in \AD~1283.\footnote + {Wüstenfeld's edition, pp.~103,~104.} + +``Section treating of certain wonderful conditions +of the sea. + +``Know that at different periods of the four +seasons, and on the first and last days of the +months, and at certain hours of the night and +day, the seas have certain conditions as to the +rising of their waters and the flow and agitation +thereof. +\PageSep{78} + +``As to the rising of the waters, it is supposed +that when the sun acts on them it rarefies them, +and they expand and seek a space ampler than +that wherein they were before, and the one part +repels the other in the five directions eastwards, +westwards, southwards, northwards, and upwards, +and there arise at the same time various winds +on the shores of the sea. This is what is said +as to the cause of the rising of the waters. + +``As for the flow of certain seas at the time +of the rising of the moon, it is supposed that at +the bottom of such seas there are solid rocks +and hard stones, and that when the moon rises +over the surface of such a sea, its penetrating +rays reach these rocks and stones which are at +the bottom, and are then reflected back thence; +and the waters are heated and rarefied and seek +an ampler space and roll in waves towards the +seashore~\dots\ and so it continues as long as +the moon shines in mid-heaven. But when she +begins to decline, the boiling of the waters +ceases, and the particles cool and become dense +and return to their state of rest, and the currents +run according to their wont. This goes +on until the moon reaches the western horizon, +when the flow begins again, as it did when the +moon was in the eastern horizon. And this +flow continues until the moon is at the middle +of the sky below the horizon, when it ceases. +Then when the moon comes upward, the flow +\PageSep{79} +begins again until she reaches the eastern horizon. +This is the account of the flow and ebb +of the sea. + +``The agitation of the sea resembles the agitation +of the humours in men's bodies, for verily +as thou seest in the case of a sanguine or bilious +man,~\&c., the humours stirring in his body, and +then subsiding little by little; so likewise the +sea has matters which rise from time to time as +they gain strength, whereby it is thrown into +violent commotion which subsides little by little. +And this the Prophet (on whom be the blessings +of God and his peace) hath expressed in a poetical +manner, when he says: `Verily the Angel, +who is set over the seas, places his foot in the +sea and thence comes the flow; then he raises it +and thence comes the ebb.'\,'' +\index{Arabian theories of tide|)}% +\index{Browne, E. G., Arabian theories of tide|)}% + +\TB + +Mr.~Magnússon has kindly searched the old +\index{Icelandic theory of tides}% +\index{Magnússon on Icelandic theories of tides}% +Icelandic literature for references to the tides. +In the Rimbegla he finds this passage:--- + +``Beda the priest says that the tides follow +the moon, and that they ebb through her blowing +on them, but wax in consequence of her +movement.'' + +And again:--- + +``(At new moon) the moon stands in the way +of the sun and prevents him from drying up the +sea; she also drops down her own moisture. +For both these reasons, at every new moon, the +\PageSep{80} +\index{Icelandic theory of tides}% +\index{Magnússon on Icelandic theories of tides}% +ocean swells and makes those tides which we call +spring tides. But when the moon gets past the +sun, he throws down some of his heat upon +the sea, and diminishes thereby the fluidity of +the water. In this way the tides of the sea +are diminished.'' + +In another passage the author writes:--- + +``But when the moon is opposite to the sun, +the sun heats the ocean greatly, and as nothing +impedes that warmth, the ocean boils and the +sea flood is more impetuous than before---just +as one may see water rise in a kettle when it +boils violently. This we call spring tide.'' + +There seems to be a considerable inconsistency +in explaining one spring tide by the interception +of the sun's heat by the moon, and the next one +by the excess of that heat. + +But it is not necessary to search ancient literature +for grotesque theories of the tides. In +1722 E.~Barlow, gentleman, in ``An Exact Survey +of the Tide,''\footnote + {``The Second Edition, with Curious Maps.'' (London: John + Hooke, 1722.)} +attributes it to the pressure +of the moon on the atmosphere. And theories +not less absurd have been promulgated during +the last twenty years. + +\TB + +The Greeks and Romans, living on the shores +of the Mediterranean, had not much occasion to +learn about the tide, and the passages in classical +\PageSep{81} +\index{Greek!theory and description of tides|(}% +\index{Roman description of tides|(}% +literature which treat of this matter are but +few. But where the subject is touched on we +see clearly their great intellectual superiority over +those other peoples, whose ideas have just been +quoted. + +The only author who treats of the tide in any +\index{Strabo on tides|(}% +detail is Posidonius, and we have to rely for our +\index{Posidonius on tides|(}% +knowledge of his work entirely on quotations +from him by Strabo.\footnote + {My attention was drawn to Strabo by a passage in Sir W. + Thomson's (Lord Kelvin's) Popular Lectures, \Title{The Tides}, vol.~ii. + I have to thank Mr.~Duff for the translations which follow from + Strabo and Posidonius. The work consulted was Bake's \Title{Posidonius} + (Leiden,~1810), but Mr.~Duff tells me that the text is very + corrupt in some places, and he has therefore also consulted a + more recent text.} + +Posidonius says that Aristotle attributed the +\index{Aristotle on tides}% +flow and ebb of the sea at Cadiz to the mountainous +formation of the coast, but he very justly +pronounces this to be nonsense, particularly as +the coast of Spain is flat and sandy. He himself +attributes the tides to the moon's influence, and +the accuracy of his observations is proved by the +following interesting passage from Strabo:\footnotemark--- +\footnotetext{Teubner's \Title{Strabo},~i.\ p.~236.} + +``Posidonius says that the movement of the +ocean observes a regular series like a heavenly +body, there being a daily, monthly, and yearly +movement according to the influence of the +moon. For when the moon is above the (eastern) +horizon by the distance of one sign of the +zodiac (\ie~$30°$) the sea begins to flow, and encroaches +\PageSep{82} +visibly on the land until the moon +reaches the meridian. When she has passed the +meridian, the sea in turn ebbs gradually, until +the moon is above the western horizon by the +distance of one sign of the zodiac. The sea then +remains motionless while the moon is actually +setting, and still more so (\textit{sic}) so long as the +moon is moving beneath the earth as far as a +sign of the zodiac beneath the horizon. Then +the sea again advances until the moon has +reached the meridian below the earth; and retreats +while the moon is moving towards the east, +until she is the distance of a sign of the zodiac +below the horizon; it remains at rest until the +moon is the same distance above the horizon, and +then begins to flow again. Such is the daily +movement of the tides, according to Posidonius. + +``As to their monthly movement, he says that +the ebbs are greatest at the conjunctions [of +the sun and moon], and then grow less until the +time of half moon, and increase again until the +time of full moon, and grow less again until +the moon has waned to half. Then the increase +of the tide follows until the conjunction. But +the increases last longer and come quicker [this +phrase is very obscure]. + +``The yearly movements of the tides he says +he learned from the people of Cadiz. They told +him that the ebb and flow alike were greatest at +the summer solstice. He guesses for himself +\PageSep{83} +that the tides grow less from the solstice to the +equinox, and then increase between the equinox +and the winter solstice, and then grow less until +the spring equinox, and greater until the summer +solstice.'' + +This is an excellent account of the tides at +Cadiz, but I doubt whether there is any foundation +\index{Polibius on tides at Cadiz}% +for that part which was derived from hearsay. +Lord Kelvin remarks, however, that it is interesting +to note that inequalities extending over +the year should have been recognized. + +Strabo also says that there was a spring near +Cadiz in which the water rose and fell, and that +this was believed by the inhabitants, and by +Polybius, to be due to the influence of the ocean +tide, but Posidonius was not of this opinion. +Strabo says:--- + +``Posidonius denies this explanation. He says +there are two wells in the precinct of Hercules at +Cadiz, and a third in the city. Of the two former +the smaller runs dry while people are drawing +water from it, and when they stop drawing water +it fills again; the larger continues to supply +water all day, but, like all other wells, it falls +during the day but is replenished at night, when +the drawing of water has ceased. But since the +ebb tide often coincides with the replenishing of +the well, therefore, says Posidonius, the idle story +of the tidal influence has been believed by the +inhabitants.'' +\PageSep{84} + +Since the wells follow the sun, whilst the tide +follows the moon, the criticism of Posidonius is +a very just one. But Strabo blames him for +distrusting the Cadizians in a simple matter of +everyday experience, whilst accepting their evidence +as to an annual inequality in the tides. + +There is another very interesting passage in +Strabo, the meaning of which was obviously unknown +to the Dutch commentator Bake---and +indeed must necessarily have been unintelligible +to him at the time when he wrote, on account of +the then prevailing ignorance of tidal phenomena +in remoter parts of the world. Strabo +writes:--- + +``Anyhow Posidonius says that Seleucus of +\index{Diurnal inequality!observed by Seleucus}% +\index{Posidonius on tides|)}% +\index{Seleucus, observation of tides of Indian Ocean}% +the Red Sea [also called the Babylonian] declares +that there is a certain irregularity and regularity +in these phenomena [the tides], according +to the different positions [of the moon] in the +zodiac. While the moon is in the equinoctial +signs, the phenomena are regular; but while she +is in the signs of the solstices, there is irregularity +both in the height and speed of the tides, +and in the other signs there is regularity or the +reverse in proportion to their nearness to the solstices +or to the equinoxes.'' + +Now let us consider the meaning of this. +When the moon is in the equinoxes she is on +the equator, and when she is in the solstices she +is at her maximum distances to the north or +\PageSep{85} +south of the equator---or, as astronomers say, in +her greatest north or south declination. Hence +Seleucus means that, when the moon is on the +\index{Seleucus, observation of tides of Indian Ocean}% +equator, the tides follow one another, with two +equal high and low waters a day; but when she +is distant from the equator, the regular sequence +is interrupted. In other words, the diurnal +inequality (which I shall explain in a later chapter) +vanishes when the moon is on the equator, +and is at its maximum when the declination is +greatest. This is quite correct, and since the +diurnal inequality is almost evanescent in the +\index{Diurnal inequality!observed by Seleucus}% +Atlantic, whilst it is very great in the Indian +Ocean, especially about Aden, it is clear that +Seleucus had watched the sea there, just as we +should expect him to do from his place of origin. + +\TB + +Many centuries elapsed after the classical +period before any scientific thought was bestowed +on the tides. Kepler recognized the +\index{Kepler!ideas concerning tides}% +tendency of the water on the earth to move +towards the sun and the moon, but he was unable +to submit his theory to calculation. Galileo +\index{Galileo!blames Kepler for his tidal theory}% +expresses his regret that so acute a man as +Kepler should have produced a theory, which +appeared to him to reintroduce the occult qualities +of the ancient philosophers. His own explanation +referred the phenomenon to the rotation +of the earth, and he considered that it afforded +a principal proof of the Copernican system. +\index{Greek!theory and description of tides|)}% +\index{Roman description of tides|)}% +\index{Strabo on tides|)}% +\PageSep{86} +\index{Kepler!ideas concerning tides}% + +The theory of tide-generating force which will +be set forth in \Ref{Chapter}{V}.\ is due to Newton, +\index{Newton!founder of tidal theory}% +who expounded it in his ``Principia'' in~1687. +His theory affords the firm basis on which all +subsequent work has been laid. + +In 1738 the Academy of Sciences of Paris +offered the theory of the tides as the subject for +a prize. The authors of four essays received +prizes, viz., Daniel Bernoulli, Euler, Maclaurin, +\index{Bernoulli, Daniel, essay on tides}% +\index{Euler, essay on tides}% +\index{Maclaurin!essay on tides}% +and Cavalleri. The first three adopted, not only +\index{Cavalleri, essay on tides}% +the theory of gravitation, but also Newton's +theory to its fullest extent. A considerable +portion of Bernoulli's work is incorporated in +the account of the theory of the tides which I +shall give later. The essays of Euler and Maclaurin +contained remarkable advances in mathematical +knowledge, but did not add greatly to +the theory of the tides. The Jesuit priest +Cavalleri adopted the theory of vortices to explain +the tides, and it is not worth while to +follow him in his erroneous and obsolete speculations. + +Nothing of importance was added to our +knowledge until the great French mathematician +Laplace took up the subject in~1774. It was he +\index{Laplace!theory of tides|(}% +who for the first time fully recognized the difficulty +of the problem, and showed that the earth's +rotation is an essential feature in the conditions. +The actual treatment of the tidal problem is in +effect due to Laplace, although the mode of +\PageSep{87} +presentment of the theory has come to differ +considerably from his. + +Subsequently to Laplace, the most important +workers in this field have been Sir John Lubbock +\index{Lubbock, Sir J., senior, on tides}% +senior, Whewell, Airy, and Lord Kelvin. +\index{Kelvin, Lord!initiates harmonic analysis}% +\index{Whewell!on tides}% +\index{Whewell!empirical construction of tide tables|(}% +The work of Lubbock and Whewell is chiefly +remarkable for the coördination and analysis of +enormous masses of data at various ports, and +the construction of trustworthy tide tables. +Airy contributed an important review of the +whole tidal theory. He also studied profoundly +the theory of waves in canals, and considered +the effects of frictional resistances on the progress +of tidal and other waves. + +Lord Kelvin initiated a new and powerful +method of considering tidal oscillations. His +method possesses a close analogy with that already +used in discussing the irregularities in the +motions of the moon and planets. His merit +consists in the clear conception that the plan of +procedure which has been so successful in the +one case would be applicable to the other. The +difference between the laws of the moon's motion +and those of tidal oscillations is, however, +so great that there is scarcely any superficial +resemblance between the two methods. This +so-called ``harmonic analysis'' of the tides is +\index{Harmonic analysis!initiated by Lord Kelvin}% +daily growing in favor in the eyes of men of +science, and is likely to supersede all the older +methods. I shall explain it in a future chapter. +\PageSep{88} + +Amongst all the grand work which has been +bestowed on this difficult subject, Newton stands +out first, and next to him we must rank Laplace. +However original any future contribution to the +science of tides may be, it would seem as though +it must perforce be based on the work of these +two. The exposition which I shall give hereafter +of the theory of oceanic tides is based on +the work of Newton, Bernoulli, Laplace, and +\index{Bernoulli, Daniel, essay on tides}% +\index{Laplace!theory of tides|)}% +Kelvin, in proportions of which it would be +difficult to assign the relative importance. + +\TB + +The connection between the moon and the +tide is so obvious that long before the formulation +of a satisfactory theory fairly accurate predictions +of the tides were made and published. +On this head Whewell\footnote + {\Title{History of the Inductive Sciences}, 1837, vol.~ii.\ p.~248 \textit{et~seq.}} +\index{History!of tidal theories|)}% +has the following interesting +passage:--- + +``The course which analogy would have recommended +for the cultivation of our knowledge of +tides would have been to ascertain by an analysis +of long series of observations, the effects of +changes in the time of transit, parallax, and +declination of the moon, and thus to obtain the +laws of phenomena; and then to proceed to +investigate the laws of causation. + +``Though this was not the course followed by +mathematical theorists, it was really pursued by +those who practically calculated tide tables; and +\PageSep{89} +the application of knowledge to the useful purposes +of life, being thus separated from the +promotion of the theory, was naturally treated +as a gainful property, and preserved by secrecy. +\dots~Liverpool, London, and other places, had +their tide tables, constructed by undivulged +methods, which methods, in some instances at +least, were handed down from father to son for +several generations as a family possession; and +the publication of new tables accompanied by a +statement of the mode of calculation was resented +as an infringement of the rights of property. + +``The mode in which these secret methods +were invented was that which we have pointed +out,---the analysis of a considerable series of +observations. Probably the best example of this +was afforded by the Liverpool tide tables. These +were deduced by a clergyman named Holden, +from observations made at that port by a harbor +master of the name of Hutchinson, who was +led, by a love of such pursuits, to observe the +tides for above twenty years, day and night. +Holden's tables, founded on four years of these +observations, were remarkably accurate. + +``At length men of science began to perceive +that such calculations were part of their business; +and that they were called upon, as the +guardians of the established theory of the universe, +to compare it in the greatest possible +\PageSep{90} +detail with the facts. Mr.~Lubbock was the +first mathematician who undertook the extensive +labors which such a conviction suggested. Finding +that regular tide observations had been made +at the London docks from~1795, he took nineteen +years of these (purposely selecting the +length of the cycle of the motions of the lunar +orbit), and caused them (in~1831) to be analyzed +by Mr.~Dessiou, an expert calculator. He thus +obtained tables for the effect of the moon's +declination, parallax, and hour of transit, on the +tides; and was enabled to produce tide tables +founded upon the data thus obtained. Some +mistakes in these as first published (mistakes unimportant +as to the theoretical value of the work) +served to show the jealousy of the practical tide +table calculators, by the acrimony with which the +oversights were dwelt upon; but in a very few +years the tables thus produced by an open and scientific +process were more exact than those which +resulted from any of the secrets; and thus practice +was brought into its proper subordination to +theory.'' +\index{Whewell!empirical construction of tide tables|)}% + +\begin{Authorities} +The history from Galileo to Laplace is to be found in the +\Title{Mécanique Céleste} of Laplace, book~xiii, chapter~i. + +The other authorities are quoted in the text or in footnotes. +\end{Authorities} +\PageSep{91} + + +\Chapter{V} +{Tide-generating Force} + +\First{It} would need mathematical reasoning to fully +\index{Centripetal and centrifugal forces|(}% +explain how the attractions of the sun and moon +give rise to tide-generating forces. But as this +\index{Forces!centripetal and centrifugal|(}% +book is not intended for the mathematician, I +must endeavor to dispense with technical language. + +A body in motion will move in a straight line, +unless it is deflected from its straight path by +some external force, and the resistance to the +deflection is said to be due to inertia. The motion +of the body then is equivalent in its effect +to a force which opposes the deflection due to +the external force, and in many cases it is permissible +to abstract our attention from the motion +of the system and to regard it as at rest, if +at the same time we introduce the proper ideal +forces, due to inertia, so that they shall balance +the action of the real external forces. + +If I tie a string to a stone and whirl it round, +the string is thrown into a state of tension. The +natural tendency of the stone, at each instant, is +to move onward in a straight line, but it is continuously +deflected from its straight path by the +\PageSep{92} +tension of the string. In this case the ideal +force, due to inertia, whereby the stone resists +its continuous deflection, is called centrifugal +force. This force is in reality only a substitute +for the motion, but if we withdraw our attention +from the motion, it may be regarded as a reality. + +The centrifugal force is transmitted to my +hand through the string, and I thus experience +an outward or centrifugal tendency. But the +stone itself is continually pulled inward by the +string, and the force is called centripetal. When +a string is under tension, as in this experiment, +it is subject to equal and opposite forces, so that +the tension implies the existence of a pair of +forces, one towards and the other away from the +centre of rotation. The force is to be regarded +as away from the centre when we consider the +sensation of the whirler, and as towards the centre +when we consider the thing whirled. A similar +double view occurs in commerce, where a +transaction which stands on the credit side in the +books of one merchant appears on the debit side +in the books of the other. + +This simple experiment exemplifies the mechanism +by which the moon is kept revolving round +the earth. There is not of course any visible +connection between the two bodies, but an invisible +bond is provided by the attraction of gravity, +which replaces the string which unites the +stone to the hand. The moon, then, whirls +\PageSep{93} +\index{Forces!tide-generating|(}% +\index{Orbit!of moon and earth|(}% +round the earth at just such a rate and at just +\index{Earth and moon!diagram}% +such a distance, that her resistance to circular +motion, called centrifugal force, is counterbalanced +by the centripetal tendency of gravity. If +\index{Centripetal and centrifugal forces|)}% +she were nearer to us the attraction of gravity +would be greater, and she would have to go +round the earth faster, so as to make enough +centrifugal force to counterbalance the greater +\Figure[0.8]{20}{Earth and Moon}{png} +gravity. The converse would be true, and the +moon would go round slower, if she were further +from us. + +The moon and the earth go round the sun in +companionship once in a year, but this annual +motion does not affect the interaction between +them, and we may put aside the orbital motion +of the earth, and suppose the moon and earth to +\index{Moon and earth!diagram}% +be the only pair of bodies in existence. When +the principle involved in a purely lunar tide is +grasped, the action of the sun in producing a +\index{Forces!centripetal and centrifugal|)}% +\PageSep{94} +solar tide will become obvious. But the analogy +of the string and stone is imperfect in one +respect where the distinction is important; the +moon, in fact, does not revolve exactly about +the earth, but about the centre of gravity of +the earth and moon. The earth is eighty times +as heavy as the moon, and so this centre of gravity +is not very far from the earth's centre. The +upper part of \fig{20} is intended to represent a +planet and its satellite; the lower part shows +the earth and the moon in their true proportions. +The upper figure is more convenient for +our present argument, and the planet and satellite +may be described as the earth and the moon, +notwithstanding the exaggeration of their relative +proportions. The point~$G$ is the centre of +gravity of the two, and the axis about which +they revolve passes through~$G$. This point is +sufficiently near to the centre of the earth to +permit us, for many purposes, to speak of the +moon as revolving round the earth. But in the +present case we must be more accurate and must +regard the moon and earth as revolving round~$G$, +their centre of gravity. The moon and earth +are on opposite sides of this point, and describe +circles round it. The distance of the moon's +centre from~$G$ is $237,000$~miles, whilst that of +the earth's centre is only $\DPchg{3000}{3,000}$~miles in the opposite +direction. The $\DPchg{3000}{3,000}$~and $237,000$~miles +together make up the $240,000$~miles which separate +the centres of the two bodies. +\PageSep{95} + +A system may now be devised so as to resemble +the earth and moon more closely than that +of the string and stone with which I began. If +a large stone and a small one are attached to one +another by a light and stiff rod, the system can +be balanced horizontally about a point in the rod +called the centre of gravity~$G$. The two weights +may then be set whirling about a pivot at~$G$, so +that the rod shall always be horizontal. In consequence +of the rotation the rod is brought into +a state of stress, just as was the string in the +first example, and the centripetal stress in the +rod exactly counterbalances the centrifugal force. +The big and the little stones now correspond to +the earth and the moon, and the stress in the rod +plays the same part as the invisible bond of +gravity between the earth and the moon. Fixing +our attention on the smaller stone or moon +at the end of the longer arm of the rod, we see +that the total centrifugal force acting on the +moon, as it revolves round the centre of gravity, +is equal and opposite to the attraction of the +earth on the moon. On considering the short +arm of the rod between the pivot and the big +stone, we see also that the centrifugal force acting +on the earth is equal and opposite to the +attraction of the moon on it. In this experiment +as well as in the former one, we consider +the total of centrifugal force and of attraction, +but every particle of both the celestial bodies is +\index{Orbit!of moon and earth|)}% +\PageSep{96} +\index{Davis, method of presenting tide-generating force}% +acted on by these forces, and accordingly a +closer analysis is necessary. + +It will now simplify matters if we make a supposition +which departs from actuality, introducing +the true conditions at a later stage in the +argument. + +The earth's centre describes a circle about the +centre of gravity~$G$, with a radius of $\DPchg{3000}{3,000}$~miles, +and the period of the revolution is of course one +month. Now whilst this motion of revolution +of the earth's centre continues, let it be supposed +that the diurnal rotation is annulled. As this +is a mode of revolution which differs from that +of a wheel, it is well to explain exactly what is +meant by the annulment of the diurnal rotation. +This is illustrated in~\fig{21}, which shows the +successive positions assumed by an arrow in revolution +without rotation. The shaft of the arrow +always remains parallel to the same direction in +space, and therefore it does not rotate, although +the whole arrow revolves. It is obvious that every +particle of the arrow describes a circle of the +same radius, but that the circles described by +them are not concentric. The circles described +by the point and by the base of the arrow are +shown in the figure, and their centres are separated +by a distance equal to the length of the +arrow. Now the centrifugal force on a revolving +particle acts along the radius of the circle described, +and in this case the radii of the circles +\PageSep{97} +described by any two particles in the arrow are +always parallel. The parallelism of the centrifugal +forces at the two ends of the arrow is +indicated in the figure. Then again, the centrifugal +force must everywhere be equal as well +as parallel, because its intensity depends both on +the radius and on the speed of revolution, and +these are the same for every part. It follows +that if a body revolves without rotation, every +part of it is subject to equal and parallel centrifugal +forces. The same must therefore be +true of the earth when deprived of diurnal rotation. +Accordingly every particle of the idealized +non-rotating earth is continuously subject to +equal and parallel centrifugal forces, in consequence +of the revolution of the earth's centre +in its monthly orbit with a radius of $\DPchg{3000}{3,000}$~miles.\footnote + {I owe the suggestion of this method of presenting the origin + of tide-generating force to Professor Davis of Harvard +\index{Davis, method of presenting tide-generating force}% + University.} + +We have seen that the total of centrifugal +force acting on the whole earth must be just +such as to balance the total of the centripetal +forces due to the moon's attraction. If, then, +the attractional forces, acting on every particle +of the earth, were also equal and parallel, there +would be a perfect balance throughout. We +shall see, however, that although there is a perfect +balance on the whole, there is not that uniformity +\PageSep{98} +which would render the balance perfect +at every particle. + +As far as concerns the totality of the attraction +the analogy is complete between the larger +stone, revolving at the end of the shorter arm +of the rod, and the earth revolving in its small +\Figure[0.7]{21}{Revolution of a Body without Rotation}{png} +orbit round~$G$. But a difference arises when we +compare the distribution of the tension of the +rod with that of the lunar attraction; for the +rod only pulls at the stone at the point where it +is attached to it, whereas the moon attracts every +particle of the earth. She does not, however, +attract every particle with equal force, for she +pulls the nearer parts more strongly than the +further, as is obvious from the nature of the law +of gravitation. The earth's centre is distant +sixty times its radius from the moon, so that the +nearest and furthest parts are distant fifty-nine +\PageSep{99} +and sixty-one radii respectively. Hence the attractions +at the nearest and furthest parts differ +only a little from the average, namely, that at +the centre; but it is just these small differences +which are important in this matter. + +Since on the whole the attractions and the centrifugal +forces are equal and opposite, and since +the centrifugal forces acting on the non-rotating +earth are equal and parallel at every part, and +since the attraction at the earth's centre is the +average attraction, it follows that where the attraction +is stronger than the average it overbalances +the centrifugal force, and where it is weaker +it is overbalanced thereby. + +The result of the contest between the two sets +of forces is illustrated in~\fig{22}. The circle +represents a section of the earth, and the moon +is a long way off in the direction~$M$. + +Since the moon revolves round the earth, +whilst the earth is still deprived of rotation, the +figure only shows the state of affairs at a definite +instant of time. The face which the earth exhibits +to the moon is always changing, and the +moon returns to the same side of the earth only +at the end of the month. Hence the section of +the earth shown in this figure always passes +through the moon, while it is continually shifting +with respect to the solid earth. The arrows in +the figure show by their directions and lengths +the magnitudes and directions of the overbalance +\PageSep{100} +in the contest between centrifugal and centripetal +tendencies. The point~$V$ in the figure is +the middle of the hemisphere, which at the moment +portrayed faces full towards the moon. It +\Figure[0.7]{22}{Tide-Generating Force}{png} +is the middle of the round disk which the man in +the moon looks at. The middle of the face invisible +to the man in the moon is at~$I$. The +point of the earth which is only fifty-nine earth's +radii from the moon is at~$V$. Here attraction +overbalances centrifugal force, and this is indicated +by an arrow pointing towards the moon. +The point distant sixty-one earth's radii from +the moon is at~$I$, and attraction is here overbalanced, +as indicated by the arrow pointing away +from the moon. + +I shall have to refer hereafter to the intensities +\PageSep{101} +of these forces, and will therefore here pause +to make some numerical calculations. + +The moon is distant from the earth's centre +sixty times the earth's radius, and the attraction +of gravity varies inversely as the square of the +distance. Hence we may take $\frac{1}{60^{2}}$ or $\frac{1}{3,600}$ as a +measure of the intensity of the moon's attraction +at the earth's centre. The particle which occupies +the centre of the earth is also that particle +which is at the average distance of all the particles +constituting the earth's mass. Hence $\frac{1}{60^{2}}$ or +$\frac{1}{3,600}$ may be taken as a measure of the average +attraction of the moon on every particle of the +earth. + +Now the point~$V$ is only distant fifty-nine +earth's radii from the moon, and therefore, on +the same scale, the moon attraction is measured +by $\frac{1}{59^{2}}$ or~$\frac{1}{3,481}$. + +The attraction therefore at~$V$ exceeds the average +by $\frac{1}{59^{2}} - \frac{1}{60^{2}}$, or $\frac{1}{3,481} - \frac{1}{3,600}$. It will be well to +express these results in decimals; now $\frac{1}{3,481}$ is +$.000,287,27$, and $\frac{1}{3,600}$ is $.000,277,78$, so that the +difference is~$.000,009,49$. It is important to +notice that $\frac{2}{60^{3}}$ or $\frac{2}{216,000}$ is equal to~$.000,009,26$; +so that the difference is nearly equal to~$\frac{2}{60^{3}}$. + +{\Loosen Again, the point~$I$ is distant sixty-one earth's +radii from the moon, and the moon's attraction +there is to be measured by $\frac{1}{61^{2}}$ or $\frac{1}{3,721}$. The attraction +at~$I$ therefore falls below the average by +$\frac{1}{60^{2}} - \frac{1}{61^{2}}$, or $\frac{1}{3,600} - \frac{1}{3,721}$; that is, by~$.000,277,78 - .000,268,75$, +\PageSep{102} +which is equal to~$.000,009,03$. +This again does not differ much from~$\frac{2}{60^{3}}$.} + +These calculations show that the excess of the +actual attraction at~$V$ above the average attraction +is nearly equal to the excess of the average +above the actual attraction at~$I$. These two +excesses only differ from one another by $5$~per +cent.\ of either, and they are both approximately +equal to~$\frac{2}{60^{3}}$ on the adopted scale of measurement. + +The use of any particular scale of measurement +is not material to this argument, and we +should always find that the two excesses are +nearly equal to one another. And further, if +the moon were distant from the earth by any +other number of earth's radii, we should find +that the two excesses are each nearly equal to $2$ +divided by the cube of that number.\footnote + {\Loosen This argument is very easily stated in algebraic notation. + If $x$~be the number of earth's radii at which the moon is + placed, the points $V$~and~$I$ are respectively distant $x - 1$ and + $x + 1$ radii. Now $(x - 1)^{2}$~is nearly equal to~$x^{2} - 2x$ or to + $x^{2}(1 - \frac{2}{x})$, and therefore $\frac{1}{(x - 1)^{2}}$~is nearly equal to~$\frac{1}{x^{2}(1 - \frac{2}{x})}$, which is + nearly equal to~$\frac{1}{x^{2}}(1 + \frac{2}{x})$. Hence $\frac{1}{(x - 1)^{2}} - \frac{1}{x^{2}}$~is nearly equal to~$\frac{2}{x^{3}}$. + By a similar argument $(x + 1)^{2}$~is nearly equal to~$x^{2}(1 + \frac{2}{x})$, + and $\frac{1}{(x + 1)^{2}}$~is nearly equal to~$\frac{1}{x^{2}}(1 - \frac{2}{x})$; so that $\frac{1}{x^{2}} - \frac{1}{(x + 1)^{2}}$~is nearly + equal to~$\frac{2}{x^{3}}$.} + +We conclude then that the two overbalances +at $V$~and~$I$, which will be called tide-generating +forces, are nearly equal to one another, and vary +\PageSep{103} +inversely as the cube of the distance of the moon +from the earth. + +The fact of the approximate equality of the +overbalance or excess on the two sides of the +earth is noted in the figure by two arrows at $V$ +and $I$ of equal lengths. The argument would +be a little more complicated, if I were to attempt +to follow the mathematician in his examination +of the whole surface of the earth, and to trace +from point to point how the balance between +the opposing forces turns. The reader must +accept the results of such an analysis as shown +in \fig{22} by the directions and lengths of the +arrows. + +We have already seen that the forces at $V$ and~$I$, +the middles of the faces of the earth which +are visible and invisible to the man in the moon, +are directed away from the earth's centre. The +edges of the earth's disk as seen from the moon +are at $D$ and~$D$, and here the arrows point inwards +to the earth's centre and are half as long +as those at $V$ and~$I$. At intermediate points, +they are intermediate both in size and direction. + +The only point in which the system considered +differs from actuality is that the earth has +been deprived of rotation. But this restriction +may be removed, for, when the earth rotates +once in $24$~hours, no difference is made in the +forces which I have been trying to explain, +\PageSep{104} +although of course the force of gravity and the +shape of the planet are affected by the rotation. +This figure is called a diagram of tide-generating +forces, because the tides of the ocean are due to +the action of this system of forces. + +The explanation of tide-generating force is +the very kernel of our subject, and, at the risk +of being tedious, I shall look at it from a slightly +different point of view. If every particle of the +earth and of the ocean were acted on by equal +and parallel forces, the whole system would +move together and the ocean would not be displaced +relatively to the earth; we should say +that the ocean was at rest. If the forces were +not quite equal and not quite parallel, there +would be a slight residual effect tending to make +the ocean move relatively to the solid earth. In +other words, any defect from equality and parallelism +in the forces would cause the ocean to +move on the earth's surface. + +The forces which constitute the departure +from equality and parallelism are called ``tide-generating +forces,'' and it is this system which +is indicated by the arrows in~\fig{22}. Tide-generating +force is, in fact, that force which, +superposed on the average force, makes the actual +force. The average direction of the forces +which act on the earth, as due to the moon's +attraction, is along the line joining the earth's +centre to the moon's centre, and its average +\PageSep{105} +intensity is equal to the force at the earth's +centre. + +Now at~$V$ the actual force is straight towards~$M$, +in the same direction as the average, but of +greater intensity. Hence we find an arrow +directed towards~$M$, the moon. At~$I$, the actual +force is again in the same direction as, but of +less intensity than, the average, and the arrow is +directed away from~$M$, the moon. At~$D$, the +actual force is almost exactly of the same intensity +as the average, but it is not parallel thereto, +and we must insert an inward force as shown by +the arrow, so that when this is compounded with +the average force we may get a total force in +the right direction. + +Now let us consider how these forces tend to +affect an ocean lying on the surface of the earth. +The moon is directly over the head of an inhabitant +of the earth, that is to say in his zenith, +when he is at~$V$; she is right under his feet in +the nadir when he is at~$I$; and she is in the +observer's horizon, either rising or setting, when +he is anywhere on the circle~$D$. When the +inhabitant is at~$V$ or at~$I$ he finds that the tide-generating +force is towards the zenith; when he +is anywhere on the circle~$D$ he finds it towards +the nadir. At other places he finds it directed +towards or away from some point in the sky, +except along two circles halfway between $V$ and~$D$, +or between $I$ and~$D$, where the tide-generating +\PageSep{106} +force is level along the earth's surface, and may +be called horizontal. + +A vertical force cannot make things move +sideways, and so the sea will not be moved horizontally +by it. The vertical part of the tide-generating +force is not sufficiently great to +overcome gravity, but will have the effect of +making the water appear lighter or heavier. It +will not, however, be effective in moving the +water, since the water must remain in contact +with the earth. We want, then, to omit the +vertical part of the force and leave behind only +the horizontal part, by which I mean a force +which, to an observer on the earth's surface, is +not directed either upwards or downwards, but +along the level to any point of the compass. + +If there be a force acting at any point of the +earth's surface, and directed upwards or downwards +away from or towards some point in the +sky other than the zenith, it may be decomposed +into two forces, one vertically upwards or downwards, +and another along the horizontal surface. +Now as concerns the making of the tides, +no attention need be paid to that part which +is directed straight up or down, and the only +important part is that along the surface,---the +horizontal portion. + +Taking then the diagram of tide-generating +forces in~\fig{22}, and obliterating the upward +and downward portions of the force, we are left +%% Plate 1 +\Figure[0.9]{23}{Horizontal Tide-Generating Force}{jpg} +%% Facing page +%[Blank Page] +\PageSep{107} +\index{Horizontal tide-generating force}% +with a system of forces which may be represented +by the arrows in the perspective picture of horizontal +tide-generating force shown in~\fig{23}. + +If we imagine an observer to wander over the +earth, $V$~is the place at which the moon is vertically +over his head, and the circle~$D$, shown by +the boundary of the shadow, passes through all +the places at which the moon is in the horizon, +just rising or setting. Then there is no horizontal +force where the moon is over his head or under +his feet, or where the moon is in his horizon +either rising or setting, but everywhere else there +is a force directed along the surface of the earth +in the direction of the point at which the moon +is straight overhead or underfoot. + +Now suppose $P$ to be the north pole of the +earth, and that the circle $A_{1}$,~$A_{2}$, $A_{3}$, $A_{4}$,~$A_{5}$ is a +parallel of latitude---say the latitude of London. +Then if we watch our observer from external +space, he first puts in an appearance on the picture +at~$A_{1}$, and is gradually carried along to~$A_{2}$ +by the earth's rotation, and so onwards. Just before +he comes to~$A_{2}$, the moon is due south of him, +and the tide-generating force is also south, but +not very large. It then increases, so that nearly +three hours later, when he has arrived at~$A_{3}$, it +is considerably greater. It then wanes, and +when he is at~$A_{4}$ the moon is setting and the +force is nil. After the moon has set, the force +is directed towards the moon's antipodes, and it +\PageSep{108} +is greatest about three hours after moonset, and +vanishes when the moon, still being invisible, is +on the meridian. + +It must be obvious from this discussion that +the lunar horizontal tide-generating force will +differ, both as to direction and magnitude, according +to the position of the observer on the +earth and of the moon in the heavens, and that +it can only be adequately stated by means of +mathematical formulæ. I shall in the following +chapter consider the general nature of the +changes which the forces undergo at any point +\index{Forces!tide-generating|)}% +on the earth's surface. + +But before passing on to that matter it should +be remarked that if the earth and sun had been +the only pair of bodies in existence the whole of +the argument would have applied equally well. +Hence it follows that there is also a solar tide-generating +force, which in actuality coëxists +with the lunar force. I shall hereafter show +how the relative importance of these two influences +is to be determined. + +\begin{Authorities} +Any mathematical work on the theory of the tides; for example, +Thomson and Tait's \Title{Natural Philosophy}, Lamb's \Title{Hydrodynamics}, +Bassett's \Title{Hydrodynamics}, article \Title{Tides}, ``Encycl.\ Britan.,'' +Laplace's \Title{Mécanique Céleste},~\&c. +\end{Authorities} +\PageSep{109} + + +\Chapter{VI} +{Deflection of the Vertical} + +\First{The} intensity of tide-generating force is to be +\index{Deflection of the vertical|(}% +estimated by comparison with some standard, and +it is natural to take as that standard the force of +gravity at the earth's surface. Gravity acts in a +vertical direction, whilst that portion of the tidal +force which is actually efficient in disturbing the +ocean is horizontal. Now the comparison between +a small horizontal force and gravity is +easily effected by means of a pendulum. For if +the horizontal force acts on a suspended weight, +the pendulum so formed will be deflected from +the vertical, and the amount of deflection will +measure the force in comparison with gravity. +A sufficiently sensitive spirit level would similarly +show the effect of a horizontal force by the +displacement of the bubble. When dealing with +tidal forces the displacements of either the pendulum +\index{Forces!numerical estimate|(}% +\index{Forces!deflection of vertical by|(}% +or the level must be exceedingly minute, +but, if measurable, they will show themselves as +a change in the apparent direction of gravity. +Accordingly a disturbance of this kind is often +described as a deflection of the vertical. + +The maximum horizontal force due to the +\PageSep{110} +moon may be shown by a calculation, which involves +the mass and distance of the moon, to +have an intensity of $\frac{1}{11,660,000}$ of gravity.\footnote + {It does not occur to me that there is any very elementary + method of computing the maximum horizontal tidal force, but it + is easy to calculate the vertical force at the points $V$~or~$I$ in~\fig{22}. + + {\Loosen The moon weighs $\frac{1}{80}$~of the earth, and has a radius $\frac{1}{4}$~as large. + Hence lunar gravity on the moon's surface is~$\frac{1}{80} × 4^{2}$, or $\frac{1}{5}$~of + terrestrial gravity at the earth's surface. The earth's radius is + $4,000$~miles and the moon's distance from the earth's centre + $240,000$~miles. Hence her distance from the nearer side of the + earth is $236,000$~miles. Therefore lunar gravity at the earth's + centre is $\frac{1}{5} × \frac{1}{240^{2}}$~of terrestrial gravity, and lunar gravity at the + point~$V$ is $\frac{1}{5} × \frac{1}{236^{2}}$~of the same. Therefore the tidal force at~$V$ + is $\frac{1}{5} × \frac{1}{236^{2}} - \frac{1}{5} × \frac{1}{240^{2}}$~of terrestrial gravity. On multiplying the + squares of~$236$ and of~$240$ by~$5$, we find that this difference is + $\frac{1}{278,480} - \frac{1}{288,000}$. If these fractions are reduced to decimals + and the subtraction is performed, we find that the force at~$V$ + is $.000,000,118,44$~of terrestrial gravity. When this decimal is + written as a fraction, we find the result to be $\frac{1}{8,450,000}$~of + gravity.} + + Now it is the fact, although I do not see how to prove it in an + equally elementary manner, that the maximum horizontal tide-generating + force has an intensity equal to $\frac{3}{4}$~of the vertical force + at $V$~or~$I$. To find $\frac{3}{4}$~of the above fraction we must augment the + denominator by one third part. Hence the maximum horizontal + force is $\frac{1}{11,260,000}$~of gravity. This number does not agree exactly + with that given in the text; the discrepancy is due to the + fact that round numbers have been used to express the sizes and + distance apart of the earth and the moon, and their relative + masses.} +Such a +force must deflect the bob of a pendulum by the +same fraction of the length of the cord by which +it is suspended. If therefore the string were $10$~metres +or $33$~feet in length, the maximum deflection +of the weight would be $\frac{1}{11,660,000}$~of $10$~metres, +\PageSep{111} +or $\frac{1}{1,166}$~of a millimetre. In English measure this +is $\frac{1}{29,000}$~of an inch. But the tidal force is reversed +in direction about every six hours, so that the +pendulum will depart from its mean direction by +\index{Pendulum!curves traced by, under tidal force}% +as much in the opposite direction. Hence the +\Figure[0.8]{24}{Deflection of a Pendulum; the Moon and +Observer on the Equator}{png} +excursion to and fro of the pendulum under the +lunar influence will be $\frac{1}{14,500}$~of an inch. With a +pendulum one metre, or $3$~ft.\ $3$~in.\ in length, +the range of motion of the pendulum bob is +$\frac{1}{145,000}$~of an inch. For any pendulum of manageable +length this displacement is so small, that it +seems hopeless to attempt to measure it by direct +observation. Nevertheless the mass and distance +of the moon and the intensity of gravity being +known with a considerable degree of accuracy, it +is easy to calculate the deflection of the vertical +at any time. +\index{Forces!numerical estimate|)}% + +The curves which are traced out by a pendulum +present an infinite variety of forms, corresponding +\PageSep{112} +to various positions of the observer on +the earth and of the moon in the heavens. Two +illustrations of these curves must suffice. \Fig{24} +shows the case when the moon is on the celestial +equator and the observer on the terrestrial +equator. The path is here a simple ellipse, +which is traversed twice over in the lunar day by +the pendulum. The hours of the lunar day at +\index{Pendulum!curves traced by, under tidal force}% +which the bob occupies successive positions are +marked on the curve. + +If the larger ellipse be taken to show the displacement +of a pendulum when the sun and +\Figure[0.8]{25}{Deflection of a Pendulum; the Moon in N. +Declination~$15°$, the Observer in N. Latitude~$30°$}{png} +moon coöperate at spring tide, the smaller one +will show its path at the time of neap tide. + +In \fig{25} the observer is supposed to be in +latitude~$30°$, whilst the moon stands $15°$~N. of +the equator; in this figure no account is taken +\PageSep{113} +of the sun's force. Here also the hours are +marked at the successive positions of the pendulum, +which traverses this more complex curve +only once in the lunar day. These curves are +somewhat idealized, for they are drawn on the +hypothesis that the moon does not shift her +position in the heavens. If this fact were taken +into account, we should find that the curve +would not end exactly where it began, and that +the character of the curve would change slowly +from day to day. + +But even after the application of a correction +for the gradual shift of the moon in the heavens, +the curves would still be far simpler than in actuality, +because the sun's influence has been left +out of account. It has been remarked in the +last chapter that the sun produces a tide-generating +force, and it must therefore produce a +deflection of the vertical. Although the solar +deflection is considerably less than the lunar, yet +it would serve to complicate the curve to a great +degree, and it must be obvious then that when +the full conditions of actuality are introduced +the path of the pendulum will be so complicated, +that mathematical formulæ are necessary for +complete representation. + +Although the direct observation of the tidal +deflection of the vertical would be impossible +even by aid of a powerful microscope, yet several +attempts have been made by more or less +\PageSep{114} +indirect methods. I have just pointed out that +the path of a pendulum, although drawn on an +ultra-microscopic scale, can be computed with a +high degree of accuracy. It may then occur to +the reader that it is foolish to take a great deal +of trouble to measure a displacement which is +scarcely measurable, and which is already known +with fair accuracy. To this it might be answered +that it would be interesting to watch the direct +gravitational effects of the moon on the earth's +surface. But such an interest does not afford +the principal grounds for thinking that this +attempted measurement is worth making. If the +solid earth were to yield to the lunar attraction +with the freedom of a perfect fluid, its surface +would always be perpendicular to the direction +of gravity at each instant of time. Accordingly +a pendulum would then always hang perpendicularly +to the average surface of the earth, and so +there would be no displacement of the pendulum +with reference to the earth's surface. If, then, +the solid earth yields partially to the lunar attraction, +the displacements of a pendulum must be +of smaller extent relatively to the earth than if +the solid earth were absolutely rigid. I must +therefore correct my statement as to our knowledge +of the path pursued by a pendulum, and +say that it is known if the earth is perfectly +unyielding. The accurate observation of the +movement of a pendulum under the influence of +\PageSep{115} +the moon, and the comparison of the observed +oscillation, with that computed on the supposition +that the earth is perfectly stiff, would afford +the means of determining to +what extent the solid earth is +yielding to tidal forces. Such +a result would be very interesting +as giving a measure of the +stiffness of the earth as a whole. + +I must pass over the various +\index{Cambridge, experiments with bifilar pendulum at|(}% +\index{Darwin, G. H.!bifilar pendulum|(}% +\index{Darwin, Horace, bifilar pendulum|(}% +\index{Deflection of the vertical!experiments to measure|(}% +\index{Pendulum!bifilar|(}% +earlier attempts to measure the +lunar attraction, and will only +explain the plan, although it +was abortive, used in~1879 by +my brother Horace and myself. + +Our object was to measure +the ultra-microscopic displacements +of a pendulum with reference +to the ground on which it +stood. The principle of the apparatus +used for this purpose is +due to Lord Kelvin; it is very +simple, although the practical +application of it was not easy. + +%[** TN: Figure wrapped in the original] +\Figure[0.2]{26}{Bifilar Pendulum}{png} + +\Fig{26} shows diagrammatically, and not drawn +to scale, a pendulum~$\Seg{A}{B}$ hanging by two wires. +At the foot of the pendulum there is a support~$C$ +attached to the stand of the pendulum; $D$~is a +small mirror suspended by two silk fibres, one +being attached to the bottom of the pendulum~$B$ +\PageSep{116} +and the other to the support~$C$. When the +two fibres are brought very close together, any +movement of the pendulum perpendicular to the +plane of the mirror causes the mirror to turn +through a considerable angle. The two silk +fibres diverge from one another, but if two vertical +lines passing through the two points of suspension +are $\frac{1}{1,000}$~of an inch apart, then when the +pendulum moves one of these points through a +millionth of an inch, whilst the other attached to~$C$ +remains at rest, the mirror will turn through +an angle of more than three minutes of arc. +A lamp is placed opposite to the mirror, and +the image of the lamp formed by reflection in +the mirror is observed. A slight rotation of the +mirror corresponds to an almost infinitesimal +motion of the pendulum, and even excessively +small movements of the mirror are easily detected +by means of the reflected image of the light. + +In our earlier experiments the pendulum was +hung on a solid stone gallows; and yet, when +the apparatus was made fairly sensitive, the image +of the light danced and wandered incessantly. +Indeed, the instability was so great that +the reflected image wandered all across the room. +We found subsequently that this instability was +due both to changes of temperature in the stone +gallows, and to currents in the air surrounding +the pendulum. + +To tell of all the difficulties encountered +\PageSep{117} +might be as tedious as the difficulties themselves, +so I shall merely describe the apparatus in its +ultimate form. The pendulum was suspended, +as shown in~\fig{26}, by two wires; the two wires +being in an east and west plane, the pendulum +could only swing north and south. It was hung +inside a copper tube, just so wide that the solid +copper cylinder, forming the pendulum bob, did +not touch the sides of the tube. A spike projected +from the base of the pendulum bob +through a hole in the bottom of the tube. The +mirror was hung in a little box, with a plate-glass +front, which was fastened to the bottom of the +copper tube. The only communication between +the tube and the mirror-box was by the hole +through which the spike of the pendulum projected, +but the tube and mirror-box together +formed a water-tight vessel, which was filled with +a mixture of spirits of wine and boiled water. +The object of the fluid was to steady the +mirror and the pendulum, while allowing its +slower movements to take place. The water was +boiled to get rid of air in it, and the spirits of +wine was added to increase the resistance of the +fluid, for it is a remarkable fact that a mixture +of spirits and water has considerably more viscosity +or stickiness than either pure spirits or +pure water. + +The copper tube, with the pendulum and mirror-box, +was supported on three legs resting on +\PageSep{118} +a block of stone weighing a ton, and this stood +on the native gravel in a north room in the laboratory +at Cambridge. The whole instrument +was immersed in a water-jacket, which was furnished +with a window near the bottom, so that +the little mirror could be seen from outside. A +water ditch also surrounded the stone pedestal, +and the water jacketing of the whole instrument +made the changes of temperature very slow. + +A gas jet, only turned up at the moment of +observation, furnished the light to be observed +by reflection in the little mirror. The gas +burner could be made to travel to and fro along +a scale in front of the instrument. In the preliminary +description I have spoken of the motion +of the image of a fixed light, but it clearly +amounts to the same thing if we measure the +motion of the light, keeping the point of observation +fixed. In our instrument the image of +the movable gas jet was observed by a fixed telescope +placed outside of the room. A bright +light was unfortunately necessary, because there +was a very great loss of light in the passages to +and fro through two pieces of plate glass and a +considerable thickness of water. + +Arrangements were made by which, without +entering the room, the gas jet could be turned +up and down, and could be made to move to and +fro in the room in an east and west direction, +until its image was observed in the telescope. +\PageSep{119} +There were also adjustments by which the two +silk fibres from which the mirror hung could be +brought closer together or further apart, thus +making the instrument more or less sensitive. +There was also an arrangement by which the image +of the light could be brought into the field +of view, when it had wandered away beyond the +limits allowed for by the traverse of the gas jet. + +When the instrument was in adjustment, an +observation consisted of moving the gas jet until +its image was in the centre of the field of +view of the telescope; a reading of the scale, by +another telescope, determined the position of the +gas jet to within about a twentieth of an inch. + +The whole of these arrangements were arrived +at only after laborious trials, but all the precautions +were shown by experience to be necessary, +and were possibly even insufficient to guard the +instrument from the effects of changes of temperature. +I shall not explain the manner in which +we were able to translate the displacements of +the gas jet into displacements of the pendulum. +It was not very satisfactory, and only gave approximate +results. A subsequent form of an +instrument of this kind, designed by my brother, +has been much improved in this respect. It was +he also who designed all the mechanical appliances +in the experiment of which I am speaking. + +It may be well to reiterate that the pendulum +was only free to move north and south, and that +\PageSep{120} +our object was to find how much it swung. The +east and west motion of a pendulum is equally +interesting, but as we could not observe both +displacements at the same time, we confined our +attention in the first instance to the northerly +and southerly movements. + +When properly adjusted the apparatus was so +sensitive that, if the bob of the pendulum moved +through $\frac{1}{40,000}$~of a millimetre, that is, a millionth +part of an inch, we could certainly detect the +movement, for it corresponded to a twentieth +of an inch in our scale of position of the gas +jet. When the pendulum bob moved through +this amount, the wires of the pendulum turned +through one two-hundredth of a second of arc; +this is the angle subtended by one inch at $770$~miles +distance. I do not say that we could actually +measure with this degree of refinement, but +we could detect a change of that amount. In +view of the instability of the pendulum, which +still continued to some extent, it may be hard to +gain credence for the statement that such a small +deflection was a reality, so I will explain how we +were sure of our correctness. + +In setting up the apparatus, work had to be +conducted inside the room, and some preliminary +observations of the reflected image of a stationary +gas jet were made without the use of the telescope. +The scale on which the reflected spot +of light fell was laid on the ground at about +\PageSep{121} +seven feet from the instrument; in order to +watch it I knelt on the pavement behind the +scale, and leant over it. I was one day watching +on the scale the spot of light which revealed +the motion of the pendulum, and, being tired +with kneeling, supported part of my weight on +my hands a few inches in front of the scale. +The place where my hands rested was on the +bare earth, from which a paving stone had been +removed. I was surprised to find quite a large +change in the reading. It seemed at first incredible +that my change of position was the cause, +but after several trials I found that light pressure +with one hand was quite sufficient to produce +an effect. It must be remembered that this was +not simply a small pressure delivered on the bare +earth at, say, seven feet distance, but it was the +difference of effect produced by the same pressure +at seven feet and six feet; for, of course, +the change only consisted in the distribution of +the weight of a small portion of my body. + +It is not very easy to catch the telescopic image +of a spot of light reflected from a mirror of +the size of a shilling. Accordingly, in setting +up our apparatus, we availed ourselves of this result, +for we found that the readiest way of bringing +the reflected image into the telescopic field +of view was for one of us to move slowly about +the room, until the image of the light was +brought, by the warping of the soil due to his +\PageSep{122} +weight, into the field of view of the telescope. +He then placed a heavy weight on the floor +where he had been standing; this of course +drove the image out of the field of view, but +after he had left the room the image of the flame +was found to be in the field. + +We ultimately found, even when no special +pains had been taken to render the instrument +sensitive, that if one of us was in the room, and +stood at about sixteen feet south of the instrument +with his feet about a foot apart, and slowly +shifted his weight from one foot to the other, a +distinct change was produced in the image of the +gas flame, and of course in the position of the +little mirror, from which the image was derived +by reflection. It may be well to consider for +a moment the meaning of this result. If one +presses with a finger on a flat slab of jelly, a sort +of dimple is produced, and if a pin were sticking +upright in the jelly near the dimple, it would tilt +slightly towards the finger. Now this is like +what we were observing, for the jelly represents +the soil, and the tilt of the pin corresponds to +that of the pendulum. But the scale of the displacement +is very different, for our pendulum +stood on a block of stone weighing nearly a ton, +which rested on the native gravel at two feet below +the level of the floor, and the slabs of the +floor were removed from all round the pendulum. +The dimple produced by a weight of $140$~lbs.\ on +\PageSep{123} +\index{Distortion of soil!by weight}% +\index{Elastic distortion!of soil by weight}% +the stone paved floor must have been pretty +small, and the slope of the sides of that dimple +\index{Dimple!in soil, due to weight}% +at sixteen feet must have been excessively slight; +but we were here virtually observing the change +of slope at the instrument, when the centre of +the dimple was moved from a distance of fifteen +feet to sixteen feet. + +It might perhaps be thought that all observation +would be rendered impossible by the street +traffic and by the ordinary work of the laboratory. +But such disturbances only make tremors +of very short period, and the spirits and water +damped out quick oscillations so thoroughly, that +no difference could be detected in the behavior +of the pendulum during the day and during the +night. Indeed, we found that a man could stand +close to the instrument and hit the tub and pedestal +smart blows with a stick, without producing +any sensible effect. But it was not quite easy to +try this experiment, because there was a considerable +disturbance on our first entering the room; +and when this had subsided small movements of +the body produced a sensible deflection, by slight +changes in the distribution of the experimenter's +weight. + +It is clear that we had here an instrument of +amply sufficient delicacy to observe the lunar +tide-generating force, and yet we completely +failed to do so. The pendulum was, in fact, +always vacillating and changing its position by +\PageSep{124} +many times the amount of the lunar effect which +we sought to measure. + +An example will explain how this was: A series +of frequent readings were taken from July +21st to~25th, 1881, with the pendulum arranged +to swing north and south. We found that there +was a distinct diurnal period, with a maximum at +noon, when the pendulum bob stood furthest +northward. The path of the pendulum was interrupted +by many minor zigzags, and it would +sometimes reverse its motion for an hour together. +But the diurnal oscillation was superposed on a +gradual drift of the pendulum, for the mean +diurnal position traveled slowly southward. Indeed, +in these four days the image disappeared +from the scale three times over, and was brought +back into the field of view three times by the +appliance for that purpose. On the night between +the 24th and~25th the pendulum took an +abrupt turn northward, and the scale reading +was found, on the morning of the~25th, nearly +at the opposite end of the scale from that towards +which it had been creeping for four days +previously. + +Notwithstanding all our precautions the pendulum +was never at rest, and the image of the +flame was always trembling and dancing, or waving +slowly to and fro. In fact, every reading of +our scale had to be taken as the mean of the +excursions to right and left. Sometimes for two +\PageSep{125} +or three days together the dance of the image +would be very pronounced, and during other +days it would be remarkably quiescent. + +The origin of these tremors and slower movements +\index{Earthquakes!microsisms and earth tremors|(}% +\index{Italian investigations in seismology|(}% +\index{Tremors, earth}% +is still to some extent uncertain. Quite +recent investigations by Professor Milne seem to +\index{Milne on seismology}% +show that part of them are produced by currents +in the fluid surrounding the pendulum, that +others are due to changes in the soil of a very +local character, and others again to changes +affecting a considerable tract of soil. But when +all possible allowance is made for these perturbations, +it remains certain that a large proportion +of these mysterious movements are due to minute +earthquakes. + +Some part of the displacements of our pendulum +was undoubtedly due to the action of the +moon, but it was so small a fraction of the whole, +that we were completely foiled in our endeavor +to measure it.\footnote + {Since the date of our experiment the bifilar pendulum has + been perfected by my brother, and it is now giving continuous + photographic records at several observatories. It is now made + to be far less sensitive than in our original experiment, and no + attempt is made to detect the direct effect of the moon.} +\index{Cambridge, experiments with bifilar pendulum at|)}% +\index{Darwin, G. H.!bifilar pendulum|)}% +\index{Darwin, Horace, bifilar pendulum|)}% +\index{Deflection of the vertical!experiments to measure|)}% +\index{Pendulum!bifilar|)}% + +The minute earthquakes of which I have +\index{Microsisms, minute earthquakes|(}% +spoken are called by Italian observers microsisms, +and this name has been very generally +adopted. The literature on the subject of seismology +is now very extensive, and it would be +out of place to attempt to summarize here the +\PageSep{126} +conclusions which have been drawn from observation. +I may, however, permit myself to add a +few words to indicate the general lines of the research, +which is being carried on in many parts +of the world. + +Italy is a volcanic country, and the Italians +have been the pioneers in seismology. Their +observations have been made by means of pendulums +of various lengths, and with instruments +of other forms, adapted for detecting vertical +movements of the soil. The conclusions at +which Father Bertelli arrived twenty years ago +\index{Bertelli on Italian seismology}% +may be summarized as follows:--- + +The oscillation of the pendulum is generally +\index{Pendulum!as seismological instrument}% +parallel to valleys or chains of mountains in the +neighborhood. The oscillations are independent +of local tremors, velocity and direction of wind, +rain, change of temperature, and atmospheric +electricity. + +Pendulums of different lengths betray the +movements of the soil in different manners, according +to the agreement or disagreement of +their natural periods of oscillation with the period +of the terrestrial vibrations. + +The disturbances are not strictly simultaneous +in the different towns of Italy, but succeed one +another at short intervals. + +After earthquakes the ``tromometric'' or microseismic\DPnote{** [sic]} +\index{Tromometer, a seismological instrument}% +movements are especially apt to be in +a vertical direction. They are always so when +\PageSep{127} +\index{Bertelli on Italian seismology}% +the earthquake is local, but the vertical movements +are sometimes absent when the shock +occurs elsewhere. Sometimes there is no movement +at all, even when the shock occurs quite +close at hand. + +The positions of the sun and moon appear to +have some influence on the movements of the +pendulum, but the disturbances are especially +\index{Pendulum!as seismological instrument}% +frequent when the barometer is low. + +The curves of ``the monthly means of the +tromometric movement'' exhibit the same forms +in the various towns of Italy, even those which +are distant from one another. + +The maximum of disturbance occurs near the +winter solstice and the minimum near the summer +solstice. + +At Florence a period of earthquakes is presaged +\index{Earthquakes!microsisms and earth tremors|)}% +\index{Tromometer, a seismological instrument}% +by the magnitude and frequency of oscillatory +movements in a vertical direction. These +movements are observable at intervals and during +several hours after each shock. + +Some very curious observations on microsisms +\index{Microsisms, minute earthquakes|)}% +have also been made in Italy with the microphone, +by which very slight movements of the +soil are rendered audible. + +Cavaliere de Rossi, of Rome, has established a +``geodynamic'' observatory in a cave $700$~metres +above the sea at Rocca di Papa, on the external +slope of an extinct volcano. + +At this place, remote from all carriages and +\PageSep{128} +\index{Rossi on Italian seismology|(}% +roads, he placed his microphone at a depth of $20$~metres +\index{Microphone as a seismological instrument|(}% +below the ground. It was protected +against insects by woolen wrappings. Carpet +was spread on the floor of the cave to deaden +the noise from particles of stone which might +possibly fall. Having established his microphone, +he waited till night, and then heard noises which +he says revealed ``natural telluric phenomena.'' +The sounds which he heard he describes as +``roarings, explosions occurring isolated or in +volleys, and metallic or bell-like sounds'' (\textit{fremiti, +scopii isolati o di moschetteria, e suoni-metallici +o di campana}). They all occurred +mixed indiscriminately, and rose to maxima at +irregular intervals. By artificial means he was +able to cause noises which he calls ``rumbling (?) +or crackling'' (\textit{rullo o crepito}). The roaring +(\textit{fremito}) was the only noise which he could reproduce +artificially, and then only for a moment. +It was done by rubbing together the conducting +wires, ``in the same manner as the rocks must +rub against one another when there is an earthquake.'' + +A mine having been exploded in a quarry at +some distance, the tremors in the earth were +audible in the microphone for some seconds +subsequently. + +There was some degree of coincidence between +the agitation of the pendulum-seismograph and +the noises heard with the microphone. +\PageSep{129} + +At a time when Vesuvius became active, +Rocca di Papa was agitated by microsisms, and +the shocks were found to be accompanied by the +very same microphonic noises as before. The +noises sometimes became ``intolerably loud;'' +especially on one occasion in the middle of the +night, half an hour before a sensible earthquake. +The agitation of the microphone corresponded +exactly with the activity of Vesuvius. + +Rossi then transported his microphone to +Palmieri's Vesuvian observatory, and worked in +conjunction with him. He there found that +each class of shock had its corresponding noise. +The sussultorial shocks, in which I conceive the +movement of the ground is vertically up and +down, gave the volleys of musketry (\textit{i~colpi di +moschetteria}), and the undulatory shocks gave +the roarings (\textit{i~fremiti}). The two classes of +noises were sometimes mixed up together. + +Rossi makes the following remarks: ``On +Vesuvius I was put in the way of discovering +that the simple fall and rise in the ticking which +occurs with the microphone [\textit{battito del orologio +unito al microfono}] (a phenomenon observed +by all, and remaining inexplicable to all) is a +consequence of the vibration of the ground.'' +This passage alone might perhaps lead one to +suppose that clockwork was included in the circuit; +but that this was not the case, and that +``ticking'' is merely a mode of representing a +\PageSep{130} +\index{Paschwitz, von Rebeur!on horizontal pendulum|(}% +\index{Pendulum!horizontal|(}% +natural noise is proved by the fact that he subsequently +says that he considers the ticking to +be ``a telluric phenomenon.'' + +Rossi then took the microphone to the Solfatara +\index{Rossi on Italian seismology|)}% +of Pozzuoli, and here, although no sensible +tremors were felt, the noises were so loud as +to be heard simultaneously by all the people in +the room. The ticking was quite masked by +other natural noises. The noises at the Solfatara +were imitated by placing the microphone +on the lid of a vessel of boiling water. Other +seismic noises were then imitated by placing the +microphone on a marble slab, and scratching +\index{Microphone as a seismological instrument|)}% +and tapping the under surface of it. + +The observations on Vesuvius led him to the +conclusion that the earthquake oscillations have +sometimes fixed ``nodes,'' for there were places +on the mountain where no effects were observed. +There were also places where the movement was +intensified, and hence it may be concluded that +the centre of disturbance may sometimes be very +distant, even when the observed agitation is +considerable. + +At the present time perhaps the most distinguished +investigator in seismology is Professor +Milne, formerly of the Imperial College of Engineering +\index{Milne on seismology}% +at Tokyo. His residence in Japan gave +\index{Japan, frequency of earthquakes}% +him peculiar opportunities of studying earthquakes, +for there is, in that country, at least one +earthquake per diem of sufficient intensity to +\index{Italian investigations in seismology|)}% +\PageSep{131} +\index{Japan, frequency of earthquakes}% +affect a seismometer. The instrument of which +he now makes most use is called a horizontal +pendulum. The principle involved in it is old, +but it was first rendered practicable by von +Rebeur-Paschwitz, whose early death deprived +the world of a skillful and enthusiastic investigator. + +The work of Paschwitz touches more closely +on our present subject than that of Milne, because +he made a gallant attempt to measure the +moon's tide-generating force, and almost persuaded +himself that he had done so. + +The horizontal pendulum is like a door in its +mode of suspension. If a doorpost be absolutely +vertical, the door will clearly rest in any +position, but if the post be even infinitesimally +tilted the door naturally rests in one definite +position. A very small shift of the doorpost is +betrayed by a considerable change in the position +of the door. In the pendulum the door is +replaced by a horizontal boom, and the hinges +by steel points resting in agate cups, but the +principle is the same. + +The movement of the boom is detected and +registered photographically by the image of a +light reflected from certain mirrors. Paschwitz +made systematic observations with his pendulum +at Wilhelmshaven, Potsdam, Strassburg, and +Orotava. He almost convinced himself at one +time that he could detect, amidst the wanderings +\PageSep{132} +of the curves of record, a periodicity corresponding +to the direct effect of the moon's action. +But a more searching analysis of his results left +the matter in doubt. Since his death the observations +at Strassburg have been continued by +M.~Ehlert. His results show an excellent consistency +\index{Ehlert, observation with horizontal pendulum}% +with those of Paschwitz, and are therefore +\index{Paschwitz, von Rebeur!on horizontal pendulum|)}% +confirmatory of the earlier opinion of the +latter. I am myself disposed to think that the +detection of the lunar attraction is a reality, but +the effect is so minute that it cannot yet be +relied on to furnish a trustworthy measurement +of the amount of the yielding of the solid earth +to tidal forces. + +It might be supposed that doubt could hardly +arise as to whether or not the direct effect of +the moon's attraction had been detected. But +I shall show in the next chapter that at many +places the tidal forces must exercise in an indirect +manner an effect on the motion of a pendulum +\index{Pendulum!horizontal|)}% +much greater than the direct effect. + +It was the consideration of this indirect effect, +and of other concomitants, which led us to +abandon our attempted measurement, and to +conclude that all endeavors in that direction +were doomed to remain for ever fruitless. I can +but hope that a falsification of our forecast by +M.~Ehlert and by others may be confirmed. +\PageSep{133} + +\begin{Authorities} +G.~H. Darwin and Horace Darwin, ``Reports to the British +Association for the Advancement of Science:''--- + +\Title{Measurement of the Lunar Disturbance of Gravity.} York +meeting, 1881, pp.~93--126. + +\Title{Second Report on the same}, with appendix. Southampton +meeting, 1882, pp.~95--119. + +E.~von Rebeur-Paschwitz, \Title{Das Horizontalpendel}. + +``Nova Acta Leop.\ Carol.\ Akad.,'' 1892, vol.~lx.\ no.~1, p.~213; +also ``Brit.\ Assoc.\ Reports,''~1893. + +E.~von Rebeur-Paschwitz, \Title{Ueber Horizontalpendel-Beobach\-tungen +in Wilhelmshaven, Potsdam und Puerto Orotava auf Tenerifa}. + +``Astron.\ Nachrichten,'' vol.~cxxx.\ pp.~194--215. + +R.~Ehlert, \Title{Horizontalpendel-Beobachtungen}. + +``Beiträge zur Geophysik,'' vol.~iii.\ Part~I., 1896. + +C.~Davison, \Title{History of the Horizontal and Bifilar Pendulums}. +\index{Davison, history of bifilar and horizontal pendulums}% + +``Appendix to Brit.\ Assoc.\ Report on Earth Tremors.'' Ipswich +meeting, 1895, pp.~184--192. + +``British Association Reports of Committees.'' + +\Title{On Earth Tremors}, 1891--95 (the first being purely formal). + +\Title{On Seismological Investigation}, 1896. + +The literature on Seismology is very extensive, and would +\index{Seismology}% +need a considerable index; the reader may refer to \Title{Earthquakes} +and to \Title{Seismology} by John Milne. Both works form volumes in +the International Scientific Series, published by Kegan Paul, +Trench, Trübner \&~Co. +\end{Authorities} +\index{Deflection of the vertical|)}% +\index{Forces!deflection of vertical by|)}% +\PageSep{134} + + +\Chapter[Distortion of the Earth's Surface]{VII} +{The Elastic Distortion of the Earth's Surface +by Varying Loads} + +\First{When} the tide rises and falls on the seacoast, +\index{Darwin, G. H.!distortion of earth's surface by varying loads|(}% +\index{Deflection of the vertical!due to tide|(}% +\index{Distortion of soil!by varying loads|(}% +\index{Elastic distortion!of earth by varying loads|(}% +many millions of tons of water are brought alternately +nearer and further from the land. Accordingly +a pendulum suspended within a hundred +miles or so of a seacoast should respond to the +attraction of the sea water, swinging towards the +sea at high water, and away from it at low water. +Since the rise and fall has a lunar periodicity the +pendulum should swing in the same period, even +if the direct attraction of the moon did not affect +it. But, as I shall now show, the problem is +further confused by another effect of the varying +tidal load. + +We saw in \Ref{Chapter}{VI}.\ how a weight resting +on the floor in the neighborhood of our pendulum +produced a dimple by which the massive +stone pedestal of our instrument was tilted over. +Now as low tide changes to high tide the position +of an enormous mass of water is varied with +respect to the land. Accordingly the whole +coast line must rock to and fro with the varying +tide. We must now consider the nature of the +\PageSep{135} +distortion of the soil produced in this way. The +mathematical investigation of the form of the +dimple in a horizontal slab of jelly or other elastic +\index{Dimple!form of, in elastic slab}% +material, due to pressure at a single point, +shows that the slope at any place varies inversely +as the square of the distance from the centre. +That is to say, if starting from any point we +proceed to half our original distance, we shall +find four times as great a slope, and at one third +\Figure[0.7]{27}{Form of Dimple in an Elastic Surface}{png} +of the original distance the slope will be augmented +ninefold. + +The theoretical form of dimple produced by +pressure at a single mathematical point is shown +in~\fig{27}. The slope is exaggerated so as to +render it visible, and since the figure is drawn on +the supposition that the pressure is delivered at +a mathematical point, the centre of the dimple +is infinitely deep. If the pressure be delivered +by a blunt point, the slope at a little distance +\PageSep{136} +will be as shown, but the centre will not be infinitely +deep. If therefore we pay no attention to +the very centre, this figure serves to illustrate +the state of the case. When the dimple is produced +by the pressure of a weight, that weight, +being endowed with gravitation, attracts any +other body with a force varying inversely as the +square of the distance. It follows, therefore, +that the slope of the dimple is everywhere exactly +\index{Slope of soil!due to elastic distortion}% +proportional to the gravitational attraction +\index{Attraction!of weight resting on elastic slab proportional to slope}% +of the weight. Since this is true of a single +weight, it is true of a group of weights, each +producing its own dimple by pressure and its own +attraction, strictly proportional to one another. +Thus the whole surface is deformed by the superposition +of dimples, and the total attraction is +the sum of all the partial attractions. + +Let us then imagine a very thick horizontal +slab of glass supporting any weights at any parts +of its surface. The originally flat surface of the +slab will be distorted into shallow valleys and +low hills, and it is clear that the direct attraction +of the weights will everywhere be exactly proportional +to the slopes of the hillsides; also the +direction of the greatest slope at each place must +agree with the direction of the attraction. The +direct attraction of the weights will deflect a +pendulum from the vertical, and the deflection +must be exactly proportional to the slope produced +by the pressure of the weights. It may +\PageSep{137} +be proved that if the slab is made of a very stiff +glass the angular deflection of the pendulum +under the influence of attraction will be one fifth +\index{Attraction!of weight resting on elastic slab proportional to slope}% +of the slope of the hillside; if the glass were +of the most yielding kind, the fraction would be +one eighth. The fraction depends on the degree +of elasticity of the material, and the stiffer it is +the larger the fraction. + +The observation of a pendulum consists in +noting its change of position with reference to +the surface of the soil; hence the slope of the +soil, and the direct attraction of the weight +which causes that slope, will be absolutely fused +together, and will be indistinguishable from one +another. + +Now, this conclusion may be applied to the +tidal load, and we learn that, if rocks are of the +same degree of stiffness as glass of medium +quality, the direct attraction of the tidal load +produces one sixth of the apparent deflection of +a pendulum produced by the tilting of the soil. + +If any one shall observe a pendulum, within +say a hundred miles of the seacoast, and shall +detect a lunar periodicity in its motion, he can +only conclude that what he observes is partly +due to the depression and tilting of the soil, +partly to attraction of the sea water, and partly +to the direct attraction of the moon. Calculation +indicates that, with the known average elasticity +of rock, the tilting of the soil is likely to +\PageSep{138} +\index{Elastic distortion!calculation and illustration|(}% +\index{Slope of soil!calculation and illustration of|(}% +be far greater than the other two effects combined. +Hence, if the direct attraction of the +moon is ever to be measured, it will first be +necessary to estimate and to allow for other important +oscillations with lunar periodicity. The +difficulty thus introduced into this problem is so +serious that it has not yet been successfully met. +It may perhaps some day be possible to distinguish +the direct effects of the moon's tidal attraction +from the indirect effects, but I am not +very hopeful of success in this respect. It was +pointed out in \Ref{Chapter}{VI}.\ that there is some +reason to think that a lunar periodicity in the +swing of a pendulum has been already detected, +and if this opinion is correct, the larger part of +the deflection was probably due to these indirect +effects. + +The calculation of the actual tilting of the +coast line by the rising tide would be excessively +complex even if accurate estimates were obtainable +of the elasticity of the rock and of the tidal +load. It is, however, possible to formulate a +soluble problem of ideal simplicity, which will +afford us some idea of the magnitude of the +results occurring in nature. + +In the first place, we may safely suppose the +earth to be flat, because the effect of the tidal +load is quite superficial, and the curvature of the +earth is not likely to make much difference in +the result. In the second place, it greatly simplifies +\PageSep{139} +the calculation to suppose the ocean to +consist of an indefinite number of broad canals, +separated from one another by broad strips of +land of equal breadth. Lastly, we shall suppose +that each strip of sea rocks about its middle line, +so that the water oscillates as in a seiche of the +Lake of Geneva; thus, when it is high water +on the right-hand coast of a strip of sea, it is +low water on the left-hand coast, and vice versa. +We have then to determine the change of shape +of the ocean-bed and of the land, as the tide +rises and falls. The problem as thus stated is +\Figure{28}{Distortion of Land and Sea-bed by Tidal Load}{png} +vastly simpler than in actuality, yet it will suffice +to give interesting indications of what must +occur in nature. + +The figure~\figref{28} shows the calculated result, the +slopes being of course enormously exaggerated. +The straight line represents the level surface of +land and sea before the tidal oscillation begins, +the shaded part being the land and the dotted +part the sea. Then the curved line shows the +form of the land and of the sea-bed, when it is +low water at the right of the strip of land and high +\PageSep{140} +water at the left. The figure would be reversed +when the high water interchanges position +with the low water. Thus both land and sea +rock about their middle lines, but the figure +shows that the strip of land remains nearly flat +although not horizontal, whilst the sea-bed becomes +somewhat curved. + +It will be noticed that there is a sharp nick at +the coast line. This arises from the fact that +deep water was assumed to extend quite up to +the shore line; if, however, the sea were given +a shelving shore, as in nature, the sharp nick +would disappear, although the form of the distorted +rocks would remain practically unchanged +elsewhere. + +Thus far the results have been of a general +character, and we have made no assumptions as +to the degree of stiffness of the rock, or as to +the breadths of the oceans and continents. Let +us make hypotheses which are more or less +plausible. At many places on the seashore the +tide ranges through twenty or thirty feet, but +these great tides only represent the augmentation +of the tide-wave as it runs into shallow +water, and it would not be fair to suppose our +tide to be nearly so great. In order to be moderate, +I will suppose the tide to have a range of +$160$~centimetres, or, in round numbers, about $5$~feet. +Then, at the high-water side of the sea, +the water is raised by $80$~centimetres, and at the +\index{Elastic distortion!calculation and illustration|)}% +\index{Slope of soil!calculation and illustration of|)}% +\PageSep{141} +low-water side it is depressed by the same +amount. The breadth of the Atlantic is about +$4,000$~or $5,000$~miles. I take then, the breadth of +the oceans and of the continents as $3,900$~miles, +or $6,280$~kilometres. Lastly, as rocks are usually +stiffer than glass, I take the rock bed to +be twice as stiff as the most yielding glass, and +quarter as stiff again as the stiffest glass; this +assumption as to the elasticity of rock makes the +attraction at any place one quarter of the slope. +For a medium glass we found the fraction to be +about one sixth. These are all the data required +for determining the slope. + +It is of course necessary to have a unit of +measurement for the slope of the surface. Now +a second of arc is the name for the angular +magnitude of an inch seen at $3\frac{1}{4}$~miles, and accordingly +a hundredth of a second of arc, usually +written~$0''.01$, is the angular magnitude of +an inch seen at $325$~miles; the angles will then +be measured in hundredths of seconds. + +Before the tide rises, the land and sea-bed +are supposed to be perfectly flat and horizontal. +Then at high tides the slopes on the land are as +follows:--- +\[ +\begin{array}{r@{\ }lc} +\multicolumn{2}{c}{\ColHead{Distance from high-water mark}} & +\multicolumn{1}{c}{\ColHead[2in]{Slope of the land measured in hundredths of seconds of arc}} \\ + 10 &\text{metres} & 10 \\ +100 &\text{metres} & \Z8 \\ + 1 &\text{kilometre} & \Z6 \\ + 10 &\text{kilometres} & \Z4 \\ + 20 &\text{kilometres} & \Z3\rlap{$\frac{1}{2}$} \\ +100 &\text{kilometres} & \Z2 +\end{array} +\] +\PageSep{142} +The slope is here expressed in hundredths of a +second of arc, so that at $100$~kilometres from the +coast, where the slope is~$2$, the change of plane +amounts to the angle subtended by one inch at +$162$~miles. + +When high water changes to low water, the +slopes are just reversed, hence the range of +change of slope is represented by the doubles of +these angles. If the change of slope is observed +by some form of pendulum, allowance must be +made for the direct attraction of the sea, and it +appears that with the supposed degree of stiffness +of rock these angles of slope must be augmented +in the proportion of $5$~to~$4$. Thus, we +double the angles to allow of change from high +to low water, and augment the numbers as $5$ is +to~$4$, to allow for the direct attraction of the sea. +Finally we find results which may be arranged +in the following tabular form:--- +\[ +\begin{array}{r@{\ }lc} +\multicolumn{2}{c}{\ColHead{Distance from high-water mark}} & +\multicolumn{1}{c}{\ColHead[1.75in]{Apparent range of deflection of the vertical}} \\ + 10 &\text{metres} & 0''.25\Z \\ +100 &\text{metres} & 0''.20\Z \\ + 1 &\text{kilometre} & 0''.15\Z \\ + 10 &\text{kilometres} & 0''.10\Z \\ + 20 &\text{kilometres} & 0''.084 \\ +100 &\text{kilometres} & 0''.050 +\end{array} +\] +At the centre of the continent, $1,950$~miles from +the coast, the range will be~$0''.012$. + +If all the assumed data be varied, the ranges +of the slopes are easily calculable, but these +\PageSep{143} +results may be taken as fairly representative, although +perhaps somewhat underestimated. Lord +Kelvin has made an entirely independent estimate +\index{Kelvin, Lord!calculation of tidal attraction}% +of the probable deflection of a pendulum +by the direct attraction of the sea at high tide. +\index{Attraction!of tide calculated}% +He supposes the tide to have a range of $10$~feet +from low water to high water, and he then estimates +the attraction of a slab of water $10$~feet +thick, $50$~miles broad perpendicular to the coast, +and $100$~miles long parallel to the coast, on a +plummet $100$~yards from low-water mark and +opposite the middle of the $100$~miles. This +would, he thinks, very roughly represent the +state of things at St.~Alban's Head, in England. +He finds the attraction such as to deflect the +plumb-line, as high water changes to low water, +by a twentieth of a second of arc. The general +law as to the proportionality of slope to +attraction shows that, with our supposed degree +of stiffness of rock, the apparent deflection of a +plumb-line, due to the depression of the coast +and the attraction of the sea as high water +changes to low water, will then be a quarter of a +second of arc. Postulating a smaller tide, but +spread over a wider area, I found the result +would be a fifth of a second; thus the two results +present a satisfactory agreement. + +This speculative investigation receives confirmation +from observation. The late M.~d'Abbadie +\index{Abbadie, tidal deflection of vertical}% +established an observatory at his château of +\index{Deflection of the vertical!due to tide|)}% +\PageSep{144} +\index{Abbadie, tidal deflection of vertical}% +Abbadia, close to the Spanish frontier and within +a quarter of a mile of the Bay of Biscay. Here +he constructed a special form of instrument for +detecting small changes in the direction of gravity. +Without going into details, it may suffice +to state that he compared a fixed mark with its +image formed by reflection from a pool of mercury. +He took $359$~special observations at the +times of high and low tide in order to see, as he +says, whether the water exercised an attraction +on the pool of mercury, for it had not occurred +to him that the larger effect would probably +arise from the bending of the rock. He found +that in $243$~cases the pool of mercury was tilted +towards the sea at high water or away from it at +low water; in $59$~cases there was no apparent +effect, and in the remaining $57$~cases the action +was inverted. The observations were repeated +later by his assistant in the case of $71$~successive +high waters\footnote + {Presumably the observation at one high water was defective.} +and $73$~low waters, and he also +found that in about two thirds of the observations +the sea seemed to exercise its expected +influence. We may, I think, feel confident that +on the occasions where no effect or a reversal +was perceived, it was annulled or reversed by a +warping of the soil, such as is observed with +seismometers. + +Dr.~von Rebeur-Paschwitz also noted deflections +\index{Paschwitz, von Rebeur!tidal deflection of vertical at Wilhelmshaven}% +due to the tide at Wilhelmshaven in Germany. +\PageSep{145} +\index{Atmospheric pressure!distortion of soil by}% +\index{Elastic distortion!by atmospheric pressure|(}% +\index{Pressure of atmosphere, elastic distortion of soil by}% +The deflection was indeed of unexpected +magnitude at this place, and this may probably +be due to the peaty nature of the soil, which +renders it far more yielding than if the observatory +were built on rock. + +This investigation has another interesting application, +for the solid earth has to bear another +varying load besides that of the tide. The +atmosphere rests on the earth and exercises a +variable pressure, as shown by the varying +height of the barometer. The variation of +pressure is much more considerable than one +would be inclined to suspect off-hand. The +height of the barometer ranges through nearly +two inches, or say five centimetres; this means +that each square yard of soil supports a weight +greater by $1,260$~lbs.\ when the barometer is very +high, than when it is very low. If we picture +to ourselves a field loaded with half a ton to +each square yard, we may realize how enormous +is the difference of pressure in the two cases. + +In order to obtain some estimate of the effects +of the changing pressure, I will assume, as before, +that the rocks are a quarter as stiff again +as the stiffest glass. On a thick slab of this +material let us imagine a train of parallel waves +of air, such that at the crests of the waves the +barometer is $5$~centimetres higher than at the +hollow. Our knowledge of the march of barometric +gradients on the earth's surface makes it +\PageSep{146} +plausible to assume that it is $1,500$~miles from +the line of highest to that of lowest pressure. +Calculation then shows that the slab is distorted +into parallel ridges and valleys, and that the +tops of the ridges are $9$~centimetres, or $3\frac{1}{2}$~inches, +higher than the hollows. Although the actual +distribution of barometric pressures is not of this +simple character, yet this calculation shows, with +a high degree of probability, that when the +barometer is very high we are at least $3$~inches +nearer the earth's centre than when it is very +low. + +The consideration of the effects of atmospheric +\index{Atmospheric pressure!distortion of soil by}% +\index{Level of sea affected by atmospheric pressure}% +\index{Pressure of atmosphere, elastic distortion of soil by}% +pressure leads also to other curious conclusions. +I have remarked before that the sea must respond +\index{Sea!level affected by atmospheric pressure}% +to barometric pressure, being depressed +by high and elevated by low pressure. Since a +column of water $68$~centimetres ($2$~ft.\ $3$~in.)\ in +height weighs the same as a column, with the +same cross section, of mercury, and $5$~centimetres +in height, the sea should be depressed by $68$~centimetres +under the very high barometer as compared +with the very low barometer. But the +height of the water can only be determined with +reference to the land, and we have seen that the +land must be depressed by $9$~centimetres. Hence +the sea would be apparently depressed by only $59$~centimetres. + +It is probable that, in reality, the larger barometric +inequalities do not linger quite long +\PageSep{147} +enough over particular areas to permit the sea to +attain everywhere its due slope, and therefore the +full difference of water level can only be attained +occasionally. On the other hand the elastic compression +\index{Elastic distortion!by atmospheric pressure|)}% +of the ground must take place without +sensible delay. Thus it seems probable that this +compression must exercise a very sensible effect +in modifying the apparent depression or elevation +of the sea under high and low barometer. + +If delicate observations are made with some +form of pendulum, the air waves and the consequent +distortions of the soil should have a sensible +effect on the instrument. In the ideal case +which I have described above, it appears that +the maximum apparent deflection of the plumb-line +would be $\frac{1}{90}$~of a second of arc; this would +be augmented to $\frac{1}{70}$~of a second by the addition +of the true deflection, produced by the attraction +of the air. Lastly, since the slope and attraction +would be absolutely reversed when the air wave +assumed a different position with respect to the +observer, it is clear that the range of apparent +oscillation of the pendulum might amount to +$\frac{1}{35}$~of a second of arc. + +This oscillation is actually greater than that +due to the direct tidal force of the moon acting +on a pendulum suspended on an ideally unyielding +earth. Accordingly we have yet another +reason why the direct measurement of the tidal +force presents a problem of the extremest difficulty. +\PageSep{148} + +\begin{Authorities} +G.~H. Darwin, \Title{Appendix to the Second Report on Lunar Disturbance +of Gravity}. ``Brit.\ Assoc.\ Reports.'' Southampton,~1882. + +Reprint of the same in the ``Philosophical Magazine.'' + +d'Abbadie, \Title{Recherches sur la verticale}. ``Ann.\ de~la Soc.\ Scient.\ +de~Bruxelles,'' 1881. + +von Rebeur-Paschwitz, \Title{Das Horizontalpendel}. ``Nova Acta +K.~Leop.\ Car.\ Akad.,'' Band~60, No.~1, 1892. +\end{Authorities} +\index{Darwin, G. H.!distortion of earth's surface by varying loads|)}% +\index{Distortion of soil!by varying loads|)}% +\index{Elastic distortion!of earth by varying loads|)}% +\PageSep{149} + + +\Chapter{VIII} +{Equilibrium Theory of Tides} + +\First{It} is clearly necessary to proceed step by step +\index{Equilibrium theory of tides|(}% +towards the actual conditions of the tidal problem, +and I shall begin by supposing that the +oceans cover the whole earth, leaving no dry +land. It has been shown in \Ref{Chapter}{V}.\ that the +tidal force is the resultant of opposing centrifugal +and centripetal forces. The motion of the +system is therefore one of its most essential features. +We may however imagine a supernatural +being, who carries the moon round the earth and +makes the earth rotate at the actual relative +speeds, but with indefinite slowness as regards +absolute time. This supernatural being is further +to have the power of maintaining the tidal forces +at exactly their present intensities, and with their +actual relationship as regards the positions of +the moon and earth. Everything, in fact, is to +remain as in reality, except time, which is to be +indefinitely protracted. The question to be considered +is as to the manner in which the tidal +forces will cause the ocean to move on the slowly +revolving earth. + +It appears from \fig{23} that the horizontal +\PageSep{150} +tidal force acts at right angles to the circle, where +the moon is in the horizon, just rising or just +setting, towards those two points, $V$~and~$I$, where +the moon is overhead in the zenith, or underfoot +in the nadir. The force will clearly generate +currents in the water away from the circle of +moonrise and moonset, and towards $V$~and~$I$. +The currents will continue to flow until the water +level is just so much raised above the primitive +surface at $V$~and~$I$, and depressed along the circle, +that the tendency to flow downhill towards +the circle is equal to the tendency to flow uphill +under the action of the tide-generating force. +When the currents have ceased to flow, the figure +of the ocean has become elongated, or egg-shaped +with the two ends alike, and the longer +axis of the egg is pointed at the moon. When +this condition is attained the system is at rest or +in equilibrium, and the technical name for the +egg-like form is a ``prolate ellipsoid of revolution''---``prolate'' +because it is elongated, and +``of revolution'' because it is symmetrical with +respect to the line pointing at the moon. Accordingly +the mathematician says that the figure +of equilibrium under tide-generating force is a +prolate ellipsoid of revolution, with the major +axis directed to the moon. + +It has been supposed that the earth rotates and +that the moon revolves, but with such extreme +slowness that the ocean currents have time +\PageSep{151} +\index{Figure of equilibrium!of ocean under tidal forces|(}% +\index{Forces!figure of equilibrium under tidal|(}% +enough to bring the surface to its form of equilibrium, +\index{Equilibrium theory of tides!chart and law of tide|(}% +at each moment of time. If the time be +sufficiently protracted, this is a possible condition +of affairs. It is true that with the earth spinning +at its actual rate, and with the moon revolving +as in nature, the form of equilibrium can +never be attained by the ocean; nevertheless it +is very important to master the equilibrium +theory. + +\Fig{29} represents the world in two hemispheres, +as in an ordinary atlas, with parallels +of latitude drawn at $15°$~apart. At the moment +represented, the moon is supposed to be in the +zenith at $15°$~of north latitude, in the middle of +the right-hand hemisphere. The diametrically +opposite point is of course at $15°$~of south latitude, +in the middle of the other hemisphere. +These are the two points $V$~and~$I$ of figs.\ \figref{22}~and~\figref{23}, +towards which the water is drawn, so that the +vertices of the ellipsoid are at these two spots. +A scale of measurement must be adopted for +estimating the elevation of the water above, and +its depression below the original undisturbed surface +of the globe. It will be convenient to measure +the elevation at these two spots by the +number~$2$. A series of circles are drawn round +these points, but one of them is, of necessity, +presented as partly in one hemisphere and partly +in the other. In the map they are not quite concentric +with the two spots, but on the actual +\PageSep{152} +\TallFig{29}{Chart of Equilibrium Tide}{png} +\PageSep{153} +\index{Semidiurnal tide!in equilibrium theory|(}% +globe they would be so. These circles show +where, on the adopted scale of measurement, the +elevation of height is successively $1\frac{1}{2}$,~$1$,~$\frac{1}{2}$. The +fourth circle, marked in chain dot, shows where +there is no elevation or depression above the original +surface. The next succeeding and dotted +circle shows where there is a depression of~$\frac{1}{2}$, and +the last dotted line is the circle of lowest water +where the depression is~$1$; it is the circle~$\Seg{D}{D}$ of~\fig{22}, +and the circle of the shadow in~\fig{23}. + +The elevation above the original spherical surface +at the vertices or highest points is just twice +as great as the greatest depression. But the +greatest elevation only occurs at two points, +whereas the greatest depression is found all along +a circle round the globe. The horizontal tide-generating +force is everywhere at right angles to +these circles, and the present figure is in effect a +reproduction, in the form of a map, of the perspective +picture in~\fig{23}. + +Now as the earth turns from west to east, let +us imagine a man standing on an island in the +otherwise boundless sea, and let us consider what +he will observe. Although the earth is supposed +to be revolving very slowly, we may still call the +twenty-fourth part of the time of its rotation an +hour. The man will be carried by the earth's +rotation along some one of the parallels of latitude. +If, for example, his post of observation is +in latitude $30°$~N., he will pass along the second +\index{Equilibrium theory of tides!chart and law of tide|)}% +\index{Figure of equilibrium!of ocean under tidal forces|)}% +\index{Forces!figure of equilibrium under tidal|)}% +\PageSep{154} +parallel to the north of the equator. This parallel +cuts several of the circles which indicate the +elevation and depression of the water, and therefore +he will during his progress pass places where +the water is shallower and deeper alternately, and +he would say that the water was rising and falling +rhythmically. Let us watch his progress +across the two hemispheres, starting from the +extreme left. Shortly after coming into view he +is on the dotted circle of lowest water, and he +says it is low tide. As he proceeds the water +rises, slowly at first and more rapidly later, until +he is in the middle of the hemisphere; he arrives +there six hours later than when we first began to +watch him. It will have taken him about $5\frac{1}{2}$~hours +to pass from low water to high water. At +low water he was depressed by~$1$ below the original +level, and at high water he is raised by~$\frac{1}{2}$ +above that level, so that the range from low +water to high water is represented by~$1\frac{1}{2}$. After +the passage across the middle of the hemisphere, +the water level falls, and after about $5\frac{1}{2}$~hours +more the water is again lowest, and the depression +is measured by~$1$ on the adopted scale. +Soon after this he passes out of this hemisphere +into the other one, and the water rises again +until he is in the middle of that hemisphere. +But this time he passes much nearer to the vertex +of highest water than was the case in the other +hemisphere, so that the water now rises to a +\PageSep{155} +height represented by about~$1\frac{4}{5}$. In this half of +his daily course the range of tide is from $1$~below +to $1\frac{4}{5}$~above, and is therefore~$2\frac{4}{5}$, whereas before +it was only~$1\frac{1}{2}$. The fact that the range of two +successive tides is not the same is of great importance +in tidal theory; it is called the diurnal +inequality of the tide. + +It will have been noticed that in the left hemisphere +the range of fall below the original spherical +surface is greater than the range of rise +above it; whereas in the right hemisphere the +rise is greater than the fall. Mean water mark +is such that the tide falls on the average as much +below it as it rises above it, but in this case the +rise and fall have been measured from the originally +undisturbed surface. In fact the mean +level of the water, in the course of a day, is not +identical with the originally undisturbed surface, +although the two levels do not differ much from +one another. + +The reader may trace an imaginary observer +in his daily progress along any other parallel of +latitude, and will find a similar series of oscillations +in the ocean; each latitude will, however, +present its own peculiarities. Then again the +moon moves in the heavens. In \fig{29} she has +been supposed to be $15°$~north of the equator, +but she might have been yet further northward, +or on the equator, or to the south of it. Her +extreme range is in fact $28°$~north or south of +\PageSep{156} +\index{Forces!those of sun and moon compared|(}% +\index{Lunar!tide-generating force compared with solar|(}% +\index{Moon and earth!tide-generating force compared with sun's|(}% +\index{Solar!tide-generating force compared with lunar|(}% +the equator. To represent each such case a new +map would be required, which would, however, +only differ from this one by the amount of displacement +of the central spots from the equator. + +It is obvious that the two hemispheres in \fig{29} +are exactly alike, save that they are inverted +with respect to north and south; the right hemisphere +is in fact the same as the left upside down. +It is this inversion which causes the two successive +tides to be unlike one another, or, in other +words, gives rise to the diurnal inequality. But +\index{Diurnal inequality!according to equilibrium theory}% +there is one case where inversion makes no difference; +this is when the central spot is on the +equator in the left hemisphere, for its inversion +then makes the right hemisphere an exact reproduction +of the left one. In this case therefore +the two successive tides are exactly alike, and +there is no diurnal inequality. Hence the diurnal +inequality vanishes when the moon is on the +equator. + +Our figure exhibits another important point, +for it shows that the tide has the greater range +in that hemisphere where the observer passes +nearest to one of the two central spots. That is +to say, the higher tide occurs in that half of the +daily circuit in which the moon passes nearest to +the zenith or to the nadir of the observer. + +Thus far I have supposed the moon to exist +alone, but the sun also acts on the ocean according +\index{Sun!tide-generating force of, compared with that of moon|(}% +to similar laws, although with less intensity. +\index{Semidiurnal tide!in equilibrium theory|)}% +\PageSep{157} +We must now consider how the relative strengths +of the actions of the two bodies are to be determined. +It was indicated in \Ref{Chapter}{V}.\ that +tide-generating force varies inversely as the cube +of the distance from the earth of the tide-generating +body. The force of gravity varies inversely +as the square of the distance, so that, as +we change the distance of the attracting body, +tidal force varies with much greater rapidity than +does the direct gravitational attraction. Thus if +the moon stood at half her present distance from +the earth, her tide-generating force would be $8$~times +as great, whereas her direct attraction would +only be multiplied $4$~times. It is also obvious +that if the moon were twice as heavy as in reality, +her tide-generating force would be doubled; +and if she were half as heavy it would be halved. +Hence we conclude that tide-generating force +varies directly as the mass of the tide-generating +body, and inversely as the cube of the distance. + +The application of this law enables us to compare +the sun's tidal force with that of the moon. +The sun is $25,500,000$~times as heavy as the +moon, so that, on the score of mass, the solar +tidal force should be $25\frac{1}{2}$~million times greater +than that of the moon. But the sun is $389$~times +as distant as the moon. And since the +cube of~$389$ is about $59$~millions, the solar tidal +force should be $59$~million times weaker than +that of the moon, on the score of distance. +\PageSep{158} + +We have, then, a force which is $25\frac{1}{2}$~million +times stronger on account of the sun's greater +weight, and $59$~million times weaker on account +of his greater distance; it follows that the sun's +tide-generating force is $25\frac{1}{2}$-$59$ths, or a little +less than half of that of the moon. + +We conclude then that if the sun acted alone +on the water, the degree of elongation or distortion +of the ocean, when in equilibrium, would +be a little less than half of that due to the moon +alone. When both bodies act together, the distortion +of the surface due to the sun is superposed +on that due to the moon, and a terrestrial +observer perceives only the total or sum of the +two effects. + +When the sun and moon are together on the +same side of the earth, or when they are diametrically +opposite, the two distortions conspire +together, and the total tide will be half as great +again as that due to the moon alone, because +the solar tide is added to the lunar tide. And +when the sun and moon are at right angles to +\index{Sun!tide-generating force of, compared with that of moon|)}% +one another, the two distortions are at right +angles, and the low water of the solar tide conspires +with the high water of the lunar tide. +The composite tide has then a range only half as +great as that due to the moon alone, because the +solar tide, which has a range of about half that +\index{Solar!tide-generating force compared with lunar|)}% +of the lunar tide, is deducted from the lunar +\index{Lunar!tide-generating force compared with solar|)}% +tide. Since one and a half is three times a half, +\index{Forces!those of sun and moon compared|)}% +\index{Moon and earth!tide-generating force compared with sun's|)}% +\PageSep{159} +\index{Spring and neap tides!in equilibrium theory}% +it follows that when the moon and sun act together +the range of tide is three times as great +as when they act adversely. The two bodies +are together at change of moon and opposite at +full moon. In both of these positions their +actions conspire; hence at the change and the +full of moon the tides are at their largest, and +are called spring tides. When the two bodies +are at right angles to one another, it is half +moon, either waxing or waning, the tides have +their smallest range, and are called neap tides. +\index{Neap and spring tides!in equilibrium theory}% + +The observed facts agree pretty closely with +this theory in several respects, for spring tide +occurs about the full and change of moon, neap +tide occurs at the half moon, and the range at +springs is usually about three times as great as +that at neaps. Moreover, the diurnal inequality +conforms to the theory in vanishing when the +moon is on the equator, and rising to a maximum +when the moon is furthest north or south. The +amount of the diurnal inequality does not, however, +agree with theory, and in many places the +tide which should be the greater is actually the +less. + +The theory which I have sketched is called +the Equilibrium Theory of the Tides, because +it supposes that at each moment the ocean is +in that position of rest or equilibrium which it +would attain if indefinite time were allowed. +The general agreement with the real phenomena +\PageSep{160} +\index{Equilibrium theory of tides!defects of}% +proves the theory to have much truth about it, +but a detailed comparison with actuality shows +that it is terribly at fault. The lunar and solar +tidal ellipsoids were found to have their long +axes pointing straight towards the tide-generating +bodies, and, therefore, at the time when the +moon and sun pull together, it ought to be high +water just when they are due south. In other +words, at full and change of moon, it ought +to be high water exactly at noon and at midnight. +\index{High water!under moon in equilibrium theory}% +Now observation at spring tides shows +that at most places this is utterly contradictory +to fact. + +It is a matter of rough observation that the +tides follow the moon's course, so that high +water always occurs about the same number of +hours after the moon is due south. This rule +has no pretension to accuracy, but it is better +than no rule at all. Now at change and full of +the moon, the moon crosses the meridian at the +same hour of the clock as the sun, for at change +of moon they are together, and at full moon +they are twelve hours apart. Hence the hour +of the clock at which high water occurs at +change and full of moon is in effect a statement +of the number of hours which elapse after the +moon's passage of the meridian up to high +water. This clock time affords a rough rule for +the time of high water at any other phase of the +moon; if, for example, it is high water at eight +\PageSep{161} +o'clock at full and change, approximately eight +hours will always elapse after the moon's passage +until high water occurs. Mariners call the clock +time of high water at change and full of moon +\index{Establishment of port!definition}% +\index{Establishment of port!zero in equilibrium theory}% +``the establishment of the port,'' because it +establishes a rough rule of the tide at all other +times. + +According to the equilibrium theory, high +water falls at noon and midnight at full and +change of moon, or in the language of the mariner +the establishment of all ports should be +zero. But observation shows that the establishment +at actual ports has all sorts of values, and +that in the Pacific Ocean (where the tidal forces +have free scope) it is at least much nearer to six +hours than to zero. High water cannot be more +than six hours before or after noon or midnight +on the day of full or change of moon, because if +it occurs more than six hours after one noon, it +is less than six hours before the following midnight; +hence the establishment of any port +cannot possibly be more than six hours before or +after. Accordingly, the equilibrium theory is +nearly as much wrong as possible, in respect to +the time of high water. In fact, in many places +it is nearly low water at the time that the equilibrium +theory predicts high water. + +It would seem then as if the tidal action of +the moon was actually to repel the water instead +of attracting it, and we are driven to ask whether +\PageSep{162} +\index{Establishment of port!definition}% +this result can possibly be consistent with the +theory of universal gravitation. + +The existence of continental barriers across +the oceans must obviously exercise great influence +on the tides, but this fact can hardly be +responsible for a reversal of the previsions of the +equilibrium theory. It was Newton who showed +that a depression of the ocean under the moon +is entirely consistent with the theory of gravitation. +In the following chapter I shall explain +Newton's theory, and show how it explains the +discrepancy which we have found between the +equilibrium theory and actuality. + +\begin{Authorities} +An exposition of the equilibrium theory will be found in any +mathematical work on the subject, or in the article \Title{Tides} in the +``Encyclopædia Britannica.'' +\end{Authorities} +\index{Equilibrium theory of tides|)}% +\PageSep{163} + + +\Chapter{IX} +{Dynamical Theory of the Tide Wave} + +\First{The} most serious difficulties in the complete +\index{Dynamical theory of tide-wave|(}% +tidal problem do not arise in a certain special +case which was considered by Newton. His supposition +was that the sea is confined to a canal +\index{Canal!critical depth|(}% +circling the equator, and that the moon and sun +move exactly in the equator. + +An earthquake or any other gigantic impulse +may be supposed to generate a great wave in this +equatorial canal. The rate of progress of such +a wave is dependent on the depth of the canal +only, according to the laws sketched in \Ref{Chapter}{II}., +and the earth's rotation and the moon's attraction +make no sensible difference in its speed +of transmission. If, for example, the canal were +$5$~kilometres ($3$~miles) in depth, such a great +wave would travel $796$~kilometres ($500$~miles) +per~hour. If the canal were shallower the speed +would be less than this; if deeper, greater. +Now there is one special depth which will be +found to have a peculiar importance in the theory +of the tide, namely, where the canal is $13\frac{3}{4}$~miles +deep. In this case the wave travels $1,042$~miles +an hour, so that it would complete the +\PageSep{164} +$25,000$~miles round the earth in exactly $24$~hours. +It is important to note that if the depth of the +equatorial canal be less than $13\frac{3}{4}$~miles, a wave +takes more than a day to complete the circuit of +the earth, and if the depth be greater the circuit +is performed in less than a day. + +The great wave, produced by an earthquake or +other impulse, is called a ``free wave,'' because +\index{Free wave, explanation and contrast with forced}% +\index{Waves!forced and free}% +when once produced it travels free from the action +of external forces, and would persist forever, +were it not for the friction to which water is +necessarily subject. But the leading characteristic +of the tide wave is that it is generated and +kept in action by continuous forces, which act +on the fluid throughout all time. Such a wave +is called a ``forced wave,'' because it is due to +\index{Forced wave, explanation and contrast with free wave}% +the continuous action of external forces. The +rate at which the tide wave moves is moreover +dependent only on the rate at which the tidal +forces travel over the earth, and not in any degree +on the depth of the canal. It is true that +the depth of the canal exercises an influence on +the height of the wave generated by the tidal +forces, but the wave itself must always complete +the circuit of the earth in a day, because the +earth turns round in that period. + +We must now contrast the progress of any +long ``free wave'' in the equatorial canal with +that of the ``forced'' tide wave. I may premise +that it will here be slightly more convenient to +\PageSep{165} +consider the solar instead of the lunar tide. The +lunar wave is due to a stronger tide-generating +force, and since the earth takes $24$~hours $50$~minutes +to turn round with respect to the moon, that +is the time which the lunar tide wave takes to +complete the circuit of the earth; but these differences +are not material to the present argument. +The earth turns with respect to the sun +in exactly one day, or as we may more conveniently +say, the sun completes the circuit of the +earth in that time. Therefore the solar tidal +influence travels over the surface of the earth +at the rate of $1,042$~miles an hour. Now this is +exactly the pace at which a ``free wave'' travels +in a canal of a depth of $13\frac{3}{4}$~miles; accordingly +\index{Canal!theory of tide wave in|(}% +in such a canal any long free wave just keeps +pace with the sun. + +We have seen in \Ref{Chapter}{V}.\ that the solar +tide-generating force \emph{tends} to make a wave crest, +at those points of the earth's circumference where +it is noon and midnight. At each moment of +time the sun is generating a new wave, and after +it is generated that wave travels onwards as a +free wave. If therefore the canal has a depth +\index{Canal!critical depth|)}% +of $13\frac{3}{4}$~miles, each new wave, generated at each +moment of time, keeps pace with the sun, and +the summation of them all must build up two +enormous wave crests at opposite sides of the +earth. + +If the velocity of a free wave were absolutely +\PageSep{166} +the same whatever were its height, the crests of +the two tide waves would become infinite in +height. As a fact the rate of progress of a wave +is somewhat influenced by its height, and therefore, +when the waves get very big, they will +cease to keep pace exactly with the sun, and +the cause for continuous exaggeration of their +heights will cease to exist. We may, however, +express this conclusion by saying that, when the +canal is $13\frac{3}{4}$~miles deep, the height of the tide +wave becomes mathematically infinite. This does +not mean that mathematicians assert that the +wave would really become infinite, but only that +the simple method of treatment which supposes +the wave velocity to depend only on the depth +of water becomes inadequate. If the ocean was +really confined to an equatorial canal, of this exact +depth, the tides would be of very great +height, and the theory would be even more complex +than it is. It is, however, hardly necessary +to consider this special case in further detail. + +We conclude then that for the depth of $13\frac{3}{4}$~miles, +the wave becomes infinite in height, in +the qualified sense of infinity which I have described. +We may feel sure that the existence +of the quasi-infinite tide betokens that the behavior +of the water in a canal shallower than +$13\frac{3}{4}$~miles differs widely from that in a deeper +one. It is therefore necessary to examine into +the essential point in which the two cases differ +\PageSep{167} +from one another. In the shallower canal a free +wave covers less than $25,000$~miles a day, and +thus any wave generated by the sun would tend +to be left behind by him. On the other hand, +in the deeper canal a free wave would outstrip +the sun, and a wave generated by the sun tends +to run on in advance of him. But these are +only tendencies, for in both the shallower and +the deeper canal the actual tide wave exactly +keeps pace with the sun. + +It would be troublesome to find out what +would happen if we had the water in the canal +at rest, and were suddenly to start the sun to +work at it; and it is fortunately not necessary +to attempt to do so. It is, however, certain that +for a long time the motion would be confused, +but that the friction of the water would finally +produce order out of chaos, and that ultimately +there would be a pair of antipodal tide crests +traveling at the same pace as the sun. Our +task, then, is to discover what that final state of +motion may be, without endeavoring to unravel +the preliminary chaos. + +Let us take a concrete case, and suppose our +canal to be $3$~miles deep, in which we have seen +that a free wave will travel $500$~miles an hour. +Suppose, then, we start a long free wave in the +equatorial canal of $3$~miles deep, with two crests +\index{Canal!theory of tide wave in|)}% +$12,500$~miles apart, and therefore antipodal to +one another. The period of a wave is the time +\PageSep{168} +between the passage of two successive crests +past any fixed point. In this case the crests +are antipodal to one another, and therefore the +wave length is $12,500$~miles, and the wave +travels $500$~miles an hour, so that the period of a +free wave is $25$~hours. But the tide wave keeps +pace with the sun, traveling $1,042$~miles an +hour, and there are two antipodal crests, $12,500$~miles +apart; hence, the time between the passage +of successive tide crests is $12$~hours. + +In this case a free wave would have a period +of $25$~hours, and the tide wave, resulting from +the action of solar tide-generating force, has a +period of $12$~hours. The contrast then lies between +the free wave, with a period of $25$~hours, +and the forced wave, with a period of $12$~hours. + +For any other depth of ocean the free wave +will have another period depending on the depth, +but the period of the forced wave is always $12$~hours, +because it depends on the sun. If the +ocean be shallower than $3$~miles, the free period +will be greater than $25$~hours, and, if deeper, +less than $25$~hours. But if the ocean be deepened +to $13\frac{3}{4}$~miles, the free wave travels at the +same pace as the forced wave, and therefore the +two periods are coincident. For depths greater +than $13\frac{3}{4}$~miles, the period of the free wave is +less than that of the forced wave; and the +converse is true for canals less than $13\frac{3}{4}$~miles in +depth. +\PageSep{169} +\index{Forced oscillation!principle of}% +\index{Free oscillation contrasted with forced}% +\index{Principle of forced oscillations}% + +Now let us generalize this conception; we +have a system which if disturbed and left to +itself will oscillate in a certain period, called the +free period. Periodic disturbing forces act on +this system and the period of the disturbance +is independent of the oscillating system itself. +The period of the disturbing forces is called the +forced period. How will such a system swing, +when disturbed with this forced periodicity? + +A weight tied to the end of a string affords +an example of a very simple system capable of +oscillation, and the period of its free swing depends +on the length of the string only. I will +suppose the string to be $3$~feet, $3$~inches, or one +metre in length, so that the period of the swing +from right to left, or from left to right is one +second.\footnote + {A pendulum of one metre in length is commonly called a + seconds-pendulum, although its complete period is two seconds.} +If, holding the string, I move my +hand horizontally to and fro through a small +distance with a regular periodicity, I set the +pendulum a-swinging. The period of the movement +of my hand is the forced period, and the +free period is two seconds, being the time occupied +by a metre-long pendulum in moving from +right to left and back again to right. If I time +the to and fro motion of my hand so that its +period from right to left, or from left to right, +is exactly one second, the excursions of the pendulum +bob grow greater and greater without limit, +\PageSep{170} +\index{Forced oscillation!principle of}% +\index{Free oscillation contrasted with forced}% +\index{Principle of forced oscillations}% +because the successive impulses are stored up in +the pendulum, which swings further and further +with each successive impulse. This case is +exactly analogous with the quasi-infinite tides +which would arise in a canal $13\frac{3}{4}$~miles deep, and +here also this case is critical, separating two +modes of oscillation of the pendulum of different +characters. + +Now when the hand occupies more than one +second in moving from right to left, the forced +period is greater than the free period of the +pendulum; and when the system is swinging +steadily, it will be observed that the excursion +of the hand agrees in direction with the excursion +of the pendulum, so that when the hand is +furthest to the right so is also the pendulum, +and vice versa. When the period of the force +is greater than the free period of the system, at +the time when the force tends to make the pendulum +move to the right, it is furthest to the +right. The excursions of the pendulum agree +in direction with that of the hand. + +Next, when the hand occupies less than one +second to move from right to left or from left +to right, the forced period is less than the free +period, and it will be found that when the hand +is furthest to the right the pendulum is furthest +to the left. The excursions of the pendulum +are opposite in direction from those of the hand. +These two cases are illustrated by~\fig{30}, which +\PageSep{171} +\index{High water!position in shallow and deep canals in dynamical theory}% +will, perhaps, render my meaning more obvious. +We may sum up this discussion by saying that +in the case of a slowly varying disturbing force, +the oscillation and the force are consentaneous, +but that with a quickly oscillating force, the +oscillation is exactly inverted with respect to the +force. + +Now, this simple case illustrates a general +dynamical principle, namely, that if a system +\Figure{30}{Forced Oscillations of a Pendulum}{png} +capable of oscillating with a certain period is +acted on by a periodic force, when the period of +the force is greater than the natural free period +of the system, the oscillations of the system +agree with the oscillations of the force; but if +the period of the force is less than the natural +free period of the system the oscillations are +inverted with reference to the force. + +This principle may be applied to the case of +the tides in the canal. When the canal is more +than $13\frac{3}{4}$~miles deep, the period of the sun's disturbing +force is $12$~hours and is greater than the +\PageSep{172} +\index{Newton!theory of tide in equatorial canal}% +natural free period of the oscillation, because a +free wave would go more than half round the +earth in $12$~hours. We conclude, then, that when +the tide-generating forces are trying to make it +high water, it will be high water. It has been +\index{High water!position in shallow and deep canals in dynamical theory}% +shown that these forces are tending to make high +water immediately under the sun and at its antipodes, +and there accordingly will the high water +be. In this case the tide is said to be direct. + +But when the canal is less than $13\frac{3}{4}$~miles +deep, the sun's disturbing force has, as before, a +period of $12$~hours, but the period of the free +wave is more than $12$~hours, because a free wave +would take more than $12$~hours to get half round +the earth. Thus the general principle shows +that where the forces are trying to make high +water, there will be low water, and vice versa. +Here, then, there will be low water under the sun +and at its antipodes, and such a tide is said to +be inverted, because the oscillation is the exact +inversion of what would be naturally expected. + +All the oceans on the earth are very much +shallower than fourteen miles, and so, at least +near the equator, the tides ought to be inverted. +The conclusion of the equilibrium theory will +therefore be the exact opposite of the truth, near +the equator. + +This argument as to the solar tide requires +but little alteration to make it applicable to the +lunar tide. In fact the only material difference +\PageSep{173} +\index{Waves!of tide in equatorial canal}% +in the conditions is that the period of the lunar +tide is $12$~hours $25$~minutes, instead of $12$~hours, +and so the critical depth of an equatorial canal, +\index{Equatorial canal, tide wave in}% +which would allow the lunar tide to become +quasi-infinite, is a little less than that for the +solar tide. This depth for the lunar tide is in +fact nearly $13$~miles.\footnote + {It is worthy of remark that if the canal had a depth of between + $13\frac{3}{4}$ and $13$~miles, the solar tides would be inverted, and + the lunar tides would be direct. We should then, at the equator, + have spring tide at half moon, when our actual neaps occur; + and neap tide at full and change, when our actual springs occur. + The tides would also be of enormous height, because the depth + is nearly such as to make both tides quasi-infinite. If the depth + of the canal were very nearly $13\frac{3}{4}$~miles the solar tide might be + greater than the lunar. But these exceptional cases have only + a theoretical interest.} + +This discussion should have made it clear that +any tidal theory, worthy of the name, must take +account of motion, and it explains why the prediction +of the equilibrium theory is so wide from +the truth. Notwithstanding, however, this condemnation +of the equilibrium theory, it is of the +utmost service in the discussion of the tides, +because by far the most convenient and complete +way of specifying the forces which act on the +ocean at each instant is to determine the figure +which the ocean would assume, if the forces had +abundant time to act. + +\TB + +When the sea is confined to an equatorial +canal, the tidal problem is much simpler than +\PageSep{174} +when the ocean covers the whole planet, and +this is much simpler than when the sea is interrupted +by continents. Then again, we have +thus far supposed the sun and moon to be always +exactly over the equator, whereas they actually +range a long way both to the north and to the +south of the equator; and so here also the true +problem is more complicated than the one under +consideration. Let us next consider a case, still +far simpler than actuality, and suppose that +whilst the moon or sun still always move over +the equator, the ocean is confined to several +canals which run round the globe, following parallels +of latitude. +\index{Latitude!tidal wave in canal in high|(}% + +The circumference of a canal in latitude~$60°$ +\index{Canal!canal in high latitude|(}% +is only $12,500$~miles, instead of~$25,000$. If a +free wave were generated in such a canal with +two crests at opposite sides of the globe, the distance +from crest to crest would be $6,250$~miles. +Now if an equatorial canal and one in latitude~$60°$ +have equal depths, a free wave will travel at +the same rate along each; and if in each canal +there be a wave with two antipodal crests, the +time occupied by the wave in latitude~$60°$ in +traveling through a space equal to its length will +be only half of the similar period for the equatorial +waves. The period of a free wave in latitude~$60°$ +\index{Waves!in canal in high latitude|(}% +is therefore half what it is at the +equator, for a pair of canals of equal depths. +But there is only one sun, and it takes $12$~hours +\PageSep{175} +to go half round the planet, and therefore for +both canals the forced tide wave has a period of +$12$~hours. If, for example, both canals were +$8$~miles deep, in the equatorial canal the +\index{Canal!tides in ocean partitioned into canals}% +period of the free wave would be greater than +$12$~hours, whilst in the canal at $60°$~of latitude +it would be less than $12$~hours. It follows then +from the general principle as to forced and free +oscillations, that whilst the tide in the equatorial +canal would be inverted, that in latitude~$60°$ +would be direct. Therefore, whilst it would be +low water under the moon at the equator, it +would be high water under the moon in latitude~$60°$. +Somewhere, between latitude~$60°$ and the +equator, there must be a place at which the free +period in a canal $8$~miles deep is the same as +the forced period, and in a canal at this latitude +the tide would be infinite in height, in the modified +sense explained earlier. It follows therefore +that there is for any given depth of canal, less +than $14$~miles, a critical latitude, at which the +tide tends to become infinite in height. + +We conclude, that if the whole planet were +divided up into canals each partitioned off from +its neighbor, and if the canals were shallower +than $14$~miles, we should have inverted tides in +the equatorial region, and direct tides in the +polar regions, and, in one of the canals in some +middle latitude, very great tides the nature of +which cannot be specified exactly. +\PageSep{176} + +The supposed partitions between neighboring +canals have introduced a limitation which must +be removed, if we are to approach actuality, but +I am unable by general reasoning to do more +than indicate what will be the effect of the removal +of the partitions. It is clear that when +the sea swells up to form the high water, the +water comes not only from the east and the west +of the place of high water, but also from the +north and south. The earth, as it rotates, carries +with it the ocean; the equatorial water is +carried over a space of $25,000$~miles in $24$~hours, +whereas the water in latitude~$60°$ is carried over +only $12,500$~miles in the same time. When, in +the northern hemisphere, water moves from north +to south it passes from a place where the surface +of the earth is moving slower, to where it is +moving quicker. Then, as the water goes to the +south, it carries with it only the velocity adapted +to the northern latitude, and so it gets left behind +by the earth. Since the earth spins from +west to east, a southerly current acquires a westward +trend. Conversely, when water is carried +northward of its proper latitude, it leaves the +\index{Latitude!tidal wave in canal in high|)}% +earth behind and is carried eastward. Hence +the water cannot oscillate northward and southward, +without at the same time oscillating eastward +and westward. Since in an ocean not +partitioned into canals, the water must necessarily +move not only east and west but also north +\index{Canal!canal in high latitude|)}% +\index{Waves!in canal in high latitude|)}% +\PageSep{177} +\index{Earth and moon!rotation of, effects on tides}% +\index{Rotation!of earth involved in tidal problem}% +\index{Vortical motion in oceanic tides}% +and south, it follows that tidal movements in the +ocean must result in eddies or vortices. The +\index{Eddies, tidal oscillation involves}% +eddying motion of the water must exist everywhere, +but it would be impossible, without mathematical +reasoning, to explain how all the eddies +fit into one another in time and place. It must +suffice for the present discussion for the reader +to know that the full mathematical treatment of +the problem shows this general conclusion to be +correct. + +The very difficult mathematical problem of +the tides of an ocean covering the globe to a +uniform depth was first successfully attacked by +Laplace. He showed that whilst the tides of a +\index{Laplace!theory of tides|(}% +shallow ocean are inverted at the equator, as +proved by Newton, that they are direct towards +the pole. We have just arrived at the same +conclusion by considering the tide wave in a +canal in latitude~$60°$. But our reasoning indicated +that somewhere in between higher latitudes +and the equator, the tide would be of an undefined +character, with an enormous range of rise +and fall. The complete solution of the problem +shows, however, that this indication of the +canal theory is wrong, and that the tidal variation +of level absolutely vanishes in some latitude +intermediate between the equator and the pole. +The conclusion of the mathematician is that +there is a certain circle of latitude, whose position +depends on the depth of the sea, where +there is neither rise nor fall of tide. +\PageSep{178} +\index{Vortical motion in oceanic tides}% + +At this circle the water flows northward and +southward, and to and fro between east and +west, but in such a way as never to raise or depress +the level of the sea. It is not true to say +that there is no tide at this circle, for there are +tidal currents without rise and fall. When the +ocean was supposed to be cut into canals, we +thereby obliterated the northerly and southerly +currents, and it is exactly these currents which +prevent the tides becoming very great, as we +were then led to suppose they would be. + +It may seem strange that, whereas the first +rough solution of the problem indicates an oscillation +of infinite magnitude at a certain parallel +of latitude, the more accurate treatment of the +case should show that there is no oscillation of +level at all. Yet to the mathematician such a +result is not a cause of surprise. But whether +strange or not, it should be clear that if at the +equator it is low water under the moon, and if +near the pole it is high water under the moon, +there must in some intermediate latitude be a +place where the water is neither high nor low, +that is to say, where there is neither rise nor fall.\footnote + {The mathematician knows that a quantity may change sign, + either by passing through infinity or through zero. Where a + change from positive to negative undoubtedly takes place, and + where a passage through infinity can have no physical meaning, + the change must take place by passage through zero.} + +\TB + +Now let us take one more step towards actuality, +and suppose the earth's equator to be +\PageSep{179} +oblique to the orbits of the moon and sun, so +that they may sometimes stand to the north and +sometimes to the south of the equator. We +have seen that in this case the equilibrium theory +indicates that the two successive tides on any +one day have unequal ranges. The mathematical +solution of the problem shows that this conclusion +is correct. It appears also that if the +ocean is deeper at the poles than at the equator, +that tide is the greater which is asserted to be +so by the equilibrium theory. If, however, the +ocean is shallower at the poles than at the equator, +it is found that the high water which the +equilibrium theory would make the larger is actually +the smaller and vice versa. + +If the ocean is of the same depth everywhere, +we have a case intermediate between the two, +where it is shallower at the poles, and where it is +deeper at the poles. Now in one of these cases +it appears that the higher high water occurs +where in the other we find the lower high water +to occur; and so, when the depth is uniform, +the higher high water and the lower high water +must attain the same heights. We thus arrive +at the remarkable conclusion that, in an ocean +of uniform depth, the diurnal inequality of the +\index{Diurnal inequality!in Laplace's solution}% +tide is evanescent. There are, however, diurnal +inequalities in the tidal currents, which are so +adjusted as not to produce a rise or fall. This +result was first arrived at by the great mathematician +Laplace. +\PageSep{180} + +According to the equilibrium theory, when the +moon stands some distance north of the equator, +the inequality between the successive tides on +the coasts of Europe should be very great, but +the difference is actually so small as to escape +ordinary observation. In the days of Laplace, +the knowledge of the tides in other parts of the +world was very imperfect, and it was naturally +thought that the European tides were fairly +representative of the whole world. When, then, +it was discovered that there would be no diurnal +inequality in an ocean of uniform depth covering +the whole globe, it was thought that a fair explanation +had been found for the absence of +that inequality in Europe. But since the days +of Laplace much has been learnt about the tides +\index{Laplace!theory of tides|)}% +in the Pacific and Indian oceans, and we now +know that a large diurnal inequality is almost +\index{Diurnal inequality!in Atlantic, Pacific, and Indian Oceans}% +universal, so that the tides of the North Atlantic +are exceptional in their simplicity. In fact, the +evanescence of the diurnal inequality is not much +closer to the truth than the large inequality +predicted by the equilibrium theory; and both +theories must be abandoned as satisfactory explanations +of the true condition of affairs. But +notwithstanding their deficiencies both these +theories are of importance in teaching us how +the tides are to be predicted. In the next chapter +I shall show how a further approximation to +the truth is attainable. +\PageSep{181} + +\begin{Authorities} +The canal theory in its elementary form is treated in many +works on Hydrodynamics, and in \Title{Tides}, ``Encyclopædia Britannica.'' + +An elaborate treatment of the subject is contained in Airy's +\index{Airy, Sir G. B.!attack on Laplace}% +\Title{Tides and Waves}, ``Encyclopædia Metropolitana.'' Airy there +attacks Laplace for his treatment of the wider tidal problem, +but his strictures are now universally regarded as unsound. + +Laplace's theory is contained in the \Title{Mécanique Céleste}, but it +is better studied in more recent works. + +A full presentment of this theory is contained in Professor +Horace Lamb's \Title{Hydrodynamics}, Camb.\ Univ.\ Press, 1895, chapter~viii. +\index{Lamb, H., presentation of Laplace's theory}% + +Important papers, extending Laplace's work, by Mr.~S.~S. +Hough, are contained in the \Title{Philosophical Transactions of the +\index{Hough, S. S.!dynamical solution of tidal problem}% +Royal Society}, A.~1897, pp.~201--258, and A.~1898, pp.~139--185. +\end{Authorities} +\index{Dynamical theory of tide-wave|)}% +\PageSep{182} + + +\Chapter{X} +{Tides in Lakes---Cotidal Chart} + +\First{If} the conditions of the tidal problem are to +\index{Lakes!tides in|(}% +agree with reality, an ocean must be considered +which is interrupted by continental barriers of +land. The case of a sea or lake entirely surrounded +by land affords the simplest and most +complete limitation to the continuity of the +water. I shall therefore begin by considering +the tides in a lake. + +The oscillations of a pendulum under the tidal +attraction of the moon were considered in \Ref{Chapter}{VI}., +and we there saw that the pendulum +would swing to and fro, although the scale of +displacement would be too minute for actual +observation. Now a pendulum always hangs +perpendicularly to the surface of water, and +must therefore be regarded as a sort of level. +As it sways to and fro under the changing action +of the tidal force, so also must the surface +of water. If the water in question is a lake, the +rocking of the level of the lake is a true tide. + +A lake of say a hundred miles in length is +very small compared with the size of the earth, +and its waters must respond almost instantaneously +\PageSep{183} +to the changes in the tidal force. Such +a lake is not large enough to introduce, to a +perceptible extent, those complications which +make the complete theory of oceanic tides so +difficult. The equilibrium theory is here actually +true, because the currents due to the changes in +the tidal force have not many yards to run before +equilibrium is established, and the lake may +be regarded as a level which responds almost +instantaneously to the tidal deflections of gravity. +The open ocean is a great level also, but sufficient +time is not allowed it to respond to the changes +in the direction of gravity, before that direction +has itself changed. + +It was stated in \Ref{Chapter}{V}.\ that the maximum +horizontal force due to the moon has an intensity +equal to $\frac{1}{11,664,000}$~part of gravity, and that +therefore a pendulum $10$~metres long is deflected +through $\frac{1}{11,664,000}$~of $10$~metres, or through $\frac{1}{1,166}$~of +a millimetre. Now suppose our lake, $200$~kilometres +in length, runs east and west, and that +our pendulum is hung up at the middle of the +lake, $100$~kilometres from either end. In \fig{31} +let $\Seg{C}{D}$ represent the level of the lake as +undisturbed, and $\Seg{A}{B}$~an exaggerated pendulum. +When the tide-generating force displaces the +pendulum to~$\Seg{A}{B'}$, the surface of the lake must +assume the position~$\Seg{C'}{D'}$. Now $\Seg{A}{B}$~being $10$~metres, +$\Seg{B}{B'}$~may range as far as $\frac{1}{1,166}$~of a millimetre; +and it is obvious that $\Seg{C}{C'}$~must bear the +\PageSep{184} +same relation to~$\Seg{C}{B}$ that $\Seg{B}{B'}$ does to~$\Seg{A}{B}$. +Hence $\Seg{C}{C'}$ at its greatest may be $\frac{1}{11,664,000}$~of half +the length of the lake. The lake is supposed +to be twice $100$~kilometres in length, and $100$~kilometres +is $10$~million centimetres; thus $\Seg{C}{C'}$~is +$\frac{1}{1.1664}$~centimetre, or $\frac{9}{10}$~of a centimetre. When +the pendulum is deflected in the other direction +the lake rocks the other way, and $C'$~is just as +much above~$C$ as it was below it before. It +follows from this that the lunar tide at the ends +of a lake, $200$~kilometres or $120$~miles in length, +has a range of $1\frac{3}{4}$~centimetres or $\frac{2}{3}$~of an inch. +The solar tidal force is a little less than half as +strong as that due to the moon, and when the +two forces conspire together at the times of +spring tide, we should find a tide with a range +of $2\frac{1}{2}$~centimetres. +\Figure{31}{The Tide in a Lake}{png} + +If the same rule were to apply to a lake $2,000$~kilometres +or $1,200$~miles in length, the range +of lunar tide would be about $17$~centimetres or +$7$~inches, and the addition of solar tides would +bring the range up to $25$~centimetres or $10$~inches. +\PageSep{185} +I dare say that, for a lake of such a +size, this rule would not be very largely in error. +But as we make the lake longer, the currents +set up by the tidal forces have not sufficient time +to produce their full effects before the intensity +and direction of the tidal forces change. Besides +this, if the lake were broad from north to south, +the earth's rotation would have an appreciable +effect, so that the water which flows from the +north to the south would be deflected westward, +and that which flows from south to north would +tend to flow eastward. The curvature of the +earth's surface must also begin to affect the +motion. For these reasons, such a simple rule +would then no longer suffice for calculating the +tide. + +Mathematicians have not yet succeeded in +solving the tidal problem for a lake of large +dimensions, and so it is impossible to describe +the mode of oscillation. It may, however, be asserted +that the shape, dimensions, and depth of +the lake, and the latitudes of its boundaries will +affect the result. The tides on the northern and +southern shores will be different, and there will +be nodal lines, along which there will be no rise +and fall of the water. +\index{Lakes!tides in|)}% + +The Straits of Gibraltar are so narrow, that +\index{Mediterranean Sea, tides in}% +the amount of water which can flow through +them in the six hours which elapse between +high and low water in the Atlantic is inconsiderable. +\PageSep{186} +\index{Waves!propagated northward in Atlantic|(}% +Hence the Mediterranean Sea is virtually +\index{Mediterranean Sea, tides in}% +a closed lake. The tides of this sea are +much complicated by the constriction formed +by the Sicilian and Tunisian promontories. Its +tides probably more nearly resemble those of two +lakes than of a single sheet of water. The tides +of the Mediterranean are, in most places, so inconspicuous +that it is usually, but incorrectly, +described as a tideless sea. Every visitor to +Venice must, however, have seen, or may we say +smelt, the tides, which at springs have a range of +some four feet. The considerable range of tide +at Venice appears to indicate that the Adriatic +\index{Adriatic, tide in}% +acts as a resonator for the tidal oscillation, in the +same way that a hollow vessel, tuned to a particular +note, picks out and resonates loudly when +that note is sounded. + +We see, then, that whilst the tides of a small +lake are calculable by the equilibrium theory, +those of a large one, such as the Mediterranean, +remain intractable. It is clear, then, that the +tides of the ocean must present a problem yet +more complex than those of a large lake. + +In the Pacific and Southern oceans the tidal +\index{Pacific Ocean, tide in, affects Atlantic}% +forces have almost uninterrupted sway, but the promontories +of Africa and of South America must +profoundly affect the progress of the tide wave +from east to west. The Atlantic Ocean forms a +\index{Atlantic, tide in|(}% +great bay in this vaster tract of water. If this +inlet were closed by a barrier from the Cape of +\PageSep{187} +\index{Pacific Ocean, tide in, affects Atlantic}% +Good Hope to Cape Horn, it would form a lake +large enough for the generation of much larger +tides than those of the Mediterranean Sea, although +probably much smaller than those which +we actually observe on our coasts. Let us now +suppose that the tides proper to the Atlantic are +non-existent, and let us remove the barrier between +the two capes. Then the great tide wave +sweeps across the Southern ocean from east to +west, and, on reaching the tract between Africa +and South America, generates a wave which +travels northward up the Atlantic inlet. This +secondary wave travels ``freely,'' at a rate dependent +only on the depth of the ocean. The +energy of the wave motion is concentrated, where +the channel narrows between North Africa and +Brazil, and the height of the wave must be augmented +in that region. Then the energy is +weakened by spreading, where the sea broadens +again, and it is again reconcentrated by the projection +of the North American coast line towards +Europe. Hence, even in this case, ideally simplified +as it is by the omission of the direct action +of the moon and sun, the range of tide would +differ at every portion of the coasts on each side +of the Atlantic. + +The time of high water at any place must also +depend on the varying depth of the ocean, for it +is governed by the time occupied by the ``free +wave'' in traveling from the southern region to +\PageSep{188} +\index{Cotidal chart}% +the north. But in the south, between the two +capes of Africa and South America, the tidal +oscillation is constrained to keep regular time +with the moon, and so it will keep the same +rhythm at every place to the northward, at whatever +variable pace the wave may move. The +time of high water will of course differ at every +point, being later as we go northward. The +wave may indeed occupy so long on its journey, +that one high water may have only just arrived +at the northern coast of Africa, when another is +rounding the Cape of Good Hope. + +Under the true conditions of the case, this +``free'' wave, generated in and propagated from +the southern ocean, is fused with the true +``forced'' tide wave generated in the Atlantic itself. +\index{Atlantic, tide in|)}% +It may be conjectured that on the coast of +Europe the latter is of less importance than the +\index{Europe, tides on coasts of}% +former. It is interesting to reflect that our tides +to-day depend even more on what occurred yesterday +or the day before in the Southern Pacific +and Indian oceans, than on the direct action of +the moon to-day. But the relative importance +of the two causes must remain a matter of conjecture, +for the problem is one of insoluble complexity. + +Some sixty years ago Whewell, and after him +\index{Whewell!on cotidal charts}% +Airy, drew charts illustrative of what has just +\index{Airy, Sir G. B.!cotidal chart}% +been described. A map showing the march +of the tide wave is reproduced from Airy's +\index{Waves!propagated northward in Atlantic|)}% +\PageSep{189} +\index{Establishment of port!shown in cotidal chart}% +``Tides and Waves,'' in~\fig{32}. It claims to +show, from the observed times of high water at +the various parts of the earth, how the tide wave +travels over the oceans. Whewell and Airy were +\index{Whewell!on cotidal charts}% +well aware that their map could only be regarded +as the roughest approximation to reality. Much +has been learnt since their days, and the then +incomplete state of knowledge hardly permitted +them to fully realize how very rough was their +approximation to the truth. No more recent attempt +has been made to construct such a map, +and we must rest satisfied with this one. Even +if its lines may in places depart pretty widely +from the truth, it presents features of much interest. +I do not reproduce the Pacific Ocean, +because it is left almost blank, from deficiency +of data. Thus, in that part of the world where +the tides are most normal, and where the knowledge +of them would possess the greatest scientific +interest, we are compelled to admit an almost +total ignorance. + +The lines on the map, \fig{32}, give the Greenwich +times of high water at full and change of +moon. They thus purport to represent the successive +positions of the crest of the tide wave. +For example, at noon and midnight (XII~o'clock), +at full and change of moon, the crest +of the tide wave runs from North Australia to +Sumatra, thence to Ceylon, whence it bends back +to the Island of Bourbon, and, passing some hundreds +\PageSep{190} +\TallFig[0.875]{32}{Chart of Cotidal Lines}{jpg} +\PageSep{191} +\index{Cotidal chart!for diurnal tide hitherto undetermined}% +of miles south of the Cape of Good Hope, +trends away towards the Antarctic Ocean. At +the same moment the previous tide crest has +traveled up the Atlantic, and is found running +across from Newfoundland to the Canary +Islands. A yet earlier crest has reached the +north of Norway. At this moment it is low +water from Brazil to the Gold Coast, and again +at Great Britain. + +The successive lines then exhibit the progress +of the wave from hour to hour, and we see how +the wave is propagated into the Atlantic. The +crowding together of lines in places is the graphical +representation of the retardation of the +wave, as it runs into shallower water. + +But even if this chart were perfectly trustworthy, +it would only tell us of the progress of +the ordinary semidiurnal wave, which produces +high water twice a day. We have, however, seen +reason to believe that two successive tides should +not rise to equal heights, and this figure does +not even profess to give any suggestion as to +how this inequality is propagated. In other +words, it is impossible to say whether two successive +tides of unequal heights tend to become +more or less unequal, as they run into any of +the great oceanic inlets. Thus the map affords +no indication of the law of the propagation of +the diurnal inequality. +\index{Diurnal inequality!not shown in cotidal chart}% + +This sketch of the difficulties in the solution +\PageSep{192} +\index{Cotidal chart!for diurnal tide hitherto undetermined}% +of the full tidal problem might well lead to despair +of the possibility of tidal prediction on our +coasts. I shall, however, show in the next chapter +how such prediction is possible. + +\begin{Authorities} +For cotidal charts see Whewell, \Title{Phil.\ Trans.\ Roy.\ Soc.}\ 1833, +or Airy's \Title{Tides and Waves}, ``Encyclopædia Metropolitana.'' +\index{Airy, Sir G. B.!\Title{Tides and Waves}}% +\end{Authorities} +\PageSep{193} + + +\Chapter{XI} +{Harmonic Analysis of the Tide} + +\First{It} is not probable that it will ever be possible +\index{Analysis, harmonic, of tide|(}% +\index{Harmonic analysis!account of|(}% +\index{Moon and earth!tide due to ideal, moving in equator}% +to determine the nature of the oceanic oscillation +as a whole with any accuracy. It is true that +we have already some knowledge of the general +march of the tide wave, and we shall doubtless +learn more in the future, but this can never suffice +for accurate prediction of the tide at any +place. + +Although the equilibrium theory is totally +false as regards its prediction of the time of passage +and of the height of the tide wave, yet it +furnishes the stepping-stone leading towards the +truth, because it is in effect a compendious statement +of the infinite variety of the tidal force in +time and place. + +I will begin my explanation of the practical +method of tidal prediction by obliterating the +sun, and by supposing that the moon revolves in +an equatorial circle round the earth. In this +case the equilibrium theory indicates that each +tide exactly resembles its predecessors and its +successors for all time, and that the successive +and simultaneous passages of the moon and of +\PageSep{194} +\index{Moon and earth!tide due to ideal, moving in equator}% +the wave crests across any place follow one +another at intervals of $12$~hours $25$~minutes. It +would always be exactly high water under or +opposite to the moon, and the height of high +water would be exactly determinate. In actual +oceans, even although only subject to the action +of such a single satellite, the motion of the water +would be so complex that it would be impossible +to predict the exact height or time of high or +of low water. But since the tidal forces operate +in a stereotyped fashion day after day, there will +be none of that variability which actually occurs +on the real earth under the actions of the real +sun and moon, and we may positively assert that +whatever the water does to-day it will do to-morrow. +Thus, if at a given place it is high water +at a definite number of hours after the equatorial +moon has crossed the meridian to-day, it will be +so to-morrow at the same number of hours after +the moon's passage, and the water will rise and +fall every day to the same height above and below +the mean sea level. If then we wanted to +know how the tide would rise and fall in a given +harbor, we need only watch the motion of the +sea at that place, for however the water may +move elsewhere its motion will always produce +the same result at the port of observation. +Thus, apart from the effects of wind, we should +only have to note the tide on any one day +to be able to predict it for all time. For by a +\PageSep{195} +\index{Satellites!tide due to single equatorial}% +single day of observation it would be easy to +note how many hours after the moon's passage +high water occurs, and how many feet it rises +and falls with reference to some fixed mark on +the shore. The delay after the moon's passage +and the amount of rise and fall would differ geographically, +but at each place there would be two +definite numbers giving the height of the tide +and the interval after the moon's passage until +high water. These two numbers are called the +tidal constants for the port; they would virtually +\index{Constants, tidal, explained}% +contain tidal predictions for all time. + +Now if the moon were obliterated, leaving the +sun alone, and if he also always moved over +the equator, a similar rule would hold good, +but exactly $12$~hours would elapse from one +high water to the next, instead of $12$~hours $25$~minutes +as in the case of the moon's isolated +action. Thus two other tidal constants, expressive +of height and interval, would virtually contain +tidal prediction for the solar tide for all +time. + +Theory here gives us some power of foreseeing +the relative importance of the purely lunar +and of the purely solar tide. The two waves +due to the sun alone or to the moon alone have +the same character, but the solar waves follow +one another a little quicker than the lunar waves, +and the sun's force is a little less than half the +moon's force. The close similarity between the +\PageSep{196} +\index{Satellites!tide due to single equatorial}% +actions of the sun and moon makes it safe to conclude +that the delay of the isolated solar wave +after the passage of the sun would not differ +much from the delay of the isolated lunar wave +after the passage of the moon, and that the +height of the solar wave would be about half of +that of the lunar wave. But theory can only be +trusted far enough to predict a rough proportionality +of the heights of the two tide waves to their +respective generating forces, and the approximate +equality of the intervals of retardation; but the +height and retardation of the solar wave could +not be accurately foretold from observation of +the lunar wave. + +When the sun and moon coëxist, but still +move in equatorial circles, the two waves, which +we have considered separately, are combined. +The four tidal constants, two for the moon and +two for the sun, would contain the prediction of +the height of water for all time, for it is easy at +any future moment of time to discover the two +intervals of time since the moon and since the +sun have crossed the meridian of the place of +observation; we should then calculate the height +of the water above some mark on the shore on +the supposition that the moon exists alone, and, +again, on the supposition that the sun exists +alone, and adding the two results together, should +obtain the required height of the water at the +moment in question. +\PageSep{197} + +But the real moon and sun do not move in +equatorial circles, but in planes which are oblique +to the earth's equator, and they are therefore +sometimes to the north and sometimes to the +south of the equator; they are also sometimes +nearer and sometimes further from the earth on +account of the eccentricity of the orbits in which +they move. Now the mathematician treats this +complication in the following way: he first considers +the moon alone and replaces it by a number +of satellites of various masses, which move +in various planes. It is a matter of indifference +that such a system of satellites could not maintain +the orbits assigned to them if they were allowed +to go free, but a mysterious being may be +postulated who compels the satellites to move in +the assigned orbits. One, and this is the largest +of these ideal satellites, has nearly the same mass +as the real moon and moves in a circle over the +equator; it is in fact the simple isolated moon +whose action I first considered. Another small +satellite stands still amongst the stars; others +move in such orbits that they are always vertically +overhead in latitude~$45°$; others repel instead +of attracting; and others move backwards +amongst the stars. Now all these satellites are +so arranged as to their masses and their orbits, +that the sum of their tidal forces is exactly the +same as those due to the real moon moving in +her actual orbit. +\PageSep{198} +\index{Interval from moon's transit to high water!in case of ideal satellite}% + +So far the problem seems to be complicated +rather than simplified, for we have to consider a +dozen moons instead of one. The simplification, +however, arises from the fact that each satellite +either moves uniformly in an orbit parallel to the +equator, or else stands still amongst the stars. +It follows that each of the ideal satellites creates +a tide in the ocean which is of a simple character, +and repeats itself day after day in the same +way as the tide due to an isolated equatorial +moon. If all but one of these ideal satellites +were obliterated the observation of the tide for +a single day would enable us to predict the tide +for all time; because it would only be necessary +to note the time of high water after the ideal +satellite had crossed the meridian, and the height +\index{Height of tide!due to ideal satellite}% +of the high water, and then these two data would +virtually contain a tidal prediction for that tide +at the place of observation for all future time. +The interval and height are together a pair of +``tidal constants'' for the particular satellite in +question, and refer only to the particular place +at which the observation is made. + +In actuality all the ideal satellites coëxist, and +the determination of the pair of tidal constants +appropriate to any one of them has to be made +by a complex method of analysis, of which I shall +say more hereafter. For the present it will suffice +to know that if we could at will annul all +the ideal satellites except one, and observe its +\PageSep{199} +\index{Moon and earth!ideal satellites replacing actual}% +\index{Partial tides in harmonic method}% +tide even for a single day, its pair of constants +could be easily determined. It would then only +be necessary to choose in succession all the satellites +\index{Satellites!ideal replacing sun and moon in harmonic analysis}% +as subjects of observation, and the materials +for a lunar tide table for all time would be obtained. + +The motion of the sun round the earth is analogous +to that of the moon, and so the sun has +also to be replaced by a similar series of ideal +suns, and the partial tide due to each of them +has to be found. Finally at any harbor some +twenty pairs of numbers, corresponding to twenty +ideal moons and suns, give the materials for tidal +prediction for all time. Theoretically an infinite +number of ideal bodies is necessary for an absolutely +perfect representation of the tides, but +after we have taken some twenty of them, the +remainder are found to be excessively small in +mass, and therefore the tides raised by them are +so minute that they may be safely omitted. This +method of separating the tide wave into a number +of partial constituents is called ``harmonic +analysis.'' It was first suggested, and put into +practice as a practical treatment of the tidal +problem, by Sir William Thomson, now Lord +Kelvin, and it is in extensive use. +\index{Kelvin, Lord!initiates harmonic analysis}% + +In this method the aggregate tide wave is considered +as the sum of a number of simple waves +following one another at exactly equal intervals +of time, and always presenting a constant rise +\PageSep{200} +\index{Moon and earth!ideal satellites replacing actual}% +\index{Prediction of tide!due to ideal satellite}% +and fall at the place of observation. When the +time of high water and the height of any one of +these constituent waves is known on any one +day, we can predict, with certainty, the height +of the water, as due to it alone, at any future +time however distant. The period of time which +elapses between the passage of one crest and of +the next is absolutely exact, for it is derived from +a study of the motions of the moon or sun, and +is determined to within a thousandth of a second. +The instant at which any one of the satellites +\index{Satellites!ideal replacing sun and moon in harmonic analysis}% +passes the meridian of the place is also +known with absolute accuracy, but the interval +after the passage of the satellite up to the high +water of any one of these constituent waves, and +the height to which the water will rise are only +derivable from observation at each port. + +Since there are about twenty coëxistent waves +of sensible magnitude, a long series of observations +is requisite for disentangling any particular +wave from among the rest. The series must +also be so long that the disturbing influence of +the wind, both on height and time, may be eliminated +by the taking of averages. It may be +well to reiterate that each harbor has to be considered +by itself, and that a separate set of tidal +constants has to be found for each place. If it +is only required to predict the tides with moderate +accuracy some eight partial waves suffice, but +if high accuracy is to be attained, we have to +\PageSep{201} +consider a number of the smaller ones, bringing +the total up to $20$ or~$25$. + +When the observed tidal motions of the sea +have been analyzed into partial tide waves, they +are found to fall naturally into three groups, +which correspond with the dissections of the sun +\index{Sun!ideal, replacing real sun in harmonic analysis}% +and moon into the ideal satellites. In the first +and most important group the crests follow one +another at intervals of somewhere about $12$~hours; +these are called the semidiurnal tides. +In the second group, the waves of which are in +most places of somewhat less height than those +of the semidiurnal group, the crests follow one +another at intervals of somewhere about $24$~hours, +and they are called diurnal. The tides +of the third group have a very slow periodicity, +for their periods are a fortnight, a month, half +a year, and a year; they are commonly of very +small height, and have scarcely any practical +importance; I shall therefore make no further +reference to them. + +Let us now consider the semidiurnal group. +The most important of these is called ``the principal +lunar semidiurnal tide.'' It is the tide +\index{Lunar!tide, principal}% +\index{Semidiurnal tide!in harmonic method|(}% +raised by an ideal satellite, which moves in a circle +round the earth's equator. I began my explanation +of this method by a somewhat detailed +consideration of this wave. In this case, the +wave crests follow one another at intervals of +$12$~hours $25$~minutes $14\frac{1}{6}$~seconds. The average +\PageSep{202} +interval of time between the successive visible +transits of the moon over the meridian of the +place of observation is $24$~hours $50$~minutes $28\frac{1}{3}$~seconds; +and as the invisible transit corresponds +to a tide as well as the visible one, the interval +between the successive high waters is the time +between the successive transits, of which only +each alternate one is visible. + +The tide next in importance is ``the principal +solar semidiurnal tide.'' This tide bears the +\index{Solar!principal tide}% +same relationship to the real sun that the principal +lunar semidiurnal tide bears to the real moon. +The crests follow one another at intervals of +exactly $12$~hours, which is the time from noon to +midnight and of midnight to noon. The height +of this partial wave is, at most places, a little less +than half of that of the principal lunar tide. + +The interval between successive lunar tides is +$25\frac{1}{4}$~minutes longer than that between successive +solar tides, and as there are two tides a day, the +lunar tide falls behind the solar tide by $50\frac{1}{2}$~minutes +a day. If we imagine the two tides to start +together with simultaneous high waters, then in +about $7$~days the lunar tide will have fallen about +$6$~hours behind the solar tide, because $7$~times +$50\frac{1}{2}$~minutes is $5$~hours $54$~minutes. The period +from high water to low water of the principal +solar semidiurnal tide is $6$~hours, being half the +time between successive high waters. Accordingly, +when the lunar tide has fallen $6$~hours +\PageSep{203} +\index{Spring and neap tides!represented by principal lunar and solar tides}% +behind the solar tide, the low water of the solar +tide falls in with the high water of the lunar +tide. It may facilitate the comprehension of +this matter to take a numerical example; suppose +then that the lunar tide rises $4$~feet above +and falls by the same amount below the mean +level of the sea, and that the solar tide rises and +falls $2$~feet above and below the same level; +then if the two partial waves be started with their +high waters simultaneous, the joint wave will at +first rise and fall by $6$~feet. But after $7$~days it +is low solar tide when it is high lunar tide, and +so the solar tide is subtracted from the lunar +tide, and the compound wave has a height of +$4$~feet less $2$~feet, that is to say, of $2$~feet. +After nearly another $7$~days, or more exactly +after $14\frac{1}{2}$~days from the start, the lunar tide has +lost another $6$~hours, so that it has fallen back +$12$~hours in all, and the two high waters agree +together again, and the joint wave has again a +rise and fall of $6$~feet. When the two high +waters conspire it is called spring tide, and when +the low water of the solar tide conspires with the +high water of the lunar tide, it is called neap +tide. It thus appears that the principal lunar +and principal solar semidiurnal tides together +represent the most prominent feature of the tidal +oscillation. + +The next in importance of the semidiurnal +waves is called the ``lunar elliptic tide,'' and here +\PageSep{204} +\index{Neap and spring tides!represented by principal lunar and solar tides}% +the crests follow one another at intervals of $12$~hours +$39$~minutes $30$~seconds. Now the interval +between the successive principal lunar tides was +\index{Lunar!elliptic tide}% +$12$~hours $25$~minutes $14$~seconds; hence, this +new tide falls behind the principal lunar tide by +$14\frac{1}{2}$~minutes in each half day. If this tide starts +so that its high water agrees with that of the +principal lunar tide, then after $13\frac{3}{4}$~days from +the start, its hollow falls in with the crest of the +former, and in $27\frac{1}{2}$~days from the start the two +crests agree again. + +The moon moves round the earth in an ellipse, +and if to-day it is nearest to the earth, in $13\frac{3}{4}$~days +it will be furthest, and in $27\frac{1}{2}$~days it will +be nearest again. The moon must clearly exercise +a stronger tidal force and create higher +tides when she is near than when she is far; +hence every $27\frac{1}{2}$~days the tides must be larger, +and halfway between they must be smaller. +But the tide under consideration conspires with +the principal lunar tide every $27\frac{1}{2}$~days, and, +accordingly, the joint wave is larger every $27\frac{1}{2}$~days +and smaller in between. Thus this lunar +elliptic tide represents the principal effect of the +\index{Elliptic tide, lunar}% +elliptic motion of the moon round the earth. +There are other semidiurnal waves besides the +three which I have mentioned, but it would +hardly be in place to consider them further +here. +\index{Semidiurnal tide!in harmonic method|)}% + +Now turning to the waves of the second kind, +\PageSep{205} +\index{Diurnal inequality!in harmonic method}% +which are diurnal in character, we find three, all +of great importance. In one of them the high +waters succeed one another at intervals of $25$~hours +$49$~minutes $9\frac{1}{2}$~seconds, and of the second +and third, one has a period of about $4$~minutes +less than $24$~hours and the other of about $4$~minutes +greater than the $24$~hours. It would +hardly be possible to show by general reasoning +how these three waves arise from the attraction +of three ideal satellites, and how these satellites +together are a substitute for the actions of the +true moon and sun. It must, however, be obvious +that the oscillation resulting from three coëxistent +waves will be very complicated. + +All the semidiurnal tides result from waves of +essentially similar character, although some follow +one another a little more rapidly than others, +and some are higher and some are lower. An +accurate cotidal map, illustrating the progress of +any one of these semidiurnal waves over the +ocean, would certainly tell all that we care to +know about the progress of all the other waves +of the group. + +Again, all the diurnal tides arise from waves +of the same character, but they are quite diverse +in origin from the semidiurnal waves, and have +only one high water a day instead of two. A +complete knowledge of the behavior of semidiurnal +waves would afford but little insight into +the behavior of the diurnal waves. At some +\PageSep{206} +time in the future the endeavor ought to be +made to draw a diurnal cotidal chart distinct +from the semidiurnal one, but our knowledge is +not yet sufficiently advanced to make the construction +of such a chart feasible. + +\TB + +All the waves of which I have spoken thus +far are generated by the attractions of the sun +and moon and are therefore called astronomical +tides, but the sea level is also affected by other +oscillations arising from other causes. + +Most of the places, at which a knowledge of +the tides is practically important, are situated in +estuaries and in rivers. Now rain is more prevalent +\index{Rivers!annual meteorological tide in}% +at one season than at another, and mountain +snow melts in summer; hence rivers and +estuaries are subject to seasonal variability of +level. In many estuaries this kind of inequality +may amount to one or two feet, and such a considerable +change cannot be disregarded in tidal +prediction. It is represented by inequalities with +periods of a year and of half a year, which are +called the annual and semiannual meteorological +\index{Annual and semi-annual tides}% +\index{Meteorological!tides}% +tides. + +Then again, at many places, especially in the +Tropics, there is a regular alternation of day and +night breezes, the effect of which is to heap up +% [** TN: "inland", "off-shore" on line breaks in the original; sole instances] +the water in-shore as long as the wind blows in-land, +\index{Wind!a cause of meteorological tides}% +and to lower it when the wind blows off-shore. +Hence there results a diurnal inequality +\PageSep{207} +\index{Estuary, annual meteorological tide in}% +of sea-level, which is taken into account in tidal +prediction by means of a ``solar diurnal meteorological +\index{Meteorological!tides}% +tide.'' Although these inequalities depend +entirely on meteorological influences and +have no astronomical counterpart, yet it is necessary +to take them into account in tidal prediction. + +\TB + +But besides their direct astronomical action, +the sun and moon exercise an influence on the +sea in a way of which I have not yet spoken. +We have seen how waves gradually change their +shape as they progress in a shallow river, so that +the crests become sharper and the hollows flatter, +while the advancing slope becomes steeper and +the receding one less steep. An extreme exaggeration +of this sort of change of shape was +found in the bore. Now it is an absolute rule, +in the harmonic analysis of the tide, that the +partial waves shall be of the simplest character, +and shall have a certain standard law of slope +on each side of their crests. If then any wave +ceases to present this standard simple form, it is +necessary to conceive of it as compound, and to +build it up out of several simple waves. By the +composition of a simple wave with other simple +waves of a half, a third, a quarter of the wave +length, a resultant wave can be built up which +shall assume any desired form. For a given +compound wave, there is no alternative of choice, +\PageSep{208} +for it can only be built up in one way. The +analogy with musical notes is here complete, for +a musical note of any quality is built up from +a fundamental, together with its octave and +twelfth, which are called overtones. So also the +distorted tide wave in a river is regarded as consisting +of simple fundamental tide, with over-tides +of half and third length. The periods of +these over-tides are also one half and one third +of that of the fundamental wave. + +Out in the open ocean, the principal lunar +semidiurnal tide is a simple wave, but when it +runs into shallow water at the coast line, and +still more so in an estuary, it changes its shape. +\index{Estuary, annual meteorological tide in}% +The low water lasts longer than the high water, +and the time which elapses from low water to +high water is usually shorter than that from +high water to low water. The wave is in fact +no longer simple, and this is taken into account +by considering it to consist of a fundamental +lunar semidiurnal wave with a period of $12$~hours +$50$~minutes, of the first over-tide or octave +with a period of $6$~hours $25$~minutes, of the second +over-tide or twelfth with a period of $4$~hours +$17$~minutes, and of the third over-tide or +double octave with a period of $3$~hours $13$~minutes. +In estuaries, the first over-tide of the +lunar semidiurnal tide is often of great importance, +and even the second is considerable; the +third is usually very small, and the fourth and +\PageSep{209} +higher over-tides are imperceptible. In the same +way over-tides must be introduced, to represent +the change of form of the principal solar semidiurnal +tide. But it is not usually found necessary +to consider them in the cases of the less +important partial tides. The octave, the twelfth, +and the upper octave may be legitimately described +as tides, because they are due to the +attractions of the moon and of the sun, although +they arise indirectly through the distorting influence +of the shallowness of the water. + +\TB + +I have said above that about twenty different +simple waves afford a good representation of the +tides at any port. Out of these twenty waves, +some represent the seasonal change of level in +the water due to unequal rainfall and evaporation +at different times of the year, and others +represent the change of shape of the wave due +to shallowing of the water. Deducting these +quasi-tides, we are left with about twelve to +represent the true astronomical tide. It is not +possible to give an exact estimate of the number +of partial tides necessary to insure a good representation +of the aggregate tide wave, because +the characteristics of the motion are so different +at various places that partial waves, important +at one place, are insignificant at others. For +example, at an oceanic island the tides may be +more accurately represented as the sum of a +\PageSep{210} +dozen simple waves than by two dozen in a tidal +river. + +The method of analyzing a tide into its constituent +parts, of which I have now given an +account, is not the only method by which the +tides may be treated, but as it is the most recent +and the best way, I shall not consider the older +methods in detail. The nature of the procedure +adopted formerly will, however, be indicated in +\Ref{Chapter}{XIII}. + +\begin{Authorities}[Authority] +G.~H. Darwin, \Title{Harmonic Analysis of Tidal Observations}: +\index{Analysis, harmonic, of tide|)}% +\index{Darwin, G. H.!harmonic analysis}% +\index{Harmonic analysis!account of|)}% +``Report to British Association.'' Southport,~1883. + +An outline of the method is also contained in \Title{Tides}, ``Encyclopædia +Britannica.'' +\end{Authorities} +\PageSep{211} + + +\Chapter{XII} +{Reduction of Tidal Observations} + +\First{I have} now to explain the process by which +\index{Reduction of tidal observations|(}% +the several partial tides may be disentangled +from one another. + +The tide gauge furnishes a complete tidal record, +so that measurement of the tide curve gives +the height of the water at every instant of time +during the whole period of observation. The +\index{Observation!reduction of tidal|(}% +record may be supposed to begin at noon of a +given day, say of the first of January. The +longitude of the port of observation is of course +known, and the Nautical Almanack gives the +positions of the sun and moon on the day and +at the hour in question, with perfect accuracy. +The real moon has now to be replaced by a +series of ideal satellites, and the rules for the +substitution are absolutely precise. Accordingly, +the position in the heavens of each of +the ideal satellites is known at the moment of +time at which the observations begin. The +same is true of the ideal suns which replace the +actual sun. + +I shall now refer to only a single one of the +ideal moons or suns, for, \textit{mutatis mutandis}, +\PageSep{212} +what is true of one is true of all. It is easy to +calculate at what hour of the clock, measured in +the time of the place of observation, the satellite +in question will be due south. If the ideal +satellite under consideration were the one which +generates the principal lunar semidiurnal tide, it +would be due south very nearly when the real +moon is south, and the ideal sun which generates +the principal solar tide is south exactly at noon. +But there is no such obvious celestial phenomenon +associated with the transit of any other of +the satellites, although it is easy to calculate the +time of the southing of each of them. We have +then to discover how many hours elapse after +the passage of the particular satellite up to the +high water of its tide wave. The height of +the wave crest above, and the depression of the +wave hollow below the mean water mark must +also be determined. When this problem has +been solved for all the ideal satellites and suns, +the tides are said to be reduced, and the reduction +furnishes the materials for a tide table for +the place of observation. + +The difficulty of finding the time of passage +and the height of the wave due to any one of +the satellites arises from the fact that all the +waves really coëxist, and are not separately +manifest. The nature of the disentanglement +may be most easily explained from a special +case, say for example that of the principal lunar +\PageSep{213} +semidiurnal tide, of which the crests follow one +another at intervals of $12$~hours $25$~minutes $14\frac{1}{6}$~seconds. + +Since the waves follow one another at intervals +of approximately, but not exactly, a half-day, it +is convenient to manipulate the time scale so as +\index{Time!lunar}% +to make them exactly semidiurnal. Accordingly +we describe $24$~hours $50$~minutes $28\frac{1}{3}$~seconds as +a lunar day, so that there are exactly two waves +\index{Lunar!time}% +following one another in the lunar day. + +The tide curve furnishes the height of the +\index{Curve, tide!partitioned into lunar time}% +water at every moment of time, but the time +having been registered by the clock of the tide +gauge is partitioned into ordinary days and +hours. It may, however, be partitioned at intervals +of $24$~hours $50$~minutes $28\frac{1}{3}$~seconds, and +into the twenty-fourth parts of that period, and +it will then be divided into lunar days and hours. +On each lunar day the tide for which we are +searching presents itself in the same way, so +that it is always high and low water at the same +hour of the lunar clock, with exactly two high +waters and two low waters in the lunar day. + +Now the other simple tides are governed by +other scales of time, so that in a long succession +of days their high waters and low waters occur +at every hour of the lunar clock. If then we +find the average curve of rise and fall of the +water, when the time is divided into lunar days +and hours, and if we use for the average a long +\PageSep{214} +succession of days, all the other tide waves will +disappear, and we shall be left with only the +lunar semidiurnal tide, purified from all the +others which really coëxist with it. + +The numerical process of averaging thus leads +to the obliteration of all but one of the ideal +satellites, and this is the foundation of the +method of harmonic analysis. The average +lunar tide curve may be looked on as the outcome +of a single day of observation, when all +but the selected satellite have been obliterated. +The height of the average wave, and the interval +after lunar noon up to high water, are the +two tidal constants for the lunar semidiurnal +tide, and they enable us to foretell that tide for +all future time. + +If the tide curve were partitioned into other +days and hours of appropriate lengths, it would +be possible by a similar process of averaging to +single out another of the constituent tide waves, +and to determine its two tidal constants, which +contain the elements of prediction with respect +to it. By continued repetition of operations of +this kind, all the constituents of practical importance +can be determined, and recorded numerically +by means of their pairs of tidal constants. + +The possibility of the disentanglement has +now been demonstrated, but the work of carrying +out these numerical operations would be +\PageSep{215} +\index{Schedule for reducing tidal observations}% +fearfully laborious. The tide curve would have +to be partitioned into about a dozen kinds of +days of various lengths, and the process would +entail measurements at each of the $24$~hours of +each sort of day throughout the whole series. +There are about nine thousand hours in a year, +and it would need about a hundred thousand +measurements of the curve to evaluate twelve +different partial tides; each set of measured +heights would then have to be treated separately +to find the several sorts of averages. Work of +this kind has usually to be done by paid computers, +and the magnitude of the operation +would make it financially prohibitive. It is, +however, fortunately possible to devise abridged +methods, which bring the work within manageable +limits. + +In order to minimize the number of measurements, +the tide curve is only measured at each +of the $24$~exact hours of ordinary time, the +height at noon being numbered $0$~hr., and that +at midnight $12$~hrs., and so on up to $24$~hrs. +After obtaining a set of $24$~measurements for +each day, the original tide curve is of no further +use. The number of measurements involved is +still large, but not prohibitive. It would be +somewhat too technical, in a book of this kind, +to explain in detail how the measured heights of +the water at the exact hours of ordinary time +may be made to give, with fair approximation, +\PageSep{216} +\index{Schedule for reducing tidal observations}% +the heights at the exact hours of other time +scales. It may, however, be well to explain that +this approximate method is based on the fact, +that each exact hour of any one of the special +time scales must of necessity fall within half an +hour of one of the exact hours of ordinary time. +The height of the water at the nearest ordinary +hour is then accepted as giving the height at the +exact hour of the special time. The results, as +computed in this way, are subjected to a certain +small correction, which renders the convention +accurate enough for all practical purposes. + +A schedule, serviceable for all time and for +all places, is prepared which shows the hour of +ordinary time lying nearest to each successive +hour of any one of the special times. The successive +$24$~hourly heights, as measured on the tide +curve, are entered in this schedule, and when +the entry is completed the heights are found to +be arranged in columns, which follow the special +time scale with a sufficiently good approximation +to accuracy. A different form of schedule is +required for each partial tide, and the entry of +the numbers therein is still enormously laborious, +although far less so than the re-partitions and +re-measurements of the tide curve would be. + +The operation of sorting the numbers into +schedules has been carried out in various ways. +In the work of the Indian Survey, the numbers +\index{Indian Survey!method of reducing tidal observations}% +have been re-copied over and over again. In +\PageSep{217} +\index{Abacus for reducing tidal observations|(}% +\index{Darwin, G. H.!tidal abacus|(}% +\index{Indian Survey!method of reducing tidal observations}% +the office of the United States Coast Survey use +\index{United States Coast Survey!method of reducing tidal observations}% +is made of certain card templates pierced with +holes. These templates are laid upon the tabulation +of the measurements of the tide curve, +and the numbers themselves are visible through +the holes. On the surface of the template lines +are drawn from hole to hole, and these lines +indicate the same grouping of the numbers as +that given by the Indian schedules. Dr.~Börgen, +\index{Borgen@Börgen, method of reducing tidal observations}% +of the Imperial German Marine Observatory +\index{German method of reducing tidal observations}% +at Wilhelmshaven, has used sheets of tracing +paper to attain the same end. The Indian procedure +is unnecessarily laborious, and the American +and German plans appear to have some +disadvantage in the fact that the numbers to be +added together lie diagonally across the page. +I am assured by some professional computers +that diagonal addition is easy to perform correctly; +nevertheless this appeared to me to be +so serious a drawback, that I devised another +plan by which the numbers should be brought +into vertical columns, without the necessity of +re-copying them. In my plan each day is treated +as a unit and is shifted appropriately. It might +be thought that the results of the grouping +would be considerably less accurate than in the +former methods, but in fact there is found to be +no appreciable loss of accuracy. + +I have $74$~narrow writing-tablets of xylonite, +divided by lines into $24$~compartments; the +\PageSep{218} +tablets are furnished with spikes on the under +side, so that they can be fixed temporarily in any +position on an ordinary drawing-board. The +compartments on each strip are provided for the +entry of the $24$~tidal measurements appertaining +to each day. Each strip is stamped at its end +with a number specifying the number of the day +to which it is appropriated. + +The arrangement of these little tablets, so that +the numbers written on them may fall into columns, +is indicated by a sheet of paper marked +with a sort of staircase, which shows where each +tablet is to be set down, with its spikes piercing +the guide sheet. When the strips are in place, +as shown in~\fig{33}, the numbers fall into $48$~columns, +numbered $0$,~$1$,~\dots~$23$, $0$,~$1$,~\dots~$23$ +twice over. The guide sheet shown in the figure~\figref{33} +is the one appropriate for the lunar semidiurnal +tide for the fourth set of $74$~days of +a year of observation. The upper half of the +tablets are in position, but the lower ones are +left unmounted, so as the better to show the +staircase of marks. + +Then I say that the average of all the $74$~numbers +standing under the two~$0$'s combined +will give the average height of water at $0$~hr.\ +of lunar time, and the average of the numbers +under~$1$, that at $1$~hr.\ of lunar time, and so forth. +Thus, after the strips are pegged out, the computer +has only to add the numbers in columns in +%% Plate 2 +\TallFig{33}{Tidal Abacus}{jpg} +%% Facing page +%[Blank Page] +\PageSep{219} +order to find the averages. There are other +sheets of paper marked for such other rearrangements +of the strips that each new setting gives +one of the required results; thus a single writing +of the numbers serves for the whole computation. +It is usual to treat a whole year of +observations at one time, but the board being +adapted for taking only $74$~successive days, five +series of writings are required for $370$~days, +which is just over a year. The number~$74$ was +chosen for simultaneous treatment, because $74$~days +is almost exactly five semilunations, and +accordingly there will always be five spring tides +on record at once. + +In order to guard the computer against the +use of the wrong paper with any set of strips, +the guide sheets for the first set of $74$~days are +red; for the second they are yellow; for the +third green; for the fourth blue; for the fifth +violet, the colors being those of the rainbow. + +The preparation of these papers entailed a +great deal of calculation in the first instance, but +the tidal computer has merely to peg out the +tablets in their right places, verifying that the +numbers stamped on the ends of the strips agree +with the numbers on the paper. The addition +of the long columns of figures is certainly laborious, +but it is a necessary incident of every +method of reducing tidal observations. + +The result of all the methods is that for each +\PageSep{220} +partial tide we have a set of $24$~numbers, which +represent the oscillations of the sea due to the +isolated action of one of the ideal satellites, during +the period embraced between two successive +passages of that satellite to the south of the +place of observation. The examination of each +partial tide wave gives its height, and the interval +of time which elapses after its satellite has +passed the meridian until it is high water for +that particular tide. The height and interval +are the tidal constants for that particular tide, at +the port of observation. +\index{Observation!reduction of tidal|)}% + +The results of this ``reduction of the observations'' +are contained in some fifteen or twenty +pairs of tidal constants, and these numbers contain +a complete record of the behavior of the sea +at the place in question. + +\begin{Authorities} +G.~H. Darwin, \Title{Harmonic Analysis, \&c.}: ``Report to British Association,'' +1883. + +G.~H. Darwin, \Title{On an apparatus for facilitating the reduction of +\index{Darwin, G. H.!tidal abacus|)}% +tidal observations}: ``Proceedings of the Royal Society,'' vol.~lii.\ +1892. +\end{Authorities} +\index{Abacus for reducing tidal observations|)}% +\index{Reduction of tidal observations|)}% +\PageSep{221} + + +\Chapter{XIII} +{Tide Tables} + +\First{A tide table} professes to tell, at a given +\index{Tables, tide|(}% +place and on a given day, the time of high and +low water, together with the height of the rise +and the depth of the fall of the water, with +reference to some standard mark on the shore. +A perfect tide table would tell the height of the +water at every moment of the day, but such a +table would be so bulky that it is usual to predict +only the high and low waters. + +There are two kinds of tide table, namely, +those which give the heights and times of high +and low water for each successive day of each +year, and those which predict the high and low +water only by reference to some conspicuous +celestial phenomenon. Both sorts of tide table +refer only to the particular harbor for which they +are prepared. + +The first kind contains definite forecasts for +each day, and may be called a special tide table. +Such a table is expensive to calculate, and must +be published a full year beforehand. Special +tide tables are published by all civilized countries +for their most important harbors. I believe that +\PageSep{222} +\index{Indian Survey!tide tables}% +\index{United States Coast Survey!tide tables of}% +the most extensive publications are those of the +Indian Government for the Indian Ocean, and +of the United States Government for the coasts +of North America. The Indian tables contain +\index{America, North, tide tables for}% +predictions for about thirty-seven ports. + +The second kind of table, where the tide is +given by reference to a celestial phenomenon, +may be described as a general one. It is here +necessary to refer to the Nautical Almanack for +the time of occurrence of the celestial phenomenon, +and a little simple calculation must then be +made to obtain the prediction. The phenomenon +to which the tide is usually referred is the passage +of the moon across the meridian of the place of +observation, and the table states that high and +low water will occur so many hours after the +moon's passage, and that the water will stand at +such and such a height. + +The moon, at her change, is close to the sun +and crosses the meridian at noon; she would +then be visible but for the sun's brightness, and +if she did not turn her dark side towards us. +She again crosses the meridian invisibly at midnight. +At full moon she is on the meridian, +visibly at midnight, and invisibly at noon. At +waxing half moon she is visibly on the meridian +at six at night, and at waning half moon at six +in the morning. The hour of the clock at which +the moon passes the meridian is therefore in effect +a statement of her phase. Accordingly the +\PageSep{223} +relative position of the sun and moon is directly +involved in a statement of the tide as corresponding +to a definite hour of the moon's passage. A +table founded on the time of the moon's passage +must therefore involve the principal lunar and +solar semidiurnal tides. + +At places where successive tides differ but little +from one another, a simple table of this kind +suffices for rough predictions. The curves marked +Portsmouth in \fig{34} show graphically the interval +after the moon's passage, and the height +of high water at that port, for all the hours of +the moon's passage. We have seen in \Ref{Chapter}{X}.\ +that the tide in the North Atlantic is principally +due to a wave propagated from the Southern +Ocean. Since this wave takes a considerable +time to travel from the Cape of Good Hope to +England, the tide here depends, in great measure, +on that generated in the south at a considerable +time earlier. It has therefore been found better +to refer the high water to a transit of the moon +which occurred before the immediately preceding +one. The reader will observe that it is noted on +the upper figure that $28$~hours have been subtracted +from the Portsmouth intervals; that is +to say, the intervals on the vertical scale marked +$6$,~$7$,~$8$ hours are, for Portsmouth, to be interpreted +as meaning $34$,~$35$,~$36$ hours. These are +the hours which elapse after any transit of the +moon up to high water. The horizontal scale is +\PageSep{224} +\index{Moon and earth!tidal prediction by reference to transit|(}% +one of the times of moon's transit and of phases +of the moon; the vertical scale in the lower figure +is one of feet, and it shows the height to +which the water will rise measured from a certain +mark ashore. These Portsmouth curves do not +extend beyond 12~o'clock of moon's transit; this +is because there is hardly any diurnal inequality, +\index{Diurnal inequality!complicates prediction}% +and it is not necessary to differentiate the hours +as either diurnal or nocturnal, the statement being +equally true of either day or night. Thus +if the Portsmouth curves had been extended onward +from $12$~hours to $24$~hours of the clock time +of the moon's passage, the second halves of the +curves would have been merely the duplicates of +the first halves.\footnote + {Before the introduction of the harmonic analysis of the tides + described in preceding chapters, tidal observations were ``reduced'' + by the construction of such figures as these, directly from + the tidal observations. Every high water was tabulated as appertaining + to a particular phase of the moon, both as to its height + and as to the interval between the moon's transit and the occurrence + of high water. The average of a long series of observations + may be represented in the form of curves by such figures + as these.} + +But the time of the moon's passage leaves her +angular distance from the equator and her linear +distance from the earth indeterminate; and since +the variability of both of these has its influence +on the tide, corrections are needed which add +something to or subtract something from the +tabular values of the interval and height, as dependent +solely on the time of the moon's passage. +\PageSep{225} +\index{Diurnal inequality!complicates prediction}% +\index{Interval from moon's transit to high water!at Portsmouth and at Aden}% +The sun also moves in a plane which is oblique +to the equator, and so similar allowances must be +made for the distance of the sun from the equator, +and for the variability in his distance from the +earth. In order to attain accuracy with a tide +table of this sort, eight or ten corrections are +needed, and the use of the table becomes complicated. + +It is, however, possible by increasing the number +of such figures or tables to introduce into +them many of the corrections referred to; and +the use of a general tide table then becomes comparatively +simple. The sun occupies a definite +position with reference to the equator, and stands +at a definite distance from the earth on each day +of the year; also the moon's path amongst the +stars does not differ very much from the sun's. +Accordingly a tide table which states the interval +after the moon's passage to high or low water +and the height of the water on a given day of +\index{Height of tide!at Portsmouth and at Aden}% +the year will directly involve the principal inequalities +in the tides. As the sun moves slowly +amongst the stars, a table applicable to a given +day of the year is nearly correct for a short time +before and after that date. If, then, a tide table, +stating the time and height of the water by reference +to the moon's passage, be computed for +say every ten days of the year, it will be very +nearly correct for five days before and for five +days after the date for which it is calculated. +\PageSep{226} + +The curves marked Aden, March and June, in +\index{Prediction of tide!example at Aden|(}% +\fig{34}, show the intervals and heights of tide, +on the 15th of those months at that port, for all +the hours of the moon's passage. The curves are +to be read in the same way as those for Portsmouth, +but it is here necessary to distinguish the +hours of the day from those of the night, and +accordingly the clock times of moon's transit are +numbered from $0$~hr.\ at noon up to $24$~hrs.\ at +the next noon. The curves for March differ so +much from those for June, that the corrections +would be very large, if the tides were treated at +Aden by a single pair of average curves as at +Portsmouth. + +The law of the tides, as here shown graphically, +may also be stated numerically, and the +use of such a table is easy. The process will be +best explained by an example, which happens to +be retrospective instead of prophetic. It will involve +that part of the complete table (or series of +curves) for Aden which applies to the 15th of +March of any year. Let it be required then to +find the time and height of high water on March~17, +1889. The Nautical Almanack for that year +shows that on that day the moon passed the meridian +of Aden at eleven minutes past noon of +Aden time, or in astronomical language at $0$~hr.\ +$11$~mins. Now the table, or the figure of intervals, +shows that if the moon had passed at $0$~hr., +or exactly at noon, the interval would have been +\PageSep{227} +$8$~hrs.\ $9$~mins., and that if she had passed at $0$~hr.\ +$20$~mins., or 12.20~\PM\ of the day, the interval +would have been $7$~hrs.\ $59$~mins. But on +March~17th the moon actually crossed at $0$~hr.\ +\Figure{34}{Curves of Intervals and Heights at Portsmouth +and at Aden}{png} +$11$~mins., very nearly halfway between noon and +$20$~mins.\ past noon. Hence the interval was +halfway between $8$~hrs.\ $9$~mins.\ and $7$~hrs.\ $59$~mins., +so that it was $8$~hrs.\ $4$~mins. Accordingly +it was high water $8$~hrs.\ $4$~mins.\ after the moon +\PageSep{228} +crossed the meridian. But the moon crossed at +$0$~hr.\ $11$~mins., therefore the high water occurred +at 8.15~\PM. + +Again the table of heights, or the figure, shows +that on March~15th, if the moon crossed at $0$~hr.\ +$0$~min.\ the high water would be $6.86$~ft.\ above +a certain mark ashore, and if she crossed at $0$~hr.\ +$20$~mins.\ the height would be $6.92$~ft. But on +March~17th the moon crossed halfway between +$0$~hr.\ $0$~min.\ and $0$~hr.\ $20$~mins., and therefore +the height was halfway between $6.86$~ft.\ and +$6.92$~ft., that is to say, it was $6.89$~ft., or $6$~ft.\ +$11$~in. We therefore conclude that on March~17, +1889, the sea at high water rose to $6$~ft.\ +$11$~in., at 8.15~\PM. I have no information as +to the actual height and time of high water on +that day, but from the known accuracy of other +predictions at Aden we may be sure that this +agrees pretty nearly with actuality. The predictions +derived from this table are markedly improved +when a correction, either additive or subtractive, +is applied, to allow for the elliptic motion +of the moon round the earth. On this particular +occasion the moon stood rather nearer the earth +than the average, and therefore the correction to +the height is additive; the correction to the time +also happens to be additive, although it could +not be foreseen by general reasoning that this +would be the case. The corrections for March~17, +1889, are found to add about $2$~mins.\ to the +\PageSep{229} +time, bringing it to 8.17~\PM, and nearly two +inches to the height, bringing it to $7$~ft.\ $1$~in. + +This sort of elaborate general tide table has +been, as yet, but little used. It is expensive to +calculate, in the first instance, and it would occupy +two or three pages of a book. The expense +is, however, incurred once for all, and the table +is available for all time, provided that the tidal +observations on which it is based have been good. +A sea captain arriving off his port of destination +would not take five minutes to calculate the two +or three tides he might require to know, and the +information would often be of the greatest value +to him. + +As things stand at present, a ship sailing to +most Chinese, Pacific, or Australian ports is only +furnished with a statement, often subject to considerable +error, that the high water will occur at +so many hours after the moon's passage and will +rise so many feet. The average rise at springs and +neaps is generally stated, but the law of the variability +according to the phases of the moon is wanting. +But this is not the most serious defect in the +information, for it is frequently noted that the +tide is ``affected by diurnal inequality,'' and this +note is really a warning to the navigator that he +cannot foretell the time of high water within two +or three hours of time, or the height within several +feet. + +Tables of the kind I have described would +\PageSep{230} +\index{Prediction of tide!method of computing|(}% +banish this extreme vagueness, but they are more +likely to be of service at ports of second-rate importance +than at the great centres of trade, because +at the latter it is worth while to compute +full special tide tables for each year. +\index{Tables, tide!method of calculating|(}% + +It is unnecessary to comment on the use of +tables containing predictions for definite days, +since it merely entails reference to a book, as to +a railway time table. Such special tables are undoubtedly +the most convenient, but the number +of ports which can ever be deemed worthy of the +great expense incidental to their preparation +must always be very limited. + +\TB + +We must now consider the manner in which +tide tables are calculated. It is supposed that +careful observations have been made, and that +the tidal constants, which state the laws governing +the several partial tides, have been accurately +determined by harmonic analysis. The analysis +of tidal observations consists in the dissection of +the aggregate tide wave into its constituent partial +waves, and prediction involves the recomposition +or synthesis of those waves. In the synthetic +\index{Synthesis of partial tides for prediction|(}% +process care must be taken that the partial +waves shall be recompounded in their proper +relative positions, which are determined by the +places of the moon and sun at the moment of +time chosen for the commencement of prediction. + +The synthesis of partial waves may be best +\index{Moon and earth!tidal prediction by reference to transit|)}% +\index{Prediction of tide!example at Aden|)}% +\PageSep{231} +arranged in two stages. It has been shown in +\Ref{Chapter}{XI}.\ that the partial waves fall naturally +into three groups, of which the third is practically +insignificant. The first and second are the +semidiurnal and diurnal groups. The first process +is to unite each group into a single wave. + +We will first consider the semidiurnal group. +Let us now, for the moment, banish the tides +from our minds, and imagine that there are two +trains of waves traveling simultaneously along a +straight canal. If either train existed by itself +every wave would be exactly like all its brethren, +both in height, length, and period. Now suppose +that the lengths and periods of the waves +of the two coëxistent trains do not differ much +from one another, although their heights may +differ widely. Then the resultant must be a single +train of waves of lengths and periods intermediate +between those of the constituent waves, +but in one part of the canal the waves will be +high, where the two sets of crests fall in the +same place, whilst in another they will be low, +where the hollow of the smaller constituent wave +falls in with the crest of the larger. If only one +part of the canal were visible to us, a train of +waves would pass before us, whose heights would +gradually vary, whilst their periods would change +but little. + +In the same way two of the semidiurnal tide +waves, when united by the addition of their separate +\PageSep{232} +displacements from the mean level, form a +single wave of variable height, with a period still +semidiurnal, although slightly variable. But +there is nothing in this process which limits the +synthesis to two waves, and we may add a third +and a fourth, finally obtaining a single semidiurnal +wave, the height of which varies according +to a very complex law. + +A similar synthesis is then applied to the second +group of waves, so that we have a single +variable wave of approximately diurnal period. +The final step consists in the union of the single +semidiurnal wave with the single diurnal one into +a resultant wave. When the diurnal wave is +large, the resultant is found to undergo very +great variability both in period and height. The +principal variations in the relative positions of +the partial tide waves are determined by the +phases of the moon and by the time of year, and +there is, corresponding to each arrangement of +the partial waves, a definite form for the single +resultant wave. The task of forming a general +tide table therefore consists in the determination +of all the possible periods and heights of the resultant +wave and the tabulation of the heights +and intervals after the moon's passage of its high +and low waters. + +I supposed formerly that the captain would +himself calculate the tide he required from the +general tide table, but such calculation may be +\PageSep{233} +\index{Machine, tide-predicting}% +\index{Predicting machine for tides|(}% +done beforehand for every day of a specified +year, and the result will be a special tide table. +There are about $\DPchg{1400}{1,400}$ high and low waters in +a year, so that the task is very laborious, and +has to be repeated each year. + +\TB + +It is, however, possible to compute a special +tide table by a different and far less laborious +method. In this plan an ingenious mechanical +device replaces the labor of the computer. The +first suggestion for instrumental prediction of +tides was made, I think, by Sir William Thomson, +now Lord Kelvin, in~1872. Mr.~Edward +\index{Kelvin, Lord!tide predicting machine}% +Roberts bore an important part in the practical +\index{Roberts, E., the tide-predicting machine}% +realization of such a machine, and a tide predicter +was constructed by Messrs.\ Légé for the +\index{Lege@Légé, constructor of tide-predicting machine}% +Indian Government under his supervision. This +is, as yet, the only complete instrument in existence. +But others are said to be now in course +of construction for the Government of the +United States and for that of France. The +Indian machine cost so much and works so well, +that it is a pity it should not be used to the full +extent of its capacity. The Indian Government +has, of course, the first claim on it, but the use +of it is allowed to others on the payment of a +small fee. I believe that, pending the construction +of their own machine, the French authorities +are obtaining the curves for certain tidal +predictions from the instrument in London. +\index{Prediction of tide!method of computing|)}% +\index{Synthesis of partial tides for prediction|)}% +\PageSep{234} + +Although the principle involved in the tide +predicter is simple, yet the practical realization +of it is so complex that a picture of the whole +machine would convey no idea of how it works. +I shall therefore only illustrate it diagrammatically, +in~\fig{35}, without any pretension to scale +or proportion. The reader must at first imagine +that there are only two pulleys, namely, $A$~and~$B$, +so that the cord passes from the fixed end~$F$ +under~$A$ and over~$B$, and so onward to the pencil. +The pulley~$B$ is fixed, and the pulley~$A$ can slide +vertically up and down in a slot, which is not +shown in the diagram. If $A$~moves vertically +through any distance, the pencil must clearly +move through double that distance, so that +when $A$~is highest the pencil is lowest, and vice +versa. + +The pencil touches a uniformly revolving +drum, covered with paper; thus if the pulley~$A$ +executes a simple vertical oscillation, the pencil +draws a simple wave on the drum. Now the +pulley is mounted on an inverted T-shaped +frame, and a pin, fixed in a crank~$C$, engages in +the slit in the horizontal arm of the T-piece. +When the crank~$C$ revolves, the pulley~$A$ executes +a simple vertical oscillation with a range depending +on the throw of the crank.\footnote + {I now notice that the throw of the crank~$C$ is too small to + have allowed the pencil to draw so large a wave as that shown + on the drum. But as this is a mere diagram, I have not thought + it worth while to redraw the whole.} +The position +\PageSep{235} +of the pin is susceptible of adjustment on the +crank, so that its throw and the range of oscillation +of the pulley can be set to any required +\Figure[0.7]{35}{Diagram of Tide-predicting Instrument}{png} +length---of course within definite limits determined +by the size of the apparatus. + +The drum is connected to the crank~$C$ by a +train of wheels, so that as the crank rotates the +drum also turns at some definitely proportional +rate. If, for example, the crank revolves twice +for one turn of the drum, the pencil will draw a +simple wave, with exactly two crests in one circumference +of the drum. If one revolution of +the drum represents a day, the graphical time +scale is $24$~hours to the circumference of the +\PageSep{236} +drum. If the throw of the crank be one inch, +the pulley will oscillate with a total range of two +inches, and the pencil with a total range of four +inches. Then taking two inches lengthwise on +the drum to represent a foot of water, the curve +drawn by the pencil might be taken to represent +the principal solar semidiurnal tide, rising one +foot above and falling one foot below the mean +sea level. + +I will now show how the machine is to be +adjusted so as to give predictions. We will +suppose that it is known that, at noon of the +first day for which prediction is required, the +solar tide will stand at $1$~ft.\ $9$~in.\ above mean +sea level and that the water will be rising. Then, +the semi-range of this tide being one foot, the +pin is adjusted in the crank at one inch from +the centre, so as to make the pencil rock through +a total range of $4$~inches, representing $2$~feet. +The drum is now turned so as to bring the noon-line +of its circumference under the pencil, and +the crank is turned so that the pencil shall be +$3\frac{1}{2}$~inches (representing $1$~ft.\ $9$~in.\ of water) +below the middle of the drum, and so that when +the machine starts, the pencil will begin to descend. +The curve being drawn upside-down, +the pencil is set below the middle line because +the water is to be above mean level, and it must +begin to descend because the water is to ascend. +The train of wheels connecting the crank and +\PageSep{237} +drum is then thrown into gear, and the machine +is started; it will then draw the solar tide curve, +on the scale of $2$~inches to the foot, for all +time. + +If the train of wheels connecting the crank to +the drum were to make the drum revolve once +whilst the crank revolves $1.93227$~times, the +curve would represent a lunar semidiurnal tide. +The reason of this is that $1.93227$~is the ratio +of $24$~hours to $12$~h.\ $25$~m.\ $14$~s., that is to say, +of a day to a lunar half day. We suppose the +circumference of the drum still to represent an +ordinary day of $24$~hours, and therefore the +curve drawn by the pencil will have lunar semidiurnal +periodicity. In order that these curves +may give predictions of the future march of that +tide, the throw of the crank must be set to give the +correct range and its angular position must give +the proper height at the moment of time chosen +for beginning. When these adjustments are +made the curve will represent that tide for all +time. + +We have now shown that, by means of appropriate +trains of wheels, the machine can be made +to predict either the solar or the lunar tide; but +we have to explain the arrangement for combining +them. If, still supposing there to be +only the two pulleys $A$,~$B$, the end~$F$ of the cord +were moved up or down, its motion would be +transmitted to the pencil, whether the crank~$C$ +\PageSep{238} +and pulley~$A$ were in motion, or at rest; but if +they were in motion, the pencil would add the +motion of the end of the cord to that of the +pulley. If then there be added another fixed +pulley~$B'$, and another movable pulley~$A'$, driven +by a crank and T-piece (not shown in the diagram), +the pencil will add together the movements +of the two pulleys $A$~and~$A'$. There must +now be two trains of wheels, one connecting $A$ +with the drum and the other for~$A'$. If a single +revolution of the drum causes the crank~$C$ to +turn twice, whilst it makes the crank of~$A'$ rotate +$1.93227$~times, the curve drawn will represent +the union of the principal solar and lunar semidiurnal +tides. The trains of wheels requisite for +transmitting motion from the drum to the two +cranks in the proper proportions are complicated, +but it is obviously only a matter of calculation +to determine the numbers of the teeth in the +several wheels in the trains. It is true that rigorous +accuracy is not attainable, but the mechanism +is made so nearly exact that the error in the +sum of the two tides would be barely sensible +even after $\DPchg{3000}{3,000}$~revolutions of the drum. It is +of course necessary to set the two cranks with +their proper throws and at their proper angles +so as to draw a curve which shall, from the noon +of a given day, correspond to the tide at a given +place. + +It must now be clear that we may add as +\PageSep{239} +many more movable pulleys as we like. When +the motion of each pulley is governed by an +appropriate train of wheels, the movement of +the pencil, in as far as it is determined by that +pulley, corresponds to the tide due to one of our +ideal satellites. The resultant curve drawn on +the drum is then the synthesis of all the partial +tides, and corresponds with the motion of the +sea. + +The instrument of the Indian Government +unites twenty-four partial tides. In order to +trace a tide curve, the throws of all the cranks +are set so as to correspond with the known +heights of the partial tides, and each crank is set +at the proper angle to correspond with the moment +of time chosen for the beginning of the tide +table. It is not very difficult to set the cranks +and pins correctly, although close attention is of +course necessary. The apparatus is then driven +by the fall of a weight, and the paper is fed +automatically on to the drum and coiled off on +to a second drum, with the tide curve drawn on +it. It is only necessary to see that the paper +runs on and off smoothly, and to write the date +from time to time on the paper as it passes, in +order to save future trouble in the identification +of the days. It takes about four hours to run +off the tides for a year. + +The Indian Government sends home annually +the latest revision of the tidal constants for +\PageSep{240} +thirty-seven ports in the Indian Ocean. Mr.~Roberts +sets the machine for each port, so as to +correspond with noon of a future 1st~of January, +and then lets it run off a complete tide +curve for a whole year. The curve is subsequently +measured for the time and height of +each high and low water, and the printed tables +are sold at the moderate price of four rupees. +The publication is made sufficiently long beforehand +to render the tables available for future +voyages. These tide tables are certainly amongst +the most admirable in the world. + +\TB + +It is characteristic of England that the machine +is not, as I believe, used for any of the +home ports, and only for a few of the colonies. +The neglect of the English authorities is not, +however, so unreasonable as it might appear to +be. The tides at English ports are remarkably +simple, because the diurnal inequality is practically +absent. The applicability of the older +methods of prediction, by means of such curves +as that for Portsmouth in~\fig{34}, is accordingly +easy, and the various corrections are well determined. +The arithmetical processes are therefore +not very complicated, and ordinary computers +are capable of preparing the tables with but +little skilled supervision. Still it is to be regretted +that this beautiful instrument should not +be more used for the home and colonial ports. +\PageSep{241} + +The excellent tide tables of the Government +of the United States have hitherto been prepared +by the aid of a machine of quite a different +character, the invention of the late Professor +Ferrel. This apparatus virtually carries out +\index{Ferrel, tide-predicting instrument}% +\index{Predicting machine for tides!Ferrel's}% +that process of compounding all the waves together +into a single one, which I have described +as being done by a computer for the formation +of a general tide table. It only registers, however, +the time and height of the maxima and +minima---the high and low waters. I do not +think it necessary to describe its principle in +detail, because it will shortly be superseded by a +machine like, but not identical with, that of the +Indian Government. + +\begin{Authorities} +G.~H. Darwin, \Title{On Tidal Prediction}. ``Philosophical Transactions +of the Royal Society,'' A.~1891, pp.~159--229. + +In the example of the use of a general tide table at Aden, +given in this chapter, the datum from which the height is measured +is $0.37$~ft.\ higher than that used in the Indian Tide Tables; +\index{Tables, tide|)}% +\index{Tables, tide!method of calculating|)}% +accordingly $4\frac{1}{2}$~inches must be added to the height, in order to +bring it into accordance with the official table. + +Sir William Thomson, \Title{Tidal Instruments}, and the subsequent +discussion. ``Institute of Civil Engineers,'' vol.~lxv. + +William Ferrel, \Title{Description of a Maxima and Minima Tide-predicting +\index{Machine, tide-predicting}% +Machine}. ``United States Coast Survey,'' 1883. +\end{Authorities} +\index{Predicting machine for tides|)}% +\PageSep{242} + + +\Chapter[Accuracy of Tidal Prediction]{XIV} +{The Degree of Accuracy of Tidal Prediction} + +\First{The} success of tidal predictions varies much +\index{Atmospheric pressure!influence on tidal prediction}% +\index{Prediction of tide!errors in|(}% +according to the place of observation. They are +not unfrequently considerably in error in our +latitude, and throughout those regions called by +sailors ``the roaring forties.'' The utmost that +can be expected of a tide table is that it shall +be correct in calm weather and with a steady +barometer. But such conditions are practically +non-existent, and in the North Atlantic the great +variability in the meteorological elements renders +tidal prediction somewhat uncertain. + +The sea generally stands higher when the +barometer is low, and lower when the barometer +is high, an inch of mercury corresponding to +rather more than a foot of water. The pressure +of the air on the sea in fact depresses it in those +places where the barometer is high, and allows it +to rise where the opposite condition prevails. + +Then again a landward wind usually raises the +\index{Wind!perturbation of, in tidal prediction}% +sea level, and in estuaries the rise is sometimes +very great. There is a known instance when the +Thames at London was raised by five feet in a +strong gale. Even on the open coast the effect +\PageSep{243} +\index{Atmospheric pressure!influence on tidal prediction}% +of wind is sometimes great. A disastrous example +\index{Wind!perturbation of, in tidal prediction}% +of this was afforded on the east coast of England +in the autumn of~1897, when the conjunction +of a gale with springtide caused the sea to +do an enormous amount of damage, by breaking +embankments and flooding low-lying land. + +But sometimes the wind has no apparent effect, +and we must then suppose that it had been blowing +previously elsewhere in such a way as to depress +the water at the point at which we watch it. +The gale might then only restore the water to its +normal level, and the two effects might mask one +another. The length of time during which the +wind has lasted is clearly an important factor, +because the currents generated by the wind must +be more effective in raising or depressing the sea +level the longer they have lasted. + +It does not then seem possible to formulate +any certain system of allowance for barometric +pressure and wind. There are, at each harbor, +certain rules of probability, the application of +which will generally lead to improvement in the +prediction; but occasionally such empirical corrections +will be found to augment the error. + +But notwithstanding these perturbations, good +tide tables are usually of surprising accuracy +even in northern latitudes; this may be seen +from the following table showing the results of +comparisons between prediction and actuality at +Portsmouth. The importance of the errors in +\index{Errors in tidal prediction|(}% +\PageSep{244} +height depends of course on the range of tide; +it is therefore well to note that the average ranges +of tide at springs and neaps are $13$~ft.\ $9$~in.\ and +$7$~ft.\ $9$~in.\ respectively. +\begin{table}[hbtp!] +\caption{Table of Errors in the Prediction of High Water at +Portsmouth in the Months of January, May, And +\index{Portsmouth, table of errors in tidal predictions}% +September, 1897.} +\[ +\begin{array}{|c|c||c|c|} +\hline +\multicolumn{2}{|c||}{\ColHead{Time}} & +\multicolumn{2}{c|}{\ColHead{Height}} \\ +\hline +\ColHead[1.2in]{Magnitude of error} & +\ColHead[0.5in]{Number of cases} & +\ColHead[1.2in]{Magnitude of error} & +\ColHead[0.5in]{Number of cases} \\ +\hline +&& \TEntry{Inches} & \\ +\Z0\mm\text{ to }\Z5\mm& 69 & \Z0\text{ to }\Z6 & 89 \\ +\Z6\mm\text{ to }10\mm & 50 & \Z7\text{ to }12 & 58 \\ + 11\mm\text{ to }15\mm & 25 & 13\text{ to }18 & 24 \\ + 16\mm\text{ to }20\mm & 10 & 19\text{ to }24 & \Z6 \\ + 21\mm\text{ to }25\mm & 11 & \Dash & \Dash \\ + 26\mm\text{ to }30\mm & \Z7 & \Dash & \Dash \\ + 31\mm\text{ to }35\mm & \Z4 & \Dash & \Dash \\ + 52\mm & \Z1 & \Dash & \Dash \\ +\hline +\Strut\Dash & \llap{$1$}77&\Dash & \llap{$1$}77 \\ +\hline +\end{array} +\] +\end{table} +\begin{table}[hbtp!] +\caption{Errors in Height for the Year 1892, +Excepting Part of July} +\[ +\begin{array}{|c|c|} +\hline +\ColHead{Magnitude of error} & \ColHead{Number of cases} \\ +\hline +\TEntry{Inches} & \\ +\Z0\text{ to }\Z6& 381 \\ +\Z7\text{ to }12 & 228 \\ + 13\text{ to }18 & \Z52 \\ + 19\text{ to }24 & \Z\Z8 \\ + 31 & \Z\Z1 \\ +\hline +\Strut\Dash & 670 \\ +\hline +\end{array} +\] +\end{table} +\PageSep{245} + +\begin{Remark} +\NB---The comparison seems to indicate that these predictions +might be much improved, because the predicted height is +nearly always above the observed height, and because the diurnal +inequality has not been taken into account sufficiently, if at +all. +\end{Remark} + +In tropical regions the weather is very uniform, +and in many places the ``meteorological +tides'' produced by the regularly periodic variations +of wind and barometric pressure are taken +into account in tidal predictions. + +The apparent irregularity of the tides at Aden +is so great, that an officer of the Royal Engineers +has told me that, when he was stationed there +many years ago, it was commonly believed that +the strange inequalities of water level were due +to the wind at distant places. We now know +that the tide at Aden is in fact marvelously +regular, although the rule according to which it +proceeds is very complex. In almost every month +in the year there are a few successive days when +there is only one high water and one low water +in the $24$~hours; and the water often remains +almost stagnant for three or four hours at a +time. This apparent irregularity is due to the +diurnal inequality, which is very great at Aden, +whereas on the coasts of Europe it is insignificant. + +I happen to have a comparison with actuality +of a few predictions of high water at Aden, +where the maximum range of the tide is about +$8$~ft.\ $6$~in. They embrace the periods from March~10 +\index{Errors in tidal prediction|)}% +\PageSep{246} +\index{Aden, errors of tidal prediction at}% +\index{Tables, tide!amount of error in}% +to April~9, and again from November~12 to +December~12, 1884. In these two periods there +were $118$~high waters, but through an accident +to the tide gauge one high water was not registered. +On one occasion, when the regular semidiurnal +sequence of the tide would lead us to +expect high water, there occurred one of those +periods of stagnation to which I have referred. +Thus we are left with $116$~cases of comparison +between the predicted and actual high waters. + +The results are exhibited in the following +table:--- +\[ +\begin{array}{|c|c||c|c|} +\hline +\multicolumn{2}{|c||}{\ColHead{Time}} & +\multicolumn{2}{c|}{\ColHead{Height}} \\ +\hline +\ColHead{Magnitude of errors} & +\ColHead[0.5in]{Number of high waters} & +\ColHead{Magnitude of errors} & +\ColHead[0.5in]{Number of high waters} \\ +\hline +&& \TEntry{Inches} & \\ +\Z0\mm\PadTxt{ and }{to}\Z5\mm & 35 & 0 & 15 \\ +\Z5\mm\PadTxt{ and }{to}10\mm & 32 & 1 & 48 \\ + 10\mm\PadTxt{ and }{to}15\mm & 19 & 2 & 28 \\ + 15\mm\PadTxt{ and }{to}20\mm & 19 & 3 & 14 \\ + 20\mm\PadTxt{ and }{to}25\mm & \Z5 & 4 & 11 \\ + 26\mm\text{ and }28\mm & \Z2 & +\multicolumn{1}{c|}{\TEntry{No high water}} & \Z1 \\ + 33\mm\text{ and }36\mm & \Z2 & \Dash & \Dash \\ + 56\mm\text{ and }57\mm & \Z2 & \Dash & \Dash \\ +\multicolumn{1}{|c|}{\TEntry{No high water}} & \Z1 & \Dash & \Dash \\ +\hline +\Strut & 117 && 117 \\ +\hline +\end{array} +\] + +It would be natural to think that when the +prediction is erroneous by as much as $57$~minutes, +it is a very bad one; but I shall show that +\PageSep{247} +\index{Tables, tide!amount of error in}% +this would be to do injustice to the table. On +several of the occasions comprised in this list +the water was very nearly stagnant. Now if the +water only rises about a foot from low to high +water in the course of four or five hours, it is +almost impossible to say with accuracy when it +was highest, and two observers might differ in +their estimate by half an hour or even by an +hour. + +In the table of comparison there are $11$~cases +in which the error of time is equal to or greater +than twenty minutes, and I have examined these +cases in order to see whether the water was then +nearly stagnant. A measure of the degree of +stagnation is afforded by the amount of the rise +from low water to high water, or of the fall from +high water to low water. The following table +gives a classification of the errors of time according +to the rise or fall:--- +\begin{table}[hp!] +\caption{Analysis of Errors in Time.} +\centering +\begin{tabular}{|c|c|} +\hline +\ColHead[1.5in]{Ranges from low water to high water} & +\ColHead{Errors of time} \\ +\hline +\Strut +Nil & \Dash \\ +$6$ in.\ to $8$ in. & $22$, $26$, $28$, $56$, $57$~minutes \\ +$13$ in. & $36$~minutes \\ +$17$ in. & $22$ \Ditto{minutes} \\ +$19$ in. & $33$ \Ditto{minutes} \\ +$2$ ft.\ $10$ in. & $22$ \Ditto{minutes} \\ +$3$ ft.\ $\Z9$ in. & $23$ \Ditto{minutes} \\ +$3$ ft.\ $11$ in. & $20$ \Ditto{minutes} \\ +\hline +\end{tabular} +\end{table} +\PageSep{248} + +There are then only three cases when the rise +of water was considerable, and in the greatest of +them it was only $3$~ft.\ $11$~in. + +If we deduct all the tides in which the range +between low and high water was equal to or less +than $19$~inches, we are left with $108$~predictions, +and in these cases the greatest error in time is +$23$~mins. In $86$~cases the error is equal to or less +than a quarter of an hour. This leaves $22$~cases +where the error was greater than $15$~mins.\ made +up as follows: $18$~cases with error greater than +$15$~mins.\ and less than $20$~mins.\ and $3$~cases with +errors of $20$~mins., $22$~mins., $23$~mins. Thus in +$106$ out of~$108$ predictions the error of time was +equal to or less than $20$~minutes. + +Two independent measurements of a tide +curve, for the determination of the time of high +water, lead to results which frequently differ by +five minutes, and sometimes by ten minutes. It +may therefore be claimed that these predictions +have a very high order of accuracy as regards +time. + +Turning now to the heights, out of $116$~predictions +the error in the predicted height was +equal to or less than $2$~inches in $91$~cases, it +amounted to $3$~inches in $14$~cases, and in the +remaining $11$~cases it was $4$~inches. It thus appears +that, as regards the height of the tide also, +the predictions are of great accuracy. This +short series of comparisons affords a not unduly +\PageSep{249} +favorable example of the remarkable success attainable, +where tidal observation and prediction +have been thoroughly carried out at a place +subject to only slight meteorological disturbance. + +If our theory of tides were incorrect, so that +we imagined that there was a partial tide wave +of a certain period, whereas in fact such a wave +has no true counterpart in physical causation, +the reduction of a year of tidal observation would +undoubtedly assign some definite small height, +and some definite retardation of the high water +after the passage of the corresponding, but +erroneous, satellite. But when a second series +of observations is reduced, the two tidal constants +would show no relationship to their previous +evaluations. If then reductions carried +out year after year assign, as they do, fairly +consistent values to the tidal constants, we may +feel confident that true physical causation is involved, +even when the heights of some of the +constituent tide waves do not exceed an inch +or two. + +Prediction must inevitably fail, unless we have +lighted on the true causes of the phenomena; +success is therefore a guarantee of the truth of +the theory. When we consider that the incessant +variability of the tidal forces, the complex +outlines of our coasts, the depth of the sea and +the earth's rotation are all involved, we should +\PageSep{250} +regard good tidal prediction as one of the +greatest triumphs of the theory of universal +gravitation. + +\begin{Authorities} +The Portsmouth comparisons were given to the author by the +Hydrographer of the Admiralty, Admiral Sir W.~J. Wharton. + +G.~H. Darwin, \Title{On Tidal Prediction}. ``Philosophical Transactions +of the Royal Society,'' A.~1891. +\end{Authorities} +\index{Prediction of tide!errors in|)}% +\PageSep{251} + + +\Chapter[Rigidity of the Earth]{XV} +{Chandler's Nutation---The Rigidity of the +Earth} + +\First{In} the present chapter I have to explain the +\index{Nutation!Chandler's|(}% +\index{Variation of latitude|(}% +origin of a tide of an entirely different character +from any of those considered hitherto. It may +fairly be described as a true tide, although it is +not due to the attraction of either the sun or +the moon. + +We have all spun a top, and have seen it, as +boys say, go to sleep. At first it nods a little, +but gradually it settles down to perfect steadiness. +Now the earth may be likened to a top, +and it also may either have a nutational or nodding +motion, or it may spin steadily; it is only +by observation that we can decide whether it is +nodding or sound asleep. + +The equator must now be defined as a plane +through the earth's centre at right angles to the +axis of rotation, and not as a plane fixed with +reference to the solid earth. The latitude of +\index{Latitude!periodic variations of|(}% +any place is the angle\footnote + {This angle is technically called the geocentric latitude; the + distinction between true and geocentric latitude is immaterial in + the present discussion.} +between the equator and +\PageSep{252} +a line drawn from the centre of the earth to the +place of observation. Now when the earth +nutates, the axis of rotation shifts, and its +extremity describes a small circle round the spot +which is usually described as the pole. The +equator, being perpendicular to the axis of rotation, +of course shifts also, and therefore the +latitude of a place fixed on the solid earth varies. +During the whole course of the nutation, the +earth's axis of rotation is always directed towards +the same point in the heavens, and therefore the +angle between the celestial pole and the vertical +or plumb-line at the place of observation must +oscillate about some mean value; the period of +the oscillation is that of the earth's nutation. +This movement is called a ``free'' nutation, +because it is independent of the action of external +forces. + +There are, besides, other nutations resulting +from the attractions of the moon and sun on the +protuberant matter at the equator, and from the +same cause there is a slow shift in space of the +earth's axis, called the precession. These movements +are said to be ``forced,'' because they are +due to external forces. The measurements of +the forced nutations and of the precession afford +the means of determining the period of the free +nutation, if it should exist. It has thus been +concluded that if there is any variation in the +latitude, it should be periodic in $305$~days; but +\PageSep{253} +only observation can decide whether there is +such a variation of latitude or not. + +Until recently astronomers were so convinced +of the sufficiency of this reasoning, that, when +they made systematic examination of the latitudes +of many observatories, they always searched +for an inequality with a period of $305$~days. +Some thought that they had detected it, but +when the observations extended over long periods, +it always seemed to vanish, as though what +they had observed were due to the inevitable +errors of observation. At length it occurred +to Mr.~Chandler to examine the observations +\index{Chandler, free nutation of earth, and variation of latitude|(}% +of latitude without any prepossession as to the +period of the inequality. By the treatment of +enormous masses of observation, he came to the +conclusion that there is really such an inequality, +but that the period is $427$~days instead of $305$~days. +He also found other inequalities in the +motion of the axis of rotation, of somewhat +obscure origin, and of which I have no occasion +to say more.\footnote + {They are perhaps due to the unequal melting of polar + ice and unequal rainfall in successive years. These irregular + variations in the latitude are such that some astronomers are + still skeptical as to the reality of Chandler's nutation, and think + that it will perhaps be found to lose its regularly rhythmical + character in the future.} + +The question then arises as to how the theory +can be so amended as to justify the extension of +the period of nutation. It was, I believe, Newcomb, +\PageSep{254} +\index{Newcomb, S., theoretical explanation of Chandler's nutation}% +of the United States Naval Observatory, +who first suggested that the explanation is to be +sought in the fact that the axis of rotation is an +axis of centrifugal repulsion, and that when it +shifts, the distribution of centrifugal force is +changed with reference to the solid earth, so +that the earth is put into a state of stress, to +which it must yield like any other elastic body. +The strain or yielding consequent on this stress +must be such as to produce a slight variability +in the position of the equatorial protuberance +with reference to places fixed on the earth. +Now the period of $305$~days was computed on +the hypothesis that the position of the equatorial +protuberance is absolutely invariable, but +periodic variations of the earth's figure would +operate so as to lengthen the period of the free +nutation, to an extent dependent on the average +elasticity of the whole earth. +\index{Elasticity of earth}% + +Mr.~Chandler's investigation demanded the +utmost patience and skill in marshaling large +masses of the most refined astronomical observations. +His conclusions are not only of the +greatest importance to astronomy, but they also +give an indication of the amount by which the +solid earth is capable of yielding to external +forces. It would seem that the average stiffness +of the whole earth must be such that it yields a +little less than if it were made of steel.\footnote + {Mr.~S. S. Hough, p.~338 of the paper referred to in the list +\index{Hough, S. S.!rigidity of earth}% + of authorities at the end of the chapter.} +But +\PageSep{255} +\index{Elasticity of earth}% +the amount by which the surface yields remains +unknown, because we are unable to say what +proportion of the aggregate change is superficial +and what is deep-seated. It is, however, certain +that the movements are excessively small, because +the circle described by the extremity of +the earth's axis of rotation, about the point on +the earth which we call the pole, has a radius of +only fifteen feet. + +It is easily intelligible that as the axis of +rotation shifts in the earth, the oceans will tend +to swash about, and that a sort of tide will be +generated. If the displacement of the axis were +considerable, whole continents would be drowned +by a gigantic wave, but the movement is so +small that the swaying of the ocean is very +feeble. Two investigators have endeavored to +detect an oceanic tide with a period of $427$~days; +they are Dr.~Bakhuyzen of Leyden and +\index{Bakhuyzen on tide due to variation of latitude}% +Mr.~Christie of the United States Coast Survey. +\index{Christie, A. S., tide due to variation of latitude}% +The former considered observations of sea-level +on the coasts of Holland, the latter those on the +coasts of the United States; and they both conclude +that the sea-level undergoes a minute +variability with a period of about $430$~days. A +similar investigation is now being prosecuted by +the Tidal Survey of India, and as the Indian +tidal observations are amongst the best in the +world, we may hope for the detection of this +minute tide in the Indian Ocean also. +\PageSep{256} +\index{Earth and moon!rigidity of|(}% +\index{Rigidity of earth|(}% + +The inequality in water level is so slight and +extends over so long a period that its measurement +cannot yet be accepted as certain. The +mean level of the sea is subject to slight irregular +variations, which are probably due to unequal +rainfall and unequal melting of polar ice in +successive years. But whatever be the origin of +these irregularities they exceed in magnitude the +one to be measured. The arithmetical processes, +employed to eliminate the ordinary tides and the +irregular variability, will always leave behind +some residual quantities, and therefore the examination +of a tidal record will always apparently +yield an inequality of any arbitrary period whatever. +It is only when several independent determinations +yield fairly consistent values of the +magnitude of the rise and fall and of the moment +of high water, that we can feel confidence +in the result. Now although the reductions of +Bakhuyzen and Christie are fairly consistent +\index{Bakhuyzen on tide due to variation of latitude}% +\index{Christie, A. S., tide due to variation of latitude}% +with one another, and with the time and height +suggested by Chandler's nutation, yet it is by no +means impossible that accident may have led to +this agreement. The whole calculation must +therefore be repeated for several places and at +several times, before confidence can be attained +in the detection of this latitudinal tide. + +\TB + +The prolongation of the period of Chandler's +nutation from $305$ to $427$~days seems to indicate +\index{Latitude!periodic variations of|)}% +\index{Nutation!Chandler's|)}% +\index{Variation of latitude|)}% +\PageSep{257} +that our planet yields to external forces, and we +naturally desire to learn more on so interesting +a subject. Up to fifty years ago it was generally +held that the earth was a globe of molten +matter covered by a thin crust. The ejection of +lava from volcanoes and the great increase of +temperature in mines seemed to present evidence +in favor of this belief. But the geologists and +physicists of that time seemed not to have perceived +that the inference might be false, if great +pressure is capable of imparting rigidity to matter +at a very high temperature, because the interior +of the earth might then be solid although +very hot. Now it has been proved experimentally +that rock expands in melting, and a physical +corollary from this is that when rock is under +great pressure a higher temperature is needed to +melt it than when the pressure is removed. The +pressure inside the earth much exceeds any that +can be produced in the laboratory, and it is uncertain +up to what degree of increase of pressure +the law of the rise of the temperature of +melting would hold good; but there can be no +doubt that, in so far as experiments in the laboratory +can be deemed applicable to the conditions +prevailing in the interior of the earth, they +tend to show that the matter there is not improbably +solid. + +But Lord Kelvin reinforces this argument +\index{Kelvin, Lord!rigidity of earth|(}% +from another point of view. Rock in the solid +\index{Chandler, free nutation of earth, and variation of latitude|)}% +\PageSep{258} +condition is undoubtedly heavier than when it is +molten. Now the solidified crust on the surface +of a molten planet must have been fractured +many times during the history of the planet, +and the fragments would sink through the liquid, +and thus build up a solid nucleus. It will +be observed that this argument does not repose +on the rise in the melting temperature of rock +through pressure, although it is undoubtedly +reinforced thereby. + +Hopkins was, I think, the first to adduce arguments +\index{Hopkins on rigidity of earth}% +of weight in favor of the earth's solidity. +He examined the laws of the precession and +nutation of a rigid shell inclosing liquid, and +found that the motion of such a system would +differ to a marked degree from that of the earth. +From this he concluded that the interior of the +earth was not liquid. + +Lord Kelvin has pointed out that although +Hopkins's investigation is by no means complete, +yet as he was the first to show that the +motion of the earth as a whole affords indications +of the condition of the interior, an important +share in the discovery of the solidity of the +earth should be assigned to him. Lord Kelvin +then resumed Hopkins's work, and showed that +if the liquid interior of the planet were inclosed +in an unyielding crust, a very slight departure +from perfect sphericity in the shell would render +the motion of the system almost identical with +\PageSep{259} +\index{Height of tide!reduced by elastic yielding of earth}% +that of a globe solid from centre to surface, +although this would not be the case with the +more rapid nutations. A yet more important +deficiency in Hopkins's investigation is that he +\index{Hopkins on rigidity of earth}% +did not consider that, unless the crust were more +rigid than the stiffest steel, it would yield to the +surging of the imprisoned liquid as freely as +india-rubber; and, besides, that if the crust +yielded freely, the precession and nutations of +the whole mass would hardly be distinguishable +from those of a solid globe. Hopkins's argument, +as thus amended by Lord Kelvin, leads +to one of two alternatives: either the globe is +solid throughout, or else the crust yields with +nearly the same freedom to external forces as +though it were liquid. + +We have now to show that the latter hypothesis +is negatived by other considerations. The +oceanic tides, as we perceive them, consist in a +motion of the water relatively to the land. Now +if the solid earth were to yield to the tidal forces +with the same freedom as the superjacent sea, +the cause for the relative movement of the sea +would disappear. And if the solid yielded to +some extent, the apparent oceanic tide would be +proportionately diminished. The very existence +of tides in the sea, therefore, proves at least that +the land does not yield with perfect freedom. + +Lord Kelvin has shown that the oceanic tides, +on a globe of the same rigidity as that of glass, +\PageSep{260} +would only have an apparent range of two fifths +of those on a perfectly rigid globe; whilst, if +the rigidity was equal to that of steel, the fraction +of diminution would be two thirds. I have +myself extended his argument to the hypothesis +that the earth may be composed of a viscous +material, which yields slowly under the application +of continuous forces, and also to the hypothesis +of a material which shares the properties +of viscosity and rigidity, and have been led to +\index{Rigidity of earth|)}% +analogous conclusions. + +The difficulty of the problem of oceanic tides +is so great that we cannot say how high the tides +would be if the earth were absolutely rigid, but +Lord Kelvin is of opinion that they certainly +\index{Kelvin, Lord!rigidity of earth|)}% +would not be twice as great as they are, and he +concludes that the earth possesses a greater average +stiffness than that of glass, although perhaps +not greater than that of steel. It is proper to +add that the validity of this argument depends +principally on the observed height of an inequality +of sea level with a period of a fortnight. This +is one of the partial tides of the third kind, which +I described in \Ref{Chapter}{XI}.\ as practically unimportant, +and did not discuss in detail. The value +of this inequality in the present argument is due +to the fact that it is possible to form a much +closer estimate of its magnitude on a rigid earth +\index{Earth and moon!rigidity of|)}% +than in the case of the semidiurnal and diurnal +tides. +\PageSep{261} +\index{Darwin, G. H.!rigidity of earth}% +\index{Earthquakes!shock perceptible at great distance}% + +It may ultimately be possible to derive further +indications concerning the physical condition of +the inside of the earth from the science of seismology. +The tremor of an earthquake has frequently +been observed instrumentally at an enormous +distance from its origin; as, for example, +when the shock of a Japanese earthquake is +perceived in England. + +The vibrations which are transmitted through +the earth are of two kinds. The first sort of wave +is one in which the matter through which it passes +is alternately compressed and dilated; it may be +described as a wave of compression. In the +second sort the shape of each minute portion of +the solid is distorted, but the volume remains +unchanged, and it may be called a wave of distortion. +These two vibrations travel at different +speeds, and the compressional wave outpaces +the distortional one. Now the first sign of a +distant earthquake is that the instrumental record +shows a succession of minute tremors. +These are supposed to be due to waves of compression, +and they are succeeded by a much +more strongly marked disturbance, which, however, +lasts only a short time. This second phase +in the instrumental record is supposed to be due +to the wave of distortion. + +If the natures of these two disturbances are +correctly ascribed to their respective sources, it +is certain that the matter through which the vibration +\PageSep{262} +\index{Darwin, G. H.!rigidity of earth}% +has passed was solid. For, although a +compressional wave might be transmitted without +much loss of intensity, from a solid to a +liquid and back again to a solid, as would have +to be the case if the interior of the earth is molten, +yet this cannot be true of the distortional +wave. It has been supposed that vibrations due +to earthquakes pass in a straight line through +the earth; if then this could be proved, we +should know with certainty that the earth is +solid, at least far down towards its centre. + +Although there are still some---principally +amongst the geologists---who believe in the existence +of liquid matter immediately under the +solid crust of the earth,\footnote + {See the Rev.~Osmond Fisher's \Title{Physics of the Earth's Crust}.} +\index{Fisher, Osmond, on molten interior of earth}% +yet the arguments which +I have sketched appear to most men of science +conclusive against such belief. + +\begin{Authorities} +Mr.~S.~C. Chandler's investigations are published in the ``Astronomical +Journal,'' vol.~11 and following volumes. A summary +is contained in ``Science,'' May~3, 1895. + +R.~S. Woodward, \Title{Mechanical Interpretation of the Variations of +\index{Woodward on variation of latitude}% +Latitude}, ``Ast.\ Journ.'' vol.~15, May,~1895. + +Simon Newcomb, \Title{On the Dynamics of the Earth's Rotation}, +``Monthly Notices of the R.~Astron.\ Soc.,'' vol.~52 (1892), +p.~336. + +S.~S. Hough, \Title{The Rotation of an Elastic Spheroid}, ``Philosoph.\ +\index{Hough, S. S.!Chandler's nutation}% +Trans.\ of the Royal Society,'' A.~1896, p.~319. He indicates a +slight oversight on the part of Newcomb. + +H.~G. van~de Sande Bakhuyzen, \Title{Ueber die Aenderung der Polhoehe}, +``Astron.\ Nachrichten,'' No.~3261. +\PageSep{263} + +A.~S. Christie, \Title{The Latitude-variation Tide}, ``Phil.\ Soc.\ of +Washington, Bulletin,'' vol.~12 (1895), p.~103. + +Lord Kelvin, in Thomson and Tait's ``Natural Philosophy,'' +on the Rigidity of the Earth; and ``Popular Lectures,'' vol.~3. + +G.~H. Darwin, \Title{Bodily Tides of Viscous and Semi-elastic Spheroids,~\&c.}, +``Philosoph.\ Trans.\ of the Royal Society,'' Part.~I. +1879. +\end{Authorities} +\PageSep{264} + + +%[** TN: Footnote mark handled by \Chapter logic] +\Chapter{XVI} +{Tidal Friction} +\footnotetext{A considerable portion of this and of the succeeding chapter + appeared as an article in \Title{The Atlantic Monthly} for April,~1898.} + +\First{The} fact that the earth, the moon, and the +\index{Friction of tides|(}% +planets are all nearly spherical proves that in +early times they were molten and plastic, and +assumed their present round shape under the +influence of gravitation. When the material of +which any planet is formed was semi-liquid +through heat, its satellites, or at any rate the +sun, must have produced tidal oscillations in the +molten rock, just as the sun and moon now produce +the tides in our oceans. + +Molten rock and molten iron are rather sticky +or viscous substances, and any movement which +agitates them must be subject to much friction. +Even water, which is a very good lubricant, is +not entirely free from friction, and so our present +oceanic tides must be influenced by fluid +friction, although to a far less extent than the +molten solid just referred to. Now, all moving +systems which are subject to friction gradually +come to rest. A train will run a long way when +the steam is turned off, but it stops at last, and +\PageSep{265} +a fly-wheel will continue to spin for only a limited +time. This general law renders it certain that +the friction of the tide, whether it consists in the +swaying of molten lava or of an ocean, must be +retarding the rotation of the planet, or at any +rate retarding the motion of the system in some +way. + +It is the friction upon its bearings which brings +a fly-wheel to rest; but as the earth has no bearings, +it is not easy to see how the friction of the +tidal wave, whether corporeal or oceanic, can +tend to stop its rate of rotation. The result +must clearly be brought about, in some way, by +the interaction between the moon and the earth. +Action and reaction must be equal and opposite, +and if we are correct in supposing that the friction +of the tides is retarding the earth's rotation, +there must be a reaction upon the moon which +must tend to hurry her onwards. To give a +homely illustration of the effects of reaction, I +may recall to mind how a man riding a high +bicycle, on applying the brake too suddenly, was +thrown over the handles. The desired action +was to stop the front wheel, but this could not +be done without the reaction on the rider, which +sometimes led to unpleasant consequences. + +The general conclusion as to the action and +reaction due to tidal friction is of so vague a +character that it is desirable to consider in detail +how they operate. +\PageSep{266} + +The circle in \fig{36} is supposed to represent +the undisturbed shape of the planet, which rotates +in the direction of the curved arrow. A portion +of the orbit of the satellite is indicated by part +\Figure{36}{Frictionally retarded Tide}{png} +of a circle, and the direction of its motion is +shown by an arrow. I will first suppose that the +water lying on the planet, or the molten rock of +which it is formed, is a perfect lubricant devoid +of friction, and that at the moment represented +in the figure the satellite is at~$M'$. The fluid will +then be distorted by the tidal force until it assumes +the egg-like shape marked by the ellipse, +projecting on both sides beyond the circle. It +will, however, be well to observe that if this figure +represents an ocean, it must be a very deep +one, far deeper than those which actually exist +on the earth; for we have seen that it is only in +deep oceans that the high water stands underneath +and opposite to the moon; whereas in +shallow water it is low water where we should +\PageSep{267} +naturally expect high water. Accepting the hypothesis +that the high tide is opposite to the +moon, and supposing that the liquid is devoid of +friction, the long axis of the egg is always directed +straight towards the satellite~$M'$, and the +liquid maintains a continuous rhythmical movement, +so that as the planet rotates and the satellite +revolves, it always maintains the same shape +and attitude towards the satellite. + +But when, as in reality, the liquid is subject to +friction, it gets belated in its rhythmical rise and +fall, and the protuberance is carried onward by +the rotation of the planet beyond its proper +place. In order to make the same figure serve +for this condition, I set the satellite backward to~$M$; +for this amounts to just the same thing, and +is less confusing than redrawing the protuberance +in its more advanced position. The planet +then constantly maintains this shape and attitude +with regard to the satellite, and the interaction +between the two will be the same as though the +planet were solid, but continually altering its +shape. + +We have now to examine what effects must +follow from the attraction of the satellite on an +egg-shaped planet, when the two constantly +maintain the same attitude relatively to each +other. It will make the matter somewhat easier +of comprehension if we replace the tidal protuberances +by two particles of equal masses, one at~$P$, +\PageSep{268} +\index{Earth and moon!rotation retarded by tidal friction}% +\index{Retardation of earth's rotation}% +and the other at~$P'$. If the masses of these +particles be properly chosen, so as to represent +the amount of matter in the protuberances, the +proposed change will make no material difference +in the action. + +The gravitational attraction of the satellite is +greater on bodies which are near than on those +which are far, and accordingly it attracts the +particle~$P$ more strongly than the particle~$P'$. It +is obvious from the figure that the attraction on~$P$ +must tend to stop the planet's rotation, whilst +\index{Rotation!retarded by tidal friction}% +that on~$P'$ must tend to accelerate it. If a man +pushes equally on the two pedals of a bicycle, +the crank has no tendency to turn, and besides +there are dead points in the revolution where +pushing and pulling have no effect. So also in +the astronomical problem, if the two attractions +were exactly equal, or if the protuberances were +at a dead point, there would be no resultant effect +on the rotation of the planet. But it is +obvious that here the retarding pull is stronger +than the accelerating one, and that the set of the +protuberances is such that we have passed the +dead point. It follows from this that the primary +effect of fluid friction is to throw the tidal +protuberance forward, and the secondary effect +is to retard the planet's rotation. + +It has been already remarked that this figure is +drawn so as to apply only to the case of corporeal +tides or to those of a very deep ocean. If +\PageSep{269} +\index{Moon and earth!retardation of motion by tidal friction}% +the ocean were shallow and frictionless, it would +be low water under and opposite to the satellite. +If then the effect of friction were still to throw +the protuberances forward, the rotation of the +planet would be accelerated instead of retarded. +But in fact the effect of fluid friction in a shallow +ocean is to throw the protuberances backward, +and a similar figure, drawn to illustrate such a +displacement of the tide, would at once make it +clear that here also tidal friction will lead to the +retardation of the planet's rotation. Henceforth +then I shall confine myself to the case illustrated +by~\fig{36}. + +Action and reaction are equal and opposite, +and if the satellite pulls at the protuberances, +they pull in return on the satellite. The figure +shows that the attraction of the protuberance~$P$ +tends in some measure to hurry the satellite onward +in its orbit, whilst that of~$P'$ tends to retard +it. But the attraction of~$P$ is stronger than that +of~$P'$, and therefore the resultant of the two is a +force tending to carry the satellite forward in the +direction of the arrow. + +If a stone be whirled at the end of an elastic +string, a retarding force, such as the friction of +the air, will cause the string to shorten, and an +accelerating force will make it lengthen. In the +same way the satellite, being as it were tied to +the planet by the attraction of gravitation, when +subjected to an onward force, recedes from the +\PageSep{270} +\index{Moon and earth!retardation of motion by tidal friction}% +planet, and moves in a spiral curve at ever increasing +distances. The time occupied by the +satellite in making a circuit round the planet is +prolonged, and this lengthening of the periodic +time is not merely due to the lengthening of the +arc described by it, but also to an actual retardation +of its velocity. It appears paradoxical that +the effect of an accelerating force should be a +retardation, but a consideration of the mode in +which the force operates will remove the paradox. +The effect of the tangential accelerating +force on the satellite is to make it describe an +increasing spiral curve. Now if the reader will +draw an exaggerated figure to illustrate part of +such a spiral orbit, he will perceive that the central +force, acting directly towards the planet, +must operate in some measure to retard the velocity +of the satellite. The central force is very +great compared with the tangential force due to +the tidal friction, and therefore a very small +fraction of the central force may be greater than +the tangential force. Although in a very slowly +increasing spiral the fraction of the central force +productive of retardation is very small, yet it is +found to be greater than the tangential accelerating +force, and thus the resultant effect is a +retardation of the satellite's velocity. + +The converse case where a retarding force results +in increase of velocity will perhaps be more +intelligible, as being more familiar. A meteorite, +\PageSep{271} +rushing through the earth's atmosphere, moves +faster and faster, because it gains more speed +from the attraction of gravity than it loses by the +friction of the air. + +Now let us apply these ideas to the case of the +earth and the moon. A man standing on the +planet, as it rotates, is carried past places where +the fluid is deeper and shallower alternately; at +the deep places he says that it is high tide, and +at the shallow places that it is low tide. In \fig{36} +it is high tide when the observer is carried +past~$P$. Now it was pointed out that when there +is no fluid friction we must put the moon at~$M'$, +but when there is friction she must be at~$M$. +Accordingly, if there is no friction it is high tide +when the moon is over the observer's head, but +when there is friction the moon has passed his +zenith before he reaches high tide. Hence he +would remark that fluid friction retards the time +of high tide. + +A day is the name for the time in which the +earth rotates once, and a month for the time in +which the moon revolves once. Then since tidal +friction retards the earth's rotation and the +moon's revolution, we may state that both the +day and the month are being lengthened, and +that these results follow from the retardation of +the time of high tide. + +It must also be noted that the spiral in which +the moon moves is an increasing one, so that her +\PageSep{272} +\index{Assyrian records of eclipses}% +distance from the earth also increases. These +are absolutely certain and inevitable results of +the mechanical interaction of the two bodies. + +At the present time the rates of increase of +the day and month are excessively small, so that +it has not been found possible to determine them +with any approach to accuracy. It may be well +to notice in passing that if the rate of either increase +of element were determinable, that of the +other would be deducible by calculation. + +The extreme slowness of the changes within +historical times is established by the early records +in Greek and Assyrian history of eclipses of the +\index{Eclipses, ancient, and earth's rotation}% +\index{Greek!records of ancient eclipses}% +sun, which occurred on certain days and in certain +places. Notwithstanding the changes in the +calendar, it is possible to identify the day according +to our modern reckoning, and the identification +of the place presents no difficulty. +Astronomy affords the means of calculating the +exact time and place of the occurrence of an +eclipse even three thousand years ago, on the +supposition that the earth spun at the same rate +then as now, and that the complex laws governing +the moon's motion are unchanged. + +The particular eclipse referred to in history is +known, but any considerable change in the +earth's rotation and in the moon's position would +have shifted the position of visibility on the +earth from the situation to which modern computation +would assign it. Most astronomical +\PageSep{273} +observations would be worthless if the exact time +of the occurrence were uncertain, but in the +case of eclipses the place of observation affords +\index{Eclipses, ancient, and earth's rotation}% +just that element of precision which is otherwise +wanting. As, then, the situations of the ancient +eclipses agree fairly well with modern computations, +we are sure that there has been no great +change within the last three thousand years, +either in the earth's rotation or in the moon's +motion. There is, however, a small outstanding +discrepancy which indicates that there has been +some change. But the exact amount of change +involves elements of uncertainty, because our +knowledge of the laws of the moon's motion is +not yet quite accurate enough for the absolutely +perfect calculation of eclipses which occurred +many centuries ago. In this way, it is known +that within historical times the retardation of the +earth's rotation and the recession of the moon +have been at any rate very slow. + +It does not, however, follow from this that +the changes have always been equally slow; indeed, +it may be shown that the efficiency of tidal +friction increases with great rapidity as we bring +the tide-generating satellite nearer to the planet. + +It has been shown in \Ref{Chapter}{V}.\ that the intensity +of tide-generating force varies as the inverse +cube of the distance between the moon and +the earth, so that if the moon's distance were +reduced successively to $\frac{1}{2}$,~$\frac{1}{3}$,~$\frac{1}{4}$, of its original distance, +\PageSep{274} +the force and the tide generated by it +would be multiplied $8$,~$27$,~$64$ times. But the +efficiency of tidal friction increases far more rapidly +than this, because not only is the tide itself +augmented, but also the attraction of the moon. +In order to see how these two factors will coöperate, +let us begin by supposing that the +height of the tide remains unaffected by the approach +or retrogression of the moon. Then the +same line of argument, which led to the conclusion +that tide-generating force varies inversely as +the cube of the distance, shows that the action +of the moon on protuberances of definite magnitude +must also vary inversely as the cube of the +distance. But the height of the tide is not in +fact a fixed quantity, but varies inversely as the +cube of the distance, so that when account is +taken both of the augmentation of the tide and +of the increased attraction of the moon, it follows +that the tidal retardation of the earth's rotation +must vary as the inverse sixth power of +the distance. Now since the sixth power of~$2$ is~$64$, +the lunar tidal friction, with the moon at +half her present distance, would be $64$~times as +efficient as at present. Similarly, if her distance +were diminished to a third and a quarter of what +it is, the tidal friction would act with $729$ and +$4,096$~times its present strength. Thus, although +the action may be insensibly slow now, it must +have gone on with much greater rapidity when +the moon was nearer to us. +\PageSep{275} + +There are many problems in which it would +be very difficult to follow the changes according +to the times of their occurrence, but where it is +possible to banish time from consideration, and +to trace the changes themselves, in due order, +without reference to time. In the sphere of +common life, we know the succession of stations +which a train must pass between London and +Edinburgh, although we may have no time-table. +This is the case with our astronomical +problem; for although we have no time-table, +yet the sequence of the changes in the system +can be traced accurately. + +Let us then banish time, and look forward to +the ultimate outcome of the tidal interaction of +the moon and earth. The day and the month +\index{Day, change in length of, under tidal friction}% +\index{Month, change in, under tidal friction|(}% +are lengthening at relative rates which are calculable, +although the absolute rates in time are +unknown. It will suffice for a general comprehension +of the problem to know that the present +rate of increase of the day is much more rapid +than that of the month, and that this will hold +good in the future. Thus, the number of rotations +of the earth in the interval comprised in +one revolution of the moon diminishes; or, in +other words, the number of days in the month +diminishes, although the month itself is longer +than at present. For example, when the day +shall be equal in length to two of our actual +days, the month may be as long as thirty-seven +\PageSep{276} +of our days, and then the earth will spin round +only about eighteen times in the month. + +This gradual change in the day and month +\index{Day, change in length of, under tidal friction}% +proceeds continuously until the duration of a +rotation of the earth is prolonged to fifty-five of +our present days. At the same time the month, +or the time of revolution of the moon round the +earth, will also occupy fifty-five of our days. +Since the month here means the period of the +return of the moon to the same place among the +stars, and since the day is to be estimated in +the same way, the moon must then always face +the same part of the earth's surface, and the +two bodies must move as though they were +united by a bar. The outcome of the lunar +tidal friction will therefore be that the moon +and the earth go round as though locked together, +in a period of fifty-five of our present +days, with the day and the month identical in +length. + +Now looking backward in time, we find the +day and the month shortening, but the day +changing more rapidly than the month. The +earth was therefore able to complete more revolutions +in the month, although that month was +itself shorter than it is now. We get back in +fact to a time when there were $29$~rotations of +the earth in a month instead of~$27\frac{1}{3}$, as at present. +This epoch is a sort of crisis in the history +of the moon and the earth, for it may be proved +\PageSep{277} +that there never could have been more than $29$~days +in the month. Earlier than this epoch, the +days were fewer than~$29$, and later fewer also. +Although measured in years, this epoch in the +earth's history must be very remote, yet when we +contemplate the whole series of changes it must +be considered as a comparatively recent event. +In a sense, indeed, we may be said to have passed +recently through the middle stage of our history. + +Now, pursuing the series of changes further +back than the epoch when there was the maximum +number of days in the month, we find the +earth still rotating faster and faster, and the +moon drawing nearer and nearer to the earth, +and revolving in shorter and shorter periods. +But a change has now supervened, so that the +rate at which the month is shortening is more +rapid than the rate of change in the day. Consequently, +the moon now gains, as it were, on +the earth, which cannot get round so frequently +in the month as it did before. In other words, +the number of days in the month declines from +the maximum of~$29$, and is finally reduced to +one. When there is only one day in the month, +\index{Month, change in, under tidal friction|)}% +the earth and the moon go round at the same +rate, so that the moon always looks at the same +side of the earth, and so far as concerns the +motion they might be fastened together by a +rigid bar. + +This is the same conclusion at which we arrived +\PageSep{278} +with respect to the remote future. But +the two cases differ widely; for whereas in the +future the period of the common rotation will +be $55$~of our present days, in the past we find +the two bodies going round each other in between +three and five of our present hours. A +satellite revolving round the earth in so short a +period must almost touch the earth's surface. +The system is therefore traced until the moon +nearly touches the earth, and the two go round +each other like a single solid body in about three +to five hours. + +The series of changes has been traced forward +and backward from the present time, but it will +make the whole process more intelligible, and +the opportunity will be afforded for certain further +considerations, if I sketch the history again +\index{History!of earth and moon|(}% +in the form of a continuous narrative. + +Let us imagine a planet attended by a satellite +which revolves so as nearly to touch its surface, +and continuously to face the same side of the +planet's surface. If now, for some reason, the +satellite's month comes to differ very slightly +from the planet's day, the satellite will no longer +continuously face the same side of the planet, +but will pass over every part of the planet's +equator in turn. This is the condition necessary +for the generation of tidal oscillations in the +planet, and as the molten lava, of which we +suppose it to be formed, is a sticky or viscous +\PageSep{279} +fluid, the tidal oscillations must be subject to +friction. Tidal friction will then begin to do its +work, but the result will be very different according +as the satellite revolves a little faster or +a little slower than the planet. If it revolves a +little faster, so that the month is shorter than +the day, we have a condition not contemplated +in~\fig{36}; it is easy to see, however, that as +the satellite is always leaving the planet behind +it, the apex of the trial protuberance must be +directed to a point behind the satellite in its +orbit. In this case the rotation of the planet +must be \DPtypo{acclerated}{accelerated} by the tidal friction, and the +satellite will be drawn inward towards the planet, +into which it must ultimately fall. In the application +of this theory to the earth and moon, it +is obvious that the very existence of the moon +negatives the hypothesis that the initial month +was even infinitesimally shorter than the day. +We must then suppose that the moon revolved +a little more slowly than the earth rotated. In +this case the tidal friction would retard the +earth's rotation, and force the moon to recede +from the earth, and so perform her orbit more +slowly. Accordingly, the primitive day and the +primitive month lengthen, but the month increases +much more rapidly than the day, so that +the number of days in a month increases. This +proceeds until that number reaches a maximum, +which in the case of our planet is about~$29$. +\PageSep{280} +\index{Instability!nature of dynamical, and initial of moon's motion|(}% +\index{Stability!nature of dynamical}% + +After the epoch of the maximum number of +days in the month, the rate of change in the +length of the day becomes less rapid than that +in the length of the month; and although both +periods increase, the number of days in the +month begins to diminish. The series of +changes then proceeds until the two periods +come again to an identity, when we have the +earth and the moon as they were at the beginning, +revolving in the same period, with the +moon always facing the same side of the earth. +But in her final condition the moon will be a +long way off the earth instead of being quite +close to it. + +Although the initial and final states resemble +each other, yet they differ in one respect which +is of much importance, for in the initial condition +the motion is unstable, whilst finally it is +stable. The meaning of this is, that if the +moon were even infinitesimally disturbed from +the initial mode of motion, she would necessarily +either fall into the planet, or recede therefrom, +and it would be impossible for her to continue +to move in that neighborhood. She is unstable +in the same sense in which an egg when balanced +on its point is unstable; the smallest mote +of dust will upset it, and practically it cannot +stay in that position. But the final condition +resembles the case of the egg lying on its side, +which only rocks a little when we disturb it. +\PageSep{281} +\index{Stability!nature of dynamical}% +So if the moon were slightly disturbed from her +final condition, she would continue to describe +very nearly the same path round the earth, and +would not assume some entirely new form of +orbit. + +It is by methods of rigorous argument that +the moon is traced back to the initial unstable +condition when she revolved close to the earth. +But the argument here breaks down, and calculation +is incompetent to tell us what occurred +before, and how she attained that unstable mode +of motion. If we were to find a pendulum +swinging in a room, where we knew that it had +been undisturbed for a long time, we might, by +observing its velocity and allowing for the resistance +of the air, conclude that at some previous +moment it had just been upside down, but +calculation could never tell us how it had +reached that position. We should of course +feel confident that some one had started it. +Now a similar hiatus must occur in the history +of the moon, but it is not so easy to supply the +missing episode. It is indeed only possible to +speculate as to the preceding history. + +But there is some basis for our speculation; +for I say that if a planet, such as the earth, +made each rotation in three hours, it would very +nearly fly to pieces. The attraction of gravity +would be barely strong enough to hold it together, +just as the cohesive strength of iron is +\PageSep{282} +\index{Forced oscillation!due to solar tide, possibly related to birth of moon|(}% +\index{Moon and earth!origin of}% +insufficient to hold a fly-wheel together if it is +spun too fast. There is, of course, an important +distinction between the case of the ruptured +fly-wheel and the supposed break-up of the +earth; for when a fly-wheel breaks, the pieces +are hurled apart as soon as the force of cohesion +fails, whereas when a planet breaks up through +too rapid rotation, gravity must continue to +hold the pieces together after they have ceased +to form parts of a single body. + +Hence we have grounds for conjecturing that +the moon is composed of fragments of the primitive +planet which we now call the earth, which +detached themselves when the planet spun very +swiftly, and afterwards became consolidated. It +surpasses the power of mathematical calculation +to trace the details of the process of this rupture +and subsequent consolidation, but we can hardly +doubt that the system would pass through a +period of turbulence, before order was reëstablished +in the formation of a satellite. + +I have said above that rapid rotation was probably +the cause of the birth of the moon, but it +may perhaps not have been brought about by +this cause alone. There are certain considerations +which make it difficult to ascertain the +initial common period of revolution of the moon +and the earth with accuracy; it may lie between +three and five hours. Now I think that such +a speed might not quite suffice to cause the +\index{Instability!nature of dynamical, and initial of moon's motion|)}% +\PageSep{283} +\index{Moon and earth!origin of}% +primitive planet to break up. In \Ref{Chapter}{XVIII}.\ +we shall consider in greater detail the conditions +under which a rotating mass of liquid would +rupture, but for the present it may suffice to say +that, where the rotating body is heterogeneous in +density, like the earth, the exact determination +of the limiting speed of rotation is not possible. +Is there, then, any other cause which might coöperate +with rapid rotation in producing rupture? +I think there is such a cause, and, although +we are here dealing with guesswork, I +will hazard the suggestion. + +The primitive planet, before the birth of the +moon, was rotating rapidly with reference to the +sun, and it must therefore have been agitated by +solar tides. In \Ref{Chapter}{IX}.\ it was pointed out +that there is a general dynamical law which enables +us to foresee the magnitude of the oscillations +of a system under the action of external +forces. That law depended on the natural or +free period of the oscillation of the system when +disturbed and left to itself, free from the intervention +of external forces. We saw that the +more nearly the periodic forces were timed to +agree with the free period, the greater was the +amplitude of the oscillations of the system. Now +it is easy to calculate the natural or free period +of the oscillation of a homogeneous liquid globe +of the same density as the earth, namely, five +and a half times as heavy as water; the period +\PageSep{284} +\index{Sun!possible influence of, in assisting birth of moon}% +is found to be $1$~hour $34$~minutes. The heterogeneity +of the earth introduces a complication of +which we cannot take account, but it seems likely +that the period would be from $1\frac{1}{2}$ to $2$~hours. +The period of the solar semidiurnal tide is half a +\index{Solar!possible effect of tide in assisting birth of moon}% +day, and if the day were from $3$ to $4$ of our present +hours the forced period of the tide would +be in close agreement with the free period of +oscillation. + +May we not then conjecture that as the rotation +of the primitive earth was gradually reduced +by solar tidal friction, the period of the solar tide +was brought into closer and closer agreement +with the free period, and that consequently the +solar tide increased more and more in height? +In this case the oscillation might at length become +so violent that, in coöperation with the +rapid rotation, it shook the planet to pieces, and +that huge fragments were detached which ultimately +became our moon. + +There is nothing to tell us whether this theory +affords the true explanation of the birth of the +moon, and I say that it is only a wild speculation, +incapable of verification. + +But the truth or falsity of this speculation +does not militate against the acceptance of the +general theory of tidal friction, which, standing +on the firm basis of mechanical necessity, throws +much light on the history of the earth and the +moon, and correlates the lengths of our present +day and month. +\index{Forced oscillation!due to solar tide, possibly related to birth of moon|)}% +\PageSep{285} +\index{Sun!possible influence of, in assisting birth of moon}% + +I have said above that the sequence of events +has been stated without reference to the scale of +time. It is, however, of the utmost importance +\index{Time!requisite for evolution of moon}% +to gain some idea of the time requisite for all the +changes in the system. If millions of millions +of years were necessary, the theory would have +to be rejected, because it is known from other +lines of argument that there is not an unlimited +bank of time on which to draw. The uncertainty +as to the duration of the solar system is +\index{Solar!possible effect of tide in assisting birth of moon}% +wide, yet we are sure that it has not existed for +an almost infinite past. + +Now, although the actual time scale is indeterminate, +it is possible to find the minimum time +adequate for the transformation of the moon's +orbit from its supposed initial condition to its +present shape. It may be proved, in fact, that +if tidal friction always operated under the conditions +most favorable for producing rapid change, +the sequence of events from the beginning until +to-day would have occupied a period of between +$50$ and $60$~millions of years. The actual period, +of course, must have been much greater. Various +lines of argument as to the age of the solar +system have led to results which differ widely +among themselves, yet I cannot think that the +applicability of the theory is negatived by the +magnitude of the period demanded. It may be +that science will have to reject the theory in its +full extent, but it seems unlikely that the ultimate +\PageSep{286} +\index{Moon and earth!rotation annulled by tidal friction and present libration}% +verdict will be adverse to the preponderating +influence of the tide in the evolution of our +planet. + +\TB + +If this history be true of the earth and moon, +\index{History!of earth and moon|)}% +it should throw light on many peculiarities of the +solar system. In the first place, a corresponding +series of changes must have taken place in the +moon herself. Once on a time the moon must +have been molten, and the great extinct volcanoes +revealed by the telescope are evidences of +her primitive heat. The molten mass must have +been semi-fluid, and the earth must have raised +in it enormous tides of molten lava. Doubtless +the moon once rotated rapidly on her axis, and +the frictional resistance to her tides must have +impeded her rotation. This cause must have +\index{Rotation!of moon annulled by tidal friction}% +added to the moon's recession from the earth, +but as the moon's mass is only an eightieth part +of that of the earth, the effect on the moon's +orbit must have been small. The only point to +which we need now pay attention is that the +rate of her rotation was reduced. She rotated +then more and more slowly until the tide solidified, +and thenceforward and to the present day +she has shown the same face to the earth. Kant +\index{Kant!rotation of moon}% +and Laplace in the last century, and Helmholtz +\index{Helmholtz!on rotation of the moon}% +\index{Laplace!on rotation of moon}% +in recent times, have adduced this as the explanation +of the fact that the moon always shows +us the same face. Our theory, then, receives a +\PageSep{287} +striking confirmation from the moon; for, having +ceased to rotate relatively to us, she has actually +advanced to that condition which may be +foreseen as the fate of the earth. + +The earth tide in the moon is now solidified +so that the moon's equator is not quite circular, +and the longer axis is directed towards the earth. +Laplace has considered the action of the earth +\index{Laplace!on rotation of moon}% +on this solidified tide, and has shown that the +moon must rock a little as she moves round the +earth. In consequence of this rocking motion or +libration of the moon, and also of the fact that +her orbit is elliptic, we are able to see just a little +more than half of the moon's surface. + +\TB + +Thus far I have referred in only one passage +to the influence of solar tides, but these are of +considerable importance, being large enough to +cause the conspicuous phenomena of spring and +neap tides. Now, whilst the moon is retarding +the earth's rotation, the sun is doing so also. +But these solar tides react only on the earth's +motion round the sun, leaving the moon's motion +round the earth unaffected. It might perhaps +be expected that parallel changes in the +earth's orbit would have proceeded step by step, +and that the earth might be traced to an origin +close to the sun. The earth's mass is less than $\frac{1}{300,000}$~part +of the sun's, and the reactive effect on the +earth's orbit round the sun is altogether negligible. +\PageSep{288} +It is improbable, in fact, that the year is, +from this cause at any rate, longer by more than +a few seconds than it was at the very birth of +the solar system. + +Although the solar tides cannot have had any +perceptible influence upon the earth's movement +in its orbit, they will have affected the rotation +of the earth to a considerable extent. Let us +imagine ourselves transported to the indefinite +future, when the moon's orbital period and the +earth's diurnal period shall both be prolonged to +$55$~of our present days. The lunar tide in the +earth will then be unchanging, just as the earth +tide in the moon is now fixed; but the earth will +be rotating with reference to the sun, and, if +there are still oceans on the earth, her rotation +will be subject to retardation in consequence of +the solar tidal friction. The day will then become +longer than the month, whilst the moon +will at first continue to revolve round the earth +in $55$~days. Lunar tides will now be again generated, +but as the motion of the earth will be +very slow relatively to the moon, the oscillations +will also be very slow, and subject to little friction. +But that friction will act in opposition to +the solar tides, and the earth's rotation will to +some slight extent be assisted by the moon. +The moon herself will slowly approach the earth, +moving with a shorter period, and must ultimately +fall back into the earth. We know that +\PageSep{289} +there are neither oceans nor atmosphere on the +moon, but if there were such, the moon would +have been subject to solar tidal friction, and +would now be rotating slower than she revolves. + +%[** TN: Not hyperlinking chapter reference] +\begin{Authorities} +See the end of Chapter~XVII. +\end{Authorities} +\PageSep{290} + + +\Chapter[Tidal Friction]{XVII} +{Tidal Friction (Continued)} + +\First{It} has been shown in the last chapter that the +prolongation of the day and of the month under +the influence of tidal friction takes place in such +a manner that the month will ultimately become +longer than the day. Until recent times no case +had been observed in the solar system in which +a satellite revolved more rapidly than its planet +rotated, and this might have been plausibly adduced +as a reason for rejecting the actual efficiency +of solar tidal friction in the process of +celestial evolution. At length however, in~1877, +Professor Asaph Hall discovered in the system +\index{Hall, Asaph, discovery of Martian satellites|(}% +of the planet Mars a case of the kind of motion +\index{Mars!discovery of satellites|(}% +which we foresee as the future fate of the moon +and earth, for he found that the planet was attended +by two satellites, the nearer of which has +\index{Satellites!discovery of those of Mars|(}% +a month shorter than the planet's day. He gives +an interesting account of what had been conjectured, +partly in jest and partly in earnest, as to +the existence of satellites attending that planet. +This foreshadowing of future discoveries is so +curious that I quote the following passage from +Professor Hall's paper. He writes:--- +\PageSep{291} + +``Since the discovery of the satellites of Mars, +the remarkable statements of Dean Swift and +Voltaire concerning the satellites of this planet, +and the arguments of Dr.~Thomas Dick and +others for the existence of such bodies, have attracted +so much attention, that a brief account +of the writings on this subject may be interesting. + +``The following letter of Kepler was written +\index{Kepler!argument respecting Martian satellites}% +to one of his friends soon after the discovery by +Galileo in~1610 of the four satellites of Jupiter, +\index{Galileo!discovery of Jupiter's satellites}% +and when doubts had been expressed as to the +reality of this discovery. The news of the discovery +was communicated to him by his friend +Wachenfels; and Kepler says:--- + +``\,`Such a fit of wonder seized me at a report +which seemed to be so very absurd, and I was +thrown into such agitation at seeing an old dispute +between us decided in this way, that between +his joy, my coloring, and the laughter of +both, confounded as we were by such a novelty, +we were hardly capable, he of speaking, or I of +listening. On our parting, I immediately began +to think how there could be any addition to the +number of the planets without overturning my +``Cosmographic Mystery,'' according to which +Euclid's five regular solids do not allow more +than six planets round the sun\dots. I am so +far from disbelieving the existence of the four +circumjovial planets, that I long for a telescope, +to anticipate you, if possible, in discovering \emph{two} +\PageSep{292} +\index{Kepler!argument respecting Martian satellites}% +round Mars, as the proportion seems to require, +\emph{six} or \emph{eight} round Saturn, and perhaps \emph{one} each +round Mercury and Venus.' + +``Dean Swift's statement concerning the satellites +\index{Swift, satire on mathematicians|(}% +of Mars is in his famous satire, `The +Travels of Mr.~Lemuel Gulliver.' After describing +\index{Gulliver@\Title{Gulliver's Travels}, satire on mathematics|(}% +his arrival in Laputa, and the devotion +of the Laputians to mathematics and music, +Gulliver says:--- + +``\,`The knowledge I had in mathematics gave +me great assistance in acquiring their phraseology, +which depended much upon that science, +and music; and in the latter I was not unskilled. +Their ideas were perpetually conversant in lines +and figures. If they would, for example, praise +the beauty of a woman, or of any other animal, +they describe it by rhombs, circles, parallelograms, +ellipses, and other geometrical terms, or +by words of art drawn from music, needless here +to repeat\dots. And although they are dexterous +enough upon a piece of paper, in the management +of the rule, the pencil, and the divider, +yet in the common actions and the behavior of +life, I have not seen a more clumsy, awkward, +and unhandy people, nor so slow and perplexed +in their conceptions upon all subjects, except +those of mathematics and music. They are very +bad reasoners, and vehemently given to opposition, +unless when they happen to be of the right +opinion, which is seldom their case\dots. These +\PageSep{293} +people are under continual disquietudes, never +enjoying a minute's peace of mind; and their +disturbances proceed from causes which very +little affect the rest of mortals. Their apprehensions +arise from several changes they dread +in the celestial bodies. For instance, that the +earth, by the continual approaches of the sun +towards it, must, in the course of time, be absorbed, +or swallowed up. That the face of the +sun will, by degrees, be encrusted with its own +effluvia, and give no more light to the world. +That the earth very narrowly escaped a brush +from the tail of the last comet, which would +have infallibly reduced it to ashes; and that the +next, which they have calculated for one-and-thirty +years hence, will probably destroy us. +For if, in its perihelion, it should approach +within a certain degree of the sun (as by their +calculations they have reason to dread,) it will +receive a degree of heat ten thousand times +more intense than that of red-hot glowing iron; +and, in its absence from the sun, carry a blazing +tail ten hundred thousand and fourteen miles +long; through which, if the earth should pass +at the distance of one hundred thousand miles +from the nucleus, or main body of the comet, it +must, in its passage, be set on fire, and reduced +to ashes. That the sun, daily spending its rays, +without any nutriment to supply them, will at +last be wholly consumed and annihilated; which +\PageSep{294} +must be attended with the destruction of this +earth, and of all the planets that receive their +light from it. + +``\,`They are so perpetually alarmed with the +apprehension of these, and the like impending +dangers, that they can neither sleep quietly in +their beds, nor have any relish for the common +pleasures and amusements of life. When they +meet an acquaintance in the morning, the first +question is about the sun's health, how he looked +at his setting and rising, and what hopes they had +to avoid the stroke of the approaching comet\dots. +They spend the greatest part of their lives +in observing the celestial bodies, which they do +by the assistance of glasses, far excelling ours in +goodness. For although their largest telescopes +do not exceed three feet, they magnify much +more than those of a hundred with us, and show +the stars with greater clearness. This advantage +has enabled them to extend their discoveries +much further than our astronomers in Europe; +for they have made a catalogue of ten thousand +fixed stars, whereas the largest of ours do not +contain above one-third of that number\dots. +They have likewise discovered two lesser stars, +or satellites, which revolve about Mars; whereof +the innermost is distant from the centre of the +primary planet exactly three of his diameters, +and the outermost, five; the former revolves in +the space of ten hours, and the latter in twenty-one +\PageSep{295} +and a half; so that the squares of their +periodical times are very near in the same proportion +with the cubes of their distance from +the centre of Mars; which evidently shows them +to be governed by the same law of gravitation +that influences the other heavenly bodies.' + +``The reference which Voltaire makes to the +\index{Voltaire, satire on mathematicians, and Martian satellites}% +moons of Mars is in his `Micromegas, Histoire +Philosophique.' Micromegas was an inhabitant +of Sirius, who, having written a book which a +suspicious old man thought smelt of heresy, left +Sirius and visited our solar system. Voltaire +says:--- + +``\,`Mais revenons à nos voyageurs. En sortant +de Jupiter, ils traversèrent un espace d'environ +cent millions de lieues, et ils côtoyèrent +la planète de Mars, qui, comme on sait, est cinq +fois plus petite que noire petit globe; ils virent +deux lunes qui servent à cette planète, et qui ont +échappé aux regards de nos astronomes. Je sais +bien que le père \emph{Castel} écrira, et même plaisamment, +\index{Castel, Father, ridiculed by Voltaire}% +contre l'existence de ces deux lunes; mais +je m'en rapporte à ceux qui raisonnent par analogie. +Ces bons philosophes-là savent combien il +serait difficile que Mars, qui est si loin du soleil, +se passât à moins de deux lunes.' + +``The argument by analogy for the existence +of a satellite of Mars was revived by writers like +Dr.~Thomas Dick, Dr.~Lardner, and others. In +\index{Dick, argument as to Martian satellites}% +\index{Lardner, possibility of Martian satellites}% +addition to what may be called the analogies of +\index{Gulliver@\Title{Gulliver's Travels}, satire on mathematics|)}% +\index{Swift, satire on mathematicians|)}% +\PageSep{296} +astronomy, these writers appear to rest on the +idea that a beneficent Creator would not place +a planet so far from the sun as Mars without +giving it a satellite. This kind of argument has +passed into some of our handbooks of astronomy, +and is stated as follows by Mr.~Chambers +\index{Chambers on possible existence of Martian satellites}% +in his excellent book on `Descriptive Astronomy,' +2d~edition, p.~89, published in~1867:--- + +``\,`As far as we know, Mars possesses no satellite, +though analogy does not forbid, but rather, +on the contrary, infers the existence of one; and +its never having been seen, in this case at least, +proves nothing. The second satellite of Jupiter +is only $\frac{1}{43}$~of the diameter of the primary, and +a satellite $\frac{1}{43}$~of the diameter of Mars would +be less than $100$~miles in diameter, and therefore +of a size barely within the reach of our largest +telescopes, allowing nothing for its possibly close +proximity to the planet. The fact that one of +the satellites of Saturn was only discovered a +few years ago renders the discovery of a satellite +of Mars by no means so great an improbability +as might be imagined.' + +``Swift seems to have had a hearty contempt +for mathematicians and astronomers, which he +has expressed in his description of the inhabitants +of Laputa. Voltaire shared this contempt, +\index{Voltaire, satire on mathematicians, and Martian satellites}% +and delighted in making fun of the philosophers +whom Frederick the Great collected at Berlin. +The `père Castel' may have been le~père Louis +\index{Castel, Father, ridiculed by Voltaire}% +\PageSep{297} +Castel, who published books on physics and +mathematics at Paris in 1743 and~1758. The +probable origin of these speculations about the +moons of Mars was, I think, Kepler's analogies. +Astronomers failing to verify these, an opportunity +was afforded to satirists like Swift and +Voltaire to ridicule such arguments.''\footnote + {\Title{Observations and Orbits of the Satellites of Mars}, by Asaph + Hall. Washington, Government Printing Office, 1878.} + +As I have already said, these prognostications +were at length verified by Professor Asaph Hall +in the discovery of two satellites, which he named +Phobos and Deimos---Fear and Panic, the dogs +\index{Deimos, a satellite of Mars}% +\index{Phobos, a satellite of Mars}% +of war. The period of Deimos is about $30$~hours, +and that of Phobos somewhat less than $8$~hours, +whilst the Martian day is of nearly the same +length as our own. The month of the inner +minute satellite is thus less than a third of the +planet's day; it rises to the Martians in the west, +and passes through all its phases in a few hours; +sometimes it must even rise twice in a single +Martian night. As we here find an illustration +of the condition foreseen for the earth and moon, +it seems legitimate to suppose that solar tidal +friction has retarded the planet's rotation until it +has become slower than the revolution of one of +the satellites. It would seem as if the ultimate +fate of Phobos will be absorption in the planet. + +Several of the satellites of Jupiter and of Saturn +present faint inequalities of coloring, and +\PageSep{298} +\index{Jupiter!satellites constantly face planet}% +\index{Saturn!satellites always face the planet}% +telescopic examination has led astronomers to believe +that they always present the same face to +their planets. The theory of tidal friction would +\index{Planets!rotation of some, annulled by tidal friction}% +certainly lead us to expect that these enormous +planets should work out the same result for their +relatively small satellites that the earth has produced +\index{Satellites!discovery of those of Mars|)}% +\index{Satellites!rotation of those of Jupiter and Saturn annulled}% +in the moon. + +The proximity of the planets Mercury and +\index{Mercury, rotation of}% +Venus to the sun should obviously render solar +\index{Venus, rotation of}% +tidal friction far more effective than with us. +The determination of the periods of rotation of +\index{Rotation!of Mercury, Venus, and satellites of Jupiter and Saturn annulled by tidal friction}% +these planets thus becomes a matter of much interest. +But the markings on their disks are so +obscure that the rates of their rotations have remained +under discussion for many years. Until +recently the prevailing opinion was that in both +cases the day was of nearly the same length as +ours; but a few years ago Schiaparelli of Milan, +\index{Schiaparelli on rotation of Venus and Mercury}% +an observer endowed with extraordinary acuteness +of vision, announced as the result of his observations +that both Mercury and Venus rotate +only once in their respective years, and that +each of them constantly presents the same face +to the sun. These conclusions have recently been +confirmed by Mr.~Percival Lowell from observations +\index{Lowell, P., on rotations of Venus and Mercury}% +made in Arizona. Although on reading +the papers of these astronomers it is not easy +to see how they can be mistaken, yet it should +be noted that others have failed to detect the +markings on the planet's disks, although they +\index{Hall, Asaph, discovery of Martian satellites|)}% +\index{Mars!discovery of satellites|)}% +\PageSep{299} +\index{Lowell, P., on rotations of Venus and Mercury}% +apparently enjoyed equal advantages for observation.\footnote + {Dr.~See, a member of the staff of the Flagstaff Observatory, + Arizona, tells me that he has occasionally looked at these planets + through the telescope, although he took no part in the systematic + observation. In his opinion it would be impossible for any one + at Flagstaff to doubt the reality of the markings. There are, + however, many astronomers of eminence who suspend their + judgment, and await confirmation by other observers at other + stations.} + +If, as I am disposed to do, we accept these observations +as sound, we find that evidence favorable +to the theory of tidal friction is furnished +by the planets Mercury and Venus, and by the +\index{Mercury, rotation of}% +\index{Venus, rotation of}% +satellites of the earth, Jupiter and Saturn, whilst +\index{Earth and moon!figure of}% +\index{Earth and moon!adjustment of figure to suit change of rotation|(}% +the Martian system is yet more striking as an +instance of an advanced stage in evolution. + +\TB + +It is well known that the figure of the earth +is flattened by the diurnal rotation, so that the +polar axis is shorter than any equatorial diameter. +At the present time the excess of the equatorial +radius over the polar radius is $\frac{1}{290}$~part of +either of them. Now in tracing the history of +the earth and moon, we found that the earth's +rotation had been retarded, so that the day is +now longer than it was. If then the solid earth +has always been absolutely unyielding, and if an +ocean formerly covered the planet to a uniform +depth, the sea must have gradually retreated +towards the poles, leaving the dry land exposed +at the equator. If on the other hand the solid +\PageSep{300} +\index{Geological evidence of earth's plasticity}% +\index{Plasticity of earth under change of rotation|(}% +earth had formerly its present shape, there must +then have been polar continents and a deep equatorial +sea. + +But any considerable change in the speed of +the earth's rotation would, through the action of +gravity, bring enormous forces to bear on the +solid earth. These forces are such as would, if +they acted on a plastic material, tend to restore +the planet's figure to the form appropriate to its +changed rotation. It has been shown experimentally +by M.~Tresca and others that even very +\index{Tresca on flow of solids}% +rigid and elastic substances lose their rigidity +and their elasticity, and become plastic under the +action of sufficiently great forces. It appears to +me, therefore, legitimate to hold to the belief in +the temporary rigidity of the earth's mass, as explained +in \Ref{Chapter}{XV}., whilst contending that +under a change of rotational velocity the earth +may have become plastic, and so have maintained +a figure adapted to its speed. Geological observation +shows that rocks have been freely twisted +and bent near the earth's surface, and it is impossible +to doubt that under altered rotation the +deeper portions of the earth would have been +subjected to very great stress. I conjecture that +the internal layers might adapt themselves by +continuous flow, whilst the superficial portion +might yield impulsively. Earthquakes are probably +due to unequal shrinkage of the planet in +cooling, and each shock would tend to bring the +\PageSep{301} +strata into their position of rest; thus the earth's +surface would avail itself of the opportunity afforded +by earthquakes of acquiring its proper +shape. The deposit in the sea of sediment, derived +from the denudation of continents, affords +another means of adjustment of the figure of the +planet. I believe then that the earth has always +maintained a shape nearly appropriate to its rotation. +The existence of the continents proves +that the adjustment has not been perfect, and we +shall see reason to believe that there has been +also a similar absence of complete adjustment in +the interior. + +But the opinion here maintained is not shared +by the most eminent of living authorities, Lord +Kelvin; for he holds that the fact that the average +\index{Kelvin, Lord!denies adjustment of earth's figure to changed rotation}% +figure of the earth corresponds with the +actual length of the day proves that the planet +was consolidated at a time when the rotation was +but little more rapid than it is now. The difference +between us is, however, only one of degree, +for he considers that the power of adjustment is +slight, whilst I hold that it would be sufficient +to bring about a considerable change of shape +within the period comprised in geological history. + +If the adjustment of the planet's figure were +perfect, the continents would sink below the +ocean, which would then be of uniform depth. +But there is no superficial sign, other than the +dry land, of absence of adaptation to the present +\PageSep{302} +\index{Moon and earth!inequality in motion indicates internal density of earth}% +rotation---unless indeed the deep polar sea discovered +by Nansen be such. Yet, as I have +hinted above, some tokens still exist in the earth +\index{Earth and moon!internal density}% +of the shorter day of the past. The detection of +this evidence depends however on arguments of +so technical a character that I cannot hope in +such a work as this to do more than indicate the +nature of the proof. + +The earth is denser towards the centre than +outside, and the layers of equal density are concentric. +\index{Density!of earth, law of internal}% +If then the materials were perfectly +plastic throughout, not only the surface, but +also each of these layers would be flattened to a +definite extent, which depends on the rate of rotation +and on the law governing the internal +density of the earth. Although the rate at +which the earth gets denser is unknown, yet it is +possible to assign limits to the density at various +depths. Thus it can be proved that at any internal +point the density must lie between two +values which depend on the position of the point +in question. So also, the degree of flattening at +any internal point is found to lie between two +extreme limits, provided that all the internal layers +are arranged as they would be if the whole +mass were plastic. + +Now variations in the law of internal density +and in the internal flattening would betray themselves +to our observation in several ways. In +the first place, gravity on the earth's surface +\index{Earth and moon!adjustment of figure to suit change of rotation|)}% +\index{Gravity, variation according to latitude}% +\index{Plasticity of earth under change of rotation|)}% +\PageSep{303} +\index{Meteorological!conditions dependent on earth's rotation}% +\index{Moon and earth!inequality in motion indicates internal density of earth}% +\index{Nutation!value of, indicates internal density of earth}% +would be changed. The force of gravity at the +\index{Gravity, variation according to latitude}% +poles is greater than at the equator, and the law +of its variation according to latitude is known. +In the second place the amount of the flattening +of the earth's surface would be altered, and the +present figure of the earth is known with considerable +exactness. Thirdly the figure and law of +density of the earth govern a certain irregularity +or inequality in the moon's motion, which has +been carefully evaluated by astronomers. Lastly +the precessional and nutational motion of the +earth is determined by the same causes, and these +motions also are accurately known. These four +facts of observation---gravity, the ellipticity of +\index{Ellipticity of earth's strata in excess for present rotation}% +the earth, the lunar inequality, and the precessional +and nutational motion of the earth---are +so intimately intertwined that one of them cannot +be touched without affecting the others. + +Now Édouard Roche, a French mathematician, +\index{Roche, E.!ellipticity of internal strata of earth}% +has shown that if the earth is perfectly plastic, +so that each layer is exactly of the proper shape +for the existing rotation, it is not possible to adjust +the unknown law of internal density so as +\index{Precession, value of, indicates internal density of earth}% +to make the values of all these elements accord +with observation. If the density be assumed +such as to fit one of the data, it will produce a +disagreement with observation in others. If, +however, the hypothesis be abandoned that the +internal strata all have the proper shapes, and if +it be granted that they are a little more flattened +\PageSep{304} +\index{Ellipticity of earth's strata in excess for present rotation}% +\index{Geological evidence of earth's plasticity!as to retardation of earth's rotation|(}% +than is due to the present rate of rotation, the +data are harmonized together; and this is just +what would be expected according to the theory +of tidal friction. But it would not be right to +attach great weight to this argument, for the +absence of harmony is so minute that it might +be plausibly explained by errors in the numerical +data of observation. I notice, however, that the +most competent judges of this intricate subject +are disposed to regard the discrepancy as a +reality. + +\DPchg{}{\TB} + +We have seen in the preceding chapter that +the length of day has changed but little within +historical times. But the period comprised in +written history is almost as nothing compared +with the whole geological history of the earth. +We ought then to consider whether geology furnishes +any evidence bearing on the theory of +tidal friction. The meteorological conditions on +the earth are dependent to a considerable extent +on the diurnal rotation of the planet, and therefore +those conditions must have differed in the +past. Our storms are of the nature of aerial eddies, +and they derive their rotation from that of +the earth. Accordingly storms were probably +more intense when the earth spun more rapidly. +The trunks of trees should be stronger than they +are now to withstand more violent storms. But +I cannot learn that there is any direct geological +evidence on this head, for deciduous trees with +\PageSep{305} +\index{Ripple mark in sand preserved in geological strata}% +stiff trunks seem to have been a modern product +of geological time, whilst the earlier trees more +nearly resembled bamboos, which yield to the +wind instead of standing up to it. It seems possible +that trees and plants would not be exterminated, +even if they suffered far more wreckage +than they do now. If trees with stiff trunks +could only withstand the struggle for existence +when storms became moderate in intensity, their +absence from earlier geological formations would +be directly due to the greater rapidity of the +earth's rotation in those times. + +According to our theory the tides on the seacoast +must certainly have had a much wider +range, and river floods must probably have been +more severe. The question then arises whether +these agencies should have produced sedimentary +deposits of coarser grain than at present. Although +I am no geologist, I venture to express a +doubt whether it is possible to tell, within very +wide limits, the speed of the current or the range +of the tide that has brought down and distributed +any sedimentary deposit. I doubt whether any +geologist would assert that floods might not have +been twice or thrice as frequent, or that the tide +might not have had a very much greater range +than at present. + +In some geological strata ripple-marks have +been preserved which exactly resemble modern +ones. This has, I believe, been adduced as an +\PageSep{306} +argument against the existence of tides of great +range. Ripples are, however, never produced +by a violent scour of water, but only by gentle +currents or by moderate waves. The turn of +the tide must be gentle to whatever height it +rises, and so the formation of ripple-mark should +have no relationship to the range of tide. + +It appears then that whilst geology affords no +direct confirmation of the theory, yet it does not +present any evidence inconsistent with it. Increased +activity in the factors of change is important +to geologists, since it renders intelligible +a diminution in the time occupied by the history +of the earth; and thus brings the views of the +\index{Earth and moon!probably once molten}% +geologist and of the physicist into better harmony. + +Although in this discussion I have maintained +the possibility that a considerable portion of the +changes due to tidal friction may have occurred +within geological history, yet it seems to me +probable that the greater part must be referred +back to pre-geological times, when the planet +was partially or entirely molten. + +\TB + +The action of the moon and sun on a plastic +and viscous planet would have an effect of which +some remains may perhaps still be traceable. +The relative positions of the moon and of the +frictionally retarded tide were illustrated in the +last chapter by~\fig{36}. That figure shows that +\index{Geological evidence of earth's plasticity!as to retardation of earth's rotation|)}% +\PageSep{307} +the earth's rotation is retarded by forces acting +\index{Earth and moon!distortion under primeval tidal friction}% +on the tidal protuberances in a direction adverse +to the planet's rotation. As the plastic substance, +of which we now suppose the planet to +be formed, rises and falls rhythmically with the +tide, the protuberant portions are continually +subject to this retarding force. Meanwhile the +internal portions are urged onward by the +inertia due to their velocity. Accordingly there +must be a slow motion of the more superficial +portions with reference to the interior. From +the same causes, under present conditions, the +whole ocean must have a slow westerly drift, although +it has not been detected by observation. + +Returning however to our plastic planet, the +equatorial portion is subjected to greater force +than the polar regions, and if meridians were +painted on its surface, as on a map, they would +gradually become distorted. In the equatorial +belt the original meridional lines would still run +north and south, but in the northern hemisphere +they would trend towards the northeast, and in +the southern hemisphere towards the southeast. +This distortion of the surface would cause the +surface to wrinkle, and the wrinkles should be +warped in the directions just ascribed to the +meridional lines. If the material yielded very +easily I imagine that the wrinkles would be +small, but if it were so stiff as only to yield with +difficulty they might be large. +\PageSep{308} + +There can be no doubt as to the correctness +\index{History!of earth and moon|(}% +of this conclusion as to a stiff yet viscous planet, +but the application of these ideas to the earth is +hazardous and highly speculative. We do, however, +observe that the continents, in fact, run +\index{Continents, trend of, possibly due to primeval tidal friction}% +roughly north and south. It may appear fanciful +to note, also, that the northeastern coast of +America, the northern coast of China, and the +southern extremity of South America have the +proper theoretical trends. But the northwestern +coast of America follows a line directly adverse +to the theory, and the other features of the globe +are by no means sufficiently regular to inspire +much confidence in the justice of the conjecture.\footnote + {See, also, W. Prinz, \Title{Torsion apparente des planètes}, ``Annuaire + de l'Obs.~R. de~Bruxelles,'' 1891.} + +\TB + +We must now revert to the astronomical aspects +of our problem. It is natural to inquire +whether the theory of tidal friction is competent +to explain any peculiarities of the motion of the +moon and earth other than those already considered. +It has been supposed thus far that the +moon moves over the earth's equator in a circular +orbit, and that the equator coincides with the +plane in which the earth moves in its orbit. But +the moon actually moves in a plane different +from that in which the earth revolves round the +sun, her orbit is not circular but elliptic, and the +\PageSep{309} +earth's equator is oblique to the orbit. We must +consider, then, how tidal friction will affect these +three factors. + +Let us begin by considering the obliquity of +the equator to the ecliptic, which produces the +seasonal changes of winter and summer. The +problem involved in the disturbance of the motion +of a rotating body by any external force is +too complex for treatment by general reasoning, +and I shall not attempt to explain in detail the +interaction of the moon and earth in this respect. + +The attractions of the moon and sun on the +equatorial protuberance of the earth causes the +earth's axis to move slowly and continuously +with reference to the fixed stars. At present, +the axis points to the pole-star, but $13,000$~years +hence the present pole-star will be $47°$~distant +from the pole, and in another $13,000$~years it +will again be the pole-star. Throughout this +precessional movement the obliquity of the equator +to the ecliptic remains constant, so that winter +and summer remain as at present. There is +also, superposed on the precession, the nutational +or nodding motion of the pole to which I referred +in \Ref{Chapter}{XV}. In the absence of tidal +friction the attractions of the moon and sun on +the tidal protuberance would slightly augment +the precession due to the solid equatorial protuberance, +and would add certain very minute +nutations of the earth's axis; the amount of +\PageSep{310} +these tidal effects, is, however, quite insignificant. +But under the influence of tidal friction, +the matter assumes a different aspect, for the +earth's axis will not return at the end of each +nutation to exactly the same position it would +have had in the absence of friction, and there is +a minute residual effect which always tends in +the same direction. A motion of the pole may +be insignificant when it is perfectly periodic, but +it becomes important in a very long period of +time when the path described is not absolutely +reëntrant. Now this is the case with regard to +the motion of the earth's axis under the influence +of frictionally retarded tides, for it is found +to be subject to a gradual drift in one direction. + +In tracing the history of the earth and moon +backwards in time we found the day and month +growing shorter, but at such relative speeds that +the number of days in the month diminished until +the day and month became equal. This conclusion +remains correct when the earth is oblique +to its orbit, but the effect on the obliquity is +\index{Ecliptic, obliquity of, due to tidal friction|(}% +\index{Obliquity of ecliptic, effects of tidal friction on|(}% +found to depend in a remarkable manner upon +the number of days in the month. At present +and for a long time in the past the obliquity +is increasing, so that it was smaller long ago. +But on going back to the time when the day +was six and the month twelve of our present +hours we find that the tendency for the obliquity +to increase vanishes. In other words, if +\PageSep{311} +there are more than two days in a month the +obliquity will increase, if less than two it will +diminish. + +Whatever may be the number of days in the +month, the rate of increase or diminution of +obliquity varies as the obliquity which exists at +the moment under consideration. If, then, a +planet be spinning about an axis absolutely perpendicular +to the plane of its satellite's orbit, the +obliquity remains invariable. But if we impart +infinitesimal obliquity to a planet whose day is +less than half a month, that infinitesimal obliquity +will increase; whilst, if the day is more +than half a month, the infinitesimal obliquity +will diminish. Accordingly, the motion of a +planet spinning upright is stable, if there are +less than two days in a month, and unstable if +there are more than two. + +It is not legitimate to ascribe the whole of +the present obliquity of~$23\frac{1}{2}°$ to the influence of +tidal friction, because it appears that when there +were only two days in the month, the obliquity +was still as much as~$11°$. It is, moreover, impossible +to explain the considerable obliquity of the +other planets to their orbits by this cause. It +must, therefore, be granted that there was some +unknown cause which started the planets in rotation +about axes oblique to their orbits. It remains, +however, certain that a planet, rotating primitively +without obliquity, would gradually become +\PageSep{312} +inclined to its orbit, although probably not to so +great an extent as we find in the case of the +earth. + +The next subject to be considered is the fact +that the moon's orbit is not circular but eccentric. +Here, again, it is found that if the tides +were not subject to friction, there would be no +sensible effect on the shape of the moon's path, +but tidal friction produces a reaction on the +moon tending to change the degree of eccentricity. +In this case, it is possible to indicate by +general reasoning the manner in which this reaction +operates. We have seen that tidal reaction +tends to increase the moon's distance from the +earth. Now, when the moon is nearest, in perigee, +the reaction is stronger than when she is +furthest, in apogee. The effect of the forces in +perigee is such that the moon's distance at the +next succeeding apogee is greater than it was at +the next preceding apogee; so, also, the effect +of the forces in apogee is an increase in the perigeal +distance. But the perigeal effect is stronger +than the apogeal, and, therefore, the apogeal distances +increase more rapidly than the perigeal +ones. It follows, therefore, that, whilst the orbit +as a whole expands, it becomes at the same time +more eccentric. +\index{Ecliptic, obliquity of, due to tidal friction|)}% +\index{Obliquity of ecliptic, effects of tidal friction on|)}% + +The lunar orbit is then becoming more eccentric, +and numerical calculation shows that in +very early times it must have been nearly circular. +\PageSep{313} +\index{Eccentricity of orbit!due to tidal friction}% +\index{Moon and earth!eccentricity of orbit increased by tidal friction}% +\index{Saint@St.\ Vénant on flow of solids}% +\index{See, T. J. J.!eccentricity of orbits of double stars}% +But mathematical analysis indicates that in +this case, as with the obliquity, the rate of +increase depends in a remarkable manner upon +the number of days in the month. I find in +fact that if eighteen days are less than eleven +months the eccentricity will increase, but in the +converse case it will diminish; in other words +the critical stage at which the eccentricity is +stationary is when $1\frac{7}{11}$~days is equal to the +month. It follows from this that the circular +orbit of the satellite is dynamically stable or +\index{Orbit!of double stars, very eccentric}% +unstable according as $1\frac{7}{11}$~days is less or greater +than the month. + +The effect of tidal friction on the eccentricity +has been made the basis of extensive astronomical +speculations by Dr.~See. I shall revert to +this subject in \Ref{Chapter}{XIX}., and will here +merely remark that systems of double stars are +\index{Stars!double, eccentricity of orbits}% +found to revolve about one another in orbits of +great eccentricity, and that Dr.~See supposes +that the eccentricity has arisen from the tidal +action of each star on the other. + +The last effect of tidal friction to which I +have to refer is that on the plane of the moon's +orbit. The lunar orbit is inclined to that of the +earth round the sun at an angle of~$5°$, and the +problem to be solved is as to the nature of the +effect of tidal friction on that inclination. The +nature of the relation of the moon's orbit to the +ecliptic is however so complex that it appears +\index{History!of earth and moon|)}% +\PageSep{314} +\index{Eccentricity of orbit!due to tidal friction}% +\index{Moon and earth!eccentricity of orbit increased by tidal friction}% +hopeless to explain the effects of tidal action +without the use of mathematical language, and +I must frankly give up the attempt. I may, +however, state that when the moon was near the +earth she must have moved nearly in the plane +of the earth's equator, but that the motion gradually +changed so that she has ultimately come to +move nearly in the plane of the ecliptic. These +two extreme cases are easily intelligible, but the +transition from one case to the other is very +complicated. It may suffice for this general +account of the subject to know that the effects +of tidal friction are quite consistent with the +present condition of the moon's motion, and +with the rest of the history which has been +traced. + +This discussion of the effects of tidal friction +may be summed up thus:--- + +If a planet consisted partly or wholly of molten +lava or of other fluid, and rotated rapidly about +an axis perpendicular to the plane of its orbit, +and if that planet was attended by a single satellite, +revolving with its month a little longer than +the planet's day, then a system would necessarily +be developed which would have a strong resemblance +to that of the earth and moon. + +A theory reposing on \textit{veræ causæ} which brings +into quantitative correlation the lengths of the +present day and month, the obliquity of the +ecliptic, the eccentricity and the inclination of +\PageSep{315} +the lunar orbit, should have strong claims to +acceptance. + +\begin{Authorities} +G.~H. Darwin. A series of papers in the ``Phil.\ Trans.\ Roy.\ +\index{Darwin, G. H.!papers on tidal friction}% +Soc.'' pt.~i.\ 1879, pt.~ii.\ 1879, pt.~ii.\ 1880, pt.~ii.\ 1881, pt.~i.\ 1882, +and abstracts (containing general reasoning) in the corresponding +Proceedings; also ``Proc.\ Roy.\ Soc.''\ vol.~29, 1879, p.~168 (in +part republished in Thomson and Tait's \Title{Natural Philosophy}), +and vol.~30, 1880, p.~255. + +Lord Kelvin, \Title{On Geological Time}, ``Popular Lectures and +\index{Kelvin, Lord!on geological time}% +Addresses,'' vol.~iii. Macmillan, 1894. + +Roche. The investigations of Roche and of others are given +in Tisserand's \Title{Mécanique Céleste}, vol.~ii. Gauthier-Villars, 1891. +\index{Tisserand, Roche's investigations as to earth's figure}% + +Tresca and St.~Vénant, \Title{Sur l'écoulement des Corps Solides}, +``Mémoires des Savants Étrangers,'' Académie des Sciences de +Paris, vols.\ 18~and~20. + +Schiaparelli, \Title{Considerazioni sul moto rotatorio del pianeta +\index{Schiaparelli on rotation of Venus and Mercury}% +Venere}. Five notes in the ``Rendiconti del R.~Istituto Lombardo,'' +vol.~23, and \Title{Sulla rotazione di Mercurio}, ``Ast.\ Nach.,'' +No.~2944. An abstract is given in ``Report of Council of R. +Ast.\ Soc.,'' Feb.~1891. + +Lowell, Mercury, ``Ast.\ Nach.,'' No.~3417. \Title{Mercury and Determination +\index{Lowell, P., on rotations of Venus and Mercury}% +of Rotation Period~\dots\ of Venus}, ``Monthly Notices +R. Ast.\ Soc.,'' vol.~57, 1897, p.~148. \Title{Further proof},~\&c., \textit{ibid}.\ +p.~402. + +Douglass, \Title{Jupiter's third Satellite}, ``Ast.\ Nach.,'' No.~3432. +\index{Douglass, rotation of Jupiter's satellites}% +\Title{Rotation des IV~Jupitersmondes}, ``Ast.\ Nach.,'' No.~3427, confirming +Engelmann, \Title{Ueber~\dots\ Jupiterstrabanten}, Leipzig, 1871. + +Barnard, \Title{The third and fourth Satellites of Jupiter}, ``Ast.\ +\index{Barnard, rotation of Jupiter's satellites}% +Nach.,'' No.~3453. +\end{Authorities} +\index{Friction of tides|)}% +\PageSep{316} + + +\Chapter[Figures of Equilibrium]{XVIII} +{The Figures of Equilibrium of a Rotating +Mass of Liquid} + +\First{The} theory of the tides involves the determination +\index{Equilibrium, figures of, of rotating liquid|(}% +\index{Figure of equilibrium!of rotating liquid|(}% +\index{Rotating liquid, figures of equilibrium|(}% +of the form assumed by the ocean under +the attraction of a distant body, and it now +remains to discuss the figure which a rotating +mass of liquid may assume when it is removed +from all external influences. The forces which +act upon the liquid are the mutual gravitation +of its particles, and the centrifugal force due to +its rotation. If the mass be of the appropriate +shape, these two opposing forces will balance +one another, and the shape will be permanent. +The problem in hand is, then, to determine +what shapes of this kind are possible. + +In 1842 a distinguished Belgian physicist, M.~Plateau,\footnote + {He is justly celebrated not only for his discoveries, but also + for his splendid perseverance in continuing his researches after + he had become totally blind.} +devised an experiment which affords +\index{Capillarity of liquids, and Plateau's experiment|(}% +\index{Plateau, experiment on figure of rotating globule|(}% +a beautiful illustration of the present subject. +The experiment needs very nice adjustment in +several respects, but I refer the reader to +Plateau's paper for an account of the necessary +\PageSep{317} +\index{Surface tension of liquids}% +precautions. Alcohol and water may be so +mixed as to have the same density as olive oil. +If the adjustment of density is sufficiently exact, +a mass of oil will float in the mixture, in the +form of a spherical globule, without any tendency +to rise or fall. The oil is thus virtually +relieved from the effect of gravity. A straight +wire, carrying a small circular disk at right +angles to itself, is then introduced from the top +of the vessel. When the disk reaches the +globule, the oil automatically congregates itself +round the disk in a spherical form, symmetrical +with the wire. + +The disk is then rotated slowly and uniformly, +and carries with it the oil, but leaves the surrounding +mixture at rest. The globule is then +seen to become flattened like an orange, and as +the rotation quickens it dimples at the centre, +and finally detaches itself from the disk in the +form of a perfect ring. This latter form is only +transient; for the oil usually closes in again +round the disk, or sometimes, with slightly different +manipulation, the ring may break into +drops which revolve round the centre, rotating +round their axes as they go. + +The force which holds a drop of water, or +this globule of oil, together is called ``surface +tension'' or ``capillarity.'' It is due to a certain +molecular attraction, quite distinct from +that of gravitation, and it produces the same +\PageSep{318} +effect as if the surface of the liquid were enclosed +in an elastic skin. There is of course no +actual skin, and yet when the liquid is stirred +the superficial particles attract their temporary +neighbors so as to restore the superficial elasticity, +continuously and immediately. The intensity +of surface tension depends on the nature +\index{Surface tension of liquids}% +of the material with which the liquid is in contact; +thus there is a definite degree of tension +in the skin of olive oil in contact with spirits +and water. + +A globule at rest necessarily assumes the form +of a sphere under the action of surface tension, +but when it rotates it is distorted by centrifugal +force. The polar regions become less curved, +and the equatorial region becomes more curved, +until the excess of the retaining power at the +equator over that at the poles is sufficient to +restrain the centrifugal force. Accordingly the +struggle between surface tension and centrifugal +force results in the assumption by the globule +of an orange-like shape, or, with greater speed +of rotation, of the other figures of equilibrium. +\index{Capillarity of liquids, and Plateau's experiment|)}% + +In very nearly the same way a large mass of +gravitating and rotating liquid will naturally +assume certain definite forms. The simplest +case of the kind is when the fluid is at rest in +space, without any rotation. Then mutual gravitation +is the only force which acts on the system. +The water will obviously crowd together +\PageSep{319} +into the smallest possible space, so that every +particle may get as near to the centre as its +neighbors will let it. I suppose the water to be +incompressible, so that the central portion, although +pressed by that which lies outside of it, +does not become more dense; and so the water +does not weigh more per cubic foot near the +centre than towards the outside. Since there +is no upwards and downwards, or right and +left about the system, it must be symmetrical in +every direction; and the only figure which possesses +this quality of universal symmetry is the +sphere. A sphere is then said to be a figure of +equilibrium of a mass of fluid at rest. + +If such a sphere of water were to be slightly +deformed, and then released, it would oscillate +to and fro, but would always maintain a nearly +spherical shape. The speed of the oscillation +depends on the nature of the deformation impressed +upon it. If the water were flattened to +the shape of an orange and released, it would +spring back towards the spherical form, but +would overshoot the mark, and pass on to a +lemon shape, as much elongated as the orange +was flattened. It would then return to the +orange shape, and so on backwards and forwards, +passing through the spherical form at +each oscillation. This is the simplest kind of +oscillation which the system can undergo, but +there is an infinite number of other modes of +\index{Plateau, experiment on figure of rotating globule|)}% +\PageSep{320} +any degree of complexity. The mathematician +can easily prove that a liquid globe, of the same +density as the earth, would take an hour and a +half to pass from the orange shape to the lemon +shape, and back to the orange shape. At present, +the exact period of the oscillation is not +the important point, but it is to be noted that if +the body be set oscillating in any way whatever, +it will continue to oscillate and will always remain +nearly spherical. We say then that the +sphere is a stable form of equilibrium of a mass +of fluid. The distinction between stability and +instability has been already illustrated in \Ref{Chapter}{XVI}.\ +by the cases of an egg lying on its +side and balanced on its end, and there is a +similar distinction between stable and unstable +modes of motion. + +Let us now suppose the mass of water to rotate +slowly, all in one piece as if it were solid. +We may by analogy with the earth describe the +axis of rotation as polar, and the central plane, +at right angles to the axis, as equatorial. The +equatorial region tends to move outwards in consequence +of the centrifugal force of the rotation, +and this tendency is resisted by gravitation which +tends to draw the water together towards the +centre. As the rotation is supposed to be very +slow, centrifugal force is weak, and its effects are +small thus the globe is very slightly flattened at +the poles, like an orange or like the earth itself. +\PageSep{321} +Such a body resembles the sphere in its behavior +when disturbed; it will oscillate, and its average +figure in the course of its swing is the orange +shape. It is therefore stable. + +But it has been discovered that the liquid may +also assume two other alternative forms. One +of these is extremely flattened and resembles a +flat cheese with rounded edges. As the disk of +liquid is very wide, the centrifugal force at the +equator is very great, although the rotation is +very slow. In the case of the orange-shaped figure, +the slower the rotation the less is the equatorial +centrifugal force, because it diminishes +both with diminution of radius and fall of speed. +But in the cheese shape the equatorial centrifugal +force gains more by the increase of equatorial +radius than it loses by diminution of rotation. +Therefore the slower the rotation the broader the +disk, and, if the rotation were infinitely slow, the +liquid would be an infinitely thin, flat, circular +disk. + +The cheese-like form differs in an important +respect from the orange-like form. If it were +slightly disturbed, it would break up, probably +into a number of detached pieces. The nature +of the break-up would depend on the disturbance +from which it started, but it is impossible to trace +the details of the rupture in any case. We say +then that the cheese shape is an unstable figure +of equilibrium of a rotating mass of liquid. +\PageSep{322} +\index{Stability!of figures of equilibrium}% + +The third form is strikingly different from +either of the preceding ones. We must now imagine +the liquid to be shaped like a long cigar, +and to be rotating about a central axis perpendicular +to its length. Here again the ends of +the cigar are so distant from the axis of rotation +that the centrifugal force is great, and with infinitely +slow rotation the figure becomes infinitely +long and thin. Now this form resembles the +cheese in being unstable. It is remarkable that +these three forms are independent of the scale on +which they are constructed, for they are perfectly +similar whether they contain a few pounds of +water or millions of tons.\footnote + {It is supposed that they are more than a fraction of an inch + across, otherwise surface tension would be called into play.} +If the period of rotation +and the density of the liquid are given, +the shapes are absolutely determinable. + +The first of the three figures resembles the +earth and may be called the planetary figure, and +\index{Planetary figure of equilibrium of rotating liquid}% +I may continue to refer to the other two as the +cheese shape and the cigar shape. The planetary +and cheese shape are sometimes called the spheroids +of Maclaurin, after their discoverer, and +\index{Maclaurin!figure of equilibrium of rotating liquid|(}% +the cigar shape is generally named after Jacobi, +\index{Jacobi, figure of equilibrium of rotating liquid|(}% +the great German mathematician. For slow rotations +the planetary form is stable, and the +cheese and cigar are unstable. There are probably +other possible forms of equilibrium, such as +a ring, or several rings, or two detached masses +\PageSep{323} +revolving about one another like a planet and +satellite, but for the present I only consider these +three forms. + +Now imagine three equal masses of liquid, infinitely +distant from one another, and each rotating +\Figure[0.85]{37}{}{png} +at the same slow speed, and let one of them +have the planetary shape, the second the cheese +shape, and the third the cigar shape. When the +rotations are simultaneously and equally augmented, +we find the planetary form becoming +flatter, the cheese form shrinking in diameter +and thickening, and the cigar form shortening +and becoming fatter. There is as yet no change +in the stability, the first remaining stable and +\index{Stability!of figures of equilibrium}% +the second and third unstable. The three figures +are illustrated in~\fig{37}, but the cigar shape +is hardly recognizable by that name, since it has +already become quite short and its girth is +considerable. +\PageSep{324} + +Now it has been proved that as the cigar shape +shortens, its tendency to break up becomes less +marked, or in other words its degree of instability +diminishes. At a certain stage, not as yet +exactly determined, but which probably occurs +when the cigar is about twice as long as broad, +the instability disappears and the cigar form just +becomes stable. I shall have to return to the +consideration of this phase later. The condition +of the three figures is now as follows: The planetary +form of Maclaurin has become much flattened, +but is still stable; the cigar form of Jacobi +has become short and thick, and is just stable; +and the cheese form of Maclaurin is still unstable, +\index{Maclaurin!figure of equilibrium of rotating liquid|)}% +but its diameter has shrunk so much that the +figure might be better described as a very flat +orange. + +On further augmenting the rotation the form +of Jacobi still shrinks in length and increases in +girth, until its length becomes equal to its +greater breadth. Throughout the transformation +the axis of rotation has always remained the +shortest of the three, so that when the length +becomes equal to the shorter equatorial diameter, +the shape is not spherical, but resembles that of +a much flattened orange. In fact, at this stage +Jacobi's figure of equilibrium has degenerated to +\index{Jacobi, figure of equilibrium of rotating liquid|)}% +identity with the planetary shape. One of the +upper ovals in \fig{38} represents the section of +the form in which the planetary figure and the +\PageSep{325} +cigar figure coalesce, the former by continuous +flattening, the latter by continuous shortening. +The other upper figure represents the form to +which the cheese-like figure of Maclaurin has +\Figure[0.85]{38}{}{png} +\index{Poincaré!figure of rotating liquid}% +been reduced; it will be observed that it presents +some resemblance to the coalescent form. + +When the rotation is further augmented, there +is no longer the possibility of an elongated Jacobian +figure, and there remain only the two +spheroids of Maclaurin. But an important change +has now supervened, for both these are now unstable, +and indeed no stable form consisting of a +single mass of liquid has yet been discovered. + +Still quickening the rotation, the two remaining +forms, both unstable, grow in resemblance to +one another, until at length they become identical +in shape. This limiting form of Maclaurin's +spheroids is shown in the lower part of~\fig{38}. +If the liquid were water, it must rotate in $2$~hours +\PageSep{326} +$25$~minutes to attain this figure, but it would be +unstable. + +A figure for yet more rapid rotation has not +been determined, but it seems probable that +dimples would be formed on the axis, that the +dimples would deepen until they met, and that +the shape would then be annular. The actual +existence of such figures in Plateau's experiment +is confirmatory of this conjecture. + +We must now revert to the consideration of +the cigar-shaped figure of Jacobi, at the stage +when it has just become stable. The whole of +this argument depends on the fact that any figure +of equilibrium is a member of a continuous +series of figures of the same class, which gradually +transforms itself as the rotation varies. Now +M.~Poincaré has proved that, when we follow a +\index{Poincaré!law of interchange of stability}% +given series of figures and find a change from instability +to stability, we are, as it were, served with +a notice that there exists another series of figures +coalescent with the first at that stage. We have already +seen an example of this law, for the planetary +figure of Maclaurin changed from stability +to instability at the moment of its coalescence +with the figure of Jacobi. Now I said that when +the cigar form of Jacobi was very long it was +unstable, but that when its length had shrunk to +about twice its breadth it became stable; hence +we have notice that at the moment of change +another series of forms was coalescent with the +\PageSep{327} +cigar. It follows also from Poincaré's investigation +\index{Poincaré!law of interchange of stability}% +\index{Poincaré!figure of rotating liquid}% +that the other series of forms must have +been stable before the coalescence. + +Let us imagine then a mass of liquid in the +form of Jacobi's cigar-shaped body rotating at +the speed which just admits of stability, and let +us pursue the series of changes backwards by +making it rotate a little slower. We know that +this retardation of rotation lengthens Jacobi's +figure, and induces instability, but Poincaré has +not only proved the existence and stability of the +other series, but has shown that the shape is +something like a pear. + +Poincaré's figure is represented approximately +in~\fig{38}, but the mathematical difficulty of the +problem has been too great to admit of an absolutely +exact drawing. The further development +of the pear shape is unknown, when the rotation +slackens still more. There can, however, be +hardly any doubt that the pear becomes more +constricted in the waist, and begins to resemble +an hour-glass; that the neck of the hour-glass +becomes thinner, and that ultimately the body +separates into two parts. It is of course likewise +unknown up to what stage in these changes +Poincaré's figure retains its stability. + +I have myself attacked this problem from an +entirely different point of view, and my conclusions +throw an interesting light on the subject, +although they are very imperfect in comparison +\PageSep{328} +\index{Darwin, G. H.!hour-glass figure of rotating liquid|(}% +with Poincaré's masterly work. To understand +this new point of view, we must consider a new +series of figures, namely that of a liquid planet +attended by a liquid satellite. The two bodies +are supposed to move in a circle round one another, +and each is also to revolve on its axis at +such a speed as always to exhibit the same face +to its neighbor. Such a system, although divided +into two parts, may be described as a figure of +equilibrium. If the earth were to turn round +once in twenty-seven days, it would always show +to the moon the same side, and the moon actually +does present the same side to us. In this +case the earth and the moon would form such a +system as that I am describing. Both the planet +and the satellite are slightly flattened by their +rotations, and each of them exercises a tidal influence +on the other, whereby they are elongated +towards the other. + +The system then consists of a liquid planet +and liquid satellite revolving round one another, +so as always to exhibit the same face to one another, +and each tidally distorting the other. It +is certain that if the two bodies are sufficiently +far apart the system is a stable one, for if any +slight disturbance be given, the whole system will +not break up. But little is known as yet as to +the limiting proximity of the planet and satellite, +which will insure stability. + +Now if the rotations and revolutions of the +\PageSep{329} +bodies be accelerated, the two masses must be +brought nearer together in order that the greater +attraction may counterbalance the centrifugal +force. But as the two are brought nearer the +tide-generating force increases in intensity with +great rapidity, and accordingly the tidal elongation +of the two bodies is much augmented. + +A time will at length come when the ends of +the two bodies will just touch, and we then have +a form shaped like an hour-glass with a very +\Figure[0.7]{39}{Hour-glass Figure of Equilibrium}{png} +\PageLabel[pg]{329}% [** TN: Used by reference on p. 356 of the original] +thin neck. The form is clearly Poincaré's figure, +at an advanced stage of its evolution. + +The figure~\figref{39} shows the form of one possible +\PageSep{330} +figure of this class; it arises from the coalescence +of two equal masses of liquid, and the +shape shown was determined by calculation. +But there are any number of different sorts of +hour-glass shapes, according to the relative sizes +of the planet and satellite which coalesce; and +in order to form a continuous series with Poincaré's +pear, it would be necessary to start with +a planet and satellite of some definitely proportionate +sizes. Unfortunately I do not know +what the proportion may be. There are, however, +certain indications which may ultimately +lead to a complete knowledge of the series of +figures from Jacobi's cigar shape down to the +planet and satellite. It may be shown---and I +shall have in \Ref{Chapter}{XX}.\ to consider the point +more in detail---that if our liquid satellite had +only, say, a thousandth of the mass of the planet, +and if the two bodies were brought nearer one +another, at a certain calculable distance the tidal +action of the big planet on the very small satellite +would become so intense that it would tear +it to pieces. Accordingly the contact and coalescence +of a very small satellite with a large +planet is impossible. It is, however, certain that +a large enough satellite---say of half the mass +of the planet---could be brought up to contact +with the planet, without the tidal action of the +planet on the satellite becoming too intense to +admit of the existence of the latter. There +\PageSep{331} +must then be some mass of the satellite, which +will just allow the two to touch at the same +moment that the tidal action of the larger on +the smaller body is on the point of disrupting +it. Now I suspect, although I do not know, +that the series of figures which we should find in +this case is in fact Poincaré's series. This discussion +shows that the subject still affords an +interesting field for future mathematicians. + +These investigations as to the form of rotating +masses of liquid are of a very abstract character, +and seem at first sight remote from practical +conclusions, yet they have some very interesting +applications. + +The planetary body of Maclaurin is flattened +at the poles like the actual planets, and the +degree of its flattening is exactly appropriate to +the rapidity of its rotation. Although the planets +are, at least in large part, composed of solid +matter, yet that matter is now, or was once, +sufficiently plastic to permit it to yield to the +enormous forces called into play by rotation and +gravitation. Hence it follows that the theory +of Maclaurin's figure is the foundation of that +of the figures of planets, and of the variation of +gravity at the various parts of their surfaces. +In the liquid considered hitherto, every particle +attracted every other particle, the fluid was +equally dense throughout, and the figure assumed +was the resultant of the battle between +\PageSep{332} +\index{Figure of planets and their density}% +\index{Gravity, variation according to latitude}% +\index{Saturn!law of density and figure}% +the centrifugal force and gravitation. At every +part of the liquid the resultant attraction was +directed nearly, but not quite, towards the +centre of the shape. But if the attraction had +everywhere been directed exactly to the centre, +the degree of flattening would have been +diminished. We may see that this must be so, +because if the rotation were annulled, the mass +would be exactly spherical, and if the rotation +were not annulled, yet the forces would be such +as to make the fluid pack closer, and so assume +a more nearly spherical form than when the +forces were not absolutely directed to the centre. +It may be shown in fact that the flattening is +$2\frac{1}{2}$~times greater in the case of Maclaurin's +body than it is when the seat of gravitation is +exactly central. + +In the case of actual planets the denser matter +\index{Planets!figures and internal densities}% +must lie in the centre and the less dense outside. +If the central matter were enormously +denser than superficial rock, the attraction would +be directed towards the centre. There are then +two extreme cases in which the degree of flattening +can be determined,---one in which the density +\index{Density!of planets determinable from their figures}% +of the planet is the same all through, giving +Maclaurin's figure; the other when the density +is enormously greater at the centre. The flattening +in the former is $2\frac{1}{2}$~times as great as in +the latter. The actual condition of a real planet +must lie between these two extremes. The +\index{Darwin, G. H.!hour-glass figure of rotating liquid|)}% +\PageSep{333} +\index{Figure of planets and their density}% +knowledge of the rate of rotation of a planet +and of the degree of its flattening furnishes us +with some insight into the law of its internal +density. If it is very much less flat than Maclaurin's +\index{Density!of planets determinable from their figures}% +figure, we conclude that it is very dense +in its central portion. In this way it is known +with certainty that the central portions of the +planets Jupiter and Saturn are much denser, +\index{Jupiter!figure and law of internal density}% +\index{Planets!figures and internal densities}% +compared with their superficial portions, than is +the case with the earth. + +I do not propose to pursue this subject into +the consideration of the law of the variation of +gravity on the surface of a planet; but enough +has been said to show that these abstract investigations +have most important practical applications. + +\begin{Authorities} +Plateau, ``Mémoires de l'Académie Royale de~Belgique,'' +vol.~xvi. 1843. + +Thomson and Tait's \Title{Natural Philosophy} or other works on +hydrodynamics give an account of figures of equilibrium. + +Poincaré, \Title{Sur l'équilibre d'une masse fluide animée d'un mouvement +de rotation}, ``Acta Mathematica,'' vol.~7, 1885. + +An easier and different presentation of the subject is contained +in an inaugural dissertation by Schwarzschild (Annals of Munich +\index{Schwarzschild!exposition of Poincaré's theory}% +Observatory, vol.~iii. 1896). He considers that Poincaré's +proof of the stability of his figure is not absolutely conclusive. + +G.~H. Darwin, \Title{Figures of Equilibrium of Rotating Masses of +\index{Darwin, G. H.!Jacobi's ellipsoid}% +Fluid}, ``Transactions of Royal Society,'' vol.~178, 1887. + +G.~H. Darwin, \Title{Jacobi's Figure of Equilibrium},~\&c., ``Proceedings +\index{Equilibrium, figures of, of rotating liquid|)}% +\index{Figure of equilibrium!of rotating liquid|)}% +Roy.\ Soc.,'' vol.~41, 1886, p.~319. + +S.~Krüger, \Title{Ellipsoidale Evenwichtsvormen},~\&c., Leeuwen, Leiden, +\index{Krüger, figures of equilibrium of liquid}% +1896; \Title{Sur l'ellipsoïde de Jacobi}, ``Nieuw Archief voor Wiskunde,'' +2d~series, 3d~part, 1898. The author shows that G.~H. +Darwin had been forestalled in much of his work on Jacobi's +figure, and he corrects certain mistakes. +\end{Authorities} +\index{Rotating liquid, figures of equilibrium|)}% +\PageSep{334} + + +\Chapter{XIX} +{The Evolution of Celestial Systems} + +\First{Men} will always aspire to peer into the remote +\index{Evolution of celestial systems|(}% +\index{Solar!system, nebular hypothesis as to origin of|(}% +past to the utmost of their power, and the fact +that their success or failure cannot appreciably +influence their life on the earth will never deter +them from such endeavors. From this point +of view the investigations explained in the last +chapter acquire much interest, since they form +the basis of the theories of cosmogony which +seem most probable by the light of our present +knowledge. + +We have seen that an annular figure of equilibrium +\index{Kant!nebular hypothesis|(}% +\index{Nebular hypothesis|(}% +actually exists in Plateau's experiment, +and it is almost certainly a possible form amongst +celestial bodies. Plateau's ring has however +only a transient existence, and tends to break up +into globules, spinning on their axes and revolving +round the centre. In this result we saw a +close analogy with the origin of the planets, and +regarded his experiment as confirmatory of the +Nebular Hypothesis, of which I shall now give a +short account.\footnote + {My knowledge of the history of the Nebular Hypothesis is + entirely derived from an interesting paper by Mr.~G.~F. Becker, +\index{Becker, G. F., on Nebular Hypothesis}% + on ``Kant as a Natural Philosopher,'' \Title{American Journal of Science}, + vol.~v. Feb.~1898.} +\PageSep{335} + +The first germs of this theory are to be found +in Descartes' ``Principles of Philosophy,'' published +\index{Descartes, vortical theory of cosmogony}% +in~1644. According to him the sun and +planets were represented by eddies or vortices in +a primitive chaos of matter, which afterwards +formed the centres for the accretion of matter. +As the theory of universal gravitation was propounded +for the first time half a century later +than the date of Descartes' book, it does not +seem worth while to follow his speculations +further. Swedenborg formulated another vortical +cosmogony in~1734, and Thomas Wright of +\index{Wright, Thomas, on a theory of cosmogony}% +Durham published in 1750 a book of preternatural +dullness on the same subject. It might not +have been worth while to mention Wright, but +that Kant acknowledges his obligation to him. + +The Nebular Hypothesis has been commonly +associated with the name of Laplace, and he undoubtedly +\index{Laplace!nebular hypothesis|(}% +avoided certain errors into which his +precursors had fallen. I shall therefore explain +Laplace's theory, and afterwards show how he +was, in most respects, really forestalled by the +great German philosopher Kant. + +Laplace supposed that the matter now forming +the solar system once existed in the form of a +lens-shaped nebula of highly rarefied gas, that it +rotated slowly about an axis perpendicular to the +present orbits of the planets, and that the nebula +extended beyond the present orbit of the furthest +planet. The gas was at first expanded by heat, +\PageSep{336} +and as the surface cooled the central portion +condensed and its temperature rose. The speed +of rotation increased in consequence of the contraction, +according to a well known law of mechanics +called ``the conservation of moment of +momentum;''\footnote + {Kant fell into error through ignorance of the generality of + this law, for he imagined that rotation could be generated from + rest.} +the edges of the lenticular mass +of gas then ceased to be continuous with the +more central portion, and a ring of matter was +detached, in much the same way as in Plateau's +experiment. Further cooling led to further contraction +and consequently to increased rotation, +until a second ring was shed, and so on successively. +The rings then ruptured and aggregated +themselves into planets whilst the central nucleus +formed the sun. + +Virtually the same theory had been propounded +by Kant many years previously, but I am not +aware that there is any reason to suppose that +Laplace had ever read Kant's works. In a paper, +to which I have referred above, Mr.~G.~F. +Becker makes the following excellent summary +\index{Becker, G. F., on Nebular Hypothesis|(}% +of the relative merits of Kant and Laplace; he +writes:--- + +``Kant seems to have anticipated Laplace almost +completely in the more essential portions +of the nebular hypothesis. The great Frenchman +was a child when Kant's theory was issued, +\PageSep{337} +and the `Système du Monde,' which closes with +the nebular hypothesis, did not appear until +1796. Laplace, like Kant, infers unity of origin +for the members of the solar system from the +similarity of their movements, the small obliquity +and small eccentricity of the orbits of either +planets or satellites.\footnote + {``The retrograde satellites of Uranus were discovered by + Herschel in~1787, but Laplace in his hypothesis does not refer to + them.''} +Only a fluid extending +throughout the solar system could have produced +such a result. He is led to conclude that the +atmosphere of the sun, in virtue of excessive +heat, originally extended beyond the solar system +and gradually shrank to its present limits. This +nebula was endowed with moment of momentum +which Kant tried to develop by collisions. Planets +formed from zones of vapor, which on breaking +agglomerated\dots. The main points of +comparison between Kant and Laplace seem to +be these. Kant begins with a cold, stationary +nebula which, however, becomes hot by compression +and at its first regenesis would be in a state +of rotation. It is with a hot, rotating nebula +that Laplace starts, without any attempt to account +for the heat. Kant supposes annular +zones of freely revolving nebulous matter to +gather together by attraction during condensation +of the nebula. Laplace supposes rings left +\index{Laplace!nebular hypothesis|)}% +behind by the cooling of the nebula to agglomerate +\PageSep{338} +in the same way as Kant had done. While +both appeal to the rings of Saturn as an example +of the hypothesis, neither explains satisfactorily +why the planetary rings are not as stable +as those of Saturn. Both assert that the positive +rotation of the planets is a necessary +consequence of agglomeration, but neither is +sufficiently explicit. The genesis of satellites is +for each of them a repetition on a small scale of +the formation of the system\dots. While Laplace +assigns no cause for the heat which he ascribes +to his nebula, Lord Kelvin goes further +back and supposes a cold nebula consisting of +separate atoms or of meteoric stones, initially +possessed of a resultant moment of momentum +equal or superior to that of the solar system. +Collision at the centre will reduce them to a +vapor which then expanding far beyond Neptune's +orbit will give a nebula such as Laplace +postulates.\footnote + {\Title{Popular Lectures}, vol.~i.\ p.~421.} +Thus Kelvin goes back to the same +initial condition as Kant, excepting that Kant +endeavored (of course vainly) to develop a moment +of momentum for his system from collisions.''\footnote + {Becker, \Title{Amer.\ Journ.\ Science}, vol.~v. 1898, pp.~107,~108.} +\index{Becker, G. F., on Nebular Hypothesis|)}% + +There is good reason for believing that the +Nebular Hypothesis presents a true statement in +outline of the origin of the solar system, and of +the planetary subsystems, because photographs +%% Plate 3 +\TallFig[0.85]{40}{Nebula in Andromeda}{jpg} +%% Facing page +%[Blank Page] +\PageSep{339} +of nebulæ have been taken recently in which we +can almost see the process in action. \Fig{40} is +a reproduction of a remarkable photograph by +Dr.~Isaac Roberts of the great nebula in the constellation +\index{Roberts, I., photograph of nebula in Andromeda}% +of Andromeda. In it we may see the +\index{Andromeda, nebula in}% +\index{Nebula in Andromeda}% +lenticular nebula with its central condensation, +the annulation of the outer portions, and even +the condensations in the rings which will doubtless +at some time form planets. This system is +built on a colossal scale, compared with which +our solar system is utterly insignificant. Other +\index{Solar!system, nebular hypothesis as to origin of|)}% +\index{Solar!system, distribution of satellites in|(}% +nebulæ show the same thing, and although they +are less striking we derive from them good +grounds for accepting this theory of evolution +as substantially true. + +\TB + +I explained in \Ref{Chapter}{XVI}.\ how the theory +of tidal friction showed that the moon took her +origin very near to the present surface of the +earth. But it was also pointed out that the same +theory cannot be invoked to explain an origin +for the planets at a point close to the sun. They +must in fact have always moved at nearly their +present distances. In the same way the dimensions +of the orbits of the satellites of Mars, Jupiter, +\index{Satellites!distribution of, in solar system|(}% +Saturn, and Neptune cannot have been +largely augmented, whatever other effects tidal +friction may have had. We must therefore still +rely on the Nebular Hypothesis for the explanation +\index{Nebular hypothesis|)}% +of the main features of the system as a +whole. +\index{Kant!nebular hypothesis|)}% +\PageSep{340} + +It may, at first sight, appear illogical to maintain +that an action, predominant in its influence +on our satellite, should have been insignificant +in regulating the orbits of all the other bodies +of the system. But this is not so, for whilst the +earth is only $80$~times as heavy as the moon, Saturn +weighs about $4,600$~times as much as its +satellite Titan, which is by far the largest satellite +in the solar system; and all the other satellites +are almost infinitesimal in comparison with their +primaries. Since, then, the relationship of the +moon to the earth is unique, it may be fairly contended +that a factor of evolution, which has been +predominant in our own history, has been relatively +insignificant elsewhere. + +There is indeed a reason explanatory of this +singularity in the moon and earth; it lies in the +fact that the earth is nearer to the sun than any +other planet attended by a satellite. To explain +the bearing of this fact on the origin of satellites +and on their sizes, I must now show how tidal +friction has probably operated as a perturbing +influence in the sequence of events, which would +be normal according to the Nebular Hypothesis. + +We have seen that rings should be shed from +the central nucleus, when the contraction of the +nebula has induced a certain degree of augmentation +of rotation. Now if the rotation were +retarded by some external cause, the genesis of +a ring would be retarded, or might be entirely +prevented. +\PageSep{341} + +The friction of the solar tides in a planetary +nebula furnishes such an external cause, and accordingly +the rotation of a planetary nebula near +to the sun might be so much retarded that a ring +would never be detached from it, and no satellite +would be generated. From this point of view +it is noteworthy that Mercury and Venus have +no satellites; that Mars has two, Jupiter five, +and that all the exterior planets have several +satellites. I suggest then that the solar tidal +friction of the terrestrial nebula was sufficient to +retard the birth of a satellite, but not to prevent +it, and that the planetary mass had contracted +to nearly the present dimensions of the earth +and had partially condensed into the solid and +liquid forms, before the rotation had augmented +sufficiently to permit the birth of a satellite. +When satellites arise under conditions which are +widely different, it is reasonable to suppose that +their masses will also differ much. Hence we can +understand how it has come about that the relationship +between the moon and the earth is so +unlike that between other satellites and their +planets. In \Ref{Chapter}{XVII}.\ I showed that there +are reasons for believing that solar tidal friction +\index{Solar!system, distribution of satellites in|)}% +has really been an efficient cause of change, and +this makes it legitimate to invoke its aid in explaining +the birth and distribution of satellites. +\index{Satellites!distribution of, in solar system|)}% + +\TB + +In speaking of the origin of the moon I have +\PageSep{342} +\index{Eccentricity of orbit!theory of, in case of double stars}% +\index{See, T. J. J.!theory of evolution of double stars|(}% +been careful not to imply that the matter of +which she is formed was necessarily first arranged +in the form of a ring. Indeed, the genesis of +the hour-glass figure of equilibrium from Jacobi's +form and its fission into two parts indicate the +possibility of an entirely different sequence of +events. It may perhaps be conjectured that the +moon was detached from the primitive earth in +this way, possibly with the help of tidal oscillations +due to the solar action. Even if this suggestion +is only a guess, it is interesting to make +such speculations, when they have some basis of +reason. + +In recent years astronomers have been trying, +principally by aid of the spectroscope, to determine +the orbits of pairs of double stars around +\index{Stars!theory of evolution|(}% +one another. It has been observed that, in the +majority of these systems, the masses of the two +component stars do not differ from one another +extremely; and Dr.~See, who has specially devoted +himself to this research, has drawn attention +to the great contrast between these systems +and that of the sun, attended by a retinue of +infinitesimal planets. He maintains, with justice, +that the paths of evolution pursued in the two +cases have probably also been strikingly different. + +It is hardly credible that two stars should +have gained their present companionship by an +accidental approach from infinite space. They +cannot always have moved as they do now, and +\PageSep{343} +so we are driven to reflect on the changes which +might supervene in such a system under the +action of known forces. + +The only efficient interaction between a pair +of celestial bodies, which is known hitherto, is +a tidal one, and the friction of the oscillations +introduces a cause of change in the system. +Tidal friction tends to increase the eccentricity +of the orbit in which two bodies revolve about +one another, and its efficiency is much increased +when the pair are not very unequal in mass and +when each is perturbed by the tides due to the +other. The fact that the orbits of the majority +of the known pairs are very eccentric affords a +reason for accepting the tidal explanation. The +only adverse reason, that I know of, is that the +eccentricities are frequently so great that we +may perhaps be putting too severe a strain on +the supposed cause. + +But the principal effect of tidal friction is the +repulsion of the two bodies from one another, +so that when their history is traced backwards +we ultimately find them close together. If then +this cause has been as potent as Dr.~See believes +it to have been, the two components of a binary +system must once have been close together. +From this stage it is but a step to picture to +ourselves the rupture of a nebula, in the form +of an hour-glass, into two detached masses. + +The theory embraces all the facts of the case, +\PageSep{344} +and as such is worthy of at least a provisional +acceptance. But we must not disguise from +ourselves that out of the thousands, and perhaps +millions of double stars which may be visible +from the earth, we only as yet know the orbits +and masses of a dozen. + +Many years ago Sir John Herschel drew a +\index{Herschel, observations of twin nebulæ}% +number of twin nebulæ as they appear through +a powerful telescope. The drawings probably +possess the highest degree of accuracy attainable +by this method of delineation, and the shapes +present evidence confirmatory of the theory of +the fission of nebulæ adopted by Dr.~See. But +since Herschel's time it has been discovered that +many details, to which our eyes must remain forever +blind, are revealed by celestial photography. +The photographic film is, in fact, sensitive to +those ``actinic'' rays which we may call invisible +light, and many nebulæ are now found to be +hardly recognizable, when photographs of them +are compared with drawings. A conspicuous +example of this is furnished by the great nebula +in Andromeda, illustrated above in~\fig{40}. + +Photographs, however, do not always aid interpretation, +for there are some which serve only +to increase the chaos visible with the telescope. +We may suspect, indeed, that the complete system +of a nebula often contains masses of cold +and photographically invisible gas, and in such +cases it would seem that the true nature of the +whole will always be concealed from us. +\PageSep{345} + +Another group of strange celestial objects is +that of the spiral nebulæ, whose forms irresistibly +\index{Nebulae@Nebulæ, description of various}% +suggest violent whirlpools of incandescent +gas. Although in all probability the motion of +the gas is very rapid, yet no change of form has +been detected. We are here reminded of a +rapid stream rushing past a post, where the form +of the surface remains constant whilst the water +itself is in rapid movement; and it seems reasonable +to suppose that in these nebulæ it is +only the lines of the flow of the gas which are +visible. Again, there are other cases in which +the telescopic view may be almost deceptive in +its physical suggestions. Thus the Dumb-Bell +\index{Dumb-bell nebula, description of photograph of}% +nebula (27~Messier Vulpeculæ), as seen telescopically, +might be taken as a good illustration of a +nebula almost ready to split into two stars. If +this were so, the rotation would be about an +axis at right angles to the length of the nebula. +But a photograph of this object shows that the +system really consists of a luminous globe surrounded +by a thick and less luminous ring, and +that the opacity of the sides of the ring takes a +bite, as it were, out of each side of the disk, and +so gives it the apparent form of a dumb-bell. +In this case the rotation must be about an axis +at right angles to the ring, and therefore along +the length of the dumb-bell. It is proper to +add that Dr.~See is well aware of this, and does +not refer to this nebula as a case of incipient +fission. +\PageSep{346} + +I have made these remarks in order to show +that every theory of stellar evolution must be +full of difficulty and uncertainty. According to +our present knowledge Dr.~See's theory appears +to have much in its favor, but we must await its +confirmation or refutation from the results of +future researches with the photographic plate, +the spectroscope, and the telescope. + +\begin{Authorities} +Mr.~G.~F. Becker (\Title{Amer.\ Jour.\ Science}, vol.~v. 1898, art.~xv.)\ +gives the following references to Kant's work: \Title{Sämmtliche +Werke}, ed.~Hartenstein, 1868 (Tidal Friction and the Aging of +the Earth), vol.~i.\ pp.~179--206; (Nebular Hypothesis), vol.~i.\ +pp.~207--345. + +Laplace, \Title{Système du Monde}, last appendix; the tidal retardation +of the moon's rotation is only mentioned in the later +editions. + +T.~J.~J. See, \Title{Die Entwickelung der Doppelstern-systeme}, ``Inaugural +Dissertation,'' 1892. Schade, Berlin. + +T.~J.~J. See, \Title{Evolution of the Stellar Systems}, vol.~i.\ 1896. +Nichols Press, Lynn, Massachusetts. Also a popular article, +\Title{The Atlantic Monthly}, October, 1897. + +G.~H. Darwin, \Title{Tidal Friction~\dots\ and Evolution}, ``Phil.\ Trans.\ +\index{Darwin, G. H.!evolution of satellites}% +Roy.\ Soc.,'' part~ii.\ 1881, p.~525. +\end{Authorities} +\index{Evolution of celestial systems|)}% +\index{See, T. J. J.!theory of evolution of double stars|)}% +\index{Stars!theory of evolution|)}% +\PageSep{347} + + +\Chapter[Saturn's Rings]{XX} +{Saturn's Rings\protect\footnotemark} + +\footnotetext{Part of this chapter appeared as an article in \Title{Harper's + Magazine} for June,~1889.} + +\First{To} the naked eye Saturn appears as a brilliant +\index{Saturn!description and picture|(}% +star, which shines, without twinkling, with a +yellowish light. It is always to be found very +nearly in the ecliptic, moving slowly amongst +the fixed stars at the rate of only thirteen degrees +per annum. It is the second largest +planet of the solar system, being only exceeded +in size by the giant Jupiter. It weighs $91$~times +as much as our earth, but, being as light as cork, +occupies $690$~times the volume, and is nine times +as great in circumference. Notwithstanding its +great size it rotates around its axis far more +rapidly than does the earth, its day being only +$10\frac{1}{2}$~of our hours. It is ten times as far from +the sun as we are, and its year, or time of revolution +round the sun, is equal to thirty of our +years. It was deemed by the early astronomers +to be the planet furthest from the sun, but that +was before the discovery by Herschel, at the +end of the last century, of the further planet +Uranus, and that of the still more distant Neptune +by Adams and Leverrier in the year~1846. +\PageSep{348} + +The telescope has shown that Saturn is attended +by a retinue of satellites almost as numerous +as, and closely analogous to, the planets +circling round the sun. These moons are eight +in number, are of the most various sizes, the +largest as great as the planet Mars, and the +smallest very small, and are equally diverse in +respect of their distances from the planet. But +besides its eight moons Saturn has another attendant +absolutely unique in the heavens; it is +girdled with a flat ring, which, like the planet +itself, is only rendered visible to us by the +illumination of sunlight. \Fig{41}, to which +further reference is made below, shows the general +appearance of the planet and of its ring. +The theory of the physical constitution of that +ring forms the subject of the present chapter. + +A system so rich in details, so diversified and +so extraordinary, would afford, and doubtless +has afforded, the subject for many descriptive +essays; but description is not my present object. + +The existence of the ring of Saturn seems +now a very commonplace piece of knowledge, +and yet it is not $300$~years since the moons of +Jupiter and Saturn were first detected, and since +suspicion was first aroused that there was something +altogether peculiar about the Saturnian +system. These discoveries, indeed, depended +entirely on the invention of the telescope. It +may assist the reader to realize how necessary +\PageSep{349} +%[** TN: Oriented vertically in the original] +\Figure{41}{The Planet Saturn}{jpg} +\PageSep{350} +the aid of that instrument was when I say that +Saturn, when at his nearest to us, is the same in +size as a sixpenny piece held up at a distance of +$210$~yards. + +It was the celebrated Galileo who first invented +\index{Galileo!Saturn's ring}% +a combination of lenses such as is still +used in our present opera-glasses, for the purpose +of magnifying distant objects. + +In July of~1610 he began to examine Saturn +with his telescope. His most powerful instrument +only magnified $32$~times, and although +such an enlargement should have amply sufficed +to enable him to make out the ring, yet he persuaded +himself that what he saw was a large +bright disk, with two smaller ones touching it, +one on each side. His lenses were doubtless +imperfect, but the principal cause of his error +must have been the extreme improbability of the +existence of a ring girdling the planet. He +wrote an account of what he had seen to the +Grand Duke of Tuscany, Giuliano de'~Medici, +and to others; he also published to the world an +anagram which, when the letters were properly +arranged, read as follows: ``Altissimum planetam +tergeminum observavi'' (I have seen the +furthest planet as triple), for it must be remembered +that Saturn was then the furthest known +planet. + +In 1612 Galileo again examined Saturn, and +was utterly perplexed and discouraged to find +\PageSep{351} +his triple star replaced by a single disk. He +writes, ``Is it possible that some mocking demon +has deceived me?'' And here it may be well to +remark that there are several positions in which +Saturn's rings vanish from sight, or so nearly +vanish as to be only visible with the most powerful +modern telescopes. When the plane of the +ring passes through the sun, only its very thin +edge is illuminated; this was the case in~1612, +when Galileo lost it; secondly, if the plane of +the ring passes through the earth, we have only +a very thin edge to look at; and thirdly, when +the sun and the earth are on opposite sides of +the ring, the face of the ring which is presented +to us is in shadow, and therefore invisible. + +Some time afterwards Galileo's perplexity was +increased by seeing that the planet had then a +pair of arms, but he never succeeded in unraveling +the mystery, and blindness closed his career +as an astronomer in~1626. + +About thirty years after this, the great Dutch +astronomer Huyghens, having invented a new +\index{Huyghens, discovery of Saturn's ring}% +sort of telescope (on the principle of our present +powerful refractors), began to examine the planet +and saw that it was furnished with two loops or +handles. Soon after the ring disappeared; but +when, in~1659, it came into view again, he at +last recognized its true character, and announced +that the planet was attended by a broad, flat +ring. +\PageSep{352} + +A few years later it was perceived that there +were two rings, concentric with one another. +The division, which may be easily seen in drawings +of the planet, is still named after Cassini, +\index{Cassini, discovery of division in Saturn's rings}% +one of its discoverers. Subsequent observers +have detected other less marked divisions. + +Nearly two centuries later, namely, in~1850, +Bond in America and Dawes in England, independently +\index{Bond, discovery of inner ring of Saturn}% +\index{Dawes, discovery of inner ring of Saturn}% +and within a fortnight of the same +time, observed that inside of the well-known +bright rings there is another very faint dark +ring, which is so transparent that the edge of +the planet is visible through it. There is some +reason to believe that this ring has really become +more conspicuous within the last $200$~years, +so that it would not be right to attribute the +lateness of its detection entirely to the imperfection +of earlier observations. + +It was already discovered in the last century +that the ring is not quite of the same thickness +at all points of its circumference, that it is not +strictly concentric with the planet, and that it +revolves round its centre. Herschel, with his +magnificent reflecting telescope, detected little +beads on the outer ring, and by watching these +he concluded that the ring completes its revolution +in $10\frac{1}{2}$~hours. + +This sketch of the discovery and observation +of Saturn's rings has been necessarily very incomplete, +but we have perhaps already occupied +too much space with it. +\PageSep{353} + +\Fig{41} exhibits the appearance of Saturn and +his ring. The drawing is by Bond of Harvard +University, and is considered an excellent one. + +It is usual to represent the planets as they are +seen through an astronomical telescope, that is +\Figure{42}{Diagram of Saturn and his Rings}{png} +to say, reversed. Thus in \fig{41} the south +pole of the planet is at the top of the plate, and +unless the telescope were being driven by clockwork, +the planet would appear to move across +the field of view from right to left. + +The plane of the ring is coincident with the +equator of the planet, and both ring and equator +are inclined to the plane of the planet's orbit at +an angle of $27$~degrees. + +A whole essay might be devoted to the discussion +of this and of other pictures, but we must +confine ourselves to drawing attention to the +well-marked split, called Cassini's division, and +\PageSep{354} +to the faint internal ring, through which the +edge of the planet is visible. + +The scale on which the whole system is constructed +is best seen in a diagram of concentric +circles, showing the limits of the planet's body +and of the successive rings. Such a diagram, +with explanatory notes, is given in~\fig{42}. + +An explanation of the outermost circle, called +\emph{Roche's limit}, will be given later. The following +are the dimensions of the system:--- +\begin{center} +\begin{tabular}{l>{\qquad}r} +Equatorial diameter of planet & $73,000$ miles \\ +Interior diameter of dark ring & $93,000$ \Ditto{miles} \\ +Interior diameter of bright rings & $111,000$ \Ditto{miles} \\ +Exterior diameter of bright rings & $169,000$ \Ditto{miles} +\end{tabular} +\end{center} +We may also remark that the radius of the +limit of the rings is $2.38$~times the mean radius +of the planet, whilst Roche's limit is $2.44$~such +radii. The greatest thickness of the ring is uncertain, +but it seems probable that it does not +exceed $200$~or $300$~miles. + +The pictorial interest, as we may call it, of all +this wonderful combination is obvious, but our +curiosity is further stimulated when we reflect on +the difficulty of reconciling the existence of this +strange satellite with what we know of our own +planet and of other celestial bodies. + +It may be admitted that no disturbance to our +ordinary way of life would take place if Saturn's +\index{Saturn!description and picture|)}% +rings were annihilated, but, as Clerk-Maxwell +has remarked, ``from a purely scientific point of +\PageSep{355} +view, they become the most remarkable bodies in +the heavens, except, perhaps, those still less \emph{useful} +bodies---the spiral nebulæ. When we have +actually seen that great arch swung over the +equator of the planet without any visible connection, +we cannot bring our minds to rest. We +cannot simply admit that such is the case, and +describe it as one of the observed facts of nature, +not admitting or requiring explanation. We +must either explain its motion on the principles +of mechanics, or admit that, in Saturnian realms, +there can be motion regulated by laws which we +are unable to explain.'' + +I must now revert to the subject of \Ref{Chapter}{XVIII}.\ +and show how the investigations, there +explained, bear on the system of the planet. We +then imagined a liquid satellite revolving in a +circular orbit about a liquid planet, and supposed +that each of these two masses moved so as always +to present the same face to the other. It was +pointed out that each body must be somewhat +flattened by its rotation round an axis at right +angles to the plane of the orbit, and that the +tidal attraction of each must deform the other. +In the application of this theory to the system of +Saturn it is not necessary to consider further the +tidal action of the satellite on the planet, and we +must concentrate our attention on the action of +the planet on the satellite. We have found reason +to suppose that the earth once raised enormous +\PageSep{356} +\index{Saturn!theory of ring|(}% +tides in the moon, when her body was +molten, and any planet must act in the same way +on its satellite. When, as we now suppose, the +satellite moves so as always to present the same +face to the planet, the tide is fixed and degenerates +into a permanent distortion of the equator +of the satellite into an elliptic shape. If the +satellite is very small compared with its planet, +and if it is gradually brought closer and closer +to the planet, the tide-generating force, which +varies inversely as the cube of the distance, increases +with great rapidity, and we shall find the +satellite to assume a more and more elongated +shape. When the satellite is not excessively +small, the two bodies may be brought together +until they actually touch, and form the hour-glass +figure exhibited in \fig{39}, \PageRef{p.}{329}. + +The general question of the limiting proximity +of a liquid planet and satellite which just insures +stability is as yet unsolved. But it has been +proved that there is one case in which instability +sets in. Édouard Roche has shown that this approach +\index{Roche, E.!theory of limit and Saturn's ring|(}% +up to contact is not possible when the +satellite is very small, for at a certain distance +the tidal distortion of a small satellite becomes +so extreme that it can no longer subsist as a +single mass of fluid. He also calculated the +form of the satellite when it is elongated as much +as possible. \Fig{43} represents the satellite in +its limiting form. We must suppose the planet +\PageSep{357} +about which it revolves to be a large globe, with +its centre lying on the prolongation of the longest +axis of the egg-like body in the direction +of~$E$. As it revolves, the longest axis of the satellite +always points straight towards its planet. +The egg, though not strictly circular in girth, is +\Figure[0.7]{43}{Roche's Figure of a Satellite when elongated +to the utmost}{png} +very nearly so. Thus another section at right +angles to this one would be of nearly the same +shape. One diameter of the girth is in fact only +longer than the other by a seventeenth part. +The shortest of the three axes of the slightly flattened +egg is at right angles to the plane of the +orbit in which the satellite revolves. The longest +axis of the body is nearly twice as long as +either of the two shorter ones; for if we take +the longest as~$\DPchg{1000}{1,000}$, the other two would be $496$ +and~$469$. \Fig{43} represents a section through +the two axes equal respectively to~$\DPchg{1000}{1,000}$ and to~$469$, +so that we are here supposed to be looking +at the satellite's orbit edgewise. +\PageSep{358} + +But, as I have said, Roche determined not +\index{Earth and moon!Roche's limit for}% +only the shape of the satellite when thus elongated +to the utmost possible extent, but also in +its nearness to the planet, and he proved that if +the planet and satellite be formed of matter of +the same density, the centre of such a satellite +must be at a distance from the planet's centre of +$2\frac{11}{25}$~of the planet's radius. This distance of $2\frac{11}{25}$ +or $2.44$~of a planet's radius I call Roche's limit +for that planet. The meaning of this is that inside +of a circle drawn around a planet at a distance +so proportionate to its radius no small +satellite can circulate; the reason being that if +a lump of matter were started to revolve about +the planet inside of that circle, it would be torn +to pieces under the action of the forces we have +been considering. It is true that if the lump of +matter were so small as to be more properly described +as a stone than as a satellite, then the +cohesive force of stone might be strong enough +to resist the disruptive force. But the size for +which cohesion is sufficient to hold a mass of +matter together is small compared with the +smallest satellite. + +I have said that Roche's limit as evaluated at +$2.44$~radii is dependent on the assumption of +equal densities in the satellite and planet. If +the planet be denser than the satellite, Roche's +limit is a larger multiple of the planet's radius, +and if it be less dense the multiple is smaller. +\PageSep{359} +But the variation of distance is not great for +considerable variations in the relative densities +of the two bodies, the law being that the~$2.44$ +must be multiplied by the cube root of the ratio +of the density of the planet to that of the satellite. +If for example the planet be on the average +of its whole volume twice as dense as the +satellite, the limit is only augmented from $2.44$ +to $3$~times the planet's radius; and if it be half +as dense, the $2.44$ is depressed to~$1.94$. Thus +the variation of density of the planet from a +half to twice that of the planet---that is to +say, the multiplication of the smaller density by +four---only changes Roche's limit from $2$ to $3$~radii. +It follows from this that, within pretty +wide limits of variation of relative densities, +Roche's limit changes but little. + +The only relative density of planet and satellite +that we know with accuracy is that of the +earth and moon. Now the earth is more dense +than the moon in the proportion of $8$~to~$5$; hence +Roche's limit for the earth is the cube root of~$\frac{8}{5}$ +multiplied by~$2.44$, that is to say, it is $2.86$~times +the earth's radius. It follows that if the moon +were to revolve at a distance of less than $2.86$~radii, +or $11,000$~miles, she would be torn to pieces +by the earth's tidal force. + +If this result be compared with the conclusions +drawn from the theory of tidal friction, it follows +that at the earliest stage to which the moon was +\PageSep{360} +\index{Saturn!Roche's limit for}% +traced, she could not have existed in her present +form, but the matter which is now consolidated +in the form of a satellite must then have been a +mere swarm of loose fragments. Such fragments, +if concentrated in one part of the orbit, would +be nearly as efficient in generating tides in the +planet as though they were agglomerated in the +form of a satellite. Accordingly the action of +tidal friction does not necessitate the agglomeration +of the satellite. The origin and earliest history +of the moon must always remain highly +speculative, and it seems fruitless to formulate +exact theories on the subject.\footnote + {Mr.~Nolan has criticised the theory of tidal friction from +\index{Nolan, criticism of tidal theory of moon's origin}% + this point of view (\Title{Genesis of the Moon}, Melbourne, 1885; also + \Title{Nature}, Feb.~18 and July~29, 1886).} + +When we apply this reasoning to the other +planets, exact data are wanting. The planet +Mars resembles the earth in so many respects +\index{Mars!Roche's limit}% +that it is reasonable to suppose that there is much +the same relationship between the densities of +the planet and satellites as with us. As with the +case of the earth and moon, this would bring +Roche's limit to $2.86$~times the planet's radius. +The satellite Phobos, however, revolves at a +distance of $2.75$~radii of Mars; hence we are +bound to suppose that the density of Phobos is +a very little more nearly equal to that of Mars +than in the case of the moon and earth; if +it were not so, Phobos would be disrupted by +\PageSep{361} +tidal action. How interesting it will be if future +generations shall cease to see the satellite Phobos, +for they will then conclude that Phobos has been +drawn within the charmed circle, and has been +broken to pieces. + +In considering the planets Jupiter and Saturn, +\index{Jupiter!Roche's limit for}% +we are deprived of the indications which are useful +in the case of Mars. The satellites are probably +solid, and these planets are known to have +a low mean density. Hence it is probable that +Roche's limit is a somewhat smaller multiple than +$2.44$~of the radii of Jupiter and Saturn. The +only satellite which is in danger is the innermost +and recently discovered satellite of Jupiter, which +revolves at $2.6$~times the planet's mean radius, +for with the same ratio of densities as obtains +here the satellite would be broken up. This confirms +the conclusion that the mean density of +Jupiter is at least not greater than that of the +satellite. + +We are also ignorant of the relative densities +of Saturn and its satellites, and so in the figure +Roche's limit is placed at $2.44$~times the planet's +radius, corresponding to equal densities. But +the density of the planet is very small, and therefore +the limit is almost certainly slightly nearer +to the planet than is shown. + +This system affords the only known instance +where matter is clearly visible circulating round +an attractive centre at a distance certainly less +\PageSep{362} +than the theoretical limit, and the belief seems +justified that Saturn's rings consist of dust and +fragments. + +Although Roche himself dismissed this matter +in one or two sentences, he saw the full bearing +of his remarks, and to do him justice we should +date from~1848 the proof that Saturn's rings +consist of meteoric stones. + +The theoretical limit lies just outside the limit +of the rings, but we may suspect that the relative +densities of the planet and satellite are such that +the limit should be displaced to a distance just +inside of the outer edge of the ring, because any +solid satellite would almost necessarily have a +mean density greater than that of the planet. + +Although Roche's paper was published about +fifty years ago, it has only recently been mentioned +in text-books and general treatises. Indeed, +it has been stated that Bond was the first +in modern times to suggest the meteoric constitution +of the rings. His suggestion, based on +telescopic evidence, was however made in~1851. + +\TB + +And now to explain how a Cambridge mathematician +to whom reference was made above, in +ignorance of Roche's work of nine years before, +\index{Roche, E.!theory of limit and Saturn's ring|)}% +arrived at the same conclusion. In~1857, Clerk-Maxwell, +one of the most brilliant men of science +who have taught in the University of Cambridge, +and whose early death we still deplore, attacked +\PageSep{363} +\index{Instability!of Saturn's ring}% +the problem of Saturn's rings in a celebrated +essay, which gained for him what is called the +Adams prize. Laplace had early in the century +considered the theory that the ring is solid, and +Maxwell first took up the question of the motion +\index{Maxwell on Saturn's ring|(}% +of such a solid ring at the point where it had +been left. He determined what amount of +weighting at one point of a solid uniform ring is +necessary to insure its steady motion round the +planet. He found that there must be a mass +attached to the circumference of the ring weighing +$4\frac{1}{2}$~times as much as the ring itself. In fact, +the system becomes a satellite with a light ring +attached to it. + +``As there is no appearance,'' he says, ``about +the rings justifying a belief in so great an irregularity, +the theory of the solidity of the rings +becomes very improbable. When we come to +consider the additional difficulty of the tendency +of the fluid or loose parts of the ring to accumulate +at the thicker parts, and thus to destroy that +nice adjustment of the load on which the stability +depends, we have another powerful argument +against solidity. And when we consider the immense +size of the rings and their comparative +thinness, the absurdity of treating them as rigid +bodies becomes self-evident. An iron ring of +such a size would be not only plastic, but semi-fluid, +under the forces which it would experience, +and we have no reason to believe these rings to +\PageSep{364} +\index{Instability!of Saturn's ring}% +be artificially strengthened with any material +unknown on this earth.'' + +The hypothesis of solidity being condemned, +Maxwell proceeds to suppose that the ring is +composed of a number of equal small satellites. +This is a step towards the hypothesis of an indefinite +number of meteorites of all sizes. The +consideration of the motion of these equal satellites +affords a problem of immense difficulty, for +each satellite is attracted by all the others and +by the planet, and they are all in motion. + +If they were arranged in a circle round the +planet at equal distances, they might continue to +revolve round the planet, provided that each +satellite remained in its place with mathematical +exactness. Let us consider that the proper place +of each satellite is at the ends of the spokes of +a revolving wheel, and then let us suppose that +none of them is exactly in its place, some being +a little too far advanced, some a little behind, +some too near and some too far from the centre +of the wheel---that is to say, from the planet---then +we want to know whether they will swing +to and fro in the neighborhood of their places, +or will get further and further from their places, +and whether the ring will end in confusion. + +Maxwell treated this problem with consummate +skill, and showed that if the satellites were +not too large, confusion would not ensue, but +each satellite would oscillate about its proper +place. +\PageSep{365} +\index{Stability!of Saturn's ring}% + +At any moment there are places where the +satellites are crowded and others where they are +spaced out, and he showed that the places of +crowding and of spacing out will travel round +the ring at a different speed from that with +which the ring as a whole revolves. In other +words, waves of condensation and of rarefaction +are propagated round the ring as it rotates. + +He constructed a model, now in the laboratory +at Cambridge, to exhibit these movements; it is +pretty to observe the changes of the shape of the +ring and of the crowding of the model satellites +as they revolve. + +I cannot sum up the general conclusions at +which Maxwell arrived better than by quoting +his own words. + +In the summary of his paper he says:--- + +``If the satellites are unequal, the propagation +of waves will no longer be regular, but the disturbances +of the ring will in this, as in the +former case, produce only waves, and not growing +confusion. Supposing the ring to consist, +not of a single row of large-satellites, but of a +cloud of evenly distributed unconnected particles, +we found that such a cloud must have a +very small density in order to be permanent, and +that this is inconsistent with its outer and inner +parts moving with the same angular velocity. +Supposing the ring to be fluid and continuous, +we found that it will necessarily be broken up +into small portions. +\PageSep{366} +\index{Stability!of Saturn's ring}% + +``We conclude, therefore, that the rings must +consist of disconnected particles; these may be +either solid or liquid, but they must be independent. +The entire system of rings must therefore +consist either of a series of many concentric +rings, each moving with its own velocity, and +having its own system of waves, or else of a confused +multitude of revolving particles, not arranged +in rings, and continually coming into +collision with each other. + +``Taking the first case, we found that in an +indefinite number of possible cases the mutual +perturbation of two rings, stable in themselves, +might mount up in time to a destructive magnitude, +and that such cases must continually occur +in an extensive system like that of Saturn, the +only retarding cause being the possible irregularity +of the rings. + +``The result of long-continued disturbance +was found to be the spreading out of the rings +in breadth, the outer rings pressing outward, +while the inner rings press inward. + +``The final result, therefore, of the mechanical +theory is, that the only system of rings which +can exist is one composed of an indefinite number +of unconnected particles, revolving round the +planet with different velocities according to their +respective distances. These particles may be +arranged in a series of narrow rings, or they may +move through each other irregularly. In the +\PageSep{367} +\index{Keeler, spectroscopic examination of Saturn's ring|(}% +first case the destruction of the system will be +very slow, in the second case it will be more +rapid, but there may be a tendency towards an +arrangement in narrow rings, which may retard +the process. + +``We are not able to ascertain by observation +the constitution of the two outer divisions of the +system of rings, but the inner ring is certainly +transparent, for the limb (\ie~edge) of Saturn +has been observed through it. It is also certain, +that though the space occupied by the ring is +transparent, it is not through the material particles +of it that Saturn was seen, for his limb was +observed without distortion; which shows that +there was no refraction, and therefore that the +rays did not pass through a medium at all, but +between the solid or liquid particles of which the +ring is composed. Here then we have an optical +argument in favor of the theory of independent +particles as the material of the rings. The +two outer rings may be of the same nature, but +not so exceedingly rare that a ray of light can +pass through their whole thickness without encountering +one of the particles.'' +\index{Maxwell on Saturn's ring|)}% + +\TB + +The last link in the chain of evidence has been +furnished by recent observations made in America. +If it can be proved that every part of the +apparently solid ring moves round the planet's +centre at a different rate, and that the speed at +\PageSep{368} +\index{Meteoric constitution of Saturn's ring}% +\index{Spectroscopic proof of rotation of Saturn's ring}% +each part is appropriate at its distance from the +centre, the conclusion is inevitable that the ring +consists of scattered fragments. + +Every one must have noticed that when a +train passes at full speed with the whistle blowing, +there is an abrupt fall in the pitch of the +note. This change of note is only apparent to +the stationary listener, and is caused by the +crowding together of the waves of sound as the +train approaches, and by their spacing out as it +recedes. The same thing is true of light-waves, +and if we could imagine a colored light to pass +us at an almost inconceivable velocity it would +change in tint as it passed.\footnote + {This statement is strictly correct only of monochromatic + light. I might, in the subsequent argument, have introduced + the limitation that the moving body shall emit only monochromatic + light. The qualification would, however, only complicate + the statement, and thus render the displacement of the lines of + the spectrum less easily intelligible.} +Now there are certain +lines in the spectrum of sunlight, and the +shifting of their positions affords an excessively +delicate measure of a change which, when magnified +enormously, would produce a change of +tint. For example, the sun is a rotating body, +and when we look at its disk one edge is approaching +us and the other is receding. The +two edges are infinitesimally of different colors, +and the change of tint is measurable by the displacement +of the lines I have mentioned. In +the same way Saturn's ring is illuminated by +sunlight, and if different portions are moving at +\PageSep{369} +\index{Spectroscopic proof of rotation of Saturn's ring}% +different velocities, those portions are infinitesimally +of different colors. Now Professor Keeler, +the present director of the Lick Observatory, has +actually observed the reflected sunlight from the +several parts of Saturn's ring, and he finds that +the lines in the spectrum of the several parts +are differently displaced. From measurement of +these displacements he has concluded that every +part of the ring moves at the same pace as if it +were an independent satellite. The proof of the +meteoric constitution of the ring is therefore +\index{Meteoric constitution of Saturn's ring}% +complete. + +It would be hard to find in science a more +beautiful instance of arguments of the most +diverse natures concentrating themselves on a +definite and final conclusion. + +\begin{Authorities} +Édouard Roche, \Title{La figure d'une masse fluide soumise à l'attraction +\index{Roche, E.!stability of ellipsoid of}% +d'un point éloigné}, ``Mém.\ Acad.\ de~Montpelier,'' vol.~i.\ +(Sciences), 1847--50. + +Maxwell, \Title{Stability of Saturn's Rings}, Macmillan, 1859. + +Keeler, \Title{Spectroscopic Proof of the Meteoric Constitution of +Saturn's Rings}, ``Astrophysical Journal,'' May, 1895; see also +\index{Keeler, spectroscopic examination of Saturn's ring|)}% +\index{Saturn!theory of ring|)}% +the same for June, 1895. + +Schwarzschild, \Title{Die Poincarésche Theorie des Gleichgewichts}, +\index{Schwarzschild!stability of Roche's ellipsoid}% +``Annals of Munich Observatory,'' vol.~iii.\ 1896. He considers +the stability of Roche's ellipsoid. +\end{Authorities} +\PageSep{370} +%[Blank Page] +\PageSep{371} +\BackMatter +\printindex +\iffalse +INDEX + +Abacus for reducing tidal observations#abacus, 217-220. + +Abbadie, tidal deflection of vertical#Abbadie, 143, 144. + +Aden, errors of tidal prediction at#Aden, 246. + +Adriatic, tide in#Adriatic, 186. + +Airy, Sir G. B.#Airy, + tides in rivers, 75; + attack on Laplace, 181; + cotidal chart, 188; + \Title{Tides and Waves}, 192. + +America, North, tide tables for#America, 222. + +Analysis, harmonic, of tide#analysis, 193-210. + +Andromeda, nebula in#Andromeda, 339. + +Annual and semi-annual tides#annual, 206. + +Arabian theories of tide#Arab, 77-79. + +Aristotle on tides#Aristotle, 81. + +Assyrian records of eclipses, 272. + +Atlantic, tide in#Atlantic, 186-188. + +Atmospheric pressure, + cause of seiches, 40; + distortion of soil by, 145, 146; + influence on tidal prediction, 242, 243. + +Atmospheric waves, Helmholtz on#Helmholtz, 48-51. + +Attraction, + of weight resting on elastic slab proportional to slope, 136, 137; + of tide calculated, 143. + +Baird, \Title{Manual for Tidal Observation}#Baird, 16. + +Bakhuyzen on tide due to variation of latitude#Bakhuyzen, 255, 256. + +Barnard, rotation of Jupiter's satellites#Barnard, 315. + +Barometric pressure. |see{Atmospheric pressure}. 0 + +Becker, G. F., on Nebular Hypothesis#Becker, 334, 336-338. + +Bernoulli, Daniel, essay on tides#Bernoulli, 86, 88. + +Bertelli on Italian seismology#Bertelli, 126, 127. + +Bifilar. |see{Pendulum}. 0 + +Borgen@Börgen, method of reducing tidal observations#Börgen, 217. + +Bond, discovery of inner ring of Saturn#Bond, 352. + +Bore, + definition, 59; + bore-shelter, 63; + diagram of rise in Tsien-Tang, 66; + pictures, 67; + rivers where found, 71; + causes, 72; + Chinese superstition, 68-70. + +Browne, E. G., Arabian theories of tide#Browne, 77-79. + +Cambridge, experiments with bifilar pendulum at#bifilar, 115-125. + +Canal, + theory of tide wave in, 165-167; + critical depth, 163-165; + tides in ocean partitioned into canals, 175; + canal in high latitude, 174-176. + +Capillarity of liquids, and Plateau's experiment#Plateau, 316-318. + +Cassini, discovery of division in Saturn's rings#Cassini, 352. + +Castel, Father, ridiculed by Voltaire#Castel, 295, 296. + +Cavalleri, essay on tides#Cavalleri, 86. + +Centripetal and centrifugal forces#Centripetal, 91-93. + +Chambers on possible existence of Martian satellites#Chambers, 296. +\PageSep{372} + +Chandler, free nutation of earth, and variation of latitude#Chandler, 253-257. + +Chinese + superstition as to bore, 68-70; + theories of tide, 76, 77. + +Christie, A. S., tide due to variation of latitude#Christie, 255, 256. + +Constants, tidal, explained#constants, 195. + +Continents, trend of, possibly due to primeval tidal friction#continents, 308. + +Cotidal chart, 188; + for diurnal tide hitherto undetermined, 191, 192. + +Currents, tidal, in rivers#tidal current, 56. + +Curve, tide#tide curve, + irregularities in, 10-16; + at Bombay, 12; + partitioned into lunar time, 213. + +D'Abbadie. |see{Abbadie}. 0 + +Darwin, G. H.#Darwin, + bifilar pendulum, 115-125; + harmonic analysis, 210; + tidal abacus, 217-220; + distortion of earth's surface by varying loads, 134-148; + rigidity of earth, 261, 262; + papers on tidal friction, 315; + hour-glass figure of rotating liquid, 328-332; + Jacobi's ellipsoid, 333; + evolution of satellites, 346. + +Darwin, Horace, bifilar pendulum#Horace, 115-125. + +Davis, method of presenting tide-generating force#Davis, 96, 97. + +Davison, history of bifilar and horizontal pendulums#Davison, 133. + +Dawes, discovery of inner ring of Saturn#Dawes, 352. + +Dawson coöperates in investigation of seiches#Dawson, 48. + +Day, change in length of, under tidal friction#day, 275, 276. + +Deflection of the vertical, 109-133; + experiments to measure, 115-125; + due to tide, 134-143. + +Deimos, a satellite of Mars#Deimos, 297. + +Denison, F. Napier, vibrations and seiches on lakes#Denison, 48-53. + +Density + of earth, law of internal, 302; + of planets determinable from their figures, 332, 333. + +Descartes, vortical theory of cosmogony#Descartes, 335. + +Dick, argument as to Martian satellites#Dick, 295. + +Dimple, + in soil, due to weight, 123; + form of, in elastic slab, 135. + +Distortion of soil + by weight, 123; + by varying loads, 134-148. + +Diurnal inequality + observed by Seleucus, 84, 85; + according to equilibrium theory, 156; + in Laplace's solution, 179; + in Atlantic, Pacific, and Indian Oceans, 180; + not shown in cotidal chart, 191; + in harmonic method, 205; + complicates prediction, 224, 225. + +Douglass, rotation of Jupiter's satellites#Douglass, 315. + +Dumb-bell nebula, description of photograph of#dumb-bell, 345. + +Dynamical theory of tide-wave, 163-181. + +Earth and moon#Earth, + diagram, 93; + rotation of, effects on tides, 177; + rigidity of, 256-260; + rotation retarded by tidal friction, 268; + figure of, 299; + adjustment of figure to suit change of rotation, 299-302; + internal density, 302; + probably once molten, 306; + distortion under primeval tidal friction, 307; + Roche's limit for, 358. + +Earthquakes, + a cause of seiches, 39; + microsisms and earth tremors, 125-127; + shock perceptible at great distance, 261. + +Ebb and flow defined#ebb, 56. + +Eccentricity of orbit + due to tidal friction, 313, 314; + theory of, in case of double stars, 342. +\PageSep{373} + +Eclipses, ancient, and earth's rotation#eclipses, 272, 273. + +Ecliptic, obliquity of, due to tidal friction#ecliptic, 308-312. + +Eddies, tidal oscillation involves#eddies, 177. + +Ehlert, observation with horizontal pendulum#Ehlert, 132. + +Elastic distortion#elastic + of soil by weight, 123; + of earth by varying loads, 134-148; + calculation and illustration, 138-140; + by atmospheric pressure, 145-147. + +Elasticity of earth#elasticity, 254, 255. + +Elliptic tide, lunar#elliptic tide, 204. + +Ellipticity of earth's strata in excess for present rotation#ellipticity, 303, 304. + +Energy, tidal, utilization of#tidal energy, 73, 74. + +Equatorial canal, tide wave in#canal, 173. + +Equilibrium, figures of, of rotating liquid#equilibrium, 316-333. + +Equilibrium theory of tides#equilibrium, 149-162; + chart and law of tide, 151-153; + defects of, 160. + +Errors in tidal prediction#errors, 243-245. + +Establishment of port, + definition, 161, 162; + zero in equilibrium theory, 161; + shown in cotidal chart, 189. + +Estuary, annual meteorological tide in#estuary, 207, 208. + +Euler, essay on tides#Euler, 86. + +Europe, tides on coasts of#Europe, 188. + +Evolution of celestial systems, 334-346. + +Ferrel, tide-predicting instrument#Ferrel, 241. + +Figure of equilibrium + of ocean under tidal forces, 151-153; + of rotating liquid, 316-333. + +Figure of planets and their density, 332, 333. + +Fisher, Osmond, on molten interior of earth#Fisher, 262. + +Flow and ebb defined#ebb, 56. + +Forced oscillation, + principle of, 169, 170; + due to solar tide, possibly related to birth of moon, 282-284. + +Forced wave, explanation and contrast with free wave#forced wave, 164. + +Forces, + centripetal and centrifugal, 91-93; + tide-generating, 93-108; + numerical estimate, 109-111; + deflection of vertical by, 109-133; + figure of equilibrium under tidal, 151-153; + those of sun and moon compared, 156-158. + +Forel + on seiches, 17-38; + list of papers, 53, 54. + +Free oscillation contrasted with forced#free oscillation, 169, 170. + +Free wave, explanation and contrast with forced#free wave, 164. + +Friction of tides, 264-315. + +Galileo, + blames Kepler for his tidal theory, 85; + discovery of Jupiter's satellites, 291; + Saturn's ring, 350. + +Gauge, tide#tide gauge, + description of, 6-11; + site for, 14. + +Geneva, + seiches in lake, 17-28; + model of lake, 28. + +Geological evidence of earth's plasticity#plasticity, 300; + as to retardation of earth's rotation, 304-306. + +German method of reducing tidal observations#German, 217. + +Giles on Chinese theories of the tide#Giles, 76, 77. + +Gravity, variation according to latitude#gravity, 302, 303, 332. + +Greek + theory and description of tides, 81-85; + records of ancient eclipses, 272. + +Gulliver@\Title{Gulliver's Travels}, satire on mathematics#Gulliver, 292-295. + +Hall, Asaph, discovery of Martian satellites#Hall, 290-298. + +Hangchow, the bore at#Hangchow, 60-70. +\PageSep{374} + +Harmonic analysis + initiated by Lord Kelvin, 87; + account of, 193-210. + +Height of tide#height + due to ideal satellite, 198; + at Portsmouth and at Aden, 225; + reduced by elastic yielding of earth, 259. + +Helmholtz + on atmospheric waves, 48-51; + on rotation of the moon, 286. + +Herschel, observations of twin nebulæ#Herschel, 344. + +High water + under moon in equilibrium theory, 160; + position in shallow and deep canals in dynamical theory, 171, 172. + +History + of tidal theories, 76-88; + of earth and moon, 278-286, 308-313. + +Hopkins on rigidity of earth#Hopkins, 258, 259. + +Horizontal tide-generating force, 107. + +Horizontal tide-generating force |see{also Pendulum}. 0 + +Hough, S. S.#Hough, + frictional extinction of waves, 47; + dynamical solution of tidal problem, 181; + rigidity of earth, 254; + Chandler's nutation, 262. + +Hugli, bore on the#Hugli, 71. + +Huyghens, discovery of Saturn's ring#Huyghens, 351. + +Icelandic theory of tides, 79, 80. + +Indian Survey, + method of reducing tidal observations, 216, 217; + tide tables, 222. + +Instability, + nature of dynamical, and initial of moon's motion, 280-282; + of Saturn's ring, 363, 364. + +Interval from moon's transit to high water + in case of ideal satellite, 198; + at Portsmouth and at Aden, 225. + +Italian investigations in seismology, 125-130. + +Jacobi, figure of equilibrium of rotating liquid#Jacobi, 322-324. + +Japan, frequency of earthquakes#Japan, 130, 131. + +Jupiter, + satellites constantly face planet, 298; + figure and law of internal density, 333; + Roche's limit for, 361. + +Kant, + rotation of moon, 286; + nebular hypothesis, 334-339. + +Keeler, spectroscopic examination of Saturn's ring#Keeler, 367-369. + +Kelvin, Lord#Kelvin, + initiates harmonic analysis, 87, 199; + calculation of tidal attraction, 143; + tide predicting machine, 233; + rigidity of earth, 257-260; + denies adjustment of earth's figure to changed rotation, 301; + on geological time, 315. + +Kepler, + ideas concerning tides, 85, 86; + argument respecting Martian satellites, 291, 292. + +Krüger, figures of equilibrium of liquid#Krüger, 333. + +Lakes, + seiches in, 17-54; + mode of rocking in seiches, 24, 25; + vibrations, 41-53; + tides in, 182-185. + +Lamb, H., presentation of Laplace's theory#Lamb, 181. + +Laplace, + theory of tides, 86-88, 177-180; + on rotation of moon, 286, 287; + nebular hypothesis, 335-337. + +Lardner, possibility of Martian satellites#Lardner, 295. + +Latitude, + tidal wave in canal in high, 174-176; + periodic variations of, 251-256. + +Lege@Légé, constructor of tide-predicting machine#Légé, 233. + +Level of sea affected by atmospheric pressure#atmospheric pressure, 146. + +Limnimeter, a form of tide gauge#limnimeter, 24. + +Lowell, P., on rotations of Venus and Mercury#Lowell, 298, 299, 315. + +Low water. |see{High water}. +\PageSep{375} + +Lubbock, Sir J., senior, on tides#Lubbock, 87. + +Lunar + tide-generating force compared with solar, 156-158; + tide, principal, 201; + elliptic tide, 204; + time, 213. + +Machine, tide-predicting#tide-predicting, 233, 241. + +Mackerel sky, evidence of air-waves#mackerel, 49. + +Maclaurin, + essay on tides, 86; + figure of equilibrium of rotating liquid, 322-324. + +Magnússon on Icelandic theories of tides#Magnússon, 79, 80. + +Marco Polo, resident of Hangchow#Marco, 70. + +Mars, + discovery of satellites, 290-298; + Roche's limit, 360. + +Maxwell on Saturn's ring#Maxwell, 363-367. + +Mediterranean Sea, tides in#Mediterranean, 185, 186. + +Mercury, rotation of#Mercury, 298, 299. + +Meteoric constitution of Saturn's ring#meteoric, 368, 369. + +Meteorological + tides, 206, 207; + conditions dependent on earth's rotation, 303. + +Microphone as a seismological instrument#microphone, 128-130. + +Microsisms, minute earthquakes#microsisms, 125-127. + +Mills worked by the tide#mills, 74, 75. + +Milne on seismology#Milne, 125, 130. + +Month, change in, under tidal friction#month, 275-277. + +Moon and earth, + diagram, 93; + tide-generating force compared with sun's, 156-158; + tide due to ideal, moving in equator, 193, 194; + ideal satellites replacing actual, 199, 200; + tidal prediction by reference to transit, 224-230; + retardation of motion by tidal friction, 269, 270; + origin of, 282, 283; + rotation annulled by tidal friction and present libration, 286; + inequality in motion indicates internal density of earth, 302, 303; + eccentricity of orbit increased by tidal friction, 313, 314. + +Moore, Captain#Moore, + illustrations of bore, 67; + survey of Tsien-Tang-Kiang, 60-70. + +Neap and spring tides#neap + in equilibrium theory, 159; + represented by principal lunar and solar tides, 204. + +Nebula in Andromeda#Andromeda, 339. + +Nebulae@Nebulæ, description of various#Nebulæ, 345. + +Nebular hypothesis, 334-339. + +Newcomb, S., theoretical explanation of Chandler's nutation#Newcomb, 254. + +Newton, + founder of tidal theory, 86; + theory of tide in equatorial canal, 172. + +Nolan, criticism of tidal theory of moon's origin#Nolan, 360. + +Nutation, + value of, indicates internal density of earth, 303; + Chandler's, 251-256. + +Obliquity of ecliptic, effects of tidal friction on#obliquity, 310-312. + +Observation, + methods of tidal, 6-14; + reduction of tidal, 211-220. + +Orbit + of moon and earth, 93-95; + of double stars, very eccentric, 313. + +Pacific Ocean, tide in, affects Atlantic#Pacific, 186, 187. + +Partial tides in harmonic method#partial tides, 199. + +Paschwitz, von Rebeur#Paschwitz, + on horizontal pendulum, 130-132; + tidal deflection of vertical at Wilhelmshaven, 144. + +Pendulum, + curves traced by, under tidal force, 111, 112; + bifilar, 115-125; + as seismological instrument, 126, 127; + horizontal, 130-132. +\PageSep{376} + +Petitcodiac, bore in the#Petitcodiac, 71. + +Phobos, a satellite of Mars#Phobos, 297. + +Planetary figure of equilibrium of rotating liquid#planetary figure, 322. + +Planets, + rotation of some, annulled by tidal friction, 298; + figures and internal densities, 332, 333. + +Plasticity of earth under change of rotation#plasticity, 300-302. + +Plateau, experiment on figure of rotating globule#Plateau, 316-319. + +Plemyrameter, observation of seiches with#plemyrameter, 19-22. + +Poincaré, + law of interchange of stability, 326, 327; + figure of rotating liquid, 325, 327. + +Polibius on tides at Cadiz#Polibus, 83. + +Portsmouth, table of errors in tidal predictions#Portsmouth, 244. + +Posidonius on tides#Posidonius, 81-84. + +Precession, value of, indicates internal density of earth#internal density, 303. + +Predicting machine for tides, 233-241; + Ferrel's, 241. + +Prediction of tide, + due to ideal satellite, 200; + example at Aden, 226-230; + method of computing, 230-233; + errors in, 242-250. + +Pressure of atmosphere, elastic distortion of soil by#distortion, 145, 146. + +Principle of forced oscillations#forced oscillations, 169, 170. + +Rebeur. |see{Paschwitz}. 0 + +Reduction of tidal observations, 211-220. + +Retardation of earth's rotation, 268. + +Rigidity of earth#rigidity, 256-260. + +Ripple mark in sand preserved in geological strata#ripple mark, 305. + +Rivers, + tide wave in, 55-59; + Airy on tide in, 75; + annual meteorological tide in, 206. + +Roberts, E., the tide-predicting machine#Roberts, 233. + +Roberts, I., photograph of nebula in Andromeda#Roberts, 339. + +Roche, E.#Roche, + ellipticity of internal strata of earth, 303; + theory of limit and Saturn's ring, 356-362; + stability of ellipsoid of, 369. + +Roman description of tides#Roman, 81-85. + +Rossi on Italian seismology#Rossi, 128-130. + +Rotating liquid, figures of equilibrium#rotating liquid, 316-333. + +Rotation + of earth involved in tidal problem, 177; + retarded by tidal friction, 268; + of moon annulled by tidal friction, 286; + of Mercury, Venus, and satellites of Jupiter and Saturn annulled by tidal friction, 298. + +Russell, observation of seiches in New South Wales#Russell, 47. + +Saint@St.\ Vénant on flow of solids#Vénant, 313. + +Satellites, + tide due to single equatorial, 195, 196; + ideal replacing sun and moon in harmonic analysis, 199, 200; + discovery of those of Mars, 290-298; + rotation of those of Jupiter and Saturn annulled, 298; + distribution of, in solar system, 339-341. + +Saturn, + satellites always face the planet, 298; + law of density and figure, 332; + description and picture, 347-354; + theory of ring, 356-369; + Roche's limit for, 360. + +Schedule for reducing tidal observations, 215, 216. + +Schiaparelli on rotation of Venus and Mercury#Schiaparelli, 298, 315. + +Schwarzschild, + exposition of Poincaré's theory, 333; + stability of Roche's ellipsoid, 369. + +Sea, + vibrations of, 44, 45; + level affected by atmospheric pressure, 146. +\PageSep{377} + +See, T. J. J., + eccentricity of orbits of double stars, 313; + theory of evolution of double stars, 342-346. + +Seiches, + definition, 17; + records of, 21; + longitudinal and transverse, 25-27; + periods of, 27; + causes of, 39, 40. + +Seine, bore in the#Seine, 71. + +Seismology, 133. + +Seleucus, observation of tides of Indian Ocean#Seleucus, 84, 85. + +Semidiurnal tide + in equilibrium theory, 153-156; + in harmonic method, 201-204. + +Severn, bore in the#Severn, 71. + +Slope of soil + due to elastic distortion, 136; + calculation and illustration of, 138-140. + +Solar + tide-generating force compared with lunar, 156-158; + principal tide, 202; + possible effect of tide in assisting birth of moon, 284, 285; + system, nebular hypothesis as to origin of, 334-339; + system, distribution of satellites in, 339-341. + +Spectroscopic proof of rotation of Saturn's ring#spectroscop, 368, 369. + +Spring and neap tides + in equilibrium theory, 159; + represented by principal lunar and solar tides, 203. + +Stability, + nature of dynamical, 280, 281; + of figures of equilibrium, 322, 323; + of Saturn's ring, 365, 366. + +Stars, + double, eccentricity of orbits, 313; + theory of evolution, 342-346. + +Storms a cause of seiches#storms, 39, 40. + +Strabo on tides#Strabo, 81-85. + +Stupart coöperates in investigation of seiches#Stupart, 48. + +Sun, + tide-generating force of, compared with that of moon, 156-158; + ideal, replacing real sun in harmonic analysis, 201; + possible influence of, in assisting birth of moon, 284, 285. + +Surface tension of liquids#surface tension, 317, 318. + +Swift, satire on mathematicians#Swift, 292-295. + +Synthesis of partial tides for prediction#synthesis, 230-233. + +Tables, tide#tide tables, 221-241; + method of calculating, 230-241; + amount of error in, 246, 247. + +Thomson, Sir W. |see{Kelvin}. 0 + +Tidal problem. |see{Laplace, Harmonic Analysis, etc.} 0 + +Tide, + definition, 1-3; + general description, 4-6. + +Tide, |see{also other headings; \eg\ for tide-generating force, |see{Force}}. 0 + +Time, + lunar, 213; + requisite for evolution of moon, 285. + +Tisserand, Roche's investigations as to earth's figure#Tisserand, 315. + +Tremors, earth#tremors, 125. + +Tresca on flow of solids#Tresca, 300. + +Tromometer, a seismological instrument#tromometer, 126, 127. + +Tsien-Tang-Kiang, the bore in#Tsien, 60-70. + +United States Coast Survey, + method of reducing tidal observations, 217; + tide tables of, 222. + +Variation of latitude, 251-256. + +Vaucher, record of a great seiche at Geneva#Vaucher, 17. + +Venus, rotation of#Venus, 298, 299. + +Vertical. |see{Deflection}. 0 + +Vibration of lakes, 41-53. + +Voltaire, satire on mathematicians, and Martian satellites#Voltaire, 295, 296. + +Vortical motion in oceanic tides#vorticial, 177, 178. +\PageSep{378} + +Waves + in deep and shallow water, 29; + speed of, 31; + composition of, 33-37; + in atmosphere, 48-50; + forced and free, 164; + of tide in equatorial canal, 173; + in canal in high latitude, 174-176; + propagated northward in Atlantic, 186-188. + +Wharton, Sir W. J., illustration of bore#Wharton, 69. + +Whewell + on tides, 87; + empirical construction of tide tables, 87-90; + on cotidal charts, 188, 189. + +Wind, + a cause of seiches, 39; + vibrations of lakes due to, 41, 42; + a cause of meteorological tides, 206; + perturbation of, in tidal prediction, 242, 243. + +Woodward on variation of latitude#Woodward, 262. + +Wright, Thomas, on a theory of cosmogony#Wright, 335. + +Wye, bore in the#Wye, 71. +\fi +\PageSep{379} +\clearpage +\null\vfill +\begin{center} +\textgoth{The Riverside Press} + +\footnotesize +CAMBRIDGE, MASSACHUSETTS, U. S. A. \\ +ELECTROTYPED AND PRINTED BY \\ +H. O. 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determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..c34386f --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #38722 (https://www.gutenberg.org/ebooks/38722) |
