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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% The Project Gutenberg EBook of General Investigations of Curved Surfaces%
+% of 1827 and 1825, by Karl Friedrich Gauss                               %
+%                                                                         %
+% 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: General Investigations of Curved Surfaces of 1827 and 1825       %
+%                                                                         %
+% Author: Karl Friedrich Gauss                                            %
+%                                                                         %
+% Translator: James Caddall Morehead                                      %
+%             Adam Miller Hiltebeitel                                     %
+%                                                                         %
+% Release Date: July 25, 2011 [EBook #36856]                              %
+% Most recently updated: June 11, 2021                         %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: UTF-8                                           %
+%                                                                         %
+% *** START OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
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+\small
+\begin{PGtext}
+The Project Gutenberg EBook of General Investigations of Curved Surfaces
+of 1827 and 1825, by Karl Friedrich Gauss
+
+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: General Investigations of Curved Surfaces of 1827 and 1825
+
+Author: Karl Friedrich Gauss
+
+Translator: James Caddall Morehead
+            Adam Miller Hiltebeitel
+
+Release Date: July 25, 2011 [EBook #36856]
+Most recently updated: June 11, 2021
+
+Language: English
+
+Character set encoding: UTF-8     
+
+*** START OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
+\end{PGtext}
+\end{minipage}
+\end{center}
+\newpage
+%% Credits and transcriber's note %%
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+Produced by Andrew D. Hwang, with special thanks to Brenda Lewis.
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+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
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+\FrontMatter
+\begin{center}
+{\LARGE\scshape Karl Friedrich Gauss} \\
+\tb
+\bigskip
+
+\LARGE\scshape General Investigations \\
+{\footnotesize OF} \\
+Curved Surfaces \\
+{\footnotesize OF} \\
+\large 1827 and 1825
+\vfill
+
+\normalsize
+TRANSLATED WITH NOTES \\
+{\scriptsize AND A} \\[4pt]
+BIBLIOGRAPHY \\[4pt]
+{\scriptsize BY \\[4pt]
+JAMES CADDALL MOREHEAD, A.M., M.S., and ADAM MILLER HILTEBEITEL, A.M. \\[4pt]
+J. S. K. FELLOWS IN MATHEMATICS IN PRINCETON UNIVERSITY}
+\vfill\vfill
+
+THE PRINCETON UNIVERSITY LIBRARY \\
+1902
+\end{center}
+\newpage
+\PageSep{ii}
+\null\vfill
+\begin{center}
+\scriptsize
+Copyright, 1902, by \\
+\textsc{The Princeton University Library}
+\vfill
+\textit{C. S. Robinson \&~Co., University Press \\
+Princeton, N. J.}
+\end{center}
+\PageSep{iii}
+
+
+\Introduction
+
+In 1827 Gauss presented to the Royal Society of Göttingen his important paper on
+the theory of surfaces, which seventy-three years afterward the eminent French
+geometer, who has done more than any one else to propagate these principles, characterizes
+as one of Gauss's chief titles to fame, and as still the most finished and useful
+introduction to the study of infinitesimal geometry.\footnote
+  {G. Darboux, Bulletin des Sciences Math. Ser.~2, vol.~24, page~278, 1900.}
+This memoir may be called:
+\Title{General Investigations of Curved Surfaces}, or the Paper of~1827, to distinguish it
+from the original draft written out in~1825, but not published until~1900. A list of
+the editions and translations of the Paper of~1827 follows. There are three editions
+in Latin, two translations into French, and two into German. The paper was originally
+published in Latin under the title:
+
+I\textit{a}. \Publication{Disquisitiones generales circa superficies curvas}
+{auctore Carolo Friderico Gauss\Add{.}}
+{Societati regiæ oblatæ D.~8.~Octob.~1827}, \\
+and was printed in: Commentationes societatis regiæ scientiarum Gottingensis recentiores,
+Commentationes classis mathematicæ. Tom.~VI. (ad~a.\ 1823--1827). Gottingæ,
+1828, pages~99--146. This sixth volume is rare; so much so, indeed, that the British
+Museum Catalogue indicates that it is missing in that collection. With the signatures
+changed, and the paging changed to pages~1--50, I\textit{a}~also appears with the title page
+added:
+
+I\textit{b}. \Publication{Disquisitiones generales circa superficies curvas}
+{auctore Carolo Friderico Gauss.}
+{Gottingæ. Typis Dieterichianis. 1828.}
+
+II\@. In Monge's \Title{Application de l'analyse à la géométrie}, fifth edition, edited by
+Liouville, Paris, 1850, on pages 505--546, is a reprint, added by the Editor, in Latin
+under the title: \Title{Recherches sur la théorie générale des surfaces courbes}; Par M.
+C.-F. Gauss.
+\PageSep{iv}
+
+III\textit{a}. A third Latin edition of this paper stands in: Gauss, \Title{Werke, Herausgegeben
+von der Königlichen Gesellschaft der Wissenschaften zu Göttingen}, Vol.~4, Göttingen,
+1873, pages~217--258, without change of the title of the original paper~(I\textit{a}).
+
+III\textit{b}. The same, without change, in Vol.~4 of Gauss, \Title{Werke}, Zweiter Abdruck,
+Göttingen,~1880.
+
+IV\@. A French translation was made from Liouville's edition,~II, by Captain
+Tiburce Abadie, ancien élève de l'École Polytechnique, and appears in Nouvelles
+Annales de Mathématique, Vol.~11, Paris,~1852, pages~195--252, under the title:
+\Title{Recherches générales sur les surfaces courbes}; Par M.~Gauss. This latter also
+appears under its own title.
+
+V\textit{a}. Another French translation is: \Title{Recherches Générales sur les Surfaces
+Courbes}. Par M. C.-F. Gauss, traduites en français, suivies de notes et d'études
+sur divers points de la Théorie des Surfaces et sur certaines classes de Courbes, par
+M. E. Roger, Paris, 1855.
+
+V\textit{b}. The same. Deuxième Edition. Grenoble (or Paris), 1870 (or 1871), 160~pages.
+
+VI\@. A German translation is the first portion of the second part, namely, pages
+198--232, of: Otto Böklen, \Title{Analytische Geometrie des Raumes}, Zweite Auflage, Stuttgart,
+1884, under the title (on page~198): \Title{Untersuchungen über die allgemeine Theorie
+der krummen Flächen}. Von C.~F. Gauss. On the title page of the book the second
+part stands as: \Title{Disquisitiones generales circa superficies curvas} von C.~F. Gauss, ins
+Deutsche übertragen mit Anwendungen und Zusätzen\dots.
+
+VII\textit{a}. A second German translation is No.~5 of Ostwald's Klassiker der exacten
+Wissenschaften: \Title{Allgemeine Flächentheorie} (\Title{Disquisitiones generales circa superficies
+curvas}) von Carl Friedrich Gauss, (1827). Deutsch herausgegeben von A.~Wangerin.
+Leipzig, 1889. 62~pages.
+
+VII\textit{b}. The same. Zweite revidirte Auflage. Leipzig, 1900. 64~pages.
+
+The English translation of the Paper of~1827 here given is from a copy of the
+original paper,~I\textit{a}; but in the preparation of the translation and the notes all the
+other editions, except~V\textit{a}, were at hand, and were used. The excellent edition of
+Professor Wangerin,~VII, has been used throughout most freely for the text and
+notes, even when special notice of this is not made. It has been the endeavor of
+the translators to retain as far as possible the notation, the form and punctuation of
+the formulæ, and the general style of the original papers. Some changes have been
+made in order to conform to more recent notations, and the most important of those
+are mentioned in the notes.
+\PageSep{v}
+
+%[** TN: Paragraph not indented in the original]
+The second paper, the translation of which is here given, is the abstract (Anzeige)
+which Gauss presented in German to the Royal Society of Göttingen, and which was
+published in the Göttingische gelehrte Anzeigen. Stück~177. Pages 1761--1768. 1827.
+November~5. It has been translated into English from pages 341--347 of the fourth
+volume of Gauss's Works. This abstract is in the nature of a note on the Paper of~1827,
+and is printed before the notes on that paper.
+
+Recently the eighth volume of Gauss's Works has appeared. This contains on
+pages 408--442 the paper which Gauss wrote out, but did not publish, in~1825. This
+paper may be called the \Title{New General Investigations of Curved Surfaces}, or the Paper
+of~1825, to distinguish it from the Paper of~1827. The Paper of~1825 shows the
+manner in which many of the ideas were evolved, and while incomplete and in some
+cases inconsistent, nevertheless, when taken in connection with the Paper of~1827,
+shows the development of these ideas in the mind of Gauss. In both papers are
+found the method of the spherical representation, and, as types, the three important
+theorems: The measure of curvature is equal to the product of the reciprocals of the
+principal radii of curvature of the surface, The measure of curvature remains unchanged
+by a mere bending of the surface, The excess of the sum of the angles of a geodesic
+triangle is measured by the area of the corresponding triangle on the auxiliary sphere.
+But in the Paper of~1825 the first six sections, more than one-fifth of the whole paper,
+take up the consideration of theorems on curvature in a plane, as an introduction,
+before the ideas are used in space; whereas the Paper of~1827 takes up these ideas
+for space only. Moreover, while Gauss introduces the geodesic polar coordinates in
+the Paper of~1825, in the Paper of~1827 he uses the general coordinates, $p$,~$q$, thus
+introducing a new method, as well as employing the principles used by Monge and
+others.
+
+The publication of this translation has been made possible by the liberality of
+the Princeton Library Publishing Association and of the Alumni of the University
+who founded the Mathematical Seminary.
+
+\Signature{H. D. Thompson.}
+{Mathematical Seminary,}
+{Princeton University Library,}
+{January 29, 1902.}
+\PageSep{vi}
+%[Blank page]
+\PageSep{vii}
+
+
+\Contents
+
+\ToCLine{Gauss's Paper of 1827, General Investigations of Curved Surfaces}
+{paper:1827} % 1
+
+\ToCLine{Gauss's Abstract of the Paper of 1827}
+{abstract} % 45
+
+\ToCLine{Notes on the Paper of 1827}
+{notes:1827} % 51
+
+\ToCLine{Gauss's Paper of 1825, New General Investigations of Curved Surfaces}
+{paper:1825} % 79
+
+\ToCLine{Notes on the Paper of 1825}
+{notes:1825} % 111
+
+\ToCLine{Bibliography of the General Theory of Surfaces}
+{biblio} % 115
+\PageSep{viii}
+%[Blank page]
+\PageSep{1}
+\MainMatter
+\Paper{1827}
+%\thispagestyle{empty}
+\begin{center}
+\LARGE
+DISQUISITIONES GENERALES
+\vfil
+{\normalsize CIRCA}
+\vfil
+\LARGE
+SUPERFICIES CURVAS
+\vfil
+{\normalsize AUCTORE} \\[8pt]
+CAROLO FRIDERICO GAUSS
+\vfil
+
+\footnotesize
+SOCIETATI REGIAE OBLATAE D.~8.~OCTOB.~1827
+\vfil
+
+\tb \\
+\medskip
+
+COMMENTATIONES SOCIETATIS REGIAE SCIENTIARUM \\[4pt]
+GOTTINGENSIS RECENTIORES\@. VOL.~VI\@. GOTTINGAE MDCCCXXVIII \\
+\tb
+\vfil
+
+GOTTINGAE \\
+TYPIS DIETERICHIANIS \\
+MDCCCXXVIII
+\end{center}
+\cleardoublepage
+\PageSep{2}
+%[Blank page]
+\PageSep{3}
+
+
+\PaperTitle{\LARGE GENERAL INVESTIGATIONS \\
+{\small OF} \\
+CURVED SURFACES \\
+{\small BY} \\
+{\large KARL FRIEDRICH GAUSS} \\
+{\footnotesize PRESENTED TO THE ROYAL SOCIETY, OCTOBER~8, 1827}}
+
+
+\Article{1.}
+Investigations, in which the directions of various straight lines in space are to be
+considered, attain a high degree of clearness and simplicity if we employ, as an auxiliary,
+a sphere of unit radius described about an arbitrary centre, and suppose the
+different points of the sphere to represent the directions of straight lines parallel to
+the radii ending at these points. As the position of every point in space is determined
+by three coordinates, that is to say, the distances of the point from three mutually
+perpendicular fixed planes, it is necessary to consider, first of all, the directions of the
+axes perpendicular to these planes. The points on the sphere, which represent these
+directions, we shall denote by $(1)$,~$(2)$,~$(3)$. The distance of any one of these points
+from either of the other two will be a quadrant; and we shall suppose that the directions
+of the axes are those in which the corresponding coordinates increase.
+
+
+\Article{2.}
+It will be advantageous to bring together here some propositions which are frequently
+used in questions of this kind.
+
+\Par{I.} The angle between two intersecting straight lines is measured by the arc
+between the points on the sphere which correspond to the directions of the lines.
+
+\Par{II.}\Note{1} The orientation of any plane whatever can be represented by the great circle
+on the sphere, the plane of which is parallel to the given plane.
+\PageSep{4}
+
+\Par{III.} The angle between two planes is equal to the spherical angle between the
+great circles representing them, and, consequently, is also measured by the arc intercepted
+between the poles of these great circles. And, in like manner, the angle of inclination
+of a straight line to a plane is measured by the arc drawn from the point which
+corresponds to the direction of the line, perpendicular to the great circle which represents
+the orientation of the plane.
+
+\Par{IV.} Letting $x$,~$y$,~$z$; $x'$,~$y'$,~$z'$ denote the coordinates of two points, $r$~the distance
+between them, and $L$~the point on the sphere which represents the direction of the line
+drawn from the first point to the second, we shall have
+\begin{align*}
+x' &= x + r \cos(1)L\Add{,} \\
+y' &= y + r \cos(2)L\Add{,} \\
+z' &= z + r \cos(3)L\Add{.}
+\end{align*}
+
+\Par{V.} From this it follows at once that, generally,\Note{2}
+\[
+\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1\Add{,}
+\]
+and also, if $L'$~denote any other point on the sphere,
+\[
+\cos(1)L·\cos(1)L' + \cos(2)L·\cos(2)L' + \cos(3)L·\cos(3)L'
+  = \cos LL'.
+\]
+
+\Par{VI.}\Note{4} \begin{Theorem}
+If $L$, $L'$, $L''$, $L'''$ denote four points on the sphere, and $A$~the angle
+which the arcs $LL'$, $L''L'''$ make at their point of intersection, then we shall have
+\[
+\cos LL''·\cos L'L''' - \cos LL'''·\cos L'L''
+  = \sin LL'·\sin L''L'''·\cos A\Add{.}
+\]
+\end{Theorem}
+
+\textit{Demonstration.} Let $A$ denote also the point of intersection itself, and set
+\[
+AL = t,\quad AL' = t',\quad AL'' = t'',\quad AL''' = t'''\Add{.}
+\]
+Then we shall have
+\begin{alignat*}{7}
+&\cos L L''  &&= \cos t\Chg{·}{}&& \cos t''  &&+ \sin t  && \sin t''  && \cos A\Add{,} \\
+&\cos L'L''' &&= \cos t' && \cos t''' &&+ \sin t' && \sin t''' && \cos A\Add{,} \\
+&\cos L L''' &&= \cos t  && \cos t''' &&+ \sin t  && \sin t''' && \cos A\Add{,} \\
+&\cos L'L''  &&= \cos t' && \cos t''  &&+ \sin t' && \sin t''  && \cos A\Add{;}
+\end{alignat*}
+and consequently,\Note{3}
+\begin{multline*}
+\cos LL''·\cos L'L''' - \cos LL'''·\cos L'L'' \\
+\begin{aligned}
+&= \cos A (\cos t  \cos t''  \sin t' \sin t'''
+         + \cos t' \cos t''' \sin t  \sin t'' \\
+&\qquad  - \cos t  \cos t''' \sin t' \sin t''
+         - \cos t' \cos t''  \sin t  \sin t''') \\
+&= \cos A (\cos t   \sin t'   - \sin t   \cos t')
+          (\cos t'' \sin t''' - \sin t'' \cos t''')\NoteMark \\
+&= \cos A·\sin (t' - t)·\sin (t''' - t'') \\
+&= \cos A·\sin LL'·\sin L''L'''\Add{.}
+\end{aligned}
+\end{multline*}
+\PageSep{5}
+
+But as there are for each great circle two branches going out from the point~$A$,
+these two branches form at this point two angles whose sum is~$180°$. But our analysis
+shows that those branches are to be taken whose directions are in the sense from the
+point $L$~to~$L'$, and from the point $L''$~to~$L'''$; and since great circles intersect in two
+points, it is clear that either of the two points can be chosen arbitrarily. Also, instead
+of the angle~$A$, we can take the arc between the poles of the great circles of which the
+arcs $LL'$,~$L''L'''$ are parts. But it is evident that those poles are to be chosen which
+are similarly placed with respect to these arcs; that is to say, when we go from $L$~to~$L'$
+and from $L''$~to~$L'''$, both of the two poles are to be on the right, or both on the left.
+
+\Par{VII.} Let $L$,~$L'$,~$L''$ be the three points on the sphere and set, for brevity,
+\begin{alignat*}{6}
+&\cos (1)L   &&= x,\quad&& \cos (2)L   &&= y,\quad&& \cos (3)L   &&= z\Add{,} \\
+&\cos (1)L'  &&= x',    && \cos (2)L'  &&= y',    && \cos (3)L'  &&= z'\Add{,} \\
+&\cos (1)L'' &&= x'',   && \cos (2)L'' &&= y'',   && \cos (3)L'' &&= z''\Add{;} \\
+\end{alignat*}
+and also
+\[
+x y' z'' + x' y'' z + x'' y z' - x y'' z' - x' y z'' - x'' y' z = \Delta\Add{.}
+\]
+Let $\lambda$~denote the pole of the great circle of which $LL'$~is a part, this pole being the one
+that is placed in the same position with respect to this arc as the point~$(1)$ is with
+respect to the arc~$(2)(3)$. Then we shall have, by the preceding theorem,
+\[
+y z' - y' z = \cos (1)\lambda·\sin (2)(3)·\sin LL',
+\]
+or, because $(2)(3) = 90°$,
+\begin{align*}
+y z' - y' z &= \cos (1)\lambda·\sin LL', \\
+\intertext{and similarly,}
+z x' - z' x &= \cos (2)\lambda·\sin LL'\Add{,} \\
+x y' - x' y &= \cos (3)\lambda·\sin LL'\Add{.}
+\end{align*}
+Multiplying these equations by $x''$,~$y''$,~$z''$ respectively, and adding, we obtain, by means
+of the second of the theorems deduced in~V,
+\[
+\Delta = \cos \lambda L''·\sin LL'\Add{.}
+\]
+Now there are three cases to be distinguished. \emph{First}, when $L''$~lies on the great circle
+of which the arc~$LL'$ is a part, we shall have $\lambda L'' = 90°$, and consequently, $\Delta = 0$.
+If $L''$~does not lie on that great circle, the \emph{second} case will be when $L''$~is on the same
+side as~$\lambda$; the \emph{third} case when they are on opposite sides. In the last two cases the
+points $L$,~$L'$,~$L''$ will form a spherical triangle, and in the second case these points will lie
+in the same order as the points $(1)$,~$(2)$,~$(3)$, and in the opposite order in the third case.
+\PageSep{6}
+Denoting the angles of this triangle simply by $L$,~$L'$,~$L''$ and the perpendicular drawn on
+the sphere from the point~$L''$ to the side~$LL'$ by~$p$, we shall have
+\[
+\sin p = \sin L·\sin LL'' = \sin L'·\sin L' L'',
+\]
+and
+\[
+\lambda L'' = 90° \mp p,
+\]
+the upper sign being taken for the second case, the lower for the third. From this
+it follows that
+\begin{align*}
+±\Delta &= \sin L·\sin LL'·\sin LL''
+   = \sin L'·\sin LL'·\sin L'L'' \\
+  &= \sin L''·\sin LL''·\sin L'L''\Add{.}
+\end{align*}
+Moreover, it is evident that the first case can be regarded as contained in the second or
+third, and it is easily seen that the expression~$±\Delta$ represents six times the volume of
+the pyramid formed by the points $L$,~$L'$,~$L''$ and the centre of the sphere. Whence,
+finally, it is clear that the expression~$±\frac{1}{6}\Delta$ expresses generally the volume of any
+pyramid contained between the origin of coordinates and the three points whose coordinates
+are $\Typo{z}{x}$,~$y$,~$z$; $x'$,~$y'$,~$z'$; $x''$,~$y''$,~$z''$.\Note{5}
+
+
+\Article{3.}
+
+A curved surface is said to possess continuous curvature at one of its points~$A$, if the
+directions of all the straight lines drawn from $A$ to points of the surface at an infinitely
+small distance from~$A$ are deflected infinitely little from one and the same plane passing
+through~$A$. This plane is said to \emph{touch} the surface at the point~$A$. If this condition is
+not satisfied for any point, the continuity of the curvature is here interrupted, as happens,
+for example, at the vertex of a cone. The following investigations will be restricted to
+such surfaces, or to such parts of surfaces, as have the continuity of their curvature
+nowhere interrupted. We shall only observe now that the methods used to determine
+the position of the tangent plane lose their meaning at singular points, in which the
+continuity of the curvature is interrupted, and must lead to indeterminate solutions.
+
+
+\Article{4.}
+
+The orientation of the tangent plane is most conveniently studied by means of the
+direction of the straight line normal to the plane at the point~$A$, which is also called the
+normal to the curved surface at the point~$A$. We shall represent the direction of this
+normal by the point~$L$ on the auxiliary sphere, and we shall set
+\[
+\cos (1)L = X,\quad \cos (2)L = Y,\quad \cos (3)L = Z;
+\]
+and denote the coordinates of the point~$A$ by $x$,~$y$,~$z$. Also let $x + dx$, $y + dy$, $z + dz$
+be the coordinates of another point~$A'$ on the curved surface; $ds$~its distance from~$A$,
+\PageSep{7}
+which is infinitely small; and finally, let $\lambda$ be the point on the sphere representing the
+direction of the element~$AA'$. Then we shall have
+\[
+dx = ds·\cos (1)\lambda,\quad
+dy = ds·\cos (2)\lambda,\quad
+dz = ds·\cos (3)\lambda
+\]
+and, since $\lambda L$~must be equal to~$90°$,
+\[
+X\cos (1)\lambda + Y\cos (2)\lambda + Z\cos (3)\lambda = 0\Add{.}
+\]
+By combining these equations we obtain
+\[
+X\, dx + Y\, dy + Z\, dz = 0.
+\]
+
+There are two general methods for defining the nature of a curved surface. The
+\emph{first} uses the equation between the coordinates $x$,~$y$,~$z$, which we may suppose reduced to
+the form $W = 0$, where $W$~will be a function of the indeterminates $x$,~$y$,~$z$. Let the complete
+differential of the function~$W$ be
+\[
+dW = P\, dx + Q\, dy + R\, dz
+\]
+and on the curved surface we shall have
+\[
+P\, dx + Q\, dy + R\, dz = 0\Add{,}
+\]
+and consequently,
+\[
+P \cos (1)\lambda + Q \cos (2)\lambda + R \cos (3)\lambda = 0\Add{.}
+\]
+Since this equation, as well as the one we have established above, must be true for the
+directions of all elements~$ds$ on the curved surface, we easily see that $X$,~$Y$,~$Z$ must be
+proportional to $P$,~$Q$,~$R$ respectively, and consequently, since\Note{6}
+\[
+X^{2} + Y^{2} + Z^{2} = 1,\NoteMark
+\]
+we shall have either
+\begin{align*}
+X &= \frac{P}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Y &= \frac{Q}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Z &= \frac{R}{\Sqrt{P^{2} + Q^{2} + R^{2}}} \\
+\intertext{or}
+X &= \frac{-P}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Y &= \frac{-Q}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Z &= \frac{-R}{\Sqrt{P^{2} + Q^{2} + R^{2}}}\Add{.}
+\end{align*}
+
+The \emph{second}\Note{7} method expresses the coordinates in the form of functions of two variables,
+$p$,~$q$. Suppose that differentiation of these functions gives
+\begin{alignat*}{2}
+dx &= a\, dp &&+ a'\, dq\Add{,} \\
+dy &= b\, dp &&+ b'\, dq\Add{,} \\
+dz &= c\, dp &&+ c'\, dq\Add{.}
+\end{alignat*}
+\PageSep{8}
+Substituting these values in the formula given above, we obtain
+\[
+(aX + bY + cZ)\, dp + (a'X + b'Y + c'Z)\, dq = 0\Add{.}
+\]
+Since this equation must hold independently of the values of the differentials $dp$,~$dq$,
+we evidently shall have
+\[
+aX + bY + cZ = 0,\quad a'X + b'Y + c'Z = 0\Add{.}
+\]
+From this we see that $X$,~$Y$,~$Z$ will be proportioned to the quantities
+\[
+bc' - cb',\quad ca' - ac',\quad ab' - ba'\Add{.}
+\]
+Hence, on setting, for brevity,
+\[
+\Sqrt{(bc' - cb')^{2} + (ca' - ac')^{2} + (ab' - ba')^{2}} = \Delta\Add{,}
+\]
+we shall have either
+\begin{align*}
+X &= \frac{bc' - cb'}{\Delta},\quad&
+Y &= \frac{ca' - ac'}{\Delta},\quad&
+Z &= \frac{ab' - ba'}{\Delta}
+\intertext{or}
+X &= \frac{cb' - bc'}{\Delta},\quad&
+Y &= \frac{ac' - ca'}{\Delta},\quad&
+Z &= \frac{ba' - ab'}{\Delta}\Add{.}
+\end{align*}
+
+With these two general methods is associated a \emph{third}, in which one of the coordinates,
+$z$,~say, is expressed in the form of a function of the other two, $x$,~$y$. This method is
+evidently only a particular case either of the first method, or of the second. If we set
+\[
+dz = t\, dx + u\, dy
+\]
+we shall have either
+\begin{align*}
+X &= \frac{-t}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Y &= \frac{-u}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Z &= \frac{ 1}{\Sqrt{1 + t^{2} + u^{2}}} \\
+\intertext{or}
+X &= \frac{ t}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Y &= \frac{ u}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Z &= \frac{-1}{\Sqrt{1 + t^{2} + u^{2}}}\Add{.}
+\end{align*}
+
+
+\Article{5.}
+
+The two solutions found in the preceding article evidently refer to opposite points of
+the sphere, or to opposite directions, as one would expect, since the normal may be drawn
+toward either of the two sides of the curved surface. If we wish to distinguish between
+the two regions bordering upon the surface, and call one the exterior region and the other
+the interior region, we can then assign to each of the two normals its appropriate solution
+by aid of the theorem derived in \Art{2}~(VII), and at the same time establish a criterion
+for distinguishing the one region from the other.
+\PageSep{9}
+
+In the first method, such a criterion is to be drawn from the sign of the quantity~$W$.
+Indeed, generally speaking, the curved surface divides those regions of space in which $W$
+keeps a positive value from those in which the value of~$W$ becomes negative. In fact, it
+is easily seen from this theorem that, if $W$ takes a positive value toward the exterior
+region, and if the normal is supposed to be drawn outwardly, the first solution is to be
+taken. Moreover, it will be easy to decide in any case whether the same rule for the
+sign of~$W$ is to hold throughout the entire surface, or whether for different parts there
+will be different rules. As long as the coefficients $P$,~$Q$,~$R$ have finite values and do not
+all vanish at the same time, the law of continuity will prevent any change.
+
+If we follow the second method, we can imagine two systems of curved lines on the
+curved surface, one system for which $p$~is variable, $q$~constant; the other for which $q$~is
+variable, $p$~constant. The respective positions of these lines with reference to the exterior
+region will decide which of the two solutions must be taken. In fact, whenever
+the three lines, namely, the branch of the line of the former system going out from the
+point~$A$ as $p$~increases, the branch of the line of the latter system going out from the point
+$A$ as $q$~increases, and the normal drawn toward the exterior region, are \emph{similarly} placed as
+the $x$,~$y$,~$z$ axes respectively from the origin of abscissas (\eg, if, both for the former
+three lines and for the latter three, we can conceive the first directed to the left, the
+second to the right, and the third upward), the first solution is to be taken. But whenever
+the relative position of the three lines is opposite to the relative position of the
+$x$,~$y$,~$z$ axes, the second solution will hold.
+
+In the third method, it is to be seen whether, when $z$~receives a positive increment, $x$~and~$y$
+remaining constant, the point crosses toward the exterior or the interior region.
+In the former case, for the normal drawn outward, the first solution holds; in the latter
+case, the second.
+
+
+\Article{6.}
+
+Just as each definite point on the curved surface is made to correspond to a definite
+point on the sphere, by the direction of the normal to the curved surface which is transferred
+to the surface of the sphere, so also any line whatever, or any figure whatever, on
+the latter will be represented by a corresponding line or figure on the former. In the
+comparison of two figures corresponding to one another in this way, one of which will be
+as the map of the other, two important points are to be considered, one when quantity
+alone is considered, the other when, disregarding quantitative relations, position alone
+is considered.
+
+The first of these important points will be the basis of some ideas which it seems
+judicious to introduce into the theory of curved surfaces. Thus, to each part of a curved
+\PageSep{10}
+surface inclosed within definite limits we assign a \emph{total} or \emph{integral curvature}, which is
+represented by the area of the figure on the sphere corresponding to it. From this
+integral curvature must be distinguished the somewhat more specific curvature which we
+shall call the\Note{8} \emph{measure of curvature}. The latter refers to a \emph{point} of the surface, and shall
+denote the quotient obtained when the integral curvature of the surface element about
+a point is divided by the area of the element itself; and hence it denotes the ratio of the
+infinitely small areas which correspond to one another on the curved surface and on the
+sphere. The use of these innovations will be abundantly justified, as we hope, by what
+we shall explain below. As for the terminology, we have thought it especially desirable
+that all ambiguity be avoided. For this reason we have not thought it advantageous to
+follow strictly the analogy of the terminology commonly adopted (though not approved by
+all) in the theory of plane curves, according to which the measure of curvature should be
+called simply curvature, but the total curvature, the amplitude. But why not be free in
+the choice of words, provided they are not meaningless and not liable to a misleading
+interpretation?
+
+The position of a figure on the sphere can be either similar to the position of the
+corresponding figure on the curved surface, or opposite (inverse). The former is the case
+when two lines going out on the curved surface from the same point in different, but not
+opposite directions, are represented on the sphere by lines similarly placed, that is, when
+the map of the line to the right is also to the right; the latter is the case when the contrary
+holds. We shall distinguish these two cases by the positive or negative \emph{sign} of the
+measure of curvature. But evidently this distinction can hold only when on each surface
+we choose a definite face on which we suppose the figure to lie. On the auxiliary sphere
+we shall use always the exterior face, that is, that turned away from the centre; on the
+curved surface also there may be taken for the exterior face the one already considered,
+or rather that face from which the normal is supposed to be drawn. For, evidently, there
+is no change in regard to the similitude of the figures, if on the curved surface both the
+figure and the normal be transferred to the opposite side, so long as the image itself
+is represented on the same side of the sphere.
+
+The positive or negative sign, which we assign to the \emph{measure} of curvature according
+to the position of the infinitely small figure, we extend also to the integral curvature
+of a finite figure on the curved surface. However, if we wish to discuss the general case,
+some explanations will be necessary, which we can only touch here briefly. So long
+as the figure on the curved surface is such that to \emph{distinct} points on itself there correspond
+distinct points on the sphere, the definition needs no further explanation. But
+whenever this condition is not satisfied, it will be necessary to take into account twice
+or several times certain parts of the figure on the sphere. Whence for a similar, or
+\PageSep{11}
+inverse position, may arise an accumulation of areas, or the areas may partially or
+wholly destroy each other. In such a case, the simplest way is to suppose the curved
+surface divided into parts, such that each part, considered separately, satisfies the above
+condition; to assign to each of the parts its integral curvature, determining this magnitude
+by the area of the corresponding figure on the sphere, and the sign by the position
+of this figure; and, finally, to assign to the total figure the integral curvature
+arising from the addition of the integral curvatures which correspond to the single parts.
+So, generally, the integral curvature of a figure is equal to $\int k\, d\sigma$, $d\sigma$~denoting the
+element of area of the figure, and $k$~the measure of curvature at any point. The principal
+points concerning the geometric representation of this integral reduce to the following.
+To the perimeter of the figure on the curved surface (under the restriction
+of \Art{3}) will correspond always a closed line on the sphere. If the latter nowhere
+intersect itself, it will divide the whole surface of the sphere into two parts, one of
+which will correspond to the figure on the curved surface; and its area (taken as
+positive or negative according as, with respect to its perimeter, its position is similar,
+or inverse, to the position of the figure on the curved surface) will represent the integral
+curvature of the figure on the curved surface. But whenever this line intersects
+itself once or several times, it will give a complicated figure, to which, however, it is
+possible to assign a definite area as legitimately as in the case of a figure without
+nodes; and this area, properly interpreted, will give always an exact value for the
+integral curvature. However, we must reserve for another occasion\Note{9} the more extended
+exposition of the theory of these figures viewed from this very general standpoint.
+
+
+\Article{7.}
+
+We shall now find a formula which will express the measure of curvature for
+any point of a curved surface. Let $d\sigma$~denote the area of an element of this surface;
+then $Z\, d\sigma$~will be the area of the projection of this element on the plane of the coordinates
+$x$,~$y$; and consequently, if $d\Sigma$~is the area of the corresponding element on the
+sphere, $Z\, d\Sigma$~will be the area of its projection on the same plane. The positive or
+negative sign of~$Z$ will, in fact, indicate that the position of the projection is similar or
+inverse to that of the projected element. Evidently these projections have the same
+ratio as to quantity and the same relation as to position as the elements themselves.
+Let us consider now a triangular element on the curved surface, and let us suppose
+that the coordinates of the three points which form its projection are
+\begin{alignat*}{3}
+&x,           && y\Add{,} \\
+&x + dx,\quad && y + dy\Add{,} \\
+&x + \delta x,\quad && y + \delta y\Add{.}
+\end{alignat*}
+\PageSep{12}
+The double area of this triangle will be expressed by the formula
+\[
+dx·\delta y - dy·\delta x\Add{,}
+\]
+and this will be in a positive or negative form according as the position of the side
+from the first point to the third, with respect to the side from the first point to the
+second, is similar or opposite to the position of the $y$-axis of coordinates with respect
+to the $x$-axis of coordinates.
+
+In like manner, if the coordinates of the three points which form the projection of
+the corresponding element on the sphere, from the centre of the sphere as origin, are
+\begin{alignat*}{3}
+&X,           && Y\Add{,} \\
+&X + dX,\quad && Y + dY\Add{,} \\
+&X + \delta X,\quad && Y + \delta Y\Add{,}
+\end{alignat*}
+the double area of this projection will be expressed by
+\[
+dX·\delta Y - dY·\delta X\Add{,}
+\]
+and the sign of this expression is determined in the same manner as above. Wherefore
+the measure of curvature at this point of the curved surface will be
+\[
+k = \frac{dX·\delta Y - dY·\delta X}{dx·\delta y - dy·\delta x}\Add{.}
+\]
+If now we suppose the nature of the curved surface to be defined according to the third
+method considered in \Art{4}, $X$~and~$Y$ will be in the form of functions of the quantities
+$x$,~$y$. We shall have, therefore,\Note{10}
+\begin{alignat*}{2}
+dX &= \frac{\dd X}{\dd x}\, dx &&+ \frac{\dd X}{\dd y}\, dy\Add{,} \\
+\delta X &= \frac{\dd X}{\dd x}\, \delta x
+          &&+ \frac{\dd X}{\dd y}\, \delta y\Add{,} \\
+dY &= \frac{\dd Y}{\dd x}\, dx &&+ \frac{\dd Y}{\dd y}\, dy\Add{,} \\
+\delta Y &= \frac{\dd Y}{\dd x}\, \delta x
+          &&+ \frac{\dd Y}{\dd y}\, \delta y\Add{.}
+\end{alignat*}
+When these values have been substituted, the above expression becomes
+\[
+k = \frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
+  - \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x}\Add{.}
+\]
+\PageSep{13}
+Setting, as above,
+\[
+\frac{\dd z}{\dd x} = t,\quad \frac{\dd z}{\dd y} = u
+\]
+and also
+\[
+\frac{\dd^{2} z}{\dd x^{2}} = T,\quad
+\frac{\dd^{2} z}{\dd x·\dd y} = U,\quad
+\frac{\dd^{2} z}{\dd y^{2}} = V\Add{,}
+\]
+or
+\[
+dt = T\, dx + U\, dy,\quad
+du = U\, dx + V\, dy\Add{,}
+\]
+we have from the formulæ given above
+\[
+X = -tZ,\quad Y = -uZ,\quad (1 - t^{2} - u^{2})Z^{2} = 1\Add{;}
+\]
+and hence
+\begin{gather*}
+\begin{alignedat}{2}
+dX &= -Z\, dt &&- t\, dZ\Add{,} \\
+dY &= -Z\, du &&- u\, dZ\Add{,}
+\end{alignedat} \\
+(1 + t^{2} + u^{2})\, dZ + Z(t\, dt + u\, du) = 0\Add{;}
+\end{gather*}
+or\Note{11}
+\begin{align*}
+dZ &= -Z^{3}(t\, dt + u\, du)\Add{,} \\
+dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3} tu\, du\Add{,} \\
+dY &= +Z^{3}tu\, dt - Z^{3}(1 + t^{2})\, du\Add{;}\NoteMark
+\end{align*}
+and so
+\begin{align*}
+\frac{\dd X}{\dd x} &= Z^{3}\bigl(-(1 + u^{2})T + tuU\bigr)\Add{,} \\
+\frac{\dd X}{\dd y} &= Z^{3}\bigl(-(1 + u^{2})U + tuV\bigr)\Add{,} \\
+\frac{\dd Y}{\dd x} &= Z^{3}\bigl( tuT - (1 + t^{2})U\bigr)\Add{,} \\
+\frac{\dd Y}{\dd y} &= Z^{3}\bigl( tuU - (1 + t^{2})V\bigr)\Add{.}
+\end{align*}
+Substituting these values in the above expression, it becomes
+\begin{align*}
+k &= Z^{6}(TV - U^{2}) (1 + t^{2} + u^{2}) = Z^{4} (TV - U^{2}) \\
+  &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}\Add{.}
+\end{align*}
+
+
+\Article{8.}
+
+By a suitable choice of origin and axes of coordinates, we can easily make the
+values of the quantities $t$,~$u$,~$U$ vanish for a definite point~$A$. Indeed, the first two
+\PageSep{14}
+conditions will be fulfilled at once if the tangent plane at this point be taken for the
+$xy$-plane. If, further, the origin is placed at the point $A$~itself, the expression for
+the coordinate~$z$ evidently takes the form
+\[
+z = \tfrac{1}{2}T°x^{2} + U°xy + \tfrac{1}{2}V°y^{2} + \Omega\Add{,}
+\]
+where $\Omega$~will be of higher degree than the second. Turning now the axes of $x$~and~$y$
+through an angle~$M$ such that
+\[
+\tan 2M = \frac{2U°}{T° - V°}\Add{,}
+\]
+it is easily seen that there must result an equation of the form
+\[
+z = \tfrac{1}{2}Tx^{2} + \tfrac{1}{2}Vy^{2} + \Omega\Add{.}
+\]
+In this way the third condition is also satisfied. When this has been done, it is evident
+that
+
+\Par{I.} If the curved surface be cut by a plane passing through the normal itself and
+through the $x$-axis, a plane curve will be obtained, the radius of curvature of which
+at the point~$A$ will be equal to~$\dfrac{1}{T}$, the positive or negative sign indicating that the
+curve is concave or convex toward that region toward which the coordinates~$z$ are
+positive.
+
+\Par{II.} In like manner $\dfrac{1}{V}$~will be the radius of curvature at the point~$A$ of the plane
+curve which is the intersection of the surface and the plane through the $y$-axis and
+the $z$-axis.
+
+\Par{III.} Setting $z = r \cos\phi$, $y = r \sin \phi$, the equation becomes
+\[
+z = \tfrac{1}{2}(T\cos^{2}\phi + V\sin^{2}\phi) r^{2} + \Omega\Add{,}
+\]
+from which we see that if the section is made by a plane through the normal at~$A$
+and making an angle~$\phi$ with the $x$-axis, we shall have a plane curve whose radius of
+curvature at the point~$A$ will be
+\[
+\frac{1}{T\cos^{2}\phi + V\sin^{2}\phi}\Add{.}
+\]
+
+\Par{IV.} Therefore, whenever we have $T = V$, the radii of curvature in \emph{all} the normal
+planes will be equal. But if $T$~and~$V$ are not equal, it is evident that, since for any
+value whatever of the angle~$\phi$, $T\cos^{2}\phi + V\sin^{2}\phi$ falls between $T$~and~$V$, the radii of
+curvature in the principal sections considered in I.~and~II. refer to the extreme curvatures;
+that is to say, the one to the maximum curvature, the other to the minimum,
+\PageSep{15}
+if $T$~and~$V$ have the same sign. On the other hand, one has the greatest convex
+curvature, the other the greatest concave curvature, if $T$~and~$V$ have opposite signs.
+These conclusions contain almost all that the illustrious Euler\Note{12} was the first to prove
+on the curvature of curved surfaces.
+
+\Par{V.} The measure of curvature at the point~$A$ on the curved surface takes the
+very simple form
+\[
+k = TV,
+\]
+whence we have the
+
+\begin{Theorem}
+The measure of curvature at any point whatever of the surface is equal to a
+fraction whose numerator is unity, and whose denominator is the product of the two extreme
+radii of curvature of the sections by normal planes.\
+\end{Theorem}
+
+At the same time it is clear that the measure of curvature is positive for concavo-concave
+or convexo-convex surfaces (which distinction is not essential), but negative
+for concavo-convex surfaces. If the surface consists of parts of each kind, then
+on the lines separating the two kinds the measure of curvature ought to vanish. Later
+we shall make a detailed study of the nature of curved surfaces for which the measure
+of curvature everywhere vanishes.
+
+
+\Article{9.}
+
+The general formula for the measure of curvature given at the end of \Art{7} is
+the most simple of all, since it involves only five elements. We shall arrive at a
+more complicated formula, indeed, one involving nine elements, if we wish to use the
+first method of representing a curved surface. Keeping the notation of \Art{4}, let us
+set also
+\begin{align*}
+\frac{\dd^{2} W}{\dd x^{2}} &= P', &
+\frac{\dd^{2} W}{\dd y^{2}} &= Q', &
+\frac{\dd^{2} W}{\dd z^{2}} &= R'\Add{,} \\
+\frac{\dd^{2} W}{\dd y·\dd z} &= P'', &
+\frac{\dd^{2} W}{\dd x·\dd z} &= Q'', &
+\frac{\dd^{2} W}{\dd x·\dd y} &= R''\Add{,}
+\end{align*}
+so that
+\begin{alignat*}{4}
+dP &= P'\,  &&dx + R''\, &&dy + Q''\, &&dz\Add{,} \\
+dQ &= R''\, &&dx + Q' \, &&dy + P''\, &&dz\Add{,} \\
+dR &= Q''\, &&dx + P''\, &&dy + R' \, &&dz\Add{.}
+\end{alignat*}
+Now since $t = -\dfrac{P}{R}$, we find through differentiation
+\[
+R^{2}\, dt = -R\, dP + P\, dR
+  = (PQ'' - RP')\, dx + (PP'' - RR'')\, dy + (PR' - RQ'')\, dz\Add{,}
+\]
+\PageSep{16}
+or, eliminating~$dz$ by means of the equation
+\begin{gather*}
+P\, dx + Q\, dy + R\, dz = 0, \\
+R^{3}\, dt
+  = (-R^{2}P' + 2PRQ'' - P^{2}R')\, dx + (PRP'' + QRQ'' - PQR' - R^{2}R'')\, dy.
+\end{gather*}
+In like manner we obtain
+\[
+R^{3}\, du
+  = (PRP'' + QRQ'' - PQR' - R^{2}R'')\, dx + (-R^{2}Q' + 2QRP'' - Q^{2}R')\, dy\Add{.}
+\]
+From this we conclude that
+\begin{align*}
+R^{3}T &= -R^{2}P' + 2PRQ'' - P^{2}R'\Add{,} \\
+R^{3}U &= PRP'' + QRQ'' - PQR' - R^{2}R''\Add{,} \\
+R^{3}V &= -R^{2}Q' + 2QRP'' - Q^{2}R'\Add{.}
+\end{align*}
+Substituting these values in the formula of \Art{7}, we obtain for the measure of curvature~$k$
+the following symmetric expression:
+\begin{multline*}
+(P^{2} + Q^{2} + R^{2})^{2}k
+  = P^{2}(Q'R' - P''^{2})
+  + Q^{2}(P'R' - Q''^{2})
+  + R^{2}(P'Q' - R''^{2}) \\
+  + 2QR(Q''R'' - P'P'')
+  + 2PR(P''R'' - Q'Q'')
+  + 2PQ(P''Q'' - R'R'')\Add{.}
+\end{multline*}
+
+
+\Article{10.}
+
+We obtain a still more complicated formula, indeed, one involving fifteen elements,
+if we follow the second general method of defining the nature of a curved surface. It
+is, however, very important that we develop this formula also. Retaining the notations
+of \Art{4}, let us put also
+\begin{align*}
+\frac{\dd^{2}x}{\dd p^{2}} &= \alpha, &
+\frac{\dd^{2}x}{\dd p·\dd q} &= \alpha', &
+\frac{\dd^{2}x}{\dd q^{2}} &= \alpha''\Add{,} \\
+%
+\frac{\dd^{2}y}{\dd p^{2}} &= \beta, &
+\frac{\dd^{2}y}{\dd p·\dd q} &= \beta', &
+\frac{\dd^{2}y}{\dd q^{2}} &= \beta''\Add{,} \\
+%
+\frac{\dd^{2}z}{\dd p^{2}} &= \gamma, &
+\frac{\dd^{2}z}{\dd p·\dd q} &= \gamma', &
+\frac{\dd^{2}z}{\dd q^{2}} &= \gamma''\Add{;} \\
+\end{align*}
+and let us put, for brevity,
+\begin{align*}
+bc' - cb' &= A\Add{,} \\
+ca' - ac' &= B\Add{,} \\
+ab' - ba' &= C\Add{.}
+\end{align*}
+First we see that
+\[
+A\, dx + B\, dy + C\, dz = 0,
+\]
+or
+\[
+dz = -\frac{A}{C}\, dx - \frac{B}{C}\, dy.
+\]
+\PageSep{17}
+Thus, inasmuch as $z$~may be regarded as a function of $x$,~$y$, we have
+\begin{align*}
+\frac{\dd z}{\dd x} &= t = -\frac{A}{C}\Add{,} \\
+\frac{\dd z}{\dd y} &= u = -\frac{B}{C}\Add{.}
+\end{align*}
+Then from the formulæ
+\[
+dx = a\, dp + a'\, dq,\quad
+dy = b\, dp + b'\, dq,
+\]
+we have
+\begin{alignat*}{4}
+&C\, dp = &&b'\, &&dx - a'\, &&dy\Add{,} \\
+&C\, dq =-&&b\,  &&dx + a\,  &&dy\Add{.}
+\end{alignat*}
+Thence we obtain for the total differentials of $t$,~$u$
+\begin{alignat*}{2}
+C^{3}\, dt
+  &= \left(A\, \frac{\dd C}{\dd p} - C\, \frac{\dd A}{\dd p}\right)(b'\, dx - a'\, dy)
+   + \left(C\, \frac{\dd A}{\dd q} - A\, \frac{\dd C}{\dd q}\right)(b \, dx - a \, dy)\Add{,} \\
+%
+C^{3}\, du
+  &= \left(B\, \frac{\dd C}{\dd p} - C\, \frac{\dd B}{\dd p}\right)(b'\, dx - a'\, dy)
+   + \left(C\, \frac{\dd B}{\dd q} - B\, \frac{\dd C}{\dd q}\right)(b \, dx - a \, dy)\Add{.}
+\end{alignat*}
+If now we substitute in these formulæ
+\begin{alignat*}{4}
+\frac{\dd A}{\dd p} &= c'\beta  &&+ b\gamma'  &&- c\beta'  &&- b'\gamma\Add{,} \\
+\frac{\dd A}{\dd q} &= c'\beta' &&+ b\gamma'' &&- c\beta'' &&- b'\gamma'\Add{,} \\
+%
+\frac{\dd B}{\dd p} &= a'\gamma  &&+ c\alpha'  &&- a\gamma'  &&- c'\alpha\Add{,} \\
+\frac{\dd B}{\dd q} &= a'\gamma' &&+ c\alpha'' &&- a\gamma'' &&- c'\alpha'\Add{,} \\
+%
+\frac{\dd C}{\dd p} &= b'\alpha  &&+ a\beta'  &&- b\alpha'  &&- a'\beta\Add{,} \\
+\frac{\dd C}{\dd q} &= b'\alpha' &&+ a\beta'' &&- b\alpha'' &&- a'\beta'\Add{;}
+\end{alignat*}
+and if we note that the values of the differentials $dt$,~$du$ thus obtained must be equal,
+independently of the differentials $dx$,~$dy$, to the quantities $T\, dx + U\, dy$, $U\, dx + V\, dy$
+respectively, we shall find, after some sufficiently obvious transformations,
+\begin{align*}
+C^{3}T &= \alpha Ab'^{2} + \beta Bb'^{2} + \gamma Cb'^{2} \\
+&\quad- 2\alpha' Abb' - 2\beta' Bbb' - 2\gamma' Cbb' \\
+&\quad+ \alpha'' Ab^{2} + \beta'' Bb^{2} + \gamma'' Cb^{2}\Add{,} \\
+\PageSep{18}
+C^{3}U &= -\alpha Aa'b' - \beta Ba'b' - \gamma Ca'b' \\
+&\quad+ \alpha' A(ab' + ba') + \beta' B(ab' + ba') + \gamma' C(ab' + ba') \\
+&\quad- \alpha'' Aab - \beta'' Bab - \gamma'' Cab\Add{,} \\
+%
+C^{3}V &= \alpha Aa'^{2} + \beta Ba'^{2} + \gamma Ca'^{2} \\
+&\quad- 2\alpha' Aaa' - 2\beta' Baa' - 2\gamma' Caa' \\
+&\quad+ \alpha'' Aa^{2} + \beta'' Ba^{2} + \gamma'' Ca^{2}\Add{.}
+\end{align*}
+Hence, if we put, for the sake of brevity,\Note{13}
+\begin{alignat*}{4}
+&A\alpha   &&+ B\beta   &&+ C\gamma   &&= D\Add{,}\NoteMark
+\Tag{(1)} \\
+&A\alpha'  &&+ B\beta'  &&+ C\gamma'  &&= D'\Add{,}
+\Tag{(2)} \\
+&A\alpha'' &&+ B\beta'' &&+ C\gamma'' &&= D''\Add{,}
+\Tag{(3)}
+\end{alignat*}
+we shall have
+\begin{align*}
+C^{3}T &=  Db'^{2} - 2D'bb' + D'' b^{2}\Add{,} \\
+C^{3}U &= -Da'b' + D'(ab' + ba') - D''ab\Add{,} \\
+C^{3}V &=  Da'^{2} - 2D'aa' + D''a^{2}\Add{.}
+\end{align*}
+From this we find, after the reckoning has been carried out,
+\[
+C^{6}(TV - U^{2}) = (DD'' - D'^{2}) (ab' - ba')^{2} = (DD'' - D'^{2}) C^{2}\Add{,}
+\]
+and therefore the formula for the measure of curvature
+\[
+k = \frac{DD'' - D'^{2}}{(A^{2} + B^{2} + C^{2})^{2}}\Add{.}
+\]
+
+
+\Article{11.}
+
+By means of the formula just found we are going to establish another, which may
+be counted among the most productive theorems in the theory of curved surfaces.
+Let us introduce the following notation:
+\begin{alignat*}{4}
+&a^{2} &&+ b^{2} &&+ c^{2} &&= E\Add{,} \\
+&aa' &&+ bb' &&+ cc' &&= F\Add{,} \\
+&a'^{2} &&+ b'^{2} &&+ c'^{2} &&= G\Add{;}
+\end{alignat*}
+\begin{alignat*}{7}
+&a &&\alpha  &&+b &&\beta  &&+c &&\gamma  &&= m  \Add{,}
+\Tag{(4)} \\
+&a &&\alpha' &&+b &&\beta' &&+c &&\gamma' &&= m' \Add{,}
+\Tag{(5)} \\
+&a &&\alpha''&&+b &&\beta''&&+c &&\gamma''&&= m''\Add{;}
+\Tag{(6)} \displaybreak[1] \\
+%
+&a'\,&&\alpha  &&+b'\,&&\beta  &&+c'\,&&\gamma  &&= n  \Add{,}
+\Tag{(7)} \\
+&a'&&\alpha' &&+b'&&\beta' &&+c'&&\gamma' &&= n' \Add{,}
+\Tag{(8)} \\
+&a'&&\alpha''&&+b'&&\beta''&&+c'&&\gamma''&&= n''\Add{;}
+\Tag{(9)}
+\end{alignat*}
+\[
+A^{2} + B^{2} + C^{2} = EG - F^{2} = \Delta\Add{.}
+\]
+\PageSep{19}
+
+%[** TN: Added parentheses around equation numbers]
+Let us eliminate from the equations (1),~(4),~(7) the quantities $\beta$,~$\gamma$, which is done by
+multiplying them by $bc' - cb'$, $b'C - c'B$, $cB - bC$ respectively and adding. In this
+way we obtain
+\begin{multline*}
+\bigl(A(bc' - cb') + a(b'C - c'B) + a'(cB - bC)\bigr)\alpha \\
+  = D(bc' - cb') + m(b'C - c'B) + n(cB - bC)\Add{,}
+\end{multline*}
+an equation which is easily transformed into
+\[
+AD = \alpha\Delta + a(nF - mG) + a'(mF - nE)\Add{.}
+\]
+Likewise the elimination of $\alpha$,~$\gamma$ or $\alpha$,~$\beta$ from the same equations gives
+\begin{alignat*}{4}
+&BD &&= \beta \Delta &&+ b(nF - mG) &&+ b'(mF - nE)\Add{,} \\
+&CD &&= \gamma\Delta &&+ c(nF - mG) &&+ c'(mF - nE)\Add{.}
+\end{alignat*}
+Multiplying these three equations by $\alpha''$,~$\beta''$,~$\gamma''$ respectively and adding, we obtain
+\[
+DD'' = (\alpha\alpha''+ \beta\beta'' + \gamma\gamma'')\Delta
+  + m''(nF - mG) + n''(mF - nE)\Add{.}
+\Tag{(10)}
+\]
+
+%[** TN: Added parentheses around equation numbers]
+If we treat the equations (2),~(5),~(8) in the same way, we obtain
+\begin{alignat*}{4}
+&AD' &&= \alpha'\Delta &&+ a (n'F - m'G) &&+ a'(m'F - n'E)\Add{,} \\
+&BD' &&= \beta' \Delta &&+ b (n'F - m'G) &&+ b'(m'F - n'E)\Add{,} \\
+&CD' &&= \gamma'\Delta &&+ c (n'F - m'G) &&+ c'(m'F - n'E)\Add{;}
+\end{alignat*}
+and after these equations are multiplied by $\alpha'$,~$\beta'$,~$\gamma'$ respectively, addition gives
+\[
+D'^{2} = (\alpha'^{2} + \beta'^{2} + \gamma'^{2})\Delta
+  + m'(n'F - m'G) + n'(m'F - n'E)\Add{.}
+\]
+
+A combination of this equation with equation~(10) gives
+\begin{multline*}
+DD'' - D'^{2} = (\alpha\alpha'' + \beta\beta'' + \gamma\gamma''
+  - \alpha'^{2} - \beta'^{2} - \gamma'^{2})\Delta \\
+  + E(n'^{2} - nn'') + F(nm'' - 2m'n' + mn'') + G(m'^{2} - mm'')\Add{.}
+\end{multline*}
+It is clear that we have
+\[
+\frac{\dd E}{\dd p} = 2m,\
+\frac{\dd E}{\dd q} = 2m',\quad
+\frac{\dd F}{\dd p} = m' + n,\
+\frac{\dd F}{\dd q} = m'' + n',\quad
+\frac{\dd G}{\dd p} = 2n',\
+\frac{\dd G}{\dd q} = 2n'',
+\]
+or\Note{14}
+\begin{align*}
+m   &= \tfrac{1}{2}\, \frac{\dd E}{\dd p}, &
+m'  &= \tfrac{1}{2}\, \frac{\dd E}{\dd q}, &
+m'' &= \frac{\dd F}{\dd q} - \tfrac{1}{2}\, \frac{\dd G}{\dd p}\Add{,}\NoteMark \\
+%
+n   &= \frac{\dd F}{\dd p} - \tfrac{1}{2}\, \frac{\dd E}{\dd q}, &
+n'  &= \tfrac{1}{2}\, \frac{\dd G}{\dd p}, &
+n'' &= \tfrac{1}{2}\, \frac{\dd G}{\dd q}\Add{.}
+\end{align*}
+Moreover, it is easily shown that we shall have
+\begin{align*}
+%[** TN: Aligning on equals sign]
+\alpha\alpha'' + \beta\beta'' + \gamma\gamma''
+  - \alpha'^{2} - \beta'^{2} - \gamma'^{2}
+  &= \frac{\dd n}{\dd q} - \frac{\dd n'}{\dd p}
+   = \frac{\dd m''}{\dd p} - \frac{\dd m'}{\dd q} \\
+  &= -\tfrac{1}{2}·\frac{\dd^{2}E}{\dd q^{2}}
+  + \frac{\dd^{2}F}{\dd p·\dd q}
+  - \tfrac{1}{2}·\frac{\dd^{2}G}{\dd p^{2}}\Add{.}
+\end{align*}
+\PageSep{20}
+If we substitute these different expressions in the formula for the measure of curvature
+derived at the end of the preceding article, we obtain the following formula, which
+involves only the quantities $E$,~$F$,~$G$ and their differential quotients of the first and
+second orders:
+\begin{multline*}
+4(EG - F^{2})k
+  = E\left(\frac{\dd E}{\dd q}·\frac{\dd G}{\dd q}
+       - 2 \frac{\dd F}{\dd p}·\frac{\dd G}{\dd q}
+  + \biggl(\frac{\dd G}{\dd p}\biggr)^{2}\right) \\
+  + F\left(\frac{\dd E}{\dd p}·\frac{\dd G}{\dd q}
+         - \frac{\dd E}{\dd q}·\frac{\dd G}{\dd p}
+       - 2 \frac{\dd E}{\dd q}·\frac{\dd F}{\dd q}
+       + 4 \frac{\dd \Erratum{E}{F}}{\dd p}·\frac{\dd F}{\dd q}
+       - 2 \frac{\dd F}{\dd p}·\frac{\dd G}{\dd p}\right) \\
+  + G\left(\frac{\dd E}{\dd p}·\frac{\dd G}{\dd p}
+       - 2 \frac{\dd E}{\dd p}·\frac{\dd F}{\dd q}
+  + \biggl(\frac{\dd E}{\dd q}\biggr)^{2}\right)
+  - 2(EG - F^{2})\left(
+      \frac{\dd^{2}E}{\dd q^{2}}
+   - 2\frac{\dd^{2}F}{\dd p·\dd q}
+    + \frac{\dd^{2}G}{\dd p^{2}}
+  \right)\Add{.}
+\end{multline*}
+
+
+\Article{12.}
+
+Since we always have
+\[
+dx^{2} + dy^{2} + dz^{2} = E\, dp^{2} + 2F\, dp·dq + G\, dq^{2},
+\]
+it is clear that
+\[
+\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}
+\]
+is the general expression for the linear element on the curved surface. The analysis
+developed in the preceding article thus shows us that for finding the measure of curvature
+there is no need of finite formulæ, which express the coordinates $x$,~$y$,~$z$ as
+functions of the indeterminates $p$,~$q$; but that the general expression for the magnitude
+of any linear element is sufficient. Let us proceed to some applications of this very
+important theorem.
+
+Suppose that our surface can be developed upon another surface, curved or plane,
+so that to each point of the former surface, determined by the coordinates $x$,~$y$,~$z$, will
+correspond a definite point of the latter surface, whose coordinates are $x'$,~$y'$,~$z'$. Evidently
+$x'$,~$y'$,~$z'$ can also be regarded as functions of the indeterminates $p$,~$q$, and therefore
+for the element $\Sqrt{dx'^{2} + dy'^{2} + dz'^{2}}$ we shall have an expression of the form
+\[
+\Sqrt{E'\, dp^{2} + 2F'\, dp·dq + G'\, dq^{2}}\Add{,}
+\]
+where $E'$,~$F'$,~$G'$ also denote functions of $p$,~$q$. But from the very notion of the \emph{development}
+of one surface upon another it is clear that the elements corresponding to one
+another on the two surfaces are necessarily equal. Therefore we shall have identically
+\[
+E = E',\quad F = F',\quad G = G'.
+\]
+Thus the formula of the preceding article leads of itself to the remarkable
+
+\begin{Theorem}
+If a curved surface is developed upon any other surface whatever, the
+measure of curvature in each point remains unchanged.
+\end{Theorem}
+\PageSep{21}
+
+Also it is evident that
+\begin{Theorem}[]
+any finite part whatever of the curved surface will retain the
+same integral curvature after development upon another surface.
+\end{Theorem}
+
+Surfaces developable upon a plane constitute the particular case to which geometers
+have heretofore restricted their attention. Our theory shows at once that the
+measure of curvature at every point of such surfaces is equal to zero. Consequently,
+if the nature of these surfaces is defined according to the third method, we shall have
+at every point
+\[
+\frac{\dd^{2} z}{\dd x^{2}}·\frac{\dd^{2} z}{\dd y^{2}}
+  - \left(\frac{\dd^{2}z}{\dd x·\dd y}\right)^{2} = 0\Add{,}
+\]
+a criterion which, though indeed known a short time ago, has not, at least to our
+knowledge, commonly been demonstrated with as much rigor as is desirable.
+
+
+\Article{13.}
+
+What we have explained in the preceding article is connected with a particular
+method of studying surfaces, a very worthy method which may be thoroughly developed
+by geometers. When a surface is regarded, not as the boundary of a solid, but
+as a flexible, though not extensible solid, one dimension of which is supposed to
+vanish, then the properties of the surface depend in part upon the form to which we
+can suppose it reduced, and in part are absolute and remain invariable, whatever may
+be the form into which the surface is bent. To these latter properties, the study of
+which opens to geometry a new and fertile field, belong the measure of curvature and
+the integral curvature, in the sense which we have given to these expressions. To
+these belong also the theory of shortest lines, and a great part of what we reserve to
+be treated later. From this point of view, a plane surface and a surface developable
+on a plane, \eg,~cylindrical surfaces, conical surfaces,~etc., are to be regarded as essentially
+identical; and the generic method of defining in a general manner the nature of
+the surfaces thus considered is always based upon the formula
+\[
+\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}},
+\]
+which connects the linear element with the two indeterminates $p$,~$q$. But before following
+this study further, we must introduce the principles of the theory of shortest
+lines on a given curved surface.
+
+
+\Article{14.}
+
+The nature of a curved line in space is generally given in such a way that the
+coordinates $x$,~$y$,~$z$ corresponding to the different points of it are given in the form of
+functions of a single variable, which we shall call~$w$. The length of such a line from
+\PageSep{22}
+an arbitrary initial point to the point whose coordinates are $x$,~$y$,~$z$, is expressed by
+the integral
+\[
+%[** TN: Round outer parentheses in the original]
+\int dw·\SQRT{
+    \left(\frac{dx}{dw}\right)^{2}
+  + \left(\frac{dy}{dw}\right)^{2}
+  + \left(\frac{dz}{dw}\right)^{2}}\Add{.}
+\]
+If we suppose that the position of the line undergoes an infinitely small variation, so
+that the coordinates of the different points receive the variations $\delta x$,~$\delta y$,~$\delta z$, the variation
+of the whole length becomes
+\[
+\int \frac{dx·d\, \delta x + dy·d\, \delta y + dz·d\, \delta z}
+          {\Sqrt{dx^{2} + dy^{2} + dz^{2}}}\Add{,}
+\]
+which expression we can change into the form\Note{15}
+\begin{multline*}
+\frac{dx·\delta x + dy·\delta y + dz·\delta z}
+     {\Sqrt{dx^{2} + dy^{2} + dz^{2}}} \\
+-\int \Biggl(
+    \delta x·d\frac{dx}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+  + \delta y·d\frac{dy}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+  + \delta z·d\frac{dz}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+  \Biggr)\Add{.}\NoteMark
+\end{multline*}
+We know that, in case the line is to be the shortest between its end points, all that
+stands under the integral sign must vanish. Since the line must lie on the given
+surface, whose nature is defined by the equation
+\[
+P\, dx + Q\, dy + R\, dz = 0,
+\]
+the variations $\delta x$,~$\delta y$,~$\delta z$ also must satisfy the equation
+\[
+P\, \delta x + Q\, \delta y + R\, \delta z = 0,
+\]
+and from this it follows at once, according to well-known rules, that the differentials
+\[
+d\frac{dx}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad
+d\frac{dy}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad
+d\frac{dz}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+\]
+must be proportional to the quantities $P$,~$Q$,~$R$ respectively. Let $dr$~be the element
+of the curved line; $\lambda$~the point on the sphere representing the direction of this element;
+$L$~the point on the sphere representing the direction of the normal to the curved
+surface; finally, let $\xi$,~$\eta$,~$\zeta$ be the coordinates of the point~$\lambda$, and $X$,~$Y$,~$Z$ be those of
+the point~$L$ with reference to the centre of the sphere. We shall then have
+\[
+dx = \xi\, dr,\quad
+dy = \eta\, dr,\quad
+dz = \zeta\, dr\Add{,}
+\]
+from which we see that the above differentials become $d\xi$,~$d\eta$,~$d\zeta$. And since the
+quantities $P$,~$Q$,~$R$ are proportional to $X$,~$Y$,~$Z$, the character of shortest lines is
+expressed by the equations
+\[
+\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}\Add{.}
+\]
+\PageSep{23}
+Moreover, it is easily seen that
+\[
+\Sqrt{d\xi^{2} + d\eta^{2} + d\zeta^{2}}
+\]
+is equal to the small arc on the sphere which measures the angle between the directions
+of the tangents at the beginning and at the end of the element~$dr$, and is thus
+equal to~$\dfrac{dr}{\rho}$, if $\rho$~denotes the radius of curvature of the shortest line at this point.
+Thus we shall have
+\[
+\rho\, d\xi = X\, dr,\quad
+\rho\, d\eta = Y\, dr,\quad
+\rho\, d\zeta = Z\, dr\Add{.}
+\]
+
+
+\Article{15.}
+
+Suppose that an infinite number of shortest lines go out from a given point~$A$
+on the curved surface, and suppose that we distinguish these lines from one another
+by the angle that the first element of each of them makes with the first element of
+one of them which we take for the first. Let $\phi$~be that angle, or, more generally, a
+function of that angle, and $r$~the length of such a shortest line from the point~$A$ to
+the point whose coordinates are $x$,~$y$,~$z$. Since to definite values of the variables $r$,~$\phi$
+there correspond definite points of the surface, the coordinates $x$,~$y$,~$z$ can be regarded
+as functions of $r$,~$\phi$. We shall retain for the notation $\lambda$, $L$, $\xi$,~$\eta$,~$\zeta$, $X$,~$Y$,~$Z$ the same
+meaning as in the preceding article, this notation referring to any point whatever on
+any one of the shortest lines.
+
+All the shortest lines that are of the same length~$r$ will end on another line
+whose length, measured from an arbitrary initial point, we shall denote by~$v$. Thus $v$~can
+be regarded as a function of the indeterminates $r$,~$\phi$, and if $\lambda'$~denotes the point
+on the sphere corresponding to the direction of the element~$dv$, and also $\xi'$,~$\eta'$,~$\zeta'$
+denote the coordinates of this point with reference to the centre of the sphere, we
+shall have
+\[
+\frac{\dd x}{\dd\phi} = \xi'·\frac{\dd v}{\dd\phi},\quad
+\frac{\dd y}{\dd\phi} = \eta'·\frac{\dd v}{\dd\phi},\quad
+\frac{\dd z}{\dd\phi} = \zeta'·\frac{\dd v}{\dd\phi}\Add{.}
+\]
+From these equations and from the equations
+\[
+\frac{\dd x}{\dd r} = \xi,\quad
+\frac{\dd y}{\dd r} = \eta,\quad
+\frac{\dd z}{\dd r} = \zeta
+\]
+we have
+\[
+\frac{\dd x}{\dd r}·\frac{\dd x}{\dd\phi} +
+\frac{\dd y}{\dd r}·\frac{\dd y}{\dd\phi} +
+\frac{\dd z}{\dd r}·\frac{\dd z}{\dd\phi}
+  = (\xi\xi' + \eta\eta' + \zeta\zeta')·\frac{\dd v}{\dd\phi}
+  = \cos \lambda\lambda'·\frac{\dd v}{\dd\phi}\Add{.}
+\]
+\PageSep{24}
+Let $S$~denote the first member of this equation, which will also be a function of $r$,~$\phi$.
+Differentiation of~$S$ with respect to~$r$ gives
+\begin{align*}
+\frac{\dd S}{\dd r}
+  &= \frac{\dd^{2} x}{\dd r^{2}}·\frac{\dd x}{\dd\phi}
+   + \frac{\dd^{2} y}{\dd r^{2}}·\frac{\dd y}{\dd\phi}
+   + \frac{\dd^{2} z}{\dd r^{2}}·\frac{\dd z}{\dd\phi}
+   + \tfrac{1}{2}·\frac{\dd\left(
+       \biggl(\dfrac{\dd x}{\dd r}\biggr)^{2}
+     + \biggl(\dfrac{\dd y}{\dd r}\biggr)^{2}
+     + \biggl(\dfrac{\dd z}{\dd r}\biggr)^{2}
+   \right)}{\dd \phi} \\
+  &= \frac{\dd\xi}{\dd r}·\frac{\dd x}{\dd\phi}
+   + \frac{\dd\eta}{\dd r}·\frac{\dd y}{\dd\phi}
+   + \frac{\dd\zeta}{\dd r}·\frac{\dd z}{\dd\phi}
+   + \tfrac{1}{2}·\frac{\dd(\xi^{2} + \eta^{2} + \zeta^{2})}{\dd\phi}\Add{.}
+\end{align*}
+But
+\[
+\xi^{2} + \eta^{2} + \zeta^{2} = 1,
+\]
+and therefore its differential is equal to zero; and by the preceding article we have,
+if $\rho$~denotes the radius of curvature of the line~$r$,
+\[
+\frac{\dd\xi}{\dd r} = \frac{X}{\rho},\quad
+\frac{\dd\eta}{\dd r} = \frac{Y}{\rho},\quad
+\frac{\dd\zeta}{\dd r} = \frac{Z}{\rho}\Add{.}
+\]
+Thus we have
+\[
+\frac{\dd S}{\dd r}
+  = \frac{1}{\rho}·(X\xi' + Y\eta' + Z\zeta')·\frac{\dd v}{\dd\phi}
+  = \frac{1}{\rho}·\cos L\lambda'·\frac{\dd v}{\dd\phi} = 0
+\]
+since $\lambda'$~evidently lies on the great circle whose pole is~$L$. From this we see that
+$S$~is independent of~$r$, and is, therefore, a function of $\phi$~alone. But for $r = 0$ we evidently
+have $v = 0$, consequently $\dfrac{\dd v}{\dd\phi} = 0$, and $S = 0$ independently of~$\phi$. Thus, in general,
+we have necessarily $S = 0$, and so $\cos\lambda\lambda' = 0$, \ie, $\lambda\lambda' = 90°$. From this follows the
+
+\begin{Theorem}
+If on a curved surface an infinite number of shortest lines of equal length
+be drawn from the same initial point, the lines joining their extremities will be normal to
+each of the lines.
+\end{Theorem}
+
+We have thought it worth while to deduce this theorem from the fundamental
+property of shortest lines; but the truth of the theorem can be made apparent without
+any calculation by means of the following reasoning. Let $AB$,~$AB'$ be two
+shortest lines of the same length including at~$A$ an infinitely small angle, and let us
+suppose that one of the angles made by the element~$BB'$ with the lines $BA$,~$B'A$
+differs from a right angle by a finite quantity. Then, by the law of continuity, one
+will be greater and the other less than a right angle. Suppose the angle at~$B$ is
+equal to~$90° - \omega$, and take on the line~$AB$ a point~$C$, such that
+\[
+BC = BB'·\cosec \omega.
+\]
+Then, since the infinitely small triangle~$BB'C$ may be regarded as plane, we shall have
+\[
+CB' = BC·\cos \omega,
+\]
+\PageSep{25}
+and consequently
+\[
+AC + CB' = AC + BC·\cos \omega
+  = AB - BC·(1- \cos \omega)
+  = AB' - BC·(1 - \cos \omega),
+\]
+\ie, the path from $A$~to~$B'$ through the point~$C$ is shorter than the shortest line,
+\QEA
+
+
+\Article{16.}
+
+%[** TN: In-line theorem, marked non-semantically]
+With the theorem of the preceding article we associate another, which we state
+as follows: \textit{If on a curved surface we imagine any line whatever, from the different points
+of which are drawn at right angles and toward the same side an infinite number of shortest
+lines of the same length, the curve which joins their other extremities will cut each of the
+lines at right angles.} For the demonstration of this theorem no change need be made
+in the preceding analysis, except that $\phi$~must denote the length of the \emph{given} curve
+measured from an arbitrary point; or rather, a function of this length. Thus all of
+the reasoning will hold here also, with this modification, that $S = 0$ for $r = 0$ is
+now implied in the hypothesis itself. Moreover, this theorem is more general than
+the preceding one, for we can regard it as including the first one if we take for the
+given line the infinitely small circle described about the centre~$A$. Finally, we may
+say that here also geometric considerations may take the place of the analysis, which,
+however, we shall not take the time to consider here, since they are sufficiently
+obvious.
+
+
+\Article{17.}
+
+We return to the formula
+\[
+\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}},
+\]
+which expresses generally the magnitude of a linear element on the curved surface,
+and investigate, first of all, the geometric meaning of the coefficients $E$,~$F$,~$G$. We
+have already said in \Art{5} that two systems of lines may be supposed to lie on the
+curved surface, $p$~being variable, $q$~constant along each of the lines of the one system;
+and $q$~variable, $p$~constant along each of the lines of the other system. Any point
+whatever on the surface can be regarded as the intersection of a line of the first
+system with a line of the second; and then the element of the first line adjacent to
+this point and corresponding to a variation~$dp$ will be equal to~$\sqrt{E}·dp$, and the
+element of the second line corresponding to the variation~$dq$ will be equal to~$\sqrt{G}·dq$.
+Finally, denoting by~$\omega$ the angle between these elements, it is easily seen that we
+shall have
+\[
+\cos \omega = \frac{F}{\sqrt{EG}}.
+\]
+\PageSep{26}
+Furthermore, the area of the surface element in the form of a parallelogram between
+the two lines of the first system, to which correspond $q$,~$q + dq$, and the two lines of
+the second system, to which correspond $p$,~$p + dp$, will be
+\[
+\Sqrt{EG - F^{2}}\, dp·dq.
+\]
+
+Any line whatever on the curved surface belonging to neither of the two systems
+is determined when $p$~and~$q$ are supposed to be functions of a new variable, or
+one of them is supposed to be a function of the other. Let $s$~be the length of such
+a curve, measured from an arbitrary initial point, and in either direction chosen as
+positive. Let $\theta$~denote the angle which the element
+\[
+ds = \Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}
+\]
+makes with the line of the first system drawn through the initial point of the element,
+and, in order that no ambiguity may arise, let us suppose that this angle is
+measured from that branch of the first line on which the values of~$p$ increase, and is
+taken as positive toward that side toward which the values of~$q$ increase. These conventions
+being made, it is easily seen that
+\begin{align*}
+\cos \theta·ds &= \sqrt{E}·dp + \sqrt{G}·\cos \omega·dq
+  = \frac{E\, dp + F\, dq}{\sqrt{E}}\Add{,} \\
+\sin \theta·ds &= \sqrt{G}·\sin \omega·dq
+  = \frac{\sqrt{(EG - F^{2})}·dq}{\sqrt{E}}\Add{.}
+\end{align*}
+
+
+\Article{18.}
+
+We shall now investigate the condition that this line be a shortest line. Since
+its length~$s$ is expressed by the integral
+\[
+s = \int \Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}\Add{,}
+\]
+the condition for a minimum requires that the variation of this integral arising from
+an infinitely small change in the position become equal to zero. The calculation, for
+our purpose, is more simply made in this case, if we regard $p$ as a function of~$q$.
+When this is done, if the variation is denoted by the characteristic~$\delta$, we have
+\begin{align*}
+\delta s &= \int \frac{\left(
+    \dfrac{\dd E}{\dd p}·dp^{2}
+  + 2\dfrac{\dd F}{\dd p}·dp·dq
+  + \dfrac{\dd G}{\dd p}·dq^{2}
+  \right) \delta p
+  + (2E\, dp + 2F\, dq)\, d\, \delta p}{2\, ds} \displaybreak[1] \\
+  &= \frac{E\, dp + F\, dq}{ds}·\delta p \\
+\PageSep{27}
+  &\qquad+ \int \delta p \left(\frac{
+    \dfrac{\dd E}{\dd p}·dp^{2}
+  + 2\dfrac{\dd F}{\dd p}·dp·dq
+  + \dfrac{\dd G}{\dd p}·dq^{2}}{2\, ds}
+  - d·\frac{E\, dp + F\, dq}{ds}\right)
+\end{align*}
+and we know that what is included under the integral sign must vanish independently
+of~$\delta p$. Thus we have
+\begin{multline*}
+\frac{\dd E}{\dd p}·dp^{2}
+  + 2\frac{\dd F}{\dd p}·dp·dq
+  + \frac{\dd G}{\dd p}·dq^{2}
+  = 2\, ds·d·\frac{E\, dp + F\, dq}{ds} \\
+\begin{aligned}
+&= 2\, ds·d·\left(\sqrt{E}·\cos\theta\right) \\ %[** TN: Added parentheses]
+&= \frac{ds·dE·\cos\theta}{\sqrt{E}}
+ - 2\, ds·d\theta·\sqrt{E}·\sin\theta \\
+%** Translator's note from corrigenda: The original and the Latin reprints **
+%** lack the factor 2; the correction is made in all the translations.     **
+&= \frac{(E\, dp + F\, dq)\, dE}{E} - \Erratum{}{2}\Sqrt{EG - F^{2}}·\Erratum{dp}{dq}·d\theta \\
+&= \left(\frac{E\, dp + F\, dq}{E}\right)
+ ·\left(\frac{\dd E}{\dd p}·dp + \frac{\dd E}{\dd q}·dq\right)
+  - 2\Sqrt{EG - F^{2}}·dq·d\theta\Add{.}
+\end{aligned}
+\end{multline*}
+This gives the following conditional equation for a shortest line:
+\begin{multline*}
+\Sqrt{EG - F^{2}}·d\theta
+  = \frac{1}{2}·\frac{F}{E}·\frac{\dd E}{\dd p}·dp
+  + \frac{1}{2}·\frac{F}{E}·\frac{\dd E}{\dd q}·dq
+  + \frac{1}{2}·\frac{\dd E}{\dd q}·dp \\
+  - \frac{\dd F}{\dd p}·dp - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{,}
+\end{multline*}
+which can also be written
+\[
+\Sqrt{EG - F^{2}}·d\theta
+  = \frac{1}{2}·\frac{F}{E}·dE
+  + \frac{1}{2}·\frac{\dd E}{\dd q}·dp
+  - \frac{\dd F}{\dd p}·dp
+  - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{.}
+\]
+From this equation, by means of the equation
+\[
+\cot\theta = \frac{E}{\Sqrt{EG - F^{2}}}·\frac{dp}{dq}
+  + \frac{F}{\Sqrt{EG - F^{2}}}\Add{,}
+\]
+it is also possible to eliminate the angle~$\theta$, and to derive a differential equation of
+the second order between $p$~and~$q$, which, however, would become more complicated
+and less useful for applications than the preceding.
+
+
+\Article{19.}
+
+The general formulæ, which we have derived in \Arts{11}{18} for the measure of
+curvature and the variation in the direction of a shortest line, become much simpler
+if the quantities $p$,~$q$ are so chosen that the lines of the first system cut everywhere
+\PageSep{28}
+orthogonally the lines of the second system; \ie, in such a way that we have generally
+$\omega = 90°$, or $F = 0$. Then the formula for the measure of curvature becomes
+\[
+4E^{2}G^{2}k
+  = E·\frac{\dd E}{\dd q}·\frac{\dd G}{\dd q}
+  + E\left(\frac{\dd G}{\dd p}\right)^{2}
+  + G·\frac{\dd E}{\dd p}·\frac{\dd G}{\dd p}
+  + G\left(\frac{\dd E}{\dd q}\right)^{2}
+  - 2EG\left(\frac{\dd^{2} E}{\dd q^{2}} + \frac{\dd^{2} G}{\dd p^{2}}\right),
+\]
+and for the variation of the angle~$\theta$
+\[
+\sqrt{EG}·d\theta
+  = \frac{1}{2}·\frac{\dd E}{\dd q}·dp
+  - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{.}
+\]
+
+Among the various cases in which we have this condition of orthogonality, the
+most important is that in which all the lines of one of the two systems, \eg, the
+first, are shortest lines. Here for a constant value of~$q$ the angle~$\theta$ becomes equal to
+zero, and therefore the equation for the variation of~$\theta$ just given shows that we must
+have $\dfrac{\dd E}{\dd q} = 0$, or that the coefficient~$E$ must be independent of~$\Erratum{g}{q}$; \ie, $E$~must be
+either a constant or a function of $p$~alone. It will be simplest to take for~$p$
+the length of each line of the first system, which length, when all the lines of the
+first system meet in a point, is to be measured from this point, or, if there is no
+common intersection, from any line whatever of the second system. Having made
+these conventions, it is evident that $p$~and~$q$ denote now the same quantities that
+were expressed in \Arts{15}{16} by $r$~and~$\phi$, and that $E = 1$. Thus the two preceding
+formulæ become:
+\begin{align*}
+4G^{2}k
+  &= \left(\frac{\dd G}{\dd p}\right)^{2} - 2G\, \frac{\dd^{2} G}{\dd p^{2}}\Add{,} \\
+\sqrt{G}·d\theta
+  &= -\frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{;}
+\end{align*}
+or, setting $\sqrt{G} = m$,
+\[
+k = -\frac{1}{m}·\frac{\dd^{2} m}{\dd p^{2}},\quad
+d\theta = -\frac{\dd m}{\dd p}·dq\Add{.}
+\]
+Generally speaking, $m$~will be a function of $p$,~$q$, and $m\, dq$~the expression for the element
+of any line whatever of the second system. But in the particular case where
+all the lines~$p$ go out from the same point, evidently we must have $m = 0$ for $p = 0$.
+Furthermore, in the case under discussion we will take for~$q$ the angle itself which
+the first element of any line whatever of the first system makes with the element of
+any one of the lines chosen arbitrarily. Then, since for an infinitely small value of~$p$
+the element of a line of the second system (which can be regarded as a circle
+described with radius~$p$) is equal to~$p\, dq$, we shall have for an infinitely small value
+of~$p$, $m = p$, and consequently, for $p = 0$, $m = 0$ at the same time, and $\dfrac{\dd m}{\dd p} = 1$.
+\PageSep{29}
+
+
+\Article{20.}
+
+We pause to investigate the case in which we suppose that $p$~denotes in a general
+manner the length of the shortest line drawn from a fixed point~$A$ to any other
+point whatever of the surface, and $q$~the angle that the first element of this line
+makes with the first element of another given shortest line going out from~$A$. Let
+$B$ be a definite point in the latter line, for which $q = 0$, and $C$~another definite point
+of the surface, at which we denote the value of~$q$ simply by~$A$. Let us suppose the
+points $B$,~$C$ joined by a shortest line, the parts of which, measured from~$B$, we denote
+in a general way, as in \Art{18}, by~$s$; and, as in the same article, let us denote by~$\theta$
+the angle which any element~$ds$ makes with the element~$dp$; finally, let us denote
+by $\theta°$,~$\theta'$ the values of the angle~$\theta$ at the points $B$,~$C$. We have thus on the curved
+surface a triangle formed by shortest lines. The angles of this triangle at $B$~and~$C$
+we shall denote simply by the same letters, and $B$~will be equal to~$180° - \theta$, $C$~to $\theta'$~itself.
+But, since it is easily seen from our analysis that all the angles are supposed
+to be expressed, not in degrees, but by numbers, in such a way that the angle $57°\, 17'\, 45''$,
+to which corresponds an arc equal to the radius, is taken for the unit, we must set
+\[
+\theta° = \pi - B,\quad \theta' = C\Add{,}
+\]
+where $2\pi$~denotes the circumference of the sphere. Let us now examine the integral
+curvature of this triangle, which is equal to
+\[
+\int k\, d\sigma,
+\]
+$d\sigma$~denoting a surface element of the triangle. Wherefore, since this element is expressed
+by~$m\, dp·dq$, we must extend the integral
+\[
+\iint \Typo{}{k}m\, dp·dq
+\]
+over the whole surface of the triangle. Let us begin by integration with respect to~$p$,
+which, because
+\[
+k = -\frac{1}{m}·\frac{\dd^{2} m}{\dd p^{2}},
+\]
+gives
+\[
+dq·\left(\text{const.} - \frac{\dd m}{\dd p}\right),
+\]
+for the integral curvature of the area lying between the lines of the first system, to
+which correspond the values $q$,~$q + dq$ of the second indeterminate. Since this integral
+\PageSep{30}
+curvature must vanish for $p = 0$, the constant introduced by integration must be
+equal to the value of~$\dfrac{\dd m}{\dd q}$ for $p = 0$, \ie,~equal to unity. Thus we have
+\[
+dq\left(1 - \frac{\dd m}{\dd p}\right),
+\]
+where for $\dfrac{\dd m}{\dd p}$ must be taken the value corresponding to the end of this area on the
+line~$CB$. But on this line we have, by the preceding article,
+\[
+\frac{\dd m}{\dd q}·dq = -d\theta,
+\]
+whence our expression is changed into $dq + d\theta$. Now by a second integration, taken
+from $q = 0$ to $q = A$, we obtain for the integral curvature
+\[
+A + \theta'- \theta°,
+\]
+or
+\[
+A + B + C - \pi.
+\]
+
+The integral curvature is equal to the area of that part of the sphere which corresponds
+to the triangle, taken with the positive or negative sign according as the
+curved surface on which the triangle lies is concavo-concave or concavo-convex. For
+unit area will be taken the square whose side is equal to unity (the radius of the
+sphere), and then the whole surface of the sphere becomes equal to~$4\pi$. Thus the
+part of the surface of the sphere corresponding to the triangle is to the whole surface
+of the sphere as $±(A + B + C - \pi)$ is to~$4\pi$. This theorem, which, if we mistake
+not, ought to be counted among the most elegant in the theory of curved surfaces,
+may also be stated as follows:
+
+\begin{Theorem}[]
+The excess over~$180°$ of the sum of the angles of a triangle formed by shortest lines
+on a concavo-concave curved surface, or the deficit from~$180°$ of the sum of the angles of
+a triangle formed hy shortest lines on a concavo-convex curved surface, is measured by the
+area of the part of the sphere which corresponds, through the directions of the normals, to
+that triangle, if the whole surface of the sphere is set equal to $720$~degrees.
+\end{Theorem}
+
+More generally, in any polygon whatever of $n$~sides, each formed by a shortest
+line, the excess of the sum of the angles over $(2n - 4)$~right angles, or the deficit from
+$(2n - 4)$~right angles (according to the nature of the curved surface), is equal to the
+area of the corresponding polygon on the sphere, if the whole surface of the sphere is
+set equal to $720$~degrees. This follows at once from the preceding theorem by dividing
+the polygon into triangles.
+\PageSep{31}
+
+
+\Article{21.}
+
+Let us again give to the symbols $p$,~$q$, $E$,~$F$,~$G$, $\omega$ the general meanings which
+were given to them above, and let us further suppose that the nature of the curved
+surface is defined in a similar way by two other variables, $p'$,~$q'$, in which case the
+general linear element is expressed by
+\[
+\Sqrt{E'\, dp'^{2} + 2F'\, dp'·dq' + G'\, dq'^{2}}\Add{.}
+\]
+Thus to any point whatever lying on the surface and defined by definite values of
+the variables $p$,~$q$ will correspond definite values of the variables $p'$,~$q'$, which will
+therefore be functions of $p$,~$q$. Let us suppose we obtain by differentiating them
+\begin{alignat*}{2}
+dp' &= \alpha\, dp &&+ \beta\,  dq\Add{,} \\
+dq' &= \gamma\, dp &&+ \delta\, dq\Add{.}
+\end{alignat*}
+We shall now investigate the geometric meaning of the coefficients $\alpha$,~$\beta$,~$\gamma$,~$\delta$.
+
+Now \emph{four} systems of lines may thus be supposed to lie upon the curved surface,
+for which $p$,~$q$, $p'$,~$q'$ respectively are constants. If through the definite point to
+which correspond the values $p$,~$q$, $p'$,~$q'$ of the variables we suppose the four lines
+belonging to these different systems to be drawn, the elements of these lines, corresponding
+to the positive increments $dp$,~$dq$, $dp'$,~$dq'$, will be
+\[
+\sqrt{E}·dp,\quad
+\sqrt{G}·dq,\quad
+\sqrt{E'}·dp',\quad
+\sqrt{G'}·dq'.
+\]
+The angles which the directions of these elements make with an arbitrary fixed direction
+we shall denote by $M$,~$N$, $M'$,~$N'$, measuring them in the sense in which the
+second is placed with respect to the first, so that $\sin(N - M)$ is positive. Let us
+suppose (which is permissible) that the fourth is placed in the same sense with respect
+to the third, so that $\sin(N' - M')$ also is positive. Having made these conventions,
+if we consider another point at an infinitely small distance from the first point, and
+to which correspond the values $p + dp$, $q + dq$, $p' + dp'$, $q' + dq'$ of the variables, we
+see without much difficulty that we shall have generally, \ie, independently of the
+values of the increments $dp$,~$dq$, $dp'$,~$dq'$,
+\[
+\sqrt{E}·dp·\sin M + \sqrt{G}·dq·\sin N
+ = \sqrt{E'}·dp'·\sin M' + \sqrt{G'}·dq'·\sin N'\Add{,}
+\]
+since each of these expressions is merely the distance of the new point from the line
+from which the angles of the directions begin. But we have, by the notation introduced
+above,
+\[
+N - M = \omega.
+\]
+In like manner we set
+\[
+N' - M' = \omega',
+\]
+\PageSep{32}
+and also
+\[
+N - M' = \psi.
+\]
+Then the equation just found can be thrown into the following form:
+\begin{multline*}
+\sqrt{E}·dp · \sin(M' - \omega + \psi) + \sqrt{G}·dq·\sin(M' + \psi) \\
+= \sqrt{E'}·dp'·\sin M' + \sqrt{G'}·dq'·\sin(M' + \omega')\Add{,}
+\end{multline*}
+or
+\begin{multline*}
+\sqrt{E}·dp·\sin(N' - \omega - \omega' + \psi)
+  - \sqrt{G}·dq·\sin(N' - \omega' + \psi) \\
+  = \sqrt{E'}·dp'·\sin(N' - \omega') + \sqrt{G'}·dq'·\sin N'\Add{.}
+\end{multline*}
+And since the equation evidently must be independent of the initial direction, this
+direction can be chosen arbitrarily. Then, setting in the second formula $N' = 0$, or in
+the first $M' = 0$, we obtain the following equations:
+\begin{align*}
+\sqrt{E'}·\sin \omega'·dp'
+  &= \sqrt{E}·\sin(\omega + \omega' - \psi)·dp
+   + \sqrt{G}·\sin(\omega' - \psi)·dq\Add{,} \\
+\sqrt{G'}·\sin \omega'·dq'
+  &= \sqrt{E}·\sin(\psi - \omega)·dp + \sqrt{G}·\sin\psi·dq\Add{;}
+\end{align*}
+and these equations, since they must be identical with
+\begin{alignat*}{2}
+dp' &= \alpha\, dp &&+ \beta\,  dq\Add{,} \\
+dq' &= \gamma\, dp &&+ \delta\, dq\Add{,}
+\end{alignat*}
+determine the coefficients $\alpha$,~$\beta$,~$\gamma$,~$\delta$. We shall have
+\begin{align*}
+\alpha &= \sqrt{\frac{E}{E'}}·\frac{\sin(\omega + \omega' - \psi)}{\sin\omega'}, &
+\beta  &= \sqrt{\frac{G}{E'}}·\frac{\sin(\omega' - \psi)}{\sin\omega'}\Add{,} \\
+\gamma &= \sqrt{\frac{E}{G'}}·\frac{\sin(\psi - \omega)}{\sin\omega'}, &
+\delta &= \sqrt{\frac{G}{G'}}·\frac{\sin\psi}{\sin\omega'}\Add{.}
+\end{align*}
+These four equations, taken in connection with the equations
+\begin{align*}
+\cos\omega  &= \frac{F}{\sqrt{EG}}, &
+\cos\omega' &= \frac{F'}{\sqrt{E'G'}}, \\
+\sin\omega  &= \sqrt{\frac{EG - F^{2}}{EG}}, &
+\sin\omega' &= \sqrt{\frac{E'G' - F'^{2}}{E'G'}},
+\end{align*}
+may be written
+\begin{align*}
+\alpha\Sqrt{E'G' - F'^{2}} &= \sqrt{EG'}·\sin(\omega + \omega' - \psi)\Add{,} \\
+\beta \Sqrt{E'G' - F'^{2}} &= \sqrt{GG'}·\sin(\omega' - \psi)\Add{,} \\
+\gamma\Sqrt{E'G' - F'^{2}} &= \sqrt{EE'}·\sin(\psi - \omega)\Add{,} \\
+\delta\Sqrt{E'G' - F'^{2}} &= \sqrt{GE'}·\sin \psi\Add{.}
+\end{align*}
+
+Since by the substitutions
+\begin{alignat*}{2}
+dp' &= \alpha\, dp &&+ \beta\,  dq, \\
+dq' &= \gamma\, dp &&+ \delta\, dq\Add{,}
+\end{alignat*}
+\PageSep{33}
+the trinomial
+\[
+E'\, dp'^{2} + 2F'\, dp'·dq' + G'\, dq'^{2}
+\]
+is transformed into
+\[
+E\, dp^{2} + 2F\, dp·dq + G\, dq^{2},
+\]
+we easily obtain
+\[
+EG - F^{2} = (E'G' - F'^{2})(\alpha\delta - \beta\gamma)^{2}\Add{;}
+\]
+and since, \textit{vice versa}, the latter trinomial must be transformed into the former by the
+substitution
+\[
+(\alpha\delta - \beta\gamma)\, dp =  \delta\, dp' - \beta\, dq',\quad
+(\alpha\delta - \beta\gamma)\, dq = -\gamma\, dp' + \alpha\, dq',
+\]
+we find\Note{16}
+\begin{align*}
+E\delta^{2} - 2F\gamma\delta + G\gamma^{2}
+  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·E'\Add{,} \\
+%
+-E\beta\delta + F(\alpha\delta + \beta\gamma) - G\alpha\gamma
+  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·F'\Add{,}\NoteMark \\
+%
+E\beta^{2} - 2F\alpha\beta + G\alpha^{2}
+  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·G'\Add{.}
+\end{align*}
+
+
+\Article{22.}
+
+From the general discussion of the preceding article we proceed to the very
+extended application in which, while keeping for $p$,~$q$ their most general meaning, we
+take for $p'$,~$q'$ the quantities denoted in \Art{15} by $r$,~$\phi$. We shall use $r$,~$\phi$ here
+also in such a way that, for any point whatever on the surface, $r$~will be the shortest
+distance from a fixed point, and $\phi$~the angle at this point between the first element
+of~$r$ and a fixed direction. We have thus
+\[
+E' = 1,\quad
+F' = 0,\quad
+\omega' = 90°.
+\]
+Let us set also
+\[
+\sqrt{G'} = m,
+\]
+so that any linear element whatever becomes equal to
+\[
+\Sqrt{dr^{2} + m^{2}\, d\phi^{2}}.
+\]
+Consequently, the four equations deduced in the preceding article for $\alpha$,~$\beta$,~$\gamma$,~$\delta$ give
+\begin{align*}
+\sqrt{E}·\cos(\omega - \psi) = \frac{\dd r}{\dd p}\Add{,}
+\Tag{(1)} \\
+\sqrt{G}·\cos \psi = \frac{\dd r}{\dd q}\Add{,}
+\Tag{(2)} \displaybreak[1] \\
+\PageSep{34}
+\sqrt{E}·\sin(\psi - \omega) = m·\frac{\dd\phi}{\dd p}\Add{,}
+\Tag{(3)} \\
+\sqrt{G}·\sin\psi = m·\frac{\dd\phi}{\dd q}\Add{.}
+\Tag{(4)}
+\end{align*}
+But the last and the next to the last equations of the preceding article give
+\begin{gather*}
+EG - F^{2}
+  = E\left(\frac{\dd r}{\dd q}\right)^{2}
+  - 2F·\frac{\dd r}{\dd p}·\frac{\dd r}{\dd q}
+  + G\left(\frac{\dd r}{\dd p}\right)^{2}\Add{,}
+\Tag{(5)} \\
+\left(E·\frac{\dd r}{\dd q} - F·\frac{\dd r}{\dd p}\right)·\frac{\dd\phi}{\dd q}
+  = \left(F·\frac{\dd r}{\dd q} - G·\frac{\dd r}{\dd p}\right)·\frac{\dd\phi}{\dd p}\Add{.}
+\Tag{(6)}
+\end{gather*}
+
+From these equations must be determined the quantities $r$,~$\phi$,~$\psi$ and (if need be)~$m$,
+as functions of $p$~and~$q$. Indeed, integration of equation~(5) will give~$r$; $r$~being
+found, integration of equation~(6) will give~$\phi$; and one or other of equations (1),~(2)
+will give $\psi$~itself. Finally, $m$~is obtained from one or other of equations (3),~(4).
+
+The general integration of equations (5),~(6) must necessarily introduce two arbitrary
+functions. We shall easily understand what their meaning is, if we remember
+that these equations are not limited to the case we are here considering, but are
+equally valid if $r$~and~$\phi$ are taken in the more general sense of \Art{16}, so that $r$~is
+the length of the shortest line drawn normal to a fixed but arbitrary line, and $\phi$~is
+an arbitrary function of the length of that part of the fixed line which is intercepted
+between any shortest line and an arbitrary fixed point. The general solution must
+embrace all this in a general way, and the arbitrary functions must go over into
+definite functions only when the arbitrary line and the arbitrary functions of its
+parts, which $\phi$~must represent, are themselves defined. In our case an infinitely
+small circle may be taken, having its centre at the point from which the distances~$r$
+are measured, and $\phi$~will denote the parts themselves of this circle, divided by the
+radius. Whence it is easily seen that the equations (5),~(6) are quite sufficient for
+our case, provided that the functions which they leave undefined satisfy the condition
+which $r$~and~$\phi$ satisfy for the initial point and for points at an infinitely small
+distance from this point.
+
+Moreover, in regard to the integration itself of the equations (5),~(6), we know
+that it can be reduced to the integration of ordinary differential equations, which, however,
+often happen to be so complicated that there is little to be gained by the reduction.
+On the contrary, the development in series, which are abundantly sufficient for
+practical requirements, when only a finite portion of the surface is under consideration,
+presents no difficulty; and the formulæ thus derived open a fruitful source for
+\PageSep{35}
+the solution of many important problems. But here we shall develop only a single
+example in order to show the nature of the method.
+
+
+\Article{23.}
+
+We shall now consider the case where all the lines for which $p$~is constant are
+shortest lines cutting orthogonally the line for which $\phi = 0$, which line we can regard
+as the axis of abscissas. Let $A$~be the point for which $r = 0$, $D$~any point whatever
+on the axis of abscissas, $AD = p$, $B$~any point whatever on the shortest line normal
+to~$AD$ at~$D$, and $BD = q$, so that $p$~can be regarded as the abscissa, $q$~the ordinate
+of the point~$B$. The abscissas we assume positive on the branch of the axis of
+abscissas to which $\phi = 0$ corresponds, while we always regard~$r$ as positive. We take
+the ordinates positive in the region in which $\phi$~is measured between $0$~and~$180°$.
+
+By the theorem of \Art{16} we shall have
+\[
+\omega = 90°,\quad
+F = 0,\quad
+G = 1,
+\]
+and we shall set also
+\[
+\sqrt{E} = n.
+\]
+Thus $n$~will be a function of $p$,~$q$, such that for $q = 0$ it must become equal to unity.
+The application of the formula of \Art{18} to our case shows that on any shortest
+line \emph{whatever} we must have\Note{17}
+\[
+d\theta = \frac{\dd n}{\dd q}·dp,\NoteMark
+\]
+where $\theta$~denotes the angle between the element of this line and the element of the
+line for which $q$~is constant. Now since the axis of abscissas is itself a shortest line,
+and since, for it, we have everywhere $\theta = 0$, we see that for $q = 0$ we must have
+everywhere
+\[
+\frac{\dd n}{\dd q} = 0.
+\]
+Therefore we conclude that, if $n$~is developed into a series in ascending powers of~$q$,
+this series must have the following form:
+\[
+n = 1 + fq^{2} + gq^{3} + hq^{4} + \text{etc.}\Add{,}
+\]
+where $f$,~$g$,~$h$,~etc., will be functions of~$p$, and we set
+\begin{alignat*}{4}
+f &= f° &&+ f'p &&+ f''p^{2} &&+ \text{etc.}\Add{,} \\
+g &= g° &&+ g'p &&+ g''p^{2} &&+ \text{etc.}\Add{,} \\
+h &= h° &&+ h'p &&+ h''p^{2} &&+ \text{etc.}\Add{,}
+\end{alignat*}
+\PageSep{36}
+or
+\begin{alignat*}{2}
+n = 1 + f°q^{2} &+ f'pq^{2}   &&+ f''p^{2}q^{2} + \text{etc.} \\
+                  &+ g° q^{3} &&+ g'pq^{3} + \text{etc.} \\
+                  &             &&+ h°q^{4} + \text{etc.\ etc.}
+\end{alignat*}
+
+
+\Article{24.}
+
+The equations of \Art{22} give, in our case,
+\begin{gather*}
+ n\sin\psi = \frac{\dd r}{\dd p},\quad
+  \cos\psi = \frac{\dd r}{\dd q},\quad
+-n\cos\psi = m·\frac{\dd\phi}{\dd r},\quad
+  \sin\psi = m·\frac{\dd\phi}{\dd q}, \\
+%
+n^{2} = n^{2}\left(\frac{\dd r}{\dd q}\right)^{2}
+          + \left(\frac{\dd r}{\dd p}\right)^{2},\quad
+n^{2}·\frac{\dd r}{\dd q}·\frac{\dd\phi}{\dd q}
+      + \frac{\dd r}{\dd p}·\frac{\dd\phi}{\dd p} = 0\Add{.}
+\end{gather*}
+By the aid of these equations, the fifth and sixth of which are contained in the others,
+series can be developed for $r$,~$\phi$,~$\psi$,~$m$, or for any functions whatever of these quantities.
+We are going to establish here those series that are especially worthy of
+attention.
+
+Since for infinitely small values of $p$,~$q$ we must have
+\[
+r^{2} = p^{2} + q^{2},
+\]
+the series for~$r^{2}$ will begin with the terms $p^{2} + q^{2}$. We obtain the terms of higher
+order by the method of undetermined coefficients,\footnote
+  {We have thought it useless to give the calculation here, which can be somewhat abridged by
+  certain artifices.}
+by means of the equation
+\[
+\left(\frac{1}{n}·\frac{\dd(r^{2})}{\dd p}\right)^{2}
+  + \left(\frac{\dd(r^{2})}{\dd q}\right)^{2} = 4r^{2}\Add{.}
+\]
+Thus we have\Note{18}
+\begin{alignat*}{3}
+\Tag{[1]}
+r^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q^{2} &&+ \tfrac{1}{2}f'p^{3}q^{2} &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2}\quad\text{etc.} \\
+      &+ q^{2} &&+ \tfrac{1}{2}g°p^{2}q^{3} &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
+      &&& &&+(\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}\Add{.}
+\end{alignat*}
+
+Then we have, from the formula\Note{19}
+\[
+r\sin\psi = \frac{1}{2n}·\frac{\dd(r^{2})}{\dd p},
+\]
+\begin{alignat*}{2}
+\Tag{[2]}
+r\sin\psi = p - \tfrac{1}{3}f°pq^{2}
+  & -\tfrac{1}{4}f'p^{2}q^{2} &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\quad\text{etc.} \\
+  & -\tfrac{1}{2}g°pq^{3} &&-\tfrac{2}{5}g'p^{2}q^{3} \\
+  &&& -(\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})pq^{4}\Add{;}
+\end{alignat*}
+\PageSep{37}
+and from the formula\Note{20}
+\[
+r\cos\psi = \tfrac{1}{2}\, \frac{\dd(r^{2})}{\dd q}\Add{,}
+\]
+\begin{alignat*}{2}
+\Tag{[3]}
+r\cos\psi = q + \tfrac{2}{3}f°p^{2}q
+  & +\tfrac{1}{2}f'p^{3}q &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\quad\text{etc.} \\
+  & +\tfrac{3}{4}g°p^{2}q^{2} &&+\tfrac{3}{5}g'p^{3}q^{2} \\
+  &&& +(\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\Add{.}
+\end{alignat*}
+These formulæ give the angle~$\psi$. In like manner, for the calculation of the angle~$\phi$,
+series for $r\cos\phi$ and $r\sin\phi$ are very elegantly developed by means of the partial
+differential equations
+\begin{align*}
+&\frac{\dd·r\cos\phi}{\dd p}
+  = n\cos\phi·\sin\psi - r\sin\phi·\frac{\dd\phi}{\dd p}\Add{,} \\
+&\frac{\dd·r\cos\phi}{\dd q}
+  = \Z\cos\phi·\cos\psi - r\sin\phi·\frac{\dd\phi}{\dd q}\Add{,} \\
+&\frac{\dd·r\sin\phi}{\dd p}
+  = n\sin\phi·\sin\psi + r\cos\phi·\frac{\dd\phi}{\dd p}\Add{,} \\
+&\frac{\dd·r\sin\phi}{\dd q}
+  = \Z\sin\phi·\cos\psi + r\cos\phi·\frac{\dd\phi}{\dd q}\Add{,} \\
+&n\cos\psi·\frac{\dd\phi}{\dd q} + \sin\psi·\frac{\dd\phi}{\dd p} = 0\Add{.}
+\end{align*}
+A combination of these equations gives
+\begin{alignat*}{3}
+&\frac{r\sin\psi}{n}·\frac{\dd·r\cos\phi}{\dd p}
+  &&+ r\cos\psi·\frac{\dd·r\cos\phi}{\dd q}
+  &&= r\cos\phi\Add{,} \\
+&\frac{r\sin\psi}{n}·\frac{\dd·r\sin\phi}{\dd p}
+  &&+ r\cos\psi·\frac{\dd·r\sin\phi}{\dd q}
+  &&= r\sin\phi\Add{.}
+\end{alignat*}
+From these two equations series for $r\cos\phi$, $r\sin\phi$ are easily developed, whose first
+terms must evidently be $p$,~$q$ respectively. The series are\Note{21}
+\begin{alignat*}{3}
+\Tag{[4]}
+r\cos\phi &= p + \tfrac{2}{3}f°pq^{2}
+  &&+ \tfrac{5}{12}f'p^{2}q^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\quad\text{etc.} \\
+ &&&+ \tfrac{1}{2}g°pq^{3}
+  &&+ \tfrac{7}{20}g'p^{2}q^{3} \\
+&&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}\Add{,} \displaybreak[1] \\
+\Tag{[5]}
+r\sin\phi &= q - \tfrac{1}{3}f°p^{2}q
+  &&- \tfrac{1}{6}f'p^{3}q
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q\quad\text{etc.} \\
+ &&&- \tfrac{1}{4}g°p^{2}q^{2}
+  &&- \tfrac{3}{20}g'p^{3}q^{2} \\
+&&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}\Add{.}
+\end{alignat*}
+From a combination of equations [2],~[3],~[4],~[5] a series for $r^{2}\cos(\psi + \phi)$, may
+be derived, and from this, dividing by the series~[1], a series for $\cos(\psi + \phi)$, from
+\PageSep{38}
+which may be found a series for the angle $\psi + \phi$ itself. However, the same series
+can be obtained more elegantly in the following manner. By differentiating the first
+and second of the equations introduced at the beginning of this article, we obtain
+\[
+\sin\psi·\frac{\dd n}{\dd q}
+  + n\cos\psi·\frac{\dd\psi}{\dd q}
+  +  \sin\psi·\frac{\dd\psi}{\dd p} = 0\Add{,}
+\]
+and this combined with the equation
+\[
+n\cos\psi·\frac{\dd\phi}{\dd q} + \sin\psi·\frac{\dd\phi}{\dd p} = 0
+\]
+gives
+\[
+\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q}
+  + \frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(\psi + \phi)}{\dd q} = 0\Add{.}
+\]
+From this equation, by aid of the method of undetermined coefficients, we can easily
+derive the series for $\psi + \phi$, if we observe that its first term must be~$\frac{1}{2}\pi$, the radius
+being taken equal to unity and $2\pi$~denoting the circumference of the circle,\Note{22}
+\begin{alignat*}{2}
+\Tag{[6]}
+\psi + \phi = \tfrac{1}{2}\pi - f°pq
+  &- \tfrac{2}{3}f'p^{2}q
+  &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q\quad\text{etc.} \\
+  &- g°pq^{2}
+  &&- \tfrac{3}{4}g'p^{2}q^{2} \\
+  &&&- (h° - \tfrac{1}{3}{f°}^{2})pq^{3}\Add{.}
+\end{alignat*}
+
+It seems worth while also to develop the area of the triangle~$ABD$ into a series.
+For this development we may use the following conditional equation, which is easily
+derived from sufficiently obvious geometric considerations, and in which $S$~denotes the
+required area:\Note{23}
+\[
+\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p}
+  + r\cos\psi·\frac{\dd S}{\dd q}
+  = \frac{r\sin\psi}{n}·\int n\, dq\Add{,}\NoteMark
+\]
+the integration beginning with $q = 0$. From this equation we obtain, by the method
+of undetermined coefficients,\Note{24}
+\begin{alignat*}{3}
+\Tag{[7]}
+S = \tfrac{1}{2}pq
+  &- \tfrac{1}{12}f°p^{3}q
+  &&- \tfrac{1}{20}f'p^{4}q
+  &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q\quad\text{etc.} \\
+  &- \tfrac{1}{12}f°pq^{3}
+  &&- \tfrac{3}{40}g°p^{3}q^{2}
+  &&- \tfrac{1}{20}g'p^{4}q^{2} \\
+  &&&- \tfrac{7}{120}f'p^{2}q^{3}
+  &&- (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
+  &&&- \tfrac{1}{10}g°pq^{4}
+  &&- \tfrac{3}{40}g'p^{2}q^{4} \\
+  &&&&&- (\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}\Add{.}
+\end{alignat*}
+\PageSep{39}
+
+
+\Article{25.}
+
+From the formulæ of the preceding article, which refer to a right triangle formed
+by shortest lines, we proceed to the general case. Let $C$ be another point on the
+same shortest line~$DB$, for which point~$p$ remains the same as for the point~$B$, and
+$q'$,~$r'$, $\phi'$,~$\psi'$, $S'$ have the same meanings as $q$,~$r$, $\phi$,~$\psi$, $S$ have for the point~$B$. There
+will thus be a triangle between the points $A$,~$B$,~$C$, whose angles we denote by
+$A$,~$B$,~$C$, the sides opposite these angles by $a$,~$b$,~$c$, and the area by~$\sigma$. We represent
+the measure of curvature at the points $A$,~$B$,~$C$ by $\alpha$,~$\beta$,~$\gamma$ respectively. And then
+supposing (which is permissible) that the quantities $p$,~$q$,~$q - q'$ are positive, we shall
+have
+\begin{align*}
+A &= \phi - \phi', & B &= \psi, & C &= \pi - \psi', && \\
+a &= q - q',       & b &= r',   & c &= r,           & \sigma &= S - S'.
+\end{align*}
+
+We shall first express the area~$\sigma$ by a series. By changing in~[7] each of the
+quantities that refer to~$B$ into those that refer to~$C$, we obtain a formula for~$S'$.
+Whence we have, exact to quantities of the sixth order,\Note{25}
+\begin{align*}
+\sigma = \tfrac{1}{2}p(q - q')
+  \bigl(1 &- \tfrac{1}{6} f°(p^{2} + q^{2} + qq' + q'^{2}) \\
+          &- \tfrac{1}{60}f'p(6p^{2} + 7q^{2} + 7qq' + 7q'^{2}) \\
+          &- \tfrac{1}{20}g°(q + q')(3p^{2} + 4q^{2} + 4q'^{2})\bigr)\Add{.}\NoteMark
+\end{align*}
+This formula, by aid of series~[2], namely,
+\[
+c\sin B = p(1 - \tfrac{1}{3}f°q^{2}
+              - \tfrac{1}{4}f'pq^{2}
+              - \tfrac{1}{2}g°q^{3} - \text{etc.})
+\]
+can be changed into the following:
+\begin{align*}
+\sigma = \tfrac{1}{2}ac\sin B
+  \bigl(1 &- \tfrac{1}{6} f°(p^{2} - q^{2} + qq' + q'^{2}) \\
+          &- \tfrac{1}{60}f'p(6p^{2} - 8q^{2} + 7qq' + 7q'^{2}) \\
+          &- \tfrac{1}{20}g°
+    (3p^{2}q + 3p^{2}q' - 6p^{3} + 4q^{2}q' + 4qq'^{2} + 4q'^{3})\bigr)\Add{.}
+\end{align*}
+
+The measure of curvature for any point whatever of the surface becomes (by \Art{19},
+where $m$,~$p$,~$q$ were what $n$,~$q$,~$p$ are here)
+\begin{align*}
+k &= -\frac{1}{n}·\frac{\dd^{2} n}{\dd q^{2}}
+   = -\frac{2f + 6gq + 12hq^{2} + \text{etc.}}{1 + fq^{2} + \text{etc.}} \\
+  &= -2f - 6gq - (12h - 2f^{2}) q^{2} - \text{etc.}
+\end{align*}
+Therefore we have, when $p$,~$q$ refer to the point~$B$,
+\[
+\beta = - 2f° - 2f'p - 6g°q - 2f''p^{2} - 6g'pq - (12h° - 2{f°}^{2})q^{2} - \text{etc.}
+\]
+\PageSep{40}
+Also
+\begin{align*}
+\gamma &= -2f° - 2f'p - 6g°q' - 2f''p^{2} - 6g'pq' - (12h° - 2{f°}^{2})q'^{2} - \text{etc.}\Add{,} \\
+\alpha &= -2f°\Add{.}
+\end{align*}
+Introducing these measures of curvature into the expression for~$\sigma$, we obtain the following
+expression, exact to quantities of the sixth order (exclusive):\Note{26}
+\begin{align*}
+\sigma = \tfrac{1}{2}ac \sin B
+  \bigl(1 &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} + 3qq' + 3q'^{2}) \\
+          &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} + 6qq' + 3q'^{2}) \\
+          &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} + \Z qq' + 4q'^{2})\bigr)\Add{.}\NoteMark
+\end{align*}
+The same precision will remain, if for $p$,~$q$,~$q'$ we substitute $c\sin B$, $c\cos B$, $c\cos B - a$.
+This gives\Note{27}
+\begin{align*}
+\Tag{[8]}
+\sigma = \tfrac{1}{2}ac\sin B
+  \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \Z9ac\cos B) \\
+          &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\
+          &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \Z9ac\cos B)\bigr)\Add{.}
+\end{align*}
+Since all expressions which refer to the line~$AD$ drawn normal to~$BC$ have disappeared
+from this equation, we may permute among themselves the points $A$,~$B$,~$C$ and
+the expressions that refer to them. Therefore we shall have, with the same precision,
+\begin{align*}
+\Tag{[9]}
+\sigma = \tfrac{1}{2}bc\sin A
+  \bigl(1 &+ \tfrac{1}{120}\alpha(3b^{2} + 3c^{2} -  12bc\cos A) \\
+          &+ \tfrac{1}{120}\beta (3b^{2} + 4c^{2} - \Z9bc\cos A) \\
+          &+ \tfrac{1}{120}\gamma(4b^{2} + 3c^{2} - \Z9bc\cos A)\bigr)\Add{,} \\
+%
+\Tag{[10]}
+\sigma = \tfrac{1}{2}ab\sin C
+  \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4b^{2} - \Z9ab\cos C) \\
+          &+ \tfrac{1}{120}\beta (4a^{2} + 3b^{2} - \Z9ab\cos C) \\
+          &+ \tfrac{1}{120}\gamma(3a^{2} + 3b^{2} -  12ab\cos C)\bigr)\Add{.}
+\end{align*}
+
+
+\Article{26.}
+
+The consideration of the rectilinear triangle whose sides are equal to $a$,~$b$,~$c$ is of
+great advantage. The angles of this triangle, which we shall denote by $A^{*}$,~$B^{*}$,~$C^{*}$,
+differ from the angles of the triangle on the curved surface, namely, from $A$,~$B$,~$C$,
+by quantities of the second order; and it will be worth while to develop these differences
+accurately. However, it will be sufficient to show the first steps in these more
+tedious than difficult calculations.
+
+Replacing in formulæ [1],~[4],~[5] the quantities that refer to~$B$ by those that
+refer to~$C$, we get formulæ for $r'^{2}$,~$r'\cos\phi'$, $r'\sin\phi'$. Then the development of the
+expression
+\PageSep{41}
+\begin{align*}
+  r^{2} + r'^{2} &- (q - q')^{2}
+    - 2r\cos\phi·r'\cos\phi'
+    - 2r\sin\phi·r'\sin\phi' \\
+    &\quad= b^{2} + c^{2} - a^{2} - 2bc\cos A \\
+    &\quad= 2bc(\cos A^{*} - \cos A),
+\end{align*}
+combined with the development of the expression
+\[
+r\sin\phi·r'\cos\phi' - r\cos\phi·r'\sin\phi' = bc\sin A,
+\]
+gives the following formula:
+\begin{align*}
+\cos A^{*} - \cos A
+  = -(q - q')p\sin A
+  \bigl(\tfrac{1}{3}f° &+ \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') \\
+  &+ (\tfrac{1}{10}f'' - \tfrac{1}{45}{f°}^{2})p^{2}
+   + \tfrac{3}{20}g'p(q + q') \\
+  &+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})(q^{2} + qq' + q'^{2})
+   + \text{etc.}\bigr)
+\end{align*}
+From this we have, to quantities of the fifth order,\Note{28}
+\begin{align*}
+A^{*} - A = +(q - q')p
+  \bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{3}{20}g'p(q + q') + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 12qq' + 7q'^{2})\bigr)\Add{.}\NoteMark
+\end{align*}
+Combining this formula with
+\[
+2\sigma = ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})- \text{etc.}\bigr)
+\]
+and with the values of the quantities $\alpha$,~$\beta$,~$\gamma$ found in the preceding article, we obtain,
+to quantities of the fifth order,\Note{29}
+\begin{align*}
+\Tag{[11]}
+A^{*} = A - \sigma\bigl(\tfrac{1}{6}\alpha
+  &+ \tfrac{1}{12}\beta
+   + \tfrac{1}{12}\gamma + \tfrac{2}{15}f''p^{2} + \tfrac{1}{5}g'p(q + q') \\
+  &+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2}) \\
+  &+ \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr)\Add{.}
+\end{align*}
+By precisely similar operations we derive\Note{30}
+\begin{align*}
+\Tag{[12]}
+B^{*} = B - \sigma\bigl(\tfrac{1}{12}\alpha
+  &+ \tfrac{1}{6}\beta
+   + \tfrac{1}{12}\gamma + \tfrac{1}{10}f''p^{2} + \tfrac{1}{10}g'p(2q + q') \\
+  &+ \tfrac{1}{5}h°(4q^{2} - 4qq' + 3q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} - 8qq' + 11q'^{2})\bigr)\Add{,}
+\displaybreak[1] \\
+%
+\Tag{[13]}
+C^{*} = C - \sigma\bigl(\tfrac{1}{12}\alpha
+  &+ \tfrac{1}{12}\beta
+   + \tfrac{1}{6}\gamma + \tfrac{1}{10}f''p^{2} + \tfrac{1}{10}g'p(q + 2q') \\
+  &+ \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr)\Add{.}
+\end{align*}
+From these formulæ we deduce, since the sum $A^{*} + B^{*} + C^{*}$ is equal to two right
+angles, the excess of the sum~$A + B + C$ over two right angles, namely,\Note{31}
+\begin{align*}
+\Tag{[14]}
+A + B + C= \pi + \sigma\bigl(\tfrac{1}{3}\alpha
+  &+ \tfrac{1}{3}\beta
+   + \tfrac{1}{3}\gamma + \tfrac{1}{3}f''p^{2} + \tfrac{1}{2}g'p(q + q') \\
+  &+ (2h° - \tfrac{1}{3}{f°}^{2})(q^{2} - qq' + q'^{2})\bigr)\Add{.}
+\end{align*}
+This last equation could also have been derived from formula~[6].
+\PageSep{42}
+
+
+\Article{27.}
+
+If the curved surface is a sphere of radius~$R$, we shall have
+\[
+\alpha = \beta = \gamma = -2f° = \frac{1}{R^{2}};\quad
+f'' = 0,\quad
+g' = 0,\quad
+6h° - {f°}^{2} = 0,
+\]
+or
+\[
+h° = \frac{1}{24R^{4}}.
+\]
+Consequently, formula~[14] becomes
+\[
+A + B + C = \pi + \frac{\sigma}{R^{2}},
+\]
+which is absolutely exact. But formulæ [11],~[12],~[13] give
+\begin{align*}
+A^{*} &= A - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(2p^{2} - q^{2} + 4qq' - q'^{2})\Add{,} \\
+B^{*} &= B - \frac{\sigma}{3R^{2}}
+  + \frac{\sigma}{180R^{4}}(p^{2} - 2q^{2} + 2qq' + q'^{2})\Add{,} \\
+C^{*} &= C - \frac{\sigma}{3R^{2}}
+  + \frac{\sigma}{180R^{4}}(p^{2} + q^{2} + 2qq' - 2q'^{2})\Add{;}
+\end{align*}
+or, with equal exactness,
+\begin{align*}
+A^{*} &= A - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(b^{2} + c^{2} - 2a^{2})\Add{,} \\
+B^{*} &= B - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(a^{2} + c^{2} - 2b^{2})\Add{,} \\
+C^{*} &= C - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(a^{2} + b^{2} - 2c^{2})\Add{.}
+\end{align*}
+Neglecting quantities of the fourth order, we obtain from the above the well-known
+theorem first established by the illustrious Legendre.
+
+
+\Article{28.}
+
+Our general formulæ, if we neglect terms of the fourth order, become extremely
+simple, namely:
+\begin{align*}
+A^{*} &= A - \tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\Add{,} \\
+B^{*} &= B - \tfrac{1}{12}\sigma(\alpha + 2\beta + \gamma)\Add{,} \\
+C^{*} &= C - \tfrac{1}{12}\sigma(\alpha + \beta + 2\gamma)\Add{.}
+\end{align*}
+\PageSep{43}
+
+Thus to the angles $A$,~$B$,~$C$ on a non-spherical surface, unequal reductions must
+be applied, so that the sines of the changed angles become proportional to the sides
+opposite. The inequality, generally speaking, will be of the third order; but if the
+surface differs little from a sphere, the inequality will be of a higher order. Even in
+the greatest triangles on the earth's surface, whose angles it is possible to measure,
+the difference can always be regarded as insensible. Thus, \eg, in the greatest of
+the triangles which we have measured in recent years, namely, that between the
+points Hohehagen, Brocken, Inselberg, where the excess of the sum of the angles was
+$14''. 85348$, the calculation gave the following reductions to be applied to the angles:
+\setlength{\TmpLen}{2in}%
+\begin{align*}
+\Dotrow{Hohehagen}{-4''.95113\rlap{\Add{,}}} \\
+\Dotrow{Brocken}{- 4''.95104\rlap{\Add{,}}} \\
+\Dotrow{Inselberg}{-4''.95131\rlap{.}}
+\end{align*}
+
+
+\Article{29.}
+
+We shall conclude this study by comparing the area of a triangle on a curved
+surface with the area of the rectilinear triangle whose sides are $a$,~$b$,~$c$. We shall
+denote the area of the latter by~$\sigma^{*}$; hence
+\[
+\sigma^{*} = \tfrac{1}{2}bc\sin A^{*}
+           = \tfrac{1}{2}ac\sin B^{*}
+           = \tfrac{1}{2}ab\sin C^{*}\Add{.}
+\]
+
+We have, to quantities of the fourth order,
+\[
+\sin A^{*} = \sin A - \tfrac{1}{12}\sigma\cos A·(2\alpha + \beta + \gamma)\Add{,}
+\]
+or, with equal exactness,
+\[
+\sin A = \sin A^{*}·\bigl(1 + \tfrac{1}{24}bc\cos A·(2\alpha + \beta + \gamma)\bigr)\Add{.}
+\]
+Substituting this value in formula~[9], we shall have, to quantities of the sixth order,
+\begin{align*}
+\sigma = \tfrac{1}{2}bc\sin A^{*}·\bigl(1
+  &+ \tfrac{1}{120}\alpha(3b^{2}+ 3c^{2} - 2bc\cos A) \\
+  &+ \tfrac{1}{120}\beta (3b^{2}+ 4c^{2} - 4bc\cos A) \\
+  &+ \tfrac{1}{120}\gamma(4b^{2}+ 3c^{2} - 4bc\cos A)\bigr),
+\end{align*}
+or, with equal exactness,
+\[
+\sigma = \sigma^{*}\bigl(1
+  + \tfrac{1}{120}\alpha(a^{2} + 2b^{2} + 2c^{2})
+  + \tfrac{1}{120}\beta (2a^{2} + b^{2} + 2c^{2})
+  + \tfrac{1}{120}\gamma(2a^{2} + 2b^{2} + c^{2})\Add{.}
+\]
+For the sphere this formula goes over into the following form:
+\[
+\sigma = \sigma^{*}\bigl(1 + \tfrac{1}{24}\alpha(a^{2} + b^{2} + c^{2})\bigr).
+\]
+\PageSep{44}
+It is easily verified that, with the same precision, the following formula may be taken
+instead of the above:
+\[
+\sigma
+  = \sigma^{*}\sqrt{\frac{\sin A·\sin B·\sin C}
+                         {\sin A^{*}·\sin B^{*}·\sin C^{*}}}\Add{.}
+\]
+If this formula is applied to triangles on non-spherical curved surfaces, the error, generally
+speaking, will be of the fifth order, but will be insensible in all triangles such
+as may be measured on the earth's surface.
+\PageSep{45}
+
+
+\Abstract{\small GAUSS'S ABSTRACT OF THE DISQUISITIONES GENERALES CIRCA \\
+SUPERFICIES CURVAS, PRESENTED TO THE ROYAL \\
+SOCIETY OF GÖTTINGEN. \\
+\tb \\[8pt]
+
+\footnotesize\textsc{Göttingische gelehrte Anzeigen.} No.~177. Pages 1761--1768. 1827. November~5.
+}
+
+On the 8th~of October, Hofrath Gauss presented to the Royal Society a paper:
+\begin{center}
+\textit{Disquisitiones generales circa superficies curvas.}
+\end{center}
+
+Although geometers have given much attention to general investigations of curved
+surfaces and their results cover a significant portion of the domain of higher geometry,
+this subject is still so far from being exhausted, that it can well be said that, up to
+this time, but a small portion of an exceedingly fruitful field has been cultivated.
+Through the solution of the problem, to find all representations of a given surface upon
+another in which the smallest elements remain unchanged, the author sought some
+years ago to give a new phase to this study. The purpose of the present discussion
+is further to open up other new points of view and to develop some of the new truths
+which thus become accessible. We shall here give an account of those things which
+can be made intelligible in a few words. But we wish to remark at the outset that
+the new theorems as well as the presentations of new ideas, if the greatest generality
+is to be attained, are still partly in need of some limitations or closer determinations,
+which must be omitted here.
+
+In researches in which an infinity of directions of straight lines in space is concerned,
+it is advantageous to represent these directions by means of those points upon
+a fixed sphere, which are the end points of the radii drawn parallel to the lines. The
+centre and the radius of this \emph{auxiliary sphere} are here quite arbitrary. The radius may
+be taken equal to unity. This procedure agrees fundamentally with that which is constantly
+employed in astronomy, where all directions are referred to a fictitious celestial
+sphere of infinite radius. Spherical trigonometry and certain other theorems, to which
+the author has added a new one of frequent application, then serve for the solution of
+the problems which the comparison of the various directions involved can present.
+\PageSep{46}
+
+If we represent the direction of the normal at each point of the curved surface by
+the corresponding point of the sphere, determined as above indicated, namely, in this
+way, to every point on the surface, let a point on the sphere correspond; then, generally
+speaking, to every line on the curved surface will correspond a line on the sphere,
+and to every part of the former surface will correspond a part of the latter. The less
+this part differs from a plane, the smaller will be the corresponding part on the sphere.
+It is, therefore, a very natural idea to use as the measure of the total curvature,
+which is to be assigned to a part of the curved surface, the area of the corresponding
+part of the sphere. For this reason the author calls this area the \emph{integral curvature} of
+the corresponding part of the curved surface. Besides the magnitude of the part, there
+is also at the same time its \emph{position} to be considered. And this position may be in
+the two parts similar or inverse, quite independently of the relation of their magnitudes.
+The two cases can be distinguished by the positive or negative sign of the
+total curvature. This distinction has, however, a definite meaning only when the
+figures are regarded as upon definite sides of the two surfaces. The author regards
+the figure in the case of the sphere on the outside, and in the case of the curved surface
+on that side upon which we consider the normals erected. It follows then that
+the positive sign is taken in the case of convexo-convex or concavo-concave surfaces
+(which are not essentially different), and the negative in the case of concavo-convex
+surfaces. If the part of the curved surface in question consists of parts of these different
+sorts, still closer definition is necessary, which must be omitted here.
+
+The comparison of the areas of two corresponding parts of the curved surface and of
+the sphere leads now (in the same manner as, \eg, from the comparison of volume and
+mass springs the idea of density) to a new idea. The author designates as \emph{measure of
+curvature} at a point of the curved surface the value of the fraction whose denominator is
+the area of the infinitely small part of the curved surface at this point and whose numerator
+is the area of the corresponding part of the surface of the auxiliary sphere, or the
+integral curvature of that element. It is clear that, according to the idea of the author,
+integral curvature and measure of curvature in the case of curved surfaces are analogous
+to what, in the case of curved lines, are called respectively amplitude and curvature
+simply. He hesitates to apply to curved surfaces the latter expressions, which
+have been accepted more from custom than on account of fitness. Moreover, less
+depends upon the choice of words than upon this, that their introduction shall be justified
+by pregnant theorems.
+
+The solution of the problem, to find the measure of curvature at any point of a curved
+surface, appears in different forms according to the manner in which the nature of the
+curved surface is given. When the points in space, in general, are distinguished by
+\PageSep{47}
+three rectangular coordinates, the simplest method is to express one coordinate as a function
+of the other two. In this way we obtain the simplest expression for the measure of
+curvature. But, at the same time, there arises a remarkable relation between this
+measure of curvature and the curvatures of the curves formed by the intersections of
+the curved surface with planes normal to it. \textsc{Euler}, as is well known, first showed
+that two of these cutting planes which intersect each other at right angles have this
+property, that in one is found the greatest and in the other the smallest radius of curvature;
+or, more correctly, that in them the two extreme curvatures are found. It will
+follow then from the above mentioned expression for the measure of curvature that this
+will be equal to a fraction whose numerator is unity and whose denominator is the product
+of the extreme radii of curvature. The expression for the measure of curvature will be
+less simple, if the nature of the curved surface is determined by an equation in $x$,~$y$,~$z$.
+And it will become still more complex, if the nature of the curved surface is given so that
+$x$,~$y$,~$z$ are expressed in the form of functions of two new variables $p$,~$q$. In this last case
+the expression involves fifteen elements, namely, the partial differential coefficients of the
+first and second orders of $x$,~$y$,~$z$ with respect to $p$~and~$q$. But it is less important in itself
+than for the reason that it facilitates the transition to another expression, which must be
+classed with the most remarkable theorems of this study. If the nature of the curved
+surface be expressed by this method, the general expression for any linear element upon
+it, or for $\Sqrt{dx^{2} + dy^{2} + dz^{2}}$, has the form $\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}$, where $E$,~$F$,~$G$
+are again functions of $p$~and~$q$. The new expression for the measure of curvature mentioned
+above contains merely these magnitudes and their partial differential coefficients
+of the first and second order. Therefore we notice that, in order to determine the
+measure of curvature, it is necessary to know only the general expression for a linear
+element; the expressions for the coordinates $x$,~$y$,~$z$ are not required. A direct result
+from this is the remarkable theorem: If a curved surface, or a part of it, can be developed
+upon another surface, the measure of curvature at every point remains unchanged
+after the development. In particular, it follows from this further: Upon a curved
+surface that can be developed upon a plane, the measure of curvature is everywhere
+equal to zero. From this we derive at once the characteristic equation of surfaces
+developable upon a plane, namely,
+\[
+\frac{\dd^{2} z}{\dd x^{2}}·\frac{\dd^{2} z}{\dd y^{2}}
+  - \left(\frac{\dd^{2} z}{\dd x·\dd y}\right)^{2} = 0,
+\]
+when $z$~is regarded as a function of $x$~and~$y$. This equation has been known for some
+time, but according to the author's judgment it has not been established previously
+with the necessary rigor.
+\PageSep{48}
+
+These theorems lead to the consideration of the theory of curved surfaces from a
+new point of view, where a wide and still wholly uncultivated field is open to investigation.
+If we consider surfaces not as boundaries of bodies, but as bodies of which
+one dimension vanishes, and if at the same time we conceive them as flexible but not
+extensible, we see that two essentially different relations must be distinguished, namely,
+on the one hand, those that presuppose a definite form of the surface in space; on the
+other hand, those that are independent of the various forms which the surface may
+assume. This discussion is concerned with the latter. In accordance with what has
+been said, the measure of curvature belongs to this case. But it is easily seen that
+the consideration of figures constructed upon the surface, their angles, their areas and
+their integral curvatures, the joining of the points by means of shortest lines, and the
+like, also belong to this case. All such investigations must start from this, that the
+very nature of the curved surface is given by means of the expression of any linear
+element in the form $\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}$. The author has embodied in the
+present treatise a portion of his investigations in this field, made several years ago,
+while he limits himself to such as are not too remote for an introduction, and may, to
+some extent, be generally helpful in many further investigations. In our abstract, we
+must limit ourselves still more, and be content with citing only a few of them as
+types. The following theorems may serve for this purpose.
+
+If upon a curved surface a system of infinitely many shortest lines of equal lengths
+be drawn from one initial point, then will the line going through the end points of
+these shortest lines cut each of them at right angles. If at every point of an arbitrary
+line on a curved surface shortest lines of equal lengths be drawn at right angles to this
+line, then will all these shortest lines be perpendicular also to the line which joins their
+other end points. Both these theorems, of which the latter can be regarded as a generalization
+of the former, will be demonstrated both analytically and by simple geometrical
+considerations. \begin{Theorem}[]The excess of the sum of the angles of a triangle formed by shortest lines
+over two right angles is equal to the total curvature of the triangle.\end{Theorem} It will be assumed here
+that that angle ($57°\, 17'\, 45''$) to which an arc equal to the radius of the sphere corresponds
+will be taken as the unit for the angles, and that for the unit of total curvature will be
+taken a part of the spherical surface, the area of which is a square whose side is equal to
+the radius of the sphere. Evidently we can express this important theorem thus also:
+the excess over two right angles of the angles of a triangle formed by shortest lines is to
+eight right angles as the part of the surface of the auxiliary sphere, which corresponds
+to it as its integral curvature, is to the whole surface of the sphere. In general, the
+excess over $2n - 4$~right angles of the angles of a polygon of $n$~sides, if these are
+shortest lines, will be equal to the integral curvature of the polygon.
+\PageSep{49}
+
+The general investigations developed in this treatise will, in the conclusion, be applied
+to the theory of triangles of shortest lines, of which we shall introduce only a couple of
+important theorems. If $a$,~$b$,~$c$ be the sides of such a triangle (they will be regarded as
+magnitudes of the first order); $A$,~$B$,~$C$ the angles opposite; $\alpha$,~$\beta$,~$\gamma$ the measures of
+curvature at the angular points; $\sigma$~the area of the triangle, then, to magnitudes of the
+fourth order, $\frac{1}{3}(\alpha + \beta + \gamma)\sigma$ is the excess of the sum $A + B + C$ over two right angles.
+Further, with the same degree of exactness, the angles of a plane rectilinear triangle
+whose sides are $a$,~$b$,~$c$, are respectively
+\begin{align*}
+A &- \tfrac{1}{12}(2\alpha + \beta + \gamma)\sigma\Add{,} \\
+B &- \tfrac{1}{12}(\alpha + 2\beta + \gamma)\sigma\Add{,} \\
+C &- \tfrac{1}{12}(\alpha + \beta + 2\gamma)\sigma.
+\end{align*}
+We see immediately that this last theorem is a generalization of the familiar theorem first
+established by \textsc{Legendre}. By means of this theorem we obtain the angles of a plane
+triangle, correct to magnitudes of the fourth order, if we diminish each angle of the corresponding
+spherical triangle by one-third of the spherical excess. In the case of non-spherical
+surfaces, we must apply unequal reductions to the angles, and this inequality,
+generally speaking, is a magnitude of the third order. However, even if the whole surface
+differs only a little from the spherical form, it will still involve also a factor denoting
+the degree of the deviation from the spherical form. It is unquestionably important for
+the higher geodesy that we be able to calculate the inequalities of those reductions and
+thereby obtain the thorough conviction that, for all measurable triangles on the surface
+of the earth, they are to be regarded as quite insensible. So it is, for example, in the
+case of the greatest triangle of the triangulation carried out by the author. The greatest
+side of this triangle is almost fifteen geographical\footnote
+  {This German geographical mile is four minutes of arc at the equator, namely, $7.42$~kilometers,
+  and is equal to about $4.6$~English statute miles. [Translators.]}
+miles, and the excess of the sum
+of its three angles over two right angles amounts almost to fifteen seconds. The three
+reductions of the angles of the plane triangle are $4''.95113$, $4''.95104$, $4''.95131$. Besides,
+the author also developed the missing terms of the fourth order in the above expressions.
+Those for the sphere possess a very simple form. However, in the case of
+measurable triangles upon the earth's surface, they are quite insensible. And in the
+example here introduced they would have diminished the first reduction by only two
+units in the fifth decimal place and increased the third by the same amount.
+\PageSep{50}
+%[** Blank page]
+\PageSep{51}
+
+
+\Notes.
+
+%[** TN: Line numbers have been omitted]
+\LineRef{1}{Art.~1, p.~3, l.~3}. Gauss got the idea of using the auxiliary sphere from astronomy.
+\Cf.~Gauss's Abstract, \Pgref[p.]{abstract}.
+
+\LineRef[1]{2}{Art.~2, p.~3, l.~2~fr.~bot}. In the Latin text \textit{situs} is used for the direction or
+orientation of a plane, the position of a plane, the direction of a line, and the position
+of a point.
+
+\LineRef[2]{2}{Art.~2, p.~4, l.~14}. In the Latin texts the notation
+\[
+\cos(1)L^{2} + \cos(2)L^{2} + \cos(3) L^{2} = 1
+\]
+is used. This is replaced in the translations (except Böklen's) by the more recent
+notation
+\[
+\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1.
+\]
+
+\LineRef[3]{2}{Art.~2, p.~4, l.~3~fr.~bot}. This stands in the original and in Liouville s reprint,
+\[
+\cos A (\cos t\sin t' - \sin t\cos t')(\cos t''\sin t''' - \sin t''\sin t''').
+\]
+
+%[** TN: One-off macro; hyperlink is hard-coded into macro definition]
+\LineRefs{Art.~2, pp.~4--6}. Theorem~VI is original with Gauss, as is also the method of
+deriving~VII\@. The following figures show the points and lines of Theorems VI~and~VII:
+%[Illustrations]
+\Figure{051}
+
+\LineRef{3}{Art.~3, p.~6}. The geometric condition here stated, that the curvature be continuous
+for each point of the surface, or part of the surface, considered is equivalent to
+the analytic condition that the first and second derivatives of the function or functions
+defining the surface be finite and continuous for all points of the surface, or
+part of the surface, considered.
+
+\LineRef[6]{4}{Art.~4, p.~7, l.~20}. In the Latin texts the notation~$XX$ for~$X^{2}$,~etc., is used.
+\PageSep{52}
+
+\LineRef[7]{4}{Art.~4, p.~7}. ``The second method of representing a surface (the expression of
+the coordinates by means of two auxiliary variables) was first used by Gauss for
+arbitrary surfaces in the case of the problem of conformal mapping. [Astronomische
+Abhandlungen, edited by H.~C. Schumacher, vol.~III, Altona,~1825; Gauss, \Title{Werke},
+vol.~IV, p.~189; reprinted in vol.~55 of Ostwald's Klassiker.---\Cf.~also Gauss, \Title{Theoria
+attractionis corporum sphaer.\ ellipt.}, Comment.\ Gött.~II, 1813; Gauss, \Title{Werke}, vol.~V,
+p.~10.] Here he applies this representation for the first time to the determination of
+the direction of the surface normal, and later also to the study of curvature and of
+geodetic lines. The geometrical significance of the variables $p$,~$q$ is discussed more fully
+in \Art{17}. This method of representation forms the source of many new theorems,
+of which these are particularly worthy of mention: the corollary, that the measure of
+%[** TN: On next two lines, replaced "Art." by "Arts."]
+curvature remains unchanged by the bending of the surface (\Arts{11}{12}); the theorems
+of \Arts{15}{16} concerning geodetic lines; the theorem of \Art{20}; and, finally, the
+results derived in the conclusion, which refer a geodetic triangle to the rectilinear triangle
+whose sides are of the same length.'' [Wangerin.]
+
+\LineRef{5}{Art.~5, p.~8}. ``To decide the question, which of the two systems of values found
+in \Art{4} for $X$,~$Y$,~$Z$ belong to the normal directed outwards, which to the normal
+directed inwards, we need only to apply the theorem of \Art{2}~(VII), provided we use
+the second method of representing the surface. If, on the contrary, the surface is
+defined by the equation between the coordinates $W = 0$, then the following simpler\Typo{ con-}{}
+considerations lead to the answer. We draw the line~$d\sigma$ from the point~$A$ towards
+the outer side, then, if $dx$,~$dy$,~$dz$ are the projections of~$d\sigma$, we have
+\[
+P\, dx + Q\, dy + R\, dz > 0.
+\]
+On the other hand, if the angle between $\sigma$~and the normal taken outward is acute,
+then
+\[
+\frac{dx}{d\sigma}X + \frac{dy}{d\sigma}Y + \frac{dz}{d\sigma}Z > 0.
+\]
+This condition, since $d\sigma$~is positive, must be combined with the preceding, if the first
+solution is taken for $X$,~$Y$,~$Z$. This result is obtained in a similar way, if the surface
+is analytically defined by the third method.'' [Wangerin.]
+
+\LineRef[8]{6}{Art.~6, p.~10, l.~4}. The definition of measure of curvature here given is the one
+generally used. But Sophie Germain defined as a measure of curvature at a point of
+a surface the sum of the reciprocals of the principal radii of curvature at that point,
+or double the so-called mean curvature. \Cf.~Crelle's Journ.\ für Math., vol.~VII\@.
+Casorati defined as a measure of curvature one-half the sum of the squares of the
+reciprocals of the principal radii of curvature at a point of the surface. \Cf.~Rend.\
+del R.~Istituto Lombardo, ser.~2, vol.~22, 1889; Acta Mathem.\ vol.~XIV, p.~95, 1890.
+\PageSep{53}
+
+\LineRef[9]{6}{Art.~6, p.~11, l.~21}. Gauss did not carry out his intention of studying the most
+general cases of figures mapped on the sphere.
+
+\LineRef{7}{Art.~7, p.~11, l.~31}. ``That the consideration of a surface element which has the
+form of a triangle can be used in the calculation of the measure of curvature, follows
+from this fact that, according to the formula developed on \Pageref{12}, $k$~is independent
+of the magnitudes $dx$,~$dy$, $\delta x$,~$\delta y$, and that, consequently, $k$~has the same value for
+every infinitely small triangle at the same point of the surface, therefore also for surface
+elements of any form whatever lying at that point.'' [Wangerin.]
+
+\LineRef[10]{7}{Art.~7, p.~12, l.~20}. The notation in the Latin text for the partial derivatives:
+\[
+\frac{dX}{dx},\quad \frac{dX}{dy},\quad \text{etc.},
+\]
+has been replaced throughout by the more recent notation:
+\[
+\frac{\dd X}{\dd x},\quad \frac{\dd X}{\dd y},\quad \text{etc.}
+\]
+
+\LineRef{7}{Art.~7, p.~13, l.~16}. This formula, as it stands in the original and in Liouville's
+reprint, is
+\[
+dY = -Z^{3}tu\, dt - Z^{3}(1 + t^{2})\, du.
+\]
+The incorrect sign in the second member has been corrected in the reprint in Gauss,
+\Title{Werke}, vol.~IV, and in the translations.
+
+\LineRef[12]{8}{Art.~8, p.~15, l.~3}. Euler's work here referred to is found in Mem.\ de~l'Acad.\
+de~Berlin, vol.~XVI, 1760.
+
+\LineRef[13]{10}{Art.~10, p.~18, ll.~8,~9,~10}. Instead of $D$,~$D'$,~$D''$ as here defined, the Italian
+geometers have introduced magnitudes denoted by the same letters and equal, in
+Gauss's notation, to
+\[
+\frac{D}{\Sqrt{EG - F^{2}}},\quad
+\frac{D'}{\Sqrt{EG - F^{2}}},\quad
+\frac{D''}{\Sqrt{EG - F^{2}}}
+\]
+respectively.
+
+\LineRef[14]{11}{Art.~11, p.~19, ll.~4,~6,~fr.~bot}. In the original and in Liouville's reprint, two of
+these formulæ are incorrectly given:
+\[
+\frac{\dd F}{\dd q} = m'' + n,\quad
+n = \frac{\dd F}{\dd q} - \frac{1}{2}·\frac{\dd E}{\dd q}.
+\]
+The proper corrections have been made in Gauss, \Title{Werke}, vol.~IV, and in the translations.
+
+\LineRef{13}{Art.~13, p.~21, l.~20}. Gauss published nothing further on the properties of developable
+surfaces.
+\PageSep{54}
+
+\LineRef[15]{14}{Art.~14, p.~22, l.~8}. The transformation is easily made by means of integration
+by parts.
+
+\LineRef{17}{Art.~17, p.~25}. If we go from the point $p$,~$q$ to the point $(p + dp, q)$, and if the
+Cartesian coordinates of the first point are $x$,~$y$,~$z$, and of the second $x + dx$, $y + dy$,
+$z + dz$; with $ds$~the linear element between the two points, then the direction cosines
+of~$ds$ are
+\[
+\cos \alpha = \frac{dx}{ds},\quad
+\cos \beta  = \frac{dy}{ds},\quad
+\cos \gamma = \frac{dz}{ds}.
+\]
+Since we assume here $q = \text{Constant}$ or $dq = 0$, we have also
+\[
+dx = \frac{\dd x}{\dd p}·dp,\quad
+dy = \frac{\dd y}{\dd p}·dp,\quad
+dz = \frac{\dd z}{\dd p}·dp,\quad
+ds = ±\sqrt{E}·dp.
+\]
+If $dp$~is positive, the change~$ds$ will be taken in the positive direction. Therefore
+$ds = \sqrt{E}·dp$,
+\[
+\cos\alpha = \frac{1}{\sqrt{E}}·\frac{\dd x}{\dd p},\quad
+\cos\beta  = \frac{1}{\sqrt{E}}·\frac{\dd y}{\dd p},\quad
+\cos\gamma = \frac{1}{\sqrt{E}}·\frac{\dd z}{\dd p}\Typo{,}{.}
+\]
+In like manner, along the line $p = \text{Constant}$, if $\cos \alpha'$, $\cos \beta'$, $\cos \gamma'$ are the direction
+cosines, we obtain
+\[
+\cos\alpha' = \frac{1}{\sqrt{G}}·\frac{\dd x}{\dd q},\quad
+\cos\beta'  = \frac{1}{\sqrt{G}}·\frac{\dd y}{\dd q},\quad
+\cos\gamma' = \frac{1}{\sqrt{G}}·\frac{\dd z}{\dd q}.
+\]
+And since
+\begin{align*}
+\cos\omega
+  &= \cos\alpha \cos\alpha'
+   + \cos\beta  \cos\beta'
+   + \cos\gamma \cos\gamma', \\
+\cos\omega
+  &= \frac{F}{\sqrt{EG}}.
+\end{align*}
+From this follows
+\[
+\sin\omega = \frac{\Sqrt{EG - F^{2}}}{\sqrt{EG}}.
+\]
+And the area of the quadrilateral formed by the lines $p$,~$p + dp$, $q$,~$q + dq$ is
+\[
+d\sigma = \Sqrt{EG - F^{2}}·dp·dq.
+\]
+
+\LineRef[16]{21}{Art.~21, p.~33, l.~12}. In the original, in Liouville's reprint, in the two French
+translations, and in Böklen's translation, the next to the last formula of this article
+is written
+\[
+E\beta\delta - F(\alpha\delta + \beta\gamma) + G\alpha\gamma
+  = \frac{EG - F\Typo{'}{}^{2}}{E'G' - F'^{2}}·F'\Add{.}
+\]
+\PageSep{55}
+The proper correction in sign has been made in Gauss, \Title{Werke}, vol.~IV, and in Wangerin's
+translation.
+
+\LineRef[17]{23}{Art.~23, p.~35, l.~13~fr.~bot}. In the Latin texts and in Roger's and Böklen's
+translations this formula has a minus sign on the right hand side. The correction in
+sign has been made in Abadie's and Wangerin's translations.
+
+\LineRef{23}{Art.~23, p.~35}. The figure below represents the lines and angles mentioned in
+this and the following articles\Chg{:}{.}
+%[Illustration]
+\Figure{055}
+
+\LineRef[18]{24}{Art.~24, p.~36}. Derivation of formula~[1].
+
+Let
+\[
+r^{2} = p^{2} + q^{2} + R_{3} + R_{4} + R_{5} + R_{6} + \text{etc.}
+\]
+where $R_{3}$~is the aggregate of all the terms of the third degree in $p$~and~$q$, $R_{4}$~of all
+the terms of the fourth degree,~etc. Then by differentiating, squaring, and omitting
+terms above the sixth degree, we obtain
+\begin{align*}
+\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} = 4p^{2}
+  &+ \left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+   + \left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+   + 4p\frac{\dd R_{3}}{\dd p} %[** TN: Omitted parentheses]
+   + 4p\frac{\dd R_{4}}{\dd p} \\
+  &+ 4p\frac{\dd R_{5}}{\dd p}
+   + 4p\frac{\dd R_{6}}{\dd p}
+   + 2p\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+   + 2p\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p},
+\intertext{and}
+\left(\frac{\dd(r^{2})}{\dd \Erratum{p}{q}}\right)^{2} = 4q^{2}
+  &+ \left(\frac{\dd R_{3}}{\dd q}\right)^{2}
+   + \left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+   + 4q\frac{\dd R_{3}}{\dd q}
+   + 4q\frac{\dd R_{4}}{\dd q} \\
+  &+ 4q\frac{\dd R_{5}}{\dd q}
+   + 4q\frac{\dd R_{6}}{\dd q}
+   + 2q\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}
+   + 2q\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}.
+\end{align*}
+\PageSep{56}
+
+Hence we have
+{\small
+\begin{align*}
+&\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} +
+\left(\frac{\dd(r^{2})}{\dd q}\right)^{2} - 4r^{2} \\
+  &= 4\left(p\frac{\dd R_{3}}{\dd p} + q\frac{\dd R_{3}}{\dd q} - R_{3}\right)
+   + 4\left(p\frac{\dd R_{4}}{\dd p} + q\frac{\dd R_{4}}{\dd q} - R_{4}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}\right) \\
+  &\quad+ 4\left(p\frac{\dd R_{5}}{\dd p} + q\frac{\dd R_{5}}{\dd q} - R_{5}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}\right) \\
+  &\quad+ 4\left(p\frac{\dd R_{6}}{\dd p} + q\frac{\dd R_{6}}{\dd q} - R_{6}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}\right)
+\displaybreak[1] \\
+  &= 8R_{3} + 4\left(3R_{4}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}\right)
+   + 4\left(4R_{5}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}\right) \\
+  &\quad+ 4\left(5R_{6}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}\right),
+\end{align*}}%
+since, according to a familiar theorem for homogeneous functions,
+\[
+p\frac{\dd R_{3}}{\dd p} + q\frac{\dd R_{3}}{\dd q} = 3R_{3},\quad\text{etc.}
+\]
+By dividing unity by the square of the value of~$n$, given at the end of \Art{23}, and
+omitting terms above the fourth degree, we have
+\[
+1 - \frac{1}{n^{2}}
+  = 2f°q^{2} + 2f'pq^{2} + 2g°q^{3} - 3{f°}^{2}q^{4}
+    + 2f''p^{2}q^{2} + 2g'pq^{3} + 2h°q^{4}.
+\]
+This, multiplied by the last equation but one of the preceding page, on rejecting terms
+above the sixth degree, becomes
+\begin{multline*}
+%[** TN: Re-broken]
+\left(1 - \frac{1}{n^{2}}\right)
+\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} \\
+\begin{alignedat}{3}
+  = 8f°p^{2}q^{2} &+ 8f'p^{3}q^{2} &&- 12{f°}^{2}p^{2}q^{4} &&+ 8h°p^{2}q^{4} \\
+                 &+ 8g°p^{2}q^{3} &&+ 8f''p^{4}q^{2}
+  &&+2 f°q^{2}\left(\frac{\dd R_{3}}{\dd p}\right)^{2} \\
+                 &+ 8f°pq^{2} \frac{\dd R_{3}}{\dd p} &&+ 8g'p^{3}q^{3}
+  &&+8 f'p^{2}q^{2} \frac{\dd R_{3}}{\dd p} + 8g°pq^{3} \frac{\dd R_{3}}{\dd p}\\
+  &&&&&+ 8f°pq^{2} \frac{\dd R_{4}}{\dd p}.
+\end{alignedat}
+\end{multline*}
+Therefore, since from the fifth equation of \Art{24}:
+\[
+\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} +
+\left(\frac{\dd(r^{2})}{\dd q}\right)^{2} - 4r^{2}
+  = \left(1 - \frac{1}{n^{2}}\right)
+    \left(\frac{\dd(r^{2})}{\dd p}\right)^{2},
+\]
+\PageSep{57}
+we have
+{\small
+\begin{multline*}
+8R_{3} + 4\left(3R_{4}
+  + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+  + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}
+\right)
++ 4\left(4R_{5}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}
+\right) \\
+  + 4\left(5R_{6}
+  + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+  + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}
+\right) \\
+\begin{aligned}
+  &= 8f°p^{2}q^{2} + 8f'p^{3}q^{2} + 8g°p^{2}q^{3} + 8f°pq^{2}\frac{\dd R_{3}}{\dd p} - 12{f°}^{2}p^{2}q^{4} + 8f''p^{4}q^{2} \\
+  &\quad+ 8g'p^{3}q^{3} + 8h°p^{2}q^{4} + 2f°q^{2}\left(\frac{\dd R_{3}}{\dd p}\right)^{2} + 8f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p} + 8f°pq^{2}\frac{\dd R_{4}}{\dd p} + 8g°pq^{3}\frac{\dd R_{3}}{\dd p}.
+\end{aligned}
+\end{multline*}}%
+Whence, by the method of undetermined coefficients, we find
+\begin{align*}
+R_{3} &= 0,\quad
+R_{4} = \tfrac{2}{3}f°p^{2}q^{2},\quad
+R_{5} = \tfrac{1}{2}f'p^{3}q^{2} + \tfrac{1}{2}g°p^{2}q^{3}, \\
+R_{6} &= (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2}
+  + \tfrac{2}{5}g'p^{3}q^{3} + (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}.
+\end{align*}
+And therefore we have
+\begin{alignat*}{3}
+\Tag{[1]}
+r^{2} &= p^{2} + \tfrac{2}{5}f°p^{2}q^{2}
+  &&+ \tfrac{1}{2}f'p^{3}q^{2}
+  &&+ (\tfrac{2}{5}f''- \tfrac{4}{45}{f°}^{2})p^{4}q^{2} + \text{etc.} \\
+%
+  &+q^{2}
+  &&+ \tfrac{1}{2}g°p^{2}q^{3}
+  &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}.
+\end{alignat*}
+
+This method for deriving formula~[1] is taken from Wangerin.
+
+\LineRef[19]{24}{Art.~24, p.~36}. Derivation of formula~[2].
+
+By taking one-half the reciprocal of the series for~$n$ given in \LineRef{23}{Art.~23, p.~36}, we
+obtain
+\[
+\frac{1}{2n} = \tfrac{1}{2} \bigl[
+  1 - f°q^{2} - f'pq^{2} - g°q^{3}
+  - f''p^{2}q^{2} - g'pq^{3} - (h° - {f°}^{2})q^{4} - \text{etc.}
+\bigr].
+\]
+And by differentiating formula~[1] with respect to~$p$, we obtain
+\begin{alignat*}{2}
+\frac{\dd(r^{2})}{\dd p} = 2 \bigl[p + \tfrac{1}{2}f°pq^{2}
+  &+ \tfrac{3}{4}f'p^{2}q^{2}
+  &&+ 2 (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{3}q^{2} \\
+  &+ \tfrac{1}{2}g°pq^{3}
+  &&+ \tfrac{3}{5}g'p^{2}q^{3} \\
+  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4} + \text{etc.}
+\bigr].
+\end{alignat*}
+Therefore, since
+\[
+r\sin\psi = \frac{1}{2n}·\frac{\dd(r^{2})}{\dd p},
+\]
+we have, by multiplying together the two series above,
+\begin{alignat*}{2}
+\Tag{[2]}
+r\sin\psi = p - \tfrac{1}{3}f°pq^{2}
+  &- \tfrac{1}{4}f'p^{2}q^{2}
+  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2} - \text{etc.} \\
+  &- \tfrac{1}{2}g°pq^{3}
+  &&- \tfrac{2}{5}g'p^{2}q^{3} \\
+  &&&- (\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})pq^{4}.
+\end{alignat*}
+\PageSep{58}
+
+\LineRef[20]{24}{Art.~24, p.~37}. Derivation of formula~[3].
+
+By differentiating~[1] on \Pageref{57} with respect to~$q$, we find
+\begin{alignat*}{2}
+\frac{\dd(r^{2})}{\dd q} = 2 \bigl[q + \tfrac{2}{3}f°p^{2}q
+  &+ \tfrac{1}{2}f'p^{3}q
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
+  &+ \tfrac{3}{4}g°p^{2}q^{2}
+  &&+ \tfrac{3}{5}g'p^{3}q^{2}
+    + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3} + \text{etc.}
+\bigr].
+\end{alignat*}
+Therefore we have, since
+\begin{gather*}
+r\cos \psi = \tfrac{1}{2}\frac{\dd(r^{2})}{\dd q}, \\
+\Tag{[3]}
+\begin{alignedat}[t]{2}
+r\cos\psi = q + \tfrac{2}{3}f°p^{2}q
+  &+ \tfrac{1}{2}f'p^{3}q
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q + \text{etc.} \\
+  &+ \tfrac{3}{4}g°p^{2}q^{2}
+  &&+ \tfrac{3}{5}g'p^{3}q^{2}
+    + (\tfrac{4}{15}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}.
+\end{alignedat}
+\end{gather*}
+
+\LineRef[21]{24}{Art.~24, p.~37}. Derivation of formula~[4].
+
+Since $r\cos\phi$ becomes equal to~$p$ for infinitely small values of $p$~and~$q$, the series
+for~$r\cos\phi$ must begin with~$p$. Hence we set
+\[
+\Tag{(1)}
+r\cos\phi = p + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.}
+\]
+Then, by differentiating, we obtain
+\begin{alignat*}{2}
+\Tag{(2)}
+\frac{\dd(r\cos\phi)}{\dd p}
+  &= 1 + {}&&\frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
+       + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p} + \text{etc.}\Add{,} \\
+\Tag{(3)}
+\frac{\dd(r\cos\phi)}{\dd q}
+  &= &&\frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
+   + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q} + \text{etc.} \\
+\end{alignat*}
+By dividing~[2] \Pageref[p.]{57} by~$n$ on \Pageref{36}, we obtain
+\begin{align*}
+\Tag{(4)}
+\frac{r\sin\psi}{n} = p - \tfrac{4}{3}f°pq^{2}
+  &- \tfrac{5}{4}f'p^{2}q^{2}
+   - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2} - \text{etc.} \\
+  &- \tfrac{3}{2}g°pq^{3} - \tfrac{7}{5}g'p^{2}q^{3}
+   - (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}.
+\end{align*}
+Multiplying (2) by~(4), we have
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(5)}
+\frac{r\sin\psi}{n}·\frac{\dd(r\cos\phi)}{\dd p} \\
+\begin{aligned}
+  = p &+ p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+   + p\frac{\dd R_{4}}{\dd p} + p\frac{\dd R_{5}}{\dd p}
+   - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q^{2} \\
+  &\begin{alignedat}{3}
+    - \tfrac{4}{3}f°pq^{2}
+    &- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+    &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+    &&- \tfrac{7}{5}g'p^{2}q^{3} \\
+    &- \tfrac{5}{4}f'p^{2}q^{2}
+    &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+    &&- (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4} \\
+    &\Typo{}{-}\tfrac{3}{2}g°pq^{3}
+    &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+    &&- \text{etc.}
+  \end{alignedat}
+\end{aligned}
+\end{multline*}
+\PageSep{59}
+Multiplying (3) by~[3] \Pageref[p.]{58}, we have
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(6)}
+r\cos\psi·\frac{\dd(r\cos\phi)}{\dd q} \\
+\begin{alignedat}{3}
+  = q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
+  & + q\frac{\dd R_{4}}{\dd q}
+  &&+ q\frac{\dd R_{5}}{\dd q}
+  &&+ \tfrac{1}{2}f'p^{3}q \frac{\dd R_{2}}{\dd q} \\
+  & + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} + \text{etc.}
+\end{alignedat}
+\end{multline*}
+Since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd(r\cos\phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(r\cos\phi)}{\dd q} = r\cos\phi,
+\]
+we have, by setting (1)~equal to the sum of (5)~and~(6),
+\begin{multline*}
+p + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.} \\
+\begin{alignedat}{5}
+= p &+ p\frac{\dd R_{2}}{\dd p}
+  &&+ p\frac{\dd R_{3}}{\dd p}
+  &&+ p\frac{\dd R_{4}}{\dd p}
+  &&+ p\frac{\dd R_{5}}{\dd p}
+  &&- (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4} \\
+%
+  &+ q\frac{\dd R_{2}}{\dd q}
+  &&- \tfrac{4}{3}f°pq^{2}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+  &&+ q\frac{\dd R_{5}}{\dd q} \\
+%
+  &&&+ q\frac{\dd R_{3}}{\dd q}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q} \\
+%
+  &&&&&- \tfrac{3}{2}g°pq^{3}
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+  &&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q} \\
+%
+  &&&&&+ q\frac{\dd R_{4}}{\dd q}
+  &&- (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} \\
+%
+  &&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &&- \tfrac{7}{5}g'p^{2}q^{3} + \text{etc.},
+\end{alignedat}
+\end{multline*}
+from which we find
+\begin{align*}
+R_{2} &= 0,\quad
+R_{3} = \tfrac{2}{3}f°pq^{2},\quad
+R_{4} = \tfrac{5}{12}f'p^{2}q^{2} + \tfrac{1}{2}g°pq^{3}, \\
+R_{5} &= \tfrac{7}{20}g'p^{2}q^{3}
+  + (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}
+  + (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}.
+\end{align*}
+Therefore we have finally
+\begin{alignat*}{2}
+\Tag{[4]}
+r\cos\phi = p + \tfrac{2}{3}f°pq^{2}
+  &+ \tfrac{5}{12}f'p^{2}q^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2} + \text{etc.} \\
+  &+ \tfrac{1}{2}g°pq^{3}
+  &&+ \tfrac{7}{20}g'p^{2}q^{3} \\
+  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}.
+\end{alignat*}
+
+\LineRef[21]{24}{Art.~24, p.~37}. Derivation of formula~[5].
+
+Again, since $r\sin\phi$ becomes equal to~$q$ for infinitely small values of $p$~and~$q$,
+we set
+\[
+\Tag{(1)}
+r\sin\phi = q + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.}
+\]
+\PageSep{60}
+Then we have by differentiation
+\begin{alignat*}{2}
+\Tag{(2)}
+\frac{\dd(r\sin\phi)}{\dd p}
+  &= &&\frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
+   + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p} + \text{etc.} \\
+\Tag{(3)}
+\frac{\dd(r\sin\phi)}{\dd q}
+  &= 1 + {} &&\frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
+       + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q} + \text{etc.} \\
+\end{alignat*}
+Multiplying (4) \Pageref[p.]{58} by this~(2), we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(4)}
+\frac{r\sin\psi}{n}·\frac{\dd(r\sin\phi)}{\dd p} \\
+\begin{alignedat}{3}
+  = p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+  &+ p\frac{\dd R_{4}}{\dd p}
+  &&+ p\frac{\dd R_{5}}{\dd p}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p} \\
+  & - \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p} - \text{etc.}
+\end{alignedat}
+\end{multline*}
+Likewise from (3)~and~[3] \Pageref[p.]{58}, we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(5)}
+r\cos\psi·\frac{\dd(r\sin\phi)}{\dd q} \\
+\begin{aligned}
+  = q &+ q\frac{\dd R_{2}}{\dd q}
+  + q\frac{\dd R_{3}}{\dd q}
+  + q\frac{\dd R_{4}}{\dd q}
+  + q\frac{\dd R_{5}}{\dd q}
+  + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
+%
+  &\begin{aligned}
+   + \tfrac{2}{3}f°p^{2}q
+   + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+   - \tfrac{3}{5}g'p^{3}q^{2} \\
+%
+   + \tfrac{1}{2}f'p^{3}q
+  &+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+   + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3} \\
+%
+   + \tfrac{3}{4}g°p^{2}q^{2}
+  &+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} + \text{etc.}
+\end{aligned}
+\end{aligned}
+\end{multline*}
+Since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd(r\sin\phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(r\sin\phi)}{\dd q} = r\sin\phi,
+\]
+by setting (1) equal to the sum of (4)~and~(5), we have
+{\small
+\begin{multline*}
+q + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.} \\
+\begin{alignedat}{4}
+= q &+ p\frac{\dd R_{2}}{\dd p}
+   &&+ p\frac{\dd R_{3}}{\dd p}
+   &&+ p\frac{\dd R_{4}}{\dd p} + \tfrac{1}{2}f'p^{3}q
+   &&+ p\frac{\dd R_{5}}{\dd p}
+     + \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+     + q\frac{\dd R_{5}}{\dd q} + \text{etc.} \\
+%
+   & + q\frac{\dd R_{2}}{\dd q}
+   &&+ q\frac{\dd R_{3}}{\dd q}
+   &&+ q\frac{\dd R_{4}}{\dd q} + \tfrac{3}{4}g°p^{2}q^{2}
+   &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+     + \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q}
+     + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q} \\
+%
+   &&&+ \tfrac{2}{3}f°p^{2}q
+   &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+   &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+     + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
+%
+ &&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+   &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+     + \tfrac{3}{5}g'p^{3}q^{2}
+     + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\Typo{.}{,}
+\end{alignedat}
+\end{multline*}}%
+\PageSep{61}
+from which we find
+\begin{align*}
+R_{2} &= 0,\quad
+R_{3} = -\tfrac{1}{3}f°p^{2}q,\quad
+R_{4} = -\tfrac{1}{6}f'p^{3}q - \tfrac{1}{4}g°p^{2}q^{2}, \\
+R_{5} &= -(\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q
+  - \tfrac{3}{20}g'p^{3}q^{2}
+  - (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}.
+\end{align*}
+Therefore, substituting these values in~(1), we have
+\begin{alignat*}{2}
+\Tag{[5]}
+r\sin\phi = q - \tfrac{1}{3}f°p^{2}q
+  &-  \tfrac{1}{6}f'p^{3}q
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q - \text{etc.} \\
+  &-  \tfrac{1}{4}g°p^{2}q^{2}
+  &&- \tfrac{3}{20}g'p^{3}q^{2} \\
+  &&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}.
+\end{alignat*}
+
+\LineRef[22]{24}{Art.~24, p.~38}. Derivation of formula~[6].
+
+Differentiating $n$ on \Pageref{36} with respect to~$q$, we obtain
+\begin{alignat*}{3}
+\Tag{(1)}
+\frac{\dd n}{\dd q} = 2f°q
+  &+  2f'pq &&+ 2f''p^{2}q &&+ \text{etc.} \\
+  &\Typo{}{+} 3g°q^{2} &&+ 3g'pq^{2} &&+ \text{etc.} \\
+  &&&+ 4h°q^{3} &&+ \text{etc.\Typo{,}{} etc.}\Add{,}
+\end{alignat*}
+and hence, multiplying this series by~(4) on \Pageref{58}, we find
+\begin{align*}
+\Tag{(2)}
+\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q} = 2f°pq
+  &+ 2f'p^{2}q + 2f''p^{3}q + 3g'p^{2}q^{2} + \text{etc.} \\
+  &+ 3g°pq^{2} + (4h° - \tfrac{8}{3}{f°}^{2})pq^{3}.
+\end{align*}
+
+For infinitely small values of $r$, $\psi + \phi = \dfrac{\pi}{2}$, as is evident from the figure on \Pgref{fig:055}.
+Hence we set
+\[
+\psi + \phi = \frac{\pi}{2} + R_{1} + R_{2} + R_{3} + R_{4} + \text{etc.}
+\]
+Then we shall have, by differentiation,
+\begin{align*}
+\Tag{(3)}
+\frac{\dd(\psi + \phi)}{\dd p}
+  &= \frac{R_{1}}{\dd p} + \frac{R_{2}}{\dd p}
+   + \frac{R_{3}}{\dd p} + \frac{R_{4}}{\dd p} + \text{etc.}\Add{,} \\
+\Tag{(4)}
+\frac{\dd(\psi + \phi)}{\dd q}
+  &= \frac{R_{1}}{\dd q} + \frac{R_{2}}{\dd q}
+   + \frac{R_{3}}{\dd q} + \frac{R_{4}}{\dd q} + \text{etc.} \\
+\end{align*}
+Therefore, multiplying (4) on \Pageref{58} by~(3), we find
+\begin{alignat*}{2}
+\Tag{(5)}
+\frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
+  = p\frac{R_{1}}{\dd p} + p\frac{R_{2}}{\dd p}
+  &+  p\frac{R_{3}}{\dd p}
+  &&+ p\frac{R_{4}}{\dd p} + \text{etc.} \\
+%
+  &-  \tfrac{4}{3}f°pq^{2}\frac{\dd R_{1}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p} \\
+%
+&&&-\tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{1}}{\dd p} \\
+&&&-\tfrac{3}{2}g°pq^{3}\frac{\dd R_{1}}{\dd p},
+\end{alignat*}
+\PageSep{62}
+and, multiplying~[3] on \Pageref{58} by~(4), we find
+\begin{alignat*}{2}
+\Tag{(6)}
+r\cos\psi·\frac{\dd(\psi + \phi)}{\dd q}
+  = q\frac{R_{1}}{\dd q} + q\frac{R_{2}}{\dd q}
+  &+  q\frac{R_{3}}{\dd q}
+  &&+ q\frac{R_{4}}{\dd q} + \text{etc.} \\
+%
+  &+  \tfrac{2}{3}f°p^{2}q\frac{\dd R_{1}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q} \\
+%
+&&&+\tfrac{1}{2}f'p^{3}q\frac{\dd R_{1}}{\dd q} \\
+&&&+\tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{1}}{\dd q}.
+\end{alignat*}
+And since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q}
+  + \frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(\psi + \phi)}{\dd p} = 0,
+\]
+we shall have, by adding (2),~(5), and~(6),
+\begin{alignat*}{5}
+0 &= p\frac{\dd R_{1}}{\dd p}
+  &&+ 2f°pq
+  &&+ 2f'p^{2}q
+  &&+ 2f''p^{3}q
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{1}}{\dd p} \\
+%
+  &\Typo{}{+} q\frac{\dd R_{1}}{\dd q}
+  &&+ p\frac{\dd R_{2}}{\dd p}
+  &&+ 3g°pq^{2}
+  &&+ 3g'p^{2}q^{2}
+  &&+ q\frac{\dd R_{4}}{\dd q} \\
+%
+ &&&+ q\frac{\dd R_{2}}{\dd q}
+  &&+ p\frac{\dd R_{3}}{\dd p}
+  &&+ (4h° - \tfrac{8}{3}{f°})pq^{3}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q} \\
+%
+&&&&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{1}}{\dd p}
+  &&+ p\frac{\dd R_{4}}{\dd p}
+  && +\tfrac{1}{2}f'p^{3}q\frac{\dd R_{1}}{\dd q} \\
+%
+&&&&&+ q\frac{\dd R_{3}}{\dd q}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{1}}{\dd q} \\
+%
+&&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{1}}{\dd q}
+  &&- \tfrac{5}{4}f°p^{2}q^{2}\frac{\dd R_{1}}{\dd p}
+  &&+ \text{etc.}
+\end{alignat*}
+From this equation we find
+\begin{align*}
+R_{1} &= 0,\quad
+R_{2} = -f°pq,\quad
+R_{3} = -\tfrac{2}{3}f' p^{2}q - g°pq^{2}, \\
+R_{4} &= -(\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q
+  - \tfrac{3}{4}g'p^{2}q^{2} - (h° - \tfrac{1}{3}{f°}^{2})pq^{3}.
+\end{align*}
+Therefore we have finally
+\begin{alignat*}{2}
+\Tag{[6]}
+\psi + \phi = \frac{\pi}{2} - f°pq
+  &-  \tfrac{2}{3}f'p^{2}q
+  &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q - \text{etc.} \\
+  &- g°pq^{2}
+  && -\tfrac{3}{4}g'p^{2}q^{2} \\
+  &&&- (h° - \tfrac{1}{3}{f°}^{2})pq^{3}.
+\end{alignat*}
+\PageSep{63}
+%XXXX
+
+\LineRef[23]{24}{Art.~24, p.~38, l.~19}. The differential equation from which formula~[7] follows
+is derived in the following manner. In the figure on \Pgref{fig:055}, prolong $AD$ to~$D'$,
+making $DD' = dp$\Chg{, through}{. Through}~$D'$ perpendicular to~$AD'$ draw a geodesic line, which will
+cut~$AB$ in~$B'$. Finally, take $D'B'' = DB$, so that $BB''$~is perpendicular to~$B'D'$.
+Then, if by~$ABD$ we mean the area of the triangle~$ABD$,
+\[
+\frac{\dd S}{\dd r} = \lim \frac{AB'D' - ABD}{BB'}
+  = \lim \frac{BDD'B'}{BB'}
+  = \lim \frac{BDD'B''}{DD'}·\lim \frac{DD'}{BB'},
+\]
+since the surface $BDD'B''$ differs from $BDD'B'$ only by an infinitesimal of the
+second order. And since
+\[
+BDD'B'' = dp·\int n\, dq,\quad\text{or}\quad
+\lim \frac{BDD'B''}{DD'} = \int n\, dq,
+\]
+and since, further,
+\[
+\lim \frac{DD'}{BB'} = \frac{\dd p}{\dd r},
+\]
+consequently
+\[
+\frac{\dd S}{\dd r} = \frac{\dd p}{\dd r}·\int n\, dq.
+\]
+Therefore also
+\[
+\frac{\dd S}{\dd p}·\frac{\dd p}{\dd r} +
+\frac{\dd S}{\dd q}·\frac{\dd q}{\dd r} = \frac{\dd p}{\dd r}·\int n\, dq.
+\]
+Finally, from the values for $\dfrac{\dd r}{\dd p}$, $\dfrac{\dd r}{\dd q}$ given at the beginning of \LineRef{24}{Art.~24, p.~36}, we have
+\[
+\frac{\dd p}{\dd r} = \frac{1}{n}\sin\psi,\quad
+\frac{\dd q}{\dd r} = \cos\psi,
+\]
+so that we have
+\[
+\frac{\dd S}{\dd p}·\frac{\sin\psi}{n} +
+\frac{\dd S}{\dd q}·\cos\psi = \frac{\sin\psi}{n}·\int n\, dq.
+\]
+\null\hfill[Wangerin.]
+
+\LineRef[24]{24}{Art.~24, p.~38}. Derivation of formula~[7].
+
+For infinitely small values of $p$~and~$q$, the area of the triangle~$ABC$ becomes
+equal to $\frac{1}{2}pq$. The series for this area, which is denoted by~$S$, must therefore begin
+with~$\frac{1}{2}pq$, or~$R_{2}$. Hence we put
+\[
+S = R_{2} + R_{3} + R_{4} + R_{5} + R_{6} + \text{etc.}
+\]
+\PageSep{64}
+By differentiating, we obtain
+\begin{align*}
+\Tag{(1)}
+\frac{\dd S}{\dd p}
+  &= \frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
+   + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p}
+   + \frac{\dd R_{6}}{\dd p} + \text{etc.}, \\
+\Tag{(2)}
+\frac{\dd S}{\dd q}
+  &= \frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
+   + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q}
+   + \frac{\dd R_{6}}{\dd q} + \text{etc.}, \\
+\end{align*}
+and therefore, by multiplying~(4) on \Pageref{58} by~(1), we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(3)}
+\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p} \\
+\begin{alignedat}{3}
+  = p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+  &  + p\frac{\dd R_{4}}{\dd p}
+  && + p\frac{\dd R_{5}}{\dd p}
+  && + p\frac{\dd R_{6}}{\dd p} + \text{etc.} \\
+%
+  & - \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{4}}{\dd p} \\
+%
+ &&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p} \\
+%
+ &&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{3}}{\dd p} \\
+%
+&&&&&- (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\frac{\dd R_{2}}{\dd p} \\
+&&&&&- \tfrac{7}{5}g'p^{2}q^{3}\frac{\dd R_{2}}{\dd p} \\
+&&&&&- (\tfrac{3}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}\frac{\dd R_{2}}{\dd p},
+\end{alignedat}
+\end{multline*}
+and multiplying~[3] on \Pageref{58} by~(2), we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(4)}
+r\cos\psi·\frac{\dd S}{\dd q} \\
+\begin{alignedat}{3}
+  = q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
+  &  + q\frac{\dd R_{4}}{\dd q}
+  && + q\frac{\dd R_{5}}{\dd q}
+  && + q\frac{\dd R_{6}}{\dd q} + \text{etc.} \\
+%
+  &+  \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{4}}{\dd q} \\
+%
+ &&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{3}}{\dd q} \\
+%
+ &&&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q} \\
+%
+&&&&&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\frac{\dd R_{2}}{\dd q} \\
+&&&&&+ \tfrac{3}{5}g'p^{3}q^{2}\frac{\dd R_{2}}{\dd q} \\
+&&&&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\frac{\dd R_{2}}{\dd q}.
+\end{alignedat}
+\end{multline*}
+\PageSep{65}
+Integrating~$n$ on \Pageref{36} with respect to~$q$, we find
+\begin{alignat*}{2}
+\Tag{(5)}
+\int n\, dq = q + \tfrac{1}{2}f°q^{3}
+  &+  \tfrac{1}{3}f'pq^{3}
+  &&+ \tfrac{1}{3}f''p^{2}q^{3} + \text{etc.} \\
+%
+  &+  \tfrac{1}{4}g°q^{4}
+  &&+ \tfrac{1}{4}g'pq^{4} + \text{etc.} \\
+  &&&+\tfrac{1}{5}h°q^{5} + \text{etc.\ etc.}
+\end{alignat*}
+Multiplying~(4) on \Pageref{58} by~(5), we find
+\begin{alignat*}{2}
+\Tag{(6)}
+\frac{r\sin\psi}{n}·\int n\, dq = pq - f°pq^{3}
+  &- \tfrac{11}{12}f'p^{2}q^{3}
+  &&- (\tfrac{13}{15}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{3} - \text{etc.} \\
+%
+  &-  \tfrac{5}{4}g°pq^{4}
+  &&- \tfrac{23}{20}g'p^{2}q^{4} \\
+  &&&- (\tfrac{7}{5}h° - \tfrac{16}{15}{f°}^{2})pq^{5}.
+\end{alignat*}
+Since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p} + r\cos\psi·\frac{\dd S}{\dd q}
+  = \frac{r\sin\psi}{n}·\int n\, dq,
+\]
+we obtain, by setting (6) equal to the sum of (3)~and~(4),
+{\footnotesize
+\begin{alignat*}{4}
+&\Neg pq
+&&- f°pq^{3}
+&&- \tfrac{11}{12}f'p^{2}q^{3}
+&&- (\tfrac{13}{15}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{3} - \text{etc.} \\
+%
+&&&&&- \tfrac{5}{4}g°pq^{4}
+&&- \tfrac{23}{20}g'p^{2}q^{4} \\
+%
+&&&&&&&- (\tfrac{7}{5}h° - \tfrac{16}{15}{f°}^{2})pq^{5} \\
+%%
+&=  p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+&&+ p\frac{\dd R_{4}}{\dd p}
+&&+ p\frac{\dd R_{5}}{\dd p} + q\frac{\dd R_{5}}{\dd q}
+&&+ p\frac{\dd R_{6}}{\dd p}
+  + \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q} + \text{etc.} \\
+%
+&+  q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
+&&+ q\frac{\dd R_{4}}{\dd q}
+&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+&&+ q\frac{\dd R_{6}}{\dd q}
+  - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\frac{\dd R_{2}}{\dd p} \\
+&&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{4}}{\dd p}
+  + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\frac{\dd R_{2}}{\dd q} \\
+%
+&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{4}}{\dd q}
+  - \tfrac{7}{5}g'p^{2}q^{3}\frac{\dd R_{2}}{\dd p} \\
+%
+&&&&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p}
+  + \tfrac{3}{5}g'p^{3}q^{2}\frac{\dd R_{2}}{\dd q} \\
+%
+&&&&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{3}}{\dd q}
+  - (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}\frac{\dd R_{2}}{\dd p} \\
+%
+&&&&&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q}
+&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{3}}{\dd p}
+  + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\frac{\dd R_{2}}{\dd q}.
+\end{alignat*}}%
+From this equation we find
+\begin{align*}
+R_{2} &= \tfrac{1}{2}pq,\quad
+R_{3} = 0,\quad
+R_{4} = -\tfrac{1}{12}f°pq^{3} - \tfrac{1}{12}f°p^{3}q, \\
+%
+R_{5} &= -\tfrac{1}{20}f'p^{4}q - \tfrac{3}{40}g°p^{3}q^{2}
+  - \tfrac{7}{120}f'p^{2}q^{3} - \tfrac{1}{10}g°pq^{4}, \\
+%
+R_{6} &= -(\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}
+  - (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
+  &\quad- \tfrac{3}{40}g'p^{2}q^{4}
+  - (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q
+  - \tfrac{1}{20}g'p^{4}q^{2}.
+\end{align*}
+\PageSep{66}
+Therefore we have
+\begin{alignat*}{3}
+\Tag{[7]}
+S = \tfrac{1}{2}pq
+  &-  \tfrac{1}{12}f°pq^{3}
+  &&- \tfrac{1}{20}f'p^{4}q
+  &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q - \text{etc.} \\
+%
+  &-  \tfrac{1}{12}f°p^{3}q
+  &&- \tfrac{3}{40}g°p^{3}q^{2}
+  &&- \tfrac{1}{20}g'p^{4}q^{2} \\
+%
+  &&&- \tfrac{7}{120}f'p^{2}q^{3}
+  &&- (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
+%
+  &&&- \tfrac{1}{10}g°pq^{4}
+  &&- \tfrac{3}{40}g'p^{2}q^{4} \\
+%
+  &&&&&- (\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}.
+\end{alignat*}
+
+\LineRef[25]{25}{Art.~25, p.~39, l.~17}. $3p^{2} + 4q^{2} + 4qq' + 4q'^{2}$ is replaced by $3p^{2} + 4q^{2} + 4q'^{2}$.
+This error appears in all the reprints and translations (except Wangerin's).
+
+\LineRef[26]{25}{Art.~25, p.~40, l.~8}. $3p^{2} - 2q^{2} + qq' + 4qq'$ is replaced by $3p^{2} - 2q^{2} + qq' + 4q'^{2}$.
+This correction is noted in all the translations, and in Liouville's reprint.
+
+\LineRef[27]{25}{Art.~25, p.~40}. Derivation of formulæ [8],~[9],~[10].
+
+By priming the~$q$'s in~[7] we obtain at once a series for~$S'$. Then, since
+$\sigma = S - S'$, we have
+\begin{alignat*}{3}
+\sigma = \tfrac{1}{2}p(q - q')
+  &-  \tfrac{1}{12}f°p^{3}(q - q')
+  &&- \tfrac{1}{20}f'p^{4}(q - q')
+  &&- \tfrac{3}{40}g°p^{3}(q^{2} - q'^{2}) \\
+%
+  &-  \tfrac{1}{12}f°p(q^{3} - q'^{3})
+  &&- \tfrac{7}{120}f'p^{2}(q^{3} - q'^{3})
+  &&- \tfrac{1}{10}g°p(q^{4} - q'^{4}),
+\end{alignat*}
+correct to terms of the sixth degree.
+%[** TN: Omitted line break in the original]
+This expression may be written as follows:
+\begin{align*}
+\sigma = \tfrac{1}{2}p(q - q')
+  \bigl(1 &- \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2}) \\
+  &- \tfrac{1}{60}f'p(6p^{2} + 7q^{2} + 7qq' + 7q'^{2}) \\
+  &- \tfrac{1}{20}g°(q + q')(3p^{2} + 4q^{2} + 4q'^{2})\bigr),
+\end{align*}
+or, after factoring,
+{\small
+\begin{multline*}
+\Tag{(1)}
+\sigma = \tfrac{1}{2}p(q - q')
+  \bigl(1 - \tfrac{1}{3}f°q^{2} - \tfrac{1}{4}f'pq^{2} - \tfrac{1}{2}g°q^{3}\bigr)
+  \bigl(1 - \tfrac{1}{6}f°(p^{2} - q^{2} + qq' + q'^{2}) \\
+  - \tfrac{1}{60}f'p(6p^{2} - 8q^{2} + 7qq' + 7q'^{2})
+  - \tfrac{1}{20}g°(3p^{2}q + 3p^{2}q' - 6q^{3} + 4q^{2}q' + 4qq'^{2} + 4q'^{3})\bigr).
+\end{multline*}}%
+The last factor on the right in~(1) can be written, thus:
+\begin{alignat*}{5}
+\bigl(1 &- \tfrac{2}{120}f°(4p^{2})
+  &&- \tfrac{2}{120}f°(3p^{2})
+  &&- \tfrac{2}{120}f'p(6qq')
+  &&- \tfrac{2}{120}f°(3p^{2})
+  &&- \tfrac{2}{120}f'p(qq') \\
+%
+  &+  \tfrac{2}{120}f°(2q^{2})
+  &&+ \tfrac{2}{120}f°(6q^{2})
+  &&- \tfrac{2}{120}f'p(3q'^{2})
+  &&+ \tfrac{2}{120}f°(2q^{2})
+  &&- \tfrac{2}{120}f'p(4q'^{2}) \\
+%
+  &-  \tfrac{2}{120}f°(3qq')
+  &&- \tfrac{2}{120}f°(6qq')
+  &&- \tfrac{6}{120}g°q(3p^{2})
+  &&- \tfrac{2}{120}f°(qq')
+  &&- \tfrac{6}{120}g°q'(3p^{2}) \\
+%
+  &-  \tfrac{2}{120}f°(3q'^{2})
+  &&- \tfrac{2}{120}f°(3q'^{2})
+  &&+ \tfrac{6}{120}g°q(6q^{2})
+  &&- \tfrac{2}{120}f°(4q'^{2})
+  &&+ \tfrac{6}{120}g°q'(2q^{2}) \\
+%
+ &&&- \tfrac{2}{120}f'p(3p^{2})
+  &&- \tfrac{6}{120}g°q(6qq')
+  &&- \tfrac{2}{120}f'p(3p^{2})
+  &&- \tfrac{6}{120}g°q'(qq') \\
+%
+ &&&+ \tfrac{2}{120}f'p(6q^{2})
+  &&- \tfrac{6}{120}g°q(3q'^{2})
+  &&+ \tfrac{2}{120}f'p(2q^{2})
+  &&- \tfrac{6}{120}g°q'(4q'^{2})\bigr).
+\end{alignat*}
+We know, further, that
+\begin{align*}
+&\,k = -\frac{1}{n}·\frac{\dd^{2} n}{\dd q^{2}}
+   = -2f - 6gq - (12h - 2f^{2})q^{2} - \text{etc.}, \\
+\PageSep{67}
+&\begin{alignedat}{4}
+f &= f° &&+ f'p &&+ f''p^{2} &&+ \text{etc.}, \\
+g &= g° &&+ g'p &&+ g''p^{2} &&+ \text{etc.}, \\
+h &= h° &&+ h'p &&+ h''p^{2} &&+ \text{etc.}
+\end{alignedat}
+\end{align*}
+Hence, substituting these values for $f$,~$g$, and~$h$ in~$k$, we have at~$B$ where $k = \beta$,
+correct to terms of the third degree,
+\begin{align*}
+\beta &= -2f° - 2f'p - 6g°q - 2f''p^{2} - 6g'pq - (12h° - 2{f°}^{2})q^{2}.
+\intertext{Likewise, remembering that $q$~becomes~$q'$ at~$C$, and that both $p$~and~$q$ vanish at~$A$,
+we have}
+\gamma &= -2f° - 2f'p - 6g°q' - 2f''p^{2} - 6g'pq' - (12h° - 2{f°}^{2})q'^{2}, \\
+\alpha &= -2f°.
+\end{align*}
+And since $c\sin B = r\sin\psi$,
+\[
+c\sin B = p(1 - \tfrac{1}{3}f°q^{2} - \tfrac{1}{4}f'pq^{2}
+              - \tfrac{1}{2}g°q^{3} - \text{etc.}).
+\]
+
+Now, if we substitute in~(1) $c\sin B$, $\alpha$,~$\beta$,~$\gamma$ for the series which they represent,
+and $a$~for~$q - q'$, we obtain (still correct to terms of the sixth degree)
+\begin{align*}
+\sigma = \tfrac{1}{2}ac\sin B\bigl(1
+  &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} +  3qq' + 3q'^{2}) \\
+  &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} +  6qq' + 3q'^{2}) \\
+  &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} +\Z qq' + 4q'^{2})\bigr).
+\end{align*}
+And if in this equation we replace $p$,~$q$,~$q'$ by $c\sin B$, $c\cos B$, $c\cos B - a$, respectively,
+we shall have
+\begin{align*}
+\Tag{[8]}
+\sigma = \tfrac{1}{2}ac\sin B\bigl(1
+  &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \Z9ac\cos B) \\
+  &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\
+  &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \Z9ac\cos B)\bigr).
+\end{align*}
+
+By writing for $B$,~$\alpha$,~$\beta$,~$a$ in~[8], $A$,~$\beta$,~$\alpha$,~$b$ respectively, we obtain at once
+formula~[9]. Likewise by writing for $B$,~$\beta$,~$\gamma$,~$c$ in~[8], $C$,~$\gamma$,~$\beta$,~$b$ respectively, we
+obtain formula~[10]. Formulæ [9]~and~[10] can, of course, also be derived by the
+method used to derive~[8].
+
+\LineRef[28]{26}{Art.~26, p.~41, l.~11}. The right hand side of this equation should have the positive
+sign. All the editions prior to Wangerin's have the incorrect sign.
+
+\LineRef[29]{26}{Art.~26, p.~41}. Derivation of formula~[11].
+
+We have
+\begin{align*}
+\Tag{(1)}
+r^{2}
+  &+ r'^{2} - (q - q')^{2} - 2r\cos\phi·r'\cos\phi' - 2r\sin\phi·r'\sin\phi' \\
+  &= b^{2} + c^{2} - a^{2} - 2bc\cos(\phi - \phi') \\
+  &= 2bc(\cos A^{*} - \cos A),
+\end{align*}
+since $b^{2} + c^{2} - a^{2} = 2bc\cos A^{*}$ and $\cos(\phi - \phi') = \cos A$.
+\PageSep{68}
+
+By priming the $q$'s in formulæ [1],~[4],~[5] we obtain at once series for~$r'^{2}$,
+$r'\cos\phi'$, $r'\sin\phi'$. Hence we have series for all the terms in the above expression,
+and also for the terms in the expression:
+\[
+\Tag{(2)}
+r\sin\phi·r'\cos\phi' - r\cos\phi·r'\sin\phi' = bc\sin A,
+\]
+namely,
+\begin{alignat*}{3}
+\Tag{(3)}
+r^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q^{2}
+  &&+ \tfrac{1}{2}f'p^{3}q^{2}
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2} + \text{etc.} \\
+%
+  &+ q^{2}
+  &&+ \tfrac{1}{2}g°p^{2}q^{3}
+  &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4},
+\displaybreak[1] \\
+%
+\Tag{(4)}
+r'^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q'^{2}
+  &&+ \tfrac{1}{2}f'p^{3}q'^{2}
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q'^{2} + \text{etc.} \\
+%
+  &+ q'^{2}
+  &&+ \tfrac{1}{2}g°p^{2}q'^{3}
+  &&+ \tfrac{2}{5}g'p^{3}q'^{3} \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q'^{4},
+\end{alignat*}
+\[
+\Tag{(5)}
+-(q - q')^{2} = -q^{2} + 2qq' - q'^{2},
+\]
+\begin{alignat*}{3}
+\Tag{(6)}
+2r\cos\phi
+  &= 2p + \tfrac{4}{3}f°pq^{2}
+  &&+ \tfrac{10}{12}f'p^{2}q^{2}
+  &&+ (\tfrac{6}{10}f'' - \tfrac{16}{45}{f°}^{2})p^{3}q^{2} + \text{etc.} \\
+%
+  &&&+ g°pq^{3}
+  &&+ \tfrac{14}{20}g'p^{2}q^{3} \\
+%
+  &&&&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})pq^{4},
+\displaybreak[1] \\
+%
+\Tag{(7)}
+r'\cos\phi'
+  &= p + \tfrac{2}{3}f°pq'^{2}
+  &&+ \tfrac{5}{12}f'p^{2}q'^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q'^{2} + \text{etc.} \\
+%
+  &&&+ \tfrac{1}{2}g°pq'^{3}
+  &&+ \tfrac{7}{20}g'p^{2}q'^{3} \\
+%
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq'^{4},
+\displaybreak[1] \\
+%
+\Tag{(8)}
+2r\sin\phi
+  &= 2q - \tfrac{2}{3}f°p^{2}q
+  &&- \tfrac{2}{6}f'p^{3}q
+  &&- (\tfrac{2}{10}f'' - \tfrac{14}{90}{f°}^{2})p^{4}q - \text{etc.} \\
+%
+  &&&- \tfrac{2}{4}g°p^{2}q^{2}
+  &&- \tfrac{6}{20}g'p^{3}q^{2} \\
+%
+  &&&&&- (\tfrac{2}{5}h° + \tfrac{26}{90}{f°}^{2})p^{2}q^{3},
+\displaybreak[1] \\
+%
+\Tag{(9)}
+r'\sin\phi'
+  &= q' - \tfrac{1}{3}f°p^{2}q'
+  &&- \tfrac{1}{6}f'p^{3}q'
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q' + \text{etc.} \\
+%
+  &&&- \tfrac{1}{4}g°p^{2}q'^{2}
+  &&- \tfrac{3}{20}g'p^{3}q'^{2} \\
+%
+  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q'^{3}.
+\end{alignat*}
+By adding (3),~(4), and~(5), we obtain
+\begin{multline*}
+\Tag{(10)}
+r^{2} + r'^{2} - (q - q')^{2} \\
+\begin{alignedat}{3}
+  &= 2p^{2} + \tfrac{2}{3}f°p^{2}(q^{2} + q'^{2})
+  &&+ \tfrac{1}{2}f'p^{3}(q^{2} + q'^{2})
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}(q^{2} + q'^{2}) + \text{etc.} \\
+  &+ 2qq'
+  &&+ \tfrac{1}{2}g°p^{2}(q^{3} + q'^{3})
+  &&+ \tfrac{2}{5}g'p^{3}(q^{3} + q'^{3}) \\
+%
+&&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} + q'^{4}).
+\end{alignedat}
+\end{multline*}
+On multiplying (6) by~(7), we obtain
+\begin{multline*}
+\Tag{(11)}
+2r\cos\phi·r'\cos\phi' \\
+\begin{alignedat}{2}
+  = 2p^{2} + \tfrac{4}{3}f°p^{2}(q^{2} + q'^{2})
+  &+ \tfrac{5}{6}f'p^{3}(q^{2} + q'^{2})
+  &&+ (\tfrac{3}{5}f'' - \tfrac{16}{45}{f°}^{2})p^{4}(q^{2} + q'^{2}) + \text{etc.} \\
+%
+  &+ g°p^{2}(q^{3} + q'^{3})
+  &&+ \tfrac{7}{10}g'p^{3}(q^{3} + q'^{3}) \\
+%
+  &&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}(q^{4} + q'^{4}) \\
+  &&&+ \tfrac{8}{9}{f°}^{2}p^{2}q^{2}q'^{2},
+\end{alignedat}
+\end{multline*}
+\PageSep{69}
+and multiplying (8) by~(9), we obtain
+\begin{alignat*}{3}
+\Tag{(12)}
+&\quad 2r\sin\phi·r'\sin\phi' \\
+&= 2qq' - \tfrac{4}{3}f°p^{2}qq'
+&&- \tfrac{2}{3}f'p^{3}qq'
+&&- (\tfrac{2}{5}f'' - \tfrac{24}{45}{f°}^{2})p^{4}qq' - \text{etc.} \\
+%
+&&&- \tfrac{1}{2}g°p^{2}qq'(q + q')
+&&- \tfrac{3}{10}g'p^{3}qq'(q + q') \\
+&&&&&- (\tfrac{2}{5}h° + \tfrac{13}{45}{f°}^{2})p^{2}qq'(q^{2} + q'^{2}).
+\end{alignat*}
+Hence by adding (11)~and~(12), we have
+{\small
+\begin{multline*}
+\Tag{(13)}
+2bc\cos A \\
+\begin{alignedat}{3}
+&= 2p^{2} &&+ \tfrac{4}{3}f°p^{2}(q^{2} + q'^{2})
+&&+ \tfrac{1}{6}f'p^{3}(5q^{2} - 4qq' + 5q'^{2})
+  - \tfrac{8}{45}{f°}^{2}p^{4}(2q^{2} + 2q'^{2} - 3qq') + \text{etc.} \\
+%
+&+ 2qq' &&- \tfrac{4}{3}f°p^{2}qq'
+&&+ \tfrac{1}{2}g°p^{2}(2q^{3} + 2q'^{3} - q^{2}q' - qq'^{2})
+\end{alignedat} \\
+\begin{aligned}
+&- \tfrac{1}{45}{f°}^{2}p^{2}(14q^{4} + 14q'^{4} + 13q^{3}q' + 13 qq'^{3} - 40q^{2}q'^{2}) \\
+&+ \tfrac{1}{10}g'p^{3}(7q^{3} + 7q'^{3} - 3q^{2}q' - 3qq'^{2}) \\
+&+ \tfrac{1}{5}f''p^{4}(3q^{2} + 3q'^{2} - 2qq') \\
+&+ \tfrac{2}{5}h°p^{2}(2q^{4} + 2q'^{4} - q^{3}q' - qq'^{3}).
+\end{aligned}
+\end{multline*}}%
+Therefore we have, by subtracting (13) from~(10),
+{\small
+\begin{multline*}
+2bc(\cos A^{*} - \cos A) \\
+\begin{aligned}
+= -\tfrac{2}{3}f°p^{2}(q^{2} + q'^{2} - 2qq')
+&- \tfrac{1}{3}f'p^{3}(q^{2} + q'^{2} - 2qq')
+ + \tfrac{4}{15}{f°}^{2}p^{4}(q^{2} + q'^{2} - 2qq') - \text{etc.} \\
+&- \tfrac{1}{2}g°p^{2}(q^{3} + q'^{3} - q^{2}q' - qq'^{2})
+ - \tfrac{1}{5}f''p^{4}(q^{2} + q'^{2} - 2qq')
+\end{aligned} \\
+\begin{aligned}
+&+ \tfrac{1}{45}{f°}^{2}p^{2}(7q^{4} + 7q'^{4} + 13q^{3}q'+ 13qq'^{3} - 40q^{2}q'^{2}) \\
+&- \tfrac{2}{5}h°p^{2}(q^{4} + q'^{4} - q^{3}q' - qq'^{3}) \\
+&- \tfrac{3}{10}g'p^{3}(q^{3} + q'^{3} - q^{2}q' - qq'^{2}),
+\end{aligned}
+\end{multline*}}%
+which we can write thus:
+\begin{multline*}
+\Tag{(14)}
+%[** TN: Re-broken, explicit narrowing]
+\qquad
+2bc(\cos A^{*} - \cos A) \\
+\begin{aligned}
+  = -2p^{2}(q - q')^{2}&
+  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
+&+ \tfrac{1}{5}h°(q^{2} + qq' + q'^{2})
+ + \tfrac{3}{20}g'p(q + q') \\
+&- \tfrac{2}{15}{f°}^{2}p^{2}
+ - \tfrac{1}{90}{f°}^{2}(7q^{2} + 7q'^{2} + 27qq')\bigr),\qquad
+\end{aligned}
+\end{multline*}
+correct to terms of the seventh degree.
+
+If we multiply (7) by~[5] on \Pageref{37}, we obtain
+\begin{alignat*}{3}
+\Tag{(15)}
+%[** TN: Re-broken]
+r\sin\phi·r'\cos\phi'
+= pq &+ \tfrac{2}{3}f°pqq'^{2}
+  &&+ \tfrac{5}{12}f'p^{2}qq'^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}qq'^{2} - \text{etc.} \\
+%
+  &- \tfrac{1}{3}f°p^{3}q
+  &&+ \tfrac{1}{2}g°pqq'^{3}
+  &&+ \tfrac{7}{20}g'p^{2}qq'^{3} \\
+%
+  &&&- \tfrac{1}{6}f'p^{4}q
+  &&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pqq'^{4} \\
+%
+  &&&- \tfrac{1}{4}g°p^{3}q^{2}
+  &&- \tfrac{2}{9}{f°}^{2}p^{3}qq'^{2} \\
+%
+  &&&&&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{5}q \\
+  &&&&&- \tfrac{3}{20}g'p^{4}q^{2} \\
+  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{3}q^{3}.
+\end{alignat*}
+\PageSep{70}
+And multiplying (9) by formula~[4] on \Pageref{37}, we obtain
+\begin{alignat*}{3}
+\Tag{(16)}
+%[** TN: Re-broken]
+r\cos\phi·r'\sin\phi'
+= pq' &- \tfrac{1}{3}f°p^{3}q'
+  &&- \tfrac{1}{6}f'p^{4}q'
+  &&- (\tfrac{1}{16}f'' - \tfrac{7}{90}{f°}^{2})p^{5}q' + \text{etc.} \\
+%
+  &+  \tfrac{2}{3}f°pq^{2}q'
+  &&- \tfrac{1}{4}g°p^{3}q'^{2}
+  &&- \tfrac{3}{20}g'p^{4}q'^{2} \\
+%
+  &&&+ \tfrac{5}{12}f'p^{2}q^{2}q'
+  &&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{3}q'^{3} \\
+%
+  &&&&&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}q' \\
+  &&&&&+ \tfrac{7}{20}g'p^{2}q^{3}q' \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}q'.
+\end{alignat*}
+Therefore we have, by subtracting (16) from~(15),
+\begin{multline*}
+\Tag{(17)}
+bc\sin A \\
+\begin{alignedat}{3}
+=p(q - q')\bigl(1 &- \tfrac{1}{3}f°p^{2}
+  &&- \tfrac{5}{12}f'pqq'
+  &&- (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{2}qq' \\
+%
+  &-  \tfrac{2}{3}f°qq'
+  &&- \tfrac{1}{6}f'p^{3}
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4} \\
+%
+  &&&- \tfrac{1}{2}g°qq'(q + q')
+  &&- \tfrac{7}{20}g'p qq'(q + q') \\
+%
+  &&&- \tfrac{1}{4}g°p^{2}(q + q')
+  &&- \tfrac{3}{20}g'p^{3}(q + q') \\
+%
+  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}(q^{2} + qq' + q'^{2}) \\
+%
+  &&&&&- (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})qq'(q^{2} + qq' + q'^{2}) \\
+  &&&&&+ \tfrac{2}{9}{f°}^{2}p^{2}qq'\bigr),
+\end{alignedat}
+\end{multline*}
+correct to terms of the seventh degree.
+
+Let $A^{*} - A = \zeta$, whence $A^{*} = A + \zeta$, $\zeta$~being a magnitude of the second order.
+Hence we have, expanding $\sin\zeta$~and~$\cos\zeta$, and rejecting powers of~$\zeta$ above the second,
+\[
+\cos A^{*} = \cos A·\left(1 - \frac{\zeta^{2}}{2}\right) - \sin A·\zeta,
+\]
+or
+\[
+\cos A^{*} - \cos A = -\frac{\cos A}{2}·\zeta^{2} - \sin A·\zeta;
+\]
+or, multiplying both members of this equation by~$2bc$,
+\[
+\Tag{(18)}
+2bc(\cos A^{*} - \cos A) = -bc\cos A·\zeta^{2} - 2bc\sin A·\zeta.
+\]
+Further, let $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, where the~$R$'s have the same meaning as before.
+If now we substitute in~(18) for its various terms the series derived above, we shall
+have, on rejecting terms above the sixth degree,
+\begin{multline*}
+(p^{2} + qq')R_{2}^{2} + 2p(q - q')
+  \bigl(1 - \tfrac{1}{3}f°(p^{2} + 2qq')\bigr)\bigl(R_{2} + R_{3} + R_{4}\bigr) \\
+  = 2p^{2}(q - q')^{2}
+  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q') \\
+  + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2})
+  - \tfrac{1}{90}{f°}^{2}(12p^{2} + 7q^{2} + 7q'^{2} + 27qq')\bigr).)
+\end{multline*}
+\PageSep{71}
+Equating terms of like powers, and solving for $R_{2}$,~$R_{3}$,~$R_{4}$, we find
+\begin{align*}
+R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
+R_{3} = p(q - q')\bigl(\tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q')\bigr), \\
+R_{4} &= p(q - q')\bigl(\tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q')
+  + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
+  &\qquad- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 7q'^{2} + 12qq')\bigr).
+\end{align*}
+Therefore we have
+\begin{align*}
+A^{*} - A = p(q - q')&\bigl(\tfrac{1}{3}{f°}^{2} + \tfrac{1}{6}f'p
+  + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{3}{20}g'p(q + q') + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 12qq' + 7q'^{2})\bigr),
+\end{align*}
+correct\Typo{,}{ to} terms of the fifth degree.
+
+This equation may be written as follows:
+\begin{align*}
+A^{*} &= A + ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + q'^{2} + qq')\bigr)
+  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') \\
+  &+ \tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q')
+   + \tfrac{1}{5}h°(q^{2} + qq' +q'^{2})
+   - \tfrac{1}{90}{f°}^{2}(2p^{2} + 2q^{2} + 7qq' + 2q'^{2})\bigr).
+\end{align*}
+But, since
+\[
+2\sigma
+  = ap\bigl(1 - \tfrac{1}{6}f°(p^{2}+ q^{2} + qq' + q'^{2}) + \text{etc.}\bigr),
+\]
+the above equation becomes
+\begin{align*}
+A^{*} = A - \sigma&\bigl(-\tfrac{2}{3}f° - \tfrac{1}{3}f'p
+  - \tfrac{1}{2}g°(q + q') - \tfrac{1}{5}f''p^{2}
+  - \tfrac{3}{10}g'p(q + q') \\
+  &- \tfrac{2}{5}h°(q^{2} + qq' + q'^{2})
+   + \tfrac{1}{90}{f°}^{2}(4p^{2} + 4q^{2} + 14qq' + 4q'^{2})\bigr),
+\end{align*}
+or
+\begin{alignat*}{3}
+A^{*} = A - \sigma\bigl(-\tfrac{2}{6}f°
+  &- \tfrac{2}{12}f° &&- \tfrac{2}{12}f° \\
+  &- \tfrac{2}{12}f'p   &&- \tfrac{2}{12}f'p \\
+  &- \tfrac{6}{12}g°q &&- \tfrac{6}{12}g°q' \\
+%
+  &-  \tfrac{2}{12}f''p^{2}
+  &&- \tfrac{2}{12}f''p^{2}
+  &&+ \tfrac{2}{15}f''p^{2} \\
+%
+  &-  \tfrac{6}{12}g'pq
+  &&- \tfrac{6}{12}g'pq'
+  &&+ \tfrac{1}{5}g'p(q + q') \\
+%
+  &-  \tfrac{12}{12}h°q^{2}
+  &&- \tfrac{12}{12}h°q'^{2}
+  &&+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2}) \\
+%
+  &+  \tfrac{2}{12}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{12}{f°}^{2}q'^{2}
+  &&+ \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr).
+\end{alignat*}
+Therefore, if we substitute in this equation $\alpha$,~$\beta$,~$\gamma$ for the series which they represent,
+we shall have
+\begin{align*}
+\Tag{[11]}
+A^{*} = A - \sigma
+  &\bigl(\tfrac{1}{6}\alpha + \tfrac{1}{12}\beta + \tfrac{1}{12}\gamma
+  + \tfrac{2}{15}f''p^{2} + \tfrac{1}{5}g'p(q + q') \\
+  &+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2})
+   + \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr).
+\end{align*}
+
+\LineRef[30]{26}{Art.~26, p.~41}. Derivation of formula~[12].
+
+We form the expressions $(q - q')^{2} + r^{2} - r'^{2} - 2(q - q')r\cos\psi$ and $(q - q')r\sin\psi$.
+Then, since
+\begin{align*}
+(q - q')^{2} + r^{2} - r'^{2} &= a^{2} + c^{2} - b^{2} = 2ac\cos B^{*}, \\
+2(q - q')r\cos\psi &= 2ac\cos B,
+\end{align*}
+\PageSep{72}
+we have
+\[
+(q - q')^{2} + r^{2} - r'^{2} - 2(q - q')r\cos\psi
+  = 2ac(\cos B^{*} - \cos B).
+\]
+We have also
+\[
+(q - q')r\sin\psi = ac\sin B.
+\]
+
+Subtracting~(4) on \Pageref{68} from~[1] on \Pageref{36}, and adding this difference to
+$(q - q')^{2}$, we obtain
+\begin{multline*}
+\Tag{(1)}
+(q - q')^{2} + r^{2} - r'^{2},\quad\text{or}\quad 2ac\cos B^{*} \\
+\begin{alignedat}{2}
+= 2q(q - q') + \tfrac{2}{3}f°p^{2}(q^{2} - q'^{2})
+  &+ \tfrac{1}{2}f'p^{3}(q^{2} - q'^{2})
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}(q^{2} - q'^{2})
+    + \text{etc.} \\
+%
+  &+ \tfrac{1}{2}g°p^{2}(q^{3} - q'^{3})
+  &&+ \tfrac{2}{5}g'p^{3}(q^{3} - q'^{3}) \\
+%
+  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} - q'^{4}).
+\end{alignedat}
+\end{multline*}
+If we multiply~[3] on \Pageref{37} by~$2(q - q')$, we obtain
+\begin{multline*}
+\Tag{(2)}
+2(q - q')r\cos\psi,\quad\text{or}\quad 2ac\cos B \\
+\begin{alignedat}{2}
+= 2q(q - q') + \tfrac{4}{3}f°p^{2}q(q - q')
+  &+ f'p^{3}q(q - q')
+  &&+ (\tfrac{4}{5}f'' - \tfrac{8}{45}{f°}^{2})p^{4}q(q - q') + \text{etc.} \\
+%
+  &+ \tfrac{3}{2}g°p^{2}q^{2}(q - q')
+  &&+ \tfrac{6}{5}g'p^{3}q^{2}(q - q') \\
+%
+  &&&+ (\tfrac{8}{5}h° - \tfrac{28}{45}{f°}^{2})p^{2}q^{3}(q - q').
+\end{alignedat}
+\end{multline*}
+Subtracting (2) from~(1), we have
+\begin{multline*}
+\Tag{(3)}
+2ac(\cos B^{*} - \cos B) \\
+\begin{alignedat}{2}
+= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} + \text{etc.} \\
+%
+  &+ \tfrac{1}{4}g°(2q + q')
+  &&+ \tfrac{1}{5}g'p(2q + q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{20}{f°}^{2})
+       (3q^{2} + 2qq' + q'^{2})\bigr).
+\end{alignedat}
+\end{multline*}
+
+Multiplying [2] on \Pageref{36} by~$(q - q')$, we obtain at once
+\begin{multline*}
+\Tag{(4)}
+(q - q')r\sin\psi,\quad\text{or}\quad ac\sin B \\
+\begin{alignedat}{2}
+= p(q - q')\bigl(1 - \tfrac{1}{3}f°q^{2}
+  &- \tfrac{1}{4}f'pq^{2}
+  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q^{2} + \text{etc.} \\
+%
+  &- \tfrac{1}{2}g°q^{3}
+  &&- \tfrac{2}{5}g'pq^{3} \\
+%
+  &&&- (\tfrac{2}{5}h° - \tfrac{8}{45}{f°}^{2})q^{4}\bigr).
+\end{alignedat}
+\end{multline*}
+
+We now set $B^{*} - B = \zeta$, whence $B^{*} = B + \zeta$, and therefore
+\[
+\cos B^{*} = \cos B \cos\zeta - \sin B \sin\zeta.
+\]
+This becomes, after expanding $\cos\zeta$ and $\sin\zeta$ and neglecting powers of~$\zeta$ above the
+second,
+\[
+\cos B^{*} - \cos B = -\frac{\cos B}{2}·\zeta^{2} - \sin B·\zeta.
+\]
+Multiplying both members of this equation by~$2ac$, we obtain
+\[
+\Tag{(5)}
+2ac(\cos B^{*} - \cos B) = -ac\cos B·\zeta^{2} - 2ac\sin B·\zeta.
+\]
+\PageSep{73}
+Again, let $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, where the $R$'s have the same meaning as before.
+Hence, replacing the terms in~(5) by the proper series and neglecting terms above the
+sixth degree, we have
+\begin{multline*}
+\Tag{(6)}
+q(q - q')R_{2}^{2} + 2p(q - q')(1 - \tfrac{1}{3}f°q^{2})
+  (R_{2} + R_{3} - R_{4}) \\
+\begin{alignedat}{2}
+= 2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
+  &+  \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} \\
+%
+  &+ \tfrac{1}{4}g°(2q + q')
+  &&+ \tfrac{1}{5}g'p(2q + q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})(3q^{2} + 2qq' + q'^{2})\bigr).
+\end{alignedat}
+\end{multline*}
+From this equation we find
+\begin{align*}
+R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
+R_{3} = p(q - q')\bigl(\tfrac{1}{4}f'p + \tfrac{1}{4}g°(2q + q')\bigr), \\
+R_{4} &= p(q - q')\bigl(\tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(2q + q')
+  + \tfrac{1}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
+&\qquad- \tfrac{1}{90}{f°}^{2}(4p^{2} + 16q^{2} + 9qq' + 7q'^{2})\bigr).
+\end{align*}
+Therefore we have, correct to terms of the fifth degree,
+\begin{alignat*}{2}
+B^{*} - B = p(q - q')\bigl(\tfrac{1}{3}f°
+  &+  \tfrac{1}{4}f'p
+  &&+ \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(2q + q') \\
+%
+  &+  \tfrac{1}{4}g°(2q + q')
+  &&+ \tfrac{1}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
+%
+  &&&- \tfrac{1}{90}{f°}^{2}(4p^{2} + 16q^{2} + 9qq' + 7q'^{2})\bigr),
+\end{alignat*}
+or, after factoring the last factor on the right,
+\begin{multline*}
+\Tag{(7)}
+%[** TN: Squeeze line so tag doesn't get pushed up]
+\scalebox{0.975}[1]{$B^{*} - B - \tfrac{1}{2}p(q - q')
+  \bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)
+  \bigl(-\tfrac{2}{3}f° - \tfrac{1}{2}f'p - \tfrac{1}{2}g°(2q + q')$} \\
+  -\tfrac{2}{5}f''p^{2} - \tfrac{2}{5}g'p(2q + q')
+  - \tfrac{2}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
+  - \tfrac{1}{90}{f°}^{2}(-2p^{2} + 22q^{2} + 8qq' + 4q'^{2})\bigr).
+\end{multline*}
+The last factor on the right in~(7) may be put in the form:
+\begin{alignat*}{3}
+\bigl(-\tfrac{2}{12}f°
+  &-  \tfrac{2}{6}f°
+  &&- \tfrac{2}{6}f° \\
+%
+  &-  \tfrac{2}{6}f'p
+  &&- \tfrac{2}{12}f'p \\
+%
+  &-  \tfrac{6}{6}g°q
+  &&- \tfrac{6}{12}g°q' \\
+%
+  &-  \tfrac{2}{6}f''p^{2}
+  &&- \tfrac{2}{12}f''p^{2}
+  &&+ \tfrac{1}{10}f''p^{2} \\
+%
+  &-  \tfrac{6}{6}g'pq
+  &&- \tfrac{6}{12}g'pq'
+  &&+ \tfrac{1}{10}g'p(2q + q') \\
+%
+  &-  \tfrac{12}{6}h°q^{2}
+  &&- \tfrac{12}{12}h°q'^{2}
+  &&+ \tfrac{1}{5}h°(4q^{2} + 3q'^{2} - 4qq') \\
+%
+  &+  \tfrac{2}{6}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{12}{f°}^{2}q'^{2}
+  && - \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} + 11q'^{2} - 8qq')\bigr).
+\end{alignat*}
+Finally, substituting in~(7) $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions which they represent, we
+obtain, still correct to terms of the fifth degree,
+\begin{align*}
+\Tag{[12]}
+B^{*} = B - \sigma&\bigl(\tfrac{1}{12}\alpha + \tfrac{1}{6}\beta
+                      + \tfrac{1}{12}\gamma + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{1}{10}g'p(2q + q')
+   + \tfrac{1}{5}h°(4q^{2} - 4qq' + 3q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} - 8qq' + 11q'^{2})\bigr).
+\end{align*}
+\PageSep{74}
+
+\LineRef[30]{26}{Art.~26, p.~41}. Derivation of formula~[13].
+
+Here we form the expressions $(q - q')^{2} + r'^{2} - r^{2} - 2(q - q')r'\cos(\pi - \psi')$ and
+$(q - q')r'\sin(\pi - \psi')$ and expand them into series. Since
+\begin{gather*}
+(q - q')^{2} + r'^{2} - r^{2} = a^{2} + b^{2} - c^{2} = 2ab\cos C^{*}, \\
+2(q - q')r'\cos(\pi - \psi') = 2ab\cos C,
+\end{gather*}
+we have
+\[
+(q - q')^{2} + r'^{2} - r^{2} - 2(q - q')r'\cos(\pi - \psi')
+  = 2ab(\cos C^{*} - \cos C).
+\]
+We have also
+\[
+(q - q')r'\sin(\pi - \psi') = ab\sin C.
+\]
+
+Subtracting~(3) on \Pageref{68} from (4) on the same page, and adding the result to
+$(q - q')^{2}$, we find
+{\small
+\begin{multline*}
+\Tag{(1)}
+(q - q')^{2} + r'^{2} - r^{2},\quad\text{or}\quad 2ab\cos C^{*} \\
+\begin{alignedat}{2}
+= -2q'(q - q') - \tfrac{2}{3}f°p^{2}(q^{2} - q'^{2})
+  &- \tfrac{1}{2}f'p^{3}(q^{2} - q'^{2})
+  &&- (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2}))p^{4}(q^{2} - q'^{2})
+    - \text{etc.} \\
+%
+  &- \tfrac{1}{2}g°p^{2}(q^{3} - q'^{3})
+  &&- \tfrac{2}{5}g'p^{3}(q^{3} - q'^{3}) \\
+%
+  &&&- (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} - q'^{4}).
+\end{alignedat}
+\end{multline*}}%
+By priming the $q$'s in formula~[3] on \Pageref{37}, we get a series for $r\cos\psi'$, or for
+$-r'\cos(\pi - \psi')$. If we multiply this series for $-r'\cos(\pi - \psi')$ by $2(q - q')$, we find
+\begin{multline*}
+\Tag{(2)}
+-2(q - q')r'\cos(\pi - \psi'),\quad\text{or}\quad -2ab\cos C \\
+\begin{alignedat}{2}
+= 2(q - q')\bigl(q' + \tfrac{2}{3}f°p^{2}q'
+  &+ \tfrac{1}{2}fp^{3}q'
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q' + \text{etc.} \\
+%
+  &+  \tfrac{3}{4}g°p^{2}q'^{2}
+  &&+ \tfrac{3}{5}g'p^{3}q'^{2} \\
+%
+  &&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q'^{3}\bigr).
+\end{alignedat}
+\end{multline*}
+And therefore, by adding (1)~and~(2), we obtain
+\begin{multline*}
+\Tag{(3)}
+2ab(\cos C^{*} - \cos C) \\
+\begin{alignedat}{2}
+= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°q'^{2}
+  &+ \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} + \text{etc.} \\
+%
+  &+ \tfrac{1}{4}g°(q + 2q')
+  &&+ \tfrac{1}{5}g'p(q + 2q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})
+       (q^{2} + 2qq' + 3q'^{2})\bigr).
+\end{alignedat}
+\end{multline*}
+
+By priming the $q$'s in~[2] on \Pageref{36}, we obtain a series for $r'\sin\psi'$, or for
+$r'\sin(\pi - \psi')$. Then, multiplying this series for $r'\sin(\pi - \psi')$ by $(q - q')$, we find
+\begin{multline*}
+\Tag{(4)}
+(q - q') r'\sin(\pi - \psi'),\quad\text{or}\quad ab\sin C \\
+\begin{alignedat}{2}
+= p(q - q')\bigl(1 - \tfrac{1}{3}f°q'^{2}
+  &- \tfrac{1}{4}f'pq'^{2}
+  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q'^{2} - \text{etc.} \\
+%
+  &- \tfrac{1}{2}g°q'^{3}
+  &&- \tfrac{2}{5}g'pq'^{3} \\
+%
+  &&&- (\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})q'^{4}\bigr).
+\end{alignedat}
+\end{multline*}
+
+As before, let $C^{*} - C = \zeta$, whence $C^{*} = C + \zeta$, and therefore
+\[
+\cos C^{*} = \cos C \cos\zeta - \sin C \sin \zeta.
+\]
+\PageSep{75}
+Expanding $\cos\zeta$ and $\sin\zeta$ and neglecting powers of~$\zeta$ above the second, this equation
+becomes
+\[
+\cos C^{*} - \cos C = -\frac{\cos C}{2}·\zeta^{2} - \sin C·\zeta,
+\]
+or, after multiplying both members by~$2ab$,
+\[
+\Tag{(5)}
+2ab(\cos C^{*} - \cos C) = -ab\cos C·\zeta^{2} - 2ab\sin C·\zeta.
+\]
+Again we put $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, the $R$'s having the same meaning as before.
+Now, by substituting (2),~(3),~(4) in~(5), and omitting terms above the sixth degree,
+we obtain
+\begin{multline*}
+q'(q - q')R_{2}^{2}
+  - 2p(q - q')\bigl(1 - \tfrac{1}{3}f°q'^{2})(R_{2} + R_{3} + R_{4}) \\
+\begin{alignedat}{2}
+= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} \\
+%
+  &+ \tfrac{1}{4}g°(q + 2q')
+  &&+ \tfrac{1}{5}g'p(q + 2q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})
+       (q^{2} + 2qq' + 3q'^{2})\bigr),
+\end{alignedat}
+\end{multline*}
+from which we find
+\begin{align*}
+R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
+R_{3} = p(q - q')\bigl(\tfrac{1}{4}f'p + \tfrac{1}{4}g°(q + 2q')\bigr), \\
+R_{4} &= p(q - q')\bigl(\tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q')
+  + \tfrac{1}{5}h°(q^{2} + 2qq' + 3q'^{2}) \\
+&\qquad- \tfrac{1}{90}{f°}^{2}(4p^{2} + 7q^{2} + 9qq' + 16q'^{2})\bigr).
+\end{align*}
+Therefore we have, correct to terms of the fifth degree,
+\begin{alignat*}{2}
+\Tag{(6)}
+C^{*} - C = p(q - q')\bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{4}f'p
+  &&+ \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q') \\
+%
+  &+ \tfrac{1}{4}g°(q + 2q')
+  &&+ \tfrac{1}{5}h°(q^{2} + 2qq' + 3q'^{2}) \\
+%
+  &&&- \tfrac{1}{90}{f°}^{2}(4p^{2} + 7q^{2} + 9qq' + 16q'^{2})\bigr).
+\end{alignat*}
+The last factor on the right in~(6) may be written as the product of two factors, one
+of which is $\frac{1}{2}\bigl(1 -\frac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)$, and the other,
+\begin{align*}
+2\bigl(\tfrac{1}{3}f° + \tfrac{1}{4}f'p
+  &+ \tfrac{1}{4}g°(q + 2q') + \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q') \\
+  &+ \tfrac{1}{5}h°(q^{2} + 3q'^{2} + 2qq')
+   - \tfrac{1}{90}{f°}^{2}(-p^{2} + 2q^{2} + 4qq' + 11q'^{2})\bigr),
+\end{align*}
+or, in another form,
+\begin{alignat*}{3}
+-\bigl(-\tfrac{2}{12}f°
+  &-  \tfrac{2}{12}f°
+  &&- \tfrac{2}{6}f° \\
+%
+  &-  \tfrac{2}{12}f'p
+  &&- \tfrac{2}{6}f'p \\
+%
+  &-  \tfrac{6}{12}g°q
+  &&- \tfrac{6}{6}g°q' \\
+%
+  &-  \tfrac{2}{12}f''p^{2}
+  &&- \tfrac{2}{6}f''p^{2}
+  &&+ \tfrac{1}{10}f''p^{2} \\
+%
+  &-  \tfrac{6}{12}g'pq
+  &&- \tfrac{6}{6}g'pq'
+  &&+ \tfrac{1}{10}g'p(q + 2q') \\
+%
+  &-  \tfrac{12}{12}h°q^{2}
+  &&- \tfrac{12}{6}h°q'^{2}
+  &&+ \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
+%
+  &+ \tfrac{2}{12}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{6}{f°}^{2}q'^{2}
+  &&- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr).
+\end{alignat*}
+\PageSep{76}
+Hence (6) becomes, on substituting $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions which they represent,
+\begin{align*}
+\Tag{[13]}
+C^{*} = C - \sigma
+  &\bigl(\tfrac{1}{12}\alpha + \tfrac{1}{12}\beta + \tfrac{1}{6}\gamma
+    + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{1}{10}g'p(q + 2q') + \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr).
+\end{align*}
+
+\LineRef[31]{26}{Art.~26, p.~41}. Derivation of formula~[14].
+
+This formula is derived at once by adding formulæ [11],~[12],~[13]. But, as
+Gauss suggests, it may also be derived from~[6], \Pageref[p.]{38}. By priming the $q$'s in~[6]
+we obtain a series for~$(\psi' + \phi')$. Subtracting this series from~[6], and noting that
+$\phi - \phi' + \psi + \pi - \psi' = A + B + C$, we have, correct\Typo{}{ to} terms of the fifth degree,
+\begin{alignat*}{2}
+\Tag{(1)}
+A + B + C = \pi - p(q - q')\bigl(f°
+  &+ \tfrac{2}{3}f'p
+  \rlap{${} + \tfrac{1}{2}f''p^{2} + \tfrac{3}{4}g'p(q + q')$} \\
+%
+  &+g°(q + q')
+  &&+ h°(q^{2} + qq' + q'^{2}) \\
+%
+  &&&- \tfrac{1}{6}{f°}^{2}(p^{2} + 2q^{2} + 2qq' + 2q'^{2})\bigr).
+\end{alignat*}
+The second term on the right in~(1) may be written
+\begin{align*}
++ \tfrac{1}{2}ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)
+  ·2\bigl(-f°
+  &- \tfrac{2}{3}f'p - \tfrac{1}{2}f''p^{2} - \tfrac{3}{4}g'p(q + q') \\
+  &- g°(q + q') - h°(q^{2} + qq' + q'^{2}) \\
+  &\qquad+ \tfrac{1}{6}{f°}^{2}(\Typo{+}{}q^{2} + qq' + q'^{2})\bigr),
+\end{align*}
+of which the last factor may be thrown into the form:
+\begin{alignat*}{3}
+\bigl(-\tfrac{2}{3}f°
+  &-  \tfrac{2}{3}f°
+  &&- \tfrac{2}{3}f° \\
+%
+  &-  \tfrac{2}{3}f'p
+  &&- \tfrac{2}{3}f'p \\
+%
+  &-  \tfrac{6}{3}g°q
+  &&- \tfrac{6}{3}g°q' \\
+%
+  &-  \tfrac{2}{3}f''p^{2}
+  &&- \tfrac{2}{3}f''p^{2}
+  &&+ \tfrac{1}{3}f''p^{2} \\
+%
+  &-  \tfrac{6}{3}g'pq
+  &&- \tfrac{6}{3}g'pq'
+  &&+ \tfrac{1}{2}g'p(q + q') \\
+%
+  &-  \tfrac{12}{3}h°q^{2}
+  &&- \tfrac{12}{3}h°q'^{2}
+  &&+ 2h°(q^{2} + q'^{2} - qq') \\
+%
+  &+  \tfrac{2}{3}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{3}{f°}^{2}q'^{2}
+  &&- \tfrac{1}{3}{f°}^{2}(q^{2} + q'^{2} - qq')\bigr).
+\end{alignat*}
+Hence, by substituting $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions they represent, (1)~becomes
+\begin{align*}
+\Tag{[14]}
+A + B + C = \pi + \sigma
+  &\bigl(\tfrac{1}{3}\alpha + \tfrac{1}{3}\beta + \tfrac{1}{3}\gamma
+    + \tfrac{1}{3}f''p^{2} \\
+  &+ \tfrac{1}{2}g'p(q + q')
+   + (2h° - \tfrac{1}{3}{f°}^{2})
+     (q^{2} - qq' + q'^{2})\bigr).
+\end{align*}
+
+\LineRef{27}{Art.~27, p.~42}. Omitting terms above the second degree, we have
+\[
+a^{2} = q^{2} - 2qq' + q'^{2},\quad
+b^{2} = p^{2} + q'^{2},\quad
+c^{2} = p^{2} + q^{2}.
+\]
+
+The expressions in the parentheses of the first set of formulæ for $A^{*}$,~$B^{*}$,~$C^{*}$
+in \Art{27} may be arranged in the following manner:
+\[
+\begin{array}{*{9}{r@{\,}}}
+&(&2p^{2} - & q^{2} + &4qq' - & q'^{2} = \bigl(& (p^{2} + q'^{2}) + & (p^{2} + q^{2}) - & 2(q^{2} - 2qq' + q'^{2})\bigr), \\
+&(& p^{2} - &2q^{2} + &2qq' + & q'^{2} = \bigl(&2(p^{2} + q'^{2}) - & (p^{2} + q^{2}) - &  (q^{2} - 2qq' + q'^{2})\bigr), \\
+&(& p^{2} + & q^{2} + &2qq' - &2q'^{2} = \bigl(&-(p^{2} + q'^{2}) + &2(p^{2} + q^{2}) - &  (q^{2} - 2qq' + q'^{2})\bigr).
+\end{array}
+\]
+\PageSep{77}
+Now substituting $a^{2}$,~$b^{2}$,~$c^{2}$ for $q^{2} - 2qq' + q'^{2})$, $(p^{2} + q'^{2})$, $(p^{2} + q^{2})$ respectively, and
+changing the signs of both members of the last two of these equations, we have
+\[
+\begin{array}{*{7}{r@{\,}}}
+ (&2p^{2} - & q^{2} + &4qq' - & q'^{2} = (b^{2} + & c^{2} - & 2a^{2}), \\
+-(& p^{2} - &2q^{2} + &2qq' + & q'^{2} = (a^{2} + & c^{2} - & 2b^{2}), \\
+-(& p^{2} + & q^{2} + &2qq' - &2q'^{2} = (a^{2} + & b^{2} - & 2c^{2}).
+\end{array}
+\]
+And replacing the expressions in the parentheses in the first set of formulæ for
+$A^{*}$,~$B^{*}$,~$C^{*}$ by their equivalents, we get the second set.
+
+\LineRef{27}{Art.~27, p.~42}. $f° = -\dfrac{1}{2R^{2}}$, $f'' = 0$, etc., may obtained directly, without the
+%[** TN: Macro expands to "Arts. 25 and 26"]
+use of the general considerations of \Arts[ and ]{25}{26}, in the following way. In the
+case of the sphere
+\[
+ds^{2} = \cos^{2}\left(\frac{q}{R}\right)·dp^{2} + dq^{2},
+\]
+hence
+\[
+n = \cos\left(\frac{q}{R}\right)
+  = 1 - \frac{q^{2}}{2R^{2}} + \frac{q^{4}}{24R^{4}} - \text{etc.},
+\]
+\ie,
+\[
+f° = -\frac{1}{2R^{2}},\quad
+h° = \frac{1}{24R^{4}},\quad
+f' = g° = f'' = g' = 0.\qquad\rlap{[Wangerin.]}
+\]
+
+\LineRef{27}{Art.~27, p.~42, l.~16}. This theorem of Legendre is found in the Mémoires (Histoire)
+de l'\Typo{Academie}{Académie} Royale de Paris, 1787, p.~358, and also in his \Title{Trigonometry},
+Appendix,~§\;V\@. He states it as follows in his \textit{Trigonometry}:
+
+\begin{Theorem}[]
+The very slightly curved spherical triangle, whose angles are $A$,~$B$,~$C$ and whose sides
+are $a$,~$b$,~$c$, always corresponds to a rectilinear triangle, whose sides $a$,~$b$,~$c$ are of the same
+lengths, and whose opposite angles are $A - \tfrac{1}{3}e$, $B - \tfrac{1}{3}e$, $C - \tfrac{1}{3}e$, $e$~being the excess of the
+sum of the angles in the given spherical triangle over two right angles.
+\end{Theorem}
+
+\LineRef{28}{Art.~28, p.~43, l.~7}. The sides of this triangle are Hohehagen-Brocken, Inselberg-Hohehagen,
+Brocken-Inselberg, and their lengths are about $107$, $85$, $69$~kilometers
+respectively, according to Wangerin.
+
+\LineRef{29}{Art.~29, p.~43}. Derivation of the relation between $\sigma$~and~$\sigma^{*}$.
+
+In \Art{28} we found the relation
+\[
+A^{*} = A - \tfrac{1}{12}\sigma(2\alpha + \beta + \gamma).
+\]
+Therefore
+\[
+\sin A^{*}
+  = \sin A\cos\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)
+  - \cos A\sin\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr),
+\]
+which, after expanding $\cos\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ and $\sin\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ and rejecting
+powers of $\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ above the first, becomes
+\PageSep{78}
+\[
+\Tag{(1)}
+\sin A^{*} = \sin A
+  - \cos A·\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr),
+\]
+correct to terms of the fourth degree.
+
+But, since $\sigma$~and~$\sigma^{*}$ differ only by terms above the second degree, we may replace
+in~(1) $\sigma$~by the value of~$\sigma^{*}$, $\tfrac{1}{2}bc\sin A^{*}$. We thus obtain, with equal exactness,
+\[
+\Tag{(2)}
+\sin A = \sin A^{*}
+  \bigl(1 + \tfrac{1}{24}bc\cos A·(2\alpha + \beta + \gamma)\bigr).
+\]
+Substituting this value for~$\sin A$ in~[9], \Pageref[p.]{40}, we have, correct to terms of the sixth
+degree, the first formula for~$\sigma$ given in \Art{29}. Since $2bc\cos A^{*}$, or $b^{2} + c^{2} - a^{2}$,
+differs from~$2bc\cos A$ only by terms above the second degree, we may replace $2bc\cos A$
+in this formula for~$\sigma$ by $b^{2} + c^{2} - a^{2}$. Also $\sigma^{*} = \tfrac{1}{2}bc \sin A^{*}$. Hence, if we make
+these substitutions in the first formula for~$\sigma$, we obtain the second formula for~$\sigma$
+with the same exactness. In the case of a sphere, where $\alpha = \beta = \gamma$, the second
+formula for~$\sigma$ reduces to the third.
+
+When the surface is spherical, (2)~becomes
+\[
+\sin A = \sin A^{*}(1 + \frac{\alpha}{6}bc \cos A).
+\]
+And replacing $2bc\cos A$ in this equation by $(b^{2} + c^{2} - a^{2})$, we have
+\[
+\sin A = \sin A^{*}\bigl(1 + \frac{\alpha}{12}(b^{2} + c^{2} - a^{2})\bigr),
+\]
+or
+\[
+\frac{\sin A}{\sin A^{*}}
+  = \bigl(1 + \frac{\alpha}{12}(b^{2} + c^{2} - a^{2})\bigr).
+\]
+And likewise we can find
+\[
+\frac{\sin B}{\sin B^{*}}
+  = \bigl(1 + \frac{\alpha}{12}(a^{2} + c^{2} - b^{2})\bigr),\qquad
+\frac{\sin C}{\sin C^{*}}
+  = \bigl(1 + \frac{\alpha}{12}(a^{2} + b^{2} - c^{2})\bigr).
+\]
+Multiplying together the last three equations and rejecting the terms containing $\alpha^{2}$~and~$\alpha^{3}$,
+we have
+\[
+1 + \frac{\alpha}{12}(a^{2} + b^{2} + c^{2})
+  = \frac{\sin A·\sin B·\sin C}{\sin A^{*}·\sin B^{*}·\sin C^{*}}.
+\]
+Finally, taking the square root of both members of this equation, we have, with the
+same exactness,
+\[
+\sigma = 1 + \frac{\alpha}{24}(a^{2} + b^{2} + c^{2})
+  = \Sqrt{\frac{\sin A·\sin B·\sin C}{\sin A^{*}·\sin B^{*}·\sin C^{*}}}.
+\]
+
+The method here used to derive the last formula from the next to the last
+formula of \Art{29} is taken from Wangerin.
+\PageSep{79}
+
+
+\Paper{1825}
+\null\vfill
+\begin{center}
+\LARGE
+NEUE \\[12pt]
+ALLGEMEINE UNTERSUCHUNGEN \\[12pt]
+{\small ÃœBER} \\[12pt]
+DIE KRUMMEN FLÄCHEN \\[12pt]
+{\normalsize [1825]} \\[12pt]
+{\footnotesize
+PUBLISHED POSTHUMOUSLY IN GAUSS'S WORKS, VOL.~VIII, 1901. PAGES 408--443}
+\end{center}
+\vfill
+\cleardoublepage
+\PageSep{80}
+%[Blank page]
+\PageSep{81}
+
+
+\PaperTitle{\LARGE NEW GENERAL INVESTIGATIONS \\
+{\small OF} \\
+CURVED SURFACES \\
+{\normalsize [1825]}}
+
+Although the real purpose of this work is the deduction of new theorems concerning
+its subject, nevertheless we shall first develop what is already known, partly
+for the sake of consistency and completeness, and partly because our method of treatment
+is different from that which has been used heretofore. We shall even begin by
+advancing certain properties concerning plane curves from the same principles.
+
+
+\Article{1.}
+
+In order to compare in a convenient manner the different directions of straight
+lines in a plane with each other, we imagine a circle with unit radius described
+in the plane about an arbitrary centre. The position of the radius of this circle,
+drawn parallel to a straight line given in advance, represents then the position of that
+line. And the angle which two straight lines make with each other is measured by
+the angle between the two radii representing them, or by the arc included between
+their extremities. Of course, where precise definition is necessary, it is specified at
+the outset, for every straight line, in what sense it is regarded as drawn. Without
+such a distinction the direction of a straight line would always correspond to two
+opposite radii.
+
+
+\Article{2.}
+
+In the auxiliary circle we take an arbitrary radius as the first, or its terminal
+point in the circumference as the origin, and determine the positive sense of measuring
+the arcs from this point (whether from left to right or the contrary); in the
+opposite direction the arcs are regarded then as negative. Thus every direction of a
+straight line is expressed in degrees,~etc., or also by a number which expresses them
+in parts of the radius.
+\PageSep{82}
+
+Such lines as differ in direction by~$360°$, or by a multiple of~$360°$, have, therefore,
+precisely the same direction, and may, generally speaking, be regarded as the
+same. However, in such cases where the manner of describing a variable angle is
+taken into consideration, it may be necessary to distinguish carefully angles differing
+by~$360°$.
+
+If, for example, we have decided to measure the arcs from left to right, and if
+to two straight lines $l$,~$l'$ correspond the two directions $L$,~$L'$, then $L' - L$ is the angle
+between those two straight lines. And it is easily seen that, since $L' - L$ falls
+between $-180°$~and~$+180°$, the positive or negative value indicates at once that $l'$~lies
+on the right or the left of~$l$, as seen from the point of intersection. This will
+be determined generally by the sign of~$\sin(L' - L)$.
+
+If $aa'$~is a part of a curved line, and if to the tangents at $a$,~$a'$ correspond
+respectively the directions $\alpha$,~$\alpha'$, by which letters shall be denoted also the corresponding
+points on the auxiliary circles, and if $A$,~$A'$ be their distances along the arc
+from the origin, then the magnitude of the arc~$\alpha\alpha'$ or $A' - A$ is called the \emph{amplitude}
+of~$aa'$.
+
+The comparison of the amplitude of the arc~$aa'$ with its length gives us the
+notion of curvature. Let $l$~be any point on the arc~$aa'$, and let $\lambda$,~$\Lambda$ be the same
+with reference to it that $\alpha$,~$A$ and $\alpha'$,~$A'$ are with reference to $a$~and~$a'$. If now
+$\alpha\lambda$~or~$\Lambda - A$ be proportional to the part~$al$ of the arc, then we shall say that $aa'$~is
+uniformly curved throughout its whole length, and we shall call
+\[
+\frac{\Lambda - A}{al}
+\]
+the measure of curvature, or simply the curvature. We easily see that this happens
+only when $aa'$~is actually the arc of a circle, and that then, according to our definition,
+its curvature will be~$±\dfrac{1}{r}$ if $r$~denotes the radius. Since we always regard $r$
+as positive, the upper or the lower sign will hold according as the centre lies to the
+right or to the left of the arc~$aa'$ ($a$~being regarded as the initial point, $a'$~as the
+end point, and the directions on the auxiliary circle being measured from left to
+right). Changing one of these conditions changes the sign, changing two restores it
+again.
+
+On the contrary, if $\Lambda - A$ be not proportional to~$al$, then we call the arc non-uniformly
+curved and the quotient
+\[
+\frac{\Lambda - A}{al}
+\]
+\PageSep{83}
+may then be called its mean curvature. Curvature, on the contrary, always presupposes
+that the point is determined, and is defined as the mean curvature of an element
+at this point; it is therefore equal to
+\[
+\frac{d\Lambda}{d\,al}.
+\]
+We see, therefore, that arc, amplitude, and curvature sustain a similar relation to each
+other as time, motion, and velocity, or as volume, mass, and density. The reciprocal
+of the curvature, namely,
+\[
+\frac{d\,al}{d\Lambda},
+\]
+is called the radius of curvature at the point~$l$. And, in keeping with the above
+conventions, the curve at this point is called concave toward the right and convex
+toward the left, if the value of the curvature or of the radius of curvature happens
+to be positive; but, if it happens to be negative, the contrary is true.
+
+
+\Article{3.}
+
+If we refer the position of a point in the plane to two perpendicular axes of
+coordinates to which correspond the directions $0$~and~$90°$, in such a manner that the
+first coordinate represents the distance of the point from the second axis, measured in
+the direction of the first axis; whereas the second coordinate represents the distance
+from the first axis, measured in the direction of the second axis; if, further, the indeterminates
+$x$,~$y$ represent the coordinates of a point on the curved line, $s$~the length
+of the line measured from an arbitrary origin to this point, $\phi$~the direction of the
+tangent at this point, and $r$~the radius of curvature; then we shall have
+\begin{align*}
+dx &= \cos\phi·ds, \\
+dy &= \sin\phi·ds, \\
+r &= \frac{ds}{d\phi}.
+\end{align*}
+
+If the nature of the curved line is defined by the equation $V = 0$, where $V$~is a
+function of $x$,~$y$, and if we set
+\[
+dV = p\, dx + q\, dy,
+\]
+then on the curved line
+\[
+p\, dx + q\, dy = 0.
+\]
+Hence
+\[
+p\cos\phi + q\sin\phi = 0,
+\]
+\PageSep{84}
+and therefore
+\[
+\tan\phi = -\frac{p}{q}.
+\]
+We have also
+\[
+\cos\phi·dp + \sin\phi·dq - (p\sin\phi - q\cos\phi)\, d\phi = 0.
+\]
+If, therefore, we set, according to a well known theorem,
+\begin{align*}
+dp &= P\, dx + Q\, dy, \\
+dq &= Q\, dx + R\, dy,
+\end{align*}
+then we have\Note{32}
+\[
+(P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi)\, ds
+  = (p\sin\phi - q\cos\phi)\, d\phi,\NoteMark
+\]
+therefore
+\[
+\frac{1}{r}
+  = \frac{P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi}
+         {p\sin\phi - q\cos\phi},
+\]
+or, since\Note{33}
+\begin{gather*}
+\cos\phi = \frac{\mp q}{\Sqrt{p^{2} + q^{2}}},\qquad
+\sin\phi = \frac{±p}{\Sqrt{p^{2} + q^{2}}};\NoteMark \\
+±\frac{1}{r} = \frac{Pq^{2} - 2Qpq + Rp^{2}}{(p^{2} + q^{2})^{3/2}}.
+\end{gather*}
+
+
+\Article{4.}
+
+The ambiguous sign in the last formula might at first seem out of place, but
+upon closer consideration it is found to be quite in order. In fact, since this expression
+depends simply upon the partial differentials of~$V$, and since the function $V$~itself
+merely defines the nature of the curve without at the same time fixing the sense in
+which it is supposed to be described, the question, whether the curve is convex
+toward the right or left, must remain undetermined until the sense is determined by
+some other means. The case is similar in the determination of~$\phi$ by means of the
+tangent, to single values of which correspond two angles differing by~$180°$. The
+sense in which the curve is described can be specified in the following different ways.
+
+\Par{I.} By means of the sign of the change in~$x$. If $x$~increases, then $\cos\phi$ must be
+positive. Hence the upper signs will hold if $q$~has a negative value, and the lower
+signs if $q$~has a positive value. When $x$~decreases, the contrary is true.
+
+\Par{II.} By means of the sign of the change in~$y$. If $y$~increases, the upper signs
+must be taken when $p$~is positive, the lower when $p$~is negative. The contrary is
+true when $y$~decreases.
+
+\Par{III.} By means of the sign of the value which the function~$V$ takes for points
+not on the curve. Let $\delta x$,~$\delta y$ be the variations of $x$,~$y$ when we go out from the
+\PageSep{85}
+curve toward the right, at right angles to the tangent, that is, in the direction~$\phi + 90°$;
+and let the length of this normal be~$\delta\rho$. Then, evidently, we have
+\begin{align*}
+\delta x &= \delta\rho·\cos(\phi + 90°), \\
+\delta y &= \delta\rho·\sin(\phi + 90°),
+\end{align*}
+or
+\begin{align*}
+\delta x &= -\delta\rho·\sin\phi, \\
+\delta y &= +\delta\rho·\cos\phi.
+\end{align*}
+Since now, when $\delta\rho$~is infinitely small,
+\begin{align*}
+\delta V &= p\, \delta x + q\, \delta y \\
+  &= (-p\sin\phi + q\cos\phi)\, \delta\rho \\
+  &= \mp\delta\rho\Sqrt{p^{2} + q^{2}}\Add{,}
+\end{align*}
+and since on the curve itself $V$~vanishes, the upper signs will hold if~$V$, on passing
+through the curve from left to right, changes from positive to negative, and the contrary.
+If we combine this with what is said at the end of \Art{2}, it follows that the
+curve is always convex toward that side on which $V$~receives the same sign as
+\[
+Pq^{2} - 2Qpq + Rp^{2}.
+\]
+
+For example, if the curve is a circle, and if we set
+\[
+V = x^{2} + y^{2} - a^{2}\Add{,}
+\]
+then we have
+\begin{gather*}
+p = 2x,\qquad q = 2y, \\
+P = 2,\qquad Q = 0,\qquad R = 2, \\
+Pq^{2} - 2Qpq + Rp^{2} = 8y^{2} + 8x^{2} = 8a^{2}, \\
+(p^{2} + q^{2})^{3/2} = 8a^{3}, \\
+r = ± a\Add{;}
+\end{gather*}
+and the curve will be convex toward that side for which
+\[
+x^{2} + y^{2} > a^{2},
+\]
+as it should be.
+
+The side toward which the curve is convex, or, what is the same thing, the signs
+in the above formulæ, will remain unchanged by moving along the curve, so long as
+\[
+\frac{\delta V}{\delta\rho}
+\]
+does not change its sign. Since $V$~is a continuous function, such a change can take
+place only when this ratio passes through the value zero. But this necessarily presupposes
+that $p$~and~$q$ become zero at the same time. At such a point the radius
+\PageSep{86}
+of curvature becomes infinite or the curvature vanishes. Then, generally speaking,
+since here
+\[
+-p\sin\phi + q\cos\phi
+\]
+will change its sign, we have here a point of inflexion.
+
+
+\Article{5.}
+
+The case where the nature of the curve is expressed by setting $y$~equal to a
+given function of~$x$, namely, $y = X$, is included in the foregoing, if we set
+\[
+V = X - y.
+\]
+If we put
+\[
+dX = X'\, dx,\qquad
+dX' = X''\, dx,
+\]
+then we have
+\begin{gather*}
+p = X',\qquad q = -1, \\
+P = X'', \qquad Q = 0,\qquad R = 0,
+\end{gather*}
+therefore
+\[
+±\frac{1}{r} = \frac{X''}{(1 + X'^{2})^{3/2}}.
+\]
+Since $q$~is negative here, the upper sign holds for increasing values of~$x$. We can
+therefore say, briefly, that for a positive~$X''$ the curve is concave toward the same
+side toward which the $y$-axis lies with reference to the $x$-axis; while for a negative~$X''$
+the curve is convex toward this side.
+
+
+\Article{6.}
+
+If we regard $x$,~$y$ as functions of~$s$, these formulæ become still more elegant.
+Let us set
+\begin{alignat*}{2}
+\frac{dx}{ds} &= x',\qquad& \frac{dx'}{ds} &= x'', \\
+\frac{dy}{ds} &= y',\qquad& \frac{dy'}{ds} &= y''.
+\end{alignat*}
+Then we shall have
+\begin{alignat*}{2}
+x'  &= \cos\phi,\qquad & y' &= \sin\phi, \\
+x'' &= -\frac{\sin\phi}{r},\qquad & y'' &= \frac{\cos\phi}{r};
+\intertext{or}
+y' &= -rx'',\qquad& x' &= ry'',
+\end{alignat*}
+\PageSep{87}
+or also
+\[
+1 = r(x'y'' - y'x''),
+\]
+so that
+\[
+x'y'' - y'x''
+\]
+represents the curvature, and
+\[
+\frac{1}{x'y'' - y'x''}
+\]
+the radius of curvature.
+
+
+\Article{7.}
+
+We shall now proceed to the consideration of curved surfaces. In order to represent
+the directions of straight lines in space considered in its three dimensions, we
+imagine a sphere of unit radius described about an arbitrary centre. Accordingly, a
+point on this sphere will represent the direction of all straight lines parallel to the
+radius whose extremity is at this point. As the positions of all points in space
+are determined by the perpendicular distances $x$,~$y$,~$z$ from three mutually perpendicular
+planes, the directions of the three principal axes, which are normal to these
+principal planes, shall be represented on the auxiliary sphere by the three points
+$(1)$,~$(2)$,~$(3)$. These points are, therefore, always $90°$~apart, and at once indicate the
+sense in which the coordinates are supposed to increase. We shall here state several
+well known theorems, of which constant use will be made.
+
+\Par{1)} The angle between two intersecting straight lines is measured by the arc [of
+the great circle] between the points on the sphere which represent their directions.
+
+\Par{2)} The orientation of every plane can be represented on the sphere by means
+of the great circle in which the sphere is cut by the plane through the centre parallel
+to the first plane.
+
+\Par{3)} The angle between two planes is equal to the angle between the great circles
+which represent their orientations, and is therefore also measured by the angle
+between the poles of the great circles.
+
+\Par{4)} If $x$,~$y$,~$z$; $x'$,~$y'$,~$z'$ are the coordinates of two points, $r$~the distance between
+them, and $L$~the point on the sphere which represents the direction of the straight
+line drawn from the first point to the second, then
+\begin{alignat*}{2}
+x' &= x &&+ r\cos(1)L, \\
+y' &= y &&+ r\cos(2)L, \\
+2' &= z &&+ r\cos(3)L.
+\end{alignat*}
+
+\Par{5)} It follows immediately from this that we always have
+\[
+\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1
+\]
+\PageSep{88}
+[and] also, if $L'$~is any other point on the sphere,
+\[
+\cos(1)L·\cos(1)L' + \cos(2)L·\cos(2)L' + \cos(3)L·\cos(3)L' = \cos LL'.
+\]
+
+We shall add here another theorem, which has appeared nowhere else, as far as
+we know, and which can often be used with advantage.
+
+Let $L$, $L'$, $L''$, $L'''$ be four points on the sphere, and $A$~the angle which $LL'''$
+and $L'L''$ make at their point of intersection. [Then we have]
+\[
+\cos LL'·\cos L''L''' - \cos LL''·\cos L'L''' = \sin LL'''·\sin L'L''·\cos A.
+\]
+
+The proof is easily obtained in the following way. Let
+\[
+AL = t,\qquad
+AL' = t',\qquad
+AL'' = t'',\qquad
+AL''' = t''';
+\]
+we have then
+\begin{alignat*}{6}
+&\cos L  L'   &&= \cos t  &&\cos t'   &&+ \sin t  &&\sin t'  &&\cos A, \\
+&\cos L''L''' &&= \cos t''&&\cos t''' &&+ \sin t''&&\sin t'''&&\cos A, \\
+&\cos L  L''  &&= \cos t  &&\cos t''  &&+ \sin t  &&\sin t'' &&\cos A, \\
+&\cos L' L''' &&= \cos t' &&\cos t''' &&+ \sin t' &&\sin t'''&&\cos A.
+\end{alignat*}
+Therefore
+\begin{multline*}
+\cos LL' \cos L''L''' - \cos LL'' \cos L'L'' \\
+\begin{aligned}
+&= \cos A \{\cos t  \cos t'  \sin t''\sin t'''
+          + \cos t''\cos t'''\sin t  \sin t' \\
+&\qquad\qquad
+  - \cos t\cos t''\sin t'\sin t''' - \cos t'\cos t'''\sin t\sin t''\} \\
+&= \cos A (\cos t \sin t''' - \cos t'''\sin t)
+          (\cos t'\sin t''  - \cos t'' \sin t') \\
+&= \cos A \sin (t''' - t) \sin(t'' - t') \\
+&= \cos A \sin LL''' \sin L'L''.
+\end{aligned}
+\end{multline*}
+
+Since each of the two great circles goes out from~$A$ in two opposite directions,
+two supplementary angles are formed at this point. But it is seen from our analysis
+that those branches must be chosen, which go in the same sense from~$L$ toward~$L'''$
+and from $L'$~toward~$L''$.
+
+Instead of the angle~$A$, we can take also the distance of the pole of the great
+circle~$LL'''$ from the pole of the great circle~$L'L''$. However, since every great circle
+has two poles, we see that we must join those about which the great circles run in
+the same sense from~$L$ toward~$L'''$ and from~$L'$ toward~$L''$, respectively.
+
+The development of the special case, where one or both of the arcs $LL'''$~and~$L'L''$ are~$90°$, we leave to the reader.
+
+\Par{6)} Another useful theorem is obtained from the following analysis. Let $L$,~$L'$,~$L''$
+be three points upon the sphere and put
+\PageSep{89}
+\begin{alignat*}{6}
+&\cos L  &&(1) = x,       &&\cos L   &&(2) = y,       &&\cos L   &&(3) = z, \\
+&\cos L' &&(1) = x',      &&\cos L'  &&(2) = y',      &&\cos L'  &&(3) = z', \\
+&\cos L''&&(1) = x'',\quad&&\cos L'' &&(2) = y'',\quad&&\cos L'' &&(3) = z''.
+\end{alignat*}
+
+We assume that the points are so arranged that they run around the triangle
+included by them in the same sense as the points $(1)$,~$(2)$,~$(3)$. Further, let $\lambda$~be
+that pole of the great circle~$L'L''$ which lies on the same side as~$L$. We then have,
+from the above lemma,
+\begin{alignat*}{3}
+&y'z'' &&- z'y'' &&= \sin L'L''·\cos\lambda(1), \\
+&z'x'' &&- x'z'' &&= \sin L'L''·\cos\lambda(2), \\
+&x'y'' &&- y'x'' &&= \sin L'L''·\cos\lambda(3).
+\end{alignat*}
+Therefore, if we multiply these equations by $x$,~$y$,~$z$ respectively, and add the products,
+we obtain\Note{34}
+\[
+xy'z'' + x'y''z + x''yz' - xy''z' - x'yz'' - x''y'z
+  = \sin L'L''·\cos\lambda L,\NoteMark
+\]
+wherefore, we can write also, according to well known principles of spherical trigonometry,
+\begin{alignat*}{2}
+  \sin L'L''·&\sin L L''&&·\sin L' \\
+= \sin L'L''·&\sin L L' &&·\sin L'' \\
+= \sin L'L''·&\sin L'L''&&·\sin L,
+\end{alignat*}
+if $L$,~$L'$,~$L''$ denote the three angles of the spherical triangle. At the same time we
+easily see that this value is one-sixth of the pyramid whose angular points are the
+centre of the sphere and the three points $L$,~$L'$,~$L''$ (and indeed \emph{positive}, if~etc.).
+
+
+\Article{8.}
+
+The nature of a curved surface is defined by an equation between the coordinates
+of its points, which we represent by
+\[
+f(x, y, z) = 0.\NoteMark
+\]
+Let the total differential of $f(x, y, z)$ be
+\[
+P\, dx + Q\, dy + R\, dz,
+\]
+where $P$,~$Q$,~$R$ are functions of $x$,~$y$,~$z$. We shall always distinguish two sides of the
+surface, one of which we shall call the upper, and the other the lower. Generally
+speaking, on passing through the surface the value of~$f$ changes its sign, so that, as
+long as the continuity is not interrupted, the values are positive on one side and negative
+on the other.
+\PageSep{90}
+
+The direction of the normal to the surface toward that side which we regard as
+the upper side is represented upon the auxiliary sphere by the point~$L$. Let
+\[
+\cos L(1) = X,\qquad
+\cos L(2) = Y,\qquad
+\cos L(3) = Z.
+\]
+Also let $ds$~denote an infinitely small line upon the surface; and, as its direction is
+denoted by the point~$\lambda$ on the sphere, let
+\[
+\cos \lambda(1) = \xi,\qquad
+\cos \lambda(2) = \eta,\qquad
+\cos \lambda(3) = \zeta.
+\]
+We then have
+\[
+dx = \xi\, ds,\qquad
+dy = \eta\, ds,\qquad
+dz = \zeta\, ds,
+\]
+therefore
+\[
+P\xi + Q\eta + R\zeta = 0,
+\]
+and, since $\lambda L$ must be equal to~$90°$, we have also
+\[
+X\xi + Y\eta + Z\zeta = 0.
+\]
+Since $P$,~$Q$,~$R$, $X$,~$Y$,~$Z$ depend only on the position of the surface on which we take
+the element, and since these equations hold for every direction of the element on the
+surface, it is easily seen that $P$,~$Q$,~$R$ must be proportional to $X$,~$Y$,~$Z$. Therefore
+\[
+P = X\mu,\qquad
+Q = Y\mu,\qquad
+R = Z\mu\Typo{,}{.}
+\]
+Therefore, since
+\begin{gather*}
+X^{2} + Y^{2} + Z^{2} = 1; \\
+\mu = PX + QY + RZ
+\intertext{and}
+\mu^{2} = P^{2} + Q^{2} + R^{2}, \\
+\intertext{or}
+\mu = ±\Sqrt{P^{2} + Q^{2} + R^{2}}.
+\end{gather*}
+
+If we go out from the surface, in the direction of the normal, a distance equal to
+the element~$\delta\rho$, then we shall have
+\[
+\delta x = X\, \delta\rho,\qquad
+\delta y = Y\, \delta\rho,\qquad
+\delta z = Z\, \delta\rho
+\]
+and
+\[
+\delta f = P\, \delta x + Q\, \delta y + R\, \delta z = \mu\, \delta\rho.
+\]
+We see, therefore, how the sign of~$\mu$ depends on the change of sign of the value of~$f$
+in passing from the lower to the upper side.
+
+
+\Article{9.}
+
+Let us cut the curved surface by a plane through the point to which our notation
+refers; then we obtain a plane curve of which $ds$~is an element, in connection
+with which we shall retain the above notation. We shall regard as the upper side of
+the plane that one on which the normal to the curved surface lies. Upon this plane
+\PageSep{91}
+we erect a normal whose direction is expressed by the point~$\L$ of the auxiliary
+sphere. By moving along the curved line, $\lambda$~and~$L$ will therefore change their positions,
+while $\L$~remains constant, and $\lambda L$~and~$\lambda\L$ are always equal to~$90°$. Therefore
+$\lambda$~describes the great circle one of whose poles is~$\L$. The element of this great circle
+will be equal to~$\dfrac{ds}{r}$, if $r$~denotes the radius of curvature of the curve. And again,
+if we denote the direction of this element upon the sphere by~$\lambda'$, then $\lambda'$~will evidently
+lie in the same great circle and be $90°$~from~$\lambda$ as well as from~$\L$. If we
+now set
+\[
+\cos \lambda'(1) = \xi',\qquad
+\cos \lambda'(2) = \eta',\qquad
+\cos \lambda'(3) = \zeta',
+\]
+then we shall have
+\[
+d\xi = \xi'\, \frac{ds}{r},\qquad
+d\eta = \eta'\, \frac{ds}{r},\qquad
+d\zeta = \zeta'\, \frac{ds}{r},
+\]
+since, in fact, $\xi$,~$\eta$,~$\zeta$ are merely the coordinates of the point~$\lambda$ referred to the centre
+of the sphere.
+
+Since by the solution of the equation $f(x, y, z) = 0$ the coordinate~$z$ may be
+expressed in the form of a function of $x$,~$y$, we shall, for greater simplicity, assume
+that this has been done and that we have found
+\[
+z = F(x, y).
+\]
+We can then write as the equation of the surface
+\[
+z - F(x, y) = 0,
+\]
+or
+\[
+f(x, y, z) = z - F(x, y).
+\]
+
+From this follows, if we set
+\begin{gather*}
+dF(x, y) = t\, dx + u\, dy, \\
+P = -t,\qquad
+Q = -u,\qquad
+R = 1,
+\end{gather*}
+where $t$,~$u$ are merely functions of $x$~and~$y$. We set also
+\[
+dt = T\, dx + U\, dy,\qquad
+du = U\, dx + V\, dy.
+\]
+
+Therefore upon the whole surface we have
+\[
+dz = t\, dx + u\, dy
+\]
+and therefore, on the curve,
+\[
+\zeta = t\xi + u\eta.
+\]
+Hence differentiation gives, on substituting the above values for $d\xi$,~$d\eta$,~$d\zeta$,
+\begin{align*}
+(\zeta' - t\xi' - u\eta') \frac{ds}{r}
+  &= \xi\, dt + \eta\, du \\
+  &= (\xi^{2}T + 2\xi\eta U + \eta^{2}V)\, ds,
+\end{align*}
+\PageSep{92}
+or
+\begin{align*}
+\frac{1}{r}
+  &= \frac{\xi^{2}T + 2\xi\eta U + \eta^{2}V}{-\xi' t - \eta'\Typo{\mu}{u} + \zeta'} \\
+  &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{X\xi' - Y\eta' + Z\zeta'} \\
+  &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\cos L\lambda'}.
+\end{align*}
+
+
+\Article{10.}
+
+Before we further transform the expression just found, we will make a few
+remarks about it.
+
+A normal to a curve in its plane corresponds to two directions upon the sphere,
+according as we draw it on the one or the other side of the curve. The one direction,
+toward which the curve is \emph{concave}, is denoted by~$\lambda'$, the other by the opposite
+point on the sphere. Both these points, like $L$~and~$\L$, are $90°$~from~$\lambda$, and therefore
+lie in a great circle. And since $\L$~is also $90°$~from~$\lambda$, $\L L = 90° - L\lambda'$, or
+$= L\lambda' - 90°$. Therefore
+\[
+\cos L\lambda' = ±\sin \L L,
+\]
+where $\sin \L L$ is necessarily positive. Since $r$~is regarded as positive in our analysis,
+the sign of~$\cos L\lambda'$ will be the same as that of
+\[
+Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).
+\]
+And therefore a positive value of this last expression means that $L\lambda'$~is less than~$90°$,
+or that the curve is concave toward the side on which lies the projection of the
+normal to the surface upon the plane. A negative value, on the contrary, shows that
+the curve is convex toward this side. Therefore, in general, we may set also
+\[
+\frac{1}{r} = \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\sin \L L},
+\]
+if we regard the radius of curvature as positive in the first case, and negative in
+the second. $\L L$~is here the angle which our cutting plane makes with the plane
+tangent to the curved surface, and we see that in the different cutting planes passed
+through the same point and the same tangent the radii of curvature are proportional
+to the sine of the inclination. Because of this simple relation, we shall limit ourselves
+hereafter to the case where this angle is a right angle, and where the cutting
+\PageSep{93}
+plane, therefore, is passed through the normal of the curved surface. Hence we have
+for the radius of curvature the simple formula
+\[
+\frac{1}{r} = Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).
+\]
+
+
+\Article{11.}
+
+Since an infinite number of planes may be passed through this normal, it follows
+that there may be infinitely many different values of the radius of curvature. In this
+case $T$,~$U$,~$V$,~$Z$ are regarded as constant, $\xi$,~$\eta$,~$\zeta$ as variable. In order to make the
+latter depend upon a single variable, we take two fixed points $M$,~$M'$ $90°$~apart on the
+great circle whose pole is~$L$. Let their coordinates referred to the centre of the sphere
+be $\alpha$,~$\beta$,~$\gamma$;  $\alpha'$,~$\beta'$,~$\gamma'$. We have then
+\[
+\cos\lambda(1)
+  = \cos\lambda M ·\cos M(1)
+  + \cos\lambda M'·\cos M'(1)
+  + \cos\lambda L ·\cos L(1).
+\]
+If we set
+\[
+\lambda M = \phi,
+\]
+then we have
+\[
+\cos\lambda M' = \sin\phi,
+\]
+and the formula becomes
+\begin{align*}
+\xi   &= \alpha\cos\phi + \alpha'\sin\phi,
+\intertext{and likewise}
+\eta  &= \beta \cos\phi + \beta' \sin\phi, \\
+\zeta &= \gamma\cos\phi + \gamma'\sin\phi.
+\end{align*}
+
+Therefore, if we set\Note{35}
+\begin{align*}
+A &= (\alpha^{2}T + 2\alpha\beta U + \beta^{2}V)Z, \\
+B &= (\alpha\alpha'T + (\alpha'\beta + \alpha\beta')U + \beta\beta'V)Z,\NoteMark \\
+C &= (\alpha'^{2}T + 2\alpha'\beta' U + \beta'^{2}V)Z,
+\end{align*}
+we shall have
+\begin{align*}
+\frac{1}{r}
+  &= A\cos^{2}\phi + 2B\cos\phi \sin\phi + C\sin^{2}\phi \\
+  &= \frac{A + C}{2} + \frac{A - C}{2}\cos 2\phi + B\sin 2\phi.
+\end{align*}
+If we put
+\begin{align*}
+\frac{A - C}{2} &= E\cos 2\theta, \\
+B &= E\sin 2\theta,
+\end{align*}
+\PageSep{94}
+where we may assume that $E$~has the same sign as~$\dfrac{A - C}{2}$, then we have
+\[
+\frac{1}{r} = \tfrac{1}{2}(A + C) + E\cos 2(\phi - \theta).
+\]
+It is evident that $\phi$~denotes the angle between the cutting plane and another plane
+through this normal and that tangent which corresponds to the direction~$M$. Evidently,
+therefore, $\dfrac{1}{r}$~takes its greatest (absolute) value, or $r$~its smallest, when $\phi = \theta$; and $\dfrac{1}{r}$~its
+smallest absolute value, when $\phi = \theta + 90°$. Therefore the greatest and the least
+curvatures occur in two planes perpendicular to each other. Hence these extreme
+values for~$\dfrac{1}{r}$ are
+\[
+\tfrac{1}{2}(A + C) ± \SQRT{\left(\frac{A - C}{2}\right)^{2} + B^{2}}.
+\]
+Their sum is $A + C$ and their product $AC - B^{2}$, or the product of the two extreme
+radii of curvature is
+\[
+= \frac{1}{AC - B^{2}}.
+\]
+This product, which is of great importance, merits a more rigorous development.
+In fact, from formulæ above we find
+\[
+AC - B^{2} = (\alpha\beta' -\beta\alpha')^{2}(TV - U^{2})Z^{2}.
+\]
+But from the third formula in [Theorem]~6, \Art{7}, we easily infer that\Note{36}
+\[
+\alpha\beta' - \beta\alpha' = ±Z,\NoteMark
+\]
+therefore
+\[
+AC - B^{2} = Z^{4}(TV - U^{2}).
+\]
+Besides, from \Art{8},
+\begin{align*}
+Z &= ±\frac{R}{\Sqrt{P^{2} + Q^{2} + R^{2}}} \\
+  &= ±\frac{1}{\Sqrt{1 + t^{2} + u^{2}}},
+\end{align*}
+therefore
+\[
+AC - B^{2} = \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}.
+\]
+
+Just as to \emph{each} point on the curved surface corresponds a particular point~$L$ on
+the auxiliary sphere, by means of the normal erected at this point and the radius of
+\PageSep{95}
+the auxiliary sphere parallel to the normal, so the aggregate of the points on the
+auxiliary sphere, which correspond to all the points of a \emph{line} on the curved surface,
+forms a line which will correspond to the line on the curved surface. And, likewise,
+to every finite figure on the curved surface will correspond a finite figure on the
+auxiliary sphere, the area of which upon the latter shall be regarded as the measure
+of the amplitude of the former. We shall either regard this area as a number, in
+which case the square of the radius of the auxiliary sphere is the unit, or else
+express it in degrees,~etc., setting the area of the hemisphere equal to~$360°$.
+
+The comparison of the area upon the curved surface with the corresponding
+amplitude leads to the idea of what we call the measure of curvature of the surface.
+If the former is proportional to the latter, the curvature is called uniform;
+and the quotient, when we divide the amplitude by the surface, is called the measure
+of curvature. This is the case when the curved surface is a sphere, and the measure
+of curvature is then a fraction whose numerator is unity and whose denominator is
+the square of the radius.
+
+We shall regard the measure of curvature as positive, if the boundaries of the
+figures upon the curved surface and upon the auxiliary sphere run in the same sense;
+as negative, if the boundaries enclose the figures in contrary senses. If they are not
+proportional, the surface is \Typo{non-uniformily}{non-uniformly} curved. And at each point there exists a
+particular measure of curvature, which is obtained from the comparison of corresponding
+infinitesimal parts upon the curved surface and the auxiliary sphere. Let $d\sigma$~be
+a surface element on the former, and $d\Sigma$~the corresponding element upon the auxiliary
+sphere, then
+\[
+\frac{d\Sigma}{d\sigma}
+\]
+will be the measure of curvature at this point.
+
+In order to determine their boundaries, we first project both upon the $xy$-plane.
+The magnitudes of these projections are $Z\, d\sigma$,~$Z\, d\Sigma$. The sign of~$Z$ will show whether
+the boundaries run in the same sense or in contrary senses around the surfaces and
+their projections. We will suppose that the figure is a triangle; the projection upon
+the $xy$-plane has the coordinates
+\[
+x,\ y;\qquad
+x + dx,\ y + dy;\qquad
+x + \delta x,\ y + \delta y.
+\]
+Hence its double area will be
+\[
+2Z\, d\sigma = dx·\delta y - dy·\delta x.
+\]
+To the projection of the corresponding element upon the sphere will correspond the
+coordinates:
+\PageSep{96}
+\[
+\begin{gathered}
+X, \\
+X + \frac{\dd X}{\dd x}·dx + \frac{\dd X}{\dd y}·dy, \\
+X + \frac{\dd X}{\dd x}·\delta x + \frac{\dd X}{\dd y}·\delta y,
+\end{gathered}
+\qquad
+\begin{gathered}
+Y, \\
+Y + \frac{\dd Y}{\dd x}·dx + \frac{\dd Y}{\dd y}·dy, \\
+Y + \frac{\dd Y}{\dd x}·\delta x + \frac{\dd Y}{\dd y}·\delta y,
+\end{gathered}
+\]
+From this the double area of the element is found to be
+\begin{align*}
+2Z\, d\Sigma
+  &= \Neg
+  \left(\frac{\dd X}{\dd x}·dx
+      + \frac{\dd X}{\dd y}·dy\right)
+  \left(\frac{\dd Y}{\dd x}·\delta x
+      + \frac{\dd Y}{\dd y}·\delta y\right) \\
+  &\phantom{={}}
+ -\left(\frac{\dd X}{\dd x}·\delta x
+      + \frac{\dd X}{\dd y}·\delta y\right)
+  \left(\frac{\dd Y}{\dd x}·dx
+      + \frac{\dd Y}{\dd y}·dy\right) \\
+  &= \Neg
+  \left(\frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
+      - \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x}\right)
+    (dx·\delta y - dy·\delta x).
+\end{align*}
+The measure of curvature is, therefore,
+\[
+= \frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
+- \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x} = \omega.
+\]
+Since
+\begin{gather*}
+X = -tZ,\qquad
+Y = -uZ, \\
+(1 + t^{2} + u^{2})Z^{2} = 1,
+\end{gather*}
+we have
+\begin{align*}
+dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3}tu·du, \\
+dY &= +Z^{3}tu·dt - Z^{3}(1 + t^{2})\, du,
+\end{align*}
+therefore
+\begin{alignat*}{2}
+\frac{\dd X}{\dd x}
+  &= Z^{3}\bigl\{-(1 + u^{2})T + tuU\bigr\},\qquad&
+\frac{\dd Y}{\dd x}
+  &= Z^{3}\bigl\{tuT - (1 + t^{2})U\bigr\}, \\
+%
+\frac{\dd X}{\dd y}
+  &= Z^{3}\bigl\{-(1 + u^{2})U + tuV\bigr\},\qquad&
+\frac{\dd Y}{\dd y}
+  &= Z^{3}\bigl\{tuU - (1 + t^{2})V\bigr\},
+\end{alignat*}
+and
+\begin{align*}
+\omega
+  &= Z^{6}(TV - U^{2})\bigl((1 + t^{2})(1 + u^{2}) - t^{2}u^{2}\bigr) \\
+  &= Z^{6}(TV - U^{2})(1 + t^{2} + u^{2}) \\
+  &= Z^{4}(TV - U^{2}) \\
+  &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}},
+\end{align*}
+the very same expression which we have found at the end of the preceding article.
+Therefore we see that
+\PageSep{97}
+
+%[** TN: Quoted, not italicized, in the original]
+\begin{Theorem}[]
+The measure of curvature is always expressed by means of a fraction whose
+numerator is unity and whose denominator is the product of the maximum
+and minimum radii of curvature in the planes passing through the normal.
+\end{Theorem}
+
+
+\Article{12.}
+
+We will now investigate the nature of shortest lines upon curved surfaces. The
+nature of a curved line in space is determined, in general, in such a way that the
+coordinates $x$,~$y$,~$z$ of each point are regarded as functions of a single variable, which
+we shall call~$w$. The length of the curve, measured from an arbitrary origin to this
+point, is then equal to
+\[
+\int \SQRT{\left(\frac{dx}{dw}\right)^{2}
+  + \left(\frac{dy}{dw}\right)^{2}
+  + \left(\frac{dz}{dw}\right)^{2}}·dw.
+\]
+If we allow the curve to change its position by an infinitely small variation, the variation
+of the whole length will then be
+{\small
+\begin{multline*}
+= \int \frac{\dfrac{dx}{dw}·d\, \delta x
+           + \dfrac{dy}{dw}·d\, \delta y
+           + \dfrac{dz}{dw}·d\, \delta z}
+            {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                 + \left(\dfrac{dy}{dw}\right)^{2}
+                 + \left(\dfrac{dz}{dw}\right)^{2}}}% \\
+%
+= \frac{\dfrac{dx}{dw}·\delta x
+      + \dfrac{dy}{dw}·\delta y
+      + \dfrac{dz}{dw}·\delta z}
+       {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+            + \left(\dfrac{dy}{dw}\right)^{2}
+            + \left(\dfrac{dz}{dw}\right)^{2}}} \displaybreak[1] \\
+\qquad- \int\left\{
+  \delta x·d\frac{\dfrac{dx}{dw}}
+    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+         + \left(\dfrac{dy}{dw}\right)^{2}
+         + \left(\dfrac{dz}{dw}\right)^{2}}}\right.
++ \delta y·d\frac{\dfrac{dy}{dw}}
+    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+         + \left(\dfrac{dy}{dw}\right)^{2}
+         + \left(\dfrac{dz}{dw}\right)^{2}}} \\
+\qquad\qquad\qquad+ \left.\delta z·d\frac{\dfrac{dz}{dw}}
+    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+         + \left(\dfrac{dy}{dw}\right)^{2}
+         + \left(\dfrac{dz}{dw}\right)^{2}}}\right\}.
+\end{multline*}}%
+The expression under the integral sign must vanish in the case of a minimum, as we
+know. Since the curved line lies upon a given curved surface whose equation is
+\[
+P\, dx + Q\, dy + R\, dz = 0,
+\]
+the equation between the variations $\delta x$,~$\delta y$,~$\delta z$
+\[
+P\, \delta z + Q\, \delta y + R\, \delta z = 0
+\]
+must also hold. From this, by means of well known principles, we easily conclude
+that the differentials
+\PageSep{98}
+\begin{gather*}
+  d·\frac{\dfrac{dx}{dw}}
+           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                + \left(\dfrac{dy}{dw}\right)^{2}
+                + \left(\dfrac{dz}{dw}\right)^{2}}}, \\
+  d·\frac{\dfrac{dy}{dw}}
+           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                + \left(\dfrac{dy}{dw}\right)^{2}
+                + \left(\dfrac{dz}{dw}\right)^{2}}}, \\
+  d·\frac{\dfrac{dz}{dw}}
+           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                + \left(\dfrac{dy}{dw}\right)^{2}
+                + \left(\dfrac{dz}{dw}\right)^{2}}}
+\end{gather*}
+must be proportional to the quantities $P$,~$Q$,~$R$ respectively. If $ds$~is an element of
+the curve; $\lambda$~the point upon the auxiliary sphere, which represents the direction of
+this element; $L$~the point giving the direction of the normal as above; and $\xi$,~$\eta$,~$\zeta$;
+$X$,~$Y$,~$Z$ the coordinates of the points $\lambda$,~$L$ referred to the centre of the auxiliary
+sphere, then we have
+\begin{gather*}
+dx = \xi\, ds,\qquad
+dy = \eta\, ds,\qquad
+dz = \zeta\, ds, \\
+\xi^{2} + \eta^{2} + \zeta^{2} = 1.
+\end{gather*}
+Therefore we see that the above differentials will be equal to $d\xi$,~$d\eta$,~$d\zeta$. And since
+$P$,~$Q$,~$R$ are proportional to the quantities $X$,~$Y$,~$Z$, the character of the shortest line
+is such that
+\[
+\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}.
+\]
+
+
+\Article{13.}
+
+To every point of a curved line upon a curved surface there correspond two
+points on the sphere, according to our point of view; namely, the point~$\lambda$, which
+represents the direction of the linear element, and the point~$L$, which represents the
+direction of the normal to the surface. The two are evidently $90°$~apart. In our
+former investigation (\Art{9}), where [we] supposed the curved line to lie in a plane,
+we had \emph{two} other points upon the sphere; namely,~$\L$, which represents the direction
+of the normal to the plane, and~$\lambda'$, which represents the direction of the normal to
+the element of the curve in the plane. In this case, therefore, $\L$~was a fixed point
+and $\lambda$,~$\lambda'$ were always in a great circle whose pole was~$\L$. In generalizing these
+considerations, we shall retain the notation $\L$,~$\lambda'$, but we must define the meaning of
+these symbols from a more general point of view. When the curve~$s$ is described,
+the points $L$,~$\lambda$ also describe curved lines upon the auxiliary sphere, which, generally
+speaking, are no longer great circles. Parallel to the element of the second line,
+\PageSep{99}
+we draw a radius of the auxiliary sphere to the point~$\lambda'$, but instead of this point
+we take the point opposite when $\lambda'$~is more than~$90°$ from~$L$. In the first case, we
+regard the element at~$\lambda$ as positive, and in the other as negative. Finally, let $\L$ be
+the point on the auxiliary sphere, which is $90°$~from both $\lambda$~and~$\lambda'$, and which is so
+taken that $\lambda$,~$\lambda'$,~$\L$ lie in the same order as $(1)$,~$(2)$,~$(3)$.
+
+The coordinates of the four points of the auxiliary sphere, referred to its centre,
+are for
+\begin{alignat*}{4}
+&L\qquad &&X\quad &&Y\quad &&Z \\
+&\lambda &&\xi    &&\eta   &&\zeta \\
+&\lambda'&&\xi'   &&\eta'  &&\zeta' \\
+&\L      &&\alpha &&\beta  &&\gamma.
+\end{alignat*}
+Hence each of these $4$~points describes a line upon the auxiliary sphere, whose elements
+we shall express by $dL$,~$d\lambda$,~$d\lambda'$,~$d\L$. We have, therefore,
+\begin{align*}
+d\xi   &= \xi'\, d\lambda, \\
+d\eta  &= \eta'\, d\lambda, \\
+d\zeta &= \zeta'\, d\lambda.
+\end{align*}
+In an analogous way we now call
+\[
+\frac{d\lambda}{ds}
+\]
+the measure of curvature of the curved line upon the curved surface, and its reciprocal
+\[
+\frac{ds}{d\lambda}
+\]
+the radius of curvature. If we denote the latter by~$\rho$, then
+\begin{align*}
+\rho\, d\xi   &= \xi'\, ds, \\
+\rho\, d\eta  &= \eta'\, ds, \\
+\rho\, d\zeta &= \zeta'\, ds.
+\end{align*}
+
+If, therefore, our line be a shortest line, $\xi'$,~$\eta'$,~$\zeta'$ must be proportional to the
+quantities $X$,~$Y$,~$Z$. But, since at the same time
+\[
+\xi'^{2} + \eta'^{2} + \zeta'^{2}  = X^{2} + Y^{2} + Z^{2} = 1,
+\]
+we have
+\[
+\xi'   = ±X,\quad
+\eta'  = ±Y,\quad
+\zeta' = ±Z,
+\]
+and since, further,
+\begin{align*}
+\xi'X + \eta'Y + \zeta'Z
+  &= \cos \lambda'L \\
+  &= ±(X^{2} + Y^{2} + Z^{2}) \\
+  &= ±1,
+\end{align*}
+\PageSep{100}
+and since we always choose the point~$\lambda'$ so that
+\[
+\lambda'L < 90°,
+\]
+then for the shortest line
+\[
+\lambda'L = 0,
+\]
+or $\lambda'$~and~$L$ must coincide. Therefore
+\begin{align*}
+\rho\, d\xi   &= X\, ds, \\
+\rho\, d\eta  &= Y\, ds, \\
+\rho\, d\zeta &= Z\, ds,
+\end{align*}
+and we have here, instead of $4$~curved lines upon the auxiliary sphere, only $3$~to consider.
+Every element of the second line is therefore to be regarded as lying in the
+great circle~$L\lambda$. And the positive or negative value of~$\rho$ refers to the concavity
+or the convexity of the curve in the direction of the normal.
+
+
+\Article{14.}
+
+We shall now investigate the spherical angle upon the auxiliary sphere, which
+the great circle going from~$L$ toward~$\lambda$ makes with that one going from~$L$ toward
+one of the fixed points $(1)$,~$(2)$,~$(3)$; \eg, toward~$(3)$. In order to have something
+definite here, we shall consider the sense from~$L(3)$ to~$L\lambda$ the same as that in which
+$(1)$,~$(2)$, and~$(3)$ lie. If we call this angle~$\phi$, then it follows from the theorem of \Art{7}
+that\Note{37}
+\[
+\sin L(3)·\sin L\lambda·\sin\phi = Y\xi - X\eta,\NoteMark
+\]
+or, since $L\lambda = 90°$ and
+\[
+\sin L(3) = \Sqrt{X^{2} + Y^{2}} = \Sqrt{1 - Z^{2}},
+\]
+we have
+\[
+\sin\phi = \frac{Y\xi - X\eta}{\Sqrt{X^{2} + Y^{2}}}.
+\]
+Furthermore,
+\[
+\sin L(3)·\sin L\lambda·\cos\phi = \zeta,
+\]
+or
+\[
+\cos\phi = \frac{\zeta}{\Sqrt{X^{2} + Y^{2}}}
+\]
+and
+\[
+\tan\phi = \frac{Y\xi - X\eta}{\zeta} = \frac{\zeta'}{\zeta}.
+\]
+\PageSep{101}
+Hence we have
+\[
+d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta
+  - (Y\xi - X\eta)\, d\zeta + \xi\zeta\, dY - \eta\zeta\, dX}
+  {(Y\xi - X\eta)^{2} + \zeta^{2}}.
+\]
+The denominator of this expression is
+\begin{align*}
+&= Y^{2}\xi^{2} - 2XY\xi\eta - X^{2}\eta^{2} + \zeta^{2} \\
+&= -(X\xi + Y\eta)^{2} + (X^{2} + Y^{2})(\xi^{2} + \eta^{2}) + \zeta^{2} \\
+&= -Z^{2}\zeta^{2} + (1 - Z^{2})(1 - \zeta^{2}) + \zeta^{2} \\
+&= 1 - Z^{2},
+\end{align*}
+or
+\[
+d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta
+  + (X\eta - Y\xi)\, d\zeta - \eta\zeta\, dX + \xi\zeta\, dY}
+  {1 - Z^{2}}.
+\]
+
+We verify readily by expansion the identical equation
+\begin{gather*}
+\eta\zeta(X^{2} + Y^{2} + Z^{2}) + YZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\
+= (X\xi + Y\eta + Z\zeta)(Z\eta + Y\zeta) + (X\zeta - Z\xi)(X\eta - Y\xi)\Add{,}
+\end{gather*}
+and likewise
+\begin{gather*}
+\xi\zeta(X^{2} + Y^{2} + Z^{2}) + XZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\
+= (X\xi + Y\eta + Z\zeta)(X\zeta + Z\xi) + (Y\xi - X\eta)(Y\zeta - Z\eta).
+\end{gather*}
+We have, therefore,
+\begin{align*}
+\eta\zeta &= -YZ + (X\zeta - Z\xi )(X\eta  - Y\xi), \\
+\xi\zeta  &= -XZ + (Y\xi   - X\eta)(Y\zeta - Z\eta).
+\end{align*}
+Substituting these values, we obtain
+\begin{multline*}
+d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)
+  + \frac{\zeta Y\, d\xi - \zeta X\, d\eta}{1 - Z^{2}} \\
+  + \frac{X\eta - Y\xi}{1 - Z^{2}}\bigl\{
+    d\zeta - (X\zeta - Z\xi)\, dX - (Y\zeta - Z\eta)\, dY\bigr\}.
+\end{multline*}
+Now
+\begin{alignat*}{4}
+&  X\, dX &&+    Y\, dY &&+     Z\, dZ &&= 0, \\
+&\xi\, dX &&+ \eta\, dY &&+ \zeta\, dZ &&= -X\, d\xi - Y\, d\eta - Z\, d\zeta.
+\end{alignat*}
+On substituting we obtain, instead of what stands in the parenthesis,
+\[
+d\zeta - Z(X\, d\xi + Y\, d\eta + Z\, d\zeta).
+\]
+Hence\Note{38}
+\begin{align*}
+d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)
+  &+ \frac{d\xi}{1 - Z^{2}}\{\zeta Y - \eta X^{2}Z + \xi XYZ\} \\
+  &- \frac{d\eta}{1 - Z^{2}}\{\zeta X + \eta XYZ - \xi Y^{2}Z\}\NoteMark \\
+  &+ d\zeta(\eta X - \xi Y).
+\end{align*}
+\PageSep{102}
+Since, further,
+\begin{align*}
+\eta X^{2}Z - \xi XYZ
+  &= \eta X^{2}Z + \eta Y^{2}Z + \zeta ZYZ \\
+  &= \eta Z(1 - Z^{2}) + \zeta YZ^{2}, \\
+%
+\eta XYZ - \xi Y^{2}Z
+  &= -\xi X^{2}Z - \zeta XZ^{2} - \xi Y^{2}Z \\
+  &= - \xi Z(1 - Z^{2}) - \zeta XZ^{2},
+\end{align*}
+our whole expression becomes
+\begin{align*}
+d\phi &= \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY) \\
+&\quad
+  + (\zeta Y - \eta  Z)\, d\xi
+  + (\xi   Z - \zeta X)\, d\eta
+  + (\eta  X - \xi   Y)\, d\zeta.
+\end{align*}
+
+
+\Article{15.}
+
+The formula just found is true in general, whatever be the nature of the curve.
+But if this be a shortest line, then it is clear that the last three terms destroy each
+other, and consequently\Note{39}
+\[
+d\phi = -\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX).\NoteMark
+\]
+But we see at once that
+\[
+\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX)
+\]
+is nothing but the area of the part of the auxiliary sphere, which is formed between
+the element of the line~$L$, the two great circles drawn through its extremities and~$(3)$,\Note{40}
+%[Illustration]
+\Figure{102}
+and the element thus intercepted on the great circle through $(1)$~and~$(2)$. This
+surface is considered positive, if $L$~and~$(3)$ lie on the same side of~$(1)\ (2)$, and if the
+\PageSep{103}
+direction from~$P$ to~$P'$ is the same as that from~$(2)$ to~$(1)$; negative, if the contrary
+of one of these conditions hold; positive again, if the contrary of both conditions be
+true. In other words, the surface is considered positive if we go around the circumference
+of the figure~$LL'P'P$ in the same sense as $(1)\ (2)\ (3)$; negative, if we go
+in the contrary sense.
+
+If\Note{41} we consider now a finite part of the line from~$L$ to~$L'$ and denote by $\phi$,~$\phi'$
+the values of the angles at the two extremities, then we have
+\[
+\phi' = \phi + \Area LL'P'P,
+\]
+the sign of the area being taken as explained.
+
+Now\Note{42} let us assume further that, from the origin upon the curved surface, infinitely
+many other shortest lines go out, and denote by~$A$ that indefinite angle which the
+first element, moving counter-clockwise, makes with the first element of the first line;
+and through the other extremities of the different curved lines let a curved line be drawn,
+concerning which, first of all, we leave it undecided whether it be a shortest line or
+not. If we suppose also that those indefinite values, which
+for the first line were $\phi$,~$\phi'$, be denoted by $\psi$,~$\psi'$ for each of
+these lines, then $\psi' - \psi$ is capable of being represented in
+the same manner on the auxiliary sphere by the space~$LL'_{1}P'_{1}P$.
+Since evidently $\psi = \phi - A$, the space\Note{43}
+\[
+\begin{aligned}[b]
+LL'_{1}P'_{1}P'L'L
+  &= \psi' - \psi - \phi' + \phi \\
+  &= \psi' - \phi' + A \\
+  &= LL'_{1}L'L + L'L'_{1}P'_{1}P'.\NoteMark
+\end{aligned}
+\qquad\qquad
+%[Illustration]
+\raisebox{-\baselineskip}{\Graphic{1.5in}{103}}
+\]
+
+If the bounding line is also a shortest line, and, when prolonged, makes with
+$LL'$,~$LL'_{1}$ the angles $B$,~$B_{1}$; if, further, $\chi$,~$\chi_{1}$ denote the same at the points $L'$,~$L'_{1}$,
+that $\phi$~did at~$L$ in the line~$LL'$, then we have
+\begin{align*}
+\chi_{1} &= \chi + \Area L'L'_{1}P'_{1}P', \\
+\psi' - \phi' + A &= LL'_{1}L'L + \chi_{1} - \chi;
+\end{align*}
+but
+\begin{align*}
+\phi' &= \chi + B, \\
+\psi' &= \chi_{1} + B_{1},
+\end{align*}
+therefore
+\[
+B_{1} - B + A = LL'_{1}L'L.
+\]
+The angles of the triangle~$LL'L'_{1}$ evidently are
+\[
+A,\qquad 180° - B,\qquad B_{1},
+\]
+\PageSep{104}
+therefore their sum is
+\[
+180° + LL'_{1}L'L.
+\]
+
+The form of the proof will require some modification and explanation, if the point~$(3)$
+falls within the triangle. But, in general, we conclude
+
+%[** TN: Quoted, not italicized, in the original]
+\begin{Theorem}[]
+The sum of the three angles of a triangle, which is formed of shortest lines
+upon an arbitrary curved surface, is equal to the sum of~$180°$ and the area of
+the triangle upon the auxiliary sphere, the boundary of which is formed by the
+points~$L$, corresponding to the points in the boundary of the original triangle,
+and in such a manner that the area of the triangle may be regarded as positive
+or negative according as it is inclosed by its boundary in the same sense as
+the original figure or the contrary.
+\end{Theorem}
+
+Wherefore\Note{44} we easily conclude also that the sum of all the angles of a polygon
+of $n$~sides, which are shortest lines upon the curved surface, is [equal to] the sum
+of $(n - 2)180° + \text{the area of the polygon upon the sphere~etc.}$
+
+
+\Article{16.}
+
+If one curved surface can be completely developed upon another surface, then all
+lines upon the first surface will evidently retain their magnitudes after the development
+upon the other surface; likewise the angles which are formed by the intersection
+of two lines. Evidently, therefore, such lines also as are shortest lines upon
+one surface remain shortest lines after the development. Whence, if to any arbitrary
+polygon formed of shortest lines, while it is upon the first surface, there corresponds
+the figure of the zeniths\Note{45} upon the auxiliary sphere, the area of which is~$A$,
+and if, on the other hand, there corresponds to the same polygon, after its development
+upon another surface, a figure of the zeniths upon the auxiliary sphere, the
+area of which is~$A'$, it follows at once that in every case
+\[
+A = A'.
+\]
+Although this proof originally presupposes the boundaries of the figures to be shortest
+lines, still it is easily seen that it holds generally, whatever the boundary may be.
+For, in fact, if the theorem is independent of the number of sides, nothing will prevent
+us from imagining for every polygon, of which some or all of its sides are not
+shortest lines, another of infinitely many sides all of which are shortest lines.
+
+Further, it is clear that every figure retains also its area after the transformation
+by development.
+\PageSep{105}
+
+We shall here consider 4~figures:
+
+%[** TN: Indented line list items in the original]
+\Par{1)} an arbitrary figure upon the first surface,
+
+\Par{2)} the figure on the auxiliary sphere, which corresponds to the zeniths of the
+previous figure,
+
+\Par{3)} the figure upon the second surface, which No.~1 forms by the development,
+
+\Par{4)} the figure upon the auxiliary sphere, which corresponds to the zeniths of
+No.~3.
+
+Therefore, according to what we have proved, 2~and~4 have equal areas, as also
+1~and~3. Since we assume these figures infinitely small, the quotient obtained by
+dividing 2~by~1 is the measure of curvature of the first curved surface at this point,
+and likewise the quotient obtained by dividing 4~by~3, that of the second surface.
+From this follows the important theorem:
+
+%[** TN: Quoted, not italicized in the original]
+\begin{Theorem}[]
+In the transformation of surfaces by development the measure of curvature
+at every point remains unchanged.
+\end{Theorem}
+This is true, therefore, of the product of the greatest and smallest radii of curvature.
+
+In the case of the plane, the measure of curvature is evidently everywhere zero.
+Whence follows therefore the important theorem:
+
+\begin{Theorem}[]
+For all surfaces developable upon a plane the measure of curvature everywhere
+vanishes,
+\end{Theorem}
+or
+\[
+\left(\frac{\dd^{2}z}{\dd x\, \dd y}\right)^{2}
+  - \left(\frac{\dd^{2} z}{\dd x^{2}}\right)
+    \left(\frac{\dd^{2} z}{\dd x^{2}}\right) = 0,
+\]
+which criterion is elsewhere derived from other principles, though, as it seems to us,
+not with the desired rigor. It is clear that in all such surfaces the zeniths of all
+points can not fill out any space, and therefore they must all lie in a line.
+
+
+\Article{17.}
+
+From a given point on a curved surface we shall let an infinite number of shortest
+lines go out, which shall be distinguished from one another by the angle which their
+first elements make with the first element of a \emph{definite} shortest line. This angle we
+shall call~$\theta$. Further, let $s$~be the length [measured from the given point] of a part
+of such a shortest line, and let its extremity have the coordinates $x$,~$y$,~$z$. Since $\theta$~and~$s$,
+therefore, belong to a perfectly definite point on the curved surface, we can
+regard $x$,~$y$,~$z$ as functions of $\theta$~and~$s$. The direction of the element of~$s$ corresponds
+to the point~$\lambda$ on the sphere, whose coordinates are $\xi$,~$\eta$,~$\zeta$. Thus we shall have
+\PageSep{106}
+\[
+\xi = \frac{\dd x}{\dd s},\qquad
+\eta = \frac{\dd y}{\dd s},\qquad
+\zeta = \frac{\dd z}{\dd s}.
+\]
+
+The extremities of all shortest lines of equal lengths~$s$ correspond to a curved
+line whose length we may call~$t$. We can evidently consider~$t$ as a function of $s$~and~$\theta$,
+and if the direction of the element of~$t$ corresponds upon the sphere to the point~$\lambda'$
+whose coordinates are $\xi'$,~$\eta'$,~$\zeta'$, we shall have
+\[
+\xi'·\frac{\dd t}{\dd\theta} = \frac{\dd x}{\dd\theta},\qquad
+\eta'·\frac{\dd t}{\dd\theta} = \frac{\dd y}{\dd\theta},\qquad
+\zeta'·\frac{\dd t}{\dd\theta} = \frac{\dd z}{\dd\theta}.
+\]
+Consequently
+\[
+(\xi\xi' + \eta\eta' + \zeta\zeta')\, \frac{\dd t}{\dd\theta}
+  = \frac{\dd x}{\dd s}·\frac{\dd x}{\dd\theta}
+  + \frac{\dd y}{\dd s}·\frac{\dd y}{\dd\theta}
+  + \frac{\dd z}{\dd s}·\frac{\dd z}{\dd\theta}.
+\]
+This magnitude we shall denote by~$u$, which itself, therefore, will be a function of $\theta$~and~$s$.
+
+We find, then, if we differentiate with respect to~$s$,
+\begin{align*}
+\frac{\dd u}{\dd s}
+  &= \frac{\dd^{2} x}{\dd s^{2}}·\frac{\dd x}{\dd\theta}
+   + \frac{\dd^{2} y}{\dd s^{2}}·\frac{\dd y}{\dd\theta}
+   + \frac{\dd^{2} z}{\dd s^{2}}·\frac{\dd z}{\dd\theta}
+   + \tfrac{1}{2}\, \frac{\dd\left\{
+       \left(\dfrac{\dd x}{\dd s}\right)^{2}
+     + \left(\dfrac{\dd y}{\dd s}\right)^{2}
+     + \left(\dfrac{\dd z}{\dd s}\right)^{2}\right\}}{\dd\theta} \\
+  &= \frac{\dd^{2} x}{\dd s^{2}}·\frac{\dd x}{\dd\theta}
+   + \frac{\dd^{2} y}{\dd s^{2}}·\frac{\dd y}{\dd\theta}
+   + \frac{\dd^{2} z}{\dd s^{2}}·\frac{\dd z}{\dd\theta},
+\end{align*}
+because
+\[
+  \left(\dfrac{\dd x}{\dd s}\right)^{2}
++ \left(\dfrac{\dd y}{\dd s}\right)^{2}
++ \left(\dfrac{\dd z}{\dd s}\right)^{2} = 1,
+\]
+and therefore its differential is equal to zero.
+
+But since all points [belonging] to one constant value of~$\theta$ lie on a shortest line,
+if we denote by~$L$ the zenith of the point to which $s$,~$\theta$ correspond and by $X$,~$Y$,~$Z$
+the coordinates of~$L$, [from the last formulæ of \Art{13}],
+\[
+\frac{\dd^{2} x}{\dd s^{2}} = \frac{X}{p},\qquad
+\frac{\dd^{2} y}{\dd s^{2}} = \frac{Y}{p},\qquad
+\frac{\dd^{2} z}{\dd s^{2}} = \frac{Z}{p},
+\]
+if $p$~is the radius of curvature. We have, therefore,
+\[
+p·\frac{\dd u}{\dd s}
+  = X\, \frac{\dd x}{\dd\theta}
+  + Y\, \frac{\dd y}{\dd\theta}
+  + Z\, \frac{\dd z}{\dd\theta}
+  = \frac{\dd t}{\dd\theta}(X\xi' + Y\eta' + Z\zeta').
+\]
+But
+\[
+X\xi' + Y\eta' + Z\zeta' = \cos L\lambda' = 0,
+\]
+because, evidently, $\lambda'$~lies on the great circle whose pole is~$L$. Therefore we have
+\[
+\frac{\dd u}{\dd s} = 0,
+\]
+\PageSep{107}
+or $u$~independent of~$s$, and therefore a function of $\theta$~alone. But for $s = 0$, it is evident
+that $t = 0$, $\dfrac{\dd t}{\dd\theta} = 0$, and therefore $u = 0$. Whence we conclude that, in general,
+$u = 0$, or
+\[
+\cos \lambda\lambda' = 0.
+\]
+From this follows the beautiful theorem:
+
+\begin{Theorem}[]
+If all lines drawn from a point on the curved surface are shortest lines of
+equal lengths, they meet the line which joins their extremities everywhere at
+right angles.
+\end{Theorem}
+
+We can show in a similar manner that, if upon the curved surface any curved
+line whatever is given, and if we suppose drawn from every point of this line toward
+the same side of it and at right angles to it only shortest lines of equal lengths, the
+extremities of which are joined by a line, this line will be cut at right angles by
+those lines in all its points. We need only let $\theta$ in the above development represent
+the length of the \emph{given} curved line from an arbitrary point, and then the above calculations
+retain their validity, except that $u = 0$ for $s = 0$ is now contained in the
+hypothesis.
+
+
+\Article{18.}
+
+The relations arising from these constructions deserve to be developed still more
+fully. We have, in the first place, if, for brevity, we write~$m$ for~$\dfrac{\dd t}{\dd\theta}$,
+\begin{alignat*}{3}
+\Tag{(1)}
+\frac{\dd x}{\dd s} &= \xi, &
+\frac{\dd y}{\dd s} &= \eta, &
+\frac{\dd z}{\dd s} &= \zeta, \\
+\Tag{(2)}
+\frac{\dd x}{\dd\theta} &= m\xi',\quad &
+\frac{\dd y}{\dd\theta} &= m\eta',\quad &
+\frac{\dd z}{\dd\theta} &= m\zeta',
+\end{alignat*}
+\begin{alignat*}{4}
+\Tag{(3)}
+&\xi^{2}  &&+ \eta^{2}  &&+ \zeta^{2} &&= 1, \\
+\Tag{(4)}
+&\xi'^{2} &&+ \eta'^{2} &&+ \zeta'^{2} &&= 1, \\
+\Tag{(5)}
+&\xi\xi'  &&+ \eta\eta' &&+ \zeta\zeta' &&= 0.
+\end{alignat*}
+Furthermore,
+\begin{alignat*}{4}
+\Tag{(6)}
+&X^{2} &&+ Y^{2} &&+ Z^{2} &&= 1, \\
+\Tag{(7)}
+&X\xi  &&+ Y\eta &&+ Z\zeta &&= 0, \\
+\Tag{(8)}
+&X\xi' &&+ Y\eta' &&+ Z\zeta' &&= 0,
+\end{alignat*}
+and
+\begin{align*}
+\Tag{[9]}
+&\left\{
+\begin{alignedat}{2}
+X &= \zeta\eta' &&- \eta\zeta', \\
+Y &= \xi\zeta'  &&- \zeta\xi', \\
+Z &= \eta\xi'   &&- \xi\eta';
+\end{alignedat}
+\right. \\
+\PageSep{108}
+\Tag{[10]}
+&\left\{
+\begin{alignedat}{2}
+\xi'   &= \eta  Z &&- \zeta Y, \\
+\eta'  &= \zeta X &&- \xi   Z, \\
+\zeta' &= \xi   Y &&- \eta  X;
+\end{alignedat}
+\right. \\
+\Tag{[11]}
+&\left\{
+\begin{alignedat}{2}
+\xi   &= Y\zeta' &&- Z\eta', \\
+\eta  &= Z\xi'   &&- X\zeta', \\
+\zeta &= X\eta'  &&- Y\xi'.
+\end{alignedat}
+\right.
+\end{align*}
+
+Likewise, $\dfrac{\dd\xi}{\dd s}$, $\dfrac{\dd\eta}{\dd s}$, $\dfrac{\dd\zeta}{\dd s}$ are proportional to $X$,~$Y$,~$Z$, and if we set
+\[
+\frac{\dd\xi}{\dd s}   = pX,\qquad
+\frac{\dd\eta}{\dd s}  = pY,\qquad
+\frac{\dd\zeta}{\dd s} = pZ,
+\]
+where $\dfrac{1}{p}$ denotes the radius of curvature of the line~$s$, then
+\[
+p = X\, \frac{\dd\xi}{\dd s}
+  + Y\, \frac{\dd\eta}{\dd s}
+  + Z\, \frac{\dd\zeta}{\dd s}.
+\]
+By differentiating~(7) with respect to~$s$, we obtain
+\[
+-p = \xi\, \frac{\dd X}{\dd s}
+   + \eta\, \frac{\dd Y}{\dd s}
+   + \zeta\, \frac{\dd Z}{\dd s}.
+\]
+
+We can easily show that $\dfrac{\dd\xi'}{\dd s}$, $\dfrac{\dd\eta'}{\dd s}$, $\dfrac{\dd\zeta'}{\dd s}$ also are proportional to $X$,~$Y$,~$Z$. In fact,
+[from~10] the values of these quantities are also [equal to]
+\[
+\eta\, \frac{\dd Z}{\dd s} - \zeta\, \frac{\dd Y}{\dd s},\qquad
+\zeta\, \frac{\dd X}{\dd s} - \xi\, \frac{\dd Z}{\dd s},\qquad
+\xi\, \frac{\dd Y}{\dd s} - \eta\, \frac{\dd X}{\dd s},
+\]
+therefore
+\begin{align*}
+Y\, \frac{\dd\xi'}{\dd s} - X\, \frac{\dd\eta'}{\dd s}
+  &= - \zeta\left(\frac{Y\, \dd Y}{\dd s} + \frac{X\, \dd X}{\dd s}\right)
+   + \frac{\dd Z}{\dd s}(Y\eta + X\xi) \\
+  &= - \zeta\left(\frac{X\, \dd X + Y\, \dd Y + Z\, \dd Z}{\dd s}\right)
+   + \frac{\dd Z}{\dd s}(X\xi + Y\eta + Z\zeta) \\
+  &= 0,
+\end{align*}
+and likewise the others. We set, therefore,
+\[
+\frac{\dd\xi'}{\dd s} = p'X,\qquad
+\frac{\dd\eta'}{\dd s} = p'Y,\qquad
+\frac{\dd\zeta'}{\dd s} = p'Z,
+\]
+whence
+\[
+p' = ±\SQRT{\left(\frac{\dd\xi'}{\dd s}\right)^{2}
+  + \left(\frac{\dd\eta'}{\dd s}\right)^{2}
+  + \left(\frac{\dd\zeta'}{\dd s}\right)^{2}}\Add{,}
+\]
+\PageSep{109}
+and also
+\[
+p' = X\, \frac{\dd\xi'}{\dd s}
+   + Y\, \frac{\dd\eta'}{\dd s}
+   + Z\, \frac{\dd\zeta'}{\dd s}.
+\]
+Further [we obtain], from the result obtained by differentiating~(8),
+\[
+-p' = \xi'\, \frac{\dd X}{\dd s}
+    + \eta'\, \frac{\dd Y}{\dd s}
+    + \zeta'\, \frac{\dd Z}{\dd s}.
+\]
+But we can derive two other expressions for this. We have
+\[
+\frac{\dd m\xi'}{\dd s} = \frac{\dd\xi}{\dd\theta},\qquad
+\left[
+\frac{\dd m\eta'}{\dd s} = \frac{\dd\eta}{\dd\theta},\qquad
+\frac{\dd m\zeta'}{\dd s} = \frac{\dd\zeta}{\dd\theta},
+\right]
+\]
+therefore [because of~(8)]
+\[
+mp' = X\, \frac{\dd\xi}{\dd\theta}
+    + Y\, \frac{\dd\eta}{\dd\theta}
+    + Z\, \frac{\dd\zeta}{\dd\theta}.
+\]
+[and therefore, from~(7),]
+\[
+-mp' = \xi\, \frac{\dd X}{\dd\theta}
+     + \eta\, \frac{\dd Y}{\dd\theta}
+     + \zeta\, \frac{\dd Z}{\dd\theta}.
+\]
+
+After these preliminaries [using (2)~and~(4)] we shall now first put~$m$ in the form
+\[
+m = \xi'\, \frac{\dd x}{\dd\theta}
+  + \eta'\, \frac{\dd y}{\dd\theta}
+  + \zeta'\, \frac{\dd z}{\dd\theta},
+\]
+and differentiating with respect to~$s$, we have\footnote
+  {It is better to differentiate~$m^{2}$. [In fact from (2)~and~(4)
+  \[
+  m^{2} = \left(\frac{\dd x}{\dd\theta}\right)^{2}
+        + \left(\frac{\dd y}{\dd\theta}\right)^{2}
+        + \left(\frac{\dd z}{\dd\theta}\right)^{2},
+  \]
+  therefore
+  \begin{align*}
+  m\, \frac{\dd m}{\dd s}
+   &= \frac{\dd x}{\dd\theta}·\frac{\dd^{2} x}{\dd\theta\, \dd s}
+    + \frac{\dd y}{\dd\theta}·\frac{\dd^{2} y}{\dd\theta\, \dd s}
+    + \frac{\dd z}{\dd\theta}·\frac{\dd^{2} z}{\dd\theta\, \dd s} \\
+   &= m\xi'\, \frac{\dd\xi}{\dd\theta}
+    + m\eta'\, \frac{\dd\eta}{\dd\theta}
+    + m\zeta'\, \frac{\dd\zeta}{\dd\theta}.]
+  \end{align*}}
+% [** End of footnote]
+\begin{align*}
+%[** TN: Re-broken]
+\frac{\dd m}{\dd s}
+  &= \frac{\dd x}{\dd\theta}·\frac{\dd\xi'}{\dd s}
+   + \frac{\dd y}{\dd\theta}·\frac{\dd\eta'}{\dd s}
+   + \frac{\dd z}{\dd\theta}·\frac{\dd\zeta'}{\dd s} %\\
+%  &\quad
+   + \xi'\, \frac{\dd^{2} x}{\dd s\, \dd\theta}
+        + \eta'\, \frac{\dd^{2} y}{\dd s\, \dd\theta}
+        + \zeta'\, \frac{\dd^{2} z}{\dd s\, \dd\theta} \displaybreak[1] \\
+%
+  &= mp'(\xi'X + \eta'Y + \zeta'Z) %\\
+%  &\quad
+   + \xi'\, \frac{\dd\xi}{\dd\theta}
+        + \eta'\, \frac{\dd\eta}{\dd\theta}
+        + \zeta'\, \frac{\dd\zeta}{\dd\theta} \displaybreak[1] \\
+%
+  &= \xi'\, \frac{\dd\xi}{\dd\theta}
+   + \eta'\, \frac{\dd\eta}{\dd\theta}
+   + \zeta'\, \frac{\dd\zeta}{\dd\theta}.
+\end{align*}
+\PageSep{110}
+
+If we differentiate again with respect to~$s$, and notice that
+\[
+\frac{\dd^{2} \xi}{\dd s\, \dd\theta}
+  = \frac{\dd(pX)}{\dd\theta},\quad\text{etc.},
+\]
+and that
+\[
+X\xi' + Y\eta' + Z\zeta' = 0,
+\]
+we have\Note{46}
+{\small
+\begin{align*}
+\frac{\dd^{2} m}{\dd s^{2}}
+  &= p\left(\xi'\, \frac{\dd X}{\dd\theta}
+          + \eta'\, \frac{\dd Y}{\dd\theta}
+          + \zeta'\, \frac{\dd Z}{\dd\theta}\right)
+   + p'\left(X \frac{\dd\xi}{\dd\theta}
+          + Y \frac{\dd \eta}{\dd\theta}
+          + Z \frac{\dd \zeta}{\dd\theta}\right) \\
+%
+  &= p\left(\xi'\, \frac{\dd X}{\dd\theta}
+          + \eta'\, \frac{\dd Y}{\dd\theta}
+          + \zeta'\, \frac{\dd Z}{\dd\theta}\right) + mp'^{2} \\
+%
+  &= -\left(\xi\, \frac{\dd X}{\dd s}
+          + \eta\, \frac{\dd Y}{\dd s}
+          + \zeta\, \frac{\dd Z}{\dd s}\right)
+      \left(\xi'\, \frac{\dd X}{\dd\theta}
+          + \eta'\, \frac{\dd Y}{\dd\theta}
+          + \zeta'\, \frac{\dd Z}{\dd\theta}\right) \\
+  &\phantom{={}}
+    + \left(\xi'\, \frac{\dd X}{\dd s}
+          + \eta'\, \frac{\dd Y}{\dd s}
+          + \zeta'\, \frac{\dd Z}{\dd s}\right)
+      \left(\xi\, \frac{\dd X}{\dd\theta}
+          + \eta\, \frac{\dd Y}{\dd\theta}
+          + \zeta\, \frac{\dd Z}{\dd\theta}\right) \\
+%
+  &= \left(\frac{\dd Y}{\dd\theta}\, \frac{\dd Z}{\dd s}
+         - \frac{\dd Y}{\dd s}\, \frac{\dd Z}{\dd\theta}\right)X
+   + \left(\frac{\dd Z}{\dd\theta}\, \frac{\dd X}{\dd s}
+         - \frac{\dd Z}{\dd s}\, \frac{\dd X}{\dd\theta}\right)Y
+   + \left(\frac{\dd X}{\dd\theta}\, \frac{\dd Y}{\dd s}
+         - \frac{\dd X}{\dd s}\, \frac{\dd Y}{\dd\theta}\right)Z.\NoteMark
+\end{align*}}
+
+[But if the surface element
+\[
+m\, ds\, d\theta
+\]
+belonging to the point $x$,~$y$,~$z$ be represented upon the auxiliary sphere of unit radius
+by means of parallel normals, then there corresponds to it an area whose magnitude is
+{\small
+\[
+\left\{
+X\left(\frac{\dd Y}{\dd s}\, \frac{\dd Z}{\dd\theta}
+     - \frac{\dd Y}{\dd\theta}\, \frac{\dd Z}{\dd s}\right) +
+Y\left(\frac{\dd Z}{\dd s}\, \frac{\dd X}{\dd\theta}
+     - \frac{\dd Z}{\dd\theta}\, \frac{\dd X}{\dd s}\right) +
+Z\left(\frac{\dd X}{\dd s}\, \frac{\dd Y}{\dd\theta}
+     - \frac{\dd X}{\dd\theta}\, \frac{\dd Y}{\dd s}\right)
+\right\}ds\, d\theta.
+\]}%
+Consequently, the measure of curvature at the point under consideration is equal to
+\[
+-\frac{1}{m}\, \frac{\dd^{2} m}{\dd s^{2}}.]
+\]
+\PageSep{111}
+
+
+\Notes.
+
+The parts enclosed in brackets are additions of the editor of the German edition
+or of the translators.
+
+``The foregoing fragment, \textit{Neue allgemeine Untersuchungen über die krummen Flächen},
+differs from the \textit{Disquisitiones} not only in the more limited scope of the matter, but
+also in the method of treatment and the arrangement of the theorems. There [paper
+of~1827] \textsc{Gauss} assumes that the rectangular coordinates $x$,~$y$,~$z$ of a point of the surface
+can be expressed as functions of any two independent variables $p$~and~$q$, while
+here [paper of~1825] he chooses as new variables the geodesic coordinates $s$~and~$\theta$.
+Here [paper of~1825] he begins by proving the theorem, that the sum of the three
+angles of a triangle, which is formed by shortest lines upon an arbitrary curved surface,
+differs from~$180°$ by the area of the triangle, which corresponds to it in the representation
+by means of parallel normals upon the auxiliary sphere of unit radius. From
+this, by means of simple geometrical considerations, he derives the fundamental theorem,
+that \Chg{``}{`}in the transformation of surfaces by bending, the measure of curvature at
+every point remains unchanged.\Chg{''}{'} But there [paper of~1827] he first shows, in \Art[1827]{11},
+that the measure of curvature can be expressed simply by means of the three
+quantities $E$,~$F$,~$G$, and their derivatives with respect to $p$~and~$q$, from which follows
+the theorem concerning the invariant property of the measure of curvature as a corollary;
+and only much later, in \Art[1827]{20}, quite independently of this, does he prove the
+theorem concerning the sum of the angles of a geodesic triangle.'' \\
+\null\hfill Remark by Stäckel, Gauss's Works, vol.~\textsc{viii}, p.~443.
+
+\LineRef[32]{3}{Art.~3, p.~84, l.~9}. $\cos^{2}\phi$, etc., is used here where the German text has~$\cos\phi^{2}$,~etc.
+
+\LineRef[33]{3}{Art.~3, p.~84, l.~13}. $p^{2}$,~etc., is used here where the German text has~$pp$,~etc.
+
+\LineRef[34]{7}{Art.~7, p.~89, ll.~13,~21}. Since $\lambda L$ is less than~$90°$, $\cos\lambda L$~is always positive
+and, therefore, the algebraic sign of the expression for the volume of this pyramid
+depends upon that of~$\sin L'L''$. Hence it is positive, zero, or negative according as
+the arc~$L'L''$ is less than, equal to, or greater than~$180°$.
+
+\LineRef[34]{7}{Art.~7, p.~89, ll.~14--21}. As is seen from the paper of~1827 (see \Pageref{6}), Gauss
+\PageSep{112}
+corrected this statement. To be correct it should read: for which we can write also,
+according to well known principles of spherical trigonometry,
+\[
+\sin LL'·\sin L'·\sin L'L''
+  = \sin L'L''·\sin L''·\sin L''L
+  = \sin L''L·\sin L·\sin LL',
+\]
+if $L$,~$L'$,~$L''$ denote the three angles of the spherical triangle, where $L$~is the angle
+measured from the arc~$LL''$ to~$LL'$, and so for the other angles. At the same time
+we easily see that this value is one-sixth of the pyramid whose angular points are
+the centre of the sphere and the three points $L$,~$L'$,~$L''$; and this pyramid is \emph{positive}
+when the points $L$,~$L'$,~$L''$ are arranged in the same order about this triangle as the
+points $(1)$,~$(2)$,~$(3)$ about the triangle $(1)\ (2)\ (3)$.
+
+\LineRef{8}{Art.~8, p.~90, l.~7~fr.~bot}. In the German text $V$~stands for~$f$ in this equation
+and in the next line but one.
+
+\LineRef[35]{11}{Art.~11, p.~93, l.~8~fr.~bot}. In the German text, in the expression for~$B$, $(\alpha\beta' + \alpha\beta')$
+stands for~$(\alpha'\beta + \alpha\beta')$.
+
+\LineRef[36]{11}{Art.~11, p.~94, l.~17}. The vertices of the triangle are $M$,~$M'$,~$(3)$, whose coordinates
+are $\alpha$,~$\beta$,~$\gamma$; $\alpha'$,~$\beta'$,~$\gamma'$; $0$,~$0$,~$1$, respectively. The pole of the arc~$MM'$ on
+the same side as~$(3)$ is~$L$, whose coordinates are $X$,~$Y$,~$Z$. Now applying the formula
+%[** TN: Omitted incorrect line number reference]
+on \Pageref{89},\Chg{ line~10,}{}
+\[
+x'y'' - y'x'' = \sin L'L''\cos\lambda(3),
+\]
+to this triangle, we obtain
+\[
+\alpha\beta' - \beta\alpha' = \sin MM' \cos L(3)
+\]
+or, since
+\[
+MM' = 90°,\quad\text{and}\quad \cos L(3) = ±Z
+\]
+we have
+\[
+\alpha\beta' - \beta\alpha' = ±Z.
+\]
+
+\LineRef[37]{14}{Art.~14, p.~100, l.~19}. Here $X$,~$Y$,~$Z$; $\xi$,~$\eta$,~$\zeta$; $0$,~$0$,~$1$ take the place of $x$,~$y$,~$z$;
+$x'$,~$y'$,~$z'$; $x''$,~$y''$,~$z''$ of the top of \Pageref{89}. Also $(3)$,~$\lambda$ take the place of $L'$,~$L''$, and
+$\phi$~is the angle~$L$ in the note at the top of this page.
+
+\LineRef[38]{14}{Art.~14, p.~101, l.~2~fr.~bot}. In the German text $\{\zeta X - \eta XYZ + \xi Y^{2}Z\}$ stands
+for $\{\zeta X + \eta XYZ - \xi Y^{2}Z\}$.
+
+\LineRef[39]{15}{Art.~15, p.~102, l.~13 and the following}. Transforming to polar coordinates,
+$r$,~$\theta$,~$\psi$, by the substitutions (since on the auxiliary sphere $r = 1$)
+\begin{gather*}
+X = \sin\theta \sin\psi,\quad
+Y = \sin\theta \cos\psi,\quad
+Z = \cos\theta, \\
+dX =  \sin\theta \cos\psi\, d\psi + \cos\theta \sin\psi\, d\theta,\qquad
+dY = -\sin\theta \sin\psi\, d\psi + \cos\theta \cos\psi\, d\theta, \\
+\Tag{(1)}
+\Typo{=}{-}\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX)\quad\text{becomes}\quad
+\cos\theta\, d\psi.
+\end{gather*}
+\PageSep{113}
+
+In the figures on \Pgref{fig:102}, $PL$~and~$P'L'$ are arcs of great circles intersecting in
+the point~$(3)$, and the element~$LL'$, which is not necessarily the arc of a great circle,
+corresponds to the element of the geodesic line on the curved surface. $(2)PP'(1)$
+also is the arc of a great circle. Here $P'P = d\psi$, $Z = \cos\theta ={}$Altitude of the zone
+of which $LL'P'P$~is a part. The area of a zone varies as the altitude of the zone.
+Therefore, in the case under consideration,
+\[
+\frac{\text{Area of zone}}{2\pi} = \frac{Z}{1}.
+\]
+Also
+\[
+\frac{\Area LL'P'P}{\text{Area of zone}} = \frac{d\psi}{2\pi}.
+\]
+From these two equations,
+\[
+\Tag{(2)}
+\Area LL'P'P = Z\, d\psi,\quad\text{or}\quad \cos\theta\, d\psi.
+\]
+From (1)~and~(2)
+\[
+-\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX) = \Area LL'P'P.
+\]
+
+\LineRef[40]{15}{Art.~15, p.~102}. The point~$(3)$ in the figures on this page was added by the
+translators.
+
+\LineRef[41]{15}{Art.~15, p.~103, ll.~6--9}. It has been shown that $d\phi = \Area LL'P'P, = dA$, say.
+Then
+\[
+\int_{\phi}^{\phi'} d\phi = \int_{0}^{A} dA,
+\]
+or
+\[
+\phi' - \phi = A,\quad\text{the finite area $LL'P'P$}.
+\]
+
+\LineRef[42]{15}{Art.~15, p.~103, l.~10 and the following}. Let $A$,~$B'$,~$B_{1}$ be the vertices of a
+geodesic triangle on the curved surface, and let the corresponding triangle on the
+auxiliary sphere be~$LL'L'_{1}L$, whose sides are not necessarily arcs of great circles. Let
+$A$,~$B'$,~$B_{1}$ denote also the angles of the geodesic triangle. Here $B'$~is the supplement
+of the angle denoted by~$B$ on \Pageref{103}. Let $\phi$~be the angle on the sphere
+between the great circle arcs $L\lambda$,~$L(3)$, \ie, $\phi = (3)L\lambda$, $\lambda$~corresponding to the direction
+of the element at~$A$ on the geodesic line~$AB'$, and let $\phi' = (3)L'\lambda_{1}$, $\lambda_{1}$~corresponding
+to the direction of the element at~$B'$ on the line~$AB'$. Similarly, let $\psi = (3)L\mu$,
+\PageSep{114}
+$\psi' = (3)L'_{1}\mu_{1}$, $\mu$,~$\mu_{1}$ denoting the directions of the elements at
+$A$,~$B_{1}$, respectively, on the line~$AB_{1}$. And let $\chi = (3)L'\nu$,
+$\chi_{1} = (3)L'_{1}\nu_{1}$, $\nu$,~$\nu_{1}$ denoting the directions of the elements at
+$B'$,~$B_{1}$, respectively, on the line~$B'B_{1}$.
+
+Then from the first formula on \Pageref{103},
+\begin{gather*}
+\begin{aligned}[b]
+\phi' - \phi &= \Area LL'P'P, \\
+\psi' - \psi &= \Area LL'_{1}P'_{1}P, \\
+\chi_{1} - \chi &= \Area L'L'_{1}P'_{1}P',
+\end{aligned}
+\qquad\qquad
+%[Illustration]
+\Graphic{1.5in}{114} \\
+\psi' - \psi - (\phi' - \phi) - (\chi_{1} - \chi)
+  = \Area L'L'_{1}P'_{1}P'
+  - \Area LL'P'P
+  - \Area L'L'_{1}P'_{1}P',
+\end{gather*}
+or
+\[
+\Tag{(1)}
+(\phi - \psi) + (\chi - \phi') + (\psi' - \chi_{1})
+  = \Area LL'_{1}L'L.
+\]
+
+Since $\lambda$,~$\mu$ represent the directions of the linear elements at~$A$ on the geodesic
+lines $AB'$,~$AB_{1}$ respectively, the absolute value of the angle~$A$ on the surface is measured
+by the arc~$\mu\lambda$, or by the spherical angle~$\mu L\lambda$. But $\phi - \psi = (3)L\lambda - (3)L\mu
+= \mu L\lambda$. \\
+Therefore
+\[
+A = \phi - \psi.
+\]
+Similarly
+\begin{align*}
+180° - B' &= -(\chi - \phi'), \\
+B_{1} &= \psi' - \chi_{1}.
+\end{align*}
+Therefore, from~(1),
+\[
+A + B' + B_{1} - 180° = \Area LL'_{1}L'L.
+\]
+
+\LineRef[43]{15}{Art.~15, p.~103, l.~19}. In the German text $LL'P'P$ stands for~$LL'_{1}P'_{1}P$,
+which represents the angle~$\psi' - \psi$.
+
+\LineRef[44]{15}{Art.~15, p.~104, l.~12}. This general theorem may be stated as follows:
+
+The sum of all the angles of a polygon of $n$~sides, which are shortest lines
+upon the curved surface, is equal to the sum of $(n - 2)180°$ and the area of the
+polygon upon the auxiliary sphere whose boundary is formed by the points~$L$ which
+correspond to the points of the boundary of the given polygon, and in such a manner
+that the area of this polygon may be regarded positive or negative according as it is
+enclosed by its boundary in the same sense as the given figure or the contrary.
+
+\LineRef[45]{16}{Art.~16, p.~104, l.~12~fr.~bot}. The \emph{zenith} of a point on the surface is the corresponding
+point on the auxiliary sphere. It is the spherical representation of the
+point.
+
+\LineRef[46]{18}{Art.~18, p.~110, l.~10}. The normal to the surface is here taken in the direction
+opposite to that given by~[9] \Pageref{107}.
+\PageSep{115}
+\BackMatter
+%[** TN: Print "BIBLIOGRAPHY" title page]
+\BibliographyPage
+\PageSep{116}
+%[Blank page]
+\PageSep{117}
+
+
+\begin{Bibliography}{%
+This bibliography is limited to books, memoirs, etc., which use Gauss's method and which treat, more or less
+generally, one or more of the following subjects: curvilinear coordinates, geodesic and isometric lines, curvature of
+surfaces, deformation of surfaces, orthogonal systems, and the general theory of surfaces. Several papers which lie
+beyond these limitations have been added because of their importance or historic interest. For want of space, generally,
+papers on minimal surfaces, congruences, and other subjects not mentioned above have been excluded.
+
+Generally, the numbers following the volume number give the pages on which the paper is found.
+
+C.~R. will be used as an abbreviation for Comptes Rendus hebdomadaires des séances de l'Académie des
+Sciences\Typo{.}{,} Paris.}
+
+\Author{Adam, Paul} \Title{Sur les systèmes triples orthogonaux.} Thesis.
+80~pp.\Add{,} Paris, 1887.
+
+\Title{Sur les surfaces isothermiques à lignes de courbure
+planes dans un système ou dans les deux systèmes.}
+Ann.\ de l'École Normale, ser.~3, vol.~10, 319--358, 1893;
+C.~R., vol.~116, 1036--1039, 1893.
+
+\Title{Sur les surfaces admettant pour lignes de courbure
+deux séries de cercles géodésiques orthogonaux.} Bull.\
+de~la Soc.\ Math.\ de France, vol.~22, 110--115, 1894.
+
+\Title{Mémoire sur la déformation des surfaces.} Bull.\ de~la
+Soc.\ Math.\ de France, vol.~23, 219--240, 1895.
+
+\Title{Sur la déformation des surfaces.} Bull.\ de~la Soc.\
+Math.\ de France, vol.~23, 106--111, 1895; C.~R., vol.\
+121, 551--553, 1895.
+
+\Title{Sur la déformation des surfaces avec conservation des
+lignes de courbure.} Bull.\ de~la Soc.\ Math.\ de France,
+vol.~23, 195--196, 1895.
+
+\Title{Théorème sur la déformation des surfaces de translation.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~23, 204--209,
+1895.
+
+\Title{Sur un problème de déformation.} Bull.\ de~la Soc.\
+Math.\ de France, vol.~24, 28--39, 1896.
+
+\Author{Albeggiani, L.} \Title{Linee geodetiche tracciate sopra taluni superficie.}
+Rend.\ del Circolo Mat.\ di Palermo, vol.~3, 80--119,
+1889.
+
+\Author{Allé, M.} \Title{Zur Theorie des Gauss'schen Krümmungsmaasses.}
+Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften zu Wien,
+vol.~74, 9--38, 1876.
+
+\Author{Aoust, L. S. X. B.} \Title{Des coordonnées curvilignes se coupant
+sous un angle quelconque.} Journ.\ für Math., vol.~58,
+352--368, 1861.
+
+\Title{Théorie géométrique des coordonnées curvilignes quelconques.}
+C.~R., vol.~54, 461--463, 1862.
+
+\Title{Sur la courbure des surfaces.} C.~R., vol.~57, 217--219,
+1863.
+
+%\Author{Aoust, L. S. X. B.}
+\Title{Théorie des coordonnées curvilignes
+quelconques.} Annali di Mat., vol.~6, 65--87, 1864; ser.~2,
+vol.~2, 39--64, vol.~3, 55--69, 1868--69; ser.~2, vol.~5,
+261--288, 1873.
+
+\Author{August, T.} \Title{Ueber Flächen mit gegebener Mittelpunktsfläche
+und über Krümmungsverwandschaft.} Archiv
+der Math.\ und Phys., vol.~68, 315--352, 1882.
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+\Author{Babinet.} \Title{Sur la courbure des surfaces.} C.~R., vol.~49, 418--424,
+1859.
+
+\Author{Bäcklund, A. V.} \Title{Om ytar med konstant negativ kröking.}
+Lunds Univ.\ Ã…rsskrift, vol.~19, 1884.
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+\Author{Banal, R.} \Title{Di una classe di superficie a tre dimensioni a
+curvatura totale nulla.} Atti del Reale Instituto Veneto,
+ser.~7, vol.~6, 998--1004, 1895.
+
+\Author{Beliankén, J.} \Title{Principles of the theory of the development
+of surfaces. Surfaces of constant curvature.} \Chg{(Russian).}{(Russian.)}
+Kief Univ.\ Reports, Nos.\ 1~and~3; and Kief, \Chg{pp.\ \textsc{ii}~+~129}{\textsc{ii}~+~129~pp.},
+1898.
+
+\Author{Beltrami, Eugenio.} \Title{Di alcune formole relative alla curvatura
+delle superficie.} Annali di Mat., vol\Add{.}~4, 283--284,
+1861.
+
+\Title{Richerche di analisi applicata alla geometria.} Giornale
+di Mat., vol.~2, 267--282, 297--306, 331--339, 355--375,
+1864; vol.~3, 15--22, 33--41, 82--91, 228--240, 311--314, 1865.
+
+\Title{Delle variabili complesse sopra una superficie qualunque.}
+Annali di Mat., ser.~2, vol.~1, 329--366, 1867.
+
+\Title{Sulla teorica generale dei parametri differenziali.}
+Mem.\ dell'Accad.\ di Bologna, ser.~2, vol.~8, 549--590,
+1868.
+
+\Title{Sulla teoria generale delle superficie.} Atti dell'Ateneo
+Veneto, vol.~5, 1869.
+
+\Title{Zur Theorie des Krümmungsmaasses.} Math.\ Annalen,
+vol.~1, 575--582, 1869.
+
+\Author{Bertrand, J.} \Title{Mémoire sur la théorie des surfaces.} Journ.\
+de Math., vol.~9, 133--154, 1844.
+\PageSep{118}
+
+\Author{Betti, E.} \Title{Sopra i sistemi di superficie isoterme e orthogonali.}
+Annali di Mat., ser.~2, vol.~8, 138--145, 1877.
+
+\Author{Bianchi, Luigi.} \Title{Sopra la deformazione di una classe di
+superficie.} Giornale di Mat., vol.~16, 267--269, 1878.
+
+\Title{Ueber die Flächen mit constanter negativer Krümmung.}
+Math.\ Annalen, vol.~16, 577--582, 1880.
+
+\Title{Sulle superficie a curvatura costante positiva.} Giornale
+di Mat., vol.~20, 287--292, 1882.
+
+\Title{Sui sistemi tripli cicilici di superficie orthogonali.}
+Giornale di Mat., vol.~21, 275--292, 1883; vol.~22, 333--373,
+1884.
+
+\Title{Sopra i sistemi orthogonali di Weingarten.} Atti della
+Reale Accad.\ dei Lincei, ser.~4, vol.~1, 163--166, 243--246,
+1885; Annali di Mat., ser.~2, vol.~13, 177--234,
+1885, and ser.~2, vol.~14, 115--130, 1886.
+
+\Title{Sopra una classe di sistemi tripli di superficie orthogonali,
+che contengono un sistema di elicoidi aventi a
+comune l'asse ed il passo.} Annali di Mat., ser.~2, vol.~13,
+39--52, 1885.
+
+\Title{Sopra i sistemi tripli di superficie orthogonali che contengono
+un sistema di superficie pseudosferiche.} Atti
+della Reale Accad.\ dei Lincei, ser.~4, vol.~2, 19--22,
+1886.
+
+\Title{Sulle forme differenziali quadratiche indefinite.} Atti
+della Reale Accad.\ dei Lincei, vol.~$4_{2}$, 278, 1888; Mem.\
+della Reale Accad.\ dei Lincei, ser.~4, vol.~5, 539--603,
+1888.
+
+\Title{Sopra alcune nuove classi di superficie e di sistemi
+tripli orthogonali.} Annali di Mat., ser.~2, vol.~18, 301--358,
+1890.
+
+\Title{Sopra una nuova classe di superficie appartenenti a
+sistemi tripli orthogonali.} Atti della Reale Accad.\ dei
+Lincei, ser.~4, vol.~$6_{1}$, 435--438, 1890.
+
+\Title{Sulle superficie i cui piani principali hanno costante
+il rapporto delle distanze da un punto fisso.} Atti
+della Reale Accad.\ dei Lincei, ser.~5, vol.~$3_{2}$, 77--84,
+1894.
+
+\Title{Sulla superficie a curvatura nulla negli spazi curvatura
+costante.} Atti della Reale Accad.\ di Torino, vol.~30,
+743--755, 1895.
+
+\Title{Lezioni di geometria differenziale.} \textsc{viii}~+~541~pp.\Add{,}
+Pisa, 1894. Translation into German by Max Lukat,
+\Title{Vorlesungen über Differentialgeometrie.} \textsc{xvi}~+~659~pp.\Add{,}
+Leipzig, 1896--99.
+
+\Title{Sopra una classe di superficie collegate alle superficie
+pseudosferiche.} Atti della Reale Accad.\ dei Lincei, ser.~5,
+vol.~$5_{1}$, 133--137, 1896.
+
+\Title{Nuove richerche sulle superficie pseudosferiche.} Annali
+di Mat., ser.~2, vol.~24, 347--386, 1896.
+
+\Title{Sur deux classes de surfaces qui engendrent par un
+mouvement hélicoidal une famille de~Lamé.} Ann.\
+Faculté des sci.\ de Toulouse, vol.~11~H, 1--8, 1897.
+
+\Author{Bianchi, Luigi.} \Title{Sopra le superficie a curvatura costante
+positiva.} Atti della Reale Accad.\ dei Lincei, ser.~5,
+vol.~$8_{1}$, 223--228, 371--377, 484--489, 1899.
+
+\Title{Sulla teoria delle transformazioni delle superficie a
+curvatura costante.} Annali di Mat., ser.~3, vol.~3, 185--298,
+1899.
+
+\Author{Blutel, E.} \Title{Sur les surfaces à lignes de courbure sphérique.}
+C.~R., vol.~122, 301--303, 1896.
+
+\Author{Bonnet, Ossian.} \Title{Mémoire sur la théorie des surfaces isothermes
+orthogonales.} \Chg{Jour.}{Journ.}\ de l'École Polyt., cahier~30,
+vol.~18, 141--164, 1845.
+
+\Title{Sur la théorie générale des surfaces.} Journ.\ de l'École
+Polyt., cahier~32, vol.~19, 1--146, 1848; C.~R., vol.~33,
+89--92, 1851; vol.~37, 529--532, 1853.
+
+\Title{Sur les lignes géodésiques.} C.~R., vol.~41, 32--35,
+1855.
+
+\Title{Sur quelques propriétés des lignes géodésiques.} C.~R.,
+vol.~40, 1311--1313, 1855.
+
+\Title{Mémoire sur les surfaces orthogonales.} C.~R., vol.~54,
+554--559, 655--659, 1862.
+
+\Title{Démonstration du théorème de Gauss relatif aux petits
+triangles géodésiques situés sur une surface courbe quelconque.}
+C.~R., vol.~58, 183--188, 1864.
+
+\Title{Mémoire sur la théorie des surfaces applicables sur
+une surface donnée.} Journ.\ de l'École Polyt., cahier~41,
+vol.~24, 209--230, 1865; cahier~42, vol.~25, 1--151,
+1867.
+
+\Title{Démonstration des propriétés fondamentales du système
+de coordonnées polaires géodésiques.} C.~R., vol.~97,
+1422--1424, 1883.
+
+\Author{Bour, Edmond.} \Title{Théorie de~la déformation des surfaces.}
+Journ.\ de l'École Polyt., cahier~39, vol.~22, 1--148,
+1862.
+
+\Author{Brill, A.} \Title{Zur Theorie der geodätischen Linie und des
+geodätischen Dreiecks.} Abhandl.\ der Kgl.\ Gesell.\ der
+Wissenschaften zu München, vol.~14, 111--140, 1883.
+
+\Author{Briochi, Francesco.} \Title{Sulla integrazione della equazione della
+geodetica.} Annali di sci.\ Mat.\ e Fis., vol.~4, 133--135,
+1853.
+
+\Title{Sulla teoria delle coordinate curvilinee.} Annali di
+Mat., ser.~2, vol.~1, 1--22, 1867.
+
+\Author{Brisse, C.} \Title{Exposition analytique de~la théorie des surfaces.}
+Ann.\ de l'École Normale, ser.~2, vol.~3, 87--146, 1874;
+Journ.\ de l'École Polyt., cahier~53, 213--233, 1883.
+
+\Author{Bukrejew, B.} \Title{Surface elements of the surface of constant
+curvature.} \Chg{(Russian).}{(Russian.)} Kief Univ.\ Reports, No.~7,
+4~pp., 1897.
+
+\Title{Elements of the theory of surfaces.} \Chg{(Russian).}{(Russian.)} Kief
+Univ.\ Reports, Nos.~1,~9, and~12, 1897--99.
+
+\Author{Burali-Forti, C.} \Title{Sopra alcune questioni di geometria differenziale.}
+Rend.\ del Circolo Mat.\ di Palermo, vol.~12,
+111--132, 1898.
+\PageSep{119}
+
+\Author{Burgatti, P.} \Title{Sulla torsione geodetica delle linee tracciate
+sopra una superficie.} Rend.\ del Circolo Mat.\ di Palermo,
+vol.~10, 229--240, 1896.
+
+\Author{Burnside, W.} \Title{The lines of zero length on a surface as
+curvilinear coordinates.} Mess.\ of Math., ser.~2, vol.~19,
+99--104, 1889.
+
+\Author{Campbell, J.} \Title{Transformations which leave the lengths of
+arcs on surfaces unaltered.} Proceed.\ London Math.\
+Soc., vol.~29, 249--264, 1898.
+
+\Author{Carda, K.} \Title{Zur Geometrie auf Flächen constanter Krümmung.}
+Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften
+zu Wien, vol.~107, 44--61, 1898.
+
+\Author{Caronnet, Th.} \Title{Sur les centres de courbure géodésiques.}
+C.~R., vol.~115, 589--592, 1892.
+
+\Title{Sur des couples de surfaces applicables.} Bull.\ de~la
+Soc.\ Math.\ de France, vol.~21, 134--140, 1893.
+
+\Title{Sur les surfaces à lignes de courbure planes dans les deux
+systèmes et isothermes.} C.~R., vol.~116, 1240--1242, 1893.
+
+\Title{Recherches sur les surfaces isothermiques et les surfaces
+dont rayons de courbure sont fonctions l'un de
+l'autre.} Thesis, 66~pp.\Add{,} Paris, 1894.
+
+\Author{Casorati, Felice.} \Title{Nuova definizione della curvatura delle
+superficie e suo confronto con quella di Gauss.} Reale
+Istituto Lombardo di sci.\ e let., ser.~2, vol.~22, 335--346,
+1889.
+
+\Title{Mesure de~la courbure des surfaces suivant l'idee commune.
+Ses rapports avec les mesures de courbure Gaussienne
+et moyenne.} Acta Matematica, vol.~14, 95--110, 1890.
+
+\Author{Catalan, E.} \Title{Mémoire sur les surfaces dont les rayons de
+courbure en chaque point sont égaux et de signes contraires.}
+Journ.\ de l'École Polyt., cahier~37, vol.~21, 130--168,
+1858; C.~R., vol.~41, 35--38, 274--276, 1019--1023, 1855.
+
+\Author{Cayley, Arthur.} \Title{On the Gaussian theory of surfaces.} Proceed.\
+London Math.\ Soc., vol.~12, 187--192, 1881.
+
+\Title{On the geodesic curvature of a curve on a surface.}
+Proceed.\ London Math.\ Soc., vol.~12, 110--117, 1881.
+
+\Title{On some formulas of Codazzi and Weingarten in relation
+to the application of surfaces to each other.} Proceed.\
+London Math.\ Soc., vol.~24, 210--223, 1893.
+
+\Author{Cesàro, E.} \Title{Theoria intrinseca delle deformazioni infinitesime.}
+Rend.\ dell'Accad.\ di Napoli, ser.~2, vol.~8, 149--154,
+1894.
+
+\Author{Chelini, D.} \Title{Sulle formole fondamentali risguardanti la curvatura
+delle superficie e delle linee.} Annali di Sci.\
+Mat.\ e Fis., vol.~4, 337--396, 1853.
+
+\Title{Della curvatura delle superficie, con metodo diretto ed
+intuitivo.} Rend.\ dell'Accad.\ di Bologna, 1868, 119;
+Mem.\ dell'Accad.\ di Bologna, ser.~2, vol.~8, 27, 1868.
+
+\Title{Teoria delle coordinate curvilinee nello spazio e nelle
+superficie.} Mem.\ dell'Accad.\ di Bologna, ser.~2, vol.~8,
+483--533, 1868.
+
+\Author{Christoffel, Elwin.} \Title{Allgemeine Theorie der geodätische
+Dreiecke.} Abhandl.\ der Kgl.\ Akad.\ der Wissenschaften
+zu Berlin, 1868, 119--176.
+
+\Author{Codazzi, Delfino.} \Title{Sulla teorica delle coordinate curvilinee e
+sull uogo de'centri di curvatura d'una superficie qualunque.}
+Annali di sci.\ Mat.\ e Fis., vol.~8, 129--165,
+1857.
+
+\Title{Sulle coordinate curvilinee d'una superficie e dello
+spazio.} Annali di Mat., ser.~2, vol.~1, 293--316; vol.~2,
+101--119, 269--287; vol.~4, 10--24; vol.~5, 206--222; 1867--1871.
+
+\Author{Combescure, E.} \Title{Sur les déterminants fonctionnels et les
+coordonnèes curvilignes.} Ann.\ de l'École Normale, ser\Add{.}~1,
+vol.~4, 93--131, 1867.
+
+\Title{Sur un point de~la théorie des surfaces.} C.~R., vol.~74,
+1517--1520, 1872.
+
+\Author{Cosserat, E.} \Title{Sur les congruences des droites et sur la théorie
+des surfaces.} Ann.\ Faculté des sci.\ de Toulouse, vol.~7~N,
+1--62, 1893.
+
+\Title{Sur la déformation infinitésimale d'une surface flexible
+et inextensible et sur les congruences de droites.} Ann.\
+Faculté des sci.\ de Toulouse, vol.~8~E, 1--46, 1894.
+
+\Title{Sur les surfaces rapportées à leurs lignes de longeur
+nulle.} C.~R., vol.~125, 159--162, 1897.
+
+\Author{Craig, T.} \Title{Sur les surfaces à lignes de courbure isométriques.}
+C.~R., vol.~123, 794--795, 1896.
+
+\Author{Darboux, Gaston.} \Title{Sur les surfaces orthogonales.} Thesis,
+45~pp.\Add{,} Paris, 1866.
+
+\Title{Sur une série de lignes analogues aux lignes géodésiques.}
+Ann.\ de l'École Normale, vol.~7, 175--180, 1870.
+
+\Title{Mémoire sur la théorie des coordonnées curvilignes et
+des systèmes orthogonaux.} Ann.\ de l'École Normale,
+ser.~2, vol.~7, 101--150, 227--260, 275--348, 1878.
+
+\Title{Sur les cercles géodésiques.} C.~R., vol.~96, 54--56,
+1883.
+
+\Title{Sur les surfaces dont la courbure totale est constante.
+Sur les surfaces à courbure constante. Sur l'équation
+aux dérivées partielles des surfaces à courbure constante.}
+C.~R., vol.~97, 848--850, 892--894, 946--949, 1883.
+
+\Title{Sur la représentation sphérique des surfaces.} C.~R.,
+vol.~68, 253--256, 1869; vol.~94, 120--122, 158--160, 1290--1293,
+1343--1345, 1882; vol.~96, 366--368, 1883; Ann.\
+de l'École Normale, ser.~3, vol.~5, 79--96, 1888.
+
+\Title{Leçons sur la théorie générale des surfaces et les applications
+géométriques du calcul infinitésimale.} 4~vols.
+Paris, 1887--1896.
+
+\Title{Sur les surfaces dont la courbure totale est constante.}
+Ann.\ de l'École Normale, ser.~3, vol.~7, 9--18, 1890.
+
+\Title{Sur une classe remarkable de courbes et de surfaces
+algebriques.} Second edition. Paris, 1896.
+
+\Title{Leçons sur les systèmes orthogonaux et les coordonnées
+curvilignes.} Vol.~1. Paris, 1898.
+\PageSep{120}
+
+\Author{Darboux, Gaston.} \Title{Sur les transformations des surfaces à courbure
+totale constante.} C.~R., vol.~128, 953--958, 1899.
+
+\Title{Sur les surfaces à courbure constante positive.} C.~R.,
+vol.~128, 1018--1024, 1899.
+
+\Author{Demartres, G.} \Title{Sur les surfaces réglées dont l'\Typo{element}{élément} linéaire
+est réductible à la forme de Liouville.} C.~R., vol.~110,
+329--330, 1890.
+
+\Author{Demoulin, A.} \Title{Sur la correspondence par orthogonalité des
+éléments.} C.~R., vol.~116, 682--685, 1893.
+
+\Title{Sur une propriété caractéristique de l'\Typo{element}{élément} linéaire
+des surfaces de révolution.} Bull.\ de~la Soc.\ Math.\ de
+France, vol.~22, 47--49, 1894.
+
+\Title{Note sur la détermination des couples de surfaces
+applicables telles que la distance de deux points correspondants
+soit constante.} Bull.\ de~la Soc.\ Math.\ de
+France, vol.~23, 71--75, 1895.
+
+\Author{de Salvert}, see (de) Salvert.
+
+\Author{de Tannenberg}, see (de) Tannenberg.
+
+\Author{Dickson, Benjamin.} \Title{On the general equations of geodesic
+lines and lines of curvature on surfaces.} Camb.\ and
+Dub.\ Math.\ Journal, vol.~5, 166--171, 1850.
+
+\Author{Dini, Ulisse.} \Title{Sull'equazione differenzialle delle superficie
+applicabili su di una superficie data.} Giornale di Mat.,
+vol.~2, 282--288, 1864.
+
+\Title{Sulla teoria delle superficie.} Giornale di Mat., vol.~3,
+65--81, 1865.
+
+\Title{Ricerche sopra la teorica delle superficie.} Atti della
+Soc.\ Italiana dei~XL\@. Firenze, 1869.
+
+\Title{Sopra alcune formole generali della teoria delle superficie
+e loro applicazioni.} Annali di Mat., ser.~2, vol.~4,
+175--206, 1870.
+
+\Author{van Dorsten, R.} \Title{Theorie der Kromming von lijnen op
+gebogen oppervlakken.} Diss.\ Leiden.\ Brill. 66~pp.\Add{,} 1885.
+
+\Author{Egorow, D.} \Title{On the general theory of the correspondence of
+surfaces.} (Russian.) Math.\ Collections, pub.\ by Math.\
+Soc.\ of Moscow, vol.~19, 86--107, 1896.
+
+\Author{Enneper, A.} \Title{Bemerkungen zur allgemeinen Theorie der
+Flächen.} Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften
+zu Göttingen, 1873, 785--804.
+
+\Title{Ueber ein geometrisches Problem.} Nachr.\ der Kgl.\
+Gesell.\ der Wissenschaften zu Göttingen, 1874, 474--485.
+
+\Title{Untersuchungen über orthogonale Flächensysteme.}
+Math.\ Annalen, vol.~7, 456--480, 1874.
+
+\Title{Bemerkungen über die Biegung einiger Flächen.}
+Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
+1875, 129--162.
+
+\Title{Bemerkungen über einige Flächen mit constantem
+Krümmungsmaass.} Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften
+zu Göttingen, 1876, 597--619.
+
+\Title{Ueber die Flächen mit einem system sphärischer
+Krümmungslinien.} Journ.\ für Math., vol.~94, 829--341,
+1883.
+
+%\Author{Enneper, A.}
+\Title{Bemerkungen über einige Transformationen
+von Flächen.} Math.\ Annalen, vol.~21, 267--298, 1883.
+
+\Author{Ermakoff, W.} \Title{On geodesic lines.} (Russian.) Math.\ Collections,
+pub.\ by Math.\ Soc.\ of Moscow, vol.~15, 516--580,
+1890.
+
+\Author{von Escherich, G.} \Title{Die Geometrie auf den Flächen constanter
+negativer Krümmung.} Sitzungsb.\ der Ksl.\
+Akad.\ der Wissenschaften zu Wien, vol.~69, part~II,
+497--526, 1874.
+
+\Title{Ableitung des allgemeinen Ausdruckes für das Krümmungsmaass
+der Flächen.} Archiv für Math.\ und
+Phys., vol.~57 385--392, 1875.
+
+\Author{Fibbi, C.} \Title{Sulle superficie che contengono un sistema di
+geodetiche a torsione costante.} Annali della Reale
+Scuola Norm.\ di Pisa, vol.~5, 79--164, 1888.
+
+\Author{Firth, W.} \Title{On the measure of curvature of a surface referred
+to polar coordinates.} Oxford, Camb., and Dub.\ Mess.,
+vol.~5, 66--76, 1869.
+
+\Author{Fouché, M.} \Title{Sur les systèmes des surfaces triplement orthogonales
+où les surfaces d'une même famille admettent la
+même représentation sphérique de leurs lignes de courbure.}
+C.~R., vol.~126, 210--213, 1898.
+
+\Author{Frattini, G.} \Title{Alcune formole spettanti alla teoria infinitesimale
+delle superficie.} Giornale di Mat., vol.~13, 161--167,
+1875.
+
+\Title{Un esempio sulla teoria delle coordinate curvilinee
+applicata al calcolo integrale.} Giornale di Mat., vol.~15,
+1--27, 1877.
+
+\Author{Frobenius, G.} \Title{Ueber die in der Theorie der Flächen auftretenden
+Differentialparameter.} Journ.\ für Math., vol.~110,
+1--36, 1892.
+
+\Author{Gauss, K. F.} \Title{Allgemeine Auflösung der Aufgabe: Die
+Theile einer gegebenen Fläche auf einer anderen gegebenen
+Fläche so abzubilden, dass die Abbildung dem
+Abgebildeten in den kleinsten Theilen ähnlich wird.}
+Astronomische Abhandlungen, vol.~3, edited hy H.~C.
+Schumacher, Altona, 1825. The same, Gauss's Works,
+vol.~4, 189--216, 1880; Ostwald's Klassiker, No.~55,
+edited by A.~Wangerin, 57--81, 1894.
+
+\Author{Geiser, C. F.} \Title{Sur la théorie des systèmes triples orthogonaux.}
+Bibliothèque universelle, Archives des sciences, ser.~4,
+vol.~6, 363--364, 1898.
+
+\Title{Zur Theorie der tripelorthogonalen Flächensysteme.}
+Vierteljahrschrift der Naturf.\ Gesell.\ in Zurich, vol.~43,
+317--326, 1898.
+
+\Author{Germain, Sophie.} \Title{Mémoire sur la courbure des surfaces.}
+Journ.\ für Math., vol.~7, 1--29, 1831.
+
+\Author{Gilbert, P.} \Title{Sur l'emploi des cosinus directeurs de~la normale
+dans la théorie de~la courbure des surfaces.} Ann.\
+de~la Soc.\ sci.\ de Bruxelles, vol.~18~B, 1--24, 1894.
+
+\Author{Genty, E.} \Title{Sur les surfaces à courbure totale constante.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~22, 106--109, 1894.
+\PageSep{121}
+
+\Author{Genty, E.} \Title{Sur la déformation infinitésimale de surfaces.}
+Ann.\ de~la Faculté des sci.\ de Toulouse, vol.~9~E, 1--11,
+1895.
+
+\Author{Goursat, E.} \Title{Sur les systèmes orthogonaux.} C.~R., vol.~121,
+883--884, 1895.
+
+\Title{Sur les équations d'une surface rapportée à ses lignes
+de longueur nulle.} Bull.\ de~la Soc.\ Math.\ de France,
+vol.~26, 83--84, 1898.
+
+\Author{Grassmann, H.} \Title{Anwendung der Ausdehnungslehre auf die
+allgemeine Theorie der Raumcurven und krummen
+Flächen.} Diss.\ Halle, 1893.
+
+\Author{Guichard, C.} \Title{Surfaces rapportées à leur lignes asymptotiques
+et congruences rapportées à leurs dévéloppables.}
+Ann.\ de l'École Normale, ser.~3, vol.~6, 333--348, 1889.
+
+\Title{Recherches sur les surfaces à courbure totale constante
+et certaines surfaces qui s'y rattachent.} Ann.\ de l'École
+Normale, ser.~3, vol.~7, 233--264, 1890.
+
+\Title{Sur les surfaces qui possèdent un réseau de géodésiques
+conjuguées.} C.~R., vol.~110, 995--997, 1890.
+
+\Title{Sur la déformation des surfaces.} Journ.\ de Math.,
+ser.~5, vol.~2, 123--215, 1896.
+
+\Title{Sur les surfaces à courbure totale constante.} C.~R.,
+vol.~126, 1556--1558, 1616--1618, 1898.
+
+\Title{Sur les systémes orthogonaux et les systémes cycliques.}
+Ann.\ de l'École Normale, ser.~3, vol.~14, 467--516, 1897;
+vol.~15, 179--227, 1898.
+
+\Author{Guldberg, Alf.} \Title{Om Bestemmelsen af de geodaetiske Linier
+paa visse specielle Flader.} Nyt Tidsskrift for Math.\
+Kjöbenhavn, vol.~6~B, 1--6, 1895.
+
+\Author{Hadamard, J.} \Title{Sur les lignes géodésiques des surfaces spirales
+et les équations différentielles qui s'y rapportent.} Procès
+verbeaux de~la Soc.\ des sci.\ de Bordeaux, 1895--96, 55--58.
+
+\Title{Sur les lignes géodésiques des surfaces à courbures
+opposées.} C.~R., vol.~124, 1503--1505, 1897.
+
+\Title{Les surfaces à courbures opposées et leurs lignes
+géodésiques.} Journ.\ de Math., ser.~5, vol.~4, 27--73, 1898.
+
+\Author{Haenig, Conrad.} \Title{Ueber Hansen's Methode, ein geodätisches
+Dreieck auf die Kugel oder in die Ebene zu übertragen.}
+Diss., 36~pp., Leipzig, 1888.
+
+\Author{Hansen, P. A.} \Title{Geodätische Untersuchungen\Add{.}} Abhandl.\ der
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+\Title{Ueber die Curven, welche sich so bewegen können,
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+\Author{Hilbert, D.} \Title{Ueber Flächen von constanter Gaussscher Krümmung.}
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+\Author{Hirst, T.} \Title{Sur la courbure d'une série de surfaces et de
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+\Author{Hoppe, R.} \Title{Zum Problem des dreifach orthogonalen Flächensystems.}
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+\Title{Principien der Flächentheorie.} Archiv für Math.\ und
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+\Title{Geometrische Deutung der Fundamentalgrössen zweiter
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+\Title{Nachträge zur Curven- und Flächentheorie.} Archiv
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+\Title{Ueber die kürzesten Linien auf den Mittelpunktsflächen.}
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+\Title{Untersuchungen über kürzeste Linien.} Archiv für
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+\Title{Ueber die Bedingung, welcher eine Flächenschaar
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+\Title{Nachtrag zur Flächentheorie.} Archiv für Math.\ und
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+\Title{Ueber die sphärische Darstellung der asymptotischen
+Linien einer Fläche.} Archiv für Math.\ und Phys., ser.~2,
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+\Title{Eine neue Beziehung zwischen den Krümmungen von
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+\Author{Jacobi, C. G. J.} \Title{Demonstratio et amplificatio nova theorematis
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+\Author{Jamet, V.} \Title{Sur la théorie des lignes géodésiques.} Marseille
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+by L.~Natani, 1890.
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+\Author{Knoblauch, Johannes.} \Title{Einleitung in die allegemeine Theorie
+der krummen Flächen.} Leipzig, Teubner, \textsc{viii}~+~267~pp.,
+1888.
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+\Title{Ueber Fundamentalgrössen in der Flächentheorie.}
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+\Title{Ueber die geometrische Bedeutung der flächentheoretischen
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+vol.~15, 249--257, 1891.
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+\Author{Königs, G.} \Title{Résumé d'un mémoire sur les lignes géodésiques.}
+Ann.\ Faculté des sci.\ de Toulouse, vol.~6~P, 1--34, 1892.
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+\Title{Une théorème de géométrie \Typo{infinitesimale}{infinitésimale}.} C.~R., vol.~116,
+569, 1893.
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+\Title{Mémoire sur les lignes géodésiques.} Mém.\ présentés
+par savants à l'Acad.\ des sci.\ de l'Inst.\ de France, vol.~31,
+No.~6, 318~pp., 1894.
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+\Author{Kommerell, V.} \Title{Beiträge zur Gauss'schen Flächentheorie.}
+Diss., \textsc{iii}~+~46~pp., Tübingen, 1890.
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+\Title{Eine neue Formel für die mittlere Krümmung und
+das Krümmungsmaass einer Fläche.} Zeitschrift für
+Math.\ und Phys., vol.~41, 123--126, 1896.
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+\Author{Köttfritzsch, Th.} \Title{Zur Frage über isotherme Coordinatensysteme.}
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+\Author{Kummer, E. E.} \Title{Allgemeine Theorie der geradlinigen
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+\Author{Laguerre.} \Title{Sur les formules fondamentales de~la théorie des
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+\Author{Lamarle, E.} \Title{Exposé géométrique du calcul differential et
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+1863.
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+\Author{Lamé, Gabriel.} \Title{Mémoire sur les coordonnées curvilignes.}
+Journ.\ de Math., vol.~5, 313--347, 1840.
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+\Title{Leçons sur les coordonnées curvilignes.} Paris, 1859.
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+\Author{Lecornu, L.} \Title{Sur l'équilibre des surfaces flexibles et inextensibles.}
+Journ.\ de l'École Polyt., cahier~48, vol.~29,
+1--109, 1880.
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+\Author{Legoux, A.} \Title{Sur l'integration de l'équation $ds^{2} = E\, du^{2} +
+2F\, du\, dv + G\, dv^{2}$.} Ann.\ de~la Faculté des sci.\ de
+Toulouse, vol.~3~F, 1--2, 1889.
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+\Author{Lévy, L.} \Title{Sur les systèmes de surfaces triplement orthogonaux.}
+Mém.\ couronnés et mém.\ des sav.\ publiés par
+l'Acad.\ Royale de Belgique, vol.~54, 92~pp., 1896.
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+\Author{Lévy, Maurice.} \Title{Sur une transformation des coordonnées
+curvilignes orthogonales et sur les coordonnées curvilignes
+comprenant une famille quelconque de surfaces du
+second ordre.} Thesis, 33~pp., Paris, 1867.
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+\Title{Mémoire sur les coordonnées curvilignes orthogonales.}
+Journ.\ de l'École Polyt., cahier~43, vol.~26, 157--200,
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+%\Author{Lévy, Maurice.}
+\Title{Sur une application industrielle du théorème
+de Gauss relatif à la courbure des surfaces.} C.~R., vol.~86,
+111--113, 1878.
+
+\Author{Lie, Sophus.} \Title{Ueber Flächen, deren Krümmungsradien durch
+eine Relation verknüpft sind.} Archiv for Math.\ og
+Nat., Christiania, vol.~4, 507--512, 1879.
+
+\Title{Zur Theorie der Flächen constanter Krümmung.}
+Archiv for Math.\ og Nat., Christiania, vol.~4, 345--354,
+355--366, 1879; vol.~5 282--306, 328--358, 518--541, 1881.
+
+\Title{Untersuchungen über geodätische Curven.} Math.\
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+\Title{Zur Geometrie einer Monge'schen Gleichung.} Berichte
+der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
+vol.~50 1--2, 1898.
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+\Author{von Lilienthal, Reinhold.} \Title{Allgemeine Eigenschaften von
+Flächen, deren Coordinaten sich durch reellen Teile
+dreier analytischer Functionen einer complexen Veränderlichen
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+\Title{Untersuchungen zur allgemeinen Theorie der krummen
+Oberflächen und geradlinigen Strahlensysteme.}
+Bonn, E.~Weber, 112~pp., 1886.
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+\Title{Zur Theorie der Krümmungsmittelpunktsflächen.}
+Math.\ Annalen, vol.~30, 1--14, 1887.
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+\Title{Ueber die Krümmung der Curvenschaaren.} Math.\
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+\Title{Zur Krümmungstheorie der Flächen.} Journ.\ für
+Math., vol.~104, 341--347, 1889.
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+\Title{Zur Theorie des Krümmungsmaasses der Flächen.}
+Acta Mathematica, vol.~16, 143--152, 1892.
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+\Title{Ueber geodätische Krümmung.} Math.\ Annalen,
+vol.~42, 505--525, 1893.
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+\Title{Ueber die Bedingung, unter der eine Flächenschaar
+einem dreifach orthogonalen Flächensystem angehört.}
+Math.\ Annalen, vol.~44, 449--457, 1894.
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+\Author{Lipschitz, Rudolf.} \Title{Beitrag zur Theorie der Krümmung.}
+Journ.\ für Math., vol.~81, 230--242, 1876.
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+\Title{Untersuchungen über die Bestimmung von Oberflächen
+mit vorgeschriebenen, die Krümmungsverhältnisse
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+der Wissenschaften zu Berlin, 1882, 1077--1087; 1883,
+169--188.
+
+\Title{Untersuchungen über die Bestimmung von Oberflächen
+mit vorgeschriebenem Ausdruck des Linearelements.}
+Sitzungsb.\ der Kgl.\ Akad.\ der Wissenschaften
+zu Berlin, 1883, 541--560.
+
+\Title{Zur Theorie der krummen Oberflächen.} Acta Mathematica,
+vol.~10, 131--136, 1887.
+
+\Author{Liouville, Joseph.} \Title{Sur un théorème de M.~Gauss concernant
+le produit des deux rayons de courbure principaux
+en chaque point d'une surface.} Journ.\ de Math.,
+vol.~12, 291--304, 1847.
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+
+\Author{Liouville, Joseph.} \Title{Sur la théorie générale des surfaces.}
+Journ.\ de Math., vol.~16, 130--132, 1851.
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+\Title{Notes on Monge's Applications}, see Monge.
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+\Author{Liouville, R.} \Title{Sur le caractère auquel se reconnaît l'équation
+differentielle d'un système géodésique.} C.~R., vol.~108,
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+\Title{Sur les représentations géodésiques des surfaces.} C.~R.,
+vol.~108, 335--337, 1889.
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+\Author{Loria, G.} \Title{Sulla teoria della curvatura delle superficie.}
+Rivista di Mat.\ Torino, vol.~2, 84--95, 1892.
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+\Title{Il passato ed il presente d.\ pr.\ Teorie geometriche.}
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+\Author{Lüroth, J.} \Title{Verallgemeinerung des Problems der kürzesten
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+\Author{Mahler, E.} \Title{Ueber allgemeine Flächentheorie.} Archiv für
+Math.\ and Phys., vol.~57, 96--97, 1881.
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+\Title{Die Fundamentalsätze der allgemeinen Flächentheorie.}
+Vienna; Heft.~I, 1880; Heft.~II, 1881.
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+surfaces.} Ann.\ de l'École Normale, ser.~3, vol.~10, 87--89,
+1893.
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+\Author{von Mangoldt, H.} \Title{Ueber diejenigen Punkte auf positiv
+gekrümmten Flächen, welche die Eigenschaft haben,
+dass die von ihnen ausgehenden geodätischen Linien nie
+aufhören, kürzeste Linien zu sein.} Journ.\ für Math.,
+vol.~91, 23--53, 1881.
+
+\Title{Ueber die Klassification der Flächen nach der Verschiebbarkeit
+ihrer geodätischen Dreiecke.} Journ.\ für
+Math., vol.~94, 21--40, 1883.
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+\Author{Maxwell, J. Clerk.} \Title{On the Transformation of Surfaces by
+Bending.} Trans.\ of Camb.\ Philos.\ Soc., vol.~9, 445--470,
+1856.
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+\Author{Minding, Ferdinand.} \Title{Ueber die Biegung gewisser Flächen.}
+Journ.\ für Math., vol.~18, 297--302, 365--368, 1838.
+
+\Title{Wie sich entscheiden lässt, ob zwei gegebene krumme
+Flächen auf einander abwickelbar sind oder nicht;
+nebst Bemerkungen über die Flächen von veränderlichen
+Krümmungsmaasse.} Journ.\ für Math., vol.~19,
+370--387, 1839.
+
+\Title{Beiträge zur Theorie der kürzesten Linien auf krummen
+Flächen.} Journ.\ für Math., vol.~20, 323--327, 1840.
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+\Title{Ueber einen besondern Fall bei der Abwickelung
+krummer Flächen.} Journ.\ für Math., vol.~20, 171--172,
+1840.
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+\Title{Ueber die mittlere Krümmung der Flächen.} Bull.\
+de l'Acad.\ Imp.\ de St.~Petersburg, vol.~20, 1875.
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+\Title{Zur Theorie der Curven kürzesten Umrings, bei
+gegebenem Flächeninhalt, auf krummen Flächen.}
+Journ.\ für Math., vol.~86, 279--289, 1879.
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+\Author{Mlodzieiowski, B.} \Title{Sur la déformation des surfaces.} Bull.\
+de sci.\ Math., ser.~2, vol.~15, 97--101, 1891.
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+\Author{Monge, Gaspard.} \Title{Applications de l'Analyse à la Géométrie};
+revue, corrigée et annotée par J.~Liouville. Paris;
+fifth~ed., 1850.
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+\Author{Motoda, T.} Note to J.~Knoblauch's paper, ``\Title{Ueber Fundamentalgrössen
+in der Flächentheorie}'' in Journ.\ für
+Math., vol.~103. Journ.\ of the Phil.\ Soc.\ in Tokio,
+3~pp., 1889.
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+\Author{Moutard, T. F.} \Title{Lignes de courbure d'une classe de surfaces
+du quatrième ordre.} C.~R., vol.~59, 243, 1864.
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+\Title{Note sur la transformation par rayons vecteurs reciproques.}
+Nouv.\ Ann.\ de Math.\ ser.~2, vol.~3, 306--309,
+1864.
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+\Title{Sur les surface anallagmatique du quatrième ordre.}
+Nouv.\ Ann.\ de Math.\ ser.~2, vol.~3, 536--539, 1864.
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+\Title{Sur la déformation des surfaces.} Bull.\ de~la Soc.\
+Philomatique, p.~45, 1869.
+
+
+\Title{Sur la construction des équations de~la forme $\dfrac{1}{x}·\Typo{\dfrac{d^{2}x}{dx\, dy}}{\dfrac{\dd^{2}x}{\dd x\, \dd y}}
+= \lambda(x, y)$, qui admettent une intégrale générale explicite.}
+Journ.\ de l'École Polyt., cahier~45, vol.~28, 1--11, 1878.
+
+\Author{Nannei, E.} \Title{Le superficie ipercicliche.} Rend.\ dell'Accad.\
+di Napoli, ser.~2, vol.~2, 119--121, 1888; Giornale di
+Mat., vol.~26, 201--233, 1888.
+
+\Author{Naccari, G.} \Title{Deduzioue delle principali formule relative
+alla curvatura della superficie in generale e dello sferoide
+in particolare con applicazione al meridiano di Venezia.}
+L'Ateneo Veneto, ser.~17, vol.~1, 237--249, 1893; vol.~2,
+133--161, 1893.
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+\Author{Padova, E.} \Title{Sopra un teorema di geometria differenziale.}
+Reale Ist.\ Lombardo di sci.\ e let., vol.~23, 840--844, 1890.
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+\Title{Sulla teoria generale delle superficie.} Mem.\ della R.
+Accad.\ dell' Ist.\ di Bologna, ser.~4, vol.~10, 745--772,
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+\Author{Pellet, A.} \Title{Mém.\ sur la théorie des surfaces et des courbes.}
+Ann.\ de l'École Normale, ser.~3, vol.~14, 287--310, 1897.
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+\Title{Sur les surfaces de Weingarten.} C.~R., vol.~125, 601--602,
+1897.
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+\Title{Sur les systèmes de surfaces orthogonales et isothermes.}
+C.~R., vol.~124, 552--554, 1897.
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+\Title{Sur les surfaces ayant même représentation sphérique.}
+C.~R., vol.~124, 1291--1294, 1897.
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+1897.
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+\Title{Sur la théorie des surfaces.} Bull.\ de~la Soc.\ Math.\ de
+France, vol.~26, 138--159, 1898; C.~R., vol.~124, 451--452,
+739--741, 1897; Thesis, Paris, 1878.
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+\Title{Sur les surfaces applicables sur une surface de \Typo{revolution}{révolution}.}
+C.~R., vol.~125, 1159--1160, 1897; vol.~126, 392--394,
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+\Author{Peter, A.} \Title{Die Flächen, deren Haupttangentencurven linearen
+Complexen angehören.} Archiv for Math.\ og
+Nat., Christiania, vol.~17, No.~8, 1--91, 1895.
+\PageSep{124}
+
+\Author{Petot, A.} \Title{Sur les surfaces dont l'élément \Typo{lineaire}{linéaire} est \Typo{reductible}{réductible}
+\Typo{a}{à} la forme $ds^{2} = F(U + V)(du^{2} + dv^{2})$.} C.~R.,
+vol.~110, 330--333, 1890.
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+\Author{Picard, Émile.} \Title{Surfaces applicables.} Traité d'Analyse,
+vol.~1, chap.~15, 420--457; first~ed., 1891; second~ed.,
+1901.
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+\Author{Pirondini, G.} \Title{Studi geometrici relativi specialmente alle
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+1885.
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+\Title{Teorema relativo alle linee di curvatura delle superficie
+e sue applicazioni.} Annali di Mat., ser.~2, vol.~16,
+61--84, 1888; vol.~21, 33--46, 1893.
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+\Author{Plücker, Julius.} \Title{Ueber die Krümmung einer beliebigen
+Fläche in einem gegebenen Puncte.} Journ.\ für Math.,
+vol.~3, 324--336, 1828.
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+\Author{Poincaré, H.} \Title{Rapport sur un Mémoire de M.~Hadamard,
+intitulé: Sur les lignes géodésiques des surfaces à courbures
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+\Author{Probst, F.} \Title{Ueber Flächen mit isogonalen systemen von
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+1893.
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+sont liés par une relation.} Bull.\ de~la Soc.\ Math.\
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+\Title{\Typo{Determination}{Détermination} des éléments linéaires doublement harmoniques.}
+Journ.\ de Math., ser.~4, vol.~10, 331--390,
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+\Title{Quelques \Typo{proprietes}{propriétés} des surfaces harmoniques.} Ann.\
+de~la Faculté des sci.\ de Toulouse, vol.~9~C, 1--44, 1895.
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+\Title{Sur les spirales harmoniques.} Ann.\ de l'École Normale,
+ser.~3, vol.~12, 145--196, 1895.
+
+\Title{Surfaces rapportées à un réseau conjugué azimutal.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~24, 51--56, 1896.
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+\Title{Leçons sur les applications géométriques de l'analyse.}
+Paris, \textsc{vi}~+~251~pp., 1897.
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+\Title{Contribution à la théorie des surfaces dont les rayons
+de courbure sont liés par une relation.} Bull.\ de~la Soc.\
+Math.\ de France, vol.~25, 147--172, 1897.
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+\Title{Sur les formules fondamentales de~la théorie des surfaces.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~25, 1--3,
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+\Title{Delle superficie sulle quali due serie di geodetiche
+formano un sistema conjugato.} Mem.\ della R. Accad.\
+dell'Ist.\ di Bologna, ser.~4, vol.~9, 765--776, 1889.
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+\Author{Reina, V.} \Title{Sulle linee conjugate di una superficie.} Atti
+della Reale Accad.\ dei Lincei, ser.~4, vol.~$6_{1}$, 156--165,
+203--209, 1890.
+
+%\Author{Reina, V.}
+\Title{Di alcune formale relative alla teoria delle superficie.}
+Atti della Reale Accad.\ dei Lincei, ser.~4, vol.~$6_{2}$\Add{,}
+103--110, 176, 1890.
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+\Author{Resal, H.} \Title{Exposition de~la théorie des surfaces.} 1~vol.,
+\textsc{xiii}~+~171~pp.\Add{,} Paris, 1891. Bull.\ des sci.\ Math., ser.~2,
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+l'une sur l'autre.} L'Inst.\ Journ.\ universel des sci.\ et
+des soc.\ sav.\ en France, sect.~I, vol.~37, 371--382, 1869.
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+\Title{Sur les surfaces orthogonales.} L'Inst. Journ.\ universel
+des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~37,
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+des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~37,
+389, 1869; C.~R., vol.~70, 330, 1870.
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+\Title{Sur la théorie des surfaces.} L'Inst.\ Journ.\ universel
+des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~38, 60--61,
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+\Title{Sur la représentation sphérique des surfaces.} C.~R.,
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+\Title{Sur les courbes enveloppes de cercles et sur les surfaces
+enveloppes de sphères.} Nouvelle Correspondance
+Math., vol.~5, 257--263, 305--315, 337--343, 385--393, 417--425,
+1879; vol.~6, 1--7, 1880.
+
+\Title{Mémoire sur la théorie générale des surfaces courbes.}
+Journ.\ de Math., ser.~4, vol.~7, 5--108, 219--270, 1891.
+
+\Author{Ricci, G.} \Title{Dei sistemi di coordinate atti a ridurre la expressione
+del quadrato dell' elemento lineaire di una superficie
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+Reale Accad.\ dei Lincei, ser.~5, vol.~$2_{1}$, 73--81, 1893.
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+\Title{A proposito di una memoria sulle linee geodetiche del
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+\end{Bibliography}
+\PageSep{127}
+
+\iffalse
+CORRIGENDA BT ADDENDA.
+%[** TN: [x] = corrected in source using \Erratum macro,
+%        [v] = verified in source, corrected by the translators]
+
+[x] Art. 11, p. 20, l. 6. The fourth E should be F.
+
+[x] Art. 18, p. 27, l. 7. For \sqrt{EG - F^2)·dp·d\theta read 2\sqrt{FG - F^w)·dq·d\theta.
+The original and the Latin reprints lack the factor 2; the correction is made in all
+the translations.
+
+[x] Art. 19, p. 28, l. 10. For g read q.
+
+[v] Art. 22, p. 34, l. 5, left side; Art. 24, p. 36, l. 5, third equation; Art. 24,
+p. 38, l. 4. The original and Liouville's reprint have q for p.
+
+[x] Note on Art. 23, p. 55, l. 2 fr. bot. For p read q.
+\fi
+
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of the Project Gutenberg EBook of General Investigations of Curved  %
+% Surfaces of 1827 and 1825, by Karl Friedrich Gauss                      %
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
+%                                                                         %
+% ***** This file should be named 36856-tex.tex or 36856-tex.zip *****    %
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% The Project Gutenberg EBook of General Investigations of Curved Surfaces%
+% of 1827 and 1825, by Karl Friedrich Gauss                               %
+%                                                                         %
+% 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: General Investigations of Curved Surfaces of 1827 and 1825       %
+%                                                                         %
+% Author: Karl Friedrich Gauss                                            %
+%                                                                         %
+% Translator: James Caddall Morehead                                      %
+%             Adam Miller Hiltebeitel                                     %
+%                                                                         %
+% Release Date: July 25, 2011 [EBook #36856]                              %
+%                                                                         %
+% Language: English                                                       %
+%                                                                         %
+% Character set encoding: ISO-8859-1                                      %
+%                                                                         %
+% *** START OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
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+}
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+
+%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+%% PG BOILERPLATE %%
+\PGBoilerPlate
+\begin{center}
+\begin{minipage}{\textwidth}
+\small
+\begin{PGtext}
+The Project Gutenberg EBook of General Investigations of Curved Surfaces
+of 1827 and 1825, by Karl Friedrich Gauss
+
+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: General Investigations of Curved Surfaces of 1827 and 1825
+
+Author: Karl Friedrich Gauss
+
+Translator: James Caddall Morehead
+            Adam Miller Hiltebeitel
+
+Release Date: July 25, 2011 [EBook #36856]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
+\end{PGtext}
+\end{minipage}
+\end{center}
+\newpage
+%% Credits and transcriber's note %%
+\begin{center}
+\begin{minipage}{\textwidth}
+\begin{PGtext}
+Produced by Andrew D. Hwang, with special thanks to Brenda Lewis.
+\end{PGtext}
+\end{minipage}
+\vfill
+\end{center}
+
+\begin{minipage}{0.85\textwidth}
+\small
+\BookMark{0}{Transcriber's Note.}
+\subsection*{\centering\normalfont\scshape%
+\normalsize\MakeLowercase{\TransNote}}%
+
+\raggedright
+\TransNoteText
+\end{minipage}
+%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
+\PageSep{i}
+\FrontMatter
+\begin{center}
+{\LARGE\scshape Karl Friedrich Gauss} \\
+\tb
+\bigskip
+
+\LARGE\scshape General Investigations \\
+{\footnotesize OF} \\
+Curved Surfaces \\
+{\footnotesize OF} \\
+\large 1827 and 1825
+\vfill
+
+\normalsize
+TRANSLATED WITH NOTES \\
+{\scriptsize AND A} \\[4pt]
+BIBLIOGRAPHY \\[4pt]
+{\scriptsize BY \\[4pt]
+JAMES CADDALL MOREHEAD, A.M., M.S., and ADAM MILLER HILTEBEITEL, A.M. \\[4pt]
+J. S. K. FELLOWS IN MATHEMATICS IN PRINCETON UNIVERSITY}
+\vfill\vfill
+
+THE PRINCETON UNIVERSITY LIBRARY \\
+1902
+\end{center}
+\newpage
+\PageSep{ii}
+\null\vfill
+\begin{center}
+\scriptsize
+Copyright, 1902, by \\
+\textsc{The Princeton University Library}
+\vfill
+\textit{C. S. Robinson \&~Co., University Press \\
+Princeton, N. J.}
+\end{center}
+\PageSep{iii}
+
+
+\Introduction
+
+In 1827 Gauss presented to the Royal Society of Göttingen his important paper on
+the theory of surfaces, which seventy-three years afterward the eminent French
+geometer, who has done more than any one else to propagate these principles, characterizes
+as one of Gauss's chief titles to fame, and as still the most finished and useful
+introduction to the study of infinitesimal geometry.\footnote
+  {G. Darboux, Bulletin des Sciences Math. Ser.~2, vol.~24, page~278, 1900.}
+This memoir may be called:
+\Title{General Investigations of Curved Surfaces}, or the Paper of~1827, to distinguish it
+from the original draft written out in~1825, but not published until~1900. A list of
+the editions and translations of the Paper of~1827 follows. There are three editions
+in Latin, two translations into French, and two into German. The paper was originally
+published in Latin under the title:
+
+I\textit{a}. \Publication{Disquisitiones generales circa superficies curvas}
+{auctore Carolo Friderico Gauss\Add{.}}
+{Societati regiæ oblatæ D.~8.~Octob.~1827}, \\
+and was printed in: Commentationes societatis regiæ scientiarum Gottingensis recentiores,
+Commentationes classis mathematicæ. Tom.~VI. (ad~a.\ 1823--1827). Gottingæ,
+1828, pages~99--146. This sixth volume is rare; so much so, indeed, that the British
+Museum Catalogue indicates that it is missing in that collection. With the signatures
+changed, and the paging changed to pages~1--50, I\textit{a}~also appears with the title page
+added:
+
+I\textit{b}. \Publication{Disquisitiones generales circa superficies curvas}
+{auctore Carolo Friderico Gauss.}
+{Gottingæ. Typis Dieterichianis. 1828.}
+
+II\@. In Monge's \Title{Application de l'analyse à la géométrie}, fifth edition, edited by
+Liouville, Paris, 1850, on pages 505--546, is a reprint, added by the Editor, in Latin
+under the title: \Title{Recherches sur la théorie générale des surfaces courbes}; Par M.
+C.-F. Gauss.
+\PageSep{iv}
+
+III\textit{a}. A third Latin edition of this paper stands in: Gauss, \Title{Werke, Herausgegeben
+von der Königlichen Gesellschaft der Wissenschaften zu Göttingen}, Vol.~4, Göttingen,
+1873, pages~217--258, without change of the title of the original paper~(I\textit{a}).
+
+III\textit{b}. The same, without change, in Vol.~4 of Gauss, \Title{Werke}, Zweiter Abdruck,
+Göttingen,~1880.
+
+IV\@. A French translation was made from Liouville's edition,~II, by Captain
+Tiburce Abadie, ancien élève de l'École Polytechnique, and appears in Nouvelles
+Annales de Mathématique, Vol.~11, Paris,~1852, pages~195--252, under the title:
+\Title{Recherches générales sur les surfaces courbes}; Par M.~Gauss. This latter also
+appears under its own title.
+
+V\textit{a}. Another French translation is: \Title{Recherches Générales sur les Surfaces
+Courbes}. Par M. C.-F. Gauss, traduites en français, suivies de notes et d'études
+sur divers points de la Théorie des Surfaces et sur certaines classes de Courbes, par
+M. E. Roger, Paris, 1855.
+
+V\textit{b}. The same. Deuxième Edition. Grenoble (or Paris), 1870 (or 1871), 160~pages.
+
+VI\@. A German translation is the first portion of the second part, namely, pages
+198--232, of: Otto Böklen, \Title{Analytische Geometrie des Raumes}, Zweite Auflage, Stuttgart,
+1884, under the title (on page~198): \Title{Untersuchungen über die allgemeine Theorie
+der krummen Flächen}. Von C.~F. Gauss. On the title page of the book the second
+part stands as: \Title{Disquisitiones generales circa superficies curvas} von C.~F. Gauss, ins
+Deutsche übertragen mit Anwendungen und Zusätzen\dots.
+
+VII\textit{a}. A second German translation is No.~5 of Ostwald's Klassiker der exacten
+Wissenschaften: \Title{Allgemeine Flächentheorie} (\Title{Disquisitiones generales circa superficies
+curvas}) von Carl Friedrich Gauss, (1827). Deutsch herausgegeben von A.~Wangerin.
+Leipzig, 1889. 62~pages.
+
+VII\textit{b}. The same. Zweite revidirte Auflage. Leipzig, 1900. 64~pages.
+
+The English translation of the Paper of~1827 here given is from a copy of the
+original paper,~I\textit{a}; but in the preparation of the translation and the notes all the
+other editions, except~V\textit{a}, were at hand, and were used. The excellent edition of
+Professor Wangerin,~VII, has been used throughout most freely for the text and
+notes, even when special notice of this is not made. It has been the endeavor of
+the translators to retain as far as possible the notation, the form and punctuation of
+the formulæ, and the general style of the original papers. Some changes have been
+made in order to conform to more recent notations, and the most important of those
+are mentioned in the notes.
+\PageSep{v}
+
+%[** TN: Paragraph not indented in the original]
+The second paper, the translation of which is here given, is the abstract (Anzeige)
+which Gauss presented in German to the Royal Society of Göttingen, and which was
+published in the Göttingische gelehrte Anzeigen. Stück~177. Pages 1761--1768. 1827.
+November~5. It has been translated into English from pages 341--347 of the fourth
+volume of Gauss's Works. This abstract is in the nature of a note on the Paper of~1827,
+and is printed before the notes on that paper.
+
+Recently the eighth volume of Gauss's Works has appeared. This contains on
+pages 408--442 the paper which Gauss wrote out, but did not publish, in~1825. This
+paper may be called the \Title{New General Investigations of Curved Surfaces}, or the Paper
+of~1825, to distinguish it from the Paper of~1827. The Paper of~1825 shows the
+manner in which many of the ideas were evolved, and while incomplete and in some
+cases inconsistent, nevertheless, when taken in connection with the Paper of~1827,
+shows the development of these ideas in the mind of Gauss. In both papers are
+found the method of the spherical representation, and, as types, the three important
+theorems: The measure of curvature is equal to the product of the reciprocals of the
+principal radii of curvature of the surface, The measure of curvature remains unchanged
+by a mere bending of the surface, The excess of the sum of the angles of a geodesic
+triangle is measured by the area of the corresponding triangle on the auxiliary sphere.
+But in the Paper of~1825 the first six sections, more than one-fifth of the whole paper,
+take up the consideration of theorems on curvature in a plane, as an introduction,
+before the ideas are used in space; whereas the Paper of~1827 takes up these ideas
+for space only. Moreover, while Gauss introduces the geodesic polar coordinates in
+the Paper of~1825, in the Paper of~1827 he uses the general coordinates, $p$,~$q$, thus
+introducing a new method, as well as employing the principles used by Monge and
+others.
+
+The publication of this translation has been made possible by the liberality of
+the Princeton Library Publishing Association and of the Alumni of the University
+who founded the Mathematical Seminary.
+
+\Signature{H. D. Thompson.}
+{Mathematical Seminary,}
+{Princeton University Library,}
+{January 29, 1902.}
+\PageSep{vi}
+%[Blank page]
+\PageSep{vii}
+
+
+\Contents
+
+\ToCLine{Gauss's Paper of 1827, General Investigations of Curved Surfaces}
+{paper:1827} % 1
+
+\ToCLine{Gauss's Abstract of the Paper of 1827}
+{abstract} % 45
+
+\ToCLine{Notes on the Paper of 1827}
+{notes:1827} % 51
+
+\ToCLine{Gauss's Paper of 1825, New General Investigations of Curved Surfaces}
+{paper:1825} % 79
+
+\ToCLine{Notes on the Paper of 1825}
+{notes:1825} % 111
+
+\ToCLine{Bibliography of the General Theory of Surfaces}
+{biblio} % 115
+\PageSep{viii}
+%[Blank page]
+\PageSep{1}
+\MainMatter
+\Paper{1827}
+%\thispagestyle{empty}
+\begin{center}
+\LARGE
+DISQUISITIONES GENERALES
+\vfil
+{\normalsize CIRCA}
+\vfil
+\LARGE
+SUPERFICIES CURVAS
+\vfil
+{\normalsize AUCTORE} \\[8pt]
+CAROLO FRIDERICO GAUSS
+\vfil
+
+\footnotesize
+SOCIETATI REGIAE OBLATAE D.~8.~OCTOB.~1827
+\vfil
+
+\tb \\
+\medskip
+
+COMMENTATIONES SOCIETATIS REGIAE SCIENTIARUM \\[4pt]
+GOTTINGENSIS RECENTIORES\@. VOL.~VI\@. GOTTINGAE MDCCCXXVIII \\
+\tb
+\vfil
+
+GOTTINGAE \\
+TYPIS DIETERICHIANIS \\
+MDCCCXXVIII
+\end{center}
+\cleardoublepage
+\PageSep{2}
+%[Blank page]
+\PageSep{3}
+
+
+\PaperTitle{\LARGE GENERAL INVESTIGATIONS \\
+{\small OF} \\
+CURVED SURFACES \\
+{\small BY} \\
+{\large KARL FRIEDRICH GAUSS} \\
+{\footnotesize PRESENTED TO THE ROYAL SOCIETY, OCTOBER~8, 1827}}
+
+
+\Article{1.}
+Investigations, in which the directions of various straight lines in space are to be
+considered, attain a high degree of clearness and simplicity if we employ, as an auxiliary,
+a sphere of unit radius described about an arbitrary centre, and suppose the
+different points of the sphere to represent the directions of straight lines parallel to
+the radii ending at these points. As the position of every point in space is determined
+by three coordinates, that is to say, the distances of the point from three mutually
+perpendicular fixed planes, it is necessary to consider, first of all, the directions of the
+axes perpendicular to these planes. The points on the sphere, which represent these
+directions, we shall denote by $(1)$,~$(2)$,~$(3)$. The distance of any one of these points
+from either of the other two will be a quadrant; and we shall suppose that the directions
+of the axes are those in which the corresponding coordinates increase.
+
+
+\Article{2.}
+It will be advantageous to bring together here some propositions which are frequently
+used in questions of this kind.
+
+\Par{I.} The angle between two intersecting straight lines is measured by the arc
+between the points on the sphere which correspond to the directions of the lines.
+
+\Par{II.}\Note{1} The orientation of any plane whatever can be represented by the great circle
+on the sphere, the plane of which is parallel to the given plane.
+\PageSep{4}
+
+\Par{III.} The angle between two planes is equal to the spherical angle between the
+great circles representing them, and, consequently, is also measured by the arc intercepted
+between the poles of these great circles. And, in like manner, the angle of inclination
+of a straight line to a plane is measured by the arc drawn from the point which
+corresponds to the direction of the line, perpendicular to the great circle which represents
+the orientation of the plane.
+
+\Par{IV.} Letting $x$,~$y$,~$z$; $x'$,~$y'$,~$z'$ denote the coordinates of two points, $r$~the distance
+between them, and $L$~the point on the sphere which represents the direction of the line
+drawn from the first point to the second, we shall have
+\begin{align*}
+x' &= x + r \cos(1)L\Add{,} \\
+y' &= y + r \cos(2)L\Add{,} \\
+z' &= z + r \cos(3)L\Add{.}
+\end{align*}
+
+\Par{V.} From this it follows at once that, generally,\Note{2}
+\[
+\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1\Add{,}
+\]
+and also, if $L'$~denote any other point on the sphere,
+\[
+\cos(1)L·\cos(1)L' + \cos(2)L·\cos(2)L' + \cos(3)L·\cos(3)L'
+  = \cos LL'.
+\]
+
+\Par{VI.}\Note{4} \begin{Theorem}
+If $L$, $L'$, $L''$, $L'''$ denote four points on the sphere, and $A$~the angle
+which the arcs $LL'$, $L''L'''$ make at their point of intersection, then we shall have
+\[
+\cos LL''·\cos L'L''' - \cos LL'''·\cos L'L''
+  = \sin LL'·\sin L''L'''·\cos A\Add{.}
+\]
+\end{Theorem}
+
+\textit{Demonstration.} Let $A$ denote also the point of intersection itself, and set
+\[
+AL = t,\quad AL' = t',\quad AL'' = t'',\quad AL''' = t'''\Add{.}
+\]
+Then we shall have
+\begin{alignat*}{7}
+&\cos L L''  &&= \cos t\Chg{·}{}&& \cos t''  &&+ \sin t  && \sin t''  && \cos A\Add{,} \\
+&\cos L'L''' &&= \cos t' && \cos t''' &&+ \sin t' && \sin t''' && \cos A\Add{,} \\
+&\cos L L''' &&= \cos t  && \cos t''' &&+ \sin t  && \sin t''' && \cos A\Add{,} \\
+&\cos L'L''  &&= \cos t' && \cos t''  &&+ \sin t' && \sin t''  && \cos A\Add{;}
+\end{alignat*}
+and consequently,\Note{3}
+\begin{multline*}
+\cos LL''·\cos L'L''' - \cos LL'''·\cos L'L'' \\
+\begin{aligned}
+&= \cos A (\cos t  \cos t''  \sin t' \sin t'''
+         + \cos t' \cos t''' \sin t  \sin t'' \\
+&\qquad  - \cos t  \cos t''' \sin t' \sin t''
+         - \cos t' \cos t''  \sin t  \sin t''') \\
+&= \cos A (\cos t   \sin t'   - \sin t   \cos t')
+          (\cos t'' \sin t''' - \sin t'' \cos t''')\NoteMark \\
+&= \cos A·\sin (t' - t)·\sin (t''' - t'') \\
+&= \cos A·\sin LL'·\sin L''L'''\Add{.}
+\end{aligned}
+\end{multline*}
+\PageSep{5}
+
+But as there are for each great circle two branches going out from the point~$A$,
+these two branches form at this point two angles whose sum is~$180°$. But our analysis
+shows that those branches are to be taken whose directions are in the sense from the
+point $L$~to~$L'$, and from the point $L''$~to~$L'''$; and since great circles intersect in two
+points, it is clear that either of the two points can be chosen arbitrarily. Also, instead
+of the angle~$A$, we can take the arc between the poles of the great circles of which the
+arcs $LL'$,~$L''L'''$ are parts. But it is evident that those poles are to be chosen which
+are similarly placed with respect to these arcs; that is to say, when we go from $L$~to~$L'$
+and from $L''$~to~$L'''$, both of the two poles are to be on the right, or both on the left.
+
+\Par{VII.} Let $L$,~$L'$,~$L''$ be the three points on the sphere and set, for brevity,
+\begin{alignat*}{6}
+&\cos (1)L   &&= x,\quad&& \cos (2)L   &&= y,\quad&& \cos (3)L   &&= z\Add{,} \\
+&\cos (1)L'  &&= x',    && \cos (2)L'  &&= y',    && \cos (3)L'  &&= z'\Add{,} \\
+&\cos (1)L'' &&= x'',   && \cos (2)L'' &&= y'',   && \cos (3)L'' &&= z''\Add{;} \\
+\end{alignat*}
+and also
+\[
+x y' z'' + x' y'' z + x'' y z' - x y'' z' - x' y z'' - x'' y' z = \Delta\Add{.}
+\]
+Let $\lambda$~denote the pole of the great circle of which $LL'$~is a part, this pole being the one
+that is placed in the same position with respect to this arc as the point~$(1)$ is with
+respect to the arc~$(2)(3)$. Then we shall have, by the preceding theorem,
+\[
+y z' - y' z = \cos (1)\lambda·\sin (2)(3)·\sin LL',
+\]
+or, because $(2)(3) = 90°$,
+\begin{align*}
+y z' - y' z &= \cos (1)\lambda·\sin LL', \\
+\intertext{and similarly,}
+z x' - z' x &= \cos (2)\lambda·\sin LL'\Add{,} \\
+x y' - x' y &= \cos (3)\lambda·\sin LL'\Add{.}
+\end{align*}
+Multiplying these equations by $x''$,~$y''$,~$z''$ respectively, and adding, we obtain, by means
+of the second of the theorems deduced in~V,
+\[
+\Delta = \cos \lambda L''·\sin LL'\Add{.}
+\]
+Now there are three cases to be distinguished. \emph{First}, when $L''$~lies on the great circle
+of which the arc~$LL'$ is a part, we shall have $\lambda L'' = 90°$, and consequently, $\Delta = 0$.
+If $L''$~does not lie on that great circle, the \emph{second} case will be when $L''$~is on the same
+side as~$\lambda$; the \emph{third} case when they are on opposite sides. In the last two cases the
+points $L$,~$L'$,~$L''$ will form a spherical triangle, and in the second case these points will lie
+in the same order as the points $(1)$,~$(2)$,~$(3)$, and in the opposite order in the third case.
+\PageSep{6}
+Denoting the angles of this triangle simply by $L$,~$L'$,~$L''$ and the perpendicular drawn on
+the sphere from the point~$L''$ to the side~$LL'$ by~$p$, we shall have
+\[
+\sin p = \sin L·\sin LL'' = \sin L'·\sin L' L'',
+\]
+and
+\[
+\lambda L'' = 90° \mp p,
+\]
+the upper sign being taken for the second case, the lower for the third. From this
+it follows that
+\begin{align*}
+±\Delta &= \sin L·\sin LL'·\sin LL''
+   = \sin L'·\sin LL'·\sin L'L'' \\
+  &= \sin L''·\sin LL''·\sin L'L''\Add{.}
+\end{align*}
+Moreover, it is evident that the first case can be regarded as contained in the second or
+third, and it is easily seen that the expression~$±\Delta$ represents six times the volume of
+the pyramid formed by the points $L$,~$L'$,~$L''$ and the centre of the sphere. Whence,
+finally, it is clear that the expression~$±\frac{1}{6}\Delta$ expresses generally the volume of any
+pyramid contained between the origin of coordinates and the three points whose coordinates
+are $\Typo{z}{x}$,~$y$,~$z$; $x'$,~$y'$,~$z'$; $x''$,~$y''$,~$z''$.\Note{5}
+
+
+\Article{3.}
+
+A curved surface is said to possess continuous curvature at one of its points~$A$, if the
+directions of all the straight lines drawn from $A$ to points of the surface at an infinitely
+small distance from~$A$ are deflected infinitely little from one and the same plane passing
+through~$A$. This plane is said to \emph{touch} the surface at the point~$A$. If this condition is
+not satisfied for any point, the continuity of the curvature is here interrupted, as happens,
+for example, at the vertex of a cone. The following investigations will be restricted to
+such surfaces, or to such parts of surfaces, as have the continuity of their curvature
+nowhere interrupted. We shall only observe now that the methods used to determine
+the position of the tangent plane lose their meaning at singular points, in which the
+continuity of the curvature is interrupted, and must lead to indeterminate solutions.
+
+
+\Article{4.}
+
+The orientation of the tangent plane is most conveniently studied by means of the
+direction of the straight line normal to the plane at the point~$A$, which is also called the
+normal to the curved surface at the point~$A$. We shall represent the direction of this
+normal by the point~$L$ on the auxiliary sphere, and we shall set
+\[
+\cos (1)L = X,\quad \cos (2)L = Y,\quad \cos (3)L = Z;
+\]
+and denote the coordinates of the point~$A$ by $x$,~$y$,~$z$. Also let $x + dx$, $y + dy$, $z + dz$
+be the coordinates of another point~$A'$ on the curved surface; $ds$~its distance from~$A$,
+\PageSep{7}
+which is infinitely small; and finally, let $\lambda$ be the point on the sphere representing the
+direction of the element~$AA'$. Then we shall have
+\[
+dx = ds·\cos (1)\lambda,\quad
+dy = ds·\cos (2)\lambda,\quad
+dz = ds·\cos (3)\lambda
+\]
+and, since $\lambda L$~must be equal to~$90°$,
+\[
+X\cos (1)\lambda + Y\cos (2)\lambda + Z\cos (3)\lambda = 0\Add{.}
+\]
+By combining these equations we obtain
+\[
+X\, dx + Y\, dy + Z\, dz = 0.
+\]
+
+There are two general methods for defining the nature of a curved surface. The
+\emph{first} uses the equation between the coordinates $x$,~$y$,~$z$, which we may suppose reduced to
+the form $W = 0$, where $W$~will be a function of the indeterminates $x$,~$y$,~$z$. Let the complete
+differential of the function~$W$ be
+\[
+dW = P\, dx + Q\, dy + R\, dz
+\]
+and on the curved surface we shall have
+\[
+P\, dx + Q\, dy + R\, dz = 0\Add{,}
+\]
+and consequently,
+\[
+P \cos (1)\lambda + Q \cos (2)\lambda + R \cos (3)\lambda = 0\Add{.}
+\]
+Since this equation, as well as the one we have established above, must be true for the
+directions of all elements~$ds$ on the curved surface, we easily see that $X$,~$Y$,~$Z$ must be
+proportional to $P$,~$Q$,~$R$ respectively, and consequently, since\Note{6}
+\[
+X^{2} + Y^{2} + Z^{2} = 1,\NoteMark
+\]
+we shall have either
+\begin{align*}
+X &= \frac{P}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Y &= \frac{Q}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Z &= \frac{R}{\Sqrt{P^{2} + Q^{2} + R^{2}}} \\
+\intertext{or}
+X &= \frac{-P}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Y &= \frac{-Q}{\Sqrt{P^{2} + Q^{2} + R^{2}}}, &
+Z &= \frac{-R}{\Sqrt{P^{2} + Q^{2} + R^{2}}}\Add{.}
+\end{align*}
+
+The \emph{second}\Note{7} method expresses the coordinates in the form of functions of two variables,
+$p$,~$q$. Suppose that differentiation of these functions gives
+\begin{alignat*}{2}
+dx &= a\, dp &&+ a'\, dq\Add{,} \\
+dy &= b\, dp &&+ b'\, dq\Add{,} \\
+dz &= c\, dp &&+ c'\, dq\Add{.}
+\end{alignat*}
+\PageSep{8}
+Substituting these values in the formula given above, we obtain
+\[
+(aX + bY + cZ)\, dp + (a'X + b'Y + c'Z)\, dq = 0\Add{.}
+\]
+Since this equation must hold independently of the values of the differentials $dp$,~$dq$,
+we evidently shall have
+\[
+aX + bY + cZ = 0,\quad a'X + b'Y + c'Z = 0\Add{.}
+\]
+From this we see that $X$,~$Y$,~$Z$ will be proportioned to the quantities
+\[
+bc' - cb',\quad ca' - ac',\quad ab' - ba'\Add{.}
+\]
+Hence, on setting, for brevity,
+\[
+\Sqrt{(bc' - cb')^{2} + (ca' - ac')^{2} + (ab' - ba')^{2}} = \Delta\Add{,}
+\]
+we shall have either
+\begin{align*}
+X &= \frac{bc' - cb'}{\Delta},\quad&
+Y &= \frac{ca' - ac'}{\Delta},\quad&
+Z &= \frac{ab' - ba'}{\Delta}
+\intertext{or}
+X &= \frac{cb' - bc'}{\Delta},\quad&
+Y &= \frac{ac' - ca'}{\Delta},\quad&
+Z &= \frac{ba' - ab'}{\Delta}\Add{.}
+\end{align*}
+
+With these two general methods is associated a \emph{third}, in which one of the coordinates,
+$z$,~say, is expressed in the form of a function of the other two, $x$,~$y$. This method is
+evidently only a particular case either of the first method, or of the second. If we set
+\[
+dz = t\, dx + u\, dy
+\]
+we shall have either
+\begin{align*}
+X &= \frac{-t}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Y &= \frac{-u}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Z &= \frac{ 1}{\Sqrt{1 + t^{2} + u^{2}}} \\
+\intertext{or}
+X &= \frac{ t}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Y &= \frac{ u}{\Sqrt{1 + t^{2} + u^{2}}}, &
+Z &= \frac{-1}{\Sqrt{1 + t^{2} + u^{2}}}\Add{.}
+\end{align*}
+
+
+\Article{5.}
+
+The two solutions found in the preceding article evidently refer to opposite points of
+the sphere, or to opposite directions, as one would expect, since the normal may be drawn
+toward either of the two sides of the curved surface. If we wish to distinguish between
+the two regions bordering upon the surface, and call one the exterior region and the other
+the interior region, we can then assign to each of the two normals its appropriate solution
+by aid of the theorem derived in \Art{2}~(VII), and at the same time establish a criterion
+for distinguishing the one region from the other.
+\PageSep{9}
+
+In the first method, such a criterion is to be drawn from the sign of the quantity~$W$.
+Indeed, generally speaking, the curved surface divides those regions of space in which $W$
+keeps a positive value from those in which the value of~$W$ becomes negative. In fact, it
+is easily seen from this theorem that, if $W$ takes a positive value toward the exterior
+region, and if the normal is supposed to be drawn outwardly, the first solution is to be
+taken. Moreover, it will be easy to decide in any case whether the same rule for the
+sign of~$W$ is to hold throughout the entire surface, or whether for different parts there
+will be different rules. As long as the coefficients $P$,~$Q$,~$R$ have finite values and do not
+all vanish at the same time, the law of continuity will prevent any change.
+
+If we follow the second method, we can imagine two systems of curved lines on the
+curved surface, one system for which $p$~is variable, $q$~constant; the other for which $q$~is
+variable, $p$~constant. The respective positions of these lines with reference to the exterior
+region will decide which of the two solutions must be taken. In fact, whenever
+the three lines, namely, the branch of the line of the former system going out from the
+point~$A$ as $p$~increases, the branch of the line of the latter system going out from the point
+$A$ as $q$~increases, and the normal drawn toward the exterior region, are \emph{similarly} placed as
+the $x$,~$y$,~$z$ axes respectively from the origin of abscissas (\eg, if, both for the former
+three lines and for the latter three, we can conceive the first directed to the left, the
+second to the right, and the third upward), the first solution is to be taken. But whenever
+the relative position of the three lines is opposite to the relative position of the
+$x$,~$y$,~$z$ axes, the second solution will hold.
+
+In the third method, it is to be seen whether, when $z$~receives a positive increment, $x$~and~$y$
+remaining constant, the point crosses toward the exterior or the interior region.
+In the former case, for the normal drawn outward, the first solution holds; in the latter
+case, the second.
+
+
+\Article{6.}
+
+Just as each definite point on the curved surface is made to correspond to a definite
+point on the sphere, by the direction of the normal to the curved surface which is transferred
+to the surface of the sphere, so also any line whatever, or any figure whatever, on
+the latter will be represented by a corresponding line or figure on the former. In the
+comparison of two figures corresponding to one another in this way, one of which will be
+as the map of the other, two important points are to be considered, one when quantity
+alone is considered, the other when, disregarding quantitative relations, position alone
+is considered.
+
+The first of these important points will be the basis of some ideas which it seems
+judicious to introduce into the theory of curved surfaces. Thus, to each part of a curved
+\PageSep{10}
+surface inclosed within definite limits we assign a \emph{total} or \emph{integral curvature}, which is
+represented by the area of the figure on the sphere corresponding to it. From this
+integral curvature must be distinguished the somewhat more specific curvature which we
+shall call the\Note{8} \emph{measure of curvature}. The latter refers to a \emph{point} of the surface, and shall
+denote the quotient obtained when the integral curvature of the surface element about
+a point is divided by the area of the element itself; and hence it denotes the ratio of the
+infinitely small areas which correspond to one another on the curved surface and on the
+sphere. The use of these innovations will be abundantly justified, as we hope, by what
+we shall explain below. As for the terminology, we have thought it especially desirable
+that all ambiguity be avoided. For this reason we have not thought it advantageous to
+follow strictly the analogy of the terminology commonly adopted (though not approved by
+all) in the theory of plane curves, according to which the measure of curvature should be
+called simply curvature, but the total curvature, the amplitude. But why not be free in
+the choice of words, provided they are not meaningless and not liable to a misleading
+interpretation?
+
+The position of a figure on the sphere can be either similar to the position of the
+corresponding figure on the curved surface, or opposite (inverse). The former is the case
+when two lines going out on the curved surface from the same point in different, but not
+opposite directions, are represented on the sphere by lines similarly placed, that is, when
+the map of the line to the right is also to the right; the latter is the case when the contrary
+holds. We shall distinguish these two cases by the positive or negative \emph{sign} of the
+measure of curvature. But evidently this distinction can hold only when on each surface
+we choose a definite face on which we suppose the figure to lie. On the auxiliary sphere
+we shall use always the exterior face, that is, that turned away from the centre; on the
+curved surface also there may be taken for the exterior face the one already considered,
+or rather that face from which the normal is supposed to be drawn. For, evidently, there
+is no change in regard to the similitude of the figures, if on the curved surface both the
+figure and the normal be transferred to the opposite side, so long as the image itself
+is represented on the same side of the sphere.
+
+The positive or negative sign, which we assign to the \emph{measure} of curvature according
+to the position of the infinitely small figure, we extend also to the integral curvature
+of a finite figure on the curved surface. However, if we wish to discuss the general case,
+some explanations will be necessary, which we can only touch here briefly. So long
+as the figure on the curved surface is such that to \emph{distinct} points on itself there correspond
+distinct points on the sphere, the definition needs no further explanation. But
+whenever this condition is not satisfied, it will be necessary to take into account twice
+or several times certain parts of the figure on the sphere. Whence for a similar, or
+\PageSep{11}
+inverse position, may arise an accumulation of areas, or the areas may partially or
+wholly destroy each other. In such a case, the simplest way is to suppose the curved
+surface divided into parts, such that each part, considered separately, satisfies the above
+condition; to assign to each of the parts its integral curvature, determining this magnitude
+by the area of the corresponding figure on the sphere, and the sign by the position
+of this figure; and, finally, to assign to the total figure the integral curvature
+arising from the addition of the integral curvatures which correspond to the single parts.
+So, generally, the integral curvature of a figure is equal to $\int k\, d\sigma$, $d\sigma$~denoting the
+element of area of the figure, and $k$~the measure of curvature at any point. The principal
+points concerning the geometric representation of this integral reduce to the following.
+To the perimeter of the figure on the curved surface (under the restriction
+of \Art{3}) will correspond always a closed line on the sphere. If the latter nowhere
+intersect itself, it will divide the whole surface of the sphere into two parts, one of
+which will correspond to the figure on the curved surface; and its area (taken as
+positive or negative according as, with respect to its perimeter, its position is similar,
+or inverse, to the position of the figure on the curved surface) will represent the integral
+curvature of the figure on the curved surface. But whenever this line intersects
+itself once or several times, it will give a complicated figure, to which, however, it is
+possible to assign a definite area as legitimately as in the case of a figure without
+nodes; and this area, properly interpreted, will give always an exact value for the
+integral curvature. However, we must reserve for another occasion\Note{9} the more extended
+exposition of the theory of these figures viewed from this very general standpoint.
+
+
+\Article{7.}
+
+We shall now find a formula which will express the measure of curvature for
+any point of a curved surface. Let $d\sigma$~denote the area of an element of this surface;
+then $Z\, d\sigma$~will be the area of the projection of this element on the plane of the coordinates
+$x$,~$y$; and consequently, if $d\Sigma$~is the area of the corresponding element on the
+sphere, $Z\, d\Sigma$~will be the area of its projection on the same plane. The positive or
+negative sign of~$Z$ will, in fact, indicate that the position of the projection is similar or
+inverse to that of the projected element. Evidently these projections have the same
+ratio as to quantity and the same relation as to position as the elements themselves.
+Let us consider now a triangular element on the curved surface, and let us suppose
+that the coordinates of the three points which form its projection are
+\begin{alignat*}{3}
+&x,           && y\Add{,} \\
+&x + dx,\quad && y + dy\Add{,} \\
+&x + \delta x,\quad && y + \delta y\Add{.}
+\end{alignat*}
+\PageSep{12}
+The double area of this triangle will be expressed by the formula
+\[
+dx·\delta y - dy·\delta x\Add{,}
+\]
+and this will be in a positive or negative form according as the position of the side
+from the first point to the third, with respect to the side from the first point to the
+second, is similar or opposite to the position of the $y$-axis of coordinates with respect
+to the $x$-axis of coordinates.
+
+In like manner, if the coordinates of the three points which form the projection of
+the corresponding element on the sphere, from the centre of the sphere as origin, are
+\begin{alignat*}{3}
+&X,           && Y\Add{,} \\
+&X + dX,\quad && Y + dY\Add{,} \\
+&X + \delta X,\quad && Y + \delta Y\Add{,}
+\end{alignat*}
+the double area of this projection will be expressed by
+\[
+dX·\delta Y - dY·\delta X\Add{,}
+\]
+and the sign of this expression is determined in the same manner as above. Wherefore
+the measure of curvature at this point of the curved surface will be
+\[
+k = \frac{dX·\delta Y - dY·\delta X}{dx·\delta y - dy·\delta x}\Add{.}
+\]
+If now we suppose the nature of the curved surface to be defined according to the third
+method considered in \Art{4}, $X$~and~$Y$ will be in the form of functions of the quantities
+$x$,~$y$. We shall have, therefore,\Note{10}
+\begin{alignat*}{2}
+dX &= \frac{\dd X}{\dd x}\, dx &&+ \frac{\dd X}{\dd y}\, dy\Add{,} \\
+\delta X &= \frac{\dd X}{\dd x}\, \delta x
+          &&+ \frac{\dd X}{\dd y}\, \delta y\Add{,} \\
+dY &= \frac{\dd Y}{\dd x}\, dx &&+ \frac{\dd Y}{\dd y}\, dy\Add{,} \\
+\delta Y &= \frac{\dd Y}{\dd x}\, \delta x
+          &&+ \frac{\dd Y}{\dd y}\, \delta y\Add{.}
+\end{alignat*}
+When these values have been substituted, the above expression becomes
+\[
+k = \frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
+  - \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x}\Add{.}
+\]
+\PageSep{13}
+Setting, as above,
+\[
+\frac{\dd z}{\dd x} = t,\quad \frac{\dd z}{\dd y} = u
+\]
+and also
+\[
+\frac{\dd^{2} z}{\dd x^{2}} = T,\quad
+\frac{\dd^{2} z}{\dd x·\dd y} = U,\quad
+\frac{\dd^{2} z}{\dd y^{2}} = V\Add{,}
+\]
+or
+\[
+dt = T\, dx + U\, dy,\quad
+du = U\, dx + V\, dy\Add{,}
+\]
+we have from the formulæ given above
+\[
+X = -tZ,\quad Y = -uZ,\quad (1 - t^{2} - u^{2})Z^{2} = 1\Add{;}
+\]
+and hence
+\begin{gather*}
+\begin{alignedat}{2}
+dX &= -Z\, dt &&- t\, dZ\Add{,} \\
+dY &= -Z\, du &&- u\, dZ\Add{,}
+\end{alignedat} \\
+(1 + t^{2} + u^{2})\, dZ + Z(t\, dt + u\, du) = 0\Add{;}
+\end{gather*}
+or\Note{11}
+\begin{align*}
+dZ &= -Z^{3}(t\, dt + u\, du)\Add{,} \\
+dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3} tu\, du\Add{,} \\
+dY &= +Z^{3}tu\, dt - Z^{3}(1 + t^{2})\, du\Add{;}\NoteMark
+\end{align*}
+and so
+\begin{align*}
+\frac{\dd X}{\dd x} &= Z^{3}\bigl(-(1 + u^{2})T + tuU\bigr)\Add{,} \\
+\frac{\dd X}{\dd y} &= Z^{3}\bigl(-(1 + u^{2})U + tuV\bigr)\Add{,} \\
+\frac{\dd Y}{\dd x} &= Z^{3}\bigl( tuT - (1 + t^{2})U\bigr)\Add{,} \\
+\frac{\dd Y}{\dd y} &= Z^{3}\bigl( tuU - (1 + t^{2})V\bigr)\Add{.}
+\end{align*}
+Substituting these values in the above expression, it becomes
+\begin{align*}
+k &= Z^{6}(TV - U^{2}) (1 + t^{2} + u^{2}) = Z^{4} (TV - U^{2}) \\
+  &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}\Add{.}
+\end{align*}
+
+
+\Article{8.}
+
+By a suitable choice of origin and axes of coordinates, we can easily make the
+values of the quantities $t$,~$u$,~$U$ vanish for a definite point~$A$. Indeed, the first two
+\PageSep{14}
+conditions will be fulfilled at once if the tangent plane at this point be taken for the
+$xy$-plane. If, further, the origin is placed at the point $A$~itself, the expression for
+the coordinate~$z$ evidently takes the form
+\[
+z = \tfrac{1}{2}T°x^{2} + U°xy + \tfrac{1}{2}V°y^{2} + \Omega\Add{,}
+\]
+where $\Omega$~will be of higher degree than the second. Turning now the axes of $x$~and~$y$
+through an angle~$M$ such that
+\[
+\tan 2M = \frac{2U°}{T° - V°}\Add{,}
+\]
+it is easily seen that there must result an equation of the form
+\[
+z = \tfrac{1}{2}Tx^{2} + \tfrac{1}{2}Vy^{2} + \Omega\Add{.}
+\]
+In this way the third condition is also satisfied. When this has been done, it is evident
+that
+
+\Par{I.} If the curved surface be cut by a plane passing through the normal itself and
+through the $x$-axis, a plane curve will be obtained, the radius of curvature of which
+at the point~$A$ will be equal to~$\dfrac{1}{T}$, the positive or negative sign indicating that the
+curve is concave or convex toward that region toward which the coordinates~$z$ are
+positive.
+
+\Par{II.} In like manner $\dfrac{1}{V}$~will be the radius of curvature at the point~$A$ of the plane
+curve which is the intersection of the surface and the plane through the $y$-axis and
+the $z$-axis.
+
+\Par{III.} Setting $z = r \cos\phi$, $y = r \sin \phi$, the equation becomes
+\[
+z = \tfrac{1}{2}(T\cos^{2}\phi + V\sin^{2}\phi) r^{2} + \Omega\Add{,}
+\]
+from which we see that if the section is made by a plane through the normal at~$A$
+and making an angle~$\phi$ with the $x$-axis, we shall have a plane curve whose radius of
+curvature at the point~$A$ will be
+\[
+\frac{1}{T\cos^{2}\phi + V\sin^{2}\phi}\Add{.}
+\]
+
+\Par{IV.} Therefore, whenever we have $T = V$, the radii of curvature in \emph{all} the normal
+planes will be equal. But if $T$~and~$V$ are not equal, it is evident that, since for any
+value whatever of the angle~$\phi$, $T\cos^{2}\phi + V\sin^{2}\phi$ falls between $T$~and~$V$, the radii of
+curvature in the principal sections considered in I.~and~II. refer to the extreme curvatures;
+that is to say, the one to the maximum curvature, the other to the minimum,
+\PageSep{15}
+if $T$~and~$V$ have the same sign. On the other hand, one has the greatest convex
+curvature, the other the greatest concave curvature, if $T$~and~$V$ have opposite signs.
+These conclusions contain almost all that the illustrious Euler\Note{12} was the first to prove
+on the curvature of curved surfaces.
+
+\Par{V.} The measure of curvature at the point~$A$ on the curved surface takes the
+very simple form
+\[
+k = TV,
+\]
+whence we have the
+
+\begin{Theorem}
+The measure of curvature at any point whatever of the surface is equal to a
+fraction whose numerator is unity, and whose denominator is the product of the two extreme
+radii of curvature of the sections by normal planes.\
+\end{Theorem}
+
+At the same time it is clear that the measure of curvature is positive for concavo-concave
+or convexo-convex surfaces (which distinction is not essential), but negative
+for concavo-convex surfaces. If the surface consists of parts of each kind, then
+on the lines separating the two kinds the measure of curvature ought to vanish. Later
+we shall make a detailed study of the nature of curved surfaces for which the measure
+of curvature everywhere vanishes.
+
+
+\Article{9.}
+
+The general formula for the measure of curvature given at the end of \Art{7} is
+the most simple of all, since it involves only five elements. We shall arrive at a
+more complicated formula, indeed, one involving nine elements, if we wish to use the
+first method of representing a curved surface. Keeping the notation of \Art{4}, let us
+set also
+\begin{align*}
+\frac{\dd^{2} W}{\dd x^{2}} &= P', &
+\frac{\dd^{2} W}{\dd y^{2}} &= Q', &
+\frac{\dd^{2} W}{\dd z^{2}} &= R'\Add{,} \\
+\frac{\dd^{2} W}{\dd y·\dd z} &= P'', &
+\frac{\dd^{2} W}{\dd x·\dd z} &= Q'', &
+\frac{\dd^{2} W}{\dd x·\dd y} &= R''\Add{,}
+\end{align*}
+so that
+\begin{alignat*}{4}
+dP &= P'\,  &&dx + R''\, &&dy + Q''\, &&dz\Add{,} \\
+dQ &= R''\, &&dx + Q' \, &&dy + P''\, &&dz\Add{,} \\
+dR &= Q''\, &&dx + P''\, &&dy + R' \, &&dz\Add{.}
+\end{alignat*}
+Now since $t = -\dfrac{P}{R}$, we find through differentiation
+\[
+R^{2}\, dt = -R\, dP + P\, dR
+  = (PQ'' - RP')\, dx + (PP'' - RR'')\, dy + (PR' - RQ'')\, dz\Add{,}
+\]
+\PageSep{16}
+or, eliminating~$dz$ by means of the equation
+\begin{gather*}
+P\, dx + Q\, dy + R\, dz = 0, \\
+R^{3}\, dt
+  = (-R^{2}P' + 2PRQ'' - P^{2}R')\, dx + (PRP'' + QRQ'' - PQR' - R^{2}R'')\, dy.
+\end{gather*}
+In like manner we obtain
+\[
+R^{3}\, du
+  = (PRP'' + QRQ'' - PQR' - R^{2}R'')\, dx + (-R^{2}Q' + 2QRP'' - Q^{2}R')\, dy\Add{.}
+\]
+From this we conclude that
+\begin{align*}
+R^{3}T &= -R^{2}P' + 2PRQ'' - P^{2}R'\Add{,} \\
+R^{3}U &= PRP'' + QRQ'' - PQR' - R^{2}R''\Add{,} \\
+R^{3}V &= -R^{2}Q' + 2QRP'' - Q^{2}R'\Add{.}
+\end{align*}
+Substituting these values in the formula of \Art{7}, we obtain for the measure of curvature~$k$
+the following symmetric expression:
+\begin{multline*}
+(P^{2} + Q^{2} + R^{2})^{2}k
+  = P^{2}(Q'R' - P''^{2})
+  + Q^{2}(P'R' - Q''^{2})
+  + R^{2}(P'Q' - R''^{2}) \\
+  + 2QR(Q''R'' - P'P'')
+  + 2PR(P''R'' - Q'Q'')
+  + 2PQ(P''Q'' - R'R'')\Add{.}
+\end{multline*}
+
+
+\Article{10.}
+
+We obtain a still more complicated formula, indeed, one involving fifteen elements,
+if we follow the second general method of defining the nature of a curved surface. It
+is, however, very important that we develop this formula also. Retaining the notations
+of \Art{4}, let us put also
+\begin{align*}
+\frac{\dd^{2}x}{\dd p^{2}} &= \alpha, &
+\frac{\dd^{2}x}{\dd p·\dd q} &= \alpha', &
+\frac{\dd^{2}x}{\dd q^{2}} &= \alpha''\Add{,} \\
+%
+\frac{\dd^{2}y}{\dd p^{2}} &= \beta, &
+\frac{\dd^{2}y}{\dd p·\dd q} &= \beta', &
+\frac{\dd^{2}y}{\dd q^{2}} &= \beta''\Add{,} \\
+%
+\frac{\dd^{2}z}{\dd p^{2}} &= \gamma, &
+\frac{\dd^{2}z}{\dd p·\dd q} &= \gamma', &
+\frac{\dd^{2}z}{\dd q^{2}} &= \gamma''\Add{;} \\
+\end{align*}
+and let us put, for brevity,
+\begin{align*}
+bc' - cb' &= A\Add{,} \\
+ca' - ac' &= B\Add{,} \\
+ab' - ba' &= C\Add{.}
+\end{align*}
+First we see that
+\[
+A\, dx + B\, dy + C\, dz = 0,
+\]
+or
+\[
+dz = -\frac{A}{C}\, dx - \frac{B}{C}\, dy.
+\]
+\PageSep{17}
+Thus, inasmuch as $z$~may be regarded as a function of $x$,~$y$, we have
+\begin{align*}
+\frac{\dd z}{\dd x} &= t = -\frac{A}{C}\Add{,} \\
+\frac{\dd z}{\dd y} &= u = -\frac{B}{C}\Add{.}
+\end{align*}
+Then from the formulæ
+\[
+dx = a\, dp + a'\, dq,\quad
+dy = b\, dp + b'\, dq,
+\]
+we have
+\begin{alignat*}{4}
+&C\, dp = &&b'\, &&dx - a'\, &&dy\Add{,} \\
+&C\, dq =-&&b\,  &&dx + a\,  &&dy\Add{.}
+\end{alignat*}
+Thence we obtain for the total differentials of $t$,~$u$
+\begin{alignat*}{2}
+C^{3}\, dt
+  &= \left(A\, \frac{\dd C}{\dd p} - C\, \frac{\dd A}{\dd p}\right)(b'\, dx - a'\, dy)
+   + \left(C\, \frac{\dd A}{\dd q} - A\, \frac{\dd C}{\dd q}\right)(b \, dx - a \, dy)\Add{,} \\
+%
+C^{3}\, du
+  &= \left(B\, \frac{\dd C}{\dd p} - C\, \frac{\dd B}{\dd p}\right)(b'\, dx - a'\, dy)
+   + \left(C\, \frac{\dd B}{\dd q} - B\, \frac{\dd C}{\dd q}\right)(b \, dx - a \, dy)\Add{.}
+\end{alignat*}
+If now we substitute in these formulæ
+\begin{alignat*}{4}
+\frac{\dd A}{\dd p} &= c'\beta  &&+ b\gamma'  &&- c\beta'  &&- b'\gamma\Add{,} \\
+\frac{\dd A}{\dd q} &= c'\beta' &&+ b\gamma'' &&- c\beta'' &&- b'\gamma'\Add{,} \\
+%
+\frac{\dd B}{\dd p} &= a'\gamma  &&+ c\alpha'  &&- a\gamma'  &&- c'\alpha\Add{,} \\
+\frac{\dd B}{\dd q} &= a'\gamma' &&+ c\alpha'' &&- a\gamma'' &&- c'\alpha'\Add{,} \\
+%
+\frac{\dd C}{\dd p} &= b'\alpha  &&+ a\beta'  &&- b\alpha'  &&- a'\beta\Add{,} \\
+\frac{\dd C}{\dd q} &= b'\alpha' &&+ a\beta'' &&- b\alpha'' &&- a'\beta'\Add{;}
+\end{alignat*}
+and if we note that the values of the differentials $dt$,~$du$ thus obtained must be equal,
+independently of the differentials $dx$,~$dy$, to the quantities $T\, dx + U\, dy$, $U\, dx + V\, dy$
+respectively, we shall find, after some sufficiently obvious transformations,
+\begin{align*}
+C^{3}T &= \alpha Ab'^{2} + \beta Bb'^{2} + \gamma Cb'^{2} \\
+&\quad- 2\alpha' Abb' - 2\beta' Bbb' - 2\gamma' Cbb' \\
+&\quad+ \alpha'' Ab^{2} + \beta'' Bb^{2} + \gamma'' Cb^{2}\Add{,} \\
+\PageSep{18}
+C^{3}U &= -\alpha Aa'b' - \beta Ba'b' - \gamma Ca'b' \\
+&\quad+ \alpha' A(ab' + ba') + \beta' B(ab' + ba') + \gamma' C(ab' + ba') \\
+&\quad- \alpha'' Aab - \beta'' Bab - \gamma'' Cab\Add{,} \\
+%
+C^{3}V &= \alpha Aa'^{2} + \beta Ba'^{2} + \gamma Ca'^{2} \\
+&\quad- 2\alpha' Aaa' - 2\beta' Baa' - 2\gamma' Caa' \\
+&\quad+ \alpha'' Aa^{2} + \beta'' Ba^{2} + \gamma'' Ca^{2}\Add{.}
+\end{align*}
+Hence, if we put, for the sake of brevity,\Note{13}
+\begin{alignat*}{4}
+&A\alpha   &&+ B\beta   &&+ C\gamma   &&= D\Add{,}\NoteMark
+\Tag{(1)} \\
+&A\alpha'  &&+ B\beta'  &&+ C\gamma'  &&= D'\Add{,}
+\Tag{(2)} \\
+&A\alpha'' &&+ B\beta'' &&+ C\gamma'' &&= D''\Add{,}
+\Tag{(3)}
+\end{alignat*}
+we shall have
+\begin{align*}
+C^{3}T &=  Db'^{2} - 2D'bb' + D'' b^{2}\Add{,} \\
+C^{3}U &= -Da'b' + D'(ab' + ba') - D''ab\Add{,} \\
+C^{3}V &=  Da'^{2} - 2D'aa' + D''a^{2}\Add{.}
+\end{align*}
+From this we find, after the reckoning has been carried out,
+\[
+C^{6}(TV - U^{2}) = (DD'' - D'^{2}) (ab' - ba')^{2} = (DD'' - D'^{2}) C^{2}\Add{,}
+\]
+and therefore the formula for the measure of curvature
+\[
+k = \frac{DD'' - D'^{2}}{(A^{2} + B^{2} + C^{2})^{2}}\Add{.}
+\]
+
+
+\Article{11.}
+
+By means of the formula just found we are going to establish another, which may
+be counted among the most productive theorems in the theory of curved surfaces.
+Let us introduce the following notation:
+\begin{alignat*}{4}
+&a^{2} &&+ b^{2} &&+ c^{2} &&= E\Add{,} \\
+&aa' &&+ bb' &&+ cc' &&= F\Add{,} \\
+&a'^{2} &&+ b'^{2} &&+ c'^{2} &&= G\Add{;}
+\end{alignat*}
+\begin{alignat*}{7}
+&a &&\alpha  &&+b &&\beta  &&+c &&\gamma  &&= m  \Add{,}
+\Tag{(4)} \\
+&a &&\alpha' &&+b &&\beta' &&+c &&\gamma' &&= m' \Add{,}
+\Tag{(5)} \\
+&a &&\alpha''&&+b &&\beta''&&+c &&\gamma''&&= m''\Add{;}
+\Tag{(6)} \displaybreak[1] \\
+%
+&a'\,&&\alpha  &&+b'\,&&\beta  &&+c'\,&&\gamma  &&= n  \Add{,}
+\Tag{(7)} \\
+&a'&&\alpha' &&+b'&&\beta' &&+c'&&\gamma' &&= n' \Add{,}
+\Tag{(8)} \\
+&a'&&\alpha''&&+b'&&\beta''&&+c'&&\gamma''&&= n''\Add{;}
+\Tag{(9)}
+\end{alignat*}
+\[
+A^{2} + B^{2} + C^{2} = EG - F^{2} = \Delta\Add{.}
+\]
+\PageSep{19}
+
+%[** TN: Added parentheses around equation numbers]
+Let us eliminate from the equations (1),~(4),~(7) the quantities $\beta$,~$\gamma$, which is done by
+multiplying them by $bc' - cb'$, $b'C - c'B$, $cB - bC$ respectively and adding. In this
+way we obtain
+\begin{multline*}
+\bigl(A(bc' - cb') + a(b'C - c'B) + a'(cB - bC)\bigr)\alpha \\
+  = D(bc' - cb') + m(b'C - c'B) + n(cB - bC)\Add{,}
+\end{multline*}
+an equation which is easily transformed into
+\[
+AD = \alpha\Delta + a(nF - mG) + a'(mF - nE)\Add{.}
+\]
+Likewise the elimination of $\alpha$,~$\gamma$ or $\alpha$,~$\beta$ from the same equations gives
+\begin{alignat*}{4}
+&BD &&= \beta \Delta &&+ b(nF - mG) &&+ b'(mF - nE)\Add{,} \\
+&CD &&= \gamma\Delta &&+ c(nF - mG) &&+ c'(mF - nE)\Add{.}
+\end{alignat*}
+Multiplying these three equations by $\alpha''$,~$\beta''$,~$\gamma''$ respectively and adding, we obtain
+\[
+DD'' = (\alpha\alpha''+ \beta\beta'' + \gamma\gamma'')\Delta
+  + m''(nF - mG) + n''(mF - nE)\Add{.}
+\Tag{(10)}
+\]
+
+%[** TN: Added parentheses around equation numbers]
+If we treat the equations (2),~(5),~(8) in the same way, we obtain
+\begin{alignat*}{4}
+&AD' &&= \alpha'\Delta &&+ a (n'F - m'G) &&+ a'(m'F - n'E)\Add{,} \\
+&BD' &&= \beta' \Delta &&+ b (n'F - m'G) &&+ b'(m'F - n'E)\Add{,} \\
+&CD' &&= \gamma'\Delta &&+ c (n'F - m'G) &&+ c'(m'F - n'E)\Add{;}
+\end{alignat*}
+and after these equations are multiplied by $\alpha'$,~$\beta'$,~$\gamma'$ respectively, addition gives
+\[
+D'^{2} = (\alpha'^{2} + \beta'^{2} + \gamma'^{2})\Delta
+  + m'(n'F - m'G) + n'(m'F - n'E)\Add{.}
+\]
+
+A combination of this equation with equation~(10) gives
+\begin{multline*}
+DD'' - D'^{2} = (\alpha\alpha'' + \beta\beta'' + \gamma\gamma''
+  - \alpha'^{2} - \beta'^{2} - \gamma'^{2})\Delta \\
+  + E(n'^{2} - nn'') + F(nm'' - 2m'n' + mn'') + G(m'^{2} - mm'')\Add{.}
+\end{multline*}
+It is clear that we have
+\[
+\frac{\dd E}{\dd p} = 2m,\
+\frac{\dd E}{\dd q} = 2m',\quad
+\frac{\dd F}{\dd p} = m' + n,\
+\frac{\dd F}{\dd q} = m'' + n',\quad
+\frac{\dd G}{\dd p} = 2n',\
+\frac{\dd G}{\dd q} = 2n'',
+\]
+or\Note{14}
+\begin{align*}
+m   &= \tfrac{1}{2}\, \frac{\dd E}{\dd p}, &
+m'  &= \tfrac{1}{2}\, \frac{\dd E}{\dd q}, &
+m'' &= \frac{\dd F}{\dd q} - \tfrac{1}{2}\, \frac{\dd G}{\dd p}\Add{,}\NoteMark \\
+%
+n   &= \frac{\dd F}{\dd p} - \tfrac{1}{2}\, \frac{\dd E}{\dd q}, &
+n'  &= \tfrac{1}{2}\, \frac{\dd G}{\dd p}, &
+n'' &= \tfrac{1}{2}\, \frac{\dd G}{\dd q}\Add{.}
+\end{align*}
+Moreover, it is easily shown that we shall have
+\begin{align*}
+%[** TN: Aligning on equals sign]
+\alpha\alpha'' + \beta\beta'' + \gamma\gamma''
+  - \alpha'^{2} - \beta'^{2} - \gamma'^{2}
+  &= \frac{\dd n}{\dd q} - \frac{\dd n'}{\dd p}
+   = \frac{\dd m''}{\dd p} - \frac{\dd m'}{\dd q} \\
+  &= -\tfrac{1}{2}·\frac{\dd^{2}E}{\dd q^{2}}
+  + \frac{\dd^{2}F}{\dd p·\dd q}
+  - \tfrac{1}{2}·\frac{\dd^{2}G}{\dd p^{2}}\Add{.}
+\end{align*}
+\PageSep{20}
+If we substitute these different expressions in the formula for the measure of curvature
+derived at the end of the preceding article, we obtain the following formula, which
+involves only the quantities $E$,~$F$,~$G$ and their differential quotients of the first and
+second orders:
+\begin{multline*}
+4(EG - F^{2})k
+  = E\left(\frac{\dd E}{\dd q}·\frac{\dd G}{\dd q}
+       - 2 \frac{\dd F}{\dd p}·\frac{\dd G}{\dd q}
+  + \biggl(\frac{\dd G}{\dd p}\biggr)^{2}\right) \\
+  + F\left(\frac{\dd E}{\dd p}·\frac{\dd G}{\dd q}
+         - \frac{\dd E}{\dd q}·\frac{\dd G}{\dd p}
+       - 2 \frac{\dd E}{\dd q}·\frac{\dd F}{\dd q}
+       + 4 \frac{\dd \Erratum{E}{F}}{\dd p}·\frac{\dd F}{\dd q}
+       - 2 \frac{\dd F}{\dd p}·\frac{\dd G}{\dd p}\right) \\
+  + G\left(\frac{\dd E}{\dd p}·\frac{\dd G}{\dd p}
+       - 2 \frac{\dd E}{\dd p}·\frac{\dd F}{\dd q}
+  + \biggl(\frac{\dd E}{\dd q}\biggr)^{2}\right)
+  - 2(EG - F^{2})\left(
+      \frac{\dd^{2}E}{\dd q^{2}}
+   - 2\frac{\dd^{2}F}{\dd p·\dd q}
+    + \frac{\dd^{2}G}{\dd p^{2}}
+  \right)\Add{.}
+\end{multline*}
+
+
+\Article{12.}
+
+Since we always have
+\[
+dx^{2} + dy^{2} + dz^{2} = E\, dp^{2} + 2F\, dp·dq + G\, dq^{2},
+\]
+it is clear that
+\[
+\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}
+\]
+is the general expression for the linear element on the curved surface. The analysis
+developed in the preceding article thus shows us that for finding the measure of curvature
+there is no need of finite formulæ, which express the coordinates $x$,~$y$,~$z$ as
+functions of the indeterminates $p$,~$q$; but that the general expression for the magnitude
+of any linear element is sufficient. Let us proceed to some applications of this very
+important theorem.
+
+Suppose that our surface can be developed upon another surface, curved or plane,
+so that to each point of the former surface, determined by the coordinates $x$,~$y$,~$z$, will
+correspond a definite point of the latter surface, whose coordinates are $x'$,~$y'$,~$z'$. Evidently
+$x'$,~$y'$,~$z'$ can also be regarded as functions of the indeterminates $p$,~$q$, and therefore
+for the element $\Sqrt{dx'^{2} + dy'^{2} + dz'^{2}}$ we shall have an expression of the form
+\[
+\Sqrt{E'\, dp^{2} + 2F'\, dp·dq + G'\, dq^{2}}\Add{,}
+\]
+where $E'$,~$F'$,~$G'$ also denote functions of $p$,~$q$. But from the very notion of the \emph{development}
+of one surface upon another it is clear that the elements corresponding to one
+another on the two surfaces are necessarily equal. Therefore we shall have identically
+\[
+E = E',\quad F = F',\quad G = G'.
+\]
+Thus the formula of the preceding article leads of itself to the remarkable
+
+\begin{Theorem}
+If a curved surface is developed upon any other surface whatever, the
+measure of curvature in each point remains unchanged.
+\end{Theorem}
+\PageSep{21}
+
+Also it is evident that
+\begin{Theorem}[]
+any finite part whatever of the curved surface will retain the
+same integral curvature after development upon another surface.
+\end{Theorem}
+
+Surfaces developable upon a plane constitute the particular case to which geometers
+have heretofore restricted their attention. Our theory shows at once that the
+measure of curvature at every point of such surfaces is equal to zero. Consequently,
+if the nature of these surfaces is defined according to the third method, we shall have
+at every point
+\[
+\frac{\dd^{2} z}{\dd x^{2}}·\frac{\dd^{2} z}{\dd y^{2}}
+  - \left(\frac{\dd^{2}z}{\dd x·\dd y}\right)^{2} = 0\Add{,}
+\]
+a criterion which, though indeed known a short time ago, has not, at least to our
+knowledge, commonly been demonstrated with as much rigor as is desirable.
+
+
+\Article{13.}
+
+What we have explained in the preceding article is connected with a particular
+method of studying surfaces, a very worthy method which may be thoroughly developed
+by geometers. When a surface is regarded, not as the boundary of a solid, but
+as a flexible, though not extensible solid, one dimension of which is supposed to
+vanish, then the properties of the surface depend in part upon the form to which we
+can suppose it reduced, and in part are absolute and remain invariable, whatever may
+be the form into which the surface is bent. To these latter properties, the study of
+which opens to geometry a new and fertile field, belong the measure of curvature and
+the integral curvature, in the sense which we have given to these expressions. To
+these belong also the theory of shortest lines, and a great part of what we reserve to
+be treated later. From this point of view, a plane surface and a surface developable
+on a plane, \eg,~cylindrical surfaces, conical surfaces,~etc., are to be regarded as essentially
+identical; and the generic method of defining in a general manner the nature of
+the surfaces thus considered is always based upon the formula
+\[
+\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}},
+\]
+which connects the linear element with the two indeterminates $p$,~$q$. But before following
+this study further, we must introduce the principles of the theory of shortest
+lines on a given curved surface.
+
+
+\Article{14.}
+
+The nature of a curved line in space is generally given in such a way that the
+coordinates $x$,~$y$,~$z$ corresponding to the different points of it are given in the form of
+functions of a single variable, which we shall call~$w$. The length of such a line from
+\PageSep{22}
+an arbitrary initial point to the point whose coordinates are $x$,~$y$,~$z$, is expressed by
+the integral
+\[
+%[** TN: Round outer parentheses in the original]
+\int dw·\SQRT{
+    \left(\frac{dx}{dw}\right)^{2}
+  + \left(\frac{dy}{dw}\right)^{2}
+  + \left(\frac{dz}{dw}\right)^{2}}\Add{.}
+\]
+If we suppose that the position of the line undergoes an infinitely small variation, so
+that the coordinates of the different points receive the variations $\delta x$,~$\delta y$,~$\delta z$, the variation
+of the whole length becomes
+\[
+\int \frac{dx·d\, \delta x + dy·d\, \delta y + dz·d\, \delta z}
+          {\Sqrt{dx^{2} + dy^{2} + dz^{2}}}\Add{,}
+\]
+which expression we can change into the form\Note{15}
+\begin{multline*}
+\frac{dx·\delta x + dy·\delta y + dz·\delta z}
+     {\Sqrt{dx^{2} + dy^{2} + dz^{2}}} \\
+-\int \Biggl(
+    \delta x·d\frac{dx}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+  + \delta y·d\frac{dy}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+  + \delta z·d\frac{dz}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+  \Biggr)\Add{.}\NoteMark
+\end{multline*}
+We know that, in case the line is to be the shortest between its end points, all that
+stands under the integral sign must vanish. Since the line must lie on the given
+surface, whose nature is defined by the equation
+\[
+P\, dx + Q\, dy + R\, dz = 0,
+\]
+the variations $\delta x$,~$\delta y$,~$\delta z$ also must satisfy the equation
+\[
+P\, \delta x + Q\, \delta y + R\, \delta z = 0,
+\]
+and from this it follows at once, according to well-known rules, that the differentials
+\[
+d\frac{dx}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad
+d\frac{dy}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}},\quad
+d\frac{dz}{\Sqrt{dx^{2} + dy^{2} + dz^{2}}}
+\]
+must be proportional to the quantities $P$,~$Q$,~$R$ respectively. Let $dr$~be the element
+of the curved line; $\lambda$~the point on the sphere representing the direction of this element;
+$L$~the point on the sphere representing the direction of the normal to the curved
+surface; finally, let $\xi$,~$\eta$,~$\zeta$ be the coordinates of the point~$\lambda$, and $X$,~$Y$,~$Z$ be those of
+the point~$L$ with reference to the centre of the sphere. We shall then have
+\[
+dx = \xi\, dr,\quad
+dy = \eta\, dr,\quad
+dz = \zeta\, dr\Add{,}
+\]
+from which we see that the above differentials become $d\xi$,~$d\eta$,~$d\zeta$. And since the
+quantities $P$,~$Q$,~$R$ are proportional to $X$,~$Y$,~$Z$, the character of shortest lines is
+expressed by the equations
+\[
+\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}\Add{.}
+\]
+\PageSep{23}
+Moreover, it is easily seen that
+\[
+\Sqrt{d\xi^{2} + d\eta^{2} + d\zeta^{2}}
+\]
+is equal to the small arc on the sphere which measures the angle between the directions
+of the tangents at the beginning and at the end of the element~$dr$, and is thus
+equal to~$\dfrac{dr}{\rho}$, if $\rho$~denotes the radius of curvature of the shortest line at this point.
+Thus we shall have
+\[
+\rho\, d\xi = X\, dr,\quad
+\rho\, d\eta = Y\, dr,\quad
+\rho\, d\zeta = Z\, dr\Add{.}
+\]
+
+
+\Article{15.}
+
+Suppose that an infinite number of shortest lines go out from a given point~$A$
+on the curved surface, and suppose that we distinguish these lines from one another
+by the angle that the first element of each of them makes with the first element of
+one of them which we take for the first. Let $\phi$~be that angle, or, more generally, a
+function of that angle, and $r$~the length of such a shortest line from the point~$A$ to
+the point whose coordinates are $x$,~$y$,~$z$. Since to definite values of the variables $r$,~$\phi$
+there correspond definite points of the surface, the coordinates $x$,~$y$,~$z$ can be regarded
+as functions of $r$,~$\phi$. We shall retain for the notation $\lambda$, $L$, $\xi$,~$\eta$,~$\zeta$, $X$,~$Y$,~$Z$ the same
+meaning as in the preceding article, this notation referring to any point whatever on
+any one of the shortest lines.
+
+All the shortest lines that are of the same length~$r$ will end on another line
+whose length, measured from an arbitrary initial point, we shall denote by~$v$. Thus $v$~can
+be regarded as a function of the indeterminates $r$,~$\phi$, and if $\lambda'$~denotes the point
+on the sphere corresponding to the direction of the element~$dv$, and also $\xi'$,~$\eta'$,~$\zeta'$
+denote the coordinates of this point with reference to the centre of the sphere, we
+shall have
+\[
+\frac{\dd x}{\dd\phi} = \xi'·\frac{\dd v}{\dd\phi},\quad
+\frac{\dd y}{\dd\phi} = \eta'·\frac{\dd v}{\dd\phi},\quad
+\frac{\dd z}{\dd\phi} = \zeta'·\frac{\dd v}{\dd\phi}\Add{.}
+\]
+From these equations and from the equations
+\[
+\frac{\dd x}{\dd r} = \xi,\quad
+\frac{\dd y}{\dd r} = \eta,\quad
+\frac{\dd z}{\dd r} = \zeta
+\]
+we have
+\[
+\frac{\dd x}{\dd r}·\frac{\dd x}{\dd\phi} +
+\frac{\dd y}{\dd r}·\frac{\dd y}{\dd\phi} +
+\frac{\dd z}{\dd r}·\frac{\dd z}{\dd\phi}
+  = (\xi\xi' + \eta\eta' + \zeta\zeta')·\frac{\dd v}{\dd\phi}
+  = \cos \lambda\lambda'·\frac{\dd v}{\dd\phi}\Add{.}
+\]
+\PageSep{24}
+Let $S$~denote the first member of this equation, which will also be a function of $r$,~$\phi$.
+Differentiation of~$S$ with respect to~$r$ gives
+\begin{align*}
+\frac{\dd S}{\dd r}
+  &= \frac{\dd^{2} x}{\dd r^{2}}·\frac{\dd x}{\dd\phi}
+   + \frac{\dd^{2} y}{\dd r^{2}}·\frac{\dd y}{\dd\phi}
+   + \frac{\dd^{2} z}{\dd r^{2}}·\frac{\dd z}{\dd\phi}
+   + \tfrac{1}{2}·\frac{\dd\left(
+       \biggl(\dfrac{\dd x}{\dd r}\biggr)^{2}
+     + \biggl(\dfrac{\dd y}{\dd r}\biggr)^{2}
+     + \biggl(\dfrac{\dd z}{\dd r}\biggr)^{2}
+   \right)}{\dd \phi} \\
+  &= \frac{\dd\xi}{\dd r}·\frac{\dd x}{\dd\phi}
+   + \frac{\dd\eta}{\dd r}·\frac{\dd y}{\dd\phi}
+   + \frac{\dd\zeta}{\dd r}·\frac{\dd z}{\dd\phi}
+   + \tfrac{1}{2}·\frac{\dd(\xi^{2} + \eta^{2} + \zeta^{2})}{\dd\phi}\Add{.}
+\end{align*}
+But
+\[
+\xi^{2} + \eta^{2} + \zeta^{2} = 1,
+\]
+and therefore its differential is equal to zero; and by the preceding article we have,
+if $\rho$~denotes the radius of curvature of the line~$r$,
+\[
+\frac{\dd\xi}{\dd r} = \frac{X}{\rho},\quad
+\frac{\dd\eta}{\dd r} = \frac{Y}{\rho},\quad
+\frac{\dd\zeta}{\dd r} = \frac{Z}{\rho}\Add{.}
+\]
+Thus we have
+\[
+\frac{\dd S}{\dd r}
+  = \frac{1}{\rho}·(X\xi' + Y\eta' + Z\zeta')·\frac{\dd v}{\dd\phi}
+  = \frac{1}{\rho}·\cos L\lambda'·\frac{\dd v}{\dd\phi} = 0
+\]
+since $\lambda'$~evidently lies on the great circle whose pole is~$L$. From this we see that
+$S$~is independent of~$r$, and is, therefore, a function of $\phi$~alone. But for $r = 0$ we evidently
+have $v = 0$, consequently $\dfrac{\dd v}{\dd\phi} = 0$, and $S = 0$ independently of~$\phi$. Thus, in general,
+we have necessarily $S = 0$, and so $\cos\lambda\lambda' = 0$, \ie, $\lambda\lambda' = 90°$. From this follows the
+
+\begin{Theorem}
+If on a curved surface an infinite number of shortest lines of equal length
+be drawn from the same initial point, the lines joining their extremities will be normal to
+each of the lines.
+\end{Theorem}
+
+We have thought it worth while to deduce this theorem from the fundamental
+property of shortest lines; but the truth of the theorem can be made apparent without
+any calculation by means of the following reasoning. Let $AB$,~$AB'$ be two
+shortest lines of the same length including at~$A$ an infinitely small angle, and let us
+suppose that one of the angles made by the element~$BB'$ with the lines $BA$,~$B'A$
+differs from a right angle by a finite quantity. Then, by the law of continuity, one
+will be greater and the other less than a right angle. Suppose the angle at~$B$ is
+equal to~$90° - \omega$, and take on the line~$AB$ a point~$C$, such that
+\[
+BC = BB'·\cosec \omega.
+\]
+Then, since the infinitely small triangle~$BB'C$ may be regarded as plane, we shall have
+\[
+CB' = BC·\cos \omega,
+\]
+\PageSep{25}
+and consequently
+\[
+AC + CB' = AC + BC·\cos \omega
+  = AB - BC·(1- \cos \omega)
+  = AB' - BC·(1 - \cos \omega),
+\]
+\ie, the path from $A$~to~$B'$ through the point~$C$ is shorter than the shortest line,
+\QEA
+
+
+\Article{16.}
+
+%[** TN: In-line theorem, marked non-semantically]
+With the theorem of the preceding article we associate another, which we state
+as follows: \textit{If on a curved surface we imagine any line whatever, from the different points
+of which are drawn at right angles and toward the same side an infinite number of shortest
+lines of the same length, the curve which joins their other extremities will cut each of the
+lines at right angles.} For the demonstration of this theorem no change need be made
+in the preceding analysis, except that $\phi$~must denote the length of the \emph{given} curve
+measured from an arbitrary point; or rather, a function of this length. Thus all of
+the reasoning will hold here also, with this modification, that $S = 0$ for $r = 0$ is
+now implied in the hypothesis itself. Moreover, this theorem is more general than
+the preceding one, for we can regard it as including the first one if we take for the
+given line the infinitely small circle described about the centre~$A$. Finally, we may
+say that here also geometric considerations may take the place of the analysis, which,
+however, we shall not take the time to consider here, since they are sufficiently
+obvious.
+
+
+\Article{17.}
+
+We return to the formula
+\[
+\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}},
+\]
+which expresses generally the magnitude of a linear element on the curved surface,
+and investigate, first of all, the geometric meaning of the coefficients $E$,~$F$,~$G$. We
+have already said in \Art{5} that two systems of lines may be supposed to lie on the
+curved surface, $p$~being variable, $q$~constant along each of the lines of the one system;
+and $q$~variable, $p$~constant along each of the lines of the other system. Any point
+whatever on the surface can be regarded as the intersection of a line of the first
+system with a line of the second; and then the element of the first line adjacent to
+this point and corresponding to a variation~$dp$ will be equal to~$\sqrt{E}·dp$, and the
+element of the second line corresponding to the variation~$dq$ will be equal to~$\sqrt{G}·dq$.
+Finally, denoting by~$\omega$ the angle between these elements, it is easily seen that we
+shall have
+\[
+\cos \omega = \frac{F}{\sqrt{EG}}.
+\]
+\PageSep{26}
+Furthermore, the area of the surface element in the form of a parallelogram between
+the two lines of the first system, to which correspond $q$,~$q + dq$, and the two lines of
+the second system, to which correspond $p$,~$p + dp$, will be
+\[
+\Sqrt{EG - F^{2}}\, dp·dq.
+\]
+
+Any line whatever on the curved surface belonging to neither of the two systems
+is determined when $p$~and~$q$ are supposed to be functions of a new variable, or
+one of them is supposed to be a function of the other. Let $s$~be the length of such
+a curve, measured from an arbitrary initial point, and in either direction chosen as
+positive. Let $\theta$~denote the angle which the element
+\[
+ds = \Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}
+\]
+makes with the line of the first system drawn through the initial point of the element,
+and, in order that no ambiguity may arise, let us suppose that this angle is
+measured from that branch of the first line on which the values of~$p$ increase, and is
+taken as positive toward that side toward which the values of~$q$ increase. These conventions
+being made, it is easily seen that
+\begin{align*}
+\cos \theta·ds &= \sqrt{E}·dp + \sqrt{G}·\cos \omega·dq
+  = \frac{E\, dp + F\, dq}{\sqrt{E}}\Add{,} \\
+\sin \theta·ds &= \sqrt{G}·\sin \omega·dq
+  = \frac{\sqrt{(EG - F^{2})}·dq}{\sqrt{E}}\Add{.}
+\end{align*}
+
+
+\Article{18.}
+
+We shall now investigate the condition that this line be a shortest line. Since
+its length~$s$ is expressed by the integral
+\[
+s = \int \Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}\Add{,}
+\]
+the condition for a minimum requires that the variation of this integral arising from
+an infinitely small change in the position become equal to zero. The calculation, for
+our purpose, is more simply made in this case, if we regard $p$ as a function of~$q$.
+When this is done, if the variation is denoted by the characteristic~$\delta$, we have
+\begin{align*}
+\delta s &= \int \frac{\left(
+    \dfrac{\dd E}{\dd p}·dp^{2}
+  + 2\dfrac{\dd F}{\dd p}·dp·dq
+  + \dfrac{\dd G}{\dd p}·dq^{2}
+  \right) \delta p
+  + (2E\, dp + 2F\, dq)\, d\, \delta p}{2\, ds} \displaybreak[1] \\
+  &= \frac{E\, dp + F\, dq}{ds}·\delta p \\
+\PageSep{27}
+  &\qquad+ \int \delta p \left(\frac{
+    \dfrac{\dd E}{\dd p}·dp^{2}
+  + 2\dfrac{\dd F}{\dd p}·dp·dq
+  + \dfrac{\dd G}{\dd p}·dq^{2}}{2\, ds}
+  - d·\frac{E\, dp + F\, dq}{ds}\right)
+\end{align*}
+and we know that what is included under the integral sign must vanish independently
+of~$\delta p$. Thus we have
+\begin{multline*}
+\frac{\dd E}{\dd p}·dp^{2}
+  + 2\frac{\dd F}{\dd p}·dp·dq
+  + \frac{\dd G}{\dd p}·dq^{2}
+  = 2\, ds·d·\frac{E\, dp + F\, dq}{ds} \\
+\begin{aligned}
+&= 2\, ds·d·\left(\sqrt{E}·\cos\theta\right) \\ %[** TN: Added parentheses]
+&= \frac{ds·dE·\cos\theta}{\sqrt{E}}
+ - 2\, ds·d\theta·\sqrt{E}·\sin\theta \\
+%** Translator's note from corrigenda: The original and the Latin reprints **
+%** lack the factor 2; the correction is made in all the translations.     **
+&= \frac{(E\, dp + F\, dq)\, dE}{E} - \Erratum{}{2}\Sqrt{EG - F^{2}}·\Erratum{dp}{dq}·d\theta \\
+&= \left(\frac{E\, dp + F\, dq}{E}\right)
+ ·\left(\frac{\dd E}{\dd p}·dp + \frac{\dd E}{\dd q}·dq\right)
+  - 2\Sqrt{EG - F^{2}}·dq·d\theta\Add{.}
+\end{aligned}
+\end{multline*}
+This gives the following conditional equation for a shortest line:
+\begin{multline*}
+\Sqrt{EG - F^{2}}·d\theta
+  = \frac{1}{2}·\frac{F}{E}·\frac{\dd E}{\dd p}·dp
+  + \frac{1}{2}·\frac{F}{E}·\frac{\dd E}{\dd q}·dq
+  + \frac{1}{2}·\frac{\dd E}{\dd q}·dp \\
+  - \frac{\dd F}{\dd p}·dp - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{,}
+\end{multline*}
+which can also be written
+\[
+\Sqrt{EG - F^{2}}·d\theta
+  = \frac{1}{2}·\frac{F}{E}·dE
+  + \frac{1}{2}·\frac{\dd E}{\dd q}·dp
+  - \frac{\dd F}{\dd p}·dp
+  - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{.}
+\]
+From this equation, by means of the equation
+\[
+\cot\theta = \frac{E}{\Sqrt{EG - F^{2}}}·\frac{dp}{dq}
+  + \frac{F}{\Sqrt{EG - F^{2}}}\Add{,}
+\]
+it is also possible to eliminate the angle~$\theta$, and to derive a differential equation of
+the second order between $p$~and~$q$, which, however, would become more complicated
+and less useful for applications than the preceding.
+
+
+\Article{19.}
+
+The general formulæ, which we have derived in \Arts{11}{18} for the measure of
+curvature and the variation in the direction of a shortest line, become much simpler
+if the quantities $p$,~$q$ are so chosen that the lines of the first system cut everywhere
+\PageSep{28}
+orthogonally the lines of the second system; \ie, in such a way that we have generally
+$\omega = 90°$, or $F = 0$. Then the formula for the measure of curvature becomes
+\[
+4E^{2}G^{2}k
+  = E·\frac{\dd E}{\dd q}·\frac{\dd G}{\dd q}
+  + E\left(\frac{\dd G}{\dd p}\right)^{2}
+  + G·\frac{\dd E}{\dd p}·\frac{\dd G}{\dd p}
+  + G\left(\frac{\dd E}{\dd q}\right)^{2}
+  - 2EG\left(\frac{\dd^{2} E}{\dd q^{2}} + \frac{\dd^{2} G}{\dd p^{2}}\right),
+\]
+and for the variation of the angle~$\theta$
+\[
+\sqrt{EG}·d\theta
+  = \frac{1}{2}·\frac{\dd E}{\dd q}·dp
+  - \frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{.}
+\]
+
+Among the various cases in which we have this condition of orthogonality, the
+most important is that in which all the lines of one of the two systems, \eg, the
+first, are shortest lines. Here for a constant value of~$q$ the angle~$\theta$ becomes equal to
+zero, and therefore the equation for the variation of~$\theta$ just given shows that we must
+have $\dfrac{\dd E}{\dd q} = 0$, or that the coefficient~$E$ must be independent of~$\Erratum{g}{q}$; \ie, $E$~must be
+either a constant or a function of $p$~alone. It will be simplest to take for~$p$
+the length of each line of the first system, which length, when all the lines of the
+first system meet in a point, is to be measured from this point, or, if there is no
+common intersection, from any line whatever of the second system. Having made
+these conventions, it is evident that $p$~and~$q$ denote now the same quantities that
+were expressed in \Arts{15}{16} by $r$~and~$\phi$, and that $E = 1$. Thus the two preceding
+formulæ become:
+\begin{align*}
+4G^{2}k
+  &= \left(\frac{\dd G}{\dd p}\right)^{2} - 2G\, \frac{\dd^{2} G}{\dd p^{2}}\Add{,} \\
+\sqrt{G}·d\theta
+  &= -\frac{1}{2}·\frac{\dd G}{\dd p}·dq\Add{;}
+\end{align*}
+or, setting $\sqrt{G} = m$,
+\[
+k = -\frac{1}{m}·\frac{\dd^{2} m}{\dd p^{2}},\quad
+d\theta = -\frac{\dd m}{\dd p}·dq\Add{.}
+\]
+Generally speaking, $m$~will be a function of $p$,~$q$, and $m\, dq$~the expression for the element
+of any line whatever of the second system. But in the particular case where
+all the lines~$p$ go out from the same point, evidently we must have $m = 0$ for $p = 0$.
+Furthermore, in the case under discussion we will take for~$q$ the angle itself which
+the first element of any line whatever of the first system makes with the element of
+any one of the lines chosen arbitrarily. Then, since for an infinitely small value of~$p$
+the element of a line of the second system (which can be regarded as a circle
+described with radius~$p$) is equal to~$p\, dq$, we shall have for an infinitely small value
+of~$p$, $m = p$, and consequently, for $p = 0$, $m = 0$ at the same time, and $\dfrac{\dd m}{\dd p} = 1$.
+\PageSep{29}
+
+
+\Article{20.}
+
+We pause to investigate the case in which we suppose that $p$~denotes in a general
+manner the length of the shortest line drawn from a fixed point~$A$ to any other
+point whatever of the surface, and $q$~the angle that the first element of this line
+makes with the first element of another given shortest line going out from~$A$. Let
+$B$ be a definite point in the latter line, for which $q = 0$, and $C$~another definite point
+of the surface, at which we denote the value of~$q$ simply by~$A$. Let us suppose the
+points $B$,~$C$ joined by a shortest line, the parts of which, measured from~$B$, we denote
+in a general way, as in \Art{18}, by~$s$; and, as in the same article, let us denote by~$\theta$
+the angle which any element~$ds$ makes with the element~$dp$; finally, let us denote
+by $\theta°$,~$\theta'$ the values of the angle~$\theta$ at the points $B$,~$C$. We have thus on the curved
+surface a triangle formed by shortest lines. The angles of this triangle at $B$~and~$C$
+we shall denote simply by the same letters, and $B$~will be equal to~$180° - \theta$, $C$~to $\theta'$~itself.
+But, since it is easily seen from our analysis that all the angles are supposed
+to be expressed, not in degrees, but by numbers, in such a way that the angle $57°\, 17'\, 45''$,
+to which corresponds an arc equal to the radius, is taken for the unit, we must set
+\[
+\theta° = \pi - B,\quad \theta' = C\Add{,}
+\]
+where $2\pi$~denotes the circumference of the sphere. Let us now examine the integral
+curvature of this triangle, which is equal to
+\[
+\int k\, d\sigma,
+\]
+$d\sigma$~denoting a surface element of the triangle. Wherefore, since this element is expressed
+by~$m\, dp·dq$, we must extend the integral
+\[
+\iint \Typo{}{k}m\, dp·dq
+\]
+over the whole surface of the triangle. Let us begin by integration with respect to~$p$,
+which, because
+\[
+k = -\frac{1}{m}·\frac{\dd^{2} m}{\dd p^{2}},
+\]
+gives
+\[
+dq·\left(\text{const.} - \frac{\dd m}{\dd p}\right),
+\]
+for the integral curvature of the area lying between the lines of the first system, to
+which correspond the values $q$,~$q + dq$ of the second indeterminate. Since this integral
+\PageSep{30}
+curvature must vanish for $p = 0$, the constant introduced by integration must be
+equal to the value of~$\dfrac{\dd m}{\dd q}$ for $p = 0$, \ie,~equal to unity. Thus we have
+\[
+dq\left(1 - \frac{\dd m}{\dd p}\right),
+\]
+where for $\dfrac{\dd m}{\dd p}$ must be taken the value corresponding to the end of this area on the
+line~$CB$. But on this line we have, by the preceding article,
+\[
+\frac{\dd m}{\dd q}·dq = -d\theta,
+\]
+whence our expression is changed into $dq + d\theta$. Now by a second integration, taken
+from $q = 0$ to $q = A$, we obtain for the integral curvature
+\[
+A + \theta'- \theta°,
+\]
+or
+\[
+A + B + C - \pi.
+\]
+
+The integral curvature is equal to the area of that part of the sphere which corresponds
+to the triangle, taken with the positive or negative sign according as the
+curved surface on which the triangle lies is concavo-concave or concavo-convex. For
+unit area will be taken the square whose side is equal to unity (the radius of the
+sphere), and then the whole surface of the sphere becomes equal to~$4\pi$. Thus the
+part of the surface of the sphere corresponding to the triangle is to the whole surface
+of the sphere as $±(A + B + C - \pi)$ is to~$4\pi$. This theorem, which, if we mistake
+not, ought to be counted among the most elegant in the theory of curved surfaces,
+may also be stated as follows:
+
+\begin{Theorem}[]
+The excess over~$180°$ of the sum of the angles of a triangle formed by shortest lines
+on a concavo-concave curved surface, or the deficit from~$180°$ of the sum of the angles of
+a triangle formed hy shortest lines on a concavo-convex curved surface, is measured by the
+area of the part of the sphere which corresponds, through the directions of the normals, to
+that triangle, if the whole surface of the sphere is set equal to $720$~degrees.
+\end{Theorem}
+
+More generally, in any polygon whatever of $n$~sides, each formed by a shortest
+line, the excess of the sum of the angles over $(2n - 4)$~right angles, or the deficit from
+$(2n - 4)$~right angles (according to the nature of the curved surface), is equal to the
+area of the corresponding polygon on the sphere, if the whole surface of the sphere is
+set equal to $720$~degrees. This follows at once from the preceding theorem by dividing
+the polygon into triangles.
+\PageSep{31}
+
+
+\Article{21.}
+
+Let us again give to the symbols $p$,~$q$, $E$,~$F$,~$G$, $\omega$ the general meanings which
+were given to them above, and let us further suppose that the nature of the curved
+surface is defined in a similar way by two other variables, $p'$,~$q'$, in which case the
+general linear element is expressed by
+\[
+\Sqrt{E'\, dp'^{2} + 2F'\, dp'·dq' + G'\, dq'^{2}}\Add{.}
+\]
+Thus to any point whatever lying on the surface and defined by definite values of
+the variables $p$,~$q$ will correspond definite values of the variables $p'$,~$q'$, which will
+therefore be functions of $p$,~$q$. Let us suppose we obtain by differentiating them
+\begin{alignat*}{2}
+dp' &= \alpha\, dp &&+ \beta\,  dq\Add{,} \\
+dq' &= \gamma\, dp &&+ \delta\, dq\Add{.}
+\end{alignat*}
+We shall now investigate the geometric meaning of the coefficients $\alpha$,~$\beta$,~$\gamma$,~$\delta$.
+
+Now \emph{four} systems of lines may thus be supposed to lie upon the curved surface,
+for which $p$,~$q$, $p'$,~$q'$ respectively are constants. If through the definite point to
+which correspond the values $p$,~$q$, $p'$,~$q'$ of the variables we suppose the four lines
+belonging to these different systems to be drawn, the elements of these lines, corresponding
+to the positive increments $dp$,~$dq$, $dp'$,~$dq'$, will be
+\[
+\sqrt{E}·dp,\quad
+\sqrt{G}·dq,\quad
+\sqrt{E'}·dp',\quad
+\sqrt{G'}·dq'.
+\]
+The angles which the directions of these elements make with an arbitrary fixed direction
+we shall denote by $M$,~$N$, $M'$,~$N'$, measuring them in the sense in which the
+second is placed with respect to the first, so that $\sin(N - M)$ is positive. Let us
+suppose (which is permissible) that the fourth is placed in the same sense with respect
+to the third, so that $\sin(N' - M')$ also is positive. Having made these conventions,
+if we consider another point at an infinitely small distance from the first point, and
+to which correspond the values $p + dp$, $q + dq$, $p' + dp'$, $q' + dq'$ of the variables, we
+see without much difficulty that we shall have generally, \ie, independently of the
+values of the increments $dp$,~$dq$, $dp'$,~$dq'$,
+\[
+\sqrt{E}·dp·\sin M + \sqrt{G}·dq·\sin N
+ = \sqrt{E'}·dp'·\sin M' + \sqrt{G'}·dq'·\sin N'\Add{,}
+\]
+since each of these expressions is merely the distance of the new point from the line
+from which the angles of the directions begin. But we have, by the notation introduced
+above,
+\[
+N - M = \omega.
+\]
+In like manner we set
+\[
+N' - M' = \omega',
+\]
+\PageSep{32}
+and also
+\[
+N - M' = \psi.
+\]
+Then the equation just found can be thrown into the following form:
+\begin{multline*}
+\sqrt{E}·dp · \sin(M' - \omega + \psi) + \sqrt{G}·dq·\sin(M' + \psi) \\
+= \sqrt{E'}·dp'·\sin M' + \sqrt{G'}·dq'·\sin(M' + \omega')\Add{,}
+\end{multline*}
+or
+\begin{multline*}
+\sqrt{E}·dp·\sin(N' - \omega - \omega' + \psi)
+  - \sqrt{G}·dq·\sin(N' - \omega' + \psi) \\
+  = \sqrt{E'}·dp'·\sin(N' - \omega') + \sqrt{G'}·dq'·\sin N'\Add{.}
+\end{multline*}
+And since the equation evidently must be independent of the initial direction, this
+direction can be chosen arbitrarily. Then, setting in the second formula $N' = 0$, or in
+the first $M' = 0$, we obtain the following equations:
+\begin{align*}
+\sqrt{E'}·\sin \omega'·dp'
+  &= \sqrt{E}·\sin(\omega + \omega' - \psi)·dp
+   + \sqrt{G}·\sin(\omega' - \psi)·dq\Add{,} \\
+\sqrt{G'}·\sin \omega'·dq'
+  &= \sqrt{E}·\sin(\psi - \omega)·dp + \sqrt{G}·\sin\psi·dq\Add{;}
+\end{align*}
+and these equations, since they must be identical with
+\begin{alignat*}{2}
+dp' &= \alpha\, dp &&+ \beta\,  dq\Add{,} \\
+dq' &= \gamma\, dp &&+ \delta\, dq\Add{,}
+\end{alignat*}
+determine the coefficients $\alpha$,~$\beta$,~$\gamma$,~$\delta$. We shall have
+\begin{align*}
+\alpha &= \sqrt{\frac{E}{E'}}·\frac{\sin(\omega + \omega' - \psi)}{\sin\omega'}, &
+\beta  &= \sqrt{\frac{G}{E'}}·\frac{\sin(\omega' - \psi)}{\sin\omega'}\Add{,} \\
+\gamma &= \sqrt{\frac{E}{G'}}·\frac{\sin(\psi - \omega)}{\sin\omega'}, &
+\delta &= \sqrt{\frac{G}{G'}}·\frac{\sin\psi}{\sin\omega'}\Add{.}
+\end{align*}
+These four equations, taken in connection with the equations
+\begin{align*}
+\cos\omega  &= \frac{F}{\sqrt{EG}}, &
+\cos\omega' &= \frac{F'}{\sqrt{E'G'}}, \\
+\sin\omega  &= \sqrt{\frac{EG - F^{2}}{EG}}, &
+\sin\omega' &= \sqrt{\frac{E'G' - F'^{2}}{E'G'}},
+\end{align*}
+may be written
+\begin{align*}
+\alpha\Sqrt{E'G' - F'^{2}} &= \sqrt{EG'}·\sin(\omega + \omega' - \psi)\Add{,} \\
+\beta \Sqrt{E'G' - F'^{2}} &= \sqrt{GG'}·\sin(\omega' - \psi)\Add{,} \\
+\gamma\Sqrt{E'G' - F'^{2}} &= \sqrt{EE'}·\sin(\psi - \omega)\Add{,} \\
+\delta\Sqrt{E'G' - F'^{2}} &= \sqrt{GE'}·\sin \psi\Add{.}
+\end{align*}
+
+Since by the substitutions
+\begin{alignat*}{2}
+dp' &= \alpha\, dp &&+ \beta\,  dq, \\
+dq' &= \gamma\, dp &&+ \delta\, dq\Add{,}
+\end{alignat*}
+\PageSep{33}
+the trinomial
+\[
+E'\, dp'^{2} + 2F'\, dp'·dq' + G'\, dq'^{2}
+\]
+is transformed into
+\[
+E\, dp^{2} + 2F\, dp·dq + G\, dq^{2},
+\]
+we easily obtain
+\[
+EG - F^{2} = (E'G' - F'^{2})(\alpha\delta - \beta\gamma)^{2}\Add{;}
+\]
+and since, \textit{vice versa}, the latter trinomial must be transformed into the former by the
+substitution
+\[
+(\alpha\delta - \beta\gamma)\, dp =  \delta\, dp' - \beta\, dq',\quad
+(\alpha\delta - \beta\gamma)\, dq = -\gamma\, dp' + \alpha\, dq',
+\]
+we find\Note{16}
+\begin{align*}
+E\delta^{2} - 2F\gamma\delta + G\gamma^{2}
+  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·E'\Add{,} \\
+%
+-E\beta\delta + F(\alpha\delta + \beta\gamma) - G\alpha\gamma
+  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·F'\Add{,}\NoteMark \\
+%
+E\beta^{2} - 2F\alpha\beta + G\alpha^{2}
+  &= \frac{EG - F^{2}}{E'G' - F'^{2}}·G'\Add{.}
+\end{align*}
+
+
+\Article{22.}
+
+From the general discussion of the preceding article we proceed to the very
+extended application in which, while keeping for $p$,~$q$ their most general meaning, we
+take for $p'$,~$q'$ the quantities denoted in \Art{15} by $r$,~$\phi$. We shall use $r$,~$\phi$ here
+also in such a way that, for any point whatever on the surface, $r$~will be the shortest
+distance from a fixed point, and $\phi$~the angle at this point between the first element
+of~$r$ and a fixed direction. We have thus
+\[
+E' = 1,\quad
+F' = 0,\quad
+\omega' = 90°.
+\]
+Let us set also
+\[
+\sqrt{G'} = m,
+\]
+so that any linear element whatever becomes equal to
+\[
+\Sqrt{dr^{2} + m^{2}\, d\phi^{2}}.
+\]
+Consequently, the four equations deduced in the preceding article for $\alpha$,~$\beta$,~$\gamma$,~$\delta$ give
+\begin{align*}
+\sqrt{E}·\cos(\omega - \psi) = \frac{\dd r}{\dd p}\Add{,}
+\Tag{(1)} \\
+\sqrt{G}·\cos \psi = \frac{\dd r}{\dd q}\Add{,}
+\Tag{(2)} \displaybreak[1] \\
+\PageSep{34}
+\sqrt{E}·\sin(\psi - \omega) = m·\frac{\dd\phi}{\dd p}\Add{,}
+\Tag{(3)} \\
+\sqrt{G}·\sin\psi = m·\frac{\dd\phi}{\dd q}\Add{.}
+\Tag{(4)}
+\end{align*}
+But the last and the next to the last equations of the preceding article give
+\begin{gather*}
+EG - F^{2}
+  = E\left(\frac{\dd r}{\dd q}\right)^{2}
+  - 2F·\frac{\dd r}{\dd p}·\frac{\dd r}{\dd q}
+  + G\left(\frac{\dd r}{\dd p}\right)^{2}\Add{,}
+\Tag{(5)} \\
+\left(E·\frac{\dd r}{\dd q} - F·\frac{\dd r}{\dd p}\right)·\frac{\dd\phi}{\dd q}
+  = \left(F·\frac{\dd r}{\dd q} - G·\frac{\dd r}{\dd p}\right)·\frac{\dd\phi}{\dd p}\Add{.}
+\Tag{(6)}
+\end{gather*}
+
+From these equations must be determined the quantities $r$,~$\phi$,~$\psi$ and (if need be)~$m$,
+as functions of $p$~and~$q$. Indeed, integration of equation~(5) will give~$r$; $r$~being
+found, integration of equation~(6) will give~$\phi$; and one or other of equations (1),~(2)
+will give $\psi$~itself. Finally, $m$~is obtained from one or other of equations (3),~(4).
+
+The general integration of equations (5),~(6) must necessarily introduce two arbitrary
+functions. We shall easily understand what their meaning is, if we remember
+that these equations are not limited to the case we are here considering, but are
+equally valid if $r$~and~$\phi$ are taken in the more general sense of \Art{16}, so that $r$~is
+the length of the shortest line drawn normal to a fixed but arbitrary line, and $\phi$~is
+an arbitrary function of the length of that part of the fixed line which is intercepted
+between any shortest line and an arbitrary fixed point. The general solution must
+embrace all this in a general way, and the arbitrary functions must go over into
+definite functions only when the arbitrary line and the arbitrary functions of its
+parts, which $\phi$~must represent, are themselves defined. In our case an infinitely
+small circle may be taken, having its centre at the point from which the distances~$r$
+are measured, and $\phi$~will denote the parts themselves of this circle, divided by the
+radius. Whence it is easily seen that the equations (5),~(6) are quite sufficient for
+our case, provided that the functions which they leave undefined satisfy the condition
+which $r$~and~$\phi$ satisfy for the initial point and for points at an infinitely small
+distance from this point.
+
+Moreover, in regard to the integration itself of the equations (5),~(6), we know
+that it can be reduced to the integration of ordinary differential equations, which, however,
+often happen to be so complicated that there is little to be gained by the reduction.
+On the contrary, the development in series, which are abundantly sufficient for
+practical requirements, when only a finite portion of the surface is under consideration,
+presents no difficulty; and the formulæ thus derived open a fruitful source for
+\PageSep{35}
+the solution of many important problems. But here we shall develop only a single
+example in order to show the nature of the method.
+
+
+\Article{23.}
+
+We shall now consider the case where all the lines for which $p$~is constant are
+shortest lines cutting orthogonally the line for which $\phi = 0$, which line we can regard
+as the axis of abscissas. Let $A$~be the point for which $r = 0$, $D$~any point whatever
+on the axis of abscissas, $AD = p$, $B$~any point whatever on the shortest line normal
+to~$AD$ at~$D$, and $BD = q$, so that $p$~can be regarded as the abscissa, $q$~the ordinate
+of the point~$B$. The abscissas we assume positive on the branch of the axis of
+abscissas to which $\phi = 0$ corresponds, while we always regard~$r$ as positive. We take
+the ordinates positive in the region in which $\phi$~is measured between $0$~and~$180°$.
+
+By the theorem of \Art{16} we shall have
+\[
+\omega = 90°,\quad
+F = 0,\quad
+G = 1,
+\]
+and we shall set also
+\[
+\sqrt{E} = n.
+\]
+Thus $n$~will be a function of $p$,~$q$, such that for $q = 0$ it must become equal to unity.
+The application of the formula of \Art{18} to our case shows that on any shortest
+line \emph{whatever} we must have\Note{17}
+\[
+d\theta = \frac{\dd n}{\dd q}·dp,\NoteMark
+\]
+where $\theta$~denotes the angle between the element of this line and the element of the
+line for which $q$~is constant. Now since the axis of abscissas is itself a shortest line,
+and since, for it, we have everywhere $\theta = 0$, we see that for $q = 0$ we must have
+everywhere
+\[
+\frac{\dd n}{\dd q} = 0.
+\]
+Therefore we conclude that, if $n$~is developed into a series in ascending powers of~$q$,
+this series must have the following form:
+\[
+n = 1 + fq^{2} + gq^{3} + hq^{4} + \text{etc.}\Add{,}
+\]
+where $f$,~$g$,~$h$,~etc., will be functions of~$p$, and we set
+\begin{alignat*}{4}
+f &= f° &&+ f'p &&+ f''p^{2} &&+ \text{etc.}\Add{,} \\
+g &= g° &&+ g'p &&+ g''p^{2} &&+ \text{etc.}\Add{,} \\
+h &= h° &&+ h'p &&+ h''p^{2} &&+ \text{etc.}\Add{,}
+\end{alignat*}
+\PageSep{36}
+or
+\begin{alignat*}{2}
+n = 1 + f°q^{2} &+ f'pq^{2}   &&+ f''p^{2}q^{2} + \text{etc.} \\
+                  &+ g° q^{3} &&+ g'pq^{3} + \text{etc.} \\
+                  &             &&+ h°q^{4} + \text{etc.\ etc.}
+\end{alignat*}
+
+
+\Article{24.}
+
+The equations of \Art{22} give, in our case,
+\begin{gather*}
+ n\sin\psi = \frac{\dd r}{\dd p},\quad
+  \cos\psi = \frac{\dd r}{\dd q},\quad
+-n\cos\psi = m·\frac{\dd\phi}{\dd r},\quad
+  \sin\psi = m·\frac{\dd\phi}{\dd q}, \\
+%
+n^{2} = n^{2}\left(\frac{\dd r}{\dd q}\right)^{2}
+          + \left(\frac{\dd r}{\dd p}\right)^{2},\quad
+n^{2}·\frac{\dd r}{\dd q}·\frac{\dd\phi}{\dd q}
+      + \frac{\dd r}{\dd p}·\frac{\dd\phi}{\dd p} = 0\Add{.}
+\end{gather*}
+By the aid of these equations, the fifth and sixth of which are contained in the others,
+series can be developed for $r$,~$\phi$,~$\psi$,~$m$, or for any functions whatever of these quantities.
+We are going to establish here those series that are especially worthy of
+attention.
+
+Since for infinitely small values of $p$,~$q$ we must have
+\[
+r^{2} = p^{2} + q^{2},
+\]
+the series for~$r^{2}$ will begin with the terms $p^{2} + q^{2}$. We obtain the terms of higher
+order by the method of undetermined coefficients,\footnote
+  {We have thought it useless to give the calculation here, which can be somewhat abridged by
+  certain artifices.}
+by means of the equation
+\[
+\left(\frac{1}{n}·\frac{\dd(r^{2})}{\dd p}\right)^{2}
+  + \left(\frac{\dd(r^{2})}{\dd q}\right)^{2} = 4r^{2}\Add{.}
+\]
+Thus we have\Note{18}
+\begin{alignat*}{3}
+\Tag{[1]}
+r^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q^{2} &&+ \tfrac{1}{2}f'p^{3}q^{2} &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2}\quad\text{etc.} \\
+      &+ q^{2} &&+ \tfrac{1}{2}g°p^{2}q^{3} &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
+      &&& &&+(\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}\Add{.}
+\end{alignat*}
+
+Then we have, from the formula\Note{19}
+\[
+r\sin\psi = \frac{1}{2n}·\frac{\dd(r^{2})}{\dd p},
+\]
+\begin{alignat*}{2}
+\Tag{[2]}
+r\sin\psi = p - \tfrac{1}{3}f°pq^{2}
+  & -\tfrac{1}{4}f'p^{2}q^{2} &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\quad\text{etc.} \\
+  & -\tfrac{1}{2}g°pq^{3} &&-\tfrac{2}{5}g'p^{2}q^{3} \\
+  &&& -(\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})pq^{4}\Add{;}
+\end{alignat*}
+\PageSep{37}
+and from the formula\Note{20}
+\[
+r\cos\psi = \tfrac{1}{2}\, \frac{\dd(r^{2})}{\dd q}\Add{,}
+\]
+\begin{alignat*}{2}
+\Tag{[3]}
+r\cos\psi = q + \tfrac{2}{3}f°p^{2}q
+  & +\tfrac{1}{2}f'p^{3}q &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\quad\text{etc.} \\
+  & +\tfrac{3}{4}g°p^{2}q^{2} &&+\tfrac{3}{5}g'p^{3}q^{2} \\
+  &&& +(\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\Add{.}
+\end{alignat*}
+These formulæ give the angle~$\psi$. In like manner, for the calculation of the angle~$\phi$,
+series for $r\cos\phi$ and $r\sin\phi$ are very elegantly developed by means of the partial
+differential equations
+\begin{align*}
+&\frac{\dd·r\cos\phi}{\dd p}
+  = n\cos\phi·\sin\psi - r\sin\phi·\frac{\dd\phi}{\dd p}\Add{,} \\
+&\frac{\dd·r\cos\phi}{\dd q}
+  = \Z\cos\phi·\cos\psi - r\sin\phi·\frac{\dd\phi}{\dd q}\Add{,} \\
+&\frac{\dd·r\sin\phi}{\dd p}
+  = n\sin\phi·\sin\psi + r\cos\phi·\frac{\dd\phi}{\dd p}\Add{,} \\
+&\frac{\dd·r\sin\phi}{\dd q}
+  = \Z\sin\phi·\cos\psi + r\cos\phi·\frac{\dd\phi}{\dd q}\Add{,} \\
+&n\cos\psi·\frac{\dd\phi}{\dd q} + \sin\psi·\frac{\dd\phi}{\dd p} = 0\Add{.}
+\end{align*}
+A combination of these equations gives
+\begin{alignat*}{3}
+&\frac{r\sin\psi}{n}·\frac{\dd·r\cos\phi}{\dd p}
+  &&+ r\cos\psi·\frac{\dd·r\cos\phi}{\dd q}
+  &&= r\cos\phi\Add{,} \\
+&\frac{r\sin\psi}{n}·\frac{\dd·r\sin\phi}{\dd p}
+  &&+ r\cos\psi·\frac{\dd·r\sin\phi}{\dd q}
+  &&= r\sin\phi\Add{.}
+\end{alignat*}
+From these two equations series for $r\cos\phi$, $r\sin\phi$ are easily developed, whose first
+terms must evidently be $p$,~$q$ respectively. The series are\Note{21}
+\begin{alignat*}{3}
+\Tag{[4]}
+r\cos\phi &= p + \tfrac{2}{3}f°pq^{2}
+  &&+ \tfrac{5}{12}f'p^{2}q^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\quad\text{etc.} \\
+ &&&+ \tfrac{1}{2}g°pq^{3}
+  &&+ \tfrac{7}{20}g'p^{2}q^{3} \\
+&&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}\Add{,} \displaybreak[1] \\
+\Tag{[5]}
+r\sin\phi &= q - \tfrac{1}{3}f°p^{2}q
+  &&- \tfrac{1}{6}f'p^{3}q
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q\quad\text{etc.} \\
+ &&&- \tfrac{1}{4}g°p^{2}q^{2}
+  &&- \tfrac{3}{20}g'p^{3}q^{2} \\
+&&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}\Add{.}
+\end{alignat*}
+From a combination of equations [2],~[3],~[4],~[5] a series for $r^{2}\cos(\psi + \phi)$, may
+be derived, and from this, dividing by the series~[1], a series for $\cos(\psi + \phi)$, from
+\PageSep{38}
+which may be found a series for the angle $\psi + \phi$ itself. However, the same series
+can be obtained more elegantly in the following manner. By differentiating the first
+and second of the equations introduced at the beginning of this article, we obtain
+\[
+\sin\psi·\frac{\dd n}{\dd q}
+  + n\cos\psi·\frac{\dd\psi}{\dd q}
+  +  \sin\psi·\frac{\dd\psi}{\dd p} = 0\Add{,}
+\]
+and this combined with the equation
+\[
+n\cos\psi·\frac{\dd\phi}{\dd q} + \sin\psi·\frac{\dd\phi}{\dd p} = 0
+\]
+gives
+\[
+\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q}
+  + \frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(\psi + \phi)}{\dd q} = 0\Add{.}
+\]
+From this equation, by aid of the method of undetermined coefficients, we can easily
+derive the series for $\psi + \phi$, if we observe that its first term must be~$\frac{1}{2}\pi$, the radius
+being taken equal to unity and $2\pi$~denoting the circumference of the circle,\Note{22}
+\begin{alignat*}{2}
+\Tag{[6]}
+\psi + \phi = \tfrac{1}{2}\pi - f°pq
+  &- \tfrac{2}{3}f'p^{2}q
+  &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q\quad\text{etc.} \\
+  &- g°pq^{2}
+  &&- \tfrac{3}{4}g'p^{2}q^{2} \\
+  &&&- (h° - \tfrac{1}{3}{f°}^{2})pq^{3}\Add{.}
+\end{alignat*}
+
+It seems worth while also to develop the area of the triangle~$ABD$ into a series.
+For this development we may use the following conditional equation, which is easily
+derived from sufficiently obvious geometric considerations, and in which $S$~denotes the
+required area:\Note{23}
+\[
+\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p}
+  + r\cos\psi·\frac{\dd S}{\dd q}
+  = \frac{r\sin\psi}{n}·\int n\, dq\Add{,}\NoteMark
+\]
+the integration beginning with $q = 0$. From this equation we obtain, by the method
+of undetermined coefficients,\Note{24}
+\begin{alignat*}{3}
+\Tag{[7]}
+S = \tfrac{1}{2}pq
+  &- \tfrac{1}{12}f°p^{3}q
+  &&- \tfrac{1}{20}f'p^{4}q
+  &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q\quad\text{etc.} \\
+  &- \tfrac{1}{12}f°pq^{3}
+  &&- \tfrac{3}{40}g°p^{3}q^{2}
+  &&- \tfrac{1}{20}g'p^{4}q^{2} \\
+  &&&- \tfrac{7}{120}f'p^{2}q^{3}
+  &&- (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
+  &&&- \tfrac{1}{10}g°pq^{4}
+  &&- \tfrac{3}{40}g'p^{2}q^{4} \\
+  &&&&&- (\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}\Add{.}
+\end{alignat*}
+\PageSep{39}
+
+
+\Article{25.}
+
+From the formulæ of the preceding article, which refer to a right triangle formed
+by shortest lines, we proceed to the general case. Let $C$ be another point on the
+same shortest line~$DB$, for which point~$p$ remains the same as for the point~$B$, and
+$q'$,~$r'$, $\phi'$,~$\psi'$, $S'$ have the same meanings as $q$,~$r$, $\phi$,~$\psi$, $S$ have for the point~$B$. There
+will thus be a triangle between the points $A$,~$B$,~$C$, whose angles we denote by
+$A$,~$B$,~$C$, the sides opposite these angles by $a$,~$b$,~$c$, and the area by~$\sigma$. We represent
+the measure of curvature at the points $A$,~$B$,~$C$ by $\alpha$,~$\beta$,~$\gamma$ respectively. And then
+supposing (which is permissible) that the quantities $p$,~$q$,~$q - q'$ are positive, we shall
+have
+\begin{align*}
+A &= \phi - \phi', & B &= \psi, & C &= \pi - \psi', && \\
+a &= q - q',       & b &= r',   & c &= r,           & \sigma &= S - S'.
+\end{align*}
+
+We shall first express the area~$\sigma$ by a series. By changing in~[7] each of the
+quantities that refer to~$B$ into those that refer to~$C$, we obtain a formula for~$S'$.
+Whence we have, exact to quantities of the sixth order,\Note{25}
+\begin{align*}
+\sigma = \tfrac{1}{2}p(q - q')
+  \bigl(1 &- \tfrac{1}{6} f°(p^{2} + q^{2} + qq' + q'^{2}) \\
+          &- \tfrac{1}{60}f'p(6p^{2} + 7q^{2} + 7qq' + 7q'^{2}) \\
+          &- \tfrac{1}{20}g°(q + q')(3p^{2} + 4q^{2} + 4q'^{2})\bigr)\Add{.}\NoteMark
+\end{align*}
+This formula, by aid of series~[2], namely,
+\[
+c\sin B = p(1 - \tfrac{1}{3}f°q^{2}
+              - \tfrac{1}{4}f'pq^{2}
+              - \tfrac{1}{2}g°q^{3} - \text{etc.})
+\]
+can be changed into the following:
+\begin{align*}
+\sigma = \tfrac{1}{2}ac\sin B
+  \bigl(1 &- \tfrac{1}{6} f°(p^{2} - q^{2} + qq' + q'^{2}) \\
+          &- \tfrac{1}{60}f'p(6p^{2} - 8q^{2} + 7qq' + 7q'^{2}) \\
+          &- \tfrac{1}{20}g°
+    (3p^{2}q + 3p^{2}q' - 6p^{3} + 4q^{2}q' + 4qq'^{2} + 4q'^{3})\bigr)\Add{.}
+\end{align*}
+
+The measure of curvature for any point whatever of the surface becomes (by \Art{19},
+where $m$,~$p$,~$q$ were what $n$,~$q$,~$p$ are here)
+\begin{align*}
+k &= -\frac{1}{n}·\frac{\dd^{2} n}{\dd q^{2}}
+   = -\frac{2f + 6gq + 12hq^{2} + \text{etc.}}{1 + fq^{2} + \text{etc.}} \\
+  &= -2f - 6gq - (12h - 2f^{2}) q^{2} - \text{etc.}
+\end{align*}
+Therefore we have, when $p$,~$q$ refer to the point~$B$,
+\[
+\beta = - 2f° - 2f'p - 6g°q - 2f''p^{2} - 6g'pq - (12h° - 2{f°}^{2})q^{2} - \text{etc.}
+\]
+\PageSep{40}
+Also
+\begin{align*}
+\gamma &= -2f° - 2f'p - 6g°q' - 2f''p^{2} - 6g'pq' - (12h° - 2{f°}^{2})q'^{2} - \text{etc.}\Add{,} \\
+\alpha &= -2f°\Add{.}
+\end{align*}
+Introducing these measures of curvature into the expression for~$\sigma$, we obtain the following
+expression, exact to quantities of the sixth order (exclusive):\Note{26}
+\begin{align*}
+\sigma = \tfrac{1}{2}ac \sin B
+  \bigl(1 &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} + 3qq' + 3q'^{2}) \\
+          &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} + 6qq' + 3q'^{2}) \\
+          &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} + \Z qq' + 4q'^{2})\bigr)\Add{.}\NoteMark
+\end{align*}
+The same precision will remain, if for $p$,~$q$,~$q'$ we substitute $c\sin B$, $c\cos B$, $c\cos B - a$.
+This gives\Note{27}
+\begin{align*}
+\Tag{[8]}
+\sigma = \tfrac{1}{2}ac\sin B
+  \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \Z9ac\cos B) \\
+          &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\
+          &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \Z9ac\cos B)\bigr)\Add{.}
+\end{align*}
+Since all expressions which refer to the line~$AD$ drawn normal to~$BC$ have disappeared
+from this equation, we may permute among themselves the points $A$,~$B$,~$C$ and
+the expressions that refer to them. Therefore we shall have, with the same precision,
+\begin{align*}
+\Tag{[9]}
+\sigma = \tfrac{1}{2}bc\sin A
+  \bigl(1 &+ \tfrac{1}{120}\alpha(3b^{2} + 3c^{2} -  12bc\cos A) \\
+          &+ \tfrac{1}{120}\beta (3b^{2} + 4c^{2} - \Z9bc\cos A) \\
+          &+ \tfrac{1}{120}\gamma(4b^{2} + 3c^{2} - \Z9bc\cos A)\bigr)\Add{,} \\
+%
+\Tag{[10]}
+\sigma = \tfrac{1}{2}ab\sin C
+  \bigl(1 &+ \tfrac{1}{120}\alpha(3a^{2} + 4b^{2} - \Z9ab\cos C) \\
+          &+ \tfrac{1}{120}\beta (4a^{2} + 3b^{2} - \Z9ab\cos C) \\
+          &+ \tfrac{1}{120}\gamma(3a^{2} + 3b^{2} -  12ab\cos C)\bigr)\Add{.}
+\end{align*}
+
+
+\Article{26.}
+
+The consideration of the rectilinear triangle whose sides are equal to $a$,~$b$,~$c$ is of
+great advantage. The angles of this triangle, which we shall denote by $A^{*}$,~$B^{*}$,~$C^{*}$,
+differ from the angles of the triangle on the curved surface, namely, from $A$,~$B$,~$C$,
+by quantities of the second order; and it will be worth while to develop these differences
+accurately. However, it will be sufficient to show the first steps in these more
+tedious than difficult calculations.
+
+Replacing in formulæ [1],~[4],~[5] the quantities that refer to~$B$ by those that
+refer to~$C$, we get formulæ for $r'^{2}$,~$r'\cos\phi'$, $r'\sin\phi'$. Then the development of the
+expression
+\PageSep{41}
+\begin{align*}
+  r^{2} + r'^{2} &- (q - q')^{2}
+    - 2r\cos\phi·r'\cos\phi'
+    - 2r\sin\phi·r'\sin\phi' \\
+    &\quad= b^{2} + c^{2} - a^{2} - 2bc\cos A \\
+    &\quad= 2bc(\cos A^{*} - \cos A),
+\end{align*}
+combined with the development of the expression
+\[
+r\sin\phi·r'\cos\phi' - r\cos\phi·r'\sin\phi' = bc\sin A,
+\]
+gives the following formula:
+\begin{align*}
+\cos A^{*} - \cos A
+  = -(q - q')p\sin A
+  \bigl(\tfrac{1}{3}f° &+ \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') \\
+  &+ (\tfrac{1}{10}f'' - \tfrac{1}{45}{f°}^{2})p^{2}
+   + \tfrac{3}{20}g'p(q + q') \\
+  &+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})(q^{2} + qq' + q'^{2})
+   + \text{etc.}\bigr)
+\end{align*}
+From this we have, to quantities of the fifth order,\Note{28}
+\begin{align*}
+A^{*} - A = +(q - q')p
+  \bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{3}{20}g'p(q + q') + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 12qq' + 7q'^{2})\bigr)\Add{.}\NoteMark
+\end{align*}
+Combining this formula with
+\[
+2\sigma = ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})- \text{etc.}\bigr)
+\]
+and with the values of the quantities $\alpha$,~$\beta$,~$\gamma$ found in the preceding article, we obtain,
+to quantities of the fifth order,\Note{29}
+\begin{align*}
+\Tag{[11]}
+A^{*} = A - \sigma\bigl(\tfrac{1}{6}\alpha
+  &+ \tfrac{1}{12}\beta
+   + \tfrac{1}{12}\gamma + \tfrac{2}{15}f''p^{2} + \tfrac{1}{5}g'p(q + q') \\
+  &+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2}) \\
+  &+ \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr)\Add{.}
+\end{align*}
+By precisely similar operations we derive\Note{30}
+\begin{align*}
+\Tag{[12]}
+B^{*} = B - \sigma\bigl(\tfrac{1}{12}\alpha
+  &+ \tfrac{1}{6}\beta
+   + \tfrac{1}{12}\gamma + \tfrac{1}{10}f''p^{2} + \tfrac{1}{10}g'p(2q + q') \\
+  &+ \tfrac{1}{5}h°(4q^{2} - 4qq' + 3q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} - 8qq' + 11q'^{2})\bigr)\Add{,}
+\displaybreak[1] \\
+%
+\Tag{[13]}
+C^{*} = C - \sigma\bigl(\tfrac{1}{12}\alpha
+  &+ \tfrac{1}{12}\beta
+   + \tfrac{1}{6}\gamma + \tfrac{1}{10}f''p^{2} + \tfrac{1}{10}g'p(q + 2q') \\
+  &+ \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr)\Add{.}
+\end{align*}
+From these formulæ we deduce, since the sum $A^{*} + B^{*} + C^{*}$ is equal to two right
+angles, the excess of the sum~$A + B + C$ over two right angles, namely,\Note{31}
+\begin{align*}
+\Tag{[14]}
+A + B + C= \pi + \sigma\bigl(\tfrac{1}{3}\alpha
+  &+ \tfrac{1}{3}\beta
+   + \tfrac{1}{3}\gamma + \tfrac{1}{3}f''p^{2} + \tfrac{1}{2}g'p(q + q') \\
+  &+ (2h° - \tfrac{1}{3}{f°}^{2})(q^{2} - qq' + q'^{2})\bigr)\Add{.}
+\end{align*}
+This last equation could also have been derived from formula~[6].
+\PageSep{42}
+
+
+\Article{27.}
+
+If the curved surface is a sphere of radius~$R$, we shall have
+\[
+\alpha = \beta = \gamma = -2f° = \frac{1}{R^{2}};\quad
+f'' = 0,\quad
+g' = 0,\quad
+6h° - {f°}^{2} = 0,
+\]
+or
+\[
+h° = \frac{1}{24R^{4}}.
+\]
+Consequently, formula~[14] becomes
+\[
+A + B + C = \pi + \frac{\sigma}{R^{2}},
+\]
+which is absolutely exact. But formulæ [11],~[12],~[13] give
+\begin{align*}
+A^{*} &= A - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(2p^{2} - q^{2} + 4qq' - q'^{2})\Add{,} \\
+B^{*} &= B - \frac{\sigma}{3R^{2}}
+  + \frac{\sigma}{180R^{4}}(p^{2} - 2q^{2} + 2qq' + q'^{2})\Add{,} \\
+C^{*} &= C - \frac{\sigma}{3R^{2}}
+  + \frac{\sigma}{180R^{4}}(p^{2} + q^{2} + 2qq' - 2q'^{2})\Add{;}
+\end{align*}
+or, with equal exactness,
+\begin{align*}
+A^{*} &= A - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(b^{2} + c^{2} - 2a^{2})\Add{,} \\
+B^{*} &= B - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(a^{2} + c^{2} - 2b^{2})\Add{,} \\
+C^{*} &= C - \frac{\sigma}{3R^{2}}
+  - \frac{\sigma}{180R^{4}}(a^{2} + b^{2} - 2c^{2})\Add{.}
+\end{align*}
+Neglecting quantities of the fourth order, we obtain from the above the well-known
+theorem first established by the illustrious Legendre.
+
+
+\Article{28.}
+
+Our general formulæ, if we neglect terms of the fourth order, become extremely
+simple, namely:
+\begin{align*}
+A^{*} &= A - \tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\Add{,} \\
+B^{*} &= B - \tfrac{1}{12}\sigma(\alpha + 2\beta + \gamma)\Add{,} \\
+C^{*} &= C - \tfrac{1}{12}\sigma(\alpha + \beta + 2\gamma)\Add{.}
+\end{align*}
+\PageSep{43}
+
+Thus to the angles $A$,~$B$,~$C$ on a non-spherical surface, unequal reductions must
+be applied, so that the sines of the changed angles become proportional to the sides
+opposite. The inequality, generally speaking, will be of the third order; but if the
+surface differs little from a sphere, the inequality will be of a higher order. Even in
+the greatest triangles on the earth's surface, whose angles it is possible to measure,
+the difference can always be regarded as insensible. Thus, \eg, in the greatest of
+the triangles which we have measured in recent years, namely, that between the
+points Hohehagen, Brocken, Inselberg, where the excess of the sum of the angles was
+$14''. 85348$, the calculation gave the following reductions to be applied to the angles:
+\setlength{\TmpLen}{2in}%
+\begin{align*}
+\Dotrow{Hohehagen}{-4''.95113\rlap{\Add{,}}} \\
+\Dotrow{Brocken}{- 4''.95104\rlap{\Add{,}}} \\
+\Dotrow{Inselberg}{-4''.95131\rlap{.}}
+\end{align*}
+
+
+\Article{29.}
+
+We shall conclude this study by comparing the area of a triangle on a curved
+surface with the area of the rectilinear triangle whose sides are $a$,~$b$,~$c$. We shall
+denote the area of the latter by~$\sigma^{*}$; hence
+\[
+\sigma^{*} = \tfrac{1}{2}bc\sin A^{*}
+           = \tfrac{1}{2}ac\sin B^{*}
+           = \tfrac{1}{2}ab\sin C^{*}\Add{.}
+\]
+
+We have, to quantities of the fourth order,
+\[
+\sin A^{*} = \sin A - \tfrac{1}{12}\sigma\cos A·(2\alpha + \beta + \gamma)\Add{,}
+\]
+or, with equal exactness,
+\[
+\sin A = \sin A^{*}·\bigl(1 + \tfrac{1}{24}bc\cos A·(2\alpha + \beta + \gamma)\bigr)\Add{.}
+\]
+Substituting this value in formula~[9], we shall have, to quantities of the sixth order,
+\begin{align*}
+\sigma = \tfrac{1}{2}bc\sin A^{*}·\bigl(1
+  &+ \tfrac{1}{120}\alpha(3b^{2}+ 3c^{2} - 2bc\cos A) \\
+  &+ \tfrac{1}{120}\beta (3b^{2}+ 4c^{2} - 4bc\cos A) \\
+  &+ \tfrac{1}{120}\gamma(4b^{2}+ 3c^{2} - 4bc\cos A)\bigr),
+\end{align*}
+or, with equal exactness,
+\[
+\sigma = \sigma^{*}\bigl(1
+  + \tfrac{1}{120}\alpha(a^{2} + 2b^{2} + 2c^{2})
+  + \tfrac{1}{120}\beta (2a^{2} + b^{2} + 2c^{2})
+  + \tfrac{1}{120}\gamma(2a^{2} + 2b^{2} + c^{2})\Add{.}
+\]
+For the sphere this formula goes over into the following form:
+\[
+\sigma = \sigma^{*}\bigl(1 + \tfrac{1}{24}\alpha(a^{2} + b^{2} + c^{2})\bigr).
+\]
+\PageSep{44}
+It is easily verified that, with the same precision, the following formula may be taken
+instead of the above:
+\[
+\sigma
+  = \sigma^{*}\sqrt{\frac{\sin A·\sin B·\sin C}
+                         {\sin A^{*}·\sin B^{*}·\sin C^{*}}}\Add{.}
+\]
+If this formula is applied to triangles on non-spherical curved surfaces, the error, generally
+speaking, will be of the fifth order, but will be insensible in all triangles such
+as may be measured on the earth's surface.
+\PageSep{45}
+
+
+\Abstract{\small GAUSS'S ABSTRACT OF THE DISQUISITIONES GENERALES CIRCA \\
+SUPERFICIES CURVAS, PRESENTED TO THE ROYAL \\
+SOCIETY OF GÖTTINGEN. \\
+\tb \\[8pt]
+
+\footnotesize\textsc{Göttingische gelehrte Anzeigen.} No.~177. Pages 1761--1768. 1827. November~5.
+}
+
+On the 8th~of October, Hofrath Gauss presented to the Royal Society a paper:
+\begin{center}
+\textit{Disquisitiones generales circa superficies curvas.}
+\end{center}
+
+Although geometers have given much attention to general investigations of curved
+surfaces and their results cover a significant portion of the domain of higher geometry,
+this subject is still so far from being exhausted, that it can well be said that, up to
+this time, but a small portion of an exceedingly fruitful field has been cultivated.
+Through the solution of the problem, to find all representations of a given surface upon
+another in which the smallest elements remain unchanged, the author sought some
+years ago to give a new phase to this study. The purpose of the present discussion
+is further to open up other new points of view and to develop some of the new truths
+which thus become accessible. We shall here give an account of those things which
+can be made intelligible in a few words. But we wish to remark at the outset that
+the new theorems as well as the presentations of new ideas, if the greatest generality
+is to be attained, are still partly in need of some limitations or closer determinations,
+which must be omitted here.
+
+In researches in which an infinity of directions of straight lines in space is concerned,
+it is advantageous to represent these directions by means of those points upon
+a fixed sphere, which are the end points of the radii drawn parallel to the lines. The
+centre and the radius of this \emph{auxiliary sphere} are here quite arbitrary. The radius may
+be taken equal to unity. This procedure agrees fundamentally with that which is constantly
+employed in astronomy, where all directions are referred to a fictitious celestial
+sphere of infinite radius. Spherical trigonometry and certain other theorems, to which
+the author has added a new one of frequent application, then serve for the solution of
+the problems which the comparison of the various directions involved can present.
+\PageSep{46}
+
+If we represent the direction of the normal at each point of the curved surface by
+the corresponding point of the sphere, determined as above indicated, namely, in this
+way, to every point on the surface, let a point on the sphere correspond; then, generally
+speaking, to every line on the curved surface will correspond a line on the sphere,
+and to every part of the former surface will correspond a part of the latter. The less
+this part differs from a plane, the smaller will be the corresponding part on the sphere.
+It is, therefore, a very natural idea to use as the measure of the total curvature,
+which is to be assigned to a part of the curved surface, the area of the corresponding
+part of the sphere. For this reason the author calls this area the \emph{integral curvature} of
+the corresponding part of the curved surface. Besides the magnitude of the part, there
+is also at the same time its \emph{position} to be considered. And this position may be in
+the two parts similar or inverse, quite independently of the relation of their magnitudes.
+The two cases can be distinguished by the positive or negative sign of the
+total curvature. This distinction has, however, a definite meaning only when the
+figures are regarded as upon definite sides of the two surfaces. The author regards
+the figure in the case of the sphere on the outside, and in the case of the curved surface
+on that side upon which we consider the normals erected. It follows then that
+the positive sign is taken in the case of convexo-convex or concavo-concave surfaces
+(which are not essentially different), and the negative in the case of concavo-convex
+surfaces. If the part of the curved surface in question consists of parts of these different
+sorts, still closer definition is necessary, which must be omitted here.
+
+The comparison of the areas of two corresponding parts of the curved surface and of
+the sphere leads now (in the same manner as, \eg, from the comparison of volume and
+mass springs the idea of density) to a new idea. The author designates as \emph{measure of
+curvature} at a point of the curved surface the value of the fraction whose denominator is
+the area of the infinitely small part of the curved surface at this point and whose numerator
+is the area of the corresponding part of the surface of the auxiliary sphere, or the
+integral curvature of that element. It is clear that, according to the idea of the author,
+integral curvature and measure of curvature in the case of curved surfaces are analogous
+to what, in the case of curved lines, are called respectively amplitude and curvature
+simply. He hesitates to apply to curved surfaces the latter expressions, which
+have been accepted more from custom than on account of fitness. Moreover, less
+depends upon the choice of words than upon this, that their introduction shall be justified
+by pregnant theorems.
+
+The solution of the problem, to find the measure of curvature at any point of a curved
+surface, appears in different forms according to the manner in which the nature of the
+curved surface is given. When the points in space, in general, are distinguished by
+\PageSep{47}
+three rectangular coordinates, the simplest method is to express one coordinate as a function
+of the other two. In this way we obtain the simplest expression for the measure of
+curvature. But, at the same time, there arises a remarkable relation between this
+measure of curvature and the curvatures of the curves formed by the intersections of
+the curved surface with planes normal to it. \textsc{Euler}, as is well known, first showed
+that two of these cutting planes which intersect each other at right angles have this
+property, that in one is found the greatest and in the other the smallest radius of curvature;
+or, more correctly, that in them the two extreme curvatures are found. It will
+follow then from the above mentioned expression for the measure of curvature that this
+will be equal to a fraction whose numerator is unity and whose denominator is the product
+of the extreme radii of curvature. The expression for the measure of curvature will be
+less simple, if the nature of the curved surface is determined by an equation in $x$,~$y$,~$z$.
+And it will become still more complex, if the nature of the curved surface is given so that
+$x$,~$y$,~$z$ are expressed in the form of functions of two new variables $p$,~$q$. In this last case
+the expression involves fifteen elements, namely, the partial differential coefficients of the
+first and second orders of $x$,~$y$,~$z$ with respect to $p$~and~$q$. But it is less important in itself
+than for the reason that it facilitates the transition to another expression, which must be
+classed with the most remarkable theorems of this study. If the nature of the curved
+surface be expressed by this method, the general expression for any linear element upon
+it, or for $\Sqrt{dx^{2} + dy^{2} + dz^{2}}$, has the form $\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}$, where $E$,~$F$,~$G$
+are again functions of $p$~and~$q$. The new expression for the measure of curvature mentioned
+above contains merely these magnitudes and their partial differential coefficients
+of the first and second order. Therefore we notice that, in order to determine the
+measure of curvature, it is necessary to know only the general expression for a linear
+element; the expressions for the coordinates $x$,~$y$,~$z$ are not required. A direct result
+from this is the remarkable theorem: If a curved surface, or a part of it, can be developed
+upon another surface, the measure of curvature at every point remains unchanged
+after the development. In particular, it follows from this further: Upon a curved
+surface that can be developed upon a plane, the measure of curvature is everywhere
+equal to zero. From this we derive at once the characteristic equation of surfaces
+developable upon a plane, namely,
+\[
+\frac{\dd^{2} z}{\dd x^{2}}·\frac{\dd^{2} z}{\dd y^{2}}
+  - \left(\frac{\dd^{2} z}{\dd x·\dd y}\right)^{2} = 0,
+\]
+when $z$~is regarded as a function of $x$~and~$y$. This equation has been known for some
+time, but according to the author's judgment it has not been established previously
+with the necessary rigor.
+\PageSep{48}
+
+These theorems lead to the consideration of the theory of curved surfaces from a
+new point of view, where a wide and still wholly uncultivated field is open to investigation.
+If we consider surfaces not as boundaries of bodies, but as bodies of which
+one dimension vanishes, and if at the same time we conceive them as flexible but not
+extensible, we see that two essentially different relations must be distinguished, namely,
+on the one hand, those that presuppose a definite form of the surface in space; on the
+other hand, those that are independent of the various forms which the surface may
+assume. This discussion is concerned with the latter. In accordance with what has
+been said, the measure of curvature belongs to this case. But it is easily seen that
+the consideration of figures constructed upon the surface, their angles, their areas and
+their integral curvatures, the joining of the points by means of shortest lines, and the
+like, also belong to this case. All such investigations must start from this, that the
+very nature of the curved surface is given by means of the expression of any linear
+element in the form $\Sqrt{E\, dp^{2} + 2F\, dp·dq + G\, dq^{2}}$. The author has embodied in the
+present treatise a portion of his investigations in this field, made several years ago,
+while he limits himself to such as are not too remote for an introduction, and may, to
+some extent, be generally helpful in many further investigations. In our abstract, we
+must limit ourselves still more, and be content with citing only a few of them as
+types. The following theorems may serve for this purpose.
+
+If upon a curved surface a system of infinitely many shortest lines of equal lengths
+be drawn from one initial point, then will the line going through the end points of
+these shortest lines cut each of them at right angles. If at every point of an arbitrary
+line on a curved surface shortest lines of equal lengths be drawn at right angles to this
+line, then will all these shortest lines be perpendicular also to the line which joins their
+other end points. Both these theorems, of which the latter can be regarded as a generalization
+of the former, will be demonstrated both analytically and by simple geometrical
+considerations. \begin{Theorem}[]The excess of the sum of the angles of a triangle formed by shortest lines
+over two right angles is equal to the total curvature of the triangle.\end{Theorem} It will be assumed here
+that that angle ($57°\, 17'\, 45''$) to which an arc equal to the radius of the sphere corresponds
+will be taken as the unit for the angles, and that for the unit of total curvature will be
+taken a part of the spherical surface, the area of which is a square whose side is equal to
+the radius of the sphere. Evidently we can express this important theorem thus also:
+the excess over two right angles of the angles of a triangle formed by shortest lines is to
+eight right angles as the part of the surface of the auxiliary sphere, which corresponds
+to it as its integral curvature, is to the whole surface of the sphere. In general, the
+excess over $2n - 4$~right angles of the angles of a polygon of $n$~sides, if these are
+shortest lines, will be equal to the integral curvature of the polygon.
+\PageSep{49}
+
+The general investigations developed in this treatise will, in the conclusion, be applied
+to the theory of triangles of shortest lines, of which we shall introduce only a couple of
+important theorems. If $a$,~$b$,~$c$ be the sides of such a triangle (they will be regarded as
+magnitudes of the first order); $A$,~$B$,~$C$ the angles opposite; $\alpha$,~$\beta$,~$\gamma$ the measures of
+curvature at the angular points; $\sigma$~the area of the triangle, then, to magnitudes of the
+fourth order, $\frac{1}{3}(\alpha + \beta + \gamma)\sigma$ is the excess of the sum $A + B + C$ over two right angles.
+Further, with the same degree of exactness, the angles of a plane rectilinear triangle
+whose sides are $a$,~$b$,~$c$, are respectively
+\begin{align*}
+A &- \tfrac{1}{12}(2\alpha + \beta + \gamma)\sigma\Add{,} \\
+B &- \tfrac{1}{12}(\alpha + 2\beta + \gamma)\sigma\Add{,} \\
+C &- \tfrac{1}{12}(\alpha + \beta + 2\gamma)\sigma.
+\end{align*}
+We see immediately that this last theorem is a generalization of the familiar theorem first
+established by \textsc{Legendre}. By means of this theorem we obtain the angles of a plane
+triangle, correct to magnitudes of the fourth order, if we diminish each angle of the corresponding
+spherical triangle by one-third of the spherical excess. In the case of non-spherical
+surfaces, we must apply unequal reductions to the angles, and this inequality,
+generally speaking, is a magnitude of the third order. However, even if the whole surface
+differs only a little from the spherical form, it will still involve also a factor denoting
+the degree of the deviation from the spherical form. It is unquestionably important for
+the higher geodesy that we be able to calculate the inequalities of those reductions and
+thereby obtain the thorough conviction that, for all measurable triangles on the surface
+of the earth, they are to be regarded as quite insensible. So it is, for example, in the
+case of the greatest triangle of the triangulation carried out by the author. The greatest
+side of this triangle is almost fifteen geographical\footnote
+  {This German geographical mile is four minutes of arc at the equator, namely, $7.42$~kilometers,
+  and is equal to about $4.6$~English statute miles. [Translators.]}
+miles, and the excess of the sum
+of its three angles over two right angles amounts almost to fifteen seconds. The three
+reductions of the angles of the plane triangle are $4''.95113$, $4''.95104$, $4''.95131$. Besides,
+the author also developed the missing terms of the fourth order in the above expressions.
+Those for the sphere possess a very simple form. However, in the case of
+measurable triangles upon the earth's surface, they are quite insensible. And in the
+example here introduced they would have diminished the first reduction by only two
+units in the fifth decimal place and increased the third by the same amount.
+\PageSep{50}
+%[** Blank page]
+\PageSep{51}
+
+
+\Notes.
+
+%[** TN: Line numbers have been omitted]
+\LineRef{1}{Art.~1, p.~3, l.~3}. Gauss got the idea of using the auxiliary sphere from astronomy.
+\Cf.~Gauss's Abstract, \Pgref[p.]{abstract}.
+
+\LineRef[1]{2}{Art.~2, p.~3, l.~2~fr.~bot}. In the Latin text \textit{situs} is used for the direction or
+orientation of a plane, the position of a plane, the direction of a line, and the position
+of a point.
+
+\LineRef[2]{2}{Art.~2, p.~4, l.~14}. In the Latin texts the notation
+\[
+\cos(1)L^{2} + \cos(2)L^{2} + \cos(3) L^{2} = 1
+\]
+is used. This is replaced in the translations (except Böklen's) by the more recent
+notation
+\[
+\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1.
+\]
+
+\LineRef[3]{2}{Art.~2, p.~4, l.~3~fr.~bot}. This stands in the original and in Liouville s reprint,
+\[
+\cos A (\cos t\sin t' - \sin t\cos t')(\cos t''\sin t''' - \sin t''\sin t''').
+\]
+
+%[** TN: One-off macro; hyperlink is hard-coded into macro definition]
+\LineRefs{Art.~2, pp.~4--6}. Theorem~VI is original with Gauss, as is also the method of
+deriving~VII\@. The following figures show the points and lines of Theorems VI~and~VII:
+%[Illustrations]
+\Figure{051}
+
+\LineRef{3}{Art.~3, p.~6}. The geometric condition here stated, that the curvature be continuous
+for each point of the surface, or part of the surface, considered is equivalent to
+the analytic condition that the first and second derivatives of the function or functions
+defining the surface be finite and continuous for all points of the surface, or
+part of the surface, considered.
+
+\LineRef[6]{4}{Art.~4, p.~7, l.~20}. In the Latin texts the notation~$XX$ for~$X^{2}$,~etc., is used.
+\PageSep{52}
+
+\LineRef[7]{4}{Art.~4, p.~7}. ``The second method of representing a surface (the expression of
+the coordinates by means of two auxiliary variables) was first used by Gauss for
+arbitrary surfaces in the case of the problem of conformal mapping. [Astronomische
+Abhandlungen, edited by H.~C. Schumacher, vol.~III, Altona,~1825; Gauss, \Title{Werke},
+vol.~IV, p.~189; reprinted in vol.~55 of Ostwald's Klassiker.---\Cf.~also Gauss, \Title{Theoria
+attractionis corporum sphaer.\ ellipt.}, Comment.\ Gött.~II, 1813; Gauss, \Title{Werke}, vol.~V,
+p.~10.] Here he applies this representation for the first time to the determination of
+the direction of the surface normal, and later also to the study of curvature and of
+geodetic lines. The geometrical significance of the variables $p$,~$q$ is discussed more fully
+in \Art{17}. This method of representation forms the source of many new theorems,
+of which these are particularly worthy of mention: the corollary, that the measure of
+%[** TN: On next two lines, replaced "Art." by "Arts."]
+curvature remains unchanged by the bending of the surface (\Arts{11}{12}); the theorems
+of \Arts{15}{16} concerning geodetic lines; the theorem of \Art{20}; and, finally, the
+results derived in the conclusion, which refer a geodetic triangle to the rectilinear triangle
+whose sides are of the same length.'' [Wangerin.]
+
+\LineRef{5}{Art.~5, p.~8}. ``To decide the question, which of the two systems of values found
+in \Art{4} for $X$,~$Y$,~$Z$ belong to the normal directed outwards, which to the normal
+directed inwards, we need only to apply the theorem of \Art{2}~(VII), provided we use
+the second method of representing the surface. If, on the contrary, the surface is
+defined by the equation between the coordinates $W = 0$, then the following simpler\Typo{ con-}{}
+considerations lead to the answer. We draw the line~$d\sigma$ from the point~$A$ towards
+the outer side, then, if $dx$,~$dy$,~$dz$ are the projections of~$d\sigma$, we have
+\[
+P\, dx + Q\, dy + R\, dz > 0.
+\]
+On the other hand, if the angle between $\sigma$~and the normal taken outward is acute,
+then
+\[
+\frac{dx}{d\sigma}X + \frac{dy}{d\sigma}Y + \frac{dz}{d\sigma}Z > 0.
+\]
+This condition, since $d\sigma$~is positive, must be combined with the preceding, if the first
+solution is taken for $X$,~$Y$,~$Z$. This result is obtained in a similar way, if the surface
+is analytically defined by the third method.'' [Wangerin.]
+
+\LineRef[8]{6}{Art.~6, p.~10, l.~4}. The definition of measure of curvature here given is the one
+generally used. But Sophie Germain defined as a measure of curvature at a point of
+a surface the sum of the reciprocals of the principal radii of curvature at that point,
+or double the so-called mean curvature. \Cf.~Crelle's Journ.\ für Math., vol.~VII\@.
+Casorati defined as a measure of curvature one-half the sum of the squares of the
+reciprocals of the principal radii of curvature at a point of the surface. \Cf.~Rend.\
+del R.~Istituto Lombardo, ser.~2, vol.~22, 1889; Acta Mathem.\ vol.~XIV, p.~95, 1890.
+\PageSep{53}
+
+\LineRef[9]{6}{Art.~6, p.~11, l.~21}. Gauss did not carry out his intention of studying the most
+general cases of figures mapped on the sphere.
+
+\LineRef{7}{Art.~7, p.~11, l.~31}. ``That the consideration of a surface element which has the
+form of a triangle can be used in the calculation of the measure of curvature, follows
+from this fact that, according to the formula developed on \Pageref{12}, $k$~is independent
+of the magnitudes $dx$,~$dy$, $\delta x$,~$\delta y$, and that, consequently, $k$~has the same value for
+every infinitely small triangle at the same point of the surface, therefore also for surface
+elements of any form whatever lying at that point.'' [Wangerin.]
+
+\LineRef[10]{7}{Art.~7, p.~12, l.~20}. The notation in the Latin text for the partial derivatives:
+\[
+\frac{dX}{dx},\quad \frac{dX}{dy},\quad \text{etc.},
+\]
+has been replaced throughout by the more recent notation:
+\[
+\frac{\dd X}{\dd x},\quad \frac{\dd X}{\dd y},\quad \text{etc.}
+\]
+
+\LineRef{7}{Art.~7, p.~13, l.~16}. This formula, as it stands in the original and in Liouville's
+reprint, is
+\[
+dY = -Z^{3}tu\, dt - Z^{3}(1 + t^{2})\, du.
+\]
+The incorrect sign in the second member has been corrected in the reprint in Gauss,
+\Title{Werke}, vol.~IV, and in the translations.
+
+\LineRef[12]{8}{Art.~8, p.~15, l.~3}. Euler's work here referred to is found in Mem.\ de~l'Acad.\
+de~Berlin, vol.~XVI, 1760.
+
+\LineRef[13]{10}{Art.~10, p.~18, ll.~8,~9,~10}. Instead of $D$,~$D'$,~$D''$ as here defined, the Italian
+geometers have introduced magnitudes denoted by the same letters and equal, in
+Gauss's notation, to
+\[
+\frac{D}{\Sqrt{EG - F^{2}}},\quad
+\frac{D'}{\Sqrt{EG - F^{2}}},\quad
+\frac{D''}{\Sqrt{EG - F^{2}}}
+\]
+respectively.
+
+\LineRef[14]{11}{Art.~11, p.~19, ll.~4,~6,~fr.~bot}. In the original and in Liouville's reprint, two of
+these formulæ are incorrectly given:
+\[
+\frac{\dd F}{\dd q} = m'' + n,\quad
+n = \frac{\dd F}{\dd q} - \frac{1}{2}·\frac{\dd E}{\dd q}.
+\]
+The proper corrections have been made in Gauss, \Title{Werke}, vol.~IV, and in the translations.
+
+\LineRef{13}{Art.~13, p.~21, l.~20}. Gauss published nothing further on the properties of developable
+surfaces.
+\PageSep{54}
+
+\LineRef[15]{14}{Art.~14, p.~22, l.~8}. The transformation is easily made by means of integration
+by parts.
+
+\LineRef{17}{Art.~17, p.~25}. If we go from the point $p$,~$q$ to the point $(p + dp, q)$, and if the
+Cartesian coordinates of the first point are $x$,~$y$,~$z$, and of the second $x + dx$, $y + dy$,
+$z + dz$; with $ds$~the linear element between the two points, then the direction cosines
+of~$ds$ are
+\[
+\cos \alpha = \frac{dx}{ds},\quad
+\cos \beta  = \frac{dy}{ds},\quad
+\cos \gamma = \frac{dz}{ds}.
+\]
+Since we assume here $q = \text{Constant}$ or $dq = 0$, we have also
+\[
+dx = \frac{\dd x}{\dd p}·dp,\quad
+dy = \frac{\dd y}{\dd p}·dp,\quad
+dz = \frac{\dd z}{\dd p}·dp,\quad
+ds = ±\sqrt{E}·dp.
+\]
+If $dp$~is positive, the change~$ds$ will be taken in the positive direction. Therefore
+$ds = \sqrt{E}·dp$,
+\[
+\cos\alpha = \frac{1}{\sqrt{E}}·\frac{\dd x}{\dd p},\quad
+\cos\beta  = \frac{1}{\sqrt{E}}·\frac{\dd y}{\dd p},\quad
+\cos\gamma = \frac{1}{\sqrt{E}}·\frac{\dd z}{\dd p}\Typo{,}{.}
+\]
+In like manner, along the line $p = \text{Constant}$, if $\cos \alpha'$, $\cos \beta'$, $\cos \gamma'$ are the direction
+cosines, we obtain
+\[
+\cos\alpha' = \frac{1}{\sqrt{G}}·\frac{\dd x}{\dd q},\quad
+\cos\beta'  = \frac{1}{\sqrt{G}}·\frac{\dd y}{\dd q},\quad
+\cos\gamma' = \frac{1}{\sqrt{G}}·\frac{\dd z}{\dd q}.
+\]
+And since
+\begin{align*}
+\cos\omega
+  &= \cos\alpha \cos\alpha'
+   + \cos\beta  \cos\beta'
+   + \cos\gamma \cos\gamma', \\
+\cos\omega
+  &= \frac{F}{\sqrt{EG}}.
+\end{align*}
+From this follows
+\[
+\sin\omega = \frac{\Sqrt{EG - F^{2}}}{\sqrt{EG}}.
+\]
+And the area of the quadrilateral formed by the lines $p$,~$p + dp$, $q$,~$q + dq$ is
+\[
+d\sigma = \Sqrt{EG - F^{2}}·dp·dq.
+\]
+
+\LineRef[16]{21}{Art.~21, p.~33, l.~12}. In the original, in Liouville's reprint, in the two French
+translations, and in Böklen's translation, the next to the last formula of this article
+is written
+\[
+E\beta\delta - F(\alpha\delta + \beta\gamma) + G\alpha\gamma
+  = \frac{EG - F\Typo{'}{}^{2}}{E'G' - F'^{2}}·F'\Add{.}
+\]
+\PageSep{55}
+The proper correction in sign has been made in Gauss, \Title{Werke}, vol.~IV, and in Wangerin's
+translation.
+
+\LineRef[17]{23}{Art.~23, p.~35, l.~13~fr.~bot}. In the Latin texts and in Roger's and Böklen's
+translations this formula has a minus sign on the right hand side. The correction in
+sign has been made in Abadie's and Wangerin's translations.
+
+\LineRef{23}{Art.~23, p.~35}. The figure below represents the lines and angles mentioned in
+this and the following articles\Chg{:}{.}
+%[Illustration]
+\Figure{055}
+
+\LineRef[18]{24}{Art.~24, p.~36}. Derivation of formula~[1].
+
+Let
+\[
+r^{2} = p^{2} + q^{2} + R_{3} + R_{4} + R_{5} + R_{6} + \text{etc.}
+\]
+where $R_{3}$~is the aggregate of all the terms of the third degree in $p$~and~$q$, $R_{4}$~of all
+the terms of the fourth degree,~etc. Then by differentiating, squaring, and omitting
+terms above the sixth degree, we obtain
+\begin{align*}
+\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} = 4p^{2}
+  &+ \left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+   + \left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+   + 4p\frac{\dd R_{3}}{\dd p} %[** TN: Omitted parentheses]
+   + 4p\frac{\dd R_{4}}{\dd p} \\
+  &+ 4p\frac{\dd R_{5}}{\dd p}
+   + 4p\frac{\dd R_{6}}{\dd p}
+   + 2p\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+   + 2p\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p},
+\intertext{and}
+\left(\frac{\dd(r^{2})}{\dd \Erratum{p}{q}}\right)^{2} = 4q^{2}
+  &+ \left(\frac{\dd R_{3}}{\dd q}\right)^{2}
+   + \left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+   + 4q\frac{\dd R_{3}}{\dd q}
+   + 4q\frac{\dd R_{4}}{\dd q} \\
+  &+ 4q\frac{\dd R_{5}}{\dd q}
+   + 4q\frac{\dd R_{6}}{\dd q}
+   + 2q\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}
+   + 2q\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}.
+\end{align*}
+\PageSep{56}
+
+Hence we have
+{\small
+\begin{align*}
+&\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} +
+\left(\frac{\dd(r^{2})}{\dd q}\right)^{2} - 4r^{2} \\
+  &= 4\left(p\frac{\dd R_{3}}{\dd p} + q\frac{\dd R_{3}}{\dd q} - R_{3}\right)
+   + 4\left(p\frac{\dd R_{4}}{\dd p} + q\frac{\dd R_{4}}{\dd q} - R_{4}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}\right) \\
+  &\quad+ 4\left(p\frac{\dd R_{5}}{\dd p} + q\frac{\dd R_{5}}{\dd q} - R_{5}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}\right) \\
+  &\quad+ 4\left(p\frac{\dd R_{6}}{\dd p} + q\frac{\dd R_{6}}{\dd q} - R_{6}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}\right)
+\displaybreak[1] \\
+  &= 8R_{3} + 4\left(3R_{4}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}\right)
+   + 4\left(4R_{5}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}\right) \\
+  &\quad+ 4\left(5R_{6}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+     + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
+     + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}\right),
+\end{align*}}%
+since, according to a familiar theorem for homogeneous functions,
+\[
+p\frac{\dd R_{3}}{\dd p} + q\frac{\dd R_{3}}{\dd q} = 3R_{3},\quad\text{etc.}
+\]
+By dividing unity by the square of the value of~$n$, given at the end of \Art{23}, and
+omitting terms above the fourth degree, we have
+\[
+1 - \frac{1}{n^{2}}
+  = 2f°q^{2} + 2f'pq^{2} + 2g°q^{3} - 3{f°}^{2}q^{4}
+    + 2f''p^{2}q^{2} + 2g'pq^{3} + 2h°q^{4}.
+\]
+This, multiplied by the last equation but one of the preceding page, on rejecting terms
+above the sixth degree, becomes
+\begin{multline*}
+%[** TN: Re-broken]
+\left(1 - \frac{1}{n^{2}}\right)
+\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} \\
+\begin{alignedat}{3}
+  = 8f°p^{2}q^{2} &+ 8f'p^{3}q^{2} &&- 12{f°}^{2}p^{2}q^{4} &&+ 8h°p^{2}q^{4} \\
+                 &+ 8g°p^{2}q^{3} &&+ 8f''p^{4}q^{2}
+  &&+2 f°q^{2}\left(\frac{\dd R_{3}}{\dd p}\right)^{2} \\
+                 &+ 8f°pq^{2} \frac{\dd R_{3}}{\dd p} &&+ 8g'p^{3}q^{3}
+  &&+8 f'p^{2}q^{2} \frac{\dd R_{3}}{\dd p} + 8g°pq^{3} \frac{\dd R_{3}}{\dd p}\\
+  &&&&&+ 8f°pq^{2} \frac{\dd R_{4}}{\dd p}.
+\end{alignedat}
+\end{multline*}
+Therefore, since from the fifth equation of \Art{24}:
+\[
+\left(\frac{\dd(r^{2})}{\dd p}\right)^{2} +
+\left(\frac{\dd(r^{2})}{\dd q}\right)^{2} - 4r^{2}
+  = \left(1 - \frac{1}{n^{2}}\right)
+    \left(\frac{\dd(r^{2})}{\dd p}\right)^{2},
+\]
+\PageSep{57}
+we have
+{\small
+\begin{multline*}
+8R_{3} + 4\left(3R_{4}
+  + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd p}\right)^{2}
+  + \tfrac{1}{4}\left(\frac{\dd R_{3}}{\dd q}\right)^{2}
+\right)
++ 4\left(4R_{5}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{4}}{\dd p}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{4}}{\dd q}
+\right) \\
+  + 4\left(5R_{6}
+  + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd p}\right)^{2}
+  + \tfrac{1}{4}\left(\frac{\dd R_{4}}{\dd q}\right)^{2}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd p}\, \frac{\dd R_{5}}{\dd p}
+  + \tfrac{1}{2}\frac{\dd R_{3}}{\dd q}\, \frac{\dd R_{5}}{\dd q}
+\right) \\
+\begin{aligned}
+  &= 8f°p^{2}q^{2} + 8f'p^{3}q^{2} + 8g°p^{2}q^{3} + 8f°pq^{2}\frac{\dd R_{3}}{\dd p} - 12{f°}^{2}p^{2}q^{4} + 8f''p^{4}q^{2} \\
+  &\quad+ 8g'p^{3}q^{3} + 8h°p^{2}q^{4} + 2f°q^{2}\left(\frac{\dd R_{3}}{\dd p}\right)^{2} + 8f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p} + 8f°pq^{2}\frac{\dd R_{4}}{\dd p} + 8g°pq^{3}\frac{\dd R_{3}}{\dd p}.
+\end{aligned}
+\end{multline*}}%
+Whence, by the method of undetermined coefficients, we find
+\begin{align*}
+R_{3} &= 0,\quad
+R_{4} = \tfrac{2}{3}f°p^{2}q^{2},\quad
+R_{5} = \tfrac{1}{2}f'p^{3}q^{2} + \tfrac{1}{2}g°p^{2}q^{3}, \\
+R_{6} &= (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2}
+  + \tfrac{2}{5}g'p^{3}q^{3} + (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}.
+\end{align*}
+And therefore we have
+\begin{alignat*}{3}
+\Tag{[1]}
+r^{2} &= p^{2} + \tfrac{2}{5}f°p^{2}q^{2}
+  &&+ \tfrac{1}{2}f'p^{3}q^{2}
+  &&+ (\tfrac{2}{5}f''- \tfrac{4}{45}{f°}^{2})p^{4}q^{2} + \text{etc.} \\
+%
+  &+q^{2}
+  &&+ \tfrac{1}{2}g°p^{2}q^{3}
+  &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4}.
+\end{alignat*}
+
+This method for deriving formula~[1] is taken from Wangerin.
+
+\LineRef[19]{24}{Art.~24, p.~36}. Derivation of formula~[2].
+
+By taking one-half the reciprocal of the series for~$n$ given in \LineRef{23}{Art.~23, p.~36}, we
+obtain
+\[
+\frac{1}{2n} = \tfrac{1}{2} \bigl[
+  1 - f°q^{2} - f'pq^{2} - g°q^{3}
+  - f''p^{2}q^{2} - g'pq^{3} - (h° - {f°}^{2})q^{4} - \text{etc.}
+\bigr].
+\]
+And by differentiating formula~[1] with respect to~$p$, we obtain
+\begin{alignat*}{2}
+\frac{\dd(r^{2})}{\dd p} = 2 \bigl[p + \tfrac{1}{2}f°pq^{2}
+  &+ \tfrac{3}{4}f'p^{2}q^{2}
+  &&+ 2 (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{3}q^{2} \\
+  &+ \tfrac{1}{2}g°pq^{3}
+  &&+ \tfrac{3}{5}g'p^{2}q^{3} \\
+  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4} + \text{etc.}
+\bigr].
+\end{alignat*}
+Therefore, since
+\[
+r\sin\psi = \frac{1}{2n}·\frac{\dd(r^{2})}{\dd p},
+\]
+we have, by multiplying together the two series above,
+\begin{alignat*}{2}
+\Tag{[2]}
+r\sin\psi = p - \tfrac{1}{3}f°pq^{2}
+  &- \tfrac{1}{4}f'p^{2}q^{2}
+  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2} - \text{etc.} \\
+  &- \tfrac{1}{2}g°pq^{3}
+  &&- \tfrac{2}{5}g'p^{2}q^{3} \\
+  &&&- (\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})pq^{4}.
+\end{alignat*}
+\PageSep{58}
+
+\LineRef[20]{24}{Art.~24, p.~37}. Derivation of formula~[3].
+
+By differentiating~[1] on \Pageref{57} with respect to~$q$, we find
+\begin{alignat*}{2}
+\frac{\dd(r^{2})}{\dd q} = 2 \bigl[q + \tfrac{2}{3}f°p^{2}q
+  &+ \tfrac{1}{2}f'p^{3}q
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
+  &+ \tfrac{3}{4}g°p^{2}q^{2}
+  &&+ \tfrac{3}{5}g'p^{3}q^{2}
+    + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3} + \text{etc.}
+\bigr].
+\end{alignat*}
+Therefore we have, since
+\begin{gather*}
+r\cos \psi = \tfrac{1}{2}\frac{\dd(r^{2})}{\dd q}, \\
+\Tag{[3]}
+\begin{alignedat}[t]{2}
+r\cos\psi = q + \tfrac{2}{3}f°p^{2}q
+  &+ \tfrac{1}{2}f'p^{3}q
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q + \text{etc.} \\
+  &+ \tfrac{3}{4}g°p^{2}q^{2}
+  &&+ \tfrac{3}{5}g'p^{3}q^{2}
+    + (\tfrac{4}{15}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}.
+\end{alignedat}
+\end{gather*}
+
+\LineRef[21]{24}{Art.~24, p.~37}. Derivation of formula~[4].
+
+Since $r\cos\phi$ becomes equal to~$p$ for infinitely small values of $p$~and~$q$, the series
+for~$r\cos\phi$ must begin with~$p$. Hence we set
+\[
+\Tag{(1)}
+r\cos\phi = p + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.}
+\]
+Then, by differentiating, we obtain
+\begin{alignat*}{2}
+\Tag{(2)}
+\frac{\dd(r\cos\phi)}{\dd p}
+  &= 1 + {}&&\frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
+       + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p} + \text{etc.}\Add{,} \\
+\Tag{(3)}
+\frac{\dd(r\cos\phi)}{\dd q}
+  &= &&\frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
+   + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q} + \text{etc.} \\
+\end{alignat*}
+By dividing~[2] \Pageref[p.]{57} by~$n$ on \Pageref{36}, we obtain
+\begin{align*}
+\Tag{(4)}
+\frac{r\sin\psi}{n} = p - \tfrac{4}{3}f°pq^{2}
+  &- \tfrac{5}{4}f'p^{2}q^{2}
+   - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2} - \text{etc.} \\
+  &- \tfrac{3}{2}g°pq^{3} - \tfrac{7}{5}g'p^{2}q^{3}
+   - (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}.
+\end{align*}
+Multiplying (2) by~(4), we have
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(5)}
+\frac{r\sin\psi}{n}·\frac{\dd(r\cos\phi)}{\dd p} \\
+\begin{aligned}
+  = p &+ p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+   + p\frac{\dd R_{4}}{\dd p} + p\frac{\dd R_{5}}{\dd p}
+   - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q^{2} \\
+  &\begin{alignedat}{3}
+    - \tfrac{4}{3}f°pq^{2}
+    &- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+    &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+    &&- \tfrac{7}{5}g'p^{2}q^{3} \\
+    &- \tfrac{5}{4}f'p^{2}q^{2}
+    &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+    &&- (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4} \\
+    &\Typo{}{-}\tfrac{3}{2}g°pq^{3}
+    &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+    &&- \text{etc.}
+  \end{alignedat}
+\end{aligned}
+\end{multline*}
+\PageSep{59}
+Multiplying (3) by~[3] \Pageref[p.]{58}, we have
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(6)}
+r\cos\psi·\frac{\dd(r\cos\phi)}{\dd q} \\
+\begin{alignedat}{3}
+  = q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
+  & + q\frac{\dd R_{4}}{\dd q}
+  &&+ q\frac{\dd R_{5}}{\dd q}
+  &&+ \tfrac{1}{2}f'p^{3}q \frac{\dd R_{2}}{\dd q} \\
+  & + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} + \text{etc.}
+\end{alignedat}
+\end{multline*}
+Since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd(r\cos\phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(r\cos\phi)}{\dd q} = r\cos\phi,
+\]
+we have, by setting (1)~equal to the sum of (5)~and~(6),
+\begin{multline*}
+p + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.} \\
+\begin{alignedat}{5}
+= p &+ p\frac{\dd R_{2}}{\dd p}
+  &&+ p\frac{\dd R_{3}}{\dd p}
+  &&+ p\frac{\dd R_{4}}{\dd p}
+  &&+ p\frac{\dd R_{5}}{\dd p}
+  &&- (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4} \\
+%
+  &+ q\frac{\dd R_{2}}{\dd q}
+  &&- \tfrac{4}{3}f°pq^{2}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+  &&+ q\frac{\dd R_{5}}{\dd q} \\
+%
+  &&&+ q\frac{\dd R_{3}}{\dd q}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q} \\
+%
+  &&&&&- \tfrac{3}{2}g°pq^{3}
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+  &&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q} \\
+%
+  &&&&&+ q\frac{\dd R_{4}}{\dd q}
+  &&- (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} \\
+%
+  &&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &&- \tfrac{7}{5}g'p^{2}q^{3} + \text{etc.},
+\end{alignedat}
+\end{multline*}
+from which we find
+\begin{align*}
+R_{2} &= 0,\quad
+R_{3} = \tfrac{2}{3}f°pq^{2},\quad
+R_{4} = \tfrac{5}{12}f'p^{2}q^{2} + \tfrac{1}{2}g°pq^{3}, \\
+R_{5} &= \tfrac{7}{20}g'p^{2}q^{3}
+  + (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}
+  + (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}.
+\end{align*}
+Therefore we have finally
+\begin{alignat*}{2}
+\Tag{[4]}
+r\cos\phi = p + \tfrac{2}{3}f°pq^{2}
+  &+ \tfrac{5}{12}f'p^{2}q^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2} + \text{etc.} \\
+  &+ \tfrac{1}{2}g°pq^{3}
+  &&+ \tfrac{7}{20}g'p^{2}q^{3} \\
+  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}.
+\end{alignat*}
+
+\LineRef[21]{24}{Art.~24, p.~37}. Derivation of formula~[5].
+
+Again, since $r\sin\phi$ becomes equal to~$q$ for infinitely small values of $p$~and~$q$,
+we set
+\[
+\Tag{(1)}
+r\sin\phi = q + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.}
+\]
+\PageSep{60}
+Then we have by differentiation
+\begin{alignat*}{2}
+\Tag{(2)}
+\frac{\dd(r\sin\phi)}{\dd p}
+  &= &&\frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
+   + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p} + \text{etc.} \\
+\Tag{(3)}
+\frac{\dd(r\sin\phi)}{\dd q}
+  &= 1 + {} &&\frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
+       + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q} + \text{etc.} \\
+\end{alignat*}
+Multiplying (4) \Pageref[p.]{58} by this~(2), we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(4)}
+\frac{r\sin\psi}{n}·\frac{\dd(r\sin\phi)}{\dd p} \\
+\begin{alignedat}{3}
+  = p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+  &+ p\frac{\dd R_{4}}{\dd p}
+  &&+ p\frac{\dd R_{5}}{\dd p}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p} \\
+  & - \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p} - \text{etc.}
+\end{alignedat}
+\end{multline*}
+Likewise from (3)~and~[3] \Pageref[p.]{58}, we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(5)}
+r\cos\psi·\frac{\dd(r\sin\phi)}{\dd q} \\
+\begin{aligned}
+  = q &+ q\frac{\dd R_{2}}{\dd q}
+  + q\frac{\dd R_{3}}{\dd q}
+  + q\frac{\dd R_{4}}{\dd q}
+  + q\frac{\dd R_{5}}{\dd q}
+  + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
+%
+  &\begin{aligned}
+   + \tfrac{2}{3}f°p^{2}q
+   + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+   - \tfrac{3}{5}g'p^{3}q^{2} \\
+%
+   + \tfrac{1}{2}f'p^{3}q
+  &+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+   + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3} \\
+%
+   + \tfrac{3}{4}g°p^{2}q^{2}
+  &+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q} + \text{etc.}
+\end{aligned}
+\end{aligned}
+\end{multline*}
+Since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd(r\sin\phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(r\sin\phi)}{\dd q} = r\sin\phi,
+\]
+by setting (1) equal to the sum of (4)~and~(5), we have
+{\small
+\begin{multline*}
+q + R_{2} + R_{3} + R_{4} + R_{5} + \text{etc.} \\
+\begin{alignedat}{4}
+= q &+ p\frac{\dd R_{2}}{\dd p}
+   &&+ p\frac{\dd R_{3}}{\dd p}
+   &&+ p\frac{\dd R_{4}}{\dd p} + \tfrac{1}{2}f'p^{3}q
+   &&+ p\frac{\dd R_{5}}{\dd p}
+     + \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+     + q\frac{\dd R_{5}}{\dd q} + \text{etc.} \\
+%
+   & + q\frac{\dd R_{2}}{\dd q}
+   &&+ q\frac{\dd R_{3}}{\dd q}
+   &&+ q\frac{\dd R_{4}}{\dd q} + \tfrac{3}{4}g°p^{2}q^{2}
+   &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+     + \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q}
+     + \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q} \\
+%
+   &&&+ \tfrac{2}{3}f°p^{2}q
+   &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+   &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+     + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q \\
+%
+ &&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+   &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+     + \tfrac{3}{5}g'p^{3}q^{2}
+     + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\Typo{.}{,}
+\end{alignedat}
+\end{multline*}}%
+\PageSep{61}
+from which we find
+\begin{align*}
+R_{2} &= 0,\quad
+R_{3} = -\tfrac{1}{3}f°p^{2}q,\quad
+R_{4} = -\tfrac{1}{6}f'p^{3}q - \tfrac{1}{4}g°p^{2}q^{2}, \\
+R_{5} &= -(\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q
+  - \tfrac{3}{20}g'p^{3}q^{2}
+  - (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}.
+\end{align*}
+Therefore, substituting these values in~(1), we have
+\begin{alignat*}{2}
+\Tag{[5]}
+r\sin\phi = q - \tfrac{1}{3}f°p^{2}q
+  &-  \tfrac{1}{6}f'p^{3}q
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q - \text{etc.} \\
+  &-  \tfrac{1}{4}g°p^{2}q^{2}
+  &&- \tfrac{3}{20}g'p^{3}q^{2} \\
+  &&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q^{3}.
+\end{alignat*}
+
+\LineRef[22]{24}{Art.~24, p.~38}. Derivation of formula~[6].
+
+Differentiating $n$ on \Pageref{36} with respect to~$q$, we obtain
+\begin{alignat*}{3}
+\Tag{(1)}
+\frac{\dd n}{\dd q} = 2f°q
+  &+  2f'pq &&+ 2f''p^{2}q &&+ \text{etc.} \\
+  &\Typo{}{+} 3g°q^{2} &&+ 3g'pq^{2} &&+ \text{etc.} \\
+  &&&+ 4h°q^{3} &&+ \text{etc.\Typo{,}{} etc.}\Add{,}
+\end{alignat*}
+and hence, multiplying this series by~(4) on \Pageref{58}, we find
+\begin{align*}
+\Tag{(2)}
+\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q} = 2f°pq
+  &+ 2f'p^{2}q + 2f''p^{3}q + 3g'p^{2}q^{2} + \text{etc.} \\
+  &+ 3g°pq^{2} + (4h° - \tfrac{8}{3}{f°}^{2})pq^{3}.
+\end{align*}
+
+For infinitely small values of $r$, $\psi + \phi = \dfrac{\pi}{2}$, as is evident from the figure on \Pgref{fig:055}.
+Hence we set
+\[
+\psi + \phi = \frac{\pi}{2} + R_{1} + R_{2} + R_{3} + R_{4} + \text{etc.}
+\]
+Then we shall have, by differentiation,
+\begin{align*}
+\Tag{(3)}
+\frac{\dd(\psi + \phi)}{\dd p}
+  &= \frac{R_{1}}{\dd p} + \frac{R_{2}}{\dd p}
+   + \frac{R_{3}}{\dd p} + \frac{R_{4}}{\dd p} + \text{etc.}\Add{,} \\
+\Tag{(4)}
+\frac{\dd(\psi + \phi)}{\dd q}
+  &= \frac{R_{1}}{\dd q} + \frac{R_{2}}{\dd q}
+   + \frac{R_{3}}{\dd q} + \frac{R_{4}}{\dd q} + \text{etc.} \\
+\end{align*}
+Therefore, multiplying (4) on \Pageref{58} by~(3), we find
+\begin{alignat*}{2}
+\Tag{(5)}
+\frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
+  = p\frac{R_{1}}{\dd p} + p\frac{R_{2}}{\dd p}
+  &+  p\frac{R_{3}}{\dd p}
+  &&+ p\frac{R_{4}}{\dd p} + \text{etc.} \\
+%
+  &-  \tfrac{4}{3}f°pq^{2}\frac{\dd R_{1}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p} \\
+%
+&&&-\tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{1}}{\dd p} \\
+&&&-\tfrac{3}{2}g°pq^{3}\frac{\dd R_{1}}{\dd p},
+\end{alignat*}
+\PageSep{62}
+and, multiplying~[3] on \Pageref{58} by~(4), we find
+\begin{alignat*}{2}
+\Tag{(6)}
+r\cos\psi·\frac{\dd(\psi + \phi)}{\dd q}
+  = q\frac{R_{1}}{\dd q} + q\frac{R_{2}}{\dd q}
+  &+  q\frac{R_{3}}{\dd q}
+  &&+ q\frac{R_{4}}{\dd q} + \text{etc.} \\
+%
+  &+  \tfrac{2}{3}f°p^{2}q\frac{\dd R_{1}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q} \\
+%
+&&&+\tfrac{1}{2}f'p^{3}q\frac{\dd R_{1}}{\dd q} \\
+&&&+\tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{1}}{\dd q}.
+\end{alignat*}
+And since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd n}{\dd q}
+  + \frac{r\sin\psi}{n}·\frac{\dd(\psi + \phi)}{\dd p}
+  + r\cos\psi·\frac{\dd(\psi + \phi)}{\dd p} = 0,
+\]
+we shall have, by adding (2),~(5), and~(6),
+\begin{alignat*}{5}
+0 &= p\frac{\dd R_{1}}{\dd p}
+  &&+ 2f°pq
+  &&+ 2f'p^{2}q
+  &&+ 2f''p^{3}q
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{1}}{\dd p} \\
+%
+  &\Typo{}{+} q\frac{\dd R_{1}}{\dd q}
+  &&+ p\frac{\dd R_{2}}{\dd p}
+  &&+ 3g°pq^{2}
+  &&+ 3g'p^{2}q^{2}
+  &&+ q\frac{\dd R_{4}}{\dd q} \\
+%
+ &&&+ q\frac{\dd R_{2}}{\dd q}
+  &&+ p\frac{\dd R_{3}}{\dd p}
+  &&+ (4h° - \tfrac{8}{3}{f°})pq^{3}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q} \\
+%
+&&&&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{1}}{\dd p}
+  &&+ p\frac{\dd R_{4}}{\dd p}
+  && +\tfrac{1}{2}f'p^{3}q\frac{\dd R_{1}}{\dd q} \\
+%
+&&&&&+ q\frac{\dd R_{3}}{\dd q}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{1}}{\dd q} \\
+%
+&&&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{1}}{\dd q}
+  &&- \tfrac{5}{4}f°p^{2}q^{2}\frac{\dd R_{1}}{\dd p}
+  &&+ \text{etc.}
+\end{alignat*}
+From this equation we find
+\begin{align*}
+R_{1} &= 0,\quad
+R_{2} = -f°pq,\quad
+R_{3} = -\tfrac{2}{3}f' p^{2}q - g°pq^{2}, \\
+R_{4} &= -(\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q
+  - \tfrac{3}{4}g'p^{2}q^{2} - (h° - \tfrac{1}{3}{f°}^{2})pq^{3}.
+\end{align*}
+Therefore we have finally
+\begin{alignat*}{2}
+\Tag{[6]}
+\psi + \phi = \frac{\pi}{2} - f°pq
+  &-  \tfrac{2}{3}f'p^{2}q
+  &&- (\tfrac{1}{2}f'' - \tfrac{1}{6}{f°}^{2})p^{3}q - \text{etc.} \\
+  &- g°pq^{2}
+  && -\tfrac{3}{4}g'p^{2}q^{2} \\
+  &&&- (h° - \tfrac{1}{3}{f°}^{2})pq^{3}.
+\end{alignat*}
+\PageSep{63}
+%XXXX
+
+\LineRef[23]{24}{Art.~24, p.~38, l.~19}. The differential equation from which formula~[7] follows
+is derived in the following manner. In the figure on \Pgref{fig:055}, prolong $AD$ to~$D'$,
+making $DD' = dp$\Chg{, through}{. Through}~$D'$ perpendicular to~$AD'$ draw a geodesic line, which will
+cut~$AB$ in~$B'$. Finally, take $D'B'' = DB$, so that $BB''$~is perpendicular to~$B'D'$.
+Then, if by~$ABD$ we mean the area of the triangle~$ABD$,
+\[
+\frac{\dd S}{\dd r} = \lim \frac{AB'D' - ABD}{BB'}
+  = \lim \frac{BDD'B'}{BB'}
+  = \lim \frac{BDD'B''}{DD'}·\lim \frac{DD'}{BB'},
+\]
+since the surface $BDD'B''$ differs from $BDD'B'$ only by an infinitesimal of the
+second order. And since
+\[
+BDD'B'' = dp·\int n\, dq,\quad\text{or}\quad
+\lim \frac{BDD'B''}{DD'} = \int n\, dq,
+\]
+and since, further,
+\[
+\lim \frac{DD'}{BB'} = \frac{\dd p}{\dd r},
+\]
+consequently
+\[
+\frac{\dd S}{\dd r} = \frac{\dd p}{\dd r}·\int n\, dq.
+\]
+Therefore also
+\[
+\frac{\dd S}{\dd p}·\frac{\dd p}{\dd r} +
+\frac{\dd S}{\dd q}·\frac{\dd q}{\dd r} = \frac{\dd p}{\dd r}·\int n\, dq.
+\]
+Finally, from the values for $\dfrac{\dd r}{\dd p}$, $\dfrac{\dd r}{\dd q}$ given at the beginning of \LineRef{24}{Art.~24, p.~36}, we have
+\[
+\frac{\dd p}{\dd r} = \frac{1}{n}\sin\psi,\quad
+\frac{\dd q}{\dd r} = \cos\psi,
+\]
+so that we have
+\[
+\frac{\dd S}{\dd p}·\frac{\sin\psi}{n} +
+\frac{\dd S}{\dd q}·\cos\psi = \frac{\sin\psi}{n}·\int n\, dq.
+\]
+\null\hfill[Wangerin.]
+
+\LineRef[24]{24}{Art.~24, p.~38}. Derivation of formula~[7].
+
+For infinitely small values of $p$~and~$q$, the area of the triangle~$ABC$ becomes
+equal to $\frac{1}{2}pq$. The series for this area, which is denoted by~$S$, must therefore begin
+with~$\frac{1}{2}pq$, or~$R_{2}$. Hence we put
+\[
+S = R_{2} + R_{3} + R_{4} + R_{5} + R_{6} + \text{etc.}
+\]
+\PageSep{64}
+By differentiating, we obtain
+\begin{align*}
+\Tag{(1)}
+\frac{\dd S}{\dd p}
+  &= \frac{\dd R_{2}}{\dd p} + \frac{\dd R_{3}}{\dd p}
+   + \frac{\dd R_{4}}{\dd p} + \frac{\dd R_{5}}{\dd p}
+   + \frac{\dd R_{6}}{\dd p} + \text{etc.}, \\
+\Tag{(2)}
+\frac{\dd S}{\dd q}
+  &= \frac{\dd R_{2}}{\dd q} + \frac{\dd R_{3}}{\dd q}
+   + \frac{\dd R_{4}}{\dd q} + \frac{\dd R_{5}}{\dd q}
+   + \frac{\dd R_{6}}{\dd q} + \text{etc.}, \\
+\end{align*}
+and therefore, by multiplying~(4) on \Pageref{58} by~(1), we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(3)}
+\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p} \\
+\begin{alignedat}{3}
+  = p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+  &  + p\frac{\dd R_{4}}{\dd p}
+  && + p\frac{\dd R_{5}}{\dd p}
+  && + p\frac{\dd R_{6}}{\dd p} + \text{etc.} \\
+%
+  & - \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+  &&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{4}}{\dd p} \\
+%
+ &&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p} \\
+%
+ &&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+  &&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{3}}{\dd p} \\
+%
+&&&&&- (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\frac{\dd R_{2}}{\dd p} \\
+&&&&&- \tfrac{7}{5}g'p^{2}q^{3}\frac{\dd R_{2}}{\dd p} \\
+&&&&&- (\tfrac{3}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}\frac{\dd R_{2}}{\dd p},
+\end{alignedat}
+\end{multline*}
+and multiplying~[3] on \Pageref{58} by~(2), we obtain
+\begin{multline*}
+%[** TN: Re-broken]
+\Tag{(4)}
+r\cos\psi·\frac{\dd S}{\dd q} \\
+\begin{alignedat}{3}
+  = q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
+  &  + q\frac{\dd R_{4}}{\dd q}
+  && + q\frac{\dd R_{5}}{\dd q}
+  && + q\frac{\dd R_{6}}{\dd q} + \text{etc.} \\
+%
+  &+  \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+  &&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{4}}{\dd q} \\
+%
+ &&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{3}}{\dd q} \\
+%
+ &&&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q}
+  &&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q} \\
+%
+&&&&&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\frac{\dd R_{2}}{\dd q} \\
+&&&&&+ \tfrac{3}{5}g'p^{3}q^{2}\frac{\dd R_{2}}{\dd q} \\
+&&&&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\frac{\dd R_{2}}{\dd q}.
+\end{alignedat}
+\end{multline*}
+\PageSep{65}
+Integrating~$n$ on \Pageref{36} with respect to~$q$, we find
+\begin{alignat*}{2}
+\Tag{(5)}
+\int n\, dq = q + \tfrac{1}{2}f°q^{3}
+  &+  \tfrac{1}{3}f'pq^{3}
+  &&+ \tfrac{1}{3}f''p^{2}q^{3} + \text{etc.} \\
+%
+  &+  \tfrac{1}{4}g°q^{4}
+  &&+ \tfrac{1}{4}g'pq^{4} + \text{etc.} \\
+  &&&+\tfrac{1}{5}h°q^{5} + \text{etc.\ etc.}
+\end{alignat*}
+Multiplying~(4) on \Pageref{58} by~(5), we find
+\begin{alignat*}{2}
+\Tag{(6)}
+\frac{r\sin\psi}{n}·\int n\, dq = pq - f°pq^{3}
+  &- \tfrac{11}{12}f'p^{2}q^{3}
+  &&- (\tfrac{13}{15}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{3} - \text{etc.} \\
+%
+  &-  \tfrac{5}{4}g°pq^{4}
+  &&- \tfrac{23}{20}g'p^{2}q^{4} \\
+  &&&- (\tfrac{7}{5}h° - \tfrac{16}{15}{f°}^{2})pq^{5}.
+\end{alignat*}
+Since
+\[
+\frac{r\sin\psi}{n}·\frac{\dd S}{\dd p} + r\cos\psi·\frac{\dd S}{\dd q}
+  = \frac{r\sin\psi}{n}·\int n\, dq,
+\]
+we obtain, by setting (6) equal to the sum of (3)~and~(4),
+{\footnotesize
+\begin{alignat*}{4}
+&\Neg pq
+&&- f°pq^{3}
+&&- \tfrac{11}{12}f'p^{2}q^{3}
+&&- (\tfrac{13}{15}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{3} - \text{etc.} \\
+%
+&&&&&- \tfrac{5}{4}g°pq^{4}
+&&- \tfrac{23}{20}g'p^{2}q^{4} \\
+%
+&&&&&&&- (\tfrac{7}{5}h° - \tfrac{16}{15}{f°}^{2})pq^{5} \\
+%%
+&=  p\frac{\dd R_{2}}{\dd p} + p\frac{\dd R_{3}}{\dd p}
+&&+ p\frac{\dd R_{4}}{\dd p}
+&&+ p\frac{\dd R_{5}}{\dd p} + q\frac{\dd R_{5}}{\dd q}
+&&+ p\frac{\dd R_{6}}{\dd p}
+  + \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{3}}{\dd q} + \text{etc.} \\
+%
+&+  q\frac{\dd R_{2}}{\dd q} + q\frac{\dd R_{3}}{\dd q}
+&&+ q\frac{\dd R_{4}}{\dd q}
+&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{3}}{\dd p}
+&&+ q\frac{\dd R_{6}}{\dd q}
+  - (\tfrac{6}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{3}q^{2}\frac{\dd R_{2}}{\dd p} \\
+&&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{2}}{\dd p}
+&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{3}}{\dd q}
+&&- \tfrac{4}{3}f°pq^{2}\frac{\dd R_{4}}{\dd p}
+  + (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q\frac{\dd R_{2}}{\dd q} \\
+%
+&&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{2}}{\dd q}
+&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{2}}{\dd p}
+&&+ \tfrac{2}{3}f°p^{2}q\frac{\dd R_{4}}{\dd q}
+  - \tfrac{7}{5}g'p^{2}q^{3}\frac{\dd R_{2}}{\dd p} \\
+%
+&&&&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{2}}{\dd q}
+&&- \tfrac{5}{4}f'p^{2}q^{2}\frac{\dd R_{3}}{\dd p}
+  + \tfrac{3}{5}g'p^{3}q^{2}\frac{\dd R_{2}}{\dd q} \\
+%
+&&&&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{2}}{\dd p}
+&&+ \tfrac{1}{2}f'p^{3}q\frac{\dd R_{3}}{\dd q}
+  - (\tfrac{8}{5}h° - \tfrac{68}{45}{f°}^{2})pq^{4}\frac{\dd R_{2}}{\dd p} \\
+%
+&&&&&+ \tfrac{3}{4}g°p^{2}q^{2}\frac{\dd R_{2}}{\dd q}
+&&- \tfrac{3}{2}g°pq^{3}\frac{\dd R_{3}}{\dd p}
+  + (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q^{3}\frac{\dd R_{2}}{\dd q}.
+\end{alignat*}}%
+From this equation we find
+\begin{align*}
+R_{2} &= \tfrac{1}{2}pq,\quad
+R_{3} = 0,\quad
+R_{4} = -\tfrac{1}{12}f°pq^{3} - \tfrac{1}{12}f°p^{3}q, \\
+%
+R_{5} &= -\tfrac{1}{20}f'p^{4}q - \tfrac{3}{40}g°p^{3}q^{2}
+  - \tfrac{7}{120}f'p^{2}q^{3} - \tfrac{1}{10}g°pq^{4}, \\
+%
+R_{6} &= -(\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}
+  - (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
+  &\quad- \tfrac{3}{40}g'p^{2}q^{4}
+  - (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q
+  - \tfrac{1}{20}g'p^{4}q^{2}.
+\end{align*}
+\PageSep{66}
+Therefore we have
+\begin{alignat*}{3}
+\Tag{[7]}
+S = \tfrac{1}{2}pq
+  &-  \tfrac{1}{12}f°pq^{3}
+  &&- \tfrac{1}{20}f'p^{4}q
+  &&- (\tfrac{1}{30}f'' - \tfrac{1}{60}{f°}^{2})p^{5}q - \text{etc.} \\
+%
+  &-  \tfrac{1}{12}f°p^{3}q
+  &&- \tfrac{3}{40}g°p^{3}q^{2}
+  &&- \tfrac{1}{20}g'p^{4}q^{2} \\
+%
+  &&&- \tfrac{7}{120}f'p^{2}q^{3}
+  &&- (\tfrac{1}{15}h° + \tfrac{2}{45}f'' + \tfrac{1}{60}{f°}^{2})p^{3}q^{3} \\
+%
+  &&&- \tfrac{1}{10}g°pq^{4}
+  &&- \tfrac{3}{40}g'p^{2}q^{4} \\
+%
+  &&&&&- (\tfrac{1}{10}h° - \tfrac{1}{30}{f°}^{2})pq^{5}.
+\end{alignat*}
+
+\LineRef[25]{25}{Art.~25, p.~39, l.~17}. $3p^{2} + 4q^{2} + 4qq' + 4q'^{2}$ is replaced by $3p^{2} + 4q^{2} + 4q'^{2}$.
+This error appears in all the reprints and translations (except Wangerin's).
+
+\LineRef[26]{25}{Art.~25, p.~40, l.~8}. $3p^{2} - 2q^{2} + qq' + 4qq'$ is replaced by $3p^{2} - 2q^{2} + qq' + 4q'^{2}$.
+This correction is noted in all the translations, and in Liouville's reprint.
+
+\LineRef[27]{25}{Art.~25, p.~40}. Derivation of formulæ [8],~[9],~[10].
+
+By priming the~$q$'s in~[7] we obtain at once a series for~$S'$. Then, since
+$\sigma = S - S'$, we have
+\begin{alignat*}{3}
+\sigma = \tfrac{1}{2}p(q - q')
+  &-  \tfrac{1}{12}f°p^{3}(q - q')
+  &&- \tfrac{1}{20}f'p^{4}(q - q')
+  &&- \tfrac{3}{40}g°p^{3}(q^{2} - q'^{2}) \\
+%
+  &-  \tfrac{1}{12}f°p(q^{3} - q'^{3})
+  &&- \tfrac{7}{120}f'p^{2}(q^{3} - q'^{3})
+  &&- \tfrac{1}{10}g°p(q^{4} - q'^{4}),
+\end{alignat*}
+correct to terms of the sixth degree.
+%[** TN: Omitted line break in the original]
+This expression may be written as follows:
+\begin{align*}
+\sigma = \tfrac{1}{2}p(q - q')
+  \bigl(1 &- \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2}) \\
+  &- \tfrac{1}{60}f'p(6p^{2} + 7q^{2} + 7qq' + 7q'^{2}) \\
+  &- \tfrac{1}{20}g°(q + q')(3p^{2} + 4q^{2} + 4q'^{2})\bigr),
+\end{align*}
+or, after factoring,
+{\small
+\begin{multline*}
+\Tag{(1)}
+\sigma = \tfrac{1}{2}p(q - q')
+  \bigl(1 - \tfrac{1}{3}f°q^{2} - \tfrac{1}{4}f'pq^{2} - \tfrac{1}{2}g°q^{3}\bigr)
+  \bigl(1 - \tfrac{1}{6}f°(p^{2} - q^{2} + qq' + q'^{2}) \\
+  - \tfrac{1}{60}f'p(6p^{2} - 8q^{2} + 7qq' + 7q'^{2})
+  - \tfrac{1}{20}g°(3p^{2}q + 3p^{2}q' - 6q^{3} + 4q^{2}q' + 4qq'^{2} + 4q'^{3})\bigr).
+\end{multline*}}%
+The last factor on the right in~(1) can be written, thus:
+\begin{alignat*}{5}
+\bigl(1 &- \tfrac{2}{120}f°(4p^{2})
+  &&- \tfrac{2}{120}f°(3p^{2})
+  &&- \tfrac{2}{120}f'p(6qq')
+  &&- \tfrac{2}{120}f°(3p^{2})
+  &&- \tfrac{2}{120}f'p(qq') \\
+%
+  &+  \tfrac{2}{120}f°(2q^{2})
+  &&+ \tfrac{2}{120}f°(6q^{2})
+  &&- \tfrac{2}{120}f'p(3q'^{2})
+  &&+ \tfrac{2}{120}f°(2q^{2})
+  &&- \tfrac{2}{120}f'p(4q'^{2}) \\
+%
+  &-  \tfrac{2}{120}f°(3qq')
+  &&- \tfrac{2}{120}f°(6qq')
+  &&- \tfrac{6}{120}g°q(3p^{2})
+  &&- \tfrac{2}{120}f°(qq')
+  &&- \tfrac{6}{120}g°q'(3p^{2}) \\
+%
+  &-  \tfrac{2}{120}f°(3q'^{2})
+  &&- \tfrac{2}{120}f°(3q'^{2})
+  &&+ \tfrac{6}{120}g°q(6q^{2})
+  &&- \tfrac{2}{120}f°(4q'^{2})
+  &&+ \tfrac{6}{120}g°q'(2q^{2}) \\
+%
+ &&&- \tfrac{2}{120}f'p(3p^{2})
+  &&- \tfrac{6}{120}g°q(6qq')
+  &&- \tfrac{2}{120}f'p(3p^{2})
+  &&- \tfrac{6}{120}g°q'(qq') \\
+%
+ &&&+ \tfrac{2}{120}f'p(6q^{2})
+  &&- \tfrac{6}{120}g°q(3q'^{2})
+  &&+ \tfrac{2}{120}f'p(2q^{2})
+  &&- \tfrac{6}{120}g°q'(4q'^{2})\bigr).
+\end{alignat*}
+We know, further, that
+\begin{align*}
+&\,k = -\frac{1}{n}·\frac{\dd^{2} n}{\dd q^{2}}
+   = -2f - 6gq - (12h - 2f^{2})q^{2} - \text{etc.}, \\
+\PageSep{67}
+&\begin{alignedat}{4}
+f &= f° &&+ f'p &&+ f''p^{2} &&+ \text{etc.}, \\
+g &= g° &&+ g'p &&+ g''p^{2} &&+ \text{etc.}, \\
+h &= h° &&+ h'p &&+ h''p^{2} &&+ \text{etc.}
+\end{alignedat}
+\end{align*}
+Hence, substituting these values for $f$,~$g$, and~$h$ in~$k$, we have at~$B$ where $k = \beta$,
+correct to terms of the third degree,
+\begin{align*}
+\beta &= -2f° - 2f'p - 6g°q - 2f''p^{2} - 6g'pq - (12h° - 2{f°}^{2})q^{2}.
+\intertext{Likewise, remembering that $q$~becomes~$q'$ at~$C$, and that both $p$~and~$q$ vanish at~$A$,
+we have}
+\gamma &= -2f° - 2f'p - 6g°q' - 2f''p^{2} - 6g'pq' - (12h° - 2{f°}^{2})q'^{2}, \\
+\alpha &= -2f°.
+\end{align*}
+And since $c\sin B = r\sin\psi$,
+\[
+c\sin B = p(1 - \tfrac{1}{3}f°q^{2} - \tfrac{1}{4}f'pq^{2}
+              - \tfrac{1}{2}g°q^{3} - \text{etc.}).
+\]
+
+Now, if we substitute in~(1) $c\sin B$, $\alpha$,~$\beta$,~$\gamma$ for the series which they represent,
+and $a$~for~$q - q'$, we obtain (still correct to terms of the sixth degree)
+\begin{align*}
+\sigma = \tfrac{1}{2}ac\sin B\bigl(1
+  &+ \tfrac{1}{120}\alpha(4p^{2} - 2q^{2} +  3qq' + 3q'^{2}) \\
+  &+ \tfrac{1}{120}\beta (3p^{2} - 6q^{2} +  6qq' + 3q'^{2}) \\
+  &+ \tfrac{1}{120}\gamma(3p^{2} - 2q^{2} +\Z qq' + 4q'^{2})\bigr).
+\end{align*}
+And if in this equation we replace $p$,~$q$,~$q'$ by $c\sin B$, $c\cos B$, $c\cos B - a$, respectively,
+we shall have
+\begin{align*}
+\Tag{[8]}
+\sigma = \tfrac{1}{2}ac\sin B\bigl(1
+  &+ \tfrac{1}{120}\alpha(3a^{2} + 4c^{2} - \Z9ac\cos B) \\
+  &+ \tfrac{1}{120}\beta (3a^{2} + 3c^{2} -  12ac\cos B) \\
+  &+ \tfrac{1}{120}\gamma(4a^{2} + 3c^{2} - \Z9ac\cos B)\bigr).
+\end{align*}
+
+By writing for $B$,~$\alpha$,~$\beta$,~$a$ in~[8], $A$,~$\beta$,~$\alpha$,~$b$ respectively, we obtain at once
+formula~[9]. Likewise by writing for $B$,~$\beta$,~$\gamma$,~$c$ in~[8], $C$,~$\gamma$,~$\beta$,~$b$ respectively, we
+obtain formula~[10]. Formulæ [9]~and~[10] can, of course, also be derived by the
+method used to derive~[8].
+
+\LineRef[28]{26}{Art.~26, p.~41, l.~11}. The right hand side of this equation should have the positive
+sign. All the editions prior to Wangerin's have the incorrect sign.
+
+\LineRef[29]{26}{Art.~26, p.~41}. Derivation of formula~[11].
+
+We have
+\begin{align*}
+\Tag{(1)}
+r^{2}
+  &+ r'^{2} - (q - q')^{2} - 2r\cos\phi·r'\cos\phi' - 2r\sin\phi·r'\sin\phi' \\
+  &= b^{2} + c^{2} - a^{2} - 2bc\cos(\phi - \phi') \\
+  &= 2bc(\cos A^{*} - \cos A),
+\end{align*}
+since $b^{2} + c^{2} - a^{2} = 2bc\cos A^{*}$ and $\cos(\phi - \phi') = \cos A$.
+\PageSep{68}
+
+By priming the $q$'s in formulæ [1],~[4],~[5] we obtain at once series for~$r'^{2}$,
+$r'\cos\phi'$, $r'\sin\phi'$. Hence we have series for all the terms in the above expression,
+and also for the terms in the expression:
+\[
+\Tag{(2)}
+r\sin\phi·r'\cos\phi' - r\cos\phi·r'\sin\phi' = bc\sin A,
+\]
+namely,
+\begin{alignat*}{3}
+\Tag{(3)}
+r^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q^{2}
+  &&+ \tfrac{1}{2}f'p^{3}q^{2}
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q^{2} + \text{etc.} \\
+%
+  &+ q^{2}
+  &&+ \tfrac{1}{2}g°p^{2}q^{3}
+  &&+ \tfrac{2}{5}g'p^{3}q^{3} \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q^{4},
+\displaybreak[1] \\
+%
+\Tag{(4)}
+r'^{2} &= p^{2} + \tfrac{2}{3}f°p^{2}q'^{2}
+  &&+ \tfrac{1}{2}f'p^{3}q'^{2}
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q'^{2} + \text{etc.} \\
+%
+  &+ q'^{2}
+  &&+ \tfrac{1}{2}g°p^{2}q'^{3}
+  &&+ \tfrac{2}{5}g'p^{3}q'^{3} \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}q'^{4},
+\end{alignat*}
+\[
+\Tag{(5)}
+-(q - q')^{2} = -q^{2} + 2qq' - q'^{2},
+\]
+\begin{alignat*}{3}
+\Tag{(6)}
+2r\cos\phi
+  &= 2p + \tfrac{4}{3}f°pq^{2}
+  &&+ \tfrac{10}{12}f'p^{2}q^{2}
+  &&+ (\tfrac{6}{10}f'' - \tfrac{16}{45}{f°}^{2})p^{3}q^{2} + \text{etc.} \\
+%
+  &&&+ g°pq^{3}
+  &&+ \tfrac{14}{20}g'p^{2}q^{3} \\
+%
+  &&&&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})pq^{4},
+\displaybreak[1] \\
+%
+\Tag{(7)}
+r'\cos\phi'
+  &= p + \tfrac{2}{3}f°pq'^{2}
+  &&+ \tfrac{5}{12}f'p^{2}q'^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q'^{2} + \text{etc.} \\
+%
+  &&&+ \tfrac{1}{2}g°pq'^{3}
+  &&+ \tfrac{7}{20}g'p^{2}q'^{3} \\
+%
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq'^{4},
+\displaybreak[1] \\
+%
+\Tag{(8)}
+2r\sin\phi
+  &= 2q - \tfrac{2}{3}f°p^{2}q
+  &&- \tfrac{2}{6}f'p^{3}q
+  &&- (\tfrac{2}{10}f'' - \tfrac{14}{90}{f°}^{2})p^{4}q - \text{etc.} \\
+%
+  &&&- \tfrac{2}{4}g°p^{2}q^{2}
+  &&- \tfrac{6}{20}g'p^{3}q^{2} \\
+%
+  &&&&&- (\tfrac{2}{5}h° + \tfrac{26}{90}{f°}^{2})p^{2}q^{3},
+\displaybreak[1] \\
+%
+\Tag{(9)}
+r'\sin\phi'
+  &= q' - \tfrac{1}{3}f°p^{2}q'
+  &&- \tfrac{1}{6}f'p^{3}q'
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4}q' + \text{etc.} \\
+%
+  &&&- \tfrac{1}{4}g°p^{2}q'^{2}
+  &&- \tfrac{3}{20}g'p^{3}q'^{2} \\
+%
+  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}q'^{3}.
+\end{alignat*}
+By adding (3),~(4), and~(5), we obtain
+\begin{multline*}
+\Tag{(10)}
+r^{2} + r'^{2} - (q - q')^{2} \\
+\begin{alignedat}{3}
+  &= 2p^{2} + \tfrac{2}{3}f°p^{2}(q^{2} + q'^{2})
+  &&+ \tfrac{1}{2}f'p^{3}(q^{2} + q'^{2})
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}(q^{2} + q'^{2}) + \text{etc.} \\
+  &+ 2qq'
+  &&+ \tfrac{1}{2}g°p^{2}(q^{3} + q'^{3})
+  &&+ \tfrac{2}{5}g'p^{3}(q^{3} + q'^{3}) \\
+%
+&&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} + q'^{4}).
+\end{alignedat}
+\end{multline*}
+On multiplying (6) by~(7), we obtain
+\begin{multline*}
+\Tag{(11)}
+2r\cos\phi·r'\cos\phi' \\
+\begin{alignedat}{2}
+  = 2p^{2} + \tfrac{4}{3}f°p^{2}(q^{2} + q'^{2})
+  &+ \tfrac{5}{6}f'p^{3}(q^{2} + q'^{2})
+  &&+ (\tfrac{3}{5}f'' - \tfrac{16}{45}{f°}^{2})p^{4}(q^{2} + q'^{2}) + \text{etc.} \\
+%
+  &+ g°p^{2}(q^{3} + q'^{3})
+  &&+ \tfrac{7}{10}g'p^{3}(q^{3} + q'^{3}) \\
+%
+  &&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}(q^{4} + q'^{4}) \\
+  &&&+ \tfrac{8}{9}{f°}^{2}p^{2}q^{2}q'^{2},
+\end{alignedat}
+\end{multline*}
+\PageSep{69}
+and multiplying (8) by~(9), we obtain
+\begin{alignat*}{3}
+\Tag{(12)}
+&\quad 2r\sin\phi·r'\sin\phi' \\
+&= 2qq' - \tfrac{4}{3}f°p^{2}qq'
+&&- \tfrac{2}{3}f'p^{3}qq'
+&&- (\tfrac{2}{5}f'' - \tfrac{24}{45}{f°}^{2})p^{4}qq' - \text{etc.} \\
+%
+&&&- \tfrac{1}{2}g°p^{2}qq'(q + q')
+&&- \tfrac{3}{10}g'p^{3}qq'(q + q') \\
+&&&&&- (\tfrac{2}{5}h° + \tfrac{13}{45}{f°}^{2})p^{2}qq'(q^{2} + q'^{2}).
+\end{alignat*}
+Hence by adding (11)~and~(12), we have
+{\small
+\begin{multline*}
+\Tag{(13)}
+2bc\cos A \\
+\begin{alignedat}{3}
+&= 2p^{2} &&+ \tfrac{4}{3}f°p^{2}(q^{2} + q'^{2})
+&&+ \tfrac{1}{6}f'p^{3}(5q^{2} - 4qq' + 5q'^{2})
+  - \tfrac{8}{45}{f°}^{2}p^{4}(2q^{2} + 2q'^{2} - 3qq') + \text{etc.} \\
+%
+&+ 2qq' &&- \tfrac{4}{3}f°p^{2}qq'
+&&+ \tfrac{1}{2}g°p^{2}(2q^{3} + 2q'^{3} - q^{2}q' - qq'^{2})
+\end{alignedat} \\
+\begin{aligned}
+&- \tfrac{1}{45}{f°}^{2}p^{2}(14q^{4} + 14q'^{4} + 13q^{3}q' + 13 qq'^{3} - 40q^{2}q'^{2}) \\
+&+ \tfrac{1}{10}g'p^{3}(7q^{3} + 7q'^{3} - 3q^{2}q' - 3qq'^{2}) \\
+&+ \tfrac{1}{5}f''p^{4}(3q^{2} + 3q'^{2} - 2qq') \\
+&+ \tfrac{2}{5}h°p^{2}(2q^{4} + 2q'^{4} - q^{3}q' - qq'^{3}).
+\end{aligned}
+\end{multline*}}%
+Therefore we have, by subtracting (13) from~(10),
+{\small
+\begin{multline*}
+2bc(\cos A^{*} - \cos A) \\
+\begin{aligned}
+= -\tfrac{2}{3}f°p^{2}(q^{2} + q'^{2} - 2qq')
+&- \tfrac{1}{3}f'p^{3}(q^{2} + q'^{2} - 2qq')
+ + \tfrac{4}{15}{f°}^{2}p^{4}(q^{2} + q'^{2} - 2qq') - \text{etc.} \\
+&- \tfrac{1}{2}g°p^{2}(q^{3} + q'^{3} - q^{2}q' - qq'^{2})
+ - \tfrac{1}{5}f''p^{4}(q^{2} + q'^{2} - 2qq')
+\end{aligned} \\
+\begin{aligned}
+&+ \tfrac{1}{45}{f°}^{2}p^{2}(7q^{4} + 7q'^{4} + 13q^{3}q'+ 13qq'^{3} - 40q^{2}q'^{2}) \\
+&- \tfrac{2}{5}h°p^{2}(q^{4} + q'^{4} - q^{3}q' - qq'^{3}) \\
+&- \tfrac{3}{10}g'p^{3}(q^{3} + q'^{3} - q^{2}q' - qq'^{2}),
+\end{aligned}
+\end{multline*}}%
+which we can write thus:
+\begin{multline*}
+\Tag{(14)}
+%[** TN: Re-broken, explicit narrowing]
+\qquad
+2bc(\cos A^{*} - \cos A) \\
+\begin{aligned}
+  = -2p^{2}(q - q')^{2}&
+  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
+&+ \tfrac{1}{5}h°(q^{2} + qq' + q'^{2})
+ + \tfrac{3}{20}g'p(q + q') \\
+&- \tfrac{2}{15}{f°}^{2}p^{2}
+ - \tfrac{1}{90}{f°}^{2}(7q^{2} + 7q'^{2} + 27qq')\bigr),\qquad
+\end{aligned}
+\end{multline*}
+correct to terms of the seventh degree.
+
+If we multiply (7) by~[5] on \Pageref{37}, we obtain
+\begin{alignat*}{3}
+\Tag{(15)}
+%[** TN: Re-broken]
+r\sin\phi·r'\cos\phi'
+= pq &+ \tfrac{2}{3}f°pqq'^{2}
+  &&+ \tfrac{5}{12}f'p^{2}qq'^{2}
+  &&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}qq'^{2} - \text{etc.} \\
+%
+  &- \tfrac{1}{3}f°p^{3}q
+  &&+ \tfrac{1}{2}g°pqq'^{3}
+  &&+ \tfrac{7}{20}g'p^{2}qq'^{3} \\
+%
+  &&&- \tfrac{1}{6}f'p^{4}q
+  &&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pqq'^{4} \\
+%
+  &&&- \tfrac{1}{4}g°p^{3}q^{2}
+  &&- \tfrac{2}{9}{f°}^{2}p^{3}qq'^{2} \\
+%
+  &&&&&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{5}q \\
+  &&&&&- \tfrac{3}{20}g'p^{4}q^{2} \\
+  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{3}q^{3}.
+\end{alignat*}
+\PageSep{70}
+And multiplying (9) by formula~[4] on \Pageref{37}, we obtain
+\begin{alignat*}{3}
+\Tag{(16)}
+%[** TN: Re-broken]
+r\cos\phi·r'\sin\phi'
+= pq' &- \tfrac{1}{3}f°p^{3}q'
+  &&- \tfrac{1}{6}f'p^{4}q'
+  &&- (\tfrac{1}{16}f'' - \tfrac{7}{90}{f°}^{2})p^{5}q' + \text{etc.} \\
+%
+  &+  \tfrac{2}{3}f°pq^{2}q'
+  &&- \tfrac{1}{4}g°p^{3}q'^{2}
+  &&- \tfrac{3}{20}g'p^{4}q'^{2} \\
+%
+  &&&+ \tfrac{5}{12}f'p^{2}q^{2}q'
+  &&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{3}q'^{3} \\
+%
+  &&&&&+ (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{3}q^{2}q' \\
+  &&&&&+ \tfrac{7}{20}g'p^{2}q^{3}q' \\
+  &&&&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})pq^{4}q'.
+\end{alignat*}
+Therefore we have, by subtracting (16) from~(15),
+\begin{multline*}
+\Tag{(17)}
+bc\sin A \\
+\begin{alignedat}{3}
+=p(q - q')\bigl(1 &- \tfrac{1}{3}f°p^{2}
+  &&- \tfrac{5}{12}f'pqq'
+  &&- (\tfrac{3}{10}f'' - \tfrac{8}{45}{f°}^{2})p^{2}qq' \\
+%
+  &-  \tfrac{2}{3}f°qq'
+  &&- \tfrac{1}{6}f'p^{3}
+  &&- (\tfrac{1}{10}f'' - \tfrac{7}{90}{f°}^{2})p^{4} \\
+%
+  &&&- \tfrac{1}{2}g°qq'(q + q')
+  &&- \tfrac{7}{20}g'p qq'(q + q') \\
+%
+  &&&- \tfrac{1}{4}g°p^{2}(q + q')
+  &&- \tfrac{3}{20}g'p^{3}(q + q') \\
+%
+  &&&&&- (\tfrac{1}{5}h° + \tfrac{13}{90}{f°}^{2})p^{2}(q^{2} + qq' + q'^{2}) \\
+%
+  &&&&&- (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})qq'(q^{2} + qq' + q'^{2}) \\
+  &&&&&+ \tfrac{2}{9}{f°}^{2}p^{2}qq'\bigr),
+\end{alignedat}
+\end{multline*}
+correct to terms of the seventh degree.
+
+Let $A^{*} - A = \zeta$, whence $A^{*} = A + \zeta$, $\zeta$~being a magnitude of the second order.
+Hence we have, expanding $\sin\zeta$~and~$\cos\zeta$, and rejecting powers of~$\zeta$ above the second,
+\[
+\cos A^{*} = \cos A·\left(1 - \frac{\zeta^{2}}{2}\right) - \sin A·\zeta,
+\]
+or
+\[
+\cos A^{*} - \cos A = -\frac{\cos A}{2}·\zeta^{2} - \sin A·\zeta;
+\]
+or, multiplying both members of this equation by~$2bc$,
+\[
+\Tag{(18)}
+2bc(\cos A^{*} - \cos A) = -bc\cos A·\zeta^{2} - 2bc\sin A·\zeta.
+\]
+Further, let $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, where the~$R$'s have the same meaning as before.
+If now we substitute in~(18) for its various terms the series derived above, we shall
+have, on rejecting terms above the sixth degree,
+\begin{multline*}
+(p^{2} + qq')R_{2}^{2} + 2p(q - q')
+  \bigl(1 - \tfrac{1}{3}f°(p^{2} + 2qq')\bigr)\bigl(R_{2} + R_{3} + R_{4}\bigr) \\
+  = 2p^{2}(q - q')^{2}
+  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q') \\
+  + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2})
+  - \tfrac{1}{90}{f°}^{2}(12p^{2} + 7q^{2} + 7q'^{2} + 27qq')\bigr).)
+\end{multline*}
+\PageSep{71}
+Equating terms of like powers, and solving for $R_{2}$,~$R_{3}$,~$R_{4}$, we find
+\begin{align*}
+R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
+R_{3} = p(q - q')\bigl(\tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q')\bigr), \\
+R_{4} &= p(q - q')\bigl(\tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q')
+  + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
+  &\qquad- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 7q'^{2} + 12qq')\bigr).
+\end{align*}
+Therefore we have
+\begin{align*}
+A^{*} - A = p(q - q')&\bigl(\tfrac{1}{3}{f°}^{2} + \tfrac{1}{6}f'p
+  + \tfrac{1}{4}g°(q + q') + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{3}{20}g'p(q + q') + \tfrac{1}{5}h°(q^{2} + qq' + q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(7p^{2} + 7q^{2} + 12qq' + 7q'^{2})\bigr),
+\end{align*}
+correct\Typo{,}{ to} terms of the fifth degree.
+
+This equation may be written as follows:
+\begin{align*}
+A^{*} &= A + ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + q'^{2} + qq')\bigr)
+  \bigl(\tfrac{1}{3}f° + \tfrac{1}{6}f'p + \tfrac{1}{4}g°(q + q') \\
+  &+ \tfrac{1}{10}f''p^{2} + \tfrac{3}{20}g'p(q + q')
+   + \tfrac{1}{5}h°(q^{2} + qq' +q'^{2})
+   - \tfrac{1}{90}{f°}^{2}(2p^{2} + 2q^{2} + 7qq' + 2q'^{2})\bigr).
+\end{align*}
+But, since
+\[
+2\sigma
+  = ap\bigl(1 - \tfrac{1}{6}f°(p^{2}+ q^{2} + qq' + q'^{2}) + \text{etc.}\bigr),
+\]
+the above equation becomes
+\begin{align*}
+A^{*} = A - \sigma&\bigl(-\tfrac{2}{3}f° - \tfrac{1}{3}f'p
+  - \tfrac{1}{2}g°(q + q') - \tfrac{1}{5}f''p^{2}
+  - \tfrac{3}{10}g'p(q + q') \\
+  &- \tfrac{2}{5}h°(q^{2} + qq' + q'^{2})
+   + \tfrac{1}{90}{f°}^{2}(4p^{2} + 4q^{2} + 14qq' + 4q'^{2})\bigr),
+\end{align*}
+or
+\begin{alignat*}{3}
+A^{*} = A - \sigma\bigl(-\tfrac{2}{6}f°
+  &- \tfrac{2}{12}f° &&- \tfrac{2}{12}f° \\
+  &- \tfrac{2}{12}f'p   &&- \tfrac{2}{12}f'p \\
+  &- \tfrac{6}{12}g°q &&- \tfrac{6}{12}g°q' \\
+%
+  &-  \tfrac{2}{12}f''p^{2}
+  &&- \tfrac{2}{12}f''p^{2}
+  &&+ \tfrac{2}{15}f''p^{2} \\
+%
+  &-  \tfrac{6}{12}g'pq
+  &&- \tfrac{6}{12}g'pq'
+  &&+ \tfrac{1}{5}g'p(q + q') \\
+%
+  &-  \tfrac{12}{12}h°q^{2}
+  &&- \tfrac{12}{12}h°q'^{2}
+  &&+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2}) \\
+%
+  &+  \tfrac{2}{12}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{12}{f°}^{2}q'^{2}
+  &&+ \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr).
+\end{alignat*}
+Therefore, if we substitute in this equation $\alpha$,~$\beta$,~$\gamma$ for the series which they represent,
+we shall have
+\begin{align*}
+\Tag{[11]}
+A^{*} = A - \sigma
+  &\bigl(\tfrac{1}{6}\alpha + \tfrac{1}{12}\beta + \tfrac{1}{12}\gamma
+  + \tfrac{2}{15}f''p^{2} + \tfrac{1}{5}g'p(q + q') \\
+  &+ \tfrac{1}{5}h°(3q^{2} - 2qq' + 3q'^{2})
+   + \tfrac{1}{90}{f°}^{2}(4p^{2} - 11q^{2} + 14qq' - 11q'^{2})\bigr).
+\end{align*}
+
+\LineRef[30]{26}{Art.~26, p.~41}. Derivation of formula~[12].
+
+We form the expressions $(q - q')^{2} + r^{2} - r'^{2} - 2(q - q')r\cos\psi$ and $(q - q')r\sin\psi$.
+Then, since
+\begin{align*}
+(q - q')^{2} + r^{2} - r'^{2} &= a^{2} + c^{2} - b^{2} = 2ac\cos B^{*}, \\
+2(q - q')r\cos\psi &= 2ac\cos B,
+\end{align*}
+\PageSep{72}
+we have
+\[
+(q - q')^{2} + r^{2} - r'^{2} - 2(q - q')r\cos\psi
+  = 2ac(\cos B^{*} - \cos B).
+\]
+We have also
+\[
+(q - q')r\sin\psi = ac\sin B.
+\]
+
+Subtracting~(4) on \Pageref{68} from~[1] on \Pageref{36}, and adding this difference to
+$(q - q')^{2}$, we obtain
+\begin{multline*}
+\Tag{(1)}
+(q - q')^{2} + r^{2} - r'^{2},\quad\text{or}\quad 2ac\cos B^{*} \\
+\begin{alignedat}{2}
+= 2q(q - q') + \tfrac{2}{3}f°p^{2}(q^{2} - q'^{2})
+  &+ \tfrac{1}{2}f'p^{3}(q^{2} - q'^{2})
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}(q^{2} - q'^{2})
+    + \text{etc.} \\
+%
+  &+ \tfrac{1}{2}g°p^{2}(q^{3} - q'^{3})
+  &&+ \tfrac{2}{5}g'p^{3}(q^{3} - q'^{3}) \\
+%
+  &&&+ (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} - q'^{4}).
+\end{alignedat}
+\end{multline*}
+If we multiply~[3] on \Pageref{37} by~$2(q - q')$, we obtain
+\begin{multline*}
+\Tag{(2)}
+2(q - q')r\cos\psi,\quad\text{or}\quad 2ac\cos B \\
+\begin{alignedat}{2}
+= 2q(q - q') + \tfrac{4}{3}f°p^{2}q(q - q')
+  &+ f'p^{3}q(q - q')
+  &&+ (\tfrac{4}{5}f'' - \tfrac{8}{45}{f°}^{2})p^{4}q(q - q') + \text{etc.} \\
+%
+  &+ \tfrac{3}{2}g°p^{2}q^{2}(q - q')
+  &&+ \tfrac{6}{5}g'p^{3}q^{2}(q - q') \\
+%
+  &&&+ (\tfrac{8}{5}h° - \tfrac{28}{45}{f°}^{2})p^{2}q^{3}(q - q').
+\end{alignedat}
+\end{multline*}
+Subtracting (2) from~(1), we have
+\begin{multline*}
+\Tag{(3)}
+2ac(\cos B^{*} - \cos B) \\
+\begin{alignedat}{2}
+= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} + \text{etc.} \\
+%
+  &+ \tfrac{1}{4}g°(2q + q')
+  &&+ \tfrac{1}{5}g'p(2q + q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{20}{f°}^{2})
+       (3q^{2} + 2qq' + q'^{2})\bigr).
+\end{alignedat}
+\end{multline*}
+
+Multiplying [2] on \Pageref{36} by~$(q - q')$, we obtain at once
+\begin{multline*}
+\Tag{(4)}
+(q - q')r\sin\psi,\quad\text{or}\quad ac\sin B \\
+\begin{alignedat}{2}
+= p(q - q')\bigl(1 - \tfrac{1}{3}f°q^{2}
+  &- \tfrac{1}{4}f'pq^{2}
+  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q^{2} + \text{etc.} \\
+%
+  &- \tfrac{1}{2}g°q^{3}
+  &&- \tfrac{2}{5}g'pq^{3} \\
+%
+  &&&- (\tfrac{2}{5}h° - \tfrac{8}{45}{f°}^{2})q^{4}\bigr).
+\end{alignedat}
+\end{multline*}
+
+We now set $B^{*} - B = \zeta$, whence $B^{*} = B + \zeta$, and therefore
+\[
+\cos B^{*} = \cos B \cos\zeta - \sin B \sin\zeta.
+\]
+This becomes, after expanding $\cos\zeta$ and $\sin\zeta$ and neglecting powers of~$\zeta$ above the
+second,
+\[
+\cos B^{*} - \cos B = -\frac{\cos B}{2}·\zeta^{2} - \sin B·\zeta.
+\]
+Multiplying both members of this equation by~$2ac$, we obtain
+\[
+\Tag{(5)}
+2ac(\cos B^{*} - \cos B) = -ac\cos B·\zeta^{2} - 2ac\sin B·\zeta.
+\]
+\PageSep{73}
+Again, let $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, where the $R$'s have the same meaning as before.
+Hence, replacing the terms in~(5) by the proper series and neglecting terms above the
+sixth degree, we have
+\begin{multline*}
+\Tag{(6)}
+q(q - q')R_{2}^{2} + 2p(q - q')(1 - \tfrac{1}{3}f°q^{2})
+  (R_{2} + R_{3} - R_{4}) \\
+\begin{alignedat}{2}
+= 2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
+  &+  \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} \\
+%
+  &+ \tfrac{1}{4}g°(2q + q')
+  &&+ \tfrac{1}{5}g'p(2q + q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})(3q^{2} + 2qq' + q'^{2})\bigr).
+\end{alignedat}
+\end{multline*}
+From this equation we find
+\begin{align*}
+R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
+R_{3} = p(q - q')\bigl(\tfrac{1}{4}f'p + \tfrac{1}{4}g°(2q + q')\bigr), \\
+R_{4} &= p(q - q')\bigl(\tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(2q + q')
+  + \tfrac{1}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
+&\qquad- \tfrac{1}{90}{f°}^{2}(4p^{2} + 16q^{2} + 9qq' + 7q'^{2})\bigr).
+\end{align*}
+Therefore we have, correct to terms of the fifth degree,
+\begin{alignat*}{2}
+B^{*} - B = p(q - q')\bigl(\tfrac{1}{3}f°
+  &+  \tfrac{1}{4}f'p
+  &&+ \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(2q + q') \\
+%
+  &+  \tfrac{1}{4}g°(2q + q')
+  &&+ \tfrac{1}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
+%
+  &&&- \tfrac{1}{90}{f°}^{2}(4p^{2} + 16q^{2} + 9qq' + 7q'^{2})\bigr),
+\end{alignat*}
+or, after factoring the last factor on the right,
+\begin{multline*}
+\Tag{(7)}
+%[** TN: Squeeze line so tag doesn't get pushed up]
+\scalebox{0.975}[1]{$B^{*} - B - \tfrac{1}{2}p(q - q')
+  \bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)
+  \bigl(-\tfrac{2}{3}f° - \tfrac{1}{2}f'p - \tfrac{1}{2}g°(2q + q')$} \\
+  -\tfrac{2}{5}f''p^{2} - \tfrac{2}{5}g'p(2q + q')
+  - \tfrac{2}{5}h°(3q^{2} + 2qq' + q'^{2}) \\
+  - \tfrac{1}{90}{f°}^{2}(-2p^{2} + 22q^{2} + 8qq' + 4q'^{2})\bigr).
+\end{multline*}
+The last factor on the right in~(7) may be put in the form:
+\begin{alignat*}{3}
+\bigl(-\tfrac{2}{12}f°
+  &-  \tfrac{2}{6}f°
+  &&- \tfrac{2}{6}f° \\
+%
+  &-  \tfrac{2}{6}f'p
+  &&- \tfrac{2}{12}f'p \\
+%
+  &-  \tfrac{6}{6}g°q
+  &&- \tfrac{6}{12}g°q' \\
+%
+  &-  \tfrac{2}{6}f''p^{2}
+  &&- \tfrac{2}{12}f''p^{2}
+  &&+ \tfrac{1}{10}f''p^{2} \\
+%
+  &-  \tfrac{6}{6}g'pq
+  &&- \tfrac{6}{12}g'pq'
+  &&+ \tfrac{1}{10}g'p(2q + q') \\
+%
+  &-  \tfrac{12}{6}h°q^{2}
+  &&- \tfrac{12}{12}h°q'^{2}
+  &&+ \tfrac{1}{5}h°(4q^{2} + 3q'^{2} - 4qq') \\
+%
+  &+  \tfrac{2}{6}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{12}{f°}^{2}q'^{2}
+  && - \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} + 11q'^{2} - 8qq')\bigr).
+\end{alignat*}
+Finally, substituting in~(7) $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions which they represent, we
+obtain, still correct to terms of the fifth degree,
+\begin{align*}
+\Tag{[12]}
+B^{*} = B - \sigma&\bigl(\tfrac{1}{12}\alpha + \tfrac{1}{6}\beta
+                      + \tfrac{1}{12}\gamma + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{1}{10}g'p(2q + q')
+   + \tfrac{1}{5}h°(4q^{2} - 4qq' + 3q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 8q^{2} - 8qq' + 11q'^{2})\bigr).
+\end{align*}
+\PageSep{74}
+
+\LineRef[30]{26}{Art.~26, p.~41}. Derivation of formula~[13].
+
+Here we form the expressions $(q - q')^{2} + r'^{2} - r^{2} - 2(q - q')r'\cos(\pi - \psi')$ and
+$(q - q')r'\sin(\pi - \psi')$ and expand them into series. Since
+\begin{gather*}
+(q - q')^{2} + r'^{2} - r^{2} = a^{2} + b^{2} - c^{2} = 2ab\cos C^{*}, \\
+2(q - q')r'\cos(\pi - \psi') = 2ab\cos C,
+\end{gather*}
+we have
+\[
+(q - q')^{2} + r'^{2} - r^{2} - 2(q - q')r'\cos(\pi - \psi')
+  = 2ab(\cos C^{*} - \cos C).
+\]
+We have also
+\[
+(q - q')r'\sin(\pi - \psi') = ab\sin C.
+\]
+
+Subtracting~(3) on \Pageref{68} from (4) on the same page, and adding the result to
+$(q - q')^{2}$, we find
+{\small
+\begin{multline*}
+\Tag{(1)}
+(q - q')^{2} + r'^{2} - r^{2},\quad\text{or}\quad 2ab\cos C^{*} \\
+\begin{alignedat}{2}
+= -2q'(q - q') - \tfrac{2}{3}f°p^{2}(q^{2} - q'^{2})
+  &- \tfrac{1}{2}f'p^{3}(q^{2} - q'^{2})
+  &&- (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2}))p^{4}(q^{2} - q'^{2})
+    - \text{etc.} \\
+%
+  &- \tfrac{1}{2}g°p^{2}(q^{3} - q'^{3})
+  &&- \tfrac{2}{5}g'p^{3}(q^{3} - q'^{3}) \\
+%
+  &&&- (\tfrac{2}{5}h° - \tfrac{7}{45}{f°}^{2})p^{2}(q^{4} - q'^{4}).
+\end{alignedat}
+\end{multline*}}%
+By priming the $q$'s in formula~[3] on \Pageref{37}, we get a series for $r\cos\psi'$, or for
+$-r'\cos(\pi - \psi')$. If we multiply this series for $-r'\cos(\pi - \psi')$ by $2(q - q')$, we find
+\begin{multline*}
+\Tag{(2)}
+-2(q - q')r'\cos(\pi - \psi'),\quad\text{or}\quad -2ab\cos C \\
+\begin{alignedat}{2}
+= 2(q - q')\bigl(q' + \tfrac{2}{3}f°p^{2}q'
+  &+ \tfrac{1}{2}fp^{3}q'
+  &&+ (\tfrac{2}{5}f'' - \tfrac{4}{45}{f°}^{2})p^{4}q' + \text{etc.} \\
+%
+  &+  \tfrac{3}{4}g°p^{2}q'^{2}
+  &&+ \tfrac{3}{5}g'p^{3}q'^{2} \\
+%
+  &&&+ (\tfrac{4}{5}h° - \tfrac{14}{45}{f°}^{2})p^{2}q'^{3}\bigr).
+\end{alignedat}
+\end{multline*}
+And therefore, by adding (1)~and~(2), we obtain
+\begin{multline*}
+\Tag{(3)}
+2ab(\cos C^{*} - \cos C) \\
+\begin{alignedat}{2}
+= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°q'^{2}
+  &+ \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} + \text{etc.} \\
+%
+  &+ \tfrac{1}{4}g°(q + 2q')
+  &&+ \tfrac{1}{5}g'p(q + 2q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})
+       (q^{2} + 2qq' + 3q'^{2})\bigr).
+\end{alignedat}
+\end{multline*}
+
+By priming the $q$'s in~[2] on \Pageref{36}, we obtain a series for $r'\sin\psi'$, or for
+$r'\sin(\pi - \psi')$. Then, multiplying this series for $r'\sin(\pi - \psi')$ by $(q - q')$, we find
+\begin{multline*}
+\Tag{(4)}
+(q - q') r'\sin(\pi - \psi'),\quad\text{or}\quad ab\sin C \\
+\begin{alignedat}{2}
+= p(q - q')\bigl(1 - \tfrac{1}{3}f°q'^{2}
+  &- \tfrac{1}{4}f'pq'^{2}
+  &&- (\tfrac{1}{5}f'' + \tfrac{8}{45}{f°}^{2})p^{2}q'^{2} - \text{etc.} \\
+%
+  &- \tfrac{1}{2}g°q'^{3}
+  &&- \tfrac{2}{5}g'pq'^{3} \\
+%
+  &&&- (\tfrac{3}{5}h° - \tfrac{8}{45}{f°}^{2})q'^{4}\bigr).
+\end{alignedat}
+\end{multline*}
+
+As before, let $C^{*} - C = \zeta$, whence $C^{*} = C + \zeta$, and therefore
+\[
+\cos C^{*} = \cos C \cos\zeta - \sin C \sin \zeta.
+\]
+\PageSep{75}
+Expanding $\cos\zeta$ and $\sin\zeta$ and neglecting powers of~$\zeta$ above the second, this equation
+becomes
+\[
+\cos C^{*} - \cos C = -\frac{\cos C}{2}·\zeta^{2} - \sin C·\zeta,
+\]
+or, after multiplying both members by~$2ab$,
+\[
+\Tag{(5)}
+2ab(\cos C^{*} - \cos C) = -ab\cos C·\zeta^{2} - 2ab\sin C·\zeta.
+\]
+Again we put $\zeta = R_{2} + R_{3} + R_{4} + \text{etc.}$, the $R$'s having the same meaning as before.
+Now, by substituting (2),~(3),~(4) in~(5), and omitting terms above the sixth degree,
+we obtain
+\begin{multline*}
+q'(q - q')R_{2}^{2}
+  - 2p(q - q')\bigl(1 - \tfrac{1}{3}f°q'^{2})(R_{2} + R_{3} + R_{4}) \\
+\begin{alignedat}{2}
+= -2p^{2}(q - q')^{2}\bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{4}f'p
+  &&+ (\tfrac{1}{5}f'' - \tfrac{2}{45}{f°}^{2})p^{2} \\
+%
+  &+ \tfrac{1}{4}g°(q + 2q')
+  &&+ \tfrac{1}{5}g'p(q + 2q') \\
+%
+  &&&+ (\tfrac{1}{5}h° - \tfrac{7}{90}{f°}^{2})
+       (q^{2} + 2qq' + 3q'^{2})\bigr),
+\end{alignedat}
+\end{multline*}
+from which we find
+\begin{align*}
+R_{2} &= p(q - q')·\tfrac{1}{3}f°,\quad
+R_{3} = p(q - q')\bigl(\tfrac{1}{4}f'p + \tfrac{1}{4}g°(q + 2q')\bigr), \\
+R_{4} &= p(q - q')\bigl(\tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q')
+  + \tfrac{1}{5}h°(q^{2} + 2qq' + 3q'^{2}) \\
+&\qquad- \tfrac{1}{90}{f°}^{2}(4p^{2} + 7q^{2} + 9qq' + 16q'^{2})\bigr).
+\end{align*}
+Therefore we have, correct to terms of the fifth degree,
+\begin{alignat*}{2}
+\Tag{(6)}
+C^{*} - C = p(q - q')\bigl(\tfrac{1}{3}f°
+  &+ \tfrac{1}{4}f'p
+  &&+ \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q') \\
+%
+  &+ \tfrac{1}{4}g°(q + 2q')
+  &&+ \tfrac{1}{5}h°(q^{2} + 2qq' + 3q'^{2}) \\
+%
+  &&&- \tfrac{1}{90}{f°}^{2}(4p^{2} + 7q^{2} + 9qq' + 16q'^{2})\bigr).
+\end{alignat*}
+The last factor on the right in~(6) may be written as the product of two factors, one
+of which is $\frac{1}{2}\bigl(1 -\frac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)$, and the other,
+\begin{align*}
+2\bigl(\tfrac{1}{3}f° + \tfrac{1}{4}f'p
+  &+ \tfrac{1}{4}g°(q + 2q') + \tfrac{1}{5}f''p^{2} + \tfrac{1}{5}g'p(q + 2q') \\
+  &+ \tfrac{1}{5}h°(q^{2} + 3q'^{2} + 2qq')
+   - \tfrac{1}{90}{f°}^{2}(-p^{2} + 2q^{2} + 4qq' + 11q'^{2})\bigr),
+\end{align*}
+or, in another form,
+\begin{alignat*}{3}
+-\bigl(-\tfrac{2}{12}f°
+  &-  \tfrac{2}{12}f°
+  &&- \tfrac{2}{6}f° \\
+%
+  &-  \tfrac{2}{12}f'p
+  &&- \tfrac{2}{6}f'p \\
+%
+  &-  \tfrac{6}{12}g°q
+  &&- \tfrac{6}{6}g°q' \\
+%
+  &-  \tfrac{2}{12}f''p^{2}
+  &&- \tfrac{2}{6}f''p^{2}
+  &&+ \tfrac{1}{10}f''p^{2} \\
+%
+  &-  \tfrac{6}{12}g'pq
+  &&- \tfrac{6}{6}g'pq'
+  &&+ \tfrac{1}{10}g'p(q + 2q') \\
+%
+  &-  \tfrac{12}{12}h°q^{2}
+  &&- \tfrac{12}{6}h°q'^{2}
+  &&+ \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
+%
+  &+ \tfrac{2}{12}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{6}{f°}^{2}q'^{2}
+  &&- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr).
+\end{alignat*}
+\PageSep{76}
+Hence (6) becomes, on substituting $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions which they represent,
+\begin{align*}
+\Tag{[13]}
+C^{*} = C - \sigma
+  &\bigl(\tfrac{1}{12}\alpha + \tfrac{1}{12}\beta + \tfrac{1}{6}\gamma
+    + \tfrac{1}{10}f''p^{2} \\
+  &+ \tfrac{1}{10}g'p(q + 2q') + \tfrac{1}{5}h°(3q^{2} - 4qq' + 4q'^{2}) \\
+  &- \tfrac{1}{90}{f°}^{2}(2p^{2} + 11q^{2} - 8qq' + 8q'^{2})\bigr).
+\end{align*}
+
+\LineRef[31]{26}{Art.~26, p.~41}. Derivation of formula~[14].
+
+This formula is derived at once by adding formulæ [11],~[12],~[13]. But, as
+Gauss suggests, it may also be derived from~[6], \Pageref[p.]{38}. By priming the $q$'s in~[6]
+we obtain a series for~$(\psi' + \phi')$. Subtracting this series from~[6], and noting that
+$\phi - \phi' + \psi + \pi - \psi' = A + B + C$, we have, correct\Typo{}{ to} terms of the fifth degree,
+\begin{alignat*}{2}
+\Tag{(1)}
+A + B + C = \pi - p(q - q')\bigl(f°
+  &+ \tfrac{2}{3}f'p
+  \rlap{${} + \tfrac{1}{2}f''p^{2} + \tfrac{3}{4}g'p(q + q')$} \\
+%
+  &+g°(q + q')
+  &&+ h°(q^{2} + qq' + q'^{2}) \\
+%
+  &&&- \tfrac{1}{6}{f°}^{2}(p^{2} + 2q^{2} + 2qq' + 2q'^{2})\bigr).
+\end{alignat*}
+The second term on the right in~(1) may be written
+\begin{align*}
++ \tfrac{1}{2}ap\bigl(1 - \tfrac{1}{6}f°(p^{2} + q^{2} + qq' + q'^{2})\bigr)
+  ·2\bigl(-f°
+  &- \tfrac{2}{3}f'p - \tfrac{1}{2}f''p^{2} - \tfrac{3}{4}g'p(q + q') \\
+  &- g°(q + q') - h°(q^{2} + qq' + q'^{2}) \\
+  &\qquad+ \tfrac{1}{6}{f°}^{2}(\Typo{+}{}q^{2} + qq' + q'^{2})\bigr),
+\end{align*}
+of which the last factor may be thrown into the form:
+\begin{alignat*}{3}
+\bigl(-\tfrac{2}{3}f°
+  &-  \tfrac{2}{3}f°
+  &&- \tfrac{2}{3}f° \\
+%
+  &-  \tfrac{2}{3}f'p
+  &&- \tfrac{2}{3}f'p \\
+%
+  &-  \tfrac{6}{3}g°q
+  &&- \tfrac{6}{3}g°q' \\
+%
+  &-  \tfrac{2}{3}f''p^{2}
+  &&- \tfrac{2}{3}f''p^{2}
+  &&+ \tfrac{1}{3}f''p^{2} \\
+%
+  &-  \tfrac{6}{3}g'pq
+  &&- \tfrac{6}{3}g'pq'
+  &&+ \tfrac{1}{2}g'p(q + q') \\
+%
+  &-  \tfrac{12}{3}h°q^{2}
+  &&- \tfrac{12}{3}h°q'^{2}
+  &&+ 2h°(q^{2} + q'^{2} - qq') \\
+%
+  &+  \tfrac{2}{3}{f°}^{2}q^{2}
+  &&+ \tfrac{2}{3}{f°}^{2}q'^{2}
+  &&- \tfrac{1}{3}{f°}^{2}(q^{2} + q'^{2} - qq')\bigr).
+\end{alignat*}
+Hence, by substituting $\sigma$,~$\alpha$,~$\beta$,~$\gamma$ for the expressions they represent, (1)~becomes
+\begin{align*}
+\Tag{[14]}
+A + B + C = \pi + \sigma
+  &\bigl(\tfrac{1}{3}\alpha + \tfrac{1}{3}\beta + \tfrac{1}{3}\gamma
+    + \tfrac{1}{3}f''p^{2} \\
+  &+ \tfrac{1}{2}g'p(q + q')
+   + (2h° - \tfrac{1}{3}{f°}^{2})
+     (q^{2} - qq' + q'^{2})\bigr).
+\end{align*}
+
+\LineRef{27}{Art.~27, p.~42}. Omitting terms above the second degree, we have
+\[
+a^{2} = q^{2} - 2qq' + q'^{2},\quad
+b^{2} = p^{2} + q'^{2},\quad
+c^{2} = p^{2} + q^{2}.
+\]
+
+The expressions in the parentheses of the first set of formulæ for $A^{*}$,~$B^{*}$,~$C^{*}$
+in \Art{27} may be arranged in the following manner:
+\[
+\begin{array}{*{9}{r@{\,}}}
+&(&2p^{2} - & q^{2} + &4qq' - & q'^{2} = \bigl(& (p^{2} + q'^{2}) + & (p^{2} + q^{2}) - & 2(q^{2} - 2qq' + q'^{2})\bigr), \\
+&(& p^{2} - &2q^{2} + &2qq' + & q'^{2} = \bigl(&2(p^{2} + q'^{2}) - & (p^{2} + q^{2}) - &  (q^{2} - 2qq' + q'^{2})\bigr), \\
+&(& p^{2} + & q^{2} + &2qq' - &2q'^{2} = \bigl(&-(p^{2} + q'^{2}) + &2(p^{2} + q^{2}) - &  (q^{2} - 2qq' + q'^{2})\bigr).
+\end{array}
+\]
+\PageSep{77}
+Now substituting $a^{2}$,~$b^{2}$,~$c^{2}$ for $q^{2} - 2qq' + q'^{2})$, $(p^{2} + q'^{2})$, $(p^{2} + q^{2})$ respectively, and
+changing the signs of both members of the last two of these equations, we have
+\[
+\begin{array}{*{7}{r@{\,}}}
+ (&2p^{2} - & q^{2} + &4qq' - & q'^{2} = (b^{2} + & c^{2} - & 2a^{2}), \\
+-(& p^{2} - &2q^{2} + &2qq' + & q'^{2} = (a^{2} + & c^{2} - & 2b^{2}), \\
+-(& p^{2} + & q^{2} + &2qq' - &2q'^{2} = (a^{2} + & b^{2} - & 2c^{2}).
+\end{array}
+\]
+And replacing the expressions in the parentheses in the first set of formulæ for
+$A^{*}$,~$B^{*}$,~$C^{*}$ by their equivalents, we get the second set.
+
+\LineRef{27}{Art.~27, p.~42}. $f° = -\dfrac{1}{2R^{2}}$, $f'' = 0$, etc., may obtained directly, without the
+%[** TN: Macro expands to "Arts. 25 and 26"]
+use of the general considerations of \Arts[ and ]{25}{26}, in the following way. In the
+case of the sphere
+\[
+ds^{2} = \cos^{2}\left(\frac{q}{R}\right)·dp^{2} + dq^{2},
+\]
+hence
+\[
+n = \cos\left(\frac{q}{R}\right)
+  = 1 - \frac{q^{2}}{2R^{2}} + \frac{q^{4}}{24R^{4}} - \text{etc.},
+\]
+\ie,
+\[
+f° = -\frac{1}{2R^{2}},\quad
+h° = \frac{1}{24R^{4}},\quad
+f' = g° = f'' = g' = 0.\qquad\rlap{[Wangerin.]}
+\]
+
+\LineRef{27}{Art.~27, p.~42, l.~16}. This theorem of Legendre is found in the Mémoires (Histoire)
+de l'\Typo{Academie}{Académie} Royale de Paris, 1787, p.~358, and also in his \Title{Trigonometry},
+Appendix,~§\;V\@. He states it as follows in his \textit{Trigonometry}:
+
+\begin{Theorem}[]
+The very slightly curved spherical triangle, whose angles are $A$,~$B$,~$C$ and whose sides
+are $a$,~$b$,~$c$, always corresponds to a rectilinear triangle, whose sides $a$,~$b$,~$c$ are of the same
+lengths, and whose opposite angles are $A - \tfrac{1}{3}e$, $B - \tfrac{1}{3}e$, $C - \tfrac{1}{3}e$, $e$~being the excess of the
+sum of the angles in the given spherical triangle over two right angles.
+\end{Theorem}
+
+\LineRef{28}{Art.~28, p.~43, l.~7}. The sides of this triangle are Hohehagen-Brocken, Inselberg-Hohehagen,
+Brocken-Inselberg, and their lengths are about $107$, $85$, $69$~kilometers
+respectively, according to Wangerin.
+
+\LineRef{29}{Art.~29, p.~43}. Derivation of the relation between $\sigma$~and~$\sigma^{*}$.
+
+In \Art{28} we found the relation
+\[
+A^{*} = A - \tfrac{1}{12}\sigma(2\alpha + \beta + \gamma).
+\]
+Therefore
+\[
+\sin A^{*}
+  = \sin A\cos\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)
+  - \cos A\sin\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr),
+\]
+which, after expanding $\cos\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ and $\sin\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ and rejecting
+powers of $\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr)$ above the first, becomes
+\PageSep{78}
+\[
+\Tag{(1)}
+\sin A^{*} = \sin A
+  - \cos A·\bigl(\tfrac{1}{12}\sigma(2\alpha + \beta + \gamma)\bigr),
+\]
+correct to terms of the fourth degree.
+
+But, since $\sigma$~and~$\sigma^{*}$ differ only by terms above the second degree, we may replace
+in~(1) $\sigma$~by the value of~$\sigma^{*}$, $\tfrac{1}{2}bc\sin A^{*}$. We thus obtain, with equal exactness,
+\[
+\Tag{(2)}
+\sin A = \sin A^{*}
+  \bigl(1 + \tfrac{1}{24}bc\cos A·(2\alpha + \beta + \gamma)\bigr).
+\]
+Substituting this value for~$\sin A$ in~[9], \Pageref[p.]{40}, we have, correct to terms of the sixth
+degree, the first formula for~$\sigma$ given in \Art{29}. Since $2bc\cos A^{*}$, or $b^{2} + c^{2} - a^{2}$,
+differs from~$2bc\cos A$ only by terms above the second degree, we may replace $2bc\cos A$
+in this formula for~$\sigma$ by $b^{2} + c^{2} - a^{2}$. Also $\sigma^{*} = \tfrac{1}{2}bc \sin A^{*}$. Hence, if we make
+these substitutions in the first formula for~$\sigma$, we obtain the second formula for~$\sigma$
+with the same exactness. In the case of a sphere, where $\alpha = \beta = \gamma$, the second
+formula for~$\sigma$ reduces to the third.
+
+When the surface is spherical, (2)~becomes
+\[
+\sin A = \sin A^{*}(1 + \frac{\alpha}{6}bc \cos A).
+\]
+And replacing $2bc\cos A$ in this equation by $(b^{2} + c^{2} - a^{2})$, we have
+\[
+\sin A = \sin A^{*}\bigl(1 + \frac{\alpha}{12}(b^{2} + c^{2} - a^{2})\bigr),
+\]
+or
+\[
+\frac{\sin A}{\sin A^{*}}
+  = \bigl(1 + \frac{\alpha}{12}(b^{2} + c^{2} - a^{2})\bigr).
+\]
+And likewise we can find
+\[
+\frac{\sin B}{\sin B^{*}}
+  = \bigl(1 + \frac{\alpha}{12}(a^{2} + c^{2} - b^{2})\bigr),\qquad
+\frac{\sin C}{\sin C^{*}}
+  = \bigl(1 + \frac{\alpha}{12}(a^{2} + b^{2} - c^{2})\bigr).
+\]
+Multiplying together the last three equations and rejecting the terms containing $\alpha^{2}$~and~$\alpha^{3}$,
+we have
+\[
+1 + \frac{\alpha}{12}(a^{2} + b^{2} + c^{2})
+  = \frac{\sin A·\sin B·\sin C}{\sin A^{*}·\sin B^{*}·\sin C^{*}}.
+\]
+Finally, taking the square root of both members of this equation, we have, with the
+same exactness,
+\[
+\sigma = 1 + \frac{\alpha}{24}(a^{2} + b^{2} + c^{2})
+  = \Sqrt{\frac{\sin A·\sin B·\sin C}{\sin A^{*}·\sin B^{*}·\sin C^{*}}}.
+\]
+
+The method here used to derive the last formula from the next to the last
+formula of \Art{29} is taken from Wangerin.
+\PageSep{79}
+
+
+\Paper{1825}
+\null\vfill
+\begin{center}
+\LARGE
+NEUE \\[12pt]
+ALLGEMEINE UNTERSUCHUNGEN \\[12pt]
+{\small ÜBER} \\[12pt]
+DIE KRUMMEN FLÄCHEN \\[12pt]
+{\normalsize [1825]} \\[12pt]
+{\footnotesize
+PUBLISHED POSTHUMOUSLY IN GAUSS'S WORKS, VOL.~VIII, 1901. PAGES 408--443}
+\end{center}
+\vfill
+\cleardoublepage
+\PageSep{80}
+%[Blank page]
+\PageSep{81}
+
+
+\PaperTitle{\LARGE NEW GENERAL INVESTIGATIONS \\
+{\small OF} \\
+CURVED SURFACES \\
+{\normalsize [1825]}}
+
+Although the real purpose of this work is the deduction of new theorems concerning
+its subject, nevertheless we shall first develop what is already known, partly
+for the sake of consistency and completeness, and partly because our method of treatment
+is different from that which has been used heretofore. We shall even begin by
+advancing certain properties concerning plane curves from the same principles.
+
+
+\Article{1.}
+
+In order to compare in a convenient manner the different directions of straight
+lines in a plane with each other, we imagine a circle with unit radius described
+in the plane about an arbitrary centre. The position of the radius of this circle,
+drawn parallel to a straight line given in advance, represents then the position of that
+line. And the angle which two straight lines make with each other is measured by
+the angle between the two radii representing them, or by the arc included between
+their extremities. Of course, where precise definition is necessary, it is specified at
+the outset, for every straight line, in what sense it is regarded as drawn. Without
+such a distinction the direction of a straight line would always correspond to two
+opposite radii.
+
+
+\Article{2.}
+
+In the auxiliary circle we take an arbitrary radius as the first, or its terminal
+point in the circumference as the origin, and determine the positive sense of measuring
+the arcs from this point (whether from left to right or the contrary); in the
+opposite direction the arcs are regarded then as negative. Thus every direction of a
+straight line is expressed in degrees,~etc., or also by a number which expresses them
+in parts of the radius.
+\PageSep{82}
+
+Such lines as differ in direction by~$360°$, or by a multiple of~$360°$, have, therefore,
+precisely the same direction, and may, generally speaking, be regarded as the
+same. However, in such cases where the manner of describing a variable angle is
+taken into consideration, it may be necessary to distinguish carefully angles differing
+by~$360°$.
+
+If, for example, we have decided to measure the arcs from left to right, and if
+to two straight lines $l$,~$l'$ correspond the two directions $L$,~$L'$, then $L' - L$ is the angle
+between those two straight lines. And it is easily seen that, since $L' - L$ falls
+between $-180°$~and~$+180°$, the positive or negative value indicates at once that $l'$~lies
+on the right or the left of~$l$, as seen from the point of intersection. This will
+be determined generally by the sign of~$\sin(L' - L)$.
+
+If $aa'$~is a part of a curved line, and if to the tangents at $a$,~$a'$ correspond
+respectively the directions $\alpha$,~$\alpha'$, by which letters shall be denoted also the corresponding
+points on the auxiliary circles, and if $A$,~$A'$ be their distances along the arc
+from the origin, then the magnitude of the arc~$\alpha\alpha'$ or $A' - A$ is called the \emph{amplitude}
+of~$aa'$.
+
+The comparison of the amplitude of the arc~$aa'$ with its length gives us the
+notion of curvature. Let $l$~be any point on the arc~$aa'$, and let $\lambda$,~$\Lambda$ be the same
+with reference to it that $\alpha$,~$A$ and $\alpha'$,~$A'$ are with reference to $a$~and~$a'$. If now
+$\alpha\lambda$~or~$\Lambda - A$ be proportional to the part~$al$ of the arc, then we shall say that $aa'$~is
+uniformly curved throughout its whole length, and we shall call
+\[
+\frac{\Lambda - A}{al}
+\]
+the measure of curvature, or simply the curvature. We easily see that this happens
+only when $aa'$~is actually the arc of a circle, and that then, according to our definition,
+its curvature will be~$±\dfrac{1}{r}$ if $r$~denotes the radius. Since we always regard $r$
+as positive, the upper or the lower sign will hold according as the centre lies to the
+right or to the left of the arc~$aa'$ ($a$~being regarded as the initial point, $a'$~as the
+end point, and the directions on the auxiliary circle being measured from left to
+right). Changing one of these conditions changes the sign, changing two restores it
+again.
+
+On the contrary, if $\Lambda - A$ be not proportional to~$al$, then we call the arc non-uniformly
+curved and the quotient
+\[
+\frac{\Lambda - A}{al}
+\]
+\PageSep{83}
+may then be called its mean curvature. Curvature, on the contrary, always presupposes
+that the point is determined, and is defined as the mean curvature of an element
+at this point; it is therefore equal to
+\[
+\frac{d\Lambda}{d\,al}.
+\]
+We see, therefore, that arc, amplitude, and curvature sustain a similar relation to each
+other as time, motion, and velocity, or as volume, mass, and density. The reciprocal
+of the curvature, namely,
+\[
+\frac{d\,al}{d\Lambda},
+\]
+is called the radius of curvature at the point~$l$. And, in keeping with the above
+conventions, the curve at this point is called concave toward the right and convex
+toward the left, if the value of the curvature or of the radius of curvature happens
+to be positive; but, if it happens to be negative, the contrary is true.
+
+
+\Article{3.}
+
+If we refer the position of a point in the plane to two perpendicular axes of
+coordinates to which correspond the directions $0$~and~$90°$, in such a manner that the
+first coordinate represents the distance of the point from the second axis, measured in
+the direction of the first axis; whereas the second coordinate represents the distance
+from the first axis, measured in the direction of the second axis; if, further, the indeterminates
+$x$,~$y$ represent the coordinates of a point on the curved line, $s$~the length
+of the line measured from an arbitrary origin to this point, $\phi$~the direction of the
+tangent at this point, and $r$~the radius of curvature; then we shall have
+\begin{align*}
+dx &= \cos\phi·ds, \\
+dy &= \sin\phi·ds, \\
+r &= \frac{ds}{d\phi}.
+\end{align*}
+
+If the nature of the curved line is defined by the equation $V = 0$, where $V$~is a
+function of $x$,~$y$, and if we set
+\[
+dV = p\, dx + q\, dy,
+\]
+then on the curved line
+\[
+p\, dx + q\, dy = 0.
+\]
+Hence
+\[
+p\cos\phi + q\sin\phi = 0,
+\]
+\PageSep{84}
+and therefore
+\[
+\tan\phi = -\frac{p}{q}.
+\]
+We have also
+\[
+\cos\phi·dp + \sin\phi·dq - (p\sin\phi - q\cos\phi)\, d\phi = 0.
+\]
+If, therefore, we set, according to a well known theorem,
+\begin{align*}
+dp &= P\, dx + Q\, dy, \\
+dq &= Q\, dx + R\, dy,
+\end{align*}
+then we have\Note{32}
+\[
+(P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi)\, ds
+  = (p\sin\phi - q\cos\phi)\, d\phi,\NoteMark
+\]
+therefore
+\[
+\frac{1}{r}
+  = \frac{P\cos^{2}\phi + 2Q\cos\phi\sin\phi + R\sin^{2}\phi}
+         {p\sin\phi - q\cos\phi},
+\]
+or, since\Note{33}
+\begin{gather*}
+\cos\phi = \frac{\mp q}{\Sqrt{p^{2} + q^{2}}},\qquad
+\sin\phi = \frac{±p}{\Sqrt{p^{2} + q^{2}}};\NoteMark \\
+±\frac{1}{r} = \frac{Pq^{2} - 2Qpq + Rp^{2}}{(p^{2} + q^{2})^{3/2}}.
+\end{gather*}
+
+
+\Article{4.}
+
+The ambiguous sign in the last formula might at first seem out of place, but
+upon closer consideration it is found to be quite in order. In fact, since this expression
+depends simply upon the partial differentials of~$V$, and since the function $V$~itself
+merely defines the nature of the curve without at the same time fixing the sense in
+which it is supposed to be described, the question, whether the curve is convex
+toward the right or left, must remain undetermined until the sense is determined by
+some other means. The case is similar in the determination of~$\phi$ by means of the
+tangent, to single values of which correspond two angles differing by~$180°$. The
+sense in which the curve is described can be specified in the following different ways.
+
+\Par{I.} By means of the sign of the change in~$x$. If $x$~increases, then $\cos\phi$ must be
+positive. Hence the upper signs will hold if $q$~has a negative value, and the lower
+signs if $q$~has a positive value. When $x$~decreases, the contrary is true.
+
+\Par{II.} By means of the sign of the change in~$y$. If $y$~increases, the upper signs
+must be taken when $p$~is positive, the lower when $p$~is negative. The contrary is
+true when $y$~decreases.
+
+\Par{III.} By means of the sign of the value which the function~$V$ takes for points
+not on the curve. Let $\delta x$,~$\delta y$ be the variations of $x$,~$y$ when we go out from the
+\PageSep{85}
+curve toward the right, at right angles to the tangent, that is, in the direction~$\phi + 90°$;
+and let the length of this normal be~$\delta\rho$. Then, evidently, we have
+\begin{align*}
+\delta x &= \delta\rho·\cos(\phi + 90°), \\
+\delta y &= \delta\rho·\sin(\phi + 90°),
+\end{align*}
+or
+\begin{align*}
+\delta x &= -\delta\rho·\sin\phi, \\
+\delta y &= +\delta\rho·\cos\phi.
+\end{align*}
+Since now, when $\delta\rho$~is infinitely small,
+\begin{align*}
+\delta V &= p\, \delta x + q\, \delta y \\
+  &= (-p\sin\phi + q\cos\phi)\, \delta\rho \\
+  &= \mp\delta\rho\Sqrt{p^{2} + q^{2}}\Add{,}
+\end{align*}
+and since on the curve itself $V$~vanishes, the upper signs will hold if~$V$, on passing
+through the curve from left to right, changes from positive to negative, and the contrary.
+If we combine this with what is said at the end of \Art{2}, it follows that the
+curve is always convex toward that side on which $V$~receives the same sign as
+\[
+Pq^{2} - 2Qpq + Rp^{2}.
+\]
+
+For example, if the curve is a circle, and if we set
+\[
+V = x^{2} + y^{2} - a^{2}\Add{,}
+\]
+then we have
+\begin{gather*}
+p = 2x,\qquad q = 2y, \\
+P = 2,\qquad Q = 0,\qquad R = 2, \\
+Pq^{2} - 2Qpq + Rp^{2} = 8y^{2} + 8x^{2} = 8a^{2}, \\
+(p^{2} + q^{2})^{3/2} = 8a^{3}, \\
+r = ± a\Add{;}
+\end{gather*}
+and the curve will be convex toward that side for which
+\[
+x^{2} + y^{2} > a^{2},
+\]
+as it should be.
+
+The side toward which the curve is convex, or, what is the same thing, the signs
+in the above formulæ, will remain unchanged by moving along the curve, so long as
+\[
+\frac{\delta V}{\delta\rho}
+\]
+does not change its sign. Since $V$~is a continuous function, such a change can take
+place only when this ratio passes through the value zero. But this necessarily presupposes
+that $p$~and~$q$ become zero at the same time. At such a point the radius
+\PageSep{86}
+of curvature becomes infinite or the curvature vanishes. Then, generally speaking,
+since here
+\[
+-p\sin\phi + q\cos\phi
+\]
+will change its sign, we have here a point of inflexion.
+
+
+\Article{5.}
+
+The case where the nature of the curve is expressed by setting $y$~equal to a
+given function of~$x$, namely, $y = X$, is included in the foregoing, if we set
+\[
+V = X - y.
+\]
+If we put
+\[
+dX = X'\, dx,\qquad
+dX' = X''\, dx,
+\]
+then we have
+\begin{gather*}
+p = X',\qquad q = -1, \\
+P = X'', \qquad Q = 0,\qquad R = 0,
+\end{gather*}
+therefore
+\[
+±\frac{1}{r} = \frac{X''}{(1 + X'^{2})^{3/2}}.
+\]
+Since $q$~is negative here, the upper sign holds for increasing values of~$x$. We can
+therefore say, briefly, that for a positive~$X''$ the curve is concave toward the same
+side toward which the $y$-axis lies with reference to the $x$-axis; while for a negative~$X''$
+the curve is convex toward this side.
+
+
+\Article{6.}
+
+If we regard $x$,~$y$ as functions of~$s$, these formulæ become still more elegant.
+Let us set
+\begin{alignat*}{2}
+\frac{dx}{ds} &= x',\qquad& \frac{dx'}{ds} &= x'', \\
+\frac{dy}{ds} &= y',\qquad& \frac{dy'}{ds} &= y''.
+\end{alignat*}
+Then we shall have
+\begin{alignat*}{2}
+x'  &= \cos\phi,\qquad & y' &= \sin\phi, \\
+x'' &= -\frac{\sin\phi}{r},\qquad & y'' &= \frac{\cos\phi}{r};
+\intertext{or}
+y' &= -rx'',\qquad& x' &= ry'',
+\end{alignat*}
+\PageSep{87}
+or also
+\[
+1 = r(x'y'' - y'x''),
+\]
+so that
+\[
+x'y'' - y'x''
+\]
+represents the curvature, and
+\[
+\frac{1}{x'y'' - y'x''}
+\]
+the radius of curvature.
+
+
+\Article{7.}
+
+We shall now proceed to the consideration of curved surfaces. In order to represent
+the directions of straight lines in space considered in its three dimensions, we
+imagine a sphere of unit radius described about an arbitrary centre. Accordingly, a
+point on this sphere will represent the direction of all straight lines parallel to the
+radius whose extremity is at this point. As the positions of all points in space
+are determined by the perpendicular distances $x$,~$y$,~$z$ from three mutually perpendicular
+planes, the directions of the three principal axes, which are normal to these
+principal planes, shall be represented on the auxiliary sphere by the three points
+$(1)$,~$(2)$,~$(3)$. These points are, therefore, always $90°$~apart, and at once indicate the
+sense in which the coordinates are supposed to increase. We shall here state several
+well known theorems, of which constant use will be made.
+
+\Par{1)} The angle between two intersecting straight lines is measured by the arc [of
+the great circle] between the points on the sphere which represent their directions.
+
+\Par{2)} The orientation of every plane can be represented on the sphere by means
+of the great circle in which the sphere is cut by the plane through the centre parallel
+to the first plane.
+
+\Par{3)} The angle between two planes is equal to the angle between the great circles
+which represent their orientations, and is therefore also measured by the angle
+between the poles of the great circles.
+
+\Par{4)} If $x$,~$y$,~$z$; $x'$,~$y'$,~$z'$ are the coordinates of two points, $r$~the distance between
+them, and $L$~the point on the sphere which represents the direction of the straight
+line drawn from the first point to the second, then
+\begin{alignat*}{2}
+x' &= x &&+ r\cos(1)L, \\
+y' &= y &&+ r\cos(2)L, \\
+2' &= z &&+ r\cos(3)L.
+\end{alignat*}
+
+\Par{5)} It follows immediately from this that we always have
+\[
+\cos^{2}(1)L + \cos^{2}(2)L + \cos^{2}(3)L = 1
+\]
+\PageSep{88}
+[and] also, if $L'$~is any other point on the sphere,
+\[
+\cos(1)L·\cos(1)L' + \cos(2)L·\cos(2)L' + \cos(3)L·\cos(3)L' = \cos LL'.
+\]
+
+We shall add here another theorem, which has appeared nowhere else, as far as
+we know, and which can often be used with advantage.
+
+Let $L$, $L'$, $L''$, $L'''$ be four points on the sphere, and $A$~the angle which $LL'''$
+and $L'L''$ make at their point of intersection. [Then we have]
+\[
+\cos LL'·\cos L''L''' - \cos LL''·\cos L'L''' = \sin LL'''·\sin L'L''·\cos A.
+\]
+
+The proof is easily obtained in the following way. Let
+\[
+AL = t,\qquad
+AL' = t',\qquad
+AL'' = t'',\qquad
+AL''' = t''';
+\]
+we have then
+\begin{alignat*}{6}
+&\cos L  L'   &&= \cos t  &&\cos t'   &&+ \sin t  &&\sin t'  &&\cos A, \\
+&\cos L''L''' &&= \cos t''&&\cos t''' &&+ \sin t''&&\sin t'''&&\cos A, \\
+&\cos L  L''  &&= \cos t  &&\cos t''  &&+ \sin t  &&\sin t'' &&\cos A, \\
+&\cos L' L''' &&= \cos t' &&\cos t''' &&+ \sin t' &&\sin t'''&&\cos A.
+\end{alignat*}
+Therefore
+\begin{multline*}
+\cos LL' \cos L''L''' - \cos LL'' \cos L'L'' \\
+\begin{aligned}
+&= \cos A \{\cos t  \cos t'  \sin t''\sin t'''
+          + \cos t''\cos t'''\sin t  \sin t' \\
+&\qquad\qquad
+  - \cos t\cos t''\sin t'\sin t''' - \cos t'\cos t'''\sin t\sin t''\} \\
+&= \cos A (\cos t \sin t''' - \cos t'''\sin t)
+          (\cos t'\sin t''  - \cos t'' \sin t') \\
+&= \cos A \sin (t''' - t) \sin(t'' - t') \\
+&= \cos A \sin LL''' \sin L'L''.
+\end{aligned}
+\end{multline*}
+
+Since each of the two great circles goes out from~$A$ in two opposite directions,
+two supplementary angles are formed at this point. But it is seen from our analysis
+that those branches must be chosen, which go in the same sense from~$L$ toward~$L'''$
+and from $L'$~toward~$L''$.
+
+Instead of the angle~$A$, we can take also the distance of the pole of the great
+circle~$LL'''$ from the pole of the great circle~$L'L''$. However, since every great circle
+has two poles, we see that we must join those about which the great circles run in
+the same sense from~$L$ toward~$L'''$ and from~$L'$ toward~$L''$, respectively.
+
+The development of the special case, where one or both of the arcs $LL'''$~and~$L'L''$ are~$90°$, we leave to the reader.
+
+\Par{6)} Another useful theorem is obtained from the following analysis. Let $L$,~$L'$,~$L''$
+be three points upon the sphere and put
+\PageSep{89}
+\begin{alignat*}{6}
+&\cos L  &&(1) = x,       &&\cos L   &&(2) = y,       &&\cos L   &&(3) = z, \\
+&\cos L' &&(1) = x',      &&\cos L'  &&(2) = y',      &&\cos L'  &&(3) = z', \\
+&\cos L''&&(1) = x'',\quad&&\cos L'' &&(2) = y'',\quad&&\cos L'' &&(3) = z''.
+\end{alignat*}
+
+We assume that the points are so arranged that they run around the triangle
+included by them in the same sense as the points $(1)$,~$(2)$,~$(3)$. Further, let $\lambda$~be
+that pole of the great circle~$L'L''$ which lies on the same side as~$L$. We then have,
+from the above lemma,
+\begin{alignat*}{3}
+&y'z'' &&- z'y'' &&= \sin L'L''·\cos\lambda(1), \\
+&z'x'' &&- x'z'' &&= \sin L'L''·\cos\lambda(2), \\
+&x'y'' &&- y'x'' &&= \sin L'L''·\cos\lambda(3).
+\end{alignat*}
+Therefore, if we multiply these equations by $x$,~$y$,~$z$ respectively, and add the products,
+we obtain\Note{34}
+\[
+xy'z'' + x'y''z + x''yz' - xy''z' - x'yz'' - x''y'z
+  = \sin L'L''·\cos\lambda L,\NoteMark
+\]
+wherefore, we can write also, according to well known principles of spherical trigonometry,
+\begin{alignat*}{2}
+  \sin L'L''·&\sin L L''&&·\sin L' \\
+= \sin L'L''·&\sin L L' &&·\sin L'' \\
+= \sin L'L''·&\sin L'L''&&·\sin L,
+\end{alignat*}
+if $L$,~$L'$,~$L''$ denote the three angles of the spherical triangle. At the same time we
+easily see that this value is one-sixth of the pyramid whose angular points are the
+centre of the sphere and the three points $L$,~$L'$,~$L''$ (and indeed \emph{positive}, if~etc.).
+
+
+\Article{8.}
+
+The nature of a curved surface is defined by an equation between the coordinates
+of its points, which we represent by
+\[
+f(x, y, z) = 0.\NoteMark
+\]
+Let the total differential of $f(x, y, z)$ be
+\[
+P\, dx + Q\, dy + R\, dz,
+\]
+where $P$,~$Q$,~$R$ are functions of $x$,~$y$,~$z$. We shall always distinguish two sides of the
+surface, one of which we shall call the upper, and the other the lower. Generally
+speaking, on passing through the surface the value of~$f$ changes its sign, so that, as
+long as the continuity is not interrupted, the values are positive on one side and negative
+on the other.
+\PageSep{90}
+
+The direction of the normal to the surface toward that side which we regard as
+the upper side is represented upon the auxiliary sphere by the point~$L$. Let
+\[
+\cos L(1) = X,\qquad
+\cos L(2) = Y,\qquad
+\cos L(3) = Z.
+\]
+Also let $ds$~denote an infinitely small line upon the surface; and, as its direction is
+denoted by the point~$\lambda$ on the sphere, let
+\[
+\cos \lambda(1) = \xi,\qquad
+\cos \lambda(2) = \eta,\qquad
+\cos \lambda(3) = \zeta.
+\]
+We then have
+\[
+dx = \xi\, ds,\qquad
+dy = \eta\, ds,\qquad
+dz = \zeta\, ds,
+\]
+therefore
+\[
+P\xi + Q\eta + R\zeta = 0,
+\]
+and, since $\lambda L$ must be equal to~$90°$, we have also
+\[
+X\xi + Y\eta + Z\zeta = 0.
+\]
+Since $P$,~$Q$,~$R$, $X$,~$Y$,~$Z$ depend only on the position of the surface on which we take
+the element, and since these equations hold for every direction of the element on the
+surface, it is easily seen that $P$,~$Q$,~$R$ must be proportional to $X$,~$Y$,~$Z$. Therefore
+\[
+P = X\mu,\qquad
+Q = Y\mu,\qquad
+R = Z\mu\Typo{,}{.}
+\]
+Therefore, since
+\begin{gather*}
+X^{2} + Y^{2} + Z^{2} = 1; \\
+\mu = PX + QY + RZ
+\intertext{and}
+\mu^{2} = P^{2} + Q^{2} + R^{2}, \\
+\intertext{or}
+\mu = ±\Sqrt{P^{2} + Q^{2} + R^{2}}.
+\end{gather*}
+
+If we go out from the surface, in the direction of the normal, a distance equal to
+the element~$\delta\rho$, then we shall have
+\[
+\delta x = X\, \delta\rho,\qquad
+\delta y = Y\, \delta\rho,\qquad
+\delta z = Z\, \delta\rho
+\]
+and
+\[
+\delta f = P\, \delta x + Q\, \delta y + R\, \delta z = \mu\, \delta\rho.
+\]
+We see, therefore, how the sign of~$\mu$ depends on the change of sign of the value of~$f$
+in passing from the lower to the upper side.
+
+
+\Article{9.}
+
+Let us cut the curved surface by a plane through the point to which our notation
+refers; then we obtain a plane curve of which $ds$~is an element, in connection
+with which we shall retain the above notation. We shall regard as the upper side of
+the plane that one on which the normal to the curved surface lies. Upon this plane
+\PageSep{91}
+we erect a normal whose direction is expressed by the point~$\L$ of the auxiliary
+sphere. By moving along the curved line, $\lambda$~and~$L$ will therefore change their positions,
+while $\L$~remains constant, and $\lambda L$~and~$\lambda\L$ are always equal to~$90°$. Therefore
+$\lambda$~describes the great circle one of whose poles is~$\L$. The element of this great circle
+will be equal to~$\dfrac{ds}{r}$, if $r$~denotes the radius of curvature of the curve. And again,
+if we denote the direction of this element upon the sphere by~$\lambda'$, then $\lambda'$~will evidently
+lie in the same great circle and be $90°$~from~$\lambda$ as well as from~$\L$. If we
+now set
+\[
+\cos \lambda'(1) = \xi',\qquad
+\cos \lambda'(2) = \eta',\qquad
+\cos \lambda'(3) = \zeta',
+\]
+then we shall have
+\[
+d\xi = \xi'\, \frac{ds}{r},\qquad
+d\eta = \eta'\, \frac{ds}{r},\qquad
+d\zeta = \zeta'\, \frac{ds}{r},
+\]
+since, in fact, $\xi$,~$\eta$,~$\zeta$ are merely the coordinates of the point~$\lambda$ referred to the centre
+of the sphere.
+
+Since by the solution of the equation $f(x, y, z) = 0$ the coordinate~$z$ may be
+expressed in the form of a function of $x$,~$y$, we shall, for greater simplicity, assume
+that this has been done and that we have found
+\[
+z = F(x, y).
+\]
+We can then write as the equation of the surface
+\[
+z - F(x, y) = 0,
+\]
+or
+\[
+f(x, y, z) = z - F(x, y).
+\]
+
+From this follows, if we set
+\begin{gather*}
+dF(x, y) = t\, dx + u\, dy, \\
+P = -t,\qquad
+Q = -u,\qquad
+R = 1,
+\end{gather*}
+where $t$,~$u$ are merely functions of $x$~and~$y$. We set also
+\[
+dt = T\, dx + U\, dy,\qquad
+du = U\, dx + V\, dy.
+\]
+
+Therefore upon the whole surface we have
+\[
+dz = t\, dx + u\, dy
+\]
+and therefore, on the curve,
+\[
+\zeta = t\xi + u\eta.
+\]
+Hence differentiation gives, on substituting the above values for $d\xi$,~$d\eta$,~$d\zeta$,
+\begin{align*}
+(\zeta' - t\xi' - u\eta') \frac{ds}{r}
+  &= \xi\, dt + \eta\, du \\
+  &= (\xi^{2}T + 2\xi\eta U + \eta^{2}V)\, ds,
+\end{align*}
+\PageSep{92}
+or
+\begin{align*}
+\frac{1}{r}
+  &= \frac{\xi^{2}T + 2\xi\eta U + \eta^{2}V}{-\xi' t - \eta'\Typo{\mu}{u} + \zeta'} \\
+  &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{X\xi' - Y\eta' + Z\zeta'} \\
+  &= \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\cos L\lambda'}.
+\end{align*}
+
+
+\Article{10.}
+
+Before we further transform the expression just found, we will make a few
+remarks about it.
+
+A normal to a curve in its plane corresponds to two directions upon the sphere,
+according as we draw it on the one or the other side of the curve. The one direction,
+toward which the curve is \emph{concave}, is denoted by~$\lambda'$, the other by the opposite
+point on the sphere. Both these points, like $L$~and~$\L$, are $90°$~from~$\lambda$, and therefore
+lie in a great circle. And since $\L$~is also $90°$~from~$\lambda$, $\L L = 90° - L\lambda'$, or
+$= L\lambda' - 90°$. Therefore
+\[
+\cos L\lambda' = ±\sin \L L,
+\]
+where $\sin \L L$ is necessarily positive. Since $r$~is regarded as positive in our analysis,
+the sign of~$\cos L\lambda'$ will be the same as that of
+\[
+Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).
+\]
+And therefore a positive value of this last expression means that $L\lambda'$~is less than~$90°$,
+or that the curve is concave toward the side on which lies the projection of the
+normal to the surface upon the plane. A negative value, on the contrary, shows that
+the curve is convex toward this side. Therefore, in general, we may set also
+\[
+\frac{1}{r} = \frac{Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V)}{\sin \L L},
+\]
+if we regard the radius of curvature as positive in the first case, and negative in
+the second. $\L L$~is here the angle which our cutting plane makes with the plane
+tangent to the curved surface, and we see that in the different cutting planes passed
+through the same point and the same tangent the radii of curvature are proportional
+to the sine of the inclination. Because of this simple relation, we shall limit ourselves
+hereafter to the case where this angle is a right angle, and where the cutting
+\PageSep{93}
+plane, therefore, is passed through the normal of the curved surface. Hence we have
+for the radius of curvature the simple formula
+\[
+\frac{1}{r} = Z(\xi^{2}T + 2\xi\eta U + \eta^{2}V).
+\]
+
+
+\Article{11.}
+
+Since an infinite number of planes may be passed through this normal, it follows
+that there may be infinitely many different values of the radius of curvature. In this
+case $T$,~$U$,~$V$,~$Z$ are regarded as constant, $\xi$,~$\eta$,~$\zeta$ as variable. In order to make the
+latter depend upon a single variable, we take two fixed points $M$,~$M'$ $90°$~apart on the
+great circle whose pole is~$L$. Let their coordinates referred to the centre of the sphere
+be $\alpha$,~$\beta$,~$\gamma$;  $\alpha'$,~$\beta'$,~$\gamma'$. We have then
+\[
+\cos\lambda(1)
+  = \cos\lambda M ·\cos M(1)
+  + \cos\lambda M'·\cos M'(1)
+  + \cos\lambda L ·\cos L(1).
+\]
+If we set
+\[
+\lambda M = \phi,
+\]
+then we have
+\[
+\cos\lambda M' = \sin\phi,
+\]
+and the formula becomes
+\begin{align*}
+\xi   &= \alpha\cos\phi + \alpha'\sin\phi,
+\intertext{and likewise}
+\eta  &= \beta \cos\phi + \beta' \sin\phi, \\
+\zeta &= \gamma\cos\phi + \gamma'\sin\phi.
+\end{align*}
+
+Therefore, if we set\Note{35}
+\begin{align*}
+A &= (\alpha^{2}T + 2\alpha\beta U + \beta^{2}V)Z, \\
+B &= (\alpha\alpha'T + (\alpha'\beta + \alpha\beta')U + \beta\beta'V)Z,\NoteMark \\
+C &= (\alpha'^{2}T + 2\alpha'\beta' U + \beta'^{2}V)Z,
+\end{align*}
+we shall have
+\begin{align*}
+\frac{1}{r}
+  &= A\cos^{2}\phi + 2B\cos\phi \sin\phi + C\sin^{2}\phi \\
+  &= \frac{A + C}{2} + \frac{A - C}{2}\cos 2\phi + B\sin 2\phi.
+\end{align*}
+If we put
+\begin{align*}
+\frac{A - C}{2} &= E\cos 2\theta, \\
+B &= E\sin 2\theta,
+\end{align*}
+\PageSep{94}
+where we may assume that $E$~has the same sign as~$\dfrac{A - C}{2}$, then we have
+\[
+\frac{1}{r} = \tfrac{1}{2}(A + C) + E\cos 2(\phi - \theta).
+\]
+It is evident that $\phi$~denotes the angle between the cutting plane and another plane
+through this normal and that tangent which corresponds to the direction~$M$. Evidently,
+therefore, $\dfrac{1}{r}$~takes its greatest (absolute) value, or $r$~its smallest, when $\phi = \theta$; and $\dfrac{1}{r}$~its
+smallest absolute value, when $\phi = \theta + 90°$. Therefore the greatest and the least
+curvatures occur in two planes perpendicular to each other. Hence these extreme
+values for~$\dfrac{1}{r}$ are
+\[
+\tfrac{1}{2}(A + C) ± \SQRT{\left(\frac{A - C}{2}\right)^{2} + B^{2}}.
+\]
+Their sum is $A + C$ and their product $AC - B^{2}$, or the product of the two extreme
+radii of curvature is
+\[
+= \frac{1}{AC - B^{2}}.
+\]
+This product, which is of great importance, merits a more rigorous development.
+In fact, from formulæ above we find
+\[
+AC - B^{2} = (\alpha\beta' -\beta\alpha')^{2}(TV - U^{2})Z^{2}.
+\]
+But from the third formula in [Theorem]~6, \Art{7}, we easily infer that\Note{36}
+\[
+\alpha\beta' - \beta\alpha' = ±Z,\NoteMark
+\]
+therefore
+\[
+AC - B^{2} = Z^{4}(TV - U^{2}).
+\]
+Besides, from \Art{8},
+\begin{align*}
+Z &= ±\frac{R}{\Sqrt{P^{2} + Q^{2} + R^{2}}} \\
+  &= ±\frac{1}{\Sqrt{1 + t^{2} + u^{2}}},
+\end{align*}
+therefore
+\[
+AC - B^{2} = \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}}.
+\]
+
+Just as to \emph{each} point on the curved surface corresponds a particular point~$L$ on
+the auxiliary sphere, by means of the normal erected at this point and the radius of
+\PageSep{95}
+the auxiliary sphere parallel to the normal, so the aggregate of the points on the
+auxiliary sphere, which correspond to all the points of a \emph{line} on the curved surface,
+forms a line which will correspond to the line on the curved surface. And, likewise,
+to every finite figure on the curved surface will correspond a finite figure on the
+auxiliary sphere, the area of which upon the latter shall be regarded as the measure
+of the amplitude of the former. We shall either regard this area as a number, in
+which case the square of the radius of the auxiliary sphere is the unit, or else
+express it in degrees,~etc., setting the area of the hemisphere equal to~$360°$.
+
+The comparison of the area upon the curved surface with the corresponding
+amplitude leads to the idea of what we call the measure of curvature of the surface.
+If the former is proportional to the latter, the curvature is called uniform;
+and the quotient, when we divide the amplitude by the surface, is called the measure
+of curvature. This is the case when the curved surface is a sphere, and the measure
+of curvature is then a fraction whose numerator is unity and whose denominator is
+the square of the radius.
+
+We shall regard the measure of curvature as positive, if the boundaries of the
+figures upon the curved surface and upon the auxiliary sphere run in the same sense;
+as negative, if the boundaries enclose the figures in contrary senses. If they are not
+proportional, the surface is \Typo{non-uniformily}{non-uniformly} curved. And at each point there exists a
+particular measure of curvature, which is obtained from the comparison of corresponding
+infinitesimal parts upon the curved surface and the auxiliary sphere. Let $d\sigma$~be
+a surface element on the former, and $d\Sigma$~the corresponding element upon the auxiliary
+sphere, then
+\[
+\frac{d\Sigma}{d\sigma}
+\]
+will be the measure of curvature at this point.
+
+In order to determine their boundaries, we first project both upon the $xy$-plane.
+The magnitudes of these projections are $Z\, d\sigma$,~$Z\, d\Sigma$. The sign of~$Z$ will show whether
+the boundaries run in the same sense or in contrary senses around the surfaces and
+their projections. We will suppose that the figure is a triangle; the projection upon
+the $xy$-plane has the coordinates
+\[
+x,\ y;\qquad
+x + dx,\ y + dy;\qquad
+x + \delta x,\ y + \delta y.
+\]
+Hence its double area will be
+\[
+2Z\, d\sigma = dx·\delta y - dy·\delta x.
+\]
+To the projection of the corresponding element upon the sphere will correspond the
+coordinates:
+\PageSep{96}
+\[
+\begin{gathered}
+X, \\
+X + \frac{\dd X}{\dd x}·dx + \frac{\dd X}{\dd y}·dy, \\
+X + \frac{\dd X}{\dd x}·\delta x + \frac{\dd X}{\dd y}·\delta y,
+\end{gathered}
+\qquad
+\begin{gathered}
+Y, \\
+Y + \frac{\dd Y}{\dd x}·dx + \frac{\dd Y}{\dd y}·dy, \\
+Y + \frac{\dd Y}{\dd x}·\delta x + \frac{\dd Y}{\dd y}·\delta y,
+\end{gathered}
+\]
+From this the double area of the element is found to be
+\begin{align*}
+2Z\, d\Sigma
+  &= \Neg
+  \left(\frac{\dd X}{\dd x}·dx
+      + \frac{\dd X}{\dd y}·dy\right)
+  \left(\frac{\dd Y}{\dd x}·\delta x
+      + \frac{\dd Y}{\dd y}·\delta y\right) \\
+  &\phantom{={}}
+ -\left(\frac{\dd X}{\dd x}·\delta x
+      + \frac{\dd X}{\dd y}·\delta y\right)
+  \left(\frac{\dd Y}{\dd x}·dx
+      + \frac{\dd Y}{\dd y}·dy\right) \\
+  &= \Neg
+  \left(\frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
+      - \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x}\right)
+    (dx·\delta y - dy·\delta x).
+\end{align*}
+The measure of curvature is, therefore,
+\[
+= \frac{\dd X}{\dd x}·\frac{\dd Y}{\dd y}
+- \frac{\dd X}{\dd y}·\frac{\dd Y}{\dd x} = \omega.
+\]
+Since
+\begin{gather*}
+X = -tZ,\qquad
+Y = -uZ, \\
+(1 + t^{2} + u^{2})Z^{2} = 1,
+\end{gather*}
+we have
+\begin{align*}
+dX &= -Z^{3}(1 + u^{2})\, dt + Z^{3}tu·du, \\
+dY &= +Z^{3}tu·dt - Z^{3}(1 + t^{2})\, du,
+\end{align*}
+therefore
+\begin{alignat*}{2}
+\frac{\dd X}{\dd x}
+  &= Z^{3}\bigl\{-(1 + u^{2})T + tuU\bigr\},\qquad&
+\frac{\dd Y}{\dd x}
+  &= Z^{3}\bigl\{tuT - (1 + t^{2})U\bigr\}, \\
+%
+\frac{\dd X}{\dd y}
+  &= Z^{3}\bigl\{-(1 + u^{2})U + tuV\bigr\},\qquad&
+\frac{\dd Y}{\dd y}
+  &= Z^{3}\bigl\{tuU - (1 + t^{2})V\bigr\},
+\end{alignat*}
+and
+\begin{align*}
+\omega
+  &= Z^{6}(TV - U^{2})\bigl((1 + t^{2})(1 + u^{2}) - t^{2}u^{2}\bigr) \\
+  &= Z^{6}(TV - U^{2})(1 + t^{2} + u^{2}) \\
+  &= Z^{4}(TV - U^{2}) \\
+  &= \frac{TV - U^{2}}{(1 + t^{2} + u^{2})^{2}},
+\end{align*}
+the very same expression which we have found at the end of the preceding article.
+Therefore we see that
+\PageSep{97}
+
+%[** TN: Quoted, not italicized, in the original]
+\begin{Theorem}[]
+The measure of curvature is always expressed by means of a fraction whose
+numerator is unity and whose denominator is the product of the maximum
+and minimum radii of curvature in the planes passing through the normal.
+\end{Theorem}
+
+
+\Article{12.}
+
+We will now investigate the nature of shortest lines upon curved surfaces. The
+nature of a curved line in space is determined, in general, in such a way that the
+coordinates $x$,~$y$,~$z$ of each point are regarded as functions of a single variable, which
+we shall call~$w$. The length of the curve, measured from an arbitrary origin to this
+point, is then equal to
+\[
+\int \SQRT{\left(\frac{dx}{dw}\right)^{2}
+  + \left(\frac{dy}{dw}\right)^{2}
+  + \left(\frac{dz}{dw}\right)^{2}}·dw.
+\]
+If we allow the curve to change its position by an infinitely small variation, the variation
+of the whole length will then be
+{\small
+\begin{multline*}
+= \int \frac{\dfrac{dx}{dw}·d\, \delta x
+           + \dfrac{dy}{dw}·d\, \delta y
+           + \dfrac{dz}{dw}·d\, \delta z}
+            {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                 + \left(\dfrac{dy}{dw}\right)^{2}
+                 + \left(\dfrac{dz}{dw}\right)^{2}}}% \\
+%
+= \frac{\dfrac{dx}{dw}·\delta x
+      + \dfrac{dy}{dw}·\delta y
+      + \dfrac{dz}{dw}·\delta z}
+       {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+            + \left(\dfrac{dy}{dw}\right)^{2}
+            + \left(\dfrac{dz}{dw}\right)^{2}}} \displaybreak[1] \\
+\qquad- \int\left\{
+  \delta x·d\frac{\dfrac{dx}{dw}}
+    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+         + \left(\dfrac{dy}{dw}\right)^{2}
+         + \left(\dfrac{dz}{dw}\right)^{2}}}\right.
++ \delta y·d\frac{\dfrac{dy}{dw}}
+    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+         + \left(\dfrac{dy}{dw}\right)^{2}
+         + \left(\dfrac{dz}{dw}\right)^{2}}} \\
+\qquad\qquad\qquad+ \left.\delta z·d\frac{\dfrac{dz}{dw}}
+    {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+         + \left(\dfrac{dy}{dw}\right)^{2}
+         + \left(\dfrac{dz}{dw}\right)^{2}}}\right\}.
+\end{multline*}}%
+The expression under the integral sign must vanish in the case of a minimum, as we
+know. Since the curved line lies upon a given curved surface whose equation is
+\[
+P\, dx + Q\, dy + R\, dz = 0,
+\]
+the equation between the variations $\delta x$,~$\delta y$,~$\delta z$
+\[
+P\, \delta z + Q\, \delta y + R\, \delta z = 0
+\]
+must also hold. From this, by means of well known principles, we easily conclude
+that the differentials
+\PageSep{98}
+\begin{gather*}
+  d·\frac{\dfrac{dx}{dw}}
+           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                + \left(\dfrac{dy}{dw}\right)^{2}
+                + \left(\dfrac{dz}{dw}\right)^{2}}}, \\
+  d·\frac{\dfrac{dy}{dw}}
+           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                + \left(\dfrac{dy}{dw}\right)^{2}
+                + \left(\dfrac{dz}{dw}\right)^{2}}}, \\
+  d·\frac{\dfrac{dz}{dw}}
+           {\SQRT{\left(\dfrac{dx}{dw}\right)^{2}
+                + \left(\dfrac{dy}{dw}\right)^{2}
+                + \left(\dfrac{dz}{dw}\right)^{2}}}
+\end{gather*}
+must be proportional to the quantities $P$,~$Q$,~$R$ respectively. If $ds$~is an element of
+the curve; $\lambda$~the point upon the auxiliary sphere, which represents the direction of
+this element; $L$~the point giving the direction of the normal as above; and $\xi$,~$\eta$,~$\zeta$;
+$X$,~$Y$,~$Z$ the coordinates of the points $\lambda$,~$L$ referred to the centre of the auxiliary
+sphere, then we have
+\begin{gather*}
+dx = \xi\, ds,\qquad
+dy = \eta\, ds,\qquad
+dz = \zeta\, ds, \\
+\xi^{2} + \eta^{2} + \zeta^{2} = 1.
+\end{gather*}
+Therefore we see that the above differentials will be equal to $d\xi$,~$d\eta$,~$d\zeta$. And since
+$P$,~$Q$,~$R$ are proportional to the quantities $X$,~$Y$,~$Z$, the character of the shortest line
+is such that
+\[
+\frac{d\xi}{X} = \frac{d\eta}{Y} = \frac{d\zeta}{Z}.
+\]
+
+
+\Article{13.}
+
+To every point of a curved line upon a curved surface there correspond two
+points on the sphere, according to our point of view; namely, the point~$\lambda$, which
+represents the direction of the linear element, and the point~$L$, which represents the
+direction of the normal to the surface. The two are evidently $90°$~apart. In our
+former investigation (\Art{9}), where [we] supposed the curved line to lie in a plane,
+we had \emph{two} other points upon the sphere; namely,~$\L$, which represents the direction
+of the normal to the plane, and~$\lambda'$, which represents the direction of the normal to
+the element of the curve in the plane. In this case, therefore, $\L$~was a fixed point
+and $\lambda$,~$\lambda'$ were always in a great circle whose pole was~$\L$. In generalizing these
+considerations, we shall retain the notation $\L$,~$\lambda'$, but we must define the meaning of
+these symbols from a more general point of view. When the curve~$s$ is described,
+the points $L$,~$\lambda$ also describe curved lines upon the auxiliary sphere, which, generally
+speaking, are no longer great circles. Parallel to the element of the second line,
+\PageSep{99}
+we draw a radius of the auxiliary sphere to the point~$\lambda'$, but instead of this point
+we take the point opposite when $\lambda'$~is more than~$90°$ from~$L$. In the first case, we
+regard the element at~$\lambda$ as positive, and in the other as negative. Finally, let $\L$ be
+the point on the auxiliary sphere, which is $90°$~from both $\lambda$~and~$\lambda'$, and which is so
+taken that $\lambda$,~$\lambda'$,~$\L$ lie in the same order as $(1)$,~$(2)$,~$(3)$.
+
+The coordinates of the four points of the auxiliary sphere, referred to its centre,
+are for
+\begin{alignat*}{4}
+&L\qquad &&X\quad &&Y\quad &&Z \\
+&\lambda &&\xi    &&\eta   &&\zeta \\
+&\lambda'&&\xi'   &&\eta'  &&\zeta' \\
+&\L      &&\alpha &&\beta  &&\gamma.
+\end{alignat*}
+Hence each of these $4$~points describes a line upon the auxiliary sphere, whose elements
+we shall express by $dL$,~$d\lambda$,~$d\lambda'$,~$d\L$. We have, therefore,
+\begin{align*}
+d\xi   &= \xi'\, d\lambda, \\
+d\eta  &= \eta'\, d\lambda, \\
+d\zeta &= \zeta'\, d\lambda.
+\end{align*}
+In an analogous way we now call
+\[
+\frac{d\lambda}{ds}
+\]
+the measure of curvature of the curved line upon the curved surface, and its reciprocal
+\[
+\frac{ds}{d\lambda}
+\]
+the radius of curvature. If we denote the latter by~$\rho$, then
+\begin{align*}
+\rho\, d\xi   &= \xi'\, ds, \\
+\rho\, d\eta  &= \eta'\, ds, \\
+\rho\, d\zeta &= \zeta'\, ds.
+\end{align*}
+
+If, therefore, our line be a shortest line, $\xi'$,~$\eta'$,~$\zeta'$ must be proportional to the
+quantities $X$,~$Y$,~$Z$. But, since at the same time
+\[
+\xi'^{2} + \eta'^{2} + \zeta'^{2}  = X^{2} + Y^{2} + Z^{2} = 1,
+\]
+we have
+\[
+\xi'   = ±X,\quad
+\eta'  = ±Y,\quad
+\zeta' = ±Z,
+\]
+and since, further,
+\begin{align*}
+\xi'X + \eta'Y + \zeta'Z
+  &= \cos \lambda'L \\
+  &= ±(X^{2} + Y^{2} + Z^{2}) \\
+  &= ±1,
+\end{align*}
+\PageSep{100}
+and since we always choose the point~$\lambda'$ so that
+\[
+\lambda'L < 90°,
+\]
+then for the shortest line
+\[
+\lambda'L = 0,
+\]
+or $\lambda'$~and~$L$ must coincide. Therefore
+\begin{align*}
+\rho\, d\xi   &= X\, ds, \\
+\rho\, d\eta  &= Y\, ds, \\
+\rho\, d\zeta &= Z\, ds,
+\end{align*}
+and we have here, instead of $4$~curved lines upon the auxiliary sphere, only $3$~to consider.
+Every element of the second line is therefore to be regarded as lying in the
+great circle~$L\lambda$. And the positive or negative value of~$\rho$ refers to the concavity
+or the convexity of the curve in the direction of the normal.
+
+
+\Article{14.}
+
+We shall now investigate the spherical angle upon the auxiliary sphere, which
+the great circle going from~$L$ toward~$\lambda$ makes with that one going from~$L$ toward
+one of the fixed points $(1)$,~$(2)$,~$(3)$; \eg, toward~$(3)$. In order to have something
+definite here, we shall consider the sense from~$L(3)$ to~$L\lambda$ the same as that in which
+$(1)$,~$(2)$, and~$(3)$ lie. If we call this angle~$\phi$, then it follows from the theorem of \Art{7}
+that\Note{37}
+\[
+\sin L(3)·\sin L\lambda·\sin\phi = Y\xi - X\eta,\NoteMark
+\]
+or, since $L\lambda = 90°$ and
+\[
+\sin L(3) = \Sqrt{X^{2} + Y^{2}} = \Sqrt{1 - Z^{2}},
+\]
+we have
+\[
+\sin\phi = \frac{Y\xi - X\eta}{\Sqrt{X^{2} + Y^{2}}}.
+\]
+Furthermore,
+\[
+\sin L(3)·\sin L\lambda·\cos\phi = \zeta,
+\]
+or
+\[
+\cos\phi = \frac{\zeta}{\Sqrt{X^{2} + Y^{2}}}
+\]
+and
+\[
+\tan\phi = \frac{Y\xi - X\eta}{\zeta} = \frac{\zeta'}{\zeta}.
+\]
+\PageSep{101}
+Hence we have
+\[
+d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta
+  - (Y\xi - X\eta)\, d\zeta + \xi\zeta\, dY - \eta\zeta\, dX}
+  {(Y\xi - X\eta)^{2} + \zeta^{2}}.
+\]
+The denominator of this expression is
+\begin{align*}
+&= Y^{2}\xi^{2} - 2XY\xi\eta - X^{2}\eta^{2} + \zeta^{2} \\
+&= -(X\xi + Y\eta)^{2} + (X^{2} + Y^{2})(\xi^{2} + \eta^{2}) + \zeta^{2} \\
+&= -Z^{2}\zeta^{2} + (1 - Z^{2})(1 - \zeta^{2}) + \zeta^{2} \\
+&= 1 - Z^{2},
+\end{align*}
+or
+\[
+d\phi = \frac{\zeta Y\, d\xi - \zeta X\, d\eta
+  + (X\eta - Y\xi)\, d\zeta - \eta\zeta\, dX + \xi\zeta\, dY}
+  {1 - Z^{2}}.
+\]
+
+We verify readily by expansion the identical equation
+\begin{gather*}
+\eta\zeta(X^{2} + Y^{2} + Z^{2}) + YZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\
+= (X\xi + Y\eta + Z\zeta)(Z\eta + Y\zeta) + (X\zeta - Z\xi)(X\eta - Y\xi)\Add{,}
+\end{gather*}
+and likewise
+\begin{gather*}
+\xi\zeta(X^{2} + Y^{2} + Z^{2}) + XZ(\xi^{2} + \eta^{2} + \zeta^{2}) \\
+= (X\xi + Y\eta + Z\zeta)(X\zeta + Z\xi) + (Y\xi - X\eta)(Y\zeta - Z\eta).
+\end{gather*}
+We have, therefore,
+\begin{align*}
+\eta\zeta &= -YZ + (X\zeta - Z\xi )(X\eta  - Y\xi), \\
+\xi\zeta  &= -XZ + (Y\xi   - X\eta)(Y\zeta - Z\eta).
+\end{align*}
+Substituting these values, we obtain
+\begin{multline*}
+d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)
+  + \frac{\zeta Y\, d\xi - \zeta X\, d\eta}{1 - Z^{2}} \\
+  + \frac{X\eta - Y\xi}{1 - Z^{2}}\bigl\{
+    d\zeta - (X\zeta - Z\xi)\, dX - (Y\zeta - Z\eta)\, dY\bigr\}.
+\end{multline*}
+Now
+\begin{alignat*}{4}
+&  X\, dX &&+    Y\, dY &&+     Z\, dZ &&= 0, \\
+&\xi\, dX &&+ \eta\, dY &&+ \zeta\, dZ &&= -X\, d\xi - Y\, d\eta - Z\, d\zeta.
+\end{alignat*}
+On substituting we obtain, instead of what stands in the parenthesis,
+\[
+d\zeta - Z(X\, d\xi + Y\, d\eta + Z\, d\zeta).
+\]
+Hence\Note{38}
+\begin{align*}
+d\phi = \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY)
+  &+ \frac{d\xi}{1 - Z^{2}}\{\zeta Y - \eta X^{2}Z + \xi XYZ\} \\
+  &- \frac{d\eta}{1 - Z^{2}}\{\zeta X + \eta XYZ - \xi Y^{2}Z\}\NoteMark \\
+  &+ d\zeta(\eta X - \xi Y).
+\end{align*}
+\PageSep{102}
+Since, further,
+\begin{align*}
+\eta X^{2}Z - \xi XYZ
+  &= \eta X^{2}Z + \eta Y^{2}Z + \zeta ZYZ \\
+  &= \eta Z(1 - Z^{2}) + \zeta YZ^{2}, \\
+%
+\eta XYZ - \xi Y^{2}Z
+  &= -\xi X^{2}Z - \zeta XZ^{2} - \xi Y^{2}Z \\
+  &= - \xi Z(1 - Z^{2}) - \zeta XZ^{2},
+\end{align*}
+our whole expression becomes
+\begin{align*}
+d\phi &= \frac{Z}{1 - Z^{2}}(Y\, dX - X\, dY) \\
+&\quad
+  + (\zeta Y - \eta  Z)\, d\xi
+  + (\xi   Z - \zeta X)\, d\eta
+  + (\eta  X - \xi   Y)\, d\zeta.
+\end{align*}
+
+
+\Article{15.}
+
+The formula just found is true in general, whatever be the nature of the curve.
+But if this be a shortest line, then it is clear that the last three terms destroy each
+other, and consequently\Note{39}
+\[
+d\phi = -\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX).\NoteMark
+\]
+But we see at once that
+\[
+\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX)
+\]
+is nothing but the area of the part of the auxiliary sphere, which is formed between
+the element of the line~$L$, the two great circles drawn through its extremities and~$(3)$,\Note{40}
+%[Illustration]
+\Figure{102}
+and the element thus intercepted on the great circle through $(1)$~and~$(2)$. This
+surface is considered positive, if $L$~and~$(3)$ lie on the same side of~$(1)\ (2)$, and if the
+\PageSep{103}
+direction from~$P$ to~$P'$ is the same as that from~$(2)$ to~$(1)$; negative, if the contrary
+of one of these conditions hold; positive again, if the contrary of both conditions be
+true. In other words, the surface is considered positive if we go around the circumference
+of the figure~$LL'P'P$ in the same sense as $(1)\ (2)\ (3)$; negative, if we go
+in the contrary sense.
+
+If\Note{41} we consider now a finite part of the line from~$L$ to~$L'$ and denote by $\phi$,~$\phi'$
+the values of the angles at the two extremities, then we have
+\[
+\phi' = \phi + \Area LL'P'P,
+\]
+the sign of the area being taken as explained.
+
+Now\Note{42} let us assume further that, from the origin upon the curved surface, infinitely
+many other shortest lines go out, and denote by~$A$ that indefinite angle which the
+first element, moving counter-clockwise, makes with the first element of the first line;
+and through the other extremities of the different curved lines let a curved line be drawn,
+concerning which, first of all, we leave it undecided whether it be a shortest line or
+not. If we suppose also that those indefinite values, which
+for the first line were $\phi$,~$\phi'$, be denoted by $\psi$,~$\psi'$ for each of
+these lines, then $\psi' - \psi$ is capable of being represented in
+the same manner on the auxiliary sphere by the space~$LL'_{1}P'_{1}P$.
+Since evidently $\psi = \phi - A$, the space\Note{43}
+\[
+\begin{aligned}[b]
+LL'_{1}P'_{1}P'L'L
+  &= \psi' - \psi - \phi' + \phi \\
+  &= \psi' - \phi' + A \\
+  &= LL'_{1}L'L + L'L'_{1}P'_{1}P'.\NoteMark
+\end{aligned}
+\qquad\qquad
+%[Illustration]
+\raisebox{-\baselineskip}{\Graphic{1.5in}{103}}
+\]
+
+If the bounding line is also a shortest line, and, when prolonged, makes with
+$LL'$,~$LL'_{1}$ the angles $B$,~$B_{1}$; if, further, $\chi$,~$\chi_{1}$ denote the same at the points $L'$,~$L'_{1}$,
+that $\phi$~did at~$L$ in the line~$LL'$, then we have
+\begin{align*}
+\chi_{1} &= \chi + \Area L'L'_{1}P'_{1}P', \\
+\psi' - \phi' + A &= LL'_{1}L'L + \chi_{1} - \chi;
+\end{align*}
+but
+\begin{align*}
+\phi' &= \chi + B, \\
+\psi' &= \chi_{1} + B_{1},
+\end{align*}
+therefore
+\[
+B_{1} - B + A = LL'_{1}L'L.
+\]
+The angles of the triangle~$LL'L'_{1}$ evidently are
+\[
+A,\qquad 180° - B,\qquad B_{1},
+\]
+\PageSep{104}
+therefore their sum is
+\[
+180° + LL'_{1}L'L.
+\]
+
+The form of the proof will require some modification and explanation, if the point~$(3)$
+falls within the triangle. But, in general, we conclude
+
+%[** TN: Quoted, not italicized, in the original]
+\begin{Theorem}[]
+The sum of the three angles of a triangle, which is formed of shortest lines
+upon an arbitrary curved surface, is equal to the sum of~$180°$ and the area of
+the triangle upon the auxiliary sphere, the boundary of which is formed by the
+points~$L$, corresponding to the points in the boundary of the original triangle,
+and in such a manner that the area of the triangle may be regarded as positive
+or negative according as it is inclosed by its boundary in the same sense as
+the original figure or the contrary.
+\end{Theorem}
+
+Wherefore\Note{44} we easily conclude also that the sum of all the angles of a polygon
+of $n$~sides, which are shortest lines upon the curved surface, is [equal to] the sum
+of $(n - 2)180° + \text{the area of the polygon upon the sphere~etc.}$
+
+
+\Article{16.}
+
+If one curved surface can be completely developed upon another surface, then all
+lines upon the first surface will evidently retain their magnitudes after the development
+upon the other surface; likewise the angles which are formed by the intersection
+of two lines. Evidently, therefore, such lines also as are shortest lines upon
+one surface remain shortest lines after the development. Whence, if to any arbitrary
+polygon formed of shortest lines, while it is upon the first surface, there corresponds
+the figure of the zeniths\Note{45} upon the auxiliary sphere, the area of which is~$A$,
+and if, on the other hand, there corresponds to the same polygon, after its development
+upon another surface, a figure of the zeniths upon the auxiliary sphere, the
+area of which is~$A'$, it follows at once that in every case
+\[
+A = A'.
+\]
+Although this proof originally presupposes the boundaries of the figures to be shortest
+lines, still it is easily seen that it holds generally, whatever the boundary may be.
+For, in fact, if the theorem is independent of the number of sides, nothing will prevent
+us from imagining for every polygon, of which some or all of its sides are not
+shortest lines, another of infinitely many sides all of which are shortest lines.
+
+Further, it is clear that every figure retains also its area after the transformation
+by development.
+\PageSep{105}
+
+We shall here consider 4~figures:
+
+%[** TN: Indented line list items in the original]
+\Par{1)} an arbitrary figure upon the first surface,
+
+\Par{2)} the figure on the auxiliary sphere, which corresponds to the zeniths of the
+previous figure,
+
+\Par{3)} the figure upon the second surface, which No.~1 forms by the development,
+
+\Par{4)} the figure upon the auxiliary sphere, which corresponds to the zeniths of
+No.~3.
+
+Therefore, according to what we have proved, 2~and~4 have equal areas, as also
+1~and~3. Since we assume these figures infinitely small, the quotient obtained by
+dividing 2~by~1 is the measure of curvature of the first curved surface at this point,
+and likewise the quotient obtained by dividing 4~by~3, that of the second surface.
+From this follows the important theorem:
+
+%[** TN: Quoted, not italicized in the original]
+\begin{Theorem}[]
+In the transformation of surfaces by development the measure of curvature
+at every point remains unchanged.
+\end{Theorem}
+This is true, therefore, of the product of the greatest and smallest radii of curvature.
+
+In the case of the plane, the measure of curvature is evidently everywhere zero.
+Whence follows therefore the important theorem:
+
+\begin{Theorem}[]
+For all surfaces developable upon a plane the measure of curvature everywhere
+vanishes,
+\end{Theorem}
+or
+\[
+\left(\frac{\dd^{2}z}{\dd x\, \dd y}\right)^{2}
+  - \left(\frac{\dd^{2} z}{\dd x^{2}}\right)
+    \left(\frac{\dd^{2} z}{\dd x^{2}}\right) = 0,
+\]
+which criterion is elsewhere derived from other principles, though, as it seems to us,
+not with the desired rigor. It is clear that in all such surfaces the zeniths of all
+points can not fill out any space, and therefore they must all lie in a line.
+
+
+\Article{17.}
+
+From a given point on a curved surface we shall let an infinite number of shortest
+lines go out, which shall be distinguished from one another by the angle which their
+first elements make with the first element of a \emph{definite} shortest line. This angle we
+shall call~$\theta$. Further, let $s$~be the length [measured from the given point] of a part
+of such a shortest line, and let its extremity have the coordinates $x$,~$y$,~$z$. Since $\theta$~and~$s$,
+therefore, belong to a perfectly definite point on the curved surface, we can
+regard $x$,~$y$,~$z$ as functions of $\theta$~and~$s$. The direction of the element of~$s$ corresponds
+to the point~$\lambda$ on the sphere, whose coordinates are $\xi$,~$\eta$,~$\zeta$. Thus we shall have
+\PageSep{106}
+\[
+\xi = \frac{\dd x}{\dd s},\qquad
+\eta = \frac{\dd y}{\dd s},\qquad
+\zeta = \frac{\dd z}{\dd s}.
+\]
+
+The extremities of all shortest lines of equal lengths~$s$ correspond to a curved
+line whose length we may call~$t$. We can evidently consider~$t$ as a function of $s$~and~$\theta$,
+and if the direction of the element of~$t$ corresponds upon the sphere to the point~$\lambda'$
+whose coordinates are $\xi'$,~$\eta'$,~$\zeta'$, we shall have
+\[
+\xi'·\frac{\dd t}{\dd\theta} = \frac{\dd x}{\dd\theta},\qquad
+\eta'·\frac{\dd t}{\dd\theta} = \frac{\dd y}{\dd\theta},\qquad
+\zeta'·\frac{\dd t}{\dd\theta} = \frac{\dd z}{\dd\theta}.
+\]
+Consequently
+\[
+(\xi\xi' + \eta\eta' + \zeta\zeta')\, \frac{\dd t}{\dd\theta}
+  = \frac{\dd x}{\dd s}·\frac{\dd x}{\dd\theta}
+  + \frac{\dd y}{\dd s}·\frac{\dd y}{\dd\theta}
+  + \frac{\dd z}{\dd s}·\frac{\dd z}{\dd\theta}.
+\]
+This magnitude we shall denote by~$u$, which itself, therefore, will be a function of $\theta$~and~$s$.
+
+We find, then, if we differentiate with respect to~$s$,
+\begin{align*}
+\frac{\dd u}{\dd s}
+  &= \frac{\dd^{2} x}{\dd s^{2}}·\frac{\dd x}{\dd\theta}
+   + \frac{\dd^{2} y}{\dd s^{2}}·\frac{\dd y}{\dd\theta}
+   + \frac{\dd^{2} z}{\dd s^{2}}·\frac{\dd z}{\dd\theta}
+   + \tfrac{1}{2}\, \frac{\dd\left\{
+       \left(\dfrac{\dd x}{\dd s}\right)^{2}
+     + \left(\dfrac{\dd y}{\dd s}\right)^{2}
+     + \left(\dfrac{\dd z}{\dd s}\right)^{2}\right\}}{\dd\theta} \\
+  &= \frac{\dd^{2} x}{\dd s^{2}}·\frac{\dd x}{\dd\theta}
+   + \frac{\dd^{2} y}{\dd s^{2}}·\frac{\dd y}{\dd\theta}
+   + \frac{\dd^{2} z}{\dd s^{2}}·\frac{\dd z}{\dd\theta},
+\end{align*}
+because
+\[
+  \left(\dfrac{\dd x}{\dd s}\right)^{2}
++ \left(\dfrac{\dd y}{\dd s}\right)^{2}
++ \left(\dfrac{\dd z}{\dd s}\right)^{2} = 1,
+\]
+and therefore its differential is equal to zero.
+
+But since all points [belonging] to one constant value of~$\theta$ lie on a shortest line,
+if we denote by~$L$ the zenith of the point to which $s$,~$\theta$ correspond and by $X$,~$Y$,~$Z$
+the coordinates of~$L$, [from the last formulæ of \Art{13}],
+\[
+\frac{\dd^{2} x}{\dd s^{2}} = \frac{X}{p},\qquad
+\frac{\dd^{2} y}{\dd s^{2}} = \frac{Y}{p},\qquad
+\frac{\dd^{2} z}{\dd s^{2}} = \frac{Z}{p},
+\]
+if $p$~is the radius of curvature. We have, therefore,
+\[
+p·\frac{\dd u}{\dd s}
+  = X\, \frac{\dd x}{\dd\theta}
+  + Y\, \frac{\dd y}{\dd\theta}
+  + Z\, \frac{\dd z}{\dd\theta}
+  = \frac{\dd t}{\dd\theta}(X\xi' + Y\eta' + Z\zeta').
+\]
+But
+\[
+X\xi' + Y\eta' + Z\zeta' = \cos L\lambda' = 0,
+\]
+because, evidently, $\lambda'$~lies on the great circle whose pole is~$L$. Therefore we have
+\[
+\frac{\dd u}{\dd s} = 0,
+\]
+\PageSep{107}
+or $u$~independent of~$s$, and therefore a function of $\theta$~alone. But for $s = 0$, it is evident
+that $t = 0$, $\dfrac{\dd t}{\dd\theta} = 0$, and therefore $u = 0$. Whence we conclude that, in general,
+$u = 0$, or
+\[
+\cos \lambda\lambda' = 0.
+\]
+From this follows the beautiful theorem:
+
+\begin{Theorem}[]
+If all lines drawn from a point on the curved surface are shortest lines of
+equal lengths, they meet the line which joins their extremities everywhere at
+right angles.
+\end{Theorem}
+
+We can show in a similar manner that, if upon the curved surface any curved
+line whatever is given, and if we suppose drawn from every point of this line toward
+the same side of it and at right angles to it only shortest lines of equal lengths, the
+extremities of which are joined by a line, this line will be cut at right angles by
+those lines in all its points. We need only let $\theta$ in the above development represent
+the length of the \emph{given} curved line from an arbitrary point, and then the above calculations
+retain their validity, except that $u = 0$ for $s = 0$ is now contained in the
+hypothesis.
+
+
+\Article{18.}
+
+The relations arising from these constructions deserve to be developed still more
+fully. We have, in the first place, if, for brevity, we write~$m$ for~$\dfrac{\dd t}{\dd\theta}$,
+\begin{alignat*}{3}
+\Tag{(1)}
+\frac{\dd x}{\dd s} &= \xi, &
+\frac{\dd y}{\dd s} &= \eta, &
+\frac{\dd z}{\dd s} &= \zeta, \\
+\Tag{(2)}
+\frac{\dd x}{\dd\theta} &= m\xi',\quad &
+\frac{\dd y}{\dd\theta} &= m\eta',\quad &
+\frac{\dd z}{\dd\theta} &= m\zeta',
+\end{alignat*}
+\begin{alignat*}{4}
+\Tag{(3)}
+&\xi^{2}  &&+ \eta^{2}  &&+ \zeta^{2} &&= 1, \\
+\Tag{(4)}
+&\xi'^{2} &&+ \eta'^{2} &&+ \zeta'^{2} &&= 1, \\
+\Tag{(5)}
+&\xi\xi'  &&+ \eta\eta' &&+ \zeta\zeta' &&= 0.
+\end{alignat*}
+Furthermore,
+\begin{alignat*}{4}
+\Tag{(6)}
+&X^{2} &&+ Y^{2} &&+ Z^{2} &&= 1, \\
+\Tag{(7)}
+&X\xi  &&+ Y\eta &&+ Z\zeta &&= 0, \\
+\Tag{(8)}
+&X\xi' &&+ Y\eta' &&+ Z\zeta' &&= 0,
+\end{alignat*}
+and
+\begin{align*}
+\Tag{[9]}
+&\left\{
+\begin{alignedat}{2}
+X &= \zeta\eta' &&- \eta\zeta', \\
+Y &= \xi\zeta'  &&- \zeta\xi', \\
+Z &= \eta\xi'   &&- \xi\eta';
+\end{alignedat}
+\right. \\
+\PageSep{108}
+\Tag{[10]}
+&\left\{
+\begin{alignedat}{2}
+\xi'   &= \eta  Z &&- \zeta Y, \\
+\eta'  &= \zeta X &&- \xi   Z, \\
+\zeta' &= \xi   Y &&- \eta  X;
+\end{alignedat}
+\right. \\
+\Tag{[11]}
+&\left\{
+\begin{alignedat}{2}
+\xi   &= Y\zeta' &&- Z\eta', \\
+\eta  &= Z\xi'   &&- X\zeta', \\
+\zeta &= X\eta'  &&- Y\xi'.
+\end{alignedat}
+\right.
+\end{align*}
+
+Likewise, $\dfrac{\dd\xi}{\dd s}$, $\dfrac{\dd\eta}{\dd s}$, $\dfrac{\dd\zeta}{\dd s}$ are proportional to $X$,~$Y$,~$Z$, and if we set
+\[
+\frac{\dd\xi}{\dd s}   = pX,\qquad
+\frac{\dd\eta}{\dd s}  = pY,\qquad
+\frac{\dd\zeta}{\dd s} = pZ,
+\]
+where $\dfrac{1}{p}$ denotes the radius of curvature of the line~$s$, then
+\[
+p = X\, \frac{\dd\xi}{\dd s}
+  + Y\, \frac{\dd\eta}{\dd s}
+  + Z\, \frac{\dd\zeta}{\dd s}.
+\]
+By differentiating~(7) with respect to~$s$, we obtain
+\[
+-p = \xi\, \frac{\dd X}{\dd s}
+   + \eta\, \frac{\dd Y}{\dd s}
+   + \zeta\, \frac{\dd Z}{\dd s}.
+\]
+
+We can easily show that $\dfrac{\dd\xi'}{\dd s}$, $\dfrac{\dd\eta'}{\dd s}$, $\dfrac{\dd\zeta'}{\dd s}$ also are proportional to $X$,~$Y$,~$Z$. In fact,
+[from~10] the values of these quantities are also [equal to]
+\[
+\eta\, \frac{\dd Z}{\dd s} - \zeta\, \frac{\dd Y}{\dd s},\qquad
+\zeta\, \frac{\dd X}{\dd s} - \xi\, \frac{\dd Z}{\dd s},\qquad
+\xi\, \frac{\dd Y}{\dd s} - \eta\, \frac{\dd X}{\dd s},
+\]
+therefore
+\begin{align*}
+Y\, \frac{\dd\xi'}{\dd s} - X\, \frac{\dd\eta'}{\dd s}
+  &= - \zeta\left(\frac{Y\, \dd Y}{\dd s} + \frac{X\, \dd X}{\dd s}\right)
+   + \frac{\dd Z}{\dd s}(Y\eta + X\xi) \\
+  &= - \zeta\left(\frac{X\, \dd X + Y\, \dd Y + Z\, \dd Z}{\dd s}\right)
+   + \frac{\dd Z}{\dd s}(X\xi + Y\eta + Z\zeta) \\
+  &= 0,
+\end{align*}
+and likewise the others. We set, therefore,
+\[
+\frac{\dd\xi'}{\dd s} = p'X,\qquad
+\frac{\dd\eta'}{\dd s} = p'Y,\qquad
+\frac{\dd\zeta'}{\dd s} = p'Z,
+\]
+whence
+\[
+p' = ±\SQRT{\left(\frac{\dd\xi'}{\dd s}\right)^{2}
+  + \left(\frac{\dd\eta'}{\dd s}\right)^{2}
+  + \left(\frac{\dd\zeta'}{\dd s}\right)^{2}}\Add{,}
+\]
+\PageSep{109}
+and also
+\[
+p' = X\, \frac{\dd\xi'}{\dd s}
+   + Y\, \frac{\dd\eta'}{\dd s}
+   + Z\, \frac{\dd\zeta'}{\dd s}.
+\]
+Further [we obtain], from the result obtained by differentiating~(8),
+\[
+-p' = \xi'\, \frac{\dd X}{\dd s}
+    + \eta'\, \frac{\dd Y}{\dd s}
+    + \zeta'\, \frac{\dd Z}{\dd s}.
+\]
+But we can derive two other expressions for this. We have
+\[
+\frac{\dd m\xi'}{\dd s} = \frac{\dd\xi}{\dd\theta},\qquad
+\left[
+\frac{\dd m\eta'}{\dd s} = \frac{\dd\eta}{\dd\theta},\qquad
+\frac{\dd m\zeta'}{\dd s} = \frac{\dd\zeta}{\dd\theta},
+\right]
+\]
+therefore [because of~(8)]
+\[
+mp' = X\, \frac{\dd\xi}{\dd\theta}
+    + Y\, \frac{\dd\eta}{\dd\theta}
+    + Z\, \frac{\dd\zeta}{\dd\theta}.
+\]
+[and therefore, from~(7),]
+\[
+-mp' = \xi\, \frac{\dd X}{\dd\theta}
+     + \eta\, \frac{\dd Y}{\dd\theta}
+     + \zeta\, \frac{\dd Z}{\dd\theta}.
+\]
+
+After these preliminaries [using (2)~and~(4)] we shall now first put~$m$ in the form
+\[
+m = \xi'\, \frac{\dd x}{\dd\theta}
+  + \eta'\, \frac{\dd y}{\dd\theta}
+  + \zeta'\, \frac{\dd z}{\dd\theta},
+\]
+and differentiating with respect to~$s$, we have\footnote
+  {It is better to differentiate~$m^{2}$. [In fact from (2)~and~(4)
+  \[
+  m^{2} = \left(\frac{\dd x}{\dd\theta}\right)^{2}
+        + \left(\frac{\dd y}{\dd\theta}\right)^{2}
+        + \left(\frac{\dd z}{\dd\theta}\right)^{2},
+  \]
+  therefore
+  \begin{align*}
+  m\, \frac{\dd m}{\dd s}
+   &= \frac{\dd x}{\dd\theta}·\frac{\dd^{2} x}{\dd\theta\, \dd s}
+    + \frac{\dd y}{\dd\theta}·\frac{\dd^{2} y}{\dd\theta\, \dd s}
+    + \frac{\dd z}{\dd\theta}·\frac{\dd^{2} z}{\dd\theta\, \dd s} \\
+   &= m\xi'\, \frac{\dd\xi}{\dd\theta}
+    + m\eta'\, \frac{\dd\eta}{\dd\theta}
+    + m\zeta'\, \frac{\dd\zeta}{\dd\theta}.]
+  \end{align*}}
+% [** End of footnote]
+\begin{align*}
+%[** TN: Re-broken]
+\frac{\dd m}{\dd s}
+  &= \frac{\dd x}{\dd\theta}·\frac{\dd\xi'}{\dd s}
+   + \frac{\dd y}{\dd\theta}·\frac{\dd\eta'}{\dd s}
+   + \frac{\dd z}{\dd\theta}·\frac{\dd\zeta'}{\dd s} %\\
+%  &\quad
+   + \xi'\, \frac{\dd^{2} x}{\dd s\, \dd\theta}
+        + \eta'\, \frac{\dd^{2} y}{\dd s\, \dd\theta}
+        + \zeta'\, \frac{\dd^{2} z}{\dd s\, \dd\theta} \displaybreak[1] \\
+%
+  &= mp'(\xi'X + \eta'Y + \zeta'Z) %\\
+%  &\quad
+   + \xi'\, \frac{\dd\xi}{\dd\theta}
+        + \eta'\, \frac{\dd\eta}{\dd\theta}
+        + \zeta'\, \frac{\dd\zeta}{\dd\theta} \displaybreak[1] \\
+%
+  &= \xi'\, \frac{\dd\xi}{\dd\theta}
+   + \eta'\, \frac{\dd\eta}{\dd\theta}
+   + \zeta'\, \frac{\dd\zeta}{\dd\theta}.
+\end{align*}
+\PageSep{110}
+
+If we differentiate again with respect to~$s$, and notice that
+\[
+\frac{\dd^{2} \xi}{\dd s\, \dd\theta}
+  = \frac{\dd(pX)}{\dd\theta},\quad\text{etc.},
+\]
+and that
+\[
+X\xi' + Y\eta' + Z\zeta' = 0,
+\]
+we have\Note{46}
+{\small
+\begin{align*}
+\frac{\dd^{2} m}{\dd s^{2}}
+  &= p\left(\xi'\, \frac{\dd X}{\dd\theta}
+          + \eta'\, \frac{\dd Y}{\dd\theta}
+          + \zeta'\, \frac{\dd Z}{\dd\theta}\right)
+   + p'\left(X \frac{\dd\xi}{\dd\theta}
+          + Y \frac{\dd \eta}{\dd\theta}
+          + Z \frac{\dd \zeta}{\dd\theta}\right) \\
+%
+  &= p\left(\xi'\, \frac{\dd X}{\dd\theta}
+          + \eta'\, \frac{\dd Y}{\dd\theta}
+          + \zeta'\, \frac{\dd Z}{\dd\theta}\right) + mp'^{2} \\
+%
+  &= -\left(\xi\, \frac{\dd X}{\dd s}
+          + \eta\, \frac{\dd Y}{\dd s}
+          + \zeta\, \frac{\dd Z}{\dd s}\right)
+      \left(\xi'\, \frac{\dd X}{\dd\theta}
+          + \eta'\, \frac{\dd Y}{\dd\theta}
+          + \zeta'\, \frac{\dd Z}{\dd\theta}\right) \\
+  &\phantom{={}}
+    + \left(\xi'\, \frac{\dd X}{\dd s}
+          + \eta'\, \frac{\dd Y}{\dd s}
+          + \zeta'\, \frac{\dd Z}{\dd s}\right)
+      \left(\xi\, \frac{\dd X}{\dd\theta}
+          + \eta\, \frac{\dd Y}{\dd\theta}
+          + \zeta\, \frac{\dd Z}{\dd\theta}\right) \\
+%
+  &= \left(\frac{\dd Y}{\dd\theta}\, \frac{\dd Z}{\dd s}
+         - \frac{\dd Y}{\dd s}\, \frac{\dd Z}{\dd\theta}\right)X
+   + \left(\frac{\dd Z}{\dd\theta}\, \frac{\dd X}{\dd s}
+         - \frac{\dd Z}{\dd s}\, \frac{\dd X}{\dd\theta}\right)Y
+   + \left(\frac{\dd X}{\dd\theta}\, \frac{\dd Y}{\dd s}
+         - \frac{\dd X}{\dd s}\, \frac{\dd Y}{\dd\theta}\right)Z.\NoteMark
+\end{align*}}
+
+[But if the surface element
+\[
+m\, ds\, d\theta
+\]
+belonging to the point $x$,~$y$,~$z$ be represented upon the auxiliary sphere of unit radius
+by means of parallel normals, then there corresponds to it an area whose magnitude is
+{\small
+\[
+\left\{
+X\left(\frac{\dd Y}{\dd s}\, \frac{\dd Z}{\dd\theta}
+     - \frac{\dd Y}{\dd\theta}\, \frac{\dd Z}{\dd s}\right) +
+Y\left(\frac{\dd Z}{\dd s}\, \frac{\dd X}{\dd\theta}
+     - \frac{\dd Z}{\dd\theta}\, \frac{\dd X}{\dd s}\right) +
+Z\left(\frac{\dd X}{\dd s}\, \frac{\dd Y}{\dd\theta}
+     - \frac{\dd X}{\dd\theta}\, \frac{\dd Y}{\dd s}\right)
+\right\}ds\, d\theta.
+\]}%
+Consequently, the measure of curvature at the point under consideration is equal to
+\[
+-\frac{1}{m}\, \frac{\dd^{2} m}{\dd s^{2}}.]
+\]
+\PageSep{111}
+
+
+\Notes.
+
+The parts enclosed in brackets are additions of the editor of the German edition
+or of the translators.
+
+``The foregoing fragment, \textit{Neue allgemeine Untersuchungen über die krummen Flächen},
+differs from the \textit{Disquisitiones} not only in the more limited scope of the matter, but
+also in the method of treatment and the arrangement of the theorems. There [paper
+of~1827] \textsc{Gauss} assumes that the rectangular coordinates $x$,~$y$,~$z$ of a point of the surface
+can be expressed as functions of any two independent variables $p$~and~$q$, while
+here [paper of~1825] he chooses as new variables the geodesic coordinates $s$~and~$\theta$.
+Here [paper of~1825] he begins by proving the theorem, that the sum of the three
+angles of a triangle, which is formed by shortest lines upon an arbitrary curved surface,
+differs from~$180°$ by the area of the triangle, which corresponds to it in the representation
+by means of parallel normals upon the auxiliary sphere of unit radius. From
+this, by means of simple geometrical considerations, he derives the fundamental theorem,
+that \Chg{``}{`}in the transformation of surfaces by bending, the measure of curvature at
+every point remains unchanged.\Chg{''}{'} But there [paper of~1827] he first shows, in \Art[1827]{11},
+that the measure of curvature can be expressed simply by means of the three
+quantities $E$,~$F$,~$G$, and their derivatives with respect to $p$~and~$q$, from which follows
+the theorem concerning the invariant property of the measure of curvature as a corollary;
+and only much later, in \Art[1827]{20}, quite independently of this, does he prove the
+theorem concerning the sum of the angles of a geodesic triangle.'' \\
+\null\hfill Remark by Stäckel, Gauss's Works, vol.~\textsc{viii}, p.~443.
+
+\LineRef[32]{3}{Art.~3, p.~84, l.~9}. $\cos^{2}\phi$, etc., is used here where the German text has~$\cos\phi^{2}$,~etc.
+
+\LineRef[33]{3}{Art.~3, p.~84, l.~13}. $p^{2}$,~etc., is used here where the German text has~$pp$,~etc.
+
+\LineRef[34]{7}{Art.~7, p.~89, ll.~13,~21}. Since $\lambda L$ is less than~$90°$, $\cos\lambda L$~is always positive
+and, therefore, the algebraic sign of the expression for the volume of this pyramid
+depends upon that of~$\sin L'L''$. Hence it is positive, zero, or negative according as
+the arc~$L'L''$ is less than, equal to, or greater than~$180°$.
+
+\LineRef[34]{7}{Art.~7, p.~89, ll.~14--21}. As is seen from the paper of~1827 (see \Pageref{6}), Gauss
+\PageSep{112}
+corrected this statement. To be correct it should read: for which we can write also,
+according to well known principles of spherical trigonometry,
+\[
+\sin LL'·\sin L'·\sin L'L''
+  = \sin L'L''·\sin L''·\sin L''L
+  = \sin L''L·\sin L·\sin LL',
+\]
+if $L$,~$L'$,~$L''$ denote the three angles of the spherical triangle, where $L$~is the angle
+measured from the arc~$LL''$ to~$LL'$, and so for the other angles. At the same time
+we easily see that this value is one-sixth of the pyramid whose angular points are
+the centre of the sphere and the three points $L$,~$L'$,~$L''$; and this pyramid is \emph{positive}
+when the points $L$,~$L'$,~$L''$ are arranged in the same order about this triangle as the
+points $(1)$,~$(2)$,~$(3)$ about the triangle $(1)\ (2)\ (3)$.
+
+\LineRef{8}{Art.~8, p.~90, l.~7~fr.~bot}. In the German text $V$~stands for~$f$ in this equation
+and in the next line but one.
+
+\LineRef[35]{11}{Art.~11, p.~93, l.~8~fr.~bot}. In the German text, in the expression for~$B$, $(\alpha\beta' + \alpha\beta')$
+stands for~$(\alpha'\beta + \alpha\beta')$.
+
+\LineRef[36]{11}{Art.~11, p.~94, l.~17}. The vertices of the triangle are $M$,~$M'$,~$(3)$, whose coordinates
+are $\alpha$,~$\beta$,~$\gamma$; $\alpha'$,~$\beta'$,~$\gamma'$; $0$,~$0$,~$1$, respectively. The pole of the arc~$MM'$ on
+the same side as~$(3)$ is~$L$, whose coordinates are $X$,~$Y$,~$Z$. Now applying the formula
+%[** TN: Omitted incorrect line number reference]
+on \Pageref{89},\Chg{ line~10,}{}
+\[
+x'y'' - y'x'' = \sin L'L''\cos\lambda(3),
+\]
+to this triangle, we obtain
+\[
+\alpha\beta' - \beta\alpha' = \sin MM' \cos L(3)
+\]
+or, since
+\[
+MM' = 90°,\quad\text{and}\quad \cos L(3) = ±Z
+\]
+we have
+\[
+\alpha\beta' - \beta\alpha' = ±Z.
+\]
+
+\LineRef[37]{14}{Art.~14, p.~100, l.~19}. Here $X$,~$Y$,~$Z$; $\xi$,~$\eta$,~$\zeta$; $0$,~$0$,~$1$ take the place of $x$,~$y$,~$z$;
+$x'$,~$y'$,~$z'$; $x''$,~$y''$,~$z''$ of the top of \Pageref{89}. Also $(3)$,~$\lambda$ take the place of $L'$,~$L''$, and
+$\phi$~is the angle~$L$ in the note at the top of this page.
+
+\LineRef[38]{14}{Art.~14, p.~101, l.~2~fr.~bot}. In the German text $\{\zeta X - \eta XYZ + \xi Y^{2}Z\}$ stands
+for $\{\zeta X + \eta XYZ - \xi Y^{2}Z\}$.
+
+\LineRef[39]{15}{Art.~15, p.~102, l.~13 and the following}. Transforming to polar coordinates,
+$r$,~$\theta$,~$\psi$, by the substitutions (since on the auxiliary sphere $r = 1$)
+\begin{gather*}
+X = \sin\theta \sin\psi,\quad
+Y = \sin\theta \cos\psi,\quad
+Z = \cos\theta, \\
+dX =  \sin\theta \cos\psi\, d\psi + \cos\theta \sin\psi\, d\theta,\qquad
+dY = -\sin\theta \sin\psi\, d\psi + \cos\theta \cos\psi\, d\theta, \\
+\Tag{(1)}
+\Typo{=}{-}\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX)\quad\text{becomes}\quad
+\cos\theta\, d\psi.
+\end{gather*}
+\PageSep{113}
+
+In the figures on \Pgref{fig:102}, $PL$~and~$P'L'$ are arcs of great circles intersecting in
+the point~$(3)$, and the element~$LL'$, which is not necessarily the arc of a great circle,
+corresponds to the element of the geodesic line on the curved surface. $(2)PP'(1)$
+also is the arc of a great circle. Here $P'P = d\psi$, $Z = \cos\theta ={}$Altitude of the zone
+of which $LL'P'P$~is a part. The area of a zone varies as the altitude of the zone.
+Therefore, in the case under consideration,
+\[
+\frac{\text{Area of zone}}{2\pi} = \frac{Z}{1}.
+\]
+Also
+\[
+\frac{\Area LL'P'P}{\text{Area of zone}} = \frac{d\psi}{2\pi}.
+\]
+From these two equations,
+\[
+\Tag{(2)}
+\Area LL'P'P = Z\, d\psi,\quad\text{or}\quad \cos\theta\, d\psi.
+\]
+From (1)~and~(2)
+\[
+-\frac{Z}{1 - Z^{2}}(X\, dY - Y\, dX) = \Area LL'P'P.
+\]
+
+\LineRef[40]{15}{Art.~15, p.~102}. The point~$(3)$ in the figures on this page was added by the
+translators.
+
+\LineRef[41]{15}{Art.~15, p.~103, ll.~6--9}. It has been shown that $d\phi = \Area LL'P'P, = dA$, say.
+Then
+\[
+\int_{\phi}^{\phi'} d\phi = \int_{0}^{A} dA,
+\]
+or
+\[
+\phi' - \phi = A,\quad\text{the finite area $LL'P'P$}.
+\]
+
+\LineRef[42]{15}{Art.~15, p.~103, l.~10 and the following}. Let $A$,~$B'$,~$B_{1}$ be the vertices of a
+geodesic triangle on the curved surface, and let the corresponding triangle on the
+auxiliary sphere be~$LL'L'_{1}L$, whose sides are not necessarily arcs of great circles. Let
+$A$,~$B'$,~$B_{1}$ denote also the angles of the geodesic triangle. Here $B'$~is the supplement
+of the angle denoted by~$B$ on \Pageref{103}. Let $\phi$~be the angle on the sphere
+between the great circle arcs $L\lambda$,~$L(3)$, \ie, $\phi = (3)L\lambda$, $\lambda$~corresponding to the direction
+of the element at~$A$ on the geodesic line~$AB'$, and let $\phi' = (3)L'\lambda_{1}$, $\lambda_{1}$~corresponding
+to the direction of the element at~$B'$ on the line~$AB'$. Similarly, let $\psi = (3)L\mu$,
+\PageSep{114}
+$\psi' = (3)L'_{1}\mu_{1}$, $\mu$,~$\mu_{1}$ denoting the directions of the elements at
+$A$,~$B_{1}$, respectively, on the line~$AB_{1}$. And let $\chi = (3)L'\nu$,
+$\chi_{1} = (3)L'_{1}\nu_{1}$, $\nu$,~$\nu_{1}$ denoting the directions of the elements at
+$B'$,~$B_{1}$, respectively, on the line~$B'B_{1}$.
+
+Then from the first formula on \Pageref{103},
+\begin{gather*}
+\begin{aligned}[b]
+\phi' - \phi &= \Area LL'P'P, \\
+\psi' - \psi &= \Area LL'_{1}P'_{1}P, \\
+\chi_{1} - \chi &= \Area L'L'_{1}P'_{1}P',
+\end{aligned}
+\qquad\qquad
+%[Illustration]
+\Graphic{1.5in}{114} \\
+\psi' - \psi - (\phi' - \phi) - (\chi_{1} - \chi)
+  = \Area L'L'_{1}P'_{1}P'
+  - \Area LL'P'P
+  - \Area L'L'_{1}P'_{1}P',
+\end{gather*}
+or
+\[
+\Tag{(1)}
+(\phi - \psi) + (\chi - \phi') + (\psi' - \chi_{1})
+  = \Area LL'_{1}L'L.
+\]
+
+Since $\lambda$,~$\mu$ represent the directions of the linear elements at~$A$ on the geodesic
+lines $AB'$,~$AB_{1}$ respectively, the absolute value of the angle~$A$ on the surface is measured
+by the arc~$\mu\lambda$, or by the spherical angle~$\mu L\lambda$. But $\phi - \psi = (3)L\lambda - (3)L\mu
+= \mu L\lambda$. \\
+Therefore
+\[
+A = \phi - \psi.
+\]
+Similarly
+\begin{align*}
+180° - B' &= -(\chi - \phi'), \\
+B_{1} &= \psi' - \chi_{1}.
+\end{align*}
+Therefore, from~(1),
+\[
+A + B' + B_{1} - 180° = \Area LL'_{1}L'L.
+\]
+
+\LineRef[43]{15}{Art.~15, p.~103, l.~19}. In the German text $LL'P'P$ stands for~$LL'_{1}P'_{1}P$,
+which represents the angle~$\psi' - \psi$.
+
+\LineRef[44]{15}{Art.~15, p.~104, l.~12}. This general theorem may be stated as follows:
+
+The sum of all the angles of a polygon of $n$~sides, which are shortest lines
+upon the curved surface, is equal to the sum of $(n - 2)180°$ and the area of the
+polygon upon the auxiliary sphere whose boundary is formed by the points~$L$ which
+correspond to the points of the boundary of the given polygon, and in such a manner
+that the area of this polygon may be regarded positive or negative according as it is
+enclosed by its boundary in the same sense as the given figure or the contrary.
+
+\LineRef[45]{16}{Art.~16, p.~104, l.~12~fr.~bot}. The \emph{zenith} of a point on the surface is the corresponding
+point on the auxiliary sphere. It is the spherical representation of the
+point.
+
+\LineRef[46]{18}{Art.~18, p.~110, l.~10}. The normal to the surface is here taken in the direction
+opposite to that given by~[9] \Pageref{107}.
+\PageSep{115}
+\BackMatter
+%[** TN: Print "BIBLIOGRAPHY" title page]
+\BibliographyPage
+\PageSep{116}
+%[Blank page]
+\PageSep{117}
+
+
+\begin{Bibliography}{%
+This bibliography is limited to books, memoirs, etc., which use Gauss's method and which treat, more or less
+generally, one or more of the following subjects: curvilinear coordinates, geodesic and isometric lines, curvature of
+surfaces, deformation of surfaces, orthogonal systems, and the general theory of surfaces. Several papers which lie
+beyond these limitations have been added because of their importance or historic interest. For want of space, generally,
+papers on minimal surfaces, congruences, and other subjects not mentioned above have been excluded.
+
+Generally, the numbers following the volume number give the pages on which the paper is found.
+
+C.~R. will be used as an abbreviation for Comptes Rendus hebdomadaires des séances de l'Académie des
+Sciences\Typo{.}{,} Paris.}
+
+\Author{Adam, Paul} \Title{Sur les systèmes triples orthogonaux.} Thesis.
+80~pp.\Add{,} Paris, 1887.
+
+\Title{Sur les surfaces isothermiques à lignes de courbure
+planes dans un système ou dans les deux systèmes.}
+Ann.\ de l'École Normale, ser.~3, vol.~10, 319--358, 1893;
+C.~R., vol.~116, 1036--1039, 1893.
+
+\Title{Sur les surfaces admettant pour lignes de courbure
+deux séries de cercles géodésiques orthogonaux.} Bull.\
+de~la Soc.\ Math.\ de France, vol.~22, 110--115, 1894.
+
+\Title{Mémoire sur la déformation des surfaces.} Bull.\ de~la
+Soc.\ Math.\ de France, vol.~23, 219--240, 1895.
+
+\Title{Sur la déformation des surfaces.} Bull.\ de~la Soc.\
+Math.\ de France, vol.~23, 106--111, 1895; C.~R., vol.\
+121, 551--553, 1895.
+
+\Title{Sur la déformation des surfaces avec conservation des
+lignes de courbure.} Bull.\ de~la Soc.\ Math.\ de France,
+vol.~23, 195--196, 1895.
+
+\Title{Théorème sur la déformation des surfaces de translation.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~23, 204--209,
+1895.
+
+\Title{Sur un problème de déformation.} Bull.\ de~la Soc.\
+Math.\ de France, vol.~24, 28--39, 1896.
+
+\Author{Albeggiani, L.} \Title{Linee geodetiche tracciate sopra taluni superficie.}
+Rend.\ del Circolo Mat.\ di Palermo, vol.~3, 80--119,
+1889.
+
+\Author{Allé, M.} \Title{Zur Theorie des Gauss'schen Krümmungsmaasses.}
+Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften zu Wien,
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+\Title{Théorie géométrique des coordonnées curvilignes quelconques.}
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+\Title{Sur la courbure des surfaces.} C.~R., vol.~57, 217--219,
+1863.
+
+%\Author{Aoust, L. S. X. B.}
+\Title{Théorie des coordonnées curvilignes
+quelconques.} Annali di Mat., vol.~6, 65--87, 1864; ser.~2,
+vol.~2, 39--64, vol.~3, 55--69, 1868--69; ser.~2, vol.~5,
+261--288, 1873.
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+\Author{August, T.} \Title{Ueber Flächen mit gegebener Mittelpunktsfläche
+und über Krümmungsverwandschaft.} Archiv
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+curvatura totale nulla.} Atti del Reale Instituto Veneto,
+ser.~7, vol.~6, 998--1004, 1895.
+
+\Author{Beliankén, J.} \Title{Principles of the theory of the development
+of surfaces. Surfaces of constant curvature.} \Chg{(Russian).}{(Russian.)}
+Kief Univ.\ Reports, Nos.\ 1~and~3; and Kief, \Chg{pp.\ \textsc{ii}~+~129}{\textsc{ii}~+~129~pp.},
+1898.
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+di Mat., vol.~2, 267--282, 297--306, 331--339, 355--375,
+1864; vol.~3, 15--22, 33--41, 82--91, 228--240, 311--314, 1865.
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+\Title{Delle variabili complesse sopra una superficie qualunque.}
+Annali di Mat., ser.~2, vol.~1, 329--366, 1867.
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+1868.
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+Veneto, vol.~5, 1869.
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+de Math., vol.~9, 133--154, 1844.
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+
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+Annali di Mat., ser.~2, vol.~8, 138--145, 1877.
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+superficie.} Giornale di Mat., vol.~16, 267--269, 1878.
+
+\Title{Ueber die Flächen mit constanter negativer Krümmung.}
+Math.\ Annalen, vol.~16, 577--582, 1880.
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+\Title{Sulle superficie a curvatura costante positiva.} Giornale
+di Mat., vol.~20, 287--292, 1882.
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+\Title{Sui sistemi tripli cicilici di superficie orthogonali.}
+Giornale di Mat., vol.~21, 275--292, 1883; vol.~22, 333--373,
+1884.
+
+\Title{Sopra i sistemi orthogonali di Weingarten.} Atti della
+Reale Accad.\ dei Lincei, ser.~4, vol.~1, 163--166, 243--246,
+1885; Annali di Mat., ser.~2, vol.~13, 177--234,
+1885, and ser.~2, vol.~14, 115--130, 1886.
+
+\Title{Sopra una classe di sistemi tripli di superficie orthogonali,
+che contengono un sistema di elicoidi aventi a
+comune l'asse ed il passo.} Annali di Mat., ser.~2, vol.~13,
+39--52, 1885.
+
+\Title{Sopra i sistemi tripli di superficie orthogonali che contengono
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+della Reale Accad.\ dei Lincei, ser.~4, vol.~2, 19--22,
+1886.
+
+\Title{Sulle forme differenziali quadratiche indefinite.} Atti
+della Reale Accad.\ dei Lincei, vol.~$4_{2}$, 278, 1888; Mem.\
+della Reale Accad.\ dei Lincei, ser.~4, vol.~5, 539--603,
+1888.
+
+\Title{Sopra alcune nuove classi di superficie e di sistemi
+tripli orthogonali.} Annali di Mat., ser.~2, vol.~18, 301--358,
+1890.
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+\Title{Sopra una nuova classe di superficie appartenenti a
+sistemi tripli orthogonali.} Atti della Reale Accad.\ dei
+Lincei, ser.~4, vol.~$6_{1}$, 435--438, 1890.
+
+\Title{Sulle superficie i cui piani principali hanno costante
+il rapporto delle distanze da un punto fisso.} Atti
+della Reale Accad.\ dei Lincei, ser.~5, vol.~$3_{2}$, 77--84,
+1894.
+
+\Title{Sulla superficie a curvatura nulla negli spazi curvatura
+costante.} Atti della Reale Accad.\ di Torino, vol.~30,
+743--755, 1895.
+
+\Title{Lezioni di geometria differenziale.} \textsc{viii}~+~541~pp.\Add{,}
+Pisa, 1894. Translation into German by Max Lukat,
+\Title{Vorlesungen über Differentialgeometrie.} \textsc{xvi}~+~659~pp.\Add{,}
+Leipzig, 1896--99.
+
+\Title{Sopra una classe di superficie collegate alle superficie
+pseudosferiche.} Atti della Reale Accad.\ dei Lincei, ser.~5,
+vol.~$5_{1}$, 133--137, 1896.
+
+\Title{Nuove richerche sulle superficie pseudosferiche.} Annali
+di Mat., ser.~2, vol.~24, 347--386, 1896.
+
+\Title{Sur deux classes de surfaces qui engendrent par un
+mouvement hélicoidal une famille de~Lamé.} Ann.\
+Faculté des sci.\ de Toulouse, vol.~11~H, 1--8, 1897.
+
+\Author{Bianchi, Luigi.} \Title{Sopra le superficie a curvatura costante
+positiva.} Atti della Reale Accad.\ dei Lincei, ser.~5,
+vol.~$8_{1}$, 223--228, 371--377, 484--489, 1899.
+
+\Title{Sulla teoria delle transformazioni delle superficie a
+curvatura costante.} Annali di Mat., ser.~3, vol.~3, 185--298,
+1899.
+
+\Author{Blutel, E.} \Title{Sur les surfaces à lignes de courbure sphérique.}
+C.~R., vol.~122, 301--303, 1896.
+
+\Author{Bonnet, Ossian.} \Title{Mémoire sur la théorie des surfaces isothermes
+orthogonales.} \Chg{Jour.}{Journ.}\ de l'École Polyt., cahier~30,
+vol.~18, 141--164, 1845.
+
+\Title{Sur la théorie générale des surfaces.} Journ.\ de l'École
+Polyt., cahier~32, vol.~19, 1--146, 1848; C.~R., vol.~33,
+89--92, 1851; vol.~37, 529--532, 1853.
+
+\Title{Sur les lignes géodésiques.} C.~R., vol.~41, 32--35,
+1855.
+
+\Title{Sur quelques propriétés des lignes géodésiques.} C.~R.,
+vol.~40, 1311--1313, 1855.
+
+\Title{Mémoire sur les surfaces orthogonales.} C.~R., vol.~54,
+554--559, 655--659, 1862.
+
+\Title{Démonstration du théorème de Gauss relatif aux petits
+triangles géodésiques situés sur une surface courbe quelconque.}
+C.~R., vol.~58, 183--188, 1864.
+
+\Title{Mémoire sur la théorie des surfaces applicables sur
+une surface donnée.} Journ.\ de l'École Polyt., cahier~41,
+vol.~24, 209--230, 1865; cahier~42, vol.~25, 1--151,
+1867.
+
+\Title{Démonstration des propriétés fondamentales du système
+de coordonnées polaires géodésiques.} C.~R., vol.~97,
+1422--1424, 1883.
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+\Author{Bour, Edmond.} \Title{Théorie de~la déformation des surfaces.}
+Journ.\ de l'École Polyt., cahier~39, vol.~22, 1--148,
+1862.
+
+\Author{Brill, A.} \Title{Zur Theorie der geodätischen Linie und des
+geodätischen Dreiecks.} Abhandl.\ der Kgl.\ Gesell.\ der
+Wissenschaften zu München, vol.~14, 111--140, 1883.
+
+\Author{Briochi, Francesco.} \Title{Sulla integrazione della equazione della
+geodetica.} Annali di sci.\ Mat.\ e Fis., vol.~4, 133--135,
+1853.
+
+\Title{Sulla teoria delle coordinate curvilinee.} Annali di
+Mat., ser.~2, vol.~1, 1--22, 1867.
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+\Author{Brisse, C.} \Title{Exposition analytique de~la théorie des surfaces.}
+Ann.\ de l'École Normale, ser.~2, vol.~3, 87--146, 1874;
+Journ.\ de l'École Polyt., cahier~53, 213--233, 1883.
+
+\Author{Bukrejew, B.} \Title{Surface elements of the surface of constant
+curvature.} \Chg{(Russian).}{(Russian.)} Kief Univ.\ Reports, No.~7,
+4~pp., 1897.
+
+\Title{Elements of the theory of surfaces.} \Chg{(Russian).}{(Russian.)} Kief
+Univ.\ Reports, Nos.~1,~9, and~12, 1897--99.
+
+\Author{Burali-Forti, C.} \Title{Sopra alcune questioni di geometria differenziale.}
+Rend.\ del Circolo Mat.\ di Palermo, vol.~12,
+111--132, 1898.
+\PageSep{119}
+
+\Author{Burgatti, P.} \Title{Sulla torsione geodetica delle linee tracciate
+sopra una superficie.} Rend.\ del Circolo Mat.\ di Palermo,
+vol.~10, 229--240, 1896.
+
+\Author{Burnside, W.} \Title{The lines of zero length on a surface as
+curvilinear coordinates.} Mess.\ of Math., ser.~2, vol.~19,
+99--104, 1889.
+
+\Author{Campbell, J.} \Title{Transformations which leave the lengths of
+arcs on surfaces unaltered.} Proceed.\ London Math.\
+Soc., vol.~29, 249--264, 1898.
+
+\Author{Carda, K.} \Title{Zur Geometrie auf Flächen constanter Krümmung.}
+Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften
+zu Wien, vol.~107, 44--61, 1898.
+
+\Author{Caronnet, Th.} \Title{Sur les centres de courbure géodésiques.}
+C.~R., vol.~115, 589--592, 1892.
+
+\Title{Sur des couples de surfaces applicables.} Bull.\ de~la
+Soc.\ Math.\ de France, vol.~21, 134--140, 1893.
+
+\Title{Sur les surfaces à lignes de courbure planes dans les deux
+systèmes et isothermes.} C.~R., vol.~116, 1240--1242, 1893.
+
+\Title{Recherches sur les surfaces isothermiques et les surfaces
+dont rayons de courbure sont fonctions l'un de
+l'autre.} Thesis, 66~pp.\Add{,} Paris, 1894.
+
+\Author{Casorati, Felice.} \Title{Nuova definizione della curvatura delle
+superficie e suo confronto con quella di Gauss.} Reale
+Istituto Lombardo di sci.\ e let., ser.~2, vol.~22, 335--346,
+1889.
+
+\Title{Mesure de~la courbure des surfaces suivant l'idee commune.
+Ses rapports avec les mesures de courbure Gaussienne
+et moyenne.} Acta Matematica, vol.~14, 95--110, 1890.
+
+\Author{Catalan, E.} \Title{Mémoire sur les surfaces dont les rayons de
+courbure en chaque point sont égaux et de signes contraires.}
+Journ.\ de l'École Polyt., cahier~37, vol.~21, 130--168,
+1858; C.~R., vol.~41, 35--38, 274--276, 1019--1023, 1855.
+
+\Author{Cayley, Arthur.} \Title{On the Gaussian theory of surfaces.} Proceed.\
+London Math.\ Soc., vol.~12, 187--192, 1881.
+
+\Title{On the geodesic curvature of a curve on a surface.}
+Proceed.\ London Math.\ Soc., vol.~12, 110--117, 1881.
+
+\Title{On some formulas of Codazzi and Weingarten in relation
+to the application of surfaces to each other.} Proceed.\
+London Math.\ Soc., vol.~24, 210--223, 1893.
+
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+Rend.\ dell'Accad.\ di Napoli, ser.~2, vol.~8, 149--154,
+1894.
+
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+delle superficie e delle linee.} Annali di Sci.\
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+\Title{Teoria delle coordinate curvilinee nello spazio e nelle
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+\Author{Codazzi, Delfino.} \Title{Sulla teorica delle coordinate curvilinee e
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+\Title{Sur les surfaces à courbure constante positive.} C.~R.,
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+\Title{Note sur la détermination des couples de surfaces
+applicables telles que la distance de deux points correspondants
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+France, vol.~23, 71--75, 1895.
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+\Author{de Salvert}, see (de) Salvert.
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+lines and lines of curvature on surfaces.} Camb.\ and
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+\Author{Dini, Ulisse.} \Title{Sull'equazione differenzialle delle superficie
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+vol.~2, 282--288, 1864.
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+\Title{Sulla teoria delle superficie.} Giornale di Mat., vol.~3,
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+Soc.\ Italiana dei~XL\@. Firenze, 1869.
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+\Author{van Dorsten, R.} \Title{Theorie der Kromming von lijnen op
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+\Title{Untersuchungen über orthogonale Flächensysteme.}
+Math.\ Annalen, vol.~7, 456--480, 1874.
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+Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
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+%\Author{Enneper, A.}
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+\Author{Ermakoff, W.} \Title{On geodesic lines.} (Russian.) Math.\ Collections,
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+\Author{von Escherich, G.} \Title{Die Geometrie auf den Flächen constanter
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+\Title{Ableitung des allgemeinen Ausdruckes für das Krümmungsmaass
+der Flächen.} Archiv für Math.\ und
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+\Author{Fibbi, C.} \Title{Sulle superficie che contengono un sistema di
+geodetiche a torsione costante.} Annali della Reale
+Scuola Norm.\ di Pisa, vol.~5, 79--164, 1888.
+
+\Author{Firth, W.} \Title{On the measure of curvature of a surface referred
+to polar coordinates.} Oxford, Camb., and Dub.\ Mess.,
+vol.~5, 66--76, 1869.
+
+\Author{Fouché, M.} \Title{Sur les systèmes des surfaces triplement orthogonales
+où les surfaces d'une même famille admettent la
+même représentation sphérique de leurs lignes de courbure.}
+C.~R., vol.~126, 210--213, 1898.
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+\Author{Frattini, G.} \Title{Alcune formole spettanti alla teoria infinitesimale
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+1875.
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+\Author{Frobenius, G.} \Title{Ueber die in der Theorie der Flächen auftretenden
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+\Author{Gauss, K. F.} \Title{Allgemeine Auflösung der Aufgabe: Die
+Theile einer gegebenen Fläche auf einer anderen gegebenen
+Fläche so abzubilden, dass die Abbildung dem
+Abgebildeten in den kleinsten Theilen ähnlich wird.}
+Astronomische Abhandlungen, vol.~3, edited hy H.~C.
+Schumacher, Altona, 1825. The same, Gauss's Works,
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+\Author{Geiser, C. F.} \Title{Sur la théorie des systèmes triples orthogonaux.}
+Bibliothèque universelle, Archives des sciences, ser.~4,
+vol.~6, 363--364, 1898.
+
+\Title{Zur Theorie der tripelorthogonalen Flächensysteme.}
+Vierteljahrschrift der Naturf.\ Gesell.\ in Zurich, vol.~43,
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+\Author{Germain, Sophie.} \Title{Mémoire sur la courbure des surfaces.}
+Journ.\ für Math., vol.~7, 1--29, 1831.
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+\Author{Gilbert, P.} \Title{Sur l'emploi des cosinus directeurs de~la normale
+dans la théorie de~la courbure des surfaces.} Ann.\
+de~la Soc.\ sci.\ de Bruxelles, vol.~18~B, 1--24, 1894.
+
+\Author{Genty, E.} \Title{Sur les surfaces à courbure totale constante.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~22, 106--109, 1894.
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+Ann.\ de~la Faculté des sci.\ de Toulouse, vol.~9~E, 1--11,
+1895.
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+\Title{Sur les équations d'une surface rapportée à ses lignes
+de longueur nulle.} Bull.\ de~la Soc.\ Math.\ de France,
+vol.~26, 83--84, 1898.
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+\Author{Grassmann, H.} \Title{Anwendung der Ausdehnungslehre auf die
+allgemeine Theorie der Raumcurven und krummen
+Flächen.} Diss.\ Halle, 1893.
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+\Author{Guichard, C.} \Title{Surfaces rapportées à leur lignes asymptotiques
+et congruences rapportées à leurs dévéloppables.}
+Ann.\ de l'École Normale, ser.~3, vol.~6, 333--348, 1889.
+
+\Title{Recherches sur les surfaces à courbure totale constante
+et certaines surfaces qui s'y rattachent.} Ann.\ de l'École
+Normale, ser.~3, vol.~7, 233--264, 1890.
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+\Title{Sur les surfaces qui possèdent un réseau de géodésiques
+conjuguées.} C.~R., vol.~110, 995--997, 1890.
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+\Title{Sur la déformation des surfaces.} Journ.\ de Math.,
+ser.~5, vol.~2, 123--215, 1896.
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+\Title{Sur les surfaces à courbure totale constante.} C.~R.,
+vol.~126, 1556--1558, 1616--1618, 1898.
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+\Title{Sur les systémes orthogonaux et les systémes cycliques.}
+Ann.\ de l'École Normale, ser.~3, vol.~14, 467--516, 1897;
+vol.~15, 179--227, 1898.
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+\Author{Guldberg, Alf.} \Title{Om Bestemmelsen af de geodaetiske Linier
+paa visse specielle Flader.} Nyt Tidsskrift for Math.\
+Kjöbenhavn, vol.~6~B, 1--6, 1895.
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+\Author{Hadamard, J.} \Title{Sur les lignes géodésiques des surfaces spirales
+et les équations différentielles qui s'y rapportent.} Procès
+verbeaux de~la Soc.\ des sci.\ de Bordeaux, 1895--96, 55--58.
+
+\Title{Sur les lignes géodésiques des surfaces à courbures
+opposées.} C.~R., vol.~124, 1503--1505, 1897.
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+\Title{Les surfaces à courbures opposées et leurs lignes
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+
+\Author{Haenig, Conrad.} \Title{Ueber Hansen's Methode, ein geodätisches
+Dreieck auf die Kugel oder in die Ebene zu übertragen.}
+Diss., 36~pp., Leipzig, 1888.
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+\Author{Hansen, P. A.} \Title{Geodätische Untersuchungen\Add{.}} Abhandl.\ der
+Kgl.\ Gesell.\ der Wissenschaften zu \Typo{Leipsig}{Leipzig}, vol.~18,
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+1896, 85--86.
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+mit constantem Krümmungsmaass.} Journ.\ für Math.,
+vol.~88, 68--73, 1880.
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+\Title{Ueber die Curven, welche sich so bewegen können,
+dass sie stets geodätische Linien der von ihnen erzeugten
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+1883.
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+Journ.\ für Math., vol.~117, 42--56,
+1897.
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+\Author{Hoppe, R.} \Title{Zum Problem des dreifach orthogonalen Flächensystems.}
+Archiv für Math.\ und Phys., vol.~55, 362--391,
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+\Title{Principien der Flächentheorie.} Archiv für Math.\ und
+Phys., vol.~59, 225--323, 1876; Leipzig, Koch, 179~pp.\Add{,}
+1876.
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+\Title{Geometrische Deutung der Fundamentalgrössen zweiter
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+\Title{Nachträge zur Curven- und Flächentheorie.} Archiv
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+1879.
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+genügen muss, um einen dreifach orthogonalen system
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+
+\Title{Nachtrag zur Flächentheorie.} Archiv für Math.\ und
+Phys., vol.~68, 439--440, 1882.
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+\Title{Ueber die sphärische Darstellung der asymptotischen
+Linien einer Fläche.} Archiv für Math.\ und Phys., ser.~2,
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+\Title{Eine neue Beziehung zwischen den Krümmungen von
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+\Author{Jacobi, C. G. J.} \Title{Demonstratio et amplificatio nova theorematis
+Gaussiani de quadratura integra trianguli in
+data superficie e lineis brevissimis formati.} Journ.\ für
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+auf die allgemeine Theorie der Flächen und Linien
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+1872; second~ed., 1881; third~ed., \textsc{x}~+~308~pp., revised
+by L.~Natani, 1890.
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+der krummen Flächen.} Leipzig, Teubner, \textsc{viii}~+~267~pp.,
+1888.
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+Ann.\ Faculté des sci.\ de Toulouse, vol.~6~P, 1--34, 1892.
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+\Title{Une théorème de géométrie \Typo{infinitesimale}{infinitésimale}.} C.~R., vol.~116,
+569, 1893.
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+par savants à l'Acad.\ des sci.\ de l'Inst.\ de France, vol.~31,
+No.~6, 318~pp., 1894.
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+\Author{Kommerell, V.} \Title{Beiträge zur Gauss'schen Flächentheorie.}
+Diss., \textsc{iii}~+~46~pp., Tübingen, 1890.
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+\Title{Eine neue Formel für die mittlere Krümmung und
+das Krümmungsmaass einer Fläche.} Zeitschrift für
+Math.\ und Phys., vol.~41, 123--126, 1896.
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+1874.
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+1860.
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+surfaces.} Nouv.\ Ann.\ de Math., ser.~2, vol.~11, 60--66,
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+1863.
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+Journ.\ de Math., vol.~5, 313--347, 1840.
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+\Title{Leçons sur les coordonnées curvilignes.} Paris, 1859.
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+Journ.\ de l'École Polyt., cahier~48, vol.~29,
+1--109, 1880.
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+\Author{Legoux, A.} \Title{Sur l'integration de l'équation $ds^{2} = E\, du^{2} +
+2F\, du\, dv + G\, dv^{2}$.} Ann.\ de~la Faculté des sci.\ de
+Toulouse, vol.~3~F, 1--2, 1889.
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+Mém.\ couronnés et mém.\ des sav.\ publiés par
+l'Acad.\ Royale de Belgique, vol.~54, 92~pp., 1896.
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+\Author{Lévy, Maurice.} \Title{Sur une transformation des coordonnées
+curvilignes orthogonales et sur les coordonnées curvilignes
+comprenant une famille quelconque de surfaces du
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+Journ.\ de l'École Polyt., cahier~43, vol.~26, 157--200,
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+eine Relation verknüpft sind.} Archiv for Math.\ og
+Nat., Christiania, vol.~4, 507--512, 1879.
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+\Title{Zur Theorie der Flächen constanter Krümmung.}
+Archiv for Math.\ og Nat., Christiania, vol.~4, 345--354,
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+der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
+vol.~50 1--2, 1898.
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+\Author{von Lilienthal, Reinhold.} \Title{Allgemeine Eigenschaften von
+Flächen, deren Coordinaten sich durch reellen Teile
+dreier analytischer Functionen einer complexen Veränderlichen
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+Oberflächen und geradlinigen Strahlensysteme.}
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+\Title{Zur Theorie der Krümmungsmittelpunktsflächen.}
+Math.\ Annalen, vol.~30, 1--14, 1887.
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+Math., vol.~104, 341--347, 1889.
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+Acta Mathematica, vol.~16, 143--152, 1892.
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+vol.~42, 505--525, 1893.
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+einem dreifach orthogonalen Flächensystem angehört.}
+Math.\ Annalen, vol.~44, 449--457, 1894.
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+\Author{Lipschitz, Rudolf.} \Title{Beitrag zur Theorie der Krümmung.}
+Journ.\ für Math., vol.~81, 230--242, 1876.
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+mit vorgeschriebenen, die Krümmungsverhältnisse
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+der Wissenschaften zu Berlin, 1882, 1077--1087; 1883,
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+mit vorgeschriebenem Ausdruck des Linearelements.}
+Sitzungsb.\ der Kgl.\ Akad.\ der Wissenschaften
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+vol.~10, 131--136, 1887.
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+\Author{Liouville, Joseph.} \Title{Sur un théorème de M.~Gauss concernant
+le produit des deux rayons de courbure principaux
+en chaque point d'une surface.} Journ.\ de Math.,
+vol.~12, 291--304, 1847.
+\PageSep{123}
+
+\Author{Liouville, Joseph.} \Title{Sur la théorie générale des surfaces.}
+Journ.\ de Math., vol.~16, 130--132, 1851.
+
+\Title{Notes on Monge's Applications}, see Monge.
+
+\Author{Liouville, R.} \Title{Sur le caractère auquel se reconnaît l'équation
+differentielle d'un système géodésique.} C.~R., vol.~108,
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+
+\Title{Sur les représentations géodésiques des surfaces.} C.~R.,
+vol.~108, 335--337, 1889.
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+\Author{Loria, G.} \Title{Sulla teoria della curvatura delle superficie.}
+Rivista di Mat.\ Torino, vol.~2, 84--95, 1892.
+
+\Title{Il passato ed il presente d.\ pr.\ Teorie geometriche.}
+2nd~ed., 346~pp.\Add{,} Turin, 1896.
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+\Author{Lüroth, J.} \Title{Verallgemeinerung des Problems der kürzesten
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+\Author{Mahler, E.} \Title{Ueber allgemeine Flächentheorie.} Archiv für
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+\Title{Die Fundamentalsätze der allgemeinen Flächentheorie.}
+Vienna; Heft.~I, 1880; Heft.~II, 1881.
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+\Author{Mangeot, S.} \Title{Sur les éléments de~la courbure des courbes et
+surfaces.} Ann.\ de l'École Normale, ser.~3, vol.~10, 87--89,
+1893.
+
+\Author{von Mangoldt, H.} \Title{Ueber diejenigen Punkte auf positiv
+gekrümmten Flächen, welche die Eigenschaft haben,
+dass die von ihnen ausgehenden geodätischen Linien nie
+aufhören, kürzeste Linien zu sein.} Journ.\ für Math.,
+vol.~91, 23--53, 1881.
+
+\Title{Ueber die Klassification der Flächen nach der Verschiebbarkeit
+ihrer geodätischen Dreiecke.} Journ.\ für
+Math., vol.~94, 21--40, 1883.
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+\Author{Maxwell, J. Clerk.} \Title{On the Transformation of Surfaces by
+Bending.} Trans.\ of Camb.\ Philos.\ Soc., vol.~9, 445--470,
+1856.
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+\Author{Minding, Ferdinand.} \Title{Ueber die Biegung gewisser Flächen.}
+Journ.\ für Math., vol.~18, 297--302, 365--368, 1838.
+
+\Title{Wie sich entscheiden lässt, ob zwei gegebene krumme
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+Krümmungsmaasse.} Journ.\ für Math., vol.~19,
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+
+\Title{Beiträge zur Theorie der kürzesten Linien auf krummen
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+\Title{Ueber einen besondern Fall bei der Abwickelung
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+\Title{Ueber die mittlere Krümmung der Flächen.} Bull.\
+de l'Acad.\ Imp.\ de St.~Petersburg, vol.~20, 1875.
+
+\Title{Zur Theorie der Curven kürzesten Umrings, bei
+gegebenem Flächeninhalt, auf krummen Flächen.}
+Journ.\ für Math., vol.~86, 279--289, 1879.
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+\Author{Mlodzieiowski, B.} \Title{Sur la déformation des surfaces.} Bull.\
+de sci.\ Math., ser.~2, vol.~15, 97--101, 1891.
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+\Author{Monge, Gaspard.} \Title{Applications de l'Analyse à la Géométrie};
+revue, corrigée et annotée par J.~Liouville. Paris;
+fifth~ed., 1850.
+
+\Author{Motoda, T.} Note to J.~Knoblauch's paper, ``\Title{Ueber Fundamentalgrössen
+in der Flächentheorie}'' in Journ.\ für
+Math., vol.~103. Journ.\ of the Phil.\ Soc.\ in Tokio,
+3~pp., 1889.
+
+\Author{Moutard, T. F.} \Title{Lignes de courbure d'une classe de surfaces
+du quatrième ordre.} C.~R., vol.~59, 243, 1864.
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+\Title{Note sur la transformation par rayons vecteurs reciproques.}
+Nouv.\ Ann.\ de Math.\ ser.~2, vol.~3, 306--309,
+1864.
+
+\Title{Sur les surface anallagmatique du quatrième ordre.}
+Nouv.\ Ann.\ de Math.\ ser.~2, vol.~3, 536--539, 1864.
+
+\Title{Sur la déformation des surfaces.} Bull.\ de~la Soc.\
+Philomatique, p.~45, 1869.
+
+
+\Title{Sur la construction des équations de~la forme $\dfrac{1}{x}·\Typo{\dfrac{d^{2}x}{dx\, dy}}{\dfrac{\dd^{2}x}{\dd x\, \dd y}}
+= \lambda(x, y)$, qui admettent une intégrale générale explicite.}
+Journ.\ de l'École Polyt., cahier~45, vol.~28, 1--11, 1878.
+
+\Author{Nannei, E.} \Title{Le superficie ipercicliche.} Rend.\ dell'Accad.\
+di Napoli, ser.~2, vol.~2, 119--121, 1888; Giornale di
+Mat., vol.~26, 201--233, 1888.
+
+\Author{Naccari, G.} \Title{Deduzioue delle principali formule relative
+alla curvatura della superficie in generale e dello sferoide
+in particolare con applicazione al meridiano di Venezia.}
+L'Ateneo Veneto, ser.~17, vol.~1, 237--249, 1893; vol.~2,
+133--161, 1893.
+
+\Author{Padova, E.} \Title{Sopra un teorema di geometria differenziale.}
+Reale Ist.\ Lombardo di sci.\ e let., vol.~23, 840--844, 1890.
+
+\Title{Sulla teoria generale delle superficie.} Mem.\ della R.
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+
+\Author{Pellet, A.} \Title{Mém.\ sur la théorie des surfaces et des courbes.}
+Ann.\ de l'École Normale, ser.~3, vol.~14, 287--310, 1897.
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+1897.
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+\Title{Sur les systèmes de surfaces orthogonales et isothermes.}
+C.~R., vol.~124, 552--554, 1897.
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+\Title{Sur les surfaces ayant même représentation sphérique.}
+C.~R., vol.~124, 1291--1294, 1897.
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+1897.
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+\Title{Sur la théorie des surfaces.} Bull.\ de~la Soc.\ Math.\ de
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+\Title{Sur les surfaces applicables sur une surface de \Typo{revolution}{révolution}.}
+C.~R., vol.~125, 1159--1160, 1897; vol.~126, 392--394,
+1898.
+
+\Author{Peter, A.} \Title{Die Flächen, deren Haupttangentencurven linearen
+Complexen angehören.} Archiv for Math.\ og
+Nat., Christiania, vol.~17, No.~8, 1--91, 1895.
+\PageSep{124}
+
+\Author{Petot, A.} \Title{Sur les surfaces dont l'élément \Typo{lineaire}{linéaire} est \Typo{reductible}{réductible}
+\Typo{a}{à} la forme $ds^{2} = F(U + V)(du^{2} + dv^{2})$.} C.~R.,
+vol.~110, 330--333, 1890.
+
+\Author{Picard, Émile.} \Title{Surfaces applicables.} Traité d'Analyse,
+vol.~1, chap.~15, 420--457; first~ed., 1891; second~ed.,
+1901.
+
+\Author{Pirondini, G.} \Title{Studi geometrici relativi specialmente alle
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+1885.
+
+\Title{Teorema relativo alle linee di curvatura delle superficie
+e sue applicazioni.} Annali di Mat., ser.~2, vol.~16,
+61--84, 1888; vol.~21, 33--46, 1893.
+
+\Author{Plücker, Julius.} \Title{Ueber die Krümmung einer beliebigen
+Fläche in einem gegebenen Puncte.} Journ.\ für Math.,
+vol.~3, 324--336, 1828.
+
+\Author{Poincaré, H.} \Title{Rapport sur un Mémoire de M.~Hadamard,
+intitulé: Sur les lignes géodésiques des surfaces à courbures
+opposées.} C.~R., vol.~125, 589--591, 1897.
+
+\Author{Probst, F.} \Title{Ueber Flächen mit isogonalen systemen von
+geodätischen Kreisen.} Inaug.-diss.\ 46~pp., Würzburg,
+1893.
+
+\Author{Raffy, L.} \Title{Sur certaines surfaces, dont les rayons de courbure
+sont liés par une relation.} Bull.\ de~la Soc.\ Math.\
+de France, vol.~19, 158--169, 1891.
+
+\Title{\Typo{Determination}{Détermination} des éléments linéaires doublement harmoniques.}
+Journ.\ de Math., ser.~4, vol.~10, 331--390,
+1894.
+
+\Title{Quelques \Typo{proprietes}{propriétés} des surfaces harmoniques.} Ann.\
+de~la Faculté des sci.\ de Toulouse, vol.~9~C, 1--44, 1895.
+
+\Title{Sur les spirales harmoniques.} Ann.\ de l'École Normale,
+ser.~3, vol.~12, 145--196, 1895.
+
+\Title{Surfaces rapportées à un réseau conjugué azimutal.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~24, 51--56, 1896.
+
+\Title{Leçons sur les applications géométriques de l'analyse.}
+Paris, \textsc{vi}~+~251~pp., 1897.
+
+\Title{Contribution à la théorie des surfaces dont les rayons
+de courbure sont liés par une relation.} Bull.\ de~la Soc.\
+Math.\ de France, vol.~25, 147--172, 1897.
+
+\Title{Sur les formules fondamentales de~la théorie des surfaces.}
+Bull.\ de~la Soc.\ Math.\ de France, vol.~25, 1--3,
+1897.
+
+\Title{Détermination d'une surface par ses deux formes quadratiques
+fondamentales.} C.~R., vol.~126, 1852--1854,
+1898.
+
+\Author{\Typo{Razziboni}{Razzaboni}, Amilcare.} \Title{Sulla rappresentazzione di una superficie
+su di un' altra al modo di Gauss.} Giornali di Mat.,
+vol.~27, 274--302, 1889.
+
+\Title{Delle superficie sulle quali due serie di geodetiche
+formano un sistema conjugato.} Mem.\ della R. Accad.\
+dell'Ist.\ di Bologna, ser.~4, vol.~9, 765--776, 1889.
+
+\Author{Reina, V.} \Title{Sulle linee conjugate di una superficie.} Atti
+della Reale Accad.\ dei Lincei, ser.~4, vol.~$6_{1}$, 156--165,
+203--209, 1890.
+
+%\Author{Reina, V.}
+\Title{Di alcune formale relative alla teoria delle superficie.}
+Atti della Reale Accad.\ dei Lincei, ser.~4, vol.~$6_{2}$\Add{,}
+103--110, 176, 1890.
+
+\Author{Resal, H.} \Title{Exposition de~la théorie des surfaces.} 1~vol.,
+\textsc{xiii}~+~171~pp.\Add{,} Paris, 1891. Bull.\ des sci.\ Math., ser.~2,
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+l'usage des candidats aux École Polyt., ser.~3, vol.~5,
+165--166, 1891.
+
+\Author{Ribaucour, A.} \Title{Sur la théorie de l'application des surfaces
+l'une sur l'autre.} L'Inst.\ Journ.\ universel des sci.\ et
+des soc.\ sav.\ en France, sect.~I, vol.~37, 371--382, 1869.
+
+\Title{Sur les surfaces orthogonales.} L'Inst. Journ.\ universel
+des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~37,
+29--30, 1869.
+
+\Title{Sur la déformation des surfaces.} L'Inst. Journ.\ universel
+des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~37,
+389, 1869; C.~R., vol.~70, 330, 1870.
+
+\Title{Sur la théorie des surfaces.} L'Inst.\ Journ.\ universel
+des sci.\ et des soc.\ sav.\ en France, sect.~I, vol.~38, 60--61,
+141--142, 236--237, 1870.
+
+\Title{Sur la représentation sphérique des surfaces.} C.~R.,
+vol.~75, 533--536, 1872.
+
+\Title{Sur les courbes enveloppes de cercles et sur les surfaces
+enveloppes de sphères.} Nouvelle Correspondance
+Math., vol.~5, 257--263, 305--315, 337--343, 385--393, 417--425,
+1879; vol.~6, 1--7, 1880.
+
+\Title{Mémoire sur la théorie générale des surfaces courbes.}
+Journ.\ de Math., ser.~4, vol.~7, 5--108, 219--270, 1891.
+
+\Author{Ricci, G.} \Title{Dei sistemi di coordinate atti a ridurre la expressione
+del quadrato dell' elemento lineaire di una superficie
+alla forma $ds^{2} = (U + V)(du^{2} + dv^{2})$.} Atti della
+Reale Accad.\ dei Lincei, ser.~5, vol.~$2_{1}$, 73--81, 1893.
+
+\Title{A proposito di una memoria sulle linee geodetiche del
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+vol.~$2_{2}$, 146--148, 338--339, 1893.
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+\Title{Sulla teoria delle linee geodetiche e dei sistemi isotermi
+di Liouville.} Atti del Reale Ist.\ Veneto, ser.~7, vol.~5,
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+\Title{Della equazione fondamentale di Weingarten nella
+teoria delle superficie applicabili.} Atti del Reale Inst.\
+Veneto, ser.~7, vol.~8, 1230--1238, 1897.
+
+\Title{Lezioni sulla teoria delle superficie.} \textsc{viii}~+~416~pp.\Add{,}
+Verona, 1898.
+
+\Author{Rothe, R.} \Title{Untersuchung über die Theorie der isothermen
+Flächen.} Diss., 42~pp.\Add{,} Berlin, 1897.
+
+\Author{Röthig, O.} \Title{Zur Theorie der Flächen.} \Typo{Jouru.}{Journ.}\ für Math.,
+vol.~85, 250--263, 1878.
+
+\Author{Ruffini, F.} \Title{Di alcune proprietà della rappresentazione
+sferica del Gauss.} Mem.\ dell'Accad.\ Reale di sci.\
+dell'Ist.\ di Bologna, ser.~4, vol.~8, 661--680, 1887.
+
+\Author{Ruoss, H.} \Title{Zur Theorie des Gauss'schen Krümmungsmaases.}
+Zeitschrift für Math.\ und Phys., vol.~37, 378--381,
+1892.
+\PageSep{125}
+
+\Author{Saint Loup.} \Title{Sur les propriétés des lignes géodésiques.}
+Thesis, 33--96, Paris, 1857.
+
+\Author{Salmon, George.} \Title{Analytische Geometrie des Raumes.} Revised
+by Wilhelm Fielder. Vol.~II, \textsc{lxxii}~+~696\Chg{;}{~pp.,}
+Leipzig, 1880.
+
+\Author{de Salvert, F.} \Title{Mémoire sur la théorie de~la courbure des
+surfaces.} Ann.\ de~la Soc.\ sci.\ de Bruxelles, vol.~5~B,
+291--473, 1881; Paris, Gauthier-villars, 1881.
+
+\Title{Mémoire sur l'emploi des coordonnées curvilignes
+dans les problèmes de Mècanique et les lignes géodésiques
+des surfaces isothermes.} Ann.\ de~la Soc.\ sci.\ de Bruxelles,
+vol.~11~B, 1--138, 1887. Paris, 1887.
+
+\Title{Mémoire sur la recherche la plus générale d'un système
+orthogonal triplement isotherme.} Ann.\ de~la Soc.\
+sci.\ de Bruxelles, vol.~13~B, 117--260, 1889; vol.~14~B,
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+1894.
+
+\Title{Théorie nouvelle du système orthogonal triplement
+isotherme et son application aux coordonnées curvilignes.}
+2~vols., Paris, 1894.
+
+\Author{Scheffers, G.} \Title{Anwendung der Differential- und Integralrechung
+auf Geometrie.} vol.~I, \textsc{x}~+~360~pp., Leipzig,
+Veit~\&~Co., 1901.
+
+\Author{Schering, E.} \Title{Erweiterung des Gauss'schen Fundamentalsatzes
+für Dreiecke in stetig gekrümmten Flächen.}
+Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
+1867, 389--391; 1868, 389--391.
+
+\Author{Serret, Paul.} \Title{Sur la courbure des surfaces.} C.~R., vol.~84,
+543--546, 1877.
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+l'Acad.\ Royale de Belgique, ser.~3, vol.~24, 467--474,
+1892.
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+\Title{Quelques formules sur la courbure des surfaces.} Bull.\
+de l'Acad.\ Royale de Belgique, ser.~3, vol.~27, 896--904,
+1894.
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+\Author{Simonides, J.} \Title{Ueber die Krümmung der Flächen.} Zeitschrift
+zur Pflege der Math.\ und Phys., vol.~9, 267, 1880.
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+\Author{Stäckel, Paul.} \Title{Zur Theorie des Gauss'schen Krümmungsmaasses.}
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+Berichte der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
+vol.~45, 163--169, 170--172, 1893.
+
+\Title{Bemerkungen zur Geschichte der geodätischen Linien.}
+Berichte der Kgl.\ Gesell.\ der Wissenschaften zu Leipzig,
+vol.~45, 444--467, 1893.
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+1896.
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+\Title{Biegungen und conjugirte Systeme.} Math.\ Annalen,
+vol.~49, 255--310, 1897.
+
+\Title{Beiträge zur Flächentheorie.} Berichte der Kgl.\ Gesell.\
+der Wissenschaften zu Leipzig, vol.~48, 478--504, 1896;
+vol.~50, 3--20, 1898.
+
+\Author{Stahl und Kommerell.} \Title{Die Grundformeln der allgemeinen
+Flächentheorie.} \textsc{vi}~+~114~pp., Leipzig, 1893.
+
+\Author{Staude, O.} \Title{Ueber das Vorzeichen der geodätischen Krümmung.}
+Dorpat Naturf.\ Ges.\ Ber., 1895, 72--83.
+
+\Author{Stecker, H. F.} \Title{On the determination of surfaces capable of
+conformal representation upon the plane in such a manner
+that geodetic lines are represented by algebraic
+curves.} Trans.\ Amer.\ Math.\ Society, vol.~2, 152--165,
+1901.
+
+\Author{Stouff, X.} \Title{Sur la valeur de la courbure totale d'une surface
+aux points d'une arête de rebroussement.} Ann.\ de
+l'École Normale, ser.~3, vol.~9, 91--100, 1892.
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+\Author{Sturm, Rudolf.} \Title{Ein Analogon zu Gauss' Satz von der Krümmung
+der Flächen.} Math.\ Annalen, vol.~21, 379--384,
+1883.
+
+\Author{Stuyvaert, M.} \Title{Sur la courbure des lignes et des surfaces\Add{.}}
+Mém.\ couronnés et autr.\ mém.\ publ.\ par l'Acad.\ Royale
+de Belgique, vol.~55, 19 pp., 1898.
+
+\Author{de Tannenberg, W.} \Title{Leçons sur les applications géométriques
+du calcul differentiel.} 192~pp.\Add{,} Paris, A.~Hermann,
+1899.
+
+\Author{van Dorsten}, see (van) Dorsten.
+
+\Author{von Escherich}, see (von) Escherich.
+
+\Author{von Lilienthal}, see (von) Lilienthal.
+
+\Author{von Mangoldt}, see (von) Mangoldt.
+
+\Author{Vivanti, G.} \Title{Ueber diejenigen Berührungstransformationen,
+welche das Verhältniss der Krümmungsmaasse irgend
+zwei sich berührender Flächen im Berührungspunkte
+unverändert lassen.} Zeitschrift für Math.\ und Phys.,
+vol.~37, 1--7, 1892.
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+\Title{Sulle superficie a curvatura media costante.} Reale
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+1895.
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+Oberflächen.} Math.\ Annalen, vol.~19, 1--26, 1882.
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+\Title{Ueber dirjenigen Flächen, auf denen zwei Scharen
+geodätischer Linien ein conjugirtes System bilden.}
+Sitzungsb.\ der Kgl.\ Bayer.\ Akad.\ der Wissenschaften
+zu München, vol.~18, 95--102, 1888.
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+\Title{Zur Theorie der Krümmung der Flächen.} Math.\
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+
+\Title{Ueber die Fundamentalgleichungen der Flächentheorie.}
+\Typo{Sitzungb.}{Sitzungsb.}\ der Kgl.\ Bayer\Add{.} Akad.\ der Wissenschaften
+zu München, vol.~22, 247--278, 1892.
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+einer Fläche.} Sitzungsb.\ der Kgl.\ Akad.\ der Wissenschaften
+zu München, vol.~27, 229--301, 1897.
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+
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+et les surfaces à courbure constante.} C.~R., vol.~116,
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+\Title{Sur les lignes géodésiques de certaines surfaces.} C.~R.,
+vol.~125, 521--523, 1897.
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+Sitzungsb.\ der Ksl.\ Akad.\ der Wissenschaften zu Wien,
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+of C.~F.~Gauss for the measure of curvature.} Quart.\
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+\Title{Exercises in curvilinear and normal coordinates.}
+Trans.\ of Camb.\ Philos.\ Society, vol.~12, 455--522, 531--545,
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+Krümmung überall gleich Null ist.} Monatsb.\ der
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+Flächen von constantem Krümmungsmaass.} Journ.\ für
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+\Title{Eine neue Klasse auf einander abwickelbarer Flächen.}
+Nachr.\ der Kgl.\ Gesell.\ der Wissenschaften zu Göttingen,
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+%\Author{Weingarten, Julius.}
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+vol.~112, 607--610, 706--707, 1891.
+
+\Title{Sur la déformation des surfaces.} Acta Mathematica,
+vol.~20, 159--200, 1897; note on same, vol.~22, 193--199,
+1899.
+
+\Author{Weyr, Ed.} \Title{Sur l'équation des lignes géodésiques.} Chicago
+Congr.\ Papers, 408--411, 1896.
+
+\Title{Ueber das System der Orthogonalflächen.} Zeitschrift
+zur Pflege der Math.\ und Phys., vol.~25, 42--46, 1896.
+
+\Author{Willgrod, Heinrich.} \Title{Ueber Flächen, welche sich durch
+ihre Krümmungslinien in unendlich kleine Quadrate
+theilen lassen.} Diss., \textsc{vi}~+~51~pp., Göttingen, 1883.
+
+\Author{Williamson, Benjamin.} \Title{On curvilinear coordinates.} Trans.\
+of the Royal Irish Acad., Dublin, vol.~29, part~15, 515--552,
+1890.
+
+\Title{On Gauss's theorem of the measure of curvature at
+any point of a surface.} Quart.\ Journ.\ of Math., vol.~11,
+362--366, 1871.
+
+\Author{Wostokow, J.} \Title{On the geodesic curvature of curves on a
+surface}, (Russian.) Works of the Warsaw Soc.\ of Sci.,
+sect.~6, No.~8, 1896.
+
+\Author{Woudstra, M.} \Title{Kromming van oppervlakken volgens de
+theorie van Gauss.} Diss., Groningen, 1879.
+
+\Author{Zorawski, K.} \Title{On deformation of surfaces.} Trans.\ of the
+Krakau Acad.\ of Sciences, (Polish), ser.~2, vol.~1, 225--291,
+1891.
+
+\Title{Ueber Biegungsinvarianten. Eine Anwendung der
+Lie'schen Gruppentheorie.} Diss., Leipzig, 1891; Acta
+Mathematica, vol.~16, 1--64, 1892.
+
+\Title{On the fundamental magnitudes of the general theory
+of surfaces.} Memoirs of the Krakau Acad.\ of Science,
+(Polish), vol.~28, 1--7, 1895.
+
+\Title{On some relations in the theory of surfaces.} Bull.\ of
+the Krakau Acad.\ of Sciences, (Polish), vol.~33, 106--119,
+1898.
+\end{Bibliography}
+\PageSep{127}
+
+\iffalse
+CORRIGENDA BT ADDENDA.
+%[** TN: [x] = corrected in source using \Erratum macro,
+%        [v] = verified in source, corrected by the translators]
+
+[x] Art. 11, p. 20, l. 6. The fourth E should be F.
+
+[x] Art. 18, p. 27, l. 7. For \sqrt{EG - F^2)·dp·d\theta read 2\sqrt{FG - F^w)·dq·d\theta.
+The original and the Latin reprints lack the factor 2; the correction is made in all
+the translations.
+
+[x] Art. 19, p. 28, l. 10. For g read q.
+
+[v] Art. 22, p. 34, l. 5, left side; Art. 24, p. 36, l. 5, third equation; Art. 24,
+p. 38, l. 4. The original and Liouville's reprint have q for p.
+
+[x] Note on Art. 23, p. 55, l. 2 fr. bot. For p read q.
+\fi
+
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of the Project Gutenberg EBook of General Investigations of Curved  %
+% Surfaces of 1827 and 1825, by Karl Friedrich Gauss                      %
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK INVESTIGATIONS OF CURVED SURFACES ***
+%                                                                         %
+% ***** This file should be named 36856-tex.tex or 36856-tex.zip *****    %
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diff --git a/36856-t/old/36856-t.zip b/36856-t/old/36856-t.zip
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diff --git a/LICENSE.txt b/LICENSE.txt
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+++ b/LICENSE.txt
@@ -0,0 +1,11 @@
+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for 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
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--- /dev/null
+++ b/README.md
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #36856 (https://www.gutenberg.org/ebooks/36856)