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diff --git a/11753-h/11753-h.htm b/11753-h/11753-h.htm new file mode 100644 index 0000000..3cec370 --- /dev/null +++ b/11753-h/11753-h.htm @@ -0,0 +1,1820 @@ +<?xml version="1.0" encoding="UTF-8" ?> +<!DOCTYPE html + PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> + +<html xmlns="http://www.w3.org/1999/xhtml"> +<head> + <meta http-equiv="Content-Type" content="text/html;charset=UTF-8" /> +<title>Experimental Determination of the Velocity of Light, by Albert A. Michelson</title> + +<style type="text/css"> + <!-- + + body { + margin .5em; + font-family: Verdana, Arial, sans-serif; + } + + h1, h2, h3, h4 { + text-align: center; + font-weight: bold; + font-variant: small-caps + } + + p.byline { font-variant: smallcaps; text-align: center; } + + .smallcaps { font-variant: small-caps } + + a { text-decoration: none; } + a:hover { background-color: #ffffcc } + + div.chapter { + margin-top: 4em; + } + + hr { + height: 1px; + width: 80%; + } + + div.note { + border-style: dashed; + border-width: 1px; + border-color: #000000; + background-color: #ccffcc; + width: 80%; + margin-left: auto; + margin-right: auto; + font-size: .8em; + } + + div.chapter div.note { + width: 40%; + float: right; + } + + div.note p { + margin: 5px; + } + + div.math { + width: 60%; + margin-left: 20%; + margin-right: 20%; + } + + img { + margin-left: auto; + margin-right: auto; + border: none; + max-width: 100%; + } + + table { + margin-left: auto; + margin-right: auto; + } + + table caption { + width: 100%; + text-align: center; + font-weight: bold; + } + + td.align-dot { + text-align: right; + text-align: "."; + } +--> +</style> +</head> + +<body> +<div>*** START OF THE PROJECT GUTENBERG EBOOK 11753 ***</div> + +<div id="tp"> +<h1 class="title">Experimental Determination of the Velocity of Light</h1> + +<h2 class="subtitle">Made at the U.S. Naval Academy, Annapolis.</h2> + +<p class="byline">By</p> + +<h2 class="author">Albert A. Michelson,<br /> +Master U.S. Navy.</h2> +</div> + + +<div id="preface"> +<h2>Note.</h2> + + + +<p>The probability that the most accurate method of determining the solar +parallax now available is that resting on the measurement of the velocity +of light, has led to the acceptance of the following paper as one of the +series having in view the increase of our knowledge of the celestial +motions. The researches described in it, having been made at the United +States Naval Academy, though at private expense, were reported to the +Honorable Secretary of the Navy, and referred by him to this Office. At +the suggestion of the writer, the paper was reconstructed with a fuller +general discussion of the processes, and with the omission of some of the +details of individual experiments.</p> + +<p>To prevent a possible confusion of this determination of the velocity of +light with another now in progress under official auspices, it may be +stated that the credit and responsibility for the present paper rests with +Master Michelson.</p> + +<p>Simon Newcomb,<br /> +<i>Professor, U.S. Navy</i>,<br /> +<i>Superintendent Nautical Almanac</i>.</p> + +<p>Nautical Almanac Office,<br /> +Bureau of Navigation,<br /> +Navy Department,<br /> +<i>Washington, February 20, 1880.</i></p> +</div> + + +<div id="toc"> +<h2>Table Of Contents.</h2> + + +<ul style="list-style-type: none"> +<li><a href="#ch01">Introduction</a></li> +<li><a href="#ch02">Theory of the New Method</a></li> +<li><a href="#ch03">Arrangement and Description of Apparatus</a></li> +<li><a href="#ch04">Determination of the Constants</a></li> +<li><a href="#ch05">The Formulæ</a></li> +<li><a href="#ch06">Observations</a></li> +<li><a href="#ch08">Discussion of Errors</a></li> +<li><a href="#ch09">Objections Considered</a></li> +<li><a href="#ch10">Postscript</a></li> +</ul> +</div> + + +<h1>Experimental Determination of the Velocity of Light.</h1> + +<h2 class="author">By Albert A. Michelson, <i>Master, U.S.N.</i></h2> + + +<div class="chapter" id="ch01"> +<h2>Introduction.</h2> + + + +<p>In Cornu's elaborate memoir upon the determination of the velocity of +light, several objections are made to the plan followed by Foucault, which +will be considered in the latter part of this work. It may, however, be +stated that the most important among these was that the deflection was too +small to be measured with the required degree of accuracy. In order to +employ this method, therefore, it was absolutely necessary that the +deflection should be increased.</p> + +<p>In November, 1877, a modification of Foucault's arrangement suggested +itself, by which this result could be accomplished. Between this time and +March of the following year a number of preliminary experiments were +performed in order to familiarize myself with the optical arrangements. +The first experiment tried with the revolving mirror produced a deflection +considerably greater than that obtained by Foucault. Thus far the only +apparatus used was such as could be adapted from the apparatus in the +laboratory of the Naval Academy.</p> + +<p>At the expense of $10 a revolving mirror was made, which could execute 128 +turns per second. The apparatus was installed in May, 1878, at the +laboratory. The distance used was 500 feet, and the deflection was about +twenty times that obtained by Foucault.[<a href="#fn01">1</a>]</p> + +<div class="note" id="fn01"><p> [Footnote 1: See Proc. Am. Assoc. Adv. Science, Saint Louis meeting.]</p></div> + +<p>These experiments, made with very crude apparatus and under great +difficulties, gave the following table of results for the velocity of +light in miles per second:</p> + +<table summary="velocity of light in miles per second"> +<tr><td> 186730</td></tr> +<tr><td> 188820</td></tr> +<tr><td> 186330</td></tr> +<tr><td> 185330</td></tr> +<tr><td> 187900</td></tr> +<tr><td> 184500</td></tr> +<tr><td> 186770</td></tr> +<tr><td> 185000</td></tr> +<tr><td> 185800</td></tr> +<tr><td> 187940</td></tr> +<tr><td> ------</td></tr> +<tr><td> Mean 186500 ± 300 miles per second,<br /> + or 300140 kilometers per second.</td></tr> +</table> +<p>In the following July the sum of $2,000 was placed at my disposal by a +private gentleman for carrying out these experiments on a large scale. +Before ordering any of the instruments, however, it was necessary to find +whether or not it was practicable to use a large distance. With a distance +(between the revolving and the fixed mirror) of 500 feet, in the +preliminary experiments, the field of light in the eye-piece was somewhat +limited, and there was considerable indistinctness in the image, due to +atmospheric disturbances.</p> + +<p>Accordingly, the same lens (39 feet focus) was employed, being placed, +together with the other pieces of apparatus, along the north sea-wall of +the Academy grounds, the distance being about 2,000 feet. The image of the +slit, at noon, was so confused as not to be recognizable, but toward +sunset it became clear and steady, and measurements were made of its +position, which agreed within one one-hundredth of a millimeter. It was +thus demonstrated that with this distance and a deflection of 100 +millimeters this measurement could be made within the ten-thousandth part.</p> + +<p>In order to obtain this deflection, it was sufficient to make the mirror +revolve 250 times per second and to use a "radius" of about 30 feet. In +order to use this large radius (distance from slit to revolving mirror), +it was necessary that the mirror should be large and optically true; also, +that the lens should be large and of great focal length. Accordingly the +mirror was made 1¼ inches in diameter, and a new lens, 8 inches in +diameter, with a focal length of 150 feet was procured.</p> + +<p>In January, 1879, an observation was taken, using the old lens, the mirror +making 128 turns per second. The deflection was about 43 millimeters. The +micrometer eye-piece used was substantially the same as Foucault's, except +that part of the inclined plate of glass was silvered, thus securing a +much greater quantity of light. The deflection having reached 43 +millimeters, the inclined plate of glass could be dispensed with, the +light going past the observer's head through the slit, and returning 43 +millimeters to the left of the slit, where it could be easily observed.</p> + +<p>Thus the micrometer eye-piece is much simplified, and many possible +sources of error are removed.</p> + +<p>The field was quite limited, the diameter being, in fact, but little +greater than the width of the slit. This would have proved a most serious +objection to the new arrangement. With the new lens, however, this +difficulty disappeared, the field being about twenty times the width of +the slit. It was expected that, with the new lens, the image would be less +distinct; but the difference, if any, was small, and was fully compensated +by the greater size of the field.</p> + +<p>The first observation with the new lens was made January 30, 1879. The +deflection was 70 millimeters. The image was sufficiently bright to be +observed without the slightest effort. The first observation with the new +micrometer eye-piece was made April 2, the deflection being 115 +millimeters.</p> + +<p>The first of the final series of observations was made on June 5. All the +observations previous to this, thirty sets in all, were rejected. After +this time, no set of observations nor any single observation was omitted.</p> +</div> + + +<div class="chapter" id="ch02"> +<h2>Theory of New Method.</h2> + + + +<div class="image"><p><img src="images/fig01.png" alt="fig 1" id="fig01" /></p></div> + +<p>Let S, Fig. 1, be a slit, through which light passes, falling on R, a +mirror free to rotate about an axis at right angles to the plane of the +paper; L, a lens of great focal length, upon which the light falls which +is reflected from R. Let M be a plane mirror whose surface is +perpendicular to the line R, M, passing through the centers of R, L, and +M, respectively. If L be so placed that an image of S is formed on the +surface of M, then, this image acting as the object, its image will be +formed at S, and will coincide, point for point, with S.</p> + +<p>If, now, R be turned about the axis, so long as the light falls upon the +lens, an image of the slit will still be formed on the surface of the +mirror, though on a different part, and as long as the returning light +falls on the lens an image of this image will be formed at S, +notwithstanding the change of position of the first image at M. This +result, namely, the production of a stationary image of an image in +motion, is absolutely necessary in this method of experiment. It was first +accomplished by Foucault, and in a manner differing apparently but little +from the foregoing.</p> + +<p><img src="images/fig02.png" alt="fig 2" id="fig02" /></p> + +<p>In his experiments L, Fig. 2, served simply to form the image of S at M, +and M, the returning mirror, was spherical, the center coinciding with the +axis of R. The lens L was placed as near as possible to R. The light +forming the return image lasts, in this case, while the first image is +sweeping over the face of the mirror, M. Hence, the greater the distance +RM, the larger must be the mirror in order that the same amount of light +may be preserved, and its dimensions would soon become inordinate. The +difficulty was partly met by Foucault, by using five concave reflectors +instead of one, but even then the greatest distance he found it +practicable to use was only 20 meters.</p> + +<p>Returning to Fig. 1, suppose that R is in the principal focus of the lens +L; then, if the plane mirror M have the same diameter as the lens, the +first, or moving image, will remain upon M as long as the axis of the +pencil of light remains on the lens, and <i>this will be the case no matter +what the distance may be</i>.</p> + +<p>When the rotation of the mirror R becomes sufficiently rapid, then the +flashes of light which produce the second or stationary image become +blended, so that the image appears to be continuous. But now it no longer +coincides with the slit, but is <i>deflected</i> in the direction of rotation, +and through twice the angular distance described by the mirror, during +the time required for light to travel twice the distance between the +mirrors. This displacement is measured by the tangent of the arc it +subtends. To make this as large as possible, the distance between the +mirrors, the radius, and the speed of rotation should be made as great as +possible.</p> + +<p>The second condition conflicts with the first, for the radius is the +difference between the focal length for parallel rays, and that for rays +at the distance of the fixed mirror. The greater the distance, therefore, +the smaller will be the radius.</p> + +<p>There are two ways of solving the difficulty: first, by using a lens of +great focal length; and secondly, by placing the revolving mirror within +the principal focus of the lens. Both means were employed. The focal +length of the lens was 150 feet, and the mirror was placed about 15 feet +within the principal focus. A limit is soon reached, however, for the +quantity of light received diminishes very rapidly as the revolving mirror +approaches the lens.</p> +</div> + + +<div class="chapter" id="ch03"> +<h2>Arrangement and Description of Apparatus.</h2> + + + +<h3>Site and Plan.</h3> + + +<p>The site selected for the experiments was a clear, almost level, stretch +along the north sea-wall of the Naval Academy. A frame building was +erected at the western end of the line, a plan of which is represented in +Fig. 3.</p> + +<p><img src="images/fig03.png" alt="fig 3" id="fig03" /></p> + +<p>The building was 45 feet long and 14 feet wide, and raised so that the +line along which the light traveled was about 11 feet above the ground. A +heliostat at H reflected the sun's rays through the slit at S to the +revolving mirror R, thence through a hole in the shutter, through the +lens, and to the distant mirror.</p> + + + +<h3>The Heliostat.</h3> + + +<p>The heliostat was one kindly furnished by Dr. Woodward, of the Army +Medical Museum, and was a modification of Foucault's form, designed by +Keith. It was found to be accurate and easy to adjust. The light was +reflected from the heliostat to a plane mirror, M, Fig. 3, so that the +former need not be disturbed after being once adjusted.</p> + + + +<h3>The Revolving Mirror.</h3> + + +<p>The revolving mirror was made by Fauth & Co., of Washington. It consists +of a cast-iron frame resting on three leveling screws, one of which was +connected by cords to the table at S, Fig. 3, so that the mirror could be +inclined forward or backward while making the observations.</p> + +<p><img src="images/fig04.png" alt="fig 4" id="fig04" /></p> + +<p>Two binding screws, S, S, Fig. 4, terminating in hardened steel conical +sockets, hold the revolving part. This consists of a steel axle, X, Y, +Figs. 4 and 5, the pivots being conical and hardened. The axle expands +into a ring at R, which holds the mirror M. The latter was a disc of plane +glass, made by Alvan Clark & Sons, about 1¼ inch in diameter and 0.2 inch +thick. It was silvered on one side only, the reflection taking place from +the outer or front surface. A species of turbine wheel, T, is held on the +axle by friction. This wheel has six openings for the escape of air; a +section of one of them is represented in Fig 6.</p> + +<p><img src="images/fig05.png" alt="fig 5" id="fig05" /></p> + +<p><img src="images/fig06.png" alt="fig 6" id="fig06" /></p> + + + +<h3>Adjustment of the Revolving Mirror.</h3> + + +<p>The air entering on one side at O, Fig. 5, acquires a rotary motion in the +box B, B, carrying the wheel with it, and this motion is assisted by the +reaction of the air in escaping. The disc C serves the purpose of bringing +the center of gravity in the axis of rotation. This was done, following +Foucault's plan, by allowing the pivots to rest on two inclined planes of +glass, allowing the arrangement to come to rest, and filing away the +lowest part of the disc; trying again, and so on, till it would rest in +indifferent equilibrium. The part corresponding to C, in Foucault's +apparatus, was furnished with three vertical screws, by moving which the +axis of figure was brought into coincidence with the axis of rotation. +This adjustment was very troublesome. Fortunately, in this apparatus it +was found to be unnecessary.</p> + +<p>When the adjustment is perfect the apparatus revolves without giving any +sound, and when this is accomplished, the motion is regular and the speed +great. A slight deviation causes a sound due to the rattling of the pivots +in the sockets, the speed is very much diminished, and the pivots begin to +wear. In Foucault's apparatus oil was furnished to the pivots, through +small holes running through the screws, by pressure of a column of +mercury. In this apparatus it was found sufficient to touch the pivots +occasionally with a drop of oil.</p> + +<p><img src="images/fig07.png" alt="fig 7" id="fig07" /></p> + +<p>Fig. 7 is a view of the turbine, box, and supply-tube, from above. The +quantity of air entering could be regulated by a valve to which was +attached a cord leading to the observer's table.</p> + +<p>The instrument was mounted on a brick pier.</p> + + + +<h3>The Micrometer.</h3> + + +<p><img src="images/fig08.png" alt="fig 8" id="fig08" /></p> + +<p>The apparatus for measuring the deflection was made by Grunow, of New +York.</p> + +<p>This instrument is shown in perspective in Fig. 8, and in plan by Fig. 9. +The adjustable slit S is clamped to the frame F. A long millimeter-screw, +not shown in Fig. 8, terminating in the divided head D, moves the carriage +C, which supports the eye-piece E. The frame is furnished with a brass +scale at F for counting revolutions, the head counting hundredths. The +eye-piece consists of a single achromatic lens, whose focal length is +about two inches. At its focus, in H, and in nearly the same plane as the +face of the slit, is a single vertical silk fiber. The apparatus is +furnished with a standard with rack and pinion, and the base furnished +with leveling screws.</p> + + + +<h3>Manner of Using the Micrometer.</h3> + + +<p>In measuring the deflection, the eye-piece is moved till the cross-hair +bisects the slit, and the reading of the scale and divided head gives the +position. This measurement need not be repeated unless the position or +width of the slit is changed. Then the eye-piece is moved till the +cross-hair bisects the deflected image of the slit; the reading of scale +and head are again taken, and the difference in readings gives the +deflection. The screw was found to have no lost motion, so that readings +could be taken with the screw turned in either direction.</p> + + + +<h3>Measurement of Speed of Rotation.</h3> + + +<p>To measure the speed of rotation, a tuning-fork, bearing on one prong a +steel mirror, was used. This was kept in vibration by a current of +electricity from five "gravity" cells. The fork was so placed that the +light from the revolving mirror was reflected to a piece of plane glass, +in front of the lens of the eye-piece of the micrometer, inclined at an +angle of 45°, and thence to the eye. When fork and revolving mirror are +both at rest, an image of the revolving mirror is seen. When the fork +vibrates, this image is drawn out into a band of light.</p> + +<p>When the mirror commences to revolve, this band breaks up into a number of +moving images of the mirror; and when, finally, the mirror makes as many +turns as the fork makes vibrations, these images are reduced to one, which +is stationary. This is also the case when the number of turns is a +submultiple. When it is a multiple or simple ratio, the only difference is +that there are more images. Hence, to make the mirror execute a certain +number of turns, it is simply necessary to pull the cord attached to the +valve to the right or left till the images of the revolving mirror come to +rest.</p> + +<p>The electric fork made about 128 vibrations per second. No dependence was +placed upon this rate, however, but at each set of observations it is +compared with a standard Ut<sub>3</sub> fork, the temperature being noted at the +same time. In making the comparison the sound-beats produced by the forks +were counted for 60 seconds. It is interesting to note that the electric +fork, as long as it remained untouched and at the same temperature, did +not change its rate more than one or two hundredths vibrations per second.</p> + +<p><img src="images/fig09.png" alt="fig 9" id="fig09" /></p> + + + +<h3>The Observer's Table.</h3> + + +<p>Fig. 9 Represents The Table At Which The Observer Sits. The Light From The +Heliostat Passes Through The Slit At S, Goes To The Revolving Mirror, &c., +And, On Its Return, Forms An Image Of The Slit At D, Which Is Observed +Through The Eye-piece. E Represents The Electric Fork (the Prongs Being +Vertical) Bearing The Steel Mirror M. K Is The Standard Fork On Its +Resonator. C Is The Cord Attached To The Valve Supplying Air To The +Turbine.</p> + + + +<h3>The Lens.</h3> + + +<p>The lens was made by Alvan Clark & Sons. It was 8 inches in diameter; +focal length, 150 feet; not achromatic. It was mounted in a wooden frame, +which was placed on a support moving on a slide, about 16 feet long, +placed about 80 feet from the building. As the diameter of the lens was so +small in comparison with its focal length, its want of achromatism was +inappreciable. For the same reason, the effect of "parallax" (due to want +of coincidence in the plane of the image with that of the silk fiber in +the eye-piece) was too small to be noticed.</p> + + + +<h3>The Fixed Mirror.</h3> + + +<p>The fixed mirror was one of those used in taking photographs of the +transit of Venus. It was about 7 inches in diameter, mounted in a brass +frame capable of adjustment in a vertical and a horizontal plane by screw +motion. Being wedge-shaped, it had to be silvered on the front surface. To +facilitate adjustment, a small telescope furnished with cross-hairs was +attached to the mirror by a universal joint. The heavy frame was mounted +on a brick pier, and the whole surrounded by a wooden case to protect it +from the sun.</p> + + + +<h3>Adjustment of the Fixed Mirror.</h3> + + +<p>The adjustment was effected as follows: A theodolite was placed at about +100 feet in front of the mirror, and the latter was moved about by the +screws till the observer at the theodolite saw the image of his telescope +reflected in the center of the mirror. Then the telescope attached to the +mirror was pointed (without moving the mirror itself) at a mark on a piece +of card-board attached to the theodolite. Thus the line of collimation of +the telescope was placed at right angles to the surface of the mirror. The +theodolite was then moved to 1,000 feet, and, if found necessary, the +adjustment was repeated. Then the mirror was moved by the screws till its +telescope pointed at the hole in the shutter of the building. The +adjustment was completed by moving the mirror, by signals, till the +observer, looking through the hole in the shutter, through a good +spy-glass, saw the image of the spy-glass reflected centrally in the +mirror.</p> + +<p>The whole operation was completed in a little over an hour.</p> + +<p>Notwithstanding the wooden case about the pier, the mirror would change +its position between morning and evening; so that the last adjustment had +to be repeated before every series of experiments.</p> + + + +<h3>Apparatus for Supplying and Regulating the Blast of Air.</h3> + + +<p>Fig. 10 represents a plan of the lower floor of the building. E is a +three-horse power Lovegrove engine and boiler, resting on a stone +foundation; B, a small Roots' blower; G, an automatic regulator. From this +the air goes to a delivery-pipe, up through the floor, and to the turbine. +The engine made about 4 turns per second and the blower about 15. At this +speed the pressure of the air was about half a pound per square inch.</p> + +<p><img src="images/fig10.png" alt="fig 10" id="fig10" /></p> + +<p>The regulator, Fig. 11, consists of a strong bellows supporting a weight +of 370 pounds, partly counterpoised by 80 pounds in order to prevent the +bellows from sagging. When the pressure of air from the blower exceeds the +weight, the bellows commences to rise, and, in so doing, closes the +valve V.</p> + +<p><img src="images/fig11.png" alt="fig 11" id="fig11" /></p> + +<p><img src="images/fig12.png" alt="fig 12" id="fig12" /></p> + +<p>This arrangement was found in practice to be insufficient, and the +following addition was made: A valve was placed at P, and the pipe was +tapped a little farther on, and a rubber tube led to a water-gauge, Fig +12. The column of water in the smaller tube is depressed, and, when it +reaches the horizontal part of the tube, the slightest variation of +pressure sends the column from one end to the other. This is checked by an +assistant at the valve; so that the column of water is kept at about the +same place, and the pressure thus rendered very nearly constant. The +result was satisfactory, though not in the degree anticipated. It was +possible to keep the mirror at a constant speed for three or four seconds +at a time, and this was sufficient for an observation. Still it would have +been more convenient to keep it so for a longer time.</p> + +<p>I am inclined to think that the variations were due to changes in the +friction of the pivots rather than to changes of pressure of the blast of +air.</p> + +<p>It may be mentioned that the test of uniformity was very delicate, as a +change of speed of one or two hundredths of a turn per second could easily +be detected.</p> + + + +<h3>Method Followed in Experiment.</h3> + + +<p>It was found that the only time during the day when the atmosphere was +sufficiently quiet to get a distinct image was during the hour after +sunrise, or during the hour before sunset. At other times the image was +"boiling" so as not to be recognizable. In one experiment the electric +light was used at night, but the image was no more distinct than at +sunset, and the light was not steady.</p> + +<p>The method followed in experiment was as follows: The fire was started +half an hour before, and by the time everything was ready the gauge would +show 40 or 50 pounds of steam. The mirror was adjusted by signals, as +before described. The heliostat was placed and adjusted. The revolving +mirror was inclined to the right or left, so that the <i>direct</i> reflection +of light from the slit, which otherwise would flash into the eye-piece at +every revolution, fell either above or below the eye-piece.[<a href="#fn02">2</a>]</p> + +<div class="note" id="fn02"><p> [Footnote 2: Otherwise this light would overpower that which forms the + image to be observed. As far as I am aware, Foucault does not speak of + this difficulty. If he allowed this light to interfere with the + brightness of the image, he neglected a most obvious advantage. If he + did incline the axis of the mirror to the right or left, he makes no + allowance for the error thus introduced.]</p></div> + +<p>The revolving mirror was then adjusted by being moved about, and inclined +forward and backward, till the light was seen reflected back from the +distant mirror. This light was easily seen through the coat of silver on +the mirror.</p> + +<p>The distance between the front face of the revolving mirror and the +cross-hair of the eye-piece was then measured by stretching from the one +to the other a steel tape, making the drop of the catenary about an inch, +as then the error caused by the stretch of the tape and that due to the +curve just counterbalance each other.</p> + +<p>The position of the slit, if not determined before, was then found as +before described. The electric fork was started, the temperature noted, +and the sound-beats between it and the standard fork counted for 60 +seconds. This was repeated two or three times before every set of +observations.</p> + +<p>The eye-piece of the micrometer was then set approximately[<a href="#fn03">3</a>] and the +revolving mirror started. If the image did not appear, the mirror was +inclined forward or backward till it came in sight.</p> + +<div class="note" id="fn03"><p> [Footnote 3: The deflection being measured by its tangent, it was + necessary that the scale should be at right angles to the radius (the + radius drawn from the mirror to one or the other end of that part of + the scale which represents this tangent). This was done by setting the + eye-piece approximately to the expected deflection, and turning the + whole micrometer about a vertical axis till the cross-hair bisected the + circular field of light reflected from the revolving mirror. The axis + of the eye-piece being at right angles to the scale, the latter would + be at right angles to radius drawn to the cross-hair.]</p></div> + +<p>The cord connected with the valve was pulled right or left till the images +of the revolving mirror, represented by the two bright round spots to the +left of the cross-hair, came to rest. Then the screw was turned till the +cross-hair bisected the deflected image of the slit. This was repeated +till ten observations were taken, when the mirror was stopped, temperature +noted, and beats counted. This was called a set of observations. Usually +five such sets were taken morning and evening.</p> + +<p><img src="images/fig13.png" alt="fig 13" id="fig13" /></p> + +<p>Fig. 13 represents the appearance of the image of the slit as seen in the +eye-piece magnified about five times.</p> +</div> + + +<div class="chapter" id="ch04"> +<h2>Determination of The Constants.</h2> + + + +<h3>Comparison of the Steel Tape with the Standard Yard.</h3> + + +<p>The steel tape used was one of Chesterman's, 100 feet long. It was +compared with Wurdeman's copy of the standard yard, as follows:</p> + +<p>Temperature was 55° Fahr.</p> + +<p>The standard yard was brought under the microscopes of the comparator; the +cross-hair of the unmarked microscope was made to bisect the division +marked o, and the cross-hair of the microscope, marked I, was made to +bisect the division marked 36. The reading of microscope I was taken, and +the other microscope was not touched during the experiment. The standard +was then removed and the steel tape brought under the microscopes and +moved along till the division marked 0.1 (feet) was bisected by the +cross-hair of the unmarked microscope. The screw of microscope I was then +turned till its cross-hair bisected the division marked 3.1 (feet), and +the reading of the screw taken. The difference between the original +reading and that of each measurement was noted, care being taken to regard +the direction in which the screw was turned, and this gave the difference +in length between the standard and each succesive portion of the steel +tape in terms of turns of the micrometer-screw.</p> + +<p>To find the value of one turn, the cross-hair was moved over a millimeter +scale, and the following were the values obtained:</p> + +<p>Turns of screw of microscope I in 1<sup>mm</sup>—</p> + +<table summary="turns of screw of microscope I in 1mm"> +<tr><td> 7.68</td><td> 7.73 </td><td> 7.60 </td><td> 7.67</td></tr> +<tr><td> 7.68 </td><td> 7.62</td><td> 7.65 </td><td> 7.57</td></tr> +<tr><td> 7.72 </td><td> 7.70 </td><td> 7.64 </td><td> 7.69</td></tr> +<tr><td> 7.65 </td><td> 7.59 </td><td> 7.63 </td><td> 7.64</td></tr> +<tr><td> 7.55 </td><td> 7.65 </td><td> 7.61 </td><td> 7.63</td></tr> +<tr><td colspan="4" style="text-align: center"> + Mean =7.65</td></tr> +<tr><td colspan="4" style="text-align: center"> + Hence one turn = 0.1307<sup>mm</sup>.</td></tr> +<tr><td colspan="4" style="text-align: center"> + or = 0.0051 inch.</td></tr> +<tr><td colspan="3"> + The length of the steel tape from 0.1 to 99.1 was found to be<br /> + greater than 33 yards, by 7.4 turns =.96<sup>mm</sup></td><td style="text-align: right"> +.003 feet.</td></tr> +<tr><td colspan="3"> Correction for temperature</td><td style="text-align: right"> +.003 feet.</td></tr> +<tr><td colspan="3"> Length</td><td style="text-align: right"> 100.000 feet.</td></tr> +<tr><td colspan="3"> </td><td style="text-align: center"> --------------</td></tr> +<tr><td colspan="3"> Corrected length </td><td style="text-align: right"> 100.006 feet.</td></tr> +</table> + + +<h3>Determination of the Value of Micrometer.</h3> + + +<p>Two pairs of lines were scratched on one slide of the slit, about 38<sup>mm</sup> +apart, i.e., from the center of first pair to center of second pair. This +distance was measured at intervals of 1<sup>mm</sup> through the whole length of the +screw, by bisecting the interval between each two pairs by the vertical +silk fiber at the end of the eye-piece. With these values a curve was +constructed which gave the following values for this distance, which we +shall call D′:</p> +<table summary="values for the distance measured at intervals of 1mm through the whole length of the screw"> +<caption> Turns of screw.</caption> +<tr><td> At</td><td style="text-align: right"> 0</td><td> of scale D′</td><td class="align-dot"> =38.155</td></tr> +<tr><td></td><td style="text-align: right"> 10</td><td> of scale D′</td><td class="align-dot"> 38.155</td></tr> +<tr><td></td><td style="text-align: right"> 20</td><td> of scale D′</td><td class="align-dot"> 38.150</td></tr> +<tr><td></td><td style="text-align: right"> 30</td><td> of scale D′</td><td class="align-dot"> 38 150</td></tr> +<tr><td></td><td style="text-align: right"> 40</td><td> of scale D′</td><td class="align-dot"> 38.145</td></tr> +<tr><td></td><td style="text-align: right"> 50</td><td> of scale D′</td><td class="align-dot"> 38.140</td></tr> +<tr><td></td><td style="text-align: right"> 60</td><td> of scale D′</td><td class="align-dot"> 38.140</td></tr> +<tr><td></td><td style="text-align: right"> 70</td><td> of scale D′</td><td class="align-dot"> 38.130</td></tr> +<tr><td></td><td style="text-align: right"> 80</td><td> of scale D′ </td><td class="align-dot"> 38.130</td></tr> +<tr><td></td><td style="text-align: right"> 90</td><td> of scale D′</td><td class="align-dot"> 38.125</td></tr> +<tr><td></td><td style="text-align: right"> 100 </td><td>of scale D′</td><td class="align-dot"> 38.120</td></tr> +<tr><td></td><td style="text-align: right"> 110</td><td> of scale D′</td><td class="align-dot"> 38.110</td></tr> +<tr><td></td><td style="text-align: right"> 120</td><td> of scale D′</td><td class="align-dot"> 38.105</td></tr> +<tr><td></td><td style="text-align: right"> 130</td><td> of scale D′</td><td class="align-dot"> 38.100</td></tr> +<tr><td></td><td style="text-align: right"> 140</td><td> of scale D′</td><td class="align-dot"> 38.100</td></tr> +</table> +<p>Changing the form of this table, we find that,—</p> +<table summary="values for the distance measured at intervals of 1mm through the whole length of the screw (alternative presentation)"> +<tr><td> For the <i>first</i></td></tr> +<tr><td style="text-align: right"> 10 </td><td>turns the <i>average</i> value of D′ is</td><td> 38.155</td></tr> +<tr><td style="text-align: right"> 20 </td><td>turns </td><td> 38.153</td></tr> +<tr><td style="text-align: right"> 30 </td><td>turns </td><td> 38.152</td></tr> +<tr><td style="text-align: right"> 40 </td><td>turns </td><td> 38.151</td></tr> +<tr><td style="text-align: right"> 50 </td><td>turns </td><td> 38.149</td></tr> +<tr><td style="text-align: right"> 60 </td><td>turns </td><td> 38.148</td></tr> +<tr><td style="text-align: right"> 70 </td><td>turns </td><td> 38.146</td></tr> +<tr><td style="text-align: right"> 80 </td><td>turns </td><td> 38.144</td></tr> +<tr><td style="text-align: right"> 90 </td><td>turns </td><td> 38.142</td></tr> +<tr><td style="text-align: right"> 100 </td><td>turns </td><td> 38.140</td></tr> +<tr><td style="text-align: right"> 110 </td><td>turns </td><td> 38.138</td></tr> +<tr><td style="text-align: right"> 120 </td><td>turns </td><td> 38.135</td></tr> +<tr><td style="text-align: right"> 130 </td><td>turns </td><td> 38.132</td></tr> +<tr><td style="text-align: right"> 140 </td><td>turns </td><td> 38.130</td></tr> +</table> +<p>On comparing the scale with the standard meter, the temperature being +16°.5 C., 140 divisions were found to = 139.462<sup>mm</sup>. This multiplied by +(1 + .0000188 × 16.5) = 139.505<sup>mm</sup>.</p> + +<p>One hundred and forty divisions were found to be equal to 140.022 turns +of the screw, whence 140 turns of the screw = 139.483<sup>mm</sup>, or +1 turn of the screw = 0.996305<sup>mm</sup>.</p> + +<p>This is the <i>average</i> value of one turn in 140.</p> + +<p>But the average value of D, for 140 turns is, from the preceding table, +38.130.</p> + +<p>Therefore, the true value of D, is 38.130 × .996305<sup>mm</sup>, and the average +value of one turn for 10, 20, 30, etc., turns, is found by dividing 38.130 +× .996305 by the values of D;, given in the table.</p> + +<p>This gives the value of a turn—</p> +<table summary="the value of a turn"> +<tr><th></th><th></th><th></th><th> mm.</th></tr> +<tr><td> For the first</td><td style="text-align:right"> 10 </td><td>turns </td><td> 0.99570</td></tr> +<tr><td></td><td style="text-align:right"> 20</td><td> turns </td><td> 0.99570</td></tr> +<tr><td></td><td style="text-align:right"> 30 </td><td>turns </td><td> 0.99573</td></tr> +<tr><td></td><td style="text-align:right"> 40 </td><td>turns </td><td> 0.99577</td></tr> +<tr><td></td><td style="text-align:right"> 50</td><td> turns </td><td> 0.99580</td></tr> +<tr><td></td><td style="text-align:right"> 60</td><td> turns </td><td> 0.99583</td></tr> +<tr><td></td><td style="text-align:right"> 70</td><td> turns </td><td> 0.99589</td></tr> +<tr><td></td><td style="text-align:right"> 80</td><td> turns </td><td> 0.99596</td></tr> +<tr><td></td><td style="text-align:right"> 90</td><td> turns </td><td> 0.99601</td></tr> +<tr><td></td><td style="text-align:right"> 100</td><td> turns </td><td> 0.99606</td></tr> +<tr><td></td><td style="text-align:right"> 110 </td><td>turns </td><td> 0.99612</td></tr> +<tr><td></td><td style="text-align:right"> 120 </td><td>turns </td><td> 0.99618</td></tr> +<tr><td></td><td style="text-align:right"> 130</td><td> turns </td><td> 0.99625</td></tr> +<tr><td></td><td style="text-align:right"> 140</td><td> turns </td><td> 0.99630</td></tr> +</table> + +<p><span class="smallcaps">Note</span>.—The micrometer has been sent to Professor Mayer, of Hoboken, to +test the screw again, and to find its value. The steel tape has been sent +to Professor Rogers, of Cambridge, to find its length again. (See page +145.)</p> + + + +<h3>Measurement of the Distance between the Mirrors.</h3> + + +<p>Square lead weights were placed along the line, and measurements taken +from the forward side of one to forward side of the next. The tape rested +on the ground (which was very nearly level), and was stretched by a +constant force of 10 pounds.</p> + +<p>The correction for length of the tape (100.006) was +0.12 of a foot.</p> + +<p>To correct for the stretch of the tape, the latter was stretched with a +force of 15 pounds, and the stretch at intervals of 20 feet measured by a +millimeter scale.</p> +<table summary="stretch intervals"> +<caption> mm.</caption> +<tr><td> At </td><td style="text-align: right">100</td><td> feet the stretch was</td><td style="text-align: right"> 8.0</td></tr> +<tr><td></td><td style="text-align: right"> 80</td><td> feet the stretch was</td><td style="text-align: right"> 5.0</td></tr> +<tr><td></td><td style="text-align: right"> 60 </td><td>feet the stretch was</td><td style="text-align: right"> 5.0</td></tr> +<tr><td></td><td style="text-align: right"> 40 </td><td>feet the stretch was</td><td style="text-align: right"> 3.5</td></tr> +<tr><td></td><td style="text-align: right"> 20</td><td> feet the stretch was</td><td style="text-align: right"> 1.5</td></tr> +<tr><td></td><td style="text-align: center"> --- </td><td></td><td style="text-align:center"> ---</td></tr> +<tr><td></td><td style="text-align: right"> 300 </td><td></td><td style="text-align: right"> 23.00</td></tr> +</table><table summary="correction"> +<tr><td style="text-align:right"> Weighted mean </td><td>=</td><td> 7.7 mm.</td></tr> +<tr><td style="text-align:right"> For 10 pounds, stretch </td><td>=</td><td> 5.1 mm.</td></tr> +<tr><td style="text-align:right"> </td><td>=</td><td> 0.0167 feet.</td></tr> +<tr><td style="text-align:right"> Correction for whole distance </td><td>=</td><td> +0.33 feet.</td></tr> +</table> +<p>The following are the values obtained from five separate measurements of +the distance between the caps of the piers supporting the revolving mirror +and the distant reflector; allowance made in each case for effect of +temperature:</p> +<table summary="distance between the caps of the piers supporting the revolving mirror and the distant reflector"> +<tr><td></td><td class="align-dot"> 1985.13 </td><td>feet.</td></tr> +<tr><td></td><td class="align-dot"> 1985.17</td><td> feet.</td></tr> +<tr><td></td><td class="align-dot"> 1984.93</td><td> feet.</td></tr> +<tr><td></td><td class="align-dot"> 1985.09 </td><td>feet.</td></tr> +<tr><td></td><td class="align-dot"> 1985.09</td><td> feet.</td></tr> +<tr><td></td><td style="text-align:center"> -------</td></tr> +<tr><td> Mean = </td><td class="align-dot">1985.082</td><td> feet.</td></tr> + +<tr><td></td><td class="align-dot"> +.70.</td><td> Cap of pier to revolving mirror.</td></tr> +<tr><td></td><td class="align-dot"> +.33.</td><td> Correction for stretch of tape.</td></tr> +<tr><td></td><td class="align-dot"> +.12.</td><td> Correction for length of tape.</td></tr> +<tr><td></td><td style="text-align:center"> --------</td></tr> +<tr><td></td><td class="align-dot"> 1986.23.</td><td> True distance between mirrors.</td></tr> +</table> + + +<h3>Rate of Standard Ut<sub>3</sub> Fork.</h3> + + +<p>The rate of the standard Ut<sub>3</sub> fork was found at the Naval Academy, but as +so much depended on its accuracy, another series of determinations of its +rate was made, together with Professor Mayer, at the Hoboken Institute of +Technology.</p> + + +<h4><i>Set of determinations made at Naval Academy.</i></h4> + +<p>The fork was armed with a tip of copper foil, which was lost during the +experiments and replaced by one of platinum having the same weight, +4.6 mgr. The fork, on its resonator, was placed horizontally, the platinum +tip just touching the lampblacked cylinder of a Schultze chronoscope. The +time was given either by a sidereal break-circuit chronometer or by the +break-circuit pendulum of a mean-time clock. In the former case the +break-circuit worked a relay which interrupted the current from three +Grove cells. The spark from the secondary coil of an inductorium was +delivered from a wire near the tip of the fork. Frequently two sparks near +together were given, in which case the first alone was used. The rate of +the chronometer, the record of which was kept at the Observatory, was very +regular, and was found by observations of transits of stars during the +week to be +1.3 seconds per day, which is the same as the recorded rate.</p> + + + +<h3>Specimen of a Determination of Rate of Ut<sub>3</sub> Fork.</h3> + + +<p>Temp.=27° C. Column 1 gives the number of the spark or the number of the +second. Column 2 gives the number of sinuosities or vibrations at the +corresponding second. Column 3 gives the difference between 1 and 11, 2 +and 12, 3 and 13, etc.</p> +<table summary="specimen of a determination of rate of Ut3"> +<caption> July 4, 1879.</caption> +<tr><td class="align-dot"> 1.</td><td class="align-dot"> 0.1 </td><td class="align-dot"> 2552.0</td></tr> +<tr><td class="align-dot"> 2. </td><td class="align-dot"> 255.3 </td><td class="align-dot"> 2551.7</td></tr> +<tr><td class="align-dot"> 3. </td><td class="align-dot"> 510.5 </td><td class="align-dot"> 2551.9</td></tr> +<tr><td class="align-dot"> 4. </td><td class="align-dot"> 765.6 </td><td class="align-dot"> 2551.9</td></tr> +<tr><td class="align-dot"> 5. </td><td class="align-dot"> 1020.7 </td><td class="align-dot"> 2552.1</td></tr> +<tr><td class="align-dot"> 6. </td><td class="align-dot"> 1275.7 </td><td class="align-dot"> 2552.0</td></tr> +<tr><td class="align-dot"> 7. </td><td class="align-dot"> 1530.7 </td><td class="align-dot"> 2551.8</td></tr> +<tr><td class="align-dot"> 8. </td><td class="align-dot"> 1786.5 </td><td class="align-dot"> 2551.4</td></tr> +<tr><td class="align-dot"> 9. </td><td class="align-dot"> 2041.6 </td><td class="align-dot"> 2551.7</td></tr> +<tr><td class="align-dot"> 10. </td><td class="align-dot"> 2297.0 </td><td class="align-dot"> 2551.5</td></tr> +<tr><td></td><td></td><td style="text-align:center"> -------</td></tr> +<tr><td class="align-dot"> 11. </td><td class="align-dot"> 2552.1 </td><td class="align-dot"> 255.180</td><td> = mean ÷ 10.</td></tr> +<tr><td class="align-dot"> 12. </td><td class="align-dot"> 2807.0 </td><td class="align-dot"> + .699</td><td> = reduction for mean time.</td></tr> +<tr><td class="align-dot"> 13. </td><td class="align-dot"> 3062.4 </td><td class="align-dot"> + .003</td><td> = correction for rate.</td></tr> +<tr><td class="align-dot"> 14. </td><td class="align-dot"> 3317.5 </td><td class="align-dot"> + .187</td><td> = correction for temperature.</td></tr> +<tr><td></td><td></td><td style="text-align:center"> -------</td></tr> +<tr><td class="align-dot"> 15. </td><td class="align-dot"> 3572.8 </td><td class="align-dot"> 256.069</td><td> = number of vibrations per second at 65° Fahr.</td></tr> +<tr><td class="align-dot"> 16. </td><td class="align-dot"> 3827.7</td></tr> +<tr><td class="align-dot"> 17. </td><td class="align-dot"> 4082.5</td></tr> +<tr><td class="align-dot"> 18. </td><td class="align-dot"> 4335.9</td></tr> +<tr><td class="align-dot"> 19. </td><td class="align-dot"> 4593.3</td></tr> +<tr><td class="align-dot"> 20. </td><td class="align-dot"> 4848.5</td></tr> +</table> +<p>The correction for temperature was found by Professor Mayer by counting +the sound-beats between the standard and another Ut<sub>3</sub> fork, at different +temperatures. His result is +.012 vibrations per second for a diminution +of 1° Fahr. Using the same method, I arrived at the result +.0125. +Adopted +.012.</p> + + +<h4><i>Résumé of determinations made at Naval Academy.</i></h4> + +<p>In the following table the first column gives the date, the second gives +the total number of seconds, the third gives the result uncorrected for +temperature, the fourth gives the temperature (centigrade), the fifth +gives the final result, and the sixth the difference between the greatest +and least values obtained in the several determinations for intervals of +ten seconds:</p> +<table summary="Résumé of determinations made at Naval Academy"> +<tr><td> July</td><td> 4 </td><td> 20 </td><td> </td><td>255.882 </td><td> 27.0 </td><td> 256.069 </td><td> 0.07</td></tr> +<tr><td></td><td> 5 </td><td> 19 </td><td> </td><td>255.915 </td><td> 26.4 </td><td> 256.089</td><td> 0.05</td></tr> +<tr><td></td><td> 5 </td><td> 18 </td><td> </td><td>255.911 </td><td> 26.0 </td><td> 256.077 </td><td> 0.02</td></tr> +<tr><td></td><td> 6 </td><td> 21 </td><td> </td><td>255.874 </td><td> 24.7 </td><td> 256.012 </td><td> 0.13</td></tr> +<tr><td></td><td> 6 </td><td> 9 </td><td> </td><td>255.948 </td><td> 24.8 </td><td> 256.087</td><td> 0.24</td></tr> +<tr><td></td><td> 7 </td><td> 22 </td><td> </td><td>255.938 </td><td> 24.6 </td><td> 256.074</td><td> 0.05</td></tr> +<tr><td></td><td> 7 </td><td> 21 </td><td> </td><td>255.911 </td><td> 25.3 </td><td> 256.061 </td><td> 0.04</td></tr> +<tr><td></td><td> 8 </td><td> 20 </td><td> </td><td>255.921 </td><td> 26.6 </td><td> 256.100</td><td> 0.02</td></tr> +<tr><td></td><td> 8 </td><td> 20 </td><td> </td><td>255.905 </td><td> 26.6 </td><td> 256.084 </td><td> 0.06</td></tr> +<tr><td></td><td> 8</td><td> 20 </td><td> </td><td>255.887 </td><td> 26.6 </td><td> 256.066 </td><td>0.03</td></tr> +<tr><td></td><td> </td><td></td><td></td><td></td><td></td><td> -------</td></tr> +<tr><td></td><td> </td><td></td><td></td><td></td><td> Mean = </td><td>256.072</td></tr> +</table> +<p>In one of the preceding experiments, I compared the two Vt<sub>3</sub> forks while +the standard was tracing its record on the cylinder, and also when it was +in position as for use in the observations. The difference, if any, was +less than .01 vibration per second.</p> + + +<h4><i>Second determination</i>.</h4> + +<p>(Joint work with Professor A.M. Mayer, Stevens Institute, Hoboken.)</p> + +<p>The fork was wedged into a wooden support, and the platinum tip allowed to +rest on lampblacked paper, wound about a metal cylinder, which was rotated +by hand Time was given by a break-circuit clock, the rate of which was +ascertained, by comparisons with Western Union time-ball, to be 9.87 +seconds. The spark from secondary coil of the inductorium passed from the +platinum tip, piercing the paper. The size of the spark was regulated by +resistances in primary circuit.</p> + +<p>The following is a specimen determination:</p> + +<p>Column 1 gives the number of the spark or the number of seconds. Column 2 +gives the corresponding number of sinuosities or vibrations. Column 3 +gives the difference between the 1st and 7th ÷ 6, 2nd and 8th ÷ 6, etc.</p> +<table summary="specimen determination"> +<tr><td class="align-dot"> 1 </td><td class="align-dot"> 0.3 </td><td class="align-dot"> 255.83</td></tr> +<tr><td class="align-dot"> 2 </td><td class="align-dot"> 256.1 </td><td class="align-dot"> 255.90</td></tr> +<tr><td class="align-dot"> 3 </td><td class="align-dot"> 511.7 </td><td class="align-dot"> 255.90</td></tr> +<tr><td class="align-dot"> 4 </td><td class="align-dot"> 767.9 </td><td class="align-dot"> 255.93</td></tr> +<tr><td class="align-dot"> 5 </td><td class="align-dot">1023.5 </td><td class="align-dot"> 255.92</td></tr> +<tr><td class="align-dot"> 6 </td><td class="align-dot">1289.2 </td><td class="align-dot"> 256.01</td></tr> +<tr><td class="align-dot"> 7 </td><td class="align-dot">1535.3 </td><td class="align-dot"> 255.95</td></tr> +<tr><td></td><td></td><td> -------</td></tr> +<tr><td class="align-dot"> 8 </td><td class="align-dot">1791.5 </td><td class="align-dot"> 255.920 </td><td>= mean.</td></tr> +<tr><td class="align-dot"> 9 </td><td class="align-dot">2047.1 </td><td class="align-dot"> - .028 </td><td>= correction for rate.</td></tr> +<tr><td></td><td></td><td> -------</td></tr> +<tr><td class="align-dot"> 10 </td><td class="align-dot">2303.5 </td><td class="align-dot"> 255.892</td></tr> +<tr><td class="align-dot"> 11 </td><td class="align-dot">2559.0 </td><td class="align-dot"> + .180 </td><td>= correction for temperature.</td></tr> +<tr><td class="align-dot"></td><td></td><td> -------</td></tr> +<tr><td class="align-dot"> 12 </td><td class="align-dot">2825.3 </td><td class="align-dot"> 256.072 </td><td>= number of vibrations per second at 65° Fahr.</td></tr> +<tr><td class="align-dot"> 13 </td><td class="align-dot">3071.0</td></tr> +</table> +<p>In the following <i>résumé</i>, column 1 gives the number of the experiments. +Column 2 gives the total number of seconds. Column 3 gives the result not +corrected for temperature. Column 4 gives the temperature Fahrenheit. +Column 5 gives the final result. Column 6 gives the difference between the +greatest and least values:</p> +<table summary="résumé"> +<tr><td class="align-dot"> 1 </td><td> 13 </td><td> 255.892 </td><td> 80 </td><td> 256.072 </td><td> 0.18</td></tr> +<tr><td class="align-dot"> 2 </td><td> 11 </td><td> 255.934 </td><td> 81 </td><td> 256.126 </td><td> 0.17</td></tr> +<tr><td class="align-dot"> 3 </td><td> 13 </td><td> 255.899 </td><td> 81 </td><td> 256.091 </td><td> 0.12</td></tr> +<tr><td class="align-dot"> 4 </td><td> 13 </td><td> 255.988 </td><td> 75 </td><td> 256.108 </td><td> 0.13</td></tr> +<tr><td class="align-dot"> 5 </td><td> 11 </td><td> 255.948 </td><td> 75 </td><td> 256.068 </td><td> 0.05</td></tr> +<tr><td class="align-dot"> 6 </td><td> 12 </td><td> 255.970 </td><td> 75 </td><td> 256.090 </td><td> 0.05</td></tr> +<tr><td class="align-dot"> 7 </td><td> 12 </td><td> 255.992 </td><td> 75 </td><td> 256.112 </td><td> 0.20</td></tr> +<tr><td class="align-dot"> 8 </td><td> 11 </td><td> 255.992 </td><td> 76 </td><td> 256.124 </td><td> 0.03</td></tr> +<tr><td class="align-dot"> 9 </td><td> 11 </td><td> 255.888 </td><td> 81 </td><td> 256.080 </td><td> 0.13</td></tr> +<tr><td class="align-dot"> 10 </td><td> 13 </td><td> 255.878 </td><td> 81 </td><td> 256.070 </td><td> 0.13</td></tr> +<tr><td class="align-dot"> </td><td></td><td></td><td></td><td> -------</td></tr> +<tr><td class="align-dot"></td><td></td><td></td><td> Mean = </td><td>256.094</td></tr> +</table> + + +<h3>Effect of Support and of Scraping.</h3> + + +<p>The standard Vt<sub>3</sub> fork held in its wooden support was compared with +another fork on a resonator loaded with wax and making with standard about +five beats per second. The standard was free from the cylinder. The beats +were counted by coincidences with the ⅕ second beats of a watch.</p> + + +<h4><i>Specimen.</i></h4> + +<p>Coincidences were marked—</p> +<table summary="specimen coincidences"> +<tr><td> At 32 </td><td> seconds.</td></tr> +<tr><td> 37 </td><td> seconds.</td></tr> +<tr><td> 43.5 </td><td>seconds.</td></tr> +<tr><td> 49 </td><td>seconds.</td></tr> +<tr><td> 54.5</td><td> seconds.</td></tr> +<tr><td> 61.5</td><td> seconds.</td></tr> +<tr><td> 61.5 - 32</td><td> = 29.5.</td></tr> +<tr><td> 29.5 ÷ 5</td><td> = 5.9 =</td><td> time of one interval.</td></tr> +</table> +<h4><i>Résumé.</i></h4> +<table summary="specimen résumé"> +<tr><td> 1 </td><td> 5.9</td></tr> +<tr><td> 2 </td><td> 6.2</td></tr> +<tr><td> 3 </td><td> 6.2</td></tr> +<tr><td> 4</td><td> 6.2</td></tr> +<tr><td></td><td> ----</td></tr> +<tr><td> Mean =</td><td> 6.13</td><td> = time of one interval between coincidences.</td></tr> +</table> +<p>In this time the watch makes 6.13×5 = 30.65 beats, and the forks make +30.65 + 1 = 31.65 beats.</p> + +<p>Hence the number of beats per second is 31.65 ÷ 6.13 = 5.163.</p> + + +<h4><i>Specimen.</i></h4> + +<p>Circumstances the same as in last case, except that standard Vt<sub>3</sub> fork was +allowed to trace its record on the lampblacked paper, as in finding its +rate of vibration.</p> + +<p>Coincidences were marked at—</p> +<table summary="specimen coincidences"> +<tr><td> 59 </td><td> seconds.</td></tr> +<tr><td> 04 </td><td>seconds.</td></tr> +<tr><td> 10.5 </td><td>seconds.</td></tr> +<tr><td> 17 </td><td>seconds.</td></tr> +<tr><td> </td></tr> +<tr><td colspan="2"> 77 - 59 = 18.</td></tr> +<tr><td colspan="2"> 18 ÷ 3 = 6.0 = time of one interval.</td></tr> +</table> +<h4><i>Résumé.</i></h4> +<table summary="specimen résumé"> +<tr><td> No. </td><td>1 6.0 </td><td> seconds. </td><td> 6.31 × 5 = 31.55</td></tr> +<tr><td> </td><td>2 6.0 </td><td> seconds. </td><td> + 1.00</td></tr> +<tr><td> </td><td>3 6.7 </td><td> seconds. </td><td> ----</td></tr> +<tr><td> </td><td>4 6.3 </td><td> seconds.</td></tr> +<tr><td> </td><td>5 6.5 </td><td> seconds. </td><td> 32.55</td></tr> +<tr><td> </td><td>6 6.7 </td><td> seconds. </td><td> 32.55 ÷ 6.31 = 5.159</td></tr> +<tr><td> </td><td>7 6.0 </td><td> seconds. </td><td> With fork free 5.163</td></tr> +<tr><td> </td><td> ---- </td><td> </td><td> -----</td></tr> +<tr><td> Mean = </td><td>6.31 </td><td>seconds </td><td> Effect of scrape = - .044</td></tr> +</table> +<h4><i>Specimen.</i></h4> + +<p>Circumstances as in first case, except that both forks were on their +resonators.</p> + +<p>Coincidences were observed at—</p> +<table summary="specimen coincidences"> +<tr><td> 21 </td><td>seconds.</td></tr> +<tr><td> 28 </td><td>seconds.</td></tr> +<tr><td> 36 </td><td>seconds.</td></tr> +<tr><td> 44 </td><td>seconds.</td></tr> +<tr><td> 51 </td><td>seconds.</td></tr> +<tr><td> 60 </td><td>seconds.</td></tr> +<tr><td> 60 - 21 = 39</td></tr> +<tr><td> 39 ÷ 5 = 7.8 = </td><td>time of one interval.</td></tr> +</table> +<h4><i>Résumé</i>.</h4> +<table summary="specimen résumé"> +<tr><td> No.</td><td> 1 </td><td> 7.8 </td><td> seconds.</td><td style="text-align: right"> 7.42 × 5 = </td><td style="text-align: right">37.10</td></tr> +<tr><td> </td><td> 2 </td><td> 7.1 </td><td> seconds. </td><td style="text-align: right"> + </td><td style="text-align: right">1.00</td></tr> +<tr><td> </td><td> 3 </td><td> 7.6 </td><td> seconds. </td><td></td><td style="text-align: center"> -----</td></tr> +<tr><td> </td><td> 4 </td><td> 7.4 </td><td> seconds. </td><td></td><td style="text-align: right"> 38.10</td></tr> +<tr><td> </td><td> 5 </td><td> 7.2 </td><td> seconds. </td><td style="text-align: right"> 38.10 ÷ 7.42 =</td><td style="text-align: right"> 5.133</td></tr> +<tr><td> </td><td></td><td> ----</td><td> </td><td style="text-align: right"> (Above) </td><td style="text-align: right"> 5.159</td></tr> +<tr><td> </td><td></td><td></td><td></td><td></td><td style="text-align: center"> -----</td></tr> +<tr><td> </td><td> Mean = </td><td>7.42 </td><td colspan="2">seconds. Effect of support and scrape =</td><td> - .026</td></tr> +<tr><td> </td></tr> +<tr><td colspan="5"> Mean of second determination was </td><td class="align-dot"> 256.094</td></tr> +<tr><td colspan="5"> Applying correction (scrape, etc.) </td><td class="align-dot"> - .026</td></tr> +<tr><td colspan="5"> </td><td style="text-align:center"> -------</td></tr> +<tr><td colspan="5"> Corrected mean </td><td class="align-dot"> 256.068</td></tr> +<tr><td colspan="5"> Result of first determination </td><td class="align-dot"> 256.072</td></tr> +<tr><td colspan="5"> </td><td style="text-align:center"> -------</td></tr> +<tr><td colspan="5"> Final value </td><td class="align-dot"> 256.070</td></tr> +</table> +<p><span class="smallcaps">Note</span>—The result of first determination excludes all work except the +series commencing July 4. If previous work is included, and also the +result first obtained by Professor Mayer, the result would be 256.089.</p> +<table summary="mean"> +<tr><td></td><td> 256.180</td></tr> +<tr><td></td><td> 256.036</td></tr> +<tr><td></td><td> 256.072</td></tr> +<tr><td></td><td> 256.068</td></tr> +<tr><td></td><td> -------</td></tr> +<tr><td> Mean = </td><td>256.089</td></tr> +</table> +<p>The previous work was omitted on account of various inaccuracies and want +of practice, which made the separate results differ widely from each +other.</p> +</div> + + +<div class="chapter" id="ch05"> +<h2>The Formulæ.</h2> + + + +<p>The formulæ employed are—</p> +<table summary="the forumulæ employed"> +<tr><td></td><td style="text-align: center"> <i>d</i>′</td></tr> +<tr><td style="text-align: right"> (1) tan φ = </td><td style="text-align: center">-----</td></tr> +<tr><td></td><td style="text-align: center"> <i>r</i></td></tr> +<tr><td> </td></tr> +<tr><td></td><td style="text-align: center"> 2592000″ × D × <i>n</i></td></tr> +<tr><td style="text-align: right"> (2) V = </td><td style="text-align: center"> -----------------</td></tr> +<tr><td></td><td style="text-align: center"> φ″</td></tr> +<tr><td style="text-align: right"> </td></tr> +<tr><td style="text-align: right"> φ = </td><td>angle of deflection.</td></tr> +<tr><td style="text-align: right"> <i>d</i>′ =</td><td> corrected displacement (linear).</td></tr> +<tr><td style="text-align: right"> r =</td><td> radius of measurement.</td></tr> +<tr><td style="text-align: right"> D =</td><td> twice the distance between the mirrors.</td></tr> +<tr><td style="text-align: right"> n =</td><td> number of revolutions per second.</td></tr> +<tr><td style="text-align: right"> α =</td><td> inclination of plane of rotation</td></tr> +<tr><td style="text-align: right"> d =</td><td> deflection as read from micrometer.</td></tr> +<tr><td style="text-align: right"> B =</td><td> number of beats per second between electric Vt₂ fork and + standard Vt<sub>3</sub></td></tr> +<tr><td style="text-align: right"> Cor =</td><td> correction for temperature of standard Vt3.</td></tr> +<tr><td style="text-align: right"> V =</td><td> velocity of light.</td></tr> +<tr><td style="text-align: right"> T =</td><td> value of one turn of screw. (Table, page 126.)</td></tr> +</table> +<p>Substituting for d, its value or d×T×sec α (log sec α = .00008), and +for D its value 3972.46, and reducing to kilometers, the formulæ become—</p> +<table summary="the formulæ becomes"> +<tr><td></td><td> </td><td style="text-align: center"> dT</td></tr> +<tr><td style="text-align: right"> (3) tan φ = </td><td style="text-align: right">c′</td><td style="text-align: center"> ----;</td><td> log c′ = .51607</td></tr> +<tr><td></td><td> </td><td style="text-align: center"> r</td></tr> +<tr><td> </td></tr> +<tr><td></td><td style="text-align: center" colspan="3"> n</td></tr> +<tr><td style="text-align: right"> (4) V = c ---; </td><td style="text-align: center" colspan="3"> log c = .49670</td></tr> +<tr><td></td><td style="text-align: center" colspan="3"> φ</td></tr> +<tr><td> </td></tr> +<tr><td colspan="4"> D and r are expressed in feet and d′ in millimeters.</td></tr> +<tr><td colspan="4"> Vt<sub>3</sub> fork makes 256.070 vibrations per second at 65° Fahr.</td></tr> +<tr><td style="text-align: right"> D = </td><td colspan="3">3972.46 feet.</td></tr> +<tr><td style="text-align: right"> tan α =</td><td colspan="3"> tangent of angle of inclination of plane of rotation = 0.02 + in all but the last twelve observations, in which it was 0.015.</td></tr> +<tr><td style="text-align: right"> log c′ =</td><td colspan="3"> .51607 (.51603 in last twelve observations.).</td></tr> +<tr><td style="text-align: right"> log c =</td><td colspan="3"> .49670.</td></tr> +</table> +<p>The electric fork makes ½(256.070 + B + cor.) vibrations per second, +and n is a multiple, submultiple, or simple ratio of this.</p> +</div> + + +<div class="chapter" id="ch06"> +<h2>Observations.</h2> + + + +<h3>Specimen Observation.</h3> + + +<p>June 17. sunset. Image good; best in column (4).</p> + +<p>The columns are sets of readings of the micrometer for the deflected image +of slit.</p> +<table summary="readings of the micrometer for the deflected image of slit"> +<tr><td></td><td class="align-dot"> 112.81</td><td class="align-dot"> 112.80</td><td class="align-dot"> 112.83 </td><td class="align-dot"> 112.74 </td><td class="align-dot"> 112.79</td></tr> +<tr><td></td><td class="align-dot"> 81</td><td class="align-dot"> 81 </td><td class="align-dot"> 81 </td><td class="align-dot"> 76 </td><td class="align-dot"> 78</td></tr> +<tr><td></td><td class="align-dot"> 79 </td><td class="align-dot"> 78 </td><td class="align-dot"> 78 </td><td class="align-dot"> 74 </td><td class="align-dot"> 74</td></tr> +<tr><td></td><td class="align-dot"> 80 </td><td class="align-dot"> 75 </td><td class="align-dot"> 74 </td><td class="align-dot"> 76 </td><td class="align-dot"> 74</td></tr> +<tr><td></td><td class="align-dot"> 79 </td><td class="align-dot"> 77 </td><td class="align-dot"> 74 </td><td class="align-dot"> 76 </td><td class="align-dot"> 77</td></tr> +<tr><td></td><td class="align-dot"> 82</td><td class="align-dot"> 79 </td><td class="align-dot"> 72 </td><td class="align-dot"> 78 </td><td class="align-dot"> 81</td></tr> +<tr><td></td><td class="align-dot"> 82</td><td class="align-dot"> 73 </td><td class="align-dot"> 76 </td><td class="align-dot"> 78 </td><td class="align-dot"> 77</td></tr> +<tr><td></td><td class="align-dot"> 76</td><td class="align-dot"> 78 </td><td class="align-dot"> 81 </td><td class="align-dot"> 79 </td><td class="align-dot"> 75</td></tr> +<tr><td></td><td class="align-dot"> 83 </td><td class="align-dot"> 79 </td><td class="align-dot"> 74 </td><td class="align-dot"> 83 </td><td class="align-dot"> 82</td></tr> +<tr><td></td><td class="align-dot"> 73</td><td class="align-dot"> 73 </td><td class="align-dot"> 76 </td><td class="align-dot"> 78 </td><td class="align-dot"> 82</td></tr> +<tr><td></td><td style="text-align: center"> -------</td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> -------</td></tr> +<tr><td style="text-align: right"> Mean =</td><td class="align-dot"> 112.801 </td><td class="align-dot"> 112.773 </td><td class="align-dot"> 112.769 </td><td class="align-dot"> 112.772 </td><td class="align-dot"> 112.779</td></tr> +<tr><td style="text-align: right"> Zero =</td><td class="align-dot"> 0.260 </td><td class="align-dot"> 0.260 </td><td class="align-dot"> 0.260 </td><td class="align-dot">0.260 </td><td class="align-dot"> 0.260</td></tr> +<tr><td></td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> -------</td></tr> +<tr><td style="text-align: right"> d = </td><td class="align-dot">112.451 </td><td class="align-dot"> 112.513 </td><td class="align-dot"> 112.509 </td><td class="align-dot"> 112.512 </td><td class="align-dot"> 112.519</td></tr> +<tr><td style="text-align: right"> Temp =</td><td class="align-dot"> 77° </td><td class="align-dot"> 77° </td><td class="align-dot"> 77° </td><td class="align-dot"> 77° </td><td class="align-dot"> 77°</td></tr> +<tr><td style="text-align: right"> B =</td><td class="align-dot"> + 1.500</td></tr> +<tr><td style="text-align: right"> Corr =</td><td class="align-dot"> - .144</td></tr> +<tr><td></td><td style="text-align: center"> -------</td></tr> +<tr><td></td><td class="align-dot"> + 1.365</td></tr> +<tr><td></td><td class="align-dot"> 256.070</td></tr> +<tr><td></td><td style="text-align: center"> -------</td></tr> +<tr><td style="text-align: right"> n =</td><td class="align-dot"> 257.426 </td><td class="align-dot"> 257.43 </td><td class="align-dot"> 257.43 </td><td class="align-dot"> 257.43 </td><td class="align-dot"> 257.43</td></tr> +<tr><td style="text-align: right"> r = </td><td class="align-dot"> 28.157 </td><td class="align-dot"> 28.157 </td><td class="align-dot"> 28.157 </td><td class="align-dot"> 28.157 </td><td class="align-dot"> 28.157</td></tr> +</table> +<p>The above specimen was selected because in it the readings were all taken +by another and noted down without divulging them till the whole five sets +were completed.</p> + +<p>The following is the calculation for V:</p> +<table summary="the calculation for V"> +<tr><th colspan="5"> 2d, 3d,</th></tr> +<tr><th></th><th> 1st set. </th><th> and 4th sets. </th><th> 5th set.</th></tr> +<tr><td> log </td><td> c′ = </td><td>51607 </td><td> 51607 </td><td> 51607</td></tr> +<tr><td> " </td><td> T = </td><td>99832 </td><td> 99832 </td><td> 99832</td></tr> +<tr><td> " </td><td> d = </td><td>05131 </td><td> 05119 </td><td> 05123</td></tr> +<tr><td></td><td></td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> -------</td></tr> +<tr><td></td><td></td><td> 56570 </td><td> 56558 </td><td> 56562</td></tr> +<tr><td> " </td><td> r = </td><td>44958 </td><td> 44958 </td><td> 44958</td></tr> +<tr><td></td><td></td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> -------</td></tr> +<tr><td> " </td><td> tan φ =</td><td> 11612 </td><td> 11600 </td><td> 11604</td></tr> +<tr><td></td><td> φ =</td><td> 2694″.7 </td><td> 2694″.1 </td><td> 2694″.3</td></tr> +<tr><td> " </td><td> c = </td><td>49670 </td><td> 49670 </td><td> 49670</td></tr> +<tr><td> " </td><td> n = </td><td>41066 </td><td> 41066 </td><td> 41066</td></tr> +<tr><td></td><td></td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> -------</td></tr> +<tr><td></td><td></td><td> 90736 </td><td> 90736 </td><td> 90736</td></tr> +<tr><td> " </td><td> φ = </td><td>43052 </td><td> 43042 </td><td> 43046</td></tr> +<tr><td></td><td></td><td style="text-align: center"> ------- </td><td style="text-align: center"> ------- </td><td style="text-align: center"> -------</td></tr> +<tr><td> " </td><td> V = </td><td>47684 </td><td> 47694 </td><td> 47690</td></tr> +<tr><td></td><td> V = </td><td>299800 </td><td> 299880 </td><td> 299850</td></tr> +</table> +<p>In the following table, the numbers in the column headed "Distinctness of +Image" are thus translated: 3, good; 2, fair; 1, poor. These numbers do +not, however, show the relative weights of the observations.</p> + +<p>The numbers contained in the columns headed "Position of Deflected Image," +"Position of Slit," and displacement of image in divisions were obtained +as described in the paragraph headed "Micrometer," page 120.</p> + +<p>The column headed "B" contains the number of "beats" per second between +the electric Vt₂ fork and the standard Vt<sub>3</sub> as explained in the paragraph +headed "Measurement of the Speed of Rotation." The column headed "Cor." +contains the correction of the rate of the standard fork for the +difference in temperature of experiment and 65° Fahr., for which +temperature the rate was found. The numbers in the column headed "Number +of revolutions per second" were found by applying the corrections in the +two preceding columns to the rate of the standard, as explained in the +same paragraph.</p> + +<p>The "radius of measurement" is the distance between the front face of the +revolving mirror and the cross-hair of the micrometer.</p> + +<p>The numbers in the column headed "Value of one turn of the screw" were +taken from the table, page 127.</p> +<table summary="big table" border="1"> +<tr><th> Date.</th> +<th> Distinctness of image.</th> +<th> Temperature, Fahr.</th> +<th> Position of deflected image.</th> +<th> Position of slit.</th> +<th> Displacement of image in divisions.</th> +<th> Difference between greatest and least values.</th> +<th> B.</th> +<th> Cor.</th> +<th> Number of revolutions per second.</th> +<th> Radius of measurement, in feet.</th> +<th> Value of one turn of the screw.</th> +<th> Velocity of light in air, in kilometers.</th> +<th> Remarks.</th> +</tr> + <tr><td>June 5</td><td class="align-dot">3</td><td class="align-dot">76</td><td class="align-dot">114.85</td><td class="align-dot"> 0.300</td><td class="align-dot">114.55</td><td class="align-dot">0.17</td><td class="align-dot">1.423</td><td class="align-dot">-0.132</td><td class="align-dot">257.36</td><td class="align-dot">28.672</td><td class="align-dot">0.99614</td><td class="align-dot">299850</td><td>Electric light.</td></tr> + <tr><td>June 7</td><td class="align-dot">2</td><td class="align-dot">72</td><td class="align-dot">114.64</td><td class="align-dot"> 0.074</td><td class="align-dot">114.56</td><td class="align-dot">0.10</td><td class="align-dot">1.533</td><td class="align-dot">-0.084</td><td class="align-dot">257.52</td><td class="align-dot">28.655</td><td class="align-dot">0.99614</td><td class="align-dot">299740</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 7</td><td class="align-dot">2</td><td class="align-dot">72</td><td class="align-dot">114.58</td><td class="align-dot"> 0.074</td><td class="align-dot">114.50</td><td class="align-dot">0.08</td><td class="align-dot">1.533</td><td class="align-dot">-0.084</td><td class="align-dot">257.52</td><td class="align-dot">28.647</td><td class="align-dot">0.99614</td><td class="align-dot">299900</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 7</td><td class="align-dot">2</td><td class="align-dot">72</td><td class="align-dot"> 85.91</td><td class="align-dot"> 0.074</td><td class="align-dot"> 85.84</td><td class="align-dot">0.12</td><td class="align-dot">1.533</td><td class="align-dot">-0.084</td><td class="align-dot">193.14</td><td class="align-dot">28.647</td><td class="align-dot">0.99598</td><td class="align-dot">300070</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 7</td><td class="align-dot">2</td><td class="align-dot">72</td><td class="align-dot"> 85.97</td><td class="align-dot"> 0.074</td><td class="align-dot"> 85.89</td><td class="align-dot">O.07</td><td class="align-dot">1.533</td><td class="align-dot">-0.084</td><td class="align-dot">193.14</td><td class="align-dot">28.650</td><td class="align-dot">0.99598</td><td class="align-dot">299930</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 7</td><td class="align-dot">2</td><td class="align-dot">72</td><td class="align-dot">114.61</td><td class="align-dot"> 0.074</td><td class="align-dot">114-53</td><td class="align-dot">0.07</td><td class="align-dot">1.533</td><td class="align-dot">-0.084</td><td class="align-dot">257.42</td><td class="align-dot">28.650</td><td class="align-dot">0.99614</td><td class="align-dot">299850</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 9</td><td class="align-dot">3</td><td class="align-dot">83</td><td class="align-dot">114.54</td><td class="align-dot"> 0.074</td><td class="align-dot">114.47</td><td class="align-dot">0.07</td><td class="align-dot">1.533</td><td class="align-dot">-0.216</td><td class="align-dot">257.39</td><td class="align-dot">28.658</td><td class="align-dot">0.99614</td><td class="align-dot">299950</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 9</td><td class="align-dot">3</td><td class="align-dot">83</td><td class="align-dot">114.54</td><td class="align-dot"> 0.074</td><td class="align-dot">114.46</td><td class="align-dot">0.10</td><td class="align-dot">1.533</td><td class="align-dot">-0.216</td><td class="align-dot">257.39</td><td class="align-dot">28.658</td><td class="align-dot">0.99614</td><td class="align-dot">299980</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 9</td><td class="align-dot">3</td><td class="align-dot">83</td><td class="align-dot">114.57</td><td class="align-dot"> 0.074</td><td class="align-dot">114.47</td><td class="align-dot">0.08</td><td class="align-dot">1.533</td><td class="align-dot">-0.216</td><td class="align-dot">257.39</td><td class="align-dot">28.662</td><td class="align-dot">0.99614</td><td class="align-dot">299980</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 9</td><td class="align-dot">3</td><td class="align-dot">83</td><td class="align-dot">114.57</td><td class="align-dot"> 0.074</td><td class="align-dot">114.50</td><td class="align-dot">0.06</td><td class="align-dot">1.533</td><td class="align-dot">-0.216</td><td class="align-dot">257.39</td><td class="align-dot">28.660</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 9</td><td class="align-dot">2</td><td class="align-dot">83</td><td class="align-dot">114.61</td><td class="align-dot"> 0.074</td><td class="align-dot">114.53</td><td class="align-dot">0.13</td><td class="align-dot">1.533</td><td class="align-dot">-0.216</td><td class="align-dot">257.39</td><td class="align-dot">28.678</td><td class="align-dot">0.99614</td><td class="align-dot">300000</td><td>P.M. Frame inclined at various angles</td></tr> + <tr><td>June 10</td><td class="align-dot">2</td><td class="align-dot">90</td><td class="align-dot">114.60</td><td class="align-dot"> 0.074</td><td class="align-dot">114.52</td><td class="align-dot">0.11</td><td class="align-dot">1.517</td><td class="align-dot">-0.300</td><td class="align-dot">257.29</td><td class="align-dot">28.685</td><td class="align-dot">0.99614</td><td class="align-dot">299980</td><td>P.M.</td></tr> + <tr><td>June 10</td><td class="align-dot">2</td><td class="align-dot">90</td><td class="align-dot">114.62</td><td class="align-dot"> 0.074</td><td class="align-dot">114.54</td><td class="align-dot">0.08</td><td class="align-dot">1.517</td><td class="align-dot">-0.300</td><td class="align-dot">257.29</td><td class="align-dot">28.685</td><td class="align-dot">0.99614</td><td class="align-dot">299930</td><td>P.M.</td></tr> + <tr><td>June 12</td><td class="align-dot">2</td><td class="align-dot">71</td><td class="align-dot">114.81</td><td class="align-dot"> 0.074</td><td class="align-dot">114.74</td><td class="align-dot">0.09</td><td class="align-dot">1.450</td><td class="align-dot">-0.072</td><td class="align-dot">257.45</td><td class="align-dot">28.690</td><td class="align-dot">0.99614</td><td class="align-dot">299650</td><td>A.M.</td></tr> + <tr><td>June 12</td><td class="align-dot">2</td><td class="align-dot">71</td><td class="align-dot">114.78</td><td class="align-dot"> 0.074</td><td class="align-dot">114.70</td><td class="align-dot">0.05</td><td class="align-dot">1.450</td><td class="align-dot">-0.072</td><td class="align-dot">257.45</td><td class="align-dot">28.690</td><td class="align-dot">0.99614</td><td class="align-dot">299760</td><td>A.M.</td></tr> + <tr><td>June 12</td><td class="align-dot">1</td><td class="align-dot">71</td><td class="align-dot">114.76</td><td class="align-dot"> 0.074</td><td class="align-dot">114.68</td><td class="align-dot">0.09</td><td class="align-dot">1.450</td><td class="align-dot">-0.072</td><td class="align-dot">257.45</td><td class="align-dot">28.690</td><td class="align-dot">0.99614</td><td class="align-dot">299810</td><td>A.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">72</td><td class="align-dot">112.64</td><td class="align-dot"> 0.074</td><td class="align-dot">112.56</td><td class="align-dot">0.09</td><td class="align-dot">1.500</td><td class="align-dot">-0.084</td><td class="align-dot">257.49</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">300000</td><td>A.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">72</td><td class="align-dot">112.63</td><td class="align-dot"> 0.074</td><td class="align-dot">112.56</td><td class="align-dot">0.10</td><td class="align-dot">1.500</td><td class="align-dot">-0.084</td><td class="align-dot">257.49</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">300000</td><td>A.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">2</td><td class="align-dot">72</td><td class="align-dot">112.65</td><td class="align-dot"> 0.074</td><td class="align-dot">112.57</td><td class="align-dot">0.08</td><td class="align-dot">1.500</td><td class="align-dot">-0.084</td><td class="align-dot">257.49</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">299960</td><td>A.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">112.82</td><td class="align-dot"> 0.260</td><td class="align-dot">112.56</td><td class="align-dot">0.06</td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">257.42</td><td class="align-dot">28.178</td><td class="align-dot">0.99614</td><td class="align-dot">299960</td><td>P.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">112.82</td><td class="align-dot"> 0.260</td><td class="align-dot">112.56</td><td class="align-dot">0.13</td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">257.42</td><td class="align-dot">28.178</td><td class="align-dot">0.99614</td><td class="align-dot">299960</td><td>P.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">112.83</td><td class="align-dot"> 0.260</td><td class="align-dot">112.57</td><td class="align-dot">0.07</td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">257.42</td><td class="align-dot">28.178</td><td class="align-dot">0.99614</td><td class="align-dot">299940</td><td>P.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">112.82</td><td class="align-dot"> 0.260</td><td class="align-dot">112.56</td><td class="align-dot">0.06</td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">257.42</td><td class="align-dot">28.178</td><td class="align-dot">0.99614</td><td class="align-dot">299960</td><td>P.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">112.83</td><td class="align-dot"> 0.260</td><td class="align-dot">112.57</td><td class="align-dot">0.11</td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">257.42</td><td class="align-dot">28.178</td><td class="align-dot">0.99614</td><td class="align-dot">299940</td><td>P.M.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">113.41</td><td class="align-dot"> 0.260</td><td class="align-dot">113.15</td><td class="align-dot">11 </td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">258.70</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M. Set micrometer and counted oscillations.</td></tr> + <tr><td>June 13</td><td class="align-dot">3</td><td class="align-dot">79</td><td class="align-dot">112.14</td><td class="align-dot"> 0.260</td><td class="align-dot">111.88</td><td class="align-dot">6 </td><td class="align-dot">1.517</td><td class="align-dot">-0.168</td><td class="align-dot">255.69</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299800</td><td>Oscillations of image of revolving mirror.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">64</td><td class="align-dot">112.83</td><td class="align-dot"> 0.260</td><td class="align-dot">112.57</td><td class="align-dot">0.12</td><td class="align-dot">1.500</td><td class="align-dot">+0.012</td><td class="align-dot">257.58</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299850</td><td>A.M.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">64</td><td class="align-dot">112.83</td><td class="align-dot"> 0.260</td><td class="align-dot">112.57</td><td class="align-dot">0.05</td><td class="align-dot">1.517</td><td class="align-dot">+0.012</td><td class="align-dot">257.60</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>A.M.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">65</td><td class="align-dot">112.81</td><td class="align-dot"> 0.260</td><td class="align-dot">112.55</td><td class="align-dot">0.11</td><td class="align-dot">1.517</td><td class="align-dot"> 0.000</td><td class="align-dot">257.59</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299900</td><td>A.M.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">66</td><td class="align-dot">112.83</td><td class="align-dot"> 0.260</td><td class="align-dot">112.57</td><td class="align-dot">0.09</td><td class="align-dot">1.517</td><td class="align-dot">-0.012</td><td class="align-dot">257.57</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299840</td><td>A.M.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">67</td><td class="align-dot">112.83</td><td class="align-dot"> 0.260</td><td class="align-dot">112.57</td><td class="align-dot">0.12</td><td class="align-dot">1.517</td><td class="align-dot">-0.024</td><td class="align-dot">257.56</td><td class="align-dot">28.152</td><td class="align-dot">0.99614</td><td class="align-dot">299830</td><td>A.M.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">84</td><td class="align-dot">112.78</td><td class="align-dot"> 0.260</td><td class="align-dot">112.52</td><td class="align-dot">0.06</td><td class="align-dot">1.517</td><td class="align-dot">-0.228</td><td class="align-dot">257.36</td><td class="align-dot">28.159</td><td class="align-dot">0.99614</td><td class="align-dot">299790</td><td>P.M. Readings taken by Lieut. Nazro.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">85</td><td class="align-dot">112.76</td><td class="align-dot"> 0.260</td><td class="align-dot">112.50</td><td class="align-dot">0.08</td><td class="align-dot">1.500</td><td class="align-dot">-0.240</td><td class="align-dot">257.33</td><td class="align-dot">28.159</td><td class="align-dot">0.99614</td><td class="align-dot">299810</td><td>P.M. Readings taken by Lieut. Nazro.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">84</td><td class="align-dot">112.72</td><td class="align-dot"> 0.260</td><td class="align-dot">112.46</td><td class="align-dot">0.08</td><td class="align-dot">1.483</td><td class="align-dot">-0.228</td><td class="align-dot">257.32</td><td class="align-dot">28.159</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M. Readings taken by Lieut. Nazro.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">84</td><td class="align-dot">112.73</td><td class="align-dot"> 0.260</td><td class="align-dot">112.47</td><td class="align-dot">0.09</td><td class="align-dot">1.483</td><td class="align-dot">-0.228</td><td class="align-dot">257.32</td><td class="align-dot">28.159</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M.</td></tr> + <tr><td>June 14</td><td class="align-dot">1</td><td class="align-dot">84</td><td class="align-dot">112.75</td><td class="align-dot"> 0.260</td><td class="align-dot">112.49</td><td class="align-dot">0.09</td><td class="align-dot">1.483</td><td class="align-dot">-0.228</td><td class="align-dot">257.32</td><td class="align-dot">28.129</td><td class="align-dot">0.99614</td><td class="align-dot">299830</td><td>P.M.</td></tr> + <tr><td>June 17</td><td class="align-dot">2</td><td class="align-dot">62</td><td class="align-dot">112.85</td><td class="align-dot"> 0.260</td><td class="align-dot">112.59</td><td class="align-dot">0.09</td><td class="align-dot">1.517</td><td class="align-dot">+0.036</td><td class="align-dot">257.62</td><td class="align-dot">28.149</td><td class="align-dot">0.99614</td><td class="align-dot">299800</td><td>A.M.</td></tr> + <tr><td>June 17</td><td class="align-dot">2</td><td class="align-dot">63</td><td class="align-dot">112.84</td><td class="align-dot"> 0.260</td><td class="align-dot">112.58</td><td class="align-dot">0.06</td><td class="align-dot">1.500</td><td class="align-dot">+0.024</td><td class="align-dot">257.59</td><td class="align-dot">28.149</td><td class="align-dot">0.99614</td><td class="align-dot">299790</td><td>A.M.</td></tr> + <tr><td>June 17</td><td class="align-dot">1</td><td class="align-dot">64</td><td class="align-dot">112.85</td><td class="align-dot"> 0.260</td><td class="align-dot">112.59</td><td class="align-dot">0.07</td><td class="align-dot">1.500</td><td class="align-dot">+0.012</td><td class="align-dot">257.58</td><td class="align-dot">28.149</td><td class="align-dot">0.99614</td><td class="align-dot">299760</td><td>A.M.</td></tr> + <tr><td>June 17</td><td class="align-dot">3</td><td class="align-dot">77</td><td class="align-dot">112.80</td><td class="align-dot"> 0.260</td><td class="align-dot">112.54</td><td class="align-dot">0.07</td><td class="align-dot">1.500</td><td class="align-dot">-0.144</td><td class="align-dot">257-43</td><td class="align-dot">28.157</td><td class="align-dot">0.99614</td><td class="align-dot">299800</td><td>P.M. Readings taken by Mr. Clason.</td></tr> + <tr><td>June 17</td><td class="align-dot">3</td><td class="align-dot">77</td><td class="align-dot">112.77</td><td class="align-dot"> 0.260</td><td class="align-dot">112.51</td><td class="align-dot">0.08</td><td class="align-dot">1.500</td><td class="align-dot">-0.144</td><td class="align-dot">257.43</td><td class="align-dot">28.157</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M. Readings taken by Mr. Clason.</td></tr> + <tr><td>June 17</td><td class="align-dot">3</td><td class="align-dot">77</td><td class="align-dot">112.77</td><td class="align-dot"> 0.260</td><td class="align-dot">112.51</td><td class="align-dot">0.11</td><td class="align-dot">1.500</td><td class="align-dot">-0.144</td><td class="align-dot">257.43</td><td class="align-dot">28.157</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M. Readings taken by Mr. Clason.</td></tr> + <tr><td>June 17</td><td class="align-dot">3</td><td class="align-dot">77</td><td class="align-dot">112.77</td><td class="align-dot"> 0.260</td><td class="align-dot">112.51</td><td class="align-dot">0.09</td><td class="align-dot">1.500</td><td class="align-dot">-0.144</td><td class="align-dot">257.43</td><td class="align-dot">28.157</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>P.M. Readings taken by Mr. Clason.</td></tr> + <tr><td>June 17</td><td class="align-dot">3</td><td class="align-dot">77</td><td class="align-dot">112.78</td><td class="align-dot"> 0.260</td><td class="align-dot">112.52</td><td class="align-dot">0.08</td><td class="align-dot">1.500</td><td class="align-dot">-0.144</td><td class="align-dot">257 43</td><td class="align-dot">28.157</td><td class="align-dot">0.99614</td><td class="align-dot">299860</td><td>P.M. Readings taken by Mr. Clason.</td></tr> + <tr><td>June 18</td><td class="align-dot">1</td><td class="align-dot">58</td><td class="align-dot">112.90</td><td class="align-dot"> 0.265</td><td class="align-dot">112.64</td><td class="align-dot">0.07</td><td class="align-dot">1.500</td><td class="align-dot">+0.084</td><td class="align-dot">257.65</td><td class="align-dot">28.150</td><td class="align-dot">0.99614</td><td class="align-dot">299720</td><td>A.M.</td></tr> + <tr><td>June 18</td><td class="align-dot">1</td><td class="align-dot">58</td><td class="align-dot">112.90</td><td class="align-dot"> 0.265</td><td class="align-dot">112.64</td><td class="align-dot">0.10</td><td class="align-dot">1.500</td><td class="align-dot">+0.084</td><td class="align-dot">257.65</td><td class="align-dot">28.150</td><td class="align-dot">0.99614</td><td class="align-dot">299720</td><td>A.M.</td></tr> + <tr><td>June 18</td><td class="align-dot">1</td><td class="align-dot">59</td><td class="align-dot">112.92</td><td class="align-dot"> 0.265</td><td class="align-dot">112.66</td><td class="align-dot">0.07</td><td class="align-dot">1.483</td><td class="align-dot">+0.072</td><td class="align-dot">257.62</td><td class="align-dot">28.150</td><td class="align-dot">0.99614</td><td class="align-dot">299620</td><td>A.M.</td></tr> + <tr><td>June 18</td><td class="align-dot">2</td><td class="align-dot">75</td><td class="align-dot">112.79</td><td class="align-dot"> 0.265</td><td class="align-dot">112.52</td><td class="align-dot">0.09</td><td class="align-dot">1.483</td><td class="align-dot">-0.120</td><td class="align-dot">257-43</td><td class="align-dot">28.158</td><td class="align-dot">0.99614</td><td class="align-dot">299860</td><td>P.M.</td></tr> + <tr><td>June 18</td><td class="align-dot">2</td><td class="align-dot">75</td><td class="align-dot">112.75</td><td class="align-dot"> 0.265</td><td class="align-dot">112.48</td><td class="align-dot">0.10</td><td class="align-dot">1.483</td><td class="align-dot">-0.120</td><td class="align-dot">257-43</td><td class="align-dot">28.158</td><td class="align-dot">0.99614</td><td class="align-dot">299970</td><td>P.M.</td></tr> + <tr><td>June 18</td><td class="align-dot">2</td><td class="align-dot">75</td><td class="align-dot">112.76</td><td class="align-dot"> 0.265</td><td class="align-dot">112.49</td><td class="align-dot">0.08</td><td class="align-dot">1.483</td><td class="align-dot">-0.120</td><td class="align-dot">257-43</td><td class="align-dot">28.158</td><td class="align-dot">0.99614</td><td class="align-dot">299950</td><td>P.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">3</td><td class="align-dot">60</td><td class="align-dot">112.94</td><td class="align-dot"> 0.265</td><td class="align-dot">112.67</td><td class="align-dot">0.07</td><td class="align-dot">1.517</td><td class="align-dot">+0.063</td><td class="align-dot">257.65</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">299880</td><td>A.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">3</td><td class="align-dot">61</td><td class="align-dot">112.92</td><td class="align-dot"> 0.265</td><td class="align-dot">112.65</td><td class="align-dot">0.09</td><td class="align-dot">1.517</td><td class="align-dot">+0.048</td><td class="align-dot">257.63</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">299910</td><td>A.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">62</td><td class="align-dot">112.94</td><td class="align-dot"> 0.265</td><td class="align-dot">112.67</td><td class="align-dot">0.07</td><td class="align-dot">1.517</td><td class="align-dot">+0.036</td><td class="align-dot">257.62</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">299850</td><td>A.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">63</td><td class="align-dot">112.93</td><td class="align-dot"> 0.265</td><td class="align-dot">112.66</td><td class="align-dot">0.03</td><td class="align-dot">1.517</td><td class="align-dot">+0.024</td><td class="align-dot">257.61</td><td class="align-dot">28.172</td><td class="align-dot">0.99614</td><td class="align-dot">299870</td><td>A.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">78</td><td class="align-dot">133.48</td><td class="align-dot"> 0.265</td><td class="align-dot">133.21</td><td class="align-dot">0.13</td><td class="align-dot">1.450</td><td class="align-dot">-0.156</td><td class="align-dot">257.36</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>P.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">79</td><td class="align-dot">133.49</td><td class="align-dot"> 0.265</td><td class="align-dot">133.23</td><td class="align-dot">0.09</td><td class="align-dot">1.500</td><td class="align-dot">-0.168</td><td class="align-dot">257.40</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>P.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">80</td><td class="align-dot">133.49</td><td class="align-dot"> 0.265</td><td class="align-dot">133.22</td><td class="align-dot">0.07</td><td class="align-dot">1.500</td><td class="align-dot">-0.180</td><td class="align-dot">257.39</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299850</td><td>P.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">79</td><td class="align-dot">133.50</td><td class="align-dot"> 0.265</td><td class="align-dot">133.24</td><td class="align-dot">0.13</td><td class="align-dot">1.483</td><td class="align-dot">-0.168</td><td class="align-dot">257.39</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>P.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">79</td><td class="align-dot">133.49</td><td class="align-dot"> 0.265</td><td class="align-dot">133.22</td><td class="align-dot">0.06</td><td class="align-dot">1.483</td><td class="align-dot">-0.168</td><td class="align-dot">257.38</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>P.M.</td></tr> + <tr><td>June 20</td><td class="align-dot">2</td><td class="align-dot">79</td><td class="align-dot">133.49</td><td class="align-dot"> 0.265</td><td class="align-dot">133.22</td><td class="align-dot">0.10</td><td class="align-dot">1.483</td><td class="align-dot">-0.168</td><td class="align-dot">257.38</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>P.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">2</td><td class="align-dot">61</td><td class="align-dot">133.56</td><td class="align-dot"> 0.265</td><td class="align-dot">133.29</td><td class="align-dot">0.12</td><td class="align-dot">1.533</td><td class="align-dot">+0.048</td><td class="align-dot">257.65</td><td class="align-dot">33.332</td><td class="align-dot">0.99627</td><td class="align-dot">299890</td><td>A.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">2</td><td class="align-dot">62</td><td class="align-dot">133.58</td><td class="align-dot"> 0.265</td><td class="align-dot">133.31</td><td class="align-dot">0.08</td><td class="align-dot">1.533</td><td class="align-dot">+0.036</td><td class="align-dot">257.64</td><td class="align-dot">33.332</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>A.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">2</td><td class="align-dot">63</td><td class="align-dot">133.57</td><td class="align-dot"> 0.265</td><td class="align-dot">133.31</td><td class="align-dot">0.09</td><td class="align-dot">1.533</td><td class="align-dot">+0.024</td><td class="align-dot">257.63</td><td class="align-dot">33.332</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>A.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">2</td><td class="align-dot">64</td><td class="align-dot">133.57</td><td class="align-dot"> 0.265</td><td class="align-dot">133.30</td><td class="align-dot">0.11</td><td class="align-dot">1.533</td><td class="align-dot">+0.012</td><td class="align-dot">257.61</td><td class="align-dot">33.332</td><td class="align-dot">0.99627</td><td class="align-dot">299820</td><td>A.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">2</td><td class="align-dot">65</td><td class="align-dot">133.56</td><td class="align-dot"> 0.265</td><td class="align-dot">133.30</td><td class="align-dot">0.13</td><td class="align-dot">1.533</td><td class="align-dot"> 0.000</td><td class="align-dot">257.60</td><td class="align-dot">33.332</td><td class="align-dot">0.99627</td><td class="align-dot">299800</td><td>A.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">3</td><td class="align-dot">80</td><td class="align-dot">133.48</td><td class="align-dot"> 0.265</td><td class="align-dot">133.21</td><td class="align-dot">0.06</td><td class="align-dot">1.533</td><td class="align-dot">-0.180</td><td class="align-dot">257.42</td><td class="align-dot">33.330</td><td class="align-dot">0.99627</td><td class="align-dot">299770</td><td>P.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">3</td><td class="align-dot">81</td><td class="align-dot">133.46</td><td class="align-dot"> 0.265</td><td class="align-dot">133.19</td><td class="align-dot">0.10</td><td class="align-dot">1.500</td><td class="align-dot">-0.192</td><td class="align-dot">257.38</td><td class="align-dot">33.330</td><td class="align-dot">0.99627</td><td class="align-dot">299760</td><td>P.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">3</td><td class="align-dot">82</td><td class="align-dot">133.46</td><td class="align-dot"> 0.265</td><td class="align-dot">133.20</td><td class="align-dot">0.05</td><td class="align-dot">1.500</td><td class="align-dot">-0.204</td><td class="align-dot">257.37</td><td class="align-dot">33.330</td><td class="align-dot">0.99627</td><td class="align-dot">299740</td><td>P.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">3</td><td class="align-dot">82</td><td class="align-dot">133.46</td><td class="align-dot"> 0.265</td><td class="align-dot">133.20</td><td class="align-dot">0.08</td><td class="align-dot">1.517</td><td class="align-dot">-0.204</td><td class="align-dot">257.38</td><td class="align-dot">33.330</td><td class="align-dot">0.99627</td><td class="align-dot">299750</td><td>P.M.</td></tr> + <tr><td>June 21</td><td class="align-dot">3</td><td class="align-dot">81</td><td class="align-dot">133.46</td><td class="align-dot"> 0.265</td><td class="align-dot">133.19</td><td class="align-dot">0.08</td><td class="align-dot">1.500</td><td class="align-dot">-0.192</td><td class="align-dot">257.38</td><td class="align-dot">33.330</td><td class="align-dot">0.99627</td><td class="align-dot">299760</td><td>P.M.</td></tr> + <tr><td>June 23</td><td class="align-dot">3</td><td class="align-dot">89</td><td class="align-dot">133.43</td><td class="align-dot"> 0.265</td><td class="align-dot">133.16</td><td class="align-dot">0.08</td><td class="align-dot">1.542</td><td class="align-dot">-0.288</td><td class="align-dot">257.32</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299910</td><td>P.M.</td></tr> + <tr><td>June 23</td><td class="align-dot">3</td><td class="align-dot">89</td><td class="align-dot">133.42</td><td class="align-dot"> 0.265</td><td class="align-dot">133.15</td><td class="align-dot">0.06</td><td class="align-dot">1.550</td><td class="align-dot">-0.288</td><td class="align-dot">257.33</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299920</td><td>P.M.</td></tr> + <tr><td>June 23</td><td class="align-dot">3</td><td class="align-dot">90</td><td class="align-dot">133.43</td><td class="align-dot"> 0.265</td><td class="align-dot">133.17</td><td class="align-dot">0.09</td><td class="align-dot">1.550</td><td class="align-dot">-0.300</td><td class="align-dot">257.32</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299890</td><td>P.M.</td></tr> + <tr><td>June 23</td><td class="align-dot">3</td><td class="align-dot">90</td><td class="align-dot">133.43</td><td class="align-dot"> 0.265</td><td class="align-dot">133.16</td><td class="align-dot">0.07</td><td class="align-dot">1.533</td><td class="align-dot">-0.300</td><td class="align-dot">257.30</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299860</td><td>P.M.</td></tr> + <tr><td>June 23</td><td class="align-dot">3</td><td class="align-dot">90</td><td class="align-dot">133.42</td><td class="align-dot"> 0.265</td><td class="align-dot">133.16</td><td class="align-dot">0.07</td><td class="align-dot">1.517</td><td class="align-dot">-0.300</td><td class="align-dot">257.29</td><td class="align-dot">33.345</td><td class="align-dot">0.99627</td><td class="align-dot">299880</td><td>P.M.</td></tr> + <tr><td>June 24</td><td class="align-dot">3</td><td class="align-dot">72</td><td class="align-dot">133.47</td><td class="align-dot"> 0.265</td><td class="align-dot">133.20</td><td class="align-dot">0.15</td><td class="align-dot">1.517</td><td class="align-dot">-0.084</td><td class="align-dot">257.50</td><td class="align-dot">33.319</td><td class="align-dot">0.99627</td><td class="align-dot">299720</td><td>A.M.</td></tr> + <tr><td>June 24</td><td class="align-dot">3</td><td class="align-dot">73</td><td class="align-dot">133.44</td><td class="align-dot"> 0.265</td><td class="align-dot">133.17</td><td class="align-dot">0.04</td><td class="align-dot">1.517</td><td class="align-dot">-0.096</td><td class="align-dot">257.49</td><td class="align-dot">33.319</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>A.M.</td></tr> + <tr><td>June 24</td><td class="align-dot">3</td><td class="align-dot">74</td><td class="align-dot">133.42</td><td class="align-dot"> 0.265</td><td class="align-dot">133.16</td><td class="align-dot">0.11</td><td class="align-dot">1.517</td><td class="align-dot">-0.108</td><td class="align-dot">257.48</td><td class="align-dot">33.319</td><td class="align-dot">0.99627</td><td class="align-dot">299850</td><td>A.M.</td></tr> + <tr><td>June 24</td><td class="align-dot">3</td><td class="align-dot">75</td><td class="align-dot">133.42</td><td class="align-dot"> 0.265</td><td class="align-dot">133.16</td><td class="align-dot">0.06</td><td class="align-dot">1.517</td><td class="align-dot">-0.120</td><td class="align-dot">257.47</td><td class="align-dot">33.319</td><td class="align-dot">0.99627</td><td class="align-dot">299850</td><td>A.M.</td></tr> + <tr><td>June 24</td><td class="align-dot">3</td><td class="align-dot">76</td><td class="align-dot">133.44</td><td class="align-dot"> 0.265</td><td class="align-dot">133.18</td><td class="align-dot">0.10</td><td class="align-dot">1.517</td><td class="align-dot">-0.132</td><td class="align-dot">257.45</td><td class="align-dot">33.319</td><td class="align-dot">0.99627</td><td class="align-dot">299780</td><td>A.M.</td></tr> + <tr><td>June 26</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot">133.42</td><td class="align-dot"> 0.265</td><td class="align-dot">133.15</td><td class="align-dot">0.05</td><td class="align-dot">1.508</td><td class="align-dot">-0.252</td><td class="align-dot">257.33</td><td class="align-dot">33.339</td><td class="align-dot">0.99627</td><td class="align-dot">299890</td><td>P.M.</td></tr> + <tr><td>June 26</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot">133.44</td><td class="align-dot"> 0.265</td><td class="align-dot">133.17</td><td class="align-dot">0.08</td><td class="align-dot">1.508</td><td class="align-dot">-0.252</td><td class="align-dot">257.33</td><td class="align-dot">33.339</td><td class="align-dot">0.99627</td><td class="align-dot">299840</td><td>P.M.</td></tr> + <tr><td>June 27</td><td class="align-dot">3</td><td class="align-dot">73</td><td class="align-dot">133.49</td><td class="align-dot"> 0.265</td><td class="align-dot">133.22</td><td class="align-dot">0.11</td><td class="align-dot">1.483</td><td class="align-dot">-0.096</td><td class="align-dot">257.46</td><td class="align-dot">33.328</td><td class="align-dot">0.99627</td><td class="align-dot">299780</td><td>A.M.</td></tr> + <tr><td>June 27</td><td class="align-dot">3</td><td class="align-dot">74</td><td class="align-dot">133.47</td><td class="align-dot"> 0.265</td><td class="align-dot">133.20</td><td class="align-dot">0.06</td><td class="align-dot">1.483</td><td class="align-dot">-0.108</td><td class="align-dot">257.44</td><td class="align-dot">33.328</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>A.M.</td></tr> + <tr><td>June 27</td><td class="align-dot">3</td><td class="align-dot">75</td><td class="align-dot">133.47</td><td class="align-dot"> 0.265</td><td class="align-dot">133.21</td><td class="align-dot">0.09</td><td class="align-dot">1.483</td><td class="align-dot">-0.120</td><td class="align-dot">257.43</td><td class="align-dot">33.328</td><td class="align-dot">0.99627</td><td class="align-dot">299760</td><td>A.M.</td></tr> + <tr><td>June 27</td><td class="align-dot">3</td><td class="align-dot">75</td><td class="align-dot">133.45</td><td class="align-dot"> 0.265</td><td class="align-dot">133.19</td><td class="align-dot">0.09</td><td class="align-dot">1.467</td><td class="align-dot">-0.120</td><td class="align-dot">257.42</td><td class="align-dot">33.328</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>A.M.</td></tr> + <tr><td>June 27</td><td class="align-dot">3</td><td class="align-dot">76</td><td class="align-dot">133.47</td><td class="align-dot"> 0.265</td><td class="align-dot">133.20</td><td class="align-dot">0.08</td><td class="align-dot">1.483</td><td class="align-dot">-0.132</td><td class="align-dot">257.42</td><td class="align-dot">33.328</td><td class="align-dot">0.99627</td><td class="align-dot">299790</td><td>A.M.</td></tr> + <tr><td>June 27</td><td class="align-dot">3</td><td class="align-dot">76</td><td class="align-dot">133.45</td><td class="align-dot"> 0.265</td><td class="align-dot">133.19</td><td class="align-dot">0.10</td><td class="align-dot">1.483</td><td class="align-dot">-0.132</td><td class="align-dot">257.42</td><td class="align-dot">33.328</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>A.M.</td></tr> + <tr><td>June 30</td><td class="align-dot">2</td><td class="align-dot">85</td><td class="align-dot"> 35.32</td><td class="align-dot">135.00 </td><td class="align-dot"> 99.68</td><td class="align-dot">0.05</td><td class="align-dot">1.500</td><td class="align-dot">-0.240</td><td class="align-dot">193.00</td><td class="align-dot">33.274</td><td class="align-dot">0.99645</td><td class="align-dot">299820</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>June 30</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot"> 35.34</td><td class="align-dot">135.00 </td><td class="align-dot"> 99.67</td><td class="align-dot">0.06</td><td class="align-dot">1.508</td><td class="align-dot">-0.252</td><td class="align-dot">193.00</td><td class="align-dot">33.274</td><td class="align-dot">0.99645</td><td class="align-dot">299850</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>June 30</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot"> 35.34</td><td class="align-dot">135.00 </td><td class="align-dot"> 99.66</td><td class="align-dot">0.10</td><td class="align-dot">1.508</td><td class="align-dot">-0.252</td><td class="align-dot">193.00</td><td class="align-dot">33.274</td><td class="align-dot">0.99645</td><td class="align-dot">299870</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>June 30</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot"> 35.34</td><td class="align-dot">135.00 </td><td class="align-dot"> 99.66</td><td class="align-dot">0.09</td><td class="align-dot">1.517</td><td class="align-dot">-0.252</td><td class="align-dot">193.00</td><td class="align-dot">33.274</td><td class="align-dot">0.99645</td><td class="align-dot">299870</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>July 1</td><td class="align-dot">2</td><td class="align-dot">83</td><td class="align-dot"> 02.17</td><td class="align-dot">135.145</td><td class="align-dot">132.98</td><td class="align-dot">0.07</td><td class="align-dot">1.500</td><td class="align-dot">-0.216</td><td class="align-dot">257.35</td><td class="align-dot">33.282</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>July 1</td><td class="align-dot">2</td><td class="align-dot">84</td><td class="align-dot"> 02.15</td><td class="align-dot">135.145</td><td class="align-dot">133.00</td><td class="align-dot">0.09</td><td class="align-dot">1.500</td><td class="align-dot">-0.228</td><td class="align-dot">257.34</td><td class="align-dot">33.282</td><td class="align-dot">0.99627</td><td class="align-dot">299740</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>July 1</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot"> 02.14</td><td class="align-dot">135.145</td><td class="align-dot">133.01</td><td class="align-dot">0.06</td><td class="align-dot">1.467</td><td class="align-dot">-0.252</td><td class="align-dot">257.28</td><td class="align-dot">33.311</td><td class="align-dot">0.99627</td><td class="align-dot">299810</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>July 1</td><td class="align-dot">2</td><td class="align-dot">86</td><td class="align-dot"> 02.14</td><td class="align-dot">135.145</td><td class="align-dot">133.00</td><td class="align-dot">0.08</td><td class="align-dot">1.467</td><td class="align-dot">-0.252</td><td class="align-dot">257.28</td><td class="align-dot">33.311</td><td class="align-dot">0.99627</td><td class="align-dot">299940</td><td>P.M. Mirror inverted.</td></tr> + <tr><td>July 2</td><td class="align-dot">3</td><td class="align-dot">86</td><td class="align-dot"> 99.85</td><td class="align-dot"> 0.400</td><td class="align-dot"> 99.45</td><td class="align-dot">0.05</td><td class="align-dot">1.450</td><td class="align-dot">-0.252</td><td class="align-dot">192.95</td><td class="align-dot">33.205</td><td class="align-dot">0.99606</td><td class="align-dot">299950</td><td>P.M. Mirror erect.</td></tr> + <tr><td>July 2</td><td class="align-dot">3</td><td class="align-dot">86</td><td class="align-dot"> 66.74</td><td class="align-dot"> 0.400</td><td class="align-dot"> 66.34</td><td class="align-dot">0.03</td><td class="align-dot">1.450</td><td class="align-dot">-0.252</td><td class="align-dot">128.63</td><td class="align-dot">33.205</td><td class="align-dot">0.99586</td><td class="align-dot">299800</td><td>P.M. Mirror erect.</td></tr> + <tr><td>July 2</td><td class="align-dot">3</td><td class="align-dot">86</td><td class="align-dot"> 50.16</td><td class="align-dot"> 0.400</td><td class="align-dot"> 47.96</td><td class="align-dot">0.07</td><td class="align-dot">1.467</td><td class="align-dot">-0.252</td><td class="align-dot"> 96.48</td><td class="align-dot">33.205</td><td class="align-dot">0.99580</td><td class="align-dot">299810</td><td>P.M. Mirror erect.</td></tr> + <tr><td>July 2</td><td class="align-dot">3</td><td class="align-dot">85</td><td class="align-dot"> 33.57</td><td class="align-dot"> 0.400</td><td class="align-dot"> 33.17</td><td class="align-dot">0.06</td><td class="align-dot">1.450</td><td class="align-dot">-0.240</td><td class="align-dot"> 64.32</td><td class="align-dot">33.205</td><td class="align-dot">0.99574</td><td class="align-dot">299870</td><td>P.M. Mirror erect.</td></tr> +</table> +<p>In the last two sets of June 13, the micrometer was fixed at 113.41 and +112.14 respectively. The image was bisected by the cross-hair, and kept as +nearly as possible in this place, meantime counting the number of seconds +required for the image of the revolving mirror to complete 60 +oscillations. In other words, instead of measuring the deflection, the +speed of rotation was measured. In column 7 for these two sets, the +numbers 11 and 6 are the differences between the greatest and the smallest +number of seconds observed.</p> + +<p>In finding the mean value of V from the table, the sets are all given the +same weight. The difference between the result thus obtained and that from +any system of weights is small, and may be neglected.</p> + +<p>The following table gives the result of different groupings of sets of +observations. Necessarily some of the groups include others:</p> +<table summary="results of different groupings of sets of observations"> +<tr><td> Electric light (1 set) </td><td> 299850</td></tr> +<tr><td> Set micrometer counting oscillations (2) </td><td> 299840</td></tr> +<tr><td> Readings taken by Lieutenant Nazro (3) </td><td> 299830</td></tr> +<tr><td> Readings taken by Mr. Clason (5) </td><td> 299860</td></tr> +<tr><td> Mirror inverted (8) </td><td> 299840</td></tr> +<tr><td> Speed of rotation, 192 (7) </td><td> 299990</td></tr> +<tr><td> Speed of rotation, 128 (1) </td><td> 299800</td></tr> +<tr><td> Speed of rotation, 96 (1) </td><td> 299810</td></tr> +<tr><td> Speed of rotation, 64 (1) </td><td> 299870</td></tr> +<tr><td> Radius, 28.5 feet (54) </td><td> 299870</td></tr> +<tr><td> Radius, 33.3 feet (46) </td><td> 299830</td></tr> +<tr><td> Highest temperature, 90° Fahr. (5) </td><td> 299910</td></tr> +<tr><td> Mean of lowest temperatures, 60° Fahr. (7) </td><td> 299800</td></tr> +<tr><td> Image, good (46) </td><td> 299860</td></tr> +<tr><td> Image, fair (39) </td><td> 299860</td></tr> +<tr><td> Image, poor (15) </td><td> 299810</td></tr> +<tr><td> Frame, inclined (5) </td><td> 299960</td></tr> +<tr><td> Greatest value </td><td> 300070</td></tr> +<tr><td> Least value </td><td> 299650</td></tr> +<tr><td> Mean value </td><td> 299852</td></tr> +<tr><td> Average difference from mean </td><td> 60</td></tr> +<tr><td> Value found for π </td><td> 3.26</td></tr> +<tr><td> Probable error </td><td> ± 5</td></tr> +</table> +</div> + +<div class="chapter" id="ch08"> +<h2>Discussion of Errors.</h2> + + +<p>The value of V depends on three quantities D, n, and φ. These will now be +considered in detail.</p> + + + +<h3>The Distance.</h3> + + +<p>The distance between the two mirrors may be in error, either by an +erroneous determination of the length of the steel tape used, or by a +mistake in the measurement of the distance by the tape.</p> + +<p>The first may be caused by an error in the copy of the standard yard, or +in the comparison between the standard and the tape. An error in this +copy, of .00036 inch, which, for such a copy, would be considered large, +would produce an error of only .00001 in the final result. Supposing that +the bisections of the divisions are correct to .0005 inch, which is a +liberal estimate, the error caused by supposing the error in each yard to +be in the same direction would be only .000014; or the total error of the +tape, if both errors were in the same direction, would be 000024 of the +whole length.</p> + +<p>The calculated probable error of the five measurements of the distance +was ±.000015; hence the total error due to D would be at most .00004. The +tape has been sent to Professor Rogers, of Cambridge, for comparison, to +confirm the result.</p> + + + +<h3>The Speed of Rotation.</h3> + + +<p>This quantity depends on three conditions. It is affected, first, by an +error in the rate of the standard; second, by an error in the count of the +sound beats between the forks; and third, by a false estimate of the +moment when the image of the revolving mirror is at rest, at which moment +the deflection is measured.</p> + +<p>The calculated probable error of the rate is .000016. If this rate should +be questioned, the fork can be again rated and a simple correction +applied. The fork is carefully kept at the Stevens Institute, Hoboken, and +comparisons were made with two other forks, in case it was lost or +injured.</p> + +<p>In counting the sound beats, experiments were tried to find if the +vibrations of the standard were affected by the other fork, but no such +effect could be detected. In each case the number of beats was counted +correctly to .02, or less than .0001 part, and in the great number of +comparisons made this source of error could be neglected.</p> + +<p>The error due to an incorrect estimate of the exact time when the images +of the revolving mirror came to rest was eliminated by making the +measurement sometimes when the speed was slowly increasing, and sometimes +when slowly decreasing. Further, this error would form part of the +probable error deduced from the results of observations.</p> + +<p>We may then conclude that the error, in the measurement of <i>n</i>, was less +than .00002.</p> + + + +<h3>The Deflection.</h3> + + +<p>The angle of deflection φ was measured by its tangent, tan φ = d/r; d was +measured by the steel screw and brass scale, and r by the steel tape.</p> + +<p>The value of one turn of the screw was found by comparison with the +standard meter for all parts of the screw. This measurement, including the +possible error of the copy of the standard meter, I estimate to be correct +to .00005 part. The instrument is at the Stevens Institute, where it is to +be compared with a millimeter scale made by Professor Rogers, of +Cambridge.</p> + +<p>The deflection was read to within three or four hundredths of a turn at +each observation, and this error appears in the probable error of the +result.</p> + +<p>The deflection is also affected by the inclination of the plane of +rotation to the horizon. This inclination was small, and its secant varies +slowly, so that any slight error in this angle would not appreciably +affect the result.</p> + +<p>The measurement of r is affected in the same way as D, so that we may +call the greatest error of this measurement .00004. It would probably be +less than this, as the mistakes in the individual measurements would also +appear in the probable error of the result.</p> + +<p>The measurement of φ was not corrected for temperature. As the corrections +would be small they may be applied to the final result. For an increase of +1° F. the correction to be applied to the screw for unit length would +be -.0000066. The correction for the brass scale would be +.0000105, or +the whole correction for the micrometer would be +.000004. The correction +for the steel tape used to measure r would be +.0000066. Hence the +correction for tan. φ would be -.000003 t. The average temperature of the +experiments is 75°.6 F. 75.6-62.5 = 13.1. -.000003×13.1 = -.00004</p> + +<p>Hence φ should be divided by 1.00004, or the final result should be +multiplied by 1.00004. This would correspond to a correction of +12 +kilometers.</p> + +<p>The greatest error, excluding the one just mentioned, would probably be +less than .00009 in the measurement of φ.</p> + +<p>Summing up the various errors, we find, then, that the total constant +error, in the most unfavorable case, where the errors are all in the same +direction, would be .00015. Adding to this the probable error of the +result, .00002, we have for the limiting value of the error of the final +result ±.00017. This corresponds to an error of ±51 kilometers.</p> + +<p>The correction for the velocity of light in vacuo is found by multiplying +the speed in air by the index of refraction of air, at the temperature of +the experiments. The error due to neglecting the barometric height is +exceedingly small. This correction, in kilometers, is +80.</p> + + + +<h3>Final Result.</h3> + +<table summary="final result"> +<tr><td> The mean value of V from the tables is </td><td style="text-align: right">299852</td></tr> +<tr><td> Correction for temperature </td><td style="text-align: right"> +12</td></tr> +<tr><td> </td><td style="text-align: center"> ------------</td></tr> +<tr><td> Velocity of light in air </td><td style="text-align: right"> 299864</td></tr> +<tr><td> Correction for vacuo </td><td style="text-align: right"> 80</td></tr> +<tr><td> </td><td style="text-align: center"> ------------</td></tr> +<tr><td> Velocity of light in vacuo</td><td style="text-align: right"> 299944±51</td></tr> +</table> +<p>The final value of the velocity of light from these experiments is +then—299940 kilometers per second, or 186380 miles per second.</p> +</div> + + +<div id="ch09" class="chapter"> +<h2>Objections Considered.</h2> + + + +<h3>Measurement of the Deflection.</h3> + + +<p>The chief objection, namely, that in the method of the revolving mirror +the deflection is small, has already been sufficiently answered. The same +objection, in another form, is that the image is more or less indistinct. +This is answered by a glance at the tables. These show that in each +individual observation the average error was only three ten-thousandths of +the whole deflection.</p> + + + +<h3>Uncertainty of Laws of Reflection and Refraction in Media in Rapid +Rotation.</h3> + + +<p>What is probably hinted at under the above heading is that there may be a +possibility that the rapid rotation of the mirror throws the reflected +pencil in the direction of rotation. Granting that this is the case, an +inspection of Fig. 14 shows that the deflection will not be affected.</p> + +<p>In this figure let <i>m m</i> be the position of the mirror when the light +first falls on it from the slit at <i>a</i>, and <i>m′ m′</i> the position when the +light returns.</p> + +<p><img src="images/fig14.png" alt="fig 14" id="fig14" /></p> + +<p>From the axis <i>o</i> draw <i>op op</i>, perpendicular to <i>m m</i> and to <i>m′ m′</i>, +respectively. Then, supposing there is no such effect, the course of the +axis of the pencil of light would be <i>a o c</i> mirror <i>c o a′</i>. That is, the +angle of deflection would be <i>a o a′</i>, double the angle <i>p o p′</i>. If now +the mirror be supposed to carry the pencil with it, let <i>o c′</i> be the +direction of the pencil on leaving the mirror <i>m m</i>; i.e., the motion of +the mirror has changed the direction of the reflected ray through the +angle <i>c o c′</i>. The course would then be <i>a o c</i>, mirror <i>c′ o</i>. From <i>o</i> +the reflection would take place in the direction <i>a″</i>, making the angles +<i>c′ o p</i>, and <i>p′ o a″</i> equal. But the angle <i>c o c′</i> must be added to <i>p +o a″</i>, in consequence of the motion of the mirror, or the angle of +deviation will be <i>a o a″ + c o c′</i>; or <i>a o a″ + c o c′ = d</i>. (1)</p> + +<p>By construction—</p> + +<blockquote class="equation"><p> c o p′ = p′ o a′ (2)<br /> + c′ o p′ = p′ o a″ (3)</p></blockquote> + +<p>Subtracting (3) from (2) we have—</p> + +<blockquote class="equation"><p> c o p′ - c′ o p′ = p′ o a′ - p′ o a″, or<br /> + c o c′ = a′ o a″</p></blockquote> + +<p>Substituting <i>a′ o a″</i> for <i>c o c′</i> in (1) we have— +<i>a o a″ + a′ o a″ = a o a′ = d</i>.</p> + +<p>Or the deflection has remained unaltered.</p> + + + +<h3>Retardation Caused by Reflection.</h3> + + +<p>Cornu, in answering the objection that there may be an unknown retardation +by reflection from the distant mirror, says that if such existed the error +it would introduce in his own work would be only 1/7000 that of Foucault, +on account of the great distance used, and on account of there being in +his own experiments but one reflection instead of twelve.</p> + +<p>In my own experiments the same reasoning shows that if this possible error +made a difference of 1 per cent. in Foucault's work (and his result is +correct within that amount), then the error would be but .00003 part.</p> + + + +<h3>Distortion of the Revolving Mirror.</h3> + + +<p>It, has been suggested that the distortion of the revolving mirror, either +by twisting or by the effect of centrifugal force, might cause an error in +the deflection.</p> + +<p><img src="images/fig15.png" alt="fig 15" id="fig15" /></p> + +<p>The only plane in which the deflection might be affected is the plane of +rotation. Distortions in a vertical plane would have simply the effect of +raising, lowering, or extending the slit.</p> + +<p>Again, if the <i>mean</i> surface is plane there will be no effect on the +deflection, but simply a blurring of the image.</p> + +<p>Even if there be a distortion of any kind, there would be no effect on the +deflection if the rays returned to the same portion whence they were +reflected.</p> + +<p>The only case which remains to be considered, then, is that given in Fig. +15, where the light from the slit <i>a</i>, falls upon a distorted mirror, and +the return light upon a different portion of the same.</p> + +<p>The one pencil takes the course <i>a b c d e f a′</i>, while the other follows +the path <i>a f g h i b a′</i>.</p> + +<p>In other words, besides the image coinciding with <i>a</i>, there would be two +images, one on either side of <i>a</i>, and in case there were more than two +portions having different inclinations there would be formed as many +images to correspond. If the surfaces are not plane, the only effect is to +produce a distortion of the image.</p> + +<p>As no multiplication of images was observed, and no distortion of the one +image, it follows that the distortion of the mirror was too small to be +noticed, and that even if it were larger it could not affect the +deflection.</p> + +<p>The figure represents the distorted mirror at rest, but the reasoning is +the same when it is in motion, save that all the images will be deflected +in the direction of rotation.</p> + + + +<h3>Imperfection of the Lens.</h3> + + +<p>It has also been suggested that, as the pencil goes through one-half of +the lens and returns through the opposite half, if these two halves were +not exactly similar, the return image would not coincide with the slit +when the mirror was at rest. This would undoubtedly be true if we consider +but one-half of the original pencil. It is evident, however, that the +other half would pursue the contrary course, forming another image which +falls on the other side of the slit, and that both these images would come +into view, and the line midway between them would coincide with the true +position. No such effect was observed, and would be very unlikely to +occur. If the lens was imperfect, the faults would be all over the +surface, and this would produce simply an indistinctness of the image.</p> + +<p>Moreover, in the latter part of the observations the mirror was inverted, +thus producing a positive rotation, whereas the rotation in the preceding +sets was negative. This would correct the error mentioned if it existed, +and shows also that no constant errors were introduced by having the +rotation constantly in the same direction, the results in both cases being +almost exactly the same.</p> + + + +<h3>Periodic Variations in Friction.</h3> + + +<p>If the speed of rotation varied in the same manner in each revolution of +the mirror, the chances would be that, at the particular time when the +reflection took place, the speed would not be the same as the average +speed found by the calculation. Such a periodic variation could only be +caused by the influence of the frame or the pivots. For instance, the +frame would be closer to the ring which holds the mirror twice in every +revolution than at other times, and it would be more difficult for the +mirror to turn here than at a position 90° from this. Or else there might +be a certain position, due to want of trueness of shape of the sockets, +which would cause a variation of friction at certain parts of the +revolution.</p> + +<p>To ascertain if there were any such variations, the position of the frame +was changed in azimuth in several experiments. The results were unchanged +showing that any such variation was too small to affect the result.</p> + + + +<h3>Change of Speed of Rotation.</h3> + + +<p>In the last four sets of observations the speed was lowered from 256 turns +to 192, 128, 96, and 64 turns per second. The results with these speeds +were the same as with the greater speed within the limits of errors of +experiment.</p> + + + +<h3>Bias.</h3> + + +<p>Finally, to test the question if there were any bias in taking these +observations, eight sets of observations were taken, in which the readings +were made by another, the results being written down without divulging +them. Five of these sets are given in the "specimen," pages 133-134.</p> + +<p>It remains to notice the remarkable coincidence of the result of these +experiments with that obtained by Cornu by the method of the "toothed +wheel."</p> + +<p>Cornu's result was 300400 kilometers, or as interpreted by Helmert 299990 +kilometers. That of these experiments is 299940 kilometers.</p> +</div> + + +<div class="chapter" id="ch10"> +<h2>Postscript.</h2> + + + +<p>The comparison of the micrometer with two scales made by Mr. Rogers, of +the Harvard Observatory, has been completed. The scales were both on the +same piece of silver, marked "Scales No. 25, on silver. Half inch at +58° F., too short .000009 inch. Centimeter at 67° F., too short .00008 cm."</p> + +<p>It was found that the ratio .3937079 could be obtained almost exactly, if, +instead of the centimeter being too short, it were too <i>long</i> by .00008 +cm. at 67°.</p> + +<p>On this supposition the following tables were obtained. They represent the +value of one turn of the micrometer in millimeters.</p> + +<p>Table 1 is the result from centimeter scale.</p> + +<p>Table 2 is the result from half-inch scale.</p> + +<p>Table 3 is the result from page 31.</p> + +<p>It is seen from the correspondence in these results, that the previous +work is correct.</p> +<table summary="the value of one turn of the micrometer in millimeters"> +<tr><th></th><th> (1) </th><th> (2)</th><th> (3)</th></tr> + +<tr><td> From 0 to</td><td style="align: right"> 13</td><td> .99563</td><td> .99562</td><td> .99570</td></tr> +<tr><td></td><td style="align: right"> 25</td><td> .99562</td><td> .99564</td><td> .99571</td></tr> +<tr><td></td><td style="align: right"> 38</td><td> .99560</td><td> .99572</td><td> .99576</td></tr> +<tr><td></td><td style="align: right"> 51</td><td> .99567</td><td> .99578</td><td> .99580</td></tr> +<tr><td></td><td style="align: right"> 64</td><td> .99577</td><td> .99586</td><td> .99585</td></tr> +<tr><td></td><td style="align: right"> 76</td><td> .99582 </td><td> .99590</td><td> .99592</td></tr> +<tr><td></td><td style="align: right"> 89</td><td> .99590</td><td> .99598</td><td> .99601</td></tr> +<tr><td></td><td style="align: right"> 102</td><td> .99596</td><td> .99608</td><td> .99605</td></tr> +<tr><td></td><td style="align: right"> 115</td><td> .99606</td><td> .99614</td><td> .99615</td></tr> +<tr><td></td><td style="align: right"> 128</td><td> .99618</td><td> .99622</td><td> .99623</td></tr> +<tr><td></td><td style="align: right"> 140</td><td> .99629</td><td> .99633</td><td> .99630</td></tr> +</table> +</div> + +<div>*** END OF THE PROJECT GUTENBERG EBOOK 11753 ***</div> +</body> +</html> + |