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diff --git a/11753-0.txt b/11753-0.txt new file mode 100644 index 0000000..278c090 --- /dev/null +++ b/11753-0.txt @@ -0,0 +1,1737 @@ +*** START OF THE PROJECT GUTENBERG EBOOK 11753 *** + +Page images provided by Case Western Reserve University's Digital +Preservation Department + + + + + + +Experimental Determination of the Velocity of Light + +Made at the U.S. Naval Academy, Annapolis. + +By + +Albert A. Michelson, +Master U.S. Navy. + + + + +Note. + + + +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. + +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. + +Simon Newcomb, +_Professor, U.S. Navy_, +_Superintendent Nautical Almanac_. + +Nautical Almanac Office, +Bureau of Navigation, +Navy Department, +_Washington, February 20, 1880._ + + + + +Table Of Contents. + + + +Introduction +Theory of the New Method +Arrangement and Description of Apparatus +Determination of the Constants +The Formulæ +Observations +Separate results of Groups of Observations +Discussion of Errors +Objections Considered +Postscript + + + + +Experimental Determination of the Velocity of Light. + +By Albert A. Michelson, _Master, U.S.N._ + + + +Introduction. + + + +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. + +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. + +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.[1] + + [Footnote 1: See Proc. Am. Assoc. Adv. Science, Saint Louis meeting.] + +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: + + 186730 + 188820 + 186330 + 185330 + 187900 + 184500 + 186770 + 185000 + 185800 + 187940 + ------ + Mean 186500 ± 300 miles per second, + or 300140 kilometers per second. + +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. + +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. + +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. + +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. + +Thus the micrometer eye-piece is much simplified, and many possible +sources of error are removed. + +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. + +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. + +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. + + + + +Theory of New Method. + + + +[Illustration: FIG. 1.] + +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. + +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. + +[Illustration: FIG. 2.] + +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. + +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 _this will be the case no matter +what the distance may be_. + +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 _deflected_ 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. + +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. + +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. + + + + +Arrangement and Description of Apparatus. + + + +Site and Plan. + + +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. + +[Illustration: FIG. 3.] + +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. + + + +The Heliostat. + + +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. + + + +The Revolving Mirror. + + +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. + +[Illustration: FIG. 4.] + +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. + +[Illustration: FIG. 5.] + +[Illustration: FIG. 6.] + + + +Adjustment of the Revolving Mirror. + + +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. + +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. + +[Illustration: FIG. 7.] + +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. + +The instrument was mounted on a brick pier. + + + +The Micrometer. + + +[Illustration: FIG. 8.] + +The apparatus for measuring the deflection was made by Grunow, of New +York. + +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. + + + +Manner of Using the Micrometer. + + +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. + + + +Measurement of Speed of Rotation. + + +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. + +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. + +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 Ut₃ 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. + +[Illustration: FIG. 9.] + + + +The Observer's Table. + + +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. + + + +The Lens. + + +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. + + + +The Fixed Mirror. + + +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. + + + +Adjustment of the Fixed Mirror. + + +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. + +The whole operation was completed in a little over an hour. + +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. + + + +Apparatus for Supplying and Regulating the Blast of Air. + + +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. + +[Illustration: FIG. 10.] + +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. + +[Illustration: FIG. 11.] + +[Illustration: FIG. 12.] + +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. + +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. + +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. + + + +Method Followed in Experiment. + + +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. + +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 _direct_ 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.[2] + + [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.] + +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. + +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. + +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. + +The eye-piece of the micrometer was then set approximately[3] and the +revolving mirror started. If the image did not appear, the mirror was +inclined forward or backward till it came in sight. + + [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.] + +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. + +[Illustration: FIG. 13.] + +Fig. 13 represents the appearance of the image of the slit as seen in the +eye-piece magnified about five times. + + + + +Determination of The Constants. + + + +Comparison of the Steel Tape with the Standard Yard. + + +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: + +Temperature was 55° Fahr. + +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. + +To find the value of one turn, the cross-hair was moved over a millimeter +scale, and the following were the values obtained: + +Turns of screw of microscope I in 1mm-- + + 7.68 7.73 7.60 7.67 + 7.68 7.62 7.65 7.57 + 7.72 7.70 7.64 7.69 + 7.65 7.59 7.63 7.64 + 7.55 7.65 7.61 7.63 + + Mean =7.65 + + Hence one turn = 0.1307mm. + + or = 0.0051 inch. + + The length of the steel tape from 0.1 to 99.1 was found to be + greater than 33 yards, by 7.4 turns =.96mm +.003 feet. + Correction for temperature +.003 feet. + Length 100.000 feet. + -------------- + Corrected length 100.006 feet. + + + +Determination of the Value of Micrometer. + + +Two pairs of lines were scratched on one slide of the slit, about 38mm +apart, i.e., from the center of first pair to center of second pair. This +distance was measured at intervals of 1mm 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′: + + Turns of screw. + At 0 of scale D′ =38.155 + 10 of scale D′ 38.155 + 20 of scale D′ 38.150 + 30 of scale D′ 38 150 + 40 of scale D′ 38.145 + 50 of scale D′ 38.140 + 60 of scale D′ 38.140 + 70 of scale D′ 38.130 + 80 of scale D′ 38.130 + 90 of scale D′ 38.125 + 100 of scale D′ 38.120 + 110 of scale D′ 38.110 + 120 of scale D′ 38.105 + 130 of scale D′ 38.100 + 140 of scale D′ 38.100 + +Changing the form of this table, we find that,-- + + For the _first_ + 10 turns the _average_ value of D′ is 38.155 + 20 turns 38.153 + 30 turns 38.152 + 40 turns 38.151 + 50 turns 38.149 + 60 turns 38.148 + 70 turns 38.146 + 80 turns 38.144 + 90 turns 38.142 + 100 turns 38.140 + 110 turns 38.138 + 120 turns 38.135 + 130 turns 38.132 + 140 turns 38.130 + +On comparing the scale with the standard meter, the temperature being +16°.5 C., 140 divisions were found to = 139.462mm. This multiplied by +(1 + .0000188 × 16.5) = 139.505mm. + +One hundred and forty divisions were found to be equal to 140.022 turns +of the screw, whence 140 turns of the screw = 139.483mm, or +1 turn of the screw = 0.996305mm. + +This is the _average_ value of one turn in 140. + +But the average value of D, for 140 turns is, from the preceding table, +38.130. + +Therefore, the true value of D, is 38.130 × .996305mm, 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. + +This gives the value of a turn-- + + mm. + For the first 10 turns 0.99570 + 20 turns 0.99570 + 30 turns 0.99573 + 40 turns 0.99577 + 50 turns 0.99580 + 60 turns 0.99583 + 70 turns 0.99589 + 80 turns 0.99596 + 90 turns 0.99601 + 100 turns 0.99606 + 110 turns 0.99612 + 120 turns 0.99618 + 130 turns 0.99625 + 140 turns 0.99630 + +NOTE.--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.) + + + +Measurement of the Distance between the Mirrors. + + +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. + +The correction for length of the tape (100.006) was +0.12 of a foot. + +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. + + mm. + At 100 feet the stretch was 8.0 + 80 feet the stretch was 5.0 + 60 feet the stretch was 5.0 + 40 feet the stretch was 3.5 + 20 feet the stretch was 1.5 + --- --- + 300 23.00 + +Weighted mean = 7.7 mm. + For 10 pounds, stretch = 5.1 mm. + = 0.0167 feet. + Correction for whole distance = +0.33 feet. + +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: + + 1985.13 feet. + 1985.17 feet. + 1984.93 feet. + 1985.09 feet. + 1985.09 feet. + ------- + Mean = 1985.082 feet. + + +.70. Cap of pier to revolving mirror. + +.33. Correction for stretch of tape. + +.12. Correction for length of tape. + -------- + 1986.23. True distance between mirrors. + + + +Rate of Standard Ut₃ Fork. + + +The rate of the standard Ut₃ 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. + + +_Set of determinations made at Naval Academy._ + +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. + + + +Specimen of a Determination of Rate of Ut₃ Fork. + + +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. + + July 4, 1879. + 1 0.1 2552.0 + 2 255.3 2551.7 + 3 510.5 2551.9 + 4 765.6 2551.9 + 5 1020.7 2552.1 + 6 1275.7 2552.0 + 7 1530.7 2551.8 + 8 1786.5 2551.4 + 9 2041.6 2551.7 + 10 2297.0 2551.5 + ------- + 11 2552.1 255.180 = mean ÷ 10. + 12 2807.0 + .699 = reduction for mean time. + 13 3062.4 + .003 = correction for rate. + 14 3317.5 + .187 = correction for temperature. + ------- + 15 3572.8 256.069 = number of vibrations per second at 65° Fahr. + 16 3827.7 + 17 4082.5 + 18 4335.9 + 19 4593.3 + 20 4848.5 + +The correction for temperature was found by Professor Mayer by counting +the sound-beats between the standard and another Ut₃ 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. + + +_Résumé of determinations made at Naval Academy._ + +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: + + July 4 20 255.882 27.0 256.069 0.07 + 5 19 255.915 26.4 256.089 0.05 + 5 18 255.911 26.0 256.077 0.02 + 6 21 255.874 24.7 256.012 0.13 + 6 9 255.948 24.8 256.087 0.24 + 7 22 255.938 24.6 256.074 0.05 + 7 21 255.911 25.3 256.061 0.04 + 8 20 255.921 26.6 256.100 0.02 + 8 20 255.905 26.6 256.084 0.06 + 8 20 255.887 26.6 256.066 0.03 + ------- + Mean = 256.072 + +In one of the preceding experiments, I compared the two Vt₃ 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. + + +_Second determination_. + +(Joint work with Professor A.M. Mayer, Stevens Institute, Hoboken.) + +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. + +The following is a specimen determination: + +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. + + 1 0.3 255.83 + 2 256.1 255.90 + 3 511.7 255.90 + 4 767.9 255.93 + 5 1023.5 255.92 + 6 1289.2 256.01 + 7 1535.3 255.95 + ------- + 8 1791.5 255.920 = mean. + 9 2047.1 - .028 = correction for rate. + ------- + 10 2303.5 255.892 + 11 2559.0 + .180 = correction for temperature. + ------- + 12 2825.3 256.072 = number of vibrations per second at 65° Fahr. + 13 3071.0 + +In the following _résumé_, 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: + + 1 13 255.892 80 256.072 0.18 + 2 11 255.934 81 256.126 0.17 + 3 13 255.899 81 256.091 0.12 + 4 13 255.988 75 256.108 0.13 + 5 11 255.948 75 256.068 0.05 + 6 12 255.970 75 256.090 0.05 + 7 12 255.992 75 256.112 0.20 + 8 11 255.992 76 256.124 0.03 + 9 11 255.888 81 256.080 0.13 + 10 13 255.878 81 256.070 0.13 + ------- + Mean = 256.094 + + + +Effect of Support and of Scraping. + + +The standard Vt₃ 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. + + +_Specimen._ + +Coincidences were marked-- + + At 32 seconds. + 37 seconds. + 43.5 seconds. + 49 seconds. + 54.5 seconds. + 61.5 seconds. + 61.5 - 32 = 29.5. + 29.5 ÷ 5 = 5.9 = time of one interval. + +_Résumé._ + + 1 5.9 + 2 6.2 + 3 6.2 + 4 6.2 + ---- + Mean = 6.13 = time of one interval between coincidences. + +In this time the watch makes 6.13×5 = 30.65 beats, and the forks make +30.65 + 1 = 31.65 beats. + +Hence the number of beats per second is 31.65 ÷ 6.13 = 5.163. + + +_Specimen._ + +Circumstances the same as in last case, except that standard Vt₃ fork was +allowed to trace its record on the lampblacked paper, as in finding its +rate of vibration. + +Coincidences were marked at-- + + 59 seconds. + 04 seconds. + 10.5 seconds. + 17 seconds. + + 77 - 59 = 18. + 18 ÷ 3 = 6.0 = time of one interval. + +_Résumé._ + + No. 1 6.0 seconds. 6.31 × 5 = 31.55 + 2 6.0 seconds. + 1.00 + 3 6.7 seconds. ---- + 4 6.3 seconds. + 5 6.5 seconds. 32.55 + 6 6.7 seconds. 32.55 ÷ 6.31 = 5.159 + 7 6.0 seconds. With fork free 5.163 + ---- ----- + Mean = 6.31 seconds Effect of scrape = - .044 + +_Specimen._ + +Circumstances as in first case, except that both forks were on their +resonators. + +Coincidences were observed at-- + + 21 seconds. + 28 seconds. + 36 seconds. + 44 seconds. + 51 seconds. + 60 seconds. + 60 - 21 = 39 + 39 ÷ 5 = 7.8 = time of one interval. + +_Résumé_. + + No. 1 7.8 seconds. 7.42 × 5 = 37.10 + 2 7.1 seconds. + 1.00 + 3 7.6 seconds. ----- + 4 7.4 seconds. 38.10 + 5 7.2 seconds. 38.10 ÷ 7.42 = 5.133 + ---- (Above) 5.159 + ----- + Mean = 7.42 seconds. Effect of support and scrape = - .026 + + Mean of second determination was 256.094 + Applying correction (scrape, etc.) - .026 + ------- + Corrected mean 256.068 + Result of first determination 256.072 + ------- + Final value 256.070 + +NOTE--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. + + 256.180 + 256.036 + 256.072 + 256.068 + ------- + Mean = 256.089 + +The previous work was omitted on account of various inaccuracies and want +of practice, which made the separate results differ widely from each +other. + + + + +The Formulæ. + + + +The formulæ employed are-- + + d′ + (1) tan φ = ----- + r + + 2592000″ × D × n + (2) V = ----------------- + φ″ + + φ = angle of deflection. + d′ = corrected displacement (linear). + r = radius of measurement. + D = twice the distance between the mirrors. + n = number of revolutions per second. + α = inclination of plane of rotation + d = deflection as read from micrometer. + B = number of beats per second between electric Vt₂ fork and + standard Vt₃ + Cor = correction for temperature of standard Vt3. + V = velocity of light. + T = value of one turn of screw. (Table, page 126.) + +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-- + + dT + (3) tan φ = c′ ----; log c′ = .51607 + r + + n + (4) V = c ---; log c = .49670 + φ + + D and r are expressed in feet and d′ in millimeters. + Vt₃ fork makes 256.070 vibrations per second at 65° Fahr. + D = 3972.46 feet. + tan α = tangent of angle of inclination of plane of rotation = 0.02 + in all but the last twelve observations, in which it was 0.015. + log c′ = .51607 (.51603 in last twelve observations.). + log c = .49670. + +The electric fork makes ½(256.070 + B + cor.) vibrations per second, +and n is a multiple, submultiple, or simple ratio of this. + + + + +Observations. + + + +Specimen Observation. + + +June 17. sunset. Image good; best in column (4). + +The columns are sets of readings of the micrometer for the deflected image +of slit. + + 112.81 112.80 112.83 112.74 112.79 + 81 81 81 76 78 + 79 78 78 74 74 + 80 75 74 76 74 + 79 77 74 76 77 + 82 79 72 78 81 + 82 73 76 78 77 + 76 78 81 79 75 + 83 79 74 83 82 + 73 73 76 78 82 + ------- ------- ------- ------- ------- + Mean = 112.801 112.773 112.769 112.772 112.779 + Zero = 0.260 0.260 0.260 0.260 0.260 + ------- ------- ------- ------- ------- + d = 112.451 112.513 112.509 112.512 112.519 + Temp = 77° 77° 77° 77° 77° + B = + 1.500 + Corr = - .144 + ------- + + 1.365 + 256.070 + ------- + n = 257.426 257.43 257.43 257.43 257.43 + r = 28.157 28.157 28.157 28.157 28.157 + +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. + +The following is the calculation for V: + + 2d, 3d, + 1st set. and 4th sets. 5th set. + log c′ = 51607 51607 51607 + " T = 99832 99832 99832 + " d = 05131 05119 05123 + ------- ------- ------- + 56570 56558 56562 + " r = 44958 44958 44958 + ------- ------- ------- + " tan φ = 11612 11600 11604 + φ = 2694″.7 2694″.1 2694″.3 + " c = 49670 49670 49670 + " n = 41066 41066 41066 + ------- ------- ------- + 90736 90736 90736 + " φ = 43052 43042 43046 + ------- ------- ------- + " V = 47684 47694 47690 + V = 299800 299880 299850 + +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. + +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. + +The column headed "B" contains the number of "beats" per second between +the electric Vt₂ fork and the standard Vt₃ 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. + +The "radius of measurement" is the distance between the front face of the +revolving mirror and the cross-hair of the micrometer. + +The numbers in the column headed "Value of one turn of the screw" were +taken from the table, page 127. + + Date. + | Distinctness of image. + | | Temperature, Fahr. + | | | Position of deflected image. + | | | | Position of slit. + | | | | | Displacement of image in divisions. + | | | | | | Difference between greatest and least values. + | | | | | | | B. + | | | | | | | | Cor. + | | | | | | | | | Number of revolutions per second. + | | | | | | | | | | Radius of measurement, in feet. + | | | | | | | | | | | Value of one turn of the screw. + | | | | | | | | | | | | Velocity of light in air, in kilometers. + | | | | | | | | | | | | | Remarks. + | | | | | | | | | | | | | | + June 5|3|76|114.85| 0.300|114.55|0.17|1.423|-0.132|257.36|28.672|0.99614|299850|Electric light. + June 7|2|72|114.64| 0.074|114.56|0.10|1.533|-0.084|257.52|28.655|0.99614|299740|P.M. Frame inclined at various angles + June 7|2|72|114.58| 0.074|114.50|0.08|1.533|-0.084|257.52|28.647|0.99614|299900|P.M. Frame inclined at various angles + June 7|2|72| 85.91| 0.074| 85.84|0.12|1.533|-0.084|193.14|28.647|0.99598|300070|P.M. Frame inclined at various angles + June 7|2|72| 85.97| 0.074| 85.89|O.07|1.533|-0.084|193.14|28.650|0.99598|299930|P.M. Frame inclined at various angles + June 7|2|72|114.61| 0.074|114-53|0.07|1.533|-0.084|257.42|28.650|0.99614|299850|P.M. Frame inclined at various angles + June 9|3|83|114.54| 0.074|114.47|0.07|1.533|-0.216|257.39|28.658|0.99614|299950|P.M. Frame inclined at various angles + June 9|3|83|114.54| 0.074|114.46|0.10|1.533|-0.216|257.39|28.658|0.99614|299980|P.M. Frame inclined at various angles + June 9|3|83|114.57| 0.074|114.47|0.08|1.533|-0.216|257.39|28.662|0.99614|299980|P.M. Frame inclined at various angles + June 9|3|83|114.57| 0.074|114.50|0.06|1.533|-0.216|257.39|28.660|0.99614|299880|P.M. Frame inclined at various angles + June 9|2|83|114.61| 0.074|114.53|0.13|1.533|-0.216|257.39|28.678|0.99614|300000|P.M. Frame inclined at various angles + June 10|2|90|114.60| 0.074|114.52|0.11|1.517|-0.300|257.29|28.685|0.99614|299980|P.M. + June 10|2|90|114.62| 0.074|114.54|0.08|1.517|-0.300|257.29|28.685|0.99614|299930|P.M. + June 12|2|71|114.81| 0.074|114.74|0.09|1.450|-0.072|257.45|28.690|0.99614|299650|A.M. + June 12|2|71|114.78| 0.074|114.70|0.05|1.450|-0.072|257.45|28.690|0.99614|299760|A.M. + June 12|1|71|114.76| 0.074|114.68|0.09|1.450|-0.072|257.45|28.690|0.99614|299810|A.M. + June 13|3|72|112.64| 0.074|112.56|0.09|1.500|-0.084|257.49|28.172|0.99614|300000|A.M. + June 13|3|72|112.63| 0.074|112.56|0.10|1.500|-0.084|257.49|28.172|0.99614|300000|A.M. + June 13|2|72|112.65| 0.074|112.57|0.08|1.500|-0.084|257.49|28.172|0.99614|299960|A.M. + June 13|3|79|112.82| 0.260|112.56|0.06|1.517|-0.168|257.42|28.178|0.99614|299960|P.M. + June 13|3|79|112.82| 0.260|112.56|0.13|1.517|-0.168|257.42|28.178|0.99614|299960|P.M. + June 13|3|79|112.83| 0.260|112.57|0.07|1.517|-0.168|257.42|28.178|0.99614|299940|P.M. + June 13|3|79|112.82| 0.260|112.56|0.06|1.517|-0.168|257.42|28.178|0.99614|299960|P.M. + June 13|3|79|112.83| 0.260|112.57|0.11|1.517|-0.168|257.42|28.178|0.99614|299940|P.M. + June 13|3|79|113.41| 0.260|113.15|11 |1.517|-0.168|258.70|28.152|0.99614|299880|P.M. Set micrometer and counted oscillations. + June 13|3|79|112.14| 0.260|111.88|6 |1.517|-0.168|255.69|28.152|0.99614|299800|Oscillations of image of revolving mirror. + June 14|1|64|112.83| 0.260|112.57|0.12|1.500|+0.012|257.58|28.152|0.99614|299850|A.M. + June 14|1|64|112.83| 0.260|112.57|0.05|1.517|+0.012|257.60|28.152|0.99614|299880|A.M. + June 14|1|65|112.81| 0.260|112.55|0.11|1.517| 0.000|257.59|28.152|0.99614|299900|A.M. + June 14|1|66|112.83| 0.260|112.57|0.09|1.517|-0.012|257.57|28.152|0.99614|299840|A.M. + June 14|1|67|112.83| 0.260|112.57|0.12|1.517|-0.024|257.56|28.152|0.99614|299830|A.M. + June 14|1|84|112.78| 0.260|112.52|0.06|1.517|-0.228|257.36|28.159|0.99614|299790|P.M. Readings taken by Lieut. Nazro. + June 14|1|85|112.76| 0.260|112.50|0.08|1.500|-0.240|257.33|28.159|0.99614|299810|P.M. Readings taken by Lieut. Nazro. + June 14|1|84|112.72| 0.260|112.46|0.08|1.483|-0.228|257.32|28.159|0.99614|299880|P.M. Readings taken by Lieut. Nazro. + June 14|1|84|112.73| 0.260|112.47|0.09|1.483|-0.228|257.32|28.159|0.99614|299880|P.M. + June 14|1|84|112.75| 0.260|112.49|0.09|1.483|-0.228|257.32|28.129|0.99614|299830|P.M. + June 17|2|62|112.85| 0.260|112.59|0.09|1.517|+0.036|257.62|28.149|0.99614|299800|A.M. + June 17|2|63|112.84| 0.260|112.58|0.06|1.500|+0.024|257.59|28.149|0.99614|299790|A.M. + June 17|1|64|112.85| 0.260|112.59|0.07|1.500|+0.012|257.58|28.149|0.99614|299760|A.M. + June 17|3|77|112.80| 0.260|112.54|0.07|1.500|-0.144|257-43|28.157|0.99614|299800|P.M. Readings taken by Mr. Clason. + June 17|3|77|112.77| 0.260|112.51|0.08|1.500|-0.144|257.43|28.157|0.99614|299880|P.M. Readings taken by Mr. Clason. + June 17|3|77|112.77| 0.260|112.51|0.11|1.500|-0.144|257.43|28.157|0.99614|299880|P.M. Readings taken by Mr. Clason. + June 17|3|77|112.77| 0.260|112.51|0.09|1.500|-0.144|257.43|28.157|0.99614|299880|P.M. Readings taken by Mr. Clason. + June 17|3|77|112.78| 0.260|112.52|0.08|1.500|-0.144|257 43|28.157|0.99614|299860|P.M. Readings taken by Mr. Clason. + June 18|1|58|112.90| 0.265|112.64|0.07|1.500|+0.084|257.65|28.150|0.99614|299720|A.M. + June 18|1|58|112.90| 0.265|112.64|0.10|1.500|+0.084|257.65|28.150|0.99614|299720|A.M. + June 18|1|59|112.92| 0.265|112.66|0.07|1.483|+0.072|257.62|28.150|0.99614|299620|A.M. + June 18|2|75|112.79| 0.265|112.52|0.09|1.483|-0.120|257-43|28.158|0.99614|299860|P.M. + June 18|2|75|112.75| 0.265|112.48|0.10|1.483|-0.120|257-43|28.158|0.99614|299970|P.M. + June 18|2|75|112.76| 0.265|112.49|0.08|1.483|-0.120|257-43|28.158|0.99614|299950|P.M. + June 20|3|60|112.94| 0.265|112.67|0.07|1.517|+0.063|257.65|28.172|0.99614|299880|A.M. + June 20|3|61|112.92| 0.265|112.65|0.09|1.517|+0.048|257.63|28.172|0.99614|299910|A.M. + June 20|2|62|112.94| 0.265|112.67|0.07|1.517|+0.036|257.62|28.172|0.99614|299850|A.M. + June 20|2|63|112.93| 0.265|112.66|0.03|1.517|+0.024|257.61|28.172|0.99614|299870|A.M. + June 20|2|78|133.48| 0.265|133.21|0.13|1.450|-0.156|257.36|33.345|0.99627|299840|P.M. + June 20|2|79|133.49| 0.265|133.23|0.09|1.500|-0.168|257.40|33.345|0.99627|299840|P.M. + June 20|2|80|133.49| 0.265|133.22|0.07|1.500|-0.180|257.39|33.345|0.99627|299850|P.M. + June 20|2|79|133.50| 0.265|133.24|0.13|1.483|-0.168|257.39|33.345|0.99627|299840|P.M. + June 20|2|79|133.49| 0.265|133.22|0.06|1.483|-0.168|257.38|33.345|0.99627|299840|P.M. + June 20|2|79|133.49| 0.265|133.22|0.10|1.483|-0.168|257.38|33.345|0.99627|299840|P.M. + June 21|2|61|133.56| 0.265|133.29|0.12|1.533|+0.048|257.65|33.332|0.99627|299890|A.M. + June 21|2|62|133.58| 0.265|133.31|0.08|1.533|+0.036|257.64|33.332|0.99627|299810|A.M. + June 21|2|63|133.57| 0.265|133.31|0.09|1.533|+0.024|257.63|33.332|0.99627|299810|A.M. + June 21|2|64|133.57| 0.265|133.30|0.11|1.533|+0.012|257.61|33.332|0.99627|299820|A.M. + June 21|2|65|133.56| 0.265|133.30|0.13|1.533| 0.000|257.60|33.332|0.99627|299800|A.M. + June 21|3|80|133.48| 0.265|133.21|0.06|1.533|-0.180|257.42|33.330|0.99627|299770|P.M. + June 21|3|81|133.46| 0.265|133.19|0.10|1.500|-0.192|257.38|33.330|0.99627|299760|P.M. + June 21|3|82|133.46| 0.265|133.20|0.05|1.500|-0.204|257.37|33.330|0.99627|299740|P.M. + June 21|3|82|133.46| 0.265|133.20|0.08|1.517|-0.204|257.38|33.330|0.99627|299750|P.M. + June 21|3|81|133.46| 0.265|133.19|0.08|1.500|-0.192|257.38|33.330|0.99627|299760|P.M. + June 23|3|89|133.43| 0.265|133.16|0.08|1.542|-0.288|257.32|33.345|0.99627|299910|P.M. + June 23|3|89|133.42| 0.265|133.15|0.06|1.550|-0.288|257.33|33.345|0.99627|299920|P.M. + June 23|3|90|133.43| 0.265|133.17|0.09|1.550|-0.300|257.32|33.345|0.99627|299890|P.M. + June 23|3|90|133.43| 0.265|133.16|0.07|1.533|-0.300|257.30|33.345|0.99627|299860|P.M. + June 23|3|90|133.42| 0.265|133.16|0.07|1.517|-0.300|257.29|33.345|0.99627|299880|P.M. + June 24|3|72|133.47| 0.265|133.20|0.15|1.517|-0.084|257.50|33.319|0.99627|299720|A.M. + June 24|3|73|133.44| 0.265|133.17|0.04|1.517|-0.096|257.49|33.319|0.99627|299840|A.M. + June 24|3|74|133.42| 0.265|133.16|0.11|1.517|-0.108|257.48|33.319|0.99627|299850|A.M. + June 24|3|75|133.42| 0.265|133.16|0.06|1.517|-0.120|257.47|33.319|0.99627|299850|A.M. + June 24|3|76|133.44| 0.265|133.18|0.10|1.517|-0.132|257.45|33.319|0.99627|299780|A.M. + June 26|2|86|133.42| 0.265|133.15|0.05|1.508|-0.252|257.33|33.339|0.99627|299890|P.M. + June 26|2|86|133.44| 0.265|133.17|0.08|1.508|-0.252|257.33|33.339|0.99627|299840|P.M. + June 27|3|73|133.49| 0.265|133.22|0.11|1.483|-0.096|257.46|33.328|0.99627|299780|A.M. + June 27|3|74|133.47| 0.265|133.20|0.06|1.483|-0.108|257.44|33.328|0.99627|299810|A.M. + June 27|3|75|133.47| 0.265|133.21|0.09|1.483|-0.120|257.43|33.328|0.99627|299760|A.M. + June 27|3|75|133.45| 0.265|133.19|0.09|1.467|-0.120|257.42|33.328|0.99627|299810|A.M. + June 27|3|76|133.47| 0.265|133.20|0.08|1.483|-0.132|257.42|33.328|0.99627|299790|A.M. + June 27|3|76|133.45| 0.265|133.19|0.10|1.483|-0.132|257.42|33.328|0.99627|299810|A.M. + June 30|2|85| 35.32|135.00 | 99.68|0.05|1.500|-0.240|193.00|33.274|0.99645|299820|P.M. Mirror inverted. + June 30|2|86| 35.34|135.00 | 99.67|0.06|1.508|-0.252|193.00|33.274|0.99645|299850|P.M. Mirror inverted. + June 30|2|86| 35.34|135.00 | 99.66|0.10|1.508|-0.252|193.00|33.274|0.99645|299870|P.M. Mirror inverted. + June 30|2|86| 35.34|135.00 | 99.66|0.09|1.517|-0.252|193.00|33.274|0.99645|299870|P.M. Mirror inverted. + July 1|2|83| 02.17|135.145|132.98|0.07|1.500|-0.216|257.35|33.282|0.99627|299810|P.M. Mirror inverted. + July 1|2|84| 02.15|135.145|133.00|0.09|1.500|-0.228|257.34|33.282|0.99627|299740|P.M. Mirror inverted. + July 1|2|86| 02.14|135.145|133.01|0.06|1.467|-0.252|257.28|33.311|0.99627|299810|P.M. Mirror inverted. + July 1|2|86| 02.14|135.145|133.00|0.08|1.467|-0.252|257.28|33.311|0.99627|299940|P.M. Mirror inverted. + July 2|3|86| 99.85| 0.400| 99.45|0.05|1.450|-0.252|192.95|33.205|0.99606|299950|P.M. Mirror erect. + July 2|3|86| 66.74| 0.400| 66.34|0.03|1.450|-0.252|128.63|33.205|0.99586|299800|P.M. Mirror erect. + July 2|3|86| 50.16| 0.400| 47.96|0.07|1.467|-0.252| 96.48|33.205|0.99580|299810|P.M. Mirror erect. + July 2|3|85| 33.57| 0.400| 33.17|0.06|1.450|-0.240| 64.32|33.205|0.99574|299870|P.M. Mirror erect. + +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. + +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. + +The following table gives the result of different groupings of sets of +observations. Necessarily some of the groups include others: + + Electric light (1 set) 299850 + Set micrometer counting oscillations (2) 299840 + Readings taken by Lieutenant Nazro (3) 299830 + Readings taken by Mr. Clason (5) 299860 + Mirror inverted (8) 299840 + Speed of rotation, 192 (7) 299990 + Speed of rotation, 128 (1) 299800 + Speed of rotation, 96 (1) 299810 + Speed of rotation, 64 (1) 299870 + Radius, 28.5 feet (54) 299870 + Radius, 33.3 feet (46) 299830 + Highest temperature, 90° Fahr. (5) 299910 + Mean of lowest temperatures, 60° Fahr. (7) 299800 + Image, good (46) 299860 + Image, fair (39) 299860 + Image, poor (15) 299810 + Frame, inclined (5) 299960 + Greatest value 300070 + Least value 299650 + Mean value 299852 + Average difference from mean 60 + Value found for π 3.26 + Probable error ± 5 + + + +Discussion of Errors. + + +The value of V depends on three quantities D, n, and φ. These will now be +considered in detail. + + + +The Distance. + + +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. + +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. + +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. + + + +The Speed of Rotation. + + +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. + +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. + +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. + +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. + +We may then conclude that the error, in the measurement of _n_, was less +than .00002. + + + +The Deflection. + + +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. + +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. + +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. + +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. + +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. + +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 + +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. + +The greatest error, excluding the one just mentioned, would probably be +less than .00009 in the measurement of φ. + +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. + +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. + + + +Final Result. + + + The mean value of V from the tables is 299852 + Correction for temperature +12 + ------------ + Velocity of light in air 299864 + Correction for vacuo 80 + ------------ + Velocity of light in vacuo 299944±51 + +The final value of the velocity of light from these experiments is +then--299940 kilometers per second, or 186380 miles per second. + + + + +Objections Considered. + + + +Measurement of the Deflection. + + +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. + + + +Uncertainty of Laws of Reflection and Refraction in Media in Rapid +Rotation. + + +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. + +In this figure let _m m_ be the position of the mirror when the light +first falls on it from the slit at _a_, and _m′ m′_ the position when the +light returns. + +[Illustration: FIG. 14.] + +From the axis _o_ draw _op op_, perpendicular to _m m_ and to _m′ m′_, +respectively. Then, supposing there is no such effect, the course of the +axis of the pencil of light would be _a o c_ mirror _c o a′_. That is, the +angle of deflection would be _a o a′_, double the angle _p o p′_. If now +the mirror be supposed to carry the pencil with it, let _o c′_ be the +direction of the pencil on leaving the mirror _m m_; i.e., the motion of +the mirror has changed the direction of the reflected ray through the +angle _c o c′_. The course would then be _a o c_, mirror _c′ o_. From _o_ +the reflection would take place in the direction _a″_, making the angles +_c′ o p_, and _p′ o a″_ equal. But the angle _c o c′_ must be added to _p +o a″_, in consequence of the motion of the mirror, or the angle of +deviation will be _a o a″ + c o c′_; or _a o a″ + c o c′ = d_. (1) + +By construction-- + + c o p′ = p′ o a′ (2) + c′ o p′ = p′ o a″ (3) + +Subtracting (3) from (2) we have-- + + c o p′ - c′ o p′ = p′ o a′ - p′ o a″_, or + c o c′ = a′ o a″_ + +Substituting _a′ o a″_ for _c o c′_ in (1) we have-- +_a o a″ + a′ o a″ = a o a′ = d_. + +Or the deflection has remained unaltered. + + + +Retardation Caused by Reflection. + + +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. + +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. + + + +Distortion of the Revolving Mirror. + + +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. + +[Illustration: FIG. 15] + +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. + +Again, if the _mean_ surface is plane there will be no effect on the +deflection, but simply a blurring of the image. + +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. + +The only case which remains to be considered, then, is that given in Fig. +15, where the light from the slit _a_, falls upon a distorted mirror, and +the return light upon a different portion of the same. + +The one pencil takes the course _a b c d e f a′_, while the other follows +the path _a f g h i b a′_. + +In other words, besides the image coinciding with _a_, there would be two +images, one on either side of _a_, 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. + +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. + +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. + + + +Imperfection of the Lens. + + +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. + +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. + + + +Periodic Variations in Friction. + + +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. + +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. + + + +Change of Speed of Rotation. + + +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. + + + +Bias. + + +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. + +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." + +Cornu's result was 300400 kilometers, or as interpreted by Helmert 299990 +kilometers. That of these experiments is 299940 kilometers. + + + + +Postscript. + + + +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." + +It was found that the ratio .3937079 could be obtained almost exactly, if, +instead of the centimeter being too short, it were too _long_ by .00008 +cm. at 67°. + +On this supposition the following tables were obtained. They represent the +value of one turn of the micrometer in millimeters. + +Table 1 is the result from centimeter scale. + +Table 2 is the result from half-inch scale. + +Table 3 is the result from page 31. + +It is seen from the correspondence in these results, that the previous +work is correct. + + (1) (2) (3) + + From 0 to 13 .99563 .99562 .99570 + 25 .99562 .99564 .99571 + 38 .99560 .99572 .99576 + 51 .99567 .99578 .99580 + 64 .99577 .99586 .99585 + 76 .99582 .99590 .99592 + 89 .99590 .99598 .99601 + 102 .99596 .99608 .99605 + 115 .99606 .99614 .99615 + 128 .99618 .99622 .99623 + 140 .99629 .99633 .99630 + + + + + + + + + +End of the Project Gutenberg EBook of Experimental Determination of the +Velocity of Light, by Albert A. Michelson + +*** END OF THE PROJECT GUTENBERG EBOOK 11753 *** |