,page images provided by ,case ,western ,reserve ,university's ,digital ,preservation ,department ,experimental ,determination of the ,velocity of ,light ,made at the ,u4,s4 ,naval ,academy1 ,annapolis4 ,by ,albert ,a4 ,michelson1 ,master ,u4,s4 ,navy4 ,note4 ,the probability that the most accurate method of determining the solar parallax now available is that resting on the measurement of the velocity of light1 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 motions4 ,the researches described in it1 having been made at the ,united ,states ,naval ,academy1 though at private expense1 were reported to the ,honorable ,secretary of the ,navy1 and referred by him to this ,office4 ,at the suggestion of the writer1 the paper was reconstructed with a fuller general discussion of the processes1 and with the omission of some of the details of individual experiments4 ,to prevent a possible confusion of this determination of the velocity of light with another now in progress under official auspices1 it may be stated that the credit and responsibility for the present paper rests with ,master ,michelson4 ,simon ,newcomb1 ,professor1 ,u4,s4 ,navy1 ,superintendent ,nautical ,almanac4 ,nautical ,almanac ,office1 ,bureau of ,navigation1 ,navy ,department1 ,washington1 ,february #bj1 #ahhj4 ,table ,of ,contents4 ,introduction ,theory of the ,new ,method ,arrangement and ,description of ,apparatus ,determination of the ,constants ,the ,formul ,observations ,separate results of ,groups of ,observations ,discussion of ,errors ,objections ,considered ,postscript ,experimental ,determination of the ,velocity of ,light4 ,by ,albert ,a4 ,michelson1 ,master1 ,u4,s4,n4 ,introduction4 ,in ,cornu's elaborate memoir upon the determination of the velocity of light1 several objections are made to the plan followed by ,foucault1 which will be considered in the latter part of this work4 ,it may1 however1 be stated that the most important among these was that the deflection was too small to be measured with the required degree of accuracy4 ,in order to employ this method1 therefore1 it was absolutely necessary that the deflection should be increased4 ,in ,november1 #ahgg1 a modification of ,foucault's arrangement suggested itself1 by which this result could be accomplished4 ,between this time and ,march of the following year a number of preliminary experiments were performed in order to familiarize myself with the optical arrangements4 ,the first experiment tried with the revolving mirror produced a deflection considerably greater than that obtained by ,foucault4 ,thus far the only apparatus used was such as could be adapted from the apparatus in the laboratory of the ,naval ,academy4 ,at the expense of #aj a revolving mirror was made1 which could execute #abh turns per second4 ,the apparatus was installed in ,may1 #ahgh1 at the laboratory4 ,the distance used was #ejj feet1 and the deflection was about twenty times that obtained by ,foucault4#a ,footnote #a3 ,see ,proc4 ,am4 ,assoc4 ,adv4 ,science1 ,saint ,louis meeting4 ,these experiments1 made with very crude apparatus and under great difficulties1 gave the following table of results for the velocity of light in miles per second3 #ahfgcj #ahhhbj #ahfccj #aheccj #ahgijj #ahdejj #ahfggj #ahejjj #ahehjj #ahgidj ------ ,mean #ahfejj #cjj miles per second1 or #cjjadj kilometers per second4 ,in the following ,july the sum of #b1jjj was placed at my disposal by a private gentleman for carrying out these experiments on a large scale4 ,before ordering any of the instruments1 however1 it was necessary to find whether or not it was practicable to use a large distance4 ,with a distance 7between the revolving and the fixed mirror7 of #ejj feet1 in the preliminary experiments1 the field of light in the eye-piece was somewhat limited1 and there was considerable indistinctness in the image1 due to atmospheric disturbances4 ,accordingly1 the same lens 7#ci feet focus7 was employed1 being placed1 together with the other pieces of apparatus1 along the north sea-wall of the ,academy grounds1 the distance being about #b1jjj feet4 ,the image of the slit1 at noon1 was so confused as not to be recognizable1 but toward sunset it became clear and steady1 and measurements were made of its position1 which agreed within one one-hundredth of a millimeter4 ,it was thus demonstrated that with this distance and a deflection of #ajj millimeters this measurement could be made within the ten-thousandth part4 ,in order to obtain this deflection1 it was sufficient to make the mirror revolve #bej times per second and to use a radius of about #cj feet4 ,in order to use this large radius 7distance from slit to revolving mirror71 it was necessary that the mirror should be large and optically true2 also1 that the lens should be large and of great focal length4 ,accordingly the mirror was made #a inches in diameter1 and a new lens1 #h inches in diameter1 with a focal length of #aej feet was procured4 ,in ,january1 #ahgi1 an observation was taken1 using the old lens1 the mirror making #abh turns per second4 ,the deflection was about #dc millimeters4 ,the micrometer eye-piece used was substantially the same as ,foucault's1 except that part of the inclined plate of glass was silvered1 thus securing a much greater quantity of light4 ,the deflection having reached #dc millimeters1 the inclined plate of glass could be dispensed with1 the light going past the observer's head through the slit1 and returning #dc millimeters to the left of the slit1 where it could be easily observed4 ,thus the micrometer eye-piece is much simplified1 and many possible sources of error are removed4 ,the field was quite limited1 the diameter being1 in fact1 but little greater than the width of the slit4 ,this would have proved a most serious objection to the new arrangement4 ,with the new lens1 however1 this difficulty disappeared1 the field being about twenty times the width of the slit4 ,it was expected that1 with the new lens1 the image would be less distinct2 but the difference1 if any1 was small1 and was fully compensated by the greater size of the field4 ,the first observation with the new lens was made ,january #cj1 #ahgi4 ,the deflection was #gj millimeters4 ,the image was sufficiently bright to be observed without the slightest effort4 ,the first observation with the new micrometer eye-piece was made ,april #b1 the deflection being #aae millimeters4 ,the first of the final series of observations was made on ,june #e4 ,all the observations previous to this1 thirty sets in all1 were rejected4 ,after this time1 no set of observations nor any single observation was omitted4 ,theory of ,new ,method4 ,illustration3 ,,fig4 #a4 ,let ,s1 ,fig4 #a1 be a slit1 through which light passes1 falling on ,r1 a mirror free to rotate about an axis at right angles to the plane of the paper2 ,l1 a lens of great focal length1 upon which the light falls which is reflected from ,r4 ,let ,m be a plane mirror whose surface is perpendicular to the line ,r1 ,m1 passing through the centers of ,r1 ,l1 and ,m1 respectively4 ,if ,l be so placed that an image of ,s is formed on the surface of ,m1 then1 this image acting as the object1 its image will be formed at ,s1 and will coincide1 point for point1 with ,s4 ,if1 now1 ,r be turned about the axis1 so long as the light falls upon the lens1 an image of the slit will still be formed on the surface of the mirror1 though on a different part1 and as long as the returning light falls on the lens an image of this image will be formed at ,s1 notwithstanding the change of position of the first image at ,m4 ,this result1 namely1 the production of a stationary image of an image in motion1 is absolutely necessary in this method of experiment4 ,it was first accomplished by ,foucault1 and in a manner differing apparently but little from the foregoing4 ,illustration3 ,,fig4 #b4 ,in his experiments ,l1 ,fig4 #b1 served simply to form the image of ,s at ,m1 and ,m1 the returning mirror1 was spherical1 the center coinciding with the axis of ,r4 ,the lens ,l was placed as near as possible to ,r4 ,the light forming the return image lasts1 in this case1 while the first image is sweeping over the face of the mirror1 ,m4 ,hence1 the greater the distance ,,rm1 the larger must be the mirror in order that the same amount of light may be preserved1 and its dimensions would soon become inordinate4 ,the difficulty was partly met by ,foucault1 by using five concave reflectors instead of one1 but even then the greatest distance he found it practicable to use was only #bj meters4 ,returning to ,fig4 #a1 suppose that ,r is in the principal focus of the lens ,l2 then1 if the plane mirror ,m have the same diameter as the lens1 the first1 or moving image1 will remain upon ,m as long as the axis of the pencil of light remains on the lens1 and this will be the case no matter what the distance may be4 ,when the rotation of the mirror ,r becomes sufficiently rapid1 then the flashes of light which produce the second or stationary image become blended1 so that the image appears to be continuous4 ,but now it no longer coincides with the slit1 but is deflected in the direction of rotation1 and through twice the angular distance described by the mirror1 during the time required for light to travel twice the distance between the mirrors4 ,this displacement is measured by the tangent of the arc it subtends4 ,to make this as large as possible1 the distance between the mirrors1 the radius1 and the speed of rotation should be made as great as possible4 ,the second condition conflicts with the first1 for the radius is the difference between the focal length for parallel rays1 and that for rays at the distance of the fixed mirror4 ,the greater the distance1 therefore1 the smaller will be the radius4 ,there are two ways of solving the difficulty3 first1 by using a lens of great focal length2 and secondly1 by placing the revolving mirror within the principal focus of the lens4 ,both means were employed4 ,the focal length of the lens was #aej feet1 and the mirror was placed about #ae feet within the principal focus4 ,a limit is soon reached1 however1 for the quantity of light received diminishes very rapidly as the revolving mirror approaches the lens4 ,arrangement and ,description of ,apparatus4 ,site and ,plan4 ,the site selected for the experiments was a clear1 almost level1 stretch along the north sea-wall of the ,naval ,academy4 ,a frame building was erected at the western end of the line1 a plan of which is represented in ,fig4 #c4 ,illustration3 ,,fig4 #c4 ,the building was #de feet long and #ad feet wide1 and raised so that the line along which the light traveled was about #aa feet above the ground4 ,a heliostat at ,h reflected the sun's rays through the slit at ,s to the revolving mirror ,r1 thence through a hole in the shutter1 through the lens1 and to the distant mirror4 ,the ,heliostat4 ,the heliostat was one kindly furnished by ,dr4 ,woodward1 of the ,army ,medical ,museum1 and was a modification of ,foucault's form1 designed by ,keith4 ,it was found to be accurate and easy to adjust4 ,the light was reflected from the heliostat to a plane mirror1 ,m1 ,fig4 #c1 so that the former need not be disturbed after being once adjusted4 ,the ,revolving ,mirror4 ,the revolving mirror was made by ,fauth ,co41 of ,washington4 ,it consists of a cast-iron frame resting on three leveling screws1 one of which was connected by cords to the table at ,s1 ,fig4 #c1 so that the mirror could be inclined forward or backward while making the observations4 ,illustration3 ,,fig4 #d4 ,two binding screws1 ,s1 ,s1 ,fig4 #d1 terminating in hardened steel conical sockets1 hold the revolving part4 ,this consists of a steel axle1 ,x1 ,y1 ,figs4 #d and #e1 the pivots being conical and hardened4 ,the axle expands into a ring at ,r1 which holds the mirror ,m4 ,the latter was a disc of plane glass1 made by ,alvan ,clark ,sons1 about #a inch in diameter and #j4b inch thick4 ,it was silvered on one side only1 the reflection taking place from the outer or front surface4 ,a species of turbine wheel1 ,t1 is held on the axle by friction4 ,this wheel has six openings for the escape of air2 a section of one of them is represented in ,fig #f4 ,illustration3 ,,fig4 #e4 ,illustration3 ,,fig4 #f4 ,adjustment of the ,revolving ,mirror4 ,the air entering on one side at ,o1 ,fig4 #e1 acquires a rotary motion in the box ,b1 ,b1 carrying the wheel with it1 and this motion is assisted by the reaction of the air in escaping4 ,the disc ,c serves the purpose of bringing the center of gravity in the axis of rotation4 ,this was done1 following ,foucault's plan1 by allowing the pivots to rest on two inclined planes of glass1 allowing the arrangement to come to rest1 and filing away the lowest part of the disc2 trying again1 and so on1 till it would rest in indifferent equilibrium4 ,the part corresponding to ,c1 in ,foucault's apparatus1 was furnished with three vertical screws1 by moving which the axis of figure was brought into coincidence with the axis of rotation4 ,this adjustment was very troublesome4 ,fortunately1 in this apparatus it was found to be unnecessary4 ,when the adjustment is perfect the apparatus revolves without giving any sound1 and when this is accomplished1 the motion is regular and the speed great4 ,a slight deviation causes a sound due to the rattling of the pivots in the sockets1 the speed is very much diminished1 and the pivots begin to wear4 ,in ,foucault's apparatus oil was furnished to the pivots1 through small holes running through the screws1 by pressure of a column of mercury4 ,in this apparatus it was found sufficient to touch the pivots occasionally with a drop of oil4 ,illustration3 ,,fig4 #g4 ,fig4 #g is a view of the turbine1 box1 and supply-tube1 from above4 ,the quantity of air entering could be regulated by a valve to which was attached a cord leading to the observer's table4 ,the instrument was mounted on a brick pier4 ,the ,micrometer4 ,illustration3 ,,fig4 #h4 ,the apparatus for measuring the deflection was made by ,grunow1 of ,new ,york4 ,this instrument is shown in perspective in ,fig4 #h1 and in plan by ,fig4 #i4 ,the adjustable slit ,s is clamped to the frame ,f4 ,a long millimeter-screw1 not shown in ,fig4 #h1 terminating in the divided head ,d1 moves the carriage ,c1 which supports the eye-piece ,e4 ,the frame is furnished with a brass scale at ,f for counting revolutions1 the head counting hundredths4 ,the eye-piece consists of a single achromatic lens1 whose focal length is about two inches4 ,at its focus1 in ,h1 and in nearly the same plane as the face of the slit1 is a single vertical silk fiber4 ,the apparatus is furnished with a standard with rack and pinion1 and the base furnished with leveling screws4 ,manner of ,using the ,micrometer4 ,in measuring the deflection1 the eye-piece is moved till the cross-hair bisects the slit1 and the reading of the scale and divided head gives the position4 ,this measurement need not be repeated unless the position or width of the slit is changed4 ,then the eye-piece is moved till the cross-hair bisects the deflected image of the slit2 the reading of scale and head are again taken1 and the difference in readings gives the deflection4 ,the screw was found to have no lost motion1 so that readings could be taken with the screw turned in either direction4 ,measurement of ,speed of ,rotation4 ,to measure the speed of rotation1 a tuning-fork1 bearing on one prong a steel mirror1 was used4 ,this was kept in vibration by a current of electricity from five gravity cells4 ,the fork was so placed that the light from the revolving mirror was reflected to a piece of plane glass1 in front of the lens of the eye-piece of the micrometer1 inclined at an angle of #de1 and thence to the eye4 ,when fork and revolving mirror are both at rest1 an image of the revolving mirror is seen4 ,when the fork vibrates1 this image is drawn out into a band of light4 ,when the mirror commences to revolve1 this band breaks up into a number of moving images of the mirror2 and when1 finally1 the mirror makes as many turns as the fork makes vibrations1 these images are reduced to one1 which is stationary4 ,this is also the case when the number of turns is a submultiple4 ,when it is a multiple or simple ratio1 the only difference is that there are more images4 ,hence1 to make the mirror execute a certain number of turns1 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 rest4 ,the electric fork made about #abh vibrations per second4 ,no dependence was placed upon this rate1 however1 but at each set of observations it is compared with a standard ,ut fork1 the temperature being noted at the same time4 ,in making the comparison the sound-beats produced by the forks were counted for #fj seconds4 ,it is interesting to note that the electric fork1 as long as it remained untouched and at the same temperature1 did not change its rate more than one or two hundredths vibrations per second4 ,illustration3 ,,fig4 #i4 ,the ,observer's ,table4 ,fig4 #i ,represents ,the ,table ,at ,which ,the ,observer ,sits4 ,the ,light ,from ,the ,heliostat ,passes ,through ,the ,slit ,at ,s1 ,goes ,to ,the ,revolving ,mirror1 c41 ,and1 ,on ,its ,return1 ,forms ,an ,image ,of ,the ,slit ,at ,d1 ,which ,is ,observed ,through ,the ,eye-piece4 ,e ,represents ,the ,electric ,fork 7the ,prongs ,being ,vertical7 ,bearing ,the ,steel ,mirror ,m4 ,k ,is ,the ,standard ,fork ,on ,its ,resonator4 ,c ,is ,the ,cord ,attached ,to ,the ,valve ,supplying ,air ,to ,the ,turbine4 ,the ,lens4 ,the lens was made by ,alvan ,clark ,sons4 ,it was #h inches in diameter2 focal length1 #aej feet2 not achromatic4 ,it was mounted in a wooden frame1 which was placed on a support moving on a slide1 about #af feet long1 placed about #hj feet from the building4 ,as the diameter of the lens was so small in comparison with its focal length1 its want of achromatism was inappreciable4 ,for the same reason1 the effect of parallax 7due to want of coincidence in the plane of the image with that of the silk fiber in the eye-piece7 was too small to be noticed4 ,the ,fixed ,mirror4 ,the fixed mirror was one of those used in taking photographs of the transit of ,venus4 ,it was about #g inches in diameter1 mounted in a brass frame capable of adjustment in a vertical and a horizontal plane by screw motion4 ,being wedge-shaped1 it had to be silvered on the front surface4 ,to facilitate adjustment1 a small telescope furnished with cross-hairs was attached to the mirror by a universal joint4 ,the heavy frame was mounted on a brick pier1 and the whole surrounded by a wooden case to protect it from the sun4 ,adjustment of the ,fixed ,mirror4 ,the adjustment was effected as follows3 ,a theodolite was placed at about #ajj feet in front of the mirror1 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 mirror4 ,then the telescope attached to the mirror was pointed 7without moving the mirror itself7 at a mark on a piece of card-board attached to the theodolite4 ,thus the line of collimation of the telescope was placed at right angles to the surface of the mirror4 ,the theodolite was then moved to #a1jjj feet1 and1 if found necessary1 the adjustment was repeated4 ,then the mirror was moved by the screws till its telescope pointed at the hole in the shutter of the building4 ,the adjustment was completed by moving the mirror1 by signals1 till the observer1 looking through the hole in the shutter1 through a good spy-glass1 saw the image of the spy-glass reflected centrally in the mirror4 ,the whole operation was completed in a little over an hour4 ,notwithstanding the wooden case about the pier1 the mirror would change its position between morning and evening2 so that the last adjustment had to be repeated before every series of experiments4 ,apparatus for ,supplying and ,regulating the ,blast of ,air4 ,fig4 #aj represents a plan of the lower floor of the building4 ,e is a three-horse power ,lovegrove engine and boiler1 resting on a stone foundation2 ,b1 a small ,roots' blower2 ,g1 an automatic regulator4 ,from this the air goes to a delivery-pipe1 up through the floor1 and to the turbine4 ,the engine made about #d turns per second and the blower about #ae4 ,at this speed the pressure of the air was about half a pound per square inch4 ,illustration3 ,,fig4 #aj4 ,the regulator1 ,fig4 #aa1 consists of a strong bellows supporting a weight of #cgj pounds1 partly counterpoised by #hj pounds in order to prevent the bellows from sagging4 ,when the pressure of air from the blower exceeds the weight1 the bellows commences to rise1 and1 in so doing1 closes the valve ,v4 ,illustration3 ,,fig4 #aa4 ,illustration3 ,,fig4 #ab4 ,this arrangement was found in practice to be insufficient1 and the following addition was made3 ,a valve was placed at ,p1 and the pipe was tapped a little farther on1 and a rubber tube led to a water-gauge1 ,fig #ab4 ,the column of water in the smaller tube is depressed1 and1 when it reaches the horizontal part of the tube1 the slightest variation of pressure sends the column from one end to the other4 ,this is checked by an assistant at the valve2 so that the column of water is kept at about the same place1 and the pressure thus rendered very nearly constant4 ,the result was satisfactory1 though not in the degree anticipated4 ,it was possible to keep the mirror at a constant speed for three or four seconds at a time1 and this was sufficient for an observation4 ,still it would have been more convenient to keep it so for a longer time4 ,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 air4 ,it may be mentioned that the test of uniformity was very delicate1 as a change of speed of one or two hundredths of a turn per second could easily be detected4 ,method ,followed in ,experiment4 ,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 sunrise1 or during the hour before sunset4 ,at other times the image was boiling so as not to be recognizable4 ,in one experiment the electric light was used at night1 but the image was no more distinct than at sunset1 and the light was not steady4 ,the method followed in experiment was as follows3 ,the fire was started half an hour before1 and by the time everything was ready the gauge would show #dj or #ej pounds of steam4 ,the mirror was adjusted by signals1 as before described4 ,the heliostat was placed and adjusted4 ,the revolving mirror was inclined to the right or left1 so that the direct reflection of light from the slit1 which otherwise would flash into the eye-piece at every revolution1 fell either above or below the eye-piece4#b ,footnote #b3 ,otherwise this light would overpower that which forms the image to be observed4 ,as far as ,i am aware1 ,foucault does not speak of this difficulty4 ,if he allowed this light to interfere with the brightness of the image1 he neglected a most obvious advantage4 ,if he did incline the axis of the mirror to the right or left1 he makes no allowance for the error thus introduced4 ,the revolving mirror was then adjusted by being moved about1 and inclined forward and backward1 till the light was seen reflected back from the distant mirror4 ,this light was easily seen through the coat of silver on the mirror4 ,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 tape1 making the drop of the catenary about an inch1 as then the error caused by the stretch of the tape and that due to the curve just counterbalance each other4 ,the position of the slit1 if not determined before1 was then found as before described4 ,the electric fork was started1 the temperature noted1 and the sound-beats between it and the standard fork counted for #fj seconds4 ,this was repeated two or three times before every set of observations4 ,the eye-piece of the micrometer was then set approximately#c and the revolving mirror started4 ,if the image did not appear1 the mirror was inclined forward or backward till it came in sight4 ,footnote #c3 ,the deflection being measured by its tangent1 it was necessary that the scale should be at right angles to the radius 7the radius drawn from the mirror to one or the other end of that part of the scale which represents this tangent74 ,this was done by setting the eye-piece approximately to the expected deflection1 and turning the whole micrometer about a vertical axis till the cross-hair bisected the circular field of light reflected from the revolving mirror4 ,the axis of the eye-piece being at right angles to the scale1 the latter would be at right angles to radius drawn to the cross-hair4 ,the cord connected with the valve was pulled right or left till the images of the revolving mirror1 represented by the two bright round spots to the left of the cross-hair1 came to rest4 ,then the screw was turned till the cross-hair bisected the deflected image of the slit4 ,this was repeated till ten observations were taken1 when the mirror was stopped1 temperature noted1 and beats counted4 ,this was called a set of observations4 ,usually five such sets were taken morning and evening4 ,illustration3 ,,fig4 #ac4 ,fig4 #ac represents the appearance of the image of the slit as seen in the eye-piece magnified about five times4 ,determination of ,the ,constants4 ,comparison of the ,steel ,tape with the ,standard ,yard4 ,the steel tape used was one of ,chesterman's1 #ajj feet long4 ,it was compared with ,wurdeman's copy of the standard yard1 as follows3 ,temperature was #ee ,fahr4 ,the standard yard was brought under the microscopes of the comparator2 the cross-hair of the unmarked microscope was made to bisect the division marked o1 and the cross-hair of the microscope1 marked ,i1 was made to bisect the division marked #cf4 ,the reading of microscope ,i was taken1 and the other microscope was not touched during the experiment4 ,the standard was then removed and the steel tape brought under the microscopes and moved along till the division marked #j4a 7feet7 was bisected by the cross-hair of the unmarked microscope4 ,the screw of microscope ,i was then turned till its cross-hair bisected the division marked #c4a 7feet71 and the reading of the screw taken4 ,the difference between the original reading and that of each measurement was noted1 care being taken to regard the direction in which the screw was turned1 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-screw4 ,to find the value of one turn1 the cross-hair was moved over a millimeter scale1 and the following were the values obtained3 ,turns of screw of microscope ,i in #amm-- #g4fh #g4gc #g4fj #g4fg #g4fh #g4fb #g4fe #g4eg #g4gb #g4gj #g4fd #g4fi #g4fe #g4ei #g4fc #g4fd #g4ee #g4fe #g4fa #g4fc ,mean #g4fe ,hence one turn #j4acjgmm4 or #j4jjea inch4 ,the length of the steel tape from #j4a to #ii4a was found to be greater than #cc yards1 by #g4d turns 4#ifmm 4#jjc feet4 ,correction for temperature 4#jjc feet4 ,length #ajj4jjj feet4 -------------- ,corrected length #ajj4jjf feet4 ,determination of the ,value of ,micrometer4 ,two pairs of lines were scratched on one slide of the slit1 about #chmm apart1 i4e41 from the center of first pair to center of second pair4 ,this distance was measured at intervals of #amm through the whole length of the screw1 by bisecting the interval between each two pairs by the vertical silk fiber at the end of the eye-piece4 ,with these values a curve was constructed which gave the following values for this distance1 which we shall call ,d3 ,turns of screw4 ,at #j of scale ,d #ch4aee #aj of scale ,d #ch4aee #bj of scale ,d #ch4aej #cj of scale ,d #ch #aej #dj of scale ,d #ch4ade #ej of scale ,d #ch4adj #fj of scale ,d #ch4adj #gj of scale ,d #ch4acj #hj of scale ,d #ch4acj #ij of scale ,d #ch4abe #ajj of scale ,d #ch4abj #aaj of scale ,d #ch4aaj #abj of scale ,d #ch4aje #acj of scale ,d #ch4ajj #adj of scale ,d #ch4ajj ,changing the form of this table1 we find that1-- ,for the first #aj turns the average value of ,d is #ch4aee #bj turns #ch4aec #cj turns #ch4aeb #dj turns #ch4aea #ej turns #ch4adi #fj turns #ch4adh #gj turns #ch4adf #hj turns #ch4add #ij turns #ch4adb #ajj turns #ch4adj #aaj turns #ch4ach #abj turns #ch4ace #acj turns #ch4acb #adj turns #ch4acj ,on comparing the scale with the standard meter1 the temperature being #af4e ,c41 #adj divisions were found to #aci4dfbmm4 ,this multiplied by 7#a 4#jjjjahh #af4e7 #aci4ejemm4 ,one hundred and forty divisions were found to be equal to #adj4jbb turns of the screw1 whence #adj turns of the screw #aci4dhcmm1 or #a turn of the screw #j4iifcjemm4 ,this is the average value of one turn in #adj4 ,but the average value of ,d1 for #adj turns is1 from the preceding table1 #ch4acj4 ,therefore1 the true value of ,d1 is #ch4acj 4#iifcjemm1 and the average value of one turn for #aj1 #bj1 #cj1 etc41 turns1 is found by dividing #ch4acj 4#iifcje by the values of ,d21 given in the table4 ,this gives the value of a turn-- mm4 ,for the first #aj turns #j4iiegj #bj turns #j4iiegj #cj turns #j4iiegc #dj turns #j4iiegg #ej turns #j4iiehj #fj turns #j4iiehc #gj turns #j4iiehi #hj turns #j4iieif #ij turns #j4iifja #ajj turns #j4iifjf #aaj turns #j4iifab #abj turns #j4iifah #acj turns #j4iifbe #adj turns #j4iifcj ,,note4--,the micrometer has been sent to ,professor ,mayer1 of ,hoboken1 to test the screw again1 and to find its value4 ,the steel tape has been sent to ,professor ,rogers1 of ,cambridge1 to find its length again4 7,see page #ade47 ,measurement of the ,distance between the ,mirrors4 ,square lead weights were placed along the line1 and measurements taken from the forward side of one to forward side of the next4 ,the tape rested on the ground 7which was very nearly level71 and was stretched by a constant force of #aj pounds4 ,the correction for length of the tape 7#ajj4jjf7 was #j4ab of a foot4 ,to correct for the stretch of the tape1 the latter was stretched with a force of #ae pounds1 and the stretch at intervals of #bj feet measured by a millimeter scale4 mm4 ,at #ajj feet the stretch was #h4j #hj feet the stretch was #e4j #fj feet the stretch was #e4j #dj feet the stretch was #c4e #bj feet the stretch was #a4e --- --- #cjj #bc4jj ,weighted mean #g4g mm4 ,for #aj pounds1 stretch #e4a mm4 #j4jafg feet4 ,correction for whole distance #j4cc feet4 ,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 reflector2 allowance made in each case for effect of temperature3 #aihe4ac feet4 #aihe4ag feet4 #aihd4ic feet4 #aihe4ji feet4 #aihe4ji feet4 ------- ,mean #aihe4jhb feet4 4#gj4 ,cap of pier to revolving mirror4 4#cc4 ,correction for stretch of tape4 4#ab4 ,correction for length of tape4 -------- #aihf4bc4 ,true distance between mirrors4 ,rate of ,standard ,ut ,fork4 ,the rate of the standard ,ut fork was found at the ,naval ,academy1 but as so much depended on its accuracy1 another series of determinations of its rate was made1 together with ,professor ,mayer1 at the ,hoboken ,institute of ,technology4 ,set of determinations made at ,naval ,academy4 ,the fork was armed with a tip of copper foil1 which was lost during the experiments and replaced by one of platinum having the same weight1 #d4f mgr4 ,the fork1 on its resonator1 was placed horizontally1 the platinum tip just touching the lampblacked cylinder of a ,schultze chronoscope4 ,the time was given either by a sidereal break-circuit chronometer or by the break-circuit pendulum of a mean-time clock4 ,in the former case the break-circuit worked a relay which interrupted the current from three ,grove cells4 ,the spark from the secondary coil of an inductorium was delivered from a wire near the tip of the fork4 ,frequently two sparks near together were given1 in which case the first alone was used4 ,the rate of the chronometer1 the record of which was kept at the ,observatory1 was very regular1 and was found by observations of transits of stars during the week to be #a4c seconds per day1 which is the same as the recorded rate4 ,specimen of a ,determination of ,rate of ,ut ,fork4 ,temp4#bg ,c4 ,column #a gives the number of the spark or the number of the second4 ,column #b gives the number of sinuosities or vibrations at the corresponding second4 ,column #c gives the difference between #a and #aa1 #b and #ab1 #c and #ac1 etc4 ,july #d1 #ahgi4 #a #j4a #beeb4j #b #bee4c #beea4g #c #eaj4e #beea4i #d #gfe4f #beea4i #e #ajbj4g #beeb4a #f #abge4g #beeb4j #g #aecj4g #beea4h #h #aghf4e #beea4d #i #bjda4f #beea4g #aj #bbig4j #beea4e ------- #aa #beeb4a #bee4ahj mean #aj4 #ab #bhjg4j 4#fii reduction for mean time4 #ac #cjfb4d 4#jjc correction for rate4 #ad #ccag4e 4#ahg correction for temperature4 ------- #ae #cegb4h #bef4jfi number of vibrations per second at #fe ,fahr4 #af #chbg4g #ag #djhb4e #ah #dcce4i #ai #deic4c #bj #dhdh4e ,the correction for temperature was found by ,professor ,mayer by counting the sound-beats between the standard and another ,ut fork1 at different temperatures4 ,his result is 4#jab vibrations per second for a diminution of #a ,fahr4 ,using the same method1 ,i arrived at the result 4#jabe4 ,adopted 4#jab4 ,rsum of determinations made at ,naval ,academy4 ,in the following table the first column gives the date1 the second gives the total number of seconds1 the third gives the result uncorrected for temperature1 the fourth gives the temperature 7centigrade71 the fifth gives the final result1 and the sixth the difference between the greatest and least values obtained in the several determinations for intervals of ten seconds3 ,july #d #bj #bee4hhb #bg4j #bef4jfi #j4jg #e #ai #bee4iae #bf4d #bef4jhi #j4je #e #ah #bee4iaa #bf4j #bef4jgg #j4jb #f #ba #bee4hgd #bd4g #bef4jab #j4ac #f #i #bee4idh #bd4h #bef4jhg #j4bd #g #bb #bee4ich #bd4f #bef4jgd #j4je #g #ba #bee4iaa #be4c #bef4jfa #j4jd #h #bj #bee4iba #bf4f #bef4ajj #j4jb #h #bj #bee4ije #bf4f #bef4jhd #j4jf #h #bj #bee4hhg #bf4f #bef4jff #j4jc ------- ,mean #bef4jgb ,in one of the preceding experiments1 ,i compared the two ,vt forks while the standard was tracing its record on the cylinder1 and also when it was in position as for use in the observations4 ,the difference1 if any1 was less than 4#ja vibration per second4 ,second determination4 7,joint work with ,professor ,a4,m4 ,mayer1 ,stevens ,institute1 ,hoboken47 ,the fork was wedged into a wooden support1 and the platinum tip allowed to rest on lampblacked paper1 wound about a metal cylinder1 which was rotated by hand ,time was given by a break-circuit clock1 the rate of which was ascertained1 by comparisons with ,western ,union time-ball1 to be #i4hg seconds4 ,the spark from secondary coil of the inductorium passed from the platinum tip1 piercing the paper4 ,the size of the spark was regulated by resistances in primary circuit4 ,the following is a specimen determination3 ,column #a gives the number of the spark or the number of seconds4 ,column #b gives the corresponding number of sinuosities or vibrations4 ,column #c gives the difference between the #ast and #gth #f1 #bnd and #hth #f1 etc4 #a #j4c #bee4hc #b #bef4a #bee4ij #c #eaa4g #bee4ij #d #gfg4i #bee4ic #e #ajbc4e #bee4ib #f #abhi4b #bef4ja #g #aece4c #bee4ie ------- #h #agia4e #bee4ibj mean4 #i #bjdg4a - 4#jbh correction for rate4 ------- #aj #bcjc4e #bee4hib #aa #beei4j 4#ahj correction for temperature4 ------- #ab #bhbe4c #bef4jgb number of vibrations per second at #fe ,fahr4 #ac #cjga4j ,in the following rsum1 column #a gives the number of the experiments4 ,column #b gives the total number of seconds4 ,column #c gives the result not corrected for temperature4 ,column #d gives the temperature ,fahrenheit4 ,column #e gives the final result4 ,column #f gives the difference between the greatest and least values3 #a #ac #bee4hib #hj #bef4jgb #j4ah #b #aa #bee4icd #ha #bef4abf #j4ag #c #ac #bee4hii #ha #bef4jia #j4ab #d #ac #bee4ihh #ge #bef4ajh #j4ac #e #aa #bee4idh #ge #bef4jfh #j4je #f #ab #bee4igj #ge #bef4jij #j4je #g #ab #bee4iib #ge #bef4aab #j4bj #h #aa #bee4iib #gf #bef4abd #j4jc #i #aa #bee4hhh #ha #bef4jhj #j4ac #aj #ac #bee4hgh #ha #bef4jgj #j4ac ------- ,mean #bef4jid ,effect of ,support and of ,scraping4 ,the standard ,vt 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 second4 ,the standard was free from the cylinder4 ,the beats were counted by coincidences with the second beats of a watch4 ,specimen4 ,coincidences were marked-- ,at #cb seconds4 #cg seconds4 #dc4e seconds4 #di seconds4 #ed4e seconds4 #fa4e seconds4 #fa4e - #cb #bi4e4 #bi4e #e #e4i time of one interval4 ,rsum4 #a #e4i #b #f4b #c #f4b #d #f4b ---- ,mean #f4ac time of one interval between coincidences4 ,in this time the watch makes #f4ace #cj4fe beats1 and the forks make #cj4fe #a #ca4fe beats4 ,hence the number of beats per second is #ca4fe #f4ac #e4afc4 ,specimen4 ,circumstances the same as in last case1 except that standard ,vt fork was allowed to trace its record on the lampblacked paper1 as in finding its rate of vibration4 ,coincidences were marked at-- #ei seconds4 #jd seconds4 #aj4e seconds4 #ag seconds4 #gg - #ei #ah4 #ah #c #f4j time of one interval4 ,rsum4 ,no4 #a #f4j seconds4 #f4ca #e #ca4ee #b #f4j seconds4 #a4jj #c #f4g seconds4 ---- #d #f4c seconds4 #e #f4e seconds4 #cb4ee #f #f4g seconds4 #cb4ee #f4ca #e4aei #g #f4j seconds4 ,with fork free #e4afc ---- ----- ,mean #f4ca seconds ,effect of scrape - 4#jdd ,specimen4 ,circumstances as in first case1 except that both forks were on their resonators4 ,coincidences were observed at-- #ba seconds4 #bh seconds4 #cf seconds4 #dd seconds4 #ea seconds4 #fj seconds4 #fj - #ba #ci #ci #e #g4h time of one interval4 ,rsum4 ,no4 #a #g4h seconds4 #g4db #e #cg4aj #b #g4a seconds4 #a4jj #c #g4f seconds4 ----- #d #g4d seconds4 #ch4aj #e #g4b seconds4 #ch4aj #g4db #e4acc ---- 7,above7 #e4aei ----- ,mean #g4db seconds4 ,effect of support and scrape - 4#jbf ,mean of second determination was #bef4jid ,applying correction 7scrape1 etc47 - 4#jbf ------- ,corrected mean #bef4jfh ,result of first determination #bef4jgb ------- ,final value #bef4jgj ,,note--,the result of first determination excludes all work except the series commencing ,july #d4 ,if previous work is included1 and also the result first obtained by ,professor ,mayer1 the result would be #bef4jhi4 #bef4ahj #bef4jcf #bef4jgb #bef4jfh ------- ,mean #bef4jhi ,the previous work was omitted on account of various inaccuracies and want of practice1 which made the separate results differ widely from each other4 ,the ,formul4 ,the formul employed are-- d 7#a7 tan ----- r #beibjjj ,d n 7#b7 ,v ----------------- angle of deflection4 d corrected displacement 7linear74 r radius of measurement4 ,d twice the distance between the mirrors4 n number of revolutions per second4 inclination of plane of rotation d deflection as read from micrometer4 ,b number of beats per second between electric ,vt fork and standard ,vt ,cor correction for temperature of standard ,vt#c4 ,v velocity of light4 ,t value of one turn of screw4 7,table1 page #abf47 ,substituting for d1 its value or d,tsec 7log sec 4#jjjjh71 and for ,d its value #cigb4df1 and reducing to kilometers1 the formul become-- d,t 7#c7 tan c ----2 log c 4#eafjg r n 7#d7 ,v c ---2 log c 4#difgj ,d and r are expressed in feet and d in millimeters4 ,vt fork makes #bef4jgj vibrations per second at #fe ,fahr4 ,d #cigb4df feet4 tan tangent of angle of inclination of plane of rotation #j4jb in all but the last twelve observations1 in which it was #j4jae4 log c 4#eafjg 74#eafjc in last twelve observations474 log c 4#difgj4 ,the electric fork makes 7#bef4jgj ,b cor47 vibrations per second1 and n is a multiple1 submultiple1 or simple ratio of this4 ,observations4 ,specimen ,observation4 ,june #ag4 sunset4 ,image good2 best in column 7#d74 ,the columns are sets of readings of the micrometer for the deflected image of slit4 #aab4ha #aab4hj #aab4hc #aab4gd #aab4gi #ha #ha #ha #gf #gh #gi #gh #gh #gd #gd #hj #ge #gd #gf #gd #gi #gg #gd #gf #gg #hb #gi #gb #gh #ha #hb #gc #gf #gh #gg #gf #gh #ha #gi #ge #hc #gi #gd #hc #hb #gc #gc #gf #gh #hb ------- ------- ------- ------- ------- ,mean #aab4hja #aab4ggc #aab4gfi #aab4ggb #aab4ggi ,zero #j4bfj #j4bfj #j4bfj #j4bfj #j4bfj ------- ------- ------- ------- ------- d #aab4dea #aab4eac #aab4eji #aab4eab #aab4eai ,temp #gg #gg #gg #gg #gg ,b #a4ejj ,corr - 4#add ------- #a4cfe #bef4jgj ------- n #beg4dbf #beg4dc #beg4dc #beg4dc #beg4dc r #bh4aeg #bh4aeg #bh4aeg #bh4aeg #bh4aeg ,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 completed4 ,the following is the calculation for ,v3 #b;d1 #c;d1 #ast set4 and #dth sets4 #eth set4 log c #eafjg #eafjg #eafjg ,t #iihcb #iihcb #iihcb d #jeaca #jeaai #jeabc ------- ------- ------- #efegj #efeeh #efefb r #ddieh #ddieh #ddieh ------- ------- ------- tan #aafab #aafjj #aafjd #bfid4g #bfid4a #bfid4c c #difgj #difgj #difgj n #dajff #dajff #dajff ------- ------- ------- #ijgcf #ijgcf #ijgcf #dcjeb #dcjdb #dcjdf ------- ------- ------- ,v #dgfhd #dgfid #dgfij ,v #biihjj #biihhj #biihej ,in the following table1 the numbers in the column headed ,distinctness of ,image are thus translated3 #c1 good2 #b1 fair2 #a1 poor4 ,these numbers do not1 however1 show the relative weights of the observations4 ,the numbers contained in the columns headed ,position of ,deflected ,image1 ,position of ,slit1 and displacement of image in divisions were obtained as described in the paragraph headed ,micrometer1 page #abj4 ,the column headed ,b contains the number of beats per second between the electric ,vt fork and the standard ,vt as explained in the paragraph headed ,measurement of the ,speed of ,rotation4 ,the column headed ,cor4 contains the correction of the rate of the standard fork for the difference in temperature of experiment and #fe ,fahr41 for which temperature the rate was found4 ,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 standard1 as explained in the same paragraph4 ,the radius of measurement is the distance between the front face of the revolving mirror and the cross-hair of the micrometer4 ,the numbers in the column headed ,value of one turn of the screw were taken from the table1 page #abg4 ,date4 ,distinctness of image4 ,temperature1 ,fahr4 ,position of deflected image4 ,position of slit4 ,displacement of image in divisions4 ,difference between greatest and least values4 ,b4 ,cor4 ,number of revolutions per second4 ,radius of measurement1 in feet4 ,value of one turn of the screw4 ,velocity of light in air1 in kilometers4 ,remarks4 ,june #ecgfaad4he #j4cjjaad4eej4aga4dbc-j4acbbeg4cfbh4fgbj 4iifadbiihej;,electric light4 ,june #gbgbaad4fd #j4jgdaad4efj4aja4ecc-j4jhdbeg4ebbh4feej 4iifadbiigdj,p4,m4 ,frame inclined at various angles ,june #gbgbaad4eh #j4jgdaad4ejj4jha4ecc-j4jhdbeg4ebbh4fdgj 4iifadbiiijj,p4,m4 ,frame inclined at various angles ,june #gbgb #he4ia #j4jgd #he4hdj4aba4ecc-j4jhdaic4adbh4fdgj4iieih cjjjgj,p4,m4 ,frame inclined at various angles ,june #gbgb #he4ig #j4jgd #he4hi,o4#jga4ecc-j4jhdaic4adbh4fejj4iie ihbiiicj,p4,m4 ,frame inclined at various angles ,june #gbgbaad4fa #j4jgdaad-ecj4jga4ecc-j4jhdbeg4dbbh4fejj 4iifadbiihej,p4,m4 ,frame inclined at various angles ,june #ichcaad4ed #j4jgdaad4dgj4jga4ecc-j4bafbeg4cibh4fehj 4iifadbiiiej,p4,m4 ,frame inclined at various angles ,june #ichcaad4ed #j4jgdaad4dfj4aja4ecc-j4bafbeg4cibh4fehj 4iifadbiiihj,p4,m4 ,frame inclined at various angles ,june #ichcaad4eg #j4jgdaad4dgj4jha4ecc-j4bafbeg4cibh4ffbj 4iifadbiiihj,p4,m4 ,frame inclined at various angles ,june #ichcaad4eg #j4jgdaad4ejj4jfa4ecc-j4bafbeg4cibh4ffjj 4iifadbiihhj,p4,m4 ,frame inclined at various angles ,june #ibhcaad4fa #j4jgdaad4ecj4aca4ecc-j4bafbeg4cibh4fghj 4iifadcjjjjj,p4,m4 ,frame inclined at various angles ,june #ajbijaad4fj #j4jgdaad4ebj4aaa4eag-j4cjjbeg4bibh4fhej 4iifadbiiihj,p4,m4 ,june #ajbijaad4fb #j4jgdaad4edj4jha4eag-j4cjjbeg4bibh4fhej 4iifadbiiicj,p4,m4 ,june #abbgaaad4ha #j4jgdaad4gdj4jia4dej-j4jgbbeg4debh4fijj 4iifadbiifej;,a4,m4 ,june #abbgaaad4gh #j4jgdaad4gjj4jea4dej-j4jgbbeg4debh4fijj 4iifadbiigfj;,a4,m4 ,june #abagaaad4gf #j4jgdaad4fhj4jia4dej-j4jgbbeg4debh4fijj 4iifadbiihaj;,a4,m4 ,june #accgbaab4fd #j4jgdaab4efj4jia4ejj-j4jhdbeg4dibh4agbj 4iifadcjjjjj;,a4,m4 ,june #accgbaab4fc #j4jgdaab4efj4aja4ejj-j4jhdbeg4dibh4agbj 4iifadcjjjjj;,a4,m4 ,june #acbgbaab4fe #j4jgdaab4egj4jha4ejj-j4jhdbeg4dibh4agbj 4iifadbiiifj;,a4,m4 ,june #accgiaab4hb #j4bfjaab4efj4jfa4eag-j4afhbeg4dbbh4aghj 4iifadbiiifj,p4,m4 ,june #accgiaab4hb #j4bfjaab4efj4aca4eag-j4afhbeg4dbbh4aghj 4iifadbiiifj,p4,m4 ,june #accgiaab4hc #j4bfjaab4egj4jga4eag-j4afhbeg4dbbh4aghj 4iifadbiiidj,p4,m4 ,june #accgiaab4hb #j4bfjaab4efj4jfa4eag-j4afhbeg4dbbh4aghj 4iifadbiiifj,p4,m4 ,june #accgiaab4hc #j4bfjaab4egj4aaa4eag-j4afhbeg4dbbh4aghj 4iifadbiiidj,p4,m4 ,june #accgiaac4da #j4bfjaac4aeaa #a4eag-j4afhbeh4gjbh4aebj4iifadbiihhj,p4 ,m4 ,set micrometer and counted oscillations4 ,june #accgiaab4ad #j4bfjaaa4hhf #a4eag-j4afhbee4fibh4aebj4iifadbiihjj,os cillations of image of revolving mirror4 ,june #adafdaab4hc #j4bfjaab4egj4aba4ejjj4jabbeg4ehbh4aebj4 iifadbiihej;,a4,m4 ,june #adafdaab4hc #j4bfjaab4egj4jea4eagj4jabbeg4fjbh4aebj4 iifadbiihhj;,a4,m4 ,june #adafeaab4ha #j4bfjaab4eej4aaa4eag #j4jjjbeg4eibh4aebj4iifadbiiijj;,a4,m4 ,june #adaffaab4hc #j4bfjaab4egj4jia4eag-j4jabbeg4egbh4aebj 4iifadbiihdj;,a4,m4 ,june #adafgaab4hc #j4bfjaab4egj4aba4eag-j4jbdbeg4efbh4aebj 4iifadbiihcj;,a4,m4 ,june #adahdaab4gh #j4bfjaab4ebj4jfa4eag-j4bbhbeg4cfbh4aeij 4iifadbiigij,p4,m4 ,readings taken by ,lieut4 ,nazro4 ,june #adaheaab4gf #j4bfjaab4ejj4jha4ejj-j4bdjbeg4ccbh4aeij 4iifadbiihaj,p4,m4 ,readings taken by ,lieut4 ,nazro4 ,june #adahdaab4gb #j4bfjaab4dfj4jha4dhc-j4bbhbeg4cbbh4aeij 4iifadbiihhj,p4,m4 ,readings taken by ,lieut4 ,nazro4 ,june #adahdaab4gc #j4bfjaab4dgj4jia4dhc-j4bbhbeg4cbbh4aeij 4iifadbiihhj,p4,m4 ,june #adahdaab4ge #j4bfjaab4dij4jia4dhc-j4bbhbeg4cbbh4abij 4iifadbiihcj,p4,m4 ,june #agbfbaab4he #j4bfjaab4eij4jia4eagj4jcfbeg4fbbh4adij4 iifadbiihjj;,a4,m4 ,june #agbfcaab4hd #j4bfjaab4ehj4jfa4ejjj4jbdbeg4eibh4adij4 iifadbiigij;,a4,m4 ,june #agafdaab4he #j4bfjaab4eij4jga4ejjj4jabbeg4ehbh4adij4 iifadbiigfj;,a4,m4 ,june #agcggaab4hj #j4bfjaab4edj4jga4ejj-j4addbeg-dcbh4aegj 4iifadbiihjj,p4,m4 ,readings taken by ,mr4 ,clason4 ,june #agcggaab4gg #j4bfjaab4eaj4jha4ejj-j4addbeg4dcbh4aegj 4iifadbiihhj,p4,m4 ,readings taken by ,mr4 ,clason4 ,june #agcggaab4gg #j4bfjaab4eaj4aaa4ejj-j4addbeg4dcbh4aegj 4iifadbiihhj,p4,m4 ,readings taken by ,mr4 ,clason4 ,june #agcggaab4gg #j4bfjaab4eaj4jia4ejj-j4addbeg4dcbh4aegj 4iifadbiihhj,p4,m4 ,readings taken by ,mr4 ,clason4 ,june #agcggaab4gh #j4bfjaab4ebj4jha4ejj-j4addbeg #dcbh4aegj4iifadbiihfj,p4,m4 ,readings taken by ,mr4 ,clason4 ,june #ahaehaab4ij #j4bfeaab4fdj4jga4ejjj4jhdbeg4febh4aejj4 iifadbiigbj;,a4,m4 ,june #ahaehaab4ij #j4bfeaab4fdj4aja4ejjj4jhdbeg4febh4aejj4 iifadbiigbj;,a4,m4 ,june #ahaeiaab4ib #j4bfeaab4ffj4jga4dhcj4jgbbeg4fbbh4aejj4 iifadbiifbj;,a4,m4 ,june #ahbgeaab4gi #j4bfeaab4ebj4jia4dhc-j4abjbeg-dcbh4aehj 4iifadbiihfj,p4,m4 ,june #ahbgeaab4ge #j4bfeaab4dhj4aja4dhc-j4abjbeg-dcbh4aehj 4iifadbiiigj,p4,m4 ,june #ahbgeaab4gf #j4bfeaab4dij4jha4dhc-j4abjbeg-dcbh4aehj 4iifadbiiiej,p4,m4 ,june #bjcfjaab4id #j4bfeaab4fgj4jga4eagj4jfcbeg4febh4agbj4 iifadbiihhj;,a4,m4 ,june #bjcfaaab4ib #j4bfeaab4fej4jia4eagj4jdhbeg4fcbh4agbj4 iifadbiiiaj;,a4,m4 ,june #bjbfbaab4id #j4bfeaab4fgj4jga4eagj4jcfbeg4fbbh4agbj4 iifadbiihej;,a4,m4 ,june #bjbfcaab4ic #j4bfeaab4ffj4jca4eagj4jbdbeg4fabh4agbj4 iifadbiihgj;,a4,m4 ,june #bjbghacc4dh #j4bfeacc4baj4aca4dej-j4aefbeg4cfcc4cdej 4iifbgbiihdj,p4,m4 ,june #bjbgiacc4di #j4bfeacc4bcj4jia4ejj-j4afhbeg4djcc4cdej 4iifbgbiihdj,p4,m4 ,june #bjbhjacc4di #j4bfeacc4bbj4jga4ejj-j4ahjbeg4cicc4cdej 4iifbgbiihej,p4,m4 ,june #bjbgiacc4ej #j4bfeacc4bdj4aca4dhc-j4afhbeg4cicc4cdej 4iifbgbiihdj,p4,m4 ,june #bjbgiacc4di #j4bfeacc4bbj4jfa4dhc-j4afhbeg4chcc4cdej 4iifbgbiihdj,p4,m4 ,june #bjbgiacc4di #j4bfeacc4bbj4aja4dhc-j4afhbeg4chcc4cdej 4iifbgbiihdj,p4,m4 ,june #babfaacc4ef #j4bfeacc4bij4aba4eccj4jdhbeg4fecc4ccbj4 iifbgbiihij;,a4,m4 ,june #babfbacc4eh #j4bfeacc4caj4jha4eccj4jcfbeg4fdcc4ccbj4 iifbgbiihaj;,a4,m4 ,june #babfcacc4eg #j4bfeacc4caj4jia4eccj4jbdbeg4fccc4ccbj4 iifbgbiihaj;,a4,m4 ,june #babfdacc4eg #j4bfeacc4cjj4aaa4eccj4jabbeg4facc4ccbj4 iifbgbiihbj;,a4,m4 ,june #babfeacc4ef #j4bfeacc4cjj4aca4ecc #j4jjjbeg4fjcc4ccbj4iifbgbiihjj;,a4,m4 ,june #bachjacc4dh #j4bfeacc4baj4jfa4ecc-j4ahjbeg4dbcc4ccjj 4iifbgbiiggj,p4,m4 ,june #bachaacc4df #j4bfeacc4aij4aja4ejj-j4aibbeg4chcc4ccjj 4iifbgbiigfj,p4,m4 ,june #bachbacc4df #j4bfeacc4bjj4jea4ejj-j4bjdbeg4cgcc4ccjj 4iifbgbiigdj,p4,m4 ,june #bachbacc4df #j4bfeacc4bjj4jha4eag-j4bjdbeg4chcc4ccjj 4iifbgbiigej,p4,m4 ,june #bachaacc4df #j4bfeacc4aij4jha4ejj-j4aibbeg4chcc4ccjj 4iifbgbiigfj,p4,m4 ,june #bcchiacc4dc #j4bfeacc4afj4jha4edb-j4bhhbeg4cbcc4cdej 4iifbgbiiiaj,p4,m4 ,june #bcchiacc4db #j4bfeacc4aej4jfa4eej-j4bhhbeg4cccc4cdej 4iifbgbiiibj,p4,m4 ,june #bccijacc4dc #j4bfeacc4agj4jia4eej-j4cjjbeg4cbcc4cdej 4iifbgbiihij,p4,m4 ,june #bccijacc4dc #j4bfeacc4afj4jga4ecc-j4cjjbeg4cjcc4cdej 4iifbgbiihfj,p4,m4 ,june #bccijacc4db #j4bfeacc4afj4jga4eag-j4cjjbeg4bicc4cdej 4iifbgbiihhj,p4,m4 ,june #bdcgbacc4dg #j4bfeacc4bjj4aea4eag-j4jhdbeg4ejcc4caij 4iifbgbiigbj;,a4,m4 ,june #bdcgcacc4dd #j4bfeacc4agj4jda4eag-j4jifbeg4dicc4caij 4iifbgbiihdj;,a4,m4 ,june #bdcgdacc4db #j4bfeacc4afj4aaa4eag-j4ajhbeg4dhcc4caij 4iifbgbiihej;,a4,m4 ,june #bdcgeacc4db #j4bfeacc4afj4jfa4eag-j4abjbeg4dgcc4caij 4iifbgbiihej;,a4,m4 ,june #bdcgfacc4dd #j4bfeacc4ahj4aja4eag-j4acbbeg4decc4caij 4iifbgbiighj;,a4,m4 ,june #bfbhfacc4db #j4bfeacc4aej4jea4ejh-j4bebbeg4cccc4ccij 4iifbgbiihij,p4,m4 ,june #bfbhfacc4dd #j4bfeacc4agj4jha4ejh-j4bebbeg4cccc4ccij 4iifbgbiihdj,p4,m4 ,june #bgcgcacc4di #j4bfeacc4bbj4aaa4dhc-j4jifbeg4dfcc4cbhj 4iifbgbiighj;,a4,m4 ,june #bgcgdacc4dg #j4bfeacc4bjj4jfa4dhc-j4ajhbeg4ddcc4cbhj 4iifbgbiihaj;,a4,m4 ,june #bgcgeacc4dg #j4bfeacc4baj4jia4dhc-j4abjbeg4dccc4cbhj 4iifbgbiigfj;,a4,m4 ,june #bgcgeacc4de #j4bfeacc4aij4jia4dfg-j4abjbeg4dbcc4cbhj 4iifbgbiihaj;,a4,m4 ,june #bgcgfacc4dg #j4bfeacc4bjj4jha4dhc-j4acbbeg4dbcc4cbhj 4iifbgbiigij;,a4,m4 ,june #bgcgfacc4de #j4bfeacc4aij4aja4dhc-j4acbbeg4dbcc4cbhj 4iifbgbiihaj;,a4,m4 ,june #cjbhe #ce4cbace4jj #ii4fhj4jea4ejj-j4bdjaic4jjcc4bgdj4iifde biihbj,p4,m4 ,mirror inverted4 ,june #cjbhf #ce4cdace4jj #ii4fgj4jfa4ejh-j4bebaic4jjcc4bgdj4iifde biihej,p4,m4 ,mirror inverted4 ,june #cjbhf #ce4cdace4jj #ii4ffj4aja4ejh-j4bebaic4jjcc4bgdj4iifde biihgj,p4,m4 ,mirror inverted4 ,june #cjbhf #ce4cdace4jj #ii4ffj4jia4eag-j4bebaic4jjcc4bgdj4iifde biihgj,p4,m4 ,mirror inverted4 ,july #abhc #jb4agace4adeacb4ihj4jga4ejj-j4bafbeg4ce cc4bhbj4iifbgbiihaj,p4,m4 ,mirror inverted4 ,july #abhd #jb4aeace4adeacc4jjj4jia4ejj-j4bbhbeg4cd cc4bhbj4iifbgbiigdj,p4,m4 ,mirror inverted4 ,july #abhf #jb4adace4adeacc4jaj4jfa4dfg-j4bebbeg4bh cc4caaj4iifbgbiihaj,p4,m4 ,mirror inverted4 ,july #abhf #jb4adace4adeacc4jjj4jha4dfg-j4bebbeg4bh cc4caaj4iifbgbiiidj,p4,m4 ,mirror inverted4 ,july #bchf #ii4he #j4djj #ii4dej4jea4dej-j4bebaib4iecc4bjej4iifjf biiiej,p4,m4 ,mirror erect4 ,july #bchf #ff4gd #j4djj #ff4cdj4jca4dej-j4bebabh4fccc4bjej4iiehf biihjj,p4,m4 ,mirror erect4 ,july #bchf #ej4af #j4djj #dg4ifj4jga4dfg-j4beb #if4dhcc4bjej4iiehjbiihaj,p4,m4 ,mirror erect4 ,july #bche #cc4eg #j4djj #cc4agj4jfa4dej-j4bdj #fd4cbcc4bjej4iiegdbiihgj,p4,m4 ,mirror erect4 ,in the last two sets of ,june #ac1 the micrometer was fixed at #aac4da and #aab4ad respectively4 ,the image was bisected by the cross-hair1 and kept as nearly as possible in this place1 meantime counting the number of seconds required for the image of the revolving mirror to complete #fj oscillations4 ,in other words1 instead of measuring the deflection1 the speed of rotation was measured4 ,in column #g for these two sets1 the numbers #aa and #f are the differences between the greatest and the smallest number of seconds observed4 ,in finding the mean value of ,v from the table1 the sets are all given the same weight4 ,the difference between the result thus obtained and that from any system of weights is small1 and may be neglected4 ,the following table gives the result of different groupings of sets of observations4 ,necessarily some of the groups include others3 ,electric light 7#a set7 #biihej ,set micrometer counting oscillations 7#b7 #biihdj ,readings taken by ,lieutenant ,nazro 7#c7 #biihcj ,readings taken by ,mr4 ,clason 7#e7 #biihfj ,mirror inverted 7#h7 #biihdj ,speed of rotation1 #aib 7#g7 #biiiij ,speed of rotation1 #abh 7#a7 #biihjj ,speed of rotation1 #if 7#a7 #biihaj ,speed of rotation1 #fd 7#a7 #biihgj ,radius1 #bh4e feet 7#ed7 #biihgj ,radius1 #cc4c feet 7#df7 #biihcj ,highest temperature1 #ij ,fahr4 7#e7 #biiiaj ,mean of lowest temperatures1 #fj ,fahr4 7#g7 #biihjj ,image1 good 7#df7 #biihfj ,image1 fair 7#ci7 #biihfj ,image1 poor 7#ae7 #biihaj ,frame1 inclined 7#e7 #biiifj ,greatest value #cjjjgj ,least value #biifej ,mean value #biiheb ,average difference from mean #fj ,value found for #c4bf ,probable error #e ,discussion of ,errors4 ,the value of ,v depends on three quantities ,d1 n1 and 4 ,these will now be considered in detail4 ,the ,distance4 ,the distance between the two mirrors may be in error1 either by an erroneous determination of the length of the steel tape used1 or by a mistake in the measurement of the distance by the tape4 ,the first may be caused by an error in the copy of the standard yard1 or in the comparison between the standard and the tape4 ,an error in this copy1 of 4#jjjcf inch1 which1 for such a copy1 would be considered large1 would produce an error of only 4#jjjja in the final result4 ,supposing that the bisections of the divisions are correct to 4#jjje inch1 which is a liberal estimate1 the error caused by supposing the error in each yard to be in the same direction would be only 4#jjjjad2 or the total error of the tape1 if both errors were in the same direction1 would be #jjjjbd of the whole length4 ,the calculated probable error of the five measurements of the distance was 4#jjjjae2 hence the total error due to ,d would be at most 4#jjjjd4 ,the tape has been sent to ,professor ,rogers1 of ,cambridge1 for comparison1 to confirm the result4 ,the ,speed of ,rotation4 ,this quantity depends on three conditions4 ,it is affected1 first1 by an error in the rate of the standard2 second1 by an error in the count of the sound beats between the forks2 and third1 by a false estimate of the moment when the image of the revolving mirror is at rest1 at which moment the deflection is measured4 ,the calculated probable error of the rate is 4#jjjjaf4 ,if this rate should be questioned1 the fork can be again rated and a simple correction applied4 ,the fork is carefully kept at the ,stevens ,institute1 ,hoboken1 and comparisons were made with two other forks1 in case it was lost or injured4 ,in counting the sound beats1 experiments were tried to find if the vibrations of the standard were affected by the other fork1 but no such effect could be detected4 ,in each case the number of beats was counted correctly to 4#jb1 or less than 4#jjja part1 and in the great number of comparisons made this source of error could be neglected4 ,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 increasing1 and sometimes when slowly decreasing4 ,further1 this error would form part of the probable error deduced from the results of observations4 ,we may then conclude that the error1 in the measurement of n1 was less than 4#jjjjb4 ,the ,deflection4 ,the angle of deflection was measured by its tangent1 tan d_/r2 d was measured by the steel screw and brass scale1 and r by the steel tape4 ,the value of one turn of the screw was found by comparison with the standard meter for all parts of the screw4 ,this measurement1 including the possible error of the copy of the standard meter1 ,i estimate to be correct to 4#jjjje part4 ,the instrument is at the ,stevens ,institute1 where it is to be compared with a millimeter scale made by ,professor ,rogers1 of ,cambridge4 ,the deflection was read to within three or four hundredths of a turn at each observation1 and this error appears in the probable error of the result4 ,the deflection is also affected by the inclination of the plane of rotation to the horizon4 ,this inclination was small1 and its secant varies slowly1 so that any slight error in this angle would not appreciably affect the result4 ,the measurement of r is affected in the same way as ,d1 so that we may call the greatest error of this measurement 4#jjjjd4 ,it would probably be less than this1 as the mistakes in the individual measurements would also appear in the probable error of the result4 ,the measurement of was not corrected for temperature4 ,as the corrections would be small they may be applied to the final result4 ,for an increase of #a ,f4 the correction to be applied to the screw for unit length would be -4#jjjjjff4 ,the correction for the brass scale would be 4#jjjjaje1 or the whole correction for the micrometer would be 4#jjjjjd4 ,the correction for the steel tape used to measure r would be 4#jjjjjff4 ,hence the correction for tan4 would be -4#jjjjjc t4 ,the average temperature of the experiments is #ge4f ,f4 #ge4f-fb4e #ac4a4 -4#jjjjjcac4a -4#jjjjd ,hence should be divided by #a4jjjjd1 or the final result should be multiplied by #a4jjjjd4 ,this would correspond to a correction of #ab kilometers4 ,the greatest error1 excluding the one just mentioned1 would probably be less than 4#jjjji in the measurement of 4 ,summing up the various errors1 we find1 then1 that the total constant error1 in the most unfavorable case1 where the errors are all in the same direction1 would be 4#jjjae4 ,adding to this the probable error of the result1 4#jjjjb1 we have for the limiting value of the error of the final result 4#jjjag4 ,this corresponds to an error of #ea kilometers4 ,the correction for the velocity of light in vacuo is found by multiplying the speed in air by the index of refraction of air1 at the temperature of the experiments4 ,the error due to neglecting the barometric height is exceedingly small4 ,this correction1 in kilometers1 is #hj4 ,final ,result4 ,the mean value of ,v from the tables is #biiheb ,correction for temperature #ab ------------ ,velocity of light in air #biihfd ,correction for vacuo #hj ------------ ,velocity of light in vacuo #biiiddea ,the final value of the velocity of light from these experiments is then--#biiidj kilometers per second1 or #ahfchj miles per second4 ,objections ,considered4 ,measurement of the ,deflection4 ,the chief objection1 namely1 that in the method of the revolving mirror the deflection is small1 has already been sufficiently answered4 ,the same objection1 in another form1 is that the image is more or less indistinct4 ,this is answered by a glance at the tables4 ,these show that in each individual observation the average error was only three ten-thousandths of the whole deflection4 ,uncertainty of ,laws of ,reflection and ,refraction in ,media in ,rapid ,rotation4 ,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 rotation4 ,granting that this is the case1 an inspection of ,fig4 #ad shows that the deflection will not be affected4 ,in this figure let m m be the position of the mirror when the light first falls on it from the slit at a1 and m m the position when the light returns4 ,illustration3 ,,fig4 #ad4 ,from the axis o draw op op1 perpendicular to m m and to m m1 respectively4 ,then1 supposing there is no such effect1 the course of the axis of the pencil of light would be a o c mirror c o a4 ,that is1 the angle of deflection would be a o a1 double the angle p o p4 ,if now the mirror be supposed to carry the pencil with it1 let o c be the direction of the pencil on leaving the mirror m m2 i4e41 the motion of the mirror has changed the direction of the reflected ray through the angle c o c4 ,the course would then be a o c1 mirror c o4 ,from o the reflection would take place in the direction a1 making the angles c o p1 and p o a equal4 ,but the angle c o c must be added to p o a1 in consequence of the motion of the mirror1 or the angle of deviation will be a o a c o c2 or a o a c o c d4 7#a7 ,by construction-- c o p p o a 7#b7 c o p p o a 7#c7 ,subtracting 7#c7 from 7#b7 we have-- c o p - c o p p o a - p o a1 or c o c a o a ,substituting a o a for c o c in 7#a7 we have-- a o a a o a a o a d4 ,or the deflection has remained unaltered4 ,retardation ,caused by ,reflection4 ,cornu1 in answering the objection that there may be an unknown retardation by reflection from the distant mirror1 says that if such existed the error it would introduce in his own work would be only #a_/gjjj that of ,foucault1 on account of the great distance used1 and on account of there being in his own experiments but one reflection instead of twelve4 ,in my own experiments the same reasoning shows that if this possible error made a difference of #a per cent4 in ,foucault's work 7and his result is correct within that amount71 then the error would be but 4#jjjjc part4 ,distortion of the ,revolving ,mirror4 ,it1 has been suggested that the distortion of the revolving mirror1 either by twisting or by the effect of centrifugal force1 might cause an error in the deflection4 ,illustration3 ,,fig4 #ae ,the only plane in which the deflection might be affected is the plane of rotation4 ,distortions in a vertical plane would have simply the effect of raising1 lowering1 or extending the slit4 ,again1 if the mean surface is plane there will be no effect on the deflection1 but simply a blurring of the image4 ,even if there be a distortion of any kind1 there would be no effect on the deflection if the rays returned to the same portion whence they were reflected4 ,the only case which remains to be considered1 then1 is that given in ,fig4 #ae1 where the light from the slit a1 falls upon a distorted mirror1 and the return light upon a different portion of the same4 ,the one pencil takes the course a b c d e f a1 while the other follows the path a f g h i b a4 ,in other words1 besides the image coinciding with a1 there would be two images1 one on either side of a1 and in case there were more than two portions having different inclinations there would be formed as many images to correspond4 ,if the surfaces are not plane1 the only effect is to produce a distortion of the image4 ,as no multiplication of images was observed1 and no distortion of the one image1 it follows that the distortion of the mirror was too small to be noticed1 and that even if it were larger it could not affect the deflection4 ,the figure represents the distorted mirror at rest1 but the reasoning is the same when it is in motion1 save that all the images will be deflected in the direction of rotation4 ,imperfection of the ,lens4 ,it has also been suggested that1 as the pencil goes through one-half of the lens and returns through the opposite half1 if these two halves were not exactly similar1 the return image would not coincide with the slit when the mirror was at rest4 ,this would undoubtedly be true if we consider but one-half of the original pencil4 ,it is evident1 however1 that the other half would pursue the contrary course1 forming another image which falls on the other side of the slit1 and that both these images would come into view1 and the line midway between them would coincide with the true position4 ,no such effect was observed1 and would be very unlikely to occur4 ,if the lens was imperfect1 the faults would be all over the surface1 and this would produce simply an indistinctness of the image4 ,moreover1 in the latter part of the observations the mirror was inverted1 thus producing a positive rotation1 whereas the rotation in the preceding sets was negative4 ,this would correct the error mentioned if it existed1 and shows also that no constant errors were introduced by having the rotation constantly in the same direction1 the results in both cases being almost exactly the same4 ,periodic ,variations in ,friction4 ,if the speed of rotation varied in the same manner in each revolution of the mirror1 the chances would be that1 at the particular time when the reflection took place1 the speed would not be the same as the average speed found by the calculation4 ,such a periodic variation could only be caused by the influence of the frame or the pivots4 ,for instance1 the frame would be closer to the ring which holds the mirror twice in every revolution than at other times1 and it would be more difficult for the mirror to turn here than at a position #ij from this4 ,or else there might be a certain position1 due to want of trueness of shape of the sockets1 which would cause a variation of friction at certain parts of the revolution4 ,to ascertain if there were any such variations1 the position of the frame was changed in azimuth in several experiments4 ,the results were unchanged showing that any such variation was too small to affect the result4 ,change of ,speed of ,rotation4 ,in the last four sets of observations the speed was lowered from #bef turns to #aib1 #abh1 #if1 and #fd turns per second4 ,the results with these speeds were the same as with the greater speed within the limits of errors of experiment4 ,bias4 ,finally1 to test the question if there were any bias in taking these observations1 eight sets of observations were taken1 in which the readings were made by another1 the results being written down without divulging them4 ,five of these sets are given in the specimen1 pages #acc-acd4 ,it remains to notice the remarkable coincidence of the result of these experiments with that obtained by ,cornu by the method of the toothed wheel4 ,cornu's result was #cjjdjj kilometers1 or as interpreted by ,helmert #biiiij kilometers4 ,that of these experiments is #biiidj kilometers4 ,postscript4 ,the comparison of the micrometer with two scales made by ,mr4 ,rogers1 of the ,harvard ,observatory1 has been completed4 ,the scales were both on the same piece of silver1 marked ,scales ,no4 #be1 on silver4 ,half inch at #eh ,f41 too short 4#jjjjji inch4 ,centimeter at #fg ,f41 too short 4#jjjjh cm4 ,it was found that the ratio 4#cicgjgi could be obtained almost exactly1 if1 instead of the centimeter being too short1 it were too long by 4#jjjjh cm4 at #fg4 ,on this supposition the following tables were obtained4 ,they represent the value of one turn of the micrometer in millimeters4 ,table #a is the result from centimeter scale4 ,table #b is the result from half-inch scale4 ,table #c is the result from page #ca4 ,it is seen from the correspondence in these results1 that the previous work is correct4 7#a7 7#b7 7#c7 ,from #j to #ac 4#iiefc 4#iiefb 4#iiegj #be 4#iiefb 4#iiefd 4#iiega #ch 4#iiefj 4#iiegb 4#iiegf #ea 4#iiefg 4#iiegh 4#iiehj #fd 4#iiegg 4#iiehf 4#iiehe #gf 4#iiehb 4#iieij 4#iieib #hi 4#iieij 4#iieih 4#iifja #ajb 4#iieif 4#iifjh 4#iifje #aae 4#iifjf 4#iifad 4#iifae #abh 4#iifah 4#iifbb 4#iifbc #adj 4#iifbi 4#iifcc 4#iifcj