LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 RECEIVED BY EXCHANGE 
 
 Class 
 
[Reprinted from the Bulletin of the Bureau of Standards, Vol. 2, No. 1.] 
 
 Talbot's Law as Applied to the 
 Rotating Sectored Disk 
 
 SUBMITTED TO THE BOARD OF UNIVERSITY STUDIES OF 
 
 THE JOHNS HOPKINS UNIVERSITY, IN CONFORMITY 
 
 WITH THE REQUIREMENTS FOR THE DEGREE 
 
 OF DOCTOR OF PHILOSOPHY. 
 
 BY 
 
 EDWARD P. HYDE. 
 
 BALTIMORE, 
 
 1906. 
 
DEPARTMENT OF COMMERCE AND LABOR 
 
 BUREAU OF STANDARDS 
 
 S. W. STRATTON, Director 
 
 BY 
 
 E. P. HYDE, Assistant Physicist 
 \^ 
 
 Bureau of Standards 
 
 REPRINT NO. 26 
 (FROM BULLETIN, VOL. 2, NO. 1, BUREAU OF STANDARDS) 
 
 WASHINGTON 
 
 GOVERNMENT PRINTING OFFICE 
 1906 
 
TALBOT'S LAW AS APPLIED TO THE ROTATING 
 SECTORED DISK. 
 
 By Edward P. Hyde. 
 
 1. Introduction. 
 
 2. Theoretical Discussion. 
 
 a. Radiation from a cylindrical source. 
 
 b. Method of mean distances. 
 
 3. Apparatus and Methods. 
 
 a. General method. 
 
 b. Sources. 
 
 c. Disks. 
 
 d. Details of methods. 
 
 White light. 
 Colored light. 
 
 4. Experimental Results. 
 
 a. Effect of speed. 
 
 b. Disks. 
 
 c. White light. 
 
 d. Colored light. 
 
 5. Conclusions. 
 
 1. INTRODUCTION. 
 
 The rotating sectored disk is one of the most valuable adjuncts in 
 photometric investigation, and yet from the first announcement by 
 Talbot 1 of the law governing its operation until the present time 
 the application of the sectored disk has been limited by a wide- 
 spread doubt of the general truth of the law announced by Talbot. 
 This law is stated by Helmholtz as follows: 2 "If any part of the 
 retina is excited with intermittent light, recurring periodically and 
 
 1 Phil. Mag., Ser. 3, Vol. 5, p. 321; 1834. 
 3 Physiolog. Optik, II Auflage, p. 483. 
 
 166189 
 
2 Bulletin of the Bureati of Standards. \.voi. 2, NO. i. 
 
 t 
 regularly in the same way, and if the period is sufficiently short, a 
 
 continuous impression will result, which is the same as that which 
 would result if the total light received during each period were uni- 
 formly distributed throughout the whole period." 
 
 Talbot's law is thus a statement of physiological rather than of 
 physical phenomena, and depends for its explanation on the action 
 of the eye. Talbot recognized this and was consequently led to 
 state, " I need hardly observe that it would be illogical to assert a 
 priori the existence of this law of optics, however simple and natural 
 it may appear, unless we were perfectly well acquainted with the 
 circumstances which accompany the action of light upon the retina, 
 which is very far from being the case. Its proof can rest on experi- 
 ment alone, and by that it seems to be most satisfactorily established." 
 
 Many experimenters since Talbot have investigated the law, nota- 
 bly Plateau, 1 Helmholtz, 2 Pick, 3 Kleiner, 4 Wiedemann and Messer- 
 schmidt, 5 and more recently Ferry, 6 and Lummer and Brodhun. 7 
 Of the earlier investigators Plateau, Helmholtz, Kleiner, and Wiede- 
 mann and Messerschmidt verified the law within their range of 
 experimental error, which was always several per cent. Pick, from 
 theoretical considerations of the complex action of the eye as shown 
 in the phenomena known as "Anklingen" and " Abklingen," con- 
 cluded that a law as simple as that of Talbot is impossible. He, 
 moreover, repeated Plateau's experiments and was led to the conclu- 
 sion that Talbot's law is not true in general, but that with more 
 intense illuminations the action of the intermittent light is stronger 
 than it should be according to the law, and that perhaps with very 
 weak illumination the effect is reversed. Aubert 8 criticised Pick's 
 conclusions on the ground that the deviations which he found were 
 of the same order of magnitude as his experimental error, and that 
 therefore his results verified the law to within the limit of accuracy 
 of his experiments. 
 
 1 Pogg. Annalen der Physik, 35, p. 457; 1835. 
 
 2 Physiolog. Optik, II Auflage, p. 483. 
 
 3 Reichert's u. du Bois-Reymond's Archiv., p. 739; 1863. 
 
 4 Pfliiger's Archiv. 18, p. 542; 1878. 
 5 \Vied. Ann. 34, p. 465; 1888. 
 Phys. Rev. 1, p. 338; 1893. 
 
 7 Zs. fiir Instrumentenkunde, 16, p. 299; 1896. 
 
 8 Physiologic der Netzhaut, p. 351. 
 
Hyde.} The Rotating Sectored Disk. 3 
 
 Of the two more recent investigations Ferry verified the law for 
 white light, but found quite large errors when the light transmitted 
 through the rotating sectored disk was of a bluer quality than that 
 incident on the other side of the photometer screen. His method 
 consisted in mounting two sources of light on the photometer bench, 
 one at each end, and interposing in the path of the rays from one 
 source a rotating sectored disk with variable openings. The disk 
 \vas mounted on a movable support so that it could readily be intro- 
 duced in the path of the rays and quickly removed again after a 
 reading had been made. 
 
 In this way, using as sources two 16 cp incandescent electric 
 lamps of approximately the same color, he found the law to be 
 verified for all openings of the sector down to a total angle of 24. 
 Moreover, as the voltage on the lamps was changed from 70 to 120 
 volts, provided the two lamps were changed together, the result was 
 the same as before. He then mounted a 50 cp incandescent lamp 
 at the disk end of the bar and a 16 cp lamp at the other end, and 
 found as before that Talbot's law is verified if the two lamps are 
 of approximately the same color. If, however, the 50 cp lamp was 
 burned at a voltage much higher than the normal, or if a piece of 
 slightly tinted blue glass was interposed in the path of the rays 
 intercepted by the sectored disk, a very appreciable deviation from 
 Talbot's law was introduced when the opening of the disk was 
 less than 180, the deviation increasing as the opening was made 
 smaller. Using successively an enclosed arc lamp and a lime light 
 on the sectored disk side of the photometer and an incandescent 
 lamp on the other side, he obtained deviations amounting to 15 per 
 cent and 10 per cent, respectively, at 24, the deviation in each case 
 being approximately zero when the opening of the disk was greater 
 than 1 80. In all cases in which deviations from the law were 
 found the disk let through less light than that demanded by Tal- 
 bot's law. 
 
 As a result of his observations Ferry concluded that although 
 Talbot's law is true for white light, " with mixed light containing 
 elements of different luminosity shining upon the retina, a rotating 
 sectored disk will appear to not cut off all the elements in equal 
 proportion, but will intercept most strongly the elements of low 
 luminosity." Further, " with any given light the error introduced 
 
4 Bulletin of the Bureau of Standards. \yoi.2, NO. i. 
 
 by the use of the rotating sectored disk increases as the aperture of 
 the disk diminishes," although "with ordinary- illuminants, the 
 error is negligible when the total aperture of the disk is more than 
 one-half the entire disk, but rapidly increases as this aperture is 
 diminished." 
 
 Inasmuch as experiments made by the writer yielded results quite 
 different from those of Ferry, it seems desirable to examine care- 
 fully his experiments and results. Unfortunately, owing to lack of 
 detail in Ferry's paper, it is difficult to subject these to a searching 
 criticism, and yet there are several points to which it may be well 
 to call attention. 
 
 First, it is to be noticed that when the two incandescent lamps 
 were varied simultaneously from 70 to 120 volts i. e., over quite a 
 large range of color no error was observed, but that when one lamp 
 was burned at a voltage somewhat higher than normal while the 
 other was kept at the normal voltage a relatively large error was 
 found. The question immediately arises: What would have been 
 the result if the second lamp also had been burned at a voltage 
 somewhat higher than normal ? Since there is nothing in the paper 
 to indicate the normal voltages of the two lamps, which were varied 
 from 70 to 1 20 volts, we are at a loss to answer the question. It is 
 probable, however, that the upper limit of such a large range would 
 represent a voltage at least as high, if not higher, than the normal. 
 On this assumption we would conclude that although the change in 
 color produced by a decrease in voltage from 120 to 70 volts is 
 accompanied by no error in Talbot's law, the color change due to a 
 rise in voltage somewhat above the normal causes a relatively large 
 error in the law for openings of small angles. In other words, the 
 color corresponding to the normal voltage is a critical color. 
 
 Since this is quite improbable, we are led to conclude that the 
 cause of the error when the lamp at the disk end of the bar was 
 raised to a voltage somewhat higher than the normal, while the 
 other lamp was kept at the normal voltage, is to be found in the 
 color difference which existed on the two sides of the screen. 
 While it is difficult to understand why the color of one side should 
 influence the effect of the rotating sector on light from the other 
 lamp, it is to be noted that in all probability a different form of 
 photometer was used when the color difference existed, and this 
 
Hyde.] The Rotating Sectored Disk. 5 
 
 might perhaps account for the error. Ferry states in his introduc- 
 tion that for lights of the same color a Lummer-Brodhun pho- 
 tometer was used, but that when a considerable color difference 
 existed a Bunsen photometer was employed. When the difference 
 in color on the two sides was small, he found the Nichols-Ritchie 
 photometer to give the best results. In describing his experiments 
 he does not state explicitly, however, which form of photometer he 
 used in each case. 
 
 Because of this lack of definiteness in Ferry's paper it seemed very 
 desirable to make further experiments on the influence of color on 
 the action of the rotating sectored disk. There are also two very 
 definite reasons why Ferry's results are open to criticism. First, 
 since the absolute illumination of the photometer screen was always 
 much greater without the disk than with it, the Purkinje effect would 
 produce errors in the results when the two sources differed to any 
 extent in color. Secondly, the inverse square law applies rigorously 
 only to point sources and can not be assumed a priori for such an 
 extended source as the filament of an incandescent lamp, particularly 
 
 Fig. 1. Lummer and Brodhun's Arrangement. 
 
 when it is surrounded by the reflecting and refracting glass bulb, 
 which alters the curvature of the emitted light to such an extent 
 that in some directions sharp images of the filament are formed sev- 
 eral meters away. The lime light and the enclosed arc lamp, apart 
 from the possible errors due to the inapplicability of the inverse 
 square law, would seem to be unsuitable for accurate work because 
 of their great unsteadiness. 
 
 The most satisfactory work that has been done on Talbot's law is 
 that of Lummer and Brodhun at the Physikalisch-Technische Reichs- 
 anstalt. These investigators first verified the law for white light 
 down to 50 total opening, by using two incandescent electric lamps, 
 thus far being merely a repetition of Ferry's work. Recognizing, 
 however, the. possible error due to the inapplicability of the inverse 
 square law, they modified their experiments in the following man- 
 ner. Three incandescent lamps were mounted on the photometer 
 bench, as shown in Fig. i. Lamp a burned continuously but lamp 
 
6 Bulletin of the Bureau of Standards. \yoi.z, NO. i. 
 
 b and lamp c were burned successively, and the current through each 
 was regulated until each produced approximately the same illumina- 
 tion of the screen. They were then both burned at the same time, the 
 rotating disk with a total opening of 180 was placed between them 
 and the photometer, and a balance was obtained by moving lamp a. 
 In this way, since the distance from lamp a to the screen remained 
 approximately constant, errors in the application of the inverse 
 square law entered only as second order effects and were hence neg- 
 ligible. In a similar way each lamp in turn was burned with the 
 disk set at 180 total opening, and then both were burned simultane- 
 ously with the disk set at 90? In every case the deviations from 
 the law were found to be of the same magnitude as the experimental 
 error, which was less than one-half per cent. It is not stated, how- 
 ever, how far the process was carried, except that it was not extended 
 to very small angular openings. 
 
 Although the results of Lummer and Brodhun are conclusive as 
 far as they go, attention should be called to the fact that because 
 the observational error was less than one-half per cent, it does not 
 necessarily follow that Talbot's law was verified to such a high 
 accuracy for all angular openings that they used. Since their 
 method of observation consisted in verifying the law step by step, 
 between 360 and 180, 180 and 90, etc., the error between 360 
 and, say, 22^ would be distributed over four intervals, so that 
 though the error within each interval may be less than one-half per 
 cent it is possible that an error of i or 2 per cent might exist between 
 360 and 22^ without being detected. This fact, together with a 
 desire to check Ferry's results for colored light, led the writer to 
 make the experiments described in the following pages. 
 
 In the investigation of Talbot's law by the use of the rotating 
 sectored disk some other law of the variation of the intensity of illu- 
 mination must be assumed. Of the several possible ways of varying 
 the intensity of illumination of the screen the method of varying the 
 distance between the source and the screen is the simplest and most 
 satisfactory, but presupposes the knowledge of the law of variation 
 of the intensity of illumination with the distance for the source to 
 be used. If the source is a point source the inverse square law holds; 
 if the source is not a point source and no actual source is strictly 
 a point source it must either be shown that within the limits of 
 accuracy of the experiments the deviations from the inverse square 
 
Hyde.} 
 
 The Rotating Sectored Disk. 
 
 law are negligible under the conditions under which the source is to 
 be used, or else the deviations from this law must be determined 
 and applied. 
 
 The source to be used must not only have the law of variation of 
 intensity of illumination with distance known, but must also be 
 constant and intense. The incandescent lamp fulfills the last two 
 conditions, but, as heretofore pointed out, it does not fulfill the first 
 condition. 
 
 It occurred to the writer to try as a source a direct current Nernst 
 glower without a globe. Although the only one of the above con- 
 ditions that could be postulated a priori was the intensity, experi- 
 
 Fig. 2. Radiating Cylinder. 
 
 ment showed that the glower remained sufficiently constant under 
 certain conditions which could be readily obtained, and the follow- 
 ing theoretical considerations led to the conclusion that the inverse- 
 square law can be applied to the Nernst glower to within negligible 
 errors if the glower is not brought closer to the photometer screen 
 than 20 or 30 cm. The Nernst glower was therefore used through 
 the experiments, and with entire satisfaction. 
 
 2. THEORETICAL DISCUSSION. 
 
 (a) Radiation from a Cylindrical Source. Let us assume that the 
 Nernst glower is a radiating circular cylinder of radius # , length 2 h, 
 
8 Bulletin of the Bureau of Standards. \_voi. 2, NO. i. 
 
 and uniform specific light intensity i. The problem then consists 
 in determining the law of variation of the intensity of illumination 
 with the distance for a uniform radiating cylinder. 
 
 Let us further assume Lambert's cosine law for the radiating 
 cylinder. Although this law holds rigorously for black bodies only, 
 the resulting error in the comparison of illuminations at different 
 distances would be quite small. If, then, we let < (Fig. 2) be the 
 angle of emission for any element of surface dS of the radiating 
 cylinder; 6 the angle of incidence of any ray on a screen at P placed 
 at right angles to OP, where OP lies in the plane perpendicular to 
 the axis of the cylinder at its middle point, O; r the distance from 
 the element dS to the screen at P; then the intensity of illumination 
 of the screen at P is 
 
 C Ci cos (f> cos 6 d S / N 
 
 /= JJ-^ (I) 
 
 taken over that part of the curved surface of the cylinder convex 
 toward P. 
 
 Expressing all the quantities involved in the above equation in 
 terms of the two cylindrical coordinates a and jy, we get for the 
 intensity of illumination at P, at a distance /, from the axis of the 
 cylinder 
 
 C f(l cos a a) (I a cos a) , x 
 
 T=i a \ ' \ ~^>dady (2) 
 
 I (a 4- 1 2atcosa4-y) 
 
 / / \ ' s / 
 
 in which the limits of y are ( ^, + h\ and the limits of a are 
 
 1# , t #\ 
 
 (cos y, + cos j> 
 
 The integral 1 of the above expression is 
 
 /Z I ' d ' tif / \ 
 
 = 7 cos f ,_ r , , a (3) 
 
 where 
 
 +*[<s+ 0^ *- ^ 
 
 1 This integral was obtained by Prof. A. S. Chessin, of Washington University, to 
 whom the writer desires to express his indebtedness. 
 
Hyde.} 
 
 The Rotating Sectored Disk. 
 
 It only remains to substitute for the constants a and h in the 
 above equation the numerical values pertaining to a Nernst glower 
 and then to compute the value of J for different distances, /. Before 
 doing this, however, it is interesting to note, in passing, the form 
 which the equation assumes when we make h approach infinity. 
 Under this condition q approaches unity, and so in the limit, equa- 
 tion (3) becomes 
 
 TT i a 
 
 In other words, the intensity of illumination due to an infinitely 
 long uniformly radiating cylinder varies inversely as the first power 
 of the distance from the axis of the cylinder, a result which also 
 follows from purely physical considerations of the normal flow of 
 energy across coaxial cylindrical surfaces. 
 
 Let us now substitute the dimensions of the Nernst glower in 
 equation (3), compute the intensity of illumination at different 
 distances, /, and compare the relative values with those obtained on 
 . the assumption of the inverse square law. Two sizes of glowers 
 were used in the investigation, 88-watt glowers and 44-watt glowers. 
 The dimensions of the former are ^=7.5 mm, a 0.6 mm. Of the 
 latter, /i = 6 mm, # = 0.4 mm. If, then, in the above equation we 
 put /iio mm and # = i mm we shall include both glowers, and 
 
 TABLE I. 
 
 Deviation from the Inverse Square Law of the Radiation of a Cylinder 20 mm long and 1 mm radius. 
 
 Distances, / 
 
 J (Direct Evaluation) 
 
 J (Inverse Square Law) 
 
 Deviation 
 
 3000 
 
 1.0000 X/3000 
 
 1.0000 
 
 s\J 3000 
 
 0.00 % 
 
 2000 
 
 2.2500 " 
 
 2.2500 
 
 II 
 
 +0.00 " 
 
 1000 
 
 9.0045 " 
 
 9.0000 
 
 M 
 
 +0.05 " 
 
 500 
 
 3.6040X10 " 
 
 3.6000X10 
 
 II 
 
 +0.11 " 
 
 200 
 
 2.2545X10 2 " 
 
 2.2500X10 2 
 
 II 
 
 +0.20 " 
 
 100 
 
 9.0081 X 10 2 " 
 
 9.0000 X 10 2 
 
 " 
 
 +0.09 " 
 
 80 
 
 1.4049X103 " * 
 
 1.4062X103 
 
 II 
 
 -0.09 " 
 
 50 
 
 3.5593X103 " 
 
 3.6000X103 
 
 (1 
 
 1.13 " 
 
 the deviations from the inverse square law which we shall find will 
 be greater than those incident to the use of either glower. The 
 
io Bulletin of the Bureau of Standards. \yoi.2,No.i. 
 
 results of this substitution for different distances, /, are shown in 
 Table I and Fig. 3. In the first column of Table I are given the 
 distances, /, expressed in millimeters, for which the values of J were 
 computed. The second column contains the ratios of the intensities 
 at the different distances, /, to that at /= 3000 mm, as obtained by 
 direct substitution in equation (3); and the third column gives the 
 same ratios obtained by the inverse square law. The percentage 
 errors are shown in the fourth column, in which (-)-) means that 
 the intensity of illumination determined by direct evaluation of the 
 integral of equation (3) is relatively greater, as compared with the 
 intensity at / 3000 mm, than the value obtained by the inverse 
 square law from the intensity at /= 3000 mm. This is shown in 
 Fig. 3, in which abscissas are distances, /, and ordinates are per- 
 
 ,,,+ 0.2% 
 
 u 0. 1% 
 5-0.2% 
 
 DISTANCE FROM AXIS OF CYLINDER IN MM 
 
 Fig. 3. Deviation of Radiation of Cylinder from Inverse Square Law. 
 
 centage deviations from the inverse square law, as deduced from the 
 value of the intensity at /= 3000 mm. It is thus seen that the value 
 of J between /= 3000 mm and I 90 mm is greater (as compared 
 withy at 7=3000 mm) than the value of J deduced on the assump- 
 tion of the inverse square law, the maximum deviation being about 
 + 0.2 per cent. At distances less than about 90 mm the intensity 
 becomes very much less than the values demanded by the inverse 
 square law, the deviation at 1= 50 mm being over i per cent. 
 
 Since in the experiments described below the extreme distances 
 used were /= 3000 mm and 1= 500 mm the maximum error on the 
 assumption of the inverse square law for the Nernst is approximately 
 o.i per cent, which is entirely negligible. Moreover, the errors 
 given in Table I and Fig. 3 were deduced for a cylinder having the 
 
Hyde.'] The Rotating Sectored Disk. 1 1 
 
 dimensions h 10 mm, a = i mm, which are somewhat greater than 
 the dimensions of an 88-watt Nernst glower, and much greater than 
 the dimensions of a 44-watt glower. Hence the true errors for the - 
 Nernsts are probably less than those given in the table. 
 
 It should be stated that the above theory was based on the assump- 
 tion of a true circular cylinder with constant specific light intensity, i. 
 The Nernst glower does not fulfill either of these conditions rigor- 
 ously. It frequently becomes warped after burning for some hours, 
 and direct experiment reveals the fact that the intensity, z, is differ- 
 ent at different parts of the filament. These deviations probably 
 remain constant, however, during a series of experiments, and so 
 only introduce differential errors in the relative intensities of illumi- 
 nation at different distances. 
 
 (b) Method of Mean Distances. In the above investigation / was 
 defined to be the distance from the center of the cylinder to the 
 photometer screen, the axis of the cylinder being parallel to the 
 screen, and the centers of both cylinder and screen lying in the same 
 plane perpendicular to the axis of the cylinder. In the experimental 
 method employed it was found difficult to measure this distance 
 directly by the use of the scale on the photometer bar. The glower 
 was therefore mounted in a horizontal position in a socket that could 
 be turned about a vertical axis through its center. The position of 
 the axis of rotation on the bar was shown by an index attached to 
 the carriage on which the socket was mounted, so that distances 
 between the axis of rotation and the photometer could be read off 
 the scale on the bar. The center of the glower was now placed 
 approximately in the axis of rotation, and readings were made first 
 with one side of the glower, [designated (+)] and then with the 
 other side of the glower [designated ( ) ] turned toward the photo- 
 meter. In each case the distance was measured to the axis of rota- 
 tion and the mean of the two readings taken. This mean distance 
 was the distance used in the comparison of intensities of illumina- 
 tion, and it must hence be shown that the square of the ratio of one 
 mean distance, x, to another mean distance, x' , gives the true value 
 of the ratio of the two corresponding illuminations, J and J' . 
 
 Before proceeding to this let us consider a different way of inter- 
 preting the deviations of the law of radiation of the cylinder from 
 the inverse square law. Since for all distances greater than / 90 
 
12 
 
 Bulletin of the Btireau of Standards. 
 
 [ Vol. 2. No. i. 
 
 mm the intensity of illumination compared with the intensity at 
 1= 3000 mm is slightly greater than that for a point source, we may 
 consider the cylinder as a point source, the effective center of radia- 
 tion of which is at the center of the cylinder for /= 3000 mm, but 
 for distances between 7=3000 mm and 7=90 
 mm it is displaced a variable small amount 
 toward the photometer screen. Hence, as the 
 deviations from the inverse square law are 
 very small for all distances used in the experi- 
 ments, we can take the percentage error in 
 distance to be one-half the percentage error in 
 intensity of illumination. If, therefore, we 
 take one-half the ordinates of Fig. 3 we obtain 
 the percentage errors in the effective distances. 
 In this way we can determine for any distance 
 the displacement, c, of the effective center of 
 radiation from the geometrical center of the 
 cylinder. 
 
 We will suppose now that the geometrical 
 center of the cylinder is at a distance, a, from 
 the axis of the socket (Fig. 4). Suppose 
 further that the photometer screen has an 
 intensity of illumination, y, the distance from the axis of the socket 
 to the screen being x l when the (+) side of the Nernst is turned 
 toward the screen, and x^ when the ( ) side is turned in that 
 direction. If now 1^ and 7 2 are the effective intensities of the (+) 
 and ( ) sides of the Nernst, respectively, considered as a point 
 source, 
 
 Jf*,. AS^f-v _!_/ r\* (5) 
 
 Fig. 4. Displacement of 
 Axis of Cylinder from Axis 
 of Socket. 
 
 a 
 
 O 2 
 
 In a similar way the equation corresponding to an illumination, 
 /' is 
 
 Tl _ ?\ . _ ^ (\ 
 
 J ~~(~l J\*~~(<r '_!_/ s-'\ Z ^ ' 
 
 Hence the true ratio of J to J' is 
 
 xac 
 
 -C 
 
 c 
 
 (7) 
 
Hyde.} The Rotating Sectored Disk. 13 
 
 which, by composition, becomes 
 
 - a 
 
 (8) 
 
 Or, if we put 
 
 X -4-X X ' -\- K ' 
 
 1 I * / *'l 1^ *< / \ 
 
 **- 4 J- t l*'= (9) 
 
 X X XX 
 
 to within negligibly small quantities. From this equation, or better 
 from equation (10), it follows that for a true point source at the cen- 
 ter of the cylinder the ratio of the intensities obtained by taking the 
 mean distances x and x' is the same as that obtained from the actual 
 distances x^ a and x\ a, or x^-\-a and x z '-\-a, since for a true 
 point source c=c'=o and from equations (7) and (10) 
 
 (12) 
 
 In the actual case of the cylinder, c and c 1 have very small values, 
 
 as pointed out above. In fact since Xioo is the percentage dis- 
 
 x 
 
 "ZC 
 
 tance correction, and x 100 is the percentage deviation of the 
 
 2,C I 
 
 intensity from the inverse square law, it follows that is of the 
 
 x 100 
 
 ordinate of the curve of Fig. 3 expressed as a decimal fraction, or is 
 equal to the ordinate of the curve expressed in per cent. Since the 
 greatest range of distance used was between V = \x' a] = 3000 mm 
 
14 Bulletin of the Bureau of Standards. [MI. 2, NO. /. 
 
 and /= [xa~\ =500 mm we see from the curve of Fig. 3 that c' = o 
 and c H-O.II per cent, so that [eq. (n)] 
 
 /= (1-0+0.11%) (13) 
 
 in which the correction is negligible. 
 
 Hence the method employed of taking the mean of the (+) and 
 ( ) readings gives absolutely true results for a point source, and 
 for a cylinder the errors are only those incident to the deviation 
 from the inverse-square law of the radiation of the cylinder. 
 
 3. APPARATUS AND METHODS. 
 
 (a) General Method. The photometer bench on which the 
 measurements were made is a Reichsanstalt precision photometer 
 bench 250 cm long, supplied with a Lummer-Brodhun contrast 
 screen. In the experiments with disks of small opening it was 
 necessary to use greater distances than could be obtained on the 
 250 cm bar. In these cases an extension was fitted to one end of 
 the bench, increasing its length by about 175 cm. In all the 
 experiments except those in which there was a decided color differ- 
 ence the contrast principle was used but in the cases of great color 
 difference the absorption strips producing the contrast were removed 
 and the setting was made for a match. 
 
 Throughout the experiments the substitution method was em- 
 ployed. Thus the source at the right end of the bar was merely 
 used as a comparison lamp. 1 The carriage on which it was mounted 
 was connected rigidly by means of adjustable links to the carriage 
 supporting the photometer screen and at a suitable distance to pro- 
 duce the desired illumination of the screen. The comparison lamp 
 thus moved with the screen, remaining at a constant distance from 
 it, except in so far as slight irregularities in the ways produced 
 small variations in the distance, depending upon the positions of 
 the carriages on the bar. These eccentricities were carefully studied 
 for all positions of both carriages on the bar and the results were 
 tabulated, so that the variations in the distance between the com- 
 parison lamp and the screen, as the latter was moved back and forth 
 
 1 Hereafter the lamp on the right will be referred to as the "comparison lamp," 
 in distinction from the ' ' test lamp ' ' on the left. 
 
Hyde. 
 
 The Rotating Sectored Disk. 
 
 on the bar, could be determined directly. Since, however, in nearly 
 every experiment the position of the photometer was approximately 
 the same with the sectored disk stationary and rotating, and since 
 regions of the bar were selected which were approximately true, in 
 general no correction was necessary for the distance between the 
 comparison lamp and the screen. Hence the illumination on the 
 right side of the screen remained constant to within the variations 
 of the comparison source itself. 
 
 On the left side of the photometer the test Nernst and the sec- 
 tored disk were mounted. Fig. 5 is a diagrammatic sketch of the 
 
 
 D 
 
 N 
 
 C S 
 
 B A 
 P 
 
 
 
 jv 
 
 
 
 
 
 (eg 
 
 aa 
 
 
 r ~\~ ] 
 
 
 
 ^) 
 
 
 
 
 
 
 
 n h 
 
 
 tef 
 
 ! 
 
 
 
 
 Fig. 5. Diagrammatic Sketch of Photometer Bench. 
 
 general arrangement, in which A, B, C, and D are diaphragms 
 covered with black velvet, which were always so arranged 
 that no stray light could reach the photometer screen except by 
 reflection from the black velvet at an angle approximately equal to 
 90? The magnitude of this stray light, when the 
 walls of the room were white and were rather 
 strongly illuminated, was found by a previous 
 investigation 1 to be entirely negligible. M is a 
 small spherical motor, which carries on its shaft the 
 sectored disk >S. NN are the two sources and P is 
 the photometer screen. 
 
 (b) Sources. Direct current Nernst glowers were 
 used throughout the investigation for the test source 
 at the disk end of the bar, and in most of the meas- 
 urements a glower was used for the comparison 
 lamp. When the effect of color difference was in- 
 vestigated an incandescent lamp at low voltage was 
 used for a comparison lamp, and in some few experiments on colored 
 light an incandescent lamp was substituted for the comparison 
 Nernst. As stated above, the glowers were burned without any 
 globes around them, since reflection, refraction, and diffusion of the 
 light by the globe might produce serious errors. Moreover the 
 
 Fig. 6. Nernst 
 Glower Mounted on 
 Extension Plug: 
 
 1 Bureau of Standards Bulletin No. 3, p. 417. 
 24353 No. I 06 2 
 
1 6 Btilletin of the Bureau of Standards. \_voi. 2, NO. i. 
 
 glowers were not provided with the customary ballast resistance. 
 They were mounted on extension plugs, as shown in Fig. 6, and were 
 operated with constant current on a storage-battery circuit of 250 
 volts, of which approximately 150 volts was taken up in rheostats. 
 This series resistance took the place of the customary ballast resist- 
 ance and served to keep the current approximately constant. 
 
 The electrical measurements were made in terms of a Weston 
 standard cell by means of a potentiometer. Thus, the current was 
 measured by the difference in potential across a standard resistance 
 and the voltage was obtained by the use of a 100,000 ohm multiplier. 
 The original plan was to leave the Nernst glowers entirely uncov- 
 ered, but it was found that due to air currents the resistance of the 
 glowers changed continually, so that it was impossible to maintain 
 the current in the glowers constant, and the fluctuations in the volt- 
 age were so great as to preclude the possibility 
 of measurement closer than to several tenths 
 of i per cent. Moreover, the effect of the 
 rotating disk in the neighborhood of the 
 Nernst was evident both in the current and in 
 the voltage. I was thus led to try the effect 
 of partially surrounding the Nernsts with 
 
 hoods. To this end two pieces of brass tub- 
 Fig. 7. -Horizontal Section Qm j and T cm diameter were each 
 of Hood through Opening. . / 
 
 provided with a rectangular opening i cm 
 
 high and 5 cm wide at a distance of 28 cm from one end of the 
 tube, so as to be opposite the Nernst when placed on the base of 
 the carriage on which the Nernst was mounted. Each of these 
 hoods was provided with a loosely fitting piece of sheet brass for a 
 cover and was painted a dead black. Lest any stray light might 
 be reflected into the photometer at approximately grazing incidence 
 from the outside of the tube, a sheet of brass 25 cm long and 15 cm 
 wide was covered with black velvet and fastened to the front of the 
 hood with the black velvet turned toward the screen. This dia- 
 phragm was provided with a rectangular opening to correspond 
 with the opening in the hood, as shown in Fig. 7, which is a hori- 
 zontal section in the plane of the opening. That part of the inside 
 of the tube which was behind the Nernst was also covered with 
 black velvet in order to prevent any possible reflection from the 
 painted metal surface. 
 
Fig. 8. Sectored Disk Mounted on Photometer Bar. 
 
Hyde.] The Rotating Sectored Disk. 1 7 
 
 Under these hoods the Nernsts operated almost as steadily as 
 incandescent lamps. The current could be maintained constant to 
 within i or 2 parts in 10,000, and the voltage could usually be 
 measured to the same accuracy. Moreover, the luminous intensity 
 remained constant to within the limit of observational error, and 
 the sectored disk in the neighborhood of the Nernst seemed to have 
 no appreciable effect on its voltage at constant current, which 
 affords a quite sensitive means of determining slight variation in 
 the temperature of the Nernst. It was for this reason that voltage 
 measurements were made with every series of observations. 
 
 One peculiar voltage effect was noticed which occurred at times 
 both when the disk was stationary and when it was rotating. The 
 voltage would suddenly change over several tenths of a volt and 
 then gradually creep back. As this change was usually accompa- 
 nied by no perceptible variation in the luminous intensity, if the 
 current was maintained constant, and as it seemed to occur more 
 frequently after the Nernsts had burned for many hours, it seemed 
 probable that it was due to the cracking of the terminals of the 
 filament and a consequent redistribution of the current at the ter- 
 minals. This hypothesis seemed to be borne out by the fact that 
 in all old Nernsts the positive ends show large and deep cracks. 
 
 At first the presence of these sudden changes in the voltage 
 caused doubt as to the accuracy of the results. They repeatedly 
 occurred both while the disk was stationary and while rotating, and 
 could not be detected at the photometer, although every other con- 
 dition was maintained constant. I was therefore led to disregard 
 them, particularly as the results obtained in sets in which there was 
 no trace of such an effect agreed with those obtained in sets in 
 which the effect was plainly noticeable. 
 
 Having shown that the Nernst could be depended upon for its 
 constancy, it remained to find some convenient way of determin- 
 ing the distance between the center of the test Nernst and the pho- 
 tometer screen. The test Nernst (as distinguished from the com- 
 parison Nernst) mounted on an extension plug, was screwed into 
 the socket of a horizontal rotator with the driving shaft removed. 
 But although the distance from the axis of the rotator to the near 
 surface of the plaster of Paris screen in the photometer-head was 
 
i8 Bulletin of the Bureau of Standards. [ vol. 2, NO. /. 
 
 known for every position of the rotator and of the photometer-head, 
 as the result of a direct calibration of the scale on the bar, the dis- 
 tance of the center of the Nernst from the axis of the rotator 
 varied slightly from time to time, due to bending of the terminal 
 wires of the Nernst on heating, etc. Hence, after each series of 
 measurements it was necessary to re-determine the distance. It 
 therefore occurred to the writer to use the method described briefly 
 in the above theory. A double pointer was attached to the rotator, 
 and by means of a corresponding pointer attached to the carriage 
 the Nernst could be turned through 180? The mean of the (+) 
 and ( ) readings as described above was taken in each case. 
 
 (c) Disks. The rotating sectored disk is shown in Fig. 8. The 
 rotating device consisted of a small spherical direct current fan 
 motor, 9 cm in diameter, mounted on a base clamped to the ways. 
 In the figure the motor and disk are turned slightly with reference 
 to the base, so as to be seen better. The distance of the motor 
 above the ways could be adjusted by means of the telescoping 
 tubes which formed the column on which the motor was mounted. 
 The frame of the motor was japanned. In order to prevent light 
 from being reflected into the photometer-head from the smooth sur- 
 face of the motor, the top of the motor was at first covered with 
 black velvet, but since the light was incident on this at nearly graz- 
 ing incidence it was thought better to substitute a small metal screen 
 placed normal to the direction of the rays. This was accomplished 
 by separating the two halves of the spherical shell of the motor and 
 inserting a blackened piece of thin sheet aluminum cut to fit. This 
 is shown in the figure. 
 
 The sectored disks were mounted directly on the shaft of the 
 motor and the height of the motor was adjusted so that the sector 
 cut the beam of light normally, the central ray of the beam lying 
 in the same vertical plane as the shaft of the motor. Instead of the 
 customary method of using a graduated variable sectored disk, it 
 was decided to use separate disks for the different angular openings. 
 This was decided upon partly because four such disks were at hand 
 and partly because there was less chance for error in the calibration 
 of the opening. The original four disks were made of aluminum 
 about 1.5 mm thick, and each contained six open and six solid sec- 
 tors equally spaced. The total angular openings (which will here- 
 
Hyde.] 
 
 The Rotating Sectored Disk, 
 
 after always be meant when reference is made to the opening of a 
 disk unless it is stated otherwise) were 288 (6x 48), 270 (6x 45), 
 240 (6x40), and 180 (6x30). Subsequently the following 
 disks were added: One steel disk, 1.5 mm thick, with total opening 
 of 60 in three sectors of 20 each; two hard brass disks, 1.5 mm 
 thick, with total openings of 210 in six sectors of 35 each, and 
 30 in six sectors of 5 each. 
 
 All of the disks were approximately 30 cm in diameter, and had 
 the same general form as that shown in Fig. 8. They were all 
 painted dead black, and the last two disks (210 and 30) had the 
 edges of the sectors beveled so as to prevent reflection from them. 
 
 With these seven disks the law 
 could be tested at seven points, 
 ranging irregularly from 30 to 
 288? In order to obtain more 
 points of the curve it was decided 
 to cover some of the openings 
 of the disks with thin pieces of 
 blackened aluminum clamped over 
 the openings (Fig. 9). The covers 
 were always placed symmetrically, 
 so that with a disk of six open 
 sectors it was possible to obtain 
 either one-half the total opening 
 by covering the alternate sectors, 
 or to obtain one-third the total 
 
 opening by covering all but two opposite sectors. Moreover, by 
 changing the covers it was possible to obtain the same opening in 
 different ways, i. e., by using different sectors of the same disk. 
 
 By this method the following total openings were obtained: 10, 
 15, 30, from the 30 disk; 60 from the 60 disk, and also from 
 the 1 80 disk; 90 from the 180 disk and also from the 2 70 disk; 
 120 from the 240 disk; 144 from the 288 disk; 180, 210, 
 240, 270, and 288 from the respective disks. 
 
 The disks were all carefully calibrated on a circular dividing 
 engine. The opening of each sector of each disk was measured at 
 seven successive radial distances 0.5 cm apart, over a range of 3 
 cm, since 3 cm was approximately the maximum height of the beam 
 
 Fig. 9. Sectored Disk with Three Openings 
 Covered. 
 
2O Bulletin of the Bureau of Standards. \_voi. 2, NO. i. 
 
 at the point where it was intercepted by the disk. The mean of the 
 seven readings was taken as the opening of the sector and the 
 height of the disk was always so adjusted that the line from the 
 center of the Nernst to the center of the photometer screen inter- 
 cepted the disk at the middle point of the region over which it had 
 been calibrated. The average deviation for different radial dis- 
 tances from the mean value for any combination of sectors that was 
 ever used was always under o.i per cent, but owing to difficulties in 
 setting on the edges, the total opening was perhaps not known to 
 an accuracy much better than o.i per cent. The values of the dif- 
 ferent sectors of the various disks are given in the " Experimental 
 Results." 
 
 The sectored disk in all cases, with one or two exceptions which 
 will be noted later, was placed between diaphragms B and C (Fig. 5) 
 and as close to B as possible. This diaphragm had a circular aper- 
 ture of 10 cm and hence screened the motor (except the upper part) 
 and its support from the photometer screen. The distance between 
 the disk and the photometer screen ranged from 20 to 25 cm. 
 Moreover, the sectored disk was always left in position with the 
 open sector in the path of the light for readings intended to give the 
 direct intensity, except when the 30 disk was used. The 5 sec- 
 tors of this disk were too narrow to permit the beam to pass. 
 Measurements made with the larger disks removed, however, failed 
 to show any effect of the stationary disks. 
 
 The complete arrangement, except the extension ways, is shown 
 in Fig. 10 which was made from a photograph. The hood was 
 removed from the comparison Nernst and is shown standing on the 
 table. 
 
 (d) Details of Methods White Light. The method of conduct- 
 ing the experiments was as follows: Two seasoned Nernst glowers 
 were mounted in the two sockets and the test Nernst was adjusted 
 so that its center was at the proper height, coinciding very nearly 
 with the axis of the rotator, and the direction of its axis was perpen- 
 dicular to the line joining its center to that of the photometer screen. 
 The two Nernsts were then brought to incandescence and allowed 
 to burn about an hour. The test Nernst was placed at some definite 
 position and the connecting links between the photometer screen 
 and the comparison Nernst were adjusted until the balance came at 
 
ffyde.} The Rotating- Sectored Disk. 21 
 
 a desirable part of the bar. While one observer maintained the 
 currents constant the other observer made the photometric settings. 
 First, with the disk stationary, a series of three or more readings 
 was made with the Nernst in the (+) position, then a second series 
 with the Nernst in the ( ) position, and then two more series, one 
 (-)-) and the other ( ), making in all four series of measurements 
 with the disk stationary. The sectored disk was then set rotating 
 and the test Nernst was moved in nearer to the photometer screen 
 to some position such that the balance came at about the same posi- 
 tion on the photometer bar as before. The carriage holding the test 
 Nernst was then clamped to the ways, and kept at this fixed position 
 while the settings for a balance were made. As explained before, 
 the settings were made by moving the photometer screen and com- 
 parison Nernst, which moved as a single system remaining at a 
 constant fixed distance apart during the entire experiment. Four 
 series of three readings each were then made as with the disk 
 stationary. This process of alternate stationary and rotating meas- 
 urements was continued until three groups of stationary readings, 
 and two groups of rotating readings had been made, making a total 
 of twelve series of stationary readings, six (-(-) and six ( ), and 
 eight series of rotating readings, four (+) and four ( ). The square 
 of the ratio of the mean distance between the photometer screen and 
 the axis of the rotator, when the disk was rotating, to the mean 
 distance when the disk was stationary, was taken as the effective 
 ratio of the light transmitted by the disk to that incident on it. 
 The comparison of this ratio with the ratio of the angular opening 
 of the disk to 360 gave the deviation from the law for the disk 
 used. 
 
 This method was used for all the disks except in some few sets 
 near the end of the investigation in which only three groups of 
 readings instead of five were made. In all cases after each series of 
 three readings the voltages on the two Nernsts were measured and 
 recorded. 
 
 Colored Light. In the measurements with colored light the 
 method was exactly the same as that used for white light except 
 that a piece of colored glass was introduced into the eyepiece 
 of the photometer. Red, green, and blue glasses were used and 
 measurements were made on three disks, the 240 disk, the 60 disk, 
 
22 Bulletin of the Bureau of Standards, \_voi. 2. NO. i. 
 
 and the 15 disk. The colored glasses used were not monochro- 
 matic, but since to the eye the light appeared distinctly red, or green, 
 or blue, as the case may be, any appreciable error due to one of the 
 colors would probably not be modified greatly by the admixture of 
 relatively small quantities of light of another color. 
 
 4. EXPERIMENTAL RESULTS. 1 
 
 (a) Effect of Speed. Before beginning the investigation of Tal- 
 bot's law, the effect of the speed of rotation of the disk on the illu- 
 mination of the screen was tested. Since numerous investigators 
 have found that the speed of the disk has no effect on the apparent 
 illumination of the screen provided the speed is greater than a critical 
 speed below which fluctuations in the intensity are visible, it seemed 
 unnecessary to make an extended study of this effect. It was desir- 
 able, however, to check the conclusion of previous investigators for 
 a range of speed likely to be used in the experiments. Hence, using 
 successively two disks the 180 disk and the 60 disk the speed 
 of the disk was varied from the "flicker" point to the maximum 
 speed attainable and no variation of the illumination within the 
 range of experimental error was detected. Subsequently, therefore, no 
 attention was paid to the speed of the disk except to be sure that the 
 speed was sufficiently high, i. e., that the number of alternations of the 
 sectors was sufficiently great to prevent the possibility of a "flicker." 
 
 (b) Disks. In order to show the accuracy of the mechanical con- 
 struction of the sectored disks, the readings on the 30 disk for the 
 six separate sectors and at different distances from the periphery of 
 the disk are given in Table II. The first column contains the dis- 
 tances, expressed in centimeters, from the periphery of the disk to 
 the point at which the measurements were made. The other six 
 columns contain the readings on the angular openings of the six sec- 
 tors numbered from I to VI. 
 
 Since the 30 disk contained the smallest sectors of all the disks 
 it had to be made and measured most carefully of all, for small 
 deviations of the edges of the sectors from straight radial lines 
 would produce relatively large errors in the angular openings. The 
 absolute angle need not be made to agree very closely with the nomi- 
 nal value, as the subsequent calibration will give the true value of 
 
 1 1 wish to express my indebtedness to my assistants, Mr. F. E. Cady and Mr. A. H. 
 Schaaf , for valuable assistance both in observations and computations. 
 
Hyde.} 
 
 The Rotating Sectored Disk. 
 
 the opening', but it is very necessary that the edges be made approxi- 
 mately radial and quite straight, so that the mean of the seven meas- 
 urements made every 5 mm may represent the mean value of the 
 opening. It is seen from Table II that the maximum range of 
 variation at different radial distances for most of the sectors is only 
 
 TABLE II. 
 
 Calibration of 30 Disk. 
 
 Distance from 
 Periphery 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 VI 
 
 cm 
 
 
 
 
 
 
 
 3.0 
 
 504 / 48 // 
 
 503'46" 
 
 503'02" 
 
 504 / 45 // 
 
 505 / 28 // 
 
 506 / 24 // 
 
 3.5 
 
 05 08 
 
 03 54 
 
 03 00 
 
 04 28 
 
 05 27 
 
 06 22 
 
 4.0 
 
 05 17 
 
 03 50 
 
 03 12 
 
 04 26 
 
 05 23 
 
 06 40 
 
 4.5 
 
 05 21 
 
 03 42 
 
 03 00 
 
 04 20 
 
 05 14 
 
 06 34 
 
 5.0 
 
 05 06 
 
 03 52 
 
 02 54 
 
 04 14 
 
 05 28 
 
 06 57 
 
 5.5 
 
 05 10 
 
 03 51 
 
 02 45 
 
 04 14 
 
 05 33 
 
 07 10 
 
 6.0 
 
 05 14 
 
 03 51 
 
 02 54 
 
 04 28 
 
 05 44 
 
 07 08 
 
 Mean 
 
 5 05 09 
 
 5 03 49 
 
 5 02 58 
 
 5 04 25 
 
 5 05 28 
 
 5 06 45 
 
 
 
 
 
 
 
 
 about 30" or i part in 600. The average deviation from the mean 
 opening would be well under i part in 1,000. 
 
 As stated above, the disks were always mounted in such a position 
 that the beam of light passed through the calibrated part of the 
 sectors. 
 
 Measurements similar to those for the 30 disk were made on all 
 the disks, the final results of which in the form in which they were 
 used are given in Table III. The first column contains the various 
 disks. The second column contains the nominal angles obtained 
 from the different disks, and the third column gives the sectors that 
 were employed to obtain these angles. In the fourth column are 
 given the ratios of the actual angles to 360? These ratios, compared 
 with the ratios of the squares of the distances obtained in the experi- 
 ments, gave the errors in the law. 
 
Bulletin of the Bureau of Standards. 
 TABLE III. 
 
 [ Vol. 2, No, i. 
 
 Disks 
 
 Nominal Angles 
 
 Sectors Used 
 
 Corresponding Angles 
 H-36o 
 
 30 
 
 30 
 
 I-VI 
 
 0.08466 
 
 
 15 
 
 I, III, V 
 
 0.04229 
 
 
 15 
 
 II, IV, VI 
 
 0.04236 
 
 
 10 
 
 I, IV 
 
 0.02822 
 
 
 10 
 
 II, V 
 
 0.02821 
 
 
 10 
 
 III, VI 
 
 0.02823 
 
 60 
 
 60 
 
 I-III 
 
 0.16914 
 
 180 
 
 180 
 
 I-VI 
 
 0.5012 
 
 
 90 
 
 I, III, V 
 
 0.2506 
 
 
 90 
 
 II, IV, VI 
 
 0.2507 
 
 
 60 
 
 I, IV 
 
 0.16697 
 
 
 60 
 
 II, V 
 
 0.16718 
 
 
 60 
 
 III, VI 
 
 0.16710 
 
 210 
 
 210 
 
 I-VI 
 
 0.5841 
 
 240 
 
 240 
 
 I-VI 
 
 0.6683 
 
 
 120 
 
 I, III, V 
 
 0.3341 
 
 
 120 
 
 II, IV, VI 
 
 0.3342 
 
 270 
 
 270 
 
 I-VI 
 
 0.7508 
 
 
 90 
 
 I, IV 
 
 0.2501 
 
 
 90 
 
 II, V 
 
 0.2504 
 
 
 90 
 
 III, VI 
 
 0.2503 
 
 288 
 
 288 
 
 I-VI 
 
 0.8000 
 
 
 144 
 
 I, III, V 
 
 0.4000 
 
 
 144 
 
 II, IV, VI 
 
 0.4000 
 
 (c) White Light. In order that the exact method of observation 
 and computation may be known, the following series of measure- 
 ments of June 2, 1905, copied from the laboratory record book, is 
 given: 
 
 SERIES XXV. 
 
 60 disk, total opening 60.89? 
 
 
 - 
 300 
 
 = 0.16914. 88-watt Nernst 
 
 No. 9 on left at o when disk is stationary, and at 71.5 when disk is 
 
Hyde.} 
 
 The Rotating Sectored Disk. 
 
 rotating. 88-watt Nernst No. 10 on right, as comparison lamp, at 
 fixed distance 119.9 cm fr m photometer screen. 
 
 60 disk stationary. 
 
 Volt. 
 
 Nernst No. 9 at o. 
 
 Volt. 
 
 122.97 
 
 .82 
 
 .56 
 
 No. 10=111.01 
 122.78 ' 9 II 7- 
 
 122.97 
 
 63 
 
 .80 
 
 122.80 
 
 No. 10=110.99 
 No. 9=117.08 
 
 75 
 
 
 
 
 
 ( ) 
 
 Volt. 
 
 ( ) 
 
 
 Volt. 
 
 122.33 
 .07 
 
 No. 10=110.98 
 122.19 No. 9=117.08 
 
 122.03 
 1.98 
 
 121.99 
 
 No. 10=111.00 
 No. 9=117.07 
 
 17 
 
 
 1.97 
 
 
 
 
 60 disk rotating. 
 
 
 Nernst No. 9 
 
 at 71.5. 
 
 ( ) 
 
 Volt. 
 
 (~ ) 
 
 
 Volt. 
 
 121.91 
 .89 
 .80 
 
 No. 10=111.01 
 121.87 No. '9=117.07 
 
 121.91 
 
 .78 
 .82 
 
 121.84 
 
 No. 10=111.00 
 No. 9=117.07 
 
 ( + ) 
 
 Volt. 
 
 ( + ) 
 
 
 Volt. 
 
 122.44 
 
 .22 
 
 23 
 
 No. 10=111.00 
 
 ,- No. 9=117.07 
 122.26 
 
 122.38 
 .16 
 17 
 
 122.24 
 
 No. 10=110.99 
 No. 9=117.04 
 
 17 
 
 
 
 
 
 
 60 disk stationary. 
 
 
 Nernst No. 
 
 9 at o. 
 
 ( + ) 
 
 Volt. 
 
 ( + ) 
 
 
 Volt. 
 
 122. 6l 
 .67 
 .80 
 
 No. 10=111.00 
 122.69 No. 9=117.03 
 
 122.97 
 71 
 99 
 
 122.85 
 
 No. 10=110.99 
 No. 9=117.06 
 
 
 
 73 
 
 
 
 (-) 
 
 Volt. 
 
 (-) 
 
 
 Volt. 
 
 122.33 
 2.10 
 1.94 
 
 No. 10=110.98 
 
 I22.II N0 " 9 = 117-02 
 
 122.02 
 
 13 
 .02 
 
 122.06 
 
 No. 10=110.97 
 No. 9=117.03 
 
 2.O7 
 
 
 
 
 
 
 60 disk rotating. 
 
 
 Nernst No. 9 
 
 at 71.5. 
 
 (_) 
 
 Volt. 
 
 (_) 
 
 
 Volt. 
 
 121.77 
 .8 7 
 .82 
 
 No. 10=110.99 
 121.82 No. 9=117.04 
 
 121.89 
 .87 
 .80 
 
 121.85 
 
 No. 10=110.99 
 No. 9=117.04 
 
 ( + ) 
 
 Volt. 
 
 ( + ) 
 
 
 Volt. 
 
 122.31 
 
 37 
 
 No. 10=110.99 
 122.30 No. 9=117.04 
 
 122.20 
 31 
 
 122.24 
 
 No. 10=110.98 
 No. 9=117.05 
 
 .22 
 
 
 .21 
 
 
 
 
 60 disk stationary. 
 
 Nernst No. 9 at o. 
 
 (+) 
 
 Volt. 
 
 ( + ) 
 
 
 Volt. 
 
 122.73 
 .81 
 
 97 
 
 No. 10=110.98 
 122.81 Na 9=117-02 
 
 122.83 
 .64 
 
 57 
 
 122.68 
 
 No. 10=111.00 
 No. 9=117.01 
 
 .72 
 
 
 
 
 
26 
 
 Bulletin of the Bureau of Standards. {voi. 2 <No.i. 
 
 60 disk stationary. 
 
 (-) Volt. 
 
 122.01 No. 10=110.99 
 
 2.29 No. 9=117.02 
 
 1.88 121.99 
 1.87 
 1.89 
 
 COMPUTATIONS. 
 Disk stationary. 
 
 Nernst No. 9 at o. 
 
 ( ) Volt. 
 
 121.97 No. 10=111.01 
 
 No. 9=117.02 
 
 i-77 
 1.93 
 
 <+)I22 io '-79 
 
 ,22.69 
 
 ,22.09 
 
 ,22.08 
 
 
 ,22. 74 
 
 (-) 121-99 I2I 6 
 .92 
 
 122.44 
 
 122.42 
 
 122.35 
 
 9 
 Disk rotating. 
 
 '" ,22.25 
 
 122.06 
 
 ( + ) 122.30 
 (-" ; g 
 
 I22.06 
 
 ,2,34 
 
 Mean stationary setting 122. 40 Mean rotating setting 122. 06 
 
 Position of Nernst carriage o. oo Position of Nernst carriage 71. 50 
 
 Distance= .............. 122. 40 
 
 Correction to bar .............. o. 41 
 
 Distance= .............. 50. 56 
 
 Correction to bar .............. o. 32 
 
 Corrected distance= ..... 121. 99 
 
 
 Corrected distance= ..... 50. 24 
 
 60.89 _o. 16914 
 360 +47=4-0.28%. Deviation from Talbot's law. 
 
 This method of observation and computation was followed 
 throughout the investigation. A great number of different Nernst 
 glowers, both of the 88-watt type and of the 44-watt type, were used. 
 The choice between an 88-watt glower and a 44-watt glower for the 
 comparison lamp was determined by convenience. For the test 
 lamp, however, measurements were made with both types with the 
 larger angular openings in order to test the effect of change in the 
 absolute illumination of the screen. Thus with the 288, 270, 
 240, and 1 80 disks measurements were made with both the 44- 
 
Hyde: 
 
 The Rotating Sectored Disk, 
 
 27 
 
 watt and the 88-watt glowers, each at two different distances, so that 
 the extreme illuminations used were in the ratio of 1:4. With the 
 60 disk illuminations in the ratio of 1:2 were used, but in no case 
 was there any evidence of an effect due to absolute intensity of illu- 
 mination. These variations in the intensity of illumination were 
 not very large, so that it would be interesting to study the effect of 
 large changes in the intensity of illumination, particularly for disks 
 with small openings. In my observations with openings smaller 
 than 60 88-watt glowers were always used, since even with them 
 the illumination of the photometer screen was quite small, because 
 of the necessarily long distance. Thus with the 10 opening the 
 distance between the test Nernst and the screen was approximately 
 300 cm when the disk was unmounted and only 50 cm when the 
 disk was rotating. Moreover, because of this great range of distance 
 but one distance for each position could be used conveniently. 
 
 Vo DEVIATION FROM 
 TALBOT'S LAW 
 
 + 0.4/o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -0- 
 
 tO. 2% 
 
 
 
 
 
 
 
 
 
 
 
 ^^' 
 
 _O_ 
 
 C 
 
 1 
 
 O 
 
 
 0.0% 
 
 300 
 
 
 
 240 
 
 
 
 180f 
 
 ,.-^ 
 
 .... ^ 
 ^^_^^ 
 
 '^- ~ 
 120 
 
 ) 
 
 
 80 
 
 
 
 
 
 -0.2% 
 
 
 O 
 
 
 
 
 
 
 TO 
 
 FAL AN( 
 
 SULAR 
 
 3PENINC 
 
 IN DE 
 
 3REES 
 
 
 
 
 -0.4% 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 11. Deviation from Talbot's Law. 
 
 The results obtained with the various angular openings are shown 
 in Table IV, in which (+) means that the disk apparently let 
 through more light than would be expected from Talbot's law. It 
 should be stated that the order in which the disks were used was 
 most irregular. Several of them, for example, were tested both 
 near the beginning and near the end of the investigation, which 
 extended over a period of ten or twelve months. 
 
 The mean values for the different angular openings are plotted 
 in the form of a curve in Fig. n, in which abscissas are angular 
 openings, and ordinates are percentage deviations from the law of 
 Talbot. The dotted curve represents the values given in Table IV. 
 
 It is seen that for all angular openings from 288 to 10 the law 
 is verified to within 0.5 per cent. Attention should be called, how- 
 ever, to the form of the curve. Since the average deviation of the 
 readings for any one angle was in no case as large as 0.2 per cent, 
 
28 
 
 Bulletin of the Bureau of Standards. 
 
 [ Vol. 2, No. i. 
 
 the probable errors of the measurements are in all cases under o.i 
 per cent, while the observed errors in the law are as large as 0.4 per 
 cent. Because the probable errors are less than o. i per cent, it does 
 not necessarily follow, however, that the results are correct to the 
 
 TABLE IV. 
 
 Percentage Deviations from Talbot's Law Using White Light. 
 
 Total Angular Open- 
 
 288 
 
 270 
 
 240 
 
 210 
 
 188 
 
 144 
 
 
 
 
 
 
 
 
 
 -0.16% 
 
 -0.05% 
 
 -0.05% 
 
 -0.06% 
 
 +0.10% 
 
 +0.24% 
 
 
 + .01 
 
 .03 
 
 - .12 
 
 - .05 
 
 + -14 
 
 .25 
 
 
 - .09 
 
 .06 
 
 - .17 
 
 
 - .15 
 
 .09 
 
 
 - .12 
 
 .16 
 
 + .01 
 
 
 - .10 
 
 
 
 
 
 
 
 + .13 
 
 
 
 
 
 
 
 + -11 
 
 
 Means 
 
 0.09 
 
 0.08 
 
 0.08 
 
 0.06 
 
 +0.04 
 
 +0.19 
 
 Average variation 
 from the mean.. 
 
 0.05 
 
 zh .04 
 
 .06 
 
 .00 
 
 .11 
 
 =h .07 
 
 Table IV, continued. 
 
 Total Angular Open- 
 ings 
 
 120 
 
 90 
 
 60 
 
 30 
 
 15 
 
 10 
 
 
 
 
 
 
 
 
 
 +0.22% 
 
 +0.37% 
 
 +0.54% 
 
 +0.35% 
 
 +0.38% 
 
 +0.14% 
 
 
 .04 
 
 .51 
 
 .15 
 
 .37 
 
 .63 
 
 .52 
 
 
 
 .39 
 
 .20 
 
 .39 
 
 .52 
 
 .38 
 
 
 
 .33 
 
 .20 
 
 
 .55 
 
 .61 
 
 
 
 .40 
 
 .20 
 
 
 .61 
 
 .39 
 
 
 
 .45 
 
 .28 
 
 
 .26 
 
 
 
 
 .03 
 
 .65 
 
 
 .26 
 
 
 
 
 .41 
 
 .44 
 
 
 .40 
 
 
 Means 
 
 +0.13 
 
 +0.36 
 
 +0.33 
 
 +0.37 
 
 +0.45 
 
 +0.41 
 
 Average variation 
 from the mean.. 
 
 zb .09 
 
 .09 
 
 .16 
 
 zt .01 
 
 .13 
 
 zb .13 
 
 same accuracy. Some constant source of error might possibly be 
 present. It will be noticed from the table that approximately the 
 same deviation was found for all the openings less than 120? Now, 
 in all the measurements on these angles, the rotating readings were 
 
Hyde.] The Rotating Sectored Disk. 29 
 
 taken with the test Nernst about 50 cm from the photometer screen, 
 at which distance the entire system was very much congested. 
 Hence, it seemed possible that some stray light might have pene- 
 trated to the screen which would not have reached the screen when 
 a longer distance was used as in the stationary readings. In par- 
 ticular the photometer screen itself reflects light back on the dia- 
 phragms and on the solid sectors of the disk, and since these were 
 necessarily very close to the screen when the test Nernst was only 
 50 cm away, the apparent deviation from the law might possibly be 
 due in part to this stray light. 
 
 An attempt was made to measure the effect of this stray light by 
 cutting off the direct rays from the Nernst, but it was found very 
 difficult to reproduce the conditions, and so this method of attack 
 was abandoned. A second method which seemed productive of 
 results, but which could not be carried as far as desired for want of 
 time, consisted in using longer distances for the 90 disk and in 
 comparing the 15 disk with the 180 disk, which has an error of 
 + 0.04 per cent. Only a few observations were made, but they 
 seemed to be in the right direction. Thus, with the 90 disk four 
 determinations at a much greater distance than that used before 
 gave as the errors +0.38 per cent, +0.14 per cent, 0.09 per cent, 
 and +0.18 per cent, with a mean of +-2O per cent, as against 
 + 0.36 per cent in the previous determinations. Similarly two 
 measurements of the 15 disk against the 180 disk, the shortest 
 distance between the test Nernst and the photometer screen being 
 about 73 cm instead of 50 cm, as was used in the previous measure- 
 ments, gave as the errors +0.22 per cent and +0.49 per cent, with a 
 mean of +0.36 per cent. This is only 0.09 per cent lower than the 
 mean value of the previous determinations, but it is in the right 
 direction. 
 
 If, now, to the above considerations we add that of the deviations 
 of the radiation of a cylinder from the inverse square law, which are 
 also in the right direction, we reduce the error for the small open- 
 ings still further. For a cylinder 20 mm long and i mm radius the 
 error for the distances used with the 10 disk would be +0.11 per 
 cent. For an 88 watt Nernst 15 mm long and 0.6 mm radius the 
 error probably would be but slightly less than + o. 1 1 per cent, or 
 about +0.08 p er cen j- or _|_ O .O9 per cent. Similar errors would be 
 
30 Bulletin of the Bureau of Standards. \_voi. z, NO. i. 
 
 present in the measurements of the other disks, so that the complete 
 curve of Fig. 1 1 would be lowered by a small amount, the lowering 
 being greatest at 10 where it would amount to about 0.08 per cent, 
 and least at 288, where it would be approximately zero. 
 
 This lowering of the curve due to the deviation from the inverse 
 square law combined with that indicated by the experiments described 
 just above would seem to justify a total lowering of the curve suffi- 
 cient to bring it within the limits +0.3 per cent and 0.3 per cent 
 as shown in the solid curve of Fig. n. 
 
 No attempt has been made to investigate the effect of diffraction 
 at the edges of the disk. This might possibly affect the result to 
 some small extent. 
 
 In order to determine whether the personal equation entered to 
 any extent into the results obtained, several sets of readings were 
 made by a second observer. With the 210 disk he made but one 
 set of measurements obtaining the result +0.04 per cent. With the 
 144 opening he obtained +0.21 per cent; with the 120 opening, 
 + 0.16 per cent and -|-o.2o per cent; with the 90 opening, -(-0.12 
 per cent, +0.26 per cent, and 0.12 per cent; with the 60 opening 
 + 0.14 per cent; and with the 15 opening -j-o.n per cent, +0.29 
 per cent, +0.33 per cent, and +0.37 per cent. His readings are 
 thus in the same direction as my own and for the most part agree 
 with them, although on the whole they show somewhat smaller 
 deviations, particularly for the smaller openings. 
 
 (d) Colored Light. In the investigation of the law with colored 
 light it was thought sufficient to determine whether for several 
 points on the curve the values obtained with colored light are in 
 approximate agreement with the corresponding values for white 
 light. The same method of observation as that for white light was 
 used, but the error of observation was very much larger, partly 
 because of the color and partly because of the reduced intensity 
 which made observations on the 15 disk very difficult. The 240, 
 60, and 15 openings were used, and each was tested with red, 
 green, and blue light by inserting pieces of glass in the eyepiece of 
 the photometer. The red glass was ordinary ruby glass and was 
 found on examination with a spectroscope to transmit very little 
 other than red, although the band in the red was quite broad. The 
 green glass was supposed to be a very high grade of monochromatic 
 
Hyde.} 
 
 The Rotating Sectored Disk. 
 
 3 1 
 
 glass, but it transmitted much light of other colors, as did also the 
 cobalt blue glass used for the blue. 
 
 In connection with the measurements with colored light it was 
 thought desirable to make a few experiments with color differences 
 on the two sides of the screen as this condition always existed in 
 Ferry's experiments when he obtained the large errors. The test 
 lamp, as in the previous experiments, was a Nernst glower at normal 
 current. The comparison lamp, however, was a 16 cp anchored 
 oval, 118 volt, 3.5 watt per candle incandescent lamp burning at 
 different voltages ranging from 116 to 102 volts. The color differ- 
 ence was very marked when the incandescent lamp was at normal 
 voltage, but in order to insure the detection of any error that might 
 exist, the incandescent lamp was burned at 102 volts in the measure- 
 ments with the 15 disk. At this voltage the contrast in color was 
 very great and yet no greater deviations than those for white light 
 on both sides of the screen were detected. 
 
 TABLE V. 
 
 Deviations from Talbofs Law Using Red, Green, and Blue Light, and Color Difference. 
 
 
 340 
 
 Means 
 
 60 
 
 Means 
 
 15 
 
 Means 
 
 Red light 
 
 +0.09% 
 
 +0.01 % 
 
 +0.37%. 
 
 +0.62% 
 
 + 1.06% 
 
 +0.44 % 
 
 
 - .07 
 
 
 .86 
 
 
 - .25 
 
 
 
 
 
 
 
 + .52 
 
 
 
 
 
 
 
 *+0.58 
 
 
 Green light 
 
 0.09 
 
 0.02 
 
 +0.51 
 
 +0.47 
 
 +0.05 
 
 +0.24 
 
 
 +0.06 
 
 
 .43 
 
 
 .42 
 
 
 Blue light 
 
 +0.01 
 
 0.06 
 
 +0.41 
 
 +0.38 
 
 +0.30 
 
 +0.50 
 
 
 - .13 
 
 
 .35 
 
 
 .70 
 
 
 
 
 
 
 
 *+0.47 
 
 
 Color difference. . . 
 
 +0.21 
 
 +0.04 
 
 +0.63 
 
 +0.46 
 
 +0.57 
 
 +0.56 
 
 
 0.14 
 
 
 .45 
 
 
 .56 
 
 
 
 
 
 .30 
 
 
 
 
 In making the measurements with color difference the absorption 
 strips were removed from the Lummer-Brodhun photometer and the 
 setting was made for a match, instead of for equal contrast. The 
 same openings were used as in the experiment with the red, green, 
 
 24353 No - 
 
32 Bulletin of the Bureau of Standards. \V<>i. 2, NO. /.] 
 
 and blue light. All of the results of the experiments with colored 
 lights, and of those in which there was a color difference on the two 
 sides of the screen are contained in Table V. The first column con- 
 tains the four different conditions under which the experiments 
 were made. In the second, fourth, and sixth columns are given the 
 corresponding results, and in the third, fifth, and seventh columns 
 are given the means of the values in the second, fourth, and sixth 
 columns, respectively. The observations marked with an asterisk 
 were made by the second observer and are not included in the 
 means. 
 
 From a comparison of Tables IV and V it is seen that within 
 the range of experimental error the deviations from Talbot's law are 
 the same for white and colored light and for difference in color 
 on the two sides of the screen. It would be well, however, to make 
 further experiments on colored light in order to reduce the probable 
 error, and obtain results more nearly comparable with the results 
 for white light. 
 
 5. CONCLUSIONS. 
 
 The results of this investigation may be summarized as follows: 
 
 (1) Talbot's law, in its application to a rotating sectored disk, is 
 verified for white light for all total angular openings between 288 
 and 10, to within a possible error of 0.3 per cent, which probably 
 expresses the limit of accuracy of the experiments. 
 
 (2) The observed deviations from the law for red, green, and blue 
 light are of the same order of magnitude as those for white light, 
 and hence Talbot's law is verified for red, green, and blue light, 
 though not to such a high accuracy as for white light. Moreover, 
 a difference in color on the two sides of the photometer screen pro- 
 duces no appreciable change in the observed deviation from the law. 
 
VITA. 
 
 Edward Pechin Hyde was born in Baltimore County, Mary- 
 land, January 3, 1879. ^ s parents were Edward I. and Caro- 
 line R. (Clemm) Hyde. He received his preparatory training 
 in the public schools of Baltimore, Maryland, and entered the 
 collegiate department of the Johns Hopkins University in 1897. 
 He was graduated from that institution in 1900 with the degree 
 of Bachelor of Arts. 
 
 In the following autumn he entered upon graduate work 
 in the Johns Hopkins University, with physics as his principal 
 subject and mathematics and physical chemistry as his subordi- 
 nate subjects. He was appointed to a Fellowship in 1901. In 
 1902 he entered the Bureau of Standards, Washington, with 
 which institution he is still connected as Assistant Physicist. 
 In May, 1904, he married Miss Virginia Getzendanner, of Balti- 
 more, Maryland. 
 
 In the progress of his graduate work he has attended lec- 
 tures given by Prof. J. S. Ames, Prof. R. W. Wood, Prof. F. Mor- 
 ley, Prof. H. C. Jones, Dr. J. B. Whitehead and Dr. A. Cohen. 
 
 The investigation leading to his dissertation on "Talbot's 
 Law as Applied to the Rotating Sectored Disk," was carried 
 out in the photometric laboratory of the Bureau of Standards. 
 He desires to express here his indebtedness to Prof. S. W. 
 Stratton, Director of the Bureau of Standards, for the courtesy 
 shown him in granting him permission to submit this investiga- 
 tion as a dissertation to the Board of University Studies of the 
 Johns Hopkins University, and to Prof. J. S. Ames, under whose 
 general direction as Director of the Physical Laboratory of the 
 Johns Hopkins University the investigation was carried out, for 
 his interest and kindness during the progress of the work. 
 
 O.- THE 
 
 [ UNIVERSITY 
 
 OF