MANUAI OF ADVANCED OPTICS 
 
 C. RIB OR G MAXX 
 
 ASSISTANT PROFESSOR OF PHYSICS IN THE UNIVERSITY OF CHICAGO 
 
 
 CHICAGO 
 SCOTT, FORESMAX AXD COMPAXY 
 
 1902 
 
. 
 
 
 
 COPYRIGHT, 1902, BY 
 SCOTT, FORESMAN AND COMPANY 
 
 BOUT. O. LAW CO., PRINTERS AND BINDERS, CHICAGO 
 
 TYPOGRAPHY BY 
 MARSH, AITKEN A CURTIS COMPANY, CHICAGO 
 
INTRODUCTORY NOTE 
 
 Anyone who has not used the methods of measurement which 
 are based upon the interference of light waves will find it difficult 
 to appreciate the high degree of accuracy which can thereby be 
 attained. That these methods are not more commonly used 
 seems to be due in large measure to the fact that they have 
 hitherto not received adequate treatment in the texts which are 
 in general use in physical laboratories. For this reason this 
 "Manual of "Advanced Optics," in which these methods are for 
 the first time presented in text-book form to students of >hysics, 
 is very timely, and should prove a valuable aid in making these 
 very practical and useful optical methods familiar to all who may 
 at any time find it necessary to make measurements of great pre- 
 cision. It is also hoped that the book will serve to promote 
 interest in the general study of experimental optics. Those who 
 desire to enter into optical investigation can not get a better foun- 
 dation for future work than by studying the optical theories here 
 presented, and performing the experimeuts described. 
 
 A. A. MlCHELSOtf. 
 
 THE UNIVERSITY OF CHICAGO, 
 November 20, 1902. 
 
 117487 
 
PREFACE 
 
 That the practical study of optics has been somewhat crowded 
 out of physical laboratories by the demands of electricity is 
 attested by the fact that there have already been published many 
 excellent manuals of the latter branch of science, whereas a prac- 
 tical treatise on optics has not yet appeared. To be sure, the 
 theory of optics needs no better treatment than it has received at 
 the hands of Mascart, Prude, Bassett, Preston, or of the authors 
 of the Winkelmann's Handbuch der Physik. But from what book 
 can the student find out how to determine, for example, that most 
 important constant of the spectrometer, the resolving power? Or 
 where can he learn to study practically the methods of using the 
 phenomena of interference for exact measurements? 
 
 This small manual is an attempt to meet the needs of the more 
 advanced students of optics. It has been written primarily for 
 the use of the author's classes in the University of Chicago, and it 
 covers the work done by them during three months of their senior 
 year. It is hoped, however, that it will be found useful at other 
 universities and will serve as a stimulus to a more extensive study 
 of this most fascinating branch of science. 
 
 Every chapter begins with a brief discussion of the theory of 
 the experiments which follow. In this discussion the attempt has 
 been made to avoid as much as possible the use of mathematics 
 to present rather the physical ideas involved, and to use those 
 ideas in building up a concrete conception of the phenomena with 
 which we are dealing. This has often resulted in a lack of rigor 
 of demonstration, e.g., in the case of the grating. In all such 
 cases references have been added so that those who wish the com- 
 plete and rigorous demonstration can satisfy themselves of the cor- 
 
 5 
 
tf PREFACE 
 
 rectness of the conclusions. This course has been followed 
 because the author believes that clear conceptions of fundamental 
 ideas are absolutely indispensable to the physicist, and that the 
 mathematical discussion, though often very elegant and conve- 
 nient, adds nothing essential to these conceptions, but tends, 
 rather, if used too freely, to cause one to forget the real essence 
 of the subject. 
 
 It is hoped that the descriptions of the manipulations are not 
 so detailed as to reduce the student to a mere mechanical copyist. 
 He should not be allowed to forget that there may be other 
 methods of adjustment and manipulation which may be better 
 than those suggested. The numerical examples are all taken from 
 the note-books of the students who have taken the course. 
 
 The apparatus needed for the experiments is not expensive. A 
 spectrometer with such accessories as a grating, a pair of Fresnel 
 mirrors and a bi-prism, a pair of Mcol prisms, and some doubly 
 refracting crystals can be used for a large number of the exercises. 
 The other necessary pieces are an interferometer and a polariscope 
 or saccharimeter. Every well equipped laboratory should have all 
 of these things, and so the course avoids the objection of requir- 
 ing elaborate and expensive apparatus. 
 
 The author's own work is so arranged that a course of lectures 
 on optical theory accompanies the laboratory work outlined in this 
 manual. The two courses are independent, and of such a nature 
 that either can be taken without the other. It is, however, very 
 advantageous to take both. For the benefit of those who take 
 the laboratory work only, the last two chapters on the develop- 
 ment of optical theory and the trend of modern optics have been 
 added. Though such chapters are not usual in a laboratory 
 manual, it is believed that they will serve the purpose of adding 
 unity to tne course, by placing each experiment in its proper con- 
 nection with the general body of the subject, so that the whole 
 will appear in good perspective. 
 
PREFACE 7 
 
 The author does not claim any great originality for his work. 
 He has followed to a large extent the method of presentation 
 developed by Mr. A. A. Michelson and used by him in his lec- 
 tures. He is also greatly indebted to the papers of Lord Ray- 
 leigh, and to the treatises mentioned above, as well as to Kayser's 
 monumental work, Handbuch der Spectroscopie. He is also under 
 obligations to his colleagues at the University of Chicago. He 
 received the course, somewhat in the shape in which he has now 
 written it out, from his predecessor, Mr. S. W. Stratton, and has 
 received during the five years in which it has been growing into 
 its present form many suggestions from the other members of the 
 Physics Department. He further wishes especially to thank Mr. 
 F. B. Jewett for making the drawings for the illustrations. 
 
 RYERSON PHYSICAL LABORATORY 
 
 The University of Chicago, 
 
 September 15, 1902. 
 
CONTENTS 
 
 PAGE 
 
 PREFACE ... . . . . . . - . . ft 
 
 I. LIMIT OF RESOLUTION 11 
 
 II. THE DOUBLE SLIT 19 
 
 III. THE FRESNEL MIRRORS ..... .30 
 
 IV. THE FRESNEL Bi- PRISM . . 44 
 
 V. THE MICHELSON INTERFEROMETER . .48 
 
 VI. THE VISIBILITY CURVES . . . . . .70 
 
 VII. THE PRISM SPECTROMETER .83 
 
 VIII. TOTAL REFLECTION . . . . . ( . . . .105 
 
 IX. THE DIFFRACTION GRATING . . . . . . 108 
 
 X. THE CONCAVE GRATING . ... . . .116 
 
 XI. POLARIZED LIGHT 127 
 
 XII. ROTATION OF THE PLANE OF POLARIZATION . 130 
 
 XIII. ELLIPTICALLY POLARIZED LIGHT 140 
 
 XIV. THE REFLECTION OF POLARIZED LIGHT FROM HOMOGENEOUS 
 
 TRANSPARENT SUBSTANCES . . . . . . 150 
 
 XV. METALLIC REFLECTION . . . . . . . 156 
 
 XVI. THE SPECTROPHOTOMETER * . 159 
 
 XVII. THE DEVELOPMENT OF OPTICAL THEORY ... 165 
 
 XVIII. THE TREND OF MODERN OPTICS . . . . . .172 
 
 APPENDIX .. .. . .. . . . . . 183 
 
 INDEX 194 
 
, 
 
 MANUAL OF ADVANCED 
 OPTICS 
 
 LIMIT OF RESOLUTION 
 
 Theory 
 
 Since diffraction phenomena play a very important part in 
 all optical instruments, it will be well to begin the study of 
 practical optics with a discussion of some of the cases of diffrac- 
 tion which occur most frequently. The simplest method of 
 observing these diffraction phenomena is that devised byFraun- 
 
 FlGURE 1 
 
 hofer.* The observation is most easily made by placing before 
 a telescope, focused for parallel rays, a screen containing the 
 opening whose diffraction pattern is to be studied, and directing 
 the telescope at a distant point or line source. The advantage of 
 
 *Fraunhofer, Neue Modification des Lichtes, Denks. d. k. Akad. d. 
 Wiss, zu Munchen, Vol. VIII., p. 1, 1821. Gesammelte Werke, p. 53; also 
 Harper's Scientific Memoirs, II, Prismatic and Diffraction Spectra, p. 
 13, 1898. 
 
 11 
 
12 MANUAL OF ADVANCED OPTICS 
 
 this procedure is apparent- because only those rays which are 
 parallel to each other in front of the screen come to a focus at a 
 given point, and therefore the measurement of the diffraction 
 pattern reduces to a measurement of angle only. 
 
 The simplest case is that of a rectangular opening. 
 
 Let AB (Fig. 1) represent a section of the screen by a plane 
 perpendicular to two of the sides of the rectangular opening, and 
 ab the section of the opening. Suppose that the source emits 
 monochromatic light only, and is a bright line perpendicular to 
 the plane of the drawing and so placed, with reference to the 
 screen, that the rays when they reach the screen form a parallel 
 beam which falls normally upon the opening. L represents the 
 objective of the telescope, and FF' its principal focal plane. 
 
 The image of the line source formed in this focal plane FF' 
 will consist of a central bright band bordered on either side by 
 alternate dark and bright bands, these latter decreasing rapidly in 
 intensity as they are more remote from the central bright band. 
 
 Let F represent the position of the center of the central 
 bright band, and F r that of the center of the first dark band. 
 We wish to find the value of the angle FLF' = subtended at the 
 center of the lens L by F and F'. Through a draw db, perpen- 
 dicular to LF, and ad, perpendicular to LF'. Then ab represents 
 the wave front of that plane wave which is brought to a focus at F. 
 All points of this wave front are in the same phase, and hence a 
 bright band is produced at F. Along ad, however, the various 
 points will be in different phases, the variation depending on the 
 angle which ad makes with ab. If now bd = \ (\ - wave length), 
 the phases of the successive points along ad will vary from A to 0, 
 that is, the phase at d will be a whole period ahead of that of the 
 wave front ab, all parts of which are in the same phase, while at a 
 the phases of the two coincide. Hence every point in one half of 
 ad will have a corresponding point in the other half of ad whose 
 phase differs from its own by A. Therefore, when the vibrations 
 
LIMIT OF RESOLUTION 13 
 
 from all the points along ad are brought together at the focus F', 
 they destroy each other in pairs, and darkness is the result. 
 
 Now the sine of the angle between the wave fronts ab and ad is 
 
 M 
 sin bad = = 
 
 ab 
 
 But, for the first dark band, M = A, and hence, placing ab = a 
 and letting the angle bad = FLF' = 0, we have 
 
 sin = - (1) 
 
 a 
 
 The illumination will again reach a minimum when bd = 2A, 
 oA, . . . wA, so that the general condition for a dark band in the 
 diffraction pattern formed in parallel light by a rectangular open- 
 ing is 
 
 a sin = wX, (2) 
 
 in which m stands for any whole number. 
 3 
 
 If, however, bd = A, the phases of the points along ad will 
 "Z 
 
 o 
 
 vary from to A, and we may conceive the beam of light to be 
 
 & 
 
 divided into three sections of equal width by planes perpendicular 
 to ad, each section containing points whose phases vary over 
 half a period. The corresponding points of two of these sec- 
 tions will be in opposite phases, and will therefore be in condition 
 to destroy each other, while the third section will transmit light 
 to the focus of the lens and produce a maximum of illumination 
 there. 
 
 Similarly, when bd is any odd multiple of half a wave, that is, 
 
 bd = - A, the phases of the consecutive points along ad will 
 So 
 
 vary from to - A, and we may conceive ad divided into 
 
 
 %m + 1 equal parts by planes perpendicular to ad. Then 2m of 
 
14 
 
 MANUAL OF ADVANCED OPTICS 
 
 these parts are in condition to destroy each other in pairs, leaving 
 one part to send light to the focus of the telescope. Hence the 
 
 condition for a maximum of illumination is bd = - A, or, 
 
 
 
 . ., 2m + 1 . 
 
 a sm = - A. (3) 
 
 4i 
 
 It is to be noted that, in determining the positions of the 
 maxima and minima of illumination, it is necessary to consider only 
 the width of the opening and the difference in phase of the two 
 rays passing *ts edges. 
 
 If the source consists of a pair of bright lines parallel to each 
 other, the diffraction pattern in the focal plane of the telescope 
 will be one which results from the superposition of the two pat- 
 terns which correspond to the two sources respectively. Let 
 <f> (Fig. 2) denote the angle subtended by the centers of the two 
 
 FIGURE 2 
 
 sources when viewed from the center of the lens L, and .Fthe 
 position of the center of the central bright fringe due to one 
 source, and G that of the center of the central bright fringe due 
 to the other source. Then FLG = <, and it is evident that if 
 <f> > 6 the central fringe at 6r, due to the second source, will be 
 separated from that at F, due to the first source, by a dark line, 
 and it will be possible to distinguish them from each other. If 
 < = 0, the two central bright fringes will just touch each other, 
 
LIMIT OF RESOLUTION 15 
 
 and it may, or it may not, be possible to perceive that we are 
 observing a double source. If <j> < 0, the two central bright 
 fringes will overlap, and it will be impossible to recognize that 
 
 the source is double. Hence <f> = 6 = is called the limit of reso- 
 lution. 
 
 It is to be noted that this limit of resolution depends only 
 on the width of the opening and the wave length of the light 
 used. Also that, excepting the central band, the angular separa- 
 tion between a maximum and its adjacent minima is 
 
 sin 0' sin = 
 
 Experiments 
 
 I. PROVE THE EQUATION sin = - 
 
 a 
 
 APPARATUS. The experiment is most easily performed with 
 a regular spectrometer. 
 
 ADJUSTMENTS. Telescope and collimator should be focused 
 for parallel rays and their axes arranged in line so that the image 
 of the slit is seen in the center of the field of view of the tele- 
 scope. The slit should be as narrow as the intensity of the 
 source will permit. The screen containing the rectangular open- 
 ing to be studied should be placed in front of the objective of the 
 telescope. The diffraction pattern will then appear in the field 
 of view. 
 
 It is, of course, not necessary to use a spectrometer. A tele- 
 scope and a sufficiently distant slit source will give satisfactory 
 results. 
 
 MEASUREMENTS. The direct measurement of A. may be avoided 
 by using a source of monochromatic light (sodium burner, mer- 
 cury or cadmium vacuum tube). However, inasmuch as all sources 
 
16 MANUAL OF ADVANCED OPTICS 
 
 of monochromatic light are comparatively faint, much better 
 results will be obtained by using sunlight or the electric arc. 
 A light which is sufficiently monochromatic for this purpose may 
 be obtained from sunlight by allowing the solar spectrum to 
 fall across the slit end of the collimator, so that the Fraunhofer 
 lines are parallel to the slit. The light which passes through the 
 slit will be sufficiently monochromatic^ and at the same time of 
 sufficient intensity to form very clear diffraction patterns. 
 
 If the solar spectrum has been so placed that one of the 
 Fraunhofer lines nearly falls upon the slit, we may assume with 
 sufficient accuracy for this purpose that the wave length which 
 forms the pattern is the same as that of the Fraunhofer line 
 which has been placed near the slit. If the slit has been placed 
 at random in the" spectrum of the sun or the electric arc, the 
 wave length may be measured by introducing between the col- 
 limator and the telescope a small diffraction grating having a 
 known number of lines to the millimeter. 
 
 The width of the opening may be measured on the dividing 
 engine or with a micrometer microscope in the usual way. In 
 order that may be large enough to be measured with accuracy, 
 a should be small, say 0.2 to 0.5 mm. 
 
 If a spectrometer has been used to perform the experiment, 
 the angle may be read directly from the graduated circle of 
 the instrument. If only a telescope has been used, the angle may 
 be measured by fastening a small mirror to the telescope and 
 measuring with a telescope and scale the angle through which this 
 mirror is turned. In case the vernier on the graduated circle of 
 the spectrometer does not read to at least 5" it is better to meas- 
 ure in any case with a telescope and scale. Since the angle 
 between two successive dark bands or between two successive 
 bright bands is also [cf. equation (2) and (3)], it is well to 
 measure the angle which corresponds to ten or more bands rather 
 than that which corresponds to a single one. 
 

 LIMIT OF RESOLUTION 1? 
 
 EXAMPLE 
 
 Angular width of 7 bands 57' 29.7" 
 
 Angular width of 1 band (9) 8' 12.8" 
 
 Wave length (A) (measured by grating). . ,0005866 mm. 
 
 Width of the opening (a) 0.245 mm. 
 
 - = .00239 
 a 
 
 sin 0=. 00239 
 
 II. DETERMINATION OF THE LIMIT OF RESOLUTION OF AH 
 OPENING OF WIDTH a. 
 
 APPARATUS. An ordinary telescope and a rather coarse 
 grating are all that is needed. 
 
 ADJUSTMENTS. Set up before a source of monochromatic 
 light the grating consisting of several slits about 1 mm. wide 
 and 1 mm. apart. Such a grating is most easily obtained by the 
 method devised by Fraunhofer. Two machine screws are set in a 
 plate of metal so as to be parallel to each other. A wire whose 
 diameter is about half the distance between the threads of the 
 screw is wound tightly about the two screws and soldered to 
 them. When the wires are cut from one side of the pair of 
 screws, the remaining wires form a very good grating. A 
 telescope, before the objective of which a screen containing an 
 opening of width a has been placed, should be focused upon the 
 grating. 
 
 MEASUREMENTS. Place the telescope with the screen 
 before its objective rather near the grating and draw it gradually 
 away, being careful to keep the edges of the slits of the grating 
 in focus. Find the point at which the observer first ceases to 
 be able to distinguish that the source at which he is looking is 
 made up of lines, i.e., the point at which the source appears to 
 be uniformly illuminated. At this point the limit of resolutidn 
 
18 MANUAL OF ADVANCED OPTICS 
 
 has been reached. If now d is the distance between the centers 
 of the line sources, and D the distance between the grating and 
 the screen before the objective of the telescope, then, according 
 to the theory given above, 
 
 .,*; 
 
 D ~ a' 
 
 The distance D may be measured with a tape, and the other 
 quantities as in the experiment above. 
 
 This experiment is largely qualitative. Different observers 
 will differ by 5% or more in determining the exact point at 
 which the grating ceases to be resolved by the telescope. 
 
 EXAMPLE 
 
 Distance between openings (d) .......... 0.138 cm. 
 
 Distance from source to screen (D) ....... 358 cm. 
 
 Width of opening in screen (a) .......... 0.157 cm. 
 
 Wave length (sodium) (A) ............... 0000589 cm. 
 
 - = .000385 
 
 - = .000375 * 
 a 
 
 *In connection with the above the student will do well to read 
 Schwerd, Die Beugungserscheinungen aus den Fundamentalgesetzen der 
 Undulationstheorie analytisch entwickelt, Mannheim, 1835. Verdet, Legons 
 d'optique physique, Vol. 1, p. 309 seq. Rayleigh, Collected Works, Vol. 1, 
 p. 488. Phil. Mag. (5) Vol. 10, p. 116, 1880. 
 
II 
 
 THE DOUBLE SLIT 
 Theory 
 
 In the preceding chapter the diffraction pattern produced by 
 a single rectangular opening was discussed. Let us now pass to 
 the case of two such openings or slits of equal width and parallel 
 to each other. It is, in the first place, quite evident that each 
 one of these slits will produce maxima and minima of illumina- 
 tion in accordance with the above conditions, namely, equations 
 (2) and (3). But in addition to these, other maxima and minima 
 
 FIGURE 3 
 
 will be formed because of the action of the rays from the 
 two slits upon each other. The conditions which determine the 
 formation of these latter fringes may be obtained by the same 
 process employed above. 
 
 The other conditions remaining the same as in Fig. 1, let db 
 and cd (Fig. 3) represent the two slits, each of width a. Call 
 the distance be between the neighboring edges of the two slits #, 
 and let a + b = d. 
 
 Let .Fbe the point at which the wave front ad is brought to a 
 
 19 
 
20 MANUAL OF ADVANCED OPTICS 
 
 focus, and F' the corresponding point for some other wave front, 
 as ED. Draw BD from B perpendicular to L F' 9 the point B 
 being determined by the condition aB = %d. If now aD = A, 3A, 
 . . . (2m + 1)A, it will readily be seen, since cc' = \aD, that the 
 points along DV will be in opposite phase to the corresponding 
 points along c'd' '. The conditions by which the minima due to 
 the interaction of the two slits are determined will, therefore, be 
 
 aD = X, 3A, . . . (2m +1) A. 
 If we call the angle between the wave fronts 0, this is equivalent to 
 
 2dsin 0= (2m + 1) A. (4) 
 
 Similarly, the conditions by which the maxima are determined 
 
 are 
 
 aD - 2mA, 
 or 
 
 2d sin 0' = 2mA. (5) 
 
 It will be noted that the angle between a maximum and its adja- 
 cent minima is sin 0' sin = / These maxima and minima 
 
 2d 
 
 which are determined by equations (4) and (5) are called those 
 of the second order. 
 
 It has thus far been assumed that the source was a bright 
 line, that is, that it was infinitely narrow. As all physically 
 attainable sources have an appreciable width, they must be looked 
 on as a series of line sources tangent to one another along their 
 length. Each of these line sources will produce its own set of 
 bands in the focal plane of the telescope, so that the diifraction 
 pattern actually observed is really made up by the superposition 
 of a large number of such diifraction bands. If the angular 
 width of the source, when viewed from the center of the lens L 
 (Fig. 3), is equal to the angle between the central bright fringe 
 and its first adjacent dark fringe as determined by equation (4), 
 the central bright fringe formed by the light which comes from 
 one edge of the source will fall on the first dark band formed by 
 
THE DOUBLE SLIT 21 
 
 light from the other edge of the source, and all trace of the 
 fringes will disappear. Hence it is seen that these interference 
 bands produced by two slits, for that is what these maxima and 
 minima of the second order really are, may be used to measure 
 small angular magnitudes. That they enable us to measure wave 
 length also is evident from equations (4) and (5). 
 
 As a clear understanding of these phenomena of interference 
 is desirable for what is to follow, it will be necessary to introduce 
 here a more detailed discussion of them. 
 
 In order that two transverse vibrations, such as the light 
 waves are, may produce interference, it is essential that they have 
 the same period. If they so interfere as to produce darkness, 
 they must in addition have the same plane of vibration and the 
 same amplitude. Since only those rays which proceed from the 
 same vibrating particle can fulfil these requirements, it is the 
 fundamental principle of all interference phenomena that only 
 those rays which proceed from the same point of the source' can 
 interfere with each other. In order, therefore, to produce inter- 
 ference, it is necessary to divide each ray into two, to lead the 
 two over paths of different lengths and to reunite them. Two 
 rays which result from this operation, and which are, therefore, 
 capable of producing interference, are called congruent rays. 
 
 Let us consider two congruent rays which are traveling in the 
 same direction along the same straight line. If we let the line be 
 the axis of #, and consider that the vibrations take place in the 
 xy plane, the displacement of a particle at the point x due to one 
 of these rays will be given by 
 
 = A COB 8r,-~ 
 
 and the displacement of the same particle due to the other ray by 
 
 It z + 
 ij z = A z cos '27T ^ --- 
 
 in which the symbols have the usual well-known significance. 
 
22 MANUAL OF ADVANCED OPTICS 
 
 The displacement of the particle due to both rays will, there- 
 fore, be 
 
 It x\ It x + 8\ 
 
 y = y\ + y* = A I cos 27r ( -f - ^ ) + A * cos %* \jr A/ 
 
 This becomes by expanding the last term 
 
 / 8\ / t x \ 8 . / t x \ 
 
 y = I A!+ A z cos 2 j cos 2ir i -= I -f AZ sin 2ir sm 2-rr i -= - j. 
 
 If in this equation we now introduce two new constants, A and 
 D, determined by the conditions 
 
 8 D 
 
 A! -F A 2 COS TT = A COS 2?r 
 A A 
 
 AZ sin STT = A sin 2-n- 
 
 A A 
 
 it reduces to 
 
 t x + D 
 
 X 
 
 This equation tells us that the superposition of two simple 
 harmonic vibrations which are traveling in the same direction, 
 results in a simple harmonic vibration which has the same period 
 as the two, but a different amplitude and phase. 
 
 If we solve the equations which define A and D f or A, we 
 obtain 
 
 cs 
 
 A 2 = A? + A* + %AiAz cos 2?r 
 
 A 
 
 In the special case in which AI and A 2 are equal, "this becomes 
 
 Hence we see that if the difference of path 8 is a whole number 
 
 ^ 
 
 of wave lengths, i.e., if = 1, 2, 3, etc., A 2 will have its maxi- 
 
 A 
 
 mum value. But if 8 is an odd number of half wave lengths, 
 
 ,8 135 
 i.e., if - = -, -, -, etc., A will be zero. 
 
THE DOUBLE SLIT 23 
 
 Let Ob (Fig. 4) represent the optical axis of a telescope, and 
 Si and S 2 two slits which are perpendicular to the plane of the 
 drawing. represents the center of a slit source parallel to the 
 two slits Si and S 2 , and separated from their plane by a distance 
 bO = d. Consider as the center of a coordinate system whose x 
 
 FIGURE 4 
 
 axis lies in the plane of the drawing and is perpendicular to bO, 
 and whose y axis is parallel to the slits, i.e., perpendicular to the 
 page at 0. Let r be any point of the source on the axis of x, 
 the angle CSiS 2 , and b the distance between the centers of the 
 slits. The difference of path D of the two rays S^' and CS 2 r, 
 which come to a focus in a direction 0, will be determined by 
 
 D=0r- Sir = CS Z + S 2 r - /Sir. 
 
 But, since Or = x, 
 
 Therefore 
 
 But S z r + Sir = 2d, nearly, and CS t = b6, therefore 
 
 But is the angle subtended by Or at the center of the lens. If 
 
24 MANUAL OF ADVANCED OPTICS 
 
 we call this angle , we shall have, as the difference of phase 
 between the two rays, 
 
 in which p represents the number of waves in the distance b 
 between the slits. 
 
 The intensity in the field of view in the direction may be 
 obtained from equation (6). If we denote by < () the intensity of 
 the light which comes from an elementary band of the source 
 at r, of angular width d, we have, as the intensity due to 
 this band alone, 
 
 I r = 2$ (I) COS>TT (0 + i) d = $ () + < () cos Ip* (0 + ) dt. 
 The total intensity in the direction is the integral of this expres- 
 sion, that is, if we let 
 
 our equation representing the intensity becomes 
 /= P + C cos 2pv0 - S sin 2 
 
 The first term of the right-hand side of this equation, when taken 
 between the proper limits, represents the total amount of light 
 emitted from the source. The values of which correspond to 
 the maximum and minimum values of / are determined by the 
 
 condition ^ = 0, or, 
 at) 
 
 C sin %pirO + 8 cos %pirO = 0. 
 
 From this we easily deduce as the equation for the intensity at 
 the maxima and minima, 
 
 I=P 
 
 It has been shown above that as the width of the source 
 changes, the fringes in the field of the telescope pass through 
 
THE DOUBLE SLIT 25 
 
 various stages of visibility, being sometimes very distinct, some- 
 times totally lost. We may define the visibility of the fringes as 
 the difference in the intensities of a maximum and a minimum 
 divided by the sum of the same intensities. Thus if /j represent 
 the intensity of a maximum, and / 8 that of a minimum, and V 
 the visibility, we may write 
 
 or, from the above, 
 
 F'-^. (8) 
 
 If the source is symmetrical with respect to the axis of y^ S - 0, 
 and we have 
 
 \ V =T (9) 
 
 If the source is, as we have supposed it above, a uniformly 
 illuminated rectangular opening or slit, <() = constant, and 
 hence, if we denote by a the angular width of the source when 
 viewed from the center of the lens, 
 
 If the linear width of the source is , a = , and we have 
 
 a 
 
 ba 
 sin TT = 
 
 The fringes will disappear whenever - - is a whole number m, 
 
 A.U 
 
 that is, when 
 
 - <> 
 
MANUAL OF ADVANCED OPTICS 
 
 Since ^ is the angular width of the source, and 7- the limit of 
 
 resolution of an opening of width #, this condition is the same as 
 the one previously mentioned (p. 20). 
 
 If we have as a source a pair of uniformly illuminated slits 
 placed symmetrically with respect to (Fig. 4), each of angular 
 width a and separated from each other by an angular distance y, 
 the expression above must be integrated twice, once between the 
 
 limits ? ?r IT + sr' an( ^ a g a i n between + ~ > + ^ + The 
 /$/$< to . a # 
 
 result is 
 
 in which p' is written to show that it differs from p. 
 
 Thus, if the linear distance between the centers of the two 
 
 s* 
 
 slits be denoted by , y = -7> and therefore the fringes will dis- 
 
 u 
 
 appear not only in accordance with equation (11) but also when 
 
 13 
 
 p y = r 2" 
 
 2m - 1 
 
 ' 1S? n 
 
 Zm - I A 
 
 where b'(= p'\] is written to distinguish it from the b in equation 
 
 (11) and m is the order of the disappearance. 
 
 Fig. 5 represents the two 
 curves expressed by equations 
 (10) and (12). The dotted curve 
 corresponds to (12), and we see 
 that from it we infer the sepa- 
 ration of the two sources. The 
 full curve corresponds to (10), 
 and from it the width of each 
 individual source may be ob- 
 FIGURE 5 tained. It is also to be noted 
 
THE DOUBLE SLIT 27 
 
 that the full curve is an envelope of the maxima of the dotted 
 curve. * 
 
 Experiments 
 
 MEASURE THE WIDTH OF A HARROW SLIT 
 
 APPARATUS. To get the best results in making measure- 
 ments with the double slit it is advisable to use a large lens of 
 rather long focus. A four or five inch telescope objective with a 
 focal length of from one to two meters answers very well. The 
 lens should be mounted fifteen or twenty meters from the source 
 whose nature is to be investigated. The double slit should be 
 mounted directly in front of the lens. These slits must either be 
 movable so that their distance apart can be varied, or they may 
 have a fixed distance apart and the nature of the source may be 
 changed. In the former case the slits must be mounted on a 
 plate and fitted with a right and left screw or with some other 
 mechanism which allows them to be moved symmetrically with 
 respect to the center of the lens, while in the latter case they may 
 be cut from a card. Slits about 1 mm. wide and 4 or 5 cm. apart 
 are very satisfactory. 
 
 ADJUSTMENTS. Sunlight or some other bright light is 
 allowed to pass through the openings whose dimensions are to be 
 determined and to fall upon the lens before which the slits have 
 been placed. The image of the source when observed with the 
 eyepiece is found to consist of a series of fringes. Care must be 
 taken to find the interference fringes, which are narrower and 
 clearer than the diffraction fringes and which lie in the center 
 of the entire pattern. 
 
 MEASUREMENTS. The observation may be made either by 
 separating the slits or by varying the nature of the source until 
 the fringes disappear. In both cases it is necessary to measure 
 
 * For similar solutions for sources of other shapes see Michelson, Phil. 
 Mag. (5) Vol. 30, p. 1, 1890. Mascart, Traite d'Optique, Vol. 3, p. 567 seq., 
 Paris, 1893. 
 
28 MANUAL OF ADVANCED OPTICS 
 
 the distance b between the slits and the width a of the source. 
 If the source is double, the distance between the two apertures 
 which compose it must also be measured. The distance d from 
 the double slit to the source must also be known. Since the wave 
 length A of the light used appears in the equation, it is well to 
 filter the light used so that the rays which reach the lens haie 
 approximately one wave length. If white light is used the disap- 
 pearance observed will be that of the fringes of greatest bright- 
 ness which correspond to the wave length .00055 mm. 
 
 EX'AMPLES 
 
 1. A single slit was set up at a distance of 2080 cm. from the 
 double slit. Sunlight was used. The width of the slit was set 
 at random and the two slits moved until the fringes disap- 
 peared. The following two sets of observations were made. 
 Only the first disappearance of the fringes was taken. 
 
 Wave length A 000055 cm. 
 
 Width of slit a as measured directly 0.049 cm. 
 
 Distance between slits b 2.335 cm. 
 
 Distance d to source 2080 cm. 
 
 ^ = .0000236 
 b 
 
 I = .0000236. 
 
 2. The width of the slit was then changed so as to be .074 cm. 
 and the distance between the double slits was found to be 1.545 
 cm. Then, as above 
 
 ^=.0000356, 
 o 
 
 | = .0000356. 
 
 3. Using sunlight as before, the two slits were set at a fixed 
 distance of 4.075 cm. apart. The width of the source was then 
 
THE DOUBLE SLIT 
 
 29 
 
 varied and measured when the fringes disappeared. In this way 
 five disappearances were observed as follows, m denoting the 
 order of the disappearance : 
 
 m = 1, 
 
 w = 2, 
 
 m = 3, 
 
 % = 
 
 .277 mm. 
 
 f-t- 
 
 .277 mm. 
 
 
 
 m 
 
 
 11 
 
 .567 
 
 it 
 
 .283 
 
 t I 
 
 .858 
 
 
 
 .286 
 
 it 
 
 1.150 
 
 it 
 
 .287 
 
 it 
 
 1.412 
 
 it 
 
 .282 
 
 mean .283 mm. 
 The distance d was in this case as above. Therefore 
 
 ^ = .0000135, 
 | = .0000136. 
 
 4. A double source was used whose two components were 
 0.053 cm. apart, i.e. , c = 0.053. The distance d remained 2080 cm. 
 Disappearance of the fringes was observed for the following 
 values of V : 
 
 1.085 cm. 
 
 " 3.324 
 " 5.580 
 
 " 
 
 8.015 
 
 b' = 2.170 cm. 
 
 2.216 
 2.232 
 2.290 
 
 mean 2.227 cm. 
 
 = 0.0000255, 
 
 l 
 
 = 0.0000247. 
 
Ill 
 
 THE FKESNEL MIRRORS 
 Theory 
 
 In the preceding chapters the nature of the measurements 
 which can be made by observing the interference phenomena 
 which arise when the light from a source is viewed with a tele- 
 scope, before the objective of which one or two slits have been 
 placed, has been discussed. This method of producing interfer- 
 ence is encumbered with two serious objections. First, the rays 
 which interfere cross at a rather large angle, which results in 
 making the fringes very narrow; and, second, the amount of light 
 received by the observer is small because the source must be very 
 narrow in order to see fringes at all. These two drawbacks may 
 be obviated by suitable changes in the form of the apparatus. 
 
 It is evident that, in the formation of these bands, that por- 
 tion of the objective which is covered plays no part. Hence, it is 
 
 FIGURE 6 
 
 possible to dispense with the entire objective excepting those 
 portions which are immediately behind the slits. These portions 
 need not, for this purpose, be curved, but may be replaced 
 either by prisms as in Fig. Ga or by mirrors as in Fig. 6J. 
 
 30 
 
THE FRESX^L MIRRORS 
 
 31 
 
 Both these forms enable us to increase the width of the fringes by 
 changing the inclination of the rays which meet at F. In 
 broadening the fringes by these methods it is necessary either to 
 
 make the distance MJF or SF large, which is objectionable on 
 practical grounds, or to bring the mirrors or prisms closer together. 
 This latter procedure changes the telescope with which we began, 
 
 
 FIGURE 8 
 
 into one of the two well-known forms of interference apparatus, the 
 Fresnel mirrors or the bi-prism. The mirrors are shown diagram- 
 matically in Figs. 7 and 8, the bi-prism in Fig. 9. It is evident 
 
 FIGURE 9 
 
 that these two forms of interference apparatus are limited in 
 their possible applications because the various parts can not be 
 moved sufficiently with reference to each other. 
 
MANUAL OF ADVANCED OPTICS 
 
 The angle between the rays which interfere may be made 
 smaller and the fringes thereby enlarged by inserting at F a plane 
 parallel plate of glass, as in Fig. 10. The ray J/i F is transmitted 
 
 FIGURE 10 
 
 and the ray M 2 F reflected by the plate F, so that by a suitable 
 inclination of this plate the angle between the rays may be made 
 as small as desired, and hence the fringes may be indefinitely 
 enlarged. 
 
 A further great improvement can be made by replacing the 
 slit by another piece of the plane parallel plate inserted at 0, as 
 in Fig. 11. With this arrangement the light is not limited to a 
 
 beam that has passed through a slit, but may come from any 
 broad source. The gain in illumination obtained by this simple 
 device is enormous. The instrument sketched in Fig. 11 is an 
 interferometer. It is capable of many variations of form, aAd can 
 be used for a large variety of delicate measurements. By thus 
 
THE FRESNEL MIRRORS 33 
 
 converting the telescope into an interferometer all definition and 
 resolution are lost, but a great gain in accuracy is the result.* 
 
 We will begin the discussion of the various forms of inter- 
 ferometer with the simpler cases of the Fresnel mirrors and 
 bi-prism.f Let J/i3/ 2 , Fig. 12, represent the projection of 
 
 FIGURE 12 
 
 the two mirrors on a plane perpendicular to their line of inter- 
 section, and S the projection of a line source of monochromatic 
 light, that source being parallel to the intersection of the mirrors. 
 Then 61 and S will represent the two virtual images of S 
 formed by J/ x and J/ 2 respectively. Let AB represent a screen 
 parallel to the line joining the virtual sources /Si and S z . 
 
 The illumination upon the screen AB, due to light reflected 
 by the mirrors, will be identical with that due to two separate 
 sources, Si and S^ each of the same intensity as S. Since P 
 is on the perpendicular erected at the middle of the line /Si$ 2 , it is 
 equidistant from Si and /Si, and therefore each pair of congru- 
 ent rays from the source will arrive at P in the same phase and 
 
 *Michelson, Am. Jour. Sci. (3) 39, p. 115, 1890. 
 
 f Fresnel: Memoire sur la Diffraction de la Lumiere, .63-64. Oeuvres, 
 VoL I, p. 329 seq. Mem. de VAcad., V., p. 414, 1826. 
 
34 MANUAL OF ADVANCED OPTICS 
 
 produce a maximum of illumination there. It is necessary to find 
 what will be the illumination at any other point P' on the screen. 
 This will clearly depend upon the difference in length of the paths 
 of the rays which arrive at P' from Si and S 2 respectively. This 
 difference of path is expressed in the notation of page 23 by 
 
 but the illumination at P' is a maximum when this difference of 
 path is a whole number of wave lengths, m\. Hence the condi- 
 tion for a maximum is 
 
 bx 
 
 mX = i 
 d 
 
 that is, we have a bright fringe at distances x from the center P 
 equal to -rA, -T-A, -T~A, etc. The distance between the successive 
 maxima is seen to be constant. If we call it e we have 
 
 e = *X. f (14) 
 
 We can use the mirrors then to determine wave lengths, for d 
 and e are easily measured directly, and if the distance from the 
 source to the intersection of the mirrors be called/, and the angle 
 between the mirrors a, it is readily seen that 
 
 b = Zf sin a ; 
 
 hence 
 
 A = Ze sin a. (14') 
 
 The measurements can be further simplified by placing the source 
 at an infinite distance by means of a lens. Then / and d become 
 infinite together and their ratio is unity. Under these con- 
 ditions we have 
 
 A = %e sin a. (15) 
 
THE FREStfEL MIRRORS 35 
 
 In order to obtain good results with the mirrors the conditions 
 assumed in the above discussion must be accurately fulfilled. The 
 two mirrors must touch each other along one edge, and that edge 
 must be made to coincide with the intersection of their reflecting 
 surfaces.* We have already shown that when the angular width of 
 the source, viewed from the intersection of the mirrors, is equal 
 to the angular width of a fringe, viewed from the same point, the 
 fringes disappear. Therefore, a narrow source is necessary for the 
 production of distinct measurable fringes. The effects of diffrac- 
 tion on the phenomena have been discussed by "Weberf and by 
 Struve.J 
 
 Let us now consider the effect of introducing a transparent 
 plate in the path of one of the rays. Let J/i and Jf 2 (Fig. 13) 
 
 FlGUBE 13 
 
 represent the mirrors, and Si and JS. 2 the virtual sources as in 
 Fig. 12. Suppose we introduce into the path of one of the rays 
 SiP a plane parallel plate Q of some transparent substance, 
 whose index of refraction for the monochromatic light used is //,, 
 
 *Feussner, Winkelmann, Handbuch der Physik, Vol. II, Pt. 1, p. 528. 
 t Weber, Wied. Ann., Vol. VIII, p. 407, 1879. 
 JStruve, Wied. Ann., Vol. XV, p. 49, 1882. 
 
36 MANUAL OF ADVANCED OPTICS 
 
 and whose thickness is t. The change D in the optical length of 
 the path SiP produced by the plate Q will be 
 
 The difference between the optical paths of the two rays at any 
 point of the screen P' will then no longer be expressed by 
 
 Sf'- /SiP'= ^ but by 
 
 S z P'~ft l P'=~-(^-l)t. 
 This difference of optical path is zero when 
 
 i\ fd 
 
 X=(fi.-l) -y> 
 
 by which the distance through which the central fringe of the 
 system is shifted from its former position P by the introduction 
 of the plate Q is determined. But at this distance x we had, 
 before the insertion of the plate, a fringe determined by the 
 equation 
 
 ft* 
 
 ^T 
 
 in which p stands for any number, whole or fractional. If in this 
 equation the value of x found above be substituted for #, we see 
 that the central fringe of the shifted system takes the place of 
 that fringe of the original system whose order is 
 
 It has thus far been assumed that the source emitted mono- 
 chromatic light only. If this is not the case, but waves of various 
 lengths are sent out, each set of waves will form its own set of 
 fringes in accordance with equation (14). Since the distance 
 between the fringes which correspond to each wave length is pro- 
 portional to that wave length, the resultant figure on the screen will 
 
THE FRESXEL MIRRORS 37 
 
 consist of the superposition of a number of systems of fringes of 
 unequal breadth. If the mirrors are so arranged, as in Fig. 12, 
 that the rays from the two sources travel entirely in air before 
 reaching the screen, the position of the central fringe, which is 
 determined by the condition, S. 2 P' SiP' = 0, will, since this 
 condition is independent of the wave length, be the same for all 
 colors. Hence all systems of fringes will agree in having a bright 
 fringe at that point P which is determined by this equation. If 
 white light be emitted from the source, the central fringe will be 
 white, free from all trace of color. Hence it is called the 
 achromatic fringe. Since the distance from P of any other 
 bright fringe is proportional to the corresponding wave length, 
 the various systems of fringes will correspond with each other at 
 no other point. Hence the adjacent fringes will be colored, their 
 color depending on how the various systems happen to overlap at 
 the point considered. 
 
 If the symmetry of the optical paths be disturbed by the 
 introduction of a transparent plate into the path of one of the 
 rays, as in Fig. 13, the central fringe of each system will be 
 
 shifted, as we have seen, an amount (/u. 1) , which will be 
 
 different for each wave length, because /n varies. In this case 
 there will, in general, be no point at which the congruent rays of 
 all wave lengths will arrive in the same phase. There will, there- 
 fore, be no absolutely achromatic fringe. Nevertheless, a system 
 of colored fringes may be obtained whose central band appears 
 nearly achromatic. The determining condition here is not that 
 the difference in the optical paths S. 2 P' and /S^P' equals zero, which 
 is manifestly impossible because of the dispersion of the plate , 
 but that the change in phase at the point P' , corresponding to a 
 change in wave length, be a minimum. Let x be the distance 
 from P, the position of the achromatic fringe when the plate Q 
 is out, to the point P', where the central fringe of the colored 
 
38 MANUAL OF ADVANCED OPTICS 
 
 system appears when the plate Q is in. The optical difference of 
 path of the two rays has been shown to be 
 
 &? -&!- 01)*. 
 
 But =- is the apparent or geometrical retardation, which we will 
 
 call D', and (//, l)t = Z>, which we will denote now by /(A). The 
 difference of phase at P' for any wave length will, therefore, be 
 
 In order that this difference of phase be a minimum, its derivative 
 with respect to X must be zero. Performing the operation we get 
 
 Now equation (16) tells us how much the central fringe of the 
 system corresponding to any wave length A. is shifted by the 
 introduction of the plate g, that is, it is displaced till it coincides 
 with a fringe of the order 
 
 But the central fringe of the colored system is thereby shifted till 
 it coincides with that fringe of the system produced by wave 
 length A, whose order is 
 
 Hence the center of the colored system is displaced from the 
 shifted center of the system corresponding to A by the number 
 of fringes expressed by the equation 
 
 This displacement can readily be calculated if we know the 
 form of the function of X for the substance of which the plate Q 
 
THE FRESNEJ, MIRRORS 39 
 
 consists. It has been found that that form of function first 
 proposed by Cauchy satisfies the experimental facts very well for 
 the visible spectrum. Assuming this equation, we have 
 
 = A + -^ 
 
 A" 
 
 but 
 
 hence 
 
 hence 
 
 Experiments 
 MEASUREMENT OF WAVE LENGTHS WITH THE MIRROR 
 
 APPARATUS. The observations with the Fresnel mirrors 
 can be made in several ways. In every case it is, of course, 
 necessary to have a pair of mirrors properly mounted. The 
 mirrors which are usually furnished with the optical bench are 
 entirely satisfactory. They are mounted as follows: A brass 
 plate, which can be fastened upon one of the uprights of the 
 optical bench, serves as a mounting for both mirrors. One 
 mirror is so fastened to this plate that it rests upon three screws, 
 and can thus be adjusted so as to be parallel to the other mirror, 
 which is rigidly fastened upon a slide and can be moved by means 
 of a micrometer screw in a direction at right angles to its surface. 
 
 Simple and effective mirrors can be made according to 
 Quincke* in the following way : Select a piece of best plate glass 
 about 10 cm. long, 2.5 cm. wide, and 3 mm. thick. Cut it in 
 the middle into two pieces each 5 "cm. long. Blacken the rear 
 
 * Quincke, Pogg. Ann. 132, p. 41, 1867. 
 
40 MANUAL OF ADVANCED OPTICS 
 
 surfaces with shellac containing lamp black, in order to destroy 
 the reflection from the rear surface. Plane a heavy block of wood 
 smooth and flat, and arrange six balls of soft wax of equal size upon 
 the block in such a way that each of the two pieces of plate glass, 
 when placed upon them, will be supported upon one ball along the 
 edge where the two touch and upon two balls along the edge 
 which is farthest from the edge of contact. Dust the surfaces 
 carefully and lay upon them another piece of carefully dusted 
 plate glass about 20 cm. long, 5 cm. wide, and 3 mm. thick. 
 Press firmly with one finger upon the larger plate of glass directly 
 over the line of contact of the two smaller pieces. The upper 
 glass will bend enough to set the lower pair of plates at a small 
 angle with each other, and will at the same time keep their edges 
 of contact together. The original apparatus used by Fresnel was 
 of this nature. 
 
 In case it is desired to perform the experiment with divergent 
 light, i.e., according to equation (14'), it is necessary to have in 
 addition to the mirrors a slit, a micrometer microscope, and a 
 telescope and scale. If the experiment is to be performed with 
 parallel light, i.e., according to equation (15), an ordinary spec- 
 trometer may be used to advantage, the mirrors taking the place 
 of the prism. The angle a can then be measured directly upon 
 the graduated circle of the instrument, and the distance between 
 the fringes can be determined by removing the objective of the 
 telescope, and measuring the angle which the fringes subtend, 
 and the distance from the line of contact of the mirrors to the 
 focal plane of the eyepiece. The description of the experiment, 
 as given below, applies to the optical bench. Its adaptation to 
 other methods is left to the student. 
 
 ADJUSTMENTS. First, the centers of the slit, the mirrors, 
 and the micrometer should be brought into the same horizontal 
 plane by measuring their distance from the table upon which the 
 apparatus rests. Second, the slit and the cross-hairs of the 
 
THE FRESNEL MIKRORS 41 
 
 micrometer should be made vertical. The slit may be made 
 vertical with the aid of a plumb line. The cross-hairs must 
 then be adjusted so as to be parallel to the slit by forming by means 
 of a lens of short focal length an image of the slit in the 
 plane of the cross-hairs and then rotating the micrometer about 
 a horizontal axis. Third, the mirrors should be adjusted so 
 that their line of intersection coincides with the edges which 
 are in contact. This can be approximately accomplished by 
 observing the image of a straight edge in the mirrors and 
 adjusting until this image is an unbroken line. The accurate 
 adjustment is made with the help of the fringes. The mirrors 
 are then set in place so as to reflect the light from the slit to the 
 micrometer. The angle between the slit and the plane of the 
 mirrors should not be less than 10 in order to avoid complica- 
 tions due to diffraction at the edges of the mirrors. On looking 
 into the micrometer, the slit being illumined with monochro- 
 matic light, and altering slowly the angle between the mirrors, the 
 fringes will appear if the preliminary adjustment has been care- 
 fully made. If the fringes do not appear, the plate which carries 
 the mirrors should be turned about a horizontal axis to bring 'the 
 intersection of the mirrors parallel to^the slit, i.e., vertical. If 
 the fringes even then do not appear it indicates that the intersec- 
 tion of the planes of the surfaces of the mirrors does not coincide 
 with their common edge. This may be caused by not having 
 adjusted the mirrors so that both are vertical, or by allowing one 
 to protrude in front of the other. The fringes may then be found 
 by turning the mirror which is supported on three screws, about 
 a horizontal axis, or by moving the other mirror, which can be 
 displaced in a direction perpendicular to its surface, or by both 
 operations. When the fringes appear they will probably not be 
 parallel to the cross-hairs of the micrometer. Their centers may 
 be made parallel to the cross-hairs by tilting the adjustable mirror 
 about a horizontal axis. They may be made parallel to the cross- 
 
42 MANUAL OF ADVANCED OPTICS 
 
 hairs throughout their entire length by tilting the plate which 
 carries both mirrors, about a horizontal axis. Having thus 
 obtained straight monochromatic fringes parallel to the cross- 
 hairs, the fringes in white light are found by displacing the 
 mirror which is movable in a direction normal to its surface. 
 
 The collimation axis of the micrometer should then be brought 
 into parallelism with the line from the intersection of the mirrors 
 to the micrometer. Open the slit rather wide and reflect, by 
 means of a small mirror, a beam of light through the lower half 
 of the slit. Turn the micrometer about a vertical axis until the 
 reflection of the slit upon the front lens of the micrometer is seen 
 through the upper half of the slit. 
 
 MEASUREMENTS. The distance between the fringes is meas- 
 ured with the micrometer. It is well to measure the distance 
 over which the thread moves in passing ten to twenty fringes. 
 
 The distance between the intersection of the mirrors and the 
 slit, and between that intersection and the cross-hairs of the 
 micrometer, can be measured with a large pair of dividers with a 
 sliding scale, or with a fine tape. 
 
 The angle between the mirrors may be measured with a tele- 
 scope and scale in the usual way. 
 
 In order to attain accuracy with the Fresnel mirrors it is 
 necessary to use a bright source of light, and to make the angle 
 between the mirrors large enough to allow the formation of 
 twenty or more fringes. As these fringes will be narrow a rather 
 high-power micrometer is necessary in order to count them 
 accurately 
 
 EXAMPLES 
 
 1. Using as a source of light a sodium burner the following 
 measurements were obtained : 
 
 Distance between the fringes e= .012 cm. 
 
 Distance from the mirrors to the slit /= 29.5 cm. 
 
 Distance from the mirrors to the micrometer . . . li = 51.1 cm. 
 
THE FRESNEL MIRRORS 43 
 
 Hence d =f+h = 80.6 cm. 
 
 Distance from mirrors to scale D = 163 cm. 
 
 Deflection, by observing the image of the scale in 
 
 first one mirror and then the other a = 2.16 cm. 
 
 Hence sin a = = .0066*25 
 
 "^= .00878 
 A = .0000582 
 
 2. The mirrors were mounted on a large spectrometer in place 
 of the prism. Sodium light was used as a source. The light 
 passed through the collimator, and was, therefore, parallel when 
 it fell on the mirrors. The following measurements were 
 obtained : 
 
 Angle between the mirrors, being % the angle between the two 
 images of the slit, a = 11' 28". 
 
 The lens of the telescope was then removed and the angle 
 subtended in the focal plane of the eyepiece by 28 fringes 
 measured. This was found to be 18' 25". The distance from 
 the focal plane of the eyepiece to the intersection of the mirrors, 
 which should coincide with the axis of the instrument, was 46.21 
 cm. Hence the angular width of a fringe was 3d". 46, and the 
 linear width e = .008835 cm. Hence A = 2e sin a = .00005897 cm. 
 
IV 
 
 THE FKESNEL BI-PEISM 
 Theory 
 
 "VVe will now pass to the consideration of the Fresnel bi-prism. 
 Attention has already been called to this form of interferometer, 
 its derivation from the telescope being sketched in Fig. 9. Since 
 measurements with the bi-prism are most easily made when the 
 source is at an infinite distance, the formulae will be developed 
 for this case only. Let ABC, Fig. 14, represent the projection 
 
 A E 
 
 FIGURE 14 
 
 of the prisms upon a plane perpendicular to their refracting 
 edges, and PP' a screen parallel to AC. Suppose a beam of 
 parallel light to fall normally upon the prism faces A C. Upon 
 leaving the prisms the beam will be divided into two whose wave 
 fronts are represented by BD and BE. From B draw BP per- 
 pendicular to PP''. The point P being equidistant from the two 
 wave fronts, will evidently be a position of maximum illumination. 
 It is necessary to find what will be the illumination at some other 
 point P' of the screen. Let PP'= x and BP'= d. As in the 
 
 44 
 
THE FRESNEL BI-PRISM 45 
 
 case of the mirrors, since all points along BD and BE are in the 
 same phase, the illumination at P' will be a maximum if 
 
 EP' -DP' = m\. 
 But EP = d sin EBP', and DP' = d sin DBP'. Hence. 
 
 EP'-DP' = 2d cos sin P'BP. 
 
 But if we denote by a the angle between the .wave fronts, 
 DBE = TT - a; and sin P'BP = -j- Therefore, since a is small, 
 
 EP' - DP' = x sin a. 
 We, therefore, have a maximum of illumination when 
 
 A t>\ 
 
 ./ = -. -- . - > etc. 
 sin a sin a 
 
 The distance between the successive maxima, denoted by e, is, 
 therefore, 
 
 A 
 
 '- - 
 
 sin a 
 from which we get for the wave length 
 
 A. = e sin. a. (18) 
 
 The same relations between the width of the source and the visi- 
 bility of the fringes hold for the bi-prism as for the mirrors. 
 
 Experiments 
 
 MEASUREMENT OF WAVE LENGTHS WITH THE BI-PRISM 
 APPARATUS. The apparatus needed for this experiment is 
 the same as that used in the previous experiment, with the excep- 
 tion of the mirrors, these being now replaced by a bi-prism. 
 
 ADJUSTMENTS. The adjustments of the bi-prism are much 
 simpler than those of the mirrors. The slit should be parallel to 
 
46 MANUAL OF ADVANCED OPTICS 
 
 the common base of the prisms, and their common face should be 
 perpendicular to the path of the light from the slit to them. 
 This latter adjustment is made by reflecting a beam of light 
 through the. lower half of the slit and revolving the bi-prism 
 about a vertical axis until the reflected light is seen through the 
 upper half of the slit. 
 
 The measurements are most easily and accurately made with 
 the help of a spectrometer. Telescope and collimator are adjusted 
 for parallel light and arranged so that the image of the slit falls 
 upon the cross-hairs of the telescope. The bi-prism is then 
 placed upon the prism table of the spectrometer and adjusted, 
 either in the way described above, by reflecting light through the 
 slit, or with the help of a Gauss eyepiece, so that it is perpendicu- 
 lar to the common collimation axis of the collimator and telescope. 
 Upon removing the objective of the telescope the fringes will 
 appear in the field of view. If the slit is not parallel to the com- 
 mon base- of the prisms the fringes will not be clear and evenly 
 spaced throughout their entire length. The slit should then be 
 rotated about a horizontal axis until the fringes are clear and 
 evenly spaced. In this case, as in the case of the mirrors, the 
 fringes are very narrow, so that a rather high-power micrometer 
 is necessary to measure them accurately, and a bright light is 
 indispensable. In both of these experiments the solar spectrum 
 across the slit will be found to give the greatest satisfaction as a 
 source of light. 
 
 MEASUREMENTS. If a spectrometer is used the only meas- 
 urements needed are the distance between the fringes and the 
 angle between the beams behind the bi-prism. To obtain the 
 former the angle through which the telescope turns when the 
 cross-hair passes over a counted number of fringes, and the distance 
 between the bi-prism and the focal plane of the micrometer are 
 measured. From these two observed quantities the linear dis- 
 tance between the fringes is at once calculated. The angle 
 
THE FKESNEL BI-PRISM 47 
 
 between the beams is measured by replacing the objective of the 
 telescope and setting the cross-hair on first one and then the other 
 of the images of the slit ; the angle through which the telescope 
 has turned is the desired angle. If the graduations upon the 
 circle of the spectrometer are not sufficiently fine to allow of 
 reading the angles accurately, a small mirror, such as is used for 
 galvanometers, should be mounted upon the telescope and the 
 angle read in this mirror with a telescope and scale in the usual 
 way. 
 
 EXAMPLE 
 
 The bi-prism was placed in the spectrometer in place of the 
 prism, and the angle between the wave fronts measured. The 
 value was 
 
 a = 18' 21". 
 
 The objective of the telescope was then removed and the angle 
 subtended in the focal plane of the eyepiece by 16 fringes in 
 sodium light was found to be 13' 10". Thus the angular width 
 of one fringe was 49 "A. The distance from the focal plane of 
 the eyepiece to the axis of the instrument was 46.21 cm. 
 Hence the linear width of a fringe was e = .01107 cm. Hence 
 
 \ .00005904 cm. 
 
 

 THE MICHELSON INTERFEROMETER 
 
 Theory 
 
 "We will now pass to the consideration of the interferometer as 
 shown in Fig. 11. This form of instrument may be further 
 simplified by making one plate perform the functions both of 
 separating and reuniting the beam. To accomplish this it is only 
 necessary to turn the two mirrors into the positions CD as shown 
 in Fig. 15. This is known as the Michelson* interferometer. Irs 
 
 
 
 FIGURE 15 
 
 its simplest and most efficient form it consists merely of four 
 glass plates. Let A, B, C, D (Fig. 15) represent the projec- 
 tions of the four plates on a plane perpendicular to their surfaces. 
 
 * Michelson, Phil. Mag. (5) 13, p. 236. 
 
 48 
 
THE MICHELSOX INTERFEROMETER 49 
 
 Light from a source S falls upon the plate A at an angle of inci- 
 dence of approximately 45. Plates A and B are polished on 
 both sides and should, to obtain the hest results, be of best optical 
 glass, free from all strains. They should, furthermore, be cut 
 from the same piece, so as to insure their having the same optical 
 thickness, and their surfaces should be as plane and as nearly 
 parallel to each other as it is possible to make them. The rear 
 surface of A is coated with a semi-transparent film of silver or 
 platinum, which should, if the clearest interference bands are 
 desired, be of such a thickness that it reflects half the light inci- 
 dent upon it to Z>, and transmits the other half to C. D and 
 C are two plane mirrors, coated on the front surface with a 
 thick coat of platinum or silver, and so adjusted as to reflect the 
 light incident upon them back over nearly the same path. These 
 two reflected beams meet again on the rear surface of the plate A 
 in a condition suitable to the production of interference bands. 
 The plate B is inserted to make the two paths optically identical. 
 
 The observer looks into the apparatus from 0. He will see 
 the plate D directly, and an image of the plate C reflected by the 
 plate A. This image of C will appear in the direction of the 
 plate D as far behind A. as the mirror C really is in front of it. 
 The instrument is thus seen to consist essentially of a film of air 
 inclosed between the plate D and the virtual image of the 
 plate C. Hence in discussing the interference phenomena pro- 
 duced by this instrument it is necessary to consider only this film 
 of air. 
 
 Let omn om. 2 (Fig. 16) represent the two plane mirrors 
 and D whose intersection is projected at o, and whose mutual 
 inclination is <. The illumination at any point P, not necessarily 
 in the plane of the figure, will depend on the mean difference of 
 phase of all the pairs of congruent rays which reach P after 
 reflection from the mirrors. 
 
 If the source of light is sufficiently broad, the illumination 
 
50 
 
 MANUAL OF ADVANCED OPTICS 
 
 FIGURE 16 
 
 at P will be independent of its distance, form, or position 
 Let us suppose that it is a plane luminous surface which coin 
 cides with om-i. Then its image in om t will coincide with 
 and its image in om z will be a plane symmetrical to om 
 with respect to om 2 . For every point p of the first surface 
 there is a corresponding point p' of the second, which is symmet 
 rically placed, and in the same phase of vibration. If w( 
 suppress now the source of light and the mirrors, and replace 
 them by the images, the eifect at any point P is unaltered 
 Consider now a pair of points p, p' . Let 8 be the angle forrne( 
 by the line joining P and^? (or //) with the normal to the surfac( 
 om 2 -> 8 and < being both supposed small. The difference of 
 optical path D will be 
 
 D = Pp' - Pp =pp' cos 8 
 
 to quantities of the second and higher order. If we denote by 2J 
 the distance between the images at the point where they are cut 
 by the line Pp, we will have to a close degree of approximation 
 
 D = 2t cos 8. 
 
 (19) 
 
 The difference of phase A at P is '2-n- - 
 
 A 
 
 Since cos 8 is nearly 
 
 equal to unity, if %t is increased by A. by gradually separating the 
 images, D is also increased by A, and the difference of phase A at P 
 passes through a complete period. 
 
THE MICHELSOX INTERFEROMETER 
 
 51 
 
 In order to find the form of the interference fringes, let cdef, 
 c'd'e'f (Fig. 17) represent the two images, and let their intersec- 
 tion be parallel to cf, and their inclination be 2<. Let P be the 
 
 FIGURE 17 
 
 )oint considered; P' the projection of P on the surface cdef\ 
 and PB the line forming with PP' the angle 8. Draw P'D 
 parallel to cf, and P'C at right angles, and complete the rectangle 
 BDP'C. Let P'PC = i, and DPP'= 0. Let PP' = P; and call 
 the distance between the surfaces at the point P', 2/ . We have 
 then, 
 
 t = t + CP' ism <f> = t + P tan <f> tan i, 
 'and 
 
 D = 2 
 
 tan < tan t) cos 8, 
 
 or 
 
 
 f + P tan < tan i 
 
 (20) 
 
 We see that in general D has all possible values; and hence 
 all phenomena of interference would be obliterated. 
 
 Bat to an eye placed at P the interference fringes will be 
 visible under certain conditions. Let db (Fig. 18) represent the 
 pupil of the eye. Since it has appreciable size it will receive light 
 
52 MANUAL OF ADVANCED OPTICS 
 
 not only from p and p' 9 . but also from other points as p l and pf. 
 Consider the ray .pa to enter the pupil at one end of a diameter, 
 and the ray pj>, parallel to pa, to enter it at the other end of the 
 
 FIGURE 18 
 
 same diameter. Since these rays are parallel they will come to a 
 focus at the same point on the retina. The difference of phase at 
 this point of the congruent rays ap and ap' will be 
 
 M COS 8 
 
 The difference of phase of the other pair of congruent rays bp t 
 and bpi will be 
 
 A 47r^ cos 8 
 
 AI= ~^~ 
 
 If now A - A 1 = -j- X, the interference phenomena which are pro- 
 duced by each pair of congruent rays are in opposite phase; that 
 is, if one pair would produce a maximum of illumination at the 
 common focus, the other would produce a minimum. The 
 resulting sensation would be a combination of the two. Hence, 
 in order that interference bands may be observed by an eye at P, 
 conditions must be so arranged that the difference in the As for 
 those pairs of congruent rays which come to a focus at the same 
 point on the retina, be less than half a wave. This may be 
 accomplished in two different ways: First, we may observe 
 through a small enough opening, the eye being focused on the 
 
THE MICHELSOX INTERFEROMETER 53 
 
 point p. By this simple device the clearness of the interference 
 phenomena can frequently be materially increased. Second, by 
 making the As the same for every pair of congruent rays which 
 are focused at the same point on the retina. This is accomplished 
 by making t = h, and focusing the eye for parallel rays. The 
 second method is generally to be preferred for reasons to be given 
 presently. 
 
 The exact investigation of the form of the fringes is a matter 
 of considerable complexity. An approximate conception of their 
 appearance to an eye placed at a point P may be obtained from 
 equation (20). Thus let cdef (Fig. 17) represent the xy plane, 
 and P' the center of a system of rectangular coordinates whose axis 
 of y is parallel to the intersection of the mirrors. Then P'C = #, 
 
 P'D = y, tan i = -=- and tan 6 = - Let tan <f> = k. Our equation 
 then becomes 
 
 D = 
 
 ^ j- j- 
 
 x u 
 
 Since -p- and -p- are small, this equation reduces to 
 
 O Ox 
 
 or 
 
 m ^^!-2P 2 . 
 
 This is the equation of a circle whose center is at the point 
 
 /2P 2 & \ 
 
 I ^ , 0\- Hence the fringes are always approximately circles 
 
 whose centers lie on the axis of x at a distance from the origin 
 determined by the inclination of the mirrors, the difference of 
 optical path, and the distance P to the point of observation. 
 
 Two cases are of special practical interest. First, if t = t 09 
 
54 MANUAL OF ADVANCED OPTICS 
 
 that is, if the mirrors are parallel, Ic = 0, and our equation 
 reduces to 
 
 In this case the center of the circles lies at the origin itself and 
 their radii are given by 
 
 If the difference of the optical paths between the successive 
 rings counted from the center be wX, we shall have 
 
 n\ = 2/ - D, 
 or 
 
 D = Zt- n\. 
 
 Hence the radii of the successive rings are given approximately by 
 
 (21) 
 
 Second, if the intersection of the mirrors passes through the 
 origin, D = at that point. In this case the distance to the 
 
 center of the circles, ~ , becomes infinite, and the fringe 
 
 through the origin is a straight line parallel to the axis of y. 
 This particular fringe, since it is the only absolutely straight one, 
 serves as a convenient mark from which to begin measurements. 
 It is called the central fringe of the system. 
 
 If the mirror om 2 (Fig. 16) is moved perpendicular to itself 
 till it passes to the other side of the mirror om ly D, since its 
 value passes through 0, changes sign, and the center of the 
 
 circles passes from H ^ to -- ^ Hence on opposite sides of 
 
 the central fringe the curvature of the fringes has opposite 
 signs; that is, .on both sides they appear convex toward the 
 

 THE MICHELSON INTERFEROMETER 
 
 55 
 
 central fringe. This fact is of great assistance in locating the 
 central fringe. 
 
 Experiments 
 
 I. MEASURE THE WAVE LENGTH OF SODIUM. LIGHT 
 APPARATUS. From the discussion above it is evident that 
 the essential parts of the interferometer are four plates of glass 
 arranged as shown in Figs. 15 and 19, and an arrangement for 
 
 moving one of the mirrors in a direction normal to its surface. 
 This motion is effected by mounting the mirror D upon a slide, 
 which can be moved by a screw along the ways EF. These ways 
 must be accurately straight, so that in its motion the mirror 
 remains strictly parallel to its original position. The screw 
 
56 MANUAL OF ADVANCED OPTICS 
 
 carries at its front end a worm wheel M which in turn is driven 
 by a worm W. The worm can be disengaged from the wheel 
 when a rapid motion of the screw is desired. The worm wheel is 
 graduated upon its front face. 
 
 The dividing plate A is mounted firmly in a metal frame 
 upon a plate of brass H which is screwed to one end of the ways. 
 That side of A which carries the silver half-film should be 
 turned toward the plate B. This plate B should be so mounted 
 upon the brass plate H as to allow of a small motion about a 
 vertical axis. This motion is necessary in order to be able to 
 adjust the two plates A and B so that they are strictly 
 parallel. It is also useful in measuring differences in phase or 
 small fractions of a wave accurately. The mirror D is rigidly 
 mounted upon the slide, while the mirror C rests upon three 
 adjusting screws, which are set in a plate, which is perpendicular . 
 to the plate H and firmly fastened to it. Small springs hold this 
 mirror in place against the three screws. 
 
 ADJUSTMENTS. Measure roughly the distance from the sil- 
 ver half -film upon the r.ear of the plate A to the front of the 
 mirror C. Set the mirror Z>, by turning the worm wheel Jf, so 
 that its distance from the rear of A is the same as that of C from 
 A. This need not be done accurately. It is suggested because it 
 is easier to find the fringes when the distance between the mirror 
 D and the virtual image of the mirror C is small. This distance 
 will hereafter be called the distance between the mirrors. 
 
 Now place a sodium burner, or some other source of monochro- 
 matic light, at Z, in the principal focus of a lens of short focus. 
 It is not necessary that the incident beam be strictly parallel. 
 Hold some small object, such as a pin or the point of a pencil, 
 between L and A. On looking into the instrument from 0, 
 three images of the small object will be seen. One image is 
 formed by reflection at the front surfaces of A and Z>; the second 
 is formed by the reflection at the rear surface of A and the front 
 
THE MICHELSON INTERFEROMETER 57 
 
 surface of D\ the third is formed by reflection from the front 
 surface of C and the rear surface of A. Interference fringes in 
 monochromatic light are found by bringing this third image into 
 coincidence with either of the other two by means of the adjusting 
 screws upon which the mirror C rests. If, however, it is desired 
 to find the fringes in white light, the second and third of these 
 images should be brought into coincidence, because then the two 
 paths of the light in the instrument are symmetrical, i.e., each is 
 made up of a given distance in air and a given thickness of glass. 
 When the paths are symmetrical, the fringes are always approxi- 
 mately arcs of circles as described above. If, however, the first 
 and third images are made to coincide, then the two optical paths 
 are unsymmetrical, i.e., the path from A to C has more glass in 
 it than that from A to D, and in this case the fringes may be 
 ellipses or equilateral hyperbolae, because of the astigmatism which 
 is introduced by the two plates A and B. It is quite probable 
 that the fringes will not appear when the two images of the 
 small object seem to have been brought into coincidence. This 
 is simply due to the fact that the eye can not judge with sufficient 
 accuracy for this purpose when the two are really superposed. To 
 find the fringes then it is only necessary to move the adjusting 
 screws slightly back and forth. As the instrument has been here 
 described, the second image lies to the right of the first. 
 
 Having found the fringes the student should practice adjust- 
 ment until he can produce at will the various forms of fringes 
 described on page 54. Thus the circles appear when the dis- 
 tance between the mirrors is not zero, and when the mirror D 
 is strictly parallel to the virtual image of C. The accuracy of 
 this adjustment may be tested by moving the eye sideways and 
 up and down while looking at the circles. If the adjustment is 
 correct, any given circle will not change its diameter, as the eye is 
 thus moved. To be sure, the circles appear to move across the 
 plates because their center is at the foot of the perpendicular 
 
58 MANUAL OF ADVANCED OPTICS 
 
 dropped from the eye to the mirror Z>, but their apparent diam- 
 eters are independent of the lateral motion of the eye. For this 
 reason it is advisable to use the circular fringes whenever possible. 
 
 To find the fringes in white light, adjust so that the monochro- 
 matic fringes are arcs of circles. Move the carriage rapidly by 
 intervals of a quarter turn or so of the wheel M. When the region 
 of the white-light fringes has been passed, the curvature of the 
 fringes will have changed sign, i.e., if the fringes were convex 
 toward the right, they will now be convex toward the left. Having 
 thus located within rather narrow limits the position of the mir- 
 ror Z), which corresponds to zero difference of path, it is only 
 necessary to replace the sodium light by a source of white light, 
 and move the mirror D by means of the worm slowly through this 
 region until the fringes appear. 
 
 These white-light fringes are strongly colored with the colors 
 of Newton's rings. The central fringe, the one which indi- 
 cates exactly the position of zero difference of path, is, as in 
 the case of Newton's rings, black. This black fringe will be 
 entirely free from color, i.e., perfectly achromatic, if the plates 
 A and B are of the same piece of glass, are equally thick, and 
 are strictly parallel. If they are matched plates, i.e., if they are 
 made of the same piece of glass and have the same thickness, 
 their parallelism should be adjusted until the central fringe of 
 the system is perfectly achromatic. When this is correctly done, 
 the colors of the bands on either side of the central one will be 
 symmetrically arranged with respect to the central black fringe. 
 
 MEASUREMENTS. An accurate scale graduated to tenths of a 
 millimeter is set upon the slide behind the mirror D. Over this 
 a micrometer microscope is placed and focused on the scale. The 
 microscope should be rigidly attached to the base of the instru- 
 ment so that it does not move relatively to the interferometer dur- 
 ing the observation. The cross-hair of the micrometer is set upon 
 one of the tenth millimeter divisions. The mirror D is then 
 
THE MICHELSON INTERFEROMETER 59 
 
 slowly moved with the worm FT, and the number of fringes which 
 pass when the cross-hair of the microscope moves over one-tenth 
 of a millimeter are counted. The circular fringes should be used 
 because, as stated above, their phase is independent of the position 
 of the eye, so that if the eye moves during the observation, no 
 error will be introduced. It will be necessary to look from time 
 to time through the microscope so as to note when the cross-hair 
 reaches the next tenth millimeter mark. Since a motion of the 
 mirror of 0.1 mm. introduces a difference of path of about 340 
 waves, it is safe to count 300 without looking at the microscope. 
 Having obtained the number of fringes which pass when the mir- 
 ror D moves through 0.1 mm., then, since the difference in path 
 introduced by this motion is 0.2 mm., the wave length sought is 
 
 2 
 A = -^ mm., in which N denotes the number of fringes counted. 
 
 EXAMPLE 
 
 Sodium light was used as a source and the number of waves 
 which passed while the carriage moved 0.3 mm. were counted. 
 
 This number was found to be 1018. Hence 
 
 
 
 A = T^ Pim. = 5894 - 10" 7 mm. 
 
 II. DETERMINE THE RATIO OF THE WAVE LENGTHS OF THE 
 SODIUM LINES D l AND D 2 
 
 Apparatus and adjustment as in Experiment I. 
 
 MEASUREMENTS. Since the yellow radiation of sodium con- 
 tains two vibrations of different periods, two different sets of 
 fringes will be formed by it. As the mirror D is moved these sets 
 of fringes, since they are formed by waves of different lengths, 
 will move at different rates. Thus in certain positions of the 
 mirror D the bright fringes of one set will fall upon the dark 
 fringes of the other set, and the field of view will be almost evenly 
 illuminated. The fringes do not disappear entirely because the 
 
60 MANUAL OF ADVANCED OPTICS 
 
 light produced by the shorter of the waves is more intense 
 than that produced by the longer. At certain other positions 
 of the mirror Z>, the bright fringes of one set will coincide with 
 the bright fringes of the other set, and there will be strong con- 
 trast between the bright and the dark fringes in the field of view. 
 This contrast in the intensity of the fringes is called their visi- 
 bility. This subject of visibility is treated at length in the next 
 chapter. Here it is sufficient to note that in the interval between 
 two positions of greatest contrast of the fringes, i.e., between two 
 positions of maximum visibility, there must be one more of the 
 shorter waves than of the longer. In order to determine the ratio 
 of the wave lengths, then, it is necessary to measure the distance 
 the mirror D moves in passing between two positions of maximum 
 or of minimum visibility. If this distance be divided by the 
 shorter wave length we obtain the number of shorter waves in the 
 interval. This number minus one will be the number of longer 
 waves in the interval, and the ratio of the wave lengths will be 
 the inverse of the ratio of the number of waves. 
 
 In making the observations it will be found impossible to 
 determine accurately the position of any one maximum or mini- 
 mum. An accurate result may, however, be obtained in the fol- 
 lowing way: Draw the mirror D as far forward as is possible 
 without causing the fringes to disappear entirely. Then move the' 
 mirror backward by jumps of about one-twentieth of a turn of the 
 worm wheel. Take the reading on the worm wheel at the points 
 which appear to be either maxima or minima. In this way about 
 twenty readings of the positions of the maxima and twenty of 
 those of the minima can be obtained. The average should then 
 be taken by subtracting the first reading in each set from the 
 eleventh, the second from the twelfth, etc., and then taking the 
 mean of these averages. In making the calculation it is to be 
 noted that the difference of path is twice the distance through 
 which the mirror has moved. 
 
THE MICHELSON INTERFEROMETER 61 
 
 EXAMPLE 
 
 The distance between the positions of the maximum clearness 
 of the fringes was determined in the way described above. In all, 
 thirty maxima and thirty minima were read. The mean value of 
 the interval was found to be 0.5802 mm. Hence 
 
 0.5802 
 
 005890 
 
 iii = n-2 - 1 = 984, 
 
 A! 085 
 A, =984' 
 
 in which A^ is the wave length of^D^ and \% that of D z . 
 
 If we wish the difference between the wave lengths in milli- 
 meters we may proceed thus : 
 
 .5802 .5802 11 1 
 
 = 1, or -- - = 
 
 A, X, A, A! .5802 
 
 Therefore, A : - A,, = --= 5.98 10~ 7 mm. 
 
 
 III. DETERMINE THE INDEX OF REFRACTION AND THE DIS- 
 PERSION OF A PIECE OF GLASS 
 
 Apparatus and adjustment as in Experiment I. 
 
 MEASUREMENTS. In order to perform this experiment in a 
 theoretically rigorous way there should be added to the interferom- 
 eter two extra metal frames similar to those used to hold the 
 mirrors, one in front of each of the mirrors D and C. These 
 frames should be set upon pivots so that they can be rotated 
 about a vertical axis. The one in front of the mirror C should be 
 arranged with a worm wheel or a tangent screw so that it can be 
 rotated slowly and steadily. Two pieces of the glass whose index 
 of refraction and dispersion are to be determined should be used. 
 These pieces should, of course, have the same thickness. One of 
 
62 MANUAL OF ADVANCED OPTICS 
 
 them is waxed to each of the movable frames so as to cover half 
 the same half of the field of view. The interferometer 
 must then be adjusted for the white-light fringes. The piece 
 of glass which is in front of the mirror D is then turned 
 through any angle, say 15 to 20. In this way extra glass is 
 introduced into the path of the light between A and D. This 
 extra glass should then be compensated for by turning the other 
 piece, that between A and (7, through the same angle. When the 
 angles through which the two pieces have been turned are the 
 same, the white-light fringes will appear in the half of the field of 
 view which is covered by the two plates. This turning of the 
 second plate of glass should be done slowly with the worm wheel, 
 and the fringes which pass during the operation should be 
 counted. The angle through which this plate turns must be 
 measured by fastening to the frame which carries it a small mir- 
 ror, and reading the angle through which this mirror turns with 
 an ordinary telescope and scale. Before measuring this angle, care 
 must be taken to have the plate perpendicular to the beam 
 passing through it. This can be done by rotating the plate 
 through the position in which it is at right angles to the beam, 
 and noting the point at which the fringes reverse their direction 
 of motion ; for it is evident that when the plate is normal to the 
 beam its optical thickness is a minimum, and therefore a turning of 
 the plate in either direction will increase the optical thickness, and 
 cause the fringes to move in one particular direction. It will, 
 however, be found that the plate can be turned through a consid- 
 erable angle before the fringes move appreciably. Therefore, to 
 obtain the scale reading which corresponds to the normal position 
 of the plate, turn the plate in one direction until two or three 
 fringes have passed, and take the reading on the scale. Then turn it 
 in the other direction until the same number of fringes has passed, 
 and take the reading. 4 The mean of these two readings will then 
 be the reading which corresponds accurately to the normal posi- 
 
THE MICHELSON INTERFEROMETER 
 
 63 
 
 tion of the plate. Having then counted the fringes which pass 
 while the plate is turning through the angle i, and having meas- 
 ured that angle, the index of refraction is obtained as follows: Let 
 t represent the thickness of the plates of glass, and 2JV the num- 
 ber of fringes counted while turning through the angle i. Let, 
 further, AB (Fig. 20) represent the direction of the light, MNOP 
 
 
 
 the plate in its position perpendicular to the beam, and M'N'O'P' 
 its position after it has been turned through the angle i. Let the 
 two surfaces OP and O'P' intersect at c, and draw through/, the 
 intersection of the surface M'N' with AB, the line//? parallel to 
 MN. The light incident along AB upon O'P' will, when the 
 plate has been turned, travel along the path cgh. It is evident that 
 before the turning the optical distance between the planes OP and /ft 
 
 consisted of a distance ce = t in glass, and a distance ef= 1 
 
 J cos i 
 
 in air. After the turning, the optical distance between these two 
 
 i 
 
 planes consists of a distance eg 
 
 cos r 
 
 in glass, and of a distance 
 
 7 t sin (i r) , . . . . , . , , 
 
 nil = - tan i in air, in which r denotes the angle of refrac- 
 
 cos r 
 
 tion. The numbers of waves in these various distances are obtained 
 by dividing the distances in air by A, the wave length, and those 
 
 in glass by , in which /w. stands as usual for the index of refrac- 
 
64 MANUAL OF ADVANCED OPTICS 
 
 tion. The difference between the number of waves in the optical 
 path between the planes OP and fli before the turning and the 
 number in the path after the turning is half of the number which 
 has been counted, because in the observation the light has passed 
 twice through the plate. Therefore the following equation is 
 obtained : 
 
 tu. t sin (i r} . t 
 
 ! -] * '- tan %- tu. ; + 1 = JV A. 
 
 cos r cos r cos i 
 
 If this equation be reduced with the help of the equation - = /n, 
 
 JV 2 A 2 
 and solved for /*, there results, neglecting the term -7 - in the 
 
 lit 
 
 numerator, 
 
 (;-^X)(l-cos/) 
 ^ #(l-cos i) - NX 
 
 The thickness t of the glass may be measured with the calipers or 
 in any other accurate^way. 
 
 It was shown in the chapter on the Fresnel mirrors that if 
 the optical symmetry of the two paths over which the light travels 
 is disturbed by the introduction of a plate of some transparent 
 substance, the fringes in white light no longer possess a truly 
 achromatic central fringe, but one which may seem fairly achro- 
 matic, and which is displaced from the true position of the central 
 fringe of the set of fringes which correspond to the wave length A. 
 by a number of fringes [cf. equation (17)] 
 
 P -?= 
 
 in which t' = 1. (23) 
 
 cos r 
 
 Now, when the glass which was added to the path AD by turn- 
 ing the plate in front of the mirror Z>, is compensated for, by 
 rotating the plate in front of the mirror (7, we add extra glass to the 
 path A C also. The two paths are thus made finally symmetrical, 
 

 THE MICHELSON INTERFEROMETER 65 
 
 so that the achromatic light fringe to which we count indicates the 
 true position of the central band of the monochromatic system. By 
 such a count, then, we obtain the true number of fringes through 
 which the monochromatic system has been shifted, namely^?. It 
 is possible, however, to compensate for the extra glass in the path 
 AD in another way, namely, by drawing the mirror D toward A. 
 If we do this and count the fringes which pass, we count to the 
 shifted position of the white-light fringes, i.e., we count the 
 number p r . If then we make the count both ways, once by turn- 
 ing the plate in front of the mirror C, and once by drawing up 
 the mirror Z>, the plate in front of C being perpendicular to the 
 beam, we determine both p and p' of the above equation. Since 
 / and X are also known, it is in this way possible to determine the 
 B of the Cauchy dispersion equation (p. 39). With this value of 
 B and the value of p determined from equation (22) , it is then 
 possible to determine the A of the dispersion equation. Thus 
 with a single source of monochromatic light it is possible to 
 determine both the index of refraction and the dispersion of a 
 plate of glass. 
 
 EXAMPLE 
 
 Two pieces of optical glass were mounted in the instrument 
 as described above. The thickness of the glass was 6.81 mm., 
 i.e., t = 6.81 mm. One of the plates was then turned through an 
 angle , which was measured with a telescope and scale, and found 
 to be i = 16 41' 30". The other piece was then turned, and the 
 sodium fringes counted until the fringes in white light appeared 
 over the whole field. This number was 342. Since the light 
 traverses the plates twice, the number N in the formula is half of 
 this, i.e., N= 171. Hence the index of refraction for sodium is 
 jt.v a = 1.5180. The compensating plate was then turned back till 
 it was normal to the path of the light, and, the first piece of glass 
 remaining inclined at the angle i, the movable mirror was drawn 
 
66 MANUAL OF ADVANCED OPTICS 
 
 up and the fringes again counted until the white fringes appeared 
 in their first position. This number of fringes was %p' = 355. 
 Hence %(p'-p) = 355-342 = 13 or/ -^ = 6. 5. 
 
 From equation (23), t' = .1254 mm. Hence from equation (17) 
 
 = 53 10- 10 , 
 and therefore (cf. p. 39), A = 1.5027. 
 
 With these values of A and B the index of refraction for the 
 green line of mercury was calculated (A = 5461 10~ 7 ). The result 
 was fA Hg = 1.5205. The fringes were then counted in mercury 
 light, and the result was 2^=370. With this N we get from 
 equation (22) p Hg = 1.5204. 
 
 IV. DETERMINE THE CHANGE or PHASE PRODUCED BY PER- 
 PENDICULAR REFLECTION AT A SILVER SURFACE 
 
 Apparatus and adjustments as in Experiment I. 
 
 MEASUREMENTS. One of the mirrors C or D of the interferom- 
 eter must be removed and freshly silvered over three-quarters of 
 its surface. This is best accomplished by keeping one-half of the 
 mirror covered with a piece of glass while it is in the silvering 
 solution. When the silver is sufficiently deposited so that the 
 film is perfectly opaque, the solution is poured off, the mirror and 
 the tray which holds it rinsed with distilled water, the piece of 
 glass on the surface of the mirror turned through 90, and a fresh 
 solution poured on. In this way the surface of the mirror is 
 coated with silver over three-quarters of its surface, as shown in 
 Fig. 21. The quarter marked a has upon it two layers of silver, 
 ~b and c each has one layer, and d has none. If now the mirror 
 be replaced in the interferometer, and the fringes found, it will be 
 noted that where the fringes cross the boundaries of these four 
 sections of the surface, they are displaced. Thus a fringe which 
 passes from a to I will not be a straight line or an arc of a circle, 
 but that portion of it which is over the surface a will be displaced 
 
THE MICHELSON INTERFEROMETER 
 
 67 
 
 with respect to that over the surface b by an amount which 
 depends upon the thickness of the film ac. If this displacement 
 is measured, we thereby determine the thickness of the film ac. 
 
 FIGURE 21 
 
 In order to measure it, the white-light fringes must first be found 
 so as to determine, by means of the central black fringe, in which 
 direction the shifting has taken place. Having thus determined 
 the direction of the shifting, its amount is measured by the com- 
 pensator. There 'should be fastened to one end of the frame 
 which holds the compensator, a small spiral spring. The other 
 end of the spring should be fastened to a string which may be 
 wound about a pin. The pin must carry a graduated drum or 
 circle, so that its position may be read and thereby the tension of 
 the spiral spring determined. The tension of this spring is 
 opposed by the elasticity of the stud by which the compensator 
 frame is fastened to the plate H. If the pin is turned so as to 
 tighten or loosen the spiral spring, the compensator will turn 
 through a small angle, and this angle will be proportional to the 
 amount of the turning of the pin. To measure the difference of 
 phase, the pin is turned until one fringe has passed, and the 
 angle through which the pin has turned is read upon the 
 graduated head. The pin should then be turned until the 
 shifted part of the fringe comes to the position of the unshifted 
 part, and the angle through which it has turned is read upon the 
 graduated head. The ratio of the angles through which the pin 
 has turned in these two operations is then the fraction of a wave 
 
68 MANUAL OF ADVANCED OPTICS 
 
 by which the fringe is shifted, and half of this fraction is the 
 thickness of the film ac in wave lengths. Of course this measure- 
 ment should be made in monochromatic light, the white-light 
 fringes being used merely to recognize the direction of the shift. 
 Having thus measured the thickness of the film ac, it is only 
 necessary to measure the shift in a fringe at the junction between 
 c and d, in order to be able to calculate the change of phase due 
 to the reflection at the silver surface. In calculating this change 
 of phase, account must be taken of the fact that half a wave is 
 lost at the reflection upon glass, and also of the direction of the 
 shift after allowance has been made for the thickness of the film 
 ac. If the shift is in the same direction as that in the fringe 
 across ab, then, since a is nearer the observer than J, the wave 
 from the silver surface is ahead of that from the glass. 
 
 The difference of phase between the light reflected at a surface 
 glass-air, and that reflected at a surface glass-silver, is very easily 
 obtained with the aid of a plane parallel plate of optical glass 
 which has been silvered over half of one surface. This plate 
 must be introduced into the interferometer in place of the mirror 
 D, with the silver side away from the observer. A second piece 
 of the same plate of glass must then be introduced in front of the 
 mirror C in order to compensate for the extra glass added by the 
 introduction of the first plate into the path AD. The white-light 
 fringes having been found upon the rear of the plate in front of 
 D, and been set perpendicular to the dividing line between the 
 silvered and the unsilvered portions of the plate, the displacement 
 of the central fringe is measured as described above. Since the 
 light is reflected from the glass-air surface without change of 
 phase, the shifting of the fringe indicates a retardation, i.e., a 
 loss of part of a wave. 
 
 EXAMPLES 
 
 1. One of the mirrors of the interferometer was coated with a 
 double film of silver as illustrated in Fig. 21. The displacement 
 
THE MICHELSON INTERFEROMETER 69 
 
 of the fringe across ab was measured in sodium light and found 
 to be 0.26. Hence the thickness of the film ac was 0.13X. The 
 fringe across cd was displaced 0.17 of a fringe. Since the reflec- 
 tion upon the glass surface d produces a change of phase of 0.5 of 
 a wave, the retardation produced hy the silver is 0.5 + 0.17 0.26 
 = 0.41X. 
 
 2. The displacement of the fringe on the glass-silver surface 
 was found to be 0.28A. 
 
 For further study, of the applications which can be made of the inter- 
 ferometer the student is referred to the following: 
 
 Michelson and Morley, "On the Relative Motion of the Earth and 
 Ether," Am. Jour. Sci. (3) 22, p. 120, 1881; 34, p. 333, 1887; Phil. Mag. (5) 
 4, p. 449, 1887. "On the Effect of the Motion of the Medium upon the 
 Velocity of Light," Am. Jour. Sci. (3) 31, p. 377, 1886. "On a Method of 
 Using the "Wave Length of Sodium Light as a Practical Standard of 
 Length," Am. Jour. Sci. (3) &#, 427, 1887; Phil. Mag. (5) 24, p. 463; Am. 
 Jour. Sci. (3) 37, p. 181, 1889. 
 
 Michelson, "Light Waves and Their Applications to Meteorology," 
 Nature, 49, p. 56, 1893. "Valeur du Metre en longeurs d'ondes lumin- 
 euse," Trav. et Mem. Bur. Internat. Poids et Mes. XI, p. 1, 1894; "On the 
 Relative Motion of the Earth and Ether," Am. Jour. Sci. (4) 3, p. 475, 1897. 
 
 Morley and Rogers, "On the Measurement of the Expansion of Metals 
 by the Interferential Method," Phys. Rev. 4, pp. 1 and 106, 1896. 
 
 Wads worth, "On the Application of the Interferometer to the Meas- 
 urement of Small Deflections of a Suspended System," Phys. Rev. 4, 
 p. 480, 1897. 
 
 Hull, "On the Use of the Interferometer in the Study of Electric 
 Waves," Phys. Rev. 5, p. 231, 1897. 
 
 Johonnott, "On the Thickness of the Black Spot on Liquid Films," 
 Phil. Mag. (5) 47, p. 501, 1899. 
 
 Earhart, "On Sparking Distances between Plates," Phil. Mag. (6) 1, 
 p. 147, 1901. 
 
 Gale, "On the Relation between Density and Index of Refraction of 
 Air," Phys. Rev. 14, p. 1, 1902. 
 
VI 
 
 THE VISIBILITY CURVES 
 Theory 
 
 In Chapter II it has been shown that it is possible to deter- 
 mine the width of a rectangular source and the distance between 
 two such sources by observations made with the double slit. In 
 these experiments the fringes disappeared when the distance 
 between the slits was such that the angle subtended by the 
 sources was one wave divided by that distance, and also when 
 that distance was such that the angular width of each single 
 source was equal to a wave length divided by it. In Chapter III, 
 the two slits have been converted into an interferometer, and in 
 Chapter V we have used the interferometer to measure the ratio 
 of the wave lengths of the two sodium lines, by determining the 
 number of waves in the change which takes place in the distance 
 between the mirrors in passing from one position of maximum 
 visibility to the next. The close similarity between the two 
 experiments must be evident at once, the difference lying in the 
 fact that in the case of the two slits we have angles to resolve, 
 while with the interferometer we have differences in wave lengths, 
 or rather in numbers of vibrations, to determine. The resemblance 
 becomes even closer if we conceive the spectral source to be 
 resolved as far as possible by an ordinary spectroscope. The 
 sodium lines, for example, would then appear as two line sources, 
 i.e., they would very much resemble the double source consisting 
 of a pair of parallel slits as treated above. 
 
 We might expect then that the equations which connect the 
 visibility curves with the distribution of light in the source would 
 
 70 
 
THE VISIBILITY CURVES 71 
 
 be very similar in the two cases, "and would differ only in the fact 
 that angles in the case of the two slits would be replaced in the 
 case of the interferometer by wave lengths or numbers of vibra- 
 tions. 
 
 The solution of a visibility curve is very difficult. It will help 
 us much in obtaining such solutions if we begin by the inverse 
 process of assuming a known distribution of light and plotting the 
 corresponding visibility curve. Fig. 22 gives a series of such 
 curves. The nature of the distribution in the source is shown at 
 the left, and the actual vibrations are plotted, the visibility curve 
 being the envelope of the curve. The abscissae of the curves rep- 
 resent distance traveled, and the ordinates intensity. Thus in 
 Fig. 22 the curve 1 represents the resultant of two trains of 
 homogeneous waves of the same amplitude but with slightly 
 different periods which start in the same phase at a. When 
 they have traveled a distance ab, they are seen to be in opposite 
 phase, and the visibility curve comes to zero. It is quite clear 
 that the distance they have to travel before they come into 
 opposite phases depends upon the difference of their periods. 
 So we can already guess that a determination of the distance ab 
 would lead to some knowledge of that difference in the periods. 
 
 In the case of the interferometer we have formed by the two 
 trains of waves two separate sets of fringes, and when the movable 
 mirror is displaced, these sets travel across the field at different 
 rates, as was shown on page 59. When a certain difference of 
 path has been introduced, represented by ab in curve 1, these two 
 sets of fringes overlap so as to present an evenly illuminated field 
 of view and the visibility curve comes to zero. As the difference 
 of path is further increased, the fringes soon come into such posi- 
 tions that one set has overtaken the other by one whole fringe, 
 and then we have a maximum of visibility as indicated at c. Thus 
 if we interpret the vibrations which unite to form the curve 1 as 
 fringes, i.e., as periodic variations of intensity, and consider that 
 
72 
 
 MANUAL OF ADVANCED OPTICS 
 
 the distance traveled is replaced in the interferometer by differ- 
 ence of path, then the envelope represents the variations in the 
 visibility of the resultant set of fringes as the two separate sets 
 pass by in the field of view. Hence the envelopes of the curves 
 in Fig. 22, are, in the case of the interferometer, the visibility 
 
 'A 
 
 L |^ A/WWV ^ 
 
 5. 
 
 j. I II ||to*^ vwwwv^^ 
 
 7. 
 
 1 
 
 FIGURE 22 
 
 curves, and from them we can draw conclusions as to the nature of 
 the source. 
 
 Thus curve 2 represents the visibility curve which corre- 
 sponds to a double source, each of whose components is broad, 
 i.e., does not send out waves of one definite period only, but waves 
 whose lengths vary between the limits X and \ + rfA. It will be 
 noted that the distance between the centers of these two sources 
 
THE VISIBILITY CURVES 73 
 
 is the same as that of curve 1, so that the positions of zero visibility 
 are not changed. The effect of broadening the source is seen to 
 be a decrease in the visibility at each successive maximum, so 
 that the fringes soon disappear altogether. 
 
 Curve 3 corresponds to two homogeneous radiations of 
 unequal amplitudes, and curve 4 represents a single, broad, uni- 
 formly illuminated source. The other curves are easily under- 
 stood. 
 
 Let us now take up the analytical discussion of the subject. 
 According to equation (6), page 22, the intensity of illumination 
 produced at any point by two congruent rays of equal brightness is 
 expressed by 
 
 A 2 = A* cos 2 TT - = 2A? (l + cos Zir - V 
 
 Now 2 A ! 2 represents twice the intensity of each of the two rays. 
 This intensity may be regarded as a function of the wave length, 
 so that we may replace %A * by $ (X) and our equation becomes 
 
 /A = A(X) + <A (X) COS27T^. 
 
 Since each of the separate vibrations is independent, the 
 resultant intensity / will be the integral of this expression taken 
 between the limits X x and X-j. Let now X<, represent a wave length 
 which is intermediate between Xj and X2. Then 
 
 in which p represents the number of waves in the distance 8. 
 The intensity may then be regarded as a function of x between 
 the limits x l and + x 2 . Hence we may replace ^ (X) d\ by 
 < (x) dx, and there results for the total intensity 
 
 x) dx + (x) cos 2/?7T (1 +#) dx. 
 
74 MANUAL OF ADVANCED OPTICS 
 
 If now we expand this cosine and introduce the notation, 
 
 C=J<}> (x) cos %pirxdx 
 
 S = /< (a;) sin %pirxdx, 
 
 there results 
 
 / = P + C cos 2/jrr S sin 2/?ir. 
 
 Now as long as the interval from Xi to x 2 is small, the variation 
 of the values of C and S corresponding to a change in p is small. 
 Hence as on page 24, the maxima and minima of / are deter- 
 mined by y- = 0, i.e., by 
 
 C sin 2jt?7r + S cos 2jt??r = 0. 
 Hence 
 
 I=P \/C' z + S 2 . 
 
 The visibility may then be denned by equations (7), (8), and 
 (9), page 25. Thus in the case of a single uniformly illuminated 
 source < (x) = const., which sends out waves for which x varies 
 
 within the limits > we have 
 
 a 
 
 \r 
 
 L 1-4 1 
 
 sn nira 
 
 V=- -~-.|ocftft*mfe. 
 
 a 1 pwa 
 
 HI 
 
 as on page 25, but in this case p is equal to the number of waves 
 in the difference of path, and a is a small fraction which deter- 
 mines the width of the source. It will be noted that the visibility 
 is equal to zero when pa = 1, 2, 3, etc., i.e., when 
 
 123 
 
 p = -, , , etc. 
 * a' a' a' 
 
THE VISIBILITY CURVES 75 
 
 The more important case is that in which the distribution of 
 light in the source is represented by 
 
 < (x) = *-*"', 
 
 which distribution is in accord with Maxwell's Law deduced from 
 the theory of probability. 
 
 In this case when k is large the value of the integral diminishes 
 rapidly with increasing x, the terms near the origin being the only 
 important ones. Hence the limits of the integration may be 
 
 taken as x. Then 
 
 I 
 
 ['- ' ( e '"*** cos 
 J -~ _ /- _ p yfc2 
 
 * r : ~^~r. V 
 
 It will be noted that the curve is not periodic, but diminishes 
 gradually as p increases. If we assume that the source is prac- 
 tically limited when < (.?;) has reached a value equal to one-half of its 
 
 maximum value, then, calling the corresponding value of #, 
 we have 
 
 I 
 
 - = e 4 , IV 
 Hence 
 
 If now q represent the value of p for which V = , then 
 
 hence 
 
 a_ _ log 2 _ .22 
 
 2 = "~ = ' 
 
76 MANUAL OF ADVANCED OPTICS 
 
 If this value of a be substituted above, there results 
 
 ff 8 log 2 
 
 V=e v* , 
 or 
 
 It will be noted that is a small fraction denoting parts of a wave 
 & 
 
 length. If we wish to get the width of the source in millimeters 
 we must multiply by X. Also both p and q are numbers of waves. 
 If we wish to have them expressed in millimeters we multiply 
 both by X expressed in millimeters. Thus letting p\ = JT, 
 q\ = A, we get 
 
 F-a-f, y-fV \ , (34) 
 
 in which W represents the width of the source in millimeters. 
 
 It has been shown on page 26, that the equation of the 
 visibility curve for a double source differs from that for a single 
 source by the addition of a cosine factor. In general let us 
 suppose we have a series of similar sources which lie about the 
 origin of coordinates and whose distribution of intensity is 
 expressed by <f> (#). The expression for the number of vibrations 
 of the waves of any source may be put in the form 
 
 1 1 
 
 l-X<*#*>v 
 
 For a symmetrical distribution (5=0), the integrals P and C 
 take the form 
 
 f<f> (x) sin %pir (d + x) = C sin Zpird, 
 
 J <j> (x) cos Zpn (d -f x) = C cos 2pird. 
 
 Hence if U represent the visibility which results from all the 
 sources 
 
 2 _ (Sff sin %mf 2 + 2C cos 2W 2 
 
THE VISIBILITY CURVES 77 
 
 Now the visibility due to each source by itself is represented by 
 
 r= 
 
 P ' 
 therefore 
 
 _ (2 VP sin Zjnrd)* + (S VP cos Zpwd)* 
 
 or 
 
 W cos 2pir (d' d) 
 
 If the law of the distribution of the light in the separate 
 sources is the same, while their intensities are proportional to 
 factors r, /, r", etc., the visibility V produced by each will be 
 the same, but P will be proportional to r ; hence, for this case 
 
 _ 
 
 cos %pir (d f - d) 2 
 
 In the case of a double source, the ratio of the intensities of 
 whose components is r : 1, this reduces, if d = d' d, to 
 
 ! 1 + r* + 2r cos ( . 
 
 (1 + r) 2 
 
 and if the two sources have equal intensities, to 
 
 U= Vcospirt1 . (26) 
 
 The value of d may be found from the positions of zero visibility, 
 i.e., the points at which U '= 0. In case the two lines have not 
 equal intensities, we may still determine the value of d from the 
 period of the curve. Thus in the case of the two sodium lines 
 the visibility curve reaches its first minimum when/; = 492 waves; 
 
 13 1 
 
 hence, since U = when pd = -> > etc. , rf = T^' So it follows 
 
 that do represents a fraction of a wave length, so that when it is 
 multiplied by AQ we obtain the difference between the wave lengths 
 of the two sources in millimeters, i.e., A^ -\z = 
 
78 MANUAL OF ADVANCED OPTICS 
 
 Since the period of the curve is the distance between two suc- 
 cessive minima which correspond to differences of path p l and p^ 
 we have, denoting the number of waves in that period by p^ 
 
 but if D represents the length of the period in millimeters, 
 
 = A,-V (27) 
 
 p = - , and therefore 
 
 In general p - ~^-> hence equation (25) becomes 
 
 A 
 
 Y 
 
 cos 2ir 
 
 Since this coefficient of V appears frequently it will be denoted 
 symbolically by cos -=j- 
 
 The inverse problem of determining the form of the distribu- 
 tion when the visibility curve is given is more difficult. The 
 general solution is shown by Kayleigh* to depend upon both C 
 and S. Now the visibility curve alone determines only C 2 + S*. 
 Hence the solution is not single valued unless we can obtain a 
 second relationship between C and S. This can be done by 
 determining the displacement of the phase of the fringes as the 
 difference in path is increased. We may, however, obtain a fairly 
 accurate idea of the distribution in quite a number of cases from 
 the visibility curve alone if we assume that we are dealing with a 
 source in which the distribution is symmetrical, for in this case 
 S = 0, and the solution is definite. 
 
 The process of determining the visibility curve of a given 
 source by observation is as follows: The light which is to be 
 analyzed is passed into the interferometer, the two mirrors being 
 
 *Rayleigh, Phil. Mag. (5) 34, p. 407. 
 
THE VISIBILITY CURVES 79 
 
 near together and adjusted to be parallel to each other. The 
 visibility of the fringes, that is, the contrast between the bright 
 and dark fringes, is called 100. The screw of the instrument is 
 then turned through a whole turn and the visibility again esti- 
 mated. It will generally be less than 100. This estimation of 
 visibility requires some practice. This practice may be obtained 
 by mounting between Nicols a convex and a concave quartz lens 
 of the same curvature. If these lenses are cut parallel to the 
 crystallographic axis and' set so that their axes are at right 
 angles to each other, circular fringes similar to those in the inter- 
 ferometer will be seen. As the lenses are rotated about the line 
 of sight as an axis, the visibility varies in a way which can be 
 calculated from the angle of inclination of the axes of the lenses 
 with the plane of polarization of the analyzer. For if a represent 
 that angle, then the two extreme values I v and L 2 of the resultant 
 intensity will be respectively 1 and cos 2 2a, and therefore 
 
 r _ /!-/ 1 ~ COS 2 2a 
 
 ~ 
 
 - / 2 ~ 1 - COS 2 -2a 
 
 Having trained the eye with such an arrangement the visibility is 
 estimated at each revolution of the screw, and these estimates are 
 plotted as ordinates, the corresponding differences of path being 
 the abscissae. Even if the entire curve is not worked out, con- 
 siderable information can be obtained from a determination of the 
 differences of path which correspond to the minima. 
 
 Experiment 
 
 DETERMINE THE DISTRIBUTION ix THE CADMIUM LINES 
 Apparatus and adjustments as in the previous chapter. 
 MEASUREMENTS. Using as a source of light a cadmium tube, 
 the light is first passed through an ordinary spectroscope so that 
 only one radiation at a time passes into the interferometer. Start- 
 ing near the position of the white-light fringes, the visibilities 
 
80 MANUAL OF ADVANCED OPTICS 
 
 which correspond to the gradually increasing differences of path 
 are observed as has been described. It is well to observe the fringes 
 through a small telescope focused for parallel rays. If a telescope 
 can not be used, a card with a small hole in it should be mounted, 
 in front of the instrument to insure keeping the eye at the same' 
 point during the observations. The observations are then plotted 
 as ordinates in a curve, the difference of path being the abscissae. 
 From the curve thus obtained we find A the difference in path] 
 which corresponds to V = 50. If the curve is periodic, corre- 
 sponding to a double source, we must take the envelope of the 
 
 maxima for making this calculation, i.e., the part of the curve; 
 
 x* 
 represented in our equation by 2 A2 . The half width of the 
 
 22A. 22 
 
 source is then = - X 2 . If the line is double, the distance 
 
 9 A 
 
 between the sources is determined from D, the period of the 
 curve, i.e., the distance between the maxima or the minima; 
 
 \ 2 
 
 for, as was shown above, \ 1 X 2 = The ratio of the intensities 
 
 of the two lines may be obtained approximately from the heights 
 of the first maximum and minimum. Thus the visibility at the 
 first maximum is always 100. If at the first minimum it is, say 
 20, and if a and b represent the two intensities, a + 1) - 100, 
 
 a 1) = 20, and, therefore, = = 0.7 approximately. 
 
 (I o 
 
 EXAMPLES 
 
 1. The red radiation from cadmium (X = 6438 10~ 7 ) was 
 observed, and the curve shown in Fig. 23 b obtained. Since this 
 
 FIGURE 23 
 
THE VISIBILITY CURVES 81 
 
 curve is iiot periodic we may conclude that the line is single. 
 The value of A is seen to be A = 138, hence the half width of 
 the source is, from equation (24) , 
 
 A 2 = .0066 - 10- 7 mm. 
 
 A 
 
 X" 
 
 The equation of the visibility curve would then be F=2 ( 138 ) a . 
 The curve marked a shows the distribution which is seen to 
 correspond to a very nearly homogeneous source. 
 
 2. The green radiation of cadmium (X = 5086 10" 7 ) was 
 observed and the curve shown in Fig. 24 I obtained. Since the 
 
 FIGURE 24 
 
 curve is periodic with a single period it corresponds to a double 
 line. The period of the curve is seen to be D = 115. Hence the 
 
 \ 2 
 
 distance between the lines is X l X 2 = jr- = .022 - 10~ 7 mm. 
 
 Further, F= 50 for X= 120, i.e., A = 120, therefore, the half 
 
 22 
 
 width of the line is ' X 2 = .0048 10- 7 mm. 
 
 A 
 
 The value of V at the first maximum is 100, at the first mini- 
 mum 66, hence = .2 nearly. Hence the equation of the curve 
 
 would be represented by 
 
 .2 
 
 F=2 (')' cos 
 
 115 
 
 The corresponding distribution is represented at the left of the 
 figure. 
 
 3. The next curve, Fig. 25, represents the envelope of the visi- 
 bility curve for sodium (X = 5890 10~ 7 ). The period which-deter- 
 
82 
 
 MANUAL OF ADVANCED OPTICS 
 
 mines the separation of the two lines D l and D 2 has already been 
 found to be .58 mm. (cf. page 61). Hence this period is omitted 
 
 FIGURE 25 
 
 
 from the curve. As it stands, the curve represents the distribution 
 of each of the sodium lines upon the supposition that it is the same 
 for both. The curve is seen to have two periods, one of 50 and one 
 
 X 2 
 of 150. Corresponding to the period 50, we have-y-=. 069 10~ 7 mm. , 
 
 A 2 
 while for the period 150 we have -=r= .023 10~ 7 mm. 
 
 The value of A is seen to be 156, and so the half width of each 
 of these lines is .0063 10~ 7 mm. ' 
 
 The ratio of the intensities corresponding to the first period is 
 found to be .7, and that corresponding to the second .2, hence the 
 equation of the curve is represented by 
 
 F=2 
 
 50 
 
 loO 
 
 In connection with this chapter the student should read Michelson, 
 Phil. Mag. (5)31, p. 338, 1891; Phil. Mag. (5) 34, p. 280, 1892; Journal de 
 Physique (3) 3, p. 5, 1894; B. A. Reports, 1892, p. 170; Trav. et Mem. du 
 Bureau Internat. des Poids et Mesures, XI, p. 1, 1894. Also some further 
 developments of the same method are given by Perot and Fabry, C. R. 
 126, pp. 34, 331, 407, 1561, 1624, 1706, 1779; Ann. de Chim. et de Phys. (7) 
 16, pp. 115, 289; C. R. 130, p. 653. 
 
 Some important applications of the method will be found as 
 follows: Michelson, "On the Broadening of Spectral Lines," Astrophys- 
 ical Journal, 2, p. 251, 1894; "Radiation in the Magnetic Field," Phil. Mag. 
 (5), 44, P- 109, 1897; 46, p. 348, 1898; A strophysical Journal, 7, p. 130, 1898. 
 
VII 
 
 THE PRISM SPECTROMETER 
 Theory 
 
 To understand the conditions which must be fulfilled in order 
 that the spectrometer may be used with the greatest efficiency, it is 
 necessary to discuss first some of the optical properties of prisms. 
 
 An optical prism is a transparent solid, two of whose faces at 
 least are plane surfaces, which intersect in a line. This line of 
 intersection is called the edge of the prism. The angle inclosed 
 by the two plane surfaces is called the refracting angle of the 
 prism, and will be denoted in what follows by A. A plane 
 passing through the prism parallel to the edge and perpendicular 
 
 FIGURE 26 
 
 to the plane bisecting the refracting angle is called the optical 
 base, and a plane perpendicular to the edge is called a principal 
 plane. 
 
 Let CAB (Fig. 26) represent the section of a prism by a prin- 
 cipal plane. Let S represent a source of light, and 800' R the 
 path of a ray from that source through the prism. Draw NOe 
 
 83 
 
84 MANUAL OF ADVANCED OPTICS 
 
 and N'O'e perpendicular to the faces CA and BA at the points 
 and 0' respectively. Continue the line SO to #, and 00' to $', 
 and draw through 0', 0' H parallel to SO. 
 
 The angle NOS between the normal NO and the direction of 
 the incident ray SO is called the angle of incidence, and is 
 denoted by i. As it is measured from the normal NO, it may be 
 either positive or negative. It is denned as positive when the 
 incident ray SO and the refracting angle A lie on opposite sides 
 of the normal. It is, therefore, negative when the incident ray 
 lies between the normal and the angle of the prism. 
 
 Similarly the angle N' 0' R is called the angle of emergence 
 and is denoted by i' . It is denned as positive when the emergent 
 ray O'R and the refracting angle A lie on opposite sides of the 
 normal N'O'. It is, therefore, negative when the emergent ray 
 lies between the normal and the refracting angle. 
 
 The angles eOO' and eO'O are called angles of refraction, and 
 are denoted by r and r' respectively. The angle r is positive 
 when the *, to which it corresponds, is positive, and negative 
 when i is negative. Similarly, r is positive or negative according 
 as the i', to which it corresponds, is positive or negative. 
 
 The angle HO' R being the angle through which the ray is 
 bent by its passage through the prism, is called the angle of devi- 
 ation and is denoted by 8. 
 
 From Fig. 25 we see that the following relations exist: 
 
 HO'R = HO'S' + S'O'R = 8 
 
 HO'S' =gOO' = i-r 
 
 S'0'R = i' -r'. 
 
 Hence 8 = i+ i' - (r + r'). 
 
 But feO' = r + r' = A, (29) 
 
 therefore, 8 = i + i' - A . (30) 
 
 The index of refraction of one medium with respect to another 
 is denned as the ratio of the velocity of light in the one medium 
 
THE PRISM SPECTROMETER 85 
 
 to that in the other. Thus, if V represent the velocity in one 
 medium, V that in the other, and ^ the index of refraction, 
 
 As is well known, this ratio is equal to that of the sines of the 
 angles of incidence and refraction, that is, 
 
 F j sin i _ sin i' 
 V i sin r sin r' 
 
 We have, then, as the fundamental equations of the prism, 
 
 
 sin I 
 
 (* = - > 
 sin / 
 
 = 
 sm r 
 
 r + r' = A. 
 
 (31) 
 
 To obtain the general equation which connects the index of 
 refraction with the angles J, i and i f we proceed as follows: 
 From equations (31) we have 
 
 sin i' = \L sin r' = /x sin (A r)' 9 
 if we expand sin (A r) and substitute for sin r its value - and 
 
 for cos r its value -v/^-sin 2 ', this equation reduces to 
 
 sin i' = sin A*/ 'f sin 2 i cos A sin i. (32) 
 
 This equation holds in general without any conditions imposed 
 upon the quantities involved. Since, however, the quantity whose 
 value is to be determined from measured values of the others is 
 usually the index of refraction /A, and since it is a matter of some 
 difficulty to measure the angle of incidence i with accuracy, it is 
 generally advisable to use the prism in one of two particular 
 positions. 
 
86 MANUAL OF ADVANCED OPTICS 
 
 One of these particular positions is determined by the condi- 
 tion 
 
 i' = 0, 
 
 that is, the ray emerges from the prism normal to its second face. 
 In this case sin *' = 0, 8 = i A, or i = 8 + A, r' = 0, and r = A. 
 Upon substituting these values in the first of equations (31), it 
 readily reduces to 
 
 sin (A + 8) 
 
 sin A 
 
 (33) 
 
 Since in this case the determining condition is i' = 0, and 
 since we also have r' = and r = A, therefore the fundamental 
 equation sin i = /u, sin r becomes, under these circumstances, 
 sin i = /u, sin A . But sin i must be less' than unity. Therefore, 
 
 sin A < . Hence, a prism can not be used in this particular 
 position unless its refracting angle falls within the limit prescribed 
 by this inequality, that is, unless sin A < 
 
 The other of these particular positions is determined by the 
 condition 
 
 i' = i. 
 
 In this case / = r = A and 8 = 2i - A, or / = (A + 8), and on 
 
 </ A 
 
 substituting these values in the first of equations (31) it becomes 
 
 sin (A + S) 
 
 (" = - V (34) 
 
 sin -A 
 
 When the prism is used under this condition, namely i' = t, 
 the deviation 8 produced by it is the smallest which can be 
 obtained with a prism of given angle and index of refraction. 
 Hence this position of the prism is known as that of minimum 
 deviation. That it is so may be proved as follows: 
 
THE PRISM SPECTROMETER 87 
 
 Since in equation (30) A is "a constant for any given prism, 
 the value of 8 will depend on that of i -f i'. Therefore 8 will be a 
 minimum when i -f i' is. But an inversion of equations (31) gives 
 
 i = sin" 1 /A sin r, and i' = sin" 1 tt sin (A r), hence 
 i + f = sin" 1 /tA sin ; + sin" 1 /A sin (J. r). 
 
 To find when this value of i + i' will be a minimum, differentiate 
 this equation with respect to r, regarding /A as constant. This 
 gives 
 
 d (i -f- /') _ /A cos r /A cos (.4 - r) 
 
 dr v/l /A 2 sin 2 r \^1 /A 2 sin 2 (^4 r) 
 
 This becomes equal to zero when r = A. A second differentia- 
 
 /v 
 
 tion with respect to r and a substitution of A for r gives 
 
 sm - 
 
 Since /x sin ^4 is* the sine of that angle of incidence which will 
 
 *Z 
 
 have a corresponding angle of refraction A, the value of the 
 
 A 
 
 right-hand side of this equation will be real and positive when 
 /* > 1. When /M < 1 its value will be negative, which means that 
 the corresponding value of i + i' is a maximum. But since, when 
 /x < 1, the prism is optically less dense than the surrounding 
 medium, the deviation will, not be given by equation (30), but by 
 
 8 = A (i -f i"). Hence in any case, the condition r = A will 
 make 8 a minimum. But r + r' = A. Hence the condition 
 r = A is equivalent to / = /' or to i = f . Therefore, under this 
 condition the deviation produced by the prism is a minimum. 
 
88 MANUAL OF ADVANCED OPTICS 
 
 There is a limit to the use of a prism in this case also; 
 
 for, under the condition i = i' we have seen that r' = A. Since 
 
 & 
 
 Bint*' < lit follows that sin A < If sin r' > , a thing which 
 
 2 p p 
 
 often occurs in practice, there is no sin i' to correspond to it. 
 Hence the ray can not leave the prism at the surface where this 
 occurs, but is totally reflected. The application of total reflection 
 to the determination of indices of refraction will be discussed in 
 the next chapter. 
 
 Consider now that instead of a single ray we have given a 
 narrow beam of light. As before let CAB (Fig. 27) represent the 
 
 FIGURE 27 
 
 section of a prism by a principal plane, and S the projection of a 
 narrow source upon that plane. Suppose that a narrow beam of 
 monochromatic light 80 A falls upon the prism in such a way 
 that one boundary of the beam passes through the edge A of 
 the prism. Let i represent the angle of incidence of the ray SO, 
 and i' the corresponding angle of emergence, and r and r' the 
 respective angles of refraction. Since the beam is supposed 
 
THE PRISM SPECTROMETER 89 
 
 *>^ 
 
 narrow, we may denote the corresponding angles for the ray SA 
 by i + di, i' + di\ etc. We then have from equations (31) 
 
 sin f = fji sin (A - r) 
 sin i = /w. sin r, 
 
 whence, by differentiation, regarding /* as constant, 
 
 cos i'di' = /A cos (A - r) dr, 
 cos idi = fjL cos 7Y/r. 
 
 Eliminating dr and substituting for A r its valne r' we have 
 
 7 ., cos /' cos /' 
 
 rfi = - ^ - tfi. (35) 
 
 cos / cos r 
 
 From this equation it follows, since the cosine terms are always 
 positive, that when di is positive, that is, i + di>i, di' is negative, 
 and therefore i' + di'<i'. Hence, the two rays O'R and AR ', 
 when prolonged backwards, will intersect at some point $', that 
 is, a virtual image of the point 8 will be formed at 8' by the 
 prism. 
 
 Equation (35) is capable of another interpretation. If 8 is 
 not a -point source, but has finite width, di may be regarded as 
 the angular width of the source when viewed from 0; di' then 
 represents the width of the virtual image when viewed from 0' ' . 
 From this it follows that, for a given i and ^', the width of the 
 image is proportional to the width of the source. Also the width 
 of the image may be altered by varying i and i'. Thus if i = 90 , 
 cos i = 0, and di' = 0, or the emergent beam is parallel. If 
 t" = 90, cos i' - 0, and di f = x . 
 
 When the prism is in the position of minimum deviation, 
 i = i', r = /, and, therefore, di = di', that is, the width of the 
 image is equal to the width of the source. Since in practice 
 prisms are most frequently used in this position, and since, as 
 will be shown later, considerations of the purity of the spectrum 
 make it desirable to have the width of the image as small as 
 
 
90 MANUAL OF ADVANCED OPTICS 
 
 possible, the slit in a . spectroscope is made infinitely narrow by 
 placing it at an infinite distance by means of a lens. Hence the 
 origin of the collimator. 
 
 If in equation (35) di = 0, then di' = 0, that is, a beam of 
 monochromatic light which is parallel before falling on the prism 
 remains so after its passage through the prism. 
 
 We have thus far been considering only those rays from the 
 source which lie in the plane CAB (Fig. 27), i.e., in a principal 
 plane. When we take into account also the rays which, coming 
 from the source S, pass through the prism in some other than 
 a principal plane, the development becomes much more com- 
 plicated ; especially if the source of light is a straight bright line 
 parallel to the refracting edge of the prism. The virtual image 
 of such a line source will, in general, be curved, owing to the 
 passing of some of the rays through the prism obliquely to 
 a principal plane.* 
 
 Up to this point the index of refraction /x has been regarded 
 as constant. While this is true for a prism of any given 
 substance so long as light of a definite wave length only is con- 
 sidered, it ceases to be so when the light from the source contains 
 more than one wave length. We have, therefore, to consider now 
 what further effects are produced by a prism when the variations 
 of the index of refraction due to changes of wave length are taken 
 into account, and to determine how the conditions under which 
 the prism is used alter those effects. 
 
 X"The question as to how the index of refraction depends on the 
 wave length has been the subject of a large amount of investiga- 
 tion. Practically it has been found that the formula first given 
 by Cauchyf expresses the relation between the two quantities to 
 
 *M. A. Bravais, Jour, de Fee. Polyt., 18, p. 79, 1845. E. Reusch, Pogg. 
 Ann., 117, p. 241, 1862. A. Cornu, Ann. ec. norm. (2) 1, p. 255, 1872. 
 G. G. Stokes, Proc. Roy. Soc. , 22, p. 309, 1874. J. L. Hoorweg, Pogg. 
 Ann., 154, p. 309, 1875. 
 
 f Cauchy, Memoir e sur la dispersion de la lumiere, Prague, 1836. 
 
THE PRISM SPECTROMETER 91 
 
 a fair degree of accuracy in all cases of normal dispersion within 
 the visible spectrum.* This formula is 
 
 K jjO 
 ^ = 4,+ 4' 
 l< K 
 
 in which 34* and B are for any given substance constants to be 
 determined by experiment. It will be noted that when 
 A = x , fj, = A, i.e., ^represents the index of refraction for 
 infinitely long waves. 
 
 In order to find the change in the angle of emergence cli' which 
 results from a variation dp in the index of refraction, it is neces- 
 sary to differentiate equations (31), regarding i and A only as 
 constant. This gives 
 
 = /A cos rdr + sin 
 
 cos i'di' = /A cos r'dr' -r sin 
 
 dr + dr = 0. 
 
 By elimination of dr and dr these reduce to 
 
 cos t cos r 
 
 This equation gives the relation between di' and dp in terms 
 of the angles A, i', and r. It is more practical to have their 
 relation in terms of the dimensions of the prism and its adjuncts. 
 This may be accomplished for the most useful case,f that in 
 which the prism is in the position of minimum deviation, as 
 follows: Since, under these circumstances we have i = f, r = r' ', 
 A = 2r, equation (37) reduces to 
 
 sin *2r _ 2 sin r 
 
 di = - - dfM = - r- dfji. 438) 
 
 cos i cos r cos i 
 
 Suppose that a parallel beam of light falls upon the prism 
 
 (* Schmidt, Die Brechung des Liclits in Glasern, Leipzig, 1874. 
 f For the general case cf. Rayleigh, Phil. Mag. (5), 7, pp. 261, 403, 477, 
 1879; 9, p. 40, 1880; Collected Works, 1, p. 415. Also Drude, Theory of 
 Optics, p. 233, Longmans, 1902. 
 
92 
 
 MANUAL OF ADVANCED OPTICS 
 
 CAB (Fig. 28) through a rectangular opening ab, two of whose 
 parallel sides are parallel to the edge -A of the prism, and whose 
 plane is perpendicular to the direction of propagation of the 
 beam. Let the prism be so placed that one boundary bA of the 
 
 beam passes through its edge A. Let db, the width of the beam, 
 be represented by a, and the distance oo', which the other bound- 
 ary of the beam travels in the prism, by t. Draw Ae bisecting 
 the angle .4, and of through o perpendicular to bA. Then, since 
 the prism is in the position of minimum deviation, oo' is perpen- 
 
 dicular to Ae, and oe = t. 
 
 From the figure we get 
 
 t 
 
 . 1 A 06 
 
 sin oAe = sin A = - = - - 
 2 oA %oA 
 
 ,. of a 
 
 cos Aof = cos i = -^r - - 
 oA oA 
 
 sm 
 
 Substituting in equation (38) it reduces to 
 
 ' 
 
 (39)- 
 
 Hence we note that the amount of separation di' of two beams 
 of light whose wave lengths are such that the difference of their 
 indices of refraction is dp, depends only on the width a of the 
 
THE PRISM SPECTROMETER 93 
 
 beam, and on the excess of transparent substance of the prism 
 traversed by one side of the beam over that traversed by the other. 
 One of the most important practical applications of the dis- 
 persive power of the prism is that of analyzing a composite beam 
 into its components. It is, therefore, of great importance to know 
 how far this analysis may be carried with any given prism. Thus 
 if we have given for analysis a beam of light containing the two 
 different wave lengths X and X + d\, we wish to know what sort 
 of prism is needed to accomplish the task, that is, to resolve the 
 beam into its component parts. Equations (39) and (1) answer 
 this question completely. Equation (39) gives the angular sepa- 
 ration due to the prism of the two beams, and if we suppose a 
 lens placed behind the prism to form real images of the source, 
 (1) tells whether the two images formed by that lens will appear 
 separated or not. Combining the two we see that the limit of 
 resolution is expressed by the following equation: 
 
 
 ,, * , x 
 
 di = dp.= > 
 a a. 
 
 or 
 
 * = -r ' ( 4 ) 
 
 dp. ^ 
 
 From this equation it appears that, if it is desired to analyze or 
 resolve with a prism two beams of light which proceed from the 
 same source and whose indices of refraction are //. and p. + dp. 
 respectively, it will be necessary to use a prism whose thickness , 
 
 as defined above, is at least equal to -j 
 
 It is often more convenient to have this expression in terms of 
 X and d\ instead of X and dp.. The value of dp. in terms of d\ 
 maybe obtained from equation (30), and when substituted in (40) 
 there results 
 
94 MANUAL OF ADVANCED OPTICS 
 
 The dispersion of a prism, denoted by .Z), may be defined as 
 the ratio of a change in deviation rfS to the corresponding change 
 in wave length dX, that is, 
 
 d& aft dp 
 Tx = TV- ' dX 
 But 
 
 $ = i + i' A, therefore, for constant A 
 
 and i, and in consideration of equation (37), 
 
 rTS _ dif_ _ sin A 
 dp~ dp ~ cos i' cos r 
 Hence 
 
 D. 
 
 cos i cos r 
 
 If the incident beam contains all possible wave lengths, that 
 part of it which corresponds to each particular wave length will, 
 on account of the dispersion of the prism, form its own particular 
 image of the source at the focus of a lens suitably placed behind 
 the prism. These images will be a series of parallel bright lines 
 which overlap and form a bright band of light. Such a band is 
 called a spectrum. The spectrum is said to be the purer, the less 
 the successive images which unite to form it overlap. Since each of 
 these elementary bands is produced by waves whose lengths vary 
 over a small interval dX, we may use this change in wave length 
 within the band as a convenient measure of the purity, that is, 
 we may define purity, denoted by P, as the reciprocal of dX. Or, 
 better, if we choose A. as the unit of measure, the purity may be 
 defined as 
 
 Let be the angular width of the elementary image due to 
 wave length X alone. The center of this image will receive light 
 
 from the neighboring images whose centers lie within a range of 
 
 </ 
 
 on either side of it, that is, from those images included within a 
 

 THE PRISM SPECTROMETER 95 
 
 region (18 = 0. Hence, since the change in wave length in this 
 interval is tfA, the illumination in the center of each elementary 
 
 image will be produced by wave lengths varying from A d\ to 
 
 X + d\. To get a value for d\ divide the equation d& = by d\. 
 We have, since the interval is small, 
 
 d\ d\ 
 But 
 
 Therefore, 
 
 and 
 
 _ ^ 
 d\~ ' 
 
 But from equation (35) 
 
 - cos /' cos / 
 
 ^ = f7i = - ^ - rZi. 
 cos i cos r 
 
 Therefore, in consideration of equations (42), (43), and (44), 
 * cos i' cos r sin A dp. 
 
 di cos i cos /' cos i cos /' d\ 
 
 
 When the source is narrow, 6 is the limit of resolution of the 
 opening, that is, = ; therefore, in this case, equation (44) 
 becomes 
 
 Under this condition is called the resolving power of the prism. 
 
 ft A 
 
 Denoting it by R we have, 
 
 R = aD. (46) 
 
96 MANUAL OF ADVANCED OPTICS 
 
 From this it appears that the resolving power of a prism of 
 given thickness is directly proportional to both the width of the 
 incident beam and the dispersion of the prism. 
 
 In the position of minimum deviation equation (42) reduces to 
 
 But R = aD, therefore, in consideration of equation (36), 
 
 From this it appears that, with a given width of beam, the 
 resolving power of a prism is proportional directly to the thickness 
 of the prism, and inversely to the cube of the wave length. 
 
 For further discussion of the properties of prisms and prism spectro- 
 scopes the student is referred to Kayser, Handbuch der Spectroscopie, 1, 
 p. 253 seq. and 490 seq. , Leipzig, 1900, where a complete bibliography will 
 be found. Also important discussions by Helmholtz, Physiologische 
 Optik, 3d Ed., p. 290 seq., Leipzig, 1896; Rayleigh, Phil. Mag. (5), 8, pp. 
 261, 403, 477, 1879; 9, p. 40, 1880; also in his Collected Works, 1, p. 415 
 seq. ; Czapski, Winkelmann's Handbuch II, 1, p. 152 seq. 
 
 Experiments 
 
 I. DETERMINE THE INDEX OF REFRACTION OF A PRISM 
 
 APPARATUS. The spectrometer consists of a collimator SA 
 (Fig. 29) and a telescope BE, both mounted at right angles to 
 a vertical axis CD and movable about that axis, and a graduated 
 circle C, upon which the angle between the telescope and colli- 
 mator can be read with the help of the vernier v. Both the 
 telescope and the collimator should be so mounted as to be 
 movable about horizontal axes aa, so that they may be adjusted 
 to have their collimation axes parallel to each other and perpen- 
 
THE PRISM SPECTROMETER 
 
 97 
 
 dicular to the axis CD. It is usually possible to clamp the 
 collimator and the graduated circle to the base of the instrument. 
 The clamp which fastens the telescope to the graduated circle 
 contains a tangent screw which permits a slow motion of the 
 telescope. The center of the graduated circle should coincide 
 
 3 
 
 FlGUBE 29 
 
 with the axis CD, and it is useful to have a small hole bored 
 through that center parallel to that axis. 
 
 ADJUSTMENTS. TJie axes of the telescope and collimator must 
 intersect the vertical axis of the instrument. If the telescope and 
 collimator are rigidly mounted so that they are not movable about 
 a vertical axis, we assume that this adjustment has been properly 
 made by the maker. In any case the adjustment can be 
 tested with sufficient accuracy by sticking a straight metal 
 rod into the hole in the center of the graduated circle, and 
 observing this rod through the slit of the collimator. To adjust 
 the telescope the eyepiece must be removed and the rod viewed 
 
98 MANUAL OF ADVANCED OPTICS 
 
 as in the case of the collimator. When the adjustment is correct, 
 the rod seems to cover the center of the objective. 
 
 Telescope and collimator must ~be focused for parallel rays. 
 The telescope may he pointed out of the window and focused on 
 a distant object, but it is better to place a prism on the prism 
 table, set the telescope perpendicular to one of its faces, and 
 observe the reflection upon the prism face of the cross-hairs in the 
 teleseope. In order to do this easily the telescope should be 
 fitted with a Gauss eyepiece, or have some other provision for 
 illuminating the cross-hairs. When the cross-hairs are illumi- 
 nated and the telescope is perpendicular to the prism face, two 
 sets of cross-hairs will be visible in the field of view. The two 
 should be brought nearly into coincidence and the telescope 
 so focused that when the eye is moved about behind the eyepiece 
 the two sets show no parallax with respect to each other. Having 
 thus focused the telescope the collimator is easily adjusted for 
 parallel rays by observing, in the telescope, the image of the slit, 
 and focusing the collimator until that image shows no parallax 
 with respect to the cross-hairs. The adjustment can also be 
 effected by placing on the prism table of the instrument a 
 plane-parallel plate of glass in such a way that the light from the 
 slit is reflected to the telescope at oblique incidence. The 
 reflected image of the slit in the telescope will appear double 
 unless both telescope and collimator are focused for parallel 
 rays. 
 
 Telescope and collimator must be perpendicular to the axis of 
 the instrument and the surfaces of the prism must be parallel to 
 that axis. To attain this the prism should be mounted on a 
 leveling tripod, so that one of its faces AB is perpendicular to 
 the line joining two of the leveling screws. The prism so 
 mounted is then set upon the prism table a little to one side of 
 the center of the graduated circle, so that when the telescope and 
 collimator are brought into line and the prism face AB is made 
 
THE PRISM SPECTROMETER 99 
 
 parallel to that line, part of the field of view of the telescope is 
 unobstructed, and it is possible to have an image of the slit 
 formed in the focal plane of the eyepiece. The slit should then 
 be turned so that it is horizontal, and the telescope and collimator 
 adjusted by means of the screws bb until the image of the slit 
 falls upon the intersection of the cross-hairs in the telescope. 
 The telescope should then be turned about the axis of the instru- 
 ment through any angle, and the prism turned through such an 
 angle that the light from the slit is so reflected from the prism 
 face AB that the image of the slit again appears in the field of 
 view of the telescope. In general this image of the slit will not 
 fall upon the intersection of the cross-hairs. It should be made 
 to do so by adjusting the leveling screws of the prism. The tele- 
 scope is then turned so as to be perpendicular to the face AB 
 of the prism. In order to set it accurately perpendicular to that 
 face, it is necessary to illuminate the cross-hairs and observe their 
 image reflected from the face AB. When this image coincides, 
 with the cross-hairs themselves, the telescope is perpendicular to 
 the face of the prism. The cross-hairs and their image should be 
 brought into coincidence by adjusting the telescope. The tele- 
 scope and prism should then be brought into their original posi- 
 tions, and the image of the slit observed directly. It will in 
 general no longer fall upon the intersection of the cross-hairs. It 
 should be made to do so by adjusting the collimator by means of 
 the screw b. This operation must be repeated until the irnaga of 
 the slit remains on the intersection of the cross-hairs in all three 
 of the positions. When this is accomplished the telescope and 
 collimator are perpendicular to the axis of the instrument, and 
 the prism face A B is parallel to that axis. 
 
 It is then necessary to make the other face of the prism, 
 namely AC, parallel to the axis CD. To accomplish this it is 
 merely necessary to set that face perpendicular to the telescope. 
 It will be remembered that the face AB of the prism was set per- 
 
100 MANUAL OF ADVANCED OPTICS 
 
 pendicular to the line which joins two of the leveling screws of 
 the tripod upon which the prism is mounted. The adjustment of 
 the face A C should be made entirely with the other screw of the 
 tripod, because a movement of this third screw of the tripod will 
 not tip the face AB, but only rotate it about an axis perpendicu- 
 lar to its plane. 
 
 MEASUREMENTS. It is first necessary to measure the angle A 
 of the prism. The telescope is set perpendicular to the face AB 
 of the prism, and its position read upon the graduated circle. It 
 m .then set perpendicular to the face AC of the prism, and its 
 position read in the same way. The angle through which the 
 telescope turns in making these two settings is the supplement of 
 the angle of the prism. To measure the angle 8 of deviation, 
 turn the slit so that it is vertical, and illuminate it with mono- 
 chromatic light, or better, with sunlight. Place the telescope 
 and collimator in line, bring the image of the slit upon the 
 cross -hair, and read the position of the telescope upon the gradu- 
 ated circle. The prism should then be introduced and placed in 
 such a position that the angle of incidence of the light from the 
 collimator upon it is nearly equal to the angle of emergence. 
 The adjustment for minimum deviation is made by rotating the 
 prism about a vertical axis while observing the spectral image of 
 the slit in the telescope. As the prism is turned that image 
 moves in the field of view, and it will be found that at one par- 
 ticular position of the prism that image is nearer the direct image 
 of the slit than in any other position. The cross-hair is set upon 
 the spectral image when the prism is in this position, and the 
 angle through which the telescope has been turned read upon the 
 graduated circle. If sunlight is used, the cross-hair is set upon 
 the particular Fraunhofer line for which the corresponding index 
 is required. Having thus measured the angle of the prism and 
 the angle of deviation, the index of refraction is easily found 
 with the help of equation (34) . 
 
LINE 
 
 WAVE LENGTH 
 
 DEVIATION INDEX (C 
 
 B 
 
 6870 - 
 
 io- 7 
 
 47 
 
 34' 
 
 24" 
 
 1 
 
 C 
 
 6563 
 
 io- 7 
 
 47 
 
 53' 
 
 0" 
 
 1 
 
 D 
 
 5890 
 
 io- 7 
 
 48 
 
 46' 
 
 46" 
 
 1 
 
 E 
 
 5270 
 
 io- 7 
 
 50 
 
 0' 
 
 12" 
 
 1 
 
 F 
 
 4861 
 
 io- 7 
 
 51 
 
 14' 
 
 46" 
 
 1 
 
 G 
 
 4308 
 
 io- 7 
 
 53 
 
 14' 
 
 28" 
 
 1 
 
 THE PRISM SPECTROMETER 101 
 
 EXAMPLE 
 
 A hollow prism filled with bisulphide of carbon was used. 
 The angle of the prism was 59 50 f = A. The following deviations 
 were observed for the lines of the solar spectrum as indicated: 
 
 (CALCULATED) 
 
 1.6060 
 1.6192 
 1.6285 
 1.6409 
 1.6532 
 1.6727 
 The temperature was 18 C. 
 
 II. DETERMINE THE DISPERSION CURVE 
 Apparatus and adjustments as in Experiment I. 
 MEASUREMENTS. It is merely necessary to measure the index 
 of refraction in the way described above, for several different 
 wave lengths, and then to plot the indices as ordinates with the 
 corresponding wave lengths as abscissae. If it is desired to deter- 
 mine the constants of the Cauchy dispersion equation [equation 
 (36)], any two values of the index of refraction with their corre- 
 sponding wave lengths may be used. It is desirable as a test of 
 the accuracy of the work to measure the indices corresponding 
 to several of the Fraunhofer lines, and to calculate the A and B 
 of the Cauchy equation from each pair of such indices. 
 
 EXAMPLE 
 
 From the values of /*, obtained in the example of Experiment I, 
 the following values of A and B of equation (36) were obtained : 
 
 INTERVAL A 
 
 B-E 1.5805 
 
 C-F 1.5786 
 
 D - G 1.5777 
 
 mean 1.5789 = A 
 
 .'. B.= 1.735 IO- 8 
 
102 MANUAL OF ADVANCED OPTICS 
 
 III. DETERMINE THE RESOLVING POWER or THE SPECTROMETER 
 Apparatus and adjustments as in Experiment I. 
 MEASUREMENTS. Having determined the B of equation (36), 
 it is in addition necessary to measure ^, the thickness of the 
 prism. If the incident beam has a rectangular cross-section, it is 
 evident from Figure 28 that the value Of t is given by 
 
 2 sin A ' 
 
 t = - ; (48) 
 
 cos ^ 
 
 If, however, the entire aperture of the collimator is used, and if 
 a' represent the diameter of that aperture, then, because the 
 cross-section of the beam is circular, 
 
 Hence the equation for the determination of t is, in the case of 
 an incident beam of circular cross-section, 
 
 2a' sin ^A 
 
 # = 0.82 - : -- (49) 
 
 cos i 
 
 It is to be noted that the resolving power of a prism is different in 
 different parts of the spectrum. 
 
 An interesting check upon equation (47) is obtained by 
 altering , and consequently t, until a given resolution is obtained. 
 This is accomplished by limiting the width of the beam by two 
 cards held upon the front face of the prism. The distance 
 between the cards should be varied until two known lines are no 
 longer resolved. The two sodium lines are a convenient pair for 
 this purpose. If d is the distance between the cards upon the 
 front face of the prism, when the sodium lines cease to appear 
 double, then it is evident that 
 
 t = 26? sin A. 
 * Verdet, Legons D'optique Physique, Vol. I, p. 301 seq. 
 
THE PRISM SPECTROMETER 103 
 
 The value of -j- in this case is about 1000. It will in general be 
 CIA. 
 
 found that the experimentally determined value of the resolving 
 power is larger than the theoretical one. This is natural, as the 
 theoretical value is the limit of resolution beyond which it is 
 impossible to go, and the fact that the observed limit is somewhat 
 higher merely goes to show the imperfection of our eyes and 
 our instruments. 
 
 EXAMPLE 
 
 Two cards were held over the objective of the collimator and 
 moved until the two sodium lines just ceased to be resolved. In 
 this position their distance apart was a = 3.5 mm. From equa- 
 tion (48), t = 5.98, since i = - (8 + A) = 54 18'. Hence, from equa- 
 
 /v 
 
 tion (47), using the value of B given on p. 101, 
 
 * = \ x 
 
 The theoretical value of R for the two D lines is -j- = 983. x 
 
 To get some idea of the reso\ying power of the instrument 
 without stops in different parts of tne spectrum, use was made of 
 equation (46) in connection with the observations in Experiment I. 
 The value of R varies over the intervals chosen, hence the num- 
 bers obtained are only averages over the interval. 
 
 INTERVAL d& rfx d\ ~ 
 
 C-D 53' 46" = 3226" = .01564 radians .0000673 232 
 F-G 1 59' 42" = 7182" = .03482 " .0000553 630 
 
 Since in this case, if a represent the diameter of the objective, 
 a = 0.82', and since a was 40 mm., a = 32.8, therefore, 
 
 INTERVAL R ( = aZ>) 
 
 C-D 7610 
 
 F- G 20700 
 
104 MANUAL OF ADVANCED OPTICS 
 
 The extreme values of R for the lines B and G are added. 
 The calculation is made from equations (47) and (49). 
 
 LINE K 
 
 B 6020 
 G 24400 
 
VIII 
 
 TOTAL REFLECTION 
 Theory 
 
 In Chapter VII we have deduced the general equation (32) for 
 the prism. Mention was also made of the fact that when the 
 
 internal angle of incidence r' becomes so large that sin / = - 
 
 the light can not escape from the prism, but is totally reflected. 
 This limiting angle of total reflection may then be used to 
 determine /A. At that angle sin i' = 1, and hence (32) becomes 
 
 1 = sin A v / 'fj? sin 2 i cos A sin i. 
 If this equation be solved for /x 2 we get 
 
 cos A + sin 
 
 Hence, to determine /* by this method it is necessary to 
 measure the refracting angle A of the prism, and the angle i of 
 incidence, 
 
 If the totally reflecting surface be covered with a liquid of index 
 
 it*', then at the limiting angle of total reflection sin i" = I = 
 and equation (32) becomes for this case 
 
 /A' = sin A v//A 2 - sin 2 i cos A sin i. (51) 
 
 Experiments 
 
 I. DETERMINE THE INDEX OF REFRACTION BY TOTAL REFLECTION 
 The apparatus and adjustments are those of the last chapter. 
 The prism used should be for convenience a total reflecting prism 
 of 90. 
 
 105 
 
106 MANUAL OF ADVANCED OPTICS 
 
 MEASUREMENTS. Place the prism on the prism table of the 
 spectrometer and allow diffused light to fall upon the face AB as 
 shown in Fig. 30. Observe the reflected light with the telescope 
 and find that position of the prism and telescope in which half of 
 the field of view is bright and the other half less bright. Set the 
 cross-hair of the telescope on the line of separation between 
 these two portions of the field and read the position of the tele- 
 scope on the circle. Then set the telescope perpendicular to the 
 face BC, and again read the circle. The difference between 
 these two readings will, because of the symmetry of the figure, be 
 
 FIGURE 30 
 
 the angle i, which corresponds to the angle i' = 90. Note that 
 in the case drawn the refracting angle of the prism is at A, and 
 therefore since it and the incident ray are on opposite sides of the 
 normal, i is positive. 
 
 If it is desired to determine the index / of a liquid (//</*) 
 wet the surface A C with the liquid. It will probably be neces- 
 sary to press a film of the liquid between the prism face and a 
 piece of glass. Observe again the limiting angle. In this case 
 the angle i will probably be negative. The index can then be 
 calculated from equation (51). 
 
 The same prism can be used to measure indices of solid sub- 
 ptances less dense than the prism itself by pressing between the 
 surface AC and the substance whose index is to be determined a 
 
TOTAL REFLECTION 107 
 
 liquid like oil of cassia, whose index is greater than those of the 
 substance and the prism. Equation (51) holds for this case also, 
 the symbols having the same meaning as in the preceding case. 
 
 EXAMPLE 
 
 Using a total reflection prism and sodium light it was 
 found that A = 45, i = 5 30'. Hence, from 
 equation (50) /u, = 1.5133. 
 
 As a check the deviation S at minimum deviation was 
 also measured, the result being 8 = 25 47'. 
 Hence, from (34) /x = 1.5134. 
 
 The surface AC was then coated with a water film, 
 and the value of i determined as i = - 25 50'. 
 Hence, from (51) //= 1.3329. 
 
 Using a denser total reflection prism and sodium light 
 it was found that A = 45, i = 11 2'. Hence;, 
 as above /x = 1.6170. 
 
 The hypothenuse of the prism was then coated with a 
 thin layer of cassia oil, and a plate of the glass 
 whose index of refraction was determined with 
 the interferometer (page 65), was placed upon the 
 oil. The value of i was then measured as 
 i = - 42 57' 20". Hence, from (51) V = 1.5189. 
 
 For other methods of determining indices from the angle of total 
 reflection, the student is referred to Kohlrausch, Wied. Ann. 4, p. Jl. 
 Abbe, "Apparate zur Bestimmung des Brechungsvermogen, " Jena, 1874. 
 Pulfrich, Zs. fur Instrk. 7, p. 55, 1887; 8, p. 47, 1888; 15, p. 389, 1895; 
 19, p. 9, 1899. Czapski, Zs. fur Instrk. 10, p. 346, 1890. 
 
IX 
 THE DIFFRACTION GRATING 
 
 Theory 
 
 The conditions under which the maxima and minima of illu- 
 mination are formed by the interference of the light waves that 
 have passed through two parallel, rectangular openings or slits, 
 have been discussed in Chapter II and are expressed by equations 
 (4) and (5), page 20. If we now consider, in the same way, a 
 large number n of such equal, equidistant, parallel slits, all of 
 which lie in one plane, we shall find, in general, that the position 
 of a maximum will be determined by an equation similar to equa- 
 tion (5) in which the 2 is replaced by n; that is, the condition for 
 
 a maximum will be 
 
 nd sin = nm\. (52) 
 
 Similarly the condition for a minimum will be 
 
 nd sin 6 = (nm + l)\ (53) 
 
 Such a series of parallel slits is called a plane grating.* 
 
 In general the positions of the maxima and minima will be 
 determined by these equations. There are certain cases, however, 
 when one or more of the maxima may be wanting. According to 
 equation (2) each individual opening has a minimum in a direction 
 determined by a sin = m\. But from equation (52) the maxima 
 are determined by d sin 0' = m'\. When the 0's are the same, 
 
 a m _.. 
 
 T = ; ' ( 4 ) 
 
 d m 
 
 * Of. Drude, Theory of Optics, p. 222 seq. , Longmans, 1902. Kayser, 
 Handbuch der Spectroscopie, /, p. 416 seq., Leipzig, Hirzel, 1900. Ray- 
 leigh, Phil. Mag. (4) 47, p. 193, 1874. Quincke, Pogg. Ann. 146, p. 1, 1872. 
 Rowland, Phil. Mag. (5) 35. Mascart, Ann. ec. norm. 1, p. 219, 1864; 
 TraiU d'optique, 1, p. 364 seq. , Paris, 1889. 
 
 108 
 

 THE D-IFFRACTION ORATING 
 
 109 
 
 Hence, when this condition happens to be fulfilled the correspond- 
 ing maximum is wanting. 
 
 We have thus far considered that the incident beam fell 
 normally upon the grating. The introduction of this condition 
 has made the explanation of the equations which determine the 
 positions of the maxima and minima very simple. It is, however, 
 immaterial whether the difference of phase nm between the 
 outside rays of the beam is introduced entirely behind the 
 grating, or part of it before the grating and the rest behind it. 
 Part of this difference of phase will be introduced before the 
 grating if the incident beam fall upon it at some angle other than 
 a right angle. Then let AB (Fig. 31) represent the grating, 
 
 'N 
 
 A ,- 
 
 N 
 
 FIGURE 31 
 
 and SABS' the incident beam, making an angle NAS = i with the 
 normal A JV to the grating. Let AR be the direction correspond- 
 ing to a maximum; call the angle which AR makes with the 
 normal AX' to the grating, the angle of diffraction, and denote it 
 as above by 0. The difference of path between the two outside 
 rays AR, BR' is now represented by DA -{-AD'. This must, 
 therefore, be substituted for AD in the deductions that lead to 
 
110 MANUAL OF ADVANCED OPTICS 
 
 equation (52). But DA = nd sin i, and AD' = nd sin 0, there- 
 fore, 
 
 DA + AD' = nd (sin * + sin 6), 
 
 and equation (52) becomes under these circumstances 
 nmX = nd (sin &' + sin 0). 
 
 Should the direction AR fall between the normal AN', and the 
 direct ray SAI, the difference of path between the outside rays 
 AR' and BR' would be given by AD BD', and the equation 
 
 reduces to 
 
 nm\ = nd (sin i sin 6). 
 
 Hence, the complete equation is, after dropping the n 9 
 
 m\ = d (sin i sin 0), (55) 
 
 whereby it is to be remembered that the negative sign is to be used 
 when the diffracted ray AR lies on the same side of the normal 
 as the direct ray A I. 
 
 The dispersion of a grating, denoted by D, may be denned as 
 in the case of the prism, as the ratio of the change in the angle 
 of diffraction dO to the corresponding change in the wave length 
 
 dX, that is, D = -j-; or, from equation (52), 
 
 llA. 
 
 (56) 
 
 d cos 
 
 From this equation it is seen that the dispersion varies directly 
 with m,_ that is, with the order of the spectrum. Hence the 
 spectrum of the second order will be twice as long as that of the 
 first order. Further, the dispersion is inversely proportional to 
 the grating space d. For this reason makers of gratings have 
 made it their aim to make d as small as possible. The dispersion 
 also varies inversely as the cos 0. If cos 0=1, i.e., = 0, the 
 diffracted ray is normal to the grating and the dispersion is a 
 minimum. Since the cosine varies very little from the value 
 unity when the arc varies through a small angle on either side of 
 
THE DIFFRACTION GRATIXG 111 
 
 zero, the dispersion curve is, under this condition, most nearly a 
 straight line. Hence the spectrum formed under this condition 
 is called a normal spectrum. 
 
 The purity of grating spectra may be measured in the same 
 
 "way as that of the prismatic spectra, that is, as before, P = 
 
 j or, when the source is infinitely narrow, the resolving power, 
 
 X X 
 
 n equals The value of -j- for gratings may be determined as 
 
 follows: By equation (52) the direction of a maximum for wave 
 length X + dX is, for normal incidence, determined by 
 
 nd sin 6 = nm (X + <7A), 
 
 and by equation (53) that of the minimum of the same order for 
 
 wave length A by 
 
 nd sin & = (nm -f 1)A. 
 
 The limit of resolution is reached when 6 and & in these two 
 equations are identical. Hence, 
 
 nm (A -f dX) = (nm + 1)A, 
 or 
 
 ^ -*-. (57) 
 
 Two conclusions can be drawn from this equation. First, if 
 m is constant, that is, if we observe spectra of the same order, 
 , the resolving power of different gratings will be proportional to 
 the number of lines which they contain. 
 
 The second conclusion may be reached as follows: From 
 equation (52) we have, for normal incidence, 
 
 nd . - 
 nm = sin 0, 
 A 
 
 but nd = 1 is the total width of the grating, therefore, 
 
 A / 
 - = sin 0. 
 ((X A 
 
112 MANUAL OF ADVANCED OPTICS 
 
 If in this equation sin 6 is constant, that is, if the spectra are 
 observed at a constant angle of diffraction, the resolving power of 
 all gratings of the same width is the same, and is consequently 
 independent of the number of lines. 
 
 Further, I cos is the width of the diffracted beam, denoted 
 by ', that is, I cos 9 = a ; or, 
 
 But by equation (56) 
 
 /i a 
 cos 6 = 
 
 _ w 1 _ nm I 
 
 d cos ~ nd a ' 
 
 or, since nd = 7, and nm = 7?, 
 
 D = ?r, or, R = a'D, (58) 
 
 exactly as in the case of the prism. It is to be noted, however, 
 that a is the width of the diffracted beam, not of the incident 
 beam. 
 
 Experiments 
 I. DETERMINE THE CONSTANT OF THE GRATING 
 
 APPARATUS. The spectrometer is used in these experiments 
 as in those described in the last chapter. The grating should be 
 so mounted that it can be placed upon the prism table of the 
 instrument and be adjustable about two horizontal axes, one of 
 which is parallel, and the other perpendicular to its surface. 
 
 ADJUSTMENTS. The grating should be so set upon the prism 
 table that its surface lies directly over the center of the graduated 
 circle of the instrument and its lower edge is parallel to the prism 
 table. The only farther adjustment is to make its surface 
 parallel to the axis of the instrument. This is done, if the 
 instrument has been adjusted as described in the last chapter, by 
 merely setting the grating perpendicular to the telescope. If the 
 
THE DIFFRACTION GRATING 113 
 
 instrument has not been so adjusted, the process there described 
 must be used, the surface of the grating taking the place of the 
 face AB of the prism. 
 
 MEASUREMENTS. Allow monochromatic light, or better, 
 sunlight, to fall upon the slit. If a transmission grating is to 
 be used, it is better to set it normal to the incident light. To 
 attain this, set the cross-hairs of the telescope on the undiffracted 
 image of the slit. Then set the grating perpendicular to the tele- 
 scope by means of the reflection of the cross-hairs in the usual 
 way. Read the position of the telescope at this point. Then 
 turn the telescope until the first spectral image falls on the cross- 
 hairs, and again read its position. The angle through which 
 it has been turned in this operation will be the angle of diffrac- 
 tion for the spectrum of the first order (m = 1). For the sake of 
 a check, this angle should be read on both sides of the central 
 undiffracted image of the slit. It is also well to determine the 
 angles of diffraction for spectra of several different orders. The 
 constant of the grating is then calculated with the help of equa- 
 tion (52), the wave length of the light used being taken from the 
 table of wave lengths. If the constant of the grating is known, 
 the same measurements can be used to determine wave lengths. 
 If the solar spectrum is used, of course a given Fraunhofer line 
 is used instead of the spectral image of the slit. 
 
 If the grating used is a reflection grating, it will probably be 
 impossible to observe with perpendicular incidence, because the 
 spectra of lower order fall so near the collimator that they 
 can not be observed in the telescope. In this case, the grating is 
 set up so that the light is incident at any angle i. The telescope 
 is then set perpendicular to the grating, and its position read 
 on the graduate circle. The angle of incidence is determined by 
 setting the cross-hair upon the directly reflected image of the slit, 
 and the angles of diffraction by setting it on the spectral images. 
 The angles of diffraction are, of course, all measured from the 
 
114 MANUAL OF ADVANCED OPTICS 
 
 normal to the grating, and their sign is minus if they lie on the 
 same side of the normal as the direct reflected image. The 
 grating space d is then easily obtained from equation (55), the 
 wave length being taken from the table. 
 
 EXAMPLE 
 
 Using sunlight as a source, a plane grating was adjusted in 
 the spectrometer. The angle of incidence was i = 48 17' 30". 
 The deviations for the first three spectra for three of the solar 
 lines were observed as follows : 
 
 LINE A j 2 3 
 
 C 6563 10~ 7 21 58' 20" 3' 30" 21 48' 10" 
 
 A 5890 10- 7 24 19' 20" 4 21' 10" 14 58' 10" 
 
 F 4861 10~ 7 28 2' 40" 11 10' 0" 4 48' 10" 
 
 Hence from equation (55) the following values of n l= ) were 
 calculated : 
 
 LINE 
 
 C 567.6 
 
 A 568.8 
 
 F 568.9 
 
 Mean 568.4 
 
 II. FIND THE RESOLVING POWER or THE GRATING 
 Apparatus and adjustments as in Experiment I. 
 Having determined the constant ^, the resolving power is 
 
 merely m , in which I is the length of the ruling and m the order 
 
 of the spectrum. As in the case of the prism the resolution can 
 be tested by finding the width of beam a which is just sufficient 
 for the resolution of a pair of known lines like the two sodium 
 lines. 
 
THE DIFFRACTION GRATING 115 
 
 EXAMPLE 
 
 A slit was placed over the objective of the collimator and 
 closed until the two sodium lines in the first spectrum were no 
 longer resolved. The width of the slit was then found to be 
 
 a = 1.28 mm. Hence, the length of grating used was : = 1.924 
 
 cos ^ 
 
 mm., and the number of lines in this width is n 1.924 = 1094 = R, 
 the resolving power according to equation (57). 
 
 7/1 
 
 The dispersion was then calculated from the equation D = -^-- 
 
 INTERVAL dd d\ D 
 
 C- D 8460" = .0410 radians 0.0000673 610 
 D-F 13400" = .0664 " 0.0001029 645 
 
 From equation (56) we get for D z in the first spectrum 
 
 . -618. 
 
 cos 
 
 The length of grating, which was just sufficient to resolve the 
 two sodium lines, was 1= 1.924 mm. For the first spectrum we 
 have = 21 58' 20". Hence, since I cos = a', the width of the 
 diffracted beam, we have as the resolving power, according ta 
 equation (58), 
 
 R = a'D = 1.784 x 613 = 1093. 
 
 The total width of the grating was 40 mm. Hence the resolv- 
 ing power in the first spectrum is 40 x 568.4 = 22736. It will be 
 noted that this number is the same for all parts of the spectrum. 
 
 For information as to the production and errors of gratings cf. Row- 
 land, Phil. Mag. (5) 13, p. 469; Nature 26, p. 211. Rayleigh, Phil. Mag. 
 (4) 47, p. 81, B. A. Reports, 1872; Nature 54, p. 332. 
 
THE CONCAVE GRATING 
 
 Theory 
 
 In using a plane grating, as in using a prism, lenses are neces- 
 sary to make the light from the source parallel and to form an 
 image for observation. Inasmuch as lenses absorb much of the 
 total energy of vibration that falls upon them, it is highly desira- 
 ble to avoid their use if possible. That this is possible has been 
 beautifully demonstrated by Eowland,* and his concave gratings, 
 by which he has been able to accomplish this end, have justly 
 become the standard instruments for spectroscopic work. 
 
 As the name implies, a grating of this kind is made by ruling 
 fine lines upon a concave mirror. The use of lenses in connec- 
 tion with the grating is thus avoided, because the concave mirror 
 will form real images by itself. 
 
 Since the series of openings thus formed are on the surface of 
 a sphere, they can not be treated as a series of parallel slits. The 
 theory of the concave grating has been fully treated by Rowland 
 and others. The following simple deduction of its leading char- 
 acteristics is abridged from a complete discussion of it by Rungeif 
 
 Suppose we have a source of monochromatic light at A 
 (Fig. 32), and a curved reflecting surface MM' at 0. Let P be 
 any point upon the surface MOM'. An image of A will be formed 
 at A' whenever the distance AP + PA' is the same for every 
 
 *Am. Jour. Sci. (3)26, p. 87; Phil. Mag. (5) 16, p. 197. 
 f Winkelmann's Handbuch, II, 1, p. 407. Kayser, Handbuch der 
 Spectroscopie, I, p. 452. 
 
 116 
 
THE CONCAVE GRATING 
 
 117 
 
 point P on the curved surface MM' , or whenever the rays 
 reflected from every point P arrive at A' in the same phase. 
 The first of these conditions will clearly be fulfilled if the reflect- 
 ing surface is an ellipsoid of revolution having A and A' as foci. 
 
 FIGURE 32 
 
 Let us now conceive a series of such ellipsoids of revolution con- 
 structed about A and A' as foci under the condition that AP + PA' 
 
 shall increase by for each such successive surface. If the mirror 
 
 4 
 
 be now supposed spherical, its surface will be divided by its inter- 
 sections with this series of ellipsoids into a large number of zones. 
 From any two adjacent zones light will reach A' in opposite 
 phase, because of the condition by which the ellipsoids were 
 drawn. If the radius of curvature of the mirror MM' and the 
 distances AP and PA are great with respect to X, these zones 
 will have very nearly equal width, and will, therefore, extinguish 
 each other's effect at A and produce there a minimum of illumi- 
 nation. If, however, this balance in the effect of the zones is 
 partly destroyed by so scratching a line on the mirror within the 
 region covered by every pair of adjacent zones that some light 
 shall arrive from each pair of zones in the same phase, we shall 
 
118 MANUAL OF ADVANCED OPTICS 
 
 obtain illumination there. It is to be noted that the condition 
 for illumination at A' is that the light from every pair of zones 
 shall arrive there in the same phase. Therefore, we may have 
 between the trains of waves from any adjacent pairs of zones a 
 difference of phase of any number of whole waves, that is, of m\. 
 
 It is now necessary to find how these lines must be drawn to 
 produce the desired effect. 
 
 Let the spherical mirror MM' now be referred to a set of 
 rectangular axes, so that its vertex is tangent to the xy plane at 
 the origin 0. Call the radius of curvature p. The equation of 
 the mirror so placed is 
 
 x z + if + z z - % P x = 0. 
 
 Let A and A' lie in the xy plane, and let their coordinates be 
 represented by , #, and a', #', respectively. Then 
 
 or, by performing the indicated operations and letting a z + b 2 = r*, 
 
 Upon substituting in this equation for 2ax its value (x* + y* + z*) 
 taken from above, it becomes 
 
 Remembering that x is of the second order with respect to y and 
 z, and neglecting terms of the third order, this reduces to 
 
 I a I a 1 \ , 1 
 --y + ( -- )# 2 + T - 
 r j 2r \r~ p / 9 r 
 
 The value of PA may be obtained in precisely the same 
 manner. It will be found to differ from that of AP only in that 
 
THE COXCAVE GRATING 119 
 
 ', b\ r', appear in it in place of , J, r, respectively. We may, 
 therefore, write 
 
 + r^- 
 
 This equation reduces to a much simpler form for certain particu- 
 lar positions of A and A'. In order to find the conditions which 
 determine these particular positions of A and A', let us suppose 
 that the mirror is so limited that the terms in z 2 may be 
 neglected. The terms in y z will vanish also if 
 
 a (a 1\ a' (a' 1 
 - + - 
 
 This will be true if r* = op and r' 2 = 'p> that is ^ ^ an( i -4' lie on 
 a circle whose center lies on the axis of x at a distance |- from the 
 
 origin. When this simple condition is fulfilled, the equation 
 reduces to 
 
 AP + PA' = r + r' 
 
 We are now in a position to answer the question as to how the 
 lines must be drawn on the mirror to produce the required effect. 
 Let e represent the difference in millimeters between the values 
 of the y's which correspond to the nth and the n + 1st lines. 
 The distance between the centers of their adjacent zones will also 
 be e. Since when A and A' are fixed, as we have supposed them, 
 r 4- r' is independent of the position of P on the mirror, the 
 question as to the illumination at A' depends for its answer upon 
 the value of the term containing y. The condition for illumi- 
 nation at A' is that the light arriving there from any zone, the 
 nth for instance, shall differ in phase from that arriving from 
 
120 MANUAL OF ADVANCED OPTICS 
 
 the next zone, the n + 1st, by a whole number of wave lengths 
 m\. This difference of phase will accordingly be determined by 
 
 or, since this must be equal to m\, by 
 
 *&+Si-**- ( 5 ) 
 
 Hence e, the difference in the values of the y's for two consecutive 
 lines, is constant. That is, the lines must be equally spaced 
 along a chord of the mirror perpendicular to the x axis. 
 
 If conditions are so arranged that the image formed at A' is 
 observed only when A' lies upon the axis of #, V becomes zero, 
 and equation (59) reduces to 
 
 e - = m\. (60) 
 
 r 
 
 But = sin i, where i is, as usual, the angle of incidence. Hence, 
 equation (60) may be written in the form 
 
 e sin i = m\. (61) 
 
 The distance A A' is, when b' = 0, equal to p sin t, or taking 
 equation (61) also into account, 
 
 AA'-?0- (62) 
 
 6 
 
 Hence A A' is proportional to the wave length X. 
 
 The use of the concave grating under the condition V = is 
 especially to be recommended, because, when the point of observa- 
 tion lies upon the normal to the grating, the dispersion is, as in 
 the case of the plane grating, very nearly constant throughout a 
 considerable range on either side of the normal. 
 
 Equations (56) and (57), expressing the dispersion and resolving 
 power of a plane grating, apply to the concave grating also. 
 
THE CONCAVE GRATING 
 
 121 
 
 Experiments 
 
 I. DETERMINE THE CONSTANT OF THE GRATING 
 APPARATUS. As shown above it is best to mount the concave 
 grating in such a way that its center of curvature coincides with 
 the point of observation. This may be accomplished in every 
 position of the grating if the slit, the center of the grating, and 
 the point of observation lie upon a semicircle whose diameter 
 passes through the center of curvature and the center of the 
 grating, and is equal in length to the radius of curvature of the 
 grating. The best method of fulfilling this condition is that 
 
 B 
 
 
 
 \ A 
 
 FIGURE 33 
 
 adopted by Rowland. The grating and the observing eyepiece are 
 firmly mounted on the ends of a rigid arm GO (Fig. 33), whose 
 length is equal to the radius of curvature of the grating. This 
 arm is supported at each end upon a small carriage, and these 
 carriages maybe moved along the tracks AB and AC, respect- 
 ively. If these two tracks are accurately perpendicular to each 
 other, and if the slit is placed at J, then in any position of the 
 arm GO the points G, and A will lie on a semicircle, which 
 fulfills the required conditions. 
 
 ADJUSTMENTS.* The tracks AB and AC must be straight, 
 
 *Cf. Ames, Phil. Mag. (5)27, p. 369. Astronomy and Astrophys. 11, 
 p. 28, 1892. 
 
MANUAL OF ADVANCED OPTICS 
 
 level, and perpendicular to each other. These tracks are usually 
 mounted upon heavy steel or wooden beams with adjusting screws 
 so that they can be raised or lowered and moved laterally. The 
 adjustment for straightness can be made by tightly stretching 
 beside the track a fine piano wire or silk thread, and then bringing 
 the track parallel to this either with the adjusting screws or by 
 filing. 
 
 The tracks may be made ^horizontal with the help of a good 
 spirit level, or with a cathetometer set up at some point, as Z>, 
 and focused on the upper edge of the track. 
 
 The tracks may be made perpendicular to each other by the 
 3-4-5 rule, i.e., by measuring from their point of intersection a 
 distance of 3 units on one track and of 4 units on the other; the 
 distance along the hypothenuse between the two points thus deter- 
 mined, should be made to equal 5 units. One of the beams 
 upon which the track is fastened is usually so mounted as to allow 
 of a lateral motion, in order to permit this adjustment to be 
 made. 
 
 The arm GO, mounted upon its carriages, should then be put 
 in place. The axes about which the arm turns with respect to 
 the carriages should be directly over the center of the track, and 
 be marked at the top by a small hole or point. It is assumed 
 that the maker of the carriages has attended to this. The 
 grating should then be set in place. The center of its surface 
 should be tangent to the axis of the carriage over which it stands. 
 The grating holder should be adjustable in every direction, i.e., 
 about each of the three rectangular axes of Figure 31. 
 
 The center of curvature of the grating should fall upon the 
 axis about which the arm GO turns at the end 0. To accomplish 
 this, a Gauss eyepiece is mounted at 0, so that its cross-hairs are 
 directly over the hole or point which marks that axis. The 
 centers of the eyepiece and the grating should lie in the same 
 horizontal plane, i.e., the distance of both from the top of the 
 
THE CONCAVE GRATIXG 123 
 
 track should be the same. The adjustment then divides itself 
 into two parts : First, the normal erected at the center of the 
 grating must intersect the axis at the observing end of the arm, 
 and must be parallel to the plane of the tracks ; and, second, the 
 cross-hairs of the eyepiece and their image formed by the grating 
 must coincide, i.e., the cross-hairs must lie at the center of curv- 
 ature of the grating. To fulfil the first of these conditions, the 
 grating should first be set by eye, so that the lines upon it are 
 approximately vertical. It should then be turned about the hori- 
 zontal and vertical axes, which are parallel to the plane tangent to 
 it at its center, i.e., about the y- and the z-axes of Figure 31, 
 until the image of the cross-hairs of the eyepiece nearly coincide 
 laterally and vertically with the cross-hairs themselves. The arm 
 GO must then be adjusted in length until the cross-hairs and 
 their reflected image show no parallax. The arm GO should 
 have been constructed to have very nearly the correct length. It 
 should, however, have a splice in it, which allows an adjustment 
 of a centimeter or so in length. 
 
 The slit should then be mounted over the intersection of the 
 tracks, with its center at the same distance from the tracks as the 
 center of the grating. If the other adjustments have been cor- 
 rectly made, the slit should be in focus in the eyepiece at 0. In 
 case it is not in focus it should be moved slightly forward or 
 backward in the direction AB. If it is found necessary to 
 move it more than a millimeter or two in order to bring it into 
 focus, it indicates that some of the other adjnstments are inac- 
 curate, and it is advisable to verify them. 
 
 To be sure that the lines of the grating are perpendicular to 
 the plane of the tracks move the arm GO back and forth. The 
 spectra will pass the point 0. The lines are vertical when the 
 spectra remain at the same height above the track at as the 
 arm is moved. 
 
 The slit should be parallel to the lines of the grating. This 
 
124 MANUAL OF ADVANCED OPTICS 
 
 adjustment is best made with the help of the solar spectrum. 
 The slit should be illuminated with sunlight, and the grating set 
 in such a position that a group of fine lines appears in the 
 eyepiece. While observing these lines the slit should be rotated 
 about a horizontal axis. When the lines are most sharply defined, 
 the slit is parallel to the lines of the grating. This adjustment 
 may be made easier by introducing a narrow opaque object, like a 
 medium-sized wire, horizontally across the slit. The images of 
 the slit in the eyepiece will then be divided into two parts, and will 
 appear to run into points on the two sides of the division. When 
 these points are vertically opposite each other the slit is parallel 
 to the lines of the grating. 
 
 MEASUREMENTS. Illuminate the slit with monochromatic 
 light of known wave length, or better with sunlight, and move 
 the arm GO until the spectral image of the slit, or a known 
 Fraunhofer line, falls upon the cross-hairs of the eyepiece. 
 Measure the distance AO and the radius of curvature OG of 
 the grating. Now the angle i of incidence is AOG, and the angle 
 of diffraction is equal to zero, hence from equation (61) 
 
 OA 
 
 m\ = e sin i = e -777^' 
 
 (J(jr 
 
 from which the value of e is readily determined. 
 
 The concave grating has been used to make absolute deter- 
 minations of wave lengths. In these measurements the grating 
 space was determined by dividing the width of the ruling by the 
 total number of lines upon it. The results obtained by different 
 observers by this method differ among themselves by one part in 
 15,000.* The grating furnishes, however, the most accurate 
 
 * The following are the most important absolute determinations of the 
 wave length D t . 5895.81, Angstrom, corrected by Thalen, Nov. Act. 
 Upsal. (3) 12, p. 1. 5896.25, Miiller and Kempf, Publ. Potsdam Obs. 5. 
 5895.90, Kurlbaum, Wied. Ann. 33, p. 159. 5896.20, Bell, Am. Jour. Sci. 
 (3) S3, p. 167; 35. p. 265; Phil. Mag. (5) 23, p. 265; 25, p. 255. 
 
THE CONCAVE GRATING 125 
 
 means of determining relative wave lengths after it has been 
 calibrated by known waves. Thus Rowland's relative determina- 
 tions, made by this method, have justly become the standard 
 work on the subject.* For fixing the absolute lengths he uses a 
 mean of the absolute determinations of the lengths of D l9 as 
 given in the footnote below. Probably the most accurate abso- 
 lute determination of wave length is that of the three cadmium 
 lines, which was made by Michelsonf by the interferometer 
 method. As a discussion of the details of the methods employed 
 to obtain accuracy in the determinations with the grating would 
 lead us beyond the scope of this book, the student is referred for 
 further information to the articles mentioned in the footnotes 
 or to Kayser's Handbuch der Spectroscopie, Vol. 1, p. 715 seq., 
 where the methods are presented at length. 
 
 EXAMPLE 
 
 To determine the constant of the grating the following 
 measurements were made upon some of the Fraunhofer lines of 
 the solar spectrum : 
 
 LINE 
 B 
 D, 
 E 
 F 
 
 The radius of curvature of the grating was 640.5 cm. Hence, 
 from equation (60) the following values of w (= ) are obtained: 
 
 AO (m = 1) 
 
 AO (m = 2) 
 
 AO (m = 3) 
 
 259.8 cm. 
 
 519.5 cm. 
 
 779.2 cm. 
 
 222.7 " 
 
 445.4 " 
 
 668.1 " 
 
 199.2 " 
 
 398.5 " 
 
 597.9 " 
 
 183.8 " 
 
 367.6 " 
 
 551.3 " 
 
 LINE 
 
 n 
 
 B 
 
 590.32 
 
 D, 
 
 590.31 
 
 E 
 
 590.30 
 
 F 
 
 590.30 
 
 mean 
 
 590.31 
 
 * Am. Jour. Sci. (3)&?, p. 182; Phil. Mag. (5) 23, p. 257; 27, p. 479; 36, 
 p. 49. 
 
 t Mem. du Bur. internal, des Poids et Mes. 11, p. 1 ; C. R. 11G, p. 790. 
 
126 MANUAL OF ADVANCED OPTICS 
 
 The width of the ruling was I = 145 mm. Hence, the total 
 number of lines upon the grating is n = In = 85,595 ; and therefore, 
 from equation (57), the resolving power in the first spectrum is 
 R = 85,595. Since 6 = at 0, the dispersion D is determined from 
 
 tTl 
 
 equation (56) as D = = mn Q = 590.31 for the first spectrum. The 
 
 6 
 
 width a of the beam is in this case the width of the ruling, i.e., 
 a = 145 mm. Hence, by equation (58), R = aD = 85,595. 
 
XI 
 
 F POLARIZED LIGHT 
 
 Quantitative experiments in polarized light presuppose some 
 general knowledge of the phenomena of polarization. Since expe- 
 rience has shown that few possess this knowledge with sufficient 
 definiteness, the following simple exercises are suggested, and the 
 student is advised to take any of the standard texts as a guide and 
 to work them out before attempting the experiments in the next 
 four chapters. 
 
 1. Make a dot on a piece of white paper and observe it perpen- 
 dicularly through a crystal of Iceland spar. The line connecting 
 the two dots seen is always parallel to the line connecting which 
 angles of a perfect rhomb? 
 
 2. Rotate the crystal and note the effect produced. Call the 
 ray which behaves extraordinarily the extraordinary ray E, and 
 the other the ordinary ray 0. 
 
 3. Toward which angle of the rhomb is E bent? 
 
 4. Prove that the two emergent rays are parallel. 
 
 5. Observe the dot through two crystals similarly placed; 
 explain the effect. 
 
 6. Observe the dot through two crystals oppositely placed; 
 explain the effect. 
 
 7. Place the crystals so that their axes inclose an angle of -45 
 and explain the effect. Call the four rays 00, Oe, Eo and Ee^ and 
 state how you distinguish each. 
 
 8. Which of the four disappear when the axes of the crystals 
 include an angle of 90? Which when the axes include an angle 
 
 of 180? Since the position of the upper crystal when Ee disap- 
 
 127 
 
128 MANUAL OF ADVANCED OPTICS 
 
 pears differs by 90 from its position when Eo disappears, these 
 two rays are said to be polarized in planes at right angles to each 
 other. 
 
 9. Observe a dot obliquely through a rhomb of Iceland spar, 
 and determine, by means of the bending of and E, which travels 
 the faster in the crystal. 
 
 10. Explain the construction of the Nicol prism. 
 
 11. Assume that vibrates in the plane perpendicular to the 
 principal plane of the wave, i.e., to the plane defined by the wave 
 normal and the optic axis. By means of this assumption deter- 
 mine the plane of transmission of the Nicol, i.e., the plane in 
 which the transmitted vibrations take place. 
 
 12. Using the Mcol to determine the plane of vibration, find 
 in which plane the vibrations of the light reflected from a glass 
 surface at the polarizing angle take place. 
 
 13. Determine the plane of maximum vibration of the light 
 transmitted by a plate of glass when the light is incident at the 
 polarizing angle. 
 
 14. Increase the number of plates and explain the effect 
 produced upon both the reflected and the transmitted light. 
 
 15. Prove, experimentally, that a Wollaston prism makes the 
 beams and E divergent, and explain the reason. 
 
 16. Observe a sodium burner through two Nicols. Shut off 
 the light by crossing the Nicols. Insert a thin crystal and explain 
 the effect. 
 
 17. Rotate the crystal and locate its axes. 
 
 18. Rotate the analyzer and explain the effects. 
 
 19. Replace the sodium burner with a source of white light. 
 Rotate the crystal and explain the effects. Rotate the analyzer 
 and explain the effects. 
 
 20. Try the same experiment without the polarizer. 
 
 21. Replace the analyzer with a Wollaston prism and prove 
 that the colors of the two images are complementary. 
 
POLARIZED LIGHT 129 
 
 22. Replace the crystal by one thinner and then by one 
 thicker and explain the effects. 
 
 23. Examine and explain the effects produced by passing con- 
 vergent or divergent plane polarized light through a crystal cut 
 perpendicular to the axis. 
 
XII 
 
 KOTATION OF THE PLAXE OF POLAKIZATIOX 
 
 Theory 
 
 It has long been known that if plane polarized light be passed 
 through certain substances, the plane of polarization is rotated 
 through an angle which is peculiar to each such substance. On 
 this account such substances are called optically active. They 
 may conveniently be divided into three classes as follows : 
 
 First, those substances which rotate the plane of polarization 
 only when they are in crystal form and lose their optical activity 
 when they are melted or brought into solution. The most 
 important of these substances is quartz, which crystallizes in the 
 hexagonal crystal system. Since the optical activity of these sub- 
 stances is lost when their crystal form is destroyed, this peculiar 
 property of theirs must depend only on the geometrical arrange- 
 ment of the molecules in the crystal, and hence its investigation 
 falls properly within the domain of physics. 
 
 Second, those substances which show this optical activity not 
 only in the crystal form but also when melted or in solution. In 
 this class belong several of the camphors and tartrates. 
 
 Third, those substances which are optically active only when 
 in the liquid form or in solution. This class contains carbon 
 compounds only. Since the members of the second and third 
 classes retain their optical activity not only in solution but also in 
 the vapor form, it follows that their power of rotating the plane of 
 polarization depends on the arrangement of the atoms in the 
 molecule, and, therefore, its study belongs properly in the 
 
 domain of chemistry. 
 
 130 
 
ROTATION OF THE PLANE OP POLARIZATION 131 
 
 For substances of the first class, the amount of rotation varies 
 with the thickness of the substance traversed by the light, with 
 the wave length of the light, and with the temperature. 
 
 It has been experimentally proved that the angle, denoted by 
 a, through which the plane of polarization is rotated by an active 
 substance is proportional to the thickness traversed.* If / 
 denotes this thickness, then 
 
 a = kl, 
 
 ' in which k denotes the rotation produced by unit thickness. For 
 quartz, the values of k for a plate 1 mm. thick cut perpendicular 
 to the optic axis are, for light of the wave lengths corresponding 
 to the Fraunhofer lines of the solar spectrum,! 
 
 FRAUNHOFER LINE B C- D E F G 
 
 k 15.75 17.31 21.72 27.54 32.76 42.59 
 
 The dependence of the angle of rotation upon the wave length 
 may be expressed by J 
 
 A B 
 
 a = TI +TT' 
 \ 2 A* 
 
 in which A and B are constants to be determined for each 
 substance by experiment. From the values of X given above for 
 quartz, A and B are found to have the following values, when A 
 is expressed in mm., 
 
 A = 7.1083 10- 6 B = 0.1477 10~ 12 . 
 
 The variation of a due to changes of temperature is, for sub- 
 stances of the first class, small. For quartz, 
 
 ar = a [1 + 0.000147(0]. 
 
 *Biot, Mem. de 1'Acad. ~ ? , pp. 41, 91, 1817; Ann. chim. phys. (2) 10, 
 p. 63. 
 
 t Soret and Sarasin, C. R. 95, p. 635, 1882. 
 
 ^Boltzmann, Pogg. Ann. Jubelbd., p. 128, 1874. For another disper- 
 sion equation cf. Lotnmel, Wied. Ann. 14, p. 523, 1881. 
 
 Gumlich, Wiss. Abh. d. Phys.-techn. Reichsanstalt, 2, p. 230, 1895. 
 
132 MANUAL OF ADVANCED OPTICS 
 
 In considering the optical activity of substances of the second 
 and third classes in solution, the idea of specific rotation, intro- 
 duced by Biot, is found convenient. The specific rotation of an 
 active substance in solution, denoted by [a], is defined as the angle 
 through which the plane of polarization is rotated by a column of 
 the solution 1 dm. long, which has in 1 cc. of its volume 1 gm. of 
 the active substance. Thus if the solution be obtained by placing 
 c gm. of the active substance in a flask and filling the flask till it 
 contain 100 gm. of solution, then by definition, 
 
 Since, as has been observed above, the optical activity of sub- 
 stances of the second and third classes is due to their molecular 
 structure, it has been found useful to introduce the molecular 
 weight into the definition. Hence, the molecular rotation is 
 defined as the specific rotation multiplied by the molecular weight 
 and divided by 100. The 100. is introduced merely to avoid large 
 numbers. Thus, if [M] represent the molecular rotation, and M 
 the molecular weight, 
 
 that is, the molecular rotation is the angle through which the 
 plane of polarization would be rotated by a column of the solution 
 of the length of 1 mm., which contained in every cc. of its vol- 
 ume 1 gram molecule of the active substance. 
 
 It was at first supposed that specific rotation was a constant 
 characteristic of a substance. Later investigation has, however, 
 brought out the fact that it varies with the concentration, with 
 the nature of the solvent, and with the temperature. These vari- 
 ations are generally small. Their cause has been shown to lie, 
 for some substances at least, in the incomplete dissociation of the 
 molecules in the solution into their ions. As a discussion of this 
 
ROTATION OF THE PLANE OF POLARIZATION 133 
 
 subject lies beyond the scope of this work the reader who wishes 
 to pursue the matter further is referred to Landolt, Optische 
 Dreltungscermogen organischer Substanzen, 2d edition, Braunsch- 
 weig, 1898. 
 
 The case of sugar, however, on account of its peculiar impor- 
 tance from a practical point of view, merits special attention. The 
 value of [a] for sugar, as determined from solutions whose concen- 
 tration varies from 5 to 30%, is essentially constant. From a 
 large number of different measurements, its value at 20 C. for 
 sodium light has been determined as 
 
 Its variation with the temperature is, according to the most 
 recent work, for values of t between 12 and 25 C,* 
 
 [a]^[a]"-0.0217 (t-'20). (64) 
 
 Sugar possesses very nearly the same dispersive power as 
 quartz. The following figures give the rotation corresponding to 
 some of the Fraunhofer lines as measured in solutions whose con- 
 centrations varied from 10 to 20%, the column of the liquid used 
 being of such length that the sodium light is rotated the same 
 amount as by a plate of quartz 1 mm. thick. f 
 
 FRAUNHOFER LINE B C D E F G 
 
 Rotation 15.20 17.23 21.71 27.64 33.08 43.14 
 
 A comparison of these values with those given above for quartz 
 shows how nearly the dispersive powers of the two substances 
 correspond. It is because of this fact that, as is done in some 
 forms of sacchari meter, it is correct to make the measurements by 
 compensating the rotation produced by the sugar solution by a 
 rotation in the opposite sense produced by quartz. 
 
 *Sch6nrock, Zs. f. Instrk. 20, p. 97, 1900. 
 t Stefan, Wien. Ber. 52, II, p. 486. 
 
 
134 MANUAL OF ADVANCED OPTICS 
 
 The specific rotation of invert sugar varies much more than 
 that of cane sugar with changes of concentration and tempera- 
 ture. Its value and its variations with the concentration and the 
 temperature are,* for values of c up to 35 and of t between 
 and 30C., 
 
 M;= [a] 2 -0.304(^-20), (65) 
 
 in which c' = c, the fraction being the ratio of the molecular 
 
 weights of the two sugars. The rotation is in this case left- 
 handed. 
 
 Cane sugar may be converted into invert sugar by adding to 
 100 cc. of the sugar solution, 10 cc. of strong hydrochloric acid, 
 and keeping the mixture at a temperature of 70 C. for ten minutes. 
 It has, however, been shownf that the value of [a']^ determined 
 from solutions thus converted is not constant, but depends some- 
 what on the relative amounts of sugar and acid in the solution. 
 The following is recommended : To every 100 parts of sugar add 
 1 part of oxalic acid and let the mixture stand for some hours at 
 a temperature of 50 to 60C. 
 
 Experiment 
 DETERMINE THE PURITY OP A SAMPLE OF SUGAR 
 
 APPARATUS. The essential parts of the polarimeter are two 
 Nicols, A and B (Fig. 34), mounted on the ends of a firm hori- 
 zontal bar about 30 cm. long. It should be possible to rotate the 
 analyzing Nicol B about a horizontal axis, and to read the angle 
 through which it has been turned upon a vertical graduated circle 
 (7, which is fastened to the base of the instrument. The accuracy 
 
 * Landolt, Optische Drehungsvermogen, Braunschweig, 1898, p. 526. 
 t Gubbe, Ber. I. deutschen Chem. Ges. 18, p. 2210. 
 

 ROTATION OF THE PLANE OF POLARIZATION 
 
 135 
 
 of the readings which can be made with this simplest form of 
 instrument is not very great. The methods employed in increas- 
 ing the attainable accuracy of setting are different in different 
 forms of instrument. The device most frequently used is that of 
 introducing directly behind the polarizer a second doubly refract- 
 
 C 
 
 B 
 
 FIGURE 34 
 
 ing object, which has the effect of dividing the field of view into 
 two or more parts whose planes of polarization make a small angle 
 with each other. Thus in the Laurent polarimeter there is intro- 
 duced behind the polarizer a thin plane-parallel plate of quartz so 
 cut as to effect a slight rotation of the plane of polarization of the 
 polarizer, and so placed as to cover half of the field of view. In 
 
136 
 
 MANUAL OF ADVANCED OPTICS 
 
 the most recent form of instrument, this alteration of the azimuth 
 of the plane of polarization is effected by a small Nicol, which 
 covers half of the field of view, and whose principal plane may be 
 set to make a small angle with that*of the polarizer. The accu- 
 racy of setting is increased by introducing this small angle 
 between the planes of polarization of the two halves of 
 the field of view, because it is then impossible so to 
 set the analyzer as to extinguish the light from both 
 halves of the field of view at the same time. When the 
 principal plane of the analyzer is perpendicular to the 
 plane which bisects the angle between the planes of 
 polarization of the two halves of the field of view, this 
 field appears uniformly illuminated, i.e., the line 
 between the two halves disappears. It has been found 
 that the eye can judge more accurately the position of 
 equal illumination of the two halves of the field, than 
 it can the position of total extinction of the whole field. 
 In order to make the line of division between the 
 two halves of the field sharp, a small telescope is usually 
 added to the observing end of the instrument. This 
 telescope is focused upon the edge of the quartz plate 
 of the Nicol which has been introduced. The arrangement of 
 the optical parts of one of the more sensitive forms of the instru- 
 ment is shown in Fig. 35. 
 
 In most instruments the polarizer can be turned through a 
 small angle, thus allowing an alteration in the angle between the 
 planes of polarization of the two halves of the field of view. 
 
 ADJUSTMENTS. If the instrument has been properly con- 
 structed, the only adjustment needed is that of the angle between 
 the planes of polarization of the two halves of the field of view. 
 Up to a certain limit, the smaller this angle the more accurate the 
 setting. It should be made as small as the brightness of the 
 source and the absorption of the solution to be examined will 
 
 FIGURE 35 
 
ROTATION OF THE PLANE OF POLARIZATION 137 
 
 permit. The student should experiment and set the angle at the 
 point at which he can set with greatest accuracy in a given 
 case. 
 
 MEASUREMENTS. It is first necessary to determine the zero of 
 the instrument. Using a sodium burner as a source of light, the 
 empty tube which is to contain the solution to be tested is placed 
 between the polarizer and the analyzer, and the analyzer is rotated 
 until the field of view is uniformly dark. The position of the 
 analyzer is then read on the graduated circle. The analyzer 
 should then be rotated through 180, and the position again read. 
 These two readings are to be used as the zero readings of the 
 instrument. The tube is then filled with the solution to be tested 
 and again placed between the polarizer and the analyzer. The 
 analyzer should then be rotated until the field of view appears 
 uniformly dark, and its position read upon the graduated circle. 
 The reading must be taken also after the analyzer has been turned 
 through 180. The mean of the differences between these two 
 last readings and the corresponding zero readings is the angle 
 through which the plane of polarization has been rotated by the 
 solution. 
 
 If the rotation is large, it may be doubtful whether the rotation 
 is to the right or to the left. This question may be settled either 
 by observing with two different sources of light and remembering 
 that the rotation for longer waves is ordinarily less than for short 
 waves, or by making a second set of observations with a more 
 dilute solution, which would*, of course, give a smaller rotation. 
 
 The polarimeter is most frequently used in analyzing solutions 
 of cane sugar. In this case the method of procedure is as follows : 
 Add 10 to 15 gm. of the sugar to be tested to about 85 cc. of 
 distilled water. After the sugar is dissolved add water till the 
 solution weighs 100 gm. Take half of this solution, i.e., 50 cc., 
 and add to it 5 cc. of strong hydrochloric acid. Warm this 
 mixture to a temperature of 70 C., and let it stand at that tern- 
 
138 MANUAL OF ADVANCED OPTICS 
 
 perature for ten minutes. The sugar will then be converted into 
 invert sugar. Before using it should be allowed to cool to the 
 temperature of the room. 
 
 Two tubes are usually supplied with the instrument, one 
 20 cm. , and the other 22 cm. long. The shorter tube should be 
 filled with the sugar solution, and the longer with the solution of 
 the invert sugar. The rotations which the two produce are 
 then determined, the sugar rotating to the right, the invert 
 sugar to the left. The temperature should be noted, especially 
 during the observations with the invert sugar. 
 
 To reduce the observations, let a represent the measured rota- 
 tion of the sugar, a! that of the invert sugar, and t' the tempera- 
 tures at which the respective observations were made. Then 
 from equations (63) and (64), 
 
 .= {[-O.Oai7(*-ao)j^ + /S/' ;., ii: 1 
 
 in which ft represents any rotation which may be produced by 
 substances other than sugar in the solution. Similarly from 
 equations (63) and (65), 
 
 a '= j [']"- 0.304 (*'-20) 
 
 in which ft is negative, because active substances other than sugar 
 are not altered by the hydrochloric acid. The sum of these two 
 equations is, in consideration of equations (64) and (65), 
 
 , ( r _, 360 r ,,,' } le 
 
 ft -f- tt == A I ft -\- ft r ' 
 
 ( L ]D 342 L J * ) 100 
 
 from which the value of c, which is the number of grams of sugar 
 in 100 gm. of solution, may be readily calculated. 
 
 EXAMPLE 
 
 13.29 gm. of cane sugar, which had been crystallized from a 
 sugar solution, was carefully dried over sulphuric acid in vacuo 
 
ROTATION OF THE PLANE OF POLARIZATION 139 
 
 and dissolved in water, so that 100 gm. of solution were obtained. 
 Half of this was converted into invert sugar as above directed. 
 The following observations were made, / being for the sugar solu- 
 tion 20 cm., and for the invert sugar solution 22 cm. : 
 
 a = 17. 35, a' = 5. 30, t = t' = 27C. 
 The reduction is then as follows : 
 
 [a']* = 19.657 + .0361c' = 20.162 
 .304 #'-20 = 2.128 
 
 = 18.034 
 
 [a]' D = 66.352 
 M' 4- 36 Fa'!'' 85 335 
 
 I Q. I "T" ""Ift I = O O O O O 
 
 .-. a + a' = 0.85335fc. 
 But, from the observations, a + a' = 22. 65 and I = 2 dm. 
 
 .-. 22.65 -1.70670 
 c = 13.27 gm. 
 
XIII 
 ELLIPTICALLY POLARIZED LIGHT 
 
 Theory 
 
 Suppose we have given a beam of plane polarized light whose 
 vibrations take place in the plane OC (Fig. 36). Let 00 (= A) 
 represent the amplitude of the vibration. Suppose, further, that 
 a plane-parallel plate of a doubly refracting crystal, cut parallel to 
 the optic axis, be introduced into the path of the plane polarized 
 beam in such a way that its faces are perpendicular to the beam, 
 
 
 
 FIGURE 
 
 while its principal planes have the directions OX and Y. The 
 incident vibration OC will be divided by the plate into two com- 
 ponents OF (= a) and OG (= &), which will be equal respectively 
 to OCcosO and 00 sin if represents the angle COP. These 
 two component vibrations travel through the plate with different 
 velocities; hence, when they emerge, though they will still be 
 vibrating parallel respectively to OF and OG, one will be in 
 
 advance of the other by an amount which depends upon the 
 
 140 
 
ELLIPTICALLY POLARIZED LIGHT 141 
 
 thickness of the crystalline plate and the difference of velocities 
 of the two components in it. If then A cos %-* represent the 
 
 original vibration, and 8 the difference in the optical paths of the 
 two components when they emerge from the crystal, tha vibra- 
 tions of the two components will be represented by 
 
 , 
 
 X = a COS TT 
 
 f-- + _l 
 
 (6(5) 
 
 respectively. After passage through the plate these vibrations 
 unite into a single one. The path which any vibrating particle 
 pursues may be determined by eliminating t from equations (66). 
 
 The result is 
 
 ^ 
 
 cos '4ir 
 
 =sin 8 27r^ (67) 
 
 a* b~ ab A. 
 
 This is the equation of an ellipse inscribed in the rectangle 
 whose sides are Ha and *2b respectively. The position of the ellipse 
 in the rectangle depends manifestly upon the value of 8. Two 
 
 cases are of especial interest: First, when 8 = 0, or- In this 
 
 8 8 
 
 . case sin HIT = 0, cos 2?r = + 1, and the path of the particle is 
 
 A A 
 
 given by 
 
 i.e., the emergent light is again plane polarized, the direction of 
 vibration being parallel to 00. 
 
 Second, when 8 = - In this case, sin 2ir = + 1, cos 2ir = 0, 
 i - A A 
 
 and the equation for the path reduces to 
 
 x* 
 
142 MANUAL OF ADVANCED OPTICS 
 
 i.e., the path of the vibration of the emergent light is an ellipse 
 whose principal axes coincide in direction with the principal 
 
 planes of the crystalline plate. When 8 = , the motion about the 
 
 ellipse is in a direction contrary to that of the hands of a watch 
 
 3 
 
 (levogyre), while when 8 = A., the rotation is in the same direction 
 
 as that of the hands of a watch (dextrogyre). If in addition it 
 should happen that a = #, the equation becomes 
 
 , * + *- ; 
 
 i.e., the path of the vibration is a circle. 
 
 Conversely it is true that if any elliptical vibration is separated 
 into components along the principal axes of the ellipse the differ- 
 
 , . TT STT 
 
 ence in phase of those components is or 
 
 # A 
 
 These two special cases of elliptically polarized light are of 
 importance, because they furnish a method of finding the posi- 
 tion of the axes of the elliptical vibration, and of measuring the 
 relative intensity of the components along those axes. The 
 method consists in reducing the difference in phase between the 
 
 two components by , and measuring the angle which the result- 
 4 
 
 ing plane polarized vibration makes with the axes of the ellipse. 
 The method is explained in detail in Avhat follows . 
 
 Experiment 
 
 ANALYZE AN ELLIPTICAL VIBRATION 
 
 APPARATUS. The most convenient instrument for deter- 
 mining the nature of elliptically polarized light is an ordi- 
 nary spectrometer to which has been added a pair of Nicols and 
 three graduated circles. The circles CCC (Fig. 37) should be 
 mounted on short tubes bbl> with their planes perpendicular to 
 
ELLIPTIC ALLY POLARIZED LIGHT 
 
 143 
 
 the length of the tubes. Two of these tubes should be made to 
 fit over the telescope and collimator at their objective ends, while 
 the third fits over the eye end of the telescope. Into the end of 
 each of these tubes fits a collar ddd which may be rotated freely 
 
 FIGURE 37 
 
 within the tube, and to which may be fitted either the Nicol 
 prisms or the Babinet compensator or the quarter-wave plate as 
 desired. These collars also carry indices Hi by which their posi- 
 tions may be read on the graduated circles upon the tubes. 
 
 The Babinet compensator consists of a pair of quartz wedges 
 having the same angle between their "faces. One wedge is so cut 
 that the optic axis of the quartz is parallel to the edge of the 
 wedge, while in the other the optic axis is perpendicular to the 
 edge and parallel to one of the faces of the wedge. These wedges 
 are mounted in a brass case, the wedge A (Fig. 38) being fastened 
 to one side of the case, while the wedge B can be moved back and 
 
144 
 
 MANUAL OF ADVANCED OPTICS 
 
 forth by the micrometer screw C in a direction perpendicular to 
 its edge. The head of this screw is graduated, and the mounting 
 of this wedge B carries a scale by which its position can be 
 determined. 
 
 Suppose light polarized in a plane which is normal to the 
 plane defined by the two optic axes of the wedges, and which 
 bisects the angle between those axes, to fall perpendicularly upon 
 the wedges as indicated by the line NN (Fig. 38). On enter- 
 
 FlGUBE 
 
 ing the wedge A, the vibration will be separated into two equal 
 components, one parallel to the edge of the wedge, and the other 
 perpendicular to it. These components will travel with different 
 velocities within the wedge, and will, therefore, emerge with a 
 difference of phase, the magnitude of which will depend upon the 
 color of the light used and the thickness of the wedge at the 
 place where the light has passed through. On entering the 
 wedge B, the light again separates into two component vibrations 
 which travel with different velocities, but because the optic axis 
 of this wedge is perpendicular to that of the other wedge, the 
 component vibration which traveled slower in wedge A will travel 
 faster in wedge B, and that which traveled faster in A will travel 
 slower in B. It is evident, then, that along that line parallel to 
 the edges of the wedges where the light has traveled through 
 equal thicknesses of the two wedges, it will emerge in exactly the 
 same condition in which it entered, while along other lines its 
 
ELLIPTICALLY POLARIZED LIGHT 145 
 
 condition will depend on the difference of the thicknesses through 
 which it has passed. 
 
 If such a pair of wedges be brought between crossed Nicols 
 and so oriented that the optic axes make angles of 45 with the 
 planes of transmission of the Xicols, the line along which the 
 thicknesses of the two wedges is the same will appear as"a dark 
 line, because along it the condition of polarization of the incident 
 light is not altered by the compensator. If the combination is 
 illuminated with white light, colored bands will be seen on either 
 side of this dark band, because wherever the difference in the 
 thicknesses is such that a difference of phase of a whole wave for 
 any particular color is introduced by the compensator, that color 
 will be wanting in the light transmitted by the analyzer. If the 
 combination be illuminated with monochromatic light, a series of 
 equidistant dark bands will appear, the distance between the suc- 
 cessive bands showing how far it is necessary to move along the 
 wedges in order to introduce a difference of phase of a whole 
 wave of the particular light used. 
 
 If the wedge B be moved by the screw (7, the bands will move, 
 and the constant of the instrument is merely the number of turns 
 which must be given the screw C in order to make the dark bands 
 move up one, i.e., the number of turns which correspond to a 
 difference of phase of a whole wave. By turning the screw C 
 through a fraction of the number of turns which correspond to a 
 difference of phase of a whole wave, a difference of phase of that 
 same fraction of a wave is introduced along the line formerly 
 marked by the central dark band. Thus the Babinei compen- 
 sator allows us to introduce any difference of phase which may be 
 desired. It may also be used to measure a difference which has 
 been introduced by other means, for such a difference of phase 
 will shift the central band, and may be measured by counting the 
 number of turns of the screw (7, which are needed to cause the 
 bands to return to their original position. This number, divided 
 
146 MANUAL OF ADVANCED OPTICS 
 
 I 
 
 by the constant of the instrument, will be the required difference 
 of phase. It is because of its use in this way to compensate a 
 phase difference which has already been introduced, that thei 
 instrument has received its name. 
 
 The quarter-wave plate is a plate of crystal so cut and of such 
 thickness that incident plane polarized light is separated by the 
 plate into two components at right angles to each other, and one 
 of these components is retarded over the other by a quarter-wave 
 of the particular light used. 
 
 ADJUSTMENTS. The tubes carrying the graduated circles 
 having been set in place upon the collimator and telescope of ] 
 the spectrometer, preferably with the zero of the circle upward, 
 the Nicols must be mounted in these tubes so that their planes of 
 transmission are either vertical or horizontal when the circles read 
 zero. To attain this, place in the tube at the end of the collima- 
 tor a double-image prism. On looking into the eyepiece, two 
 images of the slit will be seen. Rotate the double-image prism 
 until these two images are superimposed in the center of the 
 field, one projecting above the area in which they are superim- 
 posed, the other below it. If the slit of the collimator is vertical, 
 then, when the images overlap, as described, the vibrations of the 
 light in one of the images will be vertical, while those in the other 
 will be horizontal. A Nicol, suitably mounted to fit the tube, 
 should then be placed in front of the objective of the telescope, 
 and turned until it entirely extinguishes one of the images of the 
 slit. The double-image prism should then be replaced by another 
 Nicol, the index of the circle on the collimator should be set to 
 zero, and the Nicol rotated in the collar which holds it till the 
 image of the slit is extinguished. The plane of transmission of 
 the Nicol on the collimator is then either vertical or horizontal 
 when the index by which its position is read stands at zero. If 
 the plane along which the two halves of the Nicol are cemented 
 together is vertical, the transmitted vibrations are horizontal, etc. 
 
ELLIPTICALLY POLARIZED LIGHT 147 
 
 This adjustment can also be made by setting the analyzer so as 
 to extinguish the light reflected from a vertical plate of glass at 
 the angle of complete polarization. When this takes place the 
 plane of transmission of the analyzer is horizontal. 
 
 The Nicol in front of the objective of the telescope should 
 then be removed to the eye end of the telescope, and a quarter- 
 wave plate mounted in front of the objective. It is very con- 
 venient to place the Nicol at the eye end between the lenses of 
 the eyepiece, though of course it can be mounted either in front 
 of or behind the eyepiece. This Nicol is then crossed with the 
 Nicol on the collimator (the polarizer), and the quarter-wave plate 
 turned until the field is dark. The planes of transmission of the 
 quarter-wave plate are then respectively vertical and horizontal. 
 The readings of the circles which carry the quarter-wave plate 
 and the Nicol on the eye end of the telescope (the analyzer) 
 should then be taken. They are the zero positions of these two 
 parts of the instrument. 
 
 If the Babinet compensator is to be used, it should be mounted 
 at the eye end of the telescope directly in front of the eyepiece. 
 A cross-hair should be fastened to the compensator to mark the 
 position of the central black fringe, and the eyepiece focused on 
 this cross-hair. The constant of the Babinet must then be deter- 
 mined for the wave length to be used. To accomplish this, the 
 plane of vibration of the polarizer should be turned to make an 
 angle of 45 with the horizontal, and the compensator set so that 
 its optic axes are respectively vertical and horizontal. The 
 analyzer should then be crossed with the polarizer. Open the slit 
 wide and illuminate it with monochromatic light of the desired 
 color and count the number of turns of the micrometer screw on 
 the compensator which are necessary to make the dark bands in 
 the field of view move up one. 
 
 In all of these experiments with polarized light, a bright 
 source should, if possible, be used. A section of a solar spectrum, 
 
148 
 
 MANUAL OP ADVANCED OPTICS 
 
 or filtered sunlight, will, in most of these cases, be sufficiently 
 uniform. 
 
 MEASUREMENTS. Having adjusted the instrument as has 
 been described, introduce a thin piece of mica behind the polar- 
 izer. Let OH (Fig. 3y) represent the direction and amplitude 
 
 E 
 FIGURE 39 
 
 of the plane polarized light, which is incident upon the mica, and 
 OA and OB the planes of transmission of the mica. Let EEE 
 represent the resulting elliptical vibration which it is desired to 
 analyze. First consider the case in which the quarter-wave plate 
 is to be used. It is evident that the elliptical vibration will be 
 reduced to a plane one if the directions of the planes of vibration 
 of the quarter-wave plate coincide with those of the major axes of 
 the ellipse, i.e., with the directions OE and OF in the figure. The 
 plane of vibration of the restiltant plane polarized light will evi- 
 dently be represented by 0(7, and hence in order to extinguish the 
 light, the analyzer must be turned till its plane of transmission is 
 perpendicular to 0(7, i.e., till it has the direction OB'. When, 
 therefore, the quarter-wave plate and the analyzer have been set 
 so that the light is totally extinguished, the planes of vibration of 
 the plate will indicate the positions of the principal axes of the 
 elliptical vibration, and the tangent of the angle between one of 
 
ELLIPTICALLY POLARIZED LIGHT 149 
 
 those planes and the plane of transmission of the analyzer, i.e., 
 angle R 'OF (= angle COE), will determine the ratio of those axes, 
 for 
 
 (68) 
 
 If the compensator is used instead of the quarter-wave plate, it 
 is merely necessary to set it so that it will introduce a difference 
 in phase of a quarter-wave and rotate it until the central black 
 band returns to the zero position. The analyzer must then be 
 rotated until the central black fringe is blackest. The directions 
 of the optic axes of the compensator show the directions of the 
 principal axes of the ellipse, and the tangent of the angle between 
 one of those axes, and the plane of transmission of the analyzer is 
 the ratio of those axes. 
 
 EXAMPLE 
 
 The analyzer, quarter-wave plate and polarizer were adjusted 
 as described above. Filtered sunlight was used. After the intro- 
 duction of the mica it was necessary to turn the quarter-wave 
 plate through an angle of 13 20' and the analyzer through an 
 angle of 43 30' to extinguish the light. Hence, the principal 
 axes of the elliptical vibration form made an angle of 13 20' with 
 the horizontal and vertical planes respectively, and the ratio of 
 their amplitudes was tan (43 30' - 13 20') = tan 30 10' = 0.581. 
 
XIV 
 
 THE REFLECTION OF POLARIZED LIGHT FROM 
 HOMOGENEOUS TRANSPARENT SUBSTANCES 
 
 Theory 
 
 If a beam of light of amplitude A , plane polarized in the plane 
 of incidence, fall upon the surface of a homogeneous transparent 
 substance, the amplitude A' of the reflected light will, according 
 to Fresnel, * be given by 
 
 sm (i + r) 
 
 
 in which i and r denote, as usual, the angles of incidence and 
 refraction respectively. When the index of refraction of the 
 substance is greater than unity, i.e., when /.>!, i>r. Hence, 
 in this case the reflected amplitude has a sign opposite to that of 
 the incident amplitude; i.e., there is a loss of half a wave at 
 reflection, and in absolute value A'<A, becoming equal to A only 
 when i = 90. At normal incidence, i = 0, and the right-hand 
 side of the equation becomes indeterminate. The limiting value 
 of the expression may then be found with the help of the con- 
 sideration that, for small values of i and r, the sine may be 
 replaced by the angle. Thus i = nr and 
 
 If the incident light has the amplitude B, and is polarized in 
 
 * Fresnel, Oeuvres, I, p. 640, Paris, 1866. Drude, Theory of Optics, p. 
 282, Longmans, 1902. 
 
 150 
 
THE REFLECTION OF POLARIZED LIGHT 151 
 
 a plane perpendicular to the plane of incidence, Fresnel's expres- 
 sion for the amplitude of the reflected light is 
 
 tan (i + r) 
 
 When fi >1, i.e., />r, and when also i + r<90, B' is negative, 
 i.e., there is a loss of half a wave at reflection. When i + r = 90, 
 B' = 0. When i + r>90, the light is reflected without the loss 
 of half a wave. For the particular case " -t- r = 90 we have 
 
 sin r = cos /, and hence, since -. = u, 
 
 sin r 
 
 tan i = /n. (72) 
 
 The angle determined by this equation is called the angle of total 
 polarization. This equation, first experimentally established by 
 Brewster, is known as Brewster 's law.* 
 
 If the incident light is polarized neither in the plane of inci- 
 dence nor in a plane at right angles to it, but in a plane which 
 makes an angle with the plane of incidence, then, since we have 
 adopted Fresnel's assumption that the direction of vibration is 
 perpendicular to the plane of polarization, the component y of 
 the amplitude in the plane of incidence will be 
 
 y = A sin 0, 
 
 while the component x perpendicular to that plane will be 
 * 
 
 x = A cos 0, 
 
 the re-axis lying in the surface, the y-axis in the plane of incidence, 
 both perpendicular to the intersection of these two planes. The 
 values of these components after reflection will, therefore, be 
 
 sin (i-r) 
 
 x = x 
 
 y -- 
 
 sin (i + r) 
 tan (i r) 
 
 tan 
 
 * Brewster, Phil. Trans. 1815, p, 125. Jamin, Ann. de chim. et phys. 
 (3) 29, pp. 31 and 263; (3) 30, p. 257. Conroy, Proc. Roy. Soc. 31, p. 487, 
 1881. Rayleigh, Phil. Mag. (5) 33, p. 1, 1892; Collected .Works, 3, p. 49a 
 
152 MANUAL OF ADVANCED OPTICS 
 
 These two components will unite to form a plane polarized 
 beam whose inclination 0' to the plane of incidence is given by 
 
 tan 6' = ; 
 
 */ 
 
 or, substituting the values of x' and y' taken from above, 
 
 ., y cos (i + r) cos (i + r) , 
 
 tan = - - j-. (- = }-. f tan 6. (73) 
 
 x cos (i r) cos (i r) 
 
 When the light is incident normally, i = and tan = tan 0'. 
 As i increases, 0' becomes less than until i reaches the angle of 
 complete polarization, when & = 0. If i be further increased, 
 & becomes negative, reaching the value 6 when i- 90. 
 
 These reflection equations of Fresnel have been frequently 
 verified experimentally.* This is most easily done with the help 
 of equation (73). It is merely necessary to allow light which is 
 plane polarized in a plane whose azimuth with the plane of inci- 
 dence is known, to be reflected from the surface of a transparent 
 medium whose index of refraction is known, and to measure the 
 angle of incidence i, and the azimuth #', of the plane of polariza- 
 tion of the reflected light. 
 
 Experiments 
 I. VERIFY BREWSTER'S LAW 
 
 The apparatus and adjustments are those described in the pre- 
 ceding chapter. 
 
 MEASUREMENTS. A plate of crown glass is set upon the prism 
 table of the spectrometer, and adjusted so that its faces are par- 
 allel to the axis of the instrument, as described in Chapter VII, 
 page 98. Filtered sunlight is allowed to pass through the colli- 
 
 *Rood, Am. Jour. Sci. (2) 49; 50, 1870. Rayleigh, Nature 35, p. 64, 
 1886; Proc. Roy. Soc. 41, p. 275, 1886; Collected Works II, p. 522. Con- 
 roy, Proc. Roy. Soc. 35, p. 26, 1883; 37, p. 38, 1884; 45, p. 101, 1888; Phil. 
 Trans. ISO, p. 245, 1889. 
 
THE REFLECTION OF POLARIZED LIGHT 153 
 
 mator and fall upon one of the faces of the glass plate, and the 
 reflected light is received in the telescope, having the analyzer in 
 front of the objective. At the angle of complete polarization all 
 of the vibrations of the light reflected from the glass take place in 
 a vertical plane. The analyzer should, therefore, be set so that 
 its plane of transmission is horizontal, and the telescope and glass 
 plate revolved about the axis of the instrument until the position 
 is found in which the analyzer extinguishes all of the reflected 
 light. In this position, the telescope makes with the normal to 
 the glass plate an angle equal to that of complete polarization. 
 This angle is then measured in the usual way. Its tangent 
 should be equal to the index of refraction of the glass for the color 
 used. As a check this index may be determined in the usual way. 
 Care must be taken to have the surface of the glass to be tested 
 perfectly clean. 
 
 EXAMPLE 
 
 The solar spectrum was allowed to fall across the slit end of the 
 collimator and the sodium line placed on the slit. The slit was 
 then opened wide and the light allowed to fall on a piece of the 
 same glass whose index was determined with the interferometer 
 (cf. page 65). The polarizer was removed from the collimator, 
 and the analyzer set so that its plane of transmission was hori- 
 zontal. It was found that the light was completely cut off when 
 the angle of incidence was i = 56 32'; hence, tan i = 1.513 = /A. 
 
 II. VERIFY FRESNEL'S LAWS OF REFLECTION 
 
 The apparatus and adjustments are those of Experiment I. 
 
 MEASUREMENTS. As stated above this is easily done by meas- 
 uring the rotation of the plane of polarization which is produced 
 by the reflection. Using the glass plate as a reflecting surface, as 
 above, allow filtered sunlight to fall upon it through the collimator 
 and polarizer. The polarizer should then be set at a definite 
 
 
154 MANUAL OF ADVANCED OPTICS 
 
 azimuth by the graduated circle on which its position is read, 
 and the light allowed to be reflected at a measured angle i. The 
 azimuth of the reflected light is then read on the circle which 
 gives the position of the analyzer, the latter being set so as to 
 extinguish the light reflected from the prism. Having deter- 
 mined the index of refraction in the previous experiment, and 
 since i and are measured, equation (73) furnishes a check upon 
 Fresnel's reflection equations. In case it is found difficult to 
 locate exactly the angle of complete polarization these observa- 
 tions of the rotation of the plane of polarization may be used to 
 find it, for it is that angle at which the azimuth of the reflected 
 vibration is 90 from the plane of incidence. Hence, if these 
 observations be plotted with the *'s as abscissae and the 0's as 
 ordinates, the resulting curve will cross the axis of abscissae at 
 a point corresponding to the angle of complete polarization. 
 
 EXAMPLE 
 
 The polarizer was set so that its plane of transmission was 
 at an azimuth of 45 with the vertical plane. The light used 
 in Experiment I was then allowed to fall upon the reflecting glass 
 plate at measured angles of incidence i, and the following were 
 observed as the azimuths of the reflected light: 
 
 30 
 35 
 40 
 45 
 
 50 
 55 
 
 60 
 65 
 
 70 
 
 75 
 
 e (Oss.) 
 
 (CALC.) 
 
 33 24' 
 
 33 36' 
 
 28 50' 
 
 29 3' 
 
 23 50' 
 
 23 36' 
 
 17 20' 
 
 17 14' 
 
 10 20' 
 
 10 9' 
 
 2 20' 
 
 2' 31' 
 
 -5 10' 
 
 -5 16' 
 
 -13 20' 
 
 -12 56' 
 
 -20 20' 
 
 -20 13' 
 
 -27 40' 
 
 -27 0' 
 
THE REFLECTION OF POLARIZED LIGHT 155 
 
 The observed values were plotted in a curve with the z's as 
 abscissae. The curve crosses the axis at the point 56 30', which 
 corresponds to the angle of complete polarization (cf . Experiment 
 I). The calculated values were obtained with the help of equa- 
 tion (73). 
 
 
XV 
 METALLIC KEFLECTION 
 
 Theory 
 
 Experiments on the light reflected from metallic surfaces 
 furnish the following facts : 
 
 1. When plane polarized light is reflected from a metallic 
 surface, the reflected light is elliptically polarized, unless the plane 
 of polarization of the incident light is either parallel or perpen- 
 dicular to the plane of incidence. 
 
 2. Metallic surfaces do not possess the faculty of completely 
 polarizing light by a single reflection at any angle. 
 
 3. If the incident light is circularly polarized, there is one 
 particular angle of incidence for which the reflected light is plane 
 polarized. 
 
 In discussing the phenomena of metallic reflection, it is con- 
 venient to conceive that when light falls upon such a surface, the 
 incident vibration is resolved into two, one perpendicular and one 
 parallel to the plane of incidence, and that each of these two 
 components undergoes a change of phase at reflection. Experi- 
 ment shows that that component which is parallel to the plane of 
 incidence undergoes a greater change of phase than the other, and 
 that the difference in phase between the two components after 
 reflection is zero at normal incidence, and increases with the angle 
 of incidence. 
 
 Since circularly polarized light becomes plane polarized by 
 reflection at a particular angle of incidence, this angle corresponds 
 somewhat to the angle of complete polarization of transparent 
 
 substances and is called the principal angle of incidence. It is 
 
 156 
 
METALLIC REFLECTION 157 
 
 defined as the angle at which the difference in phase between the 
 component in the plane of incidence and the one perpendicular to 
 it amounts to a quarter of a wave length. This particular angle 
 will be denoted by /. The azimuth of the reflected plane polar- 
 ized light at this angle of incidence is called the principal azimuth, 
 and will be denoted by 0. 
 
 The theory of metallic reflection deduces a relation between 
 these two principal angles and the index of refraction and coeffi- 
 cient of absorption of the metal. The index of refraction /u, of the 
 metal is defined like that of a transparent body as the ratio of the 
 velocity of light in vacuo to the velocity in the metal. The 
 coefficient of absorption K is defined as follows : If A represent 
 the amplitude of the vibration at a given instant, and A' its 
 amplitude after the wave has traveled one wave length in 
 the metal, then the coefficient is defined by the equation 
 A : A' = 1 : e~ 2n *. The relation between these optical constants 
 and the angles / and 6 is expressed by the equations* 
 
 K = tan 20 1 
 
 sin / tan / 1(74) 
 
 v/iT^ 
 
 Hence /w. and K can be determined from observations of / and 0. 
 
 Experiment 
 
 DETERMINE THE OPTICAL CONSTANTS OF SILVER, GOLD, AND 
 
 PLATINUM 
 
 The apparatus and adjustments are the same as those 
 described in Chapter XIII. 
 
 MEASUREMENTS. The measurements may be made in two 
 ways: First, we may allow circularly polarized light to fall on 
 the metallic surface and determine the angle of incidence at which 
 
 *Drude, Tlieory of Optics, p. 361 seq., Longmans, 1902. Drude, Wied. 
 Ann. 36, p. 885, 1889; 39, p. 481, 1890. 
 
158 MANUAL OF ADVANCED OPTICS 
 
 the reflected light is plane polarized, and the azimuth of the plane 
 of polarization. 
 
 Second, we may allow light plane polarized at an azimuth of 
 45 to fall upon the metallic surface, and observe the angle of 
 incidence at which the central fringe of a Babinet compensator set 
 for a quarter-wave returns to the zero position. The amount 
 which the Nieol behind the Babinet has to be rotated to make 
 the central fringe black determines the principal azimuth. 
 
 The metallic surface is set upon the prism table of the spec- 
 trometer and adjusted to be perpendicular to the telescope. 
 
 EXAMPLE 
 
 Glass plates coated with opaque films of silver, gold, and 
 platinum were used, also a plate of polished steel. The following 
 observations and results were obtained, the values of /A and K 
 being computed with the help of equations (74) : 
 
 METAL, / * n 
 
 Silver 74 38' 43 20' 17.2 0.20 
 
 Gold 71 40' 41 50' 9.02 0.32 
 
 Platinum 77 30' 32 40' 2.17 2,01 
 
 Steel..... 76 5' 28 10' 1.50 2.56 
 
 Sunlight filtered through the red solution described in Appendix A 
 was used. 
 

 XVI 
 
 THE SPECTROPHOTOMETER 
 Theory 
 
 Ordinary photometers, like the Bunsen or the Lummer- 
 Brodhun, may be used to compare the intensities of the total radi- 
 ations of two sources. It is, however, often necessary to be able 
 to compare the intensities of radiation of two sources for each of 
 the separate colors. This is accomplished by the spectrophotom- 
 eter by separating each of the sources into a spectrum, and 
 arranging the optical parts so that the two spectra are adjacent 
 and can be compared at any point of their length. 
 
 One of the most convenient forms of spectrophotometer is that 
 devised by Glan.* This instrument consists of a spectrometer 
 whose slit is divided into two parts. The light from one source is 
 allowed to pass through one portion of the slit, while that from 
 the other source passes through the other portion. We thus 
 get in the field of view two adjacent spectra, one from each 
 source. A screen with a vertical slit in it in the eyepiece cuts off 
 all of these spectra but one vertical band of color, half of which 
 comes from one source, and the other half from the other. 
 
 In order to be able to measure the relative intensities of the 
 two halves of this band of light, the apparatus is so arranged 
 that the light in one half is polarized in one plane and that in the 
 other in a perpendicular plane. This is accomplished by placing 
 a double-image prism in the collimator. Such a prism will give 
 two images of each half of the slit, and these two will be polarized 
 
 *Glan, Wied. Ann. 1, p. 351, 1877. 
 
 159 
 
160 
 
 MANUAL OF ADVANCED OPTICS 
 
 in planes at right angles to each other. Thus we get four spectra 
 in the field of view, and of these the first and third are polarized in 
 one plane, while the second and fourth are polarized in a perpen- 
 dicular plane. The screen in the eyepiece cuts off the first and 
 fourth spectra and leaves the second and third, one from the upper 
 half of the slit polarized, say in a vertical plane, and one from the 
 lower half of the slit polarized in a horizontal plane. By intro- 
 ducing a Nicol befcmd the double-image prism we are able to cut 
 out either one or the other of these spectra. Thus when the 
 plane of transmission of the Nicol is vertical, the spectrum from 
 the upper half of the slit will be cut out; while when that plane 
 is horizontal, the spectrum from the lower half of the slit disap- 
 
 w 
 
 o 
 
 FIGURE 40 
 
 pears. At intermediate positions of the $Ticol, both spectra are 
 visible, the intensity of each depending on the intensities of the 
 sources and the position of the Nicol. 
 
 Let /! and / 2 represent the intensities of the two sources. 
 Let, further, the amplitude of vibration of the light from source 
 I be represented by OA (Fig. 40), and that of source 2 by OB. 
 Suppose also that OA is cut out when the index on the Nicol 
 stands at zero. The plane of transmission of the Nicol has then 
 the direction OB. Conceive the Nicol to be rotated until the two 
 
THE SPECTROPHOTOMETER 161 
 
 spectra are equally bright. This will be the case when the plane 
 of transmission OC has such a direction that the projections of 
 OA and OB upon it are equal, i.e., when it is perpendicular to 
 the line joining A and B. Call the angle through which this 
 plane has been turned a, then OB cos a = OC = OA sin a, i.e., 
 the ratio of the amplitudes is 
 
 \ 
 OB sin a 
 
 = tan a. 
 
 A cos a 
 Hence, the ratio of the intensities of the two sources is 
 
 - = tan 2 a. (75) 
 
 Experiments 
 
 I. COMPARE THE RADIATIONS OF Two DIFFERENT SOURCES 
 
 OF LIGHT 
 
 APPARATUS. The spectrometer, as used in the last chapters, 
 can readily be converted into a spectrophotometer. 
 
 ADJUSTMENTS. To convert the spectrometer into a spectro- 
 photometer it is necessary, as stated above, to place in the colli- 
 mator tube a double-image prism directly in front of the object- 
 ive. This prism should be so set that the two images off the 
 vertical slit overlap to form one line. The vibrations in one image 
 will then be vertical, while those in the other will be horizontal. 
 A narrow card should then be fastened over the middle of the 
 slit and so adjusted in width that the overlapping portions of the 
 two central adjacent images, one from the lower and one from 
 the upper half of the slit, are cut out. The center of the image 
 of the slit in the eyepiece then appears continuous, but the vibra- 
 tions of the upper half are, say vertical, while those of the 
 lower half are horizontal. The upper and lower ends of the 
 image of the slit should then be cut out with a screen in the 
 
162 MANUAL OF ADVANCED OPTICS 
 
 eyepiece, so that the light in the entire upper half of the image 
 vibrates in one plane, while that in the lower vibrates in a perpen- 
 dicular plane. 
 
 The Nicol is then put in place in the tube ~b (Fig. 37) at the 
 end of the collimator. The index is set at zero, and the Mcol 
 rotated in the collar d until one of the spectra is extinguished. 
 
 A prism is then introduced so as to produce in the field of 
 view two spectra, adjacent, 'and polarized in planes at right 
 angles to each other. 
 
 MEASUREMENTS. It is first necessary to determine a scale of 
 wave lengths. To do this allow sunlight to pass through the slit 
 and take the readings on the circle of the spectrometer for four or 
 five of the Fraunhofer lines. Then introduce into the eyepiece a 
 screen with a narrow vertical slit whose center coincides with the 
 intersection of the cross-hairs. Set the telescope so that the slit 
 in the eyepiece falls upon the red end of the spectra and take the 
 reading on the circle of the spectrometer. 
 
 Allow the two lights which are to be compared to enter 
 the two halves of the slit, and then turn the Nicol until both 
 halves of the image of the slit, appear equally bright. Take the 
 reading of this position of the Nicol. Then move the telescope 
 along a definite amount, say 20' or 30'. Again set the Nicol for 
 equal illumination, and take the reading. Proceed in this way 
 through the entire spectrum. The readings are then plotted 
 with the angles which denote the successive positions of the tele- 
 scope as abscissae, and the squares of the tangents of the 
 angles of the Isicol as ordinates. The resulting curve repre- 
 sents the intensities of the various parts of the spectrum of 
 one source, in terms of those of the other as unity. For if 
 7 2 = l,/i = tan 2 a. 
 
 Having determined the angles of the telescope which corre- 
 spond to certain Fraunhofer lines, the scale of abscissae can be 
 converted into a scale of wave lengths. 
 
THE SPECTROPHOTOMETER 163 
 
 EXAMPLE 
 
 Sunlight was used to calibrate the instrument and the follow- 
 ing readings were obtained: 
 
 ,FRAUNHOFER LINE READING 
 
 B 14 10' 
 
 C 14 0' 
 
 D 13 26' 
 
 E 12 32' 
 
 F 11 56' 
 
 G 10 36' 
 
 An incandescent lamp of sixteen candle power was then com- 
 pared with a Welsbach light. The following readings were 
 obtained, the last column giving tan 3 a, which is the ratio of the 
 intensity of the Welsbach to that of the incandescent lamp : 
 
 CORRESPONDING 
 
 READING 
 
 A - 10 
 
 a 
 
 TAN2 a 
 
 11 
 
 0' 
 
 444 
 
 52 
 
 1.64 
 
 
 20' 
 
 458 
 
 54 
 
 1.90 
 
 
 40' 
 
 473 
 
 56 
 
 2.19 
 
 12 
 
 0' 
 
 490 
 
 57 
 
 2.37 
 
 
 20' 
 
 508 
 
 5T : 
 
 2.37 
 
 
 40' 
 
 528 
 
 56 
 
 2,19 
 
 13 
 
 0' 
 
 552 
 
 55 
 
 2.04 
 
 
 20' 
 
 580 
 
 54 
 
 1.90 
 
 
 40' 
 
 614 
 
 50 
 
 1.42 
 
 14 
 
 0' 
 
 656 
 
 46 
 
 1.08 
 
 
 10' 
 
 687 
 
 38 
 
 0.61 
 
 If these values of tan 2 a be plotted as ordinates with either 
 the readings or the corresponding wave lengths as abscissae, the 
 curve will show that the Welsbach is relatively richer in blue and 
 green rays. 
 
164 MANUAL OF ADVANCED OPTICS 
 
 II. DETERMINE THE ABSORPTION OF A SOLUTION OF CYANIK 
 
 Apparatus and adjustments a's in Experiment I. 
 
 MEASUREMENTS. Sunlight is allowed to pass through both 
 halves of the slit, and then the absorbing substance is placed over 
 one of them. The method of making and plotting the observa- 
 tions is the same as that described above. The student should 
 determine several such absorption curves for substances like 
 cyaiiin, permanganate of potash, ruby glass, or a thin film of 
 silver on glass. 
 
 EXAMPLE 
 
 Light from an incandescent lamp passed into the upper half 
 of the slit directly, and into the lower half through a glass cell 
 containing a dilute solution of cyanin. The following observa- 
 tions were made: 
 
 READING 
 
 a 
 
 TAN 2 a 
 
 11 
 
 0' 
 
 42 
 
 0.81 
 
 12 
 
 0' 
 
 42 
 
 0.81 
 
 13 
 
 0' 
 
 42 
 
 0.81 
 
 
 10' 
 
 31 
 
 0.36 
 
 
 20' 
 
 14 
 
 0.06 
 
 
 30' 
 
 
 
 0.00 
 
 
 40' 
 
 16 
 
 0.08 
 
 
 50' 
 
 38 
 
 0.61 
 
 14 
 
 0' 
 
 42 
 
 0.81 
 
 If these values of tan 2 a are plotted as ordinates with the cor- 
 responding readings as abscissae, the curve will represent the 
 intensity of the light transmitted by the cyanin solution in terms 
 of the intensity of the light from the incandescent lamp. It will 
 be noted that cyanin absorbs completely the'radiation correspond- 
 ing to the reading 13 30'. From the preceding example it is 
 seen that this reading corresponds to wave length 597 10" 6 mm. 
 
XVII 
 THE DEVELOPMENT OF OPTICAL THEORY 
 
 By optical theory is meant that system of ideas or conceptions f 
 in which the various phenomena of light are unified, and by 
 which they are explained. To gain a clear idea of the meaning 
 of this statement, we must distinguish two factors which enter 
 into the formation of every science. In the first place, we 
 perceive certain external events through their effect upon our 
 senses, and, in the second place, we form conceptions by which 
 these events are systematized, harmonized, and interpreted. 
 
 It is hardly correct to apply the word development to a mere 
 increase in the number of phenomena observed ; for, though the 
 number of things which we perceive with regard to any object 
 becomes greater every year, this increase may be better described 
 by the word accretion than by the word development. . This latter 
 word seems to imply an organized increase, an evolution. Hence 
 it can appropriately be used only when it includes a reference to 
 our conceptions, for it is in the organized expansion of our 
 knowledge through these conceptions, that the life of science 
 really lies. 
 
 This distinction becomes perfectly clear if we consider an 
 example. Such an example might be taken at random from almost 
 any domain of science; but, since this book treats of optics, it will, 
 perhaps, be better to draw our illustration from this branch of 
 physics. For the sake of brevity we will begin the discussion 
 with the end of the seventeenth century, for it is then that optics 
 first became prominent. 
 
 At that time only the more conspicuous phenomena of light 
 had been noted. Thus, it was known that light seems to travel 
 
 165 
 
166 MANUAL OF ADVANCED OPTICS 
 
 in straight lines, that it is reflected from a plane mirror in such a 
 way that the angle of incidence is equal to the angle of reflection, 
 that it is bent from its straight path when it passes obliquely 
 from one medium into another of different density, and that the 
 different parts of a beam seem to be independent of one another. 
 
 In order to describe these phenomena concisely, two concep- 
 tions were formed, one by Descartes and Newton, the other by 
 Huygens and Hooke. The former considered that light consists 
 of fine particles or corpuscles which are shot out by luminous 
 bodies and travel with enormous velocity ; while the latter believed 
 that light is a form of wave motion. As is well known, the 
 former of these conceptions prevailed and was adopted by the 
 scientists of the eighteenth century as being more nearly correct. 
 Newton was unable to adopt the latter mainly because waves are 
 known to bend or be diffracted around the edges of obstacles 
 placed in their path, and the light waves did not appear to do 
 this. Hence he lent his energies to developing the conception of 
 corpuscles moving in straight lines, and carried with him the 
 world of physicists for a century or more. 
 
 It is interesting to note that of these two conceptions that of 
 particles moving in straight lines is mechanically much simpler 
 than that of wave motion, for it is characteristic of scientific 
 thought to base the first conception of a phenomenon upon some- 
 thing crudely mechanical or which is familiar and easily con- 
 ceived. This tendency is often disastrous to the healthy progress 
 of a subject, and proved to be so in the case of optics, for that 
 science made very little advance during the period in which the 
 conception of light corpuscles was held. 
 
 It finally became evident to some men of science that the cor- 
 puscular idea must be abandoned, first, because it was mechanic- 
 ally too simple to account for the exceedingly complex and varied 
 phenomena of light; and, second, because it seemed to them to 
 be internally absurd, since it was found that even so commonplace 
 
THE DEVELOPMENT OF OPTICAL THEORY 107 
 
 an event as the passage of a beam of light from air into water, 
 for example, required for its elucidation, additional assumptions 
 which were somewhat ridiculous, as shown by Xewton's "fits of 
 easy reflection, " etc. When the phenomena of interference and 
 polarization became matters of observation, largely through the 
 efforts of Young and Fresnel, then, notwithstanding the fact 
 that the corpuscular theory had to adorn itself with many gratui- 
 tous supplementary hypotheses, the strife between the rival con- 
 ceptions became fierce, and all parties to the contest sought an 
 ''exper&nenttun cruets" which would finally decide between them. 
 
 Such an experiment was found in connection with refraction ; 
 for, according to the corpuscular theory, light must travel faster 
 in the denser medium ; while, according to the wave theory, the 
 reverse is true. The controversy was then finally settled by 
 Foucault when he proved experimentally that light travels more 
 slowly in a denser medium, and so the conception that light is a 
 form of wave motion came to be generally accepted. 
 
 Thus, although the corpuscular theory is mechanically the 
 simpler, the phenomena of light are so complex that the more 
 intricate conception of wave motion has after all proved to be 
 more satisfactory in that it not only furnished simple and exact 
 descriptions of interference, diffraction, and polarization, but also 
 enabled the scientists of that day to predict effects which led to 
 the discovery of new laws. 
 
 It was, however, soon found that the idea that light is a wave 
 motion had its limitations when the phenomena of dispersion 
 were considered. In dealing with diffraction and interference it 
 is sufficient to conceive that we have to do with a wave motion, 
 but in the case of dispersion we must also form some notion of 
 the nature of the medium in which the wave motion takes place. 
 Hence the next problem which presented itself was that of form- 
 ing an adequate conception of that medium, the ether, as it has 
 been called. 
 
168 MANUAL OF ADVANCED OPTICS 
 
 The most natural solution of this problem is to assume that 
 the waves are elastic waves like those of sound, and that the 
 medium which transmits them is one which possesses mechanical 
 properties similar to those which are necessary for the propagation 
 of elastic waves. If, however, this solution is adopted, several 
 serious difficulties are at once encountered. 
 
 The first of these difficulties arises because of the enormous 
 velocity of light, 300,000 kilometers a second. Since the 
 velocity of an elastic impulse in any medium is determined by 
 the square root of the elasticity of the medium divided by its 
 density, it is necessary to assume that the medium which trans- 
 mits light has an extraordinarily high elasticity, or a very small 
 density, or both. It would not be so difficult to conceive such a 
 medium if it could be thought of as a very rare gas. But the 
 light waves are transverse waves, and the medium which transmits 
 such waves must have rigidity, i.e., must have the properties of a 
 solid. 
 
 Even this conception of a highly elastic and very rare solid 
 which fills all space might not be impossible if it were not for the 
 fact that the planets and the comets move at great velocities 
 through it without apparent resistance, and the conception of a 
 solid which yet offers no resistance to motion through it is rather 
 difficult. 
 
 But there are other conceptions involved in the assump- 
 tion of a medium which reacts to mechanical forces which are 
 even more impossible. These .conceptions depend upon the elas- 
 tic constants of the medium. The theory of elasticity demands 
 that six conditions be fulfilled when an elastic impulse passes the 
 boundary between two media. These conditions are the equality 
 on both sides of the boundary of the components of the elastic 
 displacements, and the equality of the components of the elastic 
 forces. In order to satisfy these six conditions it is necessary to 
 assume both transverse and longitudinal disturbances in the 
 
THE DEVELOPMENT OF OPTICAL THEORY 169 
 
 second medium, for the transverse alone can at best satisfy four 
 of these conditions. Hence it has been the burden of the elastic 
 theories of ether to make these four constants do the work of six, 
 since longitudinal vibrations in the ether have never been 
 detected. A detailed account of the devices employed by Cauchy, 
 Fresnel, Green, and others to surmount this difficulty will be 
 found in Winkelmann's Handbuch der Physik, Vol. II, pt. 1, p. 
 641 seq.* 
 
 The fact which is of special interest to us here, is, that 
 although the adoption of the idea that light is wave motion was 
 a step toward conceptions of greater mechanical complexity, and 
 hence served as a great stimulus to progress in the growth of 
 optical theory, yet the concomitant notion of a mechanically elas- 
 tic solid medium contained internal absurdities which placed 
 serious limitations on the usefulness of the theory as a whole. 
 Hence it appears that further development demanded an expan- 
 sion or change in the conception of the nature of the ether. 
 
 The first to suggest a new conception was Faraday, \ who, in 
 1851, while discussing the question whether the magnetic force is 
 transferred through bodies by action in a medium external to the 
 magnet or by action at a distance, wrote: "For my own part, con- 
 sidering the relation of a vacuum to the magnetic force and the 
 general character of magnetic phenomena external to the magnet, 
 I am more inclined to the notion that, in the transmission of the 
 force, there is such an action external to the magnet, than that 
 the effects are merely attraction and repulsion at a distance. 
 Such an action maybe a function of the ether; for it is not at all 
 unlikely that, if there be an ether, it should have other uses than 
 simply the conveyance of radiations." 
 
 * Cf. also Lloyd, Report on Optical Theories, B. A. Reports, 1834, p. 
 295 seq. Glazebrook, Report on Optical Tlieories, B. A. Reports, 1885, p. 
 157 seq. L'Abbe Moigno, Repertoire d'Optique Moderne, 4 Vols., Paris, 
 1847-50. 
 
 f Faraday, Experimental Researches, No. 3075. 
 
170 MANUAL OP ADVANCED OPTICS 
 
 This hint of Faraday's was taken up later by Maxwell,* who in 
 1873 published a theory of light based on the assumption that 
 the medium which transmits light is the same as that which 
 serves as a vehicle for the electric and magnetic forces. He 
 further developed a conception of the process of ether wave 
 propagation .which renders this theory free from the internal 
 absurdities and contradictions which hindered the progress of its 
 predecessor. 
 
 Thus his fundamental assumption concerning the nature of 
 the forces in the ether, i.e., his conception of displacement cur- 
 rents which produce magnetic effects, is such that it follows at 
 once from it that electromagnetic waves in the ether are trans- 
 verse, f Hence after adopting this theory we are no longer com- 
 pelled to consider the ether a solid. 
 
 Furthermore, there are only four independent conditions 
 which must be fulfilled when a train of waves passes through the 
 boundary between two media, and hence it is not necessary to 
 make any special assumptions to explain that passage. 
 
 Finally, the theory enables us to calculate optical constants 
 from electrical measurements, thus bringing two distinct fields of 
 investigation into relations which can be subjected to quantitative 
 measurements. 
 
 Now, although the attempt to describe the properties of this 
 two-sided medium, the ether, as if they were similar to the crudely 
 mechanical properties of grosser matter, has since then fre- 
 quently been made, science is fast coming to believe, if it has not 
 already reached the conclusion, that this ether belongs in a cate- 
 gory by itself, that its properties are discretely different from 
 those of perceptible matter. Thus Maxwell's theory has proved 
 to be a step from the mechanical conception of an elastic solid to 
 a wholly new and radically different idea. It thus opened at once 
 
 * Ma,xwell^Electrieity_Qnd Magnetism, Vol. II., p. 431. 
 fDruule, Theory oj Optics., p. 278. 
 
THE DEVELOPMENT OF OPTICAL THEORY 171 
 
 a vast domain for new investigation it bridged the chasm 
 between the territories of light and electricity, and has been the 
 means of adding immensely to our store of information concern- 
 ing both sciences. 
 
 It is to be noted, however, that this conclusion that the ether 
 is discretely different from ordinary matter does not by any means 
 indicate that the problem is completely solved. It merely places 
 that medium in a position in which an open discussion of its 
 properties is possible, i.e., it removes all crudely mechanical bias 
 from our minds and shows us that we have before us an almost 
 wholly unexplored domain which invites investigation. It thus 
 lends enthusiasm to optical and electrical research by offering a 
 field for the exercise of untrammeled imagination, a field unob- 
 structed by conceptions of a grossly mechanical nature. Some of 
 the results of this impetus which the electromagnetic theory of 
 light has given to optical research will be briefly discussed in the 
 next chapter. 
 
 In brief, then, optical theory has developed from a simple con- 
 ception of material particles traveling in straight lines, through 
 the more complex conception of waves in a mechanically elastic 
 medium, to the recognition of a wave motion in a medium whose 
 properties are still to be discovered and classified. And while 
 the theory has passed through these stages, the observed details 
 in the phenomena which it is invented to describe have increased 
 in number in a way commensurate with the expansion of the 
 theory. And so we return to the statement with which we 
 began, namely, that science is a vitally living thing, whose life can 
 be traced in the development of the conceptions which we form 
 of phenomena in the world about us. 
 
XVIII 
 THE TREND OF MODERN OPTICS 
 
 It remains for us to consider briefly some of the more marked 
 results to which the adoption of the electromagnetic theory has 
 led. The first important step in the development of the theory 
 was taken by Maxwell when he showed that the velocity of an 
 electromagnetic wave in a dielectric is equal to the ratio of the 
 electromagnetic to the electrostatic unit, jdivijiejiJay^b^^sqiiaxe 
 root of the dielectric constant of frhe medium. Thus if V repre- 
 sent the velocity of the wave, c the ratio of the units, and k the 
 dielectric constant,* 
 
 v -j% () 
 
 Two important results follow from this equation. The first is 
 derived from the fact that the dielectric constant of the ether is 
 defined as unity. Hence the velocity of an electromagnetic wave 
 in the ether is equal to the ratio of the electromagnetic to the 
 electrostatic units. Now this ratio of the units has been deter- 
 
 pm 
 mined by experiment and found to have the value 3 10 10 - 
 
 sec. 
 
 But this is the velocity of light in the free ether, in fact the two 
 numbers agree so closely that we can hardly regard it as a mere 
 coincidence, but are led to believe that light is an electromagnetic 
 vibration. 
 
 The second result relates to the index of refraction. In Chap- 
 ter VII this index has been defined as the ratio of the velocity of 
 light in ether to its velocity in the medium considered. Thus if 
 
 *Drude, Theory of Optics, p. 276, Longmans, 1902. 
 
 172 
 
THE TREND OF MODERN OPTICS 173 
 
 V represent the velocity in the medium, k' the dielectric con- 
 stant, and fji the index for infinitely long waves, we have from 
 equation (76), since k = 1, 
 
 i.e., the square of the index is equal to the dielectric constant. 
 This conclusion has also been tested by experiment. The first 
 results were, however, rather discouraging, for it was found that, 
 while the relation proved true for some substances, it was far 
 from correct for others. Thus for benzole we find /x, = 1.482, and 
 y/k' = 1.49; while for water, /u. = 1.33 and <STc' = 9.0. 
 
 It was some years before this discrepancy was explained in a 
 satisfactory way. It was suggested that the difficulty might be 
 due to the fact that vibrations of short period were used in deter- 
 mining the index for the visible spectrum from which the index 
 for infinitely long waves was calculated from a dispersion equa- 
 tion like that of Cauchy (cf. p. 91), while very long oscillations 
 were used in measuring k' . The objection to this explanation 
 was, that when there is a discrepancy, k' is always greater than 
 /x 2 , whereas, according to the experience of that time, the longer 
 the wave, the smaller the index ; so that for infinitely long waves 
 the index should be even smaller than it is for the visible spec- 
 trum, thus making the discrepancy worse. 
 
 It was at length discovered that the dispersion does not always 
 follow an equation like that of Cauchy; in short, it was found 
 that when the medium absorbs some of the radiations which fall 
 upon it, the dispersion law is quite different. Thus if, for exam- 
 ple, a spectrum be formed by a prism of cyanin, which absorbs the 
 yellow, it will be found that the red and orange rays are deflected 
 more than the green, blue, and violet, the two halves being separa- 
 ted by a dark band where the yellow should be. Such dispersion 
 is called anomalous. From this discovery it became at once clear 
 
174 MANUAL OP ADVANCED OPTICS 
 
 that we can not, in the case of absorbing media, calculate, with 
 the help of the dispersion equations which belong to transparent 
 media, the numerical value of the index of refraction on one side 
 of an absorption band from observations taken on the other side. 
 
 After this it was easy to surmise that substances like water, 
 for which Tc f is greater than /x 2 , have absorption bands which 
 correspond to periods of vibration between those of visible light 
 and those by which k' is measured, i.e., beyond the ultra red. 
 To prove this it is merely necessary to measure the index of 
 refraction for very long oscillations. This was done for water 
 by Drude,* and the result was a complete confirmation of 
 the theory, for he found that for such long waves the index of 
 refraction of water has the value 9.0. Hence the original state- 
 ment of Maxwell, namely /* = \/&', has been modified to read /x 
 can not be greater than */k'. If for any substance it is less, we 
 believe that we can with certainty predict that that substance 
 absorbs completely some of the radiations which correspond to 
 the ultra red. 
 
 But this discovery of the relation between absorption and 
 anomalous dispersion had an even greater result than that of 
 clearing away the discrepancy in Maxwell's original theory, for 
 the question, why should a substance absorb one particular set 
 of waves, led for its answer to the conception that the smallest 
 particles of such absorbing media must be free to vibrate, and 
 have natural periods of vibration which are the same as those of 
 the waves which they absorb. This conception was borrowed 
 from the known phenomena of absorption by vapors as mani- 
 fested in the Fraunhofer line in the solar spectrum. But the 
 assumption that the smallest particles are set in vibration by an 
 electromagnetic wave whose period coincides with their own nat- 
 ural period involves the idea that such particles must carry elec- 
 
 * Drude, Wied. Ann. 59, p. 17, 1896. 
 
THE TREND OF MODERN OPTICS 175 
 
 trie charges. Hence we reach a conception which is one of the 
 most important in modern science, namely, this: That the 
 particles of all substances which allow light to pass at all are 
 charged particles which have definite natural periods of vibration. 
 Such charged particles have been given the name ions. 
 
 This conception is not peculiar to optics, but is found to be 
 necessary in describing the phenomena of electrolysis, and has 
 proved itself equally indispensable in discussing the cathode rays 
 and the action of magnetism on light. 
 
 But how does this idea help our understanding of the facts of 
 normal dispersion? Because it can be shown* that the dielectric 
 constant of a medium composed of charged particles which have 
 natural periods of vibration, depends upon the ratio of those nat- 
 ural periods to that of the impressed vibration. When the value 
 of this ratio is not unity, i.e., when the impressed period and the 
 natural periods are not the same, the dielectric constant varies 
 continuously with the period, so that the curve which expresses 
 the relation between the two is continuous. Hence the curve 
 which expresses the relation between the index of refraction and 
 the wave length is also continuous, i.e., it agrees with the observed 
 dispersion curve. When, however, the impressed and the natural 
 periods coincide, then absorption takes place, and the dispersion 
 seems to become discontinuous at that point. 
 
 But this idea of charged particles which have natural periods 
 can be extended so as to give us some conception of the nature of 
 the particle. For it is not difficult to conceive that if by any 
 means we could set those ions into violent vibrations, they would 
 become a source of electromagnetic waves of the same period as 
 the natural periods of the particles. Now it is a well-known fact 
 that all solids, when heated to incandescence, send out waves of 
 all possible periods ; but that when the solid has been changed to 
 
 * Drude, Theory of Optics, p. 382. 
 
176 MANUAL OF ADVANCED OPTICS 
 
 a vapor, the vibrations sent out have certain particular periods 
 which are peculiar to the substance and characteristic of it. 
 These facts are the basis of spectrum analysis, and lead to two 
 conceptions which bear closely upon the general theory of optics. 
 
 For when these facts are considered in relation to the differ- 
 ence in the volumes occupied by the same quantity of a substance 
 in the solid and in the vapor state, we must recognize that the ions 
 are much closer together when they send out white light than 
 they are when they produce only certain definite vibrations. 
 Hence the conception has been formed that the ions, which by 
 their vibration produce light, are grouped into larger particles. 
 When the substance is in a solid state and heated to incandes- 
 cence, these larger particles are set into comparatively slow vibra- 
 tion. But since they are close- t-ogether, they will collide with one 
 another very frequently, i.e., each particle will not travel very far 
 before it meets another. At every such collision between two 
 particles each receives a blow which causes it to shiver so that the 
 ions which compose it are set into violent vibration. Now these 
 vibrations due to impact are not at first those corresponding to 
 the natural period of the ion, but are forced vibrations. Hence 
 they may have any period whatever. Such forced vibrations, 
 however, die down very rapidly, and the ions would then continue 
 to vibrate in their natural periods were it not for the fact that 
 they are so close together that the forced vibrations do not have 
 time to disappear between impacts. Hence white light may be 
 conceived to be due to the vibrations forced upon the ions by 
 the frequent impact of the particles in which they are grouped. 
 
 But when the substance is in the form of vapor, the particles 
 are farther apart, so that each particle travels very much farther 
 between impacts. In this case the time required for the forced 
 vibrations to die away is small in comparison with the time between 
 impacts, so that the ions send out waves of their natural periods 
 mainly. Hence the spectrum is no longer continuous but is com- 
 
THE TREND OF MODERN OPTICS 177 
 
 posed of a few lines only, namely, of those which correspond to 
 the natural periods of the ions. 
 
 Now we know that a simple sounding body, like a vibrating 
 string, sends out a series of vibrations, partial vibrations as they 
 are called, whose periods stand related to one another inversely as 
 the numbers 1, 2, 3, etc. When the geometrical form of the 
 sounding body is not so simple, the relation between the partial 
 vibrations becomes more complex; in fact, the theoretical solu- 
 tion of the case of even so simple a form as the ring has not yet 
 been worked out. Now in simple cases we can analyze the com- 
 pound sound vibrations which come to us into the partial vibra- 
 tions, and from that analysis draw certain conclusions as to the 
 nature of the sounding body. Can we do the same with the light 
 vibrations? Can we find a numerical relation between the periods 
 of vibration indicated by the bright lines in the spectrum of a 
 substance, and from that relation draw conclusions as to the 
 geometrical form of the vibrating ions? This is an extremely 
 interesting and pertinent question, for the determination of the 
 shape of what has been called an atom has been for ages one of 
 the favorite problems of the philosophers. 
 
 Now experiment has shown that there is a numerical relation 
 between the periods of vibration of the lines of each of a large 
 number of spectra. That this is not mere chance is shown by the 
 fact that this numerical relationship has led to the discovery of 
 new lines whose position in the series had been calculated but 
 which had not before been observed. Investigation has also 
 established relationships between the groups of the Mendeleeff 
 series. Thus the elements of the first and third groups (Li. , Na. , 
 K., Rb., Cs., Mg., Ca., Sr.) are characterized by doublets, while the 
 elements of the second group (Cu., Ag.) show series of triplets. 
 But, although an equation has been found* which expresses 
 
 *Kayser and Runge, Abh. d. k. Preuss., Akad. d. Wiss., 1888-1893. 
 Winkelruann's Handbuch, II, 1, p. 441 seq. 
 
178 MANUAL OF ADVANCED OPTICS 
 
 numerically the relation between the periods of vibration, the 
 geometrical form of the object that would be capable of sending 
 out such a series of partial vibrations has not yet been deter- 
 mined. 
 
 The problem is, however, by no means hopeless, and every new 
 detail discovered in the spectrum of any substance adds its share, 
 be that little or great, to the general store of information. The 
 importance of the detailed study of spectra is thus made clear, 
 for the more detailed our knowledge of the vibrations of the ions, 
 the nearer we come to being able to form a conception of their 
 size and structure. Hence the value of increasing in every way 
 the power of spectroscopic apparatus becomes manifest, and we 
 can appreciate the bearing of the chapter on visibility curves, for 
 this is at present the most refined method of light wave analysis, 
 i.e., the method which furnishes the most detailed information as 
 to the exact nature of any complex set of light vibrations. 
 
 Another factor which helps to emphasize the usefulness of 
 increased spectroscopic power is the recent discovery of the action 
 of magnetism on the vibrations of the ions. Faraday had sus- 
 pected that the magnetic field should act upon the vibrations of 
 light, and discovered the magnetic rotation of the plane of polari- 
 zation, but he was unable to find that the field produced any 
 change in the period of vibration of the light. The effect was 
 discovered a few years ago (1897) by Zeeman, who found that 
 each line of a spectrum seems, when the source of light is placed 
 between the poles of a powerful magnet, to be separated into 
 three lines, the two outer being polarized in one plane, and the 
 middle one in a plane at right angles. 
 
 The theoretical explanation of this phenomenon is as simple 
 as it is beautiful. For we have come to believe, as has been 
 stated, that the light waves are caused by vibrating ions, and, in 
 the general case, the paths which such vibrating ions follow are 
 ellipses. But an elliptical motion can be resolved into a circular 
 
THE TREND OF MODERN OPTICS 179 
 
 motion in one plane and a rectilinear motion in a plane at right 
 angles. Hence we may assume that the vibration forms of the 
 ions are thus resolved into circles in planes at right angles to the 
 lines of force of the field, and straight lines parallel to those 
 lines. Xow a moving electric charge is equivalent to an electric 
 current, and, as is well known, a magnet attracts or repels an 
 electric current which flows at right angles to the lines of force of 
 the field according to the direction of the current. Since we 
 have resolved the vibrations into circles whose planes are perpen- 
 dicular to the lines of the field and straight lines parallel to those 
 lines, only the circular vibrations will be affected by the field. 
 But since some of the ions are rotating in one direction and others 
 in the opposite direction, the effect of the field upon them will be 
 different in the two cases. For if the field act upon those rotating 
 in one direction to repel the ion, it will act with an attraction upon 
 those rotating in the opposite direction. If, therefore, in the first 
 case, the circular path of the ion is made larger, and hence the 
 period of vibration increased, in the second case the path will be 
 contracted, reducing the period of the oscillation. Since these 
 vibrations are in the plane perpendicular to the lines of force of 
 the field, they will appear to an observer looking at right angles 
 to those lines like plane vibrations. Hence, if the lines of force 
 be horizontal, these vibrations will appear to be vertical, and the 
 observer will perceive two plane polarized vibrations whose periods 
 are slightly different from that of the original vibration, one being 
 somewhat smaller, the other somewhat greater. 
 
 But the period of the vibrations parallel to the lines of force 
 of the field is not altered by the field. Hence the observer should 
 also perceive a plane polarized vibration whose oscillations are 
 horizontal. Hence we can conceive how a single line can be sep- 
 arated into three, in two of which the vibrations are vertical and 
 the period of vibration altered, while in the third the vibrations 
 are horizontal and of the same period as that of the original light. 
 
180 MANUAL OF ADVANCED OPTICS 
 
 If, however, an observer looks along the lines of force of the 
 field he should perceive two circular vibrations rotating in oppo- 
 site directions. Observation confirms this conclusion also. 
 
 But while observation confirms in general these results, it has 
 nevertheless been found that when the analysis of the vibrations 
 is carried farther, the phenomenon is not so simple. For it has 
 been shown* that a single line is not merely separated into three, 
 as has been described, but that each of the three may itself be 
 compound. Furthermore, the complexity of the lines is different 
 for different lines in the spectrum of the same element. Thus in 
 the case of cadmium the three lines into which the red radiation 
 is separated are all single ; while the green radiation breaks into a 
 triplet whose vibrations are horizontal and two quadruplets whose 
 vibrations are vertical ; and in the case of the blue radiation, all 
 the lines are doublets. These details have not yet been satisfac- 
 torily accounted for in the theory. The entire subject is one of 
 great interest, and promises well to become an important factor 
 in helping us to form conceptions of the geometrical construction 
 of the ions. Since the effects here described are so small that 
 they can be detected only by the highest spectroscopic power at 
 present available, the value of increasing the resolving power of 
 the spectroscope becomes again manifest. A very decided 
 advance has recently been made in this direction by Michelsonf in 
 the invention of his echelon spectroscope. 
 
 Another very interesting conclusion has been reached from the 
 consideration of this effect of the magnetic field upon the vibra- 
 tion of ions. For it is clear that the magnitude of these effects 
 depends upon two factors, namely, upon the strength of the 
 electric charge carried by the ion, and upon the mass of the 
 ion. Thus the greater the charge, the greater will be the 
 
 *Michelson, Phil. Mag. (5) 44, P- 109, 1897; 45, p. 348, 1898. 
 fMichelson, Am. Jour. Sci. (4) 5, p. 215; Astrophys. Jour. 8, p. 37, 
 1898. 
 
THE TREND OF MODERN OPTICS 181 
 
 effect, and the greater the mass, the smaller will be the effect. 
 Hence if e represent the charge, and m the mass of the ion, the 
 
 /> 
 
 effect will be proportional to Now there are other phenomena 
 
 in the description of which this factor - - appears, namely, elec- 
 trolysis and the cathode rays. For in the case of the former the 
 action will evidently be more rapid the greater the charge and 
 the less the mass of the ion; and the deflection of the cathode 
 rays by the magnetic field will, upon the assumption that those 
 rays are caused by ions shot out from the cathode, be greater the 
 greater the charge and the less the mass of the ion. 
 
 Unfortunately these phenomena do not permit of an inde- 
 pendent determination of either e or m, but lead only to a value 
 of their ratio. Xow the value of this ratio for hydrogen, as 
 determined from electrolysis, is about 10*, while its value, as 
 determined from the effects of the magnetic field upon the 
 cathode rays and the light vibrations, is about 1.7 10 7 . The 
 difference in the two cases may be accounted for in three ways : 
 namely, either e is the same in the three cases, while m is greater 
 in electrolysis; or the /?i's are the same, but the e's are different 
 in the three cases; or both e and m vary in the different phe- 
 nomena. Hence it is an interesting problem to find a means of 
 obtaining independent determinations of the two quantities. 
 
 This has not yet been done without making assumptions as to 
 the number of particles in a given volume at a given pressure. 
 But it is found that, if we take for that number the one to which 
 the kinetic theory of gases leads us, we find as the charge of the 
 univalent ion in electrolysis 1.29 10~ 10 .* J. J. Thomson f has 
 calculated from observations upon ions in vacuo a number of the 
 
 * Drude, Theory of Optics, p. 532. 
 
 t J. J. Thomson, Phil. Mag. (5) 44, p. 293, 1897; 46, p. 528, 1898; ?, 
 p. 547, 1899. 
 
182 MANUAL OF ADVANCED OPTICS 
 
 same order, namely, 6.7 10~ 10 , as the value of the charge upon an 
 ion under those conditions. Hence the conclusion seems probable 
 that the charges are, in the two cases, nearly the same, but that 
 the masses of th$ ions in electrolysis are greater than those of the 
 ions which take part in the phenomena of the cathode rays and of 
 the action of magnetism on light. 
 
 Hence physicists have come to believe that the particles whose 
 vibrations cause light are parts of an atom, that the time-hon- 
 ored atom, which to the chemist is indivisible, is in reality com- 
 pounded of a veiy large number of smaller ions to which the 
 special name electron has been given. The calculations are not 
 exact, but there is good reason for the belief that the hydrogen 
 atom is made up of something like one thousand of these elec- 
 trons, and that the numbers in the other chemical atoms are 
 roughly proportional to the atomic weights. According to this 
 conception an atom of mercury would be composed of about 
 200,000 electrons. 
 
 Such ideas as these tax the powers of conception of the human 
 mind to the limit. For, if it is difficult to form a mental image 
 of an atom, which is so minute that we can not even with the best 
 microscope see a group of several hundred of them, is it not bold 
 of us to attempt to realize that such atoms are themselves highly 
 complex? 
 
 Hence we can see that the science of optics is not a worn-out, 
 barren field of investigation, but that it now throbs with vigorous 
 life, and that the conceptions of Nature which it has developed 
 under the stimulus of the electromagnetic theory have led, and 
 probably will always lead, the human mind to the very outermost 
 boundaries of the knowable and the conceivable. 
 
APPENDIX 
 
 A. SOURCES OF LIGHT 
 
 One of the most useful sources of monochromatic light is the 
 sodium flame. Such a flame can readily be produced by heating a 
 piece of hard glass tubing in the flame of a Bunsen burner. The 
 tubing can be supported in the flame by a piece of iron wire, the 
 other end of the wire being wound around the burner. Or a piece 
 of asbestos which has been soaked in a strong solution of one of 
 the sodium salts can be tied about the upper end of the burner. 
 This sort of flame is very convenient in the interferometer work, 
 especially if there be mounted on the same base an ordinary white- 
 light burner with a cock, so that the sodium light can be replaced 
 by white light by merely turning the cock. 
 
 The objection to this form of sodium burner is that the light 
 which it furnishes is too faint for experiments like those of the 
 Fresnel. mirrors, in which all the light used has to pass through a 
 narrow slit. A brighter sodium light can be obtained by heating 
 the hard glass in the flame of an oxyhydrogen blow-pipe. 
 
 The most satisfactory light for these experiments is, however, 
 a portion of the solar spectrum. To obtain this it is merely 
 necessary to pass the sunlight through an ordinary spectrometer 
 and allow the solar spectrum to fall upon the slit which acts as 
 the source of light. A simple device for producing such a spec- 
 trum is shown in Fig. 41. Sunlight passes through the slit $and 
 then through the upper half of the lens L which renders it paral- 
 lel. It then traverses the prism P, is reflected by the mirror Jf, 
 and returns through the lower half of the prism and the lens, and 
 forms a spectrum below the slit. A small total reflection prism p 
 
 183 
 
184 APPENDIX 
 
 turns the light to one side so that the spectrum is formed in a 
 position convenient for observation. The advantage of this form 
 of instrument is that it gives a rather large dispersion with* only 
 
 FIGURE 41 
 
 one prism, and that it allows the different parts of the spectrum 
 to be brought upon the slit of the instrument being used, by 
 merely changing the position of the mirror M. Such a spectro- 
 scope is easily constructed, as it does not require the careful 
 adjustment usually necessary in this kind of instrument. 
 
 The vacuum tubes are useful for furnishing the cadmium and 
 mercury light,* their success depending upon their having been 
 sealed when the vapor pressure in them is just correct. Expe- 
 rience has shown that the pressure is correct when the tube, 
 excited by the electric spark, shows stratifications about 1 mm. 
 apart. The amount of cadmium or mercury needed is small, 
 about the size of a pin head. 
 
 The cadmium tubes have to be heated to about 270 C. before 
 the cadmium vaporizes. To make the heating uniform the tube 
 should be inclosed in a heavy brass box in which is a small win- 
 dow covered with mica. The mercury tubes have to be warmed 
 somewhat, not over 100 C. A metal box is not necessary with 
 them, although it is desirable to have one to keep the temper- 
 ature fairly uniform throughout the entire tube. Empty tubes 
 can be purchased from any good glass-blower. 
 
 Filtered sunlight is sufficiently monochromatic for many 
 
 4 * These can be purchased from William Gaertner, 5347 Lake Avenue, 
 Chicago, or can be readily made. 
 
APPEXDIX 
 
 185 
 
 experiments, notably those in polarized light. The following 
 solutions, recommended by Landolt,* will be found to be satisfac- 
 tory: 
 
 COLOR 
 
 THICKNESS or 
 LAYER IN mm. 
 
 AQUEOUS SOLUTION OF 
 
 GRAMS IN 
 100 cc. 
 
 AVERAGE 
 
 A -10 
 
 Red 
 
 20 
 
 Crystal violet 5 BO 
 
 005 
 
 656 
 
 Green 
 
 20 
 20 
 
 Potassium chromate . 
 Copper chloride 
 
 10 
 '60 
 
 533 
 
 Blue 
 
 20 
 
 20 
 
 Potassium chroinate. 
 Crystal violet 
 
 10 
 0005 
 
 448 
 
 
 20 
 
 Copper sulphate .... 
 
 15 
 
 
 B. SILVERING OP OPTICAL SURFACES 
 The optical surfaces used in interferometer work have to be 
 
 coated on their front faces with silver. One of the simplest 
 
 methods of doing this is the following: 
 Prepare two solutions as follows : 
 
 Silver nitrate 5 gm. 
 
 Distilled water 40 cc. 
 
 To this add ammonia slowly until the precipitate which is at 
 first formed is nearly redissolved. The success of the solution 
 depends upon leaving an excess of the precipitate. If a drop too 
 much ammonia has been added, a small crystal of silver nitrate 
 must be put in to bring back traces of the precipitate. When 
 the solution is right it will look like slightly muddy water. Then 
 dilute to 500 cc. and filter. 
 
 B 
 
 Silver nitrate 1 gm. 
 
 Rochelle salt (sodium-potassium tartrate) 0. 83 gm. 
 
 Disti lied water . . . . 500 cc. 
 
 * Landolt, Optische Drehungsvermogen, Braunschweig, 1898, p. 390. 
 
186 APPENDIX 
 
 Bring the solution to a boil and filter hot. It must be cooled 
 to the temperature of the room before it is used. 
 
 For silvering use equal parts of A and B. 
 
 The essential of success in silvering is, besides the ammonia in 
 solution A, cleanliness. The following process of cleaning the 
 surfaces to be silvered is recommended : 
 
 1. Kemove wax, if there is any, with spirits of turpentine. 
 
 2. Wash off the turpentine with soap and water. 
 
 3. Place the surfaces to be silvered in a glass or porcelain 
 tray, and remove any remaining silver with strong nitric acid. 
 
 4. Rinse well in running water. 
 
 5. Wash in a strong solution of caustic potash. During this 
 washing the plates and the dish which holds them should be 
 rubbed hard with a tuft of cotton or a piece of pure gum tubing 
 on the end of a glass rod. The success of the process depends 
 largely upon the thoroughness of this washing. 
 
 6. Pour off the solution of caustic potash and rinse well in 
 running water, being careful not to touch either the surfaces or 
 the inside of the dish with the fingers. A very minute trace of 
 grease will make the film streaked. 
 
 7. Wash in strong nitric acid. 
 
 8. Wash in running water for five minutes or more, raising 
 the surfaces with a glass rod to allow the water to run beneath 
 them. 
 
 9. Rinse in several changes of distilled water. 
 
 Now mix the two solutions and pour over the surfaces. If 
 only a thin coat is desired, the deposit must be watched, and the 
 surface removed when the film has the necessary thickness. The 
 opaque films should remain in the solution till it turns black. 
 The surfaces are then removed and set up on edge on filter paper 
 to dry. When dry they may be polished by rubbing them gently 
 on a piece of chamois skin laid on the table and covered with 
 jewelers' rouge. The transparent films can not be polished, for a 
 
APPENDIX 
 
 187 
 
 slight touch will rub the coating off entirely. If the solution is 
 successful the opaque films will be so hard that they can not be 
 rubbed off with the finger. 
 
 Table 1 
 WAVE LENGTHS OF SOME OF THE IMPORTANT LINES 
 
 SOLAR LINE 
 
 SUBSTANCE 
 
 A 107 
 
 SOLAR LIXE 
 
 SUBSTANCE 
 
 A - 107 
 
 A 
 
 
 7600 
 
 
 Tl. 
 
 5350 
 
 B 
 
 0. 
 
 6870 
 
 E 
 
 Ca. 
 
 5270 
 
 C 
 
 H. 
 
 6563 
 
 b 
 
 Mg. 
 
 5173 
 
 
 Cd. 
 
 6438 
 
 
 Cd. 
 
 5086 
 
 Di 
 
 Na 
 
 5896 
 
 F 
 
 H. 
 
 4861 
 
 D* 
 
 Na. 
 
 5890 
 
 
 Cd. 
 
 4800 
 
 
 Hg. 
 
 5790 
 
 
 Hg. 
 
 4358 
 
 
 Hg. 5770 
 
 G 
 
 Fe. Ca. 
 
 4308 
 
 
 Hg. 5461 
 
 H 
 
 H. Ca. 
 
 3968 
 
 Table 2 
 
 INDICES OF REFRACTION 
 
 SOLAS LINK 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 F 
 
 G 
 
 H 
 
 Water 
 
 1 3293 
 
 3309 
 
 3317 
 
 3335 
 
 3358 
 
 3377 
 
 3412 
 
 .3441 
 
 Alcohol 
 
 1.3586 
 
 .3599 
 
 .3606 
 
 .3624 
 
 .3647 
 
 .3667 
 
 .3705 
 
 .3736 
 
 Carbon bisulphide 
 Cassia oil 
 
 1.6103 
 1 5861 
 
 .6166 
 .5920 
 
 .6198 
 5962 
 
 .6293 
 .6053 
 
 .6421 
 .6191 
 
 .6541 
 .6340 
 
 .6786 
 .6652 
 
 .7016 
 .7010 
 
 Crown glass 
 
 1.5099 
 
 .5118 
 
 .5127 
 
 .5153 
 
 .5186 
 
 .5214 
 
 .5267 
 
 .5312 
 
 Flint glass 
 
 1.7351 
 
 .7406 
 
 .7434 
 
 .7515 
 
 .7623 
 
 .7723 
 
 .7922 
 
 .8110 
 
 Table 3 
 
 USEFUL NUMBERS 
 
 of the natural system of logarithms e = 2.7183; log e = .43429 
 Modulus of the natural logarithms M=^~^= 2.3026; log M =. 
 
 Angle whose arc is equal to the radius = 57. 2958 = 206265". 
 
 logs 1.75812 5.31442. 
 Index of refraction of air = 1.00029. 
 
188 
 
 APPENDIX 
 
 NATURAL SINES 
 
 
 
 
 
 
 
 
 
 
 
 
 Complement 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Difference 
 
 
 
 0.0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 0175 
 
 89 
 
 1 
 
 0175 
 
 0192 
 
 0209 
 
 0227 
 
 0244 
 
 0262 
 
 0279 
 
 0297 
 
 0314 
 
 0332 
 
 0349 
 
 88 
 
 2 
 
 0349 
 
 0366 
 
 0384 
 
 0401 
 
 0419 
 
 0436 
 
 0454 
 
 0471 
 
 0488 
 
 0506 
 
 0523 
 
 87 
 
 8 
 
 0523 
 
 0541 
 
 0558 
 
 0576 
 
 0593 
 
 0610 
 
 0628 
 
 0645 
 
 0663 
 
 0680 
 
 0698 
 
 86 
 
 4 
 
 0698 
 
 0715 
 
 0732 
 
 0750 
 
 0767 
 
 0785 
 
 0802 
 
 0819 
 
 0837 
 
 0854 
 
 0872 
 
 85 
 
 5 
 
 0.0872 
 
 0889 
 
 0906 
 
 0924 
 
 0941 
 
 0958 
 
 0976 
 
 0993 
 
 1011 
 
 1028 
 
 1045 
 
 84 
 
 6 
 
 1045 
 
 1063 
 
 1080 
 
 1097 
 
 1115 
 
 1132 
 
 1149 
 
 1167 
 
 1184 
 
 1201 
 
 1219 
 
 83 
 
 7 
 
 1219 
 
 1236 
 
 1253 
 
 1271 
 
 1288 
 
 1305 
 
 1323 
 
 1340 
 
 1357 
 
 1374 
 
 1392 
 
 82 
 
 8 
 
 1392 
 
 1409 
 
 1426 
 
 1444 
 
 1461 
 
 1478 
 
 1495 
 
 1513 
 
 1530 
 
 1547 
 
 1564 
 
 81 
 
 9 
 
 1564 
 
 1582 
 
 1599 
 
 1616 
 
 1633 
 
 1650 
 
 1668 
 
 1685 
 
 1702 
 
 1719 
 
 1736 
 
 80 
 
 10 
 
 0.1736 
 
 1754 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 1908 
 
 79 
 
 11 
 
 1908 
 
 1925 
 
 1942 
 
 1959 
 
 1977 
 
 1994 
 
 2011 
 
 2028 
 
 2045 
 
 2062 
 
 2079 
 
 78 
 
 12 
 
 2079 
 
 2096 
 
 2113 
 
 2130 
 
 2147 
 
 2164 
 
 2181 
 
 2198 
 
 2215 
 
 2233 
 
 2250 
 
 77 17 
 
 13 
 
 2250 
 
 2267 
 
 2284 
 
 2300 
 
 2317 
 
 2334 
 
 2351 
 
 2368 
 
 2385 
 
 2402 
 
 2419 
 
 76 
 
 14 
 
 2419 
 
 2436 
 
 2453 
 
 2470 
 
 2487 
 
 2504 
 
 2521 
 
 2538 
 
 2554 
 
 2571 
 
 2588 
 
 75 
 
 15 
 
 0.2588 
 
 2605 
 
 2622 
 
 2639 
 
 2656 
 
 2672 
 
 2689 
 
 2706 
 
 2723 
 
 2740 
 
 2756 
 
 74 
 
 16 
 
 2756 
 
 2773 
 
 2790 
 
 2807 
 
 2823 
 
 2840 
 
 2857 
 
 2874 
 
 2890 
 
 2907 
 
 2924 
 
 73 
 
 17 
 
 2924 
 
 2940 
 
 2957 
 
 2974 
 
 2990 
 
 3007 
 
 3024 
 
 3040 
 
 3057 
 
 3074 
 
 3090 
 
 72 
 
 18 
 
 3090 
 
 3107 
 
 3123 
 
 3140 
 
 3156 
 
 3173 
 
 3190 
 
 3206 
 
 3223 
 
 3239 
 
 3256 
 
 71 
 
 19 
 
 3256 
 
 3272 
 
 3289 
 
 3305 
 
 3322 
 
 3338 
 
 3355 
 
 3371 
 
 3387 
 
 3404 
 
 3420 
 
 70 
 
 20 
 
 0.3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 3518 
 
 3535 
 
 3551 
 
 3567 
 
 3584 
 
 69 
 
 21 
 
 3584 
 
 3600 
 
 3616 
 
 3633 
 
 3649 
 
 3665 
 
 3681 
 
 3697 
 
 3714 
 
 3730 
 
 3746 
 
 68 
 
 22 
 
 3746 
 
 3762 
 
 3778 
 
 3795 
 
 3811 
 
 3827 
 
 3843 
 
 3859 
 
 3875 
 
 3891 
 
 3907 
 
 67 
 
 23 
 
 3907 
 
 3923 
 
 3939 
 
 3955 
 
 3971 
 
 3987 
 
 4003 
 
 4019 
 
 4035 
 
 4051 
 
 4067 
 
 66 16 
 
 24 
 
 4067 
 
 4083 
 
 4099 
 
 4115 
 
 4131 
 
 4147 
 
 4163 
 
 4179 
 
 4195 
 
 4210 
 
 4226 
 
 65 
 
 25 
 
 0.4226 
 
 4242 
 
 4258 
 
 4274 
 
 4289 
 
 4305 
 
 4321 
 
 4337 
 
 4352 
 
 4368 
 
 4384 
 
 64 
 
 26 
 
 4384 
 
 4399 
 
 4415 
 
 4431 
 
 4446 
 
 4462 
 
 4478 
 
 4493 
 
 4509 
 
 4524 
 
 4540 
 
 63 
 
 27 
 
 4540 
 
 4555 
 
 4571 
 
 4586 
 
 4602 
 
 4617 
 
 4633 
 
 4648 
 
 4664 
 
 4679 
 
 4695 
 
 62 
 
 28 
 
 4695 
 
 4710 
 
 4726 
 
 4741 
 
 4756 
 
 4772 
 
 4787 
 
 4802 
 
 4818 
 
 4833 
 
 4848 
 
 61 
 
 29 
 
 4848 
 
 4863 
 
 4879 
 
 4894 
 
 4909 
 
 4924 
 
 4939 
 
 4955 
 
 4970 
 
 4985 
 
 5000 
 
 60 
 
 30 
 
 0.5000 
 
 5015 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 5150 
 
 59 15 
 
 31 
 
 5150 
 
 5165 
 
 5180 
 
 5195 
 
 5210 
 
 5225 
 
 5240 
 
 5255 
 
 5270 
 
 5284 
 
 5299 
 
 58 
 
 32 
 
 5299 
 
 5314 
 
 5329 
 
 5344 
 
 5358 
 
 5373 
 
 5388 
 
 5402 
 
 5417 
 
 5432 
 
 5446 
 
 57 
 
 33 
 
 5446 
 
 5461 
 
 5476 
 
 5490 
 
 5505 
 
 5519 
 
 5534 
 
 5548 
 
 5563 
 
 5577 
 
 5592 
 
 56 
 
 34 
 
 5592 
 
 5606 
 
 5621 
 
 5635 
 
 5650 
 
 5664 
 
 5678 
 
 5693 
 
 5707 
 
 5721 
 
 5736 
 
 55 
 
 35 
 
 0.5736 
 
 5750 
 
 5764 
 
 5779 
 
 5793 
 
 5807 
 
 5821 
 
 5835 
 
 5850 
 
 5864 
 
 5878 
 
 54 
 
 36 
 
 5878 
 
 5892 
 
 5906 
 
 5920 
 
 5934 
 
 5948 
 
 5962 
 
 5976 
 
 5990 
 
 6004 
 
 6018 
 
 63 
 
 37 
 
 6018 
 
 6032 
 
 6046 
 
 6060 
 
 6074 
 
 6088 
 
 6101 
 
 6115 
 
 6129 
 
 6143 
 
 6157 
 
 52 
 
 38 
 
 6157 
 
 6170 
 
 6184 
 
 6198 
 
 6211 
 
 6225 
 
 6239 
 
 6252 
 
 6266 
 
 6280 
 
 6293 
 
 51 
 
 39 
 
 6293 
 
 6307 
 
 6320 
 
 6334 
 
 6347 
 
 6361 
 
 6374 
 
 6388 
 
 6401 
 
 6414 
 
 6428 
 
 50 
 
 40 
 
 0.6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 6561 
 
 49 
 
 41 
 
 6561 
 
 6574 
 
 6587 
 
 6600 
 
 6613 
 
 6626 
 
 6639 
 
 6652 
 
 6665 
 
 6678 
 
 6691 
 
 48 13 
 
 42 
 
 6691 
 
 6704 
 
 6717 
 
 6730 
 
 6743 
 
 6756 
 
 6769 
 
 6782 
 
 6794 
 
 6807 
 
 6820 
 
 47 
 
 43 
 
 6820 
 
 6833 
 
 6845 
 
 6658 
 
 6871 
 
 6884 
 
 6896 
 
 6909 
 
 6921 
 
 6934 
 
 6947 
 
 46 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7034 
 
 7046 
 
 7059 
 
 7071 
 
 45 
 
 Complement 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 .4 
 
 .3 
 
 .2 
 
 .1 
 
 .0 
 
 Angle 
 
 NATURAL COSINES 
 
APPENDIX 
 
 189 
 
 NATURAL SINES 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 A 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Complement 
 Difference 
 
 45 
 
 0.7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7145 
 
 7157 
 
 7169 
 
 7181 
 
 7193 
 
 44 
 
 46 
 
 7193 
 
 7206 
 
 7218 
 
 7230 
 
 7242 
 
 7254 
 
 7266 
 
 7278 
 
 7290 
 
 7302 
 
 7314 
 
 43 12 
 
 47 
 
 7314 
 
 7325 
 
 7337 
 
 7349 
 
 7361 
 
 7373 
 
 7385 
 
 7396 
 
 7408 
 
 7420 
 
 7431 
 
 42 
 
 48 
 
 7431 
 
 7443 
 
 7455 
 
 7466 
 
 7478 
 
 7490 
 
 7501 
 
 7513 
 
 7524 
 
 7536 
 
 7547 
 
 41 
 
 49 
 
 7547 
 
 7559 
 
 7570 
 
 7581 
 
 7593 
 
 7604 
 
 7615 
 
 7627 
 
 7638 
 
 7649 
 
 7660 
 
 40 
 
 50 
 
 0.7660 
 
 7672 
 
 7683 
 
 7694 
 
 7705 
 
 7716 
 
 7727 
 
 7738 
 
 7749 
 
 7760 
 
 7771 
 
 39 
 
 51 
 
 7771 
 
 7782 
 
 7793 
 
 7804 
 
 7815 
 
 7826 
 
 7837 
 
 7848 
 
 7859 
 
 7869 
 
 7880 
 
 38 
 
 52 
 
 7880 
 
 7891 
 
 7902 
 
 7912 
 
 7923 
 
 7934 
 
 7944 
 
 7955 
 
 7965 
 
 7976 
 
 7986 
 
 37 
 
 53 
 
 7986 
 
 7997 
 
 8007 
 
 8018 
 
 8028 
 
 8039 
 
 8049 
 
 8059 
 
 8070 
 
 8080 
 
 8090 
 
 36 
 
 54 
 
 8090 
 
 8100 
 
 8111 
 
 8121 
 
 8131 
 
 8141 
 
 8151 
 
 8161 
 
 8171 
 
 8181 
 
 8192 
 
 35 
 
 55 
 
 0.8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 8290 
 
 34 * 
 
 56 
 
 8290 
 
 8300 
 
 8310 
 
 8320 
 
 8329 
 
 8339 
 
 8348 
 
 8358 
 
 8368 
 
 8377 
 
 8387 
 
 33 
 
 57 
 
 8387 
 
 8396 
 
 8406 
 
 8415 
 
 8425 
 
 8434 
 
 8443 
 
 8453 
 
 8462 
 
 8471 
 
 8480 
 
 32 
 
 58 
 
 8480 
 
 8490 
 
 8499 
 
 8508 
 
 8517 
 
 8526 
 
 8536 
 
 8545 
 
 8554 8563 
 
 8572 
 
 31 
 
 59 
 
 8572 
 
 8581 
 
 8590 
 
 8599 
 
 8607 
 
 8616 
 
 8625 
 
 8634 
 
 8643 
 
 8652 
 
 8660 
 
 30 9 
 
 60 
 
 0.8660 
 
 8669 
 
 8678 
 
 8686 
 
 8695 
 
 8704 
 
 8712 
 
 8721 
 
 8729 
 
 8738 
 
 8746 
 
 29 
 
 61 
 
 8746 
 
 8755 
 
 8763 
 
 8771 
 
 8780 
 
 8788 
 
 8796 
 
 8805 
 
 8818 
 
 8821 
 
 8829 
 
 28 
 
 62 
 
 8829 
 
 8838 
 
 8846 
 
 8854 
 
 s*02 
 
 8870 
 
 8878 
 
 8886 8894 
 
 8902 
 
 8910 
 
 27 8 
 
 63 
 
 8910 
 
 8918 
 
 8926 
 
 8934 
 
 8942 
 
 8949 
 
 8957 
 
 8965 
 
 8973 
 
 8980 
 
 8988 
 
 26 
 
 64 
 
 8988 
 
 8996 
 
 9003 
 
 9011 
 
 9018 
 
 9026 
 
 9033 
 
 9041 
 
 9048 
 
 9056 
 
 9063 
 
 25 
 
 65 
 
 0.9063 
 
 9070 
 
 9078 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 9135 
 
 24 
 
 .66 
 
 9135 
 
 9143 
 
 9150 
 
 9157 
 
 9164 
 
 9171 
 
 9178 
 
 9184 
 
 9191 
 
 9198 
 
 9205 
 
 23 * 
 
 67 
 
 9205 
 
 9212 
 
 9219 
 
 9225 
 
 9232 
 
 9239 
 
 9245 
 
 9252 
 
 9259 
 
 9265 
 
 9272 
 
 22 
 
 68 
 
 9272 
 
 9278 
 
 9285 
 
 9291 
 
 9298 
 
 9304 
 
 9311 
 
 9317 
 
 9323 
 
 9330 
 
 9336 
 
 21 
 
 69 
 
 9336 
 
 9342 
 
 9348 
 
 9354 
 
 9361 
 
 9367 
 
 9373 
 
 9379 
 
 9385 
 
 9391 
 
 9397 
 
 20 6 
 
 70 
 
 0.9397 
 
 9403 
 
 9409 
 
 9415 
 
 9421 
 
 9426 
 
 9432 
 
 9438 
 
 9444 
 
 9449 
 
 9455 
 
 19 
 
 71 
 
 9455 
 
 9461 
 
 9466 
 
 9472 
 
 9478 
 
 9483 
 
 9489 
 
 9494 
 
 9500 
 
 9505 
 
 9511 
 
 18 
 
 72 
 
 9511 
 
 9516 
 
 9521 
 
 9527 
 
 9532 
 
 9537 
 
 9542 
 
 9548 
 
 9553 
 
 9558 
 
 9563 
 
 17 
 
 73 
 
 9563 
 
 9568 
 
 9573 
 
 9578 
 
 9583 
 
 9588 
 
 9593 
 
 9598 
 
 9603 
 
 9608 
 
 9613 
 
 16 5 
 
 74 
 
 9613 
 
 9617 
 
 9622 
 
 9627 
 
 9632 
 
 9636 
 
 9641 
 
 9646 
 
 9650 
 
 9655 
 
 9659 
 
 15 
 
 75 
 
 0.9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 9703 
 
 14 
 
 76 
 
 9703 
 
 9707 
 
 9711 
 
 9715 
 
 9720 
 
 9724 
 
 9728 
 
 9732 
 
 9736 
 
 9740 
 
 9744 
 
 13 * 
 
 77 
 
 9744 
 
 9748 
 
 9751 
 
 9755 
 
 9759 
 
 9763 
 
 9767 
 
 9770 
 
 9774 
 
 9778 
 
 9781 
 
 12 
 
 78 
 
 9781 
 
 9785 
 
 9789 
 
 9792 
 
 9796 
 
 9799 
 
 9803 
 
 9806 
 
 9810 
 
 9813 
 
 9816 
 
 11 
 
 79 
 
 9816 
 
 9820 
 
 9823 
 
 9826 
 
 9829 
 
 9833 
 
 9836 
 
 9839 
 
 9842 
 
 9845 
 
 9848 
 
 10 
 
 80 
 
 0.9848 
 
 9851 
 
 9854 
 
 9857 
 
 9860 
 
 9863 
 
 9866 
 
 9869 
 
 9871 
 
 9874 
 
 9877 
 
 9 3 
 
 81 
 
 9877 
 
 9880 
 
 uss2 
 
 9885 
 
 9888 
 
 9890 
 
 9893 
 
 9895 
 
 9898 
 
 9900 
 
 9903 
 
 8 
 
 82 
 
 9903 
 
 9905 
 
 9907 
 
 9910 
 
 9912 
 
 9914 
 
 9917 
 
 9919 
 
 9921 
 
 9923 
 
 9925 
 
 7 
 
 83 
 
 9925 
 
 9928 
 
 9930 
 
 9932 
 
 9934 
 
 9936 
 
 9938 
 
 9940 
 
 9942 
 
 9943 
 
 9945 
 
 6 2 
 
 84 
 
 9945 
 
 9947 
 
 9949 
 
 9951 
 
 9952 
 
 9954 
 
 9956 
 
 9957 
 
 9959 
 
 9960 
 
 9962 
 
 5 
 
 85 
 
 0.9962 
 
 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 9976 
 
 4 
 
 86 
 
 9976 
 
 9977 
 
 9978 
 
 9979 
 
 9980 
 
 9981 
 
 9982 
 
 9983 
 
 9984 
 
 9985 
 
 9986 
 
 3 i 
 
 87 
 
 9986 
 
 9987 
 
 9988 
 
 9989 
 
 9990 
 
 9990 
 
 9991 
 
 9992 
 
 9993 
 
 9993 
 
 9994 
 
 2 
 
 88 
 
 9994 
 
 9995 9995 
 
 9996 
 
 9996 
 
 9997 
 
 9997 
 
 9997 
 
 9998 
 
 9998 
 
 9998 
 
 1 
 
 89 
 
 9998 
 
 9999 9999 
 
 9999 
 
 9999 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 
 
 Complement 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 .4 
 
 .3 
 
 .2 
 
 .1 
 
 .0 
 
 Angle 
 
 NATURAL COSINES 
 
190 
 
 APPENDIX 
 
 NATURAL TANGENTS 
 
 
 
 
 
 
 
 
 
 
 
 Com plement 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Difference 
 
 
 
 0.0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 0175 
 
 89 
 
 1 
 
 0175 
 
 0192 
 
 0209 
 
 0227 
 
 0244 
 
 0262 
 
 0279 
 
 0297 
 
 0314 
 
 0332 
 
 0349 
 
 88 
 
 2 
 
 0349 
 
 0367 
 
 0384 
 
 0402 
 
 0419 
 
 0437 
 
 0454 
 
 0472 
 
 0489 
 
 0507 
 
 0524 
 
 87 
 
 3 
 
 0524 
 
 0542 
 
 0559 
 
 0577 
 
 0594 
 
 0612 
 
 0629 
 
 0647 
 
 0664 
 
 0682 
 
 0699 
 
 86 
 
 4 
 
 0699 
 
 0717 
 
 0734 
 
 0752 
 
 0769 
 
 0787 
 
 0805 
 
 0822 
 
 0840 
 
 0857 
 
 0875 
 
 85 
 
 5 
 
 0.0875 
 
 0892 
 
 0910 
 
 0928 
 
 0945 
 
 0963 
 
 0981 
 
 0998 
 
 1016 
 
 1033 
 
 1051 
 
 84 
 
 6 
 
 1051 
 
 1069 
 
 1086 
 
 1104 
 
 1122 
 
 1139 
 
 1157 
 
 1175 
 
 1192 
 
 1210 
 
 1228 
 
 83 
 
 7 
 
 1228 
 
 1246 
 
 1263 
 
 1281 
 
 1299 
 
 1317 
 
 1334 
 
 1352 
 
 1370 
 
 1388 
 
 1405 
 
 82 
 
 8 
 
 1405 
 
 1423 
 
 1441 
 
 1459 
 
 1477 
 
 1495 
 
 1512 
 
 1530 
 
 1548 
 
 1566 
 
 1584 
 
 81 
 
 9 
 
 1584 
 
 1602 
 
 1620 
 
 1638 
 
 1655 
 
 1673 
 
 1691 
 
 1709 
 
 1727 
 
 1745 
 
 1763 
 
 80 
 
 10 
 
 0.1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 1944 
 
 79 M 
 
 11 
 
 1944 
 
 1962 
 
 1980 
 
 1998 
 
 2016 
 
 2035 
 
 2053 
 
 2071 
 
 2089 
 
 2107 
 
 2126 
 
 78 
 
 12 
 
 2126 
 
 2144 
 
 2162 
 
 2180 
 
 2199 
 
 2217 
 
 2235 
 
 2254 
 
 2272 
 
 2290 
 
 2309 
 
 77 
 
 13 
 
 2309 
 
 2327 
 
 2345 
 
 2364 
 
 2382 
 
 2401 
 
 2419 
 
 2438 
 
 2456 
 
 2475 
 
 2493 
 
 76 
 
 14 
 
 2493 
 
 2512 
 
 2530 
 
 2549 
 
 2568 
 
 2586 
 
 2605 
 
 2623 
 
 2642 
 
 2661 
 
 2679 
 
 75 
 
 15 
 
 0.2679 
 
 2698 
 
 2717 
 
 2736 
 
 2754 
 
 2774 
 
 2792 
 
 2811 
 
 2830 
 
 2849 
 
 2867 
 
 74 
 
 16 
 
 2867 
 
 2886 
 
 2905 
 
 2924 
 
 2943 
 
 2962 
 
 2981 
 
 3000 
 
 3019 
 
 3038 
 
 3057 
 
 73 19 
 
 17 
 
 3057 
 
 3076 
 
 3096 
 
 3115 
 
 3134 
 
 3153 
 
 3172 
 
 3191 
 
 3211 
 
 3230 
 
 3249 
 
 72 
 
 18 
 
 3249 
 
 3269 
 
 3288 
 
 3307 
 
 3327 
 
 3346 
 
 3365 
 
 3385 
 
 3404 
 
 3424 
 
 3443 
 
 71 
 
 19 
 
 3443 
 
 3463 
 
 3482 
 
 3502 
 
 3522 
 
 3541 
 
 3561 
 
 3581 
 
 3600 
 
 3620 
 
 3640 
 
 70 
 
 20 
 
 0.3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3839 
 
 69 
 
 21 
 
 3839 
 
 3859 
 
 3879 
 
 3899 
 
 3919 
 
 3939 
 
 3959 
 
 3979 
 
 4000 
 
 4020 
 
 4040 
 
 68 20 
 
 22 
 
 4040 
 
 4061 
 
 4081 
 
 4101 
 
 4122 
 
 4142 
 
 4163 
 
 4183 
 
 4204 
 
 4224 
 
 4245 
 
 67 
 
 23 
 
 4245 
 
 4265 
 
 4286 
 
 4307 
 
 4327 
 
 4348 
 
 4369 
 
 4390 
 
 4411 
 
 4431 
 
 4452 
 
 66 
 
 24 
 
 4452 
 
 4473 
 
 4494 
 
 4515 
 
 4536 
 
 4557 
 
 4578 
 
 4599 
 
 4621 
 
 4642 
 
 4663 
 
 65 21 
 
 25 
 
 0.4663 
 
 4684 
 
 4706 
 
 4727 
 
 4748 
 
 4770 
 
 4791 
 
 4813 
 
 4834 
 
 4856 
 
 4877 
 
 64 
 
 26 
 
 4877 
 
 4899 
 
 4921 
 
 4942 
 
 4964 
 
 4986 
 
 5008 
 
 5029 
 
 5051 
 
 5073 
 
 5095 
 
 63 
 
 27 
 
 5095 
 
 5117 
 
 5139 
 
 5161 
 
 5184 
 
 5206 
 
 5228 
 
 5250 
 
 5272 
 
 5295 
 
 5317 
 
 62 22 
 
 28 
 
 5317 
 
 5340 
 
 5362 
 
 5384 
 
 5407 
 
 5430 
 
 5452 
 
 5475 
 
 5498 
 
 5520 
 
 5543 
 
 61 
 
 29 
 
 5543 
 
 5566 
 
 5589 
 
 5612 
 
 5635 
 
 5658 
 
 5681 
 
 5704 
 
 5727 
 
 5750 
 
 5774 
 
 60 23 
 
 30 
 
 0.5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 5914 
 
 5938 
 
 5961 
 
 5985 
 
 6099 
 
 59 
 
 31 
 
 6009 
 
 6032 
 
 6056 
 
 6080 
 
 6104 
 
 6128 
 
 6152 
 
 6176 
 
 6200 
 
 6224 
 
 6249 
 
 58 2 * 
 
 32 
 
 6249 
 
 6273 
 
 6297 
 
 6322 
 
 6346 
 
 6371 
 
 6395 
 
 6420 
 
 6445 
 
 6469 
 
 6494 
 
 57 
 
 33 
 
 6494 
 
 6519 
 
 6544 
 
 6569 
 
 6594 
 
 6619 
 
 6644 
 
 6669 
 
 6694 
 
 6720 
 
 6745 
 
 56 25 
 
 34 
 
 6745 
 
 6771 
 
 6796 
 
 6822 
 
 6847 
 
 6873 
 
 6899 
 
 6924 
 
 6950 
 
 6976 
 
 7002 
 
 55 
 
 35 
 
 0.7002 
 
 7028 
 
 7054 
 
 7080 
 
 7107 
 
 7133 
 
 7159 
 
 7186 
 
 7212 
 
 7239 
 
 7265 
 
 54 26 
 
 36 
 
 7265 
 
 7292 
 
 7319 
 
 7346 
 
 7373 
 
 7400 
 
 7427 
 
 7454 
 
 7481 
 
 7508 
 
 7536 
 
 53 * 
 
 37 
 
 7536 
 
 7563 
 
 7590 
 
 7618 
 
 7646 
 
 7673 
 
 7701 
 
 7729 
 
 7757 
 
 7785 
 
 7813 
 
 52 28 
 
 38 
 
 7813 
 
 7841 
 
 7869 
 
 7898 
 
 7926 
 
 7954 
 
 7983 
 
 8012 
 
 8040 
 
 8069 
 
 8098 
 
 51 28 
 
 39 
 
 8098 
 
 8127 
 
 8156 
 
 8185 
 
 8214 
 
 8243 
 
 8273 
 
 8302 
 
 8332 
 
 8361 
 
 8391 
 
 50 29 
 
 40 
 
 0.8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 8693 
 
 49 30 
 
 41 
 
 8693 
 
 8724 
 
 8754 
 
 8785 
 
 8816 
 
 8847 
 
 8878 
 
 8910 
 
 8941 
 
 8972 
 
 9004 
 
 48 31 
 
 42 
 
 9004 
 
 9036 
 
 9067 
 
 9099 
 
 9131 
 
 9163 
 
 9195 
 
 9228 
 
 9260 
 
 9293 
 
 9325 
 
 47 32 
 
 43 
 
 9325 
 
 9358 
 
 9391 
 
 9424 
 
 9557 
 
 9490 
 
 9523 
 
 9556 
 
 9590 
 
 9623 
 
 9657 
 
 46 33 
 
 44 
 
 9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 9965 
 
 1.0000 
 
 45034 
 
 Complement 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 .4 
 
 .3 
 
 .2 
 
 l 
 
 .0 
 
 Angle 
 
 NATURAL COTANGENTS 
 
APPENDIX 
 
 NATURAL TANGENTS 
 
 191 
 
 Angle] .0 
 
 .1 
 
 .2 
 
 .3 
 
 A 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Dif. 
 
 45 
 
 1.0000 
 
 1.0035 1.0070 
 
 1.0105 
 
 1.0141 
 
 1.0176 
 
 1.0212 
 
 1.0247 
 
 L.0388 
 
 1.0319 
 
 36 
 
 46 
 
 1.0355 
 
 1.0392 
 
 1.0428 
 
 1.0464 
 
 1.0501 
 
 1.0538 
 
 1.0575 
 
 1.0612 
 
 1.0649 
 
 1.0686 
 
 
 
 47 
 
 1.07-24 
 
 1.0761 
 
 1.0799 
 
 L.063-3 
 
 L087B 
 
 1.0913 
 
 1.0951 
 
 1.0990 
 
 1.1028 
 
 1.1067 
 
 38 
 
 48 
 
 1.1106 
 
 1.11451.1184 
 
 1.1224 
 
 1.1263 
 
 1.1303 
 
 1.1343 
 
 1.1383 
 
 1.1423 
 
 1.1463 
 
 
 
 49 
 
 1.1504 
 
 1.1544 
 
 1.1585 
 
 1.1626 
 
 1.1667 
 
 1.1708 
 
 1.1750 
 
 1.1792 
 
 1.1833 
 
 1.1875 j 
 
 50 
 
 1.1918 
 
 1.1960 
 
 1.2002 
 
 1.2045 
 
 L2088 
 
 1.2131 
 
 1.2174 
 
 1.2218 
 
 1.2261 
 
 1.2305 
 
 51 
 
 .23491.23931.2437 
 
 1.2482 
 
 1.2527 
 
 1.2572 
 
 1.2617 
 
 1.2662 
 
 1.2708 
 
 1.2753 
 
 52 
 
 .2799 1.2846|1.2892 
 
 1.2985J1.3032 
 
 1.3079 
 
 1.3127 
 
 1.3175 1.3222 *T 
 
 58 
 
 .3270 1.3319 1.3367 
 
 1.3416 1.3465 1.3514 
 
 1.3564 
 
 1.3613 
 
 1.36631.3713 
 
 54 
 
 .3764 
 
 1.3814 
 
 1.3865|1.3916 
 
 1.3968 
 
 1.4019 
 
 1.4071 
 
 1.4124 
 
 1.4176 
 
 1.4229 
 
 53 
 
 55 
 
 .4281 
 
 1.4335 
 
 1.43881.4442 
 
 1.4496 
 
 1.4550 
 
 1.4605 
 
 1.4659 
 
 1.4715 
 
 1.4770 
 
 54 
 
 56 
 
 .4826|l.48 ..4994 l.oo.M 1.510s 1.5166 
 
 1.5224 
 
 1.5282 
 
 1.5340 si 
 
 57 
 
 .5399 1.5458 
 
 1.5517 1.5577 1.5637 1.5697 1.5757 
 
 1.5818 
 
 L0880 
 
 1.5941' 
 
 58 
 
 .60031.6066 
 
 1.6128 1.6191 1.6255 1.6319 1.6383 
 
 1.6447 
 
 1.6512 
 
 1.6577 
 
 59 
 
 .66431.6709 
 
 1.6775 
 
 1.6842 
 
 1.6909 
 
 1.6977 
 
 1.7045 
 
 1.7113 
 
 1.7182 
 
 1.7251 
 
 68 
 
 60 
 
 .732l'l.739l 
 
 1.7461 
 
 1.7532 
 
 1.7603 
 
 1.7675 
 
 1.7747 
 
 1.7820 
 
 1.7893 
 
 1.7966 
 
 72 
 
 61 
 
 .8040 
 
 1.8115 
 
 1.8190 
 
 1.8265 
 
 1.83411.84181.8495 
 
 1.8572 1.8650 
 
 1.872* " 
 
 62 
 
 . -. 
 
 :. 7 
 
 1.8967 
 
 1.9047 
 
 1.9128 1.9210 1.9292 
 
 1.9375 1.9458 
 
 1.9542 82 
 
 68 
 
 .9626 
 
 1.9711 
 
 1.9797 
 
 L9669 
 
 1. 9970 2. 0057 2. 0145 2. 0233 2. 0323 
 
 2.0413 88 
 
 64 
 
 2.0503 
 
 2.0594 
 
 2.0686 
 
 2.0778 
 
 2.0872 
 
 2.0965 
 
 2.1060 
 
 2.11552.1251 
 
 2.1348 * 
 
 65 
 
 2.145 
 
 2.154 
 
 2.164 
 
 2.174 
 
 2.184 
 
 2.194 
 
 2.204 
 
 2.215 '2.225 
 
 2.236 
 
 10 
 
 66 
 
 2.246 2.-2.J7 
 
 2.267 
 
 2.278 
 
 2.289 
 
 2.300 
 
 2.311 
 
 2.322 2.333 2.344 
 
 11 
 
 67 
 
 2.&56 
 
 2.367 
 
 2.379 
 
 2.391 
 
 2.402 
 
 2.414 
 
 2.426 
 
 2.438 2.450 2.463 
 
 :. 
 
 68 
 
 2.475 
 
 2.488 
 
 2.500 
 
 2.513 
 
 2.526 
 
 2.539 
 
 2.552 
 
 2,565 2. .57* 2.592 
 
 13 
 
 69 
 
 2.605 
 
 2.619 
 
 2.633 
 
 2.646 
 
 2.660 
 
 2.675 
 
 &680 
 
 2.703 
 
 2.718 
 
 2.733 
 
 U 
 
 70 
 
 2.747 
 
 2.762 
 
 2.77<3 
 
 2.793 
 
 1806 
 
 2.824 
 
 2.840 
 
 2,856 
 
 2.872 
 
 2.888 
 
 16 
 
 71 
 
 2.904 
 
 2.921 
 
 2.937 
 
 2.954 2.971 
 
 2.989 
 
 3.006 13.024 3.042 
 
 3.060 
 
 n 
 
 72 
 
 3.078 
 
 3.096 
 
 3:115 
 
 3.133 3.152 
 
 3.172 
 
 3.191 
 
 3.211 3.230 
 
 3.250 
 
 19 
 
 78 
 
 3.271 
 
 3.291 
 
 3.312 
 
 3.333 
 
 3.354 
 
 3.376 
 
 3.398 
 
 3.420 
 
 3.442 
 
 3.465 
 
 74 
 
 3.487 
 
 3.511 
 
 3.534 
 
 3.558 
 
 3.582 
 
 3.606 
 
 3.630 
 
 3.655 
 
 3.681 
 
 3.700 
 
 
 
 75 
 
 1.782 
 
 3.758 
 
 3.785 
 
 3.812 
 
 3.839 
 
 3.867 
 
 3.895 
 
 3.923 
 
 3.952 
 
 3.981 
 
 28 
 
 76 
 
 4.011 
 
 4041 
 
 4071 
 
 4102 
 
 4134 
 
 4165 
 
 4198 
 
 4230 
 
 4264 
 
 4297 
 
 32 
 
 77 
 
 4.331 
 
 4366 
 
 4402 
 
 4437 
 
 4474 
 
 4511 
 
 4548 
 
 4586 
 
 4625 
 
 4.665 
 
 37 
 
 78 
 
 4705 
 
 4745 
 
 4.7-7 
 
 4829 
 
 4872 
 
 4.915 
 
 4959 
 
 5.005 
 
 5.050 
 
 5.097 
 
 44 
 
 79 
 
 5.145 
 
 5.193 
 
 5.243 
 
 5.292 
 
 5.343 
 
 5.396 
 
 5.449 
 
 5.503 
 
 5.558 
 
 5.614 
 
 52 
 
 80 
 
 5.67 
 
 5.73 
 
 5.79 
 
 5.85 
 
 5.91 
 
 5.98 
 
 6.04 
 
 6.11 
 
 6.17 
 
 6.24 
 
 T 
 
 81 
 
 6.31 
 
 6.39 
 
 6.46 
 
 6.54 
 
 6.61 
 
 6.69 
 
 6.77 
 
 6.85 
 
 6.94 
 
 7.03 
 
 8 
 
 82 
 
 7.12 
 
 7.21 
 
 7.30 
 
 7.40 
 
 7.49 
 
 7.60 
 
 7.70 
 
 7.81 
 
 7.92 
 
 8.03 
 
 10 
 
 88 
 
 8.14 
 
 8.26 
 
 8.39 
 
 8.51 
 
 8.64 
 
 8.78 
 
 8.92 
 
 9.06 
 
 9.21 
 
 9.36 
 
 14 
 
 84 
 
 9.51 
 
 9.68 
 
 9.84 
 
 10.0 
 
 10.2 
 
 10.4 
 
 10.6 
 
 10.8 
 
 11.0 
 
 11.2 
 
 
 85 
 
 11.4 
 
 11.7 
 
 11.9 
 
 12.2 
 
 12.4 
 
 12.7 
 
 13.0 
 
 13.3 
 
 13.6 
 
 140 
 
 * 
 
 86 
 
 14.3 
 
 147 
 
 15.1 
 
 15.5 
 
 15.9 
 
 16.3 
 
 16.8 
 
 17.3 
 
 17.9 
 
 18.5 
 
 6 
 
 87 
 
 19.1 
 
 19.7 
 
 20.4 
 
 21.2 
 
 22.0 
 
 22.9 
 
 23.9 
 
 249 
 
 26.0 
 
 27.3 
 
 
 88 
 
 28.6 
 
 30.1 
 
 31.8 
 
 33.7 
 
 35.8 
 
 38.2 
 
 40.9 
 
 44.1 
 
 47.7 
 
 52.1 
 
 
 89 
 
 57. 
 
 64. 
 
 72. 
 
 v2. 
 
 95. 
 
 115. 
 
 143. 
 
 191. 
 
 286. 
 
 573. 
 
 
 Angle 
 
 .0 
 
 .1 
 
 9 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 
 NATURAL TANGENTS 
 
192 
 
 APPENDIX 
 
 LOGARITHMS 
 
 
 o 
 
 l 
 
 2 
 
 3 
 
 4 . 
 
 5 
 
 6 
 
 ' 7 
 
 8 
 
 9 
 
 1 2 3 
 
 456 
 
 789 
 
 10 
 
 0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 4 8 12 
 
 17 21 25 
 
 29 33 37 
 
 11 
 
 12 
 13 
 
 0414 
 0792 
 1139 
 
 0453 
 0828 
 1173 
 
 0492 
 0864 
 1206 
 
 0531 
 0899 
 1239 
 
 0569 
 0934 
 1271 
 
 0607 
 0969 
 1303 
 
 0645 
 1004 
 1335 
 
 0682 
 1038 
 1367 
 
 0719 
 1072 
 1399 
 
 0755 
 1106 
 1430 
 
 4 8 11 
 3 7 10 
 3 6 10 
 
 15 19 23 
 14 17 21 
 13 16 19 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 14 
 15 
 16 
 
 1461 
 1761 
 2041 
 
 1492 
 1790 
 2068 
 
 1523 
 
 1818 
 2095 
 
 1553 
 1847 
 2122 
 
 1584 
 1875 
 2148 
 
 1614 
 1903 
 2175 
 
 1644 
 1931 
 2201 
 
 1673 
 1959 
 
 2227 
 
 1703 
 1987 
 2253 
 
 1732 
 2014 
 2279 
 
 369 
 
 368 
 358 
 
 12 15 18 
 11 14 17 
 11 13 16 
 
 21 24 27 
 20 22 25 
 18 21 24 
 
 17 
 
 18 
 19 
 
 2304 
 2553 
 
 2788 
 
 2330 
 
 2577 
 2810 
 
 2355 
 2601 
 2833 
 
 2380 
 2625 
 
 2856 
 
 2405 
 2648 
 
 2878 
 
 2430 
 2672 
 2900 
 
 2455 
 2695 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 
 2742 
 2967 
 
 2529 
 2765 
 2989 
 
 257 
 257 
 247 
 
 10 12 15 
 9 12 14 
 9 11 13 
 
 17 20 22 
 16 19 21 
 16 18 20 
 
 20 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 246 
 
 8 11 13 
 
 15 17 19 
 
 21 
 22 
 23 
 
 3222 
 3424 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 3655 
 
 3284 
 3483 
 3674 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 3711 
 
 3345 
 3541 
 3729 
 
 3365 
 3560 
 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 3598 
 
 3784 
 
 246 
 246 
 246 
 
 8 10 12 
 8 10 12 
 7 9 11 
 
 14 16 18 
 14 15 17 
 13 15 17 
 
 24 
 25 
 26 
 
 3802 
 3979 
 4150 
 
 3820 
 3997 
 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 4031 
 4200 
 
 3874 
 4048 
 4216 
 
 3892 
 4065 
 4232 
 
 3909 
 4082 
 4249 
 
 3927 
 4099 
 4265 
 
 3945 
 4116 
 4281 
 
 3962 
 4133 
 
 4298 
 
 245 
 235 
 235 
 
 7 9 11 
 7 9 10 
 
 7 8 10 
 
 12 14 16 
 12 14 15 
 11 13 15 
 
 27 
 
 28 
 29 
 
 4314 
 
 4472 
 4624 
 
 4330 
 
 4487 
 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 4518 
 4669 
 
 4378 
 4533 
 4683 
 
 4393 
 
 4548 
 4698 
 
 4409 
 4564 
 4713 
 
 4425 
 4579 
 
 4728 
 
 4440 
 4594 
 
 4742 
 
 4456 
 4609 
 
 4757 
 
 235 
 235 
 1 3 4 
 
 689 
 689 
 679 
 
 11 13 14 
 11 12 14 
 10 12 13 
 
 30 
 
 4771 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 1 3 4 
 
 679 
 
 10 11 13 
 
 31 
 32 
 33 
 
 4914 
 5051 
 
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 5276 
 
 5024 
 5159 
 5289 
 
 5038 
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 5302 
 
 1 8 4 
 1 3 4 
 1 3 4 
 
 678 
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 568 
 
 10 11 12 
 9 11 12 
 9 10 12 
 
 34 
 35 
 86 
 
 5315 
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 568 
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 37 
 
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 39 
 
 5682 
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 567 
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 40 
 
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 345 
 
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APPENDIX 
 
 193 
 
 LOGARITHMS 
 
 55 
 
 
 
 l 
 
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 8 . 
 
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 <i:J 
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 334 
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 82i5 
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 8312 
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 8254 
 8319 
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 1 1 
 1 1 
 1 1 
 
 334 
 334 
 334 
 
 556 
 556 
 456 
 
 69 
 70 
 71 
 
 8388 
 8451 
 8513 
 
 8395 
 
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 8519 
 
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 8555 
 
 8439 
 8500 
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 8445 
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 8567 
 
 1 1 
 1 1 
 1 1 
 
 234 
 234 
 234 
 
 456 
 456 
 455 
 
 72 
 73 
 74 
 
 8573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
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 8627 
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 8745 
 
 1 
 1 
 
 1 
 
 234 
 234 
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 455 
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 455 
 
 ::> 
 
 8751 
 
 8756 
 
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 8779 
 
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 1 
 
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 77 
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 1 
 1 
 1 
 
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 1 
 1 
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 82 
 s:i 
 84 
 
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 1 
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 1 
 
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 85 
 
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 1 1 
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 233 
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 445 
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 89 
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 1 1 
 1 1 
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 223 
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 92 
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 1 1 
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 223 
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 95 
 
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 9782 
 
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 Oil 
 
 223 
 
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 96 
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 1 1 
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 1 1 
 
 223 
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 223 
 
 344 
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 344 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
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 9983 
 
 9987 
 
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 9996 
 
 1 1 
 
 223 
 
 334 
 
INDEX 
 
 Abbe, total reflection, 107. 
 Absorption, coefficient of, 157. 
 
 relation to anomalous dispersion, 
 
 173. 
 
 Achromatic fringe, 37. 
 Ames, concave grating, 121. 
 Angle of incidence, 84. 
 
 of emergence, 84. 
 
 principal, 157. 
 
 Angstrom, absolute wave length, 124. 
 Azimuth, of plane of polarization, 
 152. 
 
 principal, 157. 
 Babinet, compensator, 143. 
 Bell, absolute wave length, 124. 
 Biot, rotary polarization, 131. 
 Bi-prism, Fresnel, 44. 
 Boltzmann, rotary dispersion, 131. 
 Brewster, law, 151. 
 Cadmium, distribution in spectral 
 
 lines of, 79. 
 
 Cathode rays, 175, 181. 
 Cauchy, dispersion equation, 39, 90. 
 Concave grating, 116. 
 
 constant of, 121. 
 Congruent rays, 21. 
 Conroy, Brewster 's law, 151. 
 
 reflection of polarized light, 152. 
 Corpuscular theory, 166. 
 Czapski, total reflection, 107. 
 Deviation, minimum, 86. 
 Dielectric constant, equals square of 
 
 index, 173. 
 Diffraction, by one slit, 11. 
 
 by two slits, 19. 
 
 grating, 108. 
 
 Dispersion, Cauchy's equation for, 
 39, 90, 173. 
 
 Dispersion, determination with the 
 interferometer, 61. 
 
 of prism, 94. 
 
 anomalous, 173. 
 
 electromagnetic theory of, 175. 
 Distribution in source, determination 
 
 by visibility, 25, 71. 
 Double slit, 19, 23. 
 Drude, resolving power of prisms, 91, 
 
 diffraction gratings, 108. 
 
 reflection of polarized light, 150. 
 
 metallic reflection, 157. 
 
 electromagnetic theory, 170, 172, 
 
 174, 175, 181. 
 Electrolysis, 175, 181. 
 Electromagnetic, theory, 170. 
 Electron, 182. 
 Elliptically polarized light, 140. 
 
 analysis of, 142. 
 
 Fabry, distribution in sources, 82. 
 Faraday, electromagnetic theory, 
 
 169, 178. 
 
 Fraunhofer, diffraction phenomena, 
 11. 
 
 diffraction grating, 17. 
 Fresnel, mirrors, 30, 33. 
 
 bi-prism, 44. 
 
 reflection equations, 150. 
 Glan, spectrophotometer, 159. 
 Glazebrook, optical theories, 169. 
 Grating, plane, 108. 
 
 dispersion of, 110. 
 
 resolving power of, 111. 
 
 constant of, 112. 
 
 concave, 116. 
 
 Gubbe, rotary polarization, 134. 
 Gumlich, rotary polarization, 131. 
 Helmholtz, prisms, 96. 
 
 194 
 
IXDEX 
 
 195 
 
 Interference, general discussion of, 
 
 21. 
 Interferometer, definition of, 32. 
 
 Michelson, 48. 
 
 adjustment of, 55. 
 Ions, 175. 
 
 ratio of charge to mass, 181. 
 Jamin, Brewster's law, 151. 
 Kayser, prisms, 96. 
 
 plane grating, 108. 
 
 concave grating, 116. 
 
 absolute wave lengths, 125. 
 
 regularities in spectra, 177. 
 Kohlrausch, total reflection, 107. 
 Kurlbaum, absolute wave length, 
 
 124. 
 
 Landolt, rotary polarization, 133. 
 Lloyd, optical theories, 169. 
 Magnetism, action on light, 178. 
 Mascart, visibility curves, 27. 
 
 gratings, 108. 
 
 Maxwell, law of distribution of 
 velocities of molecules, 75. 
 
 electromagnetic theory, 170, 172. 
 Metallic reflection, 156. 
 Michelson, visibility with the double 
 slit, 27. 
 
 interferometer, 33, 48. 
 
 applications of the interferom- 
 eter. 69. 
 
 visibility curves with interfe- 
 rometer, 82. 
 
 absolute wave length, 125. 
 
 action of magnetism on light, 
 180. 
 
 echelon spectroscope, 180. 
 Mirrors, Fresnel. 30. 
 Moigno, 1'Abbe, optical theories, 
 
 169. 
 
 Molecular rotation, 132. 
 Muller, absolute --vave length, 124. 
 Optical activity, 130. 
 Perot, distribution in sources, 82. 
 Polari scope, 134. 
 
 Polarization rotation of the plane 
 of, 130. 
 
 Polarized light, qualitative experi- 
 ments in, 127. 
 
 elliptically, 140. 
 
 reflection of, 150. 
 Prism, 83. 
 
 formation of an image by, 88. 
 
 thickness of, 92. 
 Pulfrich, total reflection, 107. 
 Purity of spectrum, 94, 111. 
 Quarter- wave plate, 146. 
 Quincke, Fresnel mirrors, 39. 
 
 diffraction grating, 108. 
 Ray lei gh, limit of resolution, 18. 
 
 visibility curves, 78. 
 
 resolving power of prisms, 91-96. 
 
 plane gratings, 108, 115. 
 
 Brewster's law, 151. 
 Reflection, loss of phase on, 66. 
 
 total, 105. 
 
 of polarized light, 150. 
 
 metallic, 156. 
 Refraction, index of, definition of, 85. 
 
 determination, with the inter- 
 ferometer, 61. 
 
 with the prism spectrometer, 96. 
 
 angle of, 84. 
 
 determination of by total reflec- 
 tion, 105. 
 
 of metals, 157. 
 
 relation to dielectric constant, 
 
 173, 
 Resolution, limit of, 15, 17. 
 
 of prism, 93. 
 Resolving power, of prism, 95. 
 
 determination of, 102. 
 . of grating, 111. 
 
 determination of, 114. 
 Rood, reflection of polarized light, 
 
 152. 
 Rotation, specific, 132. 
 
 naolecular, 132. 
 Rowland, plane gratings, 108, 115. 
 
 concave gratings, 116. 
 
 relative wave lengths, 125. 
 Runge, concave gratings, 116. 
 
 regularities in spectra, 177. 
 
196 
 
 Sarasin, rotary polarization, 181. 
 Schonrock, rotary polarization, 133. 
 Schwerd, diffraction, 18. 
 Sodium, determination of wave 
 lengths, 55. 
 
 ratio of wave lengths of the 
 lines DI and D 2 , 59. 
 
 distribution of a single line, 82. 
 Soret, rotary polarization, 131. 
 Spectrometer, 83. 
 
 adjustments of, 97. 
 Spectrophotometer, 159. 
 Spectrum, purity of, 94. 
 
 order of, 110. 
 
 Spectrum, normal, 111. 
 
 absorption, 164. 
 
 Stefan, rotary polarization, 133. 
 Thomson, J. J., electron theory, 
 
 181. 
 Velocity, of light, equals ratio of 
 
 units, 172. 
 
 Visibility, with the double slit, 25. 
 with the interferometer, 60, 70. 
 Wave length, absolute determination 
 
 of, 124. 
 
 Wave theory, 166. 
 Zeeman, action of magnetism on 
 
 light, 178. 
 
 J 
 
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