447 while e remains constant. If the vibration represented by (6) be now resolved in a direction x , making an angle w with x, we have x = a cos (f> cos co + 1 cos (<f> - 8 - e) sin co = [ cos ca + b sin <a cos (S + e)] cos c/> + J sin co sin (5 + e) sin </> ; and the intensity is 2 cos 2 o> + Zr sin 2 co + 2a& cos cvsin cocos(8 + ) . . (7). Of this expression we take the mean, co and e remaining constant. Thus the apparent intensity may be written M(x- 2 ) = M( 2 ) cos 2 co + M(Zr) sin 2 co + 2M[J cos (5 + e)] cos co sin co (8). Xeces- In order now that the stream may satisfy the conditions laid down sary con- as necessary for natural light, (8) must be independent of CD and e ; ditions. so that M(a 9 )-M(i 2 ) (9), M(&cos8) = M(rti>sinS) = (10). : In thcso equations a 2 and 6 3 represent simply the intensities, or squares of amplitudes, of the x and y vibrations ; and the other two quantities admit also of a simple interpretation. The value of y may bo written 7/ = 5cos 5cos</) + Jsin 8siu0 .... (11); from which we see that ScosS is the coefficient of that part of the y vibration which has the same phase as the x vibration. Thus 6cos8 may be interpreted as the product of the coefficients of the parts of the x and y vibrations which have the same phase. Next suppose the phase of y accelerated by writing JTT + <p in place of <p. We should thus have y = - b cos S sin < + b sin 8 cos c/> , and 5sin8 represents the product of the coefficients of the parts which arc now in the same phase, or (which is the same) the pro duct of the coefficients of the x vibration and of that part of the y vibration which was 90 behind in phase. In general, if x = hcos<p + h sin </>, y <= k cos <f> + k sin. <f> . (12), the first product is hk + Ji k and the second is JiV - h Jc. Let us next examine how the quantities which we have been considering are affected by a transformation of coordinates in accordance with the formula} x = x cos ca + y sin a . ?/ = a: sin co-M/cosco . (13). We find a/ = cos0{cos co + & sin co cos 8} + sin 0.6 sin Ssinco . (14), 2/ = cos0{ -fflsinco + JcoscocosS} + sin . 6 sin 8 cos co . (15); whence amp. 2 of a; = 2 cos 2 co + i 2 sin 2 co + 2Z> cos Ssin cocos w . (16), amp. 2 of 7/ = 2 sin 2 o! + i :: cos 2 w -26cos Ssin 01 COSCD . (17). In like manner First product = (b 2 - a-) sin co cos co + ab cos 8 (cos 2 co - sin 2 CD) (18), Second product = <ib sin 8 (19). The second product, representing the circulating part of the motion, is thus unaltered by the transformation. Let us pass on to the consideration of the mean quantities which occur in (9), (10), writing for brevity M( 2 ) = A, M(Z/ 2 ) = B, M(a&cos5) = C, M(aZ>sin5) = D. From (16), (17), (18), (19), if A , B , C , D denote the corre sponding quantities after transformation, A = A cos 2 co + B sin 2 co + 20 cos co sin co .... (20), B = A sin 2 CD + B cos 2 CD -20 cos co sin CD . . . . (21), C = 0(cos 2 co-siu -co) + (B- A) cos co sin co . . . (22), D = D (23). These formulae prove that, if the conditions (9), (10), shown to be necessary in order that the light may behave as natural light, be satisfied for one set of axes, they are equally satisfied with any other. It is thus a matter of indifference with respect to what axes the retardation e is supposed to be introduced, and the con ditions (9), (10) are sufficient, as well as necessary, to characterize natural light. Reverting to (8), we see that, whether the light be natural or not, its character, so far as experimental tests can show, is deter mined by the values of A, B, C, D. The effect of a change of axes is given by (20), &c., and it is evident that the new axes may always be so chosen that C = 0. For this purpose it is only necessary to take co such that tan2 = 2C/(A-B). If we choose these new axes as fundamental axes, the values of the constants for any others inclined to them at angle co will be of the form A = Aj cos 2 co + B L sin 2 co ) B = A! sin 2 co + B x cos 2 co > (24). C = (Bj - Aj) cos co sin co ] If A! and B x are here equal, then = 0, A = B for all values of co. In this case, the light cannot be distinguished from natural light by mere resolution ; but if D be finite, the difference may be made apparent with the aid of a retarding plate. 1 Verclut, Lefons d Oplique I /tysiyue, vol. ii. p. 83. If A! and B : are unequal, they represent the maximum and minimum values of A and 13. The intensity is then a function of the plane of resolution, and the light may be recognized as partially polarized by the usual tests. If either A a or B x vanishes, the light is plane-polarized. 2 When several independent streams of light are combined, the values, not only of A and B, but also of C and D, for the mixture, are.to be found by simple addition. It must here be distinctly understood that there are no permanent phase-relations between one component and another. Suppose, for example, that there are two streams of light, each of which satisfies the relations A = B, = 0, but makes the value of D finite. If the two values of D are equal and opposite, and the streams are independent, the mixture constitutes natural light. A particular case arises when each com ponent is circularly-polarized (D=A=B), one in the right- handed and the other in the left-handed direction. The intensities being equal, the mixture is equivalent to natural light, but only under the restriction that the streams are without phase-relation. If, on the contrary, the second stream be similar to the first, affected merely with a constant retardation, the resultant is not natural, but completely (plane) polarized light. We will now prove that the most general mixture of light may Analysis be regarded as compounded of one stream of light elliptically- of generr polarized in a definite manner, and of an independent stream of case. natural light. The theorem is due to Stokes, 3 but the method that we shall follow is that of Verdet. 4 In the first place, it is necessary to observe that the values of the fundamental quantities A, B, C, D are not free from restriction. It will be shown that in no case can C 2 + D 2 exceed AB. In equations (2), expressing the vibration at any moment, let a i> ^i) a i> /3i> be the values of a, b, a, /3 during an interval of time proportional to m 1} and in like manner let the suffixes 2, 3, .... correspond to times proportional to ?>i 2 , m 3 , .... Then AB == m^a^b^ + m.ra. 2 -b. 2 2 + . . . + m l m f a-i i b.? + a.?bf} + . . . Again, by (12), C = W 1 a 1 6 1 (cos ttj cos ft 1 + sin a x sin f$ L ) + . . . = 7n 1 a 1 b 1 cos S 1 + m. 2 aj> 2 cos 8 2 + . . . , D = m 1 a 1 b l sin S 1 + m. 2 a.J} 2 sin 8 2 + . . . ; where, as before, 81 = $!-!, 8 2 = /3 2 -a 2 , .... Thus, From these equations we see that AB - C 2 - D 2 reduces itself to a sum of terms of the form m 1 m 2 [a 1 a i 3 s + a^b-f - 2 1 & 1 . J 5 2 cos (8 2 - 8j)] , each of which is essentially positive. The only case in which the sum can vanish is when 8 1 = 8 2 =8 3 = . . . , and further b { : a } = 5 2 : 2 b 3 : a 3 = . Under these conditions the light is reduced to be of a definite elliptic character, although the amplitude and phase of the system as a ivhole may be subject to rapid variation. The elliptic con stants are given by 6 2 / 2 = B/A, tan8 = D/C. . . . (25). In general AB exceeds (C 2 + D 2 ); but it will always be possible to find a positive quantity H, which when subtracted from A and B (themselves necessarily positive) shall reduce the product to equality with C 2 + D 2 , in accordance with (A-H)(B-H) = C S + D 2 ..,.. (26). The original light may thus be resolved into two groups. For the first group the constants are H, H, 0, ; and for the second A - H, B-H, 0, D. Each of these is of a simple character; for the first represents natural light, and the second light elliptically polarized. It is thus proved that in general a stream of light may be regarded Stokes s as composed of one stream of natural light and of another elliptic- theorem ally-polarized. The intensity of the natural light is 2H, where from (26) H = -i(A + B)-K/{(A-B) 2 + 4(C- + D 2 )} . . (27). The elliptic constants of the second component arc given by & 2 /a 2 = (B-H)/(A-H), tanS = D/0 . . (28), and M(a a ) = A-H ..,. ... (29). If D = 0, and therefore by (28) 8 = 0, the second component is plane-polarized. This is regarded as a particular case of elliptic polarization. Again, if A = B, = 0, the polarization is circular. The laws of interference of polarized light, discovered by Fresnel and Arago, are exactly what the theory of transverse vibrations would lead us to expect, when once we have cleared up the idea of unpolarized light. Ordinary sources, such as the sun, emit un- polarized light. If this be resolved in two opposite directions, the 2 In this case D t necessarily vanishes. y " On the Composition and Resolution of Streams of Light from Different
Sources," Camb. Phi!. Trans., 1852. * Lot. cit., p. 94.