412 PROFESSOR STOKES, ON THE COMPOSITION AND RESOLUTION OF 



oppositely polarized pencils, the intensities of the two are the same, and of course equal 

 to half that of the original group. Accordingly, in order that the group should be equivalent 

 to common light, it is necessary and sufficient that the constants B, C, D should vanish. 



18. Let us now see under what circumstances two independent streams of polarized 

 light can together be equivalent to common light. 



Let a, /3 refer to the first, and a, ff to the second stream, and let the intensities of the 

 two streams be as 1 ton; then we get from the formulas (16) 



sin 2/3 + n sin 2/3' = ; 

 cos 2a cos 2/3 + n cos 2 a' cos 2)8' = 0; 

 sin 2a cos 2/3 + n sin 2a' cos 2/3' = 0. 



Transposing, squaring, and adding, we find » 8 = 1, and therefore n = 1, since n is essen- 

 tially positive. Since /3 and /3' are supposed not to lie beyond the limits - 90° and + 90°, we 

 get from the first equation ($' = - /3, or /3' = ± 90° - /3, + or - according as /3 is positive or 

 negative. Now it is plain that any one solution must be expressed analytically in two ways, in 

 which the values of /3' are complementary, and the values of a differ by 90°, since either prin- 

 cipal axis of the ellipse belonging to the second stream may be that whose azimuth is d. 

 Accordingly, we may reject the second solution as being nothing more than the first expressed 

 in a different way, and may therefore suppose /3' = — /3. The second and third equations then 

 give cos 2 a' = — cos 2 a, sin 2 a' = - sin 2 a, and therefore a and a differ by 90°. The equations 

 are indeed satisfied by /3 = - /3' = ± 45°, but this solution is only a particular case of the 

 former. 



It follows therefore that common light is equivalent to any two independent oppositely 

 polarized streams of half the intensity ; and no two independent polarized streams can together 

 be equivalent to common light, unless they be polarized oppositely, and have their inten- 

 sities equal. 



19. We have seen that the nature of the mixture of a given group of independent 

 polarized streams is determined by the values of the four constants A, B, C, D. Consider 

 now the mixture of a stream of common light having an intensity J, and a stream, inde- 

 pendent of the former, consisting of elliptically polarized light having an intensity J ', and 

 having a for the azimuth of its plane of maximum polarization, and tan /3' for the ratio 

 of the axes of the ellipse which characterizes it. 



By the preceding article, the stream of common light is equivalent to two independent 

 streams, plane- polarized in azimuths 0* and 90°, having each an intensity equal to ^J. 

 Hence, applying the formulas (l6) and (17) to the mixture, we have 



2Jtl (c,) 2 = J + J' + J' sin 2/3' sin 2/3j + J' cos 2a' cos 2/3' cos2aj cos 2/3, 



+ J' sin 2a' cos 2/3' sin 2a x cos 2/3! ; 

 and this mixture will be equivalent to the original group of polarized streams, provided 

 J+J'=A; J' sin 2/3' = B; 1 



J' cos 2a' cos 2/3' = C; J' sin 2a' cos 2/3' = D. J 



