'} (16) 



408 PROFESSOR STOKES, ON THE COMPOSITION AND RESOLUTION OF 



9. Suppose that there are any number of independent polarized streams mixing together ; 

 let the mixture be resolved in any manner into two oppositely polarized streams, and let us 

 examine the intensity of each. 



Let us take one stream first. The intensities of its components are given by the formulae 

 (13), which become somewhat simpler in the case of opposite polarizations, since /3 2 = - /3, , 

 and a % = 90 + «i« Hence 



c 2 = { sin 2 (/3, + j8) sin 2 (a, - a) + cos 2 (/3, - j3) cos 8 (a, - a)}^ ; 

 whence we find 

 m(c 2 ) slJl + sinS/S^inS/S + cosS^cosaacosS^cosSjS + sinSaisinSacosSjS^osSjSlmCc 2 ). (15) 



There will be no occasion to write down the value of tit (c 2 ,), since c t may be taken to refer 

 to either component. 



Let us pass now to the consideration of a group consisting of any number of independent 

 polarized streams, let 



2ttt (c'O = A, 2sin2/3ltl(c s ) = 5, 2cos2acos2/3ttt (c 2 ) = C,] 



2 sin 2 a cos 2 /3 til (c 2 ) = D, 



and let c l now refer to one of the components of the whole group ; then 



21tt (cf) = A + B sin2/3 x + Ccos2ai cos 2/3, + Z>sin2a, cos2/3j. (17) 



It follows that if there are two groups of independent polarized streams which are such as 

 to give the same values to each of the four quantities A, B, C, D defined by (l6), if the 

 groups be resolved in any manner whatsoever, which is the same for both, into two oppositely 

 polarized streams, the intensities of the components of the one group will be respectively equal 

 to the intensities of the components of the other group. Conversely, if two groups of oppo- 

 sitely polarized streams are such that when they are resolved in any manner, the same for both, 

 into two oppositely polarized streams, the intensities of the components of the one group are 

 respectively equal to the intensities of the components of the other group, the quantities A, B, 

 C, D must be the same for the two groups. For, if we take accented letters to refer to the 

 second group, the second member of equation (17), and the expression thence derived by 

 accenting A, B, C, D, must be equal independently of a, and /3 15 which requires that A', B', 

 C", JJf be respectively equal to A, B, C, D. 



Definition. Two such groups will be said to be equivalent. 



10. The theoretical definition of equivalence which has just been given agrees completely 

 with the experimental tests of equivalence. One of the most ready as well as delicate modes of 

 detecting minute traces of polarization, and at the same time determining qualitatively the 

 nature of the polarization, consists in viewing the light to be examined through a plate of 

 calcareous spar, or other crystal, cut for shewing rings, followed by a Nicol's prism. The 

 plane-polarized pencils respectively stopped by and transmitted through the Nicol's prism 

 consisted, on entering the crystal of pencils elliptically polarized in opposite ways; and the 

 nature of this elliptic polarization changes in every possible manner from one point to another 

 of the field of view. If these two streams of light be equivalent according to the definition 



