410 PROFESSOR STOKES, ON THE COMPOSITION AND RESOLUTION OF 



Y, Y", or X, X" denote a pair of oppositely polarized emergent streams, which came from 

 the incident streams F, F". A particular case of this theorem was assumed in the preceding 

 article, when it was stated that the pencils, polarized in perpendicular planes, which on coming out 

 of a crystal of calcareous spar are respectively stopped by and transmitted through a Nicol's 

 prism, went into the crystal oppositely polarized. 



The theorem will evidently be true of a train consisting of any number of crystalline plates, 

 each possessing the property of resolving the incident light into two oppositely polarized 

 streams, which are propagated within the medium with different velocities. For two oppositely 

 polarized streams incident on the first plate give rise to two emergent streams which fall oppo- 

 sitely polarized on the second plate, and so on. Since the number of plates may be supposed 

 to increase and their thickness to decrease indefinitely, while at the same time the steps by which 

 the doubly refracting nature of the plates alters from one to another become separately insensi- 

 ble, the theorem will be true if the whole train, or part of it, consist of a substance of 

 which the doubly refracting nature alters continuously, as for example a piece of strained 

 or unannealed glass. If, however, the train contain a member which performs a partial 

 analysis of the light, as for example a plate of smoky quartz, or a plate of glass inclined 

 to the incident light at a considerable angle, it will no longer be true that two pencils going 

 in oppositely polarized will come out oppositely polarized. 



13. Theorem. If two equivalent groups of polarized streams be resolved in any 

 manner, which is the same for both, into two oppositely polarized groups, and these be re- 

 combined after the phase of one of the components has been retarded by a given quantity 

 relatively to that of the other, the two groups of resultant streams will be equivalent. 



Let the groups of resultants be each resolved in any manner into two oppositely polarized 

 streams, and call these 0, E. By Art. 11, if 0', E' be the streams which furnish 0, E respec- 

 tively, 0', E' are oppositely polarized. Now by Art. 3, the intensities of 0, E are the same 

 respectively as those of (/, E' ; but these are the same for the one group as for the other, by 

 the definition of equivalence. Therefore the intensities of 0, E are the same for the one group 

 as for the other ; but 0, E are any two oppositely polarized components of the resultant groups ; 

 therefore these groups are equivalent. 



Hence, if two equivalent groups be transmitted through a crystalline plate, the emergent 

 groups will be equivalent ; and by the same reasoning as in Art. 12 the theorem may be 

 extended to an optical train consisting of any number of crystalline plates, pieces of unannealed 

 glass, &c. 



14. Theorem. If two equivalent groups be resolved in any manner, the same for both, 

 into two polarized streams, the intensities of the components of the one group will be respec- 

 tively equal to the intensities of the components of the other group. 



The proof of this theorem is very easy. It is sufficient to treat the general expressions (IS) 

 exactly as in Art. 9, only that no relations are to be introduced between ec 2 , /3 2 , and a, , fi v Since the 

 coefficient of c 2 in (13) is constant, if we consider as variable such quantities only as may change 

 in passing from the one group to the other, it will be easily seen that the intensity of either 



