404 PROFESSOR STOKES, ON THE COMPOSITION AND RESOLUTION OF 



the equations (11) differ from (10) only in this, that what are regarded as the first and second 

 principal planes of the second stream when equations (10) are satisfied, are accounted respec- 

 tively the second and first when (ll) are satisfied. 



Two streams thus related may be said to be oppositely polarized. Two streams of plane- 

 polarized light in which the planes of polarization are at right angles to each other, and two 

 streams of circularly polarized light, one right-handed and the other left-handed, are particular 

 cases of streams oppositely polarized. 



In the reasoning of this article, nothing depends upon the precise relation between the two 

 polarized streams and the original stream. All that it is necessary to suppose is, that the two 

 polarized streams came originally from the same polarized source, so that the changes in epoch 

 and intensity, that is, the changes in the quantities e, c, are the same for the two streams. 

 Nothing depends upon the precise nature of these changes, which may be either abrupt or con- 

 tinuous, but must be sufficiently infrequent if abrupt, or sufficiently gradual if continuous, to 

 allow of our regarding c and e as constant for a great number of successive undulations. Our 

 results will apply just as well to the disturbance produced by the union of two neighbouring 

 streams coming originally from the same polarized source, but having had their polarizations 

 modified, as to that produced by the union, after recomposition, of the components of a single 

 polarized stream. Since the resulting intensity is independent of S, it follows that two oppo- 

 sitely polarized streams coming originally from the same polarized source are incapable of inter- 

 fering, but two streams polarized otherwise than oppositely necessarily interfere, to a greater or 

 less degree, when the difference in their retardation of phase is sufficiently small. Of course 

 the interference here spoken of means only that which is exhibited without analyzation. 



4. Two interfering streams may be said to interfere perfectly when the fluctuations of in- 

 tensity are the greatest that the difference in the intensities of the interfering streams admits of, 

 so that in case of equality the minima are absolutely equal to zero. Referring to (9), we see 

 that in order that this may be the case the maximum value of the coefficient of 2c,c 2 must be 

 equal to 1. Now the maximum value of A cos$ + /3 sin 5 is ^(A* + B 1 ), and therefore we 

 must have 



cos 2 (a 2 - a^ cos 2 (/3 2 - &) + sin 5 (a 2 - aj sin 2 (/3 2 + /3 t ) = 1 = cos 2 (a 2 - a,) + sin 8 (a 2 - a,), 

 whence, 



cos 2 (a 2 - cti) sin 2 (/3 2 - /3j + sin 2 (a 2 - aj) cos" (/3 2 + /3 t ) ■ 0, 



which lead to the very same conditions that have been already discussed in Art. 2. Hence two 

 polarized streams coming from the same polarized source are capable of interfering perfectly if 

 the polarizations are the same, not at all if the polarizations are opposite, and in intermediate 

 cases of course in intermediate degrees. 



5. When a stream of polarized light is resolved into two oppositely polarized streams, 

 which are again compounded after their phases have been differently altered, we have from (9), 

 taking account of (10) or (11), 



/ = c, 2 + c/, (12) 



so that the intensity of the resultant is equal to the sum of the intensities of the compo- 



