STREAMS OF POLARIZED LIGHT FROM DIFFERENT SOURCES. 411 



component depends on the same four constants A, B, C, D as before ; and since these have the 

 same values for equivalent groups, it follows that in the most general mode of resolution the 

 components of any two such groups have the same intensities respectively. 



15. Theorem. If two equivalent groups be each resolved in the same manner into two 

 oppositely polarized streams, and these be recombined after their vibrations have been 

 diminished in two different given ratios, the resultant groups will be equivalent. 



Let each of the resultant groups be resolved into two oppositely polarized streams 0, E, 

 and let 0', E' denote the streams which furnished O, E respectively. It is easily seen that the 

 streams O', E are both polarized, though not in general oppositely. Were there any occasion 

 to determine the nature of the polarizations, it might easily be done by following a process the 

 reverse of that by which the modified are deduced from the original groups. Thus, if it were 

 required to determine the polarization of (? , we should resolve O into streams oppositely 

 polarized in the manner originally given, augment the vibrations in ratios the inverse of those 

 in which they are actually diminished, and recombine the streams so obtained. For our present 

 purpose, however, it is sufficient to observe that the polarizations of O', E' depend only on the 

 mode of resolution, and not on the nature of the original group, and that therefore they are the 

 same for the one group as for the other. The intensity of 0' is not the same as that of 0, as 

 in the case considered in Art. 13, but bears to it a ratio depending only on the mode of resolu- 

 tion, and therefore the same for each of the two original groups. The same is true of the 

 intensity of E' compared with that of E. Now by Art. 14 the intensities of 0', E' are the 

 same for the two original groups, and the intensities of 0, E bear to those of O', E' respectively, 

 ratios the same for the two groups : therefore the intensities of 0, E are the same for the 

 two final groups; but 0, E are any two oppositely polarized components of these groups; 

 therefore these groups are equivalent. 



16. It follows from this theorem that no partial analysis of light, such, for example, as 

 would be produced by reflexion from the surface of glass or metal, or by transmission through a 

 doubly absorbing medium, can from equivalent groups produce groups which are not equivalent 

 to each other ; and we have seen already that this cannot be done by means of the alteration 

 of phase accompanying double refraction. It follows, therefore, that equivalent groups are 

 optically undistinguishable. 



In proving this property of equivalent groups, it has been supposed that the polarizations 

 of the two streams into which any group was resolved were opposite, such being the case in 

 nature. But did a medium exist such that the two streams of light which it transmitted inde- 

 pendently were polarized otherwise than oppositely, it would still not enable us to distinguish 

 between equivalent groups. 



17- The experimental definition of common light is, light which is incapable of exhibiting 

 rings of any kind when examined by a crystal of Iceland spar and an analyzer, or by some 

 equivalent combination. Consequently, a group of independent polarized streams will to- 

 gether be equivalent to common light when, on being resolved in any manner into two 

 Vol. IX. Part III. 53 



