STREAMS OP POLARIZED LIGHT FROM DIFFERENT SOURCES. 409 



given in the preceding article, they will present exactly the same appearance on being viewed 

 through a crystal followed by a Nicol's prism or other analyzer. 



11. Theorem. Let a polarized stream be resolved into two oppositely polarized streams ; 

 let the phase of vibration of one of the streams be altered by a given quantity relatively to that 

 of the other, and let the streams be then compounded. If the polarization of the original 

 stream be now changed to its opposite, the polarization of the final stream will also be changed 

 to its opposite. 



The straightforward mode of demonstrating this theorem, by making use of the general 

 expressions, would lead to laborious analytical processes, which are wholly unnecessary. For 

 the formulae which determine the components of a given stream are expressed by simple equa- 

 tions, so that the results are unique, and accordingly whenever we can foresee what the result 

 will be, it is sufficient to shew that the formula? themselves, or the geometrical conditions of 

 which the formulae are merely the expressions, are satisfied. 



For shortness' sake call the original stream X, and its components 0, E. Let p be the 

 given quantity, positive or negative, by which the phase of vibration of O is retarded relatively 

 to that of E. Let o, e denote the streams O, E after the changes of phase, and Y the stream 

 resulting from their reunion. Conceive now all the vibrations with which we are concerned to 

 be turned in azimuth through 90°. This will not affect the geometrical relations connecting com- 

 ponents and resultants. Let X', 0' , E', o, e', Y' be the streams which X, O, E, o, e, Y thus 

 become. The streams O', E" are evidently polarized in the same manner respectively as E, O, 

 except that right-handed is changed into left-handed, and vice versa ; and in passing from fl to 

 o the phase is retarded by p. Now conceive the direction of motion of a given particle reversed, 

 the motions of all other particles being derived from that of the first according to the general 

 law of wave propagation. The relations between components and resultants will evidently not 

 be violated ; and if X " , O", E", o", e", Y" denote the streams into which X', O', E", o, e', 

 Y' are thus changed, it is evident that the polarizations of X" , 0", E', o", e", Y" are respec- 

 tively opposite to those of X, O, E, o, e, V. But on account of the reversion in direction of 

 motion it is plain that there is reversion as regards acceleration and retardation of phase, so 

 that in passing from the pair 0", E" to the pair o", e" the phase of 0" is accelerated by p 

 relatively to the phase of E". 



Hence the stream X", polarized oppositely to X, is resolved into two E", 0", polarized 

 in the same manner respectively as O, E, which are recotnpounded after the phase of the 

 one polarized in the same manner as O has been retarded by p relatively to the phase of the 

 other, and the result is Y", a stream polarized in a manner opposite to Y, which proves the 

 theorem. 



12. This theorem may be applied to the case of light transmitted through a slice of a 

 doubly refracting crystal, and shews that two streams going in oppositely polarized come out 

 oppositely polarized. Also, since nothing in the demonstration depends upon the order in 

 which the decompositions and recompositions take place, it is immaterial whether X, X" 

 denote a pair of oppositely polaiized incident streams, which give rise to the emergent streams 



