ON THE HYPOTHESIS OF UNDULATIONS. 377 



Ti = in cos w \/k {.V cos - y sin 9), and t-, = m cos n \^k (.i: sin 6 + y cos 0), 



tliese two planes evidently make angles 9 and j - t) with the plane of polarization of the original 



ray. Hence putting cot for - in equation (30), we find 



— , = tan^fl, 

 m 



and we also have — = /» + m . 



o 



Therefore m = — sin'^fy, and to' = — cos'^f?. 



2 2 



The polarized ray is consequently divided in general into two unequal rays, the values of which 

 are assigned by these equations. If 9 = iS", the two rays are equal; which accords with 

 experience. 



Suppose a polarized ray to be incident at the angle of complete polarization on a reflecting 

 surface, and let 9 be the angle which the plane of incidence makes with the plane of polarization 

 -of the incident ray. Then A being the portion of the ray transmitted without bifurcation, which 

 we will suppose to' be independent of 9, and / the portion bifurcated, the transmitted ray will be 

 A + Isuf9, and the reflec'.ed ray Icos'S. If another equal ray, polarized in a plane at right 

 angles to the plane of polarization of the former be incident in the same direction, the transmitted 

 portion will be ^ + lcos'0, and the reflected portion Isuv0. These two incident rays make up, 

 according to our Theory, a ray of common light, the transmitted portion of which is 2A, and the 

 reflected portion / cos^0 + /sin^6', or /, which is independent of 9, as we know from experience 

 it should be. Respecting the law above found for the intensities of the two rays into whicii 

 a polarized ray is separated. Sir John Herschel remarks in his Treatise on Light in the Encydo- 

 pcedia Metmpolitana, (Ar(. S50), " Wc must receive it as an empirical law at present, for whicli 

 any good theory of polarization ought to be capable of assigning a reason a priori.^'' Such a reason 

 is given by the Theory I am advocating. 



Two polarized rays formed by the separation into two parts of a polarized ray derived 

 immediately from common light, possess in some respects the properties of polarized rays of the 

 latter kind, for instance, the two rays pursuing the same paths will not interfere whatever be the 

 ilifl"erence of phase. This may be proved by the very same reasoning by which it has been already 

 jiroved that two rays of first polarization do not interfere, the reasoning being purposely adapted to 

 the case when m and m' are unequal. At the same time the rays of second polarization difier in 

 this respect, that if they meet in the same phase they compose a plane polarized ray. When 

 9 = 4-5", we found that the two rays were equal. Yet their composition would form a polarized ray, 

 whilst two equal ravs of the first ])olarization meeting in the same phase would compose a ray of 

 common light. 



According to this Theory circularly and elliptically polarized light consists of two oppositely 

 polarized rays differing in phase, tjie two rays when in the same phase constituting a polarized ray o* 

 the first kind. The reason Fresnel's Rhomb does not produce elliptically polarized light, when com- 

 mon light is used, is that common light may be siqiposed to consist of two rays in opposite polariza- 

 tions, which produce exactly com|)lenniitary effects. For the .same reason common light produces no 

 coloured rings liy transmission through a thin ])late of a uniaxal or biaxal crystal cut nearly ])er- 

 l.endicularly to its axis. Each of the polarized rays, of which coninw)M liglit may l)e supi)osed to 

 Vol., VIII, Paiit III. aC 



