Crystalline Reflexion and Refraction. 1 8 1 



But, by the property of the ellipsoid (Lemma IV.), this quantity 

 is zero ;* therefore M = 0, and the equation (39) becomes 



/Ui n 2 = m rV + ,W + /x'.rV. (42) 



On each ray let a length, representing the velocity with 

 which the light is propagated along it, be measured, as before, 

 from the point 0. The distances of the plane of # 2A> from the 

 extremities of these lengths will be proportional to the coeffi- 

 cients of the squares of the transversals in the preceding equa- 

 tion. For if we take, on the incident or reflected ray, a length 

 equal to unity, its projection on the axis of s will be cos i { or p\\ 

 and if, on the refracted ray OT, we take a length equal to r, its 

 projection on the same axis will be 



r (cos i-i cos e + sin 2 sin ? 2 sin e), 



which is equal to ju 2 . Similarly, the length /, assumed on the 

 other refracted ray, will have its projection equal to juV The 

 quantities by which the squares of the transversals are multi- 

 plied, in the equation (42), are therefore the corresponding ethe- 

 real volumesf which we may conceive to be put in motion by the 

 different waves ; and as we suppose the density of the ether to 

 be the same in both media, the equation expresses a principle 

 analogous to that of the preservation of vis viva.% 



By giving a certain direction to the incident transversal, 

 that is, by polarizing the incident ray in a certain plane, we may 

 make one of the refracted rays disappear. If OT be the ray 

 which remains, we have r' 2 = 0, and the equations (34) and (38) 

 become 



* The equation M = is the same as the equation (vn.) in my former Paper. 

 Transactions, R.I. A., VOL. xvm. p. 52 (supra, p. 112). 



t Ibid., p. 48 (supra, p. 106). 



J A similar equation of v is viva holds when the light passes out of a crystal into 

 an ordinary medium. 



