540 



directions. In determining according to Mr. Stewart's principles the 

 internal radiation in any given direction within a uniaxal crystal, no 

 limitation is introduced by the restriction of equation (1) to a principal 

 plane, since we are at liberty to imagine the crystal bounded by a plane 

 perpendicular to that containing the direction in question and the 

 axis of the crystal. 



Mr. Stewart further reduces equation (1) by remarking that in an 

 isotropic medium, as we have reason to believe, A'=A, and that the 

 same law probably holds good in a crystal also, so that the equal 

 factors 1 A, 1 A' maybe struck out. Arago long ago showed 

 experimentally that light is reflected in the same proportion exter- 

 nally and internally from a plate of glass bounded by parallel sur- 

 faces ; and the formulae which Fresnel has given to express, for the 

 case of an isotropic medium, the intensity of reflected light, whether 

 polarized in a plane parallel or perpendicular to the plane of incidence, 

 are consistent with this law. In a paper published in the fourth 

 volume of the Cambridge and Dublin Mathematical Journal (p. 1), 

 I have given a very simple demonstration of Arago's law, based on 

 the sole hypothesis that the forces acting depend only on the positions 

 of the particles. This demonstration, I may here remark, applies 

 without change to the case of a crystal whenever the plane of inci- 

 dence is a plane of optical symmetry. It may be rendered still more 

 general by supposing that the forces acting depend, not solely on the 

 positions of the particles, but also on any differential coefficients of the 

 coordinates which are of an even order with respect to the time, a 

 generalization which appears not unimportant, as it is applicable to 

 that view of the mutual relation of the ether and ponderable matter, 

 according to which the ether is compared to a fluid in which a number 

 of solids are immersed, and which in moving as a whole is obliged to 

 undergo local dislocations to make way for the solids. 



On striking out the factors 1 A and 1 A', equation (1) is re- 



duced to 



' 



' 



In the case of an isotropic medium, R and R' are alike in all 

 directions, and therefore the ratio of cos i fy to cos t' 0' ought to be 

 independent of i, as it is very easily proved to be. The same applies 

 to a uniaxal crystal, so far as regards the ordinary ray. But as re- 



