476 Royal Society : — 



system polarized perpendicularly to the plane of incidence will be 

 quite independent of each other, and the equality between the radiation 

 incident externally and that proceeding in the contrary direction, and 

 made up partly of a refracted and partly of an externally reflected radia- 

 tion, must hold good for each system separately. In this case, then, 

 (2) will split into two equations, each of the form (1), R now standing 

 for half the whole radiation, and 11', A', &c. standing for R^ A x , &c, 

 or R 2 , A 2 , &c, as the case may be. It need hardly be remarked that 

 the value of A is different in the two cases, and that R' has a value 

 which is no longer, as in the case of an isotropic medium, alike in all 

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

 internal radiation in anv eiven direction within a uniaxal crvstal, 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' mav be struck out. Aras;o Ions; ago showed 

 experimentally that light is reflected m the same proportion exter- 

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

 faces ; and the formulas 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 crvstal whenever the Diane of inci- 

 deuce 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 ] —A', equation (1) is re- 

 duced to W _ cos iU 



R~am' c y' (3) 



all planes passing through the axis are alike as regards internal propagation and 

 the polarization of the refracted rays. Hence, strictly speaking, the statement as 

 to the independence of the two systems of rays should be confined to the case in 

 which the principal plane is also a plane of crystalline symmetry. As, however, 

 the unsymmetrical phenomena were only brought out when the ordinary reflexion 

 was weakened, almost annihilated, by the use of oil of cassia, we may conclude 

 that under common circumstances they would be insensible. 



