128 On the Laws of Crystalline 



theories that the more they are studied, the more simple they 

 appear to be. And we may add, that a close examination of 

 such theories always meets with its reward, in the unexpected* 

 consequences which present themselves to view. Nothing can be 

 simpler than the laws of double refraction, as they were deli- 

 vered by Fresnel; yet the properties of his wave surface still 

 continue to furnish the geometer with beautiful and curious re- 

 lations. So we may hope that a little more time, devoted to 

 the laws of reflexion, will not be spent in vain. They promise 

 to supply many other theorems, not undeserving of attention, 

 though perhaps not as simple and comprehensive as those that 

 have already been made known. 



From the nature of the rules which we have given for treating the question of 

 reflexion at either surface of the crystal, it follows that the final equation, for 

 determining the position of a transversal, is always linear, though the equation 

 of vis viva is of the second degree. This result very strongly confirms the theory ; 

 but it shows, at the same time, that the law of the preservation of vis viva is not to 

 be regarded as an ultimate principle, but rather as a consequence of some elemen- 

 tary law not yet discovered. 



It now appears that the conjectures put forward in the note, p. 93, were hasty, 

 and that there was some mistake in the calculations which gave rise to them. It 

 js scarcely necessary to mention, that the sheet in which that note is found was 

 printed off before I had obtained the result announced in the subsequent note, 

 p. 111. Various delays occurred while my Paper was going through the press; 

 and I took advantage of them to increase its value, by appending notes on some 

 of the questions which I had overlooked or omitted in the first consideration of 

 the subject. 



* As an instance of this, it may be mentioned, that the conclusion arrived at 

 in the note, p. Ill, was wholly unexpected. And in verifying the equation (viz.), 

 an unexpected and useful theorem was obtained ; for it became necessary to find 

 a manageable expression for the tangent of the angle c which the wave normal 

 makes with the ray. This expression is wanted in applying the formulae (34) and 

 (35) to biaxal crystals, and therefore I shall make no apology for introducing it 

 here. 



Having described a sphere concentric with the wave surface, let the wave 

 normal OP and the two optic axes (which are the nodal diameters of the index 

 surface) be produced from the centre to meet the sphere in the points P t , A, A t , 

 respectively, thus marking out the angles of a spherical triangle PAA t . The same 

 wave normal may belong to two different rays ; and if we select one of these 

 rays, its transversal must lie in a plane drawn through the wave normal, and 

 bisecting either the internal angle APA t of the spherical triangle, or the ex- 



