Crystalline Reflexion and Refraction. 51 



showing that the sign of A is always the same as the sign of cos S^. Now as B^ 

 differs from ^^ by a right angle, we will suppose 



0, = &,+ 9O», (33) 



and then we shall have sin 0^ = eos^g, algebraically as well as numerically. Thus 

 we see that, by adopting these conventions, the value of h in (27) will have the 

 proper sign. Therefore, substituting this value of A in formulae (13), we obtain 



sinVtane 



imO.— cos U — 1^ tan BA 2r^- — r-^ 



1 sin\tan£ 



tan e^-=. —cos (<,+t2)tan02+ 



costfgSinCt,— g 



(34) 





These formulae give the uniradial directions, or the positions of the incident 

 and reflected transversals, when the sole refracted ray is that with which we 

 have been occupied. The like directions, when the other ray exists alone, will 

 be given by the formulte 



sin^/, tan e' 





tan0'=cos(«— 4)tan0'2-l- 



cos0'2sin(£,+t'j)' 



„; / I / \. „, . sinV„tane' 



tan£''3=— cos(t,+('Jtan£i'^+ ^ 



(35) 



cos0',sin(t,-i',)' 



where all the quantities, except i^, which remains the same, are marked with 

 accents, to show that they belong to the second refracted ray. 



The uniradial directions having been found by these equations, the relative 

 magnitudes of the uniradial transversals are determined by equations (6). When 

 the incident transversal is not uniradial, it is evident, as we said before, that it 

 may be resolved* in the two uniradial directions ; that each component transversal, 

 as if the other component did not exist, will furnish a refracted ray and a partial 



* That, if an incident transversal be resolved in any tvfo directions, the reflected and refracted 

 transversals arising from it will be the resultants of those which would arise from each of its compo- 

 nents separately, is a principle which appears very evident, insomuch that we can hardly suppose it 

 to be untrue, without doing violence to our physical conceptions. Nevertheless, it is necessary to 

 prove that this principle is not contrary to the law of vis viva ; for though the vis viva may be pre- 

 served by each set of components, (as it is when these are uniradial,) yet we cannot therefore con- 



H 2 



