Crystalline Reflexion and Refraction. 1 89 



they will take the following simple form : 



in which it is remarkable that the auxiliary quantities i> rji, &, 

 are exactly, for an ordinary medium, the components of the dis- 

 placement in the theory of Fresnel. In a doubly-refracting 

 crystal, the resultant of &, j!, ?i is perpendicular to the ray, and 

 comprised in a plane passing through the ray and the wave-nor- 

 mal. Its amplitude, or greatest magnitude, is proportional to 

 the amplitude of the vibration itself, multiplied by the velocity 

 of the ray. 



The conditions to be fulfilled at the separating surface of two 

 media were given in the abstract already referred to. From 

 these it follows, that the resultant of the quantities &, 171, , 

 projected on that surface, is the same in both media ; but the 

 part perpendicular to the surface is not the same ; whereas the 

 quantities , TJ, ?, are identical in both. These assertions, analy- 

 tically expressed, would give five equations, though four are 

 sufficient ; but it can be shown that any one of the equations is 

 implied in the other four, not only in the case of common, but 

 of total reflexion ; which is a very remarkable circumstance, 

 and a very strong confirmation of the theory. 



The laws of double refraction, discovered by Fresnel, but not 

 legitimately deduced from a consistent hypothesis, either by him- 

 self or any intermediate writer, may be very easily obtained, as 

 the author has already shown, from equations (2), by assuming 



=p cos a sin 0, rj = p cos ]8 sin 0, =p cos 7 sin 0, (5) 



where 27r, 7 N 



= -^- (Ix + my + nz - sf) ; 

 A 



but the new laws, which are the object of the present supple- 

 ment, are to be obtained from the same equations by making 



g = [p cos a sin + q cos a' cos 0), 



ij = e (p cos ]3 sin + q cos ]3' cos 0), ^ (6) 



^ = c (p cos y sin + q cos A' cos 0), 



