230 Royal Irish Academy. 



and if we further put 



t—^Hl — d Jl, M=lii— Hi, P — ih.— d JHl (3.) 



dz dy' dx dz' dy dx' 



they will take the following simple form : — 



^ll = - a« X, ?5i = - fi« Y, £Ii = - c 2 Z, . . (4.) 

 rf*« d* 2 if* ' v y 



in which it is remarkable that the auxiliary quantities £„ ij„ £, are 

 exactly, for an ordinary medium, the components of the displace- 

 ment in the theory of Fresnel. In a doubly refracting crystal, the 

 resultant of £ p tj,, £, is perpendicular to the ray, and comprised in a 

 plane passing through the ray and the wave normal. Its amplitude, 

 or greatest magnitude, is proportional to the amplitude of the vibra- 

 tion 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 £„ t) lt £,, projected on 

 that surface, is the same in both media ; but the part perpendicular 

 to the surface is not the same ; whereas the quantities £, ij, £ are 

 identical in both. These assertions, analytically 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 re- 

 markable circumstance, and a very strong confirmation of the theory. 



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

 timately deduced from a consistent hypothesis, either by himself or 

 any intermediate writer, may be very easily obtained, as the author 

 has already shown, from equations (2.), by assuming 



£ = p cos a sin f , ij = p cos (3 sin <p, K—p cos y sin <p, . (5.) 

 where <p = — (Ix + my + nz — st); 



A 



but the new laws, which are the object of the present supplement, 

 are to be obtained from the same equations by making 



£ = e (p cos a sin tp + q cos a' cos <p) ~\ 



y = e (p cos /3 sin <p -f q cos |3' cos p) >• (6.) 



£ = e (p cos y sin <p + q cos y' cos tp) J 

 where <p has the same signification as before, and 



£ _ e - — (/* + 9 y + h z) 



the vibrations being now elliptical, whereas in the former case they 

 were rectilinear. In these elliptic vibrations the motion depends not 

 only on the distance of the vibrating particle from the plane whose 

 equation is 



lx + my + nz = 0, (7.) 



but also on its distance^from the plane expressed by the equation 



fx + gy + hz = 0; (8.) 



