99 



is a very remarkable circumstance, and a very strong con- 

 firmation of the theory. 



The laws of double refi-action, discovered by Fresnel, 

 but not legitimately deduced from a consistent hypothesis, 

 either by himself or any intermediate writer, may be very 

 easily obtained, as the author has already shown, from equa- 

 tions (2), by assuming 



C r: /; cos a sin ^, rj =: p cos /3 sin ^, ^rrpcosysin^, (5) 

 where 



<j> ■=! -^ (Ix + mt/ + nis — St) ; 

 A 



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

 plement, are to be obtained from the same equations by 

 making 



^ — 6{p cos a sin ^ + 5- cos a' cos ^), l 



r\ ■=. e{p cos j3 sin ^ -|- §- cos /3' cos (f), j" (G) 



^ = £ (p cos y sin ^ -1- g- cos j' cos (f), J 



where ^ has the same signification as before, and 



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 



Ix -f my -\-nss — 0, (7) 



but also on its distance from the plane expressed by the 

 equation 



fx + gy + hz = 0; (8) 



and if the constants in the equation of each plane denote the 

 cosines of the angles which it makes with the coordinate 

 planes, we shall have A for the length of the wave, and s for 

 the velocity of propagation; while the rapidity with which 



