erystalline Reflexion and Refraction. 29 
Pe at dy ae 
de dé cos a + = co sp + dé cos ¥; 
Py ae re 
dd F cosa’ + cos 6 ++ 7 C08 7/, 
ae —— a’ ce: a’ = nf! 
Te = Fp cose! + Tp os + FE cos 
attending to the relations (8), (B’), we find 
Pe ,dx ; , dy 5 dz ; 
7 ae (« qy 084 +5 iz cos 8 + c¢ re cos 7’), 
Gy dx , dy gle Ne a 
=" a cosa +b qe CB +e os 
Lage 
Gi as 
from which it appears that there is no accelerating force in the direction of a 
normal* to the wave, and consequently no vibration in that direction. Introdu- 
cing now the values of x, y, z from formule (£), the first two of these equations 
become 
~ = (a? cos °a’ + b cos *B’ + ec? cos *7’) = 
dt 
re —(a’ cos a cos a’ +b cos B cos f + c? cnr om) 
sf (6) 
a = (a@ cos*a + b’ cos *B + c° cos 2) a 
Qe 
d 
—(a® cosa cos a’ + 6? cos B cos fp’ +c’ cos + cos 7 ) — 
But as the axes of 2’, y’ are arbitrarily taken in the plane of 2’ y’, we may 
subject their directions to the condition 
@ cosa cos a’ + b? cos B cos 8’ + c* cosy cosy’ = 0; (7) 
* In the ingenious, but altogether unsatisfactory theory, by which Fresnel has endeavoured to 
account for his beautiful laws, the direction of the elastic force brought into play by the displace- 
ments of the ethereal molecules is, in general, inclined to the plane of the wave. He supposes, 
however, that the force normal to that plane does not produce any appreciable effect, by reason of 
the great resistance which the ether offers to compression.—Mémoires de l'Institut, tom. vii. p. 78. 
