crystalline Reflexion and Refraction. 47 
If w be the angle which OQ makes with OP’, and w’ the angle which OQ’ 
makes with OP, so that 
cos w = cos a, cos a’, + cos B, cos 8, + COs Y2 COS ¥;,5 
/ / 
Cos w’ = cos a, cos a,, + cos 3, cos B,, + COS Y; COS Y,, 
we find, by the formule (35), (36), (41), 
cos w = cose {cos 0, cos 6, + sin 0, sin 6, cos (7, — ¢,) + sin 6, sin (7, — %) tan €}, 
cos w = cos € {cos 6, cos 65 + sin 0, sin 6; cos (7, — %) — sin @, sin (% — 2) tan f. 
Hence, observing the relations (37), we see that the right-hand member of the 
equation (40) is equal to the quantity 
sin? 2, (7s cos w — 7's’ Cos w’). 
But, by the property of the ellipsoid (Lemma IV), this quantity is zero ;* there- 
fore mM = 0, and the equation (39) becomes 
pyr? py MEE a et Ea (42) 
On each ray let a length, representing the velocity with which the light is 
propagated along it, be measured, as before, from the point O. The distances 
of the plane of a, y, from the extremities of these lengths will be proportional to 
the coefficients of the squares of the transversals in the preceding equation. For 
if we take, on the incident or reflected ray, a length equal to unity, its projection 
on the axis of z, will be cos 2, or »,; and if, on the refracted ray OT, we take a 
length equal to r, its projection on the same axis will be 
r (cos 2, cos € + sin @, sin 2, sin €), 
which is equal to ~,. Similarly, the length 7’, assumed on the other retracted 
ray, will have its projection equal to »;. The quantities by which the squares of 
the transversals are multiplied, in the equation (42), are therefore the corres- 
ponding ethereal volumes} which we may conceive to be put in motion by the 
different waves ; and as we suppose the density of the ether to be the same in 
* The equation m = 0 is the same as the equation (vii.) in my former paper.—Transactions 
R.1. A,, vol. xviii. p. 52. 
{ Ibid. p. 48. 
