VIII. 
ON FRESNEL’S WAVE SURFACE AND SURFACES RELATED THERETO. 
By WILLIAM BOOTH, M.A., Principal of the Hoogly College, Bengal ; 
Officiating Director of Public Instruction, Assam. 
{Read November 18, 1896.] 
2 2 2 
Ir V stand for aca cs Z 5, then the equation 
2 
P+ P+2-—e +P Eye 2 — 6 +eTP +2-—¢ 
V = 0 denotes ate surface, the radius vector of which represents the wave velocity 
in Fresnel’s theory ; the equation V = 0 may be written in the form 
fe G0,0,01— Os 
where J stands for 
ex (a +B) + 2a°e?y? + v(P+0)2— (a + c)(«@ +/+2y, 
and o, for a/e—P+ae/P—C4+/S/ 8-8 (+/+), 
o» for ca Ja — 8 — ae JP 2 P+ SE = oe (2? + oy? + 2), 
o,for —cx/a@—P + oe alla 
and o, for a/e@—P+ae/P—C—-/O—eO(e@+y +2); 
from which it follows that a, =0 touches the surface V =0 where the sphere meets 
the surface /=0. Now it is clear that the curve of intersection of o, = 0 with 
I= 0 is the curve of intersection of o, = 0 with C= 0, where C@ stands for 
2(P—¢) +9 (@—¢) +2 (0—0)—22( +2) V(@—0)(8 — ce) / ae. 
Again, the intersection of o, = 0 and C= 0 is Z= 0 and M= 0 for 
ac + LM = o,ac/ a — &, 
where Z stands for ca/e—P+azVP — &, 
and MY for an J/@—-E+cee/P —C+ae/e—e. 
Similar results obtain for the other three spheres. 
The above results were at first obtained geometrically by the doctrine of 
inversion applied to the results obtained in my other Paper,* for the surface V= @ 
* Proc. Roy. Dublin Society, Vol. vuz., Pt. v., No, 48, p. 381. 
TRANS. ROY. DUBL. SOC., N.S, * VOL. VI., PART VIII, 21 
