jME. a. CAYLEY’S MEMOIE UPON CAUSTICS. 
289 
circle and the equation of the reflected ray is by the general formula, 
(bcc—a(5)(aw-\-(2y—c^)-]-{^(x,—oe^)(aa-\-b(3—c^)=0; 
or arranging the terms in a different order, 
(bx-\-a^)(oc,‘^—(D^)-{-2(by—ax)cc^—c%b-\-^)oi-\-c%a-\-w)(3=0. 
Writing now a=ccos^, (3=csin^, the equation becomes 
(bx-\-ai/) cos 20-\-(by — ax) sin 2d—{b-\-y)c cos 0-\-{a-\-x)c sin ^=0, 
where ^ is a variable parameter. 
Now in general to find the envelope of 
A cos 2^+B sin 2(f+C cos sin ^4'E=0, 
we may put e‘^=z, which gives the equation 
(A-B^y^-(C-D^y+2Ez^^-(C^-D^>+(A+B^)=0, 
and equate the discriminant to zero : this gives 
(4I)*-27(-8J)^=0, 
where 
4I=4(A^+B^)-(C^+D^)+|E^ 
-8J=A(C^-D^)+2BCD-{8(A^+B^)+(C^+D^)}P4-AE*, 
and consequently 
{4(A^+B^)-(C^+D^)+|E^}^-27{A(C^-D^)+2BCD 
-(8(A''+B’)+(C’+D=))iE+*E>}’=0; 
and substituting for A, B, C, D, E them values, we find 
{ 4(«^ + +/) — ® )^ + ( y + ^T) y—27{bx— ayjis^ -\-y^—a^—b^)—0 
for the equation of the caustic m the case of rays proceeding from a point and reflected 
at a circle : the equation was first obtained by St. Laukent. 
It will be convenient to consider the axis of x as passing through the radiant point ; 
this gives 5=0 ; and if we assume also c=l, the equation of the caustic becomes 
{{A:a^—\){x^-\-y^)—2ax—a^Y—27 c^y\x ^ + 
XIX. 
Eeverting to the equation of the reflected ray, and putting, as before; c=l, 5=0, this 
becomes 
/ n ^ . a cos 25— cos d , ^ 
(-2acos^+l>+ y+a=0. 
Differentiating with respect to we have 
( - 2« sin d)x+ ^ = 0 ; 
and from these equations 
x= 
a® cos 5(1 + 2 sin® 5) —a 
1 — 3a cos 25 + 2a® 
2 a® sin^ 5 
y 1 —3a cos 25 + 2a®’ 
2q 
MDCCCLVII. 
