359] 
A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
455 
2. The equation may be written 
A cos 20 + 5 sin 20 + G cos 0 + D sin 0 + E = 0, 
where 
A = 1, 
5 = 0, 
C = 4 y?x — 4 m, 
D = 4 y?y, 
E — — 2/U (« 2 + y 2 ) — 2/x 2 + 1 + 2m 2 , 
and which in the case of reflexion, or for /¿ = — 1, become 
-4 = 1, 
5=0, 
(7=4« — 4m, 
5 = 4y, 
E = - 2 (« 2 + y 2 ) — 1 + 2m 2 , 
viz. the equation of the variable circle is in this case 
cos 20 + 4 (« — m) cos 0 + 4y sin 0 + 2m 2 — 1 — 2 (« 2 + y 2 ) = 0. 
3. Now in general for the equation 
A cos 20 + B sin 20 + C cos 0 + D sin 0 + E = 0, 
where the coefficients are any functions whatever of the coordinates («, y), the equation 
of the envelope is 8 s — T 2 = 0, where 
S = 12 (ri 2 + 5 2 ) - 3 (C 2 + 5 2 ) + 4E\ 
- T = 27 A(C>- 5 2 ) + 545(75 - (72 (ri 2 + 5 2 ) + 9 ((7 2 + 5 2 )) E + 85 3 . 
4. Hence, substituting for A, B, G, D, E the above reflexion values, we find 
8= 12-48 ((« - m) 2 + y 2 ) + 4 (2m 2 - 1 - 2« 2 - 2y 2 ) 2 , 
-5= 432 ((« — m) 2 — y 2 ) 
— 72 (12 + 144 ((« — m) 2 + y 2 )) (2m 2 — 1 — 2« 2 — 2y 2 ) 
+ 8 (2m 2 — 1 — 2« 2 — 2y 2 ) 3 . 
Writing in these equations 
(« — m) 2 + y 2 = « 2 + y 2 — 2 mx + m 2 , 
(« — m) 2 — y 2 = 2« 2 — 2 m« + m 2 — (« 2 + y 2 ),