8 
PROFESSOR CAYLEY’S SUPPLEMENTARY MEMOIR ON CAUSTICS. 
and which in the case of reflexion, or for /a=— 1, become 
A=l, 
B = 0, 
C=4x—4m, 
D=4y, 
E = -2(x 2 -+y 2 )-l+2 m 2 , 
viz. the equation of the variable circle is in this case 
cos 29+4 (x—m) cos 9+4y sin 9+2 to 2 — 1— 2(x 2 -\-y 2 )=§. 
3. Now in general for the equation 
A cos 29 +B sin 29+C cos 9+D sin 9+E=0, 
where the coefficients are any functions whatever of the coordinates (x, y), the equa- 
tion of the envelope is 
S 3 — T 2 =0, 
where 
S=12(A 2 +B 2 ) — 3(C 2 +D 2 ) + 4E 2 , 
- T = 2 7 A(C 2 - D 2 ) + 5 4BCD - (72(A 2 + B 2 ) + 9(C 2 + D 2 ))E + 8E 3 . 
4. Hence, substituting for A, B, C, D, E the above reflexion values, we find 
S = 12 - 48(0— my+tf) + 4(2m 2 — 1 - 2x 2 -2y 2 ) 2 , 
— T=432(0 -m) 2 -y 2 ) 
- 72^12 + 144(0 -mf+y 2 )^ {2m 2 -l-2x 2 -2y 2 ) 
+ 8( 2m 2 — 1 — 2+* — 2y 2 ) 3 . 
Writing in these equations 
0 — wa) 2 +y 2 = + -Yy 1 — 2 mx + m 2 , 
0 —mf —y- = 2x 2 — 2 mx + m 2 — (x 2 +y 2 ), 
then after some simple reductions, we find 
S = 1 6 { 0 2 -\-y~ — ■ w 2 — 1 ) 2 + 6m0 — m ) } ? 
T = 32 { 2 (x 2 +y 2 — nv — l) 3 + 1 8m(x — m)(x 2 -{-y 2 — m 2 — 1) — 2 7 (x—m ) 2 } , 
and thence 
S 3 — T 2 = 1 02 40 — m) 2 U, 
U= 4(x 2J r y 2 —m 2 —lf 
+ 4m\x 2 +y 2 — m 2 — l) 2 
+ 36 m (x 2 -\-y 2 —m 2 —l)(x—m) 
-27 (x—m) 2 
-Y 32m 3 (x—m), 
where 
