1 45] A MEMOIR UPON CAUSTICS, 
379 
and the values of the coefficients are 
4 = 1, 
B = 0, 
C = 4/t 2 ir, 
D = 4/a 2 ?/, 
E = — 2/a 2 (it 2 + y 2 ) — 2/a 2 + 1. 
Substituting these values in the equation 
{12 (A 2 + B 2 ) - 3 (<7 2 + D 2 ) + 4£ 2 j 3 
- {274 (C 2 - D 2 ) + 54>BCD - (72 (4 2 + £ 2 ) + 9 (C 2 + D 2 )) E + 8# 3 } 2 = 0, 
the equation of the envelope is found to be 
16 {(1 — /a 2 + /a 4 ) — (/a 2 + /a 4 ) (it 2 + y 2 ) + /a 4 (it 2 + z/ 2 ) 2 } 3 
4 — 6/a 2 — 6/a 4 + 4/a 6 j 2 
— (6/A 2 + 3/A 4 + 6/A 6 ) (it 2 + Z/ 2 ) — 27/A 4 (it' 2 — Z/ 2 ) J 
- - [ *= o, 
— (6/A 4 + 6/A 6 ) (it 2 + y 2 ) 2 
+ 4/a 6 (it 2 + z/ 2 ) 3 
which is readily seen to be only of the 8th order. But to simplify the result, write 
first (x 2 + y 2 —1) +1, and 2if 2 — 1 — (a? Ay 2 — 1) in the place of x 2 + y 2 and x 2 — y 2 respec 
tively, the equation becomes 
4{(1 - /A 2 ) 2 - /A 2 (1 - /A 2 ) (if 2 + y 2 - 1) + /A 4 (if 2 + y 2 ~ l) 2 } 3 
' 2 (1 - /A 2 ) 3 ) 2 
- 3/A 2 (1 - /A 2 ) 2 (x 2 A y 2 - 1) - 27/aV 
—< y — 0. 
- 3/a 4 (1 -/A 2 )(if 2 + z/ 2 ~l) 2 
+ 2/a 6 (if 2 Ay 2 - l) 3 
Write for a moment l—/j? = q, y?(x 2 Ay 2 — 1) = p, the equation becomes 
4 (q 2 — qp A p 2 ) 3 — (2q 3 — Sq 2 p — Sp 2 + 2p 3 — 27/A 4 if 2 ) 2 = 0 ; 
or developing, 
4 (</ 2 — gp + p 2 ) 3 — (2q 3 — 3q 2 p — 3qp 2 + 2p 3 ) 2 
+ 54 (2^ 3 — 3q 2 p — 3qp 2 4- 2p 3 ) p}x 2 — 729/6^ = 0, 
and reducing and dividing out by 27, this gives 
q 2 p 2 (p — </) 2 + 2 (p + <?) (2p — q){p — 2q) y*x 2 — 27/A 8 it 4 = 0, 
48—2