PROFESSOR CAYLEY ON THE BICIRCULAR QHARTIC. 
451 
and in the third term 
this is 
a — 
/+« i ’ 
“(P-ft)= 
(fl-flOis 
^ + 9i ’ 
(1 + (f+60**, --T^) 
- { a(^+^)+% +yO Kw-x#) - (°t^ yi - (^ 2 +3/ 2 )=° ; 
and in this equation the coefficients of a and of 0 are separately =0 : in fact the 
coefficient of /3 is 
xx,(x - 0 - - (y +yi){xy, - +j^\ x l x2 +tf) 
— (/+^iK— (/+■ (#+% 2 i— 1 0 ; 
and similarly the coefficient of a is =0. 
And in like manner the second equation may be verified. 
21. The two equations are: 
1 — ax—fiy — (x 2 +y 2 )B! =ocx l +fiy l — (g+^yy* 
1— a^— (a$+$)R 1 =a 1 ar+/3 1 y— (/+0 )^i ; 
or, substituting for It', R x their values, these are 
x/O =ax 1 +Py 1 -(f-\-Q 1 )xx 1 -(g+Q 1 )yy 1 , */&, = — a,x — +(/+fl )x x^fa + Q % 
and similarly 
x /Q 1 =a 1 ^ 2 +/3 1 y 2 — (/+0 2 )^ 2 — (#+0 2 )^ 3 , «/ 0 2 = — + 
\/ Q 2 = «2^3 + /%3 - (/+ — (^ + to, \/^3 = - a i X 2 ~ &flj2 + (/+ + ( 9 + 5 2 )M3, 
\/^3=a 3 ^ +/3 a ,y — (f+8 )x z x ~(g+Q )y 3 y , O = — a x 3 — fi$ 9 +(f+Q a )x& +(g+Q 3 )y 3 y . 
Differentiating the equation 
(P—Pi)(x-Xi)—(*-<*i)(y-yi)+(Q-Qi)(xyi—Xiy)=o, 
we have 
[(P-p 1 )+(Q-QMdx--l(<*-<* i )+(Q-Q i )xiYy 
and writing herein 
7 (9 + 9) 7 j — (^ + ^1) 7 
dx=—^yd a , dx,~- yA, 
<%'= vs®*’ <&= -vsT*'*- 
3 s 2 
