232 
ON THE BICIRCULAR QUARTIC. 
[667 
and then, Vii denoting as above a determinate value, positive or negative as the case 
may be, of the radical, and similarly ViF, VfF, Vil 3 denoting determinate values of 
these radicals respectively, each radical having its own sign at pleasure, we further 
write 
(x 2 + y 2 ) R' = 1 — ax — ¡3 y — Vil , 
(a?! 2 + yr) R\ — 1 a x x x /3i2/i Vilj, 
(orj 4- yi) R.: = 1 - a 2 «2 - /3 2 y 2 - Vil 2 , 
+ y 3 -) -B 3 = 1 — a. A x 3 /^ 3 y 3 Vii 3 , 
O1 2 4- y/ 2 ) /¿j = 1 - a, a?! - &y r 4- Vilj, 
(& 2 2 4- y 2 2 ) -B2 = 1 — «2^2 — /3 2 y 2 4- Vil 2 , 
Os 2 + 2/.7) R s =l- a 3 x 3 - /3 3 y 3 + Vil 3 , 
(& 2 4- y 2 ) R = 1 — a x — (3 y 4- VO ; 
and this being so, we must have 
X =a+R'x=a 1 +R 1 x 1 , Y = /3 + R' y =/3 l +R 1 y 1 , R' =£(X 2 + F 2 —Jc ), 12 1 =|(X 2 4- F 2 -k x ), 
X 1 ==a 1 4-i^'#i=a 2 +-R 2 a; 2 , F = /3 X +/¿/y 1=/3,4-y,, Ri=^(Xi+Yi-k^, R 2 =^(Xi+Yi-L), 
X,=a 2 +R 2 'x 2 =a s +R 3 x 3 , Y 2 = ^+R 2 , y 2 =/3 3 +R 3 y s , R 2 '=%(X 2 2 + Y 2 2 -k 2 ), R 3 =^(Xi+ F 2 2 -& 3 ), 
X 3 =a 3 +R 3 'x 3 =a +R x , F 3 = /3 3 4-R 3 'y 3 =/3 +R y , Ri=%(X 3 2 + Y 3 2 -k :i ), R =|-(X 3 2 + Y 3 2 -k); 
and then from the values of X, F, R', R, we have 
a — a, 4- = 0, 
giving 
and similarly 
£ - Pi + R'y- Riyi = 0, 
(0 - 6»,) + R! -R, =0, 
(Æ — a?i) — (a -otiKy — yi) + (^ — $i)(&yi — X \V ) = 6 ; 
(A - &) (a?! - a? s ) - (a, - a 2 ) (y! - y 2 ) 4- (F - 0 2 ) (^y 2 - ^1) = 0, 
(/3 2 - /9 3 ) (a? 2 - a? 8 ) - (a 2 - a 3 ) (y 2 - y s ) + (0 2 - 6 3 ) (x 2 y 3 - æ 3 y 2 ) = 0, 
(& - /3 ) (x s - x ) - (a 3 - a ) (y, - y ) 4- (0 3 - 0 ) 0 3 y - « y 3 ) = 0, 
which are the relations connecting the parameters (#, y), (a^, y x ), {x 2 , y 2 ), (a? 3 , y 3 ) of the 
quadrilateral. 
19. We have thus apparently four equations for the determination of four quantities, 
or the number of quadrilaterals would be finite; but if from the first and second 
equations we eliminate (x x , yi), and if from the third and fourth equations we eliminate 
0 3 , y 3 ), we find in each case the same relation between (x, y), (x 2) yi), viz. this is 
found to be 
ililo = (1 - a x 2 - /3 y 2 ) 2 (1 - a 2 x - /3 2 y) 2 ; 
and we have thus the singly infinite series of quadrilaterals. We have, of course, between 
(x x , y0> ( x 3> Vi) the like relation, 
iliil, = (1 - a x x 3 - Ay 3 ) 2 (l - 0&! ~ /3 3 yi) 2 -