450 
PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
and then from the values of X, Y, K/, R, we have 
a — a, -j-R'je— RiX,=0, 
giving 
and similarly 
(3— pi +%—«#, = 0, 
(fl-flJ+R' — R, =0, 
(/3 -/3,)(x -Xi)-(a -Ui)(y -y,)+(0 -S,)(x^-^) = 0 ; 
(Pi - ft) (x, - x 2 ) - (a, - a 2 )(yi —y 2 ) + (0, - 5 2 )(x 1 y 2 -x 2 y 1 )= 0, 
(ft ■ - ft )(av — x 3 ) — (cc 2 - oi 3 )(y 2 — y 3 ) + (0 2 - 0 3 )(x 2 y 3 —x 3 y 2 )=0 f 
(Ps—P )(x 3 —x ) — (« 3 — a )(y 3 — y ) + (0 3 — 0 )(a?$ -xy 3 )=0, 
which are the relations connecting the parameters (x, y), (x„ y,), (x 2 ,y 2 ), (x 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 he finite ; but if from the first and second equations 
we eliminate (x,,y,), or from the third and fourth equations we eliminate (x 3 , y 3 ), we find 
in each case the same relation between ( x , y), (x 2 , y 2 ), viz. this is found to be 
00 2 = (1 - ux 2 —Py 2 )\ 1 - a 2 x-p 2 y) 2 ; 
and we have thus the singly infinite series of quadrilaterals. We have, of course, 
between (x„ y,), (x 3 , y 3 ) the like relation, 
0 , 03= (1 cL\X z fty 3 ) (1 a 3 x, ftyi) • 
20. The relation between (x,y), (x„ y,) may be expressed also in the two forms : 
l- a (^+x 1 )-/3(y+y 1 )+(/+3 1 )xx 1 + (y+5 1 )yy 1 +^^(^^ 1 y,-^H3 1 x 1 )=0, 
3-oi 1 (x+Xi)-p i (y+yi)+(f+d)xx 1 +(y-\-0)?yiy 1 -]-^^(ui-ay-pi-(3x)=0. 
In fact, the first of these equations is 
\ 1 + (/+ d P xx i + iff + bPyy* } (Wi — x <y) - j «(x ■ +■ , x, ) ■ + P(y +y, ) } (xyi—X{y) 
+ \(ot-u l )y 1 -(P-p 1 )x 1 }(x 2 +y 2 )=Q, 
which, by virtue of the original form of relation, is 
Hi+tm>*.+(s'+0 1 )w, 
-Mx+xJ+Piy+yPlixyi-Xiyj+Ka-aPyi-ip-PPxi^+y 2 )^; 
or, in the first term, writing 
_ P~Pi — a 
f+Q i 
