667] 
ON THE BICIRCULAR QUARTIC. 
233 
20. The relation between (x, y), (x X) y x ) may be expressed also in the two forms: 
1 - a 0 + x x )~ ¡3 (y + y0 + (/+ 00 xx x + (g + 0 X ) yy x + ~^— l (a - ai2/i -A- A«0 = 0, 
^2/i — x x y 
1 - cl x (x + x x ) - fi x {y + y x ) + (/ + 0) xx x + (g + 0)yyi + X '-- + Vx ( a i ~ Qy - A~ /3«)= 0. 
ir x y — xy x 
In fact, the first of these equations is 
{! + (/+ A) xx x + (g + 00 2/2/1} (xy x - «i2/) - {a O + «0 + A (2/ + 2/01 02/i ~ #i2/) 
+ {(« - *0 2/1 - (/3 - A) «1} (« 2 + y 2 ) = 0, 
which, by virtue of the original form of relation, is 
(/3 ~ A) (x ~ x i) - (a - a0 (y - y0 
- i 1 + (/+ 00 + (0 + 00 2/2/1Î 
6 — 0, 
-{«(«+ «0 + £ (2/ + 2/1)} («2/1 - <®12/) + {(a - «0 2/i — (A — A) «lî O 2 + y 2 ) = 0 ; 
or, in the first term, writing 
/3 — A A a — a, —a 
and in the third term 
0-0 x g + 00 0-0, /+0,’ 
a — a, = — 
this is 
(0 ~ 0Q « 
/+ 0i 
(1 + (/ + 00 + (y + 0, 
(/3 - A) = 
(0-00/3 
9 + 0i 
/3 (x - X x ) a (y - y0 
27+01 / + 0i 
— {a (a? + «0 + /3 (y + y0} (xy x - x x y) - ^ 2d - ®i} ^ + 2/ 2 ) = °- 
In this equation the coefficients of a and of /3 are separately = 0 : in fact, the coefficient 
of /3 is 
^1 (« - X x ) + (x~ x x ) yy x - (y + y0 («y a - ^y) + ¿»I (« 2 + y 2 ) 
= ^{ 1 -(/ + ^)^-(0 + 0Oyi 2 }-^; {!-(/+ 0)^ 2 -(y + 0)y 2 ] = O; 
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* +y 2 )R' =*x x + ,%-(/+ 6 x )xx x - (g + 6 X ) yy„ 
1 - a x x x - A2/1 - («1 2 + 2/i 2 ) A = <*ix + Ay - (/ + 0 ) xxi - (g + 0 ) yy x ; 
or, substituting for .R' and A their values, these are 
ViT = + /3y a - (/+ 00 ^1 - (9 + 00 2/2/i> Vil, = - a,« - A2/ + (/+ A ^1 + (0 + 0) yyx; 
c. x. 30