PROFESSOR CAYLEY ON THE BICIRCULAR QTJARTIC. 
449 
We have therefore 
dY =^k^m (! ' A+xVU) ’ 
and thence a value of dS which, compared with the former value, gives 
Q. + A 2 =(x 2 -\-f)b\ 
an equation which may be verified directly. 
Formulae for the Inscribed Quadrilateral. — Art. Nos. 18 to 22. 
18. We consider on the curve four points, A, B, C, D, forming a quadrilateral, 
ABCD. The coordinates are taken to be (X, Y), (X 15 YJ, (X 2 , Y 2 ), (X 3 ,Y 3 ) respec- 
tively. It is assumed that (A, B), (B, C), (C, D), (D, A) belong to the generations 
1, 2, 3, 0, and depend on the parameters (x^yj, (x 2 , y 2 ), {x 3 , y 3 ), (x, y) respectively. 
We write 
O =(1— « x — fly f— f(x 2 -\-y 2 ), 
= (1 — a,®! - - y[(x\+y\\ 
O a = (1 - a 2 x 2 - &y 2 ) 2 - y%xl +y \ ), 
Q 3 = (1 - a.,x, - P 3 y 3 f - y\(x\ +yl), 
and then, y/ O denoting as above a determinate value, positive or negative as the case 
may be, of the radical, and similarly y/ 0„ y/ 0 2 , y/ 0 3 denoting determinate values of 
these radicals respectively, each radical having its own sign at pleasure, we further 
write 
(x 2 +y 2 )B! =1 —ax —fiy —\/ O , 
(x 2 l +y 2 l )W l =l-cc l x 1 —f3 1 y l —^ O,, 
(x 2 2 +yl)B!. 2 =l-ci 2 x ;i -p. 2 y 2 — s / 0 2 , 
{x$-\~yfR > 3 — 1 cl 3 x 3 P 3 y 3 
{xl+yX)^ = 1-a^ - 0^ d-x/O,, 
(^2 "T yfR'i = i a %x 2 p 2 y 2 -(- y/ 0 2 , 
(x 2 3 +yl)R 3 =l-a 3 x 3 —fi 3 y 3 +y/ Q 3 , 
(x 2 +y 2 ) R =l-ax -Py+y/Q; 
and this being so, we must have 
X=* Y =0 +EV =@,+E*, R'=KX ! +Y s -4), R^KX’+Y’-*,), 
X,=a 1 +R , A =a a +R^ a , Y.^.+R'.y^fe+R^, R'.^Xf+Y;-*,), E 2 =i(X? + Y?-^), 
X 2 =a 2 +R^r 2 =a 3 -}-RyC 3 , Y 2 =0 2 +R^ 2 =0,+R^ 3 , R' = |(Xf+Y»-i 2 ), R,=J(X|+Y|-i,), 
X^+^.+Ei, Y,=(3„+Riy,=|3+Ry, R;=i(X|+Yl-fc), R =J(X|+Y ’-* ) ; 
3 s 
mdcoclxxvii . 
