ON THE BICIRCULAR QUARTIC. 
231 
667] 
the generating circle. If we consider the two points on the generating circle, and 
write dS' for the element at the other point, then we have 
(dS ± dS') = ± (j> ~0- 8 - dv . = 2Sdv, 
which is Casey’s formula ds' — ds = 2p dcf) (273). 
17. The foregoing forms of dX, dY are those which give most directly the required 
value of dS: but I had previously obtained them in a different form. Writing 
A — fix — ay + (/— g) xy, 
xA — fix- — axy + [(/+ 6) a? — (g -f 6) y 2 ]; 
(/+ 0) ** = i - (g + 0) y\ 
then 
or since 
this is 
xA = fix 1 — axy + [1 —(g + 0) (x 2 + y 2 )] = y (1 — ax — fiy) + (x 2 + y 2 ) (fi — (g + 0) y) 
that is, 
and similarly 
= O 2 + y 2 ) \yR' + fi - (g + 0) y} + y Vil, 
-y Vn = (# 2 + y 2 ) {yi2' + /3 - (y + 0)y}; 
— yA — x a/H = (¿r 2 + y 2 ) {aj-R' + a — (f + 0) a;}. 
We have therefore 
dX = 
dY = 
R'da) 
(x 2 + y 2 ) V© Vil 
R'dco 
(xA — y Vil), 
(yA + x Vil), 
(a; 2 + y 2 ) V© Vil 
and thence a value of which, compared with the former value, gives 
il + A 2 = (a? 2 + y 2 ) 8-, 
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, G, D, forming a quadrilateral, 
ABGD. The coordinates are taken to be (X, F), (X lf Fj), (X 2 , F 2 ), (X 3 , F 3 ) respect 
ively. It is assumed that (A, B), (B, G), (G, D), (D, A) belong to the generations 
1, 2, 3, 0, and depend on the parameters (x u y x ), (a? 2 , y 2 ), (x 3 , y 3 ), (x, y) respectively. 
il = (1 — a x -fiy) 2 — y 2 (x 2 + y 2 ), 
ilj = (1 - - fi 1 yf - y 2 (x, 2 + yf), 
11 2 = (1 - a„x 2 - fi,y.f - y 2 (x 2 + y 2 ), 
11 3 = (1 - a,x 3 - fi 3 y 3 ) 2 - y 3 2 (x 3 2 + y 2 ); 
We write