ON THE BICIRCULAR QUARTIC.
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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