667] 
ON THE BICIRCULAR QUARTIC. 
227 
theorem for the generation of the bicircular quartic. Consider the generating circle, 
If for a moment the radius is called 8, then the equation of the generating circle is 
(X —/+ Ox) 2 + (Y — g + dyf = 8 2 ; 
the condition for the intersection at right angles is 
(a —/+ 6x) 2 + (ft — g + 6yf = y 2 + 8 2 , 
and hence eliminating 8 2 , the equation of the generating circle is 
X-+ Y 2 — k —2, (X — a) (/ + 6)x-2 (F- &){g + 0)y = 0; 
and considering herein x, y as variable parameters connected by the foregoing equation 
(/+ 0) x 2 + (g0) y 2 = 1, we have as the envelope of this circle the required bicircular 
quartic. 
9. It is convenient to write R = ^ (X 2 + Y 2 — k). The equation then is 
R-(X-a)(f+0)x-(Y-{3)(g + 6)y = 0; 
the derived equation is 
(X — a) (/+ d) dx + (Y - /3) (g + d) dy = 0 ; 
and from these two equations, together with the equation in (x, y) and its deriva 
tive, we find X — a = Rx, Y — /3 = Ry ; from these last equations, and the equations 
R = \ (X 2 + Y 2 - k), (/ + 6) x 2 + (g + 6) y 2 = 1, eliminating y, R, we have 
(/ + 0)(X- a) 2 + (g + 0)(Y- ft) 2 = R\ 
(X 2 + Y 2 — k) 2 — 4 [(/+ в)(X - a) 2 +(g + 0)(Y — /3) 2 ] = 0, 
that is, 
the required equation of the bicircular quartic. 
10. We have thus X - a = Rx, Y — ¡3 = Ry, as the equations which serve to 
determine the bicircular quartic : if from these equations, together with R = \ (X 2 + Y 2 — k), 
we eliminate X and Y, we have R expressed as a function of x, y; and thence also 
X, Y expressed in terms of x, y; that is, in effect the coordinates X, F of a point 
of the bicircular quartic expressed as functions of a single variable parameter. The 
process gives 2R + k = (a + Rx) 2 + (ft -I- Ry) 2 , viz. this is 
R 2 (x 2 + y 2 ) — 2 (1 — ax — fty) R + y 2 = 0, 
or putting for shortness 
il = (1 — ax - f3y) 2 - y 2 (x 2 + y 2 ),