FOEM OF TANGENTIAL EQUATION. 
433 
and, from the same article, the double points of these pairs of lines are the four centres of 
inversion of the quartic. Since one value of p is obviously =0 in the foregoing equation, 
we see that the pair of double tangents drawn from the centre of the circle of inversion 
J {{x-f) 2 -\-{y—g) 2 =r 2 ) will, Avhen that centre is taken as origin, be «¥+% J =0. 
Hence, if the other centres be taken respectively as origin, the equations of the other 
pairs of double tangents will be 
(a'+^y+{b'Jrp,)f= 0, (277) 
(a* + + yj 3 )y —0, (278) 
{a 2 +fa)x 2 +(b 2 +fa)y 2 = 0 (279) 
Now since the pair of lines a 2 x 2 -\-b 2 y 2 =0 are the asymptotes of the reciprocal of the 
conic ^ 2 +p— 1 = 0, we infer that the pairs of lines (277), (278), (279) are the asymp- 
totes of the reciprocals of the other focal conics. Hence we have the following system 
as the equations of these conics : — 
a' + fiz 
= 1 , 
(280) 
_i_ _ y. — i 
+J*7T=1- 
a 2 H-^4 ' -hp-4 
(281) 
(282) 
Hence the four focal conics of a bicircular quartic are confocal. 
134. Since the equation (276) may be written in the form 
« 2 +j«.^6 2 + |U. ' 
and this is the discriminant of J (where J and F have the values in art. 128; 
see Salmon’s ‘ Conics,’ p. 324), we infer that the same values of p which will make 
/&F+J a pair of lines will make S + ^C+j^ 2 a pair of lines; the two pairs of lines will 
have the same double point, their equations referred to that point as origin being 
* 2 (« 2 + l *) . 2 / 2 ( 6 2 + j *) a 
a 2 "T b 2 — U ’ 
cc 2 (a 2 -\-yj) -\-y 2 {b 2 0 . 
(283) 
(284) 
Hence we have the following theorem : — If F and J be a corresponding focal conic 
and circle of inversion of a bicircular quartic, and if ^j, fa, fa be the roots of the cubic 
which is the discriminant of ^F + J, then if F be given in its canonical form 
^ 2 +p— 1 = 0, the equations of the other three focal conics are got from this by changing 
a 2 , b 2 respectively into b 2 +fa; a 2 -{-fa, b 2 —fa\ and a 2 -\-fa, b 2 +fa. 
3 p 2 
