434 
PKOFESSOR J. CASEY ON A NEW 
135. When S+^C-j-^ 2 represents a pair of lines, the coordinates of the double point 
are, by the usual process, 
—frf 
a* fpt’ m + g 
if referred to the centre of J as origin, and 
jPf_ % 
cP + g m + p 
if referred to the centre of F as origin. Hence we have the following theorem: — 
If F = r — 1 = 0 and J = (#— — g) 2 — r 2 =0 be a corresponding focal conic 
and circle of inversion of a bicircular quartic, and if [Jj v , fa, gj 3 be the three roots of the 
equation which is the discriminant of jM/F+J, then the coordinates of the centres of the 
three other circles of inversion are : — 
aj mg 
(285) 
a 2 / % 
a *+f*8* m+fa 
(286) 
aj b^g 
a*+fa F + V-s 
(287) 
136. Being given F — 1 = 0, and J = (x— g) 2 — r 2 =0, the equation 
of the quartic is 
4(ffV+% 2 )-(^ 2 -f-/+2/r+2^4-r 2 ) 2 =0 (288) 
Again, being given 
-m+fa+m+fa A_u ’ 
the equation of the same quartic is 
4{(« ! + f4l y+(4 ! +f.,)yn- 
2 a 2 / 
« 2 + ^, 
26 2 a 
* 2 +l*, 
(289) 
In order to compare the equations (288) and (289), which represent the same curve, 
they must be referred to the same origin ; we will therefore transform (288) to the same 
origin as (289), and we get 
2 
+y{ x +J^) + 2 s(y + i^r)+^f= 0 . . . (290) 
