Vol. IV, No. 4.]. A General Theory of Osculating Oonics. 173 
2 8.) 
8. To cng the axes of any conic of the system we may 
proceed as follow 
ee the form of the equation of the line of centres (23), the 
co-ordinates (X, Y), of the resin o, of any osculatiug conic of the 
system, can evidently be written 



Xana, fj at) (27). 
where A is an arbitrary constant. 
Whence, CP =3q {r? + (pr—Bq8)8} "> (28). 
and by (14) PQ oe 
Therefore by (24) OD?=CP, PQ =9q? (1+>p*). - (29). 
The equation of CD is evidently, by (27), 
(= y)—p(X—a) = (30), 
Therefore, if PM be the perpendicular from P on CD, 
8 
PM=> = »! (31). 
Hence, if a and b be the semi-axes of the oseulating conic, 
ad + b= OP2+ ODI= Be (4 (r= BOA BAD} ) 
32). 
a}? =QD2, PMt= oe se 
; The equation of the director circle follows from (27) aud (32). 
tis 
bod a 3g pr — 39?) } 
{x e+— } Y-y+ x 
% 
= SE 78-4 (pr—Bq8)2+ M+ 

or 
M(X—2)*+(¥- = . 
((X—a)8 + (¥—y)%} +99 ((X—a)r+ (¥- 0 (33). 
9. To determine the equation of any conic of the system, let 
V be any point (XY) on the conic, and é, 7 its ¢ tama refer- 
red to UP and CD, which are conjugate semi-diam 
VH and VK perpendicular from V on CD and CP, respectively. 
