Vol. IV, No, 4.) A General Theory of Osculating Coniecs. 171 
[N.S.] 
Hence, from we easily deduce the equation for the 
directrix of the osculating parabola to be 
r(X—«) + (pr- 3q*)(¥—y) —$q(1 + pt) =0 (18). 
And if (a, B) be the co-ordinates of the focus 8 of the osculating 
parabola, then, from (13), we easily deduce 
3q vir) x — Bpq* 


a=at+ a. 
2 (pr—3q*)8 +78 
pays, 2 pep) + gh(1—s) sie: 
y a (pr- Bq? )? 47% 
The aoe of the osculating parabola itself, is therefore 
(e—a)?+ (y— 
pu + (pr wen ot Cet —39*(1 + p*)} 
(pr —3q°)* + 
which, after substitutions (19) for a, 8, becomes 
{(X—a)( pr—Bq) — (Y—y)r}#=1848 {((¥-y)-p (X-2)} (21). 

(20). 
e semi latus rectum (1), of the above parabola, is the 
pendicular from the focus (a, 8) on the directrix (18). Therefore, 
2795 
ace hare (22), 
{(pr=Sqi) + 18}4 
lt be noticed here that the focal distance of P and the 
foca sal perpenditnitar He" the tangent at P, are respectively }R and 
3P, given by (14) and (15), 
6. If two canteal conics, one of them being an equilateral 
st osculate a given curve at a given point, then they 
vidently osculate each thas hence, from Theorem II of article 
(4), we draw the following conclusions :— 
(7) The ora of centres of osculating conics, to a given curve 
a given point, is a straight line. 

For, the ssid point P and the centre Q, of the gee a 
equilateral hyperbola, are, from equations (17 ), in one straigh 
line with the centre C, of any other osculating conic. The equation 
of this line of centres PQ is evidently from (13), 
( pr—39*)(X—2)—r (¥-y)=0 (23). 
(vi) The director circles of the osculating conics to a given 
of a curve form a co-axial system, having two 
real limiting points P and 
r, OP. 0Q=a* +b, from equations (17), C being the centre 
of ie santana conic and therefore of its director circle. 
