“230 Journal of the Asiatic Society of Bengal. { April, 1908. 
If R be the radius vector of the osculating equilateral hyper- 
bola, drawn from the centre to the point of osculation, then, 
from (13), 
3q (14 p?: 
Re ina (Fie 
V (pr—3qt) +r 
P be the perpendicular from centre on the tangent at the 
point of osculation, then, from (13), 
P= p(X—2)—-(¥-y)_ og8 V1 +98 (1s). 
Vite (pr-8eeee 


(14). 


The axis of the equilateral hyperbola bisects the acute angle 
between Rand P, Ifa be the length of the semi-axis, then 
4 z 
Q7q* (1+ p4)2 : (16). 
((pr—8g2)* 49338 
4. Theorem 1.—The locus of centres of equilateral hyper- 
bol sculating a given parabola, is va =e parabola, which is 
the pellesiot of the former on the directr 
a&’=R. P= 

ry 
For, taking the parabola to be y=>, we have P=5 ; 
a 
1=s) 7 =O. 
mera from (13), X=2, F=y—2a whence the theorem. 
osculating a given central conic, is the inverse of the conic with 
respect to the director circle. —— by Wolstenholme). 
For, taking the conic to be “+t 1, it is easily shewn, by 
(13), that 
_* (a* + b?) yu! (a® + 68) 
w+ yi a®+ y? At 
whence the theorem. 
rr ae al hyperbola and a parabola both osculate 
a given curve at a given point they osculate each other, for, each 
of ie eit prcle the same four consecutive points on 
the curve. 
Hence, from Theorem I, we conclude that—(i) The directrix 
O on the t eitiei at P, 
