
506 Journal of the Asiatic Society of Bengal. [ November, 1908. 
he co-ordinates of the point of intersection, of the diameter 
through point of contact with directrix, are 

ae 3Q@a2 — Rda } 
X,=2-3QP* aoe ~— RP)? (79) 
a 3BQdty -— Rdy | 
Yi=y~2QP? 9Q'+ 8QQ,- RP) 4 
f (a, 8) be the focus, then the join of (a, 8) and (X,, Yj) is 
Ssuse at right angles by the tangent at (a, y), hence 
a=X,—udy B= Y,+udz 
where ed a 
oSen- 
9Q* + (3QQ,—RP)* 
he semi-latus sated (1) is the perpendicular from focus 
on the directrix. Therefor 
s muy Pz 
3 
{9Q* + (3QQ, — RP)*}* 
he focal distance of (4, y) is equal to the distance of (a, y) 
from directrix 


=p cos Sy (81) 
1QP 
= BET GGG TEPA3 oY ae 
The axis passes through (a, 8) and is, therefore, 
(Y¥—y)(3Qd*x — Rdx) — (X —2) (3Qd*y — Rdy) 
9Q°P (3QQ,—RP) 
9Q*+ (8QQ,- RP)? 
The normal at the point of contact meets the axis (83) at 
(83) 

X=a-udy Y=y+udz (84) 
The distance of this point, from point of contact, is 
8 
uP? = ace Be =p cos*® p (89) 
9Q* + (3QQ; — RP)? 
The co-ordinates of the intersection of the directrix with the 
normal at the point of contact are 
Pdy Pdz 
Xaeb go Yey- (86) 
ae 2Q 
— the directrix of the osculating parabola meets the 
the convex side of the curve, at a distance from 
the seat of eee equal to half the radius of curvature 
inates (86) do not involve higher 
differentials than ins seqee ss we conelade that the directrices of 

