Vol. sat ay 10.] A General Theory of Osculating Conics. 505 
Thus the director circles of the system of conics of four- 
pointic contact, form a co-axial system, of jwhich the radical 
is is 
(X—a)(3Qd*a— Rdx) + (Y—y)(3Qd*y — Rdy) + 3QP=0 (75) 
This radical axis is the directrix of the osculating parabola. 
1. The condition that the osculating conic may be an 
equilateral hyperbola is a*+b?=0. Therefore, from (72) 


,- _.2@!+ (3Q,- BP)! 1 
ss + (76) 
“nd l= 27@'P3 =p* cos 3 | 
(9Q'+ (3QQ,—RP)}? 
where a is the semi-axis of the osculating equilateral hyperbola. 
he co-ordinates of the point, where the normal at the point 
of contact meets the equilateral hyperbola again, are found to be 
X=2+ =. ] 
r (77) 
_, _2Pae | 
=y Q J 
But the co-ordinates of the centre of curvature are (11) 
Pd 
X=z2-—— dy Yay+" 
Therefore, the osculating equilateral hyperbola meets the 
normal a in, towards the convex side of the curve, at a distance 
differentials than the second, we conclude that all ve oe teral 
sd ape of three-pointic contact pass through the same 
point (77). 
her, as two consecutive agit equilateral hyperbolas 
may be conceived to possess three consecutive ~s common, 
they intersect again at (77), and, antes the envelope of the 
further branch of the osculating equilateral hyperbola is the locus 
of the point given by (77). 
22. The equation of the osculating parabola, obtained from 
(56) by putting A=0 is 
{(Y-y)(3Q@2 — Rdz) — (X—2) (3Qd*y — Rdy) }8 
=18Q3{(Y—y) de—(X-—2)dy} (78) 
The diameter through point of contact is (66) 
(Y—y)(3Qd*« — Rdx) — (X—2)(3Qd*y — Edy) =Oand the directrix 
is (75) 2 
(Y¥—y)(3Qd*y — Rdy) + (X—2)(3Qd°x — Rdz) + 3QP=0. 
