172 Journal of the Asiatic Society of Bengal, (April, 1908, 
The foregoing conclusions might have been arrived at fro 
simple geometrical considerations. The system of osculating 
conics, at a given gan have been looked upon, analytically, as 
aving four consecutive points common with the curve. This is 
sary however, the best Ms of looking from the geometrical stand- 
Geometricall may consider the system of osculating 
conics as having four nainanne tangents common with the curve. 
Hence— 
(a) All osculating conics at a given point of a curve may 
be conceived as having been inscribed to the same 
vanishing quadrilateral, formed by four consecutive 
tangents. Therefore, from well-known properties of 
a system of conics inscribed to the same quadri- 
lateral, we have 
(6) The beau of centres of conics, prada a given curve 
at a given point, is a straight li ne. 
(c) The cal circles of this system of conics form a 
(d) The padioal axis of this co-axial system is the directrix 
of the osculati rabola. 
(e) The Kniting points of this co-axial system are the 
gi Pp and the centre Q of the osculating 
equilateral osha ola. 
For, the director circle vanishes only if the conic vanishes or 
is an equilateral hyperbola. 
(f) If OC be the centre of any osculating conic, then CP. 
re if is equal to the square of the radius of the director 
(g) If CD te the semi-diameter, conjugate to OP, of the 
ceonlating conic whose centre is 0, then 
OP?+ CD*=a*?+b8=C0P. CQ= OP? + OP. PQ. 
Therefore OD*=CP. PQ. (24). 
Evidently the locus of D is a parabola whose focus bisects 
, where S is the focus of the osculating parabola. 
% If we compare the values of p, R, P, a and 1 already 
obtained (12, 14, 15, 16, 22), we cra a aeunbie of eae seal 
tions, of which the most remarkable i 
at=Ip (25). 
Again if Y be the angle between the normal and line of centres 
F, 
ort t(S-() OQ) 
Therefore if y= by the 
N.B.—The angle 7 as tote caer by Transon (Liouville, 
(1+ SS dp 
ae" 
vol, vi). It is easily shewn tan y=p——>—— 
