AS 
Vol. IV, No. 10.] A General Theory of Osculating Conics.:. 503 
[N.S.] 
(—3@ - }RQ+4QR’— 3B) +(3Q4 3B} 
=18Q3{3Q+iR+3,9} 
or, 9()* + 3RQS + SRQ* = R’Q8 - 1Q? 
=9Q! + 3Q5R + 8Q°S 
or, A\X=3QS —5R?4+12QR’ 
20. Equation (56) can be written as 
{(Y—y) (3Qd?« ~ Rdx) — (X—2#)(3Qd"y — Rdy)}? 
+f (¥- ‘ia (X-e)dy-“ Ae 
whence, 
(Y-y)(8Qd?e— Rdx) —(X—2x)(3Qd*?y — Rdy) =0 (66) 
and (Y—y)de—(X—2)dy = (67 ) 
are the equations of two conjugate diameters. 
Equation (66) gives the diameter through the point of contact, 
and as it is independent of A, it represents the pe of centres of 
all conics of four r-pointic contact at the given poi 
Equation (67) gives the diameter parallel eS ‘the tangent at 
L,Y). 
The intersection of (66) and (67) is the centre, whose co-ordi- 
nates are 
Y=» p-UCeP a= — Rdz) eae —— Rdy) 
a TT 

(68) 
The osculating semi-diameter OP is given i 
OP? =o {(38Qd?a — Rdx)? + (8Qd?y — Rdy)?} 
— IQ{9Q*+ ( 3QQ, — BP)*} (69) 
AP 
For, (38Qd*2— Rdz)? + (3Qd?y — Rdy)’ 
=9Q%{(dx)?-+ (@y)9} — 6QR{dedPe+ dyity} 
+ BR? (dz? + dy*) 
Q? a Q? 
PE 
=9Q8 —6QRQ, + RP 

9Q*+(3QQ,- BP)? ( 70) 
= P 
