
i Tea 
pt eas 


Vol. IV, No. 10.] A General Theory of Osculating Oonics. 499 
[N.S.] 
or, 
Le M2 — = 2Qh LM 
0 —3q@? |=0 
6Q? Be 208 ~4QR 
or, 
(3QM— RL)? + (38QS —5R?+ 12QR’)L? = 18Q8L. 
or, 
{(Y¥—y)(3Qde— Rade) — (X—2) (3Qd?y ~ Rdy) }? 
+ (38QS —5R? + 12QR’){(Y—y)da—(X—2)dy}? (52) 
= 18Q5{(¥~y)de-(X—2)dy} 
Hence, the osculating conic is an ellipse, hyperbola or parabola, 
according as 
3QS —5.R? + 12QR’ 
is positive, negative or zero. (53) 
15. Again, whe condition that a conic may pass through six 
consecutive points on any curve, obtained as equation (49), 
namely, 
dx? dy? 
3dad?x ddyd*y 
3(d®x)*+4dad’x  3(d®y)*? + 4dydy 
10d°ad?a+5dadta 10d*yd3y + 5dyd*y 
xedy dad?y — ibe 
3(dad?y + dyd?x) pats —dydez | 0 
6d?ad?y + 4(dady? + dyd*a) dad*y — dyd*a | 
10( d’ad*y + d’ad?y) + 5(dad*y + dyd*x), oa dyd’z 
likewise transforms easily into 
0 - Q? oO Q 
0:1 7 —3@? et. 
3Q? -4QR’ —4QR 8S 
10QR —5QS’ 10QR’+5QS T 
or, 
O 3Q) 
3Q S+48’ — 4h = 
10R T+ 5S’ 10R’+58 
or, 40R3 — 45QRS +9Q°T —90QRR’ + 45Q?S' =0 (54) 
which ber Piped the Spite: form of the differential equation 
ofa co 
