254 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
then writing 
© = {a, b, c, f g, h\ 1, 0, 0 3 ) 2 
= c6 6 + 2/0 4 + 2g0 s + b& 1 + 2 hd 4- c, 
the equation of a conic having with the given cubic at a given point (1, 0, 0 3 ) 
contact of the second order, and having double contact with the given conic, is 
V U, x, y, z = 0, 
V©, 1,0, 0 s 
(V©)' . 1, 30 s 
(V©)" . . 60 
viz. in the rational form this is 
S68' 2 U — x, y, z : 2 = 0, 
V©, 1, 0, 8 s 
(V©y . i, 30 2 
(V©)" 60 | 
and this will have at the point (1, 0, 0 3 ) a contact of the third order if 0 be deter 
mined by 
V©, 1, 0, 0 s =0, 
(V©)' . 1, 30 2 
(V©)" . . 60 
(V©)"' . . 6 
viz. this is 
© (V©)'" - (V©)" = 0; 
or developing and multiplying by © 2 , this is 
6 I© 2 ©'" — f©©'©" + f©' 3 } — (@ 2 ©" — |©©' 2 ) = 0, 
or, what is the same thing, 
© 2 (0©'" _ ©") + ©©' (- |0©" + ¿0') + ©'2. £0©' = 0 ; 
and substituting for © its value, this is 
(c0 6 + 2/0 4 + 2g6 s + bd- + 2h8+ a) 2 (45c0 4 + 12/0 2 - b) 
+ (c0 8 4- 2/0 4 + 2i/0 3 4- bO- + 2hd+ a) ( 3c0 5 4- 4/0 ;: + Sg8 2 4- 60 4- h) 
(- 42c0 r> - 32/0" - 1500 2 - 2b8 + h) 
4- 30 (3c0 5 + 4/0 3 + 300 s + b0 + hf = 0.