WHICH SATISFY GIVEN CONDITIONS. 
135 
I will show also that we have 
(2x1, 1 • /)=6, agreeing with (2x1, 1 • /)=2m+2n—6, 
and (2*1, 1 //)=5, „ „ (2*1, 1 //)= m+2»— 4. 
The condition that the conic may touch an arbitrary line ax-\-(3y-\-yz—0, is in fact 
(0, -\a\ f(4^-3c 2 ), f ac, -f ab, Oja, (3, y ) 2 =0, 
which, considering therein (a, b, c, d) as coordinates, is the equation of a quadric surface 
passing through the conic a— 0, Ibd— 3c 2 =0 ; the quartic torse also passes through this 
conic ; hence the quadric surface and the torse intersect in this conic, which is of 
the order 2, and in a residual curve of the order 6 ; and the number of the conics 
(2*1, 1 • /) is equal to the order of this residual curve, that is, it is =6. 
If the conic touch a second arbitrary line a!x-\-(3'y-\-y'z = Q, then we have in like 
manner the quadric surface 
(0, -\a\ \(M-$ c 2 ), | ac, - \ab , 0 £«', /3', y') 2 =°; 
that is, we have the quartic torse and two quadric surfaces, each passing through the 
conic a— 0, Ibd— 3c 2 =0, and it is to be shown that the number of intersections not on 
this conic is =5. The two quadric surfaces intersect in the conic and in a second 
conic ; this second conic meets the torse in 8 points, but 2 of these coincide with the 
point a— 0, b= 0, <?=0, which is one of the intersections of the two conics (the point a=0, 
6=0, c=0 is in fact a point on the cuspidal edge of the torse, and, the conic passing 
through it, reckons for 2 intersections), and 1 of the 8 points coincides with the other of 
the intersections of the two conics; there remain therefore 8 — 2 — 1, = 5 intersections, or 
we have (2*1, l//)=5. 
Annex No. 5 (referred to, Nos. 22 and 71 ). — On the Conics which have contact of the third 
order with a given cuspidal cubic , and two contacts (double contact ) with a given conic. 
Let the equation of the cuspidal cubic be x 2 z—y 3 = 0 (#=0 tangent at cusp, z— 0 
tangent at inflexion, y = 0 line joining cusp and inflexion ; equation satisfied by 
x\yz— 1 :0:{) 3 ); 
and let the equation of the given conic be 
U =(a, b, c,f , ; g, hjx, y, zf= 0 ; 
then writing 
0= (a, b, c,f, g, Kfl, 6, 0 3 ) 2 
= cti 5 + 2 fP + 2 gd 3 + U 2 + 2h0 + c, 
the equation of a conic having with the given cubic at a given point (1, 6, 6 3 ) contact of 
^U, x, y, z 
x/0 1, 6 , 0 3 
(\Zey . 1 , 34 2 
Cs /©)" . . 60 
= 0 , 
