406J ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 253 
assume the equation of the conic to be (d, 3b, 0, 0, \a, §c$«, y, z) 2 = 0. The equation 
of the contact-locus then is 
a 2 d 2 + 4ac 3 + 4 b s cl — Qabcd — 36 2 c 2 = 0, 
viz. this is a developable surface, or torse, of the order 4, and we at once infer 
(2/cl, 1 :) = 4, agreeing with (2/cl, 1 :)=2m + n—5. 
I will show also that we have 
(2/cl, 1 • /) = 6, agreeing with (2/cl, 1 • /) = 2m + 2n — 6, 
and 
(2/cl, 1 //) = 5, „ „ (2/cl, 1 //) = m + 2n - 4. 
The condition that the conic may touch an arbitrary line ax + /3y + yz = 0, is in fact 
(0, -\6r, f (4bd - 3c 2 ), fac, — §a&, 0$a, /3, y) 2 = 0, 
which, considering therein (a, 6, c, cZ) as coordinates, is the equation of a quadric surface 
passing through the conic a = 0, 4bd — 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/cl, 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 + fi'y + yz = 0, then we have in like 
manner the quadric surface 
(0, -{a 2 , f (4<bd — 3c 2 ), fac, -fab, 0$a', /3', 7') 2 = 0; 
that is, we have the quartic torse and two quadric surfaces, each passing through the 
conic a = 0, 4bd — 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, 6=0, c = 0, which is one of the intersections of the two conics (the point 
a = 0, b = 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/cl, 1 //) = 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 (x = 0 tangent at cusp, 
z = 0 tangent at inflexion, y = 0 line joining cusp and inflexion; equation satisfied by 
x : y : z = 1 : 0 : 0 3 ); 
and let the equation of the given conic be 
U = (a, b, c, /, g, h^x, y, z) 3 = 0 ;