406]
that is,
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS.
247
-k ( SA — 2kB — k 2 G)
+ k*( SB-4>kC+k*D)
+ (2kA - 2k: 2 B),
or finally
— k(A — 3 Bk + 3 k 2 G - k?D) = 0,
which is satisfied by the foregoing values of A, B, C, D; hence the conic is a nodal
curve on the sextic; and by merely changing the sign of one of the radicals Va, V& (and
therefore interchanging k, k^) we obtain another conic which is also a nodal curve on
the surface, that is, we have as nodal curves the two conics
/
x : y : z : w = 6 V& : 6 Vu : 1 : — k 6 2 + j, and
rC
yx : y \ z : w = 6 ^Jb d ^Ja : 1 : - kfi 1 + ~.
*$1
'1
It is to be remarked that each of the nodal conics meets the cuspidal curve in two
, ©! = —v , for the intersec-
Aaj * Hj
x : y : z : w = © Va : © V& : 1 : £ and = — © Vu : — © \/b : 1 :
fCj
and for the intersections with the second conic
The condition of passing through any arbitrary point establishes a linear relation
between the parameters (x, y, z, w). Hence, if the conic in addition to the prescribed
conditions passes through two other given points, the point (x, y, z, w) is given as the
intersection of a line with the sextic surface; the number of intersections is = 6. If
(x, y, z, w) is situate on the cuspidal curve, then the conic instead of simply touching
the given conic will have with it a contact of the second order, and if we besides
suppose that the conic passes through a given point, then the point (x, y, z, w) is given
as the intersection of the cuspidal curve with a plane; the number is = 6. Similarly, if
the conic has two contacts with the given conic, and besides passes through a given point,
then the point (x, y, z, w) is given as the intersection of the nodal curve by a plane;
the number is = 4. Finally (observing that in the case in question of the contacts
of a conic with a conic we cannot have three simple contacts, or a simple contact and
one of the second order), a point of intersection of the nodal and cuspidal curves
answers to a contact of the third order; and the number is = 4. That is, the theory
