138
[334
334.
NOTE ON THE THEORY OF CUBIC SURFACES.
[From the Philosophical Magazine, vol. xxyii. (1864), pp. 493—496.]
The equation
AX 3 + BY 3 + 6CRST = 0,
where X+7+R + *8+T=0, represents a cubic surface of a special form, viz. each of
the planes R = 0, S = 0, T = 0 is a triple tangent plane meeting the surface in three
lines which pass through a point ( J ); and, moreover, the three planes AX 3 + BY 3 = 0 are
triple tangent planes intersecting in a line. It is worth noticing that the equation of
the surface may also be written
aaf + by 3 4- c (u 3 4- v 3 4- w 3 ) = 0,
where x + y + u + v + w — 0. In fact, the coordinates satisfying the foregoing linear
equations respectively, we have to show that the equation
AX 3 + BY 3 + GCRST = ax 3 +by 3 + c (it 3 4- v 3 + w 3 )
may be identically satisfied. We have
aaf + by 3 4- c (u 3 + v 3 + tv 3 )
= aac? + by 3 + c[(u + v + w) 3 — 3 (v + w)(w + u) (u 4- v)]
= ax? + by 3 — c (x + y) 3 — 3c (v 4- w) (w 4- u) (u 4- v),
1 The tangent plane of a surface intersects the surface in a curve having at the point of contact a
double point, and in like manner a triple tangent plane intersects the surface in a curve with three double
points, viz. each point of contact is a double point; there is not in general any triple tangent plane such
that the three points of contact come together, or (what is the same thing) there is not in general any
tangent plane intersecting the surface in a curve having at the point of contact a triple point. A surface
may, however, have the kind of singularity just referred to, viz. a tangent plane intersecting the surface in
a curve having at the point of contact a triple point; such tangent plane may be termed a ‘ tritom ’ tangent
plane, and its point of contact a ‘ tritom ’ point : for a cubic surface the intersection by a tritom tangent
plane is of course a system of three lines meeting in the tritom point. The tritom singularity is sibi-
reciprocal; it is, I think, a singularity which should be considered in the theory of reciprocal surfaces.
