34 
ON THE THEORY OE SKEW SURFACES. 
[107 
at least of the second order: assuming for a moment that it is in any case precisely 
of this order, it obviously cannot be a plane curve, and must therefore be two non 
intersecting lines. This suggests at any rate the existence of a class of skew surfaces 
of the fourth order generated by a line which always passes through two fixed lines 
and by some other condition not yet ascertained; and it would appear that surfaces 
of the second order constitute a degenerate species belonging to the class in question.) 
In particular cases a generating line will be intersected by the consecutive 
generating line. Such a generating line touches the double curve. 
Consider now a point not on the surface; the planes determined by this point 
and the generating lines of the surface are the tangent planes through the point; 
the intersections of consecutive tangent planes are the tangent lines through the 
point; and the cone generated by these tangent lines or enveloped by the tangent 
planes is the tangent cone corresponding to the point. This cone is of the w th class. 
For considering a line through the point, this line meets the surface in n points, 
i.e. it meets n generating lines of the surface; and the planes through the line and 
these n generating lines, are of course tangent planes to the cone : that is, n tangent 
planes can be drawn to the cone through a given line passing through the vertex. 
The cone has not in general any lines of inflexion, or, what is the same thing, 
stationary tangent planes. For a stationary tangent plane would imply the inter 
section of two consecutive generating lines of the surface. And since the number of 
generating lines intersected by a consecutive generating line, and therefore the number 
of planes through two consecutive generating lines, is finite, no such plane passes 
through an indeterminate point. The tangent cone will have in general a certain 
number of double tangent planes; let this number be x. We have therefore a cone 
of the class n, number of double tangent planes x, number of stationary tangent 
planes 0. Hence, if m be the order of the cone, a the number of its double lines, 
and /3 the number of its cuspidal or stationary lines, 
m = n (n — 1) — 2x, 
/3 = 3n (n — 2) — 6#, 
a = \n (w - 2) (n 2 — 9) — 2x (n 2 — n — 6) + 2x (x — 1). 
This is the proper tangent cone, but the cone through the double curve is sub modo 
a tangent cone, and enters as a square factor into the equation of the general 
tangent cone of the order n (n - 1). Hence, if X be the order of the double curve, 
and therefore of the cone through this curve, 
m + 2X — n (n — 1), and therefore X = x\ 
that is, the number of double tangent planes to the tangent cone is equal to the 
order of the double curve. It does not appear that there is anything to determine 
x\ and if this is so, skew surfaces of the w th order may be considered as forming 
different families according to the order of the double curve upon them. 
To complete the theory, it should be added that a plane intersects the surface 
in a curve of the n th order having x double points but no cusps.