630 
SURFACE. 
[797 
The tangent lines of the surface, which lie in the plane, are nothing else than the 
tangents of the plane section, and thus form a singly infinite series of lines; similarly, 
the tangent lines of the surface, which pass through the point, are nothing else than 
the generating lines of the circumscribed cone, and thus form a singly infinite series 
of lines. But, if we consider those tangent lines of the surface which are at once 
in the plane and through the point, we see that they are finite in number; and we 
define the rank of a surface as equal to the number of tangent lines which lie in 
a given plane and pass through a given point in that plane. It at once follows that 
the class of the plane section and the order of the circumscribed cone are each equal 
to the rank of the surface, and are thus equal to each other. It may be noticed that 
for a general surface (#$#, y, z, w) n , = 0, of order n without point singularities the 
rank is a, =n(n— 1), and the class is n, =n(n — l) 2 ; this implies (what is, in fact, 
the case) that the circumscribed cone has line singularities, for otherwise its class, 
that is, the class of the surface, would be a {a— 1), which is not =n(n — l) 2 . 
4. In the last preceding number, the notions of the tangent line and the tangent 
plane have been assumed as known, but they require to be further explained in 
reference to the original point definition of the surface. Speaking generally, we may 
say that the points of the surface consecutive to a given point on it lie in a plane 
which is the tangent plane at the given point, and conversely the given point is the 
point of contact of this tangent plane, and that any line through the point of contact 
and in the tangent plane is a tangent line touching the surface at the point of 
contact. Hence we see at once that the tangent line is any line meeting the surface 
in two consecutive points, or—what is the same thing—a line meeting the surface in 
the point of contact, counting as two intersections, and in n — 2 other points. But, 
from the foregoing notion of the tangent plane as a plane containing the point of 
contact and the consecutive points of the surface, the passage to the true definition 
of the tangent plane is not equally obvious. A plane in general meets the surface 
of the order n in a curve of that order without double points; but the plane may 
be such that the curve has a double point, and when this is so the plane is a 
tangent plane having the double point for its point of contact. The double point is 
either an acnode (isolated point), then the surface at the point in question is convex 
towards (that is, concave away from) the tangent plane; or else it is a crunode, and 
the surface at the point in question is then concavo-convex, that is, it has its two 
curvatures in opposite senses (see infra, No. 16). Observe that, in either case, any line 
whatever in the plane and through the point meets the surface in the points in 
which it meets the plane curve, namely, in the point of contact, which qua double 
point counts as two intersections, and in n — 2 other points; that is, we have the 
preceding definition of the tangent line. 
5. The complete enumeration and discussion of the singularities of a surface is 
a question of extreme difficulty which has not yet been solved*. A plane curve has 
* In a plane curve, the only singularities which need to be considered are those that present themselves 
in Pllicker’s equations: for every higher singularity whatever is equivalent to a certain number of nodes, 
cusps, inflexions, and double tangents. As regards a surface, no such reduction of the higher singularities 
has as yet been made.