584 
ON THE THEORY OF RECIPROCAL SURFACES. 
[416 
621. In part explanation, observe that the definitions of p and a agree with those 
given, Art. 609: the nodal torse is the torse enveloped by the tangent planes along 
the nodal curve ; if the nodal curve meets the curve of contact a, then a tangent plane 
of the nodal torse passes through the arbitrary point, that is, p will be the number 
of these planes which pass through the arbitrary point, viz. the class of the torse. So 
also the cuspidal torse is the torse enveloped by the tangent planes along the cuspidal 
curve ; and a will be the number of these tangent planes which pass through the 
arbitrary point, viz. it will be the class of the torse. Again, as regards p and a : the 
node-couple torse is the envelope of the bitangent planes of the surface, and the node 
couple curve is the locus of the points of contact of these planes; similarly, the 
spinode torse is the envelope of the ‘parabolic planes of the surface, and the spinode 
curve is the locus of the points of contact of these planes ; viz. it is the curve UH 
of intersection of the surface and its Hessian ; the two curves are the reciprocals of 
the nodal and cuspidal torses respectively, and the definitions of p', a correspond to 
those of p and <r. 
622. In regard to the nodal curve b, we consider k the number of its apparent 
double points (excluding actual double points) ; f the number of its actual double 
points (each of these is a point of contact of two sheets of the surface, and there is 
thus at the point a single tangent plane, viz. this is a plane and we thus have 
t the number of its triple points; and j the number of its pinch-points— 
these last are not singular points of the nodal curve per se, but are singular in regard 
to the curve as nodal curve of the surface ; viz. a pinch-point is a point at which 
the two tangent planes are coincident. The curve is considered as not having any 
stationary points other than the points y, which lie also on the cuspidal curve; and 
the expression for the class consequently is q = b 2 — b — 2k — 2/— Sy — 6i. 
623. In regard to the cuspidal curve c we consider h the number of its apparent 
double points ; and upon the curve, not singular points in regard to the curve per se, 
but only in regard to it as cuspidal curve of the surface, certain points in number 
6, x, co respectively. The curve is considered as not having any actual double or other 
multiple points, and as not having any stationary points except the points /3, which 
lie also on the nodal curve ; and thus the expression for the class is r = c 2 — c — 2h — 3/3. 
624. The points y are points where the cuspidal curve with the two sheets (or 
say rather half-sheets) belonging to it are intersected by another sheet of the surface ; 
the curve of intersection with such other sheet belonging to the nodal curve of the 
surface has evidently a stationary (cuspidal) point at the point of intersection. 
As to the points /3, to facilitate the conception, imagine the cuspidal curve to be 
a semi-cubical parabola, and the nodal curve a right line (not in the plane of the 
curve) passing through the cusp ; then intersecting the two curves by a series of 
parallel planes, any plane which is, say, above the cusp, meets the parabola in two 
real points and the line in one real point, and the section of the surface is a curve 
with two real cusps and a real node ; as the plane approaches the cusp, these