750] 
ON THE THEORY OF RECIPROCAL SURFACES. 
227 
601. In part explanation, observe that the definitions of p and cr agree with 
those already given. 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 the cuspidal torses respectively, and the 
definitions of p, a correspond to those of p and a. 
602. 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 f, and we thus have f —f)\ 
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/— 3y — 6k 
603. 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, w 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 the expression for the class consequently is 
r = c 2 — c — 2h — 3/3. 
604. 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 approach 
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