ON THE THEORY OF RECIPROCAL SURFACES. 
583 
6, number of its points of an unexplained singularity. 
X, number of its close-points. 
number of its off-points. 
r, its class. 
/3, number of intersections of nodal and cuspidal curves, stationary points on cuspidal 
curve. 
7, number of intersections, stationary points on nodal curve. 
i, number of intersections, not stationary points on either curve. 
G, number of cnicnodes of surface. 
B, number of binodes. 
And corresponding reciprocally to these : 
n', class of surface. 
a, class of section by arbitrary plane. 
S', number of double tangents of section. 
k, number of its inflexions. 
p', order of node-couple curve. 
a, order of spinode curve. 
b', class of node-couple torse. 
k', number of its apparent double planes. 
/', number of its actual double planes. 
t’, number of its triple planes. 
j', number of its pinch-planes. 
q', its order. 
c', class of spinode torse. 
h', number of its apparent double planes. 
O', number of its planes of a certain unexplained singularity. 
X, number of its close-planes. 
a)', number of its off-planes. 
r', its order. 
/3', number of common planes of node-couple and spinode torse, stationary planes of 
spinode torse. 
y', number of common planes, stationary planes of node-couple torse. 
i', number of common planes, not stationary planes of either torse. 
C', number of cnictropes of surface. 
B', number of its bitropes. 
In all 46 quantities.