411] 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
355 
k, number of its inflexions. 
b', class of node-couple torse. 
k', number of its apparent double planes. 
t', number of its triple planes. 
q, its order. 
p, order of node-couple curve. 
j’, number of pinch-planes. 
c', class of spinode torse. 
h', number of its apparent double planes. 
r', its order. 
a, order of spinode curve. 
O', number of off-planes. 
X, number of close-planes. 
13', number of common planes of node-couple and spinode torses, stationary planes 
of the spinode torse. 
y, number of common planes, stationary planes of node-couple torse. 
% , number of common planes, not stationary planes of either torse. 
B', number of bitropes of surface. 
C', number of its cnictropes. 
68. It is hardly necessary to recall that a spinode plane is a tangent plane 
meeting the surface in a curve having at the point of contact a spinode or cusp ; the 
envelope of the spinode planes is the spinode torse, and the locus of their points of 
contact the spinode curve. And similarly a node-couple plane is a double tangent 
plane, or plane meeting the surface in a curve having two nodes; the envelope of 
the planes is the node-couple torse, and the locus of the points of contact the node 
couple curve; the other terms made use of are all explained in the present Memoir. 
Addition, August 3, 1869. 
As in the theory of Curves, so in that of Surfaces, there are certain functions of 
the order, class, &c. and singularities which have the same values in the original and 
the reciprocal figures respectively; for convenience I represent any such identity by 
means of the symbol 2, viz. </> (n, a, &,...) = 2 denotes that the function cf)(n, a, b,...) 
is equal to the same function 4>{n', a', b',...) of the accented letters. By what precedes 
we have a = 2; and it is moreover clear that any function of the unaccented letters 
which is = 0, or which is equal to a symmetrical function of any of the accented and 
unaccented letters, or to a function of a, is = 2 ; for instance, from the equations of 
No. 5 we have 3a' — k — 3n — c, and thence 3n — c — k = 3a' — k — k, that is, 3n — c — k = 2 ; 
45—2