112 
PROCEEDINGS OF SOCIETIES. 
Jo, the number of apparent double planes of the node-couple deve- 
lope. 
t r , the number of triple planes of the node-couple develope. 
c, the class of the spinode develope. 
It', the number of the apparent double planes of the spinode deve¬ 
lope. 
/3', the number of common tangent planes of the node-couple and 
spinode developes, stationary planes of the spinode develope. 
7 ', the number of common tangent planes, stationary planes of the 
node-couple develope. 
i', the number of the common tangent planes which are not statio¬ 
nary planes of either develope. 
p, the number of the common tangent planes of the node-couple de¬ 
velope, and the tangent cone. 
</, the number of the common tangent planes of the spinode deve¬ 
lope, and the tangent cone. 
The terminology made use of is that of my paper “ On the Singula¬ 
rities of Surfaces’’ (“ Cambridge and Dublin Mathematical Journal,” 
vol. vii., 1852). To explain it, I need only remark that the term node 
is used as synonymous with double point, and the term spinode as 
synonymous with cusp; a spinode plane is a tangent plane meeting the 
surface in a curve, having a spinode at the point of contact; and a node¬ 
couple plane is a double tangent plane, or plane meeting the surface in 
a curve having two nodes; the term develope is used instead of develo¬ 
pable surface. 
To collect all the formulae, it is proper to write also— 
r, the class of the cuspidal curve. 
q, the class of the double curve. 
/, the order of the spinode develope. 
q', the order of the node-couple develope. 
"Where q' is what Mr. Salmon, who only uses it incidentally in referring 
to a result of Professor Schlafli’s, calls, after him, A. 
Mr. Salmon obtains, between the twenty-eight quantities— 
n, a, S, 
ri, a', 8, 
k, 5, Jo, t, c, Ji, {3, 7 , iy q, <t, 
ri, 5', Jo, t', c', Ji, J3', 7 ', i', p, ri, 
the twenty-one equations, 
a - ri, 
ri -n(n- 1) - 2b - 3c, 
ri = 3 n(n — 2) — 6 5 — 80, 
8 = ±n(n - 2) (ri 2 - 9)- (ri 2 - n- 6 ) (25 - 1 - 3c)+ 2 b(b - 1) + 65c + |c(c- 1), 
a (n - 2) = k + p + 2<r, 
5 (n — 2 ) = p + 2/3 + 87 + 3 1, 
c (n — 2) = 2c + 4/3 + 7 , 
a (n - 2) (n - 3) = 25 + 2ab + 3 ac -4/> - 9#, 
5 (n - 2) (n - 3) = + ah + 35c - 9/3 - 67 - 3«' - 2 p, 
c (n - 2) (n - 3) = 0>Ji + ac + 25c — 6/3 — 47 - 2 i - 3<r, 
