21 
k’, the number of apparent double planes of the node-couple deve- 
lope. 
’, the number of triple planes of the node-couple develope. 
ce’, the class of the spinode develope. 
h’, the number of the apparent double planes of the spinode deve- 
lope. 
f’, the number of common tangent planes of the node-couple and 
spinode developes, stationary planes of the spinode develope. 
y', the number of common tangent planes, stationary planes of the 
node-couple develope. 
?’, 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. 
o', 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 formule, it is proper to write also— 
r, the class of the cuspidal curve. 
q, the class of the double curve. 
yr’, 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 Schlafii’s, calls, after him, A. 
Mr. Salmon obtains, between the twenty-eight quantities— 
mG nb toh Bn ies 
n,a,6,«,0,k,t, c,h, By, 2%, ¢, &, 
the twenty-one equations, 
a=, 
a=n(n—1) - 2b - 8e, 
x = 8n(n — 2) — 6d — 8e, 
5 = 3n(n —2) (n?-9) -— (n? -n-6) (26 + 3c) + 26(b —-1) + 6be + Ye(e-1), 
a(n—-2)=xK+p+ 2e, 
b(n -2)=p+2B8 + 3y + 3b, 
e(n-2)=20+ 48 +y, 
a(n—2) (n—3) =26 + 2ab + 3ac - 4p - Yo, 
b (n-2) (n-8)=4h+ ab + 3bc —- 9B — 6y — 37 - 2p, 
¢ (n— 2) (n—8) = 6h+ae + 2be - 6B — 4y — 24-380, 
R. I. ACAD. PROC.—VOL. VIL. E 
