204 PEOEESSOE CAYLEY ON EECIPEOCAL SUEEACES. 
I attend in particular to the first of these, or rather to the reciprocal equation, which 
will be 
<r'=a-n+x'- 2/-3/-2C'-4B', 
which, writing therein a=n(n—l)—2b— 3c, and x,—3n{n— 2)— 65 — 8c, becomes 
a' =4n(n—2) —8b— 11c— 2/ — 3^' — 20 — 4B0 
The singularity a' is not explicitly defined in Salmon ; o' is the reciprocal of a, and (as 
such) it denotes the. number of common tangent planes of the spinode torse and of the 
torse generated by the tangent planes along a plane section of the surface ; or, what is 
the same thing, it is the number of the spinode planes which touch the plane section ; 
that is, it is equal to the number of points of intersection of the spinode curve and the 
plane section ; or, finally, o' is the order of the spinode curve. The spinode curve is in 
fact for a surface of the order n without singularities the intersection of the surface by 
the Hessian surface of the order 4(^—2), and is thus a curve of fhe order 4n(n— 2), 
which agrees with the formula. 
8. But the formula shows that there is in the order a reduction 8 Z>+11<? arising from 
the nodal and cuspidal curves of the surface, or, what is the same thing, that the 
Hessian surface meets the surface in the nodal curve taken 8 times, and in the cuspidal 
curve taken 11 times — a result which I had arrived at by other means, and also as appears 
post. No. 44. The formula shows further that there is a reduction 2/ +3)2 + 2(7+417, 
or say there are reductions =2, 3, 2, 4, for the reciprocals of a pinch-point, a close-point, 
a cnicnode, and a binode respectively. Geometrically this must signify that the surface 
and its Hessian partially intersect in certain curves which are not regarded as belonging 
to the spinode curve. It will at once suggest itself that for the reciprocal of a cnicnode 
this curve is a conic, and for the reciprocal of a binode it is a line counting 4 times ; 
while for the reciprocal of a pinch-point it is a line counting 2 times, and for the reci- 
procal of a close-point, a line counting 3 times. 
9. It is clear that §' will in like manner denote the order of the node-couple curve. 
10. I express in terms of 
7i, b, c, h, 7c, (3, 7, j, 0, x, C, B 
such quantities and combinations of quantities as can be so expressed. We have 
a—d— n(n—l)—2b — 3c, 
%'=3n(n-2)-6b-8c, 
B'=Xw-2)(«f-9)-(^-w-6)(25+3c)+25(5-l) + 6^+fc(e-l), 
4i — 1 2h + c(5 n — 6) — 6c 2 — 5y + 30 — 2%, 
24t— ( — 8rc+ 8)5 +(17m— 18)c + 85 2 — 18 c 2 —2{87c—18h) 
+ 20 / 3 ^- 157 + 4 /+ 95 + 6 Z , 
q=^b 2 —b—27c—3y—3t, ( t supra), 
r=c 2 — c— 2h— 3/3, 
2(t=c(*-2)-(4/5+7)-0, 
