208 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
a multiple point on the nodal curve. I apprehend that starting with a pinch-point, a 
simple point on the nodal curve, we have in the reciprocal figure a pinch-plane or torsal 
plane as above, but with some specialty in virtue of which only one of the (n— 2) points 
of intersection of the torsal line with the residual plane curve is a point of the node- 
couple curve of the reciprocal surface. In the case of a pinch-plane or torsal plane of a 
cubic surface, n — 2 is =1, and the question of multiplicity does not arise. 
26. For a surface with a pinch-plane or torsal plane as above (/— 1), the Hessian 
surface not only passes through the torsal line, but it touches the surface along 
this line, causing, as already mentioned, a reduction =2 in the order of the spinode 
curve. That the surfaces touch along the line is an important theorem*, and I annex 
a proof. 
27. Let x=0, y — 0 be the torsal line, a?=0 being the torsal plane; the equation of 
the surface therefore is x(p+y 2 -^=0 ; and if A, B, C, D be the first derived functions of 
<p, ( a , d, c, d, f, g , h , l, m, n) the second derived functions, and if (A', B', C', D'), 
(«', b\ c\ d',f, g', h', l', m', n') refer in like manner to 4s then the equation of the 
Hessian is 
0 = 
2A +xa+y 2 a! , B -\-xh +2yA! -\-y 2 h! , C -f xg -\-y 2 g ' , D -\-xl 
B+xh+2yA!+y 2 ti, xb +2^ +4?/B ' +fV , xf+2yC+y 2 f', xm+2yD'+tfm' 
C+xg+fg' , xf -\-2yC +y 2 f , xc+fd , xn +fri 
D+xl+yH 1 , xm-\-y*n' -\-2yjy , xn-\-y 2 n’ , xd ~ty 2 d' ; 
and representing this for a moment by 
A, H, G, L 
= 0 , 
IL, B, F, M 
G, F, C, N 
L, M, N, D 
then in the developed equation 
D (ABC - AF 3 - BG 2 - CPF+2FGH) 
-(BC-F 2 , CA-G 2 , AB-H 2 , GH-AF, HF-BG, FG— CH^L, M, N) 2 =0, 
observing that C, F, M, N, D are of the first order in x, y, the only terms of the first 
order are contained in B(— DG 2 — CL 2 -f 2NGL) ; and since C, D, N are of the first order, 
we obtain all the terms of the first order by reducing B, G, L to the values 2\p, C, D ; 
viz. the terms of the first order are 
2$(-C 2 dx—D 2 cx-\-2CDnx), =-2^(C 2 d+B 2 c-2CDn)x. 
Flence the complete equation is of the form 
-2^(C 2 d+D 2 c-2CDn)x+(x : y) 2 = 0, 
See Salmon, p. 218, wlicre it is only stated tliat the Hessian passes through the line. 
