336 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
[411 
at the pinch-point. In the particular case where the nodal curve is a right line the 
section is the line twice (representing the cuspidal branch), and a residual curve of 
the order n — 2, the tangent to which is the cotangent, 
24. The pinch-plane, or reciprocal singularity / = 1, is in fact a torsal plane touching 
the surface along a line, or meeting it in the line twice and in a residual curve. 
Let the line and curve meet in a point P; for the reason that the section by the 
plane is the line twice and the residual curve, the section has at P two coincident 
nodes; that is, the plane is a node-couple plane with two coincident nodes. The plane 
meets the consecutive node-couple plane in a line /x passing through P and touching 
at this point the residual curve. Considering now the reciprocal figure, the reciprocal 
of the pinch-plane is thus a point of the nodal curve, and is a pinch-point; the 
tangent plane at the pinch-point is the reciprocal of the point P; the tangent to the 
nodal curve is the reciprocal of the line ¡x, that is, of the tangent at P to the residual 
curve ; and the cotangent at the pinch-point is the reciprocal of the torsal line. 
25. There is in this theory the difficulty that for a surface of the order n, the 
torsal plane meets the residual curve of intersection in (?i — 2) points P, and if each 
of these be a point on the node-couple curve, then in the reciprocal figure the pinch- 
point would be 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 speciality in virtue of which onlv 
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 (j' = 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, x = 0 being the torsal plane; the equation 
of the surface therefore is xcf) + y 2 yfr = 0; and if A, B, C, D be the first derived functions 
of (j.>, (a, h, c, d, f, g, h, l, vi, n) the second derived functions, and if {A', B', G', D'), 
(a, h', c', d', /', g', li, l', m', n') refer in like manner to y\r, then the equation of the 
Hessian is 
0 = 
2 A + xa 4- y 2 a' , 
B + xh + ZyA' + yHi, 
G + xg + y 2 g 
D + xl + y 2 V 
B + xh + 2yA' + yHi, 
xb + 2iJr + 4 yB' + y 2 b', 
+ 2 yC' + y 2 f' 
xvi + y-n + 2 yU , 
G+xg +y 2 g', 
xf+ 2yC' + yf, 
xc + y 2 c' , 
xn + yhi , 
D +xl + y 2 l' 
xvi + 2 yD' + y 2 m' 
xn + y 2 n 
xd + y -d '; 
1 See Salmon, p. 218, where it is only stated that the Hessian passes through the line.