446 
A MEMOIR ON CUBIC SURFACES. 
[412 
188. The Hessian surface is 
(X + Z) WXZ + (X-Z'Y F 2 = 0. 
The complete intersection with the cubic surface is Y = 0, Z —0 and F=0, X=0 
(the CB-axes) each four times; F = 0, W = 0 (BB-axis) twice ; and X = 0, Z = 0 (the 
edge) twice. There is no spinode curve, a — 0 ; wherefore also /3' = 0. 
Reciprocal Surface. 
189. The equation is obtained from the binary quadric 4iv (X + Z) (Xx + Zz) + y 2 XZ, 
or say 
(8wx, 4w (x + z) + y 2 , 8wz\X, Zf. 
The equation is thus 
(y 2 + 4 wx + 4 wzf — 64 uf-xz = 0, 
or in an irrational form 
iy + 2 fwx + 2 fwz — 0. 
The section by the plane w — 0 (reciprocal of i? 4 ) is w — 0, y = 0 (reciprocal of edge) 
four times. 
The section by the plane z = 0 (reciprocal of C 2 = C) is z = 0, y 2 + 4wx = 0 (reciprocal 
of nodal cone) twice ; and similarly for the section by x = 0 (reciprocal of (X = A). 
Nodal curve. Writing the equation in the form 
y i + 8wy 2 (z + x) + 16w 2 (x — z) 2 = 0, 
we have a nodal line y— 0, x — z = 0, reciprocal of the mere line: 
and writing the equation in the form 
we have y = 0, w = 0 (reciprocal 
There is no cuspidal curve; 
1 
W = y 2 , 
4 (V# + V^) 2 
of edge), a tacnodal line counting as two lines; 
c =0. 
V = S.