434 
A MEMOIR ON CUBIC SURFACES. 
[412 
158. The planes are 
X =0, 
[1] 
7=0, 
[2] 
7 = 0, 
[056] 
7 + X = 0, 
[5] 
7 + 7 =0, 
[6] 
7-17=0, 
[34] 
X + 7 + 7 = 0, 
[12] 
17 = 0, 
[0] 
159. The transversal is facultative; 
160. The Hessian surface is 
The lines are 
X = 0, 7=0, (5) 
Z = 0, 7 = 0, (6) 
7=0, 17 = 0, (0) 
X=0, 7+7=0, (1) 
Z = 0, 7+X = 0, (2) 
W= Y= — Z, (3) 
17 = 7 = — X, (4) 
17=0; X+ 7+7 = 0, (012) 
p' = 6' = 1, t' = 0. 
17X7 (3 7 + X + 7) + 7 2 (7 — X) 2 = 0. 
The complete intersection with the surface is made up of the line 7=0, X = 0 
(05-axis) three times; the line 7=0, Z = 0 (05-axis) three times; line 7=0, W = 0 
(OO-axis) twice, and of a residual quartic, which is the spinode curve; <r' = 4. 
161. Representing the two equations by £7=0, H — 0, we have 
(3y+X + Z) U—H= 7 2 (37 2 + 47X + 477 + 4X7), = MY 2 suppose, 
and 
27 (X + Z) TJ+9H = 917X7(37 + 4X + 4>Z) + 367 2 (X 2 + X7 + 7 2 ) + 27 7 3 (X + Z); 
but 
(-9(X + 7)7+16X7)i¥ = 
64X 2 7 2 + 28 YXZ (X + Z) - 7 2 (36X 2 + 28X7 + 367 2 ) - 27 7 3 (X + 7), 
whence 
27 (X + Z) U + 9H + (- 9X7 - 9ZY + 16X7) M 
= 7X {12 7 2 + 28 YTVZ + 64XZ + 9 W (37 + 4X + 47)}; 
or, as this may also be written, 
27 7 2 (X + Z) U + 9Y 2 H 
+ (— 97X — 9YZ+ 16X7) (37 + X + Z) U + (9YX + Z- 16XZ) H, 
that is, 
{- 97(X + 7) 2 + 487X7 + 16X7(X + Z)}U + {97 2 + 97X + 7- 16X7} H 
= 7 2 7X {127 2 + 287 (7 + X) + 64X7 + 9 W(37 + 4X + 47)} = 0; 
and we thus obtain the equation of the residual quartic, or spinode curve, in the form 
37 2 + 47(X + 7) + 4X7 = 0, 
12 7 2 + 287 (X + 7) + 64X7 + 9 W (37+ 4X + 47) = 0.