A MEMOIR ON CUBIC SURFACES. 
416 
[412 
of these lines, neglecting the terms which contain x — w, y + z conjointly in an order 
above the second, the equation may be written 
viz. this is 
64w 3 (x + iv) 2 (x — wf 
+ 8w 2 (x + w) 2 (x + 3w) (z + y ) 2 
4- 8w 2 (z — y ) 2 (x — 3tv) (x — iv) 2 
— 32w 2 (z — y ) (x + w) x {x — w) (z + y) 
+ \w (z — y) 2 (lla; 2 — 27w 2 ) (z + y) 2 
— tv (z — y) 3 (3a? — 7tv) (x — tv) (z + y) 
+ \w (z — yY (x — w) 2 
+ y 3 z (z -y) (x- w) (z + y) 
— y*x (z-y) (z + y ) 2 = 0, 
(A, B, C\x — iv, z + y) 2 = 0, 
where, collecting the terms and reducing the values by means of the equations 
x — w = 0, z + y = 0, or say by writing x = w, — y — z, we have 
A = 64w 3 (x + w) 2 
+ 8w 2 (z — y) 2 (x — 3w) 
+ |w 2 (z — yY 
B = — 32 (z — y) (x + w) xiv 2 
— w(z — y) 3 (3a; — 7w) 
+ tfz (z ~ y) 
= 25 Gw 5 
— H4iW 3 z 2 
+ 4 wz A 
= 4 w (z 2 — 8 w 2 ) 2 , 
= - 128 vfiz 
+ 32 iv 2 z 3 
- 2z s 
= — 2 z (z 2 — 8 w 2 ) 2 , 
C= 8w 2 (x + w 2 )(x + 3w) = 128iv 5 
+ \w (z — y) 2 (\lx 2 — 21 w 2 ) — 32iv 3 z 2 
— xy 3 (z — y) + 2wz* 
— xy 3 (z — y) + 2ivz 4 
= 2w (z 2 — 8w 2 ) 2 . 
Hence the condition 4AC — B 2 = 0 of a pinch-point is (z 2 — 8w 2 ) 5 = 0, so that the pinch- 
points (if any) would be at the points x — w = 0, y + z = 0, z 2 — 8w 2 = 0 ; or say at x, y, z, w 
= 1, — 2V2, 2V2, 1. But these values give X, M, N = 12, 12, —16; values which 
satisfy the equations L 2 — \2iv 2 M = 0, LM + 9w 2 N=0, 4ilf 2 + 3LA T — 0, and as the points 
in question are obviously not on the line y = 0, w = 0, they lie on the ninthic 
component of the cuspidal curve, being in fact points /3', and not pinch-points. 
The line y=0, w — 0 qua nodal line would have every point a pinch-point, but 
being part of the cuspidal curve, no point thereof is regarded as a pinch-point; that 
is, in regard to this line also we have j' = 0. And therefore for the entire nodal curve 
/ = 0.