288 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
— 3 2w 2 (z —y) (x-\-w)x ( x-w)(z-\-y ) 
(z — y ) 2 {llx 2 — 27w 2 )(z-{-y) 2 
— w (z—y) 3 (3x—7w) ( x-w)(z-\-y ) 
+|w ( z—yY (x—wf 
-Ytfz ( z-y ) (x-w)(z+y) 
-y 3 x (z-y) (z+yf= 0, 
viz. this is 
(A, B, C%x-w, z+y) 3 — 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= §iiv 3 (x-\-w) 2 = 256w 5 
+ 8w\z— y)\x— 3w) — 64 w 3 z* 
+ \ w\z—yY + 4 wz* 
= 4.w(z 2 — 8w 2 ) 2 , 
B= — 32(2:— = — 128w 4 2 
— w(z— y) 3 (8x— 7w) + 32 wV 
- 2^ 5 
= -2z(z 2 ^8w 2 )\ 
C= 8v?(x-{-w 2 )(x-\-dw) — 128 w 5 
+i<z-y) 2 (lYr 2 -27<) - 32wV 
~xy\z—y) + 2wz 4 
~xy\z-y) + 2wz 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, —2^2, 2^/2, 1. But these values give L, M, N=12, 12, —16; values which 
satisfy the equations L 2 — 12w 2 M=0, LM+9w 2 N==0, 4M 2 -J-3LN=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/=0. And therefore for the entire nodal curve/ = 0. 
115, The cuspidal ninthic curve is a unicursal curve, the equations of which can be 
very readily obtained by considering it as the reciprocal of the spinode torse ; we in 
fact have 
x:y:z: w= ZW+2XY : 2YZ+X 2 : WX+Y 2 -3Z 2 : ZX, 
