314 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
or, what is the same thing, 
X 2 (YZ +YW+ZW) 
+Y 2 (ZW +ZX +WX) 
+Z 2 (WX+WY+XY) 
+W 2 (XY +XZ +YZ )=0. 
The complete intersection with the cubic surface is made up of the six axes each twice, 
and there is no spinode curve; <4=0, whence also /3'=0. 
Reciprocal Surface. 
180. The equation is immediately obtained in the irrational form 
«/ x- \-\f y ~\~\f z+\/ w—0, 
or rationalizing, it is 
(x 2 + y 2 + z 2 + uf — 2 yz — 2 zx — 2 xy — 2 xw — 2 yw — 2 zwf — Qkxyzw = 0 ; 
so that this is in fact Steinek’s quartic surface. 
Nodal curve. This consists of the lines x—y=0, z—w— 0; x— 2=0, y— w=0; 
x—iv=0, y — 2=0 ; so that V =3 
To put anyone of these, for instance the line x—y= 0, z—W= 0, in evidence, we may 
write the equation of the surface in the form 
\_{x — yf -f- (z — w) 2 — 2(x 4- y) (z + wjf — 6 ixyzw = 0, 
that is 
{ (x ~yf + (z - wf\\ {co—yf '+(?— wf - 4(a? +y)(z + w ) } 
4- i\_{x -\-y)\z + wf — 1 §xyzw] = 0, 
or finally 
{{x—yf J r{z-w)’ 2 }{{x-yY^r{z-w)‘ 2 -i{x-\-y){z-> t ic)} 
+ 4 { (x — y)\z — uff + 4 xy(z — w) 2 4- 4 zw[x—yf } = 0, 
where each term is at least of the second order in x — y f z—w. 
There is no cuspidal curve ; c'= 0. 
