PROFESSOR CAYLEY ON CUBIC SURFACES. 
265 
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ip) 
(/) 
(9) 
1 m 
whence equations may be written 
0 
0 
0 
0 
-1 
i 
(1) X=0,Y+Z=0 
0 
0 
0 
1 
0 
-i 
(2) Y = 0, Z+X=0 
0 
0 
0 
-1 
1 
0 
(3) Z=0, X+Y=0 
0 
0 
0 
0 
— n 
m 
( 4) X=0, wY+«Z=0 
0 
0 
0 
n 
0 
-l 
( 5) Y=0, rcZ+ZX=0 
0 
0 
0 
—m 
l 
0 
( 6) Z=0, lXmY=0 
1 
0 
0 
0 
0 
0 
o 
II 
£ 
o' 
II 
X 
T"H 
0 
1 
0 
0 
0 
0 
(25) Y = 0, W=0 
0 
0 
1 
0 
0 
0 
(36) Z=0, W=0 
l 
n 
n 
n 2 v 
— nlv 
0 
(15) l X+mY+j?Z=0, W+nv Z=0 
l 
in 
in 
—m 2 p 
0 
Imp 
(16) l X+mY+mZ=0, W+m^Y=0 | 
l 
in 
l 
0 
l 2 \ 
—Imp 
(26) l X+mY+l Z=0, W+& X=0 
n 
m 
n 
mnv 
—n 2 v 
0 
(24) wX+mY-fwZ=0, W -\-nv Z=oJ 
m 
m 
n 
— mnp 
0 
m 2 p 
(34) mX-\-7nX -\-n Z=0, W -f- mpY = 0 j 
l 
l 
n 
0 I 
nlk 
(35) l X+l Y+raZ=0, W + lk X=0 | 
64. The rays are not, the mere lines are, facultative; hence V=g'=9 : t'=6. 
65. The equation of the Hessian surface is 
-W(X+Y+Z)(lXL+mY+nZ)(pvX+*Y+tyZ) 
— Jc{l 2 X 4 + m 2 Y 4 + n 2 Z 4 — 2mriY 2 Z 2 — 2nlZ 2 X 2 — 2 lmX 2 Y 2 ) 
+^YZ{{l 2 +Zlm+?>ln+mn)X+(m 2 +Zmn+Zml+nl)Y+(n 2 +?>nl+?>nm+lm)7,}=(). 
The Hessian and cubic surfaces intersect in an indecomposable curve, which is the 
spinode curve; that is, spinode curve is a complete intersection 3x4; <j'—12. 
The equations may be written in the simplified form 
W(X+Y+Z)(ZX+mY+nZ)+/l;XYZ=0, 
l 2 K 4 -f m 2 Y 4 + n* Z 4 — 2mnY 2 Z 2 — 2nlZ 2 X 2 - 2 lmX 2 Y 2 
- 4XYZ { l(m + n)X + m(n+l)Y + n(l+m) Z } = 0. 
We may also obtain the equation 
^ ! (x+ Y +Z)(IX+ mY +nZ)(lX , + mV + n’tf—m -f nY Z — jT+IZX — f+iKXV) 
+ ^Y 2 Z ! + f .*Z"X ! +» ! X ! Y J -2XYZ(pX+»XY+X f .Z)=0, 
MDCCCLXIX. 2 0 
