304 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
viz. the surface consists of the uniplane X-f-Y + Z=0 twice, and of a quadric cone 
having its vertex at U 6 , and touching each of the planes X=0, Y=0, Z=0. 
The complete intersection with the cubic surface is made up of the rays each twice 
and of a residual sextic, which is the spinode curve; <r' = 6 . 
The equations of the spinode curve are 
W(X + Y + Z) 2 + XYZ = 0, 
X 2 + Y 2 +Z 2 - 2YZ - 2ZX— 2XY= 0, 
viz. the curve is a complete intersection, 2x3. 
Each of the mere lines is a single tangent (as at once appears by writing for instance 
W=0, X=0, which gives (Y— Z) 2 =0); that is, /3'=3. 
'Reciprocal Surface. 
154. The equation is found by means of the binary cubic 
4(T — #U)(T— ;yU)(T — zU) +wT 2 U, 
viz. writing for shortness 
/3 =x+y+z, 
y=yz+zx+xy, 
% =xyz , 
then the cubic function is 
(12, w-4/3, 4y, -mXT, U) 3 , 
and the equation of the reciprocal surface is found to be 
432S 2 
+ 64y 3 
— (w — 4/3) 3 S 
+ 72(w — 4/3)yS 
— (tv — 4/3 ) 2 y 2 = 0 ; 
expanding, this is 
w 3 . — & 
+w 2 .12 (3%—y 2 
-j- 8 w . — 6/3 2 §+j3y 2 +9y^ 
+ 16(4 j 3 3 ^-i3y-18(3yB + 4y 3 +27^ 2 ) = 0 ; 
or substituting for /3, y, £ in the first and last lines, this is 
w 3 . — xyz 
+w 2 . (12/3& — y 2 ) 
+ 8 w. — 6j3 2 £+j3y 2 -f9yc> 
+ 1 6 ( 3 / - z)\z - x)\x~yj = 0 
(where (3, y, o=x-\-y~j-z, yz-j-zx+xy, xyz). The section by the plane w = 0 (reciprocal 
