312 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
6 times ; X=0, Z=0 (single ray) 2 times ; and of a residual quartic, which is the spinode 
curve; a ! = 4. 
The equations of the spinode curve are XZ— Y 2 =0, XW-j-2Z 2 =0; the first surface 
is a cone having its vertex on the second surface ; and the curve is thus a nodal quadri- 
quadric. 
The mere line is a single tangent of the spinode curve; (3' = 1. 
Reciprocal Surface. 
175. The equation is obtained by means of the binary cubic 
(-% 2 > % W XX, Y) 3 , 
viz. throwing out the factor y , the equation is 
w \— 6 4# 3 ) -|- w ( — 1 6#V + 7 2xy 2 z + 2 7y 4 ) + 1 6y 2 z 3 = 0 . 
The section by the plane w=0 (reciprocal of U 7 ) is w= 0, z — 0 (reciprocal of torsal 
ray) three times, and w=0, y=§ (reciprocal of single ray) twice. 
Nodal curve. This is the line #=0, y— 0, reciprocal of the mere line : V=l. 
Cuspidal curve. The equation of the surface may be written 
(6 hr, — 1 Qz, — 3wX^ 2 +3 xw, % 2 + 4^) 2 =0, 
where 
4 . 64#( - 3w) - 256« 2 = - 256(z 2 + 3 xw). 
This exhibits the cuspidal curve £ 2 +3#w=0, % 2 -f 4;s#=0, where the surfaces are each 
of them cones ; the vertex of the second cone is on the first cone, and the two cones have 
at this point a common tangent plane ; the curve is thus a cuspidal quadriquadric. 
176. [The equation 
(64#, —16 z, — 3wX^ 2 + 3#w, % 2 +4^#) 2 =0 
resembles that of a quintic torse, viz. the equation of a quintic torse is 
( #, — 4 z, 8 iv'Jz 2 —2wx, y 2 —2zx)' 2 =Q, 
which equation, writing 9 y for y, —2# for x, and f w for w, becomes 
(—2#, —4 z, §w r fz 2 -{-?>xw, 9j 2 +42#) 2 =0, 
or, what is the same thing, 
( #, 2 z, — ?m [$V+3#w, 9^ 2 +4^#) 2 =0; 
and developing, this is 
# 3 . w 2 
+# 2 . —2 z 2 w 
+#. —18 y 2 zw+z 4 
-27y 4 w+2y 2 z 3 =Q, 
which, however, differs from the equation of the reciprocal surface, not only in the 
nu merical coefficients, but by the presence of a term xz 4 . ] 
