PROFESSOR CAYLEY OjST CUBIC SURFACES. 
297 
eliminating zw, we obtain 
' 3 (bd-c 2 ), 
< 2 i (—ad 2 —obcd J r 4c 3 ) , 
■ 6(acd-\-b 2 d—2bc 2 ), 
6 (bc—ad)b, 
a 2 d—b 3 , 
in, yY= 0 , 
which shows that the cuspidal curve lies in 4 planes, and it hence consists of four conics ; 
these are of course the reciprocals of the quadric cones which touch the cubic surface 
along the four conics which make up the spinode curve. 
134. The equation of the surface, attending only to the terms of the second order in 
y, z , iv, is 21d 2 x‘ l zw=Q ; it thus appears that the point y=0, z=0,w=0 (reciprocal of the 
plane X=0) (which is oscular along the axis joining the two binodes, or BB-axis) is a 
binode on the reciprocal surface, the biplanes being z=0,w=0, viz. these are the planes 
reciprocal to the binodes (X=0, Y=0, W=0) and (X=0, Y=0, Z = 0) of the cubic 
surface; we have thus B'=l. 
It is proper to remark that the binode y= 0, z= 0, w= 0 is not on the cuspidal curve, 
as its being so would probably imply a higher singularity. 
135. A simple case, presenting the same singularities as the general one, is when 
ci=d, b=c= 0: to diminish the numerical coefficients assume a=d=^, the cubic 
surface is thus 12XZW+X 3 +Y 3 =0, and the equation of the reciprocal surface, mul- 
tiplying it by 4, becomes 
z 3 w 3 
+ 6 x 2 z 2 w 2 
+ (9# 4 — 12 x 3 y)zw 
— Ay 3 (x 3 —y 3 )=0, 
viz. this is the surface 
4 y 6 
-4y 3 x(x 2 -{-Szw) 
+ zw( 3x 2 -f zw ) 2 = 0 
considered in the Memoir “ On the Theory of Reciprocal Surfaces.” The cuspidal curve 
is, as there shown, composed of the four conics y= 0, 3a i2 +^o = 0 and y 3 — 2a i3 =0, 
x 3, — zw= 0; and it is there shown that the two points (#=0, y= 0, ~ = 0), (.r=0, 
y= 0, w=0), each reckoned 8 times, are to be considered as off-points of the reciprocal 
surface. 
136. The like investigation applies to the general surface, and we have thus #'=16; 
the points in question are still the points ( x=0 , y— 0, 5 = 0), (a’=0, y= 0, w= 0) ; viz. 
these are the points of intersection of the surface by the line (#=0, ?/=0), which points 
are also the common points of intersection of the four conics which compose the cus- 
pidal curve, that is, they are quadruple points on the cuspidal curve ; it does not appear 
2 s 
MDCCCLXIX. 
