210 
PEOFESSOE CAYLEY ON EECIPEOCAL SUEFACES. 
when reduced by the equations P — 0, Q = 0 of the cuspidal curve, becomes simply 
F(AP) 2 =0; and we have thus the off-points F= 0, P=0, Q=0, consequently 
31. But suppose, as before, the case of a surface (A, B, CT£T, Q) 2 =0 having a cuspidal 
curve P=0, Q=0, and therefore AC— B 2 being =0 for P=0, Q=0. The equation of 
the second polar, writing therein P=0, Q=0, becomes (A, B, C)[AP, AQ) 2 =0, and if 
for any given surface this assumes the form A(MAP+NAQ) 2 =0 (observe that M, N 
may be fractional provided only the MAP+NAQ is integral), then there will be on the 
cuspidal curve the off-points A=0, P=0, Q=0. 
32. An interesting example is afforded by a surface which presents itself in the Me- 
moir on Cubic Surfaces : the surface 
4/ 
— 4y 3 x(x 2 -f 3zw) 
+ zw(3x? + zw ) 2 = 0 
has the cuspidal conic y=0, 3# 2 -f-2W=0, and (as coming under the form FP 2 -fGQ 3 =0) 
has the off-points zw= 0, y=0, 3# 2 -|-zw = 0; that is, the points (#=0, y=0, z= 0), 
(#=0, y=0, w=0) each twice; 3=4. 
But writing the same equation in the form 
(4, 6#, 8x 2 -\-zw\y 3 —2x 3 , x 2 —zw) 2 = 0, 
where 
4 . (8# 2 -f- zw ) — (6#) 2 = — - 4(# 2 — zw), 
it appears that there are also the three cuspidal conics y 3 — 2# 3 =0, x 2 —ztv = 0. Reducing 
by means of these two equations, the equation of the second polar is at first obtained in 
the form 
(4, Qx, 8x 2 -{-zwX3y 2 Ay— Go? Ax, 2xAx—zAco—uAz) 2 =0; 
hut further reducing by the same equations and writing for this purpose y=ax{a 3 =2), 
the equation becomes 
(4, 6#, 9# 2 X# 2 (3* 2 Ay-6A#), 2xAx—zAw—wAz 2 ) 2 =Q, 
that is 
x 2 \2x(2>a 2 Ay — QAx) + 3(2# A#— zAw— wAzJ] 2 =0, 
and we have thus the off-points # 2 =0, y 3 — 2# 3 =0, x 2 —zw=0, in fact the before-men- 
tioned two points each 6 times; and the complete value of 6 is 0=(4+12=)16 ; viz. 
the off-points are the points (#=0, y= 0, z— 0), (#=0, y= 0, w— 0) each 8 times. On 
account of this union of points the singularity is really one of a higher order, but equi- 
valent to 3=16. 
I am not at present able to explain the off-plane or reciprocal singularity 3'=1. 
33. As to the close-point or singularity %=1. I remark that at an ordinary point of 
the cuspidal curve the section by the tangent plane touches, at the point of contact, the 
cuspidal curve : the point of contact is on the curve of section a singular point [in the 
nature of a triple point, viz. taking the point of contact as origin, the form of the branch 
in the vicinity thereof is y 3 — # 4 =0, where y= 0 is the equation of the tangent to the 
