PROFESSOR CAYLEY ON -CUBIC SURFACES. 289 
or substituting for X, Y, Z, W their values (=165 2 , 40 + 164 5 , 165 3 , _5-80 4 -165 8 ) and 
omitting a common factor 165 2 , we find for the cuspidal curve 
x : y : z : w=30+245 5 -160 9 : 245 2 +320 6 : -4-48tf 4 : 165 3 
(values which verify the equation X#+Yy+Zz+Ww=0) ; the spinode curve being 
thus of the order =9 as mentioned. 
For 5 = oo we have the singular point (y=0, z=0, w= 0), (reciprocal of torsal biplane), 
and in the vicinity thereof x : y : z : w= 1 : — 25 -3 : 35 -5 : — 5 -6 , therefore 
For 5 = 0 we have the singular point x=0, y= 0, w = 0 (reciprocal of the other 
biplane), and in the vicinity thereof x : y : z : w= — -§0 : — 65 2 : 1 : — 40 3 , therefore 
116. The section of the surface by the plane z — 0 is an interesting curve. Writing 
z=0 in the equation of the surface, I find that the resulting equation may be written 
(64w 3 , 144 xw 2 , w 3 +7Gx 2 w+xy 2r Jw 2 +27x 2 , y 2 —32xw) 2 =0, 
where observe that 
6 4 w 3 (w 3 + 7 Qxhv -\-xy 2 ) — (72 xw 2 ) 2 
=§iw 3 \w(w 2 -\-27x 2 )-\-x(y 2 —32xw)~\ ; 
so that the curve has the four cusps w 2 +27# 2 =0, y 2 —32xw=0 ; the plane z — 0 
intersects the cuspidal ninthic curve in the point [y= 0, z— 0, w= 0) counting 5 times, 
and in the last-mentioned four points : in fact, writing in the equations of the ninthic 
z— 0, that is 1+125 4 =0, we find x, y , w=-§5, ^5 2 , 165 3 , and thence w 2 +27^ 2 = 
^5 2 (1+125 4 )=0, y 2 - 32<wi>=0. 
The curve has also nodes at the points (y= 0, x-\-w= 0 ; y= 0, x—W— 0), viz. these 
are the intersections of the plane z=0 with the nodal lines (y— z=0, #-j-w=0) and 
(y-\-z~ 0, x — w = 0), and it has at the point (y= 0, w = 0) (intersection of its plane with 
the cusp-nodal line y= 0, w= 0, and quintic intersection with the cuspidal ninthic) 
a singular point=2 cusps-j-7 nodes; hence the curve has cusps =(4 + 2 = )6 ; nodes 
(2-j-7 = )9; or 2 nodes-1-3 cusps=36 ; class =6, as it should be. 
MDCCCLXIX. 
2 R 
