412] 
A MEMOIR ON CUBIC SURFACES. 
415 
that is, the surface has the nodal lines (z + y = 0, x — tv — 0) and (z — y = 0, x + w = 0), 
which are the reciprocals of the lines 12' and 13' respectively. The nodal curve is 
made up of these two lines and of the line y = 0, w = 0 (reciprocal of edge), as will 
presently appear; so that we have b' = 3. 
112. The equations of the cuspidal curve are 
L 2 - 12w 2 M = 0, 
LAI + 9w 2 N = 0, 
UP + 3 LN = 0. 
Attending to the two equations 
L 2 — 12 w 2 AI = y 4 + 8 y 2 xw + Vdx-vf- + 2 4yzw 2 + 48ic 4 = 0, 
LAI + 9 w 2 N = y 3 z + 2 y 2 w 2 + Axtjzw + (8 — 72 =) — 64xw s + 1 Hzhu 2 = 0, 
these surfaces are each of the order 4, and the order of their intersection is =16. 
But the two surfaces contain in common the line (y = 0, w = 0) 7 times; in fact on 
the first surface this is a cusp-nodal line 4xiu + y 2 4- Ay- = 0; and on the second 
surface it is a nodal line w {Axy -I- \%zw) = 0; the sheet w = 0 is more accurately 
4xnv + y n - + By 3 ... = 0 ; whence in the intersection with the first surface the line counts 
5 times in respect of the first sheet and 2 times in respect of the second sheet; 
together (5 + 2 =) 7 times, and the residual curve is of the order (16 — 7 =) 9. 
113. I say that the cuspidal curve is made up of this curve of the 9th order, 
and of the line y = 0, w = 0 (reciprocal of the edge) once; so that c' = 10. In fact, 
considering the line in question y = 0, w — 0 in relation to the surface, the equation 
of the surface (attending only to the lowest terms in y, w) may be written 
— xz 2 (y 2 + 4xw) 2 + w (— y 3 z 2 ) + w 2 (— 36xyz 3 ) + &c. = 0, 
giving in the vicinity of the line 
5 
Axw + y 2 = Ay , 
and then 
— xz 2 A 2 + — {\ — ff) = 0, 
x 
£ 
that is A 2 = — 2 - or 4xw + y 2 = V - 2. - y f ; wherefore the line is a cusp-nodal line, 
counting once as a nodal and once as a cuspidal line; and so giving the foregoing 
results b' = 3, c = 10. 
114. I revert to the equation which exhibits the nodal lines (x — w = 0, y + z = 0), 
{x + w = 0, y — z= 0) for the purpose of showing that they have respectively no pinch- 
points ; that is, that in regard to each ol them we have j = 0. In fact for the first