296 
PEOFESSOE CAYLEY ON CUBIC SUEFACES. 
The equation is found to be 
432(<z 2 ^ 2 — 6 abed + 4 ac 3 + 4 b 3 d — Sb 2 c 2 )z 3 w 3 
+216 [(ad 2 - obed + 2c 3 )# 2 + ( - 2acd+m—2bc 2 )xy+{ - abd+2ac 2 - b 2 c)if]z 2 w 2 
+ 9 [3 d 2 x 4 — 12 cdx 3 y +(10M+ 8c 2 )#y — (4«<Z+ 8 bc)xy 3 + ( 4 ac — b 2 )if]zw 
—y 3 {dx 3 — 3 cx 2 y + 3 bxy 2 —ay 3 ) =0. 
The section by the plane w= 0 (reciprocal of B 3 =D) is the line w= 0, y= 0 (reci- 
procal of edge) three times, and the lines w= 0, dx 3 — ?>cx 2 y + obxy 2 —aif=0 (reciprocals 
of the biplanar rays). And similarly for the section by the plane 2=0 (reciprocal of 
B 3 =C). 
The section by the plane ^=0 As made up of the lines (y=0, 2=0), (^=0, w=0) each 
once, and of two conics, y— 0, 
1 6 {a?d 2 — 6 abed + 4 ac 3 +4 b 3 d— ?>b 2 c 2 )z 2 w 2 
+ 8(ad 2 — ?>bcd-\-2c 3 )x 2 zw 
+ <ZV= 0. 
131. There is not any nodal curve; b'= 0. 
132. Cuspidal curve. The equations may be written 
3# 2 +12 azw, 2xy J r 12bzw, y 1 -\-V2czw 
2xy -\-\2bzw , 3/ 2 +12c2W, 12dzw 
=0. 
Forming the equations 
( bd—c 2 ) . 1442 2 w 2 +2(^#y— cy 2 ) . 12zw— y 4 =Q, 
{ad— be ) . 1442W + (3^# 2 — 2cxy— by 2 ) . 122W— '2o?y 3 =0, 
these are two quartic surfaces having in common the lines {y= 0, w= 0), (y=0, 2=0) : 
attending to the line (y= 0, 2=0), this is on the second surface, an oscular line, 
2 = i ^i xw if ’ 011 the surface it is a nodal line, the one tangent plane being 
Q(bd — e 2 )w .z-\-dxz.y— 0, the other tangent plane being 2=0, but the line being in 
regard to this sheet an oscular line, z ~ ^dxw H ence i n the intersection of the two 
surfaces the line counts (l + 3=)4 times; similarly the line y ={ ), w = 0 counts 
( 1 + 3 = )4 times ; and there is a residual intersection of the order (16 — 4 — 4=) 8, which 
is the cuspidal curve ; c' = 8. 
133. The cuspidal curve is a system of 4 conics ; in fact from the preceding equations 
written in the forms 
(i bd—c 2 , 2 {dxy—cy 2 ), —y i 'J12zw, 1) 2 =0, 
{ad— be, Sdx 2 —2cxy—by 2 , —2xy 3r $12zw, 1) 2 =0, 
