300 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
is the two lines 2=0, y 2 —x 2 =0 (reciprocals of the nodal rays), and the conic 2=0, 
if- -f- 4xw + 4zitf = 0 (reciprocal of the nodal cone WX-f-Y 2 — X 2 =0) twice. 
142. Nodal curve. The equation shows that the line y= 0, x — 2=0 (reciprocal of the 
line W=0, X+Z = 0) is a nodal line on the surface. 
It also shows that the line y= 0, w = 0 (reciprocal of the edge) is a tacnodal line 
(=2 nodal lines) on the surface; in fact attending only to the lowest terms in y, w, we 
have 
- xfl 6 (x — 2) V + 8(a?+ z)ivf +3/ 4 ]=0, 
that is, 
4:(x- z)w +^|=-^?/ 2 = 0 , 
two values, iv=Ay 2 , w=~By 2 , which indicates a tacnodal line. 
The nodal curve is thus made up of the line y— 0, x— 2=0 once, and the line y= 0, 
w = 0 twice; V = 3. 
143. Cuspidal curve. The equations 
12 w 2 8w 2 +4m, 4w 2 +4m+4w2+?/ 2 1|=0 
| 8w 2 +4m, 4w 2 -j-4m+4w2-{-y 2 , 12 wz 
give 
(4 w + 2#) 2 — 3 (4 w 2 + 4 wx + 4 wz -j -y 2 ) = 0 , 
— 3 6 w 2 z + (2w + x)(4w 2 -f- 4m -+- 4 wz +y 2 )= 0, 
or, as these are more simply written, 
4w 2 + 4m — 12wz-{-4x 2 — oy 2 =0, 
8iv 3 +I2w 2 x — 2 8 w 2 2 + w(4:X 2 + 4 xz + 2 y 2 ) -{-xy 2 = 0 , 
so that the cuspidal curve is a complete intersection 2x3; c'=Q. 
