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A MEMOIR ON CUBIC SURFACES. 
[412 
and thence the equation is found to be 
1[y 2 — (x — zf\ 
+ 16w 3 [(2« — bz) y 2 — 2(x — 2z) (x — zf\ 
4- 8w 2 [y 4 + (a? — xz + Qz 2 ) if — 2x 2 (x — zf] 
+ 4w [fix 4- 3z) y 4 — 2x 2 (x 4- z) y 2 ] 
+ y i (y 2 ~ x *) = 0, 
where the section by the plane w = 0 (reciprocal of binode) is y i (y 2 — x 2 ) = 0, viz. this 
is the line w = 0, y = 0 (reciprocal of the edge) four times, and the lines w = 0, y 2 - x 2 = 0 
(reciprocals of the biplanar rays). 
The section by the plane z = 0 is found to be (y 2 — x 2 )(y 2 +4xw + 4w 2 ) 2 = 0, viz. this 
is the two lines z = 0, y 2 — x 2 = 0 (reciprocals of the nodal rays), and the conic 2 = 0, 
y 2 + 4xw + 4nv 2 = 0 (reciprocal of the nodal cone WX + Y 2 — X 2 = 0) twice. 
142. Nodal curve. The equation shows that the line y= 0, x—z = 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 
that is, 
— x 2 [16 (x — z) 2 w 2 + 8 (x + z) wy 2 + if] = 0, 
, , v , Va? + 
4 (x — z)w+ ~ _ y 2 = 0, 
WX -f VZ 
two values, w = 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, tu — 0 twice; b' = 3. 
143. Cuspidal curve. The equations 
j 12w 2 , 8w 2 + 4>wx, 4<w 2 + 4uvx + 4>tvz + y 2 jj = 0 
'! 8w 2 + 4wx, 4>iu 2 4- 4wx + 4wz + y 2 , 12wz 
give 
(4w -f 2x) 2 — 3 (4w 2 + 4>wx + 4wz + y 2 ) = 0, 
— 36w 2 2 + (2tv + x) (4w 2 + 4<ivx + 4wz + y 2 ) = 0, 
or, as these are more simply written, 
4nv 2 + 4 wx — 12 wz + 4x 2 — 3 y 2 = 0, 
8w 3 + 12w 2 x — 28w 2 z + w (4a; 2 + 4xz + 2y 2 ) + xy 2 — 0, 
so that the cuspidal curve is a complete intersection 2x3; c' = 6.