436 
A MEMOIR ON CUBIC SURFACES. 
[412 
The section by the plane w — 0 (reciprocal of B 3 ) is w = 0, y = 0 (the edge) three 
times; and w — 0, y — x = 0 ; w = 0, y — z = 0 (reciprocals of the CB-axes). 
163. Nodal curve. This is the line y = x — z\ wherefore b' = 1, To put the line 
in evidence, write for a moment x = y + a, z = y + y, then the equation is readily con 
verted into 
w 3 .16 (a — y) 2 
+ w 2 i—y (a 2 — lay + y 2 ) 
[+ (a + y) (2a — y) (— a + 2y) 
+ w j' y 2 (a 2 — 1 Oay + y 2 ) 'j 
; — 18yay (a + y) 
V — 27 a 2 y 2 
+ y 3 a y = 0, 
which, each term being at least of the second order in a, y (— x — y, z — y), exhibits the 
nodal line in question. 
164. Cuspidal curve. Multiplying by 27, the equation may be written 
(*ly — 3« — 3z — 5w, —y + 6w, — w\y- + 16yw — 12xw — 12ztu 4-16iv 3 , 
— 20y 2 + 24yx -f 24yz — 27xz — 8yw + 16w 2 ) 2 = 0, 
where 
4w (7y — 3« - 3z — 5w) + (— y + 6w) 2 = y 1 +1 Qyiu — 12 (x 4- z) iv + 16ty 2 ; 
and we have thus in evidence the cuspidal curve, 
y n - + 16yiv — 12 (x + z) w + 16w 2 = 0, 
— 20y n - + 24y {x + z) — 27xz — 8yw + 16w 2 = 0, 
which is a complete intersection, 2x2, or quadriquadric curve; c' = 4.