406 
A MEMOIR ON CUBIC SURFACES. 
[412 
92. This, completely developed, is 
64iv s ,ab(a + b) 2 {(a + b) y 2 — (x — z) 2 } 
+ 32w 5 .2ab [ 3 (a + b) [(a — 2b) x + (—2a + b) z] y 2 1 
\+ (x — z) 2 [(— 3a + 5b)x+ (5a — 36) z] J 
+ 16w 4 [ 3a6 (a 2 — 7a6 + b 2 ) y 4 
< + [6 (9a 2 + 26a6 — b 2 ) x 2 — 26a6 (a+b)xz + a (— a 2 + 26a6 + 96 2 ) z 2 ] y 2 
[ + (x — z) 2 [6 (— 12a + b) x 2 4- 22abxz + a (a — 126) z 2 ] 
+ 
8w 3 
r Sab [(2a — 6) x + (— a + 26) z] y 4 
< + [6 (— 2a + 56) x? + 6 (3a — 26) + a (— 2a + 36) xz 2 + a (5a — 26) 2 s ] y 2 !- 
„ + 2 (¿c — 2) 2 [— 26# 3 + 6# 2 2 + axz 2 — 2ay 3 ] J 
-4- 4w 2 / 3a6 (a + 6) y 6 ' 
I + [6 (9a — 26) x 2 + Sabxz + a (— 2a + 96) z 2 ] y 4 
+ 2 [— 66a 4 + 6ic 3 2 — (a + 6) i» 2 ^ 2 + axz 3 — 6a^] y 2 
v + 4ic 2 ^ 2 (a? — z) 2 
4- 2 w 
2a6 (a? + 2) y 6 
•< — [36a 3 + 2bx 2 z + 2axz + 3a^ 3 ] y 4 ; 
, + 4x 2 z 2 (x + z) y 2 j 
+ V i (ay 2 - x 2 ) (cy 2 - z 2 ) = 0, 
where we see that the section by the plane w = 0 (reciprocal of i? 4 ) is made up of 
the line w— 0, y = 0 (reciprocal of the edge) four times, and of the lines w = 0, 
ay 2 — x 2 = 0 ; w = 0, by 2 — z 2 = 0 (reciprocals of the rays) each once. 
93. The surface contains the line y = 0, w — 0 (reciprocal of the edge); and if we 
attend only to the terms of the lowest order in y, w, viz. 
X 2 Z 2 {1G (x — z) 2 w 2 + 8 (x + z) yhu + y 4 }, 
which terms equated to zero give 
1 
^ i / /“■ y 9 
(wx ± Wzf 
we see that the line in question (y = 0, w = 0) is a tacnodal line on the surface, the 
tacnodal plane being w — 0, a fixed plane for all points of the line: it has already 
been seen that this plane meets the surface in the line taken 4 times; every other 
plane through the line meets the surface in the line taken twice. We have in what 
precedes the d posteriori proof that in the cubic surface the edge is a facultative line 
to be counted twice. 
94. Cuspidal curve. The equation of the surface may be written 
(L 2 — 12iv 2 M) (4M 2 + SLN) — (LM + 9iv 2 N) 2 = 0,