452 A MEMOIR ON CUBIC SURFACES. [412 
viz. this is 
+ 27 cdc 2 
- 4 a 3 b 3 
+ 30 d 2 b 3 cf 
- 54 d 2 c 3 g 
+ 36 cib 3 cg 
+ 24 dbdf 2 
+ 4 b 5 h 
- ivp 
+ 18 b 2 dfg 
+ 27 c*g 2 
+ 4 c 3 f 3 = 0, 
which is the condition in order that the line {a, b, c, f, g, h) may touch the surface 
X' 2 W + XZ 2 + Y 3 = 0; and if we unite thereto the conditions that the line shall pass 
through a given point (a, fi, y, 8), we have in effect the equation of the circumscribed 
cone, vertex (a, fi, y, 8). 
Writing (/, g, h, a, b, c) in place of (a, b, c, f, g, h), we obtain 
27 f*h? 
- 4 f 3 g 3 
+ 30 f 2 g 2 ha 
- 54 f 3 h 3 b 
+ 36 fg 3 hb 
+ 24 fg№a? 
+ 4 g 5 c 
~ l^ 2 
+ 18 g 2 h 2 ab 
+ 27 frf 2 
+ 4 h 3 a 3 = 0 
as the condition that the line (a, b, c, f, g, h) shall touch the reciprocal surface 
27 (4 xw + .z 2 ) 2 + 64 y 3 w = 0 ; 
and if we consider a, b, c, f, g, h as standing for 
yy — j3z, az — yx, fix — ay, 8x — aw, 8y — fiw, 8z — yw,