PROFESSOR CAYLEY ON CUBIC SURFACES. 
323 
viz. this is 
+ 27 a 4 c 2 
- ±a 3 b 3 
+ 30 a 2 b 2 cf 
—5.4 o 2 c 3 g 
+ 3 Qab 3 cg 
+ 2Aabc 2 f 
+ 45 5 A 
- U‘f 
+18 tvfg 
+27 cy 
+ 4 +/'•=(), 
which is the condition in order that the line (a, ft, c, f, g, A) 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, &), we have in effect the equation of the circumscribed 
cone, vertex (a, (3, y, S). 
Writing (f, g, A, a , ft, c) in place of (a, b, c,f, g, A), we obtain 
27 fh 2 
- 4/V 
+ 30 fyha 
-&4:fh 3 b 
+ 36 fg 3 hb 
+24:fgh 2 a 2 
+ 4 g*c 
- 1 g A a? 
+ 18 g 2 h 2 ab 
+ 27 A 4 / 2 
+4AV=0 
as the condition that the line (a, b , c,f, g, h) shall touch the reciprocal surface 
2 7 ( ixw + z 2 ) 2 + 6 iy 3 w = 0 ; 
ana if we consider a, b , c,f, g , A as standing for 
yy—fiz , az — yx, fix— ay, lx— aw, ly—fiw, hz—yw, 
values which satisfy the relation 
( 0, A, -g, a Ja, fi, y, ft) = 0, 
-A, 0, /; ft 
-/, 0, a 
—a, —b, —c, 0 
2x2 
