PROFESSOR CAYLEY ON CUBIC SURFACES. 
271 
(a) 
(*) 
W 
r (/) 
(ff) 
4 
-4 2 
0 
0 
0 
4 
-f 2 2 
0 
0 
0 
fa 
“fa 2 
0 
0 
0 
f 4 
-£? 
0 
0 
0 
"44 
1 
-KH) 
7 
"44 
8 
44 
1 
7 
44 
8 
“44 
»(H) 
1 
-(H) 
7 
8 
"44 
*(H) 
1 
-(H) 
7 
44 
8 
“'44 
*(s + ?) 
1 
—(H) 
7 
44 
8 
44 
»(H)- 
1 1 
—(H) 
7 
"44 
(h) 
ao 
( 2 ') 
(3 r 
( 4 ') 
7 
7 
7 
7 ( 
44 \f] £ 
_i_/i 2 
f e t.U + f 3 
is(fT + fO 
X-f,Y=0, yY + f,W=0 
'fkil+l 
~<k(i + l 
±/l 
fX \ f. 
44 
k(^ + ^) 
'fK(fl + 5) 
4(fl + f0 
"4(£ + ? 
(12 . 3'4')* 
(13.2'40 „ 
(14 . 2'3') „ 
(23.U40 „ 
(24.U30 „ 
(34.T20 „ 
^equations are 
S{X-(f, +f 2 )Y} -f,f 2 Z=0, y{X- (f, +fJY} -f/ 4 W=0. 
76. To verify the equations of the line 12 . 3% observe that the two equations give 
«>w^jx(,L + .)_ ya+ , + . +a} , 
ZW= fik( X - f N ; 5Y)(X-fRY) : 
becomes"^ 0 ” * *“ b ? X observing that -yS=rf A f A , 
XW + XY=(yZ+SW) + ySY‘-jL(X-f 1 Y)(X-f a Y)(X-f 1 Y)(X-f 4 Y)=0. : 
and substituting the values just obtained, this is 
X2 (X- f, + f s Y)(X— g+fjY) + XY S (X^5+®- - Yffi o 
+f 1 f 2 faf^-(X-f 1 Y)(X-f 2 Y)(X-f 3 Y)(X-f 4 Y) == 0, 
which is in fact an identity. 
II' I! 16 facultatlve lines are the transversal and the six mere lines; V= e '=7 • t'= 3 
78. The equation of the Hessian surface is found to be 
(7Z+aW)XZW+Y 2 ( y Z-§W) 2 +3(cX+dY)XZW + 12^XY 2 («X+5Y) 
— (yZ+SW)(3«X'-f 9£X S Y + 6eXY 2 ) 
-9X»{(«_5*)X I +(« 4 _fc)XY+(M- c 2 )Y a }=0 
