270 
PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
74. Writing X(«, b,c, ^X, Y) 3 -y^Y 4 = -^(/ 1 X-Y)(/;X-Y)(/;X-Y)(/ 4 X-Y), 
the .20 planes are 
X=0, [ 0] 
X-/,Y=0, [11'] 
X-/,Y=0, [22'] 
X-/,Y=0, [33'] 
X-/ 4 Y=0, [44] 
i{X-(f,+i)Y}-fAZ =0, [12] 
S{X-(f. + f s )Y}-f 1 f»Z =0, [13] 
»{X-(f x +t)Y}-fAZ =0, [14] 
S{X-(f 2 +f 3 )Y}-£ 1 f 3 Z =0, [23] 
MX-ft + fJYJ-ttZ =0, [24] 
S{X-(f,+f 4 )Y}-f,f 4 Z =0, [34] 
y{X-(f, + f s )Y;-f,f s W=0, [1'2 '] 
7 {X-(f, + f 1 )Y>-f 1 £ ! W=0, [l'SQ 
7 {X-(f,+f 4 )Y}-f 1 f 4 W=0, [1'4'] 
y {X-ft+£,)Y)-ttW=0, [2'3] 
y {X-(f a +f 4 )Y}-£,f 4 W=0, [2'4] 
y {X-(f,+f 4 )Y}-f s f 4 W=0, [3'4] 
- yS (i+A)X+%+ y Z + W=0, [12.34] 
-yS(fj+^ 4 )x+%+ y Z + 8 w = 0 , [13.24] 
-rS(iH-i)x + %+ y Z+5W=°, [14.23] 
75. And the 12 lines are 
(*) 
m 
(*) 
(/) 
&} 
(A) 
whence equations may be written 
0 
0 
0 
0 
0 
1 
(0) X=0, Y=0 
i 
0 
0 
0 
~7 
d 
(5) X=0, <SY+yZ+&W=0 
0 
0 
0 
f? 
f> 
(1) X=f 1 Y=0, SY+f^O 
0 
0 
0 
n 
f 2 
-* 
(2) 
0 
0 
0 
fs 
A 3 
f 3 
(3) 
0 
0 
0 
f 4 2 
f 4 
-l 
(4) 
