PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
where for shortness 
a— mn—l , 
j3= nl—m , 
y= lin—n, 
't—lmn— 1, 
then taking X=0 as the equation of the plane [12], Y— 0 as that of the plane [34], 
:0 as that of the plane [56], tlfe equations of the 30 distinct planes are found to be 
X=0, 
[12] 
Y=0, 
[34] 
z=o, 
[56] 
m X+Z Y+Z=0, 
[23] 
m-'X+l Y+Z=0, 
[24] 
m X + Z-T+ Z=0, 
[13] 
m- 1 X+Z- 1 Y + Z=0, 
[14] 
X+w Y+m Z=0, 
[45] 
X+n- l Y+m Z=0, 
[46] 
X-fw Y+m -1 Z=0, 
[35] 
o' 
II 
S3 
T 
+ 
+ 
[36] 
n X+Y +1 Z=0, 
[16] 
ra^X+Y+Z Z=0, 
[15] 
w X+Y+Z-‘Z=0, 
[26] 
w _I X+Y+Z _1 Z=0, 
[25] 
w=o, 
[12 . 34 . 56] 
X+/3yW=0, 
[12.36.45] 
X-«SW=0, 
[12 . 35 . 46] 
Y -f-ayW = 0, 
[16 . 25 . 34] 
© 
II 
CQ. 
1 
[15 . 26 . 34] 
Z+«/3AV=0, 
[14.23.56] 
S3 
1 
*§, 
II 
o 
[13.24.56] 
WWiX -J“ wZY -j - Zi?2-Z -J- ctj3y()W — 0, 
[16 . 23 . 45] 
jpX+ raY+ mZ+ )3yW= 0, 
[13.26.45] 
wX+ jpY+ ZZ-f- yc/jfW — 0, 
[16.24.35] 
mX+ £Y+ pZ+ ufSfiW =0, 
[15 . 2 3 .'46] 
X+ lmY+ InZ— /3ydW=0, 
[15.24.36] 
ZmX+ Y+mwZ— ya£W=0, 
[13.25.46] 
w/X -j - WjTiY. -|- Z — e&^3cf\Y=0, 
[14.26.35] 
ZX+ ??iY+ wZ — a/3yW = 0, 
[14.25.36] 
2 N 
MDCCCLXIX. 
