264 
PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
62. Taking X+Y + Z=0 for the biplane that contains the rays 1, 2, 3, and 
ZX+mY+?&Z=0 for that which contains the rays 4, 5, 6, we may take X=0, Y=0, 
Z=0 for the equations of the planes [14], [25], [86] respectively; and then writing 
for shortness 
m—n , n — l, l — m=\ , p, v, 
and assuming, as we may do, /c=X[jijv, so that the equation of the surface is 
W(X+Y+Z)(lX+mY+nZ) + (m-n)(n-l)(l-m)XYZ=Q, 
the equations of the 17 distinct planes are 
x=o. 
[14] 
Y=0, 
[25] 
Z=0, 
[36] 
X+Y + Z=0, 
[123] 
ZX+mY-|-?iZ= 0, 
[456] 
ZX+mY+wZ=0, 
[15] 
ZX -j- nY -j-nZ = 0 , 
[16] 
ZX+mY+ZZ=0, 
[25] 
nX + mY + nZ = 0 , 
[26] 
mX +mY -f nZ — 0, 
[35] 
ZX+ZY+nZ=0, 
[36] 
w=o, 
[14 . 25 . 36] 
W+ZAX=0, 
[14 . 26 . 35] 
W-(-7W 1 «-Y=0 , 
[16 . 25 . 34] 
W+?2t'Z=0, 
[15.24.36] 
hnX + mnY -f nlZ + W= 0, 
[15 . 26 . 34] 
nlX + ImY + mnZ — W = 0, 
[16 . 24 . 35] 
63. And the coordinates of the fifteen distinct lines are 
