392 
A MEMOIR ON CUBIC SURFACES. 
[412 
62. Taking X+Y+Z= 0 for the biplane that contains the rays 1, 2, 3, and 
IX + mY + nZ = 0 for that which contains the rays 4, 5, 6, we may take JT = 0, Y=0, 
Z = 0 for the equations of the planes [14], [25], [36] respectively; and then writing 
for shortness 
m — n, n — l, l — m = \, ¡X, v, 
and assuming, as we may do, k = \/xv, so that the equation of the surface is 
W(X + Y + Z) (IX + mY + nZ) + (m — n) (n — l) (I — m) X YZ = 0, 
the equations of the 17 distinct planes are 
x = o, 
[14] 
Y = 0, 
[25] 
Z =0, 
[36] 
X+ Y + Z=0, 
[123] 
IX + mY + nZ = 0, 
[456] 
IX + nY + nZ = 0, 
[15] 
IX + nY + nZ = 0, 
[16] 
IX + mY + IZ = 0, 
[25] 
nX + mY + nZ = 0, 
[26] 
mX + mY + nZ = 0, 
[35] 
IX + IY + nZ = 0, 
[36] 
W = 0, 
[14.25.36] 
W+ l\X = 0, 
[14.26.35] 
W + 771/X Y = 0, 
[16.25.34] 
W + nvZ = 0, 
[15.24.36] 
ImX + mnY + nlZ + W = 0, 
[15.26.34] 
nlX + ImY + mnZ — W = 0, 
[16.24.35]