382 A MEMOIR ON CUBIC SURFACES.
44 We have X = 0, Y = 0, Z= 0, W = 0 for the equations of the planes
(12.34.56 = a), (42' = y), (14' = *), (12' = w);
and representing by f= IX + i Y+ i Z + W = 0 the equation of any other plane (41'= f)
the equation of the cubic surface may be presented in the several forms:
* 0 = Z7 = Wii +k %YZ,
= Wgg + k rjZX,
= Whh + k^IF,
= wee + k^s
= Ifllj + kyzx,
= TFmm! + kzxy,
= irnhj + kxyz,
= TflJ + kyzx,
= Tkm x m + kzxy,
= WnjH + kxyz,
= Tfppj + k£yz,
= W qqj + krjzx,
= TTrrj + k£xy,
= W pp! + k£y z,
= Tfqqj + kr?zx,
= IFrr, + k£xy,
which are the 16 forms containing W, out of the complete system of 120 trihedral-
pair forms.
45. The 27 lines are each of them facultative; we have therefore b' = p = 27 ;
t' = 45; moreover each of the lines is a double tangent of the spinode curve, and
therefore (3' (= 2p) = 54.
46. The equation of the reciprocal surface is not here investigated; its form is
S 3 - r P = 0,
where S = (*$#, y, z, w)\ T=(*$cc, y, z, w) s ; wherefore n' = 12.
The nodal curve is composed of the lines which are the reciprocals of the
original 27 lines (b' = 27, t' = 45 ut supra). It may be remarked that the reciprocal