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