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A MEMOIR ON CUBIC SURFACES. 
[412 
7. I have used the capital letters (X, Y, Z, W) in place of Schlafli’s (x, y, z, tu), 
reserving these in place of his (p, q, r, s) for plane-coordinates of the cubic surfaces, 
or (what is the same thing) point-coordinates of the reciprocal surfaces ; but I have 
in several cases interchanged the coordinates (X, Y, Z, W) so that they do not in 
this order correspond to Schlafli’s (x, y, z, tu) : this has been done so as to obtain a 
greater uniformity in the representation of the surfaces. To explain this, let A, B, G, D 
be the vertices of the tetrahedron formed by the coordinate planes A= YZW, B = ZWX, 
G = WX Y, D = X YZ ; the coordinate planes have been chosen so that determinate 
vertices of the tetrahedron shall correspond to determinate singularities of the surface. 
8. Consider first the surfaces which have no nodes B or U. It is clear that 
the nodes G 2 might have been taken at any vertices whatever of the tetrahedron ; 
they are taken thus: there is always a node C 2 at D; when there is a second node G. 2 , 
this is at G, the third one is at A, and the fourth at B. 
9. Consider next the surfaces which have a binode B 3 , B 4 , B 5 , or B s ; this is 
taken to be at D, and the biplanes to be X = 0, Z = 0 ( x ) (the edge being therefore 
DB), viz. in B 5 or B 6 , where the distinction arises, X = 0 is the ordinary biplane, 
Z = 0 the torsal or (as the case may be) oscular biplane. If there is a second node, 
this of necessity lies in an ordinary biplane ; it may be and is taken to be in the 
biplane X = 0, at G. I suppose for a moment that this is a node GIt is only 
when the binode is B 3 or B 4 that there can be a third node, for it is only in these 
cases that there is a second ordinary biplane Z = 0 ; but in these cases respectively 
the third node, a G 2 , may be and is taken to be in the biplane Z = 0, at A. 
10. The only case of two binodes is when each is a B 3 . Here the first is as 
above at D, its biplanes being X = 0, Z— 0 ; and the second is as above in the 
biplane X = 0, at C; the biplanes thereof are then X = 0 (which is thus a biplane 
common to the two binodes, or say a common biplane), and a remaining biplane which 
may be and is taken to be W = 0. If there is a third node, this may be either 
C. 2 or B 3 , but it will in either case lie in the biplane Z = 0 of the first binode, and 
also in the biplane W = 0 of the second binode, that is, in the line BA ; and it may 
be and is taken to be at A ; if a binode, then its biplanes are of necessity Z = 0, 
W = 0; and the plane X=0 will be the plane through the three binodes D, G, A. 
11. If there is a unode, then this may be and is taken to be at D, and its 
uniplane may be taken to be X = 0; in the surface XII = 12— U s the uniplane is, 
however, taken to be X+Y+Z= 0. There is never, besides the unode, any other 
node. 
12. The result is that the nodes, in the order of their speciality, are in the 
equations taken to be at D, G, A, B respectively ; and that (except in the case 
111 = 12 —$3) the biplanes of the first binode are X = 0, Z = 0 (for a binode B 5 or B ti , 
X = 0 being the ordinary biplane, Z= 0 the special biplane), those of the second binode 
X = 0, W = 0, those of the third binode Z = 0, W = 0, and that (except in the case 
XII = 12— U e ) the uniplane is X = 0. For example, in the surface XVII = 12 — 2B 3 — G 2 , 
as represented by its equation WXZ + Y 2 Z + X 3 = 0, we have a B 3 at D, the biplanes 
being X = 0, Z = 0, a B 3 at G, the biplanes being X = 0, W = 0 (therefore X = 0 the 
common biplane), and a G 2 at A. 
1 In the case, however, of a single B 3 , III = 12-R 3 , the biplanes are taken to be X+Y+Z = 0, lX+mY+nZ=0.