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PROFESSOR CAYLEY ON CUBIC SURFACES. 
(X, Y, Z, W) 3 ; but the equations are in the present Memoir used only as means to an 
end, the establishment of the geometrical theory of the surfaces to which they respec- 
tively belong, and the imperfection is not material. 
7. I have used the capital letters (X, Y, Z, W) in place of Schlafli’s (zc, y, z, w), 
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 (cc, y, z , w ) : this has been done so as to obtain a greater uni- 
formity in the representation of the surfaces. To explain this, let A, B, C, D be the 
vertices of the tetrahedron formed by the coordinate planes A=YZW, B=ZWX, 
C=WXY, D=XYZ; 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 
C 2 might have been taken at any vertices whatever of the tetrahedron ; they are taken 
thus : there is always a node C 2 at I) ; when there is a second node C 2 , this is at C, 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 6 ; this is taken to 
be at D, and the biplanes, to be X=0, Z=0* (the edge being therefore DB), viz. in 
B s 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 C. I suppose 
for a moment that this is a node C 2 . It 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 C 2 , maybe 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 
X), 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 Y=0 will be the 
plane through the three binodes D, C, 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 6 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, C, A, B respectively ; and that (except in the case 111=12— B 3 ) the 
biplanes of the first binode are X=0, Z=0 (for a binode B 5 or B 6 , X=0 being the 
ordinary biplane, Z=0 the special biplane), those of the second binode X=0, W=0, 
* In the case, however, of a single B 3 , 111=12— B 3 , the biplanes are taken to he X+Y+Z=0, 
2X+mY+nZ=0. 
