238 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
appears by the upper marginal line, and that of each plane by the left-hand marginal 
column (thus in diagram I, 27 X 1=27 and 45 X 1=45, 1 is the multiplicity of each line, 
and it is also the multiplicity of each plane); the frequencies of a line and plane in regard 
to each other appear by the dots in the square opposite to the line and plane in question, 
these being read, for the frequency of the line vertically, and for the frequency of the 
plane horizontally ; thus • indicates that the frequency of the line is = 3, and the 
frequency of the plane is =2. There should be and are in every line of the diagram 
3 dots, and in every column of the diagram 5 dots (a symbol • being read as just 
explained, 2 dots in the line, 3 dots in the column). 
22. For the surface 1=12, there is of course no distinction between the lines, 
but these form only a single class, and the like for the planes ; but for the other sur- 
faces the lines and planes form separate classes, as shown in the diagrams by the lower 
marginal explanation of the lines, and the right-hand marginal explanation of the 
planes. I use here and elsewhere “ ray” to denote a line passing through a single node ; 
“ axis” to denote a line joining two nodes; “ edge” (as above) to denote the edge of a 
binode ; any other line is a “ mere line.” An axis is always torsal or oscular ; when it 
is torsal, the plane touching along the axis contains a third line which is the “ trans- 
versal” of such axis ; but a transversal may be a mere line, a ray, or an axis ; in the case 
XVI =12 — 4C 2 , each transversal is a transversal in regard to two axes. 
23. In the general case 1=12, each of the 27 lines is, as already mentioned, part of 
the node-couple curve ; and the node-couple curve is made up of the 27 lines, and is 
thus a curve of the order 27. In fact each plane through a line meets the cubic surface 
in this line, and in a conic ; the line and conic meet in two points, and the plane (that 
is in any plane) through the line is thus a double tangent plane touching the surface at 
the two points in question ; the locus of the points of contact, that is the line itself, is 
thus part of the node-couple curve. But in the other cases, II to XXI, certain of the 
lines do not belong to the node-couple curve (this will be examined in detail in the 
several cases respectively) ; but I wish to show here how in a general way a line passing 
through a node, say a nodal ray, is not part of the node-couple curve. To fix the ideas, 
consider the surface 11=12 — C 2 ; there are here through C 2 six lines, or say rays; 
attending to any one of these, a plane through the ray meets the surface in the ray itself 
and in a conic ; the ray and the conic meet as before in two points, one of them being 
the point C 2 : the plane touches the surface at the other point, but it does not touch the 
surface at C 2 . (I am not sure, and I leave it an open question, whether we ought to say 
that at a node C 2 there is no tangent plane, or to say that only the tangent planes of the 
nodal cone are tangent planes of the surface ; but, at any rate, an arbitrary plane through 
C 2 is not a tangent plane.) The plane through the ray is only a single tangent plane, 
not a double tangent plane ; and the ray is not part of the node-couple curve. We say 
that a line of the surface is or is not “facultative” according as it does or does not 
form part of the node-couple curve. 
