412] 
A MEMOIR ON CUBIC SURFACES. 
367 
Article Nos. 24 to 26. Axis; the different kinds thereof. 
24. A line joining two nodes is an axis; such a line is always a line, and it 
is a torsal or oscular line, of the surface. But some further distinctions are requisite; 
using the expressions in their strict sense, cnicnode = 5, binode = 5, an axis is a 
5(7-axis joining two cnicnodes, or it is a (75-axis joining a cnicnode and a binode, 
or it is a 55-axis joining two binodes. A 55-axis is torsal, the transversal being a 
mere line, not a ray through either of the cnicnodes; a (75-axis is torsal, the 
transversal being a ray of the binode; a 55-axis is oscular. The distinction is of 
course carried through as regards the higher biplanar nodes 5 4 , 5 3 , 5 fi , and the 
uniplanar nodes U 6 , U 7 , U 8 : thus (5 3 =5) the edge of a binode 5 3 is not an axis at 
all, but (5 4 = 2C) the edge of a binode 5 4 is a (7(7-axis; (5 S = 5 + G) the edge of a 
binode 5 r> is a 55-axis; (5 (i = 3(7) the edge of a binode 5 H is a thrice-taken 55-axis; 
(U e = 35) each of the rays is regarded as a 55-axi.s; (U 7 = B + 25) the double ray is 
regarded as a twice-taken 55-axis, and the single ray as a 55-axis; (U 8 = 2B + C) 
the ray is regarded as a 55-axis + a twice-taken 55-axis. 
25. It has been mentioned that the intersection of the surface with the Hessian 
consists of the spinode curve, together with certain right lines; these lines are in fact 
the axes—viz. the examination of the several cases shows that in the complete 
intersection each 55-axis presents itself twice, each 55-axis 3 times, and each 55-axis 
4 times. We thus see that a 55-axis, or rather the torsal plane along such axis, is 
the pinch-plane or singularity j' = 1; the 55-axis, or rather the torsal plane along such 
axis, the close-plane or singularity = 1; and the 55-axis, or oscular plane along such 
axis, the bitrope or singularity B' = 1; for a cubic surface with singular lines the 
expression of a being in fact a = 12 — 2y' — 3% — 45'. There are, however, some cases 
requiring explanation; thus for the case VIII = 12 — B 5 , where the edge is by what 
precedes a 55-axis, the complete intersection is made up of the edge 4 times and of 
an octic curve; the consideration of the reciprocal surface shows, however, that the 
edge taken once is really part of the spinode curve (viz. that this curve is made up 
of the edge taken once and of the octic curve, its order being thus a = 9); and the 
interpretation then of course is that the intersection is made up of the edge taken 
3 times (as for a 55-axis it should be) and of the spinode curve. 
26. I remark in further explanation, that in the several sections, in showing how 
the complete intersection of the cubic surface with the Hessian is made up, I have 
not referred to the axes in the above precise significations; thus XIV = 12 — 5 S — 5 2 , 
the binode 5 5 is 5 + 5, and the edge is thus a 55-axis, while the axis 5 5 5 2 is a 
55-axis + a 55-axis (^' = 1 + 1, = 2, j' = 1). The complete intersection should therefore 
consist of the spinode curve, + edge (as a 55-axis) 3 times + axis (as a G B-axis + a 
55-axis) 2 + 3, =5 times: it is in the section stated (in perfect consistency herewith, 
but without the full explanation) that the intersection is made up of the axis 5 times, 
the edge 4 times, and a cubic curve—which cubic curve together with the edge once 
constitutes the spinode curve; and so in other cases: this explanation will, I think, 
remove all difficulty.