PROFESSOR CAYLEY ON CUBIC SURFACES. 
239 
Axis; the different kinds thereof. Article Nos. 24 to 26. 
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 =C, binode =B, an axis is a CC-axis joining 
two cnicnodes, or it is a CB-axis joining a cnicnode and a binode, or it is a BB-axis 
joining two binodes. A CC-axis is torsal, the transversal being a mere line, not a ray 
through either of the cnicnodes ; a CB-axis is torsal, the transversal being a ray of the 
binode; a BB-axis is oscular. The distinction is of course carried through as regards 
the higher biplanar nodes B 4 , B 5 , B 6 , and the uniplanar nodes U e , U 7 , U 8 : thus (B 3 =B) 
the edge of a binode B 3 is not an axis at all, but (B 4 =2C) the edge of a binode B 4 is a 
CC-axis; (B 5 =B + C) the edge of a binode B 5 is a CB-axis; (B 6 =3C) the edge of a 
binode B 6 is a thrice-taken CC-axis ; (U 6 = 3C) each of the rays is regarded as a CC-axis ; 
(U 7 =B + 2C) the double ray is regarded as a twice-taken CB-axis, and the single ray as 
a CC-axis; (U 8 =2B-j-C) the ray is regarded as a BB-axis -j- a twice-taken CB-axis. 
25. It has been mentioned that the intersection of the surface with the Hessian con- 
sists of the sp inode 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 CC-axis presents itself twice, each CB-axis 3 times, and each BB-axis 4 times. We 
thus see that a CC-axis, or rather the torsal plane along such axis, is the pinch-plane or 
singularity j'—l ; the CB-axis, or rather the torsal plane along such axis, the close- 
plane or singularity %'=1 ; and the BB-axis, or oscular plane along such axis, the bitrope 
or singularity B'=l ; for a cubic surface with singular lines the expression of a' being 
in fact ff r =12 — 2/— -4B'. There are, however, some cases requiring explanation; 
thus for the case VIII =12— B„ where the edge is by what precedes a CB-axis, the 
complete intersection is made up of the edge 4 times and of an octic curve ; the con- 
sideration 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 <r , =9) ; and the interpretation then of course is that the 
intersection is made up of the edge taken 3 times (as for a CB-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 — B s — C 2 , the 
binode B s is C + B, and the edge is thus a CB-axis, while the axis B 5 C 2 is a CB-axis 
+ a CC-axis (%f=l+l, =2, j'= 1). The complete intersection should therefore con- 
sist of the spinode curve, + edge (as a CB-axis) 3 times -{- axis (as a CB-axis + a 
CC-axis) 2-J-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. 
