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PROFESSOR CAYLEY ON CUBIC SURFACES. 
Explanation in regard to the determination of the number of certain singularities. 
Article Nos. 14 to 19. 
14. In the several cases I to XXI, we have a cubic surface (n= 3), with singular 
points C and B but without singular lines. The section by an arbitrary plane is thus 
a curve, order n— 3, that is, a cubic curve, without nodes or cusps, and therefore of the 
class a! = 6, having &' = 0 double tangents and z'= 9 inflexions. The tangent cone 
with an arbitrary point as vertex is a cone of the order a=6, having in the case 
1=12, c$=0 nodal lines and z=6 cuspidal lines, but with (in the several other cases) C 
nodal lines and B cuspidal lines (or rather singular lines tantamount to C double lines 
and B cuspidal lines) : the class of the cone, or order of the reciprocal surface, is thus 
w '=6.5-2(0 + C)-3(6+B)=12-2B-3C. 
15. In the general case 1=12, there are on the cubic surface 27 lines, lying by 3’s in 
45 planes; these 27 lines constitute the node-couple curve of the order g'=27, and the 
node-couple torse consists of the pencils of planes through these lines respectively, being 
thus of the class %'=b' = 27; the 45 planes are triple tangent planes of the node-couple 
torse, which has thus t’= 45 triple tangent planes. But in the other cases it is only certain 
of the 27 lines, say the “facultative lines” (as will be explained), which constitute the 
node-couple curve of the order £ : the pencils of planes through these lines constitute the 
node-couple torse of the class b'=g' ; the t' planes, each containing three facultative lines, 
are the triple tangent planes of the node-couple torse. Or if (as is somewhat more con- 
venient) we refer the numbers b', t 1 to the reciprocal surface, then the lines, reciprocals 
of the facultative lines, constitute the nodal curve of the order V ; and the points t', each 
containing three of these lines, are the triple points of the nodal curve. Inasmuch as 
the nodal curve consists of right lines, the number k' of its apparent double points 
is given by the formula 2 Jd=b’ 2 — V — Qt ' ; and comparing with the formula 
q'=b' 2 — V — 2k'— 3y'— Qt', we have £'-f-3y'=0, that is, ^'=0 ( g ' the class of the nodal 
curve), and also y'=0. 
16. In the general case 1=12, the spinode curve is the complete intersection of the 
cubic surface by the Hessian surface of the order 4, and it is thus of the order o’ =12; 
but in the other cases the complete intersection consists of the spinode curve together 
with certain right lines not belonging to the curve, and the spinode curve is of an order 
o' less than 12 : this will be further explained, and the reduction accounted for (see 
post. Nos. 24 et seg.). 
17. Again, in the general case 1=12, each of the 27 lines is a double tangent of the * 
spinode curve, and the tangent planes of the surface at the points of contact are common 
tangent planes of the spinode torse and the node-couple torse, stationary planes of the 
spinode torse ; or we have j3'=2g>'=54. In the other cases, however, instead of the 27 
lines we must take only the facultative lines, each of which is or is not a double or a 
single tangent of the spinode curve ; and the tangent planes of the surface at the points 
of contact are the common tangent planes as above — that is, the number of contacts 
gives j3', not in general =2/. 
