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A MEMOIR ON CUBIC SURFACES. 
[412 
Article Nos. 14 to 19. Explanation in regard to the Determination of the Number of 
certain Singularities. 
14. In the several cases I to XXI, we have a cubic surface (n = 3), with singular 
points G 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 8' = 0 double tangents and k — 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, 8 = 0 nodal lines and /¿ = 6 cuspidal lines, but with (in the several other cases) 
G nodal lines and B cuspidal lines (or rather singular lines tantamount to G double 
lines and B cuspidal lines): the class of the cone, or order of the reciprocal surface, 
is thus n’ = 6.5 - 2 (0 + G) - 3 (6 + B) = 12 - 2B - SG. 
15. In the general case I = 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 
p = 27, and the node-couple torse consists of the pencils of planes through these lines 
respectively, being thus of the class p = 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 p: the pencils of 
planes through these lines constitute the node-couple torse of the class b' = p ; the t' 
. planes, each containing three facultative lines, are the triple tangent planes of the 
node-couple torse. Or if (as is somewhat more convenient) we refer the numbers 
b\ t' to the reciprocal surface, then the lines, reciprocals of the facultative lines, 
constitute the nodal curve of the order b'; 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 Id of its apparent double points is given by the 
formula 2Id = b' 2 — b' — 6t'; and comparing with the formula q = b' 2 — b' — 2k' — 37' — 6t', 
we have q + 37' = 0, that is, q' = 0 (q' the class of the nodal curve), and also y = 0. 
16. In the general case I = 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 
a = 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 a less than 12: this will be further explained, and the reduction 
accounted for (see post, Nos. 24 et seq.). 
17. Again, in the general case I = 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 /3' = 2p' = 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 /3' not in general = 2p.