NOTES AND REFERENCES.
617
and the conditions are satisfied for those values of (d, h, n) against which I have set
the capacity 36*f*. I do not explain the remaining figures of the column of capacities,
but remark only that 0 means that the curve is non-existent, and that 35 refers to
the curve (9, 22, 5) which is alluded to above as not specified by Halphen.
It is important to remark that if the above-mentioned condition n (n + 5) = or
>h + 0+ 1, or restoring it to the original form (n + 1) (n + 4) — h — 6 = 3 at least, is not
satisfied, then it by no means follows, and it is not in general the case, that the curve
is non-existent : I have said only that the cone Q = 0 has in general a capacity
= ^ (n + l)(n + 4) — h— 6, but the configuration of the h + 6 lines may be such as not
to impose on the cone Q — 0 which passes through them so many as h + 6 conditions,
and the capacity of the cone may thus be greater than \{n 4-1) (w+ 4) — h— 6, and
may thus be =3 at least ; moreover supposing that in such a case the curve exists,
the capacity of the cone U = 0 instead of being = ^ d (d + 3) — h, may very well have,
and presumably has, a greater value, and the reasoning by which the capacity of the
curve was found to be = 4>d + ^(d — 2 — n) (d — 3 — n) ceases to be applicable. The
theory, as depending upon special configurations of the h lines and the 6 lines, is
a complicated and difficult one, and I do not attempt to enter upon it.
In conclusion I wish to refer to an important theorem given by Valentiner and
also by Halphen and Nöther. Considering in connexion with the curve of the order d,
a surface of the order m, then since the capacity hereof (or number of constants
contained in its equation) is = £(m + 1)(ra + 2)(m + 3) — 1 or \m (m 2 + 6m+ 11), it is
obvious that if this be greater than md, the surface can be made to pass through
more than md points of the curve, and thus that the curve will lie upon a surface
of the order m. But the condition which has really to be satisfied in order that
the curve may lie upon a surface of the order m is a less stringent one : if p be
the deficiency of the curve, = ^ (d — l)(d — 2) — h, if as before the curve is without
actual singularities, and h be the number of its apparent double points, then the
condition is ^m(m 2 + 6m+ll) greater than md—p, viz. the surface of the order m
being made to pass through md+l—p points assumed at pleasure on the curve will
ipso facto pass through p determinate points of the curve, that is in all through md + 1
points of the curve, or it will contain the curve. The theorem is true subject only
to the limitation m = or > d — 2. The most simple form of statement is perhaps that
given by Valentiner, p. 194 (changing only his letters), viz. if m be = or > d — 2, the
intersections of a surface of the order m with a curve of the order d with h apparent
double points are determined by means of
dm — I (d — 1) (d - 2) + h (= dm—p)
of these intersections.
312. The generalisation which is here given of Euler’s theorem S + F = E + 2, is
a first step towards the theory developed in Listing’s Memoir “ Census räumlicher
Complexe oder Verallgemeinerung des Euler’schen Satzes von den Polyedern.” Göttingen
Abh. t. x. (1862).
320. The transcendent i gd (— iu), with a pure imaginary argument is the function
log tan (^7T + \u) (hyperbolic logarithm) tabulated by Legendre, Exer. de Calcul Intégral,
C. V. 78
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