339]
OX SKEW SURFACES, OTHERWISE SCROLLS.
173
12. ND(m 2 , n).—Taking first a point on the curve to, this is the vertex of a
cone of the order m — 1 through the curve to, and of a cone of the order n through
the curve n; the two cones intersect in (to—l)?i lines, which are the generating
lines through the point on the curve to; that is, the curve to is a (to — 1) ?i-tuple
line on the scroll S(m 2 , ri)\ and we have thus the term to. ^ [(to — 1) n~f of ND.
Taking next a point on the curve n, this is the vertex of a cone of the order to
through the curve to; such cone has (h = ) ^ [»г] 2 + M double lines, which are the gene
rating lines through the point on the curve n; hence this curve is a [to] 2 + di)tuple
line on the surface, and we have thus the term n. f [to] 2 + Mj in ND. And therefore
ND (m 2 , n) = m. [(to — 1) tif + n.^[f [to] 2 + M] 2 ,
= n (t [to] 4 + [to] 3 + M [«г] 2 — ^) + ilf 2 . |) + [?i] 2 [to] 3 +[m] 2 ).
13. ND (m 3 ).—Taking a point on the curve m, this is the vertex of a cone of
the order m — 1 through the curve m; such cone has (h — m -f 2 = ) ^ [i??] 2 — m + 2 + M
double lines, or the curve m is a [m] 2 - m + 2 + M)tuple line on the scroll S (m) 3 ).
Hence we have
ND (m) 3 = m . ^ [да] 2 — m + 2 + ili] 2 ,
= l [да] 5 + \ [да] 4 + 2 [да] 3 + M [да] 3 + \ [«г]) + M 2 . \т.
Preparatory remarks in regard to the G formulae, the hypertriadic singularities of a
curve in space, Articles 14 to 22.
14. It is to be remarked that the generating line of any one of the scrolls
8 (m, n, p), S (to 2 , n), S (to 3 ) satisfies three conditions; and that it cannot in anywise
happen that one of these conditions is implied in the other two. Thus, for instance,
as regards the scroll S(m, n, p), if the curves to, n are given, and we take the entire
series of lines meeting each of these curves, these lines form a double series of lines,
all of them passing of course through the curves to, n, but not all of them passing
through any other curve whatever; that is, there is no curve p such that every line
passing through the curves to and n passes also through the curve p. And the like
us regards the scrolls S (to 2 , n) and 8 (to 3 ).
15. But (in contrast to this) if the three conditions are satisfied, it may very
well happen that a fourth condition is satisfied ipso facto. To see how this is,
imagine a curve q on the scroll S (to, n, p), or, to meet an objection which might be
raised, say a curve q the complete intersection of the scroll S (ni, n, p) by a plane or
any other surface. Every line whatever which meets the curves to, n, p is a generating
line of the scroll S (to, n, p), and as such will meet the curve q\ that is, in the case
in question, G(m, n, p, q), the lines which meet the curves to, n, p, q, are the entire
series of generating lines of the scroll 8 (to, n, p), and they are thus infinite in
number; so that in such case the question does not arise of finding the number of
