468 
[949 
949. 
ON HALPHEN’S CHARACTERISTIC n, IN THE THEORY OF 
CURVES IN SPACE. 
[From Crelle’s Journal d. Mathematik, t. cxi. (1893), pp. 347—352.] 
If we consider a curve in space without actual singularities, of the order (or 
degree) d, then this has a number h of apparent double points (adps.), viz. taking 
as vertex an arbitrary point in space, we have through the curve a cone of the 
order d, with h nodal lines; and Halphen denotes by n the order of the cone of 
lowest order which passes through these h lines. For a given value of d, li is at 
most = ^(d— \) (d — %), and as shown by Halphen it is at least =-W(d— l) 2 ], if we 
denote in this manner the integer part of \{d — l) 2 . For given values of d, h, it 
is easy to see that n must lie within certain limits, viz. if v be the smallest number 
such that ^v(v + 3) equal to or greater than h, then n is at most = v; and moreover 
n must have a value such that nd is at least = 2h, or say we must have nd = 2h + 6, 
where 9 is = 0 or positive. For any given value of d, we thus have a finite number 
of forms (d, h, n), and we have thus prima facie curves in space of the several 
forms (d, h, n): but it may very well be, and in fact Halphen finds, that when 
d = 9 or upwards, then for certain values of h, n as above, there is not any curve 
(d, h, n)\ thus d = 9, /¿=17 the values of n are n = 4, n — 5, but there is not any 
curve d = 9, A = 17 for either of these values of n; or say the curves (9, 17, 4) 
and (9, 17, 5) are non-existent. And in the Notes and References to the papers 
302, 305 in vol. v. of my Collected Mathematical Papers, 4to. Cambridge, 1892, see 
p. 615, I remarked that, in certain cases for which Halphen finds a curve (d, h, n), 
such curve does not exist except for special configurations of the h nodal lines not 
determined by the mere definition of n as the order of the cone of lowest order 
which passes through the h nodal lines; for instance d = 9, A = 16, n = 4, for which 
Halphen gives a curve, I find that for the existence of the curve it is not enough 
that the 16 lines are situate upon a quartic cone, but they must be the 16 lines 
of intersection of two quartic cones.