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, h is at most = ½(d - 1) (d - 2), and as shown by Halphen it is at least =[¼(d - 1)2], if we denote in this manner the integer part of ¼(d - 1)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 ½v(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 + θ, where θ is = 0 or positive.