( 530 ) 
envelope contains either a morefold infinite number of points and 
is then not a curve, or — in case it really consists of a singly in- 
finite number of points — it is situated in a space S,_. Here 
k is always 2n— 1. 
The indicated proof can be given by means of the two following 
considerations : 
a). The system. of s-+ 7 equations consisting of the equation of 
degree & 
S (thea thtathl4+....a,jt+a-=0 
and its first, second.... sth differential coefficients according to t may 
be replaced by a system of s + / equations of degree s—s in ¢, all be 
mitting coefficients that are linear forms of the coefficients of f (t) = 
b). The degree of the locus represented by s + 7 equations of ioe 
k—s in t, of which the coefficients are linear forms in the coordina- 
tes zi i= 1, 2 2,... mn), is obtained by adding to the system n—s 
entirely arbitrary equations linear in the coordinates and by elimi- 
nating the n coordinates between the so formed system of n + 1 equa- 
tions of which n—s do not contain t. The degree of the resulting 
equation in ¢ is the order of the locus we were in search of. 
The proof of these two lemmae is very simple. The first is but 
an extension of a wellknown theorem of Eurer. If we transform 
the equation f(t) = 0 by the substitution ¢ = — into the homogeneous 
v 
form g (u,v) =O, the s+ 1 indicated equations are 
And by following the method pointed out in the second lemma 
we find the number of points common to the locus of n—s dimen- 
sions, determined by the s+ 7 equations of degree k—s and the space 
Ss, being the intersection of any system of n—s spaces Sj. 
If the condition is written down, that the eliminant of the system 
of n41 equations, linear in the n coordinates, disappears, we obtain 
an equation of degree (s+1)(s—s) in ¢, which proves the theorem. 
3. It goes without saying that the lowering, which the charac- 
teristic numbers of the locus R, belonging to the skew parabola 
undergo, is closely connected with the particular structure of the 
equation. First, this equation is not complete, for ¢* is lacking ; 
secondly, not all existing terms contain the three coordinates 2, y, z 
