080 PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI, 
then 
% • • • v p )=t p e Hm) , 
the % J1 indicating that each of the p integers m 1} m. : , . . . m p is to take all integral values 
positive and negative. When the lower limits are so chosen that the sum of the 
parameters is zero for the complete intersection by any locus, this ^-function has 
remarkable properties. If we sum each of the parameters for any p — 1 points on the 
curve, the ^-function whose arguments are these sums will vanish. That is if 
then . . . Vp) = 0. If these sums are taken for any p— 2 points, not only will 
the 3 vanish, but also its differential coefficient in regard to any one of the points. 
And generally, if we take for the v the sums of the parameters for p — r points, the 3 
and its first r — 1 differential coefficients in regard to any of the points will vanish. 
Conversely, if the p quantities v are such that 3(v) and its first r — 1 differential 
coefficients vanish, then it is possible to find p — r points x such that 
([ *+f 2 -b • • • +( / ’-' S Jdv/,=v, l . 
Although here the number of equations is greater by r than the number of unknown 
quantities, yet it is possible to satisfy them all in virtue of the relations existing 
between them. 
Relation between the Order and Deficiency of a Curve. 
We shall now apply these theorems to the study of curves existing in h dimensions, 
of the order n and deficiency p. A (A: • — I ) flat cuts such a curve in n points, such 
that the sum of each of the p p ammeters, for the n points, is zero. But a (/: — l)flat 
is determined by k points ; so that, k arbitrary points being chosen on the curve, 
it is always possible to find n — k other points, so than the sum of each parameter for 
the whole n points shall be zero. Let then — v x , — v. 2 , ... — v p be the sums of the 
parameters for the given k points ; then to find the remaining?! — A; points we have 
the p equations 
(J ] + j 2 + • • • +| du t =v h 
If p is not greater than n — k, we know that these equations can be solved, although 
the solution may be indeterminate. But if p>n—k, the equations cannot be solved 
unless certain conditions are satisfied by the v. Let p — n-\-k=r, then r conditions 
must lie satisfied ; namely the v must be sums of the parameters of not more than 
p— r (or n — k) points. But they are sums of the parameters of k points ; therefore k 
is not greater than n—k, or 2k is not greater than n. We have proved then that 
