XX. On the classification of Loci

Abstract

By a curve we mean a continuous one-dimensional aggregate of any sort of elements, and therefore not merely a curve in the ordinary geometrical sense, but also a singly infinite system of curves, surfaces, complexes, &c., such that one condition is sufficient to determine a finite number of them. The elements may be regarded as determined by k coordinates; and then, if these be connected by k —1 equations of any order, the curve is either the whole aggregate of common solutions of these equations, or, when this breaks up into algebraically distinct parts, the curve is one of these parts. It is thus convenient to employ still further the language of geometry, and to speak of such a curve as the complete or partial intersection of k —1 loci in flat space of k dimensions, or, as we shall sometimes say, in a k -flat. If a certain number, say h , of the equations are linear, it is evidently possible by a linear transformation to make these equations equate h of the coordinates to zero ; it is then convenient to leave these coordinates out of consideration altogether, and only to regard the remaining k — h —1 equations between k — h coordinates. In this case the curve will, therefore, be regarded as a curve in flat space of k — h dimensions. And, in general, when we speak of a curve as in flat space of k dimensions, we mean that it cannot exist in flat space of k —1 dimensions.