[ CG3 ] 
XX. On the Classification of Loci. 
By W. K. Clifford, F.P.S., Professor of Applied Mathematics in University 
College , London. 
Received April 8,— Read May 9, 1878. 
Part I.— CURVES. 
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 /.-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. 
The whole aggregate of linear complexes may be regarded as constituting a space of 
five dimensions, in which the special complexes, or straight lines, constitute a quadric 
locus. A ruled surface, or scroll, will be thus regarded as a curve lying in a quadric 
locus in a flat space of five dimensions. If, however, the generators of the scroll all 
belong to the same linear complex, the scroll must be regarded as a curve lying in a 
quadric locus in a flat space of four dimensions. And if, further, the scroll has two 
linear directrices, so that the generators belong to a linear congruence, then the scroll 
may be regarded as a curve lying on an ordinary quadric surface in three dimensions. 
Thus, for example, quartic scrolls having two linear directrices correspond either to 
quadri-quadric curves of deficiency 1 (that is, they are elliptic curves whose coordinates 
may be expressed as elliptic functions of one variable), or to the curves of deficiency 0 
which are the partial intersections of a quadric and a cubic surface (that is, they are 
unicursal curves). 
MDCCCLXXVIII. 4 Q 
