664 
PROFESSOR W. K. CLIFFORD OR THE CLASSIFICATION OF LOCI. 
This view of ruled surfaces is made excellent use of by Voss, “ Zur Theorie der 
windschiefen Flacben,” Math. Annalen, vol. viii. p. 54. 
Similar considerations apply to surfaces. By a surface we shall mean, in general, a 
continuous two-dimensional aggregate (which may also be called a tivo-spread or two- 
way locus ) of any elements whatever, curves, surfaces, complexes, &c., defined by the 
whole or a portion of the system of solutions of k — 2 equations among Jc coordinates. 
We shall assume that none of these equations are linear, and then shall speak of the 
surface as in a flat space of k dimensions. We shall in certain cases go further, and 
speak of an /i-spread or h-w ay locus, viz., a locus determined by the whole or an alge- 
braically separate portion of the system of solutions of k — li equations among k 
coordinates ; if -none of these equations are linear, the h- way locus will be said to be 
in k dimensions. The general point of view is that of Professor Cayley, “ On the 
Curves which satisfy given Conditions,” Phil. Trans., Yol. 158 (1868), pp. 75-144; 
the methods of enumeration are those of Dr. Salmon, ‘ Solid Geometry,’ p. 261. 
Theorem A. Every proper curve of the \\ th order is in a flat space of n dimensions 
or less. For through n + I points of it we can draw a flat space of n dimensions, 
which must therefore contain the curve, since it meets it in a number of points greater 
than its order. 
Thus, for example, there is no curve of the second order, in space of any number of 
dimensions, except a plane conic. If, therefore, a system of curves, in a plane or on 
any surface, is such that two curves of the system can be drawn through an arbi- 
trary point, then the coordinates of a varying curve of the system may be represented 
by Xi-\-29y - y0\ (i= 1 , 2, 3 . . . k) , and the envelope of the system is, in the case of 
plane curves, a curve having the equation ^/U+ \fY \fW = 0, where U, Y, W are 
three curves of the system ; in the case of curves on a surface, it is the intersection of 
the surface with another having an equation of that form.* 
[* Professor Henrici has kindly written for me the following notes in elucidation of this argument : — 
“ In the first sentence of the paper it is stated that by a curve is meant any one-dimensional aggregate 
of any sort of elements. The definitions given are algebraical, hut the reasoning later on becomes more 
and more geometrical. 
“ In this note the connexion between the algebraical definition and the geometrical reasoning will be 
shown in the case where the elements are plane curves of order n. 
“ If we suppose a curve given by its equation in point coordinates we may take the coefficients as 
homogeneous coordinates of the curve. 
“ As there are ^ — - ratios of these coefficients, it follows that all curves of order n in a plane 
constitute a — - spread, and this will be a flat spread as no relation has been supposed between the 
coordinates. 
< 72 , (pi | ^ ^ 
“ To determine in this spread a A: -flat, fc< — , we have to assume a sufficient number of equations 
Z 
between the coordinates, or denoting by n v n z , . . . curves of order n we may write down the equation of 
