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PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
A cubic scroll must be of the nature of the skew cubic, because it is a curve (with 
complexes for elements) which is obliged to lie on a quadric locus (that of the special 
complexes, or straight lines). 
Theorem B. A curve of order n in flat space of k dimensions ( and no less) may be 
represented , point for point, on a curve of order n — k -f- 2 in a plane. 
The proposition is obvious when Jc=S. The cone standing on a curve of order n 
(in ordinary space of three dimensions), and having its vertex at a point of the curve, 
is of order n — 1 ; if then we cut this cone by a plane, we have the tortuous curve 
represented, point for point, on a plane curve of order n — 1. 
Now this process is applicable in general. Starting with an arbitrary point, P, of a 
curve in any number of dimensions, let us join this point to all the other points of the 
curve ; we shall thus get a cone of order n — 1 . For any flat locus of k — 1 dimen- 
sions drawn through the point P must meet the curve in n points, of which P is one ; 
and therefore it must meet the cone in n — 1 lines. Hence, if we cut this cone by such 
a flat ( k — l)way locus not passing through P, we shall get a curve of order n — 1 in 
flat space of k — 1 dimensions, which is a point-for-point representation of the original 
curve. By continuing this process we may go on diminishing the order of the curve 
and the number of dimensions by equal quantities, until we have subtracted k — 2 from 
each ; when we are left with a curve of order n — k -\- 2 in a plane. 
The reduction may, however, be effected in one step. A flat ( k — 2) way locus may 
be drawn through k — 1 arbitrary points. Suppose it to contain k — 2 consecutive 
points of the curve at P, and another variable point, Q, of the curve. Such a locus 
will meet an arbitrary plane in one point, It. As Q then moves about on the curve, B 
will trace out on the plane a curve which corresponds to it, point for point. But this 
curve is of order n — k-\- 2, for a flat (k — l)way locus, passing through k — 2 consecu- 
tive points of the original curve at P, will meet that curve in n — /.;+2 other points, 
and therefore will meet also the locus of B in n — /'fl-2 points. This locus is, therefore, 
of order n — k-\- 2, as was to be proved. 
The fixed points through which the variable (k — 2) way locus passes need not all 
be united at P, but they may be any k — 2 arbitrary points on the curve. 
We will now consider some examples of this remark. 
1. Unicursal curve of order n in n- dimensional space. 
A curve of order n in flat space of n dimensions {and no less ) is always unicursal . — 
We may prove this independently by considering a variable {n — l)flat which passes 
through n— 1 fixed points on the curve. Its equation will be of the form A+£A'=0, 
where t is a variable parameter, and it will meet the curve in one other point, which 
is thus associated with a value of t. 
The equations to such a curve may always be written in the form — 
