670 
PROFESSOR W. K. CLIFFORD OR THE CLASSIFICATION OF LOCI. 
from the corresponding numbers for its projection, the unicursal curve of order n in 
n — 1 dimensions, to which we now proceed. 
2. Unicursal curve of order n in n — 1 dimensions. 
Every curve of order n in flat space of n — 1 dimensions is either unicursal or 
elliptic. For it may be represented point-for-point on a plane cubic. 
We shall treat these two cases in succession. They are exemplified by the two 
species of quartics in ordinary tri-dimensional space. 
The coordinates of a point on the unicursal curve are proportional to rational 
integral functions of a parameter t. This representation may be simplified in a 
manner due to Rosanes, ‘ Crelle,’ vol. lxxv. p. 166. We have n binary qualities 
of order n ; now these may be linearly combined in n different ways so as to produce 
a perfect n th power. Hence the original qualities may be expressed each as a linear 
function of the n th powers of the same n linear qualities. Thus, for example, three 
binary cubics may be simultaneously reduced to the forms. 
au 3 + 6 v s fc w 3 , 
a! id -f- h' v 3 -f- c iv 3 , 
a"u 3 + h"v 3 -f c" w 3 , 
where u + r+ ic=0 identically. If the x, y, z of a point in a plane are respectively 
proportional to these cubics, we may, by solving the equations for u 3 , v 3 , w 3 , obtain 
three linear functions X, Y, Z of the coordinates, which are respectively proportional 
to u 3 , v 3 , w 3 . Transforming them to the new triangle whose sides are X = 0, Y=(), 
Z = 0 we must have the equation of a unicursal cubic expressed in the form 
X*+W+Z’ = 0. 
It is clear that the lines X=0, Y = 0, Z = 0 are tangents at the three points of inflexion. 
In general, let the n qualities be 
a 0 -\-na x t-\- . . . fay, 
byfnbyf . . . -f -by, 
h n -}~ nliyl f . . . -\-h u t", 
then the linear qualities u, v, w . . . are the factors of 
t“, — \n(n — 1)^" -3 , . . . t 
^ 1 ? 
K ^ 2 ? . . . b n 
b/Q 9 ^ 2 ? * * ’ bn 
