668 
PROFESSOR W. Iv. CLIFFORD OR THE CLASSIFICATION OF LOCI. 
If we omit the suffixes of the t in this formula we obtain the equation to the 
osculant (n — l)flat at the point t. Namely (beginning at the other end), it is — - 
0=A t n — — 1)C t n ~ 2 — . . (5) 
and we see at once that the class of such a curve is always equal to its order. 
We thus obtain a very useful representation ( Abhildung ) of the points of the 
n-dimensional space by means of groups of n points on such a unicursal curve, namely, 
each point in the space is represented by the points of contact of the n osculant 
( n — l)flats which pass through it. The use of such a representation of ordinary three- 
dimensional space by means of a skew cubic was pointed out by Dr. Hirst, and the 
corresponding representation of a plane by means of a conic has been used by 
M. Darboux (‘ Sur une classe remarquable de courbes et de surfaces algebriques,’ 
Paris, 1873, Note II., p. 183), and by me (“On the Transformation of Elliptic 
Functions,” Proc. Lond. Math. Soc., vol. vii. (1875), pp. 25-38 and 225-233). It may 
be worth while to mention that an extension to all space of the theory of the in-and- 
circumscribed polygon may be obtained by this means. 
A curve of this kind determines also a dualistic correspondence in the space 
of n dimensions. Through every point may be drawn n osculant (n — 1) flats, and 
through their points of contact another (n — l)flat, which shall be called the polar 
of the point. If the point moves along a straight line its polar will pass through 
a fixed (n — 2)flat, the polar of the line. And generally if the point lies in any 
k flat the polar will pass through a fixed (n — k — l)flat. 
When n = 2 we have the ordinary system of polar reciprocals in regard to a plane 
conic. When n= 3 we have that system in regard to a skew cubic which is described 
by Schroter, ‘ Crelle,’ vol. lxv. p. 39. These two systems are typical respectively 
of the cases in which n is even and odd. When n is even, the relation between the 
coordinates of two points, which expresses that each lies in the polar of the other, 
is a symmetrical one ; consequently those points which lie in their own polars are 
points on a certain quadric locus, and the system is merely that of the poles and 
polars in regard to this quadric locus upon which the curve lies. The equation to 
this locus is at once obtained by equating to zero the quad rin variant of the form 
(1, t)' 1 which occurs in the equation (5) of the osculant (n — l)flat, namely, it is 
0 = AL — nEK + \n (n — 1 ) CH — etc (6) 
To prove this, observe that if in the equation (5) we substitute the coordinates of 
any point p, the values of t which satisfy the equation are the parameters of the 
points of contact of the osculant (n — 1) flats which pass through the point. If 
t x , t 2 , . . . t H be these values, the equation (4) represents the (n — l)flat which passes 
through the points of contact, that is to say, the polar of the point. Now if we 
denote by A', B', . . . the results of substituting the coordinates of the point p in 
A, B, . . then we shall have- — 
