PROFESSOR W. K. CLIFFORD ON THE CLASSIFICATION OF LOCI. 
673 
secutive points of tlie curve and through n— Jc other arbitrary points is (Jc-\-l)(n—Jc) ; 
or that the &-wide locus which is traced out by (Jc— 1) fiats passing through Jc con- 
secutive points is of the order (/c -fl 1 ) (n — Jc). For the equation of an (n — 2)flat 
passing through Jc consecutive points is clearly 
0 = 
x dx d?x . . . d k l x p q ... y 
12 3 ... Jc, ... n 
where we must substitute for the X;, dx,, d 2 x i} etc., the descending powers of t-a-, 
beginning at the n th . Making Jc equal to n—1 we obtain the equation of the osculant 
(n — 2) flat at any point of the curve ; it is 
n P nUti 
where P, = product of the differences of all the a except oy. Thus the class of the 
crn’ve is 2(n— 1). 
3. Unicursal curve of order n in n — k dimensions. 
The characteristic numbers belonging to this curve may at once be obtained by 
regarding it as a projection of the full-skew curve. The number of ranks is n — Jc — 2, 
and the numerical values of them are respectively 2 (n — 1), 3(n — 2), . . . Jc(n-Jc-\-l) ; 
the class is (Jc-\- l)(n — Jc) ; and the number of points of superosculation is (Jc-\- 2 )(n — Jc — 1). 
For example, the unicursal quintic in three dimensions is of rank 2.4, =8, and of class 
3.3, =9, and it has 4.2, =8 superosculant planes. 
Convenient forms of the equations may be got by eliminating some of the variables 
from the equations of the full-skew curve ; but care must be taken to select these 
variables so that the resulting system is sufficiently general. 
4. Elliptic (or bicursal ) curve of order n in n—1 dimensions. 
We have proved already that a curve of order n in n—1 dimensions can be repre- 
sented, point for point, on a plane cubic. If, therefore, it is not unicursal, its co- 
ordinates can be expressed in terms of elliptic functions of a single parameter. Now, 
it follows from the investigations of Clebsch, ‘ Crelle,’ vol. lxiv. (1864), pp. 210-270, 
that if n points of the curve are co-flat, the sum of their parameters will differ from a 
certain constant by a sum of integer multiples of the two periods of the elliptic function. 
Let the periods be oj and of, then if t 1} t 2 , . . . t„ are the parameters of the points, 
hd~ Cff- . . . -j-C— c-\-CKo-\-bo ) , 
where c is a constant, and a, b are integers. To find the points of superosculation, we 
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