TI.—ExXtTENSIONS OF CERTAIN THEOREMS OF CLIFFORD AND OF 
CAYLEY IN THE GEOMETRY OF » Dimensions. By ELIAkim 
Hastines Moors, sr., DenvER, CoLorapo. 
I. GErnerAL THEOREMS. 
CLIFFORD, at the beginning of his) “ Classification of Loci” (Mathe- 
matical Papers, p. 305-331), proves the following theorems : 
A. Every proper curve of the n™ order is in a flat space of n 
dimensions or less. 
B.. A curve of order n in flat space of & dimensions (and no less) 
may be represented, point for point, on a curve of order »—A+2 in 
a plane, whence 
C. A curve of order m in flat space of m dimensions (and no less) is 
always unicursal. 
These theorems may be extended. Clifford’s nomenclature* and 
methods of proof are adhered to- throughout. 
* For the convenience of the reader who may not have at hand a copy of Clifford’s 
Mathematical Papers, the definitions (p. 305-6) are given here. 
“ By a curve we mean ‘a continuous one-dimensional aggregate of any sort of ele- 
ments, and therefore not merely a curve in the ordinary geometrical sense, but also a 
singly infinite system of curves, surfaces, complexes, &c., such that one condition is 
sufficient to determine a finite number of them. The elements may be regarded as 
determined by & coordinates; and then, if these be connected by A—1 equations of any 
order, the curve is either the whole aggregate of common solutions of these equations, 
or, when this breaks up into algebraically distinct parts, the curve is one of these 
parts. It is thus convenient to employ still further the language of geometry, and to 
speak of Such a curve as the complete or partial intersection of s—1 loci in flat space 
of % dimensions, or, as we shall sometimes say, in a k-flat. Ifa certain number, say 
h, of the equations are linear, it is evidently possible by a linear transformation to 
make these equations equate h of the coordinates to zero; itis then convenient to 
leave these coordinates out of consideration altogether, and only to regard the remain- 
ing :—h—1 equations between k—A coordinates. In this case the curve will, therefore, 
be regarded as a curve in a flat space of A—h dimensions. And, in general, when we 
speak of a curve as in flat space of & dimensions, we mean that it cannot exist in flat 
space of k—1 dimensions. : 
x * » By a surface we shall mean, in general, a continuous two-dimensional 
aggregate (which may also be called a two-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 kK—2 equations among & coordinates. We shall assume that 
none of these equations are linear, and then shall speak of the surface as in a flat 
Trans. Conn. Acap., Vou. VII. 2 SEpT., 1885. 
