46 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



in the sense that any product of these elements is equal to a numerical 

 multiple of a third one. In terms of this fundamental system we 

 define the angle between any two spaces. Each of the complexes of 

 the fundamental sj'stem is an infinite locus for spaces complimentary 

 to it. The entire system is invariant under a group of collineations 

 of the same order as the Euclidean group of motions. Degenerate 

 cases are obtained by taking sections having a special relation to the 

 fundamental system. 



Matrices in Three Dimensions. 



2. Progressive Matrices. We represent a point A in three dimen- 

 sions by a set of four homogeneous coordinates Oj. These coordinates 

 determine a matrix 



A = II cii a-2 os QiW = II cii II 



which may be used to represent the point. Two matrices of this 

 kind will be called equal when their corresponding elements are eciual. 

 The matrix is zero if all its elements are zero. If Oj 

 write 



A = k B. 



k hi we shall 



In this case the matrices A and B represent the same point but with 

 different magnitudes. A linear function of A and B is defined by the 

 matrix 



\A + ixB=\\ Xfl. + ^6,||. 



In a similar manner we define any linear function of points or matrices 

 A, B, C, etc. If the result does not vanish it represents a point in 

 the space determined by A, B, C, etc. If it vanishes and the coeffi- 

 cients are not all zero those points lie in a lower space than a like 

 number of points usually determine. 



The coordinates of the line joining A and B are proportional to the 

 two-row^ed determinants in the matrix 



[AB] 



(ii tto as (li 

 hi ho hz hi 



tti ttk 

 hi h, 



We shall call the elements of this matrix the two-rowed determinants 



cii ('k 



[This is not in conformity with the usual definition which makes 

 element equivalent to coordinate Oj or hi but is the only definition 



