THE INDETERMINATE PRODUCT, i 

 By H. B. Phillips. 



Presented by H. W. Tyler, November 9, 1910. Received November 12, 1910. 



The analysis considered in this paper is applicable to many linear 

 systems, such as linear forms, matrices, vectors, etc. For convenience of 

 language it is here stated in terms of linear spaces. 



A theorem involving in its statement only linear spaces can be 

 expressed in terms of two processes, the determination of the space 

 of certain spaces and the determination of the space common to certain 

 spaces. It may involve two relations, of spaces contained in a space of 

 a certain order and of spaces intersecting in a space of a certain order. 

 We develop, after Grassmann, an analysis which expresses symbolically 

 these relations and the results of these processes. 



If a, b, c are points and A, yu,, v numbers, 



Aa + /xb + vc, 



interpreted according to matrix theory, represents a point in the space 

 of a, b, c and, if A, /x, v take all values, represents all points in that 

 space. This expression vanishes only when the points lie in a lower 

 space. Thus with undetermined multipliers we can express the rela- 

 tions desired. This method is usually very clumsy, however, and the 

 peculiar excellence of Grassmann's system consists in replacing these 

 sums with unknown coefficients by products without them. 



For this purpose we consider what Gibbs called an indeterminate 

 product. It has the following properties : 



(1) A+B = B + A, 



(2) (A+B) + C=A + (B+C), 



(3) A{B ■\- C) = AB + AC, 



(4) (AB)C=A(BC), 



(5) A^ = ^A, 



(G) 0.4 = .40 = 0, 



^ The method discussed in this paper was suggested by Gibbs in his vice- 

 presidential address before the American Association for the Advancement of 

 Science (Scientific Papers, 2, 109). 



