JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. VII APRIL 4, 1917 No. 7 



MATHEMATICS. — Note on multiple algebra: The reduction of 

 real dyadics and the classification of real homogeneous strains. 

 Edwin Bidwell Wilson, Massachusetts Institute of Tech- 

 nology. (Communicated by Arthur L. Day.) 



1. About ten years ago I printed an account of some parts of 

 Gibbs's course of lectures on multiple algebra. 1 In the classifi- 

 cation, or reduction to a canonical form, which was there estab- 

 lished for dyadics no attention was paid to distinctions between 

 real and imaginary. I had in mind at that time to give the ex- 

 tension of Gibbs's work in Vector Analysis 2 needed to obtain the 

 reduction of real dyadics to type forms, but did not publish 

 my results. I desire now to show how the algebraic methods 

 used by Gibbs to find the general classification lend themselves 

 immediately to the further subdivision relative to reality. 



The sort of dyadic under consideration is (p. II): 3 



$ = a | a + (3 | 0° + . . . , 



where the antecedents a, 0, . . . are ordinary vectors and the 

 consequents a , 0°, . . . are (n-1) -dimensional vectors in n- 

 dimensional space. (The vertical bars serve as separators and 



1 Wilson*, Edwin* B. On the theory of double products and strains in hyper- 

 space. Trans. Conn. Acad. Arts Sci., 14: 1-57. 1908. 



2 Gibbs-Wilson. Vector Analysis, pp. 356-367. 



3 The page numbers in the text refer to the memoir cited in Note 1. I here 

 use an upper zero instead of a dash over a letter to represent an (n-l)-vector. 



173 



