652 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



§3. 



It has been shown by C. S. Peirce that any given hyper complex 

 number system {ei, e^, ... e.^ is a sub system of a quadrate of order n, 

 where the greatest vahie n need assume is tn -\- 1. This is, of course 

 equivalent to the theorem that any given number system can be 

 represented by a matrix whose order need not exceed m -\- \}^ 



Let now {e\, ei, ... p^) be any given number system; let 

 ^m (w> i) = 1, 2, . . . 7i) be the units of the quadrate of which 

 ifx, ei, ... %) is a sub system, when we have 



(61) 



and let 



(62) 



(w, X, t , IV = 1, 2, . . . n; v' 9^ v); 



ei= L L ^"■''''^"'' {i= 1,2, ... m). 



u=l v = l 



The units ei, ei, ... em may then be regarded as represented, respec- 

 tively, by the m linearly independent matrices E\, E-i, . . . E„, where 

 Ei, for i = i, 2, ... vi, is defined by the system of equations, 



(63) (^1', y, . . . y) = ( dn^'K dn^'K . . . ei„(') 0^1, ^2, . . . ^n), 



eni''K en2^'\ . . . dnn^^ 



and any number x = Y. -'"i ^i of (? 1, d, ... em) by the matrix of the 



linear substitution 

 (64) 



{y,y, ■ . ■ y) = ( Z '^■i&n''K L •^-.^12^'^ • • • Z ^iSirS"^ l^U ^2, ... ^n). 



i=l 1=1 t=l 



mm m 



t=l »=1 »=1 



18 Loc. cit., p. 221; also These Proceedings, 10, 392 (1875). In certain 

 cases, as shown by Peirce, we may take n < m; in other cases, n must be 

 greater than m. See § 4. 



