TABER. — SCALAR FUNCTIONS OK IIYI'KR COMPLEX NUMBERS. G67 



(1006) 0«+,,/') =0, 5m.i/'') = 1 



0', V = 1, 2, ... m; V 9^ i); 

 moreover, let 



(101a) 77,,/') = yiuv, T^um+i^*) = r?m+l.m+l^'^ = 



(i, u,v=l,2,... m), 



(1016) Vm+i/^ = 0, Vrn+i/^ = 1 



(i, V = 1, 2, ... m; v 9^ i). 



The )n matrices Ei, E-i, . . . E,„ thus obtained are hnearl\- independent, 

 as are also the ni matrices E'l, E'2, . . . E'^; and the former have the 

 same multiplication table as the units of the system {ei, ei, ... e„,), 

 while the latter have the same multiplication table as the units of 

 the reciprocal system. We now have 



in m / m m -^ 



1=1 i=l ^u = l i) = l ^ 



(102) 



TO m / m m \ 



--1' = Z "i^i = Z «»■( Z Z yiuveuv+ €m+l,i); 

 »=1 i=l \ u=l v=l ^ 



and, therefore, 



1 TO TO , I 



OiA = ~ l^ 2- Ttuu = -^ -'1 > 



(103) 



1 TO TO _| ^ 



^2^ = — Z Z Tm« = SA. 



'^ i=l u=l W 



The fundamental properties of the scalar functions given in theorem I 

 are more readily proved for the special case in which the number 

 system is a quadrate than in the general case. What precedes in this 

 section indicates how the properties of these functions may be made to 

 depend upon the properties of the single scalar function of a quadrate. 



Clark University, 

 Worcester, ]\I.\ss. 



