TABER. — SCALAR FL'NCTlONS oF 1IYPFR (uMIM.F.X N L MBERS. (3(J3 



E'l, E'2, ■ • £'m linearly iiulepentleut.^* In this case, there is some 



m 



number /^ = L ^''i ^ l> tif (<'i, c-2, . . . '',„) such that X B = for 

 »=i 



every muni )er A' = X .t,^', <>t' tliis system; and there is also a number 



i= 1 



m 



B' = Ji bic'i 5^ of the reciprocal system such that B' X' = 



1=1 

 for every munber A'' of the reciprocal system. Couxersely, if there is 



m 



some number B = Y. bi<\ 5^ of (ci, e-i, ... em) such that XB = 



1=1 



m 



for every number A' of this system (or if, for B' = Y. ^i^^'i ?^ ^'ncl 



1=1 

 for any number A'' of the reciprocal system, we have B' X' = 0) equa- 

 tions (91) are satisfied for some system of values hi, h-i, . . .hm not all 

 zero, and we cannot assign to the 0's, nor to the tj's, the values given 

 by equations (90). When equations (91) are satisfied, 



SiBei = = SiBci (i = 1, 2, . . . m); 



and, therefore, 



Ai = A-: = 0. 



When the system {ei, e-i, ... g„) contains a modulus it is not possible 

 to satisfy equations (89) nor equations (91). 



We may distinguish three cases. First, the given nimiber system 



TO 



(ci, ei, ... fm) may contain both a number .1 = H Oj^i 5^ and 



m 



a number B = Y. ^i^i 5^ such that ^A' = 0, A'i^ = for every 

 t=i 



rn 



number A' = YL ^i^i of the system, in which case the system does 



»•= 1 ' 



not contain a modulus and Ai = Ao = 0. In this case it is not possible 

 to assign to the ^'s the values given by either equations (SS) or (90), 

 nor to assign to the rj 's the values given by either of these equations. 

 Nevertheless, it may be possible in this case to put n = m, provided 

 m > 2, but not otherwise. Thus let m = 3, and let 



e-^ = fi, eic-i = 0, excz = ez, 



24 If n = m and Bux^^ = Tuir for z, u, r = 1, 2, . . . m, it follows from (5-4) 



m 



that £",£:>= 2: 7o>£'u- Cf. note 23. 



