TABER. — SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS. G37 



in which case X^''> is nilpotent, since 



r 

 u=l 



contains W{p): therefore, by theorem I, 



r r 



,^ . M=l U-1 



(25) 



S^.A^ = Z pu''SiIu-\- SoN^p) = 23 p^r>S2lu. 



u=l u=l 



If possible, let 



SiAP = SiAP^^ = ... = SiA"^'-' = 

 for some positive integer p. By (25), we then have 



Pl^^'Sl/l + P^'^SJ, + . . . + P^'^SJr = 



(/j = 0, 1,2, ...r-1); 



and since, by theorem I, neither Si/i, Syh, . . . nor Silr is zero, it fol- 

 lows that 



p,P, . . . pP+r-i = 0. 



Pr , . . . pr 



which is impossible, since by supposition the p's are distinct and other 

 than zero. A fortiore, we cannot have 



SiAP = Si JP+i = . . . = SiA""-'-^ = 



for any positive integer ^J- Similarly, we may show that we cannot 

 have 



S2AP = SiAP-"' = ... = S.AP-"'-^ = 



for any positive integer x> if A is non-nilpotent. 

 We have now the following theorem. 



Theorem III. Let {ei, co, . . . em) he any given non-nilpotent number 

 system; and let r he the maximum number of idempotcnt numbers, 

 mutually nilfactorial contained in the system. Then, if for any number 



A = aiei + 02<^2 + ... -i- <lmCm 



