658 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Then, if the number system he transformed by the substitution 



e/ = Tnei + Ti^e-i + . . . + Ttmem {i = 1, 2, . . . m), 



and if 



we have 



n f I 



oe iC j 



{ij = 1,2, ... m) 



V'= PV, 



where T is the determinant of the substitution. If V 9^ 0, the system 

 {ei, e<>, ... ej^ contains a modulus but no invariant nilpotent sub system; 

 and, in this case, if for any two numbers 



m m 



of the system we have 



SAei = SBci {i = 1,2, . . . m), 



then A = B. If V = and m' (0 < m' ^ m) is the nullity of V, the 

 system {ci, e2, ... em) contains a maximum invariant nilpotent sub 

 system with m' units constituted by any m' linearly independent solutions 

 of the equations SXci = {i — 1,2, ... m). 



In precisely the same way we may now prove the following theorem 

 of which the preceding theorem is a special case : 



Theorem VI. Let{ei, e-y, ... O be any given hyper complex number 

 system constituting a sub system of the number system ei, €2, ... e„ 

 lohose constants of midtiplication are 7^^ for u, v, w = 1, 2, ... n, so 

 that 



n 



and let 



= Z ^'"«« (^" = 1, 2, ... m). 



u=l 



For any number A = Y. «««« of the system (ei, €2, ... Cn), let 



u=l 



1^ "_ i'' '^ 



SiA = - Y T. "ujuvv, S2A = - Y Z «"7lW 



^ U=l 1 = 1 ^* M=l V=l 



