650 PROCEEDINGS OF THE AMERICAN ACADEMY. 



By (30), (33), (40), (44), and (45), we now have 



(53) wSi J„u(P) =11 muv^"'"^ = L m„„(P'^^) = ixp 



q=\ V =1 v=l 



Mp V liq _ 



v=l q=l v=l 



(p = 1,2, ... v; u = 1,2, ... fxp), 



(54) SM^^ = 0, SsJu/^) = 



(p = 1,2, ... v; u,v = 1,2, ... Up) V 9^ u). 



m 

 And since, for any number yi = Z ^'^«' we may now put 



" Mp Mp _ 



(55) ^ = Z Z Z c,/P)JJP), 



P=l M=l J!=l 



we have 



V tip t^p 



(56) Si^ = Z Z Z Cu^^S.JJ^^ 



p=l M=l t)=l 



y MP MP 



= Z Z Z C,/P)S2JJ^)= S2^. 

 p=l M=l »=! 



Therefore, in particular, 



(57) SiCiCj = SiCiCj (i,j = 1, 2, ... w), 



and thus we have Ai = A2 also in the case now considered, when the 

 system (e-i, d, ... Cm) contains no invariant nilpotent sub system and 

 neither Ai nor A2 is zero. 



From the conditions, necessary and sufficient, that the m' constants 

 7yfc (hj) ^ — 1> 2, . . . m) shall constitute the constants of multiplica- 

 tion of a hyper complex number system in m units, viz.. 



(58) Z Jukykhi = Z yikiyjhk 



k=l k=l 



(i,j, h,l = 1,2, ... m), 



