666 PROCEEDINGS OF THE AMERICAN ACADEMY, 



moreover, let 



(97 a) rjuv'-'^ = Jxiu, Vm+l/^^ = Vm+l,m+l^^^ = 



(i, u, V = 1, 2, ... m), 



(976) Vu,m+l^'^ = m,mj'^ = 1 



(i, u = 1, 2, . . . m; u ^ i). 



The m matrices Ei, E2, ... Ejn which we thus obtain have the same 

 multiphcation table as the units of the system (ci, eo, ... em) and 

 are, moreover, linearly independent. For, if 



CiEi + CoEo + • ■ • + CnEm = 0, 



then 



m 



Z CidJ'^ = {u,v^ 1,2, ... m+ 1); 

 1=1 



and, therefore, in particular 



m 

 Cu = Z Cidu,m+i^'^ =0 iu=l,2, ...m). 



Further, the 711 matrices determined by the above values of the 77's 

 are also linearly independent and have the same multiplication table 

 as the system {e'l, e'2, . ■ ■ e'^) reciprocal to (ri, ?>, . . . e„^. We now 

 have 



m m m m 



A = Y. «i^j = Z '''iCZ Z TiraCM + Cj>+i), 



t=l 4=1 M=l V=\ 



(98) 



m m m m 



A' = Y. ^^i^i ^ Z f'i(Z Z Ti'Ju fui. + ei,m+l) ; 

 t=l t=l u=l v=\ 



and, therefore, 



-.mm 11 

 Oi^ = — Z Z '''i7iu« = O^-l, 



(99) 



02^ = Z Z "'Ti/n; = '^-•1 • 



1 = 1 M=l 



We may also proceed as follows. Let// = /// + 1, and let 



(100 a) dj''^ = luiv, ^«.m+/''^ = ^m+l,m+/'^ = 



(?', 7/, i' = 1, 2, ... m), 



