630 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



and let the number system (ci, e-i. . .e^) contain at least one number 

 satisfying the' system of equations 



(6) SiXci = xiSiCiCi + X2Sie2Ci + . . . + XmSie^Ci = 



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



The resultant of this system being the determinant 



(7) Ai = SiCiCu S]_e2ei, ... Sic^ei 



SiCiCi, 816262, ... 816^62 



SyCiem, 8162 ejn, ... Sie^^Cj^ 



we then have Ai = 0. Let A' = .B be any solution of equations (6). 

 Then, by theorem I, B is nilpotent. Moreover, for any number A of 

 (ei, 62,. . Oj both BA and AB are also solutions of equations (6). 

 For, for any number 



Y = yiCi + 7/2^2 + . . . + PmCm 



of (ei, 62, ... em), we now have 



8iBY=y,8iBe,-^y28iBe2-\- ... -\- Pr^S^Be^ = 0; 

 in particular, 



Si{BA-ei) = Sy{B-Aei) = 0, 



Si{AB-ei) = 8i{A-Bci) = 8i{Bei-A) = Si{B-eiA) = 



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



Since both BA and AB are solutions of equations (6), they are both 

 nilpotent. 



Further, since, for l^i^m, Bci is nilpotent, it follows from 

 theorem I that 82Bei = 0, and thus any solution B of the system of 

 equations (6) is also a solution of the system of equations 



(8) 82Xei = xiSoCiCi -\- X2 82e2ei + . . . + avSo^m^i = 



a = 1, 2, . . . m), 

 of which the resultant is 



(9) Ao = 826161, 826261, . . . 806^61 



826162, 826262, ... 826^62 



82616m, 8-2 62 6 tn, . . . 826m e„ 



