634 PROCEEDINGS OF THE AMERICAN ACADEMY. 



SiAei = SiBei {i = 1, 2, . . . m) 



or 



S2Aei = S2Bei (i = 1, 2, . . . lyi), 



we have A = B. If Ai = 0, then A2 = 0, and conversely; moreover, 

 the nuUify of Ai is equal io the nullity of A2. Every number of the 

 system satisfying equations (a) is a solution of equations {(3), and con- 

 versely. If B is any soluiion of equations (a) {or of equations (/?)), then, 

 for any number A of the system (ei, Co ■■•€„), both B A and A B are 

 solutions of these equations. If the nullity of Ai is m', there is a set 

 of just m' linearly independent solutions of equations (a) (or equations 

 (/?) ); and any such set of m' numbers of {ci, ei, ... f^) constitute an 

 invariant nilpotent sub system containing every invariant nilpotent sub 

 system of {d, e^, ... Cm). 



Let the system (ei, eo, ... e^) contain a nilpotent sub system 

 (Ci, C2, ... Cp) such that 



p 

 CjCi = Y. 9iJhCh (i = 1,2, ... m; j = 1,2 ... p). 



For 1 ^ J ^ J), we then have, by theorem I, 



V 



SiCjCi = Y. gijhSiCh =0 (i = 1, 2, . . . m); 



A = l 



therefore, Ai = 0, and thus (ci, e-i, ... c^) contains an invariant 

 nilpotent sub system to which the sub system (Ci, C2, ... C^ belongs. 

 Similarly, we may show that, if the system (^i, ei, ... e^ contains a 

 nilpotent sub system (Ci, Ci, ... C^ such that 



V 



CiCj = Y. dijhCh (i = 1,2 ... m; j = 1,2 ... p), 



h=\ 



it then contains an invariant nilpotent sub system which includes 

 the sub system (Ci, C2, ... C^. 



If {ci, ^2, ... Cm) contains a sub system (Ci, C-2, ... C^ such that 



»SiCi =0162= . . . = SiCp = 

 or 



O2C1 = 02^2 ^^ . . . = ^2^p ^^ 0> 



this sub system is nilpotent, since then, by theorem I, every number of 

 the sub system is nilpotent. Thus, if 



C = giCi + goC2+ ... +gpCj, 



