646 PROCEEDINGS OF THE AMERICAN ACADEMY. 



when, by (34) and (39), we have 



Si{u, o) gj = SiCu, o) {o, n)i 



= Si(o, u)i {u, o) = Si(o, o)/ = 

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



and thus {u, 6) satisfies equations (6). Similarly, if 11X0^ > (1 '^u^r), 

 we may show that (ci, 62, ... Cm) contains an invariant nilpotent sub . 

 system. 



Again, if r„u (1 ^ w ^ r) contains more than one unit, that is, if 

 Wuu > 1, the system (61,^2, . . . Cm) contains an invariant nilpotent sub 

 system. For, in this case, there is a nilpotent number {u, u) of Fuu 

 whose product with any number of this aggregate is, therefore, nil- 

 potent;^^ and thus {uii) {u,u)i, for i = 1,2, . . . m, is nilpotent: 

 therefore, 



<Si(m, u)ei = Si{u, u) {u, v)i = {i = 1,2, . . . m), 



and thus (w, u) is a solution of equations (6). If, for u, v any two 

 distinct integers from 1 to r, /„ and 7» are connected, and either r^, 

 or r^ contains more than one unit; that is, if either m^,, > 1 or 

 m^ > 1, the system (ci, 62, ... %) contains a nilpotent sub system. 

 For then, by the theorem p. 645, we have ?/?„« > 1- Further, if /„ and 

 7, are not connected, and either T^v or r,a contains one or more 

 units, that is, if vi^^ > or ?«„„ > 0, the number system contains an 

 invariant nilpotent sub system. For let {u, v) 5^ 0: in this case, by 

 the theorem given, p. 642, we have 



S,{u,v)iv,u)i= (i= 1,2, ... m); 

 therefore, 



Si(m, v)ei = Si{u, v) (i), u)i = {i = 1, 2, ... m), 



and thus {u, v) satisfies equations (6). Finally, if /„ and I„ are not 

 connected and m„u > 0, {ei, 6%, ... ^m) contains an invariant nilpotent 

 sub system. 



12 Namely, when muu > 1. any number {u, u) linear in the nilpotent units 

 of Tuu is such a number. For since /„ is a modulus of the system Tuu, these 

 nilpotent units constitute an invariant nilpotent sub system of Tuu- Where- 

 fore, the products of (w, h) and any number of Tuu belongs to this nilpotent 

 sub system, and is, therefore, nilpotent. 



