656 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Wherefore, we now have 



Ai = Ao = 0.21 



m 



Conversely, if i? = X ^k^k is any solution of equations (77), 



k=i 

 Bcj (1 =i ^ m) is by theorem II then also a solution of these equa- 

 tions, and thus Bej, by theorem I, is nilpotent: therefore, by the 

 theorem of p. 654, SBcj = for j = 1, 2, ... m; that is, 5 is a solu- 

 tion of equations (75). Let the nullity of V be m', where 1 = m' ^ m. 

 There is then a set of just m' linearly independent numbers 

 Bu B-i, . . . Bm' of the system {ei, Co, ■ ■ ■ Cm) satisfying equations (75); 

 therefore, just m' linearly independent numbers of this system satis- 

 fying equations (77) : whence it follows that the nullity of Ai is m'. 

 And since each of the B's satisfies equations (77) it follows, from 

 theorem II, that Bi, Bo, . . . B^' constitute an invariant nilpotent sub 

 system of (ci, e-z, ... Cm) containing every invariant nilpotent sub 

 system of (^i, e^, ... em). 



Let now V 5^ 0. In this case, if, for any tw^o numbers 



m m 



A = Y. «» ^^i> -S = Z ^i(^i 



of (('], Co, ... em), we have 



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



then A = B; otherwise, there is a number A - B 9^ oi the system 

 satisfying equations (77). In this case, Ai ?^ and the number 

 system {ci, Co, . . . Cm) contains a nodulus l)iit no invariant nilpotent 

 sub system. 



Let now the number system (ci, e-y, . ■ . c^) be transformed by the 

 substitution 



