70 PROCEEDINGS OF THE AMERICAN ACADEMY. 



SA «! = SA e 2 = . . . = SA e n = 0, 



or there are two numbers By and B 2 of the system such that A By and 

 B 2 A are idempotent. 



We have, therefore, the following theorem : 



Theorem III. If the number 



A - 2,A-e» 



i 



of the system e u e 2 , . . . e n satisfies the equations 



Se x A = Se. 2 A = . . . = Se„ A = 0, 

 or the equations 



Se 1 A=Se 2 A=...= Se n A = 0, 



A is nilpotent. Conversely, if A is nilpotent, it satisfies both systems of 

 equations, or there are two numbers By and B 2 of the system e L , e 2 , . . . e n 

 such that both A By and B 2 A are idempotent. 



Clark University, Worcester, Mass. 



