66 PROCEEDINGS OF THE AMERICAN ACADEMY. 



fx = n S A > independent hypercomplex numbers A 1} A 2 , 1 . . A^, 

 rational in %x (R, e t ), that are idemfaciend with respect to A ; and in 

 terms of these any number idemfaciend to A can be expressed linearly. 

 By (4), (6), and (7), A lt A 2 , . . . A^ are independent, rational with 

 respect to 2ft (R, e,), and idemfacient with respect to A. Moreover, 

 there is no number A^+i, independent of A x , A 2 , . . . A^, that is, 

 idemfacient with respect to A. For, otherwise, by (4) and (7), there 

 are \x + 1 independent numbers of the system idemfaciend to A, which 

 is contrary to Theorem I. Whence it follows that there are 



n SA = n SA = ^ > 



independent numbers, rational in 2ft (R, e,), that are idemfacient to A, 

 in terms of which all numbers of the system idemfacient to A can be 

 expressed linearly. Similarly, we may show that there are n (1 — S A) 

 independent numbers of the system, rational with respect to 2ft (R, e : ), 

 that are nilfacieiit with respect to A, in terms of which all numbers nil- 

 facient with respect to A can be expressed linearly. 



If A is nilpotent, then by (4), A is nilpotent, when by Tlieorem I, 

 S A p = for any positive integer p ; and, therefore, by (1) and (3), 

 S A p = S A 1 ' = 0. If, conversely, S A p = for every positive integer 

 p, then by (1) and (3), S A p = S A p = for every positive integer p ; 

 and, therefore, by Theorem I, A is nilpotent, in which case by (4) A is 

 nilpotent. 



If there are n independent numbers A x , A 2 , . . . A n of the system 

 e x , e 2 > • • • e n such that 



SAj, = SA 2 = . . . = SA n = 0, 



then, by (1) and (7), there are n independent numbers A 1} A 2 , . . . A n 

 of the reciprocal system e x , e 2 , . . . e n such that 



SA X = SA 2 = . . . = SA n = ; 



and, therefore, by Theorem I, the system e lf e 2 , . . . e n is nilpotent, in 

 which case, by (5), e 1} e 2 , . . . e n is nilpotent. 



For a system of w = m' 2 units £,j (i,j = 1, 2, . . . n) whose multi- 

 plication table is given by the equations 



%«i* = 8*, «y*A* = (hj>&i h = 1, 2, . . . n; h %j), 



