TABER. — SCALAR FUNCTIONS OP HYPERCOMPLEX NUMBERS. 69 



That is, if A is any non-nilpotent number of the system, there is number 

 © {A), linear in powers of A, xohose product as pre-factor, or post- 

 factor into A, is equal to an idempotent number. 

 Let now 



SA ei = SAe 2 = . . . = SA e n = 0. 

 Then, for any numher 



n 



i 

 of the system, we have 



SAB= ^b t SAe t =0. 

 i 



Therefore, in particular, if A is not nilpotent, we have 



SA"f(A) =0 



for any positive integer p. Whence follows 



S(t(A)t(A)) = St(A) = 0, 



which is impossible, since f (A) is idempotent ; and hence it follows 

 that A is nilpotent. Similarly, if 



SAe 1 = SAe % = . . . =SAe n = 



we may show that A is nilpotent. 



Conversely, let A be nilpoteut. Then, either 



SAe t = SAe i = .' . . = SAe n = 0, 



or for some number 



B=%b t e, 

 i 



of the system S A B — S B A ^ ; in which case neither A B nor B A 

 is nilpotent. But, by the above theorem, if A B is not nilpotent, there 

 is a number G x such that A. B C\ = A B. C\ is idempotent ; therefore, a 

 number B^= B G x such that A B t is idempotent. And, if BA is not 

 nilpotent, there is a number C 2 such that C 2 B.A = C 2 .B A is idem- 

 potent ; therefore a number B 2 = C» B such that B 2 A is idempotent. 

 Similarly, we may show, if A is nilpotent, that, either 



