64 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Theorem (5). If the system e 1} e 2 , . . . e n is nilpotent so also is the 

 reciprocal system e l} e 2 , . . . e„; and conversely.* 



Theorem (6). If A is rational in 2ft (E, e z ), then A is rational in 

 2ft (R, e,) ; and conversely. 



Theorem (7). If ^4 X , A 2 , . . . A^ are independent, so also are A x , 

 A 2 , . . . Ap.; and conversely, f 



Assuming Theorem 7", in so far as it relates to the first scalar function, 

 S A, we are now in position to establish the theorem for S A. 



First, let 



n 



e\ = 2> T "J e i (h=l,2, . . . n), 

 i 



n 



e' h = 2j T hj V ( h = !» 2 > • • • ») 5 

 i 

 and let 



n 



e 'i e> J = 2* ?'*>'* *'* (*'».J = 1 ' 2 ' ' * * W )» 



1 



n 



c'« </ = 2* T«* ? * (•»•* = : ' 2 ' • • • w )- 



Then, by (2), 



7ifk = y% (hj, £ = 1,2,. . . w). 

 If 



^' = % a '> e '< = % a ' e > = A > 



i i 



then 



n 



2« a 'z Ty = a, 0' = 1, 2, . . . n) ; 

 i 

 and, therefore, 



A' — 2» a '* *>' — 2* a < ^' = ^' 

 i i 



But, by Theorem I, SA is invariant to any linear transformation of 

 the units of the system e u e. 2 , . . . e n ; that is, 



n n n n 



2< 2^?'* = 2 2> a < y«- 



* For, if c 1} c 2 , . . . c contains an idempotent number A, then e lf e 2 , . ■ . e n 

 contains an idempotent number A by (4). 



t This follows from the first two clauses of (3). 



