62 PROCEEDINGS OF THE AMERICAN ACADEMY. 



?. a{ '-' independent hypercomplex numbers, rational in 2ft (R, eX 

 n (1 — o A) > 



niliaciend 

 that are .. . w*7A respect to A, in terms of which any number of the 



ntliacxend 

 system .,„ . with respect to A can be linearly expressed. If A is 

 ° nilfacient 



nilpotent,^ 



SA P = 0, SA P = 



for any positive integer p; and, conversely, if either S A p = for 

 every positive integer p, or S A 1 ' = for every positive integer p, A is 

 nilpotent. Finally, if the system contains n independent numbers 



A U A. 2 , . . . A n for which 



SA 1 = SA, = . . . = SA n = 0, 

 or 



SA 1 = SA ft = . . . = SA u = 0, 



the system is nilpotent.% 



The proof of Theorem I, so far as it relates to S A, the first of the two 

 scalar functions, I have given in the paper above referred to.§ This 

 theorem may be demonstrated for the second scalar function, S A, by the 

 aid of the following theorems relating to the reciprocal systems e lf e 2 , . . . 

 e n and e x , e 2 , . . . e n . 



Theorem (1). SA = S A, S A = S A. 

 Theorem (2). If 



c'h = t*i <?i + t A2 e 2 + . . . + T hn e n (h = 1, 2, . . . n), 

 and 



e'h — t/,1 *i + r ft2 e 2 + . . . + r hn e n (k = 1 , 2, . . . n), 



* If A B = 0, B is nilfaciend with respect to A ; if B A = 0, B is nilfacient witli 

 respect to A. Peirce, loc. cit., p. 104. 



t If A m — 0, for some positive integer m, A is nilpotent. Peirce, loc. cit., p. 104. 



\ A system is nilpotent which contains no idempotent number. Peirce, loc. cit., 

 p. 115. 



§ Loc. cit., p. 614 et seq., and p. 531. 



