642 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Wherefore, there are then at least r -\- 1 idempotent numbers mutually 

 nilfactorial, namely, Z'„, /"„ and 7, for t' = 1,2, . . . u — 1, 2/ + 1, ... r, 

 which is impossible. 



The number system when regularized with respect to r idempotent 

 numbers, so that Too contains no idempotent number, and each of the 

 aggregates Fn, T22, ■ ■ ■ ^rr but a single idempotent number, is said to 

 be completely regularized. 



For 1 = i< ^ r, we may now take the m^u — 1 units other than /„ 

 of the aggregate or system r„u so that they shall all be nilpotent; in 

 which case they constitute by themselves a nilpotent sub sj^stem, 

 every number of which is, therefore, nilpotent. -^^ I shall assume that 

 in each of the aggregates ?„„ (u = 1, 2, ... r) the units have been so 

 chosen. 



Let 



r r Wuii 



(35) ^ = Z Z Z auhvJuhv. 



u = v = h = l 



By equations (29), (30), (33), and (34), and by what has just been 

 stated, we now have 



r 



(36) *Si^ = 2L ttumuuU^lJ^u 



u=l_ 



-, r r 



W 



'"' u = \ v = 



r 



(37) SiA = Y^ aumuuii^ilu 



1 r r 



M=l V=0 



I shall say that the two idempotent units /„ and /„ {1 = u = r, 

 1 ^v'^r, V 9^ ii) are connected if there are two numbers (u, v)' and 

 {v, u)' such that 



Si {u, v)' {v, n)' ^ 0; 



otherwise, not connected. If /„ and /» are not connected, then 



Si {u, v) {v, u) = 



10 This theorem is due to B. Peirce, loc. cit., p. 118. His proof is defective. 

 The first proof, I believe, of the theorem without the aid of the theory of 

 groups was given by me in the Transactions American Mathematical Society, 

 5, p. 547, by employing the generalized scalar function. 



