636 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If, for 1 £ w £ r, 



(21) " ^....M^( (''-^-)';-(°7/'^^- f, 



V — (o — Pur'' J 



— (p„ — p,)'=" 



(« = 1, 2, ... M — 1,M + 1, ... r), 

 and 



(22) /„ Cp) = <Ao("^ (p) <^l(") (p) . . . 0„-i("^ (p) c^u+i^"^ (p) . . . <^/"^ (p), 



we may write 



(23) Iu = fuiA) (u=l,2,...r).^ 



I shall denote by /• the greatest value of r for any number A of the 

 system. Then r is the greatest number of idempotent numbers, mu- 

 tually nilfactorial, contained in the system (ci, Co, ■ ■ ■ em)- For, if 

 possible, let the system contain p > r numbers Ki, K2, ■ ■ ■ Kp satis- 

 fying the conditions 



K\ = Ku ^ 0, KJ<., = 



{u, V = 1, 2, .. . p; V 9^ u). 



The K's are then linearly independent. If now 



A = XiZi + X2X2 + • • • + XpA%, 



where the X's are any p distinct scalars other than zero, the equation 

 fi (p) = has p > r distinct non-zero roots, wliich is contrary to 

 supposition. 



Let A be non-nilpotent and, for any positive integer p, let 



(24) m^) ^ Wp(A) = A^ - Z p'uL; 



u=l 



5 For then, in the first j:)lace, fu (P) contains p as a factor; therefore, fu (A) 

 is linear in powers of A. Moreover, for I <u ^r, fuip) does not contain 

 fl(p), whereas (fuiP))' — .fu(P) does contain 0(p) ; and, therefore, /« = fu{A) 9^ 0, 

 Ju^ — /u = 0. Further, for any two distinct integers u and v from 1 to r, 

 fu {P)fv (P) contains 12 (p); and, therefore, /«/» = 0. By the aid of the above 

 two equations, we may show that Ii, h, ... Ir are linearlj^ independent. Thus, if 



J ^ C./i +C2/2 + ... +Crlr = 0, 



then, for 1 < u^r, CuK = lu-J lu = 0; 



and, therefore, Cu = 0. 



