TABER. — SCALAR FUNCTIONS OF llYl'KH C0M1'LF:X NUMBERS. G35 



is any number of the sub system, we have 



C« = ^i(«)C, + ?.>WCo + . . . + g,^")Cj,; 

 therefore, 



SiC« = gi^^^SyC, + ff^^'^SiCa + . . . + ^p^'^SiCp = 0, 

 for any positive integer q. 



§2. 



For any given number 



HI 



of the non-nilpotent system {ei, 62, ■ ■ • e„i) there is a hnear relation 

 between A, A'-, . . . A'"'*'^; therefore, a smallest positive integer 

 M = »i + I for which A, A-, . . . A'^ are linearly related, and thus for 

 which we have 



(17) 9. {A) ^ A'' + ihA'^-^ + . . . + p,-iA = 0, 



where the ys are functions of the a's. Let pi, po, • • • Pr, respec- 

 tively of multiplicity ixi, ixo, • • • Mr. be the distinct non-zero roots, if 

 any, of fi (p) = 0; when we have 



(IS) fi (p) = P^O (p _ pO^l (p - po)^-2 ... (p - p,)hl>, 



where A;o ^ 1 . Further, let 



(19) W{p) ^ p{p- pi) (p - P2) . . . (p - p,). 

 Let now 



/U)= L CkA^^ 



h=l 



be any polynomial in A. If f{A) = 0, then p^ fx and/(p) contains 

 fi(p); othenvise, there is a linear relation between A, A"^, . . . A''~^, 

 which is contrary to supposition. Wherefore, if /(.4) is nilpotent, 

 /(p) contains W (p). Conversely-, if /(p) contains W (p), f(A) is nil- 

 potent; and, if/(p) contain 12 (p), then/(.i) = 0. 



Let A be non-nilpotent. Corresponding respectively to the r ^ 1 

 distinct non-zero roots of 12 (p) = 0, are r linearly independent num- 

 bers /i. I2, • • • It, linear in powers of A, wliich are severally idempo- 

 tent and mutually nilfactorial: thus we have 



(20) /„/„ = T,9^ 0, I J, = (u, V = ],2, ... r; v 9^ u). 



