302 



SIXTH RKPORT — 1836. 



lynome X, while L^'^ is a rational and integral function of x, 

 which is = if V be less than the exponent in of the degree of 

 that proposed polynome X, but otherwise is of the degree 

 V — m. In fact, if we divide the power x' by the polynome X, 

 according to the usual rules of the integral division of polynomes, 

 so as to obtain an integral quotient and an integral remainder, 

 the integral quotient may be denoted by L*^* , and the integral 

 remainder may be denoted by 



w + ./'^- 



+ J'^;i-^ + 



m— I 



and thus the identity (44) may be established. It may be no- 

 ticed that the 7n coefficients s^'\ «/'^, . . . s^^-V "^^^ ^^ consi- 

 dered as symmetric functions of the m roots x^, x^, . . • ^^ of 

 the proposed equation X = 0, which may be determined by the 

 in relations. 



-1 1 





\ (45.) 



These symmetric functions of the roots possess many other 

 important properties, but it is unnecessary here to develop 

 them. 



Adopting the notation (44), we may put, for abridgement, 



A' So'-'-'^ + A" *o^^"^ + A"' ^0^^'"^ = Po, 



m—\ m—l 711—1 ■* m—1 



W sy^ + M" .0^""^ + M'" .0^'*'") + M'^ .0^""^ = p\ 



(46.) 



(47.) 



m — 1 m—l m — 1 wt— 1 ■• m — 1' 



A'L^^'^ + A"L^^") + A"'L(^"'^ = A, .... (48.) 

 M' L^'*'^ + M" L^'*"^ + M'" L^'*"') + M'v L^''"') = M, (49.) 



