72 Mr murphy, on elimination BETWEEN AN INDEFINITE 



Another class of equations which may easily be resolved by the 

 first principle, occurs when the x^^ equation is of n dimensions, and 

 arranged according to the powers of some function of .r ; it is then 

 merely necessary to expand 



c.a:{x — l)(x-Q) (x — n) 



according to the powers of that function ; and equate the coefficients 

 of like powers in both cases. 



Example: 



Ki + SSs + «3 + + SS, = - 1, 



2a!, + 2-S2 + S'xs + + 2»ss„ = - 1, 



Sz, + 3-S5, + 3^X3 + + 3'%„ = - 1, 



w^i + w'asa + ?r%3 + + n''z^ = — 1, 



to find «„ %2 



The general or «"" equation in this case, is 



1 + x»i + x"%i + + afz^ = 0, 



the roots of which equation are x = l, 2, 3 ;/. 



Hence, the left-hand member is identical with the product 



c.{x-l){x-2){x-3) (x-tt), 



or c(-iy{S„-xS„., + x'S,_,- (-l)".x"|, 



where S„ denotes the sum of the quantities 1, 2, 3 n when taken 



in products m and m together. 



Hence, by equating, we get 



c(-lY S =1- • r = \^y .. 



- c(-1)".aS'„_,= «,; .-. x,= -aS'_,; 

 c( — 1)" .«>„_2= »2; .•. SS2=— 0,2; 

 and generally £., = S,,, 



where S.„ denotes the sum of the reciprocals of the quantities of which 

 a9„ represents the sum. 



