Resolution of Algebraical Equations. 136 



SECTION III. 



To find the sum of any specified number (m) of the roots of 

 an equation. 



Divide the given equation by x m , take the Nap. log. of the 

 quotient ; the coefficient of - , with its sign changed, will be the 

 required sum of m roots. 



Ex. (1). x" + ax + b = 0, to find the sum of two roots, take the 

 coefficient of - in - 1. (l + - + -„) viz. - a, which is evidently the 



X \ X X / 



sum of the two roots. 



Ex. (2). x-ax n — b = 0, to find the sum of two roots. By 

 the above rule, that sum 



. 1 . , /„ ax" + b\ 



coefficient of - in — 1. (1 , — ) 



X \ X / 



„ . . 1 . ax" + b (ax n + bf (ax" + bf , . 



= coefficient of - in — 2 — + — ^? + — sx 1 — + 



Hence, when » is even, this quantity is nothing which agrees 

 with truth, since the roots of the proposed equation are then of 

 the form + ^/a. 



But when n is odd, the terms of this series which involve 

 - , are at n places distance from each other, and begin at the —=— 



X ** 



term, its value therefore is 



„irr ± 8»-l-3n- 8 « 3 i^ , 5b-1 .5«-3.5»-5.5»-7 <l ^"^ &t . 

 2 . 4 3 " 2.4.6.8 ' f ' 



