136 Mr. Murphy on the 



When the term involving x m is wanting in the proposed equa- 

 tion, then, instead of dividing immediately by af, put x = x + a, 

 and the difficulty of having no term free from the unknown 

 quantity in the quotient (on which depends the expansion of 

 the log.) will be avoided. 



To prove the rule given in this Section, suppose a„ a„...a m , 

 n „, + !•••«,. to be the n roots of the equation <p(x) = 0. 



Hence (p{x) = C.{x-a,).(x-a. 2 ) (*— aj (x-a m + l ) (x-a„); 



■•■^-«(»-S)"(\-2) (»-*) 



«(l— i-)'(l~£-) (i-*), 



and taking the log. of both sides, it is manifest that 



a^a^ ... a,„ = coefficient of - in — 1. ™ - ■■ 



a; a;" 1 



This method then will give us very simply the sum of 

 any proposed number (m) of the roots of the given equation ; 

 but since there may be various combinations of m roots made 

 from the n roots of the given equation, which combination or 

 group of m roots is that given by the present rule? The answer 

 is analytically, it gives any possible group of the m roots, but 

 arithmetically the m least : that it analytically gives any is obvious, 

 since the resulting expression is manifestly a symmetrical func- 

 tion of all the roots, and therefore cannot analytically represent 

 any one combination of m roots more than any other. That it 

 gives the sum of the m least roots may be thus shewn. 



Suppose, first, that the sum of two roots of the equation 

 (p(x) = is obtained by the present method, and that o, + o„ is the 

 particular combination which it gives. 



