KEPORT ON CERTAIN BRANCHES OF ANALYSIS, 351 



and therefoi-e identical with that which arises from the deve- 

 lopement oi — -^ + V\-r ~ ^)- If^be greater than -^, the 



roots of the equation are impossible, and the series becomes 

 divergent, and gives no result- 

 Any function / {x) of the least root of an equation <p (a;) = 

 may be found " by subtracting from / (0) the coefficient of 



— m f (x) log ?-LJ," This more general theorem evidently 



includes the former. 



" The sum of any assigned number {ni) of roots of the equa- 

 tion <p {x) = is equal to the coefficient, with its sign changed, 



of — m log ^-^. 

 X ^ x"" 



The expression for the sum of m roots of an equation which 

 is thus obtained gives the arithmetical value of the sum of the 

 m least roots. In estimating the order of magnitude of such 

 roots no regard is paid to their signs of affection. 



Mr. Murphy has shown in what manner the same general 

 proposition which is employed in the deduction of the results 

 just given may be applied to the investigation of some of the 

 most general theorems which have been employed in analysis 

 for the developement and transformation of functions. Amongst 

 many others the following very remarkable theorem seems to 

 merit particular notice. 



If x^^, Xci, ^3, . . ^TO be the m least roots of the equation 



(x - a)"' - hF (x) = 0, 



d--^{f(a)F{a)} 

 = "^f^""^ + ^' l.2..{m-\)da--^ 



Jl_ d^"^-^ {f {a) {¥ {a)Y} 

 *■ 1 .2' 1.2 {2m-\)da^'"-' "^ ^' 



If in this very general theorem we make w = 1, it becomes 

 the theorem of Lagrange ; and if we make m equal to the di- 

 mensions of the equation, or greater than ^y power of x in- 

 volved in F {x), then it becomes the theorem which Cauchy 

 has given, without demonstration, in the ninth volume of the 

 new series of the Memoirs of the Institute, for the expression 

 of the sum of the different values oi f{x), when x is succes- 

 sively replaced by every root of the equation. 



The preceding conclusions, so very remarkable for their 

 great generality, and for the very simple means employed in 



then, 



