Resolution of Algebraical Equations. 141 



-". coefficient of k" + ' in u 



n 1 



= coefficient of - in - . ^ dX " + ' (:)" 

 - coefficient of s:"- 1 in i . ^ . f~Y' 



M.M + 1 dz \U/ 



(t AclX /*yj 



l.^.B+irf:"- 1 ' 



Equate this with the former value and we get the above theorem. 



To find the sum of any function of the m least roots of the 

 equation 



(a - a)'" = h.F(z) 



as /(«,) +/(«,) + /(«,„). 



Put 

 to find 



x = x + a; .'. x m = hF(a + x) 



f(a + Pi) +"/(«" + &) + &c. 

 A, &, &c. being the >» least values of x, we have, by Section 3, the 

 required sum 



= m/(a) - coefficient of - in J" {a + x) 1. {1 - —. F(a +x)}: 

 whence, if we expand the log-, this becomes 



= m f(a) + ft *" l {Sto'*m , I d—{f(a).FWf\ + . 

 l.S...«-l.<fe-' 2 ' l.'a...a^T^l.rf««— 



In this remarkable theorem if we put m = \ it becomes Lagrange's; 

 but if we put m = the dimensions of the equation, then the m 

 least are in reality all the roots, so that this case corresponds to 

 the theorem given by M. Cauchy in Vol. ix. Memoirs of the 

 Institute. 



