of Algebraic Equations. 171 



functions of the n elements a v a 2 , . . . , a n) and the roots 



Mi Kh Kb * » • {«»}j 



which we obtain by solution of 11 = 0: these I call the issuing 

 roots of U = 0. They are all irrational functions symmetric in 

 the n elements a Xi a 2) . . . , a H} or in any set of n elements of 

 which 2}« r is a symmetric function. 



I denote by a bracketed function {R} always a symmetric 

 irrational function of our variables x v a? 2 , . . . , cc n , the issuing 

 root of a given equation which we are supposed to have solved. 



(2) As our rational symmetric function 2^ can be formed, 

 and is supposed to be formed, for every system of exponents that 

 we'require, we can write for any system in which /3 only is vari- 

 able the n equations 



Z y + jZy + 2 Z y + . . . + n _iZ y = 2, u 



Z y + 1 Z y + 2 Z y + . . . + „_iZ y =2 2 , 



Zn , rjn . rjn . . rjn ^ 



>y 



where Z r , {L r > 2 Z r , . . . n _{H are ^ values of Z r made on F n . 

 From these n conditions we obtain 



{Zy}— Y n _{l 



6) 



the issuing root of a given equation of the nth degree, having 

 for coefficients rational and symmetric functions of % v a? 2 , . . . x n . 

 This symmetric and irrational function {Z y } can be written for 

 every system of exponents y, 8, . . . c } b } a, i } in which 7 only is 

 variable ; that is, we have 



Y 5 +1 Y S + 2 Y { +... + „- 2 Y { ={!,}, 



Y { * + 1 Y 5 ^ +.,*,• + ...+ n -H,*={Z t }, 



Hence 



<j -ri x « T2JL5 +• • •-r?i-2is —\^n-U' 



{Ys}=r w _ 2 x e 



is the issuing root of a given equation of the (w — l)th degree, 

 whose coefficients are irrational and symmetric functions of our 

 n variables. &nd by the same certain process which has trans- 

 formed the asymmetric sums T w _iY| and T M _ 2 Xe into the sym- 

 metric irrational functions {Z y } and {Y 5 }, we can go on through 

 the entire list of asymmetric sums of values for every system of 

 exponents, transforming each sum into the issuing root of a given 

 equation, till finally we obtain the system 



