respecting Equations of the Fifth Degree. 201 



as rational functions of Aj, . . . a^, a supposition which requires, reciprocally, that 

 n should not be greater than s, because the original quantities a^, . . . a„ are, in 

 this whole discussion, considered as independent of each other. With these 

 suppositions, which involve the equality s =^n, we may consider the n quantities 

 a,, . . . a„, and therefore also the q coefficients e„ . . . e , as being symmetric 

 functions of the n roots x^, . . . x^ of the equation 



/ + A,/ +... + A„_i a; + A„ = 0; 



we may also consider f. as being a rational but unsymmetric function of the 

 same n arbitrary roots, so that we may write 



a. =F. {Xi,.. .xj; 

 and since the truth of the equation 



F. 



+ E, F. + . . . -f E^ = 



must depend only on the forms of the functions, and not on the values of the 

 quantities which it involves, (those values being altogether arbitrary,) we may 

 alter in any manner the arrangement of these n arbitrary quantities x,, . . . x^, and 

 the equation must still hold good. But by such changes of arrangement, the 

 symmetric coefficients Ej, . . . e remain unchanged, while the rational but un- 

 symmetric function f . takes, in succession, all those p values of which it was 



before supposed to be capable ; these p unequal values therefore must all be 

 roots of the same equation of the q"' degree, and consequently q must not be less 

 than p. And since it has been shown that the former of these two last mentioned 

 numbers must not exceed the latter, it follows that they must be equal to each 

 other, so that we have the relation 



q=p: 



that is, the radical a. and the rational function f. must be exactly coexten- 

 sive in multiplicity of value. 



2e 2 



