respecting Equations of the Fifth Degree. 203 



namely, that " if a root x of the general equation of any particular degree n can 

 be expressed as an irreducible irrational function 6 "* of the n arbitrary coeffi- 

 cients of that equation, then every radical a. , which enters into the composition 



of that function b *" , must admit of being expressed as a rational, though unsym- 

 metric function f. of the n arbitrary roots of the same general equation ; and 

 this rational but unsymmetric funtion F. must admit of receiving exactly the 

 same variety of values, through changes of arrangement of the n roots on which 



it depends, as that which the radical a. can receive, through multiplications of 



itself and of all its subordinate functional radicals by any powers of the corres- 

 ponding roots of unity." 



Examples of the truth of this theorem have already been given, by anticipa- 

 tion, in the seventh and tenth articles of this Essay ; to which we may add, that 

 the radicals a/' and a/, in the expressions given above for a root of the general 

 biquadratic, admit of being thus expressed : 



«."= fs {(^. + ^2- ^3-^4)' + P^ (^ - ^2+ ^3-^4)* + Pz (^ - ^- ^3 + ^4)*} 



= h. {^.^2+^3^4 + /'3' (^, ^3 + ^2 ^4) + ft (^. ^4 + ^2 ^3)] ; 

 < = =li5 {^.^2 + ^3 ^4 + P3' (^, ^3 + ^2 ^4) + ft (^. ^4 + ^2 ^3)}' 

 — 3-S {^. ^2 + ^3 ^4 + ft' (^. ^4 + ^2 ^3) + ft (^. ^3 + ^2 ^4)}' 



= nij (ft'-ft) (^.-^2) {^r^z) (•^-•^4) (^2-^3) (^2-^4) (^3-^4)- 



But before we proceed to apply this theorem to prove, in a manner similar 

 to that of Abel, the impossibility of obtaining any finite expression, irrational 

 and irreducible, for a root of the general equation of the fifth degree, it will be 

 instructive to apply it, in a new way, (according to the announcement made in 

 the second article,) to equations of lower degrees ; so as to draw, from those 

 lower equations, a class of illustrations quite different from those which have been 

 heretofore adduced : namely, by showing, a priori, with the help of the same 

 general theorem, that no new finite function, irrational and irreducible, can be 

 found, essentially distinct in its radicals from those which have long since been 



