respecting Equations of the Fifth Degree. 247 



tric> and consequently must be a square-root, of the form deduced in the last 

 article, namely, 



a,' = 6 (x, — a;J . . . (a?, — xj, 



the factor h being symmetric. And because any other radical of the same order, 



Ojj'j might be deduced from a/ by a multiplication such as the following, a^ zz r a,', 



we see that no such other radical a^, of the first order, can enter into the ex- 



pression b , when that expression is cleared of all superfluous functional 

 radicals. On the other hand, a two-valued expression such as 



cannot represent the five-valued function x; if then the sought expression x =■ b 

 exist at all, it must involve some radical of the second order, o,", and this 

 radical must admit of being expressed as a rational function f," of the five roots, 

 which function is to have, itself, more than two values, but to have some prime 



a. " 



power, F," ' , two-valued. And since it has been proved that no such function 



F," exists, it follows that no function of the form b can represent the sought 

 root X of the general equation of the fifth degree. If then that general equation 

 admit of being resolved at all, it must be by some process distinct from any 

 finite combination of the operations of adding, subtracting, multiplying, dividing, 

 elevating to powers, and extracting roots of functions. 



[23.] It is, therefore, impossible to satisfy the equation 



+ a, -j- Ojj 6 -j- Oj -|- a^ 6 -|- O5 = 0, 



by any finite irrational function b ; the five coefficients a^, a^, a^, a^, a^ being 

 supposed to remain arbitrary and independent. And, by still stronger reason, 

 it is impossible to satisfy the equation 



b 4-a,6 -f...+a„_i6 -j-a„ = 0, 



if n be greater than five, and a^, . . .a^ arbitrary. For if we could do this, then 

 the irrational function b would, by the principles already established, have 



. VOL. XVIII. ? L 



