respecting Equations of the Fifth T^egree. 175 



x = h"'=f^{al^, <", a,", <, a,, a^, a^, aj 



the radical a/'' being defined by the equation 



«/''"=//"«"' «l"' «l'» «1' «2. 03, 04) 



= — a, + 3^3 4" ^/ » 



while o,'", a,", a,', and e^, e^, e.^, e,, retain their recent meanings. Insomuch 

 that either the function of third order h'", or the function of fourth order 6'*', 

 may be substituted for x in the general biquadratic equation ; or, to express 

 the same thing otherwise, the two equations following : 



h"" + a, b"" + a, b"" + a, b'" + a, = 0, 

 and 



b"'* + a, b"' + a, b"'' + a, 6"'+ a, = 0, 



are both identically true, in virtue merely of the forms of the irrational func- 

 tions b'" and b"', and independently of the values of the four arbitrary coeffi- 

 cients a,, a^, a J, O4. 



But for higher values of n the question becomes more difficult ; and even for 

 the case w = 5, that is, for the general equation of the fifth degree, 



a;* -|- flj or* -f- a^ at^ -\- a^ x"- -\- a^ a; -\- a^ =1 0, 



the opinions of mathematicians appear to be not yet entirely agreed respecting 

 the possibility or impossibility of expressing a root as a function of the coef- 

 ficients by any finite combination of radicals and rational functions : or, in 

 other words, respecting the possibility or impossibility of satisfying, by any 



(m) 



irrational function b of any finite order, the equation 



(m)* (m)* {my (ot)« (m) 



the five coefficients a„ a^j O3J 0S4, a^, remaining altogether arbitrary. To assist 

 in deciding opinions upon this important question, by developing and illustrating 

 (with alterations) the admirable argument of Abel against the possibility of any 

 such expression for a root of the general equation of the fifth, or any higher 



VOL. XVIH. - 2 b 



