171 



XIII. On the Argument of Abel, respecting the Impossibility of expressing a 

 Root of any Qeneral Equation above the Fourth Degree, by any finite 

 Combination of Radicals and Rational Functions. By the President. 



Read 22nd May, 1837. 



[1.] JLET a,, a.^, . . . a„ be any n arbitrary quantities, or independent 

 variables, real or imaginary, and let a\, a'^ ... a'„' be any n' radicals, such 

 that 



a,' a'„. 



«/ =/i («> • • • «»)> • • • «'«' =/n' (oi, ..•««); 

 again, let a/', . . . a"„" be n" new radicals, such that 



« 



"1 



i" =/i'(«i'> • • • «'«', o„ . . . a„ ), 



«"»"" =/'«"«, . . . a'„', a„ . . . a„) ; 

 and so on, till we arrive at a system of equations of the form 



(m) (m— l)/'(m— 1) (m— 1) (m_2) (»i_2) v 



(m) 



V =-^„W V«. '•••V-D'"' '•••«„(>«-2)"-«.'-««A 



the exponents a. being all integral and prime numbers greater than unity, and 



(*— 1) 

 the functions f. being rational, but all being otherwise arbitrary. Then, if 



C™) (m) (*) 



we represent by b any rational function/ of all the foregoing quantities a. , 



(m) (m)r (m) (m) (m— 1) (ni— 1) \ 



o =/ V", ,...a„(„), a, ,. ..a^(„_i), ....a„ . ..aj, 



