250 Sir William R. Hamilton on the Argument of Abel, 



and in like manner, 



(a, 7, 8, /3, e) = (a, ^, 7, ?, e) = (7, a, ^, 8, e) ; 



we may therefore interchange the first and second of the five elements of the 

 function, if we at the same time interchange either the second and third, or the 

 third and fourth ; and a similar reasoning shows that we may interchange any 

 two, if we at the same time interchange any two others. An even number of 

 such interchanges leaves therefore the function unaltered ; but every alteration 

 of arrangement of the five elements may be made by either an odd or an even 

 number of such interchanges : the function, therefore, is either two-valued or 

 symmetric ; it having been supposed to have fewer than five values. Indeed, 

 this is only a particular case of a more general theorem of Cauchy, which is 

 deduced in a similar way : namely, that if the number of values of a rational 

 function of n arbitrary quantities be less than the greatest prime number which 

 is Itself not greater than n, the number of values of that function must then be 

 either two or one. 



[26.] It is a necessary consequence of the foregoing argument, that there 

 must be a fallacy in the very ingenious process by which Mr. Jereabd has 

 attempted to reduce the general equation of the fifth degree to the solvible form 

 of De Moivre, namely, 



of which a root may be expressed as follows, 



because this process of reduction would, if valid, conduct to a finite (though 

 complicated) expression for a root x of the general equation of the fifth degree, 



j;^ + a, a;^ + a, «' + fls a;^ + o, * + cfj = 0, 



with five arbitrary coefficients, real or imaginary, as a function of those five co- 

 efficients, through the previous resolution of certain auxiliary equations below 

 the fifth degree, namely, a cubic, two quadratics, another cubic, and a biqua- 



