respecting Equations of the Fifth Degree. 24d 



present writer) is connected with the proof of the theorem, that a rational 

 function of five independent variables cannot have five values and five only, 

 unless it be symmetric relatively to four of its five elements ; it has been 

 thought advantageous, in this paper, as preliminary to the discussion of the 

 forms of functions of five arbitrary quantities, to establish certain auxiliary 

 theorems respecting functions of fewer variables ; which have served also to 

 determine a priori all possible solutions (by radicals and rational functions) of 

 all general algebraic equations below the fifth degree. 



[25.] However, it may be proper to state briefly here the simple and 

 elegant reasoning by which Abel, after Cauchy, has proved that if a function 

 of five variables have fewer than five values, it must be either two-valued or 

 symmetric. Let the function be for brevity denoted by (a, /3, 7, 2, e,) ; and let 

 V and V^ denote such changes, that 



O, 7» ^> e, a) = V (a, ^, 7> ^» «) » 

 {p, e, a, 7, 8) = v' (a> P, 7, ?> e). 



These changes are such that we have the two sjonbolic equations 



but also, by supposition, some two of the five functions 



V'(a, /3, 7, 8, €),... v*(a, ^, 7, ^, «) 

 are equal among themselves, and so are some two of the five functions 



V^-'Ca, /3, 7, 8, e),... V^'(«, /3, 7. S. 0; 

 we have therefore two equations of the forms 



in which r and r^ are each greater than 0, but less than 5 ; and by combining 

 these equations with the others just now found, we obtain 



V = l, v^ = l: 



that is 



(P, 7, 8, £, a) = (a, |3, 7, 8, e), and (^, e, a, 7, 8) = (a, ^, 7, 8, e) . 



Hence 



(7, a, )8, 8, e) = (^, 7, 8, £, a) = (a, ^, 7, 8, e) ; 



2l2 



