respecting Equations of the Fifth Degree. 246 



We see, at the same time, that the sum of the two functions (1, 2, 3, 4, 5). and 

 (1, 2, 3, 4, 5)j admits of at least a," different values, namely, 



V"{(1, 2, 3, 4, 5), + (1, 2, 3, 4, 5),}= f," + f^", 



VH(1. 2, 3, 4, 5). + (1, 2, 3, 4, 5),} = p^,, f/' + /)">;", 



V'""-'{(1, 2, 3, 4, 5), + (1, 2, 3, 4, 5) J = pf.r' f." + p-^:'""') f^". 



On the other hand, this sum f/' 4- F,"' cannot admit of more than a," values, 

 because it must satisfy an equation of the degree a/', with symmetric coefficients ; 

 which results from the two relations 



F.""'" + F;"«'"=2a, f/'f;" = c, 



and is either the cubic equation 



(< + f;")' - 3 c (f/' + f;") - 2 o = 0, 



or the equation of the fifth degree 



(f/' + f '"f - 5c (f," + f;")' + 5c^(f/' + f;") -2a = 0, 



according as a " is 3 or 5. We must therefore suppose that the function f," + f^^" 

 has exactly a/' values, and consequently that a^" is 5 and not 3, because no rational 

 function of five independent variables has exactly three values. And from the 

 form and properties of the only five-valued function of five variables, we must 

 suppose farther, that 



F/'+F-' = F," + ^4; = 6„ + 5.ar„ + &,^/+63^„^ + 64^„S 



x^ being some one of the five roots x^, . . . x^^ and tiie coefficients 6„ , . . . &^ 

 being symmetric ; and that conversely the root x^ may be thus expressed, 



i^. = ^0 + d, (f;' + ^,) + 4(f/'+ ^,) V • . . + ^.(fx"+ i^)'. 



the coefficients d^, . . . di being symmetric. We must also suppose that by 

 changing f,", successively, to pr, f/', p^ f,", />/ f,", p*r,", we shall obtain succes- 

 sively, expressions for the other four roots, x^, x , x^, x^, m some arrangement 



