respecting Equations of the Fifth Degree. 241 



in which a and b are symmetric. The remaining suppositions, (c)", (c)'", (d)", 

 (d)'", are easily seen to conduct only to symmetric functions ; for instance, (c)" 

 gives 



(-^5' "^l •*"« "T -^3 ^a) "~ V \^« "^3 "^5 ~r -^2 ^l) "^ \^1' ^3 ^i ~T" ^i •^i) 



= (^„ OTj a:^ + OTa X,) = (oTj, a:^ x^ + oTj o;,) =: (ar^, ^, ^3 + ^g ^4)1 



so that the condition (c)' is satisfied, and at the same time x^ is interchangeable 

 with Xt . And it is easy to see that the five- valued function <p (^^) may be put 

 under the form 



(f) . . b,Ji-b,x, + b,x,'-^hx,'-[.b,xJi 



the coefficients b^ ft, b^ b, bt being symmetric. It is clear also that neither this 

 five-valued function (f), nor the two-valued function (e), admits of any reduction 

 in respect to number of values, without becoming altogether symmetric. There 

 are, therefore, no unsymmetric and rational functions of five independent vari- 

 ables, with fewer than six values, except only the two-valued function (e), and 

 the five-valued function (f). 



Suppose now that we have the equation 



F,' being a rational but unsymmetric function ; and that 



0/ ' =/i(«i5 «2> «3, a,, a J, 



the exponent a^ being prime, and the function f being rational relatively to 

 a,, . . . ftj, and therefore symmetric relatively to ^1, . . . x^ With these suppo- 

 sitions, the function F,' must, by the principles of a former article, have exactly 

 a^ values, corresponding to changes of arrangement of the five arbitrary quan- 

 ties x^, . . . x^', the exponent a/ must therefore be a prime divisor of the product 

 120 (=1.2.3.4.5); that is, it must be 2, or 3, or 5. But we have seen that 

 no rational function of five variables has exactly three values ; and if we sup- 

 posed it to have five values, so as to put, (by what has beeu ajjeady shown,) 



we should then have three other equations of the forms 



2 k2 



