and (II. 1) gives 



respecting Equations of the Fifth Degree. ' 219 



(7, «. ^X = - (ft 7. «)i = + (a, A 7)i. 

 (a, /3, 7). = - (7, a, /3). = - (a, ft 7)., 



(«5 ft 7)i = ft (ft «. 7)i = P3^ («' ft 7)i : 



we may therefore confine our attention to the other two relations. Of these, 

 (I. 1) requires that the function should not change its value when x^ 



and Xo are interchanged, and consequently, by what was shown above, that it 



should be reducible to the form a + hx + co: ^ ; in this case, therefore, we have 

 the expression, 



(«> ft y)i - Fi {^a^ ^^y ^y) - (^„ - ^^) (« + K + c^y% 



the coefficients a, b, c, being symmetric functions of .Tj, x^, X3. Accordingly the 

 square of this function f. admits in general of three values only, while the func- 

 tion is itself in general six-valued ; because the square of the factor x^ — x^, but 

 not that factor itself, can be expressed as a rational function of x , and of the 

 quantities a,, a^, a^, which are symmetric relatively to oTj, X2, x^. It may even 

 happen that the function itself shall have only two values, and that its square 

 shall be symmetric, namely, by the factor a-{-bx -\- ex ^ being reducible to 



the form b (x^—x) (xo — x ), in which the coefficient b is some new symmetric 

 function ; but the results of the last article enable us to see that the functions 

 thus obtained, namely, those of the form 



or more simply of the form 



6(^1-^0 (^1-^3) (-^2—^3). 

 are the only two-valued functions of three variables which have symmetric 

 squares : they enable us also to see easily that the square of a three-valued 

 function of three variables is always itself three-valued. It remains, then, only 

 to consider the relation (II. 2) ; which requires that the function 



(g, i3, 7); 



