respecting Equations of the Fifth Degree. 243 



Xj, x^, x^, and />s, p^, /»/, p^* being the four imaginary fifth-roots of unity; con- 

 sequently we should have 



5 < «/ = «a + Ps' a;^ + />5' ^y + Ps' -^j + /'5 -y. ; 



a result which is absurd, the second member of the equation having 120 values, 

 while the first member has only five. We must therefore suppose that the 

 exponent a^' is = 2, and consequently must adopt the expression 



a/ == 6(a;i— OTj) {x^—x^') (x,—x^) (x^—x^) (x^—x^) (x^—x^) (x^—x^) (^3— ^4) 



the factor b being symmetric. This, therefore, is the only rational and unsym- 

 metric function of five arbitrary quantities, which has a prime power (namely its 

 square) symmetric. 



Let us next inquire whether it be possible to find any unsymmetric but 

 rational function, 



which, having itself more than two values, shall have a prime power two- valued, 



If so, the function f/' must have exactly 2a " values, and consequently the prime 

 exponent a/' must be either three or five, because it must be a divisor of 120, 

 and cannot be = 2, since no rational function of five arbitrary quantities has 

 exactly four values : so that a/' or f/' must be either a cube-root or a fifth-root of 

 the two-valued function//. And the six or ten values of f/' must admit of being 

 expressed as follows : 



(1, 2, 3, 4, 5). ; p^„(l, 2, 3, 4, 5). ; . . . />"|r '(1, 2, 3, 4, 5), ; 



a."— I 



(1,2,3,4,5)^; p^^.(l,2,3,4,5)^;...p^„ (1,2,3,4,5)^; 



in which, p „ and p „ are imaginary cube-roots or fifth-roots of unity, according 

 as a," is 3 or 5 ; while (1, 2, 3, 4, 5). and (1, 2, 3, 4, 5)^ are some two different 

 values of the function f/', which may be called f," and f,"', and correspond to 

 different arrangements of x^, x^, x^^ x^, x^, being also such that 



