respecting Equations of the Fifth Degree. 199 



A" ' = ^ (^1 + P3 ^2 + P3 ^3)' 



' admits only of two different values, in whatever way the arrangement of the 

 three roots x^, x.^, x^ may be changed ; it must therefore be itself a root of a 

 quadratic equation, in which the coefficients are symmetric functions of those 

 three roots, and consequently rational functions of a,, a^, a^ ; namely, the 

 equation 



o = {trr-ih {(^i + p' ^^ + p. ^.r + (^i + p' ^. + ?. ^^'\ {^n 



+ TT? (^1 + Pa' -^2 + p^ ^3)' (a?i + P^ 3^3 + />3 ar^)' 

 = (try + sV (2«.^-9a, a, +27^3) (^/") + (^^^) '• 



The same quadratic equation must therefore be satisfied when we substitute for 

 <j"^ the function c^ -\- a/ to which it is equal, and in which a/ is a square root ; 

 it must therefore be satisfied by both values of the function Cj± a/, because the 

 radical a/ must be subject to no condition except that by which its square is de- 

 termined ; therefore, this radical a/ must be equal to the semidifFerence of two 

 unequal roots of the same quadratic equation ; that is, to the semidifFerence of 

 the two values of the rational function t^'^ ; which semi-difference is itself a 

 rational function of x.^, x^, x^, namely, 



The same conclusion would have been obtained, though in a somewhat less 

 simple way, if we had employed the relation 



in which '. 



[11.] In general, let p be the number of values which the rational function 



W . 



F. can receive, by altering in all possible ways the arrangement of the s roots 



x^, . . . x^, these roots being still treated as arbitrary and independent quantities, 



VOL. XVIII. 2 b 



