respecting Equations of the Fifth Degree. 227 



mentioned combinations, the former alone changes, and it changes in its sign 



1,2.3 



alone, when the operation v ^ performed, so that it can enter only by its square ; 

 which square {x^ — Xo-\- x — x^Y can be expressed as a rational function of 



the product (x^ — Xn-{-x — x^) (x^ — x ) (x^ — x^), and of those symmetric 



coefficients which may enter in any manner into 0. 



By similar reasonings it appears, that in the six other cases, (I. I.) — (II. III.), 

 we have, respectively, the six following forms for f : 



(I. I.) . . F = (p (xg) - a + b x^ + c Xg^ + d Xg^ ; 



(LI.)' F=0(^a + ^^-^y-^5); 



(I. II.) . . F = (l)(x^x^+x^Xg)=a + b (x^ x^ + x^x^) + c (x^ x^ + x^ Xgf ; 



(I. III.) . . F = a ; 



(II. II.) . . F = (j)(ix^- x^.x^-x^); 

 (II. III.) . . . F = a + b(x^-Xp) {x^ - x^) (x^ - x^) {x^ - x^) {xp - x^) (x^ - x^) . 



To one or other of the ten forms last determined, may therefore be reduced 

 every rational function of four arbitrary quantities, which has fewer than twenty- 

 four values. And although the functions (I. I.)' and (11. II.) are six -valued, as 

 well as the function (IV.), yet these three functions are all in general distinct 

 from one another; the function (IV.) being one which does not change its 

 value, when the four roots x^, x^, x , x^ are all changed in some one quaternary 



cycle, but the function (I. I)' being one which allows either or both of some two 

 pairs x^, x^ and x , x^ to have its two roots interchanged, and the function 



(II. II.) being characterized by its allowing any two roots to be interchanged, if 

 the two other roots be interchanged at the same time. It may be useful also to 

 observe, that the three-valued function (I. II) belongs, as a particular case, to 

 each of these three six-valued forms, and may easily be deduced from the form 

 (I. I.)', as follows : 



F = ^{x^^X^-X^-X^)-^{x^^X^-X^~X^) = x{^a+^?- ^y-'^s) - ^K^^+^j-^j) • 



