226 Sir William R. Hamilton on the Argument of Abel, 



In the case (II.)j we may Interchange some two roots, x^, Xq, if we at the 

 same time interchange the two others ; and the function may be put under the 

 form 



(II.)-. V=<P{-'Pa + ^p — ^y — ^S'^a—^li'^y-^Sy^ 



because any rational function of the four roots may be considered as a rational 

 function of the four combinations 



or of the four following, 



of which the first may be omitted, as symmetric, and the third as being here 

 obliged to enter only by its square, which square {x^ — x^ is expressible as a 



rational function of o;^ -j" ^a" -^ ~ ^$1 involving also the symmetric coefficients 

 a„ aj, Oj, which are allowed to enter in any manner into 0. 



In the case (III.)> some three roots, x^, Xg, x , may all be interchanged, the 

 fourth root remaining unaltered ; and, on account of what has been shown 

 respecting functions of three variables, we may write 



(III.) . . F = {X^) 4- {X^ - X^) {X^ - X^) (X^ - X^) 1^ (X^) , 



the function -^ (as well as 0) being rational. 



In the case (IV.), we may change x^ to Xq, if we at the same time change 



Xo to ar , a: to Xgy and x^ to x^ ; and the function f is of the form 



(IV.).. v = (l>{x^- Xi^ + x^-x^.x^-- x^.x^-x^); 



1,2,3 1,3 



because the condition V4 = 1 involves the condition Va = 1> ^"^ consequently 

 the present function f must be rational relatively to the two combinations x^-\- x 



— Xq — x^ and x^ — x . x^ — x^\ or relatively to the two following, x^ — x^ 



-\- X — x^ and x^ — x^-\- x — x^ . x^ — x . x^ — x^; but of these two last- 



