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



for some suitable index i, belonging to the system 1, 2, 3, 4, 5, 6 ; the equation 



(«■' ^.» 7,) = («,» P.» 7,)y 

 is therefore equivalent to one or other of the two following, namely, either 



1st, . . . (a, p, y). = (^, a, y)., 



or 



2nd, ... (a, p, 7). = (/3, 7, a).; 



In the first case, the function f,. is symmetric with respect to the two quantities 

 .r^, Xg, and therefore involves them only by involving their sum and product, 

 which may be thus expressed, 



o, and Oj being symmetric functions of the three quantities or,, x^, x^, namely, 

 the following, 



a^=z —(x^-\-x^-\-Xj), a^ = XiS2-\-x^X3-\- x^x^; 



so that if we put, for abridgment, 



ttg zz Xi X2 ^3, 

 the three quantities x^, x^, x^ will be the three roots of the cubic equation 



x^-\■a^x'^-\-aiX ■\-a:^=^0. 



In this case, therefore, we may consider f. as being a rational function of the 

 root X alone, which function will however involve, in general, the coefficients 



y 



o, and a.2 ; and we may put 



_ x(^.)-x(V-'/'(^) _ .. . 



(f), x> ^^^ ^ denoting here some rational and whole functions of x , which may 

 however involve rationally the coefficients of the foregoing cubic equation. And 



