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



Attending next to conditions of the forms 



V = - 1> V = /'3> 



instead of attending only to conditions of the form 



V=l, 



we discover the forms which a rational function of four arbitrary variables must 

 have, in order that its square or cube may have fewer values than itself ; which 

 functional forms are the following : 



The general twenty-four-valued function f will have its square twelve- valued, 

 if it be either of the form ■ 



or of this other form 



P= (^a — ^/s) • "^ (^a + ^/3 — ^y - ^5' '^a " ^/3 ' ^y " ^j) ' 



The same general or twenty-four-valued function will have an eight-valued 

 cube, if it be of the form 



F =' {0 (^^) + i^a - ^H) (^a - ^y) (^/3 " ^y) ^ (^^) } (^a + P,' ^^ + P, ^) ' 



Pa being, as before, an imaginary cube-root of unity. The twelve-valued 

 function (I.) will have a six-valued square, if it be reducible to the form 



V={x^ — Xg).y^(x^ + Xp—X^ — Xg). 



The twelve-valued function (II.) will have a six-valued square, if it be either of 

 the form 



^ = (^« + ^/3 — ^y — ^s) • -f (^a — ^^ -^y - ^l)f 

 or of the form 



T = {x^- x^) {x^ -Xi).^ (^„ -^x^-x^- x^). 



The eight-valued function (III.) will have its square four-valued, if it be of the 

 form 



The six-valued functions (IV.), (I. I.)', (II. II.), will have their squares three- 

 valued, if they be reducible, respectively, to the forms. 



