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



should be of the two-valued form a-\-b {x^ — x^ i^a^^^y) (■^s — ^ ) ' because, 

 if we denote it by {x^, x^ x ), we have 



(x„, X^, X^) = (p (x^ x^, xj = (x^, X^, Xp\ 

 and 



we have, therefore, in this case, 



(a, /3, 7;) = F. (X^, X^, X^) 



= {a + bix^—Xp) (x^-x^) (Xp—x^)} ix:^-\-p3'Xj3-\-p3^y)y 



a and b being symmetric coefficients, which must not both together vanish ; and 

 accordingly we find, a posteriori, that whereas this function Fj has always itself 

 six values, its cube has only two. The foregoing analysis shows at the same 

 time, that if an unsymmetric function of three variables have fewer than six 

 values, its cube cannot have fewer values than itself; and accordingly it is easy 

 to see that the cubes of those two-valued and three-valued functions, which were 

 assigned in the last article, are themselves two-valued and three-valued. In 

 fact, the passage from any one to any other of the values of any such (two-valued 

 or three valued) function, may be performed by interchanging some two of the 

 three quantities a;,, x.^, x.^; and if such interchange could have the effect of 

 multiplying the function by an imaginary cube-root of unity, p^, another inter- 

 change of the same two quantities would multiply again by the same factor p^ ; 

 and therefore these two interchanges combined would multiply by p^^ which is 

 a factor different from unity, although any two such successive interchanges of 

 any two quantities x^, Xg, ought to make no change in the function. If, then, a 

 rational function of three arbitrary quantities have a symmetric cube, it must be 

 itself symmetric. 



The form of that six-valued function of three variables which has a two- 

 valued cube, may also be thus deduced, from the functional relation (II. 2). 

 Omitting for simplicity, the lower index i, which is not essential to the reasoning, 

 we find, by that relation, 



(|3, 7, a) = p^{a, /3, 7) ; (7, a, p) = p,^ (a, ft 7) ; 

 (7, ft a) = ,,3(0,7, /3); (ft 0,7)= p3* (a, 7,^); 



