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



above-mentioned general equation are interchanged in all possible ways, it must, 

 by still stronger reason, have fewer than six values, when only the first four roots, 

 x„ x.^, X3, X4, are interchanged in any manner, the fifth root x^ remaining un- 

 changed. 



Hence, by the properties of functions of four variables, the function f must 

 be reducible to one of the four following forms, corresponding to those which, in 

 the nineteenth article, were marked (I. HI.), (H. IH.), (I. II.), and (I. I.) : 



(a) . . (x,) I ■ ^ 



(b) . . 0(^5, ^1 X^ . X^ X3 . Xj — X^ . X^ — X^ . X^ — X^ . X^ — ^t) i 



(c). . (I>(x„ x^x^-\-j;^x,)i 

 (d) . . {x„ s,); 



or at least to some form not essentially distinct from these. In making this 

 reduction, the principle is employed, that any symmetric function of or,, x^, x^, x^, 

 is a rational function of Xj, and of the five coefficients Oj, a^, a^, a^, a^; which 

 latter coefficients are tacitly supposed to be capable of entering in any manner 

 into the rational functions 0. 



It may also be useful to remark, before going farther, that the four forms 

 here referred to, of functions of four variables, with four or fewer values, may 

 be deduced anew as folloVs. Retaining the abridged notation (a, /3, 7, 5), we 

 see immediately that if the six syntypical functions 



(1, 2, 3, 4), (2, 3, 1, 4), (3, 1, 2, 4), (1, 3, 2, 4), (3, 2, 1, 4), (2, 1, 3, 4) 



be not all unequal among themselves, they must either all be equal, in which 

 case we have the four-valued form (xj or (I. I.), or else must distribute them- 

 selves into two distinct groups of three, or into three distinct groups of two 

 equal functions. But if we suppose (1, 2, 3, 4) = (2, 3, 1, 4) = (3, 1, 2, 4), in 

 order to get the reduction to two groups, the functions (1, 2, 3, 4) and 

 (2, 1, 3, 4) being not yet supposed to be equal ; and then require that the six 

 following values of (a, j3, 7, 8), 



(1, 2, 3, 4), (2, 1, 3, 4), (1, 2, 4, 3), (2, 1, 4, 3), (1, 3, 4, 2), (3, 1, 4, 2), 



shall not be all unequal ; we must either make some supposition, such as (1, 2, 3, 4) 

 = (1, 2, 4, 3), which conducts to the one-valued form (I. III.), or else must 

 make some supposition, such as (1, 2, 3, 4) = (2, 1, 4, 3), which conducts to the 



