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



in order to depress that number below six, and thus we should still be obliged to 

 femploy one of the three conditions (d)' (d)" (d)'".) Of these twelve different 

 conditions (b)' . . (d)'", some one of which we must employ, (or at least some 

 condition not essentially different from it,) the three marked (b)' (c)' (d)' are 

 easily seen to reduce respectively the three functions (b) (c) (d) to the five- 

 valued form (a) ; they are therefore admissible, but they give no new information. 

 The supposition Ob)" conducts us to equate the function (b) to the following. 



because it allows us to interchange x^ and x^, inasmuch as s^ may previously be 

 put in the place of ^4, and Xi in the place of x^, by interchanging at the 

 same time x^ and x^, — a double interchange which does not alter the product 

 Xi — x^. . . X3 — Xt, since it only changes simultaneously the signs of the two 

 factors Xi— x^ and X3 — x^'^ or because, if we denote the function (b) by the sym- 

 bol (1, 2, 3, 4, 5), we have (1, 2, 3, 4, 5) = (2, 1, 4, 3, 5), and also, by (b)", 

 (1, 2, 3, 4, 5) = (1, 2, 3, 5, 4), so that we must have (1, 2, 3, 4, 5) = (2, 1, 4, 5, 3) 

 = (1, 2, 5, 4, 3) ; but also the condition (b)" gives (1, 2, 5, 4, 3) = (1, 2, 5, 3, 4) ; 

 we must therefore suppose (1, 2, 3, 5, 4) = (1, 2, 5, 3, 4), that is, 



\^i > ^l 3^2 • -^1 ^3 • ^1 ^i ' ^i ^3 ' "^2 '''5 • •''3 •''5/ 



— ^iyX^, X^ ' Xjj . Xj X^ , X^"^ X^, X^ X3 • X^ X^ . X3 •^5)9 



which is an equation of the form (b)', and reduces the function (b) to the form (a), 

 and ultimately to a symmetric function a, because x^ and x^ may be interchanged. 

 The supposition (b)'" conducts to a two-valued function, which changes value 

 when any two of the five roots are interchanged, so that the sum (1, 2, 3, 4, 5) 

 -|- (1, 2, 3, 5, 4), and the quotient 



(1,2, 3, 4, 5) -(1,2, 3, 5, 4) ^ 



(xi — Xi) {xi — X3)... {X4 — x^y 



are some symmetric functions, which may be called 2a and 2 J; we have there- 

 fore, in this case, a function of the form, 



(e) . . o -\-b x^—x,. x—x^ . x^—x^ . «.— «, • x,—x^ .x^ — x^. x, — x^ . x^ — x^ . 



