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



F ', obtained by changing in all possible ways the arrangement of the three roots 

 X , a;,, iCj, and therefore must be a divisor of the product 1.2.3 = 6. 



Now by the properties of rational functions of three variables, (of which an 

 investigation shall soon be given, but which it is convenient merely to enunciate 

 here, that the course of the main argument may not be too much interrupted,) 

 no three- valued function of three arbitrary quantities x^, x^,x^, can have a 

 symmetric cube ; and the only two-valued functions, which have symmetric 

 squares, are of the form 



b (iT, - x^) (x^ - X,) (x, - X,), 



b being a symmetric but otherwise arbitrary multiplier. We must therefore 

 suppose, that the radical a/ is a square-root, and that it may be thus expressed : 



«.' = F,' (^., X,, X3) = b{x^- xj (x^ - X,) (x, - X3) 



= V{P (^. - xy (a;. - x,y {x, - x.f} 



= ^/{b' (a.^ a/ - 4 a>3 - 4 a/ + I80, o, a, - 27 O3')} 



= /- 108 6* (c/-c/), 



b being here rational with respect to a^, a^, ag, as also are c, and c^, which last 



have the same meanings here as in the second article ; so that the function f is 



of the form, 



f(a^,a^,a,)=-108b'{c^'-c,'). 



No other radical of the first order, a/, can enter into the sought irreducible 

 expression b "* ; because the same reasoning would give 



aj = c{x^- xj (», - x^) (x, - x,) = y o/, 



c being rational with respect to a,, a,, a^, so that the radical aj would be super- 

 fluous. On the other hand, no expression of the form b^ + S, 0/ can represent 

 the three-valued function x ; we must therefore suppose that if the sought ex- 

 pression b '" exist at all, it is, at lowest, of the second order, and involves at least 



