230 ■ Sir William R. Hamilton on the Argument of Ahel, 



words, that the function b is irrational, with respect to the quantities a^, a^, Oj, 04, 

 if any expression of the required kind can be found at all for x. On the other hand, 



the general theorem of Abel shows that if any such expression h exist, it must 

 be composed of some finite combination of quadratic and cubic radicals, together 

 with rational functions ; because 2 and 3 are the only prime divisors of the 

 product 24 = 1.2.3.4. And the first and only radical of the first order in 



b , must be a square-root, of the form 



«/ = b {x\ - x^) (x, - ^3) {x, - X,) {x., - x^) {x^ - x;) (.r, - x,) 



- v/- 442368. 6l(e,^-e/) = /- 2". 3^ 61 (e,^ - O. 



6 being some symmetric function of x^, x.^, x^, x^, and e^, e^ having the same 



meanings here as in the second article ; because no rational and unsymmetrlc 

 function of four arbitrary quantities Xi, x.^, Xj,x^, has a prime power symmetric, 

 except either this function a/, or else some other such as a.^ which may be de- 



duced from it by a multiplication such as the following, a^ = -a,'. But a two- 

 valued expression of the form f =^b,,-\-biai cannot represent a four-valued 



function, such as x; we must therefore suppose that the sought expression b " 

 contains a radical a," of the second order, and this must be a cube-root, of the 

 form 



«i" = (Po + Pi «/) («i + /'a' ^2 4- p, u,) = ■^ (&„ + 6j O ; 

 in which, p^ is, as before, an imaginary cube-root of unity ; p„, p^, b^, b^ are 

 symmetric relatively to iT,, x^, x.^, x^, or rational relatively to a„ a^, a.^, a^; 



u^ ^^ Xj ajj ~\- x^ IC4, U2 ^^ Xy X3 -J— X2 X4, M3 ^^ x^ x^ — |— x.^ Xj f 

 and - 



6. -f b, 0/ = 1728 (p„ -f p, a,'y \e,-\-^, (p,^ - Pa) y ^ 



the rational function e^, and the radical a,' retaining their recent meanings : 

 because no rational function f," of four independent variables a;,, x^, x^, x^, 

 which cannot be reduced to the form thus assigned for a,", can have itself 

 2 a," values, a/' being a prime number greater than 1, if the number of 



values of the prime power f,""' be only 2. Nor can any other radical 



