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



for the solution of a cubic equation : it is therefore impossible to discover any 

 new irrational expression, finite and irreducible, for a root of the general cubic, 

 essentially distinct from those which have long been known ; and the only pos- 

 sible difference, with respect to the extracting of radicals, between any two 

 methods of solution which both are free from all superfluous extractions, consists 

 in the introduction of different square factors into that quantity or function y^, of 

 which, in each, the square root a/ is to be calculated ; or in the introduction of 

 different cubic factors into that other quantity or function y/, of which, in each 

 method, it is requisite to calculate the cube-root a/'. It is proper, however, to 

 remember the remarks which have been made, in a foregoing article, respecting 

 the reducibility of a certain expression, involving two cubic radicals a," and a^', 

 which is not uncommonly assigned for a root of the cubic equation. 



[15.] But it is necessary to demonstrate some properties of rational functions 

 of three variables, which have been employed in the foregoing investigation. 

 And because it will be necessary to investigate afterwards some analogous pro- 

 perties of functions of four and five arbitrary quantities, it may be conducive to 

 clearness and uniformity that we should begin with a few remarks respecting 

 functions which involve two variables only. 



Let F {x^, Xq) denote any arbitrary rational function of two arbitrary quan- 

 tities A\, oTj, arranged in either of their only two possible arrangements ; so that 

 the function f admits of the two following values 



F (or,, x^) and f (x^, a;,), 



which for conciseness may be thus denoted, 



(l,2)and(2, 1). 



These different values of the proposed function f may also be considered as 

 being themselves two different Junctions of the same two quantities or, x^ taken 

 in some determined order ; and may, in this view, be denoted thus, 



F,(a:., :rj andF,(^,,^,), 

 or, more concisely, 



(1,2). and (1,2),: 



