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



W . . - 



perceive tliat the component radicals a. , which enter into the composition of 



this irreducible function, must not be subject to, nor even compatible with, any 



equations or equation of condition whatever, except only the equations of defi- 



. W 



nition, which determine those radicals a. , by determining their prime powers 



(*) a . 



a- . For the existence or possibility of any such equation of condition 

 in conjunction with those equations of definition, would enable us to express 

 at least one of the above mentioned radicals as a rational function of others of 

 the same system, and of orders not higher than its own, or even, perhaps, as 

 a rational function of the original variables a,, . . . a , though multiplied 



in general by a root of a numerical equation ; and therefore would enable us to 



diminish the number of extractions of prime roots of functions, which would be 



inconsistent with the irreducibility supposed. 



In fact, if any such equation of condition, involving any radical or radicals of 



the order k, but none of any higher order, were compatible with the equations 



of definition ; then, by some obvious preparations, such as bringing the equation 



of condition to the form of zero equated to some finite polynomial function of 



(*) 

 some radical a of the k"' order ; and rejecting, by the methods of equal roots 



and of the greatest common measure, all factors of this polynome, except those 

 which are unequal among themselves, and are included among the factors of 

 that other polynome which is equated to zero in the corresponding form of 



. (*) 

 the equation of definition of the radical a. ; we should find that this last equa- 

 tion of definition 



(i) o. ^ (A— 1) 



must be divisible, either identically, or at least for some suitable system of 

 values of the remaining radicals, by an equation of condition of the form 



wg , m (k)g-i (k) (k) (k) 



a. + G, a. -[-... 4- Q , a. -+- g = : 



» ' « ' ' g—l ' s 



W . (*) (*) 



g being less than o. , and the coefficients Gi , . . o being functions of or- 



