186 Sir William R, Hamilton on the Argument of Abel, 



to hold good for some systems of values of the remaining radicals, of orders not 

 higher than k, we should still obtain, at least, an equation of condition between 



those remaining radicals, by raising the expression just found for a. to the power 



(*) 

 a. ; namely, the following equation of condition, 



(*) 



f. — U G /. ) • = 0, 



which might then be treated like the former, till at last an expression should be 

 obtained, of the kind above announced, for at least one of the remaining radicals. 

 In every case, therefore, we should be conducted to a diminution of the number 



of prime roots of variables in the expression of the function h , which conse- 

 quently would not be irreducible. 



For example, if an irrational function of the w'* order contain any radical 



a^ of the cubic form, its exponent a,- being z: 3, and its equation of definition 



being of the form 



if also the other equations of definition permit us to suppose that this radical may 

 be equal to some rational function of the rest, so that an equation of the form 



(m) (m) 



a 



(in which the function g, does not contain the radical a^ ,) \s compatible with 

 the equation of definition 



()b)3 (»>— 1) 



o, -f. =0; 



then, from the forms of these two last mentioned equations, the latter must be 

 divisible by the former, at least for some suitable system of values of the remain- 

 ing radicals : and therefore the following relation, which does not involve the 



radical Cj , namely, 



