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



function of the second order a root of the cubic equation. But the mere know- 

 ledge of the compatibility of the equation of condition 



with the equation of definition 



is sufficient to enable us to infer, from the forms of these two equations, that the 

 latter is divisible by the former, at least for some suitable system of values of the 

 remaining radicals a," and a,', consistent with their equations of definition ; and 

 therefore that the following relation 



and the expression 



3.- c. 



.'' = ^i ^ 



are at least consistent with those equations. In the present example, the relation 

 thus arrived at is found to be identically true, and consequently the radicals a^' 

 and a/' remain independent of each other ; but for the same reason, the radical 

 a^" may be changed to the expression just now given ; so that the proposed 

 function of the second order, 



may, by the mere definitions of its radicals, and even without attending to the 

 cubic equation which it was designed to satisfy, be put under the form 



the number of prime roots of variables being depressed from three to two ; and 

 consequently that proposed function was not irreducible in the sense which has 

 been already explained. 



[6.] From the foregoing properties of irrational and irreducible functions, it 

 follows easily that if any one value of any such function b^ , corresponding to 



