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



degree ; and by applying the principles of the same argument, to show that no 

 expression of the same kind exists for any root of any general but lower equa- 

 tion, (quadratic, cubic, or biquadratic,) essentially distinct from those which 

 have long been known ; is the chief object of the present paper. 



[3.] In general, if we call an Irrational function irreducible, when it is im- 

 possible to express that function, or any one of its component radicals, by any 

 smaller number of extractions of prime roots of variables, than the number which 

 the actual expression of that function or radical Involves ; even by Introducing 

 roots of constant quantities, or of numerical equations, which roots are in this 

 whole discussion considered as being themselves constant quantities, so that they 

 neither Influence the order of an irrational function, nor are included among 

 the radicals denoted by the symbols a,', &c. ; then it is not difficult to prove 

 that such irreducible irrational functions possess several properties in common, 

 which are adapted to assist in deciding the question just now stated. 



In the first place it may be observed, that, by an easy preparation, the 



(«) 

 general Irrational function b of any order m may be put under the form 



(m— 1) . „ . „ , 



in which the coefficient i („) („) is a function of the order tn — 1, or of a 



1 nV™) 



(>») . ... 



lower order ; the exponent /3 is zero, or any positive integer less than the prime 

 number a which enters as exponent into the equation of definition of the 



(m) 



radical a , namely, 



a 



W 



(m) a. Am—\) 



• *^ r 



t 



•^ t 



(m) (m) (m) 



and the sign of summation extends to all the a . a^ ... a terms which have 



n 

 (m) 



exponents ^ ^ subject to the condition just now mentioned. 



