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



and multiply, by it, both numerator and denominator of the recently assigned 



(™) . . 



expression for b , we obtam this new expression for that general irrational 



function. 



(«r)-(«r) ^^"^ 



( W\ / W\ 



the characteristic i denoting here some function, which, relatively to the radical 



(ot) 



a , is whole, so that it may be thus developed, 



i 



(m) / (m}\ (m) (m)' (ffl)r 



=i(a ) = io-|-ija -\-ha -\- . . .-\-i a , 



^ i ' i i r i 



r being a finite positive integer, and the coefficients !„, i,, . . . i being, in 



r 



general, functions of the m order, but not involving the radical a . And 



i 



because the definition of that radical gives 



a = a 



if 



it is unnecessary to retain in evidence any of its powers of which the exponents 



are not L 

 the form 



(m) (m) 



are not less than a ; we may therefore put the development of b under 



(m) , W , , / (<»).»^"^-l 



i 



the coefficients h^, h„ . . . being still, in general, functions of the w'* order, 

 not involving the radical a^. . It is clear that by a repetition of this process of 



