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



functions of lower orders ; and if these two cube-roots should enter only by 

 their product, so that no analysis of the foregoing kind could obtain them other- 



(m — 1) (m) (m) 



wise than in connexion, and under the form c Ui a^ \ it would then have 



been sufficient, in effecting the sjmthesis of 6 , to have calculated only the cube- 

 Cm) 3 (m)3 C"— 1) (™— 1) ^(m— 1) 



root of the product a, a^ =^ Jl :=/ , instead of calcu- 



(m)3 (m— 1) (7b)3 



lating separately the cube-roots of its two factors, a, =^ , and a^ 



(m— 1) „ . . 



= /^ : the number of extractions of prime roots of variables might, there- 



.... . . W . 



fore, have been diminished m the calculation of the function b , which would 



be inconsistent with the irreducibility of that function. 



In the cases of the irreducible functions b', b", b'", b"', which have been 



above assigned, as representing roots of the general quadratic, cubic, and 



biquadratic equations, the theorem of the present article is seen at once to hold 



good ; because in these the radicals of highest order are themselves terms of 



the developments in question, the coefficients of their first powers being already 



equal to unity. Thus in the development of b', we have a/ = ^/ ; in b", we 



have a," = t." ; in b'", we have a'" = t'" , and a'" = t'" ; and in 6 ^ we 



havea/''=i{/''. 



[9.] By raising to the proper powers the general expressions of the form 



(m) (OT-1) (m) 



T = c a , 



i i i 



we obtain a system of 'nr'"^ equations of this other form 



(m)a/'"'' (m-l)af'"'' (m-1) „' (m-\) 



T. = C. • / =/ 



• t t ( 



^ (m— 1) 



/ being some new irrational function, of an order lower than m ; and by 

 combining the same expressions with those which define the various terms 



(m) (m) 



t (^-^ , the number of which terms we shall denote by the symbol t , we ob- 

 P\ I • • • 



(m) 



tain another system of ^ equations, of which the following is a type, 



