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



a = f' (x,, ... X , Op ... a ). 



t • s n 



But before arriving at the last mentioned system of relations, another system of 

 the form 



T," = c,' a,', . . . t" = c' _ a\^ 



n n n ' 



must have been found, in which the coefficient c' is different from zero, and is 



> 



a rational function of a,', ... a' and of a,, ... a , while t" is a rational function 



n" n j 



of X, , . . . x ; we have therefore the expression 



i '^. 



and we see that every radical of the second order also is equal to a rational 

 function of Xj, . . . x and of o,, . . . o : so that we may write 



( n 



a" = v" (x, .. . X, a, . . .a). 



i i I s I n 



And re-ascending thus, through orders higher and higher, we find, finally, 



by similar reasonings, that every one of the n' -{-n"-\-...-\-n -j- ...-{- w radi- 

 cals which enter into the composition of the irrational and irreducible function 



(m) . (*) . (k) 



b , such as the radical a , must be expressible as a rational function f of 



« i 



the roots a;,, . . . x, and of the original quantities a^, . . . a : so that we have a 



> n 



complete system of expressions, for all these radicals, which are included in the 

 general formula 



(*) (A), ^ 



a = F (Xj, . . . X , o,, . . . o ). 



> < s n 



Thus, in the case of the cubic equation and the function b", when we have 

 arrived at the relation 



h — /i > , 

 in which 



^i" = :J- (^1 + P3 ^2 + p3 'I'a). and// = c, -f o/, 



we find that the rational function 



