respecting Equations of the Fifth Degree. 197 



(ffl) 



every one of the w -\- 1 functions t ' and u (m) , and therefore also 



•• /3,,... 



(m) (b») \ (m — 1) ^ (in — I) 



every one of the n ■\- t functions^ and b („) , to which they are 



respectively equal, and which have been shown to contain, among them, all the 

 radicals of orders lower than m, must be a root of some such new equation, 

 although the degree r will not in general be the same for all. Treating these 

 new equations and functions, and the radicals of the order m — 1, as the equa- 



s ^ (m) 



tion X -\- &c. = 0, the function b , and the radicals of the order m have been 

 already treated ; we obtain a new system of relations, analogous to those already 

 found, and capable of being thus denoted : 



(m— 1) (m— 2) (m— 1) 



T, = C. 



I 



(m— 1) 

 (m-1) a - im-2) 



(m-l) '(-n-Z) 



U =0 



(m— 1) (m— 1) 



j3,,... A.--- 



And so proceeding, we come at last to a system of the form, 



t/ = jc, o,', . . . t'= c o' ; 



n n n 



in which the coefficient c is different from zero, and is a rational function of 



i 



the n original quantities a,, ... a ; while x' is a rational function of the s roots 



n •' 



X, , . . . j; of that equation of the «** degree in x which it has been supposed that 



(ffl) 

 b satisfies. We have therefore the expression 



T.' 



/ < 



a = — ; 



which enables us to consider every radical o', of the first order, as a rational 

 function f' of the s roots x,, . . .x^, and of the n original quantities a,, . . . a^ : 

 so that we may write 



