respecting Equations of the Fifth Degree. 195 



(m — 1) \(m — I) 



U (m) <n,-) = i (m) (m) , 



/3. ..../3^(„) A ,...^^(„) 



if we put, for abridgment, 





and 



(m) (m) 



In this manner we obtain in general n -\-t equations, in each of which the 

 product of certain powers, (with positive, negative, or null exponents,) of the 



< terms of the development of the irrational function b , is equated to 



^ (m— 1) ^ (m— 1) 



some other irrational function, f or b , of an order lower than m. 



Indeed, it is to be observed, that since these various equations are obtained 



by an elimination of the n radicals of highest order, between their n " 

 equations of definition and the t "* expressions for the t " terms of the de- 

 velopment of b , they cannot be equivalent to more than t "" distinct rela- 

 tions. But, among them, they must involve explicitly all the radicals of lower 



orders, which enter into the composition of the irreducible function b . For if 

 any radical a , of order lower than m, were wanting in all the n -\- t 



i 



functions of the forms 



(m) 



we might then employ instead of the old system of radicals a, , ... of the 



V (m) 



order m, a new and equally numerous system of radicals a, , ■ ■ ■ according to 

 the following type, 



(m) 



according to the formula 



and might then express all the t terms of .6 , by means of these new radicals. 



