respecting Equations of the Fifth Degree. 185 



ders not higher than k, and not involving the radical a . . Now if we were to 



suppose that, for any system of values of the remaining radicals, the coefficients 



G, , . . . should all be = 0, or indeed if even the last of those coefficients should 

 thus vanish, we should then have a new equation of condition, namely the fol- 

 lowing : 



(*— 1) 



A =0' 



which would be obliged to be compatible with the equations of definition of the 

 remaining radicals, and would therefore either conduct at last, by a repetition of 

 the same analysis, to a radical essentially vanishing, and consequently superfluous, 

 among those which have been supposed to enter into the composition of the 



(m) 



function h ; or else would bring us back to the divisibility of an equation of 

 definition by an equation of condition, of the form just now assigned, and with 



coefficients g^ , . . . g which would not all be = 0. But for this purpose it would 



be necessary that a relation, or system of relations, should exist, (or at least 

 should be compatible with the remaining equations of definition,) of the form 



(*) (A) e 



%-e = -^e «.• ' 



(A) 



e being less than a. , and v^ being diffijrent from zero, and being a root of a 



numerical equation ; and because a. is prime, we could find integer numbers \ 

 and IX, which would satisfy the condition 



A a^ — /i e =: 1 ; 



(A) 



SO that, finally, we should have an expression for the radical a. , as a rational 



function of others of the same system, and of orders not higher than its own, 



though multiplied in general (as was above announced) by a root of a numerical 



equation ; namely the following expression : 



(*) M (k)—y. (k-i)\ 

 a. z=zi/ G f 



I e g—e -^i 



And if we should suppose this last equation to be not identically true, but only 



2 c 2 



