respecting Equations of the Fifth Degree. 189 



any one system of values of the radicals on which it depends, be equal to any one 

 root of any equation of the form 



s t 1 



in which the coefficients a,, . , . a^ are any rational functions of the n original 

 quantities a,, . . . a„ ; in such a manner that for some one system of values of the 

 radicals a,', &c., the equation 



1 (m) s (m) s — 1 



h +A,6 +... + A, = 



is satisfied : then the same equation must be satisfied, also, for all systems of 

 values of those radicals, consistent with their equations of definition. It is an 

 immediate consequence of this result, that all the values of the function which 



has already been denoted by the symbol h („) („) must represent roots of the 



same equation of the s'* degree ; and the same principles show that all these 



W 

 values of h ^^^ must be unequal among themselves, and therefore must represent 



s 



so many different roots x^, x^, . ■ . of the same equation x -f- &c. rr 0, if every 

 index or exponent 7 be restricted, as before, to denote either zero or some 



2 



positive Integer number less than the corresponding exponent a : for if, with 



i 



this restriction, any two of the values of b could be supposed equal, an 



equation of condition between the radicals c, , &c. would arise, which would 



be inconsistent with the supposed irreducibility of the function b . 

 For example, having found that the cubic equation 



.r^ -|- a, a;^ + a2 a: 4" «3 == 



is satisfied by the irrational and irreducible function b" above assigned,' we can 



infer that the same equation is satisfied by all the three values &„", 6/', b.^' of the 



function 6" ; and that these three values must be all unequal among themselves, 

 r." , 



