458 Mr. G. B. Jerrard on the possibility of solving Equations 



Supposing « to be any root of the binomial equation a? — 1 = 0, 

 and yfrx to be defined by 



^x=(x + a9x + a*0' 2 x+ .. +uV- l 0l 1 - l xy, ■ ■ (1) 

 he states, as the first proposition to be proved, that tyx, which is 

 obviously a rational function of x, must further admit of being 

 expressed rationally by the coefficients of §x and 6x. He then 

 substitutes m x for x in the expression for tyx, and combining 

 the equation 



e"+ v x=o v x 



with 



u4-v v 

 u r =u , 



he shows very clearly that 



y}rO m x=fx; 

 and thence 



fx=-{fx + yjr0x + ylr0 2 x+ .. +f0' l - 1 x}: . (2) 



from which he infers that tyx will be a rational and symmetric 

 function of all the roots of the equation (j)X = 0, and will therefore 

 be expressible rationally in known quantities. It is this inference 

 the truth of which has been contested. But his meaning has 

 here, as we shall see, been misapprehended. It may be briefly 

 explained thus. The expression for tyx, which, when considered 

 as a function of the coefficients of <px and Ox, may take the form 

 M + M r r + M 2 tf 2 + .. j-M^.!^- '(wherein M , M v M 2 ,.. M^_, 

 are certain rational functions of the two sets of coefficients in ques- 

 tion), will, in virtue of equation (2), be subject to the condition 



M + M i; r + M 2 # 2 + . . +M u _,^- ] = 



1 



-{/*M o +M 1 @(l) + M 8 ©(2)+..M M _ 1 @0*-l)} 



if 



&(n)=x n +(0x) n +{0*x) n + . . +{0p-Kt)"; 



and will consequently become M . For, since the proposed 

 equation <f>x = is irreducible, the quantities Mj, M 2 , ..M^_, 

 must separately vanish. 



§3. 



It is evident that, except in the case of /* = 2, M v Mg, . . M^_, 

 will not vanish of themselves, independently of the particular 

 form of the proposed equation cf>x = 0. 



If, for instance, we take /a = 3, remembering that the roots of 



theequationa 3 -l=0arel,-2 + 2^/~3 J _ 2 - 2^ / ~^' we 



