On the Algebraic Resolution of Equations of the Fifth Degree. 147 



"Such a conclusion is of course inadmissible. M. Hermite's 

 argument is given, with developments, in his " Considerations 

 sur la Resolution Algebrique de ^equation du 5 degre." See 

 pp. 326-336 of vol. i. of the Nouvelles Annates de Mathematiques 

 (par MM. Terquem et Gerono), 1842/' 



I propose to examine the argument in question. 



2. Denoting the function 



x + i n x x + i 2n x 2 -f- c 3n x 3 4- 1 4 % by f(t n ), 

 and consequently* 



x + * n x x + t 3n x 2 + i? n x 3 + t n x 4 by f(i 4n ), 

 we see that the 5x5 terms which compose the product 



will collapse into 



+ {t n +L^{x' h X l+k ) } 



2 indicating, in each case, the sum of all the terms which arise on 

 putting h successively equal to 0, 1, 2, 3, 4, and x 5 , x G being 

 such as to take the places of x 0) x x respectively in a cycle of the 

 roots arranged in the order : x , x v # 2 , x 3 , x 4 . 



3. Here we at once recognize M. Hermite's functions u and v ; 

 the former of which has for its general term x h x l+h , the latter 

 %h %2+h> Thus u and v- are linked together in the equation 



f( L n ) /( t *») = @2 + [t n -f t 4n )u + (^ + t?»)v, . . (e) 

 as will be seen on writing, in accordance with the notation of my 

 'Essay on the Resolution of Equations j*/ ©2 for X(x h )' 2 . 



4. We can, too, without the aid of this equation, prove the 

 truth of what he says respecting the rationality of u + v. 



For 



u + v = Z(x h X 1 + h ) +2(x h X 2+h ) ', 



and since any one of the ten terms which enter into the two func- 

 tions characterized by 2 differs from all the rest in both groups, 



5x4. 

 it is clear that all the - — - distinct combinations of the assigned 



1 X A 



form are exactly comprised in the expression for u-\-v. Hence 

 u + v, being a symmetrical function of the roots of the equation 

 in x, must admit of being expressed in rational terms of the co- 

 efficients Aj, A 2 , . . A 5 . 



* Observing that 



l 2(4n) = c 5n+3n ) t 3(4n) ==l 10«+2n j t 4(-l?i) = tl6«+«. 



f Published by Taylor and Francis, Red Lion Court, Fleet Street, London. 



L2 



