RESEARCHES IN THE HIGHER ALGEBRA, 113 



§ 44- 

 Now there are two conditions essential to the applica- 

 tion of these results to the theory of quintics : 



1°. The symbol 7 must be the root of an equation 



with known coefficients. 

 2°. The symbols D must be known. 



§ 45- 

 The first will be attained if we assume 



Xi = Wi + ax.2 + bx^ + cXi + dwr,, 

 for, the interchanges of x.2, x^, x^ and ^5 being equivalent 

 to corresponding interchanges of a, b, c and d, the expres- 

 sion 7 will be symmetric with respect to those four roots 

 and^ therefore, a rational function of the remaining root 

 x^, and the root of a determinate equation of the fifth 

 degree. 



§46. 



It will be remembered that 6 is symmetric in X^ but 

 that all is otherwise arbitrary. Hence, in order to attain 

 the second condition, we may give to 6 any form consistent 

 with that symmetry. One mode of seeking to attain it is 

 by giving a maximum of symmetry to 6 with respect to x, 

 in other words, by endeavouring to construct a Symmetric 

 Product. We are thus led to the sextic in 6. 



§ 47- 

 These results justify the form which Mr. Harley gave 

 to the factors of the resolvent product, and by which he 

 has so greatly simplified my discussion (compare Phil. 

 Mag. December 1852 and March 1853 ^i*^ ^i^ Memoir), 

 Viewed in the light of the present Memoir, the Method 

 of Symmetric Products has (beyond invoking the aid of 

 interchanges) no special relation to other methods in the 

 theory of equations. If we suppose the resolvent product 



SER. III. VOL. I. Q 



