[ 63 ] 



XIII. Reflections on the Resolution of Algebraic Equations of 



the Fifth Degree. By G. B. Jerrard. 



[Continued from vol. xxvi. p. 574.] 



47. rriHE remarks in No. 44-. related to a difficulty which 



-I- must arise if we can, as seems to have been proved, 



succeed in tracing the equation for W to a class of equations 



of the sixth degree, the solution of which can be effected. I 



have lately reconsidered the subject of that number, and the 



exact nature of the difficulty in question will, I think, appear 



from what follows. 



Since every symmetric function of the quantities Vi, V K » 

 V L will be such as to remain unchanged whilst one of the 

 roots, x v continues fixed, and a* 2 , X& x^ x 5 are permuted in 

 every possible way among themselves, it might easily be shown 

 that the equation for V, 



V 15 + C 1 V 14 + .. + C 15 = 0, 

 will admit of being resolved into five factors of the form 



V 3 +J- c x V 2 + t % fo) v + r 9 M = o, 



obtained by writing 1, 2, 3, 4, 5 successively for a: r 2 and r 3 

 being expressive of rational functions, and such that r n (x a ) 

 shall essentially involve x a . In this equation, therefore, we 

 cannot generally write r n (0) instead of r n {x a ). 



But the equation for V will evidently lead to twenty-five ex- 

 pressions for the five roots x {i # 2 , . . x 5i obtainable from a sy- 

 stem of functions of x a ; 



*.W ¥*(*„)» .. ¥ 8 (* fc )% 

 The question therefore suggests itself: Is it permitted, since 

 the number of distinct values of ' x cannot exceed 5, to suppose 

 that 



«% M = Vn (0), 



or that the five roots, x v X& . . .r 5 , without considering in 

 what order they will arise, may be expressed by 



%(0), ¥ 8 (0), ..^ 5 (0); 



and thus to avoid the conclusion that the equation of the third 

 degree, at which we shall arrive, will,, in the ordinary meaning 

 of the term, be simultaneous with V 15 +C!V 14 + .. +C 15 =0? 



In fine, if we can effect the resolution of algebraic equations 

 of the fifth degree, it must be possible to withdraw the terms 

 involving x a from M* (x a ) considered throughout its extent, 

 although we retain those which involve x a in r (x„). 



London, December 13, 1845. 



* See (31.). 



