of the Fifth Degree. 53 



52. Hence, if 



^i=R(^„^,), h=ne„ee)> l3=R(^3.^5), 



we may form the cubic 



the coefficients of which will be rational functions of x^. 



53. But, even were the six values of 6 independent, we could, 

 by introducing the suffixes in any three pairs, construct a cor- 

 responding cubic 



ex. qr. let 



then, using a notation analogous to that of art. 30, 



54. Consequently it is upon the special forms of the functions 

 E that the important question of the solvability of the equation 

 in 6 turns. The cubic function |'^ + «f^+ ... is always a 

 divisor of the 15-thic function fi5^5D\f'4+ ... as the 

 cubic function p + «P+ ...is of the 15-thic function f^H- 

 50,^"*+ ...; and the coefficients D\, D'g, . . . are symmetric 

 functions of d\ as D„ Dg, . . . are of 6 : but Dj, Dg, . . . are also 

 rational functions of the coefficients of the given trinomial. 



55. Let 



(©; + 07)' - (01 - ©i" )'-(©.'- ©i') (©'.' - ® i") 



= 5^(w^ —v^—w, = 5^77. 



56. Then, expressing ©4 as linear functions of ©j, we find 



50'40'; = 3©','07 + 527?, 

 5©'X'=3©;©i'-5V 



57. Consequently 



5^6'4=-(527?-3.52.d,)2 

 and 



554 = 7? — 3SiJ or ^4 = ?; — fi?i, 



if we replace 55 by t. And (g) becomes 



E-i-m5 + «5-5V + i;)V4 = (J) 



58. But (see arts. 44-46) 77 is a rational function of d,. 

 Hence ^4 is a rational function of •&,, and /4of ^,. 



59. I infer therefore that the trinomial 



r' — 5ax'^ + a^ = 



