156 -f ^'o/ Gayley, On a theorem of Abel's [March 11, 



" Si une equation du cinquieme degre dont les coefficients sont 

 des nomhres rationnels est resoluble algebriquement, on peut don- 

 ner aux racines la forme suivante : 



oil 



a = w + n v'(l + e') + VL^^ (1 + e' + V(l + e'))], 



a^ = m-n V(l + ^ + V[/^ (1 + e' - V(l + «'))], 



a, = m + Ji V(l + e') - \/[A (1 + e' + V(l + e'))], 



a^ = m-7zV(l + e')-V'[^^(l + e'-\/(l+e'))], 



J. = ^ + i:'a + K"a^ + ^'"aa,, ^1, = K^ K'a^ + /r'a3 + K"'a^a^, 



A^ = K+ K'a^+ K"a + K"'aa^, A^ = K + K'a^ + K"a^ + K"'a^a,. 



Les quantites c, h, e, m, n, K, K', K", K'" sont des nombres rationnels. 

 Mais de cette maniere I'dquation x^ + ax+h = Q n'est pas resoluble 

 tant que a et 6 sont des quantites quelconques. J'ai trouve de 

 pareils theoremes pour les equations du 7^'™®, 11^™®, 13^™% etc. degre." 



It is easy to see tbat x is the root of a quintic equation, the 

 coefficients of which are rational and integral functions of a, a^,a^,a^: 

 these coefficients are not symmetrical functions of a, a^, a^, a^, but 

 they are functions which remain unaltered by the cyclical change 

 a into a^, a^ into a^, a^ into a^, a^ into a. But the coefficients of 

 the quintic equation must be rational functions of c, h, e, m, n, 

 K, K', K'\ K"'\ hence regarding a, a^, a^, ^^ as the roots of a 

 quartic equation, (the coefficients of this equation being rational 

 functions of m, n, e, h) this equation must be such that every 

 rational function of the roots, unchangeable by the aforesaid cy- 

 clical change of the roots, shall be rationally expressible in terms 

 of these quantities m, n,e,h: or, what is the same thing, the group 

 of the quartic equation (using the term " group of the equation " 

 in the sense assigned to it by Galois) must be aa^a^f,^, a^a^a^a, 

 a^a^aa^, a^aa^a^. And conversely, the quartic equation being of this 

 form, X will be the root of a quintic equation, the coefficients whereof 

 are rational and integral functions of c, h, e, m, n, K, K' , K", K'". 



To investigate the form of a quartic equation having the pro- 

 perty just referred to, let it be proposed to find 7, 7' functions of 

 e, h, such that 7^ + 7'^ is a rational function of e, h, but that 7^ — 7'*^, 

 77' are rational multiples of the same quadric radical Jd. Assume 

 that we have 



7^ _ 7'2 = 2pJ~e, yy' = qjd; then (^ + 7")' = 4 (/ + q'jd ; 



