1878.] relating to a quintiG equation. 157 



and in order that 7^+ 7'^ may be rational, we must have p^+ (f= \^0, 



or say ^^+ q^= h^6; hence, ^=p + p i^ust be a sum of two squares, 



or, assuming one of these equal to unity and the other of them 

 equal to e^, say 6 = l+e^, we satisfy the required equation by 

 taking p — h, q = he: viz. we thus have 



7' - y" = 2 Vl + e\ 77 = hejl + e\ y' + j" = 2h (1 + e') ; 

 and thence also 



ry'=.h{l + e' + ^l+7), y" = h(l+e'-jT+V), 

 the roots of these expressions, or val.ues of 7, 7', being such that 



yy =hejl + e\ 



Taking now a rational, = m suppose, and /? a rational multiple 

 of^l+e^, =hjl + e^ suppose, it is easy to see that the quartic 

 equation which has for its roots 



a,a^,a^,a^=cf. + ^ + y, a - /3 + 7', a + /3 - 7, a- 13- y , 



has the property in question, viz. that every rational function of 

 the roots unchangeable by the cyclical change a into a^, a^ into 

 ttg, a^ into ftg, a.g into a, is rationally expressible in terms of 

 e, h, m, n. 



It will be sufficient to give the proof in the case of a rational 

 and integral function ; such a function, unchangeable as aforesaid, 

 is of the form 



^ (a, a^, a^, a^ + ^ [a^, a^, a^, a) + (j) {a^, a^, a, aj + ^ {a^, a, a^,a^) ; 



and if (f) (a, a^, a^, a^ contains a term o.^^'"'y^y"^, then the other 

 three functions will contain respectively the terms cC"'{—^y'y^{—yy, 

 a'"yS"(-7)'^(-7')^ a"'(-/3)" (-7')" (7)'; viz. the sum of the four 

 terms is 



= a'yg" [{1 + i-Y^n] 7V* + {{-y^n + {-y^n] 7VI. 



This obviously vanishes unless j9 and g are both even, or both 

 odd; and the cases to be considered are 1"; n even, j!) and q even, 

 2°; n odd, p and q even, 8"; n even,p and q odd, 4"-; n odd, p and q odd. 

 Writing, for greater distinctness, 2n or 2n + 1 for n, according as n 

 is even or odd, and similarly forp and q, the term is in the four 

 cases respectively 



