On the Theory of Equations of the Fifth Degree. 343 

 R(7i) = a + ai7i + 337,2 + 337,3 + 347,^, 



= A + A,^ + A2^2+..+A5^5, 

 and 



aud determiniug a, a,, . . , 84 so as to satisfy the five linears 



A, = B„ A2=B2, ..,A5=B5, 

 we are led to 



7,=B-A + R(7,). 



Consequently if B — A be zero, (1) is an Abelian, and a- is 

 given in terms of 7; and if B — A be finite, x is given in terms 

 of 6'. 



65. This result induces me to prefer* the sextic in 6 to that 

 in /, which in the first instance I sought to decompose into 



{t^-u,t + ^,){t^-u^t^^^){t^-u.J + ^^)=.Q. 



66. We find, moreover, 



-G, = S7 = S^,^2, 



where ^ is an unsymmetric function of 6 into which no one 

 value of 6 enters to more than two dimensions. 



67. But 6 is in general finite, and the relation 



G^+td,w^d^e,+^{e)=o, . . . . (m) 



which holds for .r-al interchanges of 6, can only subsist in so far 

 as it is an immediate consequence of the sextic in 6. This sextic, 

 considered as a function of 6^, I shall i-epresent by f[6y) =0. 



68. Eliminating between (ra) aud the system 



f{6,)=0, f{e,)=Q,...,f{d,) = Q, 

 we shall be led in a variety of ways to the relation 



where, as usual, R denotes a rational function. Therefore the 

 sextic isf an Abelian. 



* It would seem, however, from M. Kronecker's researches, that the 

 6ic in t is resolvable by means of au Abelian quintic, i. e. if the sextic in 

 / be an Abelian. 



t M. Ilermite (in Terquem's Annates) assumes that if a quintic be sol- 

 vable it is (lecomjiosable into (non-linear) factors, a restriction which Euler 

 (lid not admit, al. Wautzel's argument does not (it seems to me) make 

 due allowance for the occurrence of roots of unity in the expression for the 

 root. Mr. Jerrard's conclusions (in his recently published ' Essay,' &c.) 

 arc opposed to those of Abel and Sir W. Rowan Ilamilton. 



