of the Fifth Degree. 201 
to f be so assigned that 2; may disappear from 7,, in other words 
that 7,(7;) may be a symmetric function, say 7,(0), of a We 
may form the cubic 
V3 —r(0)V? +79(#5) V —73(25) =0, 
the roots of which will be the above three values of V. 
89. If x, does not disappear from 7, and r;, the determination 
of those functions depends upon the solution of a quintic, and 
cannot be attained. If it does disappear, the cubic becomes 
V3—r,(0)V?+7,(0) V —7,(0) =0. 
90. In the latter case, since 7(0) is not affected by any inter- 
change of the 2’s, of the fifteen values of V,,, three only will be 
distinct. But (art. 62) this involves the relation 
¥ Var = Va, 
which is equivalent, for some finite value of n, to 
6? = 67. 
91. But no such relation exists among the roots of the sextic 
in 9, and no such cubic can be formed the coefficients of which 
shall be symmetric functions of 2; and since the & of Mr. Jer- 
rard and my @ are similar functions ( fonctions semblables), | am 
constrained to conclude that the supposed cubic of that eminent 
algebraist cannot be formed, and that the supposition that the 
general quintic is soluble by an Abelian sextic involves the untenable 
supposition that the sextic in @ has equal roots, or roots some inte- 
gral powers of which are equal. 
92. In perfect accordance with this conclusion is that dedu- 
cible from the symmetric product, (9), of the sextic in 0. We 
find, by substitution in the formula which I have already* given, 
that 
r7(0) = }(108Q5— B9)*(5'°R), 
and that when 7 vanishes and cubic radicals appear, the sextic 
and the given quintic have each equal roots. 
93. So, too, although I have succeeded in obtaining unsym- 
metric functions of 6 which are symmetric in z, and therefore 
known, the doctrine of similar functions shows that these known 
quantities can only be applied to the solution of the sextic through 
the medium of a quintic. 
94. The § and « roots of the 15-ic in y can be obtained, or 
at all events verified, by a process resembling that employed in 
art. 72 for the other roots. But 
B,+83+ 84+ Bs=2QE—38Q?x,? + 2Kz,° 
is the type of the formule of verification ; and 8 and « are the 
* Phil. Mag. May 1858, p. 390 
