212 MATHEMATICS, ETC. 
the function F being unchanged by this permutation, and the values of » being 
the roots of y” = F, we have 
CM Cad ORO ea ols cc ) = a) OG Gig Bas a5dad) 
a being a [definite] mth root of unity.” 
“Tf we now interchange 7,, £, the above becomes 
B @as Bog Bao 200000) = AO Gay Bay Bag 200005 Ke 
whence, by multiplying in order, we have a2 =1. This result proves that the 
number 7, supposed prime, is necessarily 2, so that the first radical which presents 
itself in the value of the unknown must be of the second degree. This is what, in 
fact, happens in those equations which we know how to resolve.” 
The function @ having only two values, changes by any transposition whatever, 
and will not be changed by a circular permutation of three or five letters, for 
such permutations are equivalent to an even number of transpositions. (Prop. Ils 
III). Let us continue the series of operations indicated to form the value x, of . 
“ Combining the first radical with the coefficients of 7 («) = 0, (or the function @ 
with symmetrical functions of the roots) by means of the fir-t operations of algebra, 
we obtaia thus a function of the roots, susceptible only of two values, and, consequent 
ly, invariable for circular permutations of three letters. (Prop. III.) The suc- 
ceeding radicals may furnish more functions of the same kind if of the second degree, 
Suppose that we have come to a radical, for which the equivalent rational function 
is not invariable for these permutations. Denoting it by y = ¢ (a1, %, #3, .«--) 
then in the equation y” = p, we shall still have p = F (#,, 7,3, ....) but this 
function will no longer be symmetrical, but only invariable for circular permu- 
tations of three letters. If in @ we replace 2), t,, 75, by xy, #3, 1, the relation 
o” = F will still subsist; and, since F does not change by the substitution, we 
shall have 
Play egy ae args Alin = CPN Caiy as Cut eign tes wanl)s 
a being a [definite] mth root of unity.” 
“ Performing in this equation, once and again, the cireular substitution 
Das Cg, Cy, we have 
GS CBs Oa pane) eae eee n ON (oer cenie ari Hr ule) 
a) (Bas @a> Pan Cae RON CAME oe enn 2 eo) 
and, multiplying the three equations, we obtain a3 —1. Thus 7 is 3.” 
“If the number of the quantities 2,, w, 3, %4, ..». is greater than 4, or if 
the equation f (x) 0 is of a higher degree than the fourth, we can perform 
* Mr. Cockle (Phil. Mag. 1859, p. 510), remarks that this step “tacitly assumes the whole 
question, viz., that the surdis a quadratic. The only legitimate inference from 
@ (Wo; %1) vane) = OO: (2q; Borers) 1G (Ey; Tas-.-s) — alo Gost, ee) 
where g—! is the inverse of g,’’ Mr. Cockle appears to us to have misconceived Wantzel’s 
reasoning which recalls that “ every relation among the roots must be an identity,” and we 
are therefore entitled to permute the roots in any way in such a relation as the one above. 
Mr. Cockle further alludes to some objection brought by Sir W. R. Hamilton, against the ° 
validity of Abel’s proof, that every radical is a rational function of the roots. We have not 
been able to discover where Sir W. R. Hamilton’s strictures are to be found, and certainly 
can detect no flaw in the demonstration of the above in Serret’s work. 
