MATHEMATICS, ETC. 211 
If, now, the general value of the root 2, of an equation f (x) = 0 of the mth 
degree, can be expressed in algebraic functions of the co-efficients, let the above 
form be assumed for it and be substituted in the equation. The result will be of 
the form 
1 2 n-21 
A + Bon + Opn + aasapebsireiese +p = 0) 
where A, B,.... L, are rational functions of p, a, 8, ....A; and it is shown 
that this requires 
A= 0,B=0)... 0.502. Lb = 0; 
whence it follows that the above expression for x will still satisfy the equation 
a u 
when pz is replaced by tp”,tbeing any xth root of unity. We thus obtain 
m quantities, which are roots of the proposed equation,* and it is thence easily 
1 
proved that all the quantities p”, a, 8, .... A, are rational functions of the roots. 
By a similar investigation it follows that any other function which enters into any 
1 
of the quantities pz, a, 8..... A, put under the assumed general form, i: also a 
rational function of the roots; and hence :t is concluded generally that 
Prop. IV. —/f an equation is algebraically resolvible, we can give to the root 
such a form that all the algebraic functions of which it is composed are rational 
Functions of the roots of the equution. 
_ We now proceed to Abel’s demonstration as modified by Wantzel, the inverted 
commas indicating, according to Serret, the text of Wantzel’s memoir. 
Let f («) =0 be an equation of the mth degree with arbitrary co-efficients, 
and let its m roots be denoted by z,.%,,.... %m, and let us suppose them 
capable of being expressed as algebraic functions of the co-efficients. 
“Tf the equation f (x) = 0, is satisfied by the value 2,, of x, whatever be the 
co-efficients, we ought to reproduce «, identically by substituting in its expression 
the rational function [of the roots] corresponding to each radical involved in that. 
expression. Also, the roots being wholly arbitrary, every [apparent] relation 
between them must be in reality an identity, and will not cease to subsist when 
we exchange the roots one among the other in any way whatever.” 
“Let y denote the first radical following the order of calculation [7.¢., a radica 
1 
of the first order with index, m prime] which enters into the value of x,, and 
let y* = p; then p depends directly on the co-efticients of f(x) = 0, and will be 
expressed by a symmetrical function of the roots F (x1, #9, 3, --++ J;-y Will be 
a rational function, ¢ (7,, %2, 3, ....) also of the roots. (Prop. IV).” 
“Since the function ¢ is not symmetrical, (for if it were, the nth root of p would 
be exactly extracted), it ought to change when two of the roots, 7), 2g, for in- 
stance, are permuted; but the relation ¢x = F will always be satisfied. Then 
*Serret remarks that all these roots are different, but his proof of this is curiously 
erroneous ; still it is otherwise easy to see that such must be the case. He adds, however 
“Au surplus, cette remarque n’est pas indispensable pour ce qui va suivre.” 
