x^n - p1x^(n - 1) + ... = x - a.x - b. &c.
or the equivalent equations p1 = [Sigma]a, &c., we find
a^n - p1a^(n - 1) + ... = 0,
b^n - p1b^(n - 1) + ... = 0;
. . .
. . .
. . .
(it is as satisfying these equations that a, b ... are said to be the
roots of x^n - p1x^(n - 1) + ... = 0); and conversely from the
last-mentioned equations, assuming that a, b ... are all different, we
deduce
p1 = [Sigma]a, p2 = [Sigma]ab, &c.
and
x^n - p1x^(n - 1) + ... = x - a.x - b. &c.
Observe that if, for instance, a = b, then the equations a^n - p1a^(n -
1) + ... = 0, b^n - p1b^(n - 1) + ... = 0 would reduce themselves to a
single relation, which would not of itself express that a was a double
root,--that is, that (x - a)^2 was a factor of x^n - p1x^(n - 1) +, &c;
but by considering b as the limit of a + h, h indefinitely small, we
obtain a second equation
na^(n - 1) - (n - 1)p1a^(n - 2) + ... = 0,
which, with the first, expresses that a is a double root; and then the
whole system of equations leads as before to the equations p1 =
[Sigma]a, &c. But the existence of a double root implies a certain
relation between the coefficients; the general case is when the roots
are all unequal.
We have then the _theorem_ that every rational symmetrical function of
the roots is a rational function of the coefficients. This is an easy
consequence from the less general theorem, every rational and integral
symmetrical function of the roots is a rational and integral function of
the coefficients.
In particular, the sums of the powers [Sigma]a^2, [Sigma]a^3, &c., are
rational and integral functions of the coefficients.
The process originally employed for the expression of other functions
[Sigma]a^[alpha] b^[beta], &c., in terms of the coefficients is to
make them depend upon the sums of powers: for instance,
[Sigma]a^[alpha] b^[beta] = [Sigma]a^[alpha] [Sigma]a^[beta] -
[Sigma]a^([alpha] + [beta]); but this is very objectionable; the true
theory consists in showing that we have systems of equations
p1 = [Sigma]a,
p2 = [Sigma]ab,
p1^2 = [Sigma]a^2 + 2[Sigma]ab,
p3 = [Sigma]abc,
p1p2 = [Sigma]a^2 b + 3[Sigma]abc,
p1^3 = [Sigma]a^3 + 3[Sigma]a^2 b + 6[Sigma]abc,
where in each system there are precisely as many equations as there
are root-functions on the right-hand side--e.g. 3 equations and 3
functions [Sigma]abc, [Sigma]a^2 b, [Sigma]a^3. Hence in each system
the root-functions can be determined linearly in terms of the powers
and products of the coefficients:
[Sigma]ab = p2,
[Sigma]a^2 = p1^2 - 2p2,
[Sigma]abc = p3,
[Sigma]a^2 b = p1p2 - 3p3,
[Sigma]a^3 = p1^3 - 3p1p2 + 3p3,
and so on. The other process, if applied consistently, would derive
the originally assumed value [Sigma]ab = p2, from the two equations
[Sigma]a = p, [Sigma]a^2 = p1^2 - 2p2; i.e. we have 2[Sigma]ab =
[Sigma]a.[Sigma]a - [Sigma]a^2,= p1^2 - (p1^2 - 2p2), = 2p2.
13. It is convenient to mention here the theorem that, x being
determined as above by an equation of the order n, any rational and
integral function whatever of x, or more generally any rational function
which does not become infinite in virtue of the equation itself, can be
expressed as a rational and integral function of x, of the order n - 1,
the coefficients being rational functions of the coefficients of the
equation. Thus the equation gives x^n a function of the form in
question; multiplying each side by x, and on the right-hand side writing
for x^n its foregoing value, we have x^(n + 1), a function of the form
in question; and the like for any higher power of x, and therefore also
for any rational and integral function of x. The proof in the case of a
rational non-integral function is somewhat more complicated. The final
result is of the form [phi](x)/[psi](x) = I(x), or say [phi](x)
-[psi](x)I(x) = 0, where [phi], [psi], I are rational and integral
functions; in other words, this equation, being true if only [f](x) = 0,
can only be so by reason that the left-hand side contains [f](x) as a
factor, or we must have identically [phi](x) - [psi](x)I(x) =
M(x)[f](x). And it is, moreover, clear that the equation
[phi](x)/[psi](x) = I(x), being satisfied if only [f](x) = 0, must be
satisfied by each root of the equation.
From the theorem that a rational symmetrical function of the roots is
expressible in terms of the coefficients, it at once follows that it
is possible to determine an equation (of an assignable order) having
for its roots the several values of any given (unsymmetrical) function
of the roots of the given equation. For example, in the case of a
quartic equation, roots (a, b, c, d), it is possible to find an
equation having the roots ab, ac, ad, bc, bd, cd (being therefore a
sextic equation): viz. in the product
(y - ab)(y - ac)(y - ad)(y - bc)(y - bd)(y - cd)
the coefficients of the several powers of y will be symmetrical
functions of a, b, c, d and therefore rational and integral functions
of the coefficients of the quartic equation; hence, supposing the
product so expressed, and equating it to zero, we have the required
sextic equation. In the same manner can be found the sextic equation
having the roots (a - b)^2, (a - c)^2, (a - d)^2, (b - c)^2, (b -
d)^2, (c - d)^2, which is the equation of differences previously
referred to; and similarly we obtain the equation of differences for a
given equation of any order. Again, the equation sought for may be
that having for its n roots the given rational functions [phi](a),
[phi](b), ... of the several roots of the given equation. Any such
rational function can (as was shown) be expressed as a rational and
integral function of the order n - 1; and, retaining x in place of any
one of the roots, the problem is to find y from the equations x^n - p1
x^(n - 1) ... = 0, and y = M0x^(n - 1) + M1x^(n - 2) + ..., or, what
is the same thing, from these two equations to eliminate x. This is in
fact E.W. Tschirnhausen's transformation (1683).
14. In connexion with what precedes, the question arises as to the
number of values (obtained by permutations of the roots) of given
unsymmetrical functions of the roots, or say of a given set of letters:
for instance, with roots or letters (a, b, c, d) as before, how many
values are there of the function ab + cd, or better, how many functions
are there of this form? The answer is 3, viz. ab + cd, ac + bd, ad + bc;
or again we may ask whether, in the case of a given number of letters,
there exist functions with a given number of values, 3-valued, 4-valued
functions, &c.
It is at once seen that for any given number of letters there exist
2-valued functions; the product of the differences of the letters is
such a function; however the letters are interchanged, it alters only
its sign; or say the two values are [Delta] and -[Delta]. And if P, Q
are symmetrical functions of the letters, then the general form of
such a function is P + Q[Delta]; this has only the two values P +
Q[Delta], P - Q[Delta].
In the case of 4 letters there exist (as appears above) 3-valued
functions: but in the case of 5 letters there does not exist any
3-valued or 4-valued function; and the only 5-valued functions are
those which are symmetrical in regard to four of the letters, and can
thus be expressed in terms of one letter and of symmetrical functions
of all the letters. These last theorems present themselves in the
demonstration of the non-existence of a solution of a quintic equation
by radicals.
The theory is an extensive and important one, depending on the notions
of _substitutions_ and of _groups_ (q.v.).
15. Returning to equations, we have the very important theorem that,
given the value of any unsymmetrical function of the roots, e.g. in the
case of a quartic equation, the function ab + cd, it is in general
possible to determine rationally the value of any similar function, such
as (a + b)^3 + (c + d)^3.
The _a priori_ ground of this theorem may be illustrated by means of a
numerical equation. Suppose that the roots of a quartic equation are
1, 2, 3, 4, then if it is given that ab + cd = 14, this in effect
determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a = 1, b = 2 or a
= 2, b = 1, and c = 3, d = 4 or c = 3, d = 4) or else a, b to be 3, 4
and c, d to be 1, 2; and it therefore in effect determines (a + b)^3 +
(c + d)^3 to be = 370, and not any other value; that is, (a + b)^3 + (c
+ d)^3, as having a single value, must be determinable rationally. And
we can in the same way account for cases of failure as regards
particular equations; thus, the roots being 1, 2, 3, 4 as before, a^2 b
= 2 determines a to be = 1 and b to be = 2, but if the roots had been
1, 2, 4, 16 then a^2 b = 16 does not uniquely determine a, b but only
makes them to be 1, 16 or 2, 4 respectively.
As to the _a posteriori_ proof, assume, for instance,
t1 = ab + cd, y1 = (a + b)^3 + (c + d)^3,
t2 = ac + bd, y2 = (a + c)^3 + (b + d)^3,
t3 = ad + bc, y3 = (a + d)^3 + (b + c)^3:
then y1 + y2 + y3, t1y1 + t2y2 + t3y3, t1^2 y1 + t2^2y2 + t3^2y3 will
be respectively symmetrical functions of the roots of the quartic, and
therefore rational and integral functions of the coefficients; that
is, they will be known.
Suppose for a moment that t1, t2, t3 are all known; then the equations
being linear in y1, y2, y3 these can be expressed rationally in terms
of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be
known. But observe further that y1 is obtained as a function of t1,
t2, t3 symmetrical as regards t2, t3; it can therefore be expressed as
a rational function of t1 and of t2 + t3, t2t3, and thence as a
rational function of t1 and of t1 + t2 + t3, t1t2 + t1t3 + t2t3,
t1t2t3; but these last are symmetrical functions of the roots, and as
such they are expressible rationally in terms of the coefficients;
that is, y1 will be expressed as a rational function of t1 and of the
coefficients; or t1 (alone, not t2 or t3) being known, y1 will be
rationally determined.
16. We now consider the question of the algebraical solution of
equations, or, more accurately, that of the _solution of equations by
radicals_.
In the case of a quadric equation x^2 - px + q = 0, we can by the
assistance of the sign [root]( ) or ( )^1/2 find an expression for x
as a 2-valued function of the coefficients p, q such that substituting
this value in the equation, the equation is thereby identically
satisfied; it has been found that this expression is
x = 1/2{p [+-] [root](p^2 - 4q)},
and the equation is on this account said to be algebraically solvable,
or more accurately solvable by radicals. Or we may by writing x =
-1/2 p + z reduce the equation to z^2 = 1/4(p^2 - 4q), viz. to an
equation of the form x^2 = a; and in virtue of its being thus
reducible we say that the original equation is solvable by radicals.
And the question for an equation of any higher order, say of the order
n, is, can we by means of radicals (that is, by aid of the sign [root
m]( ) or ( )^(1/m), using as many as we please of such signs and with
any values of m) find an n-valued function (or any function) of the
coefficients which substituted for x in the equation shall satisfy it
identically?
It will be observed that the coefficients p, q ... are not explicitly
considered as numbers, but even if they do denote numbers, the
question whether a numerical equation admits of solution by radicals
is wholly unconnected with the before-mentioned theorem of the
existence of the n roots of such an equation. It does not even follow
that in the case of a numerical equation solvable by radicals the
algebraical solution gives the numerical solution, but this requires
explanation. Consider first a numerical quadric equation with
imaginary coefficients. In the formula x = 1/2{p [+-] [root](p^2 -
4q)}, substituting for p, q their given numerical values, we obtain
for x an expression of the form x = [alpha] + [beta]i [+-]
[root]([gamma] + [delta]i), where [alpha], [beta], [gamma], [delta]
are real numbers. This expression substituted for x in the quadric
equation would satisfy it identically, and it is thus an algebraical
solution; but there is no obvious _a priori_ reason why
[root]([gamma]+[delta]i) should have a value = c + di, where c and d
are real numbers calculable by the extraction of a root or roots of
real numbers; however the case is (what there was no _a priori_ right
to expect) that [root]([gamma] + [delta]i) has such a value calculable
by means of the radical expressions [root]{[root]([gamma]^2 +
[delta]^2) [+-] [gamma]} : and hence the algebraical solution of a
numerical quadric equation does in every case give the numerical
solution. The case of a numerical cubic equation will be considered
presently.