1. In the subject "Theory of Equations" the term _equation_ is used to
denote an equation of the form x^n - p1x^(n - 1) ... [+-] p_n = 0, where
p1, p2 ... p_n are regarded as known, and x as a quantity to be
determined; for shortness the equation is written [f](x) = 0.
The equation may be _numerical_; that is, the coefficients p1, p2^n, ...
p_n are then numbers--understanding by number a quantity of the form
[alpha] + [beta]i ([alpha] and [beta] having any positive or negative
real values whatever, or say each of these is regarded as susceptible of
continuous variation from an indefinitely large negative to an
indefinitely large positive value), and i denoting [root]-1.
Or the equation may be _algebraical_; that is, the coefficients are not
then restricted to denote, or are not explicitly considered as denoting,
numbers.
1. We consider first numerical equations. (Real theory, 2-6; Imaginary
theory, 7-10.)
_Real Theory_.
2. Postponing all consideration of imaginaries, we take in the first
instance the coefficients to be real, and attend only to the real roots
(if any); that is, p1, p2, ... p_n are real positive or negative
quantities, and a root a, if it exists, is a positive or negative
quantity such that a^n - p1a^(n - 1) ... [+-] p_n = 0, or say, [f](a) =
0.
It is very useful to consider the curve y = [f](x),--or, what would come
to the same, the curve Ay = [f](x),--but it is better to retain the
first-mentioned form of equation, drawing, if need be, the ordinate y on
a reduced scale. For instance, if the given equation be x^3 - 6x^2 + 11x
-6.06 = 0,[1] then the curve y = x^3 - 6x^2 + 11x - 6.06 is as shown in
fig. 1, without any reduction of scale for the ordinate.
It is clear that, in general, y is a continuous one-valued function of
x, finite for every finite value of x, but becoming infinite when x is
infinite; i.e., assuming throughout that the coefficient of x^n is +1,
then when x = [oo], y = +[oo]; but when x = -[oo], then y = +[oo] or
-[oo], according as n is even or odd; the curve cuts any line whatever,
and in particular it cuts the axis (of x) in at most n points; and the
value of x, at any point of intersection with the axis, is a root of the
equation [f](x) = 0.
If [beta], [alpha] are any two values of x ([alpha] > [beta], that is,
[alpha] nearer +[oo]), then if [f]([beta]), [f]([alpha]) have opposite
signs, the curve cuts the axis an odd number of times, and therefore at
least once, between the points x = [beta], x = [alpha]; but if
[f]([beta]), [f]([alpha]) have the same sign, then between these points
the curve cuts the axis an even number of times, or it may be not at
all. That is, [f]([beta]), [f]([alpha]) having opposite signs, there are
between the limits [beta], [alpha] an odd number of real roots, and
therefore at least one real root; but [f]([beta]), [f]([alpha]) having
the same sign, there are between these limits an even number of real
roots, or it may be there is no real root. In particular, by giving to
[beta], [alpha] the values -[oo], +[oo] (or, what is the same thing, any
two values sufficiently near to these values respectively) it appears
that an equation of an odd order has always an odd number of real roots,
and therefore at least one real root; but that an equation of an even
order has an even number of real roots, or it may be no real root.
If [alpha] be such that for x = or > a (that is, x nearer to +[oo])
[f](x) is always +, and [beta] be such that for x = or < [beta] (that
is, x nearer to -[oo]) [f](x) is always -, then the real roots (if any)
lie between these limits x = [beta], x = [alpha]; and it is easy to find
by trial such two limits including between them all the real roots (if
any).
3. Suppose that the positive value [delta] is an inferior limit to the
difference between two real roots of the equation; or rather (since the
foregoing expression would imply the existence of real roots) suppose
that there are not two real roots such that their difference taken
positively is = or < [delta]; then, [gamma] being any value whatever,
there is clearly at most one real root between the limits [gamma] and
[gamma] + [delta]; and by what precedes there is such real root or there
is not such real root, according as [f]([gamma]), [f]([gamma] + [delta])
have opposite signs or have the same sign. And by dividing in this
manner the interval [beta] to [alpha] into intervals each of which is =
or < [delta], we should not only ascertain the number of the real roots
(if any), but we should also separate the real roots, that is, find for
each of them limits [gamma], [gamma] + [delta] between which there lies
this one, and only this one, real root.
In particular cases it is frequently possible to ascertain the number
of the real roots, and to effect their separation by trial or
otherwise, without much difficulty; but the foregoing was the general
process as employed by Joseph Louis Lagrange even in the second
edition (1808) of the _Traite de la resolution des equations
numeriques_;[2] the determination of the limit [delta] had to be
effected by means of the "equation of differences" or equation of the
order 1/2 n(n - 1), the roots of which are the squares of the
differences of the roots of the given equation, and the process is a
cumbrous and unsatisfactory one.
4. The great step was effected by the theorem of J.C.F. Sturm
(1835)--viz. here starting from the function [f](x), and its first
derived function [f]'(x), we have (by a process which is a slight
modification of that for obtaining the greatest common measure of these
two functions) to form a series of functions
[f](x), [f]'(x), [f]2(x), ... [f]_n(x)
of the degrees n, n - 1, n - 2 ... 0 respectively,--the last term
[f]_n(x) being thus an absolute constant. These lead to the immediate
determination of the number of real roots (if any) between any two given
limits [beta], [alpha]; viz. supposing [alpha] > [beta] (that is,
[alpha] nearer to +[oo]), then substituting successively these two
values in the series of functions, and attending only to the signs of
the resulting values, the number of the changes of sign lost in passing
from [beta] to [alpha] is the required number of real roots between the
two limits. In particular, taking [beta], [alpha] = -[oo], +[oo]
respectively, the signs of the several functions depend merely on the
signs of the terms which contain the highest powers of x, and are seen
by inspection, and the theorem thus gives at once the whole number of
real roots.
And although theoretically, in order to complete by a finite number of
operations the separation of the real roots, we still need to know the
value of the before-mentioned limit [delta]; yet in any given case the
separation may be effected by a limited number of repetitions of the
process. The practical difficulty is when two or more roots are very
near to each other. Suppose, for instance, that the theorem shows that
there are two roots between 0 and 10; by giving to x the values 1, 2, 3,
... successively, it might appear that the two roots were between 5 and
6; then again that they were between 5.3 and 5.4, then between 5.34 and
5.35, and so on until we arrive at a separation; say it appears that
between 5.346 and 5.347 there is one root, and between 5.348 and 5.349
the other root. But in the case in question [delta] would have a very
small value, such as .002, and even supposing this value known, the
direct application of the first-mentioned process would be still more
laborious.
5. Supposing the separation once effected, the determination of the
single real root which lies between the two given limits may be effected
to any required degree of approximation either by the processes of W.G.
Horner and Lagrange (which are in principle a carrying out of the method
of Sturm's theorem), or by the process of Sir Isaac Newton, as perfected
by Joseph Fourier (which requires to be separately considered).
First as to Horner and Lagrange. We know that between the limits
[beta], [alpha] there lies one, and only one, real root of the
equation; [f]([beta]) and [f]([alpha]) have therefore opposite signs.
Suppose any intermediate value is [theta]; in order to determine by
Sturm's theorem whether the root lies between [beta], [theta], or
between [theta], [alpha], it would be quite unnecessary to calculate
the signs of [f]([theta]),[f]'([theta]), [f]2([theta]) ...; only the
sign of [f]([theta]) is required; for, if this has the same sign as
[f]([beta]), then the root is between [beta], [theta]; if the same
sign as [f]([alpha]), then the root is between [theta], [alpha]. We
want to make [theta] increase from the inferior limit [beta], at which
[f]([theta]) has the sign of [f]([beta]), so long as [f]([theta])
retains this sign, and then to a value for which it assumes the
opposite sign; we have thus two nearer limits of the required root,
and the process may be repeated indefinitely.
Horner's method (1819) gives the root as a decimal, figure by figure;
thus if the equation be known to have one real root between 0 and 10,
it is in effect shown say that 5 is too small (that is, the root is
between 5 and 6); next that 5.4 is too small (that is, the root is
between 5.4 and 5.5); and so on to any number of decimals. Each figure
is obtained, _not_ by the successive trial of all the figures which
precede it, but (as in the ordinary process of the extraction of a
square root, which is in fact Horner's process applied to this
particular case) it is given presumptively as the first figure of a
quotient; such value may be too large, and then the next inferior
integer must be tried instead of it, or it may require to be further
diminished. And it is to be remarked that the process not only gives
the approximate value [alpha] of the root, but (as in the extraction
of a square root) it includes the calculation of the function
[f]([alpha]), which should be, and approximately is, = 0. The
arrangement of the calculations is very elegant, and forms an integral
part of the actual method. It is to be observed that after a certain
number of decimal places have been obtained, a good many more can be
found by a mere division. It is in the progress tacitly assumed that
the roots have been first separated.
Lagrange's method (1767) gives the root as a continued fraction a +
1/b + 1/c + ..., where a is a positive or negative integer (which may
be = 0), but b, c, ... are positive integers. Suppose the roots have
been separated; then (by trial if need be of consecutive integer
values) the limits may be made to be consecutive integer numbers: say
they are a, a + 1; the value of x is therefore = a + 1/y, where y is
positive and greater than 1; from the given equation for x, writing
therein x = a + 1/y, we form an equation of the same order for y, and
this equation will have one, and only one, positive root greater than
1; hence finding for it the limits b, b + 1 (where b is = or > 1), we
have y = b + 1/z, where z is positive and greater than 1; and so
on--that is, we thus obtain the successive denominators b, c, d ... of
the continued fraction. The method is theoretically very elegant, but
the disadvantage is that it gives the result in the form of a
continued fraction, which for the most part must ultimately be
converted into a decimal. There is one advantage in the method, that a
commensurable root (that is, a root equal to a rational fraction) is
found accurately, since, when such root exists, the continued fraction
terminates.
6. Newton's method (1711), as perfected by Fourier(1831), may be
roughly stated as follows. If x = [gamma] be an approximate value of
any root, and [gamma] + h the correct value, then [f]([gamma] + h) =
0, that is,
h h^2
[f]([gamma]) + -- [f]'([gamma]) + --- [f]"([gamma]) + ... = 0;
1 1.2
and then, if h be so small that the terms after the second may be
neglected, [f]([gamma]) + h[f]'([gamma]) = 0, that is, h =
{-[f]([gamma])/[f]'([gamma])}, or the new approximate value is x =
[gamma] - {[f]([gamma])/[f]'([gamma])}; and so on, as often as we
please. It will be observed that so far nothing has been assumed as to
the separation of the roots, or even as to the existence of a real
root; [gamma] has been taken as the approximate value of a root, but
no precise meaning has been attached to this expression. The question
arises, What are the conditions to be satisfied by [gamma] in order
that the process may by successive repetitions actually lead to a
certain real root of the equation; or that, [gamma] being an
approximate value of a certain real root, the new value [gamma] -
{[f]([gamma])/[f]'([gamma])} may be a more approximate value.
[Illustration: FIG. 1.]
Referring to fig. 1, it is easy to see that if OC represent the
assumed value [gamma], then, drawing the ordinate CP to meet the curve
in P, and the tangent PC' to meet the axis in C', we shall have OC' as
the new approximate value of the root. But observe that there is here
a real root OX, and that the curve beyond X is convex to the axis;
under these conditions the point C' is nearer to X than was C; and,
starting with C' instead of C, and proceeding in like manner to draw a
new ordinate and tangent, and so on as often as we please, we
approximate continually, and that with great rapidity, to the true
value OX. But if C had been taken on the other side of X, where the
curve is concave to the axis, the new point C' might or might not be
nearer to X than was the point C; and in this case the method, if it
succeeds at all, does so by accident only, i.e. it may happen that C'
or some subsequent point comes to be a point C, such that CO is a
_proper_ approximate value of the root, and then the subsequent
approximations proceed in the same manner as if this value had been
assumed in the first instance, all the preceding work being wasted. It
thus appears that for the proper application of the method we require
_more_ than the mere separation of the roots. In order to be able to
approximate to a certain root [alpha], =OX, we require to know that,
between OX and some value ON, the curve is always convex to the axis
(analytically, between the two values, [f](x) and [f]"(x) must have
always the same sign). When this is so, the point C may be taken
anywhere on the proper side of X, and within the portion XN of the
axis; and the process is then the one already explained. The
approximation is in general a very rapid one. If we know for the
required root OX the two limits OM, ON such that from M to X the curve
is always _concave_ to the axis, while from X to N it is always convex
to the axis,--then, taking D anywhere in the portion MX and (as
before) C in the portion XN, drawing the ordinates DQ, CP, and joining
the points P, Q by a line which meets the axis in D', also
constructing the point C' by means of the tangent at P as before, we
have for the required root the new limits OD', OC'; and proceeding in
like manner with the points D', C', and so on as often as we please,
we obtain at each step two limits approximating more and more nearly
to the required root OX. The process as to the point D', translated
into analysis, is the ordinate process of interpolation. Suppose OD =
[beta], OC = [alpha], we have approximately [f]([beta] + h) =
[f]([beta]) + h{[f]([alpha]) - [f]([beta])} / ([alpha] - [beta]),
whence if the root is [beta] + h then h = - ([alpha] -
[beta])[f]([beta]) / {[f]([alpha]) - [f]([beta])}.
Returning for a moment to Horner's method, it may be remarked that the
correction h, to an approximate value [alpha], is therein found as a
quotient the same or such as the quotient [f]([alpha]) / [f]'([alpha])
which presents itself in Newton's method. The difference is that with
Horner the integer part of this quotient is taken as the presumptive
value of h, and the figure is verified at each step. With Newton the
quotient itself, developed to the proper number of decimal places, is
taken as the value of h; if too many decimals are taken, there would
be a waste of work; but the error would correct itself at the next
step. Of course the calculation should be conducted without any such
waste of work.
_Imaginary Theory_.
7. It will be recollected that the expression _number_ and the
correlative epithet _numerical_ were at the outset used in a wide
sense, as extending to imaginaries. This extension arises out of the
theory of equations by a process analogous to that by which number, in
its original most restricted sense of positive integer number, was
extended to have the meaning of a real positive or negative magnitude
susceptible of continuous variation.
If for a moment number is understood in its most restricted sense as
meaning positive integer number, the solution of a simple equation leads
to an extension; ax - b = 0 gives x = (b/a), a positive fraction, and we
can in this manner represent, not accurately, but as nearly as we
please, any positive magnitude whatever; so an equation ax + b = 0 gives
x = -(b/a), which (approximately as before) represents any negative
magnitude. We thus arrive at the extended signification of number as a
continuously varying positive or negative magnitude. Such numbers may be
added or subtracted, multiplied or divided one by another, and the
result is always a number. Now from a quadric equation we derive, in
like manner, the notion of a complex or imaginary number such as is
spoken of above. The equation x^2 + 1 = 0 is not (in the foregoing
sense, number = real number) satisfied by any numerical value whatever
of x; but we assume that there is a number which we call i, satisfying
the equation i^2 + 1 = 0, and then taking a and b any real numbers, we
form an expression such as a + bi, and use the expression number in this
extended sense: any two such numbers may be added or subtracted,
multiplied or divided one by the other, and the result is always a
number. And if we consider first a quadric equation x^2 + px + q = 0
where p and q are real numbers, and next the like equation, where p and
q are any numbers whatever, it can be shown that there exists for x a
numerical value which satisfies the equation; or, in other words, it can
be shown that the equation has a numerical root. The like theorem, in
fact, holds good for an equation of any order whatever; but suppose for
a moment that this was not the case; say that there was a cubic equation
x^3 + px^2 + qx + r = 0, with numerical coefficients, not satisfied by
any numerical value of x, we should have to establish a new imaginary j
satisfying some such equation, and should then have to consider numbers
of the form a + bj, or perhaps a + bj + cj^2 (a, b, c numbers [alpha] +
[beta]i of the kind heretofore considered),--first we should be thrown
back on the quadric equation x^2 + px + q = 0, p and q being now numbers
of the last-mentioned extended form--_non constat_ that every such
equation has a numerical root--and if not, we might be led to _other_
imaginaries k, l, &c., and so on _ad infinitum_ in inextricable
confusion.
But in fact a numerical equation of any order whatever has always a
numerical root, and thus numbers (in the foregoing sense, number =
quantity of the form [alpha] + [beta]i) form (_what real numbers do
not_) a universe complete in itself, such that starting in it we are
never led out of it. There may very well be, and perhaps are, numbers in
a more general sense of the term (quaternions are not a case in point,
as the ordinary laws of combination are not adhered to), but in order to
have to do with such numbers (if any) we must start with them.
8. The capital theorem as regards numerical equations thus is, every
numerical equation has a numerical root; or for shortness (the meaning
being as before), every equation has a root. Of course the theorem is
the reverse of self-evident, and it requires proof; but provisionally
assuming it as true, we derive from it the general theory of numerical
equations. As the term root was introduced in the course of an
explanation, it will be convenient to give here the formal definition.
A number a such that substituted for x it makes the function x1^n -
p1x^(n - 1) ... [+-]p_n to be = 0, or say such that it satisfies the
equation [f](x) = 0, is said to be a root of the equation; that is, a
being a root, we have
a^n - p1a^(n - 1) ... [+-]p_n = 0, or say [f](a) = 0;
and it is then easily shown that x - a is a factor of the function
[f](x), viz. that we have [f](x) = (x - a)[f]1(x), where [f]1(x) is a
function x^(n - 1) - q1x^(n - 2) ... [+-]q_(n - 1) of the order n - 1,
with numerical coefficients q1, q2 ... q_(n - 1).
In general a is not a root of the equation [f]1(x) = 0, but it may be
so--i.e. [f]1(x) may contain the factor x - a; when this is so, [f](x)
will contain the factor (x - a)^2; writing then [f](x) = (x -
a)^2[f]2(x), and assuming that a is not a root of the equation [f]2(x)
= 0, x = a is then said to be a double root of the equation [f](x) =
0; and similarly [f](x) may contain the factor (x - a)^3 and no higher
power, and x = a is then a triple root; and so on.
Supposing in general that [f](x) = (x - a)^[alpha] F(x) ([alpha] being
a positive integer which may be = 1, (x - a)^[alpha] the highest power
of x - a which divides [f](x), and F(x) being of course of the order n
- [alpha]), then the equation F(x) = 0 will have a root b which will
be different from a; x - b will be a factor, in general a simple one,
but it may be a multiple one, of F(x), and [f](x) will in this case be
= (x - a)^[alpha] (x - b)^[beta] [Phi](x) ([beta] a positive integer
which may be = 1, (x-b)^[beta] the highest power of x - b in F(x) or
[f](x), and [Phi](x) being of course of the order n - [alpha] -
[beta]). The original equation [f](x) = 0 is in this case said to have
[alpha] roots each = a, [beta] roots each = b; and so on for any other
factors (x - c)^[gamma], &c.
We have thus the _theorem_--A numerical equation of the order n has in
every case n roots, viz. there exist n numbers, a, b, ... (in general
all distinct, but which may arrange themselves in any sets of equal
values), such that [f](x) = (x - a)(x - b)(x - c) ... identically.
If the equation has equal roots, these can in general be determined,
and the case is at any rate a special one which may be in the first
instance excluded from consideration. It is, therefore, in general
assumed that the equation [f](x) = 0 has all its roots unequal.
If the coefficients p1, p2, ... are all or any one or more of them
imaginary, then the equation [f](x) = 0, separating the real and
imaginary parts thereof, may be written F(x) + i[Phi](x) = 0, where
F(x), [Phi](x) are each of them a function with real coefficients; and
it thus appears that the equation [f](x) = 0, with imaginary
coefficients, has not in general any real root; supposing it to have a
real root a, this must be at once a root of each of the equations F(x)
= 0 and [Phi](x) = 0.
But an equation with real coefficients may have as well imaginary as
real roots, and we have further the _theorem_ that for any such
equation the imaginary roots enter in pairs, viz. [alpha] + [beta]i
being a root, then [alpha] - [beta]i will be also a root. It follows
that if the order be odd, there is always an odd number of real roots,
and therefore at least one real root.
9. In the case of an equation with real coefficients, the question of
the existence of real roots, and of their separation, has been already
considered. In the general case of an equation with imaginary (it may be
real) coefficients, the like question arises as to the situation of the
(real or imaginary) roots; thus, if for facility of conception we regard
the constituents [alpha], [beta] of a root [alpha] + [beta]i as the
co-ordinates of a point _in plano_, and accordingly represent the root
by such point, then drawing in the plane any closed curve or "contour,"
the question is how many roots lie within such contour.
This is solved theoretically by means of a theorem of A.L. Cauchy
(1837), viz. writing in the original equation x + iy in place of x,
the function [f](x + iy) becomes = P + iQ, where P and Q are each of
them a rational and integral function (with real coefficients) of (x,
y). Imagining the point (x, y) to travel along the contour, and
considering the number of changes of sign from - to + and from + to -
of the fraction corresponding to passages of the fraction through zero
(that is, to values for which P becomes = 0, disregarding those for
which Q becomes = 0), the difference of these numbers gives the number
of roots within the contour.
It is important to remark that the demonstration does not presuppose
the existence of any root; the contour may be the infinity of the
plane (such infinity regarded as a contour, or closed curve), and in
this case it can be shown (and that very easily) that the difference
of the numbers of changes of sign is = n; that is, there are within
the infinite contour, or (what is the same thing) there are in all n
roots; thus Cauchy's theorem contains really the proof of the
fundamental theorem that a numerical equation of the nth order (not
only has a numerical root, but) has precisely n roots. It would appear
that this proof of the fundamental theorem in its most complete form
is in principle identical with the last proof of K.F. Gauss (1849) of
the theorem, in the form--A numerical equation of the nth order has
always a root.[3]
But in the case of a finite contour, the actual determination of the
difference which gives the number of real roots can be effected only
in the case of a rectangular contour, by applying to each of its sides
separately a method such as that of Sturm's theorem; and thus the
actual determination ultimately depends on a method such as that of
Sturm's theorem.
Very little has been done in regard to the calculation of the
imaginary roots of an equation by approximation; and the question is
not here considered.
10. A class of numerical equations which needs to be considered is that
of the binomial equations x^n - a = 0 (a = [alpha] + [beta]i, a complex
number).
The foregoing conclusions apply, viz. there are always n roots, which,
it may be shown, are all unequal. And these can be found numerically
by the extraction of the square root, and of an nth root, of _real_
numbers, and by the aid of a table of natural sines and cosines.[4]
For writing
/ [alpha] [beta] \
[alpha] + [beta]i = [root]([alpha]^2 + [beta]^2) ( ---------------------------- + ----------------------------i ),
\[root]([alpha]^2 + [beta]^2) [root]([alpha]^2 + [beta]^2) /
there is always a real angle [lambda] (positive and less than 2[pi]),
such that its cosine and sine are = [alpha] / [root]([alpha]^2 +
[beta]^2) and [beta] / [root]([alpha]^2 + [beta]^2) respectively; that
is, writing for shortness [root]([alpha]^2 + [beta]^2) = [rho], we have
[alpha] + [beta]i = [rho](cos[lambda] + i sin[lambda]), or the
equation is x^n = [rho](cos[lambda] + i sin [lambda]); hence observing
that (cos [lambda]/n + i sin [lambda]/n )^n = cos[lambda] + i
sin[lambda], a value of x is = [root n][rho] (cos [lambda]/n + i sin
[lambda]/n). The formula really gives all the roots, for instead of
[lambda] we may write [lambda] + 2s[pi], s a positive or negative
integer, and then we have
/ [lambda] + 2s[pi] [lambda] + 2s[pi] \
x = [root n][rho] ( cos ----------------- + i sin ----------------- ),
\ n n /
which has the n values obtained by giving to s the values 0, 1, 2 ...
n - 1 in succession; the roots are, it is clear, represented by points
lying at equal intervals on a circle. But it is more convenient to
proceed somewhat differently; taking one of the roots to be [theta],
so that [theta]^n = a, then assuming x = [theta]y, the equation
becomes y^n - 1 = 0, which equation, like the original equation, has
precisely n roots (one of them being of course = 1). And the original
equation x^n - a = 0 is thus reduced to the more simple equation x^n -
1 = 0; and although the theory of this equation is included in the
preceding one, yet it is proper to state it separately.
The equation x^n - 1 = 0 has its several roots expressed in the form
1, [omega], [omega]^2, ... [omega]^(n - 1), where [omega] may be taken
= cos 2[pi]/n + i sin 2[pi]/n; in fact, [omega] having this value, any
integer power [omega]^k is = cos 2[pi]k/n + i sin 2[pi]k/n, and we
thence have ([omega]^k)^n = cos 2[pi]k + i sin 2[pi]k, = 1, that is,
[omega]^k is a root of the equation. The theory will be resumed
further on.
By what precedes, we are led to the notion (a numerical) of the
radical a^(1/n) regarded as an n-valued function; any one of these
being denoted by [root n]a, then the series of values is [root n]a,
[omega][root n]a, ... [omega]^(n - 1)[root n]a; or we may, if we
please, use [root n]a instead of a^(1/n) as a symbol to denote the
n-valued function.
As the coefficients of an algebraical equation may be numerical, all
which follows in regard to algebraical equations is (with, it may be,
some few modifications) applicable to numerical equations; and hence,
concluding for the present this subject, it will be convenient to pass
on to algebraical equations.
_Algebraical Equations._