MATHEMATICAL MONOGRAPHS.
EDITED BY
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD.
No. 10.
THE SOLUTION OF
EQUATIONS.
BY
MANSFIELD MERRIMAN,
PROFESSOR OF CIVIL ENGINEERING IN LEHIGH UNIVERSITY.
FOURTH EDITION, ENLARGED.
FIRST THOUSAND.
NEW YORK:
JOHN WILEY & SONS. 1 T ( ^
LONDON- CHAPMAN & HALL, LIMITED.
1906.
COPYRIGHT, 1896,
BY
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD
UNDER THE T.TLE
HIGHER MATHEMATICS.
First Edition, September, 1896.
Second Edition, January, 1898.
Third Edition, August, 1900.
Fourth Edition, January, 1906.
an
ROBKRT DRUMMOND, PWINTRR, NKW VORIf.
EDITORS' PREFACE.
THE volume called Higher Mathematics, the first edition
of which was published in 1896, contained eleven chapters by
eleven authors, each chapter being independent of the others,
but all supposing the reader to have at least a^ mathematical
training equivalent to that given in classical and engineering
colleges. The publication of that volume is now discontinued
and the chapters are issued in separate form. In these reissues
it will generally be found that the monographs are enlarged
by additional articles or appendices which either amplify^ the
former presentation or record recent advances. This plan of
publication has been arranged in order to meet the demand of
teachers and the convenience of classes, but it is also thought
that it may prove advantageous to readers in special lines of
mathematical literature.
It is the intention of the publishers and editors to add other
monographs to the series from time to time, if the call for the
same seems to warrant it. Among the topics which are under
consideration are those of elliptic functions, the theory of num-
bers, the group theory, the calculus of variations, and non-
Euclidean geometry; possibly also monographs on branches of
astronomy, mechanics, and mathematical physics may be included.
It is the hope of the editors that this form of publication may
tend to promote mathematical study and research over a wider
field than that which the former volume has occupied.
December, 1905.
iii
AUTHOR'S PREFACE.
THE following pages are designed as supplementary to the
discussions of equations in college text-books, and several methods
of solution not commonly given in such works are presented
and exemplified. The aim kept in view has been that of the
determination of the numerical values of the roots of numerical
equations, and algebraic analysis has been used only to further
this end. Historical references are given, problems stated as
exercises for the student, and the attempt has everywhere been
made to present the subject clearly and concisely. The volume
has not been written for those thoroughly conversant with the
theory of equations, but rather for students of mathematics,
computers, and engineers.
This edition has been enlarged by the addition of five articles
which render the former treatment more complete and also give
recent investigations regarding the expression of roots in series.
While not designed for college classes, it is hoped that the book
may prove useful to postgraduate students in mathematics,
physics and engineering, and also tend to promote general interest
in mathematical science.
SOUTH BETHLEHEM, PA.,
December, 1905.
CONTENTS.
ART. i. INTRODUCTION Page i
2. GRAPHIC SOLUTIONS 3
3. THE REGULA FALSI 5
4. NEWTON'S APPROXIMATION RULE 6
5. SEPARATION OF THE ROOTS 8
6. NUMERICAL ALGEBRAIC EQUATIONS 10
7. TRANSCENDENTAL EQUATIONS 13
8. ALGEBRAIC SOLUTIONS . . 15
9. THE CUBIC EQUATION 17
10. THE QUARTIC EQUATION 19
11. QUINTIC EQUATIONS 21
12. TRIGONOMETRIC SOLUTIONS , . . 24
13. REAL ROOTS BY SERIES 27
14. COMPUTATION OF ALL ROOTS 28
15. ROOTS OF UNITY 31
16. SOLUTIONS BY MACLAURIN'S SERIES 33
17. SYMMETRIC FUNCTIONS OF ROOTS 37
18. LOGARITHMIC SOLUTIONS 39
19. INFINITE EQUATIONS 43
20. NOTES AND PROBLEMS 45
INDEX 47
THE SOLUTION OF EQUATIONS.
ART. 1. INTRODUCTION.
THE science of algebra arose in the efforts to solve equations.
Indeed algebra may be called the science of the equation, since
the discussion of equalities and the transformation of forms, into
simpler equivalent ones have been its main objects. The solu-
tion of an equation containing one unknown quantity consists
in the determination of its value or values, these being called
roots. An algebraic equation of degree n has n roots, while tran-
scendental equations often have an infinite number of roots. The
object of the following pages is to present and exemplify convenient
methods for the determination of the numerical values of the
roots of both kinds of equations, the real roots receiving special
attention because these are mainly required in the solution of
problems in physical science.
An algebraic equation is one that involves only the opera-
tions of arithmetic. It is to be first freed from radicals so as
to make the exponents of the unknown quantity all integers;
the degree of the equation is then indicated by the highest ex-
ponent of the unknown quantity. The algebraic solution of an
algebraic equation is the expression of its roots in terms of
the literal coefficients ; this is possible, in general, only for linear,
quadratic, curtc, and quartic equations, that is, for equations
of the first, second, third, and fourth degrees. A numerical
equation is an algebraic equation having all its coefficients real
numbers, either positive or negative. For the four degrees
2 THE SOLUTION OF EQUATIONS.
above mentioned the roots of numerical equations may be
computed from the formulas for the algebraic solutions, unless
they fall under the so-called irreducible case wherein real
quantities are expressed in imaginary forms.
An algebraic equation of the nth degree may be written
with all its terms transposed to the first member, thus:
xn + a^-i + aj*~* + . . . + an_,x + an = 0,
and, for brevity, the first member will be called f(x) and the
equation be referred to as/(;r) — o. The roots of this equa-
tion are the values of x which satisfy it, that is, those values of
x that reduce f(x) to o. When all the coefficients av , #, , . . . aH
are real, as will always be supposed to be the case, Sturm's
theorem gives the number of real roots, provided they are un-
equal, as also the number of real roots lying between two
assumed values of x, while Horner's method furnishes a con-
venient process for obtaining the values of the roots to any
required degree of precision.
A transcendental equation is one involving the operations
of trigonometry or of logarithms, as, for example, x -(- cosx = o,
or a2* -|- *&* — O. No general method for the literal solution
of these equations exists ; but when all known quantities are
expressed as real numbers, the real roots may be located and
computed by tentative methods. Here also the equation may
be designated as/(;tr) = o, and the discussions in Arts. 2-5 will
apply equally well to .both algebraic and transcendental forms.
The methods to be given are thus, in a sense, more valuable
than Sturm's theorem and Horner's process, although for
algebraic equations they may be somewhat longer. It should
be remembered, however, that algebraic equations higher than
the fourth degree do not often occur in physical problems, and
that the value of a method of solution is to be measured not
merely by the rapidity of computation, but also by the ease
with which it can be kept in mind and applied.
Prob. i. Reduce the equation (a + #)* -\- (a — x)i = 2b to an
equation having the exponents of the unknown quantity all integers.
GRAPHIC SOLUTIONS.
ART. 2. GRAPHIC SOLUTIONS.
Approximate values of the real roots of two simultaneous
algebraic equations may be found by the methods of plane
analytic geometry when the coefficients are numerically
expressed. For example, let the given equations be
x* -f-y == a\ x* — bx — y — cy,
the first representing a circle and the second a hyperbola.
Drawing two rectangular axes OX and OY, the circle is de-
scribed from O with the radius a. The coordinates of the
center of the hyperbola are found to be OA == \b and AC = \c,
while its diameter BD —<\f b* -- ?, from which the two
branches may be described.
The intersections of the circle
with the hyperbola give the
real values of x and y. If
a = i, b = 4, and c — 3, there
are but two real values for x
and two real values for y,
since the circle intersects but
one branch of the hyperbola ;
here Om is the positive and
Op the negative value of x, while mn is the positive and pq
the negative value of y. When the radius a is so large that
the circle intersects both branches of the hyperbola there are
four real values of both x and y.
By a similar method approximate values of the real roots of
an algebraic equation containing but one unknown quantity may
be graphically found. For instance, let the cubic equation
x3 -f- ax — b = o be required to be solved.* This may be
written as the two simultaneous equations
k y = x\ y= — a
*See Proceedings of the Engineers' Club of Philadelphia, 1884, V<;1. IV,
pp. 47-49
4 THE SOLUTION OF EQUATIONS.
and the graph of each being plotted, the abscissas of their
points of intersection give the real roots of the cubic. The
curve y = x* should be plotted upon
cross-section paper by the help of a
table of cubes ; then OB is laid off
equal to b, and OC equal to a/b, tak-
ing care to observe the signs of a and
b. The line joining B and C cuts
the curve at p, and hence qp is the
real root of x* -\- ax — b = o. If the
cubic equation have three real roots the straight line BC will
intersect the curve in three points.
Some algebraic equations of higher degrees may be graphic-
ally solved in a similar manner. For the quartic equation
z* -\- Az* -\-Bz- C= o, it is best to put z — A*x, and thus
reduce it to the form x* + x* -f- bx — c = o ; then the two
equations to be plotted are
y = x* + x\ y=—bx + c,
the first of which may be drawn once for all upon cross-section
paper, while the straight line represented by the second may
be drawn for each particular case, as described above. *
This method is also applicable to many transcendental equa-
tions ; thus for the equation Ax -* Bsmx = o it is best to
write ax — sinx = o ; then y = sin x is readily plotted by help
of a table of sines, while y = ax is a straight line passing
through the origin. In the same way a? — x* = o gives the
curve represented by y = a* and the parabola represented by
y = x*, the intersections of which determine the real roots of
the given equation.
Prob. 2. Devise a graphic solution for finding approximate
values of the real roots of the equation x*+ ax*-\- bx*+ ex + d =Q.
Prob. 3. Determine graphically the number and the approximate
values of the real roots of the equation arc x — 8 sin x = o.
,(Ans. — Six real roots, x = ± 159°, ± 430°, and ± 456°.)
* For an extension of this method to the determination of imaginary roots,
see Phillips and Beebe's Graphic Algebra, New York, 1882.
THE REGULA FALSI. 5
ART. 3. THE REGULA FALSI.
One of the oldest methods for computing the real root of
an equation is the rule known as " regula falsi," often called
the method of double position.* It depends upon the princi-
ple that if two numbers x^ and x^ be substituted in the expres-
sion /(V), and if one of these renders /(V) positive and the other
renders it negative, then at least one real root of the equation
f(x) = o lies between x^ and x^ . Let the figure represent a
part of the real graph of the equation y =f(x). The point X,
where the curve crosses the axis of abscissas, gives a real root
OX of the equation f(x) — o. Let OA and OB be inferior and
superior limits of the root OX which are determined either by
trial or by the method of Art. 5. ^
Let Aa and Bb be the values of
f(x) corresponding to these limits.
Join ab, then the intersection C of
the straight line ab with the axis
OB gives an approximate value
OC for the root. Now compute
Cc and join ac, then the intersection D gives a value OD which
is closer still to the root OX.
Let ^ and x^ be the assumed values OA and OB, and let
f(x^) and/(^2) be the corresponding values of/(;r) represented
by Aa and Bb, these values being with contrary signs. Then
from the similar triangle AaC and BbC the abscissa OC is
*,/(X) - *,A*,)_ , (*.-*.Vfr.) , (*.-*.)/(*.)
A*,) -A*,) *A*>y-A*,) '+ /<X) -A*,) '
By a second application of the rule to xl and x^, another value
;r4 is computed, and by continuing the process the value of x
can be obtained to any required degree of precision.
As an example let f(x) = **-}- %x* -(-7 = 0. Here it may
be found by trial that a real root' lies between —2 and — 1.8.
*This originated in India, and its first publication in Europe was by Abra-
ham ben Esra, ».n 1130. See Matthiesen, Grundziige der antiken und moder-
nen Algebra der litteralen Gleichungen, Leipzig, 1878.
6 THE SOLUTION OF EQUATIONS.
For x, = — 2,/fo) = — 5, and for *a = — i.8,/(a?2) = +4.304;
then by the regula falsi there is found x^ = — 1.90 nearly.
Again, for x^ = — 1.90, f(x^) = + 0.290, and these combined
with x^ and /(.*,) give x^ = — 1.906, which is correct to the
third decimal.
As a second example let f(x) = arc* — sin x — 0.5 = o.
Here a graphic solution shows that there is but one real root,
and that the value of it lies between 85° and 86°. For ^="85°,
/(*,) = — 0.01266, and for *„ — 86°, f(x^ = -f 0.00342 ; then
by the rule x^ — 85° 44', which gives f(x^) — — 0.00090. Again,
combining the values for x^ and ,ra there is found x^ = 85° 47',
which gives f(xt) = — 0.00009. Lastly, combining the values
for x^ and x^ there is found xb = 85" 47^4, which is as close an
approximation as can be made with five-place tables.
In the application of this method it is to be observed that
the signs of the values of x and f(x) are to be carefully re-
garded, and also that the values of f(x] to be combined in one
operation should have opposite signs. For the quickest
approximation the values of f(x] to be selected should be those
having the smallest numerical values.
Prob. 4. Compute by the regula falsi the real roots of ^B— 0.25=0.
Also those of ^a + sin 2X = o.
ART. 4. NEWTON'S APPROXIMATION RULE.
Another useful method for approximating to the value of
the real root of an equation is that devised by Newton in 1666.*
If y =f(x) be the equation of a
curve, OX in the figure represents a
real root of the equation " f(x) = o.
Let OA be an approximate value of
OX, and Aa the corresponding value
oif(x\ At a let aB be drawn tangent
to the curve; then OB is another approximate value of OX.
* See Analysis per equationes numero terminorum infinitas, p. 269, Vol. I
of Horsely's edition of Newton's works (London, 1779), where the method is
given in a somewhat different form.
NEWTON'S APPROXIMATION RULE. 7
Let Bb be the value of f(x) corresponding to OB, and at b
let the tangent bC be drawn ; then OC is a closer approxima-
tion to OX, and thus the process may be continued.
Let/(X) be the first derivative oif(x); or,/(.r) = df(x]/dx.
For x = xl = OA in the figure, the value of /(*,) is the ordi-
nate Aa, and the value of /'(*,) is the tangent of the angle
aBA ; this tangent is also Aa/AB. Hence AB = /(
and accordingly OB and <9C are found by
*•-*• /'(*,)'
which is Newton's approximation rule. By a third application
to ;r3 the closer value x^ is found, and the process may be con-
tinued to any degree of precision required.
For example, let f(x) — x* + 5^* + J = o. The first deriv-
ative is f'(x] — ^x* -f- i cur. Here it may be found by trial that
— 2 is an approximate value of the real root. For xl = — 2
/(*,) — — 5, and f'(x^) = 60, whence by the rule x^ = — 1.92.
Now for x^ = — 1.92 are found f(x^) — — 0.6599 and
f'(xj) = 29052, whence by the rule x^ = — 1.906, which is
correct to the third decimal.
As a second example let f(x] = x* -\- 4 sin x = o. Here
the first derivative is f'(x) — 2;r + 4 cos ;r. An approximate
value of x found either by trial or by a graphic solution is
#= — 1.94, corresponding to about — m°O9/. For .*•, = •— 1.94,
f(x^ — 0.03304 and f'(x^ — — 5.323, whence by the rule
xt= — 1.934. By a second application x^ — — 1.9328, which
corresponds to an angle of — 110° 54^'.
In the application of Newton's rule it is best that the
assumed value of x^ should be such as to render /(^,) as small
as possible, and also f'(x^) as large as possible. The method
will fail if the curve has a maximum or minimum between a
and b. It is seen that Newton's rule, like the regula falsi,
applies equally well to both transcendental and algebraic equa-
tions, and moreover that the rule itself is readily kept in mind
by help of the diagram.
8 THE SOLUTION OF EQUATIONS.
Prob. 5. Compute by Newton's rule the real roots of the alge-
braic equation x* — ^x -j- 6 = o. Also the real roots of the trans-
cendental equation sin x -f- arc x — 2 = 0.
ART. 5. SEPARATION OF THE ROOTS.
The roots of an equation are of two kinds, real roots and
imaginary roots. Equal real roots may be regarded as a spe-
cial class, which lie at the limit between the real and the imagi-
nary. If an equation has/ equal roots of one value and q equal
roots of another value, then its first derivative equation has
p — i roots of the first value and ^ — I roots of the second
value, and thus all the equal roots are contained in a factor
common to both primitive and derivative. Equal roots may
hence always be readily detected and removed from the given
equation. For instance, let x* — 9^ -|- 4.2: + 12 = 0, of which
the derivative equation is 4Jtr* — i8x -j- 4 = o ; as x — 2 is a
factor of these two equations, two of the roots of the primitive
equation are + 2.
The problem of determining the number of the real and
imaginary roots of an algebraic equation is completely solved
by Sturm's theorem. If, then, two values be assigned to x the
number of real roots between those limits is found by the same
theorem, and thus by a sufficient number of assumptions limits
may be found for each real root. As Sturm's theorem is known
to all who read these pages, no applications of it will be here
given, but instead an older method due to Hudde will be
presented which has the merit of giving a comprehensive view
of the subject, and which moreover applies to transcendental
as well as to algebraic equations.*
If any equation y =. f(x) be plotted with values of x as
abscissas and values of y as ordinates, a real graph is obtained
whose intersections with the axis OX give the real roots of the
* Devised by Hudde in 1659 an(^ published by Rolle in 1690. See CEuvres
de Lagrange, Vol. VIII, p. 190.
SEPARATION OF THE ROOTS.
9
equal ion f(x) = o. Thus in the figure the three points marked
JTgi'^e three values OX for three real roots. The curve which
repr/ sentsjj/ -=.f(x) has points of maxima and minima marked
A, and inflection points marked B. Now let the first deriva-
tive equation dy/dx=f'(x) be formed and be plotted in the
same manner on the axis O'X'. The condition /'(#)= o gives
the abscissas of the points A, and thus the real roots O'X' give
limits separating the real roots of f(x) =. o. To ascertain if a
real root OX lies between two values of O' X1 these two values
are to be substituted in/(^r): if the signs of f(x] are unlike in
the two cases, a real root of f(x] = o lies between the two
limits ; if the signs are the same, a real root does not lie between
those limits.
In like manner if the second derivative equation, that is,
d*y/dx? = f"(x\ be plotted on O" X" , the intersections give
limits which separate the real roots off'(x)=o. It is also
seen that the roots of the second derivative equation are the
abscissas of the points of inflection of the curve y = f(x).
To illustrate this method let the given equation be the
quintic f(x) = x* — $x* -\-6x-\- 2 = o. The first derivative
equation is f'(x] = $x* — i $x* -f~ 6 — °> tne roots of which are
approximately — 1.59, — 0.69, +0.69, 4- I-59- Now let each
of these values be substituted for x in the given quintic, as also
the values — oo , o, and + oo , and let the corresponding values
of f(x] be determined as follows :
10 THE SOLUTION OF EQUATIONS.
* = _oo, -1.59, -0.69, o, +0.69, +1.59, +00;
/(*)=- oo, +2.4, -0.6, +2, +4-7» +1-6, + GO.
Since f(x) changes sign between x0 =— oo and x1 = — 1.59,
one real root lies between these limits ; since f(x) changes sign
between x^ = — 1.59 and x^ = — 0.69, one real root lies between
these limits ; since f(x] changes sign between x^ = — 0.69 and
xz = o, one real root lies between these limits; since f(x) does
not change sign between xz = o and x^ — oo , a pair of imagi-
nary roots is indicated, the sum of which lies between -f- 0.69
and oo .
As a second example let J\x) = e* — e2*— 4 = 0. The first
derivative equation is f'(x) — e* — 2e*x = o, which has two
roots e* = -J and f = o, the latter corresponding to x = — oo .
For x = — oo , f(x) is negative; for e* = J, f(x) is negative ; for
x = -f- oo , f(x} is negative. The equation e* — e** — 4 = o
has, therefore, no real roots.
When the first derivative equation is not easily solved, the
second, third, and following derivatives may be taken until an
equation is found whose roots may be obtained. Then, by
working backward, limits may be found in succession for the
roots of the derivative equations until finally those of the
primative are ascertained. In many cases, it is true, this proc-
ess may prove lengthy and difficult, and in some it may fail
entirely ; nevertheless the method is one of great theoretical
and practical value.
Prob. 6. Show that <?* + e~Bx — 4 = 0 has two real roots, one
positive and one negative.
Prob. 7. Show that x6 -}- x -\- i — o has no real roots; also that
x' — x — i = o has two real roots, one positive and one negative.
ART. 6. NUMERICAL ALGEBRAIC EQUATIONS.
An algebraic equation of the ?zth degree may be written
with all its terms transposed to the first member, thus:
xn + a,xn~l + #X"2 + • • • -f- an-\x + an = o ;
NUMERICAL ALGEBRAIC EQUATIONS. 11
and if all the coefficients and the absolute term are real num-
bers, this is commonly called a numerical equation. The first
member may for brevity be denoted by f(x) and the equation
itself by/(V) = o.
The following principles of the theory of algebraic equations
with real coefficients, deduced in text-books on algebra, are
here recapitulated for convenience of reference :
(1) If xl is a root of the equation, /(.x) is divisible by x — x^
and conversely, if f(x) is divisible by x —xlt then xl is a root of the
equation.
(2) An equation of the nih degree has n roots and no more.
(3) If xl , #a, . . . xn are the roots of the equation, then the prod-
uct (x — xt)(x — xj . . . (x — xn] is equal tof(x).
(4) The sum of the roots is equal to — «,; the sum of the prod-
ucts of the roots, taken two in a set, is equal to -f- #2; the sum of
the products of the roots, taken three in a set, is equal to — #3; and
so on. The product of all the roots is equal to — an when n is
odd, and to -+- an when n is even.
(5) The equation f(x) = o may be reduced to an equation lack-
ing its second term by substituting y — ajn for x.*
(6) If an equation has imaginary roots, they occur in pairs of
the form/ ± qi where / represents y — i.
(7) An equation of odd degree has at least one real root whose
sign is opposite to that of an.
(8) An equation of even degree, having an negative, has at least
two real roots, one being positive and the other negative.
(9) A complete equation cannot have more positive roots than
variations in the signs of its terms, nor more negative roots than
permanences in signs. If all roots be real, there are as many posi-
tive roots as variations, and as many negative roots as permanences. f
(10) In an incomplete equation, if an even number of terms,
say 2m, are lacking between two other terms, then it has at least 2m
* By substituting j2 -\-py-\- q for x, the quantities/ and q may be determined
so as to remove the second and third terms by means of a quadratic equation,
the second and fourth terms by means of a cubic equation, or the second and
fifth terms by means of a quartic equation.
f The law deduced by Harriot in 1631 and by Descartes in 1639.
12 THE SOLUTION OF EQUATIONS.
imaginary roots; if an odd number of terms, say 2m -f- i, are lacking
between two other terms, then it has at least either 2m -f- 2 or 2m
imaginary roots, according as the two terms have like or unlike
signs.*
(n) Sturm's theorem gives the number of real roots, provided
that they are unequal, as also the number of real roots lying be-
tween two assumed values of x.
(12) If ar is the greatest negative coefficient, and if as is the
greatest negative coefficient after x is changed into — x, then all
real roots lie between the limits ar -\- i and — (as -f- i).
(13) If ah is the first negative and ar the greatest negative co-
efficient, then ar + i is a superior limit of the positive roots. If
ak be the first negative and as the greatest negative coefficient after
x is changed into — x, then ask + i is a numerically superior limit
of the negative roots.
(14) Inferior limits of the positive and negative roots may be
found by placing x = z~l and thus obtaining an equation f(z) = o
whose roots are the reciprocals of f(x) = o.
(15) Homer's method, using the substitution x = z — r where r
is an approximate value of x^ , enables the real root xt to be com-
puted to any required degree of precision.
The application of these principles and methods will be
familiar to all who read these pages. Horner's method may
be also modified so as to apply to the computation of imagi-
nary roots after their approximate values have been found. t
The older method of Hudde and Rolle, set forth in Art. 5, is
however one of frequent convenient application, for such alge-
braic equations as actually arise in practice. By its use,
together with principles (13) and (14) above, and the regula
falsi of Art. 3, the real roots may be computed without any
.assumptions whatever regarding their values.
For example, let a sphere of diameter D and specific gravity
* Established by DuGua; see Memoirs Paris Academy, 1741, pp. 435-494.
fSheffler, Die Auflosung der algebraischen und transzendenten Gleichung-
en, Braunschweig, 1859; a°d Jelink, Die Auflosung der hoheren numerischen
v. Gleichungen, Leipzig, 1865.
TRANSCENDENTAL EQUATIONS.' 13
g float in water, and let it be required to find the depth of im-
mersion. The solution of the problem gives for the depth x
the cubic equation
As a particular case let D = 2 feet and g— 0.65 ; then the
equation
*' — 3*' + 2.6 = 0
is to be solved. The first derivative equation is 3** — 6x = o
whose roots are o and 2. Substituting these, there is found
one negative root, one positive root less than 2, and one posi-
tive root greater than 2. The physical aspect of the question
excludes the first and last root, and the second is to be computed.
By (13) and (14) an inferior limit of this root is about 0.5, so
that it lies between 0.5 and 2. For xl — 0.5, /(jr,) = -j- 1.975,
and for x^ = 2, f(x^ = —1.4; then by the regula falsi ^"3=1.35.
For xz =i.35,/(^8) = — 0.408, and combining this with x, the
regula falsi gives ;r4 = 1.204 ^eet> which, except in the last
decimal, is the correct depth of immersion of the sphere.
Prob. 8. The diameter of a water-pipe whose length is 200 feet
and which is to discharge 100 cubic feet per second under a head
of 10 feet is given by the real root of the quintic equation
xb — 38.* — 101 = o. Find the value of x.
ART. 7. TRANSCENDENTAL EQUATIONS.
Rules (i) to (15) of the last article have no application to
trigonometrical or exponential equations, but the general prin-
ciples and methods of Arts. 2—5 may be always used in
attempting their solution. Transcendental equations may
have one, many, or no real roots, but those arising from prob-
lems in physical science must have at least one real root. Two
examples of such equations will be presented.
A cylinder of specific gravity g floats in water, and it is
required to find the immersed arc of the circumference. If
this be expressed in circular measure it is given by the trans-
cedental equation
f(x) = x — sin x — 27tg = o.
14 THE SOLUTION OF EQUATIONS.
The first derivative equation is I — cos x — o, whose root is
-any even multiple of 2n. Substituting such multiples in f(x)
it is found that the equation has but one real root, and that
this lies between o and 2n; substituting J/r, £TT, and n for x> it
is further found that this root lies between f TT and n.
As a particular case let g — 0.424, and for convenience in
using the tables let x be expressed in degrees; then
f(x} = x - 57° .2958 sin x — 152° .64.
Now proceeding by the regula falsi (Art. 3) let xl = 180° and
*3=I35°» giving /(X) = + 27° .36 and/(>2) = -58°.i6, whence
x^ = 166°. For x^— 166°, /(^s) = — o° .469, and hence 166° is an
approximate value of the root. Continuing the process, x is
found to be i66°.237, or in circular measure ^=2.9014 radians.
As a second example let it be required to find the horizon-
tal tension of a catenary cable whose length is 22 feet, span 20
feet, and weight 10 pounds per linear foot, the ends being sus-
pended from two points on the same level. If / be the span, s
the length of the cable, and z a length of the cable whose weight
equals the horizontal tension, the solution of the problem leads
/ L -L\
to the transcendental equation s= V* — e *zi zy or inserting
the numerical values,
( ™ -™\
f(z) = 22 — \e' — e z lz — o
is the equation to be solved. The first derivative equation is
/ ^ -i-°\ 10 / ™ -™\
f(z] = -(e* -e *)+ -(e* + e > ) = o,
and this substituted in f(z) shows that one real root is less than
about 20. Assume zl =15, then /(^J =0.486 an d /'(£,)= 0.206,
whence by Newton's rule (Art. 4) z^= 13 nearly. Next for
2^ = 13, /(#,) = — 0.0298 and f'(z^) — 0.322, whence zt = 13.1.
Lastly for zt = 13.1 f(z3) =0.0012 and/'fo) = 0.3142, whence
zt — 13.096, which is a sufficiently close approximation. The
horizontal tension in the given catenary is hence 130.96 pounds.*
*Since e9 — e~6 =• 2 sinhQ, this equation may be written n9 — 10 sinh 0,
where 6 = loz'1, and the solution may be expedited by the help of tables of
hyperbolic functions. See Chapter IV.
ALGEBRAIC SOLUTIONS. 15
Prob. 9. Show that the equation 3 sin x — 2X — 5 = o has but
one real root, and compute its value.
Prob. 10. Find the number of real roots of the equation
'2.x -\- log x — 10000 = o, and show that the value of one of them is
•* = 4995-74-
ART. 8. ALGEBRAIC SOLUTIONS.
Algebraic solutions of complete algebraic equations are
only possible when the degree n is less than 5. It frequently
happens, moreover, that the algebraic solution cannot be used
to determine numerical values of the roots as the formulas
expressing them are in irreducible imaginary form. Neverthe-
less the algebraic solutions of quadratic, cubic, and quartic
equations are of great practical value, and the theory of the
subject is of the highest importance, having given rise in fact
to a large part of modern algebra.
The solution of the quadratic has been known from very
early times, and solutions of the cubic and quartic equations
were effected in the sixteenth century. A complete investiga-
tion of the fundamental principles of these solutions was, how-
ever, first given by Lagrange in 1770.* This discussion showed,
if the general equation of the nih degree, f(x) =o, be deprived
of its second term, thus giving the equation f(y) = o, that the
expression for the root y is given by
y= cos, GD . • . G*"
in which n is the degree of the given equation, GO is, in suc-
cession, each of the nth roots of unity, I, e, e2, . . . e*-1, and
slt s9, . . . sn_, are the so-called elements which in soluble cases
are determined by an equation of the n — Ith degree. For
instance, if n = 3 the equation is of the third degree or a cubic,
the three values of GO are
* Memoirs of Berlin Academy, 1769 and 1770; reprinted in CEuvres de
Lagrange (Paris, 1868), Vol. II, pp. 539-562. See also Traite de la resolution
des equations numeriques, Paris, 1798 and 1808.
16 THE SOLUTION OF EQUATIONS.
and the three roots are expressed by
y, = *, + *, > 7, = &i + e\ , y^ — e's, + es, ,
in which s^ and ja8 are found to be the roots of a quadratic
equation (Art. 9).
The n values of GO are the n roots of the binomial equation
aon — I = o. If n be odd, one of these is real and the others
are imaginary ; if n be even, two are real and n — 2 are imagi-
nary.* Thus the roots of a? — I = o are -\- I and — I ; those
of Go3 — i = o are given above ; those of o>4 — I = o are
+ i, + /, — i, and — i where i is-v/— i- For the equation
co" — I = o the real root is -|- i, and the imaginary roots are
denoted by e, e8, e3, e4; to find these let GO* — i = o be divided
by GO— i, giving
oo* + GO* + co9 + GO + I = o,
which being a reciprocal equation can be reduced to a quad-
ratic, and the solution of this furnishes the four values,
V-
where it will be seen that e.e4 = I and ea.e3 = i, as should be
the case, since e6 = i.
In order to solve a quadratic equation by this general
method let it be of the form
x* + 2.ax + b = o,
and let x be replaced by y — a, thus reducing it to
y _ (a* -b} = o.
Now the two roots of this are yl = + ^1 and y^ = — slt whence
the product of (y — s^) and (y -\- J,) is
y - j2 = o.
Thus the value of / is given by an equation of the first degree,
* The values of GO are, in short, those of the n " vectors " drawn from the
center which divide a circle of radius unity into n equal parts, the first vector
GOi = i being measured on the axis of real quantities. See Chapter X.
THE CUBIC EQUATION. 17
£ = c? — b\ and since x= — a-\-y, the roots of the given
equation are
#, = — a + ^/~^~^b, x^ — — a — i/ a* — £»
which is the algebraic solution of the quadratic.
The equation of the n — Ith degree upon which the solution
of the equation of the «th degree depends is called a resolvent.
If such a resolvent exists, the given equation is algebraically
solvable ; but, as before remarked, this is only the case for
quadratic, cubic, and quartic equations.
Prob. n. Show that the six 6th roots of unity are + *>
ART. 9. THE CUBIC EQUATION.
All methods for the solution of the cubic equation lead to
the result commonly known as Cardan's formula.* Let the
cubic be
x* + 30** + $bx + 2c = o, (i)
and let the second term be removed by substituting y — a for
x, giving the form,
C=o, (i')
in which the values of B and C are
£=-*' + £, C=a* — \ab + c. (2)
Now by the Lagrangian method of Art. 8 the values of y are
^1 = ^1 + j. » y* = e^ + e*s* » y* = €*s* + ej3 »
in which e and e2 are the imaginary cube roots of unity.
Forming the products of the roots, and remembering that
e3 — I and e2 e i = o, there are found
For the determination of J, and ja there are hence two equa-
tions from which results the quadratic resolvent
2Cs* — J3* = O, and thus
7^. (3)
* Deduced by Ferreo in 1515, and first published by Cardan in 1545.
18 THE SOLUTION OF EQUATIONS.
One of the roots of the cubic in y therefore is
and this is the well-known formula of Cardan.
The algebraic solution of the cubic equation (i) hence con-
sists in finding B and C by (2) in terms of the given coefficients,
and then by (3) the elements 5, and s^ are determined. Finally,
Xl = — a + (s, -{- st),
fa - st), (4)
which are the algebraic expressions of the three roots.
When B* + C2 is negative the numerical solution of the
cubic is not possible by these formulas, as then both s1 and ja
are in irreducible imaginary form. This, as is well known, is
the case of three real roots, s1 + s^ being a real, while sl — st is
a pure imaginary.* When B* -\- C2 is o the elements sl and S9
are equal, and there are two equal roots, x^ = x% = — a + C$,
while the other root is x^ = — a — 2C*.
When B* + C* is positive the equation has one real and
two imaginary roots, and formulas (2), (3), and (4) furnish the
numerical values of the roots of (i). For example, take the
cubic
x* — 4.5**+ 12* — 5 = 0,
whence by comparison with (i) are found a = — 1.5, b = + 4>
c — —2.5. Then from (2) are computed B = 1.75, £T=:-)-3-I25-
These values inserted in (3) give sl = +0.9142, s^ = — 1.9142 ;
thus sl + 5, = — i.o and s,— Ja = + 2.8284. Finally, from (4)
*,= 1.5 - 1.0 = +0.5,
*f == 1.5 +0.5 + 1.4142 V~~3 = 2 + 2.44951,
*3 = 1.5 + O.5.— I.4H2 4X^1 = 2 - 2.44951,
which are the three roots of the given cubic.
* The numerical solution of this case ij possible whenever the angle whose
cosine is — C/ \/ — Bz can be geometrically trisected.
THE QUARTIC EQUATION. 19
Prob. 12. Compute the roots of x* — 2^—5= o. Also the roots
of x3 + o.6.#2 — 5.76^ -f- 4.32 = o.
Prob. 13. A cone has its altitude 6 inches and the diameter of
its base 5 inches. It is placed with vertex downwards and one fifth
of its volume is filled with water. If a sphere 4 inches in diameter
be then put into the cone, what part of. its radius is immersed in the
water ? (Ans. 0.5459 inches).
ART. 10. THE QUARTIC EQUATION.
The quartic equation was first solved in 1545 by Ferrari,
who separated it into the difference of two squares. Descartes
in 1637 resolved it into the product of two quadratic factors,
Tschirnhausen in 1683 removed the second and fourth terms.
Euler in 1732 and Lagrange in 1767 effected solutions by
assuming the form of the roots. All these methods lead to
cubic resolvents, the roots of which are first to be found in
order to determine those of the quartic.
The methods of Euler and Lagrange, which are closely
similar, first reduce the quartic to one lacking the second term,
and the general form of the roots being taken as
the values slt ^a, s9, are shown to be the roots of the resolvent,
Thus the roots of the quartic are algebraically expressed in
terms of the coefficients of the quartic, since the resolvent is
solvable by the process of Art. 9.
Whatever method of solution be followed, the following
final formulas, deduced by the author in 1892, will result.*
Let the complete quartic equation be written in the form
x* + 4^3 + 6&x* +4cx + d=o. (i)
* See American Journal Mathematics, 1892, Vol. XIV, pp. 237-245.
20 THE SOLUTION OF EQUATIONS.
First, let g, h, and k be determined from
Secondly, let / be obtained by
Thirdly, let u, v, and w be found from
u = g-\- /, v = 2g — /, w = 4«a + $k — I2£-/. (4)
Then the four roots of the quartic equation are
^ = _ a _[_ V& _L. I/?, .{- Vw,
(5)
#, = — tf — Vu -f ^ — Vw,
xt= — a — Vu — Vv — Vw,
in which the signs are to be used as written provided that
20* — $ab -\- c is a negative number; but if this is positive all
radicals except Vw are to be reversed in sign.
These formulas not only serve for the complete theoretic
discussion of the quartic (i), but they enable numerical solu-
tions to be made whenever (3) can be computed, that is, when-
ever ff -\-k* is positive. For this case the quartic has two real
and two imaginary roots. If there be either four real roots or
four imaginary roots h* -f- k* is negative, and the irreducible
case arises where convenient numerical values cannot be ob-
tained, although they are correctly represented by the formulas.
As an example let a given rectangle have the sides/ and ^,
and let it be required to find the length of an inscribed rec-
tangle whose width is m. If x be this length, this is a root of
the quartic equation
and thus the problem is numerically solvable by the above
formulas if two roots are real and two imaginary. As a special
case let / = 4 feet, q = 3 feet, and in = I foot ; then
x* — 27 x* + 48^- — 24 = o.
QUINTIC EQUATIONS. 21
By comparison with (i) are found a = o, b — — 4%, c = + 12,
and d——2^ Then from (2), £ = -{-4.%, ^=.-±J.L, and
£ — _[_ -459-. Thus k* -f- £3 is positive, and from (3) the value of I
15—3.6067. From (4) are now found, & = +0.8933, v= 12.6067,
and w= -\- 161.20. Then, since c is positive, the values of the
four roots are, by (5),
*, = — 0.945 - ^12.607 + 12.697 = — 5.975 feet,
x^ = — 0.945 + Vi 2.607+ 12.697 = + 4.085 feet,
*, = + 0.945 — 4/12.607 - 12.697 = + 0.945 — 0.301",
x^ = + 0.945 + lx 1 2.607 — 12.697 = + 0.945 + 0.302,
the second of which is evidently the required length. Each of
these roots closely satisfies the given equation, the slight dis-
crepancy in each case being due to the rounding off at the third
decimal.*
Prob. 14. Compute the roots of the equation x* + yx + 6 = o.
(Ans. — 1.388, — i. ooo, 1.194 ± 1.701*'.)
ART. 11. QUINTIC EQUATIONS.
The complete equation of the fifth degree is not algebraic-
ally solvable, nor is it reducible to a solvable form. Let the
equation be
and by substituting y — a for x let it be reduced to
/+ 5^/+ $Cf+ SDy + 2E = o.
The five roots -of this are, according to Art. 8,
in which e, ea, e* e4 are the imaginary fifth roots of unity. Now
if the several products of these roots be taken there will be
* This example is known by civil engineers as the problem of finding the
length of a strut in a panel of the Howe truss.
22 THE SOLUTION OF EQUATIONS.
found, by (4) of Art. 6, four equations connecting the four ele-
ments slt s^ s9, and st, namely,
— D =
- sfo v« + *. Vi + Vv* + J/v.) ;
but the solution of these leads to an equation of the i2Oth
degree for s, or of the 24th degree for /. However, by taking
sts4 — v3 or si* + -^a5 + *t* + -C as tne unknown quantity, a
resolvent of the 6th degree is obtained, and all efforts to find
a resolvent of the fourth degree have proved unavailing.
Another line of attack upon the quintic is in attempting to
remove all the terms intermediate between the first and the
last. By substituting y* -\- py -\-g for x, the values of p and q
maybe determined so as to remove the second and third terms
by a quadratic equation, or the second and third by a cubic
equation, or the second and fourth by a quartic equation, as
was first shown by Tschirnhausen in 1683. By substituting
y ~\~P^ ~\~ $y 4~ r f°r x-> three terms may be removed, as was
shown by Bring in 1786. By substituting y*-\-py* + qy* -\-ry-\-t
for x it was thought by Jerrard in 1833 that four terms might
be removed, but Hamilton showed later that this leads to
equations of a degree higher than the fourth.
In 1826 Abel gave a demonstration that the algebraic solu-
tion of the general quintic is impossible, and later Galois
published a more extended investigation leading to the same
conclusion.* The reason for the algebraic solvability of the
quartic equation may be briefly stated as the fact that there
exist rational three-valued functions of four quantities. There
are, however, no rational four-valued functions of five quan-
tities, and accordingly a quartic resolvent cannot be found for
the general quintic equation.
* Jordan's Trait6 des substitutions et des Equations algSbriques; Paris, 1870.
Abhandlungen iiber die algebraische Auflosung der Gleichungen von N. H.
Abel und Galois; Berlin, 1889.
QUINTIC EQUATIONS. 23
There are, however, numerous special forms of the quintic
whose algebraic solution is possible. The oldest of these is the
quintic of De Moivre,
which is solved at once by making J2 = s3 = o in the element
equations ; then — B — slst and — 2E = s^ + s<, from which
j, and st are found, and y^ = st +J4, or
* =(- E
while the other roots are y^ = es, + eV4 , yz = e"s1 -f- e*st ,
y^= e3^ -f- eV4 , and y^ — e's1 + est . If B" + E* be negative,
this quintic has five real roots; if positive, there are one real
and four imaginary roots.
When any relation, other than those expressed by the four
element equations, exists between slts9, st, s4, the quintic is
solvable algebraically. As an infinite number of such relations
may be stated, it follows that there are an infinite number of
solvable quintics. In each case of this kind, however, the co-
efficients of the quintic are also related to each other by a
certain equation of condition.
The complete solution of the quintic in terms of one of the
roots of its resolvent sextic was made by McClintock in 1884.*
By this method s^, s^, sa6, and ^45 are expressed as the roots of
a quartic in terms of a quantity / which is the root of a sextic
whose coefficients are rational functions of those of the given
quintic. Although this has great theoretic interest, it is, of
course, of little practical value for the determination of numer-
ical values of the roots.
By means of elliptic functions the complete quintic can,
however, be solved, as was first shown by Hermite in 1858.
For this purpose the quintic is reduced by Jerrard's transfor-
mation to the form x* + 5^+ ^e = o, and to this form can
also be reduced the elliptic modular equation of the sixth
degree. Other solutions by elliptic functions were made by
* American Journal of Mathematics, 1886, Vol. VIII, pp. 49-83.
24 THE SOLUTION OF EQUATIONS.
Kronecker in 1861 and by Klein in 1884.* These methods,
though feasible by the help of tables, have not yet been sys-
tematized so as to be of practical advantage in the numerical
computation of roots.
Prob. 15. If the relation stst — s^ exists between the elements
show that j,6+ s,f -f s3& + s^ — — 2E.
Prob. 16. Compute the roots of y6 -\- icy* -f- 2°y ~f~ 6 = o, and
also those of y* — icy* -f- 2oy -f- 6 = o.
ART. 12. TRIGONOMETRIC SOLUTIONS.
When a cubic equation has three real roots the most con-
venient practical method of solution is by the use of a table of
sines and cosines. If the cubic be stated in the form (i) of
Art. 9, let the second term be removed, giving
/ + &y + 2C=0.
Now suppose y=2r sin 0, then this equation becomes
B C
8 sin8 0+6-, sin 0+2-i = o,
and by comparison with the known trigonometric formula
8 sin3 0 — 6 sin 0-f- 2 sin 3$ = o,
there are found for r and sin 3$ the values
r = V^-~B, sin 30 = C/ V'=^r9
in which B is always negative for the case of three real roots
(Art. 9). Now sin 3$ being computed, 3$ is found from a table
of sines, and then 0 is known. Thus,
yv = 2r sin 0, y^ = 2r sin (.120° + 0), yt = 2r sin (240° + #)»
are the real roots of the cubic
* For an outline of these transcendental methods, sec Hagen's Synopsis der
hoheren Mathematik, Vol. I, pp. 339-344.
f When Bz is negative and numerically less than C"2, as also when £3 is
positive, this solution fails, as then one root is real and two are imaginary. In
this case, however, a similar method of solution by means of hyperbolic sines
is possible. See Grunert's Archiv fur Mathematik und Physik, Vol. xxxviii,
pp. 48-76.
TRIGONOMETRIC SOLUTIONS. 25
For example, the depth of flotation of a sphere whose diam-
eter is 2 feet and specific gravity 0.65, is given by the cubic
equation x* — $x* -{-2.6 = 0 (Art. 6). Placing x = y + I this
reduces to^s— 3jj/ -f- 0.6 = o, for which B = — i and C =-[-0.3.
Thus r = i and sin 3$ = + 0.3. Next from a table of sines,
3# = 17° 27', and accordingly 0 = 5° 49'. Then
/, = 2 sin 5° 49' — + 0.2027,
y% — 2 sin 125° 49' = + 1.6218,
.?. = 2 sin 245° 49' = -1.8245.
Adding I to each of these, the values of x are
xl = + 1.203 feet, #, = + 2.622 feet, #, =—0.825 feet ;
and evidently, from the physical aspect of the question, the
first of these is the required depth. It may be noted that the
number 0.3 is also the sine of 162° u', but by using this the
three roots have the same values in a different order.
When the quartic equation has four real roots its cubic re-
solvent has also three real roots. In this case the formulas of
Art. 10 will furnish the solution if the three values of / be ob-
tained from (3) by the help of a table of sines. The quartic
being given, g, h, and k are found as before, and the value of
k will always be negative for four real roots. Then
r = V— k, sin 30 = — h/r\
and 30 is taken from a table ; thus B is known, and the three
values of / are
4 = r sin 0, ^ = r sin(i2o° + 0), /, = r sin (240° + 8).
Next the three values of u, of v, and of w are computed, and
those selected which give #, w, and v — Vw all positive quanti-
ties. Then (5) gives the required roots of the quartic.
As an example, take the case of the inscribed rectangle in
Art. 10, and let/ = 4 feet, q — 3 feet, m = 4/13 feet; then the
quartic equation is
x* — 5 1*1 + 48 V\$ x — 156 = 0.
26 THE SOLUTION OF EQUATIONS.
Here a = o, 3 = — 8 J, c — -\- 12 Vi$, and d= — 156. Next
g- =-\- 8£, //— — £f-&, and k =— $£-' The trigonometric work
now begins; the value of r is found to be + 4J, and that of
sin 3# to be -f- 0.7476; hence from the table 3$ = 48° 23', and
0= i6°o7/40//. The three values of / are then computed
by logarithmic tables, and found to be,
/, = + !. 250, /, = + 3.1187, /4=- 4.3687.
Next the values of u, v, and w are obtained, and it is seen that
only those corresponding to /, will render all quantities under
the radicals positive ; these quantities are u = 9.75, v = 15.75,.
and w = 192.0. Then the four roots of the quartic are
^=-8.564, *• =+ 2-3'9> **=+ 1.746, x, =+4.499 feet,
of which only the second and third belong to inscribed rec-
tangles, while the first and fourth belong to rectangles whose
corners are on the sides of the given rectangle produced.
Trigonometric solutions of the quintic equation are not
possible except for the binomial x* ± a, and the quintic of
De Moivre. The general trigonometric expression for the root
kof a quintic lacking its second term isy=2rl cos#1+2r9 cos 0a,
and to render a solution possible, rt and ra , as well as cos #,
and cos#a, must be found; but these in general are roots of
equations of the sixth or twelfth degree : in fact r* is the same
as the function stst of Art. n, and r,2 is the same as s^s3.
Here cos#, and cos#a may be either circular or hyperbolic
cosines, depending upon the signs and values of the coefficients
of the quintic.
Trigonometric solutions are possible for any binomial equa-
tion, and also for any equation which expresses the division of
an angle into equal parts. Thus the roots of x* + I = o are
cos^ 30° ± i sin m 30°, in which m has the values I, 2, and 3.
The roots of x* — $x*+$x — 2 cos 5 6 = o are 2 cos (m 72°+&)
where m has the values o, 1,2, 3, and 4.
Prob. 17. Compute by a trigonometric solution the four roots of
the quartic x* + 4#8 — 24^ — 76^ — 29 = o. (Ans. —6.734, — 1.550^
-f- 0.262, + 4.022).
REAL ROOTS BY SERIES. 27
Prob. 1 8. Give a trigonometric solution of the quintic equation
x6 — 5 -f S#*x — 2e = o for the case of five real roots. Compute
the roots when b'=i and e= 0.752798. (Ans. —1.7940, — 1.3952^
0.2864, 0.9317, 1.9710.)
ART. 13. REAL ROOTS BY SERIES.
The value of x in any algebraic equation may be expressed
as an infinite series. Let the equation be of any degree, and
by dividing by the coefficient of the term containing the first
power of x let it be placed in the form
a = x + bx* + cx*+ dx* + ex* +/*> + . . .
Now let it be assumed that x can be expressed by the series
x = a + ma* -\- no* -\-pa* -\- qa* + .
By inserting this value of x in the equation and equating the
coefficients of like powers of a, the values of mt n, etc., are
found, and then
x=a - &a* + 2t>*-{a*-l>*
is an expression of one of the roots of the equation. In order
that this series may converge rapidly it is necessary that a
should be a small fraction.*
To apply this to a cubic equation the coefficients d, e,f, etc.,
are made equal to o, For example, let x* — $x -\- 0.6 = o ;
this reduced to the given form is 0.2 = x — \x*, hence a = o.2,
£ = o, c = — \, and then
x = 0.2 + i . o.23 + J . o.25 + etc. = + 0.20277,
which is the value of one of the roots correct to the fourth
decimal place. This equation has three real roots, but the
series gives only one of them ; the others can, however, be
found if their approximate values are known. Thus, one root
is about +1.6, and by placing x=y -\-i.6 there results an
equation in y whose root by the series is found to be-}- 0.0218,
and hence + 1 .62 1 8 is another root of x* - - $x -f- 0.6 = o.
*This method is given by J. B. Molt in The Analyst, 1882, Vol. IX, p. 104.
28 THE SOLUTION OF EQUATIONS.
Cardan's expression for the root of a cubic equation can be
expressed as a series by developing each of the cube roots by
the binomial formula and adding the results. Let the equa-
tion be y* -f- ^By -\- 2C = o, whose root is, by Art. 9,
then this development gives the series,
2 2.3.4 2. 3- 4-S-o
in which r represents the quantity (P? -j- C*)/$C\ If r = o
the equation has two equal roots and the third root is 2( — C )*.
If r is numerically greater than unity the series is divergent,
and the solution fails. If r is numerically less than unity and
sufficiently small to make a quick convergence, the series will
serve for the computation of one real root. For example, take
the equation x* — 6x -\- 6 = o, where B= — 2 and C = 3 ;
hence r = 1/8 1, and one root is
y = — 2.8845(1 — 0.01235 — 0.00051 — 0.00032—) = — 2.846,
which is correct to the third decimal. In comparatively few
cases, however, is this series of value for the solution of cubics,
Many other series for the expression of the roots of equa-
tions, particularly for trinomial equations, have been devised.
One of the oldest is that given by Lambert in 1758, whereby
the root of xn -\- ax — b = o is developed in terms of the
ascending powers of b/a. Other solutions were published by
Euler and Lagrange. These series usually give but one root,
and this only when the values of the coefficients are such as to
render convergence rapid.
Prob. 19. Consult Euler's Anleitung zur Algebra (St. Petersburg,
1771), pp. 143-150, and apply his method of series to the solution of
a quartic equation.
ART. 14. COMPUTATION OF ALL ROOTS.
A comprehensive and valuable method for the solution of
equations by series was developed by McClintock, in 1894, by
COMPUTATION OF ALL ROOTS. 29
means of his Calculus of Enlargement.* By this method all
the roots, whether real or imaginary, may be computed from a
single series, The following is a statement of the method as
applied to trinomial equations :
Let xn = nAxn~k + Bn be the given trinomial equation.
Substitute x = By and thus reduce the equation to the form
yn = nayn'k -\-\ where a = A/Bk. Then if Bn is positive, the
roots are given by the series
y = oo -f col~k a + Gol~2k(l —2k-\- n)a*/2 I
in which G? represents in succession each of the roots of unity.
If, however, Bn is negative, the given equation reduces to
yn = nayn~k — I, and the same series gives the roots if GO be
taken in succession as each of the roots of — I.
In order that this series may be convergent the value of a*
must be numerically less than k~k(n — k)k~n ; thus for the quar-
tic y* = Afdx + I, where n = 4 and k = 3, the value of a must
be less than 27-*.
To apply this method to the cubic equation x* *-=.$Ax '±,Bl ',
place n = 3 and k = 2, and put y = Bx. It then becomes
y = $ay ± I where # = A/B1, and the series is
in which the values to be taken for GO are the cube roots of I
or — i, as the case may be. For example, let x* '— 2x — 5 = o..
Placing y— $x, this reduces to y*=o.6%4y-\-i. Here #=0.228,
and as this is less than 4-* the series is convergent. Making
GO = i, the first root is
y = I -f- 0.2280 — 0.0039 + 0.0009 = 1.2250.
*See Bulletin of American Mathematical Society, 1894, Vol. I, p. 3; also
American Journal of Mathematics, 1895, Vol. XVII, pp. 89-110.
30 THE SOLUTION OF EQUATIONS.
Next making & = — i + J- V— 3, GO* is — J — J 4/— 3,
and the corresponding root is found to be
y = - 0.6125 + 0.3836 V^$.
Again, making & = — J — J t7— 3 the third root is found to
be the conjugate imaginary of the second. Lastly, multiplying
each value of y by 5*,
x = 2.095, x = - 1.047 ± I-13^ ^— l»
which are very nearly the roots of x* — 2x — 5 = o.
In a similar manner the cubic x* -\-2x-\- $ = o reduces to
y* = — 0.684}? — i, for which the series is convergent. Here
the three values of GO are, in succession, — I> 4 +i V— 3,
— J + J V— 3, and the three roots are y = — 0.777 an^
7 = 0.388 ± 1.137*'.
When all the roots are real, the method as above stated
fails because the series is divergent. The given equation can,
however, be transformed so as to obtain n — k roots by one
application of the general series and k roots by another. As
an example, let x9 — 243* + 330 = o. For the first applica-
tion this is to be written in the form
--* 330
~
for which n = I and k = — 2. To make the last term unity
place x = -J% and the equation becomes
whence a = 33O*/3.243*. These values of n, k, and a are now
inserted in the above general value of yt and GO made unity;
thus 7=0.9983, whence #,=1.368 is one of the roots. For
*he second application the equation is to be written
+ 243,
ROOTS OF UNITY. 31
for which n = 2 and k = 3. Placing x — 243*?, this becomes
whence a = — 1 10/243*, and the series is convergent. These
values of n, k, and a are now inserted in the formula for y,
and GO is made -f- I and — I in succession, thus giving two
values for j/, from which x^ = 14.86 and x^ = — 16.22 are the
other roots of the given cubic.
McClintock has also given a similar and more general
method applicable to other algebraic equations than trinomials.
The equation is reduced to the form yn = na . <j>y ± I, where
na . $y denotes all the terms except the first and the last.
Then the values of y are expressed by the series
in which the values of <o are to be taken as before. The
method is one of great importance in the theory of equations,
as it enables not only the number of real and imaginary roots
to be determined, but also gives their values when the conver-
gence of the series is secured.
Prob. 20. Compute by the above method all the roots of the
quartic x* -J- x -|- 10 = o.
ART. 15. ROOTS OF UNITY.
The roots of +i and — i are required to be known in the
numerical solution of algebraic equations by the method of the
last article. From the theory of binomial equations given in all
text-books on algebra, the n roots of +i are
( + i)w=cos (w/»)27r+*sin (m/ri)2n, m = i, 2, 3, . . .n, (i)
while those of — i are expressed by
m
(-i)"» =cos (m/ri)n+i sin (m/ri)n, m = i, 2, 3, ... n, (2)
32 THE SOLUTION OF EQUATIONS.
in which i represents the square root of — i. From these general
formulas it is seen that the two imaginary cube roots of + 1 are
-0.5 +0.8660254*',
= —0.5—0.8660254*,
and that the two imaginary cube roots of — i are
£/ = +J + i*\/3= +0.5+0.8660254*,
£2' = +i~i*V3= +0.5-0.8660254*.
For the first case ^+£3 + 1=0 and el£2 = ij as also e1 = £2
e22 = e1, and similar relations apply to the other case.
The imaginary fifth roots of positive unity are given in Art, 8'
expressed in radicals; reducing these to decimals, or deriving
them from the above formula (i) with the help of a trigonometric
table, there result
e =+0.3090170+0.9510565*, £2= —0.8090170+0.5877853*,
£4= +0.3090170-0.9510565*, £3= -0.8090170-0.5877853*,
while the imaginary fifth roots of negative unity are obtained
from these by changing the signs. In general, if a> is an imaginary
wth root of positive unity, — a> is an imaginary nth root of nega-
tive unity.
The imaginary sixth roots of positive unity may be expressed
in terms of the cube roots. Let £ be one of the imaginary cube
roots of +i, then the imaginary sixth roots of +i are +£, +£2.
— £, — £2; these are also the imaginary sixth roots of — i.
From (i) the imaginary seventh roots of +i are found to be
£ =+0.6234898+0.7818316*, £6= +0.6234898-0.7818316*,
£2= —0.2225209+0.9749234*, £5= —0.2225209—0.9749234*, '
£3= -0.9009688+0.4338837*, £4= —0.9009688-0.4338837*,
and if the signs cf these be reversed there result the imaginary
seventh roots of — i.
The imaginary eighth roots of +i are +»', — *, +iv/2(i±*),
and -JV2(i±*). The imaginary ninth roots of +i are the two
SOLUTIONS BY MACLAURIN'S FORMULA. 33
imaginary cube roots of +i, cos %x±i sin -§71, and cos fa±i sin |TT.
The imaginary tenth roots of +i are the five imaginary roots
of +i and the five imaginary roots of — i. For any value of
n the roots of +i may be graphically represented in a circle
of unit radius by taking one radius as +i and drawing other
radii to divide the circle into n equal parts; if unit distances
normal to +i and — i be called +i and —i, the n radii repre-
sent all the roots of + 1 . When this figure is viewed in a mirror,
the image represents the n roots of — i. Or, in other words,
the (m/n)ih roots of +i are unit vectors which make the angles
(m/ri)2x with the unit vector +i, while the (m/rif* roots of — i
are unit vectors which make the angles (m/ri)2n with the unit
vector —i.
The n roots of any unit vector cos 6+i sin 0 are readily found
from De Moivre's theorem by the help of trigonometric tables.
Accordingly the cube roots of this vector are cos %6+i sin J0,
cosj(# + 27r)+zsinj(0 + 27r) and cos"J(0-r-47r)+isin J(0+47r); the
vectors representing these three roots divide the circle into three
parts. The trigonometric solution of the cubic equation (Art. 12)
is one application of De Moivre's theorem.
Prob. 21. Compute to six decimal places two or more of the eleventh
imaginary roots of unity.
Prob. 22. Compute to five decimal places the five roots of the
equation y?— 0.8 — 0.62=0.
Prob.. 23. Compute to five decimal places the six roots of the
equation #6— 80+602' =o.
ART. 16. SOLUTIONS BY MACLAURIN'S FORMULA.
In 1903 Lambert published a method for the expression by
Maclaurin's formula of the roots of equations in infinite series.*
It applies to both algebraic and transcendental equations, and
for the former it gives all the roots whether they be real or imag-
inary. The method is based on the device of introducing a
* Proceedings American Philosophical Society, Vol. 42.
34 THE SOLUTION OF EQUATIONS.
factor x into all the terms but two of the equation /fy) =o, whereby
y becomes an implicit function of x. The successive derivatives
of y with respect to x are then obtained, and their values, as also
those of y, are evaluated for x=o. By Maclaurin's formula,
the expansions of y in powers of x become known, and if x be
made unity in these expansions, the roots of f(y)=o are found,
provided the resulting series are convergent.
To illustrate this method by a numerical example, take the
quartic equation
y4-3y2+7$y-io 000=0, (i)
and introduce an x into the second and third terms, thus,
— 10 000=0. (2)
By Maclaurin's formula y may be expressed in terms of x,
and then when x is made unity, the four series thus obtained
furnish the four roots of (i). Maclaurin's formula is
where y0, (dy/dx\, (dzy/dx2)0, etc., denote the values which y
and the successive derivatives take when x is made o. Differen-
tiating equation (2) twice in succession, and then placing x=o,
there are found
+ 10,
(dy/dx\ =-0.1125, -0.2625, + 0.1875-0.0750*, +0.1875+0.0750*,
(d*y/dx2)0= — 0.0030, +0.0030, — 0.0000+0.00392, —o.oooo— 0.0039^
in which i represents the square root of negative unity. Sub-
stituting each set of corresponding values in Maclaurin's formula
and then placing x = i, there result
y,= +9.886, 3/3=0.1875 +9.927*',
y2= -10.261, ^=0.1875 -9.927*,
which are the roots of (i), all correct to the last decimal.
SOLUTIONS BY MACLAURIN*S FORMULA. 35
This method may be readily applied to the trinomial equation
yn—nayn~k — b=o. When x is inserted in the second term, the
series obtained is
JA l-*
n) (i-
2k+n)a*/2\
($"
and each of the roots is hence expressed in an infinite series,
since bn has n values. This series is convergent when an is
numerically less than k~k(n — k)k~nbk1 and for this case the roots
can be computed. Now the condition an = k~k(n — k)k~nbk is
that of equal roots in the trinomial equation; hence for the cubic
equation the above series is applicable when one root is real and
the others imaginary, while for the quartic equation it is applicable
when two roots are real and two imaginary. For the irreducible
case in cubics and quartics the above series does not converge
and the roots cannot be computed from it; this case is treated
on the next page by inserting x in other terms. This series is the
same as that derived for trinomial equations by McClintock's
method of enlargement (Art. 14).
Asa special case take the quintic equation y5 — $ay — i=o, in
which the value of n is 5, that of k is 4, and those of b* are the
five imaginary roots of unity (Art. 15). When a is less than
4~4, or a less than about 0.33, the above series applies, and if e
designates one of the imaginary fifth roots of unity (Art. 15),
the five roots of the equation are
yl = i + a — a2 + a3 — -2-J-a5 +
36 THE SOLUTION OF EQUATIONS.
For example, let a=o.i, or y5 — \y — 1=0; then the value of yl
is found to be +1.09097, while the other roots are
y2= +0.23649 + 1.014701, }'3= -0.781975+0.48372*,
y^= +0.23649-1.01470?', ^=-0.781975-0.48372*,
which are correct in the fifth decimal place.
For the case where an is greater than k~k(n — k)k~nbn in the
trinomial equation yn — nayn~k — b = o, the roots may be obtained
by inserting x in other terms than the second. To illustrate the
method by the quintic y5 — $ay — 1=0, let x be placed in the
last term, giving y5 — $ay — #=o; obtaining the derivatives and
making n=o, there is found a series giving four of the roots,
since (50)* in this series has four values. Again, placing x in
the first term the equation is xys — $ay — i=om, and applying the
method, there is found a series which gives the other root: It
may also be shown that these series are convergent when a5 is
numerically greater than 4~4. When #5 = 4~4 the quintic has two
equal roots and the series do not apply, but in this case the equal
roots are readily found (Art. 5) and after their removal the other
three roots are found by the solution of a cubic equation.
When this method is applied to an algebraic equation of the
Mth (-[egree wnich contains more terms than three, there may be
obtained several series by inserting x in different terms, and the
series desired are those which are convergent. A general rule
for selecting the terms which are to contain x is given by Lam-
bert, and he applies the method to the solution of the quintic
equation y5 — ioyz + 6y + 1 =o. First, writing y5 — ioy* + 6xy +x = o,
the values of y0 are +3.167 and —3.167, those of (dy/dx)0 are
— i. oo and +0.090, and those of (d*y/dx*) are — 0.016 and
+0.016; inserting these In Maclaurin's formula there are found
^=+3.05 and ;y2=— 3-o6. Secondly, writing xys — io^3 + 6^ +
x=o, a series results which gives ^=+0.87 and y4= —o.6g.
Lastly, writing ocy6—ioxy*+6y+i, there is found y$= — 0.17.
This method may likewise be used for computing one of
the roots of a transcendental equation, provided the resulting
SYMMETRIC FUNCTIONS OF ROOTS. 37
series is convergent. For example, take 2^+log y — iox>oo=o.
Writing 2y+x\og y — io 000=0, there are found the values
y0= + 5 ooo, (dy/dx\ = - J log y0, and (dzy/dxz\= +0.0001 log y0.
When the logarithm is in the common system the root is 7=4998.1 5 ;
when it is in the Naperian system the root is 7=4995.74.
Prob. 24. Compute the roots of x3— 2X— 2 = 0 by the above method
and also by that of Art. 9.
Prob. 25. The equation y4— n 727^+40385 = 0 occurs in a paper
on the precession of a viscous spheroid by G. H. Darwin in Philosoph-
ical Transactions of the Royal Society, 1879, -Par^ n'> P- 5°8- Com-
pute the four roots to five significant figures.
ART. 17. SYMMETRIC FUNCTIONS OF ROOTS.
The coefficients of an algebraic equation are the simplest
symmetric functions of its roots. Let the equation be
xn-axn-l+bxn-2-cxn-3+dxn~*-. . .=o, . . (i)
and let xv x2, XB, . . . be its n roots. Then
C = XJC2XZ + X2X3X4 + . . . , d
and the last term is ±xlx2x5 . . . xn. All.symmmetric functions of
the roots may be expressed in terms of the coefficients. •
The sums of the powers of the roots are important symmetric
functions. Let Sm represent xlm+x2m+x3m+. . . ; then when
m is equal to or less than n, the following are the Newtonian
expressions for the sums of the powers of the roots :
S i = a* — ^
Let ±1 represent the coefficient of the (w + i)th term in the
general equation (i), this being + when m is even and -- when
m is odd. Then the following general formulas furnish values
of Sm for all cases:
-3+. . . ±ml =o,
— . . . ±lSm=O,
38 THE SOLUTION OF EQUATIONS.
For example, take x3 — 200-2=0, for which a=o, b= —2,
c=+2\ then from the first formula ^=0, S2 = 4, 53 = 6, and
from the second formula S4 = 8, 55 = 2O, S6 = 28, etc.
Other important symmetric functions of the roots are the
sums of the squares of the terms in the above expressions for
the coefficients b, c, d, etc. Let these be called B, C, D, etc., or
B = x*x* + x22x32 + . . . , C= x,2x22x32 + X22x32x42 + . . . ,
and let it be required to find the values of B, C, D, etc., in terms
of a, b, c, etc. For tfris purpose let (i) be written
xn+bxn~2+dxn~4+. . .=a#n-1+c#n-3+e#n-5+. . . ,
and let both members be squared and the resulting equation be
reduced to the form
. . (2)
in which y represents x2. This equation has n roots x^2, x22,
x32, . . . ; hence the value of A is x^+xf+x^-t-. . . , and the
values of B and C are the symmetric functions above written.
The algebraic work shows that
A=a2-2b, B = b2-2ac+ad, C = c2 —
and thus in general any coefficient in (2) is obtained from those
in (i) by the following rule: the coefficient of ym in (2) is found
by taking the square of the coefficient of xm in (i) together with
twice the products of the coefficients of the terms equally re-
moved from it to right and left, these products being alternately
negative and positive. 0
An equation whose roots are the squares of those of (2) may
be obtained by a similar process, the equation being
zn-A1zn~l^Blzn-2-Clzn-3+Dzn-4-. . .=o, . . (3)
in which Alt Blt Cv . . . are computed from A, B, C, in the same
manner that A, B, C, ... were computed from (i). For example,
take the equation ^7+3^4 +6=0; the equation whose roots are
LOGARITHMIC SOLUTIONS. 39
squares of those of the given equation is y1 +gy4-\-^6y2 +36=0,
and that whose roots are the fourth powers of those of the given
equation is 27+8iz4-648z3 + 194422 -25922 + 1296=0.
Prob. 26. Find an equation the roots of which are the fourth powers
of the roots of xs+x+io=o.
Prob. 27. For the cubic equation x3—ax2+bx—c=o show that the
value of x*xf+xfxf+xfoc* is b3—^abc^-^c2.
Prob. 28. For the quartic equation x4— axs+bx2— cx+ d=o show
that the value of 55 is a5— $a?b-$ab2+$a2c— $ad— $bc.
I
ART. 18. LOGARITHMIC SOLUTIONS.
A logarithmic method for the solution of algebraic equations
with numerical coefficients was published by Graffe in 1837 and
exemplified by Encke in 1841.* The method involves the forma-
tion of an equation whose roots are high powers of the roots
of the given equation; to do this an equation is first derived,
by help of the principles in Art. 17, whose roots are the squares
of those of the given equation, then one whose roots are the squares
of those of the second equations or the fourth powers of those
of the given equation, and so on. With the use of addition and
subtraction logarithms, the greater part of the numerical work
may be made logarithmic. The method is of especial value
when all the roots of the given equation are real and unequal.
To illustrate the theory of the method, let p, q, r, s, etc., denote
the roots, each of which is supposed to be a real negative number;
let [p] denote p + q+r + . . . , [pq] denote pq+qr+rs+. . . , and
so on. Then the general algebraic equation may be written
xn-[p]xn-l+[pq]xn-2-[pqr]xn-3+[pqrs]xn~*-. . . , (i)
and the equation whose roots are p2, q2, r2, . . . is, by Art. 1 7,
yn-[p2]yn-l+[p2q2]yn-2-[p2q2r2]yn-3+[p2q2r2s2]yn-*-. . . ,
in which [p2] denotes p2 +q2+r2 +. . . , [p2q2] denotes p2q2 + q2r2+. . . ,
* Crelle's Journal fur Mathematik, 1841, Vol. XXII, pp. 193-248.
40 THE SOLUTION OF EQUATIONS.
and so on. From this equation another may be derived having
the roots p4, q4, r4, . . . , and then another may be found having
the roots ps, (f, r8, . . . . This process can be continued until
an equation is derived whose roots are pm, qm, rm, . . . , where m
is a power of 2 sufficiently high for the subsequent operations.
This equation is
zn—[pm]zn~l+[pmqm]zn~2—[pmqmrm]zn~3-}-. . . .
Now let p be the root of (i) which is largest in numerical
value, q the next, r the next, and so on. Then, as m increases
the value of [pm] approaches pm, that of [pmqm] approaches pmqm,
that of [pmqmrm] approaches pmqmrm, and so on. Hence when
m is large [pm] is an approximation to the value of pm, and
[pmqm]/[pm] is an approximation to the value of qm. Accordingly
by making m sufficiently large, the values of pm, qm, rm, . . . ,
and hence those of p, q, r, . . . , may be obtained to any required
degree of numerical precision. When two roots are nearly equal
numerically, it will be necessary to make m very large; when
equal roots exist they should be removed by the usual method.
To illustrate the application of the method, let it be required
to find the roots of the quintic equation
— Six3 — 34#2 +464^ — 181 =o.
By comparison with (i) of Art. 17 it is seen that a= — 13, &= — 81,
c=+34, d= +464, e=+i8i. The equation whose roots are
the squares of those of the given quintic is now found from (2)
of Art. 17, by computing A= a2 — 2^ = 331, B = b2 — 2^+2^ = 8373,
C = c2-2bd + 2ae = 71618, D = d2- 2^ = 202988, £ = ^=32761,
and then
T5 ~33I74 +83737* — 7i6i8>>2 +202988? — 32761=0.
Taking the logarithms of the coefficients, this equation may be
written
75 ~ (2.5I983)/ + (3.92288)^ - (4.85502)7' + (5.30747)7
-(4-5I536)=o,
LOGARITHMIC SOLUTIONS. 41
in which the coefficients are expressed by their logarithms in-
closed in parentheses. The logarithms of the coefficients for the
equation whose roots are the fourth powers of the given quintic
are now found by the use of addition and subtraction logarithmic
tables, and this equation is
25 - (4.96y62)z4 + (7.36364)2* - (9.24342)22 + (10.56243)2
- (9.03072) =o.
Next the equation whose roots are the eighth powers of the
roots of the given quintic is derived from the preceding one in a
similar manner and is found to be
w5-(9-9329°X + (I4.3I934X-(i8.i4025X + (21. 12363)-^
— (18.06144) =o,
and then the equation whose roots are the sixteenth powers of the
roots of the given quintic is
vs - (19.86580)^ + (28.29778)^ - (36.I3I3I)?;2 + (42.24726)7;
- (36.12288) =o.
It is now observed that the coefficients of the second, fourth,
and fifth terms in the equation for v are the squares of those
of the similar terms in the equation for w. Hence two of the
roots are now determined as follows:
log#8 = 9-93290) log # = 1.24161, # = 17.443;
log /8 = i8.o6i44 — 21.12363, log £ = 1.61723, 1=0.4142.
These are the numerical values of the largest and smallest roots
of the given quintic, but the method does not determine whether
they are positive or negative; by trial in the given quintic it will
be found that —17.443 and +0.4142 are roots. To obtain the
others, the process must be continued until two successive equa-
tions are found for which all the coefficients in the second are
the squares of those in the first. Since in this case two roots lie
near together, the process does not terminate, with five-place
logarithms, until the 51 2th powers are reached. The three
42 THE SOLUTION OF EQUATIONS.
remaining roots are thus found to be £=+3. 230, 7=+
and s= —1.4142.
When this method is applied to an algebraic equation which
has imaginary roots, this fact is indicated by the deviation of
signs of the terms in the power equations from the form as given
in (2) of Art. 17; that is, these signs are not alternately positive
and negative. As an example of such a case Encke applies the
process to the equation
- $x +6 =o,
and deduces for the equation of the 256th powers of the roots
— (190.991 29)l>2 — (195. 21 132)^ — (199.20704) =O.
Here it is seen that the coefficients of v4 and v have signs
opposite to those of the normal , form, and hence two pairs of
imaginary roots are indicated. The real roots of the given equa-
tion are then determined as follows:
log ^256 = 74.95884, log #! =0.29281, #!=— 1.9625,
Iog^2256 = i22.8i202- 74.95886, log #2 =0.1 8693, #2=+I-5379>
log #6256 = 1 90.991 29 -i 79. 58882, log #6 =0.04454, #6=+i. 1080,
while the logarithms of the moduli of the imaginary pairs may
be obtained by taking the difference of the logarithms of v5 and
v3 and that of v2 and i*0, and dividing each by 512. It is then
not difficult to show that the two quadratic equations
x2 -0.609213; + 1.07668 =o, x2 + 1.29263 + 1 .66642 =o,
furnish the imaginary roots of the given equation of the seventh
degree.
Prob. 29. Compute the roots of x5— io#3+6#+i = o.
Prob. 30. How many real roots has the equation
Can they be advantageously computed by the above method ? What
is the best method for finding the roots to four decimal places ?
INFINITE EQUATIONS. 43
ART. 19. INFINITE EQUATIONS.
An infinite series containing ascending powers of oc may
be equated to zero and be called an infinite equation. For
example, consider the equation
in which the first member is the expansion of sin x; this equa-
tion has the roots o, TT, 271, y: , etc., since these are the values
which satisfy the equation sin #=o. Again,
X2 X4 X* X8
I-f2l+T! + 6! + 8! + '"=0
is the same as cosh x=o, and hence its roots are JTTZ, f ;rz, etc.
The series known as Bessel's first function when equated to
zero furnishes an infinite equation whose roots are of interest
in the theory of heat * ; this equation is
X2 X* XQ X8
i — ;.+•
22-42 22-42-62 ' 22-42-62-82
and it has an infinite number of real positive roots, the smallest
of which is 2.4048. The roots of equations of this kind may
be computed by tentative methods, and when they are approxi-
mately known Newton's rule (Art. 4) may be used to obtain
more precise values.
As an example take another equation which also occurs in
the theory of heat, namely,
x2 x3 x* x5
+ (^?~(3!? + (4!?~(5!72 +
It is plain that this equation can have no negative roots, for a
negative value of x renders all the terms of the first member
* Mathematical Monograph, No. 5, pp. 23, 63.
44 THE SOLUTION OF EQUATIONS.
positive. Calling the first member f(x), the first derivative is
x x2 x3 x4
By trial it may be found that one root of f(x)=o lies between
1.44 and 1.45. For # = 1.44, f(x) becomes +0.002508 and/'(#)
becomes +0.4334. Then f(x)/f(x) = 0.0058, and accordingly
^ = 1.44+0.0058 = 1.4458 is one of the roots. Another root of
this equation is #2 = 7.6178. In general equations of this kind
have an infinite number of roots.
The term infinite is sometimes applied to an algebraic equa-
tion having an infinite root, and cases of this kind are often
stated as curious mathematical problems. For instance, the
solution of the equation
when made by squaring each member twice, gives the roots x = f 0
and x=o. But x=o does not satisfy the equation as written,
although it applies if the sign of the second radical be changed.
The equation, however, may be put in the form
and it is now seen that x= °o is one of its roots. The false value
x=o arises from the circumstance that the squaring operations
give results which may be also derived from equations having
signs before the radicals different from those written in the given
equation.
Prob. 31. Differentiate the above function of Bessel and equate
the derivative to zero. Compute two of the roots of this infinite
equation.
Prob. 32. Find the roots of 2\/x— 2=\/x— 3+\/x— i.
Prob. 33. Consult a paper by Stern in Crelle's Journal fur Mathe-
matik, 1841, pp. 1-62, and explain his methods of solving the equations
cos x cosh #+ 1 = o and (4- 3#2) sin x- 4* cos x= o.
NOTES AND PROBLEMS. 45
ART. 20. NOTES AND PROBLEMS.
The algebraic, solutions of the quadratic, cubic, and quartic
equations are valid for imaginary coefficients also. In general
the roots of such equations are all imaginary. The method of
McClintock (Art. 14) and that of Lambert (Art. 16) may also*
be applied to the expression of the roots of these equations ia
infinite series.
As an illustration take the equation x5 — 3^+4^=0. By any
method may be found the roots x^= — i, x2= —0.52 + 1.936 and
#3= —0.52 — 1.936; two of the roots here form a pair in which the
imaginary part is the same for both, the real and imaginary parts
of the complex quantities having changed places. There are,
however, many equations with imaginary and complex coefficients
in which pairs of roots do not occur.
The most general case of an algebraic equation is when the
coefficients a, b, c, . . . in (i) of Art. 17 are complex quantities
of the form m+ni, p+qi, .... Such equations rarely, if ever,
occur in physical investigations, but the general methods ex-
plained in the preceding pages will usually suffice for their solution,
approximate values of the roots being first obtained by trial if
necessary. In general the roots of such equations are all com-
plex, although conditions between m and n, p and q, etc., may
be introduced which will render real one or more of the roots.
Prob. 34. Show that the equation x—ex=o has many pairs of
imaginary roots and that the smallest roots are 0.3181 ±1.3372$'.
T^ 3C^ \^ IC*'
Prob. 35- Solve -+_+_++.. .— x.
Prob. 36. Discuss the equation x— tan#=o and show that its
smallest root is 4.49341.
Prob. 37. Find the value of x in the equation e**+i = o, and also-
that in the equation e^x— i=o.
Prob. 38. Show that xz+(a+bi)x+c+di=o has one real and one
complex root when the coefficients are so related that b2c+d2—abd=o^
46 THE SOLUTION OF EQUATIONS.
Prob. 39. When and by whom was the sign of equality first used?
What reason was given as to the propriety of its use for this purpose ?
Prob. 40. There is a conical glass, 6 inches deep, and the diameter
at the top is 5 inches. When it is one-fifth full of water, a sphere
4 inches in diameter is put into the glass. What part of the vertical
diameter of the sphere is immersed in the water?
Prob. 41. When seven ordinates are to-be erected upon an abscissa
line of unit length in order to determine the area between that line
and a curve, their distances apart in order to give the most advan-
tageous result are, according to Gauss, determined by the equation
Compute the roots to five decimal places and compare them with
those given by Gauss.
INDEX.
Abel's discussion of quintic, 22.
Algebraic equations, i, 2.
solutions, 15-24.
Approximation of roots, 3, 12, 49.
rule, 6.
Bessel's function, 43.
Binomial equations, 16, 26, 31.
Cardan's formula, 17, 18, 28, 29.
Catenary, 14.
Cube roots of unity, 32.
Cubic equations, 3, 17, 28.
Cylinder, floating, 13.
De Moivre's quintic, 22, 26.
theorem, 33.
Derivative equation, 9.
Elliptic solution of quintic, 23.
Fifth roots of unity, 16, 32.
Graphic solutions, 3.
Graphs of equations, 9.
Graffe's method, 14.
Horner's process, 2, 12.
Howe truss strut problem, 21.
Hudde's method, 8, 12.
Imaginary coefficients, 45.
roots, n, 18, 20, 30, 34, 42.
Infinite equations, 43.
Lagrange's resolvent, 15.
Lambert's method, 33.
Literal equations, i, 10.
Logarithmic solutions, 39.
Maclaurin's formula, 34.
McClintock's quintic discussion, 23.
series method, 29.
Newton's approximation rule, 6.
Numerical equations, i, 10.
Powers of roots, 38, 40.
Properties of equations, n.
Quadratic equations, 16.
Quartic equations, 19, 20.
Quintic equations, 21, 36.
Real roots, 2, 3, 12, 40.
Regula falsi, 5.
Removal of terms, 22.
Resolvent, 17.
Root, i.
Roots in series, 27, 29, 31, 34.
of unity, 16, 31.
Separation of roots, 8.
Sixth roots of unity, 17, 32.
Sphere, floating, 13, 25.
Sturm's theorem, 8, 12.
Symmetric functions, 37.
Transcendental equations, 2, 4, 13.
Trigonometric solutions, 24, 26,
Trinomial equations, 29, 35.
Tschirnhausen's transformation, 22
Vectors, 16, 33.
Water-pipe problem, 13.
QA
211
M47
1906
Merriman, Mansfield
The solution of equations
4th ed., enl.
Ph> licit fc
Applied ScL
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