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AN

ELEMENTAEY TREATISE

OK THE

, THEORY OF EQUATIONS,

WITH A COLLECTION OP EXAMPLES.

^.

BY

I. TODHUNTER, M.A. F.R.S.,

HONOKABY FELLOW OF ST JOHN's COLLEGE, CAMBRIDGE.

Hontron :
MACMILLAN AND CO.

1885.

[The Eight of Translation is reserved.]

STEREOTYPED EDTTION,
3^ f ^^

PRINTED BY C. J. CLAY, M.A. k SON^
AT THE UNIVERSITY PRESS.

PKEFACE.

The present treatise contains all the propositions which
are usually included in elementary treatises on the Theory
of Equations, together with a collection of examples for
exercise.

As the Theory of Equations involves a large number of
interesting and important results, which can he demonstrated
with simplicity and clearness, the subject may advantage-
ously engage the attention of a student at an early period
of his mathematical course. The present treatise may be
read by those who are familiar with Algebra, since no
higher knowledge is assumed, ex6ept in Arts. 149, 175, 268,
308... 314, and Chapter xxxi., which may be postponed by
those who are not acquainted with De Moivre's Theorem in
Trigonometry. This work may be regarded as a sequel to that
on Algebra by the present writer, and accordingly the student
has occasionally been referred to the treatise on Algebra for
preliminary information on some topics here discussed.

In composing the present work, the author has obtained
assistance from the treatises on Algebra by Bourdon, Lefe-
bure de Fourcy, and Mayer and Choquet ; on special points
he has consulted other writers, who are named in their ap-
propriate places in the course of the work.

The examples have been selected from the College and
University examination papers, and the results have been
given where it appeared necessary ; in most cases however,
from the nature of the example, the student will be able
immediately to test the correctness of his result.

vi PREFACE.

In order to exhibit a comprehensive view of the sub-
ject, the present treatise includes investigations which are
not to be found in all the preceding elementary treatises,
and also some investigations which are not to be found in
any of them. Among these may be mentioned Cauchy's
proof that every equation has a root, Horner's method, the
theories of elimination and expansion, Cauchy's theorem on
the number of imaginary roots, the researches of Professor
Sylvester respecting Newton's Rule, and the theory of
determinants. The account of determinants has been princi-
pally taken from a treatise on that subject by Baltzer, which
was published at Leipsic in 1857 ; this is an excellent work,
distinguished for the completeness of its proofs of the funda-
mental theorems, and for the numerous applications of those
theorems which it affords.

For the parts of the Theory of Equations which are
beyond an elementary treatise, the advanced student may
consult Serret's Cours d'Alghhre Superieure : there, for
example, will be found a demonstration of the theorem,
that the general algebraical solution of an equation of a
degree above the fourth is impossible. The article Equation,
by Professor Cayley, in the ninth edition of the Encyclopce-
dia Britannica should also be noticed. Valuable historical
information, relating to the higher parts of the subject, will
be found in papers on Approximation and Numerical So-
lution, by Mr James Cockle, in the Ladys and Gentleman's
Diary for the years 1854 and 1855, and also in papers on
Equations of the Fifth Degree by the same author in the
same work, for the years 1848, 1851, 1856, 1857, 1858, and
1860.

I. TODHUNTER

St John's College,
March, 1880.

CONTENTS.

PAGE

I. Introduction 1

n. On the Existence of a Koot 15

ni. Properties of Equations 22

IV. Transformation of Equations 31

- V. -Descartes's Eule of Signs ...... 40

VI. On Equal Boots 48

VII. Limits of the Roots of an Equation. Separation of the Eoots 56

Vni. Commensurable Roots 73

IX. Depression of Equations 79

X. Reciprocal Equations 85

XI. Binomial Equations 89

XII. Cubic Equations 99

Xin. Biquadratic Equations 112

XIV. Sturm's Theorem 121

XV. Fourier's Theorem 130

XVI. Lagrange's Method of Approximation .... 135

XVn. Newton's Method of Approximation with Fourier's Additions 142

XVin. Horner's Method ^50

XIX. Symmetrical Functions of the Roots .... 165

XX. Applications of Symmetrical Funptions . . ' . . 174

XXI. Sums of the Powers of the Roots 181

XXn. Elimination .193

XXin. Expansion of a Function in Series 206

XXTV. Miscellaneous Theorems 214

XXV. Cauchy's Theorem . â€¢ 231

XXVI. Newton's Rule and Sylvester's Theorem ... . . 236

XXVII. Removal of Terms from an Equation .... 251

XXVin. Introduction to Determinants 256

XXIX. Properties of Determinants 266

XXX. Applications of Determinants ..'.... 284

XXXI. Trigonometrical Formulae . . . . . . 296

Examples 308

THEORY OF EQUATm^^^

â€¢V ^^ THE-

I. INTRODUCTION.

UlTiyBESITY

1. The reader can easily obtain a general idea ol iM Object
of the following treatise by a reference to the theory of quad-
ratic equations which he is supposed to have already studied.
The equation ax^ + bx + c = has two roots, namely,

-.h^Jb'-Aac^

2a
and with respect to these roots, we know that their sum is ,

and their product is - ; that is, their sum is equal to the coeffi-

h c
cient of the second term of the equation x^ + -x+ ~=Q. with its

^ a a

sign changed, and their product is equal to the last term of this
equation. (See Algebra, Chap, xxii.) Now it may be said that
the general object of the following pages is to establish results
with respect to equations of a higher degree than the second,
similar to those which have been established in Algebra respect-
ing equations of the second degree. The results obtained will be
useful in other branches of mathematics, and the methods of
investigation will afford valuable exercise to the student, since
they are not too difficult for a person who has gained a knowledge
of Algebra, and at the same time have sufficient variety to oc-
cupy his attention.

2. The words equation and root are already familiar to the
student from their use in Algebra; but for distinctness we will
give a definition of them.

T. E. 1

2 INTRODUCTION.

Any Algebraical expression whicli contains x may be called
a function of jr, and may be denoted by fix). Any quantity
wMcli substituted for x in f{x) makes f{x) vanish, is called a
root of the equation /(cc) = 0.

An expression of the form 4^){ -^

Â«Â«" + hx"~ ^ + ccc""^ + ...A-kx + l, <I

where 7^ is a positive integer, and the coefficients a, 6, c ...Jc, I,
do not involve x, is called a rational integral function of x of
the w*^ degree; and if we wish to find what value of x makes
this function vanish we have to find a root of a rational integral
equation of the n^^ degree; this is the kind of equation which
we shall consider in the present treatise. In such an equation
we may if we please divide by the coefficient of the highest
power of Xf so as to leave that power with only unity for its
coefficient ; the equation then takes the form

We shall say that the equation is now in its simplest form ;
as will be seen hereafter, some of the properties of equations can'
be enunciated more concisely when the equation is in this form
than when ic" has a coefficient which is not unity. If we do
not wish to suppose the coefficient of aj" to be unity, we may
conveniently denote it by 2?o j then the equation takes the form

The term p^ is called the term independent of x.

3. It must then be remembered that by equation we mean
rational integral equation ; an equation which is not of this form
may often be reduced to it by algebraical transformations; for
example, the equation ax^ + hx + c Jx =f may be reduced to a
rational integral form by transposing c Jx and f and then
squaring ; it will thus become a rational integral equation of
the fourth degree. Equations which involve logarithmic func-
tions, or exponential functions, or trigonometrical functions, or
irrational algebraical functions, will not be directly included in
our investigations ; for example, such equations as tan aj - 6""= 0,

INTRODUCTION. 3

or x\ogx â€” a = 0, will not be included. However, the theory
which will be given of rational integral equations will indirectly
throw some light on these excluded equations.

And when we speak of any function /(x) we shall always
mean a rational integral function of x, unless the contrary is
specified.

4. A remark of some importance must be made with respect
to the coefficients Pq, Pi, Pzj â€¢â€¢â€¢ i^Â«j i^ the equation

^X + P^X"'^ + jOjjCC""^ + . .. + Pn-2^^ + Pn-v"^ +Pn = 0.

In the quadratic equation ax^ + bx+ c = we are able to solve
the equation without knowing what particular numbers are de-
noted by a, b, c'j all we require to know is that a, b, c are some
numbers independent of x. If we have to solve the equation
x^ -I2x+15 = we may either transpose the 15 and complete
the square in the ordinary way, or we may take the general
formulae given in Art. 1, and put in them a=l, 6 = â€” 12, c=15.
If we wish to solve an equation without having the numerical
values of the coefficients previously assigned, we are seeking
what may be called the algebraical solution of the equation;
and if we can effect the algebraical solution of the general
equation of any degree, we may obtain a numerical solution of
an equation of that degree, by substituting the numerical values
of the coefficients in the general formula which gives the alge-
braical solution. As we proceed we shall find that the algebraical
solution of equations up to the fourth degree inclusive has been
effected; but both in equations of the third degree and of the
fourth degree, when we substitute the numerical values of the
coefficients in a specific equation in the general formula, the
result takes a form which is sometimes practically useless. And
beyond equations of the fourth degree the general algebraical
solution of equations has not been carried, and it appears cannot
be carried.

But with respect to what may be called the arithmetical solu-
tion of equations in which the coefficients are given numbers,
more success has been obtained. Thus, for example, although

1â€”2

4 INTRODUCTION.

we cannot solve algebraically the general equation of the fifth
degree, we can by numerical calculation discover any root which
an equation of the fifth degree with known numerical coefficients
may have, or at least we can approximate as closely as we please
to such a root.

"- 5. Let us denote hy/(x) the expression

p^x''+p^x''-'-i-p^''-'+... +pâ€ž.,x'+p^_^x + pâ€ž;
then the value of this expression when x = a may be denoted by
f{a). We will shew how the numerical value of f(a) may be
most easily calculated, supposing that the coefiicients of f{x), and
also a itself, are specified numbers.

Take for example an expression of the third degree j then
we wish to find the numerical value of

First obtain

Po<^;

p,a +p,;

multiply by a, this gives

p,a'+p,a;.

add P2, this gives

Poa'-^p,a+p,;

multiply by a, this gives

p,a^+p^a'+p,a;

add ^3, this gives

p,a'-\-p,a'+p^a-hp^.

We may arrange the

process in the following' way;

^0, Pi P2

Ps.

p,a . p^a' +

p,a .p,a^ + p,a' + p,a

Po<^-^Pi Po<^'+.

PlÂ« +P, Po^^ + Px^' +i^2Â« + P^

We may proceed in the same way whatever may be the
degree of /(aj). For example, required the numerical value of-
3ic^ - 2Â£c" - 5a; + 7 when a; = 3.

3-2 0-5+7

'- +9 +21 +63 +174

* +7 +21 +58 +181

(jS^ D'

Thus the result is 181.

INTRODUCTION. 5

6. If any rational integral function of x vanishes when
X = a, the function is divisible 5y x - a.

Let f{x) denote the function; then we have given that
f(a) = 0, and we have to prove that/(Â£c) is divisible hj x-a.

Divide f{x) hj x-a by common algebra until the remainder
no longer contains x; let Q denote the quotient and It the re-
mainder if there be one. Then f{x) = Q {x-a) + R. In this
identity put a for x; since ^ is a rational integral function of x
it cannot become infinite when x = a', therefore Q (x â€” a) vanishes
when x = a. A\?>of{x) vanishes when x=ahj supposition. Thus
R vanishes when a; = a ; but R does not contain (r, so that if it
vanishes when x = ait always vanishes. That is, ^ = and a? â€” Or
divides /(a;).

7. The above demonstration is important and instructive;
we may ho^vever prove the theorem in another way, which will
moreover have the advantage of exhibiting the form of the
quotient Q. Suppose

then since /(a) = we have/(a;) =f{x) -f{a)

= Po (^" - ^") -^Pt (^""^ ~ ^"~') '^Ps (^""^ ~ ^"~') '^ â€¢" +Pn-1 (^ - ^)-

Now the terms ic"-a", Â£c"~^â€” a"~\ ... are all divisible hj x â€” a
(see Algebra, Art. 483). By performing the division we obtain
for the quotiejit

(p^ {f-^+ ax""-' + a V + ... + a-'-^x + a""')

'+ pXcd"^ + ace"-' + a V"' + . . . + a"-')

+ â€¢â€¢â€¢ L

We may rearrange the quotient thus :

PX"' + {P,^ -^Pi) a^""' + (PoÂ»' +Px^-^P^ ^"'^ + '"

+py~^ +^y + ... +;?â€ž_!,

and we may denote it by

G INTRODUCTION. |

The new coefficients are therefore connected with each other
and with the old coefficients by the forinulse

that is, each new coefficient is found hy midtiplying the preceding
new coefficient hy a and then adding the corresponding old coeffir
dent. It will be observed that these new coefficients are succes-
sively determined by the process of Art. 5.

8. If x â€” Si divide f (x) which is any rational integral /unc-
tion of X, then a is a_ root of the equation f (x) = 0.

For let Q denote the quotient when f{x) is divided hj x â€” a,
then f(x) = Q (x â€” a). In this identity put a for x, then Q is not /
infinite, and therefore Q(xâ€”a) vanishes. Thus y (a;) vanishes /
when x = a, and therefore a is a root of the equation y* (a?) = 0.

^ 9. To find the remainder when any rational integral function
1 ofxis divided hy xâ€” c, where c is any constant.

'Letf{x) denote any rational integral function of a;, and divide
f{x) hy x â€” c until the remainder is independent of x; let Q
denote the quotient and E the remainder. Then

f{x)=Q{x-c)-rJR. '
In this identity put c for cc, then Q is not infinite, and therefore
Q{x-c) vanishes j thu?/(c) = R. That is, R is equal to /(c) when
a3= c, but R does not contain x, so that R is equal to f{c) always.
. For example ; if 2>x* â€” 2x^ â€” 5x + 7 is divided by xâ€”3, the
quotient is 3x^+7x^ + 2lx + 68, and the remainder is 181; see
Arts. 5 and 7.

For another example let us divide the same expression by x-i .
3- 2 0-5+7
+ 12+40+160+6^0
+ 10 + 40 + 155+627

Thus the quotient is 3x^+ 10Â£c^+ 40a3+ 155, and the remainder
is 627.

This process is a particular case of Synthetic Division; see
Algebra, Chapter lviii.

INTEODUCTION. 7

10. Let fix) be any rational integral function of x, and
suppose x-^y put for x, then we propose to arrange f{x + y)
according to powers of y, and to determine the coefficients of
the different powers.

Let fix) =p^x'' + pX~' +i?2Â«""' + ... +pâ€ž_,x +p^ ; then
f{x+y)=p,{x+yY+p^(x+yy-'+p^ix+yY-'+â€ž. +pâ€ž_,{x+y)+p^.

Expand (x + y)", {x+yy~\ ... by the Binomial Theorem, and
arrange the whole result according to powers of y; we thus
obtain for f{x + y) the following series :

P^X" + pX~'+P2^"~'+ ... +Pn-l^+Pn

+ y^npX-' + (n-l)p^x''-'+ (n-2)p^x"-'+... +p\
-f^\n{n-l)p^x-' + {n-l){n-2)p^ar-'+...+.2p^_^\

+
+

y

+
+

-Ui{n-l)...{n-r + l)p^x''^''+{n-l){n-2)...(n-r)p^x''~''~^+..X

IH-

The first line of this series is obviously y (a;). We shall denote
the coefficient of y by f'(x), the coefficient of ~-^ by f"{x)f the

y^

coefficient of r^ by /' " (x), and so on ; this notation becomes
inconvenient when the number of accents is large, and so in
general the coefficient of j- '^U ^^ denoted hjf^x). Hence

/{CO + 2/) =/(*) +lif'ix) + ^/"(x)+^f"'{x) + ...

8 INTRODUCTION.

By inspection it will be seen that tlie functions /(aj), f'{x), f"{^)i
f"'{x), .../"(x) are connected by the following general law: in
order to obtain /'"^^(a;) we multiply each term in /'"(x) by the
exponent of a; in that term and then diminish the exponent by
unity. ^

11. Let us suppose, for example, that/* (a?) is of the fourth .
degree; let

f(x) =_p,a;* + p^x^+p^+p^x+p^.
Tlien f'{x) = ^pjx^+^p^x'+2p^x+p^,
f"{x) = i,3p^x'+3,2p,x + 2pâ€ž
r'{x) = i.3.2p^x+S.2p,,
r'\x) = i.Z.2.p^-

fix + y) =f{x) + yfix) + ^^r{x) + ^/'Â» + ^^f""{x).

If we suppose numerical values assigned to p^, p^^ p.^, p^, p^, and
X, we may calculate separately /(Â£c), /'(x), ... by the method of
Art. 5 j we shall however hereafter, in explaining Horner's method
of solving equations, shew how these calculations may be most
conveniently and systematically conducted.

Eor another example suppose thaty*(aj) =p {x+ c)".

Then f{x)=pL'' + ncx''-'+'^^^-^^c'x"-^ + ...+ ;

therefore
f'{x) =p Lx-'+n{n-l)cx'-^- "" ^"^ '^"^^^ " ^^ cV-^+ ... + nc''-'^ ;

that ia /'(^) =P'^{x + c)""^ :

similarly f"i^ =pn(n â€” 1) (x + cY~^y

f"'{x) =pn {n -l){n- 2) (x + c)'-^
and so on.

Suppose that <^ (x) and ij/{x) are two rational integral functions
of x, and that f(x) = <^ (a;) + i/r (x) ; then it is easily seen that
f\x) = <}>'{x) + il/\x), and f'\x) = </>"(Â«) + ^'\x), and so on.

INTRODUCTION. â–  9

12. If we write the series for /(x + y), beginning with the
highest power of y, we shall have

( , ^, (7^-l)(n-2) 2 n{n-'l){n-'l) ^ â€ž.,
+ |;?3 + {n-2) p^x + ^ :^ -^ p^x' + -^ ^ / p^x'^j 2/"

+ ...

f / IN n(n-l)...(n-r+l) \ â€ž_,
+ U)^ + {n-r + l)p^_^x+...+-^ 'â€”r^ -Po^^y

This may be seen from the form already given for f{x + y), or by /
expanding separately every term in f{x + y), and arranging 2iC-y
cording to descending powers of y.

13. The function /' {x) is called the first derived function of
f{x)j the function /"(a?) is called the second derived function of
f{x), and so on. The reader, when he is acquainted with the ele-
ments of the Differential Calculus, will see that each derived
function is the differential coefficient with respect to x of the
immediately preceding derived function, and that the expression
fovf{x + y) in powers of y is an example of Taylor's Theorem.

Moreover, it must be observed that/''(a3) is deduced from/'(Â£c)
in precisely the same way as /'(a;) is deduced from/(aj). Thus
f"{x) is the first derived function oif{x), and /"'(a?) is the second
derived function oiff(x), and so on. Hence by the preceding
Article we have

f{x-^y)=f(x)^yr{x)^^^r{x) 4-g/'"(^) + ...

r- K ^ ^ \ n-l ^ ^ '

/ 2

Similarly fix + y) =f\x) + yf'ix) + ^^f""{^) + -"

.r-2

....-^/'(x).....^m

And so on.

10 INTRODUCTION.

^^

'14. In any rational integral function ofn arranged according
to descending 'powers of x, any term which occurs may he made to
contain the sum of all which follow it, as many times as we please,
hy taking x large enough, and any term may he made to contain
the sum of all which precede it, as many times as we please, hy
taking x small enough.

Let P(po'' +p^x"~'^+p^x''~'^+ ... +p^_^x'^ +p^_^x + p^ be any ra-
tional integral function of x; suppose for example that the r^ term
p^.^x"'""^^ occurs; that m, suppose p^_^ not zero. Let q denote the
numerical value of the greatest of the coefficients p^, p^+^^-.-p^.
The sum of all the terms which follow the r^^ term cannot exceed

x"~'"^^ â€” 1

5'(aj" '+x" ' ' + ...+X+1), that is, 5' -â€” . The ratio of the r"^

X â€” 1

term to this is P^^tz2}^^ that is, ^^->-^jâ€ž . By taking x

large enough, the numerator can be made as large as we please,
and the denominator as near to q as we please j thus the ratio can
be made as great as we please.

This proves the first part of the proposition. To prove the

second part put a; = - , then we obtain the series

2/"" {Po +Pxy+py + â€¢â€¢â€¢ +Pn-y +Pnyl'

We have now to prove that by taking x small enough, that
is by taking y large enough, any term p^y"" which occurs can be
made to bear as great a ratio as we please to the sum of the terrM*
Po + Pi'!/+ â€¢"+Pr-iy''~^ which precede it; this has been already
proved in the first part. y

15. One of the first questions which can occur in the theory
of equations is whether a root must exist for every equation ; and
we shall now give some simple propositions which establish the
existence of a root in certain cases. We shall require a theorem
which is often assumed as obvious, but which may be proved in
the manner shewn in the next Article.

16. Lety*(aj) be any rational integral function of x, andy(a).
/(5), the values oif{x) corresponding to the values a and h of o? â€¢.

INTRODUCTION. 11

then as x changes from atoh the function fix) will change from
f{a) to/(6), and will pass through every intermediate value.

Let any value c be ascribed to x, and let /(c) be the corre-
sponding value of fix) ; let c + h be another value which may be
ascribed to x ; then by taking 7b small enough f{c + h) may be
made to differ as little as we please from y(c). For

/(. + h) =/(c) +h/'{c) + j^/"(c) + ... + ^f-'i'^)-^^^/" (4

Then, by Art. 14, by taking h small enough, the first term of the

series //'(c), -^ â€” ^/"(^)* fo/'^W? â€¢â€¢â€¢ which does not vanish, can

be made to contain the sum of all which follow it as often as we
please, and by taking h small enough this term will itself be ren-
dered as small as we please. Therefore /(c -f- h) -f{c) can be made
as small as we please by taking h small enough. This shews that
as X changes, f{x) changes gradually, so that ii f(x) takes any
value for an assigned value of x, it will take another value as near
as we please to the former, by taking another value of x which is
sufficiently near to the assigned value. Hence as x changes from
a to 6, the function f{x) must pass without any interruption from
the value /(a) to the value/ (5); for to assert that there could be
interruption would amount to asserting that f{x) could take a
certain value, and could not take a second value as near as we
please to the first value.

/

17. We do not assert iii the preceding Article that f{x)
always increases from f{a) to /(&), or always decreases from f{a)
to f{b); it may be sometimes increasing and sometimes decreasing.
What we assert is, that it passes without any sudden change of
value, from the value fia) to the value f{h). The proposition is
one of great importance, and probably will appear nearly evident
to the student on reflection. It is obvious that f(x) has soTne
finite value for every finite value ascribed to x; also we have
proved that an indefinitely small change in x can only make an
indefinitely small change in fix), so that there can be no break in
the succession of values which /(cc) assumes.

12 INTRODUCTION.

/ 18. The student who is acquainted with Co-ordinate Geo-
metry will find it useful and interesting to illustrate the present
' subject by conceiving curves drawn to represent the functions.
/ Thus letf(x) be denoted byy, so that ?/=/(Â£c) maybe conceived to
' be the equation to a curve ; then by supposing this curve drawn
for the part lying between x = a and x = b, a good idea is obtained
of the necessary consecutiveness in the values assumed by /(x)
between the values /(a) and/ (6).

It must be observed that we do not restrict a, h,/(a), /{b), to
be positive quantities; and by values intermediate between /(Â«)
and /{h) we mean intermediate in the algebraical sense; that
is, any quantity z is intermediate between /(a) and /(b) which
makes z â€”/(a) and /(b) â€” z of the same sign.

19. 7/ two numbers substituted /or n in a rational integral
expression f (x) give results with contrary signs, one root at least o/
the equation f (x) =0 lies between those values o/x.

Let a and b denote the two numbers ; then /(a) and/ (5) have
contrary signs. By Art. 16, as a? changes gradually from a to b,
the expression /(a;) passes without any interruption of value from
/(a) to /(b); but since /(a) and /(b) are of contrary signs the
value zero lies between them, so that /(x) must be equal to zero
for some value of x between a and b ; that is, there is a root of the
equation /(cc) = between a and b.

We do not say that there is only one root. And we do not
say that if /(a) and /(b) are of the same sign there will be no
root of the equation /(cc) = between a and b.

20. An equation o/ an odd degree has at least one real root.
Let the equation be denoted hy/(x) = 0, where

/(x) ^p.x'' + p^x""-' + ... +pâ€ž_^X +^â€ž,

and n i^ an odd number. c .

When X is large enough the first term of /(x), namely p^x"^,

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