The principles of elementary algebra

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IN MEMORIAM
FLORIAN CAJORl

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PRINCIPLES OF ELEMENTARY ALGEBRA

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THE PEINCIPLES

ELEMENTARY ALGEBRA

BY

N. F. pUPUIS, M.A., F.R.S.C.

Protessor op Pure Mathematics in the University of Queen's
College, Kingston, Canada

BEYISKD^AITD O^EnEC'iJ^D ^VITION'

Wehj got*
THE MACMILLAN COMPANY

LONDON : MACMILLAN & CO., Ltd.
1900

All rights reserved

'S

Copyright, 1892,
By MACMILLAX AND CO.

Set up and clectrotyped August, 1892. Reprinted September,
1900.

'y:■^M^w^^.:^'\^

J. S. Cashing & Co. — Berwick & Smith
Norwood Mass. U.S.A.

PREFACE.

In the following pages I have endeavored to put into
form what in my opinion should constitute an Inter-
mediate Algebra, intermediate in the sense that it is not
intended for absolute beginners, nor yet for the accom-
plished algebraist, but as a stepping-stone to assist the
student in passing from the former stage to the latter.

The work covers pretty well the whole range of
elementary algebraic subjects, and in the treatment of
these subjects fundamental principles and clear ideas are
considered as of more importance than mere mechanical
processes. The treatment, especially in the higher
parts, is not exhaustive ; but it i^ hoped that the treat-
ment is sufficiently full to enable the reader who has
mastered the work as here presented, to take up with
profit special treatises upon the various subjects.

Much prominence is given to the formal laws of
Algebra and to the subject of factoring, and the theory
of the solution of the quadratic and other equations is
deduced from the principles of factorization.

The Sigma notation is introduced early in the course,
as being easily understood, and of great value in writ-
ill

91129?

iv PREFACE.

ing and remembering important symmetrical algebraic
forms.

Synthetic Division is commonly employed, and the
principles of its operation are extended to the finding
of the highest common factor.

Except in the case of surds, no special method is
given for finding the square root of an expression which
is a complete square, as the operation is only a case of
factoring, and a simple case at that. For expressions
which are not complete squares, the most rational method
is by means of the binomial theorem or of undetermined
coefficients, both of which are amply dealt with.

Probably the most distinctive feature of the work is
the importance attached to the interpretation of alge-
braic expressions and results. Algebra is an unspoken
language written in symbols, of which the manipulation
is largely a matter of mechanical method and of the
observance of certain rules of operation. The results
arrived at have little interest and no special meaning
until they are interpreted. This interpretation is either
Arithmetical, that is, into ideas involving numbers and
the operations performed upon numbers ; or Geometri-
cal, that is, into ideas concerning magnitudes and their
relations. Both interpretations necessitate observation
and the exercise of thought; but the geometrical offers
the wider scope for ingenuity, and is the better test
of mathematical ability. In several cases, as in that

PREFACE. V

of the quadratic equation, the solution frequently gets
its complete explanation only through its geometric
interpretation. Hence geometrical problems are freely
introduced, and the relations between the symbolism of
Algebra and the fundamental ideas of Geometry are
discussed at some length.

The Graph is freely employed both as a means of
illustration and as a medium of independent research;
and through these means an effort is made to connect
Algebra with Arithmetic upon the one hand, and with
Geometry upon the other.

The exercises are numerous and varied, and I trust
that they will be found to be fairly free from errors.

N. F. D.

CONTENTS.

Ohaptbb Page

I. Symbols, Definitions, and Formal Laws 1

II. The Four Elementary Operations 17

III. Factors and Factorization 45

IV. Highest Common Factor. Least Common Multiple 64
V. Fractions. c» and 0 74

VI. Ratio, Proportion, Variation or Generalized

Proportion 90

VII. Indices and Surds 100

VIII. Concrete Quantity. Geometrical Interpreta-
tions. The Graph 114

IX. The Quadratic 136

X. Indeterminate and Simultaneous Equations of

THE First Degree. Simultaneous Quadratics. 163
XI. Remainder Theorem. Transformation of Func-
tions. Approximation to Roots 176

XII. Progressions. Interest and Annuities 188

XIII. Permutations, Combinations, Binomial Theorem 210

XIV. Inequalities 227

XV. Undetermined Coefficients and their Applica-
tions 231

XVI. Continued Fractions 244

XVII. Logarithms and Exponentials 255

XVIII. Series and Interpolation 274

XIX. Elementary Determinants 296

For a more detailed statement consult the Index at the end of the book.

vii

CHAPTER I.
Symbols, Definitions, and Formal Laws.

1. Arithmetic is pure or concrete. Pure arithmetic
deals with abstract number or numerical quantity. Con-
crete arithmetic has relation to numbers of concrete
objects or things.

Thus 3 is an abstract number, but 3 days is concrete.

Algebra is primarily related to pure arithmetic, but its
extension to concrete arithmetic is an easy matter.

The quantities which are the subject of arithmetic are
of three kinds :

(1) Whole numbers or integers ;

(2) Symbolized operations called fractions ;

(3) Numerical quantities which cannot be exactly
expressed as integers or fractions, but whose values may
be expressed to any required degree of approximation.
Such are the square roots of the non-square numbers,
the cube roots of the non-cube numbers, etc. This third
class goes under the general name of incommensurables.

The expression numerical quantity, and frequently the
word number, will be taken to denote any of the three
classes.

2. Numbers are fundamentally subject to two opera-
tions — increase and diminution ; but convenience, drawn
from experience, has led us to enumerate four elementary

1

2 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

operations, viz. : Addition, Subtraction, Multiplication,
and Division.

All higher operations on numbers are but combinations
of the four elementary ones.

3. Algebra originated in arithmetic, and elementary
algebra is arithmetic generalized, the generalization being
effected by employing symbols, usually non-numerical,
to stand for and represent not only numbers or numeri-
cal quantities, but also the operations usually performed
upon numbers.

Thus algebra becomes a symbolic language in which
numbers and the operations upon them are written.

The symbols of algebra are thus primarily of two
kinds :

(1) Quantitative symbols, which represent numerical
quantities, and

(2) Operative symbols, which indicate operations to
be performed upon the quantity denoted by the quantita-
tive symbol.

A third class, called verbal symbols, may be enumerated,
in which the symbol is a convenient contraction for a
word or phrase.

4. The quantitative symbols are usually letters. The
operative symbols, especially in elementary algebra and
in arithmetic, are mostly marks or signs which are not
letters. Relative position is employed to denote some
operations, and in higher algebra very complex operations
are often denoted by letters.

The verbal symbols do not denote quantity, and they
cannot be said, in general, to denote operations.

SYMBOLS, DEFINITIONS, AND FOEMAL LAWS. 3

The principal verbal symbols are :

(1) = and =, either of which denotes that all that
precedes the symbol, taken in its totality, is equal to or
is the same as all that follows the symbol, taken also in
its totality.

(2) > and <. The first denotes that all that pre-
cedes the symbol, taken in its totality, is greater than all
that follows the symbol, taken in its totality; and the
second is like the first with less put for greater.

Other verbal symbols will be introduced as required.

5. From Art. 3 it is seen that operations in arithmetic
must be special cases of more general operations in
algebra. And hence it follows that arithmetic and
algebra must proceed on similar principles, and must be
subject to the same formal operative laws.

That the generalizing process of algebra should intro-
duce new ideas into arithmetic is to be expected; and
that this generalization should carry us beyond the neces-
sarily limited field of arithmetic is also to be expected.
Illustrations will occur hereafter.

6. The operative symbol -f (plus) denotes addition,
and tells us that the quantity before which it stands, and
to which it belongs, is to be added to whatever precedes.

Thus, 5 + 3 tells us that 3 is to be added to 5, and
0 -f- 3 is the same as the arithmetical number 3.

Similarly, + a is the same as a, whatever a stands for ;
and for this reason the sign + is seldom written when-
ever it can be dispensed with without producing ambiguity.

a -f 5 is the same as -j- a + 6, and indicates that the

4 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

number denoted by h is to be added to the number
denoted by a.

7. Any interpretable combination of quantitative and
operative symbols is an algebraic expression. We shall,
in the meantime, confine ourselves to expressions written
in a single line, as

3a6-f 2c-f d-, etc.

In arithmetic we know that 3 + 5 is in its sum the
same as 5-J-3, and 3 + 5 + 8 is the same as 3 + 8 + 5,
the same as 8 + 5 + 3, etc. And as this must be a par-
ticular case of algebra (Art. 5), we must have a + 6 =
h-\-a, a-\-'b-\-c = a-\-'c-{-h = h-^c-\-a = etc.

This is the Commutative Laiv for Addition, and is ex-
pressed by saying that the order of adding quantities is
arbitrary^ or the sum is independent of the order of the
addends.

8. The symbol — (minus) placed before a quantity
indicates that the quantity is to be subtracted from
whatever precedes the symbol.

Thus, 5 — 3 tells us that 3 is to be subtracted from 5 ;
and a — b tells us that the quantity denoted by b is to
be subtracted from that denoted by a. Now, a and b
denoting any numerical quantities, as long as a is greater
than b the subtraction is arithmetically possible, and the
result is an arithmetical quantity. But if a is less than
b, the operation symbolized is not arithmetically possible.
The expression a — & is then a symbolic representation
of an operation that cannot be arithmetically performed,
and the result of the operation, whatever it may be, is
not arithmetical.

SYMBOLS, DEFINITIONS, AND FORMAL LAWS. 5

Thus, 0 — 3, which is simply written — 3, and which
is called a negative number, does not belong to pure
arithmetic, but is an idea introduced into algebraic arith-
metic by generalizing the operation of subtraction.

And thus to every pure number, called now a positive
number, corresponds an algebraical negative number, the
relation between corresponding numbers being that their
algebraic sum is zero or nothing.

Negative numbers are important in their relations to
concrete arithmetic, and especially where geometric ideas
are concerned. This matter will be dealt with in Chap-
ter VIII.

9. It is said in Art. 2 that numbers are fundamentally
capable of only increase or diminution. Hence + and
— symbolize the two great operations in arithmetic and
algebra. These are distinctively the signs of algebra.

By the sign of a quantity is meant that one of these
two signs which precedes the quantity; and to change
signs is to change + to — and — to + throughout.

Also, two quantities have like signs when both are
preceded by + or both by — ; otherwise they have
unlike signs.

When no sign is written, -f- is understood.

10. That part of an expression included between two
consecutive signs is called a term.

To indicate that any portion of an expression lying
between two non-consecutive signs is to be taken in its
totality as a single term, we enclose the portion within
brackets.

Thus, in the expression a-f26c — 4(3c + 2 ab), a, 2 be

6 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

are single or simple terms, and 3c-{-2ab is to be con-
sidered as one complex term.

The sign — and the number 4 preceding the brackets,
apply to its contents in their totality.

Instead of brackets, we often employ a line called a
vinculum J drawn, above the portion indicated, as, 3 c -j- 1? a6 ;
and sometimes, for special reasons, this line is placed
beneath instead of above.

11. The symbol x or • (into or by) indicates that the
quantity following the symbol is to act as a multiplier
upon the quantity preceding the symbol.

With numerical symbols, as 4, 7, etc., it is evident that
we cannot dispense with the symbol, as 34 is not the
same as 3x4; but this difficulty does not exist with
letters, and hence we usually write ab instead of a x 6
or a • &.

In this case relative position or juxtaposition becomes
a symbol of multiplication.

12. The parts which make up a term, or an expression,
by multiplication only, are factors of the term or expres-
sion.

Thus, 3, a, &, and c are factors of Sabc, 3 being a
numerical factor, and a, b, c literal factors. So, also,
4, a, b -\- c, and c-^ a are factors of the expression
4a(6 + c)(c + a).

13. In arithmetic we know that 4x6 is the same in
value as 6x4; 3x2x5 is the same as 2x5x3, etc. ;
and as this must be a particular case of algebra, we must
agree that ab = ba, that abc = bca = etc.

This is the Commutative Law for Multiplication , and is

SYMBOLS, DEFINITIONS, AND FORMAL LAWS. 7

expressed by saying that a product is indepefident of the
order of its factors.

Thus, ab means, indifferently, that b multiplies a or
that a multiplies b.

14. Since multiplication by + 1 effects no change in
the quantity multiplied, we have

(+l)(+a) = + a;

Or, + multiplied by + gives + in the product.

Again, a — a==0 = a + (— a) = a4-(+l)(— a),as — a
may be taken in its totality by placing it within brackets.

Hence (+ 1) (— a) must be —a;

Or, + multiplied by — gives — , and, by the commuta-
tive law for multiplication, — multiplied by + gives —
in the product.

Again, a ~ a = 0,

and writing — b for a, we have

-b-{-b) = 0;
and hence —(—6), or ( — !)( — &) must be the same as +6;

Or, — multiplied by — gives + in the product.

Collecting results, we have as the Law of Signs : The
multiplication of two like signs gives +, and the multi-
plication of two unlike signs gives — in the product.

15. It is readily established in arithmetic that

3 (4 + 2 + 5) = 3x4 + 3x2 + 3x5.

And as this must be a particular case of algebra, we
must assume that

a (b -{- c -{- d) = ab + ac -\- ad.

8 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

We have here three terms, b, c, and d, which are placed
in brackets and taken as a complex term, and we have
a multiplier a which operates upon this complex term.
And we see that we may distribute this operator so as
to act separately upon each of the terms of which the
complex term is composed.

This is the Distributive Law for multiplication.

Some other operations — like multiplication — are dis-
tributive, and some are not. The case for each operation
must be worked out and learned by itself.

16. A term, such as aaabbc, containing repeated letters,
is simplified in form by writing it a^b\ in which the
small numerals placed to the right and above a and b
show how many times each of these letters enters,
respectively, as a factor.

The symbol a^ is read ^a cubed,' and b^ is read ^b
squared.'

The letters a and b are subjects or roots, and the 3 and
2 are exponents or indices.

Similarly, a"", where n denotes any integer, is the nth.
power of a, read ^ a nth-power,' or ^ a-to-the-nth/ and
denotes that a is to be taken n times as a factor.

Now if n=p-{-q, i.e. if n be separated into two
integers denoted by p and g, we have

a**= a • a • a ••• to n factors.
Or a^+* = a • a • a • • • to jp factors x a • a • a • • • to g factors
= aP . a^.

This expresses the Index Law, its statement being that
the product of any powers of the same subject is that power
of the subject ivhich is denoted by the sum of the exponents.

SYMBOLS, DEFINITIONS, AND FORMAL LAWS. 9

Thus, a'^ ' a^ z= a^ 'y a^ - a^ - a^ = a^^, etc.

Of course a is the same as a\

17. The Commutative laws for addition and multipli-
cation, the Distributive law for multiplication, and the
Index law are the great formal laws of elementary-
algebra, and its symbolism, its principles of operation,
and its results are applicable to any subject which by
any consistent process of interpretation can be shown to
be governed by these laws.

A proper conception of this fact opens the way to
important extensions in the applications of algebra.

The foregoing laws belong to elementary algebra
because this subject is a generalization of arithmetic, in
which these laws hold ; but that they are not all essential
to all kinds of algebra, we know, as we have a special
algebra. Quaternions, in which the commutative law for
multiplication does not in general hold true. But this
latter algebra is not generalized arithmetic.

18. A single letter as a quantitative symbol is said to
be of one dimension, and the number of dimensions of a
term, which consists of multiplications only, is the num-
ber of letters, either expressed or implied, which the
term contains.

Thus, ahCy a^b, a^ are each of three dimensions; and
3 a^bcdy 4 a^b% a^b^c are each of five dimensions, a numerical
factor, as 3 or 4, having no dimensions.

The number of dimensions of a term constitutes its
degree ; thus the first three of the preceding terms are
of the third degree, and the last three of the fifth degree.

Usually a term is said to be of a certain degree in
some particular letter or letters.

10 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

Tliiis, Sab^x^ is of the first degree in a, of the second
in b, and of the third in x.

When the degree of each term of an expression has
reference to the same particular letter, this letter is
called a variable, and any other letters occurring in the
expression are constants.

Thus, in the expression ax^ -^bx -{- c, ax^ is of the sec-
ond degree in x, bx is of the first, and c, not containing
Xj is the absolute or independent term. In this case x is
the variable, and a, b, c are constants.

The term dimension is derived from geometry, and its
significance will be more fully seen hereafter.

19. An expression of one dimension in each term is a
linear expression, as a + ^ + <^-

An expression which contains a variable in the first
degree only is linear in that variable.

Thus, ^abx is linear in x, and 3a?> is the coefficient of x.

Similarly, x — a and ax + be are both linear in x,
although the first is of one dimension, and the second of
two, in regard to all the letters.

An expression which is of the same dimensions in
every term is homogeneous. Such expressions are specially
important.

Thus, a + 6 4- c, ab -\- be -[- ca, a^ -f 3 a^& -f 3 ab'^ -f b^ are
each homogeneous with respect to all the letters.

20. A quantitative symbol stands for any numerical
quantity whatever, and operations upon such general
quantities can be only symbolically indicated. Thus a
and b being any quantities, we denote their sum by
a + 6, and their product by ab.

SYMBOLS, DEFINITIONS, AND FORMAL LAWS. 11

Herein lies the advantage of algebraic symbolization,
its great powers being due to two things :

(1) The universal significance of the quantitative
symbol, and

(2) That the operations performed, unlike those in
arithmetic, are not lost sight of, so that a chain of con-
secutive operations may be so reduced by transformations,
as to depend upon the smallest possible cycle of such
operations.

21. As algebra is generalized arithmetic, every alge-
braic relation, which is arithmetically interpretable, ex-
presses some general relation amongst numbers.

Thus, ah (a + h) gives a% + ab^ by distributing ah, or
ah {a -^ h) = a^h -{- ab\

This interpreted gives the arithmetical theorem :

The sum of two numbers multiplied by their product
is equal to the sum of the numbers formed by multiply-
ing each number by the square of the other.

It may be remarked that theorems like the foregoing
cannot, in general, be proved by an arithmetical process.
Repeated trials with different numbers would give a sort
of moral proof, but not a mathematical one, since we
could not possibly try all numbers. On the other hand,
the quantitative symbol standing at once for any, and
hence for every number, gives a proof which is both
rigid and universal.

EXERCISE I. a.

1. The following are identities arising from distribution ; inter-
pret them as arithmetical theorems.

i. a(a+ b) = a'^ -^ ah. ii. ab(a — b) = a^h — a6"^.

12 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

iii. (a-^b)(a-b) = a^-b^. y. (a-b)(a-b)=a-\-b^-2ah.

iv. (a+6)(a+6)=a2+62+2a&. yI (a -{- b) + (a - b) = 2a.
vii. (a + 5) -(«-&) = 2 5.
viii. (a + by + (a - 6)2 = 2(a2 + 52).
ix. (a + 6)2-(a-6)2 = 4a6.
X. (a + 6 + c)2 = a2 _|_ 52 + ^2 4 2(a& + 6c + ca).

2. Reduce to a single number —

i. l-{-2(-l + l-2)}.
ii. 3{4 - 5(6 - 7[8 - 9])}.

iii. ia-ia-Ki-i-TV-iV])}.

3. Condense as much as possible —

i. 2a — {Sa—(a — b — a)}.

ii. a-b{l -6(1 -a- 1-6)}.

4. Distribute the following —

i. {(a-6)-2(6~c)}.{(a + 6)+2(6 + c)}.

ii. {(m + l)a + (n + 1)6} • {(m - l)a + (n - 1)6}

4- {(m + l)a - (w + 1)6} . {(m - l)a - (w - 1)6}-

5. If Ti is an integer, 2 n is an even integer, and 2 n + 1 is an
odd integer.

6. The product of two odd integers is an odd integer.

7. The sum of two odd integers is an even integer.

8. The square of an odd integer is an odd integer.

9. The square of an even integer is divisible by 4.

10. What power of 2 is 2» x 22 x 2i-« x 2 ?

11. What power of a is a'"-^ - aP - a^^-p • a^-"* ?

12. According to the index law a* x a~^ = a*"i = a^.
Hence interpret a~\

SYMBOLS, DEFINITIONS, AND FORMAL LAWS. 13

22. The symbols = and =. The symbol = placed
between two expressions denotes that one of the expres-
sions may be transformed into the other by the formal
operations of algebra. The whole is then called an
Identical equation, or an Identity, and the connected
expressions are the members of the identity.

Thus, ah (a — b) = a^b — ab^ is an identity, since the
left-hand member is transformed into the right-hand one
by distribution.

The symbol = between two expressions tells us that
the expressions are to be equal in their totalities, although,
in general, no transformation can change one into the
other.

That this condition of equality may exist, some rela-
tion must hold amongst the quantitative symbols, and,
usually, one of these, called the variable, is to have its
value so adjusted as to bring about an identity.

Thus, 4a + a? — 3=6a — 1 is an equation, or a con-
ditional equation, which is true on the condition that x
takes such a value as will make the whole an identity.
We readily see that if x stands for, or is replaced by
2a-f 2, the condition is satisfied. We say then that
2a + 2 is the value of x, and that x is the variable of the
equation.

The variable is often called the unknown, and it is
manifest that any letter occurring in the equation may
be taken as the variable. If a be so taken, its value is
found to be "I a; — 1.

23. Evidently an identity is not affected by perform-
ing the same operation upon each of its members ; and,
as a conditional equation is to be brought to an identity,

14 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

the relation existing amongst its quantitative symbols is
not changed by performing the same operations upon
each member.

Thus in the equation 3aj — 2a4-l==a? + a — 4, we may
subtract x from each side, add 2 a to each side, and sub-
tract 1 from each side. We then have

2a; = 3a-5.

Then dividing each side by 2, we get as the value of x,

We notice that we transfer any term from one member
to the other by writing it with a changed sign in the
other.

Thus, x-{-a = b gives x = b — a.

The determination of the value of the variable, in terms
of the constants of an equation, is called the solution of
the equation. And although the solution of equations
does not constitute the whole of algebra, it undoubtedly
forms a very important part of it.

The following examples are given by way of illus-
tration :

Ex. 1. To find the value of x in the equation

3{a - 4(1 - x)} = 2{a + 3( a:- 1)}.

Performing all the distributions,

Sa-12-{-12x=z2a-\-6x-6.

Transferring 6 x from right to left, and 3 a — 12 from left to
riglit, we have

6x = 6 -a.

Dividing by 6, x = 1 — la.

SYMBOLS, DEFINITIONS, AND FORMAL LAWS. 15

Ex. 2. To find a number which exceeds the sum of its third and
fourth parts by 10.

Let X denote the number; the statement of the problem is alge-
braically expressed as

\x-\-\x — x — \^.

Multiplying by 12, 4 x + 3 x = 12 x - 120,

whence x = 24.

Ex. 3. A lends to B one third of a dollar more than ^ of his
money, and to C one half a dollar more than J of what he has left.
A has then $ 6. How much had he at first, and how much did he
lend to B, and to C ?

Let x denote A's money at first.

He lends to B, J x + J dollars.

He has left, x — Qx + i), or f x — | dollars.

He lends to C, 2 (§ ^ — i) + i' or ix + i dollars.

He has left, (f ^ — i) — ( 3 ic + |^) , or | x — f dollars ;

and this is $6.

Whence x = 20 ; and B's loan = C's loan = $ 7.

EXERCISE I. b.

1. Prove the following identities —
i. (a + 5)2 = a2-f h'^ + 2ab.

ii. (a2 + 52) (c2 + ^2) = ^ac + hdy + {ad - hey.

ill. (a2 _ 52)(<o2 _ (^2) ^ (^c - hdy - {ad - bey

= {ac-j-bdy-{ad-{- bey.
iv. (a + 6 + cy+{a + b - cy -^(b -{- c - ay +{c + a - by

= 4(a2-|_ 52_^c2).
V. (a2 4. ^,2)2 = (^2 _ 52)2 _|_ (2 aby.

16 SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

2. Interpret ii. and iii. of I as theorems in numbers.

3. By means of v. of 1 find two numbers such that the sum of
their squares shall be a square. Make a table of such numbers.

4. Find the value of x in each of the following —

i. x-\-ix-hix = 2x — 2.
ii. 3(l-2.3"^^)=:2{l + 2(.x-2)}.
6. Simplify l-{l-(l-x)}+l + {l-(l + a^)}+a:-{x-(x-l)}. .

6. Find x in the equation {x — 4) (a: + 6) = (x + 8) (x + 2).

7. Find a number whose half exceeds the sum of its fourth and
fifth parts by 40.

8. Find a number such that if it be increased by a, and if it be
diminished by b, one third of the first result is equal to one half
the second.

9. The sum of the ages of A, B, and C is 108 years. A is twice
as old as B, and twice C's age is equal to A's and B's together.
Find their ages.

10. After paying 2 % taxes on my income I have $ 1078 left.
What is my income ?

11. I pay 33 J % duty on the cost price of a horse. I keep him
2 months at an expense of $16, and I then sell him for $200, mak-
ing 20 % profit on the cost price. What did the horse cost ?

12. In a certain school f of the pupils are in the first form, J in
the second, ^^ in the third, and 14 in the fourth. How many
pupils are in the school ?

13. A market-woman sells to A half an egg more than half she
has, to B half an egg more than half she has left, and 10 eggs to
C, and she then has 6 eggs left. How many had she at first ?

CHAPTER 11.

The Four Elementary Operations,
addition and subtraction.

24. The addition of a and h is denoted by a+&, where
a and h stand for any quantities whatever.

If, however, a = 5 and 6= —3, the expression becomes
5 — 3, and we have a case of subtraction.

Thus, symbolically, addition and subtraction are one
and the same ; for in the expression a + 6 we cannot
know whether an addition or a subtraction is to be per-
formed, until we know something about the quantities for
which the letters stand.

Moreover, any subtraction may be put into the form
of an addition, and vice versa; for a — h is the same as
a + ( — ?>), and a + 6 is the same as a — {—h).

Thus the subtraction of one expression from another
may be expressed as an addition, by changing all the
signs of the subtrahend. Hence the rule for Algebraic
Subtraction :

Change the signs of the subtrahend^ and then perform
addition.

Ex. To subtract 3a— 2 6 + 3 from 6a+36— 4 is the same as to
add — 3a4-2 5— 3 to 6a+3 6— 4; and the result is 3 a+ 5 & — 7.

25. Since a series of terms connected by + and — signs
may be wptteix as one connected by + signs only, such

17

18 THE FOUR ELEMENTARY OPERATIONS.

a series is called an Algebraic Sum. Thus 5 — 4 + 3 — 1
has 3 as its algebraic sum, and may be written

5 + (-4) + 3 + (-l).

We are not justified in speaking of this as an Arith-
metic Sum, for — 4 and — 1 have no meaning in pure
arithmetic, and if we put it into the form S — {-{-5),
it becomes an arithmetic difference.

26. Symmetry. When the interchanging of two let-
ters of an expression leaves the expression unchanged,
except as to the order of the letters in a term, or the
order of the terms in the expression, the expression is
symmetrical in the two letters.

Thus, by interchanging a and b in

ab — ac -\- ad — be -\- bd — cdy

we get ba — be -\- bd — ac -\- ad — cd,

an expression the same as the former, except as to the
order of the terms and of the letters in some of the terms.

Hence the expression is symmetrical in a and b.

It is readily seen that the expression is not symmet-
rical in c and d.

An expression which is symmetrical in every pair of
two letters is symmetrical in all the letters.

Thus ab -\-bc-{- ca and abc + abd + acd + bed are each
symmetrical in all the letters which they contain.

Some special kinds of symmetry will be considered in
a more advanced stage of the work.

27. When we know the letters which enter into an
expression symmetrical in them all, and we are given a
type-term, we can write the full expression by building

ADDITION AKD SUBTRACTIOK. 19

up the form of the type in every possible way from
the given letters, and taking the algebraic sum of all
the terms so produced.

Thus from the letters a, b, c:

i. with type ab^- we have —

ii. with type a {he — a) :

a {he — a) + h (ca — h)+ c (ah — c).
iii. with type ahc^i

ahc^ + hca'^ + cah'^.
iv. with type (h — c) (a^ _ 5c) :

(6 - c)(a2 _ hc)-{-(ic - a)(h-^ - ca) + {a - h){c^ - ah).

V. with a, 6, c, and d, and type ah'^ :

a&2 4. 5^2 _}_ ^(.2 _|_ ca2 _|. (;r(^2 _|_ ^^2 ^ 5^2 + c62 + 6d^

+ d62 4- cd^ 4- (^c2.

It will be noticed that in examples ii. and iii. each term
is formed from the preceding one by changing a to h,
b to c, and c to a. This is called a cyclic or cir- „
cular substitution ; for if we write the letters / \
in a circle, as in the margin, we pass from one ( ,

term to the next by commencing with a letter '''^^^ — ^*
one step further around the circle until the whole is
completed.

Many substitutions, where three letters are concerned,
are of this character, the distinctive feature being that
we do not interchange any two letters without, at the
same time, interchanging every two in circular order.

A cyclic change with 4 letters and type ab gives
ab -\- be -{- cd -\- da. This lacks the terms ac and bd to
make it completely symmetrical.

20 THE FOUR ELEMEKTARY OI^ERATIONS.

In examples i. and v. a circular change is not suffi-
cient ; for from the type ab^ we must have a term &a^,
which is not given by a mere circular substitution. In
other words, we must interchange two letters without
affecting the third.

A little care and observation are all that are required
in writing out such expressions from a given type.

28. The symbol 2 (sigma), amongst other uses, is
conveniently employed to denote expressions consisting
of algebraic sums, written from a type.

These are symmetrical in all the letters employed, and
when written out are frequently of inconvenient length.

The notation ^a^b, with three letters involved, stands
for a^b 4- b^a -f b^c + c^b + c^a + a^c.

With four letters involved it stands for v. of the pre-
ceding article.

^{b — c ' a^ — be) stands for iv. of the preceding article.

As employed hereafter, 3 letters will be understood un-
less a different number is indicated, or in cases where
misunderstanding is not possible. ^4 will serve to indi-
cate 4 letters, and generally 2n to indicate n letters.

This is known as the Sigma Notation.

EXERCISE

II.

a.

1.

With 3 letters write in full —

1.

S(a2_5).

iv.

(Sa)2

ii.

2 (« + 6 -

C)2.

V.

{2 (a

-b)f.

ill.

c

vi.

2a X

^ab.

vii. (2a)-2 + 2 (a + & - c)2 - 4 2a2 + 2 Zab.
yiii. x^ — x22a + x^ab — abc.

MULTIPLICATIOK. 21

2. With 4 letters write in full —

i. Sa6. iv. (2a)2.

u. Sa&2c. V. 2 (a - &) (c - (T).

iii. Sa(5-c). vi. ^{aP' -h)(d^ - d).

MULTIPLICATION.

29. The multiplication of one expression by another
may be effected by a series of distributions, and when the
expressions are not too complex this is usually the best
method.

Thus (a -h &) (a — & + c) = a(a — & + c) + &(a — 6 + c)
= a^ — ah -{- ac -{- ah — h^ -\- be = a^ — h'^-{-ac-{- he.

By remembering and applying a few elementary pro-
duct forms, the operation may often be much curtailed.
As convenient fundamental forms we may take the fol-
lowing, although any simple form that can be remembered
may be equally useful :

(1) {a + by = a'-^h' + 2ab,

(2) (a-hy = a'-\-h'-2ah.

(3) (a-h){a + h) = a:'-h\

(4) 3 (a + &) (& + c) (c + a) = (a + 5 + c)^- a^- 6^- c^

(5) (a + h + c) (a2 ^h^ j^c^ -ah -be- ca)

= a^^b^-{-c^-3abc,
or Sa (Sa^ — 5a&) = ^a^ — S abc.

The mark .-. is a verbal symbol for 'therefore' or

hence.^

22 THE FOUR ELEMENTARY OPERATIONS.

Ex. 1. {a -\- b - c + d)(a + b + c - d)

^(a -^ b - c - d)(a -\- b + c - d)

= (a + 6)2-(c-(^)2

= a2 + 52 - c2 - (Z^ + 2 a6 + 2 c(^.

Ex. 2. (a + & + c + dy = (a + 6)2+ 2(a + 6) (c + c?) + (c + (?)2
= a2 ^ 52 ^ c2 ^ ^2 _|_ 2(a& + ac 4- a(^ + &c + 6(^ + cd).

Ex. 3. To distribute
s(s— a)(s— 5) + s(s— 5)(s— c)+-s(s— c)(s— a) — (s— a)(s— ?))(s— c),
where 2 s = a + 5 + c. s, being symmetrical in a, 6, and c, is not
altered by a circular substitution of these letters.

But this substitution brings s(s — a)(s — b) to s(s — b)(s — c),
and s(s — 6)(s — c) to s(s — c)(s — a).

Hence having the expansion of s(s — a)(s — b), the expansions
of the two following terms may be immediately written down by
a cyclic interchange of letters.

Now s(5 — a)(s — b) = s^ — s^(a -j- b) -\- s - ab.

.-. s(s -b)(s- c) = s^- s%b + c)+ S'bc,

and s(s — c) (s — «) = ^ — ^""^(^ + a) + s • ca.

The algebraic sum is

3s3_2s2(^a + 5_l_c)+s(a& + 6c+ca),
and as 2s=:a + 5 + c, this becomes

— s^ + s(a6 + bc-j- ca) A

Again the expansion of (s — a)(s — b) (s — c) is

^ — s2(a + 6 + c) + s(ab + 6c + ca) — abc,
or — s^ + s(a6 + &c + ca) — abc.

And subtracting this from A gives a6c as the final result.

This example furnishes a good illustration of the
remark in Art. 20, for the long series of operations sym-
bolized in the statement of the exercise is just equiva-
lent in its totality to the two multiplications symbolized
in the final result.

MULTIPLICATION. 28

Ex. 4. To prove the identity

8(2a)8 - S(a + by = S(2a + b + c)(2 6 + c + a)(2c + a + b).

Either of two methods may be adopted — (1) to transform one
member to the other by the rules of operation ; or (2) to transform
each to the same third expression by distribution. We shall adopt
the first way.

8(2a)3^(S2a)3= (a + b + b + c + c -\- a^.

Now put a + b = py b + c = q, c -\- a = r, and the identity
reduces to

(p + q + ry - P^ - q^ -r^ = 3(j) + r) (r + g) (g + i?),
which is true by Art. 29, (4).

30. Expansion of Symmetrical Homogeneous Expres-
sions.

This form of expression is of frequent occurrence, and
its properties of symmetricality and homogeneity enable
us to expand it with some facility.

Ex. 1. To expand (a -{- b -\- cy - (a + by - (6 + cy - (c + ay.

Being homogeneous and of 3 dimensions, the type terms in its
expansion can only be a^, a^b, and abc. Taking the type a^, we
see that its coefficient is — 1. Taking the type a'^b, its coeffi-
cient is readily found to be zero. And the coefficient of the type
abc is 6.

.-. The expansion is 6 abc — a^ — b^ — (^.

Ex. 2. To expand (Sa)^ + (« + 5 _ c) (6 + c - a) (c + a - 6).

The expansion being homogeneous of 3 dimension^ and sym-
metrical, must be of the form

m2a3 4- nSa^ft + p • abc.

The coefficient of a^ is 1 — 1 or 0 ; .*. m = 0.

The coefficient of a-5 is 3 + 1 or 4 ; .*. n = 4.

The coefficient of abc is 6 — 2 or 4 ; .-. p = 4,
and the expansion is 4 'Za'^b + 4 abc.

24 THE FOUR ELEMENTARY OPERATIONS.

EXERCISE II. b.

1. Write out the type terms in the following symmetrical and
homogeneous expressions —

i. (a-{-b + c + ay. iii. (a + 6 4- c)*.

ii. (a + by. iv. (a+b + c + dy.

V. (a + b + c + ay,

2. Show that

(a -j- b — c)(b -{- c — d)(c + a — b)

= ab(a + b)+ bc(b + c) + ca(c + a) - a^ - 6^ - c^ - 2 ahc,

3. Sa- 2a6-(a + &)(& -f-c)(c + a)=a6c.

4. Show that S{(a - 5) (2 & - c)} = (Sa)2 - 3 Sa^.

5. Show that

(a + 6 + c) (a + & - c) (6 + c - a) (c + a - 6)

= 2Srt252 _ Sa4 = a2(2&2_ a2)^ 52(202 - b'^)^d^{^aP' - d^)

= 4 52c2_(62 + c2-a2)2.

6. Expand —

i. S(a + &)(«-&).
ii. (Sa)* - S(a + &)* -f Sa*.
iii. Sa3(6_c)-Sa.Sa2(6_c).
iv. a(s — 6) (s — c) + 6(s — c) (s — a) + c(s — a) (s — 5)

+ 2(s — a) (s — 5) (s — c), where 2s=:a + & + c.

7. If 2 s = a + & + c, show that the three following expres-
sions are identical in value —

s(s — d)(b -\- c) + « (s — 5) (s — c) — 2 bcs,

s (s — 6 ) (c + a) + 6 (s — c) (s — a) — 2 cas,

s (s — c) (a + 6) + c (s — a) (s — 6) — 2 a6s.

MULTIPLICATION. 25

8. Show that (x — b)(x — c) (b — c) + (x — c) (x — a)(c — a)
-{■{x-a){x -h){a-h) + (a - 6)(6 - c)(c - a)=0.

9. Show that (2a)2 = 2^2 _|_ 2 2a&, with any number of letters.

10. Multiply 2a2 + 2 2a6 by 2a, 3 letters.

11. Multiply 2a2 + 2a& by 2a6, 3 letters.

12. Multiply 2a2 4. 2 2a6 by 2a, 4 letters.
This gives (a + & + c -f ^)^.

Prove the following theorems in numbers —

13. The difference between the square of the sum and the
square of the difference of two numbers is the product of twice
the numbers.

14. The sum of the squares of the sum and of the difference
of two numbers is one-half the sum of the squares of twice the
numbers.

15. If two numbers be each the sum of two squares, their pro-
duct is the sum of two squares.

16. If two numbers be each the difference between two squares,
their product is the difference between two squares.

17. If the sum of two numbers is 1, their product is equal to
the difference between the sum of their squares and the sum of
their cubes.

18. If the product of two numbers is 1, the square of their sum
exceeds the sum of their squares by 2.

31. The distribution of (a? + a) {x + 6) (a? + c) ..., the
product of a number of binomial factors with one letter
the same in each factor, is very important.

The dominant letter, x, is taken as the variable, and
the expansion is arranged according to the powers of x.

With three factors we readily see that taking the x
from every factor gives ^) taking it from every factor

26 THE FOUR ELEMENTARY OPERATIONS.

but one, and taking the other letter from that one gives
Qc^ {a -\- b -{- c) ', taking x from one factor and the other
letter from the other two gives a; {ab + bc-{- ca) ; and
lastly, taking the second letters only gives abc.
Thus the expansion is

a^ + aj^ (a -f 6 + c) + X (ab + ^c + ca) -f abc,

or a^ -{■ a^^a + x^ab + abc.

Similarly, with 4 factors it is readily seen that the
expansion is

aj* + a^%a + a^2a6 -f- x^abc + abed.

In a similar manner it is shown that with n factors
the expansion is

^n _|_ ^n-i^a 4- x" 22a6 + a;"-^2a6c H \-abC'".

It will be noticed in every case that the last term is
the continued product of all the letters except the variable.
This is important.

If the signs are negative in all the factors, as (x — a),
etc., then Sa, labc, etc., involving an odd number of
letters in each term, will be negative, and 2a6, etc., in-
volving an even number, will be positive.

Thus, {x — a) (x — b).(x — c) (x — d) • • • to n factors
= aj~ — a;"-^2a + x^'-^^ab — H

Ex. 1. (x + a)(x + b) (x + c)

= x^ + ic2(a 4- 2) -f c) + x(ab -\-bc -{- ca) + abc ;
and making c =zh = a gives

(x + ay=x^ + Sx'^a + Sxa'^ + a^

Ex.2. (x + a)(ix + b)(ix-\-c)(ix + d)

= X* + x^Xa + x'^Zab + x^abc + abed.

MULTIPLICATION. 27

Now 2a contains 4 terms, 2a6 contains 6, 2a&c contains 4, and
a5cd is one term.

Therefore making d = c = h = a gives

(x + a)* = X* + 4 ic% + 6 x2a2 _|- 4 ^cq^s _}. gi^

Ex. 3. Similarly, we find

{x -\-ay = x-5 + 5 x*a + 10 x^a^ + 10 x^a^^ + 5 xa^ + a'\

The coefficients of the several terms in these and
higher powers are exhibited in the following table, which
may be extended at pleasure —

The coefficients form the diagonals, up to the 8th
power.
1 ^^.-'1 ^^--'i ^^.-'1 ^.-'1 ^^^'1 ^^^'1 ^^^'1 ^^^'1

1"''' ^^'-3^'" ^-^^''' ' --''°'''' ^-■'5'"'' ^-21'''' ^.28'"'
r"' ^-'^^'^ .-'lO''" ^.^O'"' ^^.35-'"" ^,'56"""'
1"'' ^^--5"" ^--15''''" ^^35''" ^.-TO'""'
1''"' .6''"'' .21'''"'' ^,56""'

I''""' ^,-8'"'"

Ex. 4. (a; + a - 6)(x + 5 - c)(a; + c - a) = ic3 + a^s^^^ _ 5)

+ x2{(a - 5)(6 - c)} + (a - &)(& - c)(c - a).
Now 2(a - 5)= 0, 2{(a - &)(6 -c)} = 2a6 - 2a2,
and (a - 6) (6 - c) (c - a) = 2a5(6 - a),

and the expansion is

x^ + a:(2a6 - 2a2) + 2a6(& - a)

EXERCISE II. c.

1. Expand (a; _ l)(x - 2)(x - 3)(x - 4).

2. Expand (x - l)(a^ + 2)(x - 3)(ic + 4).

3. Expand (4a: + l)(3x + 2)(2x + 3)(a; + 4).

28 THE FOUR ELEMENTARY OPERATIONS.

4. Expand (x+a + & — c)(x + 5 + c — a)(x + c + a — 6).

5. Expand (a -\- b + c)^

Write it {«+(& + c)}^ and pick out the coefficients of the type
terms.

6. Expand (a + 6 + c + d)*.

Write it {(a + 6) + (c + d)}* and pick out the coefficients of the
type terms.

7. Write in S notation the expansion of (a -f 6 + c + d)^.

32. Function. An expression such as a^ + ab changes
value when a* changes value or when b changes value.
It is accordingly called a function of a and b.

When we wish to consider a alone as a variable, and
regard b as being constant, we speak of the expression
as a function of a, and we symbolize it as fa or /(a),
where /is di functional symbol.

This symbol merely denotes that a enters into an
expression as a variable, without any regard to the other
letters in the expression, and /(a;) stands for any expres-
sion in which x enters as a variable.

In general algebra the form of / is given, i.e. we are
given an expression of the form required.

Thus, if /(a) stands for a^ -{- 2 ab -\- c, then /(a;) stands
for x^ -\- 2 xb + c, where x is substituted for a in the type-
form. Similarly, f{a — x) = {a — x)^ -\- 2 (a — x) b -\- c, etc.

Ex.1. If /(a) = aH2a + l, /(a-l) = (a-l)2+2(a-l) + l=a-2.

EX.2. If /(.)=^, /(r3^)=(r^J/(i+r^)=x.

EXERCISE II. d.

1. If f(x)=x^ + 3x - 10, find /(3), also /(- 3).

2. If fix) = x2 — 5x 4- 6 and y=Z — x^ find /(?/) in terms of x.

MULTIPLICATION. 29

3. If f{a) = 1 - a, show that /{/(«)} = a,

4. If f{x) = a-x, show that p(x) = f{x),

P stands for /{/(/)}.

5. If f{x) = x^ + X + 1 find f(x - 1).

6. If fix) = 1 + X + — + ^^ + — + etc., find the

*^^ ^ 1.2 1.2.3 1.2.3.4

numerical value of /(I) to 5 decimal places.

7. If /(x) = a;4 _ 3x3 + 2x2 + 3a: - 3, find f(x + 1).

33. An expression such as

or l-2aj-3aj2 + 2a.^ + aJ*,

in which the exponents of the variable, x, are all positive
integers, is called a positive integral function of x, or
simply an integral function of x.

The first of these is written in descending powers of
the variable, and the second in ascending powers.

The coefficients may be numerical or literal.

The function is complete when all the powers of the
variable in consecutive order are represented; and any
integral function may be made complete in form by writ-
ing zero coefficients to the missing terms. Thus,

aj5 ^ Oic* + 2ar^ + Oic^ - 3a; + 1

is complete in form.

34. In multiplying together two integral functions of
the same variable, it is advantageous to operate upon the
coefficients alone, as the proper powers of the variable
are readily supplied to the result.

The functions, if not complete, should be made com-

30

THE FOUR ELEMENTARY OPERATIONS.

plete in form, and there is some advantage in writing
them in ascending order of the variable.

Ex. 1. To multiply a -\- bx + cx'^ + dx^ ^J P + qx -{- rx^.

h +c +d

■q +r

Coefficients

fa +1
\p +(

Product withf^« + ^^l^+^^^
variable supplied j ^ ^

4- ra

X2+J

+ q('
+ r6

+ re I

By observing in this typical case, how the coefficients
in the product are made up, i.e. by a sort of cross-multi-
plication, as pb + qa, pc-{-qb + ra, etc., we can perform
such multiplications with numerical coefficients with
considerable facility.

Ex. 2. To multiply 1 - 2x i- Sx'^ -{- x^ hy 2-x-{-2x^.

1-2+3 +1 Multiplicand.

2-1+2 Multiplier.

2 - 6x + 10x2 - 5x3 + 5x* + 2x5 . Product.

Ex. 3. To multiply x2 + Jx + J by x2 - Jx - |.
Arranging in ascending powers —

-i-iic-ix2 + 0x3 + x*

Ex. 4. To expand (i + x - 2 x2 + 3 x^y.

1st operation P + ^ ~ ^ + ^

ll + l_2 + 3

2d operation P + 2-3 + 2 + 10-12 + 9 1st product.

11 + 1-2 + 3

1 + 3-3-2 + 24-15-17 + 63-54 + 27
Result, l+3x-3x2-2xH24x*-15x5-17xH63x7-64x8+27x«.

MULTIPLICATION. 31

Ex. 5. To multiply x'^ - 2x^ -\- x + 1 by x'^ + 1.
Ordering in descending powers —

1+0+0-2+0+1+1

1 + 0-f 1

1+0+1-2+0-1+ 1+1+1
Result, x^ + x^ - 2x^ - x^ -{- x^ + X -{- 1.

35. A circulating decimal, or the arithmetical approxi-
mation to the value of any incommensurable, is an example
of a series of arithmetical figures which is non-termina-
ting. Similarly, in algebra we may have a series of
terms, arranged in ascending powers of the variable,
such that the series has no last term. Such a series is
called an infinite series, and is indicated by writing a few
terms at the beginning with three points, •••, with or
without ad inf. ; as

a-\-bx-{- cx^ + dar^ + • . • ad inf.

As we cannot write all the terms of an infinite series,
we cannot, in general, write all the terms of any multiple
of it. In some cases, however, certain multiples may
become finite by the vanishing of all the terms after the
first few.

We have the arithmetical analogue in a repeating or
circulating decimal, such as 1.2333---, which gives a
finite product, 3 • 7, when multiplied by 3, but another
infinite series when multiplied by 4 or 5.

To multiply two infinite series together, we take the
same number of terms in both multiplicand and multi-
plier, and retain that number of terms in the product.

32

THE FOUR ELEMENTARY OPERATIONS.

by

Ex. 1. To multiply

l-\-X + x'^4-X^ + X^ +
1-x + x^ -X^-\-X^ -

^ ll _1 + 1_1 + 1

Product, 1 + »2 + ic* +

1+0+1+0+1
, an infinite series.

Ex. 2. To multiply 1-^2 + 2 x^ - 3 x* +

by 1 + 2 X + ic2.

14-2-3...

Operation P + ^ ^

^ ll + 2+ 1

1+2 + 0 + 0+0...
Product, 1 + 2 x, a finite result as far as the series extends.

36. To square the series a -\- bx -\- cx^ -\- dx^ -{- ex^ -\- '
Operation {" +^ +' ^'^ +' + ""

Coefficients <

4-6 +c 4-^ +e +

a? -\-2ab-\-2ac

+ (

-\-2ad

-\-2ae

-{-2hc

+ 2hd

+ 0^

+

The operation is simply multiplication, but owing to
the identity of the multiplier and the multiplicand, the
coefficients in the result consist of double products and
squares ; and we notice that a square appears as the first
coefficient, and then in every alternate one.

By observing how the coefficients are made up, we
may write the square of a series with great ease.

Ex. 1. To square 1 + 2ic + Sic^ + 4x3 H to the term con-
taining x^.

1+2+3 + 4

1 + 4 + 6 1+ 8| + ...
+ 22 I + 12 I
.-. 1 + 4x + 10x2 + 20x^ + ••• is the required square.

MULTIPLICATION. 83

Ex. 2. To square 1 + hx - Ix'^ + j\x^ ■
1 + i-J +tV

/. Square = 1 + ».

EXERCISE II. e.

1. Multiply 1 + 2ic + 3x2 4-4x3 + 6x*
by l-2x + 3x2-4x3 + 6x*.

2. Multiply l-x + ix'^-7x"+ 19x* - 40x5 + ...
by l + ic-3x2,

to the term containing x^.

3. Multiply - 1 - X + 3x2 - 2x8 - X* 4- 3x5 - 2x6

by 1 + X + x2,

to the term containing x^.

4. In the complex series

1 + m(ax + 6x2 ^ cx3 + ...) -|- n (ax + hx^ + cx^ + ...)2

+ p{ax + 6x2 + 0x3+ '••y-\-'
find the coefficient of x'.

5. Find the coefficient of x^y* in the product of

(x + ax3 + 6x5 + ...) by /l _^- ^^-Lly4...y

6. Square the series, 1 + ^ " |(f )'+ |(|)'-

7. Multiply 1 + x(l - 2 x) + x2 (1 - 2 x)2 + ...
by 1 - X + 2 x2,

to the term containing x^.
Let 2/ = X — 2 x2.

34 THE FOUR ELEMENTARY OPERATIONS.

8. Find the square of x — ^ ^ ^ ?-^

2x 1-2 2-^x^ 1.2.8 23^5

The law of formation of the terms is evident, and any required
number of terms may be written down.

9. Show that

(1 - 2x + 3x2 - 4a;3 + ...)(^1 + 2a; + 3x2 + 4x3 + •••)

= (H-X2 + X* + -)^-

10. Find the coefficient of x" in the product

(1 + CiX + C2X2 + C3X3 + ...)(X~ + CjX"-! + C2X»-2 + CgX^-S + ...).

11. Find the coefficient of x"-^ in 10.

12. Find the coefficient of linear x in

l + ^^ + ^C^-l)^2_^^(x-l)(x-2) 3 ^^^
1.2 1.2.3

13. If y = ax-[- bx^ + cx^ H and x = Ay + By'^ + Cy^ -\

find A and B in terms of a and 6, on the condition that the coeffi-
cient of each power of x, in the result of substituting for ?/, is zero.

14. If/(x)^6|l + l.J-+ll^.-L + Ll3:i._L+...l

*^^^ Ix 1 3x* 1.2 5x' 1.2.3 7x10^ /

calculate /(2) to four decimal places.

15. If /(x) = x — Jx^ + ^x^ — ix^+ ..., calculate, to four deci-
mal places, the value of 8{/(i) + /(J)} + 4/(|).

In order to obtain 4 decimals exact, the calculation should be
carried to at least 5 places.

16. In any multiplication, write the terms of the multiplier in
an inverted order, and the partial products are not formed by a
cross-multiplication. How are they formed ?

DIVISION. 85

DIVISION.

37. Division, in algebra as in arithmetic, is indicated

in several ways. Thus, a-i-b, -, and a/b all mean that
a is to be divided by b.

In any case, a is the dividend and b the divisor, where
a and b stand for any numerical quantities or algebraic
expressions.

We define the operation indicated by - as the inverse

b

of multiplication, such that - xb = a.

b

Denoting - by g, we have a = bq, where q is the quo-
b

tient. Hence the theorem :

The dividend is the product of the divisor and the
quotient, and the quotient and the divisor are reciprocals
in the sense that if either be made the divisor the other
is the quotient.

Division thus consists in separating the dividend into
two factors, one of which is the divisor ; and any process
which accomplishes this effects the division.

38. Index law in division. Assume — = a^, and multi-
ply both members by a^

Then a'" = a"a^ = a"+^, by the index law.

Therefore m = n -\- p, ov p = 7n — n.

Whence — = a"*"" ;

a"

and this must hold for all integral values of m and n.

36 THE FOUR ELEMENTARY OPERATIONS.

Hence the quotient from dividing any integral power
by another integral power of the same root is that power
of the root whose index is found by subtracting the
index of the divisor from that of the dividend.

Cor. 1. If m = 71, — = 1 = a"-~ =a*^.
a"

Hence the zero power of any finite quantity is to be
interpreted as meaning + 1.

Cor, 2. Making m = 0, — = ^ = a<^ *» = a-\
a"" a""

Hence a negative exponent is to be interpreted as the
reciprocal of the same root with the corresponding posi-
tive exponent.

Thus, ? = a6-i; 1 +- + ^=1 + ^^ + ^ ^ etc.

39. The most important cases of division, where any
special process is required, are those involving a variable
in an integral function.

Let aoi? + 6a? 4- c be a divisor, and 'px^ -\-qx-{-r be the
quotient, and let Ax^ + Bd^ -{- Cx^ ■\- l)x -{- E be the
dividend.

By multiplying the divisor and quotient together, we
obtain as coefficients in the dividend,

A B C D E

ap -hbp + cp + eg -f cr

■i-aq 4- &g -\-br > .... X
-f ar

And operating upon coefficients only, when we divide
X by a 4- 6 + c we should get p -\- q -{-r] or, in other

DIVISION. 37

words, we are given the coefficients in X, and also a, 6,
and Cj and we are to obtain p, g, and r.

Let us see how it is to be done :

1. Dividing ap or ^ by a gives p, and p becomes known.

2. Multiply p by b, and subtract the product from B,
leaving aq. Divide aq by a, and we have q,

3. Multiply g by 6 and p by c, and subtract the sum of
these products from 0, leaving ar. Divide ar by a, and
r becomes known.

Thus p, q, and r are obtained.

In the foregoing, we notice — (1) that the only quantity
by which we divide is a, so that if a be 1 there is no
real division.

(2) If we change the signs of h and c, the partial
product hp, bq, br, cp, cq, and cr all become additive, so
that the only operations involved will be multiplication
and addition.

(3) That the partial products, which form any co-
efficient in the dividend, as (7, are made up by a cross-
multiplication, as explained in Art. 34, Ex. 1.

This process is known as synthetic division, because
we build up the terms of the dividend by getting the
partial products which enter into their composition,
and through this synthesis we obtain the terms of the
quotient.

The following examples will illustrate :

Ex. 1. To divide 2a:* + x^ - 8x2 + 17x - 12
by 2x2-3x4-4.

88

THE FOUR ELEMENTARY OPERATIONS.

Here a = 2, b ■■
such that —

3, c = 4 ; and we are to find p, g, and r

■3,

2p = 2, -3p + 2g = l, 4j9-3g + 2r:
whence, i? = 1, Q' = 2, r =

and the quotient is x^ + 2 cc — 3.

This operation is systematically carried out as follows —

2 + 1-8

3 6

-4

1 + 2-

+ 3-4

Divisor, signs of b and c changed.

+ 17-12 . . Dividend.

- 9 + 12^

- 8

I . . Partial products.
0 0 . . Quotient.

Here we change the sign of the 3 and 4 of the divisor,
and thus have only additions. We then divide each sum
by 2 as we proceed. Thus the multipliers in forming the
partial products are 3 and — 4, and the divisor is 2.

Ex. 2. To divide x^ - 3 ic^ + 6 x - 4 by ^2 - 2 ic + 1.

The coefficient of the first term of the divisor being 1 need not
be written. Making the functions complete in form, we have

1+0+0+0-3+0+0+0

+2+4+6+8+4+0-4

_1 -2-3-4-2+0

1+2+3+4+2+0-2-4

+ 2-

+ 6-4
-8 + 4
+ 2

0 0

and the quotient is x^ + 2 x^ + 3 x^ + 4 x* + 2 x^ — 2 x — 4.

If desired, the functions may equally well be arranged
in ascending order of the variable, as

DIVISION.

39

-4+6+0+0+0-3+0+0

^8-4+0+4+8+6+4
+ 4 -{_ 2 + 0 - 2 - 4 - 3

2-1

+ 0 + 1
+ 2 + 0.
-2-1

.4-2+0+2+4+3+2+1 0 0
■ 4 - 2 a; + 2 x^ + 4 X* + 3 x^ + 2 x"' + x^.

40. The preceding are examples of exact division. In
arithmetic, when the dividend is greater than the divi-
sor, we can obtain an integral quotient and a remainder,
where there is one ; or, we may expand the remainder
into a decimal series which, in general, is non-termina-
ting.

Now, a higher degree in algebra corresponds to a
greater quantity in arithmetic ; so that, when the divi-
dend is of a higher degree than the divisor, and the
division is not exact, we may obtain a quotient and a
remainder, or we may expand the remainder into an
infinite series.

Ex.1. To divide x^ - x^ + 5x^ + lOx^ - 5x + 1 by x*-2x8
+ x^ — 2, obtaining the quotient and the remainder.

+ 2- 1+0 + 2 . . Divisor.

1+0-1+0

+2+4+4

-1-2

+ 5 + 10-5 + 1 . . Dividend.

+ 4- 2 + 4 + 4^1

-2+4 ■ . . Partial products.

+ 2

1+2+2+2
Quotient,
x3 + 2x2 + 2a

will be noticed 1

+ 9 + 12-1 + 5 . . Result.

Remainder,
, + 2. 9x3+ 12x2 -x + 5.

bhat a vertical line is drawn to the

40 THE FOUR ELEMENTARY OPERATIONS.

left of that part of the divisor which is used in forming
the partial products.

In a case of even division, all the terms of the result
to the right of this line are zeros, and when we wish to
obtain the remainder we treat these terms as if they
were zeros in forming the partial products.

If we employ the terms to the right of the vertical
line in forming partial products, the quotient will extend
into a series, and all the terms to the right of the line
will contain negative powers of a?, and the series will thus
be arranged in descending powers of x.

If we wish the series to be in ascending powers, we
must arrange our functions in that order before begin-
ning the division.

Series so produced, like circulating decimals in arith-
metic, have their coefficients connected by a fixed law
of formation. Sometimes this law is obvious from sim-
ple inspection, and at all times it can be exactly deter-
mined.

This law is of great importance in investigations con-
nected with Recurring Series.

Ex. 2. To divide 1 + a; — x^ by 1 — 2 x + x^ to a series in
ascending powers of x.

1

+ 2-1

+ 1-1

+ 2 + 6 + 8 + 10 + 12 H

_1_3_ 4_ 5

1

+ 3 + 4 + 5+ 6+ 7 + ...

Here the law of the coefficients is obvious, and the series is

1 + 3x + 4x2 + 5x'^ + 6x* + 7x5 + ...

DIVISION. 41

If we arrange the dividend and the divisor in descending powers
of X, the quotient coefficients are —

-1-1 + 0+1 + 2 + 3 + 4 + ...
and the series is —

/v /y»0 /y^ rv*0 /y»0

%Lf JU *K/ %K/ %Kf

or - 1 - x-i + x-3 + 2 a;-* + 3 x-5 + 4 x-^ + ...

Ex. 3. To divide 1 + x by 1 — x + x^ to a series in ascending
powers.

The series is 1 + 2 x + x'^ - x^ - 2 x* - x^ + x^ + ...
Here also the law of the coefficients is readily brought out, foi
the series may be written

(1 +2X + X2)(1 _X3 + X6-X9+ ).

Ex. 4. Divide 1 by 1 - 2x + 3x2.

2-3

1

2 + 4 + 2- 8-22-20...
_3-6- 3 + 12 + 33...

1

+ 2 + 1-4-11-10 + 13...

The series is 1 + 2x + x2 - 4x3 - llx* - lOx^ + 13x6. .., and
the law is not apparent from inspection,

41. The following results are frequently required, and
should be committed to memory :

i. -^ = 1 + a: -f a;2 4- a^ 4- a;4 _j

ii. — =— = 1 — a;4-ic2 — ic^-f-ic*

l-{-x

iii. ^^ ^ =l4-2a; + 3a^ + 4a^ + 5a;^+'"

iv. -— ± — ^ = l_2a;4-3ic2-4ar» + 5a;*-

42 THE FOUR ELEMENTARY OPEKATiOKS.

14-2;
Ex. 1. To expand — — — to a series.

By iii. this is (1 ■}- z)(_l + 2 z + S z^ -\- 4 z^ + ...), which by dis-
tribution becomes

1 + 30 + 52:2 + 7^3+...

Ex. 2. To find from what division the series

1 + X - 2x^ - x^ - X* -h 2x^ + x^ + x^ - 2x^ "*
has been derived.

The series may be vsrritten

(1 + x-2x2)(l -x^ + x^-x^'");

and, by ii., 1 — x^ -\- x^ — x^ ••• =

1 + x3

/. 1 + a; - 2 a;2 .^ ^^^ division.
l + x3

EXERCISE II. f.

1. Divide x^ -\- x^ + 2x^ -2x + S by ic2 _ 2 x + 2.

2. Divide 6 a^^2 a^-\-9 a*-2 a^-a^-S a-\-l by 2a^+Sa-l,

8. Divide 1 + x- + 2x^ -2x^ + Sx^ by 1 + 2^ + 3x2 + 4x3.

4. Divide a — lbya + lto5 terms in descending powers of a.

6. Divide x^ — x^ -\- x^ — 2x — 1 by x^ + 2 x2 — 3 x + 1, giving
quotient and remainder.

6. Divide a -\- 2a^ — Sa^ -{- 4:a^'bj a — a^ + a^ — a^ to a series,
and obtain the law of the coefiicients.

7. If ?/ = 1 - 0 + (1 - s)2 + (1 -zy+ ..., and 0 = 1 + a: + a;2
+ ic3 + ..., show that y =— x.

8. Divide l+2x+3x2+4x3+... by l + 3x+5ic2+7x3+... to 6
terms of a series.

DIVISION. 43

9. Expand x -i- (x^ — 2 x -}- 1) into a series, first in ascending,
and second in descending powers of x.

10. What must be added to x'^ — x^ + x^ — x + 1 to make it an
even multiple of x^ — x + 1 ?

11. Divide 1 + fee + -\^-x'^ + H^' + l«* + fl^^ + ^
by 1 + 2x + 3x2 + 4ic3

12. Divide x^ + j/^ _j_ 3 ^.^^ _ 1 l)y a; 4. 2/ — 1.
Take x as variable and the functions are

flj3 4. 0x2 + 32/x + (2/3 _ 1) and x +(y - 1).

1 + 0 +3?/

+ (y' - 1)
- (y' - 1)

l-(2/-l) + (2/2 4-?/+l) 0

Quotient is x^ - x(y - 1) + (y^ -\- y + 1).

13. Divide a^ (b - c) + b^ (c - a) + c^ (a - b) hy a + b + c.
Take a as variable and proceed as in Ex. 12.

14. Divide x^ + 2 x ^^ - x'^y^ + a^?/* - 2/^ by x^ + xi/^ _ y^.

The functions being each homogeneous, put y = zx; this reduces
the division to x^ (1 + 2 z^ - z^ + z^ - z^) -4- (1 -f ^2 _ ^s),

15. Find the simplest division that will give the series

X + 3x2 + 2x3 - x* - 3x5 - 2x6 + x7 + ...

16. What expression added to (a + 6 + c) (ab + 5c + ca) will
make it exactly divisible by a + 5 ?

17. Multiply a + 3 + 3 a-i + a-2 by 1 - a-i.

18. Multiply (x + x-i)3 + (x + x-i)2 + (x + x-i) + 1 by x-x-i.

19. Divide 4 x2 - 7 x + 3 - x-2 + x-3 by x-i - 2 x-2 + x-^.

20. If X - a be a factor of x2 + 2 ax — 3 b'^^ then a = ±b.

44 THE FOUR ELEMENTAKY OPERATIONS.

21. If 1 -^ (a2 - ax-\- x^) be expressed as A + Bx+ Cx^ + Do(?
+ /(x), find A, B, O, D and the form of /.

22. If x^ — ax^ + &X + c be divided by x — z^ the remainder is
z^ — az^ + bz + c.

23. If the terms of the divisor are written in an inverted order,
by what arrangement of multiplication are the partial products
formed ?

CHAPTER III.
Factors and Factorization.

42. In a case of even division, we separate the dividend
into two factors (Art. 37). One or both of these might
be again separated into two factors, and so on, until the
whole expression was separated into factors which should
be linear in the variable or else numerical.

Thus a^ ^bx^ — a^x -f- a^b = (x — a){x-\- a) {x — b).

We thus see that Factorization, as an operation, is the
inverse of Distribution.

In the factored expression written above the factors
are each linear and binomial.

In 6 a (a + 6 4- c) {ab + 6c) (oj^ + £c + 1), 6 is a numerical
factor, a is linear and monomial, a-{-b -{-c is linear and
trinomial, and ab + be and oc^ -^x + 1 are both quadratic
factors.

Theoretically, any integral function of a variable can
be separated into factors linear in that variable ; but the
cases in which we can make the separation practically
are limited to but a few classes, out of all the possible
integral functions. Frequently, however, these cases are
of most importance.

43. Factorization may sometimes be effected by mak-
ing use of the standard forms of Art. 29.

Thus, because a^ — 6^ = (a + b) (a — b), we can always

45

46 FACTOKS AND FACTORIZATION.

put into factors that which can be expressed as the dif-
ference between two squares.

Ex.1. (a2 + 62)2_(a2_52)2

= (a^ + 62 + «2 _ 52) (^2 ^_ 52 _ ^2 4. 52) ^ 4 ^252.
Ex. 2. x'^-^2a-a^-l = x^-(a- 1)2= (a; - a + 1) (x + a - 1).

44. An expression of the form x^ + ax + b can be
factored at sight if we can discover two quantities such
that their sum is a and their product is b whatever a
and b may stand for. For if p and q be such quantities,
the factors are {x -\-p)(x-^q).

Ex.1. x'^-2x-S = (x-{-l)(x-S),

Ex. 2. a'^ + 2 ah + h'^ - a - b - 6 = (a + 5)2 - (a + 6) _6

= (a + 6-3)(a + 6 + 2).
Ex.3. 8ic2 + 8cc-6 = 2(2x*^ + 2.2x-3)
= 2(2x + 3)(2x-l).

No particular rules can be laid down for this kind of
factoring. Success is to be attained only by observation
and practice.

EXERCISE III. a.

1. Put (x'^ + Sx + 2') (x2 _ 3 X + 2) into four linear factors.

2. Put ab + 2a^ — Sb^ — ibc — ac — c^ into linear factors.

3. Express x"^ — (p -]- q)x^ + pqx^ + (p - q)x — 1 as the pro-
duct of two quadratics in x.

4. Factor (x + yy + (x -^ y) (a -]- b) -^ ab.

5. Factor (a + x)2 - 3(a + x) + 2.

6. Factor 6(2 x + 3 ?/)2 + 5(6x'^ + bxy - ey^)- 6(Sx - 2 yy.

7. Express 4a262 -(^2 + 52 _ ^2)2 in the form of four linear
factors.

FACTORS AND FACTORIZATION. 47

8. Factor 12 ^2 - 7 x + 1, and 6 x^ - 21 x + 18.

9. Express x^ — px^ + (^ — 1)^^ -\- px — q as two linears and a
quadratic.

45. Let f{x) be any integral function of x, and let
p, q, ?', s, etc., denote its factors, so that

f{x)=p'q'r'S'"

Now, assuming that all the factors are finite, if any
factor becomes zero, the whole expression, /(a?), becomes
zero. And, conversely, the expression cannot become
zero unless one of its factors becomes zero.

If, then, we suspect that a certain expression is a
factor of /(cc), we put that expression equal to 0; from
this we find the value of x in terms of the other quanti-
ties concerned, and we substitute this value for x in the
given function.

If the function becomes zero, or vanishes, the sus-
pected expression is a factor ; and if the function does
not vanish, the suspected expression is not a factor.

Ex. 1. Is a + 1 a factor of 6 d'h + 3 a^ + 12 aft'^ -f 12 a6 + 3 a
+ 12 6^ + 6 6 ?

Put a + 1 = 0 ; this gives a = — 1. Substitute — 1 for a in the
given function, and it vanishes.

Hence a -\- 1 is a factor.

Similarly, we find a + 2 6 to be a factor ; and by dividing by
these we get 6 & + 3 as the third factor ; and the expression
becomes (a + !)(« + 2 5)(6 6 + 3).

Ex. 2. To factor a{h + he - c)+h{c + ca - a)+ c(a -\-ab- 6).

As the expression may have a monomial factor, try a = 0, i.e.
put 0 for a.

The expression vanishes, and hence a is a factor.

But the expression is symmetrical in the three letters, and
hence b and c must also be factors.

48 FACTORS AND FACTORIZATION.

Therefore ahc is a factor.

By the index law the dimensions of the expression must be the
sum of the dimensions of its factors.

But ahc is of three dimensions, and so also is the expression ;
hence there are no other literal factors.

There may be a numerical factor, since such a factor has no
dimensions. The coefficient of the type ahc from the expression is
readily seen to be 3 ; therefore the whole expression factors to 3 ahc.

Ex. 3. To factor ah(h'^ - a^) + hc(c^ - 62) + ca(a'^ - c2).

We readily discover that there are no monomial factors.

Since h — a is a factor of one of the terms, let us try if it be a
factor of the whole.

Put h — a = 0, or write h for a in the expression. It vanishes.

Therefore 6 —a, and from symmetry, a — c, and c — 6, are
all factors.

Therefore (h — a)(a — c)(c — h) is a factor of 3 dimensions.
But the expression is of 4 dimensions. Hence there is a fourth
factor, symmetrical in a, 6, and c, and linear, a -{- h -\- c is the
only such factor that can occur here, and

(6 - a)(a - c)(c - h)(a + 6 + c)

is a factor, and comprises all the linear factors.

For the numerical factor take any type term that occurs in both
the expression and the factored result, as a'^h. Its coefficient
from the expression is — , and from the factored result it is +.
Therefore — 1 is the numerical factor, and the expression becomes

-(h- a)(a - c^iic - h)(a + 5 + c),
or (a - 6) (5 - c) (c - a) (a + 6 + c).

Ex. 4. To factor ah(c -d)+ hc(d -a)-\- cd(a -b) + da(h- c).

We find cf to be a monomial factor, and by symmetry h, c, and
d are factors.

Therefore abed is a factor of 4 dimensions.

But the expression is of only 3 dimensions, and should have
only 3 literal factors, unless it can have any number of factors.
The only expression that admits anything as a factor is 0. Hence
the expression = 0, identically.

FACTORS AND FACTORIZATION. 49

46. If we have an integral function of x, and if its
factored form be

(a; — a){x — b)(x — c) • • •,

the independent term of the function, T say, is equal to
abc'" (Art. 31).

Hence, in trying to factorize the function, since a, b, c,
etc., are all factors of T, the factors of T are the only
quantities to be substituted for x in our trial.

By substituting the rational factors of T, any rational
linear factors of the function will be discovered.

Of course it must be understood that factors are not
necessarily discoverable in this way, since it is only in
special cases that such functions have all or any of their
linear factors rational.

Ex. 1. To factorize x:^ - Sx^ - Sx^ + 7 x + 6.

Here T = Q, and its factors are ±1? ±2, ±3, and ± 6.

Put 1 for x; the function does not vanish, and x — 1 is not a
factor.

Put — 1 for X ; the function vanishes and x + 1 is a factor.

Similarly, try 2, — 2, 3, — 3, etc., successively until all the fac-
tors discoverable by this means are found.

Otherwise, having found x + 1 to be a factor, divide the func-
tion by X -\- 1. The quotient is

x^-ix^ + x + 6.

Of this new function x + 1 is again a factor. Divide by x 4- 1,
and the quotient is 2 ^ i a

which factors into (x — 3) (x — 2) .

.-. X* - 3x3 - 3x2 + 7x + 6 = (x + l)2(x - 3)(x - 2).

In employing this latter method, it will be more
expeditious to try the higher factors of T first.

50 FACTOKS AND FACTORIZATION.

EXERCISE III. b.

1. Factorize the following —

i. 2a25 + 2a5c. Iv. 2{a(65-c3)}.

ii. ^{ah'^-aT-h). v. ^{h{c - d){ah - cd)},

iii. S(a.&^-c-^). vi. S{a(6't-cO}.

vii. 2 (a62 _ a'^j^) + Sa& - 2^2 + i.

viii. (a+ 6 -c)(?) + c- a)(c + a- &)- 2a6(a + 6) + 2a^ °

ix. 2 {ah ' c — d) four letters.

2. Put 2a*6c — 2a^63 j^to quadratic factors.

3. If the 6th power of a number be diminished by 1 and the 5th
power of the same number be increased by 1 , the difference of the
results is divisible by the next greater number.

4. If any even power of a number be diminished by 1, and any
odd power of the same number be increased by 1, the results have
a common factor.

5. Factorize x* + 8^3 - 10x2 - 104 x + 105.

6. Express (x + l)(x + 3)(x + 5)(x + 7) + 15 as the product
of two linears and a quadratic.

7. Factorize x* — 5 x2 + 6.

8. Factorize (a+6 + c)3-(5 + c-a)3-(c+a-5)3-(a+6-c)3

9. Factorize ^a{bc + ac + d^ — a^} — 5 ahc.

10. Factorize

{ac + hdy^— abc{a — 5 4- c) — hcdQb — c -{■ d)— cda(c — d + a)
— dab{d — a + 6).

11. Factorize

(a + & - x) (5 + c + x) + (6 + c - x) (c + a + a;)
+ (c + a — x) (a + ^ + ic) — 3(a5 4- 5c -f ca)
4- ahc - a2 _ 52 _ c2 _^ 3 x'^.

FACTORS AND FACTOR IZATION. 5l

12. Factorize

{a ~ h){h + c)+{h - c)(c ^ a) + {c - a) {a + &) + («- c)2.

13. Factorize x^ + x{^ah - Sa^) -(^a-h){h- c) (c - «).

14. If the 4th power of a number be increased by 4 and dimin-
ished by 5 times the square of the number, the result multiplied by
the number itself is the product of 5 consecutive numbers.

47. The symbol ^ is defined by the relation -yja x V^
= a, where a denotes any numerical quantity or algebraic
expression.

Thus V(^^ + &' + 2a6) = a + &, and V^^ ^ 4.

The expression ^a is read the square root of a, or, more
concisely, the root of a, and it denotes that a is to be
separated into two identically equal factors, and that one
of these is to be taken.

Hence when it is possible to separate a quantity or an
expression into two such factors, it is possible to express
exactly the square root of the quantity or expression.

Thus Sa^ + 2 ^ah can be separated into the identically
equal factors la x 2a; and hence %a -^ -yj {%a^ -\- 2^ab) ,
for any number of letters.

48. The expression y'a requires careful consideration.

(1) If a is a positive square number it is the product
of two identically equal factors, and ^a denotes one of
these factors.

The factors may be both + or both — , since in either
case the product is -f .

Therefore ^a has two signs and is often written ± -y/a.
Thus V^ is either + x, or — a? ; and ->/16 is + 4, or — 4.

The double sign, whether written or not, must always

52 FACTORS AND FACTORIZATION.

be mentally attached to a square root, being frequently
of very great importance.

(2) If a is a positive non-square number, no two iden-
tically equal factors can be found for it.

The -y/a then symbolizes one of that class of numerical
quantities called incommensurables, or irrational quanti-
ties (Art. 1).

In this case ^a cannot have its value exactly expressed ;
but it may, under the form of a non-terminating decimal,
be expressed to any degree of approximation we please,
by the arithmetical process of ^extracting the square
root.'

Thus (1.41)2 differs from 2 by 0.0118

(1.414)2

n

" 2 " 0.000604

(1.4142)2

i(

« 2 " 0.000039

(1.41421)2

((

" 2 " 0.0000002

etc.

etc.

And the successive squares become closer and closer-
approximations to 2, the degree of approximation de-
pending upon the extent of the decimal series.

This series, unlike circulants produced by division,
has no arrangement of its digits which would indicate
any law governing the order of their succession.

(3) Let a denote a negative number.

As like signs produce only -f in multiplication, it is
not possible to find, or to approximate to, or to conceive
of a quantity, which multiplied once by itself will give
the sign — .

The symbol ^a is then called an imaginary in contra-
distinction to the quantities of (1) and (2), which are real.

Thus "y/— 3 is imaginary, while -y^3 is real.

FACTORS AND FACTORIZATION. 53

49. As in arithmetic, so in algebra, an expression may
be a complete square and be capable of having its square
root exactly expressed, or it may be a non-square and
admit only of having its root approximated to by an
infinite series.

Thus

1 x" , I'S a^ 1.3.5 a?^

^ ^2 1.2 22^1.2 2^ 1.2.3 2^^

General methods for this approximation will be con-
sidered hereafter.

It may be remarked that an algebraic expression can-
not in itself be imaginary, as the character of real or
imaginary is wholly due to the interpretation of the
quantitative symbols.

Thus Va — 6 is real, if a and h are both positive
numbers, and b is less than a, but imaginary if 5 is greater
than a. If a is positive and b negative, the expression
is always real ; and if a is negative and b positive it is
always imaginary.

50. If a denotes a positive quantity, Va can be
expressed to any degree of approximation that we please,
and hence ^a must, like other quantitative symbols, be
subject to the commutative and distributive laws.

Hence

(1) V^+ ^b=^b-\--y/a, and ya • V^= V^ ' V^-

(2) ■y/a(b-{- V^) =b^a + ^a- ^c, etc.

51. The symbol y' is introduced here as a special
operative symbol, having a relationship to the exponent,
as will appear hereafter, and it is necessary that we
should investigate the limits of its operation, and dis-

54 FACTOKS AND FACTOHIZATION.

cover in how far it obeys the great formal laws of
algebra.

(1) As (ya- V^)(V^* V^) = V<^* V^* V^' V^ = ^^
by definition ; and as Va6 • -\/ah = ah by definition,
therefore ^a • ^h = y/ah ; and the operative symbol ^
is distributive over the factors of a prodxict.

Thus V^ • V^ • "^^ = V(4 ^^^) ^ V^ • Va6 = 2 Va6.

V (45 a^ &c2) = V (^ ^'c' . 5 a&) = 3 ac V5a6.
etc. etc. etc.

(2) Since ^{a-\-b)^(a-{-b) =a -\-b by definition,
and (-^a + V^) (ya + -^b) =a-\-b-{-2 -y/ab by distribu-
tion, therefore ->/(a + 6) is not the same as ->/a + V^*

Or, the symbol ^ is not distributive over the terms of a
sum.

The statements of (1) and (2) form the working prin-
ciples of this symbol, and should be carefully remembered.

52. Since ^— a-^— a = — a by definition, whatever
a may be, we assume that ^a is subject to the same
general laws of operation whether a be positive or nega-
tive, i,e, whether the expression be real or imaginary.

■ a . -Vb = y'a • V— bj

where ^a is real.

Thus every imaginary number can be reduced to depend
upon the symbol V— 1, which is called the imaginary
unit, and is usually symbolized by i.

FACTORS AND FACTORIZATION. 55

If, then, X denotes any real number, ix denotes the
corresponding imaginary ; the relation between these
being that the square of the first is + a^, and of the
other it is —x\

This generalization introduces us to a new set of num-
bers, the symbolic numbers or imaginaries.

All whole numbers, positive, negative, and imaginary,
may be represented in the general scheme,

..._5, -4, -3, -2, -1,0,1,2, 3, 4, 5, ...
... — ^i^ — 4 1, — 3 i, — 2i, — ij 0, i, 2i, 3i, 4i, 5 i, •••

53. The powers of the symbol i, which occur very
frequently, are given in the scheme,

i^ = — 1, i^ = — i, i* = 1, f = i, i^ = — 1, etc.,

the powers repeating their values in cyclic order.

A quantity which is the algebraic sum of a real and
an imaginary is called a complex quantity; but being
arithmetically inexpressible on account of its imaginary
part, it ranks with imaginaries.

Thus, 2 + 31 is a complex, and so also is 2 — 3 i. The
square of 2 + 3i is 2^+ (3i)^4-12i, or 12 z — 5, another
complex ; but the product (2 + 3 i) (2-3 i) is 2^ - (3 i) \
or 13, a real.

a + hi represents any number whatever; for if 6=0, it
is real ; if a = 0, it is imaginary ; and if neither a nor h
be zero, it is a complex.

64. The expression x^ -\-px-\-q can always be sepa-
rated into two factors linear in x.

56 FACTORS AND FACTORIZATION.

=^(x i P + ^P'-'^^Vx I p-Vp'-4g

As these factors both contain the same square root
part, they will be both real or both complex, i.e. imaginary.
But in any case they will, upon distribution, reproduce
the original expression.

Thus the factors of a;^ + 2 a; + 3 are

both being complex quantities.

55. The expression aa^ -{-bx + c can always be sepa-
rated into a monomial factor, a, and two factors linear in x,

aoc^ + bx + c^a{x^-{--x-\--
\ a a

The part within brackets is the same expression as

ar' -f- »aj + g, if we write - for p and - for q.
a a

Making this substitution in the result of Art. 54, gives,
after reducing,

x-\ —

«{-^±tf^}{'

2a

This factorization may also be done independently as
follows :

aa^ + 2>a; + c = — { 4 a V + 4 abx + b^ -(b^ -4.ac)\

= — K2aa; + &)2-(V62-4ac)2|
4a ^ ^

FACTORS AND FACTORIZATION. 57

==^l2ax-^b-\-Vb''-4.ac\l2ax-\-b-^/b^-4.ac\

Ex. 1. The factors of 3ic2 + 5 a; - 1 are

Ex. 2. The factors of 2 ^2 - 3 x + 2 are

The two factorizations, of Art. 54 and the present
Article, are very important, and the forms of the factors
should be carefully mastered.

Taking ax^ -{-bx + c as being the most general in
form, the square root part of each factor is -Vb^ — 4 ac.
The character of the factors will depend upon that of
this part of them.

If a and c are +, and 4ac is greater than 6^, the fac-
tors will be complex quantities.

If 4 ac be less than 6^, the factors will be real and
unequal ; and if 4 ac = b^, the expression V6^ — 4ac
becomes zero, and the factors are real and equal.

Cor. The expression aa^ -{-bx-\-c is a complete square
when 6^ = 4 ac.

The finding of the square root of an algebraic expres-
sion is equivalent to the separation of the expression
into two identical factors, and hence requires no special
process.

If the expression is a complete square, the factorizar

58 FACTORS AND FACTORIZATION.

tion ranks with the simpler cases. But if it is not a
complete square, the only practical method is by means
of the binomial theorem or undetermined coefficients, to
be given hereafter, and the root is expressed as an
infinite series.

Thus, in a^ — 2 a& + a + 4 5^ — 4 & + 1, we readily see
that ± a, ±2b, and ± 1 must be terms of the root, and
the least observation shows that the root is a — 2 & -|- 1.

Ex. 3. The factors of 2 x^ — 3 x — 4 are

2{x + }(-3+ V41)}{x + K-3-V41)}.

Ex. 4. The factors ot x^ + 2x+10 are
(X + 1 + Si)(x + 1-Si).

Ex. 5. Determine the value of m so that 3 x^ + 4 mx + 12 may
be a complete square. We must have

62 = 4 ac, i.e. (4 m)2 = 4 • 3 • 12.

Whence m = 4- 3 or — 3.

56. The expression x"^ -\- bx^ -\-c can be separated into
four linear factors. Let a, /?, y, S denote these factors.
Any two of these multiplied together gives a quadratic
factor of the expression.

But these may be multiplied two together in three
different ways, namely,

afi, yS ; ay, )8S ; and a8, /?y.

Hence x'^ -{-bx^ -\- c can be separated into a pair of
quadratic factors in three different ways.

Ex. x* + 10x2 + 9= (ic2 + 9)(x2 + 1)

= (x + Si)(x — S i) (x + i) (x — i).

FACTORS AND FACTORIZATION. 59

Thence the pairs of quadratic factors are —

1. (a;2 + 9)(a:2 + l).

2. (x^ -\-iix- 3) (x2 -iix-S),

3. (^2 + 2ix-\- 3) (ic2 - 2 IX + 3).

57. The factorization of x^ — 1 is important,
and by Art. 54,

If any one of these factors becomes zero, the expres-
sion vanishes ; i.e. ar^ — 1 = 0, and qc^ = 1.

Hence, when x = l, or - ^^W^ or - hth/^ a^ = 1.
' ' 2 2

And since the cube of each of these three values of x
is 1, these are the three cube roots of unity j one of which
is real, and the other two complex.

The complex roots are generally denoted by <o and w^,
because the square of either of them is equal to the
other.

Then w^ = 1, w* = w^ . w = w, w^ = o)% w^ = 1, etc.

Since w^ — 1 = 0, its equivalent (cd— l)((o^4- a)-f-l)=0;
and as 0) — 1 is not zero, we must have

0)2 + a> + 1 = 0.

And this is the fundamental relation connecting the
cube roots of 1 ; i.e. the sum of the three roots is zero.

60 FACTORS AND FACTORIZATION.

Ex. Multiply together x + wy + w^^ and x + w^y + wz.
Distributing (x + wy + ta^z) {x-\- (ohj ■\- wz) gives

But w3 = 1, a»2 + <u = - 1, and w* + (o2 = w + w^ = - 1.

/. (x + wy H- w2;j) (x + w2?/ + wz) =:x'^ + y'^ ■\- z"^ — xy — yz — zx.

EXERCISE III. c.

1. Find V{«^ + 62 + 2 a6 - 2 a - 2 6 + 1}.

2. Separate x2 + 4y2^4a.2^_4ic_8?/4-4 into two identically
equal factors.

3. Express Vab^ + Vac* — ^74 aft^^^ as a multiple of a single
irrational factor.

4. Show that (a + hi) (c + di) has the form A + J5i, and ex-
press A and B in terms of the small letters. •

5. Distribute (x — a + 6Q (x — a — ?>i) .

6. Show that ix - I (ix)2 + J {ixY — \ (zx)* +

= i(x-Jx8+ix5-+...)+ix2(l-i^'+J^*-+-).

7. Factorize the following —

1. x2-3x-3. V. ax2 + x4-a.

ii. x2 - 2 X + 5. vi. px'^ - (p + 1) x + 1.

iii. 4 x2 - 4 X - 2. vii. {a^ - 62) x2 + 2 ax + 1.

iv. 5 x2 + 3 X - 6. viii. 2 + ax + — •

ax

8. Resolve x* — 11 x2 + 10 into four linear factors.

9. Resolve x* — 3 x2 -f 1 into linear factors.

10. Resolve x* + x2 — 2 into its three pairs of quadratic factors.

11. Resolve 4x2 — 4x — x* + l into linear factors.

12. Resolve x^ — 2 x2 — x + 2 into linear factors.

13. Distribute (1 + x)(l + t«^x)(l + w2x).

FACTORS AND ROOTS. 61

14. Show that (a + w/3 + ^27) (a + (o^^ + W7) = Sa^ - Sa/3.

15. Show that {x-\- tay -\- uiHf + (a; + ^^2/ + w^)^

= 2 2x3 -3 2x2?/ + 12x2/0.

16. Distribute (x + w?/ + w2^) (x + 0^22/ 4- iaz) (x + 2/ + «).

17. Show that (x + w?/ + ^2^)3 - (x + 0*22^ + ^^y

= - 3 V^=r3 (X - 2/) (2/ - 2;) (;? - X).

18. Find the relation between a and 6 when

(a + &)x2 - (a2 - 62)x + a26 + aW-
is a complete square.

58. The integral function of a?,

ic* _ lOa^ + 35a;2 - 50a; + 24,

factors into

(a; _ 1) (a; _ 2) (a; - 3) (a; - 4).

The numbers 1, 2, 3, 4 are the roots of the function,
because from these, and the variable x^ we may build up
the function by multiplication, or, so to speak, cause it
to grow up.

If any of these roots be put for x, the substitution will
cause the function to vanish, since it makes one of the
factors zero. And, conversely, the only single substitu-
tion that will make the function vanish must make one
of the factors vanish, or is the substitution of one of the
roots for x.

Hence the roots of an integral function of any variable
are those quantities which, when put for the variable in
the function, cause it to vanish. And reciprocally any
quantity, which put for the variable will cause the func-
tion to vanish, is a root.

62 FACTORS AND ROOTS.

Thus the roots of a^ + x^'^a + x^ab +abc are — a, — b,
and — c J for the function factors into

(x + a) (ic + b) (x + c).

59. The expression a;* — 10 ic^ + 35 ic^ — 50 a? + 24 = 0 is
a conditional equation, or simply an equation in which a;
is to have such a value as will make the expression an
identity (Art. 22).

We have seen, in the preceding article, that this will
be effected by making x equal to any one of the roots of
the function, namely, 1, 2, 3, or 4.

It is readily seen that the same principle applies to
fimctions of any degree.

Hence: (1) In the equation formed by putting an
integral function of a variable equal to zero, we obtain
the roots of the equation by separating the function into
factors linear in the variable.

The determination of any one of these factors is a
solution of the equation, and the determination of all
these factors is the complete solution.

(2) The whole number of solutions, or the number of
roots which the equation has, is the number of linear
factors into which the function is theoretically separable,
and this is the same as the degree of the function in the
variable.

The solution of an equation is thus equivalent to the
factorization of the function into factors linear in the
variable.

60. When the roots of an integral function or of the
corresponding equation are all real and all rational, they
can generally be found.

FACTORS AND ROOTS. 63

Also, the methods of factoring now at our disposal are
sufficient for the linear factorization of all integral
functions of a single variable of not more than two
dimensions ; but these methods are not sufficient for
the general factorization of functions of more than two
dimensions. They suffice, however, for many special
and particular cases.

Ex. 1. To find all the solutions of x* - x^ - 2 ic2 - 2 + 4 = 0.
By trial we readily find x — I and ic — 2 to be factors.
Dividing by these gives ic^ + 2 aj + 2 = 0.
The factors of this are (x + 1 + i) (x + 1 — i) .
And the four roots are 1,2, — 1 — ^ and — 1 + i.

Ex. 2. To solve the equation x^ — bx-\-2 = 0.

Factorization gives (x — 2) (x^ + 2 a; — 1).

The factors of ic2 + 2 x - 1 are {x + (1 + ^^)}{x + (1 - V^)}-

And the roots are 2, - (1 + ^2), - (1 - V^)-

A linear equation in any variable is simply a linear
factor of unity. By proper transformations such an
equation may always be brought to the form
Ax-B = 0,

B

or ^= .'

A

which is the only solution, x having but a single value.

EXERCISE III. d.

1. Solve the following equations —

1. x^ + 4 x^ — X — 4 = 0.

ii. x5 + x^ - x-2 - 1 = 0.
iii. x3 + 3x2 + 4x + 2 =: 0.

2. Find values of x that will make (&2 _ a2)x + ^^^ -^ equal
to (x-l)(a-6). ^

CHAPTER IV.

Highest Common Factor. — Least Common
Multiple.

61. The expressions 2a^bc and 6ab^ have 2, a, and b as
factors common to both, and the product of these, 2aby
is the highest common factor of the expressions.

The name Highest Common Factor is contracted to
H. C. F., and sometimes G. C. M. (greatest common
measure) is used in its stead.

The expressions ar^ — 7 ic + 6 and a;* + 2a^— 9a^ + 8
factor respectively into {x — l){x — 2) {x + 3) and
{x — 1) {x -\-l) {x — 2) {x -{- 4t). They accordingly have
the binomial factors (cc — 1) (x — 2) or ic^ — 3a; + 2 as
their H, G. F,

Common monomial factors, where they exist, are readily
detected by inspection. To detect binomial factors, we
may factor the expressions and pick out the common
factors, as in the preceding example, or we may proceed
upon the principle now to be established.

62. Theorem. If two expressions have a common
factor, the sum and the difference of any multiples of
the expressions have the same common factor.

Let A and B denote the two expressions, and let /
denote their common factor, so that A = Pf and B = Qf,
where P and Q denote all the factors remaining in A
and B respectively after the removal of/.
64

HIGHEST COMMON FACTOR. 65

Let a and h be any numerical multipliers.

Then, aA ±bB = aPf± bQf= (aP ± bQ)fi
and as this last expression contains the common factor
/, the theorem is proved.

63. Now let A and B be two integral functions of x,
and let them have the common factor /, which we will
suppose to be quadratic, as their H. C. F.

S J taking the sums or differences of proper mu.tipies
of A and B, Ave may reduce the dimensions of each by
unity, and obtain two new functions A^ and J5', one
dimension lower respectively than A and B, and of
which / is still a common factor.

By operating in a similar manner upon A^ and B\ we
find two functions A^^ and .B", two dimensions lower
respectively than A and B, and containing/ as a common
factor.

,_ By a continuation of this process we must eventually
reduce A and B to depend upon functions of two dimen-
sions, and having / as a common factor.

Hence these, upon rejecting all monomial factors, must
be the factor /, and must therefore be identical.

And thus the identity of the two results at any stage
of the operation indicates that the H. C. F. is obtained.

Ex.1. Let ^ = 6x3-7x2-9^-2, B = 2x^+Zx'^-\lx-Q.

Operation.

u4 .... 6x3-7x2- 9x- 2 B ... .2x3+ 3x2- llx-6

35 . . . 6x3 + 9x2-33x- 18 3^. . . ISx^ - 21x2 - 27x - 6

ZB-A . . . 16x2-24x-16 3^-5 16x3 - 24x2 - 16 x
Reject factor 8, 2 x2 — 3 x — 2 Reject factor 8x, 2x2 — 3x — 2
The results being identical shows that the highest common fac-
tor is 2x2 -3x- 2.

66 HIGHEST COMMON FACTOR.

64. In the preceding example we notice :

(1) That the presence of the variable is unnecessary,
and the operation may be carried out upon the coefficients
alone.

(2) That as we reject monomial factors wherever they
occur, all monomial factors should be removed before
beginning the operation, and that any of these that are
common to both functions should be set aside and be
multiplied by the final result to give the complete

H. a F,

Ex. 1. Let ^ = 3rK3-10ic2-|-9a;-2, 5 = 2a;8 - 7«2 + 2« + 8.

A 3-10+9-2 B 2- 7+ 2 + 8

2 A 6-20+18-4 4^ 12-40+36-8

ZB . . . ,

. . . 6 - 21 + 6 + 24

A' . . , .

19A' . . ,
14^' . .

1 + 12-28

. . . . 19 + 228 - 532

. ... 196 - 658 + 532

B' 14 - 47 + 38

14^' .... 14 + 168-392

A" 215-430

-f-215 .1-2

B" 215-430

-^215 1-2

.-. x-2 is the H. C. F.

Ex.2. Let J[ = 3x*-6x3-6x-3, 5 = 6x4 + 12x3 + 12-6.

3 being a common monomial factor, we set it aside and write

A 1-2 + 0-2-1 B 1+2 + 0 + 2-1

B 1 + 2 + 0 + 2-1 iA' 1 + 0+1

A' 4 + 0 + 4 B' 2-1 + 2-1

iA 2 + 0 + 2

-4 1 + 0 + 1

.-. 3(x2 + 1) is the H. C F.

B" 1 + 0 + 1

HIGHEST COMMON FACTOR. 67

Ex. 3. Let ^ = 3x4-55c3 + 5c2 + 4x+l, B = Sx^-nx-4.

A 3-5+1+4 + 1 B 3-11-4

B 3-11- 4 4 A" 108 + 48 + 4

B" 111 + 37

-f- 37, B'" 3 + 1

.-. Sx + lis the H.C.F.

A' . . , .
2B

. . . 6+ 5+ 4 + 1
... 6 - 22 - 8

Af

9B'" . . .

27 + 12 + 1

27 + 9

A"f 3+1

It may be remarked that the functions must be com-
plete in form before beginning the operation.

65. When two integral functions of the same variable
have a common linear factor, the corresponding equations
have a common root ; and if the functions have a com-
mon quadratic factor, the corresponding equations have
two common roots, etc.

Ex. To find the relation between the constants in order that
the equations x^ + ax -{■ b = 0 and x^ + a^x + 6i = 0 may have a
common root.

The functions x^ + ax + 6 and x^ + a^x + b^ must have a com-
mon linear factor.

A 1+ a+ b B l + «i + 6i

bj^A b^ + b^a + b^b A 1 + a + 6

bB 6 + &«! + 6i&

(a-aO + (6-6J

A', . . . (b-b,)-\-(ba,-bia)

These results must be multiples of the same Unear factor.
Hence reducing the first term to 1 in each gives —

1 A. ^cti ~ ^l<^ — 1 _L ^

&i

b — bi a — a^

.'. (b — b^y = (a — a{)(bai — b^d);
which is the required relation.

68 LEAST COMMON MULTIPLE.

EXERCISE IV. a.

1. Find the H. C. F. of each of the following —
i. a:* - 4 ^3 + 2 a;2 + 4 X + 1 and x* - 6 x^ + 1.
li. a^ _ 2 a^ + 6 a - 9 and 3 a* - 2 a3 - 8 1;;2 + 6 a - 3.
Hi. lOx^ + x2 - 9ic + 24 and 20z* - 11 x- + 48x - 3.
iv. 5ic-^(12a:3 + 4x2 + 7x-3) and 10x(24x^ - 52a:2 + 14x - 1).
V. ic* — px^ + g — 1 • x'^ -\- px — q and «* — gx^ + j? — 1 • x^ + gx — |).

vi.

«_Vl + ^and^^^^

6* «* 63 ^3

2. Find the relation between a and h when x^ _|. ^^j + 10 = 0
and x2 + 6x — 10 = 0 have a common root.

3. Find the value of c when x^ — 3 x + 2 and x^ + ex + 3 have
a common linear factor.

4. Find the condition that ax^ + 6X + c and jpx^ + qx + r may-
have a common linear factor.

6. Find the condition that

x3 + ax2 + 6x 4- 0 = 0
and x^ + ajX^ + ft^x + c^ = 0

may have a common root.

6. Find the value of a when x^ — x — 6 and x^ + x(3 — a) —3 a
have a common linear factor.

7. If X* + X + « and x* — x + 6 have a common linear factor,
show that (a - 6)* = - 8(6 + a).

66. An expression which contains two or more given
expressions as factors is a common multiple of the given
expressions, and that common multiple which is of the
lowest possible dimensions is the lowest common multiple
or the least common multiple of the given expressions, the
Letter term being more particularly applicable to num-
bers. The contraction L. C. M. is used for either.

LEAST COMMON MULTIPLE. 69

If acef and dbde be two expressions in which the
individual letters represent linear factors, their L. C. M.
is dbcdefj and we see that in order to find the L. 0. M.
of two expressions or quantities we take the factors that
are common to both, as a and e, and the factors which
are peculiar to each, as c and / from the first, and b and
d from the second, and form the continued product of
all these factors.

Evidently a similar process applies to the case of
more than two expressions.

Ex. 1. To find the L.C.M. of x'^-x{a^h)-{-ah, x^-ax-x-\-a,
and x"2 — 6x — jc + h.

The expressions factored become —

(x -a)(x- &), (x - a){x — V), and (x - 6) (x - 1) ;

and the L. C. M. is (x — a)(x — b)(x — 1); and this is evenly divisi-
ble by each of the given expressions.

Ex.2. To find the i. C itf. of

x^ — ax — bx + ab and x"^ — 2ax + a\

The expressions factored are (x — a)(x — b) and (x—a)(x—a).
Here sc — a is a common factor, x — b is peculiar to the first,
and the second ic — a to the second.
.-. the L. a M. is (x - a)\x >- 5).

Ex. 3. ro find the L. C. M. of

(x-y-{-z)(x-^y-z), {y-z-{-x){y-\-z-x), and {z-x-^y){z-\-x-y),
or a;2 - (2/ - zy, y'^-{z- x)2, and z'^-ix- y)\

The L. CM. is (x — y + z)(y — z + x) (z — x -\- y\

or — Sx^ + 'Lx-y — 2 xyz.

70 G. C. M. AND L. C. M.

67. Theorem. The product of any two quantities or
expressions is equal to the product of their H. C. F, and
their L. 0. M.

Let A and B denote the expressions, and let / be their
H. C. F. Then A=zpf and B = qf, where p and q are the
factors peculiar to A and to B respectively. Their
L. C. M. is pqf.

But A . B =pqf=pqf'f= L, (7. M. x H, C. F,

Hence we may find the L. C. M. of two expressions by
dividing their product by their H. C. F., and conversely
we may find the H. C. F. of two expressions by dividing
their product by their L. 0. M.

APPLICATION TO NUMBERS.

68. The foregoing principles apply to integral num-
bers in the same manner as to algebraic expressions.

The H. a F, or, as it is here called, the G. G. M. of
the numbers, is the greatest number which divides each
evenly.

Ex. 1. To find the G, 0. M. of 3824 and 4160. The
difference between any multiples of the two numbers
must contain their G, G. M. (Art. ^2),

Hence, from 4160 subtract the greatest possible mul-
tiple of 3824, which in this case is the number itself,
and we have 336. We have now to find the G. G. M. of
336 and 3824. Taking 11 times 336 from 3824 leaves
128, and we are to find the G. G, M, of 128 and 336. By
continuing this process we finally arrive at the factor
required.

G. C. M. AND L. C. M.

71

The whole operation appears as follows :

A

. 3824

B

. 4160

A -11J3' .

. 128 .

. A'

B - A

. 336 .

. B'

A' - B" .

. 48 .

. A"

B' ~2A'

. 80 .

. B"

A"- B"'.

. 16 .

. A'"

B" - A"
B'"— A'"

. 32 .
. 16

. B'"

The identical results show that 16 is the G, C. M.

This process may be much shortened by leaving out
the letters of reference, and by writing only remainders
in the operation.

Ex. 2. To find the G. G. M. of 10395 and 20592.

quotients.

And 99 is the G. G. M.

Explanation. — 20592 -^ 10395 gives quotient 1 and remainder
10197. 10395 H- 10197 gives quotient 1 and remainder 198. 10197
-7- 198 gives quotient 51 and remainder 99. And lastly 198 -^ 99
gives quotient 1 and remainder 99, or it is a case of even division.
Hence 99 is the common factor, and hence the G, G. M.

The quotients are important in the subject of continued
fractions.

10395

20592

1 >!

198

10197

1

99

99

51
1 -

69. Two numbers whose G. C. M. is 1 are prime to one
another ; and a number which is prime to every number,
except unity, smaller than itself is a prime number, or
simply a prime.

The following are the primes less than 100, and a table
of all primes below 1000 is given at the end of this work.

72 G.C. M. AND L. C. M.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

All numbers not primes are composite numbers.

Every composite number can be exhibited as the
product of prime factors. This is called the composition
of the number.

Let ^denote a composite number.

Divide JSf by 2, and the resulting quotient by 2, and so
on until a quotient is found which is not divisible by 2.
Call this quotient N'. Divide JV by 3, and the quotient
by 3, and so on, until a quotient, N", is found which is
not divisible by 3. Divide N^' by 5, etc., and continue
the operation by 7, 11, etc., using only primes as divisors.

Then if 2 has been used a times as a divisor, 3 b times,
5 c times, etc.,

JV^=2«.3^.5^...

And this is the composition of the number N.

Ex. 1. The composition of 8640 is 2^ . 33 . 6 j and 8640 is said to
be decomposed into its prime factors.

Ex. 2. To find the G. O. M. of 8640 and 1720.

Composition of 8640 is 2^ . S^ • 5.
Composition of 1720 is 2^ . 5 • 43.
And the G, C. M. is 23 . 5 = 40.

70. To find the L. (7. M. of two or more numbers, we
may decompose the numbers into their prime factors,
and take the highest power of each factor involved, and
form their continued product.

Ex. To find the L. C. M. of 8640, 1280, and 1560.

8640 = 26.33.5, 1280 = 28.5, 1560 = 23.3.5-13.
And the L. C. M. is 28 . 33 . 5 • 13, or 449280.

G.C.M. AND L.C.M. 73

The operation may also be conveniently carried out as
follows :

40 8640 1280 1560 40 is the G. C. M. of the three numbers.
8 216 32 39 8 is the G. C. M, of 216 and 32, and is
3 27 4 ... prime to 39.

9 ... 13 3 is the G. G. M, of 27 and 39, and is
prime to 4.

The quotients 9, 4, 13 are prime to each other.

Hence 40 x 8 x 3 x 9 x 4 x 13, or 449280 is the L. C. M.

EXERCISE IV. b.

1. Find the i. CM, of the following —
i. 12x - 36, x2 - 9, x2 - 5« + 6.

ii. ic2 - (a + h)x + ah, aj^ - (6 + c)x + &c, x'^—{c-\- a)x + ca.
iii. 1 -\-p+p\ 1 -p +i52, 1 +^2 4.^4.

2. If ax"^ + hx-\- c and cx^ + 6x + a have a linear common
factor, show that a ±h -\- c = 0,

3. Find the compositions of the numbers —

i. 72. ii. 180. iii. 824. iv. 1048. v. 25200.

4. Find the G. C. M. of 144, 840, 5040.
6. Find the L. C. M. of the 9 digits.

6. What is the least multiplier that will make 720 a complete
square ? That will make 1440 a complete cube ?

7. What is the lowest factor that will make a^ — 45c2-f-5x — 2
a complete square ?

8. What is the least multiplier that will make 144 a multiple of
64 ? That will make x3-5x2 + 5x-la multiple of x^ - 4a; + 3 ?

CHAPTER V.

Fractions. — Symbols qo and 0.

71. The expression 3/4 denotes that 3 is to be divided
by 4. As numbers, 3 cannot be divided by 4 ; and hence
we indicate this impossible arithmetical operation sym-
bolically by writing it in a form of division, and we call
the whole symbol a fraction. We then discover the laws
of transformation of these symbols, and these laws form
the working rules for fractions.

In this relation the dividend is called the numerator
and the divisor the denominator.

The expression a/h is an algebraic fraction, in which
a and h stand for any numbers or expressions ; and the
transformations of this symbol must apply to arithmetical
fractions as particular cases.

RULES OF TRANSFORMATION.

72. The fractional form ^ + ^ + ^+-'^ ^-^^^ ^^^ Ylhq

written beneath a-\-h-\-c-\ (Art. 10), shows that

a-\-h-{-c-\ is to be taken in its totality to form the

numerator.

(1) Let Q = T" , Q denoting the fraction as a whole.

Then QD = a + & (Art. 37) ; and if we take d such a

multiplier that Dd = 1, and therefore d = — , we have

74

FRACTIONS. 75

QDd = ad-{-bd, or Q = ^-\.^.

Hence the denominator of a fraction is distributive
throughout the terms of the numerator.

And conversely, the algebraic sum of any number of
fractions with the same denominator is that fraction
whose numerator is the algebraic sum of the numerators,
and whose denominator is the same denominator.

(2) Let Q = -. Then QD = ]Sr; and p being any
multiplier,

QDp = Np.

... Q = ^=^.
^ Dp D

Hence, multiplying both numerator and denominator
by the same multiplier does not alter the value of the
fraction.

(3) For p write -, and we have from (2)

Q =

N

Q
' D'

1

~d'

And hence

dividing both numerator and denominator

by the same

divisor

does not alter the value

of

the

fraction.

(4) Let

Q =

W

'd'

and

^ D'

Then

and

r. an

Q'D' = N'.

DD'

76 FRACTIONS.

Hence, to multiply two fractions together, we multiply
together the numerators for a new numerator and the
denominators for a new denominator.

(5) From (4), -^ — - = — ^, and multiplying both frac-

tions by -— gives -^ = — • — - •

Hence, to divide one fraction by another we multiply
the dividend by the inverted form of the divisor.

(6) Let Q = — ; and since p = ^,

P P

Therefore, we multiply a fraction by a given quantity
when we multiply the numerator by that quantity, or
when we divide the denominator by that quantity.

Also, by writing - for p, we see that we divide a frac-
tion by a given quantity when we divide the numerator
by that quantity, or when we multiply the denominator
by that quantity.

Co7\ If two fractions have equal denominators the
greater fraction has the greater numerator, and con-
versely.

And, if two fractions have equal numerators, the
greater fraction has the smaller denominator, and con-
versely.

FRACTIONS. 77

73. The operations with fractions are as general in
character as those with whole numbers, and beyond the
rules of transformation as now established no general
directions can be laid down. The principal operation on
fractions in themselves is the simplification of their
forms. Facility in this is the result of practice.

Ex 1 3a;2 -f x - 2_ (3a; - 2)(x + 1) _3a; - 2
2x^-x-S~(2x-S)(_x+r)~2x-S

X -f 1 . X — 1

w 2 a; - 1 X -{- l_(x + ly +(x - iy_x'^ + 1

' x + 1 X - 1~ (X -{■ ly -(x - ly^- 2x

x-l x-\-l

Ex 3 a^^-1. x + 1 _(x^-l)(Sx^ + 2x)
»2+l'3xH2x (a:2 + l)(x+l)

= Sx^-Sx^ + 2x- a^(5x2-a; + 4)^

(X2 + 1)(X + 1)

74. From Art. 72 (1), to add fractions we bring them
to have the same denominator. We then add the numer-
ators, and place the sum over the common denominator.

Subtraction being addition with changed signs follows
the same rule.

When several fractions are to be added the L. C. M. of
all the denominators is the simplest common denominator.

Ex. 1. Simplify

a , h , c

{a — 6) (a — c) (h — c)(b — a) (c — a){c — b)

The L. C. M. of the denominators is (a — h){h — c)(c — a) ;
and the numerators become respectively —

«(c — &), b(a — c), and c(b — a).

78 FRACTIONS.

And the sum is g(c^i&I+A(^' - «)+ Ha - c).
(a — b)(b — c){c — a)

And as the numerator vanishes upon distribution, and the
denominator does not, the sum of the three fractions is zero.

Ex. 2. Simplify
a

+ -7-. .., .. 7T+-

(a— 6)(a— c)(ic— a) (b — c){b — a){x—b) {c—a)(c — b){x — c)
The L. C. M. of the denominators is

(a — b){b — c) (c — a)(x — a) (x ~- b)(x — c);
and the sum of the new numerators is

— a(b — c) {x — b) (x — c) — b{c — a)(x — c)(x — a)
— c(a — b)(x — a)(x — b).
This latter expression factors into

x(a — b){b — c)(c — a).
Therefore the simplified sum of the fractions is

x

(x — a){x — 6) (x — c)

EXERCISE V. a.

1. Simplify the following —

. x'^-h jxy + y^^
ic2 + I xy - 2/2

\x a/\x aj\ x-{-aJ

.J 3x-2+ lOxy + 3y2 ^ gg + &^ + c^ - 3 abc

* 3x2 + 8x2/ -3 2/2* * a'^+b'^-^c^-ab-bc-ca

m.

w(x2 — 2/2) + (?i2 — l)x2/ * \l+x X y " \1 + X X J

^^^ a + c & + c x + c

(a — 6) (x — a) {a — b)(x — b) (x — a) (x — b)

iii. — ?^ + t + ^

ix-y)(x-z) {y-x){y-z) (z-x){z-tj)

FKACTIONS. 79

ix. s| ^ 1. xi. s| — y^^ ).

( 1 ) xii s|i3L±i^HL±Ml.

xiii. 2 \ ^^^ "^ ^^ — I , where x is not varied.

<>(« — 5) (a — c) J

I (a - 5)(a - c) i I (« _ 5)((i _ c) /

2. If a = -^— , & = -^— , c = -^-, show that

2/ + 2 z-\-x x + y

ic2 ?/2 2;2

a(l — 6c) &(1 — ca) c(l — a6)

8. Find the value of ^ + V - '^

x-y + 1

when aj = -^±A. and y = ^^±^^

«6 + 1 a6 + 1

*i2 /y2 r2

«2 ^ 62 ^ C2

a2 52 g2 ^-2 ^2 (.2

6. If 1+^ =» . 1 + ^ + ^', show that x^ = ^^^.
1— xal — x + ic2 5 + a

6. Simplify j>+I£- «U 1 1 ^fl±I2 . ?! when .2 + ^2=1.
l/c — rs c i V fc — rs c)

75. Let ic = , where A, a, b are all positive finite

a — b

quantities and b is not greater than a.

If & is < a, X has some positive finite value, and the
nearer b approaches to a in value the greater the value

80 FRACTIONS.

of X becomes. By making the difference between a and
h small enough we can make the value of x as great as
we please.

Thus, let A be 10. If a — & is 1, a; is 10 ; if a — h is
0.1, a; is 100 ; \ia-h is 0.00001, x is 1000000; etc.

When a — h\B> made smaller than any conceivable quan-
tity, X becomes greater than any conceivable quantity.

In this case h is said to approach infinitely near to a
in value ; the difference between a and h is then denoted
by 0, and the value of x is denoted by oo, read infinity.

Thus we say that ^ ^ ^ ^ any finite quantity^ ^^^
•^0 0

assuming that these symbols obey the formal laws of
quantitative symbols,

00 X 0 = any finite quantity.
But 1 is a finite quantity, and x = QO ;
therefore - X 0, or - = any finite quantity.

The expressions cx) x 0 and - are mere symbols having

no particular value except through their history ; that is,
through a knowledge of the source whence they have
come. The expression -, however, which occurs quite

often, does not necessarily mean zero, but may mean any
finite quantity whatever.

Also, since oo x 0 = ^4, 0 = — , and hence any finite

00

quantity divided by infinity gives zero as a quotient.

Moreover, we are not justified in writing go — go = 0,
or 00 ^ 00 = 1, for oo does not mean any definite quan-

FRACTIONS. 81

tity, but merely a quantity so great as to be undefinable
and inconceivable.

76. Special Roots. (1) When an integral function
contains the variable as a monomial factor in the first
degree, one of the roots is zero.

Thus ^ + 3ax^ -\-2a^x = 0 = x{x + 2a) {x-{-a);

which is satisfied by a; = 0, since the whole then vanishes.
The roots are accordingly, 0, — 2 a, — a.

If the variable is of two dimensions in the monomial
factor, two of the roots are zero.

(2) In the linear equation x = x-\- a, to which equa-
tions are sometimes reducible, we transfer x, and obtain
x — x = a.

Now we are not justified in saying that a: — a; = 0, and
therefore a = 0, for a is a given quantity whose value is
not at our disposal ; we must endeavor to find some value
for x that will satisfy the equation.

But X — x = x(l -'l) = a.

a

1-1

= oo by Art. 75.

Whence x = ck> is a symbolic root of the equation.
The meaning of the solution is that a being finite, the
larger x is, the more nearly is the equation x = x-^ a
satisfied, but that it cannot be completely satisfied by
any finite value of x. We shall return to this again in
Art. 81.

77. Let x^-{-ax-{-b^x^ -{-px + q, where a, b, p, q are
^11 finite quantities.

82 FRACTIONS.

If X is finite, a^ on one side cancels o^ on the other,
and we have a single finite value of x, namely,

q-b
a—p

(1) If q = b, and a=p, a; = - = any finite quantity
by Art. 75, and the solution is indefinite,

(2) If g — 6 is not zero, and a = p, x= x>,

(3) li q = h and a —p is not zero, a; = 0.

Again, as the equation is quadratic it must have two
roots. Art. 59 (2).

Dividing throughout by x^ gives

X X^ X x^

and the larger x becomes the more nearly does this
become an identity.

Hence ic = oo is a solution.

As the equation may be written

we infer that if the coefficient of the square term, in
a quadratic equation, becomes zero, one solution of the
equation is a; = oo.

Or more generally, if, in any integral equation, the
coefficient of the highest power of the variable becomes
zero, one root of the equation is oo.

FRACTIONS. 83

EXERCISE V. b.

1. Given ?^+_§ + ^ ^ 3x + 4, to find x.

6 3

2. Given ^-±^ + ?^ = 5, to find a;.

x + 2 3a; 3

3. Given

a c , X 1

(x — a)(x — c) (a — c) (a — x) (c — a)(c — x) a — c
to find X. What is the value of x v^rhen c = 0 ? when a = 0 ?

4. Is /'l+?Wl + ^'\ = /'l+iyi + ^'\ an equation or an
\ dj \x xj \ d J \ aj /

identity ? What value has x ?

6. What expression substituted for x will make

Zx^-l x±2^ = 1 an identity ?
a-l a + 2 -^

6. From _^— +— ^ 2(a;2 - a6) ^ ^^ ^^^ ^ general

X — a X — & {x — a){x — h)
value of X ; and also the particular value when a + & = 0.

7. Given (^-"^>^^ + ^) = ^il, to find x.

(x-2)(x+l) x-1

8. Given C^^ - 1) (^ + 2) (a; - 3) ^ i ^ q, to find all the values

(x + l)(x-2)(x + 3)
of «.

9. Given ^x^-x^-Qx-^2 4:X^ + x^-^x-2 2 ^^^

X — 1 X ic(l— x)

to find all the values of x.

10. If ^ + ? = ^ + ^, what relation holds between a and & ?

X b X a

11. Two numbers differ by 10, and one-half the less is greater
by 1 than one-sixth the greater. Find the numbers.

84 FRACTIONS.

12. One body moves about a circuit in a days, and another in
h days, and they start from the same point. How many days will
elapse between two conjunctions ?

13. The sun moves in the ecliptic 0°.9856 per day, and the moon
moves 13°. 1690 per day. Find the days elapsing between two new
moons.

14. Find a number such that if a be added to it and h be sub-
tracted from it, the difference of the squares of the results shall be
the number.

What relation must hold between a and h that the number may
be (1) zero, (2) infinity ?

15. Given ^Ji3 _ ^+_4 ^-_6 ^ x'^-2x-Vo ^^ ^^^ ^

x+\ 5C+2 ic-4 x2-9

16. Given -^- + ^^ = ^-±^ + ^^, to find all the values

X — 2 X — 1 X — 1 X — Q
of X.

17. Given a + x a-x 3a _ ^

cC^ -\- ax-\- x^ a^ — ax-\- x^ x{a^ + d^x^ + x*)
find X.

18. Divide $64 among A, B, and C, so that A may have 3 times
as much as B, and C have I as much as A and B together.

19. A person spends .|2 and then borrows as much as he has
left. He again spends $2 and borrows as much as he has left ; etc.
After his fourth spending he has nothing left. How much had he
at first ?

20. A person spends $a, and borrows as much as he has left ;
then spends $a, and borrows as much as he has left, etc., for n
times, when he has nothing left.

2n 1

Show that he had at first a dollars.

2n-l

21. In a naval battle the number of ships taken was 7 more, and
the number burnt 2 less, than the number sunk; 15 escaped, and
the fleet consisted of 8 times the number sunk. How many ships
were in the fleet ?

FBACTIOKS. . 85

78. The following properties of equal fractions are of
special importance :

I. Let ^ = |,

where a, b, c, d denote any quantities or expressions
satisfying the indicated relation.

(1) Multiplying by - gives - = -.

C C U/

(2) Adding 1 to each member

(3) Subtracting 1 from each member

(4) Dividing (2) by (3)

a-\-b _c-\- d

h " d '

a—b c—d

b d

a + b _c-}-d
a — b c — d'

Eelations (1), (2), (3), (4) are all direct consequences
of the original relations.

The number of such derived relations is unlimited;
those given are of most importance.

It will be noticed that these expressions have that
kind of correlative symmetry by which we may inter-
change a and b if we interchange c and d, or we may
interchange a and c if we interchange b and d, but in
general we cannot interchange a with c or 6 with d.

11. Let ^ = ^ = ? = ... = Q, say.

Then a = bQ, c=:dQ, e=fQ, etc.

And Q(lb -{-md-\'nf-\ ) = la-{-mC'{'ne+ "•

rK\ n ^ Za + mc 4-ne -f- •••

b lb -i- md -\- nf-\- <

86 • FRACTIONS.

By giving particular values to Z, m, n ••• an indefinite
number of special relations may be obtained.

The relations of I. and II. are frequently employed
with great advantage, the letters a, h, c, etc., being
general symbols denoting any quantities or algebraic
expressions.

^2

Ex. 1. To find X from the equation f^il^V

\a — xj

cfi -\- 2 ax + x"^ _ ah -^ ex

ah

Expanding, etc.,
Applying (4) of I.,

■2ax-\- x'^ ah

a?' -\- x^ _2 ah + ex

2 ax ex

Subtracting denominator from numerator for a new numerator,

(ia-xy^2ah
2 ax ex

.'. ex(a — xY = ^ a'^hx.
Whence x = 0, (76.1), and x = a-2aJl-Y

Ex. 2. If ^ =. ^ , then (^'^ + ?>-^) (« + 6) ^ q^,
b d (c^ + d')(ic-{-d) c3

Exercises of this kind may be solved in several ways : as (1) by
transforming one expression into the other ; (2) by using the first
relation to show that the second is an identity, etc.

m ^ = ^. . Q^^+ ^^ ^ c2 + d^ ^^^ a^+ &2 ^b^^c^

^ ^ 62 ^2* • • 5-2 cf2 ' C2 + d^ d^ C2'

by relations (1) and (2), I.

Again, «+^=L±i, and«+^ = ^ = «. ■

h d e + d d c

.: by multiplication, «^±^ • "L^^ = «! . ^ = «!. q. k. d.

FRACTIONS. 87

(2) Let - = - =p; then a = bpy c = dp,
h d

Substitute for a and c in the second expression, and it becomes
hHp'+lXp-\-^) ^ m^ an identity.

Ex. 3. If — ^— = — ^— = — - — , each fraction = 1.

2y — z 2z — X 2x — y

By (5), ^ - x + y-\-z a; + .V + g _. 2.

' 2y — z 2y — z + 2z — x-\-2x — y x -\- y + z

Ex.4. If ^ = ^ = ^^^ and £^+^ + ?!=l, to find the
X y z a^ h'^ c^

values of x, y, and z in terms of the remaining letters.
The first relation gives ^^'^^21;

^ y 1

a b c
whence, squaring each fraction and employing (5),

x2 1

an

Whence x =

^{aH^-h &2m2+ c2n2) '
with symmetrical expressions for y and z.

Question 4 furnishes an example of collateral symmetry
between the two sets of letters a, b, c and l, m, n. Thus
if we change a to & and b to c, we must, at the same time,
change Z to m and m to n. But we are never supposed
to make an interchange between letters from different
sets. In like manner we may have collateral symmetry
amongst three or even more sets of different letters.

88 FRACTIONS.

EXERCISE V. C.

1. If ^=:^ = ^=... Show —

h d f

. (a-c)2 + (6-(?)2^g2 4.;^2
'(a-c)2-(6-(?)2 a2-62*

ii g^ 4- g^ft + q&^ ■\-h^ ^ b^

... g _ V(Z2g2 + w2c2 + nV)
6 VG^^'^ + ^'^^^^ + ^T-^)*

2. If ^ = ^ and ^ = ^, then «V^ + 5V^^ gy^- fty^,

8. If ^ = ^ = ^, (a2 + 52+c2)(62 + c2 + c?2) = (a6 + &c+c(Z)2.
0 c d

4. Under the conditions of Ex. 3, show that '

V(g&) + V(&c) + y/iicd) = V{(g + & + c) (6 + c + d)}.

6. If ^ =^

then ?(y - 2j)+ |(5?-a)+ £(x -y)=0.

6. Given that ga;2 + 5^2 + 2 ^ = 0, and ^ = ^ = ^ = ?,

i m n p

show that - + — + 2 w» = 0.

a 6

7. If ^^-y^ ^ y^-zx ^ ^^^^ fraction = x + y + z.

x(l — 2/^) y(l — ^x)

8. IfgLLJ^ ^ + g = g + « ,8g + 96 + 5c = 0.

a- 6 2(6 -c) 3(c-a)

FRACTIONS. 89

9. K f±^ = ^-±^ = ^±1, then each fraction is
0 — c c — a a — h

If x^ + y^ + z^ -»

A/ \(6-c)2 + (c-a)2+(a-6)2/*

10. If -:^ = -^ = -^=2R, and a + b + c = 2S,

smA smB sm (7

then >S = i2(sin^4-sin^+ sinC).

CHAPTER VI.

Ratio, Proportion, Variation, or Generalized
Proportion.

79. The ratio of a to & is the quotient arising from
dividing a by h, where a and h denote any numerical
quantities. If the division is even, the ratio is an
integer, and is expressible ; if uneven, the ratio is a
fraction and can only be indicated.

In this relation a and h are called the terms of the
ratio, a being the antecedent and & the consequent.

The ratio is commonly symbolized as a : 6.

If a > &, the ratio is one of greater inequality.

If a = b, it is one of equality ; and if a<b, it is one
of less inequality.

When two ratios are multiplied together, after the
manner of fractions, they are said to be compounded.

Thus ac : bd is compounded of a : & and c : d.

When a ratio is compounded with itself, the terms
are squared, and the result is the duplicate ratio of the
original. Thus a^ : b^ is the duplicate of a : 6.

Similarly, a^ : b^ is the triplicate of a : 6 ; and -y/a^ : -y^b^
is, in physics, sometimes called the sesquiplicate ratio of
a: b.

80. As a ratio is virtually a fraction, all the laws of
transformation for fractions apply to ratios.

The ratio of one quantity to another does not depend
90

BATIO. 91

upon any absolute magnitudes of the quantities (for
there is no absolute magnitude), but upon the relative
magnitudes of the quantities.

It is thus that the ratio of one quantity to another
expresses the true relation of magnitude or greatness
existing between the quantities.

The ratio - : - is the same as a:b, whatever x may be.

X X

But if X is very small as compared with a and b, both
terms become very great ; and if x is very great as com-
pared with a and b, both terms are very small ; but the
relation of greatness existing between the terms remains
the same.

If x = 0, both - and - become infinite ; so that oo : oo

X X

may be any ratio whatever.

If a; = 00, both - and - becomes zero ; so also 0 : 0

X X

may be any ratio whatever. (Compare Art. 75.)

81. Theorem. The addition of the same quantity to
both terms brings the ratio nearer to unity, or to a ratio
of equality.

Let aibhe the ratio, and let x be added to each term,
making a-\'X:b + x.

Then ^ a + x_x(a-b) __^

^^^^ b~'bTx=b(b + ^'^' ^'

and ^-1^^ =Q,say.

All the letters denoting positive quantities, if a : 6 > Ij
a>b and Q and Qi are both +.

92 PROPORTION.

.*. a + xib + xKaib, and > 1 ;
or a-]-x:b'\'X lies between a : b and 1.

If a:b <lf a <b, and Q and Qi are both — .

.-. a-{-x:b + x>a:b, and < 1 ;
or a-^-xib + x lies between a : b and 1,

which proves the theorem.

Cor, 1. By writing -- x for x, we see that to subtract
the same quantity from both terms of a ratio of inequality
removes the ratio further from 1, provided the subtrac-
tion leaves both terms positive.

Cor, 2. Qi decreases as x increases, and by making x
great enough, we may make Qi as small as we please.
That is, by adding the same quantity to both terms of a
ratio we may bring the ratio as near 1 as we please.

We have here another proof of Art. 76, for if x-\- a =
x-\-b, we have x-\-a: x-{-b = lj and whatever values a
and b may have, provided they are finite, the statement
is satisfied by x — cc, since that value for x makes the
ratio to differ from 1 by a quantity less than any assign-
able quantity.

PROPORTION.

82. Pour quantities are proportional, or are in pro-
portion, or form a proportion, when the ratio of the first
to the second is equal to the ratio of the third to the
fourth.

Thus a, b, c, d are proportional when

a:b = c:d.

PROPOKTION. 93

This may be expressed as ^ = -, and a proportion,

being thus an equality of two fractions, is best dealt with
after the manner of fractions.

The proportion is evidently subject to all the trans-
formations of Art. 78, I.

83. In the proportion a:b = c:d, a and d are the
extremes and b and c the means.

But since the same proportion may be written b: a=:
d : c, the extremes and means are capable of exchanging
places.

Writing the proportion - = -, or - = -, or - = -, etc.,
b d c d a c

which all express the same relation, we may represent
the form generally by — ! — , where the letters are writ-
ten in the four corners formed by the two crossed lines.

In this form a and d, as also b and c, are opposites of
the proportion, as standing in opposite corners, and we
can make the general statement

The terms of a proportion may be written in any order,
provided the opposites are unchanged.

By cross-multiplication ad = be. That is, when four
quantities are in proportion, the product of each pair of
opposites is the same ; and conversely, if two equal
quantities be each divided into any two factors, these
factors form a proportion, of which the factors of the
same quantity are a pair of opposites.

84. If a : 6 = c : c?^ d is a fourth proportional to a, b,
and c.

94 PROPORTION.

If a:b = b : c, b is Si mean proportional, or a geometric
mean between a and c. In this case b =:^(ac).

If a:b=b : c=c: d=etc., the statement is a continued
proportion.

Ex. 1. If a + b : a = a — b : b, to find a : b.

Here ba -\- b^ = a^ — ab ; whence a — b = b^2,

and a = 5(1 +^2), or a:b = l-\- ^2.

Ex. 2. If ^2^2 _ 52^.2 _ ^252^ to find approximately the ratio
X : y when x becomes indefinitely great, and a and b are finite
constants.

Evidently y becomes indefinitely great with x. Dividing by x^
the relation gives a2 . 1^ _ ^2 _ «!^.

When ic approaches 00, - — approaches 0, and ^ approaches — •
x2 x^ a^

And when x = qo, ^ i= -j- ^, or x:y = ± a:b.
y b

Ex. 3. Two numbers are in the ratio p:q; what must be added
to each that the ratio of the new numbers may be P : Q ?
Let mp, mq be the numbers, and add x to each.
Then mp -}■ x:mq-{- x= P: Q.

whence ^^m(pQ-gP).

P-Q

Cor. When P:§=1, P=Q, and the value of a becomes
infinite.

Ex. 4. To find 4 numbers in continued proportion such that
their sum may be 66.

Let a, 6, c, d be the numbers.

Then ^=^ = ^=z,sa>y,

bed

PROPORTION. 95.

Hence a = dz^, h = dz^^ c = dz, and their sum is
d{z^ + z'^-Yz+ 1)=65.

.*. d{z'^ + 1)(^ + 1)= 65, and since we can make z what we
please, the problem is indefinite, i.e. it admits of any number of
solutions.

If ^ = 2, the numbers are ^f^, -\2_^ j2_6^ ^nd -i/-.

If 5J = I, the numbers are 8, 12, 18, and 27.

EXERCISE VI. a.

1. For what value of x will the ratio 5 + x : 8 + a: become 5 : 8,
6:8, 7:8, 8:8, 9:8?

2. In a city A a man assessed for $ 10,000 pays $72 tax, and in
a city B a man assessed for 1 720 pays $ 4.50 tax. Compare the
rate of taxation in A to that in B.

3. The ratio a — l-.h — \ is a, and that of a + 1 : 6 + 1 is /3.
Find the ratio a : 6 in terms of a and )3.

4. Two men can do in 4 days what 3 boys can do in 5 days.
Compare a man's working ability with that of a boy.

5. Find the number for which the cube root of its square is to
the square root of its cube as m to n.

6. Given 3^2 + 10a;?/ + 3?/2: 3^2 ^ gx?/ - 3x2 = 2x: ?/, to find
the ratio y : x.

7. It a:h = h'.c = cid^ show that a^ -.y^ = a:c, and a^ . 53 _
a: d.

8. If ^{p + q):^{p — q)= mm, find p : g, and also the dupli-
cate ratio of p — q : q.

9. If l^-m'^ + n'^ = 0 and z + m - n = 0, find I : m.

10. Find the ratio of (x + h)-'^ — x'^ : h when h approaches
zero.

,96 GENERALIZED PROPORTION, OR VARIATION.

GENERALIZED PROPORTION, OR VARIATION.

85. Let x be a variable, and let y be connected with x
by a constant multiplier m, so that y = mx. When x
changes its value, becoming Xi, say, y also changes value,
becoming 2/1, so that y^ = mx^.

Dividing one equation by the other,

y _ mx __ x^
yi mxi osi

Whence 2/ • !/i = ^ • ^1^

i.e. any two values of x and the corresponding values
of y are in proportion.

Also, if X takes a series of values, Xi, x^, x^, etc., and
the corresponding values of y be y^, 2/2? 2/3? ^^^v

^1*2/1 = •'^2 • ^2 = ^3 • 2/3 = ^^^*

The foregoing relations are indicated by saying that
y varies as x, or x varies as y, since the relation is
mutual, and they are symbolically expressed by writing
2/ oc i», or xccy.

Hence to say that y varies as x, is to say that one is a
constant multiple of the other, or that any two values
of X and the corresponding values of y are in proportion.

86. li y = n ' -, y varies as the inverse or reciprocal
of 2; ; or 2/ varies inversely as z.

If y = n ' -, y varies directly as x and inversely as z.

z

If 2/ = nxz, y varies conjointly as x and z.

Ex. 1. If xtxyz, and y varies inversely as 2;2^ and if 2? = 2 when
x = 10, it is required to express x in terms of z.

GENERALIZED PEOPORTION, OR VARIATION. 97

We have

x = myz, and 2/cc— •

z

And

10 = ^, or w = 20.

2

...x = ?2.
z

Ex. 2. The velocity of a body falling from rest varies as the
square root of the space passed over, and when the body has fallen
16 feet its velocity is 32 feet. Find the relation between space and
velocity.

X) = m^s, where v = velocity and s = space.

.'. I? = 32 and s = 16 gives 32 = 4 m.
,'. m = 8, and v =8 Vs, or v"^ = 64:S;
which shows that the velocity varies as the square root of the space
fallen through.

Ex. 3. The radius of the earth is r, and the attraction upon a body
without varies inversely as the square of the body's distance from
the centre. The number of beats made per day varies as the
square root of the earth's attraction upon the pendulum. How
much will a clock, with a second's pendulum, lose daily if taken to
a distance r^ from the earth's centre, r^ being greater than r.

Let n = the number of seconds in a day = 86400, and let g = the
earth's attraction at the surface.

Then g oc— , and n oc -.yg.

.*. WX-, and we write n = —, m being constant.
r r

Also, if n^ be the number of beats per day made by the pendu-
lum in its new position,

Til = — = n • — , by substituting for m.

And the clock loses n — n^ seconds daily,

z=z nil —-]= n- ^^ ~ ^ seconds.
\ rj ri

If r = 3960 and r^ = 3961, the loss is 21.81 sec.

98 GENERALIZED PROPORTION, OR VARIATION.

EXERCISE VI. b.

1. The space passed over by a body falling from rest varies
as the square of the time, and a body is found to fall 196 feet in
3J seconds. Pind the relation between the space and the time.

2. li X(xy and y = 3| when x = 6 J, find the value of y when

3. y varies inversely as x^, and z varies directly as x^. When
x = 2f y -{- z = 340 ; and when x = 1, y — z = 1275. For what
value of aj is ?/ equal to z?

4. zccu — V, uccx, and v gc x^. When x = 2, ^ = 48 ; and when
X = 5, ^ = 30. For what value of x is 2; = 0 ?

5. If x?/ GC x^ + y% and x = 3 when y = 4, find the relation con-
necting X and y.

6. The area of a rectangle is the product of two adjacent sides ;
if the area is 24 when the sum of the sides is 10, find the sides of
the rectangle.

7. If X + 2/ oc X — 2/, then x"^ + y^cc xy.

8. If X cc y, show that x^ + y^oc xy.

9. A watch loses 2 J minutes per day. It is set right on March
15th at 1 P.M. ; what is the correct time when the watch shows
9 A.M. on April 20th ?

10. The volume of a gas varies directly as its absolute tempera-
ture, and inversely as its tension. 1000 cc. of gas at 240° and
tension 800 mm. has its temperature raised to 300° and its tension
lowered to 600 mm. What volume has the gas then ?

11. The attraction at the surface of a planet varies directly as
the planet's mass and inversely as the square of its radius. The
earth's radius being 3960 miles, and the moon's 1120, and the mass
of the earth being 75 times that of the moon, compare the attrac-
tions at their surfaces.

GENERALIZED PROPORTION, OR VARIATION. 99

12. The length of a pendulum varies inversely as the square of
the number of beats it makes per minute, and a pendulum 39.2 in.
long beats seconds. When a seconds pendulum loses 30 sec. per
day, how much too long is the pendulum ?

13. When one body revolves about another by the law of gravi-
tation, the square of the time varies as the cube of the distance.
The moon is 240,000 miles from the earth, and makes her circuit
in 27 days. In what time would she complete her circuit if she
were 10,000 miles distant ?

CHAPTER VII.

Indices akd Surds.

87. The index law is the result of the convention that
when p is Si positive integer, a^ a • a-** to p factors shall
be denoted by a^. And by this law a^ - a^ = a^+*, p and q
both being positive integers.

Now, if algebra is to be consistent with itself we can
have only one index law, whatever p may denote, and
instead of making a new convention we must interpret
in conformity with this index law the cases in which p
is not positive and integral.

The interpretation of p zero, or negative and integral,
is given in Art. 38. We deal here with p fractional.

(1) If p = g = i, a^ . a'^ = a* . a* = a^ = a.

Therefore, a^ is the same as ^a, the meaning of which
is fully given in Arts. 47 and 48.
Similarly, a^ . a^ . a^ = a^"^ 3 + 3 _- ^^

And a^ means that a is to be separated into three
identically equal factors, and that one of these is to be
taken.

By an obvious extension,

- - - . n . — I 1 !-••• ton terms

a".a"-a"--- to n factors = a" " " = a;

1
and a** tells us to separate a into n identically equal
factors, and take one of these factors.
100

INDICES. 101

This factor is called the nth root of a, and is often
written ^a, the letter or figure being written in the
symbol -y/ in all cases where n is not 2. Thus -^/a de-
notes the cube root of a, etc.

Ill 1^

(2) a"" -a"" -a"" "• to m factors becomes (a")"* when we
1
consider (a") as a single quantitative symbol; and it

becomes a^ by the index law.

1 m

Therefore, (a")"* = a"; and either of these expressions
denotes that the nth root of a is to be raised to the mth
power; or that the mth power of a is to have its nth
root taken.

Thus (64)* = (64^)2, or (642)^ = 16.

Also, writing p for -, and - for m, gives —
n q

(a^)«=:a^; or (a^) » = a* = (a*)^.

(3) Making p=:- in (2) gives —

11 J. 11

or V^= V^ = V^.

Thus to get the 6th root of a quantity we may first
take the square root and then the cube root of the result,
or we may first take the cube root and then the square
root of the result.

Ex. 1. To simplify f^V^". C^Y^'J^Y

102 INDICES.

This becomes (xm-n^m+n . (^^n-iy+l . (^y.l~my+m

— x"*^-"^ . x"'^-*^ • x^^-""^ = x'"^-«^+»^-^^+^^-"»^ = x'^ = 1.
Ex.2. To simplify ^' ■ ^'^ ■ ^(i"-\
This becomes

23 . 26-3n . 24n-4 . 3-2+n . 3-2n . 3-3+3n — 25+n . 3-5 _ ?!!lL.

EXERCISE VII. a.

- 1—— 3 3

1. Simplify (a«.a "• a2)-^(«2 . ^j. ^-i)^

2. Simplify V^* • ^^^ . ^2 . ^-Ki+n).

3. Simplify (ic2)2. (xJ)n-i ^(x«)i . (x2«)«.

4. Simplify a-«+i • a"^" • a^ -^ «-«. a2. «l
6. Simplify (x^+« -^ 5C«)^ ^ (ic« -^ x^-py-^.

p q

6. Simplify V{(«~^"* "^ 2>-2«)mj«.

7. Write the square of av^2 _ (j-N/2,

8. Multiply a3 _ q^I^t 4. ^2 i3y ^ it + 5?.

9. Express the relation x = ^< ~ ~ "^ A/ (t ~ r~ ) I ' ^^ ^^ to
be free from irrationals.

10. Express the relation

Va + X + Va — aj

, . . ... Va2-x2

to be free from irrationals.

11. Find X when a* . a=" -4- a*~^ — ^^j.

12. Find n when 2" . 2«-i • 2«+i = 2i-" • 2i+« • 2i-».

=V(:-^3-

SURDS. 103

n

13. Find n when Qs . 3»»-i = 6« . 2-^ • 3«+i.

14. Find the relation between m and n when a"*" = (a™)**.

n

15. Find n when 4" • 22«-i . 8i-» = 42 • 2-»» • 163.

16. Divide x^ - Sx'^ + 2 - x-^ + x-^ by ic - 3 - 2 a;-i, giving
quotient and remainder.

17. If fx = i(a' + a-*), and 0x = |(a* - a-*), show that —
i- (fxy^-(<pxy = l. iii. 2(0aj)2=/2x-l.

ii. 2(/x)2 = 1 + /(2 X), iv. . 0(2 X) = 2(0x) . (/x).

18. If /x = J(a^ + a~**) and 0x=i(a»* - a-**), show that —

i- (/x)2+(</>X)2 = l. ii. 0(_x) = -(0X).

iii. /x=/(-x).

SURDS.

88. A surd is the incommensurable root of a com-
mensurable number (Chrystal).

Thus ^2j -^5j 3*, etc., are symbolic expressions for
surds. The arithmetical extraction of the roots indicated
would approximate to the numerical values of the surds.

The expression ^4 is an integer under a surd form.

Many known incommensurables are not surds, and
some of them are not, as far as we know, due to any
finite combination of surds. As examples, we have
3.1415926 •••, which is the ratio of the circumference of
a circle to its diameter, and which is usually denoted by
the greek letter tt; and 2.7182818 •••, which is the base
of the natural logarithms, and which is usually denote^
by c or e.

An expression such as V2 + ^2 is a surd expression,
but is not a surd according to definition.

104 SURDS.

89. Let z and u be any real positive quantities, n being
an integer.

Then a;** will pass through all values from 0 to + oo,
if X passes through all values from 0 to + oo. There-
fore for some positive value of x, a;" = z, and x =: -^z.

That is, every quantity between 0 and + oo has a
positive real nth root. This is the arithmetical root,
and is the one with which we are principally concerned. ,

If n is even, n = 2?/i ; and 2;" = 2;^"* = («'")* And since
a square root has two signs —

.-. every even root of any quantity has two values
differing only in sign.

90. Let X be negative and n be an odd positive integer.
Then a;*" is negative, and passes from 0 to — 00, while x
passes from 0 to — 00.

Therefore, for some value of a?, »" = — z, and x=-\/ —z.
That is, every real negative quantity has a real nega-
tive nth root, when n is odd.

Thus, every positive real quantity has two real square
roots, fourth roots, sixth roots, etc., and one real cube
root, fifth root, etc. And every real negative quantity
has one real cube root, fifth root, etc., and no real square
root, fourth root, etc.

91. Let a be any real positive quantity, and let r be
its arithmetical cube root.

Then, since w' = w^ = 1, r^ = wV = wV = a. And tak-
ing the cube root, r, cor, and coV are each cube roots of a.
And the three cube roots of a are -y/a, (u-^/a, in^^a,
where -^a is the arithmetical cube root.

Thus the three cube roots of 27 are 3, 3 <u, 3 w^. Hence

SURDS. 105

every real quantity has one arithmetical cube root, and
two complex cube roots.

92. Sards which are reducible to the same surd factor
are similar ; otherwise they are dissimilar.

To reduce a surd to its surd factor we proceed as
follows :

Decompose the number into its prime factors, and
then (1) for a quadratic surd take out the largest square
factor possible. The remaining factor is the surd factor.

Ex. 1. V(1350) = V(2 . 33 . 52) = 3 . 5V(2 • 3) = 15 V6,
and 6 is the surd factor.

(2) For a cubic surd take out the largest cube factor
possible. The remaining factor is the surd factor.

Ex. 2. -^(9720) = ^(2^. 3^ . 5) = 2 • 3^(3^ . 5)

= 6^(45), and 45 is the surd factor.

93. Surds of the same order are added by reducing
them to their surd factors, and adding the coefficients of
similar surds.

Ex. 3 V2 + V^^ + 2 V12 - V^S

= 3V2 + 3V2 + 4V3 -4V3 = 6V2.

With dissimilar surds, or with surds of different orders,
the addition and subtraction can only be indicated by
connecting them with the proper signs.

94. When a fraction contains a surd expression as
denominator, it becomes necessary, for the sake of ease
in calculation, to so transform the fraction as to make
its denominator rational. This is effected by multiply-

106 SURDS.

ing both parts of the fraction by some expression which
will make the denominator rational. Such a multiplier
is called a rationalizing factor of the denominator.

1 1/2/2
Ex. 1. — = ^^^=^^^, and the denominator is rational,

and -^2 is the rationalizing factor.

Ex.2. 2 ^2^2^. V^^2J/^J^^/4^ andthedenom-
inator is rational, and ■^2'^^S is the rationalizing factor.

95. Since {^a -{- ^b){-y/a— ^b) = a — b, which is
rational, the expression y'a — ^b is a rationalizing
factor for -y/a + -^/b ; and reciprocally y'a + -^b is a
rationalizing factor for -^a — -y/6.

Ex. 1. To rationalize the denominator of

3V2
^ys - v/2'

Multiplying by ^^ + V^ gives ^V^ + ^ or 3(^6 - 2).
V^ + -^2 1

14-2/2
Ex. 2. To rationalize the denominator of -^ — ^ —

H-V2

Multiplying by Liza^ gives ~^~^^, or 3 - V2.
1 — •y/2 — 1

In the present case it is better to change the order of the denom-
inator, writing it -^2 + 1, since that gives a + quantity in the
denominator of the result, and does not affect the value of the
fraction.

Ex. 3. To rationalize the denominator in

1 + V2 + V^
This may be effected at two steps.

Multiply by ^ + V^ - V^ and we get ^ + V^ - V^ or

1 + V^ - V3. And this multiplied by ^ gives V^ + ^'^g.
2-\/2 •y/2 4

SURDS. 107

Ex. 4. To rationalize the denominator in — y ^ 4- ^/g

V2 - V3 + V^

Multiplying by V^ - ^3 - V^ gj^^s ~^~V^^~V^^ And
multiplying by :^ gives " V^ -j/^Q - V^Q^

or ye + 2 V15 + 3 yio

12

A trinomial quadratic deDominator may also be
rationalized by a single multiplication, as we proceed to
show.

96. To find a rationalizing factor for ^a -\- -yjh + yc,
where the surds are all dissimilar.

Let Va =p, V^ = q, ^c = r.

Then {p -\.q^r){p ^ q — r) =p^ + g^ _ r^ + 2pg ;

and (jp^ + g^ — r^ + 2pq) (p^ + g^ — r^ — 2j9g)

= (i>^ + g^ — 7-^) ^ — 4p V-

In this final product p, g, and r appear only in even
powers, and the product is accordingly rational.

The rationalizing factor is the co-factor of (p + g + ^);
that is, (i> + g — ^) (i>^ + g^ — r^ — 2pg) .

This factor reduces to Sp^ — '^p\ + 2pqr\ or restoring
-^/a, y'ft, and y'c, to

2 (a — & — c) Va + 2 V(^^0 5
and this is the rationalizing factor.

The rationalized expression is Sa^ — 2 Sa^, and the

fraction becomes

2(a — & — c) yg + 2 V(a&c)
2a2-2 2a&

108 SURDS.

1

Ex. To rationalize the denominator of

1 + V2 + V3
a = 1, 6 = 2, c = 3.

S(a - & - c) V« + 2^(abc) = -2V2 - 4 + 2^6,
and 2a2 _ 2 2a6 = - 8.

/. the fraction becomes ^ + V ..

4

In using the form :S(a — & — c) -y^a -\- 2y'(a5c), if any
of the terms are negative, the proper signs must be
attached to the parts involving the roots only.

Thus for -yy2 — -y/S + 1 the rationalizing factor is
^ 2 V2 - 4 - 2 V6.

For V7 — V^ — V^ *^® factor is

2 V7 + 6 V3 + 8 V2 + 2 Vi2.

The rationalized result 2a^ — 2 2a&, not involving any
roots, is not affected by the signs of the terms, and
remains the same for all signs.

97. The expression S(a — 6 — c) ^a + 2-\/abc factors

into ( v« — V^ + V^) ( v^ + v^ — V^) ( v^ — v^ — V^) •

This result, although not convenient in practice, is
interesting as showing the constitution of a rationalizing
factor.

For the three factors are each derived from the expres-
sion to be rationalized, -^/a -f- -y/b + -^/c, by keeping one
term unchanged and giving to the other two all possible
variations of sign. So that of the four trinomial expres-
sions v^ + V^ + V^j v^ + v^ — v^? v^ — v^+v^^

and -^a — -y/b — ^c, the product of any three is a rational-
izing factor of the fourth.

This principle admits of great extension.

SURDS. 109

98. To find a rationalizing factor for -^a -f- -^b.

Let a = p% and 6 = ^.

Then (p + q) (p^ —pq -{- q^) = p^ + q^ = a -}- b.

.*. p^—pq-\- q^, or -^a^ — -y/ab +-^b^ is a rationalizing
factor for -^a + ^&.

Similarly, -^a^ + ^a6 + -^/ft^ is a rationalizing factor
for ^a - -^6.

Ex. 1. To rationalize ^a + ^6.

( V« + v^^)( V« - ^&) = a - ^&2= ^a^ - ^&2 ;
and this being the difference between two cube roots, the rational-
izing factor for it is —

or a2 + ab^ + &^,

and the factor required is —

( V« - W(«^ + <^Vb^ + b^b).
The rationalized expression is a^ — b^.

EXERCISE VII. b.
1. Simplify —

i. (3+V5)(2-VS). V. 5 V2(3 V4 + 6 V2).

ii. U+:^\(-3 + y^\' vi. \/48^+6\/75a.
iii. (4Vi+ViXVH2Vi). ""• V8-7V18+5V72-V50.

iv. •v/{v^8¥3}. viii. ^/iSa^b'^ -\- V^Oa^b^.

2. Rationalize the denominators of —

i. ^_. ii. :v^3^. iii. ^V^-^.

V8^ 2 - V3 3 - 2V2

110 SURDS.

iv V2(V5-1). ^. 5 4-2V2-3V.3-V6.

2V5TV5 1+V2-V3

V.

3 + 2V2-3V3-2v/6

1 + V2 - V^ - V^ 1 - V2 - V3

8. Which is the greater, ^^/^jzi or ^V^~^ ?

4. Which is the greater, ^^<!- — '^ — or -^' ~ , and what is their
difference? >/^ >/^

5. Rationalize the denominator of — '^

^2+^3

6. Rationalize the denominator of — "-^

V2+^2

99. The following theorems are important in relation
to quadratic surds.

(1) The product of two dissimilar surds cannot be
rational.

For, if p and q be their surd factors, p and q do not
contain the same factors, and hence pq is not made up
of square factors only.

Therefore Vpq is not rational.

(2) A quadratic surd is not the sum or difference of
a rational quantity and a surd.

For, if ^a = b ± -y/c, a — b^ — c = ±2b ^c, by squaring.
But a — b^ — c is rational, and therefore y^c is rational,
which is contrary to hypothesis.

Cor. The sum or difference of two quadratic surds is
not rational.

(3) A surd is not the sum or difference of two dis-
similar surds.

SURDS. Ill

For, if ■^^a = ^h + -y/c, a — b — c = 2^ he, by squaring.
But by (1), V^c is not rational, while a — b — c is
rational. .*. etc.

(4) li x-{- ^y = a + ^h, where x and a are rational,
then x = a and y = b.

For, X — a = ^/b — -y/y ; and if a? — a has any value,
it is rational, and is equal to the difference of two surds,
which is impossible by (2), Cor.

.'. x — a = 0, or x=a and b = y.

It may be well to remark here that with an equation,
such as x = a+ -y/b, we do not mean to assert that x has
not an exact finite value, but merely that its value is not
that of a quadratic surd ; or, in other words, the square
of x is not rational. So also, if a; — a = ^b—-yj'yj where
6 and 2/ are not equal, x — a has a real finite value, but
this value is not rational, and its square is not rational.
i.e. x — a is not a surd. Hence, if x and a be both
rational, x^a cannot have any value, since such a value
must of necessity be rational.

100. The results of Art. 99 enable us to separate a
surd expression, such as a + y 6 or ^a + ^b, into two
identical factors, i.e. to find its square root.

Let one factor be -y/x + -y/y.

Then a + -sJb = {-y/x-\-^yy=x-\-y -\-2^xy,

.*. Art. 99 (4), aj + 2/ = a; and Axy = b.

But x--y = ^\(x-\-yy-4xy\=^\a'-bl

x = ilx-}-y-}-x-yl = i\a-h^{a'-b)l

y = i\x + y-x-y] = i\a-^{a'-b)\

112 SURDS.

.-. V^ + V^ = V H« + Vo^"^ + VH« - Va2 - 6) ,
which is the required factor.

This factor, upon being squared, will reproduce the
expression a + -yjl) ; but the case of practical utility is
that in which a^ — 5 is a complete square. Denote it
byc^.

Then V^ + V2/ = Vi(a + c)-^^^{a - c).

Ex. 1. To find the square root of 3 + 2^2.
Here a = 3 and 6 = 8, and a^ — 6 = c^ = 1, a complete square,
whose root is 1.

/. Va^ + V2/=VF4+VF2=V2 + 1.
Ex. 2. To find V{^^ ■\-n-2 n^n}.
Here a = n^ + w, 6 = 4 n^, c^ri^ — n,

}(a + c)=n2, i(a — c)=w.

.'. V{^^ -\- n — 2 ny/n}= n — ^n.

The sign — before -^n in the result is indicated by the same
sign before 2 Uy/n in the original.

101. An expression of the form a + y^^ -f y'c + V^
can have its root found when the expression is a com-
plete square.

Assume V{^ + V^ + V^ + V^l = V^ + V2/ + V^-
Squaring,

.-. a=:x-\-y -\-z, b = 4:yz, c==4:zx, d = 4:xy.

SURDS. 113

"^-^"'■"HirMrHir^.

Ex. Root of 35 + 4^15 - 6^10 - 12 V^-
Here b = 240, c = 360, d = 864.

. * 1/360 X 864\ 4 1/864 x 240\ ^//240 x 360\
** \V 4x240'j \V 4x360 J \\ 4 x 864 j

= V18 ± V^2 ± V5
=2V3±3V2±V5.

The signs are then to be determined. It is readily seen that
they must be

2V3-3V2+V'5
in order to give those of the original.

Moreover, (2 V3)2 + (3 V2)2 + (^6y = 35.

The original expression should be reproduced by squar-
ing this, and that is the only absolute test that we have
the true root.

EXERCISE VII. C.

Find the square roots of the following perfect squares from 1 to
11 inclusive.

1. 8 + 2^15. 2. 3 + 41 3. 4+2^3. 4. l + 2uVl-u^.

5. -2i. 6. 2^2 + 2(x - 2/)\/x2 - y^ -2xy.

7. 9-4V2 + 4V3-2V6. ^q 4^10 + 8

8. 25 + 10 V6. ^^^^ ~ ^

9. I + V2. 11. a2 _ 2 + a\/^^2Tr4.

12. Simplify 1 ^— + ^

V'(16 + 2 V63) V(16 - 2 V^'^)

13. Simplify ^(3 + VO - p^) + V(3 - Vo - p^).

14. If a'^d = be, then V(^ + V^ + V^ + V^) ^^^ he put in the
form (V«5 + V2/)(V^+\/^)-

CHAPTER VIII.

Concrete Quantity. — Geometrical Interpre-
tations.— The Graph.

102. A concrete number depends upon a concrete
unit which gives to the number its name and character.
Thus 8 dollars is 8 of the concrete units called a dollar.
So a hours is a times the concrete unit known as an
hour.

The symbolism of algebra applies to these abstract
numbers, as coefficients of concrete units, but the inter-
pretation is affected by the nature of the concrete unit.

Operations with concrete quantities are in general
subject to the two following laws :

(1) To multiply 2 hours by 3, or 3 hours by 2, gives
6 hours; but to multiply 2 hours by 3 hours has no
meaning.

Hence, concrete units have no product, i.e. they do
not admit of being multiplied together.

Hence, also, concrete units have no powers and no
roots.

(2) Again, 3 hours and 4 hours make 7 hours, and
5 minutes and 8 minutes make 13 minutes, but 3 hours
and 4 minutes do not make 7 hours or 7 minutes.

Hence, concrete quantities are added only when of the
same name, and then, by adding the coefficients of the
concrete unit.

114

CONCRETE QUANTITY. 115

An important apparent exception to the foregoing laws
occurs in units of length ; this will be fully dealt with
in connection with geometrical interpretations.

Any other apparent exceptions are easily explained.

103. In many cases of the interpretation of concrete
results negative quantity has a special significance.

If an idea which can be denoted by a quantitative
symbol has an opposite so related to it that one of these
ideas tends to destroy the other or to render its effects
nugatory, these two ideas can be algebraically and
properly represented only by the opposite signs of
algebra.

If a man buys an article for h dollars, and sells it for
s dollars, his gain is expressed by s — h dollars. So
long as s > 6, this expression is +, and gives the man^s
gain.

But if s<6, the expression is — . It denotes that
whatever his gain is now, it is something exactly opposite
in character to what it was before. And as he now sells
for less than he buys for, he loses. In other words, a
negative gain means loss.

Thus gain and loss are ideas which have that kind of
oppositeness which is expressed by oppositeness in sign.
If a man gains + a dollars, he is so much the wealthier ;
if he gains — a dollars, he is so much the poorer.

Whether gain or loss is to be considered positive
must be a matter of convenience, but only opposite signs
can denote the opposite ideas.

104. Among the ideas which possess this oppositeness
of character are the following ;

116 CONCRETE QUANTITY.

(1) To receive and to give out; and hence, to buy and
to sell, to gain and to lose, to save and to spend, etc.

(2) To move in any direction and in the opposite direc-
tion; and hence, measures or distances in any direction
and in the opposite direction, as east and west, north
and south, up and down, above and below, before and
behind, etc.

(3) Ideas involving time past and time to come ; as the
past and the future, to be older than and to be younger
than, since and before, etc.

(4) To exceed and to fall short of; as, to be greater
than and to be less than, etc.

Ex. 1. A man goes 20 miles north, then 15 miles south, then 8
miles south, then 12 miles north, and lastly 18 miles south. Where
is he with respect to his starting-point ?

Denoting north by + , south becomes — .

The traveller has gone +20 — 15 — 8 + 12 — 18 miles, or — 9
miles.

That is, he is 9 miles from his starting-point, and the sign —
shows that he is south of it.

Ex. 2. A man invests $10,000. On J of his investment he
gains 25 % ; on | he loses 20 % ; and on the remaining portion he
gains 2 %. What per cent does he gain upon the whole.

Let gain be +, and let a denote the amount invested. His
gain % is

fa 25^_2_a ^ , 7a 2 \ . a _ ^ ^ p.
U * 100 5 * 100 20 * lOOy ' 100 '° '^'

Therefore, his gain is — l2^o%' ^^ ^^ loses l2^o%*

Ex. 3. A goes a miles an hour and B b miles an hour along the
same road, and A is m miles in advance of B. Wnen and where
will they be together ?

CONCRETE QUANTITY. 117

Let t be the time in hours, and j? be the distance in miles, from
B's present position to the point of meeting.

hm

Then t = _i?L_, and p =

b — a

1. Suppose a, b, and m to be all positive.
As a and b may have any values, —

(a) Let a = b. Then t and p are both oo.

Now when a = b, A and B are travelling at the same
rate in the same direction, and the values of t and p tell
us that they will be together after an infinite time, and
at an infinite distance from the present position of B,
I.e. that they will never be together. It must be noticed
that this is the only way in which the symbols of algebra
can answer the question proposed with these conditions.

(b) Let a<b. Then as t and p are both positive and
finite, the men will be together at some time in the
future, and at some distance in the positive direction
measured from B's present position.

(c) Let a > &. Then t and p are both negative.
This tells us that A and B will be together at some time

in the past, i.e. they have already been together ; for as
t denotes time to come, — t must denote time past, and
their point of meeting is at some distance in the nega-
tive direction from B's present position.

2. Let a be negative. Then A is coming backwards
to meet B, and they will meet at some time in the future,
and at some place in the positive direction from B's
position.

Other variations of signs must be left to the explana-
tion of the reader.

118 CONCRETE QUANTITY.

Ex. 4. A and B both have cash in hand and both owe debts.
B's cash is 10 times his debt. If B pays A's debt, his cash will be
2| that of A's, and if A pays B's debt, his cash will be ^^ of B's.
When all debts are paid, both together have $ 3400. How much
has A after his debts are paid ?

Let A's cash = a, and B's cash = b. Since debt is opposite to
cash capital, B's debt is capital.

Then A's debt = 3400 + ^-a-b capital.
10 ^

After paying B's debt A has a left, and this is equal to

2% b, .-. a = i b.

After paying A's debt B has 6 - ^3400 + ^-a-b\ left, and
this is equal to 2| a. '

:,^Ab = b- 3400 - A + 5 + 5.
5 4 10 4

Whence b = 4000 = B's capital,

and — 400 = B's debt, as capital.

.-. B is worth 3600 when his debts are paid, and A is worth
— 200 when his debts are paid.

That is, A owes 200 dollars more than he is worth.

105. The examples of the preceding article illustrate
the fact that a literal algebraic solution of a concrete
problem is not a solution of a particular problem, although
put in a particular form, but of every problem belonging
to a group of which the particular one may be taken
as a representative. This group includes all problems
derivable from the particular one by (1) varying the
magnitudes of the numerical quantities concerned, and
(2) changing ideas which admit of it into their opposites,
provided always that such changes do not render the
problem unintelligible.

CONCRETE QUANTITY. 119

When such a literal result is interpreted in language
as general as possible, it becomes a rule; but it seldom
happens that any rule of arithmetic or geometry can be
as broad and representative as the literal expression from
which it is derived, and of which it professes to be the
interpretation.

EXERCISE VIII. a.

1. A man buys m articles at b dollars per article and sells them
at s dollars per article ; what is his gain per cent ?

Interpret the result (i.) when he receives the articles as a gift ;
(ii.) when he gives the goods away ; (iii.) when ft > s.

Show that a person in dealing may gain any finite percentage,
but that he cannot lose more than 100 per cent.

2. One carriage wheel is m, and the other n feet in circumfer-
ence. How far has the carriage gone when one wheel has made r
revolutions more than the other.

Interpret when m = n.

3. A has a dollars more than B, but if B gives A b dollars, A
will have twice as much as B. How much has each ?

Interpret when (i.) a is negative; (ii.) b is negative; (iii.) is
there any arithmetical interpretation for both a and b negative ?

4. A^s age exceeds B's by n years, and is as much below m as
B's is above p. Find their ages.

Interpret when n > m + p.

5. The hands of a clock go around in the same direction in a
and b hours respectively. If they start from the same point
together, when will they be again together ?

Adapt your result to the case where the hands move in opposite
directions.

6. A, By Cj D are four equidistant fixed points in line ; find a
point 0, in the line, for which

3^O+5SO4-2CO + 6Z)O = 0.
On which side (the ^ or i> side) of the middle is the point 0 ?

120 GEOMETRICAL IKTERPRETATIONS.

7. Two circles have tlieir radii r and r^ and their centres lie on
a fixed horizontal line. If r>ri and d is the distance between
centres, how are the circles relatively situated when —

i. d = r -\- r^f iii. d = —r + r^,

ii. d = r — r^, iv. d =— r — r^?

8. A can run a feet in b seconds, and B can run b feet in a sec-
onds. Express the ratio of A's speed to B's.

9. Two ropes are in length as 4:5, and 6 feet being cut from
each, the remainders are as 3 : 4. Find their original lengths.

10. The lengths of two ropes are as a : 6, and c feet being cut
from each they are as a^-.b^. Eind their original lengths.

Interpret when c is negative.

11. A river flows 4 miles an hour ; a boat going down the river
passes a certain point in 20 seconds, and in going up it takes 30
seconds. Eind the speed of the boat in still water. Also the
length of the boat in feet.

12. A man walks from A to B in h hours. If he had walked
a miles an hour faster he would have been b hours less on the
road. Find the distance from A to B, and the rate of walking.

13. Change the wording of 12 to suit the case where a and b
both change signs.

GEOMETRICAL INTERPRETATIONS.

106. As we have seen in Art. 21, the substitution of
numerical quantity for the quantitative symbol gives an
arithmetical theorem for an algebraic expression, and
especially for an identity.

So, also, an expression written in the symbolism of
algebra may admit of a geometric interpretation, when
the quantitative symbol stands for some elementary

GEOMETRICAL INTERPRETATIONS. 121

geometric idea. The mode of interpretation must depend
upon the character of this idea.

By employing the quantitative symbol to stand for
different geometric ideas, mathematicians have developed
different geometric algebras, requiring different modes
of interpretation, and some of which, not being derived
from arithmetic, are not subject to all the formal laws
deduced in Chapter I. It need scarcely be said that
these latter algebras are not generalized arithmetic, and
are not generally applicable to numbers.

Only two modes of interpretation concern us here, and
these are such as belong to algebra as already defined
and developed.

The distinction between these is as follows:

I. The quantitative symbol stands for a given portion
of a straight line, or a line-segment, as a geometric
figure. The algebra then becomes a kind of symbolic
geometry, and is subject to certain restrictions arising
from the nature of geometry.

II. The quantitative symbol stands for a number, i.e.
the number of times a particular line-segment, taken as
a unit of measure, is contained in a given line-segment.

This becomes a matter of concrete quantity, admitting
of geometric interpretation, and being subject to certain
geometric limitations.

I. Symbolic Geometry.

107. The primary ideas in geometry are length and direc-
tion, or, mechanically stated, transference and rotation.

Length is denoted by a quantitative symbol, which in
this connection will be called a line-symbol. Thus a

122 GEOMETRICAL INTERPRETATIONS.

denotes a given line-segment. But it does more than
this ; it denotes transference from one end of the seg-
ment to the other in only one direction. Then — a
denotes the same amount of transference in the opposite
direction ; and thus a — a = 0 represents a neutralized
effect, and is equivalent to no transference, and conse-
quently to no line-segment.

Thus a and — a denote the same segment measured in
opposite directions, and we have thus a simple and intel-
ligible interpretation of the signs -f- and — as applied to
line-segments.

Such a segment is called a directed segment, as its
direction, being one of two opposites, is determined by
algebraic sign.

108. a-\-b is a segment as long as a and b together;
or it represents transference over length a followed by
transference over length b in the same direction.

Thus, if a = AB, b = BC, g ,, b .

a + b = AB-{-BC=Aa ^ b c

Then a — &, which is a + ( — 6) represents transference
over length a followed by transference over length b in
the opposite direction. If a is longer than 6, let a = AB,

and 6=CjB. Then -b = BG, ^^ , , ,

and a — b = AB + BC, or trans-
ference from A to B, followed by transference from B to
C. This is equivalent to transference from A to (7, and is
positive, being the remainder when the shorter segment
b is cut off from the longer a.

If b is longer than a, let b = C'B.

Then a — 6 = AB -f BC = AC, which is negative.

Similarly, na, when n is numerical, denotes a segment
n times as long as that denoted by a.

GEOMETRICAL INTERPRETATIONS. 123

109. The area of a rectangle is denoted, in the sym-
bolism of algebra, by the product form of the line-
symbols which denote two adjacent sides of the rectangle.

Thus ah means the area of the rectangle whose
adjacent sides are denoted by a and h.

Then a^ denotes the area of the square whose side is a.

If either a or 6 is negative, the area ah is negative, and
is subtractive from any other area concerned. Hence an
area is often spoken of in connection with the symbolism
of algebra as a directed area. A square, however, is
essentially positive.

110. The volume of a rectangular parallelopiped or
cuboid is symbolically expressed as the continued product
of the three line-segments which denote any three
conterminous edges, known as direction edges, of the
cuboid.

Thus ahc is the volume of the cuboid having a, h, and c
as direction edges.

Then a^, which is the same as aaa, is the volume of
the cube whose edge is a.

We thus see the meaning of the terms square and
cube as introduced from geometry into arithmetical
algebra.

111. The evidence of the legitimacy of the conventions
of the three preceding articles, or rather the proof of the
necessity of 109 and 110 as following from the funda-
mental convention of 107, is a matter for geometry rather
than for algebra.

In the elements of geometry it is also shown that,
with these conventions in regard to the geometric mean-
ings of the algebraic forms, we have

124 GEOMETRICAL INTERPRETATIONS.

(1) a + 6 = 6 + a.

(2) ab = ba,and .*. abc = acb = bac = etc.

(3) a{b -{- c) = ab -{- ac, etc.

Hence the symbols a, b, c are subject to the same
formal laws of transformation whether we consider them
as line-symbols or as quantitative symbols. And thus
every algebraic identity of proper form may be inter-
preted either as a theorem in numbers, i.e. in arithmetic,
or as a theorem in lines, areas, and volumes, i.e. in
geometry.

EXERCISE VIII. b.

Interpret the following identities as geometric theorems —

I. (a-&)2 + 2a5 = a2_^62.

(a — 6)2 is the square on the difference of two line-segments ;
ah is the rectangle on the segments ; and a'^ + &2 jg ^he sum of the
squares on the segments, therefore

The square on the difference of two segments and twice the
rectangle on the segments are together equal in area to the sum of
the squares on the segments.

2. a(a + b)=a^-\-ab. 5. (a + 6)2 -(« - 6)2 .= 4a6.

3. (a+6)(a-6)=a2_52. g. ^a+by+(ia-by=2(a^-^b^y

4. (a + 6)2 = a2_,_ 52 + 2 a6. 7. ab(ia -}- b)= a'^b -{- ab^

8. (a + 6 + c)2 = a2 + 52 + c2 + 2(a6 + 6c + ca),

9. (a+by = a^-\-b^ + Sab{a-\-b).
10. If a : 6 = c : d, then ad = be.

II. If a : 6 = 6 : c, then 62 = ac.

12. Ji a:b = b;c = c:d, then a'.d = a^:bK

GEOMETRICAL INTERPRETATIONS. 125

112. Certain restrictions must be imposed upon alge-
braic expressions if they are to be interpretable as real
geometric relations. These, besides the conditions which
render an expression arithmetically interpretable, are two
in number ; namely,

(1) In the line-symbols the expression must be of not
more than three dimensions.

This is due to the fact that there are but three dimen-
sions in space, the subject-matter of geometry, and that
by our convention each line-symbol in a product repre-
sents one of these dimensions.

(2) The expression must be homogeneous in the line-
symbols. For the adding of one species of magnitude to
another species, as a line to an area, or an area to a
volume, is not an intelligible operation.

113. Every homogeneous expression of one dimension
in line-symbols denotes a finite line-segment, or is linear.

Every homogeneous expression of two dimensions in
line-symbols denotes an area; such areas being squares, as
a^, or rectangles, as ah, or areas made up of these.

Every homogeneous expression of three dimensions
in line-symbols denotes a volume; such volumes being
cubes, as a^, or cuboids on square bases, as a^h, or cuboids
with three unequal edges, as ahc, or volumes made up of
these.

114. An expression which represents a geometric rela-
tion must always represent a geometric relation however
it is transformed, provided it is interpretable. And
hence a homogeneous expression cannot be made non-
homogeneous by any legitimit;^ tvmsformation.

126 GEOMETRICAL INTERPRETATIONS.

This fact is useful in many ways.

If at any stage in the transformations of algebraic ex-
pressions a homogeneous expression becomes non-homo-
geneous, or vice versa, some error in work is to be looked
for.

Non-homogeneous expressions are frequently made
homogeneous in form by the introduction of a unit-
variable, on account of the resulting advantages in the «
after work.

Thus 0^24. 3a;— 2=0 may be written x^+3xy -2y^=0,
where 2/ is a variable in form only, and is to be replaced
by 1 after all the necessary transformations are made.

115. In applying the symbolism of algebra to develop
metrical relations in geometry, a sufficient knowledge of
descriptive geometry is required, and in addition to the
relations already laid down in 113 and previous articles,
the following are necessary :

1. If a, c denote the sides of a right-angled triangle
and b denote the hypothenuse, 6^ = a^ -f c^.

2. Similar triangles have their homologous sides pro-
portional.

Ex. 1. Vab is linear, and denotes the side of the square whose
area is equal to that of the rectangle whose sides are a and b.

Ex. 2. Vabc is Hnear, and denotes the edge of the cube whose
volume is equal to that of the cuboid of which a, 6, c denote
direction edges.

Ex. 3. Va^bc is an area. For it is a • Vbc, and ■\/bc is linear.

Ex. 4. a^bCj being of 4 dimensions, has no geometrical inter-
pretation.

GEOMETRICAL INTERPKETATIONS. 127

Ex. 5. ah 4- he is the sum of two areas, and is therefore an area ;
but ah + c has no geometrical meaning, not being homogeneous.

Ex. 6. The base of an isosceles triangle is h and its altitude is a,
to find the perpendicular, ^, from a basal vertex to a side.

Let ABC be the triangle with B as vertex, D the foot of the alti-
tude, and AE the required perpendicular.

Then ah = AE . BC, since each expresses double the area of the
triangle.

But BDC being right-angled at Z>,

J5C2 = ^Z>2 _|. 2X72 = a2 + J 52.
.\ ah=p^(ia^ + ih^).
Whence p r- ^^^

V(4a^ + 62)

Ex. 7. If any given area be divided respectively by the areas of
the squares on the two sides of a right-angled triangle, the sum of
the quotients is equal to that obtained by dividing the given area
by the area of the square on the perpendicular from the right angle
to the hypothenuse.

Let p be the perpendicular.

Then ac=ph = twice the area of the triangle.

(Art. 115, 1)

.-. a-c2 =

i)^62;

= p%a:^

+

C2)

Whence

1

a2

-1

1

-i)2>

and multiplying by any

area,

^2 say,

W2

a2

^C2

»2'

which interpreted gives the theorem.

128 GEOMETRICAL LNTERPKETATIONS.

EXERCISE VIII. c.

1. AA' is the diagonal of a square, and is trisected at the points
C and D. Find the area of the square when the segment CD is a.

2. The sides of a rectangle are as m -. n and the diagonal is d —
i. Find the sides.

ii. Find the area.

iii. Find the perpendicular from a vertex to the diagonal,
iv. Find the distance between the feet of the two perpendiculars
upon the same diagonal.

V. Show that the rectangle on the diagonal and the line-segment
between the ieet of the perpendiculars on the diagonal, is equal in
area to the rectangle on the sum and difference of the sides.

3. The sides of a rectangle are a and b ; to find —
i. The perpendicular upon a diagonal.

ii. The distance between the feet of the perpendiculars upon
the same diagonal.

iii. Show that the volume of the cuboid, whose direction edges
are the two sides and the line-segment of ii.,is equal to that of the
cuboid, whose direction edges are the sum and difference of the
sides and the line- segment of i.

4. The side of an isosceles triangle is n times the altitude, and
the base is 2 & ; to find the area. What does the result become when
n= 1? Explain.

6. The base of an isosceles triangle is one-half the side, and the
perpendicular upon the base is J Vl5. Find the area.

6. An upright tree is broken over by the wind, and the top
touches the ground at 36 feet from the base Find the length of
the whole tree when the remaining upright part is 15 feet.

7. How large a circular disc can be cut from a triangular piece
of paper whose edges are 13, 14, and 15 inches respectively ?

GEOMETRICAL INTERPRET ATIONS. 129

8. In a right-angled triangle the median to the hypothenuse is
r times one of the sides ; find the ratio of the sides to one another.

9. In an isosceles triangle where h is the base and s the side,
the area is expressed by — ; find the ratio of the side to the
base. ^

10. If the side of a square be increased by -th of itself, where n

n
is a large number, by what part of itself is the area increased ?

11. If the edge of a cube be increased by -th part of itself,

n
where n is a large number, by what part of itself is the volume
increased ?

12. Find the ratio of the diagonal of a square to —

i. The side,
ii. The join of a vertex with the middle of a side.

13. Find the ratio of the diagonal of a cube to its edge.

14. Compare the area of an equilateral triangle on the side of a
square to one on the diagonal.

15. A boat making 10 miles an hour in still water steers directly
across a stream flowing 4 miles an hour. Compare the real velocity
of the boat with —

i. Its velocity across the stream.

ii. Its velocity down the stream.

16. A boat goes a certain distance down a stream in t seconds,
and requires t^ seconds to return. Compare the velocity of the
boat with that of the stream.

Interpret when t ^t^.

17. A street 60 ft. wide has a house 20 ft. high on one side,
and a house 30 ft. high on the other. How long a ladder is
required, and where must its foot be placed, that it may just reach
to the top of each house ?

130

THE GKAPH.

Y

II. Geometry as Concrete Quantity. — The Graph.

116. Take any point 0, and through it draw the two
lines XX\ YY' at right angles to one another.

These lines are lines of

^ reference, and are called

axes, and 0 is the origin.

To distinguish the axes,

XX' is called the avaxis,

X and YY^ the y-axis.

Measures are made
along the ar-axis from O
to the right or left, those
to the right being posi-
tive and those to the left being negative. Also measures
are made from the aj-axis parallel to the ^/-axis, upwards
or downwards, those taken upwards being positive and
those downwards negative. We have thus two sets of
measures which represent the two dimensions of the
plane, and by means of the convention of signs stated
above we may represent any point in the plane.

Let P be any point, and FM be perpendicular to OM.
Then the measures which determine the position of P
relatively to the origin and axis are OM and MP^ and
if these are known, the position of P is known.

Usually, and for the sake of uniformity, the measure

OM IS called the x of the point P, and the measure MP

is called the y of the point P, If, then, the x and y of

a point are given, the point can be laid down.

Thus, let the a; of P be 2 and its y be 3.

To get the point, we measure OM to the right equal

THE GKAPH. 131

to 2 units from any adopted scale, and then measure MP
upwards parallel to OF and equal to 3 units from the
same scale. The point thus found is an ocular represen-
tation of the given point, and is called the graph of the
given point.

If X were ~ 2, and y 3, we would take OM^ to the
left and get the point Pi ; and if the point had its
a;= — 2 and its y = — 3, we would also take J/1P2 down-
ward and get the point Pg- Thus any point is completely
determined by its x and y with their proper signs.

117. Now let y=fx be any integral function of x.
For every value of x we have a corresponding value of y,
and if x varies continuously, i.e. by infinitely small
gradations, y also varies continuously.

Let, then, a number of corresponding values of x and
y be found by giving to x any convenient arbitrary
values, and finding the resulting value of y for each.
These values form the x's and y^s for a set of points
whose graphs all lie upon the graph of the function /ic,
this graph being a line or curve passing through all the
points.

If all possible values of x could be considered, the
points would be infinite in number, and would exactly
mark out the graph of the function ; but as we cannot
practically take every value of x, we take a set of values,
usually integral, as being most convenient, and thus get
a set of points. We then connect these, as well as
possible, by a line or curve, as may be required.

The theoretical graph is an exact geometrical picture
of the function, and in itself and in its relation to the
axis represents every property of the function. The

132

THE GRAPH.

practical graph, is a more or less close approximation
to this.

Ex. 1. Let

y = 2x-l.

Take

x = -l, 0, + l, + 2,

Then

2/ = -3, -1, + 1, +3,

c/

-X-

Lay down the graphs of the points whose a;'s and i/'s

are given, as at a, h, c, d, "- in the diagram.

It is readily seen that
^ / ...

these points lie in line, and

that the graph, which is
denoted by the dotted line,
is a straight line. Hence
the reason for calling 2x — l
a linear function of x, and
^ 2a;— 1 = 0 a linear equa-
tion.

It can be readily shown
that the graph of every
function of the form ax-\-h
is a straight line.

The root . At the point
R, where the graph ad cuts
the ic-axis, we have y = 0,
and .*. 2x^1 = 0. The x of this point is OB^^^nd as
this is the value of x, which makes the function zero,
OE measures the value of the root. It is readily seen
that OiiJ = f

Thus the cutting of the ic-axis by the graph denotes a
iral root, and the distance from the origin to the point
of intersection measures the value of the root, + if to

/

/

J

THE GRAPH.

133

the right, and — if to the left. In the present example
the root is -f .

As the line ad must cut the a^axis either at a finite
point or at go, a linear equation has one root only ; this
root must be real, and may be finite or infinite in value.

Ex. 2. Let y = a^ + x^-2x-l =fx.

Take a; = -2-l 0 + 1 + 2.

Then ?/ = -l + l-l-l + 7.

The graph is given at G in the diagram, being a curve
through the points
a, 6, c, d, e ••• /

1. The graph cuts
the a.'-axis at three
points, Bj E', and
E".

Hence there are
three real roots to
the equation
a;3_^^.2_2a;-l = 0.
. Two of these, OE
and OE', are nega-
tive, and the third,
OE'', is positive.
The limits of the roots are, OE > — 2 and < — 1, OE'
> - 1 and < 0, and OE" > 1 and < 2.

2. If we move the graph bodily upwards through 1
unit, we bring it to the position G', and increase every
value of y by 1 unit. But we may increase the value of
2/ by a unit, by adding a unit to the independent term of
the function.

I

'''I

e=7

' '

^\

/'

f \

' ; !

/ \

/ 1 '

k \

/ 1

/ / 1

'^v

^\ / / 1

1 t \

\ ' 1

1 ' ^

\ / '

/ /G' 6 \

\ '' /

'' / /I

\ ^

/ / r

\

/

/
/

/'R -1 R'^

/

/

\

1 /R''2 X

/

/

/

a

c

\ y

d

I

134 THE GRAPH.

Hence, to increase the independent term of the func-
tion is equivalent to moving the graph upwards, and to
decrease it is equivalent to moving the graph downwards.

(7' is thus the graph of

in which a has come to — 2, c to 0, and d to 1.

The roots are now — 2, 0, and 1.

The two points E' and E" have come nearer together,
and the two E and E' have gone further apart ; that is, the
roots OE* and OE'^ have approached one another in value,
while the roots 0^' and OE have become farther sepa-
rated in value.

3. Add another unit to the independent term, and the
graph is moved into the position (^", which is the graph

of a^ 4-0:2 _2aj + l.

As the loop L no longer cuts the aj-axis, the points E'
and E" have become imaginary, while the point E is
still real. Hence the two roots 0^' and 0^" of G have
become imaginary in (r", while the third root, OE,
remains real. Thus, then, o^ + iB^ — 2aj + l has two
imaginary factors, i.e. complex quantities, and one real
factor.

4. In the motion of the graph from the position G' to
that of 6r'' there was an intermediate position in which
the loop L just touched the ic-axis. The points E' and
E" were then coincident, and the roots OE' and OE''
were identical.

We thus see that in passing from real to imaginary
two roots approach one another in value, become equal,
and then become imaginary ; and since two roots must
always be thus involved together, the roots must become

THE GRAPH. 135

imaginary in pairs ; or, more concisely, imaginary roots
exist in pairs.

5. The least consideration will show that similar
changes take place when the graph is lowered by sub-
tracting from the independent term of the function, the
difference being that B and R^ will then become imag-
inary, while i?" remains real.

The mode of representing a function by a graph is
due to Descartes, and its invention is one of the great
milestones in the progress of mathematics. The graph
is largely employed by statisticians, by engineers, by
physicists, by chemists, and many others who are able
to employ mathematical methods intelligently; and its
systematic discussion is the subject-matter of coordinate
geometry.

EXERCISE Vni. d.

1. Construct the graphs of 2 ?/ + 3 a; = 6, and of 3 y — 2 « = 6.

2. Construct the graph of x — y = 0. How is it situated with
respect to the axes ?

3. In the graph of ax -\- hy -{■ c = 0, what is the effect of —

i. Increasing the independent term ?
ii. Increasing the coefficient of x ?
iii. Increasing the coefficient of ?/ ?

4. Draw the graphs of ic^ — 4^ + 2 = 2/; oi x^ + ^x"^ — x — \ = y]
and oix^ — 4:X = y. How are these graphs situated in relation to
one another ?

6. What integer added to the independent term of a^ — 2 ic^ — x^
+ 2 X — 1 will make all the roots imaginary ? Will make all the
roots real ?

6. Explain from the graph why a cubic must have one re^ xoot.

CHAPTER IX.

The Quadratic.

118. The most general type of a quadratic function of
one variable is

ax^ -{-hx -{-c,

and the corresponding equation is

ax^-\-'bx-\-c = 0 {A)

In the equation we may divide through by a ; then

a a

b c

and writings for -, and q for -, the equation becomes
a a

x^-}-px + q = 0] (B)

which is the quadratic reduced to its simplest foi-m. The
roots of this are, by Art. 59,

i«i = i( — p+Vp^ — 4g), and 0^2 = i( — P — Vp^ — 4g).

On account of the double sign of the root-symbol, -yj
(Art. 48), both values are included in the one expression

^ = i(— i> ±yp^ — 4g),

and this is the solution of {B),
136

THE QUADRATIC. 137

h c

In this solution write - for p, and - for q, and re-
duce, and we obtain

Jj (I

and this is the solution of (A)

The forms of these solutions should be so mastered
that for any quadratic equation, in either of the forms
(A) or (B)y the solution may be written down at once.

Ex. 1. The roots of 3 a:2 + 2 x - 4 = 0 are

x = K-2±V4T48)=K-l±V13).
Ex. 2. The roots of 2a:2 - Sx + 2 = 0 are

X = i(S±V9^=^)=l(S ± i VT).

119. The double root, or double solution of the quad-
ratic, is frequently of the highest importance as giving
an unexpected answer to a problem, and through this
answer giving us a clearer idea of the nature of the
problem.

It is only when a problem admits, in spirit, of a
double answer, that it involves the solution of a quadratic.

A few examples will make this plain.

Ex. 1. A man buys a horse and sells him for $24, thus losing as
much per cent as the horse cost in dollars. To find the cost.

X

100*

Or x2 - 100 x + 2400 = 0.

Whence aj = 60 or 40.

This solution shows the problem to be to a certain extent
indefinite, since there is no way of determining whether the cost
of the horse was $40 or $60.

138 THE QUADRATIC.

Ex. 2. The attraction of a planet varies directly as its mass and
inversely as the square of the distance from its centre. The earth's
mass is 75 times that of the moon, and their distance apart is
240000 miles. To find a point, in the line joining them, where their
attractions are equal.

Let P be the point, and let 0 , ^ ,

EP=X. E P M Q

Attraction ofE = — ; and of If = : and these are

x-2 (240000 - xy^

to be equal. This gives

74 x2 _ 150 X 240000 a; + 76 • (240000)2 = 0.

Whence x = 215160 or 271330 miles.

The smaller of these numbers evidently gives EP ; the
larger, being greater than 240000, gives a second point,
Q, beyond the moon, and not contemplated in the prob-
lem. Our judgment tells us that there is a second point.

Cor. In the foregoing question let the masses of the moon and
earth be the same. Then we have

or ^2(1 - 1) - 480000 x + (240000)2 = 0.

x^ (240000 - x)

By Art. 77, x = 00 or 120000.

That is, one point, P, is half way between the earth and moon,
and the other is infinitely distant.

EXERCISE IX. a.

1. Solve the quadratics.

I. x^ + x-i(b-c)-ibc = 0. iv. 3a;2__2a;+l = 0.

II. x^-ax+ i(a^ - 62) = 0. v. (a2 - b^)x'^ -2ax + l=0.
ill. abx^ - (a2 + b^)x + a6 = 0. vi. x2 - x - J = 0.

THE QUADRATIC. 139

2. Find the relation between a and h in the equation
(a -\- x) (h — x) -\- abx — 1=0, when —

i. The sum of the roots is zero,
ii. The sum of the reciprocals of the roots is zero,
iii. The sum of the reciprocals of the roots is infinite.

3. If the equation ax^ + bx + c = 0 has a and p as its roots,

find the equation which has - and - as its roots.

/3

4. Show that the roots of ax^ + bx+ a = 0 are reciprocals of
one another.

5. The area of a right-angled triangle is a^ and the difference
between the two sides is d; to find the sides. Explain the double
solution, and draw figures to represent it, when a^ — 4 and d = 2.

6. ABCD is a square. P is a point on AB produced, and Q is
on AD^ and PCQ is a right angle. Determine BP so that the
triangle PCQ shall have a given area, a^. Explain the double
solution.

7. In Ex. 6, ^§ is equal to BP\ determine BP when the tri-
angle P CQ has a given area, a^. Explain the double solution.

8. AB and CD are two straight lines intersecting at right angles
in 0. AC is of a given length, I. Find AO when the triangle
ACQ has a given area, a"^. Explain the quadruple solution.

9. Find the area of the triangle of Ex. 8, when AO = 2 CO,
10. Find CO, of Ex. 8, when AO'^ = l'CO.

120. The rational part, , in the solution of (A)

Jj a

is the same for each root, the difference in the roots
being due to the part V^^ — 4 ac.

As this part may be rational, irrational, or imaginary,
both roots are alike rational, irrational, or imaginary.

(1) When V&^ — 4:ac is real, the roots are real and
different.

140

GKAPH OF QUADRATIC.

This occurs when a and c have unlike signs, or when
they have like signs and 6^ > 4 ac.

Ex. 1. The roots of x^ - 2 x - 2 are 1 di V'*^-
Ex. 2. The roots of x^ - 3 x + 1 are 3 ± V^-

(2) When -y/b^ — 4 ac = 0, the roots are real and equal.
This occurs when &^ = 4 ac, in which case the function

is a complete square.

Ex. 3. The roots of ic2 - 4 x + 4 are 2 ± 0.

(3) When V&^ — 4ac is imaginary, the roots are com-
plex numbers, unless h is zero, when they are imaginaries.

This occurs w^hen 6- < 4 ac.

Ex. 4. The roots of x^ - 2 ic + 2 are 1 ± i.

± — V— 4ac, and
2a

(4) When 6 = 0, the roots are

differ in sign only ; but they may be rational, irrational,
or imaginary.

(5) If the roots are real, and a is +, they will have
the same sign when b > V&^ — ^ac\ that is, when c is +.
The sign of the roots will be the opposite to that of b.

This takes place in Ex. 2, the roots being real, and a
and c being both +.

121. The Graph. The
graph, G, of x^—^x-\-l is
given in the margin. The
roots OR and Oi^' are both
positive (Art. 120, 5).

Let x^-{-px-{-q=0 be the
quadratic in form {B), and
let Xi and X2 be the roots.

MINIMUM AND MAXIMUM. 141

Then,

(X — Xi) {X — X^ = X^ — {Xi -f- ^2) ^ + ^1^2 = ^

is the equation ; and comparing with the former, we
have Xy + .^2 = — p, and x^x^ = g. That is —

The sum of the roots is the coefficient of linear x with
changed sign, and the product of the roots is the inde-
pendent term.

If we put for p and q their vahies in terms of a, 6, and

c, we get Xi-{-X2 — , and X1X2 = -•

a a

Since x^ + X2 is independent of g, the sum of the roots
is not affected by changing the value of g. Hence if we
move the graph upwards by adding to g (Art. 117, Ex. 2,
2) until L comes to (7, the roots become equal and their
sum is unchanged. Hence 0(7= ^ (OjR -f- Oi?').

122. Minimum and Maximum.

The y of any point of the graph expresses the value
of the function a,^ — 3 a? + 1 for the corresponding value
of X ; thus for a; = 0 the value of the function is Oa, for
a; = 01 the value is 16, and for x = OG the value of the
function is CL.

The function has then a least value (7X, called its
minimum, but it has no greatest value.

If we change the signs of the function throughout, we
do not affect the roots in any way, but we change the
sign of every value of y, and we thus reverse the graph,
putting it into the position g.

The function now has a greatest value CL\ its maximum,
but it has no least value.

Hence a quadratic function with the coefficient of x^
positive has a minimum value but no maximum ; and

142 MINIMUM AND MAXIMUM.

with the coefficient of os? negative, it has a maximum, and
no minimum.

123. To find the minimum or maximum solution.

It appears, from Art. 121, that when ic = OC, the value
of the function is either a minimum or a maximum,
according as the coefficient of o? is positive or negative.

But 0(7= 1 (0/? + OjK') = one-half the sum of the roots
= — ^p. Hence the required solution is obtained by
substituting for x one-half the coefficient of linear x with
changed sign.

Ex. 1. The minimum value of x2 - 3x + 1 is (|)2 - 3(|)+ 1,
or - f = (7/:.

Ex. 2. To divide a number into two parts such that their
product may be a minimum or a maximum, and to find its value.
Let a be the number, and x one of the parts.
Then x{a — x) is to be a minimum or a maximum.
But the function ax — x^ has a maximum solution (Art. 122).
-, and X is -, or the number is halved.

laY_a^

2

Ex. 3. Two trains A and B are on two roads crossing at right
angles and approaching the crossing. A is a miles from the cross-
ing and goes a miles an hour ; B is 5 miles from the crossing and
goes /3 miles an hour. When will they be nearest together, and
how far apart will they then be ?

Let X be the time in hours. Then —

a — ax is A' s distance from the crossing at the end of x hours,
and 6 — j3x is B's distance.

Their distance apart is V(a— ax)^ + (5— jSx)^, and this is to be
a minimum. But its square will also be a minimum.

.-. (a-ax)2 + (6-/3x)2
or x\a^ 4-/32) - 2 a;(aa + &3) -f «2 + 62

is to be a minimum.

MINIMUM AND MAXIMUM. 143

The value of x is «fL±_M.

a' + )82

If this value be substituted for x, the function reduces to

{hoL - a&y
a2 + )82 '

which is the square of the least distance.

124. We arrive at the results of Art. 123, without
using the graph, as follows :

Let x^+px + q = y.

Then a; = i( — p ± Vp^ — 4g + 4y).

Now, whatever be the value of jp^ — 4g, the expression
Vi>^ — 4g -f 4?/ cannot be made imaginary by increasing
the value of y, while it may be made so by sufficiently
diminishing the value of y. If, then, the roots are to be
real, y has a minimum value, and this minimum is
reached just as the expression p^ — 4:q -{- 4,y is passing
from + to — ; i.e. when the expression is zero.

This gives x = — ^p for the minimum solution ; and
the value of y is found either by substituting this value
of X in the function, or by putting p^ — Aq -\- 4:y equal to
zero and solving for y.

Hence y = — ^ (p^ — 4g) ; i.e. one-fourth of the quan-
tity under the sign ^, in the solution of the equation
oc^ + px -{- q =: 0, with its sign changed.

Next let — x^ -\- px -\- q z= y.

Then x=:^(p ± Vp^ -}- 4g — 4:y).

The part Vp^ + 4g — 4y may be made imaginary by
increasing the value of y, but not by diminishing it.
Hence the function now admits of a maximum value,

144 THE QUADRATIC.

but not of a minimum ; and the maximum solution as
before is given hj x = ^p, and the value of the maximum
is 2/ = i(P^ + 4g).

As the value of x, which gives a maximum or a min-
imum solution, does not involve the irrational part of the
root, the solution is independent of the nature of the
roots, as to whether they are real or imaginary.

125. By studying the graph, we see that for real roots
with x^ positive, the function has a negative value for
all values of x lying between the roots, and positive for
values lying beyond the roots ; and for a^ negative, the
value of the function is positive for all values of x lying
between the roots, and negative for all values of . x lying
beyond the roots.

Ex. For what values of 5c is 3 ic^ — 2 ic — 1 positive ?
The roots of 3 a;^ — 2 x — 1 = 0 are 1 and — i ; and the expres-
sion is positive for every value of a; > 1 and < — |.

And the expression is negative for every value of x < 1 and

EXERCISE IX. b.

1. Construct the graphs of —

i. x^ — X— 1. iii. 2 + X — x^.

ii. ^2 + X + 1. iv. 2 — X — x^.

2. Construct the graph of 4 x^ + 4 x + 1.

3. Construct the graph of \/4 — x^.

Here we put y = \/4 — x'^, and hence 2/^ = 4 — x^ ; and y has
thus two values differing in sign only for every value of x.

4. Construct the graph of \/4x.

5. Construct the graph of x(l ± 2).

THE QUADRATIC. 145

6. Construct the graph of x^ + ic + 1. • .

7. Construct the graph ot x^— x"^ — x + 1.

8. Construct the graph of ^^ " ^ + ^ .

x-{- 1

9. Find the maximum or minimum value of the following
functions :

i. x^ + x-1. iii. 3x-x2 + 2.

ii. 3 x2 - 2 a; - 1. iv. x'^-Sx.

10. Find the numerical quantity which exceeds its square by
the greatest possible quantity.

11. Divide a number into two parts such that the sum of the
squares of the parts may be a minimum.

12. Find the number which when added to its reciprocal gives
the smallest sum.

13. Divide a number a into two parts such that the square of
one part added to n times the square of the other may be the least
possible, and find the sum.

14. Divide a number into two parts such that the difference
between the sum of the squares upon the parts and the product of
the parts may be a minimum.

15. Divide 20 into two parts such that their product may be 120.
The result is x = lO ±2i ^^. The factors 2(5 + iy/5) and

2(5 — ^V5) have 20 as their sum and 120 as their product, and
thus algebraically the problem is solved. But the complex num-
bers tell us, in the only way in which algebra can do so, that the
question is arithmetically absurd or impossible. We are shown
why this is so in Ex. 2 of Art. 123.

16. The sum of a quantity and three times its reciprocal is y'S ;
is the quantity real or imaginary ?

17. Show that -^ cannot be greater than f ^3 if x is

real, x^-x + 1

146 THE QUADRATIC.

x^ A- or 1

18. If A: is a value of x which makes — — equal to 2,

x^ — X -i- 1
show that A: is a complex number. ^

19. ABCD is a square ; on AD a point P is taken, and on AB
a point §, so that AP = BQ. Find AP when the area of the tri-
angle QAP is a given quantity, a^.

Denote the side of the square by s, and
let AP = BQ = X. Then AQ = s - x, and
the area of the triangle APQ is

J x(s — x)= a^.

Thence, x = i(s± Vs^ - 8 a^) .

(1) There are two solutions, and there-
fore two positions for P. This is seen in
the diagram, in the triangle AP'Q'.

(2) a^ has a maximum, that is when 8 a^ — s2^ or the area of the
triangle is one- eighth that of the square.

(3) When the triangle has its maximum, the two solutions
become one, and x = J s, as is seen in the triangle AP"Q".

20. In Ex. 19, Q is taken on AB produced, so that BQ = AP,
Examine the case and show, (1) that there are two solutions for a
given area of triangle, (2) that the triangle has a minimum, and
(3) that for the minimum x = — J s, and a^ = — ^ s^, and explain
these negative quantities.

21. Examine Ex. 19, when BQ is so taken that the rectangle
contained by AP and BQis 2i constant, c^.

22. In Ex. 6, of IX. a, has the triangle PQC a maximum or a
minimum, and what is its value ?

23. In Ex. 7, of IX. a, has the triangle PQG a maximum or a
minimum, and what is its value ?

24. Eind the maximum value of the triangle AOC in Ex. 8 of
IX. a.

25. A rectangular field is to contain an acre of ground, and a
path from one comer to the middle of an opposite side is to be as

THE QUADRATIC. 147

short as possible. What must be the form and dimensions of the
field?

26. An isosceles triangle has its equal sides given, to find the
third side when the area is a maximum.

27. Along a road already fenced a rectangular plot of 1 acre is
to be inclosed. What must be its form that the cost of fencing the
remaining three sides may be the least possible ?

28. Two towns, A and B, are on opposite sides of a river 6
miles wide, and B is 10 miles below A. A person can walk along
the shore twice as fast as he can row across. At what point must
he leave the shore so as to get from A to B in the shortest time ?

9. In the equation —-\-—(l- ^V= i, find the relation be-
- 2 ^ ^2 I

a- (f^

tween ^, g, a, and h when the quadratic in x has equal roots.

30. What are limits between which x^ — 5 ic + 5i is negative ?

31. If a and /3 denote the roots of x'^ ■\- px -\- q = Q^ find in
terms of p and q the value of —

i. a + iS. iii. a2+i32. V. i + i-

ii. ajS. iv. - + -• vi. a^ + 0^,

a ^

32. If the height of the thermometer is expressed by the func-
tion x'^ — 2x — 29 J, where x denotes the number of days counted
from a fixed time, for how many days will the thermometer be
below zero ?

126. Every equation of the form
can be solved as a quadratic, and be put under the form

ic** = i(— i? ± Vp^ - 4g).

148 THE QUADRATIC.

For, put aj" = y, and the equation becomes

whence the solution follows.
Ex. 1. a;8-3x*- 208 = 0.

Hence a:* = |(3 ± 29) = 16, or - 13.

.-. cc2 = + 4, - 4, + V13> - Vl^-
x = +2, -2, +2i, -2 1, ± Vi'Vl^; ibV- V12;
which gives the 8 roots.

Ex. 2. xi - 7 a^t - 8 = 0.

Let y z= x^, then X2 =: 2/^.

.-. y^ -7y -S = 0, and ?/ = - 1, or 8.

/. xi =-1, or 8, and x^ = 1, or 4096.

/. X = ^1, or 16.

127. Every equation of the form
can be solved as a quadratic, and exhibited in the form

fx = i{-p±Vp^ — 4:q).

Whence, if /ic = 0 is solvable, the equation can be com-
pletely solved.

Ex. 1. Given (x^ - x ~ 1)2 + 4(a:2 _ x) - 6 = 0.
This can be put into the form —

(x2 - X - 1)2 + 4(x2 _ x - 1) - 2 = 0.

... x2 - X - 1 = K- 4 ± V24) = - 2 ± V6.
Then x^ - x + I T V^ = 0.

Whence x = i(l±V± 4 V6 - 3).

IRRATIOKAL EQUATIONS. 149

On account of the double square root, we have by permutation
of signs 4 values for x, in all, as we should have, since x rises to the
4th power in the expanded equation.

Ex. 2. x2-2x + V(^^-2a^ + 6)=6.

Add 6 to each side ; then

x2 _ 2 ic + 6 + y/{y?' - 2 ic + 6) = 12.

/. V(^^ - 2 X -f 6) = - 4, or +3.

And ic2 _ 2 a: + 6 = 16, or 9.

Whence x = 3, - 1, 1 ± V^-

IRRATIONAL EQUATIONS.

128. An equation which involves the variable under
a root sign is called an irrational equation.

These may always be freed from irrationality, and
presented as rational equations ; but the rationalizing of
them introduces certain uncertainties of solution which
it seems impossible to avoid.

A few examples will make this clear.

Ex. 1. Given y/a + -^x = ^ax.

This is readily reduced by dividing throughout by y/x, or by
treating the equation as having ^/x as its variable. Then

y/x{y/a-V) = y/a,
and ^^ a

and the solution is exact.

Ex. 2. Given Va -\- x ■{■ Va — x = 2y/x.

As it is always profitable to reduce the number of terms con-
taining Vx, where possible, we may divide throughout by y/x^ and
obtain —

ve^o-^ve-o=^-

150 IRRATIONAL EQUATIONS.

Let - — z^ and square ; then

X

Divide by 2, transpose z, and square, and

Z^-1=^-4:Z + Z^.

/. ^ = CO (Art. 77), and 5? = J.

Whence x = 0, and i^.

6

The root -— - satisfies the irrational equation ; but the
o

root x = 0 does not satisfy the equation, and although
obtained by the legitimate transformations of algebra, is
not really a root of the given equation, as the test of a
root is that it shall render the equation an identity when
substituted for the variable. The root x = 0, however,
satisfies the equation Va -{-x — Va — x = 2-yJx, which
differs from the other in a single sign.

The probable explanation of this peculiarity is that
owing to the disappearance of certain negative signs, in
rationalizing the equations by squaring, both these equa-
tions reduce to the same rational form ; and as far as
this form is concerned, there is nothing by which we can
know from which of the two irrational equations it has
come.

Hence there is no reason why the roots obtained should
not satisfy one of the irrational equations as well as the
other. But it is quite evident that both roots cannot
satisfy both irrationals.

Ex. 3. V^ + Va - ^ax + x^ = ^a.

Transposing yjx^ squaring, and cancelling a,

y/ax + ic2 = 2 y/ax -

IRRATIONAL QUADRATICS. 151

-yjx is a divisor of this. Hence (Art. 76) Vx=0, and 5c=0, and

Va + X = 2^a — ^x.
Squaring, a + a; = 4a + x- 4\/ai.

.-. X = Qo (Art. 81, Cor 2), and 3a = 4\/aa;.
Whence x = j^ a.

Of these roots, 0 and ^^ a satisfy the given irrational equation,
while ic = 00 satisfies the equation

y/x — Va + \/ax + ic2 =,^a,

in which the signs before two root- symbols are changed.

Ex. 4. Given Zx-\r VSOa;- 71 = 5.

Transposing 3 x and squaring, we obtain

ic = 4, or 2|.

Neither of these roots satisfies the given irrational equation,
they being roots of

3cc-V30x-71 = 5.

Whether the given equation has an expressible root or not, it
cannot be found by the usual methods of solution.

EXERCISE IX. c.
1. Solve the following —

i. a + « + V2ax-f «2 = &. iii. V4 a + sc = 2 V6 + x - V^i

ii. a-\-x-\- Va2 -f x^ = h. iv. ^a-\-x + Va — x = 2^0;.

V. \/aTx=2?«/{x2 + 6a& + 62).

2. Find X, when ^ " ^^ . ^/fli^'^'l = 1.

l + ax AfVl-tiK/

3. Find X, when ^^"+^ = 1+VlT^.

Vl-x l-Vl-x

152 IRRATIONAL QUADRATICS.

4. Findic, when 1+^(1 --)=J(l + -V

6. Find re, when a -\- x -}- V2 ax -\- x^ = b^{a + x — V2 ax + x^}.

6. Solve the equation 6 x — 4V6x + 1 = x^ _ 2 ^ — 4.

7. Solve the equation x(-y/x + 1)2 = 8(ic + ^x) + 240.

8. Solve the equation a + xVl -\- a^ = aVl — x^ + xVl — a"^.

9. Solve the equation 9 x + 8 + 2 xV9x-\-4 = 15 ^2 + 4.

10. Solve the equation x + V(»^^ - ax+b^)=—+b,

a

CHAPTER X.

Indeterminate and Simultaneous Equations
OF THE First Degree. — Simultaneous Quad-
ratics.

129. When a positive integral equation contains a
single vaiiable, the value or values of that variable may
be found, theoretically at least, in terms of the constants.
But if the function contains two variables, the value of
either will contain not only constants, but the other
variable, and thus this value will not be constant, but
variable, and therefore arbitrary.

Thus if 3a; -1-22/ = 6, x = 2 — ^y, and y may take as
many values as we please, and to every value of y will
correspond a single value of x ; and, conversely, to every
value of X will correspond a single value of y. Such
equations are accordingly called Indeterminate, and as we
have seen in Art. 117, they have a graph which is a
straight line.

The study of Indeterminate equations is practically the
study of their graphs, and as a consequence Indeter-
minate equations in relation to their graphs form the
subject-matter of the great body of higher geometry
known as Analytic Geometry.

130. Linear Indeterminate equations are considered
here only under the restrictions that the corresponding
values of x and y shall be positive integers.

153

154 IKDETERMINATE EQUATIONS.

The subject is best discussed in examples.

Ex. 1. To find positive integral values for x and y which shall
satisfy the equation 7 ic + 11 ?/ = 103.

This gives x = ~ ^, which is to be integral.

But ^Q^ -^^y = 14 _ y + ^-^y . and as y is to be integral,
14 — 2/ is integral, and therefore —^ — ^ is integral.

Also, as the product of integers is integral, 2 x —^ — ^, or

1--^^ — ^ is integral ; i.e. ~ ^ is integral.

The purpose in multiplying by 2 is to make the coefficient of y
greater by 1 than a multiple of 7, so that after casting out all inte-
gers the coefficient of y may be 1.

Now, put - ~ y =zp. Then y z=S + 7p; and putting this value
for y in the original equation gives x = 10 — 11^.

Therefore a; = 10 — 11^^

V IS the general solution.
y = S + 7p /

The particular solutions are got by giving to p any allowable
Integral values, provided such values do not make x or y negative.

We readily see that in the present question p can have only one
value, zero ; and x = 10, y = 3, is the only solution.

Ex. 2. Can 1 1 be paid in 9-cent pieces and 7-cent pieces, and if
so, how ?

Let X = the number of 9-cent pieces, and y = the number of
7-cent pieces.

Then 9x-\-7 y = 100 is our equation.

The solution gives y = 4: — 9p and x = S -\-7p.

When p = 0, -1,

X = 8, 1,
2/ = 4, 13.

There are thus two solutions, one by 8 9-cent pieces with 4 7-cent
pieces ; the other, by 1 9-cent piece with 13 7-cent pieces.

INDETERMINATE EQUATIONS. 155

131. It will be noticed that in the general solution
the coefficient of p in the value of x is the coefficient of
y in the equation, and the coefficient of p in the value of
y is that of x in the equation, one of the signs being
changed.

Hence in the equation ax — by = c, p will have the
same sign in the values of both x and y, and the number
of solutions will be unlimited.

Ex. To find solutions of 7 x — 5 y = 23, we easily obtain

X = 4 + 5j9, 2/ = 1 -f 7 p.
Therefore when ^ = 0, 1, 2, 3, 4...

ic = 4, 9, 14, 19, 24...

2/ = l, 8, 15, 22, 29...

132. In the equation ax ±by = c there can be no
solution in positive integers if a and b have a common
factor which is not also a factor of c.

Eor, let a = mf, and b = nf.

Then ax ±by = mfx ± nfy = c ;

and mx ±ny = —

But - is a fraction by hypothesis, and is not the sum
or difference of two integers.

133. The following problem is nearly related to one
of the three preceding articles.

Ex. To find an integral number which when divided by 3 leaves
1, by 5 leaves 4, and by 7 leaves 2.

If X denotes the number, x is evidently of any one of the forms

156 INDETERMINATE EQUATIONS.

3m + 1, 5n + 4, or 7p + 2, and these are to represent the same
number ; we have

3m + l = 5n + 4=:7i) + 2.

/. m = ^ "^ ' = an inte^?. , and - = an integ. = m,
3 ° 3 b >

and n = Sm.

7 w — 2

Again, 15 m + 4 = 7p + 2, or w == -^ = an integ.

15

» — 11
■ ^ = an integ. = m, and ^ = 15 m + H.

15

Hence x=7p + 2 = 105 m -t- 79, which is the general solution.
If m = 0, we have 79 as the lowest number satisfying the conditions.

The following method of solution is also convenient. Let x be
the number.

Then ^^^=^, ^-^, and ^^=^ are all integers. Put ^^ = »,

3 5 7 "^ 3 ^'

and X — 375 + 1. Substitute this value of x in the second fraction,

and ^ ~ ' is an integer, or ^ ~^ is an integ. = q.
5 5

,'. p = 6q -{- 1^ and x = 15 g + 4.
Substitute this new value of x in the third fraction, and
— ^ "*" is an integ., or ^^ — is an integ. = r.

Then g = 7 r — 2, and x = 105 r — 26, which is the general solu-
tion, r = 1 gives 79 as the lowest number satisfying the conditions.

EXERCISE X. a.

1. Find positive integral solutions to the following —

i. Sx+7 y = 10l. iii. 455c - 13?/ = 38.

ii. lSx+ny = 200. iv. x-ll?/ = 48.

2. Find multiples of 23 and 15 which differ by 1 ; which differ
by 2; by 3.

3. How can I measure off a length of 4 feet by means of two
measures, one 7 inches long and the otlier 13 inches long ?

LINEAR SIMULTANEOUS EQUATIONS. 157

■ 4. I have nothing but 4-pound and 7-pound weights. How can
I weigh exactly 45 pounds ?

5. A company of soldiers when arranged 4 abreast lacks 1 man,
when 5 abreast it lacks 2 men, when 6 abreast it has 3 too many,
and when 7 abreast it forms a complete block, llow many men, at
least, are in the company ?

6. A wall 27 ft. 9 in. long is to be panelled with two widths of
boards, 8 in, and 5 in. wide. How many of each kind must be
used so that —

i. The narrow boards may exceed the wide by the least number
possible ?

li. The wide may exceed the narrow by the least number
possible ?

lii. The whole number of boards may be the least ?

LINEAR SIMULTANEOUS EQUATIONS.

134. The equations ax -\-hy-^c=0 and aiX-{-biy-{-Ci=0
are both indeterminate, and being linear, both have
straight lines as their graphs (Art. 117, Ex. 1).

These straight lines have some common point, their
point of intersection, and at this point the corresponding
values of x and y must be such as to satisfy both equa-
tions ; and as the graphs have only one common point,
there is only one such set of values.

When two equations are given, and it is required to
find corresponding values of x and y that shall satisfy
both, the equations are called simultaneous, and these
particular values of a; and y form the solution to the set
of two equations.

Ex. The two equations 2x-\-Sy = 9 and Sx-h 2y = 11 are
satisfied by the values x = 3, y = 1, and by no other values.

158 LINEAR SIMULTANEOUS EQUATIONS.

135. Problem. To solve a set of two simultaneous
equations with two variables. The methods will be ex-
plained through examples.

Let 4:X — 3y = 26 and 3x -\- 5y = 5 be the equations.

First method. — By addition and subtraction.

Add 5 times the first equation to 3 times the second,
and the coefficients of y, being equg] with opposite signs,
disappear, and we have left

29 X = 145 ; whence x = B.

Again, to get rid of x, we subtract 4 times the second
equation from 3 times the first, and obtain

— 29 ?/ = 58 ; and hence y = — 2.
Second method, — By substitution.
The first equation gives 0? = — — — ^; and substituting
this for X in the second gives

I§-±-^ + 5y = 5, or 78 + 29y = 20.
4

Whence 29 y = - 58, and ?/ = - 2.

Next substitute — 2 for 2/ in either of the equations,

and we get the value of x.

Third method. — By comparison.

The values of x found from the two equations are

2Q4-3v 5 5?/

— — — ^ and ^; and as these must be equal, we have

4 3 ' ^

78 + 92/ = 20 - 20y, or 2/ = - 2.

Similarly, y = — == —

/. 20a; — 130 = 15 — Oaj, or a? = 5.

LINEAR SIMULTANEOUS EQUATIONS. 159

Fourth method. — By an arbitrary multiplier.
Multiply one of the equations, the first, by an arbitrary
multiplier m, and add to the other.

We have (4 m + 3) a; — (3 m — 5) 2/ = 26 m + 5.
As m is arbitrary, we may give to it such a value as
will make either of the brackets zero.

If m = f , (-2/ + 3) a: = 26 X 1 4- 5, or aj = 5.
If m = — -|, we similarly obtain ?/ = — 2.

EXERCISE X. b.

1. 7(a; _ 5) = 2^ _ 2, 4 ?/ - 3 = i(x + 10), to find x and y.

2. (x+5)(2/+ 7)=:(aj-l)(2/-9)+104, and2a: + 10 = 32/ + l,
to find X and y.

3. X — a = c(y — b) , a(x — a) + b(y — b)+ abc = 0, to find x
and y.

4. ^ + ^ = l=:^Z^ + ^^,tofinda:and2/.
a b b a

5. 2 s = n(« + z), d{n — 1) = 2; — a, to find a relation not con-
taining n.

6. A and B have $500 between them. A gains from B 1 of B's
money and |50. B then gains from A J of A' s original money
and $'50, and they then have the same amount. What had they
to begin with ?

7. A fraction is such that if 2 be added to its numerator it
becomes i, and if 1 be added to its denominator it becomes |.
Find the fraction.

8. A number of two digits has the sum of the digits 12, and if
6 times the first digit be subtracted from the number, the digits are
exchanged. Find the number.

9. There are two kinds of coin such that a and b pieces respec-
tively are equal to |1. How many pieces of each kind must be
taken so that the value of c pieces together may be one dollar ?

160 SETS OF LINEAR EQUATIONS.

10. A farm was taxed at 30 cents an acre, and the tenant being
allowed 10 % off his rent found the allowance to amount to 15 dollars
more than the taxes. The next year the taxes were doubled, and
the farmer was allowed 15 % off his rent, which just paid his taxes.
What was the rent of the farm, and how many acres did it
contain ?

SET OF THREE LINEAR EQUATIONS.

136. Denote the three equations by A, B, and C, and

let the variables be x, y, z.

Find from A the value of x in terms of y, z, and the
constants, and substitute this value for x in equations
^ and C. We are then said to have eliminated x between
A and B, and also between A and (7; and we have two
new equations, D and Ej which contain only y and z as
variables.

Thus in eliminating one variable we reduce the num-
ber of our equations by one.

Now eliminate y between D and E, and we are left
with a single equation F, which is just sufficient to
determine z in terms of the constants. Hence we readily
find 2/ and x.

Ex. Let ^be x + Sy — 2z = l,

Bhe Sx — 2y -\- ^ = 5,
Cbe 2x + 4y-Sz = l.

From A we have x = l — Sy-\-2z.

Substituting this value for x gives —

inB, S-9y + 6z-2y-\- z = 5, or-Uy+7z = 2 (D)

inC, 2-6i/ + 4^ + 4?/-3^ = l, or- 2?/+ z = -l (E)

And eliminating y between D and E gives
3 ^ = 15, or ^ = 5.

SETS OF LINEAR EQUATIONS. 161

Thence E gives y - ^-^ = 3, and x = l-94-10 = 2.

,'. X = 2^ y = Sj z = 6 is the solution.

This method of eliminating x and y is always sufficient,
but it may not always be the most convenient, and any
method of elimination will answer if carried out in proper
form. Thus whatever process of elimination be employed,
the same letter must be eliminated between one of the
equations and each of the others. The following method
is very convenient and gives the same results as the last.

Subtract jB from 3 A This gives 11 ?/ - 7^ =- 2 ...(D)
Subtract C from 2 A. This gives 2y - z = l . . . . (E)
Subtract 11 {E) from 2 (Z>), and - 3 2; = - 15, or z = ^.

137. From the preceding article it appears that when
we eliminate a variable from a system of equations we
lose one equation ; and conversely, that by combining
one of the equations with each of the others of the set
we can eliminate one variable from the set.

Hence the two following results :

(1) That n equations are just sufficient to determine
n variables ; and conversely, that n variables can be
determined from a system of n equations, provided the
equations be independent, i.e. such that any one cannot be
derived from the others.

(2) That with n variables, and n — 1 equations, the
final result will be a single equation containing two
variables, and be thus indeterminate. And hence a system
of 71 — 1 equations with n variables is indeterminate.

Thus the two equations with three variables,

3x-}-2y — 6z = 4., and 4:X + y-^z = 10,

162 SETS OF LINEAR EQUATIONS.

give, by eliminating x, the single indeterminate equation

5y - 27 z = 36.

If this be solved for positive integers, we may find
one or more systems of positive integral values for Xj y,
and z, which will satisfy the two given equations.

138. Let there be n variables and n + 1 equations ;
an important case requiring consideration.

As the n + 1 equations are sufficient to determine
n + 1 variables, we make up this number of variables by
taking one of the constants, provided it be literal, and
considering it as a variable. After eliminating the n
true variables, we have a single equation in which this
pseudo-variable is the only one occurring; or, in other
words, we have a single equation expressing a necessary
relation amongst the constants. This equation is called
the Eliminant of the system.

Ex. Given 3ic + 2?/ = a, 2x — 4y = &, and x + 5 ?/ = c, to find
the eliminant.

Eliminating x gives

13 2/ = 3 c — a, and 14 ?/ = 2 c — 5.

And now eliminating ?/, we have

14 a - 13 6 - 16 c = 0,

as the necessary relation between «, 5, and c that the three equa-
tions may be compatible.

If a, h, and c are numbers which do not satisfy the
eliminant, the equations are incompatible, and cannot be
satisfied by any one set of corresponding values of x and y.

On the other hand, if a, ft, and c are numbers which
do satisfy the eliminant, one of the equations is derivable

SETS OF LINEAR EQUATIONS. 163

from the other two, and thus expresses no relation but
what is already given by the other two. The equations
are then not independent, and one of them is redundant

139. An important system is one of homogeneous
equations, in which the number of variables is greater by
one than the number of equations.

Let aiX-\- h^y + Cjs; = 0 = a^x + &22/ + ^^ ^® such a sys-
tem.

Dividing through by y, these take the form

«i- + Ci- + &i = 0 = a2- + C2 - + h,

y y y y

which is a system of two equations, with the two variables
X : y and z : y.

Solving, we get - = h^i^Z^.

y Cittg — c^ai

a; _ ?/ _ ;

-, by symmetry.

61C2 — h2Ci Cia2 — c^ai a^^g — ajbi

Or x:y'.z = biC^ — ^2^1 • Cia2 — C2ai : aib2 — ajbi.

And the variables are any quantities proportional to
the denominators of the fractions.

Ex. Let 2x + 6?/-3;s=:0 = 4x-32/ + 2r.

Then x : ?/ : ^ = 3 : 14 : 30,

and the numbers 3, 14, and 30, or any multiples of these, will
satisfy the equations.

If in these equations we make 2; = a constant, both x
and y take fixed values, and we have the common case
of two equations with two variables.

164 SETS OF LINEAR EQUATIONS.

140. Problem. To solve a system of three lineal
equations with three variables by arbitrary multipliers.
Let the system be

biX + 6jj2/ + h^ = ^2?
CiX + C22/ + c^z = ds,

Multiply the equations by the arbitrary multipliers
I, m, and n, respectively, and add; then

(lai-\-mhi-\-nc^x-\- (la2-{-mh2-\-nc^y-\- {la^-\-mh^-\-nc^z

= Id^ + mdg + ^^^3-

As I, m, n are arbitrary, we may so take their values
as to make any two of the brackets zero.
Thus to eliminate y and z we must have

The solution of this is, by the preceding article,
I _ m _ n
^2^3 — ^3^2 ^2^X3 — (^^(^2 ^2^3 — ^'3^2

And Z, m, ti are any quantities proportional to the
denominators. Naturally we take as the multipliers the
denominators themselves.

The reader will find, upon trial, that these multipliers
cause the coefficients of y and z each to become zero.

We notice that the multiplier for any equation does
not contain any coefficient from that equation or any
coefficient of the variable to be determined. Thus the
multiplier for the first equation is ^2^3 — &3C2, and does
not contain a suffix 1, or an a, etc. A little observation
on the forms of these multipliers is better than any
description.

SETS OF LINEAR EQUATIONS. 165

Ex. 1. Given x + 2ij~j-Sz = 9,

2x+ y+ z = U,
Sx + 2y + ^z = S.
To eliminate y and z the multipliers are

Z = 3, m= — 4, n=— 1.
.\ Sx-Sx-Sx = 27 -56-S,
or — 8 X = — 32, and x = 4:.

To eliminate x and y the multipliers are
1 = 1, m = 4, n =— S.
.'. 3^ + 42; -16^ = 9 + 56 -9,
or — 8 ^ = 56, and z = —7,

Ex. 2. Given ax + ?/ + ^ = 0,

ic + a?/ + 2J = 1,
« + 2/ + «^ = - 1.
The multipliers for eliminating y and ^ are a2_i^ 1 _^^ and 1 — a.
Thence, {a(a^ - 1) + 2(1 - a)}x = 0, and x = 0.

Similarly, we find y = , z =

a — 1 1 — a

141. With four or more equations, multipliers may
also be found which will eliminate all the variables but
one, but these multipliers are too complex for convenient
use. In the chapter on Determinants it is shown that
all sets of linear equations are solvable upon the same
general principle.

A set of four equations may be dealt with as follows :

Ex. Let them be x + y+z-\-u=4: (A)

x + 2y-\-Sz + ^u = 10 (B)

2x-\- y + Sz+ u= 1 (C)

3x + 3?/ + ^ + 2 w = 10 (D)

166 SETS OF LINEAR EQUATIONS.

Take the first three equations, and consider the x and y part as
forming a single term.

The multipliers for eliminating z and u are — 9, 2, 1 ; and these
give 5 X + 4 2/ = 9.

Now take the last three equations ; the multipliers are 5, — 2,
— 9 ; and these give 26 x + 19 ?/ = 54.

From these two new equations we find cc = 5, ?/ = — 4. The
values of u and z are then readily found to be z =— \^ w = 4.

EXERCISE X. c.

1. Solve the set, 2x + 4 ?/ + 5^ = 49, 3x + 5?/ + 6^; = 64,
4x + 32/ + 45; = 55.

2. Solve the set, 2x — Sy + z = 2, x + y — 2z = 1^ Sx + 2y
^Sz = 6.

3. Solve the set, x — y — 2z = S, 2x + y — Sz = l\f Sx — 2y

+ z = i.

4. Solve the set, ax + by — az = b{a-\-b), bx — ay-\-z=b(b — a),
x-\-2y — 2z = 4:X — b.

5. Solve the set, x + 2/ + ^ = 0, (a + b)x + (b + c)y+(c-i-a)z=Of
abx + bey + caz = 1.

6. Solvetheset, ?-A + l = 7f, A. + J_ + 2^ioj, A__L
^ x by z 3x 2y z ox 2y

7. If 2x -\- Sy =: a, x — y = by x + 2y = c, find the eliminant.

8. Find the eliminant of ax-\-y = 1, bx + S y = 6, ex -\- b y = 10.

9. Find the eliminant of3x + 2?/ + a=0, x — Sy + b = Oj
2x + y — c = 0.

10. Solve the set, Sx-2y + 6z = U, 2x + y -Sz = \0,

Sx — Sy-]-2z = SS; and explain the cause of any difficulties.

11. If ttiX + b^y + c^z = a.^x + 6.^?/ + c.^z = a.^x -^ b.^y + c^z = 0,
show that aiC^^Cg — b.^c.^) + &i(c2«3 — c^a.^) + <^i{P"p6 — (^M = ^'

SIMULTANEOUS QUADRATICS. 167

12. When x -{- y — z — u = 2x—2y -\- z-\-u=Sx — y -{•Sz—u=0,
find four numbers having the ratios xiy.ziu.

13. What does Ex. 12 reduce to when m = 1 ?

14. Solve the set, a + 26 + 3c+4(? = 29, 4a + 6+2c+3d:=23,
3a + 45 + c + 2(Z = 25, 2a + 36 + 4c + (? = 23.

15. Solve the set, x+ by — az = -<, ax + y — z = a% -a; -f ay
— « = 1 ^ o a

16. Solve the set, a'^x + ay + z = — a^, b^x + by -^ z = — b^^
c^x + cy -}- z =— c^; and reduce the values of the variables to
lowest terms.

17. Find the eliminant of ax + by + cz = bx + cy + az = ex
+ ay + bz = 0,

18. Solve theset, If^+lVl/^^. IN 1/2 _ 6 N 1/3^ IX

x\z yz) y\x zj y\x xzj z\y xj

z\y xyj x\z yj

19. Given x(y + z)= a^, y(z-i-x)= b\ z(x + y) = c^, and

— + -- = —, to find the relation connecting a, 6, c.
a;2 2/2 ^'^

(Find the values of «, t/, and z from the first three equations,
and substitute these in the fourth.)

SIMULTANEOUS QUADRATICS.

142. A system consisting of one quadratic and one
linear can always be solved.

The most general type of a quadratic equation of two
variables may be written

aoc^ + %^ + ^^ + gx +fy + c = 0.
And any linear of two variables may be written

168 SIMULTANEOUS QUADRATICS.

If we substitute for x from the linear into the quadratic,
we have

a {py + ry + by^ — hy (py -^r)-g{py + r) -{-fy + c = 0,

a quadratic from which to determine y.

Ex. To find two numbers such that the sum of their squares
and their product is a and the sum of the numbers is 6.

We have, x^ + y"^ + xy = a, and x + y = b^ where x and y de-
note the numbers.

Substituting for x from the hnear into the quadratic,

(p _ yy _|_ 2/2 + (^ _ 2/)?/ = a,
or y^ — by = a — b'^.

Whence y = ^(b ± V4a-3 6-^),

and x = K^=FV4a-362).

If 4 a < 3 b^j the numbers are complex.

The equation a^ -\- y^ -{- xy = a and x-\-y = b are sym-
metrical in x and y ; and whenever this is the case, the
values of x and y must be interchangeable, so that having
the two values of y, we have also the two values of x.

Thus if a = 19, and & = 5, we have y = 3 or 2, and
a; = 2 or 3.

143. A system of two quadratics with two variables
does not in general admit of being solved as a quadratic,
since substituting the value of a variable from one of
the equations into the other will in general give rise to
an equation of four dimensions.

Thus the system x^ -{-y = a, and y^ -{- x=b gives, by
substituting for y^ x^ — 2ax^ -{- x = b — a^, a quartic equa-
tion.

There are, however, many cases in which a sufficient
relation exists between the forms of the equations, to

SIMULTANEOUS QUADRATICS. 169

make a solution possible without going beyond the
quadratic.

No general list of such can be given, and no very
general rules of procedure can be laid down for such
cases when they occur. Practice and observation are
the only keys to success.

The following are given by way of illustration :

144. When two quadratics have a common linear fac-
tor in the portions involving the variables, they can be
solved.

For let A be the common linear factor, and let C and
C be the independent terms.

Then the equations are of the forms AB = C and
AB' = C, and B and B' must be linear factors.

Dividing the first equation by the second, we have

B' c' a

c

And as — is a constant, one variable is linearly ex-
pressible in terms of the other. And hence by substitu-
tion we obtain a quadratic for finding one of the variables.

Ex. Given

Sx^

-4 2/2+ 4

[xy=z

-21,

12 0^2

+ 2?/2-l]

xy =

-3.

The first equation

is (Sx-

- 2y){x + :

^y) =

-21,

and the second is

(Sx-

- 2 ?/) (4 X -

-y) =

-3.

Dividing,

x + 2y
^x-y

= 7, and .•

. y =

3 a;.

Substituting in the first equation,

21 ^2 = 21 ; and x = ± 1.
Thence 2/ = ± 3.

170 SIMULTANEOUS QUADRATICS.

145. When the equations are homogeneous in the
parts involving the variables, they can often be readily
solved by putting y = ux, and then dividing one equation
by the other. This gives a quadratic for finding u, and
hence a known linear relation between the variables.

Since u will, in general, have two values, we will get
two quadratics to determine x, and hence x will have in
all four values, as it should have. So also y will have
four values.

Ex. Given sc^ + xy + 4 2/2 = 6,

3 x2 _|_ 8 2/2 = 14.

Let y = ux^ and divide equation by equation ; then

3 x2 + 8 u^x^ 7*
or 4 ^2 + 7 ^ — 2, and u = —2, or J.

.*. y = — 2x, or \x; and substituting these values in one of the
equations, the second by preference, as being the simpler,

35c2 + 32a:2 = 14, and 3x2 + | x2 = 14.

.-. x = ± WIO, =±2,
and y = Tl^lO, =Th

146. When two variables are involved symmetrically,
it frequently simplifies the solution to assume two new
variables whose sum shall be one of the original variables,
and their difference the other.

Ex. Given x^ -}- y'^ -\- x + y = S,

x-^y + xy = 6.

The variables being symmetrically involved, assume a; = m -f t,
and y = u — V,

Then ^^2 ^ ^2 _j_ ^ _ 4^

and w2 _ ^2 ^ 2 w = 5.

SIMULTANEOUS QUADRATICS. 171

Adding, 2u'^ + Su = 9; and u = f, or - 3.

Substituting these values for u in one of the new equations,
the first, we get

And ic = 2orl, =-3db i^2.

And y being symmetrical with x has the same values ; then

x = 2, y = l; x = l, y = 2; x = -S-{- i^2 ;

2/=-3-V2; x = -Z-i^2) ?/=-3+iV2;
are the four sets of corresponding values of x and y.

The present equations may be otherwise solved as
follows :

Add twice the second equation to the first, and it becomes
(x + y)2 + 3(a: + 2/)=18,
whence x + ?/ = 3, or — 6.

Then from the second, xy = 2, or 11,
and (x — yy =(x + yy — 4:xy = 1, or — 8.

... x-y = ±l, or 2iV2.
Whence x = J(x + y + x — y)=2j or — 3 + i^^ ;

y = i(x-\-y-x-y)=l, or -S-i^2;

and the values of x and y being interchangeable give the four val-
ues as before.

147. Various devices are employed to obtain solutions
of simultaneous quadratics and other simultaneous
equations. These cannot be given in detail, but will be
illustrated in the following examples.

Ex. 1. Given x^ -h V^ = 275, x-\- y = 6.

(x + ?/)5 = x5 + 5 xy (2 x'^y + 2 xy'^ -{■ x^ -\- y^) + y^ = 3125.

172 SIMULTANEOUS QUADRATICS.

Subtracting x^ -\- y^ = 276 leaves

5 xy (x^ + 2x^y + 2 xy'^ + y^) = 2850.

.-. x3 + 2 x:^y -\-2xy^ + y^ = — •
xy

But (x + ?/)3 = a:3 + 3x2y + 3ic2?/2 + 2/^ = 125.

.-. xy(x + ij)=6xy = 126- — •
xy

Whence (xyj^ — 26xy = —114,

and xy = 19, or 6.

Then having x?/ and x + ?/, we readily find

a: = 2, or 3, or J (5 ± V^l);
y = S, or 2, or J (5 T iV^l)-

Q .

Ex. 2. Given x + vx^ — y'^ = - (Vx + ?/ + Vx — ?/),

2/
and (x + ?/) 2 _ (x - ?/) -^ = 26.

Put X + ?/ = 2 s2, and x - ?/ = 2 ^2^

This reduces the equations to

(s + 0'(«'-«')=8(s + 0V2 («)

and 2(s3_^3)^2 =26 (6)

Divide (a) by s + ^, and s + i = 0, or s = — «.
Multiplying out the quotient,

s^-t^ + st{s-t)=S^2.
And substituting s^ — t^ from (5),

s^(s-0=lV2 . (c)

Dividing (6) by (c),

s^ ~ 3 *

Whence i^-+il' = 1?, and ^±-^ = ± 2.

.-. 5 = 3 «.

SIMULTANEOUS QUADRATICS. 173

From this we readily obtain

s = I V2, t = 1^2, X = 5, y = 4.
Ex. 3. Given x(y + z)= a, y(z + x)=b, z(x + y)=c.
Adding the first and second and subtracting the third,

xy + x» -{- yz + yx — zx — zy = 2 xy = a + b — c.
Similarly, 2 yz = b + c — a, 2 zx = c + a — b.

Multiplying together two of these new equations and dividing
by the third,

2xy'2yz _ 2 ^^2 _ (o^ + b - c)(b + c- a)
2zx c-\- a — b

,^ lf(a + b-c)(b + c- a)^
MX 2(c-\-a-b) r

2{c-\- a-b)
with symmetrical expressions for z and x.

Ex. 4. Given x^ — yz = a, y^ ~ zx = b, z^ — xy = c.
Then, (x2 - yz^ - (2/2 - zx) {f - xy) = a^ - be,

i.e. x(x^ + y^ + z^ — S xyz) = a^ — be.

,,^s^y3^,^_^^y^^a'-hc^b'^-ca^c'^-ab^
X y z

since the left-hand expression is symmetrical in ic, 2/, and z.

Thence each fraction ^ ^ | (^^ - ^^)^ " (b'- ^^) (^' " ^^) | .

... x = J\ a(a^-hcy I

\ I (a2 _ bey - (&2 _ ca) (c2 -ab)r

a^ - be

^(a^ + b^-i-e^-Sabey
with symmetrical expressions for y and z.

EXERCISE X. d.

1. Given x:y = S:2, and (2-xy + (l- yy = 25, to find all
the values of x and y.

2. Given x + ?/ = a, and xy(x^ + 2/*^) = b.

174 STMFLTANEOUS QUADKATTCS.

8. Given xy = 750, and x : y = 10 : S.

4. Given x + y = xy = x^ — y'^ ; that is, to find the two quanti-
ties for which the sum, product, and difference of squares may be
the same.

5. Given (x - y) {x^ - t/2) = 160 ; (x + ?/) (x'^ + y^) = 580.

6. Given x -\- y -{- xy = S4 ; x"^ + y^ -(x + y)= 42,

7. Given x^ -\- y^ -\- x -\- y = SSO ; x'^ - y^ + x - y = 150.

8. Given 4 x^ -\- y"^ + 4x + 2 y = 6 ; 2xy = \.

9. Given x + ?/ = 18 ; x* + y* = 14096.

10. Given x + y = 5 ; (x^ + ?/2) {x^ + y^) = 455.

11. Given -Vxn/ + -V^Ti= — •

X y Z

3

12. Given ^ x_^x'^-y\^ ^_^+l^y.

X X -\- y y y x x

13. Given x — Vx^ — i/^ = x(x + Vx^ — y'^) ;

X Vl — 2/ = y Vl + X.

14. Given xy = a \ yz = h ) zx = c.

15. Given that the sides of a right-angled triangle are in geo-
metrical progression, and the area is a^^ to find the sides.

16. Find three numbers such that the product of each into
the sum of the other two may be the numbers 48, 84, and 90,
respectively.

17. Given x'^yH'^ = a, x^y^z"^ = 6, x'^y'^z^ = c, to find x, y, and z.

18. Giveny^=4:ax, x~p=—2(a-\-x), y — q= ^. (q + x) to find
the relation between a, p, and q. ^ ^

19. Given x2 + xy + ^2 _ 14 ^^ x^ -\- x^y^ + y^ = 84 x2, to find x
and y, (Divide one equation by the other.)

SIMULTANEOUS QUADRATICS. 175

20. x3 4- 2/3 + xy{x -{-y) = 65, (x^ + y'^)x'"if = 468, to find x and y,
(Put x-\- y = u and xy = v.)

21. x + y + z = lS, ic2 4- 2/2 + ;s2 _ ei, jc?/ + x^ = 2 2/^, to find
X, 2/, and z.

22. The sum of the two sides of a right-angled triangle is 61,
and the hypothenuse is greater than the longer side by 3. Find
the sides.

23. The sum of the three sides of a right-angled triangle is 60,
and the sides are in A. P. To determine the triangle.

CHAPTER XI.

Remainder Theorem. — Tkansfoemation of
Functions.

149. When we divide a^ -{- ax^ -{- bx -{- c hyx—p, we
get the quotient x^ -\-(a-\-p)x-\-p^ -^ap -\-b, and the
remainder p^ + ap^ -{-bp -{-c.

We notice here that the remainder can be obtained
from the dividend by simply putting p for x ; or, in other
words, the remainder is the same function of p as the
dividend is of x.

To prove that this is always the case.

Let fx be any integral function of x, and let it be
divided by x—p. Then if Q denotes the quotient, and
B the remainder, we have

fx = (x^p)Q-\-E,

As x—p is of one dimension in x, E is independent
of X, and is not affected by any change in x.

Change xtop-, i.e. put p for Xj and x—p = Oy and

which proves the

Theorem. If an integral function of x be divided by
x — p, the remainder is the same function of p.

Ex. 1. The remainder when ic^ — 5x* + 6x — 2 is divided by
x-1 is 1» - 5- 14 + 6. 1 - 2 = 0; and x - 1 is a factor of the
given expression.

176

BEMAINDER THEOREM. 177

Ex. 2. The remainder when x^ — 6x^ + Sx — 2 is divided by

ic + 3 is

(_ 3)7 _ 6(- 3)5 + 3( - 3) - 2 = - 740.

Ex. 3. To find the result of substituting 6 for x in the function

x^-Sx^ + x^-2x-l.

To substitute 6 for x is to find the remainder when the function
is divided by x - 6.

Therefore, 6

1_3_|_ i_ 2- 1
+ 6 + 18 + 114 + 672

1 + 3 + 19 + 112 + 671
And i? = 671 is the result.

Hence to substitute a for x in an integral function of
X is equivalent to dividing the function hj x — a and
taking the remainder.

EXERCISE XI. a.

1. Eind the value of x^'^ — Sx"^ -\- x^ — bx + 6 when x = 4, when
X = — 4, when x = l.

2. Find the value of x^ - 3.6 x'^ + 4.32 x - 1.728 when x = 1.2.

3. What is the remainder when x^ — 6 x* + 5 x^ - 4 x'-^ + 3 x — 2
is divided by x + 5 ?

4. Find the remainder when (a + 6 + c) (ab + 6c + ca) — abc
is divided by a + 6.

5. Find the remainder when x^ — 7 x + 10 is divided by x — 1,
by X - 2, by X + 3.

What relation does x^ — 7 x + 6 hold to the three divisors ?

6. Find the result of substituting 1.71 for x in the function

X8-' 5.

What, approximately, is the relation between 1.71 and 5 ?

178 KEMAINDER THEOREM.

7. Find the result of substituting 1.27 for x in the function

What relation does 1.27 hold to the function ?

8. Find the remainder when x^ + 2 a:^ — 3 x^ + x + 1 is divided
by 5c2 - X + 1.

The function may be written

x(x* + 2x2+ l)_ 3x2 + 1,

and the divisor gives x2 = x — 1.

.-. x{(x - 1)2 + 2(x _ 1)+ l}~ 3(x - 1)+ 1 = i?

= x(x2)-3x+ 4 = x(x-l)-3x + 4

= -3x + 3.

We thus substitute x — 1 for x2, wherever x2 occurs, and con-
tinue the reduction until only linear x remains.

9. What is the value of x^^ - x* + x - 1 when x3 + 2x— 1 = 0?

150. Divide a^ — 3a^4-2aj — 1 by a; — 1; we get a
quotient ic^ — 2 a; + 0, and a remainder — 1.

Divide a^ — 2ic4-0 byaj — 1; we get a quotient a; — 1,
and a remainder ■— 1.

Hence a^-3a:2^2aj-l = (a:-l)(a^-2a;)-l

= (aj_l)(a;_l)(a;-l) + (x-l)(-l)-l

We have thus expressed the function of a;,
a:3__3a;2^2aj-l,
as a function of (aj — 1), viz.,

(a,_l)3_(a;_l)_l;

and we notice that the function is simplified in form as
it lacks the square term.

REMAINDER THEOREM. 179

This result tells us that to substitute any value for x
in x^ — 3oc^ -{-2x — lis equivalent to substituting a num-
ber less by unity in the function

The foregoing transformation may also be effected as
follows :

Let x — l = y; then x = y -\-l, and putting y-\-l for
X in the function, and expanding, we obtain

f-y-^',

or, since y = x — l,{x-'iy'-{x — l)^l,

161. Let it be required to express ic^ — 4a^ — 3a; + 6
as a function of x — 2.

The transformed function will take the form

(x - 2)3+ B2(x - 2)2 + E,{x -2)-{-E,

where the coefficients i?, Bi, R^ ^^^ ^o be found.

We have the identity

aj3__4aj2_3^^g^(^__2)3+i?2(^-2)2+i?,(a;-2)+i?.

Writing 2 for x gives ^ = — 8.

But to substitute 2 for x is equivalent to dividing by
a; — 2 and taking the remainder ; so that if we divide the
given function by x — 2 the remainder gives R.

Hence rejecting — 8 from each member, and dividing
throughout by a; — 2, we have from the remainder

The next similar operation gives
R, = 2.

180

EEMAINDER THEOREM.

Therefore the transformed function is

(a; _ 2)3 + 2{x - 2)2 - 7(x - 2) - 8.
The whole operation carried out in the form of divis-

2

ion is as follows :

■ 4 -3

2 -4

+ '6
-14

1 -2
+ 2

-7
+ 0

- 8

1 -0

2

-7 =

= . . .

y 2 =
1

-B

-B,

^2

We here divide by a; ~ 2, getting the remainder — S=R,

We then cut this off, and divide the quotient of — 2 a; — 7
by X — 2, and get the remainder — 7 = Ri. Cutting this
off', we divide again by a; — 2, etc., until the whole is
completed.

Ex. 1. To express x* - 12x3 + 36x2 ■
x-3.

• X — 77 as a function of

1-12
3

+ 36

-27

-1-77

+ 27 +78

1 - 9 .+ 9
3-18

+ 26

-27

+ 1

1-6-9
3-9

- 1

1-3
3

- 18

1 0
And the required functio
(X - 3)4 -

n is

18 (x-

_3)2_(a;-S

3)+l.

REMAINDER THEOREM. 181

Ex. 2. To express m^ — 3 mhi + 2 m/i^ — Zn^ as a function of
m — n'm. place of m.

Both parts being homogeneous, this is equivalent to expressing

/?» y_8(''??V+2(™U3 as a function of ™ - 1.
\n I \nj \n) n

The coefficients are readily found to be 1, 0, — 1, — 3, and the

function becomes

(m — nY — n^ (m — n) — 3 n^.

EXERCISE XI. b.

1. Express x^ — Zx'^-\-2x-\-l as a function of sc -f 1.

2. Express Za^ — a^-\-4iCfi + ^a — ^ as a function of &, where

3. Express ?/5 — 6 ?/* + 10 y^ _ 10 ?/2 + 5 ?/ — 2 as a function of
ic, when x — y — \.

4. Express x^ + ISx* + 90^3 + 270 a:2 + 404x + 241 as a func-
tion of X + 3.

5. Express a^-\-2a^h-2 a^b'^ - 12 a^b^ - 15 aS* - 5 2)5 in terms
of a + 6 and b,

6. Express x^ — 7 x + 6 as a function of sc — 1.

What relation does x — 1 hold to the function?

What relation does 1 hold to the corresponding equation?

7. If x2 -^ ax -\- b be developed as a function of x — z, for what
value of z will the function take the form

(x -zy + A(x-z)?

152. If we have to express fx as a function of
x — {a-\-b-\-C'*'), we may carry out the operation in
successive parts by putting y = x — a, and transforming
to a function of y ; then putting z = y — h, and trans-

182 BEMAINDER THEOREM.

forming to a function of 2, etc., until all the quantities
a, h, c ••• are taken in. Where 6, c, etc., are decimals,
this is generally the most convenient method, as the
whole operation is very compact.

Ex. To express x^ — 2 x^ + 3x — 4 as a function of x ~ 1.23.

In the following operation only the algebraical sums are put
down in each column.

1.23

1-2

+3

-4 L

-1

+2

-2

0

2

1

1.2

2.24

-1.552

1.4

2.52

1.6

1.63

2.5689

-1.474933

1.66

2.6187

1.69

Hence, x^ - 2^2 4. 3^ - 4 = (x - 1)3 + (x - 1)2 + 2(x - 1) - 2
= (X - 1.2)3 ^ i.6(x - 1.2)2 _|_ 2.52(x - 1.2) - 1.552
= (x - 1.23)3 + 1.69(x - 1.23)2 + 2.6187(x - 1.23) - 1.474933.

The work may be still more condensed by leaving out
the decimal points, and remembering that when begin-
ning with a new quotient figure the product must be
written one place further to the right in the first column,
two places in the second column, three in the third, etc.

153. Let /a; = a^-l-aj2-2aj-l = 2/.
When ic = - 2, - 1, 0, 1, 2, ...

2/ = ~l, +1, -1, -1, +7,...

BEMAINDEK THEOREM.

183

The graph is given
in the figure, with 0
as origin, from which
values of x are meas-
ured.

Transform to a func-
tion of z where

2; = a; + 1,
and we have

2z'-z + l = y,

And when 2; == — 1, 0, 1, 2, 3, •••

2/ = -l, 4-1, -1, -1, -fJ...

By comparing results for the same values of y, we per-
ceive that z is measured from an origin Z, one unit to
the left of the origin for x.

Of the equation ic^+ic^ — 2a; — 1 = 0, two roots, OP
and OQ, are negative, and one root, OB, is positive. Of
the equation 2:^ — 22;^ — 2; + l = 0, one root, ZP, is nega-
tive, and two roots, ZQ and ZR^ are positive.

Similarly, by transforming to a function of u where
u = x-\-2 = z-{-lj the origin is moved to U, and the
resulting equation, u^ ^ 5u^ -{-6u-- 1 = 0, has all its roots
positive.

Hence the transforming of fx to a function of a; — a is
equivalent to moving the origin with respect to the
graph, through a units along the a;-axis, to the right if a
is positive, and to the left if a is negative.

Now take the equation a^ — 5a:^ + 6a; — 1 = 0, for
which the origin is at U, and all the roots are accordingly
positive.

184 KEMAINDER THEOliEM.

For some value of a, the origin will be moved from
U to Pj and UP represents this value of a. But UP
also represents one of the roots.

Therefore the value of a which transfers the origin
from U' to P is one of the roots of the equation.

Similarly, the values of a which transfer the origin
from Uio Q, and from Uto R, are the tw^o other roots. .

154. As the roots of the equation x^ — bx^-\-^x — l=zO
are real and incommensurable, they may be approximated
to. Let us then endeavor to find the value of UR.

For this purpose we first transfer the origin through
three units from Uto F. Then, having carefully drawn
the graph, we estimate the distance VR in tenths of a
unit, as nearly as we can, and we move the origin
onwards through this estimated distance.

To know whether our estimated distance is too great
or not, we have the following test ;

As long as the origin lies between Fand R, the graph
at that point is below the ic-axis, and the independent
term is negative ; but when the origin passes R, the
graph rises above the a;-axis, and the independent term
is positive.

Hence a change in the sign of the independent term
indicates that we have caused the origin to pass R,

Now transforming cc^ — 5a^4-6aj — 1 = 0 to a function
of a; — 3 gives ?/^4-4?/2 + 3i/ — 1, where y = x^S.

Next transform to a function oi y — 0.2, and we get
z' + 4:.6z''-i-4.,72z- 0,232, where z = y -0.2', and the
independent term being — shows that our new origin is
still to the left of R.

REMAINDER THEOREM. 185

The student is advised to try the effect upon the
independent term of transforming to y — 0.3.

Again transforming the last equation to a function
of z-OM, we obtain u^ + 4.72 u^ -{- 5.0928 u- 0.035776,
where n = z — 0.04.

We have thus transferred the origin in all through
the distance 3.24, and this is to two decimals a correct
approximation to the root Uli. A repetition of the
same process will furnish as close an approximation as
may be desired.

The work is carried out practically in the following
condensed form, where only algebraic sums are written,
and decimal points are not employed :

3.24698

1-5

+ 6

-1

-2

0

-1...

+ 1

3:.

-0232...

4.

384

-0035776...

42

472..

-0005047984

44

49056

46.

50928..

464

5121336

468

5149728

472.

4726

4732

4738

It will be noticed that the independent term is being
successively reduced in value ; and as this term gives the
distance from the origin to the graph, measured parallel
to the ?/-axis, this reduction shows that the origin is ap-
proaching the point R.

We have said that the first decimal figure must be
estimated, and then tried. So, to a certain extent, must

186 REMAINDER THEOREM.

the remaining figures. But the values of the others are
readily found. Thus, after finding the 2 of the quotient,
and finishing the transformation, we add a cipher to 0232
of the last column, and divide by 472 of the second col-
umn ; this gives 4 for the next figure. Similarly, 357760
divided by 50928 gives 6 for the next, and so on.

In fact, after obtaining the three decimals 246, and
completing the transformations thus far, we may safely
obtain three more decimals by simple division. In this
way 9, 8 are obtained by dividing 5047984 by 514972.

Ex. To find the root UQ. The various transformations give
as coefficients —

for X - 1, 1? - 2, ~ 1, + 1 ;

for X - 1.5, 1, - 0.5, - 2.25, + 0.125 ;

for x-1.55, 1, -0.35, -2.2925, +0.011375;

and the approximate root is 1.554 •••

It will be noticed that the independent term for this root is +,
as the graph lies above the x-axis to the left of §, or between P
and Q,

155. The preceding methods offer an elegant means of
extracting roots of numbers.

Ex. To approximate to the cube root of 12. Let x^ — 12 = 0,
and solve this as a cubic equation. This equation has but one real
root, and that is the arithmetic cube root of 12.

0

0

-12 12.2894

2

4

- 4

4

12..

- 1.352...

6

1324

- 0.147648

62

1452..

64

150544

66

155952

668

676

•. ^2 = 2.2894...

REMAINDER THEOREM. 187

EXERCISE XI. c.

1. Determine the integral values between which the real roots
of the following equations lie —

i. x^-Sx^-\-2x-2. iii. x* - 4x2 + 3a; - 4.

ii. x3+3x2 + 2x + 2. iv. x* + 2x3-4x-2

• 2. Find to 3 decimals the greatest positive root of
x4_ 2x3-3x2 + 6x-l =0.

3. Transform x^ — 3 x^ + 3 x — 4 = 0 to an equation in (x — 1),
and thence find the roots of the given equation.

4. Transform x* — 4 x^ + 2 x- + 4x4-6 = 0 to an equation in
(x — 1), and thence find the roots of the given equation.

5. Show that if x^ — px^ -]- qx + r = 0 be transformed to an
equation in (x— ^), the equation will assume the form z^+ Qz
4- i? = 0, where the square term is wanting.

6. Remove the second term from x^ — Gx^ + 12x + 9 = 0, and
thence solve the equation.

7. In the equation x^ — 2 x^ — x — 6 = 0, move the origin 3 units
to the right, and thence find the roots.

8. Find to 5 decimals the cube root of 3.1416.

9. A gallon contains 277.273 cu. in. Find the length in inches
of the edge of a cubical box that shall hold just 10 gallons.

10. Find the fifth root of 100, to 3 decimals.

CHAPTER XII.
The Progressions. — Interest and Annuities.

m

156. A series is a succession of terms which follow
some fixed law, by means of which any term, after some
fixed term, usually not far removed from the beginning,
may be obtained from the preceding terms and from
constants.

Thus l-{-3 -\- 5 -\-7 -\- "' is a series in which each
term is got from the preceding one by adding 2.
1 + i + i + i'" ^^ ^ series in which each term is one-
half the preceding term.

The doctrine of series is a very extensive and im-
portant one, and has given rise to a distinct calculus,
that of Finite Differences ; but two series, the simplest
of their species, are of such common application as to be
treated of in elementary algebra, and even in arithmetic,
under the name of the Progressions.

A series in which each term differs from the preceding
one by a constant, as in the first of the foregoing exam-
ples, is an Arithmetic Series, or an Arithmetic Progres-
sion, and is symbolized as an A. P. ; and a series in which
each term is a constant multiple of the preceding one,
as in the second example, is a Geometric Series, or a
Geometric Progression, contracted to G. P.

A third kind, the nature of which will be explained in
the proper place, is called a Harmonic Series, or a
Harmonic Progression, contracted to H. P.
188

ARITHMETIC SERIES. 189

ARITHMETIC SERIES.

157. The quantities normally occurring here are : a,
the first term of the series ; d, the common difference ;
w, any given number of terms ; and s, the sum of n
terms.

If fn is such a function of n that the substituting of
any integral number for n gives that numbered term in
the series, fn is called the 7ith term of the series, and
is all-important, not only in an A. P., but in all series,
as expressing the law of the series.

Evidently, to know the form of fn is to know the series,
since its consecutive terms are given by the substitution
of 1, 2, 3, • • • etc., for n.

The consecutive terms of an A. P. are

a, a + cZ, a -{-2d, a + 3 d, etc.,

and it is readily seen that the nth term is a-{-(7i — l) d.
It being often convenient to denote this general term by
a single letter, z, we have

z = a-{-(n — l)d (A)

158. An A. P., like any other series of numbers, is not
necessarily limited in extent, but may be continued at
pleasure in either direction.

When we consider any portion containing n consecu-
tive terms of this unlimited series, we call the first term
a, and the nth, or last term, z. And thus any two
unequal numbers may be any two terms of an A. P.

Ex. To find the A. P. whose 5th term is 12, and whose 11th
term is 24.

190 ARITHMETIC SERIES.

There will evidently be 7 terms in this portion, and hence

24 = 12 + 6d

. •. d = 2; and the first term of the series is 12 — 4 df or 4.
Hence the nth term is 4 + 2(/i — 1) or 2 + 2 n. And the series is

4 + 6 + 8+10+ —

159. It will be noticed that the nth term of an A. P.
is a linear function of n.

Now any function of n taken as the nth term will
give rise to some series, and if it is a positive integral
function of n higher than linear, the series will be of
the same species as an A. P. but of a higher order. So
that the A. P. is the simplest series of its species.

Thus if the nth term be l(n^-\-7i) or ^n(n -f 1)? the
series is

1, 3, 6, 10, 15 ...

It is worthy of notice that the differences of the terms
in this series form an A. P.

Similarly, if the nth term be of the form n^ + n^, or a
function of three dimensions in n, the differences of the
differences of the terms of the series will be an A. P.

160. As n denotes the number of a term in a series, it
must be integral ; and hence the presentation of a non-
integral value for n indicates some absurdity or im-
possibility.

Ex. Is 100 a term of the series whose first term is 3 and whose
difference is 4 ?

As the nth term is 3+(w — 1)4 or 4n — 1, the equation
100 = 4 71 — 1 will give an integral value for n if 100 is a term of
the series.

ARITHMETIC SERIES. 191

It does not give such a value, and therefore 100 is not a term of
the series.

It is readily seen that 99 is the 25th term.

161. S being the sum of n terms of an A. P., we have

^ = a + (a + d) + (a + 2d)H (a + 7i — l-d).

Also, by reversing the order,

S=(a-\-n — l'd)-{-(a+n—2'd)-j-(a + n—3'd)-] ha.

Adding,

2S = (2a + n — l'd)-\- (2a + n — l'd)-] n terms

= n(2a + n — l'd).

.'. S=~(2a + n-l'd) (B)

The following method of investigation for the sum of
an A. P. is important.

We have 1^ = 02 +2-0 +1,

2^=12 +2.1 +1,

32 = 22 +2.2 +1,

n^ = {n -\y -\-2{n -\) ^\

(7l + l)2=n2 +271 +1.

.-. by addition,

2(l + 2 + 3+...n) = (n + l)^-(n + l);

whence S = — i — ~^ — ^ = the sum of the first n natural

numbers.

Then, the terms of an A. P. are

a, a + d, a-\-2d^ ... a-\-{n — V)d\

192 ARITHMETIC SERIES.

and summing these gives

>S'= na -h (1 + 2 4- 3 + ... n - l)cZ
= na-{- ^n(n — l)d.
And this is the same as (B),
Ex. The sum of the first n odd numbers is
^(2 + 5r^ri.2), orn2.

162. Upon multiplying out,

fv 2^1 2a — d

JS =71"^ \- n ' .•

2 2

Hence the sum of n terms of an A. P. is a quadratic
function of n, with no independent term.

And every quadratic in n, without the independent
term, is the sum of ?i terms of some A. P. The inde-
pendent term, if present, would appear as an extraneous
term which might or might not follow the law of the
series.

Ex. To find the A. P. of which 2 n^ — 3 n expresses the sum of
n terms.

Let n = I ; the sum of 1 term is — 1 = «.

Let n = 2 ; the sum of 2 terms is2 = 2a -{- d.

.'. d = i, and the 7ith term = 4 w — 5 ;
which gives the series.

Otherwise, the sum of n terms, or Sn, is 2n^ — Sn; and the
sum of w — 1 terms, or Sn-h is2(n — ly — 3(w — 1).

But Sn is got by adding the nth term to Sn-i.

.-. nth ieYm= Sn- >Sn-i=2 n^-S n-2(n-iy-{-S(n-l)=4:n-6,

163. Any positive integral function of ?i, of a higher
degree than the second, and lacking the independent

ARITHMETIC SERIES. 193

term, expresses the sum of n terms of some series of the
same species as the A. P. but of a higher order.

Ex. Let 2 n^ — 3 n^ + n be the sum of n terms.
Then Sn-Sn-i=^n^-Sn^+n - 2(n-iy+S(n-iy-{n-'i)
= 6 n2 - 12 ?i + 6,
which is the nth term. And the series is

0 + 6 + 24 + 54 + ...,
a series whose differences form an A. P. Compare Art. 159.

164. Tn problems where n is to be found from con-
ditions involving the sum of n terms, n may have two
values. If both be integral, both will satisfy the con-
ditions, but non-integral values of n must be rejected as
being inapplicable to the case.

Ex. How many terms of the series whose nth term is 27 — 2 n
will make 144 ?

The sum of n terms is - (2 a + n — 1 • d), and this is to be 144.
We readily find a = 25, and d= — 2,

.-. ^(52-2n)=144. "^

Whence n = 18, or 8.

And the sum of 18 terms = sum of 8 terms = 144.

165. When three quantities form three consecutive
terms of an A. P., the middle one is called an arithmetic
mean between the other two.

Let A be an arithmetic mean between a and b. Then
A — a = b — A] whence

A = ^{a + b).

194 ARITHMETIC SERIES.

Or the arithmetic mean between two quantities is one-
half their sum.

The following miscellaneous examples illustrate the
subject of arithmetic progression.

Ex. 1. The pth term of an A. P. is P and the qth term is Q, to
find the nth term, and the sum of n terms.

From the pth to the ^th term the difference is added q—p
times.

q-p

Also, from the first to the pth term the difference is added p—1
times.

,.a = P-(p-l).Q^z^=lil^L}l:^QLP^zJ}.
q-p q-p

Hence the nth term is P^-QP+<Q-P).

q-p

Also, s=-(ia-\- 71th. term)

_nf2(Pq-Qp) + (n+l)(Q-P}
21 q-p }

Ex. 2. In the A. P.'s 6 + 7^ + 9 + .•• and - 3 - 1 + 1 ... to
determine —

(1) If there be a common term, and if so, its value.

(2) If there be a common number of terms for which the sum
in each series is the same, and if so, to find the simi.

(1) 6 + (w — 1)1 = — 3 + (n — 1)2 gives an integral value for n
if there is a common term.

This gives n = 19, and the 19th term is common.
Its value is 6 + 18 X I or 33.

(2) /S' = ^[l2 + ^TTT. ?^ = ^ (- 6 + W^^l . 2) gives the condi-
tion for a common sum, if n is integral

ARITHMETIC SERIES. 195

This gives n = 37, and the sum of the first 37 terms is the same
in each series.

The sum is V- (12 + 36 x f) = 1221.

EXERCISE XII. a.

1. Find the nth terms in the A.P.s, two of whose terms are
given as follows —

i. 1st term = wi, 2d = ^. v. 3d = 8, 8th = 6.

ii. 1st = a + n-l -6, 2d = a. vi. 10th = 4, 4th = 20.

iii. 1st = 3, 3d = 12. vii. (w-l)th=a, (7H-l)th=&.

iv. 1st = 0, 10th = 60.

2. Sum the following A. P.s —

1. 1 + IJ + ••• to 12 terms. iv. - H h ••• to n terms.

a a-\- b

ii. i + i + — to 10 terms. v. 103 + 97 + ••• to 24 terms,

iii. 0 + 3 + .-. to 7 terms. vi. 2 + 4 + ••• to n terms.

8. Sum 100 terms of the A. P. whose 3d term is 5, and 10th
term 75.

4. Sum to n terms the series whose nth term is J (n — 1).

5. Find the sum of all the multiples of 7 lying between 200 and
400.

6. If a, 6, c are in A. P., so also are a^ (6 + c), b^(c + a), and
c2(a + 6).

7. Find the A. P. for which s = } (3 ^2 - 2 m).

8. Find the A. P. for which s = 72, a = 17, d = -2.

9. The A. P.s whose sums are 29 n — 2 n^ and J (17 n + 3 n^)
have a common term. Find it.

10. One hundred apples are placed in line at two feet apart,
and a basket is placed at an extreme apple. How far will a person
travel who takes the apples one by one to the basket ?

196 ARITHMETIC SERIES.

11. n apples are placed in line d feet apart, and a basket is
placed in the same line, m feet from the first apple. How far will
a person travel who takes the apples one by one to the basket ?

What is the difference between m positive and in negative ?

12. A debt of •$ 1000 is paid in 20 annual instalments of $ 50
with simple interest at 6% on all, due at the time of each pay-
ment. How much money is paid in discharging the debt ?

13. A person receives ^1.50 daily. He spends 15 cents the first
day, 18 the second, 21 the third, and so on.

i. When will he be worth the most, and how much will it be ?

ii. When will he be worth nothing ?
iii. When will he be worth exactly $ 21.30 ?
iv. Will he at any time have exactly 1 10 ?

14. A and B start from the same place at the same time. A
goes westward 10 miles the first day, 8 miles the 2d, etc., in A. P.
B goes eastward 3 miles the 1st day, 4 miles the 2d, etc., in A. P.

i. Where and when will they be together ?
ii. When will they be 70 miles apart ?

15. Eind the sum of 1 + 2 - 3 + 4 - 5 + 6 - + ••• to 120 terms.

16. Two sides of an equilateral triangle are each divided into
100 equal parts, and corresponding points of division are joined.
Find the total length of all these joins.

Into how many equal parts must the sides be divided, so that
the sum of the joins may be m times the side of the triangle ?

17. $ 100 is deposited annually in a bank for 20 years to be left
at simple interest at 6 %. What is the accumulated sum at the
last payment ?

18. The side of an isosceles triangle is a and the base is &, and
the altitude is an arithmetic mean between the side and base.
Show that a : 6 = 1 + ^7 : 3.

19. Insert 4 terms of an A. P. between a and b.

GEOMETRIC PROGRESSION. 197

20. Find the series for which S = in(n -\- 1) (n + 2).

21. Find the series for which S = ^ n (n -\- 1) (2 n + 1).

22. Find the series for which S = {in(n + l)}^.

23. The sums of two A. P.s to n terms each, are n^ + pn and
3 w^ — 2 n. For what value of p will they have a common nth
term ?

Show that they cannot have a common term unless j) is a mul-
tiple of 4.

24. Two sets of n lines each are drawn parallel to adjacent sides
of a parallelogram. Find the whole number of parallelograms thus
formed.

26. The natural numbers are divided into groups as follows —

(1)(2,3)(4, 6,6)(7, 8, 9, 10)...
Find the sum of the numbers in the nth group.

GEOMETRIC PROGRESSION.

166. The quantities with which we have normally to
deal in Geometric Progression are

a, the first term ; r, the common ratio ; ?i, the number
of terms ; z, the last or nth term ; and JS, the sum of n
terms.

By definition, the terms of a Geometric Progression are

a, ar, ar^, ar^ •••, and it is readily seen that the nth term

is af~^,

.-. a; = ar"-^ {A)

Again, 8 == a -{- ar -\- ar^ -{- • • • a?*"~\

and rS — ar -^an^ -\ ar""'^ + ar^

... ^(i_r)=a(l-r«),

and S^a^'^^ (J5)

1 — r

198 GEOMETRIC PROGRESSION.

Eelation (B) may also be obtained as follows :
By division,

- = a + ar + ar^ + • • • ar"~^ +

1 — r 1 — 7'

I 1^1 n-i ^ ^^" 1 — r"

1 — r 1 — r 1 —r

Ex. The population of a city increases at the rate of 5% per
annum, and it is now 20000. What will it be 10 years hence ?

It will evidently be the 11th term of the G. P. whose first term
is 20000, and whose ratio is 1.05.

.-. The population will be 20000 (1.05)io = 32578 to the nearest
Integer.

167. The finding of r or n in a geometric progression
cannot usually be conveniently done without employing
logarithms. Thus in the preceding example the labor
of raising 1.05 to the 10th power is very great. And in
the case of n, on account of its appearing as an exponent,
the common operations of arithmetic do not suffice for
finding it.

Ex. 1. In a G.P. the first term is J, and the second term is i;
to find the wth term and the sum of n terms.
Since a = \ and ar = ^. .-. r = J -;- 1 = |.

Then z = nth term = - {-Y~^ = — !•
3V2/ 2«-i

Ex. 2. How many terms of the series l + 2 + 4 + 8 + »»» will
make 127 ?

Here m ='^^^^^ = '^^^^^:^ = 2^ - 1.

r-\ 1

.•. 2» = 128, and n is evidently 7.

GEOMETRIC PROGRESSION. 199

168. When r < 1, r" diminislies as n increases, and by
taking n great enough r" may be made as small as we
please. At the limit when n = oo , r"=0, and {B) becomes

1-r

This is the limit towards which the sum of n terms of the
series approaches as n is continually increased ; and by
taking n sufficiently great we can make the sum of n
terms approach this value as near as we please.

For convenience, this limit is called the sum of the
series to infinity, although it is more properly spoken of
as the limit of the series as n approaches infinity.

Ex. 1. To find the limit of 0.333 *" ad infinitum.

And its limit is A/i + J_-|. JL + ...\ = A 1_=1.

10 V 10 102 I 10 l-yi^j 3

Ex. 2. To find the value of the circulating decimal 0.241.

10 103 V ^102^104^ I 10 103 99

_ 2 41 ^ 2 X 99 + 41 ^ 241 - 2
10 990 990 990

Hence the rule — Subtract the non-repeating part from
the whole, and write as denominator as many 9's as
there are digits in the repeating part, followed by as
many ciphers as there are digits in the non-repeating
part.

200 GEOMETKIC PROGRESSION.

EXERCISE XII. b.

1. Find the sum of n terms of the series —

i. 1 + 3 + 9 + ... ii. i_| + |-_^..

iii. y3^ + yf ^ + ••• What is this when w = oo ?
iv. ^2 + (2 — ^2) 4- -• when n = co.

2. Sum to 00 the series 1 H \- ^ '

n + 1 (n + 1)2
What are the results when n = l? =2? =S?

3. Find the nth. term of the G. P. whose first term is v +

1 V2-1

and the second is

2-V2

4. A cask of wine contains 30 gallons. 5 gallons are drawn off
and the cask is filled up with water.- After this has been done
6 times, how many gallons of the original wine are in the cask ?

5. A circle is inscribed in an equilateral triangle ; a second
circle touching the first and two sides ; a third touching the second
and the same two sides, etc. If the side be s, find the radius of
the nth circle so described. Also, find the total area of all the
circles continued to oo.

6. A square is inscribed in an acute-angled triangle having one
side of the square on the base of the triangle. A second square is
inscribed similarly in the triangle above ; a third square above
that ; etc. to x. If the base of the triangle be b and its altitude a,
find the total area of all the squares.

7. How many terms of the G. P. 1 + 2 + 4 + ... will make 1023 ?

8. Find the sum of n terms of the series whose nth term is
na + a\

9. If S, a, r, z be taken in their common acceptation, show that

r ■=

S-z

10. A country whose annual production is now 5 millions, in-
creases at the rate of 5 % per annum. What will it be 5 years hence?

GEOMETRIC PROGRESSION. 201

169. When three quantities are three consecutive
terms of a G. P., the middle quantity, is a geometric mean
between the other two. It is also called a mean propor-
tional. Art. 84.

Let (7 be a geometric mean between a and h.

Then — = — - : and hence G = Va6.

a G

Problem. To insert n terms betwen two extremes, a
and h, so that the whole may be a G. P.
Let the series be

Then ^-l = ^2^...^ = A.= ^ = ,..

a t^ t,_, ^„_i t

and ^ = ^.^...^. A..^ = ^.n+i,

« Cf' h tn-2 L^i K '

.-. r = [-Wi; ^i = ar = (a"6)»+i;

1 1

e2 = (a'»-^62)«+^; and generally, «^ = (a'»-*"+^6'")"+^

EXERCISE XII. c.

1. Insert 8 terms between 1 and 512, to form a G. P.

2. A body weighs a grams in one scale pan of a balance and h
grams in the other. Show that its true weight is Vab,

3. Three circles each touch the same two lines, and one circle
touches both the others. Show that the radii of the circles form
aG.P

4. If a, 6, c, d be in G. P., prove that

(a + 6 + c + d)2 =(a + b)2 +(c + d)'^ + 2(6 + c)2.

202 HARMONIC SERIES.

5. A right-angled non-isosceles triangle has a perpendicular
drawn from the right-angled vertex to the hypothenuse. The
larger of the resulting triangles is treated in the same manner;
and so on. Show that the perpendiculars so drawn form a G. P.

6. AB is the diameter of a circle, and CT is a tangent at any
point C on the circle, and AT is perpendicular to CT. Prove that
AC is Si geometric mean between AB and AT.

HARMONIC SERIES.

170. A number of terms form a Harmonic Series when
their reciprocals form an Arithmetic Series.

Thus 1, 2, 3, 4, 5, etc., are in A. P.

And 1, I, ^, \, ^, etc., are in H.P.

Let a, by c be three terms in H. P.

Then i, ^, i are in A. P., and ? = - + 31:.

a b c b a c

a_^a — b
c b — c

That is, three quantities are in H. P. when the first is to
the third as the difference between the first and second
is to the difference between the second and third.

The term Harmonic is derived from the property that
a string of a musical instrument stopped at lengths cor-
responding to the terms of an H. P., sounds the harmonics
in music.

In algebra itself Harmonic Progression does not play
any important part; it is in geometry that it has its
principal applications.

JS"o method of summing an H. P. is known.

INTEREST AND ANNUITIES. 203

EXERCISE XII. d.

1. Find the Harmonic Mean between a and 6.

2. If -4, G^ JET denote the Arithmetic, Geometric, and Harmonic
Means between two quantities, show that G = y/AH.

3. J., P, By Q are four points in line, and C is half-way between
A and B. If -4P, AB, AQ are in H. P., then OP, CB, CQ are
in G. P.

4. If 1, c, a are in A. P. and 1, c, 6, in G. P., can c, a, 6 be
in H. P. ?

6. If X, y, z be in H.P., a, sc, 6 in A. P., and a, z, b in G.P.,
show that

-2

'-(-'>{©'-(!)'}■

6. Three numbers are in G. P. If the first two be each increased
by 1, the result is in A. P. ; and if 2 be then added to the third, the
result will be in H. P. Find the numbers.

INTEREST AND ANNUITIES.

171. Let F denote the principal, r the rate per unit, t
the time in years, and A the sum of the principal and
interest at the end of t years.

Then Pr = the interest for 1 year.

(1) If the interest is simple, this interest remains the
same for every year, and in t years becomes Prt, And
adding the principal gives

which is the relation connecting the quantities in simple
interest.

(2) In a case of compound interest, the amount at the
end of the first year becomes the principal for the second

204 INTEREST AND ANNUITIES.

year ; the amount at the end of the second year becomes
the principal for the third year, etc.

Amount at end of 1st year = P(l + r).
Amount at end of 2d year = P(l + r) (1 + r)

= P(l + ry.

Amount at end of 3d year = P(l + ry.
Similarly,

Amount at end of t years = P(l + ry,
or A = P(l + ry,

which expresses the relation connecting the quantities
in compound interest.

172. The present value of a sum of money, payable
at some fixed future date, is that sum which put at
interest will amount to the given sum at the given date.

Ex. 1. What is the present value of a sum, S, payable t years
hence, money being worth r per unit.

Let V be the value.

Then v(l + rV = S. .: V= —

^ ^ (1 + ry

The result is given for compound interest, as in all
such cases compound interest is the only kind practically
considered.

Ex. 2. If the sum S pays yearly interest at rate r, and money
is worth rate rj, we have

S(l -\- ry = amount of S in t years, rate r,
and V(l -\- r{)* = amount of Fin t years, rate r^,

and as these must be equal,

--iHfJ-

OF ANNUITIES. 205

Ex. 3. A loan of $ 5000 pays interest annually at 4 % for 4 years,
and is to be then paid in full. What is its present value, reckoning
money at 6 % ?

V= 5000 f ^-^y= 14633.28

/L04y^,

Vi.oeJ

OF ANNUITIES.

173. An annuity is a fixed payment of money made at
stated and equidistant intervals.

If the payments continue for a definite time, it is an
annuity certain, or a fixed annuity ; if they continue only
during a person's life, it is a life annuity ; and if they
continue for all time, it is a perpetuity.

Annuities may pay annually, or semi-annually, or
quarterly, or at any other stated times ; but as the prin-
ciples are the same in dealing with all of these, we shall,
unless otherwise stated, consider the payments as being
made annually.

Problem. To find the present value of a fixed annuity.

Let P be the annual payment, r the rate per unit of
interest, t the number of years the annuity is to run,
V its present value, and let R stand for 1 + r.

Let us suppose that the annuity is paid into a bank,
and left there for t years from the time of purchase, to
accumulate at compound interest.

The 1st payment draws interest for t — 1 years, and
amounts to PR~^.

The second payment, similarly, amounts to PR*^.

The 3d payment amounts to PR^~\
etc. etc.

The last payment is PR^~^ or P.

206 OF ANNUITIES.

Therefore the whole amount is

K — 1

Now if the purchase money were deposited in the
same way, it shouhl, in t years, amount to the same sum.
But F dollars in t years amounts to F(l + r)*

7?* — 1

Whence V= P

E -1
1 - R-'

Ex. What is the present value of $ 100 paid annually for 6 years
at 10 % compound interest ?

Here 72* =(1.1)6 = 1.77156 ; and R-' = 0.56447.

Then F = 100 x ^ ~ ^'^^^^^ = $ 435.53.

0.1

Cor. When t = yo, the annuity becomes a perpetuity,
and for its present value

174. When an annuity does not begin to pay until
after the lapse of a number of years, it is said to be
deferred, or to be in reversion.

Problem. To find the present value of an annuity in
reversion.

Let p be the number of years the annuity is deferred,
and let t be the number of years through which its pay-
ments run.

OF ANNUITIES. 207

The amount of the annuity at the end of 79 + ^ years is

1-

-R

?

and the amount of V for the

same time

is

VR'^',

and these must be

equal.

R'-

1
1~

n

R

r

-"{^'l

Cor. When t = cc, we have as the present value of a
deferred perpetuity

P

rR^'

Ex. A young man, at the age of 19, will come into a property
at 23 that will pay him $ 1000 yearly during his life. If his life
probability at 23 is 40 years, how much is his annuity now worth,
money being at 6 % ?

Here P = 1000, p = 4, ^ = 40, r z^ 0.06, B = 1.06.
1000 f 1 1

And V =

{-1

0.06 C (1.06)4 (i.06)-t4.

This cannot be conveniently Avorked out without the use of
either logarithms, or tables of the powers of 1.06.
The value is $11918, to the nearest dollar.

175. An annuity which has not been paid for a number
of years is said to be foreborne. The present value of a
foreborne annuity is the cash value of all due, together
with the present value of the annuity as continued into
the future.

To find the present value of a foreborne annuity. Let
the annuity be foreborne for q years.

Its cash value is P • ^ ~ ^' or P • ^^-^-.
1-R r

208 OF ANNUITIES.

1 — R~^

And its value for the future is P , t being

r

the number of years it is to continue.

... v=p[^^].

Cor. If t=cc, we have for the present value of a
fore borne perpetuity

r

176. The following problem is of special importance.

Problem. A corporation borrows A dollars, which is
to be paid in t equal annual instalments, each instalment
to cover all interest due at the time of payment. To
find the value of each instalment.

A part of the instalment goes to pay interest, and the
remainder goes to reduce the debt.

Let a, b, c, • • • t, be the parts applied in successive years
to the reduction of the debt, and let p be one of the
annual instalments.

Then,
1st payment =p = a -{- Ar.
Eeduced debt = A — a.

2d payment =p = b + (A —a)r ; whence b = aB.
Eeduced debt = A — a— b = A — a — aR.
3d payment =2) = c-\-{A — a — aR)r ; whence c = aR^,
Eeduced debt = A — a — b — c = A — a — aR — aR\

tth payment = 2)=t-^\A—a — aR—aR^ aR-^lr,

Eeduced debt = A — \a -i- aR -\- aR^ -\ aR-^},

But after the tth payment the debt must be nothing.

,.-ix ^B'-l

OF ANNUITIES. 209

.'. A = a(l + E + B' + "^E'-') = a

But a=p — At, and eliminating a between these gives

p = A' ->

^ R-1

which is the value of the annual payment.

EXERCISE XII. e.

1. A mortgage for ^1200 pays $400 annually for 3 years with-
out interest. What is its cash value when drawn, money being
reckoned at 6 % compound interest ?

2. Find the present value of the mortgage, of Ex. 1, if it pays
interest at 4 %, while money is worth 5 %.

3. An annual annuity of $1000 is foreborne for 6 years, and
is to run 8 years in all. What is it now worth, money being
reckoned at 4 % interest ?

4. A man borrows $ 500 on a mortgage and wishes to pay prin-
cipal and interest in 5 equal instalments. What is the amount of
each instalment, calculating interest at 6 % ?

5. A corporation borrows $30000 at 4% interest, and is to repay
principal and interest in 30 equal instalments. What is the value
of an instalment ?

CHAPTER XIII.

Permutations, Combinations, Binomial
Theorem.

177. If from n different objects we form groups each
containing r objects, such that no two groups contain
the same assemblage of objects, each group is called a
Combination, and the possible number of such groups is
the number of combinations of n things r together.

This number is symbolized by "C^, and read 'n objects
combined by ?''s.'

Thus ^Cg, taking letters as objects, is 4, and the several
groups or combinations are

ahc, ahd, acd, and bed.

Similarly ^Og has for its groups a5, ac, ad, ae, he, bd,
be, cd, ce, and de ; or 10 in all.

The combination abc is the same as acb, the same as
bac, etc., since all have the same assemblage of letters.

If, however, we take relative position into considera-
tion, abc and acb are not the same, since, although they
contain the same letters, the letters are differently ar-
ranged. Distinguishing different groups in this way,
each group is called a Permutation, and the possible
number of such groups is the number of permutations
of n things r together.

This number is symbolized by "P^.
210

PERMUTATIONS. 211

The combination ahc gives 6 permutations : dbc, acb,
bac, bca, cab, and cba; and as each combination may be
treated similarly, the number of permutations of 4 letters
3 together is 24 ; or ^F^ = 24.

PERMUTATIONS.

178. Problem. To find the number of permutations
of 71 things r together, n being greater than r.

If we have r boxes, A, B, C, •••, etc., into each of which
one of the n letters, a,b,c*" is to be put, the number
of ways in which the distribution can be effected is the
number of permutations of n things r together.

In filling box A, we may choose any one of the n
letters, and we have therefore n choices.

Having filled A, we have n — 1 choices in filling B,
and any one of these n — 1 choices may be combined
with the n choices in filling A,

Hence in filling A and B we have n(n — 1) choices.

Similarly, infilling^, B, and (7, we have n(n — 1) (n — 2)
choices, and so on through the r boxes.

.-. "P^ = n{n - 1) (n - 2) ... {n-r-\- 1).

179. Multiply the value found for "P^ by the unit
fraction

(it-r)(yi-r-l)...3.2.1
1.2.3... (n-r-l)(n-r)'

and we obtain

,p^l.2.3...(n-2)(n-l)n
1.2.3... (n-r)

212 l>ERMtJTATlOKS.

The continued product of m consecutive natural
numbers, beginning at 1, is called factorial m, and is
symbolized as m !, or [m.

rt f

Hence "P. = -

{n — r)\
Cor, When r = n, we have

"P„ = 71 ! (Art. 178) = "^ (Art. 179),

and hence 0 ! must be interpreted as meaning 1.

Ex. 1. The number of permutations of 12 things 5 together is
i2p^ = 12 . 11 . 10 . 9 . 8 = 95040.

Ex. 2. The number of permutations of n things 3 together is
14| times the number of permutations of n — 2 things 2 together,
to find n.

Here

n{n-V){n-2) _U_n

(7i-2)(7i-3) 3 n-3

Whence n = 12, or 3f .

As n must be integral, its value is 12, and 3| must be rejected
as being inapplicable to the nature of the problem.

Nevertheless, (3f • 2f • If) - (If • Of) = ~%K

180. In the permutations "P^ to find how many con-
tain a particular object or letter, as a.

Putting a aside, we form a group of r — 1 from n — 1
objects, and this can be done in

(?i — 1) (n — 2) • • • (n — r + 1) ways.

In each of these groups a can have r positions; namely,
from preceding all the other letters to following them all.

PERMUTATIONS. 213

Hence the number of permutations containing a is
r(n — l)(n — 2) ... (n — r + l).

Similarly, the number of permutations containing two
particular letters, as a and 6, is

r(r-l)(n-2) ...(n-r + 1).

Containing 3 particular letters together, it is

r(r - 1) (r - 2) (n - 3) ... (n - r + 1).

etc. etc.

Ex. How many numbers can be made from 5 figures, 1, 2, 3, 4, 5,
three at a time ; and how many of these will contain 1 ? How
many contain 1 and 2 ?

1st. «P^ = 5 . 4 . 3 = 60.

2d. r(w-l)(n-2)=3.4.3 = 36.

3d. r(r- l)(n-2)=3.2.3 = 18.

181. To find the number of permutations of n things,
all together, when u of the things are alike.

Denote the number by ''F(u).

If the u things were all different, they would in them-
selves give rise to u ! permutations, each of which com-
bined with each of the permutations of ''P(u) would give

Or "P(w) = — .

ul

Similarly, if **P(u) (v) denotes the number of permu-
tations all together, when u articles are alike of one
kind, and v articles are alike of another kind,

214 PERMUTATIONS.

Ex. How many permutations can be made from the letters in
Mississippi 9

Here there are 11 letters, of which 4 are i's, 4 are s's, and 2
arep's.

.-. ^P(u) (v) (w) = —111— = 34660.

If the permutations were to be such as not to have
repeated letters, we have only 4 different letters, and
the number is

«P„ = 4 . 3 . 2 . 1 = 24.

EXERCISE XIII. a.

1. Find the values of —

2. Given «P^ = S^-iP^, to find "Pg.

3. Given "Pg = V ""^P^, to find n.

4. How many permutations can be made from the letters in
College ? in Oporto f in Amsterdam ?

In each of these how many permutations would have letters
repeated ?

5. A person writes at random 3 of the figures 1, 2, 3, 4, 5, 6.
What is the probability that they will be consecutive and in
ascending order ?

6. With four different consonants and a vowel, how many words
of 3 letters can be made having a vowel in each.

7. The figures from 1 to 9 are written down, and a person erases
3 figures at random. What is the chance that the figures erased
may be consecutive ?

8. Six points are taken on a circle. In how many different
ways may they be joined by twos ?

COMBINATIONS. 215

COMBINATIONS.

182. As a combination has no reference to arrange-
ment, each combination of r articles can give rise to r !
permutations r together.

Hence "P, = r ! x "(7^.

r ! r ! (n — r) !

or, by reduction, "0^ = ^ • ^"7 • ^^^ to r factors.

Cor. Since "(7^ is necessarily an integer, it follows
that the continued product of any r consecutive integers
is divisible by factorial r.

183. Substituting n — r for r gives

{71 — r) ! r !

Or the number of combinations of n things r together
is the same as the number n — r together.

This is quite self-evident, for every time we take an
assemblage of r things out of n things we leave an
assemblage of n — r things, and the numbers of the
two assemblages must necessarily be equal.

"Or and "C7„_r ^^re complementary combinations.

184. Since "C^= ^.^llll.^JIl^..., the number of

combinations will increase with r as long as the last
fractional factor is greater than 1. But this factor is

216 COMBINATIONS.

^"~^"^ ; and while this is >1, the number of com-
r

binations increases.

71 4- 1
.-. 71 — r + l>r, or t<--^ —

And T is to be the integer nearest to but less than
1(71 + 1). Therefore

If 71 is even, the value of r which makes "(7,. greatest is
r = ^7i; and if n is odd, the value of r is ^(ti — 1), or
1(71 + 1); the latter value giving a unit-factor.

Ex. r2a^^l2-ll-10>9^12.11.10.9.8.7.6.5^,^^^^^^^
* 1.2.3.4 1. 2. 3. 4. 5. G. 7.0 ^

12.11.10.9.8.7^^,^
: ^ 1.2.3.4.5.6

n /^ ^ 11 ' 10 ' 9 . 8 . 7 ^ 11 . 10 . 9 . 8 ♦ 7 . 6 ^ -n ^
^ 1.2.3.4.5 1.2.3.4.5.6 ^*

185. In the combinations "(7^, to find the number of
times any particular object, as a, will be present.

If we form "~^(7r_i from all the letters except a, taken
r — 1 together, we can place a with each of these groups,
and we then have all the combinations of n letters r
together containing a.
^ Thus a occurs "~^(7^_i times.

Similarly, ah occurs "^(7,._2 times, etc.

Ex. Out of a- guard of 12 men 5 are drafted for duty each night.
Relatively, how often will A be on duty ? How often will A and B
be together on duty ? How often will A be on duty without B ?
. As ^2^5 = 792, this is the total number of different drafts.,

1. A is on duty "C^ - 330 times out of 792.

2. A and B are'together ^^C^ = 120 times out of 792.

3. .'. A is present without B, 210 times out of 792.

i

BINOMIAL THEOREM. 217

EXERCISE XIII. b.

1. If a = **(73, and b = "Pj, find the relation between a and b.

2. If a = "Or, and b — "-^Pr-i, find the relation between a
and b.

3. At an election there are 10 candidates, of which 4 are to
be elected. If a man may vote for 1 or more up to 4, how many
different votes can he cast ?

4. If "C„_i = "»+i(7„»-i, then m(m + 1)= 2n.

5. Prove that "O- + "CV-i = «+iCr.

6. Prove that w{«0r + «-iCr-i}=(w + r)«(7r.

7. 3 black and 2 white balls are put into a bag ; what is the
chance of drawing 2 black balls at a single drawing of 2 balls ?

THE BINOMIAL THEOREM.

186. The expansion of (l+o;)**, with n a positive
integer, is the simplest form of the binomial theorem, or
binomial formula. The theorem is then generalized and
adapted to any value of n whatever. The simplest case
is first established.

I. w A Positive Integer.

The number of terms in 2a with n letters is "Ci; the
number of terms in 2a6 is "C2 ; and generally the number
in Sa6---r is "(7^.

These statements are self-evident.

Now (x -]- a) {x '\- b) (x -{- c) "- to n factors is

218 BINOMIAL THEOREM.

And making a = 6 = c = -..==l, we obtain

(x + 1)" = ic" -f "Cio;"-^ + "C2a;"-2 + ••• + "C„.

Or, since "(7„ = 1, "(7„.i ="(7i, "(7„_2 = "Oa, etc.

(1 + 0?)" = 1 + "Cjo; + ""C^ + - "CX + - (A)

Also writing the factor values of "Ci, "Cg, etc.,

1 • ^

(^) and (B) are common forms of the binomial the-
orem; and

iC,

r!

the (r4-l)th term from the beginning, is called the
general term.

Knowing the particular value of each coefficient, these
coefficients are often denoted by a single letter with
subscript numbers, and the theorem then becomes

where Cq = 1, Ci = n, Cg = ^^^"~ ^, etc.

Ex. 1. (a + «)« = a^(l + -V = a^(l + Cj - + c.,^ + r-\

= a« + Cia«-ix + Caa"-2x2 + ••• Cra«-*'«^ H

and thus the expansion of any binomial depends upon that of
(1 + X).

Ex. 2. (1 - iK)« = 1 + Ci(- cc) + C2(- a;)2 -f CgC- a)8 + ...
the signs being alternately + and — .

BINOMIAL THEOREM. 219

187. Since (l + x^ = l-\-''CiX-i-"C^'^ -{ h"0„a;^

making ic = 1 gives

2»=l4-"Ci + "(72 + "03+...+"a.
.-. The total number of combinations of n things taken
1 at a time, 2 at a time, and so on to n at a time, is 2" — 1,
Also, the sum of the coefficients of the expansion of
(l + a;)»is2".

Also, since (1— a;)"=l— CiX + CzX^ — c^pc^ -\ , by mak-
ing a; = 1 we have

0 = 1 4- Cg + C4 H (ci + Cg + C5 + •••)•

Or, the sum of the odd coefficients in the expansion of
(l-{-xY is equal to the sum of the even coefficients.
Ex. To find the sum of the squares of the coefficients of (1 +x)".
(1 + x)« = Cq+ CjX + C2X2 ^ (.3^3 + ... + c„x» ;
(x + 1)« = CqX^ 4- c^x'^-^ + C2a;«-2 ^ C3X"-3 -\- ,..Cn.
The coefficient of ic" from the product of the right-hand mem-
bers is

Co2+Ci2 + C22+...C„2.

But the coefficient from the product of the left-hand members is
the coefficient of x» from the expansion of (1 + x)2« ;

that is, 2«(7^ or i?^.

n! w!

.. Co -hCi +C2 + — C„ - ^^^^2

Cor. 1. This last expression on the right is the number of per-
mutations of 2 n articles, when half of them are alike of one kind,
and the other half alike of another kind.

Cor. 2. The coefficient of x"-2 from the right product is

And from the left it is

(^ny.

(n-2)!(w + 2)!
The student is left to generalize this.

220 BINOMIAL THEOREM.

EXERCISE XIII. c.

1. Write the general, or (r -f l)th, term in the expansions —

i. (a + x)". ii. (a - x)«. iii. (1 + x)2».

2. Show that

(a + x)» _ a» x^ , g^-^ ^ x^ , a^-2 . ?^ _!_... 4. «° ^.
n! w! *0! (ri-1)! * H (n-2)! * 2! 0! * n\ .

3. Find the 10th term in the expansion of (1 + xy^.

4. What is the factor which changes the (r + l)th term into the
(r + 2)th term ?

5. Find the value of r that the factor of question 4 may be the
last one greater than unity.

6. Find the greatest coefficient in the expansion of (2 4- 3 x)^.

7. Show that the 6th term has the greatest coefficient in the
expansion of (3 + 2x)i2.

8. Prove that "Cr + "Cr-i = »+^Cr.

9. In the expansion of [ x + - ] show that the coefficient of x** is

Qn-r)liin + r)\
II. n A Negative Integer.

188. Let the expression "(" + l)('^ + ^)-(^^ + ^-^)
be denoted by ""H^.

Then "H, + »+'^,_, = n(n + l)...{n + r-l)

r !

(n+1) (w+2) -(n+r-1) ^(n + l)(n + 2)-(n + r)
(r-iy. r!

= »+'ir, (D)

BINOMIAL THEOREM. 221

Now we know from division that

{1-xy

^1 ^1.2 ^1.2.3 {n-iy.

= l + 'H,x + 'Hox^ + 'B^^ + ... ^jy^.io:"-^ + ...
Therefore let us assume that
1

(I-:.-)"

= l+"iyia?+"iy2^+"^3^'+ •••+"Jy.i»'+ -. (^)

Divide both sides by 1 — a; ; the left-hand member be-
comes r, and the right as follows :

(1 - xy+^

because '^^^ + "+^^,_i = "+W„ from (D) .

.-. __^^ ^^^ = 1 + "-^^J^io? + -+^^2^^ + "^^^3^^

Hence if the expansion is true for any value of n, it is
true for the next greater value. But it is true for n = 2,
and therefore for n = 3, for n = 4:, etc. : that is, it is
generally true. And the expansion, (E), is true.

XT /I \n -I I n(n—l) n(n—l)(n^2) ,
Now {l—xy=l—nx+ ^ ^ ^^ — ToQ '^

222 BINOMIAL THEOREM.

Change n to —n, and this becomes

^ ^ 1.2 1.2.3

= 1 + ".^laJ + "^^2^^ + "^3^^ + •••, agreeing with (jEJ).

And we see that the general form of the binomial
theorem holds good for all integral values of n, positive
or negative.

With n positive, the series terminates when ^~^ "^ ■

r !
becomes zero ; i.e. when r = n + 1. Or the series con-
tains n + 1 terms.

With n negative, however, the series is infinite, as
w(n4-l)(n4-2)... cannot become zero by extending the
number of factors.

189. To interpret "5;.
1

1

1-hx

1

1 — ex

= 1 + 5a; 4. 52^ + 53^ _f. ... + ^r^r ^ ..

= 1 4- ex + c^x^ 4- cV H h e^'x*' + ...

By multiplying n such equations together, the coeffi-
cient of X*", on the right, is the sum of all the homogeneous
terms of r dimensions that can be made out of n letters.
But if we make a = & = c=... = l, the left-hand product
becomes (1 — x) ~", and the coefficient of x"" is ""11^.

BINOMIAL THEOREM. 223

.•. "jET^ = the number of homogeneous terms of r dimen-
sions which can be formed from n letters, and their
powers.

Thus, if n = 4 and r = 2, ^Hr = ^H^ = j^ = 10 ; that is, there

are 10 homogeneous terms ; namely, a^, 6"^, c-^, c2^ ab, ac, ad, be, bd,
and cd.

It is well to notice that if we denote ""Hi by /ij, "^2 by
^2j 6tc.,

(1 + xy = 1 -h Cio; 4- CgO^ + CgiB^ H signs all +.

(1 — a;)~" = 1 + hiX + ^2^ + /igO^ H signs all +.

(1 — xy = 1 — CiX 4- Cgo;^ — CgO^ H signs alternate.

(1 4- a?) ~" = 1 — /ii^; + ^2^ — h^a^ -\ — • • • signs alternate.

EXERCISE XIII. d.

1. Expand (1 — jc)-*, and show that the coefficients are the
sums of the coefficients of (1 — x)-\

2. Expand ^t_^ — in ascending powers of x, and find the

1 + x-\- x'^
coefficient of x" in the expansion.

(1 4. x\*»
l-xj

4. Prove that (l+x)«=2» \ l-/^^ .lli-?+ ^ Yln^V-^- ... \.

( 1+x \i-tx/ y

5. Find the coefficient of x^^^ in ^-^^ ,

(l-x)2

6. Find the coefficient of x" in the expansion of

(1 -2x4-3x2-4x3 + )-«.

7. Show that if n is a positive integer, (5 + 2^Qy is odd in its
integral part.

224 BINOMIAL THEOREM.

Since 52 -(2^6)2 = 1 and 5 + 2^6 > 1, .-. 6 - 2^6 < 1, and
is accordingly a proper fraction.

. •. 5« - Ci 5"-i . 2 V^ + Cj 6"-2 22 . 6 - + ...=/' = proper fraction.
5« + Ci 5^-1 . 2 V6 + C2 5«-2 22 . 6 ++ — = /+/ = an integral
part + a proper fraction.

. •. 2{5« + C2 • 5«-222 . 6 + ...} = / + / + /= an integer.

. *. f + f must = 1 ; and as / + / + / must be even, /must be odd.

8. Show that the coefficient of x'^ in the expansion of (1 + xy^
is double the coefficient of x" in the expansion of (1 + x)2«-i.

9. By the Binomial theorem find 99^.

10. Prove that c^ - 2 Cj + 3 Cj - + ••• + (- l)^(w + l)c„ = 0.

11. Show that »/r^ = «+^-iCV.

III. n A Fraction.

190. With n fractional there are certain difficulties in
the Binomial theorem which we cannot here explain ;
and no very satisfactory proof of the theorem with
n fractional can be given without involving higher con-
siderations than occur in this w^ork.

Several methods, however, will furnish proofs which
are morally sufficient.

The following is Euler's.

(1 + a;)'' is a function of n ; denote it hj fn.
Then (l-\-x)"'=fm,Vind (l+a^)"*+"=/(m+n).

But (1 + x)"^ ' (1 -f xY = (1 + .1')'"^" by the index law.

.-. fm'f)i=f{m-\-n).
Similarly, fm -fn -fp =f{m + 71 -\-p) j
and generally,

fm 'fn 'fp'"k factors =f{m + n -i-p'-'Jc terms).

BINOMIAL THEOREM. 225

h

Now let m = n=p= '*' —

k

and f(^={fh)\ = {l+x)i.

But, fn = l + nx + ^^^^^^x' +

1 • z

h

i-^).

Or (l + a;)Ll+|a;4-^^^^a^ +

And the form of the Binomial theorem holds good for
n fractional.

-1

Cor. If we make k = — 1, f(—h) = (fh)
But

and (//()-'= ^ ^

fh (l+x)"
1

. = (! + «')•

-ft

■• {l + x)

which proves the theorem for negative indices

226 BINOMIAL THEOREM.

EXERCISE XIII. e.

1. To find VT^^, This is (1 - x)*,

and (1 - a;)i = 1 - Jx + Ki^niix^ - i(L—0(isz^y.9 ^ ...
^ ^ ^1-2 1-2.3 ^

= 1 -_ 1 a; - J_x2 - -1^x3 - -i-lil^x* -- ...
^ 2.4 2.4.6 2.4.6.8

2. Write the general term, (r + l)th, of Ex. 1.

3. Expand (I -\- x)^ in ascending powers of x.

4. Find the approximate value of a(l — x) Vl + x^ when sc^
is so small as to be rejected.

5. Expand (14- x)% and find the result when x = 0.

6. Expand ( 1 + - ) , and find the result when x = oo.

(-i)-

7. Find the value of $ 1 compounded every moment for t years
at r % per annum.

1

8. By expanding (1 + x)" and making x = 3 and n = 2, show
that 2 is the limit of the series

2 1.2V2/ 1.2.3V2>/

9. Find the limit of the series to infinity —

l + }-2-J.22 + /^.23-^VV.244--

10. If e = ( 1 + - ) where x = oo, show

that

^21 3! n\

CHAPTER XIV.
Inequalities.

191. When two unequal expressions are compared,
particularly with the purpose of showing that the ex-
pressions are not equal, the whole is called a non-equatlop^
or inequality.

An inequality employs the signs > and < between
its members, and sometimes the signs ^fc, read not equal
to, >, read not greater than, and <, read not less than.

It usually happens that some values of the variables
will change an inequality to an equality, i.e. an identity.

The working rules for inequalities being in some
respects different from those for equations, must be here
established.

I. Let CL>b,

and let all the quantitative symbols denote positive
quantities.

1. Let a = b + /3, and add p to both sides.
.-. a+p = b-^p-\-p, ov a-{-p>b+p.

2. Subtract p from both sides, and

a —p = b—p-\- 13, OT a —p >b—p.

Hence the same quantity may be added to or subtracted
from both members of an inequality ; and hence a term
may be transposed from one member to the other by

227

228 INEQUALITIES.

changing the sign of the transposed term, without affect-
ing the character of the inequality.

3. Subtract both members from p.

Then p — a=p — h — P] ov p — a<p — b, and the
character of the inequality is changed.

Therefore, if both members be subtracted from the "
same quantity, the character of the inequality is reversed.

4. ma = mb + w/5 ; or ma > mb.

m m m mm

Hence, if both members be multiplied or be divided
by the same quantity, the character of the inequality is
unchanged.

/> m mm mB ^„ '^ ^ "^

a b + p b b{b + py a b

Hence, dividing the same quantity by both members
changes the character of the inequality.

7. To multiply or divide both sides by a negative
quantity is equivalent to exchanging the members, and
therefore it reverses the character of the inequality.

II. Let a>b and c>d.

Put a = 6 + A andc = d + S.

8. a-{-c = b-{-d + (3-\-S; ora + c>6 + d.

9. a — c = & — d + /5 — S; from which we cannot infer
whether a — c>b — dov <6 — d.

If /5 > S, a - c> & - d; but if ^ < S, a — c < 6 — d

IKEQUALITIES. 229

Hence, inequalities of the same character may have
corresponding members added; but they do not in
general admit of being subtracted.

192. Inequalities are usually referred to certain stan-
dard forms, or determined by fixed relations.

(1) Eor all values of x and y^ except equality,

a^ + 1/^ > 2 a^i/.

Proof. (x — yY is essentially positive, and .-. >0.
... r^^f_2xy>0,

and x^ + y^':>2xy (A)

Ex. 1, The sum of a number and its reciprocal is greater than 2.

x + - >2,

X

if a;2+i2^2x.l.

And this latter is true {A). /. the former is.

Ex. 2. To show that 1 -{- a^ -\- a'^> ^(a + a^).
If a is negative, this is evidently true, since the left-hand mem-
ber is essentially positive.
Let a be positive.
To prove that 2 + 2a2 + 2a* > Sa + Sa^.

(a — 1) (a^ — 1) = a* — a — a^ + 1 ; and is + when a is +.

.-.1 -\-a^>a + aS.

Also 1 +a2->2a . . . (Ex.1)

and a^ + a^>2 aK

.-. Adding, 2 + 2a'^ + 2a^>Sa + Sa^.

193. (2) (x~ — 2/") (^•"' — 2/"") > 0, if m and n are both
odd or both even positive integers.

230 INEQUALITIES.

Proof. If X and y have the same sign or opposite
signs, both factors have the same sign, and the product
is positive.

Ex. 1. sfi + ^^ x^y + xy^ according as
x^ — x^y + i/^ — xy^ -^ 0,

as (a^ - 2^) (aJ — 2/) ^ 0.

But (x5-2/5)(x-?/)>0.

. •. x^ + t/^ > x^y -\- xyK

EXERCISE XIV

1. If a, ft, c ••• a be any unequal quantities forming a cycle, show
that Sa2 > 2a6.

2. Showthata2 + 3 52^2 6(a + &).

3. Show that a% + ah^>2 a^h'^.

4. Show that {aP' + h^) (a* + h^) > {a^ + &^)2.

5. With three letters, ^a% > 6 ahc.

6. L±^ + ^i«-f «±^>6, unlessa = 6 = c.

ahc

7. Which is the greater —

1. (a2 + 62) (c2 4- ^2) or (ac + 6(^)2 ?
ii. m2 + m or m^ + 1 ?

8. If X is real, x2 - 8x + 22 < 6.

9. Under what circumstances isx + ->or<4?

X

10. An isosceles triangle is greater in area than a scalene
tri.mgle with the same perimeter and base.

CHAPTER XV.

Undetermined Coefficients and their
"Applications.

194. Theorem. If an integral function of aj of n
dimensions is satisfied by more than n different quanti-
ties, it is satisfied by all quantities, or its coefficients are
severally zero.

Let fx = ax"" + 6aj"-^ + cx""'^ -\ sx + t = 0

be satisfied by the n values, a, (3, y** t.

Then /a; = a(a; — a) (x — /?) (a; — y) ... (a? — t) = 0.

Now, if it is satisfied by an (n -\- l)th value z,

fz = a(z -a){z- p) (z - y)"'(z - r) = 0.

But z is different from a, and fi, and y, etc., so that
none of the binomial factors are zero.

.-. a = 0. And rejecting aa;", we can show in like
manner that 6 = 0; thence that c = 0, etc. And the
coefficients being severally zero, the function is satisfied
by all values for x, since it is zero identically.

196. Let

Ax'^+Bx'^-^+Cixf-^-] j-^=aa;"+6a;»-^+caj«-2H \-t.

Then

{A - a)x- + {B- 6)a;~-i + {C- c)x--'-{- ... + ( 2"- 0 = ^•

231

232 UKDETEKMINED COEFFICIENTS.

And if this equation is to be true independently of
the value of x, that is, for all values of x, we must have

A = a, B=b, C=C"'T=L

And this establishes the principle of undetermined
coefficients for functions of finite dimensions.

The statement of the principle is, that if a positive
integral function of x of finite dimensions be zero for all
values of x, the coefficients of the several powers of x
are each equal to zero.

The extension to functions of infinite dimensions will
be established hereafter.

We shall now consider applications of this prolific
principle.

I. Partial Fractions.

196. The sum of the fractions and is

■ . : and with reference to this latter fraction, the
1 — ar

parts which make it up by addition are called its partial
fractions. It is often necessary to separate a fraction
into its partials, it being understood that the denomina-
tors of the partials shall be linear whenever practicable,
but at any rate be less complex than that of the original.

Ex. 1. To separate "*" ^ into its partials.
1 — x'^

Since the denominator is (1 — x)(l -\- x),

.«i.Tr.o S±x _ A ^ B .

l-x2 \-x 1 + x'
where A and B are coefficients to be determined.

PARTIAL FRACTIONS. 233

Then, S-\-x = A(l -i-x)+ B{1 - x).

And as this is to be true for all values of x, we apply the prin-
ciple of undetermined coefficients, which gives

S = A + B, and 1 = A - B,

Whence A = 2, and B=l;

and 3 + x^_J_^. 1

1— X'^ 1 — X 1 -\- X

Ex. 2. To separate into its partials.

^ (X- l)(x-2)(x-3)

Assume ^ _=^_ + ^_+_^.

{x-l)(x-2)(x-S) x-1 x-2 x-S

Then

x^ = A(x — 2)(x-S)-hB(x-l)(x-S) + C(x-l)(x-2).

We might now equate coefficients ; but the following method is
simpler.

Since this equation is to hold for all values of a;,

Make x = 1 ; then 1 =2A^ and ^ = J.
Make x = 2; then 4 = — 5, and 5 = — 4.
Make x = S; then 9 = 2 (7, and (7 = f .
/ x2 1 4.9

(x - l)(aj - 2)(x - 3) 2 (X - 1) X - 2 2 (« - 3)

/v2 x 4- 1

Ex. 3. To separate — into its partials.

(X- l)2(x + 2)

In forming this fraction by addition there may have been a frac-
tion of the form — - — and another of the form , and in

x-l (x- ly

our assumption we make provision for these. Therefore we assume

x^-x-\-l ^ A B C

(x-l)2(xH-2) {x-iy x-l x-{-2

Then x'^ - x + 1= A(x + 2)+ B (x - l)(x + 2) + C (x - ly.

Let x = l ; then 1 = 3^, and A = ^,

234 PARTIAL FRACTIONS.

Substitute J for A, and

x'i-ix + i = B(ix- l)(x + 2)+ C(x - 1)2.
Let x = -2', then 7 = 9 (7, and C = J.

Substitute | for C, and equate the coefficients of x^j which gives

1 = ^ + J, or ^ = f .
x^ - X + 1 _ 1 , 2 7

ix-\y\x + '2) 3(ic-l)2 9(ic-l) 9(a:4-2)

3 'v^ I 3. _ 1

I'x. 4. To separate — i— H- ^ into partials.

X'^ — 1

The denominator is (x — l)(x^ -\- x + 1), and the quadratic fac-
tor is not separable into real factors.

But a proper fraction with a quadratic factor in its denominator
may have a linear factor in its numerator. We make provision
for this by assuming

Sx'^ + x-1 _ ^ ^ Bx + J7

x^-l x-l x^ + x + l

Then, 3x2 + X - 1 = A(x^ + ^ + 1) + (^x + C)(a; - 1),

whence we readily find A = lj B = 2, (7=2.
. 3x2 + x-l_ 1 __^_ 2x + 2

X^ — 1 X — 1 X2 + X + 1

For a fuller discussion of this subject the student is
referred to works on Higher Algebra, and to the Calculus.

EXERCISE XV. a.

1. Separate into partial fractions the following —
ic 4- 2 . ax + b

(x - l)(x - 2) (a - x)(& - x)b

oc+l „ 7x

x2 _ 5x + 6 (2x + 3)(x + 2)2

3x — 2 . ax

(x-l)(x-2)(x-3)* ' a2-x2'

PARTIAL FRACTIONS. 235

197. We shall now extend the principle of undeter-
mined coefficients to the case of an integral function of
X of infinite dimensions.

Theorem. In a positive integral function of x of
infinite dimensions, and arranged in ascending powers,
any term may be made greater than the sum of all that
follow by making x sufficiently small.

Let a -{- bx -{- coc^ -{- daP -i- ex^ -\- fx^ -\ be the function,

and let ca^ be the term chosen.

Also let k be greater than any coefficient following c.

Then kx^{l -{- x -\- x^ -\- '")>dx^ + ex^ -\-fx^ -f •••

Or kx^ ' -^— >dx^-{- ex^ -^fx^ + ...

1 — x

But cx'>kj^'-^— if c>:^^

1—x 1—x

And since — — = 0 when a; = 0, and c and k are con-
1 — x

stants, — — can be made less than c by taking x suffi-

1 —X

ciently small.

.*. cx^ can be made > dx^ + ex'^ +fx^ H

Now let A + Bx-\-Cx^-\ = a + bx + cxr-\ be true

for all values of x. Then

ji^a + {B-b)x + (C-c)x^-\--* = 0

is true for all values of x.

But when x is sufficiently sm^ll, ^ — a is greater than
all that follows, and its sign controls that of the series ;
but the whole series is zero ; therefore A — a = 0,

,', A = a.

236 EXPA]srsTo:tT of functions.

And by striking out A and a as being equal, we prove
in like manner that B=b'j thence C =c, etc.

II. .Expansion of Functions.

198. If a function of x which has but one value for
each value of x be expanded in powers of x, it must take
the form

a -{- bx -\- cx^ -}- dx^ -\

where every exponent is a positive integer.

For if there be a term of the form gx"\ this term will
become infinite when x = 0, and therefore the inde-
pendent term a must be infinite, and the expansion is
impossible. ^

Again, if there be a term of the form hx"", this term
has n values, and therefore the expansion has at least n
values for each value of a;, which is contrary to the
hypothesis.

Ex. 1. To expand

I — X — x:^ -^ x^

Assume ^ "*" ^ — p^ = I + ax -^ bx^ -]- cx^ + dx* + ...
1 — X — x^ -{- x^

Then I + x- Sx^ = 1 + a\x + b

-1 -a

x'^ + c

~b

1 -a

+ 1
And equating coefficients,

x^^d
— c
-b
+ a

x^ + .

l = a — 1, —3 = 5 — a — 1, 0 = c— 6 — a + 1, 0 = d— c — & + a, etc.,
whence a = 2, 6 = 0, c = 1, c? = — 1, e = 0, etc.

And the expansion is

14-2x4- 0x2 4-x3-x* + 0x5...

EXPANSIOlSr OF FUNCTIONS. 237

Ex. 2. To expand Vl + x.

Assume y/\ -\- x = 1 + «x + hx'^ + cx^ + c?x* + •••

Then l+x = l + 2aa: + 25|aj2 + 2c \x^-\-2d »* + ■

aH 2ah\ 2ac

whence a = }, 6 = — J^, c = jV, <? = — yf j, etc.

.-. vrTx = i + ix-ix^ + j\x^-jijx^'''

^ 2-4 2.4.6 2.4.6-8

EXERCISE XV. b.

1. Expand ^(1 +2x + Sx'^ + ix^ + "- wx«-i + •.-).

2. Expand y/(l + x -{- x'^) to x*.

3. Expand -jf^— ^) to x*.

4. Expand ^(1 + x) to x*.

5. Given (1 + a*)" = ^— = 1 + CjX + CjX^ + CgX^ + ..., to

(1 + x) "^
find the coefficients of the expansion of (1 + x)-" in terms of
Ci» C2, c.^, etc., up to the fourth.

6. It y = a^x + a.^x^ + a^x^ + •••, find x in terms of y.
Assume x = Ay + Bif + Cy^ + •" ; write for y, in this assump-
tion, the value given, and equate coefficients.

7. If v = l4-- + -- + — + •••! find X in terms of z, where

/ 1 2! 3!

z = y - 1.

8. If X = ?/ - 2 ?/2 + ?/3, develop ?/ as a function of x.

9. If (a + 7)x + rx2 + ...)2 = (a + 2 6x + 22cx2 +..-), find the
values of a, &, c, etc.

238 SUMMATION OF SERIES.

III. Summation of Series.

199. It will be noticed in Article 163 that the expres-
sion for the nth term, as a function of n, is one dimen-
sion lower than the expression for the corresponding
sum, and this can be shown to be true for all series of
that species.

For suppose aS'„ = a'nP + hn^~^ + cn^~^ -\

Then S,_^= a{n - iy-\- b(n - iy-^+ c{n - iy-'+ ...

And the difference, >S" — /S"~^, is the nth term ; and on
expanding n^ disappears.

The coefficient of n^"* is b — ap — b, or ap, which can-
not be zero unless a or p is zero, both of which suppo-
sitions are contrary to the assumption.

.-. S'' — aS"-^ is of the (p — l)th dimension.

Ex. 1. To find the sum of the series of squares of the natural
numbers, viz., 12 4- 22 + 3*^ + ••• w^.

Smce the nth term is of two dimensions, assume

Sn = an^ + hn^ + en.
Then Sn-\ = a(n - \y + h{n - 1)2 + c(n - 1).

.'. 8n — Sn-^1 = 3an2— (3 a — 2 5)w+a — & + c = ?ith term
= n2.

And equating coefficients,

a = J, 5 = J, c = i.

/. Sn = in^ + in'^ -^ in = "^(71 + l)(2n + 1).
o

Ex. 2. To find the nth term, and the sum of n terms, of the series

1 + 4 + 8 + 14 + 23 + 30 + .-.

SUMMATION OF SERIES. 239

Taking first differences, we have

3 + 4 + 6 + 9 + 13 + ...

and for second differences,

1 + 2 + 3 + 4 + ..., an A. P.

Now as the nth term of an A. P. is linear, the nth term of the
first difference is quadratic, and of the original series is cubic.
Therefore, assume the nth term = a + 6w + c/i^ -}- dn^.

When n = l, l = a+ 6+ c+ d,
n = 2, 4 = a + 2&+ 4c+ 8d
n = 3, 8 = a + 36+ 9c + 27d.
n = 4, 14 = a + 46 + 16c + 64d.

3 = 6 + 3c+ Id.

4 = 6 + 5c + 19d
6 = 6+7c + 37df.

l = 6d /. (? = J.

Thence 1 = 2 c + 12 (^,

2 = 2 c + 18 (?.

Thence c = - J, 6 = %^, a = - 2.

And the nth term = \{n^ - 3 n^ + 20 n - 12).

Next, for the sum, assume

^„ = an + hn^ + cn^ + (^n*.

Sn-i = a(n - 1) + 6(n - 1)2 + cCn - 1)^ + d(n - 1)*.

Then J (n^ - 3 n^ + 20 n - 12) = i^n - Sn-\ = the nth term

= a-6 + c-(2+(2 6-3c + 4d)n + (3c-6(2)n2 + 4 dn^.

And equating coefficients, we get

d = Jj, c = - ^\, 6 = If, and a = —\%,

. Q _ n{n3 - 2 n2 + 35 n- 10}
• " 24

EXERCISE XV. C.

1. If the nth term of the series l + 8x+4x2+5x3+6a^+7x5+.
is of tbe form 1 + ax + &x"^, find the nth term.

240 MISCELLANEOUS.

1 j_ 2 x x'^

2. Develop — , and show what two series it is the

sum of. (l-x)(l + x)-^

(Separate into partial fractions and develop each.)

3. Sum to n terms the series whose nth term is 1 — n + n^.

4. Sum the series whose nth term is | (n^ + n) .

5. Sum to ri terms, 1 . 2 + 3 . 4 + 5 . 6 + ••»

6. Sum to n terms, IJ + 2 + IJ + 0 - 2J ...

7. Sum ton terms, 1.2.3 + 2.3.4 + 3.4.5+ ...

8. Find the series whose nth term is the sum of the natural
numbers from 1 to n.

IV. Miscellaneous.

200. The following are miscellaneous applications of
the principle of undetermined coefficients to problems
which fall under none of the previous heads.

Ex. 1. To find the condition under which ax"-* + 6x + c shall be
a complete square.

Assume ax^ + 6x + c = (px + g)^.

Then expanding (px + 5)2, and equating coefficients,

a = p'^, b = 2pq^ c = q'^.

But (2i)g)2 = 4j)2g2.

.-. 52 — 4:ac
is the required condition.

Ex. 2. To find the condition that the equation x^-\-ax'^-{-bx-\-c=0
may have two equal roots.

This is the condition that x^ + ax^ + 6x + c may have a square
factor.

Assume x^ + ax^ + 6x + c = f x + — ] (x + p)^.

V py

MISCELLANEOUS. 241

Expanding, and equating coefficients,
from which we must determine p.

1st

9*i2 j_ C

2 r)2 I i^ — ^29

p r«p2_26p4-3c = 0,

whence { „ , ^

^24.2c^^. I 3p2_2ap+6 = 0.

p
Eliminating linear p, p^ —

3ac

-62

a^-

35

9c

— ah

Eliminating square p, -^ ' 2ra2 - 3 6)

... 4 (3 ac - 62) ((^2 _ 3 6) = (9 c - a6)2
IS the required relation.

Cor. For the equation x^ -\- hx + c =^ 0^ we get by making
a = 0, 46"^ — 27 c2, as the condition.

Ex. 3. To find the condition that

ax^ 4-62/2 + 2 hxy + 2 (/x + 2./>/ + c (A)

may be the product of two factors, rational in x and y ; and to find
the factors.
Assume
ax""- + by'^ + 2hxy + 2 gx + 2fy + c = (ax + ^ + s\ (x + py + ■

and equate coefficients of xy and x, and we obtain

ap2 _ 2 /ip + 6 = 0, and s'^ — 2 gs + ac = 0.

Whence p = -(h -{■ V/i^ — ab), s = g -^ Vg'^ — ac.
a

Denote Vh^ — ab by H, and ^/g'^ — ac by G, and the factors
become

{ «^ + rz^+ ^ + '^ I ^ + ^^^^"^^ + ^1'

I h + H Jl a g -\-GfJ

or i{ax + (;i - JET)?/ + ^ + G'}{ax + (/i + ^)?/ + ^ - G^},

a

242 MISCELLANEOUS.

As these factors do not contain /, we equate the coefficients of
linear y from the function and from its factored equivalent, and
obtain

2/= 1{(^ ^G){h-^H) + {g- G)(h - IT)} = ? (gh + GH).
a a

And putting for G and H their values, and reducing, we obtain

ahc-\-2fgh-aP-hg'^-ch'^ = () {B)

which expresses the necessary condition.

This very important function, B, is called the dis-
criminant of the function A,

Ex. 4. To find a number such that if 1 be added to it the sum
shall be a square, and if 1 be subtracted from it the difference shall
be a square.

Let X denote the number.

Then ic + 1, and x — 1, and consequently x^ _ |^ are all to be
squares.

Assume x'^ — \ ={x — pY = x"^ — 2px ■\- p\

Then x = l-±^,

2p

and ^+1^(1 +P)^,

2p

which will be a square if 2 p is a square.

Let 2p = s2 ; then p = -, and x = i±-^;

where s may be any quantity whatever.
When s = \,\, 1, 2, 3, 4...

and the problem is thus indeterminate.

EXERCISE XV. d.

1. Find the condition that x'^ — ahx-\- he may be a complete
square.

MISCELLAKEOUS. 243

2. If x^ + ax2 + 6a; + c is a complete cube, show that 27 c = a^,
and 3 & = a^.

3. Find the condition that ax^ -\- bx + c and a^x^ + b^x + Cj
may have a common linear factor.

2 A X 2 B

4. Determine X so that the equation 1 1 = 0 may-
have equal roots in a;. x-^a x x-a

6. Show that 1 +^ + i?^ + i^^+ ... is the square of
12! 3! ^

6. Find the value of m that y — mx — S = 0 may be compati-
ble with y — X — 1 =0 and y — 2x — 2 = 0.

7. Find the value of c in order that 2x^-}-y'^ — 4:xy + 6y + c
may be put into rational factors in x and y, and find the factors.

8. Show that ax^ + 2 hxy + ftz/^ can always be rationally fac-
tored, whatever be the values of a, h, and b ; and find the factors.

9. Find a formula for numbers which put for x make x"^ -\- b a>
complete square.

10. Determine the fraction of form — ^ "^ — , which expands

1 + ex + dx'^ ■

into 1 +3x + 4x2+7^3 + llx*+ 18x^ + 29x6 + ...

11. Find the relation between a and b that (x — a)2 + (x — 6)
may be a square.

12. Express x*-4 x3H-x2+2 x in the form (x2+ax+ 6)2- C^+c)2 ;
and thence show how the corresponding equation can be solved.

13. Put x2 + x?/-2?/2 4-2x + 72/-3 into factors.

14. Find the value of m that 2x2 — 3xy4-2x — y + m may be
put into rational factors in x and y.

15. If ax2 + 6?/2 + 2 hxy -\- 2 gx + 2 fy -\- c is expressible in fac-
tors rational in x and ?/, show that h has two values for given val-
ues of a, 6, /, ^, and c, unless /- = 6c, or g'^ = ac.

CHAPTER XVI.
Elementary Continued Fractions.

201. Take any proper fraction, preferably in its lowest
terms, as ^.

Then,
11^ l^l^l^l^ 1

25 25 2 + A 2 + — 2H — 2+ ^

11 '11 'll 3^2 3^ 1

This expanded result is a continued fraction, and is
often written

1

2 + ^- 1

2

Or, for the purpose of saving space, it is more generally
written

1111

2 + 3 + 1+2*

From the nature of the expansion, it is evident that
every fraction can be expressed as a finite continued
fraction.

Ex ^ = '^ 1 1 1 1.

Ill 2 + 1+3 + 2+4

244

CONTINUED FRACTIONS.

245

202. Problem. To express any vulgar fraction as a
continued fraction.

Proceed as in finding the G. C. M. of the numerator
and denominator of the given fraction, and write the
quotients in order as denominators of the continued
fraction, the numerators being 1.

Ex. 1. To change j\% to a continued fraction.

56

103

11

9

47

1

1

2

5

4

2J

- quotients. .-. — -

56_.
103"

11111

1 + 1 + 5 + 4 + 2*

Ex. 2. To change 3.1416 to a continued fraction.

This is 3 + ^^^1 and we change xVoVo ^^ ^ continued fraction.

1416 I 10000

7
16
11

1416 .

10000 ■

1 Jl i_

7 + 16 + 11'

and 3.1416 = 3 + -

1 1

Ex. 3. To express

X2 + X + 1

a;2 + X + 1 _ 1 a;+ 1

X2 11

7 + 16 + 11
as a continued fraction,

— X . X + 1 _

-^ — = x + ,

5C+ 1 X+ 1

X X

X'^ + X+l 1 + X+ (-1)+ (-X)

203. The quotients obtained by the process of finding
the G. O. M. are called partial quotients.

Let the partial quotients be denoted by aj, ag, a^, etc.,
all the quantities being affected with the sign +.
Then the continued fraction is

ill 1

ai + ag + agH ha„'

246 CONTINUED FRACTIONS.

which in its totality is equal to the fraction from which
it is derived, and which we shall denote by x.

— is the 1st convergent, which denote by Vi ;

— — is the 2d convergent, which denote by Vj 5
tti -f ttg

— — — is the 3d convergent, v^ ;
ai + ag + «3

etc. etc. etc.

Then

— >x, since its denominator a^ is too small.

Vg? or — — <x, since its denominator is too great.

«! + ^2

Vg, or — — — > cc ; ^4 or — — — —<x)

«! + «2 + «3 «! + ^2 + tta + ^4

etc. •••

Thus, all the odd convergents are greater than x, and
all the even convergents are less than x\ so that the
value of X lies between that of any two consecutive con-
vergents, until we reach the last convergent, which is the
value of X itself.

Illustration. — The convergents to \\ (Art. 201) are

Vl = J, ^2 = f , V3 = 1, n=l\ = «.

Now \\ = x = 0.44.

Vj = 0.50, and is too great by 0.06.

^2 = 0.42857 .-. and is too small by 0.02142 ...

Vj = 0.4444 ••• and is too great by 0.0044 •••

CONTINUED FllACTIONS. 247

Thus the consecutive convergents, while being alter-
nately too great and too small, approximate more and
more nearly to the value of a?; and hence the names
convergent and converging fractions.

We thus see that one obvious application of continued
fractions is to find a fraction, with few figures, which
shall be a close approximation in value, to a given
fraction with so many figures as to be unwieldy.

Thus f and Jf are close approximations to f fif.

204. The first convergent Vi = —, and the second,
V2 = ; so that to get the second convergent from

the first we write ai H — ■ for ai in the first.

Similarly, to get v^ from V2 we write a2-\ — for ag in

a

the value of V2; and generally, to get v„^i from v„, we
write a„ H for a^ in the value of i;„.

P P P

Now let Vi = — ^ V2 = -?, ^3 = -?, etc.
Vi ^2 Vs

Then Vi = -^ = —', V2 = -^ =

^ Q3 a3(a2ai + l) + «i a3Q2+Qi'

-^4 =

(^a3 + ^y2 + P,
P4 _ V CI J

' ^^ (-3 + ^)&+Q. ^^^^+«^

etc.

248 CONTINUED FRACTIONS.

We here see that the forms for % and v^ are exactly
alike, the only difference being that the subscripts are
each increased by unity.

To prove that this is always the case.

Assume v,, = ^ = ^^^^-' +^-2.

Qn CtnQn-1 + Qn-2

an + — ]Pn-l+Pn-

Then v . _ ^n+i _ V «n+i.

_ an+l(<^nPn-l + Pn-2) + Pn-l _, ^n+l^n + -Pn--1.

Which shows that the form is true for v^+i if it is true
for Vn' But it is true for v^ and V4, and therefore for
Vsj Vqj etc. ; i.e. it is generally true.

205. The result of the preceding article furnishes a
convenient means of finding all the consecutive con-
vergents, when we have any two consecutive ones, and
the partial quotients.

P P

Taking _?i=l^ —^^ a„+i, we get the (n + l)th convergent

Qn-l Qn

as follows :

Multiply P„ by a„^i and add P„_i, for P„^i ; and mul-
tiply Q„ by a,+i and add Q„_i, for Q„+i.

This operation gives ^^ = ^^+^^^ + Pn-i. ^hich is

correct. ^"+1 <^n+lQn+6n-l

Eor convenience we assume a fictitious convergent,
Vq = ^, and carry out the operation as in the following
example :

CONTINUED FRACTIONS. 249

Ex. Let 2, 1, 3, 1, 2 be partial quotients.

0 1 i ^ 6_U

1 2 1 '^ 1 2

11 14 39

The partial quotients after the first are written between
the lines, and the parts of the corresponding convergents
are written above and below the lines.

Thus the convergents are

Vi = h '^2 = h ^3 = IT* n = T4> and ^5 = if.
206. Taking the convergents of the preceding example,
111 14-1

v,-v,=---= ] V,,-Vo = -- — ^

2 3 2x3' ^ 3 11 3 X 11

4 5 1 ^.^

v« — v.= = : etc.

^ * 11 14 11 X 14 '

Thus the difference between two consecutive conver-
gents is the fraction whose numerator is 1, and whose
denominator is the product of the denominators of the
two convergents. And this difference, taken in regular
order, is alternately positive and negative.

To prove that this is always the case.

But, Art. 204,

= — {■Pn-lQn-2 — Qn-l^n-2)'

Similarly,

- {Pn-lQn-2 - Qn-lPn-2)=Pn-2Qn-S " Q«-2^n-3 = etC,

SO that PnQn-i — QnPn-i cvidcutly has, with the
exception of sign, the same value for all values of n.

250 CONTINUED FRACTIONS.

But when n = 2, P2Q1 — Q2P1 = ^2% — (^A + 1) = — 1 5
and for 71 = 3, P3Q2 - QsA = + 1-

Hence ?;„ — v^-i =

(-1)"-',

Q„Qn-l

Which proves the theorem. And as the difference
between any two consecutive convergents is a fraction
with unity as numerator, every convergent is in its low-
est term, i.e. its parts are prime to one another.

207. As the value of x lies between those of two con-

P

secutive convergents, —~ differs from x in value by less

1 Qn 1

than — -— — , i.e. by less than -— .

QnQn+l Qn

Thus y*i, a convergent to |f , differs from the latter fraction by
less than y^y ; and ^s^, another convergent to the same, differs from
it by less than jj^.

Kow if a„ is relatively large, the quantities — Pn-29
and — . Q„_2 are relatively small, and the whole fraction
differs* but little from ^^.

Qn-l

That is, if a„ is relatively large, the difference between
v^_i and Vn is relatively small, and hence v^-i is a close
approximation to x.

Hence, the last convergent preceding a large partial
quotient is a close approximation to the value of the
fraction.

CONTINUED FRACTIONS. 251

Thus, if the partial quotients be

1, 2, 1, 3, 15, 2,
the convergents are

Vj = 1, 1^2 = 2, v^ = 1, V, = \\, Vg = Iff, vg = a; = f f J ;

and v^, preceding the quotient 15, differs from x by less than

^ or-^.
15 X 229 3435

EXERCISE XVI. a.

1 . Find all the convergents to Jf f .

P a

2. If - be the convergent preceding -, show that Ph differs

q b

from Qa by unity.

Thence form a rule for finding multiples of two numbers prime
to one another, so that such multiples shall differ by a given integer.

3. Find multiples of 23 and 31 that shall differ by 6.

4. Find an improper fraction to express 3.1416 to a near
approximation.

6. Express 1.4142 by a vulgar fraction, each of whose parts are
less than 100.

209. A continued fraction may be non-terminating;
i.e, its partial quotients may be an endless series of
numbers.

The convergents approximate in the same way whether
the fraction is terminating or not, but no convergent,
however high its order, can express exactly the quantity
denoted by the continued fraction. Such a fraction has
in general an incommensurable for its value.

If the partial quotients exist in recurring periods, like
the figures in a circulating decimal, the fraction is a

252 CONTINUED FKACTIONS.

periodic continued fraction, and every such fraction is the
development of a square root or quadratic surd.

An infinite continued fraction, in which the partial
quotients are not periodic, may be the expansion of a
cubic or higher form of surd expression, but, in general,
the equivalent surd expression cannot be found.

Ex. 1. Let the partial quotients be 2, 2, 2 •••, and let x be the
equivalent fraction.

Then a. = l 1 1 ~ '

2 + 2 + 2+ 2-\-x

.'. x2 + 2 X = 1, and ic = V^ - 1-

Ex. 2. Let the partial quotients be 1, 2, 3, 1, 2, 3 •••

, Then x = 1 . = 1±^,

l+_i_ 10 + 3x

2+-i-

S + x

.-. 3a:2 + 8x = 7, and X = K">/37 - 4).

And the method applies to all, however great the
periodic part may be.

210. To expand the square root of a non-square num-
ber into a periodic fraction.

We give the method of operation by means of examples.

Ex. 1. To develop ^1.

Since 2 is the highest integer in the root of 7, we subtract 2 from
^7, and throughout the operation no number greater than 2 is to
be thus subtracted.

3

7 = 2+ V7 -2 = 2 +
multiplying ^/l^ by ^^.

V7 + 2'

V7 + ^^i +

3

CONTINUED FEACTIONS

2

253

^^1^ = 1+- -,

3 V7 + 1

adding 1 to 2 to get an integral quotient, 1, and subtracting 1
from y/1 so as to keep the whole unchanged.

2 2 V7 + 1'

3 ^3 V7 + 2'
V'^ + ^ = 4 + ^7 _ 2, etc.

The partial quotients obtained are 2, 1, 1, 1, 4; and
as we have now to begin again with ^1 — 2, the same
quotients will constantly recur. We notice then that
when a quotient which is double the first one appears,
the period is complete.

^ ^1+1+1+4+

Ex. 2.

VIS = 3 + V13 -3 = 3 +

V13 + 3

a/13 4-3^.^ V13-l^. 3

4 "^4 V13 + 1

V13 + 1^1 1 V^-^-

■ = 1 +

V13 + 2

Vl3 + 2^ V13-1^ 4

3 3 V13 + 1

4 ^4 ^Vl3 + 3
V13 + 3 = 6+ V13 - 3, etc.

.,^13 = 3 + 1 1 1 1 1 ...

254 CONTINUED FRACTIONS.

The convergents to the fractional part are

Vj = 1, ^2 = h ^3 = h v^ = f , etc.
/. the convergents to ^13 are

4, 3}, 3|, 3|, 3ff, etc.

EXERCISE XVI. b.

1. Find the value of 1 + - - - i —

2+3+2+3

2. Find the value of 1 - - - L.

1+2+3+4+1

8. Show that the C. F. whose partial quotients are 1, — 2, 3,
— 4, forming a period, has 3 for its total value, and find the first
10 convergents.

4. Find the value of 2 + - - - - ...

2+4+2+4+

6. Expand ^2 and y^6 into periodic C. F.'s.
6. Expand y'l? and y'19 into periodic C. F.'s.

CHAPTER XVIL
Logarithms and Exponentials.

211. In the expression a^=h, x is called the logarithm
of h to the base a; and this relation is otherwise indi-
cated by writing x = log„&.

The base, a, being some fixed positive number, to
every value of b there is a corresponding value of x.

If these corresponding values be tabulated in opposing
columns, the 6-column is a column of numbers, and the
ic-column is a column of logarithms, and the whole forms
a table of logarithms to the base a.

As will be shown hereafter, the general properties of
logarithms are the same for all bases, and any positive
number, commensurable or incommensurable, may be
taken as a base; but certain numbers offer special ad-
vantages as bases in working with logarithms, and in
calculating them.

As a consequence logarithms are, in practice, taken
to one of two bases ; namely, 10, as being the base of
our numerical system ; and a certain incommensurable,
usually denoted by e, and called the Napierian or natural
base.

Logarithms to the base 10 are decimal or common
logarithms, and those to the base e are Napierian or
natural logarithms.

255

256 GENERAL PROPERTIES OF LOGARITHMS.

GENERAL PROPERTIES OF LOGARITHMS.

212. Let a" = h, and a^ = c.

Then x = log J), and y = log„c.

(1) bc = a''a^ = a'+^,
and log^(bc) = X + y,

or log„ (&c) = log„6 + log„c.

That is, the logarithm of the product of two numbers
is the sum of the logarithms of the numbers.

(2) ^ = ^ = a^-y.

log/-j=a;-2/,

or log„/^- j = log,6 - log„c.

That is, the logarithm of the quotient of two numbers
is the logarithm of the dividend diminished by the loga-
rithm of the divisor.

(3) (a-)" = 6-= a--.

.*. loga(&") = nx = nlogj).

That is, the logarithm of the nth power of a number is
n times the logarithm of the number.

(4) Writing - for n,

n

log.(&»)=-logA

n

EXPONENTIAL EQUATIONS. 257

or the logarithm of the nth root of a number is one-nth
of the logarithm of the number.

These four relations form the working rules of loga-
rithms in their applications to quantity.

213. The results of the preceding article show that
multiplication in numbers corresponds to addition in
logarithms ; division in numbers, to subtraction in loga-
rithms ; the raising of a number to a power, to the mul-
tiplication of a logarithm by a number ; and the extracting
of the root of a number, to the division of a logarithm by
a number.

There are in arithmetic, as confined to numbers, no
known operations which correspond to the multiplication
or division of one logarithm by another, and hence to
the raising of a logarithm to a power, or to the extrac-
tion of its root.

Such operations upon logarithms can correspond only
to some hyper-arithmetical processes. Thus logarithms
not only facilitate the more difficult arithmetical opera-
tions ; they also, by an extension of processes, give rise
to a sort of transcendental arithmetic.

EXPONENTIAL EQUATIONS.

214. An exponential equation is one in which the
variable appears as an exponent.

Thus a' = b, with x variable, is an exponential equation.
The method of solution is obvious ; for taking the
logarithms of both members,

log(a'=) = a^loga = log 6.

loga

258 EXPONENTIAL EQUATIONS.

And the operation which gives x is the transcendental
one which corresponds to the division of one logarithm
by another.

Ex. 1. Given a'' + a-' = 26; to find x.

Multiply by a*,

a^x -2ba' + 1=0.

Whence a' = b± Vb^ - 1, (Art. 126)

and ^^logjb+Vb^^l

log a

Ex. 2. To express the logarithm of \' ^ ' — ^, in terms of the
logarithms of 2, 3, 5, and 7. 2^(216)*

5.9

The given expression reduces to (3i ^ • 7^) h- (2^3 , 56^ . and hence
its logarithm is

fflog3 + 61og7 - 131og2-61og5.

EXERCISE XVII. a.

1. If 3 be taken as a base, what are the logarithms of 9, of 81,
of 729, of 2V, of ^^3 ?

2. If 6 be the base, show that | is the logarithm of 14.697.

3. If J be the base, of what numbers are 1, 2, 0, — -,

the logarithms ? ^

4. Prove that, with any positive base, 1 is the logarithm of the
base, 0 is the logarithm of 1, and — 00 is the logarithm of 0.

5. If logfc a = n loga 6, show that logfe a = ^ ' ^^ , where the
logarithm without a suffix is taken to any base. ^^ ^

6. Solve the exponential equations —

i. 20^ = 100. iv. oc^ = y'^ and x^ = y\

ii. (23)a:.(32)« = 4.9. V. 2a^+ a^^ = a^,

iii. 32*. 53*-^ = 7=^-1112-*. vi. ai.a2.a3...Qjx_yj,

OF THE TABLE OF LOGARITHMS. 259

7. Express the logarithm of (8 V3 • ^12) ^ ( V^ • v^l^) in terms
of the logarithms of 2, 3, and 6.

8. Given log 2 = 0.30103 ; find log 64, log 256, log vl28, log J,
log 25. (Base = 10.)

9. Express in terms of log 2 and log 3 the logarithms of 6, 18,
72, ji^, 0.25, 0.0416.

10. How many terms of the G. P. 1 + § + | + ••• will make

18915 9

11. How long will it take a sum of money to double at 5%
compound interest ?

OP THE TABLE OP LOGARITHMS.

216. In a' = b, iib is greater than a and less than a%
0? is > 1 and < 2 ; i.e. oj = 1 + a proper fraction.

If 6 is > a^ and <a^, a? = 2 + a proper fraction ; etc.

Thus a logarithm consists of an integral part, called
the characteristic, and a fractional part, called the man-
tissa. Either of these may, however, become zero.

Taking 10 as a base, every integral power of 10 con-
sists of 1 followed by ciphers only, and the logarithm of
such power is an integer, or characteristic, being the
index of the power.

Thus log 100 = 2, log 1000 = 3, etc.

For any number between 100 and 1000 the logarithm
is 2 + a decimal ; for a number between 1000 and 10000,
it is 3 + a decimal ; etc.

Hence one convenience of decimal logarithms is that
we know the characteristic at sight, and it is not nec-
essary to tabulate it.

The following rule gives the characteristic for decimal
logarithms :

260 OF THE TABLE OF LOGARITHMS.

Call the units' place of the number zero, and count to
the significant figure farthest upon the left ; the number
of this figure is the characteristic, positive if counted
leftward, and negative if counted rightward.

Thus the characteristic of the logarithm of 0.000074 is

- 5, of 386.50 it is 2, and of 430070 it is 5.

216. The Mantissa. Let n be a number, and let c and
m be the characteristic and mantissa of its logarithm.

Then log n=:zc-\-m.

To divide n by 10* we subtract log lO'' from log n. But
log 10"= = 07; and dividing a number by an integral power
of 10 has no effect other than moving the decimal point.

Therefore log(n-^ 10'') = (c — a?) + m,

and since x is integral, the mantissa is unchanged.

Hence the mantissa of a logarithm to base 10 does not
depend upon the position of the decimal point, but only
upon the arrangement of figures in the number ; so that
the same arrangement always corresponds to the same
mantissa, and vice versa.

The characteristic, on the other hand, is determined
wholly by the position of the decimal point.

Thus the logarithms of 0.0024, 0.24, 240, 24000, etc.,
all have the same mantissa, while the characteristics are

— 3, — 1, 2, and 4 respectively.

217. As the logarithms of integral numbers are mostly
incommensurable, the approximation to their mantissse
is carried to 4, 5, 6, 7, etc. decimal places, thus giving rise
to tables of 4-place, 5-place, 6-place, or 7-place logarithms.

OP THE TABLE OF LOGARITHMS.

261

Portions of a table of 5-place logarithms. A, from number 1780
to 1889 ; B, from number 5700 to 5769 ; and C, from number 7320
to 7429.

N.
178

0

1

2

3

4

5

6 1 7

8

9

D.

25042

066

091

115

139

164

188: 212

237

261

9

285

310

334

358

382

406

431

455

479

503

24

180

527

551

575

600

624

648

672

696

720

744

I

768

792

816

840

864

888

912

935

959

983

2

26007

031

055

079

102

126

150

174

198

221

3

245

269

293

316

340

364

387

411

435

458

4

482

505

529

553

576

600

623

647

670

694

5

717

741

764

788

811

834

858

881

905

928

6

951

975

998

021

045

068

091

114

138

161

23

7

27184

207

231

253

277

300

323

346

370

393

8

416

439

462

485

508

531

554

577

600

'"'.

570

75587

595

603

610

B.

618

626

633

641

648

656

I

663

671

679

686

694

702

709

717

724

732

2

740

747

755

762

770

778

785

793

800

808

3

815

823

831

838

846

853

861

868

876

884

8

4

891

899

906

914

921

929

937

944

952

959

5

967

974

982

989

997

005

012

020

027

035

6

76042

050

057

065

072

080

087

095

103

no

732

.86451

457

463

469

C.

475

481

487

493

499

504

3

510

516

522

528

534

540

546

552

558

564

4

570

576

581

587

593

599

605

611

617

623

5

629

635

641

646

652

658

664

670

676

682

6

688

694

700

705

711

717

723

729

735

741

7

747

753

759

764

770

776

782

788

794

800

6

8

806

812

817

823

829

835

841

847

853

859

9

864

870

876

882

888

894

900

906

911

917

740

923

929

935

941

947

953

958

964

970

976

I

982

988

994

999

005

on

017

023

029

035

2

87040

046

052

058

064

070

075

081

087

093

r24

2

5

7

10

12

14

17

19

22

P. \ g

2

5

7

9

II

14

16

18

21

I

2

2

3

4

5

6

6

7

I 6

I

I

2

2

3

4

4

5

5

262 OF THE TABLE OF LOGARITHMS.

The larger tables are mostly 7-place, but 5-place loga-
rithms are sufficiently accurate for the majority of arith-
metical applications.

We give on page 261, and merely for purposes of illus-
tration, portions of a 5-place table, in which, as is usual,
only mantissse are registered.

218. The working a table of logarithms consists in two
operations inverse to one another ; namely,

(a) to find the mantissa corresponding to a given
arrangement of figures in a number, and

(h) to find the arrangement corresponding to a given
mantissa.

(a)

A complete 5-place table gives the mantissse for every
arrangement of 4 figures from 1000 to 9999 ; the three
right-hand figures being taken from column N, and the
fourth from the horizontal line at the top of the table.

Thus, for 1854 the mantissa is 26811 ; for 1864 it is
27045, the last three figures 045 being in distinctive type
to show that the first two figures of the mantissa are to
be taken from the first column and the line below, being
27 instead of 26.

Ciphers occurring before or after an arrangement do
not affect the mantissa.

Thus, 23, 2300, 23000, .023, etc., have the same man-
tissa.

Ex. 1. To find the mantissa of 18347.

Mantissa for 18340 is 26340.
Mantissa for 18350 is 26364.
Difference 10 24.

OF THE TABLE OF LOGARITHMS. 263

Thus each unit between 18340 and 18350 adds f ^ to the man-
tissa, and hence 7 adds 7 x f J, or 17 nearly.

.-. 26340 + 17 = 26357 is the mantissa required.

The column marked D (differences) and the row at the bottom
marked P (proportional parts) are intended to facilitate this
operation.

Thus, for 1834 we find D to be 24, and in line with 24, in the
row P, we have 17 in the column having 7 at the top. This quan-
tity, 17, is to be added to the mantissa of 1834 to give the mantissa
of 18347.

(&)

Ex. 2. To find the arrangement corresponding to the mantissa
26845.

The tabular mantissa next below this is 26834, and the cor-
responding arrangement is 1855.

The excess of 26845 is 11, and D being 23, we find in line with
23, in row P, that 11 is in the column having 5 at the top. Then 5
is to be attached to 1855, giving 18555 as the arrangement cor-
responding to 26845.

219. It must be remembered that the mantissa is
always positive, while the characteristic is negative for
numbers less than 1, zero for numbers from 1 to 10, and
positive for numbers above 10.

To mark the negative characteristic the minus sign is
written above the characteristic instead of before it.

Ex. 1. To find the value of (1.8471)*^.

logl.8471 = 0.26649

7

log (1.8471)7 = 1.86543
/. (1.8471)7 = 73.355...

Ex. 2. To find the value of (18.71)i

logl8.71 = 1.27207

264 OF THE TABLE OF LOGARITHMS.

Divide by 5. log (18. 71)* = 0.25441

/. (18.71)^ = 1.7964...

Ex. 3. To find the value of (0.185)^.

log (0.185) =1.26717
7

.-. log (0.185)7 = 6.87019

.-. (0.185)7 = 0.0000074162 ...

Ex. 4. To find the value of (0.001836)TT.
log (0.001836) =3.26387
log (0.001836)5= 14.31935

= 22 + 8.31935
.-. log (0.001836)TT ^ 2.75631

and (0.001836)A = 0.057056 ...

Notice that to divide the negative characteristic, 14,
by 11, we make it evenly divisible by subtracting 8 from
it and adding 8 to the mantissa, so as to keep the whole
unchanged.

EXERCISE XVII. b.

(All the exercises here given can be worked by means of the
portions of logarithmic table given.)

1. Find the continued product of 1.783, 1.791, and 1.799.

2. Find the value of (18.43 x 18.65 x 1.876 x 5736) -- (1854
X 186.6 X 5766).

3. Find the value of (0.1866)^ x (7.365)^

4. Find the value of (1.8337)3-3037.

6. Given that 17.80 x 17.977 = 320, to find the logarithm of 5.
6. To what power must 74 be raised to give 57 ?

KAPIEHIAN BASE, AND EXPONENTIAL SERIES. 265

NAPIERIAN BASE, AND EXPONENTIAL SERIES.

220. Definition. The quantity which we have denoted
by e, and called the Napierian base in Art. 211, is the

limiting value of (1 + 7i)" as n approaches the value zero.

By the Binomial theorem

(i + ^0" = i + --^+-Y:2^^+^'nT2T3 — ''^'

^1 I 1 I ^(^-^) I Ul-^0(l-2r0 _
1.2 ^ 1.2.3

= i + i + J_+ _!_ + ..

1.2^1.2.3 '
when 71 = 0.

By adding a sufficient number of terms, we find for
the approximate value of e,

e = 2.7182818 ... to 7 decimals.

This peculiar incommensurable quantity is one of the
most important constants in mathematics.

221. From the definition of e in Art. 220,

1
e' = the limit of ^1 + n)"}*, as n approaches zero.

266 NAPIERIAN BASE, AND EXPONENTIAL SERIES.

But

i(i+n)V=(i+^)"

xfx _^\ X

1-^-^

1.2 1.2.3

and when n = 0,

^" = ' + f! + S + S+l] + - • • (^>

Any power of e is got in the form of a series, by writ-
ing the index of the power in the place of x in the series
for e'. Thus,

- = 6-^ = 1-1 + --- + --+ -
e 2! 3! 4!

and Ve = e* = 1 + - + -i- + -^ + -^ + • • •

^ 2 2 ! 22 3 ! 2^ 4 ! 2*

222. Let a = e\ Then c = log^ a ; or, denoting, in
future, Napierian logarithms by the single italic I fol-
lowed or not by a point, c = l - a.

And a'=e«=H-cx + ~ + ^+-

This last series is called the exponential series ; and it
expresses any power (x) of a given number (a) in terms
of the exponent and the Napierian logarithm of the
number.

NAPIERIAN BASE, AND EXPONENTIAL SERIES. 267
Coi\ Making x = l,

and this series which expresses a number (a) in terms
of its Napierian logarithm is sometimes called the anti-
logarithmic series.

EXERCISE XVII. c.

1. Show that e = limit of ( 1 + - j as n approaches oo.

90 92 94

2. Prove that (e2 - 1)2 - 8e2 = |^ + ^ + ^+ -

1

3. If X is positive, then e > x'.

4. The series — + — ^— ^ — + ^'^'^, ^ + ... is the ex-

1.2 1.2.3.4 1.2.3.4.6.6

pansion of (^^e — 1).

5. Find the sum of e + e-i.

6. Show that I (e'' + e-«) =1-^ + ^-^ + ...

2! 4! 6!

7. Show that — (e'* - e-'*) =x- — + — - — +

2i ^ 3! 5! 7!

8. Expand ^

e*-l
Assume the expansion to be a + 5ic + cx^ + ...

Then a; = (a + 6x + cx2 + ...) ^x + ^ + ^ 4- .-.]

Distribute and equate coefficients of like powers of a, and find
a, &, c, etc.

9. Show that l^ + ?i + §!. + ^+... =2e.
1 ! 2 ! 3 ! 4 !

10. Find the value of i^ + — + — + — 4- -

11 21314!

268 LOGARlTHxMIC SERIES.

LOGARITHMIC SERIES.

223. In the exponential series (O, Art. 222) the ]^a-
pierian logarithm of a is the coefficient of linear x in the
expansion of a^ And as a; is arbitrary it follows from
the principle of undetermined coefficients that if we
expand a^ in ascending powers of x, by any means, the
coefficient of linear x in the expansion will still be the
Napierian logarithm, of a.

But a' = {l + a-iy

1.2.3 ^ ^

by the Binomial Theorem.

And picking out the terms which form the coefficient
of linear x, we have

Z . a = (a - 1) - i(a - l)^ + -i(a - 1)^- + ... (E)

which gives Z . a in terms of the number less by unity.
Writing 1 -{-x for a,

Z.(l4-aj) = a^-ia^ + iaj3-^aj^ + (F)

which is the logarithmic series.

224. Writing 1 for a; in i^ gives a series for I -2 ; but
this series is so slowly convergent (see limits of a series.
Chap. Xyill.) as to be of no practical utility in compu-
tations.

We transform the logarithmic series as follows :

LOGARITHMIC SERIES. 269

Til (F) write — x for x, and we get

6 • ( X — X) — ~~ X — -n" X "~~" -5- 3/ — -T U/ ~"~ * * * >

and subtracting this latter series from (F),

l.l±1^2{x + ^x' + lx^ + ...) ((?)

1 — X

Now make x = ; then "*" ' = , and this

-, .^. . 2z — l 1 — x z — 1

reduces (O) to

l.z=l(z-l)

\2z-l 3 {2z-iy 5 (2z-iy I ^ ^

This makes l-z depend upon I (z — 1), and a function
of z which makes up the difference between the two
logarithms.

Ex. 1. Let z = 2. Then since M = 0,

^. 2 = 2(1 + 1. -1+1. I+.4
l3 3 33^5 35^ /

= 0.69315... to five places.

Ex. 2. Let z = 6. Then since Z . 4 = 2 ? . 2,

Z.6 = 2Z.2 + 2/l + l.i + l.i + ...\
193 93 5 95 /

= 1.60944 ... to five places.
Ex. 3. Z . 10 = Z . 2 + Z . 5 = 2.30259 to five places.

225. The series now obtained furnishes a practical
method for computing logarithms to the base e.

270 LOGARITHMIC SERIES.

Now let a' = 6, and take the Napierian logarithms of
both members of the equation ; then

ha' = xl'a = l'b,
and X = log J).

.-. log.& = -^.Z.&.
Z-a

Hence is a multiplier which changes I • b into log„5.

ha

And a being given, T a is also given and constant, being
the Napierian logarithm of the given base.

The multiplier, {l-a)~\ is called the modulus of the
system of logarithms having a as base.

The modulus for base 10, or (MO)-^ is 0.43429448...
to 8 decimals.

Thus the Napierian logarithm of any number is
changed into the decimal logarithm of the same number
by being multiplied by 0.43429448 •«. ; and the decimal
logarithm of any number is changed into the Napierian
logarithm, by being divided by 0.43429448 ... or by being
multiplied by the reciprocal of this quantity, namely,

2.30258509..., which is Z.IO.

The logarithms of any two systems are thus connected
by a constant multiplier, the modulus of one system
Avith respect to the other.

Napierian logarithms are most convenient in analysis,
and decimal logarithms in practical applications, and the
change from one system to the other is easily effected.

LOGAKITHMIC SERIES. 271

EXERCISE XVII. d.

1. Find X from the equation a(h + 1) = I • (a + ic).

2. Show that a; = e«-* = a^« = a^°^«"'.

1 1 1

3. li y = e^-^-", z= ei-«-% then x= e^-^-y,

4. Prove that Z.a-?..x= ^^^ -^f ^:^YV-V^^^V- + •••
6. Prove that

ii{(x -V)-\{x- \y + K^ - 1)' - •••}

^x^-\ i/x^-i\2 i/x^-iy
x« 2V x» y 3V X" y

l+ae 1-x 2 4 S

6. Show that ?.{(! + a:) 2 . (i _ x) 2 } ^z -^ + ^- + ^- + ...

^^ ^ ^ ^ ^ 1-2 3.4 5.6

7. Prove that the ratio log a : log &, where a and ?> are given
numbers, is the same for all systems.

8. The modulus which changes from base 10 to base 3 is
log 10 : log 3, taken in any system.

9. Show that log,o9 = 1 - m fi - i . — +-. — - + -^

^^ Vio 2 102 3 103 ^ y

where m is the modulus to base 10.

10. Show that the logarithm of a number cannot be developed
in terms of the number itself.

226. The exponential and the logarithmic series can
be obtained by other methods besides the ones already
employed. We give some of these as examples and
exercises.

Ex. 1. Assume a^ — \^-h^x-\- \x^ + h.^x'^ + ...

Then a^=\ + 5i(2x) + &2(2^)2 + &3(2a;)3 + "• (1)

272 LOGARITHMIC SERIES.

Equate coefficients of like powers of x in the two expressions
for d^.

Ex. 2. In Ex. 1, &i is indeterminate ; what does it mean?

Ex. 3. How do we know that the expansion of a^ must begin
with 1?

Ex. 4. If a = e*S show from the result of Ex. 1 that

Ex. 6. Assume a^ = \ -\- h^x -\- h.^x'^ + ... hnX"" + •••

Then ay = I -\- h^y -\- h.^y'^ -\ 6„?/" + ...

And a^ - av = h^{x -y)^- h^ipcP' - y'^) + — 6„(x" - ?/«) + ... (3;
But w"- aM = ayia^'-y - 1)

= ay{b,{x-y) + h^{x - ?/)2+ ••. 6„(x - yy^- ...} (4)

Make (3) = (4) as they are equivalents; divide throughout by
X — y, which is a factor ; put y = X] and equate coefficients of x'^.

Then 6„^i = _k^.6

Erom this relation obtain the coefficients in terms of 5^, which
is indeterminate.

Ex. 6. In Ex. 6, why is it necessary to divide hjx — y before
making y = x?

Ex. 7. If e« = 6, cc = Z . 6.

But 5 _1 = 5C + —+- + . ..

2! 3!

Assume x = A(b - l)-\-B(b - 1)^ + C(b - ly + -" (5)

In (5) put the value of 5 — 1 taken from the preceding line, and
equate coefficients of like powers of x. Then find A, B, C, etc.
This gives L 6 in terms of 6 — 1.

LOGARITHMIC SERIES. 273

Ex. 8. Starting from a* = 6, proceed as in Ex. 7 to find loga b
in terms of b'— 1. What is the meaning of the indeterminate co-
efficient in the result ?

Ex. 9. Assume log (1 + x)= ax+ bx'^ + cx^ + •..
Then n being an arbitrary quantity,

n\og(l+x')=n{ax+bx^ + cx^ + ''-) . • . (6)
But n log (1 + x) = log (1 + x)«

= log (1 + «(7iX + "02^2 +...)
= a(^C,x + »a,x^ + ...)+ bi^C^x + ^a,x^ + "-y + - . (7)
by taking the expression ^C^x + ^C^x^ + ••• as a variable.

(6) and (7) are equivalents. Make /i = 0, and equate coefficients
of like powers of x.

The result contains the indeterminate a. What is this, and
what is the effect of making a = l?

Ex. 10. In the assumption of Ex. 8, how do we know that the
first term of the expansion must contain x ?

CHAPTER XVIII.
Of Series.

227. Series are too varied in character to be rigidly-
classified, but the greater number of them have a relation
to the geometric series, or to the arithmetic series, or to
both.

In series related to the geometric, any term is con-
nected with one or more of the preceding terms by
constant multipliers, or by multipliers which vary with
the number of the term in the series.

Thus in 1 + 3 + 7 + 15 + 31 + 63 + ...

63 = 3 X 31 - 2 X 15, 31 = 3 x 15 - 2 x 7, etc..

with constant multipliers, 3 and — 2.

In i + £4-^ + ^%...

1! 2! 3!

the wth term is got from the (w — l)th term by multiplying by -,
and the multiplier is a function of the position of the term in the

228. The nth term of a series is such a function of n
that after some particular term, usually near the begin-
ning of the series, any term is got by writing the number
of the term for n in the function of n.

Usually, however, when the series contains a variable,
X, in ascending powers, the absolute term is not con-
274

BECURRING SERIES. 275

sidered in the counting, so that the term counted as the
nth is the (n + l)th from the beginning. This usage
makes the nth. term contain a?" ; and in purely numerical
series a unit variable is frequently introduced for this
and other purposes.

Thus (2'»+i — l)x" is the nth. term of the series

RECURRING SERIES.

229. A recurring series is generally the expanded form
of a proper fraction, and is analogous to the circulating
decimal in arithmetic.

Thus — — =:l-\-x + x^ + x^-] — , a geometric series.

1 —X

h±ll =l + 5a;+13«2 + 29x3 + 61cc*+...,

1 — 3 X + 2 ic2

a recurring series of the second order.

1j=l^ =l_2a + 4x2-7x3 + 13a:*-23«5_f-...

1 + X-2X2-X3 '

a recurring series of the 'third order.

etc. etc. etc.

In this relation the fraction which by its expansion
produces the series, is called the Generating Function
(G) of the series ; and the denominator of the fraction
is the Scale of Relation (E).

When the E is binomial and linear, the series is
geometric, and is of the first order ; and generally, the
order of the recurring series is the same as the dimen-
sions of the K.

276

RECITERIKG SERIES.

230. Problem. G-iven a recurring series and its order,
to find its E; and its G-.

Take the recurring series of the 2d order given above,
viz. :

Assume

N

= l + 5aj+13a;2+29a^+61a;*+.

1 -\- ax -\- bo(^

Multiply by the denominator. Then

JV^= 1 + 5
a

ic + 13

aj2 + 29

0^3 + 61

5a

13 a

29 a

b

5b

13 b

x^ + •

But N cannot be higher than linear in x ; and there-
fore all the coefficients after linear x must vanish.

That is,

13 -}_ 5a+ & = 0;

29 + 13a + 5& = 0;
61 + 29a + 136 = 0, etc.

The first two equations give a = — 3, 5 = 2, and these
values satis.fy the third equation.

This compatibility shows that the series is of the
second order and the R is quadratic.

If the third equation were not satisfied, the E, would
be of higher dimensions, and the series would be of a
higher order.

TheEis l-3a; + 2a;2.

By putting — 3 for a in the terms up to the linear
inclusive, we get JV= 1 + 2a;.

l + 2a;

G =

l_3a; + 2ic2

BECURRING SERIES. 277

231. We see from the foregoing article, that if z be
the order of the series, it requires z terms to find the
E, and z terms to find N. Hence the number of
terms required to determine completely a recurring
series of any order is twice the number of the order;
and if z be the order, 2; — 1 terms at most, counting from
the beginning, may not follow the law of the rest of the
series.

232. Problem. To find the 7ith term of a recurring

series.

Taking the series of Art. 230, its R factors into
{l — x)(l—2x)j and going to partial fractions,

l + 2x ^4 3

l^^x-^2x'~l-2x 1-x

Then expanding these partials, we get the equivalent
geometric series :

= 4(l + 2a;4-2V+...2"a;"+...).
= 3(1 + aj + a;2 4-...aj« + ...).

l-2a;
3

And confining ourselves to the coefficient of a;",
(2«+2 _ 3)aj^ = the nth term.

This shows that the terms of a recurring series are, in
general, the algebraic sums of the corresponding terms
of two or more geometric series.

And in finding the nth term we need not write out the
geometric series ; for if

z , -^ -^, z , etc.,

\ — ax 1 — f3x 1 — yx

278 BECURRING SERIES,

be the partial fractions,

is the 72th term.

233. Our ability to find the nth term depends upon
our ability to factor the E-. If the R rises only to a
quadratic, or is separable into factors none of which are
higher than quadratic, it is possible to find the rith term ;
but when the linear factors are irrational, the operation
may be laborious.

We give one example.

Ex. The series 1 - 4 ic + 19 a^^ - 17 x^ + 265 x*

has 1 + 4 a; + ic2 as its R, and N = 1 — x.

1 + 4x + X2 =: {1 + (2 + ^S)X}{1 + (2 - ^S)x],

Assuming — ^-^^^ — = ^ + - ^

l + 4a: + ic2 i+(2-f^3)x l-i-(2-y/S)x
we obtain ^ =: J (1 + -y/S), B=i(l - ^S).
Thence the nth term is
(-)** H(2 + V3)" + (2 - V^)" + V[(2 + V^)" - (2 - V3)"]}aJ«.

EXERCISE XVIII. a.

1. Find the R and the G of 1 + 2 x + 3 cc2 + 4^3 + ..., it being
of the second order.

2. Eind the R and G of 1 + 3 x + 4 x^ + 5 x^ + •••, a recurring
series of the second order.

3. Find the next two terms of the series of Ex. 2.

1+ 2 X - x2

4. Develop the series whose G is

(l-x)(l + x)2

DIFFEKENCE SERIES. 279

5. The terms 1 + x — 2 x^ + 3 ic^ are the first four terms of a
recurring series of the second order, and also of one of the third
order. Find the G's of the series, and the 5th term in each.

6. The first four terms of a recurring series of the third order
are 1 — x + 2x^ — 2x^. Find an expression for the nth term, and
thence find the 99th term.

7. Find the nth term of the series of Ex. 4.

8. Find the nth term of the series of the second order,

1 + 5x+ 19x2 + 65x3+ ...

9. If there be n terms given, N may contain any number of

terms from 1 to - — 1 if w is even, and from 1 to ^ ~ if n is odd.
Z 2

10. If n terms be given, they may belong to a series of any
order from w — 1 to - + 1 if w is even, and from n — 1 to ILtA
if n is odd. ^ 2

DIFFERENCE SERIES.

234. These have an alliance with arithmetic series.
Take the series

2 + 3 + 6 + 12 + 22 + 37 + —
1st differences 1 + 3 + 6 + 10 + 15 +
2d « 2 + 3 + 4 + 5 +

3d " 1 + 1 + 1 +

4th " 0 + 0 +

By subtracting each term from the following, we obtain
a set of series similar to the first, but of successively
lower orders, called the series of 1st differences or Aj-
series, the series of 2d differences or Aj-series, etc.

280 DIFFERENCE SERIES.

In the example given, the Ag-series is arithmetic, and
the A4-series vanishes ; and for any true difference series
some A-series is arithmetic, and the second one there-
after vanishes.

Thus in the series of cubes 1, 8, 27, 64, 125, etc., the
Ag-series is arithmetic. In 1^, 2^, 3^, etc., the A4-series
is arithmetic, etc.

Evidently if any general relation exists between the
original series and its A^-series, a similar relation must
exist between each two consecutive A-series.

235. Let Uq + Ui-\- U2-\- •*- be a difference series in
which the suffix serves the purpose of the exponent of a
variable.

Then, -w-series t^o + ^^i + '^2 + ^s + ^4 -f- •••
Ai-series Ai A/ A/' A^'"
Ag-series Ag Ag' A2"

Ag-series A3 A3'

A4-series A^

Now, Ui = Uo-{-Ai, A/ = Ai4-A^

A2' = A2 + ^3? ^3' = A3 + A4, etc.
Again, U2 = Ui-\- A/ = 2^0 + 2 Aj -[- Ag.
.-. Ai" = Ai + 2A2 + A3.
A2" = A2 + 2A3 + A4.
Again, ^(3 = U2-{- A/' = -i^o + 3 Ai + 3 Ag + A3.

.-. A/" = Ai + 3A2 + 3A3 + A4.
Again, u^=zUs + Aj'" = i^o + 4 Ai + 6 Ag + 4 A3 + A4.

And obviously, from the mode of formation of the
terms,

^„ = ^0 + "CiAi + "C2A2 + "C3A3 + ...

INTERPOLATION.

281

which is an expression for the nth. term of a difference
series.

Ex. To find the nth term of 2 + 3 + 6 + 12 + 22 + •••
u, = 2, \ = 1, A, = 2, A, = l, A, = 0.

/. u^ = 2 -\-n + n{n - 1) + | n(w - 1) (n - 2)

INTERPOLATION.

236. Take the difference series whose first 5 terms are

7, 2, 1, 4, 11.
The expression for the nth. term is

u^ = 7 -^n(2n — T).
Eegarding ti as a variable of the function whose value
is denoted by u^y we draw, as in the figure, the graph (G)
of the function, in which 7i takes the place of x^ and u^ of y

Then

Oa:

^ = 7, lb = u, = 2,

2c = U2 = l, 3 d = % = 4, etc.
And the points a, 6, c, d, etc., represent, by their

4

V

ordinates, Oa, lb, etc., the
terms of the series.

Hence we may define a
series as a set of point values
of a function of a variable,
corresponding to equidis-
tant values of the variable;
the equidistant values being
generally regarded in all
series as the consecutive in-
tegers from zero upwards, or the numbers of the succes-
sive term.

Xm'

Li

I 1 m 2 ^ 3

282 INTERPOLATION.

Now n = I gives the point value W corresponding to
the middle point of 01; w = f gives mm' corresponding
to the middle point of 1 2 ; etc. And the points 0, Z,
1, m, 2, etc., being equidistant, we have a, Z', 6, m\ c, etc.,
as terms of a new series, such that every alternate term,
counting from the first, belongs to the original series.

We are then said to have interpolated single mean
terms in the original series.

Our unit on the ic-axis being arbitrary, we may make
0 1 the unit by writing ^ n for n in the function, and leav-
ing u^ unchanged.

This gives ^^^ = 7 4. ^ (n - 7)

for the new series ; and the series itself is

7, 4, 2, 1, 1, 2, 4, 7, 11 ...

So that IV — 4, mm^ = 1, nn' = 2, etc.

In a similar manner by writing - for n in the nth

o

term of the original series, we obtain the nth term of a
series in which two mean terms are interpolated between
each two consecutive terms of the given series, etc.

In like manner, if any real value whatever be given to
n, the resulting value of u^ is the ordinate corresponding
to that particular value of n.

237. The least consideration will show that interpola-
tions can be made accurately whenever the nth term can
be accurately expressed, and that the last condition is
satisfied for any series in which an order of differences
becomps zero. Also, that if no order of differences is
zero, the ?ith term can be expressed only approximately,

INTERPOLATION. 283

and the interpolated terms will be only approximately
eorrect.

As an illustration of the latter statement consider a
t ible, such as that of logarithms, for example.

The logarithmic series is a function of a variable n,
and the tabulated logarithms are the point values of this
function corresponding to consecutive equidistant values
of the variable, as 30, 31, 32, etc., say.

These logarithms do not form a proper difference series,
and the rith term cannot be exactly expressed.

Thus

log 30 = 1.47712

Ai

1424
log 31 = 1.49136 ^ - 45

1379
log 32 = 1.50515 -43

log 33 = 1,51851

1336

But A2 is small as compared with A^, and nearly con-
stant, so that

u, = 1.47712 + 1424 n - -V-n(n - 1)

is approximately true for small values of n, as from 0 to
1 ; i.e. the result will be practically correct for the loga-
rithm of any number lying between 30 and 31.

Thus log 30.3 = 1.47712 + -j-% . 1424 + S^- • y'lr • tV

= 1.47712 +427 + 5

= 1.41844,

which is true to the last figure.

This example shows that in a case like the present
proportional parts are not always sufficient.

284 SUMMATION OF SERIES.

EXERCISE XVIII. b.

1. Find the orders of differences of the difference series, 50, 52,
50, 45, 38, 30, etc.

2. Find the expression for the nth term of Ex. 1.

3. Find the nth. term of the difference series of which the first
four terms are 1 + 7 + 11 + 13.

4. Find the nth term of li, 2, 3, 4|, 8 ...

5. Interpolate mean terms in the series 4, 1, 2, 7.

6. If a, 5, c be three consecutive terms of a difference series,
and m and n be mean terms between a, b and 6, c, respectively,
show that m = ^(3 a + 6 6 — c), and n = J(3 c + 6h — a).

7. If in Ex. 6, m, n be two interpolated mean terms between
a and 6, and j?, q be two between b and c, show that, upon the
supposition that A3 = 0, m = i(5 a+56 — c), n = |(2a + 8& — c),
p =z 1 (2 c + 8 & - a), and g' = i(5 c + 5 6 - a).

8. The expectation of life at 10 years of age is 48.8, at 20 it is
41.5, at 30 it is 34.3, and at 40 it is 27.6, What is it at 15 ? at 25 ?

9. At 9 o'clock the distance of a star from the moon is 42', at
10 it is 19', and at 11 it is — 3'. How were they situated at
10 h. 52 m. ?

10. Given sin 240 =0.40674, sin 25°= 0.42262, sin 26°= 0.43837,
sin 27°= 0.45399; find sin 24° 25'.

SUMMATION OF SERIES.

238. The Sum of a Series is a somewhat indefinite
expression, as the following statements will show.

(1) If the series be numerical, the sum of its first n
terms is intelligible, whether a general expression for
such a sum can be found or not.

Thus the sum of n terms of the series 1-1-2 + 3+ ••• is Jw(ri+1),
for all values of n ; and the sum of any given number of terms of

SUMMATION OF SERIES. 285

the series l + i + i + i+--- may be found, although no general
expression for the sum of n terms has ever been obtained.

(2) In many numerical series the sum of the series to
infinity may be given as a finite expression ; but as we
cannot properly speak of summing an infinite number of
terms, this expression is more correctly spoken of as the
limit of the series, i.e. the value towards which the sum
of the terms approaches, as more and more of the terms
are included in the summation, and to which the sum
may be made to approach as near as we please.

Thus 2 is the limit of 1 + J + i + J H ad inf.

(3) If a series contains a variable in ascending or
descending powers, what is called the sum is in reality
the Generating Function of the series.

Thus e"" is the G. of 1 + a; + — + . • . ad inf., and can-

Z I

not be spoken of as the sum of the series without extend-
ing the meaning of the word sum quite beyond that
usually given to it.

Similarly, — is frequently spoken of as the sum of

1 + ^ + ^ • • • 4- ^'"~^ because, when developed by division,
it produces the series. But it is evidently a generating
function rather than a sum.

Thus in reference to series, the word sum applies
properly to a finite number of terms of a numerical series.

The word limit applies to an infinite numerical series ;
and the term generating function to a series, finite or
infinite, containing ascending or descending powers of a
variable.

We shall not, however, always apply these distinctions
rigidly.

286

SUMMATION OF SERIES.

SERIES TO n TERMS.

I. Generating Functions.

239. As a particular case take the recurring series of
the second order ;

l + x

is the G- of

l-2x + x^
Assume

l-2x-{-x'
Then iV:==

(2n+l)aj'»+...adinf.

1+3

-2

x-i-5
-6
+1

aj2_^...(2n-fl)

-2(2w-l)

(2ri-3)

--2(2n+l)
(2n-l)

.n+l

+(2n+l)a;'*+2

*'* """ 1-2X + X'

is the generating function required; and this fraction,
by division, gives the series to n terms, and no more.

The variable x may take any value except 1 (as for
this value the G„ becomes indeterminate), and the G»
becomes the sum of n terms of a numerical series.

Thus, putting a? = 2, we have

S,= 3 + (2n-l)2^-^\
as the sum of the first n terms of the series

1 4- 6 + 20 -f 56 +... (2 n + l)2^

SUMMATION OF SERIES.

287

Similarly, for a; = ^ we get ~ ^^ — — ^ as the

sum of n terms of the series

240. Now take the general case, and let

l+px-{- qx^

tto + ^1^ + • • • €inX\

■^n = «0 + «1

pao

x-\ a„

PCtn-l

a;"+^..

+ga"

v.n+2

And from the property of the R, that every column
with three terms is zero,

^n = «o + {cti -i-pcto)^ + lpc('n + Q^tn-i + got„a;Ji»"+\
Also a„+i + pa„ + g«n_i = 0.

.-. iV; = ao + (cti +i>ao)a; — (a„^i — ga„a;)aj"+\
And the required G is

1 -j- px + ga;^

In a similar manner the G„ can be found for a recur-
ring series of any order.

241. To find the sum of n terms of the series whose

nth term is •

n{n-{-l){n-\-2)

This series is allied to a recurring series, but the scale
of relation is not of finite dimensions.

288
We have

SUMMATION OF SERIES.

n{n + l)(n + 2) 2n 2{n^V) 2(n + 2)

And by putting n = 1, 2, 3, etc., we express the given
series as the sum of three series, viz. :

^r.

^ + 2- + l+'

2
2'

2_
3

4-

1

n
2

n

71 + 1
1

+ -

= '}-■

2(2

2 (^ + l)(n + 2))
Cor. If n = 00, we obtain as the limit of the series to oo,

s=\.

Series of the form of the foregoing can always be
summed when the numerator of the Tith term is constant,
and the denominator has its factors of the form

(n + A:)(^ + 2^0(^ + 3^0••• etc.

II. Difference Series.

242. Let Uq + xi^ -f ^2 + • • • u^ be a given difference
series, and let U'o + t/i + t/'o + ••• be the series of which
^0 + '^1 + "^^2 + *•• is the first series of differences.

Then Uo = 0, t/j = Wq? t/g = n^ + ^^u C/3 = i^o + 'ih + Wg,
and generally JJ^ = '?^o + ^^1 ••• + ^n-i = ^n-x-

But the 7ith term of the C/'-series is given by

U"^ = To + "Oii^o + "^2^1 + ^C^^2 + • • • (Art. 235)

SUMMATIOI^ OF SERIES. 289

and •.• Uo = 0,

Sn-i = nuo + "O^Ai + "(73A2 + -

which gives the sum of ?i — 1 terms counting from the
2d term, or of n terms counting from the 1st.

Hence for the sum of n terms from the beginning of
the series,

^n = nUo + "C2A, + "(73A2 + • • •

Ex. The sum of the 5th powers of the first n natural numbers is —
n + y n(n - 1) + 30 n(n - 1) (?i - 2) + -^^ w(w - 1) (n - 2) {n - 3)
+ Sn(n - IXn - 2)(n - S)(n - 4)
+ ^(n - l)(n - 2)(n - 3)(n - 4)(n - 6) ;

which reduces to

j\n\n + ly (2n^ + 2n - 1).

LIMIT OF A SERIES.

243. The limit of a series is either finite or infinite.
When the limit is finite, it is often called the sum of the
series to infinity, and the series is said to be convergent.
In general, series which are not convergent are classed
together as divergent, and cannot be said to have any sum.

The limit of a converging series may be rational, or in-
commensurable ; but in either case the rational value,
or a sufficiently close approximation to the incommen-
surable, may be employed in place of the series in com-
putations.

Such is the case with logarithms, with e, with trigono-
metrical functions, etc.

To know whether a series has a sum or not, we must
determine whether it is convergent or not.

290 CONVERGENCY OF SERIES.

CONVERGENCY OP SERIES.

When a series contains a variable, its convergency
or divergency is usually dependent upon the numerical
value assumed by the variable.

Thus X being positive, the series 1 + ^ + ^^ + ••• is
convergent only when cc < 1.

Some series of this kind, however, and especially such
as have increasing factorials in the denominators of their
terms, are convergent for all numerical values of the
variable.

244. It is shown under geometric series that the series

1 -}- X -{- x^ -\- a^ -\- * * ' ad inf. has as its limit when

1— X

X is positive and less than 1, and hence that under these

conditions the series is convergent.

Now let Uq -\- Ui -{- U2 -}- Us -\- • • • be an infinite series.

Then

^ ( . Ui Uo Ui Uo Uo Ua ')

;S=^0il +- + -•- + -•-•- + ••• ['

( Uq Ui Uq U2 Ui Uq )

And if each ratio —, —, — ? etc., be < x,

0 ^1 2

S<Uo\l+x + x'-{-a^+'''l',
which is convergent ii x<l.

Therefore the series Uq + Wi + 1^2 + • • • is convergent
if, after some finite term, ^^^^ is less than a quantity
which is less than 1 for all values of n.

CONVERGENCY OF SERIES. 291

Ex. 1. The Binomial series

1 + „x + ?L(»^Liix-^ + ... n(n-l)-(n-r+l)^, ^ ...
2 ! r !

where n is negative or fractional, is infinite in extent.
To show that it is convergent if a: be < 1.

Un±i ^ n(n- l)»>-(yt-r) ^^+i r\ 1^

Un (r + 1)! n{n — 1)'" (n — r + 1) x^

\l-hr 1 + rJ

But ^ = 0, and — - — = 1, when r = oo. And the whole is
1 +r 1 +r

< 1 if a: < 1, which proves that the series is convergent if x < 1.

Ex. 2. The series 1+-^ + ^ + ^+... is convergent for all
numerical values of ic. ...

^^^^^ = — ^ ; and for all finite values of x, this is < 1, when n
Un 7^ + 1
is great enough, and is zero when n = ao.

245. The series Wo — Wi + ^2 — '^3 H — • • •? with alternat-
ing signs, is convergent when each term is greater than
the following one.

For S = Uo — (ui — U2) — {uq — u^)— "'

And as Ui > Wg, u^ > u^, etc., every bracket is positive,
and

Again, S = {uo — u{) + {U2 — ^^3) + {u^ — %) H

And every bracket being positive, S >Uo — Ui,
.', S lies between Uq and Uq — w^, and is finite.

Ex. The logarithmic series

x-ix'^ + ix^-lx^-h

is convergent when a; < 1.

For if a; be < 1, the condition is evidently satisfied.

292 CONVERGENCY OF SERIES.

246. When the sum of a few of the first terms of a
series is a close approximation to its limit, the series is
rapidly convergent.

If the ratio u^^^ : u^ approximates to zero as n ap-
proaches 00, the series is rapidly convergent. But even
when this ratio approaches 1 as ii approaches oo, we are
not justified in saying that the series is not convergent,
as it may even then be slowly convergent, and may require
further examination.

247. Theorem. The series 1 + ^ + ^ + — + ••• — + •••

IS convergent it p > 1.

Separate the terms of the given series, after the first,
into groups of 2, 2^, 2^, etc., terms ; then,

l>i4.i. £>lj_i4.l4.i. I>i4....etc
.*. US be the limit of the given series,

I.e.

<^+2fe + (2^J+(2^J+'

But this latter series is convergent if — — < 1 ; that is,
ifi>>l ^'~'

•*• ^ + 2^ + 3^ + ^ + ••• ^^ convergent if p > 1.

248. The series Wq + Wi + ^2 • • * + ^^n + • • • is convergent
if u^ = — , and _p > 1.

CONVEEGENCY OF SERIES. 293

Then, J^ = i^±l)!=A+lY

= l+£ + K£^ + 4 + ...

Every term of the right-hand member except the
first vanishes when n = oo. Hence the series

^o + Wi + WgH u^^

is convergent if the function n \ — ^ — 1 f- > 1, when
n = 00.

Ex. To examine the series whose nth term is — i-^.

1 + ^3

t^«+i^l+(r^ + l) .l + n3^ n^+2n3 + n4-2 ^ .^ ^henn = oo
Un 1+ (n + 1)^ 1 + n w* + 4 n3 + ... + 2 '

and this test is not sufficient.

Again, n{^-\\= 2/i^ + 6n3... ^ 2, when n =. oo.
l Un^\ y n* + 2 n^ + n + 2

.'. the series is convergent.

The tests here given are the most useful tests of
convergency. Eeaders who wish to make themselves
acquainted with other tests will find such in larger
Algebras, where special attention is given to the subject,
or better, in works on Finite Differences.

249. When the sum of n terms of a series can be
found, and is of such a form as to become finite when n

294 CONVERGENCY OF SERIES.

is infinite, the series is convergent, and the finite expres-
sion found by making n = oo is the limit of the series.
This is quite self-evident.

Ex. The sum of n terms of 1 1 f- .•• is

3.4 4-5 5.6 3(n + 3)'

and when w = qo, the limit of the series is i.

EXERCISE XVIII. c.

1. Find the G„ of 1 + 2 ic + 3a;2 + ... (n + 1) x^.

2 3 4

2. Sum n terms of the series 1 -{ 1-^ + — + •••

3. Find the Gn of 1 + 3 x + 6 0^2 + ... ^ (w) (n + 1) x^-K

4. Sum n terms of the series 1 + 2.2 + 3 2^ + 4 . 23 + ...

5. Find the G„ of 1 - 2 ic + 3^2 - 40:"^ +

6. Sum n terms of the series 1 — 2 + 3 — 4+5 — 6 + «»

7. Sum n terms of -— - + — — + — — + —

1.2 2 • o 0.4

8. Sum n terms of — - + -—- + — — H —

1.3 3.5 5.7

9. Sum n terms of 1 h 1

1.4 2.5 3.0

10. Show that Sn | ^^ zA = sJ-^] ^, where

l(n+l)(w + 2)i \n + iy n + 2

Sn is the sum of n terms from the beginning.

11. Find the limit of i- (I2 + 22 + 32 + ... ^2), when w = 00.

12. Find the limit of i (l^ + 2^ + 3^ + ... w^), when n = cc.

CONVERGENCY OP SERIES. 295

13. Find the limit of -J- + -i- + -1- + ... ad inf.

1.4 2.5 3-6

14. Is the series 1 + - + llll^ + 1-S.6x^ convergent ?

2 2.4 2.4-6 ^

15. Under what condition is 1 + 2'^ + S^x^ + 4^^^ -f ••• conver-
gent ?

16. Show that x — \x^ + ^x^ — ^-x"^ + ... is convergent if x < 1;
thence find to four decimals the approximation to the value of

l5 3.58 5.55 / 1239 3(239)8 J

17. Find the limit of the series in Ex. 2.

18. Find the limit of 1 - - + — - -^ + —

CHAPTER XIX.

Determinants.

250. Two figures, as 3 and 4, are in order when the
less precedes the greater, and they form an inversion
when the greater precedes the less.

Thus 12 3 4 are in order ; 1 3 2 4 has 1 inversion, 3 2 ;
14 2 3 has 2 inversions, 4 2 and 4 3 ; and 4 13 2 has 4 inver-
sions, 4 1, 4 2, 4 3, and 3 2.

For illustration, take any arrangement of figures,

3 14 2 5 6 8.

This contains 3 inversions. Interchange any two con-
secutive figures, as 2 and 5; the arrangement becomes

3 14 5 2 6 8, and contains 4 inversions. Or interchange

4 and 2, and the new arrangement has 2 inversions.
Thus the interchange of any two consecutive figures

increases or decreases the number of inversions by one.

This is readily seen to be always the case; for if the
figures be in order before the interchange, they form an
inversion afterwards, and vice versa, while their relations
to the figures which precede or follow them are unchanged.

251. Starting with any arrangement, as

3 14 2 5 6 8,

let us interchange two figures which are not consecutive,
as 1 and 6. To do this, we must move 1 through 4
296

DETERMINANTS. 297

places to the right, and then move 6 through 3 places
to the left ; or we must move 1 through 3 places to the
right, and 6 through 4 places to the left. In either case
we make 7 consecutive changes in all; and the num-
ber of inversions is thus increased or decreased by an
odd number. The new arrangement, 3 6 4 2 5 18, has
10 inversions, or 7 more than the original.

Similarly, if the orders of any two figures be denoted
by m and 7i, where m<7i, to interchange m and n requires
us to move m through n — m places to the right, and then
to move n through n — m — 1 places to the left ; or, to
make 2 (71 — m) — 1 consecutive interchanges in all. And
this being an odd number gives the important

Theorem. — To interchange any two numbers in an
arrangement ^increases or decreases the number, of inver-
sions by an odd number,

252. Consider the four-dimensional term «! 62 ^3 ^4;
composed of four letters with attached suffixes, and in
which both the letters and suffixes are in order.

Keeping the letters in order, let us permute the four
suffixes in every possible way, as aj 63 Cg c?4, ag &i c^ d^, etc.
As we can permute the four suffixes in ^P^ or 24 ways,
we shall have 24 terms in all, of which no two have the
same suffixes attached to the same letters, or are wholly
alike.

The term ai b^ C3 d^, being the one from which the others
are derived, is called the principal or leading term ; but
as the whole set of terms may be derived from any one
of them, any term may be taken as a principal term.

Let us take as our principal term that one having no
inversions, and calling this positive, let us agree that in

298 DETERMINANTS.

forming the other terms every inversion is to be accom-
panied by a change of sign. Then a term with one in-
version in its suffixes, as aj b^ Cg d^, is negative ; a term
with two inversions, as ai b^ c^ d^y is positive ; and gener-
ally, a term is -\- or — according as its suffixes contain an
even or an odd number of inversions.

These considerations apply to a leading term of any
number of letters, and its derived terms.

253. A Determinant is the algebraic sum of all the
terms that can be derived from a leading term, by per-
muting the suffixes without changing the letters, each
term being taken with its proper sign.

A letter with its attached suffix is an element of the
determinant, and as each letter takes in turn each suffix,
if there are n letters, there are also n suffixes and n^
elements.

A determinant with n letters and n suffixes is of the
nth order, and contains n ! terms.

The determinant of the second order is afi^ — ^2^1? ^^^
of the third order it is

254. A letter with its attached suffix, standing as an
element of a determinant, is symbolic, and may be
replaced by any quantitative symbol whatever.

But it is only through this symbolic and symmetrical
notation that we are enabled to discuss with any facility
the general properties of determinants. Moreover, owing
to their unwieldiness when written at length, it becomes
necessary to employ some symbolic or contracted form
for the whole expression.

The symbols :S±ai&2C3--- and |aAc3---|, where the

DETERMINANTS.

299

«! 61

Ci

^1

a2 62

C2

d2

ag 63

C3

ds

a^ 64

C4

d.

leading term is written after 2 ± or between straight-line
brackets, are both employed. But the working form
known as a matrix is made by writing the elements in a
square between parallel vertical lines, in
such a manner that all the same letters
stand in the same column, and all the same
suffixes are situated in the same row.

The determinant of the 4th order is
written as a matrix in the margin.

255. The diagonal of the matrix from the upper left-
hand corner to the lower right-hand corner, namely,
tti 62 C3 d^, is the principal or leading diagonal, as giving
the principal term of the determinant.

The sign of the matrix as a whole depends upon that
of its principal diagonal.

The matrix being a symbolic form for a determinant
must be capable of being expanded so as to give the
determinant, and with a matrix having symbolic elements
this expansion can be effected by permuting the suffixes
of the leading diagonal according to the definition of
Art. 253. Hence two matrices containing the same sym-
bolic elements can differ only in sign, and the signs of
two such matrices will be the same or opposite according
as the number of inversions in their principal diagonals
differ by an even or by an odd number, Art. 2b2.

256. Consider the matrices

0)

a^ hi Cj di

a2 &2 ^2 ^2

«3 ^3 ^3 ^3

, (2)

a^ 64 0*4 d^

ci| 0^-2 (X3 a^
61 &2 h h

Ci C2 C3 C4

di ^2 r/g ^4

«! hi Cx c?i

, (3)

«3 ^3 C3 ^3

^2 62 C2 do

a^ 64 C4 d^

300 DETERMINANTS.

(1) is the standard matrix of the 4th order, the suf-
fixes being in order in the columns, and the letters being
in order in the rows. Its principal diagonal is aj h^ c^ d^.

(2) differs from (1) in having the rows of (1) for its
columns and the columns of (1) for its rows, the letters
and suffixes still being in order. The principal diagonal
of (2) is «! &2 ^3 ^4? ^11 d. being the same as that of (1), the
expansions give the same determinant.

Therefore, a matrix is not changed in value by chang-
ing its rows to columns and its columns to rows, provided
the letters and suffixes maintain the same order. Hence
whatever is true for a matrix with respect to its columns, is
true also ivith respect to its rows, and vice versa,.

(3) differs from (1) in having its 2d and 3d rows in-
terchanged. This introduced one inversion into its prin-
cipal diagonal, and hence changes the sign of the matrix.
And it is readily seen from the principles of Art. 252,
that the interchange of any two rows, or of any two
columns in a matrix changes the sign of the matrix,
since it increases or decreases the number of inversions
in the principal diagonal by an odd number.

257. Theorem. If tivo columns or two rows of a matrix
he identical, the value of the matrix is zero.

For, by interchanging the identical columns or the
identical rows, the matrix changes sign. But the columns
being identical leaves the matrix unchanged. The only
quantity or expression which remains unchanged when
you change its sign is zero. Hence the value of the
determinant is zero.

Thus the matrix with two a-columns, as in the mar-
gin, expands into a^a.fi.^ — aia.^b.^-\- a./i.^b^ — a^a^h^-^ a.^ai^b.2
— a-/(:J)^, which is identically zero.

a.

a, b,

«2

a, b.

a.

«j h

DETERMINANTS.

301

258. Expansion of the Matrix. Let us take a matrix
of the third order to begin with.

As every term in the determinant contains each letter
once and each suffix once, the terms which contain a^ can-
not contain any other a or any other letter with suffix 1.
Hence the coefficient of aj is the sum of all the terms
that can be made from the remaining letters and the
remaining suffixes. But this, for a determinant of the

3d order, is the expression denoted by the matrix

Similarly, the coefficient of ag is
ho Co

h C3

&2 ^2

&3C3
and of ttg it is

Hence the expansion takes the form

±0^1

&2 C2

±^2

5i c/

±a.

h Ci

h cs

^3 C3

&2 ^2

But taking the principal diagonals, ai &2 C3 is -}-, (Xg 61 Cg
is — , and a^ hi c^ is +. And the expansion is

«i

&2 C2

-a2!^'^

+ a3

61 Ci

&3 C3

1^3 C3

^>2 C2

(^)

and the matrix of the third order is made to depend
upon matrices of the second order.

Similarly, the expansion of the matrix of the 4th
order is

&2 C2 d^

hi Ci di

&1 Ci di

61 Ci di

hs C3 ^3

-a^

hs C3 ds

+ «3

ho C2 d2

--a4

&2 C2 C^2

54 O4 ^4

&4 C4 d^

64 C4 ^4

63 C3 C?3

which makes it depend upon matrices of the 3d order.

302

DETERMIKANTS.

So also a matrix of the nth. order may be expanded to
depend upon matrices of the (n — l)th order ; and these
again upon those of the (n — 2)th order; and so on.

259. Reducing the matrices of (A), the determinant
of the third order becomes —

(^ih^s — ai^gCg + ctgftgCi — a2&iC3 + a^hiC2 — agftgCi-

Comparing this with the matrix here written, in which
the first two rows are repeated in order below
the matrix, we see that the three terms aj 62 ^3,
ttg 63 Ci, and ag bi Cg, read in the direction of
the principal diagonal, are +, and the three,
Us 62 ^1) ^1 ^3 ^2? ^^^ ^2 ^1 ^3? I'Gad in the direc-
tion of the other diagonal, are — .

This is the rule of Sarins for expanding a
matrix of the third order ; in practice the portion with-
out the matrix is not written, the operation being carried
on mentally.
2

Ex.1. 121 =2.2.6 + 4.1.3 + 7.1-5-3.2.7
-2. 5. 1-1. 4.6 = - 5.

ai

&i

Cl

a2

b.

C2

%

h

C3

a,

h

Cl

Ct2

h

C2

2 4 71

121

3 5 6

1 4 7

2 5 8

3 6 9

a 1 a

a a 1

\ a a

Ex.2. 2 58 =1.5.9 + 2.6.7 + 3.4.8-3.5.7
-6.8.1-9.2.4 = 0.

Ex. 3. a a\ = a^ ■\- a^ + \ - d:^ - a!^ - a^ = 2a^ -^a^ -Ji-l.

EXERCISE XIX. a.
1. Find the value of the following matrices of the third order —

1 2 3

7

1

6

a h g

x\\

2 3 1

ii.

1

-2

4

iii.

hh f

iv.

0 X 1

3 1 2

3

-5

-1

Q f ^

0 0 a:

DETERMINANTS.

303

1 X y

a b c

1 «2 y2

vi.

b c a

vii.

1 X3 2/3

cab

1 1

1 -1

-1 1

a b c
bed
ode

2. Expand

12 3 4

13 2 4
12 4 3

14 3 2

8. Show that

X

X

1

1

X

1

X

1

X

1

1

X

X

1

1

1

(Expand to depend upon matrices of
the third order, and then expand
these. )

= x(l - x)8.

260. We see from the preceding article that as soon
as a matrix is reduced to depend upon matrices of the
third order, we can write out its expansion.

We turn our attention now to the investigation of
those properties of the matrix which enable us to expand
or reduce it more readily.

OPERATIONS ON THE MATRIX.

Let D denote a determinant, of any order, with sym-
bolic elements, and let A^ be the coefficient of ai. Then
(Art. 2bd>) Aiy which is called sl first minor of D, is a
determinant of the next order lower than D, and contains
no a and no letter with suffix 1. Similarly, let A2 be
the coefficient of a2, A^ of a^, etc.

Then (Art. 258)

D = a^Ai - a^A^ + a^Ao^ - + ••• (-)«-^a„^„.

261. To multiply the matrix by any quantity, m.

mD = maiA^ — -wiagztg + ma^A^ — f- • • •
But this is the expansion of the matrix in which mai is
written for a^ mag for ag, etc., throughout the a-column.

304

DETERMINANTS.

Therefore, to multiply a matrix by m we multiply every
element of a column or of a roiv by m.

Also, as multiplying by a fraction with unit numerator
is equivalent to dividing by the denominator,

Therefore, to divide a matrix by m we divide every
element of a column or of a row by m.

= etc.

«i ^1

Cj 1

mai

h, c.

mai

mbi mc^

Ex. 1. '

m

^2 ^2 ^2

a, h, c.

-

ma.2 ^2 ^2
ma.^ 63 C3

=

C?2 ^2 ^2

«3 ^3 C3

Jl 1

13 4

Ex.2.

1 i 1

= t\

2 14 =111.

1 1 i

2 3 1

8 4 2

2 2 2

1 1 1

Ex.3.

12 4 3

= 8

3 2 3

= 16

3 2 3

= -64.

4 6 5

]

3 6

1

2 5

262. Let each element of a column be the algebraic
sum of two quantities.

Thus, let ai=pi-\- q^, ag = P2 + Q2) ^^c.

Then D = (Pi+gO^i- (P2+q2)A+(P3+q3)A- + '"

= PiA -P2A +P3^3 - + •••

+ \q^A^ - q^A^ + q^A^ -+•••}

= the matrix with p written for a, + the
matrix with q written for a.

And the matrix thus becomes the sum of two matrices
of the same order.

2 3 1

1 + 13 1

1 3 1

1 3 1

1 3 1

Ex.1.

5 2 1
3 4 1

-

4 + 12 1
2 + 14 1

-

4 2 1

2 4 1

+

1 2 1
1 4 1

"

4 2 1
2 4 1

= 4,

since the s

econd 1

na

trix, having

tv

*^o colur

nn

3 ahke,

vai

lishes.

DETERMINANTS.

305

2 3 1

0 + 231

0 3 1

5 2 1

=

3 + 2 2 1

=

3 2 1

3 4 1

1 + 2 4 1

1 4 1

Ex.2.

as the second matrix will vanish after dividing by 2.

263. Let «! be changed to ai + nbi, ag to ^2 + 7ib2, % to
% + ^^^3? etc., throughout the a-column ; and let the value
of the new matrix be noted by D'.

Then

D'= (ai + nbi)Ai — (as + ^^2)^2 + («3 + ^^3)^3 — + •••

= aiJ.i — a2^2 + ^3^3 — + •••

+ ^{Mi-M2 + M3-+---^
= D + 71 times the matrix with b put for a.

But the matrix with b written for a has two 6-columns,
and therefore vanishes.

Hence D' = D, and we have the following important

Theorem. The value of a matrix is not changed by
increasing or diminishing any column by a midtiple of any
other column, or any row by a multiple of any other row.

In the examples which follow we shall denote the
columns from left to right by (7i, C2, C^, etc., and the
rows from above downwards by E^ E2, B^, etc. Then
R2 + Ri indicates that we are to add the second row to
the iirst row, and C2 — nC^ denotes that we are to subtract
n times the first column from the second column : etc.

Ex. 1.

15 3 6

3 14 1

4 6 17

—

2 2 5 2

1 5

3 6

7

5 17

0 14

5 17

=r2

1

-9 3

0 2

-9 3

4

1 10

0 8

1 10

= -24.

306

DETERMINAKTS.

1

a

h

c

1

a2

62

C2

1

a^

63 c3|

1

a*

6*

cH

Here we write S R^ — R^ for i?.^ ; R.^ — 2R^ for 7?3 ; and 2 i^j— i?^
for jR^. The C^ of the new matrix being all ciphers except 1, the
matrix is at once reduced to one of the third order.

1. Divide by abc.

2. i?i, i?2 — Rv ^3 ~ ■^2» -^4 ~ -^3-
Ex. 2. C ", r, ", . 3. Divide by (1 - a)(l - 6)(1 - c).

4. Cj, Ci — Cg, 0.2 — Cg.

5. Divide by (a— 6) (5 — c), and reduce.

Result, - abc(l - a) (1 - 6) (1 - c) (a - 6) (6 -c)(c-a).
Ex. 2 may also be reduced as follows —

1. If a, &, or c = 0, the matrix vanishes. .*. a, 6, c are mono-
mial factors.

2. If a, or 6, or c = 1, two columns are alike, and the matrix
vanishes. .-. a — 1, 6 — 1, c — 1 are binomial factors.

3. If a = 6, or 6 = c, or c = a, two columns are alike, and the
matrix vanishes. .-. a — b, b — c, c — a are binomial factors.

Since the expansion cannot have any term higher than of 9
dimensions, these are all the literal factors. And the expansion is
readily seen to be abc (a — 1) (6 — 1) (c — 1) (a — b) (b —c)(c — a).

EXERCISE XIX. b.

1. Evaluate the following determinants :

3 13 1

12 12
0 10 1

13 13

12 3 4
4 12 3
3 4 12
2 3 4 1

1111

0 111

0 0 11

10 0 1

2. lta,\-\-b,\-\ + c,\...\-d,\...
a 4th-order determinant, fill in the brackets.

be the expansion of

3. If a matrix be rotated through ninety degrees, the rows
become columns, and the columns rows, but they are not disposed
in the original order. How does this affect the sign ?

DETEKMINANTS. 807

4. If the rotated matrix of 3 be turned over in the plane, so that
the 4th column may become the 1st, the 3d, the 2d, etc. , how does
this affect the sign ? How does the matrix now compare with the
original ?

5. Show that to keep the suffixes in order and permute the let-
ters is equivalent to keeping the letters in order and permuting the
suffixes.

6. If Pi and p^ denote the diagonals of a matrix of the nth. order,
show that the sum of the inversions in p^^ and p.^is ^n{n — 1).

7. A matrix of the nth order has both diagonals of the same
sign when n or n — 1 is a multiple of 4 ; and of opposite signs in
other cases.

DETERMINANTS IN THEIR RELATION TO
EQUATIONS.

264. Take the set of linear equations, with symbolic
coefficients —

a^ + b^y + 0^=92,

Let Ai be the first minor of | a^ 62 ^3 1 with respect
to ai, A2 with respect to a^j etc.

Multiply the 1st equation by Ai, the 2d by A2, the 3d
by A^, and add ; then —

(ai^i — a^A^ + CLzA)^ + (Ml — M2 + hA)y

+ (ci^i - C2A2 + 03^3)2; = giA^ — g^A^ + ^3^3-

But the coefficient of a? is | ai &2 ^3 1 .

The coefficient of ?/ is | % 62 ^3 1 with b put for a, and
it therefore vanishes.

308

DETERMINANTS.

The coefficient of 2; is | ai 62 Cg | with c for a, and
it vanishes.

The independent term is | ^1 62 ^3 1 , i.e. \ ai &2 C3 1 with
g put for a.

I 91 h C3 1

.'. X -.

I c^i h C3

Similarly, y = !-^l|^, and . = j ^^ ^^ ^^

^1 ^2 ^3

ai &2 ^3

Thus, in the solution of the set of 3 linear equations
in 3 variables, each variable appears as a fraction whose
parts are matrices. The denominator is common to all,
and is the matrix formed by taking the coefficients of
the variables in order as elements of the matrix.

The numerators are the denominators in which one
column is replaced by the independent terms of the
equations, the coefficients of x being replaced in finding
the value of x, those of y in finding y, and those of z in
finding z.

Ex. To solve the set 2x + ^y + 4:Z = 16,

x + ^y + 2z = lS,

Sx +

Here

16 3 4

13 4 2

7 11

2 3 4

14 2

3 11

y +

2 16 4
1 13 2

3 7 1

z= 1.

3 4

4 2
1 1

2 3

16 1

1 4 13|

3 1

7

2 3

4 '

1 4

2

3 1

1

whence

x = l, y = 2, z = 2.

265. Prom the nature of the investigation of Art. 264,
it is evident that the same method of solution applies to
the case of n linear equations in n variables.

DETERMINANTS.

309

So that in the case of any number of linear equations
in the same number of variables, we can, at sight, write
down the values of the variables in the forms of frac-
tions whose parts are matrices. The rest of the solution
consists in the evaluation of the matrices.

266. Take the set of four homogeneous linear equa-
tions in four variables —

QiX -\- hiU + CiZ + diu = 0
a2X + 622/ + ^2^ + ^2^ = 0
a^x + % + CqZ -\-dsU = 0
a^x + b^y + C42; + d^u = 0 .
Divide the last three equations throughout by w, and
solve for the ratios -, ^, and

(^)

Then, - = -. -

u I a2 63 a

u u

hc^d^\ y

u

■y

u

! 0^2 ^3 ^4 I

I a^ bs C4 1
z

z
u

I <^2 h ^4 I

I ag &3 C4
- u

(B)

" I 62 C3 ^4 I I a2 C3 ^4 I I a2 &3 ^4 I 1 a2 63 C4 ^

which gives the ratios x: y : z : u.

Ex. Given Sx-\-2y — z = 4:X — 6y + z = 0, to find the ratios
x:y:z.

X

y _ ^ ora^_2/_2r

2-1

3-1 3 2 ' ^' 4 7 26

-6 1

4 1 4-6

..

. x:y:z = 4::7:26.

267. Since the denominators, in (B) of Art. 266, are
proportional to the numerators, write these denominators
for X, y, z, and u in the first equation of {A) of the same
article, then —
tti I &2 C3 (^4 1 — &i I (^2 C3 ^4 1 4- Ci I ag ?>3 ^4 1 — ^1 1 ^2 &3 C4 1 = 0.

810

DETERMINANTS.

Biit this expression is the expansion of the matrix

I di h C3 di |.

Hence the Eliminant (Art. 138) of a set of homogene-
ous linear equations, having the same number of equa-
tions as of variables, is the determinant formed from the
coefficients taken in order.

The vanishing of the eliminant indicates that the set
of equations is consistent, or that the equations are
compatible.

Ex. 1. Given the equations

2x + 32/-l=:3x-2?/ + 4 = ccH-62/-m = 0,
to find the value of m that the equations may be compatible.
Writing a unit- variable, z^ in the independent term, we have

' Sx — 2y + 4:z = 0 whence
. x + Qy — mz^O

2 3

3 ^2
1 6

-1

4

— m

= 0,

from which m = 4/^.

Ex. 2. Given al + bm : hm + en : en -^ al = x
ratios a:b: c and l-.m-.n.

V'^y

to find the

We have «^ + ^^ = ^"^ + ^^= en -\- al ^ ^^
X y z

say.

Then, al -\- hm —x

1^ = 0

bm -h en — yu = 0
al + e7i — zu = 0

And treating a, 6, c, and u as variables,
a — b c

(Art. 266)

Whence a:b: e=mn(x-y-{-z) : nl^y-z+x) : lm(z-x+y').
Similarly, l:m: n = bc(x-y-hz) : ca(y-z-^x') : ab(z-x+y).

m 0 X

I 0 X

I m X

m n y

0 n y

0 m y

0 n z

I n z

I 0 z

DETERMINANTS.

311

EXERCISE XIX. c.

1. Express the condition that y = niiX + h^, y = rrioX + h^, and
y = m.^x + h^ may be satisfied for the same values of x and y.

How would you interpret this fact with respect to the graphs of
the given functions ?

2. Express the condition that Ax-^ + By^^ 4- C = Ax^ + By^ + C
= Ax.^ + By^ + 0 = 0 may be true, A, B^ and C being considered
as variables.

3. If 3 sheep, 1 cow, and 4 horses are worth ^ 318, 4 sheep, 3
cows, and 1 horse are worth -$137, and 6 sheep, 2 cows, and 5
horses are worth $ 420, what is the value of 2 sheep, 5 cows, and
3 horses ?

4. If a -{-\(hm + gn)= b -^ - (hi -{- fn)= c + - (gl + fm)=\,

I m n

eliminate Z, m, and w, and find a cubic equation for determining X.

5. Find the condition that x^ -{• ax = b and x^ + a^x = b^ may
have a common root.

6. By means oi Ax -{- By + Cz = 0 and Px+Qy-\-Bz=0 elimi-

inate x, y, and z from the statement - = ^ = -.

I m n

7. Determine m, n, and p from the set

Wq — 4m + 6Wi — 4n + 2^2 = wi — 4wi + 6n — 4w2+i)

= Wj — 4 n + 6 1/2 — 4 J9 + ^3 = 0.
(If Wq, i^i, u^^ ^«3, be four terms of a difference series, w, n, and
p are intermediate equidistant terms.)

8. Find the eliminant of a%-2 + b'^y-'^' + c^z-^ = alx-^ + bmy-^
+ cn^~2 = a?'x~2 + bm'y^ + cri'^"^ = 0.

9. Show that

a -h b — c
a
b

10. Solve the equation in x,

c c

a b c

- c — a a

=

b c a

b c + a — b

cab

«! + b^x Ci di

a^ + b^x C2 (?2

= 0.

«3 + h^ <^3 ^3

312 MISCELLANEOUS EXERCISES.

MISCELLANEOUS EXERCISES.

1. If X + y = a, y-\- z=:h, z-\-x=c, and a"^ -\- h'^ ■{■ d^ = 0,
then xy + yz -\- zx = I {ah + &c + ca).

2. Extract the fifth root of 78502725751.

3. Expand ^-^^ — |- ^-i_ into an ascending series.

x-\-l X — 1

5. Given x -\- by — az = -^ aic + 2/ — - = «^, - -{- ay — z = 1,
to solve the set. ^ a a

6. Find all the positive integral solutions of 17 x + 31 ?/ = 100.

7. Expand —^ in ascending powers of x.

1 — x-{- x^

8. Divide 100 into two parts such that twice the square of one
added to three times the square of the other may be a minimum.

x^ — or 4- 1

9. Determine if ^^— - has a maximum or a mmmium

X2 + X + 1

value, or both for real values of x.

10. Factor 3 x^ — 10 x — 25 and thence show for what values of
x the expression is positive.

11. Show that f^ + ^ + ^U^ + 5 + ^W 9.

\a b cj \x y z]

12. Find the value of Vx^ + ax — x when x = cc.

* ( CL -4- x^(b -4- x^

13. Find the maximum or minimum value of ^^ — — — ^-^ — — — ^—

X2

14. Write the nth term of 1 - 2 x + 5 x2 - 8 x^ + H x* ...

15. Solve the set, x^ + ^/^ + x + 2/ = 14, 2(x2 + t/^) + 3x?/ = 29.

16. A person distributes 50 cents among beggars, giving 7 cents
to some, and 12 cents to others. How many were there ?

MISCELLANEOUS EXERCISES. 313

17. A and B play four games, each having the same sum at
starting. In the first game A wins ^ of B's money. In the second
B wins .f 20. In the third A wins J of what B has ; and in the
fourth B wins 1 5, and has then J as much as A. What did each
start with ?

18. Find three numbers in A. P. whose sum is 21, while the first
and second terms are together equal to f of the second and third.

19. Sum to n terms the series whose wth term is n'^ — n + \.

20. A falling body descends \f feet the first second, f / feet the
second second, |/ feet the third, and so on. How far will it fall
in the wth second ? How far in n Seconals ?

21. Solve 2x3 — 5c2 — 2ic+l=0 by putting it under the form

x^ — (x^ -\- px -\- qY = ^.

22. If 2 w — 10 is the nth term, how many terms will make
1*2 _ 3 6 ?

23. Find three numbers in G. P. whose sum is 13, and the sum
of whose squares is 91.

24. The sums of n terms of two A. P. s are n(n-\- 1) and
- ( - — 1 ) . Determine if they have a common term.

2^2 y

For what same numbered term is the term in one series 8 times
that in the other ?

25. A sum of $ 1000 is compounded annually for 4 years and
amounts to $ 1464. 10. What is the rate ?

26. Divide 1 into 5 parts in A. P. so that the sum of the squares
of the parts may be ^^.

27. Factor 2 ^2 - 21 xy - 11 ?/2 - x + 34 ?/ - 3.

28. If a'^-\-h'^'.a^-c = a^-h'^'.a- c, then a : 6 = 6 : c ; and
a-2h^-c={a- hY/a = (6 - c)2/c.

29. If zcc(x-\- y)^ and y oc ^2, and if when x = J, 2/ = J» ^ = h
express z in terms of x.

314 MISCELLANEOUS EXERCISES.

31. At a game of cards 3 are dealt to each person, and each can
hold 425 times as many different hands as there are cards in the
pack. How many cards are in the pack ?

32. If the number of combinations of n things 10 together be
the same as the number 5 together, what is the number 2 together?

33. The sums of n terms of two A. P. s are as 11 — 5 w to 11 + 3 n.
Find the ratio of their sixth term.

34. How many different guards of 6 soldiers can be formed from
a company of 30 ? And relatively,

1. How often will A be on duty ?

ii. How often will A and B be on duty ?
iii. How often will A be with B, without C ?
iv. How often will A be without B or C ?

35. At what time after T o'clock will the hands make an angle
of a° with one another ?

36. A, B, and C start from the same point at the same moment
to travel around an island 34 miles in circumference. A goes 13,
B, 7, and C, 4 miles an hour. When and where will they first be
together again ?

37. Find the expansion of ( 1 — ^ ~ j when x = 0,

38. In a G. P., s = 73.5682, r = 1.1, a = 3, to find the number of
terms.

39. Expand vTl as a periodic C. F.

40. Find values of x and y so that 314 x ~ 451 y = 13.

42. Divide a^ (b -c)+ b^ (c-d)+ c^ (a - 6)
by a2 (6 - c)+ b^ (c - a)+ c2 (a - 5).

MISCELLANEOUS EXERCISES. 315

43. Express a;^ — 1 in terms of u when u = x — \,

44. li s = a + \ prove that a* + i = §2 (^2 _ 4)+ 2.

a a*

45. Put 4 ic^ - 4 ic3 + 5 x2 + 3 under the form A^ - B^.

46. Factorize 2x'^ + Qy'^ -%xy + xz -Zyz '-2x -[-^y - z,

A

47. Expand in ascending powers of x.

3 — X (1 — x)

48. If -^ = a, -^— = 6, -^— = c, show that

y-\-z z-\-x x-\-y

x^ _ y2 _ g2

a(l — 6c) 6(1 — ca) c(l — a6)

49. From each of two towns 45 miles apart a traveller sets out,
at 9 o'clock, towards the other town, and the rate of one is 5 miles
more than | that of the other. They meet at 12 : 45. Where ?

50. If ^ = 1 + ix+ |x2 + i5c5+ ... and z = \ - x + Ix"^ - \x^
+ Jxi ..., show that x = 2 (1 — 2/^)+ | (1 - yz)'^ + V (1 - yzf-\-'"

51. P is any point on the bisector of the angle AOB, and A, P, B
are in line. Show that the sum of the reciprocals of OA and OB
is constant.

52. A gives to B and C as much as each one has. B then gives
to A and C J as much as they have, and then C gives to A and B
1 of what they then have, when A*has 72 cents, B 104 cents, and
C 28 cents. What had they at first ?

53. xy = c{x-{- ?/), yz = a(y -\- ^), zx = b(z -{- x), to solve in x,
y^ and z.

54. If x(^B-^b)=h-^b, find the value of x(B -b)+ Bh
in terms of P, 6, and h.

55. If a carriage wheel, 16 1 feet in circumference, took one sec-
ond longer to revolve, the rate of the carriage would be 1| miles
less per hour. How fast does the carriage go ? Explain the double
solution.

316 MISCELLANEOUS EXERCISES.

66. Prove by inequalities that the triangle of greatest area, with
given base and perimeter, is isosceles.

57. If xy + yz + zx = 0, and xyz = a, find y and z in terms of
a and x.

58. Resolve a;^ + 6a;-2 .^^^ partial fractions.

(x2-x + l)(x-l)2

59. By the expansion of (1 + 1 — x)»» show that
Ci(l + Ci 4- C2 + ...)= 2ci + 4c2 + 6C3 + .

60. By the expansion of (1 + x — 1)^ show that

i. C0-C1 + C2- Cb ..• Cn = 0.
ii. ci ~ 2 C2 + 3 C3 — 4 C4 ••• = 0.

61. If Pand Q are the pth. and gth terms of an A. P., what is
the (p + g)th term ?

62. Divide a number into two parts such that the product of
the parts shall be equal to the difference of the squares of the
parts.

63. Three numbers are in A. P. If 1 be taken from each of the
first two, the three terms will be in G. P. ; and if the last term be
increased by the first, they will be in H. P. Find the numbers.

64. Eind the square root of 2x + 2^{y(2 - y)+ x'^ - 1}-

65. How many sums can be made of a farthing, a penny, a six-
pence, a shilling, a half-crown, a crown, a half-sovereign, and a
sovereign ?

66. A stream runs 2 miles an hour, and a rower who rows 4
miles an hour wishes to go directly across it. At what angle up
the stream must he direct his boat ?

67. Find the approximate value of

{ Vr+^ + v'(l + iC)2} ^ {1 + X + VTTx]

when first powers of x only are retained.

MISCELLANEOUS EXERCISES. 317

2 S or -A- x^

68. Expand — to the term containing x^.

69. Find the fraction which expanded gives

2 - Sx -\- x'^ + 2 x^ - S x^ + x^ + 2 x^

70. Show that 1 + ^ + ^^ + ^^+- ^ 1±^.

1—X + X^ — X^+"' 1—x

71. Express x^ — 5 x^^y 4- 9 x^y^ — 7 x^y^ + 2 ic?/* — 2/^ as a func-
tion of ?/ and ^, when z = x — y.

72. A rectangle is inscribed in a triangle, and has a side coinci-
dent with the base of the triangle. Show that the area of the
rectangle is a maximum when its altitude is one-half that of the
triangle.

73. A cubic foot of lead weighs 715 pounds. If 10 pounds of
lead is formed into a cubic block, how long is the edge ?

74. What three linear expressions divided into x^ — 7 x -1- 10
will each give a remainder 4 ?

75. Express x^ + S x'^y — 6 xy^ — 2 ?/3 as a function of y and ^,
when z = x-\- y.

76. If ic2 - 2 a; - 1 = 3, find the values of x^- 5 xH 3 x'^- x+l
as a linear function of x.

77. If Bac — 5A, Cab = cA, find the value of

a^BC

2(B^/l - (72 + CVl - ^2)
in terms of abc and A.

78. Put x^ -\- x^(a + h)— x{a^ — ah) — a'^b into linear factors.

79. If £+^ + «^« = ^lL^+6^^, provethat a = 6.

a — X a -{- X b — X b -\- x

80. If ^ = ^ = ^ , then

lx{ny — mz) my(lz — nx) nz(mx — ly)

^(Z-x)+— (m-2/) + — (n-c) = 0.
Ix my nz

318 MISCELLANEOUS EXERCISES.

81. Two circles, whose diameters are D and df, are made to over-
lap until their common chord is c. Find the distance between the
centres, and explain the quadruple solution.

82. Two perpendiculars from a point to adjacent sides of a
square are 6 and 7, and the line to the vertex not related to those
sides is 6. Find the side of the square, and explain the double
solution.

83. The wheel of a barrow is 20 in. in diameter, and its axle is
1 in. out of the centre. If the wheel turns uniformly at the rate
of 1 revolution per second, find the greatest and least velocity of
the barrow in feet per second.

84. A V-shaped trough has an angle of 60°, and is 6 feet long,
A sphere 12 inches in diameter is placed in it and rolls throughout
its length. How many revolutions does it make?

85. If a + X Vl + a^ = a Vl - x^ + x Vl - a^ fmd x.

86. From a log 20 in. diameter and 20 ft. long, the largest beam
is to be cut so as to be twice as wide as thick. How many cubic
feet will it contain ?

87. Given x2 -f i = 12 + 6 f x + -V to find x.

x^ \ xj

88. Given ^^(x^ -f y"^) =c^(p — Ix — myy, to find n in terms of c,
when the terms of two dimensions in x and y form a square.

89. To go through a rectangular field along a diagonal is 10 rods
shorter than going around, and one side of the field is twice the
other. Find its area.

90. Divide a into two -parts, such that the difference of their
squares divided by the sum of their squares is a max. or a min.

91. Given x(i-^x + iy= 102(x + V^) - 2576, to find x.

92. If two chords of a circle intersect at right angles, the sum
of the squares on the segments is equal to the square on the
diameter.

MISCELLANEOUS EXERCISES. 319

93. Find the nth term of the A. P. whose sum to n terms is

94. A pair of wheels, whose diameters are d and d'^ are rigidly-
fixed upon an axle, a feet apart. How large a circle will the
smaller wheel describe when the set rolls on a plane ?

95. ABCD is a square with side s. AE = BF= CG=DH=ns,
E being on AB, F, or BC^ etc. Determine n so that the square
formed by AF, BG, CH^ and DE may have a given area cfi.

Explain the double solution, and thence find the maximum and
minimum values of a^.

96. A farmer bought some sheep for |72. If he had bought
6 more for the same money, he would have paid $ 1 less for each.

How many did he buy ? Explain the double solution, and change
the wording of the question accordingly.

97. A G. P. and an A. P. each has its first term = a, and the
sums of the first three terms are equal.

Find r in terms of a and d, and show that both values of r satisfy
the conditions.

98. If a, &, c are all positive, ?i + 1 + 1 > ^'^ -f ^2 _|_ ^2 ^^^
^Va+V6+Vc a h c abc

Vabc

99. If u = l + x + x^+'" Siiid v = l-x + x'^-+ ..-, 1 + 1 = 2.

u V

100. Show that l-(ll' + m7n' + nn'y^=(mn'-m'nyi-{nV-Vny
+ (Im' - Vmy\ if P + m2 -\- n^ = V^ + m'--^ + ?z'^ = 1.

101. X and Y are towns 20 miles apart. A person goes a certain
number of miles in a certain direction, and then changing his course
through a right angle arrives at Y after travelling 1 mile further
upon the second course than the first. How long was he in going
from X to Y at 4 miles an hour ?

102. On the rectangle ABCD, with sides AB = a and BC= 6,
P, Q, i?, S are taken, so that AP = na = CB, and BQ= nb =DiS,
Show that

320 MISCELLANEOUS EXERCISES.

area of Rectangle : area of Parallelogram PQBS = 1:2 n^— 2/1+1;
and explain the result when n = 1. When n is negative.
Has the parallelogram any maximum ? any minimum ?

103. n /i = l + ic + - + ^ + ..., prove that

^2 + 1_^ /. , X2 , CC* X6

/l+^%^ + ^% ...I.

1 2! 4! 6! /

104. A ladder 20 feet long leans against an upright wall, and has
its foot 3 feet from the wall. If a person pulls the foot outward,
compare the rates with which the foot and the top begin to move.

105. A room 18 by 24 is to be so carpeted as to have its floor
two-thirds covered, and the width of the uncovered part is to be
uniform around the room. Find the size of the carpet.

106. In a rectangular garden, 40 by 60 feet, a flag pole 50 feet
high is placed at 6 feet from a longer side of the garden and 15 feet
from an adjacent side. How far is the top of the pole from each
corner of the garden ?

107. If «&2 _ 5c2 = a, hc^ - ca"- = 6, ca2 - ah^ = c, show that

(ab + 6c + ca)(a'^ + 52 + ^2) = - (a* -f 6* + c^).

108. Find all the values of x from /x + iV+1^:--

V xj x^

109. Two wheels, in the same plane, are 3 feet and 1 foot in
diameter, and their centres are 4 feet apart.

Find the length of the belt which envelops the wheels and
crosses between them.

110. Show that l+x + x2 + ic3^--fl + l + -l+ ...V

x\ x x^ J

111. If (^^±yiy= lizl!, then each fraction = 1.

\y + zxj 1 - x^

112. Express as a fraction 1 — 2 ic (1 — x) (1 + ic2 + x* + •••) +
I 3 I.2V3J 1.2.3^3/ r

MISCELLANEOUS EXERCISES. 321

113. Given
a

{x — d)(x — c) {a — c){a — x) {c — x)(c — a) a — c
to find X; and also the ratio (c + a)2 : (ex + a"^).

114. Determine the isosceles triangle in which the altitude is
equal to one-third the whole perimeter.

115. The radius of a circle being r, find the angle between two
radii when the triangle formed by them and the chord through their
extremities is a maximum.

116. Through a point P to draw a line so as to form with two
given intersecting lines the minimum triangle, show that there are
two minima, and explain.

117. A man walks from A to B. If he had walked m miles an
hour faster he would have been h hours less on the road, and if he
had walked m^ miles an hour slower he would have been h^^ hours
longer.

Find the distance from A to B, and the rate of walking.

118. If X = re, y = rs, r = as, and s^ + c^ = 1, eUminate c, s,
and r.

119. How many sets of positive integers satisfy the system

120. A rectangular garden, sides a and ?>, is to be bordered with
a walk of uniform width which shall occupy one-half the plot.
Find the width of the walk ; and explain the double solution.

121. If f(x) ~l+^ + f + f^ ..., find /(O + /(_ ^), where

1 Z\ o!

122. If s, p, g, be the sum, product, and quotient of two num-
bers, ;? = S2(g _ 2g2^ 3g3_4g4 _| ),

123. Two wheels, A and B, geared together should move, as
nearly as practicable, with relative velocities of 1401 and 1945.

322 MISCELLANEOUS EXERCISES.

The number of teeth in a wheel being limited to not more than 120,
find the numbers to be employed.

After 100 revolutions of A how much will B be in advance of,
or behind, its true place ?

124. The resistance to sliding a stone on the ground varies as
the weight, and the weight varies as the cube of the diameter.
The power of a running stream to move a stone varies conjointly
as the square of the diameter and the square of the velocity of the
stream. Show that if the velocity of a stream be doubled it can
move a stone 2'^ times as heavy as before.

125. If a,, a.^, a.^, a^ be perpendiculars from the vertices of a
square to any line, and p be the perpendicular from the centre of
the square to the same line, show that 2a^ = 4j)2 _(- §2^ where s is
the side of the square.

126. If &p ^21 &3, ?>4 be line-segments from any point to the ver-
tices of a square, and p be the line-segment to the centre, show
that 26-2 = 4j9'2 + 2 si

127. The natural numbers are grouped as follows :

(1) (2, 4) (3, 5, 7) (6, 8, 10, 12) (9, 11, 13, 15, 17) ...
Show that the sum of the numbers in the nth group is

128. If xy :=1, show that a;" + ?/" :

(x+2/)«-4 0. 1

TABLE or PRIME NUMBERS UNDER 1000.

The hundreds are found in the top row, and the two remaining figures in the
body of the table.

0

1

2

3

4

5

6

7

8

9

I

01

II

07

01

03

01

01

09

07

2

03

23

II

09

09

07

09

II

II

3

07

27

13

19

21

13

19

21

19

5

09

29

17

21

23

17

27

23

29

7

13

33

31

31

41

19

33

27

37

II

27

39

37

33

47

31

39

29

41

13

31

41

47

39

57

41

43

39

47

17

zi

51

49

43

63

43

51

53

53

19

39

57

53

49

69

47

57

57

67

23

49

63

59

57

71

53

61

59

71

29

51

69

67

61

77

59

69

63

77

31

57

71

73

63

87

61

73

77

^Z

37

63

77

79

67

93

73

87

81

91

41

67

81

83

79

99

77

97

^3

97

43

73

83

89

87

^^

87

47

79

93

97

91

91

53

81

99

59

91

61

93

67

97

71

99

n

79

83

89

97

323

ANSWERS TO EXERCISES.

I.

a. 2. i. - 3. ii. - 183. iii. 30521/415800. 3. La- b.
ii. a-h + b^- ab'^ + ab^. 4. i. a^ -9b^ + ^c^ + 4: ac.
ii. 2(m2 - l)a2 4 2(n2 - 1)62. lo. 4th. 11. pth.

b. 4. i. 12. ii. 4 J. 5. 1-x. 6. -5. 7. 800. 8. 2a + 3 6.

9. 48,24,36. 10. $1100. 11. $120. 12. 120. 13. 67.

II.

b. 6. i. 0. ii. UabcZa. iii. 0. iv. abc. 10. 'Za^ + S'Za'^b
-^-Qabc. 11. 2a35+2a252_^3 2^25c. 12. Sa3_f3 Sa^d+O 2a&c.

C. 1. a:4_i0x3+35ic2-50ic+24. 2. a:H2x3-13 x2-14 x+24.

3. 24a:* + 154a:H269x2+154x+24. 4. x3 + x2Sa + x(2 Sa6-2«2)
+(Sa25 _ 2a3 _ 2 a^c). 5. Sa^ -}. 5 Sa^?, + lO Sa^ft^ 4. 20 Sa^^c
+ 30 Sa262c. 6. 2a* + 4 Sa^^ + 6 Sa262 + 12 l^a^bc + 24 a6cd.

d. 1. 26, -46. 2. x'^-x. 6. 2.71828. 7. ic* + a:3-ic2 + 2 x.

e. 1.14-2x2 + 3x^+14x0 + 25x8. 2. 1. 3. x2-2x-l.

4. mc-^2nab-\-pa^, 5. -J6(8a+1). 6. 1+x. 7. 1. 8. x^-l.

10. 12 + Ci2 + C22+--.. 11. C2 + C1C3 + C2C4 + C3C5+---. 12. m-^m'^

4.1^3 _+.... 13. ^ = i, J5=r--^, C = ?^^^^-^, etc.

14. 3.1416. 15. 3.1416. a a a

f. 1. x3 + 2 x2 + 3 X + 4. 2. 3 a3 + a2 _ 1. 3. 1 - 2 x + 2 x2.
4. l-2a-i+2a-2-2a-3.... 5. Q=x2-2 x+6, i2=-18x2+18x-7.
6. l + (3a-a2 + «3+5^4)(i4.Qj4_|.^8 + ...). 8. l-x+x2-x3+ -••..
9. x+2x2+3x3+4x* + ..., and x-i+2 x-2+3x-3+.... 10. x-1.

325

326 AKSWERS.

11. l-{-ix-\-ix^ + ixK 13. a2(5_c)-a(62_c2) + 6c(5-c),
or -(a-&)(6-c)(c-a). 14. x^-hy\ 15. x(l + 2x)/ {l-x-hx^).
16. 6%. 17. a + 2 - 2 a-2 - a-s. 18. a:* + x^ + 3 x2 + 2 a;

- 2 x-i - 3 x-2 - x-3 - X-*. 19. 4 x3 + x2 + X + 1. 21. J. = a-2,
^ = a-3, 0 = 0, i> = - a-5, /x = - ic4(a - x)/a%a^ - ax + x^).

III.

a. 1. (x-l)(x+l)(x-2)(x+2). 2. (2a + 3 5 + c)(a-5-c).
3. (x^ - px + l)(ic2 -qx-1). 4. (x + y + a)(x + ?/ + 5).
5. (a + x-l)(a + x-2). 6. 169 xy. 7. (a + b -\- c){a + b - c)
(& + c-a)(c+a-6). 8. (3x-l)(4x-l) and 3(x-2)(2 x-3).

9. (x-l)(x + l)(x'^ -px + q).

b. 1. i. (6 H-c)(c + a)(a + 5). ii. (& - c)(c - «)(a - 5).
iii. (5 — c)(c — a)(a — 6). iv. (a — b)(b — c)(c — a)(a -{- b + c).
V. vanishes. vi. (a — b)(b — c){c — a)(1>a'^ + 'Zab),
vii. (a — 6 + 1)(6 — c + l)(c — a + 1). viii. — 2 abc. ix. van-
ishes. 2. (a2 _ be) (b^ -ca) (c^ - ab). 5. (x - 1 ) (x - 3) (x + 5)
(x + 7). 6. (x-f-2)(x + 6)(x2 + 8x + 10). 7. (x + V3) (:k - V^)
(x+V2)(a:-V2). 8- 24a?)c. 9. (a + 5-c)(?) + c-a)(c+rt-6).

10. (a-&)(6-c)(c-(^)((^-a). 11. abc. 12. (a-6)(6-c).
13. (x - a + 5) (x - & + c) (x - c + a) .

c. 1. a + 6 - 1. 2. (X + 2 1/ - 2)2._ 3. (b - c)\/a.
5. x2-2«x + a2+52. 7. i. {x - J(3 + V21)}{x - i(3-V21)}.
ii. (x-l + 20(x-l-20. iii. (2x - 1 + V3)(2 x - 1 - V^)-
iv. 5(x+x%4-tV^129)(x+A-tVv'129). v. a(x+-i+J-Vr=4^2]

/a; + J__ J_Vl--4a2y vi. (x-l)(i9x-l). vii. (o^-x+l)

(a + 6 • X + !)•

IV.

a. 1. i. x2 - 2 X - 1. ii. x2 - 3. iii. 2 x2 - 3 x + 3. iv. 5 x.
V. x2 - 1. vi. a26-2 - 1 + 52^-2. 2. a2 - 62 = 40. 3. c = - 4,
= -3J. 4. (cp-ar)2=:(6;)-«^)(cg-6r). 5. {BG - Aq)
iCQ - BB) = {C^ - ABY, where ^ = a-ai, B = b-bu
C=c — Ci, § = cai — aci, i? = c6i — 6ci. 6. a = ^, = — 2.

ANSWERS. 327

b. 1. i. 12(x-2)(x-3)(a: + 3). ii. (x - a)(x - b)(x ~ c).
iii. i^p^^pi, 3. i. 2332. ii. 22325. iii. 23103. iv. 23131.

V. 2^32527. 4. 223. 5. 2520. 6. 5, 150. 7. x-2,

8. 4, X - 3.

V.

a. 1. i. (2x + i/)/(2x-y). ii. (Sx -{■ y)/(Sx -y).
iii. (iix -\- y) / (nx — y) . iv. 2(a — x)(a2 + x2)/ax2. v. a + h + c,
vi. l/(2x- — 1). vii. 0. viii. 1. ix. a + 5 + c. x. 1. xi. 0.
xii. —1. xiii. x. xiv. a-\-h-\-c. xv. —2. xvi. 0. 3. a. 6. r/f.

b. 1. - f. 2. 00. 3. 00, (c2 + 2 ac - a2)/2 c ; - 00 ; -.

5. (a2-f2)/(4a + 5). 6. 00, 2ab/(a + b); 00. 7. 00. 8. 0, V^-

9. 0, 1, i(l±Vl7). 10. a = b. 11. 8, 18. 12. ab/(a-b).
13. 29.5.... 14. X =(b'^ - a^) / (2 a -{-2 b -I), 15. 00, 1.
16. X, 4. 17. 3/2 a2. is. 36, 12, 16. 19. $3.75. 21. 32.

VI.

a. 1. 0, 4, 16, 00, - 32. 2. 144 : 125. 3. (2 ajS - a - /3)
/(a + ^-2). 4. 15:8. 5. (n/m)^. 6. 1, -3, f.
8. (?7i2+n2) : (m2-n2), {2 n2 : (m2 - n2)}2. 9. 0, -1. 10. -1 : x2.

b. 1. s = 16 ^2. 2. 145/416. 3. X = ± 4. 4. x = 0, 6.
5. (x-yy = ^\xy. 6. 6,4. 9. 10 hrs. 29.74 min. 10. 1666|.
11. © : D =75 X (28)2 : (99)2. 12. 0.027 in. 13. 5 hrs. 31 m. nearly.

VII.

2n-5 _^ PI

a. 1. a. 2. -^z. 3. X 2 . 4. a ^. 6. x^?. 6. (ft^/a"*)"*".
7. ggy/g -f a-2v^2 _ 2. 8. a^ + b^. 9. x^ + rx^ + 3fV ^' = 0-
10. (aT^^-2a)2 = 4(a2_a;2). n. 2. 12. n = {. 13. w = -6.
14. m"-i = n. 15. n = S. 16. x2 + 2 + 8 x-i + 27 x"^ ;
i2 = 97 x-2 + 55 x-3.

b. 1. i. 1 - V5. ii. -24J. iii. ^38 + 13^6). iv. \/2a".
V. 30V2_+_3V2. vi. 9bVJa. vii. 8V2. viii. a?)(3 a + 5 6) \/2a'.
2. i. 3V2a/4a. ii. l+V^. iii. V^- iv. ^(3V5-5)VlO+2v5.

328 ANSWERS.

V. 1 + v/2. vi. 2 - ^3. vii. 1(2 - ^2 - ^6). 3. The former.

4. The latter, by JO'^V^ - ^V^)- 5. K^^ - ^v^^ + 2^2).
6. fv2(V2-^2)(2+^4 + ^2).

C. 1. V^ + \/3- 2. 2 + i. 3. 1 + V'3. 4. tf + V(l - w^)-

5. l-^. 6. a^-?/+V(^-2/''): 7. 2+V3-V2. 8. Vl^+Vl^.

9. l + iV2. 10. 1(V16 + VQ). 11. (a + Va2-4)/V2. 12. 3.
13. V(6 + 2i)).

VIII.

a. 1. 100(s-h)/b, 2. mnr/{m-n). 3. ^ = 2 a + 3 &,
^ = a+3&. 4. A's = J(m + w +p), B's = i(m-n +^).

5. ah /{a ~h). ^. AO = l\\a^ where a is the distance be-
tween consecutive points. 8. a'^/d^. 9. 24 and 30.

10. ac{hi — ai)/(abi — ai&), bc(bi — a{) / {ahi — aib). 11. 20
miles an hour and 704 ft. long. 12. dist. = ha(h — b)/h ;
rate = a(h — b)/b.

c. 1. I a2. 2. i. dm/ Vm^ + n'^, dn/Vm^n^.
ii. d2m?i/(m2+n2). iii. dmn/(m'^+n^). iv. d{m'^—n^)/(m'^-\-)v^),
3. i. ab/Va^ + b^ ii. (a2 - 62) / Va^ + 6^. 4. 62/V^i2Tn".

6. iVl5. 6. 64. 7. 4in. rad. 8. ^(ir^-1). 9. iV^-

O Q

10. _ths nearly. 11. _ths nearly. 12. i. w2. ii. |Vl07
71 n

13. V3- 14. 1:2. 15. i. V29 : 5. ii. \/29 : 2. 16. t + ti'.t-h.
17. ladder = 39J ft. nearly ; position, 34J ft. from wall.

d. 5. 3, 1.

IX.

a. 1. i. —\b, }c. ii. lia — b), J(a + 5). iii. a/b, b/a.
iv. HiV2, i-iV2. V. l/(a-6), l/(a + 6). vi. i-iV2,
1+1^2. 2. i. b = a/(\-\-a). ii. 6=:a/(l + a). iii. a6 = 1.
3. cx2+6a:+a=0.^_5^J<V#+8^±(^). 6. BP=^(2 a^-s^).

7. J5P=i(s±V5s2_8a2). 8. ^0 =^{4^^ ±^^* - 16 a*)}-

9. Z2 = 5«2. 10. 00=iZ(V5-l).

b. 9. i. min. — |. ii. min. — |. iii. max. 4J. iv. min. — 2J.

10. |. 12. 1. 13. a/(l + w), n«/(l + n), wa2/(i + n).

14. J a, J a. 16. imag. 22. min. J s'^. 23. max. |ths of
the square. 24. J Z2. 25.4^5,8^5. 26.5^2- 27.4^5,8^5.

ANSWERS. 329

28. 10 - 2 v3 from A. 29. p'^/a'^ + q^/b'^ = 1. 30. IJ, 3^.

31. i. —p. ii. q. iii. p^ — 2 q. iv. —p/q. v. {p^ — 2q)/q'^
vi. iipq—p^. 32. 11 days.

C. 1. i. (b-ay/2b, 00. ii. oo, b(b - 2 a)/2(b - a).

iii. 00, (b-ay^/(2a-b). iv. 0, fa. v. oo, (5 a6 + 6'^ - a-^)/2a.

2. 0, V{(2«-&)/«2&}. 3. 0, JV3. 4. faV^- 5. ia(5^-6"^)2.
6. 0, 4. 7. 16, 25. 8. 0, - 2 a( Vl + a2_ Vl - a^)

/{a2 + 2 - 2\/l - a*}. 9. f , - f 10. 0, «, K« ± VSa^-Baft).

a, 1. 1. a; = 29 + 7p, ?/ = 2-3jt). ii. a; = l + 17p, y=ll-lSp.
iii. x = 13p + 2, ?/ = 45^ + 4. iv. ic = 48 + llp, 2/=i). 2. 2 and
3, 4 and 6, 6 and 9. 3. 5 7-in. and 1 13-in. 4. 6 4-lbs. and 3
7-lbs. 5. 63 ; general formula 420^9 - 357. 6. i. 21 wide, 33
narrow, ii. 26 wide, 25 narrow, iii. 41 wide, 1 narrow.

h, I. x = 6, y = 2. 2. x = S, y = 6. 3. x = a — abc^/(b -\- ac),
y = b-i/{b + ac). 4. X = aV(a - 6), y= b^/(b - a),

5, 2sd=(z + a) (z-a + d). 6. A = 200, B = 300. 7. xV

8. 93. 9. alb - c) / (b - a) of the first, b{a - c) / (a - b) of
the second. 10. 150 acres, rent = ^ 600.

C. 1. X = 7, 2/ = 5, ^ = 3. 2. X = 2, 2/ = 0 = 1. 3. a; = 3,
y = 2, ^=— 1. ^.x = y = b,z = 0. 5. x=(b — a)

/{a — b){b — c)(c — «), with symmetrical expressions for y and z.

6. ic = J, 2/ = i, ^ = i- 7. 3 a = 6 + 5 c. 8. 5 6 = 3 c.

9. 7 a + 6 + 11 c = 0. 12. x:y:z:u = 1:S : I: S. IZ. x = z = ^^,
y = 1. 14. a = 1, 5 = 3, c =r 2, d = 4. 15. x = a, y = a"^,
z = l. 16. X = - Sa, 2/ = 2a6, ^ = - abc. 17. 3 a6c = Sa^.
18. X = 3, 2/ = 1, 2; = - 2. 19. Sa* + 2 2:a2&2 = 8 ^252.

d. 1. x = 6, -2xV 2/ = 4, - IxV 2. X = i(a + \/a*-8 5),
2/= 1 (a -</«*- 8 6). 3. x = ±50, y = ±16. 4. x=:J(3±v5),
2/ = i(l ± V^)- 5._x = 7, 2/ = 3. 6. X = 4, i(- H + *V59),
2/ = 6, i(_ 11 _ iV59). 7. X = 15, - 16, 2/ = 9, - 10. 8. x = i,
K V3 - 2), 2/ = 1, V3 - 2. 9. X = 10, 2/ = 8. 10. x = 3, 2/ - 2.
11. x= 1.786..., 2^ = 1.731.... 12. x = -i,y=:-j\. 13. x = i,
2/ = i. 14. x = y/{ac/b), y=^(ab/c), z=:^(bc/a).

330 ANSWERS.

15. 2a/\/2T2V^, qv^2 + 2V5, j q\/(2 + 2 V5)3. 16. x r= 3,
y = 7, z = 9. n. x= Vbc^'^a-^, y = '\/ab^^c-^ z = Vca>^h-\

18. 27 a^2 = 4(i) - 2 a)3. 19. x = 2, y = 4. 20. x = 2, ?/ = 3.
21. X = 4, ?/ = 6, ;? = 3. 22. 15, 36. 23. 15, 20, 25.

XI.

a. 1. 999666, 1098010, 0. 2. 0. 3. - 7617. 4. 0. 5. 4 in
each case. 6.0.000211. 7.-0.0114-... 9. 12^2 - 31 x + 11.

b. 1. (x+ l)3-6(x + l)2+ll(x + l)-5. 2.3 65+30 6*
+ 119 63+238 62 + 249 6 + 106. 3. x^-l. 4. (x + 3)5-(x+3)+l.

5. (a + 6)5-3 6(a + 6)* - 4 h^{a + 6)2 + 6^.

6. (x - 1)3 + 3(x - 1)2 - 4(x - 1).

C. 1. i. 2 and 3. ii. — 2 and — 3. iii. — 3 and — 2, 1 and 2.
iv. Oand-1, land2. 2. 2.2284. 3. 1+^3. 4. \+ ^(2-{-i^^).
6. 2-^17. 7. 3, J(-l + V")- 8. 1.46460.... 9. 14.0487 in.
10. 2.5119....

XII.

a. 1. i. m +(n — l)(j9 — w). ii. a + 6(n — 1)(2 - n).
iii. 1(3 n - 1). iv. -^^(n - 1). v. \{Q\ _ 3 6 + n • 6 - 8).
vi. K92-8w). vii. K«+^)- 2. i. 45. ii. 12i. iii. 63.
iv. w(2a + 3 6-?i6)/2a(a + 6). v. 816. vi. n(n+l). 3. 48000.

4. \ 7i{n - 1). 5. 8729. 7. nth = i(6 w - 5). 8. 6 or 12 terms.
9. 11. 10. 19800. 12. $1630. 13. i. 46 days. ii. 91 days.
iii. 20 or 71 days. iv. no. 14. i. 27 days. ii. 7 or 20 days.
15. 62. 16. 49J s, n = 2 m + 1. 17. $ 3140. 19. diff. == ^(b - a).
20. nth = i 71(^ + 1). 21. wth=rn2. 22. nth= n^ 23. i)=4(n-l).
24. i{n + l)2(w + 2)2. 25. J w(n2 + 1).

b. 1. i. K-^" - 1). ii- lU -(- I)"}- iii- f {1 -(A)"}» f-
iv. V2 + i. 2. n + 1. 3. 1{(V2-1)/V2}«-^. 4. 12.056 ... gals.

5. s^^/2'S% ■j\7rs2. 6. a26/(2a+6). 7. 10. 8. iaw(n + l)
+ («"+! -a)/(a-l). 10. 6381407.

C. 1. 2, 4, 8, 16, 32, 64, 128, 256.

d. 1. 2a6/(a + 6). 6. 1, 2, 4.

e. 1. $1069.20. 2. $1156. 3. $8192, 4. $118.75.
5. $1734.90.

ANSWERS. 831

XIII.

a. 1. i. 6. ii. 6. iii. 840. iv. (n-l)\/(n-l - m)l 2. 720.
3. 10. 4. 1260, 120, 90720. 5. 1 in 30. 6. 36. 7. 1 in 72. 8. 60.

b. 1. Qa={n-2)b. 2. nb = a(r)\. 3. 385. 7. 3 in 10.

C. 1. i. {n(w - 1) ••• (n-r-\- l)a«-^x^}/r !. ii. i x (- l)*".

iii. {2 n(2 n - 1) ••. {2 n - r + l)xr}/r !. 3. ^^CqX^.

^. {(n-r)x}/{air + l)}. 5. ^^^^=-^>r >?^^^:^-l. 6. 1080.

d. 2. 1, 0, ~1, accordingas wisof formSm, 3m + 1, or3m+2.

3. coho + cihi + C2h2 + ... + Cnhn. 5. - 197. 6. ^~?"i'^'^~,\^'-

n !(2 n — 1)!

I 4.9 4.8V9/ 4.8.12\9/ /

e. 2. 1 .3.5...(2r-3)af/2.4.6...2r. 3. l+^x + -^x^
x3 4- -. 4. a(l-x+ I x2 - ix3 - Jx*). 9. V^.

2.4.6

XIV.

7. i. The first, ii. The second, iii. The first. 9. Less than 4
for all values of x from 1 to 3.

a. 1.

XV.
4 3

X-2X-1 X-3X-2

_-t i-+ ' ■

2(x-l) x-2 2(x-3)

a-\-l a^ + b

(a— 6)(6 — x) 6(a — 6)(a — x)
21 42 ^ 14 vi ^ ^

x + 2 2x + 3 (x + 2)2 2(a-x) 2(a + x)

b. 1. l+x-hx'^ + x^+-", 2. l + ix + fx2-TVa;^+Tl^a:*.-..
3. i_a;+ 1x2-1 a-3+fx*.... 4. l + ix-ix24-A^^-]j?%-'^*---

6. (1 +X)-« = 1 -CiX+ (C12-C2) X2- (Ci3-2 C1C2 + C3) X3+ (Ci*-3 Ci2C2

332 ANSWERS.

+ 2ciC3 4-C22-C4)ic*.-.. 6. x=l/ai'y—a2/ai^'y'^-\-(2a2^-asai)/

9. a = 1, 6 = 6, c = 62/2 \, d = h^/^ !, etc.

C. 1. (nH-l)x«-i. 2. (2-x+2x2 )_(l_2ic+3x2-4x3...).

3. Jn(n2 + 2). 4. J n(n + l)(w;2+ n + 2). 5. | n(w + l)(4 w-1).

6. 4(11 -2w)(l + n). 7. in(n + l)(ri + 2)(n + 3).
12

d. 1. a^h = 4 c. 3. (ca' - c'aY = {ah' - a^h) (be' -h'c).

4. \ = 2VAB-{A + B). 6. 3. 9. (b-p^)/2p. 10. (l+2x)
/(l_aj-x2). 11. a -6 = 1 12. (x2 - 2 x - l)2-(x + 1)2.
13. (x + 2?/ - l)(x - y + 3). 14. m = f

XVI.

a.

1.

1, i li

If , AV, if?.

3. 558,

552.

4.

¥, m-

5.

b.

1.

i(Vi5-

-1).

2. K2V39-

9).

4.

V6.

5. 1 +

1

'94-

2 + -
2

1 1

+ 4 + "•

6.

'-sV

-., 4-f

1 1 1 1 1 1 '

2+ 1+ 3+ 1+2+ 8 +

XVII.

1 ^

a. 1. 2, 4, 6, - 3, - 5. 3. 1, i, 1, 23, 2« 6. i. 1.537....
ii. 0.8379 .... iii. 1.242. iv. x = 2l,y = 3f. v. log( ^2 + 1)/
21oga. vi. iV(l + 81ogn/loga)-i. 7. -^^log2+ilog3-ilog5.
8. 1.80618, 2.40824, 1.05361, 1.69897, 1.39794. 9. log2 + log 3,
log2 + 21og3, 31og2 + 21og.3, -(2 log 2 + log 3), -21og2,
-(31og2 + log3). 10. 8. 11. « = log 2/log 1.05.

b. 1. 6.7449. 2. 0.0018542. 3. 0.57122. 4. 7.4123....
6. 0.69897. 6. 0.93936.

c. 10. Se.

d. 1. x = e«(»+i)-a.

XVIII.

a. 1. i? = 1 - 2 X + x2, G^ = 1/i?. 2. i? = 1 - 2 X + x2.

G^rz:(l + x-x2)/jR. 3. 6 x* + 7 x^. 4. 1 + X - x2 + 3 x^ - 3 X*
+ 5x5-5x6.... 6. (3 + 8x)/(3 + 5x+x2), l/(l-a;+3x2-8x3),

ANSWERS. 333

~4ix4,17a^4. 6. i{l+(_)«(2n + 3)}, -50.

7- i{l + (-)"(! - 2 n)}. 8. (3«+i - 2«+i)x^

b. 2. 50 + -(^2 _ 15 ;^ + 26). 3. 1 + 7 n - n2.

4. -^^(n^ _ 4 w2 + 13 n + 6). 6. 4, 2, 1, 1, 2, 4, 7. 8. 45.14,
37.89. 9. -0'.13. 10. 0.41337.

C. 1. {l-(n+2)x«+H(w+l)x«+2}/(i_x)2. 2. (2"+2-n-3)/2«

3. {l-i(w + l)(w + 2)x» + n(n + 2)x»+i-| n(^+_l)a;«+2}/(l-x)8.

4. l + w.2"+i. 5. {H-(-)"(7i+2x«+Hw + l^"+2)j/(i+aj)2.
6. i{l+(-)"(2n + 3)}. 7. n/(n+l). 8. n/(2nH-l),
9. 7i/3(/i + 1) + n/6(n + 2) + n/9(?i + 3). 11. f 12. J.
13. a. 14. conv. ifx<l. 15. x<l. 17. 4. 18. |.

XIX.

a. 1. i. -18. ii. 173. iii. abc + 2 fgh - af^ - hg^ - ch^.
iv. x^. V. xz/(y — x)(l — ic)(l — y), vi. 3 a6c — Sa^ vii. — 4.
viii. c(ae - c^) + 6(c(Z - 6e) + (^(^c - ad), 2. 0.

b. 1. i. 0. ii. - 160. iii. 1.

C. 3. $301. 4. X8 - X22a + A(2a& - 2/2) - (ahc ■\-2fgh- ap
— hg'^ — ch'^) = 0, wlifTe ahc and fgh are collateral systems of
letters. 5. (a - aiy\abi - a^h) = (b - b^y. 6. mn{BB - CQ)
= jil(CP-AB) = lm(AQ-BP). 7. m = j\(6uo + lbui-{-U3-6u2)y
^ = T6 (^ ^1 + 9 ^2 — Wo — Wa), p = yV (^ W3 + 15 W2 + Wo — 5 Ui),
8. a6c I a??i?i' |. 10. x=— \ aic^dz |/| b\C2,dz |.

MISCELLANEOUS.

2. 151. 3. -2(1+2x2 + 2x4 + 2x6+...). 4. x = a/\/a234.
5. x=a, y = \/a^ z = \. 6. none. 7. l + 2(x— x^— x^+x^+x^—).
8. 60 and 40. 9. max. 3, min. \. 10. x must not lie between
-fandS. 12.1a. 13. min. -(a- 6)2/4 a?). 14. (-)"(3 7i-l)x^
15. x=:3, i/ = l. 16. 5. 17. $120. 18. 5, 7, 9. 19. in(w2 + 2).
20. J/(2 n - 1), 1 n2/. 21. 1, - 1, i- 22. i 5. 23. 1, 3, 9.
24. no, 5th. 25. 0.1 per unit. 26. ^, iV» ^^•> tui ij%.

27. (x-lly + l)(2x + ?/-3). 29. z=j\x+^x^ 30. 4.

334 ANSWERS.

31. 52. 32. 105. 33. -1. 34. 593775; i. 118755; ii. 20475;
iii. 17550; iv. 80730. 35. -Vr(30r±a°) seconds after

hour T. 36. 34 hrs., at starting point. 37. Put I — z for x,

expand, and then make z = \. 38. 13. 39. 3 + i - ....

3 + 6
40. 1027, 715. ^l. {ah{h-a)-\-hc{c-h)-\-ca{a-c)}/ahc,

42. a-\-b-^c, 43. w5 + 5 w^ + lO i^H 10 wHS w. 45. (2x2-a: + 2)2
-(2x-l)2. 46. (2x-2^ + ;2)(x-3?/-l). 47. t(l + ir.-fa:2
— ^jX^ + ••.). 49. 15 miles from one town. 52. A, 120 ;

B, 60 ; C, 24. 53. x = 2 ahc/(ab — be -\- ca), with sym. ex-

pressions for y and z. 54. h{B -}- b -{- VBb). 55. 5| miles.

67. y = KVa2-4ax3 - a)/x'^, z = i(- Vo^r^Tox^ - a)/x^,

58. 5/(x-l)2 + 3/(x- l)_(3x + 4)/(x2-x+ 1).
61. {qQ-pP)/{q~p). 62. greaterpart = J a(V5 - 1).

63. 2, 3, 4. 64. Vx - 2/ + 1 + Vx -f ?/ - 1. 65. 255.

67. 1 - i X. 68. 2 - X + 5 x2 - 7 x3 + 17 x^ - 31 x^ + 65 x^ ....

69. (2 - x)/(l + x)(l + x;^). 71. z^ - z^y^ - y^. 73. 2.8912 in.
74. X - 1, X - 2, X + 3. 75. z^-^zy'^^b ?/^. 76. 437 x + 557.
78. (X + 5) {x + i «(1 - V5)} . {x + 1 a(l + yS)}. 81. i(v'X>2_c2
4. Voji _ ci^, 83. 22 7r/12, and 18 7r/12. 84. 3.82. 85. 0,
a(Vr=r^-\/l + a2)/(l + irt2_Vl- «4). 86. 22fcu. ft.

87. 4 ± Vli, -l±L 88. 7i2 z= c\P + m2). 89. 25(7jf 3^5)
sq. rds. 90. x = J a, a min. 91. yx = 7, - 8, ^"^185 — 1).
93. i2pn-p-q)/(p-q'). 94. md. = d' ^(a'^-^id^'^)/ (d-d'y,
95. w=^±aV2s2-a2)/(s2_^2). 93. 18, or -24. ^01. iV799.
104. V39i:.3. 105. 14.234 by 20.234. 106. V2761, V4561,

\/388l, V568I. 108. x =(- 3 ± V21)V V^- 109- 4v3 + f7r.
112. (1 -x + x.l + x3)/(l + x). 113. x=(c2 + 2ac-a2)/2c;
ratio is 2. 114. side : base = 13 : 10. 115. 90°. 117. rate

= mmi(h + hi)/(mhi — mih); dist. — mmihhi(m + mi)(Ji + ^1)
/(mhi-mihy. IIS. x^-]-y^ = ay. 119. 12 solutions, 1 + 3 p, 12 -p,
10+5/). 120. i(a + 6±Va2TP). 121. 2^-i+l-+...y
123. 85 and 118 ; + yi^th, nearly.

INDEX.

The Numbers refer to the Articles,

ART.

Algebraic expression 7

Algebraic subtraction 24

A Igebraic sum 25

Annuity 173

Annuity in reversion 174

Annuity foreborne 175

Antilogarithmic series 222

Arbitrary multipliers 135, 140

Arithmetic mean 165

Binomial theorem 186

Characteristic 215

Coefficient 19

Coefficients of (a+a;)« 31

Combination 177

Commutative law 7, 13

Complementary combination 183

Complex number 53

Composite number 69

Concrete quantity 102

Constant 18

Convergent 203

Convergency of series 243

Cube root of 1 57

Cube root of a number 91

Cyclic substitution 25

Degree 18

Determinant 253

Difference series 234

Dimension 18

Directed segment 107

Discriminant 200

Distributive law 15

EliramanJ; 138

ART.

Euler's proof of binomial theorem 190

Equation 22

Expansion of homogeneous sym-
metrical functions 30

Exponent 16

Exponential series 222

Factor 12

Factorial 179

Fraction 71

Function 32

G. C. M.,H. C. F 61

Grenerati ng function 229

Geometric mean 169

Graph. , 116

Graph of quadratic 121

Highest common factor 61

Homogeneous 19

Identity 22

Imaginary quantity 48

Imaginary unit 52

Incompatible equations 138

Indefinite solution 77

Independent equations 137

Indeterminate equations 129

Index 16

Index law 16, 38

Infinite series 35

Infinity, qo 75

Integral function 33

Interpolation 236

Interpretation of "fl^. 189

Irrational quantity 48

Irrational equation 128

335

336

INDEX.

ART.

Law of signs 14

Least common multiple 66

Linear expression 19

Line symbol 107

Linear factor 42

Limit 168, 238

Logarithm 211

Logarithmic series 223

Mantissa 215, 216

Matrix 255

Minimum and maximum 122

Napierian base 211, 220

Negative number 8

Operative symbol 4

Partial fractions 196

Partial quotient 203

Permutation 177

Periodic continued fractions 209

Perpetuity 173

Prime number 69

Progressions 156

Proportion 82

Quadratic, The 118

Quadratic factor 42

Quadratic surds, Theorems on. . . 99

Quantitative symbolfl 4

ART.

Ratio 79

Rationalizing factor 94

Real quantity 48

Recurring series 229

Redundant equation 138

Remainder theorem 149

Roots 58, 117

Rule of Sarrus 259

Scale of relation 229

Sigma notation 28

Similar surds 92

Simultaneous equations 134, 136

Solution 23, 59, 134

Special roots 76

Summation of series 199

Sum of a series 238

Surd 88

Symbolic geometry 106

Symmetry 26, 78

Synthetic division 39

Term 10

Term, nth 157, 228

Undetermined coeflScients 195

Unit-variable 114

Variable 13

Verbal symbols 4

THE PRINCIPLES

OF

ELEMENTARY ALGEBRA,

BY

NATHAN F. DUPUIS, M.A., F.E.S.C,

CB in the University
Canada.

i2mo. $1.10.

Professor of Pure Mathematics in the University of Queen's College, Kingston,
Canada.

FROM THE AUTHOR'S PREFACE.

The whole covers pretty well the whole range of elementary algebraic
subjects, and in the treatment of these subjects fundamental principles
and clear ideas are considered as of more importance than mere mechan-
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exhaustive; but it is hoped that the treatment is sufficiently full to
enable the reader who has mastered the work as here presented, to take
up with profit special treatises upon the various subjects.

Much prominence is given to the formal laws of Algebra and to the
subject of factoring, and the theory of the solution of the quadratic and
other equations is deduced from the principles of factorization.

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think he has succeeded. I commend the work to all teachers of Algebra
as a science." — Prof. C. H. Judson, Furman University, S.C.

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$1.10.

FROM THE AUTHOR'S PREFACE.

" The whole book has been thoroughly revised, and the early chapters
remodelled and simplified; the number of examples has been very
greatly increased ; and chapters on Logarithms and Scales of Notation
have been added. It is hoped that the changes which have been made
will increase the usefulness of the work."

From Prof. J. P. NAYtOR, of Indiana University.

*' I consider it, without exception, the best Elementary Algebra that
I have seen."

PRESS NOTICES.

"The examples are numerous, well selected, and carefully arranged.
The volume has many good features in its pages, and beginners will
find the subject thoroughly placed before them, and the road through
the science made easy in no small degree." ~ Schoolmaster.

** There is a logical clearness about his expositions and the order of
his chapters for which schoolboys and schoolmasters should be, and
will be, very grateful." — Educational Times.

*' It is scientific in exposition, and is always very precise and sound.
Great pains have been taken with every detail of the work by a perfect
master of the subject." — School Board Chronicle.

"This Elementary Algebra treats the subject up to the binomial
theorem for a positive integral exponent, and so far as it goes deserves
the highest commendation." — Athenseum.

" One could hardly desire a better beginning on the subject which it
treats than Mr. Charles Smith's ' Elementary Algebra.' . . . The author
certainly has acquired — unless it * growed ' — the knack of writing
text-books which are not only easily understood by the junior student,
but which also commend themselves to the admiration of more
matured ones." — Saturday Review.

THE MACMILLAN COMPANY,

66 FIFTH AVENUE, NEW YORK.

ELEMENTAKY ALGEBRA FOU SCHOOLS. By H. S.

Hall, M.A., and S. R. Knight, B.A. 16mo. Cloth. Withon* an-
swers, 90 cents. With answers, ^1.10.

NOTICES OF THE PRESS.

** . . . We confidently recommend it to mathematical teachers, who,
we feel sure, will find it the best book of its kind for teaching pur-
poses.'' — Nature.

*' We will not say that this is the best Elementary Algebra for school ,
use that we have come across, but we can say that we do not remember
to have seen a better. . . . It is the outcome of a long experience of
school teaching, and so is a thoroughly practical book. . . . Buy or
borrow the book for yourselves and judge, or write a better. ... A
higher text-book is on its way. This occupies sufficient ground for the
generality of boys." — Academy,

HIGHER ALGEBRA. A Sequel to Elementary Alge-
bra for Schools. By H. S. Hall, M.A., and S. R. Knight, B.A.
Fourth edition, containing a collection of three hundred Miscella-
neous Examples which will be found useful for advanced students.
12mo. il.90.

OPINIONS OF THE PRESS.
** The * Elementary Algebra ' by the same authors, which has already
reached a sixth edition, is a work of such exceptional merit that those
acquainted with it will form high expectations of the sequel to it now
issued. Nor will they be disappointed. Of the authors' ' Higher Alge-
bra,' as of their ' Elementary Algebra,' loe unhesitatingly assert that
it is by far the best work of its kind with which we are acquainted.
It supplies a want much felt by teachers." — The Athenxum.

" . . . Is as admirably adapted for college students as its predecessor
was for schools. . . . The book is almost indispensable and will be
found to improve upon acquaintance." — The Academy.

"... The authors have certainly added to their already high repu-
tation as writers of mathematical text-books by the work now under
notice, which is remarkable for clearness, accuracy, and thorough-
ness. . . . Although we have referred to it on many points, in no
single instance have we found it wanting." — The School Guardian,

THE MACMILLAN COMPANY,

66 FIFTH AVENUE, NEW YORK.

ALGEBRAICAL EXERCISES AND EXAMINATION

Papers. By H. S. Hall, M.A., and S. R. Knight, B.A. 16mo.
Cloth. GO cents.

This book has been compiled as a suitable companion to the ** Ele-
mentary Algebra" by the same Authors. It consists of one hundred
and twenty Progressive Miscellaneous Exercises followed by a compre-
hensive collection of Papers set at recent examinations.

ARITHMETICAL EXERCISES AND EXAMINATION

Papers. With an Appendix containing questions in Logarithms
and Mensuration. By H. S. Hall, M.A., and S. R. Knight, B.A
Second edition. IGmo. 60 cents.

KEY TO THE ELEMENTARY ALGEBRA FOR SCHOOLS.

12mo. $2.25.

KEY TO THE HIGHER ALGEBRA FOR SCHOOLS.

12mo. $2.60.

iit*^ These Keys are sold only upon a teacher's written order.

HALL and STEVENS. — A Text-Book of Euclid's Ele-

MENTS. Including Alternative Proofs, together with additional
Theorems and Exercises, classified and arranged. By*H. S. Hall,
M.A., and F. H. Stevens, M.A.

Books I.-VI. Globe 8vo. $1.10.

Also sold separately :

Book I. 30 cents. Books I. and II. 50 cents.

Books I.-IV. 75 cents. Books III.-VI. 75 cents.

Book XI. 30 cents. Books V., VI., XI. 70 cents.

*' This is a good Euclid. The text has been carefully revised, the defini-
tions simplified, and examples added illustrative of the several propo
sitions. We predict for it a very favorable reception." — Academy.

LEVETT and DAVISON. — The Elements of Trigonom-

etry. By Rawdon Levett and A. F. Davison, Masters at King
Edward's School, Birmingham. Crown 8vo. $1.60.

This book is intended to be a very easy one for beginners, all diffi-
culties connected with the application of algebraic signs to geometry
and with the circular measure of angles being excluded from Part I.
Part II. deals with the real algebraical quantity, and gives a fairly
complete treatment and theory of tlie circular and hyperbolic functions
considered geometrically. In Part III. complex numbers are dealt with
geometrically, and the writers have tried to present much of De Mor-
gan's teaching in as simple a form as possible.

THE MACMILLAN COMPANY,

66 FIFTH AVENUE, NEW YORK.

ARITHMETIC FOR SCHOOLS. By Rev. J. B. Lock,

Fellow and Bursar of Gonville and Gains College, Cambridge ; for-
merly Master at Eton. Third Edition, revised. Adapted to Ameri-
can Schools by Prof. Charlotte A. Scott, Bryn Mawr College,
Pa. 70 cents.

The Academy says: "'Arithmetic for Schools,' by the Rev. J. B.
Lock, is one of those works of which we have before noticed excellent
examples, written by men who have acquired their power of present-
ing mathematical subjects in a clear light to boys by actual teaching
in schools. Of all the works which our author has now written, we^
are inclined to think this the best.'*

TRIGONOMETRY FOR BEGINNERS, AS FAR AS

THE Solution of Triangles. By Rev. J. B. Lock. Third Edi-
tion. 75 cents. Key, supplied on a teacher's order only, ^1.75.

The Schoolmaster says : '' It is exactly the book to place in the hands
of beginners. . . . Science teachers engaged in this particular branch
of study will' find the book most serviceable, while it will be equally
useful to the private student."

TO BE PUBLISHED SHORTLY.
INTRODUCTORY MODERN GEOMETRY OF THE

Point, Ray, and Circle. By William B. Smith, Ph.D., Profes-
sor of Mathematics in Missouri State University.

THE ELEMENTS OF GRAPHICAL STATICS. A Text-

Book for Students of Engineering. By Leander M. Hoskins,
C.E., M.S., Professor of Theoretical and Applied Mechanics, Univer-
sity of Wisconsin.

Correspondence from professors and teachers of mathematics, re-
garding specimen copies and terms for introduction, respectfully
invited*

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FEB 27 1957

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911397

THE UNIVERSITY OF CALIFORNIA LIBRARY