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http://www.archive.org/details/cu31924031285806
Cornell University Library
^V19314 Ray's
new
higher algebra
3 1924 031 285 806 olin,anx
ECLECTIC EDUCATIONAL SERIES.
RAY'S
NEW HIGHER ALGEBRA. ELEMEI^TS OF
ALGEBRA, FOR
COLLEGES, SCHOOLS, AND PRIVATE STUDENTS.
By JOSEPH RAY,
M.
D.,
LATE PROFESSOR OF MATHEMATICS IN WOODWARD COLLEGE.
Edited by DEL.
KEMPER, "T
A. M., Prof, of Mathematics,
VAN ANTWERP, BRAGG & 137
Hamden Sidney
OOLIiEOE.
WALNUT STREET,
28
CO.,
BOND STREET,
NEW
CINCINNATI.
®
YORK.
— Ray's Mathematical Series.
ARITHMETIC. Bay's Bay's Bay's Bay's
New Primary Arithmetic New Intellectual Arithme New Practical ArithmetJ*. New Higher Arithmetic.
TWO-BOOK Bay's Bay's
New New
\
SERIES.
g-\
^ S
^*
:
^-:
Elementary Arithmette:^ Practical Arithmetic
ALGEBRA. Bay's Bay's
New New
Elementary Algebra. Higher Algebra.
HIGHER MATHEMATICS. Bay's Bay's Bay's Bay's Bay's Ray's
Plane and Solid Geometry. Geometry and Trigonometry. Analytic Geometry.
Elements of Astronomy. Surveying and Navigation. Differential
and Integral Calculus.
Bnterpd according to Act of Congress, Clerk's Office of the District Court of
in the year tiie
United
IS52,
by W. B. Smith, in the
St.ites, for
the District of Ohio.
Entered according to Act of Congress, in the year Iftfifi, bv Saboebt, Wtlson HiNKLE, in the Clerk's Office of the District Court o'f the United States for the Southern District of Ohio.
&
PREFACE. Algebra
is
justly regarded one of the
most interesting and and an acquaintatice with it is now who advance beyond the more common elements.
useful branches of education,
sought by
To of
those
all
who would know Mathematics, a knowledge
elementary principles, but also of
its
its
essen-
is
while no one can lay claim to that discipline of mind which
tial;
education confers, It is
who
is
not familiar with the logic of Algebra.
both a demonstrative and a practical science
of truths and reasoning, from which
is
are of the highest possible utility in the arts of
The
object of the present treatise
science in a brief, clear,
and
is
means of understanding
of every process he
is
required to perform.
of which
an outline of
The aim
form.
principle,
nish the student the
to simplify subjects
many life.
to present
practical
throughout has been to demonstrate every
made
system
an endless variety of
the student, but
to
—a
derived a collection of
Ilulee that maj' be used in the solution of
problems, not only interesting
this
not merely
higher parts,
and
to fur-
clearly the rationale
I^o
by omitting that which
eflfort
has been
is difficult,
but
rather to present them in such a light as to render their acquisition within the reach of all
To
fix
who
the principles in the
their bearing
and
utility,
will take the pains to study.
mind of
the student, and to
show
great attention has been paid to the
preparation of practical exercises.
These are intended rather
illustrate the principles of the science,
than as
to torture the ingenuity of the learner, or
difficult
amuse
to
problems
the already
skillful Algebraist.
An
effort
has been made throughout the work
to
observe
a.
natural and strictly logical connection between the different •parts, 80 that the learner
may
not be required to rely on a
prin-'
PREFACE.
iV ciple,
or 'employ a process, with the rationale of which he
The
already acquainted.
is
not
reference by Articles will always en-
him to trace any subject back to its first principles. The limits of a preface will not permit a statement of
the
who
are
able
peculiarities of the work, nor
interested
however, received
know
to
proper
more than usual
themselves.
attention.
It
is,
Equations have
The same may be
said of
Theorem, and of Logarithms,
all
of
useful in other branches of Mathematics.
.so
On some
for
it
that Quadratic
remark,
Kadicals, of the Binomial
which are
necessary, as those
examine
will
to
is it
subjects
work within suitable
it
was necessary
of the Theory of Equations, outline of the
more
wliich alone
is
by reference
to the
practical
to be brief, to bring the
For example, what
limits.
to
is
and
is
here given
be regarded merely as an
interesting parts of the subject,
sufficient for a distinct treatise, as
works of Young or Hymers
may
be seen
in English, or
of DeFourcy or Reynaud in French.
Some
topics
and
exercises,
deemed both
will be found here, not hitherto dents.
But
useful
and
interesting,
presented to the notice of stu-
these, as well as the general
manner
of treating the
subject, are submitted, with deference, to the intelligent educa-
tional public, to
whom
the author
is
already greatly indebted for
the favor with which his previous works have been received.
WooDWAKD
College, Jhnj,
Publishers' Notice. Ray's Algebra, Part Potter.
Del.
II.,
1S,J2.
— This was
work, originally published as revised, in lS(i7,
by Dr. L. D.
Portions of the work were revised in 1875, by Prof.
Kemper.
—
CONTENTS. I.— FUNDAMENTAL RULES. articles.
Definitions and Notation
Exercises on the Definitions and Notation
Examples Addition
to
be written in Algebraic SjTnbols
—General Rule.
Bracket, or
Subtbaction
.
— Rule
Vinculum
Observations on Addition and Subtraction
.
.
—Preliminary principle Rule of Coefficients — of Exponents Rule of the Signs — General Rule
Multiplication
.
.
,
.
Multiplication by Detached Coefficients
Remarks on Algebraic Multiplication Division
— Rule
of Signs
Division of a
— Coefficients-— Exponents Monomial by a Monomial by Monomials
Division of Polynomials
Division of one Polynomial by another Division by Detached Coefficients
.
.
CONTENTS.
III.— ALGEBRAIC
FRACTIONS. ARTICLES.
Definitions
— Proposition — Lowest terms
To reduce a Fraction
to
.
.
114
.
an Entire or Mixed Quantity
To reduce a Mixed Quantity Signs of Fractions
.
to
121
the form of a Fraction
122
.
123
.
To reduce Fractions
to a
To reduce a Quantity
Common Denominator
to a Fraction
...
— 126 127 — 128 129 — 131 — 132 125
.
with a given Denominator
Addition and Subtraction of Fractions
.
L'^0
.
Multiplication and Division of Fractions
Reduction of Complex Fractions
I33
Resolution of Fractions into Series
Miscellaneous Propositions
Theorems
in Fractions
— 119
I34
Fractions
135
137
— Miscellaneous exercises
138
139
140
149
in
IV.— SIMPLE EQUATIONS. Definitions
and Elementary
Transposition
— Clearing
of
principles
....
.
— 151 152 — 153
Fractions
Solution of Simple Equations
150
— Rule
Questions involving Simple Equations
154
Simple Equations with two unknown quantities Elimination by Substitution— Comparison— Addition, .
... etc.
.
155
156-158
Problems producing Simple Equations containing two un-
known
quantities
.
....
.
.
...
159
Simple Equations involving three or more unknown quan^-'ties .
.
160
.
Problems producing Simple Equations containing three or more unknown quantities
161
v.— SUPPLEMENT TO SIJIPLE EQUATIONS. Generalization
— Formation of Rules.— Examples .... — Discussion of Problems — Couriers
Negative Solutions
Cases of Indeterminatinn and Impossible Problems
A
Simple Equation hos but One Root
.
162
163
.
.
164--166
.
.
167—169 170
CONTENTS.
vii
POWEES— EXTRACTION OF ROOTSRADICALS— INEQUALITIES.
VI.— FORMATION OF
Involution or Formation of Powers
— Newton's Method
—Of Fractions — Theorem
Squakk Eoot of Numbers
.
autici.es. .
.
.
.
172
173
—179
Approximate Square Roots Square Root of Monomials
180
— Of Polynomials
—184 .... 185— 189 182
—Approximate Cube Roots
Cube Root of Numbers
Cube Root of Monomiols—Of Polynomials
190—191
— Sixth Root— Nth Eoot, etc Signs of the Roots — Nth Root of Monomials Radical Quantities — Definitions — Reduction
193—194
Fourth Root
192
of Eadicals
.
195
—203
Addition and Subtraction of Radicals
204
Multiplication and Division of Radicals
To render Rational the Denominator
...
of a Fraction
...
205
....
Powers and Eoots of Radicals— Imaginary Quantities
206
.
.
.
207
—210
,
.
.
212
—213
Theory of Fractional Exponents
211
Multiplication and Division in Fractional Exponents
Powers and Roots of Quantities with Fractional Exponents
214
.
Simple Equations containing Radicals Inequalities
—Propositions
I to
216
Y— Examples
217
— 224
224
— 229
VII.— QUADRATIC EQUATIONS.
—Pure Quadratic Equations— Problems ....
Definitions
....
Affected Quadratic Equations
Completing the Square
—General Eule—Hindoo Method
230
.
.
2:il— 232
.
.
234
233
Problems producing Affected Equations Discussion of General Equation
— Problem of Lights
.
— Definitions — Theorems— Square Eoot of A dry.... Varieties of Trinomials — Form of Fourth Degree
240
Trinomial Equations Binomial Surds
ii
.
.
241
.
242—243 244
Simultaneous Quadratic Equations
Pure Equations
— Affected Equations
Questions producing Simultaneous Quadratic Equations
—
FormulsE
General Solutions
Special Artifices and
Examples
—239
245 .
.
— 250 251
252
253
CONTENTS.
viii
VIII— RATIO— PROPORTION-—PROGEESSIONS. Ratio
—Kinds — Antecedent
and Consequent
.
.
— MuHii)liealion and Division of ...
Ratio
— 256
...
.
— — Compound — Duplicate— Triplicate ....... ... Ratios — Corajiarison of Proportion — Definitions Of greater and less Inequality
Ratio of Equality
ARTICLES.
2o4
.
.
.
259 260
.
261
Ratio
262
.
.
Product of Means equal
to
.
.
.
—
203
Product of Extremes
Proportion from two Equal Products
Product of the Extremes equal Proportion by Alternation
268
to the Sqiiare of
— By
Mean
the
Inveision
.
270
.
273
....
Like Powers or Roots of Proportioniils are
...
Proportion
.
275
in
277 .
27S
IIarmonical Proportion .
281
.
.
m Arithmetical Means
Geometrical Progressiox
292—29-1
between two numbers, Ex.
— Increasing and
Decreasing
last
Term— How
Sum
of Decreasing Infinite Geometrical Series
to find
it— Sum of Series— Rule
Table of General Formulte
To
find a Geometric
To
insert
m
.
...
.
.
.
.
.
.
Geometrical Means between two numbers,
18
16
29-1
295
.
296—297
.
.
Mean between two numbers, Ex.
299
.
300
.
300
E.x. 19
300
Harmomcal
— To find the value of PROGRrssioN — Proposition
Problems
Arithmetical and Geometrical Progression
Circulating Decimal?
in
— 290 291
.
.
insert
— 279 280
— Prnpositions— Exercises ... Aritiimktical Progression — Increasing and Decreasing last Term — Rule — Sum of Series — Rule — Table "\'ariatiox
To
— 274 276
.
Proportion .... — Exerciteb— Problems ....
Products of Proportionals are
Continued Pro]tortion
in
— 271 272
.
— By Division
Proportion by Composition and Division
269
.
Proportion from equality of Antecedents and Consequents Proportion by Composition
2fi6
267
301 .
.
.
.
302
— 303 304
.
IX.— PEE.MUTATIONS- COMBIXATIOXS— BINOMIAl
THEOREM. Permutations Combinations
.
....
305 308
—307 —309
CONTENTS.
ix AIlTirLES.
Binomial Theokem when the Exponent Binomial Theorem
is
a Positive Integer
310
ajiplied to Polynomials-
311
X.— INDETERMINATE COEFFICIENTS—BINOMIAL THEOEEM —GENERAL DEMONSTRATION— SUMJIATION AND INTERPOLATION OF SERIES. Indeterminate Coefpicients *
— Theorem — Evolution
Decomposition of Rational Fractions
Binomial Theorem
for
any Exponent
E.^traction of Roots by the Binomial
Limit of Error
in a
.
.
.
314
.
.
.
319
317 318
.
—Application
of
Theorem
—321 322
Converging Series
323
— Orders of Differences To find the nth term of a Series — The sum of n terms Differential Method of Series
.
.
.
Piling of Cannon Balls and Shells
324
.
.
Interpolation of Series
—327 328— 331 —33i 336 — 338 339 — 344 — 346 326
3.33
Summation of Infinite Series
Recurring Series
3-13
Reversion of Series
XL— CONTINUED FRACTIONS— LOGARITHMS-EXPONENTIAL EQUATIONS— INTEREST AND ANNUITIES. Continued Feacttotis .
.
.
.
.
.
.
Logarithms of Decimals— Of Base— Of
— Logarithmic Series .... — Computation of Common Logarithms— Computation of by Series .... Computation of Logarithms
—356 — 359 360 — 361 347
.
—Definitions— Characteristic Table Properties of Logarithms— Multiplication — Division Formation of Powers — Extraction of Boots Logarithms
367
362
— 363
364—368 370
— 373
Naperian Logarithms
375
Single Position
380
377
Double Position
381
Exponential Equations Interest and ANNUiTiES^Simple Interest
Compound
Interest
— Increase of Population — Formula:— Annuities
Compound Discount
— 383 381 — 385 386 — 387 388 — 391 382
CONTENTS.
XII.— GENERAL
THEORY OF EQUATIONS. ARTICLES.
Definitions
An An
— General
Form
Equation whose Root
is
of Equations
a
is
divisible by x
393— 39i
.
.
—a
395
.
Equation of the nth Degree has n and only n roots
396—397
Relations of the Roots and Coeificients of an Equation
398
What Equations have no
399
Fractional Roots
.
.
To change the Signs of the Roots of an Equation
400
Number
401
of
Imaginary Roots of an Equation must be
Descartes' Rule of the Signs
— A method
Limits of a Root
402 of finding
Transformation op Equations Synthetic Division
Equal Roots
.
.
— Transformation of
Derived Polynomials
.... .
403
404—403
.
Equations by
409—410
— Law of— Transformation by
411—413 414
.
Limits of the Roots op Equations
.
.
415
.
Limit of the greatest Root— Of the Negative Roots
XIII.— KESOLUTION OF Eational Roots
416^18
....
Sturm's TuEORiiii
— Rule
420—427
XUMEEICAL EQUATIONS.
for finding
....
429
Horner's Method op Appeoxim ition
430
434
Approx]mation' hy Double Position
436
Newton's Method of Approximation
437
Cardan's Rule
for
Solving Cubic Equal
438
ioijs
Ekciprooal or Recurring Equations Binomial Equations
441
442
. ,
443
444
:
:
HIGHER ALGEBRA. DEFINITIONS.
I.
Article
Mathematics is the Magnitude
1.
relations of Quantity as to
science of the exant
or Form.
2. Quantity, as the subject of mathematical investicjaany thing capable of being measured, or about
tions, is
which the question 1.
How much? may
Geometric, involving
3. Number
Form;
2.
be asked.
tude of the quantity
When
is
indicated
are represented
be,
by
;
and the magni-
its ratio to
the unit.
by conventional symbols.
the symbols used are general, as distinguished from
the arithmetical symbols,
process
may
quantity considered as composed of equal
is
parts of the same kind, each called the unit
4. Numbers
It
Number.
investigation
of
viz.,
the Arabic
called
is
numerals, the
Hence, we
Algebraic.
have the following definitions
5. Algebra
is
the method of investigating the relations
of numbers by means of general symbols.
Remark. — It whenever used
G. The
should be remembered that the word "quantity"
in algebra, is
synonymous with "number."
algebraic symbols are of two kinds:
of numbers
Numbers
;
2.
Symbols of
are usually represented
sometimes, of course,
T. The symbols of
1.
Symbols
relation.
by
when known, by
letters as, a, 5, x, y the Arabic numerals. ;
relation, usually called Signs, are the
representatives of certain phrases, and are used to express
operations with precision and brevity. braic signs are
;
=
-|-
—
X
The
principal alge-
-^ V^11
;
RAY'S ALGEBRA, SECOND BOOK.
12
8. The Sign of Equality, =,
is
read equal
It de-
to.
placed are
notes that the quantities between which Thus, x=b, denotes that the quantity represented equal. it
by X equals
5.
9. The Sign
+,
of Addition,
that the quantity to which
Thus, a-\-h denotes that
10. The Sign
it
is
is
h is to
Thus, a
subtracted.
—
is
be added to
—
of Subtraction,
It denotes
read plus.
prefixed
denotes that the quantity to which
from
is
,
it
is is
to be added. a.
read minus. prefixed
It
to be
is
b denotes that h is to be subtracted
a.
11. The former
signs
-(-
and
—
are called
the
The
signs.
called the posi/ivf, the latter the negatloe sign
is
they are said to be contrary, or opposite.
12. Every
quantity
tive or negative sign.
fixed to
it,
supposed to have either the
When
a
posi-
quantity has no sign pre-
Thus, a=-\-a.
understood.
is
-|-
is
Quantities having the positive sign are called positive;
those having the negative sign, negative.
13.
Quantities having the same sign are said to have
Jihe signs
Thus, while
those having different signs, unlilce signs.
;
and
-)-a
-j-c
and
-\-h,
— d have
14. The Sign
or
—a
and
—
I),
have like
of Multiplication,
denotes
that
the
X,
is
read
quantities
raultipUed
hy.
which
placed are to be multiplied together.
it is
It
signs.;
unlike signs. into,
or
between
The product of two
or more letters is also expressed by by writing the letters in close succesThe last method is generally to be preferred. sion. Thus, the continued product of the numbers designated a dot or period, or
by
a, h,
and
c,
is
denoted by aX^'X^j Of
a.L.c, or ahc.
DEFINITIONS.
15. Factors tors,
are quantities that are to be multiplied
Thus, in
together.
product ah, there are two
the
a and 6; in the product
factors, 3, 5,
13
and
16. The Sign
3x5x7,
^.
of Division, -^,
Division
is
read divided hy.
is
denotes that the quantity preceding that following
fac-
there are three
it is to
It
be divided by
it.
also expressed
by placing the dividend
as the
numerator, and the divisor as the denominator of a fraction.
Thus,
a-=-6, or
17. The Sign
-,
signifies that
of Inequality,
a
>,
the two quantities between which
The
than the other.
ojjcning
is to
it
be divided by
6.
denotes that one of is
placed
of the sign
is
greater
is
toward the
greater quantity.
Thus, a^h, denotes that a read a greater than
than
and
d,
A
18.
is
greater than
is
Also, c uX"X«=oi*-
Sign, ]' or
quantity, denotes that
to denote
since
and
^-''«
32. An Algebraic
signify the
it,
2
is 3.
is 2.
is
under-
same thing.
Quantity, or an Algebraic Exqjrcs-
— DEFINITIONS.
15
any quantity written in algebraic language, that by means of symbols. Thus,
sion, is
5a, is the algebraic expression of 5 times the
number a; number 6
36-)-4c, is the algebraic expression for 3 times the
creased by 4 times
3a2
tlie
number
Tab, for 3 times the square of a, diminished by 7 times the product of the number a by the number b.
A
23.
Monomial
a quantity not united to any other
is
;
monomial
24.
A
is
d'hc,
often called a simple quantity, or term.
Polynomial, or Compound Quantity,
braic expression composed of two or more terms c
4a,
as,
etc.
4a;!/,
A
in-
r;
by the sign of addition or subtraction
—
is,
—
an alge-
is ;
as, a-\-h,
x-\-y, etc.
25m A Sinomial
a quantity having two terms
is
;
as,
a-j-t, x'-j-y, etc.
A Residual Quantity which is negative as, a 26. A terms;
Trinomial
as, a-\-h
—
is
—
;
is
a binomial, the second term of h.
a quantity consisting
2T. The Numerical Value is
the
of three
c.
of an algebraic expression
number obtained by giving
a particular value to each
and then performing the operations indicated. Thus, in the algebraic expression 4a 3f, if a=5 and
letter,
—
c=6,
the numerical value
25. The
is
4x5—3x6=20—18=2.
value of a polynomial
is
not affected by chang-
ing the order of the terms, provided each term retains sign.
Thus,
P — 2a6-|-c
29. The Degree of literal
is
evidently the same as
of any term
factors which
it
is
contains.
h'-\-c
its
— 2ah.
equal to the number
——
—
RAY'S ALGEBRA, SECOND BOOK.
16 Thus,
ba
is
of
ax
is
of the second degree;
&« first
degree;
Za^lj-c='iaaabbc.
30. A of
polynomial
terms
its
is
x^'
— ~xy-
is
ixy'^
is
X-
—
31. An
said to be homogeneous
homogeneous; each term being of the first degree. homogeneous; each term being of the third degree. not homogeneous.
algebraic quantity
said to be arranged ac-
is
cording to the dimensions of any letter llic
when each
Thus,
of the same degree.
is
a—b —3c
contains one literal factor. contains two literal factors.
it
of the sixth degree.
is
is
it
exponents of that
it
contains,
when
occur in the order of their
letter
magnitudes, either incnd^ing or decreasing. Thus, ax--{-a-x^a'':c",
powers of a; and
6.c^'
arranged according
is
to the
ascending
arranged according
b^'x'--\-b'^x, is
to the de-
scending powers of x.
A
32.
Parenthesis,
is
( ),
terms of a polynomial which
used
it
to
show that
all
the
incloses are to be consid-
ered together, as a single term.
Thus,
—
— — 6)
means that a b is to be subtracted from 10. 5{a-if-b c is to be multiplied by 5. c) means that a^b LKi^(b—c) means that b c is to be added to 5a. 10
(«
—
—
—
When is
the parenthesis
generally omitted.
is
used, the sign of multiplication
Thus, (a
h)y^(a-\-h),
is
written
i"-l)Ca+h).
A
Vinculum,
,
is
sometimes used instead
of a
Thus, a-\-byc5 means the same as b(a-\-h). Sometimes the vineulum is placed vertically it is then parenthesis.
;
called a hnr.
Thus,
a
X-,
is
the same as (a
h-\-c)x''.
—h 33.
Similar, or Like Quantities, are such as contain
the same' letter or letters with the
same exponents.
:
DEFINITIONS.
17
Thus, 3o6 and —2ab, BaPb and 5a't, 3a'6 and —^a'b, are similar.
Unlike Quantities are such as contain
diflferent
letters
or differeitt puwers of the same letter.
Thus, 5a and
3Z>,
Zah' and
are unlike or dissim-
3a'^6,
ilar.
Remark.— An taken
ters are
to
exception must be made in those cases where letrepresent coefficients. Thus, ax- and bx- are like
when a and 6 are taken
quantities,
34. The Reciprocal
of a quantity
unity divided by
is
Thus,
that quantity.
The
as coSfficieuts of a^.
reciprocal of a
is
—
;
of 3,
is tj
;
of
j,
is
1-^|=^
;
Hence, Tlie reciprocal
of a fraction
35. The same quantities
letter,
is the
accented,
fraction inverted.
is
often used to denote in different equa-
which occupy similar positions
tions or investigations.
Thus,
and
a,
a", a'", read, a, a prime,
a',
n second, a
third,
so on.
36. The
following signs are also used, for the sake of
brevity a quantity indefinitely great, or infinity.
oo, .
•
• .
,
.
•
.J
,
,
signifies therefore, or consequently.
signifies since, or because. is
tities, as
used to represent the difierenee between two quanc—'d, when it is not known which is the greater.
EXERCISES. First,
copy each example on the
and then read Second,
it,
find
a=2, b^3, c=4, 2d Bk.
slate
or blackboard;
common
language. is, express supposing each, in value numerical the that
a;==5,
2
it
y^6.
in
— RAY'S ALGEBRA, SECOND BOOK.
18 1.
76+x—y.
2.
d'hij-^ixK
ceo— ay
Ans. 20. Ans. —3. Ans. 10.
3.
c4-aX''^a-
4.
(c-|-a)(c— a.)
Ans. 12.
5.
——
Ans.
Ans. o6(c
„ ^.
jJ
Ans. 4.
l}-,.
— a)
y—c
-j/a6y. Ans.O.
Find the difference between alix, and a-|-6-|-a:, when x=3 and when a^5, i=*7, a:^12. Ans. 11 and 396.
9.
a=4, 6=i,
;
10. Required the values of a''-{~2ab-\-V, and a^
a=7
when
the value of
is
7t=4, and when j!=10 12.
— 2ah-\-¥,
Ans. 121 and
and Zi=4.
What
11.
when
5.
?
9.
n{n~l) (n— 2) (n— 3), when Ans. 24 and 5040.
Find the difference between 6o5c 2ab, and Qobc^2ab, i, f, are 3, 5, and 6 respectively. Ans. 492.
rt,
— when a=5
13. Find the value of
,
and 6^3. Ans.
Verify the following-, by giving
whatever
to
5^.
each letter any value
:
14. a(jn^n')(in ,-,3
15.
— 92)=om' —
an^.
)/
^x'''[-xy-\-y^.
TO BE EXPRESSED IN ALGEBRAIC SYMBOLS. 1.
Five times
2.
X,
3.
X plus
-t.
5.
a,
plus the second power of
plus y divided by
3.3.
divided by
3-.
y,
h.
3 into X minus n times y, divided by in minus n. a third power minus x third power, divided by a sec-
ond power minus x second power. minus the square root of n. 6. The square root of 7.
The square
root of
m m
minus
n.
— BAY'S ALGEBRA, SECOND BOOK.
20
39. Second Case.— Let of +9«, —5a, +4a, and Here, -\-9a+4a
Now
since
which one evidently
— 7a
0,
tity -\-13a,
two
of
will
+6a
and leave
sum sum of
In like manner, to obtain the
is
OPERATION.
of
-I-
and +2a, we find —13a, and the sum
the
4a,
—
— 9a,
+ia —2a +6a
-\-5a,
— 9a and —
and
of -|-5a
of
-la
-f 2a is -f 7a.
OPERATION.
—
—9a
7a in the quantity 13a; Now, -j-7a will cancel 6a for the aggregate. Therefore, which leaves
—
9a
— 5a
for the aggregate, or re-
sult of the four quantities.
—
—7a.
is
equal quantities,
and the other negative, is cancel -\-7a in the quan-
positive
is
be required to find the sum
+130; and —5a- 2a
is
sum
tlie
it
—2cx^ Ibbcx^ ~' ZbxX^cx'
2a _ ~ Sbx'
Or, dividing
Or,
by
nnp 3ai'
36a;
3d'
Ans.
8.
Ans
— 3a6
Ans
5.
a+b a
—
2ax iax'' 1-2.T. Ans, ~"3~6ax 5a,''-{-bax
—
9,
cf
b'
1
1+x'
Ans.
6.
10
Ans.
x'+l
to be solved
x—1 x+2-
'x'—x-\-l'
Ans.
27a;*+63a;^— 12x^—2 8x
5a
—X
a
a;^+5a;+6'
—
The following examples are
Ans.
x'
a:^+2a;—
a'
12.
m.-\-p
—
ex'
3a^-|-3a6 4.
Ans. m''p-\-7np''
a-|-a;
3.
— m'p
7.
5
3a'6x' a.-c-f-a;'
^^^'
bcx'^,
Ans.
2.
61
Sa'+l 9x'— 4a;"
by factoring, but the
process requires care and practice.
-
„
_
,
16. Keduce
— —^^-
x''-\-(a-{-c)x-i-ac ,
;,
(
.
,
to its lowest terms.
x'-\-(b-\-c)x-\-bc x^-\-{a-\-c)x-\-ac=x^-\-ax-\-cx-^ao
^x[x-\-a)-\-c{x-\-a)^x-\-c){x-\-a). Also,
a;2+(6+c)a;+6c=(a;+c){a;+6); .-.
^
the fraction becomes
a/+2bx+2ax+bf
(a;+c)(a;-|-a)
(x+cjjx+b)
x-\-a
~ x+b'
,
Ans.
/-|-2a;'
——
—
:
RAY S ALGEBRA, SECOND BOOK.
62
xM-xy+rry+y
j^
-.
^^^_
—y*
X*
'
— —
a}+(a-\-l')ax-\-lx'
r,
Ans.
^
a*
120.
y''
'
,
ox
a
O'x''
.
g"-'
j^ ax'"—hx"'+'
^^^
bV
a'ia;
—
a-\-x
.
i—
16.
a^'+y x'
6(a-j-6a;)'
Exercises in Division, in whieh the quotient
is
a fraction, and capable of being reduced 1.
2aV
Divide
axA-x' by ' ohx
2.
Ans. t^.
by bd^x^b
—ex
Ans.
a>-V
3.
Ans.
by a^-b^ •'
Case
II.
121.
— To
'-
c
?!!±^^. a-\-b
a»— t' by fa— by
4.
——
-r-,
6b
'
Ans.
+
7
.
reduce a Fraction to an Entire or Mixed Quantity.
Since the numerator of tbe fraction
may be
re-
garded as a dividend, and the denominator as the divisor, Hence, this is merely a case of division.
— Divide
the numerator by the denominator, for the If there he a remainder, place it over the denominator, for the fractional part, and reduce it to its lowest
Rule.
entire part.
terms.
1.
Reduce
—^ — a'
a^—ax
'— to an entire or
mixed quantity.
a.c
a^—ax
a—x
1
—
6
ALGEBRAIC FRACTIONS. Reduce the following
to entire or
63
mixed quantities Ans. X
^-
a
a— i;—
O.
A.
^-
5-
„
l+2a:
A
,
c
ISa:^ ,
Ans. x^
— bx ,
xV—z'-i-xz—2
III.
'a— r
^°^- l+^"+II=3-x-
x'
Case
1
.
a
a+oH ^
Ans.
l=3i x^,
:
x-\-l
—
.
,
,
X z
—
—1
x-^1
— To reduce a Mixed
Quantity to the torm
or A Fraction.
122. we have Rule,
This
is,
obviously, the reverse of Case II.
Hence,
the following
—
1.
Multiply (he entire pari hy the denominator of
the fraction. 2.
Add
the
numerator
3.
Place the
the product, if the sign
to
fraction he plus, or subtract
it,
of
the
if the sign he minus.
result over the denominator.
Before applying this rule,
it is
necessary to consider
—
123. The Signs of Fractions. Each of the several terms of the numerator and denominator of a fraction is preceded by the sign plus or minus, expressed or understood and the fraction, taken as a whole, is also preceded by the sign plus or minus, expressed or understood. ;
Thus, in the fraction
—
,
the sign of a^is plus; of 6^, minus;
x-\-y
while the sign of each term of the denominator sign of the fraction, taken as a whole, is minus.
is
plus; but the
—
—
—
:
RAY'S ALGEBRA, SECOND BOOK.
64
134. It is often convenient to change the signs of the numerator or denominator of a fraction, or both. By
the rule for the signs, in Division (Art. 69),
^^-|-&; or, changing the signs of both terms,
If
we change
If
we change the sign
the sign of the numerator,
we have
we have, ^377-=+^-
— ——ab --
-\-ab
Tlie signs
1.
without altering 2.
we have
,
=
Hence,
o.
of both terms of a fraction may be changed, its value or changing its sign, as a whole.
the sign
If
of the denominator,
^ —o.
of either term be changed, the sign of Hence, also,
the
fraction will be changed. Tlie signs
3.
of either term of a fraction
ivithont altering its value, if the sign
same
at the
Thus,
,
,
And,
be changed,
time.
a^x
^=
a—x
=
—a-\-x
=
l~a—x)=za-\-x. ' ^
— a-A-x'- —a-\--=o.A
a2— a;2
a
may
of the fraction be changed
,
a:^—x''-
!
—
"
-a~\-x
Applying the above principles, the
wade
Reduce the following quantities 1.
2
+ '
A.
and
3 a
a-YxA
3. a^
2— a
DO
Ans. M- and
x
— ax-^x?
——
.
Ans. '
{a—xf
2a-;r+^-^^-^.
o.
a'
a
-}
a+6
I.
——
Ans.
X
'
a-\-x
Ans
X r
be
form
to a fractional
a-\-x 4.
may
sign of the fraction
plus, in all cases, if desired.
a'
X' ,
Ans.
— -y. a+6 ab
:
ALGEBRAIC FRACTIONS. 0.
7.
a
65
—
—X
Ans.
a+x
a-\-x'
l-^^H^ ^+y
^^•y
Ana
^/-
Case rV.— To reduce Fractions of Different Denominators TO Equivalent Fractions having
A Common Denominator.
125. — 1.
Let
,
a
it
be required to reduce ^
.
common aonominator. If
we multiply both terms
of the first fraction
— —
m
,
,
and
-,
n
to
r
hy nr, of the
sec-
ond by mr, and of the third by tnn, we have
anr ,
'mnr
As
bmr mnr
,
,
and
cmn mnr
.
the terms of each fraction have thus been multiplied
same quantity, the value
by the
of the fractions has not been changed.
Hence,
(Art. 118.)
TO REDUCE fractions TO A COMMON DENOMINATOR, !Rule.
— Multiply hoth terms of each fraction
uct of all the denominators, except
its
own.
hy the prod-
Or,
Multiply each numerator
hy the product of all the own, for the new numerators. 2. Multiply all the denominators together for the common 1.
denominators except
its
denominator.
com-
fractions in each of the following to a
Keduce the
mon denominator „
1
2
2.
-,
-,
ah ha
X 3.
T-
.
Ans.
Ans. .,
and
2d Bk.
xyz'
a
x+a 6
.
Ans.
——
a?-irax -!
^
a''
,
and -
x—a
yz 2xz Zxy -^ ^, —, ary«
a;^z
2
y
X 4.
3 , and -
r-
x^—a^
b^
and -j ab
-=-
ab ,
and
— — a'
ax
x'—a^
;
RAY'S ALGEBRA, SECOND BOOK.
66
denominators of
It frequently happens, that the
12G.
the fractions to be reduced contain a
common
mon
denominator. it
he required to reduce '
least
ah— run
— m
,
Let
1.
common
.
denominator.
,
-
com-
least
c
and ^,
,
In
factor.
such cases the preceding rule does not give the
to their
nr
Since the denominators of these fractions contain only three prime factors,
m,
and
n,
r,
it is
least common denominaand no others; that is, it will
evident that the
tor will contain these three factors,
be mnr, the L.C.M. of m, mn, and nr.
now remains
It
to
to
reduce each fraction, without altering
its
value,
another whose denominator shall be mnr.
we must multiply both terms, of the first fraction by r, and of the third by to. But these multipliers will evidently be obtained by dividing mnr by m, mn, and nr; that is, by dividing the L.C.M. of the given denominators by To
effect this,
nr, of the second by
the several denominators.
Hence,
TO REDUCE FRACTIONS OF DIFFERENT DENOMINATORS TO
EQUIVALENT FRACTIONS HAVING THE LEAST COMMON DENOMINATOR, Rule.
—
will he the 2.
Find
1.
Dividp the L.C.M. hy the
and multiply the
product
3.
the L.C.M.
of all the denominators;
first
tcill
Reduce the
to
find each of the other numerators.
fractions, in each of the following, to equiv-
common denominator
_^, A, _! 3a;'
the given denominators,
of the given numerators
he the first of the required numerators.
Proceed thus
6xy'
of
the quotient hy the first
alent fractions having the least
2.
this
common denominator.
Ans.
2y
a+6' a—h' a'—h''
'
'
a:'—h'
'
_^ ?^ Sa-y'
:
3c^
Qxy' Qxy'
d'—h'
'
d'—h'-
—
—
.
ALGEBRAIC FRACTIONS.
m— n
^
mV —
m-\-n
m-\-n'
m —n
ni'
— —
(m '
n''
67 {m-\-ny
n)*
ni'
li'
m'
'
—
ni'n' '
ii'
m'
—
n''
Other exercises will be found in Addition of Fractions.
Note
.
— The two following Articles may be of frequent use.
137. To
reduce an entire quantity to the form of a
fraction having a given denominator,
RulCc tor,
and
1.
Multiply the entire quantity hy the given denominawrite the product over
Reduce
a;
to a fraction
it.
whose denominator
a.
is .
Ans. 2.
Reduce 2az
to a fraction
whose denominator
ax — a
is z'.
2a2»
Ans. z'
3.
Reduce
x-\-y to a fraction
whose denominator Ans.
128. To
is
x
—y.
^ x—y
convert a fraction to an equivalent one hav-
ing a given denominator,
Rule.
Divide the given denominator hy
the given fraction,
of
the
denominator
and multiply both terms by
the (juu-
tienl.
1.
Convert |
to
an equivalent fraction, having 49 for
denominator.
its
a
Convert = and 3 denominator 9c^ 2.
Ans. |^. 5
—
to equivalent fractions having; the
c
3^p2
^^^ and =
Ans. -7^—
-
yc' 3. •
ins:
Convert -^,- and
"~, „ ^ the denominator
.
a^
——
—
"if b\
j-
yc'
to equivalent fractions havA Ans.
(p+w /. ^
,
i—q .
n
—1
4.
n 5.
R
^-
=
1—x
'^—
r^
Ans.
n
,
and n
and
:;
1
—
—— — a'
P+q
"
,
Ans.
=-
,
Ans.
1
1
(z+l)(^+2) ^°
A
-,=
^
1
a^
=-
b'
,.
p'—q' 1 — 2™ — TO
k"
-.
,=
Ans.
and —r-r
Say
5ar
,
Ans.
n
.
— i-j— 1+x
1
{x+l)(x+2Xx+Sy ^'''-
(x+l)(*+3)-
——
—
RAT'S ALGEBRA, SECOND BOOK.
70 tj
a
.
(ad
— hc)x
a~^ Ix
.
'
c(c-\-dx)
c
8m+2n.
„
1
o.
Ti.is
o y.
2"3m——^r2ji «+c ;
(a
,
and
1
c-\-dx'
3™ — 2re
;=-._
.
Ans.
p^-
2'39»+2)i
~ —a) — h)(x r
J and
— —— — h)(x 6+c
,
(a
9m'— in^x-\-c
.
.
A.
h)'
12mn
'
(x
— a)(x— 6)'
Find the value 1
r.
10.
/->n
Of
4»i „y^
o(l
ah
Sn
m-\-Bn
,
^+11 K—Q7T 6(1 — n)
n)
ac
he
2n
.
•
71
•
•
•
Ans.
m 1-
:
ALGEBRAIC FRACTIONS. 2d. If either of the factors is a
71
mixed quantity, reduce
it to
an
improper fraction.
When the numerators and denominators have common such factors be first separated, and then canceled.
3d. let
factors,
a+b Th„. _gg!_x.^(g+^)'_ 2a^X(a+6)(a+6) 'a^—b^'^ 4a^b {a+b)(a—b)4a'b ~2b{a^Find the products of the fractions T
1.
3.1;
^
4.T
,
^
by
and
—
8a'b
,
by
c^d g-,.
in the following
.
a
x^
^-
6
a'x
J^
Ans.
and
Ans.
iUFpJ
j-^
rr'+3x+2 a;'+5a;+4 a;^+2x+l '^"'^ 5^+7^+12 «'—^' 6c-|-&a;
i
a;+2
XX
x-\-l+- by
x—l+~
„
4a 6x
3x Ab
26
3a!
da;
4a
,
-^
,Q pr+(jJg+g?-)a;+gV
p
—
%y{c~xy
—x—y
Ans.
r^
} (a;— ^)"
8.
iqiB-
4a;(a+a;) °^"
o?—ax
a—oTt
J
a>+!/
j^,.
^°^-
1^
——
bed
^.
a*~x*
.
I_^and2+^
36y' c''— a;^' I-
and
Ans.
>
4-
-
Ans.
,x -+"— — a X a 3.
x'
.
.
Ans. ,
8a6
-^
a;^-f 1
,
„
,
+ x' 4. 9.i;^
8a6
9a;2
ps+(yi!— gs)a;— gte ' p+ga;
ga;
Ans.
rs-\-(rt-)rqs)x-\-qtx^.
Find the value
»-«'(i+5)(->;')-(^s)f3-')^+2a^ Ans ad
fee
——
c
.
BAY'S ALGEBRA, SECOND BOOK.
72
Case VII.— DIVISION OF FRACTIONS.
133. — 1.
Required
a Here, as in arithmetic, the quotient of ,.
,
a ^
„
tient of
Or
.
a
^
a thus,
ad
cd"
be
and -,=
.
is -;-
b
d'
-7-
-1 is -7-,
and the quo-
ad
,
dividea by '
ab-^ and cd-'.
(Art. 81)
ad
1
by
-z.
C,
or y-—
be
'
Dividing,
we have
„ Hence,
—3-,=-=—.
Rule.
ad
c -,,
c
-J-
afr-' 1
1
.
by c times —„ or
b
by
to find the quotient of j
Invert the divisor,
and proceed
as in 'middplication
of fractions.
Remark. — To
divide a fraction by an integral quantity, reduce
the latter to the form of a fraction, by writing unity beneath or,
multiply the denominator by the integer. Remarks 2 and 3, Art. 131, apply equally well
it;
to division of frac-
tious.
Required, in their simplest forms, the quotients ,
1.
2.
_ 3.
„„
Of
a^Vc
aZ»^t'
xy
Of -J—-; a-\- c a ^„ Of
——— -
—
Of
t)
r.
6.
^„ Of
—
(('
"•-'S+f X X*
„
n+x
a''x-\-a^ ,
,
.
Ans.
V.Of(jl^^+j^J-.(ll,-I^^).
:
7-
a
A»s.
—y
^— T— -^
^a —
x'-\-ax'
.
Ans.
X
y
a' ---
—
-;
tt^ —
x'
Ans.
T
ax
d'x
.t'
c^ii
a'x
a'
5.
.
Ans. -f.
;
xy
.
1.
{a'+ax+x').
.
Ans.l.
1
1
:
ALGEBRAIC FRACTIONS. 1x
3ar
Of
8.
73
2a;— 2
•
.j:— 1"
Ans
.
Ans.
3
a-— 5' X
—
To REDUCE A Complex Fraction to a Simple one.
133.
This
is
merely a case of division,
which the
in
dividend and divisor are either fractions or mixed quantities.
6
Thus, '
«+^—^
n
m
is tlie
same as
— by vx
aA
to divide
^c
.
r
•'
r ac-\-b
mr^n
ac-\-b
X:
acr-^br
Or, the following method, obviously true,' will generally be found
more convenient. Multiply both terms of the complex fraction hy the product of the denom,inators, or hy their L. CM.
we
Thus, in the above, multiplying by cr,
acr-\-br
— en'
have, at once.
cnir
Solve the following examples by both methods a-(-l
2x-
Ans. _
1.
x—1 a
f
g-
-f
a-\-l ,
1
— -1.
a—1 a+1
Ans.
TT— —2a
a+h+ b'
c
ili
^- e
3.
a
Ans ^°^-
fK^'^+^c) hd{eh-fgy
h 2d Bk.
7*
.
a+b+t
Ans.
.
RAY'S ALGEBRA, SECOND BOOK.
74
Resolution of Fractions into Series.
IS^. An ber of terms
Law
The
Infinite Series consists of an unlimited
which observe the same of a Series
terms, such as that
may
others
a relation existing
is
num-
law.
when some of them
are
between
known
its
the
be found.
Thus, in the infinite series 1
1
^
etc.,
.,-1-,
he found by multiplying the preceding term by
any term may .
Any proper algebraic fraction, whose denominator is a polynomial, may, by division, be resolved into an infinite series. 1 1.
l-x\l 1-J
X
Convert the fraction I
1
=
X
.
into an infinite series.
.r
— 2.t:+2.(;--2.r'-i
,
etc.
It is
evident that the
this series
after the
is,
laio
of
that each term,
second,
is
equal to
the preceding term, multiplied
by —X.
llesolvo the followinp; fractions into infinite series 1
2
ALGEBRAIC FRACTIONS. the
75
of the numerator and denominator may be by reason of some suppositions as to the values of the known numbers involved in the qviestion, thus giving rise to anomalous results requiring explanation. values
chan^"^ three times the square of the first two terms, plus three times the product of the first two terms by of the third, all three midtiplied by
By
reversing this law,
we
the third, plus the sqvare the third,
and
so on.
derive the following
Rule for Extracting the Cube Root of a Polynomial. Arrange the polynomial with reference to a certain letter.
1st.
2d. Extract the cube root of the
of
the root,
and
subtract
its
cube
first
from
term for the
first
term
the given polynomial.
BAY'S ALGEBRA, SECOND BOOK.
156
3d. Tahc thrre times the square of the first term of the root,
and
call
trial divisor
a
it
for finding each of the remaining
Find how
terms of the root.
the trial divisor
oftin
the first tirm of the nnieiinder ;
tainiil in
con-
Then form a complete divisor hy
srrimd term aj the root.
adding together
is
this tcill givr the
three times the square
of the
first
term of the
root, plus tJtrer times the
product of the first term hi/ the second, plus the square of the second. Multiply these l>y the second term of the root, and subtract the product fromi the firxt remiiiiider.
4th. Again, fiml t]te
of the
how
often the trial divisor is contained in
term of the remainder ;
first
this
uiU
give the third term
Then form a complete divisor as
root.
hcfore,
ing tngellicr three times the square of the first terms, plus
terms
hi/
three times the
term of the remainder.
these hy the third
from
product of the
the third, p>hts the square
the last
5th. Con/iiiue thus
till
root,
of
and
hi/
add-
and second and second
first
MultipAy
the third.
subtract the product
all the terms of the root are found.
Find the cube root of j-«--6.r'+12.r'+3(r'.r*— 8.c' 2„V-\- 1 2a'.r'-\- 3aV-— Go^x+a".
1.
1
2J5_63-''+12a;'' 6561,
index.
\5'5T2,
y 15625.
ADDITION AND SUBTRACTION OF RADICALS.
S04. It 13
Required
to find the
sum of
S^a
and b-^a.
evident that 3 times and 5 times any quantity, must
make
8 times that quantity; therefore, 3fa-\-5^a=S^a. But, if it were required to iind the sum of 3|/a and 5^'o, since j/a and fa are different quantities, we can only indicate their addition; thus,
3^a-{-5^a.
RAYS ALGEBRA, SECOND BOOK.
168 Similarly,
S^/2+Tj/^—iy'^=^v"^-
But 3,/5"and 4, 3 -^V^'Hi,/ 3. _ So also 3^/5 and 4(5/5-3, 6+4f 5.
_
Radicals that are not similar, may often be made so; thus, i/12 and , -" are equal to 2) '3 and 3^/3, and their sum is 5, 3.
The same principles apply
to the
From
the following
the above
Rule
subtraction of radicals.
for the Addition of Radicals.
radicals
mon
we derive
to
—
1st.
Reduce
their simplest forms, and, if necessary, to
the
a com-
index.
2d. If the radicals are coefficients, and.
prefix
it
to
similar, find the
common
the
sum of
their
radical; hut if they
are not similar, connect them by their proper signs.
Rule of
for the Subtraction of Radicals.
and
the subtrahend,
1.
Find the sum of
y 448
44S:^,/(j4
-
112=,
By
2.
Find the sum of
8-
Vv/iS-
12.
^128— 1^686— if 16+4^250
13.
2f r+8f
14.
6^4a2-|-2f 2a+?/8a^.
Ans.
3f 2 9,yM
15.
2v/3-iv/12+4/27—2/3.3^ ^re+fST— f^=5l2+f 192— 7f 9
Ans.
^V^^
16.
loF
17,
Ans. ibfyl
^
Ans.
Ans.
1
10.
Ans.g^ab.
^j'^ + i^V{(i'f>-ia^b^+4ab>) ,
MULTIPLICATION AND DIVISION OF RADICALS.
205. The
rule
multiplication
the
for
of radicals
ia
founded on the principle (Art. 200) that The product of the n"' root of two or more quantities equal to the n* root of their product. That
is,
is
VaXv^5'=Va5. (See Art. 198.) a'{/bXoVd=aXcX\'bX'l/d=aci/bd.
Hence, (Art. 53,)
The
rule for division
Tlie quotient
is
founded on the principle that
u"* roots
of the
of two quantities
is
equal
to
the n'* root of their quotient.
That
is,
—
nia
—iVa ^Mi-; which
Raising both sides
,
.
,
to the ruh
.
is
,
thus proved:
power,
that the previous equation is true.
we have
Hence,
j-
=
we have
j-,
which shows
the following
Rules for the Multiplication and Division of Radicals.
— If the radicals
the sam,e index. I.
have different indices, reduce them
To Multiply.
— Multiply
the coefficients together
of the product, and also the parts under for the radical part of the product. 15* 2d Bk. coefficient
to
Then,
for the
the radical
RAY
170 II.
To Divide.
ALGEBRA, SECOND BOOK.
S
— Divide
the coefficient
the coefficient of the divisor
for the
of the dividend by
coefficient
of the
quotient^
the radical part of the dividend hy the radical part of
and
the divisor
1.
for the radical part of
the quotient.
Multiply 2y'aZ by Sa-j/oic.
^ab
2
Za^abc 6ay/a262c=6a/a262xc=6aXa&l/c=6a26/c. 2.
Divide ^a\/ah by 2|/ac. 4a, cib
'^
2
3.
2
2^3
Multiply liji
ia \ab
„
Vc
by 3i
2a
]bc
„
|6
Vac
j-
c "
Vc2
2.
3=2(3)^=2(3)e=2|/P=2|/'9.
3/2=3(2)2=3(2)B=3^23=3"i/"8. J
Multiplying,
4.
.
6
.
Ans.
Divide 6i 2 by 3,^2. 6,
2=6f/23=6j/8.
(1.)
3,5
2=3," 22=3,"
(2.)
Dividing
(1)
by
(2),
4.
we have
2|/2.
6.
Multiply 3i'12 by 5, 18. Multiply 4f 12 by 3,r4.
Y.
Multiply together Sj
8.
Multiply
9.
Multiply together
5.
'72,
3^5
by
3,
il
11. Divide 12. Divide
.
^
i/40 by ; 2. 6i/54 by 3v^2.
y
Ans. 24
f
6.
and |/2. Ans 140 Ans.
.
if 3, and
10. Multiply together '{7,
Ans. dOi/E.
....
7]
4|'''a. ^
...
^i
7\ and
5. 'f
A. i?.
12^^^.
\='
648000.
Ans.
'^^a;'.
Ans. 2y'^.
Ans. Gy'S.
RADICALS.
171
70f 9 by 7^18 ^72 by ^^2.
Ans. b^i.
15. Divide
4f
Ans.
16. Divide
^3 f ^Uy^jl
13. Divide 14. Divide
17. Divide
9 by
Ans. f3.
2^3.
72 by
Polynomials containing radicals
18. Multiply
3+^/5 by
may
2^W.
Ans.
y2.
Ans.
^
also be multiplied
;
thus,
2— 1/5.
3+ /F 2- 1/5
6+2/F —31/5—5
6— ,/5—5=1— y/5,
Ans.
|/2+l by i/2— 1 llv/2—4y'15 by |/6+i/5.
19. Multiply 20.
Ans. Ans.
21. Raise ^/g'+y'S to the 4th power. Ans.
22. Multiply a/l2+-i/19 by 23. Multiply
206. To
49+20|/6.
4ll2— ^19.
Ans.
5.
»'— a;|/2+l by a)2+xi/2+l. Ans. x*+l.
24. (a;^+l)(a;^— a;|/3+l)(x^+X|/3+l).
radicals, to
1.
2|/3— ^/lO.
Ans. ai'+l.
reduce a fraction whoso denominator contains
an equivalent fraction having a rational denom-
inator.
When tional if
we multiply both terms by Thus,
^=
yb Again,
—-^
the denominator is a monomial, as
if the
wall
become ra-
j/6.
""^v^^ayb '
ybx-^/b
denominator
^a^, the denominator
it will
is
-^a, if
become
b
we
multiply both terms by
fa'^fa'^^fa^^a.
:
RAY'S ALGEBRA, SECOND BOOK.
172
In like manner, if by multiplying
When
denominator is i/ct", a"'-"by Therefore,
will
become
ra-
")'
'
denominator of
the
it
tlie it
tional
the
fraction
a monomial,
is
multiply hoth terms by such a factor as will render the ex-
ponent of of
under the radical equal
the quantity
to the
index
the radical.
Since the
equal
sum
of two quantities, multiplied
of the form
^
by
their difference, is
of their squares (Art. 80); if the fraction
to the difference
and we multiply both terms by b
,
—
is
,/c, the de-
nominator will be rational.
a Thus,
a(h—^'c]
6+,
'
(6^, c){6-,
c
If the denominator is b
the denominator if
it 13
If
I
the
b
—
is
e,
I
|
—
b---^
j
r,
c,
is
of
b-~c
the multiplier will be b-\-y'c.
the multiplier will be
the multiplier will be
denominator
ah—a-^ c)
the form
),-
yb —
form
C;
If
and
b-\-y/c.
i'f'+)/i+^c,
rendered rational by two successive multiplications. result in a quantity of the
j
m — yn, which
may
it
The
may
be
first will
be made ra-
tional as before.
Reduce
tlie
following fractions to equivalent onea hav-
ing rational denominators 1
8
1.
4.
3
8—5/2
Ans.
r6
6
o
Ans. f f
3, 5"-2,^
0.
—
:
EADICALS.
173
1
+„x—yx'^—1
x-\-^x'''—l
the preceding transformations,
the process of
finding the numerical value of a fractional radical o
abridged.
Thus, to find the value of
^,
which
5,
value of which
very
much
divide 2 by the
2.2360679+.
But
—2= = 2/5 --L_,
the
true
O yo found by multiplying 2.2360679 by 2, and divid-
is
ing the result by
is
we may
is
^
1
square root of
Ans. 2x.
yx'+l+yx^—i
yx-^+l—/x-^—l
Remark. — By
,
5.
Rfeduce each of the following fractions to
its
simplest
form, and find the numerical value of the result
-='
—S
=""1
12.
"
Ans. .894427+, and .707106+.
|/ji
j/5 •
Ans. 15.745966+.
/5— y'd
POWERS 207.
Let
it
OF RADICALS.
be required to raise -j^Sa to the 3d power.
Taking y/3a as a factor three J
So,
times,
we have
'3ax f^X \/~Sa= ^Tfo?.
"l/ayy aiyCv^oT
.
.
to
n
factors,
=v'a^
Hence,
Sule for raising a Radical Quantity to any Power. the quantity under the radical to the given power, and
Raise
affect the result If the quantity
given power.
by reduction,
with the primitive radical have a
coefficient, it
Thus, the 4th power of
must
sign.
also be raised to the
2fZa^ is l&fWafi. becomes 16^27a6x3a2=48a2f So^:
This,
— BAY'S ALGEBRA, SECOND BOOK.
174
If the index of the radical is a multiple of the exponent of the power, the operation may be simplified. Thus,
(
J/
Za)'={J ^2af=y/2.a,
In general, ('^Va)''=y !i/y'a
If
index of the radical
the
may perform
(Art. 192.)
"="/(?.
)
Hence,
divisible hy the exponent of
is
the
power, we
tity
under the radical sign unchanged.
and
this division,
leave the quan-
Thus, to raise f 3a to the 4th power, we have f 8YcF—'\ dividing 8 by 4, we obtain at once ySa.
fSla*
=-^/'iia, or,
1.
Raise f^2a to the
4tli
power.
.
.
Ans. 2af^2a.
.
2.
Zf^2aV
3.
-^^'ac^
to the
2d power.
4.
1 ^ac-'
to the
4th power
Ans.
5.
1
3d power
Ans. cy
6.
-i/x
to the
"6c^ to the
—y to the
4th power.
.
Ans. lQ2ah'-^2ab\
.
3d power.
.
.
.
.
Ans. {x
Ans. c^
a.
a'c*.
3.
—y)\/x —y.
ROOTS OF RADICALS.
208.
Since
y'a^Ty'a
"ll
tract the roots of radicals,
Rule.
(Art. 192), therefore, to ex-
we have the following
Multiply the index of the radical by the index of
the root to be extracted,
and
leave the quantity
under the radi-
cal sign unchanged. Thus, the square root of f2a.
is
\
f2d=y-^a.
If the radical has a coefficient, its root If the
quantity under the radical
degree as the root
to
is
must also be extracted.
a perfect power of the same
be extracted, the process
Thus, \j i/Scfiis equal (Art. 192)
to
-^
may
be simplified.
^SaS^f/SaT
RADICALS. 1.
175
Extract the cube root of y'a'b.
16aY^-
Ans. 2d' f2^. Ans. -^la.
2.
The 4th
3.
The square
4.
The cube
5.
The cube root of (rn-\-n)i/m-\-n
root of
root of
root of
Ans. [/a'b.
.
^49a\
.
64^^80^
Ans. 4,^2^.
.
Ans. y'm-\-n.
IMAGINARY, OR IMPOSSIBLE QUANTITIES.
309. An
imaginary quantity (Arts. 182, 193)
is
an
even root of a negative quantity.
—
—
y a, and ^ 6'', are imaginary quantities. The rules for the multiplication and division of radicals (Art. 205) require some modification when imaginary quantities are to be mulThus,
tiplied or divided.
Thus, by
the rule
|/a2=±a. by
(Art.
^"^^Xi/— «=l/— «X— «—
205),
any quantity multiplied must give the original quantity, (Art. 198,)
But, since the square root of
the square root itself,
,/^aXi/^a=— a.
therefore,
SIO.
Every imaginary quantity
and
may
he resolved into two
factors, one
a real
pression, |/
— 1, or an expression containing
This
is evident, if
quantity,
we
the other the
imaginary ex-
it.
consider that every negative quantity
be regarded as the product of two factors, one of which Thus,
— a=aX —
1,
— 6^=6^X—
1.
is
may
—
1.
''"^ so on.
|/— a2=^a2X— l=v'«^Xl/^T=d=a/^.
Hence,
Since the square root of any quantity, multiplied by the square root itself,
must give
the original quantity;
^^-TxV'^ =—^-
Therefore,
(i/^=T?=
Also,
(/:=ij3=(v/-=Tpx^^r=-v^r=-/^T. (i/-i)^=(i/-if (v-i)'=(-i)(-i)=+i-
Attention to this principle will render tions,
all the
algebraic opera-
with imaginary quantities, easily performed.
Thus,
y'l^S
X
(i/=IF=-i/a&.
V^ = >/« X i/=i X V'^X i/^ = -/"Sx
—
—
1
RAY'S ALGEBRA, SECOND BOOK.
176
OPEKATION. If
be required to find the product of a by^ 1, the operation is
it
^
'
— —
a-\-b^/^—i by
c
performed as in the margin.
f>^/
—
a^4-ab\/—[
— a6i/^ + 62 Since
a'-\-b-
= {a-\-b^/ —
terms are positive is
the
sum and
may
b^/
l)(ffl
— 1),
any binomial whose
be resolved into two factors, one of which
and an imaginary
the other the difference of a real
quantity.
Thus, m^n={-y/'m-\-yriy^—r){-^/7rv—y''n-/—l).
—
—
1.
Multiply
2.
Find the 3d and 4th powers of a^/
^
ii'
by
i
b'K
.
Aus.
a-
2,-'^ by
—
"-=2:
3.
Multiply
4.
Divide
5.
Simplify the fraction
6.
Find the continued product of
—
and
(I,
1.
6y
^
.r—lr-^/
—
by 2^
8i
^^.
::j
—
a^^
.
.
;^=^
.
or what numlier
are
ah.
and
a*.
— 6^/
6.
¥.
^^
Ans. y^—lx-f«^'
—
—
a*.
Ans.
x*
1,
24+Y-i/— I, and 24— t-j/^, Ans. 625.
the imaginary factors?
VI.
Ans.
x-\-a,
1.
—
1.
— 1, .Ans.
.
.
Ans.
.
.
THEOEY OP FRACTIONAL EXPONENTS.
211. The
rules
for integral exponents in multiplica-
involution, and
evolution,
(Arts. 56,
iO,
tion,
divi.'iion,
1'72,
and 104.) are equally applicable when the exponents
are fractional.
Fractional exponents have their origin (Art.
196)
in the
.
•
FRACTIONAL EXPONENTS. extraction of roots,
Thus, the cube root of a^
4-"' and a
2
The forms aS, aS,
a
to the
nent
power of
|,
is
is
not
root.
a'.
",
1 and a
a exponent
|,
the exponent of the power
when
by the index of the
divisible
177
So the
may be
to the
a exponent
read
a"
root of
n'l'
a
to the
power of minus
— —m
m —
;
is
a"
power of or,
a
§,
expo-
MULTIPLICATION AND DIVISION OF QUANTITIES WITH FRACTIONAL EXPONENTS. shown (Art. 56) that the exponent is equal to the sum of its expoIt will now be shown that the nents in the two factors. exponents are fractional. the when same rule applies
212.
of any
It has been
in the product
letter
2
1.
Let
4
be required to multiply a* by a*.
it
a^=fa^='{/aP, a^=^a^=^^ai^,
But
this result is the
same as
(Art. 205.)
by adding the expo-
that obtained
nents together.
2,4
Thus,
22 2 4 in+12 a3Xa5=a3 s—^is iz^cii".
Hence, where the exponents of a quantity are fractional,
To Multiply, Rule. 2.
Let
Adding
it
—
|
213. By preceding
— Add
the exponents.
be required to multiply a
and
|,
we have
J-,.
by
Hence, the product
an explanation similar
article,
^
to
a^. is
aT2^ or ^p'a.
that given in the
we derive the following
rule.
Where
the
exponents of a quantity are fractional.
To Divide, Rule.
from
— Subtract
the exponent
the exponent of the dividend.
of the divisor
6
RAYS ALGEBEA, SECOND BOOK.
178
Perform the operations indicated ing examples
in each of the follow-
:
and a
J
Ans. Ans.
J/+1/3) (a;?—2/4). 1.
l_
2
1
i
and
2,
a;'"J/"^a;"j/"'
.
Ans.
.
8.
(a?— 64)-j-(a4_64).
9.
(a— 62)^(a4+a262+a46+62.)
.
.
.
,
a—
x^y—y^ Tn-^n
1_
1_
(a+6)™X(«+*)"X («— ^rXC"— ^)"-
7. a;3-4~2;4,
I
Ans. a^, and a^
(a3-f a363^_63)(a3— 63)
5. (a;4
6.
3
2xa3
112 11
2 4.
_i
2
1
1. a^Xo.'^,
Ans.
5 a;T5,
(a2— 62) m» an— gm
and
a;
'""
2/"-"*
Ans. a5_|_a?64_|_62 Ans.
a^— 62
POWERS AND ROOTS OF QUANTITIES WITH FRACTIONAL EXPONENTS.
S14. of
m
Since the m"" power of a quantity
Therefore, to raise
L
a"
L
to the nith
Hence,
Rule.
to
the product
power, we have
L
a"X«"X3
as,
;
and 7>4.
inequalities are said to subsist in a contrary sense,
when the greater stands on the in the other
319.
added
an inequality, same sense. Thus,
5>1
as,
;
Proposition
quantities, he
I.
to
— If
...
.
.
;
Any
a>6,
quantity
1,
and
may
Proposition
sense, the
,^9.
3^1.
.
— 96 — c.
from one
7^,6,
side of
If two inequalities
may
Hence,
an
in-
sign he changed.
its
exist
in the
added together, same sense.
be
the resulting inequality will exist in the if
4,
7.
same time
corresponding members
Thus,
in the
7~_:.5.
11
then a-|-c>6-(-c, or
eqiiality to the other, if at the
3SO.
.
and by adding and subtracting
— 1< + Similarly, if
both members of
Jrom
inequality will continue
to
— 54; then, or 12>10.
7+5>6+4,
When subtract
two inequalities exist in the same sense, if we corresponding members, the resulting in-
the
— INEQUALITIES.
183
equality will exist, sometimes in the same, and sometimes in a contrary sense. First,
7>3 4>1
By
we
subtracting,
find the resulting inequality
exists in the same sense.
3>2 Second,
10>9
In
8>3 n^o In general, values of a, or
this case, after subtracting,
we
find the
resulting inequality exists in u contrary sense.
if
a>6
b,
c,
and e>d, then, according to the particular and d, we may have a c^b—d, a c4
and
8X3>4x3,
or
inthe
24>12.
8--2>4-i-2, or 4>2.
Also,
This principle enables us to clear an inequality of fractions.
If the multiplier be a negative number, the resulting inequality will exist in a contrary sense. Thus,
—3—IX— we
2,
or
6>2.
derive
Proposition IV.
— The signs of
all the terms of both
members of an inequality may be changed, if at the same time we establish the resulting inequality in a contrary sense.
For
this
333. equality
is
the same as multiplying both
Proposition V.
may
be raised
root e'Aracted,
same
and
to
members by
2-16,
1.
— Both the
members of a positive insame power, or have the same
the resulting inequality will exist in the
sense.
Thus,
—
22', the radical, one,
+p
and
cases.
and fourth forms, where q
in the third
If,
]/
—q-YP't becomes
0,
is
negative,
and
x^—p
we iu
in the other.
then said, the two roots are equal.
It is
In fact,
we
if
substitute
p-
for q, the equation in the
3d form
oecomes x---2px--2J-=0. (x-fp)-, or, (xJj5)(a;+p)=0.
Hence,
The
first
member
placed equal
is
to zero,
the product of two equal factors, either of which, gives the same value for X. A like result is
obtained by substituting p- for q in the fourth form. 2d.
If,
in the general equation,
the two values of
x reduce
x=—p In
fact, the
x-~ 2px=q, we suppose q~0,
to,
fp=0, and x=—p—p:=—2p.
equation
is
then of the form
x-^2px~0, wliich can be satisfied only
or a-(x-|-2p)=0,
by making
a;=0, ora'-)-2p=0; whence, a;r=0, or a;= 3J.
we
If,
in the general equation, x--\-2px^q,
'2p.
we suppose Sp^O,
liave
X'=q; whence, In this case,
the
x^±y q.
two values of x are equal and Imve con-
trary signs, real, if g
is jiosilive,
forms, and imaginary, if q
as in the first
is negative,
and second
as in the third
and
fourth forms.
Under belongs
this supposition the equation contains only
to the class
4th. If
treated of in
.\rt.
2p=.0, and
/)
F,
,r.
CP =a—x.
then,
at the distance of /)
the two lights.
B
1
foot, is 6, at 2,
•
4 9 It) distance of X and of
3,
4,
.
.
B and
hence, the intensity of
;
a
—X
feet,
feet, it
/ must be - and X-
B
must be
of (1 at the '
.
(a — xj-
.
But, by the conditions of the problem, these two intensities are cciual.
Hence, wo have for the equation of the problem, ^ ,
r-
M'Ih
c = [a^x)-
r„
,
.
,
, which reduces
,
to
("— .r)2 ,
r-
—
e ==- ;
QUADRATIC EQUATIONS. Let i>c.
I.
The
first value
—=,'6
=.
IS
b+yc
1
—^—
of X,
also,
.
that
P
and, consequently,
is
since
—^
=\_
—
...
—=
and evidently
is positive,
than
less
a -r.
Z
second value of X,
— — — -,/6>^6-^c; ^ ^
—
_/
=,
yb~yc
.
positive,
is
—= b
->1, and
^'b—^
c
\'
.
This value gives a point
and
AVe perfor,
2y/byyl}+yc, and
or
a,
,
;
yb+yC^2-
yb+y'o
for
B and C. C than B
to
manifestly correct, for the required point must be nearer of less intensity. The corresponding value of a x,
di/c
Tlie
value gives for the point
tliis
nearer
-^ =>2, l/6+l/e This
than a, for
less
situated between
is
y 6-(- y^ 6> y/ 6+
the light
and
positive,
Hence,
P
the point
is
we have
6>c,
••
-,
a pi-oper fraction.
illuminated equally, n point ceive,
209
,
,
P', situated
B
in the same direction from
and greater than a;
a-,,1)
-
_ >«.
y/b—yo
on the prolongation of BC,
In
as before.
fact,
since the two
lights emit rays in all directions, there will be a point P', to the
right of C, and nearer the light of less intensity, which
illumin-
is
ated equally by the two lights.
The second value of a
— X, —^_-'—
be,
^,
is
negative, as
it
ought
to
6-|/c
1
and represents the distance CP', in the
opposite direction
from C,
(Art. 47.)
II.
p//
The
_
first value
B of
X,
P
dy/ b —r~-
_
Vb+yc>,/b+yb; 2J Bk.
Let 6|/e— ^6
This represents CP", and
rt.
^
is
-.
—
1
:"^
.-.
It is
';
r^l;
the suvi of the dis-
^/o-^'b
tances
CB and
BP'', in the same direction from
III.
The
first
values of
x and
of
C as
before.
Let h=c.
a — x,
reduce
to -^
which shows that
the point illuminated equally is at the middle of the line BC, a result manifestly true,
two lights are
upon the supposition that the
The other two values This result
is
are
reduced
to
which
is
first
is
system of values of
dently correct, for
BP and CP
(Art. 136.)
.
no point at any
also
finite
distance, except
equally illuminated by both.
IV. Let h=:c, and The
—^—=00
manifestly true, for the intensities of the two lights
being supjiosed equal, there the point P,
intensities of the
equal.
when
become
a^O.
x and a — .T, become
the distance
BC
becomes
0. 0,
This
is evi-
the distances
0.
The second system of values of X and symbol of indetermination, (Art. 137.)
a — x, become
=;
this is the
;
QUADRATIC EQUATIONS.
211
This result is also correct, for if the two lights are equal, and placed at the same point, every point on either side of them will be illuminated equally by each.
a=0,
V. Let All the values of
X and a
—X reduce
to
^c.
hence, there
is no point In other words, the solution of the
equally illuminated by each.
problem
6 not being
fails in this case, as it
;
evidently should.
This might also have been inferred from the original equation for if
we put a=0, -.,=
7
to
becomes -^
(x—a)^ x^ be true except when 6=c, as in Case IV. X-
23d\ 1.
Examples for
discussion
and
= —x" s,
illustration.
Required a number such, that twice
creased by 8 times the
number
which can never
itself,
its
Ans.
How may
the question be changed,
square, in-
shall be 90. 5, or
—
9.
that the negative answer,
taken positively, shall be correct in an arithmetical sense?
2.
The
uct 21.
diiference of
two numbers
Required the numbers. Ans. +3,
4,
is
-1-7,
or
and their prod-
—3
and
—7.
3. A man bought a watch, which he afterward sold for $16. His loss per cent, on the first cost of the watch, was the same as the number of $'s which he paid for it. What Ans. $20, or $80. did he pay for the watch ?
4. Required a number such, that the square of the number increased by 6 times the number, and this sum, in-
creased by 7, the result shall be 2. Ans.
What do
x= — 1,
or
—
5.
X show ? How may the question be changed an arithmetical sense ?
the values of
to be possible in
Divide the number 10 into two such parts, that the Ans. 4 and 6, or 6 and 4. product shall be 24. 5.
Is there
more than one solution?
Why?
RAY'S ALGEBRA, SECOND BOOK.
212
Divide the number 10 into two sucli parts that the |/ Ans. b-\-y 1. 1, and 5
6.
—
product shall be 26.
What
— —
do these results show?
7. The mass of the earth is 80 times that of the moon, and their mean distance asunder 240000 miles. The at-
traction
of gravitation being directly as the quantity of
matter, and inversely as the square of the distance from
the center of attraction,
it is
required to find at what point
on the line passing through the centers of these bodies, the forces of attraction are equal. Ans. 2158G.j.5-|- miles from the earth,
—
"
"
"
moon.
Or,
270210.4+
"
"
"
earth,
and
30210.4+
"
and
24134.5
beyond the moon from the
earth.
This question inyolves the same principles as the Problem of the and may be discussed in a similar manner. The required
Lights,
may be obtained directly from the values of page 208, calling a=240000, 6=80, and c=l.
results, -however,
X,
TRINOMIAL EQUATIONS. 240. A
Trinomial Equation is one consisting of form of which is ax'"-\-hx"=c.
three
terms, the general
Every trinomial equation of the form
that
is,
every equation of three terms containing only two
powers of the unknown quantity, and exponents
is
manner
an affected equation.
as
in
which one of the
double the other, can be solved in the same
As an example,
let it
be required to find the value of x
in the equation X*
— 2px^:^q.
QUADRATIC EQUATIONS. Completing the square,
X*
213
— 2.px--\'p'^=zq^p".
x'^—p=y' q-\-pK
,.x=±^p±y/q-^P''-
341. Binomial Surds. — Expressions Aij/B, like the value of x^ just found, |/A=ty/B, The first
of
form
the
or of the form
are called Binomial Surds.
of these forms,
A±y/B,
viz.,
frequently re-
from the solution of trinomial equations of the fourth
sults
and as it is sometimes possible to reduce it to a more simple form by extracting the square root, it is necesdegree
;
sary to consider the subject here.
We
show that
shall first
tract the square root of
it is
sometimes possible
A±|/B,
to ex-
or to find the value of
A±^B.
VLet
us inquire
how such binomial
surds
may
ai-ise
from
involution. If
we square 2±y'3, we have 4±4^/3-|-3, which, by
becomes 7±4|/3. it
may
Hence,
be shown that
-.
I7d=4^3=2±^3.
reduction,
In the same
way
.^5±2)/6=i/2±^/3.
It thus appears that the form A±-[/B may sometimes result from squaring a binomial of the form a±|/6, or j/a±j/6, and uniting
the extreme terms, which are necessarily rational, into one.
such cases,
A
and |/B
twice their product.
is
To find the ceed to find
is
the
sum
root, therefore,
X and
y, the
put a;2_[_j,2_A and 2a;2/=i/B^ and pro-
terms of the root.
Extract the square root of
Put
.
a;2
.
Thus,
7-]-4|/3.
+2/2=7
(1),
and
2xy=i^Z. Adding, we have Subtracting,
In
of the squares of the two terms of the root,
x^-[-2xy-|^y'^=l-\A^/'Z.
we have x2__2:r!/+2/2=7—4/37
RAY'S ALGEBRA, SECOND BOOK.
214
Extracting the
root,
x+y^Jl+iy'S
(2).
^•-.?/=A '-^1 3
(3).
(4).
By adding and subtracting 2.1/--— 6 y=•^W. Hence,
and
.
.
(1)
and
2— y-3
(4), is
we have 2a;-^8
Extract the square root of 15-)-6;
1.
6.
Of 34~24i/2 Of 14±4i/6.
2.
3.
We
shall
form,
when A'
Theorem
I.
—B
^
.T=a — j
'5;
.
111 o^
Theorem
II.
-T
X
.
squaring both
— a-~b ^
;
in rational quantity, which
— Jn any equation
,
that
sides,
.
IS,
...
an irrational
is impossible.
of the form
x±| y=art
the rational quan/i/iis on opposite sides are equal,
b,
is
irrational.
X^n-~'ln, b^b\ ..1
1
this it
;
value of a quadratic surd can not he
and partly
to a
W.
To do
a perfect square.
is
— The
equal
2±2v
-,
demonstrate more fully that
to
-,-,-!-,
is
6.
2.
A±| B may always he found in a simple
For, if possible, let
quantity
x^2
4-3^
Ans.
necessary to prove the following theorems
partly rational
.
Ans. 3-\-y Ans.
now proceed
the square root of
.
the root to be found.
and
also the trra/ional quantities.
For
if
X does not =a,
Therefore, that
is,
a—m—i
let
.i/=-a
x=a+m; — /d; m+i/3/=T/6; .-.
the value of a quadratic surd is partly rationaj
irrational,
which has been shown by Th.
x=a, and
1
.i/=|
I,
to
and partly
be impossible; hence,
6.
AYe shall now proceed to find a formula for extracting the square root of A-|-|/B.
C
C
.
QCADRATIC EQUATIONS.
....
Assume
215
^A+yii=y'x+yy, A.+y^li=x+y^2^xy, by
By
Th.
II,
CE+3/=A(l); and
Squaring equations
(1)
and
squaring.
2/^=^B(2); we have
(2),
a:2+2a;3/+y2=A2
=B;
4xy
x^—2xy+y2=A^—B;
Subtracting,
A2— B
Let
Therefore,
.
.
(x—y)^—C^,
=C^ or
....
But,
wv AVhence, And
be a perfect square
.
.
.
;
C=t/A2— B.
a;+2/=A;
—L_ x= A+C
,
;
and
/A+C ^x==tz'\~2~;
.
then,
{x~yy^=A^—'B.
x—2/=C;
A—
y= ——
_
—
.
or,
*"-0. {x^—2axY—6a'{x-—2ax)=iea*.
Hence, the given equation
Or,
.
.
.
Proceeding with the solution, we find a;=4a, —2a, or adizay'^.
even,
of the equation
by
RAY'S ALGEBRA, SECOND BOOK.
220
x*—2x'—2x'-\-Sx=10S.
2.
—3,
Ans. a;=.4, 3.
X*— 2a;'+a:=30.
4.
.^^'_6a;'+lla;— 6=0.
x^B, —2,
Ans.
.
.
Ans.
or
14
12+Ax _
2,
or 3.
-^(3±i/-15> 1,
or
2±|/5.
i.(l±,/=43).
1
~2x'^
3x
|,
x=1,
or
—
°-
T.-^:'"*"
.i(l±,/_19).
—
43;*+|=4a;'+83. Ans.a;=2,
30
or
x^b,
7.
X
^(1±^— 35).
Ads.
.
x«— 6.x'+5x^+12a:=60. Ans. x^b, —2,
5.
or
Ans.
^'
x=4,
3, or
-i(7±v'69).
SIMULTANEOUS QUADRATIC EQUATIONS CONTAINING TWO OR MORE UNKNOWN QUANTITIES.
244. unknown and
ftuadratic Equations, containing two or more quantities,
may
be divided into two classes, j^ure
affected.
Pure Equations embrace those that may be solved without completing the square. Affected Equations embrace those in the solution
which
it is
of
necessary to complete the square.
The same equations may sometimes be solved by both methods.
PURE EQUATIONS. 345.
Pure equations may
in general
be reduced to the
solution of one of the following forms, or pairs of equations. .1 ^
a;+2/=a 1
^
x~2/=c
)
x'+f=a \
QUADRATIC EQUATIONS.
We
221
method of solution
shall explain the general
in
each
of these cases.
To find
x-\-i/=^a
solve
X
—y.
Squaring Eq.
(1),
.
Multiplying Eq.
(2)
Subtracting,
.
(2),
we
must
x^-\-2xy-\-y^=a'^;
.
by
xy=h
and
(1),
ixy
4,
=46;
x-—2xy^y^=d^— 46,
.
(a;— 2/)2=0(2— 46;
Or,
....
Whence,
a:— 2/=±|/a2— 46;
x-\-y=a;
But,
Adding, and dividing by
x^\a±\^/a'^
2,
Subtracting, and dividing by
The pair
of equations (2)
that in finding x-\-y,
square of the
The pair
3/=Jaq=J|/a2 — 46.
2,
is
— 46.
solved in the same manner, except
we must add 4
times the second equation to the
first.
equations (3) is solved merely by adding and subby 2 and extracting the square root.
ot
tracting, then dividing
Given
1.
a::'-(-y=:25,
Squaring the 2d
and
Eq.,
But,
a;2
Subtracting,
Taking
(2)
Whence,
.
.
(1).
=24,
(2).
x^—2xy->^y'^=
(1),
....
But,
+2/2=25
^y
.
from
.-c-f-j/^T, to find
(3)
and
(4),
(3). (4).
and dividing by
a;=4, or 3; and 2/=3, or
Given a;^+a;^+/=91(l), and
2.
X and
x—^/l^-\-y=
(2),
But,
By
subtracting,
Whence
.
.
.
2,
4.
a;+|/^+^=13(2),
y.
Divide Eq. (1) by
y.
1
x—2/=±l x+y=l
.
Adding and subtracting
to find
x and
x'^-\-2xy-\-y^= 49;
7.
(3).
a:+T/gy+y=13.
(2).
2^/xy=&.
l/^=3, and xy=%.
(4).
RAY'S ALGEBRA, SECOND BOOK.
222 By adding Squaring,
(2)
and
(3),
.
(5),
Multiplying (4) by
.
^+2/=
.
10.
(5).
x^--i^2xy-\-y^-=lQ0;
.
= 36;
Ajcy
4,
x^~'lxy-\-y~^ But, x-)-2/=10; whence,
x=%,
or 1
64,
.
.
x—y=±&.
and 2/=l, or
;
9.
Equations of higher degrees than the second, that can be solved
by simple methods, are usually classed with pure equations of the second degree. J
Given
3.
In
x'^
3
i
3
-\-y"^Q, and a;''-(-^^^126, to find x and y.
all cases of fractional
exponents, the operations
may
be simpli-
by making such substitutions as will render the exponents integral. To do this, put the lowest power of each unknown quantity equal to the first power of a new letter,
fied
11
a
.3
X'J^P, andj/S^Q; then, a;4=p3 and 2/*=Q^. The given equations then become, In this example,
let
P+Q= p3-(
Dividing Eq,
(2)
Squaring Eq.
(1),
by
Subtracting,
(1),
(2).
.
P=- PQ+Q==21; P2+2PQ+Q2=36;
.
3PQ=15,
(1),
.
6
Q3=i26
.
PQ=5.
.
Having P+Q=6, and PQ=5, by the method explained and Q=l, or 5. (1), we readily find P=5, or I
in form
;
Whence, a:=625, or 4.
1
;
and 3/=!, or 3125.
Given (cB—^)(.'b'—^')=160
(1),
(a;+!/)(a-.^+3/^)=580
Add
x3—a;22/—a:)/2 +2/3=1 60
(1),
a;3_|_a;22/-(-a:t/2+2/'=580
(2),
2a:=2/+2a;)/2=420
(3),
(3) to (2),
Extract cube
From
(3)
(2), to find
7.
y.
by multiplying. "
"
by subtracting.
x3+3a;22/4-3x?/2+2/3=1000.
root,
.
a;-{-?/=10.
xy{x^y)=2\Q;
.
From a;+2/=10, and xy=2\, we readily 2/=3, or
x and
.xy=2\. find a;:=7,
or 3;
and
))
;;
QUADRATIC EQUATIONS.
223
Solve the following by the preceding or similar methods 5.
x—y=1,
6.
a:^+/=13,
Ans.
")
x=15, ^^13,
or
Ans.
!:} 7.
2x+3/=7,
Ans. I
8.
a;^—/=:16,
a:—y=2. 9.
a;+y=
.
.
.
.
.
.
j
11,)
—y^x=\.^.
.}^f =24.
14.
.
.
.
.
Ans.
15.
Ans.
=16.
y=7a;y, x'—f=.lxy,
-y=2.
\ ;
Ans.
.
.
xy -j-^'^13.
X—y=i/x4-j,-'y,
Ans. J
J+/^= 3
x=7, y=4,
or 4;
Ans.
x=4; ^=2.
or 7.
x=±7;'
5,
x=5, y=Z,
x=12,
or 3;
or 5. or
or
— 2;
3/=2, or
—4.
x=4,
x^±3,
or
±1
or
±3.
Ans.
x=16,
y^ Ans.
or
x=16,
or 81;
2
x^+y^^ X
9;
9, or 16.
3^=27, or 18.
4;
4, or 12.
3/=±l,
x^—^^=37. 17.
x=5;
y=
I
x*+xy+/=9l, x^-\-
16.
}
x2+y^-|-x^=208 =208,
X
Ans.
Ans.
—xy-|-^^=l 9
x+3/
or | or 4.
J
12. x'+2/'=152,
13.
x=2, y^^,
y=3.
Ans.
11. x''+a;y=84,
x°
x:^±3; 3'=±2.
I
10. Y(a»+2/»)=9(a;»-y), x?y
— 13; — 15.
or
-|-y
5,
^35.
Ans. x:=
8,
y=27,
8.
or 27; or
8.
;
;;
RAYS ALGEBRA, SECOND BOOK.
224 19.
»2+3/2=
Ans. a;=9, or
4,
y=l,
21.
14. }
a=+3/=
4,)
2/^2, or
:
.
-/-
Ans. X-
_
z(x+y)=c.
/
"^^
2(6+f— a) )— c)(5+c-
y^ \
y(x+z)=h,
or 3.
;_|_C— i)(a-)-6— c)
'
x(y+z)^a,
7.
Ans. a;=3, or 1
.
y^l, 22.
or 9.
Ans. a;=7, or 2;
20. a^-\-f=3bl,
xy=
1
2(a+c— J(a-|-c— 6)
'
(;,+c— g)(a+c— t)
\
2(a+?+cy+rf^+e=0
(1),
a:'+(aV+y)a^+cy+'^'3/+«'=0
(2)-
unknown quantities, we must now show that this operation pro-
find the values of either of the
eliminate the other.
We
shall
duces an equation of the fourth degree.
By subtracting
a—a'=a",
b
Whence, x=
the second equation from the
— l/=b'^,
etc.,
first,
we have
(a"y^b")xJr c"y^-\- d"y + e"=0. &'y'^^ d"y^€^'
_
a"y+b
and making
:
QUADRATIC EQUATIONS. As
this ralue of
X
225
contains y^, that of x^ will evidently con-
tain y*, which value of X-, substituted in the first equation, necessarily gives rise to
an equation of the fourth degree.
Hence,
The solution of two quadratic equations, containivg two unknoicn quantities, depends upon the solution of an equation of the fourth degree, containing one unknown
qiiantity.
As there are no direct methods of solving equations of any higher degree than the second, those of the class now under consideration can not be solved except in particular cases, and then only by indirect methods, or special artifices.
We
now proceed
to point out
some of these
special cases,
in addition to those already referred to in Arts. 242, 243,
and 245, with some of the more common
em-
artifices
ployed.
247. There are two cases in quadratics which may always be solved as equations of the second degree, viz. Case
I.
— When
one of the equations rises only to the
first degree.
ax-\-hy^c
Given
dx''-\-exy-\-fy'^-\-gx-\-hy=k
From
eq. (1),
we may X and
Case
II.
(2), to find
y and
— When
x'^
in (2),
x and
X in terms of the new equation
obtain a value of
stituting this value, for
dently contain only
(1),
2/.
y.
Sub-
will evi-
y'^.
both equations are homogeneous.
(See
Art. 30.)
Given
ax''-\-h
xy-\-cy''^d
a'x^-\-h'xy-\-c'y''=d'
(1), (2), to find
x and
y.
Put y^=tx, where < is a third unknown quantity, termed an Substituting this valu^ of y in the two equaauxiliary quantity. tions,
we have
a a;2+6 te^+e t^x'^=x\a
+6
t-\-e
P)=d
a'x^+b'tx^+c'tix^—x^{a'+b't+&t')=d^
(3),
(4).
RAY'S ALGEBEA, SECOND BOOK.
226 From
eq. (3),
we
find
...
«'= „,..,
From
eq. (4),
we
find
...
^''= '^
(5)-
,,.2
d' (^)-
„,,M^n'f2 +b't-^c'P a'
d Therefore,
Or,
.
,
d{a'-\-b't\c't-)=d'{a-\-bt^cfi),
.
a quadratic equation, from which the value of (Art. 231
)
and thence X from
or (B),
(5)
and
t
y
may he found, from the equa-
tion y^^tx.
34S. When
two quadratic equations are symmetrical
with respect to the the two
may
unknown
unknown
quantities
;
tliat is,
frequently be solved by substituting for the
quantities the 1.
t»wo
sum and X
Given
when
quantities are similarly involved, they
-\-y
=a
(1), (2), to find
x^-\-if'=h
Let x=S-\-z, and y^=s—z; then, x^=8^'-\- 5s^z
unknown
difference of two others.
+
1
Os-c-
s=^
X and
y.
(3),
+ 1 0.s2z3+ 5S£H -f z\
7/"=.s''— 5s'i2:+10s%2_10.s«+5s4rt— z'';
By
substituting the value of s=i^,
and reducing, we
find
Completing the square, we find the value of z; and from of
X and
(3),
that
y.
S49. An
artifice
that
consists in adding such a
equation as will render
is
often used with advantage,
number
it
to
both members of an
a trinomial equation that can
be resolved by completing the square, (Art. 240). I'ollowing is an
example;
The
QUADRATIC EQUATIONS.
^V^. + ^ + ^ = ^
Given
2.
a;^+y=20 Since
then
J^
?+
y
to
j
and
(2), to find
=2._|_2+|2; add 2
x and
y.
each side of eq.
(o
and
(1),
complete the square.
Whence, ?
Let
|
(1),
227
+ ?=±3-i = 5
-+^=5;
Whence,
xy^=^8,
—2.
or
_XA
tlien,
or
— =».
and 2a'2/=16.
From the equation a;2-l-?/2— 20, and 2a;2/=16, -we readily find a;=±4, and 2/=±2. x y By taking - -\- - = — |, two other values of X and y may be found.
330.
It
is
often of advantage to consider the sum,
difference, product, or quotient of the tities
as
a single
unknown
quantity,
two unknown quan-
and find
its
value.
Thus, in example 9, following, the value of xy should be found from the first equation, and in example 10, the value of
-.
y Other auxiliaries and expedients may frequently be employed with advantage, but their use can only be learned
by experience, judgment, and
Note.
—In
tact.
some of the examples
all the
values of the
unknown
quantities are not given; those omitted are generally imaginary.
8. x''-\-y''-\-x-\-y=2>ZQ,'\
a;2_y_)-a;—^=150. 4.
a;-|-4y=14,
/+4.^=23/+ll.
j 3
J
.
.
.
.
.
.
Ans. a;=15, or
y=
— 16;
9, or
—10.
Ans. a:=2, or
—46;
y==3, or
15.
RAY'S ALGEBRA, SECOND BOOK.
228 5.
2y— 3x=14,
Ans. x^=
.4.}
3:^^+2(3,-11)' 6.
2, or
y=10,
x—y=.2,
Ans. x=^b, or ^= =3, or
1.
8.
Sx'+ Xt/=1S,\ 4/4-3a;y=54. j x''+x>/=10,
Ans.
.
I
a:3-+2/=:24. I 9.
4,-ry=96— a:y,
.
I
x-]-y= X'
10. :l
y 11.
+ i^ =
11;
or 8f.
85
y
x~y=2. a;y=180— 8*^, a;+3y=ll. =12,
Ans.
x=±2, ^=±3,
x^ y^
3
•
4
'
-11.
or =t2i. 3
or rfiSy
S.
;
QUADRATIC EQUATIONS.
229
QUESTIONS PRODUCING SIMULTANEOUS QUADRATIC EQUATIONS INVOLVING TWO OR MORE
UNKNOWN SSI. — 1. plied
by the
QUANTITIES.
There are two numbers, whose sum multiless
is
equal
4 times
to
the
greater, but
whose sum multiplied by the greater is equal to 9 times the less. What are the numbers? Ans. 3.6, and 2.4. 2. There is a number consisting of two digits, which being multiplied by the digit in the ten's place, the product is 46 but if the sum of the digits be multiplied by ;
the same digit, the product
is
only 10.
Required the
number. 3.
Ans. 23.
What two numbers
are those
whose
plied by the difference of their squares
sum
multiplied by the
sum of
difference multiis
32, and whose
their squares
is
272
?
Ans. 5 and 4.
The product of two numbers
their cubes 133.
Note
.
is
10, and the
Required the numbers.
3.
sum of
Ans. 2 and
5.
—The preceding problems may be solved by pure equations.
5. What two numbers are those whose sum multiplied by the greater is 120, and whose difference muliiplied by Ans. 2 and 10. the less is 16? 6.
Find two numbers whose sum added to the sum of and whose product is 15. Ans. 3 and 5.
their squares is 42,
Find two numbers such, that their product added to sum shall be 47, and their sum taken from the sum Ans. 5 and 7. of their squares shall leave 62. 7.
their
8. Find two numbers such, that their sum, their product, and the difference of their squares, shall be all equal to each Ans. i+^-^/b, and i+iy^b. other.
:
RAY'S ALGEBRA, SECOND BOOK.
230
Find two numbers whose product
9.
ference of their squares, and the
is
equal to the dif-
sum of whose squares
is
equal to the difference of their cubes.
Ans.
A
10.
B
and
lyb, and
-|(5
gained by trading $100.
+ |/'5).
Half of A's
stock was less than B's by $100, and A's gain was
Supposing the gains in proportion required the stock and gain of each. B's stock.
tPq
of
to the stock,
Ans, A's stock $600, B's $400; A's gain |60, B's $40. 11. is
23
;
The product of two numbers added to their sum and 5 times their sum taken from the sum of their
squares leaves
Required the numbers.
8.
1950;
5
is
find the
S53.
their
;
a
sum
44, and continued product
is
numbers.
Formulse.
producing
^.
numbers, the difference of whose
12. There are three differences
Ans. 2 and
—A
Ans. 25, 13,
General Solution
to
a
6.
problem
quadratic equation, like one in simple equa-
tions, gives rise to
formula, (Art. 162,) which expressed
a,
We
in ordinary language, furnishes a rule.
shall illustrate
the subject by a few examples.
Express each of the following formulae rule,
1. 2/,
and
-solve the
numerical example by
in the
form of a
it
Investigate a formula for finding two numbers, x and
of which the
of the squares
sum
of their squares
Ans, x-^ly
Example.
is
s,
and difference
d.
W^+d)
y^.i-^, ;
2(^=7).
—
Find two numbers such that the sum and difference of their squares are respectively 208 and 80. Ans. 12 and 8. 2. 1/,
Investigate a formula for finding two numbers,
of which the difference
Ans. x=l,(d+j/d^
is
d,
+ 4p)
x and
and the product p. ;
j,=J(-rf+,/rf^+4p).
— QUADRATIC EQUATIONS.
231
—
Ex. A man is 8 years older than his wife, and the product of the numbers expressing the age of each is
How
2100. 3.
old are they?
Ans.
Man
50, wife 42.
Investigate a formula for finding a number, x,
which the sum of the number and
square root
its
Ans. x=s-|-^—
—
Ex. The sum of a number and what is the number? 4. its
The same when
square root
Ex.
it,
ys+f
square root
is
272;
Ans. 256.
the difference of the
number
x,
and
Ans. x^d^i,-\-i/ d-\-\.
is d.
— Find a number such
tracted from
its
of
is s.
that if
its
square root be sub-
the remainder will be 132.
Ans. 144.
Given x-\-y=.s, and xy^p, to find the value of ^-\-ift snd x*-\-y*, in terms of s and p. Ans. x^-fy=s'^' 2p]
5.
^^-\-y'i
a;' -)-^';=s'
— Zps
;
a*-\-y^^s'^—Aps^^2p''.
Ex.
—The
sum
of two numbers
is
5,
and their prod-
Required the sum of their squares, of their cubes, and of their fourth powers. Ans. 13, 35, and 97.
uct 6.
333.
Special Solutions and Examples.
— If an
equa-
tion can be placed under the form
(a;+a)X=0, in
which
X
represents an expression involving x, at least
one value of the unknown quantity
For since the factor ;=0, we a;= a, is one values of x will
—
tion
may
be found.
equation will be satisfied by making either
X=0.
have x-\-a^O,
and
solution
equation, and the other
of the
be found by solving,
Therefore,
if possible, the
equa-
X=0.
Thus, the equation x^
form (X
— — 4a;-|-4=0, may be placed Hence, — 2=0, ora;^-)-2; x'^
—2)[x''-\-x— 2)=0.
the other factor,
we
find a;=-)-l, or
a;
—
2.
under the and, from
5
RAY'S ALGEBRA, SECOND BOOK.
232
such an equation into
Skill in separating
factors
its
must be
acquired by practice. 1.
2
.-c— 1=2-1
Given
=, to find x.
V
-^
2
x-l=(yx^l){y7^1)
Since
2
2+":=;= -^Ux+I);
and O
Therefore, {y Therefore,
,
F+l)(/i^l )=-— =(i
_
J>i-1);
•*-
1
.r+l^rO, and
x=(— 1)^=1.
o
/«^1=-^,
Also,
Whence, '
,
by dividing by
—1; and x~A,
or
.r^-2,
— 3.f=2.
\/x-\-\.
(Add
2a: to
or 1.
each side.)
Ans. a;=
—
1, or 2.
9
3
r'
— =li
Trauspose
^.
and
,]
I
j
'(lii^OO J(— l=h^ _7).
Ans. ..=-^, or 4.
Ans. a;=l, or
2.c='^,r^=l.
5. rr-''— 3.s'+.r+2=0. 6. 7.
a;-|-7a;^=22.
ar+Ya:^— 22=(2-— 8)
+ 7(a;^— 2).
39z=81. Ans. ar=±3,
artifice that is
or
^(1±^
29±7v^^a;^— 2
number
is
a divisor.
1(— 13±^— 155).
frequently employed, consists
each side of the equation, such a
in
^. (jriven
x^^
]2+8i/z
—— — i?
X
Clearing of fractions,
Add
x'^
x-\-^ to each side,
^ ,
-
,
to find x.
— 5a;^12^8j
adding
to
or quantity as will render
both sides perfect squares. „ y.
S).
— 3)-
^(_3±v
Ans. a:=8, or
8. a^'+L^a-''—
An
Ans, a;=2, or
Ans. a:=3, or
a;'=6.r+9.
x.
and extract the square
root.
a;-2=±(4+v71-). From which we easily find x=9, 4, or J( — 3dz( —7).
QUADRATIC EQUATIONS.
10.
233
a:-3^HV^ X
Ans. a;=4(7±|/r3), -^(— liy-^F).
48
493-'
11-
,„ 6 ^ *" " -. 5—49=9+-.
,
^7
H
a-+-^
34a:=16.
Ans. a;=3, Divide by
14.
—8,
or ->(—
Ans. a;=±2,
7'K
(1—
^
each side.
to
x'
Ans. x=-2, 12.
1 Add i
X
3±i/93).
—8,
or
—h
\ 2 I
to
each
—3',
or
side.
J(— l±y_251).
9
ar",
and add j—
j
to
each
_-|--g— = —^^-.
Multiply by
2,
8x and add n^+Si
^°
Ob
841 IT— 232 2x'+-^ Sx
15 ^'"^ ^^- 27x'
-
I
Ans.
x^2,
Multiply both sides by
3,
(See Ex. 15, Art. 242.)
side.
Ans.
to
9,
—4,
or
—9.
each side.
1
-Lf^
37^+^-
—V,
transpose
or
—^
J(-2±i/— 266). and —j, and add
1
to
each side to complete the square.
We
shall
now present
a few solutions giving examples of
other artifices.
16.
pr-T'^i'^Cj to find
(1+xy
X.
l+x^=a{l+xY=a{l+4x-\-ex^+4x^-\-x^), 2d Bk.
(1— a)(l+a;#-
— KAY S ALGEBRA, SECOND BOOK.
240
Remark. —The
words
and proportion should not be con-
ratio
founded.
Thus, two quantities are not in the proportion of 2 to
but in the
ratio of
ii
2
A
to 3.
3,
between two quantities,
ratio subsists
proportion between four.
364. Each
of the four quantities in a proportion
The
3alled a lerm.
and
first
last
is
terms are called the ex-
tremes; the second and third terms, the means.
263. Of
proportion, the first and and the second and fourth, and the last is said to be a
four quantities in
third are called the antecedents, the consequents (Art.
257)
;
fourth proportional to the other three taken in their order.
S66.
Three quantities arc
has the same ratio
The middle term
third.
proportion when the
in
mean
a
is
first
second has to the
to the second, that the
proportional between
the other two. Thus,
then 6
a b
if
is
:
a mean proportional between
third proportional to
When
b
:
a and
c,
:
a and
e;
and c
is
called a
b.
have the same ratio between each
several quantities
two that are consecutive, they are said
to
form a continued
proportion.
367.
Proposition
of the means Let
.
is
equal
.
.
Since this
.
is
.
I.
to the
a b
.
:
by numbers.
c
:
:
a true proportion,
Clearing of fractions, Illustration
In every jjroportion, (he product product of the extremes. :
we must have
a
c'
bc
= ad.
2
6
:
d.
:
5
;
15;
and 6X5=2X15-
ad J 6c Taking oc=aa, we hnd a= c=-^, _
,
.
,
,
,.
,
— Or
,
O
(Art. 263)
o=ad —G ,
,
be a=-j-. CL
Hence,
RATIO AND PROPORTION. If any
three terms of
term
may
he found,
1.
The
first
—y;
241
a proportion he given, the remaining
three terms of a proportion are x-\-y,
what
the fourth?
is
—
^',
Ans. x^
1st,
in'-
or"-
— Ixy^y'^. — 2 The 3d, and 4th terms of a proportion are — and m-\-n required the 2d. — Ans.
and X
in
;
The
3.
1st,
——'—=, — a'
a—yh
h)',
{inx
m',
6,
n.
2d,
and 4th terms of a proportion are
and
^
^
—i-\=
i ;
required the 3d.
0+1/6 Ans.
This proposition furnishes a more convenient
test
1.
of proportion-
ality tlian the metliod given in Art. 263.
Thus, 2
equal
to
:
3
:
:
5
S68.
8,
is
not a true proportion, since
Proposition
tiBO quantities is
them,
:
3X5
is
not
2x8.
may
he
equal
made
the
II.
— Conversely,
to
If the product of product of two others, two of
the
means, and the other two the extremes
of a proportion. bc
Let
= ad.
Dividing each of these equals by ac, we have acl
ac'
6 d - = -.
Or,
a a
'
That
By
6c
ac
is (Art. 263),
o
:
b
:
dividing each of the equals by
:
at),
:
d.
cd, bd,
etc.,
we may have
the proportion in other forms. Or, since one
member of the equation must form we have the following
the extremes
and
the other the means,
Rule.
— Take
either factor
on either side of
the equation
term of the proportion, the two on the other side for the second and third, and the remaining factor for the
for the
first
fourth.
2d
BIc.
21*
242
RAY'S ALGEBRA, SECOND BOOK.
Thus, from each of the equations
may have
the eight following forms
a:b:
:
bc=ad, and 3/12^4X9, we
RATIO AND PROPORTION,
SYl.
— If four
Proposition V.
tion, they will he in
second will be
a b :
.
.
.
•
That
is,
,
b
If
....
It follows
:
G
j;=3:
5:10: :6:
a
:
:
d
c.
:
10:5: ;12;
12; then,
from this proposition, that the equation
6.
— = — may
converted into a proportion in either of two ways, thus
a b: :
QT^. ati
c
and
and
b
d, or
:
Proposition VI.
antecedent
cedent
:
the
d.
:
:
d .
.
is,
- = -;
Inverting the fractions, (Art. 263),
that
;
as the fourth to the third.
Let .
quantities are in propor-
proportion by Inversion
to the first
Then, (Art. 263),
243
•
a
— If two
:
:
d
sets
:
be
:
o.
of proportions have
consequent in the one, equal
an
to
ante-
consequent in the other, the remaining terms will
be proportional. Let
a b
And
a:b:.e.f c d e f.
.
Then
will
From
(1),
.
.
-=-;
Which gives 4
If
:
8
373.
:
:
.
10
.
:
.
.
:
from .
C
20 and 4
:
:
:
:
-='-.
d 8
d
:
:
:
Proposition VII.
(1),
(2);
:
(2),
:
c:
:
e 6
:
:
Hence,
f.
12; then, 10
— If four
the
second,
fourth.
:
20
:
:
6
:
12.
quantities are in pro-
Composition that sum of the first and second will he to the first or as the sum of the third and fourth is to the third or
portion, they will he in proportion hy is,
-=-;
;
244
RAY'S ALGEBRA, SECOND BOOK. a: b
:
:
RATIO AND PROPORTIOX. a: b:
Lei
Then
will
From
(1),
.
.
.a-{-0:a
.
By
... .
a
.
From which, Or, If
a-f 6
.
by
:12:3;
;
376.
:
:
b
:
c-\-d
c
:
a+6
6+2
then,
Proposition X.
—
c
:
—d :
a-]-b
alternation,
6:2
:
:
b
:
c~\-d
a —b :
:
(/.
(Arts. 273, 274,)
:
d;
:
d.
a—b
:
:
:
:
:
c-\-d
:
c
— d.
c—
cZ.
12+4:12—4, or8:4::16:8.
— If four
powers or roots of those
tion, like
—
d.
:
b
c
:
d;
:
(/
:
-.
6 -2
:
(1),
C-\-d
c^d
:
:
—b
(Art. 2712),
b
:
a—b
.
alternation,
And
c:d
:
:
:
by Composition and Division, a-f 6
And
—b
245
quantities are in proporqiiantities will also
he in
proportion.
a b
Let
:
....
Then
will
From
the 1st
That
If
.
.
:
6
8:27:
277.
10
:
:
:
b"
:
:
64:216; then,
:
:
C
:
c"
=—
6"
either a whole
30; then, 22
:
:
d,
:
rf".
:
Raising each of these equals
.
6" — — = d"
.
a"
is,
Where n may be If 2
:
— power,
to the n'"
a"
6^
:
o"
:
d",
:
number :
10^
:
:
or a fraction. 302, or
fW: ^27:: ^64:
Proposition XI.
—If two
4
;
36
f 216; or
100
:
:
900.
:
2:3:: 4:6.
of quantities are in
sets
proportion, the products of the corresponding terms will also he in proportion. Let
....
...
Then
a
m
And
am
will
For from
(1),
Multiplying
|=|
(3)
by
(3);
(4)
:
:
:
b
:
:
c
:
n
:
:
r
:
:
:
cr
bn
and from
~ =—
;
:
d
(1),
s
(2),
ds.
(2),
^=^
this gives,
am
(4).
:
bn
:
:
cr
:
ds.
RAYS ALGEBRA, SECOND BOOK.
246 3
If
9
:
2TS.
:
2
:
:
15
:
sum of
:
;
4
;
12; then, 15
135
:
:
8
:
:
72.
—
In any number of proportions any antecedent is to its consequent as
Proposition XII.
having the same the
and 5
6,
ratio,
sum of
all the antecedents is to the
all the con-
sequents.
Let
.
.
(I
.
a
Then
a b a b
Since
:
Since
:
:
:
:
c
:
m:
:
:
:
b
:
b
:
C
:
m
n, etc.
:
;
bc=:ad
we have
b-\-d^n. (Art. 267).
The sum of these equal-
ab=zab.
ab^bc-\'hm^ab-\-ad^an.
+ c4-m)=a(6+rf-|-?i).
6(a
This gives (Art. 268),
a:b:
:
:
we have bm^=an,
Factoring,
or 5
:
n,
....
ities gives
5
cl
d,
Also,
If
:
a+c+wi
:
:
10
10 :
;
:
2
:
10
4
:
3
:
;
6,
etc.
.
a+c+?n b^d^n. then, 5 10 5+2+3 10+4+6, :
:
;
:
:
:
20.
;
EXERCISES IN RATIO AND PROPORTION. 1.
Which
the greater ratio, that of 3 to 4, or 3' to
is
4'? 2.
Ans.
Compound
ratio of 3 to
4
;
the duplicate ratio of 2 to 3
and the subduplicate
;
last.
the triplicate
ratio of
64
to 36.
Ans. 1 3.
What
of the ratio
m
:
n,
that
it
may become
4.
5.
to 6, 6.
If the ratio of a to i
and of
3a. to
—a
to
If the ratio of
to 6ot,
and also
to
2^, what
is
?>
is
13,
what
is
a?
m
:
p—q
the ratio of la
Ans. IJ, and 3g.
If the ratio of a to
and of h
is
4i?
p ql mq — np
equal to
Ans.
to i,
to 4.
quantity must be added to each of the terms
to n
bn!
the ratio of a-\-h
Ans. is
-^,
what
is
g,
and
the ratio of
|.
m —n
Ans. 14, and 6|.
RATIO AND PROPORTION. If the ratio of
7.
ratio of
to'
a;
What
8.
5^^
—
bt/ is
what
6,
Ans. 7
the
is
to 11.
eight proportions aro deducible from the equa-
Ans. a a b
If x''-\-7/^^2ax, what
9.
—
yl
ab=a'—x'.
tion
1x
8a; to
247
is
:
:
:
a-^.c
a
—x
a-\-x
:
:
:
:
:
a
:
—x
:
h,
a-\-x
:
b,
a
:
a, etc.
—x
the ratio of x to ^?
Ans. X
:
y
:
y
:
:
2a
—
x.
10. Four given numbers are represented by a, b, c, d; what quantity added to each will make them proportionals? .
Ans.
is
be — ad — — — c-\-d ;
a
^.
b
11. If four numbers are proportionals, show that there no number which being added to each, will leave the
resulting four numbers proportionals.
12. Find X in terms of ^ from the proportions x:y::o?
and a
:
b
:
f^c-\-x
:
:
-.b^,
-^d-\-y.
13. Prove that equal multiples of two quantities are
each
to
ma mb :
other :
:
a
the
as
quantities
themselves,
or
that
b.
:
14. Prove that like parts of two quantities are to each other as the quantities themselves, or that
-.-.:
15. If a also that
:
ma
b :
:
:
nb
c :
:
:
d,
mc
nd,
m
:
b.
ma mb nc nd, and and n being any multiples.
prove that :
a
n
n :
:
:
:
16. Prove that the quotients of the corresponding terms
of two proportions are proportional.
379. The
following examples are intended as exercises
in application of the principles of proportion. 1.
sum as
Resolve the number 24 into two of their cubes
35
to 19.
may
factors, so that the
be to the difference of their cubes
;
;
BAY'S ALGEBRA, SECOND BOOK.
248 Let
X and y denote
the required factors; then, a;i/=24,
x^--y' Therefore, (Art. 275),
:
Or, (Art. 270),
35; 19;
2y^
54: 10;
Ix--
...
Or,
x-—y
.
:
x^
:
y
X
.
y'
27-
8;
3:
2.
From which 2/=j^; then, substituting the value we find X—dizG; hence, y=±4:.
of
y
and
in the equEi-
tioQ j'y_24,
2.
''c+l+^
Gn
-I
^2,
to find X.
Resolving this equation into a proportion, we have
X+l-^'x^l
f .-.
Or,
Or, (Art. 276),
Whence,
.r+y
:
:
;
:
:
2
:
.
;
.
:
:
1
:
2;
x—l 3 1 J-1 ::3:1;
if ^
:
x—\
:
J;— 1
2S
:
:
.'11:1; :
26
52.r^5G, or a;=lJ_.
.
x~y .r
2x
.
.
3. x-[-ij
a;_|_l
.
(Art. 275),
4.
2f ;r^l f^qri
(Art. 275),
.
.
^anJ + f
.
:
3
:
7
:
1, I
:
5, I
xy-\-y''=-\'2G.
.
.
.
.
.
Ans.
Ans.
x=A, y=2.
a;=±15,
]
Ans. a;=±9, .-rj/=63.
1
6.
a
-\- ^
a'
y=±n.
J
—
Ans
-h
2ay
b
3-=
x'
2ah .
1
8.
It
"+3;+i is
Ans. xz
~h'
a-
required to find two numbers whose product
320, and the difference of whose cubes their difference, as
61
is
to 1.
is
to
is
the cube of
Ans. 20 and 16.
RATIO AND PROPORTION.
—
380. Harmouical tities are in
249
Proportion. Three or four quanHarmonical Froportion when the first has the
same ratio to the last, that the diflereuce between the first and second has to the diiFerence between the last and the last
except one.
Thus, a,
b
—
1.
6, c,
and
c;
Let
it
tional X, to
We
are in harmonical proportion
when a d
a, b, c, d,
:
a—b
two given numbers a and
have,
2.
:
when a
c
:
— d.
:
c
:
:
a— b
:
be required to find a third harmonical propor-
.
Therefore, (Art. 267),
Whence,
:
.
.
b.
a x a — 6 b —x a(b—x)=x{a—b); :
x=
:
:
'2a—
:
;
b'
Find a third harmonical proportional
to
3 and
5.
Ans. 15. 3.
Find a fourth harmonical proportional a, b, and c.
eiven numbers,
Ans.
281.
Variation,
eral Proportion,
is
x, to
x=
three
ac
2a—
b'
or, as it i.s sometimes termed, Genmerely an abridged form of common
Proportion.
Variable Quantities are such as admit of various values in the
same computation.
Constant, or Invariable Quantities have only one fixed value.
One quantity
is
said to vary directly as another,
wlen
the two quantities depend upon each other in such a manner, that if one be
changed the other
is
changed in
the
same ratio. Thus, the length of a shadow varies directly as the height of the object which casts it^ Such a relation between A and B is expressed thus,
BAY'S ALGEBRA, SECOND BOOK.
250
A oc B,
the symbol
being used instead of
oc
varies,
or
varies as.
282.
There are four
A
I.
simply
Here
B.
tx
A
A
which
diiferent kinds of Variation,
are distinguished as follows is
:
said
to vary directly as B, or,
varies as B.
—
Ex. If a man works for a certain sum per day, the amount oi" his wages varies as the number of days in which he works. II.
A
Ex.
— The
cc
Here
=.
A
is
said to vary inversely as B.
time in which a
man may perform
a journey
vary inversely as the rate of traveling.
will
A
III.
oc
Here
BC.
A
is
said
to
vary as
B
and C
jointly.
Ex.
— The wages
jointly as the
to be received by a workman will vary number of days he works, and the wages
per day. T>
IV A
oc
—
Here
A
is
said to vary directly as B, and
inversely as C.
Ex.
— The
time occupied in a journey varies directly as
the distance, and inversely as the rate of travel.
These four kinds of variation may be otherwise modithus, A may vary as the square or cube of B, in-
fied
;
versely as the square or cube, directly as the square and inversely as the cube, etc.
—
Ex. The intensity of the light shed by any luminous body upon an object will vary directly as the size of the luminous body, and inversely as the square of its distance from the
object.
(See Art. 238.)
.
RATIO AND PROPOKTION. In the following articles, A, B, of
any variable
and
quantities,
corresponding values
C, represent
a, b,
251
any other corresponding
c,
values of the same quantities.
2S3>
If one quantity vary as a second, and
tliat
second
as a third, the first varies as the third.
A
Let
A
A
:
;
:
:
a
:
:
6,
and
B
c
that
is,
:
C
:
B
and
B,
oc
B
a
;
:
and
B
oc
284.
A
that
-=f,
If each of
:
:
A
In a similar manner
then
cc C,
t
cc
C
For
cc C.
272),
C.
may
it
A
shall
therefore, (Art.
c;
:
be proved that if
A
cc
B,
oc ^j.
vary as a third, their
tioo quantities
siim,
or their difference, or the square root of their product, will
vary as the third.
A a C, and B oc C
Let
By
the supposition,
.
then,
;
A a
.
.
:
Alternately, (Art. 270),
By Composition
A B :
.
or Division,
Alternately,
:
A
B 6
:
a:
:
Therefore, (Art. 277), (Art. 276),
.
.
....
a similar
may
285.
:
V
:
:
:
;
C
:
.
C
:
AB a6 :
|/AB
:
:
:
:
:
:
6;
6
:
:
C
:
e;
e;
e;
C^
/a6l
:
c^;
:
C
:
c;
C.
method of reasoning, the following propo-
be proved:
If one quantity vary as another,
is,
C.
b;
:
it
will also vary
as any multiple, or any part of the other.
That
oc
6;
:
.
l/AB"a
is,
sitions
a
:
•.
And
By
:
|/AB
also,
;
A±B B a±b A±B a±6 B A±B oc C.
Again,
That
C
:
:
is,
And,
C
:
C
cc
A:a::B:6;
Therefore,
That
A±B
if
A
oc
B:
then,
A
oc
mB,
or
cc
—
.
RAY'S ALGEBRA, SECOND BOOK.
252
3S6.
If one quantity vary as another, any poircr or root former will vary as the same jioiei r or root of (he
the
of
la Iter.
A
Let
oc
B
A"
then,
;
n being integral or
B",
oc
frac-
tional.
287.
If one quantity vary as another, and each of them any quantity, variable or invari-
be multiplied or diiiilcd by
products or quotients
the
able.,
A
Let
cc
B
q\
then,
;
v:itl
vary as each other.
oB, and
cc
A — B — oc 2
2SS. the
of
If one quantity vary as
latter
tivo
2
others jointly, either
varies as the first directly,
and
(he other in-
versely.
Let
A
cc
BC
then,
;
B
tx
A —
,
and C
A cc yr-.
B
\j
3S9. If A vary as B, Some constant quantity.
A
Let If
cc
B
;
i'.s
equal
to
B
we know any corresponding values of
200.
m may
A
and B, the
be found.
In general, the simplest method of treating varia-
them
to convert
is
multiplied by
A=n(B.
then,
constant quantity
tions,
A
into equations.
1. Given that y cc the sum of two quantities, one of which varies as x, and the other as :r, to find the corre-
sponding equation. Because one part
and
the other part
Therefore,
where
m
oc x, lot (his
oc X-, .
.
X and
'•
=ma", ^?(.r'-.
2/- mx-\-nx'^,
and n are two unknown invariable, quantities which know two pairs of corresponding values
can only be found when we of
"
y.
RATIO AND PROPORTION.
If y^r-f-s, where
2.
r cc
x and
s cc
y^6, and when x=^2, y=9, what
is
253
— and ,
if,
when x=l,
the equation between
X and y? n
n
r=mx, and s=-
Let
..
a;
But
if
And
a;=l, 2/=6,
if
m=4,
Hence,
6=m-f »i;
.-.
a;=2, 2/=0,
.
«=-?na;-|--. ^ ^x
9=2m4--.
.
2 n-=2, and 2/=4a;+-.
3. If y cc a;, and when x=r2, between x and y.
If y
4.
X
find the equation
;
Ans. j/=2aa;.
and when a;=^, " 3/=8
oc -,
between x and
y=:4ffl
find the equation
;
A
y=-
Ans.
y.
If y= the sum of two quantities, one of which varies and the other varies inversely as x^ and when x=-\, find the equation between 2/^6, and when x=^2, y=^b 5.
as X.
;
;
x and
,
Given that
6.
the 1st
is
y
i
is
s cc
constant;
Remarks. — 1. Ex.
5,
sum of three 2d varies as
s,
quantities, of x,
Ans. y^S-\-2x-\-x''.
/ is constant and s cc /, 2s=/, when ^=1. Find the Ans. s=-}^f('. and t.
P,
when
;
also,
The above examples may
we put x=:l
which
and the 3d varies
2, 3, y=^G, 11, 18, respectively;
a;.
equation between /,
if in
the
when x=l,
in terms of
Given that
7.
when
y^
invariable, the
Also,
as x'.
find
4 i o Ans. y^2x-\-—.
y. ^
in the answer,
y
all
be proved.
will equal 6.
If
Thus,
we put
x=2, y=b. 2.
The Principles of Variation are extensively applied
ical philosophy.
in
mechan-
RAYS ALGEBRA, SECOND BOOK.
254
ARITH:\tETICAL rROGRESSION.
291. An tities
Arithmetical Progression is a series of quanwhich iuorease or decrease by a common difference.
Thus,
3,
1,
5,
1,
9,
a-\-d, a-|-2(Z, etc., a, a
—
etc.,
d,
a
.,r
—
12, 9,
2t?, etc.,
6,
3, etc.,
and
a,
are in Arithmeti-
cal Progression.
The
series is said to be iiicrcasinr/ or decreasing, accord-
ing as d
is
positive or negative.
292. To
investigate a rule for finding
any term of an
arithmetical progression, take the following series, in which
the
first line
denotes the
an incrtasing arithmetical ing arithmetical series. 1
number of each series,
term, the second
and the third a decreas-
ARITIIMKTICAL PROGRESSION.
S93.
255
Having given the first term a, the common difnumber of terms n, to find S, the sum
ference d, and the
of the If
the
series.
we take any
same
arithmc(ical series, as the following, and write
under
series
it
in
an inverted
S=ll+9 Adding,
.
.
Whence,
.
.
To render
this
series both iu
u,
we have
order,
S= 1+3 + 5+ 7+
9+11,
+ 7+ 5+ 3+
1.
28=12+12+12+12+12+12. 2S=12x the number of terms, =12x6--72. S^J of 72^36, the sum of the series. method general, let l^ the and inverted order.
last term,
S=a+(a+d) + (a+2d) + (a+3d).
Then,
the
-\-l,
.
S=l + {l—d) + (l~2d) + {I— 3d). 2s={lJ,a)+(l+a)+{l+a)+(l+a). 2S^(^+a) taken as many times as there
And,
and write
direct
.
.
.
.
+a. +(?+«), are terms («) in
the series.
Hence,
....
2S=(^+fe)/i;
S=(;+a)"=r-t^)n. Rule
for finding the
Sum
of
Hence,
an Arithmetical
Series.
—
Multiply half the sum of (he two extremes by the number of terms. It also appears that
The sum of the extremes is equal to the svm of any other two terms equally distant from the extremes.
S04. The
equations ?=a+(»i
—
and
l)c7,
^:=(a-\-l)-^,
lurnish the means of solving this general problem
Knowing any which
enter into
three
of
the five quantities,
an arithmetical
series, to
a, d,
:
I,
n, S,
determine the other
two.
The following table contains the results of the solution diiferent cases.
As, however,
it
is
not possible
of all the
to retain these
in
a RAY'S ALGEBRA, SECOND BOOK.
256 the
memory,
best, in
it is
ordinary cases,
to solve all
examples
itt
Arithmetical Progression by the above two formulaj:
Ruquireil.
a, d,
n
=a-\-[n—l)d,
a,
d,
S
=-id±,/|2dS+(a-:irfr},
n, n,
S
d, n,
S
2S
(n— l)d
S
a,
d,
n
'7.
0.
10.
l-—a^
l-^u
a, n,
I
d,
n,
I
=
a,
71,
I
I — "n—V
a, n,
S
ln[2.l—{n—\)d}.
2(S-an) ~ ni^ii—l) I-
11.
a,
I,
S
12.
n,
I,
S
13.
a, d, I
+ (n-l)d\,
=i,i[2a
a, d, I
*
~>
I
'
— a"
"2S— i!— a' 2(nl—%)
I—
14.
a, d,
S
15.
a,
I,
S
IG.
d,
I,
S
17.
d,
11,
I
18.
d, n, S
+1,
_±,/(2a— (i)-+8dS— 2a+c{
M
"
2S
_
_2^+d±;/'(2;+d)^-8dS 'Id
=1
— [n—V)d,
S
(n— l)d 2
''n
19.
d,
I,
S
20,
n,
I,
S
2S
,
'
'
;
ARITHMETICAL PROGRJilSSION. Find the IS"" term of the
1.
257
series 3, 7, 11, etc.
Ans. 59.
a=3, n — 1=14, and d^4.
Here,
formula
2.
(1),
Substituting these values in
we have ;=3+14x4=3+5G=59.
Find the 20"' term of the
series 5, 1,
—3, etc. —71.
Ans. 3.
Find the
8"'
term of the series
|,
-^^,
J,
etc.
Ans. 4.
Find the
30"' term of the series
-jL
—27, —20, —13, Ans. 176.
etc. 5.
Find the
Of
2
«,"'
term of 1
+ 8 + 5 + 7.
+ 2J + 2H
Ans. J(«+5)
Of 13+12I+12J + Find the sum of l + 2-(- 3+4, .
.
6.
2n— 1,
Ans.
.
.
Ans. J(40-^i),
etc., to
50 terms
From formula (1), we find ^^50. Substituting this in formula we have S=(l-|-50j25=1275, Ans. Or, use formula 5.
(2),
7.
Of 'J+Y + ^i+'
etc., to
8.
Of 12 + 8+4+,
etc., to
9.
Of 2+21+2^+,
10.
Of
A— 5 — V—
,
etc.,
16 20
terms. terms.
Ans. 142. Ans.
—520.
ton terms. Ans.^07
etc., to
11. Or
+ 11).
n terms. A. ^^(13—770hi etc., to
n terms. Ans.
«-l 2
12. If a falling hody descends 16^L feet the 1st sec,
3 times
how
this distance the next, 5 times the next,
far will
it fall
the 30th sec, and
Two hundred
a straight
22
ft.
stones being placed on the ground in
line, at the distance
2d Bk.
and so on,
far altogether in
Ans. 948}^, and 14475
half a min.?
13.
how
of 2 feet from each other
RAY'S ALGEBRA, SECOND BOOK.
258
how
who shall bring them sepawhich is placed 20 yards from the first from the spot where the basket stands? Ans. 19 miles, 4 fur., 640 ft.
person travel
far will a
rately to a basket, stone, if he starts
14. Insert 3 arithmetical means between 2 and 14.
o=2, ?=14, and
Here,
From formula 8, and 11.
Ji=5.
Hence, the three means will be
To solve
(Ij,
we
d=3.
obtain
5,
m
problem generally, let it be required to insert a and I. Since there are terms between a and ?, we shall have n^m-\-2, this
arithmetical means between
m
and formula
becomes
(1)
l:=a
{m~
-^
tZ=
Hence,
l)d.
,
Therefore,
TTie
common
difference will he equal to
cxtnmes divided hy
the
tlie
difference of the
number of means plus
one.
15. Insert 4 arithmetical means between 3 and 18.
Ans.
6,
12, 15.
9,
16. Insert 9 arithmetical means between 1 and
Ans. 17.
How many
amount
From
to
i,
-§,
to
etc.,
—
terms of the series 19, lY, 15,
91?
Ans. 13, or
(2) and (1), find n, or use formula 14.
1.
—
i.
etc.,
7.
Explain
this result.
18. etc.,
How many
amount
19.
to
terms of the series .034, .0344, .0348,
2.748?
The sum of the
progression
is
4,
Ans. 60. first
and the
two terms of an arithmetical
fifth
term
is
Ans. 20.
The
two
first
being together
sum
of r terms
;
3,
find the series. 5,
7, 9, etc.
and the next three terms
be taken to
21. In the series 1, 3, 5, the
9
term.s of an arithmetical progression
=18,
many terms must
1,
:
:
x
:
1
;
make 28? etc.,
the
Ans.
=12, how 4, or 7.
sum of 2r terms:
determine the value of
Ans.
x.
4.
GEOMETRICAL PROGRESSION.
A
22.
B
sets out for a certain place,
and travels 1 mile the Five days afterward
day, 2 the second, and so on.
first
and must B
sets out,
how
far
12 miles
travels
259
a day.
How
long and
A?
travel to overtake
Ans. 3 days, or 10 days; and travel 36 miles, or 120 miles. Explain these results.
GEOMETRICAL PROGRESSION.
393. A Geometrical Progression is a series of terms, each of which is derived from the preceding, by multiplying
by a constant quantity, termed the
it
Thus,
2, 4,
1,
16,
8,
Also, 54, 18, 6, 2,
In general, sion,
when
when
r is
less
is 2.
a decreasing geometrical prois J.
ratio
geometrical progres-
a, ar, ar', ar', etc., is a
whose common
series
ratio
etc., is
whose common
gression,
an increasing geometrical
etc., is
common
progression, whose
r
is
than
ratio
is
ratio.
r,
and which
greater than 1
;
is
an increasing
but a decreasing series
It is evident that
1.
In any given geometrical series, the common ratio ivill be found by dividing any term, by the term next preceding.
S96.
To
term of a geometrical progression.
find the last
Let a denote the
first
term, r the
term, and S the
sum
of n -terms
n'*
common ;
ratio,
I
the
then the respective
terms of the series will be 1,
2,
3,
4,
5,
.
.
.
n_3,
a,
ar,
ar^,
ar^,
ar*
.
.
.
ar"-^,
That
is,
the exponent of
r,
n"'
term of the series
1,
n,
ar"-', ar"-'.
in the second term,
is
1, in
and so on. Hence, will be lz^ar"~'. Hence,
the third term 2, in the fourth term
the
n— 2, w— ar"-^,
3,
3
1
—
•
RAY'S ALGEBRA, SECOND BOOK.
260
Rule for finding the last Term of a Geometrical Multiply the first term by the ratio raised tu a power whose exponent is one less than the number of terms.
Series.
—
Required sion
whose
term of the geometrical progresand common ratio 2.
to find the 6""
term
first
is 7,
2^=32; and 7x32=224, SOT'. To
find the
sum
of
the G'* term.
the terms of a geometri-
all
cal progression.
If we take the sum by S; then,
series, 1, 3, 9, 27, 81,
Multiplying by the ratio Subtracting (a) from
To generalize
this
3S—S=243— 1
method, its
.
.
be any
+a?'''-'-f ar".
whence,
Since
;=ar''-',
™.
„
S=
etc.,
air^
—S=ar"^a;
Therefore,
ar^^
we have
rS=ar+ar-+•— l)Sr
'
"-'
a(r"—
rl
—a
>=P
_
S= lr"—l
r,
n.
r,
n, I
10.
r,
n,
S
11.
r,
I,
S
12.
n,
I,
S
I
Jr-VjS r»— 1
'
a—rl~(r—l)S, a(S-a)"-i— «(S--Z)"-i=0.
13.
14.
a, n,
S
15.
a,
S
16.
n,
!,
I,
S
—
S
S-a
_
=0,
S— rt r"— ^^_
17.
,
r4
s
„
,
fog.
;
,
=0.
S—
"^s-r I
-log,
a
18.
a, r,
S
Jogr.
^ '
log. r
[a+{r—l)^']—log.a
"
log. r log.
19.
I— log. a
'
..
(S—a)—log. (S— ^)+ l-log. [^r-(r-l)S] ^ Jog.
'
"log.
20.
r,
!,
S
log. r
^ '
a
GEOMETRICAL PROGRESSION. By
observing, in any particular example, what are given and re-
may
quired, the proper formula
Nos.
263
3,
12, 14,
be selected from the above table.
and 16 may require
higher than the second degree.
the solution of
an equation
Nos. 17, 18, 19, and 20 are obtained
by solving an exponential equation, (Art. 382) but are introduced The two formulae
here to render the table complete. l^ar'^-^ are,
(1),
and 8=^^-=:^",
(Art. 298,)
or,
however, sufficient for the solution of
cal Progression, and
1.
Find the
2.
The
may
S""
all
examples
i,i
(2),
Geometri-
memory.
easily be retained in the
term of the series
±^
5, 10, 20, etc.
Ans. 640. '7*
term of the series 54, 27, 13^,
etc.
Ans. ||. 3.
The
6'A
4.
The
7"'
term of the series 3|, 2i, 1^,
etc.
Ans.
term of the series
—21,
14, — 9J,
Ans.
^.
etc.
— 4|f. 3>i—
5.
The
6.
Find the sum of 1
From 7. 8.
9.
10. 11.
(1),
n."'
Z=1X38=6561.
+ 4+16 + Of 8 + 20 + 50+, Of 1 + 3+9+, Of
1
J,
term of the series
,
+ 3+9 + From
etc., to
(2),
etc., to
etc., to
;^^j.
9 terms.
S=?^^^^^-^^=9841 Ans. ,
Ans. 21845.
7 terms.
n terms.
1— 2+4— 8+, etc., Of a;—v+— — C+, etc.,
Of
,
8 terms.
etc., to
Ans.
^, |, etc.
Ans. 3249|.
Ans. ^(3"— 1).
to n terms. Ans. J(1:::f2").
°^'
to
n terms.
a;+3/l
\
x)
)'
12. The first term is 4, the last term 12500, and the wumber of terms 6. Required the ratio and the sum of Ans. Eatio =5; sunt ^15624. all the terms.
RAY'S ALGEBRA, SECOND BOOK.
264
Find the sum of an the following series
+ + +
13.
Of
f
14.
Of
9+6+4+,
15.
Of I-J + 1-,
16.
Of
i
J
Ans.
etc
,
a+i+- + a
number of terms of each of
infinite
;
j.
Ans. 27.
etc
Ans.
etc
i.
"'^
^]+,
Ans.
etc
'
a'
'
a
—b
The sum of an infinite pcometric series is 3, and the sum of its first two terms is 2^ find the series. Ans. 2+1 1+ ... or 4-|+^-. 17.
;
+
18. Find a geometric Here,
.
mean between 4 and 16.
a=4, 1=1Q, and n=3;
19. The
or,
term of a geometric series
first
term 96, and the number of terms 6
;
Ans.
mean
(Art. 269) the
is
.
4^T(x
=r,,
3,
8.
the last
find the ratio,
and
the intermediate terms.
Ans. r=2. If
it
be required
numbers,
r^^+V —
a and .
Or,
I,
to
insert
m
Int. terms, 6, 12, 24, 48.
geometrical means between two
we have n^Tn^2;
we may employ formula
n — l=m+l, and
hence,
(1).
20. Insert two geometric means between i§ '
and
Ans.
2.
I,
|.
21. Insert 7 geometric means between 2 and 18122.
Ans.
301. To is,
6,
18, 54, 162, 486, 1458, 4374.
find the value
of Circulating
Decimah; that
decimals in which one or more figures are continually
repeated. In such decimals the ratio
is
more figures recur.
i
-JL, 1
one, two, or
I
Thus,
00
__'
,
1000'
etc., '
accordinff ° aa
HARMONICAL PROGRESSION. The part within the parenthesis ^'"1
«=T^Vn
....
S=.25313131 100008=2531.3131
Let
Dividing by 100,
.
99008=2506
.
are said to be in
.
25.3131
Find the value of .636363. Find the value of .54123123.
303. Harmonical titles
100S=
.
....
Subtracting,
2.
having
be performed more simply, as follows:
Multiplying by 10000,
1.
series,
.... =Tf5+^|k=M8e=JM§-
Therefore, .253131
may
infinite
Hence, (Art. 299,) S=^|J,g.
'•=Tk-
This operation
an
is
265
.
.
.
S=2506.
.-.
.
.
.
.
Ans. j\.
;
Ans. J|fg-3.
—
Three or more quanProgression. Harmonical Progression, when their
reciprocals are in arithmetical progression. Thus,
1,
1,
4,
^,
and
etc.;
1,
1,
f,
etc.,
f,
are in harmonical progression, because their reciprocals 1,
3,
5,
7,
etc.;
and
4, 3J,
3, 2J,
etc.,
are in arithmetical progression.
303.
Proposition.
— If three
cal progression., the first term
ence of the first
and
and second
is
is
quantities are in to
harmoni-
the third as the differ-
difference of the second
to the
third.
For
if a, 6,
c,
are in harmonical progression,
are in arithmetical progression
;
—
,
j, -,
therefore,
= —o Hence, multiplying by abc, ac — hc=rah — ac; or c(a — h)=a(h — a — h b—.c; therefore, gives (Art. 268), a .
h
a
c
c).
This
A
:
c
Harmonical Progression
:
:
is
:
a series of quantities in
'harmonical proportion (Art. 280)
;
three consecutive terms be taken, the as the difference of the first
ence of the second and third. 2d Ek.
23*
or such that if
and second
is
any
to the third
first is
to the differ-
— RAT'S ALGEBRA, SECOND BOOK.
266 Hence, ical
problems with respect
all
may be
progression,
to
numbers
barmon-
in
solved by inverting tbem, and
considering tbe reciprocals as quantities in arithmetical progression.
We
give, however, below, two formulae of frequent use
Given the
1.
a and
sion,
Here, a, -, ,,
and
,
two terms of a harmonical progres-
to find the n"' term.
h,
and
b,
first
the first two
I,
and
1
become (Art. 302),
n"" terms
in formula (l) (Art. 294).
Also,
d=v
^=
L+(._i)^*=(!^z:llfi^(!^=2)6 ab a 'ad
Therefore I
Whence,
:
'
'
^
—
=-•
'
ab
1=
(n
— l)a —
(».
-'l)b'
By means of this formula, when any two successive terms of a harmonical progression are given, any other term may be found. 2.
Insert
Here, since
T
m
harmonic means between a and
m^^n
—
2,
and ??i-(-l=n
= —\-(n— 1)«>
and
—
a=
whence, the arithmetical progression
is
1,
we have, =
I.
as above,
,:=
•
_^
found; and by inverting
it«
terms, the harmonicals are also found.
3.
Insert two harmonic means between 3 and 12.
Ans. 4 and 4.
Insert two harmonic means between 2 and
6.
i.
Ans. I and 5.
6'* is
The j't;
;
first
term of a harmonic series
^.
i,
and the
B'
B!
find the intermediate terms.
^°^6. a, h,
is
c,
4'
and
are in arithmetical proc^ression,
are in harmonical progression
;
prove that a
:
h
:
TOc,
6, :
c
:
d.
d,
AEITHMETIC AND GEOMETRIC PROGRESSION.
PROBLEMS
IN
267
ARITHMETICAL AND GEOMETRICAL PROGRESSION.
304. — 1. The sum gression
of 6 numbers in arithmetical pro-
35, and the
is
sum
Let x—2y, X
Ans.
—y,
335
of their squares.
the numbers. x, x-{-y, x-\-2y,
find
;
10, 13.
1, 4, 7,
he the numbers.
There are 4 numbers in arithmetic progression, and the squares of the extremes is 68, and of the
2.
sum of means 52 the
Ans.
find them.
;
x~y,
let x—Zy,
x-\-y, x-\-Zy,
2, 4, 6, 8.
be the numbers.
Suggestion. — When the number of terms in an arithmetic is odd, the common difference siiould be called y, and the middle term X; but when the number of terms is even, the common difference must be 2y, and the two middle terms X —y and x-\~y.
progression
The sum of 3 numbers in arithmetical progression sum of their squares 308 find them.
3. is
30, and the
;
Ans.
10, 12.
8,
There are 4 numbers in arithmetical progression, find them. their sum is 26, and their product 880 Ans. 2, 5, 8, 11. 4.
;
There are 3 numbers in geometrical progression, whose and the sum of the 1st and 2d sum of 1st is 31
5.
sum
;
and 3d
:
:
3
a;=: 1st
Let
.
:
13
Ans.
find them.
;
term and
y^
ratio; then,
xy and x^-
5,
1,
=
25.
2d and 3d
terms. 6.
The sum
of the squares of 3 numbers in arithmetical
83 and the square of the mean is greater product of the extremes find them. 4 than the by progression
is
;
;
Ans. 7.
3,
5,
7.
Find 4 numbers in arithmetical progression, such that
the product of the extremes
^27
;
of the means
Ans.
^35.
3, 5,
7 9.
,
BOOK.
RAYS ALGEBRA, SECOND
268
8. There are 3 numbers in arithmetical progression, whose sum is 18; but if you multiply the first term by 2, the second by 3, and the third by 6, the products
be in geometrical progrcstion
will
them.
find
;
Ad.s. 3, 6, 9.
The Fum of
9.
natural numbers
the fourth powers of three suL-ccf-^ivc
9G2
is
;
find them.
Aus.
3, 4,
5.
10. The product of four successive natural numbers
840;
fiud them.
Ans.
The product
11.
gression
is
of four
numbers
is
4, 5, 6, 7.
in arithmetical pro-
280, and the sum of their squares 166; find
them.
Ans.
10.
1, 4, 7,
The sum of 9 numbers in arithmetical progression 45, and the sum of their squares 285; find them.
12. is
Ans. 18. The is
3, etc., to 9.
1, 2,
sum of 7 numbers in arithmetical progression sum of their cubes 1295; find them.
35, and the
Ans.
2, 3, etc., to 8.
14. Prove that whcTi the arithmetical mean of two numbers
is
mean
to the geometric
;
5
:
:
4
;
that one of tliem
4 times the other.
is
The sum
15.
7
is
;
of 3
numbers
in geometrical progression
and the sum of their reciprocals
is
;
]
find them.
Ans.
SuGCE-STiON
r,
)/,
In
—
for 3 numbers, use X.
— all
;
for five,
'—
,
.c-,
j
xu, y-,
is
equal
the ratio in each case, divide
16. There the
sum of
are
the
or,
>/,
' ;
to
.r-,
:ri/.
for six, '—„
y-\ for four,
—,
and third
X, y, —-,
of
any
the square of the second.
any expression by
—
—.
three,
To find
the preceding.
4 numbers in geometrical progression,
first
second and fourth
2, 4.
express them hy other forms. ^„ ,
to
—
.iif,
these cases the product of the first
taken consecutively,
1,
solring difficult jrirobloms in geometrical pro-
111
sometimes preferable
grcs.sion, it is
Thiis,
.
is
and third 30;
is
10, and the
find them.
Ans.
sum
of the
1, 3, 9, 27.
PERMUTATIONS AND COMBINATIONS.
269
lY. There are 4 numbers in geometrical progression, the
sum
of the extremes
35, the sum of the means
is
Ans.
find them.
is
30;
12, 18, 21.
8,
18. There are 4 numbers in arithmetical progression, which being increased by 2, 4, 8, and 15 respectively, the sums are in geometrical progression; find them. Ans. 6, 8, 10, 12. 19. There are 3 numbers in geometrical progression, whose continued product is 64, and the sum of their Ans. 2, 4, 8. cubes 584; find them.
IX.
PERMUTATIONS, COMBINATIONS, AND BINOMIAL THEOREM.
303. The
Permutations of quantities are the
different
orders in which they can be arranged. Quantities
may
be arranged in sets of one and one, two
and two, three and three, and so on. Thus, if we have three quantities, a,
them
h,
r,
we may arrange
in sets of one, of two, or of three, thus
Of Of Of
one,
a,
h,
two,
ab, ac;
ha, he;
abc, ach;
three,
306. To
find the
formed out of n three together.
.
.
c.
ea, ch.
hac, hca; cah, cba.
number of permutations
letters, .
:
and
taken
singlj/,
that can
be.
taken two together,
r together.
Jc, be the n letters; and let P, denote the Let a, b, c, d, whole number of permutations where the letters are taken sinffly; Pj the whole number, taken 2 together .... and Pr the number .
taken r together.
.
RAY'S ALGEBRA, SECOND BOOK.
270
The number of permutations of n
number
to the
dently equal
taken singly,
letters
of letters; that
evi,
is
is,
Pi=n. The number of permutations of n letters, taken two n{n 1). For since there are n quantities,
together,
is
Writing
a
—
a,
we remoTe
if
— 1)
ad
— 1) we
quantities,
ab, ac, is,
Tc,
.
there will remain (n
,!
(\—xY=l—nx-\-
.-.
Corollary
,
3.
„
n
1
the
r"*
—1
—^—
,. ,
"-^
^
— Since
which forms the
dd term
n(n—V\'
,
x^
the
——^^
n{n—\){n I'' -3 ^
last
factors,
coefficient, are for the ,
tor the
^
,
4th term
term they will be
n
m— q— —
a"-''x, in
®'^''-
the
in
fraction
2d term ^, etc.
;
for the
.
„
thereiore, lor
-
^-=:
we
shall
Hence, the general term of the
nfn-l)(n-2) 1-2 -3
+'
— (r——2)
the 3d term a"-^a:^ in
therefore, in the r* term,
,
, ,
Also, for the exponents of a and x,
term
2")
we have
have o""'"— 'x''-'.
series is
{n^ r+2) (T-\)
in the
2d
the 4th term a"-^a?;
""
^"^
—
— RAY
278 This
is
Example.
(?i
by making r=2,
the others can be deduced from
— Required
r=5, and 71=7]
term of (a
5"'
the
—
it.
a;)'.
therefore, the term required
— If n be a — ''+2) becomes
Corollary then,
ALGEBRA, SECONB BOOK.
called the general term, because
3, 4, etc., all
Here,
S
and r^n-{-2 and the («-|-2) term vanishes
positive integer,
4.
0,
therefore, the series consists of (;i-|-l) terms altogether
that
is.
The number of (frms the
power
to
Corollary tive integer,
5.
one rjrcaler than the exponent of
is
binomial
irhicjb the
— When
is
to
be raised.
the index of the binomial
is a
posi-
the coefficients of the terms taken in an in-
verse order from the end of the series, are equal to the coefficients
of the corresponding terms taken in a direct
order from the beginning. If
we
compiire the expansion of (a4 -T)", and (X -«)",
(a-)-a;)"z-a" -na"-i.r-| -4j— ., 'a"—x--]
——^—
71 In {x-\-a)''=x" -\-7ix"-^a^- A: ,
,
1)
,
,
,
—-— —n(n^-— ^-„— ;;
l)i;7
.r"--a--i
we have
'a"--a:3 -|-, etc.
-2)
,
,
.r"-''n--L,etc.
Since the binomials are the same, the series resulting from their
expansion must be the same, except that the order of the terms will be inverted.
It
is
clearly seen that the coefficients of the corre-
sponding terms are equal. Hence, in expanding such a binomial, the latter half of the exp.insioD
may
be taken from the
Example.
— Expand
(a
first half.
b)^-
Here the number of terms (»+l) Bary
is
6; therefore,
it is
only ncces-
to find the coefficients of the first three, thus:
—^a^b'^—lOa-b^+bab*-/,':
a
(a—b)^=a'^—5a^b+-
4
—
— BINOMIAL THEOREM. Corollary
6.
— The
terms are positive, For
is
sum of
311. From (a-f a;)",
If the
it is
the coefficients, where both 2".
always equal to
we make a;=a^l
if
(hen,
;
279
(a;-(-a)"=(l-fl)»=i2».
.
an inspection of the general expansion of
evident that
of any term he multiplied hy the expoof the binomial in that term, and the
coefficient
nent of the first
letter
product be divided by the number of
the term, the quotient
will be the coefficient of the next term.
For examples,
see Newton's Theorem, Art. 172.
313. To expand exponents,
or
a binomial affected with coefficients
—
(2a^
as
3i')*,
Newton's
see
Theorem-,
Art. 172.
313. By means of the Binomial Theorem, we can raise any polynomial to any power. Thus, let it be required to
— to — b^m,
a Let a
raise
1.
b-\-c
the third power.
already explained, Art. 172.
etc., as
Expand (a+6)», («— 6)', and (5— 4a:)*. (1) Ans.
a'^-f
8a'6
+ 28a''i'+56o^6=+'70a*5*+56a'i5
+ 28a^6''+8a6'+?-«. (2) Ans. a'~la%-\-2\a^b'—?,ba'h'-J^Zba'h'-~2\a:'h^
+ laV--b\ 625— 2000^+ 2400a:'— 1280a;=+256x*.
(3) Ans.
»
2.
Required the
coefficient
of a? in the expansion of
Ans. 210.
(x-\-yy. 3.
Find the
5""
term of the expansion of
(c^
d^y^.
Ans. 495c'W.
ScGGESTiON. substitute c^,
4.
.
Find the
— (See Cor.
dr, 12,
7"*
and
3,
Art. 310.)
Instead of o,
a;, re,
and
5.
term of (a'+3ai)».
Ans. 61236a'56«.
r,
:
RAYS ALGEBRA, SECOND BOOK.
280
7.
Find the middle term of (a"'+a;'')". A. 924a«'"a^". ^us. 330.?;'. Find the 8"' term of (!+.;-fCa-'+Da-'+, etc etc.,
contaiBing
x,
have A=^A', Tillill
=A'+B'a-+CV-f
,
for every possible value of
x (A, B,
DW+,
A', B', etc., not
and x being a variable quantity) we shall
B=B',
coefficients
of
C— C,
etc.;
that
is,
the terms invoicing the
same
poiC( is of
:r
the lico series, are respectively equal. For,
by transposing
all the
terms into the
A— A'+(B— B')I^-(C— CO.r--|-{U— If A — A' is not equal to 0, let it then,
we have
D')a,-3-|-,
first
member, we have
etc.,
=0.
be equal to some quantity p;
(B— B')x-f(C— C')a:H(D— iy)x3-|-,
etc.,
=—p.
— INDETERMINATE COEFFICIENTS.^
281
Now, since A and A' are constnnt quantities, their must be constant; but p=(B— B')a;-)-(C C^)x^-\-,
—
may
tity whicli
upon X] which is
etc.,
evidently have various values, since
therefore, the
same quantity [p)
it
both fixed and
is
p,
a quan-
depends variable,
impossible.
Hence, there
no
is
A— A';
difference
Hence,
[p) which can express the
possible quantity
or, in
other words,
A— A'=0 By
difference,
,
A=A'.
.
(B—B')a;-f-(C— C')a:2-)-(D— D')a:3_|__
dividing each side by
x,
=0.
etc.,
we have
B-B'+(C— C')a;+(D-D')a:2+,
=0.
etc.,
Reasoning as before, we may show that B=B'; and so remaining coefficients of the like powers of X.
Corollary.
— If we have
an equation of the form
A-f-B2:+Ca)2-|-Da;'+Ea;*+,
which
is
B=n=0,
C=0,
true
for
=0,
etc.,
avy value whatever of x
etc.; that
is,
on, for the
each
;
then,
A:^0,
coefficient is separately
equal
to zero.
For the right hand member 0-(-Ox-fOX"+Ox'-|-,
powers
of X,
313.
etc.;
then,
may
evidently be put under the form
comparing the
we have A^O, B=0, 0=0,
Let
it
cotfficients of the like
etc.
=— into a series
be required to develope
a-\-bx
without a resort
to division.
The series will consist of the powers of X multiplied by certain undetermined coefficients, depending on either a or 6, or both of them, and
X
will not enter into the
first
term;
^-^=A+Ba:+Ca:2+Da;3+,
therefore,
assume
etc.
Multiply both sides by the denominator a-\-bx, and arrange the to the powers of X; we thus obtain
terms according
a=Aa-f Ba +A6i
I
2d Bk.
24
x-\-Ca a;2-|-Da I
I
-fBfil
+C6|
x^-\-, etc.
—
;
RAYS ALGEBRA, SECOND BOOK.
282
But by the preceding theorem and
a=Aa
;
corollary,
A^l
hence,
Ba+A6=0;
B=
"
;
Ca+E6=0; Da+Cb=0;
63
D=
"
-,
etc.
Substituting these values in the assumed scries,
a
^
b
1
a
a-\-bx
X-
—
'
b'-
b^
.,
—a
find
Z>'
,
;X
.,x-
a-
wc
,
•-{
a-'
• , , -X* etc, the
same aa would
be obtained by actual division.
316. A
series
with indeterminate coefficients
is
gener-
assumed to proceed according to the ascending integral and positive powers of a;, beginning with x" but in ally
;
many
The
series this is not the case.
tion will then be shown, either
error in the assump-
by an impossible
result,
or by finding the coefficients of the terms which do not exist in the actual series, equal to zero. Thus,
if it
be required
to
develope ^
series to be A-|-Bic-l-C'^"H-I'-f"-
Ex^^,
and we assume the
—,,
oX
X-
we
etc.,
have, after clearing
of fractions,
l=3Ax+(3B— A)a:2+(3C— E)x3+, from which, by equating the
coefficients of the
1=0, 3A=0, The
first
equation,
etc.;
same powers of
etc.
1=0, being absurd, we
infer that the expres-
sion can not be developed under the assumed form.
S^-l^^ clearing
of fractions,
powers of
X,
1
\
we
l\
and equating the
X
X?
B=^,
coefficients
C=^i^, D=gT-,
x^
3i^P=x\ 3 + 9 + 27 + 81+'
But,
3^=A+B. + C:r=+,e.c.,
^""'°S
find A=-t,
X,
\
,
"'" )
2-1
a"
etc.
X
of the
like
Hence, x^
= ^+9+27 + 8i+'^"'-
INDETERMINATE COEFFICIENTS. Or, since the division of 1 by
gives 5-, or
3a;~',
we ought
to
tlie
first
Again,
if
we assume
317.
Assume
hecoming
—
2.i--|-3ar*
— Extract
—
sides,
the square root of
we
the
coefficients.
a'-\-x''.
y {a^+x^)=A+Sx+Cx^+'Dx^-\-Ex*-\-,
Squaring both
etc.,
5a;''-f-,
zero.
Evolution by indeterminate
Example.
etc.
^^A-\-Jix-j-Cx^-\-'Dx^-{-, etc.;
.,
shall find the true series to he 1
coefficients B, D, F, etc.,
term of the denominator
have assumed
5-^^,=Aa:-'+B+Ca;+Da;2+,
we
283
etc.
have,
a2+x^^=A^+2AJ^x+2AC a;2+2AD a;3+2AE x*+, 1
+B2
etc.
I
+2BC
1
+2BD
I
+C2 .-.
A2=a2, 2AB=0, 2AC+B2=1, 2AD+2BC=0,
And,
A=a, B=0, C==-, D=0,
Therefore, l/(«^-f-a;^)=a4-2^—
318.
E=— =-^— gp+,
etc.
etc.
,
etc
Decomposition of Rational Fractions.
tions vyhose denominators can be separated factors,
may
often
whose denominators factors.
To
Decompose
be
— bx-\-o^ „
other
—Fraccertain
fractions
more of these
by an example.
14
5.^
into
shall consist of one or
illustrate
x'
decomposed
into
into
two other fractions whose
denominators shall be the factors of x'
— 6x-\-8.
Since x^—6x-\-8={x—2)(x—'i), (Art. 284), as.sume
5a;— 14 __A^ B id9"r;i x^—6x-\-83 X— 2^x^-4'
—
,
RAY S ALGEBRA, SECOND BOOK.
284
Reducing the fractions
to
5x-U
common denominator,
A(a;-4)+B (r
-2)
oa;— 14=A(a^-4)4-B(a;— 2)=(A-|-B)a;—4A— 2B.
Or,
Now, since
may
a
this equation is true for
any value whateTer of
equate the coefficients (Art. 314);
A4-B^5;
— 4A— 2B=— 14;
whence,
.5X-14
And
X-
— 6a;-i-8
2
,
x~^2
3
x—i'
the method of Indeterminate Coefficients,
1.
:r^,^ 1
—
show
o.c
=l + 5a:-f-15.r2+452^-|-.
etc.
-^-~^r-^.l + 3x-l-4x:'-]-1x'-{-U.i-*Jr'i-8x'+,
3.
-i-i^=l^+2'^x+3'-.t^+4W+5V-(-,
4.
y
1— .t^l—o— 5-4
1
(1+^+^0 = i+'5+:f-ig+.
:r_
^
l-J-a;
1
—
X
.<
that
•
2.
5.
we
A=2, and B=3.
By
1-1-2
x,
this gives
.1--
.T+l
3..
we have
i-j-X,
M-f-2Ba;+3Ca;2+4rte3+,
+
and
coefficients of z in (A)
(B).
etc.
I
etc. J
=n(l+a;)» =n(l+?ia;+Ba;2-f Ca;'-}-,
equating the coefficients of the same powers of 2iii-\-n='n?
.
2B--iin-
.
^_ "-"
3C+2B=Bn
..
— «=n(?i—
n(w-l) 1-2
X,
etc.)
we have
1).
_ '
3C=B(n—2);
B(n-2) _n(ra— l)(n— 2)
~
3
4D+3C=nC
Also,
1
.-.
2-
3
C(»—3) __/i(n— l)(n—2)(n— 3) 1 •2-3-4
.-.
putting
b
for X,
,
(a+6)»=a"( 1+-
nfn—1)62
'
4D=C(n.-3); and so on
for E, F, G, eto.
)",
n{n—l)(n—2)b^
=a"+na'-i6+^^a'-26=+?^i^^==^^(^«»-363+, If
— 6 be put for
(Art. 193)
and
6,
then since the odd powers of
— b are negative
the even powers positive,
^a'^^b^ —n{n~l)
7^n « n 17. na"-'6-J (a— &)"=a"—
„
,
eto.,
etc.
^
,.,
—
nfw—11(n— 2)-a'^^tfi , .,., S-^i
which establishes the Binomial Theorem in
J—;, its
,
+,
most general
form.
Remark. —From Art. 310,
the preceding, corollaries, similar to those in
may be drawn, but
it is
The following additional proposition
not necessary to repeat them. is
sometimes useful.
:
RAY'S ALGKBRA, SECOND BOOK.
288
330. sion of
If m
I.
From
find the numerically/ greatest
To
(tt -{-
a positive integer
is
Cor.
—r
\
1
v?iiie of
greatest term
\
—
1
——
r-
each term
whiili
that
;
<
nature
the
1
the
=
_
in
^
be the greatest.
will
+ l)b
then the
,
a+b
If n
II.
— >
It
But
diver. ic.
or
increases;
i
necessarily
is
first
--7—
= ;
integral
—
integer
—
If
term
p—
a~b r
if
;
and the r"
,
^j^ij
an integer, and we take
is
the
(r-t-l)'*,
for
this
and each
of these
only occur
can
is
when
a positive fraction
is
there
1
r'"
as
he
it
JO'->
the
any other term
greater than
>
case,
a-\-h
term m
may
—-,
less than 1, indicates the term will be the greatest when
>"'
or r first
of
fractional, take r
IS
term
greater than the preceding;
is
makes
first
the
is,
IS first
I/a
r
From
r,
diminishes
multiplier
tliis
is [greater tlian 1,
it
and the
I
and
)-,
:
apiicars that the (r-|-l)"'
it
r* by multijilying the latter by
tlie
Ja
wliile
Art. 310,
3,
formed from ,
term in the expan-
b)".
<
-
if
no
is
1
greatest term, for the series will evidently
the series will have
its
greatest term (or terms)
n
may
wliose jKisition III. If
be ascertained as in
Tlie multiplier that changes the
— ?i—r+1 —
b
,
'
term
is
.,,
be written
.- nia\'
a greatest
I.
n be negative, either an integer or a fraction
—
/'v
term into the
+ r—l\h
(
\
?'''
r
J-, /a
and,
in
the
I,
r'-'
term will be the greatest wdien
r-
,
firat
As
in
numericatly
b
.-
is
a
first
<
1
~. > Mn—1) a—b I, if
a
—
;
be a whole
number
there are two equal terms
each greater than any other; and, as in
no
tJie
+ T —1 T
or r
„
.
.,
sought, disregard the sign of the multiplier: then. li
;is
as
> 1:
(r-j-l)"*, viz.,
greatest term.
II, it
— be >
1,
there
is
:
BINOMIAL THEOREM. IV.
If
greatest; alt
<
n be negative and
1,
and
-
<
1,
289
the
for in tliis case tiie multiplier
values of
r,
that
each term
is,
is less
first
term is
;
is
<
the
1 for
than the preceding.
—
Note. If h is negative, since it is tlie numerical value ot the term tliat is to be considered, we may disregard tlie sign ot b and apply the appropriate one of the preceding rules. [Cf. Todhunter's Algebra, Art. 520.]
Examples.
— Find
the greatest term in each of the following
expansions 1.
6
.
5
.
(2+1)°.
Here
4
20000
_
5^
(ii+l)6 _35
a+b
r=4
~n
gives the greatest term:=
-.93.-
1.2.3^
3"
81
(1+J) I.
2.
3.
(l+f)^
4.
(1+^)-".
Ans.
Here^!^'=Ll a
12-13 1-2 5.
—
5"'
Ans.
2''.
and
6'".
=3 gives the greatest term=:
4
]_ •
52
"25
(l+?)-3.
HereM^=5 a—
5"'=6">,
and each
is
greater
than any other term. 6.
(1-AH.
Ana.
3''''.
331. In the application of the Binomial Theorem, it is merely necessary to take the general formula (a + 6)"^=a"4no"~^6-f-, etc., and substitute the given quantities in the formula, and then reduce each term to its most simple form Example.— 1. Find
the expansion of (l+.r)*
Here, o=l, b^x, n=\-
\(\-\)
{l+x?
,
j(i-l)(i-2)
1
1
1-1-3. 1-1-3-5 —
^l+ia;— 2-7^2:2+2- ^r-g2d Bk.
etc.
1-2-3
1-2
25^
2
•
4
6
„.T^+, etc.
S'
RAY'S ALGEBRA, SECOND BOOK.
290 Example.Here, .-.
As
—
—
:
o=l,
—
2.
6=
Develope (1 X,
n=—
the general
formula (l^.r)"=l±Ka;-|-
Develope
Here,
— -.
n
= \-
(l+.rj»=l-,-»,r+
•I ^+5 6
+-a
Hence,
2.
—^x^zh,
.^-
reduce the quantity
etc.,
complicated,
is less
to
\
.
.
be expanded
y7^PJ=^/'E{ 1+* U_
and since
Yro-* +^T-r.Tr3— ^'+,
-^ +=5+ T~2~V+
/
]_ 62
_ i;
1
3
1-2-3
etc.,
-53+'
''^'=-'
^,3
4a2 + 2-4-6"a3~ *'"
^a+6^,
«(1+,^ _g_^ + _______^+,
=(1—,r)-'=l+.r+.-r'+.r'+.j* +
,
etc.).
etc.
^j-:^^-,=(l-.i-)-'=l+2-^+3x''+4x'+5.T*-h,
=1,
2.r
a
3.r^
\-— ,
a^
4:r' 5.T* -— -f---_ ,
a*
a^
etc;.
is
it is
\' a-\-b into a series.
a^b^al l+_ ?:
to
etc..
Thus
this form.
Since
memory, and
in the
generally most convenient
1.
J.
(l-a;)-*=l-^(-a;)+tzi)|:=^)(_a;)2+,
more easily retained
~
x)~^.
etc.
to
— BINOMIAL THEOREM. a? t.
.-«—
TTi
g
-L
X
x^
ba?
g
g^
,
291
etc.
X
5.
v/a^+.^a+2-„-8^3 +
6.
(a3-^)^^a-g^_y-,_g^3
5a:'
7.
o
o.
(l
5-
/-^;
x'=a
—
x* Tt—;
a;''
-71
2o
8a^
Ix
9.
10.^.X' etc.
24ba"
+ 2x)-=l+a;— Aa;^+-ix'— |a;*+,
i/a' '
etc.
Ty;^-128^,+.
etc.
x^ Sx" — Iba^ — 128a' „„ — T
Ix^
,.
^
-
5x'
,
etc.
, '
10,'X'
fa+x^ra(l+^^-^^ + ^-^^^+, ,
etc.).
10.
(a'+x3)^=a(l+£3-g^,+ ^^,-,
11.
r9=^8+r=2+|. J-|-l, + l-;-l,-,
etc.
12.
(a'-x')^=a(l-3-^3-3^,-3^Q^-,
etc.).
a'
Here,
—
,
E^!_^ (a3-x3)^
2x'
,
2 5x«
etc.).
2 5 8x' •
= a3(a3 — x3)"*= a3 X (a^r^i
1-
^
"^'
"
)"*=«' '
-"V-l)-^-('-S) 333. To
find the
approximate roots of numbers by the
Binomial Theorem.
Let
N
required
represent any proposed ;
take a such that a"
is
number whose
?i'*
the nearest perfect
ji'*
root
is
power
;
RAY'S ALGEBRA, SECOND BOOK.
292 to
N
and
N^a"±'>,
so that
+
—
or
,
l=t-7; txeneral
formula
a i 1 ±1
.
N>
|"=j by
or
-writing
—
for
h
n
'la
\
a"
I
h
'Zn
a", will
.
2-_-i / 'Sn
when
\
A V_ a" J
etc ''' i '
6 is small with
14. Required the approximate cube root of 128. 1
the
give the required root to a considerable
degree of accuracy.
Here, f
in
;
this series a few terms only,
regard to
witli a,
of balls in the respective
courses will be as follows: Z'i.
3''.
4'*.
••
••• ••
•••• ••• ••
••••• •••• •mo
•
•
••
1''.
•
and
•
Hence, the number of balls in the respective courses
so on.
1-1-24-3,
1-1-2,
is 1,
S'*.
13
1-I-2-I-3-I-4,
l+2+3+4-|-5, and
Hence, to find the number of balls in a triangular the
sum
of the series
1,
3,
so ou;
or,
15
10
6
6,
10, 15, etc., to as
pile, is to find
many
terms (n) as
there are balls in one side of the lowest course.
By applying of the series
sum of n terms we have a=l, Di=2, 0^=1, and
the formula (Art. 327) to finding the
1,
3,
6,
10,
etc.,
D3=0. ^
,
,
Hence, the formula na-\-
n+n^—n+
330.
A
—
nln—i)^
^—
n3_3n2+2«
^—l)(n-2)^ —n(n— ^2 gi^es .
,
?^^^^-\
ir^-g
«(n-^l)(w+2)
(A)
^
To find the nnviber of balls in a square
square
pile, as
V — EFH, is
pile.
formed
of successive square horizontal courses,
such that the number of balls in the sides
of
tinually
these
courses decreases con-
by unity, from the bottom
to
the single ball at the top. If
we commence
at the top, the
number
of balls in the respective
courses will be as follows: I".
2'*.
S'l.
4'*.
RAY'S ALGEBRA, SECOND BOOK.
300 and
Hence, the number of balls in the respective courses
so on.
1=,
is
2-,
42,
32,
52,
number
to find the
the squares of
etc.,
1,
3, etc., to
2,
1,
or
4,
9,
16, 25,
and
so on.
Therefore,
of balls in a square pile, is to find the
as
many
terms
(71)
sum
of
as there are balls
in one side of the lowest course.
But
331.
A
nfn-|-l) (2n-|-l)
sum
this
(Ex.
is
(B)
pp. 297, 298) is
To find the number of balk in a rectangular pile.
rectangular
BCA,
2,
EFD
pile, as
formed of successive
rectangular courses, the number of balls
each
in
sides decreasing
of the
i«ll^^reV\%V«A^
by unity from
the bottom to the single row at the top. If
we commence
the
number
is
1,
2,
8,
number in the length of the and so on. Hence, the commencing with the top, will
of balls in the top row, the
second row will be
number
number of balls in the breadth and so on. Also, if m^l denotes
at the top, the
of the successive rows
»i-(-2, in the third, »?i-|-3,
in the respective courses,
be l(m-|-l), 2(m+2),
3(m-|-3j,
and
in the n"' course n(m-|-n).
Or,
S=l(m+l)+2(TO+2)+8(TO-f8)+ =OT(1 +2+3+4 but the
sum
827, 330,)
is
of
n
wi+n
+n2);
and
—
-1)
"i!^
.
Hence,
6
mn{n^\)
Here,
J^n[m^n)
.
terms of the series in the two parenthoses (Arts.
"i!^\
lowest course.
.
+n)+(12+22+32+42+
.
n(n+l)(2n+l)
n{n+l)
(3m+2ra+l)
number of balls in the length of the we put TO+n=?, we have 3m+2n=8? n; sub8771+2/1, in (C), we have
represents the If
stituting this for
(C).
—
SERIES—PILING OF BALLS. It is
301
evident that the number of courses in a triangular or square
pile is equal to the
number
and in the rectangular
of balls in one side of the base course,
pile to the
number
of balls in the breadth of
the base course.
,332S. Collecting together the results of the three prearticles, we have I'or the number of balls in a
ceding
Triangular pile
-,n{n-\-l)(n+2)
....
(A);
Square pile
^n(n+l'){2n+l)
....
(B);
Rectangular pile 77n(H-i-l)(3?
— »+!)
•
(C).
•
.
In (A) and (B), n denotes the number of courses, or number of balls in the base course. In (C), n denotes the number in the breadth, and I the number in the length, of the base course.
The number of balls in an incomple/e pile is evidently found by subtracting the number in the pile which is wanting at the top, from the whole pile considered as complete.
1.
Find the number of
balls in a triangular pile of
Here,
15
Ana. 680.
courses.
n=15.
Substituting this value in (A),
we
~]5(15+l)(15+2) _15Xl6Xl7_^sn 6
find the
number
Ans
2x3
2.
Find the number of
pile of
From
15
courses, having
balls in an incomplete triangular
21
balls in the
the illustrations in Art. 329,
it is
of balls in one side of the upper course
have been removed from the
pile.
upper course.
evident that the is
number
6; therefore, 5 courses
From formula
(A),
we
find that
the pile, considered as complete, would contain 1540 balls, and that the left.
removed
pile contains 35.
Hence, 1540—35=1505, the number
RAYS ALGEBRA, SECOND BOOK.
302
Find the number of
3.
a square pile
balls in
courses.
Find the number of
4.
balls in a rectangular pile, the
length and breadth of the base containing 52 and respectively.
6.
How many
balls
an incomplete triangular
in
25
a side of the base course having
balls,
of the top 13.
15
34
Ans. 24395.
Find the number of balls
5. pile,
of 15
Ans. 1240.
and a side
Ans. 2561. balls in an incomplete triangular pile of
38
courses, haviog
balls in a side of the base?
Ans. Y580. 7.
Find the number of balls
pile, a side
an incomplete square
in
of the base course having
44
balls,
of the top 22. 8.
The whole number of balls in 1521 and 169
the base and top courses
of a square pile are
respectively;
are in the incomplete pile? 9.
and a side
Ans. 26059.
how many
Ans. 19890.
The number of balls in a complete rectangular pile is 6440 how many balls are in its base?
of 20 courses
;
Ans. 740.
10 The number number in a square
of balls pile
in the side of the base
each
in
11.
a triangular pile
is
to the
having the same number of balls as
6 to 11
;
required the number
Ans. 816, and 1496.
pile.
How many
pile of 8 courses,
ll
in
balls are in an incomplete rectangular
having 36 balls
in the shorter side
in
the longer side, and
of the upper course?
Ans. 6520.
INTERPOLATION OF SERIES. 333. diate tables.
Interpolation
numbers
in
is
the process of finding interme-
mathematical,
astronomical,
Its object is to furnish a shorter
pleting such tables
culated by formula.
or
other
method of com-
when portions of them have been
cal-
INTERPOLATION OF SERIES. Thus,
logarithms of 5, 6, and 8, are respectively
if the
0.7'782, and
0.6989,
303
0.9031,
may be
it
required from
these data to find the logarithm of 7.
The
numbers are sometimes
latter
called functions of
the former, and the former arguments of the functions.
As
the functions constitute a series, the principle upon
which interpolation that
is
founded
certain terms of a
is,
is
explained in Art. 326;
series being
known,
it
is
re-
quired to find the «" term.
Three cases may
Case
I.
arise,
— When the
which we
will
now
consider.
differences of the functions are pro-
portional, or nearly proportional, to the diiferences of the
arguments, or the functions are in arithmetical progression.
Ex. of
— Given
86,
89,
the Dip of the Sea Horizon at the heights 95, and 98 feet, viz., 9'08", 9'17",
92,
9'26", 9'36", and 9'45"; required that of
101
feet.
Ans. 9'54". Here, the
first
differences being
9^45" for the Dip at 101
9'',
or nearly
so,
we add 9"
In all practical examples, there is no common first and it becomes necessary to employ the second, third, ences.
If in the series
to
feet.
difference, etc.,
differ-
composing the functions, we can obtain an
order of differences equal
to zero,
In most cases, however, Dj, Dj, small after Dj or Dj that they
the interpolation will be exact.
etc.,
may
do not vanish, but become so be omitted without sensible
error.
334.
Case
II.
— When
the differences of the functions
are not proportional to the differences of the arguments, and the term to be interpolated is one of the equidistant functions.
Ex.— Given ^25=2.92401,^26=2 ^"29=3.07231,
to find the
96249,
cube root of 28.
fW=S,
— RAY'S ALGEBRA, SECOND BOOK.
304
la such examples, "when three quantities are given, we pose D3
the equation
or d,
may
We
vanish or become very small.
to
may
sup-
then have (Art. 326)
— a-{-3b — 3c-(-£^=0, and any of the quantities
be found, when the other three are given.
a, b, 0,
Similarly, if
the fourth diiferenoes vanish, theu
a_46-(-6c— 4rf+e=0. In the above example, four quantities are given to find a we have a 46-(-6c 4d-\-e=0, where d is the term
—
therefore,
interpolated;
4d = a-|-6c+e — 46 =2.92401
hence,
to
be
+ 18+3.07231
where d, or ,^28=3.03669, which
-11.84996=12.14636,
fifth;
is
true
to .00001.
333.
Case
III.
— When the
differences are
2d, and the term to be interpolated
is
as in Case
inlermediate to any
two of the functions.
Ex.— Having and 105,
let
given the logarithms of 102, 103, 104, be required to find the logarithm of
it
103.55. Taking the formula. Art. 326, put intervals,
in
of the required term
which c.'\sep=n
the
number
—
of terms.
1,
(t)
since the
p
to
represent the distance, in
from a, the
number
first
term of the
o{ intervals
is
series,
one less than
Then,
t=a+pV,+Pl^^,+PAP^l^\+,
etc.
The intervals between the given numbers is always to be considuntil/, and p is to be reckoned in parts of this interval;
ered as hence,
p
will be fractional.
Sufficient accur.acy is generally obtained
D2 only, in the above formula. 1q practice, however, the following
is
by making use of D^ and
generally adopted:
Take the i'^o functions of the series which precede, and the two which follow the term required, and find from them the three first differences, and the two second differences. Call the second of the three first differences d, the mean of the two second differences d', the fractional part of the interval p',
and second term
have from the above formula,
t=b+p'(d^^szLd'),
6.
We
theu
INFINITE SERIES. Applying
Nos.
this
formula
to the
above example, we have
305
RAYS ALGEBRA, SECOND BOOK.
306
not be exceeded by adding together any number of terms
whatever.
A
Convergent Series
is
whose
a convergent series,
any number of
A
one which has a sum or
is
l+^+J+H-..,+3i, + gL+,
Thus,
its
limit
terms can not exceed
Divergent Series
is
Ascending Series
sum of
2.
one which has no sum or limit;
1-1-2+4+8+16+32+, An
etc,
since the
is 2,
limit.
as,
etc.
one in which the powers of the
is
leading quantity continually increase
;
as,
a-\-hx-\-cx^-\-da?-\-.
A
Descending Series
one in which the powers of the
is
leading quantity continually diminish a-\-hx'^'^-\-cx~''-\-dx~^-\-,
337.
or a-\
as,
;
X
+ -T+h— x' of 7
There are four general methods of converting an
algebraic expression into an infinite series of equivalent
which has been already exemplified;
value, each of
By
1st.
Art.
Division, Art.
183;
and, 4th.
3d.
By
By
134;
By
Extraction of Roots,
Coefficients,
Arts.
315-7;
the Binomial Thtorcm, Art. 321.
338. The Summation finite
2d.
Indetirminate
viz.,
of a Series
is
the
finding a
expression equivalent to the series.
The General Term of a Series is an expression from which the several terms of the series may be derived according to some determinate law. Thus, in the series
term
is
—
-^- + — -|--—
,
the general
]-
1
a
1-
o
4
because by making a;=l,
2,
3,
etc.,
each term of the
scries is found.
Again, in the series 2 general term
is
2(.r+l).
2+2 3+2 4+2 -5+ •
•
the
^ j
i
INFINITE SERIES.
As
3O7
different series are in general
governed by different methods of finding the sum, which are applicable
laws, the
to one class, will not apply universally.
We
present two methods of most general application.
First Method.
— In
whose
term
series,
first
regular decreasing geometrical
a
is a,
and
ratio
the
r,
sum
is
=
(Art. 299).
Second Method. Ex.
1.
— Find the sum of the
I
Then,
2.
5
etc.,
2V3 + 3T4 + 4T5 + 5T6+'
Tten,
infinite series =
2
In such
^^'^'
—+—
1
= S; = S -1. 2
2
^-^ + ^-^ + ^_^+,
Subtracting
+ 577+'
=
i'
etc,
=
1,
able and
Since
o
e
11 + 3—
^°^-
series, the first factor in the successive
general term The ^
p
5
o
and j-^
denominators
variable, while the second factor exceeds the first by
quantity. •' ^
•
9
T+H3+4+. «*«•' 1+1+^+^+ etc.,
^""^
3
etc.
-, •
'
= h Ans.
«tpfCg;
(1) (2) etc.
(3)
The values of p and q may be found by eliminating between any two of these equations. Taking the first two, (Art. 158.)
EC— AD Ex.
BD—C2
,
—Find the scale of the
series l-|-2a;+3a;''4-4x'-|-5a;*,
etc.
Here,
A=l, B=2, C=3, D=4,
^^ P= 2X3—1X4 2^-1X3
Now, by we please :
=5;
, , 2X4—32 =^"'^«=25=IX3=-1-
„
•
the use of the scale, the 5th coefficient
the 6th
etc.
we may extend
coefficient=2X5— 4=6; the
the ascending powers of
X
the series as far as
=pX the 4th-|-g'X the 3d^2X4 —
are wanted,
7th=2X6— 5=7,
we have 6x'
and as
for the 6th
term, 7x^ for the 7th, etc.
34S.
In a recurring series of the third order, the law
of the series involves three terras, which
by p,
q,
and
Fa^^+Gx*,
r,
we
will represent
the series being A-{-Bx-\-Cx^-\-Da?-\-'Ex''-\-
etc.
Then, by the law of the
series,
D=Cp+Bg+A>-;
E=Dp+Cg+Br; F=Ep+Dg+Cr;
etc.,
And, by combining these equations, the values of p,
q,
determined in
series
and r are
In a similar manner the scale of the higher orders.
readily found, (Art. 158.)
In finding the scale of a series, we must by inspection whether the series is in G. P.
first ;
may
be
ascertain
if not,
then
—
—
RAY'S ALGEBRA, SECOND BOOK.
312
make
of a scale containing two terms, then one of and so on, until a correct result is obtained.
trial
three, four,
We
must be careful not to assume too mamj terms; for in that ease every term of the scale will take the form f
343.
To find
scale of relation
is
sum
the
an
of
infinite
Let A+B.T+C.i:'+D.i;''+E.r', of the second order, Then,
.
.
recurring series whose
known.
p and
be a recurring series
etc.,
q being the terms of the scale.
A=A;
.
Ba;=Ba;; D.r'=C)W'-f-Bg,r'';
etc.,
ad
infinitum.
Represent by S the required sum, and add together the corresponding members of the preceding equations, observing that Bx-fCr-'+Dx^+j etc., =S A then, we have
—
;
S=A+Bz+(S—A) 2->x+Sqx^ S—Sp.-c—Sga;2=A+Ba;— A^AC; s= ^+S'-A;y — 1 ;
.-.
Or,
If
...
.
px
qx-
we make q=0, (remembering that B=Ap),
comes
S=; 1
—pK'
,
which
is,'
as
it
ought ^
the formula be-
with the
to be, identical '
formula of Art 299.
—
Rejiark. Every recurring development of a rational
tlie
may
series
fraction,
be supposed to arise from
and
tlie
summation
of sncli
a series niny be regarded as a return to the generating fraction. There are several methods of accomplishing this return of these the preceding is regarded as the most suitable for an elementary :
work. 1.
Find the sum of
Here,
+ 3a;+5.r+7.c'+9.i;*,
A=l, B=3, C=5, D=7,
And, hence, „, J. hen, '
l
„ >5
(Art. 341.)
p=2,
etc.
etc.
g= — 1.
——2.rT^nl+.r —= 1+ar—
A-LB)-^A;).r
= 1—p.c ^,
;
(jx-
1
-Ix+x-
^•
(1—xr
EEVERSION OF SERIES. In each of the following
series, find the scale
and the sum (S) of an
tion,
of rela-
number of terms
infinite
+ 6aj+12x'+48x»-fl20a;*+,
2. l
3 13
:
etc.
Ans.p=l,,=6;S=jl±^, l+2a;+3a;^+4a:'+5a;'+6a;5+,
3.
p=2,
Ans. a
.
abx
,
abV
Ans.
The
2=-l
S=--J_.
;
ah'a?
c-^+-?
*•
etc.
?-+.«*«•
series
is
in G. P.
pz=
S=r
•
c
c+to
x-\-x'-\-3?-\-, etc.
5.
Ans. The X
6.
—
x'-\-a^
—
G. P.
series is in
l+Sx+bx''+7x'+9x*+,
7.
G. P.
X
~l—x
»= — 1 S=
^ .
;
—
1,
l^+2^x+3V+4V+5V+6V+, Ans.
8-
;
l-\-x
etc.
Ans. J3=2, gz= 8.
p=l
etc.
.i;*-(-,
Ans. The
series is in
1+x S= l—2x-\-x'' etc
p=3, 3=-3, r=l
;
S— S=^i±^,. -(1-x)
EEVERSION OF SERIES. 344. To
Eevert a Series is to express the value of in it by means of another series involving the powers of some other quantity. Let X and y represent two undetermined quantities, and express the value of 1/ by a series involving the powers the
unknown quantity
of X
;
thus, y^ax-\-bx^-\-ca:?-\-da^-\-, etc.,
in
which
a,
b,
c,
d,
etc.,
are
known
revert this series is to express the value of
2d Bk.
27*
(1),
quantities
x
;
then, to
in a series
1
RAYS ALGEBRA, SECOND BOOK.
314
containing the
powers of
known
quantities a,
b,
d, etc.,
c,
and the
y.
To revolt
this series,
assume x^Ay-\-By--\-Cy^-\-T)y*,
•which the coefficients A, B, C
Find the values of
y^,
ifi,
.
y^
.
.
.
from
y^=
a^x"+Ba^l)z*+
3/"=
a'x'-i-
0=Aa x+Ab
x^-\~
(2),
Ac
in
(1), thus,
y-=a''-x'^-^2abx^-\-{Jf-+1ai:)X^-\-
Substituting these Talues in
etc. (2),
aie undetermined.
.
.
.
.
.
.... ....
etc.
and arranging, we have x^-\-
Ad
a;^+, etc. I
-1
Ba-
+2Ba6
+
Ca3
+ Bi^l + 2Bac + 3Ca26 I
Since this of X, x^, z^,
Aa—
is true, etc.,
whatever be the value of
will each
=0,
x,
(Art. 314, Cor.),
=0,
.
and
the coefficients
we have
a b
— REVERSION OF SERIES.
[Art. 344], the result will be the required development of x; y—a' being substituted for 2, the result is
(3,)
and
then,
=^=jXy-9
and 1st.
^=1
;
(a'Xlf^)=3|
log. a.
Let us resume the equation a^^N.
we make ar^l, we have
If
that
a'^N^a
;
hence, log.
is,
Wliatever he the base of the system,
system
(«+x)j.
log.
logarithm in that
its
1.
is
we make x^Q,
2d. If
o''=N=-l Jn any system
36S.
;
a*=N, we have
in the equation
hence, log.
the logarithm
1^0
of 1
;
that
is 0.
In the equation a"^=N, consider
common and
is,
a>l,
as in the
x negative
the Naperian systems, and
;
we
then have
a~'=--;=N, and -^^-=0'^ =0, or a"
Hence, than
1, is
'
of 0, in a system whose number and negative.
the logarithm
an
infinite
In a similar manner,
base
is less
loor.
a
^han
1,
it
may be shown
the logarithm of
3I»9. As the
positive
0= — oo
base
is
.
greater
that in a system whose
is infinite
and
positive.
and negative characteristics are
taken to designate whole numbers and fractions, there re-
mains no method of designating negative quantities hy or, as N, in each of the equations a''=:N and
logarithms
o"^^N,
is
.Negative
;
positive,
numbers have no real logarithms.
RAYS ALGEBRA, SECOND BOOK.
332
COMPUTATION OF LOGARITHMS. 370.
Before proceeding to explain
the methods of
computing logarithms, we may observe that sary
compute the logarithms of
to
the
For, the logarithm of every composite
of the logarithms of 5,
its factors.
4=22
1.
to
site 2.
4=2
hence, log.
6=2x3
"
log.
8=23
"
log.
9=32
"
log.
"
log.
10=2x5
7
number
equal
is
6= 8=3 9=2 10=
4,
6,
1,
2,
3,
Thus,
etc.
8,
sum
to the
Hence, the logarithms of
being known, we can find those of
7, etc.,
only neces-
it is
prime mimhers.
log. 2, (Art. 362); log. 2-j- log. 3; log. 2; log. 3;
log.
2+
log. 5.
Suppose the logarithms of the numbers
2, 3, 5, and show how the logarithms of the componumbers from 12 to 30 may be found.
known
be
;
Of what numbers between 30 and 100, may
rithms be found from those of 2, 3,
5,
and
<
Ans. Of 23 different numbers, from 32
371*
In the
common
the loga-
and why?
;
to 98.
system, the equation a'^^N (Art.
357j becomes 10'=N. If
we multiply both
Also,
.
.
Hence, in the
become
common
system, the logarithm of any
the logarithm of 10 times,
increasing the characteristic by
advantage of Briggs'
100 times,
1,
2
etc.
etc.,
From
number
this results
ia
0.477121,
"
1.477121,
300 "
2.477121.
"
30
"
will
that number, by
system.
Thus, the log. of 3 "
we have
sides by 10,
10^>h''-\-,
will be the
Now,
etc.
taken so small that etc.,
371
the sign
same as
may
evident that h
of the
mm
X7i-|--^
be
/t'^^-l-,
the sign of the first term X7i.
X'7i+^X"7i-+,
Eor, since
it is
etc.,
=A(X'+JX"/i+,
etc.),
if
h
be
taken so small, that ^X"h-\- ^X"'h'^-\-, etc., becomes less than X' (their magnitudes alone being considered), the sign of the
sum
of these two expressions
must be the same as the sign
of the
greater X'.
413. By comparing the transformed equation in Art. 406, with the development of Xj in Art. 411, it is easily seen that
Xj may be considered
y corresponding
to x,
and
the transformed equation,
r to h.
Hence, the tranformed equation may be obtained by substituting the values
Let less
of
X, X',
etc.,
the development
be required to find the equation whose roots are
it
by 1 than those of the equation
Here,
in
As an example,
of Xj.
.
.
.
X
=a;3— 7a;+7,
x'
— 7a;+Y=0.
—
—
372
RAYS ALGEBRA, SECOND BOOK.
EQUAL ROOTS. To determine the equal roots of an equation.
414.
We
have already seen (Art. 390, Keni.)
may have two
tion
"We now propose and how
roots,
more of
determine when an equation has equal
to
we
Hence,
3(.f
is
— 2)'^0
(1), its first derived
— 2/-=^0.
see that if .any equation contains the
times, its fiist derived
same
In general, a)"'(X
its
first
of
the given equation,
(X
— a)°'~'(a;— 6)"-', and
equal
and n
to a,
takem
divisor of the
we have an equation X=0, containing
if
— b)",
the factors
derived polynomial will contain the fac-
m(x— a)^~hl(x— 6)"~'
tors
factor
polynomial will contain the same factor
taken twice; this last factor is, therefore, a common given equation, and its iirst derived polynomiaL
(.r
an equa-
tliat
roots equal to each other.
its
to find them.
we take the equation (x
If
polynomial
tlirpr
or
that
;
and
its
the
is,
first
greatest
the given equation will
equal
roots, eacli
to
common
divisor
derived polynomial, will be
have 7n
roots,
each
b
Therefore, to determine whether an equation has equal roots,
Find
the greatest
common
diiisor hetiveen the equation
dirici'd jiojynom i)
it
has two
and so on.
— — 8.r-)-12=i0, .(;-
if so, to find
to
determine
them.
derived polynomial (Art. 411), ox-
and the given eqtiation (Art. 108)
-2=0. and x=-\-2.
to 2.
.r''
—
contains a factor of the
to h,
has e(|ual roots, and
The G.C.D. of Hence,
if
three roots equal to a,
it lias
Given the equation
1.
whether
We
a;
has a factor of the form (x
equal to
'coots
to
and
dicisor,
— 2.r — S. is
x
—
2.
Therefore, the equation has two roots
LIMITS OF THE ROOTS OF EQUATIONS. Now, since the equation has two by (x—2){x—2), or (x-2)-'.
ible
373
2, it must be divisWhence,
roots equal lo (Art. 395).
x3—x--Sx+12=(X^2)\x+3)^0,
and 3-+3=0, or
.t
=
—
3.
Hence, ivhcn an equation contains other roots besides the equal degree of the equation
roots, the
may
be depressed by division, and
the unequal roots found by other methods.
The following equations have equal
roots
;
find all the
roots. 2.
.t'— 2.r^— 15.r+36=0.
3.
x*_9.r=+4.r+l'2=0.
4. a;^—6a;=+12a;-^— 5. 6. 1.
.
.
.
Ans.
.
Ads.
.
102+3=0.
2, 2,
Ans.
af—2x'+dx^—1x'+8x—S=.-.0.
9.
x^-\~8x'—6x*—6x'-\-9x'+Sx~'i=^0.
Ans.
Ans.
Suggestion. — In example, the G.C.D.
is
common measure
x^ of
may
— x^ —x-\-l. tliis
thus resolve into factors;
1,
1,
solution of
the
the principles above explained
and
1,
1,
1, 1,
—\±ly—n. —1, —1, —4.
equations of high degree, Thus, in the last
be extended.
Proceeding, we may,
its first
If it is of the
form X
factors of the original equation will evidently be (X tlielst method,
(.T-)-l);
we
find x^
by the 2d, (x — 1)^
hence, [X
— 1)-
is
1st, find
derived polynomial, and 2d, find the G.C.D. of the first and
or,
second derived polynomials.
By
1, 1, 1, 3.
x'—1x'-\-9x'-\-21x—b4=0. Ans, a:=3, 3, 3, —2. x*+2.r'— 3x2— 4^+4=0. Ans. —2, —2, +1, +1. a:*— 12.r'-f50a;^— 84.i'+49=0. A.3±;/2, 3±v 2^.
8.
the
—4. —1, —3.
3, 3,
— a,
one of the
— o)^,
etc.
—x^—x-\-l={x— l)(x^— l)=(a;— 1)2
is
a factor of x^
a factor of the original equation;
— x-—x-\-\.
LIMITS OF THE ROOTS OF EQUATIONS.
415. Limits to a Root of an Equation are any two numbers between which that root lies.
A
Superior Limit to the positive roots
is
a
number
numerically greater than the greatest positive root.
1
] 1
RAYS ALGEBRA, SECOND BOOK.
374
Its characteristic
thun
An
is,
when it, or any number jrreater x in the equation, the result is
that
substituted for
is
it,
Inferior Limit to the negative roots,
number The
a
is
numerically greater than the greatest negative root. substitution of
The
number greater than
or any
it,
produces a negative
it,
for
a-,
is
to
result.
object of ascertaining the limits of the roots
diminish the labor necessary in finding them.
416.
Proposition
increased hy unity,
is
I.
— rAe
negative coefficient,
greatest
greater than the greatest root of the
equation.
Take the general equation
....
a;»-|-Aa;"-i-|-Ba;''-'
A
and suppose
The reasoning
-|-T.t-|-V=0,
be the greatest negative coefficient.
to
will not be affected if
we suppose all the and each equal to A. find what number substituted for x will
coefficients to be negative, It is required to
make By
;c">A(a;"-i-|-x"^"-|-a^"-3. Art. 297
„^ havea;">A( .
But render
if
/
^
x"= „
x'=
\x"
X
—
A.r"
X
sum
the
a;"— 1
—
^,
1,
we
_|_a:-)-l).
.
.
in purentlicsis
is
a—
=-
;
hence,
we must
A
A.r"
\
or a;">
,.
find a;=A-|-l;
.
.
A-fl
substiluteil for
will
A
A.r"
conseqnentlv, a"'> '
x
'
'
,
-. .and,
By considering
.
'
,
.r— 1
-.
.1-1
all the cotfficients after the first negative,
we have
taken the most unfavorable case; if any of them, as B, were positive, the quantity in parenthesis would he less.
417.
Proposition
coeffieient,
II.
— If
extract a root of
number of tenns preceding crease
it
it
icc
tahe the greatest negatice
vhose index
hy unity, the result will he greater
positive root of the equation.
is
equal
the first negative term, titan
to
the
and
in-
the greatest
;
:
LIMITS OF THE ROOTS OF EQUATIONS.
Let
Cx"^'' be the first negative term,
est negative coefiicient; then,
render
positive
C
;
the
+a;+l)
because
negative,
of the
first
proposed
supposes
this
and each equal
the great-
any value of x which makes
x''>C(a;»-'--|-x"-'-' will
C being
375
(1)
equation
>0,
or
the coefiioients after
all
which
to the greatest,
evi-
is
dently the most unfavorable case.
By
a:">c(
'
or,
^_^
x"^
Cx"~''+^ =
>C .-.
—
x-1
by X —
by multiplying both members
.
Hence,
X n-
•'
when (a;— l)a;'--i=C, or Buta;— 1 is vTn
x=l + v
or
>! + {
Or
iT,
CL
Find superior limits of the roots of the following equations 1.
:
a;*— 5a;'-f37x'— 3.x
Here, 2.
By
.-.
1+^^0=1+5x^6,
l+^C^l-\-f49^1-\-1=.S,
Ans.
Ans.
a-^+llx'— 25a;— 67=0. supposing the second term +0a;', we have
hence, the limit 4.
r=l
and
x'+'7x*~12x'—49x'+b2x—lS^0.
Here, 3.
C=5,
+ 39=0.
is
1+^^6*7, or
6.
3a:'— 2a;^— llx+4=0.
Dividing by
3,
Here, the limit
x'—fcc^— Va;+|=0. is
1
+ Vi
or 5.
r^3
— RAY
370
—
.
ALGEBRA, SECOND BOOK.
S
418. To
determine the inferior limit to the negative this will change the signs of tlie alternate terms
roots,
;
change the signs of the roots (Art. 400)
The
superior
changing
then,
;
by
of the roots of this equation,
limit
its sign, will
lo the inferior limit of the roots of
the proposed equation.
419. taken in
Proposition III. tlie
order of
being greater tlian
a
series
than
a, b'
b and
b,
of numbers,
e,
—
//' the real roots of an equation, magnitude, he a, b, c, d, etc., a b grmter than c, enid so on; then, if
tlicir
a',
b',
and
which
in
d', etc.,
c',
a nundjcr between a and
b,
c'
x in
so on, be sidtstitutedfor
tlie
tion, the results will be alternately positice
is
greater
prajiused equa-
and
first member of the proposed equation =.0. {.r—a)(x—b)(x—e')(.x—d).
The to
;i'
a number between
negative.
is
equivalent
.
X
Substituting for
we
obtiiin the
{a'
—a)(a'—b){a' — c)i«' the factors are
(6'
—«)(&'
^)[b'
d), etc.
— C)(6'— d),
etc.
c)(o'--(}). etc.
—
,
b)(d'
and the
.r,
in
1.
=+ product,
since
all
.
.
.
^ — product,
since only
is
=+
.
product, since two
+.
rest
— c)(cl'—d),
odd number of factors
Corollary
.
is
factors are
for
a', b', c', etc.,
-j-.
— (ej'—a\ic' — b){e' — one factor
{d'—ei][d'
numbers
the proposed series of
following results:
etc.
—
,
and
r^
.
— product,
since an
so on.
— If two numbers be successively substituted
any equation, and give results with contrary
there must bo one, three, fee, or some odd
signs,
number of
roots
between these numbers. Corollary
2.
— If
results with the
two numbers, substituted for
same
sign,
there must be two, four, or some even
or no roots at
all.
a-,
give
then between these numbers
number of
real roots,
THEOREM OF STURM.
—
Corollary than
q,
377
3. If a quantity q, and every quantity greater render the results continually positive, q is greater
than the greatest root of the equation. Corollary
4.— Hence,
be changed, and
if p,
the signs of the alternate terms
if
and every quantity greater than p,
—p
renders the result positive, then
is less
than the least
root of the equation. Illustration.
— If we form the equation -whose roots are
—
—3, the result is x^ 422_lla;^30=0. number whatever for x, greater than 5, put a;^5, the result If 2,
we
is zero,
Substituting a
From
Cors. 3
number
and
number less
4, it is
real roots, either in the
the result is positive.
less than
less
than
easy
and any
5, 2,
substitute If
we
should be.
—
Substituting
is positive.
we
if
than
2,
and greater than
5,
Putting x=2, the result
is negative.
Substituting for X, any
the result
it
any number
substitute for X,
the result
as
Now,
is zero.
and greater than
—
3,
3, it is zero.
—
3,
to find
the result
is
negative.
when we have
ascending or descendiug
passed
all
the
scale.
STURM'S THEOREM.
430.
To find
the
number of real and imaginary
roots
of an equation. In 1834, M. Sturm gained the mathematical prize of the French
Academy
beautiful theorem, by
nation of
all
of Sciences, by the discovery of a means of which the number and sit-
the real roots of an equation can, with cer-
tainty, be determined.
This theorem we shall now proceed
to explain.
_|-Ta;+V==0, be Let X=x»-f Ax-'-'-f Ba:''-^ any equation of the re"" degree, containing no equal roots ;
for if the given equation contains equal roots, these
be found (Art. 414), and vision.
2d Bk.
32
its
may
degree diminished by di-
:
RAYS ALGEBRA, SECOND BOOK.
378 Let the
Divide
first
X
by
X
derived function of
of a lower degree with respect to
is
(Art. 411) be denoted
X|)X
Xi until the remainder x
(Qi
XjQi
than the divisor, and call this remainder X^; that is, let the remainder,
,-
q1_
,,
—
wiih
lis
and
ncr,
X2
in the
^
^:)Xi (Q2 X0Q2
be denoted by X^.
sign changed,
Divide Xi by
by Xj.
same man-
so on, as in the margin, de-
XoQo^
y^
X.,
noting the successive remainders, with their signs
until
we
by X3,
changed,
X^,
"
etc.,
"'I'
arrive at a i-emainder ^vhich
does not contain X, which must always
its
sijrn
its first
derived function.
changed, be called
In these divisions,
X3Q3^
Xi
happen, since the equation having no equal roots, there can be no factor containing equation and
(^'^
-
-^^^M?.
x.
common
X, the
to
Let this remainder, having
X,._j,j.
we may,
to
avoid fractions, cither multiply or
divide the dividends and divisors by ^Tiy positive number, as this will not affect the signs of the functions X, Xj, Xo, etc.
By
this operation,
we
ol)tain the series of quantities
X, Xi, X,, X3.
Each member of .r
.
X,+i
.
this series is of a
(1).
lower degree with respect
primitive function,
and Xi, Xo,
431. Lemma
I.
— Tico
etc.,
to
X
the
Xi, X,,
/()/•
than the preceding, and the last does not contain X.
Call
auxiliary functions.
C077secnt!vc functions,
example, can not both cauhJi for the savie value of x.
From
the process
by which Xi, Xo,
are obtained,
etc.,
we have
the
following equations
X,
=X,Qi-X, =X,Q;-Xo
X.
- XiQ.;-
X
(1) (1^)
X.|
X,_i=X,.Q,-X,+i.
(3)
.
.
.
^r).
If possible, let Xi^O, and Xj^O; then, by eq. (2) we have X..^0; hence, by cq. (3) we have X4^0; and proceeding in the same way, we shall find X-^0, X5=0, and finally Xr-|^] 0. But
=
this is impossible, since X^-i-i does not contain X,
not vanish for any value of x.
and therefore can
—
h
THEUKEM OF STURM.
—
43S. Lemma ishes for
379
II. If one of the auxiliary functions vanany particular value of x, the two adjacent functions
must have contrary signs for the same value of Let us suppose
— X4,
X3=0, when x—a;
tliat
and X3=0;
X2=— X4;
therefore,
x.
then, because
that
is,
Xo— X3Q3
Xj and X4 have
contrary signs.
433. Lemma vanishes
of any of
and
III.
— If
any of
the
when x=:a, and h be taken
auxiliary functions
small that no root
so
the other functions in series (1) lies between a
—
number of variations and permah and a-(-h are substituted for x in this
a-|-h, then will the
nences, series,
when
a
—
be precisely the same.
a
Suppose, for example, the substitution of function X3 to vanish
for
X
causes
the
by Art. 421, neither of the functions Xo or X^ can vanish for the same value of X; and since when X3 vanishes, Xj and X4 have contrary signs, (Art. 422); therefore, the substitution of a for x in X2, X3, X4, must give
Xj
;
then,
,
X3
,
And
X4
or Xj, X3, X4.
,
-
+
-
,
+
h is taken so small that no root either of X2^0, or between a h and Ct-|-''', the signs of these functions will continue the same whether we substitute a — h or a-\-h for X (Art. 419). Hence, whether we suppose X3 to be -f- or by the
S.^=0,
since
—
lies
—
substitution of
OS
one permanence.
h and a-l"" ^"^ ^t there will be Thus, we shall have either
X2
+
,
X3
,
X4
,
or X_.
± -
,
X3
,
one variation
and
X^,
- ± +
So that no alteration in the number of variations and perma-
nences can be made in passing from
424. Lemma IV.— If
a
is
a
—h
to a-]-h.
a root of
X, Xj, X,, in passing from a h
then the series of functions
variation of signs
—
taken so small that no root of the function a
—h
and
a-l-h.
X=0,
the equation etc.,
to
will lose one
a-(-h
X,=0
;
lies
li
being
between
RAY
380
For X substitute
by H.
ALGEBRA, SECOND BOOK.
a-(-/t in tlie
Also, put. A, A',
when
tions
S
A"
equation
X^O, and
for the values of
a-}-/i is substituted for x\
a
But, since
X^O, we
a root of the eq.
is
0,
since the eq.
H=A'7i-|-JA'"7!2+,
X^O
denote the result
and
derived func-
its
then (Art. 411),
H=A+A'/t-|-LA"A=+,
K' can not be
X
etc.
shall have
A^O,
has no equiil roots.
\\'hile
Hence,
=7i(A'-|-JA''A-|-, etc)-
etc.,
Now, A may be talien so small that the quantity within the parenhave the same sign as its first term A', (since .V ex-
thesis shall
presses the
derived function of X, corresponding
first
therefore, the sign of X,
Art. 4121;
when a:=a^n,
to X^,
in
will be the same
as the sign of Xj. If
wo
result
substitute
by
H',
a— A
we then
for
X
in the equation
X^O, and
have, by changing A into
—
A,
denote the
in the expres-
sion for H,
n'=- A| A'— l.VA-f
,
etc).
Now, it is evident that for very small values of A, the sign of 11' A'A, and, consequently, will be depend upon the first term
—
will
contrary to that of A'.
of signs in the
x=a-\-hj there
first is
Hence, when
X^Q — A,
there
a continuation of the
same
sign.
a variation
is
two terms of the series X, Xj
;
and when
Therefore, one
is lost in pnssmg from x^^a — li to a-\^h. any of the auxiliary functions should vanish at the same time by making x=^a, the number of variations will not be affected on this account (Art. 423), and therefore, one variation of signs will Btill be lost in passing from a— A to a-|-A.
variation
If
425.
Sturm's Theorem.
— If any
tico
numlers, p and
q,
than q) he siihslil;ili:d for x in the scrips of functions X, X], X2, etc., the substitution of p for x giving (p Jicing
less
k farialions, and that of q for k
— k'
uill he the rxact
X=0,
lohich
He.
hetnven
Lot us suppose that pose that degrees
reaches
u.
of
x, gicing k' variations
number of real p and q.
;
tJun,
roots of the ri^uatiou
—
oo is substituted for x, and supcontinually iticreases and passes through all
magnitude
-|- gc.
till
it
becomes
0,
and
finally
THEOREM OF STURM. Now,
381 minus
evident, that so long as X, with its
it is
than any of the roots of
X=0, Xi=0,
sign, is less
no alteration will take
etc.,
place in the signs of any of these functions (Art. 419); but when X becomes equal to the least root (with its sign) of any of the
auxiliary functions, although a change
we have seen
this function, yet
may
occur in the sign of
(Art. 423) that
it
is
the order only,
and not the number of variations which is affected. But when x becomes equal to any of the roots of the primitive function, then one variation of signs Since, then,
passes through
is
always
variation
a,
lost.
always
is
lost
whenever the value of X X=0, and since a,
root of the primitive function
u,
variation can not be lost in any other vpay, nor can one be ever introduced,
it
by
above that given by
a;==p,
number
of real roots of
Corollary.
equation
number of
number of imaginary
(p
