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j^ g PROPERTY OF ^
Wimtjjof
ARTES SCIENTIA VERITAS
N
ELEMENTARY
MATHEMATICAL ANALYSIS
\
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McGraw-Hill Y>oo\i(jDmpar^
PuSGsAers qfjBoo/^/br
Electrical World I1ie£jig|neeringaiid>&mig Journal
Engineerii^ Record Engineering News
I2aitv\ray Age Gazette Americdn Machinist
Signal EtigiriQer AmericanEt^gpieer
Electric Railway Journal Coal Age
Metallurgical and Chemical Lngineering Power
MODERN MATHEMATICAL TEXTS
Edited by Charles S. Slighter
ELEMENTARY
MATHEMATICAL ANALYSIS
A TEXT BOOK FOR FIRST
YEAR COLLEGE STUDENTS
BY ^,
CHARLES Sl'SLICHTER,
PROFSBSOR OF APPLIED MATHEMATICS
UNIVEB8ITT OF WISCONSIN
First Edition
McGRAW-HILL BOOK COMPANY, Inc.
239 WEST 39TH STREET, NEW YORK
6 BOUVERIE STREET, LONDON, E. C.
1914
€NGINEERIN6
IIBRARY
ecu
Copyright, 1914, by the
McGraw-Hill Book Company, Inc.
THB.MAPLK. PRES3*TOBK*PA
PREFACE
This book is not intended to be a text on "Practical Mathe-
matics" in the sense of making use of scientific material and of
fundamental notions not already in the possession of the student,
or in the sense of making the principles of mathematics secondary
to its technique. On the contrary, it has been the aim to give
the fundamental truths of elementary analysis as much prominence
as seems possible in a working course for freshmen.
The emphasis of the book is intended to be upon the notion of
functionality. Illustrations from science are freely used to make
this concept prominent. The student should learn early in his
course that an important purpose of mathematics is to express and
to mterpret the laws of actual phenomena and not primarily to
secure here and there certain computed results. Mathematics
might well be defined as the science that takes the broadest view of
all of the sciences — an epitome of quantitative knowledge. The
introduction of the student to a broad view of mathematics can
hardly begin too early.
The ideas explained above are developed in accordance with a
two-fold plan, as follows:
Firstj the plan is to group the material of elementary analysis
about the consideration of the three fundamental functions:
1. The Power Function y = ox** (n any number) or the law
"asx changes by a fixed multiple j y changes by a fixed multiple also.''
2. The Simple Periodic Function t/ = a sin mx, considered as
fundamental to all periodic phenomena.
3. The Exponential Fimction, or the law "asx changes by a fixed
increment f y changes by a fixed multiple.''
Second, the plan is to enlarge the elementary functions by the
development of the fundamental transformations applicable to
273376
vi PREFACE
these and other functions. To avoid the appearance of abstruse
ness, these transformations are stated with respect to the graph
of the functions; that is, they are not called transformations ^ bu
' ' motions ^ ^ of the loci. The facts are summarized in several simpl
'^Theorems on Loci," which explain the translation, rotation, sheai
and elongation or contraction of the graph of any function in tl:
xy plane.
Combinations of the fundamental functions as they actuaU
occur in the expression of elementary natural laws are also di
cussed and examples are given of a type that should help to explai
their usefulness.
Emphasis is placed upon the use of time as a variable. Th
enriches the treatment of the elementary functions and brin^
many of the facts of "analytic geometry" into close relation t
their application in science. A chapter on waves is intended t
give the student a broad view of the use of the trigonometric f un<
tions and an introduction to the application of analysis to per
odic phenomena.
It is difficult to understand why it is customary to introduc
the trigonometric functions to students seventeen or eighteen yeai
of age by means of the restricted definitions applicable only to tl
right triangle. Actual test shows that such rudimentary methoc
are wasteful of time and actually confirm the student in narrov
ness of view and in lack of scientific imagination. For that reasoi
the definitions, theorems and addition formulas of trigonometr
are kept as general as practicable and the formulas are give
general demonstrations.
The possibilities and responsibilities of character building in tl
department of mathematics are kept constantly in mind. It
accepted as fundamental that a modern working course in math<
matics should emphasize proper habits of work as well as prop<
methods of thought; that neatness, system, and orderly habi
have a high value to all students of the sciences, and that a tex
book should help the teacher in every known way to. develop thej
in the student.
Chapters V, VI and VII contain material that is required f(
admission to many colleges and universities. The amount of tin
devoted to these chapters will depend, of course, upon the loc
requirements for admission.
PREFACE vii
The present work is a revision and rewriting of a preliminary
form which has been in use for three years at the University of
Wisconsin. During this time the writer has had frequent and
valuable assistance from the instructional force of the department
of mathematics in the revision and betterment of the text. Ac-
knowledgments are due especially to Professors Burgess, Dresden,
Hart and Wolff and to Instructors Fry, Nyberg and Taylor.
Professor Burgess has tested the text in correspondence courses,
and has kindly embraced that opportunity to aid very materially
in the revision. He has been especially successful in shortening
graphical methods and in adapting them to work on squared paper.
Professor Wolff has read all of the final manuscript and made
many suggestions based upon the use of the text in the class room.
Mr. Taylor has read all of the proof and supplied the results to the
exercises.
Professor E. V. Huntington of Harvard University has read the
galley proof and has contributed many important suggestions.
The writer has avoided the introduction of new technical terms,
or terms used in an unusual sense. He has taken the liberty, how-
ever of naming the function ax*», the "Power Function of x," as a
short name for this important function seems to be an unfortu-
nate lack — a lack, which is apparently confined solely to the
English language.
It is with hesitation that the writer acknowledges his indebted-
ness to the movement for the improvement of mathematical in-
struction that has been led by Professor Klein of Gottingen;
not that this is not an attempt to produce a text in harmony with
that movement, but for fear that the interpretation expressed
by the present book is inadequate.
The writer will be glad to receive suggestions from those that
make use of the text in the class room.
Charles S. Slighter.
University of Wisconsin
July, 25, 1914
I
CONTENTS
Pbefacb V
Introduction xi
Mathematical Signs and Symbols xiv
Chapter
I. Variables and Functions of Vabiables 1
II. Rectangular Coordinates and the Power Func-
tion 25
III. The Circle and the Circle Functions. 94
IV. The Ellipse and Hyperbola 137
V. Single and Simultaneous Equations 162
VI. Permutations, Combinations, the Binomial
Theorem 182
VII. Progressions 198
VIII. The Logarithmic and Exponential Functions . . . 214
IX. Trigonometric Equations and the Solution of
Triangles 282
X. Waves 321
XI. Complex Numbers 341
XII. Loci 381
XIII. The Conic Sections 398
XIV. Appendix — A Review of Secondary School
Algebra 452
Ii^EX * 483
ix
ELEMENTARY
MATHEMATICAL ANALYSIS
xii INTRODUCTION
n. Materials. All mathematical work should be done on one
side of standard size letter paper, 8J X 11 inches. This is the
smallest sheet that permits proper arrangement of mathematical
work. There are required:
(1) A note book cover to hold sheets of the above named size and
a supply of manila paper "vertical file folders" for use in submit-
ting work for the examination of the instructor.
(2) A number of different forms of. squared paper and computa-
tion paper especially prepared for use with this book. These sheets
will be described from time to time as needed in the work. Form
Af 2 will be found convenient for problem work and for general
calculation. M2 is a copy of a form used by a number of public
utility and industrial corporations. Colleges usually have their
own sources of supply of squared paper, satisfactory for use with
this book. The forms mentioned in the text, printed on 16 lb.,
St. Regis Bond, cost about 25 cents per pound in 100 lb. lots
(12,000 sheets) from F. C. Blied & Co., Madison, Wis.
(3) Miscellaneous supplies such as thumb tacks, erasers, sand-
paper-pencil-sharpeners, etc.
m. General Directions. All drawings should be done in
pencil, unless the student has had training in the use of the ruling
pen, in which case he may, if he desires, "ink in" the most im-
portant drawings.
All mathematical work, such as the solutions of problems and
exercises, and work in computation should be done in ink. The
student should acquire the habit of working problems with pen
and ink. He will find that this habit will materially aid him in
repressing carelessness and indifference and in acquiring neatness
and system.
TO THE INSTRUCTOR
The usual one and one-half year of secondary school Algebra
including the solution of quadratic equations and a knowledge of
fractional and negative exponents, is required for the work of this
course. In the appendix will be found material for a brief review
of factoring, quadratics, and exponents, upon which a week or ten
days should be spent before beginning the regular work in this
text.
)
INTRODUCTION
Xlll
The instructor cannot insist too emphatically upon the require-
ment that all mathematical work done by the student — whether
preliminary work, numerical scratch work, or any other kind
(except drawings)— «haU be carried out with pen and mk upon
paper of suitable size. This should, of course, include all work
done at home, irrespective of whether it is to be submitted to the
instructor or not. The "psychological effect" of this requirement
will be found to entrain much more than the acquirement of mere
technique. If properly insisted upon, orderly and systematic
habits of work will lead to orderly and systematic habits of
thought. The j&nal results will be very gratifjdng to those who
sufficiently persist in this requirement.
At institutions whose requirements for admission include more
than one and one-half units of preparatory algebra, nearly all of
Chapters V, VI, and VII may be omitted from the course.
An asterisk attached to a section number indicates that the
section may he omitted during the first reading of the book.
GREEK ALPHABET
Capitals
Lower
case
Names
Capitals
Lower
case
Names
A
B
r
A
E
z
H
G
I
K
A
M
a
/3
7
S
€
r
e
K
X
Alpha
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Theta
Iota
Kappa
Lambda
Mu
N
V
E
^
0
0
II
IT
P
P
s
<r
T
T
T
V
*
0
X
X
^
^
12
(0
' '
Nu
Xi
Omicron
Pi
Rho
Sigma
Tau
Upsilon
Phi
Chi
Psi
Omega
XIV INTRODUCTION
• • •
read
=
read
J^
read
•
read
^^
read
>
read
<
read
>
read
(a, 6)
read
i?
read
n!
read
lim ., .
read
X = 00
read
|a|
read
lOgoX
Igx
read
read
In X
read
MATHEMATICAL SIGNS AND SYMBOLS
and 80 on.
is identical with.
is not equal to.
approaches.
is approximately equal to.
is greater than.
is less than.
is greater than or equal to.
point whose coordinates are a and b.
factorial n.
factorial n or n admiration.
limit off(x) as x approaches a.
X becomes infinite,
absolute value of a.
logarithm of x to the base a.
common logarithm of x.
natural logarithm of x.
i» — r
Zw„ read summation from n = 1 to n = fof u
n ^ TO
ELEMENTARY
MATHEMATICAL ANALYSIS
CHAPTER I
VARIABLES AND FUNCTIONS OF VARIABLES
1. Scales. If a series of points corresponding in order to the
numbers of any sequence^ be selected along any curve, the curve
with its points of division is called a scale. Thus in Fig. 1 (o)
the points along the curve OA have been selected and marked in
order with the numbers of the sequence:
0,1/4,1/2,1,2^3,5,7,8
Thus primitive man might have made notches along a twig
and then made use of it in making certain measurements of
A
3 «
(a) A Non Uniform Scale
li II I li I nil 1 1 il I 1 1 il I 111 In 1 1 h I I I h 1 1 I li I I il III 1 1
0 12 3 4 5
I \b) k Uniform Arithmetical Scale
I I I I I I I I I I I 1 1 I I 1 II I I I I I I 1 I I I I I I ! I I I I 1 1 I I I I I I I I I I II I I
-B -4 -S -2 -1 0 +1 +2 +3 +4 +5
(c) A Uniform Algebraic Scale
Fig. 1. — Scales of Various Sorts.
interest to him. If such a scale were to become generally used by
others, it would be desirable to make many copies of the original
scale. It would, therefore, be necessary to use a twig whose shape
could be readily duplicated; such, for example, as a straight stick;
and it would also be necessary to attach the same symbols in-
variably to the same divisions.
A sequence of numbers here means a set of numbers arranged in order of
iQagnitude.
1
ELEMENTARY MATHEMATICAL ANALYSIS
[§1
Certain advantages are gained (often at the expense of others,
however) if the distances between consecutive points of division
are kept the same; that is, when the intervals are laid off by repe-
tition of the same selected distance. When this is done, the scale
piiiiii|i
Fig. 2. — An Ammeter Scale.
is called a uniform scale. Primitive man might have selected for
such uniform distance the length of his foot, or sandal, the breadth
of his hand, the distance from elbow to the end of the middle
finger (the cubit), the length of a step in pacing (the yard), the
amount he can stretch with both
arms extended (the fathom), etc.,
^'^ etc.
We are familiar with many
scales, such as those seen on a
yardstick, the dial of a clock, a.
thermometer, a sun-dial, a steam- ■
gage, an ammeter or voltmeter, |
the arm of a store-keeper's scales,
etc., etc. The scales on a clock, a yardstick, or a steel tape are
uniform. Those on a sun-dial, on an ammeter or on a good
thermometer, are not uniform.
One of the most important advantages of a uniform scale is
the fact that the place of beginning or zero may be taken at any
one of the points of division. This is not true of a non-uniform
12 u
Fig. 3. — Sun-dial Scale
i
VARIABLES AND FUNCTIONS OF VARIABLES 3
scale. If the needle of an ammeter be bent the instrument cannot
be used. It is always necessary in using such an instrument to
know that the zero is correct; if a sun-dial is not properly oriented,
it is useless. If, however, a yardstick or a steel tape be broken,
it may still be used in measuring. The student may think of
many other advantages gained in using a uniform scale.
2. Formal Definition of a Scale. If points be selected in order
along any curve corresponding, one to one, to the numbers of
any sequence, the curve, with its divisions, is called a §cale.
The notion of one to one correspondence, included in this
definition, is frequently used in mathematics.
In mathematics we frequently speak of the arithmetical scale
and of the algebraic scale. The arithmetical scale corresponds to
the numbers of the sequence:
*
U, 1, 4b, o, 4, O, . . .
and such intermediate numbers as may be desired. It is
usually represented by a uniform scale as in Fig. 1 (6). The
algebraic scale corresponds to the numbers of the sequence:
... -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, . . .
and such intermediate numbers as may be desired. It is usually
represented by a uniform scale as in Fig. 1 (c). The arithmetical
scale begins at 0 and extends indefinitely in one direction. The
algebraic scale has no point of beginning; the zero is placed at any
desired point and the positive and negative numbers are then
attached to the divisions to the right and the left, respectively, of
il/e zero so selected. The scale extends indefinitely in both
rections.
Exercises
1. Show that the distance between two points selected anywhem
m the algebraic scale is always found by subtraction.
8. If two algebraic scales intersect at right angles, the commoe
>oint being the zero of both scales, explain how to find the distance
rom any point of one scale to any point of the other scale.
8. What points of the algebraic scale are distant 5 from the
w)int 3 of that scale? What point of the arithmetical scale is
listant 5 from the point 3 of that scale?
ELEMENTARY MATHEMATICAL .^'ALYSIS
li
9.
= ^S
E^3
2.
.=2
3
3
9 ;:
3-
= -S
5-^r 3
?5
3 —
c—
3 IZZ.
3 4
9S ^
3 H
S-
i1 i
s-
ai
^ -
"3
= ^ i -
1 3^
< Z
^
«
o
IS
— + s =
2 ** w
53] VARIABLES AND FUNCTIONS OF VARIABLES
5
3. Two Scales in Juxtaposition or Double Scales. The relation
between two magnitudes or quantities, or between two numbers,
may be shown conveniently by placing two scales side by side.
Thus the relation between the number of centimeters and the
number of inches in any length may be shown by placing a centi-
meter scale and a foot-rule side
by side with their zeros coin-
ciding as in Fig. 4.
A thermometer is frequent-
ly seen bearing both the
Fahrenheit and the centi-
grade scales (see Fig. 5).
It is obvious that the double
scale of such a thermometer
may be used (within the
limits of its range) for convert-
ing any temperature reading
Fahrenheit into the corres-
ponding centigrade equiva-
lent and vice versa. The con-
struction of scales of this
sort may be made to depend
upon the solution of the fol-
lowing problem in elementary
geometry: To divide a given
liu into a given number of equal parts.
To construct a double scale showing the relation between speed
expressed in miles per hour, and speed expressed in feet per second,
we may proceed as follows : A mile contains 5280 feet; an hour con-
tains 3600 seconds. Hence, one mile per hour equals 5280 /3600
or 22 /15 feet per second. On one of two intersecting straight
lines, OA (see Fig. 7), lay off 22 convenient equal intervals (say 1 /4
inch each). On the second of the intersecting lines, 05, lay off
15 equal intervals (say 1/2 inch each). Join the 15th division
of OB with the 22nd division of OA and draw parallels to the
line AB through each of the 15 divisions of OB. Then the 22 and
the 15 equal subdivisions stand in juxtaposition along OA and
constitute the double scale required. Labelling the first scale
Fig. 7. — Method of Construction
of Double Scale showing Relation
between "Miles per Hour" and "Feet
per Second."
6 ELEMENTARY MATHEMATICAL ANALYSIS [§3
"feet per second" and the second scale "miles per hour," the
double scale may be used for converting speed expressed in
either unit into speed expressed in the other.
By annexing the appropriate niunber of ciphers to the numbers
of each scale, the range of the double scale may be considered
220 and 150 or 2200 and 1500, etc., respectively.
The lengths of the various units selected for the diagram are, of
course, arbitrary. As, however, the student is expected to prepare
the various constructions and diagrams required for the exercises
in this book on paper of standard letter size (that is, 8i by 11
inches), the various units selected should be such as to permit a
convenient and practical construction upon sheets of that size.
Exercises
The student is expected to carry out the actual construction of only
two of the double or triple scales described in the following exercises.
1. Construct a double scale ten inches long expressing the relation
between fractions of an inch expressed in tenths and fractions of an
inch expressed in sixteenths.
To draw this double scale it is merely necessary to lay oflP the
intervals directly from suitable foot-rules. On the scale of tenths
indicate the inch and half inch intervals by longer division lines than
the others. On the scale of sixteenths represent the quarter inch inter-
vals by longer division lines than those of the sixteenths, and represent
the half inch and inch intervals by still longer lines, as is usually done
on foot rules.
2. Draw a double scale showing pressure expressed as inches of
mercury and as feet of water, knowing that the density of mercury
is 13.6 times that of water.
These are two of the common ways of expressing pressure. Water
pressure at water power plants, and often for city water service, is
expressed in terms of head in feet. Barometric pressure, and the
vacuum in the suction pipe of a pump and in the exhaust of a con-
densing steam engine are expressed in inches of mercury. The
approximate relations between these units, i.e., 1 atmosphere = 30
inches of mercmy = 32 feet of water = 15 pounds per square inch,
are known to every student of elementary physics. To obtain, in
terms of feet of water, the pressure equivalent of 1 foot of mercury,
the latter must be multiplied by 13.6, the density of mercury. This
§3] VARIABLES AND FUNCTIONS OF VARIABLES 7
result when divided by 12 gives the pressure equivalent of 1 inch of
mercury, which is 1.13 feet of water.
If we let the scale of inches of mercury range from 0 to 10, then the
scale of feet of water must range from 0 to 11.3. Hence draw a line
OA 10 inches long divided into inches and tenths to represent inches
of mercury. Draw any line OB through 0 and lay off 11.3 uniform
intervals (inch intervals will be satisfactory) on OB, Connect the
end division on OA with the end division on OB by a line AB. Then
from 1, 2, 3, . . . inches on OB draw parallels to BAj thus forming
adjacent to OA the scale of equivalent feet of water. Each of these
intervals can then be subdivided into 10 equal parts corresponding
to tenths of feet of water.
3. Draw a triple scale showing pressure expressed as feet of water,
as inches of mercury, and as poimds per square inch, knowing that the
density of mercury is 13.6 and that one cubic foot of water weighs
62.5 pounds.
To reduce feet of water to pounds per square inch, the weight of one
cubic foot of water, 62.5 pounds, must be divided by 144, the number
of square inches on one face of a cubic foot. This gives 1 foot of
water equivalent to 62.5/144 or 0.434 pounds per square inch. To
obtain the pressure given by 1 foot of mercury, the pressure equiva-
lent of 1 foot of water must be multiplied by 13.6, the density of
mercury. This result when divided by 12 gives the pressure equiva-
lent of 1 inch of mercury, or 0.492 pounds per square inch.
One pound per square inch is equivalent, therefore, to 1/0.434 or
2.30 feet of water or to 1/0.492 or 2.03 inches of mercury. If we let
the scale of pounds range from 0 to 10, we may select 1 inch as the
equivalent of 1 pound per square inch, and divide the scale OA into
mches and tenths to represent this magnitude. Draw two inter-
secting lines OB and OC through 0, and lay off 23 uniform intervals
on OB and lay off 20.3 uniform intervals on OC, 1/2 inch being a
convenient length for each of these parts. Connect the end divisions
of OB and OC with A and through all points of division of OB draw
lines parallel to BA and through all points of division of OC draw lines
parallel to CA, and subdivide into halves the intervals of the scales
last drawn. The range may be extended to any amount desired by
annexing ciphers to the numbers attached to the various scales.
Extending the range by annexing ciphers to the attached numbers
is obviously practicable so long as the various intervals or units are
decimally subdivided. The method is impracticable for scales that
are not decimally subdivided, such as shillings and pence, degrees and
minutes, feet and inches, etc.
8 ELEMENTARY MATHEMATICAL ANALYSIS [§3
4. Draw a triple scale showing the relations between the cubic foot,
the gallon and the liter, if 1 cubic foot = 7i gallons = 28i liters.
Divide the scale of cubic feet into tenths, the scale of gallons into
quarts, and the third scale into liters.
It is obvious that it is always necessary first to select the range of
the various scales, but it is quite as well in this case to show the equiva-
lents for 1 cubic foot only, as numbers on the various scales can be
multiplied by 10, 100, or 1000, etc., to show the equivalents for larger
amounts.
Select 10 inches = 1 cubic foot for the scale {OA) of cubic feet.
Draw two intersecting lines OB and OC. On OB lay off 7§ equal
parts (say, 7i inches) and on OC lay off 28i equal parts (say, 28i
quarter inches). Connect the end divisions with A and draw the
parallel lines exactly as with previous examples. The intervals of the
scale of gallons can then be subdivided into the four equal parts to
show quarts.
5. The velocity in feet per second of a falling body is given by the
formula v = gt/m. which g — 32.2 and t is measured in seconds. Draw
a double scale showing the velocity at any time.
It is obvious that the reading 32.2 on the v-scale must be placed
opposite the mark 1 on the ^-scale. First, select the range for the
t-scale, say from 1 to 10 seconds. Then a convenient scale for i is 1
inch equals 1 second, which scale can readily be subdivided to show
1/5 or 1/10 seconds. If the general method be followed, it would be
necessary to lay off 322 equal parts on a line (OB) intersecting the
/Hscale (OA). As this is an inconveniently large number, it is better
to lay off 3.22 divisions on the construction line OB. Each of these
divisions may be 2 inches in length, so that 6.44 inches will represent
the terminal or end division on the intersecting line OB. From the
6.44 inch mark on OB draw a line to 10 on the /-scale OA . Then from
2, 4, 6 inches on OB draw parallels to BA, thus locating v = 100,200,
md 300. These intervals can then be subdivided into 10 equal parts
;o show V = 10, 20, 30, . . . If values of v are wanted for t > 10,
seros may be annexed to the numbers attached to both scales.
6. Select sections from any of the double scales described above and
iscuss the relation of the number of units on one side to the number of
Qits on the other side. Show that the ratio in different sections of
le number of units on the two sides of the same double scale is not
nstant if one scale be a non-uniform scale.
7. If a double scale be drawn on a deformable body, as, for example,
a rubber band, would the double scale still represent true relations
4] VARIABLES AND FUNCTIONS OF VARIABLES 9
^hen the rubber band is stretched? What if the stretching were
ot uniform?
4. Functions. The relation between two magnitudes expressed
raphically by two scales drawn in juxtaposition, as above, may
ometimes be expressed also by means of an equation. Thus,
' y is the number of dollars, and x is the number of pounds sterling
1 any amount, then:
y = 4.87 X (1)
Iso. if F be the reading Fahrenheit, and C the reading centigrade
f any temperature, then:
F = |C + 32 (2)
Iso,
U = 13.67/12 = 1441F/62.5 (3)
rhere U, V, and W are pressures measured, respectively, in feet of
rater, inches of mercury, or in pounds per square inch.
Note. The letters x, y, Fy C, C7, F, IF in the above equations
tand for numbers; to make this emphatic we sometimes speak of them
js pure or abstract numbers. These numbers are thought of as arising
rem the measurement of a magnitude or quantity by the application
)f a suitable unit of measure. Thus from the magnitude or quantity
3f water, 12 gaUons, arises, by use of the unit of measure the gallon, the
abstract number 12.
Algebraic equations express the relation between number s, and it
should always be understood that the letters used in algebra stand
jor numbers and not for quantities or magnitudes.
Quantity or Magnitude is an answer to the question: "How
much?" Number is an answer to the question: "How many?''
An interesting relation is given by the scales in Fig. 6. This
diagram shows the fee charged for money orders of various
amounts; the amount of the order may first be found on the upper
scale and then the amount of the fee may be read from the lower
scale. The relation here exhibited is quite different from those
previously given. For example, note that as the amount of the
order changes from $50.01 to S60 the fee does not change, but
remains fixed at 20 cents. Then as the amount of the order
changes from $60.00 to $60.01, the fee changes abruptly from 20
cents to 25 cents. For an order of any amount there is a cor-
10 ELEMENTARY MATHEMATICAL ANALYSIS [§4
responding fee, but for each fee there corresponds not an order of
a single value, but orders of a considerable range in value. This is
quite different from the cases described in Fig. 5. There for
each reading Fahrenheit there corresponds a certain reading
centigrade, and vice versa^ and for any change^ however amaUy id. one
of the temperature readings a change, also small, takes place in
the other reading. For this reason the latter quantity is said to be
continuous.
The relation between the temperature scales has been expressed
as an algebraic equation. The relation between the value of a
money order and the corresponding fee cannot be expressed by a
similar equation. If we had given only a short piece of the centi-
grade-Fahrenheit double scale, we could, nevertheless, produce it
indefinitely in both directions, and hence find the corresponding
readings for all desired temperatures. But by knowing the fees
for a certain range of money orders one cannot determine the fees
for other amounts. In both of these cases, however, we express
the fact of dependence of one number upon another number by
saying that the first number is a function of the second number.
Definition. Any number, u, is said to be a function of another
number, f, if, when t is given, the value of u is determined. The
number t is often called the argument of the function u.
Illustrations. The length of a rod is a function of its tempera-
ture. The area of a square is a function of the length of a side.
The area of a circle is a function of its radius. The square root
of a number is a function of the number. The strength of an iron
rod is a function of its diameter. The pressure in the ocean is a
function of the depth below the surface. The price of a railroad
ticket is a function of the distance to be travelled.
It is obvious that any mathematical expression is, by the abov©
definition, a function of the letter or letters that occur in it-
Thus, in the equations:
u = f2 + 4i + 1
t- 1
u = VJ+I + ^^ - 7
u is in each case a function of f.
t4] VARIABLES AND FUNCTIONS OF VARIABLES 11
Temperature Fahrenheit is a function of temperature centigrade.
?he value of the fee paid for a money order is a function of the
mount of the order.
Goods sent by freight are classified into first, second, third,
Durth, and fifth classes. The amount of freight on a package is
function of its class. It is also a function of its weight. It is
Iso a function of the distance carried. Only the second of these
mctional relations just named can readily be expressed by an
Igebraic equation. It is possible, however, to express all three
raphically by means of parallel scales. The definition of the func-
'on is given (for any particular railroad) by the complete freight
iriff book of the railroad.
The fee charged for a money order is a function >3f the amount of
he order. The functional relation has been expressed graphically
1 Fig. 6. Note that for orders of certain amounts, namely,
2i, $5, $10, $20, $30, $40, $50, $60, $75, the function is not de-
ned. The graph alone cannot define the function at these
alues, as one cannot knpw whether the higher, the lower, or an
itermediate fee should be demanded. One can, however, define
he function for these values by the supplementary statement (for
xample) : "For the critical amountSy always charge the higher fee.**
is a matter of fact, however, the lower fee is always charged.
A function having sudden jumps like the one just considered, is
laid to be discontinuous.
Exercises
In the following exercises the function described can be represented
by a mathematical expression. The problem is to set up the expres-
sion in each case.
1. One side of a rectangle is 10 feet. Express the area A as a
function of the other side x.
2. One leg of a right triangle is 15 feet. Express the area A as a
function of the other leg x.
3. The base of a triangle is 12 feet. Express the area as a func-
tion of the altitude I.
4. Express the circumference of a circle as a function (1) of its
radius r; (2) of its diameter d.
5. Express the diagonal d oi a. square as a function of one side x.
12 ELEMENTARY MATHEMATICAL ANALYSIS [§4
6. One leg of a right triangle is 10. Express the hypotenuse h
as a function of the other leg x.
7. A ship B sails on a course AB perpendicular to OA. If OA = 30
miles, express the distance of the ship from 0 as a function of AB,
8. A circle has a radius 10 units. Express the length of a chord
as a function of its distance from the center.
9. An isosceles triangle has two sides each equal to 15 cm., and
the third side equal to x. Express the area of the triangle as a
function of x.
10. A right cone is inscribed in a sphere of radius 12 inches. Ex-
press the volume of the cone as a function of its altitude I.
11. A right cone is inscribed in a sphere of radius a. Express the
volume of the cone as a function of its altitude I,
12. One dollar is at compound interest for 20 years at r per cent.
Express the amount A as a function of r.
Functional Notation. The following notation is used to ex-
press that one number is a function of another; thus, if w is a
function of t we write:
u = m
Likewise,
y = fix)
means that 2/ is a function of x. Other symbols commonly used to
express functions of x are:
<l>(x), X(x), fix), F{x), etc.
These may be read the "0-f unction of x," the "X-functionof x,"
etc., or more briefly, "the <t> of a:,*' "the X of a:," etc.
Expressing the fact that temperature reading Fahrenheit is a
function of temperature reading centigrade, we may write:
F = f{C)
This is made specific by writing:
F = f (7 + 32
Likewise the fact that the charge for freight is a function of class,
weight, and distance, may be written:
r = f{c, w, d)
§5] VARIABLES AND FUNCTIONS OF VARIABLES 13
To make this functional symbol explicit, might require that we be
furnished with the complete schedule as printed in the freight tariff
book of the railroad. The dependence of the tariff upon class and
weight can usually be readily expressed, but the dependence upon
distance often contains arbitrary elements that cause it to vary
irregularly, even on different branches of the same railroad. A
complete specification of the functional symbol / would be con-
sidered given in this case when the tariff book of the railroad was in
our hands.
5. Variables and Constants. In elementary algebra, a letter is
always used to stand for a number that preserves the same value
in the same problem or discussion. Such numbers are called
constants. In the discussion above we have used letters to stand
for numbers that are assumed not to preserve the same value but
to change in value; such numbers (and the quantities or magnitudes
which they measure) are called variables.
If r stands for the distance of the center of mass of the earth from
the center of mass of the sun, r is a variable. In the equation
8 — igt* (the law of falling bodies), if t be the elapsed time, s the
distance traversed from rest by the falling body, and g the acceleration
due to gravity, then a and t are variables and g is the constant 32.2
feet per second per second.
The following are constants : Ratio of the diameter to the circumfer-
ence in any circle; the electrical resistance of pure copper at 60° F.;
the combining weight of oxygen; the density of pure iron; the breaking
strength of mild steel rods; the velocity of light in empty space.
The following are variables: the pressure of steam in the cylinder of
an engine; the price of wheat; the electromotive force in an alternating . -
current; the elevation of groundwater at a given place; the discharge
of a river at a given station. When any of these magnitudes are
assumed to be measured, the numbers resulting are also variables.
The volume of the merciuy in a common thermometer is a variable;
the mass of mercury in the thermometer is a constant.
6. Algebraic Functions. An expression that is built up by
operating on x a limited number of times by addition, subtraction,
multiplication, division, involution and evolution only, is called
an algebraic function of x. The following are algebraic functions
of x:
14 ELEMENTARY MATHEMATICAL ANALYSIS
(1) x2. (4) 2x + 5. (7) x» - 6x* + llo; - 6.
(2) xn. (5) 1/x. (8) ^-:^,
(3) 3\/x: (6) x2 - 5. (9) (x - a) (x - 6) (a; - c).
The expression x^ is an algebraic function of x but 2* is not an
algebraic function of x. The fee charged for a money order is not
an algebraic function of the amount of the order.
It is convenient to divide algebraic functions into classes. Thus
x^ + 2 is said to be integral; (x + 1) /(2 — x*) and 2 + ic~* are
said to be fractional; likewise x^ + 2 and {x + 1) /(2 — a;*) are
said to be rational; \/i — x and 3 — x^^ are said to be irrationaL
These terms may be formally defined as follows:
An algebraic function of x is said to be rational if in building up
the expression, the operation of evolution is not performed upon
Xf or upon a function of x; otherwise the function is irrational.
Thus,, expressions (1), (4), (5), (6), (7), (9), above, are rational
functions of x. Expressions (3) and (8) are irrational. Ex-
pression (2) is rational if n is a whole number; otherwise irrational.
A rational function is said to be integral if in building up the
function the operation of division by a:, or by a function of x,
is not performed; otherwise the function is fractional.
Thus expressions (1), (4), (6), (7), (9), above, are integral func-
tions of X, Expressions (1), (4), (6), (7), (9) are both rational
and integral and may therefore be called rational integral
functions of x.
Exercises
Classify the following functions of r, <, or x, answering the following
questions for each function : (A) is the function algebraic or (B) non-
algebraic? If it is algebraic, is it (a) rational or (6) irrational; if it
is rational, is it (1) integral or (2) fractional? The scheme of classifi-
cation is as follows :
A. Algebraic.
. . ^. 1 / (1) integral
(a) rational j ^2) fractional
(6) irrational .
B. Non-algebraic.
1. IQ.W; -yJa^ - x^; ^Jax*; ^a/x.
i7| VARIABLES AND FUNCTIONS OF VARIABLES 15
i.ax> + bx-+ex+d.
«. 2=6+s«+2-+|
8.^'; (1+0(1-0; a + Vi)(l-VO.
7.mx+ Vo' - *''; 3.37s'" ('■".
g.-:!.. g' - x\ a' + l'
». {a-a)Ca' + ai+;i;'); (o* - il)(ot +aM +i»).
1Q_ -y:^ — ■ Write an equal integral expression.
7. Graphical Computation. The ordinary operationB of arith-
iietjc, Buch as multiplication, division, involution and evolution.
8. — Graphical Multi-
PJwttion by Properties of
SMir Triangles.
c
,
,
/
(..4
A
'^•C
A-
B
(B
U
/'I
fl
I-*
t G e T B
Fia. 9.— Method of Graphical Mul-
tiplication Bud Diviaion carried out on
Squared Paper, The figure shows 1.9
X 4.4 - 8.4.
"'^ be performed graphically as explained below. The graphical
wnatniotion of products and quotients is useful in many problems
of Kience. The law of proportional sides of similar triangles is
"le fundamental theorem in all graphical computation. Its
application is very simple, as will appear from the following work.
16 ELEMENTARY MATHEMATICAL ANALYSIS [§7
Problem 1: To compvie graphicaUy the product of two
numbers. Let the two numbers whose product is required beo
and b. On any Hne lay off the unit of measurement, 01, Fig. 8.
On the same line, and, of course, to the same scale, lay off OA
equal to one of the factors a. On any other line passing through
1 lay off a line 1J5 equal to the other factor 6. Join OB and
produce it to meet AC drawn parallel to IB. Then AC is the
required product. For, from similar triangles:
AC :IB = 0A: 01 (1)
or,
AC = OA X IB (A)
It is obvious that the angle OAC may be of any magnitude.
Hence it may conveniently be taken a right angle, in which case the
work may readily be carried out on ordinary squared paper. Many
prefer, however, to do the work on plain paper, lajdng off the
required distances by means of a boxwood triangular scale. The
squared paper, form Ml, prepared for use with this book is suitable
for this purpose. On a sheet of this paper, draw the two lines
OX and OF at right angles and the unit line IC/, as shown in Fig.
9, Then from the similar triangles OIB and OAC the proportion
(1) and the formula (A) above are true. Hence to compute
graphically the product of two numbers a and b count off (Fig. 9)
OA = a to the OX-scale and IB = b to the OF-scale. Lay a
straight edge or edge of a transparent triangle down to draw OC
It is not necessary to draw OC, but merely to locate the point C.
Then count off AC to the OF-scale. Then AC = a X 6 by (A).
The figure as drawn shows the product 4.4 X 1.9 = 8.4.
All numbers can be multiplied graphically on a section of
squared paper 10 units in each dimension by properly reading the
OX and OY scales. Any product ab can be written aibi X 10» **
Ci X 10", where ai and 6i each have one digit before the decimal
point, and ci ^ 100.
Thus:
440 X 19 = 4.40 X 1.9 X 10^ = 8.40 X 10?
also
37 X 73 = 3.7 X 7.3 X 10^ = 27 X 10*
To proceed with the product of ai X &i, we first determine by
§7] VARIABLES AND FUNCTIONS OF VARIABLES 17
inspection whether Ci > or < 10. If Ci < 10, we read the scales
IS they are in Fig. 9 when counting off ai, hi and Ci. If Ci > 10, we
ead the OX scale as it stands when counting off ai but read the
)Y scale 0, 10, 20, 30, etc., in counting off the numbers 6i and Ci.
Exercises
In using form Ml for the following exercises take the scale OF at
le left marginal line of the sheet and use 2 cm. as the unit of measure.
Compute graphically the following products: Check results:
1. 2.5 X4.8. 4. 78.5 X 16.5.
2. 4.15 X 6.25. 6. 2.14 X 0.0467.
3. 3.14 X 7.22. 6. 2140 X 0.0467.
Problem 2: To compute graphically the quotient of two
imbers a and h. Formula (A) above can be written:
IB = ^ (5)
:om this it is seen that the quotient of two numbers a and h can
adily be computed graphically by use of Figs. 8 or 9. In Fig. 9
>unt off OA = 6, the divisor, to the OX scale, and AC = a, the
ividend, to the OF scale. Lay the triangle down to draw OC.
'0 not draw OC, but mark the point B and count off IB to the
Y scale. Then IB = a/b by (J5). Fig. 9 shows the quotient
.4 -r 4.4 = 1.9. Any quotient a/b may be written
^ X 10« = Q X 10"
here iV, D, Q are each ^ 10 but > 1. Hence, the OX and OY
;ales may always be read as they stand in Fig. 9.
Exercises
Compute graphically the following quotients : Check results
1. 6.2/2.5. 4. 7.32/1.25.
2. 1.33/6.45. 6. 872/321.
3. 234/0.52. 6. 128/937.
IS
ELEMENTARY MATHEMATICAL ANALYSIS
IP
Pboslem 3: To eompuU grapkicaHy the fqman root of oAJf
n^uTnber S, In Fig. 10 count 6R \A = S to tlie OX scale, and
draw a semicircle on OA as a diameter. Then \C ^ \/N to
the OY scale. Another construction is to piace the triani^ m
the portion shown in Fig. 10, so that the two edges pass throng
O and A and the vertex of the right an^e lies on the line lU.
Fig. 10 shows the construction for \ 7. The reacfings on the OX
Fig. 10. — Graphical Method of the Extractioa of Square Root&. Tht
figure shows \'7 ^ 2.65.
scale may be multiplied by 10^ and those on the OY scale by W
where n is any integer positive or negative.
State the two theorems in plane geometry on which the proof ol
these two constructions depends.
Phoblem 4: To comptUe graphically the square of any iwm-
ber y. This is a special case of Problem I, when a = 6 = y.
1. Compute the square roots of 2, 3, 5, and 7.
2. Compute the square roots of 3.75, 37.5, 0.375.
3. Compute the squares of 1.23 and 3.45.
4b Compute the squares of 7.75 and 0.S95.
6. Show that x^ is neariy 10.
VARIABLES AND FUNCTIONS OF" VARIABLES 19
BLEM 5: To compute graphically the reciprocal of any
r N. This ia a epecial case of Problem 2, when a = 1 and
BLEM 6: To compute graphically the integral powers of
itnber N. This problem is solved by the successive apphca-
f Problem 1 to construct N^, N', N*, etc., and of Problem 2
J
I]
7
7
t
h7
1/
it
til
it
itr
JU^'-
5/7.
mP' ^
m^w
struct iV""', JV~', N~\ etc. This construction is shown for
■wers of 1.5 in Fig. 11.
Exercises
bmpute the redprocal of 2.5; of 3.33; of 0.75; of 7.5.
bmpute {1.2)', (0.85)'. (1.15)*,
20
ELEMENTARY MATHEMATICAL ANALYSIS
3. Show that (1.05)" = 2.08, so that money at 5 percent com-
pound interest more than doubles itself in fifteen years.
Note: The work is less if (1.05)* is first found and then this result
cubed.
4. From the following outline the student is to produce a complete
method, including proof, of constructing successive powers of any
number.
Let OA (Fig. 12) be a radius of a circle whose center is 0. Let
OB be any other radius making an acute angle with OA, From B
drop a perpendicular upon OA, meeting the latter at Ai. From A\
drop a perpendicular upon OB meeting OB at A%. Prom A% drop a
perpendicular upon OA meeting OA at As, and so on indefinitely.
Then, if OA be unity, OAj is less than unity, and OA2, OA3, OAi
. . . are, respectively, the square, cube, fourth power, etc., of Oii.
Fig. 12. — Graphical Computation of Powers of a Number.
Instead of the above construction, erect a perpendicular to OB m©^^
ing OA produced at ai. At ai erect a perpendicular meeting OB p^
duced at 02, and so on indefinitely. Then if OA be unity, ai *
greater than unity and 02, as, 04, . . . are, respectively, the squaJf^
cube, etc., of ai. As an exercise, construct powers of 4/5 and of 3-^
6. Show that the successive "treads and risers" of the steps ^
the "stairways" of Figs. 13 and 14 are proportional to the pow^''
of r. The figures are from Milaukovitoh, Zeitschnft fur Ma/^
nnd Nat. Unterricht, Vol. 40, p. 329.
8. Double Scales for Several Simple Algebraic Functions. W^*
may make use of the graphical method of computation explayiC^
VARIABLES AND FUNCTIONS OF VARIABLES 21
e to construct graphically double scales representing simple
raic relations. For example, we may construct a double
for determining the square of any desired number.
Fig. 14.
Computation of ar, ar^, ar^, . . . for r < 1 and for r > 1.
OA (see Fig. 15) the scale on which we desire to read the
ber; call OB the scale on which we read the square. Let
gree to lay off OA as a uniform scale, using 01 as the unit of
sure. Since we desire to read opposite 0, 1, 2, 3, of the
16. — Method of Constructing a Double Scale of Squares or of Square
Roots.
orm scale, the squares of these numbers, the lengths along the
3 OB must be laid off proportional to the square roots of the num-
0, 1, 2, 3, . . , that is, the square root of any length, when
12 ELEMENTARY MATHEMATICAL ANALYSLS [§4
6. One leg of a right triangle is 10. Express the hypotenuse h
as a function of the other leg x.
7. A ship B sails on a course AB perpendicular to OA. If OA = 30
miles, express the distance of the ship from 0 as a function of AB,
8. A circle has a radius 10 units. Express the length of a chord
as a function of its distance from the center.
9. An isosceles triangle has two sides each equal to 15 cm., and
the third side equal to x. Express the area of the triangle as a
function of x.
10. A right cone is inscribed in a sphere of radius 12 inches. Ex-
press the volume of the cone as a function of its altitude L
11. A right cone is inscribed in a sphere of radius a. Express the
volume of the cone as a function of its altitude L
12. One dollar is at compound interest for 20 years at r per cent.
Express the amount A as a function of r.
Functional Notation. The following notation is used to ex-
press that one number is a function of another; thus, if w is a
function of t we write:
u = m
Likewise,
y = /(^)
means that y is a function of x. Other symbols commonly used to
express functions of x are:
<t>(x), Xix), fix), F{x), etc.
These may be read the "0-f unction of x/' the "X-f unction of x"
etc., or more brieifly, "the 0 of x," "the X of x/^ etc.
Expressing the fact that temperature reading Fahrenheit is a
function of temperature reading centigrade, we may write:
F = f(C)
This is made specific by writing:
F = |C + 32
Likewise the fact that the charge for freight is a function of class,
weight, and distance, may be written:
r = /(c, w, d)
§5] VARIABLES AND FUNCTIONS OF VARIABLES 13
To make this functional symbol explicit, might require that we be
furnished with the complete schedule as printed in the freight tariif
book of the railroad. The dependence of the tariff upon class and
weight can usually be readily expressed, but the dependence upon
distance often contains arbitrary elements that cause it to vary
irregularly, even on different branches of the same railroad. A
complete specification of the functional symbol / would be con-
sidered given in this case when the tariff book of the railroad was in
our hands.
6. Variables and Constants. In elementary algebra, a letter is
always used to stand for a number that preserves the same value
in the same problem or discussion. Such numbers are called
constapts. In the discussion above we have used letters to stand
for numbers that are assumed not to preserve the same value but
to change in value; such numbers (and the quantities or magnitudes
which they measure) are called variables.
If r stands for the distance of the center of mass of the earth from
the center of mass of the sun, r is a variable. In the equation
s — igt* (the law of falling bodies), if t be the elapsed time, s the
distance traversed from rest by the falling body, and g the acceleration
due to gravity, then s and t are variables and g is the constant 32.2
feet per second per second.
The following are constants : Ratio of the diameter to the circumfer-
ence in any circle; the electrical resistance of pure copper at 60° F.;
the combining weight of oxygen; the density of pure iron; the breaking
strength of mild steel rods; the velocity of light in empty space.
The following are variables: the pressure of steam in the cylinder of
an engine; the price of wheat; the electromotive force in an alternating . '
current; the elevation of groimd water at a given place; the discharge
of a river at a given station. When any of these magnitudes are
assumed to be measured, the numbers resulting are also variables.
The volume of the mercury in a common thermometer is a variable;
the mass of mercury in the thermometer is a constant,
6. Algebraic Functions. An expression that is built up by
operating on a: a limited number of times by addition, subtraction,
multiplication, division, involuti6n and evolution only, is caUed
an algebraic function of x. The following are algebraic functions
of i:
24 ELEMENTARY MATHEMATICAL ANALYSIS [§9
of the perpendiculars erected at corresponding values. The
result is shown in Fig. 17.
In the same manner any of the double scales may be opened
about any point as pivot. If the angle between the scales is
made 90°, the relation between the fimction and its argument is
shown by points on a straight line making an angle of 45** with each
scale. If one of the scales be non-uniform, it may, after it is
turned about the selected pivot, be made a uniform scale, in which
case the straight line just mentioned becomes, in general, a curved
line. We see, therefore, that instead of showing the relation
between a function and its variable by means of two scales in
juxtaposition, we may use two imiform scales intersecting at an
angle, and connect corresponding values of the variable and its
function by perpendiculars erected at these corresponding points.
The pairs of perpendiculars intersect at points which, in general,
lie upon a curve. This curve is obviously characteristic of the
particular functional* irelation under discussion. The respresenta-
tion of functional relations in this manner leads to the considera-
tion of so-caUed coordinate systems, the discussion of which is
begun in the next chapter.
10. Rectangular CoSrdinates. Two intersecting algebraic
scales, with their zero points in common, may be used as a system
of latitude and longitude to locate any point in their plane. The
student should be familiar with the rudiments of this method from
the graphical work of elementary algebra. The scheme is illus-
Y
2
.3
t
a
)
,
X
O
X
-
«
1)
■d
K
J
-
-
I
(-
2.-1)
1
1
J'l t
P
(
)
Fio. 18.— Rectangular Coordinates.
trated in its simplest form in Fig. 18, where one of the horizontal
lines of a sheet of squared paper has been selected as one of the
algebraic scales and one of the vertical lines of the squared paper
has been selected for the second algebraic scale. To locate a given
point in the plane it is merely necessary to give, in a suitable unit
of measure (as centimeter, inch, etc.), the distance of the point to
the right or left of the vertical scale and it« distance above or below
26 ELEMENTARY MATHEMATICAL ANALYSIS [§10
the horizontal scale. Thus the point P, in Fig. 18, is 2J units to
the right and 3J units above the standard scales. P2 is 3 units to
the left and 2 units above the standard scales, etc. Of course
these directions are to be given in mathematics by the use of the
signs " + " and " — " of the algebraic scales, and not by the use
of the words ^^right" or "left," '*up" or "down." The above
scheme corresponds to the location of a place on the earth's
surface by giving its angular distance in degrees of longitude east
or west of the standard meridian, and also by giving its angular
distance in degrees of latitude north or south of the equator.
The sort of latitude and longitude that is set up in the manner
described above is known in mathematics as a system of rectangu-
lar coordinates. It has become customary to letter one of the
scales XX\ called the X-axis, and to letter the other YY', called
the Y-axis. In the standard case these are drawn to the right
and left, and up and down, respectively, as shown in Fig. 18.
The distance of any point from the F-axis, measured parallel to
the X-axis, is called the abscissa of the point. The distance of
any point from the X-axis, measured parallel to the F-axis, is
called the ordinate of the point. Collectively, the abscissa and
ordinate are spoken of as the coordinates of the point. Abscissa
corresponds to the longitude and ordinate corresponds to the
latitude of the point, referred to the X-axis as equator, and to
the F-axis as standard meridian. In the standard case, abscissas
measured to the right of YY^ are reckoned positive, those to the
left, negative. Ordinates measured up are reckoned positive,
those measured down, negative.
Rectangular coordinates are frequently called Cartesian co-
ordinates, because they were first introduced into mathematics
by Ren6 Descartes (1596-1650).
The point of intersection of the axes is lettered 0 and is called
the origm. The four quadrants, XOY, YOX\ X'OY\ YVX, are
called the first, second, third, and fourth quadrants, respectively.
A point is designated by writing its abscissa and ordinate in a
parenthesis and in this order: Thus, (3, 4) means the point whose
abscissa is 3 and whose ordinate is 4. Likewise ( — 3, 4) means the
point whose abscissa is (— 3) and whose ordinate is {+ 4).
Unless the contrary is expHcitly stated, the scales of the co-
Sm RECTANGULAR COORDINATES 27
Ordinate axes are assumed to be straight aad uniform and to inter-
sect at right angles. Exceptions to this are not uncommon,
however, of which examples are given in Figs. 19 and 22.
The use of two intersecting algebraic scales to locate individnal
points in the plane, as explained above, is capable of immediate
enlargement. It will be explained below that a suitable array, or
set, or locus of such points may be used to exhibit the relation
between two variables laid off on the two scales, or between a
variable laid oft on one of the scales and a function of the variable
laid off on the other scale. This fact has already been explained
from another point of view at the close of the preceding chapter.
11. Statistical Graplu. From work in elementary algebra the
student is supposed to be familiar with the construction of statis-
tical graphs similar to those presented in Figs. 19 to 32. The
student will study each of these graphs and the following brief
descriptjons before making any of the drawings required in the
exercises that follow.
Fig. 19 is a barograph, or autographic record of the atmospheric
pressure recorded November 24, 1907, during a baUoon journey
from Frankfort to Marienburg in West Prussia. One set of scales
consists of equal circles, the other of parallel straight lines. The
zero of the scale of pressure does not appear in the diagram.
Note also that the scale of pressure is an inverted scale, increasing
downward. The scale of time is an algebraic scale, the zero of
which may be arbitrarily selected at any convenient point. The
scale of pressure is an arithmetical scale. The zero of the baro-
metric scale corresponds to a perfect vacuum — no less pressure
28 ELEMENTARY MATIIEMATICAI. ANALYSIS 1511
A.M.
v„,
P.»
-Woodi™.k
',
/\
\
/
\
HBSnrd
—
T
—
—
—
—
-/
^
-^
J«.™.I1I«
\
■f
/
\
'
\
\
1
¥
^
/
\\
\ /
/
\/
/^
\
\
/
/'
\
\
/
W2
/
\
\
/'
\
^/
f
3H2
/
\
k
/
K/
fi,"^'!
\ A
/
^>
«.
?'=.„
Ha
P.M
gf i_
M
-i
iifcijiAjy
ill]
RECTANGULAR COORDINATES
29
Fig. 20 is a graphical tim^-table of certain passenger trainB be-
tween Chicago and Minneapolis. The curves are not continuous,
as in the case of the barc^raph, but contain certain sudden jumps.
What is the meaning of these? What indicates the speed of the
traios? Where is the fastest track on this railroad? What shows
the meeting point of trains?
// the diagram, Fig. 20, be wrapped around a vertical cylinder of
mch size that the two midnight lines just coincide, then each Irain line
may be traced through continuously from terminus to terminus.
30
ELEMENTARY MATHEMATICAL ANALYSIS
[§11
Functions having this remarkable property are said to be periodic.
In the present case the trains run at the same time every day,
that is, periodically. In mathematical language, the position of
the trains is said to be a periodic function of the time.
Fig. 21 is the graphical time-table of "limited" trains between
Chicago and Los Angeles. The schedule of train No. 1, a very
heavy passenger train, is placed upon the chart for comparison.
The periodic character of this function is brought out very clearly
by using time as the abscissa. The student should discuss the
S
o
I
r-60
1-60
1-40
1-80
j-20
1-10
^
•i'"!""! I 1 I I I L
J_
±
J L— J
10 20 30 40 bO 60 70 * 80
Amount of the Money Order in Dollars
90 100
Fig. 23. — The Graph of a Discontinuous Function.
discontinuities and the various speeds as shown from the diagram.
The track profile is given at the right of the diagram for purposes
of comparison.
Fig. 22 represents the fluctuation of the elevation of the ground-
water at a certain point near the sea-coast on Long Island. The
fluctuations are primarily due to the tidal wave in the near-by
ocean. Here the scale of one of the coordinates (elevation) is
laid off on a series of equal circumferences similar to those of Fig.
19. The scale of the other coordinate (time) is laid off on the mar-
gin of the outer or bounding circle. The curve is continuous.
Is the curve periodic? What indicates the rate of change in the
elevation of the ground- water? When is the elevation changing
most rapidly? When is it changing most slowly?
§12] RECTANGULAR COORDINATES 31
Fig. 23 represents the functional relation between the amount of
a domestic money order and the fee. Two arithmetical scales
were used in making the diagram, as in ordinary rectangular co-
ordinates, except that the vertical scale is ten-fold the horizontal
scale; that is, lengths that represent dollars on the one scale rep-
resent cents on the other. This is an excellent illustration of a
discontiniLous function. On account of the sudden jumps in the
values of the fee, the fee, as explained in the preceding chapter, is
said to be a discontinuous function of the amount of the order.
12. Suggestions on the Construction of Graphs. Two kinds of
rectangular coordinate paper have been prepared for use with this
book. Form Ml is ruled in centimeters and fifths, and permits
two scales of twenty and twenty-five major units respectively to
be laid off horizontally and vertically on a standard sheet of letter
paper 8i X 11 inches. Form Af 2 is ruled without major divisions
in uniform 1 /5-inch intervals. This form of ruling is desirable for
general computation and for graphing functions for which non-
decimal fractional intervals are used, such as eighths, twelfths,
or sixteenths, which often occur in the measurement of mass
or time.
It is a mistake to assume that more accurate work can be done
on finely ruled than on more coarsely ruled squared paper. Quite
the contrary is the case. Paper ruled to 1 /20-inch intervals does
not permit interpolation TVithin the small intervals while paper
ruled to 1 /lO or 1 /5-inch intervals permits accurate interpolation
to one-tenth of the smallest interval. Form Ml is ruled to
2-mm. intervals, and is fine enough for any work. The centi-
meter unit has the very considerable advantage of permitting
twenty of the units within the width of an ordinary sheet of letter
paper (8§ X 11 inches) while seven is the largest number of inch
units available on such paper.
In order to secure satisfactory results, the student must recognize
that there are several varieties of statistical graphs, and that
each sort requires appropriate treatment.
1. It is possible to make a useful graph when only one variable
is given. Thus the following table gives the ultimate tensile
strength of various materials:
32
ELEMENTARY MATHEMATICAL ANALYSIS
ULTIMATE TENSILE STRENGTH OF VARIOUS MATERIALS
Material
Tensile strength,
tons per square inch
Hard steel '
50.0
30.0
25.0
21.5
16.0
12.0
11.0
10.0
5.0
Structural steel
Wroucht iron
Drawn brass
Drawn copper
Cast brass
Cast copper
Cast iron
Timber, with erain
A graph showing these results is given in Fig. 24. Thei"
two practical ways of showing the numerical values perta
to each material, both of which are indicated in the diagram; e
rectangles of appropriate height may be erected opposite
name of each material, or points marked by circles, dots or cr
may be located at the appropriate height. It is obvious in
case that a smooth curve should not be drawn through these p
— such a curve would be quite meaningless. In this case
are not two scales, but merely the single vertical scale. The
zontal axis bears merely the names of the different mat<
and has no numerical or quantitative significance. The i
is obviously not the graph of a function, for there are not
variables, but only one. The graph is merely a convenien
pression for certain discrete and independent results arra
in order of descending magnitude.
2. It is possible to have a graph involving two variabL
which it is either impossible or undesirable to represent the g
by a continuous curve or line. For example. Fig. 25 is a g
representing the maximum temperature on each day of a ce
month. Because there is only one maximum temperatur
each day, the value corresponding to this should be shown t
appropriate rectangle, or by marking a point by a circle, or
dot or cross, as in the preceding case, since a continuous (
through these points has no meaning. The horizontal scale
RECTANGULAR COORDINATES
33
Sr
ked by the names of the days of the week or by numbers,
ither case the horizontal line is a true scale, as it corresponds
lapse of the variable time. Sometimes, as in Fig. 25,
of this kind are represented by marking the appropriate
jy dots or circles and then connectii^ the successive points
ight lines. These lines have no special meaning in such
but they wd the eye in
ig the succession of sepa-
raph be made of the noon- S
iperatures of each day of f
ne month referred to in |
one of the same methods ^
id above would be used S
esent the results; that is, ^
cctanglra, marked points, J
rked points joined by ^
Although a smooth curve 3
.hrough the known points ^
have a meaning (if cor-
is obvious that the noon-
nperatures alone are not
it for determining its
In all such cases a smooth
hould not be drawn.
26 shows the monthly
and gross earnings of a
company during its first
of operation; the fixed
illlliiil
Fio. 24. — Graph Showiug Ten-
sile Strength of Certain Structural
Materials,
are also shown upon the same diagram. (See also Figs.
84,)
the data are reasonably sufiicient, a smooth curve may,
en should, be drawn through the known points. Thus if
perature be observed every hour of the day and the results
ted, a smooth curve drawn carefully through the known
will probably very accurately represent the unknown
ttures at intermediate times. The same may safety bo
exercises (1) and (2) below. In scientific work it is desir-
ELEMENTARY MATHEMATICAL ANALYSIS [jlS
r
f]
/
zo
^[
\
A I
V
1/
\h
A
r
V
f
J
I
D«ys of Month
Fra. 25.— Maximum Daily Temperatures, Madiaoa, Wis., February, 191-
*1MM
:pctjo E
„,{,.i,ii,jj.i..
a»
L»W
iHi-J'.T
12IUD
s a..i..-:
i::4^^ilj:
--.. i
DM
It: :x
; = -i
i"" —
+:"!
//
- "" i
j=/A
f '"^
-
""""" P^ n
r""TTTfT"
=
'r'4qi
tlill
' s
"Z J
""•^fpr
s
'1 ""
tU'i"
1
"ji
'1
iffr
■; ,'%
1" -Jitiifi
1
§12] RECTANGULAR COC)RDINATES 35
able to mark by circles or dots the values that are actually given
to distinguish them from the intermediate values "guessed" and
represented by the smooth curve.
In addition to the above suggestions, the student should adhere
to the foUowing instructions:
4. Every graph should be marked with suitable numerals along
both numerical scales.
5. Each scale of a statistical graph should bear in words a
description of the magnitude represented and the name of the unit
of measure used. These words should be printed in drafting let-
ters and not written in script.
6. Each graph should bear a suitable title telling exactly what is
represented by the graph.
7. The selection of the units for the scale of abscissas and ordi-
aates is an important practical matter in which common sense must
sontrol. It is obvious that in the first exercise given below 1 cm.
= 1 foot draft for the horizontal scale, and 1 cm. = 100 tons for
the vertical scale will be units suitable for use on form Ml .
Further instruction in practical graphing is given in §33.
Exercises
1. At the following drafts
a ship has the displacenients stated:
Draft in feet, h
15
12
9
6.3
Displacement in tons, T . . .
2096
1512
1018
586
Plot on squared paper. What are the displacements when the
drafts are 11 and 13 feet, respectively?
2. The following tests were made upon a steam turbine generator*
Output in kilowatts, X ! 1,190| 995| 745| 498i 247
Weight, pounds of steam con- | 23,1201 20,040 16,630, 12,560 8,320
sumed per hour, W. i i
Plot on squared paper. What are the probable values of K when
IT is 22,000 and also when W is 11,000?
36 ELEMENTARY MATHEMATICAL ANALYSIS [§13
3. Make a graphical chart of the zone rates of the Parcel Post
Service for the first three zones, using weight of package as abscissa
and cost of postage as ordinate.
4. The average temperature at Madison from records taken at
7 a. m. daily for 30 years is as follows:
Jan. 1, 14.0. F. July 1,67.5. F.
Feb. 1, 15.1. Aug. 1, 64.0.
Mar. 1, 35.2. Sept. 1,55.4.
Apr. 1, 40.0. Oct. 1, 44.1.
May 1, 53.9. Nov. 1, 30.0.
June 1, 63.2. Dec. 1, 18.3.
Make a suitable graph of these results on squared paper.
13. Mathematical, or Non-statistical Graphs. Instead of the
expressions ^* abscissa of a point" or ^'ordinate of a pointy" it has
become usual to speak merely of the "x of a point," or of the "y of
a point," since these distances are conventionally represented by
the letters x and 2/, respectively. If we impose certain conditions
upon X and y, then it will be found that we have, by that very fact,
restricted the possible points of the plane located by them to a
certain array, or set, or locus of points, and that all other points
of the plane fail to satisfy the conditions or restrictions imposed.
It is obvious that the command, "Find the place whose latitude
equals its longitude," does not restrict or confine a person to a par-
ticular place or point. The places satisfying this condition are
unlimited in number. We indicate all such points by drawing
a line bisecting the angles of the first and third quadrants; at ail
points on this line latitude equals longitude. We speak of this
line as the locus of all points satisfying the conditions. We might
describe the same locus by saying "the y of each point of the
locus equals the a;," or, with the maximum brevity, simply write
the equation "2/ = x." This is said to be the equation of the
locus, and the line is called the locus of the equation.
It is of the utmost importance to be able readily to interpret any
condition imposed upon, or, what is the same thing, any relation
between variables, when these are given in words. It will greatly
aid the beginner in mastering the concept of what is meant by the
term function if he will try to think of the meaning in words of the
§13] RECTANGULAR COORDINATES 37
relations commonly given by equations, and vice versa. The
very elegance and brevity of the mathematical expression of rela-
tions by means of equations, tends to make work with them formal
and mechanical unless care is taken by the beginner to express in
words the ideas and relations so briefly expressed by the equa-
tions. Unless expressed in words, the ideas are liable not to
be expressed at all. •
The equation of a curve is an equation satisfied by the co-
ordinates of every point of the curve and by the coordinates of no
other point.
The graph of an equation is the locus of a point whose coordi-
nates satisfy the equation.
Exercises
1. Draw and discuss the following loci:
The ordinate of any point of a certain locus is twice its ab-
scissa; the X of every point of a certain locus is half its y; the y of
a point is 1/3 of its x; a point moves in such a way that its lati-
tude is always treble its longitude; the sum of the latitude and
longitude of a point is zero; a point moves so that the difference
in its latitude and longitude is always zero.
2. Draw this locus: Beginning at the point (1, 2), a point moves
so that its gain in latitude is always twice as great as its gain in
longitude.
3. A point moves so that its latitude is always greater by 2 units
than three times its longitude. Write the equation of the locus
and construct.
4. A head of 100 feet of water causes a pressure at the bottom of
43.43 pounds per square inch. Draw a locus showing the relation
between head and pressure, for all heads of water from 0 to 200 feet.
Suggestion: There are several ways of proceeding. Let pounds
per square inch be represented by abscissas or x, and feet of water be
represented by ordinates or y. Then v/e take the point x = 43.43,
y = 100 and other points, as a; = 86.86, y — 200, etc., and draw the
line. Otherwise produce the equation first from the proportion
100
X :y :: 43.43 : 100, or, 43.432/ = 100 a: or y = x and then draw the
100
graph from the fact that the latitude is always ~~,^ of the longitude.
38 ELEMENTARY MATHEMATICAL ANALYSIS [§13
Be sure that the scales are numbered and labeled in accordance with
suggestions (4), (5) and (6) of §12.
6. A pressure of 1 pound per square inch is equivalent to a column
of 2.042 inches of mercury, or to one of 2.309 feet of water. Draw a
locus showing the relation between pressure expressed in feet of watier
and pressure expressed in inches of mercury.
Suggestion: Let x = inches of mercury and y = feet of water.
First properly number and label the X-axis to express inches of mer-
cury and number and label the F-axis to express feet ot water. Since
negative numbers are not involved in this exercise, the origin may be
taken at the lower left-hand corner of the squared paper. First locate
the point x = 2.042, y = 2.309 (which are the corresponding values
given by the problem) and draw a line through it and the origin. This
is the required locus since at all points we must have the proportion
X :y :: 2.042 : 2.309, which says that the ordinate of every point ot the
locus is 2309/2042 times the abscissa of that point.
6. A certain mixture of concrete (in fact, the mixture 1:2: 5) con-
tains 1.4 barrels oi cement in a cubic yard of concrete. Draw a locus
showing the cost oi cement per cubic yard of concrete for a range ot
prices of cement from $0.80 to $2.00 per barrel.
Suggestion: Let x be the price per barrel of cement and y be the
cost of the cemeut in 1 cubic yard of concrete. Number and label
the two scales beginning at the lower left-hand corner as origin. Since
prices between $0.80 and $2.00 only need be considered, the first
'Vision on the X-axis may be marked $0.80 instead of 0. Each
centimeter may represeut $0.10 on each scale. The cost of cement per
cubic yard of concrete must, by the condition of the problem, be 1.4
times the price per barrel of cement. Hence the first point located
on the vertical scale must correspond to 1.4 X $0.80, or to $1.12 cost
per cubic yard. As this is the lowest cost to be entered, it is desirable
not to start the vertical scale at $0.00, but at $1.00. Thus the lower
left-hand corner of the coordinate paper may be taken as the point
(0.80, 1.00) in a system in which the unit of measure is 1 cm. = 10
cents.
7. Draw a locus showing the cost per cubic yard of concrete for
various prices of cement, provided $2. 10 per yard must be added to the
results of example 6 to cover cost of sand and crushed stone.
8. Cast iron pipe, class A (for heads under 100 feet), weighs, per
foot of length: 4-inch, 20.0 pounds; 6-inch, 30.8 pounds; 8-inch, 42.9
pounds. For each size of pipe construct upon a single sheet of
il41
RECTANGULAR CObRDINATES
39
squared papat & locus showing the cost pec foot for all variations
in market price between S20.00 and $40.00 per ton.
Sugobbtion: If the horizontal scale be selected to represent
price per ton, the scale may begin at 20 and end at 40, as this covers
the range required by the problem. Therefore let 1 cm. represent
(1.00. Since the range of prices is from 1 cent to 2 cents per pound,
the cost per foot will range from 20 cents to 40 cents for 4-inch
pipe and from 42.9 cents to 85.8 cents for 8-inch pipe. Hence
for the vertical scale 10 cents may be represented by 2 cm. In this
case the vertical scale may quite as well begin at 0 cents instead of
at 20 cents, as there is plenty of room on the paper.
II ' 'I 1^ _. !
- -\i'4i,- !--- /C4f'-- -
H A ^ ' ' SL/L ^
"^ \ i'" /■■ 1-i -jL
-%±\M h^-^-~
^^-mj^-^-^^i.
itAi 1 Ui- A \^
FiQ. 27.— Lines of Slope (1.5) and of Slope ( —2).
14. Slope. The slope of a straight line is defined to be the
change in y for an increase in x equal to 1. It will be represented
in this book by the letter m. Thus in Fig. 27 the line A has the
slope m = 1.5, for it is seen that at any point of the line the
ordinate y gains 1,5 units for an increase of 1 in x. The Une B,
parallel to the line A, is also seen to have the slope equal to 1.5.
The equation of the line A is obviously y = l.bx. In the same
figure the slope of the line C is — 2, for at any point of this line
40 ELEMENTARY MATHEMATICAL ANALYSIS [§15
the ordinate y loi^es 2 units for an increase in x equal to 1. The
equation of the line C is obviously t/ = — 2x. Line Z>, parallel to
line Cj also has slope ( — 2)
If h be the change in y for an increase of x equal to fc, then the
slope m is the ratio h jk.
The technical word slope differs from the word slo-pe or slant in
common language only in the fact that slope, in its technical use,
is always expressed as a ratio. In common language we speak of a
"slope of 1 in 10/' or a "grade of 50 feet per mile/' etc. In mathe-
matics the equivalents are "slope = 1/10/' "slope = 50/5280,"
etc.
As already indicated, the definition of slope requires us to speak
in mathematics of positive slope and negative slope. A line of pos-
itive slope extends upward with respect to the standard direction
OX and a line of negative slope extends downward with reference
to OX.
In a similar way we may speak of the slope of any curve at a
given point on the curve, meaning thereby the slope of the tangent
line drawn to the curve at that point.
Exercises
1. Give the slopes of the lines in exercises 1 to 8 of the preceding
set of exercises.
2. Draw y = x\ y = 2x; y = 3x; y = 3 ; 2/ = 2^ 2/ = 4) 2/ == - 2^5
y = — Sx; y = Ox.
3. Prove that y = mx always represents a straight line, no matter
what value m may have.
15. Equation of Any Line. Intercepts. — In Fig. 28, the line
MN expresses that the ordinate y is, for all points on the line, always
3 times the abscissa a:, or it says that y = 3x. The line HK states
that "?/ is 2 more than 8x." Thus the hne HK has the equation
?/ = 3x + 2.
In general, since y = mx is always a straight line,^ then y =
mo; + 6 is a straight line, for the y of this locus is merely, in each
case, the y of the former increased by the constant amount b (which
1 See exercise 3, §14, above.
f
RECTANGULAR COORDINATES
41
f course, bo positive or negative). Therefore, y = nix + b
e parallel to y = mx. The distance OB (Fig. 28) is equal
The distance is called the y-intercept of the locus. The
e OA is equal to — fc Im, for it is the value of x when y is
It is called the x-intercept of the locus.
■/
K
Ji
r
B
^
1
J
1
tl
^
/
7
'«
/
J
^
Exercises
ketch, from inspection of tlie equations, tlie lines given by:
(•) V ■
m y ■
(c) y -
M y -
(.) V -
ij) y-
42 ELEMENTARY MATHEMATICAL ANALYSIS [§16
2. Sketch, from inspection of the equations, the lines given by:
(a) 2/ = ix. (/) y = ~ jx.
(6) y = ix. {g) y ^ - x.
(c) y = X. (h) y = — 2x.
(d) y = 2x. {%) 2/ = - 3x.
(6) y = 3x. 0') y = \/2x.
3. Sketch the lines given by :
(a) a; = 3. (d) 2/ = 1. (fir) 2/ = 0.
(6) a; = 5. (e) 2/ = 5. {h) a; = 0.
(c) X = -2. (/) 2/ -= -3. (i) a;2 = 4.
4. Sketch from inspection of the equations, the following:
(a) 2/ = a; + 1.
(6) 2/ = iaJ + 1.
(c) 2/ = -2a; + 4.
{d) 2/ = 5x + 3.
(e) y = —5a; — 2.
6. Sketch, from inspection of the equations:
(a) 2/ = a; + 4.
(6) 2/ - 2a; - 3 = 0.
(c) 2/ + ?a; + 1/3 = 0.
(d) ax + 62/ = c.
(6) x/a + y/b = 1.
6. The shortest distance between y = mx and 2/ = mx + 6 is not ft.
Show that it equals 6/\/l + m*.
16. Additive Properties. Sometimes a useful result is obtained
by adding (or subtracting) the corresponding ordinates of two
graphs. Thus in Fig. 26, operating expenses of a power plant
may be added to ordinates representing various rates of divi-
dends, and compared (by subtraction) with monthly revenue.
Sometimes, however, it becomes necessary to determine a result
by adding two functions corresponding to different values of the
variable or argument. Fig. 29 is an excellent illustration of this.
This diagram enables one to find the cost of a cubic yard of
"1:2:4" concrete (except cost of mixing) by knowing the prices
of the constituent materials. The information necessary to con-
6]
RECTANGULAR COORDINATES
43
ruot the locd is given in the first line of Table I, p. 44. The
QOUDt of cement in I cubic yard of 1:2.4 concrete is seen to be
58 barrels. The pricf per barrel of cement may be considered
variable changing with the condition of the market and with
e locality where sold.
ailing xi the price of
ment, the cost j/i of the
ment in 1 cubic yard of
2:4 concrete is then, for
.1 market prices of
:ment, expressed by the
{uation:
^ '^iSSBr ^----
I'^^wm
l-="=Sg^^^S="
i^Sfffl
'. - - l/|uJ^-IiJ^-
■ i^"^ - -- - J^--
!/i = 1.58x,
bia is graphically repre-
inted in Fig. 29 by the
Qe of slope 1.58. Note in
lis case that the slope
[ the line has a "physi-
U" meaning, namely it is
le cost of the cement in 1 •aceml^^'i'si^^nTDlnln"'"
ibic yard when the price fio. 29.
; tl.OO a barrel. In the
ime way the cost of the sand and of the crushed stone in 1
ubic yard of concrete for various market prices of these com-
lodities is expressed by the lines of Fig. 29 of slopes 0.44 and
.88 respectively.
Example: Let the price x, of cement be $1.20 per barrel; let
he price ij of atone be $1.75 per cubic yard, and thepricexjofsand
le $1.10 per cubic yard. Find the cost of the materials necessary
iO make 1 cubic yard of 1:2:4 concrete. Then, from Fig. 29:
X, = $1.20 then j/i = $1.90
xi = 1.75 J/, = 1.54
I) = 1.10 y, = ^.48
Total, or cost of material for 1
cubic yard of concrete = $3.92
The coat of concrete, ^, is a function of three variables, xi, xi,
44 ELEMENTARY MATHEMATICAL ANALYSIS l§16
Xzf all of which, for convenience' sake, have been measured on the
same scale or axis OX. The representation of several variables
on the same scale need not cause any confusion.
Since in this case the prices of the constituents of the concrete
are not the same, the total cost of 1 cubic yard of concrete cannot
be found by adding the ordinates at the same abscissa of the
three graphs, because the abscissas or the various market prices
of the ingredients are not the same.
The second line of the table may be used by the student as the
basis of construction of another diagram similar to that of Fig. 29.
TABLE I
The quantities of material required to make 1 cubic yard of concrete
(based on 33 J percent voids in the sand and 45 percent voids in
the broken stone).
Mixture
Quantities of materials in 1 cubic yard, of concrete
Cement,
barrels *
1.58
1.33
Sand,
cubic yards
Stone,
cubic yards
1:2:4 concrete
1:2J:5 concrete
1:3:6 concrete
0.44
0.46
0.88
0.92
* A barrel (4 bags) of cement weighs 380 pounds and contains 3} cubic feet of
cement.
. Note : The student may be interested to know how the figures in
the first line of the table are obtained. The explanation will best be
understood if the figures as given are first verified. First the 1.58
barrels of cement should be reduced to cubic yards. It gives 0.22
cubic yard. A part of this must be used to fill the 33 J percent of
voids in the 0.44 cubic yard of sand. The cement required for this is
0.146 cubic yard. Thus the sand and cement combine to make
0.44 + 0.22 - 0.146 or 0.514 of mixed material. A part of this mix-
ture is used to fill the 45 percent voids in the 0.88 cubic yard of stone,
which equals 0.396 cubic yard. Hence the total volume of stone,
sand and cement is 0.88 + 0.514 - 0.396, which equals 0.998, or the
cubic yard required.
To find the numbers in the table, the above process needs to be
reversed and stated algebraically. Thus, to make a cubic yard of
1:2:4 concrete let
RECTANGULAR COORDINATES 45
X = cubic yards cement required.
y = cubic yards sand required,
a = cubic yards stone required.
'rom the given porosities, or percent of voids,
X — iy = surplus of cement after filling voids in sand.
{x — iy) + y = volume of mixed sand and cement.
Ux — iy) + y] — 0.463 = surplus of mixed sand and ce-
ment after filling voids in atone .
z + [(x — iy) + y] — 0A5z - 1, the total volume,
0.55a + ly + x = l
T rrrrn-T-rr fi TT rrr
^"rV^I'o
T.'^.T
-/-
f-
//
-<
--
// ,^^. , i 1
1
3
ecause the mixture is 1:2:4:
ix = I
give:
4.53i = 1
T = 0.22 cubic yard
ter reasoning is as follows: As the voids in the crushed stone
be completely filled in the finished concrete, the z cubic yards
46 ELEMENTARY MATHEMATICAL ANALYSIS [§17
of stone counts as only 0.55'Z cubic yards in the final product. As the
voids in the sand are to be completely filled in the final mixture, the
y cubic yards of sand counts as only i-y cubic yards in the final
product. As there are no voids to be filled in the cement, it counts
as X cubic yards in the final result. Hence the equation
^ + iy + 0.55z = 1, etc.
Exercise
From the diagram, Fig. 30, determine and insert in a table like
Table I, the quantity of each sort of material in 1 cubic yard of
1:3:6 concrete.
THE POWER FUNCTION
17. Defimtion of the Power Function. The algebraic function
consisting of a single power of the variable, such for example as the
functions x^, x^j 1 /x, 1/x*, x^, etc., stand next to the linear
function of a single variable, mx + 6, in fundamental impor-
tance. The function x« is known as the power function of x.
18. The Graph of x^. The variable part of many functions of
practical importance is the square of a given variable. Thus the
area of a circle depends upon the square of the radius; the distance
traversed by a falling body depends upon the square of the elapsed
time; the pressure upon a flat surface exposed directly to the wind
depends upon the square of the velocity of the wind; the heat
generated in an electric current in a given time depends upon the
square of the number of amperes of current, etc., etc. Each of
these relations is expressed by an equation of the form y = ax\ in
which X stands for the number of units in one of the variable quan-
tities (radius of the circle, time of fall, velocity of the wind, amperes
of current, respectively, in the above named cases) and in which
y stands for the other variable dependent upon these. The num-
ber a is a constant which has a value suitable to each particular
problem, but in general is not the same constant in different prob-
lems. Thus, if y be taken as the area of a circle, y = tz^, in which
X is the radius measured in feet or inches, etc., and y is measured in
square feet or square inches, etc. ; or if s is the distance in feet
traversed by a falling body, then s = 16.U*, where t stands for the
§19] RECTANGULAR COORDINATES 47
elapsed time in seconds. In one case the value of the constant a
is 3. 1416 and in the other its value is 16. 1 .
Let us first graph the abstract law or equation y = x*, in which
a concrete meaning is not assumed for the variables x and y but
in which both are thought of as abstract variables. First form a
suitable table of values for x and x^ as follows:
-3 -2-10 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 3
z* or y
9 4 10 0.04 0.16 0.36 0.64 1.0 1.44 1.96 2.56 3.24 4 9
Here we have a series of pairs of values of x and y which are asso-
ciated by the relation y = x^. Using the x of each pair of values
as abscissa with its corresponding y there can be located as many
points as there are pairs of values in the table, and the array of
points thus marked may be connected by a freely drawn curve.
To draw the curve upon coordinate paper, form Ml, the origin
may be taken at the mid-point of the sheet, and 2 cm. used as the
unit of measure for x and y. If the points given by the pairs of
values are not located fairly close together, it is obvious that a
smooth curve cannot be satisfactorily sketched between the points
until intermediate points are located by using intermediate values
of x in forming the table of values. The student should think of
the curve as extending indefinitely beyond the limits of the sheet of
paper used; the entire locus consists of the part actually drawn and
of the endless portions that must be followed in imagination beyond
the range of the paper. If the graph oi y = x^ be folded about
the F-axis, OF, it will be noted at once that the left and right
portions of the curve will exactly coincide. The student will
explain the reason for this fact.
19. Parabolic Curves. The equations y = x, y = x^, y = x^\
y = x^ should be graphed by the student on a single sheet of coor-
dinate paper, using 2 cm. as the unit of measure in each case.
Table II may be used to save numerical computation in tlie con-
struction of the graphs of these power functions. As in the case
of y = x^y a smooth curve should be sketched free-hand through
the points located by means of the table of values, and intermediate
values of x and y should be computed when doubt exists in the mind
of the student concerning the course of the curve between any two
nt\\rt4-a
48 ELEMENTARY MATHEMATICAL ANALYSIS [jl
3 -2 -i \j ^^ \
Fio. 31.— Parabolic Curv
RECTANGULAR COORDINATES
49
e graphs of the above power functions are observed to be
luoits lines, without breaks or sudden jumps. A formal proof
TABLE II
x^
x'
Vx
Vi
x^
1/x
l/x»
0.04
0.008
0.447
0.585
0.089
5.000
25.000
0.16
0.064
0.632
0.737
0.252
2.500
6.250
. 0.36
0.216
0.775
0.843
0.465
1.667
2.778
; 0.64
1
0.512
0.894
0.928
0.715
1.250
1.563
»' 1.00
1.000
1.000
1.000
1.000
1.000
1.000
:: 1.44
1.728
1.095
1.063
1.312
0.8333
0.6944
1 . 96
2.744
1.183
1.119
1.657
0.7143
0.5102
» 2.56
4.096
1.265
1.170
2.034
0.6250
0.3906
; 3.24
5.832
1.342
1.216
2.415
0 . 5556
0.3086
), 4.00
8.000
1.414
1.260
2.828
0.5000
0.2500
5 4.84
10.65
1.483
1.301
3.263
0.4545
0.2066
I 5.76
13.82
1.549
1 . 339
3.717
0.4167
0.1736
) 6.76
17.58
1.612
1.375
4.193
0.3846
0.1479
y 7.84
21.95
1.673
1.409
4.685
0.3571
0.1276
) 9.00
27.00
1.732
1.442
5.196
0.3333
0.1111
I 10.24
32.77
1.789
1.474
5.724
0.3125
0.0977
I 11.56
39.30
1.844
1.504
6.269
0.2941
0.0865
\ 12.96
46.66
1.897
1.533
6.831
0.2778
0.0772
i 14.44
54.87
1.949
1.560
7.407
0.2632
0.0693
) 16.00
64.00
2.000
1.587
8.000
0.2500
0.0625
I 17.64
74.09
2.049
1.613
8.608
0.2381
0.0567
I 19.36
85.18
2.098
1.639
9.229
0.2273
0.0517
5; 21 . 16
97.34
2.145
1.663
9.866
0.2174
0.0473
^ 23.04
110.6
2.191
1.687
10.42
0.2083
0.0434
) 25.00
125.0
2.236
1.710
11.18
0.2000
0.0400
l\ 27.04
140.6
2.280
1.732
11.85
0.1923
0.0370
I 29.16
157.5
2 . 324
1.754
12 . 66
0.1852
0.0343
3 31.36
175.6
2 . 366
1.776
13.25
0.1786
0.0319
i 33.64
195.1
2.408
1.797
13.97
0.1724
0.0297
f
50
ELEMENTARY MATHEMATICAL ANALYSIS [fW
TABLE
II. — \S^ont\nutd\
X
X2
X3 1
2.449
Vx
x^^
l/x
l/x^
6.0
36.00
216.0
1.817
14.70
0 . 1667
0.0278
6.2
38.44
238.3
2.490
1.837
15.44
0.1613
O.O260
6.4
40.96
262 . 1
2.530
1.857
16.19
0 . 1563
0.0244
6.6
43.56
287.5
2.569
1.876
16.96
0.1516
O.O230
6.8
46 . 24
314.4
2.608
1.895
17.33
0.1471
0.0216
7.0
49.00
343.0
2.646
1.913
18.52
0.1429
0.0204
7.2
51.84
373.2
2.683
1.931
19.32
0.1389
0.0193
7.4
54.76
405.2
2.720
1.949
20.13
0.1351
0.0183
7.6
57.76
439.0
2.757
1.966
20.95
0.1316
0.0173
7.8
60.84
474.6
2.793
1.983
21.79
0.1282
0.0164
8.0
64.00
512.0
2.828
2.000
22.63
0.1250
0.0156
8.2
67.24
551.4
2.864
2.017
23.48
0.1220
0.0149
8.4
70.56
592.7
2.898
2.033
24.35
0.1190
0.0142
8.6
73.96
636.1
2.933
2.049
25.22
0.1163
0.0135
8.8
77.44
681.5
2.966
2.065
26.11
0.1136
0.0129
9.0
81.00
729.0
3.000
2.080
27.00
0.1111
0.0123
9.2
84.64
778.7
3.033
2.095
27.91
0.1087
0.0118
9.4
88.36
830.6
3.066
2.110
28.82
0 . 1064
0.0113
9.6
92.16
884.7
3.098
2.125
29.74
0.1042
0.0109
9.8
96.04
941.2
3.130
2.140
30.68
0 . 1020
0.0104
10.0
100.00
1000.0
3.162
2.154
31.62
0.1000
O.OIOO
that X" is a continuous function for any positive, rational value of
n will be given later.
All of the graphs here considered have one impor tant prop"
erty in common, namely, they all pass through the points (0, 0) |
and (1,1). It is obvious that this property may be affirmed of any
curve of the class y = a;", if n is a positive number. These curva
are known collectively as curves of the parabolic family, or simpbF
parabolic curves. The curve y = x^ is called the paraboli*
y = x^ is called the cubical parabola, y = x^^ is called the semi-
cubical parabola, etc. Curves for negative values of n do not p«»j
through the point (0, 0) and are otherwise quite distinct. Thcyf
i20] RECTANGULAR COORDINATES 51
ire known as curves of the hyperbolic type, and will be discussed
ater.
The student should cut patterns of the parabola, the cubical
)arabola and the semi-cubical parabola out of heavy paper for
ise in drawing these curves when required. Each pattern should
lave drawn upon it either the x- or y-axis and one of the unit lines
,0 assist in properly adjusting the pattern upon squared paper.
20. Symmetry. In geometry a distinction is made between two
dnds of symmetry of plane figures — symmetry with respect to a
ine and symmetry with respect to a 'point, A plane figure is
{ymmetrical with respect to a given line if the two parts of the
igure exactly coincide when folded about that line. Thus the let-
iers M and W are each symmetrical with respect to a vertical line
irawn through the vertex of the middle angles. We have already
aoted that 2/ = x* is symmetrical with respect to OY.
A plane figure is symmetrical with respect to a given point when
the figure remains unchanged if rotated 180° in its own plane about
an axis perpendicular to the plane at the given point. Thus the
letters N and Z are each symmetrical with respect to the mid-point
of their central line. The letters H and 0 are symmetrical both
with respect to lines and with respect to a point. Which sort of
symmetry is possessed by the curve y = x^? Why?
Another definition of symmetry with respect to a point is per-
haps clearer than the one given in above statement: A curve is
said to be symmetrical with respect to a given point 0 when all
lines drawn through the given point and terminated by the curve
are bisected at the point 0.
What kind of symmetry with respect to one of the coordinate
axes or to the origin (as the case may be) does the point (2, 3) bear
to the point (-2, 3)? To the point (-2,-3)? To the point
(2, -3)?
Note that symmetry of the first kind means that a plane figure is
unchanged when turned 180° about a certain line in its plane, and
that symmetry of the second kind means that a figure is unchanged
when turned 180° about a certaii^ line perpendicular to its plane,
21. The curves in the diagram, Fig. 31, are sketched from a
limited number of points only, but any number of additional
values of x and y may be tabulated and the accuracy, as well as
I
52 ELEMENTARY MATHEMATICAL ANALYSIS [§22
the extent, of the graph be made as great as desired. A num-
ber of graphs of power functions are shown as they appear in the
first quadrant in Figs. 32 and 35. The student should explain how
to draw the portions of the curves lying in the other quadrants
from the part appearing in the first quadrant.
In the exercises in this book to "draw a curve" means to conr
struct the curve as accurately as possible from numerical or other
data. To "sketch a curve" means to produce an approximate or
less accurate representation of the curve, including therein its
characteristic properties, but without the use of extended numer-
ical data.
Exercises
1. On coordinate paper draw the curves y = x*, y = a;', y = x^,
y = x^, using 4 cm. as the unit of measure. On the same sheet
draw the lines a;= ± 1, y — ± 1, y = ± x.
2. On coordinate paper sketch the curves x = y*, a; = y', a; = y^*
X = 2/'. Compare with the curves of exercise 1.
3. Sketch and discuss the cxirves y = Vx, y = \/ ^, y = Vx.
Can any of these curves be drawn from patterns made from the
curves of exercise 1? Why? Explain the graphs of the first and
last if the double sign " + " be understood before the radicals, and
compare with the graphs when the positive sign only is to be ^mde^
stood before the radicals.
4. Draw the curve y^ = x*. Compare with the curve y = »•.
5. Name in each case the quadrants of the curves of exercises
1-4, and state the reasons why each curve exists in certain quad-
rants and why not in the other quadrants.
22. Discussion of the Parabolic Curves. Draw the straight
lines X = l,x = — 1, 2/=l, 2/= ~1 upon the same sheet upon
which a number of parabolic curves have been drawn. These
lines together with the coordinate axes divide the plane into a
number of rectangular spaces. In Fig. 33 these spaces are shown
divided into two sets, those represented by the cross-hatching,
and those shown plain. The cross-hatched rectangular spaces
contain the lines y = x and y= —x and also all curves of the para-
bolic type. No parabolic curve ever enters the rectangular siript
shown plain in Fig. 33.
S22)
RECTANGULAR COORDINATES
53
The line y = x divides the spaces occupied by the parabolic
curves into equ^ portions. Why does the curve y = x* (in the
first quadrant) lie below this line in the interval i = 0 to x = 1,
but above it in the interval to the right of a; = 1? On the other
hand, why does the curve y = \/x, or j/' — x (in the first quad-
rant), lie above the line y — x'va the interval j;= 0 to a: = I and
below y = a; in the interval to the right of a; = 1?
One part of the parabolic curve y — x" always lies in the first
quadrant. If n be an even number, another part of the curve lies
Fta. 33. — The Regiona of the Parabolic and the Hyperbohc Curvea.
All parabolic curves lie within the cross-hatched region Ail hyperbolic
curves lie within the legion shown plain
in which quadrant? If n be an odd number, the curve hea in which
quadrants?
If the exponent n of any power function be a positive fraction,
may m[r, the equation of the curve may be written y = x". If
in this case both m and r be odd, the curve lies in which quadrants?
If m be even and r be odd, the curve lies in which quadrants? If
m be odd and r be even, the curve lies in which quadrants? If
both m and r be even the curve lies in which quadrants?
A curve which is symmetrical to another curve with respect to a
Hne may be spoken of figuratively aa the reflection or image of the
second curve in a mirror represented by the given line.
I
54 ELEMENTARY MATHEMATICAL ANALYSIS [§23
Exercises
Exercises 1-5 refer to curves in the first quadrant only.
1. The expressions x^, x^^y x^, x^ are numerically less than x for
values of x between 0 and 1. How is this fact shown in the diagram,
Fig. 31?
2. The expressions x^, x^^, x', x^ are numerically greater than x for
all values of x numerically greater than unity. How is this fact
pictured in the diagram, Fig. 31?
3. For values of x between 0 and 1, o;^ < x' < x* < x^^ <x.
For values a; > 1, x* > x' > x^ > x^^ > x. Explain how each of these
facts is expressed by the curves of Fig. 32.
4. Show that the graphs y = x^, y = x^^, y — x^y y — x^ are the
reflections of y = a:^, y = x^, y = x^, y — x^, in the mirror y — x.
5. Sketch without tabulating the numerical values, the following
loci: y = x^^, y == a;°S V = ^^°°> 2/ = ^°°^-
The following are to be discussed for all quadrants.
6. Sketch, without tabulating numerical values, the following loci
2/2= X*, y* = x^, y^ = x^j y^ — x^j y^ = x^.
7. Sketch the following: y^^ = x^", y^^^ = a;»», y^^^^ = x^^^K
8. Sketch the following: y = — x^, y — — x^,y^ = — x*.
23. Hjrperbolic Tjrpe. Loci of equations of the form yx"*^ = 1^
ov y = l/x", where n is positive, have been called hjrperbolic
curves. The fundamental curve xy = Ij or y = 1 /x is called the
rectangtilar hyperbola. Its graph is given in Figs. 34 and 35,
but the curve should be drawn independently by the student, using
2 cm. as the unit of measure. Its relation to the x- and t/-axes is
most characteristic. For very small positive values of x, the value
of y is very large, and as x approaches 0, y increases indefinitely.
But the function is not defined for the value x = 0, for the prod-
uct xy cannot equal 1 if x be zero. For numerically small but
negative values of x, y is negative and numerically very large, and
becomes numerically larger as x approaches 0. The locus thus
approaches indefinitely near to the F-axis, as x approaches zero.
Instead of saying that ^'y increases in value without limit," it
is equally common to say **y becomes infinite;" in fact, ''infinite"
is merely the Latin equivalent of "no limit." It is often written
2/ = 00 . This is a mere abbreviation for the longer expressions,
RECTANGULAR COORDINATES
55
»ecome8 infinite" or "y increases in value without limit."
student must be cautioned that the symbol «• does not stand
, number, and that "y = >» " must not be interpreted in the
! way that "y = 5 " is interpreted.
I X increases from numerically large negative values to 0,
atinually decreases and becomes negatively infinite (abbre-
«i J/ = — CO ). As a: decreases from numerically large positive
■i
^ 1
1"
"\
J
/
^
I'w-»
.
^=^
fli_
"*Jr7
"^
^
[.....•
\ .
\
. 1
y'
—Hyperbolic Curves.
SI to 0, K continually increases and becomes infinite. Thus,
le neighborhood of x = 0, ^ is discontiiiuous, and, in this case,
liscontinuity is called an infinite discontinuitT.
1 account of the symmetry in xy = 1, if we look upon a; as a
tion of y, all of the above statements may be repeated, merely
ehai^ng x and y wherever they occur. Thus, there is an
ite discontinuity in x, as y passes through the value 0.
le lines XX' and YY' which these curves approach as near as
56
ELEMENTARY MATHEMATICAL ANALYSIS
we please, but never meet, are called the asTmptotes of the
hyperbola.
All other curves of the hyperbolic family, such as yx* = I,
xy'^ — 1, y'x'^ = 1, y*x^ — J and the like, approach the X- and
F-axes aa asymptotes. The rates at which they approach the
axes depends upon the relative magnitudes of the exponents of the
powers of x and y; the quadrants in which the branches lie depend
upon the oddness or evenness of these exponents.
1^1
u
,..-
'^
FiQ. 35. — Hyperbolic Curvea
First Quadrant, y " l/x''"'
Adiabatic Curve for Air.
1 the Fia. 36. — A Hyperbola
the Formed by Capillary Action of
Two Converging Plane Plates.
Exercises
1. Draw accurately upon squared paper the loci, xy = 1, xy^ = 1,
x'y = l,xy^ = 1.
2. Show that the curves of the hyperboUc type lie in the rectangular
regions shown plain, or not crosa-hatched, in Fig. 33.
8. In what quadrants do the branches of i"y' = 1 lie?
4. How does the locus of x'y* = 1 differ from that ot xy = 1?
B. Sketch, showing the esHential character ot each locus, the curves
iV = 1, i'°j/ = 1, x^o'-hf = 1.
6. Show that xy = a passes through the point (Va, \/a); that
x^ = a* passes through (a, a) and can be made from xy = 1 by
§24] RECTANGULAR COORDINATES 57
"stretching'* (if a > 1) both abscissas an(J ordinates oi xy = I in the
ratio 1 :a.^
24. Curves Symmetrical to Each Other. Some of the facts of
symmetry respecting two portions of the same parabola or hyper-
bola may be readily extended by the student to other curves.
First answer the following questions:
How are the points (a, h) and (—a, 6) related to the F-axis?
How are the points (a, b) and (a, —b) related to the X-axis?
How are the points (a, b) and (6, a) related to the line y = x?
Prove the result by plane geometry.
The following may then be readily proved by the student:
Theorems on Loci .
I. If X be replaced by (—a:) in any equation containing x and y,
the new graph is the reflection of the former in the axis YY\
IT. If y be replaced by (—y) in any equation containing x and 2/,
the new graph is the reflection of the former in the axis XX' ,
HI. If X and y be interchanged in any equation containing x
and y, the new graph is the reflection of the former one in the line
y = X.
2$. The Variation of the Power Ftmction. The symmetry of
the graphs of the power function with respect to certain hues and
points, while of interest geometrically, nevertheless does not con-
stitute the most important fact in connection with these functions.
Of more importance is the law of change of value or the law by which
the function varies. Thus returning to a table of values for the
power function x^ for the first quadrant,
X
0 1/2 1 3/2 2 5/2 3
x^\0 1/4 1 9/4 4 25/4 9
we note that as x changes from 0 to 1/2 the function grows by the
small amount 1 /4. As x changes from 1 /2 by another increment of
1/2 to the value 1, the function increases by 3/4 to the value 1.
As x grows by successive steps or increments of 1 /2 unit each, it
is seen that x^ grows by increasingly greater and greater steps,
until finally the change in x^ produced by a small change in x
^To "elongate '* or "stretch " in the ratio 2:3 means to change the length
of a line segment so that (original length): (new or stretched length) » 2: 3.
58 ELEMENTARY MATHEMATICAL ANALYSIS [§26
becomes very large. Thus the step by step increase in the function
is a rapidly augmenting one. Even more rapidly does the func-
tion x^ gain in value as x grows in value. On the contrary, for posi-
tive values of x the power functions 1 /x, 1 /a;^, 1 /x', etc., decrease
in value as x grows in value. Referring to the definition of the
slope of a curve given in §14, we see that the parabolic curves
have a positive slope in the first quadrant, while the hyperbolic
curves have always a negative slope in the first quadrant.
The law of the power function is stated in more definite terms
in §34. That section may be read at once, and then studied
a second time in connection with the practical work which
precedes it.
26. Increasing and Decreasing Ftmctions. As a point passes
from left to right along the X-axis, x increases algebraically.
As a point moves up on the F-axis, y increases algebraically and
as it moves down on the F-axis, y decreases algebraically. An
increasing function of x is one such that as x increases algebraically,
y, or the function, also increases algebraically. By a decreasing
function of x is meant one such that as x increases algebraically,
y decreases algebraically. Graphically, an increasing function is
indicated by a rising curve as a point moves along it from left to
right. The power function y = x^ (n positive) is an increasing
function of x in the first quadrant. The power function y =
x~^ (— n negative) in the first quadrant is a decreasing function
of x.
The power function y = x^ is an increasing function for all values
of X while y = x^ isB. decreasing function in the second quadrant
but an increasing function in the first quadrant. In a case like
y = ± x^^f where y has two values for each positive value of x, it
is seen that one of these values increases with x while the other
decreases with x.
Exercises
1. Consider the function y = + x^ and construct its locus. As x
grows by successive steps of one unit each, does the function grow by
increasingly greater and greater steps or not? Why? Is the slope
of the curve an increasing or a decreasing function of a;?
§27] RECTANGULAR COORDINATES 59
2. Does the algebraic value of the slope of xy = 1 increase with x
in the first quadrant?
8. As a; changes from — 5 to +5 does the slope of y — x^ always
increase algebraically?
4. Express in the language of mathematics the fact that the
curves y = a?*, when n is a rational number greater than unity, are
concave upward.
Answeb: "When n is greater than unity, the slope of the curve
increases as x increases/'
Express in a similar way the fact that the curves y = x^^^ are
concave downward.
27. The Graph of the Power Ftmction when x** has a Coeffi-
cient. If numerical tables be prepared for the equations
y = x^
and y' = 3x^
then for like values of z each ordinate, y', of the second curve
will be three fold the corresponding ordinate, y, of the first curve.
It is obvious that the curve
2/' = OX" (l)
and the curve
2/ = a;« (2)
are similarly related; the ordinate y^ of any point of the first locus
can be made from the corresponding ordinate y (i.e., the ordinate
having the same abscissa) of the second by multiplying the latter
by a. If a be positive and greater than unity, this corresponds to
stretching or elongating all ordinates of (2) in the ratio 1 : a; if o
be positive and less than unity, it corresponds to contracting or
shortening all ordinates of (2) in the ratio 1 : a.
For example, the graph of y' = ax** can be made from the graph
of y = a;* if the latter be first drawn upon sheet rubber, and if
then the sheet be uniformly stretched in the y direction in the ratio
1 : a. If the curve be drawn upon sheet rubber which is already
under tension in the y direction and if the rubber be allowed to
contract in the y direction, the resulting curve has the equation
y = ax"* where a is a proper fraction or a positive number less than
unity.
The above results are best kept in mind when expressed in a
00 ELEMENTARY MATHEMATICAL ANALYSIS [§27
slightly different from. The equation y* — a-x*» can, of course, be
written in the from (s/'/a) = a;'*. Comparing this with the equa-
tion y = a:", we note that (s/'/a) = y or 2/' = ay, therefore we may
conclude generally that substituting {y'/a) for y in the equation of
any curve multiplies all of the ordinates of the curve by a. For
example, after substituting (2/V2) for y in any equation, the new
ordinate y' must be twice as large as the old ordinate y, in order
that the equation remain true for the same value of z.
In the same manner changing the equation y = a;» to y =
( — ) , that is, substituting (a^'/a) for x in any equation multiplies
all of the abscissas of the curve by a. Multiplying all of the abscis-
sas of a curve by a elongates or stretches all of the abscissas in
the ratio^ 1: a if a > 1, but contracts or shortens all of the abscis-
sas if a <1. As the above reasoning is true for the equation of
any locus, we may state the results more generally as follows:
Theorems on Loci
IV. Substituting l-]for x in the equation of any locus multiplies
all of the abscissas of the curve by a.
V. Substituting ( ~ ) for y in the equation of any locus multiplies all
of the ordinates of the curve by a.
Note: It is not necessary to retain the symbols a:' and y' to
indicate new variables, if the change in the variable be otherwise
understood.
Exercises
1. Without actual construction, compare the graphs y = x* and
x^ 1 2
y = 5x^; y =x^ and y = }^; y= - and y = -; 2/==a;« and y=2x^;
Ji X X
y ^ x^ and 2/ = 2 '
2. Without actual construction, compare the graphs y = x* and
©2 "U [ X\ 8 1/
; 2/=a;'and2=a;3; y = x^ and y = U ) ; y= x^and 2=3;*.
iSee footnote, p. 67.
§281 EECTANGULAR COORDINATES 61
3. Compare y* = x' and y' — Is) ; y' = x^ and Is I = x*; v' = x'
28. Orthographic Projectioii. In elementary geometry we
learned tliat the projection of a given point P upon a ^ven line or
plane is the foot of the perpendicular dropped from the given point
upon the given line or plane. Likewise if perpendiculars be
dropped from the end points A and B of any line segment AB upon
a given line or plane, and if the feet of these perpendiculars be
called P and Q, respectively, then the line segment PQ is called the
projectioii of the line AB. Also, if perpendiculars be dropped
from all points of a given curve AB upon a given plane MN, the
Fia. 37. — Orthographic Proieetion of Line SeRmentH
locus of the feet of all of the perpendiculars so drawn is called the
projection of the given curve upon the plane MN.
To emphasize the fact that the projections were made by using
perpendiculars to the given plane, it is customary to speak of them
as orthogonal or orthographic projections.
The shadow of a hoop upon the ground is not the orthographic
projection of the hoop unless the rays of light from the sun strike
perpendicular to the ground. This would only happen in our lat-
itude upon a non-horizontal surface.
The shortening by a given fractional amoimt of all of a set of
62 ELEMENTARY MATHEMATICAL ANALYSIS l§29
parallel line segments of a plane may be brought about geometric-
ally by orthographic projection of all points of the linfe segments
upon a second plane. For, in Fig. 37, let AiBi^ AiBt, AsBj,
etc., be parallel line segments lying in the plane MN, Let their
projections on any other plane be A'lC'i, A\C\^ ii'jC'j, etc.,
respectively. Draw A%C% parallel to A '2^2 and A\C\ parallel to
A'lC'i, etc. Then since the right triangles A\BxC\y AJ^%C\^
AiBsCsf etc., are similar,
AiJBi ^ AJB2 ^ AiBi
AiCi A2C2 A'zd
Call this ratio a. It is evident that a > 1. Substitute the
equals: A'ld = AiCi, A\C\ = A^C^^ etc. Then:
A\B\ _ A2B2 _ AzB 3 _ _ a
A/Wi " A\C2 " A^zCz " * ' ' " r
The numerators are the original line segments; the denominators
are their projections on the plane MO. The equality of these
fractions shows that the parallel lines have all been shortened in
the ratio a : 1.
The above work shows that to produce the curve y = (x/a)*,
(a < 1), from 2/ = x» by orthographic projection it is merely neces-
sary to project all of the abscissas of 2/ = x" upon a plane passing
through YOY' making an angle with OX such that unity on OX
projects into a length a on the projection of OX, To produce
the curve y = ax^* (a <1) from y = x'^hy orthographic projection
it is merely necessary to project all of the ordinates of y = x* upon
a plane passing through XOX' making an angle with OY such that
unity on OF projects into the length a on the projection of OY,
To lengthen all ordinates of a given curve in a given ratio,
1 : a, the process must be reversed; that is, erect perpendiculars to
the plane of the given curve at all points of the curve, and cut them
by a plane passing through XOX* making an angle with OY such
that a length a (a >1) measured on the new F-axis projects into
unity on OF of the original plane.
29. Change of Unit. To produce the graph of y = lOx* from
that oi y = x^, the stretching of the ordinates in the ratio 1: 10
need not actually be performed. If the unit of the vertical scale
RECTANGULAR COORDINATES
63
of V "= X* be taken 1/10 of that of the horizontal scale, and the
proper numerical values be placed upon the divisions of the
scales, then obviously the graph oi y = x* may be used for the
graph of J = l(te'. Suitable change in the unit of measure on one
or both of the scales oi y = x" ia often a very desirable method of
representing the more general curve y = ax-.
An interesting example is given in Fig. 38. The period of vi-
bration of a simple pendulum is given by the formula r = it\/lfg-
When g = 981 cm. per
second per second (abbre-
viated cm. /sec.') this gives
T = O.lOOSvT, which for
many purposes is suffici- :
ently accurate when writ-
ten T = O.lOVl- In this '
equation Tmust be in sec-
onds and I in centimeters.
Thus when I = 100 cm., T
= 1 sec, so tliat the graph
may be made by drawing
the parabola y = y/x from
the pattern previously
made and then attaching the proper numbers to t
shown in Fig. 38.
80. Variation. The relation between y and x expressed by the
equation y = ax", where n is any positive number, is often expressed
by the statement "y varies as the nth power of x," or by the
statement "y is ■proportional to x"." Likewise, the relation
y = ajx; where n is positive, is expressed by the statement
"y varies inversely as the nth power of x." The statement "the
elongation of a coil spring is proportional to the weight of the sus-
pended mass" tells us:
y = mx (1)
where y is the elongation (or increase in length from the natural
or unloaded lei^th) of the spring, and x is the weight suspended by
the spring, but it does not give us the value of m. The value of m
may readily be determined if the elongation corresponding to a
given weight be given. Thus if a weight of 10 pounds when aus-
i scales, as
64 ELEMENTARY MATHEMATICAL ANALYSIS [§30
pended from the spring produces an elongation of 2 inches in the
length of the coil, then, substituting a: = 10 and y = 2 in (1),
2 = mlO
and hence m = 1/5
If this spring be used in the construction of a spring balance, the
length of a division of the uniform scale corresponding to 1 poimd
will be 1 /5 inch.
A special symbol, oc , is often used to express variation. Thus
y oc 1/^2
states that y varies inversely as rf*. It is equally well expressed by :
k
where A; is a constant called the proportionality factor.
The statements "y varies jointly as u and v," and "y varies
directly as u and inversely as v," mean, respectively:
y = auv
au
y = —
V
Thus the area of a rectangle varies jointly as its length and breadth,
or,
A = kLB
If the length and breadth are measured in feet and A in square feet,
k is unity. But, if L and B are measured in feet and A in acres,
then & = 1 /43560. If L and B are measured in rods and A in
acres, then & = 1 /160.
From Ohm's law, we say that the electric current in a circuit
varies directly as the electromotive force and inversely as the
resistance, or:
C «: EIR or C = kE/R
The constant multiplier is unity if C be measured in amperes, E
in volts, and R in ohms, so that for these units
C = EIR
§31] RECTANGULAR COORDINATES 65
31. lUustrations from Science. Some of the most important
laws of natural science are expressed by means of the power func-
tion^ or graphically by means of loci of the parabolic or hyperbolic
type.
The linear equation y = mxia, of course, the simplest case of the
power function and its graph, the straight line, may be regarded as
the simplest of the curves of the parabolic type. The following
Jlustrations will make clear the importance of the power function
n expressing numerous laws of natural phenomena. Later the
jtudent will learn of two additional types of fundamental laws of
jcience expressible by two functions entirely different from the
power function now being discussed.
The instructor will ask oral questions concerning each of the
following illustrations. The student should have in mind the
general form of the graph in each case, but should remember that
the law of variation, or the law of change of value which the func-
tional relation expresses, is the matter of fundamental importance.
The graph is useful primarily because it aids to form a mental pic-
ture of the law of variation of the function. The practical graph-
ing of the concrete illustrations given below will not be done at
present, but wiU be taken up later in §33.
(a) The pressure of a fluid in a vessel may be expressed in either
pounds per square inch or in terms of the height of a column of
mercury possessing the same static pressure. Thus we may write:
p = 0A92h (1)
in which p is pressure in pounds per square inch and h is the height
of the column of mercury in inches. The graph is the straight
line through the origin of slope 492 /lOOO. The constant 0.492 can
be computed from the data that the weight of mercury is 13.6 times
that of an equal volume of water and that 1 cubic foot of water
weighs 62.5 poimds.
In this and the following equations, it must be remembered
that each letter represents a number , and that no equation can
he used until all the magnitudes involved are expressed in terms
of the particular units which are specified in connection with
that equation.
^For brevity ax^ as well &ax^ wiU frequently be called a power function of x.
5
66 ELEMENTARY MATHEMATICAL ANALYSIS [§31
(b) The velocity of a falling body which has fallen from a state
of rest during the time t, is given by
V = 32. 2t (2)
in which t is the time in seconds and v is the velocity in feet
per second. If < is measured in seconds and v is in centimeters per
second, the equation becomes^ t; = 981^. In either case the
graph is a straight Hne, but the lines have different slopes.
• (c) The space traversed by a falling body is given by
s = igt' (3)
or, in English units (s in feet and t in seconds) :
s = 16.1<« (4)
(d) The velocity of the falling body, from the height h is:
V = \/2gh = \/6iAh (5)
The resistance of the air is not taken into account in formulas
(2) to (5).
The formula equivalent to (5) :
^mv^ = mgh (6)
where m is the mass of the body, expresses the equivalence of
^v^j the kinetic energy of the body, and mghj the work done
by the force of gravity m^, working through the distance h-
1 A full discussion of the process of changing formulas like the ones in the pres-
ent section into a new set of units should be sought in text-books on physics and
mechanics. The following method is sufficient for elementary purposes. First,
write (for the present example) the formula v = 32.2 t where v is in ft./see. and
t is in seconds. For any units of measure that may be used, there holds a general
relation v » ct, where c is a constant. To determine what we may call the
dimensions of c, substitute for all letters in the formula the names ot the units ia
which they are expressed, treating the names as though they were algebraie
numbers. From v = ct write, ft. /sec. »» c sec. Hence (solving for ditnenaiana of e),
c has dimensions ft./sec' Thereiore in the given case, we know c ^ 32.2 ft./see*.
To change to any other units simply substitute equals for equals. Thus 1 ft. *
30.5 cm., hence c = 32.2 X 30.5 cm./sec* = 981 cm./sec*
To change velocity from mi./hr. to ft./sec. in formula (19) below, we have
R - 0.003 V^ where R is in Ib./sq. ft. and V is in mi./hr. Write the general
formula R = cV^. The dimensions of c are (lb. / ft.«) -^ (mi.Vhr.«) or (lb. / ft.«)X
(hr.Vmi.s). In the given case we have the value of c = 0.003 (Ib./ft.*)X
(hr.'/mi*)- To change V to ft./sec, substitute equals for equals, namely Ihr."
3600 sec, 1 mi.« 5280 ft., or merely (approximately) mi./hr. « § ft./sec.
RECTANGULAR COORDINATES 67
le intensity of the attraction exerted on a unit mass by the
y any planet varies inversely as the square of the distance
3 center of mass of the attracting body. If r stand for
^ance and if / be the force exerted on unit mass of the
i body, then
m
•^ = r' (7)
3tant m is the value of the force when r is unity.
le formula for the horse power transmissible by cold-rolled
is:
- - 'I («
r is the horse power transmitted, d the diameter of the
inches, and N the number of revolutions per minute,
ipid variation of this function (as the cube of the diameter)
3 for some interesting facts. Thus doubling the size of the
•erating at a given speed increases 8-fold the amount of
bat can be transmitted, while the weight of the shaft is
d but 4-fold.
be constant, N varies inversely as d^. Thus an old-f ash-
)-h.p. overshot water-wheel making three revolutions per
requires about a 9-inch shaft, while a DeLaval 50-h.p.
urbine making 16,000 revolutions per minute requires a
shaft but little over 1 /2 inch in diameter,
he period of the simple pendulum is
T = Wljg (9)
' is the time of one swing in seconds, I the length of the
m in feet and g = 32.2 ft./sec.,^ approximately,
he centripetal force on a particle of weight W pounds,
in a circle of radius R feet, at the rate of N revolutions
ind is '7
F = ^'^'^^^ (10)
9
= 32.16 ft./sec.2,
F = 1.227QWRN^ (11)
68 ELEMENTARY MATHEMATICAL ANALYSIS [§31
where F is measured in pounds. If iV^ be the number of revolutions
per minute, then
= 0.000341 TT/JiV* (13)
(i) An approximate formula for the indicated horse power
required for a steamboat is:
I.H.P.= -^ (14)
where S is speed in knots, D is displacement in tons, and C is a con-
stant appropriate to the size and model of the ship to which it is
applied. The constant ranges in value from about 240, for finely
shaped boats, to 200, for fairly shaped boats.
(j) Boyle's law for the expansion of a gas maintained at
constant temperature is
pv =^ C (15)
where p is the pressure and v the volume of the gas, and C is a con-
stant. Since the density of a gas is inversely proportional to its
volume, the above equation may be written in the form
p = cp (16)
in which p is the density of the gas.
(k) The flow of water over a trapezoidal weir is given by
q = S.S7Lh^^ (17)
where q is the quantity in cubic feet per second, L is the length of
the weir^ in feet and h is the head of water on the weir, in feet.
(0 The physical law holding for the adiabatic expansion of
air, that is, the law of expansion holding when the change of
volume is not accompanied by a gain or loss of heat,* is
expressed by
p = cp^''^^ (18)
1 The instructor is expected fully to explain the meaning of the technical termi
here used.
2 Note that when a vessel containing a gas is insulated by a non-conductor of
heat, so that no heat can enter or escape from the vessel, that the temperatun of
the gas will rise when it is compressed, or fall when it is expanded. Adiabatic ezpan*
sion may be thought of, therefore, as taking place in an insulated veasel.
§31] RECTANGULAR COORDINATES 09
This is a good illustration of a power function with fractional expo-
nent. The graph is not greatly different from the semi-cubical
parabola
y = cx^^
(m) The pressure or resistance of the air upon a flat surface per-
pendicular to the current is given by the formula
R = 0.00372 (19)
in which V is the velocity of the air in miles per hour and R is the
resulting pressure upon the surface in pounds per square foot.
According to this law, a 20-mile wind would cause a pressure of
about 1.2 pounds per square foot upon the flat surface of a building.
One foot per second is equivalent to about 2 /3 mile per hour, so
that the formula when the velocity is given in feet per second
becomes:
R = 0.001372 (20)
(n) The power used to drive an aeroplane may be divided into
two portions. One portion is utilized in overcoming the resistance
of the air to the onward motion. The other part is used to sustain
the aeroplane against the force of gravity. The first portion does
"useless" work — work that should be made as small as possible by
the shapes and sizes of the various parts of the machine. The
second part of the power is used to form continuously anew the
wave of compressed air upon which the aeroplane rides. Calling
the total power^ P, the power required to overcome the resistance
Pr, and that used to sustain the aeroplane P„ we have
P==Pr + P. (21)
We learn from the theory of the aeroplane that Pr varies as the
cube of the velocity, while P, varies inversely as 7, so that
Pr = cV^ (22)
and
P. = y (23)
Thus at high velocity less and less power is required to sustain the
aeroplane but more and more is required to meet the frictional
> Power ( » work done per unit time) is measured by the unit horse power, which is
550 foot-pounds per second.
70
FXKMEXTARV MATHKMATIC.U, ANALYSIS 1131 1
resistance of the medium. The law espresaed by (23) that Im
and less power is required to sustain the aovplane as the speed la
increased is known as Langley's Law. From this law Langjey w»
convinced that artificial flight was possible, for the whole matta
seemed to depend primarily upon getting up sufficient speed. It
is really this law that makes the aeroplane possible. An analogous
case is the well-known fact that the faster a person skates, the
thinner the ice secessarj- to sustain the skat«r. In this case
1 1 l\J / .
H-
J -/ / /
/^■/A,
H-ff-/-
i+f-i
-/-/-/7
-M^.
?*'
r ///
■Sm
±HM/t/y
//y/X/C'y'^
?i5
—[fffttt/f/
V/Z^^^
/ // If./ (7///
''!
Ul m////////yy>'< 1
///////f
k68^
!
f!
J/^^ K!!«4g!ig^
' !
V
W \ 1
SISS
iiiii
ISSJiJislH
II Oh Foot Depth
FiO. 39. — Cat)acitj- of Reclanpilsr and Circular Tanks pw Foot of Depth-
part of the energj- of the skater is continually forming anew on
the thin ice the ware of depressioa which sustains the akatw,
while the other part overcomes the frictional reststance of the
skates on the ice and the resistance of the air.
(o) The capacity of cast-tron pipe to transmit watn- n ttftoi
given by the formula:
q>-'" = LftSAti"-'^ (24)
§32] RECTANGULAR COORDINATES 71
in which q is the quantity of water discharged in cubic feet per
second, d is the diameter of the pipe in feet and h is the loss of
head measured in feet of water per 1000 linear feet of pipe.
This is a good illustration of the equation of a parabolic curve
with complicated fractional exponents. The curve is very
roughly approximate to the locus of the equation
y=cVhx^'' (25)
(p) The contents in gallons of a rectangular tank per foot of
depth, b feet wide and I feet long, is
q = 7MI (26)
The contents in gallons per foot of depth of a cylindrical tank d
feet in diameter is
q = 7.5 Trd 2/4 (27)
Fig. 39 shows the graph of (26) for various values of b and also
shows to the same scale the graph of (27).
32. Rational and Empirical Formulas. A number of the
formulas given above are capable of demonstration by means of
theoretical considerations only. Such for example are equations
(1), (2), (3), (4), (5), (7), (8), (9), (10), etc., although the constant
coefficients in many of these cases were experimentally deter-
mined. Formulas of this kind are known in mathematics as
rational formulas. On the other hand certain of the above for-
mulas, especially equations (14), (17), (19), (22), (23), (24),
including not only the constant coefficients but also the law of
variation of the function itself, are known to be true only as the
result of experiment. Such equations are called empirical
fonnulas. Such formulas arise in the attempt to express by an
equation the results of a series of laboratory measurements.
For example, the density of water (that is, the mass per cubic
centimeter or the weight per cubic foot) varies with the tem-
perature of the water. A large number of experimenters have
prepared accurate tables of the density of water for wide ranges
of temperature centigrade, and a nimiber of very accurate empirical
formulas have been ingeniously devised to express the results, of
which the following four equations are samples:
72 ELEMENTARY MATHEMATICAL ANALYSIS (532 K
Empirical formulas for the density , d, of wcUer in terms of im- \ ■
perature centigrade^ d,
96(^ - 4)*
(a) d = 1 -
(6) d= 1 -
10^
(c) a = 1
(d) d = 1 +
10»
0.485g« - 8L3g' + 602g - 1118
107
Exercises
1. Among the power functions named in the above illustrations, pick
out examples of increasing functions and of decreasing functions.
2. Under the same difference of head or pressure, show by formula
(24) that an 8-inch pipe will transmit much more than double the
quantity of water per second that can be transmitted by a 4-inch pipe.
3. Wind velocities during exceptionally heavy hurricanes on the
Atlantic coast are sometimes over 140 miles per hour. Show that the
wind pressure on a flat surface during such a storm is about fifty
times the amount experienced during a 20-mile wind.
4. Show that for wind velocities of 10, 20, 40, 80, 160 miles per hour
(varying in geometrical progression with ratio 2), the pressure
exerted on a flat surface is 0.3, 1.2, 4.8, 19.2, 76.8 pounds p6r
square foot respectively (varying in geometrical progression
with ratio 4).
6. A 300-h.p. DeLaval turbine makes 10,000 revolutions per min-
ute. Find the necessary diameter of the propeller shaft.
6. A railroad switch target bent over by the wind during a tornado
in Minnesota indicated an air pressure due to a wind of 600 miles per
hour. Show that the equivalent pressure on a flat surface would
be 7.5 pounds per square inch.
7. Show that a parachute 50 feet in diameter and weighing 50
pounds will sustain a man weighing 205 pounds when falling at the
rate of 10 feet per second.
Suggestion: Use approximate value ir = 22/7 in finding area of
parachute from formula for circle, irr^* and use formula (20) above.
8. Show that empirical formulas (a) and (6) for the density of
water reduce to a power function if the origin be taken at ^ = 4^ (i = 1.
§33] RECTANGULAR COORDINATES 73
33. Practical Graphs of Power Functions. The graphs of
the power function
2/ = a;2, 2/ = x^ t/ = lA, V = ^^^, etc., (1)
can, of course, be made the basis of the laws concretely expressed
by equations (1) to (27) of §31. If, however, the graph of a
scientific formula is to serve as a numerical table of the function
Tor actual use in practical work, then there is much more labor
in the proper construction of the graph than the mere plotting
of the abstract mathematical function. The size of the unit
bo be selected, the range over which the graph should extend,
the permissible course of the curve, become matters of practical
importance.
If the apparent slope^ of a graph departs too widely from
-|- 1 or — 1, it is desirable to make an abrupt change of unit in
}he vertical or the horizontal scale, so as to bring the curve back
to a desirable course, for it is obvious that numerical readings can
Dest be taken from a curve when it crosses the rulings of the co-
ordinate paper at apparent slopes differing but little from ± 1.
The above suggestions in practical graphing are illustrated by
the following examples:
Graph the formula (equation (8), §31), for the horse power
transmissible by cold-rolled shafting
^ d'N (2)
n which d is the diameter in inches and N is the number of
revolutions per minute. The formula is of interest only for the
range of d between 0 and 24 inches, as the dimensions of ordinary
shafting lie well within these limits. Likewise one would not
ardinarily be interested in values of N except those lying between
10 and 3000 revolutions per minute. Fig. 40 shows a suitable
graph of this formula for the range 1 <d < 10 for the fixed
value of iV = 100. In order properly to graph this function, three
different scales have been used for the ordinate Hj so that the
slope of the curve may not depart too widely from unity.
^ Ot course the real slope of a curve is independent of the scales used. By
apparent slope » 1 is meant that the graph appears to cut the ruling of the
squared paper at about 45*.
74 FXEMENTARY MATHEMATICAL ANALYSIS [(33
If aimilar graphs be drawn for N = 200, N =- 300, i*? » 400,
etc., a set of parabolas is obtained from which the horse poTW
of Hhafting for various speeds of rotation as well as for various
diuiiiotcrs may be obtained at once. A set of curves systematically
eonHtnictcd in a manner similar to that just described, is often
(nulled a family of cuives. Fig. 39 shows a family of straight hues
expressing the capacity of rectangular tanks corresponding to
the various widtha of the tanks.
Iriasiiiucli as niuny of the formulas of science are used only for
positive values of the vari-
ables, it is only necessary in
these cases to graph the
function in the first of the
four quadrants. For such
problems the origin ma; Ik
taken at the lower left cor-
ner of the coordinate paper
so that the entire sheet be-
comes available for the
curve in the first quadrant.
The above iUustrations
are sufficient to make clear
the importance inscienceof
the functions now being
discussed. The following
exercises give furth» pnu-
lico in tho \isefu1 application of the properties ot the funcli<»i9.
Exercises
'I1t« (traplta for iho foUowinn la-ablems are to be conatnieted upon
i«4t«nKuUr c<.>(^tinat« p»pw. The in?tfucuoDa aie for centimeter
Ivxi^M ^form .Ul^ ruled intii 20 X «1 cm. squares. In each c«K
Iho uni<» f>,ir nlvii'Msa and for i.ifiiiaAt«^ arr t<> be ao ariected as best
tvi rvhiliii tW fun^-livws. <v>nsitl«nng both ibe workable range li
v»lu<v .\( th*- vsrUMes and iht suitable sli>pe of the ranres.
Ilt<' »tutli>ui »hv>uK) r<(«tl $U a !«»>t:<j time before prooeediDg
«il)) iW f>4to«iit)t <'vrf\^is>(«. ^vin$ «;pmal fare to instruetiooi
v*\ ,J" attd vS' siwn in that i«si;oa.
§33] RECTANGUT.AR COORDINATES 75
1. Classify the graphs of formulas (1) to (27), §31, as . to
parabolic or hyperbolic type.
2. Graph the formula w* = 2ghj or t; = v^^ = 8.02^^, if h range
between 1 and 100, the second and foot being the units of measure.
See formula (5), §31.
The following table of values is readily obtained :
h 1 5 10 20 _30 _40_50 ^_J0 80 90 100
t; 8.02 17.9 25.3 35.8 43.9 50.7 56.7 62.1 67.1 71.7 76.0 80.2
Use 2 cm. = 10 feet as the horizontal unit for hj and 2 cm. = 10
feet per second as the vertical unit for v. The graph is then readily
constructed without change of unit or other special expedient.
3. Graph the formula q = 3.37LA^^ for L= 1, for h= 0, 0.1, 0.2,
0.3, 0.4, 0.5. See formula (17), §31. Use 4 cm. = 0.1 for hori-
zontal unit for h and 2 cm. =0.1 for vertical unit for q.
4. Draw a curve showing the indicated horse power of a ship
I.H.P. = S^D^^/C for C = 200 if the displacement D = 8000 tons, and
for the range of speeds /S = 10 to /S = 20 knots. See formula (14),
§31.
For the vertical unit use 1 cm. = 1000 h.p. and for the horizontal
unit use 2 cm. = 1 knot. Call the lower left-hand corner of the paper
the point (S = 10, 1, H.P. = 0).
6. From the formula expressing the centripetal force in pounds of a
rotating body,
F = 0.000341 Tri2iV2
draw a curve showing the total centripetal force sustained by a 36-inch
automobile tire weighing 25 pounds, for all speeds from 10 to 40 miles
per hour. See formula (13), §31.
Miles per hour must first be converted into revolutions per minute
by dividing 5280 by the circumference of the tire and then dividing
the result by 60. This gives
1 mile an hour = 9i revolutions a minute
If V be the speed in miles per hour the formula for F becomes
F = 0.000341 (1.5) 25 (9i)«F2 = l.llF^
For horizontal scale let 4 cm. = 10 miles an hour and for the vertical
scale let 1 cm. = 100 pounds.
6. Draw a curve from the formula / = m/r^ showing the accelera-
tion of gravity due to the earth at all points between the surface of
76
ELEMENTARY MATHEMATICAL ANALYSIS (S34
the earth aad a point 240,000 miles (tiie distance to the moon) from
the center, it / = 32.2 when radius of the earth - 4000 miles.
It is convenient in conirtructing this graph to take the radius of
the earth as unity, so that the graph will then be required of
/ - 32.2/r' from r = ltor = 60. In order to construet a suitftble
curve several changes of unite are desirable. See Fig. 41. Ooe
centimeter represents one radius (4000 miles) from r = 0 to r = 10,
afterwhich the scale is reduced to I cm. = lOr. In the vertical direc-
tion the scale is 4 cm. = 10 feet per second lor 0 < r < 5, 4 cm. =
1 foot a second for 5 < r < 10, and 4 cm. = 0.1 foot a second for
w
_ ""
.,
1
„
15
1
\
\
\
^
\
"N
^
V
1
1
^^
MM
*t
6
■ ^
0«
su
.'j
2
i_
104
OE
0&
Fio. 41. — Gravitatioaal Acceleration at Various Distances from the
Eaith'a Center. The moon is distant appcoiimately 60 earth'e radii from
the center of the earth.
10 < r < 60. Even with these four changes of units just used the
first and third curves are somewhat steep. The student can readily
improve on the scheme of Fig. 41 by a better selection of units.
Si. The Law of the Power Functioiis. Suf&cient illufitratioDH
have been given to show the fundamental character of the power
function as an expression of numerous laws of natural phenomena.
How may a functional dependence, of this sort be expressed in
words? If a series of measurements are made in the laboratory,
so as to produce a numerical table of data covering certaio phe-
§34] RECTANGULAR COORDINATES 77
nomena, how can it be determined whether or not a power function
can be written down which will express the law (that is, the
function) defined by the numerical table of laboratory results?
The answers to these questions are readily given. Consider first
the law of the falling body
s = 16.U2 (1)
Make a table of values for values of i = 1, 2, 4, 8, 16 seconds, as
follows:
t
8
8 16
16.1 64.4 257.6 1030.4 4121.6
The values of t have been so selected that t increases by a fixed
multiple; that is, each value of < in the sequence is twice the pre-
ceding value. From the corresponding values of s it is observed
that s also increases by a fixed multiple, namely 4.
Similar conclusions obviously hold for any power function.
Take the general case
y = ax" (2)
where n is any exponent, positive, negative, integral or fractional.
Let X change from any value Xi to a multiple value mxi and call
the corresponding values of y, yi and 2/2. Then we have
2/1 = axi^ (3)
and
= a(mxiy = am'^Xi** (4)
Divide the members of (4) by the members of (3) and we have
^^ = mn ^ (5)
2/1 ^
That is, if x in any power function change by the fixed multiple
m, then the value of y will change by a fixed multiple m\ Thus
the law of the power function may be stated in words in either of
the two following forms:
In any power function, if x change hy a fixed muUiple, y mil
change by a fixed multiple also.
In any power function, if the variable increase by a fixed percent,
the function will increase by a fixed percent also,
4
78 ELEMENTARY MATHEMATICAL ANALYSIS [§35
This test may readily be applied to laboratory data to determine
whether or not a power function can be set up to represent as a
formula the data in hand. To apply this test, select at several
places in one column of the laboratory data, pairs of numbers
which change by a selected fixed percent, say 10 perceut, or 20
percent, or any convenient percent. Then the corresponding pairs
of numbers in the other column of the table must also be related by
a fixed percent (of course, not in general the same as the first-
named percent), provided the functional relation is expressible by
means of a power fimction. If this test does not succeed, then
the fimction in hand is not a power function.
Since the fixed percent for the function is w" if the fixed percent
for the variable be m, the possibility of determining n exists,
since the table of laboratory data must yield the numerical values
of both m and m».
35. Simple Modifications of the Parabolic and of the Hyperbolic
Types of Curves. In the study of the motion of objects it is
convenient to divide bodies into two classes: first, bodies which
retain their size and shape unaltered during the motion; second,
bodies which suffer change of size or shape or both during the
motion. The first class of bodies are called rigid bodies; a mov-
ing stone, the reciprocating or rotating parts of a machine, are
illustrations. The second class of bodies are called elastic bodies;
a piece of rubber during stretching, a spring during elongation or
contraction, a rope or wire while being coiled, the water flowing in
a set of pipes, are all illustrations of this class of bodies.
When a body changes size or shape the motion is called a
strain.
Bodies that preserve their size and shape unchanged may possess
motion of two simple types: (1) Rotation, in which all particles
of the body move in circles whose centers lie in a straight line
called the axis of rotation, which line is perpendicular to the plane
of the circles, and (2) translation, in which each straight line of
the body remains fixed in direction.
We have already noted that the curve
^' = X- (1)
a
RECTANGULAR COORDINATES 79
lade from the curve
2/ =a;» (2)
plying all the ordinates of (2) by a. The effect is either to
or to contract all of the ordinates, depending upon whether
»r a < 1 respectively. The substitution of (2/1 /a) for y
jf ore produced a motion or strain in the curve y = x*», which
ise is the object whose motion is being studied. Likewise
y={xila)n (3)
lade from
2/ = a;» (4)
iiplying all of the abscissas of (4) by a. The effect is
o stretch or to contract all of the abscissas, depending
lether a > 1, or a < 1 respectively,
leral, if a curve have the equation
y = fix) (5)
y=Kxi/a) (6)
from curve (5) by lengthening or stretching the XY-
liformly in the x direction in the ratio 1 : a,
statement just given is made on the assumption that
If a < 1 then the above statements must be changed
itituting shorten or contract for elongate or stretch.
•easons for the above conclusions have been previously
jubstituting (-- j everywhere as the equal of z multiplies
3 abscissas by a. That is, if ( — 1 = x, thenaji = ax, so that
)ld the old X.
[lall now explain how certain other of the motions men-
,bove may be given to a locus by suitable substitution for
ranslation of Any Locus. If a table of values be prepared
of the loci
y = x^ (1)
y = (^1 - 3)« C2)
i
80 ELEMENTARY MATHEMATICAL ANALYSIS [§36
as follows:
X I - 2 - 1 0 12 3 4
y\ i' i 0 1 4 9 16
a;ii-2 -10123456
y \ 25 16 9410149
and then if the graph of each be drawn, it will be seen that the
curves differ only in their location and not at all in shape or size.
The reason for this is obvious: if (xi — 3) be substituted for x
in any equation, then since (xi — 3) has been put equal to x, it
follows that Xi = X + 3, or the new x, namely a;i, is greater
than the original x by the amount 3. This means that the new
longitude of each point of the locus after the substitution is greater
than the old longitude by the fixed amount 3. Therefore the
new locus is the same as the original locus translated to the right
the distance 3.
The same reasoning applies if (xi — a) be substituted for x,
and the amount of translation in this case is a. The same reason-
ing applies also to the general case y = f(x) and y = /(a;i— a),
the latter curve being the same as the former, translated the dis-
tance a in the x direction.
As it is always easy to distinguish from the context the new x
from the old x, it is not necessary to use the symbol Xi, since the
old and new abscissas may both be represented by x. The
following theorems may then be stated:
Theorems on Loci
VI. // (x — a) be substituted for x throughout any equaiion, the
locus is translated a distance a in the x direction.
VII. If (y — b) be substituted for y in any equation, the locusts
translated the distance b in the y direction.
These statements are perfectly general: if the signs of a and
b are negative, so that the substitutions for x and y are of the form
X + a' and y + b\ respectively, then the translations are to
the left and down instead of to the right and up.
Sometimes the motion of translation may seem to be disguised
by the position of the terms a or b. Thus the locus j/ = 3x + 5
RECTANGULAR COORDINATES 81
lame as the locus y = 3x translated upward the distance 5,
first equation is really y — 5 = 3a;, from which the conclu-
obvious.
Exercises
ompare the curves: (1) y = 2x and y = 2(x — 1); (2) y = x^
= (x — 4)8; (3) y = a;» and y — 3 = a;»; (4) y = x^ and
- 5)^^; (5) y = 5x^ and y = 5(a; + 3)*; (6) y = 2x* and
r - ky; (7) y = 2x» and y = 2a;» + fc; (8) y + 7 = x^ and
and y - 7 = a;«; (9) 3y« = 5a;» and 3(y - by = 5(a; - a)».
ompare the curves: (1) y ^ x^ and y = (a;/2)'; (2) y = a;'
=a;V8; (3) y = a;^ and y/2 = x»; (4) y =^ x^ and y = 2a;»; (5)
c» and (y/5)« = 3(x/7)»; (6) y« = a;» and y« = (3a;)»; (7)
and y = 4a;* (note: explain in two ways); (8) y = a;' and
» and y = 27a;«.
ranslate the locus y = 2a;'; (1) 3 units to the right; (2) 4 units
(3) 5 imits to the left.
longate three-fold in the x direction the loci: (1) y* = a;; (2)
«; (3) y« = 2x'] (4) y = 2a; + 7.
[ultiply by 1/2 the ordinates of the loci named in exercise 4.
low that y = — r-r and y = 7 are hyperbolas.
X -\- 0 X — 0
X
low that y = — j-j is a hyperbola.
s: Divide the numerator by the denominator, obtaining the
a; +a
low that y = — 7^ is a hyperbola, namely, the curve xy =0—6
ted to a new position.
Shearing Motion. An important strain of the XF-plane
if we derive
y = f{^) + wx (1)
2/' = fix) (2)
ically, the curve (1) is seen to be formed by the addition
ordinates of the straight line y" = wa; to the corresponding
bes of y' = fix). Thus, in Fig. 42, the graph of the func-
i
82 ELEMENTARY MATHEMATICAL ANALYSIS [S37
tion x' 4- X iB made by addii^ the corresponding ordinates of
J/' ■= x* and y" = x. Mechanically, this might be doae by drawing
the curve on the edge of a pack of cards, and then elippii^ the
cards over each other uniform amounts. The change of die
shape of a body, or the strain of a body, here illustrated, is
called lamellar rootioit or shearing motion. It is a form of
motion of very great importance.
™3B
2 44--, ^
We shall speak of the locus y = /(i) + mx aa the shear qf Ifc
curve y = f(x) in Ihe line y = mx.
Theorems on Loci
VIIl. The addition of the term mx to the right side of y = i{i)
shears the locus y=f{x) in Ihe line y = tnx.
The locus
RECTANGULAE COORDINATES
83
sde from y = x^by& combination of (1) a uniform elongation
2} a slieariug motion [m], and (3) a translation [b]. Either
ion may be chained in sense by changing the sign of a, m,
, respectively,
he student may easily show that the effect of a shearing motion
1 the straight line y = mx + b is merely a rotation about
fixed point (0, b). The hne is really slrelched in the direction
.8 own length, but this does not change the shape of the line
does it change the line geometrically. A line segment {that
t hne of finite length) would be niotlified, however,
he parabola y = x^ ia transformed under a shearing motion
. moat interesting way. For, after shear, y = x'' becomes:
' + 2vi
(3)
84
ELEMENTARY MATHEMATICAL ANALYSIS
where, for convenience, the amount of the shearing moti(
represented by 2m instead of by m. Writing this in the for
or.
y = x^ + 2mx + m* — m*
y = {x + m)* — m*
y + m^ = {x + m)*
we see that (4) can be made from the parabola y = x^hyt
lating the curve to (h
the amoimt m and
the amount w*. (See
44.)
Shearing motion, t
fore, rotates the str
line and translates th<
rabola. The effect on •
curves is much more
plicated, as is seen
Figs. 42 and 43.
The parabola y =
identical in size and s
with y = x^ + mx
Likewise, y = ax^ + k
is a parabola differing
in position from y = c
Exercises
Fig. 44. — The Shear of y = x^ in the line
2/ = 0.6a;. 1 Explain how the (
y — X* -\- 2x may be ]
from the curve y = x^. How can the curve y = 2x^ + 3a; be i
from the curve y = 2x^1
2. Find the coordinates of the lowest point of y = x^ -
that is, put this equation in the form y — 6 = {x — a)\
3. Compare the curves y = x^ -\- 2x and y = x* — 2x. (D(
draw the ciurves.)
4. Explain the curve y = l/x + 2a; from a knowledge of y =
and of y = 2x.
\
\
4
^ 1
\
v\
2
7/2
//
\\
' /
/y;
^
\
^
I
^
-3
-2
^
0
-1
:
I
3
/
^
.2
-a
-4
m RECTANGULAR COORDINATES 85
38. Rotation of a Locus. The only simple type of displace-
aent of a lociis not yet considered is the rotation of the locus
bout the origin 0. This will be taken up in the next chapter
Q the discussion of a new system of coordinates known as polar
oordinates. The rotation of any locus about the X-axis or about
he 7-axis is readily accomplished, however, as previously ex-
ilained. For substituting {— x) for x changes every point that is
0 the right of the F-axis to a point to the left thereof, and vice
ersa. It is equivalent, therefore, to a rotation of the locus about
he 7-axis. Likewise, substituting (— y) for y rotates any locus
80® about the X-axis. It is preferable, however, to speak of the
3CUS formed in this way as the reflection of the original curve in
he y-axis or in the x-axis, as the case may be.
39. Roots of Functions. The roots or zeros of a function are
he values of the argument for which the corresponding value of
he function is zero. Thus, 2 and 3 are roots of the function
;' — 5a; + 6, for substituting either number for x causes the
unction to be zero. The roots of x^ — x — 6 are + 3 and — 2.
The roots of x^ - Qx^ + llx - 6 are 1, 2, 3.
The word root, used in this sense, has, of course, an entirely
iifferent significance from the same word in "square root," "cube
'oot," etc. But the roots of the function x^ — 5a; — 6 are also the
'oots of the equation a; ' — 5a; — 6 = 0.
In the graph of the cubic function y = x^ — x in Fig. 42, the
5urve crosses the X-axis at a; = — 1, a; =0, and a; = 1. These are
he values of x that make the function a;' — a; zero, and are, of
iourse, the roots of the function a;^ — x. No matter what the func-
ion may be, it is obvious that the intercepts on the X-axis, as OA,
)5, Fig. 42, must represent the roots of the function.
Exercises
1. From the curve y = x^ sketch the curves i/ — 4 = a;^; y = 4a;2;
^y = a;2; y = (re - 4)*.
(x — S')*
2. Sketch y = a;V2; y = x' - 1/4; y = xy2 - 4; y = -- -- '
3. Sketch the curves y = Vx; y = Vx] 2/= ^Vx; y = y/x^2]
/ - 2 = Va; - 2, and 2/ = Va; - 3.
ff
ELEMENTARY MATHEMATICAL ANALYSIS [fU
= {X - 3)>; (y - 2)' - x\ and (y-2]'
x' and thence y = x + x'.
+ 8 =.0, from the graph of
A. Sketch the curves j
= (I - 3)'.
5. Graph 1/1 ^ X and j/i
6. Find the roots of
y " x'' - 6x + S.
7. Find the roots of the fi
8. Compare the curves V ^ I'and
B - ar + 3 and v = ~ 2i + 3.
9. Graph j/i = a and yt — l/x and thence y — a; + l/i
10. Compare y = lA, y - \/{x - 2), y = l/{i + 3).
11. Compare y = l/x, y = l/(2i), y = Ijx.
40. ""Grapliical Constnictioii of Power Functions and of oflMi
Functions.' The graphical computation of products and quo
- I'i y — a:' ai
- !/=!•;
y
V
1
/
B
^
/
/
1,1
/
f
(/,
/
h
\
/
M
fA
1/
\
/r
a.
if
1
J
7
1
i
/
X
/
--
/
/
\
/
r
1 the Product of Tw
tients, etc., explained in §7, may be applied to the construction"
the power functions. For this purpose it is desirable to elaborate
slightly the previous method so as to provide for finding pro't
ucts, etc., of lines that are parallel to each other, instead of at ri^
angles as OA and IS, Ftg. 9.
' The lemsinder of this chapter (eicept the review exeician} m^y Iw ODiitudiH^
out loeg of continuity.
§40]
RECTANGULAR COORDINATES
87
The constructions can be carried out on plain paper by first
drawing the axes, the unit lines and the line y = x, without the
use of scales or measuring device of any sort. The work is more
rapidly done, however, on squared paper, as then the use of a
T-square and triangle may be dispensed with. A unit of measure
equal to 2 inches or 4 cm. will be found convenient for work
on standard letter paper 8J X 11 inches.
Note that the following constructions give both the magnitude
and the proper algebraic sense of the results.
(1) To construct an ordinate
equal to the product of two ordi-
nates: Let XX', 77', Fig. 45, b e
the axes, C/i, U2 the unit lines,
and OR the line y = x, which
Fig. 46. — Construction of an Or-
dinate Equal to the Quotient of
Two Given Ordinates.
/I I
/ 1 /
/ 1
/
^ •/
/
/ 1
/
/ 1 y
/
{ jf^
/
y \
/
X "*
/
/
/ 1 f'-
/ y
/ /
/ >!-.
/ y^/
\ \ ■ X
0
^ 'z>
1
< X
1
1
I
Fig. 47. — Construction of an Or-
dinate Equal to the Square of a
Giv
'en Qrdinat<
we shall call the reflector. Let a and h be two ordinates whose
product is required. Move one of the two given ordinates as h
until, in the position l^lDy its end touches the reflector OR.
Move the second of the two ordinates to the position IMon the
unit line t/i. Draw OMF. The point P at which DiV^ is cut by
OM (produced if necessary) determines DF^ which is the prod-
uct a X 6. This result follows by similar triangles from the pro-
portion
DP : IM = 0T> : 01
88
ELEMENTARY MATHEMATICAL ANALYSIS [§4fl
Substituting a for IM and h for OD ( = DN = h) and unity for
01 y we obtain
DP:a = b:l
or
DP = aXb
The same diagram shows the construction of the products cXd
and a X c for cases in which one or both of the factors are negative.
Note that by the above construction the ordinate representing
the product is always located at a particular place^ D, at which the
abscissa of the product a X 6 is either equal to a or to 6, depending
upon which of the ordinates was moved to the reflector OR.
Y
Ui
A Ui
i
^.--'
-^
1
X
Fig. 48. — Construction of the Reciprocal of x.
(2) To construct an ordinate equal to the quotient of two ordi-
nates: This is done by use of the second unit line 172 as shown in
Fig. 46. The ordinate representing the quotient is located at D
where OD equals the dividend b.
(3) The special case of (1) when a = b leads to the construction
of x^ as shown in Fig. 47. The figure shows the construction of
x* at D where OD = x and of Xi^ at Di where ODi = xi.
(4) The special case of (2) where 6=1 leads to the construc-
tion of 1/x as shown in Fig. 48.
(5) To construct the graph of y = x^j it is merely necessary to
make repeated applications of (3) to the successive ordinates of
KECTANGULAH. COORDINATES 89
ae y = X, &a shown in Fig. 49. Thus from any point A of
; move horizontally to the unit line Ui locatii^ B, then if
leets DA &tP,P is k point of the curve y = x*. The figure
I the construction for a number of points, lettered Ai, At,
To construct the graph of y = x\ first cut out a pattern of
;' of heavy paper, marking upon it the lines OY and Ui
'J
(T
-
\v
/
/
I
»<
/
\/
/u
\W
'-
=
p.
s
^
'
I'
.-
X
/
\
^
/
/
r
— Conatruefcion of the Curve 1/ =■ a:
leane of this pattern draw the curve y = x* upon a fresh sheet
.per as shown in Fig. 50. Then multiply each ordinate of
c' by X by moving it horizontally from any point Aoty = x*
e unit line XJi at B, then locating P on DA by drawing OB
it cuta DA at P. The result is the cubical parabola
To draw the hyperbola y = \/x, make repeated application
) above to successive values of x. To draw y= l/z*, repeat
ion by a: to the ordinates of y = \/x, etc. -
90 ELEMENTARY MATHEMATICAL ANALYSIS [f«
(8) To construct the graph 0/ y =■ x^'': Firet, from the pattern
of J/ = a;' draw the curve y = \/ x- From a psttem draw the
curve y = x^ upon the same axes. Then from any point A\
of y = x^^ proceed horizontally to B, on the reflector; then ver-
tically to Ci on the curve y = x', then horizontally to Pi on the
ordinate DAi first taken. Then Pi is on the curve y = x^'.
For, call i)-4,= yi; WC, = )/,; DPi=y; OD = x; OH ^ ii.
Then by construction (Fig. 51)'
of the Curve » - i' from the ou
rvB K - 1"-
OH -DA-y,- «»
(11
P. - » - HC, - s, - I,'
(!)
Hence, by (3) and (2):
and by (1)
RECTANGULAR COORDINATES
91
9) Function of a function: The construction and reasoning
t given applies to a much more general case. Thus if the curve
1, Fig. 51, has the equation
y = /(x)
1 if the curve OCi has the equation
V = Fix)
1
1
L
r
/
z
h
/
/
it
■A
/
7
-
-
-
■i'
'
J
.
^
/
\
/
y
a the curve 0P| has the equation
y = F[f(x)]
13, if OA I be the curve
. OCi the curve
a 0P\ is the curve
!/ = {!- a;'l'*
92 ELEMENTARY MATHEMATICAL ANALYSIS
For constructions of the function
y = a + aix + a^* + . . . + ana;*
see "Graphical Methods" by Carl Runge, Columbia University
Press, 1912.
Miscellaneous Exercises
1. Define a function. Explain what is meant by a discontinuous
function. Give practical illustrations.
2. Define an algebraic function; rational function; fractional
function. Give practical illustrations in each case.
3. Give an illustration of a rational integral function; of a
rational fractional function.
4. Write a short discussion of the Cartesian method of locating a
point. Explain what is meant by such terms as "axis," *'af of a
point," "quadrant," etc.
6. What is meant by the locms of an equation?
6. Write the equations of the lines determined by the following
data:
(a) slope 2 F-intercept 5
(&) slope —2 F-intercept 5
(c) slope 2 F-intercept —5
(d) slope —2 F-intercept —5
(e) slope —2 X-intercept 4
7. Make two suitable graphs upon a single sheet of squared paper
from the following data giving the highest and lowest average clos-
ing price of twenty-five leading stocks listed on the New York Stock
Exchange for the years given in the table :
Year Highest Lowest
1913 94.56 79.58
1912 101.40 91.41
1911 101.76 86.29
1910 111.12 86.32
1909 112.76 93.24
1908 99.04 67.87
1907 109.88 65.04
1906 113.82 93.36
1905 109.05 90.87
1904 97.73 70.66
1903 98.16 68.41
1902 101.88 87.30
RECTANGULAR COORDINATES 93
Should smooth curves be drawn through the points plotted from this
table?
8. Define a parabolic curve. What is the equation of the parabola ?
Of the cubical parabola? Of the semi-cubical parabola?
9. What is the definition of an hyperbolic curve ? Of the rectangu-
lar hyperbola?
10. Draw on a sheet of coordinate paper the lines a; = 0, a? = 1,
a; = — 1, y = 0, y = 1, 2/ = —1. Shade the regions in which the
hyperbolic curves lie with vertical strokes; and those in which the
parabolic curves lie with horizontal strokes. Write dovm all that the
resulting figure tells you.
11. Consider the following: y =* x*, i/ = a;"', y = ^x^t xy — — 1,
y = — x', y* = a;*, y* = a;*, xy = 1, a;* = — y*, a;* = — y*. Which
are increasing functions of x in the first quadrant? For which
does the slope of the curve increase in the first quadrant? For
which does the slope of the curve decrease in the first quadrant?
12. Which of the curves of exercise 11 pass through (0, 0)?
Through (1, 1)? Through (-1, -1)?
13. Find the vertex of the curve y = x* — 24x + 150.
Note: The lowest point of the parabola y = x* may be called
the vertex.
Suooestion: It is necessary to put the equation in the form y — h
= (x — a)*. This can be done as follows: Add and subtract 144 on
the right side of the equation, obtaining
y = x* - 24x + 144 - 144 + 150
or,
y =(x- 12)2 + 6
or,
y - 6 = (x - 12)2
Then this is the curve y = x* translated 12 units to the right and 6
units up. Since the vertex of y = x* is at the origin, the vertex of the
given curve must be at the point (12, 6).
14. Find the vertex of the parabola y = x* — 6x +11.
16. Find the vertex of y = x* + 8x + 1.
16. Find the vertex of 4 + y = x* — 7x.
17. Find the vertex of y = Ox* + 18x + 1.
18. Translate y = 4x2 — 12x + 2 so that the equation may have
the form y = 4x2,
CHAPTER HI
THE CmCLE AND THE CDtCULAR FUHCTIOHS
41. Equation of the Circle. In rectangular cowdinates the
abscissa x, and the ordinate y, of any point P (as OH and Jff,
Fig. 52) form two sides of a right triangle whose hypotenuse
squared is z' + ;/'. If the point P move in such manner that the
length of this hypotenuse remains fixed, the point P describee a
circle whose center is the origin (see Fig. 52). The equation of
this circle is, therefore:
X' + y> :
(t)
if a stand for OP, Fig. 52, namely
the fixed length of the hypote-
nuse, or the radius of the cirde.
It is sometimes convenient to
write the equation of the circle
solved for y in the fonn
This gives, for each value of J,
the two corresponding equal and
opposite ordinates.
To translate the circle of radius a so that its center shall be the
point (A, k), it is merely necessary t« write
(x-h)'+(y-k)' = a» (3)
This is the general equation of any circle in the plane xy, for it
locates the center at any desired point and provides for any
desired radiiL-^ a.
1. Write the equations of the circles with center at the origin
having radii 3, 4, 11, V^ respectively.
§43] THE CIRCLE AND THE CIRCULAR FUNCTIONS 95
2. Write the equation of each circle described in exercise 1 in
bhe form y = ± \/a* — x*.
3. Which of the following points lie on the circle x^ + y^ — 169
(5, 12), (0, 13), ( - 12, 5), (10, 8), (9, 9), (9, 10)?
4. Which of the following points lie inside and which lie outside
3f the circle x« + 1/« = 100: (7, 7)^ (10, 0), (7, 8), (6, 8), ( - 5, 9),
( -7, - 8), (2, 3), (10, 5), (V40, Vso), (Vl9, 9)?
42. The Equation, x^ _|_ y2+ 2gx + 2fy + c =o (1)
may be put in the form (3). For it may be written
z^ + 2gx + g^ + y^ + 2fy+P = g^ + P - c,
or,
(X +gy + (y+f)'= {Vg'~+'P - cV (2)
which represents a circle of radius x/gr^ +p — c whose center is at
the point (— fi', — /). In case g^ + p — c < 0, the radical
becomes imaginary, and the locus is not a real circle; that is,
coordinates of no points in the plane xy satisfy the equation. If
the radical be zero, the locus is a single point.
43. Any equation of the second degree y in two variables , lacking
the termzy and having like coefficients in the terms x^ andy^, repre-
sents a circle y redly null or imaginany. The general equation of
the second degree in two variables may be written:
ax^ + by^ + 2hxy +2gx + 2fy + c=0 (3)
for, when only two variables are present,there can be present three,
terms of the second degree, two terms of the first degree, and one
term of the zeroth degree. When a = b and A = 0 this reduces
to (1) above on dividing through by a.
Exercises
Find the centers and the radii of the circles given by the following
equations :
1. a;2 + t/2 = 25. Also determine which of the following points
are on this circle: (3, 4), (5, 5), (4, 3), (-3, -4), (-3, 4), (5, 0),
(2, V21) .
2. a;2 + 2/^ = 10.
3. x* + 2/^ - 4 = 0.
96 ELEMENTARY MATHEMATICAL ANALYSIS (§44
4. x« + 2/* - 36 = 0.
6. x^ +y^ +2x = 0.
6. y = ± V169 — X*. Also find the slope of the diameter through
the pomt (5, 12). Find the slope of the tangent at (5, 12).
7. 9 - a;* - !/« = 0.
8. x^ + y^ -6y = 16.
9. x* - 2x + 2/« - 6y = 15.
10. (x + a)« + (y - 6)« = 50.
11. x2 + 2/* + 6x - 21/ = 10.
12. x2 + t/2 - 4a; + 62/ = 12.
13. X* + t/2 - 4x - 82^ + 4 = 0.
14. 3x« + Si/* + 6x + 122/ - 60 = 0.
16. Is X* + 2y* + 3x — 42/ — 12 = 0 the equation of a circle?
Why?
16. Is 2x* + 2y* — 3x + 41/ — 8 = 0 the equation of a cirde?
Why?
44. Angular Magnitude. By the magnitude of an anfi^e is
meant the amount of rotation of a line about a fixed point. If
a line OA rotate in the plane XY about the fixed point 0 to the
position OP, the line OA is called the initial side and the line OP is
called the terminal side of the angle AOP. The notion of angular
magnitude as introduced in this definition is more generid than
that used in elementary geometry. There are two new and very
important consequences that follow therefrom:
(1) Angular magnitude is unlimited in respect to size — ^that is,
it may be of any amount whatsoever. An angular mag^tude of
100 right angles, or twenty-five complete rotations is quite as
possible, under the present definition, as an angle of smalli^
amount.
(2) Angular magnitude exists, under the definition, in two
opposite senses — ^f or rotation may be clockwise or anti-clockwise.
As is usual in mathematics, the two opposite senses are distin-
guished by the terms positive and negative. In Fig. 53, AOPi,
AOP2, AOPzf AOP4 are positive angles. In designating an
angle its initial side is always named first. Thus, in Fifr 53,
AOPi designates a positive angle of initial side OA. P\OA
designates a negative angle of initial side OPi.
151 THE CIRCLE AND THE CIRCULAR FUNCTIONS 97
Fig. 53. — Triangles of Reference
(ODiPi, ODiPi, etc.) for Angles 0
of Various Magnitude.
In Cartesian coordinates, OX is usually taken as the initial
ne for the generation of angles. If the terminal side of any angle
ills within the second quadrant, it is said to be an "angle of the
3cond quadrant/' etc.
Two angles which differ by y
ny multiple of 360** are called
ongruent angles. We shall
nd that in certain cases con-
ruent angles may be substi-
Lited for each other without —
lodifying results.
The theorem in elementary
eometry, that angles at the
3nter of a circle are propor-
onal to the intercepted arcs,
olds obviously for the more
eneral notion of angular mag-
itude here introduced.
46. Units of Measure. Angular magnitude, like all other
lagnitudes, must be measured by the application of a suitable
nit of measure. Four systems are in common use:
(1) Right Angle System. Here the unit of measure is the right
Qgle, and all angles are given by the number of right angles and
'action of a right angle therein contained. This unit is familiar
3 the student from elementary geometry. A practical illus-
ration is the scale of a mariner's compass, in which the right angles
re divided into halves, quarters and eighths.
(2) The Degree System. Here the unit is the angle corre-
;>onding to ^l^j- of a complete rotation. This system, with the
3xagesimal sub-divisions (division by 60ths) into minutes
nd seconds, is familiar to the student. This system dates back
3 remote antiquity. It was used by, if it did not originate among,
be Babylonians.
(3) The Hour System. In astronomy, the angular magnitude
bout a point is divided into 24 hours, and these into minutes
nd seconds. This system is familiar to the student from its
nalogous use in measuring time.
(4) The Radian or Circular System. Here the unit of measure
7
i
98 ELEMENTARY MATHEMATICAL ANALYSIS t|4S
is an angle such that the length of the arc of a oircle deecribed about
the vertex as center is equal to the lei^th of the radius of Hie
circle. This system of angular measure is fundamental in m&-
chanics, mathematical phj^ics and pure mathematics. It most
be thoroughly mastered by the student. The unit of measure in
this system is called the radian. Its size is shown in Fig. 54.
Fio. 54.— Definition of the Radian. The Au^e AOP u
Inasmuch as the radius is contfuned 2x times in a circumfereDU,
we have the equivalents:
2t radians = 360°.
or, 1 radian = 57° 17' 44".8 = 57° 17'.7 = 57°.3.
1 degree = 0.01745 radians.
The following equivalents are of special importance:
a straight angle = w radians.
a right angle = ^ radians.
60° = „ radians.
45° = , radians.
30° = - radians.
I] THE CIRCLE AND THE CIRCULAR FUNCTIONS 99
ere is no generally adopted scheme for writing angular magni-
le in radian measure. We shall use the superior Roman letter
" to indicate the measure, as for example, 18** = 0.31416'.
>ince the circumference of a circle is incommensurable with its
meter, it follows that the number of radians in an angle is
'-ays incommensurable with the number of degrees in the angle.
The speed of rotating parts, or angular velocities, are usually
en either in revolutions per minute (abbreviated "r.p.m.")
in radians per second.
L6. Uniform Circtilar Motion. Suppose the line OP, Fig. 52,
revolving counter-clockwise h^ per second, the angle AOP
radians is then kt, t being the time required for OP to turn from
3 initial position OA, If we call B the angle AOPj we have B = kt
the equation defining the motion. The foUowing terms are
common use:
1. The angular velocity of the uniform circular motion is k
wiians per second).
2. The amplitude of the uniform circular motion is a .
3. The period of the uniform circular motion is the number of
conds required for one revolution.
4. The frequency of the uniform circular motion is the number
' revolutions per second.
Sometimes the unit of time is taken as one minute. Also the
totion is sometimes clockwise or negative.
Exercises
!• Express each of the following in radians: 135**, 330**, 225**, 15°,
50^ 75°, 120°. (Do Thot work out in decimals; use tt).
2. Express each of the following in degrees and minutes: 0.2'',
3. How many revolutions per minute is 20 radians per second?
4« The angular velocity, in radians per second, of a 36-inch
utomobile tire is required, when the car is making 20 miles per hour.
5. What is the angular velocity in radians per second of a 6-foot
rive-wheel, when the speed of the locomotive is 50 miles per hour?
^' The frequency of a cream separator is 6800 r.p.m. What is
* period, and velocity in radians?
100 ELEMENTARY MATHEMATICAL ANALYSIS t§47
7. A wheel is revolving uniformly 3(F per second. What is its
period, and frequency?
8. The speed of the turbine wheel of a 5-h.p. DeLaval steam turbine
is 30,000 r.p.m. What is the angular velocity in radians per
second?
47. The Circular or Trigonometric Functions. To each point
on the circle x^ + y^ = a^ there corresponds not only an abscissa and
an ordinate, but also an angle 6 < 360®, as shown in Figs. 52 and 53.
This angle is called the direction angle or vectorial angle of the
point P. When 6 is given, x, y and a are not determined, but the
ratios y/a, x/a, y /x, and their reciprocals, a/y, a/x, x/y are de-
termined. Hence these ratios are, by definition, functions of i
They are known as the circular or trigonometric functions of ^,
and are named and written as follows:
Function of 6, Name. Written.
y/a. sine of 0. sin d.
z/a. cosine of 6. cos 6.
y/x. tangent of 6. tan 6.
z/y. cotangent of 6. cot 6,
a/z. secant of 6. sec 6.
a/y. cosecant of 6. esc 6.
The circular functions are usually thought of in the above order:
that is, in such order that the first and last, the middle two, and
those intermediate to these, are reciprocals of each other.
The names of the six ratios must be carefully committed to
memory. They should be committed, using the names of x, y,
and a as follows:
Ratio. Written,
ordinate /radius. sin 6.
abscissa /radius. cos 0.
ordinate /abscissa. tan 6.
abscissa /ordinate. cot 6.
radius /abscissa. sec 6.
radius /ordinate. esc 6.
The right triangle POD of sides x, y and a, whose ratios give the
functions of the angle XOP, is often called the triangle of reference
i
KS] THE CIRCLE AND THE CIRCULAR FUNCTIONS 101
or this angle. It is obvious that the size of the triangle of refer-
nce has no effect of itself upon the value of the functions of the
ngle. Thus in Fig. 53 either PiODi or PiODi may be taken as
le triangle of reference for the angle 6i. Since the triangles are
milar we have.
ODi ODi' OPi OPi'
X5., which shows that identical ratios or trigonometric functions of
are derived from the two triangles of reference.
48. Elaborate means of computing the six functions have been
3vised and the values of the functions have been placed in
mvenient tables for use. The functions are usually printed
) 3, 4, 5 or 6 decimal places, but tables of 8, 10 and even 14 places
tist. The functions of only a few angles can be computed by
ementary means; these angles, however, are especially important.
(1) The Functions of 30°. In Fig. 55a, if angle AOB be 30°,
ngle ABO must be 60°. Therefore, constructing the equilateral
•iangle BOB\ each angle of triangle BOB' is 60°, and
y = AB = i'BB' = i-a
'herefore,
sin 30° = ^ = ^- = 1/2
a a '
Jso:
OA = V"o52 - AB'' = Va2 - ia2 = \a Vs
.^herefore,
sin 30° =1/2.
cos3o° = '^-v^^ = v;^
a 2
i^ _ ^ V3
I a V3 3 "
tan 30° = -
2
^^^ 30° = tanW = v/3
sec 30° =
o 1 ^ 2\/3
cos 30° 3'
CSC 30° = .-^^o = 2
sm 30
I
102 ELEMENTARY MATHEMATICAL ANALYSIS
(2) Functions of 46°. In the diagram, Fig. 556, the tria
OAB is isosceles, so that y = x^ and a^ ^ x^ + y^ = 2x^.
follows that a = x\/2 = y\/2.
Fig. 56.--Triangles of Reference for Angles of 30°, 45° and 60°.
Therefore:
y
2/V2
V2
2
x
V2
X'\/2 ""
2
X
1
= 1
tan 45° "
1
cos 45° "
-V'2
1
V2
sin 45° =
cos 45° =
tan 45°
cot 45° =
sec 45°
esc 45° =
(3) Functions of 60°. In the diagram, Fig. 55c, construe
equiangular triangle OBB'\ then it is seen that, as in case
above,
OA = \'0R' = i-a
and
y = Va* -i-a^ = i*a\/3
Therefore: . «^o i*a'\/3 __ y/Z
sin 60° =
a
?nE CIRCLE AND THE CIRCUTAR FUNCTIONS 103
cos60° = ^=l/2
a
tan 60°= "^'^^ = \/3
cot 60° =
tan 60° 3
sec 60° = ^^ = 2
cos 60
af,o 1 2\/3
CSC 60 = —-^7^ = — ^ —
sin 60° 3
Graphical Computation of Circular Functions. Approximate
aination of the numerical values of the circular functions of
ven angle may be made graphically on ordinary coordinate
Locate the vertex of the angle at the intersection of any
nes of the squared paper, form Af 1. Let the initial side of
igle coincide with one of the rulings of the squared paper
,y off the terminal side of the angle by means of a protractor.
sine or cosine is desired, describe a circle about the vertex
t angle as center using a radius appropriate to the scale of
luared paper — for example, a radius of 5 cm. on coordi-
paper ruled in centimeters and fifths (form Af 1) permits
reading to 1/25 of the radius a and, by interpolation, to
of the radius a. The abscissa and ordinate of the point
3rsection of the terminal side of the angle and the circle may
)e read and the numerical value of sine and cosine computed
/^iding by the length of the radius.
he numerical value of the tangent or cotangent be required,
mstruction of a circle is not necessary. The angle should
d off as above described, and a triangle of reference con-
ed. To avoid long division, the abscissa of the triangle of
nee may be taken equal to 50 or 100 mm. for the determina-
►f the tangent and the ordinate may be taken equal to 50
) mm. for the determination of the cotangent.
) following table (Table III) contains the trigonometric
ons of acute angles for each 10° of the argument.
104 ELEMENTARY MATHEMATICAL ANALYSLS
Table III
Natural Trigonometric Functions to Two Decimal Places
0
0^
0.00
sin 6
cos 6
1.00
tan e
cot e
sec e
CSC 0
0.00
0.00
00
1.00
00
10
0.17
0.17
0.98
0.18
5.67
! 1.02
5.76
20
0.35
0.34
0.94
0.36
2.75
1.06
2.92
30
0.52
0.50
0.87
0.58
1.73
1.15
2.00
40
0.70
0.64
0.77
0.84
1.19
1.31
1.56
50
0.87
0.77
0.64
1.19
0.84
1.56
1.31
60
1.05
0.87
0.50
1.73
0.58
2.00
1.15
70
1.22
0.94
0.34
2.75
0.36
2.92
1.06
80
1.40
0.98
0.17
5.67
0.18
5.76
l.<)2
90
1.57
1.00
0.00
00
0.00
00
1.00
The most important of these results are placed in the following
table:
0°
30°
1/2
1 V3
2
: V3
3
1 45°
60°
90°
Sine
Cosine
Tangent
0
1
0
V2
2"
V2
2
1
V3
2
1/2
V3
1
0
00
V2 =
1.4142
1
V3 =
= 1.7321
Exercises
1. Find by graphical construction all the functions of 15°.
Note. — A protractor is not needed as angles of 45° and 30° may be
constructed.
2. Find tan 60°. Compare with the value found above in §48.
3. Lay off angles of 10°, 20°, 30°, 40°, with a protractor and deter-
mine graphically the sine of each angle, and record the results in a
suitable table.
4. Find the sine, cosine, and tangent of 75°.
6. Which is greater, sec 40° or esc 60°?
6. Determine the angle whose tangent is 1/2.
7. Find the angle whose sine is 0.6.
§50] THE CIRCLE AND THE CIRCULAR FUNCTIONS 105
8. Which is greater, sin 40'' or 2sin 20°?
9. Does an angle exist whose tangent is 1,000,000? What is its
approximate value?
60. Signs of the Functions. The circular functions have, of
course, the algebraic signs of the ratios that define them. Of
the three numbers entering these ratios, the distance or radius
a may always be taken as positive. It enters the ratios, there-
fore as an always signless, or positive number. The abscissa
and the ordinate, x and 2/, have the algebraic signs appropriate
to the quadrants in which P falls. The student should deter-
mine the signs of the functions in each quadrant, as follows:
(See Fig. 53.)
First
quadrant
Second
quadrant
Third
quadrant
Fourth
quadrant
Sine
+
+
+
+
Cosine
Tangent
Of course the reciprocals have the same signs as the original
functions.
The signs are readily remembered by the following scheme:
Sine
Cosine
+
+
+
Tangent
+
Cosecant
+
Secant
+
Cotangent
The following scheme is of value in remembering the circular
functions and their signs in the different quadrants: Place on the
same line the variables and functions of the same algebraic signs,
thus :
Ordinate . . y . . sin ^ . . esc ^
Abscissa , . x . . cos d . . sec 6
Slope . . . m . . tan 6 . . cot d
106 ELEMENTARY MATHEMATICAL ANALYSIS (§51
The above scheme associates the signs of the functions with the
coordinates (Xy y) of the point P and the slope of the line OP
for each of its four positions in Fig. 53.
61. Triangles of reference, geometrically similar to those in
Fig. 66 for angles of 30°, 45°, and 60° exist in each of the four
quadrants, namely, when the hypotenuse and a leg of the triangle
of reference in these quadrants are both either parallel or perpen-
dicular to a hypotenuse and leg of the triangle in the first quad-
rant— then an acute angle of one must equal an acute angle of
the other and the triangles must be similar. The numerical
values of the functions in the two quadrants are therefore the
same. The algebraic signs are determined by properly taking
account of the signs of the abscissa and the ordinate in that
quadrant. Thus the triangle of reference for 120° is geometri-
V3
cally similar to that for 60*
cos 120° = - 1/2 and tan 120'
Hence, sin 120 =
= -V3.
but
Exercises
1. The student is to fill in the blanks in the following table with
the correct numerical value and the correct sign of each function:
Function
120°
135°
150°
210° 225°
240°
300°
315°
330**
Sin
1
1 , ...
Cos ' i ' ; ■ 1 1
Tan i
1
i '
i
Cot
1
1 !
Sec
1 1
1
i '
1 !
Csc ! ' ,
i 1
1
2. Write down the functions of 390° and 405°.
3. The tangent of an angle is 1. What angle < 360° may it be?
4. Cos ^ = — 1/2. What two angles < 360° satisfy the equation?
6. Sec e = 2. Solve for all angles < 360°.
6. Csc ^ = - \/2. Solve for e < 360°.
/
§52] THE CIRCLE AND THE CIRCUT.AR FUNCTIONS 107
62. Functions of 0° and 90°. In Fig. 52 let the angle AOP
decrease toward zero, the point P remaining on the circumference
of radius a. Then y or PD decreases toward zero. Therefore,
sin 0° = 0. Also, x or OD increases to the value a, so that the
ratio x/a becomes unity, or cos 0° = 1. Likewise the ratio
y fx becomes zero, or tan 0° = 0.
The reciprocals of these functions change as follows; As the angle
AOP becomes zero, the ratio a/y increases in value without limit,
or the cosecant becomes infinite. In symbols (see §23)
CSC 0® = ». Likewise, cot 0° = », but sec 0° = 1.
In a similar way the functions of 90° may be investigated. The
results are given in the following table:
Angle
1
From From
From
From
0° to 90°
90° to 180°
180° to 270°
270° to 360°
Sin
Oto + l
H- 1 to 0
Cos
-flto 0
0 to- 1
Tan
Oto+oo
-ooto+ 1
Cot
+ 00 to 0
0 to -00
Sec
+ 1 to+ 00
— 00 to — 1
Csc
H- ooto + l
+ 1 tO+ 00
The student is to supply the results for the last two columns.
63. Fundamental Relations. The trigonometric functions are
not independent of each other. Because of the relation x^ + y^
= a* it is possible to compute the numerical or absolute value of
five of the functions when the value of one of them is given. This
may be accomplished by means of the fundamental formulas de-
rived below:
Divide the members of the equation:
by a2. Then
x2 + 1/2 = a2
(1)
1
or.
sin2 e + cos2 ^ = 1
(2)
108 ELEMENTARY MATHEMATICAL ANALYSIS I §53
Likewise divide (1) through by a;^: then
or.
sec2 ^ = 1 + tan2 6
Also divide (1) through by y^: then
or,
^y/ \y
csc2 ^ = 1 + cot* e
Also, since
y.
a _ y
X X
a
we obtain,:
. . sin d
tan B = — -
cos 9
and likewise
cos e
cot e = -.—:
sin e
sin =1 /esc
cos— l/sec
tan =l/cot
mn
OS
tan
It
Bec2s=l
+ tan«
81 iC
(3)
(4)
(5)
(6)
CIC
Hin
cos
tan
08^*=! + cot'
Fig. 56. — Diagram of the Relations between the Six Circular Functions.
Formulas (2) to (6) are the fundamental relations between the six
trigonometric functions. The formulas must be committed to
memory by the student.
The above relations between the expressions may be illustrated
§53] THE CIRCLE AND THE CIRCULAR FUNCTIONS 109
by a diagram as in Fig. 56. The simpler or reciprocal relations are
shown by the connecting lines drawn above the functions.
The reciprocal equations and the formulas (2), (3) and (4) are
sufficient to express the absolute or numerical value of any function
of any angle in terms of any other function of that angle. The
algebraic sign to be given the result must be properly selected in
each case according to the quadrant in which the angle lies.
Exercises
All angles in the following exercises are supposed to be less than
ninety degrees.
1. Sin e = 1/5. Find cos 6 and tan 6,
Draw a right triangle whose hypotenuse is 5 and whose altitude is
1 so that the base coincides with OX. In other words, make o = 5
and y = 1 in Fig. 57. Calculate x = V25 — 1 = 2V^ and write
down all of the functions from their definitions.
O X A
Fig. 57. — Triangle of Reference for 6 and Complement of 0.
2. Cos e = 1/3. Find esc e.
Take o = 3 and a; = 1 in Fig. 57. Find y and then write down the
functions from their definitions.
3. Tan d = 2. Find sin e.
Take a; = 1 and y = 2 in Fig. 57, and calculate a and then write
down the functions from their definitions.
4. Sec e = 10. Find esc 0.
Take o = 10 and x = \ and compute y.
5. Find the values of all functions of d if cot ^ = 1.5.
6. Find the functions of ^ if cos 6 — 0.1.
7. Find the values of each of the remaining circular functions in
each of the following cases :
no KLKMENTARY MATHEMATICAL ANALYSIS [§53
ig) tan ^ = m.
Qi) sin e •■
a
Va« +6«
(a) »in 0 - 6/13. (d) tan ^ = 3/4.
(6) COM 0 - 4/5. (c) sec d = 2.
(fi) HOC 0 - 1.26. (/) tan B = 1/3.
Hliow that the following equalities are correct:
8. Tan 9coh B » sin B.
9. Sin doot d'Hco ^ = 1.
10. (Sin » -V cos d)» = 2Hin dcos 0 + 1.
11. Tan 0 -j- cot B = sec dcsc 0.
12. ICxprrHH each trigonometric function in terms of each of the
othorHj i.e., fill in all blank spaces in the following table:
\^\\\
tM«
tan
W!
*i^*
sin
COS '
tan j
cot
sec
CSC
sin
1
CSC
Cl>3
1
sec
tan
1
cot
■ '
1
tan
cot
■
1
COS
sec
\"^*
\
CSC
t"^" :\>JV^wuvp c.\vro4«t? w^MT :o Ar^&o$ <360* <>f any quadrant:
13^ If $;n ^ -« - ^ 4 And i;ju:i f ^^ ^\$;:)iv\^, find Uie remaining five
I4s Vr x\>E!i ^ - t» 1^ Ar,d $;r, f i? i^^iai*JT^, find the remaining
Ilk V? ^Ai"- f - \S a:n^ A\jt f ^ r««iiw. fed the ranttniing f unc-
Hk Vr *xxs^ ^ - - * <;? *sixX :j^, ^ is; rvta::v!f. fed the lemaining
54] THE CIRCLE AND THE CIRCULAR FUNCTIONS 111
(.'k,h) Ps
Pi ik.k)
P(h,k)
17. If tan ^ = 5/12 and sec d is negative, find the remaining
unctions of 0,
18. If sin ^ = 3/5 and tan d is negative, find the remaining f unc-
ions of 0,
64. Functions of Comple-
lentary Angles. Complementary
ngles are defined as two angles
irhosesumisOO®. Supplementary
ng^es are two angles whose sum
3 180**.
Let 6 be an angle of the first
[uadrant, and draw the angle
90**— 6) of terminal side OPi, as
hown in Fig. 58. Let P and Pi
ie on a circle of Tadius a. Let
,he coordinates of the point P be
h, k), then Pi is the point (fc, h),
aence PiDJOPi = h/a =
(k,'h)
Pi C*.-A)
FiQ. 58. — Triangles of Reference
for 6 , and 0 combined with an
dn (90® — 6), But from the tri- Odd Number of Right Angles.
ingle PDO, h /a = cos 6, Hence
Likewise,
sin (90° — 6) = cos 6
tan (90° - 6) = cot 6
sec (90° — ^) = CSC 6
These relations explain the meaning of the words cosine, cotangent,
zosecant, which are merely abbreviations for complements sine,
zomplemenVs tangent, etc. Collectively, cosine, cotangent, cosecant
are called the co-functions. Likewise from Fig. 58:
cos (90'
cot (90*
CSC (90'
e)
e)
sin 6
tanO
sec 6
Later it will be shown that the above relations hold for all
values of 6, positive, or negative.
66. Graph of the Sine and Cosine. In rectangular coordinates
we can think of the ordinate t/ of a point as depending for its value
upon the abscissa or x of that point by means of the equation y =
sin X, provided we think of each value of the abscissa laid off on
112 ELEMENTARY MATHEMATICAL ANALYSIS [(55
the X-axis as standing for some amount of angular magnitude.
Therefore the equation !/=ein x must poBsees a graph in reotai^-
lar coSrdinates. In order to produce the graph ofy = sinzittiB
best to lay off the angular measure x on the X-axis in such amannef
that it may conveniently be thought of in either radian or degree
measure. If we suppose that a scale of inches and tenths is in the
hands of the reader and that a graph is required upon an ordinary
sheet of unruled paper of letter size (84 X 11 inches), then it wiU
be convenient to let 1 /5 inch of the horizontal scale of the X-azia
correspond to 10° or to t/18 radians of angular measure. To
Fia. 59. — Construotion of the Sinusoid.
accomplish this.the length of one radian must be 1.16 inches (lA
18 /Stt inch) , which length must be used for the radius of the cirde
on which the arcs of the angles are laid off. Hence, to graph
y = sin X, draw at the left of a sheet of (unruled) drawing paper a
circle of radius 1.15 inches, as the circle OPB, Fig. 59. Take 0 U
the origin and prolong the radius BO for the positive portion OJ of
the X-axis. Subdivide this into 1 /5-inch intervals, each corre-
sponding to 10° of angle; eighteen of these correspond to tiie
length X, if the radius BO (1.15 inches) be the unit of measure.
I^ext divide the F-axis proportionately to sin x in the f<^offiiig
maimer: Divide the semicircle into eighteen equal divisions u
shown in the figure, thus making the length of each small »rc
exactly 1/5 inch. The perpendiculars, or ordinatea, dropped
upon OX from each point of division, divided by the radius a,
are the sines of the respective angles. Draw lines parallel to
OX through each point of division of this circle. These cut the
F-axis at points Ai, .Ji, . . ., such that OjIi, OAj, . . . sn
S] THE CIRCLE AND THE CIRCULAR FUNCTIONS 113
roportional to sin OBPi, sin OBP2, sin OBP^y . . . or in the
ineral case, proportional to sin x (for lack of room only a few
■ the successive points Pi, P2, Ps, • • • , of division of the
Liadrant OP^P^, are actually lettered in Fig. 59). These are the
iccessive ordinates corresponding to the abscissas already
iid off on OL, The curve is then constructed as follows:
irst draw vertical lines through the points of division of OX;
aese, with the horizontal lines already drawn, divide the
lane into a large number of rectangles. Starting at 0 and
ketching the diagonals (curved to fit the alignment of the points)
f successive "cornering'' rectangles, the curve OCNTL is approxi-
nated, which is the graph oiy = sin x. This curve is called the
inusoid or sine curve. The curve is of very great importance for
t is found to be the type form of the fundamental waves of science,
!uch as sound waves, vibrations of wires, rods, plates and bridge
nembers, tidal waves in the ocean, and ripples on a water surface.
?he ordinary progressive waves of the sea are, however, not of
his shape. Using terms borrowed from the language of waves, we
lay call C the crest, N the node, and Tthe trough of the sinusoid.
It is obvious that as x increases beyond 27r'^, the curve is re-
peated, and that the pattern OCNTL is repeated again and again
»oth to the left and the right of the diagram as drawn. Thus it is
sen that the sine is a periodic function of period 27r^, or 360°.
The small rectangles lying along the X-axis are nearly squares.
'hey would be exactly equilateral if the straight line OAi was equal
3 the arc OPi. This equahty is approached as near as we please
3 the number of corresponding divisions of the circle and of OX is
idefinitely increased. In this way we arrive at the notion of the
lope of a curve in mathematics. In this case we say that the
lope of the sinusoid at 0 is + 1 and at JV is — 1, and at L is + 1.
^e say that the curve cuts the axis at an angle of 45° at 0 and
b an angle of 315° (or, — 45° if we prefer) at JV. The slope at C
ad at T is zero.
The curve y = a sin a; is made from y = 8m x by multiplying
11 of the ordinates of the latter by a. The number a is called
tie amplitude of the s^'nusoid.
56. Cosine Curve. If 0' be taken as the origin, the curve CNTL
I the graph of y = cos x. Let the student demonstrate this by
8
114 ELEMENTARY MATHEMATICAL ANALYSIS l§57
showing that the distances BDi, BD2, . . . , BD4 . . . in the
semicircle at the left of Fig. 59 go through in reverse order the
same sequence of values as PiDi, PiD2, . . . , and that if the
origin be taken at 0\ the successive ordinates of the sinusoid to the
right of O'C are equal to BDj, BZ)2, . . . respectively, and hence
are proportional to cos x.
It is best to carry out the construction of the sinusoid upon
unruled drawing paper as described above. The curve can readily
be drawn, however, upon form M2, which is already ruled in
1 /5-inch intervals, or upon form Ml if the radius of the circle be
taken as 2.3 cm. and if 2/5 cm. be used on OX to represent
an angle of 10°. A much neater result is obtained when
unruled paper is used for the drawing.
67. Complementary Angles. The graph 7/2 = sin ( — x) is
made from ?/i = sin x by substituting (— x) for a: in the function
} V- sin X y^y* sin (-«)
Fig. CO. — Shows the Relation Between y = sin z and y — sin ( — x) and
Between y = sin (90° — z) and y = cos x, etc.
sin x\ that is, by changing the signs or reversing the direction of
all of the abscissas qf the sinusoid 2/ = sin x; or, in other words,
2/2 = sin(— x) is the reflection of i/i = sinx in the F-axis.
This is merely a special case of the general Theorem I on Loci,
§24. The former curve has a crest where the latter has
a trough and vice versa, as is shown by the dotted and full
curves in Fig. 60. Now, if the curve 1/2 = sin ( — x) (the dotted
curve in Fig. 60) be translated to the right the distance ir/2,
the resulting locus is the cosine curve y = cos x. To translate
2/2 = sin (— x) to the right the. distance 7r/2, the constant ir/2
must be subtracted from the variable x in the equation of the
curve, as already learned in tlie last chapter. Performing this
operation we liave, for the translated curve,
?/2 = sin (- [x- 2])
§58] THE CIRCLE AND THE CIRCULAR FUNCTIONS 115
(Note that 7r/2 is subtracted from x and not from — x.) Or,
removing the brackets,
2/2 = sm
(i-)
sm
But, as stated above, the curve in its new position is the same as
the cosine curve
y = cos a:
Hence, for all values of x:
{^ -x^ =cosx (1)
In the same manner it can be proved that cos («— ^) = sina:,
and the other results of §64 follow for all values of x.
58. Trigonometric Fmictions of Negative Arguments. First
compare the curves yi = rni x and 1/2 = sin (— x) as has been
done in the preceding section, and as is illustrated by Fig. 60.
The curve y^ = sin (— x) was described as the reflection of the
sinusoid yi = sin x in the F-axis. It is obvious from the figure,
however, that the dotted curve may also be regarded as the
reflection of the original curve in the X-axis; for the one has a
crest where the other has a trough and the ordinates of the two
curves are everywhere of exactly equal length but opposite in
direction. This means that t/2 = — 2/i, or,
sin (— x) = — sin X (1)
for all values of x.
If the origin be taken at the point 0', Fig. 60, the full curve
is the graph of y = cos x. In this case the crest of the curve lies
above the origin and the curve is symmetrical with respect to the
K-axis. This means that changing x to ( — x) in the equation
1/ = cos X does not modify the locus. Hence we conclude that
cos (— x) = cosx (2)
' for all values of x. Hence by division
tan (— x) = — tan x (3)
59. Odd and Even Functions. A function that changes sign
but retains the same numerical value when the sign of the variable
is changed is called an odd function. Thus sin x is an odd function
of Xj since sin (— x) = — sin x. Likewise x' is an odd function
116 ELEMENTARY MATHEMATICAL ANALYSIS [§59
of Xf as aro all odd powers of x. Geometrically, the graph of an
odd function of x is symmetrical with respect to the origin 0;
that is, if P is a point on the curve, then if the line OP be pro-
duced backward through 0 a distance equal to OP to a point
P\ then P' lies also on the curve. The branches of y = x* k
the first and third quadrants arc good illustrations of this
property.
A function of x that remains unaltered (both in sign and
nuinoric^al value) when the variable is changed in sign, is called
an even ftmction of x. Examples are cos x, x^, x^ — 3a;*, . . .
Most functions are neither odd nor even, but mixed, like
x^ + sin Xf x' + x', a; + cos x, . . .
Exercises
Bin 2
1. Show from (1) and (2) §68 and the relations esc x = - — '
» (an X, etc., that
co« X * *
1
2
{a) t'sc ( — x) = — esc X
{h) see ( — j) = sec x
(r) tan ( — j) = — tan x
(d) cot ( — x) = — cot X.
!• Is sin* X an inid or an even function of x? Is tan' x an odd or aa
even function of x?
3* Is the function sin x -f - tan x an odd or an even function? h
sin X \ \\\Si X an ixid or an even function of x?
ea The Defining Equations cleared of Fractions. The student
should iHMumit to momorj- the equations defining the trigonometMjt
functions uhtrn ckarK\i of fractions. In this form the equations
an^ quite juj useful a;s t-ho original ratios. They are written:
y = asinH z = y cot 6
X - a cos H a = X sec 6
y ^ X tan t^ & = 7 esc 6
As appliixl to tb,o rii:h: anijlevl triangle, they may be stated in
wv^rvts as folW*^:
yM\tT ;,Y ,^/,) H",-*.; ?r;\:':s:.V !\< ;V%.:J ?o thf hypoimvse muUipM
>^ jW ,^*50 s/ trc H\r :v<R><\ ,'" >j J.v ,Vv*5*5^ o/ the adHaterU^ angU*
•V«f ^*^ ^<V .''/ ,5 "^V^^ ?^"s:*^3^V sV ^As:-' V t^,f cihtr leg iiiiitttp{M5jf A«
^^^5,^^J v/ J^«* /t «\\w\v. /- S ;.v A>fs:.%^Y^; of the adjacent, angl^
i
\
»1] THE CIRCLE AND THE CIRCULAR FUNCTIONS 117
The hypotenuse of a right triangle is equal to either leg mvUiplied
I the secant of the angle adjacent, or by the cosecant of the angle
:>posit€ that leg.
These statements should be committed to memory.
61. Projections. In Fig. 52 the projection of OP in any of its
ositions, such as OPi, OP2, OP 3, . . . , is ODi, OD2, ODz, . • . ,
r is the abscissa of the point P. Thus for all positions:
X = a cos d
ihe sign of x gives the sign, or sense, of the projection. In each
sase 6 is said to be the angle of projection.
The above definition of projection is more general in one
espect than that discussed in §28. By the present definition
he projection of a line is negative if 90° < ^ < 270® (read,
'if d is greater than 90° but is less than 270°"). This con-
sept is important and essential in expressing a component of a
lisplacement, of a velocity, of an acceleration, or of a force.
The cosine of 6 might have been defined as that proper fraction
►y which it is necessary to multiply the length of a line in order to
►reduce the projection of the line on a Hne making an angle 6
nth it.
Exercises
1. A stretched guy rope makes an angle of 60® with the horizontal.
Vhat is the projection of the rope on a horizontal plane? What is
be projection of the rope on a vertical plane?
2. Find the lengths of the projections of the line through the origin
nd the point (1, \/3) upon the OX and OY axes, if the line is 12
iches long.
3. A force equals 200 dynes. What is its component (projection)
n a line making an angle of 135® with the force? On a Hne making
n angle of 120® with the force?
4. A velocity of 20 feet per second is represented as the diagonal
f a rectangle the longer side of which makes an angle of 30® with the
iagohal. Find the components of the velocity along each side of the
Bctangle.
6. Show that the projections of a fixed line OA upon all other
nes drawn through the point 0 are chords of a circle of diameter OA.
\ee Fig. 63.
6. Find the projection of the side of a regular hexagon upon the
hree diagonals passing through one end of the given side, if the
Fig. 61. — Polar Coordinates.
118 ELEMENTARY MATHEMATICAL ANALYSIS [|62
numerical value of cos 30° = 0.87, and if each side of the hexagon
is 20 feet.
62. Polar Co5rdmates. In Fig. 61, the position of the point
P may be assigned either by giving the x and y of the rectangular
coordinate system, or by giving the vectorial angle 6 and the
distance OP measured along the terminal side of 0. Unlike
the distance a used in the preceding work, it is found conven-
ient to give the line OP a sense or direction as well as length;
such a line is called a vector. In the present case, it is known as
the radius vector of the point P,
and it is usually symbolized by
the letter p. The vectorial or
direction angle 0 and the radius
P vector p are together called the
polar codrdinates of the point?,
and the method, as a whole, is
known as the system of polar
coordinates. In Fig. 61 the
point P' is located by turning
from the fimdamental direction
OX, called the polar axis, through
an angle 6 and then stepping
backward the distance p to the
point P'; this is, then, the point {—p, 6), P' has also the coordi-
nates (p, ^2), in which ^2 = ^ + 180°; likewise Pi is (+ p', ^1) and
P'l is (~ p', ^1). Thus each point may be located in the polar
system of coordinates in two ways, i.e., with either a positive or a
negative radius vector. If negative values of 6 be used, there
are four ways of locating a point without using values of ^ >
360°. In giving a point in polar coordinates, it is usual to name
the radius vector first and then the vectorial angle; thus (5, 40°)
means the point of radius vector 5 and vectorial angle 40°.
63. Polar Co5rdinate Paper. Polar coordinate paper (form AfS)
is prepared for the construction of loci in the polar system. A re-
duced copy of a sheet of such paper is shown in Fig. 62. This
plate is graduated in degrees, but a scale of radian measure is given
in the margin. The radii proceeding from the pole 0 meet all of
the circles at right angles, just as the two systems of straight lines
i63] THE CIRCLE AND THE CIRCULAR FUNCTIONS 119
meet each other at right angles in rectangular coordinate paper.
For this reaaon, both the rectangular and the polar systema are
called orthogonal BysteniE of coordinates.
We have learned that the fundamental notion of a function
implies a table of corresponding values for two variables, one called
the argument and the other the function. The notion of a graph
i
^
^gg^
:
Jlffff^p
^SnTiTTl"
^m
^mw
:
^SSflp
rPc>^X//
Fra. 62. — Polar CoOradioate Squared Paper. (From Jtf3.)
implies any sort of a scheme for a pictorial representation of this
tabie of values. Therearethree common methods in use: thedouble
scale, the rectangular coordinate paper, and the polar paper. The
polar paper is most convenient in case the argument is an angle
measured in degrees or in radians. Since in a table of values for a
functional relation we need to consider both positive and negative
valuee for both the argument and the function, it is necessary to
use on the polar paper the convention already explained. The
argument, which is the angle, is measured counter-clockwise if
positive and clockwise if negative from the tine numbered 0°,
120 ELEMENTARY MATHEMATICAL ANALYSLS [§6-1
Fig. 62. The function is measured outward from the center along
the terminal side of the angle for positive functional values and
outward from the center along the terminal side of the angle
produced backward through the center for negative functional values.
In this scheme it appears that four different pairs of values are
represented by the same point. This is made clear by the points
plotted in the figure. The points Pi, P2, P3, Pa are as follows:
Pi : (6.0,40°); (6.0, - 320°); (- 6.0,220°); (- 6.0, - 140°).
P2 : (10, 135°); (10, - 225°); (- 10, 315°); (- 10,- 45°).
Pz : (5, 230°); (5, - 130°); (- 5, 50°); (- 5, - 310°).
P4 : (6.0,330°); (6.0, - 30°); (- 6.0, 150°); (- 6.0, - 210°).
The angular scale cannot be changed, but the fimctional scale
can be changed to suit the table of values by multiplying or
dividing it by integral powers of ten.
In case the vectorial angle is given in radians, the point may be
located on the polar paper by means of a straight edge and the
marginal scale on form MS.
Exercises
1. Locate the following points on polar coordinate paper; (1, t/2);
(2, t); (3, 60°); (4, 250°); (2i, 1.8x).
2. Locate the following points: (0, 0°); (1, 10°); (2, 20°); (3, 30°);
(4,40°); (5,50°); (6, 60°) ; (7, 70°) ; . . . (36, 360°). Use 1 cm. =10
units.
3. The equation of a curve in polar coordinates is ^ = 2. Draw
the curve. The equation of a second curve is p = 3. Draw the
curve.
Notice that p = a constant is a circle with center at 0, while
^ = a constant is a straight line through 0.
4. Draw the curve p — 0 using 2 cm. as unit for p. Note that the
curve p = d is a spiral while the curve y = xia a. straight line.
64. Graphs of p = a cos d and p = a sin ^. These are two
fundamental graphs in polar coordinates. The equation
p = a cos 6 states that p is the projection of the fixed length a
upon a radial line proceeding from 0 making a direction angle S
with a, or, in other words, p in aU of its positions must be the side
adjacent to the direction angle ^ in a right triangle whose hypote-
nuse is the finite length a. (See §61.) It must be remem-
i64l THE CIRCLE AND THE CIRCULAR FUNCTIONS 121
bered that the direction angle 6 is always measured from the fixed
direction OA. Hence, to construct the locus p = a cos 6, draw
aa many radii vectores as desired, asin Fig. 63. Project on each
of these the fixed distance OA or a. This gives OP, orp, in numer-
ous positions as shown in the diagram. Since P is by construction
the foot of the perpendicular dropped from A upon OP, it is always
at the vertex of a right triangle standing on the fixed hypotenuse a,
and therefore the point P is on the semicircle AOP; for, from plane
geometry a right triangle is always inscribable in a semicircle.
-The Graph of p
When 0 is in the second quadrant, as 6i, Fig. 63, the cosine is
negative and consequently p is also negative. Therefore the point
Pi is located by measuring backward through 0. Since, however,
Pi is the projection of a tlirough the angle 6j (see §S1), the
angle at Pi must be a right angle. Thus the semicircle
OPtA is described as 0 sweeps the second quadrant. When
0 is in the third quadrant, as $>, the cosine is stiU negative and
p is measured backward to deacribe the semictTcle AP,0 a second
time. As 8 sweeps the fourth quadrant, the semicircle OPiA is
described the second time. Thus the graph in polar coordinates
of p — a cos 6 is a circle twice drawn as 8 varies from 0° to 360°.
Once around the circle corresponds to the distance from crest to
trough of the "wave" y = a cos x, in Fig. 59 (0' is origin).
The second time around the circle corresponds to the distance
122 ELEMENTARY MATHEMATICAL ANALYSIS
[|64
from trough to crest of the cosine curve. Trough and crest of all
the successive "wave lengths "fall at the points. The nodes are
all at 0.
The polar rcprcscnlatJoii of the cosine o! a variable by means
of the circle la more useful and important in science than the
Cartesian representation by means of the sinusoid. The ideas
here presented must be thoroughly mastered by the student.
The graph of p ^ n sin 0 is also a circle, but tiie diameter iE
the line OB makii^ an ai^e of 90° with OA, as shown in Fig. 64.
Since p = a sin 9, the radius vector,
as 9 increases to 90°, must equal
the side Ij'ii^ opposite the an^je S
in a right triangle of hypotenuse <i.
Since angle AOPi = ai^e OBPi,
the point P may be the vertex of
any right triangle erected on OB or
a as a hypotenuse. The semicircle
£PiO is described as 0 increases
from 90° to 180°. Beyond 180° the
sine is negative, so that the radius
vector p must be laid off backward
for such angles. Thus Ft is the
point corresponding to the angle 0i, of the third quadrant. As 9
sweeps the third and fourth quadrants the circle OPtBPfi is
described a second time. Therefore, the graph of p = a sin S
is the circle ttn'ce drawn of diameter a, and tangent to OX at 0.
The first time around the circle corresponds to the crest, the
second time around corresponds to the trough of the wave or
sinusoid drawn in rectangular coordinates. The points corre-
sponding to the nodes of the sinusoid are at 0 and the points
corresponding to the maximum and minimum points are at B.
We have st-en that the graph of a function in polar coordinates
is a ver\- different cur\-e from its graph in lectangular coiirdi-
nates. Thus the cosine of a variable if graphed in rect&ngular
coordinates is a sinusoid: but if graphed in pcdar codrdi-
nat(« the graph is a circle (.twice drawnV Thoe is in this caw
a ver}.- great difference in the ease with which these curvea can be
const ructeil: the sinusoid requires an elabonte method, while the
Fra. 64.— The Graph a( „ = a
§65] THE CIRCLE AND THE CIRCULAR FUNCTIONS 123
circle may be drawn at once with compasses. This is one reason
why the periodic or sinusoidal relation is preferably represented
in the natural sciences by polar coordinates.
66. Graphical Table of Sines and Cosines. The polar graphs
of p = a sin ^ and p = a cos 6 furnish the best means of construct-
ing graphical tables of sines and cosines. The two circles passing
through 0 shown on the polar coordinate paper, form M3, Fig. 62,
are drawn for this purpose. A quantity of this coordinate paper
should be in the hands of the student. If the diameter of the
sine and cosine circles be called 1, then the radius vector of any
point on the lower circle is the cosine of the vectorial angle, and
the radius vector of the corresponding point on the upper circle
is the sine of the vectorial angle. As there are 50 concentric cir-
cles in Form MS, it is easy to read the radius vector of a point
to 1 /lOO of the unit. Thus, from the diagram, we read cos
45° = 0.70 ; cos 60° = 0.50 ; cos 30° = 0.866. These results are
nearly correct to the third place.
66. Graphical Table of Tangents and Secants. Referring to
Fig. 62, it is obvious that the numerical values of the tangents of
angles can be read off by use of the uniform scale of centimeters
bordering the polar paper (form M3). The scale referred to
lies just inside of the scale of radian measure, and is numbered
0, 2, 4, . . . , at the right of Fig. 62. Thus to get the numerical
value of tan 40° it is merely necessary to call unity the side OA
of the triangle of reference OAP, and then read the side AP = 0.84;
hence tan 40° = 0.84. To the same scale (i.e., OA = 1) the dis-
tance OP = 1.31, but this is the secant of the angle AOP, whence
sec 40° = 1.31. By use of the circles we find sin 40° = 0.64 and
cos 40° = 0.76.
In case we are given an angle greater than 45° (but less than
135°) use the horizontal scale through B, Starting from B as
zero the distance measured on the horizontal scale is the cotangent
of the given angle. The tangent is found by taking the reciprocal
of the cotangent.
Exercises
Find the unknown sides and angles in the following right triangles.
The numerical values of the trigonometric functions are to be taken
i
124 ELEMEN TARY MATHEMATICAL ANALYSIS [§66
from the polar paper. The vertices of the triangles are supposed
to be lettered Ay Bj C with C at the vertex of the right angle. The
small letters a, h, c represent the sides opposite the angles of the same
name.
By angle of elevation is meant the angle between a horizontal Une
and a line to the object, both drawn from the point of observation,
when the object lies above the horizontal line. The similar angle
when the object Ues below the observer is called the angle of depression
of the object.
The solution of each of the following problems must be checked.
The easiest check is to draw the triangles accurately to scale on form
Ml and use a protractor.
1. When the altitude of the sun is 40°, the length of the shadow cast
by a flag pole on a horizontal plane is 90 feet. Find the height of the
pole.
Outline of Solution. Call height of pole a, and length of shadow 6.
Then A == 40° and B = 50°. Hence:
o = 6 tan 40°
Determining the numerical value of the tangent from the polar paper,
we find:
a = 90 X 0.84 = 75.6 ft.
which result, if checked, is the height of the pole. To check, either
draw a figure to scale, or compute the hypotenuse c, thus;
c = 90 sec 40°
From the polar paper find sec 40°. Then:
c = 90 X 1.31 = 117.9
Since a^ + 6^ = c^, we have c^ - 6^ = ««, or (c - 6) (c + 6) = a*.
Hence if the result found be correct,
(117.9 - 90) (117.9 + 90) = 75.62
5800 = 5715
These results show that the work is correct to about three figures, for
the sides of the triangle are proportional to the square roots of the
numbers last given.
2. At a point 200 feet from, and on a level with, the base of a tower
the angle of elevation of the top of the tower is observed to be 60°.
What is the height of the tower?
3. A ladder 40 feet long stands against a building with the foot of
the ladder 15 feet from the base of the wall. How high does the
ladder reach on the wall?
f •
§66] THE CIRCLE AND THE CIRCULAR FUNCTIONS 125
4. Prom the top of a vertical cliff the angle of depression of a point
on the shore 150 feet from the base of the cliff is observed to be 30°.
Find the height of the cliff.
5. In walking half a mile up a hill, a man rises 300 feet. Find the
angle at which the hill slopes.
If the hill does not slope uniformly the result is the average slope
of the hill.
6. A line 3.5 inches long makes an angle of 35° with OX. Find the
lengths of its projections upon both OX and OY.
7. A vertical cliff is 425 feet high. From the' top of the cliff the
angle of depression of a boat at sea is 16°. How far is the boat
from the foot of the cliff?
8. The projection of a line on OX is 7.5 inches, and its projection
on OF is 1.25 inches. Find the length of the line, and the angle
it makes with OX.
9. A battery is placed on a cliff 5 10 feet high. The angle of depres-
sion of a floating target at sea is 9°. Find the range, or the distance
of the target from the battery.
10. From a point A the angle of elevation of the top of a monument
is 25°. From the point 5, 110 feet farther away from the base of the
monument and in the same horizontal straight line, the angle of eleva-
tion is 15°. Find the height of the monument.
11. I^d the length of a side of a regular pentagon inscribed in a
circle whose radius is 12 feet.
12. Proceeding south on a north and south road, the direction of a
church tower, as seen from a milestone, is 41° west of south. From
the next milestone the tower is seen at an angle of 65° W. of S.
Find the shortest distance of the tower from the road.
13. A traveler's rule for determining the distance one can see from
a given height above a level surface (such as a plain or the sea) is as
follows : "To the height in feet add half the height and take the square
root. The result is the distance you can see in miles." Show that
this rule is approximately correct, assuming the earth a sphere of
radius 3960 miles. Show that the drop in 1 mile is 8 inches, and
that the water in the middle of a lake 8 miles in width stands 10} feet
higher than the water at the shores.
14* Observations of the height of a mountain were taken at A and
B on the same horizontal line and in the same vertical plane with the
top of the mountain. The elevation of the top at -4 is 52° and at J? is
36°. The distance AB is 3500 feet. Find the height of the mountain.
16. The diagonals of a rhombus are 16 and 20 feet, respectively.
Find the lengths of the sides and the angles of the rhombus. ^
126 ELEMENTARY MATHEMATICAL ANALYSIS
1867
Fig. 65. — Diagram for
Exercise 17.
16. The equation of a line is y = f x + 10. Compute the short-
est distance of this line from the origin.
17. Find the perimeter and area of ABCD, Fig. 65.
67. The Law of the Circular Functions. It will be emphasized
in this book that the fundamental laws of exact science are three in
number, namely: (1) The power function
expressed by y = ax" where n may be
either positive or negative; (2) the har-
monic or periodic law y = a sin nx, which
is fundamental to all periodically occurring
phenomena; and a third law to be dis-
cussed in a subsequent chapter. While
other important laws and functions arise
in the exact sciences, they are secondary
to those expressed by the three funda-
mental relations.
We have stated the law of the power
function in the following words (see §34) :
In any power function, if x change by a fixed multiple j y is
changed by a fixed multiple also. In other words, if x change by
a constant factor, y will change by a constant factor also.
Confining our attention to the fundamental functions, sine
and cosine, in terms of which the other circular functions can
be expressed, we may state their law as follows:^
The circular functions, sin 6 and cos d, change periodically in
value proportionally to the periodic change in the ordinate and
abscissa, respectively, of a point moving uniformly on the circle
X2 + 1/2 = a2.
The use of the periodic law in natural science is, of course,
very different from that of the power function. The student will
find that circular functions similar toy = a sin nx wiU be required
in order to express properly any phenomena which are recurrent
or periodic in character, such as the motion of vibrating bodies,
all forms of wave motion, such as sound waves, light waves, electric
waves, alternating currents and waves on water surfaces, etc.
Almost every part of a machine, no matter how complicated its
motions, repeats the original positions of all of the parts at
^ Chapter X is devoted to a discussion of these fundamental periodic laws.
§e8l THE CIRCLE AND THE CIRCULAR FUNCTIONS 127
stated intervals and these recurrent positions are expressible in
terms of the circular functions and not otherwise. The student
will obtain a most limited and unprofitable idea of the use of the
circular fimctions if he deems that their principal use is in numer-
ical work in solving triangles, etc. The importance of the
circular functions lies in the power they possess of expressing
natural laws of a periodic character.
68. Rotation of Any Locus. In §36 we have shown that
any locus y = f{x) is translated a distance a in the x direction by
substituting {x — a) for x in the equation of the locus. Likewise
the substitution of {y — h) for y was found to translate the locus
the distance h in the y direction. A discussion of the rotation of
a locus was not considered at that place, because a displacement
of this type is best brought about when the equations are expressed
in polar coordinates.
If a table of values be prepared for each of the loci
p = cos ^ (1)
p = cos (^1 - 30°) (2)
as follows:
e i 0° 30** 60° 90° 120° 150° 180° .
1 ^3" 1/2 0 -1/2 - §\/3 -1
^1 I 30** 60° 90° 120° 150° 180°
1 Wi 1/2 0 -1/2 - W^ .
and then if the graph of each be drawn, it will be seen that the
curves differ only in their location and not at all in shape or size.
The reason for this is obvious: The same value of p is given by
^1 = 90** in the second case as is given by ^ = 60° in the first
case, and the same value of p is given by ^i = 60° in the second
case as is given by ^ = 30° in the first case, etc. The sets
of values of p in the two cases are identical, but like values corre-
spond to vectorial angles B differing by 30°. In more general
terms the reasoning is that if (^i — 30°) be substituted for 0 in any
polar equation, then since (^i — 30°) has been put equal to ^, it
follows that ^1 = (^ + 30°), or the new vectorial angle Bi is greater
than the original B by the amount 30°. Since all values of B
128
ELEMENTARY MATHEMATICAL ANALYSIS
in the new locus are increased by 30°, the new locus is the
same as the original locus rotated about O (positive rotation)
by the amount 30°.
The above reasoning does not depend upon the particul&r con-
stant angle 30° that happened to be used, but holds juat aa well
if any other constant angle, say a, be used instead. That ie,
substituting (9i — a) for 6 does not change the size or shape of
the locus, but merely rotates it through an ai^e a in the positive
sense. The same reasoning
applies also to the general
case : If p = /(ff) be the polar
equation of any locus, then p
= /(Si — «) is the equation
of the same locus turned
about the fixed ptoint 0
through the an^e a; for if
(ffi — a) be every where subili-
tuted for the vectorial angle
9, 9i must be a greater than
the old e. That is, each
point is advanced the at^ar
amount a, or turned that
0 = amount about the point 0-
The rotation is positive, or
anti-clockwiae, if a be posi-
tive— thus, substituting (ff — 30°) for S in p = o cos fl turns th«
circle p = a cos fl through 30° in the anti -clockwise sense, as is
shown in Fig. 66, but substituting (S + 30°) for ff in p = o eoe S
turns the circle ft = a cos fl through 30° in the clockwise direc-
tion of rotation, as shown in the same figure.
The four circles
p = a cos (0 -I- a) (3)
p = a cos (0 - a) (4)
p = o sin (fl -)- «) (5)
p = a sin (0 - a) (6)
are shown in Fig. 66. Each has diameter a. The student must
carefully distii^iuish between the constant angle a and the variable
§69] THE CIRCLE AND THE CIRCULAR FUNCTIONS 129
angle 6, just as he must distinguish between the constant distance
a and the variable vector p.
The above result constitutes another of the
Theorems on Loci
IX. If (6 — a) he substituted for 6 throughout the polar equation
of any locus, the curve is rotated through the angle a in the positive
sense.
Note that the substitution is (^ — a) for ^ when the required
rotation is through the positive angle a, and that the substitution
is (^ + a) for ^ when the required rotation is through the negative
angle a.
The rotation of any locus through any angle is readily accom-
plished when its equation is given in polar coordinates. Rota-
tions of 180** and 90° are very simple in rectangular coordinates.
Let the student select any point P in rectangular coordinates and
draw the radius vector OP and the abscissa and ordinate OD and
DP; then show that the substitutions x = —Xi,y= —yi will turn
OP through 180° about 0 in the plane xy, and that the substitutions
X = y^y = — ^1 will turn OP through 90° about 0 in the plane rcy.
Exercises
Draw the following circles:
1. p = 3 cos (^ - 30)°. 4. p = 2 sin (^ + 135°).
2. p = 3 cos (^ + 120°). 6. p = 4 cos (d + yj •
3. p = 2 sin (^ - 45°). 6. p - 5 sin (| - 0)) •
7, Show that p== a sin d is the locus p= a cos 0 rotated 90°
counter clockwise.
Solution: Write p= o cos (^— 90°), then
p = o cos (90°-^) by (2) §68, then p= a sin ^ by §67.
69. Polar Equation of the Straight Line. In Fig. 67 let MN be
any straight line in the plane and OT be the perpendicular dropped
upon MN from the origin 0. Let the length of OT be a and let
the direction angle of OT be a, where, for a given straight line,
a and a are constants. Let p be the radius vector of any point
9
130 ELEMENTARY MATHEMATICAL ANALYSIS
P on the line MN and let its direction angle be 6. Then, by
definition,
- = cos (6 — a)
Therefore the equation of the straight line M'N is
a ^ p cos {B — a) (1)
for it is the equation satisfied by the (p, 0) of every point of the
line. This is the equation of any straight line, for its location is
perfectly general. The
constants defining the line
are the perpendicular dis-
tance a upon the given line
from 0 and the direction
angle a of this perpendic-
ular. The perpendicular
or or a is called the nor-
mal to the line MN and
the equation (1) is called
the normal equation of the
straight line.
* The equation of the cir-
cle shown in the figure is
Pi = a cos {$ — a) (2)
in which pi represents the
radius vector of a point Pi
on the circle. From plane geometry OT or a is a mean propor-
tional between the secant OP and the chord OPi, or,
p :a = a: Pi
or,
ppi = a2 (3)
This gives the relation between the radius vector of a point on the
line and the corresponding radius vector of a point on the circle.
Now if on the radius vector p = OP, drawn from the fixed origin
0 to any curve, we lay off a length OPi = pi = — (where a is a
constant), then Pi is said to describe the inverse of the given curve
with respect to (?t In this special case the circle is the inverse of
Fig. 67. — The Circle p = a cos (^ — a)
and its Inverse, the line MN or a =
p cos (d — a).
I] THE CIRCLE AND THE CIRCULAR FUNCTIONS 131
straight line and vice versa. If a = 1 we note that OPi and
are reciprocals of each other. •
t is important in mathematics to associate the equation of the
;le and the equation of its inverse with respect to 0, or the line
gent to it. Thus
, circle
p = 10 cos Is - ^]
10 = p cos le - ^)
. straight line tangent to it.
0. Relation between Rectangular and Polar Co5rdinates.
ink of the point P whose rectangular coordinates are (a;, y).
bhe radius vector OP be called p and the direction angle be
ed Bj then the polar coordinates of P are (p, B), Then x and
or any position of P are the projections of p through the
;le B, and the angle (90° — ^), respectively, or,
X = p cos B (1)
y = p sin (9 • (2)
3se are the equations of transformation that permit us to express
equation of a curve in pplar coordinates when its equation in
bangular coordinates is known, and vice versa. Thus the
light line a; = 3 has the equation
p cos ^ = 3
>olar coordinates. The line x + y = 3 has the polar equation
p cos ^ + p sin ^ = 3.
3 circle x^ + y^ = a^ has the equation
p2cos2 ^ + p2 sin2 B = a^
p^ = a^
p = a
132 ELEMENTARY MATHEMATICAL ANALYSIS [§71
To solve equations (1) and (2) for ^, we write
• X
B = the angle whose cosine is —
P
y
0 = the angle whose sine is —
P
The verbal expression "the angle whose cosine is," etc., are
abbreviated in mathematics by the notations "cos"^," read
"anti-cosine," and "sin~^," read "anti-sine," as follows:
d = cos-i (x/p) (3)
d = sin-i (y/p) (4)
Dividing the members of (2) by the members of (1) we obtain
y
tan 6 = \J which, solved for d, we write
6 = the angle whose tangent is —
which may be abbreviated
e = tan-i (y/x) (5)
and read "^ = the anti-tangent of y/x"
The value of p in terms of x and y is readily written
p = \/x2 + y2 (6)
Exercises
1. Write in polar coordinates the equation x* -\- y* -{- Sx = 0.
The result is p^ + 8p cos ^ == 0, or p = —8 cos 6,
2. Write in polar coordinates the equations (a) a;* + y* — 4y = 0;
(h) a;2 + 2/2 - 6a; - 42/ = 0; (c) x^ + y^ - 6y =^ 4.
3. Write in polar coordinates the equations (a) x -{-y = l; (6) »+2|f
= l;(c) x+ V^2/ = 2.
4. Write in rectangular coordinates (a) p cos 0 + p sin ^ = 4; (&)
p cos ^ — 3p sin ^ = 6.
5. Write in polar coordinates x^ + 2y' — 4a; = 0.
71. Identities and Conditional Equations. It is useful to make
a distinction between equalities like
(a - x){a + x) = a^ - x^ (1)
which are true for all values of the variable x and equalities like
x^ -2x = 3 (2)
i
t
71] THE CIRCLE AND THE CIRCULAR FUNCTIONS 133
hich are true only for certain particular values of the unknown
umber. When two expressions are equal for all values of the
iriable for which the expressions are defined, the equality is
lown as an identity. When two expressions are equal only for
rtain particular values of the unknown number the equality is
loken of as a conditional equation. The fundamental
rmula
• sin* (t> + cos* <^ = 1
an identity.
2 sin ^ + 3 cos ^ = 3.55
a conditional equation. Sometimes the symbol = is used to
stinguish an identity; thus
a^ — x' = (a — x){a^ + ax + x^)
Exercises
The following exercises contain problems both in the establishment
trigonometric identities and in the finding of the values of the un-
own number from trigonometric conditional equations.
The truth of a trigonometric identity is established by reducing
ch side to the same expression. This usually requires the applica-
>n of some of the fundamental identities, equations (1) to (5),
3. Facility in the establishment of trigonometric identities is
•gely a matter of skill in recognizing the fundamental forms and of
jenuity in performing transformations. In verifying the identity
two trigonometic expressions it is best to reduce each exp ression
aarately to its simplest form. Unless the student writes the
>rk in two separate columns, transforming the left member alone
one column, and the right member alone in the other column,
is very liable to get erroneous results. All results should be
zcked. The following worked exercises will aid the student.
(a) Prove that
(1 — sin u cos u) (sin u + cos u) = sin' u + cos' u
le sum of two cubes is divisible by the sum of the numbers them-
ves so that after division we have:
1 — sin w cos u = sin^ w — sin w cos u + cos^ u
loe sin* u + cos* w = 1, this equation is true and the original iden-
y is established.
;b) Show that
sec* X — 1 = sec* x sin* x
134 ELEMENTARY MATHEMATICAL ANALYSIS [|71
Substituting sec' x s on the right side
cos' a;
sin' a; ^ .
sec' X — \ = — = tan' x
cos'x
or sec' a; s 1 4-tan' x
which is a fundamental identity.
Solutions to exercises in trigonometric conditional equations similar
to exercises 1, 4, 5, 9 below must be checked. The necessity for a
check is made apparent by the following illustration:
(c) Solve for all angles less than 360°
2 sin a; + cos a; = 2 (1)
Transposing and squaring we get:
cos' X =4 — 8sina; + 4 sin' x (2)
since sin' x + cos' x = 1.
1 — sin' X =4 — 8sina; + 4 sin' x (3)
5 sin' rr - 8 sm x + 3 = 0 (4)
sin x = 1, or 0.6 (5)
X = 90°, 37°, or 143° (6)
Check: 2 sm 90° + cos 90° = 2 + 0 = 2 (7)
Check: 2 sin 37° + cos 37° = 1.2 + 0.8 = 2 (8)
Does 2 sin 143° + cos 143° = 1.2 - 0.8 = 0.4 = 2? (9)
The last value does not check. The reasons for this will be dis-
cussed later in §§93 and 94. Therefore the correct solutions are
90° and 37°.
1. Solve for all values of ^ < 90° : 6 cos' ^ + 5 sin ^ = 7.
Suggestion: Write 6(1 — sin' ^) + 5 sin ^ = 7 and solve the
quadratic in sin B.
6 sin' ^-5 sin ^+1=0
or,
(3 sin ^ - 1 )(2sin ^ - 1) = 0
sin ^ = 1/3 or 1/2
e = 19° or 30°.
The results should be checked.
2. Prove that for all values of B (except 7r/2 and 3ir/2, for which
the expressions are not defined)
sec* e — tan* Q s tan' B + sec' B.
§71] THE CIRCLE AND THE CIRCULAR FUNCTIONS 135
3. Show that
sec^ u — sin* u e tan* u + cos* u,
for all values of the variable u except 90** and 270% for which the
expressions are not defined.
4. Find u, if
tan u + cot u = 2.
5. Find sec d, if
2 cos ^ + sin ^ = 2.
6. Find the distance of the end of the diameter of
p == 8 cos (^ - 60°)
from the line OX,
7. If pi = a cos Of and P2 = a sin d, find pi — P2 when 0 = 60°
and o = 5.
8. Find the polar equation of the circle x* + 2/* + 6a; ~ 0.
9. For what value of d does p = 3.55, if p = 2 sin ^ + 3 cos 6?
Result: $ = 23° 30' and 43° 30'.
10. Prove that
sin A _ 1 + cos A
1 — cos A sin A
11. Prove that 2 cos* w — 1 s cos* u — sin* u.
12. Prove that
sec u + tan u =
sec u — tan
13. Prove that sec* u + esc* u s esc* u sec* u.
14. Show that (tan a + cot a)* s sec* a esc* a.
Va* + 6*
15. Find sin ^ if esc ^ =
a
16. A circular arc is 4.81 inches long. The radius is 12 inches.
What angle is subtended by the arc at the center? Give result in
radians and in degrees.
17. Certain lake shore lots are bounded by north and south lines
66 feet apart. How many feet of lake shore to each lot if the shore-
line is straight and runs 77° 30' E. of N.?
18. If y = 2 sin A + 3 cos A - 3.55, take A as 20°; as 23°; as 26°.
Find in each case the value of y. From the values of y just found
approximate the value of A for which y is just zero. This process is
known as "cut and try."
136 ELEMENTARY MATHEMATICAL ANALYSIS [§71 I
19. The line y = (3/2) x is to coincide with the diameter of the \
circle :
p = 10 cos {$ — a)
Find a.
20. The line y — 2xia to coincide with the diameter of the circle :
p = 10 sin (e + a)
Find a.
21. To measure the width of the slide dovetail shown in Fig. 68,
two carefully ground cylindrical gauges of standard dimensions are
placed in the F's at A and 5, as shown, and the distance X carefully
Fig. 68. — Diagram to Exercise 21.
taken with a micrometer. The angle of the dovetail is 60°. Find
the reading of the micrometer when the piece is planed to the required
dimension MN = 4 inches. Also find the distance Y, (Adapted
from "Machinery," N. Y.)
22. Show that:
p— sin d -\- cos d
is a circle.
23 Draw the curve:
24 Sketch
and
and then
and discuss.
y = sin X -\- cos x.
X
2/ = sin X
X
y = 2 + sinx
CHAPTER IV
THE ELLIPSE AND HYPERBOLA
72. The Ellipse. If all ordinates of a circle be shortened by
the same fractional amount of their length, the resulting curve
is called an ellipse. For example, in Fig. 69, the middle points
of the positive and negative ordinates of the circle were marked
and a curve drawn through the points so selected. The result
is the ellipse ABA'B'A.
If
x= + j,= = a' (1)
is the equation of a circle, then
X' + {myY = o' (2)
in which m is any constant > 1 , is the equation of an ellipse; for
substituting my for y divides all
of the ordinates by m. by Theo-
rem V on Loci, |27. The ellipse
may also be looked upon as the
orthographic projection of the
circle. See §28.
It is easy to show, as a con-
sequence of the above, that the
shadow cast on the floor by a
circular disk held at any angle
in the path of vertical rays of
light is an ellipse.
The curve made by elongating ro. 69.— Definition of the Ellipse,
by the same fractional amount
ot their lei^h all of the abscissas or ordinates of a circle is also
ui ellipse, as the following considerations will show.
First let the ordinates of the circle (1) be shortened as before.
The reeult is
x' + (my)' = o' (2)
137
c
/i ^ r\
c
138 ELEMENTARY MATHEMATICAL ANALYSIS [|72
If the abscissas of the same given circle be multiplied by m
to make another curve, the result is
' + 2/2 = a* (3)
ii)
where m is supposed to be > 1 in both cases. If equation (3)
be multiplied through by w* we get:
x^ + imyy = a^w} (4)
This shows that the second curve can be made by dividing by m
all of the ordinates of a circle of radius ma. That is, (3). is an
ellipse made from a circle of radius ma in the same manner
that the ellipse (2) is made from a circle of radius a. Hence (3)
is an ellipse whose dimensions are m-fold those of (2).
Thus an elhpse results if all of the ordinates or if all of the ab-
scissas of a circle be multiphed or divided by any given constant m.
It is usual to write the multiplier m in the form a/6, so that
equation (1) may be written:
x^-\- {ay /by = a*
or:
x'/a' + y'/'b*=l (5)
which is the equation of the ellipse in a symmetrical farm. Apply-
ing the principles of §27, the locus (5) may be thought of as
made from the unit circle x* -h !/* = 1 by multiplying its abscissas
by a and its ordinates by 6.
When written: '
y = ±{b /a) Vo» - x« (6)
y = ± Va^ - x» (7)
the ellipse and circle are placed in a form most useful for many
purposes. It is easy to see that (6) states that its ordinates are
the fractional amount b/a of those of the circle (7).
In Fig. 69 the points A and A' are called the vertices and the
point 0 is called the center of the ellipse. The line A A' is called
the major axis and the line BB' is called the minor axis. It is
ob^dous that A A' = 2a, and from equation (5) or (6) it follows
BE' = 26.
The definition of the term function permits us to speak of y as a
function of x, or of x as a function of y, in cases like equation (5)
THE ELLIPSE AND HYPERBOLA
139
above; for when i is given, y ia detennined. To distir^uish thia
from the case in which the equation is solved for y, as in (6), y, in
the former case, is said to be an implicit function of x, and in the
latter case y Is said to be an explicit function of x.
If a circular cylinder be cut by a plane, the section of the
cylinder is an ellipse. For select any diameter of a cir-
cular section of the cylinder aa the x-axis. Let a plane be passed
through this diameter making an ai^le a with the circular section.
Then if ordinates (or chords perpendicular to the common x-axis)
be drawn in each of the two planes, all ordinates of the section
made by the cutting plane can
be made from the ordinates of
the circular section by multiply-
Hence any
if a cylinder is an
ing them by s
plane section o
ellipse.
73. To Draw the EUipse. A '
method of drawing the ellipse is
shown in Fig. 70. Draw con-
centric circles of radii a and b
respectively, a > b. Draw any
number of radii and from their
intersections with the larger
circle draw vertical linea, and
from their intersections with ,the smaller circles draw horizontal
hnes. The points of intersection of the corresponding horizontal
and vertical lines are points of the ellipse.
Proof. In the figure, let P be one of the points just described.
Then:
P^, : P,D = P^ : P,0
or, substituting PD for the equal PjDs
PD : PiD = PiO : PiO
Now OPi = a and OPi = b and PiD is the ordinate of the circle
of radius a or is equal to V a' — x'. Substituting these in the
last proportion and solving for PD we obtain:
''.n — ;
PD =
140 ELEMENTARY MATHEMATICAL ANALYSIS . [§74
This is the equation of an ellipse. Hence the curve APB is an
ellipse.
The two circles are called the major and minor auxiliary circles.
The vectorial angle d of Pi is called the eccentric angle or the
eccentric anomaly of the point P.
Exercises
1. Draw an ellipse whose semi-axes are 5 and 3, and write its
equation.
2. From what circle can the ellipse y = ± i\/9 — x* be made by
shortening of its ordinates?
3 Write the equation of the ellipse whose major axis is 7 and minor
axis is 5.
4. Find the major and minor axes of the ellipse x*/7 -h y'/17 = 1.
6. What curve is represented by the equation a;*/9 + y'/46 = 1?
74. Parametric Equations of the Ellipse. From Fig. 70,
OD and PD^ the abscissa and ordinate of any point P of the
ellipse, may be written as follows:
X = a cos d (1)
y = b sin ^
for OD is the projection of OPi = a through the angle d and DP
is the projection of OP 2 = h through the angle t 12 — 6. The
pair of equations (1) is known as the parametric equations of tiie
ellipse. The angle 6, in this use, is called the parameter. Writ-
ing (1) in the form:
X
- = cos B
y • a ^ }
r = sm ^ • '
0
squaring, and adding, we eliminate 0 and obtain:
the symmetrical equation of the ellipse.
If the abscissa and ordinate of any point of a curve are ex-
pressed in terms of a third variable, the pair of equations are
called the parametric equations of the curve. Thus:
X = U
y = t+i
§74] THE ELLIPSE AND HYPERBOLA 141
are the parametric equations of a certain straight line. Its
ordinary equation can be found by eliminating the parameter t
between these equations.
From equations (1) we see that the ellipse might be defined as
follows: Lay off distances on the X-axis proportional to cos d,
and distances on the F-axis proportional to sin 6. Draw
horizontal and vertical lines through the points of division, thus
dividing the rectangle 2a, 26 into a large number of small rec-
tangles. Starting at the point (a, 0) and drawing the diagonals
of successive cornering rectangles, a line is obtained which ap-
proaches the ellipse as near as we please as the number of small
rectangles is indefinitely increased. The student should draw
this diagram. See Fig. 129.
Exercises
1. Draw the curve whose parametric equations are:
X = cos $
y — sin e,
2. Write the equation of the ellipse whose major and minor axes
are 10 and 6, respectively.
3 Find the axes of the ellipse whose equation is :
4. Write the parametric equations of the ellipse :
y = ± Wsi -x\
6. Discuss the curve :
a; = ± iV4 -y\
6. Discuss the curves:
x« + 42/2 = 1 .
4a;« -h 2/« = 1
(l/4)a:» + 2/« = 1.
7. Write the Cartesian equation of the curves
whose parametric
equations are:
, . fa; = 2 cos 0 ,,. fa; = 6 cos d , .
Ly = sm ^ ^ ^ \_y = 2 am e ^
X = \/3 cos e
jy = y/2 sin B.
8. What locus is represented by the parametric
equations
X ^2t + l
y = 3« + 5?
142 ELEMENTARY MATHEMATICAL ANALYSIS [§75
9. Show that
X — at
y ^ht
is a line of slope h/a.
10. Write the equation of an ellipse whose major and minor axes
are 6 and 4 respectively.
11. What curve is represented by the parametric equations:
X = 2 + 6 cos ^
2/ = 5 + 2 sin ^?
12. Show that the curve
a; = 3 + 3 cos ^
2/ = 2 + 2 sin ^
is tangent to the coordinate axes.
13. The sunlight enters a dark room through a circular aperture of
radius 8 inches, in a vertical window and strikes the floor at an angle
of 60°. Find the dimensions and the equation of the boundary of
the spot of light on the floor.
14. The ellipse
2/ = ± IVO - x»
is the section of a circular cylinder. Find the angle a made by the
cutting plane and the axis of the cylinder.
76.^ Other Methods of Constructing an Ellipse. The following
methods of constructing an ellipse of semi-axes a and b may be
explained by the student from the brief outlines given:
1. Move any line whose length is a + 6 (see Fig. 71) in such a
manner that the ends A and B always lie on the X- and F-axes,
respectively. The point P describes an ellipse.
2. Mark on the edge of a straight ruler three points P, Af, N,
Fig. 72, such that PM = h and PN = a. Then move the ruler
keeping M and N always on AA^ and BB^ respectively. P
describes an ellipse. The elliptic "trammel" of "ellipsograph" is
constructed on this principle by use of adjustable pins on PMN and
grooves on A A' and BB\
3. Draw a semicircle o f radius a about the center C, Fig. 73,
and produce a radius to 0 such that CTO = a + b. From C draw
iThis section may be omitted altogether or assigned as problems to variotts
members of the class.
76] THE ELLIPSE AND HYPERBOLA 143
Dy number of linea to the tangent to the cirijle at T. From 0
raw lines meeting the tai^ent at the same points of TN. At the
oints where the lines from C cut the semicircle, diaw parallels
Fio. 73. — A Graphical Conatniotion of
» Cr. The points of meeting of the latter with the lines radiating
torn 0 determine points on the ellipse.
144 ELEMENTARY MATHEMATICAL ANALYSIS [§76
To prove the above, note that OD = a cos d, PD = ODt&a d\
also that tan 0 : tan ^' : : a : 6. Discuss the latter case when 6 = c
and also when h > a,
76. Origin at a Vertex. The equations of the ellipse (5) and (6)
§72 and (1) §74 are the most useful forms. It is obvious
that the ellipse may be translated to any position in the plane
by the methods already explained. The ellipse with center
moved to the point (A, k) has the equation:
a» "^ 62 -A [i)
Of special importance is the equation of the ellipse when the origin
is taken at the left-hand vertex. This form is best obtained from
equation (6), §72, by translating the curve the distance o in
the X direction. Thus:
or,
y = ± — Va* — {x — a)2
V = — X — 7, x^
or, letting I stand for the coefficient of x,
y^ = lx- -' x^ = Ixil - x/2a) (2)
a2
For small values of x, x /2a is very small and the ellipse nearly
coincides with the parabola y^ = Ix,
11, Any equation of the second degree, lacking the term xy and
having the terms containing x^ and y^ both present and wiih
coefficients of like signSy represents an ellipse with axes paraUel to
the coordinate axes. This is readily shown by putting the equation
ax^ -f by^ + 2gx + 2fy+ c = 0 (1)
in the form (1) of the preceding section. The procedure is as
follows:
a{x^ + 2 Jx) + b(y^ + 2^^y) = - c (2)
<^' + 2^ + 1)+ '{y' + H-^ + ft) = 'i +f - ^(')
7S] THE ELLIPSE AND HYPERBOLA 145
etM stand for the expression in the right-hand member of (3),
tien we get:
I-ilMLtil^,
a h
This shows that (1) is an ellipse whose center is at the point
[ > — t) and which is constructed from the circles whose cen-
ters are at the same point and whose radii are the square roots of the
denominators in (4). The major axis is parallel to OX or OF ac-
cording as a is less or greater than 6. The cases when the locus
is not real should be noted. Compare §42.
Illustration: Find the center and axes of the ellhpse
x2 + 4y2 + 6x - 82/ = 23
Write the equation in the form
x^ + 6x + 4:y^ - 82/ = 23
Complete the squares
X* + 6a; + 9 + 42/2 _ 82/ + 4 = 36
Rewriting (a; + 3)^ + 4(2/ - l)^ = 36
or {x + 3)V36 + (2/ - 1)V9 = 1
This is seen to be an ellipse whose center is at the point (— 3, 1)
and whose semi-axes are a = 6 and 6 = 3.
The rotation of the ellipse through any angle about 0 as a
center will be considered in another place. It should be noted,
however, that thie ellipse is turned through 90° by merely inter-
changing X and y.
78. Limiting Lines of an Ellipse. It is obvious from the
equation
y= ± - Va2 -
X
2
a
that X = a and x = — a are limiting hnes beyond which the curve
cannot extend; that is, x cannot exceed a in numerical value
^thout y becoming imaginary. The same test may be applied
'0 equations of the form:
X* + 4x + 92/2 - 62/ + 4 = 0
10
«
146 ELEMENTARY MATHEMATICAL ANALYSIS [S78
Solving for y in terms of x:
32/ = 1 ± VF- {x + 2)«
The values of y become imaginary when:
{x + 2)2 > 1
or,
x + 2>+lor<-l
or,
x> -lor< -3
These, then, are the Hmiting lines in the x direction. Finding
the limiting lines in the y direction in the same way, the rectangle
within which the ellipse must lie is determined.
In cases like the above the actual process of finding the limiting
lines and the location of the center of the ellipse is best carried
out by the method of §77.
Exercises
Find the lengths of the semi-axes and the codrdinates of the center
for the six following loci and translate the curves so that the terms in
X and y disappear, by the method of §77.
1. 12a;2 - 48x + 3y^ + 62/ = 13.
2. 2/2 - 82/ + 4x2 + 6 = 0.
3. x^ -Qx + 42/2 + 82/ = 5.
4. x^ + 92/2 - 12a; + 62/ = 12.
6. 4a;2 + y^ - 12x + 2y - 2 =^ 0.
6. x* + 22/2 - x - y/2y = 1/2.
7. Show that x^ + ix + 9y^ — 62/ = 0 passes through the origin.
8. Show that a;2 — 4a; + 42/2 + 82/ + 4 = 0 is an ellipse.
9. Discuss the curves :
9 "^4
9 "^ 4
4 "^ 9
9^4 •
r9] THE ELLIPSE AND HYPERBOLA 147
10. Discuss the following parabolas:
y = 2px*
y = — 2px^
2/ = - 2paj2 + h.
What are the roots of the last function?
11. Write the symmetrical equation of the ellipse if its parametric
ijuations are :
X = (3/2) cos e
y = (2/3) sin d.
12. Discuss the curve y* = (18/5)a; - (9/25)aj«.
13. Compare the curves y* =^ x — x^ and y^ = x.
li. Find the center of the curve y^ = 2a; (6 — x) .
79. Graph of y = tan x. If this graph is to be constructed on a
heet of ordinary letter paper, 81 inches X 11 inches, it is desirable
0 proceed as follows : Draw at the left of the sheet of paper a semi-
ircleof radius 1.15 . . . inches, (that is, of radius = IS/Stt), so
hat the length of the arc of an angle of 10° or t /18 radians will be
/5 inch. Take for the x-axis a radius COX prolonged, and take for
he y-axis the tangent OY drawn through 0, as in Fig. 74. Divide
he semicircle into eighteen equal parts and draw radii through the
loints of division and prolong them to meet OF in points Ti, T2,
"1, Tiy . . , Then on the y-axis there is laid off a scale
^Y' in which the distances OTi, OT2, . . . are proportional to
he tangents of the angles OCSi, OCS2, . . . ; for the tangents
»f these angles^ are OTi /COy OT2 /CO, ... and CO is the unit of
neasure made use of throughout this diagram. Draw horizontal
ines through the points of division on OF and vertical lines through
he points of division on OX, thus dividing the plane into a large
mmber of small rectangles. Starting at 0, tt, 27r, . . . — tt,
-27r, . . . and sketching the diagonals of consecutive cornering
ectangles, the curve of tangents is approximated. Greater pre-
cision may be obtained by increasing as desired the number of
livisions of the circle and the number of corresponding vertical
nd horizontal lines.
It is observed that the graph of the tangent is a series of similar
ranches, which are discontinuous for x = ir/2, —ir 12, (3/2)7r,
148 ELEMENTARY MATHEMATICAL ANALYSIS [M
— (3/2)ir, . . . For these values of x the curve has vertiol
asymptotes, as shown at AB, A'B', in Fig. 74.
If the number of corresponding vertical and horizontal lino;
be increased sufficiently, the elope of the diagonal of any rectangle
gives a close approximation to the true slope of the curve at thit
point.
It haa already been noted that all of the trigonometric functions
are periodic functions of period 2ir. It is seen in this case, howevH,
r M A M' a'
n\
\ 1 \
//J , \ /
w
rrJJ/S'^i
x
~'j\. ::::::::::.
/ __\
' L-.
'-"V\\ ' h
t ,,.
V ' ^
/ \
\ T 1 1
\\IV
y B N
FiQ. 74.— Graphical Construction at
For lack of room only a few of the points i
in the diagram. The dotted curve ia v
B N
he Curve of Tangents v = taiu-
,.S,.-..T,,T, arelettwri
that tanx has also the shorter period tt; for the pattern MH,
M'N', M'N', of Fig. 74 is repeated for each interval ir of the
variable x.
80. Ratio (sin x)/x and (tan x)/x for Small Values of x. Pre-
cisely as in the case of the locus oiy = sin x, the rectangles along,
and on both sides of, the x-axis in the graph of y = tan x, are
nearly squares. In Fig. 74, the x-sides of theee rectai^les ut
M THE ELLIPSE AND HYPERBOLA 149
5 inch, but the y-sides are slightly greater, since OTi is slightly
eater than the arc OSi of the circle. To prove this, note that OTi
half of one side of a regular 18-sided polygon circumscribed about
le circle; since the perimeter of this polygon is greater than the
rcumference of the circle, OTi > O/Si, for these magnitudes are
/36 of the perimeter and circumference, respectively, just named,
likewise in Fig. 59, DSi < OSi, for DSi is one-half of the side of
n 18-sided regular polygon inscribed in the circle and OSi is
/36 of the circumscribed circumference,
lence:
sin a; < x < tan x (1)
)r dividing by sin x,
1 < -T^ < sec X (2)
sm X '
Sow as X approaches 0, the last term of this inequaUty approaches
inity. Hence the second term, whose value always lies between
ihe first and third term of the inequahty, must approach the same
/alue, 1. This fact is expressed in mathematics by the statement
the limit of = 1 as x approaches 0
3r, in symbols:
lim sin z
X = 0 z
Dividing (1) by tan x,
= 1 (3)
Now as X approaches 0, the first term of this inequaUty approaches
inity. Hence the second term, whose value always lies between
:;he first and third term of the inequality, must approach the same
iralue, 1. This fact is expressed by the statement
the limit of = 1 as x approaches 0
)r, in sjonbols:
lim tan z
liquations (3) and (5) express very useful and important facts,
jeometrically they state that the rectangles along the !c-axis in
«
150 ELEMENTARY MATHEMATICAL ANALYSIS [J81
Figs. 59 and 74, approach more and more nearly sqitares as the
number of intervals in the circle is increased. Each of the ratios
in (2) approaches as near as we please to unity the smaller x is
taken, but the limits of these ratios are unity only when the angles
are measured in radians.
The word " limit *' used above stands for the same concept that
arises in elementary geometry. It may be formally defined as
follows:
Definition: A constant, a, is called the limit of a variable,
(, if, as t runs through a sequence of numbers, the diflference
(a — 0 becomes, and remains, numerically smaller than any pre-
assigned number.
81. Graph of cot x. In order to lay off a sequence of values of
cot ^ on a scale, it is convenient to keep the denominator con-
stant in the ratio (abscissa) (ordinat-e) which defines the cotangent.
11
Pio
P^ P% Pi
P.P. Pz
Pt
^
^
M
^
/^i.^^
^s
i
/V\^
1
Fig. 75. — Construction of a Scale of Cotangents.
Dx
The denominator may also, for convenience, be taken equal to
unit 3'. Thus, in Fig. 75, the triangles of reference DiOPi, DfiPi,
. . .for the various values of d shown, have been drawn so that
the ordinates PiDi, PiDi^ . . . are equal. If the constant ordi-
nate be also the imit of measure, then the sequence ODi, ODj, ODi,
. . . OD7, ODg, represents, in magnitude and sign, the cotan-
gents of the various values of the argument 6. Using ODi, ODt^
... as the successive ordinates and the circular measure of
^ as the successive abscissas, the graph of y = cot x is drawn, as
shown by the dotted cur\'e in Fig. 74.
The sequence ODi, OD*. . . . Fig. 75 is exactly the same as the
sequence OTi, OT*, . . . Fig. 74, but arranged in the reverse
order. Hence, the graph of the cotangent and of the tangent are
alike in general form, but one curve descends as the other ascends,
so that the position, in the plane xy, of the branches of the curve
THE ELLIPSE AND HYPERBOLA
151
are quite different. In fact, if the curve of the tangents be rotated
about OK as axis and then translated to the right the distance
t/2, the curvra would become identical: Therefore, for all values
of z:
tan()r/2 - x) = cot a; (1)
This IS a result previously known
Fio. 76. — Graphical Conatruction of y " sec x.
82. Graph of y = sec x. Since sec 0 is the ratio of the radius
divided by the abscissa of any point on the terminal side of the
angle 6, it is desirable, in laying off a scale of a sequence of values
of sec 6, to draw a series of triangles of reference with the abscissas
in all cases the same, as shown in Fig. 76. In this figure the angles
were laid off from CQ aa initial line. Thus;
CTilCSi = secQCSs
or, if CSi be unity, the distances like CTt, laid off on CQ, are th^
152 ELEMENTARY MATHEMATICAL ANALYSIS [§83
secants of the angles laid off on the arc QS^O or laid off on the axis
OX.
The student may describe the manner in which the rectangles
made by drawing horizontal lines through the points of division on
CQ and the vertical lines drawn at equal intervals along OX, may
be used to construct the curve. If the radius of the circle be 1.15
inches, what should be the length of Ox in inches?
The student may construct and discuss the locus of y = esc x.
Compare with the locus
y = secx
Exercises
1. Discuss from the diagrams, 59, 74, 76, the following statements:
Any number, however large or small, is the tangent of some angle.
The sine or cosine of any angle cannot exceed 1 in numerical value.
The secant or cosecant of any angle is always numerically greater
than 1 (or at least equal to 1).
2. Show that sec (o "" ^ ) = esc a? for all values of x,
3. If tan 0 sec tf = 1, show that sin ^ = }( V5 — 1) and find 9
by use of polar coordinate paper, Form.AfS.
4. Describe fully the following, locating nodes, troughs, crests,
asymptotes, etc.:
y =sin (x-^)
y = cos (a; + 1^)
y = tan ^^x + ^J
y =i tan (j + 1).
83. Increasiiig and Decreasing Ftmctioiis. The meanings of
these terms have been explained in §26. Applying these terms to
the circular functions, we may say that y = sinx, jf = tanx,
y = secx are increasing functions for 0 < x < x/2. The co-
functions, y = cos J, y = cot X, y = CSC x, are decreasing functions
within the same inter\-al.
Exercises
Dbcuss the following topics from a consideration of the graphs of
the fimctions:
m]
THE ELLIPSE AND HYPERBOLA
153
I. Id which quadiante is the ajne an inoreaaing function of the
iin^7 In which a decreaaing function?
S. In which quadrants is the tangent an increasing, and in which a.
d^nreaait^, function of its variable?
3. In which quadrants are the cos 6, cot B, sec 8, esc 8, increasing
and in which are they decreasing functions of 0?
1. Show that all the co-functions of angles of the first quadrant are
decreasing functions.
Fio. 77. — Construction of the Reclangular Hyperbola.
U. The Rectangular Hyperbola. We have Been that the circle
8 the locus of a point whose abscissa is a cob 9 and whose ordinate
s a sin S. The rectangular, or equilateral, hyperbola may be
lefined to be the locus of a point whose abscissa is o sec fl and whose
■rdinate is a tan fl. To construct the curve, divide the X-axis pro-
>ortionaUy to sec 6, and the y-axis proportionally to tan 6, as
hown in Fig. 77. The scale OX of this diagram may be taken
rom 0 r of Fig. 76, and the scale OY may be taken from 0 K of Fig. ,
154 ELEMENTARY MATHEMATICAL ANALYSIS
74. The plane of xy may be divided into a large number of rec-
tangles by passing lines through the points of division perpendicu-
lar to the scales and then, starting from A and A', sketching the
diagonals of the successive cornering rectangles.
The parametric equations of the curve are, by definition:
X = a sec B\
y = a tan ^ J
(1)
The Cartesian equation is easily found by squaring each of the
equations and subtracting the second from the first, thus eUminat-
ing B by the relation sec^ B — tan^ ^ = 1 :
a;2 - 2/* = aHsec2 B - tan* 6)
or,
x2 - y2 = a* (2)
This is the Cartesian equation of the rectangular hyperbola.
The equation of the rectangular hyperbola may also be written in
the useful form:
y
= ± Vx2 - a* (3)
Compare (1) and (3) with the equations of the circle.
The rectangular hyperbola here defined will be shown, in §86,
to be the curve 2xy = a* rotated 45° clockwise about the origin.
85. The Asymptotes. Let GV be the line y = x, Fig. 77. The
slope of OP^ is PD /OD or y jx or
a tan e . ^
~1,^~^ = sm B.
a sec e
The value of B corresponding to the point P is AOH. As the point
P moves upward and to the right on the curve, the angle 0, or
AOHy approaches 90° and sin B approaches unity. Hence the
line OP approaches OG as a limit, and P approaches as near as we
please to OG. The same reasoning applies to points moving out
on the curve in the other quadrants. The lines GG' and //' are
called asymptotes to the hyperbola.
86. The Curves 2xy = a^ and x^ - y2 = a*. In Fig. 78, let
the curve be the locus 2x1^1 = a^, referred to the axes X/Xi and
YiYi, This curve has already been called the rectangular hyper-
bola. (See §23.) We desire to find the equation of the curve
1 To avoid an excessive number of construction lines, OP is not shown in the
figure.
187]
THE ELLIPSE AND HYPERBOLA
155
Tefeired iff the axes XtX't and YtT't- In the figure, yi is the sum
of the projectiooB of X) and yi on PDi. The angle of projection is
45°, whose cosine is i\/2- Hence,
yi^iV2(y, + x,) (1)
Likewise, Xi is the difference in the projections through 45° of xt
and !/j on XiX'i. Or:
X, = W2(xi - y,) (2)
Hence, multiplying the
members of (1) and (2):
2xiyi = Xi' - J/,* (3)
Since by hypothesis 2xiyi
= a\ the equation of the
curve referred to the axes
X,y, is
xi' - yi' = o* (4)
Thus, 2xy = a' is the
curve !' — {;' = a' turned
anti-clockwise tlirough an
angle of 45°.
By §27, the curve 2xy
= 0° may be made from
xy = 1 hy multiplying
both the abscissas and the ordinates by a/\/2-
Are the curves xy = 1 and i' — y' = 1 of the same size ?
87. Hyperbola of Ssmi-axes a and b. The curve whose ab-
scissas are proportional to sec S and whose ordinates are pro-
portional to tan 9 is called the hyperbola. Its parametric
equations are, therefore:
x^asecel
y = b tan flj ^ '
where a and b are constants.
To construct the curve, draw two concentric.circles of radii a and
6, respectively, as in Fig, 79. Divide both circumferences into
the same number of convenient intervals. Lay off, on XOX',
distances equal to a sec 6 by drawing tangents at the points of
division on the circumference of the a-circle; also lay off disia
istanMta
156 ELEMENTARY MATHEMATICAL ANALYSIS IS87
equal to b taa 8 on the vertical tangent to the ^-circle by prolong-
ing the radii of the latter through the points of division of the cir-
cumference. Draw horizontal and vertical lines through the
points of division of MN and XX' respectively, dividing the
plane into a lai^e number of rectangles which are used exactly
as in Fig. 77 for the construction of the curve.
In the above construction, there is no reason why the diameter erf
the ii-ciTcle may not exceed that of the a-cirele.
f
\
[[
A
^^^
\Y
^r
^y
\
y On
/
~^3>
^r^ M.
■
-^^
^^Jj
/
5 At /
Vva;/
S '
A^
n^
'-W
-^x
Y
1
A
X'^
?
Fio. 79.— The Hyperbola I'/o' - wV6* - 1-
Writing (1) in the form:
- = seed
^ = tan I
and eliminating B as before we obtain :
§87] THE ELLIPSE AND HYPERBOLA 157
the Cartesian equation of the hyperbola. This is also called the
syxmnetrical equation of the hyperbola.
The line AA' =: 2a is called the transverse axis, the line BB'
is called the conjugate axis, the points A and A' are called the
vertices, and the point 0 is called the center of the hyperbola.
Let the line G'OG be the line through the origin of slope 6 /a and let
J'OJ be the line of slope — h \a. The slope of the radius vector
OP is:
PD y 6 tan ^ & . ^
/Tp. = - = 'k — - sin d
• OD X a sec B a
The limit of this ratio as the point P moves out on the curve away
from Oiabfa; for 6 approaches 90° as P moves outward, and hence
sin 6 approaches 1. Hence, the line OP approaches in direction
OG as a limit. Points moving along the curve away from 0 in the
other quadrants likewise approach as near as we please to G'G or
JV. The lines G'G and J^J are called the asymptotes of the hyper-
bola. The equations of these lines are
y= ± \^ (3)
Solving the equation (2) for y, the equation of the hyperbola may
be written in the useful form
y = ± ^Vx2 - a^ (4)
Compare this equation with the equation of the ellipse, (6) § 72.
It is easy to show that the vertical distance PG of any point of
the curve from the asymptote G^G can be made as small as we please
by moving P outward on the curve away from 0.
Write the equation of the hyperbola in the form
yi = -\/x^-a^ (5)
and the equation of the asymptote GG' in the form
y2 = -X (6)
a
Then:
PG = y2-yi = ^(x - \/x^ - a2) ^ (7)
a
a^
a x+ \/x^ - a^
(8)
158 ELEMENTARY MATHEMATICAL ANALYSIS [§88
by multiplying both numerator and denominator in (7) by
X + \/x* — a*. Now, as x increases in value without limit the
right side of (8) approaches zero. Whence:
P(j A 0 as X ^ 00
Exercises
1. Write the symmetrical equation of the hyperbola from the
parametric equations x = 5 sec d^ y — Z tan d,
2. Find the Cartesian equation of the hyperbola from the relations
X — 1 sec ^, y = 10 tan^. Note that the graphical construction of
the hyperbola holds if 6 > a.
3. What curve is represented by the equation
{X - 3)2 (y + 2)2
25 16
= 1?
4. What curve is represented by the equation y = iVx* — o*?
6. Write the equation of a hyperbola having the asymptotes
y =* ± (3/4) x, and transverse axis = 24.
6. Show that the curves
x* + 6a; - 2/2 - 4?/ + 4 = 0
and
(x + 3)2 - (y + 2)2 = 1
are the same, and show that each is a hyperbola.
7. What curve is represented by the equations
X = h + a sec 6
y = k + htsine?
8. Discuss the curve x^ — Sx — 2y^ — 12y = 0.
88. Orthographic Projections. When the equation of the
hyperbola is written in the useful form
y = ± -Vx^ - a2 (1)
it is seen that the hyperbola may be looked upon as generated from
the equilateral hyperbola
y = ± Vx^^^^ (2)
by multiplying all of its ordi nates hyb/a.
89. Conjugate Hyperbolas. Consider the hyperbola
f8B] THE ELLIPSE AND HYPERBOLA 159
Interchanging x and y in this equation gives, by Theorem III on
Loci, $2i, a new locus which is the reflection of (I) in the line
y ^ X. The new equation may be written in the form
: _1
(2)
in which all signe have been changed after interchangii^ x and y.
Since (2) is the same curve as (1) but in a new position, it is still
a hyperbola; its vertices are located on the y-axis instead of on
the X-axis. The asymptotes of (I) have been found to be
Therefore the asymptotes of
!2) may be found by reflecting
3) in the line y = x; hence
hey must be given by:
y= ±lx (4)
Now, if the constants o
nd 6 in equation (2) be in-
^rchanged giving thereby the
quation
hen the shape of the hyper-
ola (2) will be changed but „ T""- f-~^ family of Conjugate
' ' , , , , Pairs of Hyperbolas with Common
tS position will be unaltered, AsymptoWa. (An interference pat-
hat is, its vertices will still tern made from a glasa plate under
_ I , J i.i,„ V „ ■„ compression. From R, Strauble-
e located on the F-axis. ..^^^^^ ^.^ Elaaticitftts-^aillen und
"he asymptotes of (5) are modiiln des Glaaes." Wied. Ann.
oiind, of course, by inter- Bd. 68, 1899, p. 381.)
hating a and b in (4), which
jves an equation exactly like (3). Hence the hyperbola (5) has
he same asymptotes as the original hyperbola (1). When a
lyperbola with vertices on the y-axis has the same asymptotes
s a hyperbola with vertices on the X-axis, and of such size that
he transverse axis of one hyperbola is the conjugate axis of the
ther, then the two hyperbolae are said to be conjugate to each
160 ELEMENTARY MATHEMATICAL ANALYSIS [§89
other. Thus (1) and (5) are two hjrperbolas which are conju-
gate to each other. Obviously a hyperbola and its conjugate
completely bound the space about the origin, except the cuts or
lines represented by the common asymptotes.
Fig. 80 shows a family of pairs of conjugate hyperbolas.
Exercises
1. Sketch on the same pair of axes the four following hyperbolas and
their asymptotes :
(1) x2 - 2/2 = 25
(2) a;2 - 2/2 = - 25
3. Compare the curves :
and
4. Compare the curves :
rE2
(3) 25
9 ^
rr2
(4)25
y2
"9 ^•
hyperbola y
= ± l^x^ - 64.
x^ y^
a2 62 ^
re* y^
a2 62
1.
x^ y^
9 16 ^
x2 y^
16 9 ~ ^•
and
5. Write the equation of the hyperbola conjugate to
2/ = ± } Va;2 - 64.
6. Compare the graphs of:
2/ = ± J Va;2 - 64
2/ = ± i Va;2 - 16
2/ = ± i Vx2 - 4
y ^ ± I Va;2 - 1
2/ = ± i Vx2"-~1/16
2/ = ± J Vx2 - 0.
7. Show that 3a;2 - 42/* - 7x + 52/ + 2 = 0 is a hyperbola. Find
the position of the center and of the vertices. The vertices locate
the so-called "limiting lines" of the hyperbola.
{89] THE ELLIPSE AND HYPERBOLA 161
8. Show that x* — 4a; — 4^^ + 4^/ = 4 is a hyperbola. Find the
limiting lines and center.
9. Discuss the graphs:
a;2 - 2/2 = 1
and
2/2 - a:* = 1.
10. Discuss the graph 16a;2 — y^ — 40a; — Qy =2, and find the
limiting lines.
11. Li Fig. 77, show that DS — PD and, hence, from the triangle
DSO, a;2 - 2/* = a*.
12. In Fig. 77, show that PK = a; - 2/, PK' == a; + y, and that the
rectangle PK X PK^ is constant for all positions of P and equal to the
square on OA.
11
CHAPTER V
SINGLE AND SIMULTANEOUS EQUATIONS
90. The Rational Integral Function of z. The general form of
a polynomial of the nth degree is: ^
> aox* + aix""^ + a2X"7^ +-<^ • • + (U-iX
where the symbols, ao, ai, at, ... , stand for any real constants
whatsoever, positive or negative, integral or fractional, rational or
irrational, and where n is any positive integer. The number of
terms in the rational, integral function of the nth degree is (n + 1).
91. The Remainder Theorem. If a rational integral function oj
X be divided by (x — r) the remainder which does not contain x U
obtained by writing, in the given function, r in plaice of x: This
theorem means, for example that the remainder of the division:
(x» - 6x2 + iia; _ 6) ^ (a; _ 4) is 4J _ 5(4)2 + 11(4) _ gorG
Also that the remainder of the division:
(x» - 6x2 + iia- _ 6) ^ (a- + 1)
^ i-iy - 6(-l)2 + ll(-l)-6= - 24
The theorem enables one to write the remainder without aotuafly
performing the division.
To prove the theorem, let
f(x) = aox* + aix*-! + ajx'*-* + . . . + Gn-ix + a» (1)
and: /(r) = aor** + air*-^ + ajr*-* + • . . + a»-ir + a» (2)
then:
f{x) —fir) = OoC-c* — r") + ai(x'»-^ — r--^ +'...-
+ an^iix - r)\ (3)
The right side of this equation is made up of a seri^irf t^^ con-
taining differences of like powers of x and r, and, hence, by the
well-knowTi theorem in factoring,^ each binomial term is exactfr
divisible by (x — r). The quotient of the right side of (3) by
' See Appendix.
162
§92] SINGLE AND SIMULTANEOlJS EQUATIONS 163
(a; - r) may be written out at length, but it is sufficient to
abbreviate it by the symbol Qix) and write:
m^ = Q(,) (4)
or:
^^ = 0W+^^- (5)
Now if iV be any dividend, D any divisor, and Q the quotient and
R the remainder, then:
NID ^Q+ttlD (6)
This form applied to (5) shows that fir) is the remainder when
][x) is divided by {x — r). Thus the Remainder Theorem is
established.
92. The Factor Theorem. If a rational integral function of
I becomes zero when r is written in the place of x, (x — r) is a fac-
tor of the function: This means, for example, that if 3 be substi-
tuted for X in the function x^ — 6x^ + llx — 6 and the result
]i - 6(3)2 + 11(3) -6 = 0, then (x - 3) is a factor of
c'-6x2 + 11a; - 6.
This theorem is but a corollary to the remainder theorem,
''or if the substitution x = r renders the function zero, the
emainder when the function is divided by {x — r) is zero, and the
heorem is established.
The value r of the variable x that causes the function to take
n the value zero has already been named a root or a zero of the
unction. The factor theorem may, therefore, be stated in the
inn: A rational integral function of the variable x is exactly
ivisible by (x — r) where r is any root of the function.
The familiar method of solving a quadratic equation by f actor-
ig is nothing but a special case of the present theorem. Thus if:
^2 - 5x + 6 = 0
hen:
(x - 2)ix - 3) = 0
nd the roots are x = 2 and x = 3. The numbers 2 and 3 are
uch that when substituted in x^ — 5x + 6 the expression is
ero; and the factors of the expression are x — 2 and x — 3
•y the factor theorem,
d
164 ELEMENTARY MATHEMATICAL ANALYSIS [|93
Exercises
1. Tabulating the cubic polynomial x' — 6a;* + llx — 6, we
obtain :
X _ -3^ -2-1-01 1.5 2 2.5 34
~JLx), - 120, - 60, - 24, - 6, 0, + 0.375, 0, - 0.375, 0, 6
What is the remainder when the function is divided by a; - 4?
By X + 2? By a; + 3? By a; - 1.5? By a; - 3?
Name three factors of the above function.
2. Find the remainder when a;* — 5a;' + 12x* + 4x — 8 is divided
by X - 2.
3. Show by the remainder theorem that x" + o* is divisible by
X + a when n is an odd integer, but that the remainder is 2a" when n
is an even integer.
4. Without actual division, show that x* — 4x* — 7x — 24 is
divisible by x — 3.
6. Show that a* -\- d^ — ab* — 6* is divisible by a — 6.
6. Show that (6 - c)(6 + c)* + (c - a){c-\' a)« + (o - 6)(a +W'
is divisible by (fe — c){c — a) {a — h).
7. Show, that (x + l)*(x - 2) - 4(x - l)(x - 5) + 4 is divisible
bv X - 1.
8. Show that (6 - c)» + (c - a)» + (a - 6)» is divisible by
(fe — c){c — a)(a — 6).
9. Show that 6x» - 3x* - ox» + 5x* - 2x - 3 is divisible by
J + 1.
98. It follows at once from the factor theorem that it is possible
to set up an equation with any roots desired; for example, if we
de^re an equation with the roots 1, 2, 3 we have merely to
write:
(x - l)(x - 2)(x - 3) = 0 (1)
Forming the product:
j» - 6x- + llJ - 6 = 0
or transposing the terms in any manner, as:
x»+ llJ = 6x» + 6
in no way esssentiaUy modifies the equation. If, howev^, the
equation (1) be multiplied through by any function of x, the
number of roots of the equation may be increased. Thus, mul'
tiplying (1) by (x + 2) introduces a new root x = — 2. likewisei
INGLE AND SIMULTANEOUS EQUATIONS 166
3quation (1) through by. the factor {x — 2), leaves an
{z - l)(a: - 3) = 0 (2)
cs the root x = 2.
principles or axioms of algebra, an equation remains
) unite the same number to both sides by addition or
n; or if we multiply or divide both members by the
aber, not zero; or if like powers or roots of both
be taken. But we have given sufficient illustrations to
. these operations may affect the number of roots of the
This is obvious enough in the cases already cited.
3, however, the operation that removes or introduces
0 natural and its effect is so disguised that the student
to take due account of its effect. Thus, the roots of:
S{x - 5) = x(x - 5) + x2 - 25 (3)
ind 5, for either of these when substituted for x will
B equation. Dividing the equation through by a; — 5,
Jig equation is:
3 = x + x + 5 (4)
bion is not satisfied by x = 5. One root has disappeared
msformation. This is easy to keep account of if (3)
1 the form:
{x - 5){x + 1) = 0, (5)
Lct that a factor has been removed may be overlooked
equation is written in the form first given,
important effect upon the roots of an equation results
iring both members. The student must always take
count of the effect of this common operation. To il-
ake the equation:
X + 5 = 1 - 2a: (6)
fied only by the value x = — 4/3. Now, by squaring
1 of the equation, we obtain:
x2 + 10a; + 25 = 1 - 4a; + 4a;2 (7)
itisfied by either a; = 6 or a; =— 4/3. Here, obviously,
eous solution has been introduced by the operation of
Doth members.
166 ELEMENTARY MATHEMATICAL ANALYSIS [§93
It is easy to show that squaring both members of an equation
is equivalent to multiplying both sides by the sum of the left and
right members. Thus, let any equation be represented by:
L{x) = R(x) (8)
in which L{x) represents the given function of x that stands on
the left side of the equation a.ndR(x) represents the given function
of x that stands on the right side of the equation.
Squaring both sides:
lL(x)]^ = lR(x)V (9)
Transposing:
[L(x)]^ - lR{x)Y = 0 (10)
or factoring;
[L(x) + R{x)] lL{x) - R(x)] = 0 (11)
But (8) may be written:
L{x) - R{x) =^0 (12)
Thus, by squaring the members of the equation the factor
L(x) + R(x) has been introduced.
The sum of the left and right members of (6), above, is 6 - 1.
Hence, squaring both sides of (6) is equivalent to the introduction
of this factor, or, the operation introduces the root 6, as already
noted.
As another example, suppose that it is required to solve:
sin a cos a = 1 /4 (13)
for a < 90°. Substituting for cos a:
sin aVl - sin2 a = 1/4 (14)
squaring:
sin^a (1 - sin^a) = 1/16 (15)
completing the square:
sin4 a - sin2 a + 1 /4 = 3 /16 (16)
Hence :
sin a = ± \/l/2 ± (1/4) \/3
= ± 0.9659 or ± 0.2588 (17)
Only the positive values satisfy (13); the negative values were
introduced in squaring (14). If, however, the restriction a <Vf
be removed, so that the radical in (14) must he wriOen with the doM
sign, then no new solutions are introduced hy squaring.
N SINGLE AND SIMULTANEOUS EQUATIONS 167
94. Legitimate and Questionable Transfonnations. If one
quation is derived from another by an operation which has no
ffect one way or another on the solution, it is spoken of as a
egitimate transfonnation; if the operation does have an effect
ipon the final result, it is called a questionable transformation,
leaning thereby that the effect of the operation requires ex-
mination.
In performing operations on the members of equations, the
Bfect on the solution must be noted, and proper allowance
lade in the result. It cannot be too strongly emphasized that
e test for any soliUion of an eqvxjAion is that it satisfy the original
juation. "No matter how elaborate or ingenious the process
y which the solution has been obtained, if it do not stand this
»t it is no solution; and, on the other hand, no matter how simply
Dtained, provided it do stand this test, it is a solution."^
Among the common operations that have no effect on the solu-
on are multiplication or division by known numbers, or addition
: subtraction of like terms to both members; none of these intro-
iice factors containing the unknown number. Taking the
(uare root of both numbers is legitimate if the double sign be
ven to the radical. Clearing of fractions is legitimate if it be done
) as not to introduce a new factor. If the fractions are not in
leir lowest terms, or if the equation be multiplied through by an
cpression having more factors than the least common multiple
I the denominators, new solutions may appear, for extra factors
re probably thereby introduced. Hence, in clearing of fractions
18 multiplier should be the least common denominator and the
•actions should be in their lowest terms. This, however, does not
anstitute a sufficient condition, therefore the only certainty lies
% checking all results.
Exercises
Suggestions: It is important to know that any equation of
[le form
ax^"* + 6a;» + c = 0
an be solved as a quadratic by finding the two values of x^,
'requently equations of this type appear in the form
1 Chrystal's Algebra.
168 ELEMENTARY MATHEMATICAL ANALYSIS [{94
Likewise any equation of the form
af(x) + b ^Jf(^) + c = 0
can be solved as a quadratic by finding the two values of V/W
and then solving the two equations resulting from putting V/(a;)
equal to each of them. One of these usually gives extraneous
solutions.
These two types occur in the exercises given below.
Since operations which introduce extraneous solutions are
often used in solving equations, the only sure test for the solution
of any equation is to check the results by substituting them in
the original equation.
Take account of all questionable operations in solving the following
equations :
3a;_ 6 9
X - 3 "" x + a"^ X - 3*
2. (x» + 5x+6)/(x - 3) + 4a: - 7 = - 15.
8. 3(x - 5){x - l)(x - 2) = (x - 5)(x + 2)(x + 3).
Note: Divide by (x — 5), but take account of its effect.
4. x^/a + ax = x^/h + hx,
6. ax{cx - 36) = 5a (36 - ex).
6. x^ — w* = » — X.
7. (X - 4V + IX - 5y = 3l[(x - 4)« - (x - 5)*]. Divide by
U- 4) + U-5) or 2x - 9.
J** — *\t 1
8. , , + 2 + ., = 0. If the fractions be added, multi-
X* — 1 X — 1 '
pHo^Uion is unnooossary. There is only one root.
9. X » 7 - \ X* - 7.
10. N X -h 20 - \ X - 1 - 3 = 0.
11, N 15 4 + X = 3 2 -f X X.
18. 2lVr \ 10x~9 ~ \ 10x~9 = IS \ 10j-9 +9.
18. ^ = * -.>• Consider as a proportion and take
NX-\x-3 '-^
by oivn^Hv^ition and division.
li. X** -h 5 2 - vl3 4V'\
1^ \ x» - 2\ X -" X « 0. Divide by ^ x.
le. 2n X* 5 X r 2 ~ X* -r :sx = ar - 6. CaU x« - 5x + 2 = tt«.
SINGLE AND SIMULTANEOUS EQUATIONS 169
* - 4a; + 20\/2 a;« - 5 x + 6 = 6a; + 66.
» - 2a;-i = 8.
ii _ 5ajHx +4=0.
Ox-* + 1 = 21x-».
'a: + 4a;->^ = 5.
•^ -Sx'^^ = 63.
— a)* - 3(a; - a)-« = 2.
^ - 3a;^ + a; = 0.
Ltersection of Loci. Any pair of values of x and y that
an equation containing x and y locates some point on
)h of that equation. Consequently, any set of values of
that satisfies both equations of a system of two equations
ng X and y, must locate some point common to the
3f the two equations. In other words, the coordinates
it of intersection of two graphs is a solution of the equations
raphs considered as simultaneous equations,
id the values of x and y that satisfy two equations, we
em as simultaneous equations. Hence, to find the points
section of two loci we must solve the equations of the
ves. There will be a pair of values or a solution for each
intersection.
the intersection of the lines y = 3x — 2 and y = x/2 + 3
oint (2, 4) and a; = 2, 2/ = 4, is the solution of the simul-
equations.
id the points of intersection of the circle x^ + y^ = 25
straight line x + y = 9 we solve the equations by the
ethod, as follows:
x^ + y^ =25\ (1)
X +y ^ If (2)
phs are a straight line and a circle, as shown in (1),
Squaring the second equation, the system becomes:
x^ + y^ = 25\ (3)
a;2 + 2xy + ^/^ = 49 J (4)
ond equation represents the two straight lines shown in
(2). The effect of squaring has been to introduce two
)us solutions corresponding to the points Pz and P4.
170 ELEMENTARY MATHEMATICAL ANALYSIS [1
Multiplying (3) by 2 and subtracting (4) from it, the last
of equations becomes:
x« + 2xj/ + y« = 49 J 0
which gives the four straight lines of Fig. 81, (4). Taking tl
square root of each member, but discarding the equation z +y
— 7, because it coi
sponds to the extraneoi
solutions introduced b]
the questionable operatioi
we have:
x-y=±l\ (8)
x + y^7 j (9)
By addition and subtrac-
tion we obtain the results:
5
1
a
x = S\
y = 4/
a; = 4l
2/ = 3/
(10)
(11)
D
Fig. 81. — Graphic Representation of
the Steps in the Solution of a Certain set
of Simultaneous Equations.
represented by the inter-
sections of the lines parallel
to the axes shown in Fig.
81, (5).
This is a good illustra-
tion of the graphical
changes that take place
during the solution of sim-
ultaneous equations of the
second degree. The ordinary algebraic solution consists, geo-
metrically, in the successive replacement of loci by others of an
entirely different kind, but all passing through the points of in-
tersection (as Pi, P2, Fig. 81) of the original loci.
Exercises
1. Find the points of intersection of the circle and parabola:
X'
y'
5
4x,
SINGLE AND SIMULTANEOUS EQUATIONS 171
Note that of the tioo lines parallel to the ynixisj given by the equation
X* -{■ 4x — 5 =0, one does not cut the circle : x* + I/* =* 5.
2. Find the points of intersection of x* + y* = 5 and the hyperbola
a;» - 2/8 = 3.
3. Solve, by graphical means only, to two decimal places:
y = X* + X — 1
xy = 1. •
4. Solve in like manner :
a;« + 2/2 = 16
x* — 2xy + y* = 9.
Reason out what each equation represents before attempting to
graph.
6. Solve in like manner:
x^-^y^+x+y = 7
2x« + 22/» - 4a; + 42/ = 8.
These loci should be graphed without tabulating numerical values
of the variables.
6. Solve graphically:
u2 + t;2 = 9
tA* - V* = 4.
Note. Draw the lines x + y = 9, and x — y — 4:, The values
of X and y determined by the intersection of these lines are the
values of w* and v* respectively, from which u and v can be computed.
7. Solve the system :
x^ + 2/' = 10
xV16 + yV9 = 1.
96. Qixadratic Systems.^ Any linear-quadratic system of
simultaneous equations, such as:
y ^ mx + k
ax^ + hy^ + 2hxy + 2gx + 2fy + c = 0
can always be solved analytically; for y may readily be eliminated
by substituting from the first equation into the second. A
system of two quadratic equations may, however, lead, after
elimination, to an equation of the third or fourth degree; and,
hence, such equations cannot, in general, be solved until the
solutions of the cubic and biquadratic equations have been
explained.
1 A large part of the remainder of this chapter can be omitted if the students
have had a good course in algebra in the secondary school.
172 ELEMENTARY MATHEMATICAL ANALYSIS [§97
A single illustration will show that an equation of the fourth
degree may result from the elimination of an unknown number
between two quadratics. Thus, let:
x2 — y = 5
x^ + xy = 10
From the first, y = x^ — 5. Substituting this value of y in the
second equation, and performing the indicated operations, we
obtain:
x* + x^ - 5x + 10 = 0.
While, in general, a bi-quadratic equation results from the
process of elimination from two quadratic equations, there are
special cases of some importance in which the resulting equation
is either a quadratic equation or a higher equation in the quadratic
form. Two of these cases are:
(1) Systems in which the terms containing the unknown num-
bers are homogeneous; that is, systems in which the terms con-
taining the unknown numbers are all of the second degree with
respect to the unknown numbers, such, for example, as:
x^ — 2xy = 5
(2) Systems in which both equations are symmetrical; that is,
such that interchanging x and y in every term does not alter the
equations; for example:
X2 + y2 - X - y = 7S
xy + X + y = 39
97. Unknown Tenns Homogeneous. The following work
illustrates the reasoning that will lead to a solution when applied
to any quadratic system all of whose terms containing x and y
are of the second degree. Let the system be:
x^ — xy = 2
2x2 -f 2/2 = 9 (1)
Divide each through by x^ (or y^), then:
1- iy/x) = 2/a;2
2-f (2//x)2 = 9/a;2 (2)
Since the left members were homogeneous, dividing by x^ renders
them functions of the ratio (y/x) alone; call this ratio m. Then
SINGLE AND SIMULTANEOUS EQUATIONS 173
»iis (2) contain only the unknown numbers m and x^.
bter is readily eliminated by subtraction, leaving a quad-
>T the determination of m. When m is known, substituting
determines x, and the relation y — mx determines the
•ending values of y,
above illustrates the principles on which the solution is
In practice, it is usual to substitute y — mx 2iX once, and
liminate x'^ by comparison; thus, from the substitution
c in (1), we obtain:
x^ — mx^ = 2
2x^ + rn^x^ = 9 (3)
\'
a:2 = 2/(1 - m)
0^2 = 9/(2 + m2) (4)
e:
2/(1 - m) = 9/(2 + m2) (5)
2m2 + 9m = 5 (6)
ing:
(2m - 1) (m + 5) = 0 (7)
m = 1 /2 or - 5 (8)
x= ±2or ± (l/3)v/3
2/= ± lorq: (5/3)\/3 (9)
solutions should be written as corresponding pairs of values
•ws:
a; =-2 x= (l/3)\/3 x=-(l/3)v^
2/ = - 1 2/ = - (5/3)^^ 2/ = (5/3)^/3
system can readily be solved without the use of the mx
ution by merely solving the first equation for y and sub-
ig in the second.
)hically (see Fig. 82), the above problem is equivalent to
the intersections of the curves:
x{x - 2/) = 2
(\/2a;)2 + 2/« = 9
first is a curve with the two asymptotes a^ = 0 and x — y
J
174 ELEMENTARY MATHEMATICAL ANALYSIS [lie
■^ 0. As a matter of fact, the curve is a hyperbola, although pKxJ
that sucti is the caee cannot be given untO the method of rotating
any curve about the origin hae been explained. The second curve
is obviously an ellipse generated from a circle of radius 3 bj
shortening the abscissaa in the ratio y/2: 1- The two curves
intersect at the points:
X = 2 - 2 0.557 ... - 0.557 ...
y=l - 1 -2.887 ... +2.887 ...
\
1
y
y
/ /\
/(
/>'^*''
/ / '
/'"I
X-
X
a/
S 1
'""'//
^
V
//
'T.
-«--'/■ YT
/
r
1\
Fia. 82 .^Solutions of a. Set of Simultaaeous Quadratics given gtapli'
ically by the coordinates of the points of Intersection of the EtlipM Kii
Hyperbola.
The auxiliary lines, y = ^x and y = — 5x, made use of in the
solution are shown by the dotted lines.
98. Symmetrical Systems. Simultaneous quadratics of this
type are always readily solved analy ticaUy by seeking for the valua
of the binomiftla x + v and x — y. The ingenuity of the student
|»8l SINGLE AND SIMULTANEOUS EQUATIONS 175
will usually show many short cuts or special expedients adapted
to the particular problem. The following worked examples
point out some of the more common artifices used.
1. Solve
x + y:=Q (1)
xy = 5 (2)
Squaring (1)
x^ + 2xy + 2/2 = 36 (3)
Subtracting four times (2) from (3) :
x^ - 2xy + y^ = IQ
whence:
X - y = ±4
But from (1):
x + y = Q
Therefore:
X = 5 X = 1
y = 1 2/ = 5
2. Solve
a:2 + 2/2 = 34
(1)
xy = 15
(2)
A^ddmg two times (2) to (1):
x^ + 2x2/ + 2/* = 64
(3)
Subtracting two times (2) from (1):
x^ — 2xy + 2/* = 4
(4)
^ence, from (3) and (4) :
x + y = ±8
x-y = ±2
therefore :
x = 5
X = 3 X = - 5
X
= -3
2/ = 3
2/ = 5 2/ = - 3
y
= - 5
rhe hyperbola and circle represented by (1) and (2)
should be
IrawH by the student.
o*
x8 + 2/3 = 72
(1)
X +y = 6
(2)
Cubing (2) :
x» + 3aj2y + 3x2/2 + 2/» = 216
(3)
4
176 ELEMENTARY MATHEMATICAL ANALYSIS (§98
Subtracting (1) and dividing by 3:
xy{x + 2/) = 48 (4)
whence, since
« + 2/ = 6
we have xy = S (5)
From (2) and (5) proceed as in example 1, and find:
a; = 4 X = 2
y = 2 2/ = 4
Otherwise, divide (1) by (2) and proceed by the usual method.
4. Solve
x^ + xy = {7/3)(x + y) (1)
y^ + xy=^ {ni3)(x + y) (2)
adding (1) and (2):
{x + yy - 6(0: + 2/) = 0 (3)
whence:
X + 2/ = 0 or 6 (4)
Now, because x -\- y m b. factor of both members of (1) and (2),
the original equations are satisfied by the unlimited number of
pairs of values of x and y whose sum is zero, namely, the coor-
dinates of all points on the line x + y = 0.
Dividing (1) by (2), we get:
xly = Tin
This, and the line a; + y = 6, from (4), give the solution:
X = 7/3
y = 11/3
Graphically, the equation (1) is the two straight lines:
{x-imx + y) =0
Equation (2) is the two straight lines:
{y- limx + y) =0
These loci intersect in the point (7/3, 11/3) and also intersect
everywhere on the line x + 2/ = 0.
Exercises
1. Show that:
X* + 2/2 = 25
X + 2/ = 1
It] SINGLE AND SIMULTANEOUS EQUATIONS 177
as a solution, but that there is no real solution of the system:
a;« + yj = 25
ic +2/ == 11.
2. Do the curves:
ic2 + 2/« = 25
xy = 100, intersect?
Do the curves : x* + y^ = 25
xy = 12, intersect?
3. Solve:
(iC«+2/')(x+2/) =272
35* + 2/* + aJ + 2/ = 42.
Note: Call aj* + j/* = w, and x -\- y = v.
4. Show that there are four real solutions to :
a;2 + 2/2 - 12 = x + 2/
a;2/ + 8 = 2(a; + y).
6. Solve: x* + 2/* + a; + 2/ = 18
xy — Q.
99.. Graphical Solution of the Cubic Equation. The roots of a
ibic x^ + ax^ + 6x + c = 0 (where a, 6, and c are §iven known
umbers) may be determined graphically as explained in §39,
r we may. proceed as follows: The next highest term in the
ibic may be removed by the substitution x = xi — a /3, as may
tadily be shown by trial. Hence, it is merely necessary to con-
der cubic equations of the form:
x^ + ax + h = Q (1)
onsider the system of equations:
2/ = x» I
y = — ax — 0 ]
raphically, the curves consist of the cubic parabola (Fig. 83)
id a straight line. The intersections of the two graphs give the
>lutions of the system. Eliminating y by subtraction, we obtain
x> + ax + 6 = 0
hich shows that the values of x that satisfy the system (2)
ive the roots of equation (1). Hence (1) may be solved by means
f the graph of (2). In this graph the cubic parabola y = x^
I the same for all cubics; hence if the cubic parabola be once
rawn accurately to scale, then all cubic equations can be solved
12
178 ELEMENTARY MATHEMATICAL ANALYSIS [S99
by properly drawing the appropriate straight line, or by properiy
laying a straight edge across the graph of the cubic parabola.
In drawing the graph of the cubic parabola, it is desirable to
use, for the ^-scale, one-tenth of the unit used for the x-acale, so as
to bring a greater range of values for y upon an ordinary sheet of
coftrdinate paper. The cubic parab-
ola graphed to this scale is shown
in Fig. 83. The diagram gives the
solution of a:' — X — I =- 0. The
graphs y = X* and y = a; -J- 1 are
seen to intersect at a; = 1.32. This,
then, should be one root of the
cubic correct to two decimal places.
The line y = x + 1 cuts the cubic
parabola in but one point, which
shows that there is but one real root
of the cubic. To obtain the imagi-
nary roots, divide s' — x — 1 by
X - 1.32. The result of the divi-
sion, retaining but two places of
decimals in the coefficients, is:
x'+ 1.32r-f 0.7424 (3)
Putting this equal to zero and solv-
ing by completing the square, we
find:
= - 0.66 ± V^O.3068
X 0.66 ± OMV- I (4}
in which, of course, the coefficients are not correct to more than
two places.
The equation:
x' - 10^ - 10 = 0 (5)
illustrates a case in which the cubic has three real roots. The
straight line y '= lOx + 10 cuts the cubic parabola (see Fig. 83)
at a; = - 1.2, x = - 2.4, and x = 3.6. These, then, are the
approximate roots. The product:
^ ix + 1.2)(x -I- 2A)(x ~ 3.6) ='x' - lO.O&c - 10.37 1
§99] SINGLE AND SIMULTANEOUS EQUATIONS 179
should give the original equation (5) . This result checks the work
to about two decimal places.
It is obvious that a similar process will apply to any equation
of the form
x» + ax + 6 = 0
The a;-scale of Fig. 83 extends only from — 5 to + 5. The
same diagram may, however, be used for any range of values by
suitably changing the unit of measure on the two scales; thus, the
divisions of the a;-8cale may be marked with numbers 5-fold the
present numbers, in which case the numbers on they-scale must be
marked with numbers 125 times as great as the present numbers.
These results are shown by. the auxiliary numbers attached to
the y-scale in Fig. 83.^
Exercises
Solve graphically the following equations checking each result
separately.
1. a;» - 4a? + 10 = 0.
2. x^ -12a; -8=0.
3. x^ + X - 3 = 0.
4. x» -15x -5=0.
6. a;8 - 3a; + 1 = 0.
6. x3 - 4a; - 2 = 0.
7. 2 sin ^ + 3 cos $ = l"i5.
Note: Construct on polar paper the circles p = 2 sin d and
p = 3 cos 0.
8. 2a; + sin a; = 0.6.
Note: Find the intersection of y = sin x and the line
y = — 2x + 0.6. If 1.15 inches is the amplitude of t/ = sin a;, then
1.15 must be the unit of measure used for the construction of the
line 2/ = - 2a; + 0.6.
9. x^ +X + 1 +l/x = 0.
10. Show that x^ -^ ax -{-h =0 can have but one real root if a > 0.
11. (a) Show that the graph oi y = x^ -{-hx is symmetrical with
respect to the origin. (See §37, equation (1).)
^For other graphical methodic of solution of equaitons, see Range's " Graph-
ical Methods," Columbia Univerrity Press, 1912.
180 ELEMENTARY MATHEMATICAL ANALYSIS [§100
(6) Show that the graph oi y ^ x^ -{- hx -{- cia symmetrical with
respect to the point (0, c).
(c) If the substitution a; = Xi — a/3 removes the term ox* from the
equation y — x* -{- ax^ + bx -{- c, show that the graph of this last
equation must be symmetrical with respect to some point.
12. On polar paper, draw a curve showing the variation of local or
mean solar time with the longitude of points on the earth's surface.
If it be noon by both standard and mean solar (local) time at Green-
wich, longitude 0°, construct a graph on polar paper showing standard
time at all other longitudes, if the longitude of a point be represented
by the vectorial angle on polar paper and if time relative to Greenwich
be represented on the radius vector using 1 cm. =2 hours, and also if
it be assumed that the changes of standard time take place exactly at
15° intervals beginning at 7i° west longitude.
If it be noon at Greenwich, write an equation which will express the
local time of any point in terms of the longitude of the point. Does the
expression hold for points having negative longitude? Does this
function possess a discontinuity?
Can a similar expression be written giving the standard time at any
point in terms of the longitude of the point?
If £ be standard time and B longitude, and if the functional relation
by expressed by /, so that :
t = m
is / a continuous or a discontinuous function? Is the function /
defined for 0 = 15°, 30°, 45°, etc., and why?
In actual practice, how is the function / given?
100. Method of Successive Approximations. The graphic
method of solving numerical equations, combined with the method
explained below, is the only method which is universally ap-
plicable. It therefore possesses a practical importance exceeding
that of any other method. An example will illustrate the method.
Suppose that it is required to find to four decimal places one
root of x3 - x - 1 = 0. See §99 and Fig. 83. The graphic
method gives x = 1.32. This is the first approximation. A
second approximation is found as follows: Build the table
of values fory = x^ — x — l
X y
1.32
1.33
- .0200
+ .0226
0.01 . 0426 Differences.
§100] SINGLE AND SIMULTANEOUS EQUATIONS 181
Now reason as follows: The actual root lies between 1.32 and
1.33, and the zero value 'of y corresponds to it. This zero is
200/426 of the way between the two values of y; hence if the
curve be nearly straight between x = 1.32, and x = 1.33,
the desired value of x is approximately 200/426 of the way
between 1.32 and 1.33 or it is x = 1.324694. This value is
probably correct to the fourth decimal place.
To find a third approximation we build another table of
values:
X I y
1.3247i - .0000766
1.3248' + .0003499
0.0001 .0004265 Differences.
Reasoning as before, we get x = 1.324718 which is very likely
true to the last decimal place.
The above method is applicable to an equation like exercise
8 above. In fact it is the only numerical method that is
applicable in such cases.
CHAPTER VI
PERMUTATIONS AND COMBINATIONS;
THE BINOMIAL THEOREM
101. Fudamental Principle. // one thing can be done in n
different ways and another thing can he done in r different ways^
then both things can be done together, or in succession, in n X r
different ways. This simple theorem is fundamental to the work
of this chapter. To illustrate, if there be 3 ways of going from
Madison to Chicago and 7 ways of going from Chicago to New
York, then there are 21 ways of going from Madison to New
York.
To prove the general theorem, note that if there b6 xynly one
way of doing the first thing, that way could be associated with
each of the r ways of doing the second thing, making r ways
of doing both. That is, for each way of doing the first, there are
r ways of doing both things; hence, for n ways of doing the first
there are n X r ways of doing both.
Illustrations: A penny may fall in 2 ways; a common
die may fall in 6 ways; the two may fall together in 12 ways.
In a society, any one of 9 seniors is eligible for president and any
one of 14 juniors is eligible for vice-president. The number of
tickets possible is, therefore, 9 X 14 or 126.
I can purchase a present at any one of 4 shops. I can give it
away to any one of 7 people. I can, therefore, purchase and give
it away in any one of 28 different ways. '
A product of two factors is to be made by selecting the first
factor from the numbers a, 6, c, and then selecting the second factor
from the numbers x, y, z, u, v. The number of possible products
is, therefore, 15.
If a first thing can be done in n different ways, a second in r
different ways, and a third in s different ways, the three things
can be done in n Xr X s different ways. This follows at once
from the fundamental principle, since we may regard the first
two things as constituting a single thing that can be done in nr
182
§102] PERMUTATIONS AND COMBINATIONS 183
ways, and then associate it with the third, making nr X s ways
of doing the two things, consisting of the first two and the third.
In the same way, if one thing can be done in n different ways, a
second in r different ways, a third in s, a fourth in t, etc., then all
can be done together in nXrXsXt. . . different ways.
Thus, n different presents can be given to x men and a women
in (z + ay different ways. For the first of the n presents can
be given away in (x + a) different ways, the second can be given
away in (x + a) different ways, and the third in {z + a) different
ways and so on. Hence, the number of possible ways of giving
away the n presents to (x + a) men and women is:
{z + a){z + a){z + a) . . . to n factors, or (z + a)"
102. Definitions. Every distinct order in which objects
may be placed in a line or row is called a permutation or an
arrangement Every distinct selection of objects that can be
made, irrespective of the order in which they are placed, is called
a combination or group.
Thus, if we take the letters a, 6, c, two at a time, there are six
arrangements, namely: afe, acy ha, he, ca, ch, but there are only
three groups, namely: afe, ac, be.
If we take the three letters all at a time, there are six arrange-
ments possible, namely: ahc, act, hca, hac, cab, cba, but there is
only one group, namely: abc.
Permutations and combinations are both results of mode of
selection. Permutations are selections made with the understand-
ing that two selections are considered as different even though
they differ in arrangement only; combinations are selections made
with the understanding that two selections are not considered as
different, if they differ in arrangement only.
In the following work, products of the natural numbers like
1X2X3; 1X2X3X4X5; etc.
are of frequent occurrence. These products are abbreviated by
the symbols 315! and read "factorial three," "factorial five"
respectively.
103. Formula for the Number of Permutations of n Different
Things Taken All at a Time. We are required to find how many
possible ways there are of arranging n different things in a line.
184 ELEMENTARY MATHEMATICAL ANALYSIS [§104
Lay out a row of n blank spaces, so that each may receive one of
these objects, thus:
.Ll_i LAj 1:3 _i i±j |_5_; . . . i^j
In the first space we may place any one of the n objects; therefore,
that space may be occupied in n different ways. The second
space, after one object has been placed in the first space, may be
occupied in (n — 1) different ways; hence, by the fundamental
prirTciple, the two spaces may be occupied in n{n — 1) different
ways. In like manner, the third space may be occupied in (n — 2)
different ways, and, by the same principle, the first three spaces
may be occupied in n(n — 1) (n — 2) different ways, and so on.
The next to the last space can be occupied in but two different
ways, since there are but two objects left, and the last space
can be occupied in but one way by placing therein the last re-
maining object. Hence, the total number of different wajrs of
occupying the n spaces in the row with the n objects is the product:
n(n - 1) (n - 2) . . . 3 • 2 • 1
or,
n!
If we use the symbol Pn to stand for the number of permutations
of n things taken all at a time, then we write:
Pn = n! (1)
104. Formula for the Number of Permutations of n Things
Taken r at a Time. We are required to find how many possible
ways there are of arranging a row consisting of r different things,
when we may select the r things from a larger group of n different
things.
For convenience in reasoning, lay out a row of r blank spaces,
so that each of the spaces may receive one of the objects, thus:
111121131 Ir-lllr.
. In the first space of the row, we may place any one of the n objects;
therefore, that space may be occupied in n different ways. The
second space, after one object has been placed in the first space,
may be occupied in (n — 1) different ways; hence, by the fun-
damental principle, the two spaces may be occupied in n{n — 1)
different ways. In like manner, the third space may be occupied
§104) PERMUTATIONS AND COMBINATIONS 185
in (n — 2) different ways, and hence, the first three may be
occupied in n(n — 1) (n — 2) different ways, and so on. The
last or rth space can be occupied in as many different ways as there
are objects left. When an object is about to be selected for the
rth space, there have been used (r ~ 1) objects (one for each of
the (r — 1) spaces already occupied). Since there were n objects
;o begin with, the number of objects left is n — (r — 1) or
I — r + 1, which is the number of different way^ in which the
ast space in the row may be occupied. Hence, the formula:
P...= n(n - l)(n - 2) . . . (n - r + 1) (1)
II which Pn,r stands for the number of permutations of n things
aken r at a time.
The formula, by multiplication and division by l^jlT? becomes:
t(n - 1) . . . {n-'r+ l)(n - r){n -r-1) . . . 3-2-1
{n-r){n - r - 1) . . . 3-21
Pn.r= — ^ (2)
(n-r)!
This formula is more compact than the form (1) above, but the
raction is not in its lowest terms.
Formula (1) is easily remembered by the fact that there are
ust f factors beginning with n and decreasing by one. Thus we
lave:
Pio,7 = 10X9X8X7X6X5X4
Exercises
1. How many permutations can be made of six things taken all at
ktime?
2. How many different numbers can be made with the five digits
., 2, 3, 4, 5, using each digit once and only once to form each number?
3. The number of permutations of four things taken all at a time
>ears what ratio to the number of permutations of seven things taken
Jl at a time?
4. How many arrangements can be made of eight things taken
hree at a time?
6. How many arrangements can be made of eight things taken
ive at a time?
6. How many four-figure numbers can be formed with the ten
ligits 0, 1, 2, . . . 9 without repeating any digit in any number?
186 ELEMENTARY MATHEMATICAL ANALYSIS [§105
7. How many different ways may the letters of the word algebra
be written, using all of the letters?
8. How many different signals can be made with seven different
flags, by hoisting them one above another five at a time?
9. How many different signals can be made with seven different
flags, by hoisting them one above another any number at a time?
"^>^0. How many different arrangements can be made of ninQ ball
players, supposing only two of them can catch and one pitch?
106. Formula for the number of combinations or groups of n
different things taken r at a time.
It is obvious that the number of combinations or groups con-
sisting of r objects each that can be selected from n objects, is
less than the number of permutations of the same objects taken
r at a time, for each combination or group when selected can be
arranged in a large number of ways. In fact, since there are r
objects in the group, each group can be arranged in exactly r!
different ways. Hence, for each group of r objects, selected from
n objects, there exists r! permutations of r objects each. There-
fore, the number of permutations of n things, taken r at a time, is
r! times the number of combinations of n objects taken r at a
time. Calling the unknown number of combinations x, we have:
xri = Pnw =
(n — r) !
or, solving for x:
n\
X =
rl{n — r)\
This is the number of combinations of n objects taken rat a time,
and may be symbolized:
r ^ ?:! (1)
"" rl(n-r)l
This fraction will always reduce to a whole number. It may be
written in the useful form:
^ ^ n(n-l)(n-2) . . . (n - r + 1) (2)
1X2X3 ... r
It IS easily remembered in this form, for it has r factors in both
the numerator and the denominator. Thus for the number of
J105] PERMUTATIONS AND COMBINATIONS 187
combinations of ten things taken four at a time we have four
factors in the numerator and denominator, and
_ 10 X 9 X 8 X 7
^10,4- 1X2X3X4
Exercises
1. How many different products of three each can be made with the
five numbers a, 6, c, d, e, provided each combination of three factors
gives a different product.
2. How many products can be made from twelve different num-
bers, by taking eight numbers to form each product?
3. How many products can be made from twelve different num-
bers, by taking four- numbers to form each product?
4. How many different hands of thirteen cards each can be held
at a game of whist?
6. In how many ways can seven people sit at a round table?
6. In how many ways can a child be named, supposing that there
are 400 different Christian names, without giving it more than three
names?
''^. In how many ways can a committee of three be appointed
from six Germans, four Frenchmen, and seven Americans provided
each nationality is represented?
8. There are five straight lines in a plane, no two of which are
parallel; how many intersections are there?
9. There are five points in a plane, no three of which are collinear;
how many lines result from joining each point to every other point?
10. In a plane there are n straight lines, no two of which are parallel ;
how many intersections are there?
11. In a plane there are n points, no three of which are collinear;
how many straight lines do they determine?
12. In a plane there are n points, no three of which are collinear,
except r, which are all in the same straight line; find the number of
straight lines whch result from joining them.
13. A Yale lock contains five tumblers (cut pins), each capable of
being placed in ten distinct positions. At a certain arrangement of the
tumblers, the lock is open. How many locks of this kind can be made
so that no two shall have the same key?
14. In how many ways can seven beads of different colors be strung
so as to form a bracelet?
188 ELEMENTARY MATHEMATICAL ANALYSIS [§106
16. How many different sums of money can be formed from a dime,
a quarter, a half dollar, a dollar, a quarter eagle, a half eagle, and an
eagle?
106.* The Arithmetical Triangle. In deriving by actual mul-
tiplication, as below, any power of a binomial x + a from the
preceding power, it is easy to see that any coefficient in the new
power is the sum of the coefficient of the corresponding term in the
multiplicand and the coefficient preceding it in the multiplicand.
Thus:
x3 + Sax^ + 3aH + a^
X + a
X* + Sax^ + Sa^x* + a^x
ax^ + Sa^x^ + SaH + a*
x^ + 4aa;« + Qa^x^ + 4a»ic + a*
or, erasing coefficients, we have:
1+3+3+1
1 + 1
1+3+3+1
1+3+3+1
1+4+6+4+1
from which the law of formation of the coefficients 1, 4, 6, . . .
is evident. Hence, writing down the coefficients of the powers
of X + a in order, we have:
Powers Coefficients
]
2
3
4
5
6
7
8
9
10
11
0 1
L
1 ]
L 1
2 ]
L 2
1
3 ]
L 3
3
1
4 ]
L 4
6
4
1
5 ]
L 5
10
10
5
1
6 ]
L 6
15
20
15
6
1
7 ]
L 7
21
35
35
21
7
1
S ]
L 8
28
56
70
56
28
8
1
9 ]
L 9
36
84
126
126
84
36
9
1
10 ]
L 10
45
120
210
252
210
120
45
10
1
[§107 PERMUTATIONS AND COMBINATIONS 189
In this triangle, each number is the sum of the number above it
and the number to the left of the latter. Thus 84 in the 9th line
equals 56 + 28, etc. The triangle of numbers was used previous
to the time of Isaac Newton for finding the coefficients of any de-
sired power of a binomial. At that time it was little suspected
that the coefficients of any power could be made without first
obtaining the ocefficients of the preceding power. Isaac Newton,
while an undergraduate at Cambridge, showed that the coefficients
of any power could be found without knowing the coefficients of
the preceding power; in fact, he showed that the coefficients of
any power n of a binomial were functions of the exponent n.
The above triangle of numbers is known as the arithmetical
triangle or as Pascal's triangle.
107. Distributive Law of Multiplication. The demonstration
of the binomial theorem may be based upon the following law of
multiplication: The product of any number of polynomials is
the aggregate of all the possible partial products which can he made
hy taking one term and only one from each of the polynomials.
This statement is merely a definition of what is meant by the
product of two or more polynomials. (See appendix.) Thus:
{x + a){y + h){z + c) =
xyz + ayz + hxz + cxy + ahz + hex + cay + ahc
Each of the eight partial products contains a letter from each
parenthesis, and never two from the same parenthesis. The
number of terms is the number of different ways in which a letter
can be selected from each of the three parentheses. In the present
case this is, by §101, 2X2X2 = 8.
108. Binomial Formula. It is required to write out the value
of {x + a)», where x and a stand for any two numbers and n is a
positive integer. That is, we must consider the product of the n
parentheses:
{x + a){x -\- a){x + a) . . . {x -\- a)
by the distributive law stated above.
First, Take an x from each of the parentheses to form one of
the partial products. This gives the term a;» of the product.
Second, Take an a from the first parenthesis with an x from
each of the other {n — 1) parentheses. This gives ax»~^ as
190 ELEMENTARY MATHEMATICAL ANALYSIS
another partial product. But if we take a from the second paren-
thesis and an x from each of the other (n — 1) parentheses, we get
ax'^'^ as another partial product. Likewise by taking a from any
of the parentheses and an a: from each of the other (n — l) paren-
theses, we shall obtain ax*-^ as a partial product. Hence, the
final product contains n terms like aa;»~^, or nax"^^^ is a part
of the product
Third. We may obtain a partial product like a'x'*^* by taking
an a from any two of the parentheses, together with the x's from
each of the other (n — 2) parentheses. Hence, there are as many
partiial products like a*a:»"* as there are ways of selecting two a's
from n parentheses; that is, as many ways as there are groups or
combinations of n things taken two at a time, or:
n{n — 1)
Hence, - ^ fl^JJ""' is another part of the product.
Fourth, We may obtain a partial product like a^»~* by taking
an a from any three of the parentheses together with the x's from
each of the other (n — 3) parentheses. Hence, there are as many
partial products like o'x**-' as there are ways of selecting three a's
from n parentheses, that is, as many ways as there are combina-
tions of n things taken three at a time, or — ^^ T^r^i '*
l'Z'6
n(n — l)(n — 2) , , . ,. , . ,,
Hence, i'2~i o'J?""' is another part of the product.
In general f we may obtain a partial product like a''a;'»~' (where r
is an integer < n) by taking an a from any r of the parentheses
together with the .r^s from each of the other (n — r) parentheses.
Hence, there are as many partial products like a^'x"-' as there are
waj's of selecting r a's from n parentheses; that is, as many ways
as there are combinations of n things taken r at a time, or
, * , • Hence, ,- * - ., a'^jr*-' stands for any term
r! {n — r)\ ' r! (n — r)! "^
in general in the product (x + a)".
Finally, we may obtain one partial product like a» by taking an
a troin each of the parentheses. Hence, a" is the last term in the
proiiuct.
§109] PERMUTATIONS AND COMBINATIONS 191
Thus we have shown that:
(x + a)» « x» + nax*-^ + . a'x**-* + . . .
rl (n — r)!
This is the binomial formula of Isaac Newton. The right side is
called the expansion or development of the power of the binomial.
It is obvious that the expansion of (x — a)" will differ from the
above only in the signs of the alternate terms containing the odd
powers of a, which, of course, will have the negative sign.
109. Binomial Theorem. The binomial expansion is a series,
that is, each term may be derived from the preceding term by a
definite law. This law is made up of two parts which may be
stated as follows:
(1) haw of Exponents, In any power of a binomial, x + a, the
exponent of x commences in the first term with the exponent of the
required power, and in the following terms continually decreases by
unity. The exponent of a commences with 1 in the second term and
continually increases by unity,
(2) Law of Coefficients, The coefficient in the first term is 1,
that in the second term is the exponent of the power; and if the
coefficient in any term be multiplied by the exponent of x in that
term and divided by the exponent of a, increased by lyit will give the
coefficient in the succeeding term.
Exercises
1. Expand (m + 3y)'. Here x — u and a — Zy. By the formula
we get:
u^ + 5ii*(3|/) + 10M»(3y)> -f 10m«(32/)» + 5it(32/)*+ (32/)»
Performing the indicated operations, we obtain :
ti» + Ibu^y -f 90M>y« + 270t*2y»+ 405m2/* + 2432/'^
Expand each of the following by the binomial formula :
2. (r» - 2)*. 8. (1/2 + x)\
3. (36 - 1/2) ». 9. (6« - c«)^
4. (c + xy. 10. (3a + l/2)«.
5. (2x» - xY. 11. {bd - Zy)K
6. (1 - o)». 12. (3x3 _ 1)4,
7. (- a; -h 2ay, 13. (Va + x)«.
192 ELEMENTARY MATHEMATICAL ANALYSIS IJllO
14. (x^ -h x^ )•. 17. (a -h [a; + y])».
15. (a-» - b^ )*. 18. (a + 6 - y)».
16. (VoS - i^)«. 19. (a;» -f- 2aa; -f- a»)».
110. Binomial Theorem for Fractional and Negative Exponents.
It is proved in the Calculus that:
,1 , N , , , n (n - 1) , , n (n - 1) (n - 2)
(1 ± a;)» = 1 ± nx H ^-^ — - x^ ± -^ ~^ -x^ + . . .
is true for fractional and for negative values of n, provided x is
less than 1 in absolute value. The number of terms in the expan-
sion is not finite, but is unlimited^ and the series or expansion
converges or approaches a definite limit as the number of terms of
the expansion is increased without limit, provided |a;| < 1.
By the above formula, we have:
vr+i = . + ( 1/2) . + "^" y - "»■
, (1/2) (1/2 - 1) (1/2 - 2) .
"i o I X ~\~ ...
= 1 + (l/2)a; - (l/8)a:* + (l/16)x' - (5/128)2*
If
X = 1/2
this becomes:
V372 = 1 + 1/4 - 1/32 + 1/128 - 5/2048 + . . .
Therefore, using five terms of the expression:
The square root, correct to four figures, is really 1.2247. Thus the
error in this case is less than one-tenth of 1 percent if only five
terms of the series be used. The d^ree of accuracy in each case
is dependent both, upon the value of n and upon the value of x.
Obviously, for a given value of n, the series converges for small
values of x more rapidly than for larger values.
As another example, suppose it is required to expand (1 — x)'*.
By the binomial theorem:
PERMUTATIONS AND COMBINATIONS 193
-1( - 1 -1) , , ,,
1 _ j;)-i = 1 + (_ i)( -x) + — ^ -- — ( - xr
==l-\-x + x^ + x^+ . . .
J terms of the series be used, the error is 1 /16 f or x = 1 /2,
>ut 3 percent.
. Approximate Formulas. If x be very small, the expansion
troximately:
(1 + x)» =? 1 + nx (1)
x^, x^ and all higher powers of x are much smaller than x.
using the symbol =f to express "approximately equals," we
for example:
(1.01)3 =r 1.03
(1 + 1/100)3 =r 1 + 3/100
rue value of (l.Ol)^ is 1.030301, so that the approximation is
?ood.
ewise:
(1 - x)» =F 1 - nx (2)
e small.
r, 2/, and z be small compared with unity, the following
ximate formulas hold:
(l + x)(l + y)=Fl+x + y (3)
(i;+x)/(l + y) =Fl + x-y (4)
(1 + x)(l + y)(l + z) =F 1 + X + y + z (5)
e approximation formulas are proved as follows:
x){l + y) = "^ + X + y + xy "^ I + X + yy for xy is small
ared to x and y,
X^ V^ — XV
— V = 1 + X — 2/ + --— 1 — -' =? 1 + X — y, for the fraction is
y) *^ ' 1 + 2/
compared to x and y.
x){l + y)(l + z) ^ {I + X + y) (l + z) ^ I + X + y + z
13
194 ELEMENTARY MATHEMATICAL ANALYSIS 11113
112. * The ProgreBsive Mean. In using scientiiic data it is often
desirable to determine the so-called progresdve mean of a h^;hly
fluctuating magnitude. Thus if we wish to detra'mine whethff
or not the rainfall at New York has on the average been increasing
or decreasing in the last !00 years, we form an average for eaeli
successive group of five or six or seven or other convenient numbv
of years, and tabulate and compare these averages. In finding
these averages, however, the various years are weighted ss
1
n
1
A '
'l r
'•' Vn
/
j
' 7^-
l-::-l.
-^ Ui
\
V> u-
-
H
.
■:
I
_
HI
1
■
.
ii!
m iiii
1
fallows: If the tuimbers whose progressive means are desired be
(ii. ti!. Hj. <i« then the pn^ressive mean correspondmg
U> <ii« would Ih'. for tive-yoar inter\als.
fi = y,.ii-t- 4,!,-i-tJii;,-r 4o„ + Oil) ,'16
and for !ioveu-yt\-»r intorvak, •
-I - V.I: • IV.,* l.\!,-2l.Vi;*-r 15on + 6a„ + 0,^/64
l« Ouv*o oxptxvfsious (ho rtH>flioion»s aw the binomial coefficimts
and lh« divisors arv ih* sum of the co^doits. See Fig. 84.
ill2] PERMUTATIONS AND COMBINATIONS 195
Exercises
1. Explain the following approximate formulas, in which xl < 1
Vl H-a;^ 1 +(l/2)x
Vl - X ? 1 - (l/2)x
(1 +xr^=r 1 - (l/2)x
■ ^1 -f X ? 1 + (l/3)x
^1 - X =7= 1 - (l/3)x
(1 + x)-^ =F 1 - (2/3) X
(1 H-x2)''^=? 1 +(l/3)x2.
2. Compute the numerical value of :
(1.03)^ (1.05)^
(1.02) (1.03) 1.02/1.03
(1.01)(1.02)/(1.03)(1.04).
3. The formula for the period of a simple pendulum is :
T = WT/g
I'or the value of gravity at New York, this reduces to
^ 6.253
in which ly the length of the pendulum, is measured in inches. This
pendulum beats seconds when
I = (6.253)2 or 39.10 inches.
^at is the period of the pendulum if I be lengthened to 39.13 inches?
Hint:
VT
^ 6.253
^ - -6.253~ " 6:253 ^1 + ^/^
Take I = 39.10, and h = 0.03
Then:
• r = 1 4- 0.03/78.20
= 1.00038.
A day contains 86,400 seconds. The change of length would, there-
fore, cause a loss of 32.8 seconds per day, if the pendulum were attached
to a clock.
196
ELEMENTARY MATHEMATICAL ANALYSIS [(lU
how tar can one see at an elevation of h feet «bon
4. On the
ita Burface?
Call the radius of the earth a{ = 3960 miles), and the diatanta
one can see d, which is along a tangent from the point of observatira
to the sphere. Since A is in feet, and a +■ fmTji d, and a are the «da
of a right triangle, we have (o + ft/5280)» = d* + a'
on a'(I + ft/5280a)" = d' + a'.
■fir'
pit:!'"
liiii::
■ i
, , , j ..- ;: ; . .
/
I
■ 1 1
11' "
° "' '' T T "!;:'
1
Fig. 85.— Graphical Representalion of the Values of the Bin
CoefficieDte in the 999th power of a Binomial. The middle coefficien
taken equal to 5, for convenience, and the others are eipresBed tr
scale also.
Expanding by the approximate formula:
o'(l + 2A/6280O) =d' + a'
or:
d' = 3oA/5280
= 3 X 3960ft/5280
= (3/2)A
d = VlS/2)h '
where d is expressed in miles and A ia in feet. See §««, exercise 1
§113] PERMUTATIONS AND COMBINATIONS 197
6. How much is the area of a circle altered if its radius of 100 cm.
he changed to 101 cm.?
6. How much is the volume of a sphere, ^a^, altered if the radius
be changed from 100 cm. to 101 cm.?
7. If the formula for the horse power of a ship is I.H.P. = "onoT
where S is speed in knots and D is displacements in tons, what in-
crease in horse power is required in order to increase the speed from
fifteen to sixteen knots, the tonnage remaining constant at 5000?
What increase in horse power is required to maintain the same speed
if the load or tonnage be increased from 5000 to 5500?
113. * Graphical Representation of the Coefficients of any Power
of a Binomial. If we erect ordinates at equal intervals on the
X-axis proportional to the coefficients of any power of a binomial,
we find that a curve is approximated, which becomes very striking
as the exponent is taken larger and larger. In Fig. 85, the ordi-
nates are proportional to the coefficients of the 999th power of
(x + a). The drawing is due to Quetelet.
The limit of the broken line at the top of the ordinates in Fig. 85
is, as n is increased indefinitely, a bell-shaped curve, known as
the probability curve; its equation is of the form y = ae~^'^, as
is shown in treatises on the Theory of Probability.
CHAPTER VII
PROGRESSIONS
114. An Arithmetical Progression or an Arithmetical Series,
is any succession of terms such that each term differs from that
immediately preceding by a fixed number called the common
difference. The following are arithmetical progressions:
(1) 1, 2, 3, 4, 5. •
(2) 4, 6, 8, 10, 12.
(3) 32, 27, 22, 17, 12.
(4) 2i, 31, 5, 6i, 7i.
(5) (u — v)j u, {u + v).
(6) tty a + d, a + 2dy a + 3d, . . .
The first and last terms are called the extremes, and the oth^
terms are called the means.
Where there are but three numbers in the series, the middle
number is called the arithmetical mean of the other two. To find
the arithmetical mean of the two numbers a and 6, proceed as
follows:
Let A stand for the required mean; then, by definition:
A — a = h — A
whence:
A = {a + h)/2
Thus, the arithmetical mean of 12 and 18 is 15, for 12, 15, 18 is an
arithmetical progression of common difference 3.
By the arithmetical mean or arithmetical average of several
numbers is meant the result of dividing the sum of the numbers by
the fiumber of the numbers. It is, therefore, such a number that
if all numbers of the set were equal to the arithmetical mean, the
sum of the set would be the same.
The general arithmetical progression of n terms is expressed by:
Number of
term: 12 3 4 . . . n
Progression: a, {a + d), {a + 2d), (a + Sd), . . . (a + [n — Ijd)
198
§1151 PROGRESSIONS 199
Here a and d may be any algebraic numbers whatsoever, integral
or fractional, rational or irrational, positive or negative, but n
must be a positive integer. If the common difference be negative,
the progression is said to be a decreasing progression; otherwise,
an increasing progression.
From the general progression written above, we see that a for-
mula for deriving the nth term of any progression may be written:
1 = a + (n - l)d (1)
in which I stands for the nth term.
115. The Sum of n Terms. If s stands for the sum of n terms
of an arithmetical progression, and if the sum of the terms be
written first in natural order, and again in reverse order, we have:
s = a + {a + d) + (a + 2d)+ . . . +Xa + [n - l]d) (1)
s= I +(l -d) + (1 -2d) + . . . + (/ - [n - l]d) (2)
Adding (1) and (2), term by term, noting that the positive and
negative common differences nullify one another, we obtain:
2s = (a + 0 + (a + 0 + (a + 0 + . . . + (a + 0 (3)
or, since the number of terms in the original progression is n, we
may write:
2s = n{a + I)
or: s = n(a+l)/2 (4)
In the above expression, (a + I) 1 2 is the average of the first and
T^th terms. The formula (4) states, therefore, that the sum equals
the number of the terms multiplied by the average of the first and
last.
116. An arithmetical progression is a very simple particular in-
stance of a much more general class of expressions known in mathe-
matics as series. A series is any sequence of terms formed accord-
ing to some law, such as:
(a;+ 1) + {x + 2y+ix + 3y, . . .
cos X + cos 2x + cos 3x + . • •
It is only in a very limited number of cases that a short expression
can be found for the sum of n terms of a series. An arithmetical
progression is one of these exceptions.
200 ELEMENTARY MATHEMATICAL ANALYSIS 1(117
117. The formulsB (1) and (4) above are illustrated graphically
I)y Fig. 86. Ordinate^ proportional to the terms of a progression
are laid off at equal intervals on the
line OX. The ends of these lines, be-
cause of the equal increments in the
terms of the series, lie on the straight
line MA'. By reversing terms and
adding, the sums lie within the rec-
tangle OK whose altitude is {a -I- I).
The sum of an arithmetical pro-
gression is readily constructed. On
0 Y, lay off the unit of measure 01;
and, to the same scale, n. On OX,
lay off (a -I- /)■ From 2 on OF draw
a line to (a -|- 0 on OX. From n on
OY draw a parallel to the latter, out-
ting OX in s, the required sum. This
construction has little value, except
that it illustrates that s, for all
values of a and d, increases indefi-
nitely in absolute value as n increases
without limit, or, using the equiva-
lent terms already explained, that
: becomes infinite as n becomes in-
finite.
118. Formula (I), $114, enables us to obtain the value of any'
one of the numbers, I, a, n, d, when three are given. Thus:
(1) Find the 100th term of:
3 -I- 8 -I- 13 + . . .
; 7 a 9
6.— Graphical Deter i
■not the Sum of and. P.
therefore.
l = Z-\
J X 5 =
(2) Find the number of terms in the progression:
5-1-7 + 9-1- . - . -1-39
Here: a= 5,d = 2, i = 39
whence: 39 = 5 -1- (n - 1)2
Solving for n: n = IS
§118] PROGRESSIONS 201
(3) Find the common difference in a progression of fifteen terms
in which the extremes are 1/2 and 42 1:
Here: a = 1/2, Z = 42^ n = 15
whence: 42^ = 1/2 + (15 - l)d
Solving: d = 3
Formula (4) , §115, enables us to find the value of any one of the
numbers s, n, a, Z, when the values of the other three are given.
Thus:
(5) Find the number of terms in an arithmetical progression in
which the first term is 4, the last term 22, and the sum 91.
Here: a = 4, ^ = 22, s = 91
whence: 91 = n(4 + 22) /2
solving forn: n = 7
The two formulas, (1) §114 and (4) §116, contain five letters:
hence, if any two of them stand for unknown numbers, and the
values of the others are given, the values of the two unknown
numbers can be found by the solution of a system of two equa-
tions. Thus:
(6) Find the number of terms in a progression whose sum is
1095, if the first term is 38 and the difference is 5.
Here: s = 1095, a = 38, and d = 5
whence: / = 38 + (n - 1)5 (1)
1095 = n(38 + I) /2 (2)
From (1) : Z = 33 + 5n (3)
From (2) : 2190 = 38n + nl (4)
Substituting the value of I from (3) in (4), we get:
2190 = 7 In + 5n2 (5)
Solving this quadratic, we find:
n = 15, or - 29.2
The second result is inadmissible, since the number of terms
cannot be either negative or fractional.
Exercises
Solve each of the following :
1. Given, a = 7, d = 4, w = 15; find I and s,
2. Given, o = 17, Z = 350, d = 9; find n and s.
a
202 ELEMENTARY MATHEMATICAL ANALYSIS [§119
3. Given, a = 3, n = 50, «.= 3825; find I and d.
4. Given, « = 4784, a = 41, d = 2; find I and n.
6. Given, s = 1008, d = 4, i = 88; find a and n.
6. Find the sum of the first n even numbers.
7. Find the sum of the first n odd numbers.
8. Insert nine arithmetical means between — 7/8 and + 7/8.
9. Sum (a + by + (a* + b^) + (a - by to w terms.
10. Find the sum of the first fifty multiples of 7.
11. Find the amount of $1.00 at simple interest at 5 percent for
1912 years.
12. How long must $1.00 accumulate at 3 J percent simple inter-
est until the total amounts to $100?
13. How many terms of the progression 9 + 13 + 17+ . . .
must be taken in order that the sum may equal 624? How many
terms must be taken in order that the sum may exceed 750?
14. Show that the only right triangle whose sides are in arithmetical
progression is the triangle of sides 3, 4, 5, or a triangle with sides pro-
portional to these numbers.
119. Geometrical Progression. A geometrical progression
is a series of terms such that each term is the product of the
preceding term by a fixed factor called the ratio. The following
are examples:
(1) 3, 6, 12, 24, 48.
(2) 100, -50, 25, -12i
(3) 1/2, 1/4, 1/8, 1/16, 1/32.
(4) a, ar, ar^y ar^, ar* . . .
The geometrical mean of two numbers, a and 6, is found as
follows: Let G stand for the required mean. Then, by the
definition of a geometrical progression:
G;a =b;G
whence:
or: G' = ah
G = \ab
Thus, 4 is the geometrical mean of 2 and 8. The arithmetical
mctvk of 2 and S is 5. The geometrical mean of n positive num-
bers is tho vahio of the nth root of their product. Thus the geo-
nxotriciU n\i\u\ of:
S, l> and 24 is 12 = \ S X 9X^4
§120] PROGRESSIONS 203
120. The nth Term and the Sum of n Terms. If a represents
the first term and r the ratio of any geometrical progression, the
progression may be written:
Number of term: 123 4...n— In
Progression: a, ar^ ar^, ar^, . . . af^'^y ar""'^
Therefore, representing the nth term by l, we obtain the simple
formula:
I = ar«-i (1)
Representing by s the sum of n terms of any geometrical pro-
gression, we have:
s = a + ar + ar^ -{• . . . + ar""'^ + ar""'^
Factoring the right member:
s = a(l + r + r2 + . . . + r"-^ + r^-O
But, by a fundamental theorem in factoring, ^ the expression in the
parenthesis is the quotient of 1 — r" by 1 — r. Hence:
s = a(l - r») /(I - r) (2)
Another form is obtained by introducing I by the substitution:
s = (a - rl) /(I - r) (3)
121. Formula (1), or (2), enables one to find any one of the four
numbers involved in the equations when three are given. The
two formulas (1) and (2) considered as simultaneous equations
enable one to find any two of the five numbers a, r, n, /, s, when the
other three are given. But if r be one of the unknown numbers, the
equations of the system may be of a high degree, and beyond the
range of Chapter VII, unless solved by graphical means. If n be
an unknown number, an equation of a new type is introduced,
namely, one with the unknown number appearing as an exponent.
Equations of this type, known as exponential equations, will be
treated in the chapter on logarithms. The following examples
illustrate cases in which the resulting single and simultaneous
equations are readily solved.
(1) Insert three geometrical means between 31 and 496-
Here:
a = 31, / = 496, and n = 5
1 See Appendix.
i
204 ELEMEXTJO^Y 31ATHEMATICAL AXALY^I^ 'fia
11
j]
496 = 31 X r« 1
or: tfc
r*= 16
tbcrefore: ^
r = ±2
axseqaentiy, the required means are ehher 6Z 124. and 24S,
or - 62, -H 124, and - 248.
(2) Find the sum of a geometrical progreaacxL <^ fi^e tsne,
the exteemes being 8 and 10^368.
a = 8, / = 10,368, and ji = 5
whence:
10,368 = Sr* (1)
9 = (10,368r - 8) ;(r - 1) i2)
From the first,
r = 6
whence, irmn the second,
8 = 12,440
(3) Find the extremes ol a geome^cal progres^on whoge jooi
19 635, if the ratio be 2 and the numbar of terms be 7.
Here:
8 = 635, r = 2, and n = 7
whence:
/ = a-2« (1)
635 = (2/ -a) /I i2)
Substituting / from (1) in (2), we get:
635 = 128 a - a
whence:
a = 5, hence, / = 320
(4) The fourth term of a geometrical progression is 4, and the
sixth t^m is 1. What is the tenth term?
Here:
ai^ = 4 (1)
and:
or* = 1 (2)
1122] PROGRESSIONS 205
v^hence, dividing (2) by (1) :
r* = 1/4, or r = ± 1/2
herefore, from (1):
a = 4/r» = ± 32
Then the tenth term is:
+ 32( + l/2)»= 1/16
Exercises
1. Find the sum of seven terms of44-84-164-. • .
2. Find the sum of — 4 + 8 — 16 + • • . to six terms.
3. Find the tenth term and the sum of ten terms of 4 — 2 +
L - . . .
4. Find r and s; given a = 2, i = 31,250, w = 7.
6. Insert two geometrical means between 47 and 1269.
6. Insert three geometrical means between 2 and 3.
7. Insert seven geometrical means between a* and 6*.
8. Show that the quotient (a* — b**)/ (a — 6) is a geometrical
progression.
9. Sum a;* " ^ + a?" ~ * y + a;** ~ ' y* + . • . to n terms.
10. Sum x*» " ^ — x" " * y + x** " ' y* — . . . to n terms.
11. Sum a-\-ar~^-\-ar^^-\-. . . to n terms.
12. If a, bj c,d, . . . are in geometrical progression, then a* + 6*,
>* + c^t c^ + d^j ' . . are also in geometrical progression.
13. If any numbers are in geometrical progression, their differences
ire also in geometrical progression.
14. A man agreed to pay for the shoeing of his horse as follows:
L cent for the first nail, 2 cents for the second nail, 4 cents for the third
lail, and so on until the eight nails in each shoe were paid for. What
lid the last nail cost? How much did he agree to pay in all?
122. Compound Interest. Just as the amount of principle and
nterest of a sum of money at simple interest for n years is ex-
)ressed by the (n + l)st term of an arithmetical progression, so,
n the same way, the amount of any sum at compound interest for
I years is represented by the (n + l)st term of a geometrical pro-
gression. Thus, the amount of $1.00 at compound interest at
I: percent for twenty years is given by the expression:
1(1.04)20
The amount of d dollars for n years at r percent is:
d
hm)
i
206 ELEMENTARY MATHEMATICAL ANALYSIS [§123
The present value of $1.00, due twenty years hence, estimating
compound interest at 4 percent, is:
1/(1.04)20
The value of $1.00, paid annually at the beginning of each year
into a fund accumulating at 4 percent compound interest, is, at
the end of that period :
(1.04)1+ (1.04)2+ , (1.04)20
which is the sum of the terms of a geometrical progression of
twenty terms.
l^roblems of this character in compound interest and in com-
pound discount, and the more compHcated problems that proceed
therefrom, are basal to the theory of annuities, life insurance arid
depreciation of machinery and structures. The computSation of
the high powers involved necessitates the postponement of such
problems until the subject of logarithms has been explained.
123. Infinite Geometrical Progressions. If the ratio of a
geometrioal progression be a proper fraction, the progression is
said to be a decreasing progression. Thus:
1, 1/2, 1/4, 1/8, 1/16, and 1/3, 1/9, 1/27, 1/81
are decreasing progressions. If we increase the number of terms
in the first of these progressions the sums will always be less than 2;
but the difference 2 — s will become and remain less than any
preassigneii number. By definition, 2 is, therefore, the Umit of
this sum.* The sum of n terms of this particular progression
should bo written down by the student for a number of successive
values for >i, thus:
Number of terms:
h 2, 3. 4. 5, ... 10,
Sum: M + I 2. 1 + ;i 4, 1 + 7 S. 1 + 15 16, ... 1 + 511/512,
The nth term ditTors from 2 by only 1 2* - *.
It is eas\' to show that the sum of everj- decreasing geometrical
proftrt^on appn^aohos a fixeii limit as the number of i&vas
bwomw infinite. For, write the formula:
a — or'
^'= l-r
^SkM 4«ttmaim« $••.
124] PROGRESSIONS 207
I the form:
S = :; :j (1)
1 — r 1 — r ^ ^
' we suppose that r is a proper fraction and that n increases with-
it limit, then r" can be made less than any assigned number, for
le value of any power of a proper fraction decreases as the ex-
3nent of the power increases. As the other parts of the second
action in (1) do not change in value as n changes, the fraction
I a whole can be made smaller than any number that can be
signed. Hence, we write:
^ s = -^ (2)
n=oo 1 — r ^ ^
Exercises
As w = 00 , find the limit of each of the following :
1. 1/2 - 1/4 + 1/8 - 1/16 + . . .
;re:
a = 1/2, r = - 1/2
1/2
lence, the limit « = TTr^ITTToV = 1/3.
2. 0.3333 . . .
jre:^ a = 3/10, r = 1/10
lence, the limit: s = j .T^iTin =1/3.
3. 9 - 6 +4- . . .
4. 0.272727 ...
6. 0.279279279 ...
6. 1/3 - 1/6 + 1/12 - . . .
7. 4 H- 0.8 H- 0.16 + . . .
8. Express the number 8 as the sum of an infinite geometrical
egression whose second term is 2.
124. Graphical representation of the terms and of the sum of a
ometrical progression: If lines proportional to the terms
an arithmetical progression be erected at equal intervals normal
any line, the ends of the perpendiculars will lie on a straight
le, as already explained in §117. We shall now explain
corresponding construction for a geometrical progression.
1st, note that all the essentials of ^ gepjnetrical progression may
i
208 ELEMENTARY MATHEMATICAL ANALYSIS [§124
be studied if we assume the first term to be unity, for the number
a occurs only as a single constant multiplier in each term, and
also occurs in the same manner in the formulas for I and s. There-
fore, by taking a-fold these expressions in a geometrical series
whose first term is 1, the results are obtained for the more
general case.
To represent the geometrical series l-|-r+r?+r'+. . .+
r«-i graphically, lay off OM = 1 on OF, OSi = 1 on OX, SiPi =
r on the unit line, and draw MPi. Draw the arc P1S2 and erect
O Si s, ^8 Sa Ss
Fig. 87. — Graphical Construction of the Sum of a, G. P. r > 1.
P2S2. Draw the arc P2>S2 and erect P^Ss. Continue this con-
struction until you draw the arc Pn-iSn and erect PnSn- The
series of trapezoids OMSiPi, S1P1P2S2J S2P2P3SZ, . • •>
Sn-iPn-iPnSn arc simllar and, since PiSi = r XOM, it follows
that P2S2 = rPiSt, PsSz = tPzaSz, • . • , PnSn = rPn-iSn-i-
Hence we have:
OM = OSi =1
PiiSi = S1S2 = r .-. OaS2 = 1 + r = sum of 2 terms
P2S2 = SiSz = r2 .-.0^3 = 1 + r + r2 = sum of 3 terms
P3/S3 = SzSa = r3 .'.OSa = 1 + r + r2 + r^ = sum of 4 terms
Pn-lSn-l = Sn-lSn = r--^ .'.OSn = 1 + f + r^ + . . . f*"! =
sum of n terms.
•
Fig. 87 shows the series whose ratio is r = 1.2. Fig. 88 shows
the series whose ratio is 0.8.
The line MPi has the slope (r — 1) in Fig. 87 and the slope
— (1 — r) in Fig. 88. In both, its F-intercept is 1. Its equation
1-2/
is, in both cases, y = {r — l)x -^ 1 or x ^
1 -r
In both
124]
PROGRESSIONS
209
gures^ when y »= PJSn «= r^'jZ = OSn- Substituting these values
1 — f "
)r X and y, we get for the sum of n terms, S ■« yzL — ^^8*
7 shows that when the number of terms is allowed to increase
rithout limit, the sum OSn also increases without limit. Fig.
8 shows that when the number of terms is made to increase
dthout limit, the sum OSn approaches OL as a Umit. Now the
alue of OL is the value of x when 2/ = 0. Hence the limit
»f the sum of the progression, or OL = .. _ •
Consult also §7, problem 6, exercise 5 and Figs. 13, 14.
In Figs. 87 and 88 the ordinates OM, Si Pi, /Si2P2, . . . repre-
lenting the successive terms of the geometrical progressions, were
lot erected at equal intervals along OX, If the ordinates repre-
senting the successive terms of the progressions be erected at equal
ntervals along OX, the line MPiPJPz . . . passing through
ihe ends of the ordinates will be a curve and not a straight hne.
Y
U
M
Pi
p.
1 ■■!
P,
1
' \
^
..8 \
P4
Pb
^
]
r" \
r4
y^^
'^^'--^^
O Sx St Ss. Si Se L
Fig. 88. — Graphical Construction of the Sum oi & G. P. r < 1.
To construct this curve, a geometrical construction different from
ihat given above is to be preferred. Near the lower margin of a
(heet of 8i X 11-inch unruled paper lay off a uniform scale of
nches and draw vertical hues through the points of division, as
shown in Fig. 89. Select one of these for the 2/-axis, and on the
unit line lay off the given ratio of the progression IN = r. Then
divide the j^axis proportionally to the successive powers of r,
either by the method of problem 6, §7, Fig. 11, or by the
method shown in Fig. 89. Through the points of division od the
y-axis draw lines parallel to the x-axis, thus dividing the plane
into a large number of rectangles. Starting at the point M
(0, 1) sketch free hand the diagonals of successive cornering
14
i
210 ELEMENTARY MATHEMATICAL ANALYSIS [il24
rectai^lee, rounding the results into a smooth curve as ehowD.
Then the relation between ordinate!/ and abscissa I for the vahfls
of X = — 2, —1,0, 1, 2, 3, etc., is given by the equation y = r".
Fig, 89 is drawn f or r - 3 /2 so that the curve isy = (3 /2) ■,
The method used in Fig. 89 may be explained as follows:
Draw the hnes y = x and y = tx. From the point (1, r)
on y = rx draw a horizontal hne to y = x, thence a vertical line
S
y
a
i
^
-
/
q/
f
■
J
/
Z
N
/,
^
v'
u
^/
V
r
3 -
B
N
to J/ = n, etc., thereby forming the "stairway" of hne segments
between y = x and ;/ = ra as shown in the figure. Then the
points, iV, P, Q, etc., have the ordinates r, t', H, etc., as required,
for, to obtain the ordinate of P, or PD, the value of x used w»s
OD = r, hence P is the point on y = tx for x = r, M" J =
PD = r'. likewise Q is by construction the point on y = n
for X ■ r', hence the y of the point Q = r X f* — r*, etc.
The figure shows the process for findir^ r-', r~*, etc. In
Chapter VIII a method will be explained for locating intraroediate
points on the curve.
125] PROGRESSIONS 211
The curve generated by. the method described above is one of
be most important curves in mathematics. In general, it is seen
tiat the points located on the curve MN always satisfy an
q[uati6n of the form
y =: r'
here r is a constant. This is called an exponential equation
nd the curve is known as the exponential or compound interest
arve.
Note that the ordinates y to the right of M increase rapidly as x
icreases and that the ordinates to the left of M decrease very
owly as X decreases; that is, the curve rapidly leaves the positive
-axis, but slowly approaches the negative x-axis as an asymp-
>te. These results are exactly reversed in case r < 1.
125. * Harmonical Progressions. A series of terms such that
leir reciprocals form an arithmetical progression are said to form
n harmonical progression. The following are examples:
(1) 1/2,1/3,1/4,1/5.
(2) 1,1/5,1/9,1/13.
(3) lKx-y\ llx,l/{x + y).
(4) 1/3,1, -1,-1/3.
(5) 4, 6, 12.
(6) 1/a, l/(a + d), l/{a + 2ci), . . .
Although harmonical progressions are of such a simple character,
o simple expression has been found for the sum of n terms. Our
aowledge of arithmetical progressions enables us to find the
alue of any required term and to insert any required number
r harmonical means between two given extremes, as in the
samples below.
(1) Write six terms of the harmonical progression 6, 3, 2.
We must write six terms of the arithmetical progression,
/6, 1/3, 1/2. The common difference of the latter is 1/6, so
hat the arithmetical progression is 1/6, 1/3, 1/2, 2/3, 5/6, 1, and
he harmonical progression is 6, 3, 2, 1.5, 1.2, 1.
(2) Insert two harmonical means between 4 and 2.
We must insert two arithmetical means between 1/4 and 1/2;
lese are 1/3 and 5/12, whence the required harmonical means
'e 3 and 2.4.
i
212 ELEMENTARY MATHEMATICAL ANALYSIS til2B
126.* Harmonica] Mean. The harmonical mean is found u
follows: Let the two numbers be a and b and let H stand for the
required mean. Then we haver
l/H - lla= \jb-\IH
That is:
2/ff = l/a+l/6 = {a + b)!ah
whence:
H = 2ab/(a + b) (1) i
Thus the harmonical mean of 4 and 12 is 96/(4 + 12) = 6. J
By the harmonical mean of several numbers is meant the redproad J
of the arithmetical mean of their reciprocals. Thus the har-
monica) mean>of 12, 8 and 48 is 13Vt.
]""
A' ^
1
Lx
■? ^,
« c
-y.
127. * Relation between A, G, and H. As previously fouDd:
A = ia + b)l2, G = Voft^^ = 2ab/(a + b)
whence:
AH = ab
but:
ab = G'
or:
Vah
§1271 PROGRESSIONS 213
That is to say, the geometrical mean of any two positive numbers
is the same as the geometrical mean of their arithmetical and
harmonical means.
The arithmetical, geometrical and harmonical means may be
constructed graphically as in Fig. 90. Draw the circle of diameter
(a + 6) = OM + MK. Then the radius is the arithmetical
mean A. Erect a perpendicular at M. Then MG is the geomet-
rical mean. MakeOG' = MG and draw CG'. Dt&wG^H perpen-
dicular to CG\ Then OH is the harmonical mean, since
OG' = ^|0C X OH
Now A >G > H;foT from the figure, MG < CA. Therefore,
the angle G^CO is less than 45° and also its equal HGV is less
than 45°. Therefore, HO < OG' which establishes the in-
equaUty.
Exercises
1. Continue the harmonical progression 12, 6, 4.
2. Find the difference: (1.8 + 1.2 + 0.8 + . . . to 8 terms)
- (1.8 -h 1.2 + 0.6 + . . . to 8 terms).
3. If the arithmetical mean between two numbers be 1, show that
the harmonical mean is the square of the geometrical mean.
{
CHAPTER VIII
THE LOGARITHMIC AND THE EXPONENTIAL
FUNCTIONS
128. Historical Development. The almost miraculous power
of modern calculation is due, in large part, to the invention of
logarithms in the first quarter of the seventeenth century by a
Scotchman, John Napier, Baron of Merchiston. This invention
was founded on the simplest and most obvious of principles, that
had been quite overlooked by mathematicians for many genera-
tions. Napier' s invention may be explained as follows : ^ Let there
be an arithmetical and a geometrical progression which are to be
associated together, as, for example, the following:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Now the product of any two numbers of the second line may be
found by adding the two numbers of the first progression above
them, finding this sum in the first line, and finally taking the num-
ber lying under it; this latter number is the product sought. Thus,
suppose the product of 8 by 32 is desired. Over these numbers
of the second line stand the numbers 3 and 5, whose sum is 8.
Under 8 is found 256, the product desired. Now since but a
limited variety of numbers is offered in this table, it would be
useless in the actual practice of multipHcation, for the reason
that the particular numbers whose product is desired would
probably not be found in the second Hne. The overcoming
of this obvious obstacle constitutes the novelty of Napier's inven-
tion. Instead of attempting to accomphsh his purpose by ex-
tending the progressions by continuation at their ends, Napier
proposed to insert any number of intermediate terms in each
progression. Thus, instead of the portion
0, 1, 2, 3, 4
1, 2, 4, 8, 16
of the two series we may write:
^ Merely the fundamental principles of the invention, not historical details. 9X9
given in what follows.
214
I LOGARITHMIC AND EXPONENTIAL FUNCTIONS 215
a 1/2, 1, Ih 2, 2i, 3, 3i 4
1, \/2, 2, \/8, 4, \/32, 8, \/i28, 16
iserting arithmetical means between the consecutive terms
lie arithmetical series and by inserting geometrical means
een the terms of the geometrical series. Let these be
)uted to any desired degree of approximation, say to two
nal places. Then we have the series
A.P. G.P.
0.0 1.00
0.5 1.41
^ 1.0 2.00
1.5 2.83
' 2.0 4.00
2.5 5.66
3.0 8.00
•
n inserting arithmetical and geometrical means between the
s of the respective series we have:
A.P.
G.P.
0.00
1.00
0.25
1.19
0.50
1.41
0.75
1.69
1.00
2.00
1.25
2.38
1.50
2.83
1.75
3.36
2.00
4.00
2.25
4.76
r continuing this process each consecutive three figure number
finally be made to appear in the second column, so that, to
degree of accuracy, the product of any two such numbers
be found by the process previously explained. The decimal
ts of the factors may be ignored in this work, as for example,
product of 2.38 X 14.1 is the same as that of 238 X 14.1
pt in the position of the decimal point. The correct position
16 decimal point can be determined by inspection after the
4
>
216 ELEMENTARY MATHEMATICAL ANALYSIS [§129
significant figures of the product have been obtained. Using
the above table we find 2.38 X 14.1 = 33.6.
The above table, when properly extended, is a table of loga-
rithms. As geometrical and arithmetical progressions different
from those given above might have been used, the number of
possible systems of logarithms is indefinitely great. The first
column of figures contains the logarithms of the numbers that
stand opposite them in the second column. Napier, by this
process, said he divided the ratio of 1.00 to 2.00 into "100 equal
ratios," by which he referred to the insertion of 100 geometrical
means between 1.00 and 2.00. The *^ number of the ratio"
he called the logarithm of the number, for example, 0.75 opposite
1.69, is the logarithm of 1.69. The word logarithm is from two
Greek words meaning " The number of the ratios.*^ In order to
produce a table of logarithms it was merely necessary to compute
numerous geometrical means; that is, no operations except multi-
plication and the extraction of square roots were required. But
the numerical work was carried out by Napier to so many decimal
places that the computation was exceedingly difficult.
The news of the remarkable invention of logarithms induced
Henry Briggs, professor at Gresham College, London, to visit
Napier in 1615. It was on this visit that Briggs suggested the ad-
vantages of a system of logarithms in which the logarithm of
1 should be 0 and the logarithm of 10 should be 1, for then it would
only be necessary to insert a sufficient number of geometrical
means between 1 and 10 to get the logarithm of any desired
number. With the encouragement of Napier, Briggs undertook
the computation, and in 1617, published the logarithms of the
first 1000 numbers and, in 1624, the logarithms of numbers* from
1 to 20,000, and from 90,000 to 100,000 to fourteen deciwd
places. The gap between 20,000 and 90,000 was filled by a Hol-
lander; Adrian Vlacq, whose table, pubUshed in 1628, is the source
from which nearly all the tables since published have been
derived.
129. Graphical Computation of Logarithms. In Fig. 89 the
terms of a geometrical progression of first term 1 and ratio IN = r
are represented as ordinates arranged at equal intervals along OX.
Fig. 89 is drawn to scale for the value of r = 1.5. Fig. 91 is
I LOGARITHMIC AND EXPONENTIAL FUNCTIONS 217
ailar figure drawn for r = 2, in which a process is used for
ing intermediate points of the curve, so that the locus may
:etched with greater accuracy. The lines y = a; and y = rx
his case y = 2i) are drawn as before, and the "stairway"
tructed as before (see §124). Vertical lines drawn
^h X = — 2, — 1, 0, 1, 2, 3, . . . and horizontal lines drawn
r V
ft
7
G
\''/V
/'/>'
_ . T/
/ /
. Jj
/
----"W
p^"=-=^-
Pia. 91. — Graphical ConstructioQ of the Curve y = 2'.
Ugh the horizontal tread of each step of the stairway divides
plane into a large number of rectangles. Starting at M
sketching the diagonals of successive cornering rectangles
snaooth curve MNP is drawn. Intermediate points of
iurve are located by doubbi^ the number of vertical hues by
:tii^ the distances between each original pair, and then
tcreasing the number of horizontal hues in the foUowii^ man-
Draw the line y = Vri (in the ease of the Fig., j/ = V2 x).
218 ELEMENTARY MATHEMATICAL ANALYSIS [§129
At the points where this line cuts the vertical risers of each step
of the "stairway" (some of these points are marked A, 5, C
in the diagram) draw a new set of horizontal lines. Each of the
original rectangles is thus divided into four smaller rectangles.
Starting at M and sketching a smooth curve along the diagonals
of successive cornering rectangles, the desired graph is obtained.
By the use of the straight line !/ = Vr re another set of intermedi-
ate points may be located, and so on, and the resulting curve
thus drawn to any degree of accuracy required. In explaining
this process, the student will show that the method of construc-
tion just used consists in the doubling of the number of horizontal
linos of the figure by the successive insertion of geometrical means
between the terms of a geometrical progression, while at the same
time the number of vertical lines is successively doubled by
insertion of arithmetical means between the terms of an arith-
metical series. Thus the graphical work of construction of the
curve corresponds to the successive insertion of geometrical and
arithmetical means in the two series discussed in the preceding
section.
As explained above, the ordinate y of any point of the curve
MXP of Fig. 91 is a term of a geometrical progression, and the
abscissa x of the same point is the corresponding term of an
arithmetical progression. Since, when y is given, the value of x
is determined, we say, by definition, that a: is a function of y
(§4). This particular functional relation is so important
that it is given a special name: x ts called the logarifhm of y,
and the statement is abbreviated by writing
J^ = log y,
but to distinguish from the case in which some other geometrical
prv>gn>ssiou might have been used, the ratio of the progression
may bo written as a subscript, thus:
X = logr y
whioh is ri\*id: '*x is the logarithm of y to the base r."
If wo assiuno that tho process of locating the successive sets of
intoTu\i\liato points by tho constmction <rf successive geometrical
nx^vius will KvHd« if Cv^nvinuod indefinitdy, to the generation of
§129] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 219
the curve MNP without breaks or gaps, then we may say that in
the equation:
x = logry (1)
the logarithm is a function of y defined for all positive values of y
and for all values of x.
As a matter of fact, both the arithmetical and the geometrical
method given above defines the function or the curve only for
rational values of x ; that is, the only values of x that come into
view in the process explained above are whole numbers and
intermediate rational fractions like 2|, 2}, 2f , 2 1\, 2^, . . .
It is seen at once from the method of construction used in Fig.
91 that the values of y at x = 1, 2, 3, 4, . . . , are respectively
y = r, r^, r', r\ . . . , and the values of !/ at x = 1/2, 3/2, 5/2, . . . ,
are y = r^^, r^, r^^, . . . , respectively, and similarly for other inter-
mediate values of x. In other words, the equation connecting
the two variables a; and y may be written
y = r* (2)
Thus, when the values of a variable x run over an arithmetical
progression {of first term 0) while the corresponding values of a
variable y run over a geometrical progression (of first term 1), the
relation between the variables may be written in either of the forms
(1) or (2) above. Equation (2) is called an exponential equation
and ^ is said to be an exponential function of x, while in (1) x
is said to be a logarithmic function of y. The student has fre-
quently been called upon in mathematics to express relations
between variables in two different or "inverse" forms, analogous
to the two forms y = r' and x = logr y. For example, he has
written either
y = x^
or:
X = ± \/y
and either
y = x^^^
The graph of a function is of course the same whether the equation
be solved for x or solved for y.
i
220 ELEMENTARY MATHEMATICAL ANALYSIS [§130
130. The student is required to construct the curves described
in the following exercises by the method of §129. The
inch, or 2 cm., may be adopted as the unit of measure; the curv^
should bo drawn on plain paper within the interval from x =
- 2 to X = + 2.
If tangents be drawn to the curves at x = — 2, — 1, 0, 1, 2,
it will be noted, as nearly as can be determined by experiment,
that the several tangents to any one curve cut the X-axis at the
same constant distance to the left of the ordinate of the pomt
of tangency. This distance is greater than unity if r = 2 and less
than unity if r = 3. The value of r for which the distance is exactly
unity is later shown to he a certain irrational or incommensurable
number, approximately 2.7183 . . . , represented in mathematics
by the letter e, and called the Naperian base. This number, and
the number t, are two of the most important and fundamental
constants of mathematics.^
^ It is not easy to locate accurately the tangent to a curve at a given pdnt
of the curve. To test whether or not a tangent is correctly drawn at a pdnt
P, a number of chords parallel to the tangent may be drawn. If the two end
points AB of the chord tend to approach the point of tangency P as the chord
is taken nearer and nearer to P (but always parallel to AB) then the tangent
was correctly drawn. If the two points A and B do not tend to coalesce at the
point P when the chord is moved in the manner described, then the tangent
was incorrectly drawn.
A number of instruments have been designed to assist in drawing tangents to
curves. One of these, called a "Radiator." will be found listed in most catalogs
Fiv^. 92. — Mirr\"»Twi Ruler for Drawing the Normal (and hence the Tan-
irent^ to anj* C^irre.
\>f drawing inMrumeni«. Another instrument consists of a straight edge provided
with a vertivNikl mirror as shown in Fig. 9:2. When the straight edge is placed
acrwMt a curve the T«>Ae^iiv>n v«i the curve in the mirror and the carve itself can
both be 9e<en and usually <he curve and image meet to form a coqp or an|k-
The straight edge may be turned, however, until the image forms a smooth
<r\vfttiauatic«a of the given curve. In this position the straight-edge is pcipeadica-
lar to the t^r^gent asxi the tasgeat eaa thea be aeeurateiy drawn. See Gnmr
berg. TVvh?-i*f5w M<N»u:xgert. 1911.
§131] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 221
Exercises
Draw the following curves on plain paper using 1 inch as the unit
of measure; make the tests referred to in the second paragraph of
§130.
1. Construct a curve similar to Fig. 91, representing the equation
X = log2 Vy from x = — 2toa; = +2, and draw tangents at a; = — 1,
a; = 0, a; = 1, a; = 2.
2. Construct the curve whose equation is a; = logs y from a; = — 2
to a; = + 2, and draw tangents at a; = — 1, a; = 0, a; = 1, a; = 2.
3. Construct the curve whose equation is a; = log2.7 y, and show by
trial or experiment that the tangent to the curve at a; = 2 cuts the x-axis
at nearly a; = 1, that the tangent at a; = 1 cuts the a;-axis at nearly
a; = 0, that the tangent at x = 0 cuts the x-axis at nearly a; = — 1,
etc.
4. Draw the curve x = logo, i y and show that it is the same as the
reflection of a; = log2 y in the mirror x = 0.
Notb: The student must remember that the experimental testing
of the properties of the tangents to the curves called for above does not
constitute mathematical proof of the usual deductive sort familiar to
him. The experimental tests have value, however, in preparing the
student for the rigorous investigation of these same properties when
taken up in the calculus.
131. The Exponential Function. The expression a', where a
is any positive number except 1, has a definite meaning and
value for all positive or negative rational values of x, for the
meaning of numbers affected by positive or negative fractional
exponents has been fully explained in elementary algebra. The
process outlined above likewise defines logr x for all rational
values of x, but the process would not lead to irrational values
of Xy such as V2, ^5, etc. As a matter of fact the expression a*
has as yet no meaning assigned to it for irrational values of x;
•v/2~
thus 10 has no meaning by the definitions of exponents pre-
viously given, for \/2, is not a whole number, hence IC^^ does
not mean that 10 is repeated as a factor a certain number of
times; also V^ is not a fraction, so that 10 cannot mean a
power of a root of 10. But if any one of the numbers of the
following sequence
1.4 1.41 1.414 1.4142 1.41421
• • •
i
222 ELEMENTARY MATHEMATICAL ANALYSIS [|131
be u^ed as the exponent of 10, the resulting power can be com-
puted to any desired number of decimal places. For example,
10»-*i is the 141th power of the 100th root of 10; to find the 100th
root we may take the square root of 10, find the square root of
tliis result, then find its 5th root, finally finding the 5th root
of this last result.
If the various powers be thus computed to seven places we find:
101*
= 25.11887 . .
101.41
= 25.70396 .
101.414
= 25.94179 .
101.4142
= 25 . 95374 .
101.41421
= 25 . 95434 .
101.414213
= 25 . 95452 .
101.414213 5
= 25.95455 .
Now the so<iuonce of exponents used in the first coliunn are
found by extracting the square root of 2 to successive decimal
j)lacos. If the sequence in the second colunm approaches a limit,
this limit is taken by definition as the value of 10 . It is shown
in higher mathematics that such a limit in this and similar cases
always oxist.s and consequently that a niunber with an irrational
ox]>onont has a moaning. In this book we shall assume, without
a formal prtH)f, that a* has a meaning for irrational values of x.
To summarize: In order logically to complete the definition of
<!' ioT irrationjU vjUues of j, and to set forth other important
propertio:^» wo would be required to proceed as follows:
1,0 It must bo shown that if x be an alwaj's rational variable
npproaohinjr an irrational number n as a limit, that the limit of
ti* f',rl>^^^ Tho notation a', where j is rational, is understood to
n\oan tho positive value of o', so that the limit of a*, when it is
shown to exist, will niH*o:«isarily be a pasitire number.
\2^ Tho above dosoriboil limit of a' must be taken as the defini-
(jon i»/ a** whon^ fi is tho irrational mmiber approached by x as
a limit.
vo"^ It must bo shown that kJ* is a continuous function of :t.
v-l"^ It must bo shown that the fundamental laws of exponents
appl>' to uumboj^ aiTootod with irrational e3qx>nmits.
\Vhoj\ it is shown. v>r when it is assumed, that a value of x
rl32] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 223
Jways exists which will satisfy the equation a* = y^ where a
nd y are any given positive numbers, then the expression a*
3 called the exponential function of x with base a; otherwise a"" is
lefined only for rational values of x.
132. Definitions. In the exponential equation a' = y:
The number a is called the base.
The number y is called the exponential function of x to the base
, and is sometimes written y = expa x.
The number x is called the logarithm of y to the base a, and
} written x = logot/. Thus in the equation a' = y, x may be
ailed either the exponent of a or the logarithm of y.
The two equations:
y = a'
^ = loga y
Kpress exactly the same relations between x and y; one equation
solved for x, the other is solved for y. The graphs are identical,
1st as the graphs of y = x^ and x = ± \/y are identical.
See also Anti-logarithm, §142.
133. Common Logarithms. In the equation 10'' =^ y, x is
illed the common logarithm of y. It is also called the Brigg's
»garithm of y. Thus, the comi^aon logarithm of any number is
le exponent of the power to which 10 must be raised to produce
le given number. Thus 2 is the common logarithm of 100,
nee 10^ = 100; likewise 1.3010 will be found to be the common
igarithm of 20 correct to 4 decimal places, since 10^'°^°
= 20.0000 to 4 decimal places.
134. Systems of Logarithms. If in the exponential equation
= a*, where a is any positive number except 1, different values
e assigned to y and the corresponding values of x be computed
nd tabulated, the results constitute a system of logarithms.
'he number of different possible systems is unlimited, as already
oted in §128. As a matter of fact, however, only two
ystems have been computed and tabulated; the natural or
faperian or hyperbolic system, whose base is an incommensurable
umber, approximately 2.7182818, and the common or Briggs'
ystem, whose base is 10. The letter e is set aside in mathematics
0 stand for the base of the natural system.
i
224 ELEMENTARY MATHEMATICAL ANALYSIS (§135
Natural logarithms of all numbers from 1 to 20,000 have
been computed to 17 decimal places. The common logarithms
arc usually printed in tables of 4, 6, 6, 7 or 8 decimal places.
It will be found later that the graphs of all logarithmic functions
of the form x = log« y can be made by stretching or by contract-
ing in the same fixed ratio the ordinates of any one of the logarith-
mic curves. For that reason numerical tables in more than
one system of logarithms are unnecessary.
In the following pages the conmion logarithm of any number n
will be written log n, and not logi© n; that is, the base is supposed
to be 10 unless otherwise designated; In x for log. x and Ig x for
logio X are also used.
Exercises
Write the following in logarithmic notation.
1. 10» = 1000.
2. 10-» = 0.001.
8. 10» = 1.
4. IV = 121.
6. 16» « = 2.
6. c* = y-
7. 10»« = 1.7783.
8. loo.'oio = 2.
9. a» = a,
10. 10 ^"^f^,.^ = y.
Express the following in exponential notation:
11. logic 4 = 0.6021.
12. log 10000 = 4.
18. log 0.0001 = - 4 .
14. logs 1024 = 10.
16. log. a = 1 .
16. logi^lOO = 2 3.
17. logs: (1/3) = -1/3.
18. logioolO = 1/2.
19. log 1 =0.
20. loga 1=0.
185. Graphical Table. In Fig. 93 is shown the graph of Uie
function defined by the two progressions whose use was suggested
by Briggs to Napier, and which are referred to in the last para-
3S] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 225
aph of §128. By inserting means three times between 0
id 1 in the arithmetical progression and between 1 and 10 in the
ometrical progression, we get
A. P. or I G. P, or j Exponential
Lc^aritbms Numbers Form of G. P.
0.000
1.000
0.125
1.334
0.250
1 , 778
0.375
2.371
0.500
3 162
0.625
4.217
0.750
5,623
0.875
7.499
I-OOO
10-000
IQOOt
^ L - Login W 7
Vi,r-»TTW"i;rtiilriir J7 1 1 1 J9 Li
Fia. 9
—The Curvet - logmiV.
If we let L stland for the logarithm of the number N, the
actional relation is obviously L = Ic^ioJV or JV = 10^. The
rve (Fig. 93) may now be used as a graphical table of logarithms
im which the results can be read to about 3 decimal places.
226 ELEMENTARY MATHEMATICAL ANALYSIS [§136
The logarithms of numbers between 1 and 10 may be read directly
from the graph. Thus, logio 7.24 = 0.860. If the logarithm is
between 0 and 1, the number is read directly from the gr^ph.
Thus if the logarithm is 0.273, the number is 1.87.
If we multiply the readings of the iV-scale by 10**, we must add
n to the readings on the Ir-scale, for IQ^N = 10^ + **.
If we divide the readings on the iV-scale by 10**, we must
subtract n from the readings on thelr-scale, for N jlQi^ = 10^ "" ^
This fact enables us to read the logarithms of all numbers from
the graph, and conversely to find the number corresponding to
any logarithm. Thus we have, log 72.4 = 1.860, log 724 = 2.860,
log 0.724 = 0.860 - 1, log 0.0724 = 0.860 - 2.
If the logarithm is 1.273, the number is 18.7.
If the logarithm is 2.273. the number is 187.
If the logarithm is 0.273 — 1, the number is 0.187.
If the logarithm is 0.273 - 2, the number is 0.0187.
We observe that the computation of a three place table of
logarithms would not involve a large amount of work: such a table
has actually been computed in drawing the curve of Fig. 93.
The original tables of Briggs and Vlacq involved an enormous
expenditure of labor and extraordinary skill, or even genius in
computation, because the results were given to fourteen places
of decimals,
136. Properties of Logarithms. The following properties of
logarithms follow at once from the general properties or laws of
exponents.
(1) The logarithm of 1 is 0 in all systems. For a® = 1, that
is, logo 1 = 0. In Fig. 91, note that the curve passes through
(0, 1).
(2) The logarithm of the base itself in any system is 1. For
a^ = 1, that is, logo a = 1. In Fig. 91, by construction iV is always
the point (1, r), where r is the ratio of the first or fundamental
progression; in the present notation, this is the point (1, a).
(3) Negative numbers have no logarithms. This follows at
once from §131, (1). In Figs. 89, 91, and 93, note that the
curves do not extend below the X-axis.
Note : While negative numbers have no logarithms, this does not
prevent the computation of expressions containing negative factors
i 137] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 227
md divisors. Thus to compute (287) X (— 374), find by logarithms
287) X (374) and give proper sign to the result.
137. Logarithm of a Product. Let n and r be any two positive
Lumbers and let:
loga n = X and logo r = y (1)
Chen, by definition of a logarithm:
n = a* atid r = av (2)
ilultiplying:
nr = a'ay = a*+»'
Therefore, by definition of a logarithm §132:
loga nr ^ X + y
^r, by (1)
logo nr = logo n + logo r (3)
Hence, the logarithm of the product of two numbers is equal to
he sum of the logarithms of those numbers.
In the same way, if loga s =2 , then:
nrs = a*+»+*
liat is,
loga nrs = logo n + loga r + loga s
Exercises
Find by the fornaulas and check the results by the curve of Fig. 93.
1. Given log 2 = 0.3010, and log 3 = 0.4771; find log 6; find log 18.
2. Given log 5 = 0.6990 and log 7 = 0.8451; find log 35.
3. Given log 9 = 0.9542, find log 81.
4. Given log 386 = 2.5866 and log 857 = 2.9330; find the logarithm
>f the product.
6. Given log llx = 1.888 and log 11 = 1.0414; find log x.
138. Logarithm of a Quotient. Let n and r be any two
Positive numbers, and let:
logon = X and logaf = y (1)
Trom (1) by the definition of a logarithm,
n = a* r — ay
Dividing,
n/r — a' -T- a'' = a* "^
{
228 ELEMENTARY MATHEMATICAL ANALYSIS [§139
Therefore by definition of a logarithm,
loga(n/r) = a; - 2/
or by (1)
loga(n/r) = loga n - logo r (2)
therefore, the logarithm of the quotient of two numbers equah tk
logarithm of the dividend less the logarithm of the divisor.
Exercises
Check the results by reading them off the curve of Fig. 93.
1. Given log 5 = 0.6990 and log 2 = 0.3010; find log (5/2); find
log 0.4.
2. Given log 63 = 1.7993, and log 9 = 0.9542; find log 7.
3. Given log 84 = 1.9243 and log 12 = 1.0792; find log 7.
4. Given log 1776 = 3.2494 and log 1912 = 3.2815; find log
1776/1912; find log 1912/1776.
6. Given log a:/12 = 0.4321 and log 12 = 1.0792, find log x.
139. Logarithm of any Power. Let n be any positive number
and let:
logo^ = X (1)
From (1), by the definition of a logarithm,
n = a'
Raising both sides to the pth power, where p is any number what-
soever y
UP = ap'
therefore, by definition of a logarithm,
log o(np) = px
or by (1):
loga(np) = p logaH (2)
therefore the logarithm of any power of a number equals the logarithm
of the number multiplied by the index of the power.
The above includes as special cases, (1) the finding of the
logarithm of any integral power of a number, since in this case
p is a positive integer, or (2) the finding of the logarithm of any
root of a number, since in this case p is the reciprocal of the index
of the root,
140] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 229
Exercises
1. Given log 2 = 0.3010; find log 1024; find log v/2; find logj/2.
2. Given log 1234 = 3.0913; find log V1234. Find log v^l234.
3. Given log 5 = 0.6990; find log 5^^; find log 5^^.
4. Simplify the expression log 30/V210 .
Ixpress by the principles established in §§137-139 the following
►garithms in as simple a form as possible :
6. log (V9 -5- VS)_>
6. log (Vl2 -^ V6).
7. log (u^ -5- u^).
8. log (lOa^hya^ h^).
9. Show that log (11/15) + log (490/297) - 2 log (7/9) = log 2.
10. Find an expression for the value of a: from the equation 3* =567.
Solution : Take the logarithm of each side
x log 3 = log 567
Mi log 567 = log (3^ X 7) = 4 log 3 + log 7
lerefore:
x log 3 = 4 log 3 H- log 7
r:
X = 4 + (log 7)/(log 3).
11. Find an expression for x in the equation 5* = 375.
12. Given log 2 = 0.3010 and log 3 = 0.4771, find how many
igits in 6^°.'
13. Find an expression for x from the equation :
3* X 2'+i = V5i2.
14. Prove that log (75/16) - 2 log (5/9) + log (32/243) = log 2.
140. Characteristic and Mantissa. The common logarithm
F a number- is always written so that it consists of a positive
ecimal part and an integral part which may be either positive
r negative. Thus log 0.02 = log 2 - log 100 = 0.3010 - 2.
og 0.02 is never written — 1.6990.
When a logarithm of a number is thus arranged, special names
re given to each part. The positive or negative integral part is
lUed the characteristic of the logarithm. The positive decimal
art is called the mantissa. Thus, in log 200 = 2.3010, 2 is
le characteristic and 3010 is the mantissa. In log 0.02 =
.3010 — 2, ( — 2) is the characteristic and 3010 is the mantissa.
Since log 1 = 0 and log 10 = 1, every number lying between 1
\
230 ELEMENTARY MATHEMATICAL ANALYSIS [§140
and 10 has for its common logarithm a proper fraction-4hat
is, the characteristic is 0. Thus log 2 = 0.3010, log 9.99 =
0.9996, log 1.91 «= 0.281. Starting with the equation:
log 1.91 = 0.2810
we have, by §137,
log 19.1 = log 1.91 + log 10 = 0.2810 + 1
log 191 = log 1.91 + log 100 = 0.2810 + 2
log 1910 = log 1.91 + log 1000 = 0.2810 + 3, etc.
Likewise, by §138,
log 0.191 = log 1.91 - log 10 = 0.2810 - 1
log 0.0191 = log 1.91 - log 100 = 0.2810 - 2
log 0.00191 = log 1.91 - log 1000 = 0.2810 - 3, etc.
Since the characteristic of the common logarithm of any number
having its first significant figure in units place is zero, and since
moving the decimal point to the right or left is equivalent to
multiplying or dividing by a power of 10, or equivalent to adding
an integer to or subtracting an integer from the logarithm,
(§135): (1) the value of the characteristic is dependent merely
upon the position of the decimal point in the number; (2) the
value of the mantissa is the same for the logarithms of all
numbers that differ only in the position of the decimal point.
In particular, we derive therefrom the following rule for finding
the characteristic of the common logarithm of any number:
The characteristic of the common logarithm of a number equak
the number of places the first significant figure of the number is
removed from units' place, and is positive if the first significant
figure stands to the left of units' place and is negative if it stands
to the right of units' place.
Thus in log 1910 = 3.2810, the first figure 1 is three places from
units' place and the characteristic is 3. In log 0.0191 = O.2810
— 2 the first significant figure 1 is two places to the right of units'
place and the characteristic is — 2. A computer in determining
the characteristic of the logarithm of a number first points to
units place and counts zero, then passes to the next place and
counts one and so on until the first significant figure is reached.
Logarithms with negative characteristics, like 0.3010-1)
§141] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 231
0.3010 — 2, etc., are frequently written in the equivalent form
9.3010 - 10, 8.3010 - 10, etc.
Exercises
1. What numbers have 0 for the characteristic of their logarithm?
What numbers have 0 for the mantissa of their logarithms?
2. Find the characteristics of the logarithms of the following
numbers: 1234, 5,678,910, 212, 57.45, 345.543, 7, 7.7, 0.7, 0.00000097,
0.00010097.
3. Given that log 31,416 = 4.4971, find the logarithms of the
following numbers: 314.16, 3.1416, 3,141,600, 0.031416, 0.31416,
0.00031416.
4. Given that log 746 = 2.8727, write the numbers which have the
following logarithms: 4.8727, 1.8727, 0.8727 - 3, 0.8727 - 1, 3.8727,
0.8727 - 4.
141. Logarithmic Tables. A table of logarithms usually con-
tains only the mantissas of the logarithms of a certain con-
venient sequence of numbers. For example, a four place table
will contain the mantissas of the logarithms of numbers from
100 to 1000; a five place table will usually contain the mantissas of .
the logarithms of numbers from 1000 to 10,000, and so on. Of
course it is unnecessary to print decimal points or characteristics.
A table of logarithms should contain means for readily obtaining
the logarithms of numbers intermediate to those tabulated, by
means of tabular differences and proportional parts.
The tabular differences are the differences between successive
mantissas. If any tabular difference be multiplied successively
by the numbers 0.1, 0.2, 0.3, . . . , 0.8, 0.9, the results are called
the proportional parts. Thus, from a four place table we find
log 263 = 2.4200. The tabular difference is given in the table
as 16. If we wish the logarithm of 263.7, the proportional
part 0.7 X 16 or 11.2 is added to the mantissa, giving, to four
places, log 263.7 = 2.4211. This process is known as interpola-
tion. Corrections of this kind are made with great rapidity after
a little practice. It is obvious that the principle used in the
correction is the equivalent of a geometrical assumption that
the graph of the function is nearly straight between the successive
values of the argument given in the table. The corrections
232 ELEMENTARY MATHEMATICAL ANALYSIS [§142
should invariably he added mentally and all the work of interpolalwn
should he done mentally if the finding of the proportional park
hy Tnental work does not require multiplication heyond the range of
12 X 12. To make interpolations mentally is an essential practice,
if the student is to learn to compute by logarithms with any skill
beyond the most rudimentary requirements.
A good method to follow is as follows: Suppose log 13.78 is
required. First write down the characteristic 1 ; then, with the
table at your left, find 137 in the number column and mark the
corresponding mantissa by placing your thumb above it or your
first finger below it. Do not read this mantissa, but read the
tabular difference, 32. From the p. p. table find the correction,
26, for 8. Now return to the mantissa marked by your finger,
and read it increased by 26, i.e., 1393; then place 1393 after
the characteristic 1 previously written down.
The accuracy required for nearly all engineering computations
does not exceed 3 or 4 significant figures. Four figure accuracy
means that the errors permitted do not exceed 1 percent of
1 percent. Only a small portion of the fundamental data
of science is reliable to this degree of accuracy.^ The usual meas-
urements of the testing laboratory fall far short of it. Only
in certain work in geodesy, and in a few other special fields of
engineering, should more than four place logarithms be used.
142. Anti-logarithms. If we wish to find the number which
has a given logarithm, it is convenient to have a table in which
the logarithm is printed hefore the number. Such a table is known
as a table of anti-logarithms. It is usually not best to print
tables of anti-logarithms to more than four places; to find a number
when a five place logarithm is given, it is preferable to use the
table of logarithms inversely, as the large number of pages required
for a table of anti-logarithms is a disadvantage that is not com-
pensated for by the additional convenience of such a table.
^Fundamental constants upon which much of the calculation in applied
science must be based are not often known to four figures. The mechanical
equivalent of heat is hardly known to 1 percent. The specific heat of super-
heated steam is even less accurately known. The tensile, tortional and com-
pressive strength of no structural material would be assumed to be known to a
greater accuracy than the above-named constants. Of course no oalculated
result can be more accurate than the least accurate of the measurements upon
which it depends.
H43l LOGARITHMIC AND EXPONENTIAL FUNCTIONS 233
143. Cologarithms. Any computation involving multiplica-
ion, division, evolution and involution may be performed by
he addition of a single column of logarithms. This possibility
3 secured by using the cologarithm, instead of the logarithm, of
.11 divisors. The cologarithm, or complementary logarithm,
►f a number n is defined to be (10 — log n) — 10. The part
10 — log n) can be taken from the table just as readily as log n,
y subtracting in order all the figures of the logarithm, including the
haracteristiCf from 9, except the last figure, which must he taken
rom 10. The subtraction should, of course, be done mentally.
7hus log 263 = 2.4200, whence colog 263 = 7.5800 - 10. It
3 obvious that the addition of (10 — log n) — 10 is the same
.s the subtraction of log n.
The convenience arising from this use may be illustrated as
ollows:
Suppose it is required to find x from the proportion
37.42 :a; ::647 : Vo.582.
Ve then have
2 log 37.4 = 3.1458
(1/2) log 0.582 = 9.8825 - 10
colog 647 = 7.1891 - 10
log [1.650] = 0.2174
therefore x = 1.650.
It is a good custom to enclose a computed result in square
►rackets.
144. Arrangement of Work. All logarithmic work should be
rranged in a vertical column and should be done with pen and
ok. Study the formula in which numerical values are to be
ubstituted and decide upon an arrangement of your work in the
ertical column which will make the additions, subtractions, etc.,
f logarithms as systematic and easy as possible. Fill out the
ertical column with the names and values of the data before
uming to the table of logarithms. This is called blocking out
tue work. The work is not properly blocked out unless every
ntry in the work as laid out is carefully labelled, stating exactly
he name and value of the magnitude whose logarithm is taken.
{
234 ELEMENTARY MATHEMATICAL ANALYSIS [§144
and unless the computation sheet bears a formula or statement
fully explammg the purpose of the work.
Computation Sheet, Form M7, is suitable for general logarithmic
computation.
Exercises
1. From a four place table find the logarithms of the following
numbers: 342, 1322, 8000, 872.4, 35.21, 0.00213, 3.301, 325.67,
2i 3.1416, 0.0186, 250.75, 0.0007, 0.33333.
2. Find the numbers corresponding to each of the following
logarithms: 0.3250, 2.1860, 0.8724, 1.1325, 3.0075, 8.3990 - 10,
9.7481 - 10, 4.0831, 7.0091 - 10, 0.5642.
3. Compute by logarithms the value of the following: 2.56 X3.ll
X 421; 7.04 X 0.21 X 0.0646; 3215 X 12.82 -5- 864.
4. Compute the following by logarithms: 81* -f- 17*; 158 V^;
(343/892)8; Vl893 Vl912/446^
6. Compute the following by logarithms: (2.7182)i-"S; (7.41)"^;
(8.31)0-".
6. Solve the following equations: 5* = 10; 3* " ^ = 4; log, 71 = 121
logx 5 = logio 4.822.
7. Find the amount of $550 in fifteen years at 5 percent com-
pound interest.
8. A corporation is to repay a loan of $200,000 by twenty equal
annual payments. How much will have to be paid each year, if
money be supposed to be worth 5 percent?
Let X be the amount paid each year. As the debt of $200,000 is
owed noWf the present value of the twenty equal payments of x dollars
each must add up to the debt or $200,000. The sum of x dollars
to be paid n years hence has a present worth of only
X
7^05)^
if money be worth 5 percent compound interest. The present value,
then, of X dollars paid one year hence, x dollars paid two years hence,
and so on, is
XXX X
1.05 ^ (1.05)» ^ (1.05)3 -r . • . -r (i,o5)»o
This is a geometrical progression.
The result in this case is the value of an annuity payable at the
end of each year for twenty years that a present payment of $200,000
will purchase.
§146] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 235
9. It is estimated that a certain power plant costing $220,000 will
become entirely worthless except for a scrap value of $20,000 at the
end of twenty years. What annual sum must be set aside to amount
to the cost of replacement at the end of twenty years, if 5 percent
compound interest is realized on the money in the depreciation
fund?
Let the annual amount set aside be x. In this case the twenty
equal payments are to have a value of $200,000 twenty years hencef
while in the preceding problem the payments were to be worth
$200,000 now. In this case, therefore,
a;(1.05)i» + x(1.05)i« + a:(1.05)i^ + . . .
+a;(1.05)« + a:(1.05) +x = $200,000.
The geometrical progression is to be summed and the resulting
equation solved for x.
10. The population of the United States in 1790 was 3,930,000 and
in 1910 it was 93,400,000. What was the average rate percent in-
crease for each decade of this period, assuming that the population
increased in geometrical progression with a uniform ratio for the entire
period.
11. Find the surface and the volume of a sphere whose radius is
7.211.
12. Find the weight of a cone of altitude 9.64 inches, the radius
of the base being 5.35 inches, if the cone is made of steel of specific
gravity 7.93.
13. Find the weight of a sphere of cast iron 14.2 inches in diameter,
if the specific gravity of the iron be 7.30.
14. In twenty-four hours of continuous pumping, a pump discharges
450 gallons per minute; by how much will it raise the level of water in
a reservoir having a surface of 1 acre? (1 acre = 43560 sq. ft.)
146. Trigonometric Computations. Logarithms of the trig-
onometric functions are used for computing the numerical value
of expressions containing trigonometric functions, and in the
solution of triangles. The right triangles previously solved by
use of the natural functions are often more readily solved by
means of logarithms. (See §66.) The tables of logarithmic func-
tions contain adequate explanation of their use, so that de-
tailed instructions need not be given in this place. Two new
matters of great importance are met with in the use of the loga-
rithms of the trigonometric functions that do not arise in the use
i
236 ELEMENTARY MATHEMATICAL ANALYSIS [§145 ^^
of a table of logarithms of numbers, which, on that account, require ^
especial attention from the student:
(1) In interpolating in a table of logarithms of trigonometric
functions, the corrections to the logarithms of all co-functions mv^
be subtracted and not added. Failure to do this is the cause of
most of the errors made by the beginner.
(2) To secure proper relative accuracy in computation, the
S and T functions must be used in interpolating for the sine and
tangent of small angles.
In the following work, four place tables of logarithms are
supposed to be in the hands of the students.
i
Exercises '^
1. A right prism, whose base is a square 17.45 feet on a side, is
cut by a plane making an angle of 27° 15' with a face of the prism, h
Find the area of the section of the prism made by the cutting plane, j
2. The perimeter of a regular decagon is 24 feet. Find the area of s
the decagon.
3. To find the distance between two points B and C on opposite |
banks of a river, a distance CA is measured 300 feet, perpendicular '
to CB. At A the angle CAB is found to be 47° 27'. Find the
distance CB.
4. In running a Une 18 miles in a direction north, 2° 13.2' east,
how far in feet does one depart from a north and south line passing
through the place of beginning?
6. How far is Madison, Wisconsin, latitude 43° 5', from the earth's
axis of rotation, assuming that the earth is a sphere of radius 3960
miles?
6. Find the length of the belt required to connect an 8-foot and a
3-foot pulley, their axes being 21 feet apart.
7. A man walking east 7° 15' north along a river notices that after
passing opposite a tree across the river he walks 107 paces before he
is in line with the shadow of the tree. Time of day, noon. How far
is it across the river?
8. Solve the right-angled triangle in which one leg = 2V3 and the
hypotenuse = 2x.
9. The moon's radius is 1081 miles. When nearest the earth, the
moon's apparent diameter (the angle subtended by the moon's disk as
seen from the position of the earth's center) is 32.79'. When farthest
i
238 ELEMENTARY MATHEMATICAL ANALYSIS [1148
the number r. See Fig. 94. In a system of exponential curves
y = T* passing through the point (0, I) or the point M oi Fig.
94, we shall assume that there is one curve passing through U
with slope I. The equation of this particular curve we shall call
y = e', thereby defining the number e at that valite of r Jor vihieh
the curve y = r' passes through the point (0, 1) urilA slope 1. Thia
is a second definition of the number e; we shall show in this section
that it is consistent with the first definition of e given in |130.
Fio. 94. — DefioitioQ of Tangent
The exercises of §130 developed experimentally the charac-
teristic property of the exponential curve to the base e"
The slope of the curve y = e' at any point is equal to the ordinal*
of that point. This fact, developed experimentally in §130,
will now be shown to follow necessarily from the definition of e
just given.
Select the point P on the curve i/ = «' at any point desired.
Draw a line throi^h P cutting the curve at any neighboring
point Q. (Fig. 94.) A line like PQ that cuts a curve at two points
is called a secant line. As the point Q is taken nearer and neam
to the point P {P remaining fixed), the limiting position ap-
il46] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 239
>roached by the secant PQ is called the tangent to the curve at
.he point P. This is the general definition of the tangent to any
jurve.
The slope of the secant joining P to the neighboring point Q
s HQ/PH, As the point Q approaches P this ratio approaches
he slope of the tangent to y = e' at the point P. Let OD
= X and PH = h; then OE = x + h, also DP = e* and EQ =
;*+*. Since HQ is the y of the point Q minus the y of the
>oint P, we have:
HQ _ e'+^-e' _ e*-l
PH~ h *" ^' A
S^ow the slope of y = e* at P is the Umit of the above expression
is Q approaches P or as ^ approaches zero. That is:
slope of e* at P = e', ^ ^ — ^ — (1)
Ne now seek to find
limit e* — 1
;i = 0 h
Z such Umit exists. Since P is any point, consider the point M
vhere a; = 0. The slope there is:
« Umit 6* — 1
^ /i = 0 /i
Chat is, the slope of i/ = 6* at M is:
limit e^ — 1
iut by the definition of 6, the slope of y = e* at M is 1. Hence
p'e must conclude that the required limit exists and that
limt e^--l __ . .
h^O h "^ ^^^
ubstituting this result in equation (1), we have
Slope at P = e' (3)
?his expresses the fact that the slope of y = e* at any point is e*,
T is the ordinate y of that point, a fact that was first indicated
xperimentally in §130. At that same place the approxi-
mate value of e was seen to be 2.7. A more exact value is known
0 be 2.7183, as wiU be computed later.
I
240 ELEMENTARY MATHEMATICAL ANALYSIS HM
In Fig. 94 the slope of j/ = 6' at P ia given by PD meaaured by
the unit OM. The distance TD, called the subtangent, U
constant for all positions of the point P.
The slope of i; = r* at any point is readily found. That
exists a number m such that e** = r. Hence j/ = r* may be
written y = (e'*)' = e"". Now this curve is made from y = P
by substituting mx for x, or by multiplying all of the abscissds
of the latter by I /m. Therefore the side TD of the triangle PDT
in Fig. 94 will be multiplied by I /m, the other side DP remaining
- J-l"--?
:|:r,:|:E:::::::::::
rFgrrf ■'
:";:;;aK:::::::::
T r ' 1 '•
nffFP'l 1 Hj
::::!:; j_:::::::::::
Mm illi mra
•"'•-"*gsi:h^^^'-Yja»
^-^.F-T--^--3^-
Fia. 95. — Esponential ai
o the Natural Base e
the same. Therefore the slope of the curve, or DP/TD will be
multiplied by m, since the denominator of this fraction is multi
plied by 1 fm. Hence the slope of y = r' at any point is m times
the ordinate of that point, where m satisfies the equation e"
The curve y = e~' is, of course, the curve y = e' reflected in
the K-axis." This curve, as well as the curve y = log, x and its
symmetrical curve, are shown in Fig. 95. Sometimes the curve
y = e' is called the exponential curve and the curve y = logi i
is called the logarithmic cmre. This distinction, howevw, has
■ See SS4.
(146] LOGABITHMIC AND EXPONENTIAL FUNCTIONS 241
little utility, as the equation of either locus can be expressed in
either notation.
The notation y = In x is often used to indicate the natural hga-
rilkm of X and the notation y = Ig x or y = log x is used to stand
for the common logarithm of x.
TABLE IV.
The following table oE powers ot
e is useful i
1 sketching e
curves.
e'-' = 1.2214
e-"
> =0.8187
e-* = 1.4918
e""
* =0,6703
e" = 1.8221
e-»
« - 0.5488
go.. = 2. 2255'
e""
« = 0.4493
e = 2.7183
e"'
= 0.3679
e' = 7.3891
6-'
= 0,1353
«' = 20.0855
e~'
= 0.0498
e' = 54,5982
e-'
= 0.0183
e« = l
.6487
e^ = 1
.3956
e« = 1
.2840
gH = 1
.2214
4lif
^. V \ 1 III ^
X* \ W I / / ■v
— X — V~\ r li/ — y~~
^^^ '\\\\ \l / y" '•i^
■"-0 ^""^^ ^^^^^^"^^ ""°
Fio. 96. — A Family of ExponentiaU, j/ — l"'.
Exercises
1. Draw the curve y = ^ + e~'. Show that yiaane
of X, that is, that y does not change when the sign of i: is
ven function
changed. ,
242 ELEMENTARY MATHEMATICAL ANALYSIS [§147
2. Draw the curve y — e' — «"*. Show that this is an odd func-
tion of X, that is, that the function changes sign but not absolute value
when the sign of x is changed.
3. Draw the graphs oi y ^ e*/', and y = c*/'.
4. Draw the graphs of i/ = e*^*, and y = e"'^^.
6. Compare the curves: y = e*/*, y = e*/', y =* e*, y = c^.
6. Sketch the curves 2/ = 1*, y = 2*, y = 3*, y = 4*, y = 5',
2/ = 6', 2/ = 8«, 2/ = 10*, from x= -3toa; = +3.
7. From the graphs of
.2
and
solve the equation
y = X'
y = log loa; + 1.8
x2 - log x - 1.8 = 0.
8. Solve graphically the equation
5 log a; - (\/2)x + 2=0.
9. Solve graphically :
10* = x^,
10. Solve graphically :
(1/2)* = log X.
11. Solve graphically:
12. Solve graphically:
13. Solve graphically:
14. Solve analytically:
10* = 5sin X.
sin a; = X — 0.1.
cos X = x"^ — 1.
ex-i = 10*.
147. The Exponential Curve and the Theorems on Loci. It has
already been shown (§ 145) that the curve y = a' can be derived
from the curve y = e' (a>e) by multiplying the abscissas of
the latter curve by 1 /m (m > 1), that is, by orthographic projection
of 2/ = e* upon a plane passing through the F-axis. There exists
a number m (m > 1) such that a = e'». Hence, y = a' may be
written y = e"»* and, by §27, the latter curve may be made
from 2/ = e* by multiplying its abscissas by 1 /w. Also note that
the slope of the curve y = e' Sit any point is equal to the ordinate
of the point, and that the slope of 2/ « a* at any point is m
times the ordinate of that point. The number 1 /m is called the
modulus of the logarithmic system whose hose is a.
§147] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 243
The modulus of the common system is the reciprocal of the value
of m that satisfies e"" = 10, or it is the value of M that satisfies
ei/^ = 10, or that satisfies e = 10^. That is, the modulus M of
the common system is the logarithm of e to the base 10, or, to four
figures, equals 0.4343. The value of w or 1 /M = 2.3026. Thus
we have the fundamental formulas:
100.4343 — a]
~ (1)
^2.3026 =10] ^ ^
and
logioAT = 0.4343 log JV 1 ^2^
logei\r = 2.3026 logioAT J ^ ^
Another remarkable property of the logarithmic curve ap-
pears from comparing the curves y = a' and y = a*+^, or, more
generally, the curves y = a' and y = a*-^^. The second of these
curves can be derived from y = a'hy translating the latter curve
the distance 1 (in the general case the distance h) to the left.
But y = a«+^ may be written y — aa'y and y = a'^^ may be
written y = aH', From these it can be seen that the new curves
may also be considered as derived from y = a'hy multiplying all
ordinates oiy = a' by a, or in the general case, by a*.
Translaiing the exponential curve in the x-direction is the same as
mvUi'plying all ordinates by a certain fixed number, or is equivalent
to a certain orthographic projection of the original curve upon a plane
ihrov>gh the X-axis,
Changing the sign of h ch^lnges the sense of the translation and
changes elongation to shortening or vice versa.
The exponential curve might be defined as the locus that
possesses the above-described fundamental property. There are
numerous ways in which this property may be stated. Another
form is this: Any portion of the exponential curve included within
any interval of x, may be made from the portion of the curve
included within any other equal interval of x, by the elongation
(or shortening) of the ordinates in a certain ratio, or, in other
ivords, by orthographic projection upon a plane passing through
the a;-axis. This is illustrated by Fig. 97, which is a graph of an
exponential curve drawn to base 2. If the portions of the curve
^1^2, PiPiyPzP Ay • . . correspouduig to equal intervals 1 of x
I
244 ELEMENTARY MATHEMATICAL ANALYSIS HW
be changed by shortening all ordinatea of PiPi measured above the
height of Pi in the ratio 1 /2, by shortenii^ all ordinates of PtPi
measured above Pi in the ratio 1 /4, by shortening all ordinates of
P1P4 measured above Pi in the ratio 1 /8, . . : the results ara*
the curves PiFi, PiPs, PjPj, . . . which are identical with thes
portion PJ'i of the ordinal curve.
~
"1
— 1
^
"
.
//
,
J
/
,
•'"
,
/
J
1/
Pi
'.'i
r/
Pi
L
k
if
_
_
_
FiQ.'97.— Illuatralion ot s
This is also illustrated by Fig. 93, which is a small portion of the
curve X = logio y drawn on a large scale, and, for convenience, with
the vertical unit 1 /lO the horizontal unit. From this small portion
of the curve we may read the logarithms of all numbers. For the
distances along the x-axts may be designated 0.0, 0.1, 0.2, . . .
or I.O, 1.1, 1.2, . . . or 2.0, 2.1, 2.2, , , ,, etc., in which case
wereadl,2,3, . . . or 10,20,30, . . . or 100,200,300, . . .
etc., respectively, along the y-axis. This, it will be observed, is
merely a geometrical statement of the fact that a table of man<
tissas for the numbers from 1.000 to 9.999 is sufficient for deter-
mining the logarithms of all four-figure numbers.
§148] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 245
Exercises
1. State the difference between the curves 2/ = e* and y = 10*.
2. Graph y = e-o» where e = 2.7183.
3. Graph the logarithmic spiral p = e*, 0 being measured in
radians.
Note: The radian measure in the margin of Form Af3 should be
used for this purpose.
4. Graph p = e~*.
6. The pressure of the atmosphere is given in millimeters of mer-
cury by the formula:
y = 760-e-* 8000
where the altitude x is measured in meters above the sea level. Pro-
duce a table of pressure for the altitudes a; = 0; 10; 50; 100; 200; 300;
lOOO; 10,000; 100,000.
6. From the data of the last problem, find the pressure at an alti-
tude of 25,000 feet.
7. Show that the relation of Exercise 5 may be written :
X = 18,421 (log 760 - log y).
In X
8. Determine the value of the quotient i — for the following
values of x: 2, 3, 5, 7.
9. How large is e°°°^ approximately?
10. What is the approximate value of 10° °°^?
148. Logarithmic Double Scale. The relation between a num-
ber and its logarithm can be shown by a double scale of the sort
discussed in §§3 and 8. In constructing the double scale,
one may select for the uniform scale either the one on which the
numbers are to be read, or the one on which the logarithms are to
be read. A scale having a most remarkable and useful property
results if the logarithms are laid off on a uniform scale and the
corresponding numbers are laid off on a non-uniform scale, as
shown in the double scale of Fig. 98. This scale is constructed
for the base 10. The distances measured on the 5-scaIe, although
it is the scale on which the numbers are read, are 'proportional to
the common logarithms of the successive numbers; that is, if the
total length of the scale be called unity, the distance on the B
scale from the left end to the mark 2 is 0.3010, the distance' to the
mark 3 is 0.4771, etc.; also the distance on this scale from the left
end to the mark 6 is the sum of the distance from the left end to
246 ELEMENTARY MATHEMATICAL ANALYSIS [§148
the mark 2 and the hke distance to the mark 3; also the distance
to 8 is just treble the distance to 2.
Since log lOx « 1 + log x, it follows that, if the scales A and 5,
Fig. 98, were extended another unit to the right,
this second unit would be identical to the first
one, except in the attached numbers. The
numbers on the A-scale would be changed from
0.0, 0.1, 0.2, ... 1.0 to 1.0, 1.1, 1.2, . . .,
2.0, while those on the non-uniform, or 5-scale,
would be changed from 1, 2, 3, . . . , 10 to
10, 20, 30, . . . 100.
Passing along this scale an integral number
•^ of unit intervals corresponds thus to change of
Si characteristic in the logarithms, or to change
3 of decimal point in the numbers.
o It is not, however, necessary to construct
o more than one block of this double scale, since
J we are at liberty to add an integer n to the
•^ numbers of the uniform scale, provided at the
M same time we multiply the numbers of the
■t" non-uniform scale by 10". In this way we
00 may obtain any desired portion of the extended
. scale. Thus, we may change 0.1, 0.2, 0.3, . .,
g e 1.0 on A to 3.1, 3.2, 3.3, . . ., 4.0, by adding
3 to each number, provided at the same time
= 1 we change the numbers on the 5-scale 1, 2, 3,
^ si 4, . . ., 10 to 1000, 2000, 3000, 4000, . . .
I 10,000 by multiplying them by 10'. If n is
negative (say — 2) we may write, as in the
case of logarithms, 8.0 - 10, 8.1 - 10, 8.2 -
^ 10, . . . , 9.0 — 10, or, more simply, — 2, -
ft^ -^ I^ ^ 1.9, - 1.8, - 1.7, . . ., - 1.0, changing the
numbers on the non-uniform scale at the same
o
K'w4
fH
—
» ^^
^n
o»-
—~
■=
—
o>
00
—
-s?
"^
___
r-.
~
—
QO
•
o
—
to
^^^
—
—
^—
r-
»o
—
-c5
—
—
—
to
*
'•f
—
-<^
—
^^
__
_
t!i
— ^
m^\
—
3
CO
__
—
O n
J) i-«
o
°'3
time to 0.01, 0.02, 0.03, . . .,0.10.
To produce the scale of distances proportional to the logarithms
of the successive numbers as used above, it is merely necessary to
draw horizontal lines through the points 1, 2, 3, . . . of the
2/-axis in Fig. 99, and then draw vertical lines through the points
5149] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 247
P], Pi, P« . . . where the horizontal lines meet the curve; the
intercepts on the z-axis are then proportional to 1(^ x.
149. The Slide Rule. By far the most important application
of the non-uniform scale ruled proportionally to log x, is the com-
puting device known as the slide rule. The principle upon which
the operation of the slide rule is based is very simple. If we have
1 k
1
ly
",
/'
m
>
f
/
e
/
■ U
/■
/'
/
y\
'
f-
I
i
>r
t
_
_
-\-
_
^_
\-
—
-
--
"■?. Ill"
11
S 1
!
1:r;i^r'
Fia. 99. — A Method of Construtting the Logarithmic Scale-
two scales divided proportionally to log x (A and B, Fig. 100),
so arranged that one scale may slide along the other, then by shd-
ing one scale (called the slide) until its left end is opposite any
desired division of the first scale, and, selecting any desired division
of the slide, as at R, Fig. 100, taking the reading of the original
scale beneath this point, as N, the product of the two factors
whose logarithms are proportional to AB and BR can be read
directly from the lower scale at N; for AN is, by construction,
the sum of AB and BR, and since the scales were laid off propor-
tionally to log X, and marked with the numbers of which the dis-
tances are the logarithms, the process described adds the logarithms ■
meohanically, but indicat«s the results in terms of the numberif
248 ELEMENTARY MATHEMATICAL ANALYSIS [§149
themselves. By this device all of the operations commonly carried
out by use of a logarithmic table may be performed mechanically.
Full description of the use of the slide rule
need not be given in detail at this place, as
complete instructions are found in the pamph-
lets furnished with each slide rule. A very
brief amount of individual instruction given to
the student by the instructor will insure the
rapid acquirement of skill in the use of the
instrument. In what follows, the four scales of
the slide rule are designated from top to bottom
of the rule, Ay By C, D, respectively. The ends
of the scales are called the indices.
o An ordinary 10-inch slide rule should give
pt5 results accurate to three significant figures,
y which is accurate enough for most of the pur-
cQ poses of applied science.
rg An exaggerated idea sometimes prevails con-
o cerning the degree of accuracy required by work
b in science or in applied science. Many of the
^ fundamental constants of science, upon which a
^ large number of other results depend, are known
^ only to three decimal places. In such cases
I. greater than three figure accuracy is impossible
o even if desired. In other cases greater accuracy
^ is of no value even if possible. The real desid-
S eratum in computed results is, first, to know by a
suitable check that the work of compviation is correcty
and, second, to know to what order or degree of
accuracy both the data and the result are dependable.
The absurdity of an undue number of decimal
i places in computation is illustrated by the orig-
« inal tables of logarithms, which if now used
^ would enable one to compute from the radius
" of the earth, the circumference correct to 1 /10,000
2 part of an inch.
•^ ^ The following matters should be emphasized
in the use of the slide rule:
eo
la
n
ci
ft^-.
.(M
00
19] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 249
1) All numbers for the purpose of computation should be con-
jred as given with the first figure in units place. Thus 517
1910 X 0.024 should be considered as 5.17 X 1.19 X 2.4 X
X 10' X 10~2. The result should then be mentally approxi-
ted (say 24,000) for the purpose of locating the decimal point,
[ for checking the work.
2) A proportion should always be solved by one setting of the
e.
V) A combined product and quotient like
g X6Xc Xd
rXs Xt
uld always be solved as follows:
Place runner on a of scale D.
Set r of scale C to a of scale D ;
Runner to 6 of C;
s of C to runner;
Runner to c of C;
i of C to runner;
at d of C find on D the significant figures of the result.
4) The runner must be set on the first half of A for square
ts of odd numbered numbers, and on the second half 6i A for
square roots of even numbered numbers.
3) Use judgment so as to compute results in most accurate
iner — thus instead of computing 264/233, compute 31 /233 and
ice find 264/233 = 1 + 31/233.1
3) Besid^ checking by mental calculation as suggested in (1)
ve, also check by computing several neighboring values and
phing the results if necessary. Thus check 5.17 X 1.91 X 2.4
computing both 5.20 X 19.2 X 2.42 and 5.10 X 1.90 X 2.38.
Exercises
Compute the following on the slide rule.
L 3.12 X 2.24; 1.89 X 4.25; 2.88 X 3.16; 3.1 X 236.
2. 8.72/2.36; 4.58/2.36; 6.23/2.12; 10/3.14.
3. 32.5 X 72.5; 0.000116 X 0.00135; 0.0392/0.00114.
L 3,967,000 -^ 367,800,000.
6.54 X 42.6. 8.75 X 5.25
^' 32.5 ' 32.3
Show by trial that this gives a more accurate result.
250 ELEMENTARY MATHEMATICAL ANALYSIS [§149
78.5 X 36.6 X 20.8
6.
7.
5.75 X 29.5
6.46 X 57.5 X 8.55
3.26 X 296 X 0.642
8. Solve the proportion
x:1.72: = 'A.14:V2gh
where g == 32.2 and h = 78.2.
9. Compute ^^^-
^ 166.7 X 4.5
10. The following is an approximate formula for the area of a seg-
ment of a circle:
A = h*/2c + 2ch/Z
where c is the length of the chord and h is the altitude of the segment.
Test this formula for segments of a circle of unit radius, whose arcs
are 7r/3, x/2, and w radians, respectively.
11. Two steamers start at the same time from the same port; the
first sails at 12 miles an hour due south, and the second sails at 16
miles an hour due east. Find the bearing of the first steamer as seen
from the second (1) after one hour, (2) after two hours, and compute
their distances apart at each time.
The following exercises require the use of the data printed herewith.
An "acre-foot" means the quantity of water that would cover 1
acre 1 foot deep. '* Second-foot" means a discharge at the rate of 1
cubic foot of water per second. By the "run-off " of any drainage area
is meant the quantity of water flowing therefrom in its surface stream
or river, during a year or other interval of time.
1 square mile = 640 acres
1 acre = 43,560 square feet.
1 day = 86,400 seconds.
1 second foot = 2 acre feet per day.
1 cubic foot = 7i gallons.
1 cubic foot water = 62J pounds water.
1 h.p. = 550 foot pounds per second.
450 gallons per minute = 1 second foot.
Each of the following problems should be handled on the slide rule as
a continuous piece of computation.
12. A drainage area of 710 square miles has an annual run-off of
120,000 acre feet. The average annual rainfall is 27 inches. Find
what percent of the rainfall appears as run-off.
13. A centrifugal pump discharges 750 gallons per minute against
il50] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 251
A total lift of 28 feet. Find the theoretical horse power required.
Also daily discharge in acre feet if the pump operates fourteen hours
per day.
14. What is the theoretical horse power represented by a stream
discharging 550 second feet if there be a fall of 42 feet?
16. A district containing 25,000 acres of irrigable land is to be sup-
plied with water by means of a canal. The average annual quantity
of water required is 3i feet on each acre. Find the capacity of the
canal in second feet, il the quantity of water required is to be delivered
uniformly during an irrigation season of five months.
16. A municipal supply amounts to 35,000,000 gallons per twenty-
four hours. Find the equivalent in cubic feet per second.
17. A single rainfall of 3.9 inches on a catchment area of 210 sq[uare
miles is found to contribute 17,600 acre feet of water to a storage
The run-off is what percent of the rainfall in this ease?
I^ U-U--
B/
P
Fig. 101. — The Theory of the Use of Semi-logarithmic Paper.
160. Semi-logarithmic CoSidlnate Paper. Fig. 101 i
a sheet of rectangular coQrdinate paper, on which ON has been
chosen as the unit of measure, Aloi^ the r^ht-hand ei^e of this m
252 ELEMENTARY MATHEMATICAL ANALYSIS [§150
sheet is constructed a logarithmic scale LM of the type discussed
in § 148, i.e., any number, say 4, on the scale LM stands opposite
the logarithm of that number (in the case named opposite 0.6021)
on the uniform scale ON.
Let us agree always to designate by capital letters distances
measured on the uniform scales, and by lower case letters dis-
tances measured on the logarithmic scale. Thus Y will mean the
ordinate of a point as read on the scale ON, while y will mean the
ordinate of a point as read on the scale LM. In other words, we
agree to plot a function, using logarithms of the values of the
function as ordinates and the natural values of the argument or
variable as abscissas.
Let PQ be any straight line on this paper, and let it be required
to find its equation, referred to the uniform a;-scale OL and the
logarithmic y-scale LM. We proceed as follows:
The equation of this line, referred to the uniform X-axis OL
and the uniform F-axis ON, where 0 is the origin, is
Y = mX + B
m being the slope of the line, and B its y-intercept. Now, for the
line PQ, m = 0.742 and B = 0.36, so that the equation of PQ is
Y = 0.742X + 0.36 (1)
To find the equation of this curve referred to the scales LM and
OL, it is only necessary to notice that
Y = \ogy
so that we obtain:
log y = 0.742X + 0.36 (2)
The intercept 0.36 was read on the scale ON, and is therefore the
logarithm of the number corresponding to it on the scale LM.
That is, 0.36 = log 2.30. Substituting this value in equation (2)
we obtain:
log y = 0.742a; + log 2.30
which may be written
log y - log 2.30 = 0.742a;
or.
1°8 2I0 = ^-^^^
ilSOl LOGARITHMIC AND EXPONENTAIL FUNCTIONS 253
On changing to exponential notation this becomes:
y = 2.30(10<"*^)
—
—
— _
— ^
/
/
,
/
,
y
,
y
, .
/■
y
4
o,.
In general, if the equation of a straight line referred to the
Males OL and Oi^ is
Y = mX -VB (4)
its equation referred to the scales Oh and LM may be obtained by
Replacing Y by Ior y and B by log h in the manner described above,
giving
log y = ntr + log h (5)
254 ELEMENTARY MATHEMATICAL ANALYSIS ((MO
which, as above, may be reduced to the form
y - blO" (6)
This is the general equation of the exponential curve. Hence:
Any exponentwl curve can be represented by a sb^ighl line, provided
ordinates are read from a suitable hgarOhmic scale, and abscmai
are read from a uniform scale.
SemL lAoarlthiQlo Paper
Fia. 103.— Exponential Curves on Form MS. The curve — ... is
y = 10-"; — . isy= 10""; — . . ia v - 10".
Fig. 102 represents the same line PQ (y = (2.30)10"'"'), as
Fig. 101. The two figures differ only in one respect: in F%. 101
the nilir^s of the uniform scale ON are extended across the page,
while in F^. 102 these rulings are replaced by those of the scale
LM.
Co3rdinate paper such as that represented by F^. 102 isknon
il50] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 255
ks semi-logarithmic paper. It affords a convenient coordinate
(ystem for work with the exponential function.
Every point on PQ (Fig. 102) satisfies the exponential equation
y = 2.30(100^"')
Thus, in the case of the point R,
3.98 = 2.30(100^") 0 320
= 2.30(100- "8)
The slope of any line on the semi-logarithmic paper may be read
or determined by means of the uniform scales BC and AB of form
M5. The scale AD of form Af5 is the scale of the natural loga-
rithms, so that any equation of the form y = c"»* can be graphed
at once by the use of this scale. Thus, the line y = e' (Fig.
103) passes through the point A or (0, 1), and a point on BC op.
posite the point marked 1.0 on AD, Note that 1.0 on scale AD
2.718 on the non-uniform scale of the main body of the paper
and 0.4343 on the scale BC all fall together, as they should.
To draw the line y = 10"*, the comer D of the plate may be
taken as the point (0, 1). On the line drawn once across the sheet
representing y = 10"**, y has a range between 1 and 10 only.
To represent the range of y between 10 and 100, two or more sheets
of form M5 may be pasted together, or, preferably, the continua-
tion of the line may be shown on the same sheet by suitably
changing the numbers attached to the scales AB and BC. Thus
Fig. 103 shows in this manner y = 10^* and y = 10'*.
Remember that the line
y = blO'»* (7)
passes through the point (0, 6) with slope m. Note that
L = IQMx - a) (g)
D
passes through the point (a, h) with slope w.
Exercises
On semi-logarithmic paper draw the following :
1. y « 10**, y - 10«*, y » 10', t/ = 10-*, 2/ = lO'^*, y = lO"**.
2. y — e**, y « e*, y « e""*, 2/ « e"**.
3. 3x « log y, (l/2)x = log y.
256 ELEMENTARY MATHEMATICAL ANALYSIS l§151
4. 2/ = 10*/«, y = 10*/".
5. Graph!/ = 2(10)' and ^ = 10* "3.
161. The Compound Interest Law. Logarithmic Increment.
The law expressed by the exponential curve was called by Lord
Kelvin the compotmd interest law and since that time this name
has been generally used. It is recalled that the exponential cum
was drawn by using ordinates equal to the successive terms of
a geometrical progression which are uniformly spaced along the
X-axis; since the amount of any sum at compound interest is given
by a term of a geometrical progression, it is obvious that a sum at
compound interest accumulates by the same law of growth as is
indicated by a set of uniformly spaced ordinates of an expo-
nential curve; hence the term ''compound interest law," from
this sui>erficial view, is appropriate. The detailed discussioo
that follows will make this clear:
Let $1 be loaned at r percent per annum compound interest
At the end of one year the amount is: (1 + r/100).
At the end of two years the amount is: (1 + r/100) ^
and at the end of t years it is: (1 + r/100)'.
If interest be compounded semi-annually, instead of annually,
the amount at the end of t years is: (1 + r/200)2'
and if compounded monthly the amount at the end of the same
period is: (1 + r/1200)i2,
or if compounded n times per year i/ = (1 + r/100n)»'
where t is expressed in years. Now if we find the limit of this
expression as n is increased indefinitely, we will find the amount of
principle and interest on the hypothesis that the interest was
compounded continuously. For convenience let r/100» =!/«•
Then:
2/ = (1 + l/u)«'-^/ioo (1)
where the limit is to be taken as it or n becomes infinite. Calling
(1 + llu)-=f{u) (2)
and expanding by the binomial theorem for any integral value of
Uj we obtain:
= 1 + 1 + (1 - 1/m)/2! + (1 - 1/m)(1 - 2/m)/3! + . . . (3)
"i
^
151] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 257
Q the calculus it is shown that the limit of this series as u becomes
ifinite is the limit of the series
1 + 1 + 1/2! + 1/3!+ ... (4)
he limit of this series is easily found; it is, in fact, the Napierian
use e. It is shown in the calculus that the restriction that u
lall be an integer may be removed, so that the limit of (3) may
5 found when w is a continuous variable.
It is easy to see that the limit of (4) is > 2} and < 3. The sum
the first three terms of the series (4) equals 2}; the rest of the
rms are positive, therefore e > 2|. The terms of the series (4),
ter the first three, are also observed to be less, term for term, than
16 terms of the progression:
(1/2)^ +(1/2)3+ . _ (5)
uttMs is a geometrical progression the limit of whose sum is 1/2.
herefore (3) is always less than 2 J + | or 3. The value of e is
jadily approximated by the following computation of the first
terms of (4) :
2.00000 =1+1
3 10. 50000 = 1/2!
4 0.16667 = 1/3!
5|0. 04167 = 1/4!
6|0^q833 = 1/5!
7 0.00139 = 1/6!
0_^0020 = 1/7!
um of 8 terms = 2 . 71826
"he value of e here found is correct to four decimal places.
Returning to equation (1) above, the amount of $1 at r
ereent compound interest compounded continuously is:
y = gr«/100
hus $100 at 6 percent compound interest, compounded annually,
mounts, at the end of ten years, to
y = 100(1.06)10 = $179.10
he amount of $100 compounded continuously for ten years is
y = 100eO.« = $182.20
!ie difference is thus $3.10.
17
258 ELEMENTARY MATHEMATICAL ANALYSIS l|151
The compound interest law is one of the important laws of
nature. As previously noted, the slope or rate of increase of the
exponential function
y = oe**
at any point is always proportional to the ordinate or to the value
of the function at that point. Thtis when in nature we find any*
function or magnitude that increases at a rate proportional to itsdf
we have a case of the expontential or compound interest law.
The law is also frequently expressed by saying, as has been re-
peatedly stated in this book, that the first of two magnitudes varies
in geometrical progression while a second magnitude varies in ariih-
meticdl progression, A familiar example of this is the increased
friction as a rope is coiled around a post. A few turns of the haw-
sers about the bitts at the wharf is sufficient to hold a large ship,
because as the number of turns increases in arithmetical preces-
sion, the friction increases in geometrical progression. T)ius the
following table gives the results of experiments to determine what
weight could be held up by a one-pound weight, when a cord
attached to the first weight passed over a round peg the numbo"
of times shown in the first column of the table:
n = number of
turns of the cord
on the peg
w — weight just held
in equilibrium by
one-pound weight
Logs of preceding d « logarithmie
numbers ^increment
1/2
1
li
2
2i
3
1.6
3.0
6.1
8.0
14.0
23.0
0.204
0.477
0.709
0.903
1.146
1.362
0.273
0.531
0.195
0.243
0.216
Average logarithmic increment =
0.23
If the weights sustained were exactly in geometrical progression,
their logarithms would be in arithmetical progression. The test
for this fact is to note whether the differences between logarithms
of successive values are constant. These differences are known
as logarithmic increments or in case they are negative, as logft-
rithmic decrements. In the table the logarithmic increments
fluctuate about the mean value 0.23.
1152] LOGARITHMIC AND EXPONENTAIL FUNCTIONS 259
The equation connecting n and w is of the form
w = 10" /"* OT n — m log w
By graphing columns 1 and 3 on squared paper, the value of m is
determined and we find
w = IQo^n or n = 2.2 log 2
Another way is to graph columns 1 and 2 on semi-logarithmic
paper.
An interesting example of the compound interest law is Weber's
law in psychology, which states that if stimuli are in geometrical
progression, the sense perceptions are in arithmetical progression.
152. Modulus of Decay, Logarithmic Decrement In a very
large number of cases in nature the "compound interest" law
appears as a decreasing function rather than as an increasing
function, so that the equation is of the form
y = a€~^' (1)
where — 6 is essentially negative. The following are examples of
this law:
(1) If the thickness of panes of glass increase in arithmetical
progression, the amount of light transmitted decreases in geo-
metrical progression. That is, the relation is of the form
L = ae-*< (2)
where t is the thickness of the glass or other absorbing material
and L is the intensity of the light transmitted. Since when i = 0
"the light transmitted must have its initial intensity, Lo, (2)
becomes
L = Loe-^' (3)
The constant b must be determined from the data of the problem.
Thus, if a pane of glass absorbs 2 percent of the incident light,
Lo = 100, L = 98 for e = 1,
then: 98 = lOOe"*'
or log 98 - log 100 = - 6 log e.
r«i. i. r 0.0088 ^^^
Therefore: h = Q4343 = 0.02
The light transmitted by ten panes of glass is then
Lio = 100e-"<oo2^ = lOOe-o-2
260 ELEMENTARY MATHEMATICAL ANALYSIS l|162
or, by the table of §146,
Lio = 100/1.2214 = 82 percent
(2) Variation in atmospheric pressure with the altitude is
usually expressed by Halley's Law:
p = 760e-*^8ooo
where h is the altitude in meters above sea level and p is the at-
mospheric pressure in millimeters of mercury. See §147, Exer-
cises 5, 6, 7.
(3) Newton's law of cooling states that a body surrounded by a
medium of constant temperature loses heat at a rate proportional
to the difference in temperature between it and the surrounding
medium. This, then, is a case of the compound interest law.
If d denotes temperatiu-e of the cooling body above that of the
surrounding medium at any time t, we must have
d = ae-^'
The constant a must be the value of 6 when ^ = 0, or the initial
temperature of the body.
Exercises
1. A thermometer bulb initially at temperature 19°.3 C. is exposed
to the air and its temperature 6 observed to be 14°.2 C. at the end of
twenty seconds. If the law of cooling be given by 0 = Ooe'^ where
t is the time in seconds, find the value of 0 and 6.
Solution: The condition of the problem gives 0 = 19.3 when
t = 0, hence ^o = 19.3. Also, 14.2 = 19.36"**. This gives
log 19.3 - 206 log e = log 14.2
from which h can be readily computed.
2 If li percent of the incident light is lost when light is directed
through a plate of glass 0.3 cm. thick, how much light would be
lost in penetrating a plate of glass 2 cm. thick?
3. Forty percent of the incident light is lost when passed through
a plate of glass 2 inches thick. Find the value of a in the equation
L = Loe "<*' where t is thickness of the plate in inches, L is the pe^
cent of light transmitted, and Lo = 100.
4. As I descend a mountain the pressure of the air increases eadi
foot by the amount due to the weight of the layer of air 1 foot thidc
As the density of this layer is itself proportional to the pressure, show
that the pressure as I descend must increase by the compound intoest
law.
1163] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 261
5. Power is transmitted in a clock through a train of gear wheels
n in number. If the loss of power in each pair of gears is 10 percent,
draw a curve showing the loss of power at the nth gear.
Note: The graphical method of §124, Figs. 88, 89, may appro-
priately be used.
6. Given that the intensity of light is diminished 2 percent by
passing through one pane of glass, find the intensity / of the light
after passing through n panes.
7. The nimiber of bacteria per cubic centimeter of culture increases
under proper conditions at a rate proportional to the number present .
Find an expression for the number present at the end of time t if there
are 1000 per cubic centimeter present at time zero, and 8000 per
cubic centimeter present at time 10.
8. The temperature of a body cooling according to Newton's law
fell from 30° to 18° in six minutes. Find the equation connecting the
temperature of the body and the time of cooling.
163. Empirical Curves on Semi-logarithmic Co5rdinate Paper.
One of the most important uses of semi-logarithmic paper is in
<letermining the functional relation between observed data, when
such data are connected by a relation of the exponential form.
Suppose, for example, that the following are the results of an
experiment to determine the law connecting two variables x and y:
X
0.04
0.18
0.36
0.51
0.685
0.833
0.97
y
5.3
4.4
3.75
3.1
2.6
2.33
1.9
If the equation connecting x and y is of the exponential form, the
points whose coordinates are given by corresponding values of x
and y in the table will lie in a straight line, except for such slight
errors as may be due to inaccuracies in the observations. Plotting
the points on semi-logarithmic coordinate paper, we find that they
lie nearly on the line PQ (Fig. 104) . Assuming that, if the data
^ere exact, the points would lie exactly on this line,^ we may pro-
> We would not be at liberty to make such an assumption if the variation of the
tx>int8 away from the line was of a character similar to that represented by
the dots near the top of Fig. 104. These points, although not departing greatly
Troin the line shown; depart from it systematically. That is, they lie below it at
ftach end, and above it in the center, seeming to approximate a curve, (such as the
>ne shown dotted) more nearly than the line. The points arranged about the line
^Q depart as far from that line as do the points above the higher line, but they
4o not depart systematically, as if tending to lie along a smooth curve. When points
^jrange themselves as at the top of Fig. 104, one must infer that the relation con-
E^ecting the given data is not exponential in character.
262 ELEMENTARY MATHEMATICAL ANALYSIS. (§15
ceed to determine the equation of this line as approxiniately repn
senting the relation between x and y.
It is easy to find the equation of such a line referred to the ud
form scales AB and BC of form M5. We may imagine that s
rulings are erased and replaced by extensions of the uniform A
scale, as in Fig. 101. The equation of the line PQ is then
Y = mX + B
(■
D^M 2
[K-. 1
1 1 1 1 1 1 1 1 1
1 1 ■ 1 1 1 1 1 1
1 1 1 1 1 1 10
»
*^^^
« :
^»k
.
8
^^v^
8:
■
-
. T
"^^>
*^
I :
■
"^
6
^^
a :~
P
«
ks.=
^•"^"v^ !
1
"
^^E
4
•"Sv^
1
4 :
^
"
. !
^">^
8 :
1 • 1
'^^
;
1
1 1
1
"^"^..1
■
;
I
Q:
«
I :
*-*^
■ ■ ■ :"■' 11
A L *^ «.> «.3 «.4 «l5 tLt %.z t^ t^ utB
S«ai LocMithmie P»per
Fig. 104. — Emiuriead Equations Determined by TTae of Form Mh.
wh«^ m i$ the slope, and B is the y-intercept. Now, for P
w « - 0.447 and B = 0.730 = log 5.37. Equation (1) of i
becometss thetnefore:
)■ = - 0.447A: -r 0.730
SIH] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 263
or, replacing F by log ^ and 0.730 by log 5.37, in order to refer
the curve to tbe scales AB and LM,
log y - log 5.39 = - 0.447X
J, = 5.39(10-''."'0 (2)
If it is desired to express the relation to the base e instead of
base 10, we may note 10 = e**"*' (S^W, equation (1)), or, sub-
stituting in (2),
y= 5.39 (e> >•»)-<>■'«'
= 5.39 e-'°"' (3)
The same result might have been obtained directly by use of
the uniform scale AD, at the left of form M5. This scale is so
constructed that the length 1 on AB corresponds to the length
2.3026 on AD. Now, we know that e>"i2« = 10, hence we may
replace 10 in 10-" by e if we make m in 10"' 2.3026 times as
great as before. This is readily done by measuring the slope of
J'Q by the use of the uniform scale AD instead of the uniform
scale BC. Computing the slope of PQ by use of the scale AD
Tce find;
r of Q = 0.653
y ofp = 1.6^1
Difference = - 1.028
Smce AB — 1, this is the slope of the line, measured to the scale
■AD, and is therefore the value of m in the equation
y = oe- (4)
Sence the equation of PQ is
y = 5.39e-'°"'
vhich agrees with tbe equation previously obtained.
lU. Change of Scale on Semi-logaiithmic Paper. A sheet of
semi-logarithmic paper, form M5, is a square. If sheets of this
paper be arranged "checker-board fashion" over the plane, then
the vertical non-uniform scale will be a repetition of the soaleLjlf ,
Fig. 104, except that the successive segments of length LM must
be numbered 1, 2, 3, . . . , 9 for the original LM, then 10, 20,
30, . . . , 90 for tbe next vertical segment of the checker-board,
then 100, 200, 300, . . . , 900, for the next, etc. It is obvious,
therefore, that the initial point A ot b, sheet of semi-lt^arithmic
e . 19.3 14.2 10.4 7.6 5.6 4.1 3.0
Plot these results on semi-logarithmic paper and test whether or not
$ follows the compound interest law. If so, determine the value of
$0 and b in the equation 9 = 9(ie~^, Note that the last i)oint given
by the table, namely f = 120, ^ = 3.0, goes into a new square if the
scale AB be called 0^100. K the scale AB be called 0—200 then all
entries can appear on a single sheet of form M5.
2. Graph the following on semi-logarithmic paper:
n 1/2 1 U 2 2} 3
w 1.6 3.0 5.1 8.0 14.0 23.0
Show that the equation connecting n and w is v — lO^***.
Suggestion: The scale AB, form J/5, may be called 0 — 5 for the
purpose of graphing n.
264 ELEMENTARY MATHEMATICAL ANALYSIS [|1M
paper may be said to have the ordinate 1, or 10, or 100, etc., or ^
10"^ 10-*, etc., as may be most convenient for the particular
graph under consideration. The horizontal scale being a uniform
scale, any values of x may be plotted to any convenient scale on
it, as when using ordinary squared paper. However, if the hori-
zontal unit of length (the length AB, form MS) be taken as any
value different from unity, then the slope m of the line PQ drawn
on the semi-logarithmic paper can only be found by dividing its
apparent slope by the scale value of the side AB. That is, the
correct value of m in
y = al0"»'
is, in all cases,
_ apparent slope of PQ
"" scale value of AB
*
The "apparent slope" of PQ is to be measured by applying any
convenient uniform scale of inches, centimeters, etc., to the
horizontal and vertical sides of a right triangle of which PQ is the
hypotenuse.
Exercises
1. A thermometer bulb initially at temperature 19.3** C. is exposed
to the air and its temperature $ noted at various times t (in seconds)
as follows:
t 0 20 40 60 80 100 120
5155] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 265
6. Graph the following on semi-logarithmic paper, and find the
equation connecting n and w.
n
0.2
0.4 0.6
0.8
1.0
1.2
1.4
1.6
w
2.60
3.41 4.45
5.75
7.56
9.85
1.30
16.6
4. A circular disk is suspended by a fine wire at its center. When
at rest the upper end of the wire is turned by means of a supporting
knob through 30°. The successive angles of the torsional swings of
the disk from the neutral point are then read at the end of each swing
as follows:
Swing number
1
2
3
4
5
6
7
Angle
26°.4
23°.2
20^5
18°.0
15°.9
14^0
12°.3
Show that the angle of the successive swings follows the compound
interest law and find in at least two different ways the equation
connecting the number of the swing and the angle. Show by the
slide rule that the compound interest law holds.
166. The Power Function Compared with the Exponential
Function. It has been emphasized in this book that the funda-
mental laws of natural science are three in number, namely: (1)
the parabolic law, expressed by the power function y = ax"*
where n may be either positive or negative; (2) the harmonic or
periodic law, y = a sin nx, which is fundamental to all periodically
occurring phenomena; and (3) the compound interest law dis-
cussed in this chapter. While there are other important laws and
functions in mathematics, they are secondary to those expressed
by these fundamental functions. The second of the functions
above named will be more fully discussed in the chapter on waves.
The discussion of the compound interest law should not be closed
without a careful comparison of power functions and exponential
functions.
The characteristic property of the power function
y = ax'
(1)
is that as x changes by a constant factor ^ y changes by a constant
factor also. Take
y = axn = /(X) (2)
i
266 ELEMENTARY MATHEMATICAL ANALYSIS I §165
Let X change by a constant factor m, so that the new value of %
is mx. Call y' the new value of y. Then
2/' = a{yn£)'' — f{mx) (3)
That is:
y a{mxY ...
, - ~ — ~ = m" (4)
y' ax"
which shows that the ratio of the two y^s is independent of the value
of X used, or is constant for constant values of m.
Another statement of the law of the power function is: As a;
increases in geometrical progression ^ y, or the power function, in-
creases in geometrical progression also.
Let m be nearly 1, say 1 + r, where r is the percent change in x
and is small, then we have:
y' = f(^+M = a{x+jx_y ^ (1 + ,)_ 1 + „, (51
y f{x) ax» \ • /
by the approximate formula for the binomial theorem (§111).
Hence, replacing 1 on the right side of (5) by -'jr-^ and
transposing:
y'-y fix + fx) - fix) , .
y fix)
The fraction in the first member is the percent change in y or infix).
The number r is the percent change in the variable x. Therefore
(0) states that for small changes of the variable the percent of
change in the function is n times the percent of change in the variabk.
Let the exponential function be represented by
y = oe*' = Fix) (7)
As already noted in the preceding sections, increasing x by a con-
stant term increases y or the function by a constant factor. Thus
V' _Fix + h)^ 0.*^^^^ •
y " Fix) oe*» ^ ^ ^
which is independent of the value of x or is constant for constant h.
The expression e^ is the factor by which y or the function is in-
creased when X is increased by the term or increment h. See also
§147 and Fig. 97.
In other wonis, as x increases in arithmetical progression, y
§156] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 267
ot the exponential function increases in geometrical progression.
The percent of change is:
F{x) ^ . ^^^
which is constant for constant increments h added to the variable x.
If X change by a constant percent from x to a;(l + r), it will be
found that the percent change in the function is not constant, but
is variable.
The above properties enable one to determine whether measure-
ments taken in the laboratory can be expressed by functions of
either of the types discussed; if the numerical data satisfy the
test that if the argument change by a constant factor the function
also changes by a constant factor, then the relation may be repre-
sented by a power function. If, however, it is found that a change
of the argument by a constant increment changes the function
by a constant factor, then the relation can be expressed by an
equation of the exponential type.
We have already shown how to determine the constants of the
exponential equation by graphing the data upon semi-logarithmic
paper. In case the equation representing the fuiiction is of the
form:
y = ae** + c (10)
then the curve is not a straight line upon semi-logarithmic paper.
If tabulated observations satisfy the condition that the function
less (or plus) a certain constant increases by a constant factor as
the argument increases by a constant term, then th« equation of
the type (10) represents the function and the other constants can
readily be determined.
The determination of the equations of curves of the parabolic
and hyperboUc type is best made by plotting the observed data
upon logarithmic coordinate paper as explained in the next
section.
156. Logarithmic Coordinate Paper. If coordinate paper be
prepared on which the uniform x and y scales are both replaced
by non-uniform scales divided proportionately to log x and log y
respectively, then it is possible to show that any curve of the 'para-
bolic or hyperbolic type when drawn upon such coordinate paper mil
f
268 ELEMENTARY MATHEMATICAL ANALYSIS HIM
be a straight line. This kind of squared paper la called logaritfamic
paper, and is illustrated in Fig. 105.
To tind the equation of a linePQ on Buch paper, we imagine, as
ia the case of semi-logarithmic paper, that all rulings are erased
and replaced by continuations of the uniform scales ON and MN,
on which the length ON or MN is taken as unity. Denotii^, as
_^-:
p
L
SlTig)« Lagarlthmic, S<:a]« o£ Oommdn Lo^rithnu Iv Marffliu
Fia. 105. — Logarithmic Coordinate Paper, Form Mi. The finer rulings
of form Mi are not reproduced.
before, distances referred to these uniform scales by capital letters,
we may write as the general equation of a straight line:
Y = mX + B (1)
In the case of the line PQ, m = 0.505, B = 0.219, and hence
Y = 0.505X + 0.219
But, y = log V, X = log X, where y and x represent distances
.OGARITHMIC AND EXPONENTIAL FUNCTIONS 269
•ed on the scales LM and IX) respectively, and 0.219 =
5. Hence:
log y = 0.505 log X + log 1.65
log y — log 1.65 = 0.505 log x
may be written in the form
logj^ = logx0-606
= x^'^°^
1.65
y = 1.65x0-50* (2)
meral, if B = log 6, we may write the equation (1) in the
y = bx"^ (3)
le straight line on logarithmic paper passes through the
1, 1) its Cartesian equation is
y = mX (4)
rred to the logarithmic scales,
log y = m log X
y = a;«. (5)
;traight line on logarithmic paper passes through the point
ith slope m, its equation referred to the logarithmic scales is
I = [I]- <«>
)garithmic paper, form M4, the numbers printed in the lower
the left margin refer to the non-uniform scale in the body
paper. By calling the left-hand lower corner the point
(10, 10), (10, 1), (10, 100), (1, 100) or (100, 100), . . .,
of (1, 1), these numbers may be changed to 10, 20, 30,
Dr to 100, 200, 300, . . . , etc.
le following exercises the graphs are to be carefully con-
1 upon logarithmic paper, and the values of the various
iions and all other necessary information indicated on the
a terms of the proper concrete units.
I
270 ELEMENTARY MATHEMATICAL ANALYSIS [§156
If the range of any variable is to extend beyond any of the single
decimal intervals, 1—10, 10— -100, 100—1000, . . ., the
''multiple paper," formM6, maybe used, or several straight lines
may be drawn across form M4 corresponding to the value of the
function in each decimal interval, 1 — 10, 10 — 100, . . ., so
that as many straight lines will be required to represent the func-
tion on the first sheet as there are intervals of the decimal scale to
be represented. However, if the exponent w in y = 6a5"» be a
rational number, say nlr^ then the lines required for all decimal
intervals will reduce to r different straight lines.
One of the most important uses of logarithmic paper is the de-
termination of the equation of a curve satisfied by laboratory
data. If such data, when plotted on logarithmic paper, appear
as a straight line, an equation of the parabolic type satisfies the
observations and the equation is readily found. The exponent
m is determined by measuring the slope of the line with an ordi-
nary uniform scale. The equation of the line is best found by
noting the coordinates of any one point (o, 6) and substituting
those and t he slope in in the equation
y _ rx-|m
h - Ya\
Exercises
Draw tlie following on single or multiple logarithmic paper, forms
.\/4or A/6:
!• y « x» y « 2x» y = 3x, y = 4x, . . .
2. y - X, y =« x«, y = x», y = x*,
8. y - I x. y = 1 x\ y = 1 'x», . . .
4. y « x^*, y= x^'', y = x*\ . . .
5. .4 - rr*.
6. ;> — 0.(X>3r*, whwt> p is the pressure in pounds per square foot
on ji fljit surfaw oxjx^sed to a wind velocity of v miles i)er hour.
T. r - c\ r.< for c = 1 10 and r = 1.
a* / - \ iV^i for 5; = 32.2.
9, (' - /T R \vhoTt> *: = 110 volts.
10. .< ,1 2\c;* whoix- V- = 5-—
11, V ' T\ I ,:. whtNrt^ ,7 = o2.2.
S156) LOGARITHMIC AND EXPONENTIAL FUNCTIONS 271
IS. p/po ■?■ (*>/pii)'"', whMe pt = 0.07S, the weight of 1 cubic
foot of air in pounds at 70° P. and at pressure po of 14.7 pounds per
square inoh.
13. ff = ^^, for D = 5000, 10,000, 15,000 20,000, where C =
100, 200, 300, 400, 500, 600, 700, 800,
225, D is displacement in tons and iS is speed
900, 1000. d is the diameter of cold rolled shafting in inchee; the
line should be graphed for values of d between d = 1 and d = 10.
b^
4^-Mj^
»ii»
■ittS
gjjatf U [ [|] [i-||/"|^^^|rp[^--
tJlilTJ'fn
Mf f-t : ; 1 -
H
1 fti li;'
H
11 If
Fio. 106, — A Weir Formula Graphed on Multiple LoKarithmic Paper.
16. F = O.O0O341IFflJV', where N is revolutions per minute, R is
r&dius in feet, W is weight in pounds, and F is centrifugal force in
pounds.
16. q = 3.37tft" for L = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, O.S, 0.9,
1.0. See Fig. 106.
17. H — — J, „ ■ , where V is the velocity of water in feet per
Second under the head of H feet per 10,000 feet in clean cast-iron pipe
of diameter d feet. See Fig. 108.
U. The relation between electrical resistance and amount of total
SoUds in solution for Arkansas River Valley water at 70° F. is given
by the following table:
272 ELEMENTARY MATHEMATICAL ANALYSIS [S15T
S — total solids in solution aa parts per 1,000,000:
1,000, 800, 600, 400, 300, 200
R = resiatanoe in ohms: 215, 260, 340, 480, 615, 860.
Plot the results on form Mi and find from the graph the equation
coimeoting S and R.
"t-'
^^;«
1
m
11
fiLllU
lit
//
II
120
110
ijll
//
1
'III
/,
III
//
ill
/
r"
'II
1 ™
6D
jl
d
K
is
••■
1/
d
//
,
III.
20
w
#''
^
6<-
19. Replot the curves of Fig. 107. On the new diagram draw the
lines corresponding to slopes of 7, 8, and 9 feet per 10,000 respectively.
30. Explain the periodic character of the rulings on Figs. 106
and 108.
167. Sums of Exponential Functions. Functions consisting of
the sum of two different exponential functions are of frequent
occurrence in the application of mathematics, especially in elei>-
5157] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 273
trical science Types of fundamental importance are e" +e-"
and c" — e~" whicli are so important that the forms (e" +e~") /2
and (e" — e-")/2 ha\e been given special names and tables
of their values ha\e been computed and pnnted The first of
IW
iliese is called the hyperbolic cosine of u and the second is called the
^perbolic sine of u; they are written in the following notation:
= (e- + e-") 12, sinh u = (e-
274 ELEMENTARY MATHEMATICAL ANALYSIS [II67
U X = a cosh u and y = a sinh u, then squEuing and Bubtracting
X* — y* = o'(cosh' « — sinh* u)
Therefore the hyperbolic functions
X = a cosh u,y = a sish u
k
■i
\
f
V
/
•
M
-
!N
^/
ir
/
-■■
.;
-2.
-l! J^
0
I
/
,
/
/
.J
/
1
'.,
1 ;
1
i 1
1 1
Fig. 109.— The Curves of the Hyperbolic Sine and Cosine.
, appear in the parametric equations of a rectangular hyperbolt
just as the circular functions
X = acos0,y = a sin d
appear in the parametric equations of the circle
x' + y* = a*
The graplis oS y = a cosh x and y = a sinh x were called for i"
exercises 1, 2, §146. They are shown in F«. 109. The firat I
of these curves is formed when a chain is suspended between ^ I |l
points of support; it is called the cfttenuy. These two cuttm 1
LOGARITHMIC AND EXPONENTIAL FUNCTIONS 275
est drawn by averaging the ordinates of y = c* and y =e"*,
;he ordinates of y = e* and y = — e~'.
rves whose equations are of the form y = ae"»* + 6e»* take
lite a variety of forms for various values of the constants. A
idea of certain important types can be had by a comparison of
irves of Fig. 110 whose equations are:
1.75
1.5
1.25
1
.76
.5
.26
0
..25
-.6
10. — Combinations of Two Exponential Curves. After Steinmetz.
y = e-* + 0.5e-2*
y = e-' + 0.2e-2«
y = e-'
y = e~' — 0.2e-2a5
y = e-' — 0.5e~^'
y = e-' — 0.8e-2*
y = g-x _ g-2*
y = €-' — 1.56"2»
5 student should arrange in tabular form the necessary
rical work for the construction of these curves.
\"
(l)l/-e"*+0.5«-2«
(2) !/-«'*+ 0.2 e -2 «
(3) !/-«'*
(4)l/-e-'- 0.2tf-2«
(5) !/-«-*- 0.5tf-2«
(6) i/.e'*- 0,8tf-2*
{8)l/-e-*- 1.5e-2«
i
^
^
k
(6)^
^
■ —
J
5
1
L5
2
2.5
\
f
276 ELEMENTARY MATHEMATICAL ANALYSIS [§]
If the second exponent be increased in absolute value, the poi
of intersection with the y-axis remain the same, but the regioD
close approach of the curves to each other is moved along the cii
y = 6~* to a point much nearer the t/-axis. To show this the i
lowing curves have been drawn and shown in Fig. 111.
175
L5
*
1.25
1
(1)
\<2)
(l)l^-e"'+0,5e"*®*
(2) V^e"'
C3)l/-e-*-0.1e"W*
{4)l/-e"*-0.5e-M«
(6)i/.e-»-1.5«-w*
.75
AV<3)
.5
fhh)
\
.25
m)
\
\^
^
0
.25
5
.
5
1
1.5
2
2.5
Fig. 111. — Combinations of Two Exponential Curves. After Steinm
y
y
y
y
y
y
- O.le-iO'
158.* Damped Vibrations. If a body vibrates in a medium 1
a gas or liquid, the amplitude of the swings are found to get sma
and smaller, or the motion slowly (or rapidly in some cases) c
out. In the case of a pendulum vibrating in oil, the rate
5168] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 277
decay of the amplitude of the swings is rapid, but the ordinary
rate of the decay of auch vibrations in air is quite slow. The ratio
between the lengths of tha successive Umplitudes of vibration is
called the damping factor or the modulus of decay.
The same fact is noted tn case the vibrations are the torsional
vibrations of a body suspended by a fine wire or thread. Thus a
nscometer, an instrument used for determining the viscosity of
lubricating oils, provides means for determining the rate of the
decay of the torsional vibration of a disk, or of a circular cylinder
/
-\
^
1
\
'
\
4
^
\
,,..r
"
\
>
^
'
Fig. 1 12.— The Curve y = e"' " aio (.
suspended in the oil by a fine wire. The "amplitude of swing"
ia in this case the angle throi^h which the disk or cylinder turns,
measured from its neutral position to the end of each swing.
In all such cases it is found that the logarithms of the successive
amplitudes of the swings differ by a certain constant a-mount or, as
it ia said, the logarithmic decrement is constant. Therefore the
amplitudes must satisfy an equation of the form
where A is amplitude and t is time. The actual motion is given
by an equation of the form
y = ofi-*' sin ct
A study of oscillations of this type will be more fully taken up in J
278 ELEMENTARY MATHEMATICAL ANALYSIS [S15g
the calculus, for the present it will suffice to graph a. tew examples
of this type. Let the expression be
y = e-'/'sin( (1)
A table of values of t and y must first be derived. There are three
ways of proceeding: (1) Ass^;ti successive values to I irrespective
of the period of the sine (see Table V and Fig. 112). (2) Select
for the values of t those values that give aliquot parts of the period
2t of the sine (see Table VI and f%. 113). (3) Draw the sinu-
soid y = sin t carefully to scale by the method of §5S; then draw
upon the same codrdinate axes, using the same units of measure
\
/
^
t
J,
-i
^
\
,
,-r^
^
kr-
—
—
*7r
\
!A '^
r^
LJ!
»>
<
/
Fio. 113.— The Curve j/ = «-"• fdn i.
adopted for the sinusoid, the exponential curve y = e~''^; finally
multiply together, on the slide rule, corresponding ordinates taken
from the two curves, and locate the points thus determined.
The first method involves very much more work than the seeond
for two principal reasons: First, tables of the Ic^aritbms of the
trigonometric functions with the radian and the decimal divisioDS
of the radian as argument are not available; for this reason 57.3°
must be multiplied by the value of t in each case so that an ordinaiy
tr^onometric table may be used; second, each of the values written
58l LOGARITHMIC AND EXPONENTIAL FUNCTIONS 279
column (3) of the table must be separately determined, while
the periodic character of the sine be taken advantage of, the
Lmerical values would be the same in each quadrant.
The second method, because of the use of aUquot divisions of
e period of the sine, such as tt /6 or ^ /12 or tt /18 or t /20, etc.,
assesses the advantage that the values used in column (3) need
I found for one quadrant only and the values required in column
) are quite as readily found on the slide rule as in the first
ethod.
TABLE V
Table of the function y = e~'/* sin t
in radians
log c~V« ■=
- (0.0869)«
log sin t or log
sin 57.3£ if t is
in degrees
logi/
0.0
- 0.0000
+ 0.000
0.5
- 0.0434
9.6807
9.6372
1 ^^ • ^^^^^^
+ 0.434
1.0
- 0.0869
9.9250
9.8381
+ 0.689
1.5
-0.1303
9.9989
9.8686
+ 0.739
2.0
- 0.1737
9.9587
9.7850
+ 0.610
2.5
-0.2172
9.7771
9.5599
+ 0.363
3.0
-0.2606
9.1498
8.8892
+ 0.077
3.5
- 0.3040
9.5450
9.2410
-0.174
4.0
- 0.3474
9.8790
9.5312
- 0.340
4.5
- 0.3909
9^9901
9.5992
- 0.397
5.0
- 0.4343
9.9818
9.5475
- 0.353
5.5
- 0.4777
9.8485
9.3708
-0.235
6.0
- 0.5212
9.4464
8.9252
-0.084
6.5
- 0.5646
9.3322
8.7679
+ 0.059
7.0
- 0.6080
9.8175
9.2095
+ 0.162
7.5
- 0.6515
9.9722
9 . 3207
+ 0.209
8.0
- 0.6949
9.9954
9.3005
+ 0.200
8.5
- 0.7383
9.9022
9.1634
+ 0.146
9.0
-0.7817
9.6149
8.8332
+ 0.068
9.5
- 0.8252
8.8760
8.0508
-0.011
10.0
- 0.8686
9.7356
8.8670
- 0.074
10.5
- 0.9120
9.9443
9.0323
-0.108
11.0
- 0.9555
9.9999
9.0444
-0.111
11.5
- 0.9989
9.9422
8.9433
- 0.088
12.0
- 1.0424
9.7296
8.6872
-0.049
280 ELEMENTARY MATHEMATICAL ANALYSIS [§158
^.
TABLE VI
Table of the function y
= e"* /^ sin t
1
2
3
4
5
n ■■ ( in
units of
ir/6 radians
loge-^V" *»
- (0.0465) n
iog sin nT/6
logy
V
0
1
- 0.0000
-0.0455
0.000
+ 0.450
9 . 6990
9.6535
2
-0.0910
9.9375
9.8465
+ 0.702
3
-0.1364
0.0000
9.8636
+ 0.731
4
- 0.1819
9.9375
9 . 7556
+ 0.570
5
-0.2274
9.6990
9.4716
+ 0.296
6
- 0.2729
+ 0.000
7
- 0.3184
9 . 6990
9.3806
-0.240
8
- 0.3638
9.9375
9.5737
- 0.375
9
- 0.4093
0.0000
9.5907
-0.390
10
- 0.4548
9.9375
9.4827
-0.304
11
- 0.5003
9 . 6990
9 . 1987
-0.158
12
13
- 0.6458
- 0.5912
0.000
+ 0.128
9 . 6990
9 . 1078
14
- 0.6367
9.9376
9.3008
+ 0.200
15
- 0.6822
0.0000
9.3178
+ 0.208
16
- 0.7277
9.9375
9.2098
+ 0.162
17
-0.7732
9 . 6990
8 . 9258
+ 0.084
18
19
- 0.8186
-0.8641
0.000
-0.068
9 . 6990
8 . 8349
20
- 0.9016
9 . 9375
9 . 0279
- 0.107
21
-0.9551
0.0000
9 . 0449
- 0.111
22
- 1.0006
9 . 9375
8 . 9369
-0.087
23
- 1.0460
9 . 6990
8.6530
- 0.045
24
- 1 . 0915
0.000
The third method is perhaps more desirable than either of the
others if more than two figures accuracy is not required. The
curve can readily be drawn with the scale units the same in
both dimensions, as is sometimes highly desirable in scientific
applications.
In Figs. 112 and 113 a larger unit has been used on the vertical
scale than on the horizontal scale. In Fig. 113 the horizontal unit
is incommensurable with the vertical unit. To draw the curve
to a true scale in both dimensions it is preferable to lay off the
LOGARITHMIC AND EXPONENTIAL FUNCTIONS 281
linates on plain drawing paper and not on ordinary squared
r. Rectangular coordinate paper is not adapted to the proper
iruction and discussion of the sinusoid, or of curves, like the
snt one, that are derived therefrom.
irves whose equations are of the form y = Je-'/* sin t or
3e~'/^ sin i, etc., are readily constructed, since the constants
3, etc., merely multiply the ordinates of (1) by 1/2, 3,
as the case may be. Likewise the curve y = e'^' sm ex is
ly drawn since sin ex can be made from sin x by multi-
g all abscissas of sin x by 1 /c.
CHAPTER IX
TRIGONOMETRIC EQUATIONS AND THE SOLUTION OF
TRIANGLES
A. FURTHER TRIGONOMETRIC IDENTITIES
169. Proof that p = a cos ^ + b sin ^ is a Circle. J. Geomet-
rical Explanation. We know (§64) that pi = a cos ^ is the polar
equation of a circle of diameter a, the diameter coinciding in
direction with the polar axis OX; for example, the circle Oi,
Fig. 114. Likewise, p2 =
b sin 0 is a circle whose dia-
meter is of length b and
makes an angle of + 90°
with the polar axis OX, as
P the circle OB, Fig. 114.
Also, p = c cos (^ — ^i) is
a circle whose diameter c
has the direction angle ^i.
See equation (4), §68. We
shall show that if the radii
vectores corresponding to
any value of ^ in the equa-
tions pi — a cos d and pi =
b sin 6 be added together to
Fig. 114.-Combination of the Cir- ^^^ ^ ^^^ j^^jj^g ^e^tor
cles p = a cos 0 and p = h sin 0 into a i i? n i £ a
Single Circle p = o cos ^ + & sin 0. P, then, for all values Of t/,
the extremity of p hes on
a circle (the circle OC, Fig. 114) of diameter s/a'^ + 6*. I^
other words we shall show that:
p = a cos ^ + 6 sin ^ (1)
is the equation of a circle.
282
§159] TRIGONOMETRIC EQUATIONS 283
In Fig. 114, pi = a cos 6 will be called the a-circle OA; p2 =
b sin d will be called the h-cirde OB. For any value of the angle
B draw radii vectores OM, ON, jneeting the a- and 6-circles
respectively at the points M and N, If P be the point of inter-
section of MN produced with the circle whose diameter is the
diagonal OC of the rectangle described on OA and OB, we shall
show that OM + ON = OP, no matter in what direction OP be
drawn.
Let the circle last mentioned be drawn, and project BC on OP,
Since ONB and OPC are right angles, NP is the projection of
BC (= a) upon OP, But OM also is the projection of a (= OA)
upon OP, Hence NP = OM because the projections of equal
parallel lines on the same line are equal. Therefore, for aU values
of e, NP = pi and OP = ON + NP = p2 + pi, which is the fact
that was to be proved.
Designating the angle AOC by di, the equation of the circle OC is
by §68
p = Va^ + h^ cos {$ - di) (2)
The value of Oi is known, for its tangent is . It should be observed
that there is no restriction on the value of d. As the point P
moves on the circle OC, the circumference is twice described as d
varies from 0° to 360°, but the diagram for other positions of the
point P is in no case essentially different from Fig. 114.
The above reasoning and the diagram involve the restriction
that both a and b are positive numbers. While it is possible to
supplement the reasoning to cover the cases in which this restric-
tion is removed, it will be unnecessary as the analytical proof at
the end of this section is applicable for all values of a and 6.
Example: From the above we know that the equation
p = 6 cos ^ -f- 8 sin ^ is a circle. The diameter of the circle is
V^oM-^ = VP +^^ = 10» so that the equation of the circle
may also be written in the form p = 10 cos {d — ^i), in which ^i
is the angle whose tangent is - = ^ = 1.33. From a table of tan-
gents di = 53° 8', so that the equation of the circle may be
written p = 10 cos {$ - 53° 8').
II, Analytical Proof. We shall prove analytically that p =
284 ELEMENTARY MATHEMATICAL ANALYSIS [§159
a cos ^ + 6 sin ^ is a circle, without imposing conditions upon the
algebraic signs of a and b. Multiply both members of
p = a cos ^ 4- 6 sin ^ (4)
by p, and obtain
p2 = ap cos 6 + bp sin d (5)
By §70, the expression p cos d is the value in polar co-
ordinates of the Cartesian abscissa x; also p sin ^ is the value in
polar coordinates of the ordinate y. Likewise p^ = x^ + y^.
After the substitution of these values, (5) becomes
a;2 + 2/2 = ax + by (6)
Transposing and completing the squares:
r aV . r ^V a^ + b^ mx
This is the Cartesian equation of a circle with center at the point
\^y 2 r ^^^ ^^ radius i\/a^ + b^. The circle passes through
the origin, since the coordinates (0, 0) satisfy the equation, and
also passes through the point (a, b) since these coordinates
satisfy (6).
Since (4) is now known to represent a circle passing through the
origin, its polar equation can be written in any of the forms (3)-
(6) of §68. Calling ^i the direction angle of the diameter
of (7), (no matter what direction OC actually occupies) we can
write the eqation of the circle in the form
P = \/a2 4- 62 cos {$ - ^i) (8)
in which the direction angle di is the angle AOC, Fig. 114, or the
angle whose tangent isb -r- a. If a and b are not both positive,
the angle ^i is still easily determined. For example, if a = — 1,
and 6 = — 1, then ^i = angle of third quadrant whose tangent
is 1, or = 225°, so that equation (8) becomes:
p = \/2 cos (0 - 225°)
This may also be written
p = V^ cos (^ + 135°)
since the resulting circle may be thought of as p = \/2 cos 9
rotated negatively through 135°.
§169] TRIGONOMETRIC EQUATIONS 285
The equation of the circle OC in any position, that is, for any
values of a and 6, positive or negative, may also be written in
the form
p = Va^ + b 2 sin {$ + $2) (9)
in which 62 is the angle BOC in Fig. 114. See §68, equations
(5) and (6).
It has been emphasized above that ^, ^1, ^2, are any angles —
that is, angles not restricted in size or sign. The distinction be-
tween them need not be lost sight of, however, d is any angle be-
cause it is the variable vectorial angle of any point of the locus, and
ranges positively and negatively from 0° to any value we please.
01 is an) angle (positive or negative) because it is the direction
angle of the diameter OC. It is a constant, but a general, or
unrestricted, angle, but would usually be taken less than 360°
in absolute value. By construction, 62 is also any constant
angle.
The result of this section should also be interpreted when the
variables are x and y in rectangular coordinates, and not p and
d of polar coordinates. Thus, y = a cos x is a sinusoid with
highest point or crest at x = 0, 27r, 47r, . . . Likewise, y =
6 sin X is a sinusoid with crest at x = ^^ ~^* ~^* • • • The
above demonstration shows that the curve
y = a cos X + bsin X
is identical with the sinusoid
y = Va^ + b^ cos (x — hi) = \/a^ + b^ sin (x + /12)
of amplitude \/a^ + b^ and with the crest located at x = hi, or at
rt — ^2, where hi is, in radians, the angle whose tangent is -, and
^ a
^2 is, in radians, the angle whose tangent is r*
Exercises
1. Put the equation p = 2 cos 0 + 2\/3 sin ^ in the form
(x — hy -\- (y — k)^ = /i^ + A;*; also in the form p = a cos {0 — ^1)
and find the value of Oi. See equation (7) above.
2. Find the value of ^1 if p = cos d — -y/s sin d.
286 ELEMENTARY MATHEMATICAL ANALYSIS
3. Put the equation p = 4 cos ^ + ^y/S sin d in the forms
(x - /i)* + (y - A;)* = /i' + A;* and p = a cos (0 - di).
4. Put the equation p = — 4 cos ^ — 4 sin d in the form
(x " hy + (y — ky = /i^ H- fc' and find the value of 0% when the
given equation is written in the form p =» a sin {0 -\- O2).
6. Put the equation p = 2\/3 cos $ -{■ 2 sin ^ in the form
(x — h^) -\- {y — ky == h^ -{■ k*; also in the form p = a cos (^ — ^1).
6. Put the equation p = 3 cos $ -{■ A sin $ in the form p =
a sin (d + ^2). Put the same equation in the form p = o cos
(e - ^1). (^1 is the angle AOC, Fig. 114.
7. Put the equation p = 5 cos d + 12 sin 0 in the form p =
a sin (6 + ^2); also in the form p ^ a cos ($ — $1),
8. Put p = 3 cos d + 4 sin 6 in the form {x - h)^ + (2/ - A;)* =
9. Put p = 5 cos 0 + 12 sin $ in the form (x - hy + (y - ky =
h^ + A;2.
10. Put the equation (x— l)* + (2/ — 1)^=2 in the form p =
a sin {$ + a) and determine a and a.
11. Put the equation {x + 1)^ + (2/ - \/3)^ = 4 in the form
p == a sin (d — a) and determine a and a.
12. Put the equation (x + 1)^ + (y + Vs)* = 4 in the form p =
a sin (d — a) and determine a and a.
. 13. Put the equation {x + 1)^ + {y + ly ^ 2 in the form p =
a cos ($ + cr) and determine a and a.
14. Put the equation (x + 1)^ + (2/ + Vs)^ = 4 in the form p =
o cos (d + cr) and determine a and a.
16. Find the maximum value of cos 0 — Vs sin dj and determine
the value of 0 for which the expression is a maximum.
Suggestion : Call the expression p . The maximum value of p is the
diameter of the circle p = cos 0 — V 3 sin 0. The direction cosine of
the diameter is the value of a when the equation is put in the form
P — a cos {0 — a).
16. Find the value of 0 that renders p = Ws cos ^ — J sId tf a
maximum and determine the maximum value of p.
17. Find the maximum value of 3 cos t + 4 sin t.
160. Addition Formtilas for the Sine and Cosine. From the
preceding section, equations (1), (8) and (9), we know that the equa-
1160]
TRIGONOMETRIC EQUATIONS
tion of the circle OC, Fig. 115, may be written in any c
forms:
p = acoafl + tsinfl
p = c&in (B — 9i)
p = c COS (ff - 9,)
Heace, for all values of 0, 8,, and 8i,
sin (fl - fl») = - cos ff + - sin 0
287
e,of the
(1)
(2)
(3)
008 (0 - 0,) = -- COS 0+ - sin 8 (5)
In each of these equations c =■ ■\/a* + b'. The letters a and 6
stand for the eoflrdinates of C irrespective of their signs or of
the position of C.
h /^'~~
7
V ^
Y
Fig. 115. — The Circle |. = ■: cos (fl— «,) at f - ain (« - »j,) used in
the Proof of the Addition Formulas. Note that ti — 90° + »i which is
abo true for neoaUve analet, namely Ji » 90° -}- S%
Since (4) and (5) are true for all values of 6, they are true when
fl = 0° and when 8 = 90°.
First, let e = 0= in (4) and (5).
then from (4): a/c = sin (- fl^) sin 9, by §68 (6)
From (6): a/c = cos (- fl,) = cos tfi by §68 (7)
288 ELEMENTARY MATHEMATICAL ANALYSIS [§160
Second, let ^ = 90** in (4) and (5).
then from (4): 6 /c = sin (90° — ^2) = cos 62 (8)
From (5) : b/c = cos (90** - $0 = sin Si (9)
Substituting (6) and (8) in (4); also (7) and (9) in (5), we have
sin {d — ^2) = sin d cos 62 — cos 6 sin 62 (10)
cos (6 — ^0 = cos 6 cos 61 + sin 6 sin ^1 (11)
Since these are true for all values of 61 and ^2, put ^1 = (— ei)
and $2 = (-€2). Then by §68, (10) and (11) become
sin (6 + €2) = sin 6 cos €2 + cos 6 sin €2 (12)
cos (6 + €1) = cos 6 cos €1 — sin ^ sin €1 (13)
To aid in committing these four important formulas to memory,
it is best to designate in each case the angles by a and j8, and
write (12) and (13) in the form
sin (a + jS) = sin a cos fi + cos asm ff (14)
cos {a + P) = cos a cos jS — sin a sin jS (15)
and also write (10) and (11) in the form
sin (a — jS) = sin a cos jS — cos a sin j8 (16)
cos (a — jS) = cos a cos jS + sin a sin j8 (17)
The four formulas (14), (15), (16) and (17) must be committed to
memory. They are called the addition formulas for the sine and
cosine. The above demonstration shows that the addition
formulas are true for all values of a and p.
By the above formulas it is possible to compute the sine and cosine
of 75° and 15° from the following data:
sin 30° = 1/2 sin 45° = W2
cos 30° = W3 cos 45° = W2
Thus:
sin 75° = sin (30° + 45°) = sin 30° cos 45° + cos 30° sin 45°
= i-W2 + W3-W2
- iV2 (V3 + 1)
Likewise:
sin 15° = sin (45° - 30°) == iV2 (Vs - 1)
§161] TRIGONOMETRIC EQUATIONS 289
161. Addition Fonnttla for the Tangent Dividing the
members of (14) §160 by the members of (15) we obtain:
^ , , ^. sin (a + /S) sin a cos /S + cos a sin /9 ,^.
tan (a-f-p) = 7 ; — rr = ; 1 — - (1)
^ '^^ COS (a + /S) COS a COS /S — sm a sm /3 ^ ^
Dividing numerator and denominator of the last fraction by
COS a cos P
sin a cos /3 , cos a sin /3
. ^^ cos a COS /3 COS a COS /3 ,-.\
tan (a + B) = ^ ^ r— ^ (2)
V« -r K/ COS a COS /3 __ Sm a sm /3 ^
COS a COS /S COS a COS /3
or:
, . ^. tan a + tan /3
tan (a + j8) = ^j ^ ' "^^ (3)
^ ' ^' 1 — tan a tan /3 ^ ^
Likewise it can be shown :rom (16) and (17), §160, that:
, ^. tan a — tan fi
tan (a — j8) = . , . 7 r (4)
^ '^^ 1 + tan a tan /3 ^ ^
Equations (3) and (4) are the addition formulas for the tangent.
Exercises
1. Compute cos 75° and cos 15**.
2. Compute tan 75° and tan 15°.
3. Write in simple form the equation of the circle
p = sin ^ H- cos 0.
4. Put the equation of the circle p = 3 sin d H- 4 cos 6 in the form
p = c sin ($ + ^1) and find from the tables, or by the slide rule, the
value of ^1.
6. Derive a formula for cot (a + /8).
6. Prove cos (s + 0 cos (s — 0 = cos^ s — sin* L
7. Express in the form c cos (a — h) the binomial 3 cos a +
4 sin a.
8. Express in the form c sin (a + &) the binomial 5 cos a + 12 sin a.
9. Find the coordinates of the maximum point or crest of the sinu-
soid y = sin X -\- yS cos x, [First reduce the equation to the form
y = c sin (a; H- «)].
10. Prove the addition foimulas in the following manner: (1)
In cos (0 — di) = — cos ^ + 7 sin $, show that a/c = cos Bi, h/c =
19
290 ELEMENTARY MATHEMATICAL ANALYSIS [|I62
BJn fli, for all vBluea of 9i. (2) Find coa (* + tfj) by repUcmg «i
by (-Si)- (3) Plnd sin («+0i) by the substitution in (1) of
S ■■ {*/2 - *). (4) Find sin (0 - e,) by replacing «i by (- tfj).
16S. Functions of Composite An^es. The sine, cosine, or
tangent of the angles (OC - e),{90'' + tf), (180° - 9), (180° + 9},
(270° - ff), (270° + 0) can be expressed in terms of functions
of 6 alone by means of the addition formulas of §$160 and 161.
If 0 be an angle of the first quadrant, it is easy, however, to obtain
all the relations by drawing the triangles of reference for the
various angles and then comparing homologous sides of the similar
1-
k.k) P,
p,(k.h)
^
V ^'
/
\
lA
\\\ X
T
/
\
\
/
\
-t,-fc)
(ft.-*)
right triangles of reference. Let the terminal side of the angle 9
be OP (Fig. 116 B), and let P be the point (A, k). Let the
terminal sides of the angles (90° - $), (90° + ff), (270° - ff), etc.,
be cut by the circle of radius a at the points Pi, Pi, P., . . .
Then the coSrdinatea of Pi are {k, A); of Pj are (— ft. A); of P| are
(- k, —A), etc. Hence sine = kja, cos 9 = A /a, sin (90° + fl) =
A /a, cos (90° +9) ft /a, sin (270° -0) A/a, cos (270° -«)
= — kja, etc., which lead to the equalities:
sin (90° + fl) = cos fl (1)
cos (90° + ff) = - sin fl (2)
sin (270° - fl) = - cos fl (3)
cos (270° - e) = - sin 9 (4)
etc.
§162}
TRIGONOMETRIC EQUATIONS
291
By division of (1) by (2) and (3) by (4),
tan (90** + ^) = - cot ^ (5)
tan (270** - ^) = cot 6 (6)
Also from Tig. 116 A, cos (180° -^ B) = -hla, sin (180° + B)
= - A; /a, cos (180° + ^) = - hja, sin (- ^) = - A; /a, cos (- 6)
= A /a, whence there results:
sin (180° - ^) = sin 6
(7)
and by division
ol o.f\
cos (180° - ^) = - cos ^
tan (180° - B) = - tan ^
(8)
(9)
alaxj
sin (180° + ^) = - sin ^
(10)
and by division
cos (180° + B) ^ - cos ^
(11)
tan (180° + ^) = tan ^ (12)
In the above work the angle B is drawn as an angle of the first
quadrant. The proof that the results hold for all values of B is
best given by means of the addition formulas of §§160 and 161.
The method will be outlined in the next section.
The results are brought together in the following table. No
effort should be made to commit these results to memory in this
form. The statements in the form of theorems given below offer a
ready means of remembering all of the results.
TABLE VII
Functions of 0 Coupled vrith an Even or with an Odd Number of
Right Angles
- e
90°- d
90° ^-e
180** - e
180° + e
270°- B
270°+ B
sin
— sin $
cos d
cos 0
sin e
— sin B
— cos B
— cos B
cos
cos $
sin 0
— sin 0
— cos B
— cos B
— sin B
sin B
tan
— tan^
cot e
— cot e
— tan^
tand
cot B
— cot 9
For completeness of the table the functions of ( — B) and of
(90° — B) have been inserted in columns 1 and 2.
All of the above results can be included in two simple state-
ments. For this purpose it is convenient to separate into different
I
292 ELEMENTARY MATHEMATICAL ANALYSIS [§163
classes the composite angles that are made by coupling 6 with
an odd number of right angles, as (90° + ^), (^ - 90**), (270° - 6),
(450° + 6) J etc., and those composite angles that are made by
coupling 6 with an even number of right angles, as (180° + 6),
(180° - 6), (360° - ^), ( - 6), etc. Note that 0 is an even num-
ber, so that ( — ^) or (0° — 6) falls into this class of composite
angles. We can then make the following statements:
Theorems on Functions of Composite Angles
Think of the original angle ^ as an angle of the first quadrant:
I. Any function of a composite angle made by coupling 6 {by
addition or subtraction) with an even number of right angles^ is
equal to the same function of the original angle dj with an algebraic
sign the same as the sign of the function of the composite angle in
its quadrant.
II. Any function of a composite angle made by coupling d [by
addition or subtraction) with an odd number of right angles, is equal
to the co-function of the original angle 6, with an algebraic sign
the same as the sign of the function of the composite angle in Us
quadrant.
For example, let the original angle be B, and the composite angle
be (180° + ^). Then any function of (180° + ^), say
tan (180° + ^), is equal to + tan B, the sign + being the sign of the
tangent in the quadrant of the composite angle (180° + B) or
third quadrant. Likewise cot (270° + B) must equal the negative
co-function of the original angle, or — tan B, the algebraic sign
being the sign of the cotangent in the quadrant of the composite
angle (270° + 0), or fourth quadrant. In the above work it has
been assumed that the angle B is an angle of the first quadrant.
The results stated in italics are true, however, no matter in
what quadrant B may actually lie.
163. Functions of Composite Angles. General Proof: All
of the results given by Table VII or by theorems I and II above
can be deduced at once from the addition formulas, with the
especial advantage that the proof holds for all values of the angle
B, Thus, write
sin (a + jS) = sin a cos jS + cos a sin jS (1)
cos (a + jS) = cos a-cos jS — sin a sin jS (2)
§164]
TRIGONOMETRIC EQUATIONS
293
Put a = 180^, and jS = ± ^; then (1) and (2) become, re-
spectively:
sin (180° ± ^) = T sin ^ (3)
cos (180° ± ^) = - cos ^ (4)
Also in (1) and (2) put a = 90°, and jS = ± ^, then (1) and (2)
become, respectively:
sin (90° ± ^) = cos % (5)
cos (90° ± ^) = + sin ^ (6)
In a similar manner aU of the results given in the table may be
proved to be true.
164. Angle that a Given Line Makes with Another Line. The
slope m of the straight line y = mx + 6 is the tangent of the
Y
iWTf.
y^.
rVv.
V.
V
p
A
o
\Ax
^1
\L2
Fig. 117. — The Angle 0 that a Line L\ makes with Lt.
direction angle, that is,the tangent of the angle that the line makes
with OX. // L\ and L2 are any two lines in the plane, the angle
that Li makes with L2 is the positive angle through which Li
mvst he rotated about their point of intersection in order that L%
may coincide with Li, Represent the direction angles of two
straight lines
y = miX + bi (1)
y = mix + bi (2)
by the symbols di and ^2. Then, through the intersection of the
lines pass a line parallel to the OX-axis, as shown in Fig. 117.
Call <l> the angle that the line Li makes with L2; that is, the positive
angle through which L2, considered as the initial line, must be
turned to coincide with the terminal position given by Lv. II
294 ELEMENTARY MATHEMATICAL ANALYSIS [§164
^1 > $2, then 0 = ^1 - 02, but if ^j > ^i, then 0 = 180° -(^j-
^i). In either case (by equations (9), §162, and (3), §58):
tan 0 = tan (^i - ^2) (3)
That is:
tan<^=;^^-;^f^',
^ 1 + tan ^1 tan $2
or,
tan 0 = . , _ _ (5)
1 i- mim2
The condition that the given lines (1) and (2) are parallel is
obviously thd,t
mi = mj (6)
Thus the lines y = 5x + 7 and y = 5a; — 11 are parallel.
The condition that the given lines (1) and (2) are perpendicular
to each other is that tan 0 shall-become infinite; that is, that the
denominator of (5) shall vanish. Hence the condition of perpen-
dicularity is
1 + tnim2 = 0
or,
mi = - ;^ (7)
m2
Therefore, in order that two lines may he perpendicular to each
other, the slope of one line miLst he the negative reciprocal of the slope
of the other line.
Thus the lines y = (2/3)x - 4 and y = - (3/2)x + 2 are
perpendicular.
Exercises
1. Find the tangent of the angle that the first line makes with
the second line of each set :
(a) 2/ = 2x + 3, y ^ X f2.
(6) y = 3a;- 3, y = 2x^+ 1.
(c) y = 4a; + 5, y = 3a; — 4.
(d) y ^ 10a; + 1. 2/ = 11a; - 1,
2. Find the angle that the first line of each pair makes with the
second:
§165] TRIGONOMETRIC EQUATIONS 296
(a) y = a; + 6, y = - x + 6.
(6) y = {X/2)x + 6, y ^ -2x.
(c) 2/ = 2a; + 4, y = x + 1.
(d) 2a; + 32/ = 1, (2/3)a; + 2/ = 1.
(6) 2a: + 42/ = 3, 3a: + 62/ = 7.
Cf) 2a: + 42/ = 3, 6a: - 32/ = 7.
3. Find the angle, in each of the following cases, that the first line
makes with the second:
(a) 2/ = x/\/Z +4, 2/ = a/3 X + 2.
(6) 2/ = a:/\/3 + 1, 2/ = a/3 a: - 4.
(c) 2/ = V3 a: - 6, 2/ = V3 a; - 3.
4. Find the angle that 2y — 6a: + 7 =0 makes with 2/ + 2a: +
7 » 0 and also the angle that the second line makes with the first.
166. The Functions of the Double Angle. The addition
formulas for the sine, cosine and tangent reduce to formulas of
great importance for the special case fi — a.
Thus: sin (a + a) = sin a cos a + cos a sin a
or: sin 2o: = 2 sin a cos a (1)
Also: cos (a + a) = cos a cos a — sin a sin a
which can be written in the three forms:
cos 2 a = cos* a — sin* a (2)
cos 2 a = 2 cos* a - 1 (3)
cos ^ a = 1 — 2 sin* a (4)
Forms (3) and (4) are obtained from (2) by substituting,
respectively, sin* a = 1 — cos* a and cos* a = 1 — sin* a.
Equations (3) and (4) are frequently useful in the forms:
- 1 + cos 2 a ,.,
cos* a = r (5)
. , 1 - cos 2a ,^.
sm* a = 2 (6)
Again:
, V _ tan a + tan a
or:
^ 2 tan a ,^.
tan 2 a = r— ^— (7)
1 — tan* a
i
296 ELEMENTARY MATHEMATICAL ANALYSIS [§166
166. The Functions of the Half Angle. From (6) and (5) of
§166 we obtain, after replacing a by u/2 and extracting the
square root,
sin (u/2) « ±V(1 - cosu)/2 W
cos (u/2) = ± \/(H- cos u) /2 (2)
Dividing (1) by (2), we obtain:
X / /o\ . jl-cosu , 1-cosu sinu ,„.
tan (u/2) = ± ^ h—^ = ± — -. = ± T—f (3)
\ / / -^^i + cosu sinu 1 + cosu ^'
Formulas (1), (2) and (3) have many important appb'cations in
mathematics. As a simple example, note that the functions of 15°
may be computed when the functions of 30** are known. Thus:
cos 30° = (1 /2) ^/^
therefore: sin 15° = \/(l - cos 30°) /2 = Vl/2 - (l/4)\/3
Also: cos 15° = \/l/2 + (1/4) \/3"
Likewise by (5) :
tan 15° = i:q^= 2 - ^A3
Exercises
1. Compute sin 60° from the sine and cosine of 30®.
2. Compute sine, cosine, and tangent of 22i°.
3. If sin X — 2/5, find the numerical value of sin 2a;, and cos 2z
tan 2x, if x be the first quadrant.
4. Show by expanding sin {x + 2a;) that sin 3a; = 3 sin x -
4 sin' a;.
• _ -^ . o 3 tan X — tan' x
5. Prove tan 3x = — = ^tl — s •
1—3 tan^ X
6. Show that sin 2^/sin B — cos 20/cos ^ = 2 sec 6.
7. Show that:
I sms + cos 2) = 1 H- sm ^.
8. Show that: cos 2^(1 + tan 2e tan e) = 1.
9. If sin A = 3/5, calculate sin (A/2).
10. Prove that tan (^/4 + e) = \ ^ ^^^ ^
' ^ 1 — tan e
11. Prove that tan (ir/4 - d) = (1 - tan d)/(l + tan a).
§167] TRIGONOMETRIC EQUATIONS 297
1 H- sin ^
12. Show that sec 6 + tan 6 =
13. Show that
cos B
1+2 sin a cos a cos a + sin a
cos' a — sin* a cos a — sin a
14. Show that sec e + tan e = tan t + « '
16. Show that — t—. — j t-b = tan A tan 5.
cot A + cot -D
16. Prove that cos (s + <) cos (s — 0 + sin (s + 0 sin (s — 0 =
cos 2L
167. Sums and Differences of Sines and of Cosines Expressed
as Products. The following formulas, which permit the substi-
stution of a product for a sum of two sines or of two cosines, are
important in many transformations in mathematics, especially in
the calculus. They are immediately derivable from the addition
formulas; thus, by the addition formulas (14) and (16), §160, we
obtain:
sin {a + h) + sin (a — 6) =2 sin a cos h
Likewise by subtraction of the same formulas:
sin (a + 6) — sin (a — 6) =2 cos a sin h
By the addition and subtraction, respectively, of the addition
formulas for the cosine there results:
cos {a + h) + cos (a — 6) = 2 cos a cos h.
cos (a + &) — cos (a — 6) = — 2 sin a sin h.
Represent (a + &) by a and (a — h) by j3.
Then a = (a + jS) /2 and 6 = (a - jS) /2
Hence the above formulas become:
sma + smp = 2sm — ^ — cos — ^ (1)
. ^ ^ 0L + 3 . ot — 3 /^v
sm a — sm p = 2 cos — - — sm — ^-^ (2)
^ ^ Qj + j8 a — 3 ,„v
cos a + COS jS = 2 cos — r-^ cos — 2"^ (3)
cosa-cos^=-2sm^-sm-2- (4)
The principal use of these formulas is in certain transformations
in the calculus. A minor use is in adapting certain formulas to
logarithmic work by replacing sums and differences by products.
298 ELEMENTARY MATHEMATICAL ANALYSIS I|169
168. Graph of y = sin 2z, y «= sin nx, etc. Since the substi-
tution of nx for X in any equation multiplies the abscissas of the
curve by 1 /n, or (n > 1) shortens or contracts the abscissas of all
points of the curve in the uniform ratio n : 1, the curve y = sin 2x
must have twice as many crests, nodes or troughs in a given
interval of x as the sinusoid y = sin x. The curve y = sin 2a; is
therefore readily drawn from Fig. 59 as follows: Divide the axis
OX into twice as many equal intervals as shown in Fig. 59 and
draw vertical Unes through the points of division. Then in the
new diagram there are twice as many small rectangles as in the
original. Starting at 0 and sketching the diagonals (curved to
fit the alignment of the points) of successive cornering rectangles,
the curve y = sin 2x is constructed. It is, of course, the ortho-
graphic projection of y = sin x upon a plane passing through
the y-axis and making an angle of 60** (the angle whose
cosine is 1/2) with the xy plane. The curve y = cos 2x is simi-
larly constructed. In each of these cases we see that the
period of the function is tt and not 2t,
169. Graph of p = sin2^, p = cos 2^, etc. The curve p = cos^
is the circle of diameter unity coinciding in direction with the axis
OX, We have already emphasized that as 6 varies from (f to
360° the circle is twice drawn, so that the curve consists of two
superimposed circular loops. Now p = cos 20 will be found to
consist of four loops, somewhat analogous to the leaves of a four-
leafed clover, but each loop is described but once as 6 varies from
0° to 360°. The curve p = cos 3^ is a three-looped curve, but each
loop is twice drawn as 0 varies from 0° to 360°. Also p = cos US
has eleven loops, each twice drawn, while p = cosl2^ has
twenty-four loops, each one described but once, as 6 varies from
0° to 360°.
The curves p = cos 26, p = sin SB, p = ainO /2 should be drawn
by the student upon polar coordinate paper.
By changing the scale of the vectorial angle, the circle of diame-
ter unity may be used as the graph of the equation p = sin nd.
However if two such equations are to be represented at the same
time, this expedient is not available, for the vectorial angles of the
points of each curve, for the purpose of comparison, must be
drawn to a true scale.
U70)
TRIGONOMETRIC EQUATIONS
299
170. Graph of y = sJn" x, y = cos' i. The graphs y ^ sin' x
and y ^ cos' x have important applications in science. The io\-
lowing graphical niethod offers an easy way of constructing the
curves and it illuatrates a number of important properties of the
functions involved. We shall first construct the curve y = cos* x.
At the left of a sheet of S^ X ll-inch paper, draw a circle of radius
36
— (= 2.30) inches, (OA, Fig. 118). Lay off the angles 8 from
OA, Pig. 118, as initial line, corresponding to equal intervals (say
10° each) of the quadrant APE as shown in the figure. Let the
point P mark any one of these equal intervals. Then dropping
the perpendicular AB from A upon OP, the distance OB is the
A S
u I It
Fio. lis,— The Graph of i/ = cos" i.
cosine of 9, if OA be caUed unity. Dropping a perpendicular from
B upon OA, the distance OC is cut off, which is equal to OB'
or cos' 6, anee in the right triangle OBA, OB^ = OCOA = OCl.
Making similar constructions for various values of the angle 8,
say for every 10° interval of the arc APE, the line OA is divided at
a number of points proportionally to cos' S. Draw horizontal
lines through each point of division of OA. Next divide the axis
OX into intervals equal to the intervals of 8 laid off on the arc APE;
since the radius of the circle OA was taken to be (36 /Sir) inches, an
interval of 10° corresponds to an arc of length 2/5 inch, which
therefore must be the length of the equal intervals laid off on OX.
Through each of the points of division of OX draw vertical lines.
300 ELEMENTARY MATHEMATICAL ANALYSIS [§171
thus dividing the plane into a large number of small rectangles.
Starting at A and sketching the diagonals of successive cornering
rectangles, the locus ARB oiy ^ cos* x is constructed.
From Fig. 118, it is seen that B always lies at the vertex of a
right-angled triangle of hypotenuse OA, Thus as F describes the
circle of radius OA, B describes a circle of radius OA /2. Therefore
the curve ARSX is related to the small circle ABO in the same
manner that the curve of Fig. 59 is related to its circle; conse-
quently the curve ARSX of Fig. 118 is a sinusoid tangent to the
X-axis. Thus the graph y = cos* x is a cosine curve of ampUtude
1 /2 and wave length or period tt, lying above the x-axis and tangent
to it.
In Fig. 118, OC = OH + OC = OH + HB cos 26 =
1/2 -h (1/2) cos 26. Therefore the curve ARS has also the
equation:
y = l/2 + {l 12) cos 2x (1)
Hence we have a geometrical proof that
cos* X = 1 /2 + (1 12) cos 2x (2)
which is formula (5) of §166. Note that (1) is the curve y =
cos 2x with its ordinates multiplied by 1/2 then translated 1/2
unit upward.
The curve y = sin* x is readily drawn in a manner similar to that
above, by laying off the angle 6 from OX as initial line. The curve
IT
is the same as that of Fig. 118, moved the distance j to the right.
B. PLANE TRIANGLES : CONDITIONAL EQUATIONS
171. Law of Sines. The first of the conditional equations pe^
taining to the oblique triangle is a proportion connecting the sines
of the three angles of the triangle with the lengths of the respect-
ive sides lying opposite. Call the angles of the triangle A, B, C,
and indicate the opposite sides by the small letters a, 6, c, respect-
ively. From the vertex of any angle, drop a perpendicular p
upon the opposite side, meeting the latter (produced if necessary)
at D. Then, from the properties of right triangles, we have, from
either Fig. 119 (1) or 119 (2)
p = c sin DAB = a sin C (1)
5172]
TRIGONOMETRIC EQUATIONS
301
Therefore:
Or:
But,
sin DAB = sin A Fig. 119 (1)
= sin (180° - A) Fig. 119 (2)
= sin A
p = c sin A = a sin C (2)
a/sin A = c/sin C (3)
In like manner, by dropping a perpendicular from A upon a, we
can prove:
b/smB = c/sin C (4)
Therefore: a/sin A = b/sin B = c/sin C = 2R (5)
Stated in words, the formula says: In any oblique triangle the
sides are proportional to the sines of the opposite angles.
(2)
Fig. 119. — Derivation of the Law of Sines and the Law of Cosines.
Geometrically: Calling each of the ratios in (5) 2/2, it is seen
from Fig. 119 (2) that R is the radius of the circumscribed circle,
and that c/sin C = 2R can be deduced from the triangle BAE^
Similar construction can be made for the angles B and A,
172. Law of Cosines. From plane geometry we have the theo-
rem: The square of any side opposite an acute angle of an oblique
triangle is equal to the sum of the squares of the other two sides di-
minished by twice the product of one of those sides by the projection
of the other side on it. Thus in Fig. 119 (1) :
a« = 6» + c« - 2bd (1)
Now: d = c cos A
Therefore: a' = 6» + c' - 26c cos A (^\
302 ELEMENTARY MATHEMATICAL ANALYSIS (§172
Likewise we learn from geometry that the square of any side opp-
site an obtuse angle of an oblique triangle is equal to the sum of
the squares of the other two sides increased by twice the product of one
of those sides by the projection of the other on it. Thus in Fig. 119
(2):
a2 = 6« + c« + 2bd (3)
Now: d = c cos DAB = c cos (180 — A) = — c cos A
Therefore (3) becomes:
a* = b* + c« - 2bc cos A (4)
This is the same as (2), so that the trigonometric form of the geo-
metrical theorem is the same whether the side first named is oppo-
site an acute or opposite an obtuse angle.
In the same way we may show that, in any triangle:
b2 = c2 + a* - 2ca cos B (5)
c2 = a2 + b2-2abcosC (6)
Independently of the theorem from plane geometry, we note from
Fig. 119(1):
a2 = (6 - dy + p2 = (5 « ^)2 + c2 - d«
= 62 + c2 - 2bd
= 62 + c2 - 26c cos A
From 119 (2) : a^ = (6 + dy + p^ = (b + dy + c^ - d*
= 62 + c2 + 2bd
= 62 + c2 + 26c cos DAB
= b^ + c^ - 26c cos A
since DAB = 180° - A and cos (180*' - A) = - cosi
Second Proof: Since any side of an oblique triangle is
the sum of the projections of the other two sides upon it, the
angles of projection being the angles of the triangle, we have:
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A
Multiply the first of these equations by a, the socond by h
the third by c, and subtract the second and third from the first
The result is:
TRIGONOMETRIC EQUATIONS 303
|2 «. 52 _ ^2 -5 06 cos C + ca cos B
— be cos A — ab cos C
— ca cos B " be cos A
= — 26c cos A
a2 = 6« + c2 - 26c cos A
3. Law of Tangents. An important relation results if we
formula (5) §171 by composition and division. First
5 the law of sines in the form:
a _ sin A
6 "" Sn B ^^^
1, by composition and division, the sum of the first anteced-
.nd consequent is to their difference as the sum of the second
cedent and consequent is to their difference; that is:
a + 6 _ sin A + sin B .
a — 6 ~ sin A — sin B ^ '
•essing the sums and difference on the right side of (2) by
ucts by means of the formulas (1) and (2) of §167, we
in:
a + b _ 2 sin jjA + B) cos jjA - B)
a - 6 " 2 cos i(A + B) sin i{A - B) ^^^
mplifying and replacing the ratio of sine to cosine by the
ent, we obtain:
a + b tan i(A + B) ^
ke
manner
a - b tan i(A - B)
it follows that:
b + c tan i(B + C)
b - c "■ tan i(B - C)
c + a tan KC + A)
(5)
c - a ^ tan i(C - A) ^^^
'essed in words: In any triangle, the sum of two sides is to
difference^ as the tangent of half the sum of the angles opposite
the tangent of half of their difference.
soMETRiCAL Proop: From any vertex of the triangle as
3r, say fit, draw a circle of radius equal to the shortest of the
sides of the triangle meeting at C, as in Fig. 120. Let
circle meet the side a Sii R and the same side produced at
\
304 ELEMENTARY MATHEMATICAL ANALYSIS [§174
E. Draw A^, AR. Call the angles at A, a, jS, as shown. Then
BE = a + 6 and BR = a — b. Also:
a+P = A
and: Z CRA = fi + B (the external angle of a triangle
RAB is equal to the sum of the two interior opposite angles),
OT a — P = B.
Therefore:
a = h(A+ B)
/3 = i(A - B)
Z EAR = Z ARS = 90**
BE/BR = A^/^/e
"AR ' AR
But B^ = a + 6 and 5i2 = a-h,
while
AE ^ .SR . f.
= tan a and -rri = tan p
Draw 725 || to EA,
By similar triangles:
AR
Therefore:
AR
a + b tan i{A+B)
a-b tan i(A - B)
Fig. 120.— Geometrical Deriva- 174. The following Special fonn-
tion of Law of Tangents. ulas are readily deduced from the
sine formulas and are sometimes
useful as check formulas in computation. They are closely re-
lated to the law of tangents. From the proportion:
a:b:c = sin A: sin 5: sin C
by composition:
c _ sin C
a + 6 "" sin A + sin jB
Now by §165 (1) and §167 (1) this may be written:
c 2 sin iC cos |(7
a + 6 ~ 2 sin i(A + B) cos ^(A - B)
Since C = 180° - (A + J5), therefore:
C/2 = 90° - HA + B), and cos C/2 = sin i(A + B)
c . sin JC _ cos i(A + B) yj
Hence:
a + b cos i(A - B) cos J(A - B)
TRIGONOMETRIC EQUATIONS 306
3 manner it can be proved that:
c ^ sinMA + B) ^
a - b " sin i(A - B) '^^^
(1) and (2) can be readily deduced geometrically from
20.
. The s-formulas. The cosine formula:
a2 = 62 + c2 - 26c cos A
) written in the forms:
a2 = (6 + c)2 - 26c(l + cos A) (1)
a* = (6 - c)2 + 26c(l - cos A) (2)
ding (+ 26c) and (— 26c) to the right member in each case
w% know from §166, (1) and (3), that:
1 + cos A = 2 cos2 {A /2)
1 - cos A = 2 sin2 {A /2)
fore (1) and (2) above become:
a2 = (6 + c)2 - 46c cos2 (A /2) (3)
a» = (6 - c)2 + 46c sin2 {A /2) (4)
g these in the form:
46c sin2 {A /2) = a^ - (6 - c)^ (5)
46c cos2 (A /2) = (6 + c)2 - a' (6)
lividing the members of (5) by the members of (6), we
i:
tanMA/2)=f^r=^ (7)
ring the numerator and denominator we obtain:
t,^, (X 12) = (^+A^;«-^A+4 (8)
^ ' ^ (6 + c + a)(6 + c — a) ^ '
le perimeter of the triangle be represented by 2s, that is,
a + 6 + c = 2s
3 subtracting 2c, 26, and 2o in turn:
a + 6 — c = 2s — 2c (subtracting 2c)
a — 6 + c = 2s — 26 (subtracting 26)
6 + c — a = 2s — 2o (subtracting 2a)
>fore equation (8) becomes:
tan' (A 12) = ^' ~/^^^ T '^ (9)
20
306 ELEMENTARY MATHEMATICAL ANALYSIS l§176
Let:
then:
or:
Likewise:
^ *.«
(« — o)(« — 6)(« — c)l8 = r
tan«U/2) =rV(« - «)'
tan (A/2) = r/(8 - a)
tan (B /2)
tan (C /2)
r/(s-b)
r/(s-c)
(10)
(11)
(12)
(13)
Fig. 121. — Geometrical Derivation of the s-Formulas.
Geometrically: These formulas may be found by means
of the diagram Fig. 121. Let the circle 0 be inscribed in the
triangle ABC; its center is located at the intersection of the bi-
sectors of jthe internal angles of the triangle. Let its radius be f-
TRIGONOMETRIC EQUATIONS 307
I ATi = AT3, BT2 = 5^8, CTi = CT2, and since 2s =
? + c, it follows that one way of writing the value of s is :
s = BT2 + T2C + ATi
efore:
ATi = s - a
)e it follows that:
tanU/2) =r/(s-a) (14)
! this result is the same as (11) above, it proves that the
equation (10) is the radius of the inscribed circle, and there-
)roves that the radius of the inscribed circle may be expressed
le formula
,=J«Z«)(i.^«^) (15)
t that is usually proved in text books on plane geometry.
r."** Miscellaneous Formulas for Oblique Triangles. The fol-
g formulas are given without proof. They are occasionally
1 for reference, although no use will be made of them in
Dook. The following notation is used: The three sides of
►blique triangle are named a, 6, c, and the angles opposite
Ay B, C, respectively. The semi-perimeter of the triangle
)r, 2s = a + 6 + c. The radius of the circumscribed circle
that of the inscribed circle is r, and the radii of the escribed
s are ro, r^, Vej tangent, respectively, to the sides a, 6, c
le given triangle. K stands for the area of the triangle.
s = 4R cos JA cos ^B cos iC
8 — c = 45 sin iA sin ^B cos iC
inalogs f or s — a and s — b.
r = 4R sin JA sin iB sin JC
Tc = 4fl cos JA cos ^B sin ^C
malogs for r. and n.
fa = s tan JA, Tb = s tan JB, r, = s tan JC
2K = ab sin C = be sin A <= ca sin B
K = 2R^ sin A sin B sin C_= 1^"
iC (1)
(2)
(3)
(4)
(5)
(6)
(7)
\
308 ELEMENTARY MATHEMATICAL ANALYSIS (§178
K = ^Js{8 - a)(8 - 6)(s - c) (8)
K = rs = ra{s — a) = nis — 6) = rds — c) (9)
K^ = rrar^Tc (10)
iiC2 = (s _ a) tan JA = (s - 6) tan iB =
(s - c) tan iC (11)
C. NUMERICAL SOLUTION OF OBLIQUE TRIANGLES
178. An oblique triangle possesses six elements; namely, the
three sides and the three angles. If any three of these six
magnitudes be given (except the three angles), the triangle is
determinate, or may be constructed by the methods explained
in plane geometry; it will also be found that if any three of these
six magnitudes be given, the other three may be computed by the
formulas of trigonometry, provided, in both instances, that the
given parts include at least one side.
It is convenient to divide the solution of triangles into four
cases, as follows:
I. Given two angles and one side.
II. Given two sides and an angle opposite one of them.
III. Given two sides and the included angle.
IV. Given the three sides.
The solution of these cases with appropriate checks wiU now
be given. The best arrangement of the work of computation
usually consists in writing the data and computed results in the
left margin of a sheet of ruled letter paper (8 J inches X 11 inches)
and placing the computation in the body of the sheet. Every
entry should be carefully labeled and computed results should be
enclosed in square brackets. All work should be done on ruled
paper and invariably in ink. Special calculation sheets (forms
M2 and M7) have been prepared for the use of students. Neatness
and systematic arrangement of the work and proper checking
are more important than rapidity of calculation.
179. Computer's Rules. The following computer's rules are
useful to remember in logarithmic work:
Last Digit Even: When it becomes necessary to discard a
5 that terminates any decimal, increase by unity the last digi*
§180] TRIGONOMETRIC EQUATIONS 309
retained if it be an odd digit, but leave it unchanged if it be an
even digit; that is, keep the last digit retained even. Thus log ir
= 0.4971; hence write (1/2) log tt = 0.2486. Also log sin
18** 5' = 9.4900 + (correction) 19.5 = 9.4920.
Of course if the discarded figure is greater than 5, the last
digit retained is increased by 1, while if the discarded figure is
less than 5, the last digit retained is unchanged.
Functions of Angles in Second Quadrant: In finding
from the table any function of an angle greater than 100° (but
< 180°) replace hy their sum the first two figures of the number
of degrees in the angle and take the cof unction of the result. The
method is valid because it is equivalent to the subtraction
of 90° from the angle. By §162 this always gives the cor-
rect numerical value of the function. The algebraic sign should
be taken into account separately. Thus: sin 157° 32' 7" =
cos 67° 32' 7\ In case of an angle between 90° and 100°>
ignore the first figure and proceed in the same way:
tan 97° 57' 42" = - cot 7° 57' 42"
180. Case I. Given two angles and one side, as A, B, and c.
1. To find C, use the relation A + B + C = 180°.
2. To find a and 6, use the law of sines, §171.
3. To check results, apply the check formula (1) or (2) §172.
Example: In an oblique triangle, let c = 1492, A = 49°
52', B = 27° 15'. It is required to compute C, a, b.
The following form of work is self explanatory. This arrange-
ment, while readily intelligible to the beginner, does not conform
to the proper standards of calculation explained above. It
should be noted, however, that the process of work and the meaning
of each number entering the calculation is properly indicated or
labeled in the work.
To find C: C = 180° - (A + B) = 103° 53'
To find a:
As sin C (103° 53') colog 0.0129
: c (1492) log 3.1738
: : sin A (49° 52') log 9 . 8834
a [1175] log 3.0701 {
310 ELEMENTARY MATHEMATICAL ANALYSIS [§181
To find h:
Check :
As sin C (103° 53') colog 0.0129
c (1492) log 3.1738
sin B (27° 15') log 9.6608
b [703.9] log 2.8475
As sin i(A +B) (38° 33.5') colog 0.2053
: sin i(A - B) (11° 18.5') log 9.2924
: c (1492) log 3.1738
[469.4]
w.
i\
I
a -h [469.4] log 2.6715
Also a — h from first computations = 471.1 which checks 469.4,
as computed, within 1.7.
The above work arranged in compact form appears as follows:
Computation of Triangle
c, A and B given
Data and results
To find a
To find b
Check
.
c sin A
"" sinC
- c sin B
sin C
a -
csiniU - B)
' ^ ~ sin i(A + B)
log 3.1738
log sin 9.8034
log 3.1738 log 3. 1738
€ = 1492
A = 49° 52'
B = 27° 15' log sin 9.6608
C = (103° 53') colog sin 0 .0129 colog sin 0.0129
a = [1175]
6 = [7^3~.9y
log
3 . 0701
log 2.8475
(A - B)/2 = (11° 19.5')
{A +B)/2 = (38° 33.5')
log sin 9 . 2924
colog sin 0.2053
a-b = (471.1)
Check !
[469.4] 2.6715
Examples
Find the remaining parts, given:
1. A = 47° 20', B = 32° 10',
2. B = 37° 38', C = 77° 23',
3. B = 25° 2', C = 105° 17',
4. C = 19° 35', A = 79° 47',
a = 739.
b = 1224.
b = 0.3272.
c = 56.47.
181. Case n. Given two sides and an angle opposite one of
therriy as a, 6, and A.
1. To find B, use the law of sines, §171.
2. To find C, use the equation A + B + C^ 180°.
3. To check, apply the check formula (1) or (2), §174.
When an angle as By above, is determined from its sine, it admits
§181]
TRIGONOMETRIC EQUATIONS
311
of two values, which are supplementary to each other. There
may be, therefore, two solutions to a triangle in Case II. The
solutions are illustrated in Fig. 122.
In case one of the two values of B when added to the given
angle A gives a sum greater than two right angles, this value
of B must be discarded, and but one solution exists. If a be
less than the perpendicular distance from C to c, no solution
is possible.
,c
Fig. 122. — Case II of Triangles, for One, Two, and Impossible Solutions*
Example: Find all parts of the triangle if a = 345, b = 534,
and A = 25^ 25'.
The solution is readily understood from the following work. •
To find B: As a (345)
h (534)
sin A (25** 25')
sin B [41** 37']
B' [138** 23']
C - 180** - (A + B) = 112° 58'.
To find c:
AssmA (25^25')
: sin C (IW 58')
: : a (345)
: c [740.1]
colog 7.4622
log 2.7275
log 9.6326
log 9.8223
Check:
As c (740.1)
: b -a (189)
: : sin i(fi + A) (33° 31')
: sin i{B - A) [8° 6']
Check!
colog 0.3674
log 9.9641
log 2.5378
log 2.8693
log 2.8693
colog 7 . 7235
colog 0.2579
log 9.1489
9.9996
312 ELEMENTARY MATHEMATICAL ANALYSIS [§181
The sum of the logs should be 0. The discrepancy is 4 in the last
decimal place.
To find c': C = 180^ - A
- B' = 16^ 12'
As sin A (25** 25')
colog 0.3674
: sin C (16*^ 12')
log 9.4456
: : a (345)
log 2.5378
: c' [224.3]
log 2.3508
To Check:
As c' (224.3)
log 2.3508
: 6 - a (189)
colog 7.7235
:: 8inJ(^'+^) (8^54')
colog 0.0043
: 8inJ(^' -A) (56^29')
log 9.9210
Check! 9.9996
The following arrangement of the work satisfies the require-
ments of properly arranged computation and is much to be pre-
ferred to the arrangement given above.
Computation of Triangle
a, 6, and A given
Data and
results
To find B
To find c Check
• » ^
sin .
A asinC c sini(B + -4)
sm B —
a
"~ sin A 6 — a"~8in J(B — il)
a
= 345 colog
7.4622 log 2.5378
b
= 534
log
2.7275
A
= 25° 25' log
sin
9.6326 colog sin 0.3674
B
= [4r 37']
9 . 8223
c
= (112° 58')
log sin 9.9641
c
= [740.1]
log 2.8693 log 2.8693
(B - A)/2
= (80° 6')
log sin 9.1489
(B + A)/2
= (33° 31')
colog sin 0.2579
b — a
= (189)
colog 7.7235
Check! 9.9996
B'
= (138° 23')
C
= (16° 12')
log sin 9.4456
c'
= [224.3]
log 2.3508 log 2.3508
(B' - A)/2
= (56° 29')
log sin 9.9210
(B' + A)/2
= (81° 54')
colog sin 0.0043
b -a
= (189)
colog 7.7235
Check! 9.9996
§182] TRIGONOMETRIC EQUATIONS 313
Examples
Compute the unknown parts in each of the following triangles:
1. o = 0.8, b = 0.7, B = 40° 15'.
2. a = 17.81, h = 11.87, A = 19° 9'.
3. 6=81.05, c = 98.75, C = 99° 19'.
4. c = 50.37, a = 58.11, C = 78° 13'.
6. o = 1213, b = 1156, B = 94° 15'.
182. Case III. Given two sides and the included angle, as
a, 6, C.
1. To find A + 5, use A + B = 180° - C.
2. To find A and B, compute (A — B) /2 by the law of tangents,
§173, equation (4), then A = {A + B) /2 + {A - B) /2 and
B = (A +B)/2 - {A - B)/2,
3. To find c, use law of sines, §171.
4. To check, use the check formula (2) §174.
Example: Given a = 1033, b = 635, C = 38° 36'
A + B = 180° - 38° 36' = 141° 24'
To find A and B:
Aaa + b (1668) colog 6.7778
: o - 6 (398) log 2 . 5999
: : tan J(A + B)/ (70° 42') log tan 0.4557
: tan i{A —
B)/
[34° 16']
A = 104°
B = 36°
58'
26'
log tan
9 . 8334
To find c:
As sin A
(104° 58')
colog
0.0150
: sin C
(38° 36')
log
9.7951
: : a
(1033)
log
3.0141
: c
[667.1]
log
2 . 8242
Check:
As sm i(A - B)
(34° 16')
colog
0.2495
: sin i(A + B)
(70° 42')
log
9 . 9749
:: a — 6
(398)
log
2 . 5999
: c [667.2] log 2.8243
Check!
An experienced computer would arrange the above work as
follows:
314 ELEMENTARY MATHEMATICAL ANALYSIS [§183
Computation of Triangle
a, h, C given
Data and results To find A - 5 To find C Check
t&n\iA-B) ^ o-b a Bine _ c_ sinjU-i-B)
taniU + -B) " a+6 *' " sin i a - b "siniU-B)
a = 1033 log 3.0141
6 = 635
a - 6 = (398) log 2.6999 log 2.5999
a + 6 = (1668) colog 6.7778
C = 38° 35' log sin 9.7951
(A + B)/2 = (70° 42') log tan 0.4557 log sin 9.9749
(A - B)/2 = [34° 16'] log tan 9.8334 colog sin 0.2495
A = (104° 58') colog sin 0. 0150
B = (36° 26')
c = [667.1] log 2.8242 log 2.8243
Check!
Examples
Compute the unknown parts in each of the following triangles.
1. a = 78.9, 6 = 68.7, C = 78° 10'.
2. c = 70.16, a =39.14, B = 16° 16'.
3. 6 = 1781, c = 982.7, A = 123° 16'.
4. a = X 6 = t/2, C = ir/3.
183. Case IV. Given the three sides.
1. To find the angles, use the s-formulas, §176, (11), (12)
and (13).
2. To check, use A +B + C = 180^
Example: Given a = 455, b = 566, c = 677, find A, B
and C.
The following work is self explanatory. The work is arranged
in final compact form, which, in this case, is as simple as any
other possible arrangement.
§183] TRIGONOMETRIC EQUATIONS 315
Computation of Triangle
a, bf c given
Data and results To find A, B, C
r^ = (s — a)(s — 6)(s — c)/s
tan A/2 = r/(s — a). . .
o = 455
6 =566
c = 677
2s = 169'8
i s = 849
colog
7.0711
s - a = 394
log
2 . 5955
s - 6 = 283
log
2.4518
s - c = 172
log
2 . 2355
r«
log
4 . 3539
r
log
2.1770
A/2 = [20° 53']
log tan
9.5815
B/2 = [27° 68']
log tan
9 . 7252
C/2 = [4r 91
log tan
9.9415
A = 41° 46'
B = 55° 56'
C - 82° 18'
—
Check! 180° 0'
Exercises
Find the values of the angles in each of the following triangles:
1. a = 173, b = 98.6, c = 230.
2. a = 8.067, b = 1.765, c = 6.490.
3. a = 1911, b = 1776, c = 1492.
Miscellaneous Problems
The instructor will select only a limited number of the following
problems for actual computation by the student. The student should
be required, however, to outline in writing the solution of a number
of problems which he is not required actually to compute, and, when
practicable, to block out a suitable check for each one of them.
1. From one corner P of a triangular field PQR the side PQ bears
N. 10° E. 100 rods. QR bears N. 63° E. and PR bears N. 38° 10' E.
Pind the perimeter and area of the field.
316 ELEMENTARY MATHEMATICAL ANALYSIS [§183
2. The town B lies 15 miles east of A, C lies 10 miles south of A.
X lies on the line BC, and the bearing of AX is S. 46** 20' E. Find
the distances from X to the other three towns.
3. To find the length of a lake (Fig. 123), the angle C = 48° lO^,
the side a = 4382 feet, and the angle B = 62° 20' were measured.
Find the length of the lake c, and check.
4. To continue a line past an obstacle
L, Fig. 124, the line BC and the angles
marked at B and C were measured and
found to be 1842 feet, 28° 15', and 67"
24', respectively. Find the distance CD,
and the angle at D necessary to continue
the line AB] also compute the distance
BD.
6. Find the longer diagonal of a par-
allelogram, two sides being 69.1 and 97.4
and the acute angle being 29° 34'.
What is the magnitude of the single
force equivalent to two forces of 69.1 and
97.4 dynes respectively, making an angle of 29° 34' with each other?
6. A force of 75.2 dynes acts at an angle of 35° with a force F.
Their resultant is 125 dynes. What is the magnitude of F?
7. The equation of a circle is p = 10 cos 6. The points A and
B on this circle have vectorial angles 31° and 54° respectively. Find
the distance AB, (1) along the chord; (2) along the arc of the circle
Fig. 123. — Diagram for
Problem 3.
Fig. 124. — Diagram for Problem 4.
8. Find the lengths of the sides of the triangle enclosed by the
straight lines :
e = 26°; e = 115°; p cos (6 - 45°) = 50.
9. A gravel heap has a rectangular base 100 feet long and 30 feet
wide. The sides have a slope of 2 in 5. Find the number of cubic
yards of gravel in the heap.
§183] TRIGON-OMETRIC EQUATIONS 317
10. A point B is invisible and inaccessible from A and it is necessary
to find its distance from A. To do this a straight line is run from A
to P and continued to Q such that B is visible from P and Q. The
following measurements are then taken: AP = 2367 feet; PQ = 2159
feet; APB = 142° 37'.3; AQB = 76° 13'.8. Find AB.
11. To determine the height of a mountain the angle of elevation
of the top was taken at two stations on a level road and in a direct
line with it, the one 5280 yards nearer the mountain than the other.
The angles of elevation were found to be 2° 45' at the further station
and 3° 20' at the nearer station. Find the horizontal distance ol the
mountain top from the nearer station and the height of the
mountain above it. Use S and T functions.
12. Explain how to find the distance between two mountain peaks
Ml and M2, (1) when A and B at which measurements are taken
are in the same vertical plane with Mi and M2', (2) when neither A
nor jB is in the same vertical plane with Mi and ikf 2.
13. The sides of a triangular field are 534 yards, 679 yards and
474 yards. The first bears north, and following the sides in the
order here given the field is always to
the left. Find the bearing of the other
two sides and the area.
14. From a triangular field whose sides
are 124 rods, 96 rods, and 104 rods a
strip containing 10 acres is sold. The
strip is of uniform width, having as one
of its parallel sides the longest side of
the field. Find the width of the strip. _ ^^^ t^.
•t r rni • 1 J. n J. riQ. 125. — Diagram for
16. Three circles are externally mutu- Problem 16
ally tangent. Their radii are 5, 6, 7 feet.
Find the area and perimeter of the three-cornered area enclosed by
the circles and the length of a wire that will enclose the group of
three circles when stretched about them.
16. To find the distance between two inaccessible objects C and Z),
Fig. 125, two points A and B are selected from which both objects are
visible. The distance AB is found to be 7572 feet. The following
angles were then taken:
ABD
^
122°
37'
ABC
=
70°
12'
BAG
=
80°
20'
BAD
^s
27°
13'
Find the distance DC and check.
318 ELEMENTARY MATHEMATICAL ANALYSIS l§183
17. A circle of radius a has it« center at the point G>i) ^i)- Find
its equation in polar coordinates. (Use law of cosines.)
18. A surveyor desired the distance of an inaccessible object 0
from A and B, but had no instruments to measure angles. He
measured AA' in the line AO, BB' in the line BO; also AS, BA\ AB'.
How did he find OA and OB?
19. Prom a point A a distant object C bears N. 32° 16' W. with
angle of elevation 8° 24'; from B the same object bears N. 50° W.
AB bears N. 10° 38' W. The distance AB is 1000 yards. Find the
distance AC.
Fio. 126.— Diagram for Problem 20.
20. The angle of elevation of a mountain peak is observed to be
19° 30'. The angle of depression of its image reflected in a lake 1250
feet below the observer is found to be 34° 5'. Find the height of the
mountain above the observer and the horizontal distance to it. (See
Fig. 126.)
21. One side of a mountain is a smooth eastern slope inclined at an
angle of 26° 10' to the horizontal. At a station A a vertical shaft is
sunk to a depth of 300 feet. From the foot of the shaft two horizontal
tunnels are dug, one bearing N. 22° 30' E. and the other S. 65° E.
These tunnels emerge at B and at C respectively. Find the lengths
of the tumiels and the lengths of the sides of the triangle ABC.
§183} TRIGONOMETRIC EQUATIONS 319
22. A rectangular field ABCD has side AB = 40 rods; AD = 80
rods. Locate a point P in the diagonal AC so that the perimeter of
the triangle APB will be 160 rods. {Hint: Express perimeter as a
function of angle at P.)
X i -
23. Find the area enclosed by the lines 2/ = o' 2/ ~ V 3 x, and the
circle x* — lOx -f 2/^ = 0. {Hint: Change to polar coordinates.)
24. The displacement of a particle from a fixed point is given by
d — 2.5 cos t -f 2.5 sin t.
What values of t give maximum and minimum displacements; what
is the maximum displacement?
26. A quarter section of land is enclosed by a tence. A farmer
wishes to make use of this fence and 60 rods of additional fencing in
making a triangular field in one corner of the original tract. Find the
field of greatest possible area. Show that it is also the field of maxi-
mum perimeter, under the conditions given.
26. A force Pi = 100 dynes makes an angle of 0° with the horizontal,
and a second force Pa = 50 dynes makes an angle of 90° with Pi.
Determine 6 so that (1) the sum of the horizontal components of Pi
and Pa shall be a maximum; (2) so that the sum of the vertical com-
ponents shall be zero.
27. Find the area of the largest triangular field that can be enclosed
by 200 rods ©f fence, if one side is 70 rods in length.
28. Change the equation of the curve xy = 1 to polar coSrdinates,
rotate through — 45° and change back to rectangular coordinates.
29. A particle moves along a straight line so that the distance
varies directly as the sum sin t + cos t. When t = ^/4, the distance
is 10; find the equation of motion.
SO. From the top of a lighthouse 60 feet high the angle of de-
pression of a ship at anchor was observed to be 4° 52', from the
bottom of the lighthouse the angle was 4° 2'. Required the horizon-
tal distance from the lighthouse to the ship and the height of the
base of the lighthouse above the sea.
31. A vertical square shaft measuring 3 feet 6 inches on a side
meets a horizontal rectangular tunnel 6 feet 6 inches high by 3 feet
6 inches wide. Find an expression for the length of a line AB
shown in Fig. 127 when the angle 6 is 37°.
32. University Hall casts a shadow 324 feet long on the hillside
on which it stands. The slope of the hillside is 15 feet in 100 feet,
and the elevation of the sun is 23° 27'. Find the height of the
building.
320 ELEMENTARY MATHEMATICAL ANALYSIS [1183 |
33. To determine the distance of a fort A from a place B, a line BC
and the angles ABC and BCA were measured and found to be 3225,5
yards, 60° 34', and 56° 10' respectively. Find the distance AB.
34. A balloon is directly over a straight level road, and between
two points on the road from which it is observed. The points sre
15,847 feet apart, and the ai^lea of elevation are 49° 12' and 53° £9'.
Find the height.
"^ m"
*T>ii.
B
'X
k
\ .
'
runnel
A
FiQ. 127.— Diaitram for Problem 31.
36. Two trees are on opposite sides of a pond. Denoting the trees
by A and B, we measure AC = 297.6 feet, BC = 864.4 feet, and the
Migle ABC = 87° 43'. Find AB.
36. Two mountains are 9 and 13 milea respectively from a town,
and they include at the town an angle of 71° 36'. Find the distanc«
between the mountains.
37. The sides of a triangular field are, in clockwise order, 531
feet, 679 feet, and 474 feet; the first bears north; find the beaiinga
of the other sides and the area.
38. Under what visual angle is an object 7 feet long seen when
the eye is 15 feet from one end and 18 feet from the other?
SS. The shadow of a cloud at noon is cast on a spot 1600 leet
west of an observer, and the cloud bears S., 76° W., elevation 23*.
Find its height.
CHAPTER X
WAVES
181. Simple Haimonic Motion. Let P be any point on a circle,
and let D be the projection of P on any straight line in the plane
of the circle. Then if the point P move uniformly (that is, so that
equal distances are described in equal times) on the circle, the
^
k
^
ir"
JrtK.
^:
'G,
I'M
if
^'-^-
■1
i
\'-w,
i||
1
— ,'-----±
::::-!:::::ir_
^--
£
c Molion. and of a,
back-and-forth motion of the point D on the given strait hne
ia c^ed simple baimonic motion. On account of the frequency
with which this term will occur, we shall abbreviate it by the
symbols S.H.M. Fig. 128 illustrates a way in which this
motion may be described by mechanical means. Let the uni-
formly rotating wheel OAB be provided with a pin M attached
to its circumfer^ice, and free to move in the slot of the cross-
head as shown, the arm of the cross-head being restricted to
21 321
322 ELEMENTARY MATHEMATICAL ANALYSIS [§184
vertical motion by suitable guides GiGi. Then, as the wheel
rotates, any point P of the arm of the cross-head describes simple
harmonic motion in a vertical direction. The amplitude of
the S.H.M. is the radius of the circle, or 0B\ its period is the
time required for one complete revolution of the wheel.
In elementary physics it is explained that the motion of a simple
pendulum is nearly simply harmonic. Also that the motion of
a point of a vibrating violin string, or of a point of a tuning fork, is
S.H.M. S.H.M. is a fundamental mode of motion of the particles
of all elastic substances, and is therefore of great importance.
The motion of the point Dt can readily be expressed by an equa-
tion, if the value of the angle B be expressed in terms of the elapsed
time t Since the rotation is uniform, B = kt^ where k is the angle
described in one second, or the angular velocity of P. Let the
radius 03/ of the circle be a.feet. If 0 be taken as origin, and if
the angle AOM be called B, then if the point M was at A when
< = 0, the displacement ODi = y is given at any time t by
y = a sin ^ = a sin A:< (1)
In a similar way, the point Di, the projection of M on OA, de-
scribes a S.ILM., and the displacement ODi = x may be written
X = a cos ^ = a cos A^^ (2)
If the point 3/ was at E when i — 0, the displacements ODt - y
and ODi = ar are given by
y = a sin (kt - c) (3)
X = a cos (kt — c) (4)
when € stands for the angle £"0*4 ; for H = angle EOM and B ^ \
EOM — EOA, In this equation A^ — c is called the phase an^^
and € is called the epoch angle of the S.H.M.
These expressions may also be written in terms of the linear
velocity 1' of 3/ instead of the angular velocity k of OAf. Let
the uniform velocity of 3/ be r feet per second. Since the radius
OM is a feet, OM rotates at the rate of r/a = k radians per second.
This value of k may be substituted in equations (1) to (4).
It is ob^'ious that .
y = a cos kt
represents a S.H.M. ^ in advance of y = o sin i<, since
186] WAVES 323
IT
in (A;* + o) ^ ^^s kt. A pair of S.H.M.'s possessing this prop-
rty are said to be in quadrature. (1) and (2), or (3) and (4)
lay be said to be in quadrature.
The period of the S.H.M. y ^ asinkt is the time T required
>r a complete revolution. If h be the time at which M is at any
iven position, and if t2 be the time at which M is next at the
une position, then, since the angular velocity multiplied by
lie elapsed time gives the angular displacement, we have,
k{t2 - ti) = 27r
'herefore, since the difference t2 — h = T is the period :
'he number of complete periods per unit time is:
^^ T 27r
I is called the frequency of the S.H.M.
It is obvious that all points of the moving cross-head, Fig.
28, describe S.H.M., and that (1) may be regarded as the
quation of motion of any point of the cross-head if a suitable
rigin be selected. Thus (1) is the equation of motion of P
of erred to the origin Oi, where Oi is the middle point of the up-
nd-down range of motion of P.
185. If P, Fig. 128, be a tracing point attached to the vertical
urm of the cross-head and capable of describing a curve on a uni-
formly translated piece of smoked glass, HK, then when P de-
scribes S.H.M. in the vertical Une OP, the curve NiCTN^P traced
on the plate HK is a sinusoid, for the ordinates on HK measured
^th respect to the median line OiNi are proportional to sin d and
by h3rpothe3is the abscissas or horizontal distances vary um'f ormly.
If the plate HK move with exactly the same speed as the point M,
the undistorted sinusoid of Fig. 59 is described, whose equation is
y = a sin — i = a sin — (1)^
1/ X
^ The student should note that - <» sin - is of exactly the same shape as y <» sin x,
for multiplying both ordinates and abscissas of any ourye by a is merely constructing
1/ X
Ae eurve to a different scale. However, » — sin „ is a distorted sinusoid, for the
Mrdinates of y « sin « are multiplied by 3 while the abscissas are muLtip\\«d oxA-^ >a^ ^«
324 ELEMENTARY MATHEMATICAL ANALYSIS [|1»
where 3. is the abBcissa of any poiDt of the Binusoid referred to M
origin (aa N) moving with the plate. If, however, the vehxu^ rf
the plate be v' instead of v, then the equation of the curve on HK,
referred to axes moving with the plate, is of the form
a sin — (" a sin -
„??.
a sin Ax
(2)
= p/a. Changing the reU-
tive speed of the wheel and
plate corresponds to stretch-
ii^ or contracting the Bine
curve in the x direction.
186. Composition of Tm
S.H.M.'8 ftt Right An^
We have shown if a point Jf.
moving uniformly on a circle,
be projected upon both the
Z- and y-axes, two S.H.M.'i,
result. The pbaae at^ee d
tiiese two motions differ from
each other by | or 90°, The
converse a! this fact, namel;
that uniform nwtion in a ei^
cte may be the resultant of
two S.H.M. in quadrstui^
is easily proved, for that**
equations of S.H.M.:
X — a coakt
y ~ asinkt •
are ob\-ious!y the parametne
equations of a circle. Hence
the Ihoviivm:
I'niform fh^ii'n in a cinle mat/ be rtgarded aa the regvlUaii of
lm> S.H.M.'s of t\'ifiil a-ipJituJts and equal perioda and differinf
by X '2 in ftii.'V inijjfc-
Thi^ tiuixvi.-uti iruth is illustrated by Fig. 129. Let thev-
and y-!KW K> divided pn^irtionallj- to the
I'lO.li-a.— Tlwlir,!..,
§186] WAVES 325
as in Fig. 69. Through the points of division of the two axes
draw lines perpendicular to the axes, thus dividing the plane into
a large number of small rectangles. Starting at the end of one
of the axes, and sketching the diagonals of successive cornering
rectangles, the circle ABA'B' is drawn.
If the same construction be carried out for the case in which
tiie 2/-axis is divided proportionally to h sin kt and in which the
X-axis is divided proportionally to a sin hty the ellipse A\BiA\B\
results. These facts are merely a repetition of the statements
made in §74.
Exercises
1. Draw a curve by starting at the intersection of any two lines of
Fig. 129, and drawing the diagonals of successive cornering rectangles,
and write the parametric equations of the curve.
2. Find the periods of the following S.H.M. :
(a) 2/ = 3 sin 2t.
(h)y = 10 sin (1/2)^.
(c) y — 7 cos 4^
(d) y — a sin 2irt.
(e) y == a sin (lOt - ir/3).
(/) 2/ = a sin (2//3 - 2^/5).
(g) y ^ a sin (bt + c).
3. Give the amplitudes and epoch angles in each of the instances
given in example 2.
4. The bob of a second's pendulum swings a maximum of 4 cm.
each side of its lowest position. Considering the motion as rectilinear
S.H.M. write its equation of motion. ^
Write the equation of motion of a similar pendulum which was
released from the end of its swing 1/2 second after the first pendulum
was similarly released.
6. A particle moves in a straight line in such a way that its dis-
placement from a fixed point of the line is given by d = 2 cos* t.
Show that the particle moves in S.H.M. , and find the amplitude
and period of the motion.
6. A particle moves in a vertical circle of radius 2 units with
angular velocity of 20 radians per second. Counting time from the
1 The term period is used differently in the case of a pendulum than in the
ease of S.H.M. The time of a atnng is the period of a pendulum; the time of
a nnng'Swang is the period of a S.H.M.
326 ELEMENTARY MATHEMATICAL ANALYSIS [§187
instant when the particle is at its lowest positioni write the equa-
tion of motion of its projection (1) upon the vertical diameter;
(2) upon the horizontal diameter; (3) upon the diameter bisecting
the angle between the horizontal and vertical.
187. Waves. The curve described on the moving plate HK
of Fig. 128, if referred to coordinate axes moving with the plaU,
is the sinusoid or sine curve, which for the sake of greater generality
we shall suppose is of the type (§185, equation (2))
y = aainhx (1)
If, however, we consider this curve as referred to the fixed origin
Oi, then the moving sinusoid thus conceived is called a simile
progressive sinusoidal wave or merely a wave. Under the con-
ditions represented in Fig. 128, it is a wave progressing to the
right with the uniform speed of the plate HK, At any singk
instant f the equation of the curve is:
y = asiahix- OiN) (2)
where OiN is the distance that the node N has been translated
to the right of the origin Oi. If V be the uniform velocity of
translation of HK, then:
OiN = Vt &'
and the equation of the wave is:
y = asin h{x — Vt)
or,
y = a sin (hx — kt) (4)
if k be put for hV, so that
Because of the presence of the variable t, this is not the equation
of a. fixed sinusoid, but of a moving sinusoid or wave.
Applying the same terms used for S.H.M., the expression
(hx — kt) is the phase angle, the expression { + kt) ia the epoch
angle and a is the amplitude of the wave. See Fig. 130a and c.
The expression (Jix — kt) is a. linear function of the variables
^ In what follows, t is not the time elapsed sinoe Mt Fig. 128, was at A, as used in
§184, but is the elapsed time since N was at Oi. These values of t differ by the tiffl*
of half a revolution or by r/k.
urn
WAVES
327
X and t. The sine or coaine of thia functioa is called a simple
harmonic function of x and t.
188. Wave Length. Since the period of the sine is 2ir, if t
remain constant and the expression hx be changed by the amount
Fio. 130.— Wave Forms, (a) ot Di£ferent Amplitude; (5) of Different
Wave LenEths; (e) of Different Phase or Epocb Anglee.
2ir, the curve (4) is translated to the left or right an amount such
that trough coincides with trough and crest coincides with crest,
and the curve in its second position coincides with the curve in
328 ELEMENTARY MATHEMATICAL ANALYSIS [§189
its first position. Call X2 the abscissa of any point of the curve
in its second position whose original abscissa was Xi. Then:
hx2 — hxi = 2ir
or:
X2 — a;i = 2t /h
Calling the distance 2:2 — a;i = L, we have:
L = 27r //i (1)
L is called the wave length. It is the distance from any crest
to the next crest or from any trough to the next trough or from
any node to the second succeeding node, or from any point of
the wave to the next similar point. See Fig. 1306.
The wave length can also be determined in the following manner:
The wave length of
y = ain X (2)
is obviously 27r, the length of the period of the sine. The sine
curve
y = sinhx (3)
can be made from the above by multiplying the abscissas of all
1 27r
points by r. Therefore the wave length of the latter is -j-. The
wave length of
y = sin (hx — kt) (4)
must also be the same as that of (3), since the effect of the term
kt is merely to translate the curve as a whole a certain distance
to the right.
189. Period or Periodic Time. If we fix our attention upon
any constant value of x, and if kt in (4) above be permitted to
change by the amount 27r, then since the period of the sine is
27r, the curves at the two instances of time mentioned must coin-
cide. Calling the two values of t, ti and <2, we have by hypothesis
kt2 — kti = 27r
Writing:
«2 - «i = T
we find
T = 27r/k (1)
The expression T is called the periodic time or period of the
wave. It is the length of time required for the wave to move one
§190] WAVES 329
wave length, or the length of time that elapses until trough again
coincides with trough, etc. To contrast wave length and period,
think of a person in a boat anchored at a fixed point in a lake. The
time that the person must wait at that fixed point (x constant)
for crest to follow crest is the periodic time. The wave length
is the distance he observes between crests at a given instant of
time (< constant).
190. Velocity or Rate of Propagation. The rate of movement
V of the sinusoid on the plate HKj Fig. 128, is shown by equa-
tion (5), §187, to be k/h units of length per second. This is
called the velocity of the wave or the velocity of propagation.
The equation of the wave may be written:
y = asm h{x — Vt)
From equations (1) §188 and (1) §189 we may write
whence
-I
h _ L
h ~ T
k
Since 7 = t> we have:
h
V = ^ (1)
This equation is obvious from general considerations, for the
wave moves forward a wave length L in time T, hence the speed
of the wave must be ^•
191. Frequency. The number of periods per unit of time is
called the frequency of the wave. Hence, if N represent the
frequency of the wave,
N = ^ = £ (2)
There is no name given to the reciprocal of the wave length.
192. L and T Equation of a Wave. If we solve equations (1)
330 ELEMENTARY MATHEMATICAL ANALYSIS [§193
§188 and (1) §189 for h and k respectively, and substitute
these values of h and k in the equation
y = a sin (hx — kt)
we obtain
[1-^]
y = asin27r j^ - ^ (1)
From this form it is seen that the argument of the sine increases
by 27r when either x increases by an amount L or when t increases
by the amount T, By use of (1), §190, the last equation may
also be written:
y =:asin^(x-Vt) (2)
193.' Phase, Epoch, Lead. Consider the two waves:
y = a sin J- {x — Vt) (-1)
2/ = a sin ~^(x - F« - E) (2)
The amplitudes, the wave lengths and the velocities are the
same in each, but the second wave is in advance of the first by
the amount E (measured in linear units), for the second equation
can be obtained from the first by substituting (x — E) forx, which
translates the curve the amount E in the OX direction. In this case
E is called the lead (or the lag if negative) of the second wave
compared with the first.
The lead is a linear magnitude measured in centimeters, inches,
feet, etc. The epoch angle is measured in radians. In the
present case the epoch angle of (2) is 2ir{yt + J^) /L.
The terms phase and epoch are sometimes used to designate
the timBi or, more accurately, the fractional amount of the period
required to describe the phase angle and epoch angle respectively.
In this use, the pha^e is the fractional part of the period thai has
elapsed since the moving point last passed through the middle poif^
of its simple harmonic motion in the direction reckoned da positive.
See Fig. 130c.
The tidal wave in mid ocean, the ripples on a water surface,
the wave sent along a rope that is rapidly shaken by the hand,
are illustrations of progressive waves of the type discussed above.
§193] WAVES 331
Sound waves also belong to this class if the alternate condensations
and rarefactions of the medium be graphically represented by
ordinates. The ordinary progressive waves observed upon a lake
or the sea are not, however, progressive waves of this type. The
surface of the water in this case is not sinusoidal in form, but
is represented by another class of curves known in mathematics
as trochoids.
Exercises
1. Derive the amplitude, the wave length, the periodic time, the
velocity of propagation of the following waves:
2/ = a sin (2x — 3/).
2/ = 5 sin (0.75a; - lOOOO-
y = lOsin (I - I)
2ir
y = 50 sin yCx — 30-
y = 100 sin ||(x - 2Qt - 4).
2/ = 100 sin (5a; + 4^).
y = 0.025 sin ^(a; + ^3).
2. Write the equation of a progressive sinusoidal wave whose height
is 5 feet, length 40 feet and velocity 4 miles per hour.
3. Write the equation of a wave of wave length 10 meters, height
1 meter and velocity of propagation 3.5 miles per hour. (Note:
1 mile = 1.609 kilometers.)
4. Sound waves of all wave lengths travel in still air at 70° F. with
a velocity of 1130 feet per second. Find the wave length of sound
waves of frequencies 256, 128, 600 per second.
6. The lowest note recognizable as a musical tone was found by
Helmholtz to possess about 40 vibrations per second. The highest
note distinguishable by an ordinary ear possesses about 20,000 vibra-
tions per second. If the velocity of sound in air be 1130 feet per
second, find the wave length in each of these limiting cases, and write
the equation of the waves if the amplitude be represented by the
symbol a.
332 ELEMENTARY MATHEMATICAL ANALYSIS [§194
194. Stationary Waves. The form of a violin string during its
free vibration is sinusoidal, but the nodes, crests, troughs, etc.,
are stationary and not progressive as in the case of the waves
just discussed, and is therefore called a stationary wave. The
water in a basin or even in a large pond or lake is also capable of
vibrating in this way. Fig. 131 may be used to illustrate the
stationary waves of this type, either of a musical string or of the
water surface of a lake, but in the case of a vibrating string, the
ends must be supposed to be fastened at the points 0 and N.
The shores of the lake may be taken at / and iiC or at J and H,
etc. As is well known, such bodies are capiable of vibrating in
segments so that the number of nodes may be large. This
Fig. 131. — A Stationary Wave.
explains the "harmonics" of a vibrating violin string and the
various modes in which stationary waves may exist on a water
surface. A stationary wave on the surface of a lake or pond is
known as a seiche, and was first noted and studied on Lake
Geneva, Switzerland. The amplitudes of seiches are usually
small, and must be studied by means of recording instruments
so set up that the influence of progressive waves is eliminated.
The maximum seiche recorded on Lake Geneva was about 6 feet,
although the ordinary amplitude is only a few centimeters.
The equation of a stationary wave may be found by adding the
ordinates of a progressive wave:
y = a sin {hx — kt)
(1)
traveling to the right (A; > 0), to the ordinates of a progressive
wave:
y = asm {hx + kt) (2)
traveling to the left.
§194] WAVES 333
Expanding the right members of (1) and (2) by the addition
formula for the sine, and adding:
y = 2a cos kt sin hz (3)
or in terms of L and T
y = 2a cos {-7^) sin /-j^ j (4)
In Fig. 131, the origin is at 0 and the X-axis is the line of nodes
ONX. If we look upon 2a cos kt as the variable amplitude of
the sinusoid
y = sin hx
we note that the nodes, etc., of the sinusoid remain stationary,
but that the amplitude 2a cos kt changes as time goes on. When
t = 0, the sine curve has amplitude 2a and wave length 2T/h.
When t = T/2k or T/4 the sinusoid is reduced to the straight line
y = 0. When t = w /k or T /2 the curve is the sinusoid: •
y = — 2a sin hx
which has a trough where the initial form had a crest, and vice
versa.
Exercises
In the following exercises the height of the wave means the maxi-
mum rise above the line of nodes. When a seiche is uninodal, the
shores of the lake correspond to the points / and X, Fig. 131. When
a seiche is binodal, the points / and H are at the lake shore.
1. From the equation of a stationary wave in the form y —
2a sin 2tx/L cos 2irt/T, show that X, Fig. 131, is at its lowest depth
for t = T/2, ST/2, 5T/2, ...
2. Henry observed a fifteen-hour uninodal seiche in Lake Erie, which
was 396 kilometers in length. Write the equation of the principal
or uninodal stationary wave if the amplitude of the seiche was 15 cm.
3. A small pond 111 meters in length was observed by Endros to
have a uninodal seiche of period fourteen seconds. Write the equa-
tion of the stationary wave if the amplitude be a.
4. Forel reports that the uninodal longitudinal seiche of Lake Geneva
has a period of seventy-three minutes and that the binodal seiche has
a period of thirty-five and one-half minutes. The transverse seiche
has a period of ten minutes for the uninodal and five minutes for
the binodal. The longitudinal and transverse axes of the lake are
334 ELEMENTARY MATHEMATICAL ANALYSIS [(lU
45 mileB and 5 miles reapeotively. Write the equation of tbew
different seiches.
fi. A standing wave or uninodal seiche ensta on. Lake Mendots
of period twenty-two minutes. If the maximum height is 8 inoliH
and the distance across the lake is 6 miles, write the equation of
the seiche.
196. Compound H&imonic Motion and Compound Waves.
The addition of two or more simple harmonic functions of dif-
ferent periods gives rise to compound haimonic motion. Thus:
1/ = a sin fc( + 6 sin 3 ft(
corresponds to the superposition of a S.H.M. of pmod 2]r/3t
and amplitude b upon a fundamental S.H.M. of period 2r/k
and amplitude a. To compound motions of this type, there cor-
respond compound waves of various sorts, such as a fundamental
sound wave with overtones, or tidal waves in restricted bays or
harbors. The graphs of the curves:
3j: and the Compound Curve
should ba constructed by the student. They may be drawn by
adding the ordinates of the various sinusoids constructed on the
same axis, as in F^. 132. To compound the curves, first draw
the component curves, say y = sin x and y = sin 3x of F^. 132.
Then use the edge of a piece of paper divided proportionally
to sin X (that is, like the scale OB, Fig. 132) and use this as t
scale by means of which the successive ordinates of a given x
WAVES
335
>e added. For example, to locate the point on the composite
correspondii^ to the abscissa OD, Fig. 132, we must add
ad DQ. Hence place vertically at P the lower ond of the
scale just mentioned. The sixth scale division above P
s scale will then locate the required point M of the oompoaite
Fig. 133 the curves:
1/ = sin a; + sin (2^ + 2s-n/16)
j; = sin ac + sin (3x + 2Tn/16)
Lown for values of n = 0, 1, 2, . . ., 15 in h
), for successive phase differences corresponding to o
I of the wave length of the fundamental y = sin x.
ve forms compounded from the odd harmonics only are
ally important, as alternating-current curves are of this type.
Ig. 134.
336
ELEMENTARY MATHEMATICAL ANALYSIS [JIB?
196. Harmonic AnaljsiB. Fourier showed in 1822 in iiis
"Analytical Theory of Heat" that a periodic sin^e-valued
function, say y = /(x), under certain conditions of continuitf,
can be represented by the sum of a series of sines and cosines of
the multiple angles of the form:
!/ = tio + «i cos I + tti cos 2i + at cos 3a; + . . .
+ 6, sin I + 6, sin 2a; + 6) sin 3a: + . . .
This means, for example, that it ia always possible to represent
the complex tidal wave in a harbor, by means of the sum of &
' y'' V
. t A
-, i^
t ^^
t ,'^
t
- t 4
^- /
number of simple waves or harmonics. The term hannonic
analysis is given to the process of determining these sinusoidal
components of a compound periodic curve. In §196 we
have performed the direct operation of finding the compound
curve when the component harmonica are given. The inverse
operation of finding the components when the compound curve
is given is much more difficult, and its discussion must be post-
poned to a later course.
197.* Test for a Sinusoidal Function. Squared paper, knowo
as semi-sinusoidal paper, has been prepared (see Fig. 135)
with the horizontal scale divided proportionally to sin z and the
vertical scale divided uniformly. The divisions are precisely
the same as those in Fig. 59, except that the number of divisioiu
971
WAVES
337
greatly increased. On thk paper, the sine curve is represented
a straight line drawn diagonaUy across the papa: Since the
le curve appears aa a straight hne on this paper (just as the
;arithiiiic curve appears as a straight line on semi-logarithmic
per) it is eaay to teat whether or not observed periodic data
low the law expressed by the sine curve. Thus the times of
arise at Boston, Massachusetts, for the first day of each month
ve been plotted upon the sheet shown in Fig. 136. The points
J;^ ::
* ^^^;~^^^ -c
, -^— ^
rresponding to the various dates do not form a straight line,
:hough it is obvious that the sine curve is a first approximation
the proper curve.
The times of sunrise plotted in Fig. 135 are given in exercise
) below.
Ezerdses
L The times of sunset and sunrise at Boston, Massachusetts, for
i first day of each month are as fallows:
JFMAMJJASOND
nset 4:37 5:13 5:49 6:25 6:59 7:29 7:40 7:21 6:37 5:44 4:55 4:28
nrise 7:30 7:16 6:38 5:44 4:56 4:26 4:25 4:515:23 5:56 6:317:09
ngth of day
Graph the times of sunset upon semi-sinusoidal paper.
Note: The earliest sunset tabulated is December 1, which should
used for the date of the trough of the wave.
338 ELEMENTARY MATHEMATICAL ANALYSIS ((198
S. Determine the length of the day trgm the data of enerciBe 1, and
graph the same upon aemi-ainusoidal paper.
3. Determme whethE^r the curves of Fig. 136 ere sinusoidal or not,
198. Connecting Rod Motion. If one end of a stra^ht line B
be required to move on a circle while the other end of the line i
moves on a straight line passing throi^h the center of the aide,
" 1 "
< 1 ' ' 1 ' i 1
—
—
17*;;-^+
—
E
--H
S:
=^-f^l^
: =
=:
%
■^mf'
:E
—
?jk
>\\ wi -
- —
zz
~
i
=^=
I~
=
E
=
=1
=^=
: =
T
ji
=
=E
=p-W
V —
;2
—
-
=
i
V-^
1
Mm.
^
E
ii
=
I
z^
:p[ik
|b
r:
the resulting motion is known as connecting rod motion. The
connecting rod of a steam engine has this motion, as the end at-
tached to the crank travels in a circle while the end attached to
the pifiton travels in a straight line. The motion of the end k
of the connecting rod is approximately S.H.M. The approii-
mation is very close if the connecting rod be very long in compwi-
aon with the diameter of the circle.
A second approximation to the motion of the point A can be
WAVES
339
1 to introduce the second harmonic or octave of the funda-
d. In Fig. 137, let the radius of the circle be a and the
1 of the connecting rod be L The length of the stroke MN
and the origin may conveniently be taken at the mid point
I stroke, 0. When B was at H, A was at M and when B
It K, A was at N. Then MH = NK = ly and OC = I.
X = CA-CO = CA-l = CD + DA -I
CD = a cos d
(1)
(2)
b:
DA = VZ* - BD^
= ^l^-a^sin^e
t^'""t""^
O.EJil.-i-- 3
FiQ. 137. — Connecting Rod Motion.
(3)
X = a cos ^ + Z Vl - (a^/l^) sin* ^ - Z (4)
—I Jif
jximating the radical by §111 (Vl — a; = 1 — «/2) we
i:
- , , /. a* sin2 ^v
X = a cos^ + Mi 272 — ) ■" ^
(5)
iin« ^ = (1 - cos 26)12, hence:
a; = a cos ^ + "77 cos 2^ — -jy
(6)
L is approximately true as long as I is much greater than a.
a seen from the above result that the second approximation
nnecting rod motion contains as overtone the octave or
r2
I harmonic, -jy cos 26, in addition to the first or fundamental
>nio a cos 6.
340 ELEMENTARY MATHEMATICAL ANALYSIS §198
Exercises
1. Draw the curve corresponding to equation (5) above if a = 1.15
inches, and 2 — 3 inches.
2. The motion of a slide valve is given by an equation of the form:
2/ = ai sin (d + e) + at sin (20 + 90°).
Draw the curve if ai = 100, aa = 25, e = 40®.
3. Graph:
2/ = 5 sin (d + 30°) + 2 sin (20 + 90**).
4. Graph:
2/ = sin x + (1/3) sin dx + (1/5) sin 5x.
CHAPTER XI
COMPLEX NUMBERS
199. Scales of Numbers. To measure any magnitude, we
apply a unit of measure and then express the result in terms of
aumbers. Thus, to measure the volume of the liquid in a cask
we may draw off the liquid, a measure full at a time, in a gallon
measure, and conclude, for example, that the number of gallons
is 12J. In this case the number 12^ is taken from the arith-
metical scale of numbers, 0, 1, 2, 3, 4, . . . If we desire to meas-
ure the height of a stake above the ground, we may apply a
foot-rule and say, for example, that the height in inches above the
ground is 12J, or, if the positive sign indicates height above the
ground, we may say that the height in inches is + 12J. In
this latter case the number + 12^ has been selected from the
algebraic scale of numbers . . . — 4, — 3, — 2, — 1, 0, + 1,
+ 2, + 3, + 4, . , .
The scale of numbers which must be used to express the value of a
magnitude depends entirely upon the nature of the magnitude. The
attempt to express certain magnitudes by means of numbers taken
rom the algebraic scale may sometimes lead, as every student of
Jgebra knows, to meaningless absurdities. Thus a problem involving
ihe number of sheep in a pen, or the number of marbles in a box, or
ihe number of gallons in a cask, cannot lead to a negative result, for
ihe magnitudes just named are arithmetical quantities and their meas-
irement leads to a number taken from the arithmetical scale. The
ibsurdity that sometimes appears in results to problems concerning
,hese magnitudes is due to the fact that one attempts to apply the
lotion of algebraic number to a magnitude that does not permit of it.
Science deals with a great many different kinds of magnitudes, the
neasurement of some of which leads to arithmetical numbers while the
neasurement of others leads to algebraic numbers; the remarkable
fact is that two different number scales serve adequately to express
magnitudes of so many different sorts. The magnitudes of science
are so various in kind that one might reasonably expect that the num-
ber of number systems required in the mathematics of these sciences
would be very great.
341
342 ELEMENTARY MATHEMATICAL ANALYSIS [§199
The arithmetical scale, which includes integral and fractional
numbers, is itself more general than is required for the e]q>refi8ion of
some magnitudes. For some magnitudes /rac<ion« are absurd — quite
8s absurd, in fact, as negative values are for other magnitudes. Thus
the number of teeth on a gear wheel cannot be a fraction. The
solution of the following problem illustrates this: ''How many teeth
must be cut on a pinion so that when driven by a spur gear wheel
of fifty-two teeth it will revolve exactly five times as fast as the gear
wheel?"
The arithmetical scale is used when we enumerate the number of
gallons in a cask and say: 0, 1, 2, 3, . . . If we observe 3 gal-
lons in the cask, and then remove one, we note those remaining and
say two; we may remove another gallon and say one; we may remove
the last gallon and say zero; but now the magnitude has come to an
end — no more liquid may be removed.
Another conception of numerical magnitude Is used when we meas-
ure in inches the height of a stake above the ground and say tAree
We may drive the stake down an inch and say two; we may drive the
stake another inch and say one; we may drive the stake another inch
and say zero, or *' level with the ground;" but, unlike the case of the
gallons in the cask, we need not stop but may drive the stake another
inch and say one hdow the ground, or, for brevity, fniniLs one; and so on
indefinitely, buf always prefixing ''minus" or "below the ground"
or some expression that will show the relative position with respect to
the zero of the scale. In this case we have made use of the algebraic
scale of numbers.
Likewise, in estimating timcy there is no zero in the sense of the
gallons in the cask from which to reckon; we cannot conceive of an
evoDt so far past that no other event preceded it; we therefore select a
standard event, and measure the time of other events with reference
to the lapse before or after that; that is, we measure time by means ol
the algebraic scale; the symbols "B.C." or "A.D." could quite as well
be replaced by the symbols "minus" and "plus" of the algebraic scale.
The zero used is an arbitrary one and the magnitude exists in reference
to it in two opposite senses, future and past, or, as is said in algebra,
positive and negative. We are likewise obliged to recognize quantity
as extending in two opposite senses from zero in the attempt to measure
many other things; in locating points along an east and west line, no
point is so far west that there are no other points west of it, hence the
points could not be located on an arithmetical scale; the same in
measuring force, which may be aUraxiive or repulsive; or motion, which
may be toward or from^ or rotation, which may be clockwiae or aatir
§199] COMPLEX NUMBERS 343
{iockwisBf etc. Because of the necessity of nxeasuring such magDi-
tudeS; our notion of algebraic number has arisen.
Many of the magnitudes considered in science are completely ex-
pressed by means of arithmetical numbers only; for example, such
magnitudes as density or specific gravity; temperature;^ electrical re-
sistance; quantity of energy; such as ergs, joules or foot-pounds;
power, such as horse power, kilowatts, etc. All of the magnitudes
just mentioned are scalar, as it is called; that is, they exist in one
sense only — not in one sense and also in the opposite sense, as do
forces, velocities, distances, as explained above. The arithmetical
scale of numbers is therefore ample for their expression.
The distinction, then, between an algebraic number and an arith-
metical number is the notion of sense which must always be associated
with any algebraic number. Thus an algebraic number not only
answers the question *^how many" but also aflSrms the sense in which
that number is to be understood; thus the algebraic number
* -h 12}, if arising in the measurement of angular magnitude, refers to
an angular magnitude of 12} units (degrees, or radians, etc.) taken
in the sense defined as positive rotation.
Exercises
Of the following magnitudes, state which may and which may not
be represented adequately by an arithmetical number:
1. 10 volts. 15. 10° centigrade.
2. 16 calories. 16. 272° absolute temperature.
3. 25 dynes. 17. 16 feet per second (velocity).
4. 2 kilograms. 18. 32.2 feet per second per sec-
6 20 miles per hour. ond (acceleration).
6. 4 acre-feet. 19. 200 gallons per minute.
7. 180 revolutions per second. 20. 20 pounds per square inch.
8. 6-cylinder (engine). 21. 60 horse power.
9. 3 'atmospheres. 22. 1.16 radians per second.
10. 20 light-years. 23. 30° latitude.
11. 27° visual angle. 24. 14° angle of depression.
12. Atomic weight of oxygen. 26. 18 cents per gallon.
13. 28 amperes. 26. 60 beats per minute.
14. 7} pounds per gallon. 27. 6360 feet above sea level.
28. 312 B.C.
1 Temperature is an arithmetical quantity, since there is an absolute zero of
temperature. Temperature does not exist in two opposite eensea, but in a single
344 ELEMENTARY MATHEMATICAL ANALYSIS [§201
200. Algebraic Number Not the Most General Sort Algebraic
numbers, although more general than arithmetical numbers, are
themselves quite restricted. Sir William Hamilton, in order to
emphasize the restricted character of an algebraic number, called
algebra the ** science of pure time,*' That is, algebraic magnitude
exists in the same restricted sense that time exists — ^because if we
fix our attention upon any event, time exists in one sense (future)
and in the exactly opposite sense (past), but in no other sense at all.
Likewise, with the algebraic numbers, each number corresponds to
a point of the algebraic scale (see §1); but for points not on
the scale, or for points side wise to the same, there corresponds no
algebraic number. This is a way of saying that the algebraic
scale is one-dimensional; Sir William Hamilton desired to emphasize
this restriction by speaking of the "science of pure time," for it
is of the very essence of the notion of time that it has one dimension
and ODO dimension only. It is thus seen that there is an opportu-
nity of enlarging our conception of number if we can remove the
restriction of one dimension — that is, if we can get out of the line
of the algebraic scale and set up a number system such that one
number of the system will correspond, for example, to each point of
a plane, and such that one point of the plane will correspond to
each number of the system. We will seek therefore an exten-
sion or generalization of the number system of algebra that will
enable us to consider, along with the points of the algebraic scale,
those points which he without it.
201. Numbers as Operators. The extension of the number
system mentioned in the last section may be facilitated by changing
the conception usually associated with symbols of number. The
usual distinction in algebra is between symbols of number and sym-
bols of operation. Thus a symbol which may be looked upon as
answering the question ''bow many'' is called a number, while a
symbol which tells us to do something is called a symbol of opera-
tion, or, simply, an operator. Thus in the expression \/2, V is
a symbol of operation and 2 is a number. A symbol of operation
may always be read as a verb in the imperative mood; thus we
may read \/x: "Take the square root of a;.'* Likewise log x,
and cos 6 may be read : "Find the logarithm of x," " Take the cosine
of Of*' etc. In these expressions "log" and "cos" are symbols of
5201] COMPLEX NUMBERS 345
>peration; they tell us to do something; they do not answer the
question "how many" or "how much" and hence are not numbers.
Eere we speak of \/, log, cos, as operators; we speak of x as the
>peraiid, or that which is operated upon.
It is interesting to note that any number may be regarded as a
symbol of operation; by doing so we very greatly enlarge some
original conceptions. Thus, 10 may be regarded not only as ten,
Euiswering the question "how many," but it may quite as well be
regarded as denoting the operation of taking unity, or any other
operand that follows it, ten times; to express this we may write
10-1, in which 10 may be called a tensor (that is, "stretcher"),
or a symbol of the operation of stretching a unit until the result
obtained is tenfold the size of the unit itself. In the same way
the symbol 2 may be looked upon as denoting the operation of
dovMing unity, or the operand that follows it; likewise the tensor
3 may be looked upon as a trebler, 4 as a quadrupkrf etc.
With the usual understanding that any symbol of operation
operates upon that which follows it, we may write compound
operators like 2*2*3*1. Here 3 denotes a trebler and 3*1 denotes
that the unit is to be trebled, 2 denotes that this result is to be
doubled and the next 2 denotes that this result is to be doubled.
Thus representing the unit by a line running to the right, we have
the following representation of the operators:
The unit ->
3-1 ->->-^
231 _ ) >
2-2-31 ) >
Notice the significance that should now be assigned to an expo-
nent attached to these (or other) symbols of operation. The
exponent means to repeat the operation designated by the operator;
that is, the operation designated by the base is to be performed,
and performed again on this result, and so on, the number of opera-
tions being denoted by the exponent. Thus 10^ means to perform
the operation of repeating unity ten times (indicated by 10) and
then to perform the operation of repeating the result ten times,
that is, it means 10 (10-1). Also, 10^ means 10[10(101)]. Like-
wise log' 30 means log(log 30) which, if the base be 10, ec\u^a.U
346 ELEMENTARY MATHEMATICAL ANALYSIS [§202
log 1.4771, or finally 0.1694. An apparent exception occurs in
the case of the trigonometric functions. The expression cos* %
should mean, in this notation, cos (cos «), but because trigonometry
is historically so much older than the ideas here expressed, the
expression cos* x came to be used as an abbreviation for (cos x)',
or (cos a;) X (cos a;).
To be consistent with the notation of elementary mathematics,
the expression \/4, looked upon as a symbol of operation, must
denote an operation which must be performed tvdce in order
to be equivalent to the operation of quadrupling; that is,
such that {\/iy = 4. Likewise Vi denotes an operation which
must be performed three times in succession in order to be
equivalent to quadrupling. But we know that the operation
denoted by 2, if performed twice, is equivalent to quadrupling;
therefore \/4 = 2, etc. Just as 4*, 4^, etc., may be called stronger
tensors than a single 4, so \/4i ^i niay be called weaker tensors
than the operator 4.
202. Reversor. The expression ( — 1), looked upon as a
symbol of operation, is not a tensor, as it leaves the size unchanged
of that upon which it operates. If this operator be applied to
any magnitude, it will change the sense in which the magnitude
is then taken to exactly the opposite sense. Thus, if 6 stands
for six hours after y then ( — 1)(6) stands for six hours before
a certain event, and ( — 1) is the symbol of this operation of
reversing the sense of the magnitude. Also if (6) stands for a
line running six units to the righJt of a certain point, then
( — 1)(6) stands for a line running six units to the left of that
point, so that ( — 1) is the symbol which denotes the opera-
tion of turning the straight line through 180°. As 2, 3, 4, when
looked upon as symbols of operations, were called tensors, the
operator ( — 1) may conveniently be designated a reversor.
Exercises
Show graphically the effect of the operations indicated in each of
the following exercises. Take as the initial unit-operand a straight
line 1/2 inch long extending to the right of the zero or initial point.
Explain each expression as consisting of the operand unity and sym-
§203] COMPLEX NUMBERS 347
bols of operation — tensors, reversors, etc., which operate upon it
one after the other in a definite order.
1. 2-3-1. 8. (>/i)'*(-l)'l.
2. 3-31. 9. (-l)3-2«-31.
3. -1-31. 10. 3-3«l.
4. 2»1. 11. (-1)«-2*1.
6. V31. 12. 3(-l)V21.
6. (V2)«-l. 13. (V2)-(-l)""-l.
7. V9V41. 14. >7l0'2(-l)-l.
16. A tensor, if permitted to operate seven times in succession,
will jifst double the operand. Symbolize this tensor.
16. A tensor, if permitted to operate five times in succession, will
quadruple the operand. Symbolize this tensor.
203. Versors. The expression \/— 1 canDot consistently,
with the meaning already assigned to \/ and ( — 1), be looked upon
as answering the question
"how many," and therefore
is not a number in that sense;
yet if we consider \/— 1 as
a symbol of operation, it can
be given a meaning consistent
with the operators already
considered. For if 2 is the
operator that doubles, and
\/2 is the operator that when
ijLsed twice doubles, then since
( — 1) is the operator that
reverses, the expression Yiq, 138.— The Integral Powers of
\/— 1 should be an operator V- i.
which, when used twice, re-
verses. So, as ( — 1) may be defined as the symbol which
operates to turn a straight line through an angle^ of 180°, in a
similar way we may define the expression \/^ 1 as a symbol
which denotes the operation of turning a straight line through an
angle of 90° in the positive direction. The restriction of positive
rotation is inserted in the definition merely for the sake of
convenience.
t
B
Q
1
isr^fa
O a ^A
c
iv^y-
-X
0
^
1-t
1
J
1
r
348 ELEMENTARY MATHEMATICAL ANALYSIS [§204
The symbols ( — 1) and \/— 1 are not tensors. Thoy do not
represent a stretching or contracting of the operand. Their ^ect
is merely to turn the operand to a new direction; hence these
symbols may be called versors, or '' turners.''
204. Graphic Representation of \/--^- In Fig. 138, let a be
any line. Then a operated upon by \/— 1 (that is, \/— 1 a) is
turned anti-clockwise through 90°, which gives OB. Now, of
course, \/— 1 can operate on V— 1 ^ just as well as on a.
Then \/— 1 IV— 1 fl)> or OC, is V— 1 ^ turned posi-
tively through 90°. Similarly, \/^^ IV^^iy/^^ a)] is
V— 1 (V— 1 fl) turned through 90°, etc.
As we are at liberty to consider two turns of 90° as equivalent to
one turn of 180°, therefore, \/^^(\/-- 1 a) = (-l)a. Now
OP = (-1) OB, 02) =(-1) (V-ia); but also OD =^
V — 1 (— a), therefore, ( — 1 ) V— 1 ^ = V— 1 ( — <*)•
Thus the^student may show many like relations.
The operator \/ — 1 is usually represented by the symbol i and
will generally be so represented in what follows
Exercises
Interpret each of the following expressions as a symbol of operation:
1. 2,3,4, -1.
2. 3^_2^ 4_o, (-1)2, (-1)^
3. V2, V3, V - 1, V2, V - 1.
Select a convenient unit and construct each of the following expres-
sions geometrically, explaining the meaning of each operator:
4. 2-3'51. 7. (-lyW^^'h
6. 23-(-l)-l. 8. 2=-(-l)3-(V-~i)«-l.
6. 3V- 1'21. 9. 3V- l-(-l)V- 11.
206. Complex Numbers. The explanation of the meaning oi
the symbol (a + hi) will be given in the following section. It
will be shown in subsequent theorems that any expression made
up of the sum, product, power or quotient pf real numbers and
imaginaries may be put in the form a + hi, in which both a and
b are real. The expression a + bi is therefore said to be the
typical form of the imaginary. An expression of the form a +bi
i206]
COMPLEX NUMBERS
349
13 also called a complex number, since it contains a term taken
from each of the following scales, so that the unit is not single
but double or complex:
. . - 3, - 2, - 1, 0, + 1, + 2, + 3, . . .
• • • — 3i, — 2i, — i, 0, + i, + 2iy + 3i, . . .
It is important to note that the only element common to the two
series in this complex scale is 0.
206. Meaning of a Complex Number. Any real number, or
any expression containing only real numbers, may be consid-
ered as locating a point in a line.
Thus, suppose we wish to draw the expression 2 + 5. Let 0 be
the zero point and OX the positive direction. Lay off OA = 2 in
the direction OX and at A lay off AB = 5 in the direction OX.
Then the path OA + AB is the geometrical representation of
2+5.
0 A B X
X
Any complex number may be taken as the representation of the
pK)sition of a point in a plane. For, suppose c + di is the complex
number. Let 0 be the zero
pK)int and OX the positive
direction. Lay off OA = + c
in the direction OX and at
A erect di in the direction
OY, instead of in the direction
OX as in the last example.
See Fig. 139. It is agreed
to consider the step to the
right, OA, • followed by the
step upward, AP, as the
meaning of the complex num-
ber c + di. Either the broken path OA + AP or the direct path
OP may be taken as the representation of c-\- di, and either path
ccynstitvies the definition of the sum of c and di.
In the same manner c — di, ^ c — di and — c -\- di may be
constructed.
The meaning of some of the laws of algebra as applied to imagi-
naries may now be illustrated. Let us construct c + di + a -V ^•
Fig. 139. The Geometrical Construc-
tion of a Complex Number, c + di.
350 ELEMENTARY MATHEMATICAL ANALYSIS [§207
The first two terms, c + di, give OA + AB, locating B (Pig.
140). The next two terms, a + W, give BC + CP, locating P,
Hence the entire expression locates the point P with reference to 0.
Now if the original expression be changed in any manner allowed by
the laws of algebra, the result is merely a different path to the same
point. Thus:
c + a +di + bi is the path OA, AD, DC, CP
(c + a) + id + h)i is the path OD, DP
a+di + c +bi is the path OE, EH, HC, CP
a + di +bi +c is the path OE, EH, HF, FP, etc.
The student should consider other cases. Are there any method
of locating P with the same four elements, which the figure does not
illustrate?
Y
a
c
y\_
^
t
1
F
' G\
^A
P
'«»
""^
/_
_B
B
a
C
*e
1
••5
i
, o
E
1
A
D
o
t
FiQ. 140. Illustration of the Application of the Laws of Algebra to the
Expression c -^ di -\- a '\- hi,
207. Laws. It can be shown by simple geometrical construc-
tion that the operator i, as defined above, obeys the ordinary
laws of algebra. We can then apply all of the elementary laws of
alegbra to the symbol i and work with it just as we do with any
other letter. The following are illustrations of each law:
Commutative Law:
c + di + a+hi^C'jrO' + di + hi = di + c + hi + a, etc.
di = ia, iai = iia = aii, etc.
The equation lOV^ = V^'IO, or better, 10\/^-l =
V— 1101 may be said to mean that the result of performing the
operation of turning unity through 90° and performing upon the
§208] COMPLEX NUMBERS 351
result the operation of taking it ten times, is the same as the result
of performing the operation of taking unity ten times and per-
forming upon this result the operation of turning through 90**.
Associative Law:
(c + di) + (a + hi) = c + {di + a) + hi, etc.
{ah)i = a{bi) = ahi, etc.
Distributive Law:
(a + h)i = ai + hi, etc.
The expression \/— a, where a is any number of the arith-
metical scale, is defined as equivalent to \/ — l*fl;that is, \/— a
= i\/a- ^or example, V— 4 = 2i, \/— 3 = t\/3, etc. In
what follows it is presupposed that the stiident will reduce expressions
of the form \/— a to the form i s/a hefore performing algebraic op-
eraiions. From this it follows that \/— av'— &= "" V^ <*^^
not \/ah.
The relation •%/— 4 = 2 -v/ — 1 may be interpreted as follows: ( — 4)
is the operator that quadruples and reverses; then V — 4 is an operator
which used twice quadruples and reverses. But 2 \/— 1 is an opera-
tor such that two such operators quadruple and reverse. That is,
208. Powers of i. We shall now interpret the powers of i by
means of the new significance of an exponent and by the commu-
tative, associative and other laws. First:
i« or t» 1
= + 1
i* = iH
*
= I
i* or i* 1
•
= I
I's = iH
= - 1
i»
= - 1
t^ = iH
•
t» = iH
•
t's = m
= + 1
i* = iH*
= +1
etc.
etc.
Whence it is seen that all even powers of i are either + 1 or — 1,
and all odd powers are either i or — i. The student may reconcile
this with Fig. 138. The zero power of i must be unity, for the
exponent zero can only mean that the operation denoted by the
symbol of operation is not to he performed at all; that is, unity is to
be left unchanged; thus 10° or 10° • 1 = 1, 2° = 1, log^ x = x,
sin® X = X, etc.
352 ELEMENTARY MATHEMATICAL ANALYSIS [§210
Exercises
Select as unit a* distance 1/2 inch in length extending to the right
and represent graphically each of the following expressions:
1. i + 2i2 + 3i3 + 4i* + . . .
2. i + t* + i* + i« + i« + . . .
3. i + i* + t' + i« 4- i« + t" + . . .
4. ^(i + ^* + ^^ + ^«^-^« + ^"+ . . . ).
6. i + t2 + i3 4- 2i* + i'^ + i« + i' + 3i« 4- . . .
209. Two complex numbers are said to be conjugate if they
differ only in the sign of the term containing \/— 1. Such are
X + iy and x — iy.
ConjiLgate imaginaries have a real sum and a real product.
For: (x + yi) + (x - yi)
= X + yi + X — yi, by associative law
= X + X + yi — yi, by commutative law
= 2x + (yi — yi), by associative law
= 2x + (y — y)i, by distributive law
Likewise: {x + yi){x — yi)
= x{x — yi) + yi{x — yi), by distributive law
= x* — xyi + yix — yiyi, by distributive law
= x^ — yH^ + xyi — xyi, by commutative law
= x^ + y2 _|_ (j-y __ ^.y)^-^ |jy distributive law and by
substituting i* = — 1
= x2 + 2/*
It is well to note that the product of two conjugate complex
numbers is always positive and is the sum of two squares.
This fact is very important and will be frequently used. Thus '
(3 - 4i)(3 + Ax) = 32 + 42 = 25; (1 + i){l - i) = 2
(cos ^ + i sin 0)(cos B — i sin B) = cos^ d + sin^ ^ = 1, etc.
210. The sum, product, or quotient of two complex numbers is,
in general, a complex number of the typical form a + M,
Let the two complex numbers he x + yi and u + vi,
(1) Their sum is (x + yi) + (u + vi)
= X + yi + u + vi
= X + u + yi + vi
= (x + u) + (y + v)i
§210] COMPLEX NUMBERS 353
by the laws of algebra. This last expression is in the form a + bi,
(2) Their product is (a; + yi) {u + vi)
= x{u + vi) + yi{u + vi)
= xu + xvi + yiu + yivi
= xu + yvi^ + xvi + yui
= (xu — yv) + (xv + yu)i •
by the laws of algebra. This last expression is in the form a + H,
(3) Their quotient is
X + yi __ {x + yi){u — vi)
u'+ vi ~ {u + vi)(u — vi)
By the preceding, the numerator is of the form a' + 6'i. By
§209, the denominator equals u^ + v^. Then the quotient equals
a' + h'i a' 6' .
u^ + t;2 u^ + v^ ^ u^ + v^
by distributive law. This last expression is of the form a + bi.
Exercises
Reduce the following expressions to the typical form a -{- bi;
the student must change every imaginary of the form V - a to the
form i Vo-
1. V-25 +V-49 4- V^i2i - V-64 - 6i.
2. (2V^ 4- 3 V^)(4V^ - SV^).
3. (x - [2-\-di]){x - [2 -3i]).
4. (- 5 + 12V'^)l 6. (Vr+~i)(Vr -1).
6. (3 - 4V^)2. 7. (Ve -V^)*.
a 1
8. -F^f • 12.
V- 1 ' (1 -^)«
2 1 - t3
9. o . /-=^' 13.
3 + V-2 (I -i)«
56
10. r=- 14.
1 - 2V -3
1 - V^^7 ""' 1 + 2V -^'
1+i ,^ (2+3V'^l)^
l-i* 2 + V-l
^^ a -\- xi a — xi
16. : - —-—.'
a — XI a + XI
99
354 ELEMENTARY MATHEMATICAL ANALYSIS [§211
211. Irrational Numbers. A rational number is a number that
can be expressed as the quotient of two integers. All other real
numbers are irrational. Thus \/2> \/5) \/7,'Jr, c, are irrational
numbers. An irrational number is always intermediate in value
to two rational numbers which differ from each other by a number
as smalt as we please. Thus
1.414 < \/2 < 1.415
1.4142 < \/2 < 1.4143
1.41421 < v/2 < 1.41422, etc.
It is easy to prove that \/2 cannot be expressed as the quotient
of two integers. For, if possible, let
V2=f (1)
where a and b are integers and t is in its lowest terms. Squaring
the members of (1) we have
2 = p (2)
This cannot be true, since 2 is an integer and a and b are prime
to each other.
An irrational number y when expressed in the decimal scale j is
never a repeating decimal. For if the irrational number could be
expressed in that manner, the repeating decimal could be evalu-
ated by §123 in the fractional form y~z — ' which, by definition
of an irrational number, is impossible. On the contrary, every
rational number when expressed in the decimal scale is a repeating
decimal Thus 1/3 = 0.33 ... and 1/4 = 0.25000. . . .
The proof that x and e are irrational numbers is not given in
this book.
See Monographs on Modern Mathematics j edited by J. W. A.
Young.
The student should not get the idea that because irrat'onal numbers
are usually approximated by decimal fractions, that the irrational
number itself is not exact. This can be illustrated by the graphical
construction of -\/2. Locate the point P whose coordinates are (1,1)-
Call the abscissa OD and the ordinate DP, Then OP — V2 ^^
OD = 1, DP = 1. It is obvious that the hypotemise OP must^be
i212] COMPLEX NUMBERS 355
sonsidered just as exact or definite as the legs OD and DP. The
lotion that irrational numbers are inexact must be avoided.
The process of counting objects can be carried out by use of the
jrimitive scale of numbers 0, 1, 2, 3, 4, . . . The other numbers
nade use of in mathematics, namely,
(1) positive and negative numbers
(2) integral and fractional numbers
(3) rational and irrational numbers
(4) real and imaginary numbers
nay be looked upon as classes of numbers that permit the opera-
ions subtraction, division and evolviion, to be carried out under all
sircumstances. Thus, in the history of algebra it was found that in
)rder to carry out subtraction under all circumstances, negative num-
)ers were required; to carry out division under all circumstances, frac
Jons were required; to carry out evolution of arithmetical numbers
inder all circumstances, irrational numbers were required; finally to
jarry out evolution of algebraic numbers under all circumstances,
maginaries were required. It will be found that it will not be neces-
lary to introduce any additional form of number into algebra; that
s, the most general number required is a number of the form a + 6i,
irhere a and 6 are positive or negative, integral or fractional, rational
•r irrational. This is the most general number that satisfies the f ol-
3wing conditions:
(a) The possibility of performing the operations of algebra and
he inverse operations under all circumstances.
(6) The conservation or permanence of the fundamental laws of
Igebra: namely, the commutative, associative, distributive and index
iws.
Further extension of the number system beyond that of complex
lumbers leads to operators which do not obey the commutative law in
nultiplication; that is, in which the value of a product is dependent
apon the order of the factors, and in which a product does not neces-
sarily vanish when one factor is zero. Numbers of this kind the
student may later study in the introduction to the study of electro-
magnetic theory under the head of "Vector Analysis" or in the
subject of "Quaternions." Such numbers or operators do not belong
to the domain of numbers we are now studying.
212. // a complex number is equal to zero, the imaginary and
real parts are separately equal to zero.
Suppose X + y yl -1 = 0
then « = — 2/ V — 1
356 ELEMENTARY MATHEMATICAL ANALYSIS [§214
Now it is absurd or impossible that a real number should equal '
an imaginary, except they each be zero, since the real and imagi-
nary scales are at right angles to each other and intersect only at
the point zero.
Therefore: x = 0 and y = 0
If two complex numbers are equal y then the real and the im^aginary
parts must he respectively equal.
For if
X + yi = u + vi
then {x — u) + iy — v)i = 0
Whence, by the above theorem,
x — u = 0 and y — v = 0
That iS; X ^ u and y = v
213. Modulus. Let the complex number x + yihe constructed,
as in Fig. 141, in which OA = x and AB = yi. Draw the line
OP, and let the angle AOP be called e.
The numerical length of OP is called the modulus of the complex
number x + yi. It is algebraically represented by V«^ + y'>
or by \x + yi\. Thus, mod (3 + 4i) = V9 + 16 = 5.
The student can easily see
that two conjugate complex
numbers have the same modu-
lus, which is the positive value
of the square root of their
prodv/^t.
If y_= 0, the mod (x + yi)
= Vx^ = \x\, where the
Fig. 141. — Modulus and Amplitude of „«^;^„i i* ^« • j- . au«*
a Complex Number vertical hnes mdicate that
merely the numerical or
absolute value of x is caUed for. Thus the modulus of any real
number is the same as what is called the numerical or dbsduie
value of the number. Thus, mod (—5) = 5.
214. Amplitude. In Fig. 141 the angle AOP or ^ is called the
argument or amplitude or simply the angle of the complex number
X + yi.
215] COMPLEX NUMBERS 357
Hitting r = Vx* + y* = mod (x + yi), we have
sin ^ = — cos ^ = -
T T
therefore,
X + yi = r cos 6 + ir sin 6 = r(cos ^ + i sin 6)
a which we have expressed the complex number x + yi in terms of
bs modulus and ampUtude.
To put 3 — 4i in this form, we have:
mod (3 - 4i) = V9 + 16 = 5; sin ^ = y = - g; cos ^ = -^ = g
?herefore,
(3-40=5[|-|i]
The amplitude 6 is tan"^ { ~ ^) » ^^^ ^ "^ the/owr^/i quadrant. Why ?
It is well to plot the complex number in order to be sure of the ampli-
ide 0, It* avoids confusion to use positive angles in al^ cases. For
cample, to change 3 — V 3 i to the polar form, plot the point
\ - V3) and find from the triangle that r = 2 V 3 and ^ = 330°.
[ence
3 - V 3 t = 2 V 3(cos 330° -f i sin 330°)
The ampUtude of all positive numbers is 0, and of all negative
umbers is 180°. The unit expressed in terms of its modulus and
miplitude is evidently l(cos 0 + t sin 0).
215. "Vector. The point P, located by OA + AP or a; + jfi,
lay also be considered as located by the line or radius vector OP]
[lat is, by a Une starting at 0, of length r and making an angle 6
rith the direction OX, A directed line, as we are now considering
[P, is called a vector. When thus considered, the two parts of the
ompound operator
r (cos 6 + i sin 6) (1)
eceive the following interpretation : The operator (cos d + i sin 6) ,
rhich depends upon 6 alone, turns the unit lying along OX
hrough an angle dy and may therefore be looked upon as a
ersor of rotative power 6, The versor (cos 6 + i sin 6) is often
abbreviated by the convenient symbol cis 6, The operator r
3 a tensor, which stretches the turned unit in the ratio r : 1. The
esult of these two operations is that the point P is located r units
rem 0 in a direction making the angle d with OX.
358 ELEMENTARY MATHEMATICAL ANALYSIS [§216
The operator (1) above is also represented by the notation
(r, 6), for example (5, /30**). Expression (1) is called the polar
fonn of the complex number (x+iy).
Thus, the operator (cos 6 + i sin d) is simply a more general
operator than i, but of the same kind. The operator % turns
a unit through a right angle and the operator (cos 6 + i sind)
turns a unit through an angle 0. If 6 be put equal to W,
cos d + i sin d reduces to i.
For: ^ = 0, cos 6 + i sin 6 reduces to 1
6 = 90°, cos 6 + 1 sin 6 reduces to i
6 = 180**, cos 6 + i sin 6 reduces to — 1
6 = 270°, cos ^ + I sin ^ reduces to — i
Since 3 — 4i = 5(| — ji) the point located by 3 — 4i may be
reached by turning the unit vector through an angle d =
sin"^ (—4/5) = co8~^ 3/5 and stretching the result in the
ratio 5:1.
// a complex number vanishes, its modtdus vanishes; and con^
versely, if the modtUits vanisfies, the complex number vanishes.
If X + yi = 0, thwi X = 0 and y = 0, by §212. Therefore,
Vx* + y« = 0. Also, if Vx« + y« = 0, then x» + y* = 0, and
since x and y are real, neither x* nor y* is negative, and so their
sum is not zero unless each be zero.
// two complex n umbers are eqiud, their moduli are equalf but if
tico moduli are equal j the complex numbers are not necessarily equal.
If X + yi = w + «*« » then X = t4 and y = r by §212.
Therefore, Vx« + y* = Vu» + r«
But if Vx* + y' = \ m' + r', obviously x* need not equal
u' nor y' = r*.
21& Sum of Comi^ex Numbers. Let a given complex number
locate the point .1, Fig. 142, and let a second complex number
locate the point B. Then if the first of the complex numbers be
represented by the radius vector OA and if the second complex
number be represented by ihe radius vector OB, the sum of the two
complex numbcars will be represented by the diagonal OC of the
parallelogram constructed on the lines OA and OB. This law of
addition is the well-known laic of addition of vectors used in physics
when the resultant of two forces or the resultant of two vdocities,
§216]
COMPLEX NUMBERS
359
two accelerations, or two directed magnitudes of any kind, is to be
found.
The proof that the sum of the two complex numbers is repre-
sented by the diagonal OC is very simple. Let the graph of the first
complex number be ODi + DiA and let that of the second be OD2
+ D2B. To add these, at the point A construct AE = OD2 and
EC = DuB, Then the sum of the two complex numbers is geo-
metrically represented by ODi + DiA + AE + EC, or by the
radius vector OC which joins the end points. Since, by construc-
tion, the triangle AEC is equal to the triangle OD2B, therefore AC
Fig. 142. — Sum of Two Complex Numbers. »
must be equal and parallel to OB, so that the figure OACB is a
parallelogram, and OC, which represents the required sum, is the
diagonal of this parallelogram, which we were required to prove.
Exercises
find algebraically the sum of the following complex numbers, and
construct the same by means of the law of addition of vectors.
1. (1 + 2t) + (3 + 4i).
2. (1 4- i) + (2 + 1).
3. (1 - i) + (1 + 2i).
4. (3 - 4i) - (3 + 4t).
5. (- 2 +i) + (0 - 4i).
6. (- 1 + i) + (3 + i) + (2 + 2i).
360 ELEMENTARY MATHEMATICAL ANALYSIS [§217
7. (2 -i) +(-2 4-i) + (1 +i).
8. Find the modulus and amplitude (in degrees and minutes) of
2 (cos 30° + i sin 30°) + (cos 45° + i sin 45°).
9. By the parallelogram of vectors, show that the sum of two con-
jugate complex numbers is real.
10. If 12 be the sum of the complex numbers zi = xi + iyi, «» =
X2 + iytj «8 = xa + Vih etc., show that — R, zi, zt, «»,... form the
sides of a closed polygon.
217. Polar Diagrams of Periodic Functions. Three methods of
representing simple periodic phenomena have already been ex-
plained: (1) Crank or clock diagrams as shown in Fig. 128 and
explained in §184; (2) the sine curve or sinusoid in rectangular
coordinates, as shown in Fig. 59 and explained in §55;
(3) polar diagrams^ in which the circle (twice drawn) corresponds
to a crest and trough of the sine curve, as shown in Fig. 63 and
explained in §64. As the principal application of these methods
is to phenomena that vary with the time, one of the variables
may conveniently be taken to represent time or a constant multi-
ple of time. Thus the angle 6 in the crank and polar diagrams or
the abscissa in the Cartesian diagram, may be represented by
a constant multiple of ^ as a)^
The difference between a clock diagram and a polar diagram
of a simple periodic fimction may be stated as follows: In a
clock diagram, a rotating vector of fixed length OP is continu-
ously projected upon a fixed line OX; in a polar diagram, a
statiorfery line of fixed length OA is continuously projected
upon a rotating radius vector OP. See Figs. 52 and 63.
Each of the three methods possesses a peculiar advantage of its
own, but probably the best insight with regard to periodic phe-
nomena is given by the polar diagram. In each, the complete
period of the phenomena is represented by one complete revolution
of the radius vector. The polar method is not only well adapted
to represent simply varying periodic phenomena, in which case the
polar diagram is a circle passing through the origin, but it is
equally well adapted to represent cases in which the periodic
motion is compounded from a number of simple harmonics. In
many important cases in science, especially in the phenomena of
alternating electric currents, only the odd harmonics are commonly
present as components of the resulting motion. The equation of
§2171 COMPLEX NUMBERS 361
compound harmonic motion in rectangular coordinates, in which
only odd harmonics appear, is of the form:
y = ai sin cat + az sin Scot + a^ sin 5o}t + • . • (1)
in which ai, as . . . are any constants and in which cjt has re-
placed x. If the epochs^ of the various harmonics be ^1, ^3, ^6, . . .
the proper form of the equation would be:
2/ = ai sin o)(t — ti) + as sin 3co(< — tz) + a^ sin 5co(< — h)
+ . . . (2)
A curve of type (1) or (2) must represent a pattern within the inter-
val cot = T to cat = 27r which is the opposite of the pattern pre-
sented within the interval cat = 0 to cat = t; for increasing cat
by the amount t in each of the components:
sin (atf sin dcat, sin 5o)t , . .
of the compound motion has the effect of changing the algebraic
sign of each term, but leaves the absolute value unchanged. This
is because the sine curve, and all of the odd harmonics of the sine
curve, are just reversed in sign by adding a straight angle (180°)
or an odd number of straight angles to the original angle. Hence
y has the same sequence of values, but of opposite signs, within
each of the two half-intervals of the period 2t, Fig. 143 illustrates
this. The curve A is the graph of an alternating current wave
(after Fleming) in rectangular coordinates, while the same func-
tion is shown in polar coordinates by curve B, It is observed that
the second portion of the Cartesian curve is exactly similar to the
first portion, except that its position with reference to the a;-axis
is reversed. In the polar diagram this truth is brought out by
the fact that the loop that represents the "trough" of the wave
is identical with the loop that represents the "crest" of the wave,
that is, the curve is twice drawn to represent the interval of a
complete period from cat = 0 to cat = 2t.
If only even harmonics are present, the equation of the curve in
rectangular coordinates is of the form:
y = ao + a2 sin 2(at + cla sin 4coi + . . . (3)
or, if the epochs are not zero,
y = ao + Gi sin 2co(< — ^2) + a^ sin 4co(< — ^4) + . • . (4)
iThiB expression insteed of "epoch angle" is the proper term in this case as t is
measured in units of time and not in angular measure. The epoch anglea «.x^ tA\i\<»
uttt etc. '
362 ELEMENTARY MATHEMATICAL ANALYSIS 1(217
Beoauee of the factor 2 in each harmonio term, the period of the
function may be considered it instead of 2^, bo that the soquence of
values of y are repeated in each interval 0 to r, r to 2ir, etc., and
not reversed in sign aa in the case of the odd harmonics.
The fact that the pattern for each interval r is repeated right
eide up and not reversed is illustrated by the graph of
V = 1 + sin 2ioi + sin 4io( (5)
1
I
>rr7?\\
my
1
^\
I
V
Flu. 143.— Rectangular
(Afler
shown in Fig. 144. The effect of the constant term Is, of cour3«>
merely to raise the graph a distance of one unit.]
In/orni, curves with only even harmonics present do not differ
from curves with Iwjth odd and even, for substituting f = a, the
general case (eii»ation (3)1 becomes:
y - Ob + nt sin uA' + n, sin 2u*' + <i« sin 3wf + . . . (6)
whii.^ contains both odd and even harmonics in C, aod is of period
St. The curve (3) is of the same shape as (6) but of period r.
S2171
COMPLEX NUMBERS
A curve made up of both odd and even harmomos may have any
form consistent with itsone-valuedandcontinuouscharacter. The
portion of the curve above the x-axis (if any) need not have the
same form as the part below; the only essential is that the curve for
Fig. 144. — Graph o£ i/ =
each Bucce^ive interval of 2t be a repetition of the curve in the
preceding interval.
In polar coSrdinates, a curve made up of only even harmonics
is described but once as 6 varies from 0° to 360°. In general such
FiQ. 145.— Graph of p = sin 2s + gin 49.
curves have more "loops" than curves made up of odd harmonica,
for the loops of the odd harmonics are tvnce drawn as 6 varies
from C to 360°. Thus the curves:
p = sin e, p -= sin 36, p = sin M, . . .
364 ELEMENTARY MATHEMATICAL ANALYSIS [§217
have 1, 3, 5, . . . loops respectively each twice drawn. The
curves
p = sin 26, p = sin 4^, p = sin 6^, . . .
have 4, 8, 12, . . . loops respectively, each once drawn.
Also the curve:
p = sin 26 + sin 4^ (7)
is represented by the heavy curve of Fig. 145 as 6 varies from (f to
180** and by the dotted curve as 6 varies from 180° to 360**. The
/r^VHfHf
Fig. 146. — Curves made up of Odd Harmonics only, of Even
Harmonics only, and of Both.
numbered points 1, 2, 3, 4, . . . of Fig. 145 correspond to the
similarly numbered points in Fig. 144. The curves of Fig. 144
and Fig. 145 correspond, except that the constant term was
omitted from the equation in constructing Fig. 145.
Exercises
1* If / be the frequency of the fundamental harmonic, show that:
2/ = sin 2jrft + sin Qirjt + sin 10ir/< + . • .
contains odd harmonics only. *
§218] COMPLEX NUMBERS 365
2. Write an expression containing even harmonics only, using the
frequency / as in the last exercise.
3. How many loops has:
p = cos 60; p = sin Q0;
p = cos 7 ( 0 — ^ ) ; p = sin 40?
4. In the diagram, Fig. 146, pick out curves made up of odd
harmonics only, of even harmonics only, and those made up of both
odd and even harmonics.
218.'*' Simple Periodic Variation Represented by a Complex
Nmnber. Fluctuating magnitudes exist that follow the law of
S.H.M. although, strictly speaking, such magnitudes can be said
to be "simple harmonic motions" in only a figurative sense. For
example we may think of the fluctuations of the voltage or amper-
age in an alternating current as following such a law. Thus if
E represent the electromotive force or pressure of the alternating
current, then the fluctuations are expressed by
E = Eo sin o)t
or by
E = Eq sin 2Tft
where / is the frequency of the fluctuation. Instead of S.H.M.
such a variable is more accurately called a sinusoidal Yaiying
magnitude, although for brevity we shall often call it S.H.M.
The graph in rectangular coordinates of such a periodic function
is often called a "wave," although this term should, in exact
language, be reserved for a moving periodic curve, such as y =
a sin (kx — kt).
If the polar representation
p ^ a sin ct){t — ti) (1)
of the sinusoidal varying magnitude be used, then, as noted in the
last section, the graph of (1) is a circle of diameter a inclined the
angular amount o)ti to the left of the axis OF, as is seen at once
by calling cat = 6 and o)ti = a in the equation of the circle p =
a sin (^ — a). The circle can be drawn when the length and di-
rection of its diameter are known; that is, the circle is completely
specified when a and the direction of a (told by a) are given.
Therefore the simple harmonic motion is completely aymbolxxaiiL
366 ELEMENTARY MATHEMATICAL ANALYSIS l§218
by a vector OA of length a drawn from the origin in the direction
given by the angle (ah; the direction angle of the vector OA is
a + - orco<i + -.
The circle on the vector OA is located or characterized equally
well if the rectangular coordinates (c, d) of the end of the diameter
of the circle be given. But the complex number c + di la repre-
sented by a vector which coincides with the diameter a of this
circle. Hence we may represent the circle by the complex num-
ber c + di. Its modulus is a = Vc* + d* and its amplitude is
a + rt- Therefore if in (1) we take a = Vc' + d^ cah = a and
the variable angle cat = 6, we can completely determine the S.H.M.
by the complex number c + id. In the theory of alternating cur-
rents the sinusoidal varying current or voltage can conveniently be
represented by a complex number, and that method of repre-
senting such magnitudes is in common use.
One of the advantages of representing S.H.M. by a vector or by
a complex number is the fact that two or more such motions of
like periods may then be compounded by the law of addition of
vectors. This method of finding the resultant of two sinusoidal
varying magnitudes of like periods possesses remarkable utility
and simplicity.
To summarize, we may say:
(a) A sinibsoidal varying magnitude is represented graphicdUy
in polar coordinates by a vector, which by its length denotes the
amplitude and by its direction angle with respect to OY denotes the
epoch angle.
(6) Sinusoidal varying magnitudes of like periods may he
compounded or resolved graphically by the law of parallelogram of
vectors.
If two sinusoidal varying magnitudes of like periods are in
quadrature (that is, if their epoch angles differ by 90°), their rela-
tion, neglecting their epochs, can be completely expressed by a sin-
gle complex number. Thus let two S.H.M. in quadrature
^, = 113 sin cait - ti) (2)
and
Ec = 40 cos o){t - ti) (3)
§218]
COMPLEX NUMBERS
367
be represented by the circles and by the vectors marked OEo and
OEc, Fig. 147. CaU the resultant of these E.-. Then
^i = 113 sin o)(t - ti) + 40 cos o)(t - h) (4)
= V402 + 1132 sin o){t - ^2)
= 120 sin o)it - <2) (5)
where co<2 is measured as shown in Fig. 148. Instead of represent-
ing (2) and (3) in the polar diagram by OEo and OEc and their
Fia. 147. — Composition of Two S.H.M. in Quadrature by Law of Addi-
tion of Vectors.
resultant by OEij we may represent (2), (3) and (4) in the complex
number diagram, Fig. 148, by Eo, lEc and Eo + lEe, respectively.
Since the modulus and amplitude of Eo + lEc are ^JEo^ + Ee^
and a, respectively, and since the epoch angle of the resultant in
Fig. 147 is call = cati — a, we can state the resultant as follows:
// we have given two S,H.M.'s in quadrature and take the ampli-
tude of the one possessing the greater epoch angle as c and the
amplitude of the other S.H.M. as d, and construct the complex
number c + di, then this complex number c + di completely
characterizes both of the S,H,M*s. and their resultant. For, we can
368 ELEMENTARY MATHEMATICAL ANALYSIS [J219
Fig. 148. — Complex Number
Representation of the facts shown
by Polar Diagram, Fig. 147.
determine the modulus p and the amplitude a of c + dt and then
if call is the epoch angle of the motion with amplitude c, the epoch
angle of the resultant is <ati — a.
If we consider the two harmonic motions:
p = ai sin (»){t — h)
and
p = 02 sin <a(t — ^2)
then if ti be greater than t2 the first S.H.M. reaches its maximuid
value after the second reaches its maximum. The first S.H.M. is
therefore said to lag the amount
(ti — ^2) behind the second
S.H.M. That is, a S.H.M. rep-
resented by a circle located
anticlockwise from a second
circle represents a S.H.M. that
la^s behind the second.
219.* Illustration from Alter-
nating Currents* The steady
current C flowing in a simple electric circuit is determined by
the pressure or electromotive force E and the resistance R ac-
cording to the equation known as Ohm's law:
E
R
or,
E = CR
E is the pressure or voltage required to make the current C flow
against the resistance R, If the current, instead of being steady,
varies or fluctuates, then the pressure CR required to make the
current C flow over the true resistance is called the ohmic voltage
or ohmic pressure. ♦But a changing or fluctuating current in an
inductive circuit sets up a changing magnetic field around the cir-
cuit, from which there results a counter electromotive force or
choking effect due to the changing of the current strength. This
electromotive force is called the reactive voltage or reactive pres-
sure. The choking effect that it has on the current is known as the
inductive reactance. In case of a periodically changing current it
acts alternately with and against the ciurent. Opposite to the
(7 =
I
9] COMPLEX NUMBERS 369
ctive voltage there is a component of the impressed voltage
,t is consumed by the reactance. See Fig. 149.
The pressure which is at every instant applied to the circuit
oa without is called the impressed electromotive force or vol-
e. Of the three pressures — namely, the impressed voltage, the
nic voltage (consumed by the resistance) and the reactive vol-
e consumed by the inductive reactance, any one may always be
arded as the resultant of the other two. Hence if in a polar
gram the pressures be represented in magnitude and relative
ise by the sides of a parallelogram, the impressed voltage may
regarded as the diagonal of a parallelogram of which the other
) pressures are sides. Since, however, the reactance or the
inter inductive pressure depends upon the rate of change of the
rent, it lags, in the case of a sinusoidal current, 90° behind the
e or ohmic voltage, which last is always in phase with the
rent. The pressure consumed by the counter inductive pres-
e therefore leads the current by 90°. Thus, in the language of
nplex numbers
Ei ^ Eo + iEc (1)
which
Ei = impressed pressure
Eo = ohmic pressure, or pressure consumed by the
resistance
Ee == counter inductive pressure, or the pressure con-
sumed by reactance
s found that the counter inductive pressure depends upon a cou-
nt of the circuit L called the inductance and upon the angular
xyUy or frequency of ' the alternating impressed pressure, so that:
Ec = 27r/LC = wLC
nee (1) may be written:
Ei = JKC + i27r/LC (2)
= JKC + i(»iLC (3)
6 modulus of the complex number on the right of this equation is
C ^Ri + co2L2
24
370 ELEMENTARY MATHEMATICAL ANALYSIS [§219
Considering, then, merely the absolute value |J^o| and \C\ of
pressure and current, we may write:
. \C\ =
From the analogy of this to Ohm's law:
E
R
(4)
C =
the denominator Vi^^ + cj^L^ is thought of as limiting or restrict-
ing the current and is called the impedance of the circuit.
Let there be a condenser in the circuit of an alternator, but let
the circuit be free from inductance. Then besides the pressure
consumed by the resistance, an additional pressure is required at
any instant to hold the charge on the condenser. If K be the ca-
pacity of the condenser, it is found that that part of the pressure
C C
consumed in holding the charge on the condenser is o-jrfK ^^ "K
and is in phase position 9(f be-
hind the current C. The chok-
ing effect of this on the cunent
may be called the condensive
reactance. When a condenser
is in the circuit i7i addition to in-
ductance ^ the total pressure con-
FiG. 149.— Complex Number Dia- sumed by the reactance has the
gram of Equation 5, §219 -
^""^^ " 27rfK
and the complex number that symbolizes the vector is
Ei = RC + i2TfCL - 2^^ (5)
(see Fig. 149).
Further illustrations of the applications of complex numbers to
alternating currents is out of place in this book. The illustrations
are merely for the purpose of emphasizing the usefulness of these
numbers in applied science. An interesting application of the use
of complex numbers to the problem of the steam turbine will be
found in Steinmetz's "Engineering Mathematics," page 33.
COMPLEX NUMBERS
371
Exercises
raw the polar diagram and complex number representation
f iJ = 5, C =21, / = 60, L = 0.009, K = 0.005.
raw a similar diagram if J?» = 100, J^o = 90, / = 40, L =
K = 0.003.
Product of Complex Numbers. The product of two or
omplex numbers is a complex number whose modulus is the
)t of the moduli and whose
ude is the sum of the am-
js of the complex numbers,
.6 complex numbers be:
xi + yii
ri (cos ^1 + i sin ^i)
x% + Vzi
r2 (cos 02 + i sin ^2), etc.
actual multiplication, we
(2 + 2t) ( vT+ i)
Fig. 150.— Product of Two Com-
plex Numbers.
5 = rifi [(cos 01 COS 02
— sin 01 sin ^2)
n 01 cos ^2 + cos^i 8m02)i]
= rir2 [cos (^1 + ^2)
+ i sin (^1 + ^2)]
ce it is seen that rir2 is
odulus of the product and
^2) is the amplitude.
above theorem is illustrated
g. 150. If the two given complex numbers be represented
}ir vectors OPi and OP2, their product will be represented
5 vector OPs whose direction angle is the sum of the ampli-
of the two given factors, and whose length OPs isthe product
lengths OPi and OP2.
figiu*e represents the product (2 + 2t)(\/3 4*^). Expressed
ns of modulus and amplitude these may be written:
V3 + t = 2 (cos 30° 4- i sin 30°)
2 + 2i = 2 V2(cos 45° + i sin 45°)
n =2, fj = 2V2, Oi = 30°, 02 = 45°
:ore: (2 + 2i)(V3 + t) = 4V2 (cos 75° -f ^ am15*'^
372 ELEMENTARY MATHEMATICAL ANALYSIS [§221
Exercises
Find the moduli and amplitudes of the following productSi and
construct the factors and products graphically. Take a positive angle
for the amplitude in every case.
1. (1 +\/3t)(2V3 +2i).
2. (2 4-|V3i)(2 4-2i).
8. (V3+3i)(2-2i).
4. (l+i)«.
5. (2 - 2V30 (\/3 + 3i).
6. (1 - t)*.
7. (1 4- t)'(l - i)'.
8. 2(cos 15^+ i sin 15°) X 3(cos 25° + i sin 25'').
Find numerical result by use of slide rule or trigonometric tables.
9. 2(cos 10° + i sm 10°) X (l/3)(cos 12° + i sin 12°) X 6(cos8"
+ i sin 8°).
10. Find the value of 4V2(cos 75° +i sin 75**) -^ (V3 +»)•
221. Quotient of Two Complex Numbers. T?ie quoHerd of
two complex numbers is a complex number whose modulus is the
quotient of the moduli and whose amplitude is the difference of the
amplitudes of the two complex numbers. Let the complex numbers
be:
zi = xi + yii = ri(cos ^i + i sin ^i)
Z2 = X2 + y^i = r2 (cos B% + i sin ^2)
We have:
Zi _ ri(cos Bi + tsin gi)(cos 62 — ism $2)
Z2 ~ r 2(008 02 + i sin ^ 2) (cos ^2 — t sin ^2)
^ ri[cos (01 - 62) + i sin (^1 - 62)]
~ r2 (cos2^2 + sin*^2)
= p[cos (^1 - ^2) + i sin (^1 - ^2)]
'2
Whence it is seen that — is the modulus of the quotient and
(^1 — ^2) is the amplitude.
In Fig. 151, the complex number represented by the vector OPi
when divided by the complex number represented by OPt yields
the result represented by OP3, whose length ri/tt is found by dividing
the length of OPi by the length of OPt, and whose direction angle
§222]
COMPLEX NUMBERS
373
is the difference (Oi — 62) of the amplitudes of OPi and OPt. The
figure is drawn to scale for the case:
5(cos60°+tsin60°) .
2(cos20°+i8in20°) = ^^'^^ ^^^' ^^ +^ «^^ ^^ >
Exercises
Find the quotient and graph the result in each of the following
exercises. Always take amplitudes as positive angles and if
$% < $1, take 01 4- 360** instead of Oi.
Fig. 161. — Quotient of Two Complex Numbers.
1. (1 + Vsi) -^ (2 + V2i).
2. (i + J Vsi) ^ ( V2 - V2i).
8. (3 Vs - 3i) -i- ( - 1 + V3 i).
4. (1 - V3 i) -^ i.
5. 2(cos 36** 4- i sin 36*») -^ 5(cos 4** + i sin 4**).
6- 12(co8 48** + t sm 48°) ^ [2(cos 15° + i sin 15°)
3 (cos 9° 4- i sin 9°)].
. [4+(4/3)V3i](2 + 2V3t)
'• 8 4- 8i
8. Express in terms of o, 6, c, d the amplitude of (a ■{- H) -h (c 4- di).
222. De Moivre's Theorem. As a special case of §220 consider
the expression:
(cos e + i sin BY
374 ELEMENTARY MATHEMATICAL ANALYSIS [§222
This being the product of n factors like (cos ^ + t sin 6), we write,
by means of §220 :
(cos 6 + i sin ^)(cos 6 + i sin 6) . . .
= [cos {6 + 6+ , . . ) + t sin(^ + 6+ . . . )]
or:
(cos ^ + I sin ^)» = (cos n6 + i sin n6) (1)
which relation is known as De Moivre's theorem.
De Moivre's theorem holds for fractional values of n. For, first
consider the expression:
(cos ^ + I sin 6)'^'
where the power l/t of cos ^ + i sin ^ is, by definition, an oper-
ator such that its tth power equals cos 6 + i sin 0.
Put 6 = i<t)f so that <f> = .
Then: (cos 6 + i sin 6)'^' = (cos t4> + i sin <</>)'/*
= [(cos <t> + i sm 4>yp' by (1)
= cos <i> + i sin <t>
= cos T + i sin r (2)
Next consider the case in which n = -, We know:
(cos 6 + i sin 6)'^* = [(cos 6 + i si^x 6)')]"^'
= (cos s6 + i sin s^)'/' by (1)
= cos r + I sin y by (2) (3)
Likewise the theorem may be proved for negative values of n.
The following examples illustrate the application of De Moivre's
theorem.
(1) Find (3 + i Vs)*.
write: 3 + Ws = 2 V3(cos 30° + i sin 30**)
Theii, by De Moivre's theorem:
(3 4- WS)* = 144 (cos 120° + i sin 120°)
= 144 ( - 1/2 4- iVsi)
= - 72 4- 72 V3i
(2) Find (2 + 2i)".
§223] COMPLEX NUMBERS 375
Write: 2 + 2t in the form: _ _
2 + 2i = 2 V2(iV2 + § V2i)
(2 + 2i)" = (2 V2)"(cos 45** + i sin 45*»)"
= (2 V2)"(cos 495^ 4- i sin 495*')
= (2 V2)"(cos 135** + i sin 135**)
= (2 V2)"( - J V2 4- i V2i)
= 2i«(- 1 +i)
Exercises
Evaluate the following by De Moivre's theorem, using trigonometric
table or slide rule when necessary.
1. (8 + 8 ^l^^y\
2. <^27(cos 75** - i sin 75**).
3. Vl25 i.
4. [cos 9** + i sin 9**]".
5. (3 + V3i)».
6. [1/2 + (1/2) V3il*.
7. (1 + i)8.
8. (-2 + 2i)H.
9. 1(1/2) V3 - (l/2)il«.
10. Find value of ( - 1 + V- 3)« + ( - 1 - V- 3)« by De
Moivre's theorem.
11. Find the value of x^ - 2a; + 2 for a; = 1 + i.
12. If ii = - 1/2 + (1/2) V - 3 and ji = -1/2 - (1/2) V - 3,
jhow that j i» = 1, ^2* - 1, ii* = J2, J2^ =ii, ii*" = i2»* = 1,
223. The Roots of Unity. Unity may be written:
1 = cos 0 + i sin 0
1 = cos 2t + i sin 2jr
1 = cos 4nr + i sin 47r
1 = cos Gw + i sin Gtt
and so on. By De Moivre's theorem the cube root of any of these
i
376 ELEMENTARY MATHEMATICAL ANALYSIS IJ223
is taken by dividing the amplitudes by 3. Therefore, from the
above expressions in turn there results:
Vi = cos 0 + I sin 0 = 1
Vl = cos {2k IS) + i sin (27r/3) = cos 120** + i sin 120°
= -l/2 + i*(l/2)v3
i/i = cos (47r/3) + i sin (4t/3) = cos 240° + % sin 240°
= -1/2 -i(l/2)V3
Vl = cos Gtt /3 + i sin Gtt /3 = same as first, etc.
Fig. 152.— The Cube Roots of Unity.
Therefore there are three cube roots of unity. Since these are the
roots of the equation x^ — 1 = 0, they might have been found by
factoring, thus:
ar3 - 1 = (x - l)(x2 + X + 1)
= (x - l){x + 1/2 + WSi)(x + 1/2 - i>/3i)
The three roots of unity divide the angular space about the point
0 into three equal angles, as shown in Fig. 152. In the same
way, it can be shown that there are four fourth roots, five fifth
roots, etc., of unity and that the vectors representing them have
modulus 1 and amplitudes that divide equally the space
about 0.
§223] COMPLEX NUMBERS 377
To find all of the roots of any complex number, proceed as in the
following illustrative examples.
(1) Find VV3 + 3i.
Write V3 + 3i in the form:
V3 + 3i = 2V3(cos 60° + i sin 60°)
Hence, by De Moivre's theorem:
(Vs + 3i)^^ = \/l2 (cos 30° + i sin 30°)
= \/i2 [(1/2) V3 + (1/2)^
= (1/2)^108 + (l/2)\/l2 i
A second root can be found by writing:
Vs + 3i = 2V3[cos (60° + 360°) + i sin (60° + 360°)]
since adding a multiple of 360° to the amplitude does not change the
value of the sine and cosine. In applying De Moivre's theorem
there results:
(V3 + 3i)^^ = V^12 (cos 210° + i sin 210°)
= V^j - (1/2)V3 - (l/2)i] ^
(2) Find the cube root of - V2 + V2i.
We write:
- V2 + t V2 = 2(cos 135° + i cos 135°)
- 2[cos (135° + n360°) + i cos (135° + n360°)]
in which n is any integer. Hence:
(- V2 + i ^|2)^ = '?/2[cos (45° + nl20°) + i sin (45° + n 120°)]^
= V2(cos 45° + i sin 45°) for n = 0
= V2(cos 165° + i sin 165°) forn = 1
= V2(cos 285° -f i sin 285°) forn = 2
These are the three cube roots of the given complex number. For
n = 3 the first root is obtained a second time.
Exercises
Find all the indicated roots of the following:
1. (8 -h 8 V3 i)^.
2. ^27 (cos 75° - i sin 75°).
8. Vl25i.
378 ELEMENTARY MATHEMATICAL ANALYSIS I §224
4. ( - 2 + 2i)^.
5. (2 + 2i)^.
6. 32 ^.
7. v/512;
8. Find to four places one of the imaginary 7th roots of + 1«
Note: Cos 51** 25.7' + i sin 51** 25.7' = 0.6236 + 0.7818 i.
224. Inverse Functions. The exponent ( — 1) attached to a sym-
bol of operation signifies the "undoing" of the operation denoted
by the symbol of operation. The number of different operations
in mathematics is an even number; that is, for every operation we
define, we may, and usually do, define the operation that "un-
does" the given operation. Thus if we define addition, we at once
follow it by defining the undoing of addition, or svhtraction; if
we define multiplication, we follow it with the concept of
the undoing of multiplication, or division; if we define in-
volution, or the raising to powers; we also define the undoing of this
operation, namely evolviion, or the extraction of roots. The
second of each of these pairs of operations is called the inverse
of the first operation, and vice versa.
The exponent ( — 1) attached to any symbol of operation is defined
to mean the inverse of the operation called for by the symbol to
which it is attached. Thus 2-^ is not a doubler; the operation
called for is the "undoing" of doubling, or haMng. The symbol
log-^ X, read the "anti-logarithm of x" calls for the number of
which X is the logarithm. Thus, if the base be 10, log-^ 2 = 100,
log-i 3 = 1000, log-i 0.3010 = 2, log-^ 1 = 10, log-^ 0=1, etc.
Since "log~*" is the symbol of undoing the operation indicated
by "log," the double symbol (log"^ log) must leave the oper-
and unchanged. The operator that leaves an operand unchanged
is unity. Hence a double symbol like (log-^ log) can always be
replaced by 1; thus log~^ log 467 = 467; also log log-^ 467 = 467.
Likewise 3-1 -3 -1 = 1; (V3)-i(V3)-l = 1, etc.
An important use of the present notation is in the symbols sin'^x,
cos-i X, tan ~^ x, etc., used in §70. These are read "anti-sine of
X," "anti-cosine of a," etc., or "the angle whose sine is x" "the
angle whose cosine is x," etc. Thus sin-* (1 /2) = 30**, tan"^
§224] COMPLEX NUMBERS 379
= 45®, COS"* 0 = t/2, etc. Note that log"* x must be care-
fully distinguished from (log .u)"*, which means 1/log x\ simi-
larly, sin~* X must be distinguished from (sin x)"*. A notation
like log X"* is ambiguous, and should never be used.
If we write r = cos 0, y = log x,y = tan x, the same func-
tional relations may be expressed in the inverse notation by
6 = COS"* r, X = log"* y, X = tan-* y. Thus y = a', x = logay,
y = log«"* x» and y = expaX are four ways of expressing the
same relation between x and y.
Any relation expressed by means of the direct functions may also
be expressed in terms of the inverse functions. Thus we know:
log (xy) = log x + log y (1)
Let log a; = a, log y = h, then it follows that:
X = log"* a, y — log-* h
Hence (1) becomes:
log Gog~* CL log-* h) = a + &
or:
log"* a log"i h = log-i (a + h) (2)
Likewise consider:
sin {a '\- $) = sin a cos jS + cos a sin jS (3)
Let sin a = a and sin ^ = h
then: a = sin"* a, /3 = sin~i&
Also since sin a "= a
cos a = Vl — a^
Likewise:
cos /3 = Vl - &2
Hence (3) may be written:
sin (sin"ia + sin'^ h) = aVl - &« -f &Vl - a"
or:
sin"* a + sin"* & = sin"* (aVl ~ &« + b Vl - o*)^
Since there are many angles whose sine is equal to a given
number x, it is desirable to specify by definition which angle
is meant. The following conventions are therefore useful:
sin"* x means the angle between — 90° and -|- 90° whose
sine is x.
I
380 ELEMENTARY MATHEMATICAL ANALYSIS [§224
cos~^ X means the angle between 0® and 180** whose
cosine is x.
tan~^ X means the angle between — 90® and + 90° whose
tangent is x.
Exercises
1. Show that sin-i (1/2) + sin-i Vl/2 - 5ir/12. %
2. Show that sin~^ x -\- cos~^ x + cos~^ x — x/2.
3. Is there any difference between the graph of y = /(x) and the
graph of x = /~Ky)?
4. Prove that tan-^x -f tan-i (1/x) = ir/2.i
6. Find the value of x in the equation sin"^ x -|- sin"^ 2x = ir/3.
6. If /(x) = x\ find /-Hx).
Let y = /(x) = x^ Then x = /"Hy) = x/l/. Hence if /"Kv) =
Vy, then /~H^) = V X.
7. If/(x) = e«, find/-Ux).
8. What is the inverse of j{&) = 1 - d?
Let y = /(e), so that B = /"Ky), etc.
9. Show that the function
X + 1
y = 7
X — 1
is its own inverse.
iThe symbol (=) may here be interpreted as meaning "congruent to."
CHAPTER XII
LOCI
225. Parametric Equations. The equation of a plane curve is
ordinarily given by an equation in two variables, as has been amply
illustrated by numerous examples in the preceding chapters. It
is obvious that a curve might also be given by two equations con-
taining three variables, for if the third variable be eliminated from
the two equations, a single equation in two variables results.
When it is desirable to describe a locus by means of two equations
in three variables the equations are known as parametric equations,
as has already been explained in §74. Two of the variables usu-
ally belong to one of the common coordinate systems and the third
is an extra variable called the parameter. In applied science the
variable time frequently occurs as a parameter.
The parametric equations of the circle have already been written.
They are:
X = a cos 6 (1)
2^ = a sin ^
where the parameter 6 is the direction angle of the radius vector
to the point (x, y). Likewise the parametric equations of the
ellipse have been written:
X = a cos 0 (2)
y = b sind
and those of the hyperbola have been written:
X = a sec d (3)
y = b tan 6
In harmonic motion, the ellipse was seen to be the resultant of
the two S.H.M. in quadrature:
X = a cos o)t (4)
2/ = 6 sin o)t
Here the parameter t is time,
226. Problems in Loci. It is frequently required to find the
equation of a locus when a description of the process of its genera-
tion is given in words, or when a mechanism by means of which the
381
382 ELEMENTARY MATHEMATICAL AXALTSES lV»\
curve is generated is fully describecL There is only one way to
gain facility in obtaining the equations of curres thus described)
and that is by the solution of numerous {HoUems. Sometimtt
it is b^t to seek the parametric equations of the canre, bat
sometimes the ordinary pdar or Cartesian equation can be ob-
tained directly. The following problems are iBasiimtive:
(1) A straight line of constant length a + 6 moves with its ends
always sliding on two fixed lines at right ang^ to eadi other. Find
the equation of the curve described by any p^unt of tiie moving
Ime. (See §76.)
Fig. 153.— Generation of So-called *' Elliptic Motion."
In Fig. 153, let AB be the b'ne of fixed length, and let it so m( :e
that A remains on the a^-axis and B remains on the s^-axis. Let any
point of this line be P whose distance from A is 6 and whose distaije
from B is a. If the angle X^AB be called 6, then PD, the ordinate of
P, is:
y = 6 sin 0
and OD, the abscissa of P, is:
X = a cos 6
Therefore P describes an ellipse of semi-axes a and h.
(2) A circle roUs without slipping within a circle of twice the
diameter. Show that any point attached to the moving cirde'
describes an ellipse.
^ "Circle" is here used in the sense of a "disc" or oiroolar area and not in
the sense of a "circumference."
§226] LOCI 383
Draw the smaller rolling circle in any position within the larger
circle, and call the point of tangency T, as in Fig. 153. Since
the smaller circle is half the size of the larger circle, the smaller
circle always passes through 0, and the line joining the points of
intersection of the small circle with the coordinate axes is, for all
positions, a diameter, since the angle AOB is a right angle.
If we can prove that the arc AT — the arc HT for all positions
of T, then we shall have shown that as the small circle rolls from an
initial position with point of contact at H, the end A of the diameter
AB slides on the line OX. Since B lies on OF and since AB is of
fixed length, this proves by problem (1) that any point of the small
circle lying on the particular diameter AB describes an ellipse.
To prove that arc AT — arc HT, we have that the angle HOT
arc HT
is measured in radians by — 7yu~' The angle AO'T is measured in
arc A T
radians by qT^ ' Since Z AO'T = 2 Z HOT, we have
arc AT _ arc HT
O'A "" ^ OH
But, OH = 20' A. Hence arc AT = arc HT.
We can now prove that any other point of the rolling circle de-
scribes an ellipse. Let any other point be Pi. Through Pi draw
the diameter JO'K. The above reasoning applies directly, replacing
A by J and H by N.
It is easy to see that all points equidistant from the center such
as the points P, Pi, of the small circle, describe ellipses of the same
sP'^^i-axes a and 6, but with their major axes variously inclined to
c .
' v3) Determine the curve given by the parametric equations:
jj X = a cos 2<at (1)
y = a sin «^ (2)
To eliminate t, the first equation may be written
x = a (1 - 2 sin2 «0 (3)
y
From the second equation, sin cat = — Substituting for sin <at in
(3),
or,
2/' = ^ o ^ + o- (6)
384 ELEMENTARY MATHEMATICAL ANALYSIS [§226
This curve is the parabola y^ = mxy the special location of which the
student should describe.
(4) Construct a graph such that the increase in y varies directly
as X.
If y varied directly as x, then y would equal kXy where k is any
constant. In the given problem the increase in 2/ (and not y itself)
must vary in this manner. Let the initial value of y be represented
by yo. Then the gain or increase of 2/ is represented by y — yo.
Hence, by the problem:
y — yo = kx (1)
Since 2/0 is a constant, (1) is the equation of the straight line of slope
k and intercept on the 2/-axis = yo, which ordinarily would be written
in the form:
y ^ kx + yo
(5) Express the diagonal of a cube as a function of its edge, and
graph the function.
If the edge of the cube be x, its diagonal is Va;* + x^-\' x^ or a;V3.
If the diagonal be represented by y, we have y = V 3a;, which is *
straight line.
(6) A rectangle whose length is twice its breadth is to be in-
scribed in a circle of radius o. Express the area of this rectangle
in terms of the radius of the circle.
Let the rectangle be drawn in a circle whose equation is x* + 2/' = o'-
At a corner of the rectangle we have x = 2y, The area A of the
rectangle is 4x2/, or since a; = 2y, is 8y^. From the equation of the
circle we obtain Ay^ + y^ ^ a- or y^ = a^/5. Hence:
A = (8/5)a2
If A and a be graphed as Cartesian variables, the graph is a parabola.
(7) A rectangle is inscribed in a circle. Express its area as a
function of a half of one side.
Here, as above:
A — 4:xy = 4x'Sa^ — x^
The student should graph this ciu-ve, for which purpose a may be
put equal to unity. First draw the semicircle y = \ a* - x^
For x = 1/5 take one-fifth of the ordinate of this semicircle. For
X = 2/5 take two-fifths of the ordinate of the semicircle, and so on.
The curve through these points is y = xya^ — x\ from which
y = Ax^Ja^ — x^ can be had by proper change in the vertical imit of
measure.
§226] LOCI 385
Exercises
1. In polar coordinates, draw the curves:
r = 2 cos 6 r — 2 cos ^ + 1
r = 2 cos d — 1 r = 2 cos d + 3.
2. A curve (polar coordinates) passes through the point (1, 1).
(This means the point whose coordinates are one centimeter^ and one
radian,) Starting at this point, a point moves so that the radius
vector of the point is always equal to the vectorial angle. Sketch
the curve. Write the polar equation of the curve.
8. A point moves so that one of its polar coordinates, the radius
vector, varies directly as the other polar coordinate, the vectorial
angle. Write the polar equation of such a curve. Does the curve
go through the point (1, 1)?
4. A polar curve is generated by a point which starts at the point
(1, 2) and moves so that the increase in the radius vector always
equals the increase in the vectorial angle. Write the equation
of the curve.
6. A polar curve is generated by a point which starts at the point
(1, 2) and moves so that the increase in the radius vector varies directly
as the increase in the vectorial angle. Write the equation of the curve.
6. A ball is thrown from a tower with a horizontal velocity of 10
feet per second. It falls at the same time through a variable distance
given by 8 = 16. li*, where t is the elapsed time in seconds and s is
in feet. Find the equation of the curve traced by the ball.
7. The point P divides the line AB^ of fixed length, externally in
the ratio a :6, that is, so that PA/PB = a/b. If the UneAB move
with its end points always remaining on two fixed lines OX and OY
at right angles to each other, then P describes an ellipse of semi-axes
a and h,
8. If in the last problem the lines OX and OY are not at right
angles to each other, the point P still describes an ellipse.
9. A point moves so as to keep the ratio of its distances from
two fixed lines AC and BD constant. Prove that the locus consists
of four straight lines.
10. A sinusoidal wave of amplitude 6 cm. has a node at + 5 cm.
and an adjacent crest at + 8 cm. Write the equation of the curve.
11. The velocity of a simple wave is 10 meters per second. The
period is two seconds. Find the wave length and the frequency.
12. A polar curve passes through the point (1, 1) and the radius
vector varies inversely as the vectorial angle. Plot the curve and
write its equation. Consider especially the points where the vectotYbi.
386 ELEMENTARY MATHEMATICAL ANALYSIS I §227
angle becomes infinite and where it is zero. Sketch the same func-
tion in rectangular coordinates.
13. Rectangles are inscribed in a circle of radius r. Express by
means of an equation and plot : (a) the area, and (&) the perimeter
of the rectangles as a function of the breadth.
14. Right triangles are constructed on a line of given length h
as hypotenuse. Express and plot: (a) the area, and (b) the per-
imeter as a function of the length of one leg.
15. A conical tent is to be constructed of given volume, V, Express
and graph the amount of canvas required as a function of the radius
of the base.
16. A closed cylindrical tin can is to be constructed of given vo lume,
V. Plot the amount of tin required as a function of the radius of the
can.
17. A rectangular water-tank lined with lead is to be constructed
to hold 108 cubic feet. It has a square base and open top. Plot
the amount of lead required as a function of the side of the base.
18. An open cylindrical water-tank is to be made of given volume,
V. The cost of the sides per square foot is two^-thirds the cost of
the bottom per square foot. Plot the cost as a function of the
diameter.
19. An open box is to be made from a sheet of pasteboard 12 inches
square, by cutting equal squares from the four corners and bending
up the sides. Plot the volume as a function of the side of one of the
squares cut out.
20. The illumination of a plane surface by a luminous point varies
directly as the cosine of the angle of incidence, and inversely as the
square of the perpend'cular distance from the surface. Plot the
illumination of a point on the floor 10 feet from the wall, as a func-
tion of the height of a gas burner on the wall.
21. Using the vertical distances between corresponding points on
the curves y = sin t and y = — sin ^ as ordinates and the vertical
distances between corresponding points of y = 2t and y = <* as
abscissas, find the equation of the resulting curve.
227. Loci Defined by Focal Radii. A number of important
curves are defined by imposing conditions upon the distances of
any point of the locus from two fixed points, called foci.
(1) A point moves so that the product of its distances from two
fixed points is constant. Find the equation of the path of the particle.
Let the two fixed points Fi and F2, Fig. 154, be taken on the a^-axis
the distance a each side of the origin. Call the distances of P from
27] LOCI 387
e fixed points n and rj. Then the variables ri and r2 in terms of z
d y are:
ri* = 2/* + (a; - a)2
rj* = 2/* + (x + a)» ^^^
3nce:
riVs* = [2/2 + (x - a)2][y2 + (x + a)2] (2)
klling the constant value of rir2 = c*, we have as the Cartesian
uation of the locus:
[2/2 + (x - a)2][y2 + (x + a)2] = c4 (3)
Fig. 154. — The Lemniscate.
aich may be written:
(yi 4- a;2 4. ct2)2 - 4a2 rc2 = c* (4)
(a;2 + 2/^)* + 2a2x* + 20*2/2 + a* - 4a2a;2 = c* (5)
(a;2 + 2/*)^ = 2a2(x2 - y^) + c* - a^ (6)
c = a the curve* is called the lemniscate, and the Cartesian equa-
}n reduces to:
(x2 + 2/2)2 = 2a2(x2 - 2/2) (7)
For other values of c the curves are known as the Cassinian ovals.
Tien c <, a the curve consists of two separate ovals surrounding the
ci, and f or c > a there is but a single oval. The curves are shown
Fig. 157. These curves give the form of the equipotential surfaces
a field around two positively or two negatively charged parallel
ires. To construct the curves proceed as follows: In Fig. 155
b the circle have a radius c and in 156 let the circle have the
ameter c.
In Fig. 155 we can use the theorem: "// from a point without a
rde a tangent and secant he drawn j the tangent is a mean proportional
the entire secant and the part without the circle.** In Fig. 156 we can
(6 the theorem: "// from the vertex of the right angle of any right
langle a perperuUcviar Ife dropped upon the hypotenuse, then eiih^
388 ELEMENTARY MATHEMATICAL ANALYSIS [§227
leg V the triangle is a mean proportional between the hypotenuse and
the adjacent segment.^ ^
Then in either Pig. 155
or 156, SPi X Spi = c» and
likewise SPt X Spt = c*,
etc. Therefore SPo, SPi,
SPif . . . are the values
of ra that correspond to ri =
Spof Spi, Spi, . . ., re-
spectively, and the ovals
may be constructed by the
intersection of arcs described
about F\ and Ft as centers,
using pairs of these values
as radii.
(2) Construct the curve
such that the ratio of the
distances of any point of
the curve from two fixed
Fig. 155. — Construction of the Constant
Products SPn X Spn = c*.
points is constant.
Let the two fixed points be A
and B, Fig. 158; let the constant
ratio of the distances of any
point of the curve from the two
fixed points be ri/r2 = m/n.
To find one point of the locus,
draw circles from A and B as
centers whose radii are in the
ratio m/n. Let these circles in-
tersect at the point P. At P
bisect the angle between PA and
PB internally and externally by
the lines PM and PN respectively.
The line AB \b then divided at M
internally in the ratio MA/ MB
— m/n and externally at N in
the ratio NA/NB = m/n, because
the bisectors of any angle of a
triangle divide the base into
segments proportional to the adjacent sides. Since the external
and internal bisectors of any angle must be at right angles to each
other, PM is perpendicular to P'N for any position of P. Hence
Fig. 156. — Construction of the Con-
stant Products SPn X 5p» = c'.
the locus of P is a circle, since it is the vertex of a right trian^e
described on the fixed hypotenuse MN.
Fro. 157. — The Lemnisoate and the Ci
Fio. 168. — Construction of the Curva n/r
the circle JWPJV.
If a large number of circles be drawn for different values of e, and
if simOar circles be described about B, then these series of circles ore
known as the dipolar circles. See Fig, 159. In physics it is found
390 ELEMENTARY MATHEMATICAL ANALYSIS [!228
that these circles are the equipotential lines about two parallel wires
perpendicular to the plane of the paper at A and B and carryinj
electricity of opposite sign.
Fia. 159.— The Dipolar
ri/r, = e
Exercises
1. Draw the loous satisfying the condition that the ratio of the
distances of any point from two fixed points ten units apart is 2/3.
2. IJraw the two circles which divide a line of length 14 internsll)'
and externally in the ratio 3/4.
228. The Cycloid. The cycloid is the curve traced by a point
on the circumference of a circle, called the generating circle, {
Definition ot the Cycloid.
which rolls without slipping on a fixed line called the base. To
find the equation of the cycloid, let OA, Fig. 160, be theba8e,Pthe
tracing point of the generating circle in any one position, and 6 the
angle between the radius SP and the line SH to the point of con-
tact with the base. SinceP was at 0 when the circle began to roll,
OH = 0$
§229] LOCI 391
if a be the radius of the generating circle. Since x = OB and
y = PD, we have:
x = OH - SP sin e = aid -sine) (1)
y = HS - SPcosB = a(l - cos d) (2)
These are the parametric equations of the curve. For most
purposes these are more useful than the Cartesian equation.
It is readily seen from the definition of the curve, that the locus
consists of an unlimited number of loops above the x-axis, with
points of contact with the x-axis at intervals of 27ra (the circum-
ference of the generating circle) and with maximum points at
X = ira, 37ra, etc.
From the second of the parametric equations we may write:
1 - cos^ = y/a (3)
The expression (1 — cos 0) is frequently called the versed sine of
0, and is abbreviated vers 6, Hence we have:
0 = vers"^ y /a (4)
Also from (3) : cos ^ = (a — y) /a
Hence: sin^ = -v/1 - cos^^ = ~V2ay - y^ (5)
Cv
whence substituting (4) and (5) in the first of the parametric equa-
tions we have:
X = a vers -1 (y/a) — V2ay - y^ (6)
which is the Cartesian equation of the cycloid, with the origin 0
at a cusp of the curve.
229. Graphical Construction of the Cycloid. To construct the
cycloid. Fig. 161, draw a circle of radius 1.15 inches and divide the
circumference into thirty-six equal parts. Draw horizontal lines
through each point of division exactly as in the construction of
the sinusoid, Fig. 59. Lay off uniform intervals of 1 /5 inch each
on the X-axis, marked 1, 2, 3, . . . Then from the point of
division of the circle pi lay off the distance 01 to the right.
Prom p2 lay off 02 to the right, from ps lay off 03 to the right,
etc. The points thus determined lie on the cycloid. The number
of divisions of the circumference is of course immaterial except
that an even number of division is convenient, and except that
i
392 ELEMENTARY MATHEMATICAL ANALYSIS l|230
tha divisions laid off on the base OA must be the same lei^th ts
the area laid off on the circle.
Note that by the process of construction above, the vertical
distances from OX to points on the curve are proportional to
(1 — cos 9} and that the horizontal distances from OY to points
on the curve are proportional to (5 — sin 9.)
of the Cycloid.
The analogy of the cycloid to the sine curve is brought out by
Fig. 162. A set of horizontal lines are drawn as before and also a
sequence of semicircles spaced at horizontal intervals equal to
the intervals of arc on the circle. The plane is thus divided
into a large number of small quadrilaterals having two sides
straight and two sides curved. Starting at 0 and sketching the
Fig. 162.— Analogy of the Cycloid
di^onals of successive cornering quadrilaterals the cycloid is
traced. If, instead of the sequence of circles, unifornoly spaced
vertical straight lines had been used, the sinusoid would have been
drawn. The sinusoid on that account is frequently called tbe
" com'panion to the cycloid."
230. Epicycloids and Hypocycloids. The curve traced by a
point attached to the circumference of a circle which rolls without
§230] LOCI 393
slipping on the circumference of a fixed circle is called an epi-
cycloid or a hjrpocycloid according as the rolling circle touches
the outside or inside of the fixed circle. If the tracing point
is not on the circumference of the rolling circle but on a radius
or radius produced, the curve it describes is called a trochoid if
the circle rolls upon a straight Hne, or an epitrochoid or a hjrpo-
trochoid if the circle rolls upon another circle. These curves will
be discussed in the calculus.
Exercises
1. Construct a cycloid by dividing a generating circle of radius
1.15 inches into twenty -four equal arcs and dividing the base into
intervals 3/10 inch each.
2. Compare the cycloid of length 2ir and height 1 with a semi-
ellipse of length 2t and height 1.
3. Write the parametric equations of a cycloid for origin C,
Fig. 160.
4. Write the parametric equations of a cycloid for origin B, Fig. 160.
5. Find the coordinates of the points of intersections of the cycloid
with the korizontal line through the center of the generating circle.
6. Show that the top of a rolling wheel travels through space
twice as fast as the hub of the wheel.
7. By experiment or otherwise show that the tangent to the cycloid
at any point always passes through the highest point of the generating
circle in the instantaneous position of the circle pertaining to that
point.
Exercises for Review
1. Simplify tlie product:
(a; - 2 - ^^)ix- 2 - i ^J^)(x - 2 + V3)(x - 2 + Vi3).
2. Express in the form c cos (o — b) the binomial:
30 cos 0 + 40 sin a.
3. Find tan 6 by ixeans of the formula for tan (A + B), if e =
tan-i 1/2 + tan-i 1/3.
4. Find am 6 ii $ = ^m-^ 1/5 + sin-^ 1/7.
5. Find the equationof a circle whose center is the origin and
which passes through the point 14, 17.
6. The first of the following tests was m,ade in 1875 with the
automatic air brake on a ttain composed of cars weighing 30,000
394 ELEMENTARY MATHEMATICAL ANALYSIS [§230
pounds. The second in 1907 with the "LN'' brake on a train
composed of cars weighing 84,000 pounds. Find by use of loga-
rithmic paper the equation connecting the speed and the distance
run after application of the brakes.
Distance run after application of brake Corresponding speed
1875
1907
0 feet
Ofeet
57 . 3 miles
56 . 0 miles
per hour
50 feet
70 feet
55 . 0 miles
(C
200 feet
220 feet
50.0 miles
it
350 feet
360 feet
45 . 0 miles
tl
500 feet
500 feet
40.0 miles
tl
820 feet
770 feet
25 . 0 miles
It
950 feet
880 feet
15 . 0 miles
u
980 feet
922 feet
10.0 miles
n
1,010 feet
940 feet
5 . 0 miles
11
1,020 feet
954 feet
0.0 miles
tl
7. Discuss the curve:
X
= aJO
y
= a(l — cos e).
8. Graph on
polar
paper:
P^
= a* cos 2e.
9. A fixed point located on one leg of a carpenters "square''
traces a curve as the square is moved, the two arms of the square,
however, always passing through two fixed points A »nd B. Find
the equation of the curve.
10. Find the parametric equations of the oval traced by a point
attached to the connecting rod of a steam engine.
11. The length of the shadow cast by a tower varies inversely as
the tangent of the angle of elevation of the sun. Graph the length
of the shadow for various elevations of the sun.
12. From your knowledge of the equations of ^he straight line and
circle, graph:
y = ax -\- Vo* — x^.
(See Shearing Motion, §37.)
13. In the same manner, sketch:
y = a -{- X + "yJa^ — x*.
14. Graph the curve:
y = a/x + bxl
Has this curve a minimum value for all positive values of a and 6?
§230]
LOCI
395
IB. Find by use of logarithmic paper the equations of the curveB
of Fig. 163. Theae curves give the amounts in cents per kilowatt-
hour that must be added to price of electric power to meet fixed
charges of certain given annual anxouuts for various load factors.
16. The angle of elevation of a mountain top seen from a certain
point ii 29° 4'. The angle of depression of the image of the mountain
:
lXy\
_A\
\\
i: \\ \
3„ \ \ S
\ v,W T
~~^^ "^^"-'^iu. ,,
CmtBpcr K,W.H<nir
top seen in a lake 230 feet below the observer is 31° 20'. Find the
height and horizontal distance of the mountain top, and produce a
single formula for the solution of the problem.
17. Find the points of intersection of the curves:
i> + 3/' = 4
y' = ix.
18. Solve 1102-' + 1 = 2Ii-'.
19. Solve 3(x - 7)(_x - l)(i - 2) = (i + 2)(i - 7)(i + 3).
50. Solve sin X cos z = 1/4.
51. State the remainder theorem and illustrate by an example.
306 ELEMENTARY MATHEMATICAL ANALYSIS l|230
ft. I^d the compound interest on SIOOO for twenty-five jt»a at
5 per ceat. S/unp htm to aolve by means of pragreeeMHis.
S3. The curve y* ^ x* appenn in which quadnuata? Inwliat
quadrants is y* = t*T Compare the curree xV ** 1 o°<l 'V ~ 1-
U. Which trigoDometric functions of * inereMC M t ina«aaea in
the first quadrant? Which decrease?
U. GK-en sin 30° = 1/2, cos 45° > Vl/2. Find the foUowing:
sin 150°, oos 135°, sin 225°, cos 300°, ain 330°, ain <- 30^.
M. Which ia greater, t»n 7° or sin 7°, and whv? Whidi is Rreattf,
sec 5° w CSC 5", aod why?
ST. Sketch the curves:
(a) X* + 4jc + y' - 6f 12
ft) *' + 4*' + 6* - 21.
IS. n«in the graph of y = z^ obtftin the graph of 4v — z* and aS
t-ixK
1 ! 1 U,
N 1 ■
' y^
\i
: / 1
Is
/ 1
\
/ ' V\ ' 1
/ : ift
/ ' \ ■
/ ' \\
/ \'-
A^ :A' '■
/\ M \
Fig. 164.— Trajrctorr of i
cnnui .^rmrBuUel for ■ Ranee of 1000
Meiers.
M. Girm, cos 0 = 25: find sin 9. tan ( and cot (.
Sft. nnd the equations of the six straight lines determined by the
inlnseciions of:
X' -i- *' = 25
x' - jr' - 7.
31. In Fig. 164 the full drawn cmve is the trajectory ot the pro-
jeetile of a German Army buUet fcr a range of 1000 noetas. The
dotted curve is the tbeoreiiral tntjevtocy that would have been de-
scribed by the bullet if there had be«i no ak renrtaneo. The dotted
§230]
LOCI
397
curve is a parabola (of second degree). Find its equation, taking
the necessary numerical data from the diagram.
32. Find the maximum value of p if p = 3 cos 0 — 4 sin 0.
33. Find the maximum value of y ii y =^ V3 cos x — sin x, and
find the value of x for which y is a maximum.
34. In Fig. 165 let ABCO be a square of side a. Show that
for all positions c^QN, CM >^|k^ = a'f and hence show how to use
this diagram in iSrconstructW of a lemniscate.
B N N N N N
Fig. 165. — Construction of a Constant Product CM X AN = AB*.
CHAPTER XIII
THE CONIC SECTIONS
i cKn
231. The Focal Radii of the
major and minor circles of radii cKad h respWfcively, as in Fig.
se. DrQ^^ny ellipse with
p^Ri\
166. Draw tangents, IT and KK', to the minor circle at the
extremities of the minor axes and complete the rectangle Il'KK!.
The points Fi and F2, in which IK and FK' cut the major axis, are
Fig. 166. — Properties of the Ellipse.
called the foci of the ellipse. From any point on the ellipse draw
the focal radii PFi = ri and PF2 = r2, as shown in the figure.
Represent the distance OFi or its equal OF 2 by c. Then it follows
from the triangle OIFi that:
a2 = b2 + c^
(1)
This is one of the fundamental relations between the constants of
the ellipse.
From the triangles PFiD and PF2D there follows:
ri2 = (c - xY + 2/2 (2)
n^ = (c + xY + 2/2 (3)
398
§231] THE CONIC SECTIONS 399
But the equation of the ellipse is
b
a
or
y «= — Va^ -
X
y'^^.ia^-x^) (4)
Substituting this value of y^ in (2)
r? = c2 - 2ox + x^ + -^(a2 - x^) (5)
= c^ -2cx + x^ + b^ 1 x2
= a2 - 2cx + x^ [l -— 2] (6)
or by (1)
Substituting
we obtain
a2 " a2 " a2
c^x^
n^ = a2 - 2caj + —2- (7)
(8)
Therefore:
-["-?]
r,-.-f (9)
Likewise, from (3), by exactly the same substitutions, there
follows:
r^ = a + f (10)
From (9) and (10) by addition:
ri+r2 = 2a (11)
Hence in any ellipse the sum of the focal radii is constant and equal
to the major axis.
The converse of this theorem, namely, if the suni of the focal
radii of any locus is constant, the curve is an ellipse, can readily
be proved. It is merely necessary to substitute the values of ri
and r2 from (2) and (3) in equation (11), and simplify the resulting
equation in x and y; or first square (11) and then substitute ri and
Ti from (2) and (3). There results an equation of the second degree
lacking the term xy and having the terms contaiv^ing gc^ and y^ both
d
400 ELEMENTARY MATHEMATICAL ANALYSIS [§231 ;
present and with coefficients oj like signs. By §77, such an equation
represents an ellipse.
Hence the ellipse might have been defined as the locus of
a pointy the sum of the distances of which from two fixed points w
constant.
An ellipse can be drawn by attaching a string of length 2a by
pins at the points Fi and F2 and tracing the curve by a pencil so
guided that the string is always kept taut. Or better, take a
string of length 2a + 2c and form a loop enclosing the two pins;
the entire curve can then be drawn with one sweep of the
pencil.
The focal radii may also be evaluated in terms of the parametric
or eccentric angle 6, The student may regard the following
demonstration of the truth of equation (11) as simpler than that
given above:
Since x = a cos dy and y = 6 sin ^
ri2 = 52 sin2 ^ + (c - a cos ^)2 (12)
= 62 sin* ^ + c* - 2ac cos 6 + a^ cos* e (13)
To put the right side in the form of a perfect square, write
6* = a* — c*. Then:
ri* = a* sin* ^ — c* sin* ^ + c* — 2ac cos ^ + a* cos* B
= a* - 2ac cos ^ + c* cos* B (14)
Whence:
ii = a — c cos B (15)
Likewise:
12 = a + c cos B (16)
Whence:
^1 + ^2 = 2a
232. The Eccentricity. The ratio cja measures, in terms of a as
unit, the distance of either focus from the center of the ellipse.
This ratio" is called the eccentricity of the ellipse. In the triangle
IFiOy the ratio c /a is the cosine of the angle F\OIy represented in
what follows by j8. Calling the eccentricity c, we have:
e = c /a = cos 0 (1)
The ellipse i? made from the major circle by contracting its ordi-
§233] THE CONIC SECTIONS 401
nates in the ratio m = b/aj or by orthographic projection of the
circle through the angle of projection:
a = cos~^ b /a
Hence, as companion to (1) we may write:
m = b/a = cos a = sin j8 (2)
233. The Ratio Definition of the Ellipse. In Fig. 166, let the
tangents to the major circle at I and /' be drawn. Draw a
perpendicular to the major axis produced at the points cut by
these tangents. These two lines are called the directrices of the
ellipse.
We shall prove that the ratio PFi /PH (or PF2 IPW) is constant
for all positions of P. From §231, equation (9) or (15),
ri = a — c cos 6 (1)
From the figure, ON = a sec ION = a sec P (2)
But:
Hence:
But
Therefore
Hence from (1) and (4) :
cos P = c/a
ON = a2 /c (3)
PH = ON - X
PH = a^lc - a cose (4)
^ _ per /PIT _ a-ccose
PH " ^^1/^^ - a^/c - a co§ 6
__ c a_— c cos_^
^ a a — c cos d
or
PFi/PH = c/a = e = cos /3 (5)
A similar proof holda for the other focus and directrix. Thus,
for any point on the ellipse the distance to a focus bears a fixed
ratio to the distance to the corresponding directrix. From (5),
the ratio is seen to be less than unity.
Assuming the converse of the above, the ellipse might have been
defined as follows: The ellipse is the locus of a point whose distance
from a fixed point (called the focus) is in a constant ratio less than
unity to its distance from a fixed line (called the directrix).
26
402 ELEMENTARY MATHEMATICAL ANALYSIS [§233
If, in any ellipse, c = 0, it follows that h must equal a and the el-
lipse reduces to a circle. If c is nearly equal to a, then from the
equation:
a2 = 6* + c»
it follows that the semi-minor axis h must be very small. That is,
for an eccentricity nearly unity the ellipse is very slender.
If the sun be regarded as fixed in space, then the orbits of the
planets are ellipses, with the sun at one focus. (Th^'s is '^Kepler^s
First Law J*) The eccentricity of the earth's orbit is 0.017. The
orbit of Mercury has an eccentricity of about 0.2, which is greater
than that of any other planet.
Exercises
Find the eccentricities and the distance from center to foci of the
following ellipses :
1. xy9 4- 2/V4 = 1- _ 4. 2y = Vl - x\
2. y = (2/3) V36 - x\ 6. 9x^ + IQy^ = 14.
3. 25x2 4- 41/2 = 100. 6. 2x^ + Syi = l.
Find the equation of the ellipse from the following data:
7. e = 1/2, a = 4. Draw this ellipse.
8. c = 4, a = 5.
9. ri = 6 - 2x/3, rj = 6 + 2a;/3.
10. ri = 5 — 4 cos dy r2 =5+4 cos 6.
Solve the following exercises :
11. Find the eccentricity of the ellipse made by the orthographic
projection of the circle x^ -\- y^ = a^ through the angle 60**.
12. The angle of projection of a circle x"^ + y^ = a' by which an
ellipse is formed is a. Show that the eccentricity of the ellipse is
sin a.
13. A circular cylinder of radius 5 is cut by a plane making an
angle 30° with the axis. Find the eccentricity of the elliptic section.
14. If the greatest distance of the earth from the sun is 92,-
500,000 miles, find its least distance. (Eccentricity of earth's orbit
= 0.017.)
15. In the ellipse xV25 + t/V16 = 1, find the distance between
the two directrices.
16. Write the equation of the ellipse whose foci are (2,0), ( — 2,0),
and whose directrices are a; = 5 and x = — 5.
§234} THE CONIC SECTIONS 403
17. Prove equation 11 §231 by transposing one radical in:
squaring, and reducing to an identity.
234. The Latus Rectum. The double ordinate through the
focus is called the latus rectum of the ellipse. The value of the
semi-latus rectum is readily formed from the equation
by substituting c for x. If I represents the corresponding value
of y,
I = {bla)4a^ -c^ = b^/a (1)
since a^ — c^ = b^. Hence the entire latus rectum is represented
by:
21 = ^-^ (2)
a
Eiquation (1) may also be written:
Z = 6Vl - c^/a^
= 6Vl - e^ (3)
In Fig. 166 the distances AF, AN, ON, OB, OF, FN may
readily be expressed in' terms of a and e as follows in equations (4)
to (10). The addition of the formulas (11), (12), (13) brings into
a single table all the important formulas of the ellipse.
AFi = a — c = a(l — e)
(4)
^^.lF..^a(l-_ei
e e
(5)
0N= osec/3 =*
e
(6)
e = cos /3
(7)
OB = b = a sin /3 = a Vl - e^
(8)
OFi = c = ae
(9)
FiN = OiV - c = a(l - e') /e
(10)
1 = bVa = a(l - e'')
(11)
ri = a — ex = a — X cos j8
(12)
r2 = a + ex = a + x cos /3
<.\3.\
404 ELEMENTARY MATHEMATICAL ANALYSIS [§235
Exercises
1. Find the value in miles of OF for the case of the earth's orbit.
2. Find the value of /3 for the earth's orbit. (Use the S functions
of the logarithmic table.)
3. In the ellipse y = (2/3) V36 - x« find the length of the latus
rectum and the value of e.
4. The eccentricity of an ellipse is 3/5 and the latus rectum \& 9
units. Find the equation of the ellipse.
6. In (o) x^ -h 4y« = 4 and (6) 2x« + 3y« = 6 find the latus
rectum, the eccentricity and the distances ON and AF,
6. Determine the eccentricities of the ellipses,
(o) y« - 4x - (l/2)a;« (6) y« = 4x - 2a:'.
7. Find the equation of an ellipse whose minor axis is 10 units
and in which the distance between the foci is 10.
8. Find the equation of an ellipse whose latus rectum is 2 unit« and
minor axis is 2.
9. The distance from the focus to the directrix is 16 units. An
ellipse divides the distance between focus and directrix externally
and internally in the ratio 3/5. Find the equation of the ellipse.
10. The axes of an ellipse are known. Show how to locate
the foci.
11. In an ellipse a = 25 feet, e — 0.96. What are the values of
r and 6?
12. For a certain comet (Tempel's) the semi-major axis of the
elliptic orbit is 3.5, and c = 1.4 on a certain scale. For another
comet (Enke*s) a = 2.2, e — 0.85. Sketch the curves, taking
3 cm. or 1 inch as unit of measure.
IS. If Z = 7.2, € = 0.6, find c, a, 6.
14. An ellipse, with center at the origin and nugor axis coinciding
with the X-axis, passes through the points (10, 5) (6, 13). Find
the axes of the ellipse.
23& Focal Radii of the Hyperbola. Construct a hyperbola from
auxiliary circles of radii a and 6, then the transverse axis of the
hyperbola is 2a and the conjugate axis is 26. Unlike the case of
the ellipse* 6 may be either j^^ater or less than a. As previously
explained, the asymptotes are the extensions of the diagonals of the
rectangles BTAO, BT'A'O, From the points I, F, in which the
asymptotes cut the o-circle, draw tangents to the a<-cirde. The
points Fu Ft in which the tangents cut the axis of the h3rptf-
bola are called the loci. See Fig. 167.
THE CONIC SECTIONS
405
The difltance OFi or OFs is represented by the letter c. Then,
since the triangles FJO and OAT are equal, FJ must equal b, so
that we have the fundamental relation between the constants of
the hyperbola:
a' + b' = c= (I)
Fig, 167. — Properties of the Hyperbola.
From ai^ point on either branch of the hyperbola draw the focal
radii PFi and PFi, represented by u and n respectively. Then
from the figure:
ri' = (I - c)= + y' (2)
But from the equation of the hyperbola;
r.'- (i-e)> + 6'(i=-i.')/i.'
(4)
= {a*x* - 2o'M + a^c* + 6V -
-a'6')/«'
(5)
= (cV - 2a»CT + a') ja^
(6)
- (ra - o')'/o'
(7)
Hence: n = (c/a)i - a
(8)
In like manner it may be shown that
r, -(c/o)i + «
(9)
406 ELEMENTARY MATHEMATICAL ANALYSIS I§236
Hence from (8) and (9) it follows:
r2 - ri = 2a (lO)
Hence in any hyperbola, the difference between the distances of any
point on it from the foci is constant and eqxwl to the transverse am.
The above relation may be derived in terms of the parametric
angle B. Thus, since in any hyperbola x = a sec 6 and y =b tan^,
ri2 = 62 tan2 ^ + (a sec ^ - cY
= 62 tan* ^ + a* sec^ ^ - 2ac sec ^ + c*
To put the right-hand side in the form of a perfect square, write
6« = c* - a2. Then
r)} — c2 sec* 6 — 2ac sec 6 + a^
Therefore: ri = c sec ^ — a (11)
and: r2 = c sec ^ + a (12)
236. The Ratio Definition of the Hyperbola. Through the
points of intersection of the a-circle with the asymptotes, draw
IKf rK\ These lines are called the directrices of the hyperbola.
It will now be proved that the ratio of the distance of any point of
the hyperbola from a focus to its distance from the corresponding
directrix is constant. Adopt the notation:
c a = sec fi — e * (1)
Then from the figure:
PFy ;PH = ri .(j- - OX) = ri /(a sec ^ - a cos 0) (2)
Substituting ri from (11) above:
PFi ;PH = (r sec 6 - a) /(a sec ^ - a cos P) (3)
= {a sec /3 sec 6 — a) '{a sec ^ — a cos j8) (4)
sec ^J sec ^ - 1 . , ,^.
= ^ ^ = sec^ ^ e = c^a (5)
sec p — cos 6 ' ^ '
which proves the theorem. The constant ratio e is called the
eccentricity of the h>T>erbola, and, a? shown by (5), is always greater
than unity.
Assuming the converse of the above, it is obvious that the hyper-
bola might have been defined as foUows: The hyperbola is tiie
loru^ of a point who$e distance from a fixed point {called the focus)
1$ in a constant ratio greater than unity to its distance from a fixed
line {called the directrix^.
§237] THE CONIC SECTIONS 407
237. The Latus Rectum. The double ordinate through the focus
is called the latus rectum of the hyperbola. The value of the
semi-latus rectum is readily found from the equation:
y = (b/a) Vx2 - a2
by substituting c for x. If I represents the corresponding value of
y-
I = (h/a) Vc^"^^ = h^/a (1)
Hence the entire latus rectum is represented by:
21 = 2b Va (2)
Equation (1) may also be written:
I
-'VF
I = b ^e^ -1 (3)
In Fig. 167 the distances AFi, AN, ON, OB, OFi, FiN may
readily be expressed in terms of a and e, as follows in equations
(4) to (8). Collecting in a single table the other important for-
mulas for the hyperbola, we have:
AFi = c - a = a(e - 1) (4)
AN = AFi/e = a(e - l)/e (5)
ON = acosiS = a/e (6)
e = sec j8 (7)
0B=b = atani8 = a Ve^ - 1 (8)
OFi = c = ae
FiN = c - OiV = ae - a/e = a(e2 - 1) /e (9)
1 = bVa = b Ve2 - 1 = a(e2 - 1) (10)
Ti = ex — a = X sec j3 — a (11)
r2 = ex + a = X sec jS + a (12)
The important properties of the hyperbola are quite similar
to those of the ellipse. It is a good plan to compare them in
parallel columns.
408 ELEMENTARY MATHEMATICAL ANALYSIS [§237
Ellipse Hyperbola
1. Definition of Foci and Focal
Radii
2. a2 = 6^ + c« 2. a« + 6« = c*
3. ri + r2 = 2a 3. r2 — ri = 2a
1. Definition of Foci and Focal
Radii
. . c c
4. Eccentricity, e = — = cos /3 4. Eccentricity, e = — = sec /S
5. Definition of Directrices 5. Definition of Directrices
PFi PFi
6. The Ratio Property, p^ = e ; 6. The Ratio Property, pfv - «
262 i 262
7. The Latus Rectum = — 7. The Latus Rectum = —
a a
Exercises
1. Find the eccentricity and axes of x^/4. — y^/16 — 1.
2. Find the eccentricity and latus rectum of the hyperbola con-
jugate to the hyperbola of the proceeding exercise.
3. A hyperbola has a transverse axis equal to 14 units and its
asymptotes make an angle of 30° with he x-axis. Find the equation
of the hyperbola.
4. Find the latus rectum and locate the foci and asymptotes of
4x2 _ 36^2 = 144,
5. Locate the directrices of the hyperbola of the preceding exercise.
6. In Fig. 167 show that rz = GK' and ri = GI and hence that
r2 — ri — IK' or 2a.
7. Find the equation of the hyperbola having latus rectum 4/3
and a = 26.
8. The eccentricity of a hyperbola is 3/2 and its directrices are
the lines x = 2 and x = — 2. Write the equation and draw the
curve with its asymptotes, o-circle, 6-cirele, and foci.
9. Find the eccentricity and axes of 3x2 — Sy^ = — 45.
10. Find the eccentricity of the rectangular hyperbola.
11. Describe the shape of a hyperbola whose eccentricity is nearly
unity. Describe the form of a hyperbola if the eccentricity is very
large.
12. Describe the hyperbola if b/a = 2, but a very small.
13. Write the equation of the hyperbola if (1) c = 5, a = 3; (2)
c = 25, a = 24; (3) c = 17, 6=8.
14. Describe the locus:
{X + 1)2/7 - (t/ - 3)2/5 = 1.
§238]
THE CONIC SECTIONS
409
15. Find the equation of the hyperbola whose center is at the
origin and whose transverse axis coincides with the x-axis and which
passes through the points (4.5, — 1), (6, 8).
238. The Polar Equation of the Ellipse and Hyperbola. In
mechanics and astronomy the polar equations of the ellipse and
hyperbola are often required with the pole or origin at the right focus
in the case of the ellipse and at the left focus in the case of the hyper-
bola. In these positions the radius vector of any point on the
Fig. 168. — Polar Equation of a Conic.
curve will increase with the vectorial angle when 6 < 180°. To
obtain the polar equation of the ellipse and hyperbola, make use of
the ratio property of the curves, namely: that the locus of a point
whose distances from a fixed point (called the focus) is in a constant
ratio e to its distances from a fixed line (called the directrix), is an
ellipse if 6 < 1 or a hyperbola if e > 1. In Fig. 168 letF be the
fixed point or focus, IK the fixed line or directrix, P the moving
point, and FL = I the semi-latus rectum. Then the problem is
to find the polar equation from the equation
PH = e (!)
410 ELEMENTARY MATHEMATICAL ANALYSIS [§238
If e is left unrestricted in value, the work and the result wiU apply
equally well either to the ellipse or to the hyperbola.
When the point P occupies the position L, Fig. 168, we hive
PF = I and PH = FN, whence from (1)
FN = ^ (2)
e
Take the origin of polar coordinates at F, and also take FP = p
and the angle AFP = 6. Then:
PH = FN - FD (3)
FD = p cos e (4)
Hence from (2), (3) and (4)
PH = - -p cos e (5)
e
Substituting these values of FP and PH in (1), clearing of frac-
tions and solving for p, we obtain
" = l + eco80 (^^
which is the equation required.
When e < 1, (6) is the equation of an ellipse with pole at the
right-hand focus. When e > 1, (6) is the equation of a hyperbola
with the pole at the left focus; in both cases the origin has been so
selected that p increases as 6 increases.
Note: Calling FN (Fig. 168) = n, equation (1) above may be
written in rectangular coordinates:
n — X
or,
^^^ ^y'^e (7)
x2 + 2/2 = eKn - xy (8)
which may be reduced to the form:
r ^ 1 - eV ^ 1 - e^ - (1 - 6^)2 W
By §§77 and 87 this represents an ellipse if e < 1 or a hjrper-
bola if e > 1. Thus starting with the ratio definition (7) we have
proved that the curve is an ellipse or a hyperbola ; that is, we have
proved the statements in italics at bottom of pp. 401 and 406.
§239] THE CONIC SECTIONS 411
Exercises
1. braph on polar paper, form Af3, the curve p = .. , ^
for e = 2; also for e = 1/2, also for e = 1.
It will be sufficient in graphing to use 6 = 0% 30°, 60°, 90°, 120°,
150°, 180°, 210°, . . . , 360°.
2. Write the polar equation of an ellipse whose semi-latus rectum
is 6 feet and whose eccentricity is 1/3.
3. Write the polar equation of an ellipse whose semi-axes are 5
and 3.
4. Discuss equation (6) for the case e = 0.
5. Write the polar equation of a hyperbola if the eccentricity be
V2 and the distance from focus to vertex be 4.
6. Write the polar equations of the asymptotes of
^ 6 .
^ ~ 4 + 5 cos ^
See §87.
9 9
7. Compare the curves p = . , , and p = j ^ ^'
^ 4+5 cos B 4—5 cos B
8. Discuss the equation p = =— ; ,- \» ^ which a is a
^ '^ 1 + e cos (B — ay
oonstant.
239. Ratio Definition of the Parabola. Among the curves of the
parabolic type previously discussed, the one whose equation is of
the second degree is of paramount importance. On that account
when the term parabola is used without qualification, it is under-
stood that the curve is the parabola of the second order, whose
equation may be written, y^ = ax or x^ = ay.
The locus of a point whose distance from a fixed point is always
equal to its distance from a fixed line is a parabola. In Fig.
169, let F be the fixed point and HK the fixed line. Take the ori-
gin at A half way between F and HK. Let P be any point satis-
fying the conditionPF = PH. Call OD = x, PD = 2/, and represent
the given distance FK by 2p. Then, from the right triangle PFD :
PF^ = 2/2 + FZ)2 (1)
= y^ + (a: - 0F)2
= 2/2 + (x - p)2
412 ELEMENTARY MATHEMATICAL ANALYSIS [§240
Since PF by definition equals PH or x + p, we have:
{x + vy = y^ + {x-vY ' (2)
whence :
y2 = 4px (3)
which is the equation of the parabola in terms of the focal distance,
OF or p.
The double ordinate through F is called the latus rectum.
The semi-latus rectum can be computed at once from (3) by
placing X = p, whence:
1 = 2p (4)
where I is the semi-latus rectum. Hence the entire latus rectum is
4p, or the coefficient of x in equation (3).
Fig. 169. — Properties of th6 Parabola i/* — 4px.
In Fig. 169, the quadrilateral FLIK is a square since FL and
FK are each equal to 2p.
240. Polar Equation of the Parabola. In accordance with the
ratio definition of the parabola, its polar equation is found at !
once from equation (6), §238, by putting e = I. Hence the polar \
equation of the parabola is I
§241] THE CONIC SECTIONS 413
For this equation we may make the following table of values:
e
P
0°
1/2
90°
I
180°
00
270°
I
This shows that the parabola has the position shown in Fig. 168.
This is the form in which the polar equation of the parabola is
used in mechanics and astronomy.
241. The Conies. It is now obvious that a single definition
can be given that will include the ellipse, hyperbola and parabola.
These curves taken together are called the conies. The definition
may be worded: A conic is the locus of a point whose distances
from a fixed point {called the focus) and a fixed line {called the
directrix) are in a constant ratio. The unity between the three-
curves was shown by their equation in polar coordinates. Moving
the ellipse so that its left vertex passes through the origin, as in
§76, and writing the hyperbola with the origin at the right ver-
tex (so that both curves pass through the origin in a comparable
manner), we may compare each with the parabola as follows:
The ellipse:
1/2 = 2lx - {¥la^)x^
(1)
The parabola:
2/2 = 2lx
(2)
The hyperbola:
2/2 = 2lx + (62/a2)aj2
(3)
In these equations I stands for the semi-latus rectum of each
of the curves. These equations may also be written:
2/2 = 2lx - {lla)x^ (4)
2/2 = 2lx (5)
2/2 = 2te + {l/a)x^ (6)
whence it is seen that if I be kept constant while a be increased
without limit, the ellipse and hyperbola each approach the para-
bola as near as we please. Only for large values of x, if a be large,
IS there a material difference in the shapes of the curves.
i
414 ELEMENTARY MATHEMATICAL ANALYSIS [§241
Exercises
1. Write the equation of the circle in the form (1) above.
2. Write the equation of the equilateral hyperbola in the form (3)
above.
Fig. 170. — A Hyperbola Translated at an Angle of 45** to OX.
■r^
' / / "r
/ / /
A 10 C, 10 Co
Fig. 171. — A Parabola Translated Fig. 172. — Bridge Truss in Form of
at an Angle of 60° to OX, Circular Segment.
3. Describe the curve:
I
p =
1 + cos (^ — a)
N
where a is a constant.
i242}
THE CONIC SECTIONS
415
4. In Fig. 170 translate the curve zy — J by auitable change
in the equation to the poaition shown by the dotted curve, it the
translation of each point is unity.
5. In Fig. 171 translate the curve y* = ipx by suitable change in
the equation to the position shown by the dotted curve, if the
distance each point ia moved be 3p.
6. A bridge truss has the form of a circular segment, as shown in
Kg. 172. It the total span be 80 yards and the altitude BS
be20yarda, findtheordinatesCiOi, CiD, erected at uniform intervals
of 10 yards along the chord AAi.
Fid. 173. — Bridge Truss in the Form of a Parabolic Segment.
T. A bridge truss has the form of a parabolic segment, as shown in
Fig. 173. The span AAi is 24 yards and the altitude OS is 10
yards. Find the length of the ordinates DC, DiCi, . . . erected at
uniform intervals of 3 yards along the line AAi.
242.* The Conies are Conic Sections. The curves now known
as the conicB were originally studied by the Greelc geometers as the
sections of a oircular cone cut by a plane. At first these sections
were made by passing a plane perpendicular to one element of a
right circular cone. If the angle at the apex of the cone was a
right angle, the section was called (Ae section of the right angled
cone. If the angle at the apex of the cone was less than 90°, the
section made by the cutting plane was called the section of the
acute angled cone. Likewise a third curve was named the section
of the obtuse angled cone. Thus the curves of three different
l^pes now called the parabola, ellipse, and hyperbola were studied.
The present names were not introduced until much later, and
until it was shown that the three classes of curves could be made
respectively by cutting any cone: (1) by a plane parallel to an
element; (2) by a plane cutting opposite elements of the same nappe
416 ELEMENTARY MATHEMATICAL ANALYSIS li212
of the cone; (3) by a plane cutting both nappes of the cone. The
two nappes of a conical surface, it will be remembered, are the
two portions of the surface separated by the apex.
In Vig. 174, let the plane NDN'D', called the cuttii^ plane, cut
the lower nappe of a right circular cone in the curve VPV. We
shall prove that this curve is an ellipse.
Let the plane VA V pass through the axis of the cone. It is
then possible to fit into the cone two spheres which will be tangent
to the elements of the cone and also tangent to the cutting plane.
FlQ. 174. — Section of a Circular Coi
For it is merely necessary to locate by plane geometry the circle
inscribed in the triangle A TF', and the escribed circle Bf'fi', and
then to rotate these circles about the axis AB to describe tbo
required spheres while the line AR describes the conical surface.
Let the points at which the cutting plane touches the two
spheres be called F and F'.
From any point P on the curve VP V draw lines PF and PP'
to the points F and F'. These lines are tangent to the spheres,
since each lies in a tangent plane and passes through the point of
tangency. Throughi'draw an element of the cone AHPK. The
§242] THE CONIC SECTIONS 417
lines PH and PK are also tangents to the upper and lower spheres
respectively. Since all tangents to the same sphere from the same
external point are equal:
PF ^PH
PF = PK
Hence:
. pp + pp^ =PH + PK
But PH + PK is an element of the frustum <Sii//S' RKR\ and hence
preserves the same value for all positions of P, Hence:
PF + PF' = a constant sum
Therefore the section is an ellipse with foci F and F',
Let the upper and lower circle of tangency of the spheres and
conical surface, namely SHS' and RKR\ be produced until they
cut the cutting plane in the straight lines ND and N'D\ DPD'
is a perpendicular at P to the parallel lines ND and N'D\ We
shall show that the parallel lines ND and N'D' are the directrices
of the ellipse.
Since:
PF = PH
we have
PF/PD =PH/PD
The two intersecting lines DD' and HK are cut by the parallel
planes DNS and D'N'R', Hence we have the proportion:
PHjPD = PKjPD' = HK/DD'
This last ratio, however, has the same value for all positions of
P, since HK is an element of the frustum and since DD* is the
fixed distance between the parallel lines ND and AT'D'.
Therefore with respect to the points F and F' and the lines
ND and AT'D' the ratio definition of the ellipse applies to the
curve VPV\ It is easy to show that the ratio HK/DD is less
than unity.
If the cutting plane be passed parallel to the element AR\
it is easy to prove that the curve of the section satisfies the
ratio definition of the parabola. In case the cutting plane cuts
both nappes, one of the tangent spheres lies above the apex and
it is^easy to show that PK - PH is constant.
27
418 ELEMENTARY MATHEMATICAL ANALYSIS [§243
243. Tangent to the Parabola. Let us investigate the condition
that the line y = mx + b shaXL he tangent to the parabola y' =
4px. First find the points of intersection of these loci by solving
the two equations for x and y:
y = mx + b (1)
2/2 = 4px (2)
as simultaneous equations.
Eliminating y by substituting the value of y from (1) in (2)
m^x^ + 2mbx + ¥ - ipx = 0 (3)
or
mV + 2{mb - 2p)x + 6^ = 0 (4)
Solving for x (see formula for quadratic, Appendix.
Therefore there are in general two values of x or two points of
intersection of the straight line and the parabola. By the defini-
tion of a tangent to a curve (§ 146) the line becomes a tan-'
gent to the parabola when the two points of intersection be-
come a single point; that is, when the radical in (5) vanishes.
This condition requires that:
p^ — pmb = 0
or:
b = p/m (6)
Therefore when b of equation (1) has this value, the line touches
the parabola at but a single point, or is tangent to it. The
equation of the tangent is therefore:
y = mx + p/m (7)
This line is tangent to the parabola y^ = 4px for all values of
m. Substituting in (5) the value of 6 = p/m, we may find the
abscissa of the point of tangency:
xi =p/m2 (8)
Substituting this value of x in (7) the corresponding ordinate of
this point is found to be:
yi = 2p/m (9)
244] THE CONIC SECTIONS 419
244. Properties of the Parabola. In Fig. 169, F is the
3C11S, HK is the directrix, PT is a tangent at any point P.
'he perpendicular PN to the tangent at the point of tangency is
ailed the normal to the parabola. The projection DT oi the
angent PT on the x-axis is called the subtangent and the pro-
se tion DN of the normal PN on the x-axis is called the sub-
orxnaL The line through any point parallel to the axis, asP/2,
t known as a diameter of the parabola.
(a) The subtangent to the 'parabola at any point is bisected by
le vertex. It is to be proved that OT = OD for all positions of P.
Tow OD is the abscissa of P, which has been found to be p /m^.
'rom the equation of the tangent:
2/ = mx + p/m
he intercept OT on the x-axis is found by putting y = 0 and
olving for x. This yields:
X = — p /m^
Chis is numerically the same as OD, hence the vertex 0 bisects
DT.
(b) The subnormal to the parabola at any point is constant and
iqual to the semi'laiv^ rectum.
The angle DPN has its sides mutually perpendicular to the
lides of the angle DTP, hence the angles are equal. Since the
;angent of the angle DTP = m, therefore:
tangent DPN = m
From properties of the right, triangle PDN:
DN = PD tangent DPN
= PDm
= {2p lm)m = 2p
Since KF also equals 2p, we have
KF = DN
(c) PFTH is a rhombus. By hypothesis PF = PH. To prove
he figure PFTH a rhombus it is merely necessary to show that
T = PH.
Tow:
FT = F0 + OT
PH = DK = DO + OK
)
420 ELEMENTARY MATHEMATICAL ANALYSIS l§246
But:
OD = OT and OK = FO
therefore:
FT = PH
and the figure is a rhombus.
It follows that the two diagonals of the rhombus intersect at
right angles on the y-axis.
(d) The normal to a parabola bisects the angle between the focd
radius and the diameter at the point. We are to show that:
Z NPF = Z NPR
Since FPHT is a rhombus:
Z FPT = Z TPH
But:
Z TPH = Z /2P5
being vertical angles. From the two right angles NPT and MPS
subtract the equal angles last named. There results:
Z FPN = Z NPR
It is because of this property of the parabola that the reflectors
of locomotive or automobile headlights are made parabolic.
The rays from a source of light at F are reflected in lines parallel
to the axis, so that, in the theoretical case, a beam of light is sent
out in parallel lines, or in a beam of undiminishing strength.
245. To Draw a Parabolic Arc. One of the best ways of de-
scribing a parabolic arc is by drawing a large number of tangent
lines by the principle of §244 (c). Since in Fig. 169 the tan-
gent is for all positions perpendicular to the focal line FH at
the point where the latter crosses OF, it is merely necessary to
draw a large number of focal lines, as in Fig. 175, and erect
perpendiculars to them at the points where they cross the t/-axis.
The equations of the tangent lines in Fig. 175 are of the form:
y = mx + p Im (1)
in which p is the constant given by the equation of the parabola,
and in which m takes on in succession a sequence of values appro-
priate to the large number of tangent lines of the figure. These
lines are said to constitute a family of lines and are said to envelop
1451 THE CONIC SECTIONS 42
le curve to which they are tangent. The curve itself is calloi
e envelope of the family of lines.
The curve of the supporting surface of an aeroplane as well a
to. 175.— Graphical CoQBtruction of a Parabolic Arc "by Tangents."
e curve of the propeller blades is a parabolic arc. The curve of
e cables of a suspension bri^e is also parabolic.
1. Write the equation of the parabola which the family y =
5 + 7/2m envelops.
422 ELEMENTARY MATHEMATICAL ANALYSIS [§246
2. Draw an arc of a parabola if p » 3 inches.
3. At what point ia y = mx + 3/w tangent to the parabola
y« = 12x?
4. At what point is y = wx + 11/m tangent to y* = 44x7
6. Draw the family of lines y = mx + 1/w for m = 0.4, m = 0.6,
m = 0.8, w = 1, w = 2, w = 4, w = 8.
246. Tangent to the Circle. An equation of a tangent line to
a circle can be found as in the case of the parabola above by finding
the points of intersection of:
y = mx + b (l)
and
x2 + 2/2 = o2 (2)
and then imposing the condition that the two points of intersection
shall become a single point. The value of b that satisfies this
Fig. 176. — The Equation of a Line of Given Slope, Tangent to a Given
Circle.
condition when substituted in (1) gives the equation of the re-
quired tangent. It is easier to obtain this result, however, by the
following method. In Fig. 176 let the straight Une be drawn
tangent to the circle at T. Let the slope of this line be m.
Then m = tan ONT = tan a, if a be the direction angle of the tan-
gent line. The intercept of the line on the y-axis can be expressed
in terms of a and a:
6 = aseca = a^Jl + m^ (3)
§247] THE CONIC SECTIONS 423
Hence the equation of the tangent to the circle is:
- y = mx ± aVl + m^
The double sign is written in order to include in a single equation
the two tangents of given slope m, as illustrated in the diagram.
Exercises
1. Find the equations of the tangents to x^ + y^ — 16 making an
angle of 60** with the x-axis.
2. Find the equations of the tangents to x^ -\- y^ = 25 making an
angle of 45** with the x-axis.
3. Find the equation of tangents to x^ -\- y^ =25 parallel to
y = 3x - 2.
4. Find the equation of tangents to x^ -\- y^ = 16 perpendicular
to y = (l/2)x + 3.
6. Find the equations of the tangents to (x — 3)^ + (2/ — 4)^ = 25
whose slope is 3.
247. Normal Equation of Straight Line. The normal equation
of the straight line was obtained in polar coordinates in §69.
The equation was written:
p cos {6 — a) = a (1)
In this equation (p, 6) are the polar coordinates of any point on
the line, a is the distance of the line from the origin and a is the
direction angle of a perpendicular to the line from the origin.
(See Fig. 177.) Expanding cos {6 — a) in {1) we obtain:
p cos 0 cos a + p sin 6 sin a = a (2)
But for any value of p and 6, p 60s 6 = x and p sin 0 = y.
Hence (2) may be written in rectangular coordinates:
X cos a + y sin a = a (3)
This also is called the normal equation of the straight line.
If an equation of any line be given in the form:
ax + hy = c (4)
it can readily be reduced to the normal form. For dividing this
equation through by Va^ + b^:
a , b c
, X + -^ 7 = — _- =-^ (5)
Va2 + b2 Va2 + b2 Va^ + b^
Now aMd^ + b^ and bHa^ + b^ may be regarded as the cosine
424 ELEMENTARY MATHEMATICAL ANALYSIS [§248
and sine, respectively, of an angle, for a and b are divided by a
number which may be represented by the hypotenuse of a right
triangle of which a and 6 are legs. Calling this angle a, equation
(5) may be written:
X cos a + y sin a = d (6)
which is of the form (3) above. Inasmuch as the right side of the
equation in the normal form represents the distance of the Une
from the origin, it is best to keep the right side of the equation
positive. The value of a and the quadrant in which it lies is
then determined by the signs of cos a and sin a on the left side of
the equation. The angle a may have any value from 0® to 360**.
Illustrations:
(1) Put the equation 3x — 42/ = 10 in the normal form. Here
a' + 6» = 25. Dividing by 6 we obtain:
(3/5)x - (4/5)2/ = 2
The distance of this line from the origin is 2. The angle a is the angle
whose cosine is 3/5 and whose sine is — 4/6. Therefore from the
tables:
a = 306° 52'
(2) Put the equation — dx + 4y = 20 in the normal form.
Here cos a = - 3/5, sin a = 4/5, a = 4. Hence a = 126° 52'.
(3) What is the distance between the lines (1) and (2)? The lines
are parallel and on opposite sides of the origin. Their distance
apart is therefore 2 + 4 or 6.
Exercises
1. The shortest distance from the origin to a line is 5 and the direc-
tion angle of the perpendicular from the origin to the line is 30**.
Write the equation of the line.
2. The perpendicular from the origin upon a straight line makes
an angle of 136° with OX, and its length is 2^2. Find the equa-
tion of the line.
8. Write the equation of a straight line in the normal form if
a « 60** and a = V3.
248. To Translate Any Point a Given Distance in a GivenDirec-
tion. To move any point the distance d to the right we sub-
stitute {xi — d) for X. To move the point the distance d in the
§249] THE CONIC SECTIONS 425
y direction we substitute (t/i — d) for y. To move any point
the distance d in the direction a we substitute:
X = X\ — d cos OL
y = yi — d sin a (1)
which must give the desired position of the new point. It is
not necessary to use the subscript attached to the new coordinates
if the distinction between the new and old coordinates can be
kept in mind without this device.
The circle x^ + y^ = a^ moved the distance d in the direction
a becomes:
(x — d cos ay + (y ^ d sin a)^ = a^
which may be simplified to:
x^ — 2dx cos a + 2/^ — 2dy sin a = a^ — d^
249. Distance of Any Point From Any Line. Let the equation
of the line be represented in the normal form:
X cos a + y sin a = a (1)
and let {xi, yi) be any point P in the plane. (See Fig. 177.)
If the point (xi, yi) is on the same side of the line as the origin,
the point can be moved to the line by translating the point
the proper distance in the a direction. Let the unknown amount
of the required translation be represented by d. To translate
the point P the amount d in the a direction, we must substitute
for Xi and yi the values:
Xi = X — d cos a ,rt\
yi = y — dsin a
By hypothesis the point now lies on the line, and therefore the
new coordinates (x, y) of the point must satisfy the equation of
the line. Hence, solving (2) for x and y and substituting their
values in (1) we have:
(xi + d cos a) cos a + {yi + d sin a) sin a = a (3)
Performing the multiplications and solving for the unknown
number d, we have:
d = — (xi cos a + yi sin a — a) (4)
This is the distance of (xi, yi) from the line. Since this distance
would ordinarily be looked upon as a signless or arithmetical
426 ELEMENTARY MATHEMATICAL ANALYSIS (§249
number, the algebraic sign may be ignored, and only the absolute
value of the expression be used. The negative sign means that
the given point lies on the origin side of the line.
Equation (4) may be interpreted as follows:
To find the distance of any point from a given line, put the equa-
tion of the line in the normal form, transpose all term^s to the left
Fig. 177. — Normal Equation of a Line, and the Distance of Anj- Point
from a Given Line.
member and substitute the coordinates of the given point for x and
y. The absolute ralue of the left member is the distance of P from
the line.
If the given point P and the origin of coordinates lie on op-
jKisite sides of the given line, then the point P (Fig. 177) must
bo translated in the direction (ISO® -h a) to reach the Hne.
Hence the substitutions are
J: = X — (/ cos vlSO** + a)
t/i = t/ — d sin use + a)
§250] THE CONIC SECTIONS 427
or,
Xi = X -{- d cos a
yi = y + d sin a
Solving these for x and y^ substituting in the equation of the line,
and solving for d we obtain:
d = Xi cos a + yi sin a — a (6)
The absolute value is of the same form as before. Hence only
the one formula (4) is required. When the result in any problem
comes out negative it merely means the given point lies on the
origin side of the Hne.
The above facts may be stated in an interesting form as follows:
Let any line be:
x cos a + y sin a — a = 0
If the coordinates of any point on this line be substituted in this
equation, the left member reduces to zero. If the coordinates of
any point not on the line be substituted iorx and y in the equation,
the left member of the equation does not reduce to zero, but
becomes negative if the given point is on the origin side of the line
and positive if the given point is on the non-origin side of the
line. The absolute value of the left member in each case
gives the distance of the given point from the hne. Thus every
line may be said to have a ** positive side" and a "negative
side." The "negative side" is the side toward the origin.
Exercises
1. Find the distance of the point (4, 5) from the line 3x + 4^ = 10.
2. Find the distance from the origin to the line x/S — 2//4 = 1.
8. Find the distance from (—3, — 4) to:
12(a; + 6) = 5{y - 2)
4. Find the distance from (3, 4) to the line x/d — y/4: = 1.
5. Find the distance between the parallel lines y = 2x + 3,
2/ = 2x + 5.
6. Find the distance between 2/ = 2x — 3, y = 2x + 5.
7. Find the distance from (0, 3) to 4x — 32/ = 12.
8. Find the distance from (0, 1) to x + 2 - 22/ = 0.
250. Tangent to a Circle at a Given Point. The equation of
the tangent to the circle obtained in §246 is the equation
428 ELEMENTARY MATHEMATICAL ANALYSIS [§251
of the tangent line having a given or required slope m. We
shall now find the equation of the line that is tangent to the circle
at a given point (xo, t/o).
The line:.
a = p cos (6 — a) (1)
or its equivalent:
X cos a + y sin a = a (2)
is tangent to the circle of radius a, and the point of tangency is at
the end of the radius whose direction angle is a. The pK)int of
tangency is therefore (a cos a, a sin a). Hence, multiplying (2)
through by a, we obtain:
x{a cos a) + y{a sin a) = a* (3)
or:
xox + y^y = a' (4)
which is the equation of the line tangent at the point (xo, yo) to
the circle of radius a.
Thus 3x -h 4y = 25 is tangent to x' + i/' = 25 at the point
(3, 4),
Fig. 178. — Tangent to the Ellipse at a Given Point.
851. Tangent to the EUqise at a Given Point. It is easy to
draw the tangent to the ellipse at any desired point. In Fig. 178,
let Po be the point at which a tangent is desired. Then draw the
migor circle, and let Pi of the circle be a point on the same ordinate
§252] THE CONIC SECTIONS 429
as Po. Draw a tangent to the circle at Pi and let it meet the x-axis
at T, Then when the circle is projected to form the ellipse, the
straight line PiT is projected to make the tangent to the ellipse.
Since T when projected remains the same point and since Po is
the projection of Pi, the line through Po and T is the tangent to
the ellipse required.
The equation of the tangent PoT is also readily found. The
equation of PiT is;
xxq + yy\ = a2 (1)
To project this into the line P^T it is merely necessary to multiply
the ordinates y and y\hy b/a; that is, to substitute y = aylb and
t/'o = ayo/b. Whence (1) becomes:
Xific + a^y^fb^ = a^ (2)
or dividing by a^,
Xox/a2 + yQy/b2 = l (3)
which is the tangent to:
x^ /a2 + !/2 /62 = 1
at the point (xo, 2/o).
Exercises
1. Find the equations of the tangents to the ellipse whose semi-axes
are 4 and 3 at the points for which x = 2.
2. Find the equations of the tangents to x^/l6 + 2/^/9 = 1 at the
ends of the left latus rectum.
3. Required the tangents to x^/9 + 2/^/4 = 1 making an angle of
45° with the x-axis.
4. Find the equations of the tangents to x^/lOO + y^/25 = 1 at
the points where y = S.
6. Find the equations of the tangents to x^/SQ + y^/lQ = 1 at
the points where x = y.
252. The Tangent, Normal, and Focal Radii of the Ellipse. In
the right triangle PiOT, Fig. 178, the side PiO is a mean propor-
tional between the entire hypotenuse OT and the adjacent
segment OD, That is:
a^ = XoOr
But: FiT = OT - OFi
= aVa?o — ae
430 ELEMENTARY MATHEMATICAL ANALYSIS [§253
Likewise: F^T = OT + OF2
= a^ /xo + ae
Therefore: FiT IF2T = (aVa^o - ae) /(a* /xo + ae)
= (a — exo) l(a + cxq)
But by §231 this last ratio is equal to ri/r2. Therefore we
may write: FiT/FiT = PoFi/PoFi.
Hence T, which divides the base F2F1 of the triangle Po^aFi
externally at T in the ratio of the two sided PF2 and PFi of the
triangle, lies on the bisector of the external angle FiPfQ of the
triangle F2P0F1, This proves the important theorem:
The tangent to the ellipse bisects the external angle between the
focal radii at the point.
This theorem provides a second method of constructing a
tangent at a given point of an ellipse, often more convenient
than that of §251, since the method of §251 often runs the
construction off of the paper.
The normal PoN, being perpendicular to the tangent, must
bisect the internal angle F^PoFi between the focal radii F2P0 and
FiPo.
Since the angle of reflection equals the angle of incidence for
light, sound, and other wave motions, a source of light or sound at
Fi is ^'brought to a focus *' again at F2, because of the fact that the
normal to the ellipse bisects the angle between the focal radii.
253. Additional Equations of the Straight Line. The equations
of the straight line in the slope form:
y = mx + b (1)
and in the normal forms:
p cos (0 — a) = a (2)
x cos a + y sina = a (3)
and the general form :
ax + by + c — 0 (4)
have already been used. Two constants and only two are neces-
sary for each of these equations. The constants in the first
equation are m and b; in the second and third, a and a; in the
fourth a /c and b /c, or any two of the ratios that result from divid-
ing through by one of the coefficients. Equation (4) appears to
contain three constants, but it is only the relative size of these that
§253] THE CONIC SECTIONS 431
determines the particular line represented by the equation, since
the line would remain the same when the equation is multiplied
or divided through by any constant (not zero).
These facts are usually summarized by the statement that two
conditions are necessary and sufficient to determine a straight
line. The number of ways in which these conditions may be given
is, of course, unlimited. Thus a straight line is determined if we
say, for example, that the line passes through the vertex of an
angle and bisects that angle, or if we say that the line passes
through the center of a circle and is parallel to another line, or if
we say that the straight line is tangent to two given circles, etc.
An important case is that in which the line is determined by the
requirement that it pass through a given point in a given direc-
tion. The equation of the line adapted to this case is readily
found. Let the given point be (xi, yi). The line through the
origin with the required slope is
y = mx
Translate this line so that it passes through (xi, yi) and we have
y - yi = m(x - Xi) (5)
Another way of obtaining the same result is: substitute the
coordinates (a;i, yi) in (1) :
2/1 = mxi + b (6)
Subtract the members of this from (1) above, so as to eliminate
b. There results:
2/ - 2/1 = m{x - xi) (7)
This is the required equation; the given point is (a^i, yi) and the
direction of the line through that point is given by the slope m.
Another important case is that in which the straight line is
determined by requiring it to pass through two given points.
Let the second of the given points be {x2, 2/2). Substitute these
coordinates in (5) :
2/2 — 2/1 = M^2 — xi) (8)
To eliminate m, divide the members of (7) by the members of (8) :
2/ - 2/1 ^ _?jr_^i (9)
y2 - yi X2 — xi
432 ELEMENTARY MATHEMATICAL ANALYSIS [§254
or, as it is usually written:
y - yi 72 - yi
(10)
X — Xi X2 — Xi
This is the equation of a Une passing through two given points.
Since (10) may be looked upon as a proportion, the equation may
be written in a variety of forms.
254. The Circle Through Three Given Points. In general, the
equation of a circle can be found when three points are given.
Either of the general equations of the circle:
(x - hy + (2/ - ky = a2 . (1)
or:
x^ + y^ + 2gx + 2fy + c = 0 (2)
contains three unknown constants, so that in general three
conditions may be imposed upon them. It is best to illustrate
the general method by a particular example. Let the three
given points be (— 1, 3), (0, 2), and (5, 0). Then since the co-
ordinates of these points must satisfy the equation of the circle,
we obtain from (2) above:
1 + 9 - 2^ + 6/ + c = 0 (3)
4 + 4/ + c = 0 (4)
25 + lOgr + c = 0 (5)
Eliminating c from (3) and (4) and from (4) and (5), we obtain:
6 - 2^ + 2/ = 0
21 + lOgr - 4/ = 0
\
Eliminating /:
whence:
and
c = 30
So the equation of the circle is:
a;2 + 2/2 - Ux - I7y + 30 = 0
Exercises
1. Find the equation of the line passing through (2, 3) witfi
slope 2/3.
2. Find the equation of the line passing through (2, 3), (3, 5).
§2551 THE CONIC SECTIONS 433
8. Find the line passing through (2,-1) making an angle whose
tangent is 2 with the x-axis.
4. Find the line through (2, 3) parallel to 2/ = 7x + 11.
6. A line passes through ( — 1, — 3) and is perpendicular to
y — 2a; = 3. Find its equation.
6. Find the line passing through ( - 2, 3), ( - 3, - 1).
7. Find the equation of the line which passes through ( — 1, — 3),
(-2,4).
8. Find the slope of the line that passes through ( — 1, 6), ( — 2, 8).
9. Find the equation of the line passing through the left focus and
the upper end of the right latus rectum of a;*/25 + y^/9 — 1.
10. Find the equation of the circle passing through (2,8), (5, 7),
and (6, 6).
11. Find the equation of the circle which passes through (1, 2),
( - 2, 3), and ( - 1, - 1).
12. Find the equation of the parabola in the form y' = 4px which
passes through the point (2, 4).
255. Change from Polar to Rectangular Coordinates. The
relations between x, y of the Cartesian system and p, 0 of
the polar system have already been explained and use made of
them. The relations are here brought together for reference:
X = p cos d (1)
y = p sin ^ (2)
By these we may pass from the Cartesian equation of any locus
to the equivalent polar equation of that locus. Dividing (2)
by (1) and also squaring and adding, we obtain:
^ = tan_;;iy/x (3)
P = \/x2 + y2 (4)
These may be used to convert any polar equation into the Cartesian
equivalent.
256. Rotation of Any Locus. It has already been explained
that any locus can be rotated through an angle a by substituting
(^1— a) for 6 in the polar equation of the locus. It remains to
determine the substitutions for x and y which will bring about
the rotation of a locus in rectangular coordinates. Let us consider
any point P of a locus before and after rotation through the given
single a. Call the coordinates of the point before rotation
28
434 ELEMENTARY MATHEMATICAL ANALYSIS [§256
(Xj y) in rectangular coordinates and (p, 6) in polar coordinates.
Then, from (1) and (2), §255,
X = p cos 6 (1)
y = p sin 6 (2)
Call the coordinates of the point after rotation (xi, yi) and
(pi, ^i), but note that the value of p is unchanged by tha rotation.
Then for the point P', Fig. 179, we may write:
P(Pi.di) or
Cxi.i/i)
P(P,0) or
Fig. 179. — Rotation of Any
Locus.
Fig. 180. — Effect of Rotation on the Special
Forms x* + y*, 2xy, and x* — |/».
xi = p cos ^1 (3)
2/1 = p sin di (4)
Since, however, the rotation requires that
e = $1- a (5)
equations (1) and (2) become:
x = p cos (di — a) = p cos ^1 cos a + p sin ^i sin a (6)
y = p sin (^1 — a) = p sin ^i cos a — p cos ^i sin a (7)
But, from (3) and (4), p cos ^i and p sin 6i are the new values of
X and y; hence, substituting in (6) and 7) from (3) and (4) we
obtain:
X = Xi cos a + yi sin a (8)
y = yi cos a — Xi sin a (9)
Hence if the equation of any locus is given in rectangular co-
^66] THE CONIC SECTIONS 435
)rdinates, it is rotated through the positive angle a by the sub-
ititutions
z cos ce + y sin a f or X
y cos a — X sin a for y (10)
n which it is permissible to drop the subscripts, if the context
hows in each case whether we are dealing with the old x and y
»r with the new x and y*
K the required rotation is clockwise, or negative, we must
eplace a by ( — a) in all of the above equations.
Whenever convenient, the equation of a curve should he taken in
he 'polar form if it is required to rotate the locus.
Important Facts: The following facts should be remembered
)y the student:
(1) To rotate a curve through 90®, change xtoy and y to (— x),
This fact has been noted in §68.
(2) Rotation through any angle leaves the expression x^ + y^
or any function of it) unchanged. This is obvious since the circle
:« + 2^2 = a* is not changed by rotation about (0, 0).
(3) Rotation through + 45° changes 2xy to y^ — x^.
Rotation through — 45° changes 2xy to x^ — y^,
(4) Rotation through + 45° changes x^ — y^ to 2xy.
Rotation through — 45° changes x^ — y^ to — 2xy.
Itatements (3) and (4) follow at once from consideration of the
t][uations
x2 - 2/^ = a2 (1)
2xy = a^ (2)
2/2 - x2 = a* (3)
- 2xy = a2 (4)
of the four hyperbolas bearing corresponding numbers (1), (2),
(3), (4) in Fig. 180. The proper change in any case can be
remembered by thinking of the four hyperbolas of this figure.
(5) The degree of an equaiion of a locus cannot he changed hy
% rotation* This follows at once from the fact that the equations
of transformation (8) and (9) are linear.
Exercises
In order to shorten the work, use statements (1) to (4) whenever
9ossible.
436 ELEMENTARY MATHEMATICAL ANALYSIS [§257
1. Turn the locus ar' — y* = 4 through 45°.
2. Turn a;« + y* « a^ through 79°. Turn 4txy = 1 through 45°.
3. Turn x cos a + ^ sin a « a through an angle /S. (Since this locus
is well known in the polar form, transformation formulas (6) and (7)
above may be avoided.)
4. Rotate x» - y« = 1 through 90°.
6. Rotate x^ — y^ — a^ through — 45°.
6. Change the equation (x — a)* + (y — 6)* = r* to the polar
form.
7. Change p cos 2d = 2a, one of a class of curves known as Cote's
spirals, to the Cartesian form.
8. Write the equation of the lemniscate in the polar form.
9. Show that p* — 2ppi cos (0 — 0i) + pi* = a* is the polar equa-
tion of a circle with center at (pi, ^i) and of radius a.
10. Write the Cartesian equation of the locus p* = 16 sin 26.
11. Turn p' = 8 sin 20 through an angle of 45°.
12. Rotate x* - 2y^ = 1 through 90°.
13. Rotate (x^ ^ y^)H -{. (x« - y»)94 = 1 through 46°.
14. Rotate log (x« -f y^) = tan (x* - y«) through 45°.
257. EUipse with Major Axis at 45'' to the OX Axis. The
ellipse frequently arises in applied science as the resultant of the
projection of the motion of two points moving uniformly on two
circles, as has already been explained in §186. Thus the
parametric equations:
X = a cos t (1)
y = bsint (2)
define an ellipse which may be considered the resultant of two
S.H.M. in quadrature. We shall prove that the equations:
X = a cos t (3)
y = a sin (t + a) (4)
define an ellipse, with major axis making an angle of 45° with OX.
The graph is readily constructed as in Fig. 181. The Car-
tesian equation of the curve is found by eliminating t between
(3) and (4). Expanding the sin {t + a) in (4) and substituting
from (3) we obtain:
y = X sin a + v a* — x^ cos a (5)
Transposing and squaring:
x^ — 2xy sin a -\- y^ = a^ cos^ a (6)
THE CONIC SECTIONS
437
By $266 rotate the curve through an angle of (— 45°.) We
know that (x* + y*) is unchanged and that 2xy is to be replaced
by (x* — y*). Therefore (6) becomes:
I'd - sin a) + !/'(! + sin a) = a' cos= a (7)
/
(
\
l£.
■^
J \p
,-'
y\
/
y\' X
o
/
A
T
/
1
•i
_.
y
y'
Fio. 181— The E
Replacing cos' a by 1 —
right member, we obtain;
a'(l + sin «) ^
which may be written:
, and dividing through by the
2o' cos' ^ 2a' sin' ^
where ^ is the complement of a. Equation (8) or (9) proves
that the locus is an ellipse. It is a.ny ellipse, since by properly
choosing a and a. the denominators in (S) can be given any desired
values. Hence the pair of parametric equations (3) and (4), or
the Cartesian equation (5) represents an ellipse with its major axis
inclined + 45" to the OX-axis.
268. General Equation of the Second Degree. The general
equation of the second degree in two variables may be written in
the standard form:
ox' + 2Axj/ + fey' + 2sx +'?f)/ + c = 0 (1)
438 ELEMENTARY MATHEMATICAL ANALYSIS [§259
In the next two sections we shall show that the general
equation of the second degree in two variables represents a conic.
We shall be able to distinguish three cases as follows:
The general equation of the second degree represents:
an eUipse ]ih^ - ab <0 (2)
a parabola if h^ — ab = 0 (3)
a hyperbola ifh^ — ab > 0 (4)
To render the above classification true in all cases we must classify
the "imaginary ellipse," — ^ + t^ = — 1, as an ellipse, and other
a 0
degenerate cases must be similarly treated. The expression
h^ — ab is called the quadratic invariant of the equation (1), so
called because its value remains unchanged as the curve is moved
about in the coordinate plane. In other words, as the locus (1)
is translated or rotated to any new position in the plane, and while
of course the coefficients of a:^, xy, and y^ change to new values, the
function of these coefficients, h^ — a6, does not change value, but
remains invariant. This fact is not proved in this book, but it can
readily be proved by comparing the value of h'^ — ah before and
after the substitutions:
X cos a + y sin a — m for x
y cos a — x sin a — n f or y
where m and n indicate the amount of the translation, and a the
angle of rotation.
259.* Conies with Their Axes Parallel to the Codrdinate Axes.^
Let us comider the equation
ax^ + by^ + 2gx + 2/2/ +. c = 0 (1)
If we solve this equation for y in terms of x, we get
- / + V- abx^ - 2bgx -bc+p ,ox
y = ^ W
1. We saw in completing the squares, §77, that (1) is the
equation of an ellipse when a and b are alike in algebraic signs.
We can now restate this condition by saying that (2) is the equa-
tion of an ellipse when the coefficient of x^ is negative. Note
^§§269 and 260 are from the correspondence course prepared by Professor
H. T. Burgess.
§260] THE CONIC SECTIONS 439
that the equation of a circle is included as a special case when
a = b.
2. We saw in completing the squares, §87, that (1) is the
equation of a hyperbola when a and b have unlike signs. We
can restate this condition by saying that (2) is the equation of
a hyperbola when the coefl&cient of x^ is positive.
3. We observe that (1) is the equation of a parabola
when a = 0. We can restate this condition by saying that
(2) is the equation of a parabola when the coefficient of x^ is
zero.
260. * The General Case. Write the quadratic in two variables
in the standard form:
ax^ + 2hxy + by^ + 2gx + 2fy + c =^ 0 (1)
I. We have already seen, §43, that when and only when
h = 0 and a = 6 the locus of (1) is a circle.
II. When h is not equal to zero, we have as yet no knowledge
of the nature of the locus represented by (1), except that it is
not a circle.
Let us rotate this locus clockwise through an angle a and see if
the equation can be simplified so that the character of the locus
represented by (1) can be recognized. Substituting in (1) from
§256, we get
a(x cos a — 2/ sin a) 2 + 2h{x cos a — y sin a) (x sma + y cos a)
+ b{x sijxa + y cos a)^ + 2g{x cos — y sin a)
+ 2f{x sin a + 2/ cos a) + c = 0 (2)
If we simplify (2), we find that the coefficient of the term in xy is:
2(6 — a) sin a cos a + 2h(cos^ a — sin^ a) (3)
This term will drop out of (2), if we can find a value for the angle
a that will make (3) zero.
Substituting in (3) from equations (1) and (2), §165, we get:
(b - a) sin 2a + 2h cos 2a = 0 (4)
From this we find :
tan 2a = , (5)
a — b ^
440 ELEMENTARY MATHEMATICAL ANALYSIS [§260
Hence if we choose a as half of the angle whose tangent is _ , >
equation (2) will have no term in xy, and it will be of the form:
Ax^ + By^ + 2Gx + 2Fy + C=-0 (6)
where Ay B, (r, F, and C stand for long expressions in terms of
the coefficients of equation (1).
Since the loci of equations (1) and (6) are identically the same
curve, we now see from §258 that the locus of (1) must be
an ellipse, hyperbola, or parabola.
III. We can now devise a test by which we can tell immediately
which curve is represented by equation (1). If we solve (1) for
y in terms of x, we get
-ihx+f)± V{h^ - ah)x^ + 2ihf - gb)x +P-bc
y^ _
Let us now consider the two equations :
(7)
^^=-6^-6 • («)
^ Vjh^ - ab)x^ + 2{hf - gh)x + P - be .^^
2/2 = ± ^ W
It is obvious that the locus of (7), whatever it is, may be obtained
by shearing the locus of (9) in the line (8). We must consider
the three following cases:
1. When h^ < ah the coefficient of x^ in (9) is negative and the
locus of (9) is an ellipse. Hence the locus of (7) is a locus made
by shearing an ellipse in a line, and is therefore a closed curve.
The locus of (7) is in this case an elUpse, for it must be either an
ellipse, a hyperbola, or a parabola by II, and it cannot be either
a hyperbola or a parabola since it is a closed curve.
2. When A^ > ah the coefficient of x^ in (9) is positive and the
locus of (9) is a hyperbola. Hence the locus of (7) is a locus made
by shearing a hyperbola in a line, and is therefore an open curve
with two branches. The locus of (7) is in this case a hyperbola,
for it cannot be an ellipse or a parabola since it has two open
branches.
3. When h^= ah the coefficient of x^ in (9) is zero and
the locus of (9) is a parabola. Hence the locus of (7) is a
locus made by shearing a parabola in a line, and is therefore an
(261]
THE CONIC SECTIONS
441
open curve with one branch. The locus of (7) is in this case a
parabola, for it cannot be an ellipse or a hyperbola since it has
one open branch.
We now state the results in this form : The locus of the general
equation of the second degree in two variables is for
A' < ab an ellipse
k^ > ab a hyperbola
h^ = ab a parabola
If we shear the locus of (7) in any line y = mx + b, the form
of the equation is not changed. Hence the following important
facts:
The shear of an ellipse in a line is on ellipse.
The shear of a hyperbola in a line ts a hyperbola.
The shear of a parabola in a tine is a parabola.
If we put (mx) for x and (ny) for y in (7), no change will be
made in the sign of the coefficient of x^; hence Ihe elongation or
contraclioh (orthographic projection) of an ellipse, hyperbola, or
parabola in any direction is an ellipse, hyperbola, or parabola.
26L Shear of Hie Circle. The effect of the addition of the term
mx to f(x), in the equation y = fix), has been shown in |37 to
be to change the shape of the locus by lamellar or shearing motion
of the xy plane. We usually speak of this process as "the shear of
tJie locus y = f(x) in the line y = mx." Wlien applied to the cirde
442 ELEMENTARY MATHEMATICAL ANALYSIS [§262
2/ = ± Va^ — x^ the effect is to move vertically the middle point
of each double ordinate of the circle to a position on the line
y = mx. The result of the shearing motion is shown in Fig. 182.
The area bounded by the curve is unchanged by the shear.
The equation after shear is:
y = mx ± Va^ — x^ (1)
This is the same form as equation (5) of §257, if we put
m = and then multiply all ordinates by cos a. There-
cos a ^ •^ "^
fore the curve of Fig. 182 is an ellipse.
The straight line y = mx passes through the middle points of
the parallel vertical chords of the ellipse
y = mx
+ Va2 - x^ (2)
The locus of the middle points of parallel chords of any curve is
called a diameter of that curve. We have thus shown that the
diameter of the ellipse is a straight line. Since the same reasoning
applies to
y = mx+ (b/a) Va^ - x^ (3)
which may be regarded as any ellipse in any way oriented with
respect to the origin, the proof shows that the mid-points of arbi-
trarily selected parallel chords of an ellipse is always a straight
line.
262. A Second Proof. The generality of the preceding fact
may seem clearer if the ellipse be kept fixed in position while the
direction of the set of parallel chords is arbitrarily selected. Con-
sider first the circle
X^ + yi = a^ (1)
and draw any set of parallel chords. Let the slope of these chords
be s. Then the equation of the chords is
y = sx + p (2)
in which p is an arbitrary parameter, to various values of which
correspond the different chords of the family of parallel,chords.
§263] THE CONIC SECTIONS 443
The equation of the bisectors of aU of the chords is a Hne through
the origin perpendicular to (2), or:
X
y=-- (3)
Now if the circle (1) and the chords (2) and the diameter (3) be
changed by orthographic projection upon a plane through the
X-axis, then the circle (1) becomes an ellipse, while the parallel
chords and the line through their mid-points remain straight lines,
but with modified slopes. Let the given orthographic projection
multiply all ordinates of (1), (2) and (3) by - • Then the equation
Cv
of the ellipse is:
S + g=l (4)
The parallel chords now have the equation
shx hp
y = h (5)
The equation of the locus of the mid-points of the parallel chords
or the diameter is:
sb
Representing — , the slope of (5), by m, equation (6) takes the form:
Cv
which is the equation of the diameter of (4) that bjsects the family
of parallel chords of slope m.
263. Confocal Conies. Fig. 183 shows a number of ellipses
and hyperbolas possessing the same foci A and B. This family
of curves may be represented by the single equation:
x^ y^
in which the parameter k takes on any value lying between 0
and a*, and in which a > b, li k satisfies the inequality:
0 <k <b^
the curves are ellipses. If k satisfies the inequality:
b^ <k <a^
444 ELEMENTARY MATHEMATICAL ANALYSIS IS263
the curves are hyperbolas. The ellipaes of Fig. 183 may be
regarded as repreaeQting the successive positions of the wave front
of sound waves leavii^ the sounding body AB; or they may be
regarded as the equipotential lines around the magnet AB, of
which the hyperbolas represent the lines of magnetic force.
Exercises
1. Sketch the curve:
y = 2x + \'i - X'.
3. Draw the curve:
X - 2COB0
y-2am(9+T/6).
5. Find the axes of the ellipBe:
X = 3cosfl
y = 3 ain (9 +
4. Draw the curve:
y = x± ■>J&x -
6. Draw the curve:
y = X ± -^x'
§2631 THE CONIC SECTIONS 445
6. Show that:
y = X ± V 6x is a parabola.
7. Sketch the curve:
y = (l/2)a: + Vl6 - x^.
8. Sketch the curve:
y = 5x sin 60° + cos 60°V25 - x\
9. Discuss the curve:
x^/a^ + y^/h^ — 2{xy/ab) cos a = sin^ a.
Show that the locus is always tangent to the rectangle x
= ± a, 2/ = ± 6, and that the points of contact from a parallelo-
gram of constant perimeter ^Vd^ -\- h^ for all values of a.
10. Show that x = a cos (0 — a), y = h cos (0 — fi) represents an
ellipse for all values of a and /3.
11. Prove from equation (13), §257, that the distance from the
end of the minor to the end of the major axis of the resulting ellipse
remains the same independently of the magnitude of a.
12. Show that the following construction of the hyperbola
xy^ = a' is correct. On the — x-axis lay off OC = a. Connect C
with any point A on the y-&xis. At C construct a perpendicular to
AC cutting the y-axis in B. At B erect a perpendicular to BC cutting
the + X-axis at D. Through A draw a parallel to the x-axis and
through D draw a parallel to the 2/-axis. The two lines last drawn
meet at P, a point on the desired curve.
13. Explain the following construction of the cubical parabola
a^y = x^. Lay off OB on the — ^/-axis equal to a. From B draw a
line to any point C of the x-axis. At C erect a perpendicular to BC
cutting the y-axis at D. At D erect a perpendicular to CD cutting
the X-axis at E. Lay off OE on the 2/-axis. Then OE is the ordinate
of a point of the curve for which the abscissa is OC.
14. Explain and prove the following construction of the semi-
cubical parabola, ay^ = x'. Lay off on the — x-axis OA = a.
Prom A draw a parallel to the line y ^ mx, cutting the y-axis in B.
Erect at B a perpendicular to AB cutting the x-axis at C, and at C
erect a perpendicular to OC. The point of intersection with y = mx
is a point of the curve.
446 ELEMENTARY MATHEMATICAL ANALYSIS [§263
Problems for Review
1. Find the approximate equations for the following data:
(o) Steam pressure: v = volume, p = pressure.
(b) Gras-engine mixture: v = volume, p = pressure.
(a)
(h)
V
t
P
t;
P
2
68.7
3.54
141.3
4
31.3
4.13
115.0
6
19.8
4.73
95.0
8
14.3
5.35
81.4
10
11.3
5.94
71.2
6.55
63.5
7.14
54.6
2. Show that p^- = a* cos 2d is the polar equation of a lemniscate.
3. When an electric current is cut off, the rate of decrease in the cur-
rent is proportional to the current. If the current is 36.7 amperes
when cut off and decreases to 1 ampere in one-tenth of a second,
determine the relation between the current C and the time t.
4. Write four other equations for the circle p = 2 V3 sin 6—2 cos 6.
6. Write four other equations for the sinusoid y = sin x — V 3 cos x.
6. Find the angle that 3x + 4y = 12 makes with 4x — 3y = 12.
7. From the equation
a = 6 sin (2i - V)
determine the amplitude, period, and frequency of the S.H.M.
8. A simple sinusoidal wave has a height of 3 feet, a length of 29
feet, and a velocity of 7 feet per minute. Another wave with the
same height, length, and velocity lags 15 feet behind it. Give the
equation of each.
9. Interpret ri(cos ^i -f t sin ^i) as an operator upon
rs(cos 6i 4- i sin Ot).
10. Give a rule for writing down the value of i".
11. Calculate:
(3V3 - 30^ - 1 + \^3t)» (co836°-f tsin36**)(cos20^+tffln20°)'
(2 -f 2V2i)
12. CiUcidate: (1 - v^3t)*i.
2(co8ir + tsinir)
§263] THE CONIC SECTIONS 447
13. Write the inverse functions of the following :
(a) y = ox", (6) y = sin x, (c) y = e*, (d) 2/ = log, x,
14. Plot the amount of tin required to make a tomato can to hold
1 quart as a function of the radius of its base. Determine approxi-
mately from the graph the dimensions requiring the least tin.
16. Find the axes of the ellipse whose foci are (2, 0) and (— 2, 0),
and whose directrices are x = ± 5.
16. Write the polar equation for the ellipse in problem 15.
17. Find the equation of the hyperbola whose foci are (5, 0) and
(— 6, 0), and whose directrices are x — ±2.
18. Write the equation of the hyperbola in 17 in polar coordinates.
19. Discuss the curve p(l +cos^) = 4. Write its equation in
rectangular coordinates.
20. Find the foci of the hyperbola 2xy = a*. Also its eccentricity.
21. What property of the parabola is useful in designing automobile
headlights?
22. How do you draw a tangent to an ellipse? To a parabola?
23. Find the equation of a point whose distance from the point
(3, 4) is always twice its distance from the line 3a; + 42/ = 12. What
is the locus?
24. Give the type of each of the following conies:
(a) 2x^ + 21/2 + 3a; - 4y + 3 = 0.
(6) a;2 + 4a;y + 42/2 + a: - 32/ + 8 = 0.
(c) x^ + 3x2/ - 32/2 -f 3x - 22/ - 3 = 0.
(d) a;2 - 5x2/ + 7y^ + 2x + 32/ + 28 = 0.
26. Solve each of the equations in problem 24 for y and explain
how the graphs may be constructed by shear.
26. A point moves so that the quotient of its distance from two
fixed points is a constant. Find the equation of the locus of^the
point.
27. Evaluate:
log 10 - log2 8 + log7 492.
28. Find the maximum and minimum value of (3 sin x — 4 cos x).
What values of x give these maximum and minimum values?
29. Find the equation of a circle passing through the points (1, 2),
(-1,3) and (3, -2).
30. A sinusoidal wave has a wave-length of ir, a period of t, and an
amplitude of t. Write its equation.
31. Compute graphically the following :
(l+i)(l-i); (1 H-i) + (1 -i);
448 ELEMENTARY MATHEMATICAL ANALYSIS [§263
f^^; 7cis47**X6ci8(-ir);
(7 -f 6i)"; V^7i -f 31.
32. Prove by the addition formulas that:
sin (90° - t) = cos r
sin (90° -\-t) ^ cos t
sin (360° - t) = - sin r
tan (r + 270°) = - cot r.
33. Sketch the curves:
y = 2'
y =3«
What property of the exponential function do these curves illustrate?
34. Sketch y = 2* and y = 3».
36. Solve: x* + 6x -f Vx* + 6x -fl = 1.
36. Find graphically the product of3-2iby -2+t.
37. Find all the values of:
(cos ^ + » sin $)*; (cos ^ + i sin $)H; \/i; y/f,
38. Write a short theme on operators, making mention of (a) the
integers; (6) (— 1); (c) V — 1; {d) cis $. Develop the rules for addi-
tion, subtjraction, multiplication, and division of vectors, and state
them in systematic form.
39. Show that
sin (a + 6+ c) = sin a cos b cos c + oos a sin b cos c
+ cos a COS b sin c — sin a sin & sin c.
40. Sketch the curves
y =3';
? = 3';
on the same sheet of pap^-. What property of the e3q[>onential func-
tion do these curves illustrate?
41. Draw upon squared pi4>^, using 2 cm. = 1, the curve y* - x.
By counting the small squares of the pap^ find the area bounded by
the eur\*eandthe ordinatesx » 1 2, 1, 1}, 2, 2}, 3,3i, 4, ... By
plotting these points upon some form <rf oo<3rdm&te paper, find the
functional negation esdsting between the x oodrdinale and the area
under the curve.
§263]
THE CONIC SECTIONS
449
42. The latitude of two towns is 2T 31'. They are 7 miles
apart measured on the parallel of latitude. Find their difference in
longitude.
43. Solve 3*'"^ = 2*+*. Be very careful to take account of all
questionable operations . There are two solutions.
44. Find (three problems) the equation connecting :
X
y
6.8
19.0
14.2
21.6
21.8
23.2
32.0
26.3
46.5
31.5
65.0
39.1
78.0
47.0
X
002
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
y
18.8
19.3
19.6
19.7
20.2
20.5
21.0
21.4
21.8
21.9
22.1
X
1.3
2.0
2.8
3.7
4.3
5.3
y
21
25
29
33
35
38
46. Find the wave length, period, frequency, amplitude and velocity
for
y = 10 sin {2x - 30-
46. Prove that:
CSC* A
sec 2 A.
CSC* A - 2
47. Find the parametric equations of the cycloid.
48. Find the equation of the ellipse, center at the origin, axes coin-
ciding with coordinate axes, passing through the point ( — 3, 5) and
having eccentricity 3/5.
49. Define the "logarithm of a number."
60. Prove:
CSC 2x(l — cos 2x) = sin x sec x.
CSC xiX — cos x) = ?
61. A S.H.M. has amplitude 6, period 3. Write its equation if
time be measured from the negative end of the oscillation. State the
difference between a S.H.M. and a wave.
62. Find by inspection one value of x satisfying the following equa-
tions:
29
450 ELEMENTARY MATHEMATICAL ANALYSIS [J263
(o) cos 45** cos (90** - x) - sin 45* sin (90** - x) = cos a?.
(6) COS (45* - x) cos (46* + a;) - sin (46* - x) sin (46* + x) = cosx.
63. Sketch on Cartesian paper:
y = 2» y = logj X
y = 3» y = logi X
y = 6» y = logj X
2/ = 10' y = logio X,
64. Solve 3* + 2a; = 1.
66. Sketch:
p = Oy p = sec $, p = a sin d, p = — o sin 6,
p = -, p = o cos dj f> =^ — a cos ^, p = a — a cos tf.
p = 2( — cos tf), p = 2 cos tf — 3, p = COB ^ + sin ^.
66. Simplify the expression:
sin (i - ^ ) sec ( j + t j - sin (j + t j sec ^j - r j
67. A point moves so that the product of its distance from two fixed
points is a constant. Find the equation of the locus. Discuss the
curve.
68. Simplify and represent graphically :
(«) 54-^ W (l+t)(l+2t).
69. Find the velocity and frequency of the wave of problem 30.
60. Find the coordinates of the center, the eccentricity, and
the lengths of the semi-axes of: (a) a;' + 3x + y' = 7,
(6) a;' + 2x + 4y2 - 3y = 0, (c) a;« - a; - y* - y = 0,
(d) a;2 + a; + y + 3 = 0.
61. Find the amplitude, period, frequency and epoch of the fol-
lowing S.H.M.:
y = 7 sin 6^
y = 6 sin 2-^1.
y = a sin (co^ + o).
62. Find cis^ 0. Hence show that:
cos 5a; = cos^ a; — 10 cos' x sin* a; + 5 cos x sin* x,
63. Find graphically (on form M3) the fifth roots of 2* cis 35*..
64. Complete the following equations:
sin (a ± 6) = ? tan 2x = ?
cos (a ± 6) = ? cot 2x = ?
J263] THE CONIC SECTIONS 451
tan (a ± J5) = ? sin |^ = ?
sin 2x = ? cos o" ~ ^
cos 2a; = ? cot |^ = ?
66. Change the equations of exercises 33 and 40 to logarithmic
form. What properties of logarithms are illustrated by these
equations?
66. Solve 2* - a;» + 7a; + 6 = 0.
67. Show that the sum of the two focal radii of the ellipse is constant.
68. y — — 3i* + 4^ — 5 and x = 5^ are the parametric equations of a
curve. Discuss the curve.
69. Show that [r(cos 6 -^ i sin $)] [r'(cos ^' + i sin $')] =
n^Icos (d + d') + i sin {6 + e')].
70. Two S.H.M. have amplitude 6 and period two seconds. The point
executing the first motion is one-fourth of a second in advance of
the point executing the second motion. Write the equations of
motion.
71. Show that:
sin 5a; = sin^ a; — 10 sin' x cos^a; + 5 sin a; cos* x,
72. Prove that:
tan (45° + r) - tan (45° - r) = i^T^^*
73. Show that the difference of the two focal radii for the hyperbola
is constant.
74. Find graphically the quotient of 6 — 2i by 3 + 75i.
76. Solve by inspection, for y\
sin (90° + \y) cos (90° - \y) + cos (90° + \y) sin (90° - Jy) = sm y.
76. Write the parametric equations for the circle, the ellipse, the
hyperbola.
CHAPTER XIV
A REVIEW OF SECONDARY SCHOOL ALGEBRA
300. Only the most important topics are included in this
review. From five to ten recitations should be given to this
work before beginning regular work in Chapter I.
With the kind permission of Professor Hart, a number of the
exercises have been taken from the Second Course in AlffebrOi
by Wells and Hart.
301. Special Products. The following products are fundamental:
(1) The product of the sum and difference of any two numbersi
{x + y){x — y) = x* - I/*
(2) The square of a binomial:
{x ± I/)* = x« ± 2xy 4- y*
r
If the second term of the binomial has the sign ( — ), then the mid-
dle term of the square has the sign ( — ).
(3) The product of two binomials having a common term:
(x + a){x + &) ^ x^ + {a + b)x + ab
thus (x + 5)(x - 11) = x« + (5 - n)x + 5(-n)
= x^ - Qx - 55
(4) The product of two general binomials:
{ax + b){cx + d) = acx^ + (jbc + ad)x + bd
thus
(3a - 46) (2a + 76) = (3a) (2a) + (- 8 + 21)o6 + (-4 6) (76)
= 6a« + 13a6 - 286*
Exercises
Find mentally the following products:
1. {5x - 2y)\ 4. (2m + 3)(m + 4).
2. (a + 116)(a + 36). 5. (y* + 4z)(y« + Aa).
3. (a - 2tO(a + \2v), 6. &xy - 7)*.
452
SECONDARY SCHOOL ALGEBRA
453
\uh) -4t)(Suh) + 4).
Ix - 6){x + 4).
Ir* - 7)(3r2 + 5).
»'-3?)(p« + 7g).
» + i)(a - i).
b + 52/) (Jx - 5y).
i - i)(t^ - i).
Ix + S){ix + 1)\
lx^ + ^hc){3x* - 4bc).
/ - S)(y + 5).
c - i){ic - i).
L -6s) (3 + 2s).
It - 7w^)(St - 4w;2).
h* - i)(«w + i).
ir -70(5r + 20.
Llx« - l)(12a;2 + 1).
5* -6)(z« + 12).
c + 3i/2)(a; - 22/2).
)m« - 6s2)(5m3 + s^)
)aj + i)(5x - i).
Ix + 7)(x -5).
ia - 36»)2.
I + 6) (a -6)(o2 + h^)(a*
I + & + c)2 = ?
I + 6)» = ?
I - 6)» = ?
c* - [32/2 + {2x2 - (2/2 +
b - [462 - (2a2 _ 62) - {
5,2 _ [22/2 + (92 - 2yz)] =
29. (2 - 3si)(5 +2st).
30. (o26 + 6c)(a26 - 13c).
31. (Zp + 5)(Zp-4). ,
32. (a8 + 7)(a3 - 11).
33. (3o + 5) (7a -8).
34. (1 + 8w)(l - 9w).
35. (2a - 6^) (2a + 36*).
36. (I2x - i)(9x - i).
37. (20 - 16z)(3 +2z).
38. (r3 + 16s) (r 3 - s).
39. (a -6x2) (a .^3.2).
40. (4r + wv) (4r — 5wv) .
41. (6X2 _ 1)2.
42. (1 + 23n)(5 - n).
43. (x* -y')(x*+y').
44. (5a2 - 46)(6a2 - 56).
46. (x^y + y^x)(x^y — y^x).
46. (Ja + 10)(2a+i).
47. (9r + 2s)(3r -4s).
48. (12x2 + 5) (4x2 _ 3),
49. (a26* + 4x2)2.
50. (a« - 6«)(a6 + 6«).
+6*).
55. (a + 6)* = ?
56. (a - 6)* =- ?
57. (a + 6 + c + d)^ = ?
3X2) 4-51/2} _ ^2] ^ ?
- 5a2 + 2a6 -362}] = ?
?
I
;. Factoring. A rational and integral monomial is one that is
up of the product of two or more arithmetical or literal number 3.
10, 7x, 4o6c, 6a262/2 are rational and integral, but 2a/6, 36v'x
)t.
3 algebraic sum of any number of rational and integral mono-
is called a rational and integral polynomial,
factor an algebraic expression is to find two or more rational
ntegral expressions which will produce the given expression
multiplied together.
ct to the removal of a common monomial factor from all of the
of a polynomial, as, for example, na + nh -[■ nc = w(o + 6 + c),
454 ELEMENTARY MATHEMATICAL ANALYSIS
the most fundamental cases of factoring are those depending u
the special products of the preceding section. Thus,
(1) %rhe difference of two squares equals the product of the sum
the difference of their sqiuure roots:
x« - y« = (x - y)(x 4- y)
Thus
81a« - 6« = (9a* - 6»)(9a* + 6»)
(2) A trinomial is a perfect sqtiare when, and only trA^n, two c
terms are perfect squares and the remaining term is twice the prodi
their square roots.
To find the square root of a trinomial perfect square, take the sq
roots of each of its two perfect square terms and connect them h]
sign of the remaining term.
Thus, 9a' — 24a6 + 166* is a perfect square, since v^9<^ =
v'li5b> = 46 and 24a6 = 2(3a)(46).
Also 9a' + 30a + 16 is not a perfect square, for 30a does not e
2(3a)(4).
(3) Trinomials of the form x' + px + 9 can he factored vhen
numbers can be found whose product is q and whose sum is p.
Thus x« - 4j - 77 = (x + 7)(x - 11), for 7(- 11^ = - 77
(+7) + (-11^ = -4.
(4) Trinomials of the form ax* + 6x + c, if facUj^rahU, wtay ht
tared in accordance with the properties of the special product 4' . !
In the product
ox + 6
ex -^d
acz^ + (6c + ad)x + bd
the terms acx^ and bd are called end products and hex and ^ii
called cro^ jMroducis, This most important case of factoiisg ^
learned from the consideration of actual examples.
Factor 21x5 _^ 5j. _ 4
From the term 21x*, consider as possible first tenns 7x sad
thus (7x )(3x ). For factors of (— 4), try 2 and 2, wink m
signs> and signs so arranged that the cross product with larger sbec
value shall be positive: thus (7x — 2)(3x + 2). This gives mi
term &r; incorrect. For ( — 4) try 4 and 1, with signs seieet»£ *
fore; thus^ (7x — l)(3x -h 4>. Middle term 25x; ixfeeorrecti.
(7x + 4)(ax - 1). Middle term an ; correct.
SECONDARY SCHOOL ALGEBRA 456
(5) The difference of two cubes: x^ — y^ =^ {x — |/)(x* + xy -\-y*).
Thus 27a;» - !/• - (3a;) » - (|/«)»
= (3a; - |/«)(9x« + 3x2/2 + y*)
(6) The sum of two cubes: x' + |/' = (a; + |/)(a;* — xy + |/^).
Thus 125o» + 6» = (5a) » + (5»)»
= (5a + 68)(25a« - 5a6» + 6«)
303. To factor a poljmomial completely, first remove any monomial
factor present; then factor the resulting expression by any of the type
forms which apply, until prime factors have been obtained throughout.
Thus,
•
(a) 5a« - 66« = 5(a« - 6«) = 5(a3 - b^)ia^ + 5«)
- 5(a - 6)(a« + ab + b*)(a + h)(a^ - ah + 6«)
(6) 42aa;« + lOaa; - 8a = 2a(21x^ + 6a; - 4)
= 2o(7a; + 4)(3a; - 1)
Exercises
Factor the following expressions:
1- tV** - IfV*- 22. a;2 + 6a; - 27.
2. 9a;« - Ay\ 23. c» -64i».
8. 25a;* - 1. 24. Sx^ - 1.
4. 81 - 4a;«. 26. 1 - 13« - 68i«.
6. 1 - 64a26*c«. 26. a;* - 6a;*6 - 556«.
6. X* — y*. 27. aw* — 4awt; — 45ai;*.
7. 225 - a«. 28. 28a« - a - 2.
8. 121a;2 - 144y«. 29. 3s* - 17«« + 24^*.
9. 49m* - SQx^HK 30. 15r« - r - 6.
10. 169 -:^ a*a;2. 31. 4ty^ - 3?/ - 7.
11. 4a;* - 20a; + 25. 32. 64w« - 27a;«.
12. 9a* + 6a6 + 5*. 33. 6ar - 3a« + 4a^
13. a*6* - 17a5c - 60c*. 34. a* + 2a - 35.
14. r* - llr* + 30. 35. 9a;* + 12xy - 322/*.
16. 165* + 306 + 9. 36. a* + 10a6 + 256*.
16. 81w* + ISOui; + lOOv*. 37. 625a;*2/* - ^.
17. 36a* - 132a + 121. 38. ^cdy^ - 9cdy - 30cd.
18. a;*2/* - 4a;2/* + 4. 39. 4aa;* - 25ay*.
19. a*6* - 2a6 - 35. 40. 3y^ +24.
20. u« + t*» - 110. 41. 4a;* - 27a; + 45.
21. a*6* - 14a*6 + 49. 42. 6a;* + 7a; - 3.
466 ELEMENTARY MATHEMATICAL ANALYSIS
43. ^^z^ - 1. 58. 2am« - 60a.
44. lOoj'y - 5a;«y2 - hxy^, • 69. 72 + 7a; - 49a;«.
46. mhi^ + 7mn - 30. 60. 31a;« + 2Zxy - 8y*.
46. x* - 3x2/ - 702/«. 61. 24a2 + 26a - 5.
47. mx« 4- 7mx - 44m. 62. 1 - 3a;y - lOSx^y*.
48. x^ - 3a:« - 108x. 63. x* - 14mx + 40w*.
49. x« - 2/». 64. 26 + lOafe - 28a«5.
50. X* - 5x«y - 242/«. 66. c» + 27d».
61. 8w« + 18n - 5. 66. 3x«y - 27x2/».
52. 3x* - 12. 67. i^x'^y'^ - ^x^y*.
63. 9m2 - 42w^ + 49i^ 68. 49w*2/ - 196w*2/».
64. lOx* - 39x + 14. 69. x« - 16x + 48.
66. 12x« + llx + 2. 70. x« + 23x - 50.
66. 36x2 4_ 12a; _ 35. 71. ahi^ + 31aW + 30.
67. x» - Sy\ 72. 9x« + 37xy + 4y«.
304. General Distributive Law in Multiplication. From the mean-
ing of a product, we may write
{a + 6 + c4-. . .)(x +2/4-2+. . .)=ax+5x + cx+. . .
+ ay + hy + cy +, . .
+ a2 + 62 + C2 + . . . ,
etc.
Stating this in words : The product of one polynomial by another is the
sum of all the terms found by multiplying each term of one polynomial
by each term of the other polynomial.
To multiply several polynomials together, we continue the above
process. In words we may state the generalized distributive law of
the product of any number of polynomials as follows :
The product of k polynomials is the aggregate of all of the possible
partial products which can be made by multiplying together k terms^ of
which one and only one must be taken from each polynomial.
Thus,
(a + 6+c + . . .)(x+2/+z+. . .)(w+v + ip+. . .)
= axu + axv + . . . + ayu + ayv + . . . + azu + azv + . . .
+ bxu + bxv + . . . + byu + byv + . . . + bzu + bzv + . . .
+ cxu + cxv + . . .
"^ . • • , exc.
// the number of terms in the different polynomials be n^ r, Sy t. . .
respectively, the total number of terms in the product will be nrst . . .
The student may prove this.
SECONDARY SCHOOL ALGEBRA 467
306. The Fundamental Theorem in the Factoring of x» + a».
The expression (x" — a") is always divisible by (x — a).
p Write x** — a** = x* — ox""* + ax"~* — a*»
= x'^'^ix — a) + a(a;"~i — a""*)
Now if (a;*~* — a"~*) is divisible by (x — a), then plainly
x^~^(x — o) 4" aCx**""* — a"""*) is also divisible by (x — a). But
this last expression equals (x* — a"), as we have shown. Therefore,
if {x — a) exactly divides (x""* — a**"*), it will also exactly divide
(x* — a").
But (x — a) will exactly divide {x^ — a^), therefore it will divide
(x* — a*)f and since {x — a) exactly divides {x* — a*) it will exactly
divide (x* — a*), and so on.
Therefore, whatever positive whole number be represented by
n, {x — o) will exactly divide (x** — a").
We see that (x — a) is one factor of (x** — a"). The other factor
of (x* — a**) is found by actually dividing (x** — a**) by {x —a).
Thus
(x* - a'») = (a; - a)(x»»-i + ax»-2 + a^x"-^ + . . . + a^-^^ 4. a^-i)
The student may show that (x + o) divides x« + a" if w be odd, and
divides x" — a** if w be even.
306. Quadratic equations are usually solved (a) by factoring, (6)
by c«>mpleting the square, or (c) by use of a formula.
(a) To solve by factoring, transpose all terms to the left member of
the equation and completely factor. The solution of the equation is
then deduced from the fact that if the value of a product is zero, then
one of the factors must equal zero. Thus
(1) Solve the equation
x^ 4- 54 = 15x
Transposing x* — 15x + 54 = 0
Factoring (x - 9)(x - 6) =0
x-9=0ifx = 9
X - 6 = Oif X = 6
Hence the roots of the equation are 9 and 6.
Check: Does (9)* + 54 = 15 X 9?
Does (6)2 + 54 = 15 X 6?
(6) To solve by completing the square, use the properties of
(x ± ay = x* ± 2ax + a*, as follows:
458 ELEMENTARY MATHEMATICAL ANALYSIS
(2) Solve x^ - 12a; = 13.
Add the square of 1/2 of 12 to each side
x^ - 12a; + 36 = 49
Take the square root of each member
a; -6 = ±7
Hence
a; = 6 + 7 = 13
a; = 6 -7 = -1
Check: Does (13)* - 12 X 13 = 13?
Does (-1)2 - 12 X (-1) = 13?
Smce in general {x — a){x — h) = x* — (a + ^)^ + o&i we can check
thus:
Does 13 + (-1) = - (-12)?
Doesl3(- 1) = -13?
(3) Solve x^ - 20a; + 97 = 0.
Transpose 97 and add the square of 1/2 of 20 to each side:
x^ - 20x + 100 = - 97 + 100 = 3
Take the square root of each number:
x - 10 = ± \/3"
Hence
xi = 10 + a/3^
X2 = 10 - \/3
Check: Does xi + X2 = — (— 20)?
Does X1X2 = 97?
(c) To solve by use of a formula, i&rst solve
0x2 4- 6x + c = 0 (X)
The roots are
X =
2a
(2)
For a particular example, substitute the appropriate values of a, 6,
and c. Thus:
(4) Solve 2x2 - 3a; - 5 = 0.
Comparing the equation term by term with (1) we have
a = 2, b=-3, c=-5
SECONDARY SCHOOL ALGEBRA 469
Substitute these values in the formula (2)
Therefore
_ -(-3)±V(-3)«~4(2)(-5)
2(2)
3 + 7
Xi = 5/2, 072 = — 1
Check: Does xi + a;2 = — h/a = 3/2?
Does a; 1X2 = c/a = — 5/2?
Exercises
Solve the following quadratics in any manner:
1. a;« + 5a; + 6 = 0. 29. Zx^ - 12ax = BSo*.
2. x^ -\-4tx ^ 96. 30. 4a;2 - 12aa; = lQa\
3. a;2 = 110 4- x. 31. a;* - a; = 6.
4. a;« + 5a; = 0. 32. a;* + 7a; = - 12.
6. 6a;« + 7a; + 2 = 0. 33. a;^ - 5a; = 14.
6. 8a;2 - 10a; + 3 = 0. 34. x^ + x ^ 12.
7. a;* 4- ma; - 2m* = 0, 36. a;* - a; = 12.
8. 3<« - « - 4 = 0. 36. a;« = 6a; - 5.
9. 10r2 + 7r = 12. 37. a;* = - 4a; + 21.
10. a;* + 2aa; == h. 38. a;* = - 4a; + 5.
11. a;« + 4a; = 5. 39. a;* + 5a; + 6 = 0.
12. x2 + 6a; = 16. 40. x^ + Hx = - 30.
13. 2a;« - 20a; = 48. 41. a;* - 7x + 12 = 0.
14. x^ + 3a; =18. 42. x^ - 13a; = 30.
16. a;2 + 5a; = 36. 43. 3a;2 + 4aj = 7
16. 3a;« + 6x = 9. 44. 3a;2 + 6a; = 24. '
17. 4x2 _ 4a; = 8. 46. 4x2 - 5x = 26.
18. x« - 7x = - 6. 46. 5x* - 7x = 24.
19. X* - ax = 6a2. 47. 2x2 - 35 = 3x.
20. x2 - 2ax « 3a2 48. 3x2 - 50 = 5x.
21. x2 - X = 2. 49. 3x2 _ 24 =6x.
22. x2 + X = a2 + a. 60. 2x2 _ 3^; = 104.
23. x2 - lOx = - 9. 61. 2x2 + lOx = 300.
24. 2x2 _ 15a. = 50. 62. 3x2 _ lOa; = 2OO.
26. x2 + 8x = -15. 53. 4x2 - 7x + i = q.
26. 3x2 4. i2x = 36. 64. |x2 - f x = - f^.
27. 2x2 + lOx = 100. 66. 9x2 + 6x - 43 = 0.
28. x2 - 5x = - 4. 56. 18x2 - 3x - 66 = 0.
i
460 ELEMENTARY MATHEMATICAL ANALYSIS
57. |x* - 3x + li = 0. M. 2x« - 22x == - 60.
X* ar 60. 3x« + 7x - 370 = 0.
^:i -^ -^^-0, 61.5r*-ix-T3y=0.
X* X 1
•*-3~2+6=^
68. X* + 2r + 1 = ex + 6. 69. «* = 5« + 6.
64. x« - 49 = 10(x - 7). 70. r« + 3r = 4.
66. 2x* + 60r = - 400. 71. 2»* + 4iw - c = 0.
66- a* + 7a + 7 = 0. 72. x* + 6ar - 5 = 0.
67. 2* = 32 + 2. 78. X* - lOax = - 9o^
66- r = r« - 3. 74. cr» + 2d:r + « = 0.
76. 2r* + 6x - n = 0.
7«. y* + fy = |. 88. 4x« - 3x = 3.
77. x« = 5 + |x. 64. 9(> + 41 == 6.
78. i*» - f u - 1 = 0. 86. 5(x* - 25) - X - 5-
79. «» + i< = |. 86. 9i«» + 18« + 8 = 0
80. r« - I = |r. 87. x« + JPtr + «.
81. «* - |« = Y- 86. X* - 8x« + 15 = 0.
81. 3r* - 2r ^ m. 89. m^ - 29m* + 100 = 0.
X* * 3x •'•5-x^8-x "*■
5 ^. n — 3n+4
91. 2y ^ f = J-- 94.
_ %
4y n — 2 n
* 4x 4x* x X
24 24
96. — - ^ -r 1 = 0.
X X — 2
97. x« - 3ar» -h 216 = 0.
100.ii-r-=a-r-
307. The Defiiiitioiis of Es^onents.
Vl^ » a ptusitive integer: a* = oocx . . . to » factors.
V2^ n and r pcssitive integers: a' ' = \^ and o* ^ = kv'^;'
= V^.
(3> a* = 1.
(4) » any number, poatire or negative, integral or fractional:
a~« = 1 a*.
308. The Laws of EipOAeBts. For n and r any numbers, positive
or negative, integral or fractional :
Vl) o*tt^ = a*^« or law for multiplication and divisioa.
SECONDARY SCHOOL ALGEBRA
461
(2) (a^y = a"', or law for involution.
(3) a^b* = (a6)», or distributive law of exponents.
Note: The student must distinguish between — o" and (
Thus - 8^ = - 2, and (- 8)^ = - 2, but (- 3)* = 9 and -
-9.
a)^
32=
Exercises 1
Use the definitions of exponents (1), (2), (3), (4) §307, and the laws
of exponents (1), (2), (3), §308, and find the results of the indicated
operations in the following exercises.
1. x^^x^K
2. a^^'a^'',
3. a;«»+ix«.
4. h^b^+K
6. a»-"2a8+n,
7. aj^-'+V.
8. fn^'*m~*''.
9. a;" -^ a;8.
10. X8 -h X^\
11. a3« -h a».
12. e^^** -^ e3.
13. 10^+3 -5- lO^
14. n''+8 -^ n''+3.
16. W"-^ -^ w"-'.
16. x""-^-^^ -^ x\
2«- ©"•
17. (a7)3.
18. (a*)«.
19. {- ah^y,
20. (a22/2)^
21. (6"»)2.
22. (- a»6'-)3.
23. (a36«)«.
24. (r"»s»)P.
27.
Exercises 2
Write each of the following sixteen expressions, using fractional
exponents in place of radical signs :
1. Vo.
6. v^a5.
2. y/aK
6. (v^)«.
3. y/a\
7. \/a^
4. \/aK
8. {</^y.
9. V^x'.
«/— N
10. (Vx)».
11. v^:
12. (v'J)*.
x'.
13. Va- 5.
14. (v^5^^)'.
15. Va* - 62.
16. V'Ca+b)'.
Find the numerical value of each of the following sixteen
expressions:
17. 4*.
21. 625*.
26. 81*.
29. 256*.
18. 27*.
22. 64*.
26. 125*.
30. 64*.
19. 9*.
23. 216*.
27. 32*.
31. 512*.
20. 16*.
24. 16*.
28. 81*.
32. 128*.
Write each of the following expressions %n two waysj using radical
signs instead of fractional exponents :
462 ELEMENTARY MATHEMATICAL ANALYSIS
as. ai
37. n*.
34. 1*.
38. 6*.
•
36. m*.
39. e*.
36. 2*.
40. hi.
41. r*.
46. a<.
9
r
42. x«.
46. h*\
43. yi
IH-I
47. X ' .
•
rf«
44. a^
48. a < •
Exercises 3
Perform the indicated operations in each of the following examples
by means of the laws of exponents.
1. a* X o*.
2. X* X x^, 4. xi X x^, 6. xT* X aTi
3. X* X x^. 6. o* X o*. 7. a^ X oi.
8. o^ -^ a*.
9. ;i* -J- ;i*. 11. 8o»6* -r 4o«6i 13. 6a* -^ 3ai.
10. m* -¥ m^, 12. 9o* -5- o*. 14. ah^ -^ a* 67;.
16. (a*)*.
16. (a*) A. 18. (a*)*. 20. [(x»)T]f.
17. (^*)*. 19. (o*)*. 21. {xiky.
22. (a^x^yi)^.
23. (a26*)i. 25. (36a*x*2/»)*. 27. (32x^2/*)*.
24. (adi)i 26. (a^xV)"- 28. {\a^h^c)^.
a'
36. (a* + a* + l)(a* + a - a*).
SECONDARY SCHOOL ALGEBRA 463
We arrange the work thus:
a* + o* + 1
o^ 4- a — a^
a^ + a^ + a^
a^ + a!^ + a
— a^ — a — a^
37. (a; + 2yi + Syi){x - 22/* + Sy*).
38. (x* + yhixi - 2/*).
39. (a* - 3o*6* + 4a?6 - ah^){J - 2a*6*).
* i- * 1 i-
40. (o^ — 2a»»+ 3o«)(2o'» — a»).
Exercises 4
Find the numerical value of each of the following:
1. 2-1. 4. 10-». 7. 2-*. 10. 1024-*.
2. 4-2. 6. l-» 8. 16-*. 11. 512-i
3. (-2)-3. 6. 2-«. 9. 81-*. 12. 625"*.
1 5 5-2 16-|
13. ^r::rr 15. (-. ±\-t' 17. r- 19. -^:^'
o 1-8 32-t 7-1
14. ^. 16. I^i- 18. ^sn^. 20.
F^ *-• 8-1 — 2-1 --• 49-i
Write each of the following expressions without using negative
exponents:
21. a;-2. 26. So-". 29. {x + y)-\ 33. 2a»a;-22/-*.
22. x^y-^. 26. Ba-^b"*. 30. (- x)-^. 34. (- a2)-3.
«« 1 o., 2a-2 ^^ a;* ^^ a-M
23. — r^- 27. ^T^-:i^- 31. -^k* 35.
x-^ — 362X-3 — y-5 -"• 3a;45-i
«.. ^"' «« «^&"* •« 3of6-i ^^ 3a26-2c-*
24. ^ri- 28. _, .• 32. i — 36. g _,,_. _..•
Write each of the following expressions in one line:
37.1. 39.4?,- 41. ?5^- 43. ^"^'-
38. — • 40. o -n -,• 42. —7-^- 44.
o'' 3a-22/-3 M*2-' x-iz/*^'
46. — ^ i ^' 46. — i + -i H h — i?'
464 ELEMENTARY MATHEMATICAL ANALYSIS
Exercises 6
Perform the indicated operations in each of the following by means
of the laws of exponents.
1. a8 X a-6. 4. 8a-* X 3a«. 7. m""* X w*.
2. r" X r-io. 6. u-^ X w*. 8. 6aa;-« X hhxK
3. c-8 -r c-^ 6. x» -5- x-". 9. a-86-« 4- oft-'.
10. ( - 7a-»6-«) ( - 4o26-J ) (a-^^-' ) .
11. (2a*6-*)(o-*6*- io*6* + a*6-*).
^-65-10) -f .
- 1x8)-*.
- a-»)-».
- a»)-* .
\ xy^ /'
a«6-3 \_j.
12. 7a-i6-2c-3-f
■8a-
-26-3c-
-*. 23. 0
13. 5Qx^y-h* -f
Tx-
'I7/-82-
-*. 24. (.
14. 18o-4&tc-»
-^ Qahh-
■\ 26. (
16. Qx^y-^zi -f-
2x-
■*y»2"
*• 26. (.
16. (o-3)2.
27. 0
17. (a-^)-K
18 (a«)-».
28. (
19. (n*)-3.
29. (
20. (r-l)-*.
30. (
21. (c-8)l.
31. (
22. {abc)-^.
32. (
33. f):^
36.
m-'
»- (S)-
37.
/ a-^b^ \ 2
[x-'y-y '
35. g-e).
38.
/a-^bX-i
\x^y-y '
39.
41.
x-^y ^
V
42. (a2x-i+3a3x-2)(4a-i - 5x-i + Qax-^)
4a-^ -- 5x~^ + 6aa;"*
aH-^ 4- Sa^x-^
4ax-i - 5a2a:-2 + Ga^x'^
12a^a;-^ - ISa^^-^ -f ISa^x-*
Aax-^ + 7a2x-2 - 9aH-^ + ISa^x'*
43. (2x-^ -Sx + 4a;^)(x-^ - 2x-* + Sx"^).
44. (x-* - 2x-h^ + 2/*)(x-^ - 2/*).
45. (3x* - fxi + 4) X 2x-i.
46. (x-* + x-J + l)(x-* - 1).
47. (x-*+ 2/-2)(x-i - 2/"')-
48. (x^y + yhixi - 2/-i).
SECONDARY SCHOOL ALGEBRA 465
49. (2a*- 3ox*)(3a-i + 2x-*)(4a*a;* + 9a-^x*).
60. (x-* - x-^y^ + x-^y - ?/*) -J- (x-^ - y*).
X-* - x'^y^
x~^y — y^
61. (x-^ + 2a;-2 - 3x-i) -h (a;-^ + 3x-i).
309. Reduction of Surds or Radicals.
1. // any factor of the number under the radical sign is an exact
power of the indicated root, the root of that factor may he extracted and
written as the coefficient of the surdy while the other factors are left
under the radical sign.
(1) Thus, a/8 = \/4 X 2
= VW2
= 2\/2
(2) Also, V81 = i/27 X_3
= V27i/3
=3^3
(3) Also, Vl^ax* = VSx^ X2ax
= V3x^V2ax
= 2xV2ax
2. The expression under the radical sign of any surd can always be
made integral.
(DThus ^b^lxl = ^:
18
27
= a/£-X18
27
= Vi8
3
(2) Also A/rA/^^l = A/i^^"
30
466 ELEMENTARY MATHEMATICAL ANALYSIS
3. TFe may change the index of some surds in the following manner:
(1) Thus, \/4 = VVi
(2) Also, v'rdoo = Wiooo
= Vio
(3) Also, \/256c«a8 = \/\/256c«o<^
A surd is in its simplest form when (1) no factor of the expression
under the radical sign is a perfect power of the required root, (2) the
expression under the radical sign is integral, (3) the index of the surd is
the lowest possible.
Methods of making the different reductions required by this defini-
tion have already been explained. We give a few examples.
(1) Simplify -V —
ify -\/ —
\86«
\86« \265
lify ^^
2&
(2) Simplify ^P-.
^/400 _ /20
(3) Simplify
.. 5^/512
= |V15
5\125 2\5
2\125 2\5
I
= Vio
SECONDARY SCHOOL ALGEBRA 467
' In any piece of work it is usually expected that all the surds will
finally be left in their simplest form.
Exercises
Reduce each of the following surds to its simplest form :
81*
9. Simplify V12 + iVV75 + 6V^-
10. Simplify 1 + Vs + V2 - V27 - Vl2 + V75.
11. Simplify i^21 +7V2 X ^21 - 7V2.
12. Find the value of a;^ - 6a; + 7 if a; = 3 - V3.
13. Find the value when x = \/s of the expression
2a;- 1 _ 2a; +1
(x-iy (x + iy
14. Find the value of
(SSVlO + 77\/2 + 63\/3)(VlO + V2 + V3).
Solve and check each of the following equations :
16. Va;+4 = 4.
16. \/2x + 6 J= 4.
17. VlOx + ie = 5^^
18. V2a; + 7 = V5a; - 2.
19. 14 -M/4a; - 40 =_10.
20. Viea; + 9 = 4V4a; - 3.
21. \/¥ + X = I + Vi.
22. \/a; - Va; - 5 = Vg.
23. Va;-7 = Vx-U + 1.
24. Va; - 7 = Va; + 1 - 2.
26. X = 7 - Va;2 - 7.
«. \/a;+20 - Vx-J -3=0.
■37. ^/x+_S_+ Vdx-2 = 7.
^8. ^/2x + 1 + Va; - 3 = 2 Vi.
20a; , 18 . ^
29. -/— ^ - VlOa; - 9 = /.^ ^ + 9.
V 10a; - 9 VlOa; - 9
a?- 1 Vx + 1
30.
31.
Vx- 1 "a;- 3
Vx + \/x — 3 _ 3
\/5 — Vx^ 3 a; —3
aoAj'OoSfi oiaS
ofiSi 0710O155
.38 iO»
'J "33
38J0 3SjE j8;6
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4030' 46 s 4 466g
J4S00 4S14
48-'B U»43 4*SV
18 454* 4TSi
4871 488(1 40 or
5-iIi S034SO38
6083 600616107 6
436JJS
:ns;;
4 64 84 0493
'1 fisW 6sgo
678567041
6S75 6884 6S03
6964 6973 6g8r
1 B
. 1 =. 1 a 1 4 1 S . 6
T 1 8 1 9 1.2 3,4 S 6.7 8 0
SI
7185
7367
;i;ps!iii
7I3S|7i43l7!Sa
72I8|7326,733S
7J0D|7308|73l6
11 3' ] t •
a ? ?
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7314
733' 7340
7348
7356
7J64!7375
7443:7451
738o;7388 7396
7459 7466,7474
11 i\i t I
t tl
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7634
7642,764!
ii
7S"3'7S30 7528
7S89 7597'76n4
766476727679
7535 7S43'7S51
7613 7619 7627
7686,7694:7701
■ 3 2I 3 4 5
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77I6|77S3
77BOi779(
7803
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?8B3,7889'7896
77607767 7774
7833 7839 7846
: ; 2 1 1 ;
I il
a
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ii
793l!7938,7B45
Bono 8007:8014
8060 807SS08>
7953 7959' 7966
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8080,8096 8103
7973! 708o' 7987
Tltl 8?56,EI3 3
: ; =j j 3 4
I 11
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Btie
81368141 B149
Bis6!8l62,8.69
81768183,8189
66
ii!
8jO?|8j?4'aiBo
8331,8338,8344
ill';
B3S1
8328 8335
830318390
8357.8363
8420' S426
8483 848 B
8543:8540
i603 8609
8722:8727
8341 834B|B254
B306 8313 S3I9
8370 8376 8382
5 n ! H
69
7<
ill
B39S 8401:8407
84s 7 8463 S 470
8510 BS'S 8531
B537
8433 8439' B 445
8494 8500.8506
8SS5 85618567
" S !i 1 s S
!
i
8570 858S'8S9I
863986458631
B597
S6S-J
t6lS8621;B637
8733 8739 874s
i nl ! y
87S6 8763:8768
8774
8779'8!8S
8837 8841
88938899
8949 80S4
8791,a7!l7|Bao3
8848 8S54 8859
8904 8010 89 15
8960 8965,8071
~.i ' j 4 ; 5
i
Off
8814 88208815
8871 8876;8883
S037 893a;8938
IssJ
E
s
8983 8987,8993
903630439047
0090 9096 9101
8998 9004 9000
0OS3 9058 906 J
B.BB
Will iiu
1
9196 0301I9306
93489253,9258
!>I50:9l65'9nO
9263 9369 9374
917S9l8o'9ia6
H'l 9384 "so
n |j n
85!^
86l93«
8a|9494
9400 940s '041°
94509455; 9460
9595 9600 9605
946s 9469 9474
951395180333
9609,0614 9619
938Q 9385 9390
l-r';
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9S38 9533 9S3f
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1 1 11 ti
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5638
968s
9643 9647:9652
9689 0604.9699
9736,9741 ]9745
9657 '9661 '9666
9703:9708,9713
9671 9675 9680
9763 976a 977;
III'''
95
0777
O783I9786
9926
970S 9800 980s
084198439850
98B6 9a9t.!9894
9939 903419939
9809,96149818
0 1 I
= '3344
IS
a8>3
9868
9837 98 J I
0872|9S7T
98549859986;
9943.9948 00S3
:i i
' ' i, 3 44,
m'boss
9961 996s '9969
9974^99789983
99879091 0096
0 1 .
1
tv
S^fiiS
that
Ki
a of tfae t.
the proM
boTo table whi
rty of Mm««.
M=
lABitBif.
Ib&dfor
LooABiTHus OF Teioonometbic Functions
;o6s8
■3088
!>6g
IS
5B0
.3668
5!!
:S
=63
:W
248
.764s
III
T.U
■ 85
:!ffi
il
V51S
1ST j 18B 1 134 130 US 1 UT
13. 7i 13,5 13,4 13.0 12.9 IS.T
us
12.8
ii
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11:1
47. (
n;.
107 :■
llT
70,;
ii
111
:5J 71:0' ei'.i
iliiaioiio:;
IlilillsliS
liliisltiisliK
50,1
iM:i
bsi
Formula! for
iBi' + S
igi" + T
I - log i" + T
LOGARITBUS OF Trioomouetric Functions
loE-m
d
log t«n
dc log cot
.ogco.| 1 „ \
s ij
I
9S4S
J37
8
9563
I38
;
0580
I
9983 0 Sj
9.8>|so
11?^
;;:.
109
30
I
&
9945
;^3
I
11
il^
;
a
I
958140
9980^30
m
«'i
«'.6
fi 'J
I
"s
I
0336
\ll
:
sii
I
67.8! 6g,6
6^4
ji
I
0436
gi
ii
I
i?l
loa
9«i
I
99?!
i
loB
":6
;:r.
T ^t
I
1060
J "a
!
I
109(
08
g
I
IT.
9566
9964
9963
9961
0 S3
i
64:2
SI
41.0
IV,
S ^J
1
;«^
87
°
138;
is6a
E
asai
I
iir-
'§',
iii
i
I
I78I
at
J
i65(
IB3I
t:
8345
I
Es
u°:
1:1
9 ^o
I
1863
So
78
^
Wi
tl
S0S5
800;
I
094S 10
9946I 0 81
ii ;i^:
E:
40
I
if
76
1
1236
78
768,
I
994' 40
0938 30
I
J
i
ill
ibIj
—
'.
SS!
73
S
^89
1463
74
I
761 1
I
?.&•'.>.
1
80
4|S^
3;*
lis
;j
8:1
S8.1
38,0
lo
COS
"^
log CQt, dc
.OgU.
logsta ' "
£isi
m; «__] 9^
,!•;
,"!,'
?J"=:'^.
^J*o'.!^M
TS
".
;'.i;».i;'.u"J
2 If
a' 27;*! s!
3 36
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il:J
SI lU
5 15.2
3i
■ ! ..M ».7»,.
2!
:
u
30. 0!B. 8^9,3
S:!::|!:!
4 "'
8 M
1 37.3 8|
11
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1.8 6
6 42 ;o
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>, J.,<!.1..11,!
ill!: !!«:!
40
1
i
7 66
S 7fi
e 84
ji|ji
9 80
ii
H?
S| Tslfl
73
4 M.7 B5.3'64.a
1 M-8| 83.2M.;
1
1
a
SKBl!
mulas fa- HJiKg roble inwrjily
■!«[ Job I
V log I
z\i
■tani:-
It llliiii}:i|SiE|,T
H I MlOB I
LoQARiTHMB OP Tbigonometric Functions
•■l-l'
10, .» idc
loi cot
loicoi
"
79
10 o
40
31
J.
I
nil
3674
ag70
iS
I'm
3«6
34ft6
36ag
36Sa
37Sfi
'Ml
3986
4083
64
flo
i
48
^
J463
JOSs
3458
3634
3601
3148
3S04
38S9
13
4T78
69
07
fis
64
(•3
63
Ol
56
•
S980
si;;
S78B
SI
660J
ly;
64J4
6366
6309
67SJ
6106
6.41
603J
SBJfi
5873
S8«
;
0896
9B93
9890
iSl
B
9B63
nil
9BS3
9849
i
0 So
0 »6
1!
k
:|
8 6
J->6
646
iii
i
IDI CO. d
iDCCDt
..[...„.
logidi.
! - =
!i|-
1
^4
aS.fi
11:1
1^:1
6a
36:°
4s:o
n'kt
s
1 55
3ai4
37.8
f;i
16:1
36.4
a
4
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so
1 48
14"
14-
g:
LOQABITHMS OF TRIQONOUBTRIC FUNCTIONS
■ log rin
"
log tan dc ! log cot
log CDS
•II" 1
I
till
4S76
46 IS
till
48II
4861
5163
S306
46
45
i
u
36
11
9.4381
D-43S1
4B
J
S66g
5619
S!38
4B6l
4835
4671
ttil
4469
44 J 9
4389
5,5849
e.p846
5.9843
9.0839
9.5836
9.9Bj.ii
9.9817
9.9814
9.9810
5:9801
9.5198
9.5786
9.9782
9.9778
9-9761
9.9748
:
0 7S
0 T4
0 73
40
40
0 70
1
\
I
I
4s;o;m:?
Mil
■ 1 ,",
A ft
!!:S;!
10|U»
d
lotcot
dc
laiua
log tin
.|. •
9'At'.t,\40-'s
lit
43 1 41
1
,1
■41 *■
!i J'i
6-i'i6.o
6:036:0
39
;1
3
1!
i
4 ';
V.I
331
6
(
JJ:S
31. S
Logarithms of Trigonometric Functions
o /
log sin
d
log tan
do
log cot
log COS
d
PP
ao o
lO
9. 5341
9.S37S
34
34
9.5611
9.5650 ,
39
39
0.4389
0.4350
9.9730
9.9725
1
S 1 0 70
4 ;so
i4
1 0.4
2 0.8
3 1.2
20
30
40
9-5409
9.5443
95477
34
34
33
9.5689
0.5727
9.5766
38
39
38
0.43II
0.4273
0.4234
9.9721
9.9716
9.9711
1
S 40
5 30
5 20
so
3Z 0
10
9.SSIO
95543
9.5576
33
33
33
9.5804
9.5842
9.5879
38
37
38
0.4196
0.4158
0.4I2Z
9.9706
9.9702
9.9697
4 10
5 0 69
5 SO
1
4 1.6
5 2.0
6 .2.4
20
30
40
9.5609
9.5641
9.5673
32
32
31
9.5917
9.5954 '
9.5991 !
1
37
37
37
0.4083
0.4046
0.4009
9.9692
9.9687
9.9682
S 40
5 30
5 ;20
7 2.8
8 3.2
9 3.6
SO
33 0
10
9.5704
9.5730
9.5767
32
31
31
9.6028 '
9.6064 :
9.6100 ■
36
36
36
0.3972
0.3936
0.3900
9.9677
9.9672
9.9667
1
5 'xo
5 0 68
6 so
1
! 5
1 0.5
2 I.O
3 IS
1
4 2.0
s 2.5
6 3.0
1
7 3.5
8 4.0
9 .4.5
20
30
40
9.5798
9.5828
9.5859
30
31
30
9.6136
9.6172 1
9.6208 j
1
36
36
35
0.3864
0.3828
0.3792
9.9661
9.9656
9.9651
S 140
s '30
S l20
1
SO
23 0
10
20
30
40
9*5889
9.5919
9.5948
9.5978
9.6007
9.6036
30
29
30
29
29
29
9.6243 1
9.6279 '
9.6314 i
1
9-6348
9.6383
9.6417
36
35
34 ;
35
34 ■
35
0.3757
0.3721
0.3686
0.3652
0.3617
0.3583
9.9646
9.9640
9.963s
9.9629
9.9624
9.9618
6 10
5 j 0 67
6 ISO
s 140
6 30 1
S 20
so
24 0
10
9.6065
9.6093
9.6121
28
28
28
9.6452
9.6486
9.6520
34
34
33
0.3548
0.3514
0.3480
9.9613
9.9607
9.9602
6 10
5 0 66
6 so
6
1 0.6
2 1.2
3 1.8
4 2.4
s 3.0
6 3-6
7 4.2
8 4.8
9 5-4
38 37
2.8 2.7
5-6 5.4
8.4 8.1
20
30
40
9.6149
9.6177
9.620s
28
28
27
9.6553
9.6587
9.6620
34
33
34
0.3447
0.3413
0.3380
9.9596
9.9590
9.9584
6 40
6 30
5 20
so
25 0
9.6232
9.6259
27
9.6654
9.6687
33
0.3346
0.3313
9. 9579
9.9573
A 10 *
^ 0 6s
log cos
d
log cot
dc
log tan
log sin
d 1' •
39
1 3.9
2 7.8
3 II. 7
38
3.8
7.6
II. 4
37 36
3-7 3-6
7.4 7.2
II. I 10.8
35
3.5
7.0
10.5
34 33 32 31
3-4 3.3 3.2 3.
6.8 6.6 6.4 6.
10.2 9.9 9.6 9.
30 29
1 3.0 2.9
2 6.0 S.8
3 9.0 8.7
4 15.6
5 19.5
6 23.4
15-2
19.0
22.8
14.8 14.4
18.5 18.0
22.2 21.6
14-0
17.5
21.0
13.6 13.2 12.8 12.
I7-0 16.5 16.0 IS-
20.4 19.8 19.2 18.
4 12.0 II. 6
5 is-o 14. S
6 18.0 17.^
) II. 2 10.8
14-0 13-5
[ 16.8 16.2
7 27.3
8 31.2
935.1
26.6
30.4
34-2
25.9 25.2 24. 5 23.8 23.1 22.4 21.
29.6 28.8 28.0 27.2 26.4 25.6 24.
33-3 32.431-5 30.6 29-7 28.8 27.
7 21.0 20.3 19.6 18.9
8 24.0 23.3 22.4 21.6
9 27.0 26.1 25.2 24.3
474
Logarithms or TniaoNOMETRic Fdnctions
loiiln 1 d
log tui j dc
log cot
togcw
d
„ 1
3<
40
so
1^
as
6444
S470
esji
6546
6S9S
Hi
6691
6763
683J
6856
6878
6901
"4
^
nil
v&
is
;i;
7196
7378
7408
;&•
7SS6
7SBS
7614
33
30
30
30
30
30
19
J,
"11
3l8|
3086
i960
1835
JB04
1683
B.3S61
Hi!
9.94B6
9.9466
9.9446
0.9383
6
0 fis
SO
'0 64
40
30
'o6j
30
1
1
1
).8
1:!
S.6
if
tog COB
d
log cot
it
togun
logidn
*
1
I'i
161
1
30 1 a9
pi
17 ifi
iS
1
■
*
6
&
LooABiTHMS OP Trigonometric Fdnctions
-h
logUn
dc
log cot
10|CO>
d
pp
" ,1
34^0
3S 0
9.6ago
'■'Is?
','Z
9:7380
9:74J8
9-7S6fl
9.75BS
:i
9.7614
9.5644
9.7S73
l;!SS
Ki
O.B097
9.8J17
S:Si!J
9.8J98
9.84iS
9.84S1
19
19
i
j8
0.3386
0.53S6
0.1847
0:1683
:■■!•
0.9161
%s
9.0338
i'ii
9.9a6B
9.056O
ill
Hi;;
9-9169
9:9134
i
i
s
I
\
9
0 60
1 **
0 *'
'0 S6
1
i
I
ii
9
log CDS
-
log cot
ac
lOgUB
Iqgsin d
1
1 30 1 19
;S::3:
.;t;S s;r,si;;;i;;i 1;;,
ill h:1i;iiii ;H-
u
8
5:1
LoGARiTBMS OF Trigonoubtric Functions
= ' loi rin
d
losUo
ds
log™*
loKCOi
d
" 1
40
40
so
9.7586
9:7640
'M
'M
9^7893
9.70>e
9:7989
O.B004
Sisojs
9.BC.S0
9.8066
:!
il
IS
ii
lb
il
9.845a
9-84J9
9-8S06
liSI
?;K
9.8630
S:869J
il;;
g-8876
g.890>
9.8918
9.8954
9.8980
9:903"
IB
9:9187
9:9138
16
>6
O.IS48
o:l3B7
o,lj6t
0,1J76
0:1114
0,1098
0:1046
0.086s
o:o8!3
0.0788
9:9098
9.9085
9,906[
0:904a
0:9014
0.8975
9.896J
0.8955
0.894s
9.893s
O-891S
9.891s
9.890s
9 . 889s
lH',i
9.8864
9-8853
0-8843
;:
I
"
1
1
1
1::
1:1
j log CO.
d
lot oot dc
lOltUB
lOEtiu
d
I
i:i
I'i
14.3
16
I:! 3
IS
.o.yo.j 6
;i:t:;:? ^
16.115.3 9
16
iiljiS
LOOARITHMB
6
4 1 S 6 17.815.
12 314 5 6;t 8g||
10 0000
D043 0086:0118
0194^3340374
ii\i
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0569
0607
0969
064s
1038 -oil 1 106
4 7 "
iiii
lis
=
i*Sr
Itl3
iao6ii3o
ISJ3 I5S3
1S84
'.'"
,644
1367 1395 1430
1 673 1703 1731
1 ' 10
!; if i
13 26 30
11 J5«j
"-
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1B47
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'9S9US7
1
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lou
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19 Jl IJ
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106 S
13SS
3S33
3380
1148
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i8
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1S78I
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17.«. 741 1765
1945 19S7 19B9
: J 1 1 ii ii
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lO
30333054
307s
30B6,3Iia
3139
3l6o'3l8l3aoi
2 4 6 B II ,3
tS 17 1!
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ii
3543'3aS3
"36 3^s;
3184
33043314
35-Jll3SlJ
3691 37 1 1
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37S4
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li
41S0
38Jo'3a3a
3597,4014
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3856
4631
3874^3891
404814065
408?
391713945
4009 4116
4265|4lBl
3962
429S
Hl|;ii°liii!i!
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6398 700717016
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70597067
I. 3I. 4 ...Jl
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Logarithms
. 1 . 1 3 1 « : 5 1 6
7 1 B 1 B 1. 3 3 1 « S 6.7 80
!|ti!S
Si
719;
7.84
'VI
S!
V,SV.il
73o8|V3l6
\: iii t 1
6 6 '
A
s;;3S
7348
7437
73S6 736;
m'
73B0
7388,! 7396
74661 7 474
\: ',
I
isi;
s-
75IJ;7S30 7S38
7S8B'7S97i7604
7t,t,i7b71 7679
7S36|7S43 7S51
76861769417701
\ I I
3 S 4 S 6 ?
?i
?B6o 7868 78?;
773B 7745l77Si
JB10.781S783S
7a83,7a8e'7896
7903|79>ol79n
; ; ;
3 4 4I 5 67
!i
8o&0 So75.8aS^
79S3'79S9'7966
SoSQ'Bogfi 8103
70731 7980 7987
;; '■
3 3 4I S 6 6
■■0
ai36 8r4!:aiw
8I56'8I63 8169
8l76'8ia2;8l89
II >' 3 3 4I S 5 6
'.I
8331 8,138 83+.
Bi55 840l'8407
Hs^ s*6j 8470
SSI9 8saS,BS3I
8j87'Bjg3|8309
8351,8357.8363
8>41
B3D6
8370
8«J
B494
BsS5
8348:8354
83116319
8376 8383
'l ! 5J 3 ll 3 n
i?
8414
B476
BS37
S4"'8436
8482,8488
a543]SS49
860318609
8663 8669
B7SJ|S7a7
8439 844s
8500 BS06
8561 '8S67
ii M I :: Vt
il
8579 8585 8S91
sead 8045 »6si
S60S 8704,8710
875687638768
B871 8871^888)
8937 893 s 8038
8597
8716
86l5la631 18617
8675,8681 8686
8733,8739'874S
B791 B797aBoj
'-LJ ' ' '
i'
8774'B775
883T 8837
8887 S8q3
8943 8940
878s
)8
is
8 843
Ba4a:a8s4'8Ss9
8B04,B9Io;891S
89606965:8971
i ; !^ 5 !
;6
11
BliBj'as,ST'8(.53
0036 9041 904-
899 8 9O0i
lo^i
9069 9074 '«079
ii I
I I I
18
9106 9301 gjot
9348,9553 oasB
931i
9163
916;
9>09
9374
1
OiSo;bt86
038419389
1 ; 2
'. 1 3
94
l>399;!)3O4,0305
931S
0335,9340
■ 1 >
> 3 3
'i
mso'mssWo
MOO940S.9410
B4SO 04s S 0460
9365 0370
9465,9469
947'
9380,93859390
9430 9435 9440
9479 948419489
:i i
I • '
BO
!)499!5S04
0S47;9S5J
E^96^
9736|a74'
97aj,!l786
98379833
9871987;
96oi
9S63 9566
9609 9614
ii
9S3B95339S38
963496389633
li i
; ; I
11
9653
9690
9657 9661:9666
9703, 07UB 0713
9671
98DO
0675 O6B0
9733]9737
0768.9773
0 T 1
i I 3
77
0814 9818
0 1 I
J 1 3
3 4 4
u
'it.
0936
lltl
9930
984519850
9890! 080*
99789983
9854
989^
998^
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Van o-nA
LooARiTHMs OF Tbiqonometeic Functions
' ! Idl Hb
TT"
S.30SCJ sSo
B.3669 I Sia .
I'o &g la
6.4636-6
giriis :
i 0 8t 180 I
9 . 998s o 85 3QO I 0.46316
us 141 ISB 1 UT It [ IM ; ISO 11* . 117 1» lU 111 ■ IIB
ill
^l Hi 41:4 41:! 4o;s; 40:2 39:0 zi.i KA 37.°; 36:9, sl'.t 35:7
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23.1
fli (1
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114. 4 113. 6 110. 4100. 0 108.0107. 2 1040103. 2 101. 6 100 0 98,4 97.6 93 2
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"s
ioi:f
I - log !• + T « i lo8 =ot I - log {{
LoGAaiTHiia or TsiaoHOUETRic Functions
loE tux log sin '
U I U I »1 ! S9 I ST I M I » : U I U I 31 | T«
I 18. S 18.0 18.3
S.7' 8. el E.S.
78 I TT TS I TS I Tt I Tl
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30.0138.1 SB. 03T.SS7.0M.
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a: sg.e es.sl es.oi 07,21 «
log i:- - lOK .
col^ r' - log c
LoQARiTHMS OF Teioonometeic FnNCTioNS
let tan dc | !■>■ c(
LooAHiTHMs or Trigonometric Functions
4«9a 4S
1d( cot I dc IDE tan
41
40
as
i\
I ^^
36,
35
i..
t?n
J»
7.61 l\ 7.4
7.
6.,
16.0
1S.6
.ji :!,,:
.4.4
.4.
4.6
J4.0
33-4!!". 8
6 11. a
J1.6
g-7'.li.O
n.3'»6,6
f..
"■»
»4.
6-9
36.0
34.3
olsaia
3^::
LooARiTHMs OF Trioonometric Functions
lOKtin
■J
lottu
dc
lofCOt
log™
0
n
40
IS o
I
S375
S477
is
SOoo
5073
sssa
5889
6036
606s
3>
9.S680
0.5804
9.5a«
9.SB70
9:6064
9.6136
9.6143
9^6314
9.645=
9-6SS3
9-6654
g.6687
38
38
'A
i
36
i
36
33
0.4389
ill
0:3686
0.3583
0.3S48
0:3480
0:^380
0-3346
%■'&
9.96B3
9-9677
9:9667
9.9661
9.9656
9.9651
9.9646
9:963s
9.9639
9.9613
9:9603
9.9596
9.9584
1
\
I
0 TO
0 69
'0 68
50
'o6t
30
'0 66
'0*5
3.»
6
1:6
lOICM d
loE cot { dc lOE tu
lot sin
d
39 33
37 1 36^! 35 34
i
aa
U li n
ivh
174
LoOARiTHHS OF Trigonometric Functions
9.6546
9.6570
S>.6S95
9.6833
9.6856
9- 6878
lot cot d lo( col dc , li>( tan log al
a;
•.ii
lil
1
1
S!
1: :!:
8,
LOGARITHUS OF TRIGONOMETRIC FONCTIONS
loCfiii
d
IO(tU
dc lofCOt
1«I co> 1 d
„ 1
JO <
if
Jl J
40
I
6090
T016
'S
h
IB
0.7614
9.7644
9.7613
O.lSli
0.T84S
sill
0.8014
p. 8097
I'.i'll
0.8180
0^8J35
US'
I'B
9:8«j
59
=8
o:«56
0:1184
oijooB
0.19B6
O.I95S
0.1847
0.1656
'
9361
ii
9«8
0169
91S1
8
0 60
30
0 SS
0 56
0 SS
I
3
I
I
1
I;!
s
3.6
'"-]'
las cat
dc
log tan
Icf^ d
1
B
Logarithms op Trigonometric Functions
• 'h-
d
■«-l-
lofcot
-"■
pp
3<
3<
" i',
3'
3f
I
mi
nil
7C-S7
is
7778
7SaS
7861
7877
7803
75 j6
8004
80:10
Sojs
80SO
8066
18
t
•
8470
8so6
HS
8Sg6
8613
B639
a69J
874s
8771
assS
Sgoi
89=8
8354
•3
9187
9338
27
0.1548
sis:
i;iS;
i-ii
o!i(a4
0,109s
oimjft
iii
0.086s
0.Q839
0.07S8
;
9089
9080
9061
si
8975
8965
89SS
8945
8935
89=5
89 IS
SK
S8S3
8843
■;
30
0 50
i
I
1
1
ft
ti
It
1:1
-™|-
log cot
H —
los^
d
1
I
is'p
16
i
1
'S
■a
ii:
'7
iS
Logarithms of Tbigonometbic Functions
log sin
log tan
dc
log cot
log COS
pp
40 O
10
20
30
40
50
41 o
10
20
30
40
so
4a o
10
20
30
40
so
43 o
10
20
30
40
50
44 o
10
• 20
30
40
SO
45 o
9.8081
9.8096
IS
IS
9.8III
9.812s
9.8140
14
IS
IS
9.8ISS
9.8169
9.8184
14
IS
14
9.8198
9.8213
9.8227
IS
14
14
9.8241
9.8255
9.8269
14
14
14
9.8283
9.8297
9.83II
14
14
13
9.8324
9.8338
9.8351
14
13
14
9.836s
9.8378
9.8391
13
13
14
9.8405
9.8418
9.8431
13
13
13
9.8444
9.8457
9 . 8469
13
12
13
9.8482
9.8495
13
log COS
9.9238
9.9264
9.9289
9.93IS
9.9341
9.9366
9.9392
9.9417
9.9443
9 . 9468
9.9494
9. 9519
9. 9544
9.9S70
9. 9595
9.9621
9.9646
9.9671
9.9697
9.9722
9.9747
9.9772
9.9798
9.9823
9.9848
9.9874
9.9899
9.9924
9.9949
9.9975
0 . 0000
26
25
26
26
25
26
25
26
25
26
25
25
26
25
26
25
25
26
25
25
25
26
25
25
26
25
25
25
26
25
0.0762
0.0736
0.07II
0.0685
0.0659
0.0634
0.0608
0.0583
0.0557
0.0532
0.0506
0.0481
0.0456
0.0430
0.040s
0.0379
0.0354
0.0329
0.0303
0.0278
0.0253
0.0228
0.0202
0.0177
0.0152
0.0126
O.OIOI
0.0076
0.0051
0,0025
o , 0000
9.8843
9.8832
9.8821
9.8810
9.8800
9.8789
9.8778
9.8767
9.8756
9.874s
9.8733
9.8722
9.8711
9 . 8699
9.8688
9.8676
9.866s
9.8653
9.8641
9.8629
0.8618
9.8606
9.8594
9.8582
9.8569
9.8557
9.854s
0.8532
9.8520
9.8507
9.8495
log cot
dc
log tan
log sin
12
o 50
so
40
30
20
10
o 49
SO
40
30
20
10
o 48
SO
40
30
20
10
o 47
50
40
30
20
10
o 46
SO
40
30
20
10
o 45
I
2
26
2.6
5.2
7.8
25
2.5
so
7.5
10.4
13.0
6 15.6
18.2
20.8
23 -4
4 lo.o
S'i2.S
6 is.o
7'i7.S
8|20.0
9;22.s
15
1. 5
3.0
4.S
6.0
7.5
9.0
10.5
12.0
135
14
1.4
2.8
4.2
S.6
7.0
8.4
9.8
II. 2
12.6
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
13
1.3
2.6
3.9
5.2
6.5
7.8
91
10.4
II. 7
10
I.O
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
II
I.I
2.2
3.3
4.4
55
6.6
7.7
8.8
9.9
12
1.2
2.4
3.6
8
.0
7.2
8.4
9.6
10.8
t:
478
«
Natural Trigonombtbic Functions
Deg.
Radians
n sin
n CSC
n tan
n cot
n sec
n cos
o
0.0000
.000
.000
1. 000
1. 00
1.5708
90
I
2
3
0.0I7S
0.0349
0.0524
.017
.035
.052
57.3
28.7
19. 1
.017
.035
.052
57.3
28.6
19. 1
1. 000
1. 001
1. 001
1. 00
.999
.999
1.5533
1.5359
I. 5184
89
88
87
4
5
6
0.0698
0.0873
0.1047
.070
.087
.105
14.3
II. 5
9.57
.070
.087
.105
14.3
II. 4
9.51
1.002
1.004
1.006
.998
.996
.995
I. 5010
1.4835
I. 4661
86
85
84
7
8
9
0.1222
0.1396
0.IS7I
.122
.139
.156
8.21
7.19
6.39
.123
.141
.158
8.14
7.12
6.31
1.008
1. 010
1. 012
.993
.990
.988
I . 4486
I. 4312
I. 4137
83
82
81
zo
0.174s
• 174
5.76
.176
5.67
1. 015
.985
1.3963
80
II
12
13
0.1920
0 . 2094
0.2269
.191
.208
.225
5.24
4.81
4-45
.194
.213
.231
5.14
4.70
4-33
1. 019
1.022
1.026
.982
.978
.974
1.3788
I. 3614
1.3439
79
78
77
14
IS
i6
0.2443
0.2618
0.2793
.242
.259
.276
4.13
3.86
3.63
.249
.268
.287
4.01
3.73
3.49
1. 031
1. 035
1.040
.970
.966
.961
1.326s
1.3090
1.291S
76
75
74
17
i8
19
0.2967
0.3142
0.3316
.292
.309
.326
3.42
3.24
3.07
.306
.325
.344
3-27
3 08
2.90
1.046
1. 051
1.058
.956
.951
.946
I. 2741
1.2566
1.2392
73
72
71
ao
0.3491
.342
2.92
.364
2.75
1.064
.940
I. 2217
70
31
32
23
0.366s
0.3840
0.4014
.358
.375
.391
2.79
2.67
2.56
.384
.404
.424
2.61
2.48
2.36
1. 071
1.079
1.086
.934
.927
.921
I . 2043
I . 1868
I . 1694
69
68
67
24
-25
26
0.4189
0.4363
0.4538
.407
.423
.438
2.46
2.37
2.28
M66
.488
2.25
2.14
2.05
1.095
1. 103
1. 113
.914
.906
.899
1.1519
I . 1345
1.1170
66
65
64
27
28
29
0.4712
0.4887
0.5061
.454
.469
.485
2.20
2.13
2.06
.510
.532
.554
1.96
1.88
1.80
1. 122
1. 133
1. 143
.891
.883
.875
I . 0996
I. 0821
I . 0647
63
62
61
30
0.5236
.500
2.00
.577
1.73
1. 155
.866
1.0472
60
31
32
33
0.S4II
0.5585
0.5760
.515
.530
.545
1.94
1.89
1.84
.601
.625
.649
1.66
1.60
1.54
1. 167
1. 179
1. 192
.857
.848
.839
1.0297
I. 0123
0 . 9948
59
58
57
34
35
36
0.5934
0.6109
0.6283
.559
.574
.588
1.79
1.74
1.70
.675
.700
.727
1.48
1.43
1.38
1.206
1. 221
1.236
.829
.819
.809
0.9774
0.9599
0.9425
56
55
54
37
38
39
0.6458
0.6632
0.6807
.602
.6x6
.629
1.66
1.62
1.59
.754
.781
.810
1.33
1.28
1.23
1.252
1.269
1.287
.799
.788
.777
0.9250
0.9076
0.8901
53
52
51
40
0.6981
.643
1.56
.839
1. 19
1. 30s
.766
0.8727
50
41
42
43
0.7156
0.7330
0.7505
.656
.669
.682
1.52
1.49
1.47
.869
.900
.933
1. 15
I. II
1.07
1.325
1.346
1.367
.755
.743
.731
0.8552
0.8378
0.8203
49
48
47
44
45
0.7679
0.7854
.695
.707
1.44
1. 41
.966
1. 00
1.04
1. 00
1.390
1. 414
.719
.707
0.8029
0.7854
46
45
n cos
n sec
n cot
n tan
n CSC
n sin
Radians
Deg.
479
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INDEX
(The numbers refer to the pages)
Abscissa, 26
Absolute value of complex num-
ber, 356
Addition formulas for sine and
cosine, 286-288
for tangent, 289
Additive properties of graphs,
42, 273-276
Algebraic function, 13, 14
scale, 3, 342
Alternating current curves, 362
et seq.
represented by complex
numbers, 368
Amplitude of complex number,
356
of S. H. M., 322
of sinusoid, 113
of uniform circular motion,
99
of wave, 326
Angle, 96
depression, 124
direction, 100
eccentric, 140
elevation, 124
epoch, 322, 326
phase, 322, 326
that one line makes with an-
other, 293
vectorial, 100
Angular magnitude, 96
units of measure, 97
velocity, 99, 322
Anti-logarithm, 232
Approximation formulas, 193
Approximations, successive, 180
Argument of fimction, 10
of complex number, 356
Arithmetical mean, 198
progression, 198-201
triangle, 188
Asymptotes of hyperbola, 154, 157
Auxiliary circles, 140
Axes of ellipse, 138
of hyperbola, 157
Binomial coefficients, graphical
representation of, 196,
197
theorem, 189 et seq.
Briggs, Henry, 216
system of logarithms, 223
Cartesian coordinates, 26
Cassinian ovals, 387, 389
Catenary, 274
Change of base, 242, 243
of unit, 62, 73 et seq., 263
Characteristic, 229, 230
Circle and circular functions,
Chap. III.
Circle, dipolar, 389
equation of, 94, 95
sine and cosine, 120, 121
tangent to, 422, 427
through three points, 433
Circular functions, 100 et seq.
graphical computation of^
103, 111
483
484
INDEX
Circular fundamental relations,
107, 282-297
motion, 99
Cologarithm, 233
Combinations, 183, 186, Chap.
VI.
Common logarithms, 223
Complementary angles, 111, 114
Completing square, 457
Complex numbers, Chap. XI,
341 et seq.
defined, 348
laws of, 350
polar form, 358
typical form, 348
Composite angles, functions of,
290-293
Composition of two S. H. M.'s,
324
Compound harmonic motion, 334
interest, 205, 21*1
law, 256
Computers rules, 309
Conditional equations, 132, 300-
315
Conies, 413, 415
con-focal, 443
sections. Chap. XIII, 398 et
seq.
Conjugate axis, 157
complex numbers, 352
hyperbola, 158
Connecting rod motion, 338
Constants and variables, 13
Continuous function, 10
compounding of interest, 256
Coordinate paper, 26, 119, 267-
274
Coordinates, Chap. II, 26 et seq.
Cartesian, 26
orthogonal, 119
polar, 118, 433
Coordinates, rectangular, 26, 27
et seq.
relation of polar and rectan-
gular, 131, 433
Cosine, 100
curve, 113, 120
law, 301
Crest of sinusoid, 113
Cubical parabola, 50
Cubic equation, 177 et seq.
*'CutandTry," 135
Cycloid, 390
Damped vibrations, 276
Damping factor, 277
Decreasing function, 58
geometrical series, 206
DeMoivres theorem, 373
Descartes, Ren6, 26
Diameter of any curve, 442
of ellipse, 442
of parabola, 419
Direction of ellipse, 401, 413
of hyperbola, 406, 413
of parabola, 411, 413
Discontinuous function, 11, 31, 55
Distance of point from line, 425
Distributive law of multiplica-
tion, 189, 351
general, 456
Double angle, functions of, 295
scale, 5-8, 20, 21, 245-265
of algebraic functions, 20
of logarithmic functions,
245-265
''e," 220, 223, 238^ 257
Eccentric angle, 140
Eccentricity of earth's orbit, 402
of ellipse, 400
of hyperbola, 406
of parabola, 411, 413
INDEX
485
Ellipse, 137 et seq.; 398 et seq.
Chaps. IV and XIII.
axes of, 138
construction, 142, 143
directrices, 401, 413
eccentricity, 400
focal radii, 398, 429
foci, 398
latus rectum, 403
parametric equation, 140
polar equation, 409
shear of, 436
symmetrical equation of, 138
tangent to, 428
vertices, 138
Elliptic motion, 325, 382
Empirical curves, 261, 274
formulas, 71
Envelope, 421
Epicycloid and epitrochoid, 393
Epoch angle, 322, 326, 330
Even function, 115
Exponential curves, 237-244
equation, 211, 219
function Chap. VIII, 214 et
seq.
defined, 219, 221, 223
compared with power, 265
Exponents, definition of, 460
irrational, 222
laws of, 460
Factorial number, 183
Factoring, 453-455
fundamental theorem in, 457
Factor theorem, 163
Family of curves, 74
of lines, 420
Focal radii and foci, 386
of ellipse, 398, 429
of hyperbola, 404
radius of parabola, 412
Frequency of S. H. M., 323
of sinusoidal wave, 329
uniform circular motion, 99
Function, of a function, 91
periodic, 30, 113, 360
power, 46 et seq., 265
rational, 14, 162
S. H. M., 327
trigonometric, 100
Functions, 9, 10
algebraic, 13, 14
circular. Chap. Ill, 94 et
seq., 100
continuous, 10
discontinuous, 11, 31, 55
even and odd, 115
explicit and implicit, 139
exponential, 219, 221, 223,
265
increasing and decreasing,
58, 152
integral, 14, 162
General equation of second de-
gree, 437-440
Geometrical mean, 202
progression, 202 et seq.
Graphical computation, lb et seq.
of integral powers, 19
of logarithms, 217
of product, 16, 388
of quotient, 17, 87, 88
of reciprocals, 89
of sq. roots, 18, 21
of squares, 18, 21, 87
solution of cubic, 177
simultaneous equations,
174 et seq.
Graph of arithmetical series, 200
of binomial coefficients, 196,
197
of complex number, 349
486
INDEX
Graph of cycloid, 392
of ellipse, 142, 143
of equation, 37
of functions of mutilple an-
gles, 298, 299
of geometrical series, 207-
210
of hyperbola, 153-156
of hyperbolic functions, 274
of logarithmic and exponen-
tial curves, 216, 217,
237 et seq.
of parabolic arc, 420
of power function, 46, 48, 59,
73,86
of sinusoid, 112
of tangent and secant curves,
147-151
Half-angle, functions of, 296
HaUey*s law, 260
Harmonic analysis, 336
cur\'es, 363, 364
functions, 327
motion, Chap. X, 321 et seq.
compound, 334
Hyperbola, Chap. IV and XIII.
asymptotes, 154, 157
axes, 155, 157
center, 157
conjxigate, 158
eccentricitv, 406
foci and focal radii, 404
latus, rectum, 407
parametric equations, 154
polar equation, 409
, rectangular, 54, 153
symmetrical equation, 151
vertices, 157
Hyperbolic cur\-es, 51, 54
sine and cc^sine, 273
sj-^tiMw of lograrithms, 223
Hypocycloid and Hypo-tro-
choid, 393
i = V~l, 348
Identities, 107, 108, 132, 133,
282-297
Image of curve, 53
Increasing function, 58, 152
progression, 199
Increment, logarithmic, 256
Infinite discontinuity, 55
geometrical progression, 206
Infinity, 54, 55
Integral function, 14, 162
Intercepts, 40, 41
Interest, compound, 205, 256
curve, 211
Interpolation, 231
Intersection of loci, 169
Inverse of curve, 130
of straight line and circle,
130
trigonometric functions, 132,
378, 379
Irrational function, 14
numbers, 334
Lamellar motion, 82
Langley's law, 70
Latitude and longitude of a point,
26
Latus rectum of ellipse, 403
of hyp»bola, 407
of parabola, 412
Law of circular functions, 126
of complex numbers, 350
of compound interest, 256
of exp<Hiential function, 267
of power function, 76
of sines, cosines, and tan-
gnats, 301-303
Lead or lag. 330, 368
INDEX
487
Legitimate transformations, 167
Lemniscate, 387, 389
Limit, 150
Limiting lines of ellipse, 146
Loci, Chap. XII, 381 et seq.
defined by focal radii, 386
Theorems on, 57, 60, 80, 129,
242, 292
Locus of points, 36
of equation, 36
Logarithmic and exponential
functions. Chap. VIII,
214 et seq.
coordinate paper, 267-274
curves, 237-244
double scale, 245-265
functions, 219, 223
increment and decrement,
256, 258, 259, 277
tables, 229-233
Logarithm of a number, 216,
223
Logarithms, common, 223
graph, 216-217
properties of, 226-229
systems of, 223
Mantissa, 229
Mean, arithmetical, 198
geometrical, 202
harmonical, 212
progressive, 194
Modulus of complex number,
356
of decay, 259, 277
of logarithmic system, 242
Motion, circular, 99
compound harmonic, 334
connecting rod, 338
eUiptic, 325, 382
shearing, 81
S. H. M., 321 et seq.
Naperian base, 220, 223, 228, 257
system of logs., 223
Napier, John, 214
Natural system of logarithms,
223
Negative angle, 96
functions of, 115
Newton's law, 260
Node, 113
Normal, 130
equation of line, .130, 423
to ellipse, 429
to parabola, 419
Oblique triangles, 300-315
Odd functions, 115
Operators, 344
Ordinate of point, 26
Origin, 26
at vertex, 145, 413
Orthogonal systems, 119
Orthographic projection, 61, 117,
137, 158, 243
Paper, logarithmic, 268 et seq.
polar, 118 et seq.
rectangular, 26 et seq.
semi-log, 251, 261 et seq.
Parabola, 50, 411
cubical, 50
polar equation, 412
properties of, 419
semi-cubical, 50
Parabolic curves, 47 et seq. 267
Parameter, 140, 381
Parametric equations, 140, 381
of cycloid, 391
of ellipse, 140
of hyperbola 154, 155
Pascal's triangle, 188, 189
Periodic functions (see trig.-
fens.), 30, 113, 360
488
INDEX
Period of S. H. M., 322
of simple pendulum, 325
of uniform circular motion,
99
of wave, 328
Permutations, 183, 184
and combinations, Chap. VI,
182 et 8eq.
Phase angle, 322, 326, 330
Plane triangles, 300-315
Polar codrdinates, 118, 433
diagrams of periodic func-
tions, 120, 298, 360
equation of ellipse, 409
of hyperbola, 409
of parabola, 412
of straight line, 129
form of complex number, 358
relation to rectangular, 131,
433
Polynomial, 162
Positive and negative angle, 96,
115
coordinates, 26
side of line, 427
Power function, 46 et seq.
compared with exponen-
tial, 265
law of, 76, 77
practical graph, 73
variation of, 57
Probability curve, 197
Products, special, 452
Progressions, Chap. VII, 198 et
seq.
arithmetical, 198-201,215
decreasing, 206
geometrical, 202-210, 215
harmonical, 211, 212
Progressive mean, 194
Projection, orthographic, 61, 117,
137, 158, 243
Proportionality factor, 64
Quadrants, 26
Quadratic equations, 457
systems of equations, 171
Questionable transformations,
167
Radian unit of measure, 97, 98
Radicals, reduction of, 465
Radius vector, 118
Ratio definition of conies, 413
Rational formulas, 71
functions, 14, 162
numbers, 354
siji X t&n X
Ratio of and 148,
X X *
Rectangular coords, {see Coordi-
nates), Chap. II, 26
et seq.
Reflection of curve, 53
Reflector, 87
Remainder theorem, 162
Reversors, 346
Right angle system, 97
Root of any complex number, 377
of equation, 85
of function, 85, 163
of unity, 376
Rotation of locus, 78
polar coordinates, 127-129
rectangular, 433-435
of rigid body, 78
Scalar numbers, 343
Scale, 1, 3
algebraic, 3, 342
functions, 20
arithmetical, 3, 342
double, 5 et seq.
logarithmic, 245-265
uniform, 2
INDEX
489
Scientific laws and formulas, 65
et seq.
Seiche, 332 et seq.
S-formulas, 305
Semi-cubical parabola, 50
Semi-logarithmic paper, 251-261
Series, (see progressions), 19
Shearing motion, 82 et seq'.
Shear of circle, 441
of ellipse, 436, 441
of hyperbola, 441
of parabola, 441
of straight line, 81 et seq.
Simple harmonic function, 327
motion. Chap. X, 321 et
seq
pendulum, 63, 195, 325
Sine, 100
law, 301
Sinusoid, 112, 113
Sinusoidal varying magnitude,
365
wave, 326
Slide rule, 149 et seq.
Slope of line, 39
of curve, 40, 113
Stationary waves, 332
Statistical graphs, 27
Straight line, 40, 130, 423, 430
Strain, 78
Sub-normal, 419
Sub-tangent, 240, 419
Supplementary angles. 111
Surds, reduction of, 465
Symmetrical equation of ellipse,
138
of hyperbola, 157
systems of equations, 174
Symmetry, 51, 57
with respect to point, 51
to line, 51
to curve, 57
Tables, damped vibrations, 279,
280
logarithms, 231
material in concrete, 44
natural trig, functions, 104,
123
powers, 49, 50
of^'e,". 241
Tangent, 100
graph, 147
law, 303
to circle, 422, 427
to curve, 239
to ellipse, 428, 429
to parabola, 418
Theorems, binomial, 189 et seq.
factor, 163
remainder, 162
functions of composite an-
gles, 292
on loci, 57, 60, 80, 129, 242,
292
Transformations, legitimate and
questionable, 167
Translation, 78, 79, 424
of any locus, 79, 80
of point, 424
of rigid body, 78
Transverse axis, 157
Triangle of reference, 100, 106
Triangles, solution of, 123, 300-
315
oblique, 300-315
right, 123-125
Trigonometric curves, 112, 120,
147-151, 298
functions, 100 et seq.
Trochoid, 393
Trochoidal wave, 331
Trough of sinusoid, 113
Uniform circular motion, 99 |ri
490
INDEX
Unit, change of, 62, 73 et seq., 263
of angular measure, 97
Variables and constants, 13
and functions of variables,
Chap. I
Variation, 63
of power function, 57
Vector, 118, 357
radius, 118
Vectorial angle, 100, 118
Velocity, angular, 99, 322
of wave, 329
Versors, 347
Vertices of ellipse, 138
of hyperbola, 157
Vibrations, damped, 276
Waves, Chap. X., 326 et seq.
compound, 334
length of, 327
sinusoidal, 326 et seq.
stationary, 331
trochoidal, 331
Zero of function, 85
^
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univ. of mich.
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