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AN ELEMENTARY TEXT-BOOK

ON THE

DIFFERENTIAL AND INTEGRAL
CREO ULES

BY

WILLIAM H» ECHOLS

Professor of Mathematics in the University of Virginia

 

NEW YORK
HENRY HOLT AND COMPANY
1902

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Copyright, 1902, iahtat
BY . °
HENRY HOLT & CQ,

ROBERT DRUMMOND, PRINTER, NEW York.
PREFACE.

Tus text-book is designed with special reference to the needs of
the undergraduate work in mathematics in American Colleges.

The preparation for it consists in fairly good elementary courses
in Algebra, Geometry, Trigonometry, and Analytical Geometry.

The course is intended to cover about one year’s work. Experi-
ence has taught that it is best to confine the attention at first to func-
tions of only one variable, and to subsequently introduce those of
two ormore. For this reason the text has been divided into two
books. Great pains have been taken to develop the subject con-
tinuously, and to make clear the transition from functions of one
variable to those of more than one. The ideas which lie about the
fundamental elements of the calculus have been dwelt upon with
much care and frequent repetition.

The change of intellectual climate which a student experiences in
passing from the finite and discrete algebraic notions of his previous
studies to the transcendental ideas of analysis in which are involved
the concepts of infinites, infinitesimals, and limits is so marked that
it is best to ignore, as far as possible on first reading, the abstruse
features of those philosophical refinements on which repose the foun-
dations of the transcendental analysis,

The Calculus is essentially the science of numbers and is but an
extension of Arithmetic. The inherent difficulties which lie about its
beginning are not those of the Calculus, but those of Arithmetic and
the fundamental notions of number. Our elementary algebras are
beginning now to define more clearly the number system and the
meaning of the number continuum. This permits a clearer presen-
tation of the Calculus, than heretofore, to elementary students.

As an introduction and a connecting link between Algebra and
the Calculus, an Introduction has been given, presenting in review
those essential features of Arithmetic and Algebra without which it is
hopeless to undertake to teach the Calculus, and which are unfor-
tunately too often omitted from elementary algebras.

The introduction of a new symbolism is always objectionable,
lii
iv PREFACE.

Nevertheless, the use of the ‘‘ English pound ’’ mark for the symbol
of ‘‘ passing to the limit’’ is so suggestive and characteristic that
its convenience has induced me to employ it in the text, particularly
as it has been frequently used for this purpose here and there in the
mathematical journals.

The use of the ‘‘ parenthetical equality’’ sign (=) to mean
‘* converging to ’’ has appeared more convenient in writing and print-
ing, more legible in board work, and more suggestive in meaning than
the dotted equality, —, which has sometimes been used in American
texts.

An equation must express a relation between finitenumbers. The
differentials are defined in finite numbers according to the best mod-
ern treatment. In order to make clear the distinction between the
derivative and the differential-quotient, I have at first employed the
symbol Df, after Arbogast, or the equivalent notation /’ of Lagrange
exclusively, until the differential has been defined, and then only has
Leibnitz’s notation been introduced, After this, the symbols are
used indifferently according to convenience without confusion.

The word guanfily is never used in this text where number is
meant, True, numbers are quantities, but a special kind of quantity.
Quantity does not necessarily mean number,

The word ratio is not used as a relation between numbers. It is
taken to mean what Euclid defined it to be, a certain relation between
quantities. The corresponding relation between numbers is in this
book called a quotient. The quotient of a by 4 is that zumdser whose
product by 4 is equal to a.

In preparing this text J have read a number of books on the
subject in English, French, German, and Italian. The matter pre-
sented is the common property now of all mankind. The subject
has been worked up afresh, and the attempt been made to present it
to American students after the best modern methods of continental
writers.

I am especially indebted to the following authors from whose
books the examples and exercises have been chiefly selected: Tod-
hunter, Williamson, Price, Courtenay, Osborne, Johnson, Murray,
Boole, Laurent, Serret, and Frost.

My thanks are due Dr, John E. Williams for great assistance in
reading the proof and for working out all of the exercises.

W. H. E.

UNIVERSITY OF VIRGINIA, October, 1902.
CONTENTS .
INTRODUCTION.

FUNDAMENTAL ARITHMETICAL PRINCIPLES.

PAGE
SecTion I. ON THE VARIABLE....... MVE Gaweeeeweeee OL
The absolute number. The integer, reciprocal integer, rational num-
ber. The infinite and infinitesimal number. The real number system
and the number continuum. Variable and constant. Limit of a vari-
able. The Principle of Limits. Fundamental theorems on the limit.

Section IJ. FUNCTION oF A VARIABLE........... 00000000 19

Definition of functionality. Explicit and implicit functions. Continu-
ity of Function. Geometrical representation. Fundamental theorem of
continuity.

BOOK I.
FUNCTIONS OF ONE VARIABLE.

PART I.
PRINCIPLES OF THE DIFFERENTIAL CALCULUS.

CuapTer I. On THE DERIVATIVE OF A FUNCTION............ 35

Difference of the variable. Difference of the function. Difference-
quotient of the function. The derivative of the function. Ab initio
differentiation.

.

CuHapTeR II, RULES FOR ELEMENTARY DIFFERENTIATION...... 41
Derivatives of standard functions log x, xa, sin x. Derivative of sum,
difference, product and quotient of functions in terms of the component
functions and their derivatives.
Derivative of function of a function. Derivative of the inverse function.
Catechism of the standard derivatives.

CuarTer III. ON THE DIFFERENTIAL OF A FUNCTION......... 55

Definition of differential. Differential-quotient. Relation to difference.
Relation to derivative.

CuHapTerR IV. Own SUCCESSIVE DIFFERENTIATION............. 62

The second derivative. Successive derivatives. Successive differentials.
The differential-quotients, variable independent. Leibnitz’s formula for
the zth derivative of the product of two functions. Successive derivatives
of a function of a linear function of the variable.
vi CONTENTS.

CHAPTER V. ON THE THEOREM OF MEAN VALUE...... iesow er 74

Increasing and decreasing functions. Rolle’s theorem. Lagrange’s
form of the Theorem of Mean Value, or Law of the Mean. Cauchy’s form
of the Law of the Mean.

CuarpTer VI. ON THE EXPANSION OF FUNCTIONS........-+--> 82

The power-series. Taylor’s series. Maclaurin’s series. Expansion of
the sine, logarithm, and exponential. Expansion of derivative and primi-
tive from that ot function.

CuapTer VII, ON UNDETERMINED FORMS .......-....006-5 92

Cauchy’s theorem. Application to the illusory forms 0/0, % /o,
0X «0, 0 —0, 0%, 0 % 1%,
CuapTeR VIII. On Maximum anpD MINIMUM..... een ee aie LOS
Definition. Necessary condition. Sufficient condition. Study of a
function at a point at which derivative iso. Conditions for maximum,
minimum, and inflexion.

PART II.
APPLICATIONS TO GEOMETRY.

CHAPTERIX. TANGENT AND NORMAL.............. ieeeaves. EE?

Equation of tangent. Slope and direction of curve at a point. Equa-
tion of normal. Tangent-length, normal-length, subtangent and subnor-
ma]. Rectangular and polar coordinates.

CHAPTER X. RECTILINEAR ASYMPTOTES..............02-00> 121
Definitions. Three methods of finding asymptotes to curves. Asymp-
totes to polar curves.

CuarTer XI, Concavity, CONVEXITY, AND INFLEXION..... pee)

Contact of a curve and straight line. Concavity. Convexity. In-
flexion, concavo-convex, convexo-concave. Conditions for form of curve
near tangent.

CuapTER XII. CONTACT AND CURVATURE. .......0.0. 000008 130
Contact of two curves. Order of contact. Osculation. Osculating circle,
circle of curvature, radius, and center.
CuHaprer XU) CENVECOPES)..02:i.2 so 4a Sas wide bas Gch wee ee 138
Definition of curve family. Arbitrary parameter. Enveloping curve
of a family. Envelope tangent to each curve.
CuHapTerR XIV. INVOLUTE AND EVOLUTE............000000. 144
Definitions. Two methods of finding evolute.

CHAPTER XV, EXAMPLES OF CURVE TRACING............. o+. 147

Curve elements. Explicit and implicit equations. Tracings of simple
curves,
CONTENTS. vii

PART III.
PRINCIPLES OF THE INTEGRAL CALCULUS.
PAGE
CHAPTER XVI. ON THE INTEGRAL OF A FUNCTION........... 165

/ ae Definition of element. Definition of integral, Limits of integration.
Integration tentative. Primitive and derivative. A general theorem on
integration, The indefinite integral, The fundamental integrals by ab
initio integration.

CHAPTER XVII. THe SranparD INTEGRALS. METHODS OF
INTEGRATION yeh) e eh ce bee w bev ida Ree ee

The irreducible form f du, The catechism of standard integrals. Prin-

ciples of integration. Methods of integration. Substitution (transforma-
tion, ratiunalization). Decomposition (parts, partial fractions).

CHAPTER XVIII. Some GENERAL INTEGRALS. ............... 193

Binomial differentials. Reduction by parts. Trigonometric integrals.
Rational functions. Trigonometric transformations. Rationalization.
Integration by series.

CHAPTER XIX. ON DEFINITE INTEGRATION. .........0.2005- 215

Symbol of substitution. Interchange of limits. New limits for change
of variable. Decomposition of limits. A theorem ofmean value. Exten-
sion of the Law of Mean Value. The Taylor-Lagrange formula with the
terminal term a definite integral. Definite integrals evaluated by

series.
PART IV.
APPLICATIONS OF INTEGRATION.
CHAPTER XX. ON THE AREAS OF PLANE CURVES ..... ics ee eons 226

Areas of curves, rectangular coordinates, polar coordinates. Area
swept over by line segment. Elliott’s extension of Holditch’s theorem.
CHAPTER XXI, ON THE LENGTHS OF CURVES...... ps.cie was BAS
Definition of curve-length. Length of a curve, rectangular coordinates,
polar coordinates. Length of arc of evolute, Intrinsic equation of curve.
CHAPTER XXII, Own THE VOLUMES AND SURFACES OF REVOLUTES. 255

Definition of rotation. Revolute. Volume of revolute. Surface of
revolute.

CHAPTER XXIII. ON THE VOLUMES OF SOLIDS,.......... .. 264

Volume of solid as generated by plane sections parallel to a given
plane.
viii CONTENTS.

BOOK II.
FUNCTIONS OF MORE THAN ONE VARIABLE.
PART V.
PRINCIPLES AND THEORY OF DIFFERENTIATION,
PAGE
CHAPTER XXIV. THE FUNCTION oF Two VARIABLES......... 273

Definition. Geometrical representation. Function of independent
variables. Function of dependent variables The implicit function of
several variables. Contour lines. Continuity of a function of two vari-
ables. The functional neighborhood.

CHAPTER XXV. PARTIAL DIFFERENTIATION OF A FUNCTION OF
SS WOt VARIABLES pho ve. - clase when va eeteriineacs ~ facet ray apg han nag wee ic wea ee 282

On the partial derivatives. Successive partial differentiation. Theorem
of the independence of the order of partial differentiation.

Cuaprer XXVI. Torat DIFFERENTIATION................ - 290

Total derivative defined. Total derivative in terms of partial deriva-
tives. The linear derivative. Total differential. Differentiation of the
implicit function.

CuHaprerR XXVII. Successive ToTAL DIFFERENTIATION........ 299

Second total derivative and differential of z = f(x, y). Second deriva-
tive in an implicit function in terms of partial derivatives. Successive
total linear derivatives.

CHAPTER XXVIII. DIFFERENTIATION OF A FUNCTION OF THREE
WARTABEES 9 av eine eisasial 4 tarhen ae icc auiaasihad aula earn Riasawio aban 306

The total derivative. The second total derivative. Successive linear
total differentiation.

CHAPTER XXIX, EXTENSION OF THE LAW oF MEAN VALUE TO
FUNCTIONS OF TWo AND THREE VARIABLES............... 309

CHAPTER XXX. Maximum anD Minimum. Functions or SeEv-
ERAL VARIABLES: vce ave ee ows 84s be aeae be ewe catgea 314

Definition. Conditions for maxima and minima values of T(x, y) and
S(x,y, 2). Maxima and minima values for the implicit function. “Use of
Lagrange’s method of arbitrary-multipliers.

CHAPTER XXXI. APPLICATION TO PLANE CURVES

Definition of ordinary point. Equations of tangent and normal at an
ordinary point. The inflectional tangent, points of inflexion. Singular
point. Double point. Node, conjugate, cusp-conjugate conditions,
Triple point. Equations of tangents at singular points. Homogeneous
coordinates. Curve tracing. Newton’s Analytical Polygon, for separat-
ing the branches at a singular point. Envelopes of curves with several
parameters subject to conditions. Use of arbitrary multipliers.
CONTENTS. ix

PART VI.
APPLICATION TO SURFACES.

PAGE
CuapTER XXXII. Srupy or THE ForM OF A SURFACE AT A POINT. 347

Review of geometrical notions. General definition of a surface. Gen-
eral equation of a surface. Tangent line to a surface. Tangent plane
to a surface. Definition of ordinary point. Inflexional tangents at an
ordinary point. Normalto a surface. Study ofthe form of a surface
at an ordinary point, with respect to tangent plane, with respect to
osculating conicoid. The indicatrix. Singular points on surfaces.
Tangent cone. Singular tangent plane.

CHAPTER XXXIII. CURVATURE OF SURFACES...... ise ores ai en 395

Normal sections. Radiusofcurvature. Principal radii of curvature.
Meunier’s theorem. Umbilics. Measures of curvature of a surface.
Gauss’ theorem.

CHAPTER XXXIV. CURVES IN SPACE......0-00 000 eee eee ee 375

General equations of curve. Tangent to a curve ata point. Oscu-
lating plane. Equationsofthe principal normal. The binormal, Circle
of curvature. Tortuosity, measure of twist. Spherical curvature.

CHAPTER XXXV. ENVELOPES OF SURFACES.........-0-00005 385

Envelope of surface-family having one arbitrary parameter. The
characteristic line. Envelope of surface-family having two independent
arbitrary parameters. Use of arbitrary multipliers.

PART VII.

INTEGRATION FOR MORE THAN ONE VARIABLE. MULTIPLE
INTEGRALS.

CHaPpTER XXXVI. DIFFERENTIATION AND INTEGRATION OF IN-
TEGRAES: 4. 4c ccgas ee ax g aptins do tionane siesta ees aeteae povaon ee 391

Differentiation under the integral sign for indefinite and definite inte-
grals. Integration under the integral sign for indefinite and definite
integrals.

CHAPTER XXXVII. APPLICATION OF DoUBLE AND TRIPLE
INTEGRALS o 5 oes ba vee ev ee Meee a ee eR GES A Me ea ee 396

Plane Areas, double integration, rectangular and polar coordinates.
Volumes of solids, double and triple integration, rectangular and polar
coordinates. Mixed coordinates. Surface area of solids. Lengths of
curves in space.

CHAPTER XXXVIII. DiFFeRENTIAL EQUATIONS OF FIRST ORDER
AND. DEGREBs 3365.55 35s 1s Meo Se Coed wate og ee Be 409
Rules for solution. Exact and non-exact equations. Integrating
factors.
CHAPTER XXXIX. Exampies OF DIFFERENTIAL EQUATIONS OF
THE FirsT ORDER AND SECOND DEGREE.......-...2-2-0-05 428

Rules for solution. Orthogonal trajectories. The singular solution.
c- and f-discriminant relations. Redundant factors not solutions. Node,
cusp, and tac- loci.
x CONTENTS.

PAGE

CHAPTER XL. ExXaMpLes OF DIFFERENTIAL EQUATIONS OF THE
SECOND ORDER AND FIRST DEGREE... ......0.0 000 eee ere 439

The five degenerate forms. The linear equation, and homogeneous
linear equation having second member o.

APPENDIX,

Note 1. Weierstrass’s Example of a Continuous Function which has no deter-

minate derivative..........0 cece cece cece aeeeeeeee diet sieie eels seed aveisi'sionene 451
Note 2. Geometrical Picture of a Function of a Function...........02e0+00> 453
Note 3. The th Derivative of the Quotient of Two Functions.........++-++ 454
Note 4, The wth Derivative of a Function of a Function.....+....+es--.e0s 455
Note 5. The Derivatives of a Function are infinite at the same points at which

the Function is infinite... 0... ... 0... cee eee eee Jose aera aes és sige s 456
Note 6. On the Expansion of Functions by Taylor’s Series....... ei Bie eee, 457
Note 7. Supplement to Note 6. Complex Variable..........+eeeeeeeeeeeee 465

Note 8. Supplement to Note 6. Pringsheim’s Example ofa Function for which
the Maclaurin’s series is absolutely convergent and yet the Function and

Series are diftereni tis: és <se\eisie gaia ae tede saree Ube ws eres ache a Bet age Aer 467
Note 9. Riemann’s Existence Theorem. Proof that a one-valued and con-
tinuous function is integrable... .... 6... eee cece eee cet eee eens 468
Note 10. Reduction formule for integrating the binomial differential........ 470
Note 11. Proof that a curve lies between the chord and tangent, when the
chord i1staken Short Cnoug yy sia, sce é.9¢ 01514: 4. piereue ana a a a, troassa rg Betbua eee 471
Note 12. Proof of the properties of Newton’s analytical polygon for curve-
acing’ fos eksacmatene< ia SeReie ae PEE Maeae ee eorD Discs ou qiniaiey 471

INDEX........ sah ovlera lela. insalin ‘019 (oi eye ens) ile Sree SiS ee pleRidies Seea ta Sew elaie ess eel eo dteniaene ATG
INTRODUCTION TO THE CALCULUS.

SECTION I.
ON THE VARIABLE.

1. Calculus, like Arithmetic and Algebra, has for its object the
investigation of the relations of Numbers. It is necessary to under-
stand that the symbols employed in Analysis either represent numbers
or operations performed on numbers.

2. The Symbols.
1, 2s Be Aya ee (i)
are symbols used to represent the groups of marks which we call in/e-
gers. Thus*

The system of integers (i) extends indefinitely toward the right,
as indicated by the sign of continuation. This system is called the
table of integers. Each integer has its assigned place, once and for
all, in the table. Any integer in the table is, conventionally, said to
be greater than any other integer to the left of it, and less than any
integer to the right of it in the table (i).

3. Definition of Infinite Integer.—When an integer is so great
that its place in the table of integers cannot be assigned in such a
manner that it can be uniquely distinguished from each and every
other integer, that integer is said to be unassignably great or infinite.
Mathematical infinity has no further or deeper meaning than this.

4. The Inverse Integer.

 

The reciprocals of the integers

oe ee Gi)

constitute an extension of the table (i) to the left of the integer 1,

which number is its own reciprocal. As before, any number in this

table is said to be greater than any number to the left of it, and less
than any number to the right of it.

Corresponding to each number in (i) there is a number in (ii),

 

* The symbol = is to be read, ‘‘7s ¢dentical with,” or ‘‘is the sameas.”
I
2 INTRODUCTION TO THE CALCULUS. [Sec. I.

and conversely. Those numbers in (ii) which are the reciprocals of
the infinite or unassignably great integers, are said to be zmjinzlesimals
or unassignably small.*

5. The Absolute Number. The Absolute-Number Continuum.

When in the table of numbers

we ey H HT, 2, Boe ee (iii)
the gap between each pair of consecutive numbers is filled in with all
the rational (fractional) and irrational numbers that are greater than
the lesser and less than the greater of the pair, we construct a table of
numbers which is called the absolute-number continuum. Each number
in this system has its assigned place. It is said to be greater than any
number to the left of it and less than any number to the right of it.
Each number in the absolute-number continuum is called an adsoluée
number.

Any and all numbers in the table that are greater than any integer
that can be uniquely assigned, as in § 3, are said to be zmfimife or unas-
signably great. In like manner any number in the table that is less
than any reciprocal-integer that can be uniquely assigned a place in
the table is zfinztesimal or unassignably small.

The absolute continuum is thus divided into two classes of num-
bers: the uniquely assigned or simply the asszgned numbers, which we
call the fzzve numbers ; and the numbers which cannot be uniquely
assigned or “vansfinife numbers.

The transfinite numbers greater than 1 are called infinite, those less
than 1 infinitesimal numbers.

6. Zero and Omega.—The absolute-number system, as con-
structed in § 5, extends indefinitely both ways, in the direction of the
indefinitely great and in that of the indefinitely small. In this sys-
tem there is no number greater than all other numbers in the system,
nor is there any number that is less than all others in the system.

The system is conventionally c/osed on the left by assigning in the
table a number zero whose symbol is 0, which shall be less than any
number in the absolute system. Since now there is no number greater
than 1 to correspond to the reciprocal of this number o, we design
arbitrarily a number omega whose symbol is Q. as the reciprocal of o,
and which is greater than any number in the absolute system.

The number o is the familiar naught of Arithmetic. The num-
ber O is the whmafe number of the Theory of Functions, and with
which we shall not be further concerned in this book.

The number o is not an absolute number, but is the inferior boundary
number of that system. In like manner the number Q is not an abso-

lute number, but is the superior boundary number of the absolute
system.

 

_ *The words ‘great’ and ‘small’ have in no sense whatever a magnitude mean-
ing when applied to numbers. They are mere conventional phrases and the words
‘right’ and ‘left’ or ‘in* and ‘out,’ might just as well be employed,
ART. 7.] ON THE VARIABLE. 3

7. The conventional symbol for the whole class of unassignably
great or infinite numbers is «. There has been adopted no conven-
tional symbol for the class of infinitesimals; the symbol most com-
monly used is the Greek letter iota, 2.

8. The Real-Number System.—When in the algebraic system
of numbers

— O,...,—-3,-2,—-1, o,+1,+2,+3,...,++0,
the gap between each consecutive pair is filled in with all the rational
and irrational numbers that are greater than the lesser and less than
the greater of the pair, the system thus constructed is called the rea/-
number continuum. :

It is understood that any number in this table is greater than any
number to the left of it and less than any number to the right of it.

The modulus of any real number is its arithmetical or absolute
value. Thus, the modulus or absolute value of + 3 or —3 is the
absolute number 3. If we employ the symbol x to represent any
number in the real continuum, then its modulus or absolute value is
represented by |x| or mod x.

In this book we shall be directly concerned only with real num-
bers and their absolute values. Hereafter when we speak of a numéer,
we mean a real number unless otherwise specially mentioned.*

9. Geometrical Picture of the Real-Number System.—We assume a cor-

respondence between the points on a straight line and the numbers in the real
continuum.

foto tte ag

A O A B PC D
Fic. 1,

tot 8
&

Select any point O on a straight line. Choose arbitrarily any unit length; with
which construct a scale of equal parts, 4, &, C,.. . starting at O proceeding

 

* The real-number continuum is a closed system of numbers to all operations
save that of the extraction of roots. When we consider the square root of a nega-
tive number we introduce a new number, The complex or complete number of
analysis is

x+y,

where « and y are any two real numbers, and 7 is a conventional symbol represent-

ing + ¥—1. Corresponding to any real number y there are as many complex
numbers as there are real numbers x; and corresponding to any real value x
there are as many complex numbers as there are real numbers y. The complex
system is a double system. In the theory of functions of complex numbers,
which includes that of real numbers as a special case, the ultimate number Q, is
conventionally a number common to all systems in the same way as is 0.

The student is already familiar with the impossibility of solving all questions
in analysis with real numbers only. For example, in the theory of equations
when seeking the roots of equations. All the more so is this true in the Calculus,
for we cannot solve the fundamental problem of expanding functions in series with.
out the use of complex numbers, except in a very few particular cases.

If z is any complex number x + zy, its modulus or absolute value is

Jat Vxtty.

 
4 INTRODUCTION TO THE CALCULUS. [Sec. I.

toward the right, and 4’, B’, C’, .. . toward the left. Mark the points of divi-
sion, 0 at the origin O, and + 1, +2, etc., toward the right; —1, — 2, etc.,
toward the left. Then, corresponding to any real number x there is a point Pon
the line to the right of O if x is positive ; and P’ to the left of O if x is negative.
The number x is the measure of the length OP with respect to the unit length
chosen. Conversely, corresponding to each point Pon the line there is a number
in the real-number system.

to. Variable and Constant.—In the continuous number system,
as designed in § 8, it is convenient to use letters as general symbols
to represent temporarily the numbers in that system. Thus, we
can think of a symbol x as representing any particular number in that
system. Further, we can think of a symbol x as representing any
particular number, say + 3, and then representing continuously in
succession every number between + 3 and any other number, say
+ 5, and finally attaining the value + 5. We speak of sucha symbol
x, representing successively different numbers, as a number, and
we speak of any particular number which it represents, as its
value.

Definition.—A number x is said to be variable or constant ac-
cording as it does or doesnot change its value during an investigation
concerning it.

We shall frequently be concerned with symbols of numbers which
are variable during part of an investigation and are constant during
another part.

Generally, variables are represented by the terminal letters x, ,
w, x,y, 2, etc., and constants by the initial letters a, 4, c, etc., of
the alphabet. This is not always the case, however, as the context
will show.

11. Interval of a Variable.—We shall sometimes confine our
attention toa portion of the number system. For example, we may
wish to consider only those numbers between a and 4. We shall employ
the symbol (a, 4), 2 being less than 4, to represent the numbers a, 4
and all numbers between them. If we wish to exclude from this
system 4 only, we write (a, 6(; if @ only, we write ) a, 5); when we
wish to exclude a and 4 and consider only those numbers greater than
a and less than 4, we represent the system by )a, 4(.

If x is a general symbol representing any number in sucha portion
of the number system, or interval, defined by a and 2, we have the
equivalent notations,

 

A variable x is said to vary continuously through an interval (a, 4),
when % starts with the value @ and increases to 4 in such a manner as
1o pass through the value of each and every number in (a, 4). Or, x

, 2 ’
ART. 12.] ON THE VARIABLE. 5

passes in like manner from 4 to a. The number ~ is said to increase
continuously from a to 4, or decrease continuously from 4 to a.

 

oO a 2 6
O A PB
Fic. 2.

In the geometrical picture, of § 9, illustrating the number system, if the points
A, P, B, correspond to the numbers a, «, 6, the segment AZ represents the inter-
val (a, 6). As x varies continuously from a to 4, or through the interval (a, 4), the
point P corresponding to the number x generates the segment 42.

12. The Limit of a Variable.

Definition.—When the successive values of a variable x approach
nearer and nearer to the value of an assigned constant number a in such
a manner that the absolute value of the difference « — @ becomes and
remains less than any given assigned consfant absolute number € what-
ever, we say that the number x has a for its /:mit.

In symbols, under the above conditions we write

L (*) = 14,
which is read, ‘‘ the limit of xis a.’’ The variable is said to converge
to its limit.

EXAMPLES.

Arithmetic furnishes examples of a limit .

1, In the extraction of roots of numbers. Whenever a number has no rational
number for a root, its root, if real, is an 2xra¢/onal number called a sxrd, which is
the limit of a sequence of rational numbers constructed according to a certain law.

2. In general, the definition of a number is: *

Any sequence of rational numbers

Oy Egy 8 4 Ong os
defines and assigns a number, when it is constructed according to any law which
requires each number in the sequence to be finite and such that, whatever assigned
number ¢€ be given (however small), we can always assign an integer # for which
|@n — an+4l <6,
for any assigned value of the integer # (however great).

The very definition of a number, on which all analysis is founded, is a limit.
The number assigned by the above vegz/ar sequence of numbLers is but the limit to
which converges the element a, of that sequence, as 7 increases indefinitely. If a
be the symbol of the number thus defined, then in symbols we write,

a= &(4r)-

It should be observed that irrational numbers having been thus defined, the
numbers a, in a regular sequence can be any numbers rational or irrational. The
regular sequence defines and assigns a number inits place in the table of numbers.

Algebra furnishes a useful and an interesting example of a limit in the evalua-
tion of the infinite geometrical progression.

3. The identity

r—attiz(i—x\itsetai+... +x)
is established by multiplying the two factors on the right.

 

* Due to Cantor and Weierstrass.
6 INTRODUCTION TO THE CALCULUS. [Sec. I.

Therefore, in compact symbolism, which we shall frequently employ,

I antt

ren
= =
r=0

 

 

I—-x I-x

If x is any number such that |x| <1, we can make and keep x«* +1, and there-
fore also the second term of the member on the right, less than any assigned
number e, by making » sufficiently great. Therefore, the limit of the sum of the
series on the left is 1/(1 — x), or in symbols

L BS oesitetet 3.48%

N=om r=0

 

In this example the variable is
Spsi+tat+... +2
If now xis any assigned number in )o, 1(, x is positive, and S, continuously
increases as y increases. The variable 5, is always less than the limit. If x is in

 

)— 1, O(, it is negative, say x= — a; then
n
S(— rar =1 —a4ae—... 4+( —1)Ka4,
r=0 a I (—1)tart3
= fet I+a

When x is even the variable S, is greater than its limit; when » is odd
the variable 5S, is less than its limit. Therefore, as 2 increases through integral
values, the variable converges to its limit, changing from greater than the limit to
less as % changes from even to odd and vice versa. ‘

It is to be observed that if |xj > 1, the sum of the series and the equivalent
member on the right increase indcfinitely with 2, in absolute value, and can be
made greater than any assigned number and therefore become infinite. Under
these circumstances the serics has no limit; its value becomes indeterminately
great.

Geometry furnishes numcrous illustrations ofthe limit. The most notable being:

4. The evaluation of the area of the circle as the limit to which converge the
areas of the circumscribed and inscribed regular polygons as the number of sides
is indefinitely increased.

5. The evaluation of the irrational and transcendental number 7 representing
the ratio of the circumference of a circle to its diameter.

Trigonometry furnishes an illustration of a limit which will be found useful
later:

6. To evaluate the limit of the quotient sin x ~ x as x diminishes indefinitely
in absolute value.

A Draw a circle with radius 1. Draw 1/4 =
MB perpendicular to O7. Then
1 Area quadrilateral O4 7B = tan x,
: Area triangle OAMB = sin zx,
Area sector OANB = x,
where x is, of course, the circular measure of
Z AOT.
O Then, obviously, from geometrical consider-
ations,
sina < x < tan x,
x I
or Eg Se

sin.x ~ cos x’

cs sin x “
ote — cos x.
Fic. 3: %
ART. 13.] ON THE VARIABLE. 7

When x diminishes indefinitely in absolute value, cos x becomes more and more
nearly equal to 1, and has the limit 1 as « converges too. Consequently the quo-
tient (sin x)/x converges to the limit 1 as x converges too. In our symbolism,

sin x
—-f=I
x

2t(=)o

13. Definition.—When a symbol x, representing a variable num-
ber, has become and subsequently remains always less, in absolute
value, than any arbitrarily small assigned absolute number, ~ is said
to be ifinitesimal.

When a variable becomes and remains greater, in absolute value,
than any arbitrarily great assigned number, the variable is said to be
infinite.

When a variable x is infinitesimal, we write * x(=)o. It follows
from the definition that when a variable becomes infinitesimal it has
the limit o, or assigns the number o.

When «x has the limit 2, or £4 = a, then by definition

L(x — a) =0.

When + — a is infinitesimal, we write

x —a(=)o.
This same relation we shall frequently express by the symbol
x(=)e,
meaning that the absolute value of the difference between x and a is
infinitesimal. When a is the limit of x, the symbol x(=)a is to be
read, ‘‘as x converges to a,’’ or ‘‘.x converges to a.”’

We shall frequently use the symbol e (efsz/om) to represent an
arbitrarily small assigned absolute number. We then speak of the
interval (2 — €, a+ e€) as the xeighberhood of an assigned number
a. The symbol «(=)a@ means that ‘‘ x is in the neighborhood of a.’’
All numbers that are in the neighborhood of an assigned number are
said to be consecutive numbers.

When a variable x becomes infinite we write + = oo. Such a
variable has no limit, it simply becomes indeterminately great. The
symbol « = co merely means that x is some number in the class of
unassignably great numbers.

14. The Principle of Limits.
I. A variable cannot simultaneously converge to two different
limits.

 

*The equality sign in parenthesis (=) may be read ‘parenthetically equal
to,” the word ‘parenthetically ’ carrying with it the explanation of the nature of the
approximate equality. It is simply another way of saying that the difference
between two numbers is infinitesimal.

ja —aj=z and x — a(=)o
mean the same thing. The symbol = has been used for (=), but appears less con-
venient, expressive, and explicit.
8 INTRODUCTION TO THE CALCULUS. [Src. I.

It is impossible for a (one-valued) variable « to converge to two
unequal limits a and 4. For, the differences | — a@| and |» — d| can-
not each be less than the assigned constant number 4|4 — a@| for the
same value of x.

The direct proof of this statement rests on this:

The number x must be either greater than, equal to, or less than
the number 4(a -+ 4), where say a < 0.

If «= d(a +4 4), ww. w—a=4(b—a).
If x >4(a+ 9), ww. x —a>t(b—a).
Ifx<t(¢+4), wv. 6 —x>4(6—<2).

II. If two variables x and _y are always equal and each converges to
a limit, then the limits are equal.

If £x =a, and £y=4, and x = y =z, then, by I, the variable
z cannot converge to two unequal limits simultaneously. Therefore
a=b.

15. Theorems on the Limit.*

I. If the limit of x is 0, then also the limit of cx is o, where c is
finite and constant.

For, whatever be the assigned constant absolute number e, we can
by definition of a limit make and keep || less than the constant
|¢/c|, and therefore cx less than € in absolute value. Consequently,

by definition
& (02) = 0=6£(2).
2(=)o
II. If each of a fimzte + number of variables x,, *,,..., 1, has
the limit o, then the algebraic sum of these variables has the limit o.

Let x be the greatest, in absolute value, of the # variables. Then

|x, ta, ...4+x,|< 2x.

Since 7 is finite, the limit of this sum is 0, by I.

III. If fx, =4,, Ax, = 4,,..., Ax, =4,, then when 2 isa
finite integer

Z£A(a4+*,4+. oF + x) = Lx, + 4%, 4+. % & + Lx.

For, put x, =4,+a@,,...,4,=4,+ a, By definition, the
limits of a,, ..., @,areo. Hence

abate. tere (Qt... +a) + (a+... +a,),
by II, gives

Ala t..-+am)=a+... +4,

Therefore the limit of the sum of a jimzfe number of variables is

equal to the sum of their limits.

 

* The theorems of this article are of such fundamental importance and so
absolutely necessary for the foundation of the Calculus that it will, in general, be
assumed hereafter that they are so well known as to require no further reference to
them.

+ If the number of variables is 2ot finite, this theorem does not hold in general.
ART. 15.] ON THE VARIABLE. 9

IV. The limit of the product of two variables x, and x, which
have assigned limits a, and a, , is equal to the product of their limits.

Let, as in III, 4,=a4,+a4,, x%,=4,+ 4,.

5 Hye, = 4a, + aa, + 4,0, + aa,
By III, we have
B(%%) = 44, + GL, + Ly, + £(% 04).
But, La, =0, fa, =o, and a fortiori £(a,a,)=0. Therefore
A(%y%) = 242, = (4x,)(4%,)-

Cor. The limit of the product of a fimzve number of variables having

assigned limits, is equal to the product of their limits. In symbols *
& MU (x,) = T£(%,).
r=1 r=1

V. The limit of the quotient, x,/x,, of two variables is equal to

the quotient of their limits, provided the limit of the denominator is

not o.
With the same symbolism as in IV,

Ae Sie Os Bi er

— = ?

Xs a, ot a, a, 4, + a, a,
gq 4,a, — 2%,

a, a,(4, a,)

By hypothesis, £a,=0, fa,=o0, anda,#0. Therefore the
denominator of the second term on the right is always finite, while,
by III, the limit of the numerator is o. The limit of this term is

o, by Lt . ‘ ge
woo eGe

VI. If «x and y are two variables and a is a constant, such that _y
always lies between x and a, then if £x =a, also Ly=a.

* As the symbol & is used to indicate the sum, so JJ is used to indicate the

product of a set of numbers. Thus,
n

Sep HSx, + %+... 444,
1

nn

Tix, =x, % 4%. XK... X XH.
I

The advantage of such symbolism is in compactness of the formule.

+ Notice particularly the provision that £7, 0. For, when fx, =0 and
4£*, # 0, the quotient «,/x, incieases beyond all limit or becomes infinite as x,
and x, converge to their limits. An infinite number cannot be a limit under the
definition.

Again, if 4x, = 0 and also £x, = 0, the quotient of the limits 0/o is com-
pletely indeterminate, while the quotient x,/x_. = g may or may not converge to
a determinate limit. The value of this quotient as x, and xr, converge to 0 depends
on the law connecting the variables x, and x, as they converge too. This case is
one of profound importance and is the foundation of the Differential Calculus.
10 INTRODUCTION TO THE CALCULUS. [Sec. I.

The truth of this is obvious, since |x —a@|>|y—a@|, and x —a@
has the limit o.

In like manner, it follows that if « and g have the common limit
a, and y isa third variable between x and z, then also must y= a.
For, |_y — a| must at all times be less than one or the other of the
differences |x —a@| and |z— aj, and each of these differences has
the limit o.

VII. If one of two variables is always positive and the other is
always negative, and they have a common limit, that limit is o.

Let @ be the common limit of x and y, where x is always positive
and y is always negative. Then

+l2|=ata, and —|y|=a+f,
' where La=o, £8 =o. Subtracting,
|z|+|s|=a—8.
Since f(a — 8) =0, ... a@4+a=2a=0, anda, the com-
mon limit of « and_y, is o.

VIII. Ifa variable x continually increases and assumes a value
a but is never greater than a given constant A, then there must exist
a superior limit of x equal to or less than A.

(1). No number such as a which x once attains can be a limit of
x. For, since x continually increases, it must subsequently take some
value a’ > a, and it is never possible thereafter for x — a to be less
than the constant a’ — a.

(2). The variable x cannot attain the number A, since if it did, x
continually increasing must become greater than A, which is contrary
to hypothesis.

(3). Divide the interval A — a= into 10 equal parts. The vari-
able x after attaining @ must either attain a+ 44 or remain always
less than a+ 352. Ifxattainsa + 7,4, it must either attain @ + yh
or remain always less than a+ 2,4. We continue to reason thus

until we find a digit p, such that x must attain @ + Ay and remain
10

always less than a +

I 5
AT? 5, That is, x must enter and always

remain in one of the ro intervals.
In like manner, divide the interval

(e+e «+ ht)

into 10 equal parts. In the same way we find that x must enter and
always remain in one of these intervals, and that there is a digit p,
such that

Ay

Io

A, h

Py B
at Pope oe Ser reo oe
ArT. 15.] ON TIIE VARIABLE. II

In like manner, continue this process ” times. Then

* p n”
a Pr

10” Se Oe i 10”
I I

This process can be carried on indefinitely. Consequently the
construction leads to the constant number
o

by

10”

I
frum which x can be made to differ by a number less than #/10* which
can be made and kept less than any given number e, for all values of
n greater than m, where 2/10” < €.

Therefore the constant @ is the limit of +, and is either equal to
or less than A.

In the same way, we prove the theorem: If a variable x always
diminishes and attains a value a, but is never less than an assigned
constant number A, then the variable x has an inferior limit that is
equal to or greater than A.

h
ath ae

a=ath

IX. If there be two variables x and_y, such that y is always greater
than x, and if x continually increases and_y continually decreases, and
the difference _y — x becomes less in absolute value than any assigned
absolute number ¢, then there is a constant number greater than +
and less than _y which is the common limit of x and y.

By Theorem VIII, x hasa superior limit a, and y has an inferior limit
6. For, any particular value_y, of y fixes a constant than which x
cannot be greater, and any particular value x, of x fixes a constant
than which y cannot be less. Hence, if we put

x=a—a, andy=db4 £,

y—x=(a—))— (a+).
But, £ (vy —*) =0, £(a + B) =0; .. a—b =o. This defines
the equality ofa and 4, Therefore x and y converge to a common
limit.

we have
12 INTRODUCTION TO THE CALCULUS. [Sec. I.

EXERCISES.

1. The successive powers of any assigned number greater than 1 increase
indefinitely and become infinite as the exponent becomes infinite.

Let @ be any absolute number, and any integer.

Then (a+ ay > t+ma. (1)
In fact, (I+ ayvs=i4+2at+ta?>1+2a.
The formula (1) is true when # = 2. Assume it to be true whenm =. Then
(I+ a)" >1+ 2a.
Multiply both sides by 1+ @.
(rf attr > 14 (n+ 1a + na’,
>1i14+(”4+ 1Da.

(1) is true also for#-+ 1. But, being true for m = 2, it is also true for m = 3,
and therefore for m= 4, etc., and generally. Therefore, since mq@ and conse-
quently (1 + a) can be made greater than any assigned number, the proposition
is demonstrated.

2. The successive powers of any assigned absolute number less than 1 diminish

indefinitely and have o for limit.
Any number less than 1 can be written as the quotient 1/(1+ a). By Ex. 1,

I I I
G+ ay ~1+ma gee
This can be made less than any assigned number e, by sufficiently increasing 7.
3. The successive roots of an aéso/ute number greater than 1 continually
diminish ; those of an absolute number less than 1 continually increase 3 and in

either case have the limit 1.
Whatever be the absolute number a,

ai #41) I I mM+I
a” =a mn+1)— | ana+t) 7
a oe z #
atti —@ Bt) =) gnmtn
Therefore, by Exs. 1, 2, whatever be the integer 7,
L I
a*>a"tl ifa>1;
& rT
a ca hisa<cr.

I

If@>1, then a® > I.
I
Let a@=>1-+a, and a*=14 8;
<
then (I+ a)" =1+,
or (+a =(1-+ fy >14 nf.
$< a/n, and we have
ms
= a
a ee
a SPs
ae
Hence "atm.
ART. 15.] EXERCISES. 13

Leta <1, say a= 1/(1 + a).

a a
Also, @*<1I, say a* = 1/(1 + f).
Then, as before, ( < a/n, and

rT

— I
L> at > ——__
; I+a/n
which shows again that
I
b gt = Is
h=00
4. Show that when a is any assigned positive number,
oh a = I,
x(=)o

whatever be the way in which x converges to 0.
(1). Let m, x, ~, g be any positive integers. Then
ee ee ht
a@at= at . vy,
vb
If@> 1, then @ >1,

BoP. IE a( £) a

at 8 ae and a “as ca,
Therefore a* continually increases as .v increases by rational numbers.

zs

Ifa@ <1, then a? <1.
Mm é m @ ii p

a teat, and eg eS BE,
Therefore @* continually diminishes as x increases by rational numbers,

When |} is rational and less than 1, there can always be assigned two con-
secutive integers 7 and m + 1 such that

I pee
x =e
mers! m

The above results show that whether a be greater or less than 1, a* lies between
r t I I

a” +1 and am, When m=, a”+? and a” converge to 1, Ex. 3, and there-
fore also does a*; and £a* = 1, when 2(=)o.

(2). When x is irrational there can always be assigned two rational numbers @&
and differing from each other as little as we choose, such that a <x < f.
The number a* is defined by its lying between ae and a8. Since x(=)o when
a( =) B(=)o, we have, as before, a* converging to I along with a and a8.

5. Show that fax = a£* = a8, if {x = f.

We have aB — ax = aB(I — a*—B8),

Passing to limits, we have, by Ex. 4,

ap — Lu == 10;
6. If @ and f# are positive numbers, and £x = /, show that
£ loga + = logs, £(x) = logy f.
3
We have log, 6 — log, x = log, e

AG
The above exercises show that however + converges to 7, £ loga (@/x) = 0.
Therefore
logg PB — £ loggx = 0.
14 INTRODUCTION TO THE CALCULUS. [Sec. I.

7. Utilize Ex. 6, to prove IV, V, from III, § 15.
8. Use Ex. 6, to show that
B(a*) = (Ar),
where »’ has a positive limit, and the limit of x is determinate.

9. Aset of numbers a,, a@,. +, @r,.. , arranged in order is called a se-
quence. Any number of the sequence, @,, is called an element of the sequence ; the
number ~ is called the order of the element a,. Any sequence is said to be known
when each element is finite and known when its order is known.

If a,, dy, ++ +5 @y, -+~ be a sequence of numbers such that a, is finite when
ry is finite, then will £a,, when z = #, be O or according as

:

z=
is less or greater than I, respectively.
Let, when 2 = 0, £(@n43/@n) = &, and& > 1. Then, by the definition of a
limit. we can always assign a number #’ such that 1 < & < #, whence corre-
sponding to 4’ we can find an integer # for which we have, for all values of 7,

Qn+1
an

 

 

Qntmt+i > PB.
anim
we dy ty > RB! Om J
'
eymt2 > Rk! anti > k amy
Omntn > Bay.
By hypothesis, a,, is finite. Since we can make #” greater than any assigned
number by sufficiently increasing z, we have fa, = 0.
In like manner, if £(@, 4 1/an) = & < 1,
Qn in <P;
which can be made less than any assigned number by increasing 7, when as before
I>k <k o £4, = 0, when z=.
In order that the element a, may have a finite-limit different from 0, it is neces-
sary that *

The quotient, wv, ,,/an, of each element by the preceding one will hereafter
be called the convergency quotient of the sequence. This theorem is of importance
and will be used later.

10. The series of numbers
atato.. tant... (i)
is said to be absolutely convergent when the corresponding series of the absolute
values of the terms is convergent.
That is, when
Su = |4| + la} 0 ee lan
has a determinate limit when 727 =o.
Show that (i) is absolutely convergent if

Gntr \<] 3;
an :

and if this limit is greater than 1, the sum of the series is 20.

 

* When the symbols |=], |>|, |<] are used, they mean that the equality or in-
equality of the aéso/ute values of the two members of the equation is asserted.
Arr. 15 ] EXERCISES. 15

Let the letters in (i) represent absolute numbers, and let

Fat
—=h .
Tas

Then there can always be assigned an integer m corresponding to any number
2' such that & < #' <1, for which

Umtntt < RB,
amtn
for all values of #. As in Ex. 9, we have
intn < kin am
Hence the sum of the series after a,, is less than
Bieta. Ply eh es SA As og PRO e535,
RB
OT sie

This is finite, since & # 1. Therefore S., must be finite. Also, by Ex. 9,
£%m = 0, when m=. Consequently we can always assign an integer 7 such
that

= 4.

Snim — Sy <6,
for all values of 7, where € is any assigned number. Hence S$, has a determi-
nate limit. Otherwise, the existence of the limit of 5,, follows at once from VIII,
§ 15. For S, continually increases, but can never exceed
BR
a+tut.ewe + Im TP tm
Again, if £(@n41/an) > I, say equalto# > 1. Then, as before, we can assign
2 between & and 1, and have the sum of the series after @,, greater than

hpi WEE pens hy RE Bs 5 ns
which iso.
The number £(an41:/a@n) is called the convergency quotient of the series.

11. The arithmetical average, or mean value of a sequence of x numbers,
Diy tgs 8S ey Ons
is one #th of their sum, or

1”
a= —- D4,
oy

Show that when the number of elements in a sequence increases indefinitely
according to any given law, the mean value has a determinate limit, if all the ele-
ments are finite.

Since

Lean< tM,

where Z and J are the least and greatest elements respectively, the mean value
must remain finite. Also,

ntp 1%
An+p— Mm = se p>" - aoe

I n+p p n
aeRO aa Fe

 
16 INTRODUCTION TO THE CALCULUS. [Sec. I.

 

But
PR - ee _ Pe
nat Pn |S! nin p) = 2 FB
n+p
pG
app ol Sla ee

G being an assigned number, than which no element can be greater in absolute
value. Whatever be the assigned integer , we can always assign an integer
that will make Qn+p— less than any assigned number e«. The mean value
therefore converges to a determinate limit. The value of this limit depends on the
law by which the sequence is formed.

12. Find the limit of
(4)
z

when z becomes infinite in any way whatever.
Divide both numerator and denominator in

mH
xm —tT
x—tI

 

?
x

where m is a positive integer, by x —1. Whence results

I

m

rtam4... 4(x") Cag
ie oe cto” 4. pare

(1). Ifa=i1+ art , then each of the m terms in the denominator of the frac-
m

 

tion on the right is less than I.

 

m+tr
am —!1 Io m+t
rs .
x—I m m
Hence mt I
Te me, Le Sey,
m
J mr a m
or I ——- >{I+—}.-
(4 m

Therefore, the value of the expression continually increases with m, and is
always greater than 2, by Ex. 1.

 

 

I ‘ ,
(2); ikv= r= 7 each of the 7 terms in the denominator of the same
m
fraction is greater than 1.
m +1
I-—-x« ™ I m I
a KL + == + fe
I—<« m m
mat
mM T
Hence 1—-x <5
m
m+

or a eS se,
m
ART. 15,] EXERCISES. 17

I m+i I\"

or («- 5) > («- 5)"
I —(m+1)° 1\-m
See en

Therefore the expression continually diminishes as the positive integer m in-
creases.
(3). Whatever be the positive number x, we have

x2> 42 — 1.

 

 

A i eS
x—I1 x
Hence I ae I ny",
-5) > ae ,

whatever positive value + may have.

(4). Also,
1\-* x —I1\-* x \%
boa) =(S) = Gap

= eye Bey
= i) A 3)"
ifwe putx =y+1.

These results (1), ... , (4), show that

(3

continually increases as z increases by positive integers, and continually de-
creases as z decreases by negative integers, and that the latter set of numbers is
always greater than the former, by (3). Also, these ascending and descending
sequences have a common limit,* by (4).

The value of this limit lies somewhere between

(1 + 1/6)6 = 2.521... and (1 — 1/6)-6 = 2.985...

We represent it, conventionally, by the symbol «. More accurately computed,

its value is

 

e = 2-7182818285 ...

A more convenient method of computing ¢ will be given later. It only remains
now to show that the limit is the same, whether z increases by rational or irrational
values, or continuously.

If z isany positive number, rational or irrational, we can always find two con-
secutive integers 7 and m +- 1, such that

maozcm+i,
I m t\z y\mt+1
and (+—45] < (r+ 2)<(:+3) ;

or («+ wm) (tag) ’< (+ + ze (: +3) (3 + =):

This shows that when m = o, thenz = oo, and

flee

zZ=0

 

*Put (I — m-1)-” = am, [1+ (m — 1)-']}*—-1 = 4,,. Then assigning to
m the values I, 2, 3,..., we have two sequences of positive numbers. The
sequence a@,, always diminishes, the sequence 4,, always increases. The difference
2m — 6, is a positive number converging too when z = 0. The two sequences
therefore define a common limit e.
18 INTRODUCTION TO THE CALCULUS. [Sec. I.

The result in (4) shows this is true whether z be positive or negative.* This
limit is the most important one in analysis.

I.
13. Show that £(itz)* =e,
x(=)o
14. Show that

I
L loga( +i) £ loga(t + x)* = logae,
n=O x x(=)o

andis1,if@=e. Use Ex. 6.
x =
15, Show that £(14+2) = Lu0+0)% =o.
x= 0 x“ x(=)o
16. Show that 4 (a* — 1)/x = log, a.
x(=)o
Hint. Puta*= 144,
17. If m is a positive integer, show that
am — qm
— = marl,
x—a
a(=)a
18. Show that Ex. 17 also holds true when m is a negative integer, ase if m
is any positive or negative rational number. ee
Hint. Put m = P/9. Divide the numerator and denominator by rr _- aq,
to obtain the quotient in determinate form for evaluation.

19. Show that £ Sig = sie = cosa. Use Ex. 6, § 12.
(sje *— a
20. Let , represent some particular one of the digits 0,1, . . . , 9, fora par-
ticular value of x. Show that the periodic decimal
2Py- ++ Pipiar+ ++ PltemPi+1- ++ Pltmsss

has for its limit the rational number
N

Me 10(10" — 1)’

where M=a-p,... pi, and NE=Pipitr1-+-Pi4+my and pity = pitamers
g being any integer, and 7 any integer less than or equal to g.

 

* The evaluation here given is a modification of one due to Fort, Zettschrift fiir
Mathematik, vii, p. 46 (1862). See also Chrystal’s Algebra, Part Il, Pp: 77-
SECTION II,
ON THE FUNCTION OF A VARIABLE.

16. Definition.—When two variables x and y are so related that
corresponding to each value of one there is a value of the other they
are said to be functions of each other.

If we fix the attention on yas the function, then x is called the
variable ; if on x as the function, then _y is called the variable.

Such functions as x and y defined above are not amenable to
mathematical analysis until the Jaw of connectivity between them can
be expressed in mathematical language.

CLASSIFICATION OF FUNCTIONS.

Functions are classed as exp/icit or implicit functions according as
the law of connectivity between the function and the variable is direct,
explicit, or indirect, implied, implicit.

17. Explicit Functions.—The simplest form of a function of a
variable «x is any mathematical expression containing x. Such a
function is called an explicit function of x, because it is expressed
explicitly in terms of the variable.

Our attention will be confined in Book I principally to explicit
functions of one variable.

The three standard or elementary functions,

wt, sin x, log, *,
and their inverse functions,
x sin x, a*,

represent the three fundamental classes of functions called algebraic,
circular, and logarithmic or exponential. All the elementary explicit
functions of analysis are formed by combining these standard functions
by repetitions of the three fundamental laws of algebra,

Addition, Multiplication, Involution,
and their inverses,
Subtraction, Division, Evolution.

Explicit functions are classified as algebraic or transcendental accord-
ing as the number of operations (including only
addition, multiplication. involution,
subtraction, division, ‘ evolution,
by which the function is constructed from the variable), is jiwife or
inpinite.
19
20 INTRODUCTION TO THE CALCULUS. [Sec. ll.

18. The Explicit Rational Functions.

I. The Explicit Integral Rational Function.

The function of the variable vr,

@,tax+tace4+ ... $a,x",
where the numbers a,, . . . , @, are independent of ~, and x is a finite
integer, is called an explicit integral rational function of .v, or briefly a
polynomial in x.

This is the familiar function which is the subject of inquiry in the
Theory of Equations. Its place and properties in the system of func-
tions correspond in many respects to the place and properties of the
integer in the system ofnumbers. It can advantageously be expressed
by the compact symbolism :

n
2 Oy Xt,
r=o0

meaning the sum of terms of type a,x” from r= 0 tor =n.

II. The Explicit Rational Function.
The quotient of two explicit integral rational functions of a vari-
able xv,
Q- ae... fp ayr”
Btoaet.. . +4,0"’
is called an explicit rational function of ., or simply a rafonal function
of x.
Its place in the system of functions corresponds to that of the
rational or fractional number in the number system.

III. The Explicit Irrational Algebraic Function.

Any expression involving a variable x, or an integral or rational
function of x, in which evolution a finite number of times (fractional
exponents) is the only irrational part of the construction, is said to be
an explicit irrational algebraic function of a.

Such a function in the function system corresponds to those irra-
tional numbers in the number system called surds.

For example,

1
Va? — x, a+ 6x3, Vital V1 05
are irrational algebraic functions.

19. Explicit Transcendental Functions.—Any expression which
is constructed by an zfinife (and cannot be constructed by a finite)
number of algebraic operations on a variable x is said to be an ex-
plicit transcendental function of x.

Examples of such functions are sin x, ex, log x, tan—' x, etce., which can only be

constructed from x by an infinite number of operations, such as infinite series or
products, or continued fractions.

 

20. Implicit Functions.—Whenever we have any equation involv-
ing two variables, x and_y, this equation is an expression of the law of
ART. 21.] ON THE FUNCTION OF A VARIABLE. 21

connectivity between the two variables and defines one of them as a
function of the other. The functional relation is zmplred by the
equation and is not explicit until the equation is solved with respect
to one or the other of the variables.
For example, the equation
ax* +. by? —c=0

defines x as a function of y, and, just as much so, yas a function of x. These
functions can be expressed explicitly by solving for x andy. Thus we have

c— by ¢— ax
r=] aot and y= opt

or x and y are expressed as explicit irrational algebraic functions of each other.

 

In general, any algebraic polynomial in two variables x and_y
when equated to zero defines _y as an algebraic function of x, and x
as an algebraic function of y. The explicit algebraic functions of
§ 18 are but particular cases of this more generally defined algebraic
function.

21. Conventional Symbolism for Functions.—We frequently
have to deal with a class of functions having a common property
or common properties, and with functions of complicated form,
which makes it convenient to adopt abbreviated symbols for func-
tions. Thus, we frequently represent a function of the variable x
by the symbol /(x), or #(), O(x), o(x), etc., when it is necessary
or advisable to indicate the variable and the function in one com-
pact symbol. When the variable is clearly understood, the paren-
thesis and the variable are frequently omitted and the function symbol
written 7, /, d or 7, etc.

We frequently employ the symbols y, 2, #, v, etc., as functions of x.

In like manner we write a function of two variables x, y as p(x, _y)
or f(x, y), etc., meaning a mathematical expression containing x and
y. The equation

 

P(x, 7) =0

implies, as said before, a functional relation between x and y, and
defines yas an implicit function of x, or x as an implicit function ofy.

If (x) is a function of x, and if @ is any particular assigned value
of .v, we write /(a) as the value of the function when x = a, or, as we
say, the value of /(x) at a.

For the present, when we use the word function we mean an
explicit function of one variable.

A function, /(), is said to be wnzform or one-valued at a when the
function has one determinate value at a.

For example,

avt+t bx+e, ex, sin x,

are one-valued functions for any value of x.

If /(x) has two, three, etc., distinct values corresponding to a
22 INTRODUCTION TO THE CALCULUS. [Sec Il.

value of the variable, it is said to be a two-, three- valued, etc., func-
tion.
For example, ax#, 4/a? — x*, are two-valued functions of x.

Frequently a function does not exist (in real values or finite values)
for certain values of the variable. Then, it is necessary to define the
interval of the variable in which the function does exist and in which
the investigation is confined.

For example, the function 4/a? — x? exists as a real function only in the inter-
val ( — w,-+ a); the function represented by the series

It+etxet...
exists as a determinate finite function only in the interval ) — 1, + 1(.

22. Continuity of a Function.

Definition : /() is said to be a continuous function of x atx = a,
when /(.x) converges to /(a) as a limit, at the same time that x con-
verges continuously to @ as a limit.

The definition and condition of continuity of /(x) at @ are com-
pactly expressed in symbols by

AM) =NS*)-

The function /(x) is said to be continuous in an interval (a, #)
when it is continuous for all values of x in (a, f).

The definition of continuity of /(*) at x asserts that whatever
absolute number ¢ is assigned, we can always assign a corresponding
absolute number 4 such that for all values of ~, satisfying the inequality

| xy x| <A,

I7) — A+) | < 4.

Since, by definition, the limit of /(x,) is (x), we can make and

keep

- |Ax) —Ax)|
less than any assigned absolute number 6 for all values of J(*,) sub-
sequent to an assigned value /(.’).

If « + h is the value of the variable corresponding to /(x’), then
all the values of the function corresponding to the values of the vari-
able in (x, x + 2) satisfy the inequality above.

An important corollary to the above and a principle which will
constantly be employed later is: If (a), the value of A(x) at a, is
different from o and is finite, then we can always assign a finite num-
ber 4 such that for all values of x in the interval (2 — h, a + A) the
function /(x) has the same sign as /(2).

The above definition shows that a continuous function must change
its value gradually as the variable changes gradually, and that the dif-
ference of the values of the function

J(%,) —SK%,);
must be arbitrarily small in absolute value when the difference of the
corresponding values of the variable, x, — %,, 1s arbitrarily small.

we have
ArT. 23.] ON THE FUNCTION OF A VARIABLE. 23

It also shows that a function cannot be infinite at. a value of the
variable for which the function is continuous, and vice versa.
In symbols, when /() is continuous at a, we must have simulta-

neously
£(e—a=o and £LAx) —Aa)] =o.

The definition and condition of continuity at 2 can be expressed in

the compact symbol
4 M4) = So).

a(=)a

23. Fundamental Theorem of Continuity.—If /(x) is a uniform
(§ 21) and continuous function of x in an interval (a, 4), then what-
ever number JV be assigned between the numbers /(a) and _/(4), there
is a value & of x in (a, 4) such that at & we have

Ag) = N.
The proof of this theorem falls under two heads.

I. Ifa function /(x) is one-valued and continuous throughout an
interval (a, 4), and /(a) and /(é) have contrary signs, then there is a
number & in (a, 4), at which we have

Fé) =0.

Suppose /(a) is negative and /(4) positive. Then_/(x*) cannot be
o or + arbitrarily near to.x =a, nor can f(x) be o or — arbitrarily
near to x = 4, by definition of continuity of /() at @ and 0.

Let8—a=A, Divide this interval into 10 equal parts by the
numbers

a,a+tqyh, ... ,a+a%5h, 4.

Either /(x) is o for « equal to one of these numbers, in which
case the theorem is proved, or it isnot. In the latter case let a, be
the last of these numbers, proceeding from the left, at which /() is
negative, and 4, the first at which it is positive.

Proceed in exactly the same way, subdividing the interval (a, , 4,)
into to equal parts. Then if /(+) is not o at one of the new division
numbers, let a, be the last at which it is negative and 4, the first at
which it is positive.

Continuing this process 2 times, we find that either /(.) is o at
one of the interpolated numbers, or that (2,) is negative and /(4,) is
positive (see § 15, VII, 3), and

n n h
Q,=ath fe , b6,=a+h Sa

Io” Io

where each f,(r = 1, 2, . . . , #) represents some one of the digits
o, I,..., 9 If f(x) is o for some one of the interpolated

numbers obtained by continually subdividing (a, 4), the theorem is
proved; if not, then the two numbers a, and 4,, the former always
24 INTRODUCTION TO THE CALCULUS. (Sec. II.

increasing, the latter always diminishing, converge to the common

limit
= Pr
§& =a+h a

Meanwhile /(a,) and /(4,) converge to the common limit 7(&),
by the definition of continuity. The first of these /(@,) is always
negative, the second /(4,) is always positive. Also, since 6, —a, =
4/10”, we must have

LiF On) —S(@)} = Lt F (Gn) | + |S (4) |

 

=2|7(8)|-
But this limit is 0, by definition of continuity.
ite) =p.

In like manner we prove the theorem when / (a) is positive and
TF (4) is negative.

II. The general theorem now follows immediately. For, what-
ever be the numbers / (a) and /(4), if V lies between them, then

f(x)-N
must have contrary signs whenx =a, += 46. Therefore, by I, there
must be a number & in (a, 4) at which
f(E) -N=0.

The important fact demonstrated by this theorem is this: Ifa
function / (x) is uniform and continuous in an interval (a, 4) of the
variable, then as the variable x varies continuously through the inter-
val (a, 6), the function must vary continuously through the interval
determined by the numbers /(a) and _/(4). That is, the function /(x)
must pass through every number between /(a) and /(4) at least
once.

24. General Theorems.—The following general theorems result
immediately from the theorems on the limit, § 15, and the definition of
a continuous function.

I. The sum ofa finite number of continuous functions is a con-
tinuous function throughout any common interval of continuity of
these functions.

If A(«), A(x), . - ,7,(*), are continuous at x, then

AxyEf7(ay+t... +A(*)
is a continuous function at v. For we have
APO) = LAE) + FACE
= LAO) +... SAX),
=S(4°)+ --- SAL),
=f)... +A(2) = O(2).
ArT. 24.] ON THE FUNCTION OF A VARIABLE. 25

II. The product of a finite number of continuous functions is a
continuous function in any common interval of continuity of these
tunctions.

If P(*)EL(*) - fe - ++ Sul%)s
then wf ae)= = LA) - - - A];

= LI")... SAA’);
=S(L*') - PAL#),
=/\(*) . + - Pde = o(*).
Corollary. Any finite integral power of a continuous function is a
continuous function in the same interval of continuity.

III. The quotient of two continuous functions is a continuous
function in their common interval of continuity, except at the values
of the variable for which the denominator is zero.

If A(x) = (x) /(*), then we can consider (x) as the product of
@(x) and 1/(x). The theorem is then true by the reasoning of the
preceding theorem.

Otherwise,

P(x’) _ £O(*’)
“y= =>.
war = 2 we ee?
_ (2)
p(x)’
provided p(x) #0. If p(x) =o and G(x) # 0, then /(x) = « and
is not continuous at x If ~(x) =o and also d(x) =o, then /(+)
may or may not have a determinate value as the limit of /(x’), a case
which we shall investigate later.

 

IV. It has been shown, Exercises, Sec. I, Ex. 5, that
Lal) = ahh) = ahh,

when a is positive. Therefore 2” is continuous when /(x) is con-
tinuous.

V. In like manner, Ex. 6, Exercises, Sec. I,

4 log. Ax) = logs £/(*) = log. 4);

(x) being positive. Therefore, log, /(*) is continuous.
VI. Again, if (x) and (x) are continuous and /(-x) is positive,

we have
=[/i«~)]™.
: ney P(*) log /(x),

£ log vy = £[P(*) log Ax)],
= £9(x):£ log ((*).

log Ly = HL) log KL),
= log [A £42) 194”.

LU) = (A4*)]1™,
26 INTRODUCTION TO THE CALCULUS. [Sec. IL.

and the function y is continuous when (x) is continuous and (+) is
continuous and positive.

SPECIAL THEOREMS.

Since y=. is a continuous function of x, the product a,x”, where
a, is independent of x, and ¢ is any finite integer, is continuous for all
finite values of x. Also the sum of any finite number of terms of this
type is continuous. Therefore the algebraic polynomial in x is a
continuous function for all finite values of x.

By the theorem for the quotient, it follows that the algebraic frac-
tion or rational function is continuous everywhere, except at the roots
of the denominator.

By Trigonometry, since

sin x’ = sin x + 2.cos £(x’ + 4) sin 4(x’ — x),
and 4 sin («’ — x) =0, when «’(=)x,
we have Z% sin x = sin x = sin £x.

Therefore sin x is everywhere continuous.

In like manner we show that cos x is everywhere continuous, and
by § 24, III, tan x, cot x, sec x and csc x are continuous functions
everywhere except at the roots of their denominators, cos x, sin x.

The continuity of any algebraic function of

a a", sin x, log x,
can now be easily determined.
25. Geometrical Illustration of Functions.—If we adopt the method of rep-

resenting the variable, in §$ 8, 11, by points on a straight line, such as Ox, then at
any point AZ on Ox corresponding to + = @ we can represent the corresponding

 

 

P
i y
; Py
B R,
| R,
Msg IL MM, ae
[24 MM, © | M,M M,
B
8 1 B
Fic. 4.

value of a uniform function /(x) by a point Pina plane xOy. The point P is con-
structed by laying off a perpendicular A7P to Ox, such that the distance AP is
equal to the number /(a), and is measured upward if /(2) is positive, and down-
ward if /(a) is negative.

For each and every value of x for which /(x) is a defined function, such as aq,
@,, ..., we can construct corresponding points P,, P,, ..., representing /(2,),

gle 4 8

- *This is the familiar method of Analytical Geometry, invented by Descartes.
If we put y = fix), then Oy perpendicular to Ox can be called the axis of the func-
tion, corresponding to Ox, the axis of the variable; and x, y are the Cartesian coor-
dinates of the point P representing the functional form /(x).
ART. 25.] ON THE FUNCTION OF A VARIABLE. 27

If the function f(x) is continuous in any interval (4;, 45), then corresponding to
any point J, in AZM, there is a point P, inthe plane xOy, at a finite distance
from Ox, representing the function. Moreover, any two such points 2’, P” cor-
responding to AZ’, A/” can be brought as near together as we choose by bringing
M’ and M" sufficiently near together. Can we say that the assemblage of a// the
ae /, representing a continuous function in a given interval (4, 6) of x, isa

ine

To answer this question it is necessary to consider the question: What con-
stitutes a line, or in general a curve?

Geometrically speaking, the older definitions, now antiquated, required a line
to have in the first place a determinate /ength corresponding to any two arbitrarily
chosen points on the line, and also to have direction at any point. This requires
a definition of direction and of length, concepts themselves abstruse. The old
definition, ‘‘a line has length without breadth or thickness,” is now taken to mean
that a line is simply extension in one dimension.

In order that the assemblage of points in the plane «Oy representing a con-
tinuous function 7x) can be defined as a line, this assemblage must have some
analytical property at each point that will define a determinate direction, and cor-
responding to any two points some analytical property that will define a determi-
nate length. These properties must be inherent in the function J(x) of which the
assemblage of points is the geometric picture.

To define the first of these properties, i.e., a determinate direction, is the prov-
ince of the Differential Calculus ; the second, which gives meaning toa definite
length, is furnished by the Integral Calculus.

At our present stage of knowledge, then, we cannot say that the assemblage of
points which represents a continuous function is a line. But it will be demonstrated
in what follows that such continuous functions as those with which we shall be con-
cerned can be represented by curves, and we shall in the course of our work
develop an analytical definition of a line, and find means of measuring its direc-
tion, length, and curvature, and many other properties that are unattainable save
through the Calculus. _

In order to take advantage of the intuitive suggestiveness of geometrical pictures
as illustrations of the text, we shall assume for the present that the assemblage of
points 7, ,..., “4 representing values of a continuous function in the interval
M, M,,, has the following properties :

q

 

 

 

 

 

O| M, M, Mert M,,
Fic. 5.

Join the consecutive points by straight lines. Consider the broken polygonal
line PR, P,... Pn. Then, if 44 and JZ, correspond to two fixed values a, 4 of «x,
and we increase the number of points, 47, between JZ, and J/, indefinitely, in such
a manner that the distance between any two consecutive points JZ, and J/, + ; con-
verges to zero, we shall have:

First. The distance between the corresponding points P, and P, 4; converges
to o. For, /, P,+4, is the hypothenuse of a right-angled triangle, P, VP, +.,
whose sides P, Vand VP, 4 ;=M,M, +, converge too together, when J7,(=)1/,4,,
since the function (x) is continuous.*

* The point JV, not shown in the figure, is the point in which a straight line
through P, ,, parallel to Ox cuts AZ, P,.
28 INTRODUCTION TO THE CALCULUS. [Sec, II.

Second. We assume that the angle P,_, P, P,4, between any pair of con.
secutive sides of the polygonal line, such as P, _, P, and P, F, + ;, converges to two
right angles as a limit.

Third. We assume that the sum of the lengths of the sides of this polygonal
line P, P,, converges to a determinate limit length.

The first consideration secures continuity, the second determinate direction, and
the third determinate length.

The three together constitute the necessary conditions that the assemblage of
points shall be a curve.

The analytical equivalents of the second and third conditions will be developed
later. That for the first has already been established in the definition of a con-
tinuous function.
ART. 25.] EXERCISES. 29

EXERCISES.

1. If Ax) = 2x8 — x? — 12x41, show that the function has a root in
each of the intervals (0, 1), (2, 3), (— 3, — 2).

2. If P(x) =(* — 1)/(« +1), show that

62) — os) _a—s

1+ Ga) pb) 1 fab"
3. If p(t) =e + e-4, show that

(34) = [YO] — 340),

Ye +I) X Oe — 3) = Hex) + Y(2y).

 

 

4. If A(x) = log : + -, show that
Fp) + Ag) = ae

5. What functions satisfy the functional equations
Ke +9) =N) Ss
P(*) + O(¥) = G(x),
Hx) — YH) = Wx/y),
Mx — y) = Mx)/A9):
6. If f(x) = ax? — bx+ 06, write /(sin x).

7 If y= 2? +x —5, write x as a function of y.
I

8. Show that e* is discontinuous at x = 0. Examine the behavior of this func-
tion as x increases through 0, ;

9. If y = log (x + x? + 1), show that
x= her —e-).
This last function is called the hyperbolic sine of y and is written
sinh y = $(ey — e—9).

10. If y = log (x + x? — 1), x is called the hyperbolic cosine of y and
written cosh y. Find this as a function of y.

11. Show that ey = sinh y + coshy.
12. Let x be any assigned real number. Consider the function
_ xn
F(a) = ar
where x takes only positive integral values, Show that A(z) has the limit o when
7 = 0, whatever may be the finite value of x. .
13. Show that
_aa—1)...(a—r+1
for) = a OE er,

in which a and x are assigned real numbers, has the limit o when 7 = o, pro-
vided |x| <1. What is the value of /(oo ) when |x| > 1?

14. Investigate

for |x| ZI.
3° INTRODUCTION TO THE CALCULUS. [Sec. II.

2 = f\2
cece ace
2 2
shows that the geometrical mean, 4/aé, of two unequal numbers lies between them

and is less than their arithmetical mean 3(@ + 4).
Finding the square root of any absolute number / can be reduced to finding

the square root of a number between 1 and 100. For, we can always assign an
integer ” such that 1074 = a, where I < a < 100; m being + or — according
as # is less or greater than i. Then

VB = 10-*Wa.
Consider any given number between 1 and 100. Choose x, from one of the
integers 2, ..., 10, such that

(4-1? <a<x

15. The identity

 

Then <2 Va<n;

a
a — I a
=, < Va <Ziat Z)=s.

Show that if this construction be continued to x,,, then

_— I
Xm — Va Sia

and therefore the sequence of numbers x,, x,, ... defines the square root of a,
and
M=o0

16. Apply 15 to show that 4/5, to six decimal places, is
__ 2207

4, = —— = 2.2360680.
4 987 3 9
17. Show that the cubic function of x,
BR RS ge =f)
ho, b= x, Ff
g : Tt 5 ¢—x

always has three real roots.
Expanding with respect to the first row,

Se) & (4 — #8 — xe — 2) — f?] — [Ae — 2) — 2 fgh + 26 — x).
Let 2, g, of which p is not less than g, be the two roots of the quadratic
function
(6 — xe ~ x) —-fPHxe?— (64 0x4 be ~ f?.
Then P+qa=b+e, and fg = be — f?.

Therefore neither nor g can be between 4 and ¢ or equal to 4 or c, and pis
greater and g is less than either d orc. But

K+e“)=—o0, —
KA) = + [AVP — c+ ¢ Vp — bP,
Ag =—[AV¥e—9—- gVb— Gg},
f—0)= 400.

Hence, by § 23, I, f(x) vanishes between + 0 and f, between p and g, also
between g and — «, and the three roots are real. This exercise will be needed
in subsequent work.
ART. 25.] EXERCISES. 31

18. Determine the condition that the function
ax* + 26x +
shall retain its sign unchanged for all values of the variable x.

The function can be written
6\2 ¢ by”
e{(=+5) aoa

(ax + 6) + (ac — 0%)

= a

a (#425 +5)

In order that this shall retain its sign unchanged for all values of x, it is
necessary and sufficient that ac — 4? shall be positive. This condition being satis-
fied, the function has the same sign as a for all values of x.

19. Determine the condition that the function

ax* +. 2hxy + by?
shall retain its sign unchanged for all values of the variables + and y.
By completing the square, the function can be written
(ax + Ay)? +3%(ab — )
Me ee ee
a

which, when ad — #2 is positive, has the same sign as @ for all values of x and y.
20. Determine the condition that the function
ax? + by? 4 cz? + 2 fyz 4 2aguz + 2hay

shall keep its sign unchanged for all values x, y, 2.
By completing the square, the function can be written

EA (ax + ge + Ft (ab — Wy" + alfa — hg ys + (ae — gre f

The function will keep its sign unchanged and have the same sign as @ what-
ever be the values of x, y, 2, provided the quadratic function
(ab — yy? 4- 2( fa — hg) yz + (ae — g*)2
is always positive. This will be the case, by Ex. 19, when
ab — hk? and (ab — (ac — g?) — (fa —Agy
are both positive. Or, what is the same thing, ad — #? and

a(abe + 2fgh —af? — bg? —ch*)=alahg

hof
ist ¢

 

 

must be positive. :
Exercises 18, 19, 20 will be drawn on in the sequel.
33

BOOK I.
FUNCTIONS OF ONE VARIABLE.
PARI 1,

PRINCIPLES OF THE DIFFERENTIAL CALCULUS.

CHAPTER 1,

#S

ON THE DERIVATIVE OF A FUNCTION.

26. The Difference of the Variable.—The difference oi a variable
x is a technical term, which means the result obtained by subtract-
ing a particular value of the variable, say x, from an arbzirarily
assigned value of the variable, say 2.
Or, in symbols,
x, —
We use the characteristic letter J to represent the symbol of this
operation, and write
Ax=x,— %.
This difference, 4x, is of course positive or negative according
as 1, is greater or less than x.
We sometimes for convenience write

Ax=x,—x=h,
so that
x= x+,
and call 4 the increment of the variable x.

27. The Difference of the Function.—The difference of the func-
tion is a corresponding technical term, which means the result
obtained by subtracting the value of a function at a particular value
of the variable, say x, from the value of the function at an arédrarily
chosen value of the variable, +,. In symbols

J(*,) —S(*)

is the difference of the function /(~) at .v.
As in the case of the difference of the variable, we use the letter

4 as the symbol of this operation, and write

AK x) =/(%,) — 7).

 

35
36 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [Cu.I.

28. The Difference-Quotient of a Function.

A difference of a function and a difference of the variable are
said to ‘‘ correspond ’? when the same values of the variable occur
in the same way in these differences.

Definition. —The quotient obtained by dividing a difference of
the function by the corresponding difference of the variable is called
the difference-guotient of the function at the particular value of the
variable,

Thus, in symbols,

4K %) _A%) —/%) _ 4
Be ae
is the difference-guotient of the function /(v) at x.
For an assigned particular value +, the number g, depends on

the value assigned to the arbitrary number ~,.

29. The Derivative of a Function.

Definition.—Whenever the function /() is such that when we
assign to the arbitrary value of the variable successive arbitrarily
chosen values

Msg Ne so, oko. Sys Cayesi Sor Je

in such a manner that this sequence converges to the particular value
x as a limit, and the corresponding sequence of difference-quotients,

Ax)—fe) 4 Aa-f®)_, AA) _,

a, % Bt Hig — 2

es

has a determinate number as a limit, this mz is called the deriva-
ave of the function /(x) at x,

In other words, the function /(.x) is said to be differentiable at x
when the difference-quotient

Spx) _ Ax!) — fx)
LR BP eae

converges to a determinate limit as x’ converges to x asa limit in
any arbitrary manner whatever,

In symbols,
SJ’) =S(*)
x’ — x
xl (=)x
is called the derivative of f(x) at x.

This derivative is, in general, a function of x, and we shall
represent it, after Lagrange, by the symbol /’(x), a convenient and
characteristic symbolism because it shows the association of the
derivative /’(x) with the primitive function (x) from which it has
been derived.
i

ART. 30.] ON THE DERIVATIVE OF A FUNCTION. 37

We shall also use sometimes another symbolism, to represent the
operation by which this limit is derived, instead of the cumbersome
one employed above representing the limit of the difference-quotient.

We use the characteristic letter D as a symbol to represent the
operation gone through of dividing the difference of the function by
the corresponding difference of the variable, and determining the
limit of this difference-quotient when the arbitrary value of the
variable converges to the particular value of the variable as a limit.

In compact symbols, we write

y= f £290 -—7)
Dix) = 7 =
x a)x
But we have already agreed that this limit, the derivative, shall
be represented by f(x). Hence we have the equivalent symbolism
Df(x) =/'().
Or, the operation D performed on the function /(.x) results in
the derivative /’(.v).
This operation is called differentiation.
30. Observations on the Derivative.—We observe that in order
that a function /(x) may be differentiable (have a derivative), it must
be continuous. For, unless we have

4) AYO) -—/)I = 2

a'(=)x x= )x

 

as is required by the definition of a continuous function, then, since

we do have
Lv’ — +) = 9,
aN =)x
the value of the corresponding difference-quotient would be «, or
no limit exists.

Hence the Differential Calculus deals directly with none but
continuous functions.

The converse of the above statement is not true, i.e., a function
that is uniform and continuous is not always differentiable. There
exist functions that are uniform and continuous and yet the limit of
the difference-quotient is completely indeterminate for all values of
the variable in certain finite intervals.* We shall not have occasion
to meet any of these highly transcendental functions in this book,
and the functions with which we deal will, in general, be differentia-
ble. Only for isolated values of the variable will the derivatives of
these functions be found indeterminate. Such values are singular
values and receive treatment in their appropriate places.

The evaluation of the derivative of a function falls under the
case specially excepted in § 15, V. Here, the limit of the numerator

 

* See Appendix, note I.
38 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [Cu. I.

(the difference of the function, 4/7), and the limit of the denomi-
nator, 4x, are each o. :

The quotient of the limits o/o is always indeterminate.

We are not concerned, in evaluating the derivative, with the
quotient of the limits, but only with the limit of the quotient.
We are not concerned or interested in the difference-rato but with
the difference-quotient.

This is a variable number which does or does not have a limit
according as the function is or is not differentiable at the particular
value of the variable considered.

The derivative of any comstant is necessarily o by the definition.
For, the quotient of differences is constantly o and remains o for
4x(=)o.

EXAMPLES.

1. Differentiate the function .?.
We have the difference-quotient,

The limit of this number when 2’(=)x is 2x.
Dx? = 2x5

2. Differentiate the function x?
We have aa = td

eax toot

the limit ot which is x—?/2 when x’(=)..
Dxt = gat
3. If f(x) =sinx, show that D sin x = cos «.
We have, by Trigonometry,
sin «” — sin x = 2 cos }(x’ 4 x)sin h(x’ — x),
sin en eee oe a.

Ex. sin $(7 — x)
But, by § 12, Ex. 4, fe =4;

x= )zx
7'(x) = cos x.
4. Show that the derivative of any constant is zero.
If A is any constant, it keeps its value unchanged whatever be the value of x.
Therefore the difference-quotient is

A-—A _ ,
x 4x
for all values of x, # x and when x,(=)x. Consequently DA =o.

_ 5. Show that the derivative of the product of a constant and any function of x
is equal to the product of the constant and the derivative of the function.

 
ART. 31.] ON THE DERIVATIVE OF A FUNCTION, 39

Let abe constant and y a function of x. Let y take the value J, when x takes
the value x, The difference-quotient of ay is

M-VY_ io nA-y
yr x, — x’

the limit of which is aDy. os Day = aDy.

31. Geometrical Picture.—We have seen that a differentiable
function is necessarily continuous. We shall now see that the as-
semblage of points taken to represent it possesses the characteristic
property of a determinate direction at each point and can be considered
as a curve,

 

 

 

 

y
| -L
P.
Gx |M M’
Oo a ae" a
Fic. 6.

In the figure, if P, P’ represent /(x), /(x’), then
4% =x’ —x« = MM’,
A(x) = f(x’) —/(*) = NP’,
Af(x) _ NP’
4x ~ PN
where 6’ = / NPP’ is the angle which the secant PP’ makes with Ov.
By the definition of a tangent to a curve, the limiting position of the
secant PP’ as the point P’ moves along the curve and converges to P
as a limit is the tangent PZ'to the curve at P. At the same time 6’
converges to # as a limit, 6 being the angle which the tangent P7’
makes with Ox, But tan @ is the limit of tan 6’, and is therefore the
limit of the difference-quotient, or is the derivative of /(x) at «.
Therefore we have

= tan &’,

 

 

Df(x) = f"(x) = tan 6.

Hence the derivative of a function is represented by the slope of
the tangent to the curve which represents the function. The direc-
tion of a curve at any point on it is the direction of the tangent there,
and the slope or declivity of the curve is that of its tangent at the
point.
40 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [Cu. IL.

32. Ab Initio Differentiation.—The process of differentiating a
given function directly from the definition by evaluating the limit of
the difference-quotient is, in general, a complicated and tedious pro-
cess. We shall in the next chapter deduce certain rules of differentia-
tion by which, when once we have differentiated log » and sin x by
the ad initio process, we can write down directly the derivatives of all
the elementary functions in terms of these derivatives and those
which follow. Meanwhile, in order not to lose sight of the ad znztio
process and the rationale of differentiation which is at bottom always
the evaluation of a limit, the limit of the difference-quotient, the fol-
lowing exercises are set for solution by this method.

EXERCISES.
Differentiate the following functions :

1. 32? — 6x. 6(x — 1).
2. 7x* — 13. 2823.
3. (« — 1)(2x + 3) 4x+1.
4. x. — x2,
5. ax-3, — 3ax-4,
6. (x — a)/(x + a). 2a(x + a)-2.
7. x? gach.
8. (x? — 2)4, x(x? — 2)-4,
9. 2(x + 1)7}. — («+ 1)73.
10. x3. fa-3,
11. x. (2 any finite integer. *) nx",

b é
12. x7. (p and g positive integers. *) Pad,
13. cos x. = sin x,
14, tan «. sec? x.
15. log x. (See § 15, Ex. rr.) a1,
16. sec x. sec x tan x,
17. a*. (Use Ex. 15, § 15.) a* log, ua.
18. x2. (x positive, a rational.) axa-t,

 

* Divide numerator and denominator of the diff.-quot. by x, — x in Ex. 11
z I

’

and by 4, — x? in Ex. 12.
CHAPTER II.
RULES FOR ELEMENTARY DIFFERENTIATION.

33. As was stated in Chapter I, when we have once differentiated
x7, sin x, log x, by the ad zntfio process, we can differentiate directly
any elementary function of these functions by certain rules for
differentiation, without recourse to the aé initio process directly.*
These rules are themselves deduced by that process, and their appli-
cation to differentiation is but a short method of evaluating the
limits which we call derivatives. We shall see that the direct differ-
entiation of only two, sin x and log .x, are necessary, for 2* can be
differentiated by means of log 1. Independent proofs, however, are
given in each case.

34. Derivative of log, x.—We have for the difference-quotient,
writing +, —.1 = 4,

log, (* +) —log,x _1 ( *)
a log,\1 a ’

tl
|
—
°
gq
a
os’
4
als

x
writing +/4 = 2. When A(=)o, z= 0.

I 1\?

.. Diog, * = f zis. (1+ 2)’
= ole ( ay, § Ex. 6
= = log. Oi as Q15, Ex. 6.

z=00

 

* Asa matter of fact, the evaluation of only one of these functions, log x, by
the ad inztzo process is necessary. That is, the differentiation of all functions can
be reduced to the evaluation of the single limit, (1 + 1/x)*, when x =o,
§ 15, Ex. 12. For, the differentiation of log x gives that of ex, and we have

1s , : — Fi
sin x = —(e'* — e—?*), where = + y/—1. Wedo not, however, recognize com-
22

plex numbers in this book directly, which necessitates an independent differentia-
tion of sin x, and restricts us to a geometrical definition and differentiation of that
function.

41
42 PRINCIPLES OF TIIE DIFFERENTIAL CALCULUS. [Cu. I.
The evaluation of this limit is effected in § 15, Ex. 12, and is

the number
I Zz
fb+3) SOS Be F182 os

o. Dlog, «= = log. é

In particular, if a= e, then log, e = 1, and

x

D log x = =.

According to common usage, when the base of the logarithm
employed is e we omit writing the base and put log ~ for log, ~.

35. Derivative of x4.—Let a = p/g, where p and ¢ are positive
integers.

Dividing the numerator and denominator of the difference-
quotient

2: 4
Xt — x9
x,

I I
by x,7 — <9, the difference quotient-becomes

len + foal ae SEs ee a ele) + lay
1\¢—2

baa) leg) wets tele! aha)

In the numerator there are f terms each of which has the limit

aye
(7) , and in the denominator there are g terms each of which

I

q-1
has the limit (x ) , when a\(=)a. Therefore the limit of the

difference-quotient is
1\p-1

Dexa a p (x7) 7 Dey?
g

 

 

é
= Lue
qg
If a = — p/g, then the difference-quotient is
at at 2 of
XI Ko I Kt x8
BP ge ge?

meee
the limit of which for «,(=). is, by the above,
b é
eae 2 = _ fat

%
xT 7 7
Arr. 36.) RULES FOR ELEMENTARY DIFFERENTIATION. 43

Therefore, whatever be the rational number a,
Da = ¢a3-',
Rule: Multiply by the exponent and diminish the exponent by 1.

36. Derivative of sin x, cos x.
§ 30, Ex. 3, that

 

It has been shown in Chapter I,

D sin 1. = cos x.

The derivatives of all the other circular functions can and should
be deduced in like manner. They can, however, as we shall see, all
be deduced from that of the sin x.

For immediate use we have, from Trigonometry,

cos x’ — cos x = — zsin $(a” 4 2) sin d(1’ — x).
cos. — cos v sin 4(4’ — x)

= SS = = sin Hx! 42)

Hence, on passing to the limit,

F(x" = 3)
D cos a = — sin x.

RuLes FOR DIFFERENTIATION.

37. We proceed to establish rules for the derivative of the (1) sum,
(2) product, (3) quotient, (4) inverse function, and (5) function
of a function, in terms of the derivatives of the functions involved.

These are the general rules for the differentiation of all functions
with which we shall be concerned. It is necessary to know them
perfectly, for they are the tools with which the Differential Calculus
works.

38. Derivative of an Algebraic Sum.

Let yo=utov+w,
where uz, v, ware differentiable functions of 1. Let the differences
of these functions be 4y, 4u, Jv, Aw, respectively, corresponding
to the difference Jv of the variable x. Then, if y, uw, v, w take the
values _y,, #,, 2,, w, when .v takes the value 1, we have

I, —I = Ay, I =I + AY,
and so for w, v, w.
A= +444,
J; —y=(%4—4)+(%,—-%7+(%,-»),

Ay = Au + 4vo+ dw.

4y fu 4 Aw
8 ae aie Ae? ae

or

The student should observe the detail with which the difference-
quotient is worked out here, as this detail will be omitted hereafter

and he will be expected to supply it.
44 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  ([Cu. II.

Since the limit of a sum is equal to the sum of the limits, we have
for Jx(=)o, on passing to limits,

Dy = Du + Dv + Du, (1)

or Du +v+w) = Du+ Dv+ Dw.
Corollary. What has been proved for three functions here is
equally true for any finite number of functions #4, . . . #,, and it can

be proved in the same way that
n n
D2u, = 2Du,;
I ¥
hence the rule:
The derivative of the algebraic sum of a finite number of differ-
entiable functions is equal to the sum of their derivatives.
In all cases in which we pass from an equation in difference-

quotients to one in derivatives, the student is required to quote the
corresponding theorem of limits, § 15, which justifies the equality.

EXAMPLES.
1. The derivative of any polynomial in w,
. dy tax tart... + anx,
is
a + 24,% + Zax? +... 4manxt-l,

This can be expressed in the following sites

Strike out every term independent of «x, since its derivative is zero, and
multiply each remaining term by the exporient of that term and diminish that
exponent by 1.

2. If ¥ = 2xt+ log x3 — 3 sin x,
show that Dy = 5x? 4 5/* — 3 cos x.
2+ ob
3. If f= a, show that

SJ'(x*) = ¢ — ax,
4. Make use of the identity
sin (2 + x) = sina cos x + cosa sin x,
to show that D sin (a + x) = cos (a + +).
39. Derivative of a Product of Functions.
Let y= uv,
Then, with notation as in § 40, we have
Ay = (u + Au)(v + Av) — uv,
= a + rig Mu. Av,
Ay 4u- dv :
Ax iat S gt oa Ax ° (3)

Since, by ec

Au do
Du= f and Do= f
ART. 40.] RULES FOR ELEMENTARY DIFFERENTIATION. 45

are finite, the last term on the right of (i) has the limit o when
Ax(=)o; for it can be written either

Au

(35) 4

Av )
aul
and 4u(=)o, 4v(=)o, when 4x(=)o, the functions being con-
tinuous,
Therefore, in the limit, (i) becomes
D(uv) = vDu+u Do, (II)
In particular, if v is constant, v = a, then Da = 0, and
D(au) = a Du.
Corollary. Show that
D(uow) = uv Dw + uw Dv + vw Du,
and, in general, that the derivative of the product of a finite number of

functions is equal to the sum of the derivative of each function multi-
plied by the product of the others.

EXAMPLES.

. Show that D(«* sin x) = wx”—1 sin « + x” cos x.
D(x# log x) = «4-1 (log x*-+4 1).
. Show that £ (D log xsinx — cos x log x)= 1.

2(=)o
. Show that JD sin? x = sin 2x.

2
If y = (log x)?, showthat Dy = log (x)*.
. If f(x) = log x, then f(x) = 2/x.
. Show that D sin 2x = 2 cos 2x.
. Show that D cos 2x = — 2 sin 2x.
Use cos 2x = (cos x + sin x)(cos « — sin x).
9. Show that JD (log x*) = log x +1.

ONO FP ON =

40. Derivative of a Quotient.
u

Let Se

Then y, wu, v, become y+ Jy, u+ du, v+ dv, when x becomes
x + 4x, and we have
u+ du
B= v +. do
v4u — udv

= v(v + dv) *

Au dv

4y "Ax ” Ax
Ax u(v+ 4v) *

u
a
46 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [Cu. II.

Since 4z(=)o when 4x(=)o, we have, provided v 30, on
passing to limits

 

u v Du — u Dv
Dyess o(%) Es oe (IIT)
v U
In particular, if «=a, any constant, then Du = o, and
a Do
2(¢) = = cP. IV
= P (IV)
EXAMPLES.
1. Show that D tan x = sec? x,
We have Dtanx = D sau =z
cos x
__ cos? + sin? x
= cos? x
= sec? x.
2. Show that D cotx = — csc?x, using both
cotx = cosx/sinx and cot x = I/tan x.
3. Show that D sec x = sec x tan «,
using both sec x = I/cosx and secx = tan x/sin x.
4. Show that D csc x = — csc x cot x.
5. Show that D vers x = sin x.
atx b—a
t ee bets eee
6. Show tha D (As eat
2
7. Show that el a
a—x (2 — x)

x
8. Show that Jee = bee is
log ax ~~ (log ax)

41. Derivative of the Inverse Function.—If y is a continuous
function of x, we must have 4y(=)o when 4x(=)o, by the defini-
tion of continuity. Therefore for any particular value of x at which
y is a continuous function of + we can always make 4y converge to
o continuously in any manner we choose, such that simultaneously
we have dx = 0, Also, for corresponding differences Jy and 4x,
we have

Ay 4x
Ax ay

If we represent the derivative of y with respect to x, by D,y, and
the derivative of x with respect to y, by D,x, then whenever y is a
differentiable function of x and D, y # 0, we shall have xa differ-
entiable function of _y, and the relation

Dy Dx = 1
always exists.
Therefore, if_y and -v are functions of each other and the deriva-
ART. 41.] RULES FOR ELEMENTARY DIFFERENTIATION. 47

tive of the first with respect to the second can be found, then the
derivative of the second with respect to the first is the reciprocal or
inverse of the first derivative.

If = f(x), then v = (y), obtained by solving y = /(*) for x,
is the inverse function of /(.).

GEOMETRICAL ILLUSTRATION.

If the curve ABZ represents the function 1 = /(*), and we con.
sider v as the function and 1 as the variable, we have + = ey)

 

 

 

Fic, 7.

represented by the same curve, except that now Oy is the axis of
the variable and Ox the axis of the function. Fora particular .,
the point .Y represents /(v) and ¢(y’), and we have
AY = /S(%)i pX = O(y).
Again, if 6, @ are the angles made by the tangent to AB at X,
with Ox, Oy respectively, measured according to the conventions of
Cartesian Geometry, we have

D1 = f'(*x) = tan 6,
Dx = $'(y) = tan ¢.
But, since we always have tan @ tan @ = 1,
» DV DX = 1,

EXAMPLES.

1. If. y = 2? 4 2ax 4 6, then
Dy = 2x + a).

T
OF = se Fa
If we solve for .v, we get the inverse function
eae —at Vi +y— 32,

a function which we do not yet know how to differentiate, but we know its deriva-
tive must be the value D,.x obtained above.

2. Ify = x8, find Dy, Dyx, and verify Dy Dx = 1.
48 PRINCIPLES OF THE DIFFERENTIAL CALCULUS,  [Cu. II-

3. Differentiate sin-1.
Ify = sin-%x, then x = siny.
Dyx = cosy = VI — x3
I
VI— at
We know from Trigonometry that the angle whose sine is x, sin-tx, is
multiple-valued and that

Hence Diy =

sin [zw + (—1)"6] = sin §,
where 2 is any integer. In the derivative of sin~'z above, the radical shows its
value is ambiguous as to sign. But if we agree to take sin—1x to mean that angle
between — 37 and + 47 whose sine is x, there is no ambiguity, since then cos y is
positive.

Then we have
I

D sin-!x = ——____.,
+ Vi — x?
4. Show in like manner that
I
D cosx = —————-
— VI — x
where oO < cosy < Zz.

This can also be shown immediately by differentiating the identity
sin-ix + cos-Ix = 4m.

5. Show that Dtantx = oi
I++
Put y = tan—tx, then x = tan y, and
Dyx = sey = 14+ 2x. Ex. 1, § 41.
Dtan-tx = ie,
1+ x

where we take tan—!x to be that number such that
— 3m < tan < + 4x.
6. Show in like manner that

D cot-iw = esr
1+
where Oo <cottx < zm.

Also, by § 38, from
tan—ix + cot-lr = $m.

7. Show that D secrtx = ar
xxt—t
If y = sec-ix, then x = secy, and
Dyx = secytany = x 7x? — 1, Ex. 3, §41.
Dyy = aT
he x2 =f
8. Show, as in Ex. 7, that
. Desertx = ee

x fxr — r
Also, by § 38, using the identity
csc—tx + sec—tx = im.
ART. 42.] RULES FOR ELEMENTARY DIFFERENTIATION. 49

9. Differentiate a~.
Put y= az, then x = logy.
Therefore, by § 34, we have

I
Dyx = i loga ¢.

Soy ise
Dyy = me a* log, a.

 

In particular, if @ = e, then
Dax = ax log a
becomes
De* = ex,
or the function e* is not changed by differentiation.

42. Differentiation of a Function of a Function.—We come
now to consider one of the most powerful methods of differentiating
certain classes of functions.*

Let z be a function of the variable y, say s = /(v), and let_y be a
function of +, say y= d(x). We require the derivative of z with
respect to the variable x.

If zis a differentiable function of the variable», and y is a dif-
ferentiable function of the variable x, for corresponding values of z,
y and x, then we shall have

D2 = Dz Dy, (VI)
or SAI) =S5( 9): Ded
For, we have
Az 42 Ay
Lx Ay Ax’

and since by hypothesis D,z and D,y are determinate limits, D,2 is a
determinate limit equal to their product, and (VI) is true.

Corollary. If w is a function of v, v a function of w, wa function
of s, 2 a function of y, and finally ya function of x, then the difference-
relation

Au 4u 4v dw Az Ay

4x — Av fw dz Ay Ax
leads to the derivative
Du = Dy Dp. Dw: Dz Diy
whenever the derivatives on the right are determinate. Hence the
following rule: The derivative of a function of a function, etc., is
equal to the product of the derivatives of the functions, each derivative
taken with respect to its particular variable.

EXAMPLES.

1. Differentiate xe, when «x is positive and @ irrational.
Put y = xa, then taking the logarithm or, as we shall say, ‘‘logarating,” + we

have
logy = a@ log x.

 

* For a geometrical picture of a function of a function, see Appendix, Note 2.
+ The term ‘‘taking the logarithm ”’ is the meaning of an operation so frequently
used that it seems to deserve a verb ‘‘to logarate.””
PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [Cu. IL.

50
Differentiate with respect tox. We have
I _ a
ee =>
zy
Dy = «a a axa-t,

the same formula as when @ is rational.

2. Differentiate (a + dx)*.
Put fui) =e, where y = a-+ dx.
fil) = aye Dy, and Dy =4.
Dia + bx)e = bala + bx) a,
3. To find Dcos« from Dsinx = cos x.
We have cos x = sin ($a — +).
D cos x = D sin (4a — x),
= cos (lm — x) Dida — x),
=>— sins.
4, Deduce in like manner D cot «, D csc «, given the derivatives of tan x
and sec a.
5. Ify = cosstx, then x = cosy.
Differentiate both sides with respect to x.
I= — siny Dy.
D cos—1x, as before.

6. Find in like manner D cot-1x, Dcsc-tz, from Dtansx, Dsec x.

7. Ify = a, then logy = x log a.
Differentiating with respect to x, we have
D, log y-Dzy = log a,

I
or ~D,yv = log a.
ye gs

Dy = a* log a, as before
8. Differentiate 4/a? — 2% Put « = a? — x2,
Dw = Dybv,u,
= a a 2x),
— <x
~ Var ae
9. As an example of the differentiation of a complicated function of functions,

differentiate
log sin ¢ ©0s (a—2z)°,

Let

yvyoa— bx. 5 Dg = a bs
z= (a — bx} =7 = Dyg = 392)
u = cos (a — me = cos z, a. Dw = — sins.
uv = @ C08 (a—bx)" — eu, wt. Dyv = ev
Sous ip RB! oe
w = sin e Cos (a—6x)" — sin v, - Dw == cos v,
Therefore the required derivative is the function :
36 :
— ye" sin 2 ccs v,
w

which can be expressed as a function of . directly.
“ART. 43.] RULES FOR ELEMENTARY DIFFERENTIATION. et

43. Examples of Logarithmic Differentiation.—The differentia-
tion of products, quotients, and exponential functions are frequently
simplified by taking the logarithm before differentiation.

 

EXAMPLES.
1. Show that

Diww*) Du Du
Sage an et
ust u v
the upper signs going together and lower signs going together. Put y = wv +1,
then taking the logarithm,
log vy = log w + log v.
Dy Du | Dv
—_— aS rt
y u v
This expresses compactly the formule for differentiating the product and the
quotient of two functions.

2. Show that if 7, #, .. . wn is the product of 2 functions of x, the derivative of
the product is given by

n
nt.
DIT yet, Du,
ii” thy Uy
I
3. Differentiate #%, where z and v are functions of x.

Put y = wv and take the logarithm.
log y = v log w.

 

Differentiating,
om = Dy-log wu + pee

“a
Dur = Ww (ls u Du + iDe
z

4. Differentiate log, .
Put y = log, w, then v? = w. Logarate this with respect to the base e. and
we have
y log v = log w.

Differentiating with respect to «x,

 

 

D
log 7 Dy + Du = —.
Du log wu Du I
eee (= ~ Jog v FT log v"

44. For general reference in differentiation a table or catechism of
the standard rules and elementary derivatives is compiled and should
be memorized.

In this table w or v is any differentiable function of a variable with
respect to which the differentiation is performed.
52 PRINCIPLES OF THE DIFFERENTIAL CALCULUS, ([Cu. 1.

Tue DERIVATIVE CATECHISM,

 

1. D(cu) =cDu.
2. Du+v) = Dut Do,
3. D(uv) —uDv+oDu,
u v Du — u Do
ols) ats sPe
I Dv
5. D5) =- =:
v v
6. Dut = aur Du,
D
7. Diogu = = 10g. e.
Du
8. D log “LO a
9. Da” = a“ log a Du.
10, De” = e* Du.
11. Du = uv” log u Du + vu Du.
iz. D sinw = + cos uDu,
13. Dcosu = — sinu Dau,
14, Dtanu = + sec? uw Du.
15. Dcot wu = — csc? uw Du.
16. Dsecu = + sec u tan x Du.
17. Desc u = — csc u cot u Du.
18 Dsintw = Shia Ee,
Vi — 2&2
19. Dceosu = Se,
VI-w“
Du
. D tan = —
20 an" + rw
21. Deot™% = — ae
1+ 2
22. Dsec™ = + as eas)
uVu® — 1
23. Descu% = — ef as
uVu? — 1
24. Dverstu = ay

Vou — 2
ART. 44.] EXERCISES. 53

EXERCISES.

_ 1, Differentiate by the aé initio process, and check by the catechism, the follow-
ing functions : ,

(1), 2. (2), cx. (3), 22%. (4), cr, (5), 2-1. (6), ax-4, (7), #7 — 2x. (8
58 —4r+ 7. (9) I/f(ax+ 4). (10), et — 34 — 24-7 (II), (w — 1)(34 i.
(12), (% — 3)/e + 5). (13), 24. (14), 8. (15), 73,

The solution in each case depends on the fact that a* — 4” is divisible bya—6
when » is an integer.

(16), cos. (17), sin ax. (18), tanax. (19), csc ax.

2. Draw the curve y = 3? and find the slope of the tangent where x = 2.

3. Draw the curve y = x? + 2x — 3, and find the angle at which it crosses
the Ox axis.

4. Use the relation of the derivatives of inverse functions to find the derivatives
T

of xt, 2t, +4, x *, and check the results by the rule for differentiating a function
of a function.

5. Show that the equation to the tangent to any curve y = J) is
¥ =fhayt+(~— af"),
the point representing /(@) being the point of contact.
6. Differentiate Vat — x, Vx? — a, Va? + 2bx.
Ans. — x(a? — x?)~4, x(x? — a2) 4, b(a? + 2bx)74,
(1) Da + xe = ea + x).
(2) Dia + 2°)8 = Ox(a + x),
(3) Dle 4 423 )t = 12b2%(6 + 4x3),
(4) Dax? + bx +c)? = 5(ax* + bx + c)i(2ax + 4).
(5) D(a? — x*)§ = — rox(a? — x?)A,
(6) Dax + bx*)' = 7(a?x + bx7)6(a? + 2bx).
(7) DIO + cx™)* = mncxm—(b + cx),
(8) Dt tax)? = — ax(1 + axty*.
(9) D(a — x) = — 32% a? — xy“)
(10) D sin? ax = — D cos? ax = a sin 2ax.
(11) D sin” ax = na sin*—lax cos ax.
(12) D sin (sin x) = cos «x cos (sin x).
7. Show that the equation to the tangent at « = a, y = f,, for the curve

()ttytae is rat yp aa.
2 2

ee & ee.

(3) yp = 4px is yp = 2p(x + @).

8. Given sin 3x =3sina —4sin'w, find cos 3x.
9. Given cos5x = 16 cos’ x — 20 cos? x + § cos, find sin 5x,
10. Verify cos x = 1 — 2 sin? 4x, by differentiating.
11. Obtain new identities by differentiating
sin 3a + sin 2a — sin 2 = 4 sin a cos 4a cos $a,
sin 4 sin (ta — 4) sin (a + 6) = f sin 36,
a and 4 being variables.
54 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IL.

12. Differentiate the identity
cos* 2x — 3 cos 2x = 4(cos§a — sin§ x).
13. Differentiate x? sin x, 2? 4/a + dx, (ax + b)3.
14. D[(w + 12" — 1)8] = (16x + 31)(% + 1)(2x — 1)
4 2x? — 1
TU he SS eee
16, D{(1 — 2x + 347 — 481 + xP} = — 2023(1 + x).
17. Di(z — 32% + Oxty(x 4 +°)8} = 6ox(1 + 2?)
18. Show that

 

 

pte = z ; pits _ 3-64 — 27
I—*  (—a2)Y¥1—- 2 Pw43 ~ @ +e
an es nxt x _ a\--8
Danae = Wa on
ge a, p2ve __34-#) .
Vi + x2 +x) eee B+ xP yx’
pete t Veae _ e+tavfa— x
Oe VR eee
19. Show that
D fie ee ee ee Dsin-3 pall ee b

 

 

yr—-e@ r—x2’ V3 ~ Yt—3e— a
20. Differentiate sin-"(x/a), tan—"(ax + 4), cos—!

 

x
Yaqe’ sec — \(a/x),
sect + cx’), ;

21. Dlog sinx = cotx; Dlogcosx =?

22. Differentiate ¢2%, e—*, e”*, esinx, glog x,

23. Differentiate a*, asinax, qlogx, gtanz,

24. Differentiate log x, log (a+), log (ax-+ 4), xtex, axex, 2%,
ex log (x + a), log (x- e*), ex/log x, log (xe*), sin (e*) log x, 08 log (cos x),
log, tan x, lee 5sinax, log,2(cos ax).

25. D sin[cos(ax + 4)"] = — na(a + bx)*-1sin (ax + 6)* cos [cos(ax + 4)"].

26. If y = }(e* — e-*), show that

x=lg(y+7r +7),

and that D,y Dyx = 1.

: 27. In Ex. 1, § 41, differentiate + as a function of y and check the result there
given.
CHAPTER III.
ON THE DIFFERENTIAL OF A FUNCTION.

45. Definition.—The differential of a function is defined to be
the product of the derivative of the function and an arbitrary differ-
ence of the variable.

If /(x) represents any function of x, and x, — x any difference
of the variable, then
(4, — *)/"(2)

is the differential of /(x) at x.

The value of the differential at a particular value x depends on
the value assigned to the arbitrary number .x,.

We use, after Leibnitz, the characteristic letter d to represent the
differential, and write d/(x) to represent the differential of the func-
tion Ax) at x Thus

U(x) = (%, — x) f"(#);
=f’ (x) Ax.
46. Theorem.—The differential of a function is equal to the

product of the derivative of the function into the differential of the
variable.

For, let (x) =, then //’(x) = 1, and
dx = Dx. 4x
Sk
Therefore we can write dv for 4x, and have
(x) = f(x) dx.

The differential of the variable is then any arbitrary difference or
increment of the variable we choose to assign. In writing the
differential of a variable we choose to assign to it a/ways a finite
number as its value. In fact we cannot asszgn to it any other value.

47- The Differential-Quotient of a Function.*—Since the differ-
ential of the variable is a finite number we can divide by it, and have

LE) = 7129,

 

 

* By some writers the derivative /’(x) is called the dferential-cocficient of
the function /(x), because of its relation to the differentials in the equation

af(x) = f'(x) dx.
55
56 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. ([Cu. III.

or, the dzferential-guotent of a function, which is the quotient of the
differential of the function by the differential of the variable, is equal
to the derivative of the function,

This furnishes another notation, due to Leibnitz, for the value of
the derivative, and expresses that number as the quotient of two
numbers. The advantages of this notation will appear continually
in the sequel, in the symmetry of the equations, and in the analogy
and relation of differentials to differences.

We frequently abbreviate the differential-quotient into

df 2p ay
dx re St , or ax’
where y = /(x). Also, when _/(x) is a complicated function we fre-

quently write
a(x dq

dx ~ ax

48. Geometrical Illustration.—We have seen, § 31, that if
J =/\(*) is represented by a curve PP,, then the derivative /’(x) or

 

 

 

 

 

‘ a
P
ML
cx
O x x ae,
Fic. 8.

Dy is represented by tan 6, where 6 is the angle made by the tangent
PT to the curve at P with the axis Ox.
Assign any arbitrary number .,, and let P, represent J(*,), and

7 the corresponding point on the tangent to the curve at P. Then
we have

PM=x,—x = dx = dx.
Gfx) = (x, — x) f"(),
= PMtan MPT,
= MT.

MT therefore represents the differential of the function J(x) at x
corresponding to x,. While

MP, =/(*,) —/(*) = Af(x).

dfand Afare more nearly equal when Jx or dx is a small numbcr.
ART. 49.] ON THE DIFFERENTIAL OF A FUNCTION 57

Observe that for a particular x the differential-quotient

Ze) = f(x) = tan 6

is constant for all aie - X.

49. Relation of Differentials to Differences.—Since the differ-
ence-quotient has the derivative for its limit, we can put

LO = f(x) +,

where a(=)o, when oe Therefore
dfx) =f"(x)4x + a Ax,
= f'(x) dx + a dx.
4f{*)

a
anx) ~** Pay
Hence, when /’(x) 4 0, we have
f AK*) _
a(x)
Ax(=)o
This substantiates the remark made in § 48 that the difference

and the differential of a function are more nearly equal the smaller
we take dx.

50. Differentiation with Differentials.— Observe that all the
formule in the derivative table, § 44, are immediately true in differ-
entials when we change D into d. For we need only multiply such
derivative equation through by dx in order to make it read differen-
tials instead of derivatives.

We have
af(u) = D,f(u) ax.

aft) = D, flu) du,
= D,/(u)- Du dx,
= D,f(u) ax,
since D,f(u)=/f'(u) Du, and du = D.udx.
w. D,f(u) du = D,/(u) ax,
or the first differential of a function is the same whatever be the

variable. ae
More generally, let u, v, and w be functions of x. Distinguish-
ing differentials like derivatives by subscripts, we have

d, flu) = D,f(u) do = D, flu) Dy do,
= D, fu) du = d, flu).
In like manner, d,,/(“) = d,/(u). Therefore
af (u) = dy JH);

For, by definition,
58 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. III.

or the differential of a function is independent of the variable
employed. It is not necessary, therefore, to indicate the variable by
subscripts or in any other way; in fact the variable need not be
specified. It is due to this that frequently the use of differentials
has marked advantages over that of derivatives.

51. We add a further list of exercises in differentiation, using in-
differently the notations of derivatives, differential-quotients, and
differentials in order to insure familiarity in their use. The sequel
will show the advantage of each in its appropriate place,
ART. 51.] EXERCISES. 59

EXERCISES.
1. If, y, are the coordinates of a point on a curve, show that
dy
VYoya=(X — x)
y= ( “i
or (Y—y) dx = (X — x)dy

is the equation of the tangent at x, y, where X, Yare the current coordinates
on the tangent. This equation can also be written

Y—y X—«x
i ae"
2. Show, with the above notation, that
(Y—y)dy +(X — x)dx =0
is the equation of the normal at x, y.
3. Show that d(23 logx) = «(log «3 + 1) dx.
4. d(cos mx cos mx) = — m cos nx sin mx dx — n cos mx sin nx dx.

dad. ‘
5. Te sin? « =  sin*—1! x cos «.
se

6. dsin (1 + x?) = 2x cos (1 + +’) dx.
7. If y= sin x sin mx, show that

q . :
sin? x = = msinmtiyx sin(m + 1)x.
ix

8. D(a sin? x + 4 cos? «)* = n(a — 4) sin 2x(a sin? x + 4 cos? x)#—1,
9. dsin(sin x) = cos x cos(sin x) dx.
10. f(x) = sin-1(x*), show f’(x) = mxe—u(1 — 2?)-4
H. dsin-(1 — yf = — (1 — 7x,
b+tacssx Va—B ;
a+técosx” a+ sbcosx
13. dsect x = msec” x tan x dx.
2ax

xVet— 1
15, da? + x%)} = x(a? 4 x?)-hdx.
16, d(a? — x?)~t= x(a? — 22) “3dx.

ad —} a
FE ee

18. Dx(2ax — x*)t = (2 — x)(2ax — x)-}.
19. If f(x) =4e—1sin2x, then /(x) = sin’ x,
20. Show that d(j« + Lsin 2x) = cos’ x dx.

12. = cos!

14. d sec—1(x?) =

 

21. 2 ra — cos x )= sin’ x.
ax
‘ sins h WD cogs
22. If y=sinx—jfsin’+, then oe
23. dlog cos « = — tan dx.

24. J log sin « = cot x.
25,y =tanx— x. dy = tan? x dx.
60

26.
27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.
38.

39.

40.

41.

42.
43.

46.
47.

48.

PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. III.

P(t) = cot? + ¢, then f(t) = — cot? Z.

If z= logtaniy, show that

j= D log {E08 ¥
dy I+ cosy
Dy, log tan (ta + 4) = sec m.

a i+sina _

da 8 \roes ee

d@ sin—(36 — 40%) = 3(1 — 62)—? 46.

 

@ apr 20 +4 _ vac - 2
dp Yac—- 2 D+ 2wp+e

dy “w—1\t 1
aa 8 uti)” @— 1

 

 

d (1 (s + 13 I 28 1). 8

“ds 16 sar Ta V3 t= i>
V@+C—a_ ak

d log z = Cyr Te x z

=| Var — Pp at sin 2b a yar ap

an 7 n et an.

in Vo=5 = ee,

? e@- Bo 2a pip — 8B)
in. Gigget eae Eee ee = eee ue :

dz a+tébcosz” a+ 6 cosz

dex(y — x3) = ex(1 — 3x7 — 25) dx.

d (sin mv)" _ mn (sin mv)"—! cos (nv — mv)
‘du (cos nvy* (cos mujer! :

 

d@ sin™ § _ sin™—16
dO cos" 6 cos*#1
dx* = x*(1 + log x) dx.

De® = e** xx(1 + log x).

qd fy\ _  (y\” Jy
Sey =a

‘

(m cos? 6 + x sin? 4).

d@ ee 4

* dx ce + e-* = (e# + ex)
45.

ex — e-*%
eo oe

De(atx)® sin x = elatx)” [2(@ + x) sin x + cos x].

d log (e* + e-*) = dx.

@ 2 — e&1-2)—1
deaz—1 (2 — 17

 

 

3 z Med n ( t ”
dt i “tyr +e —
ART. 51.] EXERCISES. 61

49.

50.
51.

52.
53.
54.

55.
56.
57.
58.
59.
60.

—t
Tf y(t) = ai*—2) "show that

t =e
vy (é) = ——, a'?-*) “log a.
(a — a
a a i
dtana* = —a* u-? sec? a™ log adu.

d [6 + log cos (3a — 6)] = 2(1 + tan 6)— @6.
Dw sin—'y) = sin p + pI — py.
D(tan @ tan—! 6) = sec? 6 tan—16 + (1 + 6?)—1 tan 6.

De-*"** cos bx = — eV (2a2x cos 6x + 4 sin bx).
I t
dx® — x® (1 — log x) dx.

de = & ex dx,

Dut = x” xx [x—1 + log x + (log )?].
dat = x8" ex x (1 + x log x) dx.

Dit — tan x) cos x = — cosa — sin x.

D log (log ¢) = 1/log ¢#.

61. If (4) = e7¢ sin d¢, show that

where

62.

63.

64.

65.

66.

67.

g(t) = ext (a? + BF sin (Be 4 8),

tan 6 = 4/a.
If siny = xsin(a+ yy), prove that
ay __ sin? (a + y)

ax sin @
If «(1 +y) +ya+ xy = 0, show that
Day =—(1+ 4)? or kL
If y=/(4) and «= (2), show that
_ Kh
2 =

If xy = ex-y, show that

dy log x

dx (1 + log x)¥
d (sin x)* = (sin x)* (log sin x + x cot x) ax.
log tan ¢

a 2
a (log tan ¢)? = 4 aah
CHAPTER IV.
ON SUCCESSIVE DIFFERENTIATION

52. The Second Derivative.—The derivative /’(x) of a function
/(x) is itself a function of x, which is, in general, also differentiable.

The derivative of the derivative /’() of a function /(x) we call
the second derivative of /(x), and write it /’’(x).

Thus

me

 

x1(=)x
For example, if /(x) =r”, the first derivative /’(x) is mx"—!, and
in the same way we find the second derivative
SJ’ (x) = n(n — 1).
Again, if /(x) =sin x, then
J (x) =cosx and f(x) = — sin x.
If we use the symbol D/(.x) to represent the operation of differen-

tiation performed on /(.x), then two successive differentiations of
/(*), which result in the second derivative, are represented by D?/(x).

. DIDAx)]= DA) =S'" (2).

EXAMPLES.
1. Dla + bx + cx?) = b 4 20x,
Da + bx + cx’) = D(b + 2x),

= 26,

2. Dcos ax = — asin ax,
DP cos ax = — aD sin ax = ~ a? cos ax.
3. Dlogax=a/x; D* log ax = — a/x*.
4. Da — x = — x(a? — x?)=3,
D? Ya? — x? = — (a? — x2)73 + x? 'a? — xy},

53. Successive Differentiation.—The second derivative like the
first is, in general, a differentiable function. Its derivative is called
the third derivative of the function, and written

7% a= fess =)

Vy
*1(=)x

= Dif\x).
62
ART. 54.] ON SUCCESSIVE DIFFERENTIATION. 63

In general, if the operation of differentiation be repeated » times
on a function /(x), we call the result the zth derivative of the func-
tion, We write the mth derivative in either of the equivalent symbols

DY" f(x) = f(x).

It is customary to omit the parenthesis in /” (x), including the
index of the order of the derivative attached to the functional symbol
/ when there is no danger of mistaking it for a power, and write

D" f(x) =f"(x).
The index of either D or / in D”, 7” denotes merely the order of
the derivative and number of times the operation is performed.

54. Successive Differentials.—In defining the first differential
of a function, the differential of the independent variable was taken
to be an arbitrary number. In repeating this operation it is con-
venient to take the same value of the differential of the independent
variable in the second operation as that in the first, In other words,
we make the differential of the independent variable constant during
the successive differentiations.

Thus the second differential of /(.x) is

a@* fx) = d[d/(x)],
= a[ f(x) ae], |
= d[f'(x)] 2x, (i)
since dx is constant. But, by the definition of the differential,
aL f'(x)] = DUF(x)] a,
= f'"(x) dx. (ii)
Substituting in (i), we have for the second differential
ad? f(x) = f(x) (ax),
or the second differential of a function is equal to the product
of the second derivative into the sgware of the differential of the
variable.

It is customary to write the square of the differential of the
variable in the conventional form ¢x* instead of (@x)*, whenever there
is no danger of confounding

ax? = (dx)?
with d(x), the differential of the square of x. We shall write then
@?f(x) =f" (x) dx’.
In like manner for the third differential of /(x)
d[d3flx)) = dL f(x) ax,
= aL f"(x)]-d2",
since dx is constant; and since by definition
ale" (a)] = DL"(x)] ax,
= f(x) dx,
64 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IV.

we have for the third differential
dfx) = f(x) ax,
and so on.

In general, the th differential of a function is equal to the
product of the wth derivative of the function into the wth power of
the differential of the independent variable. In symbols

d* f(x) = f(x) ax",
where it is always to be remembered that dx” means (dxv)", and @”,

f” indicate the number of operations and order of the derivative
respectively.

EXAMPLES.
1. We have d sin x = cos x dx, and 7
@ sin x = d(cos x dx) = d(cos x)-dx = — sin x dx},
2. da + bx*) = d(2bx).dx = 2b dx’.
2
3. @ log x= a(z).ae = il
x x

 

55. The Differential-Quotients.—The mth differential-quotient
of a function is the quotient of the #th differential of the function by
the mth power of the differential of the independent variable.

In symbols we have, from § 54,

a” f(x
This symbol is also written, for convenience, in the forms
a" f(x a"
2 eS) Aa,

dx”
all of which notations are equivalent to either of

D'fx) efx),

and are used indifferently according to convenience.

 

56. Observations on Successive Differentiation.—In practice
or in the applications of the Calculus we require, in general, only
the first few derivatives of a function for solving the ordinary
problems that are proposed. But, in the theory of the subject, ie.,
the theory of functions, we are required to deal with the general or
nth derivative of a function in order to know all the properties of the
function.

The formation of the wth derivative of a given function presents
no theoretical difficulty, but owing to the fact that differentiation,
in general, produces a function of more complicated form (owing to
the introduction of more terms) than the primitive function from
which it was derived, the successive derivatives soon become so
ART. 56.] ON SUCCESSIVE DIFFERENTIATION. 65

complicated that the practical limitations (of our ability to handle
them) are soon reached.

The Differential Calculus as an instrument for investigating func-
tions finds its limitations fixed by the complexity of the general or
ath derivative of the function whose properties we wish to investigate.

There are a few functions whose wth derivatives can be obtained
in simple form, as will be shown below.

We are aided in forming the wth derivatives of functions by the
following:

(1). The wth derivative of the sum of a finite number of functions
is equal to the sum of their zth derivatives.

(2). The wth derivative of the product of a finite number of func-
tions can be determined by a formula due to Leibnitz, which we shall
deduce presently.

(3). The th derivative of the quotient of two functions can be
expressed in the form of a determinant and in a recurrence formula,
directly from Leibnitz’s formula. This is done in the Appendix,
Note 3.

(4). The mth derivative of a function of a function can be
expressed in terms of the successive derivatives of the functions
involved. This is also given in the Appendix, Note 4. ,

In the application of the Calculus to the solution of ordinary
geometrical questions, we need the first, frequently the second, and
but rarely the third derivative of a function. When the function is
given explicitly in terms of the variable, these derivatives are found
by the direct processes as heretofore applied. If the derivatives are to
be found from an implicit relation, such as (x, y) = 0, we can of
course solve for_y, when possible, and differentiate as before. It is
generally, however, better to differentiate p(x, y) with respect to +
and then solve for Dy. If we wish D*y, we can either differentiate
Dy with respect to x, or differentiate P(x, y) = O twice with respect
to v and solve the equations for Dy.

In illustration,
203 — 373 — axy =O.
wt. 6x? — ay — (gy? + ax)Dy = 9,
12x — aDy — (18y Dy + a)Dy — (gy? + ax)D*y = 0.

 

Therefore Z
_ 6x*— a
oe Tag!
Dy= 12x(gy? + ax)? — 2a(6x? — ay) oy? + ax) — 18y(6x? — ayy
os (oF + ax

Again, we frequently require the derivatives Dy y and Dz y, when we have
given the polar equation ¢(p, 6) = 0, where x = pcos 6, y = p sin 6.
We have

Dey
Dry = Day DA = Die?
__ sin 6 Deo + p cos 8 (1)

™~ cos 6 Dap — p sin &”
66 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IV.

Also,
Diy = DDzy) = Dg Diy) * DA,
_ Djy + Dox — Day Dix
== (Dox)? :
_ 0 + 2(Dep)? —pD5p
™ (cos 6 Dep — p sin 4)>
In which Dgp and Djp must be determined from the polar equation O(p, 8) = 0.*

EXAMPLES.
1. The wth derivative of x7, a being constant.
(1). Let z= m be a positive integer. Then
. Dx" = mxm—1,
Dx = m(m — 1x",
Dixm = mm —1)... (m— n+ I)x,
for all values of 2 <m. If xn=m, then
Duxm =mm—1)...3:-21=m!
This being a constant, all higher derivatives are 0.
Dutixm = 0
for all positive integers Z.

Also, when x = 0,

Dix" = 0, n<m,

(2). Let the constant @ be not a positive integer. Then, as before,

Dixa = ala —1)... (a —n + I1)xe,

Whatever be the assigned constant 2, we can continue the process until 2 > a,
when the exponent of x will be negative and continue negative for all higher deriv-
atives.

Consequently, when + =0,

‘ D"x4 = 0, nou
Drx4 = 0, n> a.

 

* The differentiation of an implicit function @(x, vy) = Ois, properly speaking,
the differentiation of a function of two variables, and a simpler treatment will be
given in Book I.

It will be shown in Book II that the derivative of y with respect to x, when

P(x, V) = 9, is

Ip
ay ax
di ap”
y
ap a 3
where 3_ means the derivative of O(x, y) with respect to x, x being the on/y vari-

able ; a means the derivative of @ with respect to y, y being the only variable.
For example, if (x, v7) = 2*° — 37° — axy = 0,
then —=62%-—-a; —=-— 9 — ax.

Therefore, as in the text,
dy 62° —a

dx oy + ax"
ART. 56.] ON SUCCESSIVE DIFFERENTIATION. 67

2. Deduce the binomial formula for (1 + x)", when the exponent 7 is a posi-
tive integer.

We have
(Ef a*)r1fe)= (1 + aeP 1 tar4 x,
(I 2)1 + af = (1+ xP = 14 Ze 4 3x2 4 8.

By an easy induction we see that (1 + x)* must be a polynomial in x of degree
w. It is our object to find the numerical coefficients of the various powers of x in
this function. Let

(E> x) = a + ax t aye? +... 4 a,x,
Differentiating this x times with respect to x, we have
nim—1)... —rf Ilia rar! a +... 4 n(n—1)... (W—r+1)an xr,
This equation is true for all assigned values of x and y, and when x = 0,

nn —t)...(#-7+1)

ne r!

n!
~ r\a— ryt?

a number which it is customary to represent conventionally by either of the

symbols
Cr, or é )

This number is of frequent occurrence in analysis. In Algebra, when x is an
integer, it represents the number of combinations of » things taken ¢ at a time.
Hence we have the binomial formula of Newton,

a+t+eaf= = Ca, rx" (1)

Corollary. If we wish the foeewatty expression for (a2 + y)*, then
(e+ 9)"=a" ( +2)
Put y/a for x in (1), and multiply both sides by a.

n

(a+tyy*= S Cu, pan—rar,

r=o0
This can be written more symmetrically thus:
Gry. 3 a a
n!} “(arr
3. The th derivative of log x. We have

Diog x = Es ax,
w
Therefore, by Ex. I,
I
D* log « = (— 1)*-\(” — 1)! aa
4, The wth derivative of ax. We have
7 Dax = a* log a.
wt. Dta* = a* (log a)*.
In particular, Dex = e*; Dex = e*. This remarkable function is not changed
by differentiation.
68

PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IV.

5. The zth derivative of sin x and cos x.
We observe that

D sin x« = + cos x; Dcosx = — sin x;
D*sinx = —sinx; DD cosx = — cos x;
Disinx = —cosx; DM cosx = + sin x;
MH*sinzxg =-+sinz; Dtcosx = + cos x.

Thus four differentiations reproduce the original functions and therefore the higher
derivatives repeat in the same order, so that

D2» sin x = (— 1)*— cos x;
D sin x = (— 1)* sin x;

D>"! cos x = (— 1)" sin x;
D2 cos x = (— 1)" cos x.
In virtue of the relations
cos x = sin (4a + ~), sin x = — cos(4a-+ 4),
these formule can be expressed in the compact forms

D* sin x = sin (« + 3);
n
D* cos x = cos (= + 7)

6. Given find Diy, Diy.

x?
gta
Differentiating with ee to x,

ate dx

ay
= se oe az) 7

dy as
ae ~~ y
ot

ay?

Differentiating again,

Differentiating again, we can find
dy

a ~~

7.iTf +% show that

dy
dx y'
show that

= 4ax,
2a dy

8. If »?

— axy = a,
dy y dy
de ~y—x’ ade
9. From the relation «3 + y3 — 3axy
dy _
dx ~

a

 

x — ay

vy = ax’
show that
d’y

10. If secx cosy =a,

dy
aa

tan x
tan y’”

dx

Or = a)

ae

da

ED ae

y dy

Bde ~

stale) }>

ex
ae

ay

since
dx

308x

4a?
a

ay 3a%x
Ze fy
show that

2a3xy
(7? — ax"

= 0,

__ tan? y — tan? x
dx

tan3 y
ART. 57.] ON SUCCESSIVE DIFFERENTIATION. 69

57. Leibnitz’s Formula for the nth Derivative of the Product
of Two Functions.—Let u, v be any two functions of x. For sake
of brevity, let us represent the successive derivatives of wand v by
these letters with indices, thus :

u', ul’, ul” eka. We

v, vw, or, as

 

Then
D(uv) = u’v + v'n,
D (uv) = uly 4 uo! 4 u'o! + "nu,
= uly + 2u'o! + uo",

In like manner, differentiating again this sum of products, we find

on simplification
D (uv) = uly 4 3u!'v! + 3u'0"” 4 ww!”

Observing, when we use indices to indicate the derivatives, the
symbols D°u, (x), v, mean that no differentiation has been
performed and the function itself is unchanged,

Du=zwW=u, and (x) =/(x).

In the above successive derivatives of wv we observe that the
indices representing differentiation follow the law of the powers of
uw + vwhen expanded by the binomial formula, and the numerical
coefficients are the same as those in the corresponding formula of that
expansion

In order to find if this law is generally true, let us assume it true
for the wth derivative and then differentiate again to see if it be true,
in consequence of that assumption, for # + 1,

Assume that (see Ex. 2, § 56)

n

D"(uv) = 2C,,, wv’,
r=0

=wv+C,., ww +. tC, utror +... tu",
Differentiating this, we have
DD (uvy=u V+, wt, weortort...u’o
+ wo +... +0, 4 WOOL... nu'v Lut,
=uttyt Cae, - uty’ + ee + Quay ue tort, Ly suv",
- S Cons yrti-r vu,

r=0
in virtue of the relation* C, ,-+ Cy, ,41 = Crs.»
Therefore, when the law is true for any integer , it is also true
forz-+ 1. But, being true for 7 = 2, 3, it is true for any assigned
integer whatever.

 

*

n! n! oa) ni - I _ (#+1)!
rin—r)! ab (r—)\(e—r+1)! (r7—1)!(2—7)! (Z ¥ aa) ~ riaz+i—r)!
7oO PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IV.

We can express the wth derivative of the product wv symbolically,

thus:

D" (uv) = (u + 0)",
in which (w + v)" is to be expanded by the binomial formula, and
the powers of w and vin the expansion are taken to indicate the orders
of differentiation of these functions. Remembering that when the

index is a power we have z’=1, but when it means differentiation,
oO —
“w=t4,

EXAMPLES.
1. To differentiate the product of a linear function by any function /(x).
Let u = (ax + b)f(x).
Then Dax + 6) = a, Dax + 6) = 0,

Diu = (ax + b)f"(x) + na f*-1(x).
2. In like manner show that the zth derivative of the product of a quadratic
function of x, say y, by any other function /, is

IP + wy/fO + tale — ry" fr.
3. Show, if p(x) and 7(-x) are differentiable functions of x,
Dl Gixyb(x)] _ S g(x) (2)
n! = Zoa@—n ort
58. Function of a Function.—A formula for the th derivative
of a function of a function will be deduced in the supplementary
notes.* However, the simple case of a function of a linear function

of the independent variable is so useful and of such frequent occur-
rence that we give it here.

Let u=ax-+4, and /(z) be any differentiable function of w. Then
DJ(¥) = Side) Dit,

 

= af,(u).

Di f(u) = aD, [A(z)],
= af; (u) Du,
=a fix),

and generally
De) = FY).
EXAMPLES.
1. Show that
D* sin ax = a” sin G + on)

n
D* cos ax = a* cos G + o*):
2

2. Dieax = ateax,

3. Show that D” ao) ats
x a (« — a)ttt

* Appendix, Note 4.
ArT. 58.] EXERCISES. 71

EXERCISES.
Show that

r!

I
1. >(%) =(— 1)" me
>(S) eae arg iee I)... (w#tr—1)
xn attr i
3. >(,) =r! athe ay
¢—x (¢ — x)rtt
4. D* log (i + x) = (— 1)"

5. Dt(x3 log x) = 6x-1.
6. D(«t + a sin 2x) = 32a cos 2x.

 

ed

(7 — 1)!

Pa

7. Di{xu) = x Du +r Dr-u, where w is any function of x.
8. Dia — x)e = (a — x) Dru — r Dr-mu,

9. DA (xt log x) = — 4!x-2,

10. D(x log «) = (— 1)*(” — 2)! at,

I. Dixcx = xx(1 + log x)? + xx—1,

12. D3 log (sin x) = 2 cos x csc3 x.

13. D5(2t log 8) = 24-1,

14, Dracx = ax (log a)".

 

 

 

 

THEO at! ac+é ac—b
i a oe ooo!
Observe that by the method of partial fractions we can write

ax +b _ 1 §ac+s , ac—db
eae eet ee
(ie ut (etree =:
(+ — pie — 9) BO ENE PY Eg

17. Make use of the method of partial fractions, to find the mth derivative of
aettbxte_ 1 fat pote ag? + bg te
aa er er eet xp #-9 )+e

Ey soeqa= n! :

dx} x? — 5x +6 (2 — xyttt

19. If y = a(1 + 2*)4, show that

(1 4 2?) y() 4 2nx yD) + n(x — 1)y(*-2) = 0,

20. If Ax) = cos (log x) + sin (log x), show that

P(x) + af'(a) + f(x) = 0.
21. Show in 20 that the following equation is true :
xfnte 4 x(am + 1) feet + (w+ If =o.
22, If y =e2sin '%, show that
(1 — 2)a’¢y — x dy dx = ay dx’,

thence find, asin 21, an equation in y*+2, y*+1, vn

18. Show that (
72 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IV.

23. lf y=(«+ Vx? —1)”, show that

(x? — 1)d*y + x dy dx — my dx? = o.
24. If y = sin(sin x), show that

dy + tan x dy dx + y cos?x dx? = o.
25. If y =Acosmx + Asin mx, then

Dy + ny = 0.
n
26. Dea cos bx = (a? + 5%)? c4% cos (6x +n@),where tan @ =4/a. Differ-
entiate once, twice, and observe that the law follows directly by induction.
27. D* tantax-!t = (— 1)*(” — 1)! sin* (tan—tx—') sin m(tan—tx—}),
Put y=tan-tx1, Then x=coty, and
Dey = —(1+ 2) = — sin? y.
Diy = — D,sin? y = — D, sin? y Dy = sin’ y sin 2y

The rest follows by an easy induction.
28. If x = ¢(¢) and y = p(Z), then y is a function of x
Required D,y, Diy.

We have gawd, dx = g'(t)d. e
a _ vo
dx @(t)”

Also,

dy dy dw dt

“dx? ax of = dt of ax

’

= did! — vi bi! dt = =
= (g'P F dx i *

dy dx dy d*x
aa” dt de

ate

29. If « =sin 3/4, y = cos 37, show that

D3 = = 7-3.
30. D* tan—ix = (— 1)"-(” — 1)! Sin_(m tanta~1)

(x + x2)"

This follows immediately from Ex. 27, since
tan—'x = Iw — tan—1x—-1,
31. If » = tan-'x, show that
(1 + 2?) yt) + any + n(n — 1jy"-1)— 0,
32. Dea? + 2?)-1 = (— 1)*2! a2 sin4+1@ sin (xn + 1),
where tan @ = a/x. Hint. Use Ex. 30, and
D tan-(a/x) = — a(a? + 2x),
33. Drx(a? + x?)-t = (— 1)"a-*-m! sint+1 cos (27+ 1)
where tan @ =a/x, Use Ex. 32 and Leibnitz’s Formula.
34. ei ate

— ey
dx} ttx (1 + xjyari*
ART. 58.] EXERCISES. 73

a6. meee FAS ave
2x Vx

36. If yr + 2%) = (1 — x + x?)?, then
d*y 1+ 3x4 +7
Wa oF (ib xi :
37. If y = sin (m sin—1x), prove
2
(1 — aed = aD my = oO.

38. If y =sin—1%, deduce

(1 — 2) y" — xy’ = 0,

 

duty ant at
ns (a) Fae Get Oe — ge =o

by applying Leibnitz’s Formula to the above. The deduction of such differential
equations is of fundamental importance for the expansion of functions in series.

39. Show that
(a — x)"tt é i: F (*)

n! dx} a—x

n ( SS r
e =2% — 20" fr)

Apply Leibnitz’s Formula to the product /(x)-(@ — x)-1.
40. Show that

(eH (9S) =F _ pay SEIN
gh ee xy =f (*) — rl Fy),
0

where, in the differentiation indicated by By? x is constant and y the variable.
The result follows at once when Leibnitz’s Formula is applied to the product of the
two functions f(x) — f(y) and (x — y)—!.

This is one of the most important formule in the Calculus. Observe that it is
obtained by successive differentiation of the difference-quotient.

41. Show that the derivative of the right member of the equation in Ex. qo,
with respect to y (x being considered constant during the operation), is
ea
= Ga f nti (y).
Hint. Differentiating each product in the sum, we find that the terms all can-
cel out except the last.
CHAPTER V.
ON THE THEOREM OF MEAN VALUE.

59. Increasing and Decreasing Functions.

Definition.—A function /(x) is said to be an increasing func-
tion when it zzcreases as its variable izcreases. A function is said to
be a decreasing function when it decreases as its variable zmcreases.

In symbols, /(x) is an increasing function at x = a when

I(x) — Aa) (1)
changes from negative to positive (less to greater) as x increases
through the neighborhood, (¢ — e, a+ e), of a. In like manner
J(x) is a decreasing function at a when the difference (1) changes
from positive to negative (greater to less) as x increases through the
neighborhood of a.

60. Theorem.—A function /(x) is an zncreasing or decreasing
function at @ according as its derivative /’(a) is posztive or negative

respectively.
Proof: If /() is an increasing function at a, the difference-

quotient
A*) =)

x—a

 

is always positive for x in the neighborhood of a, consequently its
limit /’(a) cannot be negative. If /’(a) is a positive number, then
for all values of x in the neighborhood of a the difference-quotient
must be in the neighborhood of its limit /’(a), and therefore posi-
tive. The function is therefore increasing at a.

In like manner, if /(-c) is decreasing at a, the difference-quotient
is negative for all values of x in the neighborhood of a2 and therefore
its limit cannot be positive. Hence, if /’(a) is a negative number,
the difference-quotient must be negative for x in the neighborhood
of a, and therefore /(x) is decreasing at a.

GEOMETRICAL ILLUSTRATION.

Let y = f(x) be represented by the curve 4,4,. The function is increasing at
A, and decreasing at 4,.
We have
7'(a,) = tan 6, = 4,

for 6, is acute, while
f(a) = tan§, = —,

74
ArT. 61.] ON THE THEOREM OF MEAN VALUE. 75

since 9, is obtuse. Remembering that, under the convention of Cartesian coordi-
nates, the angle which a tangent to a curve makes with the x-axis is the angle
between that part of the tangent above Ox and the positive direction of Ox.

y
E

A, A,

 

 

Hh oa
O J ay & Ag

Fic. 9.

 

 

 

a

 

61. Rolle’s Theorem.—If{ a function /(x) is one-valued and
differentiable in (a, 8), and we have f(a) = /(f), then there is a
value & of x in (@, #) at which we have

J'(&) = 9,
provided /’(x) is continuous in (a, f).

If /(x) is constant in any subinterval of (a, (), its derivative
there is o and the theorem is proved.

If /(x) is not constant in (a, f), then at some value «’ in (a, f)
we shall have /((x’) # f(a). If /(x’) > /(a#) =/(), the function
must increase between a and x’ and decrease between x’ and f, in
order to pass from /(@) to the greater value /(’), and from /(x’) to
the lesser value (8). Also, if /(x’) << /(a) =/(A), then the func-
tion must decrease in (a, x’) and increase in (x’, #), for like
reasons. In either case the derivative //(x,) at some point ~, in
(a, x’) must have contrary sign, § 60, to the derivative /’(.x,) at
some value x, in («’, f).

Since /’(x,) and /’(x,) have opposite signs, and /’(x) is, by
hypothesis, continuous in (x,, «,), then there is, § 23, I, a number
& in (x,, x,), and therefore in (a, #), at which we have

J'(é) =0.

In particular, if (a) = 0 and {(A) = 0, then there is a number

& between a and £ at which
f'(6) =0.

Rolle’s Theorem is usually enunciated: If a function vanishes for
two values of the variable, its derivative vanishes for some value of
the variable between the two. Or, the derivative has a root between

each pair of roots of the function.
The figure in § 60 illustrates the theorem.

62. Particular Theorem of Mean Value.—lIf /(x) is a one-
valued differentiable function having a continuous derivative in
(a, @), and if a and 4 are any two values of x in (a, f), then

Jo) — Ka) = 6-9/8),

where § is some number in (a, 4).
76 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [CH. V.

The truth of this theorem follows immediately from Rolle’s
Theorem.
Let & represent the difference-quotient

h _/) —J(@)
b—a

Then
S) — (a) = (4 — 24,

J (6) — kb = f(a) — ha. (1)
The function /(x) — 4x is equal to the number on the left of
(1) when x = 4, and to the number on the right when + =a.
Therefore, by Rolle’s Theorem, having equal values when + = a and
when x = 4, its derivative must vanish for x = & between a and 4.
Differentiating, /(x) — kx, & being independent of x, we have at &
J'(8) —# = 9,
which proves the theorem.
Another way of establishing the result is to observe that the

function

(2 — 4) Ax) — (x — 4) (a) + (x — a) /)
vanishes when x = a, also when x = 4. Therefore its derivative
must vanish for some value of x, say &, between a and 4.

ts (@—3)f(&) — fla) +18) =0.

GEOMETRICAL ILLUSTRATION.

or

 

Fic. I0.

Each of these processes admits of geometrical illustration.

(1). & is the trigonometrical tangent of the angle which the secant 42 makes
with Ox. Draw OA’S’ parallel to 4B. Then

BB’ = f() — kb = AA’ = f(a) — ka,

NX’ = f(x) — x is equal to AA’ when x = u, and to BB’ whenx = 3. The
theorem asserts that there is a point Z on the curve y = f(x) having abscissa é at
which /’(é) = &, or the tangent at Z is parallel to the chord 42.

(2). The function

(2 — 6) fx) — (* — 4) fla) + (« — 2) fd)
is nothing more than the determinant
Ke), -%, 1
J(@), a,
Ko), 8 1
ArT. 63.] ON THE THEOREM OF MEAN VALUE. 77

which is the well-known formula in Analytical Geometry for twice the area of the
triangle 4X, in terms of the coordinates of its corners. This vanishes when X
coincides with 4 or 8. It attains a maximum when the distance of Y from the
base AZ is greatest, or when -X is at £, where the tangent is parallel to the chord.

This theorem amounts to nothing more than Rolle’s Theorem when the axes of
coordinates are changed.

63. Lemma.—Ex. 39, § 58, forms the basis of the most important
theorem in the Differential Calculus, i.e., the Theorem of Mean Value
for a function of one variable. On account of its usefulness, we inter-
polate its solution here.

The starting point of the Differential Calculus is the difference-
quotient. On that is based the derivative of the function. We shall
now use it in presenting the Theorem of Mean Value.

Let _/(x) be a one-valued successively differentiable function of v in
a given interval (@, #). Let x represent any ardirary value of the
variable, and _y some particular value of the variable at which the
derivatives of / are known.

(1). Consider the difference-quotient

S(*) —/Y)

x—y -

If we hold x constant while we differentiate this 2 times with
respect to the variable y by Leibnitz’s Formula, § 57, and then
multiply both sides by

(x — yy}
n} ;

we shall obtain
Ax) —/0) — @ ry —.-. E27 wy
ae (2)'( (x) =f),

n! oy x—y
For, we have
DI) =F = — 70);
Di (x — yt = (1 — 1)! (x ym,
which values substituted in the form of Leibnitz’s Formula in Ex. 3,
§ 57, give the result.
(2). On account of the importance of this formula we give
another deduction which does not use Leibnitz’s Formula directly.

Let
SAMI) — o,
xy

Ae) —/) = (# — NO

Then
78 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [CH. V.

To introduce the known derivatives at y, let x be constant and
differentiate this last equation successively with respect to y. Thus

AA -/) =F HNO (1)
=—$'D) = — 2g, —Q, (2)
—F"'(9) = (* — IQ" — 205, (3)

=P) = Foye? = a (7+ 1)
2 I ;
Multiply (2) by (* —_y), (3) by = (x —y)*, ..., and (z+ 1)
by a(« — y)", and add the x + 1 equations. There results

A2)-2)— DS") —. Bip) =", @)

the same formula as in (1).

64. The Theorem of Mean Value. Lagrange’s Form.—The
Theorem of Mean Value, which we now present, is the most impor-
tant theorem in the Differential Calculus. The applications of the
Differential Calculus depend on it as do also its generalizations. It
is but a direct modification of the differential identity (g) established
in § 63, and consists in the evaluation of the mth derivative, Q™, of
the difference-quotient Q in a different form.

Consider the arbitrarily laid down function of z,

—2)"t1

x —2Z)* x
FQ) =f) fe) —(@—#)'(@) — «. — FEM ppg) Em,
in which, as in § 63, , :
») = pf (2) — SV
oe (=A)
does not contain z and is constant with respect to z.

Observe that this function F(z) is o when zg = x, because the first
two terms cancel and all the others vanish whenz = x. Also, F(z)
is o when z = y, by reason of the identity (¢).

Consequently, by Rolle’s Theorem, § 61, the derivative F'’(z)
must be o for some value & of z between x and y. Differentiating
with respect to z, and observing that the terms on the right, after
differentiation, cancel except the last two, we have

Fi(2) = — — Ps) b(n + 1) Ba a Je Oo,

Hence, when z = &, at which #'’(&) = 0,

fi TS)
on = A,
ART. 65.] ON THE THEOREM OF MEAN VALUE. 79

Substituting this value in we have Lagrange’s f
Theorem of Mean value, * (9); grange’s form of the

SEPM DI OV. SP yy + Ca pay,

(xy . xo ‘i
= Se Sr rn (L)

65. Theorem of Mean Value. Cauchy’s Form.—Cauchy has
given another form to the evaluation of the difference

fe) = a GM ry,

r=o
which for some purposes is more useful than that of Lagrange. Its
deduction is somewhat simpler.

Let + be constant and ga variable. Consider - function
Fe) =/e) + (*-97'@)+...42= Bow,
By the Theorem of Mean Value, § 62,
f(x) — F(a) = (« — a) FS), (ii)

where & is some number between x and a.
When z = x, we have from (i)

F(x) = /(*).
When z = a, then from (i)
F@) =/@) + (a7) +... 4 Fo TV prea),
Differentiating (i), ,
Fie) = Fa pote,
and

——

Bi) = FV pave,

Substituting in (ii), we have ae s form

fe) = FSV pr) +(e ype), ©

 

* In order that this result shall be true, it is necessary that the function /(.r)
and its first 2+ 1 derivatives shall be finite and determinate at x and at y, and
also for a// values of the variable between x andy. This important formula will
be presented in another form in the Integral Calculus, Chapter XIX, § 152.

For a proof of the Theorem: If a function becomes oo at a given value of the
variable, then all its derivatives are oo there, and also the quotient of the deriva-
tive by the function is 0 , see Appendix, Note 5.
80 PRINCIPLES OF THE DIFFERENTIAL CALCULUS.  [CH. V.

The numbers represented by & in (C) and in (L) are not equal
numbers. All we know about & in either case is that it is some
number between certain limits.

 

66. Observations on the Theorem of Mean Value.—The formula
(L) or (C) is a generalization of the theorem of mean value stated
in § 62; that theorem corresponds to the particular value z = o.

The Theorem of the Mean is the basis of the expansion of a
function in positive integral powers of the variable. When this
expansion in an infinite series is possible, it solves the problem:
Given the value of a function and of its derivatives at any one par-
ticular value of the variable, to compute the value of the function
and of its derivatives at another given value of the variable.

The Theorem of Mean Value is the basis of the application of the
Differential Calculus to Geometry in the study of curves and of sur-
faces, as will be amply illustrated in the sequel.

It solves the problem: To find a polynomial in the variable which
shall have the same value and the same first derivatives at a given
value of the variable as a given function. This polynomial, therefore,
has the same properties as the given function at the given value of
the variable, so far as those properties are dependent on the first 2
derivatives. This is a most important and valuable property of the
formula, for it enables us to study a proposed function by aid of the
polynomial, and we know more about the polynomial than about any
other function.

67. In Chapters I, ..., IV, we may be said to have designed
the tools of the Differential Calculus, for functions of one variable,
in the derivatives on which the properties of functions depend.

In the present chapter this design may be said to have culminated
in the presentation of the Theorem of Mean Value.

The subject has been developed continuously and harmoniously
from the difference-quotient. The difference-quotient is the founda-
tion-stone from which the derivatives have been evaluated, and by
successive differentiation of the difference-quotient we have been led
to the Theorem of Mean Value.

It is not necessary to add here any exercises or examples of the
application of the Theorem of the Mean, since it will be employed
so frequently in what follows. We merely notice other forms under
which the formula may be expressed.

68. Forms of the Theorem of Mean Value.

(1). It is customary to write R, as a symbol of the difference
between the functions

F(a) and - Ca a),
ArT. 68,] ON THE THEOREM OF MEAN VALUE. 81

so that

 

F(x) =) = 2 Fra) la Pe

Or, more briefly,
J(*) = Si + Ri,
where S,, represents the tunetion:

(2). ‘In particular, if @ = 0, and (x) is differentiable, le I
times at o and in (0, x), we eve

Fe) =f (0) + ef"(0o) +... “F"(0) cre.

where, using Lagrange’s form,

xt 7
R= (27+ ae 6), & in (9, ),
or, using Cauchy’s form,
R= aS)" pott(é, & in (0, x).

(3). If we write the difference + — y = h, so that x = y + &,
h®
PEPE SPI POT ee ers Oy Be
(4). Again, since / is arbitrary we can put A = dy. Then

SO +0) =/0) +90) +... +727 +2,

 

 

or
a? d"
ApS GES cp ER,
EXERCISES.
1. Iff(x) = owhenx =«a,, .., * = ay, where

AS ag See Say,
and f(x) and its first 2 derivatives are continuous in (a;, @4), show that

e ey

 

SI(#) = (* — 4). + + (* — an)

where & is some number between the greatest and the least of the numbers
Hy Oyy eens Oe

2. In particular, ifa,=4,=... =a, = 4, then
x — a)r
fay = My,

where & lies between x and a.
CHAPTER VI.
ON THE EXPANSION OF FUNCTIONS.

69. The Power-Series.—To expand a proposed function, in
general, means to express its value in terms of a series of given func-
tions. This series has, in general, an infinite number of terms, and
when so must be convergent.

We confine our attention here to the expansion of a proposed
function in a series of positive integral powers of the variable, based
on the Theorem of Mean Value.

The problem of the expansion of a proposed function in an
infinite series of positive integral powers of the variable does not
admit of complete solution in general, when we are restricted to real
values of the variable, for the reason that the values of the variable
at which the function becomes infinite enter into the problem,
whether these values of the variable be real or imaginary. In the
present chapter we shall confine the attention to those simple func-
tions whose expansions can be readily demonstrated in real variables,
relegating to the Appendix * a more complete discussion of the gen-
eral problem.

70. Taylor’s Series.—If in the formula of the Theorem of Mean

Value,

7 (a = a" oy

fe= ) Spa) + Ry, (1
the derivatives /”(a), y= 1, 2,..., at a, are such that the series
va x — a)”
8,2) CS ma),
has a finite limit when 7 = ws , and we also have
L Ra = oO,

then for the values of x and a involved we have
x — a)?
fe) =fa) + (ea f(a) +2 — Verma g. (ay
This is called Zaylor’s formula or series.

* See Appendix, Notes 6, 7, 8.
82
ART. 71.] ON THE EXPANSION OF FUNCTIONS. 83

We may use any of the different forms of &, we choose in show-
ing LR, = 0.

71. Maclaurin’s Series,—Under the same conditions as in § 70,
ifa=o,

Ae) =S(0) +4f"(0) FESO) + (M)

This is called Maclaurin’s formula * or series,

The series (M) generally admits calculation more readily than
does Taylor’s (T), because usually the derivatives at o are of simpler
form than those at an arbitrarily selected value of the variable a,

EXAMPLES.

1, Any rational integral function or polynomial (x) can always be expressed as
; x — a)"
fa) + — a) f(a) +... + 2F =" pa,
where x is the degree of the polynomial /(x).
For, since f is of the wth degree, all derivatives of order higher than / are o.
Consequently the theorem of mean value gives

fa) = S&S" pe,

whatever values be assigned to x and a.
In particular, we may put a2 = 0, and have

He) = fo) + af'0) ++ +H),

and this must be the polynomial ‘considered when arranged according to the
powers of x.

2. We may define as a transcendental integral function one such that a// of its
derivatives remain determinate and xon-infinite for any assigned value of the
variable.

Any such function can be calculated by either Taylor’s or Maclaurin’s series
for any finite value of the variable, whatever.

For if # be such a function, then, whatever be the assigned number a, we have

f ee * frt1(2) =o,

n=a
since f*+1(é) is finite for any & between x and a, for all values of ”. Also,
(x — ayttt/(m + 1)! has the limit o when 2 ='00 (see § 15, Ex. 9). ;
Moreover, the series is absolutely convergent (Introd., § 15, Ex. 10), since

© (* — ar ela — alr
SPT pas 32a,
r=0 r=0
where J/ is a finite absolute number not less than the absolute value of any deriva-
tive of fat a. The serrés on the right is absolutely convergent, since

|x — a|
i ee

N=

see § 15, Ex. 10.

 

* This formula is really due to Stirling.
84 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. ([Cu. VI.

Therefore, if f(x) be any ¢ranscendental integral function, we have for any
assigned value of x or a

A*) = f(2) + (# — )F'(4) +' ge po) +s
Also,

Ax) = fo) + 2/0) + ef "O) +.
Such functions are sin x, cos x, e*.

3. Show that if /(x) is any transcendental integral function as defined in Ex. 2,
then (px + ¢) can be expanded in Taylor’s series for any assigned values of J, g,
x and a.

This follows immediately from 2, since

a\n
(Ze) Aor + 0) = Pryor + 0)

4. To expand e* by Maclaurin’s formula.
We have Drex = e~ forall values ofr. Ato we have
Drex = e°= 1. Also,
ant
oe é —
(z#+1)!
n=O
Hence, substituting in Maclaurin’s a we oe
ex=itnx +5 +5 nen
oo xT
=i
r=0

In particular, when + = I,
I
e=iti+s+ weg “+.

which gives a simple and easy method of aes ¢ to any degree of approxima-
tion we choose.

5. To compute sin x, given x, by Maclaurin’s formula.
sino = 0, D—1 sino = (— 1)", and J sino = 0,

by Ex. 5, § 56. Therefore
a oS ot

sinx=r— ay re
6. To compute in the same way cos x, given x.
By Ex. 5,§ 56, coso=1, D%-1coso=0, D2" coso = (— 1)*,
Be) Gk a8
cosx =~ > Aare

The derivatives of sin x and cos x being always finite, these functions are trans-
cendental integral functions and it is unnecessary to examine the terminal term Ry.
The limit of 2, , however, is very readily seen to beo, since we have respectively

XAFT 3 n :
7D! sin (: + or), for sin x,

(7+

=a (+3 tn), for cos x.

a
ArT. 71.] ON THE EXPANSION OF FUNCTIONS. 85

7. The binomial formula for any real exponent.

Consider the expansion of (1 + x)* by Maclaurin’s series, when @ is any
assigned real number.

We have
Dur 4+ xe = a(a—1)...(@—n 4 11 + xe-%,
[Onl + *)*]e20= aa — 1)... (a—n+ 1).

Substituting in Maclaurin’s series, we have

 

ieee Ss Ae) ys + Ba
The quotient of convergency, ni 15, Ex. 9, of this series is
a—n
me x! = |4|. (1)

 

n=a0
Therefore the series is absolutely convergent when |x| < 1, or for all values of x
in)—1, + 1(, For |x| > 1, the series is 0.
Also, by (C), § 65, or § 68, (2),
(x — &)" aa — 1)... (a—n)

   

 

 

i= n! (I+ Sta (2)
Whatever be the value of between x and 0, so long as Ix] < 1 we have
a—-nx—€& iat
Aft Le (3)
n=0 *
For this limit is the same as
f r+ é
which is less than 1 wheno <x <1. If« <0, putxr = — x and&é= — &.

Then the limit is equal to

 

satel

f=

But «/ —& <1—&, since 0 <4’ <1 and Oo<b <x’,
Inequality G) being true, £2, =0, in (2). Therefore the series is equal to the
function for the same values of x for which the series is absolutely convergent.

(if xt = 1-fax + Meat + ERE as

for all values of x in )— 1, + 1(, and the equality does not exist for any value of
x for which |z| > 1.

8. Expand log (1 + x) by Maclaurin’s series.

Let I(x) = log (1 + : ae
aT

» Sx) = (- xyes { iF + r+ x)’
and fr(0) = (— 1)*43(2 —1)!.

Substituting in Maclaurin’s series, we get
Ae Be oh ge eee

The convergency quotient of this is

= |s|.

 

 

n
—-——«
n+I1
86 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VI.
The series is therefore absolutely convergent for |x| < 1, and is 0 for |x| > 1.
Also, we have, by (C), § 65,

 

 

 

= ip eS ee
Rn = (— 1) “a + ea
Whatever may be & between x and o when |x| < 1, we have, as in Ex. 7,
x—é
I.
- t+e| <
Therefore £Rx =o, and
log (1+ x) =x —}e?4+ 48 -—fet+... (1)

This series converges too slowly for convenience, that is, too many terms have
to be calculated to get a close approximation to the value of log (1 + +).
By changing the sign of «x,

 

 

 

log (I —x) = —4—42°—-—43-... (2)
By subtracting (2) from (1),
1+
log 2 t* = ae tart tat +...) (3)
If 2 and m are any positive numbers, put
Et oe then eee ’
I1—x n 2an+m

Substituting in (3),

atm\ mo tt ms I m?
log ( n )=oG Satie pert ieee t’)
a series which converges rapidly when x > m, and gives the logarithm of m+

when log x is known.
The logarithms thus computed are of course calculated to the base e, To find

the logarithm to any other base, we have

 

 

loge y
log, a

 

loga y =

72. Observations on the Expansion of Functions by Taylor’s
Series.—The expansion of a given function by the law of the mean
is rendered difficult, in general, because of the complicated character
of the #th derivative which it is necessary to know in order to get the
law of the series and test of its convergency.

Still more difficult is the investigation of the limit of R,. This
latter investigation is usually more troublesome than the question of
convergency of the series because of the uncertainty regarding the
value of the number &. The only information we have with regard
to & is that it is some number which lies between two given num-
bers. Moreover, we know that & is a function of # and in general
changes its value with z, It is therefore necessary that we should
show that £2, = 0 for all values of & between x and a, in order to
be sure that £2, is o for the particular value & involved in the law
of the mean whatever may be that number & between x anda. In
the deduction of the form #,, in the Integral Calculus, Chapter XIX,

 
ART. 73.] ON THE EXPANSION OF FUNCTIONS. 87

§ 152, it is there shown that not only is it sufficient that we should
consider all values of & in the interval (a, x), but it is also necessary.
The equality of the function and the series depends on R#,, vanishing
for a// values of & in (a, .v).*

It is desirable therefore, that we should have such general laws
with regard to the expansion of functions as -will enable us, as far as
it is possible, to avoid the formation of the mth derivative and the
investigation of the remainder term 2,,, and which will permit us to
state for certain classes of functions determined by general properties
that the equivalence of Taylor’s or Maclaurin’s series with the func-
tion is true for a certain definite interval of the variable. The
general discussion of this subject is too extensive for this course.
We give in the next article some observations which will be of assist-
ance in simplifying the problem. In the Appendix a more general
treatment of the question is discussed.

73. Consider a function /(x) and its derivative J'(x). We can
state certain relations between a primitive and its derivative, with
regard to the corresponding power series as follows:

Cauchy’s form of the law of the mean value applied to each of
the functions /(x) and_/’(x) gives

Aa) =f0) +(e —af@t...+ 2S" mat Re, (1

PEL) += OF) + FT map Ri, (2)
where

Ry = (2 — a) EEN pera, (3)

Re = (@ — JA peg, (4)

I. We observe that the quotients of convergency of (1) and (2),
as obtained by taking the limit of the quotient of the (# + 1)th
term to the mth term, have the same value, for

xa fra) (x—a fra)
x api F(a) = a Fay’

 

 

 

 

os aa pe I ‘e
={ mee n+t1/ °

 

* In the theorem of the mean, (I), § 70, the series
2) (x — a)
SM 40)
oC

may be absolutely convergent and yet not equal to the function /(x). For Prings-
heim’s example, see Appendix, Note 8.
88 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [CuH. VI.
Therefore, if
a
J |_p (5)

Ll PKa)|

is a finite determinate number, then the two series

Aa) + (2 — af") + FSV pay 4.

 

and
S'(a) + (# — @S'"(@) + (x = a)" pra) “Pi

are absolutely convergent in the common eee, ja—R,a+R(;
and are both o for any value of x outside of this interval.

The number a is called the base of the expansion, or the centre
of the interval of convergence. The number 2& is called the radius
of convergence.

II. We observe that if, for a// values of §& between x and a, we

have
x—& 7")
fe st ”

n=n
in which & has the same value wherever it occurs, then, § 15, Ex. 9,
must (3) and (4) be o when z= whatever be the value of &
between .v and a in (3) or (4).

Consequently, if we have determined (5) for any function and
shown that (6) is true for values of x in the interval of convergence,
then this function, its derivative or its primitive is equal to the
corresponding Taylor’s series in the common interval

ja—R, a+ Ri.

 

EXAMPLES.

1. Having proved that the requirements in § 73 are satisfied for (1 + x)*, and
this function is equal to its Maclaurin’s series for all values of x between — 1 and
1, and for no values of x outside these limits, it follows immediately, in virtue
of § 73, that log (I -+ x) is equal to its Maclaurin’s series in the same interval,
since
Diog (i +2) = (1+ +).

2. The function tan—'x is equal to its Maclaurin’s series for x2 <1. For

Dtan-tx = ae
I+ x?
and x? < 1 is the interval of equivalence of (1 + x?)-1 with its Maclaurin’s series.
Moreover, since
Q+y7=1—-xr+et+—-x6+...,

and the primitive of (1 + x?)-' is tan—tx, and tan—10 = 0, we have, by $73,

tan-lx = x — f+ 45 —deT+ ..
for -I<r<4+t.
ArT. 74.] ON THE EXPANSION OF FUNCTIONS. 89

We can verify this result directly, for
(2 — 1)!

(r+ 22)

[D* tan—x], = (— 1)*-(”2 ~ 1)! sin(4nz).

Dd tan-ty = (— 1)" sin (# tan—t—),

: : nu
Also, sin (2 =) = 0, sin (2% +4 1) —= (— 1)",
2 2
Therefore the Maclaurin’s series for tan—tr is
BP GS iat hey

which has the interval of absolute convergence )— 1, + 1(.
For &,, in Lagrange’s form, we have

__ x sin (7 tan—iw -1)
“a Ges eye
the limit of which, for 2 = 0, is oO when |x| < 1,
In particular, if 2+ =: tan iw = 1/3, then
1 11 ri wa
Sten e+ fsb w:
2V3 S35 sy re ;

which can be used to compute the number z. A better method, however, is given
below

Rn

 

3. For all values of |x| < 1 we have shown that

(ha athe ppt Seat 4 O59

 

OP

But a primitive of (1 — x*)-+ is sin—tx, and since sin-io = 0, we have, by
§ 73,
5 Il I-31
sin-ix = xt —— #8 BSS pik g 425
23 2-45
forxin)—1, +1.
In particular, since 37 == sin~! 4, we have
ae 11 I-3 1
fa 2328+ 3-4-5 25
from which z can be computed rapidly.
4. Determine the Maclaurin’s series for cos—%x, cot-!x. sec—tx, csc—tx. In
each case determine the interval for which the function is equal to the series.

74. We can find the #th derivative of sin-‘v without difficulty,
but it would be difficult to evaluate the corresponding limit of 2, by
the direct processes of Maclaurin’s formula.

Observe that the coefficients in the power series for sin~'x can be
determined from Ex. 38, § 58, where we have

(1 — x?) D"*? sinty — (am + 1)a DD" sin — nD" sin“ = o.
. D*? sino = 27D" sino.

When we have found D sin~'o, J? sin~'o, the other derivatives at
o can be found directly, and the interval of the convergence of the
series established. The interval of equivalence of the function and
the series by evaluating £A,, is a matter of considerable difficulty.
go PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VL

In the text we go no further into this matter of the expansion of
functions by Taylor’s formula. We have made use of it to show how
the tables of the ordinary functions and of logarithms can be com-
puted, and the numbers e and z evaluated.

We add a few exercises in the application of the formula. The
cases in which the remainder term &, is inserted are those for which
we have not established either the convergence of the infinite series or
its equivalence with the function ; they may be regarded as exercises
in differentiation or as applications of the Law of Mean Value. Some
of these results will be useful later in the evaluation of indeterminate
forms and approximate calculations,

We observe that for the purpose of approximate calculations, if JZ
be the greatest and m the least absolute value of the (# + 1)th deriv-
ative in the interval (a, x), the error committed in taking

fe) =e = oY rca)

lies in absolute value between

oo gy | ett Re a a+r

eee. and pers My,

(z+ 1)! (2+ 1)!
by Lagrange’s form of &,. When we know the zth derivative of the
function to be calculated, we can thus determine beforehand how many
terms of the series will have to be taken in order that the error shall
not exceed a given number.

EXERCISES.

J. If cis the chord of a circular arc @, and 6 the chord of half the arc, show
that the error in taking

I
= — (86 —c
3 ( )
is less than com where a < radius of the arc.

2. If dis the distance between the middle points of the chord ¢ and arc a, in
Ex. 1, show that the error in taking

8 d@?
¢=>a—— —
3 4
‘4
is less than 77 ve
3 4
3. The series 1+ %-+ x7 ... is convergent for |x| <1. It is infinite

when x = 1, and also o when x < — 1. Show that we can make x converge
to — 1 in such a way as to make the sum of the series equal to any assigned
number we choose.

a a
“a+r

the sum of z + 1 terms of the series

Let

 

— 1, where a is any assigned number. Then we have for
ArT. 74.] ON THE EXPANSION OF FUNCTIONS, gt

e a n+
ee eee (*-,4,)

I—x a
——
n+1
If 2=2m or 2m+1, and m= oo, this sum is respectively equal to
3(I + 74) or 4(1 — 4),

one or the other of which can be made equal to any given number by properly
assigning a.
Show that

4 tanx=x+ti34 204 R.

 

 

5. secx = I+ dx? 4 Satt Sat + R,.
6. log (1 + sin x) = x — Ja? 4 48 — At + RK.
7. ex secx =I px x? 4+ 284 det Be + RK,
8. Show that for |x| < 1 we have

— 1-30

belpyise lege 2 oa.
re) 23 204 5
Hint. Dlg (e+ fre) =(14+ 2)?
2
9. Expand  sin—1 ss 3 and tan wy in powers of x, determin-
I x I—x

ing the intervals of equivalence, §§ 72, 75.

10. Expand x/x?+ a? + a? log (x + /x*+-a*), in powers of x and deter-
mine the interval of equivalence.

Hint. The derivative is 2 #/a? + 2%,
11. Expand in like manner
I tig r+ Y24 27 Ye OV
42 eye 24/2 1-7
by using its derivative (1 + -+)—.
12, Show that the zth derivative of (x? + 6x + 8)—' at ois

I I
(— 1)"n! at - ro

Expand the function in integral powers of «+ and determine the interval of
equivalence.

13. Show by Maclaurin’s formula that
(i+ +a)F= eft —tr+ He — Welt &,
= log (1+ x
int IF +S ee Ge thea ee ——
"lL y=, Oe) St—detet—-dt+...,
and the first few derivatives can be found.

14. Compute the following numbers to six decimal places: e, 2, log 2, log, 10,
sin 10°.

 
CHAPTER VII.
ON UNDETERMINED FORMS.

75. When w and v are functions of x, they are also functions of
each other, If, when +(=)a, we have u(=)o and o(=)o, the guotent

u

v

will in general have a determinate limit when x(=)a. This limit
will depend on the law of connectivity between wu andv. The evalua-
tion of the derivative is but a particular and simple case of the
evaluation of the limit of the quotient of two functions which have
a common root as the variable converges to that root. For, in the
derivative, we are evaluating the limit of the quotient

SJ\x) —/(a)
when J(x) —f(a)(=)o and x —a(=)o.

The evaluation of the quotient «/v when « converges to the
common root a of w and 2, is but a generalization of the idea
involved in the evaluation of the derivative. For, let O(x) and 7()
be two functions which vanish when x = a, or, as we say, have a
common root a. Then

(2) =o and (a) =0y

We wish to evaluate the limit of the quotient

(2)

(x)
when x(=)a.
Since @(2)=0, (2) =0, we have
Hx) _ H(2) — o(@)
ex) ex) — ¥(a)’

 

(x) — G(2)
= x—a

~ £(x) = 4(a
x—a

Consequently if O(x) and 7(.) are differentiable functions at a,

@?
ART. 76.] ON UNDETERMINED FORMS. 93

and the member on the left has a determinate limit when 2(=)a, we

have
P(x) _ (a)

wy PO) (ay

 

For example,
log x
vt

 

=1.
x(=)r
It may happen that @ is a common root of #/(x) and #/(x), then
¢'(a) = 0 and y’(a2) = 0. In this case we shall require a further
investigation in order to evaluate the quotient ¢/#. For this pur-
pose we require the following theorems:

76. A Theorem due to Cauchy.—Let (x) and (x) be two
functions which vanish at a, as also do their first x derivatives, but
the (7 +- 1)th derivatives of both @() and (x) do not vanish at a.
Then we shall have

P(x)P"(S) = Payers),

where & is some number between + and a.

Let z be a variable in the interval determined by the two fixed
numbers x anda. Then the function

S(2) = $2) b(*) — ¥@) PO),
=o when s=a, also whenz=.2,.

By the law of the mean, § 62, /’(z) =o for some number z= &,
between x and a. But, in virtue of the fact that g(a) = ’(a) = 0,
we have /’(2) =0. Consequently /’’(z) = o for some number &,
between &, and a.

In like manner /’’’(z) = 0 for z= &, between &, and a, and so
on until finally we have

T(E) = PVE) P(x) — HE) O(a) = 0,

where & is some number between x and a.
If p**+1(z) is not o between xv and a, we can divide by it. Hence

Ha) _ gtt(é)
p(x) pers)
This theorem is of great generality and usefulness.
For example, the functions (« —@)*+!/(#-+ 1)! and
Ray=fay— 35S" pre)

are such that they and their first 7 derivatives vanish at x = a, while the (# + 1th
derivative of the first function is 1. Therefore, by the theorem just proved, we

have
(x — a)jtt1

Fa) = (ap ay urn),

which is Lagrange’s formula for the law of the mean.
94 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VIL

This theorem can be utilized for finding many of the: different forms of the
remainder in the law of the mean. It has, however, its chief application in:

77. The Theorem of 1’Hépital.—If f(x) and (x) are two func-
tions which vanish at a, as also do their first ~ derivatives, then we

shall have
oe) _ fer) _ (ee)
Ha (x) = pr(é) 7 prrt(xy’
= e”*"(a)
ea)”

For, by Cauchy’s theorem, § 76,

(x) _ o*(8)

Pe) PFE)
where & lies between x and a. Hence, since & and x converge to
a together, we have for +(=)a

Hx) _ g(a).
Pa) Pa)

Moreover, Cauchy’s theorem shows that the quotients

$'(2)

b(%) |
all have this same limit.
Therefore, to find the value of the undetermined form, we
evaluate successively the quotients of the successive derivatives until
we arrive at a quotient no longer indeterminate.

 

 

(FS 1y 2p 0555. %)

 

EXAMPLES.
1. Evaluate, when x(=)1, the quotient

x? — 3x 42
ay *

x? —3x+2=0, whens =1.

D(x? — 3x 42) = 2x -3, =—1, whenxe =
2—1=0, whenx=1.
D(x*— 1) = 2x, = 2, when x =1.
xi 3x42 2x —- 3 I
a 2x ~~—-2t

a(=)r a(=)r

2. Show that
Art. 78.] ON UNDETERMINED FORMS, 95

3. Evaluate, when x(=)o, the following:

am ‘

or oa ee : x sin x ‘
i — ’ ae
sim + x —2sinx

+(=)o a(=)¢

4. Show that for x(=)o, we have

GP Venn ene att ext ex —2
ae =2; —____— = 2.
x — sin « - vers x

5. Evaluate, when +(=)o,

x — sin-ly I ax — bx \ a ane 2
— —_ og = ; oe =
sin’x 6? x Bz? x — sine ie

6. Find the limits, when 2(=)o,

x—sinx I sin 3x 3 x 2
—-=-7; ——_— =--; Fi ss
3 6 x —$sin 2x a I — cos mx ~ m?

78. The Illusory Forms.—When z and v are two functions of .v,
which are such that the functions
u/d, uv, “u— Dv, u”
tend to take any of the forms
o/o, 0/o, OX, wo —m, 0% © 1%,
as x converges to a; then when these functions have determinate
limits for +(=)a, the theorem of I’ H6pital will evaluate these limits.
All these forms can be reduced to the evaluation of the first, o/o,
as follows:
(1). c/o and oXo _ reduce directly to o/o.

For, if uj=0o, v¥,=0, then

 

 

 

 

 

?

©

1/v, 0
ru, O°

 

’
oe
oe)

< |S

and we evaluate
1/V,
1/t, ‘
If u,=0, v,=0, then
ug oO

“ = => —

Vv = =
oe 1/v, 0

 

2

and we evaluate

uy

 

1/0,
(2). In like manner, if u,=0, w=, then
v,\ _ 1—,/u,_ ©
) = 3,
provided £(v,/u,) = 1, otherwise this form has no determinate finite
limit and is o.

u,— 0g = u(x —
96 PRINCIPLES OF THE DIFFERENTIAL CALCULUS

[Cu. VIL
This illusory form can also be reduced to the evaluation of the
form o/o when +(=)a, thus

which takes the form o/o when x =a. Therefore, if Se?” =
L(¥—?v) =, for x(=)a.
(3). The last three forms, 0°, 0°, 1%, arise from the function
zu”, which can be reduced to o/o, thus

Since “= ee,

wae log u_

In each of the cases 0°, 09, 1”, the function v log u takes the
form o X «©, which can be turned directly into o/o and evaluated as
in (I).

EXAMPLES OF 0 /oo and 0/co

The evaluation of w/v, when «= 0, v=. for x = a, is carried outin the
same way as for 0/o. For we have

gan f ee = =—W eye,
w=) LL /eon” LD -¢

 

 

PV LP)P
= {eos
¥(z) $ O@)
when 2(=)a. If now @(x)/y(x) has a determinate limit 4 ~ 0, when «(=)a,
then
v2)
A=/4 ee
P'(2)
Therefore, for x(=)a, when @(x) = w, Wx) = w,
0) 4 {o)_ (2)
V(2) W(x) Pa)’
if p'(a)/y’(a) is determinate.

 

seca tanx x tan x

7 (= 2 Sg ne ee

Or immediately, by Trigonometry, ae =sin x.

fee tan x sec? x sec x

 

2. Show that

!
a n=o0,
ex ee

=O

Also when » is not an integer.
3. Show that £ «(log x)" = 0.
2(=)o

when x is a positive integer.
ART. 79.] ON UNDETERMINED FORMS. 97

4. Show that f log (6 — 37) aie,

 

 

tan 6
O(=)hr
5. Evaluate, when «(=)iz,
tan | log tan 21 _ I —sinx + cos x |
tan 3x ’ logtanx ’ sine + cosx—1?
log sin x — sec x | tan x
(w — 2x7’ sec 3x’ tan 5x
6. Show that £ (1 — x) tan 1(zx) = a
x( =) 7

EXAMPLES OF 00 — oo.
1. £(secx —tanx)=0, for x(=)iz.
8 £(2-' — cotx) = 0, for x(=)o,
9. x tan «— $m secx(=)—1, when 2x(=)ym °
x —sinxr
10. a
MW. (a* — 1)/x(=) loga, when x(=)o.

EXAMPLES OF #,

(De when x(=)o.

Wata (= A=)
1B (t+ 24)(=)1, (=.
14, (ee 4 rietk x= Ow.
15. (cos sen es x(=)o.

16. x1-*(=)e-1, x(=)L.

79. General Observations on Illusory Forms.—In evaluating
illusory forms, we may at any stage of the process suppress any com-
mon factors in the numerator and denominator, and evaluate indepen-
dently any factor which has a determinate limit. We can frequently
make use of algebraic and trigonometric transformations which will
simplify and sometimes permit the evaluation without use of the

Calculus.
In illustration consider the limit of

(e — 1)85""™*, when a(=)r.
This takes the form 0°. To evaluate, equate the function to_y and

take the logarithm,
log (x — 1)

1 = 2
nee ee log sin wx

I

log (*# — 1) x—T I sin wx
ee = ff = Sec 2x.
log sin wx 1M COSMK 7 x—I

sin mx

 
98 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VIL

But £sec ax = —

and
(a sin mx = {+ cos Hx
—_———_ = -7
x— I I
» Llogy=log Ly =a.
Hence Ly =e, when x(=)1.
Frequently the evaluation can be simplified by substituting for the

functions involved their values in terms of the law of the mean.
For example, evaluate for +(=)o,

(1 + x)*—e
- .
Differentiating numerator and denominator, we have

x X— (14+ 4) log (x + «)
a

AaH x)= = e, and the limit of the other factor is, by the ordinary
process, readily found to be — 4. Hence the limit is — e/a.

I

Otherwise, put for (1 + x)? its oe Ex 13, Chap. VI,
sen ge 5 7 a8 t,
| I = + rr + &,

and the result appears immediately i differentiation,

GEOMETRICAL ILLUSTRATIONS.

(1). If{f(2)=0, P(2) = 0, f'(@) #0, P'(2) #0, consider the curves rep-
resenting y = f(x), y = O(* ).
y ys (a)

re y=o(x)

O| wae x

Fic. 11.

”
Le

These curves cross Ox at x =: a at angles whose tangents are equal to /“(a),

Ces /'(a) = tan6,, (2) = tan §,.

S(*) xA, «A, \ _ tan,
“O(x) ecel a7 ~ tan 65°

The limit of the quotient /(x)/@(x) is represented by the quotient of the slopes
of these curves at their common point of intersection with Ox.
ART. 79.] ON UNDETERMINED FORMS. 99

(2). Consider the functions x and y in
(x7 + pF = a*(x? — 2), (i)
Differentiate with respect to x and solve for Dy.
28 + 2x? — alx
axty - 2y> aly’
_ Dy takes the form 0/o when x = 0, for then also y = 0 by (i). To evaluate
this, differentiate the numerator and denominator with respect to x.

6x? + ay?+ gay Dy — a

o> Dy = oe
au aay + (2x? + 6? + a2)Dy’
— a*
=> @£LDy

ZOoy=1, or £Dy= +t.

This means that the curve whose equation is (i) in Cartesian coordinates has
two branches passing through the origin x = 0, y = 0, which is a singular point.
There the slopes of the two branches to Ox are + 1 and —1. The curve is the
demniscate.

y

el

0

Fic. 12.

We can find Dy at x = 0, y = 0 for the curve (i), without indetermination by
differentiating the equation (i) twice with respect to x. Thus
(2a? — 12x? — 4 y*) = 16xy Dy + (4x? + 129? + 20°) (Dy)? + (422 +495 + 20%) Dy,
which gives, as before, Dy = +1, when x=0, y=0oO.,
(3). We know, from trigonometry, that the radius g of the circle circumscrib-
ing a triangle 48C with sides a, 4, ¢ having area 5, is
_ abe
A= ria
Also, from Analytical Geometry, we have
2S =\|4 yy 41
yy, Vy,» 2F
voy J Ir
where x, 93 1,1} %g, V2) are the coordinates of the corners of ABC. Show that
if A, B, C are three points on a curve y= /(x), then the radius of the circle through
these three points, when +,(=)«, +,(=)-, is
3
joe
Dy

’

 

 

We have
a= (4 — xP +(% — 7,
B=, — «P+ (%— 7%
a = (x, — *)?+ (7, — )*
Ioo PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VII.

Also,

xy I= yy — 94, +I — *) — HI. — 9)

xX Viel

X_ Vg I
Substitute these values in the expression for p. Observe, when x,(=), 9 is of

the form 0/o. Divide the numerator and denominator by x, — x and let x,(=).x.

 

 

 

 

y
a
O}
Fic. 13.
To evaluate
XY, — I*
xx’

for x,(=)x, differentiate the numerator and denominator with respect tox, and then
let x,(=)«.

 

WY, IX ee a
ae £(«Dy,—y) = x Dy -J-
X1(=)x
Therefore, when B(=)A,
I (7, — x? ; wa 2\t 2yt
=— I 1 Dyy
p =p eo +(e) [+ (Ov

The first factor takes the form 0/o when x, = «. To evaluate it, differentiate
the numerator and denominator with respect to x,, and we have

I 2x, — x),
2 Dy, — Dy’

this isagain 0/o when «, = «. To find its limit when x,(=)«, differentiate the
numerator and denominator with respect to x,, and there results

 

I
DP? :
which has the limit 1/D? vy when x,(=)«.

Therefore when the points B and C converge to 4 along the curve, the circle
ABC converges to a fixed circle passing through 4 which has the radius

 

; ay)?
tory 11+ (z) j
“Fe ae

ax?

This circle is called the circle of curvature of the curve y = f(x) at the point
x,y, and & is called the radius of curvature. Observe that when x,(=)x and
x, x x, the circle and curve have a common tangent at 4, or, as we say, are
tangent at 4. When this is the case the curve and circle both lie on the same side
of the tangent at 4. Also the circle lies on the same side of the curve in the neigh-
ArT. 79.] ON UNDETERMINED FORMS. ol

borhood of A. But when also +x,(=)x the circle crosses over the curve at 4.
The circle of curvature is said to cut a curve in three coincident points at the point
of contact, in the same sense that a tangent straight line to a curve is said to cut
the curve in two coincident points at the point of contact. Remembering that all

points in the same neighborhood are consecutive, the above statement has definite
meaning.

Much shorter ways of finding the expression for the radius of curvature will be
given hereafter, but none more instructive.

EXERCISES.
1. Evaluate, when x(=)o,

ex — 2c0s x + e-* 3 sin 2x -+ 2 sin’x — 2 sinx x
x sin x = cos x — cos? x ae
2. Also, for the same limit of x,
sin 4x cotx __ 8. sin $x Cos 2%
vers 2x cot? 2x” vers x cotx
3. Show, when x(=)o,
msinx —sinmx m tanzx —natanx _
x(Cos x — cos mx)” 3’ nsinx —sinax
x(=)r

4. If x(=)o, then

(x — 2)ex + x2 _
LF

5, Evaluate for x = 0,

mx 2
(I — x) tan — = —.
2 a

a .

x(=)I

2 8

a\*= a\*x a\~* a \log x
(<>: <) ; (cos =) ; (cos =) 4 (:vs *) ‘
Be x x x
6. Find the limits, when x(=)o, of

q \tan x yr \sinx : 2 7
(=) , 5 , (sin x)sinx, (sin x)tan~,

x x

7. Find the radius of curvature of the parabola y? = 4px at any point x, y, and
show that at the origin it is equal to 29.

8. Evaluate

(@—or+(a—6)) 2a
(8 —6)'+ (2-6) 1+ 473

GS 8 ia log a, when 2(=)}7.
log sin «

ex — 4+ e-* + 2 cos x I
10. Se EES
x(=)o

i

UW. £ xe*= mw.

x(=)o

(1 4+ x)*—et fer i

12. a a

(= Jo
102 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VII.

13. f i “(i + “~ exd log (1+ = { = ae

x( = )oo
14 d fax? + bx +e\_ @
‘ az pe+y PS

15. Evaluate, when x 7 ‘
Yxta—Yx+6, Yx* 4+ ax —-x, a* sin (c/a*),
16. Find where the quadratrix
= + 2
Jy = x cot =

crosses the y axis.

I I I
17. Show that (as = aeae)= -

x(=)o
CHAPTER VIII.

ON MAXIMUM AND MINIMUM.

 

80. Definition.—A function /(1) is said to have a maximum
value at . = @ when the value of the function, J(@), at ais greater
than the values of the function corresponding to all other values cf
x in the neighborhood of a.

The function is said to have a minimum value at a when J(a) is
fess than /(.v) for all values of « in the neighborhood of a.

In symbols, /(x) is a maximum or a minimum at a according as

S(*) —S(@)
is negative or postive, respectively, for all values of x a in
(2 — €, a+ €) the neighborhood of a,

81. Theorem—At a value a of the variable for which the func-
tion /(x) is differentiable and has a maximum or a minimum, the
derivative /’(a) is o.

Ata value @ at which /() is a maximum ora minimum, by defini-
tion the differences

SJ’) — f(a) and f(x”) — f(a);
where +’ <a < +”, have the same sign.
Consequently the difference-quotients

7 =) a and g* =i) — oe

have opposite signs for all values of x’ and x” in the neighborhood
of a, since +’ — a is negative and «’’ — a is positive. Therefore,
since g’ and g’’ have the common limit /’(2) when «’(=)a and

x’’(=)a, we have
IA@~ 2) =A (al + led,
= 2|/"(2)| =0.

Hence J (a) =0.

Notice that at a maximum value of the function the derivative is o,
and since, by definition, the function must increase up to its maxi-
mum value and then decrease as . increases through the neighbor-
hood of a, the derivative on the inferior side of a is positive and on
the superior side is negative, § 60.

Hence, at a maximum, a, the derivative, /’(a), is o and /’(x)
changes from fosifive to negative as x increases through a.

103
104 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VIII.

In like manner, at a minimum, + = 4g, the derivative, /’(a), is 9,
and /’(x) changes from negative to positive as x increases through a.

Conversely, whenever these conditions hold, then the function
has a maximum or a minimum value at a, accordingly.

For example:

1. Let SJ(*) Sx? — 2x 4 3.

ote f(x) = 2(x — 1).

We have f(1)=0. Also for x < 1, we have /’(x) negative, and for x > 1,
J (x) positive.

Hence /(1) = 2 is a minimum value of f(r).

2. Let K(x) = — 2x7 + 8x — 9.
(2) = 4(2 — 2).
We tiave J(2)=0, f(2-—e)=t, f(2te)=—
oo. (2) = — Lisa maximum.

82. The condition /’(a) = 0 is necessary, but it is not sufficient,
in order that the function /(~) shall have a maximum or a minimum
value at a. For the derivative /’(x) may not change sign as +
increases through a. It may continue positive, in which case /()
continues to increase as x increases through a; or /’(x) may be
negative throughout the neighborhood of a, in which case the func-
tion continually diminishes as x increases through a. These condi-
tions can be illustrated geometrically thus:

GEOMETRICAL ILLUSTRATION.

Represent y = /(x) by the curve ABCDE. Then /’(x) is represented by the
slope of the tangent to the curve to the x-axis. At a maximum or a minimum,
Jj'(x) = © or the tangent to the curve is parallel to Ox. In the neighborhood of

A 4 Q
fe Fi,
oO
B 4|/

D

 

 

 

 

 

Fic. 14.

a maximum point, such as 4 or C, the curve lies helow the tangent, and the
ordinate there is greater than any other ordinate in its neighborhood. In like
manner at a minimum point, such as B or DJ, the points 4, D are the lowest points
in their respective neighborhoods. At a point Z the tangent is parallel to Ox,
and /’(x) =0, but the curve crosses over the tangent and is an increasing function
at £, also the derivative J’ (x) is positive for all values of x in the neighborhood.

It will frequently be impracticable to examine the signs of the
derivative in the neighborhood of a value of + at which /’(x) = 0.
A more general and satisfactory investigation is required to discrimi-
nate as to maximum and minimum at such a point.
ART. 83.] ON MAXIMUM AND MINIMUM. 105

83. Study of a Function at a Value of the Variable at which
the First n Derivatives are Zero.

(1). Let /(x) be a function such that /’(2) ~ 0. Then by the
law of the mean, §§ 62, 64,

I*) — Aa) = (x — OFS).

By hypothesis, /’(a) x 0 is the limit of /’(v) and of /’(&) as
x(=)a, since & lies between x and a. Consequently we can always
take « so near @ that throughout the neighborhood of a we have
J'(&) of the same sign as /’(a) for all values of x in that neighbor-
hood. Hence, as + increases through the neighborhood of a, the
difference /(x) — /(a) changes sign with + — a; and by definition
/(*) is an increasing or decreasing function at @ according as /’(a)
is positive or negative respectively.

(2). Let /’(2) =o and /’(a) 40. Then

x — a) ‘
fe) — Aa) = SEF prey,

Throughout the neighborhood of a, /’’(&) has the same sign as
its limit /’(a2) 0, and therefore does not change its sign as +
increases through a. But, as (v — a)* also does not change sign as
x passes through a, we have the difference

J (*) — /(2),

retaining the same sign for all values of x in the neighborhood of a,
and having the same sign as /’’(a2). Consequently, by definition, the
function /(x) has a maximum or a minimum value f(a) at a according
as f'’(a) is negative or positive respectively.

(3). Let /’(2) = 90, f"(a@) = 0, f(a) # 0. +=Then

(¥ ae a)° v1}
J(*) -/(9) = a (6).

As before, in the neighborhood of a, /’’’(&) has the same sign as
its limit f’’’(a) ~ 0. But (v — a)’ changes its sign from — to +
as xv increases through a. ‘Therefore the difference

Ke) —S()
must change sign as x increases through a, and /(x) is an icreasing
or decreasimg function at a according as /’’’(q) is postive or negative.

(4). Let f(a) =/"(2) =... = /"(2) = 9, but /*7(a) # ©.
Then, by the law of the mean,
: a eer a6
J(*) -/(®) = “oe (8).

In the neighborhood of a, /**1(&) has the same sign as /"*1(a).
If 2 +1 is odd, then (x — a)"*! and therefore /(x) — /(2) change
sign as x increases through a; and J (x) isan mcreasing or decreasing
106 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cn. VIII.

function at @ according as /"*}(a) is positive or negakve. If, how-
ever, z+ 1 is even, then (x — a)"*! does not change sign, nor does
the difference /(x) —_/(a), as x increases through a; consequently
J(%x) is a maximum or a minimum at a according as /"*1(a) is nega-
five or positive. Hence the following

84. Rule for Maximum and Minimum.—To find the maxima
and minima values of a given function /(x), solve the equation
f'(x)=0. If abe a root of the equation /’(x) = o, and the first
derivative of /(x) which does not vanish at @ is of even order, say
(a) £ 0, then /(2) is a maximum if f*"(a) is negative, or a minimum
if 7*"(a) is postive.

 

EXAMPLES.

1. Find the max. and min. values, if any exist, of

P(x) = 2 — ox? 4 240 — 7.

We have @' (x) = 3(x? — 6x + 8) = 3(% — 2)(x — 4).
g'(2) = 9, (4) =0.
Also, p(x) = 6(* — 3).
(

i 2 .
-. G2) =+4+ 13 isa maximum, @(4) = 9 a minimum.
2. Investigate for maxima and minima values the function
P(*)= e* + e-* + 2 cos a.

Wehave 90) = #0) =H") =, GMO) = 4.
~(0) = 4 isa minimum. Show that 0 is the only root of (x).

3. Investigate 25 — 5at+-5x7 1, at x= 1, r= 3.
. Investigate +3 — 327+ 3r-+7, at x= 1.
5. Investigate for max. and min. the functions

xi — 30 + Ox +7, #8 — ox + 15x — 3.

305 — 12543 + 21607, x + 327 + 6x — 15.
6. Show that (1 — x 4 x*)/(1 ++ *— x’) ismin. atx = 1,
7. It ay(y — x) = 2a*, show that y has a minimum value when x = a.

>

8. If 30%? + xy + 4ax8 = 0, show that when x = 3a@/2, then y = — 3a
isa maximum. Jy being then — ‘

9. If 2x5 +4 3ayt — x*y3 — 0, thena = 53

@ makes y = sia a minimum.

85. Observations on Maximum and Minimum,

(1). We can frequently detect the max. or min. value of a func-
tion by inspection, making use of the definition that there the neigh-
boring values are greater or less than the min. or max. value
respectively.

For example, consider the function
ax? + bx +e
Substitute y — 4/2a for x, The function becomes
fai

4a
Art. 35.] ON MAXIMUM AND MINIMUM. 107

which is evidently a maximum when y = 0 and a is negative, and a minimum
when y = Oand a is positive.

(2). Labor is frequently saved by considering the behavior of the
first derivative in the neighborhood ofits roots, instead of finding the
values of the higher derivatives there.

For example, see Ex. 6, § 85, and also

P(x) = (% — 4)P(@ + 2),

Here O'(«) = 3(34 — 2)(x — 4)(x + 2)%

9’ passes through 0, changing from + to — as x increases through — 2; there-
fore @(— 2) is a maximum.

@’ passes through 0, but is always positive as x increases through 4 ; therefore
(4) is an increasing value of @(.x). Also @’ passes through 0, changing from
— to + as x increases through 2/3, and the function is a minimum there.

(3). The work of finding maximum and minimum values is fre-
quently simplified by observing that

Any value of x which makes /(%) a maximum or a minimum also
makes C/(x) amaximum ora minimum when C is a positive constant,
and a minimum or a maximum when C is a negative constant.

J(x) and C+ /() have max. and min. values for the same values
of x.

(4). If~ is an integer, positive or negative, /(x) and { /(x)}” have
max, and min. values at the same values of the variable. In particu-
lar, a function is a maximum or a minimum when its reciprocal is a
minimum or a maximum respectively.

(5). The maximum and minimum values of a continuous function
must occur alternately.

(6). A function /(x) may be continuous throughout an interval
(a, 8), and have a maximum or a minimum value at + = a in the
interval, while its derivative /’(x) is o at a, but continuous for all
values of (x) on either side of a.

In this case, to determine the character of (x) at a, we can use
(1) or (2) as a test. Otherwise we can consider the reciprocal
1/f'(x), which passes through o and must change sign as x passes
through a, for a maximum or a minimum of /(*) at a.

EXAMPLES.
1, Consider (x) = (x — 2) +1.

@ is a one-valued and continuous function and is always positive. It clearly
has a minimum at x = 2, where @(x) = 1. We have

 

 

ga) => —
Sr
3 (« — 2)8 y a
and @(2) =o. Also, @~’(2 — %) is negative and
g’(2 + A) is positive.
2. In like manner ae
P(x) = 1 — (x — 2)! O| 2

has a maximum at x = 2.
108 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VIII

3. Consider (x)= 14 (x — 2)?

y which is also uniform and continuous.
We have
, 3 I
v2) =2 —+,
os 5 (x — 2)
O 2 which is + c when x = 2, but is always + in the neigh-
Fic. 16. borhood of 2. Therefore, at x = 2, (+) is an increasing
function.

2 : : :
In like mamer 1 — (x — 2) is a decreasing function at x = 2.

(7). In problems involving more than one variable we reduce the
conditions to a function of one variable by algebraic considerations,
Otherwise, we can frequently make a problem involving more than
one variable depend on one which can be solved by elementary con-
siderations.

For example, the sum of several numbers is constant; show that their product
is greatest when the numbers are equal.
First, take twonumbers, and let

ety=e
Then acy = (x + yP — (x —yPame—(x—y),
which is evidently greatest when x = y.
Let xatytere

Then, as long as any two of x, y, z are unequal, we can increase the product
xyz without changing the third, by the above result. Therefore xyz is greatest
whenx =y =z. The method and result is general, whatever be the number
of variables.

EXERCISES.

1. Find the maximum and minimum values of y, where
y = (x — I)(x — 2)
2, Find the max. and min. values of
(1). 2x3 — 15x? + 36x 4+ 6.
(2). (« — 2)" — 3)
(3). 23 -- 3x? + 6x + 3.
(4). 30° — 25x5 + 60x.
3. Show that (x? + « + 1)/(*? — x + 1) has 3+) for max. and 3-1 for min.
4, Find the greatest and least values of
asinx-+écosx and asin’x + 3b cos?x.
5. Investigate («? + 2x — 15)/(x — 5), and also
x? — 7x + 6
x—I0 ’
for maximum and minimum values.

 

6. The derivative of a certain function is
(@ — 1% — 27% — 3) — 4);
discuss the function at x = I, 2, 3, 4.
7. Find the max. and min. values of

(a). (x — I)(~ — 2)(x —3), (e). (I — x\(r — x),
(3}. «* — 823 + 22%% — 24x, (f). (2 — H/o? + 3),
(Oe (« — a(x — 4), (g). sin x cos8x,

(2). (x — ajt(x — 4)8, (2). (log x)/x.
ART. 85.] ON MAXIMUM AND MINIMUM. 10g

8. Show that the shortest distance from a given point to a given straight line is
the perpendicular distance from the point to the straight line.

9. Given two sides a and 4 of a triangle, construct the triangle of greatest
area.

10. Construct a triangle of greatest area, given one side and the opposite
angle.

11. If an ova/ is a plane closed curve such that a straight line can cut it in only
two points, show that if the triangle of greatest area be inscribed in an oval, the
tangents at the corners must be parallel to the opposite sides.

12. The sum of two numbers is given; when will their product be greatest ?
The product of two numbers is given; when will their sum be least ?

13, Extend 12 by elementary reasoning to show that if
n
Zar) Ssxyt..- fa ay
x

tye Xn

then TI(x,)
I

is greatest when 4) = «4, =... = 4x.

14. Apply 13 to show that if. + y+ 2 =, the maximum value of xy?25 is
0/432.

15. Show that ifx + y+ z= ¢, the maximum value of x! yz is

Lmmynclimtn
(2m + nyetmte

16. Find the area of the greatest rectangle that can be inscribed in the

ellipse. (Use the method of Ex. 15.)

 

2 2
< + 5 = 1; [Ans. 2ad.]
17, Find the greatest value of 8xyz, if
BEE ee it abe
+ G+5er [ ans 8 |

This is the volume of the greatest rectangular parallelopiped that can be
inscribed in the ellipsoid.

18. Show that the greatest length intercepted by two circles on a straight line
passing through a point of their intersection is when the line is parallel to their
line of centres.

19. From a point C distant ¢ from the centre O of a given circle, a secant is
drawn cutting the circle in 4 and B. Draw the secant when the area of the
triangle 4 OZ is the greatest. [With C as a centre-and radius equal to the diagonal
of the square on ¢, draw an arc cutting the parallel tangent to OC in D. Then
DC is the required secant. Prove it.]

20. A piece of wire is bent into a circular arc. Find the radius when the seg-

mental area under the arc is greatest and least. [y= a/x, r=~0.]
21. Find when a straight line through a fixed point P makes with two fixed
straight lines 4C, AZ, a triangle of minimum area. [P bisects that side. ]

22, The product xy is constant; when is « + y least?

23. An open tank is to be constructed with a square base and vertical sides,
and is to contain a given volume ; show that the expense of lining it with sheet lead
will be least when the depth is one half the width,

24. Solve 23 when the base is a regular hexagon.
II0 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. VIIL

25. From a fixed point 4 on the circumference of a circle of radius a, a perpen-
dicular 4 Y is drawn to the tangent at a point ?; show that the maximum area of

the triangle 4P Vis 3 ¥3 a?/8.

26. Cut four equal squares from the corners of a given rectangle so as to con-
struct a box of greatest content.

27. Construct a cylindrical cup with least surface that will hold a given
volume.

28. Construct a cylindrical cup with given surface that will hold the greatest
volume.

29. Find the circular sector of given perimeter which has the greatest area.

30. Find the sphere which placed in a conical cup full of water will displace
the greatest amount of liquid.

31. A rectangle is surmounted by a semicircle. Given the outside perimeter
of the whole figure, construct it when the area is greatest.

32. A person in a boat 4 miles from the nearest point of the beach wishes
to reach in the shortest timea place 12 miles from that point along the shore; he
can ride 10 miles an hour and can sail 6 miles an hour : show that he should
land at a point on the beach 9g miles from the place to be reached.

33. The length of a straight line, passing through the point a, 4, included be-
tween the axes of rectangular coordinates is 7. The axial intercepts of the line are
a, 8, and it makes the angle 4 with Ox. Show that

(a). Zis least when tan@= (6/a)*.
(3). a + fis least when tan 9 = (6/2).
(c). af is least when tan 6 = b/a.

34. Find what sector must be taken out of a given circle in order that the
remainder may form the curved surface of a cone of maximum volume.

[Angle of sector = 24(1 — ¥2/3).]
35. Of all right cones having the same slant height, that one has the great-
est volume whose semi-vertical angle is tan—1 V2.
36. The intensity of light varies inversely as the square of the distance from
the source, Find the point in the line between two lights which receives the least
illumination.

37. Find the point on the line of centres between two spheres from which the
greatest amount of spherical surface can be seen.

38. Two points are both inside or outside a given sphere. Find the shortest
route from one point to the other via the surface of the sphere.

39. Find the nearest point on the parabola y? = 4px to a given point on the
axis. ;

40. The sum of the perimeters of a circle and a square is 7. Show that when
the sum of the areas is least, the side of the square is double the radius of the circle.

41. The sum of the surfaces of a sphere and a cube is given. Show that when
the sum of the volumes is least, the diameter of the sphere is equal to the edge of
the cube. :

42. Show that the right cone of greatest volume that can be inscribed in a given
sphere is such that three times its altitude is twice the diameter of the sphere.

Also show that this is the cone of greatest convex surface that can be inscribed
in the sphere.

43. Find the right cylinder of greatest volume that can be inscribed in a given
right cone.
ArT. 85.] ON MAXIMUM AND MINIMUM. Tit

44. Show that the right cylinder of given surface and maximum volume has its
height equal to the diameter of its base.

45. Show that the right cone of maximum entire surface inscribed in a sphere
of radius @ has for its altitude (23 — ¥17)a/16; while that of the corresponding
right cylinder is (2 — 2/ ¥5)#a.

46. Show that the altitude of the cone of least volume circumscribed about a
sphere of radius a is 4a, and its volume is twice that of the sphere.

47. The altitude of the right cylinder of greatest volume inscribed in a given
sphere of radius a is 2a/ V3.
48. The corner of a rectangle whose width is ais folded over to touch the

other side. Show that the area of the triangle folded over is least when ja is
folded over, and the length of the crease is least when a is folded over.

49. Show that the altitude of the least isosceles triangle circumscribed about an
ellipse whose axes are 2a and 2d, is 36. The base of the triangle being parallel to
the major axis,

50. Find the least length of the tangent to the ellipse x*/a? +.y?/é? = 1, inter-
cepted between the axes. [Ans. @ + 4.]

51. A right prism on a regular hexagonal base is truncated by three planes
through the alternate vertices of the upper base and intersecting at a common point
on the axis of the prism prolonged. The volume remains unchanged. Show that

the inclination of the planes to the axis is sec—t¥3 when the surface is least.
[This is the celebrated bee-cell problem. ]

52. Show that the piece of square timber of greatest volume that can be cut
from a sawmill log Z feet long of diameters D and d at the ends has the volume

2. LU
27D —da
53. A man in a boat off shore wishes to reach an inland station in the shortest
time. He can row «miles per hour and walk v miles per hour. Show that he
should land at a point on the straight shore at which
cosa :cosf=u: v,

approaching the shore at an angle a@ and leaving it at an angle f.
[ This is the law of refraction. ]

54. From a point O outside a circle of radius x and centre C, and at a distance
a from C, a secant is drawn cutting the circumference at X and A&’. The line OC
cuts the circle in A and &.

Show that the inscribed quadrilateral 4R2A’ZB is of maximum area when the
projection of RA’ on ABZ is equal to the radius of the circle.

55. Design a sheet-steel cylindrical stand-pipe for a city water-supply which
shall hold a given volume, using the least amount of metal. The uniform thickness
of the metal to be a.

If His the height and & the radius of the base, then H= R,

56. If a chord cuts off a maximum or minimum area from a simple closed curve
when the chord passes through a fixed point, show that the point must bisect the
chord.
PART II.
APPLICATIONS TO GEOMETRY.

CHAPTER IX.
TANGENT AND NORMAL.

86. The application of the Differential Calculus to geometry is
limited mainly to the discussion of properties at a point on the curve.
Of chief interest are the contact problems, or the relations of a pro-
posed curve to straight lines and other curves touching the proposed
curve at a point. The application of the Calculus to curves is best
treated after the development of the theory for functions of two
variables.

87. The Tangent (Rectangular Coordinates).—Let y = /(x), or
P(x, v) = 0, be the equation to any curve. The equation to the
secant through the points +, y and
x,, y, on the curve ts

VF — Dee Vg iy
X= 2° 2, — 3 (1)
X, F being the coordinates of an
arbitrary point on the secant. By
definition, the tangent to a curve
at P is the straight line which is
the limiting position of the secant
Fic. 17. PP, when P(=)P. But when
P(=)P we have +,(=)x and y,(=)y. The member on the right of
equation (1), being the difference-quotient of_y with respect to x, has
for its limit the derivative of y with respect to 2. At the same time
the arbitrary point X, FY on the secant becomes an arbitrary point
on the tangent. Therefore we have for the equation to the tangent
at P

 

 

 

 

AP OP as.
Kae ae oo (2)

 

in terms of the coordinates 1. y of the point of contact.
112
ArT. 88.] TANGENT AND NORMAL. 113

The equation to the tangent (2) can be written
| Foy=(x-9%, (3)
or in differentials :
(¥ — y)dx — (X — x)dy = 0, (4)
or in the symmetrical form
X—x F-y

ax oF, dy % (5)

 

EXAMPLES.

1. Find the equation to the tangent to the circle x? + y? = a%
Differentiating, we have
2x + 2y Dy =0.

Dy = — x/y, and the tangent at x, y is
x
BOD ae et a a aa So
or Yy + Xx = a’.
2. Find the tangent at x, y to «?/a? + 7?/s? = 1.
3. Find the tangent at x, y to 2?/a? — y?/? = 1.
4. Find the tangent at x, y to y? = 4px.
5. Find the tangent ata, y to 22+ 7? +4 2f7 + 2aex+d=0,
6. Show that the equation to the tangent at x, y to the conic
P(x, ¥) = ax? + by? 4 vhay + afx + 2gy +d=0
is (ax fly + f/\X+ (ize toy t+ g¥+(etotd) =o.
7. Show that the equation to the tangent at 2, y to the curve
ar yn 1
qu bm
‘i xm X yu ay Se
qu bm
8. Find the tangent at x, y to 25 = ay’, [5X/* — 2Y/y = 3.]

9. The tangent at x,y to «3 — 3axy +75 =O is
(9? — ax)V 4+ («7 — ay)X = axy.
10. Find the equation to the tangent to the hypocycloid
a +7 5a a,
and prove that the portion of the tangent included between the axes is of constant
length.

88. If the equation to a curve is given by
x=), v= 4);
then, since dx = @’(/)dt, dy=y'(t)dt, we have for the equation

to the tangent
(F—y)0'(t) = (X — x) b"(). (1)
114 APPLICATIONS TO GEOMETRY. (Cu. IX.

EXAMPLES.

1. If the coordinates of any point on a curve satisfy the cycloid
x = a(@ — sin 6), y = a(I — cos 6),
show that the tangent at x, y makes an angle 39 with Oy, and has for its equation
Y—y =(X — x) cot 6.
2. In like manner, if
x = c sin 26(I + cos 20), y = ¢ cos 26(1 — cos 26),
the tangent makes the angle 6 with Ox, and its equation is
Y—y=(X — x) tan 6.

89. The angle at which two curves intersect is defined as the angle
between their tangents at the point of intersection.

Ify = $(*) and_y = (2) are two curves, and these equations be
solved for x and_y, we find the coordinates of the points of intersection.
If the curves intersect at an angle w, then since #’(..) and #’(x) are
the tangents of the angles which the tangents to the curves make with
Ox, we have

be — Ps

r+ OM,’ (x)

The two lines cut at right angles when Oi) = — 1.

tan w@ =

Ex. Show that x? + y? = 8ax and y*(2a — x) = x? cut at right angles and
at 45°.

90. The Normal (Rectangular Coordinates).—The normal at a
point of a curve is the straight line perpendicular to the tangent at
that point.

If 4, and @,, are the angles which the tangent and normal at a point
make with Ox respectively, then since one is always equal to the sum

 

of $7 and the other, we have tan 6, tan 6, = —1. Therefore
ax
tan 6, = =a = — D,x.
Hence the equation to the normal at x, y to a curve is
Foy+(X—«)Dx=0, (1)
or (P—y)Dy+X-x=0, (2)
or in differentials
(¥—y)& + (X ~ x)dx = 0, (3)

where D,.y or D,x must be found from the equation to the curve.

EXAMPLES.
1, The equation to the normal at x, y to x*/a? + y?/d? = Lis
eX FY
— — —= a — 8,

mG y
ArT. 91.] TANGENT AND NORMAL. II5

2. The normal at x, y toy = ax* is
ny VY + mxX = ny* + mx*.
3. Show that the tangent and normal to the cissoid
y(2a— x)= 93, at x =a, are,
at (a, @), Yrow-—a zwtxes= 3a;
at(@,—@), y+2x=>a, 2= x — 3a.
4. In the Witch of Agnesi, y(4a? + x”) = 83, the tangent and normal at
x = 2a, are
x+2= 4a, y = 2x — 3a.
5. Show that the maximum or minimum distance from a point to a curve is
measured along the normal to the curve through the point.
Let a, # be a point in the plane of a curve O(x, y) = 0.
If 6 is the distance from a, f to a point .r, y on the curve, then
oF = (a — xP + (6 — yf.
When this is a maximum or minimum,
d3? = — 2a — x)dx — 2(8 — y)dy = 9,
which is the equation (3), § 90, to the normal through a, f.

or. Subtangent and Subnormal (Rectangular Coordinates).—
The portion of the tangent, PZ, included between the point of
contact, P, and the x-axis, is called
the /angent-length. The portion of Y
the normal between the point of con- P
tact and the x-axis is called the zor-
mal-length. The projections 7'/ and
ALN of the tangent-length and nor-
mal-length on the v-axis respectively 2
are called the subtangent and subnor- o| 7 Mu N
Fic. 18.

mal corresponding to the point P.
If 4, 2, S;, S, represent the tangent-length, normal-length, sub-

tangent, and subnormal respectively, then we have directly from the

figure
spf. ge \*? | y
say [Z, pay +(#) Te

d ay \?
Sa IS, nash +(#).

S; is measured from 7’ to the right or left according as S;, is +

 

or —, and S,, is measured from / to the right or left according as
S, is + or —.
EXAMPLES.
1. Show that the subnormal in the ellipse x?/a? + y?/0? = 1 is
Sa = — Bx/a’.

2. Show that 5S; in y = a~ is constant.
116 APPLICATIONS TO GEOMETRY. [Cu. IX.

. In y? = 2mx, show that S, = m is constant.

3
eS ee
4. In the catenary y = ja & +e | , naa=y/a.
5. Show that (x, vy) = o must be a straight line if S,/S,, is constant.
6. Show, in the cissoid «3 = (2a@ — x)y?, that
St = (2ax — x*)/(3a — x).
7. Show that the circle x? + y? = a? has » constant.

92. Tangent, Normal, Subtangent, Subnormal (Polar Coor-
dinates).—Let /(p, ¢) = 0 be the equation to any curve in polar coor-
dinates, # the angle which the tangent at any point makes with the
radius vector, and @ the angle which the tangent makes with the initial
line. From the figure we have

 

Fic. Io.

PM | OP,=p,=p+ 4p,

psin 46
tan JZ, a
ae p+ 4p — pcos 46’
sin 46
_ ace 46
4p 46 1 —cos 40°
rea 46
When 46(=)o, we have, passing to limits,
do
tan y =p dp’ (1)
: I —cosx :
since {= =Z#sinx=o, when +(=)o.

Also, since @=6+%4, we have
p D,@ + tan @
1—ptané@D,6’

_ pt tan 4 Dep
~ Deo — ptand (2)

tan d=
ART. 92.] TANGENT AND NORMAL. 17

Observe that (2) is the same value as that obtained for D,y in
§ 56.

 

Draw a straight line through the origin perpendicular to the
radius vector, cutting the tangent in Zand the normalin WV. We
call PN and P7, the portions of the normal and tangent intercepted
between the point of contact, P, and the perpendicular through the
origin, O, to the radius vector, OP, the polar xormal-length and polar
langent-length respectively ; and their projections, OV and OT, on this
perpendicular are called respectively the polar suénormaé and subtangent,

We have directly from the figure

f=psecp = pt + p(D,6), (3)
n= pescp =V p> + (Dep)”. (4)
S,= ptany = °D,9, S, =p cot > = Dep. (5)

When J, is positive (negative), S, is to be measured from O to
the right (left) of an observer looking from O to P.

Putting p’=D,p, we have for the perpendicular from the origin
on the tangent

p
= ——— 6
PT Vere a
since ff = pS, This can be written
Tate
if we put e = 1/4, for then
dp 1 du
dowd"
EXAMPLES.

1. In the spiral of Archimedes g = a9, show that tany=— 6, and Sy is
constant.

2. Show that S; is constant in the reciprocal or hyperbolic spiral p§ = a.
APPLICATIONS TO GEOMETRY. [Cu. IX.

118
3. In the equiangular spiral » = ae@cota, show that p= a, S; = tana,

Sn = pcot a.
4. If p = a®, show that tan % = (log a)—.
5. Show that the perpendicular from the focus to the tangent in the ellipse
(I — ¢ cos 6)o = a(1 — é)
gh ee
Pee

 

is
6. Determine the points in the curve p = a(1 + cos 6), the cardioid, at which

the tangent is parallel to the initial line.
7. If p =a{t — cos 6), show that
S_ = 2a sin? 39 tan 36.

y=, p£ = 2a sin} 3,

EXERCISES.
1. Show that in ‘x, vy) = 0, the intercepts of the tangent at any point x, y on
the axes are
Ay= x —yDyx, VYi=y —xDzy.
2. The length of the perpendicular from the origin on the tangent is
_ «Dy—y
V+ (Dy
3. Show that when the area of the triangle formed by the tangent to a given
curve and the axes of coordinates is a maximum or a minimum, the point of con-
tact is the middle point of the hypothenuse.
Indicate D,y by y’, and the area by Q. Then

t= 7
-

Also,
2 22 _ (9 — Ny + wy"
y? ,
where y” = Dzy. For a maximum or a minimum DQ. = 0. The conditions
VY #0 y" #0, y—ay #0, y+ay =O
show, by Ex. 1, that X; = 2x, Y; = 2y.
4. Find when the area of the triangle formed by the coordinate axes and the

tangent to the ellipse
a a
at aot

is a minimum.
5. Show that the tangent at the point (2, — 1) of the curve
4 2xy — 3 +4arty—4=0
is 8x -- sy = 1.
6. The line ex + y = e(1 4 7) is tangent to the curve
sinx —cosx=logy, at (Z, e).

7. The line y + 1 = 0 is tangent at (+ 1, — 1) to
at — 2x°y? — 398 4 gry + 4e+ 5y+3=0,

8. Determine the points at which the tangents to
B+ yy = 3x
are parallel to the coordinate axes. (*=O7y=0) (St Ly = t+ Y2).
ART. 92.] TANGENT AND NORMAL, 11g

9. At what point of «4 + 4y — 9 = 0 is the tangent parallel tox — y = 0?
2 (+= —-—1,y = 2.)
10. The tangents from the origin to
at — y + 3x2y 4 2ay? = 0
are yao, 3x —y = 0, x+y =o.
11. The perpendicular from the origin to the tangent at x, y of the curve
xi + yi =ai is p= Vaxy.
12. Show that the slope of the curve xy? -- a3(” + y) to the x-axis is 37
at 0, Oo.

13. If x, y are rectangular coordinates and p, @ the polar coordinates of a point
on a curve, show geometrically that when D,y =0 we have Dop = p tan 6,
and verify from the formulz in the text.

14, Show that the curves
ax y
al Re
cut at right angle if a? — 6? = a’ — 4/2,
15, In the parabola Fog - y =a
Yt+ Vet = (axy)?,
and that the sum of its intercepts is constant and equal to a.
16. The tangent at x, y to (x/a)? + (y/6)8 = lis
Xxfa? + (Y+ 2y)/30% 4 =1.
Also find the normal.

17, The tangent and normal to the ellipse
x*+ 2y? — 2xy —x =0

2 2
= a
I and sat ja =t

, Show that at x, y the tangent is

at x = I are,
at (I, 0), awox—t, VY 2x = 23
at (1, 1), 2y=x--1, y+ 2x = 3.
18. In the curve y(« — 1)(+ — 2) = x — 3, show that the tangent is parallel
to the x-axis atx = 3 + 7/2.
19. In the curve (a/a)t + (y/6)8 = 1, show that (see Ex. 1.)

AT, OY}
+tp=Hh

 

a
20. Show that the tractrix
as c ess Vo — yt
at fe — y? = — log ——~___=—
2 c + Vo - yy
has a constant tangent-length.
21. In the curve y+ = a*—1x, find the equation to the tangent; and determine
the value of 7 when the area included between the tangent and axes is constant.
22. In (ae + be-6) = ab, show that
Sy = — ab/(aed — be-8),
23. If p? cos 28= a’, show that sin = a?/p?,
24. If two points be taken, one on the curve and one on the tangent, the points
being equidistant from the point of contact, show that the normal to the curve is the

limit of the straightline passing through the two points as they converge to the
point of contact.
120 APPLICATIONS TO GEOMETRY. . [Cu. IX.

25. If Q, P, R are three points on a curve, P the mid-point ot the arc QA. and
V the middle point of the chord QA, show that the normal at P is the limit of the
line PV as Q(=)P, R(=)P.
26. Prove that the limit of any secant line through any two points X, Qona
curve is the tangent at a point Pas R(=)P, QO(=)P.
27. Show that as a variable normal converges to a fixed normal, their intersec-
tion converges, in general, to a definite point, and find its coordinates.
Let (Y—yy' +X -—x =0
and (V-yyy +X - 1=9,
where y’, yj represent D,y at x, y and x,, 7’, be the equations of a fixed normal at
x, y and a variable normal at x,, 7,. Eliminating \, we have
Y(t 9") =I — BI" +
=I IV +I — I) + *)
1444 mt y
ote YS eae! ewe ,
: Kay
vy — x

72
=y+ : aed » When 2,(=)*.
J

 

 

Also,
1+ 9" _@y
a ” a ;
X=a—y y » Where yy” =—— 2

This point is the center of curvature of the curve for x, y.

 
CHAPTER X.
RECTILINEAR ASYMPTOTES.

93. Definition.—An asympiole to a curve is the limiting position
of the tangent as the point of contact moves off to an infinite dis-
tance from the origin.

Or, an asymptote is the limiting position of a secant which cuts
the curve in two infinitely distant points on an infinitely extended
branch of the curve.

94. We have the following methods of determining the asymptotes
to a curve f(x, y) = 0:

I. The equations to the tangent at x, y and its axial intercepts
are

d
Foy=(X—2)%,

gd
=I) — ese
ax
Xia x — y—.

If we determine, for x =y =o,
LX; = 4, Bf; = 4,
then the equation to the asymptote is

e F

ay

Or, if we determine, for x =y =o,
qs

oe = my,
ax

and either a or 4 as above, we have for the asymptote
y=mx+6 or xa pa,

This method involves the evaluation of indeterminate forms,
which must be evaluated either by purely algebraic principles or by
aid of the method of the Calculus prescribed for such forms. The
algebraic evaluations are of more or less difficulty, and another

method will be given in III for algebraic curves.
I21I
122 APPLICATIONS TO GEOMETRY. [Cu. X.

EXAMPLES.
: xy?
1. Find the asymptotes to the hyperbola ao g=t
We have X; = a@?/x, YV;=— é*/y. These are o whenx=>y=.
Therefore the asymptotes pass through the origin. Also,
dy Px 6 ae ees
dz ay * a Yraatja’

the limits of which are + 6/a whenx = 0. The equations to the two asymptotes
are ay = + bx.

2. Find the asymptotes to the curves

(2). y = log x. x=0.
(4). yrex. (Fig. 33.) y=oO
(¢)- Y=, (Fig. 34-) y=o.
(Z). pre== x? — 1, y=o.
(). I +y = e*. x=0 y=0,

(Ff) yotanax, y=cotax, y = secax.
3. Show that y = x is an asymptote of 23 = (x? + 3a7)y.
4. x+y = 2 is an asymptote of y3 = 62? — x3.
5. x = 2a is an asymptote of «8 = (2a — x)y.
6. «8 + y3 = ai has y + x = 0 for an asymptote.
7. The asymptotes of (x — 2a)y? = x$ — a are
x= 2a and at+xex=+typ.

x = 2a is readily seen to be an asymptote. For the others express Dy in terms
of « and make + = o ; the result is + 1. Find the intercept in same way.

8. Find the asymptote of the Folium of Descartes
x + 8 = 3axy.
See Fig. 49. The asymptote isx+y-+oa=0. Puty = mx in theequation
to determine slope and intercept.

II. We can sometimes find the asymptotes to curves by expansion
in a series of powers. Thus, if

a, a.
SS aa ae ee

then y =a. -+ a, is anasymptote. For, evaluating as in I, we
have m=a,, V;= 4,.
Observe also, if we have

Y= O@)+I4+54...,

then when x = © the difference between the ordinate to this curve
and that of the curve _y = (x) continually decreases as x increases.
We say the two curves are asymptotic to each other.
ArT. 94.] RECTILINEAR ASYMPTOTES. 123

EXAMPLES.
9. In Ex. 1, I, we have

é + 6
y= + 5+(:-5) =ar(1-55

As rag aaa indefinitely, the point x, y converges to the straight-line asymptote
ay = + OX.

10. Solve Ex. 3, I, by expansion.

11. Solve Ex. 6, I, by expansion. Here we find that the given curve and the
hyperbola .

y? = x? + 2ax + 4a?
have the same asymptotes.

III. We pass now to the most convenient method of determining
the asymptotes to algebraic curves.

If the given curve is a polynomial, /(2, 7) = 0, in x and y, or
can be reduced to that form, we can always find its asymptotes as
follows:

Rule 1. Equate to o the coefficients of the two highest powers

of x in
S(x, mx + 6) = 0,

These two equations solved for m and é furnish the asymptotes
oblique to the axes.

Rule 2. Equate to o the coefficients of the highest powers of
and of y in f(x, vy) =o. The first furnishes all the asymptotes parallel
to the x-axis, the second those parallel to the y-axis.

Proof: (A). The straight line

y=mx tb (1)
J(% 9) = 0 (2)

in points whose abscisse are the values of x obtained from the solu-
tion of the equation in 1,
S(x, mx + 6) = 0. (3)
If (2) is of the mth degree in x and y, then (3) is of the th
degree in x, and will furnish, in general, ” values of x (real or
imaginary).
Let (3), when arranged according to powers of x, be
A," +A, I+... Axr+A,=0. (3)
If one of the points of section of (1) and (2) moves off to an
infinite distance from the origin, then one root of (3) is infinite, and
the coefficient, A,, of the highest power of + must be 0, or A, = 0.
This is readily seen to be true by substituting 1/z for x in (3),
and arranging according to powers of z. Then when z= 0, we have
x =O, and 4, =0.
In like manner if a second point of intersection of (1) and (2)
moves off to an infinite distance on the curve, a second root of (3)

cuts the curve
124 APPLICATIONS TO GEOMETRY. [Cu. X.

is infinite and we must have the coefficient of «*-1 equal to 0, or
af =105

u-]
When (1) and (2) intersect in two infinitely distant points, then
(1) is an asymptote of (2), and we have for determining the
asymptotes the two equations

Hoo Op Ag =o:
These two equations when solved for m and 4 give the slopes and

intercepts on the y-axis of the oblique asymptotes of (2).

EXAMPLES.
12. Consider 2x3 = (x* + 3a*)y, see Ex. 3.
Here eB — xy — 307 =O
becomes (1 — m)x3 — bx* — 3a°mx — 30°76 = 0,
when mx + 4 is substituted for y. Hence
I—m=-=0 and —4=0

give y = x as the oblique asymptote.
13. In «4+ 73 = 3axy, Ex. 8, put y = mx + 4.

(1 + m)23 + 3m(mb — a)? +...=0,
it being unnecessary to write the other terms.
Hence m= —1, 6=—a. Therefore the oblique-asymptote is
yourox-a.

14. Show that y= 2x-+4a isan asymptote of y? = ax? + x.
15. The asymptotes of y* — xt 4 2ary = Px
are ya«—ta and y+rtta=o.

16. <9 + 3x°y — ay? — 399 + x? — 207 + By? +4445 =0
has for asymptotes

Yttrt+t=o ys=rth vVt+r =}

(B). If the term 4,_,x* 1 is missing in (3), or if the value of m
obtained from A, = o makes 4,,_, vanish, then (3) has three infinite
roots when

A,=o and 4, ,=0,
which equations give the values of m and 4 which furnish the
asymptotes. <A, _, will be of the second degree in 4, furnishing two
é’s for each m, and there will be for each m two parallel oblique
asymptotes, which we say meet the curve in three points at oo.

If also the term A,_,*%”-? is missing, or if 4,_, vanishes for the
value of m obtained from A, = o, then the equations

A, =O, Ay.
furnish three parallel oblique asymptotes, in general, for each m.

gO

EXAMPLES.
V7, TE (@ + 9 P(X? + 9? + xy) = ay? + B(x — ¥),
then 4,45 (1+ mp1 + m+ m,
Ags = 0,
An = BP — a.
o. m=o=—I, 6= +a giveasymptotes y= —u +a.
ART. 95.] RECTILINEAR ASYMPTOTES.. 125

18. In xy + x)? + 2ay?%(x + 9) + 8aray + aby = 0,
the asymptotes are ytx=2a, y+u+4a=0,
19. Find the asymptotes to the curves:
(a). xy? — xy = a(x + yy) + 2B. CSO, GSO eS
(4). v8 — x3 = atx. youn.
(¢). at — yt = Gay + By x+y=0, xr=y.
(C). For the asymptotes parallel to the coordinate axes, the fol-
lowing simple process determines them:
Arrange /(x, v) = o according to powers of, thus:

Ay” -+ (But Oy" t+ (Ft + Get A)y"?+...=0. (4)

If the highest power of y is, 2, the degree of the curve, there will

be no asymptote parallel to Oy, since then 4 x 0. If, however, the
term Ay” is missing, or 4 = o, then for any assigned x one root in
the equation (4) in y will be o. If, now, Bx + C = 0, a second

root of (4), inv, is oat « = — C/B, and this will be an asymptote
to the curve, since D,y is «© for the same value of x which makes
Pare

If the terms involving the two highest powers of yin (4) are
missing, then

Ft Gxt H=o
makes three roots of (4), in_y, infinite, and this is the equation to two
asymptotes parallel to Oy, and so on generally.

In like manner, arranging /(x, y) according to powers of x, we
find the asymptotes parallel to Ox by equating to o the coefficient of
the highest power of x.

Therefore the coefficients of the highest powers of x and _y in the
equation to the curve, equated to zero, give all the asymptotes parallel
to the axes. Of course, if these coefficients do not involve x or v
they cannot be o and there are no asymptotes parallel to the axes.

EXAMPLES.
20. Find the asymptotes to the following curves:
(2). y?x — ay? = + ax? + 6. x=a, yo=xut+a yt+utaz=o.
(6). v(x? — 36x 4 28) = x8—3axtta r= b, x= 26, ytjaazuxt 30.
(¢). x?y? = a%(x? + y?). x= +a yoda.
(d). x2y? = a(x? — y*). yta=o0o y-—az=0.
(e) a= yx + x3, x=a

(Ff): (2 — oP = a? by = 0.
(g). (x — yy? — 22? + y?) =O.
(2). x<?(a? 4+ a?) = (a? — x*)y?,
(2). ey? = Baty.
Cpe SP ae y- BPB
(2). x(~ — yP = We — 9) + 2.
95. Asymptotes to Polar Curves.—If /(0, 6) = 0 is the equa-
tion to a curve in polar coordinates, then, when it has an asymp-

 
126 APPLICATIONS TO GEOMETRY. [Cu. X,

tote, that asymptote must be parallel to the radius vector to the point
at co on the curve, if the asymptote passes within a finite distance of
the origin.

The distance of the asymptote from the origin is the limiting value
of the polar subtangent when the point of contact is infinitely distant.

To determine the polar asymptotes to /(p, 7) = 0, determine
the values of 6 which make p = «. These values of 6 give the
directions of the asymptotes.

If the equation can be written as a polynomial in p, the values of
# are furnished by equating to o the coefficient of the highest power
of p.

To construct the asymptote when, 6 = a, the direction has been
found ; evaluate for 6(=)a@ and p = o the subtangent

ag ag
Zs = onion a ie ae
Pep Sed Pa Liv
#(=)o
where pz = 1. The perpendicular on the asymptote is to be laid off
from the origin to the right or left of an observer at the origin look-
ing toward the point of contact, according as p is + or — respec-
tively.
EXAMPLES.
21. Let p = asec 6 + 4 tan 8.
p =o when § = 173 also,
9 (a+ sin 6)

dp” asing+6’

the limit of which isa + 4. The asymptote is then perpendicular to the initial
line at a distance 2 + # to the right of O. Also, when @ = 87, p = w, and the
corresponding value of the subtangent gives 2 — 4 and another asymptote.

22. Show that p? sin (6 — cr) + ap sin (6 —2a@)+a?=0 has the asymp-
totes psin(@—a)= +asina.

23. Find the asymptotes of sin @ = a6.

24. Find the straight asymptotes of sin 49 = a sin 36.

25. Show that pcos § = — a isan asymptote of cos @ = a cos 26.
26.4=(9—a)sin@ has psinf = 4 for asymptote.

27. Determine the asymptotesof cos 26 = a.

Polar curves may have asymptotic circles or asymptotic points.

EXAMPLES.

28. Find the asymptotes of pf =a, for §6=0, Goo. Fig. 57.
29. Find the circular asymptotes of p(@ + a) = 66, and of
ae? aye 6+ cos 4
P= pee P~Gpsne = Frenne
CHAPTER XI.
CONCAVITY, CONVEXITY, AND INFLEXION.

96. On the Contact |jof a Curve and a Straight Line.—Let
y = f(x) be the equation to a curve. The equation to the tangent,
§ 87, at x = a (¥ being the ordinate corresponding to x) is

F=/f(a) + («—af"(a).

 

T P,
y T
A P A
0} a x ea
Fic. 21,

The difference between the ordinates of the curve and tangent at

any point (by the theorem of mean value) is
Ax) — F = 4% — a7'(8)-

If /’(a) ¥ 0, this difference will retain its sign unchanged for
all values of x in the neighborhood of a2. Therefore throughout this
neighborhood the curve will lie wholly on one side of the tangent.
It will lie below the tangent when /’’(2) is —, and above it when
J (a) is +.

The curve y= /(.v) is said to be concave at a when /’’(a) is
negative, or the curve lies below the tangent there; and is said to be
convex at a when /’’(a) is postive, or the curve lies above the tan-
gent there.

EXAMPLES.
1. The curve y = e* is always convex, since D?e* = ex is always positive.
2. The curve y = log x is always concave, since D? log y = —.x~? is always
negative.

3. The curve y = x3 + ax is convex when x is positive and concave when x
is negative, since D?y = 6x.
127
128 APPLICATIONS TO GEOMETRY. [Cu XL

Points oF INFLEXION.

 

 

Fic. 22.

Suppose, at + = a, we have f’"(a) = 0, but /’"(a2) #0. Then
the difference between the ordinates of the curve and tangent at a is
foy— F=\e- oy").

Since /’’’(a) ¥ 0, then throughout the neighborhood of a, /’’’(&)
keeps the same sign as its limit /’’"(a), But (« — a)? changes from
— to + as x increases through a. Consequently the corresponding
point 7 on the curve crosses over from one side of the tangent to the
other as / passes through 4.

The curve is convex on one side of 4 and concave on the other.
The curve is said to have a point of imflexton at x, 9’ when at this
point we have D,y determinate and D*y = 0, D*y # o.

At a point of inflexion «=a a curve is said to be convexo-
concave when it changes from convex to concave as x increases
through a, and to be concavo-convex when it changes from concave
to convex as .v increases through a. See the points 4 and 4’ in
Fig. 22.

EXAMPLES.
4. If yu2ux«—apo qr—i,
oy" = 12A*—a)=09, when x = a,
and yy" = 12.

The curve has a concavo-convex inflexion at x = a.
5. Show that every cubic
Ax) Sate bx textd
has an inflexion aud classify it.
Again, suppose at + = a we have

S"@=o f"(a=o, f(a) #0.

x —a)}
fa) — F=P Tyna,
In the neighborhood of a, /'”(&) keeps its sign unchanged, as also
does (x — a)‘. Consequently the curve lies wholly on one side of

the tangent, and is convex or concave according as /i¥(a) is + or —.
In general, if,7"(2) = .. +. =/"(2) = 6, f"*(a) = 0, then

ies pe (x — ay" pik
fe) — ¥= FE yung),

Then
ART. 96.] CONCAVITY, CONVEXITY, AND INFLENION, T29

If m+ 1 is even, the curve is concave or convex at a according as
SJ” *\a) is negative or positive.

If m + 1 is odd, the curve has an inflexion at + = a, and is concavo-
convex if f™**(a) is 4+, and is convexo-concave if f"+(a) is —.

The tangent at a point of inflexion is sometimes called a séa/ion-
ary tangent, since D,6 = o there. For, 6 being the angle which the
tangent makes with Ox, wehave tan 0 = D,y, etc.

The conditions for a point of inflexion given above, for /(x, vy) = 0,
are exactly those which have been previously given fora maximum ora
minimum of D,y. For y = f(x) has a convexo-concave inflexion
whenever /’() is a maximum, and a concavo-convex inflexion whenever
SJ (x) is a minimum. The investigation of y = f(x) for points of
inflexion amounts to the same thing as investigating the maximum
and minimum values of y = /'(x).

It is not necessary to give many examples of finding points of
inflexion, since it would be but repeating the work of finding the
maximum and minimum values of functions.

EXAMPLES.

6. Show that «3 = (a* + x*)y has an inflexion at the origin. What kird of
inflexion ?

7. Show that ay = dxy + cx? + dx inflects at 0, o.

8. The origin is an inflexion on a—1y = x”, if m > 2 is an odd integer.

9. When is the origin an inflexion on y# = Lum?

10. Find the point of inflexion on x8 — 34x? + a®y = 0, and classify it.

[x = 4, y = 203/a?.]
11. Show that the inflexions on f(g, 6) = 0 are to be determined from

ake dle
ze oe os =e
+2(2) Page =

See § 56. If we put p = 1/z, this takes the simpler form
u ob ug = 0.

oO.

The polar curve is concave or convex with respect to the pole according as
w- ug ist+or—. The curve in the neighborhood of the point of contact is con-
cave or convex with respect to the pole according as it does or does not lie on the
same side of the tangent as the pole.

12. Find the inflexion on sin 6 = a@.
13. In p§ =a there is an inflexion when @ = 4/m(1 — m).
14, Find the points of inflexion on the curves:

(a). tanax=y. (d). y= eax’,
(4). y = sin ax. (e). vy = (x — I) — 2)(x — 3).
(c). vy = cot ax. (7). p(@ — 1) = a6?

15. Show that the curve x(x? — ay) = a‘ has an inflexion where it cuts Ox.
Find the equation to the tangent there.

16. Show that «3 + 73 = a3 has inflexions on O, and O,.

17. The inflexions of x°y = @(*—y) areatx =o, x= +03.
18. x = log(y/x) inflectsat x= —2, y= — 2e-%,

19. 963 = a@ has an inflexionat p= uz
CHAPTER XII.
CONTACT AND CURVATURE.

97. In the preceding chapter we have studied the character of the
contact of a curve with its straight-line tangent. Now we propose
to study the nature of the contact of two curves which have a
common tangent at a point.

98. Contact of Two Curves.

I. Let_y = f(x) and y = #(x) be two curves, the functions @
and w having determinate derivatives at a.

If we solve y= G(x) and y= (x) for x and y, we find the
points of intersection of the curves.

(1). If d(2) = (2) and ¢'(2) ¥ (a), the curves cu/ at a, and
cross there. For, by the law of the mean applied to the function

F(x) = d(x) — Pr),

P(x) — (*) = (* — a)[9'(S) — $'(8)].- (§ 62)

The derivatives @’(&), ’(&) are arbitrarily nearly equal to #’(a)
and ’(a2) for x in the neighborhood of a. Therefore, since
'(a) # (a), the difference @’(&) — w’(&) keeps its sign unchanged
in the neighborhood of a, and « —a changes sign as x passes
through a.

(2). If we have O(a) = (a), $'(a2) = (a), but f(a) ¥ (a),
then the curves have a common tangent at a, and are said to be tan-
gent to each other, and to have a contact of the first order at a.

By the law of the mean, the difference

P(x) — HC) = 3% — a [6'(S) — H'"(6)]
shows that this difference does not change sign as x increases through
a, and therefore the curves do not cross at a.
(3). If da) = $@), O(a) = (a), $"(a) = H"(a), but Ha)
~ w’’(a), then the curves have a contact of the second order at a,
and we have

P(x) — P(e) = ee — aL) — H"'"(8)).
This shows that the curves do cross at a, since the difference of their
ordinates changes sign as x increases through a.

(4). In general, if @(x) and (+) and their first » derivatives at a
are equal, but their (~-+ 1)th derivatives are unequal, then the

130

we have
ART. 97.] CONTACT AND CURVATURE. 131

curves are said to have an mth contact at a, or a contact of the th
order. They do or do nof cross at the point of contact according as
nm + 1 is odd or even.

For we have, by the ay of the mean,

(ea) att n+l
(2) — He) = FI fore) — oe),
which changes sign or does not according as z-+ 1 is odd or even
when x increases through a.

Two functions are said to have a contact of order z at a value of
the variable when for that value of the variable the corresponding
values of the functions and their first ” derivatives are equal.

II. The character of the contact of two curves is made clear by
the following theorem:

If two curves y = G(x) and y= w(x) intersect in » distinct
points at @,, @,,..., @, then when these # points of intersection
converge to ‘one point, the curves have a contact of order 7 — 1.

To prove this the following lemma * will be established:

If #(x) vanishes at a,, @,,..., @,, then

(w — a)... (4% — an) py
F(x) = SE),
where & is some oar between the greatest and least of the num-
bers x, 2,
Consider the ancien of gz,

J(2) = (¥ = &) 6% = GW) FB) — @ — 4%). = Gn) A()-

We have /(z) = = oat the z + 1 values of z equal tov, a,, ..., &,
By Rolle’s theorem, /’(z) vanishes 7 times, once between each con-
secutive pair of these numbers, Also by the same theorem /’’(z)
vanishes 2 — 1 times, once between each consecutive pair of numbers
at which /’(z) vanishes; and so on, until finally /”(z) vanishes once
between the pair of values for which J*\(z) vanishes. This value,
say &, at which /”(z) vanishes is certainly between the greatest and
least of x, a,,...,@, Hence

y(é) = =(x—a)...(*—4a,)F"(§) — 1! F(x) = 9,
and the lemma is proved.
Now let F(x) = @(*) — (x). Then

o(x) — pa) =F 9- i (=O) gma) — (2).
This shows that when 2, =a, =... = 4, =a, we have
(x) — (2) = R=" rangy — prey,

where & lies between x and a.

 

® Due to Ossian Bonnet. -
132 APPLICATIONS TO GEOMETRY. (Cu. XII.

This last equation shows that @() and 7(.v) and their first z — 1
derivatives at @ are equal, or the two curves have a contact at a of
order x — 1, Therefore, when two curves have a contact of the th
order, it means that they have z-+ 1 coincident points in common
at @; or, as we sometimes say, they intersect in 2 + 1 consecutive
points. A curve which cuts another 2 times in the neighborhood of
a point, leaves that curve on the same side it approaches it when z
is even and leaves on the opposite side when z is odd. Thus we see
why it is that curves having even contact cross, while those having
odd contact do not cross, at the point of contact.

99. To find the order of contact of two given curves, we must
solve their equations for the points of intersection, and compare
their corresponding ordinate derivatives at these points.

EXAMPLES.

1. Find the order of contact of the curves
y= and y = 3x7 — 344+ 1.

Solving the equations, we find that « = 1, y = Lis a point common to both
curves. Also, their first derivatives, Dy, are equal to 3 there, and their second
derivatives, D’y, are equal to 6; while their third derivatives, D*y, are not equal
to each other. Therefore, at the point 1, 1 the curves have a contact of the second
order.

2. Show that the straight line y = x — x and the parabola 4y = x? have a
first-order contact.

3. Find the order of contact of

oy = x8 — 3x7 +4 27 and oy + 3x = 28. [Second. ]
4. Find the orders of contact of the curves:
(q) y=log(x—1) and 2? —6x+2yt8=0. [Second. ]
(4). 4y = x*— 4 and 2? + y? — ay = 3. [Third. ]
(c), xy = a and (« — 2a)? + (y —2a)? = axy, (Third. ]

5. Find the value of a in order that the hyperbola xy = 3x — 1 and parabola
y =x -+1-+ a(x — 1)? may have contact of the second order.

too. Osculation.—(1). We can always find a straight line which
has a contact of the first order with a given curve y = P(x) at a given
arbitrary point. In general, at any point of ordinary position, a
straight line cannot have a contact with a curve of order higher than
the first.

For, let _y = mx + 6 be the equation to a straight line, in which
mand 6 are arbitrary and are to be so determined that the straight
line shall have the closest possible contact with the curve y = (x)
atx =a.

Then we must have

 

ma +b = $(a),
m= P'(a).
These two conditions completely determine m and 4, and give

¥ = P(a) + (¥ — 2)9"(a)
ART. 100.] CONTACT AND CURVATURE. 133

as the equation to the required line, which has contact of the first
order with the curve at a. This is the familiar equation to the tan-
gent to the curve at a.

The line can have no higher contact with the curve at a@ unless
we have @’(a) = 0, and so on, see § 98. At an ordinary point of
inflexion the tangent has a contact of the second order, and cuts the
curve there in three coincident points crossing the curve.

(2). Consider the equation to the circle

(X — a)? + (FP — fp) = R* (1)

This is the most general form of the equation to the circle, and
can be made to represent any circle whatever, by assigning proper
values to the arbitrary constants a, /, &, the coordinates of the
centre and the radius.

Let us determine a, (, and &, so that the circle shall have the
closest possible contact with a given curve y = (x) at a given point
x, y of general position on the curve.

Differentiating (1) twice with respect to X,

V—a+(¥—£)DV=0, (2)
1+ (¥— p)D¥ + (DP) = 0. (3)
The conditions for the contact are
Fay, D¥ = $'(x), PY = $''(x).
The values of a, #, & determined from the three equations

(x — a)? + (9 — BP = RB, (4)
x—a+(y— f)p'(x) =9, (5)
1+ (vy — £)¢'(x) + [0"(2)] =o (6)

determine the circle of closest contact, of the second order, at x, y
on the curve. Solving these equations and writing »’, y’’ for d’, p”,
we have for the coordinates of the centre of curvature

 

 

Ly r+y"
Bayt 77? aax—y’ yy , (7)
and for the radius of curvature
1+ y/?)3
R= i a (8)

J

Whenever the coordinates x, y are given, we can substitute in
these formulz and compute a, 6, and &, and write out the equation
to the circle.

Observe that the three equations completely determine the circle,
and the circle at a point of ordinary position on the curve can have
no closer contact with the curve than that of second order. Observe
that this is the same circle obtained in § 79, Ill. (3), where we con-
sidered the circle which was the limiting position of a circle through
three points on the curve when these three points converge to +, y
134 APPLICATIONS TO GEOMETRY. [Cu. XII.

as a limit. Having a contact of the second. order with the curve,
the circle of curvature crosses over the curve at the point of contact,

This circle is called the circle of curvature of the curve at the
point +, y, and # is called the radius of curvature, the point a, @
is called the centre of curvature of the curve at x, y.

(3). In general, when the equation of a curve y = (*) con-
tains a number, 2 + 1, of arbitrary constants, we can determine the
values of these constants so that the curve shall have a contact of the
mth order with a given curve y= (+), at a given point of arbitrary
position and no higher contact. For, if we equate the values of the
function @ and its first 2 derivatives to the corresponding values of
@ and its first 2 derivatives, we shall have z + 1 equations between
the 2 -+ 1 arbitrary constants in 7. These equations serve to deter-
mine the values of these constants which will make y = p(x) have
an zth contact with y= (x) at the point under consideration,
This is the highest contact such a curve y = @ can have with a given
curve y = @ at a point of ordinary position. Then y = y is said to
osculate the curve y = @ at x, y.

At certain singular points an osculating curve can have a contact
of higher order with a given curve than that which it has at a point
of ordinary position—as, for example, the tangent line to a curve at
an inflexion.

tor. Construction of the Circle of Curvature.—Since Dy is the
same, at the point of contact, for the circle and the curve, they have
a common tangent and normal there; also, the centre of curvature
is on the normal to the curve. They have the same convexity or
concavity at the point of contact. The radius of curvature, involv-
ing the radical sign, is ambiguous; we remove the ambiguity by
taking & as positive when y” is positive, or when the curve is convex;
and negative when y’’ is negative or the curve is concave. Conse-
quently the value of # is
pelt
ye
The center of curvature is to be constructed by measuring off R

from the point of contact along the normal, upward or downward
according as # or y"’ is + or —.

EXAMPLES.
1. Find the radius of curvature at any point on the parabola x? = 4my.
Here amy = 4, amy" =1, 14+ 7% = 1+y/m;
2(m 3
Vm

2. Find the radius of curvature in the catenary

f= wola+ ee),
ART. 101.] CONTACT AND CURVATURE, £35

x x

Here y=t (a oo y= y/a; oo p=ty/a.

Show that the radius of curvature is equal and opposite to the normal-length.

3. In the cubical parabola 3a2y = x3 we have

ey = 2, aly! = ax, (1 + y'2)2 = (at + xt)2 Jas.
(at + xt)?
p = ap .
2atx

4. Newton’s Rule for the Radius of Curvature. At any point Pon a given
curve draw a circle tangent to the curve and cutting it in a third point Q at dis-
tances # and g from the common normal and tangent respectively.

Let 7 be the radius of the circle. Then, by elementary geometry, the products
of the segments of the secants are equal, and we have

P= 9(27 — 9);
=f .%
or r= 2g + =
When Q(=)/, the circle becomes the circle of curvature at Pand £r = 2.
ee «ia
R= Bigg?

when 4(=)o, g(=)o.
5. If Q, P, Rare three points on any curve, such that Vis the middle point
of the chord QA, and / is the mid-point of the arc QA, show that

oY?
a A2PV ?
when QO(=)P, R(=)P.
EXERCISES.

1. Find the parabola y = 4x? + Bx + C which has the same curvature as a

given curve y = f(x) at a given point x, y.
V= fix) + (XK -— 2) f'(x) + UX — af'"(a).

2. Show that a straight line has contact of second order with a curve at a point
of ordinary inflexion.

3. Show that the radius of curvature isoo at a point of inflexion, and explain
geometrically.

4. Show that the circle of curvature has a contact of third order at a maximum
or a minimum value of A, and therefore does not cross the curve at such a point.

At a max. or min. value of R we have D,X? = 0. Differentiating (8), § 100, and
solving, we find for the curve
ayy"
apy?

Computing Dy, for the circle, from (5) and (6), we find it has the same value.

5. Show from (5), § 100, that the normal passes through the center of curva-
ture.

6. Find the radius of curvature for the ellipse

 

x y
2 Ga
(a? sin? p+ 8? cos? p)? _ PNG ony
Sa) = ae 3 a6?,

@ being the eccentric angle.
136 APPLICATIONS TO GEOMETRY. (Cu. XIL

7. 23 + y = a is satisfied by x= a cos’ g, 7 =asin'@. Show that
Rea=— 3(axy)*.
a
8. Show that the radius of curvature of e+ = sec (x/a) is
R = asec (x/a),
9. The coordinates of a point on a curve are
x = csin 241 + cos 2¢), y = € cos 2¢(1 — cos 2¢);

show that R = 4ce cos 34.

10. Find 2& for «8 = ay’.

11. Show that, when y = sin x isa maximum, R= 1.

12. Find the center and radius of curvature of xy = a?.

a= (34? + y7)/2x, B= + 3)/z, R= + 9)8/202.
13, Show that if a variable normal converges to a fixed normal as a limit, their

intersection converges to the center of curvature as a limit.
The equations to the normals at ,, 9, and «, y are

ay, dy
— — = SS Voc — = =0.
(Y Fiat * 4, =9, ( Na+ -#x=0
The ordinate of their intersection is
ay dy
i, ee
i dy ay, 4
dx dz,

which takes the illusory form 0/o for x, = x.
When evaluated in the usual way, we have, when x,(=)z,

I y"?
Ysy + tH ’
which is the ordinate of the center of curvature.
Substitution of Y — y in the equation of the normal gives Y as the abscissa of

the center of curvature.
14. Find the radius of curvature at the origin for
247 + 32y — 4? + 8 — Gy =O.
Using Newton’s method,
a 3
R ons Bs => a

15. Find the radius of curvature at the maximum ordinate of VY = eax?,
What is the order of contact of the circle of curvature ?
16. If /(, 6) = 0 is the polar equation to any curve, show that at any point
p, @ the radius of curvature is

 

(p? + pF
p? + 2p — pp"”
where for brevity we write p’ = Dep, p” = Djo.

This follows immediately from substituting for Dy and Dy, (1) and (2), & 56,
in (8), § 100. .

17. Show that if pw =1, w' = Dou, uw” = Dju, the value of the radius in
Ex 16 becomes

(+ ul)!
~ (ae + ul’) ”
ART. IOI] CONTACT AND CURVATURE. 137

18. Since at a point of inflexion y’” = 0, we have there R = «. Therefore the
inflexion condition for a polar curve is, as found before, # + 2” = 0.

19. If p= a0, show that RF = a(t + 6%)3/(2 + 6%).

20.1f p= a9 then R= [t+ (log apy’.

21. If p = 29 — 11 cos 20, Roo at cos2) = +.

22. Show that R= 4as, for p—=asin 46, at the origin.

23. Find the radius of curvature for the hyperbola
xa? — 3/0 = I.
24, Find the radius of curvature of:
The circle o = a@ sin 6; the lemniscate p? = a? cos 26;

spiral o = ¢78; the trisectrix = 2a cos @ — a.

25. If 2 is the radius of curvature of /(, y) = 0, show that

Rp ee + a
~ adtydx — dy d*x’
regardless of the independent variable.
Differentiate the equation of the circle of curvature,
R= (x — aft (y — by.

0 = (x — ajdx + (y — 8)ay,

0 = ax? + (x — a)d*x + dy? + (y —d)d.
The elimination of « — a and y — & gives the result.

the logarithmic
CHAPTER XIII.
ENVELOPES.

102. If (x, y) = o is the equation of a certain line containing a
constant a, then we can implicitly indicate that the position of this
curve depends on the value of a by including it in the functional
symbol, thus:

I(%, I a) = 0.

If we change a by substituting for it another number a,, we get
another curve,
J(% I) %) = 9,

which will, in general, intersect the first curve.

The arbitrary constant a in /(x,v, a) = 0 is called a parameter.
All the curves obtained by assigning different values to @ are said to
belong to the same family of curves, of which a@ is the variable

parameter. Thus
I\% I, &) =0 ()

is the equation of a family of curves when we regard a as a variable, and
any curve obtained by assigning a particular value to a@ isa particular
member of that family.

Thus, in the figure, let the curves 1, 2, 3, .. . be the particular
curves of the family (1), obtained by assigning to @ the particular
values @,, a, .. . taken in order.

 

 

Q¢d 2 458 73
Fic. 23.

Two curves of this family are said to be consecutive when they
correspond to consecutive values of a. The sequence of curves corre-
sponding to a,, @,, ..., as drawn in the figure, intersect in points
A, B,G...

138
ART. 103.] ENVELOPES. 139

ILLUSTRATIONS.

The arbitrary constant or parameter being a:

(a). y = mx + a@ is the tamily of parallel straight lines sloped m to the axis
of x. Consecutive members of this family do not intersect in the finite part of the
plane.

(4). y = ax + 6 is the family of straight lines passing through the point 0, d.

(c). x cosa + ysin a = / is the family of straight lines tangent to the circle
xt yt =p

(2). y = ax +6/a is a family of straight lines tangent to a parabola y* = 46x,
and

¥ = ax — 2ba — bai
is the family of normals to the same curve.

(e). (w — a? + (vy — 6 = a? is the family of circles with center u, 6 and
variable radii. The curves of the family do not intersect.

(f). 2 + y* — 2ax 4+ 7? = 0 is the family of circles with radius 7 having
their centers on Ox. Two curves of the family do intersect, provided we take
their centers near enough together.

 

103. The Envelope of a Family of Curves.—lf
I(x, 9, @) = 0 (1)
and Irs a) =o (2)

are two curves of the same family which intersect at a point x, y, let
us seek to determine the limiting position of the point of intersection
x, y when a,(==)a. When a,(=)a all points on curve (2) converge
to corresponding points on (1), and in the limit curve (2) passes over
into curve (1) and they have an infinite number of points in common.
Therefore the attempt to determine the limiting position of the point
x, y of intersection of (1) and (2), by solving (1) and (2) for the
coordinates and then making a,(=)a, leads to indeterminate forms.

We shall proceed to find the limit to which converges the point
x,y of intersection of (1) and (2), %
by finding a third line which also \
passes through their intersection, andy \P
which does not coincide with (1) oe
when a,(=)a. (1) \

Assign to .v and 1’ the numbers @ \
and 4, the coordinates of the inter- \
section of (1) and (2), and let a be (2) (4)
a variable number. Then /(a, 4, a) O|
is a function of the single variable a,
and we have, by the law of the mean,

Ala, b, a) —Ma, 6, a) = (a, — afl 6), (3)
where ps is some number between a, and a.
But, a, 6 being on (1) and (2), we have

J(a, 6, a) =0 and f(a, 4,a)=0.

Therefore
Si; 4, B) =0a (4)

 

8

Fic. 24.
140 APPLICATIONS TO GEOMETRY. (Cu. XII.

For the particular value yu assigned in (3) we have //(4, y, 4) = 0
as the equation to some curve passing through the intersection of (1)
and (2), in virtue of equation (4). We do not know the number yu
in (3) and (4), since all we know about it is that it lies between @
and @,.

But, whatever be the number y satisfying (4), we know that the
curve

A I; BSP (5)
passes through the intersection of (1) and (2). Now, when a,(=)a,

then u(=)a. If, therefore, when a,(=)a, the two curves (1) and
(2) intersect in a point which converges to a fixed point as a limit,

then (5) becomes
Jal 3, a) = 0, (6)

the equation to a curve which passes through the limit of the inter-

section of (1) and (2) as (2) converges to (1). Moreover, (6), being

a curve distinct from (1), has in general a definite intersection with (1).
If, between the equations

I%, ¥, &) =, (1)
Sa(%, ¥) @) = 0, (6)

the variable parameter @ be eliminated, we obtain the locus
E(x, vy) =0 (7)

of all points in which the consecutive curves of the family /(x, y, a)
= 0 intersect as @ varies continuously.
The curve (7) is called the envelope * of the family (1).

ILLUSTRATION OF THE ENVELOPE.

As the parameter @ varies continuously, the curve /(x, 7, @) = 0 sweeps over
or generates a certain portion of the surface of the plane «Oy, and leaves unswept
a certain portion. The envelope may be regarded as the line which is the bound-
ary between these two portions of the plane «Oy.

104. The envelope, “(x, vy) = 0, is tangent to each member of
the family (x,y, a) = 0 which it envelops.

We are not prepared to give a rigorous proof of this statement
now. This provf requires a knowledge of functions of several
variables. We can, however, give a geometrical picture which will
illustrate the general truth of the statement. For this proof see

227.
Let (A), (2), (C) be three contiguous curves of the family, (4)

 

* Strictly speaking, the equation of the envelope is the equation gotten by
equating to o that factor of Z(x, vy) which occurs only once in Z(x, 9). See
Chapter XXXIX.
ART. 105] ENVELOPES. 141

and (C’) intersecting the fixed curve (#) in points P and @Q re-
spectively. When (A) and (C) converge to coincidence with (2),

g.

 

a4 @ ©
Fic. 25.

the points P and Q converge to each other and to two coincident
points on the envelope. ‘The straight line PQ converges to a common
tangent to (&) and the envelope.

EXAMPLES.
The variable parameter being a, find the envelopes of the following curve
families:
1. xcosa+ysina—pro=f(x,y, a).
J, = —xsina+ycosa@. Square andadd. Hence
2+ 7? p?, a circle with radius 2.
2. Envelope the family f= y —ax — b/a=0.

fo=—a4t bar. 1. a = Yb/x. Hence y® = 4bx.
3. Envelope the family / = y — ax + 2ba@ +4 ba’3.
Jez — x +264 360% .. a? = (x — 26)/36. Hence

2796 = a(x — 265%.
4. Find the envelope of (« cos a)/a + (y sin a)/d = 1.
. Find the envelopes of y = ax + yaa? + B [x2/a? + 77/2 = 1.]
6. Envelope the family 2? + y? — 2ax =7*.

ao

105. Envelopes when there are Two Connected Parameters.

Let (x, ¥, a, B) = 0 (x)
be the equation to a curve, involving two arbitrary parameters a and
f which are related by the condition

(a, B) =o. (2)

I. When we can solve (2) with respect to a or # and substitute

in (1), we reduce that equation to that ofa family with one parameter.
The envelope is then found as before.
142 APPLICATIONS TO GEOMETRY. (Cu. XIII.

EXAMPLE.
Find the envelope of the ellipses
x x
a + B =, (1)

when @i+ fac.
Wehave 6=c—a. Therefore

 

x y i
a Gap
Differentiating with respect to a, and
solving for a,

 

 

 

 

cx! of
a= and = -— ;
a+ yA zi yf
which substituted in (1) give
xt + yi =e

Fic. 26.

II. Otherwise, when it is inconvenient to solve (2), it is generally
simpler to proceed as follows :

Let x, y be constant, and differentiate (1) and (2) with respect to
any one of the parameters, say 6. Eliminate a, 6 and a’ = da/df,
between the four equations.

P+, a, B) = 0, (z)
Pa(x, ¥, & 2) = 0, (2)
p(a, B) = 0, (3)

ta(a, f) = o. (4)

The result is the envelope Z£(x, y) = 0.

For example, take the same question proposed in I.
We have for (1), (2), (3), (4),

x J
we + B =I, (1)
ey
aS + B =090, : (2)
a + p =F (3)
a+izo. (4)
The elimination gives the same result as before.
EXERCISES.
1. Find the envelope of a straight line of given constant length, whose ends
move on fixed rectangular axes. [x3 +73 = |
2. Find the envelope of the ellipses
x? yt
at ga"

when the area is constant. [2xy = ¢.]
ART. 105.] ENVELOPES. 143

3. Find the envelope of a straight line when the sum of its intercepts is con-

stant. [2 + yi = ¢. ]
4. One angle of a triangle is fixed; find the envelope of the opposite side when
the area is given. [Hyperbola. ]

5. Find the envelope of x/a + y/8 = 1 when a+ 6" = cm,

m m m
mt fe ym aa
6. Show that the envelope of 2/7 +. y/m = 1, wherg Z/a + m/b =1 is the
parabola (x/a) + (7/8) =i;

7. Froma point Pon the hypothenuse of a right-angled triangle, perpendiculars
PM, PN are drawn to the sides; find the envelope of the line AZ.

8. Find the envelope of the circles on the central radii of an ellipse as diameters.
9. Find the envelope of y = 2ax + a’, [16y3 + 2744 = 0.]
10. Find the envelope of the parabola y? = a(x — a). [4y? = x?.]

11. Find the envelope of a series of circles whose centers are on Ox and radii
proportional to their distances from 0.

12. The envelope of the lines « cos 3a@ + 4 sin 3a@ = a(cos 2a)iis the
lemniscate (x? + y?)? = a(x? — 4).

13, Find the envelope of the circles whose diameters are the double ordinates

of the parabola y? = 4ax. [o? = 4a(a + x).J
14. Find the envelope of the circles passing through the origin, whose centers
are on 3” = 4ax. [(x + 2a)? + 8 = 0.]

15. Find the envelope of x*/a? + 9?/# = 1, when a? + #? = 2.
[(7 +97? = #.]

16. Circles through O with centers on «x? — y? = a® are enveloped by the
lemniscate (x? + y?)? = 4a°(«? — y?).

17. Show that the envelope of

La? + 2Ma-+-N=0,
es L, M, N are functions of x and y, and @ is a variable parameter, is
= 7.

18. In Ex. 17 show that if Z, JZ, Mare linear functions of x and 4’, the envelope
is a conic tangent to Z = 0, VW = oand having M/ = 0 for chord of contact.
Differentiate ZV — 47? = o with respect to x,

L’N+ ML = 2MaM'.
At the intersection of Z = oand JZ = owe have LZ’ = 0; and since there
N #0, wehave Z’=0,. The Dyy from this is the slope of the tangent to the

envelope. Hence ZL’ = ois the tangent at the intersection of Z = 7 = 0 to the
envelope, etc.
CHAPTER XIV.

INVOLUTE AND EVOLUTE.

 

106. Definition.—When the point of contact, P, of the circle of
curvature of a given curve moves along the curve, the center of curva-
ture, C, describes a curve called the evolu/e of the given curve.

The evolute of a given curve is the locus of its center of curva-
ture. The given curve is called an zvolule of the evolute.

107. There are two common methods of finding the evolute of a
given curve.
I. If d(x, _v) = 0 is the equation of the given curve, and a, ( are
the coordinates of the center of curvature, then we have, § 100, (2),
aay ae

a-en TT, poya ite. (x)

If we eliminate + and _y from these two equations and the equation
to the curve, A(x, vy) = 0, we leave a and £# tied up with constants
in the equation to the evolute.

Eliminations are, in general, difficult and no general rule can be
given for effecting them. Another method of finding the evolute will
be given in II, which frequently simplifies the problem.

EXAMPLES.

|. Find the evolute of the parabola y? = 4px.
Wehave y= pes y= qin 3. Hence
a—x=Ax+p), Boy = —2(p ted + pet),
-_ 3

a=2p+ 34, B= —2p ren

Eliminating x, we have for the equation to the evolute (§ 112, Ex. 17, Fig. 44):
ala — 2p) = 27p

2. To find the evolute of the ellipse x?/a? + y*/s2 = 1.
We can differentiate directly, or use the eccentric angle

x=acosg, y= 4sin @, and find

 

y= —Bxfay, xy" = — b/ary,
at — $% a — p
2= — 3, p =- —jr

tice (aac) + (08)% = (a? — 028, (Fig. 43.)
144
ArT. 108.] INVOLUTE AND EVOLUTE. 145

U. The evolute ofa given curve /(x, vy) = ois the envelope of
the nermals to the curve.

The equation to the normal to f= 0 at +, y is

XY —+«+(F—y)y' =0. (1)

But » and »’ are functions of +, from the equation /= 0 to the
curve. Therefore x is a parameter in (1), by varying which we
get the system or family of normals. Hence the required locus is to
be found by differentiating (1) with respect to ., keeping .Y, VY con-
stant. Thus

—14(F—p\y"—st=0. (2)
Eliminating .. between (1) and (2), we have
= ty 7 rity”?
I ee ges and Y—x= —y’ a

in which Y and Fare the coordinates of the center of curvature,
S$ 100, (2). This proves the statement.

EXAMPLES.

1, Find the evolute of the parabola y* = 4px.
The equation to the normal is

 

y = ax — 2pa — pai, (a)
O= a4 — 26 — 3pa’.

_ (x — 2p\3
«=( 3P ie

which substituted in (1) gives as before in I, 4(x — 29)8 = 2797’.
2. Find the evolute of the ellipse x?/a? + y?/? = 1.
Taking the equation to the normal
ax sec a — by csca = a* — 6,
ax sec atana-+ by csc a cota =o.

Hence tan @ = — (ay /ax)', which leads to the same result as in I,
(ax) + (dy)8 = (a? — oh
108. The normal to a curve is a tangent to the evolute.
Let (x— a)? +(y—- BY =F (1)
be the equation of the circle of curvature at x, y. Then, letting x, 1
vary on the circle, R remaining constant, we have, on differentiation

with respect to .v,
x—at(y— fp) =0, (2)
rts + (y= flr’ =o. (3)

Now let .v, ” vary along the curve, R being variable. The num-
bers a and # are also functions of x. Differentiate (2), which is the
equation to the normal to the curve at +, 1", with respect to x.

ry? + (vy — Bly" — a’ — ply’ =0, (4)
146 APPLICATIONS TO GEOMETRY. [Cu. XIV.

Subtract (4) from (3).

ap dy _
Eas dx dx
d d.
or ip. = hie
da dy
which proves the statement.
EXERCISES.

1, Find the centre of curvature of y* = a®x.
a + i5yt aty — oy®
- » pat.
~ 6a4y 2a
These equations are the equations of the evolute, a and @ being expressed in
terms of y, a third variable.

 

2. Find the coordinates of the center of curvature of the catenary, Fig. 38,
= a
yah (3 cee)

wax Pra, B= 2.

3. Find the center of curvature and the evolute of the hypocycloid,

xt + yi = a

aa=axt3ety Baytartyt, (a+ pi + (a — A = 2al,
4. In the equilateral hyperbola 2xy = a,

eis ee potas

a- 3
a?

(a + fp) — (a — f)§ = 204.
5. In the parabola a +7 = a’, a+ fP=3(e+y).
CHAPTER XV.
EXAMPLES OF CURVE TRACING,

10g. Until functions of two variables have been studied we are
not in position to consider the general problem of curve tracing in
the most effective manner. Nevertheless it will be advantageous to
apply the properties of curves which have been developed for func-
tions of one variable to finding the forms of a few simple curves,
whose figures will be useful in the sequel, before we study functions
of more than one variable.

110. Principal Elements of a Curve at a Point.—We collect
here for handy reference the principal elements of a curve at a point,
as deduced in the preceding pages. The notations are the same as
there used.

I. Rectangular Coordinates. D,y =’, Diy =v".
1. Equation of the tangent:
(F—9) = (X— xp",
2. Equation of the normal:
(Y —y)y' = — (X—2).
3. Subtangent and subnormal:
S,=1', S, =p’.
4. Tangent-length and normal-length:
ee ee ee
5. Tangent intercepts on the axes:
Lae, Faso.
6. Perpendicular from origin on the tangent:

Jay
i=
Vip y
7. Radius of curvature:
I 218
gaa
ve
8. Coordinates of center of curvature:
t 2 I 72
waaay ty, Bayt te.

147
148 APPLICATIONS TO GEOMETRY. (CH. XV.

Il. Polar Coordinates. Dp =p’, Dep=p". up=t.
1. Angle between the tangent and radius vector:

tan > = £.,
2. Angle between the tangent and the initial line:
tan dg = ee
3. Subtangent and subnormal:
s=$= oe S.= p= %.

4. Tangent-length and normal-length:
ie av p+ p?, n= Vp? + p?

5. Perpendicular from the origin on the tangent:
p I i ‘s
= ——_, =z — 4 ae
Vere FON
6. Radius of curvature:
pn. e+ 0
~~ ge 2p'* — pe’
_ (e+ us
~ Bia + wu’)
111. Explicit One-valued Functions.—If the equation to a

curve can be solved with respect to the ordinate or the abscissa so as
to give

 

 

y= P(x) or x= HV)
as its equation, in which either @(x) or ¢:(y) is a one-valued func-
tion, or if more than one-valued the branches can be separated, we
have the simplest class of curves for tracing.

Given any value of the variable, we can compute the value of the
function. We thus obtain the coordinates of a point on the curve.
By finding the first and second derivatives, y’, y’’, we can compute
all the elements of the curve at this point. y’ gives the direction
and y”’ the curvature at the point.

A regular method of procedure for tracing a curve Is:

1, Examine the equation for symmetry.

If the equation is unchanged when the sign of y is changed, the
curve is symmetrical with respect to Ox.

If the equation is unchanged when the sign of x is changed, the
curve is symmetrical with respect to Oy.

If the equation is unchanged when the signs of x and y are
changed, the curve is symmetrical with respect to the origin which
is a center of the curve.
ART. 111] EXAMPLES OF CURVE TRACING. 149

If the equation is unchanged when .v and » are interchanged, the
curve is symmetrical with respect to the line y = -.

if the equation is unchanged when x and y are interchanged and
the signs of both x and y changed, the curve is symmetrical with
respect to x+y =0,

2, Examine for important points.

These are: the origin, the points where the curve cuts the axes,
maximum and minimum points, and points of inflexion.

If += 0, y = 0 satisfy the equation, the curve passes through
the origin. Put + =o and solve for y to find the intercepts on Oy;
put = o and solve for x to find the intercepts on Ox.

Find the maximum and minimum and inflexion points by the
regular methods of the text.

3. Determine the asymptotes, if any.

4. Compute a sufficient number of points on the curve to give a
fair idea of the locus, and sketch the curve through the points.

(In the following examples all details, omitted in the hints, must be supplied.)

EXAMPLES.

1, Trace the common parabola y = x*. The curve is symmetrical with respect
to Oy. It passes through O and cuts neither axis else-
where. Since y’ = 2x is 0 at O, Ox is tangent.

Also, y' is positive as x continually increases from o, \, y
and y, the ordinate, continually increases. Since
y”’ = 2 is always +, the curve is everywhere convex.
Investigation shows that the curve has no asymptote.
The form of the curve is as in the figure. (Fig. 27.)
2. Trace y = 2x7 — 3x4 4.
Y= 4e—3, "= 4.

The curve is always convex. It has a minimum| O
ordinate, y = 23, atx = 3. Its slope + according as
«< =, It cuts Oy at y = + 4, and neither axis else-
where. It is symmetrical with respect to the line x = 8. The curve is the com-
mon parabola. (Fig. 28.)

Fic. 27.

 

 

ty
ee

=,

 

 

FIG. 29.
3. Trace x = 3 — 2y — 3%. Here x is a one-valued function of y.
Dyx=-2-%, Djx=—6. x is a maximum at y = — 4, when x = 3t.
The curve cuts Ox at y = 0, x = 3, and Oyat y= + 0.78, y = — 1.44. It is

everywhere concave to Oy. x continually diminishes from its maximum value,
150 APPLICATIONS TO GEOMETRY. [Cu. XV.

and the curve has no asymptote at a finite distance. It is as before the common
parabola. (Fig. 29.)

y 4. Trace the cubical parabola y = x3. Here
y’ = 3x, y= 6x. The curve passes through O. It
is symmetrical with respect to O, since the equation is
unchanged when the signs of x and y are changed, The
ordinate is + when x is +, and y is — when « is —.
The curve lies in’ the first and third quadrants. In the
gw first quadrant it is everywhere convex, in the third every-
O where concave to Ox. It changes its curvature at the
origin where there is concavo-cunvex inflexion. There
are no asymptotes and the absolute value of y is oo when
that of x isoo. (Fig. 30.) (Read foot-note, p. 164.)
5. Trace the semi-cubical parabola y? = x. Ox is
an axis of symmetry. The origin is on the curve. When
x is —, y is imaginary and the curve does not exist in the
Fic. 30. plane to the left of Oy. x= yf is a one-valued function
ofy. Dyx = By —3 shows the slope oo at O with respect to Oy, and this slope is
+ for y + respectively, Djx = — 2y—3 is always negative, or the curve is con-

cave to Oy. There are no asymptotes and x, y become o together. (Fig 31.)

Fic. 31. FIG. 32.

6. Trace the logarithmic curve y = log x. We adopt the convention that the
logarithm of a negative number is imaginary. Then the curve does not exist as a
continuous function to the left of Oy. The ordinate is negative and infinite for
x = 0, positive and infinite for x = + «, and iso when x = I where it cuts the
axis Ox. The derivative y’ = 1/x is infinite for x = 0, which line is an asymp-
tote. y’is always positive and decreases as x increases. The ordinate continually

increases. y’’ = — x? is always —, hence
the curve is everywhere concave and as in y
the figure. (Fig. 32.)
7. Trace the exponential curve y = e*.
y is always +, by convention e* is +.
y = + 0 when + = + ©; y(=)o when
x=—o. Also y’ = y"’ = ex. The curve
is always convex and increasing, and since 2
; :
y'(=)o when « = — 0, Ox is an asymptote. O
When x = 0, y = 1, where the curve cuts
Oy. If we agree with some authors that y Fic. 33.

has negative values for + = (22 + 1)/2m, sj
m and 7 being integers, then there will be a corresponding series of points
representing the function lying below Ox on a curve represented by a dotted
ArT. I11.] EXAMPLES OF CURVE TRACING. 151

line symmetrical with that above Ox. The exponential curve, however, is con-
ventionally taken to be the locus of the equation

ae. oe
Pee Ae ay ay ee

The curve y = e* is identically the same as the curve in Ex. 6 if we inter-
change x andy. (Fig. 33.)

8. Trace the probability curve y = e-**, The ordinate is always +; it has a
maximum at 0, I; it iso when x is + 0. There is a concavo-convex inflexion at

 

yy
ol =
Fic. 34.
‘
eof 1/V2 and a convexo-concave inflexion at x = — 1/2. Ox is an

asymptote in both directions, and Oy an axis of symmetry. Show that the curve
is as in the figure. (Fig. 34.)

9. Trace the céssoid of Diocles, (22 — y)x? = y®. The curve has Oy as an
axis of symmetry, and passes through O, and cuts the axes nowhere else, Since

y

 

oO
Fic. 35.

 

ys + x2y = 2ax?, y cannot be negative if ais positive. We find that y = 2a is
an asymptote in both directions, since « = + oo when y = 2a.

Again, corresponding to an assigned y, there are only two equal and opposite
values of x. Therefore, for an assigned x, there is only one value of y. Also,

2a —
# 2 = 2
3
is + according as xis +. The curve is decreasing for « negative and increasing

for x positive. To find y’ at the origin, the above value of y’ is indeterminate.
But we have directly from the equation to the curve

i Oei 7 =

x(=)o
which is the slope of the curve at O. Therefore Oy is tangent to the curve. The
origin, like that in Ex. 5, is a singular point which we call a cus. By plotting a
sufficient number of points, we find the curve to have the form as drawn in the

figure. (Fig. 35-)

Yo 2

 
152 APPLICATIONS TO GEOMETRY. (Cu. XV.

10. Trace the witch of Agnesi, y = 8a°/(x? +. 4a”). The ordinate is always
an

O
Fic. 36.

 

+, and has a maximum y = 2a, atx =o. Oy is an axis of symmetry, and Ox
an asymptote. There are inflexions atx = + 2a/ 73. (Fig. 36.)

WW. Trace the cubic a®y = 4x3 — ax? + 203, in which a is positive. There is
a maximum y = 2a atx = 0; anda minimum y = ja, atx=2a. An inflexion
occursat x =a. For x < a the curve is concave, and for x > @ convex. There
are no asymptotes. The curve crosses Ox between x = — @ and x = — 2a,
Also, y = + 0 whenx = +0. (Fig. 37.)

y
y
OD a
O
Fic. 37. Fic. 38.

x x
12. Trace the catenary, y = ha (a + =) , in which a isa positive constant.
The curve is the form of a heavy flexible inextensible chain hung by its ends.
The ordinate y is a minimum and equal toa when x = 0, and is +4 for all values
ofx. The curve is convex everywhere. y—=-+ 0 when x = + 0, and there are
no asymptotes. The slope continually increases with x. (Fig. 38.)
13. Trace the cudical parabola x? = y>( y — a), where a is positive.
Since x= tyVWy—a,

y the point 0, O is on the curve. But xo other point
in the neighborhood of the origin ts on the curve,
since for such values of v, x is imaginary. The
origin is therefore a remarkable point, it is an 7so-
Zated point of the curve, and such points are called
conjugate points. For each value of y greater than
@ there are two equal and opposite values of 2.
The curve is symmetrical with respect to Oy. The
ordinate y is « minimum at x = 0, where the tan-

a gent is horizontal. yy’ = 0 gives inflexions at

 

O
FIG. 39.

convexo-concave and for x — is concavo-convex. There is no asymptote, and
y=+o forx= +0. (Fig. 39.)

4 4. 4
fp gs Mwa e: which for 2 + is
ART, I11.] EXAMPLES OF CURVE TRACING.

153

14. Trace y = (x? — 1)% The curve lies above Ox and has Oy for an axis

¥y

 

Oo

 

Fic. 40.

of symmetry. y has a maximum at «= 0, and minima atx = +1.

inflexions atx = + 1/ 73.
The infinite branches have no asymptote. (Fig. 40.)

15, Trace the curve y -= (: + ) ‘
x

The ordinate has the limit e when y
x= +0. This is the important
limit on which differentiation was
founded. y has the limit 1 when x =o

There are

 

and continually increases with x. For
— 1<.« <o the curve does not exist.
The point 0, 1 is what is called a stop
point, the branch ending abruptly e
there. For « < — 1, and converging

to— 1, y is greater than ¢ and iso. 1
As x decreases to — », the curve 1

 

 

decreases continually and becomes Oo
asymptotic toy =e. (Fig. 41.)

 

Fic. 41.
EXERCISES.
Trace the curves y= sinx, y = cosa.
. Trace y=tanx, y = cot x,

. Trace y= secx, y = csc x.

PON =

. Trace y = vers x= I — cos x,

1

. Trace yme *.

oO oo

. Trace the curves
w= (*—I(y —2)=3, We -Y(e—-2)= 4
7. Trace the curve y(* — I)(x — 2) = ( — 3)(x — 4).
8. Trace y(a? + x*) = aXa — x),
9. Trace xy — a) = @B — xy’,
10. Trace a%x = (x — a).
WW. Trace y? = x(2a — x).
12. Trace (#? + 4) = yx’.
154 APPLICATIONS TO GEOMETRY. [CH. XV.

13. Trace 34(1 — xy = 1 — 5x.
a(a — x)

14. Trace the guadratrix y = x tan z
a

15. Trace the curve y = sin (7 sin x).

16. Trace y = (2x — a) (x — af

112. Implicit Functions.—In general, when the equation to a
curve is given in the implicit form /(x, vy) = 0, and we cannot solve
for either variable, the investigation requires more advanced treatment
than we are prepared to give here. This subject will be taken up
again in BookII. The ordinates to such curves are, in general, several-
valued functions of the variable.

We give here simple examples of important curves. The student
will do well to study the hints given in tracing such curves.

15. Trace the hypocycloid of four cusps,
at + yt = a,

The curve is symmetrical with respect to O, Ox, and Oy. There are two
equal and opposite values of y to each x, and two of x to eachy, for either variable

7
Q

Fic. 42.

 

 

less than a2. The curve does not exist for values of x or y greater than a. We have
in the first quadrant

zh
?

x

 

LS

and the curve is tangent to Or at x = a, and to Oyvatx=o0,y=a. ¥y” being
positive in the first quadrant, the curve is convex at any point on it. The curve
is sometimes called the asteroid. It is the locus of a fixed point on the circumference
of a circle as that circle rolls inside the circumference of another circle whose radius
is four times that of the rolling circle. (Fig. 42.)

16. Trace the evolute of the ellipse

(ax) 4 (dy)t = (a? — oF
in the same way as above. (Fig. 43.)
ART. 112.] EXAMPLES OF CURVE TRACING. 155

Show by inspection that four normals can be drawn to the elli i
inside the evolute. eee

 

 

From what points can 1, 2, or 3normals be drawn? y
y
O ze
Fic. 44.
xz
o
Fic. 43. Fic. 45.

17. Trace the parabola y? = 4x, and its evolute, 4(« — 29)3 = 27Ay%. Show
that the curves are as drawn. Find the angle at which they intersect. Show from
which points in the plane can be drawn I, 2, or 3 normals to the parabola. (Fig. 44.)

18. Trace the curve (y — +’)? = 25,
pS x(a + 2).
There are two branches,
y= 22(1 + 2), yr (1 - 2),
The first continually increases as x increases from 0. The second increases,

attains a maximum, and then descends indefinitely, crossing Ox atx = 1. Both
branches are tangent to Ox at O since
5

y = 2x + bx
iso when x =o. The curve does not exist in the plane to the left of Oy, Ex-

amine for asymptotes. Find the inflexion and the maximumordinate. The origin
is a singular point called a cusp of the second species. (Fig. 45.)

19. Trace in the same way the curve g
xt — 2ax*y — axy + ay =o.

20. Trace the curve y* = (x + 1)2*%.
y isatwo-valued function of x,

YrtECYK AL O
Ox is an axis of symmetry.
The curve passes through the origin in two branches, Fic. 46.

yotaeyYx ti yor Yet

The curve does not exist in the plane to the left of x = — 1. Between — 1
156 APPLICATIONS TO GEOMETRY. (Cu. XV.

and 0 the ordinate is finite, having a maximum and a minimum. We have for the
slopes of the two branches passing through O at x = 0,

fens frre

x(=)o
As x increases positively, y increases without limit in absolute value. Are there
asymptotes? (Fig. 46.)
The point in which two branches of the same curve cross each other, having
two distinct tangents there, is called a zode. In this curve the origin is a node.

21. Trace the curve (dx — cy = (x — a),
y Clearly, x=a, y = ab/e,
is on the curve. But these values make the deriva-

tive y’ indeterminate. Differentiate the equation
twice.

(8 ey" — (bx — gay" — 10(x — aj" =o,
and atthe point x =a, y=ad/c,
x (8 — oP) =0,
gives y’ = 6/c. Since y is imaginary when x < a

4 ——n
Fic. 47. and yore + * V(x — a),

the curve is asin the figure. The pointa, ab/c, isa cusp of the first species. (Fig. 47.)
22. Trace the curve 4y? = 4x3 + 12x? 4 ox.
y

Fic. 48.
23. Trace the lemniscate,
(2 + 92)? = a 2? — y’),
PL Pa= pave —
shows that y cannot be greater than x and only equal tox when they are both o at
the origin. The curve is symmetrical with respect to O, Ox, Oy. Also,

y 2,
«+ 2y=eq)r— (2)

and since y Sx, we have, when x = 0, y = 0,

f: = a (See Ill. (2), § 79.)

which are the slopes of the two branches of the curve passing through the origin.

Again,
pis xa 2(x? +3")
“aE a +
ART. 113.] EXAMPLES OF CURVE TRACING. 157

In the first quadrant y’ is + from . =o to the point determined by
2(2x? + 7”) = a, 4(x? ae yt) Sa
where it changes sign, giving y 4 maximum, and jy’ decreases until y' = o at
= a yp =o,
Being symmetrical with respect to the axes the curve is as in the figure. No

part of the curve exists for x > a, since the equation is of the fourth degree and a
straight line cannot cut the curve in more than four points.

Put y = mx, and plot points on the curve by assigning different values to m.
Thus, in terms of the third variable #, we have
Yi — ne = Yr — mm

rz ae (Fig. 48.)

113. General Considerations in Tracing Algebraic Curves.-—
The equation of any algebraic curve when rationalized is of the form
of a polynomial of the th degree in x and _y. It can always be

written
o=uytuyt...+% =F, eo)

where z, is the constant term (independent of x and y), 4, %,,
etc., are homogeneous functions or polynomials in x, y of respective
degrees 1, 2, etc.

If ~, = 0, the origin is a point on the curve.

(1). To find the tangent at the origin when uw, # o,

When x, = 0, the line y = mx intersects the curve at O.

Substitute mv for y in the equation to the curve. Then, if
u, = px + y, the equation (1) becomes

1
(P+ met T+... =0, (2)
where the terms 7), etc., contain higher powers of x than the first.
Divide the equation (2) by .+, which factor accounts for one 0 root.
Then let x = 0, and (2) becomes

ptm=o, or m= —p/y.

This value of m is the slope of the curve at the origin, since now
the line y = mx cuts the curve in two coincident points at the origin,
and

“u=px+y=o
is the equation of the tangent at the origin.
li #, = 0, #4 = 6, and #, = ra* + ayy +- by
Then, as before, put mx for_y and the equation becomes
(r + sm + tm)? 4 Ti+... =9, (3)
where the terms Z,, etc., contain higher powers of + than 2. —

Divide by 2°, which accounts for two zero roots of (3); in the
result put + = 0.

ot. m+ smtr=0 (4)
is a quadratic giving two values of m, the two slopes of the curve
at O. The equation to the two tangents at O is

“= rxtt sav + Wao.
158 APPLICATIONS TO GEOMETRY. [Cu. XV.

These are real and different, real and coincident, or imaginary,
according as the roots of the quadratic (4) in m are real and unequal,
equal, or imaginary. The origin being a double point called a node,
cusp, or conjugate point accordingly.

In like manner if also ~, = 0, the equation of the three tangents
at Ois

u, =O,
and the origin is a triple point.

Hence, when the origin is on the curve, the homogeneous part
of the equation of lowest degree equated to o is the equation of the
tangents at O.

Further discussion of singular points and method of tracing the
curve at a singular point will be given in Book II.

(2). A straight line cannot meet a curve of the zth degree in more
than z points. For, if we put mx -+ 6 for yin U =o, we have an
equation of the zth degree in ~ for finding the abscissze of the points
of intersection of y= mx +4and V=o0.

If now uw, is the term of lowest degree in U, and we put mx for
yin U, then x” is a factor and represents 7 roots equal too. The
line y = mx cuts the curve V= 0, yr times at the origin, and there-
fore cannot cut it in more than 2 —r other points. This will fre-
quently enable us to construct a curve by points, when otherwise the
computations would be quite difficult.

(3). Singular Points. A point through which two or more
branches of a curve pass is called a simgular point. Illustrations
have been given of nodes, cusps, and conjugate points.

At a singular point on a curve D,y is indeterminate. Points at
which D,y is determinate and unique are called points of ordinary
position, or ordinary points.

To find a singular point on a curve @(x, y) = 0, differentiate
with respect to x. The result will be

M+ MN’ =0, (1)
where Jf and J are functions of x and _y. At a singular pointy’ is
indeterminate and /7=0, V=o0. Any pair of values of x, y satis.
fying the equations

@=0, M=0, N=-=0
is a singular point. If (1) be differentiated again, we have
P+ Qy' + Ry? + Ny" =o.

At the singular point V = o, leaving a quadratic in y’ for deter-
mining the slopes of the curve, if the point is a double point. Ifa
triple point, another differentiation will give a cubic in y’ for deter-
mining the slopes, etc.

If the curve has a singular point whose coordinates are a, 6, and
we transform the origin to the singular point by writing «+ a,
y + # for x and yin the equation to the curve, the construction of
the curve will be simplified as in (1), (2).
ART. 113.] EXAMPLES OF CURVE TRACING. 159

EXAMPLES.
24. Trace the Zemmniscate, Ex. 23.

(7 ++ 9? — a? — y*) = 0.
Here 7, = x? — y? = o is the equation to the tangents at 0, or y = + «x, as
before in Ex. 23.
Put y = mx in the equation and compute a number of points. Clearly m
cannot be greater than I.

25, Trace the foium of Descartes,
2+ y3 — 3axy = 0.
The equation of the tangents at the
origin is 3xy = 0, or x =O, y= 0. We

find that
x+y+a=o0o

is the only asymptote. Put y = mz, then

3am 3am? x
r+»? an +m

x, y are finite for 0o< m< +o. Com- \

pute a number of points corresponding to
assigned values of m. Observe that if we
change # imto I/m, x and y interchange
values. The curve is symmetrical with
respect to the line y = x. In the first Fic. 49.
quadrant there is a loop, the farthest point \
from the origin being x = y= 8a. Determine the maximum values of x and y for
this loop. For negative values of 7 we construct the infinite branches above the
asymptote, since y = mx cuts the curve before it does the asymptote. (Fig. 49.)

y

26. Trace the curve (y — 2)%(a — 2)x = (x — 1){x* — 2x — 3).
Examining for singular points, we find
rafy + 3 x—T
= x(a — 27?) y— 2
Therefore « = 1, y = 2 is a singular point. Transform the origin to this
point by writing x + 1 for x, y + 2 for y.
Then the equation becomes
yet — 1) = 24a? — 4).
Examining for asymptotes, we find the
asymptotes x = +1, y= +4. The
equation to the tangents at O is 7? = 4x’,
oY = + 2x. When y= 0; x = + 2,
41 = 0. ‘he curve is symmetrical with
respect to Ox, Oy, and O. We need there-
fore trace it only in the first quadrant, in
ae order to draw the whole curve.
The line y = mx cuts the curve in points
whose coordinates are

A we

4— mm? 4—

xm = =m =<.
i! ae I—

These increase continually as # increases
from 0 to 1, and the branch approaches the
asymptote as drawn. The coordinates are
imaginary for 1 << m < 2, and when m= 2;

Fic. 50. x=0, y=o0. As'm increases from 2 to

+o, x and y are real and increasing. and

m= © gives x = + I, ¥ = 00, the curve approaches the asymptote as drawn.
The origin is an inflexional node. (Fig. 50.)

 

 

 

 

 
160 APPLICATIONS TO GEOMETRY. [CH. XV.

27. Trace the curve (x + 3? = x(x — 1)(x — 2).

28. Trace the curve ay? = bxt + x’.

29. Trace the dumb-bell aty® = a?xt — x8,

30. Show that 25 + y5 = 5ax*y? has the form given in Fig. 51.

 

Fic. 51. FIG. 52.

31. Trace xt = (x? — yy.

The lowest terms are of third degree. The origin is a triple point. The
tangents there being y = 0, y = + x. Oyis an axis of symmetry. There are
no asymptotes. The line y = mx cuts the curve in one point, besides the origin,
whose coordinates are

x= mI — ml), y = m1 — me).

This shows that there are two loops, in the first and fourth octants, and infinite
branches in the sixth and seventh octants. The curve is a double bow-knot and has
no asymptotes. (Fig. 52.)

 

 

Fic. 53. Fic. 54.
32. Trace the curves
Paar xt, Poa— x8, P(x — a) = (x — 42.
33. Trace the conchoid of Nicomedes,
(2+ 726 — vy? = ey, when b=, <, >a.

34. Trace the curves

y = (x — I(x — 2% — 3), Cx = oP 4 x) xt — yt 2axy? = O.

35. Show that xy? + at = a%(x? — y*) consists of two loops and find the form
of the curve.

36. Show that the scarabeus

4a? + y? + 203)%(2? + 92) = B(x? — y?

has the form given in Fig. 53.
ART. 114.] EXAMPLES OF CURVE TRACING. 161

37. Show that the devz7
yt — x*+ + ay? + bx7° = 0, where 2 = — 24, 6= 25,
has the figure given (Fig. 54).

114. Tracing Polar Curves.—As in Cartesian coordinates, no fixed
rule can be given for tracing these curves. The general directions
are the same as before. The particular points are :

(1). Compute values of p corresponding to assigned values of 0, or
vice versa, according to convenience. Plot a sufficient number of
points to give a fair idea of the general position of the curve.

(2). Determine the asymptotes, by finding values of 6 which make
yo =o for the directions of the asymptotes. Place the asymptote in
position by evaluating the limit of p*?2,6 = — D.6, for the perpen-
dicular distance of the asymptote from the origin, as previously
directed. Examine for asymptotic points and circles.

(3). The direction of a polar curve at any computed point is given
by tan # = p/p’.

(4). Examine for axes or points of symmetry.

(5). Examine for maximumand minimum values of p and for points
of inflexion.

(6). Examine for periodicity.

 

115. Inverse Curves.—If /(p, #) =o is the polar equation to
any curve, then /(p', @) = 0 is the polar equation of the verse
curve.* We have been accustomed to put p~? = z, so that /(u, #) =o
is the equation of the inverse curve.

1. Show that if x, y are the rectangular coordinates of a point ona curve, the
equation to the inverse curve is obtained by substituting

% 2
wae FTF
for x and y in the equation to the given curve.
2. Show that the asymptotes of any curve are the tangents at the origin to
the inverse curve.
3. Show that a straight line inverts into a circle and conversely. Note the case
when it passes through the origin.

4. Show that the inverse of the hyperbola with respect to its centre is the
lemniscate.

EXAMPLES.

38. Trace the spiral of Archimedes, p = a6. The distance from the pole is
proportional to the angle described by the radius vector. tan ~=6. The curve
is tangent to the initial line at O. The intercept PQ between two consecutive
revolutions is constant and equal to 27a. Therefore we need only construct one
turn directly. The dotted line shows the curve for negative values of §, which

 

* More generally two polar curves are the inverses of each other, when for the
same 9 their radii vectores are connected by pp, = 4% & = constant.
162 APPLICATIONS TO GEOMETRY. [Cu. XV.

is the same as the heavy line revolved about a perpendicular to the initial line
through O. (Fig. 55.)

O

Fic. 55. Fic. 56.

6
39. Trace the eguiangular spiral p = e. We can write the equation

6 = blog p,
if we prefer. tana = 4, or the angle between the radius and tangent is constant.
p = @ for 6 = 0, and increases as @ increases. (=)o for? = — 0.
[Ss The pole O is an asymptotic point. (Fig. 56.)
P

40. Trace the hyperbolic or reciprocal spiral
p§ = ua. The pole O is an asymptotic point.
a A line parallel to the initial line at a distance a
above it is an asymptote. For negative values
of 6, rotate the curve through z about @ normal

 

G 0 to OA at O. (Fig. 57.)
41. Trace the /emniscate p? = 2a? cos 26.
Fic. 57.
42. Trace the conchoid p = asec § + 4,
or (7 + ya — a)? = Bx?

When a < 4, there is a loop; when a = 4, a cusp; when a>4, there are two
points of inflexion. (Fig. 58.)

a :
i |

Fig. 58. Fic. 59.

 

 
ART. 115.] EXAMPLES OF CURVE TRACING. 163

43. Trace the cardioid p = a(1+ cos 6). The curve is finite and closed,
symmetrical with respect to Ox. p= 2a, uw, 0, for 6 = 0, $7”, mw, and
diminishes continually as 9 increases from o to 7. Also, tan y = — cot 49. As
6(=)x, o(=)z, or the curve is tangent to Ox at the pole, which point is a cusp.
The rectangular equation is

we Pax = +4 x2 $57, (Fig. 59.)
44. Trace the three-leaved clover p = a cos 39.
45. Trace the curves :

(1). p = asin 26, p = acos 29,
(2). p = asin 36, p = asin 46.
(3). p= asec? iO, p=asec 6,

(4). p = asin 6, p = asin 19,

46. Trace the curve (6? — 1) = af.
47. Trace p=avers@ and p =a(1 — tan 6).
48. Trace the evolutes of y = sin x and y = tan x.

49. The Cyc/oid. The path described by a point on the circumference of a
circle which rolls, without sliding, on a fixed straight line is called the cycloid.

 

 

 

 

YU) ON LL Vv
VA,
L A
Ore M D A .
Fic. 60.

(t). Let the radius of the rolling circle A7PL be a, the point P the generating
point, 47 the point of contact with the fixed straight line Ox which is called the
base. Take J7O equal to the arc A/P; then Ois the position of the generating”
point when in contact with the base. Let O be the origin and w, 3’ the coordinates
af P, £ PCM = 6.

Then we have

x= OM — NM = a(§ —sin6), vy = PN=a(i — cos 6).

The coordinates are then given in terms of the angle @ through which the rolling
circle has turned. OA = 2m< is called the base of one arch of the cycloid. The
highest point Vis called the vertex. Eliminating §, we have the rectangular equation

Ceay
x = a cos
a

 

— y2ay — vy. (Fig. 60.)

cycloid when the vertex is the origin,
the tangent and normal there are the
axes of x and y, we have directly from
the figure 6
x=a)+asin0, y =a—acos 6.
Eliminating 6 for the rectangular P
equation,
ea: ee at M

x = a cost + Y2ay — y. O

a
(Fig. 6r) Fic. 61.

The cycloid is one of the most important curves.

(2). To find the equations to the ‘

 

 

 

 

 
164 APPLICATIONS OF GEOMETRY. (CH. XV.

50. The Zrochoids. When a circle rolls on a fixed straight line, any point
rigidly fixed to the rolling circle traces a curve called a trochoid. The curve is
called the epitrochoid or hypotrochoid according as the tracing point is outside or
inside the rolling circle.

Their equations are determined directly from the figure,

 

 

Fie. 62,

Let CM=a, CP=)7, CP’=7', £ MCP=4.

Then x= ON=a6—psiné, y= PV=a— psin§,
for a point Pon the hypotrochoid PV. Replacing ~ by 7’, the same equations give
the epitrochoid. (Fig. 62.)

51. Epicycloids and hypocycloids.

The curve traced by any point on a circle which rolls on a fixed circle is called
an epicycloid or a hypocycloid, according as

 

 

 

 

 

the circle rolls on the outside or on the inside
y of the fixed circle. (Fig. 63.)
— . Let Obe the center of the fixed circle of
C radius a, and C the center of the rolling circle
b of radius 4, and Pthe tracing point. Then
g with the notations as figured, we have
a Phix arc AM = arc PM, or af = bg. Hence
x=ON= OL ~— WML,
O 4] wn = (a+ 6) cos 6 — bcos (6+ @),
Fic. 63 = (a + 5) cos 6 — bcos +“;
y = PN=CL— CK = (a+ 4) sin @ — dsin(6 + ¢),
éb
= (a + 4) sing — bsin 2+ “p,

b
for the coordinates of the epicycloid. For the hypocycloid change the sign of 4.

 

In this book convexity or concavity of a curve at a point is fixed by the sign of
the second derivative of the ordinate representing the function. Diy = + or
Djx = + means convexity with respect to O, or O, respectively. This is the
equivalent of viewing the curve trom the end of the ordinate at — oo, instead of
from the foot of the ordinate as is sometimes done.
PART III.
PRINCIPLES OF THE INTEGRAL CALCULUS.

CHAPTER XVI.
ON THE INTEGRAL OF A FUNCTION,

116, Definition.—The product of a difference of the variable
x, — V, into the value of the function /(x) taken anywhere in the
interval (v,, 2) is called an element.

In symbols, if z is either of the numbers x, or x,, or any assigned
number between ., and .,, the product

(x, 4 x) (2)

is the element corresponding to the interval (x,, ~,).

GEOMETRICAL ILLUSTRATION.

If_y = /(x) is represented by the curve 42 in any interval (2, 4),
and x,, v, are any two values of + in
(a, 4), then the element corresponding y P B
to (x,, «,) is represented by the area M,Z -
of any rectangle «,JZJfy,, whose Me
base is the interval +, — x,, and alti- A
tude is the ordinate zZ to any point
on the curve segment P,P.

117. Definition.—The iegral of
a function /() corresponding to an | a @,2 @ ob
assigned interval (a, 6) of the variable Fie. 64,
is defined as follows:

Divide (a, 6) into partial or sub-intervals (a, x,), (*,, %,),

2) (Xy2> Yu), (%4., 4), by interpolating between @ and é the
numbers 1, ..., ¥, , taken in order from @to 4, And for con-
tinuity of expression let x, = a, x, = 4.

The ztegral of a function is the “mz of the sum of the elements
corresponding to’the z sub-intervals, when the zumder of these sub-
intervals is increased indefinitely and at the same time each sub-
interval converges to zero.

 

 

 

 

 

 

165
166 PRINCIPLES OF THE INTEGRAL CALCULUS, [Ci XVI.
In symbols, we have for the integral of /(x) corresponding to the
interval (a, 4),

tr(=)rr—y r=n

> (, — %)/e).

n=o r=
In which z, is either x,, +,_, or some number between x, and x,_,,
or as we say, briefly, some number ofthe interval (v,_,, x,). At
the same time that 7 — co we must have x, — +,_,(=)o.

GEOMETRICAL ILLUSTRATION.

If y = /(*) is represented by a continuous and one-valued ordinate to a curve,
then the integral of /(x) for the interval (a, 4) is represented by the area of the
surface bounded by the curve, the x-axis, and the ordinates at @ and 4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y W, 8
Biss N, mu, P
7 P
N;, B WA
A
M, R e
O a fy, He2 ws ad
Fic. 65.

For, any elementary area, such as (x3 — x,)/(23), lies between the areas of the
rectangles +,2,J/,x, and x,V,/,x, constructed on the subinterval (x,, +,), or is
equal to one of them, according as 2, = +,, z,= 43. Also, the corresponding area
x,P,P,x, bounded by the curve P,P,;, Ox, and the ordinates at x, and x, lies
between the areas of the same rectangles, in virtue of the continuity of /(x), when
ty — x, is made sufficiently small.

Hence the sum of the integral elements and the fixed area of the curve lie
between the sum of the rectangular areas

Max, + Nyy be. + Md (1)
Mya + Myx, +... 4 Myra. (2)

Let RQ be not greater than the greatest of the subintervals into which (a, 4) is
divided. The difference between the areas (1) and (2) is not greater than the area
of the rectangle BDQA, whose base is RQ and whose altitude BA is equal to the
difference (4) — fia) and to the sum of the altitudes of V,JZ,, NV,4Z,,..., MM,
This rectangle 2Q has the limit 0, since each subinterval has the limit 0; and so
also has RQ, while its altitude is finite and constant, or does not change with 7.

Consequently the areas (1) and (2) converge to the constant area of the curve
which lies between them, and so also must the area represented by the sum of the
elements of the integral.

Hence the integral of /(r) for (a, 4) is equal to the area of the curve, as enun-
ciated.

and
ArT. 118.] ON THE INTEGRAL OF A FUNCTION. 167

118. Evaluation of the Integral of a Function.*—In order that
a function shall admit of the limit which we call the integral for a
given interval, the function must, in general, be finite and continuous
throughout the interval.

Should the function be finite and continuous everywhere in the
interval (a, 4) except at certain isolated values of the variable, at
which singular points it is discontinuous, either infinite or indeter-
minately finite, then special investigation is necessary for such singular
values, and we omit the consideration of them,

We shall assume that the functions considered are uniform,
finite, and continuous throughout the interval, unless specially
mentioned otherwise.

The process of evaluating the mt defined as the integral, in
§ 117, is called w/egratron.

In evaluating the limit

£ 2x, — % Ie), = Hy =).

we are said to integrate the function /from a= x,tod=.x,. The
numbers a and @ are called the boundaries or limits of the integration
or integral. The lesser of the numbers a and 4 is called the iz/erzor,
the greater the sugerzor, limit of the integration. +
In the differentiation of the elementary functions
x*, a, log x, sin x,

and like functions of them and their finite algebraic combinations,
we have seen that the derivative could always be evaluated in terms
of these same functions. Not so, however, is the case in evaluating
the integrals of these functions. The integral cannot be always
expressed in terms of these same functions, and when this is the case,
the integral itself is a new function in analysis which takes us beyond
the range of the elementary functions such as we have defined them
to be.

We shall be interested, in this book, directly with only those
functions whose integrals can be evaluated in terms of the elementary
functions.

It can be stated in the beginning that there is no regular and
systematic law known by which the integral of a given function can
be determined as a function of its limits in general,

The process of integration is therefore a tentative one, dependent
on special artifices.

 

* For Riemann’s Theorem: A one-valued and continuous function in a given
interval is always integrable in that interval; see Appendix, Note 9. .

+The word Zimit as here employed does not in any sense have the technical
meaning “mit of a variable as heretofore defined. It is an unfortunate use of the
word, retained out of respect for ancient custom. It is contrary to the spirit of
mathematical language to use the same word with different meanings, or in fact to
use two words which have the same meaning.
168 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVI.

The systematizing of the artifices of integration is the object of
this part of the text.

119. Primitive and Derivative.—I{ we have two functions (+)
and /(x), so related that /(v) is the derivative of F(x), then F(x) is
called a primitive of f(x). The indefinite article is used and /(+x) is
called a primitive of /(x), because if

DF(x) = (x),

D(F(x) + Cl =f),
where C is any assigned constant.
Any one of the functions
F(x) + C,
obtained by assigning the constant C, is a primitive of /(x). The

primitive of /(x) is the family of functions containing the arbitrary
parameter C.

then also we have

GEOMETRICAL ILLUSTRATION.

¥ = Me) + CQ, (2)
y= AD+G, (2)
are so related that at any point x their tangents at P, and P, are parallel, and each
curve has for the same abscissa the same slope. Their ordinates differ by a con-

The two curves

 

m0)
: A (2)
P,
0 0 i

Fic. 66.

stant. Each curve represents a primitive of f(x). Any particular primitive is
determined when we know or assign any point through which the curve must pass.

120. A General Theorem on Integration.—If a primitive of a
given function can be found, then the integral of the given function
from a to X can always be evaluated. The given function being
continuous in (a, X).

Let (x) be a continuous function in (a, X), and let F(x) bea
primitive of /(.x).

Let +, =a, x, =X. Interpolate the numbers x,,..., 4.4
between @ and Y in the interval (a, Y), in order from a to .X, sub-
dividing the interval (2, X) into the subintervals

(%5 x), (85 %,); ae = y (4535 Xy-1)s Css X,,).

 
ART. 120.] ON THE INTEGRAL OF A FUNCTION. 169

We have the sum
2 (%, = a=) = X — a,
rat

whatever be 2.
Since /(v) is the derivative of F(x),

F(x) = f(a).
By the law of the mean value applied to each of the subintervals,
we have the z equations

F(X) | Xn )= a (4, Ts Naa )NEn);
F441) — F(%y2) = es = SEs)

Fe,) aa F(x,) = —e, (a, aaa «,) AB),
FY) — (4) = (%, = H)AE)-
Adding, we have

FUX) — Fla) = 3 (x, — = AE), (1)

in which &, is some particular number 7 the interval (v,, %,_,).
The sum on the right, in the above equation, is equal to the
member on the left. The left side of the equation is independent
of x. The equation is true whatever be the integer 7, and when
n=co. The sum onthe right remains constant as we increase 2,
and being finite when 2 = o,
r=n
F(X) 6 (a) — b = (x, = +, V(S,)-
N=O rat
Now let z, be any number whatever of the subinterval (v,, -,_,),
for each subinterval. Then

NE) =) +

where €,(=)0, when +, — x,_,(=)o, by reason of the continuity of

SJ(*)-
Therefore
Sa, — aE) = 3, — 2) 2) + €],

n n
= 2 (x, ee X12) + 2 (x, a Xx) E+
Let e¢ be the greatest, in absolute value, of the numbers
Bj os en Sys Then
a nn
Sie ~ 26 |2)e2u,— X,-1) = e(¥ — a),
the limit of which is o, when 7 = oo ; provided each subinterval
x, — ¥,_,(=)o
whenz=o.
170 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVI.

Therefore, when x = oo, and at the same time each subinterval
x, —~ «,_,(=)o, we have

te 2 jes See Seye «

n=O Y=1 H=O r=.
z, being any number of the interval (x,, x,_,); that is, z, may be
X,, OF X or any number we choose to assign between x, and
x,

r-1?

hie member on the right in (2) is, by definition, the integral of
/(x) from a to X, and we therefore have for that integral

n
» aU — +,_,)f(2,) = F(X) — F(a),
which is evaluated whenever we know a primitive of /(x), and can
calculate its values at a@ and X.

Observe that it is not necessary that we should know the values
of the primitive anywhere except at the limits a@and Y. The integral
is therefore a function of its limits.

121. In the preceding articles of this chapter we have fixed no
law by which the values x,, ..., 1,-; were interpolated between a
and .Y. The integral has been defined and evaluated for any distri-
bution of these numbers whatever, subject to the sole condition that
the intervals between the consecutive numbers must converge to 0
at the same time that the number of the subintervals becomes
indefinitely great.

Since it makes no difference how we subdivide the interval of
integration, we shall generally in the future subdivide the interval of
integration into 2 equal parts, so that

x, — 4, = dx, =h = (X —a)/n,

and we shall take the value of the function to be integrated at x,_,,
the lower end of each subinterval.
The integral of /(x) from a to .X is then

Ax(=)o #

f(X) — F(a) bs 2I(%,) Am.

But observe that
J(x,)4x = F'(x,)dx = dF(x,).

Hence the integral of /(x) from + = a to « = N is the limit of the
sum of the differentials of the primitive function.

122. Leibnitz’s Notation.—The notation previously used to
represent the integral, while valuable as indicative of the operation
ab initio performed in evaluating this limit, is cumbersome, and when
once clearly assimilated it can be replaced by a more convenient and
abbreviated symbolism. We replace the limit-sum symbol by a
ART. 123.] ON THE INTEGRAL OF A FUNCTION. 171

I
more compact and serviceable symbol designed by Leibnitz. Thus
in future we shall write in the suggestive symbolism :

[Avex = f Sftu\dy,

as the symbol for the integral of /(x) from a to X.
The characteristic symbol f is a modification of the letter S, the

initial of swm, and is taken to mean /imit-sum, or f = £2. The

ea J(x) dx represents the 4#e of the elements whose sum is
taken.
If #(x) is a primitive of (x), then

F(X) — F(a) = [re dx,
= [Fo dx,

This, then, is the final reduction of the integral; and whenever the
expression to be integrated, /(x) dx, can be reduced to the differen-
tial d# (x), then 7(x) is recognized as a primitive of /(x) and the
integral can be evaluated when the limits are known.

123. Observations on the Integral. —Differentiation was founded
on the exceptional case in the theorems in limits, wherein we sought
the limit of the quotient of two variables when each converged to o.

_ We found that the theorem stating: the limit of the quotient is
equal to the quotient of the limits, did not hold, § 15, V (foot-
note) in the case when the limit of the numerator and of the denomi-
nator was o, but that the limit sought or defined was the limit of
the guofient of the variables.

Integration is founded on another exceptional case in the theorems
in limits. Here we seek the limit of the sum of a number of terms
when the number of terms increases indefinitely and also each term
diminishes indefinitely. The limit we seek is the “mit of the sum.
The theorem which states: the limit of the sum of a number of
variables is equal to the sum of their limits, was only enunciated and
proved for a finite number of variables, and does not necessarily hold
when that number is infinite. The sum of. the limits of an infinite
number of variables, each having the limit 0, is o and nothing else.

The important point in the definition of the integral which makes
it a matter of indifference where in the subinterval of the integral
element we take the value of the function, is an example of an
important general theorem in summation, which can be stated thus:
172 PRINCIPLES OF THE INTEGRAL CALCULUS.  [CH. XVI.

Lemma. If the sum of ” variablesw,,..., u, has a determinate
limit A when each converges to o for 7 = o, so that

Blyton te) =A,
and there be any other ~ variables 7,,. .. , ¥,, such that each con-
verges to o for 7 = o, and at the same time

“ uy i
—=H I, . + ey —=iI,
yy, On

Lt... +) =A.

For, whatever be 7,

then also

v,
— = 1+ €,,

u,
where €,(=)o, when 2 = 0. Also,
L2v, = £L2(u,+ ¢4,) = LBu,+ feu,
If € is the greatest absolute value of €,,..., €,, then
Z€eu,| S |esu, = €A,
the limit of which is 0, and, § 15, III,
Lav, = £2u, = A.
This principle is of far-reaching importance in integration, and

will be frequently illustrated and applied in the applications of the
Calculus.

GEOMETRICAL ILLUSTRATION.

Let 1 = /(x) be represented by a curve, and let #’(x) = f(x). Then /(x) is
the slope of the curve or of its tangent at x,
We have SQ equal to

 

 

 

 

 

 

 

 

 

 

 

 

 

L,
Ag MX) — Ma) = MP + MP, +
T, B -. +48, (1)
y f, < Ee = SAF.
9 Wa Also, the sum of the differentials of 7
a FE 7 Bt ey ayy w aaa) IS
9 Mz Q Sah = MT, 44,7, 4 ..-+MnT (2)
The difference between this sum and
that in (1) is
DdF—-ZAFHP,T. APT, +... +87),
O a @y hy ws X a But we know that the limit of
Fic. 6 a eete
1G. 67. dF = MT,

is 1 when z = 0 and 4x(=)o. Hence, by the lemma above, we have
F(X) — F(a) = fA 4F = L2dF,
= £2 F'(«) dx,
= fas (x) ax,
which is another illustration of the integral.
ART. 124.] ON THE INTEGRAL OF A FUNCTION. 173

124. The Indefinite Integral.—When we know a primitive of a
given function we can integrate that function for given limits. It is
therefore customary to call a primitive of a given function the
indefinite integral of that function.

Indefinite integration is therefore a process by which we find a
primitive of a given function. A primitive /(x) of a given function
J(x) is called the mdefinite integral of f(x), and we write conven-
tionally, omitting the limits,

[Ae dx = F(x).

This, of course, becomes the definite integral
x
f K(x) dx = F(X) — F(a)

when the limits of integration a and X are assigned.
The indefinite symbol

f I(x) dx

proposes the question: Find a function which differentiated results
in f(x); or, find a primitive of /(«). ;

Before we can solve questions in the applications of the integral
calculus, we must be able, when possible, to find the primitive of a
proposed function. ‘The next few chapters will be devoted to this

object.
125. The Fundamental Integrals.—The two integrals

xX Xx
fea and fosin x ax

are called the /undamen/al integrals. They can be determined
directly by the ad zm/io process, and all other functions that can be
integrated in terms of the elementary functions can be reduced to

the standard form
paw =4

by means of these fundamental integrals.
1. We have, where (.V — a)/n = 4,

fe dx = £ A[e* + a eer)

h(=)o

eT #4 h
SAM re = £(e es

 

 

= e* — e,
174 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVI.

2. Also,
[isn xdx= LA{sna+sin (a+ 4) +... 4sin (e+2—14)],

h(=)o
sin{a + $(” — 1)A] sin 42h

=? sin gh :

by a well-known trigonometrical summation.*
But the expression under the limit sign is equal to

eo See biG vy
4

sin $h’

which, when 2(=)o, has the limit cos a — cos X.

= {cos (¢ — $2) — cos (X — $f)}

x
[sin x ae = —cosX + cosa,
a

 

* See Loney’s Trigonometry, Part I, § 241, p. 283.
CHAPTER XVII.
THE STANDARD INTEGRALS. METHODS OF INTEGRATION.

126. As stated in the preceding chapter: if /(«) is the derivative
of F(x), then F(x) is a primitive of /(*), or an indefinite integral
of (x). This and the next chapter will be devoted to finding primi-
tives of given functions.* This process is nothing more than the
mmverse operation of differentiation. The word integrate, when used
unqualified, for the present means ‘‘ find a primitive.’’

If we choose to work in derivatives, then in the same sense that
D/(x) means, find the derivative of /J(*); the symbol D-*/(x) means,
find a primitive of /(x).

It is usually preferable to work with differentials and employ the

symbol [7 dx to mean, find a primitive of A(x), or simply,

integrate /(x).
If « is any function of x, then

w= [du

and is the solution of the integral.
The solution of

{Ax ax

invariably consists in transforming /(x) dx into the differential dz
of some function w of x, and when this is done the integral or primi-
tive w is recognized.

But, inasmuch as every function that has been differentiated in
the differential calculus furnishes a formula, which when inverted by
integration gives the corresponding integral of a function, we do not
consider it necessary that we should always reduce an integral com-

pletely to the irreducible form f du. ‘There are certain standard

functions, such as those in the Derivative Catechism, which we select
as the standard forms whose integrals we can recognize at once, and
thus save the unnecessary labor of further and ultimate reduction to

fa.

* This is the starting-point of the theory of differential equations, an extensive
branch of the Calculus.
175
176 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVII.

THE INTEGRAL CATECHISM.

1 feuds =e fide.
2 futojde= fuds + fod.
3. fu do = ww — fv du,

Uart

4 [urdu = - oF aexex—t.
5 f# = log z.

6. fe du = en

1. fa du = az

 

 

 

 

 

 

 

 

i‘ cos au sin
8. fsin audu = — 3 feos audu = oe
a a
tan au cot au
9. [sec au du = 7 fest audu = — -
‘i a a
10. ffsec utan udu = sec u. fos u cot udu = — csc
du ‘ u u“
Tl. — = sin-?— = — cos-!_.
Ya—w a a
du :
12. — = log (u + i? + a).
Ve +a
du u I u
13. f =——, = — tan-!¥—, or ~——cot-1—
wtart a a’ a a

See r tog * = fi*
oO" Sata 2a “eae a ee aaa a — 4
15. fan du = log sec w. cot w du = log sin 2,

16. f sec u du = log tan (jz + 3m). f csc u du = log tan fe.
17. [vara du = tu Va ae + 4a? sin

18, f YF Ea du = 46 Vesa + fa log (ut Yu + a?).
19. if sint 1 de = yu — sin 2u. ff cost u du = 4u+ sin 2u.
20. frog x de = u(log « — 1).

2 du ae! gy 7 1 gee

| ase a a a’

du
22. ————_- _ = vers“'4u = — covers! 4,
ru — wv

 
ART. 127.] METHODS OF INTEGRATION. 177

These standard forms are certain elementary functions of frequent
occurrence, and they constitute the Integral Catechism, which should
be memorized, and to which must be reduced all other functions
proposed for integration.

In the formulz, ~, v, etc., are functions of x.

127. Principles of Integration.—The first two formule in the
Catechism enunciate two fundamental principles of integration.

I. Since ¢ du = d(cu), where ¢ is any constant, we have

fedu= fd(cu) = cu = cf du,

or the integral of the product of a constant and a variable is equal to
the product of the constant into the integral of the variable. There-

fore a constant factor may be transposed from one side of J to the
other without changing the integral.

 

EXAMPLES.
1. foaragfoararferar = +f det) = pat,

2. fein ax ax =-if(- @ sin ax)dx = — * | aoos mas

a

 

II. Since d(u +v-+w) =du+ dv+ dw,
a | (de + do + deo) =fdutotw),

It follows, therefore, that the integral of the sum of a fimzve number
of functions is equal to the sum of the integrals of the functions, and
conversely.

EXAMPLES.

1 fax + ex3)dx = fax ax + fet ax,

safsde te fads,

=a fdas) + ef da)
= fax? + fox

2. ff (cos » — sin ax) dx = f cos xdx — [sin ax de,

= fa sia x) + fe(=),
I

= Sine — 008: ae,

 
178 PRINCIPLES OF THE INTEGRAL CALCULUS. ([Cu. XVIL

128. Methods of Integration.—The first and simplest method of
integrating a given function is, when possible, to

Complete the Differential.

This means, to transform the integral into J du by inspection, and

thus recognize w. Except for the simplest functions this cannot be
done directly, and we have recourse to the following.
The methods employed by which we reduce a proposed function

to be integrated to the irreducible fundamental form [aw or to the
recognized form of one of the standard tabulated functions in the

Catechism, are
I. Substitution.

(1) Zransformaton., (2) Rationalization.

II. Decomposition.
Parts. 4) Partial Fractions.
3

129. While nearly all the standard integrals in the catechism are
immediately obvious by the inversion of corresponding familiar
formulz in the derivative catechism, we shall deduce them by aid of
the principles of § 127 and the methods of § 128, and the two
fundamental integrals

[ease, [sin x = — cos x,

established in § 125, in order to illustrate the methods of integration
laid down in § 128, and to fix the standard integrals in the memory.

130. Transformation (Sudstifuton).—This is a method by which
we transform the proposed integral into a new one by the substitu-
tion of a new variable for the old one. The object in view being to
so choose the new variable that the new integral shall be of simpler
form than the old one.

Thus, if the proposed integral is

[ J(*) &x,

and we put x= A(z), then dx = @’(z) dz. The integral is trans-
formed after substitution into the new integral

{Ae@]¢'@) @.

This when integrated appears as a function of z, which is retrans-
formed into a function of x by solving « = @(z) for z and substituting
this value z= (x). The final result is the proposed integral
ART. 130.] METHODS OF INTEGRATION, 179

EXAMPLES.

1. Use a substitution to find {= ae

Put #=e%, then du= e? av,

au
fos fe =v=log zu.

2. Make use of fe du=e" to find fu du.

 

 

Put wt e% 1. aua—idu = erdv. Hence
a+l atl
I I —v I <8 a I
utdu=—eudv=—e% dy = e4 ae UV).
a a a+1 a
arly
ee att
fu au = —_- = ——__
a+iI a+1
3. Integrate [ cosx dx, given f sin «du = — cos u.
We have cosxdx = — sin (4a — x)d(4a — x).

Hence, if w= 4a — 2,
foosede = — f'sin udu = cosu = sina.
4. Integrate tan x dx. We have, by Ex. 1,
sin x a(cos x)
tanx dx = if = = — log cos x.
cos ee cos x

. Integrate [ cot x dx.
foot x ax = fs # = log sin x,

sin x

 

 

oa

 

6. Show that

‘ I
[sin ax ax = — = bore

Te
cosax dx = ae sin ax.

I
7. Show that fia ax dx= 2 log sec ax.

ax
8. Integrate f Vee
Substitute «= sinz. .. dx = cosz dz,
dx 4

: {pga fear a sete

: Yi — x
9. Integrate f (a + bx) dx.
Put atdxray. .. dx=dy/b.

fF epic Pe 3 lea
ie — c i
fete) cosy fy Y= Fer) Oe 1)

Put ¢=atan@. Then

 

at

10. ete

n= =a fae 6 “ ae

t
180 PRINCIPLES OF THE INTEGRAL CALCULUS. ([Cu. XVII

11. fo du. Puty=a*% .. dy =aloga du.

I y au
iu a ay = —
fe va log a { = log a loga

12. Integrate the functions

 

 

e ae
3%, at — 2%, atbeter y\lez—22),
x ia? Kerk
13. Integate eit ope! xm a’

14. [22* eS _ fo + sin x) =o lee (2:42 20. 3),

I + sin x I + sin «
15. [sine x dx = afc — cos 2xj)dx = fx — Isin 2x.

sin x ax

16. Find the integrals f [cost x dex.

I—cosx’ ,
17, Given the definite integral
x gett x axetil — gett
f ial = =) ee
a c+1 c4+1

a

 

at
deduce the integral / — = log.
t
In the value of the definite integral, let c(=)— 1, then (see § 75, 0/o form),
xetl — gett

FE = £(x«tt log x — att log a),

e(=)-1
= log x — log a.
log a is the constant of integration and we have

{= =
= = logs.

18. ——— Put «w= asec,
uve —a

 

19. f*} This can be written
2s — s®
i ds
YrI-(i— spe
Put 1—s=cos@. .- ds = sin @ dG, and the integral becomes

fa = § = cos-(1 — s) = vers—ts,
20. foe a tanxdax = {= Hin fas a)

cos? x cos? x
= sec x.

21. fesc x cot x ar =?

 

Z— xX

az.

 

&

ax ——,
22. aS Put Vetaemz— 4. “ as

—=-ft = log z = log(x + x? + a?)
ART. 130.] METHODS OF INTEGRATION. 181

23. Show by a like substitution that

 

ax ey
J pran e+ v=,
24, Integrate | ae

sin x cosx

f ax =/[= x ax d (tan x)
sin x cos x tan x =} = log (ana).

tan +

 

25. fo =(ai > a4) =
sin x sin 4x cos 4x og (tan 32), by Ex. 24,

‘4 v.
26. To integrate {= put x«=4r—2z

 

 

 

fz z= -{< mae log (tan 42) = log (cot 42),

= log[cot (ta — 4x)] = log tan (4x + 4a).
These results can be identified with 16 in the table.

27. Observing that we can, by inspection, write

I I I I
x—a@ 2a\x—a «tal?
f ax 4) eee
—>— = — 10: .
x? —_@ 2a Sa

This process is a particular case of the general method of decomposition into

 

 

we have

 

partial fractions.

Integrate this case, using the substitution (wv — a) = (x + a)z.
Also, integrate the more general integral

=

by means of the transformation « — @ = (x — 4)s
28. We can make use of Ex. 27 to obtain the integrals in Exs. 25, 26. For
we have
ax eee _ f fe (sin x) és 1+ sin «
feel cos? I aa =P S\r sin x
Show in like manner that
dx I lo I — cos x
Ise; 8 1 + cos x/~

dx
29. Integrate jf ——__, Put ex = sec 6.
pier Vex —1

Then dx = tan 6 dO, and the integral becomes
fa = 6 = cos" (e-4).

sin 0 a6
30. Integrate f Sie

We can complete the differential by inspection, for the integral becomes

aa — bcos6) _ 1 3
r + f= TER 3 log (a cos 4).

 

 

 
182 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVII.

Otherwise, puta — dcos@=z .. dsin@ dQ = as.
The integral is therefore
I fds 1

I
Z Sm ples? = Flog (@ — 4 cos 6).

 

31. la Put a? + a= 2, Then x dx =z dz,
+ a
dx az dx + dz a(x + 2)

e x eta x42

- {Ss Fa = = {t= { el ee (x + 2)
= log (*+ x? + xt + a®).

131. Rationalization (Sudstitution).—The object of this process
is to rationalize an irrational function proposed for integration, by the
substitution of a new variable.

Rationalization by substitution is but a particular case of trans-
formation by substitution. But, since the direct object in view in
rationalization is not generally to reduce the function directly to a
standard integral, but to first transform it into a rational function
which can be subsequently integrated by decomposition into partial
fractions, the process demands separate and distinct recognition.

Only a few simple examples will be given here in illustration.
The subject will be considered more generally in the next chapter.

or

 

EXAMPLES.
1. Integrate if (a+ bx8)8x5 ax.
Put a+ dx? = 28... bx*dx=2* dz. Onsubstitution the integral becomes

I 1 /28 2b
R fe = aes oh o=),

e pr (a + 4x5)8(50x3 — 3a).

2. Integrate f fs $
(a + 4x78
Put atdsxtaed +. x dx = 327% dz/26,
The integral is 3 a4 bx -
3. Put @-+ dx = 5, and show that
x ax 3 2
[FR = Ga - 30 + beh

(a + bx)i
4. Put a? — x? = 23, and show that
(ea ea ioe 3 (307 4 2x2)(a? — x2)
(a — x2) 20

5. To integrate f = ‘
ay + x
ART. 132.] METHODS OF INTEGRATION. 183

Put 14 t/x? = 2% 1. dx = — gx8 dz,
The integral becomes
I 2x7 — 1
Tenge ey ee oe aE
fi 2)dz = 2 ao ae
6 f ax P C 3
. Sie ut 1/7? —1r = 2% 1. dx = — x2 dz.

After substitution the integral becomes

i I
— fas —2=-L yin

7. j= ) dx. Putx= zt, .. dx = 423 dz.
1 — xt
The integral becomes

Loses fe e cial ot,

 

33

 

         

since
z—
ta
apes eee
1— x? 3
ia Putx—=asin@. .-. dx = acos 6 do.

f Yar — xtdx = at f cost 6 do = 4a? fo + cos 20)26,
= 40% + 4sin 26),
= 4a? sin—t + ge a — x2
Rationalization by trigonometrical substitutions will be considered more gener-
ally later,

132. Parts (Decomposition).—This important method of decom-
posing an integral into two parts, one of which is immediately inte-
grable by definition and the other is an integral of more simple form
than the original integral, is one of the most powerful methods of
integration we possess. It is based on the formula for the differentia-
tion of the product of two functions,

diuv) =u dv+vdu,
u dv = d(uv) — v du.
Integrating, we have the formula for integration by parts,

[ud =w—fodu.

EXAMPLES.
1. Integrate f log x dx.

Decompose the differential log x dx, so that
uatlogx and w= dx,

aa? and vo
x
184 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVII.

Hence
flog «ax = «x log x — [as =xlgxr—-x,

2. Integrate f tan—'x dx.

Put u=tan-lz, dv = dx.
Then du a vox.
= I+ oe?

x ax
v fitan-e ax = «x tan-tx — {= ay
b= #

=a«tan- — log ¥1 + 2,
3. Integrate j x e* ax.

Put ui x, du = e* dx.
Then du = dx, v= ex,

f* ede a xer— ferdeaer(e— I).

4, Integrate f x4 log x dx.

 

Put u = log x, dv = x,
+
Wom ly v= ae
x a+I
att xt dx
» fxtbog x dx = hog = aa"
xett 1 att
: pa =" ie ap
5. Integrate f Vx? + a® dx.
Put w= Yt ai, dv = ax.
du = a. vos,
x2 + a
Hence
a SP ee 2
{ vere dx =x Y¥xttat -/( = a .
Vor a

But

 

7 2 is xt + a @ dx x dx
PPR) ees pe) pe
Adding, we have
of VETS dx = 2 BFE + of OO
7 f Vx2 at dx = 4x x? +a? + fat log(x + WP fa),

by Ex. 22, § 130, or Ex. 31, § 137.
ART. 133.] METHODS OF INTEGRATION. 185

6. Show, in like manner, that

f VX —@ dx = tx VP —@ — jot log (x + fx? =a).

_1. We can frequently determine the value of an integral by repeating the process
of integrating by parts. Thus, integrate

f et sin bx dx.
Put u = sin dx, dv = e%% dx,
I

du =bcosbxdx, v = —eax,
a

3 I
few sin dx dx = — 4x sin bx — few cos bx dx.
a a
But, in the same way, we have
I b :
fox cos bx dx = — eax cos bx 4 — fo sin bx dx.
» a a

Substituting and solving, we get the integrals

 

 

 

iy . eax ki
fer sin dx de = "(a sin bx — 6 005 bx),
ax oe i
fe cos 6x dx = LE (a2 cos 6x + 34 sin dx),
Put 6/a = tan a, then these integrals can be written
eae 2 eax
=> sin (64 — a@) and ———— cos (6x — a)
Ya + 62 ya? + Be
respectively.
8. Use Exs. 5, 6 to integrate

1 Vx + a xt — a
9. Show that f sin-txdx = x sin-1x + 4/1 — x? by putting # = sin—tz,
dv = dx.
10. Use the method of Ex. 5 to show that

if Ya — 2 dx = fra — x + fa? sin,

11. Use the work of Ex. 10 to get
x dx

————. i x
f- —=_ = -# Yar — xt + Ja? sin.
ya — a

133. Rational Fractions (Decomposition).—Whenever the func-
tion to be integrated is a rational algebraic function, we know from
algebra (see C. Smith’s Algebra, § 297) that it can always be decom-
posed into the sum of a number of partial fractions, each of which is
simpler than the proposed function. (See Chapter XVIII.)

We do not propose to consider here the general process of inte-
grating rational fractions, but merely consider a few elementary
examples illustrating the process.

If the function to be integrated is the rational fraction

(2)

pry’
186 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVH.

and the degree of @ is higher than that of #, we can always divide
@ by %, so as to get
P(x) _ F(*)

= f(x) +
ney ee ace
in which the quotient /(%) is a polynomial in « and can be integrated
immediately. The remainder /()//'(~) is a rational function in which
F(x) is a polynomial of one lower degree than y(), the general
integration of which will be considered later.

 

EXAMPLES.

= La — Lt $ x — log (1 + 2).

 

 

 

3
2— 3x = ax Se _ 4)
2, [iar a2 fa- wae
i e-2 3 oo
ar 7 oe 4).
a ee = 3°47 5
wet 4
we — Be tI 7x —1 I 7x
eg aa Ee

 

x — 3x -+ 1 Ax + 4) + 4)
. fee Boe ax “fret fae a “weg, *

xT
= = +5 — tan— ae 2 (x? + 4).

4. To integrate f a

We can always write

 

I _ ot I I
(x — a(x — 6)” a—b\x —a -=)
by inspection. Therefore

 

f a 1 x—a
J @—ae@e—-h a—6 83-8

134. Observations on Integration.—The processes of Sudsti/ution

and Decomposition, in their four subdivisions:

1. Substitution,

2. Rationalization,

3. Parts,

4. Partial Fractions,
constitute the methods of finding a primitive of a given function by
reduction to a recognized or tabular form. These may be regarded
ArT. 134.] METHODS OF INTEGRATION. 187

as the rules of integration in general form corresponding to the rules
of differentiation. With this difference, however, that in integra-
tion there are no regular methods of applying these rules to all
functions as is the case in differentiation.

The successful treatment of a given function depends on practice
and familiarity with the processes of the operation.

Sometimes different processes of reduction lead to apparently
different results. It must be remembered, in this connection, that
the indefinite integral found is but @ primitive of the function pro-
posed, and both results may be correct. They must, however, differ
only by a constant.

Frequently, in reducing an integral to a standard form, we shall
have to use all four of the methods of reduction. Experience soon
teaches the best methods of attack.

In the next chapter we shall consider the subject more generally

and make more systematic the methods of reduction to the standard
forms.

EXERCISES.

Integrate Exs. 1 to 10 by the primary method of completing the differential
by inspection.

1 / xt dx, f ax dx, / 2x—2dx.

2. fo + ib xde = Hat tr)

3 [Ss See = log (3 — 3a2x)t,

— 3a°x

4. / (roe? — +4)at = f8 4 yr.

5. je + a )ax, fe — 1)ds/s, fe du/(v? — 1).

ut1
a? + 20
7. fi (2 — 2)8-¥ dt = 2t-+ — 6t-* + 12 — log &,

8. f (a — 22/8 /x dx, f{ (va — x)*dx, i; (x + 17x.

2ax + 6 d Sa 2ax + 6
- Sater Jexpe*™ aa be ee”

ex sect
fe ++ a)tex dx, az” {<= dx.

(1 4 x?)-1 ie (1 — x?) dx

is — >:
tan—“w ; log x*

TT du = log Yt? 4 2u.

sin
10. Write immediately the integrals of
I x x x et
epi’ xr’ x41’? S47 aan

cos? $x, cos*x sin x, tan%x sec®r.
188 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIL

 

11. i cos Vx 4, = 2sin Vx. Put + = 2%,
Vu

12. fe cos e*dx =? Put et*#=2.

13. fax cos x* dx =? Put «2% = 2.

cos (log x)

14. feces ese Putlos a= wi
2x

15. [eae Put es
3% =, > ez,

16. ae dx =? Put #=2.

 

17. ie ax i ax udv +vdu
yi — 4x? yi — 2 Vt — wrt

we. a ee oss i SO
I+ 4? gx? + 4 xox? — 1

i frin 3x ax, free 46 29, fos 49 dp.

 

_
o

xl dx I
20. f 7 log (a + dx*).

at bx =

21. {Se= Va to-+(+4)?).

22. [CREE = log x $2 + ae

23. — syed ne

24. f we bdo = Bane f mips?
25. ffsin 26 d0 =? foos 26d =?

sin (m+ n)x | sin (m — n)x
2(m + 2) 2(m — 2) ~
sin (m — n)x sin (m + 2)x
2(m — Nn) 2(m +n) ~
Use cos w cos 6 = cos (a + f) + $08 (a@ — 8), etc.
cos (72 + nm) ae

2(m + n)
28. f'sin 4x cos dx dx =? foes 3x cos 5xdx =?

26. feos mx Cos wx ax =

fsin mx sin nx dx =

cos (m — 2)

27. — fein Mx COS nx AX = 2(m — n)

 

log x
29. / . dx = i(log x).
30. [\F@ =a sin — Va — x.

a—x
ART. 134.] METHODS OF INTEGRATION. 189

Multiply the numerator and denominator by Ya+x.

 

31. [ove + adx = Xx + a)! — a(x + a)i, Put x+a= 27.
32. te dx = ex(x? — 2x + 2). Parts.
33. [tex dx = ex(x3 — 3x? 4 6x — 6). Parts.
ax 2 2x —a
34, (| == to] ; Put 2ax — a? = 23,
xV2ax—a? @ a

35. Jcot-tx dx = x cot-tx + } log (1 + x’).

36. fe tan-ix dx = 4(x? +1) tan-iy — fx.

37. fotsins de = 2 cos x + 2x sin « — x? cos x.
38. ffx 08 x dx = x? sin x + 2x cos x — 2 sin x.
39. fos x log sin x dx = sin x (log sin « — 1).
ax—1

40. | xeex dx = e9% 5
a

ax dx ax
ih le — 1x + 3) te + 3x — 10’ I —3«-—6

Hint. Complete the square.
ax ae
42. = og (x a + x — fp).
. V(x — a(x — B)

Put «—a=2", then dx = 22 dz.

 

i dx a f dz
iss ——_———_——_—_————_ = 2 pS
V(x — a(x — B) VF +a—B
= 2 log (2 + ¥? F a— Bf).
ax it  @&
ee

Put «x — a@= 27, as above, and the integral becomes

 

 

f dz
2.4 >
V¥B-—a—#
44, 1 Ya + 2bx + ex? dx.
a+ 2bx + ex? = cae + 2bex 4 cx),
= c[(ex + 6)? — (2? — ac)].
Put ex+6=4. .. dx = dz/c, and the integral becomes

a f V2 —(6? — ac) dz,
a
the standard form 18, § 126.
Igo PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVII

a. sipieee “33
45. Integrate { a ee where m and » are positive integers,
or m + 2 is a positive integer greater than I.
Put + — a = (x — 6)z, then
a — bz (a — 4)z a—s a—b
x ce re =, « —b= x = ——dz;
I—2 1-2 I—2 (1 — 2)

 

 

and the expression transforms into
(1 — 2)™+%—-2 dz
(a — b)ymtn—i gm °

Expand the numerator by the binomial formula and integrate directly.

46. Integrate f sinéx cos’xdx, whenever » + g is an even negative

integer.
Let +9 = — 2x. Then
sinéx cos?x = sinéx cos—A—2%x = tanbx sec?x,
= tanéx(x + tan?’x)*—1 sec?x.
Put tanx =7, Then

[sintx costxdx = fee + f)"—-1dt,
Expand by the binomial formula and integrate directly.

47. Integrate sinéx cos7xdx, whenever # or g is an odd positive integer.
Let p = 27 +1, then

/ sin2*t+ly cos7rdx = — [ (sin?x)" cos? d(cos x),
= i (1 — cos*x)r cos7x d(cos x),

= fo — e)rer de.
Expand by the binomial formula and integrate.

48. f sin’@ d6 = $ cos*@ — cos 6.

49. f cos*@ d§ =? Check by putting 47 — x for x.
50. { cost) di = sin 6 — } sin’ + } sin®6.

51. f sin°6 cos’@ d@ = 7,cos!°§ — tcos*.

52. f sindx cos—2x dx = sec x + 2 cos x — $ cos®x.

53. f sin x costx dx = }sinix — 2 sintx,

54. f cos8x cschx dx = 3 sintx — $ sin?x.

55. f eschx secix dx = $ tanix — 2 cot?y.
AR. 134.] METHODS OF INTEGRATION. Igt
56. fsints secox dx = } tandy -+ 1 tandy,
57. fonts secix dr = $ tant,
58. [esctx sectx dx = 2 tan'x (1 + 2 tan2x),

59. [ tan") a) = f tan™—2 (sect — 1) dO,

 

 

—
= Va - fran do.
60. Show that A ene dy a SE f cot™29 dg
n—t ‘

61. f tan‘ a = }tan%@ — tan 6 + 6.

62. [oor dj = — } cot*@ — log (sin 6).

63. [cov a = — gcot*é + cotd+ 6.

64. / cots do = — } cot + 4 cot®9 + log (sin 8).

65. fsin x C08 x (a? sintx + 3? costx) tar.

Note, da? sin?x + 6? cos’x) = 2(a? — 62) sin x cos x dx. Hence the integral

7 (2? sintx + 8? cos®x)?
3(@? — #) es
dx I é
66. =a aye *(G tan).

Divide the numerator and denominator by cos?x.

 

e ax 2h ; es
67. : | ws laa Divide the numerator and denominator by V a 6,
and put tan @ = a/b. Then we have
I ip ax a I nets
Yi pR Ga) gape EON Ge — ie + ae
ax
- I; + 6 cos x* were

a+ bcos x = a (sin? fx + cos’ tx) + 4 (cos? 4x — sin? 4x)
= (a + 4) cos? 4x + (a — 4) sin? fx,
which reduces the integral to the form of Ex. 66.
Divide the numerator and denominator by cos? 4x, and put z = tan $x. Then
the integral becomes
dz.
(a + 6) + (a — 3b)”

which is standardized. Hence

 

 

ax 2 = a—8 x
{mes - eS ag nS 5 a> 3s;
a+tébcosx Va —P a+sé
I Vota + Vb ~a tan fe wee

 

2 eae” Vota — Ys—atan tx’
192 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVII.

69. — == ins es oe
J5S+t4snx 3 3
I + cos x I I

70. Int 4 sina) 2 (w@ + sin x}
0. Integrate ic oP t 2 (x + sin x)"

71. fx sinx de = sin x — x 008 x.

 

72. [Hee log (1 + x)? — x.

Ix

2 dx 2. I
13: [Se ee
f (@+x 3 (at + 2)

ax
a $$ =| tan— .
” Wes + x?) tan oe ae)
75. —— ==:2 unl + : = cos-! ea
V5 -- 4x — x? 6 3
76. fee = sin (log x). Put « = log z.
x
71. a
4— 5sinx

5 — 4 COS 2x

 

 

tan 4c — 2

log —_——___.,
“8 tan dx — 1

wile wien

tan—t (3 tan x).
CHAPTER XVIII.
GENERAL INTEGRALS.
GENERAL Forms DirectLty INTEGRABLE.

135. The Binomial Differentials.—Expressions of the type
x*(a + bx8)y dx, (A)
where a, (, y are any rational numbers, are called d:nomial differen-

tals,
This expression is directly integrable in /wo cases.

att
p

The substitution is @+ 4x8 =z. Then

T I
—-1

zg —a\B 1 /z — a\8

I, When

 

is a positive integer.

 

hence
eee
= B
x(a + bx8)\¥ dx = eo 7.
poe

 

ati. bh
Consequently, when + is a positive integer, the transformed

expression can be expanded by the binomial formula and immediately
integrated.

II. When ea + y is a negative integer.

B
The substitution is a + 4x8 = 2x8,
For, if we substitute + = 1/y in the differential x*(a + 6x8)’dv,
it becomes
= Ee aah a Otay,

 

which, by I, is integrable when — meet is a positive integer,
or, what is the same thing, when
a I
ety

193
194 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIII.

isa negative integer. Also, the transformation a + 6x8 = z becomes
5 -\- ay = 298,
Hence, under the transformation,
ayl

Only _(*Fta,
x*(a + bx8)\¥dx = a Br (6 — 2) ( Bie derde,

In working examples it is better to make the transformations than
to use the transformed general formule, which are too complicated
to be remembered.

When a, £, y do not satisfy the conditions in I, II, the binomial
differential must be reduced by parts.*

 

EXAMPLES.
2 at ae
1. a> Ans. yar at = + a? log (a? — x*),
we dx I a
“(a+ ext)? Be 4A(a + cx? mr 6c%(a + cx?)
© dx I I I
3. aq Ans. eae We a + z log (x? + 1).
dx x cx?
4. | ———... Ans, ———____ ee
fe + cx?) i aa+ ait : 3(@ + cx?) §
x? dx x I cx?
. | ——_- Ans, ———___. 9 2. — et
, f (a + «x2yh a + ah 3. 5(@+ at
3 dx 2a? + 3x?
——;. Ans, — ———_,
ef (@ + =i 50a a8
x dx $34
7. ae Ans. 8(1 + x3)#(« 2).
ax x
feretclene e Ans. ———_.
B fe + +5)5 ne (1 + 23)
ax I
— aS t
9. Some + >E Ans. P (2 H+ tye
dx 2x
—_—_———;. Ans. ————_.,
uo: len + #8 i (1 + yt
6. Integration of Bes ts B
Bg aHES (A + Ce*)(a+ cx?*\¥ (B)
The substitution is @ + cx? = x%2?,
ax dz
— — =
cdx = 2 dx + 2xdz, or ee
ax az

 

(4+ Ce)(a + cx*)t ~ (Ae — Ca) — A”
which is standardized, being 13 or 14 (§ 126) according as (Ac—Ca)/A
is negative or positive.

 

* For formule of reduction see Appendix, Note ro.
ART. 137] GENERAL INTEGRALS. 195

If (de — Ca)/A = —, the integral is
I tan 4 Ca — Ac

VA(Ca— Ae) Vala + ex)
If (Ac — Ca)/A = +, the integral is

 

 

I log Ala + cx*) + 24/Ac — Ca — Ca
2/A(A4c — Ca) ~ Ala + ca) — xp/Ac — Ca
EXAMPLES
1. f 6 Ans. aes tan? _ 1 :
(x + x?*)\(1 — x24 V2 Vi — x
ax
Bo Ans, —— tan-1-__34_
f 3+ 42)4— 32) ii 5 3 ee oe
i ad ; oes 2(3 ++ 4x°)t + 5x
(4 — 32°)(3 + 4x) 20 ' °F 3G +42) — 5x
137. Integration of Ee x (C)

This is a particular and simple case of the rational fraction which
will be treated generally in § 148. On account of its special impor-
tance we give it separate treatment here,

Let Z represent the /zear function p + gx.

Let Q represent the quadratic function @ + 26% + cx.

ax
I. Consider So
Q

Completing the square in Q, we have

 

ax = cdx
a+ 2dx+ cx? J (cx + 6) — (# — ac)
Put cx + 6=2. Then the integral becomes

dz
Jz — ( — acy

This is standardized, and depends on whether 2? — ac is positive
or negative. If zegafve, the roots of the denominator are zmaginary
and the integral is an angle, the standard 13. If poszhve, the roots of
the denominator are rea/ and the integral is a logarithm, the standard

14 (§ 126).
If ac > B,
a= ca eee ex $b 4 (x)
= = war Vac —
If ac < 3%,
dx I cx +b — fB — ae

 

oC 24/8 — ac se cx b+ fh —ac (2)
196 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIII.

II. Consider fz dx.

Since the derivative, Q’, of Q is a linear function, we can always
determine two constants A and #, such that

L=A+BQ,
or ptrgqx =A 2B 2cBx.
Equating the constant terms and coefficients of x,
B= q/20, A=p— b¢/e.

L _ pe— gb f(dx q dQ
Jo EI Oa oO

The first integral has been reduced in (1), (2), and the second
is log Q.

In working examples, carry out the process and do not substitute
in the general formula.

 

EXAMPLES.

ax wc —2 I 2x+4
: eel eee ee fas,
= — 2 tan-(x + 2) + $ log (x7 + 4x + 5).

 

 

 

2 (Sa =- aa log (2% + 2% + 3).
3. fsSS= apiece 1).
west = 5 log (2? + 4x + 5) — tan(x + 2).
5. at.- — log (3 — +).
Bi fsa dx-= 2x — log (x? + 6x + 10)! + 11 tan-(x 4 3).
7, foe een log (x2 + 2x +2)? +3 tame pi.

Ax)
Ey Q until the remainder is of the form Z/Q, and integrate.
xe)
eee Sar a (D)
a+ 2bx 4 cx?
Let, as in § 137, Z and @Q represent the linear and quadratic
functions respectively.

J. Consider or

Complete the square in the quadratic, and then

lo" jaa

8. To integrate dx, where F(x) is any polynomial in x, divide F(x)

138. Integration of
ART. 138.] GENERAL INTEGRALS. 197

which is the standard 11 or 12 according as 8? is greater or less than
ac. If a and ¢ are both negative and ac > 6, the function is
imaginary.

We have, according as the roots of Q are real or imaginary,

ve log [cx + 6+ c(a + 26x 4+ cx*)],
in a b
Vi ac +6 + ee

as the corresponding values of the integral.

II. Consider & ax.
Write, as in II, § 137, 2 = A + BQ’, and determine 4 and Z.

Then
Se dQ
Sae-4fart2fo

The first integral on the right was reduced in IJ, the second is

2@Q3.

 

 

; ax
III. Consider Lor
gdx dz I — pz
Put = 1/2. a‘ =, = :
Papel P+ 8 2

Substitute in the integral and it transforms into

a dz
a’ + 2b'2 + ce?’

which can be integrated by I, then replace z by 1/(p-+ gx).

EXAMPLES.

1. a= 2log (¥x-+ x — a).

2. [ae = i = = sina ( - :).
Yax — x2 a a

3. /—_— = 2sin—14/x — I = sin-(2x — 3).
3x — x? — 2

 

 

ef ewe eet year.
Ss [ QE Rjee = VEF AEFI + (0 Oleg WEF A+ FED,
fen

6. S75 V5

 

 
198 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu, XVIII.
Z {SS = Z cost,
x x2 —@qe @--  «

8 {———=- ee
“Japayi-a T+

 

 

9 (= Bs a+ Vay x
=e sy~erta «@ 8 x .
ax a a, Ve
m lage" 7" rf «
i. f dx eee Vx? 28 $3442
(«—DYe — ar +3 2 x—1
1 | BSS HIF RFS owe ETS

RepuctTion By Parts.
139. Integration of Powers of Sine and Cosine.
i. sin*x dx = f sin”“x sin « dx.

Put u = sin” x, dv = sin x dx;
.. du = (m— 1) sin**x cos xdx, v= — cos x.
Hence, applying the formula for parts,

fein "x dx= — sin*-“1x cos x + (4 — 1) f sint*x cos? x dx,
= — sin*"'v cos « + (2 — 1) f sin x(x — sin* x) dx,
=—sin" x cos «+ (n— 1) f sintardx— (n—1 ) f sinnx ax.

f sin” 2x dx. (1)

When » is a positive integer this reduces the exponent by 2, and

sin*"x cos x , a— 1

 

», fsintxdx= =
nN

leads to fax or fin x dx according as # is even or odd.

Since integration by parts depends only on the differential equa-
tion d(uv) = udv + udu, the formula is true when ~ is any positive
or negative rational number.

Change z into — z + 2 in (1),.and we have

ax — cos x n—2 ax
sin*x  (”% — 1) sine T n—1/ sin*?x (2)

In (1) and (2) change x into 47 — x, then

cos" xsinx  ”m—I
in —_ ‘ n—2
[os x ax = Sg [oo x ax, (3)

ax sin x n— 2 ax
cos*x ~ (# — 1)cos*""x teat J cose (4)

 

 

 
ART. 139.] GENERAL INTEGRALS. 199

These formule are important. They reduce the integrals to stand-
ard forms whenever ~ is an integer.

Formule (1), (2), (3), (4) can be obtained directly and in-
dependently by integration by parts. In practice this is the better
method. The separation into the parts «and dv is indicated in each
case in the formule below.

sin” sta" y sin
ne IX = x ax
ie co J cos"! * cos i
sec” sec”~? sec?
adx = x dx.
oh csc” ve esc? * * esc?

In the part f vdu use sintx + cos*x = 1, sec’x = 1 4 tan*x,
or csc2x = 1 + cot, as the case requires.

EXAMPLES.
1. [sints de = a tes ed sinae.
2. f sinkz dx = — $sin’x cosr — } cos x = } cos’x — cos x.

3. [sintx dx = —} cos x sin x (sin’x + 8) + $x.
4. f sindx dx = —} sintx cos x + 4 i sintx dx.

5. f sin'x dx = — }cos x (f sindx + 55 sin’x + § sin x) + yx.

6. Find the corresponding values for cos x, integrating by parts. Check the
result by putting $2 — x for x.

 

7. f as = log tan 4x = log (csc x — cot x).
sin x

= — cot x.
8. J sive

 

 

 

 

cos x I x
=—- , =1 —.
9. {<& sin3.x 2 sin?x He 2 8 tan 2
I cos x 2
10. ae Sots —3- — — cot ex.
sintx 3 sin’x 3
I cos x 3 cosx 3 x
=-—- = -5 > log tan —.
MN. f= —~ 4 sintx 8 sin?x +3 8 2
dx I cosx 4 cosx 8
1 ee - = — oot «.
sin®x 5 sinmtx 75 sin’x 15

13. Deduce the corresponding integrals of cos «, and check the result by
putting 4” — x for x.
200 PRINCIPLES OF THE INTEGRAL CALCULUS. ([Cu. XVIIL.

140. Integration of / sin™x cos*x dx,

We have for all positive or negative rational values of mand 2

d@ sin*™—"x =F jon + ( joe sin”

— —_—- m— 1 n— 1 ‘

ax cos" x cos"? COs ".x
Therefore

 

 

sin”. I sin™!y m—I sin” 24"
dx = = = dx. ( 5)
cos".x* m— 1 cos* tx n—tI cos"—?7.¢
In particular, when m= 2,

tan” tx
ae tantx dx = ————_ _ fe tan"—*x dx. (6)
nu—i!I

Put 4a — x, for x in (5) and (6). Then
cos" — I cos™ ly m—tI cos” 2.4
: an = - _ : a@x, (7)
sin”. m— isin” "x n— I sin”?

tt"
/ cot™: dx = — _ J cot” x ax. (8)
a—1

The same results are obtained immediately by changing the signs
of m and 2,
Change the sign of in (5), then

; sin™—x cos*t1y am —1 i
sin”x cos*.x dx = — —______ —__|. —___ sin™—*x cos"t2x dx.

n+ n+1

But sin” x cos*t27 = sin” *x cos*«(1 — sin*x),

 

 

 

= sin™—x cos"x — sin”™x cos*.x
Substituting and solving, we have
: m—i ‘ sin” 1x cos**x
sin"x cos"x dx = sin™—*x cos"*x dx ———_—_—_—_—. (9)
m+n mtn

In like manner, change the sign of z in (7) and write 1 — cos*x
for sin?w in the last integral. Then

 

m—T jn #t+1
oe cosx sintx dx ==" [ cogm—x sintx 2 alta (10)
n+m m+n

These formulz serve to integrate sin*x cos”x dx whatever be the
integers 7 and 2.

It is well to be able to integrate the functions of this article in-
dependently. The forms below show the separation into the parts «
and do which effect the integration directly when the trigonometrical
relations sin’ + cos’x = 1, sec’x = 1 + tan®x, csc’x = 1+ cot®x

are used in the integral f vdu,
ART. I41.] GENERAL INTEGRALS. 201

ji sin”x cos*«dx—= f sin”—“x cos**Xxsinadv= J sin” cos""1xxcosxdx,

tan” d tan*~? tan? a
cot” Xax = cot*—2 ae OX cot? Xax.
EXAMPLES.

1. f cos*x sintx dx = } sin x cos x(t sinte — ¥, sintx — 1) + Jw,

2. 2.
aa + log tan tx.

sin x ii ea =

I cos x 3 x
3. I= —— — ——— + = log tan.
si ae cog = cos x 2 sintxr ab 28 2

4. rants dx =} tan'xy — tane + x,

5. f cottrdx = — 4 cot8x + cot « + x.

6 ax 1. ’
A la =~ tants ~ [0g (sin 2).

dx 23%
7. Sa = Fane tT 7 + log (sin x),

INTEGRATION OF RATIONAL FUNCTIONS.

141. General Statement.—Any rational function of x whose
numerator is a polynomial V and denominator a polynomial D can
by division be decomposed into

N R
p=elt+p

where Q is a polynomial, and the degree of & is that of D less x,

We then have
R
lx = =f Odx +f3 dx.

The first integral on the right can be written out directly. The
second integral demands our attention. We know from the theory
of equations (C, Smith’s Algebra, § 436) that every polynomial in x
of degree has # roots, real or imaginary, and can be written

A(x —a)(x —a,)... (x —4,).

If there is no second root equal to a,, then a, is said to bea
single root. If, however, there is another root equal to a,, say
a,=a,, then the two factors can be written (+ — a,)*, and we say
that @, is a double root, or that the polynomial has two equal roots.
In like manner, if there are 7 equal roots equal to a, the correspond-
ing factor is (x — a)", and we say that @ is a multiple root of order
r, or the polynomial has 7 equal roots of value a.
202 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIIL

Again, we know that if the coefficients in the polynomial are all
real, then imaginary roots must occur in conjugate pairs (C. Smith,
Algebra, § 446). Therefore, if there is an imaginary root a + dY — 1,
there must be another a — 4¥ — 1. Now the product of the factors
corresponding to these two roots is

(4 —a—bY — 1)(4—a+b¥ — 1) = (e —aP +B,
=x — 2ax4+ 04 2,
which can be written = xt pet yg.

Moreover, if a + i (¢ = ¥ — 1) isa multiple root of order 7, so
also is a —2b, and we have the corresponding factor in the
polynomial

 

(x? + px + 9)".
Hence any polynomial in x is composed of factors, linear and
quadratic, of the types
x—a, (x— 5), P+pxtg, (P+ pet gy
F(x)
Ax)
be a rational function, in which F(x) is of a degree at least 1 lower
than that of /(), we can always decompose the function into the
sum of partial fractions corresponding to the roots of /(x), as follows:
For each single real root a there is a fraction
A .
x—@

If

 

for each multiple real root 4 of order 7 there are 7 fractions
B B. B,

I poe ens
@-jte-HT Tete
for each pair of conjugate imaginary roots there is a fraction
C+ Dx |
Bo pee
for each pair of conjugate multiple imaginary roots of order s there
are s fractions of the types
etaxr+ Bo (ee +ax+ BP oO T OF + ax + pf)

In these partial fractions the numbers A, B, C, D, £, F, etc.,
are constants. Since there are exactly as many of these constants as
there are roots of /(x), they are z in number.

If now we equate /(x)//(x) to the sum of the partial fractions
and multiply the equation through by /(), we shall have F(x)
equal to a polynomial in x of degree x — 1. When we equate the
constant terms and the coefficients of like powers of x on each side

 
ART. 141.] GENERAL INTEGRALS. 203

of this equation, we have z linear equations in the constants 4, B,
C, etc., which serve to determine their values.*
The integral of the rational function then’ depends on

ax / (Z + Fx) dx
fee si tet b px +9)"

The first of these can be integrated immediately, the second is
always of the type

(E44 Ff) dx : zdz
tenarp ep th) feat? f wpa

wherein ..=a-+<z,. The last integral on the right is

latr ade 1 d(z*) _ I —1

(2+ Ry 2 J @+A” arr) #4 Rr

To integrate the first integral on the right, put z = 4 tan @.
dz = b sec*6 dé.

dz 4 27—2
Then lepry = pf 00s 0d,

which can always be integrated by parts, § 139.
Hence the rational function can always be integrated.

EXAMPLES.
pf eee
a — 4x
We have here single real roots; hence
x? + 6x — 8 x?+ 6x — 8 A B Cc
x8 — 4x -pane eo a ean! x+2°

Clearing of fractions,
x? + 64 — 8 = A(x — 2)(x + 2) + Bx(x + 2) + C(x — 2)x, (1)
=(A4 B+ C)x?+ 1B C)x — 44.
Equating coefficients,
A+8+C=1, 2aB—C)=6, — 44 = — 8
A = 2; B=t1, C= —2.

Hence the integral is

fas x = 2 log x + log (x — 2) — 2 log (x + 2),

x — 4x
= x(x — 2)
(ebay
If we assign particular values to v in (1), we can find 4, B,C
more easily. Thus put «=o, then — 44d = — 8; put+ = 2, then

 

* Provided these 2 equations are independent, which they are.
+See also Ex. 88, at the end of the chapter.
204 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVII.

8B = 8; put x = — 2, then 8C= — 16, which give the constants at
once. The general principle involved in this abbreviated process is:
when there are only single roots, put « equal to each root in turn, and
the constants are immediately determined.

 

 

og = Flee te — 3) + Hog ( + 2)
8. f aS = Boe @ + 3) + Flog (x =»)
4. [etree dx = 4 log x + 4log (x — 2) + log (x + 3).
5. [epee =
6. fot gd = lo [le +3" @ — 2).
1 bee ae Te a Fo eet dow ET tM log (= +2)

f ite dx. Here there is one single root, 0, and a triple root, x = 1.

Hence Sam, o ‘ ; ‘
sont) a Goa Gow” aor

Clearing of fractions, we have

x41 Sire ee ne +(34+8-C4+D)cx—A
1=>A+D
o=C— 34 — 2D,

 

 

=—A.
Whence A-=-—1, B=2, C=1, D=2
aos = - E+ op tas tee
{ate = ae : i ~S + 2 log (x — 1),
: log @ —1P | is e ¥
s. [8 2 ge

— 40° + 4x 1 — 2

10. f{ Cry pt = = Grae ae +3 log (x 42).

pen 2 es
it f= 8x? — 4x ti we=es (x ee ee 4c +1

 

xt — 2x5 4 x? x(a — 1)
x dx
12. {aS Here there are a pair of imaginary roots.
G+ ite + ay . ae
x A Lx + M

 

@PnedPy ~r4¢e° Tee"
ART. 141.] GENERAL INTEGRALS. 205

Clearing of fractions,
x = A(t + x2) + (Lx + MI + x),
=(A4M)4(L4 Myx + (4 + Dx.
Equating coefficients,
ie L4M=1, A+M=o.
= M = 4, AS ph

xdx I+ x? .
Desi +1) = Flo crests tan—Ix,

13. ft We have T+ x3 = (14 x\(r — « + 2%),

 

: I Joel Lx + MM
a toe (pte foe
Clear the fractions and puta = —1. Then 4 =}. Substituting this, we
et
A 3(Ze + M) = 2— «x.

 

ax _i (2 — x)dx
far =3 [5+ +if, r— xt x?’
= Flog (1+ x) — J log (haat a) + ton

3 V3
ef Sse APE oe
8. fee:

eet esata
at = Alt $ 2) + Bee — Hla? 4) + (Le + Me — 1p

Equating coefficients,

 

A-~BtM=o0,
A—B—-2L+4+M=1,

 

 

 

B+Ll=o,
Bo+l~2M@=0.
“ M=0 B=Az=}4, L=—f.
Hence the integral is :
I
ea kaa <1 = a 24
Paeeaia oe og (« — 1) og (x? +
St+ 12 4 = pili ott ce tie
AB, x(x? + 4) eee yeta 2 _ 2
(2x? — 3x — 3)dx Io (x? — 2x + 5)! Sg GO
ti Jeane one es ;
= I et 3 = x
as de S42 Flog? ES ae
18. {seee ast j 8 —a V3 tan i

ax = * i, x4
1 faaaen a" ee

I
— —tanlx.
2
206 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIII.

20. f ee as ae dx. Here there is a double pair of imaginary roots. Hence
x

we put
284tet3 Ax4+B Cx+D
@piy “Gi ape
o 28+ta4+3=C¥4 D+ (44+ Cee4+84D.
ow A=—1, B=3 C=2, D=0.
a faxt+3 _ —*#+3, 2%
@+iy ~@pit ayer

To integrate f oa put x = tan 9, then the integral becomes

 

 

 

 

 

 

ff cos dO = 49 +2 sin 26 = Ptane + A,
2 ae = RE + tants + log (0 + 0.
le eee ae ae
a £ ea an os qe aa ee 2) — ; “a tan- a ;
23. fs a ee a =% Sos 5 tee 7 4tanmte:

142. Trigonometric Transformations.—On account of the simple
character of the reduction formule in §§ 139, 140, it is often advan-
tageous to transform many algebraic integrals to these forms, and con-
versely many trigonometrical formule can be transformed into useful
algebraic forms.*

 

EXAMPLES.

1. Put « =atan6, then
xm dx

(a? + x)"

2. Put x =asin6, then

if am dx = anne f =e a6.
(@ — xt cos*—19
3. Put x =asec§, then

xm dx peas f cos*#—m—29 a.

(a? — ay sin’—19
4. Put x = 2asin’@, then
xm ax

(2ax — xy

= quant f sin™§ cos*—m—26 a6.

 

jn2"—#+1
= amcmtzgmnts f SOS ag.
cos*—19

5. Make the same transformations in the above integrals when m or x is
negative.

 

* The reduction formule for the binomial differentials are given in the Ap-
pendix, Note ro.
ART. 144.] GENERAL INTEGRALS. 207

The general integral
xm dx

(@ + cx*)™
can always be transformed to the trigonometric integral when the signs
of a and ¢ are known, whatever be the signs of mand a.

EXAMPLES.

1, Integrate by trigonometrical transformations

f Ve — Pee: { ye? we, i VF +e dx,
ax a:
I Va — x?” I a I ie

RATIONALIZATION.

 

143. Integration of Monomials.—If an algebraic function con-
tains fractional powers of the variable x, it can be made rational by
the substitution + = 2", where is the least common multiple of the
denominators of the several fractional powers,

+
For example, furs ne Jay
1+ at

Put «= 24, The transformed integral is
23(1 + 2) dz
af page
Consequently the integral is

xt — oxt — get + g tan—at — 2 log (1 + 2°).
Again, any algebraic function containing integral powers of x along
with fractional powers of a linear function @ + 4x can be ration-
alized by the transformation a + dx* = 2”, in the same way as above.

 

EXAMPLES.
. f wes (5x8 + 6x? 4 8x 4 16) YF —T.
2. f = 2 oe,
(atbxit 8 Vat bx

Complete the differential, integrate and compare results.
3. — = log(x + x —1) — 2 tan
a+ yx-—1 V3

(x? + I)dx pee f
i xy¥efatpr 9 ¥Ppak2
144. Observations on Integration.—As we have remarked be-

fore, comparatively few functions have primitives which can be
expressed in a finite form of the elementary functions, For example,

by a+ bx = 27,

2Yx—141.
"3,

 

Pat «> 2? =e
208 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIII.

[virdx, when y isa polynomial in « of degree higher than the second,
is not, in general, an elementary function and cannot be expressed in
finite form in terms of the elementary functions. Ify is of the third

or fourth degree, the integral defines a new class of functions called
elliptic functions.

Functions that are non-integrable in terms of the elementary
functions can frequently be expanded by Taylor’s series and the integral
evaluated by means of the infinite series.

Any rational algebraic function of x and 4/ax® + 6x + ¢ can be
rationalized and integrated as follows:

Factor out the coefficient of 2? and lety = 42+ px +g.

The rational function (x, y) is rationalized in +:

I. When the coefficient of x? in y is positive, by the substitution
V~etprtearse— x.

 

 

 

Then watat, i gi ET gi 2(z°+ pe+9) 5
Poe p+ 22 “(p+ 22)?

— 7 spe g\s pe +9

of Mey\te = a fr Gs aad pe

II. When the coefficient of x? is negative and the roots of the
quadratic aw, # are real, then

— att pe tg = (x — a)(6 — 2).
The function F(x, y) is rationalized by either of the substitutions
/— F px +9 = ¥@—a)(B— x) = (x — ade or (6 — 2),
Then _ av B 22(a— f) (8 —a)z

fo ee, OS dz, (x —a)jg= a

1+ 22’ “(G+ 2)?
= aft 8 (B—a)z\ 2dz
f Poses a0) fr EE SS ae

When the roots of — x? + px + g are imaginary the radical is
imaginary.

 

 

 

145. Integration by Infinite Series.—We know that if a function
AX) HQ taxtaet+...
in an interval ) — H, + H(, then also its primitive is equal to the
primitive of the series for this same interval (§ 72). Hence

[Aa)de = age + 40,0? + Jas? +
EXAMPLES.
Txf Lez a 1.3.5 x6
te aS

I
1 sintx 1-3 sinty
2 f= =2 ae + — 7)
sin x v 2 ; 3 :

 

 

 

 

 

 

 
ART. 145.] GENERAL INTEGRALS. 209

Put sing =z. .. dx = dz/cos., and the integral is

dz
{a yi gh
b 2 2
igi Ee peo x Pip -— 9)? x?"
3. fo A cxt)a xm—tde = xm C +e ae cet ete z caer rer +...
For what values of x is this true?
4. Show that

f ax x 1 x5 1-349 re
SSS SS ie Fe gy 2h ae i.
Yr + xt Ro2 5 "249 : 7
I rt
Se aes a> I.

x ' 2 545 2-4 9x9
5. Show that

 

eax ab te x @ (64x)
dy See — e-ab so
ae zie = a) og (6 +2) +4 i = +...
Determine the values of x for which this is true.
Puté+x=o%2 1. ct = e—abeaz, etc.

6. The elliptic integral y (1 — # sintxsidx, # <1, can always be ex.

panded by the binomial formula, and the general term f sin’ dx integrated.

3 2 4
7. [Weel G -p ata hn .}.
3 7a OS git
EXERCISES.
1 f ~< ee = Ssin-ix — Le 71 — 27 (3 + 2x).
Gy}
- —~VWt— x2 T— x
2 f=... pi a
Rou To. 42 G: 2x

g = x _ Fa
‘7 (at + xt)F ata? + x*)# — 3at(at + at)?

xt dx — x 3 _*
4. leper or = wage ti (e- ate =).

a as =
e La = 2b — (2ax — x) (4x + $a) + 3a? sot
on 3 6 6
6, fier dx = (8 — Sa + Se — 5).

7. fx (log x)? dv = fxt[(log x? — 4 log x + 3).
8. fs cos x dx = 8 sin x + 3x% cos x — 6x sin x — 6 cos x.
9. fx sin x dx = — xt cos x + 4x3 sin x + 1247 cos x — 24 (x sin x + Cos x).

sin?@ 29
10. Je aa = Gi a= = 2tan 16 — 4.
210 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIII.

11. cos sin 26 d6 = — 2 cos*§.

12. if sin’s cos*@ d@ = 3 sin’@ — 3} sind.

13. if sinto cos’@ dg = — 35 (cos 26 — 2 cos’ 26 + 1 cos® 26).
14, f costx csc x dx = 1 cos’x + cos x + log tan fx.

15.

costx cschx dx = (cos8x — § cos x) csc?x — $ log tan fx.

f
16. tay V4 2x— V2—%
pe

arama 4V2 V4paxpVa—a-

a = (fet fet st na(n fat
(+ a
ax
To + xt Lxit z= + .
rs f [(a? + 2%)t + ax]? =o (Pe oe

Show that f x™[(a? + 2°) + x]*dx can be integrated by the same sub-

stitution when m is a positive integer.

 

(1 + 7) + aT is I o
9. [ar ag let at ale
xt dx 4 }
20. Jaa ee $ log ( + 1).
ax ge +5) 1)°
[aa

 

at4iy 6 12
=— st — Ht log 2 thy
22. I" a 3 ur og 24 log (+17 +- 1).

: + 4 tan-tt.

 

23. ff = a Fett 2 log
xb — xb

j= dx a
aYxtir — Yx+141
f xwidx _ Ox* + ox+1

24.

25.

(Gxt xi 2(4e+1

eee + . : .
26. fa (* + ¥) 3(@ + 1)? + 3 log (1 + Wr + =)

 

dx 4p 4 i
27. tas = a 2a)(at + a)}.
33 dx I j.——;
ag ree
2 dx I fa Sass
29. era = 7" — 2x7 + 2) 227 FT.
 

 

 

ART. 145.] GENERAL INTEGRALS. 211
30. j (a? — x2bdx = Bylo — atx? — 5a4)(a? — x28,
31. (—S = ies Ma ew? i
“ xx? + at 2a oa aae
x dx as
ai =. t
32, tame ‘eae = log (73 — +? + 1)t + log (4/3 — a? — 3)8.
a3, ax pl Se
nepaae “7 ee Vee
a. ax Vroxpetxe— 2
; Waa ee Vi Ot ee aes
dx s ee
35. as («+ 7x + ex — 1).
36. | vee a” a it

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

              

 

 

 

Ox — x? ss : \=
5 =— —— ets 2 tant... |.—_—_-.
tae ae co =
fi ax - Yr — 2x +2
(« — 1) x? =e ca ET .
x 3x — 5
fa dx = log (x ~ 2) — ES, Put
wide
@tye a
f—— = * og Reece Put
xYarsxt? 4 at a? + x?
2 dx
: J @igh 8 et ry. Put
(#24 ee (x +a) cos a —sin a log sin(x-+a@), Put
e2% ax
ae x 2
Saag gp ROO 9 9 Put
ax
—— ee oa e+e log (e* — 2).
eee 3).
feo log x dx = ~ x (le x 3)
frsinx ds = — x cos x + sin x.
flo (x + 2) dx = (x? — 4) log Wx +2 —F +x.

xa—-2>2.

~a+t+iIn=s

xz a,
e@4trms,
xeta=z.
w+ 2,
212 PRINCIPLES OF THE INTEGRAL CALCULUS. [CxH. XVII.

50. f x tan-tx dx = $(x? + 1) tan-lw — dex.

51. Integrate
fre — 2) dx, fi — x8 dx, fe (2? + x?) dx,

x? 4/a? — x? dx, x? at + x? dx, fe — «Pde.

52. f sintx cos’x dx = 4cos x(} sin'x — x, sinix — 2 sin x) + pyr.

 

pe ee

of 2-5 COs a 4 tan tx — 2

dx I
54, {== S — tan-1(2 tan $x)
pa. ff ___* _. Put «2 =1.
(@+ cx)? aat exr)#

56 dx = b+ cx

te + 26x 4+ cx)? (ac — 8 )(a + 26x + ext)

Complete the square and put cx + %4—=2. The integral reduces to 55.
57. f (Ptgx)de —ag+(p — bg)x
(a -- 26x 4+ cx*)8 (ac — B Ya 4+ 26x + cx)

For xz = 1 transforms

 

 

 

ax dx 2 — dz
3 Into ra
(a + 2bx + cx)? (az* + 262 +e?
f x ax “ a+ bx
(a+ 2bx-+ex)! (ae — BY(a + 26x + ex?)

Combining with Ex. 56, the result follows at once.

 

 

 

fe a
1 — 2x 207)" (1 — 2x 4 2x2)?

on Je — ala — on —a—) (a = bp log Ge a

0 eaE= NETH ma “oH ee 5

a f Bo 0. at = =7b a ay

62. feat Secs =e + Flog OE 3"

_ ty oe — 2) + 3)

63. — wea 6 ee weg
x? dx (30? — 7 +7)dx

i faites) oe
ee ee i — 2h 2)

@—

 

’
Arr. 145.] GENERAL INTEGRALS, 213

dx
65. Sas =i
ay oe i = =a

 

at dx a 81 206
66. fia Fy ~ ie ORD + Glos ait 2 log (e+ 4)
ax Tod x
e lasoaz = ave ae
d.
6 a 2 log ~ aes

(x — es — 2)
rs i 2 a wb I ax —a—b
(x — its — BF ~ (a — op 8 era (a — b) (x — ay(x — b)

dx I I x
10. f sarpay =~ ag tots

69.

 

 

 

a te ao Ge ae a 18 Pet ae
Notice wt get 3 = (x — 1a? 4+ 2x 4 3).

& (x — ae + 2) = = + log oF : 3 + 2 tan“ — 1).

3. foto papain it+ tate

coe att, Put 23 =z,

a Jaze gin SES,

”. foo a+ OTR cae a

e inmate to

79. Tf fix) = (« — a)... (« — ay), and A(x) is a polynomial of degree less

than 2, show that
Fe) y,  & Far)
SFR? = > Fla F(a,) 8 (x — @,).

80. Show that any algebraic aa involving integral powers of x and frac-
tional powers of
_ a+ bx

~ p+ qx

can be rationalized by putting y = 2”, where is the least common multiple of
the denominators of the fractional powers. Apply to Exs. 81, 82.

V¥x—b+Ye—@
¥xr—b—-Yx—a

 

 

 

a = (« — ax — 4) — “(a — 6) log

82. f a* f = = 5) oe f = =‘) vi

 

 

 
214 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cu. XVIII.

83. If f(x) is a rational function of sin x, cos x, then /(x) dx is rationalized
by the substitution tan 4x = 2.

I— 2 d 2d
= —— +
ae 1+ 22’ 1+ 2?
In particular, when # — 1, 2 — 1, or m+ is even, say 27, we get for these
respective cases

Then sin oS cos x =

sin”x cos¥x dx = — cX(1 — ¢*)r de,
= + smi — s°)rds,
_ _ tmdt
~ OF apr?
where S=sinx, c=cosx, ¢= tans.
: a. : :
84. To integrate (=, where Q,, Q, are any quadratic functions of 4.

Write out Q,— in partial: ‘fractions, This reduces the integral to § 136, (B), or
to § 138, (D).

85. In general, if /(x, y) is any rational function of x and y, where
yaat rx + cxt= cx — alle — f),
then any one of the following substitutions will rationalize f(x, y) dx:

y = at xz,
= 24 xet,
= a(x — a).
a8 . : Put aes,
86. lam (@ + + =({ = OF atx yn maa dy for the other factor.

» fate mena erent *- feret-
87. Given the signs of the constants a and 4, transform the binomial differential
x(a + bx8)¥ dx
into the trigonometrical differential
C sin”x costx dx,
determining the constants C, m and x for each case.
CHAPTER XIX.
ON DEFINITE INTEGRATION.
146. The Symbol of Substitution.—We use the symbol
x=b
F (*)] >
or, in the abbreviated form when the variable is understood,
B
ff (x)]5

to mean that the number a is to be substituted for x in the function
and the result subtracted from the value of the function when 2 is sub-
stituted for x. Thus

Fx)], = F(b) — F(a).
If F(x) is a primitive of f(x), then we have
[70 a= FO] = Fe) — Fo).

The definite integral is a function of its limits. If one limit is
constant the definite integral is a function of one variable, the other
limit.

147. Interchange of Limits.

Since [fl Ax) & = FO) — Fa),
i " f(x) dx = F(a) — F(s),
ef Fea) de = — [Ae ax.

That is, interchange of the limits is equivalent to a change of sign
of the definite integral.

This is also at once obvious from the original definition of an
integral. For dx has opposite signs in the two limit-sums

[Ae dx and Ax) de,

while they are equal in absolute value.
215
216 PRINCIPLES OF THE INTEGRAL CALCULUS.  [Cu. XIX.

148. New Limits for Change of VWariable.—If we transform
the integral

i Z (x) dx

by the substitution of a new variable for x, then we have to find the
corresponding new limits.

Let the substitution be x = (2), which solved for 2 gives
z= p(x). Then, when x = x,, we have z, = y:(x,), and when « = X,
Z=7(X). Also,

Ix) dx =f [P(2)] O'(2) #2 = F(z) ae.
ie Ax) dx = if ” F(a) ds.

For example, put « = a tan z. Whence z = tan . When x =0, then z =0;

when x = a, thenz = 4. Consequently
dar

2 dx I I
f =i f/ cos saz = =
9 (a@ tery Td, a 2

since feos zaz=sin z.

 

149. Decomposition of the Definite Integral Limits.
If

f . Nx) dx = F(X) — F(x,),
then a

- * Ax) dx = F(a) — F(x),
f Ax) dx = F(X) — Fia).

Whence, on addition,
7 xX x
if J(*) dx +f J(x) dx =f SJ (x) dx,

Therefore a definite integral is equal to the sum of the definite
integrals taken over the partial intervals. This is also immediately
evident from the definition of the definite integral.

EXAMPLES.
Evaluate the following definite oa

Eyes) g-i=

2 fs = log = ]'= ee

3. fvsinadr =f" cosa dx = 1.
0 0
Art. 149.] ON DEFINITE INTEGRATION. 217

 

 

b 43
4. f (Px — Bide = 40. 5. f ao de i:
0
3 x ax — © 8a5 dx 3
6. f cag PE Ys 7. f wage ee
a ar
8. f Bee 22, 9. f[ vers’§ d§ = 37.
0 Yar—xt 2 0
iw dm
10. fl sh dO = 4. 11. [ cos 26 dj = }.
0 0
dn 5
12. f cos’x sin x dx = }. 13. i (3 Vt — f 2)dt = 2y5—5.
0 0
ee ax we 30 Ave _ I
4. [ eae a a 5. [e de ==.
7 4
16. t: ui a, 17. ae gh
np «cSt 3 7 COS x V3
o

1. ax =f dx _ @
. [rpicare m2 y £+2xrcse Gtx 2sing

oO
A o
19. [o sin mx ax = 20. jes cos mx dx =
0 0

m
at me atm?

pee dx x '
eee eg ee
22. Show, by putting + = 1 — gz, that

1 1
fore — x)itdx = fare — x) dx.
0 0

This is called the First Eulerian Integral. Integrating by parts,

fea a x yuri ax= mua + mas fant 3 x)? dx.

Use this to show that the value of the above integral is
i (g — 3)!

PONT EI ee

i Pet... (P+tg—I

when g is a positive integer, and therefore whenever # or g is a positive integer
the integral can be evaluated.

213
3711-13) 5+7-9+13-17—

1
230), feo — x)idx =
0
oo
24. The integral fe e—*x"dx is called the Second Eulerian Integral or the
0

1
232), fc —x) ax
0

Gamma-function, D'(n + 1).
We have, by parts,

e-*xh dx = — e-*yn +n f ean! dx,
Since ¢~*x* = Oo when x = Oand whenu = 0
+

ro) 30
if e-*y"dx =n a e—*y"—-l dx,
0 0

I(2 + 1) = xT (x).
218 PRINCIPLES OF THE INTEGRAL CALCULUS.  [CH. XIX.

Also, when z is an integer,
Ta+i1j=2!
The Eulerian Integrals are fundamental in the theory of definite integrals.
1 (— 1)*”! 1 1\*
25. fem 24 dx = ——__——-__— = —y f cd —) dx.
, (log x)* dx Le (-1) | = log =) ax
Hint. Put ¢e-* =< in Ex. 24.
© !
26. f e—azg" dz = = Put x» = az in Ex. 24.

. a0 {7
27. Show that 1 costs dx = f sintx dx,
0 0

 

and that
$0
f sin, dx = ea ee Sa Sue
0 2-4-6...2m 2’
i 2-4:6... 2m

sint+1y dx — —— "+
f EN By a «= (EE Ay

when m is a positive integer.

fig an 8\F
28. i, eae) ax = 6. Put xz le
(a — 2)tdx —
29. =8+8 ra Put —2= 2%,
i. AWS nner ca
log 5 ae on
30. f € es Yer I dx=4—2. Put e*—1= 2.
+3
ay nu
SS EE Wh b
af a+6cos6” gi — # ete ee

in ;
32. x sin «dx = I.
y
150. A Theorem of Mean Value.—Since in
x
[Cfo a

dx keeps the same sign throughout the summation,

a x Ax) dx <M 3
m fr a< ff (x) dx < im x,

where mand J/are the least and greatest values respectively of the func-
tion /(x) in (%,, X). Therefore the integral lies in value between
m(X — x,) and M(X — x,).

Since Nea) i is continuous in the interval, there must bea value of x,
say , in (x,, X), for which

[ 70) = (Xa) AB),

J(&) being a value of the function between m and J, its least and
greatest values.
ART. 150. ] ON DEFINITE INTEGRATION. 219
The value
Aa) = ye f Ax) ax

Vy x,
is called the mean value of the a in (x, X).
If /(x) is a primitive of (x), then

F(X) — F(x) = (X — ») /(S),
= (X — x,) F'(8),
since #’(x) = /(x). This is the familiar form of the Law of the
Mean as established in the Differential Calculus.

The theorem of mean value for the Integral Calculus can be estab-
lished directly from the definition of a mean value. For, if

4x = (NX — x,)/n,
then

ie Ax) dx 2 £" 3 Ax fe),

n=O r=l

= 5) {@ tA. tA)

If the limit of the arithmetical mean of the z values of the function
at the points of equal division of (~,, X) be indicated by /(&), the
result is the same as above indicated.

GEOMETRICAL ILLUSTRATION.

If 7 = f(x) is represented by the curve 48, then

 

 

ie
ydx = area (x,AZBX). B
*o . y 8

This area lies between the rectangles Rk CG

«47-1 and x,SBX, constructed with +4
as base and the least and greatest ordinates
to the curve respectively as altitudes.
There is evidently a point € between xy
and X at which the ordinate €Z = /(é)
is the altitude of a rectangle +,AQ4X, inter-

 

 

 

 

 

mediate in area between the greatest and Ol x 2 xX x

least rectangles, whose area is equal to that :

bounded by the curve. Fic. 68.
EXAMPLES.

1, Find the mean value of the ordinate of a semi-circle, supposing the ordinates
taken at equidistant intervals along the diameter,

Let x? + y? = a? be the circle. Then
I
xf. Ya — x? dx = ja,

viz., the length of an arc of 45°,
220 PRINCIPLES OF THE INTEGRAL CALCULUS.  [Cu. XIX.

2. In the same case, suppose the ordinates drawn through equidistant points
measured along the circumference. Then the arc length is the variable, and the

mean ordinate is
I 7 wt 2
=f asin §d9 = —a.
ao u

We shall see later that this is the ordinate of the centroid of the semi-
circumference.

3. A number z is divided at random into two parts; find the mean value of
their product.

I 135
5 fee — alder = 5 0

4, Find the mean value of cos x between — wand + z.

5. If M(9) is the mean value of y = f(x) in (x,, x,), show that:
(a). M"(2x* + 3” — 1) = 8}.
(3). Mi(2 — 3x4 52% — 8) =
(). M(x + x + 2) = 124.
(d). M3"(sin 6) = 2/7.

6. Find the mean distance of the points on the semi-circumference of a circle of
radius 7, from one end of the semi-circumference, with respect to the angle.

Mu 2 for cos do — ©,
0 do "

By the mean value of z numbers is meant the wth part of their
sum. To estimate the mean value of a continuous variable between
assigned values, we take the mean of 2 values corresponding to equi-
distant values of some independent variable and find the limit of this
average when the number of values is increased indefinitely. The
mean value depends on the variable selected. See Exs. 1 and 2 above.

If_y is a function of /, then the mean value of y with respect to ¢
for the interval (4, 4) is

I
4 4 z

 

ty
y dt. ,

1

151. An Extension of the Law of the Mean.—If f(x) and 7(x)
are two continuous functions of x, one of which, (x), has the same
sign for all values of 1 in (v,, ), then we shall have

[Eee dx = G8) [7 p(2 ax,

where & is some number between .v, and X.
For if m and JZ are the least and greatest values of A(x) in
(«,, X), then the integral must lie between the numbers

m [oe dx and M 1. “b(2) dx,

since y() dx does not change sign in (x,, X). Therefore there
ART. 152.] ON DEFINITE INTEGRATION. 221
must be a number & in (x,, X) for which the integral has the value
proposed, since #(-) is a continuous function.

152. The Taylor-Lagrange Law of Mean Value.—Integration
by parts furnishes a simple and an elegant method of deducing the
important formula of Lagrange, and gives the form of the remainder
in a much more useful form than that of the Differential Calculus.

Let z be a variable in the fixed interval (a, x). Then

f(x) —Aa) = [P'(@) de = — [7 dx — 2).
Put w#=/"(z), dv=d(x —2z), and integrate by parts.
os fix) —Na) = —(@-— YOR + [we — a7" &,
= (x — a)f"(a) — [@ — 2) f" (2) d(x — 2).

Put w= /'(z), dv = (x — 2z)d(x — 2), and integrate the
integral on the right by parts. Then

Fe) —Ma= oar 42S p(ay—_ [ES pr ayat—2)

Continue to integrate by parts in the same way, and there results

(Oa Le On a a eS

This is Lagrange’s theorem with the terminal term expressed as
a definite integral. This form of the terminal term shows that the
difference between the function /(x) and the series vanishes when
n= ©, provided

 

(x = 2)"

fGirro =e (2)

n!
Sor all values of z in (a, x); and moreover, if this limit is not zero
for any finite subinterval of (a2, +), however small, the terminal
term does not vanish and the series, although convergent, cannot be
equal to the function. *
The law of the mean expressed in § 151 enables us to transform
the definite integral in (1) directly into the forms of the terminal

 

* The reader should be warned against the language of many writers who con-
found the remainder of Taylor’s series with the terminal term of the law of the
mean, for they may be quite different. In fact, if Taylor’s series S. is convergent
and S, = S,+ &y, then we should write

Ae) = Sat Rat Te
The terminal term being &, + 7,. In order that /(x) = S, it is necessary that
both ane =0, £7, =0. £Ry = O does not ensure £ 7, = 0. See Appendix
Note 8.
\222 PRINCIPLES OF THE INTEGRAL CALCULUS.  [Cu. XIX,

term given in the differential calculus. For, since (« — 2)? keeps
its sign unchanged for all values of z in (a, x), we have

ae (x — 0" pea) a= a Sy pee) (x —2)?dz, (3)

where & is some number between a and x. This result, (3), takes
Lagrange’s form when p = z, and Cauchy’s when p = 0. The more
general form (3), where g is any integer, is due to Schlémilch and
Roche.

153. The Definite Integral Calculated by Series.—If /(z) can
be expressed in powers of (2 — a) by Taylor’s series, for all values
of z in (a, x), then also can the primitive of /(z), and the definite
integral of the function is equal to that of the series, taken term by
term, between a and x. Hence, ee between a and +,

=

Ae) = Aa) + (@ — f(a) + ESF pray +
we have
[fo @ = (apa) + FSF (a vatece ee

In particular, put + = 0, shen we have

[194 = 219 -SP@+ 57" - (

a formula due to Bernoulli.
Knowledge of the derivatives at a serve therefore to compute the
integral. When a = o in (1), then

a 2 3
[7e) dz = 40) + 5/0) + we"(0) Aes. Gy

which is Maclaurin’s form, and is more convenient, in general, for
computation than (2).

t2

)

EXAMPLES.

1, Deduce Bernoulli’s formula (2), § 153, by using the formula for parts,

[pe ) dx = aflx) — f af" (x) dx,

II
2. o fest dv =a (1 +2 toa ta -):
ae eo re 1x7
fe Cae yg ace

tr I I I
4. fee tian oe =—(1-54+5-n+---).

154. Observations on Definite Integration.—In order that a
function may admit of definite integration in an interval (a, () it
must, in general, be one-valued and continuous throughout the
ART. 154.] ON DEFINITE INTEGRATION. 223

interval. If the function is not one-valued, then generally the
branches must be separated so that each may be taken as a one-
valued function. If the function becomes infinite for anv value of
the variable between the limits of integration, tnen for such particu-
lar values of the variable the integral must receive special investiga-
tion, a case which we do not consider in this text.

In definite integration when one of the limits is infinite, we
consider the integral

[ fe ax
as the limit to which converges the integral

f[ f(x) ax,

when x = o, provided there be such a limit. The same remark
holds when one limit is — o and the other + o.

All continuous one-valued functions are integrable in the interval
of continuity, as demonstrated in the Appendix, Note 9. But all
continuous one-valued functions are not differentiable (see Appendix.
Note 1).

The study of definite integrals will be taken up again in Book II.

EXERCISES.
[f° = en 2. [San
0 VYa—x 0 vax — x
1
3. f= 4, [sine de = ye — 1,
1 exe —1 0
i ax u tm
8. f —_—_—__ = ——.. 6. f tandx dx = log 2t — }.
op 2+ cosx 343 0 8 t

1 ["__# ae a (i __# _ 6
ag 1+ cos@cos x sin@ -f 1+ cos @cosx*” sin 6"

9 jr ax _ 2

: f a sintx + 6 costx — 24d

10. i ax _ x(a® + 8?)
0

(@ sin’x + 6? cos’x)? ~ —4a%8

B dx = iv sin x dx = _ 1
tl esas © ha ers hg,
13. Show that, when # < 1,

ft" id ae r+ () e+ ($3) + (533) # +
0 i—esin® 2 2 2-4 2-4-6 |
This is an elliptic integral.
14. Show that
n n n x
df artape? te) >

 

 

n 1 n+ 2? 2H)
224 PRINCIPLES OF THE INTEGRAL CALCULUS,  [Cu. XIX.

Put dx = 1/n. The limit of the sum is then

1 dx _
I Se-7

15. Show that the limit of the sum
I I I
Vee T gon °° Gece

when z = 0, is }z.
16. Show that

T TT
f sin mx sin nx dx and f cos 72x cos wx ax
0 0

are zero when m and 7 are unequal integers, and are equal to $a if m and # are
equal integers.

i
17. sin’x cos’x dx — 2..
, TS

+ sing + cos 9 ‘
Se are — . a 6=
18. f 3 4 sin 20 a6 = } log 3 Put sin 6 ; cos x
2+ V5 (x!
19. f ie ee = log 3. Put x—at= sz,
1 ax xt 7? +1
7 ax a 0.
on f I — 2acosx ta? 1—a ZI. f RES
3 x dx tvs ax La
22. f —— = |] : 23. f SOS
ae ars 0 ¥2—3%7 443
= ax rf 1 ax
a4. ae eee sara =} log
Oo ada a
26. Pas gy za = thee. 27. is aE ee
. ax x dx m—2
2 fo a
+
30. f[ "tan des log 4/2, 31. ft sectx dx = 1.
0
7 dy n a an
ads f a+ b°—2ab cos§ — 2 a

        

1 3 1
34. f[ xr — x)'dxe = 2 fe sin5§ cos4@ d§@ — 345.
0 0

0

35. f er — xd = [ost costo of =~.
0 32

36. [ x8 oy — 3

. 1— =25
fd x?( x)? dx = 158"
37. Putting e* —1 = y?, show that

= 1 2 d, 4-2
eS rde=2 f J Ee .
0 v fp I+yv 2

log 2

 

 
ART. 154.] ON DEFINITE INTEGRATION. 225

38. if = 4- I=y,
1 2 sae
fo vise dem fo yi wy = y3—dbogl2 + 43).

39. Putting +. =a sinQg,
4
"x Yo — 2 dx = af" sin’@ cos*@ d@ =
0 0
40. If «= actané,
a dx ind 5
[tno hae Ee
0 0

a x) cos? @ 2
PART IV.
APPLICATIONS OF INTEGRATION.

CHAPTER XX.
ON THE AREAS OF PLANE CURVES.

155. Areas of Curves. Rectangular Coordinates.—The sim-
plest method of considering the area of a curve is to suppose it
referred to rectangular coordinates. The area bounded by the
curve, the x-axis, and two ordinates corresponding to the values
x,, x, of x, is represented by the definite integral

A= * y dx.

x9
This has been shown to be true in Chapter XVI, as an illustra-
tion of the definite integral. It has been shown that the definite
integral is independent of the manner in which the ordinates are dis-

tributed in making the summation.
We demonstrate again that the definite integral gives the area in
question. For simplicity we divide the interval (x,, x,) into 2
B equal parts, each equal
f Sto 4x. Let AB be the
Q curve representing the
D equation y= /(x), and
x,ABx, the boundary of
the ‘area required. Let
A IR MN be one of the sub-
divisions of x,x,, Draw
ordinates to the curve at
each of the points of
division, and construct
| es ae ey, “1 the z rectangles such as
Fic. 69. MPN, and also the x
rectangles such as QM. Since the curve is continuous, we can
always take 4x or ZN so small that for each corresponding pair of
rectangles the curve PQ lies inside the rectangle PsQg, and therefore
the area Z/PQN of the curve lies between the areas of the rectangles
MP N and MpQN. Hence the whole area x,ABx, for the curve

226

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
ART. 155,] ON THE AREAS OF PLANE CURVES. 227

lies between the sum of the rectangles represented by A7P¢gN and the
sum of those represented by A4QN. The difference between the
sums of these rectangles is the sum of # rectangles of type PpQ¢.
Which sum is equal to a rectangle represented by B&R, whose base
BS is Ax and altitude RS is y, —y,, where y, = xB, y, = x,A.
(¥,,.%,) being the greatest and least ordinates in the interval. When
the number of rectangles, #, is increased indefinitely, the difference
between the sums of the rectangles, the one greater, the other less,
than the curved area, converges to zero. Therefore the sum of
either set of rectangles has for its limit, when x = o, the area of
the curve, or
“ x,
ZL Sydx= ff *y dx.
n=n I Xo
If y = /(x) is the equation of a curve, the area A included
between the curve, the ordinates y,, y, at x,, x,, and the x-axis is

= f Ax) dx.

EXAMPLES.
1, Area of the circle.

Taking x? + y? = a? as the equation of the circle,
you VRE

ae
B

 

P
PB

 

 

zw

 

O| xo vwmyA
Fic. 70.

Taking the oe value of the radical, we have for the area x, P,P,

2 x2)\h a ca
" Vat — x wt dx = a -+- = sin-1 =| :

ie 2 2 a jx,

If x, =a, we get the area of the semi-segment x,P,Ad. If x) =0, and x, =a,
we have the area of the quadrant OBA equal to

ce Yar — x? dx = yma’.

me is the angle POA, then y = asin 6, « = a cos 6.
= — asin d. The area, 4, of the circular quadrant is then given by

0
A= [yds saat f sinto dg = at [sina a,
0 iv

= 409 — sin 6 cos 6) |" = 42a’.

The area of the entire circle is therefore ra’.
228 APPLICATIONS OF INTEGRATION. [CH. XX.

2. The area of the edipse.
4 i—_—
From the equation of the ellipse x?/a? + y?/d? = 1, we gety = 2 Yar — x,

Consequently, as in Ex. 1, the area of the elliptic quadrant is

 

 

 

 

6 pe
y 4=—-f Ya — x? dx,
a
iP which is 5/a times the corresponding area of a circle
of radius a2. Hence the area of the entire ellipse is
nab.
O IN « 3. Area of the parabola.
Taking y? = px as the equation of the curve,
' P’ and the positive value of the radical iny = /px, we
' M have the curve OP. The area OPM is then
ee os 3*
A= f YVpx ax = apt],
Fic. 71. a
= ay

But xy is the area of the rectangle OVP. The area of the segment POP’ of
the parabola cut off by a chord perpendicular to the
diameter is two thirds the rectangle J7PP'1Z'. y

4, Area of the hyperbola.

Let x2/a? — y?/b? = 1 be the equation to the P
curve. Then the area of AP is

A= fy dz, éIN 4\N =

b ei

m Vx? — a? dx,

b ss ee a wa) x

=F = xt a — — log (# + V7 — a ; Fic. 72.
a

a@\|2

b ——, «a xf V¥r—a
ae 2 =. Gea EE a aE ESS
mat V* a 7 108 - ‘

= tay — 4ab log E +3) .

5. Area of the catenary.
The equation to the curve is

pad ees
y=ta (G +e 2) .
The area OV PV is

: A= fie (a aes) dx,

x x

=i Ge =) =H=avy—a’.
If VZ is pexpendicular to the tangent at P, show that the above area is twice
that of the triangle PLM. Observe that tan LVP = Dy, LN = y cos LNP, etc.
6. Show that the area of a sector of the eguzlateral hyperbola x* — y* = a
included between the x-axis and a diameter through the point x, y of the curve is

4a’ log a,

 

Fic. 73.

7. Find the entire area between the wztch of Agnesi and its asymptote.
The equation is (2? + 4a*)y = 8a. Ans. 47a’.

 
ART. 155.] ON THE AREAS OF PLANE CURVES. 229

8. Find the area between the curve y = log x and the x-axis, tounded by the
ordinates at « = I and x. Ans, x(logx —1)+41.

9. Find the area bounded by the coordinate axes and the paradola xt + y* = at.
Ans. ia’.

10. Find the entire area within the curve 3) es (3) Ps I. Ans. 4nab.
a

11, Find the entire area within the Aypocycloid xt + y? = al.
Hint. Put «= asin*9, y = 2 cos*9. Ans. $n’.

12, Find the entire area between the cissotd (2a — x)y? = x3, and its asymp-
tote « = 2a. Ans, 37a’.

13. Find the area included between the parabola x* = 4ay and the wttch
y(x? + 4a?) = 8a. Ans. a(2m — 4).
The origin and the point of intersection of the curve give the limits of the
integral.
14. Find the area of the loop of the curve y
cy? = (x — al(x — By.

(6—aye
Hint. Let x — @ = 2%. Ans. ay" 2) .
15 ¢

 

Fic. 74.
15, Find the whole area of the curve a*y? = x3(2a — x). Ans. 1a".
y 16. Find the area of the loop c. ‘he curve
aby? — x4(b + x).
The area of the loop is
x 2 po 325%
A= =/ 2 4/k 2 ye es a

oO aol? Yot+uxdx= greegal

Put 6+x= 2%

Fic. 75.

17. Show that ify = /(x) is the equation of a curve referred to oblique coordi-
nate axes inclined at an angle @, then the area bounded by the curve, the x-axis,
and two ordinates at x), x, is

. 1
A = sin of y ax,
ro

18. The equation to a parabola referred to a tangent and the diameter through
the point of contact is y? = kx.

Show that the area cut off by any chord parallel to the tangent is equal to two
thirds the area of the parallelogram whose sides are the chord, tangent, and lines
through the ends of the arc parallel to the diameter.

19. The equation to the hyperbola referred to its asymptotes as coordinate axes
is xy = c. If wis the angle between the asymptotes, show that the area between
the curve, x-axis, and two ordinates at x), x, is

x.
2 sin log (=) :
*o

20. If y = ax* is the equation to a curve in rectangular coordinates, show that
the area from x = 0 to x is
xy
n+
230 APPLICATIONS OF INTEGRATION. [CH. XX.

156. If the area bounded by a curve, the axis of_y, and two abscis-
se x,, ¥,, corresponding to the ordinates _y,, y,, is required, then that
area 1S

Wy
A= x dy.
yo

EXAMPLES.

1, Find the area of the curve y? = px between the curve and the y-axis from
y=otoy=y.

2. Find the area of the curve y = e* between the curve, the y-axis, and ab-
scisse at y = 1, y = a. Check the result by finding the area between the curve
and the x-axis for corresponding limits.

Also find the area bounded by the curve, the y-axis, and the negative part of the
X-axis.

157. Observe that in the examples thus far given the portion of
the curve whose area was required has been such that the curve was
wholly on one side of the

y axis of coordinates.
+ It is evident that if
+ the curve crosses the axis

 

O “between the limits of in-

| tegration, then, y being
positive above the x-axis
and negative below it,
those portions of the area above Ox are positive, those below are

negative. The integral
f " y dx
+o

is then the algebraic sum of these areas, or the difference of the area
on one side of Ox from that on the other side.

Fic. 76.

EXAMPLE.
Find the area of y = sin x from x = 0 to x = 37,
We have
am, ba
A =[ sin x dx = — cos x] = %
0 0

But [os x dx = 2,
0

an
i sin x dx = — 1.
0

 

3.
A™ = ATH AM a 2-151.
: or = 2—1 I

158. It is evident that the area considered can be regarded as the
area generated or swept over by the ordinate moving parallel to a fixed
direction, Oy.
ART. 160.] ON THE AREAS OF PLANE CURVES, 231

If we have to find the area between two
curves y P,
I, = $(*), =H); L
and two ordinates at @ and J, such as the N
area LIVNVR in the figure, that area can be Be DB
computed by finding the area of each curve zi .
separately. Butif it is more convenient, thed 2 ob >

‘ a
area 1s Fic. 78.

[or —J,) dx = fee) — P(x)] ae.

The area in question is generated by the line P,P,, equal to the
difference of the ordinates y, — y,, moving parallel to Oy from the
position RZ to WAL.

=

 

 

 

 

EXAMPLE.

Find the area bounded by the curves

x(y —e*) = sinx and 247 = 2sinx + 23,

the y-axis, and the ordinate at x = I.
1
A= (e — 40°) dx >e —2=1-55 4.
0

It would not be so easy to find the areas of each curve separately.

159. If it be required to find the
whole area of a closed curve, such as that
represented in the figure, we may proceed
as follows:

Suppose the ordinate A/P to meet the
curve again in Q, and let MP=¥y,,
1 _7MQ=y,. Leta and 4 be the abscissze

5 "of the extreme tangents aA and 4B.
Fic. 79. Then the area of the curve is

2
A= ['(y,-4,) &.

This result also holds if the curve cuts the axis of x.

   

EXAMPLE.
Find the whole area of the curve (y — mx)? = a? — x*,
Here
a — y

Yr=mxt Ya — x.

Jy = me + VFA,

Jy = mx — Ya — x

+a
4=f" ( — 9) dx, Oo ©

+a
=2 Vo — 2 dx = na
—a

160. The area of any portion of the

curve
oe
r(z, $= & (1) Fic. 80.

 
232 APPLICATIONS OF INTEGRATION. (Cu. XX.

is ad times the area of the corresponding portion of the curve
J(%, v) = 6. (2)
For (x) is transformed into (2) by putting + = ax’, y = dy’ in (1);
and hence y dx, from (1), becomes aé y’ dx’, and we have
xy xy!
[va@= ab f* 9! de.

EXAMPLES.

1. The entire area of the circle x? + y? = 1 is w. Hence that of the ellipse
x?/a? + y?/b* = 1 is aban.

$ 4
2. Find the whole area of the curve (=) + (5) —

In Ex. 11, § 155, it is shown that the area of
xf +9 =t
is x. Hence that of the proposed curve is §zaé.
3. Check the result in Ex. 2 by putting « = a sind, y = 6 cos*@.
Then ydx = 3a6 sin’ costp dp.

A= r2ab sin?d cost@ dp = $2ad.
0

161. Sometimes the quadrature of a curve is to be obtained when
the coordinates are given in terms of a third variable, or is facilitated
by expressing the coordinates in terms of a third variable. Thus if

x= O4), y= 4,

the element of area is
ydx = wt)’ (s)dt.

 

EXAMPLES.
1, Find the area of the loop of the folium of Descartes,
y x 4 73 = 3axy.
Put y =¢x; then
_ 3at _ 3ae
sp pt ?— pee
I — 28
dx = a+5p + ep 3adt, and
(1 — 2t%)dt 6a? ga?
dx = 9a? = — ;

Fic. 81. fo ae § (1+ 4) 1+8 a2a(1+ 8A)

The limits for ¢areoando. Hence 4 = 8a’.
2. In the cycloid,
x= a(t—sin?t), y = a(I — cos 24),

dx — a* | vers*t at = 4a? | sint 44 d¢.
y fosinty

Taking ¢ between 0 and z, we get 37a? for the entire area between one arch of
the cycloid and its base.
ART. 162.] ON THE AREAS OF PLANE CURVES. 233

3. Find the area of the ellipse using x*/a? + y?/d? = 1, where x = a cos @,
y = bsin &.

4. Find the area of the hyperbola x?/a? — y*/d? = 1, fromx =u to r= 4x,
using x=asec gd, y= étan ©.

162. Areas in Polar Coordinates.—Let p = /(6) be the polar
equation to a curve. We require the area of a sector, bounded by
the curve and two positions of the radius vector, corresponding to
G=a, d= £.

 

 

Let AB represent p = /(6), OJ the initial line. 0A =a,
ZI0OB = 8. Then OAS is the sector whose area is required.
Divide the angle AOB = 6 — a@ into » equal parts each equal to
406, and draw the corresponding radii cutting 42 in corresponding
points P, Q, etc.; dividing the curve ABZ into z parts, such as PQ.
Through each of the points of division draw circular arcs with center
O, such as Qp, gP, etc. From the continuity of p = /(6), we can
always take J@ so small that the sector OPQ of the curve lies
between the corresponding circular sectors OPg and OpQ, and there-
fore the area of the whole sector OAB lies between the sum of the
circular sectors of type OPg and the sum of the circular sectors of
type OfQ. But the difference between these sums of circular sectors
is equal to the area

ALNM = 3(0B? — OA*)46,

which has the limit o when J6(=)o, or when 7 = 00. Therefore
the sum of either the external or internal circular sectors converges
to the area of the sector OAB as a limit when 2 = o.

Putting p,= OA, p, = OB, and p,(7=1, 2,...), for the
radii to the points of division of AB, the area of the curvilinear
sector OAB is

LD tpnp = [i sea0.

R=0 ral
234 APPLICATIONS OF INTEGRATION. (CH. XX.

EXAMPLES.

1. Find the area swept out by the radius vector of the spiral of Archimedes,
p = a6, in one revolution,

20 217
We have 4=4f pas f a’? do = 4n%a2.
0 0

2. Find the area described by the radius vector of the eee spiral

p = &, from 6 = 0 to 0 = 4. Ans. iu (e™ — 1).
3. Show that the area of the circle p = a sin 6 is faa’.
4. Find the area of one loop of p = a sin 26. Ans, 41a".
5. Find the entire area of the cardioid g = a(1 — cos 6). Ans. $n’,
6. The area of the parabola p = a sec? 49 from 6 = 0 to 6 = 4a is 4a’.
7. Show that the area of the /emmniscate p? = a® cos 26, is a’.

8. In the hyperbolic spiral p§ = a, show that the area bounded by any two
radii vectores is proportional to the difference of their lengths.

9. Find the area of a loop of the curve p? = a? cos 76. Ans, a/n.
10. Find the area of the loop of the foltwm of Descartes,
+ ys = Zaxy.
Transform to polar coordinates. Then
3a cos @ sin 9
= ‘sin3@ - cos°6"
Therefore the area is

ga? = sin’6 cos’6.d6 __ 9a? u? du a
“2 Jy (sin®@ + cos%9)? — =f at T+ yp ;
where w = tan 4.

11. Show that the whole area between the curve in Ex. 10 and its asymptote is
equal to the area of the loop.

12, Find the area between the curves
m\? ‘\ 2
ety () and p?+ 6? = () ‘

13. The area of p = a cos 36, from o to iz, is s, 2a’.
14. Show that the area of g = a(sin 26 + cos 26), from 0 to 27, is wa’.
15. The area of cos 6 = a cos 26, from 0 to ja, is $(2 — 47)a?.

163. We come now to consider the area generated by a straight-
line segment which moves ina plane, under certain general conditions.
In rectangular coordinates we have considered the area generated by
the moving ordinate toa curve. In polar coordinates the area con-
sidered was generated by a moving radius vector. In the former case
the generating line moves parallel to a fixed direction, in the latter it
passes through a fixed point.

A point Q is taken on the tangent at P to a given curve PP’, such
that PQ=4. To find the area bounded by the given curve, the curve
QQ’ described by Q, and two positions PQ, P’Q’ of the generating
line.
ART. 163.] ON THE AREAS OF PLANE CURVES. 235

Let PQ=4,P'V=/4 44, PI=61, P’/I= 6't, and 6 be the
angle which the tangent at P makes with a fixed direction. Let 4A
represent the area swept over by PQ in moving from PQ to P’Q’ through

 

 

Fic. 83.

the angle 46. Draw the chord PP’ and the circular arcs QM, QO’ AL’
with Jas a center. Then 4A is equal to the area of the circular
sector Q/M, plus a fraction of the area of the triangle PZP’, plus a
fraction of the area QMZ Q/M". Or, in symbols,

4A = 4 — O46
+ List. 6't sin 40 +2e4 At + 6’t)? — (¢— 64)"] 40,

where 7,, /, are proper fractions. Observing that J/, d/, and 6’
converge to o when 46(=)o, divide by 46 and let 46(=)o. Then

aA
fig
ag ame at ’
or dA = 11? 6.
Hence between the limits 6 = a, 6 = # the area swept over by
is
A= af"? aa,

When the law of change of /, the length-of the tangent, is given
asa function of @, the area can be evaluated. If /= /(@) be this
relation, the curve ¢ = _/(6), considering / as a radius vector and @
the vectorial angle, is called the directing or director curve of the
generating line.

EXAMPLES.

1. Show that the area swept over by a line of constant length a laid off on the
tangent from the point of contact is 7a?, when the point of contact moves entirely
around the boundary of a closed plane curve.

2. The ¢ractrix is a curve whose tangent-length is constant. Find the entire
area bounded by the curve. (Fig. 84.)

The area in the first quadrant is generated by the constant length PZ = a
turning through the angle 47 as the point P moves from / along the curve /PS
asymptotic to Ox. Therefore the area in the first quadrant is ja, and the whole
area bounded by the four infinite branches is 7a’.
236 APPLICATIONS OF INTEGRATION. [CH. XX.

3. Check the above result by Cartesian coordinates and find the equation to the
tractrix. :
We have directly from the figure

 

WS api a ee ee ydx = — Ya— yay.
dx ya — x
aT
P
a
8
a
o- oN T o
Fic. 84.

Hence the element of area of the tractrix is the same as that of a circle of
radius a. It follows directly that the whole area of the tractrix is 2a’, This
gives an example of finding the area of a curve without knowing its equation. To
find the equation of the tractrix, we have

ve
w

dx = — dy.
Integrating, we get
x= — Ya— y+ alog

since x = Owheny =a. This is said to be the first curve whose area was found
by integration.

Ciera F
#

4. Show that the area bounded by a curve, its evolute, and two normals to the

curve is
+ * 0? a
J. ;

where 9 is the radius of curvature of the curve, and § the angle which the normal
makes with a fixed direction.

164. Elliott’s Theorem.—Two points P, and P, on a straight
line describe closed curves of areas (P,) and (P,). The segment
PP, moves in such a manner as to be always parallel and equal
to the radius vector of a known curve p = /(6) called the director
curve.

It is required to find the area of the closed curve described by a
point P on the line P,P, which divides the segment P,P, in constant
ratio.
ArT. 164.] ON THE AREAS OF PLANE CURVES. 237

Let (P), (P,), (P,), (A) be the areas of the closed curves
described by the corresponding points as shown in the figure. Let

  

Fic. 85.

P,P, and P,/P,', Fig. 86, be two positions of the segment. Produce
them to meet in C.

 

Lett p=PP,,

 

PP,=— PP —ko,

 

a m, + m, I= 4
where 4, + 4, = 1.
The element of area, PPP?’ is, $ 163, if CP =,
@(P,) — @(P,) = 3(p + 7)? ad tate ir a6,
= pr dé + 40 dao, (1)
In like manner the element of area P,PP’P,’ is
UP) —aP,) =t4&p+7) db —1r db,
= k,pr db + 1k? dé. (2)
Multiply (1) by 4, and eliminate 4,07 26 between (1) and (2),
remembering that £4, + 4,= 1. Then
UP) = ka P,) + 4a(P,) — 2,4,d(A).
Integrating for a complete circuit of the points P, and P, about
the boundaries of the curves, we have
(P) = k(P,) a2 &(P,) — hk(A), (3)
where the area of the director curve is given by

(A)=4f p* 28,
the limits of the integral being determined by the angle through
which the line has turned.
238 APPLICATIONS OF INTEGRATION. [CH. XX.

In particular, if P,P, = pis constant and equal to a, we have
Holditch’s theorem,

(P) = A(P,) + A(P,) — Haha? f a.

Ifa chord of constant length @ moves with its ends on a closed
curve of area (C), the area of the closed curve traced by the point
P which divides the chord in constant ratio m : # is

Bah
= (C) — ¢,¢,7,

if P is distant c, and c, from the ends of the chord.

EXAMPLES.

1. A straight line of constant length moves with its ends on two fixed intersect-
ing straight lines; show that the area of the ellipse described by a point on the line
at distances a and 4 from its ends is mad.

2. Achord of constant length ¢ moves about within a parabola, and tangents
are drawn at the ends of the chord; find the total area between the parabola and
the locus of the intersection of the tangents. Ans. tne.

The area between the parabola and the curve described by the middle point of
the chord is the same.

3. It can be shown that the locus of the intersection of the tangents in Ex. 2
to the parabola y? = 4ax is

(9? — 4ax)(¥? + 4a?) = 70?
Check the result in Ex. 2 by the direct integration

f zady = jen
from y = — © toy = +o. 2 being half the distance from the intersection of
the tangents to the mid-point of the chord.

4. Tangents to a closed oval curve intersect at right angles in a point P; show
that the whole area between the locus of P and the given curve is equal to halt
the area of the curve formed by drawing through a fixed point a radius vector
parallel to either tangent and equal to the chord of contact.

5. If p,, 6, and p,, 8, are the polar coordinates of points P, and /, on a straight
line, then the: radius vector p of a point on this straight line which divides the
segment FP, = Aso that PA, = 4,A, PP, = k,A, is determined by

p? = 2p? + 2p? — 2h. (1)

This is Stewart’s theorem in elementary geometry. If @ is the angle which p
makes with ?,/,, then

py = p? + 7A? — 2h, Ap cos @,
py = p? + APA? + 2hAp cos @.

The elimination of cos @ gives (1) at once.
Multiply (1) through by 4¢6, then

40? db = b, 4p, 0 + ky 4p2do — bk, gl? dB, (2)
or UP) = hyd Py) + hy UP:) — 4h, aA),
and Elliott’s theorem follows immediately on integration.

‘
ART. 164.]} ON THE AREAS OF PLANE CURVES. 239

The geometrical interpretation of (2) is as follows: Let A = P,P, be constant.
Construct the izstantaneous center of rotation 7 of A as PP, turns through 46.
Then P,P’, PP’, P,P, (Fig. 86) subtend the angle 46 at ze "The center / being
considered as origin or pole, (2) follows at once. The extension to the case when
A is variable is immediately evident.

6. Theory of the Polar Planimeter.

In Fig. 86, let PP, = /be constant. At P let there be a graduated wheel
attached to the bar /,/, in such a manner that the axle of the wheel is rigidly
parallel to P,/,. This wheel can record only the distance passed over by the bar at
right angles to the bar.

Let PP=h, PP=h. let CP =r.

Then with the symbolism of § 164 we have

aP,) — dP) = ir +4) do — 477 2,
= rl, dQ + 42,7 do.

aP) — a P,) = 477 do — 4r — LY a,
= rl, dj — 42 dd.

Adding these two equations,

AP,) — AP.) = 2-r do + 442 — 22) do.

But + d@ = @R is the wheel record for a shift of the bar.
Integrating, we have for the area bounded by the curves traced by P, and P,
and the initial and terminal position of the bar
(P,) — (Py) = AR, — Ry) + HA? — 47)(8, — 6),

6,,9, being the initial and terminal angles which the bar makes with a fixed
direction, and #,, #, the initial and terminal records of the wheel.
Notice that when the wheel is attached to the middle of the bar
(Pz) — (A) = 4&2 — &).

The path of P, is a circle in Amsler’s instrument.

EXERCISES.
1. Find the area of the limagon = a cos @ + 4, when J > a.
Ans. (8 + 4a?)x,
2. Show that the area of a segment of a parabola cut off by any focal chord in
terms of /, the chord length, and 9, the parameter, is 37 ip.
3. Show that the area of the curve xy? = (a — x(x — 4)is x(at — a).

4. Show that the whole area between the curve y(a? + x?) = ma® and the
x-axis is ma’.

5. Show that the whole area between the curve y?(a? — x*) = 64 and its
asymptotes is 276%.

6. Show that the area between the curve and the axes in the first quadrant for
(x/a)t + (y/6)t = 1 is ab/20.
7. Show that the area of a loop of the curve y* — 2c?y? + ax? = 0 is 243/34,

8. The locus of the foot of the perpendicular drawn from the origin to the tan-
gent of a given curve is called the peda/ of the given curve.
(1). The pedal of the ellipse (v/a)? + (y/d)? = 1 is

p? = 2 cos’ + 2 sin’@.
240 APPLICATIONS OF INTEGRATION. [Cu. XX.

Show that its area is {x(a + 0?
(2). The pedal of the em (x/a)? — (9/6¥ = ris

p? = a? cos’*@ — 2? sin?6.
Show that its area is ad + (a? — 6?) tan—(a/d).
9. If ¥,, ¥, 7, be three ordinates, y, being midway between y, and y,, of the

curve
y =a + bx? + cx +d,
show that the area bounded by the curve, the x-axis, and the ordinates y, and y,is

4% — NI. + Is + 42).

If we transfer the origin to x,, 0, and put +, = — 4, x, = + 4%, the equation
of the curve can be written

yo=atet Prtyxt+ 6.
We have for the area

i ydx = 2h(482 + 6),

and 44(y, +3 + 4y,) has this same value. This is called Newton’s rule.
10. Show that the area of any parabola

yroaert betes,
from x = — h, tox = + 4, can be expressed in terms of the coordinates au Wn
and ., , /, of any two points on the curve, whose abscissz satisfy x74, = — }h?
Ans, 2s eo,
ty = He

The mean ordinate in the interval is

I +h ;
Im = ae ydx = gah? +.
Let p and g be two undetermined numbers. Then

PV + Wo-Im = A Px? + 7%? — AM) + O pay + 9%) + (P+ 9 — Ve.
The three equations in J, ¢,

Px? + 9x? = Fh, . (1)
px, + px, = 9, (2)
2p + q =1, (3)

give determinate values of # and g, provided

2 2 21
xy, 47, gh
1) %X_,, O

 

 

1, 1, I
or yx, = — FH.
Then Im =P, +ADWes

and the values of # and g from (2), (3) give the result.

11. In Elliott’s theorem, § 164, (3), show that the mean of the areas of the curves
described by all points on the segment P,P, is 4[(P,) + (B)] — 4).

12. A given arc of a plane curve turns, without changing its form, around a
fixed point in its plane; what is the area swept over by the arc?
ArT. 164.] ON THE AREAS OF PLANE CURVES. 241

13. If a curve is expressed in terms of its radius vector 7 and the perpendicular
from the origin on the tangent me prove that its area is given by

pr dr dr
Vr? —p o= ro
14, Lagrange’s Interpolation Formula.
We have seen, in the decomposition of rational fractions, that when

Y(x) = (x — a(x — a). . . (4 — Gn),
and A(x) is a polynomial in « of degree less than #,
Aix) 4 1 FKa,)
Wa) = pape ay (ay)

See § 133, and Ex. 79, Chapter XVIII.
If Aix) is any differentiable function of x, then, since

 

 

 

2 W(x) Ka,)
Fix) — ee Pas)

vanishes at x =4@,,..., @, and the second term is a polynomial of degree x — 1,
we have, § 98, II, lemma,

 

 

 

 

5) Fla, a
n(€), I
A= 3S gat a )
where <= is some number ee the greatest and least of the numbers +,
a +a
"The formula
W(x) Far)
A(x) =
(4) = 72-6, 8%)

is called Lagrange’s interpolation formula. The member on the left computes the
value at « of an unknown function when its values at @,,'. . . , @4 are known,
with an error which is represented by

(eo @) ae a GR
n!

- 4+ - an) Fu).

15. Gauss’ and Jacobi’s theorem on areas.

If A(x) is any polynomial of degree 2% — 1, then the exact area of the curve
y = Ax) between x = p, x = g can be computed in terms of # properly assigned
ordinates.

Let

‘, Wx) Fier)
12% y'(a,)?
where, as in Ex. 14, (x) = (x — 4%)... (x — an).

Then /(x) = Ar) — L(x) i is a polynomial of degree 2% — 1, in which Ax)
is of degree 2x — 1, L(x) of degree x — 1. Also, /(x) vanishes when x = a,

- .) 4. Hence
F(x) — L(x) = A P(x) V(x),

where 4 is some constant and (x) some polynomial of degree z — 1, since y(x)
is of degree x.
Integrating between f and g,

[ro ax -f" L(x) dx = 4 foe) Wx) dx.

L2)= 3

 

 
242 APPLICATIONS OF INTEGRATION. (CH. XX.

Jacobi has shown as follows that we can always assign a,, . . . , &,, So that

[oven =o.

For, integrating by parts successively,
f obae = oY, — P+... — (— IG",

where @(r) denotes the result of differentiating @ r times, and #, the result of
integrating ~ + times, remembering that @(*—» is a constant.

It we take, after Jacobi, for the values a,, ..., u,, the # roots of the equa-
tion of the zth degree

(zz) We — ale - a = 0,

then the integrals y,, ..., between g and ¢ are all o, since each contains
(x — p(x — g) as a factor.
Therefore, for these values of a,, ..., ¢n, we have

’ @* Fay) W(x)
a)dx = f
Lr ) ? 2 Vans — a,’
or the proposition is established. *

If the degree of F(x) is 2”, then the area can be expressed in terms of # + 1
ordinates taken at the roots of

 

 

(x) Ue — pix — g)l*tt = 0.

The area of y = F(x) can be expressed in a singly infinite number of ways if
one more than the required number of ordinates be used, in a doubly infinite
number of avays if two more than the required number be used, and so on.

16. Show that the area of
YHH+ ax + a,x? + a,23,
from — 4 to + 4, is equal to
My, + Is),

where y, and y, are the ordinates at « = + Af 4/3. Give a rule and compass
construction for placing these ordinates.

 

* See Boole’s Finite Differences, p. 52.
CHAPTER XxXI.
ON THE LENGTHS OF CURVES.
RECTANGULAR COORDINATES.

165. Definition of the Length of a Curve.—A mechanical con-
ception of the length of a curve between two points on it can be
obtained by regarding the curve as a flexible and inextensible string
without thickness, which when straightened out can be applied to a
straight line and its length measured. The curvilinear segment is
then said to be rectified.

' The rigorous analytical definition of a curve and of its length is
a more difficult matter.

If_y is a function of x such that y, Dy, D*y, are uniform and
continuous functions in an interval + = a, + = #, then the assem-
blage of points representing ,

I =S(*)

 

in (a, (8) is called a curve.

We can demonstrate * that if P and P, are any two points on this
curve, we can always take Vv
P and P, so near together y
that the curve between P
and P, lies wholly within
the triangle whose sides are
the tangents at P and P,
and the chord PP, And
also, if Q, & be any other
two points on the curve
between P and P,, then,
however near together are O|
Q and &, the same property
is true for Q and R.

If we divide the interval (a, 4) into subintervals and at the
points of division erect ordinates to the points 4, Z,..., B, etc.,
on the curve, then draw the chords through these points, and the
tangents to the curve there, we shall have two polygonal broken lines
ALMNB inscribed, and A 7RSVB circumscribed, to the curve AB.

 

 

 

Fic. 87.

 

* Appendix, Note 11.
243
244 APPLICATIONS OF INTEGRATION. (Cu. XXI.

Let c, represent the length of the 7th chord, and 4 that of the 7th
side of the circumscribed line.

Clearly, whatever be the manner in which (a, 4) is subdivided or
to what extent that subdivision be carried, we shall always have

fp =0 t,=0 n

Soe Sh and £ Se, —£ 2f=o.

If we interpolate more points of division in (a, 6), then 2¢
decreases while 2c increases. Consequently 2¢ and 2c converge
’ to acommon limit. This /zmzt we define to

 

- be the length of the curve between A and J.

4 be ZAP. 166. Let P bea point x, y on a curve,
cy the length of which between A and FP is s.

P 4 Let P, be a point on the curve having

(A coordinates x -+ dx, y+ Jy, and let the

length of the curve between P and P, be
As, the length of the chord PP, be Je.
O| “ @ Draw the tangents at P and P,. Let the
Fic. 88 angle which PZ makes with Ox be 6, and
joneee the angle between Z’P and 7P, be 46.
Let 7P =4, TP, = fis then, by § 165,
Ac< 4s<t+ih.
But, from the triangle P7P,,
(4c)? = #7? 4 42+ 24, cos 48,
= (¢+4)? — 44%, sin? 146.
2
(<5) = a as ve
PEL (+4y

4,/(¢-+ 4)? can never be greater than 1, and when A6(=)o,
Ac(=)o, 4+ 4(=)o, also sin?446(=)o. Therefore when Jx(=)o,

 

 

 

 

we have
4c \ | ‘
+4)
Ax(=)o
Since, by definition, Js lies between 4¢ and ¢-++ 4, we also have
Ac _
Age I.
Ax(=)o
Now, ;
(Ac)? = (4x)? + (4y)?,
or

(as) = (Zi) (as) =* + Ge):
ART. 166.] ON THE LENGTHS OF CURVES, 245

Therefore, in the limit, for Jx(=)o,
ds\2 dy\?
a) =*4 (2
w=r+ (2) me) ae (1)

Hence the length of the arc of the curve from 4 to P is

rfl VE :+ (2) ax, (2)

In like manner, using Jy instead of Jx, we obtain

afro °

dst = dx® + dy’.

Since dy/dx = tan 6, 6 being the slope of the curve at x, y, we
have

or

In differentials

dx = cos @ds, dy=sin Ods.

dx da ae :
Therefore a o are the direction cosines of the tangent to the

curve at 4, y.
EXAMPLES.

1. Rectify the semz-cubical parabola ay* = 3x3.

x dy 3 /x\t ads =.
Here tee oe =i G)" mai(HS Z

foo) = (4)

the arc being measured from the vertex. This was the first curve whose length
was determined. The result was obtained by Wiliam Neil in 1660.

2. Rectify the ordinary parabola y® = 2ax.
We have Dx = y/a.

rai f va? + 9? dy,
a
=a y¥P te pe ee ee

Ss

the arc being measured from the vertex.

x x
3. Rectify the catenary y= a a te =) é

x

x
We have Dy = alee - oe) ‘

pana f(@ 4%) de = te (eae)
246 APPLICATIONS OF INTEGRATION. (Cu. XXI.

Show that s = PZ. Fig. 73. Also, WZ =constant. The catenary is therefore
the evolute of the tractrix represented by the dotted line in the figure.
4. Rectify the evolute of the ellipse
(ax)! + (by = (a — 2).
Write the equation in the form

put «-asinid, y = f cos*.

s= = f (a sired + 3? cos? p)ta(a® sin?’ + f? cos’) .

Measuring the curve from x = 0, y = f, we get

ce (a? sin'd + f? cos’)? — 3

ae — 3?
5. Find the length of the curve 9ay? = x(x — 3a)’, from x =0 to x = 3a.
Ans, 2a 3.
6. Find the entire length of the Aypocyclord xt + yt = at. Ans. 6a.

7. Find the length of the arc of the circle x? + y? = a’, from x = oto x = 4,
and the whole perimeter.

8. Find the length of the logarithmic curve y = ca*.
We have Dyx = 6/y, where 6b = I/log a.

2 2\t 2 2\b
mm 8 =f dy =(# +5248 log P@t IF — 6
9. Find the length of the ¢ractrix (see § 163, Fig. 84)
qd
Sa af S= — a logy + const.

Measured from the vertex, « = 0, y = a,

S = a log (a/y).

xt 4?
10. Length of an arc of the ed/ipse gata=t
Put x= asing, y= 6cos@. Then

Sx fo cos’? + 6? sin?p)* dQ,

aig f (1 — 2 sin’)! d@.

where ¢ is the eccentricity. This is an elliptic integral and cannot be integrated
in finite elementary functions. The length of the elliptic quadrant corresponds to
the hmits @ = 0, @= 42. Since e? sin’@ is always less than I, we can expand
the radical in a series of powers of sin @, and integrate, obtaining the length of the
quadrant (see Ex. 27 § 149)

me tx (=)'¢ z\'e 1-3-5\ 2 6 }
S - a) S - ()'$ - GS ----

 
ART. 167.] ON THE LENGTHS OF CURVES. 247

POLAR COORDINATES.
167. If p = f(@) is the equation to a curve, and p, 6 are the
coordinates of P; p+ 4p, 6+ AG, those of P,, then, calling 4c
the chord PP,, we have

(4c)? = (p + Ap)? + p? — 20(9 + Ap)cos 44,
= (4p) + 2(p + 4p)(1 — cos 46),

4dc\? oa — cos 46

But when eo we have 4p(=)o, 4c(=)o, and

Hence

 

I — cosx I sin x I
xe ~ 2 x 2°

x(=)0 a(=)o

= G+

 

As 4c Ac _ ads a ; ds ee
Aba ae? ag ae since = 1, §1 ‘
de = dp pdf,

en ep. | ——779\2
cof eet (BY a0= fr + (By ep

Otherwise we can deduce the formula for the length of an arc in
polar coordinates directly from the corresponding formula in rect-
angular coordinates.

For x=pcos0, y= psin 6.
dx = dpacos@— psin6dé, dy= desing + pcosé dé.
@aodettdyaode+ pid.

Also,

or
248 APPLICATIONS OF INTEGRATION. [CH. XXI.

EXAMPLES.

1, Find the length of the cardio¢d =p = a(t + cos 6).
Dep = — asin 6, and therefore

s= af'{a + cos 9)? + sin?6]} da = 2a f cos 46 d@ = qa sin »|
The entire length is 8a.

4,
1

2. Show that the length of the arc of the spiral of Archimedes, p = a6, from
the pole to the end of the first revolution, is

ala Yi + 4x? + 4 log (22 + Yt + 4n*)],
3. Logarithmic spiral p = ao. Put 6 = r/log a.
Then s= [a+ ap -0+ ea), — 7).
ay
4. Show that the length of the arc of log » = a6, from the origin to (, @), is
B: Ve + 1.
a
. Find the length of p + @? = a’.
. Find the length of p= asin6, from oto $7.

. Find the length of » = asec6, from o to 4z.
. Show that the entire length of p = asin?19 is $a.

OoOmAnN DO

. Show that the entire length of the epicycloid
4(0? — a?)} = 27a4p? sin’,
which is traced by a point on a circle of radius 4a rolling on a fixed circle of radius
a, is 12a.
10. Find the entire length of the curve p = a sin 26.
11. Show that the length of the hyperbolic spiral ph=a is

[ AER — oop EEE]

Po
12. Show that the length of the parabola p= asec?i6, from 6 = — 4x
to 6=-+42, is 2a(sec $a + log tan 27).

168. Geometrical Interpretations of the Differential Equations
ds? = dx*-+ ad and ds? = dp? + prdg?.
T I. In Cartesian coordinates, if we take x
as the independent variable, then we have
PM = dx. Also, since :
Dy =tan6 = tan AZPT,
= tan@ dx = MT.
a3 = PM? 4 MT? = PT,
Hence ds = PT, while the correspond-
ing difference of the arcis PP,. Also, we
have the relations
0 dx a dx = dscos0, dy = ds sin 6.
O It is easily shown geometrically that the
limit of the quotient of ds = PT, by either
Fic. 90. As = PP, or dc = PP,, is 1, when dx = 4x
converges too. See definition of the length of a curve, § 165.

So
&

q P

 

 

 

 
Art. 168.] ON THE LENGTHS OF CURVES. 249

II. We can in the case of polar coordinates exhibit ds, dp, and p d@ as the
lengths of certain circular arcs as follows:

Let OA be the initial line and Pa ’
point on the curve /(p, 6) = 0, PT the Cc
tangent at P. Draw OC perpendicular “
to p = OP, cutting the normal at P
in C. Then 2 = PC is the normal
length, and S,, = OC is the subnormal. C
Let 6 be the independent variable, then
@)j = 46 is an arbitrarily chosen angle.
We have the differentials ds, dp, p dO
proportional to the sides of the triangle
POC, or to n, Sy, p, respectively. For
we have

= (6) +

   

O
Fic. oI.

But do = S,d6, by §92, (5). Also, S,?+p?= n?. Hence ds = ndé.
Draw PC’ parallel and equal to OC. Strike the arcs PN, PL, and P/V with centers
C’ radius Sn, C radius , O radius p, having the common central angle 46 = dé.
Then

ds= PL, da= PN, pd= PM.

It is interesting to notice that the rectilinear triangle PZ/Vis a right-angled tri-
angle similar to PCO; the sides of which, PZ, PV, VL, are equal to the chords
L,  Ssubtending the arcs PZ, PN, and PAZ respectively.

Therefore, in the triangular figure PZ whose sides are
the circular arcs PZ, PN, and an arc LN with radius p equal
3 to PA, we have the sides (PV) = dp, (PL) = ds, (LN) = pao.
&

& Also, the angles between the circular arcs are
(L)=4e-% (P)=4, (VY) = 4x4 a
zB and ds* = dp? + p? d6?,
do NV dp=dscosy, pd =dssiny.
Fic. 92. In order to prove these statements, it is only necessary to

show that the rectilincar segment ZV is equal to the chord
subtending PAZ Let x, y be the chords subtending PZ, PM. Then from the
rectilinear triangle PVZ we have

LN? = x* 4 4? — 2xy cos LPN.
But ZLIPN=Y+440—446= y.
Also, x= 2nsin}40, y = 2S, sin $46.
we LN? = 4(S)? + n? — 2nSy cos p) sin? 446,
= 4p sin? 446 = (chord A/P)?.
The remainder follows easily.
Observe that if we draw PM perpendicular to OP, as
in Fig. 93, and put PAZ = 69, AZP, = dp, then we have,
for J6(=)o,

As | Op 6p
Lawn fan=% La>*

Therefore the difference equation
Ac = 6p* + Sp?
leads at once to the differential equation
ds? = dp? + p? de’.

 
250 APPLICATIONS OF INTEGRATION. [Cu. XX].

169. Radius of Curvature and Length of Evolute.
If f (x, y)= o is the equation of a curve, then
dy dy a, 20
ag ane, Fa = COT.
Hence if & is the radius of curvature at x, y,
_ (1 + 3 eS dx ds
R= gfe er ae
since ds = sec 0 dx. Therefore ds = RF dé.
The angle 46 = @@ is the angle between the tangents at P and P,,
and is equal to the angle between the normals at P and P,.

170. The length of the arc of the evo-
lute of a given curve is equal to the differ-
ence of the corresponding radii of curvature
of the involute.

Let x, y be a point on the involute cor-
responding to the point a, 6 on the evo-
lute.

Then we have for the radius of curvature

R= (a= 2) + (6 — 9)

 

Fic. 94. Differentiating, we get
RdR = (a — x)(da — dx) + (6 —y)(@B — &),
= (a — x)da + (B—y) ap, (1)

since (a — x)dx + (8 —y)dy =o, this being the equation of 2,
the normal to the involute. If@ is the angle which the tangent to the
involute at x, y makes with Ox, then, since F is tangent to the evolute,
R makes with Ox the angle @ = 4” + 6, and we have
a—x=—Rsind=Rceosd, B—y= Reosd= Rsin ¢.
Hence, on substitution in (1),
@dR=dacos@+ df sin d,
= do,
if o is the length of the arc of the evolute.
Integrating between a,, 6, and a,, £,, we have
KR, SS 6, — Oy
This can be shown otherwise, for we have ;
(x —a)da+(y— fB)dB=o, (2)
the equation to the normal to the evolute at a, /.
The perpendicular £, from +, y on the involute to the line (2), is

pn Sede + (9 = Bap
Vda? + dp? ,
or (x— a)da+ (y — B)dB= R do.
Equating with (1), we get dR = do as before.
ART, 171,] ON THE LENGTHS OF CURVES. 251

It is to be particularly observed that the theorem as enunciated ap-
plies only to an arc of the involute such that between its ends the
radius of curvature has neither a maximum nora minimum value.
For when & passes through a maximum or a minimum value dX

changes sign. dR would be zero when taken between two points
8 P

at which £ has equal values.

In applying the theorem one should be careful to determine the
maximum and minimum values of the radius of curvature for the invo-
lute, and add the corresponding adso/uée values of the lengths of the
evolute, when the radius of curvature has maximum or minimum values
between the ends of the arc under consideration,

From a mechanical point of view, since the evolute is the envelope
of the normals of the involute, we can regard the involute as a point
described by a point in the tangent, as the tangent is unrolled from
its contact with the evolute; the arc being considered as a flexible
inextensible string wrapped on the curve. ‘The truth of the above
theorem from this point of view is made evident.

The theorem of this article rectifies any curve which is the evolute
of a known curve whose radius of curvature can be found.

EXAMPLES.

1. Find the length of 27ay? = 4(x — 2a)’, the evolute of the parabola y* = qax.

We have for the coordinates x’, y’ of the center of curvature and A, the radius
of curvature of the parabola at x, y,

 

a
3 a+tx\i
x = 2a + 34, vat R = 20 ( =

Measuring the arc of the evolute from the cusp, x = 24, y=0, to x’, y’,

we have
3
S = 2a (=) — 2a,

3a

 

2. Find the length of

(ax) + (dy)} = (a — 38,
the evolute of the ellipse.

3. Show that the catenary is the evolute of the tractrix, and find the length of
an arc of the catenary as such.

171. The Intrinsic Equation of a Curve.—The length s measured
from the point of contact O ofa curve with a fixed tangent, and the
angle @ which the tangent at the end P of the arc makes with the
fixed tangent, are called the z#/rimsic coordinates of a point on the
curve.

The equation /(s, @) = 0, which expresses the relation between s
and @, is called the zmérzmstc equation of the curve.

To find the intrinsic equation of a curve (x, v)=o or /(0, @)=0,
we have to find the length of the arc from a fixed point to an arbitrary
252 APPLICATIONS OF INTEGRATION. [Cu. XXL

point on the curve, then the angle @ between the tangents there.
Eliminating the original coordinates between these three equations,
the result is the intrinsic equation.

EXAMPLES.

1, Find the intrinsic equation of the cazenary.
Take the vertex as the initial point, then

yale 4).

Dy=4 & - | = tan @

ae ae
Also, ale —e |
Eliminating x, we have 5s =a tan @.
2. Find the intrinsic equation of the involute of the circle.

Let «? + y? = @ be the equation to the circle. Unwrap the arc beginning at
the point a, 0, and let the radius to the point of contact make the angle @ with
Ox. Then @ is also the angle which the tangent to the involute makes with Ox.
The radius of curvature is the unwrapped tangent length, or R= aq. But

ds= Rdp=—apdgp. .. s=iag’.

172. General Remark on Rectification.—The problem of find-
ing the length of any curve whose equation is given involves the
integral of a function which is in irrational form. ‘This in general
does not admit of integration in finite form, and cannot generally be
expressed in terms of the elementary functions. There are, generally
speaking, but few curves that can be rectified, in terms of elementary
functions.

 

EXERCISES.
1. Show that the length of 8a3y = x* + 6a%x? measured from the origin is
x(a? 4a*)* /8a3.
2. Show thatthe whole length of 4(x? + 72) — a? = 303? is 6a.

3. Show that ay" = «+ is a curve whose length can be obtained in finite
n n I. :
terms when Se + 3 isan integer.
4. Show that the intrinsic equation to y3 = ax? is
5 = fa (sec’d — 1).
5. Show that p” = a cos m@ can be rectified when 1/m is an integer.
6. If «= Ot) »y = ¥(4), then

ds\* dx\ ? ady\?
(@) = (@) + (@) =leor + wor
7. In the cycloid x = a(6 — sin#), y = avers 6, show that

ds = 2a sin 46 dQ.
Hence the length of one arch is 8a.
ART. 172.] ON THE LENGTHS OF CURVES. 253

8. Show that the length of the cycloid

a

r= acost 4 4 W/2ay — yp

 

a
from the origin to x,y is 4/8ay.
9. Show that the intrinsic equation of the cyc/oid in Ex. 8 is
5 = 4a sin g,
the tangent at the origin being the initial tangent.

6

10. In the equiangular spiral = ae”. show that s = p sec y, where tan p
= m, measuring s from the pole.

11. Find the length of the reciprocal spiral from § = 27 tu @ = 4m, the equa-
tion being p§ = a.

12. Show that the whole perimeter of the Zemmiscat:
p” = a’ cos 20

‘ I 1-3 1-3-5
is 4a (+ 54584 23554. : .).

13. Show that the length of y? = +3 between x = 0, x = 1 is gi (13? — 8).

 

14. Designating by TAGS) the length of the curve /(x, y) = 0, from x = @
to x = 4, show that:

(a). LE2%(@4 — 6xy 43) = Hh
(8). LE23(28 + y3 — a3) = 6a.
———— 2 3.
(c). LE23 [ay — x fx? —1 — log(x — Wx? —1)] = fe dx = 4.

2 _ vy |
nae = a log Bs
(e). Eig — log sin x) = log 3.

(f). LE y — fx — x — sin x) = [\xtds = 2.

(g). Zy21(9* + 40 — 2 logy) = He log 2 + 3).
15. Use the binomial theorem to evaluate

2 = xt + at
Leal XY — a?) =] — ja

(2). Hi| — 7a — y* + a log

16. Show that

L2=%(y —sinz)=mi+t—P_t+ aby —---)
17, LP= (0% — 2ap8 + a?) = [ (3 + log 4).

o=1 e9—I\ 2
18. Z8>! (0 -533) =
aia
19. yin(¥e=* - cost _ s) = 4a.
20. A hawk can fly wv feet per second, a hare can run v’ feet persecond. The

hawk, when a feet vertically above the hare, gives chase and catches the hare when
the hare has run 4 feet. Find the length of the curve of pursuit.
254 APPLICATIONS OF INTEGRATION. [Cu. XXI.

Take O, the starting-point of the hawk, as origin, the line OA drawn to the
starting-point of the hare Has y-axis, and a parallel Ox to the hare’s path as x-axis.
When the hawk has flown a distance s to P, the hare will have run a distance o to
fF’ in the tangent to the curve at P.

 

 

 

 

WT] ,
4 P
se ee
al 7{P
O
FIG. 95.

Let PP’ =¢4. Wehaved =v T=HC. S=vT, the length OPC. T being
the time of pursuit.

vy!
= — = £ = constant,
U

In like manner,

do =kds. Also,
A= (0 — a + (a 97
tat = (o — a)\(do — dx) — (a — y)dy.

But o—x=fcos#, @a—y=fsin§ Also, docos@is&dscos§@ or
k dx, and

dx cos @ + dy sin@ = ds.
dt = k dx — ds.

b 0
Hence Saf dx—[odatte
0 a
2
=Ste.

a a
saf4+afere

21. If & and vare the radii of the fixed and rolling circles in the epicycloid and
hypocycloid

 

Rtr r

 

 

«=(R +r) cos @ Fr cos ay ? y=(Rtr)sing F peat = Q,
Py
show that the lengths of the curves from cusp to cusp are respectively
8r(R + r)/R.

22. In § 164 (Elliott’s theorem), if s,, 5. s are the corresponding lengths of
the arcs described by the points P,, 7,, P respectively and o the corresponding
length of the director curve, show that

ds? — hy ds? + hy ds,? — bk, do
CHAPTER XXII.
ON THE VOLUMES AND SURFACES OF REVOLUTES.

173. Definition.—A point is said to revolve about a straight line
as an axis when it describes the arc of a circle whose plane is per-
pendicular to the straight line and whose center is on the straight
line.

A plane figure is said to revolve about a straight line in its plane
as an axis when each point of the figure revolves about the line as an
axis. 4
The solid geometrical figure generated by the revolution of a
plane figure about a straight line in its plane as axis is called a
revolute. The surface of the figure revolved generates the volume,
and the perimeter of the figure revolved generates the surface of the
revolute.

Examples of revolutes are familiar from the three round bodies
of elementary geometry, the cylinder, cone, and sphere.

 

A

 

 

 

 

 

 

 

Fic. 96.

AB being the axis of revolution, the cylinder is generated by the
revolution of the rectangle ABCD, the cone by that of the right-
angled triangle ABC, the sphere by that of the semicircle APB.
The volumes of the revolutes are generated by the surfaces, and the
surfaces of the revolutes by the perimeters of the revolving figures.

We know from elementary geometry that the volume of the
cylinder is equal to the area of the circular base multiplied by the
altitude or by DC the length of the line generating the curved surface.
Also, the curved surface of the cylinder has for its area the product
of the circumference of the base into CZ, the length of the generating
line.

255
256 APPLICATIONS OF INTEGRATION. (Cu. XXII.

It 1s evident from the definition of a revolute that any section of
a revolute by a plane perpendicular to the axis AZ is a circle, such
as ODD’. The circular sections cut out of the surface by planes
perpendicular to the axis are called parallels. In
like manner the section of the surface of a revolute
cq by any plane passing through the axis is a line
identically the same as the generating line. For if
in the figure the surface is generated by the revolu-
KOS tion of the line ACDB about the axis 4A, then the
section AD’ is nothing more than one position of
the generating line ACB. Again, the revolute can
always be regarded as being generated by a circle
B moving in such a manner that its center moves
Fic along the axis to which its plane is perpendicular,
. 97. - : : :
and its radius changes according to a given law.
174. Volume of a Revolute.—Let y= /(x) be a curve AB.
We require the volume of the solid generated by the figure aA BS
revolving about Ox as axis B
of revolution. , M
Divide (2, 4) into n 4%
subintervals, and pass PB
planes through the points
of division cutting the
solid into 2 parts, such as
the one generated by the
revolution of w«PP'x”.
We can always take 2’ —
x = 4x so small that the
curve PP’ will lie inside
the rectangle PAZP’M/’, if
J(x) is continuous. Let
y be an increasing one- Fic. 98.
valued function from «x =eto x«=408. The volume, JV, of the
solid generated by xPP'x’, lies between the volumes of the cylinders
generated by the rectangles xPM/'x’ and xM/P’x’, Hence for each
subdivision of the solid we have

,

 

Av

 

gQ
==—— Tah
Co

 

be
ano oe eee

 

 

myPAx << AV < my’* dx. (1)

The whole volume of the revolute, therefore, lies between the
sum of the # interior cylinders and that of the exterior cylinders, or

SmpAx<V < S ayl*dAx, (2)

But if we interpolate more points of subdivision in (a, 6), we increase
ArT. 174.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 257

the sum of the interior volumes and decrease that of the exterior; and

since
my? Ax aan
my * Ax t
4x(=)o

these sums have a common limit, which is V.

Ax(=)o #
“. KW, = £ Sy’ dAx,

N=0 1

= f ” ay ax, (3)
Again, we have directly from the inequality (1)

4 AV mT
ay Saye -

Hence, for 4x(=)o, we have

= = my,
since fy’ = v, when Jx(=)o.
dV = ny dx, (4)
and as before
a xf 9 ax. (5)

In like manner we show that if x is a one-valued function of »,
say x = (y), then the volume generated by the revolution of the
curve about Oy as an axis, included between two planes perpendicular
to Oy at y= pand y = g, is

q
Vy aap ray. (6)

EXAMPLES.

1. Find the volumé of the cove of revolution generated by the revolution of the
triangle formed by the lines x = 0, y =0, x/a + y/b = 1, about Ox as axis.

a a, b “
Vaan ["yde=n f (+ - = *) he,
0 0 a
3
= ~ 5 (s- =e) [ase
3 6 a s

But a is the altitude and 4 the radius of the base of the cone. Therefore the
volume is equal to one third the product of the area of the base into the altitude.

2. Find the volume of the sphere generated by the revolution of a semi-circle
of x? + y? = a about Oy.

+a +a
Y= af xwdy = x (2 — v*) dy = 4a.
—a —a,

3. The prolate spheroid is the revolute generated by the revolution of an ellipse
about its long axis, sometimes called the odlongum.
258 APPLICATIONS OF INTEGRATION (Cu. XXIL

Let @ be the semi-major axis of x?/a? + y?/s? = 1.
Then we have for the volume of the oblongum

a HOE as 2) dx — 4mab?
4 a — x")dx = 4nab?,

4. The oblate spheroid or oblatum is the revolute obtained by revolving the
ellipse about its minor axis; show that its volume is 47a74, where 4 is the semi minor
axis.

5. Show that the volume of the revolute obtained by revolving the parabola
y” = 4ax about Ox, from x = 0 to « = 4a, is 27a5.

This is the paraboloid of revolution.

2

6. If the hyperbola z -5 = I revolves about Oy, the revolute is called the
hyperboloid of revolution of one sheet. Show that the volume from y =0 toy =y

. 1 a
a (9° + 32).
If the curve revolves about Ox, find the volume from x = a to x = 2a. This
surface is called the Ayperboloid of revolution of two sheets.
7. Find the entire volume generated by the revolution about Ox of the
hypocycloid x? + y? = af. Ans. B3,ma°.
8. The surface generated by the revolution of the tractrix about its asymptote is

called the pseudo-sphere. This important surface has the property of having its
curvature constant and negative. Find its volume.

Here y? dx = — (a? —y?)ty dy. Hence the volume from x = 0 tox = x is
a
V,=%0 f (a? — y*)ty dy = gn(2? — 9%},
Y

The volume of the entire pseudo-sphere is $2ra3, or one half that of a sphere
with radius a,

9. Find the volume generated by the revolution of the catenary
x x

y=tha (awe: Za) about Ox from 0 to x. Ans. 4na(ys + ax),
10. The volume generated by revolving the witch (x? + 4a?) y = 823 about its
asymptote is 477a°,
y 175. To find the volume of the
revolute generated by a closed curve
revolving about an axis in its plane, but
external to the curve.

We take the difference between the
volumes of the revolutes generated by
MABCWM and MADCN. Hence the
volume of the solid ring generated by
ABCD revolving about Oy is

 

 

ari :

_| x V, a mf (x3— x) dy,

Oo where x, = RD, x, = RB, and the
Fic. 99. limits of the integral are y = OM,

Y= ON. A corresponding integral gives the volume about Ox.
ArT. 176.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 259

EXAMPLES.

1. The solid ring generated by the revolution of a circle about an axis external
to it is called a ¢orus. Show that the volume of the torus generated by the circle

(att yar
(a =r) about Oy is 27a.
We have x= at Vr— yy,

a, =a ray
+r
N= xf 4a fr? — 7 dy = 20a,
—r

Observe that the volume is equal to the product of the area of the generating
circle into the circumference described by its centre.

2. Show that the volume of the elliptic torus generated by
@-f
ae ee
(¢ >a) about Oy is 27adc.

176. The Area of the Surface of a Revolute.—We know, from
elementary geometry, that the curved surface of a cone of revolution
is equal to half the product of the slant height into
the circumference of the base.

The area of the curved surface of the frustum
included between the parallel planes AD and BC is
therefore

 

a(VD.AD — VC-BC).
Since BC/AD = VC/VD, we deduce for the
surface generated by the revolution of CD about
BA the area

 

2nxMN-CD,

where JZN joins the middle points of AB and CD.

In the figure of § 174, Fig. 98, subdivide, as before, the interval
(a, 6) into # parts; erect ordinates to the curve 42 at the points of
division. Join the points of division on the curve by drawing the
chords of the corresponding arcs, thus inscribing in the curve 4B
a polygonal line AB with 2 sides. Let PP’ be one of the sides
of this polygonal line. The curved surface of the frustum of a cone
generated by the chord Jc = PP’ revolving about Ox has for its
area

/
a2 TY go = an(y + $y) Ac,

We define the surface generated by the revolution of the arc of
the curve 4B about Ox to be the limit to which converges the sur-
face generated by the revolution about Ox of the inscribed polygonal
line, when the number of the sides of the polygonal line increases
indefinitely and at the same time each side diminishes indefinitely.
260 APPLICATIONS OF INTEGRATION. | (Cu. ‘XXIL.

To evaluate this limit, we have for the area of the surface gen-
erated by the curve AB

Ox(=)o #

S.= 4 = 2n(y + 4y)Ac,

n=o
bx(= ts

= £ Se2myds.

n=O 1.
Since for each pair of corresponding elements of these two sums
we have

am(y + 44y) de _
2my ds ~

4x(=)o
Hence we have, by definition of an integral,

Sy = anf yas,

nn nis eae fNeG)™ 6

In like manner, if 42 revolves about Oy, we have for the area of
the surface ee

S, = anf * eds,

nef rm afl 0

EXAMPLES.
1. Find the surface of the sphere generated by the revolution of the circle
y? = a — x? about Ox.
ay x ds _a
I ey re, yes
We have ee ae
Sn = ax fy ds = 20(x, — x,)a

Hence the area of the zone included between the two parallel planes is equal
‘ to the circumference of a great circle into the altitude of the zone. If «, = + a,
| 4, = — 4, we have the whole surface of the sphere 47ra*.

2. Show that the curved surface of the cone generated by the revolution of
ji =x tana about Ox, from x = otox =h, is rh? tan asec a. Verify the for-
mula deduced for the surface of a frustum in 8 176,

3. Surface area of the paraboloid of revolution.
Let y? = 2mx revolve about Ox. Then

=F fort my oy = 2 (t+ mi mt

4. Let 2a be the major axis of ay? + J?x? — @?4?, and e¢ its eccentricity. : Then
_we have for the surface ofthe prolate spheroid

+
5.572 f " Je - 8 er = ona (yr 4 ‘).
ART. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 261

5. Show that the surface of the pseudo-sphere is
Se = ana [ dy = 2mala — y).
y

Its entire surface is 27a?,
6. In the catenary show that
S_ = (ys + ax),
Sy = 2n(a? + xs — ay),
from x = oOtox = x.
7. Show that the surface of the Aypocycloid of revolution generated by
xt + yt = al about Ox is 3270,
8. A cycloid revolves around the tangent at the vertex. Show that the whole
surface generated is 3277.
9. The cardioid p = a(1 + cos6) revolves about the initial line. Show that the
area of its surface is 52a.

177. Ifa plane closed curve having an axis of symmetry revolves
about an axis of revolution parallel to
the axis of symmetry and at a distance 4
a from it, then we shall have for the p
volume and surface of the revolute gen-
erated, respectively,

V=e2maA, S=2nal, NOT Jat
where 4 is the area and Z the length of
the generating curve.

Let . = a be the axis of symmetry O a a oy
and Oy the axis of revolution. Then Fic. 101.
for the volume

 

 

 

 

 

 

 

 

= Nid 2) J
v= f(x — ») a.
Butif CM =CN= 2, x, = a+x, x4 =a’,
é
aoe V, = ana 2x' dy = 27aA,

For the surface
-
S, = ox f (x, + +) ds,

bp
= ona ff ads == 2naL,
q

The results obtained assume that the axis of revolution does not cut
the generating curve.

EXAMPLES.
1. The volume and surface of the torus generated by the revolution of a circle
of radius a about an axis distant ¢ from the center (¢c = a) are respectively 27?a?c
and 47°ac. :
2. The volume generated by the revolution of an ellipse, having 2a, 26 as major
and minor axes, about a tangent at the end of the major axis is 277a?4,
262 APPLICATIONS OF INTEGRATION. (CH. XXII.

EXERCISES.

1. Show that the segment of the parabola y? = 2px, made by the line x = a,
when rotated about Ox, generates the volume

2np fre ax = npa’.
0

2. The figure in Ex. 1 rotated about the y-axis generates the volume

2pa y* a2
on f (« = B) dy = 8na* s/2pa.

3. The volume generated by the closed curve x* — a?x* + a%y? = o about the
x-axis is

20 *
ae f (@x* — xt) dx = a.

4. The curve x? + yt = 1 rotating about the y-axis generates a solid whose
volume is 47.

5. The volumes generated by y = e* about Ox and Oy are respectively
0 1
a f exdx = 41, xf (log v)? dy = 2m.
=o 0

6. The curve y = sin x rotating about Ox and Oy, respectively, for x = 0,
x = 7, generates the volumes

dr
n ffsinte dx =47, 2 f (x? — 2x) cos x dx = 277.
0

7. The volume generated by one arch of the cycloid
x = a(@ —sin6), y = a(I —cos 9), rotating about Ox, is

3274 f "sins 46 d(40) = 572°.
8. The same branch rotating about Oy gives the volume
gra? 1 "(x — 6+ sin 6) sin 6 29 = 6m5a8,
9. Show that the whole surface of an oblate spheroid is

I1— 2 Ite
ana (x + se

log
¢ being the eccentricity and a the semi-major axis of the generating ellipse.

 

2e r=

10. The curve y°(x — 4a) = ax(x — 3a), from x = 0 to x = 3a, revolving
around Ox generates the volume 47a*(15 — 16 log 2),

11. The curve y?(2a — x) = + revolves around its asymptote. Show that the
volume generated is 27a,

12. The curve xy? = 4a7(2a — x) revolves around its asymptote. Show that the
volume generated is 477a°,

13, Find the volume and the surface generated by revolving y? = 4ax about Oy,
from x = Otox = a. Ans, V=2na. S=147a? [64/2—log(3 +24/2)].
14. Show that the volume generated by revolving the part of the parabola
xt + yt = a? between the points of contact with the axes about Ox or Oy is ja’.
ART. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 263

15. The surface generated by y = +’, from x = 0 to x = 1, rotating about Ox,
is

ax fi VI + ox 8 dx = ( 4/1000 — 1).

16. The surface generated by «4 — a’x?4 8a*y? = 0, about Ox, from « = 0
to x = a, is

nw a
Ht f (3@x — 2x3)dx = tna?
17. Ifa circular arc of radius @ and central angle 2a < @ revolves about its
chord, the volume and surface of the spindle generated are respectively
2ma*(2 sin w@ + Lsin ct cosa — acos a), 4ra%(sin a — a cos a).

18. The surface generated by x4 + 3 = 6xy turning about Oy, from x« = I to
x = 2, has for area 2(45 + log 2), and 424 when turned about Ox,

19, The surface generated by y?+ 4x = 2 log y, rotating about Ox, from y = 1
toy = 2, is 10m,
20. The area of the surface of revolution of
ay = x xe? — 1+ log (x — x? — 1),
about Oy, from « = 2 to x = 5, is 787.

21. The surface of the cycloid of revolution is §47a?, and its volume is 57723,
the base being the axis of revolution.

22. When the tangent at the vertex is the axis of revolution, in Ex. 21, the
surface and volume are 427a? and 7a’.

23. When, in Ex. 21, the normal at the vertex is the axis of revolution the sur-
face and volume are'respectively

8ra%(n — 4), wad($n? — 8).
24. Show that when the Zemniscate p? = a? cos 26 is revolved about the polar
axis, the surface generated is

4a? fl sin § d@ = 2na*(2 — 4/2).
0

25. Show that if the curve y? = ax? + dx + ¢ be revolved about Ox, the
volume generated between x,, x, is

a y 2
Vy= @ (2 = ey IP +I? + 4m)

where y,, is the ordinate at 4(%, + x).

This curve can be made any conic whose axis coincides with Ox, by properly
assigning the numbers a, 6, . The result then gives the volume of any conicoid
of revolution around one axis of the generating curve.

26. Show that the volume of the egg generated by
xi? = (x — al(s — x),
revolving about Ox as an axis, is

4
n} (a + 8) log = — 26 —2)}.
27. The volume of the heart-shaped solid generated by revolving » = a(1-++cos 6)
about the initial line is 87a’,

28. Find the volume of the Aour-glass generated by revolving the curve
yt — 2cy? + a*x? = o about Oy.
CHAPTER XXIII.
ON THE VOLUMES OF SOLIDS.

178. We have seen that the volume of a revolute is generated by
a circular section moving with its center on a straight line and its
plane always perpendicular to that straight line. If Mis the distance
between any two circular sections A, and A,, and A the area of the
circular section at a distance 4 from 4,, then the volume included
between the sections 4, and A, is

r= fra dh. $174, (3).

We propose to generalize this and to show that this same formula
gives the volume of any solid included between two parallel planes
whenever the area A of a section of the solid by a plane parallel to
the two given planes can be determined as a continuous function of
its distance from one of them.

In the first place, we observe that if the plane of any plane curve

of invariable shape moves in such a manner that

the plane of the curve remains parallel to a fixed

plane and the curve generates the surface of a

cylinder, then the volume of the solid generated
| is equal to the area of the generating curve multi-

plied by the altitude of the cylinder generated.

For we can always inscribe in the curve a polygon

of ~ sides which will generate a prism as the

curve moves in the manner described. If P is the

area of the polygon and @ its altitude, then PH

is the volume of the prism. When 2 = o and
each side of the polygon converges to 0, the area of the polygon
converges to A, the area of the curve, and the prism and cylinder
have the same altitude H The volume of the cylinder is the limit
of the volume of the prism and is therefore 4Z.

179. Volume of a Solid.—Consider any solid bounded by a
surface. Select a point O and draw a straight line Ox ina fixed
direction. Cut the solid by two planes perpendicular to Ox at points
X,, X, distant X, and X, from 0.

Whenever the area A of the section PJ of the solid by any plane
PAM perpendicular to Ox, distant v from O, is a continuous function

264

 

Fic. 102.
ART. 179.] ON THE VOLUMES OF SOLIDS. 265

of x, then the volume of the solid included between the parallel
planes at .Y, and X, is

Va fade.

To prove this, divide the interval between X, and X, into a large
number of parts, 7. Draw planes through the points of division
perpendicular to Ox, thus dividing the solid into » thin slices, of

 

 

 

 

Fic. 103.

which M@PP,M, isa type. Let A be the area of the section PJ, and
A, that of section PAZ, ata distance x, from O. Let 4V be the
volume of the element of the solid included between the sections at
x and x,, and +, — x = 4x the perpendicular distance between the
sections.

We can always take 4x so small that we can move a straight
line, always parallel to Ox, around the inside of the ring cut out of
the surface by the planes at x and x, in such a manner as to always
touch this part of the surface and not cut it, and thus cut out of the
element of the solid a cylinder whose volume is less than 7V. Let
the area of the curve traced by this line on the plane PAZ be A’.
Then the volume of this cylinder is

OV’ = A’ Ax.

In like manner, we can move a straight line parallel to Ox around
the ring externally, always touching and not cutting it. Thus
cutting out between the planes of the sections at + and +, a cylinder
of which the element of volume of the solid is a part. Let this
straight line trace in the plane PAZ a curve whose area is 4’, The
volume of this external cylinder is A’’ dx.

Hence we have

A’Ax << 4V < A" Ax,
or
/ 4 4 a
Ae< An = A”,

Also, necessarily, from the manner of construction of the lines

bounding the areas 4’ and A”,

Ac A< A”,
266 APPLICATIONS OF INTEGRATION. (Cu. XXIII,

If now the surface of the solid is such that the boundary of the
section P,JZ, at x, converges to the boundary of the section PA/ at
x, when x,(=).x, then also A’(=)A, A’’(=)A, and we have

ce
a4

Xx X dV x
v= fr ave fi Ze = fae

When A is determined as a function of x, say A = @(«), then
the evaluation of Vis a matter of integration, and we have

v= f. (x) ax.

Therefore

EXERCISES.
1. If the parallel plane sections of any solid have equal areas, then
X
v= "Adz = (X,— XA.
*y

Therefore, if a plane figure moves in any manner without changing its area
or the direction of its plane, the volume generated is equal to that of a cylinder or
prism whose base is equal in area to that of the generating figure and whose alti-
tude is equal to the distance between the initial and terminal positions of the
generating plane.

2. The general definition of a cone is as follows:

A straight line which passes through a fixed point and moves according to any
law generates a surface called a cone. In general, the cone is defined by a straight-
line generator passing through a fixed point, the verfex, and always intersecting a

Vi given curve, called the directrix.
A cone is generated by a straight line passing through
a fixed point VY, and always intersecting a closed plane
curve of area 8. Find its volume.

Draw a perpendicular V4/ = # to the plane of the
curve. Draw a plane parallel to # cutting the surface in a
curve of area A, ata distance VW = from V. Then we
shall have

 

A #
B~ HH
For, inscribe any polygon in the curve ZB and join the
corners to V. The edges of the pyramid thus formed intersect the parallel plane
containing 4 in the corners of a similar polygon inscribed insection 4. If Pandp
are the areas of these polygons, we have
Boo
PH?
from elementary geometry. But 4 and Z are the respective limits of and P. The
volume of the cone is then

Gal fithe f° Be
= |, = Boy dh =4BH.

Fic. 104.

3. A conoid is the surface generated by a straight line moving in such a
manner as to always intersect a fixed straight line and remain parallel to a fixed
ArT. 179.] ON THE VOLUMES OF SOLIDS. 267

plane. If the generating line is always perpendicular to the fixed straight-line di-
rector and traces a curve ina plane parallel to the directing straight line, the conoid
is said to be a right conoid, and the curve is called its base.
‘ Find the volume of a right conoid having a closed plane curve of area B for its
ase.

Let AVC be the straight-line director at a dis- 4 V Cc
tance VD = # from the plane of the base.

Any plane VWVAZ perpendicular to VC cuts out
of the surface a triangle of constant altitude HW, and
base ZV = y. This triangle moving parallel to
itself generates the volume required. Hence

AC etc
v= f Tide =] Ly dex, /
0 0 }

where 7=area MVN, x = OD=AVP. oO
But f'y dx = B, the area of the base. Therefore
Pa Lee. FIG. 105.

The volume of the conoid is therefore half that of a cylinder on the base B
having the same altitude 7.

This is at once geometrically evident by constructing the rectangle on A/V as
base with altitude 7.

4. On the ordinate of any plane curve, of area B, as base a vertical triangle is
drawn with constant altitude H. Show that whatever be the curve traced by the
vertex Vin the plane parallel to the base, as the ordinate generates the area of the
base, the triangle generates a volume 4/78.

5. A rectangle moves parallel to a fixed plane. One side varies directly as the
distance, the other as the square of the distance of the rectangle from the fixed
plane.

If the rectangle has the area 4 when at distance A, show that the volume gen-
erated is #4.

6. The axes of two equal cylinders of revolution intersect at right angles. The
solid common to them both is called a grozm. Find its volume.

Let Ox and Oy be the axes of the two cylinders at right angles. The quarter-
circles OAC and OBC are one fourth of
their bases. The plane xOy cuts the
surfaces of the cylinders in the straight
lines 4Z and BE. The surfaces inter-
sectin CWE. A plane DLAN parallel
to x Oy cuts the cylinders and the vertical
planes xOC, yOC in a square, which
moving parallel to «Oy generates one
eighth of the groin. Let x be a side of
this square, whose distance from QO is 4.
Then x? = a? — #*, Hence

ra fat — Rah = 2a.

The volume of the groin is 48a,
Fic. 106. where a is the radius of the cylinders.

Knowing that any figure drawn on a

cylinder rolls out into a plane figure, show that the entire surface of the groin
is 16a?.

7. Ox, Oy, Oz are three straight lines mutually at right angles to each other.

A cylinder cuts the plane xOy in an ellipse of semi-axes OA = a, OB = 4; and

the plane xQz in an ellipse with semi-axes OA = a, OC =c. The generating

 
268 APPLICATIONS OF INTEGRATION. [Cu. XXIII.

lines of the cylinder are parallel to BC. Show that the volume of the cylinder
bounded by the three planes x Oy, yOz, zOx is tabc.

8. A right cylinder stands on a horizontal plane with circular base. Show that
the volume cut off by a plane through a diameter of the base and making an angle
a with the plane of the base is 2a5 tan a.

9. On the double ordinates of the ellipse 6?x? +. a*y? = a?J*, and in planes per-
pendicular to that of the ellipse, isosceles triangles of vertical angle 2a@ are con-
structed. Show that the volume of the solid generated by the triangle is 42d? cot a,

10. Two wedge-shaped solids are cut from a right circular cylinder of radius a2
and altitude 4, by passing two planes through a diameter of one base and touching
the other base. Show that the remaining volume is (7 — 4)a?z.

11. Twocylinders of equal altitude 4 havea circle of radius a for their common
base; their other bases are tangent to each other. Show that the volume common
to the cylinders is $a°A.

12. A cylinder passes through two great circles of a sphere which are at right
angles. The volume common to the solids is (I+-}7)/z times that of the sphere.

13. Two ellipses have a common axis and
their planes are at right angles. Find the
volume of the solid generated by a third ellipse
which moves with its center on the common
axis, its plane perpendicular to that axis, and
its vertices on the other two curves.

Let AOC and AOB represent quadrants of

Cc
7 j
[: the given ellipses.
ee O4 =a, OB=6b, OC=«.

Then ZA/N represents a quadrant of the
L moving ellipse, having z andy as semi-axes.
Let x = OM be the distance of the plane ZN
from O. The area of the moving ellipse is zyz.
Also, xt + az? = arc? and d?x? 4+ ay? = a5,
Hence we have for the volume

+a
v= f (myz) dx = 4nabe.
=.

B

Fic. 107.

The surface is called the e///fsoid with three unequal axes.

14. Two parabole have a common axis and vertex. Their planes are at right
angles. Find the volume generated by
an ellipse which moves with its center
on the common axis, its plane perpen-
dicular to that axis, and its vertices
on the parabole.

Let OM and OM be the two parabole
whose equations referred to AOL, BOL
as axes are x? — 2az and y? = 2622,

MLN is the position of a quadrant
of the generating ellipse at a distance
z= OL from O, The area of the
ellipse is wxy. The volume generated
from 2 = Otoz = cis

 

V= ff (wea = mabe.
0

The surface generated is called the Fic. 108.
elliptic paraboloid.
ART. 179.] ON THE VOLUMES OF SOLIDS. 269

15. Volume of the hyferdolatoid.

Given two parallel planes at a distance apart /7. The solid cut out between
the planes by a straight line intersecting them and moving in such a manner as to
return to its initial position is called the hyperdbolatoid,

If in one of the planes a fixed point P be taken, then a straight line through
P, moving always parallel to the line generating the curved surface of the hyper-
bolatoid, cuts out a cone between the planes, called the director cone of the hyperbo-
latoid. Show that the volume of the hyperbolatoid between the parallel planes is

equal to
rae (A> -§)
2 6/]’

where &,, 8, are the areas of the sections of the solid by the parallel planes dis-
tant apart H, and C is the area of the base of the director cone.

Hint. Any plane parallel to the given planes cuts the generating line in seg-
ments that are in constant ratio. Therefore the area Z of any such section is

B= kB, + 2B, — kk,C
(projecting on a plane parallel to the bases), by Elliott’s theorem, § 164, (3).

&, and &, can be expressed in terms of 4, the distance of the section B from
either base B, or B,. Then
H
ps i Bah,
0

where # is a quadratic function of 4, and the result follows directly.

Since # is a quadratic function of 4, the results of Exercises 10, 11, 16, Chapter
XX, apply also to the hyperbolatoid, when ordinates are read sectional areas.

An important general case is: Ifthe generating straight line moves in such a
manner as to remain always parallel toa fixed plane, then C = oand

V = 4B, + 4).
16. Find the section of minimum area in a given hyperbolatoid, and show that
sections equidistant from the least section have equal areas.

17. On the double ordinate of x? + y* = a’, as a central diagonal, is con-
structed a regular polygon of z (even) sides, whose plane is perpendicular to that
of the circle. Show that the volume generated by the polygon is

. 20
Sie
A4ggs
37a

 

20?
n
and therefore the volume of the sphere is 47a.

18. Show that the hyperbolic paraboloid passing through any skew quadrilateral
divides the tetrahedron having for vertices the corners of the quadrilateral into two
parts of equal volume.

19. On a sphere of radius & draw two circles whose planes are parallel and

distant R/ 4/3 from the center of the sphere. Draw tangent planes to the sphere
at the ends of the diameter perpendicular to the planes of the circles.

Show that any ruled surface passing through the circles cuts outa solid between
the tangent planes whose volume is equal to that of the sphere.
BOOK II.

FUNCTIONS OF MORE THAN ONE
VARIABLE.

271
PART V.
PRINCIPLES AND THEORY OF DIFFERENTIATION.

CHAPTER XXIV.
THE FUNCTION OF TWO VARIABLES,

180. Definition.— When there is a variable z related to two other
variables x and yin such a manner that corresponding to each pair
of values of +, y there is a determinate value of 2, then z is said to
be a function of the variables x and y.

We represent functions of two variables x, y by the symbols
J(x, ¥), P(*, ”), etc., in the same sense that we employed the corre-
sponding symbols /(x), d(x), etc., to represent functions of one
variable .v.

When it is so well understood that we are considering a function
T(x, y) of the two variables x and y that it is unnecessary to place
the variables in evidence, we frequently omit the variables and the
parenthesis and represent the function by the abbreviated symbol 7
In like manner we frequently consider the single letter z as represent-
ing a function of the variables x and_y, and write

z= S (x,y ) :
181. Geometrical Representation.—Let z be « function of two variables x
and y. Let the value ¢ of 2 correspond to the values @ of x and 4 of. Through

a point Oin space draw three straight lines
Ox, Oy, Oz mutually at right angles, in such a
manner that Oz is vertical as in the figure. We
then have a system of three planes «Oy, yOz,
zOx mutually at right angles, of which xOy is
horizontal. These planes divide space into eight
octants. The plane xOy we take as the plane
of the variables x and y, in which we represent,
any pair of values of the variables + and y by a
point having these values as coordinates referred
respectively to Ox, Oy as axes, as in plane ana- N
lytical geometry. y

We take, as in the figure, Ox drawn to the
right as positive, drawn to the left as negative; Oy drawn in front of the xOz plane
as positive, drawn behind that plane as negative; Oz drawn upward above the hori-
zontal plane as positive, drawn downward as negative,

 

Fic. 109.

273
274 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXIV.

To represent the value z = ¢ of the function corresponding to the values + = a,
y = 6 of the variables: Construct the point Vin the plane, x Oy, of the variables,
having for its coordinates OM = a, MN = 6. The value c of the function z can
then always be represented by a point P, which is constructed by drawing a per-
pendicular VP to the plane of the variables at V, such that VP = cis drawn
upwards or downwards according as ¢ is positive or negative.

The representation is nothing more than the Cartesian system of coordinates in
analytical geometry. The numbers a, 4, ¢, or in general +, y, 2, are the coor-
dinates of the point P with respect to the orthogonal coordinate planes xOy, Os,
zOx.

We can then always represent any determinate function (x, 7) of two variables
by a point in space whose distance from a plane is the value of the function.

182, Function of Independent Variables.—Let z = /(.v, y) bea
function of the two variables x and_y. When there is no connection
whatever between x and _y, then z is said to be a function of the two
independent variables x and_y.

This means that, within the limits for which z is a function of x
and y, whatever be the arézrarily assigned values of x and_y there
corresponds a value of 2.

GEOMETRICAL ILLUSTRATION.

Consider the function of two independent variables

pie = a oF
This function has no real existence for values of x and y such that x? + y?> a?
Also, for x? + y? = a? the function iso, while for any arbitrarily assigned values
z of x and y whatever, such that x? + y? < a, the func-
tion has a unique determinate positive value. Geo-
metrically speaking, the function exists for any point
on or inside the circumference of the circle x? + y? = a?

 

 

\

\

\ P in the plane xQy, and the point representing the

! function for any such assigned pair of values of x, y,

-- po 7 is a point on the surface of the hemisphere
O x x? + 7? + gt — q2

Y which lies above +Oy. The circle x? + y? = a? is
Fic. 110. called the boundary of the region of the variables for

which the function

z=tYei—- x ~
is defined, or exists in real numbers.

In general, a function z = /(x, y) of two independent variables is represented
by the ordinate to a surface of which z = /(x, y) is the equation in Cartesian
orthogonal coordinates. The study of a function of ‘wo independent variables
corresponds, therefore, to the study of surfacés in geometry, in the same sense that
the study of a function of ove variable corresponds to the study of plane curves as
exhibited in BookI.

183. Function of Dependent Variables.—Let z = /(x, y) be a
function of two independent variables x and y. Since x and y are
independent of each other, we can assign to them any values we choose
in the region for which z is a defined function of x and y.
Art. 183.] THE FUNCTION OF TWO VARIABLES. 275

I. In particular, we can hold y fixed and let x alone vary. In
which case z is a function of the single
variable x. For example, let_y = 4 be
constant, then

z= /(x, 4) (1)
is a function of the single variable .v.
If z = /(x,_v) be represented by a sur-
face, then equation (1), which is nothing
more than the two simultaneous equa-
tions

 

(2) Fic. 111.

z=/(x, 4), t

y=,
is represented by a curve AZ ina plane .’O’s’, parallel to and at a
distance 4 from the coordinate plane 10z. Or, is the curve of inter-
section of the surface z = /(+,y) and the vertical plane y = 4, as
exhibited by the simultaneous equations (2). The equation s = /(., 4)
of this curve is referred to axes O’x', O’z’ of x and z respectively, in
its plane x’O2’.

II. In like manner, if we make x remain constant, say + = a,
and let y vary, then z = /(x, y) becomes

z= /(4, 9); (3)
a function of y only, and is represented
by a curve AZ in a plane y’O's’, Fig.
112, parallel to and at a distance a from
the coordinate plane yOz. Or, it is the
curve of intersection of the surface
a 2 =/(x, y) and the plane + = a, whose
equations are

  

2=/(*, 9), :
= (4)
Fic. 112. III. Again, since x and y are inde-

pendent, we can assign any relation we
choose between them. For example, instead of making, as in I, II,
xand y take the values of coordinates of points on the line x = a
or y = din xOy, we can make them take the values of coordinates
of points on the straight line

x—-a y—b
cos@ sing

 

 

=7; (5)

which is a straight line through the point a, din xOy and making
an angle a with the axis Ox.
Substituting

x=s=a+rcosa, y=btrsina
276 PRINCIPLES AND THEORY OF DIFFERENTIATION, [Cu. XXIV.

inz=/(x, y) for « and y respectively, and observing that 7 is the
distance of x, y from a, 4 measured
on the line (5), we have

z=/f(a+rcosa, d+ rsin a). (6)

If wis constant, (6) is a function of
the single variable 7, and is the equa-
tion of a curve AP# cut out of the
surface zg = /(x, y) by a vertical plane
through (5), and the curve has for its
equations

 

 

 

z= f(x, 7),
xe a y—s (7)
cos@. sina’

The curve (6) is referred in its own plane, 7O’z’, to O'r, O'z’ as
coordinate axis. The coordinates of any point P on the curve being
Ticks

IV. In general, + and_y being independent, we can assume any
relation between them we choose. ,

For example, we may require the
point x, y in «Oy to lie on the curve

P(*, 7) = 0.

Then, as in III, s= f(x,y) is a
function of the dependent variables x and
y which are connected by the functional
relation @(x, y) =o. The geometrical
meaning of this is: The point P repre-
senting the function z must lie in the 9 (ay)?
vertical through the point P’ represent- Fic. 114.
ing x, y on the curve A(x, y) =o. Or,
the function z of the dependent variables .v, y is represented by the
ordinate to a curve in space drawn on the vertical cylinder which has
‘the curve A’P’ for its base. The curve A’P’, whose equation in yOx
is @(x, y) = 0, is the horizontal projection of the curve in space AP
representing the function.

Geometrically speaking, the function z = /(x, y) of two depend-
ent variables x and y, connected by the relation (x, y) = 0, is
represented by the space curve which is the intersection of the surface
z= /(x, y) and the vertical cylinder ¢(, y) = 0, whose equations

are
z= /(%, 4); \
o= P(x, 9).

 

(8)
ART. 185.] THE FUNCTION OF TWO VARIABLES. 257

If we solve A(x, y) = 0 for y and get y = (x), then substitut-
ing for y in /(x, y), we express g as a function of x only, thus :

z= /[x, ¢()]. (9)
This equation (9) is the equation of the projection of the space
curve AP (8) on the plane 20x.
In like manner we can express zg as a function of y only, and get
the equation of the orthogonal projection of (8) on the plane yOz.

184. The Implicit Function.—We saw in Book I how the
functional dependence of one variable on another was expressed by
the implicit functional relation, or equation in two variables,

I, J) = Oo,
and that this implied or defined either variable as a function of the
other. We also saw that this functional relation could be repre-
sented by a plane curve having x and y as coordinates of its points.
The implicit function of two variables is a particular case of a func-
tion of two independent variables. For, in such a function,

2=/(*, 7),

of the two independent variables x and_y, if we make z constant, say
= ¢, we have the implicit function in two variables
S(%; I) = 6. (1)
Geometrically, this is nothing more than the equation to the
curve LIN, Fig. 115, cut out of the
surface z = /(x, y) by the horizontal
plane z = ¢, at a distance c from «Oy.
Its equations are
z=S/(x, ¥),
d =A ? y) \ (2)
The lines cut on a surface by a
series of horizontal planes are called
the contour lines of the surface. In
particular, if z = o, then (x, vy) = 0
is the equation in the xOy plane of the horizontal trace of the surface
z= /(x, ¥), or the curve ABC cut in the horizontal plane by the
surface.
In the same way that the implicit equation in two variables
defines either variable as a function of the other, the implicit function

Is; J, 2) =O
is an equation defining either of the three variables as a function of the

other two as independent variables, and can be represented by a
surface in space having x, y, 2 as the coordinates of its points.

185. Observations on Functions of Several Variables.—The
gencral method of investigating a function of two independent
variables is to make one of the variables constant and then study the

P

 

Fic. 115.
278 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXIV.

function as a function of one variable. Geometrically, this amounts
to studying the surface represented by investigating the curve cut
from the surface by a vertical plane parallel to one of the coordinate
planes.

Or, more generally, to impose a linear relation between the
variables x and y, and thus reduce the function to a function of one
variable, as in § 183, III, which can be investigated by the methods
of Book I. Geometrically, this amounts to cutting the surface by
any vertical plane and studying the curve of section.

As we have seen in § 184, and as we shall see further presently,
the study of functions of two variables is facilitated by reducing them
to functions of one variable, and reciprocally we shall find that the
study of functions of two or more variables throws much light on the
study of functions of one variable.

186. Continuity of a Function of Two Independent Variables.
Definition.—The function z = /(x, y) is said to be continuous at
any pair of values .v, y of the variables when corresponding to x, y
we have /(, v) determinate and

BIN y) =N% 9)
for «,(=)x, y,(=)y, independent of the manner in which +, and y, are
made to converge to their respective limits x and _y.
The definition also asserts that
L [4 91) —S(*,¥)] = 0,
for «,(=)x, ,(=)y.

In words: The function z = f(x, y) is continuous at x, y when-
ever the number z, = /(x,, y,) converges to gz as
a limit, when the variables x,, y, converge
simultaneously to the respective limits .., y in
an arbitrary manner.

Geometrically interpreted, the point P,,
representing +,, 7,, 2,, must converge to P,
representing ., y, 2, as a limit, at the same time
that the point J, representing «,, y,, con-
verges to JZ, representing x, 7; whatever be the
path which 4 is made to trace in xOy as it
converges to its limit JZ,

A function /((x, ) is said to be continuous in a certain region A
in the plane «Oy when it is continuous at every point x, y in the
region A,

An important corollary to the definition of continuity of /(x,_y)
at x, y is this: Whatever be the value of /(x, y) different from o, we
can always take x,, y,so near their respective limits x, y that we
shall have (x, , _¥,) of the same sign as /(x, 9).

187. The Functional Neighborhood.—A consequence of the
definition of continuity of z= /(x, y) is as follows:

 

 

Fic, 116,
Art. 187.] THE FUNCTION OF TWO VARIABLES. 279

If /(x, 1) is continuous in a certain region containing a, 4, we can
always assign an absolute number ¢€ so small that corresponding to
e there are two assigned absolute numbers 4 and &, such that for all
values of . and y for which

jx — aj <4, |y—b| <&,

IA 7) — Ne 4)| <é
The proof of this is the same as that given for a function of one
variable. For, let_y and a be fixed numbers, and let « vary. Then
whatever number te be assigned, we can always assign a correspond-
ing number / # o, such that for |. — a|< 4 we have

| J) —N4 3) | < $6,
since /(x, y) is a continuous function of one variable x, and its limit

is /(4, 8’).

In like manner for | — 4| < & we have

[7a ¥”) —/(% 4)| < 46
[7 ¥) —J(4, 8) | <eé

for all values of x, y, such that
jx —al <4, |y—3b| <&.

Geometrically speaking, whatever be the value c = /(a, 4), we
can always assign an arbitrarily
small number ¢, corresponding to
which there is a rectangle AZJZNV
in the plane xQy, the coordinates
of whose corners are K, (a — 4,
+h; L, (@th, +4);
(a+ 4, b—2); N, (a—h, b>),
such that, whatever be the point
x, v in the rectangle AZAZN, the
corresponding point x, y, 2 on the
surface z = /(x,_1’) lies between the
parallel planes z = c— e, (STUV)
and z=c+e, (WXYZ). The
point P representing a, 6, ¢. i

Such a region KZA/N is called 1Ge TEs
the neighborhood of the point a, 4, The point is called its center. In
like manner the corresponding parallelopiped STUV-WXFZ is
called the zeighborhood of the point P in space.

The above results may be stated thus: When the variables 2, y
are in the neighborhood of a, 4, then must the continuous function
S/(x, ”) be in the neighborhood of SG 4).

An important consequence is this: If (x, y) is continuous in the

we have

and on addition

 
280 PRINCIPLES AND THEORY OF DIFFERENTIATION. (CH. XXIV.

neighborhood of /(a, 4) # 0, then we can always assign a neighbor-
hood of a, 6 such that for all values of x, y in this neighborhood
the value /(x, y) of the function has the same sign as /(a, 4).

EXERCISES.

1. Trace the surface representing the function

Ae, y) Sy — mx + b.
Put z = vy — mx +6. When z = 0, the surface cuts xOy in the straight line
v=mx —b. If « =a, we have for the section of the surface by the plane x =a
the straight line

z=y—ma+s.

Whatever be a, this line is sloped 45° to the plane xOy. As x = a varies, this
line moves parallel to itself, intersecting the fixed line y = mx — 6 in xOy, and
therefore generates a plane.

In like manner it can be shown that the implicit function of the first degree in

xX, IV, 4,
Ks, 7, #) = Ax+ By 4+ Cz 4+ D=0,
is always represented by a plane.
2. Show that the function
ya? — xt — 3?
can be represented by a sphere, by showing that it can be generated by a circle
whose diameters are the parallel chords of a fixed circle, and whose planes are per-
pendicular to that of the fixed circle.
3. Trace the surfaces representing the implicit functions
x y? gt x y
atpta-t=% [tG-2=0
by their plane sections.
4. Trace by sections the surface representing
(x? — az)%(a? — 2%) — xy? =o.
5. Find the maximum value of the function
ax? y2
eB
when the variables are subject to the condition x + y = 1.

Let z = fix, v). Then zis immediately reduced to a function of one variable
by substituting 1 — x for y.

Ax, 9) 21

x? y?
eS Be
dz 2x (I — x)
a Ma oa

gives x= @/(a? +0), y= /(a? +H) 2 =1 — 1/(a? + 8%), which is a
maximum value of z since D%z is negative.

Consider the geometrical aspect of this problem. We have
x yf

BM a aa) (1)

the equation of the elliptic paraboloid whose vertex is 0, 0, 1, and which cuts Oy
in the ellipse x?/a? + y?/é? = 1.
ArT. 187.] THE FUNCTION OF TWO VARIABLES. 281

We wish the highest point on the curve cut out of the surface by the plane
x+y=1. Take O’r, the horizontal trace of this plane, as the positive axis
of ry, and O’z’, its vertical trace on yQz, as axis
of z in the plane 7O’z’. Then for the equation
to the curve in the plane x + y = 1, or

ila ee ill
a
we substitute x= r 4/2, y = 1 — r 4/2 in (1).

Hence the equation to the curve of section in its
own plane is

sl ( 3) +72 (“a )*.

 

 

 

arp?
Dz = 0 gives r = a®/(a? + 8%) 4/2, and
Dir =—. Hence the values of x, y, z as before.

The first method, in which we substitute for y in terms of x, is only possible
when we can solve the condition to which the variables are subject, with respect to
one ofthem. The second method, in which we express x and y in terms of a third
variable, is always possible, although perhaps cumbersome.

The class of problems such as the one proposed and solved here should be care-
fully considered, for we propose to develop more powerful methods for attacking
them. But it should not be forgotten that those methods themselves are developed
inthe same way as is the solution of this particular problem. The student should
accustom himself to seeing curves referred to coordinate systems in other planes
than the coordinate planes, for in this way a visual intuition of the meaning of the
change of variables, and a concrete conception of the corresponding analytical
changes which the functions undergo, is acquired, :
CHAPTER XXV,

PARTIAL DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES.

188. On the Differentiation of a Function of Two Variables.—
A function of two independent variables has no determinate deriva-
tive. It is only when the variables are dependent on each other that
we can speak of the derivative of a function of two variables. The
derivative of a function of two variables is indeterminate unless the
variable is mentioned with respect to which the differentiation is
performed and the law of connectivity of the variables given.

189. The Partial Derivatives of a Function of Two Independ-
ent Variables.—Among all the derivatives a function of two variables
can have, the simplest and most important are the partial derivatives,

Let 2 = /(x, y) be a function of the two independent variables
xand y. The simplest relation we can impose between x and y is
to make one of them remain constant while the other varies. We
then reduce the function gto a function of one variable, to which
we can apply all the methods of Book I for functions of one variable.

For example, let_y be constant and x variable. Then z = _/(x,y)
is a function of x only, and it can be differentiated with respect to
x by the ordinary method, and we have

re yf)
x, — x

x(=)%
= SAX, ¥)-

This is called the fartial derivative of the function z or / with
respect to x. To obtain the partial derivative of /(x, y) with respect
to x, make y constant and differentiate with respect to x.

Correspondingly, the partial differential of f(x, y) with respect to
x is the product of the partial derivative with respect to x, D,/%, and
the differential of « or x,—«= 4x. If we represent the partial
differential of / with respect to x by d,f, then we have

a,f =SAx, Jy) ax,
and the corresponding partial differential quotient is

4S = fe, J) = Dz.

It is customary to employ the peculiar symbolism designed by
282

 
ART, 190.] THE FUNCTION OF TWO VARIABLES. 283

Jacobi for representing the farfza/ differential quotient or derivative
of A(x, ¥) with respect to x. Thus the above will hereafter be
written (the symbol @ is called the round 2)

oF

dx —~ dx”
The symbol 0 being used instead of d to indicate the partial
differential as distinguished from what will presently be defined as
the total differential, which will be represented as formerly by d.

In the same way, if we make x constant, then /(x, vy) becomes a
function of one variable _y, and has a determinate derivative with
respect to y. This derivative we call the partial derivative of /(x, y)
with respect to y, which is written and defined to be

Alo 9) = fie 3) — M3)
dy ey, I,—-d :

190. Geometrical Illustration of Partial Derivatives.—lIf
z= /(*, v) is represented by the ordinate to a surface, then at any
point P(x, y, 2) on the surface
draw two planes P.1/Q and PAR
parallel respectively to the coor-
dinate planes «Oz and yOz. These
planes cut out of the surface the
two curves, PK and P/ respectively,
passing through P.
z= f(x, 7) (y constant)

is the equation of the curve P& in
the plane PAZQ.
z= /(*, 7) (x constant) Fic, 119.
is the equation of the curve P/ in the plane PAR.

Draw the tangents P7 and PS to the curves PX and P7/ in their
respective planes, and let them make angles @ and w with their
horizontal axes, as in plane geometry. Then we have

dz dz
3y oo tan qd, oo .

 

Therefore the partial derivatives of /(.v, 1’) with respect to x and
_y are represented by the slopes of the tangent lines to the surface
z= /(x, y), at the point +, 1, 2, to the horizontal plane xOy. These
tangents being drawn respectively parallel to the vertical coordinate
planes Oz, yOz.

Also, draw PT’ parallel to 4ZQ, and PU parallel to AZ. Then
we have

VT =(x,—~x)tang¢, US=(y’ —y) tan y,
284 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CH. XXV-

if Qis .x,, y, and Ris x,y’. Or

af ae
VT= an US = met
represent the corresponding partial differentials of / with respect to
xand y at P(x, y, 2).

Thus the partial derivatives and differentials of /(x, _y) are
interpreted directly through the corresponding interpretations as
given for a function of one variable.

191. Successive Partial Derivatives.—lf 2 = /(x,_y) is a func-
tion of two independent variables + and y, then, in general, its
partial derivative with respect to «x,

Y= fx, 9),

is also a function of x and_y as independent variables. This deriva-
tive can also be differentiated parfadly with respect to either x or y,
as was /(x, vy). Thus, differentiating again with respect to x, y being
constant, we have the second partial derivative of / with respect to x.

In symbols
Of(x, Vv
PAHS) = 72x, 9).
In like manner /,(~, y) can be differentiated partially with respect
to y, «being constant. Thus we have for the second partial differen-
tial quotient of / with respect first to + and then to y
OPI) en
“ayax my (Xs ¥)-

Similarly, differentiating /j(., ) partially with respect to_y, we

~~ have

Ox, ¥) _ AF (%,Y)
a
and with respect to + we have
OKI) _— SNH I) og
“Oxby ae Fel 9).

Thus we see that the function z = /(x,_y) has two firs? partial
derivatives,

=S il 9),

Oz 80g
ax’ ao’
and four second partial derivatives,
0°s Ag 0°72 022
aa?” a? Gyan’ Geos
Each of these give rise to two partial derivatives of the third
ART. 192.] THE FUNCTION OF TWO VARIABLES. 285

order, and generally the function has 2* partial derivatives of the
nth order, of the forms

On Oz
Ox? Ay? Oy? Bat?
where f and ¢ are any positive integers satisfying +g =. These
nth derivatives, however, are not all different, for we shall demon-

strate presently that 0x¢ and 0)” in the denominators are interchange-
able when the partial derivatives are continuous functions, and that

Wz
aah oy = Sra”
or the order of effecting the partial differentiations is indifferent
The number of partial derivatives of /(x, _v) of order is then # + 1

EXAMPLES.
If z= 27+ axy + cosxsiny,
a
= = 2x + ay — sinx siny,
dz
ay = ax + cos x cos y.

x? 2
21 fx, 7) = St+5 —1,

. F ax Of wy
eae ee ap ae
Oz 0?z

8. In Ex. 1, =a—sinxcosy=

 

dy Ox dx dy.’
de : ez .
aga eS COS SID Yi ae cos x sin y.
af af
4. In Ex. 2, show that i ip an Oy

 

192. Theorem.—The partial derivatives are independent of the
order in which the operations are effected with respect to x and y.
In symbols, if z = /(*, y), we have
Wz dO
az op By Ox"
Consider the rectangle of the
four points
M, (%, 9); AL, (4%);
Q, (ep 9)3 Re (HN)
The theorem of mean value applied
toa function of one variable -v gives
Af _ Ax) 9) —N% 9)
Ax x, :
=f (8,9); (z)

where is some number between +, and x. (See Book I, § 62.)

 
286 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CuH. XXV.

Form the difference quotient of (1) with respect to_y,

4A fy) —S% 9) ~% 4) Te)

 

Ly Ax (1 — 9), — *)
= AEWA BEM p08, 9), @)

where 7 is some number between y, and y. The value (2) is therefore
equal to the second partial derivative of 4, taken first with respect to
x, then with respect to_y, at a pair of values &, 7 of x, y. Geomet-
rically, at a point &, 7 in the rectangle 7 QAZR.

In like manner, taking the difference-quotient of 7, first with respect
to y, we have

Af _ f(x, ) —SI(*, 9) ee’ / ,
ay = sees ey eee =Sy(*, 17); (3)

where 77’ is some number between _y, and y.
Now taking the difference-quotient of (3) with respect to x, we

 

have
4s = SA) —S(*; Hi) —Sf(x,, y) +/(*, J)
Ax Ay (x, — x)(y, —)
_— Silty 7) Si (% 1’) Wo et
See =Sae (5% 1’), (4)

where &’ lies between x, and +, 7’ between y, and y. The value of
(4) is then equal to the second partial derivative of /, taken first with
respect to _yand then with respect to x at some point &’, 7’, also
inside the rectangle 7 QILR.

But (2) and (4) are identically equal. Hence we have

SiS ”) = 785" 1’);
ané,n) _ aAé',7)
or “On 0E = 8s’ On! (5)

This relation is true whatever be the values +,, y,.
If now the functions

ay as
dude ae oy

 

are continuous functions of « and yin the neighborhood of x, y, then
since &’, n’ and &, 7 converge to the respective limits +, y when

x(=)*, y(= yy, the two members of (5) converge toa common limit
at the same time, and therefore

OF gs Or

aydx Ox dy" (6)
ART. Ig2.] THE FUNCTION OF TWO VARIABLES. 287

Incidentally, equations (2) and (4) show that the difference-
quotients

44f_ 4 4 af _ AS
Ay Ax = dy Ax’? Ax dy = Bx dy
converge to a common limit whatever be the manner in which
Ax(=)o, 4y(=)o, and that common limit is
2 2
el, Agee LE
Oy Ox Ox Oy

 

 

Observe that in the ee
aay =le
the operations are performed in the order of the proximity of the vari-

able to the function.
In like manner, making use of the result in (6), we have

2(@4)=22%=2 aaf a ay

7 Ox

Bx Oy) — dx dy Ox Oy Ox Ox By Bak
of oF
Ox? dy” Oy 02?’
and similarly for other cases. Hence, in general,
artery abras
ian Heda?
in whatever order the differentiations be made.

EXERCISES.
1. If 2 = tan i show that
dz Oz = Pay
ax dy — ~ ay dx Ox =a + 2)? *
ay
2.1f :=—=—;, find Diz, Dix.

qi ee

3. Verify in the following functions the equation

 

ef af
ax dy ~ dy Ox"
xsiny + y sin x, log tan (y/x),
x log y, (ay — bx)/(by — ax),
x), y log (« +29).
4.If <= ea » show that
at =
Oz I hi I5xy

 

 

ax dy = (1 x? + yd "Ax? ay? ( +t 4 yt
5. If « = x3y? — 2xy* + 3x75, show that
288 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CH. XXV.

or ot
6. If ¢ = log (ea + 8), then ae B =I.

Oc | Oc
2) ees ge foeee a =
7. If c(ea+ ¢) = ea, then a5 Tag =(a+6— 1t)e.

8 If z= e*siny + &sin x, show that
(=) : + (=) : = 62% 4 er + 2e%+¥ sin (x + y).
Ox ay
9. From z=. y*, show that

Oz Oz
Fae TD 5g a ae 1 a

10. Show that if 7 is the angle between the plane «Oy and the tangent plane to
the surface z = f(x, y) at x, y, then

vers (6) @):

Let P, (x, y, 2) be on the surface, and
PRS the tangent plane.

Draw MN = p perpendicular to &S.
Then y = PNM.

af - PM af _ PM
ae RM? oy Sa
Since RS-VM = RM-MS,
and RS? = RM? + MS,
FiG:. ¥2i; ws (MN )-2 = (RM)-2 + (MS),
and therefore

wir = Bes (+ (= (B)'+ BY

af\* af\ 2
se?y = I+ 4) + (F) S
11. In Fig. 121, let W be axy point in the trace of the tangent plane with xOy.

Let V7 make an angle 6 with Ox, and the tangent line VP to the surface make an

angle @with the horizontal plane xOy. Then the triangle RS/Z is the sum of the
triangles RAZN, NALS, or

RM. SM = SM.NM cos 6 + RM.NM sin 6,
PM PM PM

Wat = Rar °° + ar

 

Also,

or sin 6.

 

Therefore the slope of a tangent line to the surface at x, y, s, whose vertical
plane makes the angle 6 with Ox, is

me x . oan =
tang@= a5 608 6 Tay sin 0. (1)
12, Find the tangent line to a surface at P which has the steepest slope.

From Ex. 11 we have

*

d Ff af
_onee— a, 8 Tig oe
ART. 192.] THE FUNCTION OF TWO VARIABLES. 289

The values of sin 6, cos § from this equation put in (1), Ex. 11, give for the
tangent line of steepest slope

Observe that this is the slope of the tangent plane in Ex. 10.
13. If @(«, v) = 0 is the equation of any plane curve, show that

ID(x, ¥)
dy Ox
dx G(x, ¥)"

oy

Let zs = (x,y) be the equation of a sur-
face cutting the horizontal plane in the curve
P(x: 1) =O.

Let P, (x, ¥) and A, (x, ,) be two
points on the curve @(x, vy) = 0. Draw the
vertical planes through / and /, parallel
respectively to xOz and yOz, cutting the sur- N»
face in curves PQ, 2,0. Then Q isa point
+, 7, 2on the surface. The derivative of y
with respect to x in @(x, vy) = o is the limit of the difference-quotient

 

Fic. 122.

 

y—-—y _ MP, MQcotMPQ_ tanMPQ tan MPO
x,—« PM” MQcoMPQ” tanMPO”~ ~~ tan VPRO
Also, tan MPO = ee) , tanWVPOQ = eet,

& being between x and x, 7 between y and y, (by the theorem of the mean).
Therefore, when +,(=)x, v,(=)y,

ap
dy nay ax
ax ae 5D
ay
This usually saves much labor in computing the derivatives of implicit functions

in x and y.

The important results of Exs. 10, 11, 12, and 13 are deduced here geometrically
to serve as illustrations of the usefulness of partial differentiation. They will be
given rigorous analytical treatment later.

14. Employ the methods of Book I, and also that of Ex. 13, to find D,y in the
following curves:
x/a? — 7/6 —1 = 0, xsiny —ysinx« = 0,
axty +. byx — 4xy = 0, ex sin y — log y cos« = 0.
15. Show that the slope of the tangent at x, y on the conic
ax? + by? + 2hxy 4+ 2ux 4+ 2vy-+d=0
ay ax + hy +u

s ax Ax bby ty
CHAPTER XXVI.

TOTAL DIFFERENTIATION.

193. In the fartial differentiation of /(x, y) we made x or y
remain constant during the operation, and differentiated the function
of the one remaining variable by the ordinary methods of Book I.

We now come to consider the differentiation of /(x, v) when both
x and y vary during the operation of evaluating the derivative. Such
derivatives are called /ofa/ derivatives.

‘In order to make clear the nature of the total derivative of a

function
z=/(%, 9),

consider the simple case when there is a linear relation between x
ond y,

wa x yy’
tm

 

 

= T,
where / = cos 6, m = sin 6, and the differentiation is performed with
respect to 7, Let x’, »’; 4 m; be constant. Then 7 varies with «
and y, and
= (xa (9 — 9p
Also, x and y are linear functions of 7, and

xa’ tl, yoy + aor.
Substituting these values of x and yin /(x, y), we reduce that
function to a function of the one variable 7, and it becomes

(x! + Ir, yy’ + mr), (1)

The derivatives of this function with respect to 7 can now be

formed by the methods of Book I. Thus we get by the ordinary
process of differentiation

a oe ee
dr’ dr®’— dr”
for the successive derivatives of / with respect to 7. These are
called the /ofa/ derivatives of f with respect to 7. Both variables x
and y vary with 7.

We can give a geometrical interpretation to this total derivative

as follows: The equation
x— a yy
a: - “ao (2)

290

etc

“9

 

 
ART. 194.] TOTAL DIFFERENTIATION. 291

is the equation of a straight line through x’, y’ in the horizontal
plane xOy, making an angle 6 with Ox. 7 being the distance
between the points .x’, y’ and .v, y on the line. Let O’ be the point
x,y’. Draw O’z’ vertical, The vertical plane rO’z’ through the

 

Fic. 123.

line (2) cuts the surface representing z = /(x, y) in a curve PP,,
whose equation in its plane, referred to O’r and O’z’ as axes of
coordinates 7 and gz, is

e=f(x +l,’ 4+ mr). (3)
Let P, be a point on this curve whose coordinates in space are
x,,%,, %, and in rO’z' are 7,, 2. Let 7,—r= Ar. Then, by

definition, the derivative of z with respect to r at x, y is the limit
of the difference- quotient, when 7,(=)r,

ep Ss S(#) 1) — Ie, J).

nor 1—T

 

Hence we have
dz i
rr = tan @,
where oo is the angle which the tangent P47” to the curve PP, at
f, and therefore to the surface, makes with O’r, or the horizontal
plane «Oy.
Observe that as +,, y, converge to x, y, the point JZ, converges
to M along the line A/,A7.
By assigning different values to 6 we can get the slope of any
tangent line to the surface, at P, with the horizontal plane.
In particular, when the line (2) is parallel to Ox or Oy, or, what
is the same thing, when 6 = z or 47, the total derivative becomes
a partial derivative, as considered in the preceding chapter.

194. The Total Derivative in Terms of Partial Derivatives.—
It is in general tedious to obtain the total derivative, after the
manner indicated in § 193, by reducing the function directly to a
292 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVI.

function of one variable, and generally it is impracticable. We now
develop a method of determining the total derivative in terms of the
partial derivatives. Let z= /(x,_”), where + and y are connected
by any relation @(x, vy) = 0. To find the derivative of z with respect
to 4, where /is any differentiable function of x and y.

Let z take the value z,, and ¢ become 4, when x, y become

Bey Di

Let be constant and x, be a variable. Then the law of the
mean is applicable to the function /(x,, y) of the one variable ~,,
and we have

Aes 3) I 9) = — Dae NE 9), 6)

where & is some number between x, and +.
In like manner, let x be constant and y, vary, then, by the law of
the mean,

Ae) Ae) = A-Dz Ov Mr

where 77 is some number between y, and y.
Adding (1) and (2), we have

a a
NA I )-(* 9) = (%, a *)ae5s9) - (, 5S 7). (3)
Therefore the difference-quotient with respect to / is
2-2 ; 4x ; Ay

Ax, Ay, 42, Af converge to o together, and at the same time
x, (=)*, 6(=)*, (=, o(=)y. Also,
ang, 9) a(x, .7)
ae and a

 

have the respective limits
0 -
YORI) oq Pes
Ox oy

if these latter functions are continuous in the neighborhood of x, y.
Passing to the limit in (4), we have for the total derivative of /(x, 9)
with respect to 4, at x, y,
if _odx ofa
a@-aata a (5)
The geometrical interpretation of (1) is this: In Fig. 123 we
have M, (x, ¥); AL,, (%,,54)3 Q (%, 9); RB, (%%).

Iso,
SX) I) —/(% 9) = OK = PQ' tan O'PK.
ART. 195.] TOTAL DIFFERENTIATION. 293

But, since on the curve PA there must be a point X, (&, y, 2) at
which the tangent is parallel to the chord,

tan Q'/PK = aeNés Jy).
In like manner for equation (2),
N&,39;) —AAp 9) = LP, = — LE tan LKP.,
But, since there is a point 7, (x,, 7, 2) on the curve AP, at which
the tangent is parallel to the chord, we have
— tan ZAP, = af, 7).

195. The Linear. Derivative.—An important particular total
derivative is the case considered in § 193. Suppose there is a linear
relation between x and _y, such as

x—a_y—b
lm
Then x =a+/,y=64+ mr. To find the total derivative of
J(x, ¥) with respect to the variable 7, we have

 

== 7.

ae Z a m
oe ae
Z= cos 6, m =sin 0, being constant. Therefore
yf pe of
ae ae (x)

This isa much simpler way evaluating this derivative than that
proposed in § 193.
As before (see Ex. 11, § 192, § 193),

0
of F co 50+ 7 sin 8 (2)

is the slope to the horizontal plane of a oe line to the surface,
in a vertical plane making an angle @ with xOz.

Again, suppose, as in §194, that x and y are related by
@(x,_”) = 0, and we wish the derivative of * with respect to s, the
length of the curve @(*, _v) = 0, measured from a fixed point to x, y.
Then, putting ¢ = s in (5) § 194,

yaa, Sw
a dxds' By as (3)

tan w@ =

q
But = = cos 0, ; = sin 6, where @ is the angle which the tan-
gent to (x, vy) =o at x, y makes with Ox. Hence we have the
same value of the derivative as in (1),

g

fF =-7. os 0+ 2 sin 4, (4)
294 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVIL.

which is also the slope to the horizontal plane of the tangent line to
the surface.

196. The Total Differential of /(x, y).—By definition, the
differential of a function is the product of the derivative into the
differential of the variable. Hence, mutiplying (5), § 194, through
by dé, we have for the total differential of fat x, y

_ of af
f= ax ax + By?” (x)
Observe that

OF jes Sa OF yas
| HI BV =aS
are the partial differentials of f Hence
f= 9,f + 4,f; (2)

or, the total differential of fat x, y is equal to the sum of the partial
differentials there.

The value of the differential at a fixed point depends on the
values of dx and dy, which are quite arbitrary.

The geometrical interpretation of the differential is as follows: In
Fig. 123, let dv = MQ anddy= MR. Draw PR’, QM’, OS
parallel to JZR. Then

a.7= VO" =MS and If= RR’ = SM",
df = M'S + SM" = M'M";
or, the differential of the function is represented by the distance
from a point in the tangent plane to the surface at P from a hori-
zontal plane through P.

197. The Total Derivatives with respect to x and _y.—If, in the

total derivative
af_oax yw
a be a ay at”
we take / = x, then the total derivative of / with respect to x is

FF aT @

dx — ax T Gy dx’ (1)
If we take 7 = y, then

af _afdax af

a sea et Ele (2)

Equations (1) and (2) represent the /o/al derivatives of / with
regard to x and y respectively. These derivatives are quite distinct
and different from the partial derivatives, as is shown by the formule,
and as is exhibited in their geometrical interpretations as follows:

The total derivative of z = /(x, y) with respect to x is the limit
of the difference-quotient

2-8

 

?
an,
ArT. 197.] TOTAL DIFFERENTIATION. 295

x and _y varying as the coordinates of a point on some curve /H in
the horizontal plane.

 

If, P, is x,, 1, 2,, then, in Fig. 124,
4-2 =/P,=JP{, 5 —* = NM = JP’,
a ‘ ye ,
Therefore = = tan q@ is the total derivative of z with respect
to x. That is, the total derivative of f with respect to x is repre-
sented by the slope to Ox of the projection P’7” of the tangent P7’

to the surface on the vertical plane xOz. The tangent PZ being
in a vertical plane through P which makes with «Oz the angle 6

determined b a = tan 6, as determined from ¢(x, y) = 0. That
VR P(%, I

is, os is the slope to Ox of the horizontal projection AZM of the
tangent PT.

In like manner the total derivative of / with respect to_y is equal
to tan f#, this being the slope to Oy of the projection of the same
tangent PZ on the perpendicular plane _yOz.

Equations (1) and (2) are immediately determined from the total

differential
_ f af
af = 5K + we?

by dividing through first by dv and then by dy.
In Fig. 124 we have

af = JT afr” = JUL,
and equations (1) and (2) can be verified by the differential quotients
taken from the figure directly.
296 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVI.

198. Differentiation of the Implicit Function /(x, y) = o.—An
important and valuable corollary to the total differentiation of the
function z = /(x, y) is that which results in giving the derivative of
y with respect to x in the implicit function /(x, v) = 0.

Since z= Oinz=/(x, _y) gives /(x, vy) = 0, andin /(x, y) =0
are admissible only those values of x and_y which make 2 constandly
zero, the derivative of z with respect to any variable must be o.

Therefore, from (5), §194, or (1), § 196, § 197,

F
dy Ox
ae OOF

y

This has been geometrically interpreted in Ex. 13, Chap. XXV.

In general, the plane z = ¢, c being any constant, cuts the surface
z = f(x, y) in a contour line, or curve in a horizontal plane, at dis-
tance c from the horizontal plane xOy. The equation of this curve
in its plane is (x, vy) =c. In the same way as above,

Ofdx afd df _ de

i et wa aa”
dy af
. &. 4. &
dx ax af’
ne

which corresponds to the slope of the tangent to the contour (at the
point x, y, c) to the vertical plane Oz.

 

EXERCISES.
If «5+ 73 — 3axy =, find Dy.
af =< 2 af = 2
Here ig = 3" — = 307 — a2)
dy wv#-—w
dx ax—y

2. Find Dyy in x™/am + yu/bm = 1.
af 3 MAM af myn , dy _ BN Poh at
ae ae a ee VaR ON) IG)

dy y log y*¥ — y

3. If x logy —ylog x =0, then te | wee

 

 

4. Let «x =pcos6. Find the total differential of x.

Ox Ox 7
Taal 7 = —psin 6,

dx = cos 0 dp — psin@ do.
ART. 108. | TOTAL DIFFERENTIATION, 297

5. Find the slope to the horizontal plane of the curve

xt?
HEatR x —

ae! Be Ans. (z- =) V2.
=axt+y.

6. Find the slope to xOy (the steepness) of the curve cut from the hyperbolic
aa 2 = x*/a* — y*/6? by the parabolic cylinder y? = 4px.
e have

dz ds dx , dz ay
Bo aa as me
s being the length of the parabola y? = 4px. Here

tan w =

Oz ax Os oy
dx at? ay BE

afrt @)]-S BaP

 

which is the declivity of the curve in space at x, y, 2.
Find the points at which the tangent to this curve is horizontal.

7. 1f w= tan-(y/x), du = («dy — y dx)/(x* 4-9).
8. If 2= 49, dz = yx-t dy + xy log x dy.
oe oo the locus of all the tangent lines to a surface z = f(x, y) at a point
o Three FP draw a vertical plane, Fig. 123, r4/P, whose equation is
xa yb
Lo hh

 

(1)

Then the equation to the tangent line, P47’, to the surface at FP, in the plane
rMP in terms of its slope at a, 4, ¢, is
gm 6 af
+r ar”

 

z and r being the cvordinates of any point on the tangent line. But at a, 4, ¢

 

af | ae _ xa, 6) da | Af(a, 6) db
ar da dr a6. dr’
= ge + may y

Therefore the equation to the tangent line to the surface at a, 4, c, whose hori-
zontal projection makes 7 @ with Ox (where / = cos rie m = sin 6), is
tla ae

Z—-¢=>

ae rm L (2)
Eliminating 77 and rm between (1) and (2), we have

a
s—ea(e-gZyyy— %, (3)
an equation of the first degree in x, y, 2, which is the locus in space of the tangent
lines at a, 4, con the surface. This locus is a plane, Exercise 1, Chap. XXIV,
touching the surface at a, 4, c, and is defined to be the tangent plane to the surface
at a, 6, ¢
298 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVI.

10. Show that the equation to the tangent plane to the surface z = ax? + dy?
at any point x’, y’, 2’ on the surface is

of 2! = 2(axx’ + dyy').
11. Use the equation to the tangent plane

@—aZ 49 - oF Axo

to verify Ex. 12, § 192.

af a
The direction cosines of the plane are proportional to is z — 1. Hence
if 7, m, m are these cosines, ‘
a $e
Fe” F =i” F\? (fv?
baa Jt (=) (=)
Also, sec?y = 1/n*, giving the same result as Ex. 12.
12. Show that when
OF 4.
ae = ae 9, (1)

the tangent plane to the surface is horizontal at values of x, y satisfying z= /(x, y)
and (1).

13. Show that the curve on the surface zs = /(x, y) at all points of which the
tangent plane to the surface makes the angle 45° with «Oy is the curve cut on the

surface by the cylinder
af\ 2 af\ 2
(7) + (J) =1.

14. Apply Ex. 13 to show that the cylinder 2? + y? = 4a? cuts the sphere
x? + y?+ 2? = a? in a line at every point of which the tangent plane to the sphere
is sloped 45° to the horizontal plane. Draw a figure and verify geometrically.

15. The equation x? + y? = a® represents a vertical cylinder of revolution whose
axis is Oz and radius isa. Find the equations of the path of a point which starts
at + = a, y = 0, 2 = O and ascends the cylinder on a line of constant grade 4.
This curve is the helix, a spiral on the cylinder, having for its equations
z
za’

z

% = acos
ka

y =asin
CHAPTER XXVII.
SUCCESSIVE TOTAL DIFFERENTIATION.

199. Second Total Derivative and Differential of z = /(x, »).
It has been shown in § rg4, (5), that
df fd aw is
“dt” 0x dt ay dt’

where + and y are any differentiable functions of 4.
If we differentiate again with respect to 4, then

Pf = ase: ats ate 4),

 

dP adt\ax dt dy at)’
_dxd(f\ Y ex , wd[S\, H ty
~ a ale )+ 3 ax dP a d\ay | tay ae)
Since a ; i are functions of x and _y to which (1) is applicable,

in the same way we have

an (5c) = az (Se): ae + (as) ar’

_fax , Wy
~axt dt ay dx di (3)

d(af\ _ Of dx of ey
o(}) = ax oy di | a at’ (4)
Substituting in (2) and remembering that
OF = UF
ax Oy Oy Ox’
we have finally for the second total derivative of /(x,y) with respect
to?

Pf oF (ax\*, | OF dud OF (dH Px Fy
ae gel ae) “bey OF ae Ot ae) |) Ge de a ae

Multiplying through by d/, we have the second total differential
of (x, 9);

 

Also,

 

299
300 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVII.

=F ata ye Vt Be + Lows Lay, (6)

In (5), 7 is taken as the independent variable, and while d¢ is per-
fectly arbitrary in (1) in actual value, we agree, as in Book I, that d
shall be taken as having a constant value in the successive differen-
tiations,

Thus if we take x as the eee variable instead of /, then dx

ax dx
is taken constant, in which case —; < —]=o0, and we have for

dx ax
the second total derivative of f with 5 to x
af _ of oF ww eo of ay
de® — a2  ? Ox dy ot 5a ay ay dx® (7)

In like manner taking y as the oe aa changing / toy
in (5), we have dy constant, and the total derivative of f with respect

 

to_y is
af af (dx\? Of dx Of Px 8
ap 32a) + ox oy dy + OPT ax TE (8)
dy
200. 7 when /(*, 7) = 0.

The formule of the preceding article furnish means of expressing
the second derivative of_y with respect to x in an explicit function
J(x, ¥) = ©, in terms of the partial derivatives of /(x,y). This gen-
erally saves much labor in computing this derivative when / is a com-
plicated function.

For brevity, represent the partial derivatives of / with respect to x

and y by
To Fes es as Sass » ete.,

and the first and second derivatives of y with respect to x by y’, vy”.
Putting = z = 0 in (7), § 199, we have

Lb SGI ALI ALI"
Buty’ = Hib Substituting this and solving for_y’’,
dy fff — FSP —- LES?
Aaa (4) ;
In like manner we get, by interchanging x and _y, the second
derivative Djx. Otherwise deduced from (8), § 199.

 

201. Higher Total Derivatives.—We shall not have occasion to
use the higher total derivatives of z = /(x, y) above the second.
They, however, are deduced in the same way as has been the second,
by repeated applications of the formula for forming the first derivative.
For the third total derivative of / with respect to / see Exercise 35 at
the end of this chapter. ~
ART. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 301

The higher total derivatives of /(x, y) with respect to an arbitrary
function ¢ of x and y become very complicated and are seldom
employed in elementary analysis. There is, however, an important
particular case in which the higher derivatives of /(.x, y) require to be
worked out completely—that is, when x and y are connected by a
linear relation. This case we now consider and call it linear dif-
ferentiation.

202. Successive Linear Total Derivatives.—To find the #th
derivative of /(x, y) with respect tov, when x and y are linearly
related by

x—a —3é
= =a Ws
a, 6, 7, m being constants.
The first derivative is, as found before,

 

    

 

Oey of
dr ae Dl Oy (7)
Differentiating again with respect to 7, we have
a _ pW, Bf | df
ar Ox Ox ane ay” (2)
Otherwise this ae immediately from (5), § 199, wherein
- dx dy ax dy _
{= Tr; ae = Z, ay =m, ae == ar —

Differentiating (2) again with respect to 7, and rearranging the
terms, we have
Bf _ pF a7 ar 30S
ge > ag +P aay Te” gop tM ap
We observe that (1), (2), (3) are formed according to a definite
law. The powers of /, m, and their coefficients follow the law of the
binomial formula.

 

0. (0 ; ;
If we consider the symbols ie* Gy as operators on /, and write

ort 0\*/ A \q
ax? ay = (=) (=) :

then we can write

conventionally

af _/,a a
ae (: Bae Sh (4)
ay a a \2

ae = (5 + ms) fh (5)

ay a a \3
an = (5. +m \h 6)
302 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVII.

in which the parentheses are to be expanded by the binomial formula

a
and the indices of the powers of 5 = iy and ay taken to mean the num-

ber of times these operations are ne
We can demonstrate that this law is general and that we shall

 

have
af a a \*
i =(' ag t @ zs) Sf; (7)
as follows.
First, observe that
dof a7 daf 90 a 3
won ger! aan ee ae (8)
For
d af dF dx ar a
dr ax ax° dr | dy dx dr’
OY Oe
=n
ia dy Ox"
Also,
ad _ 24 I ar ma
dx dr ax \ Ox Oy )?
eee Of
x? ae Ox dy’

which proves the first ae in (8), and the second is proved in the

same way.
Now assume (7) to be true. Differentiating again with respect to 7,

we have
r= Flac ta) h
=(ge +" 3) $e
(dsb) idea)
iE (5. dar) ay )s

The memoria technica (7) being true for 2 = 3, it is true for 4,
and so on generally.
ArT. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 303

EXERCISES.
1. Given x? + y? = a’, find Dy.
2. 1f 23 + xy? — ay? =0, 9 = (3x2 + 9)/2v(a — x).
du ta? — & — w)
dua tP+aey
xte dz _ 227
x2 ax 22x — mx? — 2)
dv _ i2uv? 4 20 4

it wtf ageas. =

du 3074 — yanty

3. If (2 + uw)? = 20%” — w*),

4. If 2=

 

6. If /(x, v) = 0, is the equation to any curve, show that

Aafia, y ax,
BED 4 9) ey y)

(X — *) ay Ss

 

 

(X — x) wee =(¥— ya),

are the equations to the tangent and normal at x,y. The running coordinates
being X, Y.
7. Show by Ex. 6 that the equations of the tangent and normal to the ellipse
x*/a? + y?/b? = Tare
XX

a

xX
+ wy =1 and pe 2s a — 8,
oe a y
8. Show that the second derivative of y with respect to x, in f(x, v) = 0, can
be expressed in the form

ya a a\2
ay ep rea)

. (m)
we

9. Show that the ordinate of the curve f(x, vy) = oO is a maximum or a minimum
when /, = 0, according as fz, and /;, are like or unlike signed.
For a maximum value of y we must have

wy Ff f/f
a ~ ax /
or fi = 0, fy #0. When this is the case, by § 200,
dy af sa
dx*~ — dx® [ ay’

which gives a maximum when /; and /; are like signed and a minimum when
unlike signed.

10. Show that the maximum and minimum ordinates of the conic

Sf = ax? + by + 2hxey + ager + fy +d=o0

are found by aid of
Te=axtht+g=0.
If fy = by + hx +f, _ is positive, the ordinate isa maximum; if negative, a
minimum.
11. Find the maximum ordinate in the folium of Descartes,
yy —3axy +x = 0 = f(x, 7).
He=—av+ x, Vf a2? — ax.
304 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVIL.

Eliminating y between f = 0, /{ = 0, we have
a6 — 2083 = 0.
x 220, «=a 4/2. These values givey=0, y= 274. For x =0,

y = 0, we have f,’ = 0, but for x= aV2,7 =2 V4, we have a maximum y if
a= 4, since

dy offs 2

dx? ~ dxt | dy ~~ a’

12. If two curves G(x, vy) = 0, P(x, vy) = O intersect at a point x, y, and if w
be their angle of intersection, prove that

2 By — Dy be
2 Py + oy bx
13. Show that two curves @ = 0, » = O cut at right angles if at their point
of intersection
ap Op ap Op s

tan o =

Ox dy “Oy ax ~
14. Apply this to show that the ellipses
w/a? + /h = 1, xt/att y/fr=1
willcut at right angles if a? — 6? = a? — /.

15. Show that the length of the perpendicular p from the origin on the tangent
to the curve @(%, y) = Oat x, y is

= 20+)
V(b2) + (DP
16. Show that the radius of curvature of /(x, vy) = 0 at x, y is
R= A? + (Fy?

 

Sid TY — 2hay Shy + y(Tay
17. If f(x, v) = 0, show that

 

 

of a lg af dy \ & 0 dy 0\83
dat 3 (sap tap ae) att (s+ iH) PEO
18. If y? = 2xy + a’, show that
ay ay a By 3a dx a
dx y— a! ast ~ (yay a ~~ (yap? a ~~
Also, that x = + a@are maximum and minimum values of x.
19. Investigate y = sin (x + y) for maximum and minimum y.
ay cos(x + y) dy -—y

dx ~ 1 —cos(x +)’ dx? ~ [x — cos(x + y)P
20. If z= xy? — 2xy* + 32%, show that

dz dz
Bae te oy = 5%,

az az Oz
Bio dee, pose Se,
at a ay a Bed ae noe
Oz dz
2.If <= AO(y+ax)4+ Wy — ax), .. aa ae”

i ee ae | Oo Te

‘ =

way bx (x yP

 
ART. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 305

dz dz
23. If y —az = f(x — mz), then m ae + a=
24. If «= y*, prove wil, = y*-(1 + log y*) = wy,
ee a a\2
25. If z= fx? +7%, prove («5 —+y =) z=0.

a 2
If 2 = 7x +73, prove (3 +I a) # = Bz;

26. The curve x3 + y3 — 3x =0 has a maximum ordinate at the point

aa oa . eo
I, ¥2, and a minimum ordinate at — 1, — 4/2.

27. The curve p(sin*@ + cos*#) = a sin 20 has a maximum radius vector at the
point @ 4/2, 47.

28. The curve 2x*y + y*? + 4 — 3 = ohas no minimum ordinate, it has a
maximum ordinate at the point — 4, 2

29. The curve xt + y* — gry3 — 2 = ohas neither maximum nor minimum
ordinate.

30. Show that (0, 2) gives y a maximum, and + } 4/3, — }.a minimum, while
$ 3, 3 makes x a maximum, and — $ 4/3, 2 gives x a minimum in the cardioid
(+ 9? — 2? + 9?) — = 0.
31. In x4 + 2ax°y — ay’ = 0, y isa minimum at x = 4+ a.
32. In 3a*y? + ay? + 4qax3 = 0, y isa maximum for + = 3a/2.

33. Investigate the conic ax? + 2hkxy + dy? = 1, for maximum and minimum
coordinates,

34. If & is the radius of curvature of /(x, v) = 0, and @the angle which the tan-
gent makes with a fixed line, show from ds = & d@ and @ = tan— dy/dx, that

pate _ _@e+ ay
~ oy" ~ Py dx — dy Px’
The first when x is the independent variable, the second when the independent
variable is not specified and dx, dy are variables.
35. The third total derivative of /(x, y) with respect to any variable ¢ is

af (dx d | dy a YF dx dy
as (F oe Tae alte se ge Pay “ay dts
Of Pxdx — PF (dx dy | dy dx af ay dy
Bian wmatngy Batma)tym a |
CHAPTER XXVIII.
DIFFERENTIATION OF A FUNCTION OF THREE VARIABLES.

203. We are particularly interested here in the differentiation of

a function

w= f(x, ¥, 2)
of three independent variables, for the reason that when w = 0 we
have

Ha V5 2). = 0;

the implicit function of three variables, which can be represented by
a surface in space, and also because the treatment of the function of
three variables assists in the discussion of the implicit function of
three variables.

We do not attempt to represent geometrically a function w of
three independent variables.

However, corresponding to any triplet « =a, y = 6, z =¢, there
is a point in space which represents the three variables x, y, z for
those particular values.

When, corresponding to any triplet v, y, 2, the function /(a,_¥, 2)
has a determinate value or values it is defined as a function of
X,Y, &

The function / is a continuous function of x, y, z at 1, y, 2 when
for all values of x,, _y,, 2, in the neighborhood of +, y, z we have
the number /(x,, ¥,, 2,) in the neighborhood of A(x, y, 2).

204. Differentiation of w = /(x, 9, 2).—Let +, y, 2 and x, »,,
z, be represented by two points P, P,
in space. Complete the parallelopiped
PROP, with diagonal PP,, by drawing
parallels to the axes through P and P,.
Then in the figure we have the coordi-
nates of FR, (x,, y, 2), and of Q, (x,
¥,, 2). Let PP, = dr, and let J, m,
n be the direction cosines of the angles
which PP, makes with the axes Ox,
Oy, Oz, respectively.

 

Fic, 125,
Then we have

xX = lAr,
Y,—yv =mdr,
2,— 2 = Ndr.
306
ART. 205.] FUNCTIONS .OF. THREE VARIABLES. 307

Applying the theorem of mean value for one variable, letting
z, 1’, «in succession alone vary, we have
S(*,I2;) —S(*,92) = (2, ae 2)fe(x,9,0),
N%I2) — N92) = (9, — Ia 12);
S92) — Sz) = (%, — *) (E92),
where &, y, 2; X,, ™ 2; x,, ¥,, €, are points such as Z, JZ, N,
respectively, on the segments PA, RQ, QP,. By addition, we have
w, — w= Sy(Gy2) 4x + fy(n2)4y + fe (%,0) 42.
Now let ¢ be any differentiable function of x, y, 2, such that
7=4, when x, y, 2 become x,, 9,, 2, Then for the difference-
quotient of w with respect to 4

w,— w Ax Ay Az
oor =Si(a)F; + Say) +SU% IS) FZ;
If now the partial derivatives = a es 7 are continuous func-

tions throughout the ee of Xx, y, 2, we have, on passing to
limits in the above equation, the total derivative of 7 with respect

to4
YF _afax | fw | ¥ &
Qo ie we Oy ae ae a (1)
The process is obviously general for a function of any number of
variables, and if #’is a function of 2 independent variables v,, . . . ,
v,, then the derivative of / with respect to 4, a function ‘of these

variables, is
dF OF dv,
a dv, dt”
1
Second Total Derivative of w= /(x, y, z).—We can differ-
entiate (1) with respect to and obtain in the same way
af fax a , wa pes a Of ax dy | Fas
we (5 ax day! aw a)! tie a a+ a ar + Os a te)
205. Successive Linear Differentiation.—Of chief importance
are the successive linear total derivatives of /(x, y, 2) with respect to
r when
x—a_y—b_2-€c
Ll mH
where a, 4, c, 4 m, m are constants. Then
x=a+h, y=bt+mr, 2=c+n7,

 

 

and
ax dy dz

— —/] <= m ac
dr > dr > adr
are constants, their higher derivatives are o.
308 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXVIII.

Equation (1), § 204, becomes

YF _ of mee a
Go Om + oy oe (7)
We can differentiate (1) again with ree to vy and get.
we = (ag t ma tg) A (2)
dr? ~ \ ax dz

or obtain the result directly from es equation (2) in § 204.
We can show, as for two variables, that the zth linear total
derivative can be expressed Pe

ay a a\*
ae lat ma ae las (3)
where the parenthesis is to be expanded by the multinomial theorem

and the exponents of the operative symbols indicate the number of
times the operation is to be performed on 7

EXERCISES.

0
Li w=y—je—ae—y, #44 Fa

Oz a(z — xe)

2. If xey +logz—yz=0, a c=;
3. If w= log(x* + 9% + 33 — 3xv2), ul tutu = 3(x ty + 2)7.
4, If w = log(tana + tany+ tanz), wi sin2x-+ w)sin2y-+ w}sin2z = 2.
5B. If w= (+ y? 4+ 2)"#, show that

ew Cw Pw

aa aye ae

08

6. If w= en, tan = (1+ Zaye + xty?atyers,

Te Tf we = a824 4 exy2z8 4 xty2g?,  wiY | = Gexyz? + Byz.

xxyZ

a
8. Show that = = oo at the point (3, 4, 2)on the surface
x? 327+ ay — 2ayz — 3x — 42 = 0,
a2
9. Show that me = 0 at the point (— 2, — 1, 0) on the surface

4c? 2? — 5xez + gyz + y — 22-15 = 0.
Oz 50 :
10. co Ee at the point (1, 2, — 1) of the surface
x? — y+ 22% + 2xy — Axe a —y 2-5 =0,
11. Show that the second total derivatives of w= /(x, y, 2) with respect to x, y, 2
are respectively
Pw dy 8 aes a af dy 4 of az
ax? = (s+ dx dy Va z)- +s aa dz dx?’
Py dx 0 dz a Of ax faz
= (gutyten lt ax dy? © az dy??
Pw dxd ada af dx of dy
age (Fut +a5tH)- ot goa Oy az?"

 
CHAPTER XXIX.

EXTENSION OF THE LAW OF THE MEAN TO FUNCTIONS OF TWO
AND THREE VARIABLES.

206. Functions of Two Variables.—Let z = /(x, y) be a function
of two independent variables.

When x = a,y = 4, let z become c = fa, 4). Also, let

x—a y—b
7 = > i) es (1)

Then z= f(x,y) =/(a + 7,5 + mr) (2)
is a function of the one variable 7, if a, 4, 4, m are constants. This
function becomes ¢ = /(a, 6) whenr =o. If this function of r and
its first 2 +1 derivatives with respect to 7 are continuous for all
values of y from r = o tor = ¢, then, by the Law of the Mean for
functions of one variable,

dz rm (a"z pert gtttg
pao (F),+ ee ee (i=) + NE 5g ys (3)
iB ‘
Here (=) means the /th derivative of z with respect to r
0

drb
q'tty

taken at 7 = o, and (=). means the (7 + 1)th derivative of 2

with respect to 7 taken at some value o of r between o and ¢.
Also, since these derivatives are linear derivatives of z, we have

(25) Man

 

i ae when + = a, y = 4,

= (’55 + ™ 54) Med

I

=5)O-9F + -DF{ Nad,
since 7 = (x — a)/r, m =(y — 6)/r, from (1). Hence
(Bare -9GtU-DgbAed —

p! \dr? 2!
309
310 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXIX.

In like manner let x = &,y =n, when r=; &, 7 being num-
bers respectively between a and x, dandy. Then

coon (Gm'),_.= Gan lO ae ODay FE.)

Substituting the values of (4) and (5) in (3), we have the Law of
the Mean Value extended to functions of two variables, or

Ax 9) = (© — a) 24 (y— 05 ba, 8)
pao

I 0 Qo) “t+
+ (+ 1)! | (« —Yaet v= x KE. (6)

207. The geometrical interpretation of § 206 is as follows :
Given the ordinate to a surface at a particular point a, 4, and the
partial derivatives of the ordi-
nate at that point. To find the
ordinate at an arbitrary point
Hy P
Let z = /(*, 9) be the equa-
tion to a surface on which
A,(a, 6, ¢) is the point at which
the coordinates and partial de-
rivatives of g are known. Let
P be the point on the surface
at which x, y are given and z or
J(*, ¥) is required.

Fic. 126. 7 Pass a vertical plane through
A and P, cutting the surface in the curve AP and the horizontal plane
in the straight line BAZ, whose equation is

 

we—a yb |
pe ag a
The equation of the curve AP cut out of the surface by this verti-

cal plane is

z= /f(a+lr,b4 mr),

referred to axes Br, Bz’ and coordinates 7, z, in its plane rBz’ The
law of the mean is applied to this function of the variable 7, resulting
in (3). Then, since these derivatives are linear, they can be ex-
pressed in terms of the partial derivatives of z at a, 4, and (3) is trans-
formed into (6).

208. Expansion of Functions of Two Variables.—Whenever
the function (2), § 206, of the one variable ~ can be expanded in
ARI. 210] EXTENSION OF THE LAW OF THE MEAN. 311

powers of 7 by Maclaurin’s series as given in Book I, then we can
make z = oo in (6), and we have
oO

fe = SF Le + V-Day bed)
és

and the function /(x, y) can be computed in terms of /(a, 4) and the
partial derivatives at a, 4.

209. Functions of Three Variables.— Following exactly the same
process as in § 206, for
w= f(x; 4,2);

we have the law of the mean for three variables,
Ty a a a)?
Aesad= SZ | eae + (—Dayt eed "Aa, bse)
‘pro

L a 0 QO pat.
tpn (~—a) get ag tee Her Os)

where &, 7, € are the coordinates of some point on the straight-line
segment joining the points in space whose coordinates are .v, y, 2 and
a, b, ¢.

Whenever the function of one variable 7,

(a+, b4 mr, c+ ar),
can be cxpanded in an infinite series of powers of ry by Maclaurin’s
series, Book I, then we can make 7 = oo in (1), and have

Aes v= De 5p tUDagp + eg, | Mb). (2)

210. Implicit Functions.—The law of the mean enables us to
express the equation of any curve or surface in terms of positive
powers of the variables, and permits the study of the curve or surface
as though its equation were a polynomial in the variables.

Thus if z = f(x,y) is constant and o, then /(x, v) = 0 is the
equation of a curve in the plane xOy. The equation of any such curve
can, by (6), § 206, be written in the form

n

I a a |?
o= SE fe -aE + Dy Aa, B)+ Re (2)
p=0
In like manner, by (1), $209, the equation to any surface
J(x, ¥,%) = 0 can be written

° =) 5 | (ea) +B) 5H (e—2) 521 “f(a, b,c) +Ru (2)
p=a
312 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXIX.

&, being (5), § 206, for equation (1) above, and the correspond-
ing value in (1), $209, for equation (2).

211. The law of the mean as expressed in this chapter is funda-
mental in the theory of curves and surfaces. It permits the treatment
of implicit equations in symmetrical forms, which is a far-reaching
advantage in dealing with general problems whose complexity would
otherwise render them almost unintelligible.

A most useful form of the equations for two and three variables is
obtained by putting

x—-a=h, y—b=hkh, z-c=l,

and in the result changing a, 8, ¢ into x, y, z
Thus for two variables

Feng = ale +A jy) 9) (10)

For three variables

0 0 0\7
Seth sth otl)= ) East bs $05) Aen 8) (11)

EXERCISES.

1, Show that the equation of any algebraic curve of degree # can be written as
either

n a
o= S569 +0-9 Ft Aad (0
or
n 0 0 r
° = 25 (* 35: +y a) JO, 0). (2)

2. Show that any algebraic surface of zth degree can be written in either of the
ae,

0 r
{@-9F+-OZte-92 bai, a)

Sef

2
n 0 r
oe St (2 fort? Foyt * ae) Moo) (2)

i

3. The function (a5 +9 =) A(x, ¥) is called a concomitant of f(x, ¥).

Find the concomitants of a homogeneous function f(x, v) of degree x.
In (10), § 211, put 2 = gx, & = gy, then

fetenyto)= 38 (25 +95) An.

Since / is homogeneous in x and y - degree x,

Sete yto=sf{itox it+a)y} = G+ "Kx y).
(er fe) = 35 (er +75) Aon.
ART. 211.] EXTENSION OF THE LAW OF THE MEAN. 313

This equation is true for all values of g including 0, Therefore, equating like
powers of g, we have
9, if

0
gia

af 20°F =
ina?” ae i mh

 

02
eae ee

(= +95) "Fame

In the same way, if /(x, y, z) is homogeneous of degree #, we find, by putting
h = gx, k = gy, = gz in (11), § 211, as above, the concomitants of /(x, 7, z),

a a a\r
eee) fann—1)...@—r4+f
fory = 1,2,...,%.

The concomitant functions are important in the theory of curves and surfaces.
They are ixvariant under any transformation of rectangular axes, the origin
remaining the same.
CHAPTER XXX.
MAXIMUM AND MINIMUM. FUNCTIONS OF SEVERAL VARIABLES,

212. Maxima and Minima Values of a Function of Two Inde-
pendent Variables.

Definition.—The function z= /(x, 1’) will be a maximum at
x =a, y= 6, when (a, 4) is greater than /(x, y) for a// values of x
and y in the neighborhood of a, 4.

In like manner /(a, 4) will be a minimum value of /(x, 7) when
(a, 6) is less than /(x, y) for ad// values of x, y in the neighborhood
of a, 6.

In symbols, we have /(a, 6) a maximum or a minimum value of the

function /(x, y) when
Js; J) —K(4 b)

is negative or positive, respectively, for all values of x, y in the
neighborhood of a, 4.

Geometrically interpreted, the point P, Fig. 115, on the surface
representing z = /(.v, _y) is a maximum point when it is higher than
all other points on the surface in its neighborhood. Also, P is a
minimum point on the surface when it is lower than all other points
in its neighborhood.

This means that all vertical planes through P cut the surface in
curves, each of which has a maximum or a minimum ordinate z at P
accordingly.

Also, when / is a maximum point, then any contour line ZAZN,
Fig. 115, cut out of the surface by a horizontal plane passing through
the neighborhood of P, below P, must be a small closed curve; and
the tangent plane at P is horizontal, having only one point in
common with the surface in the neighborhood. Similar remarks
apply when P is a minimum point.

When the converse of these conditions holds, the point P will be
a maximum or minimum point accordingly.

213. Conditions for Maxima and Minima Values of /(x, y).—
Let z = /(*, _y), « and_y being independent. To find the conditions
that z shall be a maximum or a minimum at x, y.

I. Any pair of values x’, y’ in the neighborhood of x, y can be
expressed by
e=axt+h, y =y+nmr,
314
ART. 213.] MAXIMA AND MINIMA VALUES. 315

where 7= cos 6, m= sin 6. Then

z=af(x+ lr, y+ mr)
is a function of the one variable ~, if @ is constant.
If z is a maximum or a minimum, we must have, by Book I,
dz az
a? aR

respectively, for all values of 6. That is,

negative or positive,

dz of v=
ar = 098 oe aor sin Oe

This must be true for - ia of 7 But when @ = o and
= iz, we have

qf 9) = 0 and S/a,3) =o (x)

respectively. Equations (1) are necessary conditions in order that
x, y which satisfy them may give z a maximum oraminimum. But
they are not sufficient, for we must in addition have

dz ,0f ay Oy
2
dr ae Ox? 2 Oe ay a Be ; (3)

different from o and of the same sign for all values of 6. When (2)
is negative for all values of 4, then zat 1, y is a maximum; and when
(2) is positive for all values of 6, then z is a minimum.

 

 

oF OF ar
ae ae Oe Fay age
The quadratic function in /, m (see Ex. 19, § 25),
Al? + 2Him + Bm’, (3)

will keep its sign unchanged for all values of the variables /, m, pro-
vided
AB — ff?
is positive. Then the function (3) has the same sign as 4.
(a). Therefore the function /(x, y) is a maximum or a minimum
at ., y when

a Or OF ee
Ge ag = OP “Bee ep) (4)
a ar .
and is a maximum or a minimum according as either a Of ay Re is

negative or positive respectively.

(4). If AB — H? = —, then will (2) have opposite signs when
m = oand m/l = — A/H; also when /= oandm// = — H/B. The
function cannot then be either a maximum or a minimum (see Ex,

19, § 25).
316 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CuH. XXX.

(c). If AB— H*? =o, and A, B, Hare not all o, then the right
member of (2) becomes

(A 4+ mH)? (mB4+1H)?
— B d

 

A
and has the same sign as A or # for all values of 6, except when
m/i= — A/H. Then (2) iso. This case requires further examina-

tion, involving higher derivatives than the second; as also does the
case when 4, B, Hare allo.

To sum up the conditions, we have /(x, y) a maximum ora mini-
mum at x, y when

a i
Ts = 09; Fy = 0;
Hi os max. Sh
xx aa min., ay | +.
We as max. v7
or a - min., xy SI

 

 

If the determinant is negative, there is neither maximum nor
minimum; if zero, the case is uncertain.*

To find the maximum and minimum values of z = /(x, y),
solve 71, = 0, Sy = == 0, to find the values of +, y at which the maxi-
mum or minimum values may occur, then substitute x, y in the
conditions to determine the character of the function there.

The value of the function is obtained by either substituting +, y
in /(x, v), or by eliminating +, y between the three equations

s=fx, 9), flzo y=

for the maximum or minimum value z.

This method employed for finding the conditions for a maximum
or a minimum value of z = /(x, y) has been that which corresponds
geometrically to cutting the surface at .v, y by vertical planes and
determining whether or not a// these sections have a maximum or
a minimum ordinate at x, y.

II. Another way of determining these conditions is directly by
the law of mean value. We have

NOI) — Kes) = (# Y=)

For all values of x’, y’ in the a of x, y we have a n also
in the neighborhood of x, y. Ifthe values 7, 7; are different from
o, then the values /;, /;, are in the neighborhoods of their limits and
have the same signs as those numbers for all values of x’, y’ in the
neighborhood of x,y. Therefore the difference on the left of the
equation changes sign when x’ == x, as_y’ passes through ¥y, if /, # o.
In like manner this difference changes sign when_y’ = y, as x’ passes

pees ne. 7) ee m)

 

* For examples of the uncertain case in which the function may be a maximum,
a minimum, or neither, see Exercises 22, 25, at the end of this chapter.
ART. 213.] MAXIMA AND MINIMA VALUES. 317

through x, if f; #0. Hence it is impossible for /(.x, y) to be a
maximum or a minimum unless “/ = o and /; = o.
When fj = 0, /j = 0, we have

Ke J Vs J)= (x’ — «yas, SL +2( (x '—x)(y’ Nae —yhF

If the member on the right of this equation retains its sign unchanged
for all values of x’, y’ in the neighborhood of x, y, the function will
be a maximum oraminimum at, y. But in this neighborhood the
sign of the member on the right is the same as that of its limit,

2 2
of + (= 35%

(x’ — aoe + a(x’ — x)\(0" — »)—
Ox? Ox
This gives the same conditions as in I, and leads to the same results,

an

ay

EXAMPLES.

1. Find the maximum value of z = 3axy — x3 —
This is a surface which cuts the horizontal plane i in the folium of Descartes.
Here

Oz oz
ie ath gee (1)
Oz az 822
aga = — 6x, a = — 6, Onan y = 3a. (2)
The equations (1) furnish
3ay — 3x2 =0, jax — 3y7=0, (3)

for finding the values of x, y at which a maximum ora minimum may occur.
Solving (3), we have
x=0, y=0, and x=a, y=a.

For *=0, y=0,

0?z dz 0?z 2 ,
523 52 ~ \j>9,) =~ 9%
Ox? dy? Ox Oy
and there can be neither maximum nor minimum at 0, 0.
For x«=a, y=a4,
Oz 02z a2z \2 oom
Sf at ae ey) es
and since =, = — 6a, we have the conditions for a maximum value of z at a, u
x
fulfilled. Hence at @, 2 the function has a maximum value a’,
2, Show that 3/27 is a maximum value of
(4 - xKa—y\(x +7 — a).

3. Find the maximum value of x? -} «y + y? — ax — by.
Ans. }(ab — a? — 6),

4. Show that sinx + siny + cos(«+y) is a minimum when « = y = $a, a
maximum when « =y =z.

5. Show that the maximum value of

(ax + by + cPf(P?t+ty? +1) is #404,
318 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXX.

6. Find the greatest rectangular parallelopiped that can be inscribed in the
ellipsoid. That is, find the maximum value of 8xyz subject to the condition

fa + 9? /P 4 2/c% = I. (1)
Let «= xyz. Substituting the value of z in this from (1), we reduce toa

function of two variables,
x? wy?
waa sty" (1 aah):
Ou? O20?

From Pee a = 0, we find the only values which satisfy the con-
a

ditions x = a/ 73,7 = 6/3. These give » =c/ 4/3, and the volume
required is 8adc/3 4/3.
7. Show that the maximum value of x?y3z!, when 2x + 3y + 42 = 4, is (a/9)°.

8. Show that the surface of a rectangular parallelopiped of given volume is
least when the solid is a cube.

9. Design a steel cylindrical standpipe of unifurm thickness to hold a given
volume, which shall require the least amount of material in the construction. [Ra-
dius of base = depth. ]

10. Design a rectangular tank under the same conditions as Ex.9. [Base
square, depth = } side ot base.]

11. The function z = 2? 4 ay + 9? — 5x —4y+1has a minimum for x = 2,
yo.

12. Show that the maximum or minimum value of

 

 

 

zxax? +t by? + 2hxy + 2gx + 2fy te (1)
is
z=l|ahg|+ | ah
hb f | 2b
gfe
We have
rds h = 1 02 —h by =
3 aE =ax+hy+ g=09, ap ee (2)
Multiply the first by x, the second by y, subtract their sum from (1), and we
get
z=gxtfy+e (3)

Eliminating x and y between (2), (3), the result follows.
The condition shows that when aé — /? is positive, the above value of z is a
maximum or minimum according as the sign of ais negative or positive. If

ab — h? = —, then z is neither maximum nor minimum. We recognize the
surface as a paraboloid, elliptic for ad — /? positive, and hyperbolic when
ab -P=—.

13. Investigate z = x? + 37? — xy + 3x — 7y + 1 for maximum and min-
imum values ot s.

14, Investigate max. and min. of x + y# — 4? + ay — y?,
“=0y=0,max; x=ys>ti,min; «= —y= + 473, min.
15. The function (+ — y)? — 4y(x — 8) has neither maximum nor minimum.

16. The surface x? + 27? — 4x + ay + 32 + 15 = O has a maximum
z-ordinate at the point (2, —1I, —3).

17. The function +4 + y4 — 2x? + 4x) — 2y? has neither maximum nor minimum
for « = 0, y = 0; but is minimum at (+ 4/2, — 4/2), (—V2, + 72).
ArT. 214.] MAXIMA AND MINIMA VALUES. 319

18. Show that cos x cos a+ sin xsin a cos(y — #) is a maximum when
x= a y= fp.

19. Show that x? — 6xy? + oy at 0, o is minimum if ¢ > 9, and is neither
maximum nor minimum for other values of c. Hint. Complete the square in «.

20. Show that (1 + x? + y*)/(I — ax — dy)has a maximum and a minimum
respectively at

Soe otk Pie ae
a a a + b
21. Show that 3, 2 make «3y6 — + — y) a maximum.

22. Show that a, 6 make (2ax — x*)(26y — y?) a maximum.
23. Show that 3 + 4 4/2 is a maximum, — 6 — 4 4/2 a minimum, value of
yt — 88 + 18y? — By + x3 — 3x? — 30.

214. Maxima and Minima Values of a Function of Three
Independent Variables.

Let u = f(x,y, 2),
a—valra=h, y—-y=mrakh, 2,-2=nr=g.
As before, if # is a maximum or a minimumat.1v, y, 3, we must have
ua=f(v+ler, yvtmr, z+nr),

a maximum or a minimum for all values of 7, m, 7, or

 

du Ou
ag ag ae
ata ee 2, — 2 Ou
r Ox roy r @2’

must be o for all values of /, m,” or of x,, _y,, 2, in the neighborhood
of x, y, 2, OF

Ou Ou Ou
(4 - a+ Oda tG —)R =O
Hence the necessary conditions
Ou Ou Ou
— =0; av =o, as = 0. (2)

Now when the relations (1) hold, and for all values of x,, ¥,, 2, in
the neighborhood of x, 1, 2, we also have

Ce SPSL MLL mf + alm fl + anf + amn fy,
or, what is the same thing,
AP 4+ BRA Co? + 2Fke + 2Ghg + 2Hhk (2)

(wherein An fi Be fh Ca fh i= fi Caf b= fe)
negative (positive) “ all values of 4, 4, g, then will w be a maximum
(minimum).

The condition that (2) shall keep its sign unchanged for all values
320 PRINCIPLES AND THEORY OF DIFFERENTIATION, [CH. XXX,

of 2, &, g has been determined in Ex. 20, § 25, where it is shown that
when
A Aj and A|A HG
ie Z| HBF
GFC

are both positive (2) has the same sign as A for all values of 4, &, g.
Therefore /(x, _y, 2) is a maximum or a minimum at 4, y, 2,

determined from
Re cas a
4, = 09, Sy =o, Jf) =09,

when we have

—_— max,

ty vf
ax min.? Tie yx
VI v1

xy S yy

=+, |AA Ae Su| = F at
Sy Sy Sy
Fea Sa Sst
The conditions for maximum or minimum can be frequently
inferred from the geometrical conditions of a geometrical problem,

without having to resort to the complicated tests involving the second
derivatives.

 

 

 

EXAMPLES.
1 JS =Erty+e4 4-22 — xy.
fe=2x-y+1=0, ff=2ay—x a0, fp =22-2=0,
. £=—h y=—-f, 2=1, give f= —4.
Also, fl, = 2, ay eh I=) fe=—Ty f=) yz =O
|2i =3, [2 10|/=6.
I2 120
002

 

 

Therefore — 4/3 is a minimum value of 7.
2. Find the maximum and minimum values of
ax + by? 4 cz? 4 afys + 2gxz + 2hay + 2ux + avy + 2wz+d.
Here Ix = 2(ax +hy+-gzt+u)=0,
fy =2hx + + fetv) =o, (1)
fi = 292 + fy +2 $ w) = 0.

Multiply the first by x, the second by y, the third by z. Add together and
subtract the result from the function

 

 

J =ux+outwe+d. (2)
Eliminating x, y, z between (1) and (2), we have
fo=|¢ch guj+rlahg |,
hbfou Aobf
few jgfe
“vw

which is a maximum or a minimum according as

a=fF, aki= ahg = F
53 W hoy '

efe

 

 

the upper and lower signs going together.
ART. 215.] MAXIMA AND MINIMA VALUES. 321

3. Find a point such that the sum of the squares of its distances from three
given points is a minimum.
Let 41, Jy) 24) + + + %g) g) 23, be the given points. Then
f= Bz [e —areP + — yr? + 2 — 2);
Je = 23 (x — x,) =O = 3x — Bx,,
Sy = 22(¥ — Ir) =O = 3Y — Bry
Se = 232 — 2%) =O = 32 — Bt,
=H tet ey) Y= RMI, + Ish % = HAH 42 + 25)
The point is ee the centroid of the three given points.
ay Si 6, Sey xz = Syz = 0. Show that the solution is a min-
imum.
Extend the problem to the case of # given points.
4. If w = ax? + byx 4 dz? + Ixy + myz, show that x = y = 2 = 0 gives
neither a maximum nor a minimum.

215. Maximum and Minimum for an Implicit Function of
Three Variables.—To find the maximum or minimum values of z in

I(% I, 2%) =O
Since total sna: of f are o, we have
df= (5, pth a tea 7 )S=0 (1)

a a
ay =(dax 5 2 oda d\ aL aes Yar L tre (9
Also, at a maximum or a minimum value of z we must have
pom Ox oy
2 aap =o
dz

for all values of dy and dx. It is therefore necessary that

Oe Sa, EE
ig = ayo ag oO (3)

Substituting these values in (2), we have at the values of x, y which
satisfy (3), and make dz = 0,

(e+ 5) 7
ey Ox oy
af 9
ae
dh SLA dy de fl PALE
zm 4

In order that this shall retain its sign for all values of dy and dx,

we must have :
da = Sel => (4)
322 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CH. XXX.

Then the sign of dz is that of A[//2. (See Ex. 19, p. 31.)
Hence z will be a maximum (minimum) at +, y, z, determined

from
Sn =o; Jy =; J = 0;
when //7/// is positive (negative), provided (4) is true.

EXAMPLES.

1, Find the maximum and minimum of z in
2a? + 5y? + 2? — 4gxy — 2x -—Gy —$F = 0.
Je=4e —-4y —2=0, fy = loy—4x—4=0,
give a= yol = 42.
fi=2wz=+4, os oa (fay)? = 24.
z is therefore a maximum and a minimum at 8,

2. Show that * in cs? + 327 — qxy + yz = has neither a maximum nor a
minimum atx = — §,y=—73p 27 = —

216. Conditional Maximum and Minimum. — Consider the
determination of the maximum or minimum value of z= /(x, 9),
when .v and _y are subject to the condition ¢(1, y) =o.

Geometrically illustrated, 2 = /(x, y) and (%,_y) = o are the
equations of the line of intersection of the surface = /and the ver-
tical cylinder @ = o. We seek the highest and lowest points of this
curve.

Since, at a maximum or minimum value of z,

ed a ag oy dy = 09, (1)
a a
also  ieaatey °, (2)

we have, eliminating dy, dx, the equation
Jz by — Sy Px = © (3)

to be satisfied by x, y at which a maximum or minimum occurs.
Equation (3) together with @ =o determines x and y for which a
maximum or minimum may occur.

Usually the conditions of the problem serve to discriminate be-
tween a maximum, minimum, or inflexion at the critical values of «x, 1’.

The test of the second derivative, however, can be applied as
follows: We have

Pas fl de + fb dedy + fi P+ fi@x + fidy, (4)
which must keep its sign unchanged for all values of x, y satisfying

e ae in the neighborhood of the x, y also satisfying (3). But we
also have

wn OX? + 2G, dxdy + Py P+ O.Pxt+ody=o. (5)
ArT. 217.] MAXIMA AND MINIMA VALUES. ‘ 323

To eliminate the differentials from (4), (5), multiply (4) by #}, (5)
ry Sys and subtract, having regard for (3). In the result substitute
for dy/dx from (2).

dx)? ?

=r We ae PLP +

y xx xx
When this is negative es we ane a maximum ae

value of z. The form of the test (6) is too complicated to be very use-
ful, and it is usually omitted.

ye a | ty

py

 

 

BF onl. (6)

 

 

 

EXAMPLES.

1, Find the minimum value of x? + 7? when x and y are subject to the condi-
tion ax + by +d=0.
Condition (3) gives 6x = ay. Therefore, at

ad ba
Sn eae
we have
&
e+y= Le BR?

which can be shown to be a minimum by (6). Otherwise we see at once from the
geometrical interpretations that this value of x? + y? must be a minimum.

First. 4/x? + y? is the distance from the origin, of the point x, y which is on
the straight line ax + 6y + d=0, and this is least when it is the perpendicular
from the origin to the straight line, which was found above.

Second. z= 2?-+ y? is the paraboloid of revolution. The vertical plane
ax + by 4+.d=0 cuts it ina parabola, whose vertex we have found above, and
which is the lowest point on the curve.

 

2. Determine the axes of the conic ax? + dy? + 2hxy = 1.

Here the origin is in the center, and the semi-axes are the greatest and least
distances of a point on the curve from the origin. We have to find the maximum
and minimum values of x?-+ y?, subject to the above condition of x, y being on
the conic.

Let w=x?+ty? and @ = ax?+ by? 4+ 2hxy-—1=0.
Condition (3) gives
x ax-+thy

yy hx
Multiply both sides by x/y and compound the proportion, and we get
(a —w)x + by =o,
he + (6 —u-Hy =o.
Eliminating x and y, there results
Oe | a ee 8) = + ab ft =0
for determining the maximum and minimum values of zw.

217. The whole question of condizonal maximum and minimum is
most satisfactorily treated by the method of undetermined multinliere
of Lagrange.

The process is best illustrated by taking an example sufficiently
general to include all cases that are likely to occur and at the same
time to point out the general treatment for any case that can occur.
324 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXX.

To find the maximum and minimum values of

“= f(x,y, 2, w) (z}
when the variables x, y, 2, w are subject to the conditions

P(x, J; %, W) = 0, (2)

a(x, J, 4, w) =o. (3)

Since, at a maximum or minimum value of z, we must have du = 0,
the conditions furnish

fidet flb+ fl ds+ fi dw =o,
b1 dx +b dy + $i ds + by dw =o, (4)
ws dx + oy dy + oi dz -+ tp dw =o.
Multiply the second of these by A, the third by yw, A and yu being
arbitrary numbers. Add the three equations.

(Pet Abit Mbadee +S) + ADH Mb) +
(fi + Ads mpds + (Fa + Aa + Mpu)dw = 0. (5)

Since A and yw are perfectly arbitrary, we can assign to them
values which will make the coefficients of dx and dy vanish; moreover,
since equations (2) and (3) connect four variables, we can take two of
them, say z and wz, independent, and therefore dz and dw are arbitrary.
Consequently, in (5), after assigning A and uw as above, we must have
the coefficients of dz and dw equal too. Therefore

SL + AGL + ph, = 0, |
fy +46; + hy = 0, | ;
Si +A + MEE = 0, | (6)
Su + AGL A whs =o. |

The six equations (2), (3), (6) enable us to determine x, y, 2, w,
A, #, which furnish the maxima and minima values of w.

The discrimination between a maximum and a minimum by means
of the higher derivatives is too complicated for our investigation. In
ordinary problems this discrimination can generally be made through
the conditions of the problem proposed.

EXAMPLES.
1, Find the maximum value of « = x? + y? + 2? when a, y, 2 are subject to
the condition
@a=axtbytetd=o.
Here we have, as in equations (6),
Ou 0p
at tae =O= 2x+ Aa,
a a
ce
oy ay

Ou ap
a 3 = oS tac

= = 2y+ 24,
ART. 217.] MAXIMA AND MINIMA VALUES.

325
Multiply by a, 6, ¢ and add. Also, transpose and square. Then
2(ax + by +ez) + (P@4P+4 AA = —2d+(P?+PVP4 ce) =0,,
A(x? + y? + 2%) — (a? +674 A)? = qe — (P4824 2)? = 0,

= d
Vu SS eee
fVertLete
The problem is to find the perpendicular distance from the origin to a plane
2. Find when « = x? + y? + 2? is a maximum or minimum, -, y, 2 being sub-
ject to the two conditions
e/a 2/4 2/AR= 1, let m+n =o.

Geometrically interpreted: Find the axes of a central plane section of an ellip-
soid.

Equations (6) give

2x
ax At + ee =0.
ay fax Y 1 um =o,

22
22 +A + un = 0.

Multiply by x, y, 2 and add. We getA = —w. Therefore

hari?

 

a ee mt _ pent
= aay wee oe ey
Hence the required values of « are the roots of the quadratic
azl? b2m2 nr
s ——<— —, = 2.
u—a’ u— u—e

3. Find the maximum and minimum values of
wu arxt + byt 4 c223,
, ’, 2 being subject to the conditions
e+ yieoryi dx&tmtn=o,.
The required values are the roots of the quadratic
P/ (ue — a) + m/(u — 6) + n2/(u — ) = 90,
4, Find the maximum and minimum values of
us xtt y? + 2?
when «, y, 2 are subject to the condition
ax? +. by® 4 co? + afys 4+ 2agxz + 2hxy = 1.

Geometrically interpreted: Find the axes of a central conicoid
The conditions (6) give

x4 (ax + ly 4g = 0,
¥+ Ze +4 fe)A =9,
eti(gx+ fy + cz)A = 0.

Multiply by x, y, zandadd. .. A= —uw.
Eliminating x, y, 2 from the above equations,
@—uw, A, & = 0.
kh , 6—u-y,
& ’ f » eS u-t

The three real roots of this cubic, see Ex. 17, § 2 5s furnish the squares of the
semi-axes of the conicoid.
326 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CH. XXX.

5. Show how to determine the maxima and minima values of x? + y? + 2?
subject to the conditions

(22 fy? + at = ata? + by? + cat,
lx + my + nz = 0.

EXERCISES.

1. Show that the area of a quadrilateral of four given sides is greatest when it
is inscribable in a circle.

2. Also, show that the area of a quadrilateral with three given sides and the
fourth side arbitrary is greatest when the figure is inscribable in a circle.

3. Given the vertical angle of a triangle and its area, find when its base is
least.

4. Divide a number « into three parts x, y, 2 such that xy*zb may be a

maximum, x g a
Ans. Boe Ma Pe ae iy
m

np _ mtn+tp
5. Find the maximum value xy subject to the condition x?/a? + y?/2 = 1.
This finds the greatest rectangle that can be inscribed in a given ellipse.

6. Find a maximum value of xy subject to ax + dy = ¢, and interpret the result
geometrically.

7. Divide @ into three parts x,y, z, such that xy/2 + x2/3 + y2z/4 shall bea
maximum. Ans. x/21=y/20 = 2/6 = a/47.

8. Find the maximum value of xyz subject to the condition
x /a + y?/8 4 22/8 = 1
by the method of § 216.
9. Show that « + y-+ 2 subject to a/x-+ 6/y4 ¢/z=1 is a minimum when
x/Ya=y/Vo=2/Vo= Yat Yo He.

10, Find a point such that the sum of the squares of its distances from the
corners of a tetrahedron shall be least.

ii. If each angle of a triangle is less than 120°, find a point such that the sum
of its distances from the vertices shall be least. [The sides must subtend 120° at
the point. ]

12. Determine a point inthe plane of a triangle such that the sum of the squares
of its distances from the sides a, 4, cis least. 4 being the area of the triangle.
z
a c
13, Circular sectors are taken off the corners of a triangle. Show how to leave

x y 24
the greatest area with a given perimeter. [The radii of the sectors are equal. ]

oe a 4+ 4 ot

14. In a given sphere inscribe a rectangular parallelopiped whose surface is
greatest; also whose volume is greatest. [Cube. ]

15. Find the shortest distance from the origin to the straight line,
Axe+tmy+uz=p,, ;
1x + my + 2,2 = py.
‘The equations of the planes being in the normal form.
Arr. 217.] MAXIMA AND MINIMA VALUES. 327

We have, if «* = x? y? + 22,
2x4 4A + AM =0,
2y+mA+ mu =0,
22+ nA + nu =O.

Multiply these by x, y,z in order and add. Multiply by 4, m,, , in order
and add. Multiply by /,, m,., in order and add, Whence the equations

2+ Plt py =o,
2p, + A+ cosOu=0,
2p, + cos 6A + Mao.

Since 4? + m2 + a? = 1? + me + n= 1, Al, + mm, + nn, = cos 6, where
6 is the angle between the normals to the planes. Eliminating A and yz, we have

 

 

u2 Ay Po = 0,
Dy I cos 6
Pp, cos@ 1
or 2 sin’?@ = p,? + p,? — 2f,P, cos 8,

which result is easily verified geometrically as being the perpendicular from the
origin to the straight line.

16. A given volume of metal, 7, is to be made into a rectangular box; the sides
and bottom are to be of a given thickness a, and there is no top.
Find the shape of the vessel so that it may have a maximum capacity.
If x, y, z are the external length, breadth, depth,
a ee
— ; t= he
3a
17. Find a point such that the sum of the squares of its distances from the faces

of a tetrahedron shall be least. If Vis the volume of the solid, x, y, z, w the per-
pendicular distances of the point from the faces whose areas are A, B, C, D, then

oY ww. BV
A BCD #ARPLCLD
18. Of all the triangular pyramids having a given triangle for base and a given
altitude above that base, find that one which has the least surface.

The surface is #(@ + 6+ c) ¥7*?+/F*, where a, 4, c are the sides of the base, 7
the radius of the circle inscribed in the base, 4 the given altitude.

19. Show that the maximum of (ax + dy + cz)e— 024? Biy?— 32 is given by

a é é a oe 2
fet pena! (att 5):

20. Show that the highest and lowest points on a curve whose equations are

 

4eyoa+

P(x, J, 2) =9, Yr I, 2) = 9, (1)
are determined from these equations and
go: + Ap’ =0, ,' +Ay,’ = 0. (2)

21. Show that the maximum and minimum values of r? = x? + y?+ 2%, where
x, ¥, s are subject to the two conditions

ax? + by* + czt 4 afys + 2gxz + 2hxy = 1, le + my+nz=o,

are given by the roots of the quadratic,

a—r-2 h g Z| =0
h 6—r2 ft m
g St ¢—-r2n
Z m n °
328 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CH. XXX.

Geometrically, this finds the axes of any central plane section of a conicoid with
origin at the center. It also solves the problem of finding the principal radii of
curvature of a surface at any point.

The following four exercises are given to illustrate the uncertain case of max-
imum and minimum conditions.

22. Investigate 2 = 2x7 — 3xy? + y* = (y* — x)(y* — 22).

At 0, 0 we have 22 = 4 = ay = 2, = 0, 22, = 4 The conditions
aii 2, — (2x,)* = 0 makes the case uncertain. The function z vanishes along
each of the parabole y? = x, y? = 2x. It is positive for all values x, y in the
plane z = 0, except between the two parabolz, where itis negative. The function
is neither a maximum nor a minimum at 0, 0, since it has positive and negative
values in the neighborhood of that point. In fact z is negative all along y? = 3.4/2

except at 0, 0.
23. z= ay? — 2ax*y + x4 4 oy.
At 0, o the case is uncertain. Put y = mx, then
z= x [(1 + m')x? — 2amx + am].
When x ory iso the function is positive. For all values of # the quadratic

factor in the brace is positive.
Hence z is a minimum at 0, o.

24. za y — xy? — 2axty + xt
As in 23, the condition is uncertain ato, 0. Puty = mx. Then
2 = x(x? — m(m + 2)x 4- m*].
The function is positive when. ory iso. For any value of m not arbitrarily
small z is positive for all arbitrarily small values of x. But since

m(3m? — lm —1)

is negative for all arbitrarily small values of m, the quadratic function of x in the
brace has two small positive roots for each such value of 7. Between each pair of
these arbitrarily small roots the quadratic factor, and therefore z, is negative. The
function is neither a maximum nor a minimum ato, 0. In fact along the curve
x? = y the function is z= — x°.

25. Consider the function z defined by the equation
(¢ — a + (yy — a)? = a,

or 2=a— Y2ap — p,
wherein the positive value of the radical is taken and p? = x? +y%. This is the
lower half of the surface generated by revolving the circle (x — a)? + y? = a? about
the y-axis.
Here
dz x a-p Oz y a—p

TG Yee Bh Vaan
At all points satisfying +*-+ y? = p? = a® these derivatives are o. Also at
such points
we S/o, ay = Pah, 24) = xy /a’.
ate Zyy — (225)? =o.

The function z is 0 at each point x, y satisfying +? + y? = a®, and is positive for
every other x,y. It is neither a maximum nor a minimum, nor does it change
sign in the neighborhood of any x, y in x? + y? = a? We shall see later that the
plane z = Ois a simgudar tangent plane to the surface.

&
CHAPTER XXXI.
APPLICATION TO PLANE CURVES.

I. Orpinary Points.

218. We have seen that when the equation of a curve is given
in the explicit form y = /(x), and the ordinate is one-valued, or
two-valued in such a way that the branches can be separated, the curve
can be investigated by means of the derivatives of y with respect to x,
or through the law of the mean, as given in Book J, for functions of
one variable.

In the same way, when the equation of the curve is given in the
implicit form /(«,y) = 0, we can investigate the curve through the
partial derivatives and the law of the mean for functions of two variables.
This amounts, geometrically, to considering the surface z = Lx, y),
whose intersection with the plane z=o is the curve we wish to
investigate. *

219. Ordinary Point.—If F(x, v) = o is the equation to a
curve, then any point x, y at which we do not have both
a and =
ie ° wy °
is called a stg/e point on the curve, or a point of ordinary position, or
simply an ordinary point.
By the law of the mean,

 

F(x, 9) = Fa, )b wma + uae

If F{x, v) = 0, and a, dis an ordinary point on this curve, then
#(a, 6) = 0. Hence
or or
= (x — a) == —6)—_.
Pee) oer hy ee
From this we derive for x(=)a, 1(=)é,

dy aF / OF
dx ax ay"

 

* For convenience of notation we shall generally write the explicit equation to
a curve in the form y = /(x), and the implicit equation as A(x, 7) = 0.
329
3309 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CuH. XXXL.

Therefore the curve (x, y) = 0 and the straight line
OF ar
(it) a (i) a (1)

have a contact of the first order at a, 4, or (1) is the equation of the
tangent to the curve at a, 4.

We propose to deduce the equation to the tangent at length, in
order to lead up to the general methods which are to follow.

Let /(+,_v) = o be the equation to a curve, then

oF oF
Se) is ee) ee Se) a

$5 {8D g tl gl F (2)

is the equation to the curve in the form of the law of the mean. The
straight line
X—x Fy

FO age (3)

 

intersects this curve in points whose distances from x, y are the roots
of the equation in 7,

a a 2/ a a \2
o= Maa) +7 (25 +m) PE (tiem o'r (4)

If the point x, y is on the curve, this is one point of intersection,
and one root of (4) is 0, for (x, y) =o.
If in addition we have

oF oF
ip Oe (5)

then two roots of (4) are 0, and the line (3) cuts the curve in two
coincident points at x, y, and is by definition a tangent to the curve
at x,y.

Eliminating 7, m between the condition of tangency (s) and the
equation to the straight line (3), we have the equation to the tangent
at x, y,

(KN + (FN Z =o, (6)

the current coordinates being X, F.
The corresponding equation to the normal at x, y is

X—x  F—y
af ar (7)
ax ey

 
ART, 219.] APPLICATION TO PLANE CURVES. 331

EXAMPLES.

1. Use Ex. 3, § 211, to show that if A(x, y) = ¢ is the equation toa curve, in
which F(x, y) is homogeneous of degree x, then the length of the perpendicular
from the origin on the tangent is

- ne
2. If Mx, y) S Un + Un +... + 4, + uy = Ois the equation of a curve
of mth degree, in which w, is the homogeneous part of degree 7, show that the
equation of the tangent at x, y is
ar ar
to + ag tte Ft +...- mm =0.

If X, VY isa fixed point, this is a curve of the (z — 1)th degree in x, y which
intersects A(x, vy) = O in x(z — 1) points, real or imaginary. These points of
intersection are the points of contact of the 2(z — 1) tangents which can be drawn
from any point X, Y toa curve / = 0 of the wth degree. a

3. If X, Ybe a fixed point, the equation of the normal through X, Yto F=o0
at x, y is

oF OF
_ —=—(Y-—y)—.
(X- 5 = (YZ
This is of the zth degree in x, vy, which intersects “= 0 in #? points, real or
imaginary, the normals at which to *= 0 all pass through X, Y. There can

then, in general, be drawn x? normals to a given curve of the zth degree from any
given point.

4. Show that the points on the ellipse 2x?/a? + y?/J2?—1 at which the
normals pass through a given point a, @ are determined by the intersection of
the hyperbola

ay(a — 3) = aay — Box
with the ellipse.
5. If A(x, vy) =0 isa conic, show that its equation can always be written

a a a a)*
o= Fa d+ Je-ae+y-) 5b F+a) OE +0-D gl Fa)
(a2). Show that the straight line whose equation is
tr, (2)

where / = cos 6, # = sin 6, cuts the curve in two points whose distances from a, 6
are the roots of the quadratic

a a\2
o= Fa, +r (tg +m a) Fb ae (et 5) £, (3)
(2). Show that

Z m

 

 

xa yb

oF
(e-aH 4(y 4) & =0 (4)

is the equation of a secant of which a, 4 is the middle point of the chord.’
(¢). Show that the equations
OFS Be oF 8
ox — 7 ay tad 7
solved simultaneously, give the coordinates of the center of the conic.
(d). Show that
x—-a_ y—bs
=

OF oF
and tie +m ofS oO
332 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXXI,

are the equations of a pair of conjugate diameters of the conic / = 0, whose center
is a, 6.

6. If #2 < 1, show that the tangent to x*/a? + y?/d? = # cuts off a constant
area from «?/a? + y?/? = 1.

7. In Ex. 5, show how to determine the axes and their directions in the gee
f = 0, by finding the maximum and minimum values of 7 in the quadratic (3),
a function of 6, the center of the conic being a, 4.

                                       

220.
the curve /(x, y) = 0, the straight line

X—x F-y

Z aa a 7 (1)

yon

 

cuts the curve in points whose distances from x, y are the roots of the
equation in 7,

a é \3
o=r(i7: — +m a)ete ?(igetm 5) Pte *(Getms.) (2)

If we have
oF ge
the line (1) cuts the curve in two coincident points at x, y, and is tan-

gent to the curve there.
If, in addition to (3), 7 and m satisfy

a a\?
(atm y) Fao (4)

then the line cuts the curve /’ = o in three coincident points at x, y,
provided

(Cae tag) P xe. (5)

In this case the line (1) has a contact of the second order with
=o at x, y, and this point is an ordinary point of inflexion. This
means that the value of 7/m = tan @ in (3) must be one of the roots of
the quadratic in Z/m (4).

Eliminating 7 and m between (3) and (4), we have a condition that
x, y may be a point of inflexion,

Fi BP? —2FY Fl Fi 4 By FE =o. (6)

To find an ordinary point of inflexion on /’= 0, solve (6) and
f=oforx andy. If the values of x, y thus determined do not
make both FY and FY vanish, and do satisfy (5), the point is an
ordinary point of inflexion.

The solution of equations (6) and # =o is generally difficult.

In general, if x, y is an ordinary point satisfying / = o, and

os
ART. 221.] APPLICATION TO PLANE CURVES. 333

PEE 2, By 5 ea g MS Ty and
or a oF a\*
ac ae Se ah
Oy Ox Ox Oy
then when 2 is odd we have a point of inflexion at which the tangent
cuts the curve in 7 coincident points at 1, 7. When z is even x, 3'is
called a point of undulation and the curve there does not cross the
tangent but is concave or convex at the contact.
am he conditions for concavity, convexity, or inflexion at an ordinary
point on #’= 0 can be determined as in Book I. For, differentiating
/’=0 with respect to v as independent variable,
_ OF , OF dy
~ On av ax’
a dy a\? Of d*y
ae Se ee eh ee
. (z “1 aie x) Beh ea
At an ordinary point 0,7 ~ o or d,/ 0. Hence the curve is
convex, concave, or inflects at +, y according as

0 dy a\*_. oF a ar a \?
Ep epee WY 2 eee
(a ax 7) = AG Ox Ox 5)?

aF = aF\s
ay ay

is positive, negative, or zero.

 

d*y
ax

EXAMPLES.

1. Show that the origin is a point of inflexion on
ay = bxy + cx + dx.
2. Show that x = 46, y = 268/a? is an inflexion on
2B — 36° Lay =o.
3. Show that the cubical parabola y? = (« — @)*(x — 6) has points of inflexion

determined by 3x + @ = 4d.
Hint. Solve the conditional equation for (« — a)/(y — 4).

4. If 1 = f(x) be the equation toa curve, prove that the abscissze of its points
of inflexion satisfy
afix) f(x) = | F'(4)}?.
II. Srncuar Ponts.

221. If at any point x, y on a curve /(x, y) =0

OF =~ ar :
ay =0 4 wy =o,
the point x, y is called a singular point.
Since wy =— ae coll the direction of a curve ata singular point
ax ox / ay

s .
is indeterminate.
334 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CuH. XXXI.

222. Double Point.—If at a singular point the second partial
derivatives of /’ are not all o, we shall have

Fr er er
oS (f= 2) oar lx — aE Si gerg, cs (FoI ae

Divide through by (X — x)? and let X(=)x. Then

_ PF or aq OF / dy \2
© = Ga® T 7 ax ay Or \dx
This quadratic furnishes, in general, two directions to the curve at
x, y. Such a point is called a double point. The two straight lines

er er ee
(xa ot 7 eA B= Dare yaa 8

pass through the point x, y and have the same directions there as the
curve, and are therefore the two tangents to the curve at the double
point.
The coordinates of a double point on F(x, y) = o must satisfy the
equations
F=0, Fly=0, F/=0o. (1)

The slopes of the tangents there are the roots 4 and 4 of the
quadratic
PRY + eR 4+ Fo. (2)
(A). Node. If the roots of the quadratic (2) are real and different,
then

Hot, HL = (3)
the curve has two distinct tangents at x, y, and the point is called a
node. ‘The curve cuts and crosses itself at a node.
(B). Conjugate. If the roots of the auaaene © in /(2) are imaginary,
or

Fa ky FB = 4, (4)
the point is a conjugate, or isolated point of the curve. The direction
of the curve there is wholly indeterminate. ‘There are no other points

in the neighborhood of a conjugate point that are on the curve. For
the equation to the curve can be written

={@-) 5 +0-ngl'r

{Xa +—y S PE

For arbitrarily small values of Y — x and Y — y the sign of the
second member is that of the first term, and (4) is the condition that
ART. 222.] APPLICATION TO PLANE CURVES. 335

this term shall keep its sign unchanged. Therefore the equation can-
not be satisfied for XY, Y in the neighborhood of x, y.
(C). Cusp-Conjugate. If the roots of (2) are equal, or

fy Py — FIP =o, (5)

the point may be either a conjugate point or a cusp. The curve has
one determinate direction there and a double tangent. Equation (5)
assumes that #,, 7 J%, F,/ are not independently o. Further con-
sideration of the cusp-conjugate class is postponed.

ILLUSTRATIONS.

1, The following example, taken from Lacroix, serves to illustrate the distinc-
tion and connection between the different kinds of double points.
(a). Let y= (x — al(x — byx — 0), (1)
where a, 4, ¢ are positive numbers, and a < 6 < «. y
The curve is real, finite, two-valued, and sym-
metrical with respect to Ox fora < x < 4, It does
not exist for x < aord < x <c; itis finite and sym- O| q@ bc x
metrical with respect to Ox for all finite values of
x > c. The ordinate isco when x=. The curve
consists of a closed loop from a to 4, and an infinite
branch from con. The curve is shown in Fig. 127. Fic. 127.
y (6). Let ¢ converge to 4.
Then the loop and open branch tend to come together,
and in the limit unite in
op acs se gate ale ah, @)
giving at 6a node. (See Fig. 128.)
(c). Let 6 converge to a in (1), The closed oval

Fic. 128. continually diminishes, shrinking to the point a.
In the limit we have
y= (x — a(x — ¢), (3)

which consists of a single isolated or conjugate point
x = a, and an open branch for x > ¢. (Fig. 129.)

(2). Let ¢ and 4 both converge to a, The oval
shrinks to a, and the open branch elongates to a also,
resulting in

 

y= (x — af, (4) Fic. 129.
which has a cusp ata. (Fig. 130.)

y 2. A clear idea of the meaning of singular points
on a curve is obtained when we consider the surface
z = Ax, y), which for any constant value of z is a
curve cut out of the surface by a horizontal plane.

oO a x For example, using (1), Ex. 1, we have the surface
z= (x — al(x — dx — 6) — 7,
Fic. 130. which is symmetrical with respect to the «Oz plane,

and cuts the xOz plane in the cubic parabola
# = (4 — a(x — Bx — 0),
and the horizontal plane in the curve
yt = (x — al(x — Hx — 0).
336 PRINCIPLES AND THEORY OF DIFFERENTIATION. [CH. XXXI.

A moving horizontal plane cuts the surface in curves of the same family. For
example, DD is an open branched curve; BF is a curve with a node as in Fig. 128;
AA is a curve with a closed oval and one open branch asin Fig. 127; so also is CC.

As the horizontal cutting plane rises until it reaches a maximum point 7 on the

 

surface the closed oval shrinks until it becomes the point of contact of the horizon-
tal tangent plane, which plane cuts the surface again in the open branch 7, The
point of touch Z'is a conjugate of the curve 77 and part of the intersection of the
surface by the plane. If the cutting plane be raised higher, to a position /, the
oval and conjugate point disappear altogether and the section is only the open
branch /.

oa that the tangent plane at the node of B& is also horizontal, but the
ordinate to the surface is there neither a maximum nor a minimum.

The node of BB is a saddle point on the surface.

To illustrate the cusp, consider the surface

z=(« —ap—y’

This cuts «Oz in z = (x — a), and the
horizontal plane in y? = (« — a). All
planes parallel to yOz cut the surface in
aoa ~ordinary parabole. All sections of the
surface by horizontal planes are open
branched curves, none having cusps except
Pe that one in xOy. All horizontal sections for
a *% negative have inflexions in the plane x=a,
and their tangents there are parallel to Or.
The horizontal sections above xOy have no
inflexions. As the plane of the horizontal
section below xOy rises, the inflexional tan-
Fic. 132. gents unite in the z#zgze double tangent at
the cusp in the plane «Oy.

3. The above considerations will always enable us to discriminate between a
conjugate point and a cusp of the first species,* when the singular point is of the
cusp-conjugate class under condition (5). For, let A(x, v) = 0 have a point of
this class, and let Aix, vy) = 0 be the equation of the curve referred to the singular
point as origin and the tangent there as x-axis. The point isa cusp of the first
species if F{x, 0) changes sign as x passes through o. If F(x, 0) does not change
sign as x passes through 0, the point is either a conjugate or a cusp of the second
species. Ifin the neighborhood of such a point no real values of x, y satisfy the
equation, the conjugate point is identified. Also, the conjugate points on * = 0
are the values of x, y which make zg = F'a maximumor a minimum.

   

 

* A cusp is of the first spectes when the branches of the curve lie on opposite
sides of the tangent there. If both branches lie on the same side of the tangent,
the cusp is of the second species,
ART. 223.] APPLICATION TO PLANE CURVES. 337

The only forms that double points on an algebraic curve can have,
besides the conjugate point, are nodes and cusps. (See Fig. 133.)

ZZ

Node. Cusp, first species. Cusp, second species.
Fic. 133.

In fact, all other singular points of algebraic curves are but combi-
nations of these, together with inflexions.

EXAMPLES.
1. Show that the origin is a node of y%(a? + x*) = x(a? — x*), and that the
tangents bisect the angles between the axes.
2. Show that the origin is a cusp in ay? = x3,
3. Find the singular point on y3 = x%(x + a). [Cusp.]
4. Investigate d(x? + y?) = 23 at the origin.
5. Investigate x3 — 3axy + y3 = 0 at the origin.

6. Find the double point of (4+ — cy)? = (« — a)§, and draw the curve there.
[«¥=4, y =ab/c. Cusp.]

7. The curve (y — ¢)? = ( — a)(x — 4) has a cusp at a, ¢, if@ 2 4; conju-
gate ifa < 4

8. Investigate y? = x(x +a)? and xt 4 yt = a? for singular points.

9. Investigate at the origin the curve

f= ay — 2x7" + 3yx? —ax 4 FF + 461+ 75 = 0,

Here #4, =0, Hy=0, Fy? — Fy Fy, =0, at the origin, and the third
partial derivatives are not allo. The origin is a point of the cusp-conjugate class,
and y? = 0 is the double tangent.

Since Aix, 0) = — ax + xt changes sign as x passes through 0, the origin is
a cusp of the first kind.

223. Triple Point.—If x, y satisfy the equations
and do not make all the third partial derivatives of # vanish. Then
we have at any point X, VY on the curve

o= | age (rs be

Divide by (X — x)? and make X(=)x. We have the cubic in /
for finding the three directions of the curve at x, y,

o = FL + Fy + 3h Py + Ay. (2)
The solution of this gives, in general, three values of ¢ = dy/dx,
furnishing the three directions in which the curve passes through x, y,
338 PRINCIPLES AND THEORY OF DIFFERENTIATION. (CH. XXXI.

which is a /riple port on the curve. ‘The equation of the three tan-
gents at x, y is

{ag + (F-9) 5) F=o.

Some forms of triple points are shown in Fig. 134.

Pc

Fic. 134.

EXAMPLES.
1. Show that x4 = (x? — y?)y has atriple point at the origin.
2. Investigate at O the curve 24 — 3axy? + 2ay3 = 0.
224. Higher Singularities.—In general, if
or
aa? Ot =
for all values of +9= 7, and 7 =0,1,2,...,%—1, then

the curve # = o has an z-ple point at x, ¥, and in general passes
through the point times.

The equation of the z tangents there is
ee a1
es) a as, Fao,

Their slopes are the roots of

0 0\*
(s +45) 7 = 0

Examples of multiple points are shown in Fig. 135.

EXAMPLES.
1. Investigate 25 + 75 = Sax%y?, ato, o,
2. Investigate (vy — x7)? = x8, ato,o.
3. In x5 + dxt — aty?= 0, the origin is.a double cusp.
4. Determine the tangents at the origin to

y= eI — x),

[Ix ty =0.]
ART. 225. | APPLICATION TO PLANE CURVES. 339

5. Show that 2 — 3axy +74 = 0 touches the axes at the origin.
6. Investigate «+ — ax’y + dy>~o ato,o.
7. Show that 0, 0 is a conjugate point on
ay? — 84+ d22=0
if a and é are like signed, and a node when not.
8. Show that the origin is a conjugate point on
(x? — a) = a4, andacuspon (y — 22)? = x3,
9. Investigate (yv — x?)? = x* ato, 0, for 2 Sas
10. Investigate (x/a)i + (v/6)§ = 1, where it cuts the axes.
11. Find the double points on
x4 — gax? + ga%x? — By? 4. 258y — at — bb = 0,
12. Also on xt — 2ax*y — axy* + ay? = 0.
13. Find and classify the singular points on
xt — 2ax*%y — axy*+ ay =0
when Ge o> 1, «<i

14. Show that no curve of the second or third degree in x and y can have a cusp
of the second species.

Show that if 7(x, 7) = o is any equation of the third degree, having a point of
the cusp-conjugate class at the origin and the x-axis as tangent, the origin is a
cusp or conjugate point according as A(x, 0) does or does not change sign as x
passes through 0, that is, according as the lowest power of x is odd or even.

15. If (x, y) = ois any curve of the fourth degree, having at the origin a
double point of the cusp-conjugate class, and the tangent there as x-axis, then the
origin is a cusp of the first species if the lowest power of x in F(x, 0) is odd;
otherwise, it is a cusp of the second kind or a conjugate point according as the
co-factor of x? in /(x, mx) has real or imaginary roots for arbitrarily small values
of m.

16. Show that the origin is a cusp of the second kind in
xt + yt — ay’ — 2ax*y 4+ axy® + ay? = 0;
is a conjugate point in
at yt — ay — axty + axy? + aly? = 0;
and a cusp of the first kind in
e+ yt — a — bxty + cry? + ay? =o.

225. Homogeneous Coordinates.—The study of homogeneous
functions is very much simpler than that of heterogeneous functions,
owing to the symmetry of the results. This is exemplified in the con-
comitants. It is therefore of great advantage, in the study of curves,
to make the equations homogeneous by the introduction of a third
variable. While we do not propose to follow up this method, it is so
necessary and so universally employed in the higher analysis that it is
mentioned here in order to give a geometrical interpretation to the
meaning of the process and to illustrate what has been said about the
study of surfaces facilitating the study of curves.

In the present chapter we have been really studying a curve as the
section of a surface by the plane z= 0. If now we make the equa-
34° PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXXI.

tion to any curve /(x, y) = o homogeneous in x, y, 2, by writing the

equation
ey
F(—,=)=0 I
(; : *) m (1)
and clearing of fractions, then we have the homogeneous equation in
three variables x, y, 2,
F(x, y, 2) = 0. (2)

F =o becomes / = 0 when we make z = 1.

But /, = o being homogeneous in 1, y, 2, it is the equation of a
cone with vertex at the origin, and which cuts the horizontal plane
z= in the curve /’= 0, which curve is the subject of investigation.

Consequently any investigation of /, = o carried on for a homo-
geneous function in .v, y, z is applicable to the curve / = o when in
the results of that investigation we make z = 1.

III. Curve TRacinc.

226. In the tracing of algebraic curves, the following remarks are
important.

(1). If the origin be taken on a curve of the mth degree, at an or-
dinary point, the straight line_y = mx can meet the curve in only 2 —1
other points.

If a curve has a singular point of multiplicity m, and this be taken
as origin, the line y = mx can meet the curve in only ~ — m other
points.

‘Therefore, if any curve of the zth degree has at the origin a sin-
gularity of multiplicity 7 — 2, the line y = mx can meet it in only
two other points besides the origin, and by assigning different values
to m we can plot the curve by points conveniently.

(II). If any curve has a rectilinear asymptote, and we take the
y-axis parallel to this asymptote, we lower the degree of the equation
in y by 1. Ifthere be m parallel asymptotes, and we take the y-axis
parallel to them, we lower the degree of the equation in y by m. If
the degree of the equation in_y can thus be made quadratic or linear
in y, then by assigning different values to x, the curve can be plotted
by points conveniently.

(III). In any algebraic equation of a curve #= 0, when the
origin is on the curve, the coefficients of the terms in x, y are the re-
spective partial derivatives of the function Fat o, o. Therefore the
homogeneous part of the equation of lowest degree equated too is the
equation of the tangents at the origin. The origin is a singular point
whose multiplicity is that of the degree of the lowest terms ; it is an
ordinary point if this be 1.

(IV). The Analytical Polygon.—Newton designed the follow-
ing method of separating the branches of an algebraic curve at a
singular point, and tracing the curve in the neighborhood of that
ART. 226.] APPLICATION TO PLANE CURVES. 341

point. The method also determines the manner in which the curve
passes off to oo.

Let F(x, y) be any polynomial in x and _y which contains no con-

stant term. Then
F(x,” = SCrxty =o
is the equation of a curve passing through the origin.

Corresponding to each term
C_x'y!, plot a point with reference
to axes Op, Og, having as abscissa
and ordinate the exponents pg and ¢
of «andy respectively, Thus lo-
cating points 4,,..., A, draw
the semple polygon 4,4,4,4,4,4,A,
in such a manner that no point
shall lie outside the polygon.

Such a polygon is determined by
sticking pins in the points and
stretching a string around the sys-
tem of pins so as to zwclude them all. Fic. 136.

The properties of the polygon are :*

(1). Any part of the equation / = 0, corresponding to terms
which are on a side of the polygon cutting the positive parts of the
axes Op, Og, and such that no point of the polygon lies de/ween that
side and the origin, when equated to o is a curve passing through the
origin in the same way as does #'= 0.

Thus, if we strike out of /’ = o all terms except those correspond-
ing to terms on the side 4,4,, we have left a simple curve which
passes through the origin in the same way as #' = 0. In like manner,
if we strike out all terms save those corresponding to points on the side
A,A,, we have another simple curve passing through the origin in the
same way as does /’= o, and so on,

(2). Any part of the equation #’=o corresponding to points
which lie on a side of the polygon cutting the positive parts of the
axes Op, Og, and such that no point of the polygon lies on the opposite
side of this line from the origin, when equated to o gives a simple curve
which passes off to zz/inity in the same way as does /' = o.

Thus the part of # =o corresponding to the side 4,A, gives
such a curve. Again, the part corresponding to 4,4,, gives another
such curve.

(3). Any side of the polygon which cuts the positive part of one
axis and the negative part of the other merely gives one of the axes
Ox or Oy as the direction of an asymptote to ¥ = 0, and these are
more simply determined by equating to o the coefficients of the
highest powers of x and of yin #=o0. Sucha side is 4,A,.

(4). Any side of the polygon which is coincident with one of the

 

 

 

* For a demonstration of these properties see Appendix, Note 12.
342 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXXI.

axes Og, Op, as A,A,,, merely gives the points of intersection of
& = 0 with Ox or Oy accordingly.

(5). Any side of the polygon which is parallel to one of the axes
Op, Og gives rectilinear asymptotes parallel to an axis, or the axis as
a tangent to the curve according as the side falls under conditions (2)

or (1).

EXAMPLES.

1. Trace x® + 2a%x8y — By3 = o.
Numbering the terms in the order in which
they occur, we have 4, , 4,, A,, in the polygon
corresponding to the terms of the equation.
The curve passes through O in the same
way as does the curve
x84 20%x3y = x8( x54 202) =0,
corresponding to 4,4,, or
5 as shown in Fig. 138.
Also, the curve passes
Ege N32: through O in the iis way Fic, 138.

 

as does
2a2xty — yi = y(2a%x — by”) = 0,
corresponding to 4,.4,, as shown in Fig. 139.
The curve passes off tooo in the same way as
does the curve

x6 — Jy} — 0, or x? = by,
Fic. 139. corresponding to 4,4,, Fig. 140.

The form of the curve is therefore as in Fig. 141. en
y
oO
Fic. 141.
Trace the following curves:
2. x4 — 2ax*y — axy? + ay? = 0. 3. at — axty + axy? + ayo.
4, a? —xy—ayxtaxt—x*=0 for @a=1I, «>t, ach
5. xt —adxy + By? — 0, 6. y = a(x? — 1).
T. x* — 3axy* + 2ay3 = 0. 8. x5 — 203 x? 4+ Sa2xy —2a5y?+ yi =0.
9. at + oxy — yt =0. 10. a(x«? — 7?) 4+ 24+ yt = 0.
MN. 2(x?+y?)—2a(a—ySettyt=o. 12. af y? — x2\(y — 2x) = yt
13. a(y — xP(y + x) = yt + #4. 14, 24 — axy? + 74 = 0.
15. ax(y — xP =. 16. xt — axy + 34? = o.
17. Trace 25 — axty — axy' + a’y3 = 0, near the origin.
18. x4 — a®xy? = ay. 19, x4 — axy = ay’.
20. x4 + ax*y = ay’, 21. x(y — xP = Oy.
22. 8 4 95 = Saxy. 23. (x — 33 = (y — 1x8.
24, (x — 2)y? = 4x. 25. (« — 1)(x — 2)? = x.

26. (y—x)(y — 4x)\y + 2")= Bax 27. (y — x)(y + x)(y + 2x) = 1604
Art. 228.] APPLICATION TO PLANE CURVES. 343

IV. ENVELopEs,

227. Differentiation of functions of several variables affords a
method of treating the envelopes of curves, which in general simplifies
that problem considerably and gives a new geometrical interpretation
of the envelope.

For example, we can supply the missing proof, in § 104, that the
envelope is tangent to each member of the curve family. When «x, y
moves along a curve of the family

F(x, y, @) =o, (1)
a is constant, and we have on differentiation
or OF
ag + wy dy =o. (2)

But if +, y moves along the envelope, a@ is variable, and on dif-
ferentiation of (1)

or or oF
ig ay Ot ag (3)
oF dy
Also, on the envelope So Therefore Te from (2) and (3),

are the same at a point x, y common to the curve and its envelope.

228. Again, let a, 8, y be variable parameters in the equation

L(xY; a, f, y) = 0, (1)
where a, £, y are connected by the two relations

(a, f; vy) Ss (2) p(a, B, vy) =0. (3)

We require the envelope of the family of curves (1) when a, £, y
vary. Obviously, if we could solve equations (2), (3) with respect to
two of the parameters and substitute in (1), or, what is the same thing,
eliminate two of the parameters between equations (1), (2), (3), we
could reduce the equation to the family of a single parameter and pro-
ceed as in Book I. This is not in general possible, and it is generally
simpler to follow the process below.

Differentiating (1), (2), (3), the parameters being the variables,

or or OF

aa 2% +38 oh ag
0g 0@ 0d e
ia 2% Peg eee
dy dup Ob oy —
ie ag gy

Multiply the second of these by A, the third by yw, andadd. Deter-
mine A and yw so that the coefficients of da and d@ are zero. Then
344 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cu. XXXI.

if we take dy as the independent variable parameter, the differentials
d®, dy are arbitrary and we can assign them so that the remainder of
the equation shall be zero. ea

or 0

ge tase + uh =o, (4)
ar 8b, ae

ap t hag + Mag = ° ©
or dee’ Oy

ay +A or ue =o. (6)

The envelope is the result Sie by eliminating a, 6, y, A, u
between the six numbered equations.

If we have only two parameters and one equation of condition, the
particular treatment is obvious; as is also the treatment of the gen-
eral case when we have z variable parameters connected by 2 —1
equations of condition.

229. We can get a concrete geometrical intuition of the relation
of curves of a family and their envelope, by letting z be a variable
parameter and considering

(x,y, 2) = 0
as the equation of a surface in space. Then the curves of the family
are the projections on the horizontal plane «Oy of horizontal plane
sections of the surface, obtained by varying z= a.

EXAMPLES.

1, Find the envelope of a line of given length, 7, whose ends move on two fixed
rectangular axes.
We have to find the envelope of
x/a+y/6=1 when a?4 8? = 7?,
x/a = ra, y/P = XO.

Hence Asa, and a= (/2zx)t, b= (/%y)},
and the envelope is wit yi = 78,
2. Find the envelope of concentric and coaxial ellipses of constant area.
Here w/a? + y?/ = 1 and ab=e,
x/G — Ab, 7?/B =a. .. 2A = 1.
The required envelope is the equilateral hyperbola 2xv = ¢.

3. Find the envelope of the normals to the ellipse.
Here a@x/a— vy/B =a — 0 and a®/a4 H/P = 1.

xfer =dAa/a, By/P=—-AP/. « AHH BK,
t p by t
a ax ry
Hence pe a Se p) a ed (ae = 7) 5

give the required envelope
(ax)i + (dy)8 = (a? — 8)8,
ART. 229.] APPLICATION TO PLANE CURVES. 345

4. Show that the envelope of x/a + y/b = 1, where a and 4 are connected by
™m

m mt
qm + ému om, is amet Lymer = cmt,

5. Show that the envelope of x// + y/m = 1, where the variable parameters
/, m are connected by the linear relation 7/a + m/b = 1, is the parabola

t t
x J
a
6. Show that if a straight line always cuts off a constant area from two fixed

intersecting straight lines, it envelops an hyperbola.

7. Show that the envelope of a line which moves in such a manner that the sum

a the squares of its distances from x fixed points x,, y, is a constant 4%, is the
locus

Sx7— kh, Sey, , Sx, x| =o.
2x, Ir 2 Sy; aa e, 2,5 Jy

ty ’ Vr » hy 2

“ , of » To, 0

Let the line be 2x + my +%-=0. Then
B= PS xp + mdy} + np? + 2mp Sy, + 2p Sx, Ht 2lm SxrVry
= aP + bm? + cp? + 2fmp + 2aglp + 2him.

Also, ? + m? = 1, /and m being direction cosines of the line.
Hence we have .

altim+ p+ +Iux =0,
hl + bm + fp + Am + 4uy = 0,
gsitint potty =o

lI

Multiply by 7, m, gin order andadd. ... A= — #
Eliminating /, m, ~, « between the equations
(@-BV4 im + gh + Aux =0,

Al + (6 — Bn + fp + ty = 9,
gl + Jnu+pt+te =09,
al ym+p+o =9,

we have the result.

8. Show that the envelope of a straight line which moves in such a manner that
the sum of its distances from z points x,, y, is equal to a constant 4, is a circle
whose center is the centroid of the fixed points and whose radius is one zth the
distance 2.

Let Ze t+my+ty=0 betheline. Then ?+4+m*=1, and

k= l3Sx, + may, + np,
=al+t bm op.

ataAax+ 2ul/ =09,
b+ Ay + 20m = 0,
e+taA +o =0.

.. Asa—c=—nxn. Multiply these three equations by 4, m, p in order and
add. Hence 4+ 2u= 0.

Here we have
346 PRINCIPLES AND THEORY OF DIFFERENTIATION. (CH. XXXI.

The equations a — mx = kl, 6 — ny = km, squared and added, give the

envelope
2x, \? Zy,\? A\?
Pa Gs) =):
9. Find the envelope of a right line when the sum of the squares of its distances

from two fixed points is constant, and also when the product of these distances is
constant.

 

10. A point on a right line moves uniformly along a fixed right line, while the
moving line revolves with a uniform angular velocity. Show that the envelope is
a cycloid.

11. Show that the envelope of the ellipses x?/a? + y?/s? =1, when @-+ 3? = #,
is a square whose side is 4 4/2.

12. Show that the envelope of line xa™ 4 yom = cmt, when a" 6% = da”, is

n
n n

aim 4. yumm = (S=) cash
13. Find the envelope of the family of parabolz which pass through the origin,
have their axes parallel to Oy and their vertices on the ellipse «?/a? + y?/82 = x.
[4 parabola. |
14. The ends of a straight line of constant length a describe respectively the
circles (xv + ¢)* + y? = a. Show that the envelope of the curve described by the
mid-point of the line, ¢ being a variable parameter, is
ala? + 92 — 4aP)x? + aly? = 0.
15, Find the envelope of a family of circles having as diameters the chords of a
given circle drawn through a, fixed point on its circumference. [A cardioid. |

16. In Ex. 14 show that the area of each curve of the family is $ma* when
¢ > 4a. Also, show that the entire area of the envelope is 4a°[4x — ¥3 if
PART VI.
APPLICATION TO SURFACES.

CHAPTER XXXII.

STUDY OF THE FORM OF A SURFACE AT A POINT.

230. We shall in the present chapter use /(x, v) and F(x, », 2),
when abbreviated into fand /, to mean a function of /wo and /hree vari-
ables respectively.

The functions immediately under consideration are

z=/(*,y) and f(x, 7, 2) =o.

The first expresses z explicitly as a function of x and y, and is to
be regarded as the solution of the implicit function ¥’= o with respect
to z.

It is to be observed generally that since

F=f—z,

results obtained from the investigation of # = o are translated into
those for z = /by

arta ortaf oF Ott

oS soy = 0

Ax? Oy? — Axh dy? Az > Oger
231. In the present article we recall a few fundamental principles
of solid analytical geometry which will be needed subsequently.*

I. Zhe Plane. The equation of the first degree in x, 1, 2 can
always be represented by a plane.
We repeat the proof of this from geometry as follows:

Let Ax + By tCz+ D=o0, (1)
A, B, C, D being any constants. Assign to x and _y any values -,, y,
whatever, and compute z,, so that x,, y,, 2, satisfy (1). In like man-
ner assign arbitrarily x,, y,, and compute 2, so that.x,, y,, 2, also
satisfy (1).

Represent, as previously, x, y, 2 by a point in space with respect to

 

* For a more detailed discussion, see any solid analytical geometry.
347
348 APPLICATION TO SURFACES. (Cu. XXXII

coordinate axes Ox, Oy, Oz. Then, whatever be the numbers m and 2,
the point whose coordinates are
re Mx, + NX, ¥ _ + wy, oh ye + 22,
m+n’ min’ m+n’
is a point on the straight line through the points x,, y,, 2, and x,, y,, 2,,
which divides the segment between these points in the ratio of m to x.

By varying m and 2 we can make +’, y’, 2’ the coordinates of any
point whatever on this straight line. But the point x’, y’, 2’ must
be on the surface (1), since, on substitution, these values satisfy (1).
Therefore, whatever be the two points whose coordinates satisfy (1),
the straight line through these points must lie wholly in the locus
representing (1). This is Euclid’s definition of a plane surface.

‘Lhe intercept of the plane on the axis Oz is —D/C. Therefore,
when C= 0, the intercept is o, or the plane is parallel to Oz.
Hence (1) becomes

Ax+ By4+ D=o0,
the equation of a plane parallel to Oz, cutting the plane xOy in the
straight line whose equations are z= 0, Ax+ By+ D=o0.

We use orthogonal coordinates unless otherwise specially men-
tioned. If 2, m, are the direction cosines of the perpendicular from
the origin on the plane and # is the length of that perpendicular, the
equation of the plane can be written in the useful form

Ix"t+ my + nz —p=0, (2)
where PA m+ n= 1,
Il. The Straight Line. Since the intersection of any two planes is

a straight line, the eguafions of a straight line are the s¢mulfaneous
equations

Ax+By+Cz2+ D,=09, (3)

Ax+By+Cz+D,=0. 2

The equations (3) of a straight line can always be transformed
into the symmetrical form

x—a_ y—b _ 2-€¢

 

 

lm n ei (4)

where a, 4, ¢ is a point on the line; 7, m, ~, the direction cosines of
the line; and A is the distance between the points 4, y, z and a, 4, ¢.

III. Zhe Cylinder. A cylinder is any surface which is generated
by a straight line moving always parallel to a fixed straight line and
intersecting a given curve. The moving straight line is called the
element or generator, and the fixed curve the d:recirix of the cylinder.

With reference to space of three dimensions and rectangular coor-

dinates, any equation
I(%, 9) = 0 (5)

is the equation of a cylinder generated by a straight line moving
ArT. 233.] STUDY OF THE FORM OF A SURFACE AT A POINT. 349

parallel to Oz and intersecting the plane «Oy in the curve /(x, 7’) = 0.
For /(*, _v) = o is nothing more than the equation

I(*, Is 2) =o
in three variables, in which the coefficients of z are zero, and whichis
therefore satisfied by any +, yon the curve /(x, vy) =o in Oy and
any finite value of z whatever.
In like manner /(y, 2) = 0, /(x, 2) = o are cylinders parallel to
the Ox, Oy axes respectively.

IV. Zhe Cone. Acone is a surface generated by a straight line
passing through a fixed point, called the ver/ex, and moving according
to any given law, such as intersecting a given curve called the directrix
or dase of the cone.

Any homogeneous equation of the mth degree in x, 3’, 2, such as

F(x, 4, 2) =0, (6)
is the equation of a cone having the origzn as vertex.

Let a, 6, y be any values of x, ¥, 2 satisfying (6). Then, since
(6) is homogeneous, ka, £8, ky will also satisfy (6), and we shall have
(kx, ky, kz) = RE (x, 1, 2) = 0
whatever be the assigned number &. The coordinates of any point
whatever on the straight line through the origin and a, #, y can be
represented by ka, £8, ky. Therefore all points of this straight line
satisfy (6). When the point a, f, y describes any curve, the straight
line through O and a, #6, y generates a surface whose equation is (6),

and this is by definition a cone.

If we translate the axes to the new origin — a, — 4, — c, by writ-
ing x — a, y — 6,2 — ¢, for x, y, 2 in (6), we have

F(x —a,y —b,2—c)=0, (7)
a homogeneous equation in « — a, y — 4, z — ¢, which is the equa-
tion of a cone whose vertex is a, 4, c.

232. General Definition of a Surface.—If /(~, y, 2) is a continn-
ous function of the independent variables , 1’, z, and is partially dif-
ferentiable with respect to these variables, we shall define the assemblage
of points whose coordinates +, y, 2 satisfy

F(x, 9, 2) = 0 (2)
as a surface, and call (1) the equation of the surface.

233. The General Equation of a Surface.—Let /(x, y,z) = 0
be the equation of any surface.
Then, by the law of the mean, we can write

F(x,y, 8) = P(e’, Jw, 2’)
: I Oe ro itd "7K
+ Efe St Nagy te-—apy

 
350 APPLICATION TO SURFACES. {[Cu. XXXII-

in which the summation can be stopped at any term we choose, provided
we write &, 7, € instead of x’, y’, 2’ in the last term, where &, 7, €
is a point on the straight line between x, y, zand x’, y’,2’. Wecan
therefore always write the equation to any surface in the standard
form

L(x',9", 2’)
a 0 0)”
+ i \ (em eye POSS) aor B=?) a7 } F=o0. (1)

‘This enables us to study the function asa rational integral function
of x, y, 2.

If the equation of the surface be given in the explicit form
z = f(x,y), then in like manner, by the law of the mean, we have
for the equation to the surface

£2 Gh +)'s [@-+0—N glnA ©

in which the summation stops at any term we choose, provided in the
last term we write & for x’ and 7 for_y’; &, 7 being a point on the line
joining x, vy to x’, ’.

234. Tangent Line to a Surface.—A tangent straight line to a
surface at a point A on the surface is defined to be the limiting posi-
tion of a secant straight line 4Z passing through a second point B
on the surface, when #2 converges to 4 as a limit along a curve on
the surface passing through 4 in a definite way.

To find the condition that the straight line

/ / ie
a ee (x)

ey eee

 

 

shall be tangent to the surface F(x, y, 2) = 0.
The equation of a surface in implicit form is, § 233, (1),

t 7 / , oF Vor f oF
Fass" 2) + @ 2) 55+ Ugg + — 2) gt Rao. ()

Substitute JA, mA, nA for « — x’, y —y’, 2 — 2’, from (1) in (2).
We have the equation in A,

ae oF  aF ar
o= Mei 24+ (Mote a tag lAtR (3)

for determining distances from x’, y’, 2’ to the points in which (r)
intersects the surface (2). If F(x’, y’, 2’) = 0, or x’, 9’, 2’ is on the
surface, one root of (3) iso. If in addition

OF oF or
hast t @ pope A ae = Bs (4)
ART. 235.] STUDY OF THE FORM OF A SURFACE AT A POINT. 351

two values of A are o, or two points in which the secant (1) cuts the
surface (2) coincide in x’, y’, 2’, and the line will be tangent to the
surface at .v’, y’, 2’, and have the direction determined by /, m, x.

Observe that in conditions (4) and 7? + m*? + n%= 1 we have
only two relations to be satisfied by the three numbers /, m, 2, and
therefore there are an indefinite number of tangent lines to a surface
at a point 2’, y’, 2’.

If the equation to the surface be in the explicit form z = /(*,¥),

or
oS

x

Ste G+, (s)

then, as before, the straight line (1) meets the surface (5) in x’, y’, 2’
when z = 2’ and other points whose distances from x’, y’, 2’ are the
roots of the equation in A,

on ((Lind-a)rase

g=a +(x — x') 5

The condition of tangency is that a second point of intersection
shall coincide with x’, y’, 2’, or

Ff oy =
Rp agg

235. Tangent Plane to a Surface.—When the locus of the tan-
gent lines at a point on a surface is a plane, that plane is called the
tangent plane to the surface at that point. The point is called the
point of contact.

Tangent plane to F(x, y, 2) = 0 at x’, 9, 2’.

The straight line

—y z— 2
jf =t = (2)

m n

 

 

 

is tangent to the surface = at x’, 9’, 2’ when F(x’, y’, 2’) = o and
OF mo tas =. (2)
If now at x’, y’, 2’ the derivatives
ar ar ar
ax’? ay” — Az’
are not all o, we obtain the locus of the tangent lines to "= 0 at

x’, y’, 2’ by eliminating /, m, ” between (1) ) and (2). Therefore this
locus is

ar OF
wo) B tun B+e-Nge ae)

Equation (3) is of the first ce in x,y, 2,and therefore is a
plane tangent to # = oat x oD
352 APPLICATION TO SURFACES. [Cu. XXXII.

Tangent plane to z= /(x,y).
Eliminating /, m, 2 between (1) and
af of
Fgyh age = (4)
we have
s— @ sale a") 5 +l ie (5)

as the tangent plane to z = fat’, y’.

236. Definition of an Ordinary Point on a Surface.—We have
just seen that when at any point on a surface /' = o the first partial
derivatives,

OF OF <aF

Ox > ay > az’
are not all zero, the surface has at that point a unique determinate
tangent plane. Such a point is called a povnt of ordinary position, or
simply an ordinary point.

On the contrary, ifat x, , 2 we have

0,F¥=0 aF=o0, IF=o0,
the point is called a simgular point on the surface. We shall see

presently that the surface does not have a unique determinate tangent
plane at a singular point.

EXAMPLES.

1. Find the conditions that the tangent plane to z = f(x, y) shall be parallel to
xOy. Ans. 0.f = df= oO.

2. Find the conditions that the tangent plane to F(x, y, z) = 0 shall be hori-
zontal. Ans. 0,F =0,F =0, I,F # 0.

3. Show that the tangent plane at 1’, y’, s’ to the sphere x? + y? 4 2? = a? is

xa’ + yy! + 22 = a.
4. Find the tangent plane to the central conicoid = Ae eas + . =

of yy fee =
5. Show that the tangent plane to the paraboloid ax? . by? = 22 at x’, 9’, 2!
is axa! + byy'’ = 2+ a,
6. Show that the tangent plane to the cone Aix, y, 2) = 0, having the origin as
vertex, is XO + I yt P+ sd. = oO.

This follows directly from the fact that His homogeneous, and therefore the
ee plane is

oF oF OF
sey teary Fort! oe ame’, 2) 0,

where x is the degree of the cone.
ART. 236.] STUDY OF THE FORM OF A SURFACE AT A POINT. 353

7. Find the equation to the tangent plane at any point of the surface

ato. yt 4 ot — a’, and show that the sum of the squares of the intercepts on the
axes made by the tangent plane is constant.

8. Prove that the tetrahedron formed by the coordinate planes and any tan-
gent plane to the surface xys = a? is of constant volume.

9. Show that the equation of the tangent plane to the conicoid
ax? + by + cx? + afys + 2g¢x2 4 2hxy + 2ux + 20y + 20s +d=0,
at x’, 9, 2’) 18
(ax! + hy! + gs! + ue + (x 4 by! + fe! +o +
(gx + Jy’ 4-02! + w)s 4 ux’ toy + we fdao.
10. Show that

2 0 ¥
3}, [@ ag tay +Gaoa. | Fe

is the general equation of any conicoid, and that
fF ar _aF

dx ~ ay — oz —
are the equations of the center of the surface.

11, Show that the plane
af oF ar
ie wore Pigg t= 2s =0
cuts the conicoid “ = o in a conic whose center is a, #6, y, and therefore this is
the tangent plane when a, , y is on the surface.
12. Show that the locus of the points of contact of all tangent planes to the
surface # = 0, which pass through a fixed point a, 6, y, is the intersection of
# = o with the surface

OF oF oF
(@-#)s-+6-N + ( -9z =O

13. This surface is of degree 2 — 1 when / = 0 is of degree 2.

For, let F= U,+ .. + U,-+ U,, where U, is the homogeneous part of
degree x. Then, as in two variables, we have the concomitant

ov”, aU, aU,
x os +y 7 z oe tO os
Therefore the tangent plane at x, y, 2 may be written

OF oF aF
Jee Sy ae ee 1Up+1 +... 4+ U,, or
OF oF OF
xe + va teat Uy bag ves oe — Oo =o
since Unt... 44 4=0.
14. Find the condition that the plane Zr + my -+ nz = 0 shall be tangent to

the cone
Fes ax? + by? + es? 4 2afys + ages + 2hxy = 0
at REV 5 Se
The equation to the tangent plane at x’, y’, 2’ is, Ex. 6,

ar OF , aF _,
eo TP gf sf =o
354 APPLICATION TO SURFACES. fC. XXXII.

In order that this shall be identical with 2x + my + nz = 0, the coefficients
of x, y, 2 must be proportional.

oF 7_ oF _ oF ae
or = ay mage [n= » Say.

Hence ax’ + hy’ + ge =,
‘hal + by 4 fe! = ma,
el + fi’ +o! =a,
Ze’ + my'+ nz’ = 0,
In order that these equations shall be consistent we have
ah ig des oO,

hid ff mM
Cf COR
+ we # Oo

the required condition.

15. Generalize Ex, 14 by finding the conditions that the plane may cut, be tan-
gent to, or not cut the cone except in the vertex.

Eliminating s, the horizontal projection of the intersection of the plane and cone
is two straight lines

(an? 4 el? — 2gin)x* 4 2(hn® + clm — gmn — fln)xy + (bn? + cm? — 2fmn)y* = o.
These will be real and different, coincident or imaginary, according as ;

(an? + cl? — 2¢ln)(bn® 4+ cm® — 2fmn) — (hn® + clan — gmn — finy,
which can be written as the determinant

As=lah g li,
Ab f m
eof (6K

lJ mn o
is negative, zero, or positive, respectively.

16. Show that the projections of the two lines in 15 can be written

04 o4 0d
— x7? — — xy +— y2 —
ab ~ an 3a? %

with similar equations for the projections on the other two coordinate planes.

17. Show that J in Ex. 15 can be written

aD 0D aD aD aD 0D
psa pele 2 a i — — ——
gg ee et ae Oe i ee ge i eo ay
where Dz=l|a h gi.
hob Ff
gf ¢

 

 

237. Conventional Abbreviations for the Partial Derivatives.
The elementary study of asurface is usually confined to those properties
which depend only on the first and second derivatives, that is, on the
quadratic part of the equation to the surface when the equation is
expressed by the law of the mean.

This being the case, it is of great convenience in printing and
writing to have compact symbols for the first and second partial deriv-
atives. These derivatives being the coefficients of the first and second
powers of x, y, 2 in the equation, it is customary to represent them by

 
ArT. 238.] STUDY OF THE FORM OF A SURFACE AT A POINT. 355

the same letters as are conventionally employed as the coefficients of

the terms in the general equation of the second degree in three vari-
ables.

We shall hereafter frequently write:

 

When F(x, y, 2) =0,
oF or or
L= M= — = _—
Ox’ ay’ sl az’
_ OF _ oF _ ar
A= a 3 ap C= oa
ee _ ar
~~ Oy O2’ ~ Ox a2’ ~~ ax dy’
When = fix);
BOF gS pe ges OF gee
~ ax? 2 ay? 7 a? 9 day? 7

 

238. Inflexional Tangents at an Ordinary Point.—We have seen,
SS 234, 235, that there are an indefinite number of tangent lines to a
surface at an ordinary point, lying in the tangent plane and passing
through the point of contact. If the second partial derivatives of
Ff =o are not allo, there are two of these tangent lines that are of
particular interest.

(1). Let s = /(x,.v) be the equation of a surface.

The straight line

 

 

 

pag og (1)

cuts the surface z = fin points whose distances from the point x, 1, 2
on the surface are the roots of the equation in A

0 A/,9 0 \?
on (Lt me — 4S tmg) P+ R= 0. (2)

2
a a
If es of

ax + m ay —n=od0, (3)
we have seen that (1) is tangent to z = fat x, y, 2.
If in addition we have /, m, x satisfying the condition
a a \? ar arr as
ee ens ape 2 =
(a msc) PSV a et ee aoa tae AD
two roots of (2) are o, and the line (1) cuts the surface in three coin-

cident points at +, ), 2.
The conditions

 

 

Pom? tn =1,
ple +qm —n =0,
rl? +. 2slm +- tm? = 0,
356 APPLICATION TO SURFACES. [CH. XXXII.

determine two straight lines, in the tangent plane, tangent to the
surface s = fat the point of contact. Each cuts the surface in three
coincident points there.

These are called the inflexional tangents at x, y, z. They are real
and distinct, coincident, or imaginary, according as the quadratic
condition

rl? + 2slm + tm* =o,

in 2/m, has real and different, double, or imaginary roots, or accord-

ing as
2 ar \:
nan (BY
Ox? ay? Ox Oy
is negative, zero, or positive.

Since any straight line, such as (1), cuts any surface of the mth
degree in points, the straight lines in any plane cut the curve of sec-
tion of asurface ofdegree 7 in ” points. ‘Therefore a plane cuts a sur-
face of the zth degree in a plane curve of degree z.

The tangent plane toa surface of degree 7 cuts the surface ina
curve of degree 2 passing through the point of contact. But each of
the inflexional tangents to the surface cuts this curve in three coincident
points at the point of contact. Each is therefore tangent to the curve
of section at the point of contact of the tangent plane, which is there-
fore a singular point on the curve of section. ‘This point is a node,
conjugate point, or cusp according to the value of condition (5). Com-
pare singular points, plane curves.

Eliminating 7, m, 2 between (1), (3), (4), we have for the equa-
tions of the inflexional tangents at x, y, 2

Z—2= (X—«)p+ (F—y)9,
(X — x)? + 2(X— x)(P—y)s + (F—pPt=o.

The second is the equation of two vertical planes cutting the first,
the tangent plane, in the inflexional tangents.

(2). If the equation of the surface is “= o, then the straight line
(1) cuts the surface in points whose distances from x, y, z are the roots
of the equation in A,

 

(5)

ar or OF\ d?/ 9 a a\2
rea(egeen ay t"a)+ + (lgetmaet nar) F4IR=0.
If x, y, 2 is on the surface, or F(x, y, z) = o, and
Li+ Mm +Nn=o0,

the line (1) is tangent at x, y, 2. If in addition /, m, x satisfy the
condition

0 0 0\?
(atm Zt*5) F= 0;
ART. 239.] STUDY OF THE FORM OF A SURFACE AT A POINT. 357

the line (1) cuts the surface in three coincident points at x, y, z. The
conditions

Pimtnvotd, (6)
Ll+ Mm + Nn = 0, (7)
AP + Bm? + Cr? + 2Fmn + 2Gin + 2Him = 0, (8)

determine the directions of the two inflexional tangents.

Eliminating /, m, 2 between (1), (7), (8), we have the equations of

the inflexional tangents at x, y, 2,

|X-a +P et Zag Pao, (9)

[Xe tPdgt+e-agl Fao. (10)

The first is the tangent plane, which cuts the second, a cone of the
second degree with vertex , y, 2, in the two inflexional tangents.

These tangents will be real and different, coincident, or imaginary,
according as the plane (9) cwés the cone (10), is tangent to it, or
passes through the vertex without cutting it elsewhere. That is, ac-
cording as the determinant (see Ex. 15, § 236)

AHGL
HEFKFM (11)
GFCWN
LMNo
is negative, zero, or positive,
239. Should the second partial derivatives also be separately o at
xv, 1’, z, and x the order of the first partial derivatives thereafter which
do not all vanish at x, y, 2, then there will be at x, y, zg on the sur-

face 7 inflexional tangents, which are the 7 straight lines in which the
tangent plane at x, y, 2 cuts the ~ planes

{\e-9e+r-ngl rao,

or the cone of the rth degree,
a a a)"
}X= 9 P+ (Fs + (2-2) 5 Po.

These 7 inflexional tangents to the surface are the 7 tangents to
the curve cut out of the surface by the tangent plane at the point of
contact, which point is an 7v-ple singular point on the curve of
section.

EXAMPLES.
1. Show that the inflexional tangents at any point x’, y’, 2’ on the hyperboloid

x?/a? + y?/6? — 2°/c? = 1, lie wholly on the surface and are therefore the two
right-line generators passing through the point. Show that their equations are
x— x’ y —y Res 2

@ bx'2! + acy’ a ay's! F bcx’ =A 4 2/2
358 APPLICATION TO SURFACES. [Cu. XXXII.

2. Show that the inflexional tangents at a point x, y, zon the hyperbolic parab-

 

 

oloid x?/a? — y?/b? = 22 lie wholly on the surface, and that their equations
are
X—«x Y-y Z-—2
2 £8 #.y9
ze ROY
a 6b

the upper signs going together and the lower together.
3. Show that the inflexional tangents to the cone
ax? +. by* 1 cz? 4 2fyez + agxz + 2hxy = 0
are coincident with the generator through the point of contact.
4. Show that at a point on a surface at which any one of the coordinates is 4
maximum or a minimum the inflexional tangents are imaginary.

240. The Normal to a Surface at an Ordinary Point.—The
straight line perpendicular to the tangent plane at the point of contact
is called the zorma/ to the surface at that point.

Since the equation to the tangent plane at x, y, 2 is

oF or 0.
(Xa 4 (Py) 4-9 =o,
0
or Z-2= (X= + (FZ

the coefficients of X, FY, Zare proportional to the direction cosines
of the normal, and we have for the equation to the normal at x, y, 2

X—x F-y  Z-2

 

 

ar .—s—iF”~‘“t«C
Ox “ay Oz
X—x F-y Z2-8
or ee .
ee aed ee
Ox oy
EXAMPLES.

1. Show that the normal at x, y, z to xyz = ais
Xx — xt = Yy—y? = Ze — 2
2. Find the equations of the normal to the central conicoid ax* + dy? + cx? = 1.
X=% Vay. Zas

ax —COY””:stCt RS

 

 

8. Show that the normal to the paraboloid ax? + dy? = 2z has for its equations
X—x V-y

ae => by = gs Ze

 

 

241. Study of the Form of a Surface at an Ordinary Point.
—We may study the form of a surface at an ordinary point by
examining it (1) with respect to the /angent plane, (2) with respect to
the conicord of curvature, (3) with respect to the plane sections parallel

e
,
ART. 242.] STUDY OF THE FORM OF A SURFACE AT A POINT. 359

to the tangent plane, (4) with respect to the plane sections through the
normal,

242. With respect to the tangent plane:

(1). Let 2z=/(x, y). Then the equation of the surface is

0 3
Z—2—(K— 9) Ly) Pat) (We HP ag

Let X, FV, Z, be a point in the tangent plane inthe neighborhood
of the point of contact x, y, 2. Then the difference between the
ordinate to the surface and the ordinate to the tangent plane is

9

tk Megas. tere a)e
22,24 (K— ae t(P—I) goth

This difference is positive for all values of X, Y in the neighbor-
hood of x, y when

aay (ay 4
ax? aF (x ) ae x8
are positive (Ex. 19, §25). Then inthe neighborhood of the point
of contact the surface lies wholly above the tangent plane, and is said
to be convex there.

In like manner Z— Z, is negative throughout the neighborhood
when 7/ — s* is positive and.y is negative at the point of contact.
Then the surface in the neighborhood of the point of contact lies
wholly below the tangent plane and is said to be concave there.

(2). Let (x, y, z) = 0. Inthe same way we have the equa-
tion of the surface,

(Naa) PLY (Py) Fj 4 (Z-a) Fl =
-1{anZrrwd seo be
2 0g an 0g ;
and for that of the tangent plane at x, y, 2,

(X —a)FL+ (Py) /4+ (4, -—2)F = 0.
On subtraction,

: 0 0 a)?
(2-2) Fiat | (Xa) get Y5, + (2—aae |

Therefore, at x, y, 2, by Ex. 20, $25, when

A H| and Ald HG
lx B HBF
GFC

are positive, the surface is convex when A and F” are unlike signed,
concave when A and /Jare like signed.

 
360 APPLICATION TO SURFACES. [Cu. XXXII.

Observe that a surface is concave or convex at a point when the
inflexional tangents there are imaginary, and conversely. Whena
surface is either concave or convex at a point, its form is said to be
synclastic there. When the inflexional tangents are real and different
the surface does not lie wholly on one side of the tangent plane in the
neighborhood of the point of contact, but cuts the tangent plane ina
curve having a node at the point of contact and tangent to the inflex-
ional tangents. At such a point the form ofthe surface is said to be
anticlastic, and the surface lies partly on one side and partly on the
other side of the tangent plane in the neighborhood of the contact.

The conditions that a surface may be synclastic or anticlastic at a
point are, (11), § 238,

AHGL

WRFM= + synclastic,
. a - ao | = — anticlastic.

The hyperboloid of one sheet and the hyperbolic paraboloid are the
simplest examples of anticlastic surfaces, these being anticlastic at
every point of the surfaces. The surface generated by the revolution
ofa circle about an external axis in its plane generates a /orus. This
surface is anticlastic or synclastic at a point according as the point is
nearer or further from the axis of revolution than the center of the
circle,

243. With Respect to the Conicoid of Curvature.
(1). The explicit equation z = /(x, ¥’), or

2 af af 1 \ S fC) 0)?
Z=24(X—2) 5! +P) $3] A—aget Py) 5 OA
shows that in the neighborhood of x, y, z the surface differs arbitra-
rily little from the paraboloid

This is called the paraboloid of curvature of the surface at x, y,z. It
has the same first and second derivatives at x, y, 2 as has the surface
z= /, and therefore, at that point, has, in common with the surface,
all those properties which are dependent on these derivatives.

Obviously, the surface is synclastic or anticlastic according as the
paraboloid is elliptic or hyperbolic.

From analytical geometry, the discriminating quadratic of the
paraboloid

rei t+ tr + esxy 4 2px-+ egy — 22 +k =0
is Ae (r+ AA + (r# — #) = 0.
ART. 245.] STUDY OF THE FORM OF A SURFACE AT A POINT. 361

This gives the elliptic or hyperbolic form according as r/ — s? is
positive or negative.
(2). In the same way, the implicit equation I(x, 9, 2) = 0, or

a oF - OF oF
(T—2)5-+(2 —I) gz, + (4 — 2) as

I

: 0 0)?
+ }W=9 5+ (V9) + 2—2) 5h =k,

2
shows that in the neighborhood of x, y, z the surface differs arbitrarily
little from the conicoid of curvature whose equation is the same as the
left member of the equation above when equated to o. The form
of the surface at .v, 1, z is the same as that of the conicoid of curvature
there, and they have the same properties there as far as these proper-
ties are dependent on the first and second derivatives of /.

The discrimination of the conicoid can be made through the
discriminating cubic (see Ex. 17, p. 30)

 

 

a—A, ,G = 6,
H , B~A, F
G yt », G—2r
and the four determinants
1d AG Li,
HBFM
GFCWN

 

 

 

 

as in analytical geometry.*

244. The Indicatrix of a Surface.—At an ordinary point x, y, 2
on a surface, at which the second derivatives are not all o, a section
of the surface by a plane parallel to and arbitrarily near the tangent
plane differs arbitrarily little from the section of the conicoid of
curvature made by this plane. Such a plane section of the conicoid
of curvature is called the zndica/rzx of the surface at xv, y, 2.

Points on a surface are said to be circudar (umbilic), elliptic, para-
bolic, or Ayperbolic according as the indicatrix is a circle, ellipse,
parabola (two parallel lines), or hyperbola (two cutting lines).

245. Equation to Surface when the Tangent Plane and Normal
are the z-plane and z-axis.—If the equation is z= A(x, y), then since
z=0,p= 0,¢g= Oat the origin, the equation is

err? asxyt P+ 2R,

The equation of the indicatrix at the origin is

 

gore t+ asxcy + Hy’,

 

* See Frost’s, Charles Smith’s, or Salmon’s Analytical Geometry.
362 APPLICATION TO SURFACES. [Cu. XXXII.

z being an arbitrarily small constant, This is an ellipse or hyperbola
according as 7/ — s® is positive or negative, giving the synclastic or
anticlastic form of the surface there accordingly.

246. Singular Points on Surfaces.—If, at a point x, y,z ona
surface / = o, we have independently

ar oF oF 2
ss =O) — = ———
ax , ay QO; ae , ()
the point is said to be a simgular point.
If the second derivatives are not all zero, then all the straight lines
whose direction cosines 7, m, m satisfy the relation

a a a \?
Z7—tm>=—-+t2,)F=0 2
( oe av a a) (2)
will cut the surface in three coincident points at x, y, 2, and are called
tangent lines, Eliminating /, m, 2 by means of the equation to the
line and (2), we obtain the locus of the tangent lines at x, y, 2,

{(X-) gt Ig +@-IzbP=o Gs)

This is the equation of a cone . the second degree, with vertex
x, ¥, 2, which is tangent to the surface /"= 0 at the point x, y, 2
The form of the surface at 1, y, 2 is therefore the same as that of this
cone. Such a point is called a conical point on the surface.

When this cone degenerates into two planes, then all the tangent
lines to the surface at x, 1, zg lie in one or the other of two planes.
The point is then called a zoda/ point. The condition for a nodal
point is that equation (3) shall break up into two linear factors, or

dA H Gl= (4)
H B F
G F C

 

 

A line on the surface #’= 0 at all points of which (4) is satisfied
is called a zodal line on the surface. Such a line is geometrically
defined by the surface folding over and cutting itself in a nodal line,
in the same way that a curve cuts itself ina nodal point.

If r is the order of the first partial derivatives which are not
all zero, then the surface has a conical point at a, y, 2 of order 7, and
a tangent cone there of the rth ee whose equation is

: d "7
(X34) +P —N + 2-2) gz) FH. 6)
247. A singular fangent plane is a plane which is tangent to a

surface all along a line on the surface. For example, a torus laid on
a plane is tangent to it all along a circle. The torus has two singular
Art. 247.] STUDY OF THE FORM OF A SURFACE AT A POINT. 363

tangent planes. All planes tangent to a cylinder or cone are
singular.

EXERCISES.

1. The tangent plane to yx? = a’z at x1, ¥,, 2, is
2xxyV, + y xt — az = 2072).
Find the equation to the normal there.
2. The tangent plane to 2(4? + y?) = 2hxy at 2, 9,, 2, is
2x(xy5, — AY) + 2v (942, — 4x,:) + 241 + 97) — 2h4a,y, =O.

The tangent plane meets the surface in a straight line, and an ellipse whose
projection on the «Oy plane is the circle

(x* + y(t — 97) + Tt bi), — 7) = 0.
Show that the z-axis is a nodal line.
3. The tangent plane to ay? = x°(c? — 27) at x4, 91, 2% is
xxy(C — 22) — ayy, — 22x + xis? =O.

At any point on Oz, F,/ = F,' = F;/ = 0, show that at any such point there

are two tangent planes
Bo ey
a \ a

4, Show that the tangent plane at x,, y,, 2, to
w+ 78 + 38 — 3xy2 = a8
is a(x? — 4%) + (Vt — 1%) + 2? — 41) = 2
5. The tangent plane at x,, 7,, 2, to «"y¥zt =a is

 

m n on Fu
a ar x Te Pit OTE?

6. Show that (2a, 2a, 2a) is a conical point on
xyz — a(x* + y? + 2?) + 4a® = 0,
and find the tangent cone at the point.
Ans. x? + y? + 2? — aye — 220 — 2x9 = O
7. Show that the surface

2 yh gh\ 2 ae? ® gt I
= tata —3\atp ra
has two conical points.
The tangent cone at 0, 0, 0 is 3.42/a? + 39/8? 4 22/2 = 0,
8. Determine the nature of the surface

ay? + bat + a(x? $y + ot) = 0
at the origin.

The origin is a singular point, the tangent cone there is ay? + 62? = 0. Ifa
and 4 are like signed, the origin is a cuspal point around the x-axis.

9. A surface is generated by the revolution of a parabola 2 = 4mx about an
ordinate through the focus; find the nature of the points where it meets the axis of

revolution.
364 APPLICATION TO SURFACES, (Cu. XXXIL

Hint. The equation of the surface can be written
16m?(x? + y?) = (2? — gm’)?
The two right-angled circular cones x? + y? = (z + 2m)? are tangent to
the surface at the singular points.
10. If tangent planes are drawn at every point of the surface
a( yz 4 2x ay) = x2,

where it is cut by a sphere whose center is the origin, show that the sum of the
intercepts on the axes will be constant.

11. Show that the general equation of surfaces of revolution having Oz for axis

x ty? = f(z).

Thence show that the normal to the surface at any point intersects the axis of
revolution.

12. Show that at all points of the line which separates the synclastic from the
anticlastic parts of a surface the inflexional tangents must coincide.

13. The equation of an anchor-ring or torus is

(F +94 4 A — att = glx? +7"),
Show that the tangent plane at .’, 9”, a8 is
(7 — oer’ + yy") + rss’ = [P+ er — 1,

where 7? = x/? 4 yy"?

The tangent plane at any point on the circle «? + y? = (¢ — a)? cuts the surface
in a figure 8 curve whose form is given by the equation

(7? + 2 — aac? + dele — a)s? = 0.

14, When the tangent plane passes through the origin it cuts two circles out of

the torus which intersect in the two points of contact.

15. Show that the cylinder 2? + y? = ¢ cuts the torus in two parts, one of which
is synclastic, the other is anticlastic.

is
CHAPTER XXXIII.
CURVATURE OF SURFACES,

248. Normal Sections. Radius of Curvature.—The normal
section of a surface at a point is the curve cut on the surface by a
plane passing through the normal to the surface at the point.

To find the radius of curvature of a normal section.

Let the tangent plane and normal at
an ordinary point on the surface be taken
as the z-plane and z-axis respectively.
Then the equation to the surface can be
written

= 4(reF + acy +)+ 2, (2)
since at the origin zs =0, P=o0, g=0.

Cut the surface by a plane passing
through Oz and making an angle 4 with
Ox. At every point of this plane let

 

x=pcos#, y=psin?d. Fic. 142.
z= tp"(r cos’d + 25cos 6 sin @ +7 sin®6) + 7,
where 7’ contains as a factor a higher power of p than p?.

The radius of curvature # of this normal section PO is, by New-
ton’s method, § ror, Ex. 4, given by

22
ru (2)

I
R
pi=)o
= rcos*6+ 2s5cos Osiné + / sind,
=4(r +4) +24(r — A cos 26+¢ sin 29, (3)
The directions of the normal sections in which the radius of cur-
vature is a maximum or a minimum are given by the equation

2S

 

tan 26 = (4)
If a is the least positive value of @ satisfying (4), the general solu-
tion is 427 + a, showing that the normal sections of maximum and
minimum curvature are at right angles. These sections are called the
principal sections of the surface at the point considered. Their radii
of curvature at the point are called the princpal radii of curvalure.
365

Rad
366 APPLICATION TO SURFACES. [Cu. XXXIII.

If the principal sections be taken for the planes xOz, yOz, the ex-
pression for the radius of curvature of any section will be

= 7 cos’d + /sin?d, (5)

|

since then s = 0, by (4).
Let 2, and &, be the radii of the principal sections.
Then when 6=0, Rot =7; G=42, Rot =~s, in (5).
2 aa)
a = = a = a (6)
1 2
Also, if &’ is the radius of curvature of a normal section perpen-
dicular to that of &, then

 

I sin?@ cos*@
BR TR,” (6)
I I i, I
tag at ee, (7)

The sum of the reciprocals of the radii of curvature of normal
sections at right angles is constant. This is Euler’s Theorem.

249. Meunier’s Theorem. —To find the relation between the
radii of curvature of a normal section
and an oblique section passing through
the same tangent line.

Take «Oz as the normal plane, and
let the oblique plane 1?O@ make the
angle @ with +Ox.

Then the equation of the surface is

ag=ra*t asay ty?

 

I 3

0 0
+3(*a¢ +739)

At any point P in the oblique sec-
tion y = 2 tang.

 

: : v/a d \3
Se a 2s— tan @+f a tan’ . (Get eee OS

2
x
But since Ox is tangent to the curve OP at o,

zsec dp zg
=o _—,
x x

x(=)o

f= bx)...

as P converges to O along PO. Also, in the xQz section, if
MR = x,, we have y = o, and

aan ea
x, i aa x= 6

 
ART. 250.] CURVATURE OF SURFACES. 367

Let #,, & be the radii of the normal and oblique sections. Then,

for 2(=)o,
= x We
R= f cos 4, R= as

Hence R= k, cos d.
This is Meunier’s theorem.

250. Observe, in the equation to the surface (1), § 248, the equa-
tion of the indicatrix is

2g = ra? t asay tA Qa)

The principal sections of the surface at O pass through the axes

of the indicatrix conic, whose equation is
ager? +1 (2)
when «Oz and_yOz are the principal planes.

Equation (1) shows that the radius of curvature of anormal section
varies as the square of the corresponding central radius vector of the
indicatrix. All the theorems in central conics which can be expressed
by homogeneous equations in terms of the radii and axes furnish
corresponding theorems in curvature of surfaces.

We shall adopt the convention that the radius of curvature of a
normal section of a surface is positive or negative according as the
center of curvature of the section is above or below the tangent
plane.

When the indicatrix is’an ellipse the principal radii have like
signs, and have opposite signs when the indicatrix is the hyperbola.
The inflexional tangents are the asymptotes of the indicatrix.

251. At any point of a surface to find the radius of curvature of a
normal section passing through a given tangent line at the point,

Let #’= o be the equation of the surface. Let P be the given
point .v,9', 2, and /, m, m the direction cosines of the tangent line
there. Let Q be another point X, Y, Z on the surface and in the
normal section.

Let QR be the perpendicular from @Q on the tan-
gent line PR.

Then for &, the radius of curvature of the sec- e
tion, we have

a PEO PR (PR? PEC
= 20R 20k \PQ) 20R°
The tangent plane at P is
(Y—a)L+(F—-y)\M+ (Z-2a)N= 0. 2 g
The distance of Q from this plane is Fic. 144.
I y
OR= = ((X—a)L 4+ (YM + (Z— aN},
where e= Pt Met N°.

 

 

 

 
308 APPLICATION TO SURFACES. (Cu. XXXIII.

Also, Q being a point on the surface,
(X — a)Z 4(F — »)\M+ (Z—2)N

( 0 0 0)?
+7 {8-4 £4 e+ 2-93 | PET Ho,

 

a a a)?
< pP{k— Dat Det e—agt Fear
R =f (fa) 4 (Fy + 2 2) :
= AP + Bm 4+ Cn? + 2Fmn + 2Gin + 2Him, (1)

since £7 =o for Q(=)P, and
Aas Fay 2-8 _
lL” mo

 

 

rd

is the equation of the tangent P#. The derivatives Z, A, etc., of
course being taken at P.

252. If the equation of the surface be /(+,y) —z = F=o0,
then since Z=/, A= 7, N= —1,C=F=G=o, (1), $251,
becomes

I rl + 2slm + tm?

REE PTA ©

253. To Find the Principal Radii at Any Point on a Surface.
—wWe have only to find the maximum and minimum values of & in

(1), § 251, § 252.
I. In (1), $251, let 2, m, » vary subject to the two conditions
LimMtnrN=0, P+mitinr=t.
Then, by the method of § 217,
Al+ Hm+ Gn+AL + pl =o,
Hl + Bm + Fn + AM+ pm = 0,
Gi+ Fm + Cn +AN + pn = 0.

Multiply by Z, m, 2, respectively and add. .°. pe= — «/R.
(A — k/RV+ Hm +- Gun +AL=o0,
fl + (B —k/R)m + fn + AM= o,
Gi + Fm +(C—k/R)n+AN = 9,
Li +4 Mm + Nn =0.
Eliminating 7, m, n, A, we get the quadratic
A —k/Rk, H ,G , L/=9,
HT , B—k/R, F ,
G , » C—k/Rk, N

L , p lV , Oo
ART. 254.] CURVATURE OF SURFACES. 369

the roots of which are the principal radii of curvature at the point at
which the derivatives are taken.

Il. Ifz = /(x, y) be the equation to the surface, then in (1),
§ 252, we have Z, m, 2 subject to the two conditions
pltqm—n=o0, and ?+ m+nt=1,
which reduce to the single condition :
(z+ YE + apgim + (1 + gm? = 1.
Applying the general method for finding the maximum and
minimum values to (1), § 252,

ri + sm + A[(r + f°) + pom] = 0,
sl + im + AL pgl + (2 + Gm] = 0.
Multiply respectively by Zand mandadd. WhenceaA = — k/R.
Eliminating 7 and m from

[7R— (1+ P*)KV + (sk — pyx)m = 0,
(sR — pox +[R — (1+ ¢)K]m = 0,
there results the quadratic

[rR — (2+ pP)«] [ER — (14+ 9°)K] — (CR — pox)? = 0,
r
(ré— 8) KR — [rr +9") + 41 +f) — 2pgs]KR + KA = 0,

for finding the radii of principal curvature. In this equation
earth + ¢

The problem of finding the directions of the principal sections and
the magnitude of the principal radii of curvature is the same as that of
finding the direction and magnitude of the principal axes of a section
of the conicoid

Ae +t BPA Ce 4A 2hyz + 2Gx2 4 2Hxy =1,

made by the plane Zx + Jy + Nz = 0.

254. To Determine the Umbilics on a Surface.—At an umbilic
the radius of normal curvature is the same for all normal sections.
Consequently equation (1), § 251, for any three particular tangent
lines will furnish the conditions which must exist at an umbilic.

Through any umbilic pass three planes parallel to the coordinate
planes cutting the tangent plane there in three tangent lines whose
direction cosines are 1, m,, 0; /,, 0, 2,; 0, m,, 7,, respectively. Then
equating the corresponding values of «/R in (1), § 251,

Al?+ Bm? + 2m, = Al? + Cn? + 2Gln, = Bm + Cn?+ 2im,n,.

Also, since these three tangent lines are parallel to the tangent

plane, the equations
Li, + Mm, = Li,+ Nn, = Mm, 4+ Nn, = 0

0.

give ai Be

ee, Oe
f= 7g = eae
“37° APPLICATION TO SURFACES. (Cu. XXXTIL

and /,, m, have opposite signs. The same equations give like values
for 7,, #,, etc. On substitution we obtain the conditions which must
exist,at an umbilic,

AM*4+ BL?—2HIM  AN*+CL?—2GLN

 

; LD? 4 M2 ~ + N?
BN? + CM? — 2FMN
SNe NR

These two equations in x, y, 2, together with the equation to the
surface, give the points at which umbilics occur.

If the equation of the surface is (x,y) — 2 = 0, results are cor-
respondingly simplified and the conditions which must exist at an
umbilic are immediately obtained from the fact that «/2 is constant
for all values of /, m, 2, satisfying the identical equations

= = rl? + aslm + tm’,
t= (r+ Pf)? + rpg lm + (1 + 9?) m".
Whence results, from proportionality of the constants,
K ee tf (2)

RiP fe te

255. Equations (2), § 254, are very simply obtained by seeking
the point on the surface z = /(x,_y) at which the sphere

P(%,9, 2) = (x — a)? + (v¥— YP + — VP — fp? =0

osculates the surface z= /. The first and second partial derivatives

of z in @ are the same as those for fat the point of osculation. Dif-
ferentiating @ = o partially with respect to x and _y, we get

x —a+(z—y)p=9,
¥—B+(-—Y)V=9,
1+pf+(e—y)r=o0,
1+¢+(¢@—y)l=o0,

ba+(@—y)s=o.

 

aft 22 See
gg gt
Also, R= — (:—y)V1 +7? + @, since the direction secant of

the normal with the z-axis is — (1 + ?-++ g?)*-

256. Measure of Curvature of a Surface.—The measure of cur-
vature of a surface is an extension of the measure of curvature of a
curve in a plane, as follows:
ART. 256.] CURVATURE OF SURFACES. 371

The measure of entire curvature of a curve ina plane is the amount
of bending. Let P, and P, be two points on a
curve whose distances, measured along the curve,
from a fixed point are s, and s,. Let @, and @, be
the angles which the tangents at P|, P, make with
a fixed line in the plane of the curve. Then the
whole change of direction of the curve between P,
and P, is the angle ¢, — ¢,. This angle is also
the angle through which the normal has turned as a
point P passes from P, to P, along the curve.

This angle between the normals is called the entire curvature of
the curve for the portion P,P,. It can also be measured ona standard
circle of radius 7, as the angle between two radii parallel to the nor-
mals to the curve at P,, P,. If P,’P,’ be the subtended arc in the
standard circle (Fig. 145), the whole curvature of P,P, is proportional
to P,’P,’, or

 

Boge?

S
Py ra P, = gee
If the standard circle be taken with unit radius, the entire curvature
of P,P, is measured by the arc s,’ — s,’ on the unit circle.
The mean curvature, or average curvature, of P,P, is the entire cur-
vature divided by the length of the curve P, P,,

oy p, = 5, an

Sy 7 f,— 3,"
or, is the quotient of the corresponding arc on the unit circle divided
by the length of curve P,?,.

The specific curvature of a curve, or the measure of curvature of a
curve at a point /, is the limit of the mean curvature, as the length
of the arc converges to zero. It is therefore the derivative of @ with
respect tos. But since ds = Rdd, where & is the radius of cur-
vature of the curve at a point, we have for the specific curvature

dp _ 1
ds” R’

The curvature of a curve at a point is therefore properly measured
by the reciprocal of the radius of curvature.

To extend this to surfaces, we measure a solid or conical angle by
describing a sphere with the vertex of a cone as center and radius r.
Then the measure of the solid angle o is defined to be the area of the
surface cut out of the sphere by the cone, divided by the sguare on

the radius, or
A

R
The unit solid angle, called the s/eradian, is that solid angle which

Qo=
372 APPLICATION TO SURFACES. [Cu. XXXII.

cuts out an area A equal to the square on the radius. In particular,

if we take as a standard sphere one of unit radius, then ' :
: Q= Al

or, the area subtended is the measure of the solid angle.

Definition.—The entire curvature of any given portion of a curved’
surface is measured by the area enclosed on a sphere surface, of unit
radius, by a cone whose vertex is the center of the sphere and whose
generating lines are parallel to the normals to the surface at every
point of the boundary of the given portion of the surface.

Horograph.—The curve traced on the surface of a sphere of unit
radius by a line through the center moving so as to be always parallel
to a normal to a surface at the boundary of a given portion of the sur-
face is called the Aorograph of the given portion of the surface.

Mean or average curvature of any surface. ‘The mean or average
curvature of any portion of a surface is the entire curvature (area of the
horograph), divided by the area of the given portion of the surface.
If S be the area of the given portion and @ the entire curvature, the

mean curvature is a

S .

Specific Curvature of a surface, or curvature of a surface at a point.
The specific curvature of a surface at any point, or, as we briefly say,
the curvature of a surface at a point on the surface, is the limit of the
average curvature of a portion of the surface containing the point, as
the area of that portion converges too. In symbols, the curvature at
a point is ‘doo

as’

Gauss’ Theorem. The curvature of a surface at any point is equal
to the reciprocal of the product of the principal radii of curvature of
the surface at the point, or

d@ I
ads RR,

Let S be any portion of a sur-
face containing a point P.

Draw the principal normal sec-
tions PM= 4s,, PN = As,
AS=As,-ds,= Rig, R44,

4a= Ao,-4do, = 49,-49,.
Ao,, 4o, being the arcs of the
horograph corresponding to 4s,,
As,, on the surface.
CO’ _ 40 1
gh deed
In the limit
doo I

aS RR

 
ART. 256 .} CURVATURE OF. SURFACES. 373

EXERCISES.
1. Find the principal radii of curvature at the origin for the surface
Be = Ox? — Say '— By. Ans. yy — Py

2. A surface is formed by the revolution of a parabola about its directrix; show
that the principal radii of curvature at any point are in the constant ratio.1: 2.

3. Find the principal radii of curvature, at 2, y, 2, of the surface

 

: 2 2 ¥
Jee = ws SS. Ans. + sae
a a a
. 8
4. Show that at all points on the curve in which the planes z= + _ 3 cut

the hyperbolic paraboloid 22 = ax? — éy? the radii of principal curvature of the
latter surface are equal and opposite. This curve is also the locus of points at
which the right-line generators are at right angles.

5. Show from (6), § 248, that the mean curvature of all the normal sections of
a surface at a point is ;
1/1 I
i(a+z):

6. Show that at every point on the revolute generated by a catenary revolving
about its axis, the principal radii of curvature are equal and opposite.

_ 7. Show that at. every point on a sphere the specific curvature is constant

and positive. _ : - ; :

8. Show that at every point of a plane the specific curvature is constant and o.

9. Show that at every. point on the revolute generated by the tractrix revolving
about its asymptote, the specific curvature is constant and negative. This surface
.is,called the pseudo-sphere.. .

10. If the plane curve given by the equations

x/a = cos 6-+'log tan}6, ‘y/a = sin 6,

revolves about O., the surface generated has its’specific curvature constant.

i. If &,, XR, are the principal radii of curvature at any point of the ellipsoid
on the line of intersection with a given concentric sphere, prove that

: (A, 8)
‘a4 REX

12. Prove that the specific curvature at:any point of the elliptic or hyperbolic
paraboloid y?/é + 27/c = x varies as (~/z)', p being the perpendicular from the
origin on the tangent plane. : : ;

13. In the helicoid y= x tan (z/a) show that the principal radii of curvature,
at every point at the intersection, of the helicoid with a coaxial cylinder, are con-
stant and equal in magnitude, opposite in sign. ~ i

14. Prove that the-specific curvature at.every point of the elliptic paraboloid
22 = x*/a + y*/b, where it is cut by the cylinder x?/a? 4 y?/6? = 1, 4s (qab)—1.

15. Prove that the principal curvatures are equal and opposite’in the surface
x*(y + 2) + ayz = o where it is miet by’ the cone (x? + 6yz)yzi =. (y.— z)h

16. The principal radii of curvature at the points of the surface

oH

= const.

- ; ax? = 2x2 y%), ‘where x=y=%,
aregivenby aR 4.2 V3 aR — ga = 0.
374 APPLICATION TO SURFACES. (CH. XXXIIL.

17. Prove that the radius ot curvature of the surface «+ ym +. zm = am at
m—2
an umbilic is 3 2% a@/(m — 1).

18. Show that ~= 2% = 2 is an umbilic on the surface
a ¢
Bfa+y73/b + B/e = ke,
19. Show that x = y = z = (aéc)# is an umbilic on the surface xyz = adc and
the curvature there is }(adc)—t.

: aoe er ee See se
20. Find the umbilici on the ellipsoid 3 + pta=t
2g? — f2 2( 42 _ ¢2
Ans. The four real umbilics are x? = eae , 2a late
ae at —¢

21, At an ordinary point on a surface the locus of the centers of curvature of all
plane sections is a fixed surface, whose equation referred to the tangent plane as
z-plane and the principal planes as the x- and y-planes, is

ey
+4) (Ete) =e +9.
22. Show that an umbilicus on the surface

(x/ayt + (y JP + (2 Jot =x

. . I [x 4 I —t I /z -t
is given by (=) =e (3) => (=) é

23. If # = 0 is the equation of a conicoid, show that the tangent cone to the
surface drawn from the vertex a, §, y touches a surface along a plane curve which
is the intersection of # = o and the plane

oF OF aF
(4 - aa ty “Slag te =e + Fla, B, vy) = 0.

24. Find the quadratic equation for determining the principal radii of curva-
ture at any point of the surface

P(x) + WY) + X(2) = 9,

and find the condition that the principal curvatures may be equal and opposite.
25. Show that the cylinder
(a? 4 c2)b2x2 4. (6% 4 e*)a2y? = (a? + B)a70?
cuts the hyperboloid x?/a? + y?/é? — 2?/c? = 1 in a curve at each point of which
the principal curvatures of the hyperboloid are equal and opposite.

26. Show that the principal radii of curvature are equal and opposite at every
point in which the plane x = a cuts the surface

x(x? + yt + 28) = 2a(a? + y?),
27. In the surface in Ex. 24 show that the point which satisfies
_ bx) = Wy) = x"2)
is an umbilic.
28. Find the umbilici on the surface 22 = x?/a + y?/b,
An. x =0Qyru— V (ab — 8), z=4a—5), if a>sb
29. Show that z = /(x, y) is generated by a straight line if at all points
af OF ar \e
at a= (aa i)
This is also the condition that the inflexional tangents at each point of the sur-
face shall be coincident. ‘Such a surface is called a forse or developable surface.
CHAPTER XXXIV.
CURVES IN SPACE.

257. General Equations.—A curve in space is generally defined as
the intersection of two surfaces. A curve will in general have for its
equations

P(*,y, 2) = 0, (x,y, 2) = 0. (2)

If between these two equations we eliminate successively x, y, 2, we
obtain the projecting cylinders of the curve on the coordinate planes,
respectively,

#,(y, 2) =o, (4%, 2) =o, o,(*,¥) = 0.
Any two of these can be taken as the equations of the curve.

258. Acurve in space is also determined when the coordinates of
any point on the curve are given as functions of some fourth variable,

such as /,
= $4), y= 40, 2 =X(O. (2)
The elimination of ¢ between these equations two and two give the
projecting cylinders of the curve.

259. Equations of the Tangent to a Curve at a Point.—If the
equations of the line are (1), the equations of the tangent line to (1) at
x, y, 2 are the equations to the tangent planes to ¢, = 0, ¢, = 9,
taken See or

(X = 4) 28 4 (Py) 4 (2 - ae
r (a)

(x — x) 4 (F352 Bt (Z— 2) Fe = 0. |

Since the tangent line is a ueals to the normals to these
planes, the direction cosines /, m, 7 of the tangent line are given by

 

Z m n I

MN, — UN, ~ NL,— NL, ~ L,M,—LjM.~ ke’

 

 

where
= (ALN, — M,N,)* + (NL, — N,L,)* + (LM, — LM)
L,, M,, ™, being the first partial Seats of @, at x, ¥, 2, and
similarly i M,, 7, are those of ¢,.
375
376 APPLICATION TO SURFACES, (CH. XXXIV.

260. Ifs is the length of a curve measured from a fixed point
to x, y, 2, then the direction cosines of the tangent to the curve at

x,y, 2 are
{= = m = ay = ge
ds’ ds’ ds’
and the equations of the tangent are
X—-x F- y Z—-2

ax ie dy mo ies (2)
ds. ds ds”
If the equations to the curve be given by (2), § 258, then

 

 

d. dt
-- = (4) Ze etes and the equations (2) become

X—x F-—y _Z-2
PM) #4) xX”
In general the equations to the tangent are
A— x YF, Z-2
“ae ee ae ie
without specifying the independent variable.

 

(3)

261. The Equation to the Normal Plane to a Curve at x, y, 2 is
ax dy dz
[aa ep (haere 2) = © (2)

the normal plane being defined as the plane which is perpendicular
to the tangent at the point of contact.
Regardless of the independent variable, (1) becomes

(X — x)dx 4 (F — y)dy + (Z— 2)d2 = 0. (2)

EXAMPLES.

1, Find the tangent line to the central plane section of an ellipsoid.
The equations of the curve are

x? y gt '
ay wa Ss Ax+ By + Cz =0,
The equations of the tangent at x, y, z are
X—x VY-y _ 2-28

y Be Pete ae y°
C583 Agee Ba-AR

 

2. Trace the curve (the helix)
x=acost, y=asint, z= ot.
Show that the tangent makes a constant angle with the x, y plane, and that the

¢urve is a line drawn on a circular cylinder of revolution cutting all the elements.
at a constant angle. :
Art, 262,] CURVES IN SPACE. 377

3. Find the highest and lowest points on the curve of ‘intersection of the
surfaces ; /

22 = ax? by, Ax+By+Cz+D=0,
from the fact that at these points the tangent to the curve must be horizontal.

4. Show that at every point of a line of steepest slope on any surface / = 0
we must have

ar OF
a = pe oO.

J

5. Show that the lines of steepest slope on the right conoid « =y/(z) are cut
out by the cylinders x? + y? = 7?, r being an arbitrary radius.

262. Osculating Plane.—IfP, Q, R be three points on a curve,.
these three points determine a plane. The limiting position of this
plane when P, Q, & converge to one point as a limit is called the
osculating plane of the curve at that point.

The coordinates x, y, 2 of any point on a curve are functions of the

length, s, of the curve measured from some fixed point to x, y, 2.
Therefore, if s, be the length to x,, ¥,, 2, ©

1?

 

ax Px ax
t= et (5, iam N\A tH =a) aa +e ce 8) ado’
where o is the length to some point between x, y, 2 and +,, 4, 2,
Put ds=s,—s, x = Dx, x’ = Dix, etc., then
w= at Osx’ 4 dds” + 1659. ael!,
Let P, Q, FR bex, ¥, 23 %, 15 213 %yyHo) 2 Then
H=Haet Osx, 9, = yt Osi, 2, = at Os-35. (1)
x, = x + hdsx’ 4 LROS*. xg,
2 J + kds+ + AROS? yf},
z, = 2+ hose 2’ + UROS*. 2f,

The equation to the plane through P can be written
A(X — 2) + BY — 9) + CZ — 2) = 0. (3)
If this passes through Q and &, then
A(x, — x) + Bly, —») + C2, —2z)= et (4)
A(x, — 2) + BU, —9) + Cl, — 2) = 0.
Substitute the values of the coordinates from (1) and (2) in (4).
Divide by 6s, 05°, and let ds(=)o.
ing Ax’ + By + Cy =0,
Ax” + By” + C2” =0., (5)

Eliminating 4, B, C between (3) and (5), we have the equation to
the osculating plane at x, y, 2,

(2)

HU Ul

X—x«, F—y, ae (6)
ao, ly 2”
wt YG 2! |

* J
378 APPLICATION TO SURFACES. [CH. XXXIV.

Or, regardless of the independent variable,

X—x«, F-y, Z—2\=0o. (7)
ax dy dz
ax d’y az

263. To Find the Condition that a Curve may be a Plane
Curve.—lIf acurve is a plane curve, the coordinates of any point must
satisfy a linear relation

Ax+hy+Cz2+ D=o0,
where 4, B, C, Dare constants. Differentiating,
Adx + Bly + Caz =,
Ad*x + Bd*y + Ca*z = 0,
Adx + Bday +. Cd®z = 0.
Eliminating A, B, C, we have the condition

 

dx, dy, ads|=o,
ax, dy, dz
ax, dy, dz

 

 

which must be satisfied at all points on the curve.

264. Equations of the Principal Normal.—The principal normal
to a curve at a point is the intersection of the osculating plane and the
normal plane at the point.

Let 4, m, ~ be the direction cosines of the principal normal at
x,y, 8 Then, since this line lies in the normal and osculating planes,

f pt
iy B x
+m +x
yy" gl’ ae eyyti

dx’ + my’ + nz’ = 0.
These conditions are satisfied by 7= x”, m=y", n= 2", since

a! x’

gl! al”

/ /

Z

=o,

 

 

 

 

 

 

IT AIT

 

 

 

 

 

 

vt
yg! Z tt vy sz =0.
ae is 3! ty” 2”! i +2” ry = |x’ y’ 2’
xe’? y!" 3
Also differentiating x’? + y/?+ 2/2 = 1,
a! + yy” + 2/2" =o.
Therefore the equations of the principal normal are
X—x Fry Z—2
ge ge ar (1)
X—x F-y_ Z-2
ee Pa ay Pa (?)

265. The Binormal.—The binormal to a curve at a point is the
straight line perpendicular to the osculating plane at the point.
ART. 267.] CURVES IN SPACE. 379

Its equations are therefore, from (6), § 262,
X—x Y—y Z—2
yal! = yy” g! = ax” ce a’ x! = x’ y"” e= xy”

 

(4)

Dividing through by ds%, the equations can be written without
specifying the independent variable.

266. The Circle of Curvature.—The circle of curvature at a given
point of a space curve is the limiting position of the circle passing
through three points on the curve when the three points converge to
the given point.

Clearly, the circle of curvature lies in the osculating plane and is
the osculating circle of the curve.

To find the radius of curvature. Let a, 6, y be the coordinates
of the center, and g the radius of the circle of curvature at x, y, 2.

Then
te ay ele = 6) a yt =
Let x,y, z vary on the circle. Differentiate twice with respect to s.
Then

(x — axe” + (y — By" t(e-—y)"+xeP+tyP+2%=0.
But x’? + yp’? + 2/%= 1, Also, the line through x, y, z and

a, 8, y is the principal normal, whose direction cosines, by (1),
§ 264, are

axl”

er
with similar values for mand”. Since
x—-a=lh, v—fB=mp, z—-y=n9,
I
oe Bae ae pe

The center of the circle is a = x — Jp, etc.

Z

267. The direction cosines of the binormal are
Z= p(y’2"” — ey"), m= p(2'x"” — x'2"), n= p(x'y’ —y'x").
For, by (4), § 265,
Z m n
y's” =. gy Ta ax! aa, x'g!! = xy” — yx" °
Also differentiating x’? + y’? + 2’% = 1,
xe A oyly” + 2/2” = 0,
The sum of the squares of the denominators in (1) is
(x"? + y/? + gz! (a7 2 4 yy” 2: + gl’ *\—(a'x” + yy” + 2’2"’) = 1/p.

Hence the results stated.

 

(1)
380 APPLICATION TO SURFACES. (CH. XXXIV.

268. Tortuosity.. Measure of Twist...
Definition,—The measure of torsion or twist of a space curve is the
rate per unit length of-curve at which the osculating plane. turns
around the tangent to the curve, as the point of contact moves along
‘the curve.
If the osculating plane turns Bitoagh the angle 47 as the point of
contact P moves to Q through the are 4s, the measure of torsion at P is
. dr Ar ,
as As’
when 4s(=)o. The number o = D,s is sometimes called the radius
“of torsion.
Let 7, ™,, 7,; /,, m,, m,, be the. direction cosines of two planes
including an angle 6. Then
sin?@ = (mn, — n,m,)* + (nl, — 1n,)? + (4m, — m,z,)%
Let 7, m, ” be the direction cosines of the osculating plane at P,
and/ + Al,m+ Am, 2 + An those at Q.
’ Let 47 be the angle between these planes. Then
sin’ dr = (mdn — nAm)? + (ndl — iaay + ¢4m — mal).
Divide by 4s and write
sint?4r__ sin’ dr /Ar\*
Ae Ar? \As}*
Let 4s(=) 0. Then, in the limit,

dr\2 dn dm\2 [_ dl_dn\® [jdm dl\2
(z)'= (ma 9G) + (rae) + 02-92)" ©
= dl dm
} ya 2 2 ‘ . i m— es
Since P4mtrnt=i,° Bag oe tibet Te
Square this last equation and subtract, from (1).

dx dl dm\? (dn\2
a) alt) + Gy
269. The measure of torsion can be i a in’ another form, as
follows, 7

Let 2, m, # ‘te the denon cosines of the Pieniel and
A =y' 2" — 2’ y", etc., asin § 267. Then

= 0.

pine fo. mn =o
Li me —  @
Whence ; P4M+N = 1/p. : (2)
Since the binormal is perpendicular to the tangent 4 and principal
normal, ,
ix’ +-my’ + nz’ =0, (3)

x! + my” + nz” = 0. (4)
ART. 270.] CURVES IN SPACE. 381

Differentiating (3) and using (4),

 

Vx! + my’ +n's' =0. (5)
Differentiating, P44 mt n= 1. (6)
i! +. mm’ + nn’ = 0. (7)
From (5) and (7) we get
Z’ m! n’
ma — ny nx! —Ta’ Ty — max”? (8)

and each of these is equal to
Ux"! m'y"" n'z!!
max” — ny'x"  axy” — lay” Ty's" — mx”
ish L'x! + my" + n'g!!
~ TE + mM + nN (9)

 

Differentiating (4),
Lx!’ Ae m'y" An’! Ada!” my!” 4 ns!” = 0,
Therefore (8) is equal to
dx!” + my" + ng!" = xn"'L + y/"M + ZN
L+mM@t+tnN © LPP4+ M*?+ Ne

Remembering that /, m, 2; x’, y’, 2’ are the direction cosines of two
lines at right angles,

(m2! — ny’)? + (nx' — be’ + (b! — mx’)? = sin? da = 1,
Therefore, by (2), § 268, and (8),
adt\? ("LA MH a! N\2
(@)= (aera)
or

/ / ,

1_d@r_sésy ae
a Pat yt ’ (10) |
att

J fll

 

RRR
Ree

z
gl!
2

 

by (2), and the determinant form of
e'L + "MT + ZN.
270. Spherical Curvature.—Through any four points on a space
curve can be passed one determinate sphere. The limit to which
converges this sphere when the four points converge to one as a limit

is called the osculating sphere, or sphere of curvature.
Differentiating the equation of the sphere,

eae? a Be pea yy eae
@- ae +0=2¥ PeH nf! =o,
(4 — ax” + (9 — Bly” +(e — ype” = —1,
(=e pO BY Fe Vie = ©
382 APPLICATION TO SURFACES. [Cu. XXXIV.

Eliminating between the last three equations,

 

 

, , é t
Zz . |e 2 > QB alot! Tay.
(x = a) = Z| tl ae Sn 2” | op (92 — ay )i
JIT yf ght
pen

 

 

y— p = op*(2'x/" — "x x’); z— y= = op (xly'" =: xy"),
Squaring and adding,
R = o* pt (aly — — yl") + (92 gl! — ay"? + (2'x all! — x'2!”)"],
Clearly the circle of curvature lies on the sphere of curvature.
Let P, Q, &, / be four points on a curve and in the same neighbor-
hood, # and 9 the radii of spherical and circular curvature.
Then, C being the center of the circle through P, Q, &, and S
S that of the sphere through
P, Q, R, J, we have directly
from the figure,

_ 4p 2— p24 (4P\*
sc=“2, Reapt+-(SP)"

eoos(2)

 

af
= AZ 2,
= P+ oF .
_ dp _ _ ap
Sh pe

Fic. 147.

271. The expressions for the value of the radius of curvature and
measure of torsion in § 266, and (10), $269, have been worked out
with respect to s, the curve length, as the independent variable. These
can be written in differentials, regardless of whatever variable be
taken as the independent variable.

Represent by
dx dy dz
| ax dy @&z
the sum of the squares of the three determinants
(dy Pz — dzd*y)* + (dzd’x — dx d’z) + (dx Py — dy dx)*.
Then, regardless of the independent variable employed,
GA + Pte oe (x)
Al dx dy dz
ee | ax dy Pz
dx dy dz
ax dy az
_| ex dy Ba)

oe oe | dy dz
x ay Pe |i

 

 

 

 

 

 

 

[ (2)

 
ART. 271.] CURVES IN SPACE. 383

(1) comes immediately from § 267, and (2) from putting the value
of p* from (1), § 271 in (10), § 269.

EXERCISES.
1. Show that in a plane curve the torsion is 0.

2. The equations of the tangent at x, y, z to the curve whose equations are
ax? +. by? + cz? = 1, bx? + cy? + az? = 1, are
aX — 2x) W¥—y) _ AZ — 2)
ab—2 ” be —a ~ ac—

 

3. The equations of a line are
x? ty? + 52 = 4a? and x? + 2? — 2ax,
Show that the equations of the tangent line and normal plane are

ByUtestyh £-2=(-2 (2-2)

Cope x Z y
4. The equation of the normal plane to the intersection of
w/a - y/b+ 2/e=1 and 27+ y? 4+ 2? = ad?
3 Xx Y Z
is pee ate aot ae eee.
5. Show that the curve 2(4 + z)(4 —a)= 4, o y+ 2)(y — a) = a, isa
plane curve.

6. If the osculating plane at every point of a curve pass through a fixed point,
prove that the curve will be plane.

7. Prove that the surface +* + y* + 24 = 324 cuts the sphere

xP y2 tg? = gi
in great circles.

8. Show that the equations of the tangent to the curve
y= ax — x, 2 = at — ax,

2y
a — 2x

 

are A= LS (Y-y)=—=(Z—»),

9. Find the osculating plane at any point of the curve
x=acot y=dbsint, z= ct.
Ans. (Xy — Yx) + ab(Z — 2) = 0.
10. Find the radius of circular curvature at any point of
afht+v/kRa=w 2P+2%= at
Ans. ener aeaety .
arth? fh? 1B
11. Show that the curves of greatest slope to xOy on the surfaces xyz = a? and
cz = xy are the lines in which these surfaces are cut by the cylinder x? — y? = const.

12. Find the osculating plane at any point of the curve
x=acos6+dsiné, y=asin6+4cosé, z=c¢sin 26.
384 APPLICATION TO SURFACES. [Cu. XXXIV.

13. Find the principal normal at any point of
at yX= a aza x? ~ yy,
Hint. Express x, y in terms of z as the independent variable.

14. Given the ketix x = acos6, y= asin6@, 2 = 46, show that

(1). The tangent makes a constant angle with the xy plane.
(2). Find the normal and osculating planes, principal normal.
(3). Locus of principal normals.
(4). Coordinates of center and radius of curvature.
(5). Radius of torsion.
Ans. (2). aX sin @ —aY cos § — 4% — 8) = 9,

6X sin @ — bY cos 6+ a(z— 68) = 0.
Y = tan =
(3). = = tan a

(4). p = a(t + 4/a?).
(5). o = (a? + 8/6,

15. Show that # = 0, Fy == oare the equations of the line of contact of the
vertical enveloping cylinder of * = 0, and that the horizontal projection of this
line is the envelope of the horizontal projections of parallel plane sections of 7= o.

16. Show that the equations of the level lines and lines of steepest slope on the
surface * = 0 are

F=0, Fydx+ Fdy=o and F=0, fydy— Fydx =o
respectively, and that they cross each other at right angles.

17. Find the lines of steepest slope on the surfaces

ax* + by? + c22> = 1 and 2 = ax? + by,

18. A line of constant slope on a surface is called a Loxodrone. Find the loxo-
drone on the cone x? + y? = A(z — c). Show that its horizontal projection is a
logarithmic spiral.

19. Find the loxodrone on the sphere x? + y? + 2? = a’.

20. A dine of curvature on a surface is a line at every point of which the tangent
to the line lies in a principal plane of the surface. Show that through every
ordinary point on a surface pass two lines of curvature at right angles.

21. A geodesic line on a surface is a line whose osculating plane at any point
contains the normal to the surface at that point. Use Meunier’s Theorem to show
that between two arbitrarily near points on the surface the geodesic is the shortest
line that can be drawn on the surface. Show that at every point on a geodesic on
the surface # = 0, we have

Ff dx = Fj/d*y = Fj/d*z.
CHAPTER XXXV.

ENVELOPES OF SURFACES.

272. Envelope of a Surface-Family having One Variable
Parameter.—When F(-, 7, 2) = 0 is the equation of a surface con-
taining an arbitrary parameter a, we can indicate the presence of this
arbitrary parameter w by writing the equation

L(x, 4, 2, @) = 0. (1)
The position of the surface (1) depends on the value assigned to a.
By assigning a continuous series of values to a@ we have asingly infinite
family of surfaces whose equation is (1).
If we assign to a@ a particular value a,, we have another position of
the surface (1) whose equation is

 

F(x, y, 2, 04) = 0. (2)
The two surfaces (1) and (2) will in general intersect in a curve.
When a,(=)qa@ the surface (2) converges to coincidence with the sur-
face (1), and their line of intersection may converge to a definite posi-
tion on (1). At any point on the intersection of (1) and (2) the
values of x, y, gare the same in both equations. By the law of the
mean,
L(x, I; z, a,) = L(x, 9, g, a) + (a, >= alu (xd; &, a’),
a’ being a number between @ and ay.
At any point of intersection of (1) and (z)
E(x, 9, 2, a&) = F(x, y, 2, a) = 0.
Therefore at any such point we have
F(x, J, 5, a") = 0. (3)

If, when a,(=)a, the line of intersection of (1) and (2) converges
to a definite position on (1), then the coordinates of all points on this
line must satisfy, by (3), the equation

0
Oa (49; a, a) QO, (4)

and the surface (4) passes through the limiting position of (1) and (2).
If from equations (1) and (4), i.e.,
L(x, I; a, a) =O, FUX DI; 8, a) = 09, (5)
385
386 APPLICATION TO SURFACES. [CH. XXXV.

a@ be eliminated, the result is an equation @(x, y, z) = 0, which
is the surface generated by the line whose equations are (5), or @= 0
is the locus of the ultimate intersections of consecutive surfaces of the
family (1). This locus is called the envelope of the family (1). The
line whose equations are (5) is called the characteristic of the envelope.

273. Each Member of a Family of One Parameter is Tangent
to the Envelope at all Points of the Characteristic.—The parameter
a being assigned any constant value, the tangent plane to

 

f(x,y, 2, @) = 0, at x,y, 2, is

ar

Ox

But in (x, v, 2, @) = 0, as x, y, 2 vary along the envelope, @ also
varies, and the equation to the tangent to the envelope is

oF oF
Oe ag dg (2)

ar or oF or
a ae ee ae (2)

Since at any point x, y,z common to the surface “= 0 and the
envelope, that is all along the characteristic, we have #{—= 0, the
planes (1) and (2) coincide.

EXAMPLES.

1. Show that the envelope of a family of planes having a single parameter is a
torse (developable surface).

Let z= x(a) + yW(a) + x(a).

dz dz iu

ag = PMs oe = U(a); ve xPy + yb. + Xo = 0.
Also,
az a2

ppp iy OM ez da ee eee da
7 = P(A) a age ay we ae ax’
Hence vt — 5s? = 0. See Ex. 29, § 256.

a+ty

2. Envelop

 

+ 2a 2.
Ans. Hyperbolic cylinder, xz + yz = 1.

3. Envelop x+y — 2az = a’.
Ans. Parabolic cylinder, x + y + 22 = 0.

4, Generally if @, ~, x are linear functions of x, y, 2, then the envelope of the

plane
oe +2patyzyo=o
is y? = py, a cone or cylinder having @ = 0, y =O as tangent planes, and
w = Oisa plane through the lines of contact.
5. Find the envelope of the family of spheres whose centers lie on the parabola
x? + 4ay = 0, 2=0, and which pass through the origin.
Ans. x? 4 y® 4+ 2? = 20x/y,
ART. 274.] ENVELOPES OF SURFACES. 387

6. Find the envelope of a plane which forms with the coordinate planes a
tetrahedron of constant volume.
Ans. xyz = const.
7. Find the envelope of a plane such that the sum of the squares of its
intercepts on the axes is constant.

Ans. xt 4 yb 4 zt = const.

274. Envelope of a Surface-Family with Two Variable Param-
eters.

If F(a, 8) = Fle, 9, 2, a, B) = 0 (1)
is a surface of the family, then
F(a, B;) = E(x, 9; z, a> 1) =o0o (2)

is a second surface of the family.
At any point x, y, z where (1) and (2) meet,

Flay f,) = Flay B) + (a — 55+ (Bi — Daw (3)

where a’ is between a, and a, fi’ between f, and f.
In virtue of (1) and (2), (3) gives

(a, = 4) + (B= A 5g = ©. (4)

This is the equation of a surface passing through the intersection
of (1) and (2). But for any fixed values x, y, 2, a, @ satisfying (1)
and (2) there are an indefinite number of surfaces (4) obtained by
varying a,, (,, all of which cut (1) in lines passing through x, y, z
Consequently there are of these surfaces (4) two particular surfaces,

OF ar
a age =
which cut (1) in lines passing through x, y, 2

If now the point x, y, z has a determinate limit when a,(=)a,
£,(=) 8, then the three surfaces

f(a, B) =0, Fi(a, 8) = 0, F(a, Bf) = 0,
pass through and determine that point.
These surfaces (5) intersect, in general, in a discrete set of points.
If, however, we eliminate between them a@ and #, we obtain the equa-

tion to the locus of intersections. This locus is a surface called the
envelope of the family (1).

275. The Envelope of the Family /(x, y,2, a, 6) = o is Tan-
gent to Each Member of the Family.—The tangent plane to any
member of the family is

on de + ay Ut oe OF ie =o. (1)
388 APPLICATION TO SURFACES, [Cu. XXXV.

As the point «, y, 2 moves on the envelope, a and # vary. The
plane tangent to the aoe is

joist Fa + 5 Fate + Gada + SUB =o. (2)

At a point x, y, zg common to the envelope and one of the surfaces,
we have 7, =, #g = 0, and therefore the planes (1) and (2)
coincide, Since this point is the intersection of the line whose
equations are /, = 0, #g = o with the surface # = o, the envelope
is tangent to the surface at a point, and not along a line.

276. Use of Arbitrary Multipliers.—If 7(x, y, z, a, 6) = 0,
where a, fi are two arbitrary parameters connected by the relation
g(a, P) = o, then # = o is a family of surfaces depending on a
single variable parameter. The equation of the envelope is found by
the elimination of a, 6, da, dB between

F=0, ¢=06, FidatFldp=o, g¢idat odB=o.

This is best effected, as in the corresponding problems of maximum
and minimum, by the use of arbitrary multipliers. Thus the equation
of the envelope is the result obtained by eliminating a, #, A from

F=0, g=0, MtAgi=o, y+ Ags=o.

The family of surfaces, represented by “= 0, containing x
parameters which are connected by 2 — r or ” — 2 equations is
equivalent to a family containing one or two independent parameters
respectively. Such a family, in general, has an envelope. The
problem of finding the envelope is generally best solved by intro-
ducing arbitrary multipliers to assist the eliminations.

If more than two independent variable parameters are involved,
there can be no envelope. For in this case we obtain more than
three equations for determining the limiting position of the intersec-
tion of one surface with a neighboring surface. From these three
equations +, y, could be eliminated, and a relation between the
parameters obtained, which is contrary to the hypothesis that they
are independent.

In general, if #' = 0 contains » arbitrary parameters a,, . . . , Qn
connected by the #— 1 equations of condition @,=0,...,
Qn—1 = ©, the equation of the envelope is found by eliminating the
2m — 1 numbers @,,..-, @,A gig between the 2”
equations

 

19 nl?

FR o, Pi =% +1 oy Pur =
F"(a,) ae A, ,'(,) tee ef An Pn—s (@) = 0,

. . . . . . . . . . . . . .

F'n) + AD (An) How ee An Pri(An) = 0
ART. 276.] ENVELOPES OF SURFACES. 389

EXERCISES.
1. Find the envelope of (9 being the variable parameter)
xsin@ — ycos@ = a8 — cz.
a + VF pea a a+yPtyoet
Ans. x sin Sg ee xt fy? =a.

2. Find the envelope of a sphere of constant radius whose center lies on a circle
in the xy-plane.
dns. Ifx* + y? = is the circle, and the sphere has radius a, the envelope is

the torus wt y? = [e+ (a? — 22)
. ee ee ee
3. Find the envelope of the ellipsoids ai + B + aah wherea +6+c¢=4

Ans. xt yy zt = 2,
4. Find the envelope of the ellipsoids in Ex. 3 when they have a constant
volume.
5. Find the envelope of the spheres whose centers are on the x-axis and whose
radii are proportional to the distance of the center from the origin.
Ans. y® 4 2? = max? + yy? 4 2%),
6. Find the envelope of the plane ax + fy + yz = 1 when the rectangle
under the perpendiculars from the points (@, 0, 0) and (— «, 0, 0)on the plane is
equal to 2.

 

2 2 2
Ans. a, £ positive. ae SS =i;
1

7. Find the envelope of the planes — +2 + = = 1 when a” + J". ct = fe,

n n

n n
Ans, itt 4 yttt oath — ports,

8. Spheres are described having their centers on < = 2 = — , and their radii

proportional to the square root of the distance of the centers from the origin ; show
that the equation of the envelope is (/, m, 2 being direction cosines)
xt yt ot = (de + my + nz + 6). ¢ = const.

9. Envelop the family of spheres having for diameters a series of parallel chords
of an ellipsoid. fi

10. If A(a~) = 0 is the equation of a family of surfaces containing a single
arbitrary parameter a, then the equations of the characteristic line on the envelope
are Fia~) = 0, F’(~) = 0. As e& varies this line moves on the envelope; it will in
general have an enveloping line onthat surface. The envelope of the characteristic
is called the edge of the envelope. Show that the equations of the edge of the enve-
lope are obtained by eliminating a between

Fa) =0, F(a) = 0, (a) = 0.
11, Find the equations of the edge of the envelope of the plane
xsin@ — ycos@ = af — cz.

Ans. xttyt aa, y= xtan<

12. Envelop a series of planes passing through the center of an ellipsoid and
cutting it in sections of constant area.
390 APPLICATION TO SURFACES. (Cu. XXXV.

Let /x-+ my + nz =0 be the plane; the parameters are connected by
24m tnt=1, Pa + mo? ie pict = dg.

Ans. a—F oS Soo 7 =O.

13. Spheres are described on chords of the circle «? + y? = 2ax, z=0 which
pass through the origin, as diameters, show that they are enveloped by

(2? + 98} oF — ax) = aa + 9%)

14. Show that the envelope of planes cutting off a constant volume from the
cone ax? + dy? + cz? = o is a hyperboloid of which the cone is the asymptote.

15. Find the envelope oo ee lx + my ae = d, when P+mw?tn? =,

qr ae ta OF

Ans. Fresnel’s Wave-surface,
azz by? c2g2
rpypp eo mr eye Be als P+yPte—A
16. Find the envelope of a plane passing through the origin, having its direc-

tion cosines proportional to the coordinates of a point on the line in which intersect
the sphere and cone

et ter rt, x?/a?+ 77/0? + 22/2 =o.

17. Find the envelope of a plane which moves in such a manner that the sum of
the squares of its distances from the corners of a tetrahedron is constant.

18. Show that the envelope of a plane, the sum of whose distances from fixed
points in space is equal to the constant 4, is a sphere whose center is the centroid
of the fixed points and whose radius is one wth of &.

19. Show that the envelope of a plane, the sum of the squares of whose distances
from # fixed points in space is constant, is a conicoid, Find the equation of the
envelope.

20. If right lines radiating from a point be reflected from a given surface, the
envelope of the reflected rays is called the caustic by reflexion,

_ Show that the caustic by reflexion of the sphere 2? + y? + 2? = 7%, the radiant
point being 4, 0, 0, is

[4220 2 — 7p? + 2hx + A?) 8 = 27h y2 + 27)(9? = hy,
in which p? = x? + y? + 2%

 
PART VII.

INTEGRATION FOR MORE THAN ONE VARIABLE.
MULTIPLE INTEGRALS.

CHAPTER XXXVI.
DIFFERENTIATION AND INTEGRATION OF INTEGRALS.

277. Differentiation under the Integral Sign. Indefinite In-
tegral.—Let /(x, y) be a function of two independent variables x, y.
Let

F(x, 9) = i Nx, vax,

the integration being performed for y constant. This integral is a
function of y as well as of x. On differentiating with respect to x,

 

 

or
Ox =x, J):
Again, differentiating this with respect to _y,
OF WG)
ay dx oy
But
CF oF _ OF
axdy dyax oy’
or af
oe da iy =a
Consequently
: OF _ SS. be
ay ~ J

or

a [rx Wee Li nd) 9

Therefore, to differentiate with respect to y the integral taken with
respect to x of a function of two independent variables x, y, differ-
entiate the function under the integral sign.

391
392 INTEGRATION FOR MORE THAN ONE VARIABLE. [CuH. XXXVI.
In like manner we have

0” = ane
x [te ydx = | ya

This process is useful in finding new integrals, from a known inte-
gral, of a function containing an arbitrary parameter.

EXAMPLES.
1. We have the known integral

eax
essa =—.
a

Differentiating with respect to a, we find

frosts =< (x-2).
a a

And generally, differentiating 2 times,

ff srersde = e7x (« +5) i G) .

‘ ‘ cos ax
2. Since foo ax dx= —

 

’
a

asinax | cosax

 

[x 008 ax dx =

(a + bxyett
(4+ 1)d

bx — a) bx\e

fx + bx)"dx = Aer Be aa" ee zl s

watt

a+I1
+
flog rar = (log « - 5).

278. Differentiation of Definite Integrals when the Limits are
Constants.

Let “= [fe y)dx,
where a and 4 are independent of x. Then the result of § 277 holds

as before, and
Ou b af
aah ee

On account of the importance of this an independent proof is
added. Let 4u denote the change in w due to the change Jy in y.
Then, the limits remaining the same,

Au = f'[/a9 + 49) — Ax, ») ex.
Au [fx y+ 4”) — (9)
Pe ge

a?

3. From f (a 4 bx)"de = show that

show that

 

4. From i xtdx =

 
ART. 279.] DIFFERENTIATION OF INTEGRALS. 393

Hence, when 4y(=)o, we have

Ou = oof
af ee

au ’ ie

EXAMPLES.
1. If f gta ee
0 a

be differentiated # times with respect to a, we get

ee nt
xNe—axdx = —-_,
0 aqntt

and, generally,

 

 

mo dx a

2. From f eS
0 (+4) 2at
_ 1:3+5-++-(2%—1) 2
ip (x? + ajttt 2-4-6... 20 aattt
The value of a definite integral can frequently be found by this method. Thus:
1(xa — 1)
3. Let t= f ee ee
“log x
du x2 log x

Then aol “orn =a poles a

 

aos log (2 + 1),

no constant being added since # = 0 when a = o.
4, Find We log (1 -L a cos 6)d8.
Ans. 7 log (1 + ar)

279. Integration under the Integral Sign.
I, Indefinite Integral.

Let F(x, 9) =f Ax, dx
Then will
[[fasoax |= f[f As xe lex
Let v= i I(x, y)dy.
Then = /(*, 4):

Also, si ax _se ax = [ x, 9)dx = F(x, y).
fe ax = f F(x, 9a,

or f{fponghar=f} fresnel a
394 INTEGRATION FOR MORE THAN ONE VARIABLE. (Cu. XXXVI

Hence the order in which the integration is performed is indif-
ferent. This shows that in indefinite integration when we integrate
a function of two independent variables, first with respect to one vari-
able and then with respect to the other, the result is the same when
the order of integration is reversed. This being the case, we can
represent the result of the two integrations by either of the compact

symbols
[ffeas ff fara.

As in differentiation, the operation is to be performed first with
respect to the variable whose differential is written nearest the func-
tion, or integral sign.

II. The same theorem is true for the definite double integral of a
function of two independent variables when the limits are constants,

Let
[Ae ax = Xx”, [Ax ry = P95
f [Arex ay =f [Ae ray ax = F(x, 9).
Then
[Fee = Xe 9) — X49),

fh dy = P(x, A) a F(%,.)3

os [eo sax ee F(X ys In) ive M(x, +;) m F(%) I) + F(x, I),
i “4 [Cro tdi = Flex, 9) — Flas d4) — Fp I) + Flap I)

The last two values are the same. Hence
"9 x4 xq 9

ax { dy = ' ay i ax,

I i A 3 i ie ;

f . f ic s)dx dy = f ig T(x, y)dy dx.

The integral sign with its appropriate limits and the corresponding
differential are written in the same relative position with respect to
the function.

or

EXAMPLES.
1. [owe = 2; Hence
a a

f [ose ax ={"2 = log =,
0 Jao Boo %

lya@y—-I _ gar
[Feces
0 log x %
ArT. 280.] DIFFERENTIATION OF INTEGRALS. 395
Put x =e,

20 g—42__ p42 a
ove i, ——__—. dz = log.
0 @ %

co) - }
mip ees = .
2. f e—**sin bx eS ae
f foerersin bx dadx = hep ee)
0 ao a @ + 6
00 o — Ls gm

or f din ae aS IF bx dx = tan—-t fa tan—-! 20,
0 x. & &

If as=0, a, =, then

i. sin dx meee
0

 

x
3. Evaluate f eT ae,
0
Put = i e7* dx.
0
ots f cP a dx =k
0
[re e7 Or +2?) a2,
and adx = ke
~ f° [re en P0429 da dx = ie dg = k,
Als, [Coretta oats
—-I Ge —_ ok,
and +f" Spa titan | =jir=F.
0
ae [Cora =3 7x,
0
ae — a
and f dx = oF Vn.

This gives the area of the probability curve.

280. If F(x, y, 2) is a function of three independent variables, the
same rules as for a function of two independent variables govern the

triple integral
f f / F dx dy dz.

Examples of double and triple integrals will be given in the next
chapter.
CHAPTER XXXVI.

APPLICATIONS OF DOUBLE AND TRIPLE INTEGRALS.

PLANE AREAS.

DouBLE INTEGRATION.

281. Rectangular Formule.—lIf x,y are the rectangular coor-
dinates of a point in a plane xOy, then

dw = 4x 4 ly = ax ay
is an element of area, being the area of the rectangle whose sides are

M

Fic. 148.

 

Ax and dy.

Let the entire plane xOy be
divided into rectangular spaces
by parallels to Ox and Oy, of
which 4x Ay is a type. The
area of any closed boundary
drawn in the plane is the limit
of the sum ofall the entire rectan-
gular elements of type 4y 4x
included in the boundary, when
for each rectangle 4x(=)o,
4y(=)o. For the area within
the closed boundary A is equal to

A= Ay Ax
plus the sum of the fractional

rectangles which are cut by the
boundary. This latter sum can

be shown to be less than the length of the boundary multiplied by the
diagonal of the greatest elementary rectangle, and therefore has the

limit zero. Hence

A= £2 Ay Ax,

taken throughout the enclosed region, when 4x(=)4y(=)o.

The summation is effected by summing first the rectangles in a
vertical strip PQ and then summing all the vertical strips from & to 7;
or, first sum the elements in a horizontal strip PZ, then sum all the
horizontal strips in the boundary from Sto U. These summations are
clearly represented by the double integrals

La
=o”

Be [ae ay
y= Cy)

396
ArT. 282.] APPLICATIONS OF DOUBLE AND TRIPLE INTEGRALS. 397

In the first integration in either case the limits of the integration
are, in general, functions of the other variable which are to be deter-
mined from the given boundary.

EXAMPLES.

1. Required the area between the parabola y? = ax and the circle v? = 2ax — x?,
in the first quadrant.
The curves meet at the origin and at x = «a.

(1) A ae aces py ax = [v2 3 - 20x] dx

 

= Vax
_ ma? 2a?
~ 4 3
Q). A= hee ie dx dy
y=o dx=a— Var-x?
a ( y?2
= yp nat Vea ly
0 a
_ ma? 2a?
cra 3

‘ 2. Find the area outside the parabola y? = q4a(a — x) and inside the circle
92. 4a? =. x4

3. Find the area common to the parabola 3y* = 25x and 5x7 = gy. Aus. 5.

282. Polar Coordinates.—The surface of the plane is divided into
checks bounded by rays drawn
from the pole and concentric
circles drawn with center at the
pole.

The exact area of any check
PQ bounded by arcs with radii p,
p + Ap, and these radii includ-
ing the angle 46, is
+{(0 + 4p)” — p?} 40

=p4p4d+4 4p? 46,

The entire area in any closed
boundary is the limit of the sum
of the entire checks in the 0° :
boundary. The sum of the par- Bes AD:
tial checks on the boundary being o when 4p(=)46(=)o, as in § 281.

But, since
pApA6+i4dp?4e _ 4p
7 pdp 40 Prat p

= iI

 

 

when 4p(=)46(=)o, the area within any closed boundary is equal to
when 4p(=)46(=)o.
398 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVII.

This summation can be effected in two ways:

(1). We can sum the checks along a radius vector #S, keeping
44 constant, then sum the tier of checks thus obtained from one value
of @ to another.

(2). We can sum the checks along the ring UV, keeping p and
4p constant, then sum the rings from one value of p to another.

These operations clearly give the double integrals

9 pp=w(6) Po PO=u(p)
dp dé a0 p dp.
i law paper i: dias re

EXAMPLES.

1. Find the area between the two circles p = acos6, p=4cos@, b> a.

2 pb cosé
a 4= f° f eo8 "a dp dB,
0 a

cos 6

= f * (0 — a2) cost6 dh =F (2 — 2),
2 7

—~1? rh
5 Ss cos ZB
gas ff Fis pap+ ff » 4 p dp,
av0 9 cos—'4

which gives the same result as (1).

The double integration is not necessary for finding the areas of
curves; it is given here as an illustration of a process which admits of
generalization.

VOLUMES OF SOLIDS. DOUBLE AND TRIPLE INTEGRATION.

283. Rectangular Coordinates.—
Let x,y, 2 be the coordinates of a
point in space referred to orthogonal
coordinate axes.

Divide space into a system of rectan-
gular parallelopipeds by planes parallel
to the coordinate planes. Let Jx, Jy, dz
be the edges of a typical elementary
parallelopiped. Then the volume

Ax Ay dz
is the elementary space volume.
The volume of any closed surface is
the limit of the sum of the entire elemen-
Fic, 150. tary parallelopipeds included by the
surface when 4.x(=)4y(=)42(=)o.
V= £2 Ax Ay Az,
taken throughout the enclosed space.
(1). Let x,y, Jy, 4x be constant. Sum the elementary volumes

 
ArT. 283.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 399

between the two values of z, obtaining the volume of a column JZS of
the solid. The result expresses z as a function of x and y given by
the equation or equations of the boundary.

(2). Let x, dx be constant. Sum the columns between two
values of y for 4y(=)o. The result is the slice of the solid on the
cross-section *« = constant, having thickness 4.x.

(3). Sum the slices between two values of x for Jx(=)o. The
result is the total sum of the elements, expressed by the integral

"= eh I) Je dy dx,
z

xy dy=o(~) Jz=Alx, ¥)
= [” ["% dy dx,

ey OL
= [°A, dx.

x

Clearly, if more convenient we may change the order of integra-
tion, making the proper changes in the limits ot integration.
EXAMPLES.
1. Find the volume of one eighth the ellipsoid
2 y? ge
A Ra

W 2 Mae z

J at Pe dy de,
0

x

a
ref]
0 0
2
a a ep
4. ¢ 1-5 Ry &
2
=m f" (0-3) ae = trate

0

2. Find the volume bounded by the hyperbolic paraboloid xy = az, the «Oy
plane, and the four planes « = *4) 4 = +) ¥ =p) = Ip:

xy
V 0 [oa ae
x, U¥1 40

= ene dy ax,
xy dy, @

 

2 2
a [4s x dx,
xy 2a

I
= Pra — Vilag — Ds

: (#2 — (Yo — WAN H+ eo %2 + He + %2Ii)9

4a

=F — IVA + 4+ 23 + 24).

*
400 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVII.

The volume is therefore equal to the area of the rectangular base multiplied by
the average of the elevations of the corners. This is the engineer’s rule for calcu-
lating earthwork volumes.

284. Polar Coordinates.—The polar coordinates of a point P in
space are ~, the distance of the point from
the origin; @, the angle which this radius
vector OP makes with the vertical Oz; and @,
the angle which the vertical plane POz makes
with the fixed plane «Oz.

Through P draw a vertical circle PAZ
with radius p. Prolong OP to Rk, PR = Jp.
Draw the circle RQ in the plane POA/ with
radius p+ 4p. If 4A is the area PROS,
then

 

AA
Fic. 151. App do I.

We may therefore take dA = pdpd@. This area revolving around

Oz generates a ring of volume
27 psin 6dA.

Therefore the volume generated by @A revolving through the arc
ds = sin # d¢ is in the same proportion to the volume of the ring as
is the arc to the whole circumference, or the element of volume is

p'sin 6 dp dp db.

We divide space into elementary volumes by a series of concen-
tric spheres having the origin as center, and a series of cones of
revolution having Oz for axis, and a series of planes through Oz

The volume of any closed surface is the hmit of the sum of the
entire elementary solids included in the surface when

Ap(=)49(=)46(=)o.
Or, the volume is equal to the triple integral
V =f ff resin dd dp do,

taken with the proper limits as determined by the boundaries of the
surface.
EXAMPLES.

1. Find the volume of one eighth the sphere p = «a.

o=7 = p=a . 2
V= f 2 I 2 7 p’dp-sin 6 d.dq,
g=o0 8=0 p=o
7 T
3 apa pose
=<f" ‘f 2 sin 6 dd.d@
3 J¢=0 0=0

5
HS [40 = 400
3 Jo
ART. 285.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 401

The first integration gives a pyramid with vertex O and spherical base
a’sin§ di dg. The next integration gives the volume of a wedge-shaped element
of a solid between two vertical planes determined by @and @+ 4q@. The last
integration sums up these wedges.

2. A right cone has its vertex on the surface of a sphere, and its axis coincident
with the diameter of the sphere passing through the vertex; find the volume
common to the sphere and cone.

Let a be the radius of the sphere, a the semi-vertical angle of the cone. The
polar equation of the sphere with the vertex of the cone as origin is = 2a cos 6.

2% Pa f2acosé ‘
ae val f f pi dp-sin§ d-d@.
oO v o

3. The curve p = a(1 + cos 9) revolves about the initial line; find the volume
of the solid generated.

mT fPanw fa(1 + cos 6) f
v=f f f p?dp-dp-sin 6 a8,
fe ° °

2na> fr ‘
= ar f (1 + cos 6)8 sin 6 do = 87°.
285. Mixed Coordinates.—Instead
of dividing a solid into columns stand-
ing on a rectangular elementary basis,
as in the method

V= f feaxra,

it is sometimes advantageous to divide
it into columns standing on the polar
element of area. Thus the elementary
column volume is

2p 20 dp.
Therefore for the volume of a solid yy

we have
b= | f [rddp,

= | fz dd,

taken between the proper limits.

   

EXAMPLES.

1. Find the volume bounded by the surfaces z = 0,
xt ty?— gaz and y® = 2ex — x.

Here z = p?/4a and the limits of g and @ must be such as to extend the inte-
gration over the whole area of the circle y? = 2cx — 27, Let _p, = 2¢ cos 6;

then el
V= ee i dp di =— ” cost 2,
-trdo 4a a tw

aa 26 [costo e263 Io
ado a4q422 8a
402 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIL

2. Find the volume of the solid bounded by the plane z = o and the surface

 

_ #47
z= ae a,
ae
Here p?=x?ty% .-. vaaf fe ° 9 dp do.

2

oe ae
f e “pdp=—4cre ? = #7.
° °

27
iL do = 27.
°

ow Va mac. [See Todhunter, Int. Cal. p. 181.]

SuRFACES OF SOLIDS,

286. When the plane through any three points on a surface (the
points arbitrarily chosen) converges to a tangent plane asa limit when
the three points converge to a fixed point as a limit, then a definite
idea of the area of the surface can be had, as follows:

Inscribe in a given bounded portion of the surface a polyhedral
surface with triangular plane faces. The area of the given portion of
the surface is the limit to which converges the area of the polyhedral
surface when the area of each triangular face converges to zero.

To evaluate the limit of the sum of the triangular areas inscribed in
the surface we proceed as follows:

Let P bea point x, y, z on a surface,
and Qa pointy + 4x,y+ dy, 24+ Az.

The prism A/7NUV on the rectangle
whose sides are 4x, Jy cuts the surface
in an element of surface PRQ.S. Draw
the diagonal JZ and the two inscribed
triangles PRQ and PSQ. Let perpendic-
ulars to the planes of the triangles PRQ,
PSQ at the point P make angles y,, y,
with Oz respectively. The angles y,, v,
are then the angles which these planes make with the horizontal plane
xOy. Since the area of the orthogonal projection of a plane triangle is
equal to the area of the triangle into the cosine of the angle between
the plane of the triangle and the plane of projection, we have the areas

PRQ=MTN sec y,, PSQ= MUN sec y,.
Also, MTN = MUN.

 

Fic. 153.

PRO+ PSO = SEIT CMs Ay Ax.
ArT. 286.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 403

By hypothesis, if 4*,$ is the area of the element of surface PRQS,

then

Laan
sec y, + sec ¥, , aye ;
5 y

But when Q(=)P the perpendiculars to the planes PRQ, PSQ
have the normal to the surface at P as a limit, since the planes PRQ,
PSQ converge to the tangent plane at P as a limit.

If y is the angle which the tangent plane at P makes with the
plane xOy, then

ey sec y, +sec y, _ ey

2

dz \? az \?
=yr+ (5) + (3):
as

ay dx = sec Y,;

s= ff sc y wy ax,

a

=ff ft +(%) + (5) oes

taken between the limits determined by the boundary of the portion
of the surface whose area is required.

and

EXAMPLES.
1, Find the area of the sphere-surface x? + y? + 2 = a.
dz x Oey

,

ax ay z
x= a = Vai
eee v= VaI—x dy dx 2
x=0 =O ya* — x -y

=< oe sin— oe, ee

x=o Va — x ° ,

are i. a na’.
2/0

2. The center of a sphere whose radius is a is
on the surface of a cylinder of revolution whose
radius is $a. Find the surface of the cylinder
intercepted by the sphere.

(1). Let the equations of the sphere and
cylinder be

z

eye = a,
xt y? = ax,
as in the figure.

 
404 INTEGRATION FOR MORE THAN ONE VARIABLE, [Cu. XXXVIL.

da; ae 2= Vat—ax ay \2 ay
ae saa f™ Le J+ (%)'+ @iae
= I mre bey *) a— = \e es
Va?—ax da de ax
Vax — oer
iS
a 94 : ae rom fi (rea
0 Yax—x 0 me
' (2). Let s be the length of the arc of the base of the cylinder measured from

the origin. Then
S=4 fe ds,

taken over the semi-circumference. Let y be the angle alien the sphere radius to
P makes with Oz, and 6 the angle which OA = p makes with Ox. Then

z=acsy=asing. sai, ds=adg.
6=
saa f # 9 sin dO == aut.
@=0

(3). Otherwise, immediately from the geometry of the figure,

2= Var—ax  dzdx _ dz ax
= dx = 4a?
S = a f~ ye re = zo f” Ve x = 4a*,

as in (I).
3. Find the surface of the sphere intercepted by the cylinder i in Ex, 2.
From the figure,

 

 

q.
ya —— yp

“=a be ra Vax—x? “2 _dydx ax

= y= Va — x? — 72?
=F e "ties = ax.
= x=0 ate

Integrate directly, or put sin?@ = x/(a-+ x) and integrate
Hence S = 2a — 2).

 

 

(1). sec y = <=

 

(2). Again,
45 = [" pisecy dp.
p=acst=asiny. .. y= jtn-—9
45 = — at ['6c0s 6 ao = — a? [6 sind + cos 6].” = — ain — 1)

LENGTHS OF CURVES IN SPACE.

-287. Asin plane curves, the length of a curve in space is defined
to be the limit to which converges the sum of the lengths of the sides
of a polygonal line inscribed in the curve.
Akt. 287.| APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 405

Since (=) As = Ax + AP 4 Ae,

tale +(2)+ (2) 2 (dz\2

z= Ni +(¢:) + (a)

with similar values for the derivatives = =
Pade dy dz

= fora)

with corresponding values for s when y or 2 is taken as the indepen-
dent variable.

If the coordinates of a point on the curve are given in terms ofa

variable /, then
ds\2__ (dx\2, (dy\* dz\2
(7)'= (@)'+ @) + @)

= f'N(G) + Ga) + Ga) *

EXAMPLES.
1, Find the length of the helix

an

z : es
x« = @cos-s, yzasin-e,

measured from z = oO.
“'" Take zg asthe independent variable. Then
- oF ss ‘a 2 dy .a z

7 ny ery

= a “eB ‘+ (3 ye dz,
“Fpetenbe

2. Find the length, measiined from the origin, of the curve.

 

 

2ay = x*, 602 = x8.

saf(- 4 Byars f(t+B)eaet araete

3. Show that the length, measured from the origin, of. __
yoasing, 42 = @(x + cosx sin x),
is x+2.
4. Find the length of Ss ee
De a ee es
Y= 2Vax — x; == so

measured from the origin. Ans. s=uty—%
406 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIL

5. Find the length, measured from the horizontal plane, of the curve

Zz z

at v si, # =io(# 42°).

a BF
SS RANE
ieee + * Vx? — at.

288. Observations on Multiple Integrals. — The problem of
integration always reduces ultimately to the irreducible integral

J dE,

dE being the element of the subject to be integrated. Or this may
be taken as the starting-point and considered as the simplest element-
ary statement of the problem for solution. This, in simple cases, may
be evaluated directly, otherwise it may be necessary to integrate par-
tially two or more times with respect to the different variables which
enter the problem. There may be several different ways in which the
elements can be summed. A careful study of the problem in each
particular case should be made in order to determine the best way of
effecting the partial summations, with respect to the limits at each
stage of the process.

One is at perfect liberty to take the elements of integration in
geometrical problems in any way and of any shape one chooses, as the
limit of the sum is independent of the manner in which the subdivision
is made (see Appendix). This should be verified by working the
same problem in several different ways.

The applications of multiple integration in mechanics are numerous
and extensive. Further application beyond the elementary geometrical
ones given here is outside the scope of the present work.

EXERCISES.

In these exercises the results should be obtained by double and triple integra-
tion, and also by single integration whenever it is possible.

1. Find the volume bounded by the surfaces
xvtyt—a,z=0, z=extana.

2_ yf t
Ans. ff He oi a0" de dy dc = 4a tan a.
° ° °o

2. Find the volume bounded by the plane z = 0, the cylinder
(x — a)? + (y -6)) = RY
5
and the hyperbolic paraboloid xy = cz. Ans. xe Re,
3. Find the volume bounded by the sphere and cylinders

et ytpataat, xtfy? = 0, p? = a? cos + sin’,
Ans, 4(16 — 32)(a? — 82).
Arr. 288.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 407

4. A sphere is cut by a right cylinder whose surface passes through the center
of the sphere; the radius of the cylinder is one half that of the sphere a. Find the
volume common to both surfaces. Ans. 3(m — $)a5.

5. Show that the volume included within the surface
xy 8
oe ) =
is adc times the volume of the surface
Fx, V, 2%) =O.
6. Show that the volume of the solid bounded by the surfaces
g=0, x+y? = gaz, x? + y? = 2er, is Sact/a.

7. Find the entire volume bounded by the positive sides of the three coordinate
planes and the surface

; a\t y\t z\t abe
(=) + (3) + (=) = f Ans. eo
8. Find the volume bounded by the surface
at 4 yh + st = af, Ans. pra.
9. Find the volume of the surface
t
(=) + (5) ee (2) t I. Ans. Asmadbe.

10. Show that the volume included between the surface of the hyperboloid of
one sheet, its asymptotic cone, and two planes parallel to that of the real axes is
proportional to the distance between those planes.

11. Find the whole volume of the solid

x fa* + y?/b? 4+ zt/ct = 1. Ans. § mabe.
12, Find the whole volume of the solid bounded by
(2? + yy? 4 2?)8 = 270% xyz. Ans. 3a’.

13. Use § 285 to show that the volume of the torus
(2 fy +P 4 2 — a)? = g(x? + 9%) is 2%".

14. Find the volume of the solid bounded by the planes x = 0, y = 0, the sur-

face (x + y) = 4az, and the tangent plane to the surface at any point /, g, 4.
Ans. tah’,

15. Show that the surfaces y? + 2? = qax, and x — 2 = a, include a volume
87a.

16. Show that the volume included between the plane z = 0, the cylinder
y? = 20x — x, and a paraboloid ax? + by? = 2s is {act(Sa—t + 6-1).

17. Show that the whole volume of the surface whose equation is

(x? + y? + 27)? = cxyz is equal to 3/360.
18. Show that the volume included between the planes y = + 4 and the surface
atx? + 8%? = 2(ax + bz)y? is 4nh5/5ad.
19. Find the form of the surface whose equation is
(xtja? by? /B® 4 22/c%)? = x2/a® 4 2/82 — 24/e,

and show that the volume is 2*adc/4 V2.
408 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVII.

20. Find the entire surface of the groin, the solid common to two equal cylin-
ders of revolution whose axes intersect at right angles.
Ans. 16K. # being the radius of the cylinders.

21. Find the area of the surface
22+ (xcosa-+ysin a)? = a?

in the first octant. Ans. 2a? csc 2a.
22. Find the volume of the solid in the first octant bounded by xy = az and
xty+tez=a4. Ans. (43 — log 4)a*.

23. Find the surface of the sphere x? + y? + 2? = a? in the first octant inter-
cepted between the planes -0,y =O,x = dy = 4.

 

 

2
Ans. a(26 ses oes —a@sin—- g ):

Va _— B a — 8

24. A curve is traced on a sphere so that its tangent makes always a constant
angle with a fixed plane. Find its length from cusp to cusp.
CHAPTER XXXVIII.
INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS.

289. Classification.—A differential equation is an equation which
involves derivatives or differentials.

An ordinary differential equation is one in which the derivatives
are taken with respect to ome independent variable. These are the
only kind that we shall ‘consider.

Differential equations are classified according to the order and
degree of the equation. The order of a differential equation is the
order of the highest derivative contained in the equation. The degree
of the equation is the highest power of the highest derivative involved.

_ 290. We shall consider in this text only examples of ordinary dif-
ferential equations of the first and second degree in the first order,
and a few particular cases of the first degree in the second order,
291. Examples of Equations of the First Order and First
Degree.—The derivative equations of the first order and first degree
dy q_ y

; gy
ee eae eae Ay ee Be
=cosx, 2% ay— xy, axyl- = 2x ds

when multiplied by dx, are equivalent to the differential equations of
the first order and first degree
dy=cosxdx, wzdy=(3v—xy)dx, axdy = 2x dy — y dx.

@
In general, any linear function of = ;

@
Go +H =o,

in which ¢@ and. are constants, or functions of x or_y, or of x and y,
is a derivative equation of the first degree and order. When multi-
plied by dx it becomes the general differential equation of the first

degree and order
o dy +p dx =o.

292. Examples of Equations of the First Order and Second
Degree.—The equations

dy\* _ aq? (aw =
(%) = ax, «(% 2y\ +ax=0,

are of the second degree and first order. Written differentially,
dy? = axtdx®, x dj — 2y dy dx+ax dx* =o,
, 409
410 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIII.

In genéral, the type of an equation of the first order and second

degree is
dy \? @
(2) 40(2)4x=6

where @, 7, y are functions of 2, y or x and _y, or constants.

293. Equations of the Second Order and First Degree.—Such
equations as
ad’y

ay gy
Za MY = met Poe 26%,

ay fan 2, OY dy :
a (z) =y" log y, ao a Me gt |) Oy
are of the second order and first degree.

294. Solution of a Differential Equation.—To solve a given
differential equation
F(x; 9, 9) = 9,
= ea , isto find the values « and _y which satisfy the equa-
tion. Thus, if the values of x and _y which satisfy the equation

P(%, ¥) = 0
satisfy a differential equation /' = o, then @ = o is a solution of
Foo.

The solution of a given differential equation may be a particular
solution or it may be the gezeral solution. The general solution in-
cludes all the particular solutions. Or the solution may be a sizgular
solution, which is not included in the general solution. The complete
solution of a differential equation includes the general solution and the
singular solution, The meaning of these solutions will be developed
in what follows.

The solution of a differential equation is considered as having
been effected when it has been reduced to an equation in integrals,
whether the actual integrations can be effected in finite terms or not.

where y’”

EQUATIONS OF THE First DEGREE AND FIRST ORDER.

295. The simplest type of an ordinary differential equation of the
first order and degree is

dy = A\x)de. (1)
Integrating, we obtain the solution
J = Mx) +6, (2)

where (x) is a primitive of /() and ¢ is an arbitrary constant. For
a particular assigned value of c¢, (2) is a particular solution of (1),
Arr. 295.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 4it

and is the equation of a particular curve in a definite position. At
each point of the curve (2),

dy
3, = JP)
is the slope, or direction of the curve (2). For different values of
c we have different curves. The ordinates of any two such curves
differ by a constant. Equation (2) is then the equation of a family
of curves having the arbitrary parameter c. This singly infinite sys-
tem of curves, or family of curves with a single parameter, is the
general solution of the differential equation (1).

296. Every equation of the first order and first degree can be
written

Mdx+ Ndy=o, (2)
where, as has been said before, JZand Mare either constants, functions
of x or y, or functions of x and_y.

297. Solution by Separation of the Variables.—This solution
consists in arranging the equation
idx + Ndy=o0, (1)
so that it takes the form

Pla)dx + p(y)dy = o. (2)

The process by which this is effected is called separation of the

variables, When the variables have been thus separated the solution
is obtained by direct integration. Thus, integrating (2),

[oe ax + po) a =e,

where ¢ is an arbitrary constant, and is the parameter of the family
of curves representing the solution.

I. Variables Separated by Inspection.—A considerable number of
simple equations can be solved directly by an obvious separation of
the variables. The process is best illustrated by examples which
follow.

EXAMPLES.

1. Find the curve whose slope to the x-axis is — x/y, and which passes through

the point 2, 3. ; : ; : ;
The geometrical conditions give rise to the differential equation

ay x

“= @ xax= 0.

ae y’ or yay +

The solution of which, obtained by integration, is the family of circles

et ya
The particular curve of the family through 2, 3 is
xt y? = 13.
412 INTEGRATION FOR MORE THAN ONE VARIABLE. [CuH. XXXVIIL

, 2. Find the line whose slope is constant.
“a = m gives the family of parallel straight lines y = mx + ¢.
3. Find the curves whose differential equation is
xdy+tydx =o.

_The variables when. separated give
ax dy
—_ — =0.
x a y

logx+logy =e, or =k.
Otherwise we may write the solution xy = e-. This is a family of hyperbole
having for asymptotes the coordinate axes.
If we observe that x dy + y dx is nothing more than a(xy), the solution xy = ¢
is obvious.

4. Find the curve whose slope at any point is equal to the ordinate at the point.

d
Here ea Zo eh ° = dx,
‘Hence logy =ate or y = exte = etex = aex,

which is the exponential family of curves,

5. Find the curve whose slope is proportional to the abscissa.

Ans, The family of parabole y = ax? + ¢, in which ¢, the constant of integra-
tion, is the parameter.

6. Find the curve whose slope at x, y is equal to xy, Ans. y = cet”.

7. Find the curve whose subtangent is proportional to the abscissa of the point
of contact.
a. a: d
Here if = ax, ot = =a : gives
logx =alogy+tc, or y? = hx.
8. Find the curve whose subnormal is constant.

q;
y = =a gives y? = 2ax-+c, the parabola,

x
9, Find the curve whose subtangent is constant. Ans. y = ce*,

10. Find the curve whose subnormal is proportional to the th power of the
ordinate. What is the curve when x is 2?

11, Find the curve whose normal-length is constant.
_.. Here the geometrical conditions give the differential equation

Vottt aay . Bacto.
Sey ee a = ae

Integrating, « — c= — (a? — y%)#, or the family of circles
(«-cPtya=e,
with radius a, having their centers on the x-axis.

12. Find the curve in which the perpendicular on the tangent drawn from the
foot of the ordinate of the point of contact is constant and equal to a.
ArT. 297.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 413

The differential equation of condition is

ee SE ae ae es: ee
(eV? Vy — a
oN de

The solution is therefore the family of curves
cx = alog (y+ Vy = a7).
When c = 0 this is the catenary with Oy as axis.
13. Find the curve in which the subtangent is proportional to the subnormal.

14, Determine the curve in which the length of the arc measured from a fixed
point to any point Pis proportional to (1) the abscissa, (2) the square of the
abscissa, (3) the square rot of the abscissa of the point P.

(1).-A straight line,

2
(2). The condition is s= =.
a

ast 2 ae
ot. @S* = dx? + ay =a,
or ady = Vx*—@ dx.
The solution of this is
e+ av = te x? — a? — 4a? log [x + Vx? — a],

(3). The geometrical condition can be written s = 2 4/ax,

ow. a= ax? + dy? = ds? = oe gives
x

Vz ax,

wa

Put « = 2? and integrate. The result is the cycloid

 

2 ax.

 

cty= VsE=H) +a sins FE

Ex. 14, really leads to a differential equation of the first order and second
degree, which furnishes two solutions which are the same.

15, Find the curve in which the polar subnormal is proportional to (1) the radius
vector, (2) to the sine of the vectorial angle. (1). @ = cea@. (2). @=¢—acosf.
16. Find the curve in which the polar subtangent is proportional to the length

of the radius vector, and also that curve in which the polar subtangent and polar
sub-normal are in constant ratio. Ans, p= cead,

17. Determine the curve in which the angle between the radius vector and the
tangent is one half the vectorial angle. Ans. p = 1 — cos 6).

18. Determine the curve such that the area bounded by the axes, the curve, and

any ordinate is proportional to that ordinate.
x

If Q isthe areaa Qoay. .. di=ydx =ady 1 y= cet,

19. Determine the curve such that the area bounded by the +x axis, the curve,
and two ordinates is proportional to the arc between two ordinates.
ey

Qa=as. .. ydx = ads, Coe
414 INTEGRATION FOR MORE THAN ONE VARIABLE. [CuH. XXXVIIL._

This gives, on integration, the catenary
cx =a log(y+ ¥7 — a).

20. Find the curve in which the square of the slope of the tangent is equal to
the slope of the radius vector to the point of contact.

The parabola att yt= ct, or (x —y)? — 2x +7) +2 =0.
21. Solve Madx+ NMdy, when Mx + My =o.
(1) Mx +My =0 gives M/N= — y/x.

‘i a
Substituting in the equation, j = < ea eS
(2). Mx — My =o gives M/N=y/x.
5 id 3 ‘ dx | @
Substituting in the equation, x a =0 a syou.

II. Solution when the Equation is homogeneous in x and y,—When
the equation
MdaxtNdy=o
is such that W= (x, vy), V = (x,y) are homogeneous functions
of x and_y and of the same degree, the solution can be obtained by
the substitution y = 2x.
We have
DE Gy,
Ni (x9)
Divide the numerator and denominator by +”, 2 being the degree

of ¢ or yp.

 

dy dz
as nt pea = = HEL
Hence
ax dz

2° s+
and the variables are separated. The integration of this gives an

equation in x and z. On substituting y/x for z the solution of the
original equation is obtained.

EXAMPLES.

1. Solve the equation (2x7 — y*)dy — 2xy dx =0.

dz 22
Put yowx .. of a sh OF

_ 2

a = Z 3 2 dz.

a 2
Integrating,

I
log x = = log 2.

Replacing z by y/x, we have
xt = yc — log y).
ArT. 298.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 415

2. Determine the curve in which the perpendicular from the origin on the
tangent is equal to the abscissa of the point of contact.
Ans. The circles x? + y? = 2cex.

3. Find the curve in which the intercept of the normal on the x-axis is propor-
tional to the ordinate of the point of contact.

x+y o r= my. .. (x — my)dx +ydy =o, ete.
4, Find the curve in which the subnormal is equal to the sum of the abscissa
and radius vector.

5. Find the curve whose slope at any point is equal to the ratio of the arith-
metic to the geometric mean of the coordinates of the point.

6. Solve y?dx + (xy + x?)dy = 0. Ans. xy? = Ox + 2y).
7. Solve x2y dx = (x3 + y3)dy. Ans. log = z.
298. Solution when J/ and J are of the First Degree.—The
equation
(a,x + dy +¢,)dx = (a,x + 4,9 + 6,)dy (t)

can always be solved as follows :
Put «=x +h, y= y' +4, where / and & are arbitrary
constants. Then (1) becomes

dy ax’ +bhy +ah+ hhc,

 

dx’ ax’ + by +ah+bk+e,’ (2)
I, If a4, # a,4,, assign to 2, & the values which satisfy
ah+bk+¢,=0,) (3)
ah+bk+e, = 0.
Then (2) becomes .
ad’ ax’ +bhy es

de agate hy
This is homogeneous and can be solved by § 297.
If f(x’, 1”) = o is the solution of (4), then f(x — 4, y — &) = 0
is the solution of (1).
Il. Ifa, = a,b

a é
.0,, let Pe Sh ny
a

b

1 1
Then (1) becomes

wy _ ax+tay+e, : (s)
ax m(av + 69) + ¢,
Put z = a,x +4,y. Then (5) becomes
dz zs+o,

= é
a = tmz + ¢,’

in which the variables can be readily separated.
416 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIL

EXAMPLES.

1. Solve (3y — 7% +7)@x + (77 — 34+ 3)ay = ©.
Ans. (y 2 Loy be— hon
2. Solve (2x + y +1)dx 4 (4% + 2y — I)dy =0.
Ans. x+2y 4+ log(jae+y —1) =e
3. Solve. (7y + 2 + 2)dx — (34 + 57 4+ 6)dy = 0.
Ans. x+5y+2=cee—y-+ 2).
299. The Exact Differential Equation.—The differential equation
Mdxt Ndy=o
is said to be an exact differential equation when it is the zmmedia/e result
of differentiating an implicit function /(x, y) = 0.
In fact, if
“u=/(x, y) = 0,
iat

then ge =o

gives an exact differential es

300. Condition that 1/7 dx + N dy = o be Exact.—Since 1/ must
be the first partial derivative with respect to x, and WV the first partial
derivative with respect to_y of some function /(x, y), then

wa, va¥

x ~~ Oy"
But since
vw a
dy Ox Ax dy’
we must have the relation
om oN
a be (1)

existing between JZand (Vin order that Mdx+ Ndy =o shall be
exact. This condition is also sufficient, and when (1) is satisfied
Max + Nay is an exact differential.

For,* let V= i. Max.

oY 9g OM en

Ox? Oy Ax ay Ox’
aN #8
ee

j 78 i,
Hence v= f= (Hara Sr + P'(¥);

* This is due to Professor James McMahon.

 
ART. 301.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 417

where the constant of integration @’(_v) is some function of y or a
constant independent of x. Therefore

OV OV i
Mdx + Nady = age eo p'(s)e,

=a[(V + ¢)],

an exact differential.

301. Solution of the Exact Equation.

M aN
If — = =, there exists a function u, of x and_y, such that
oy Ox
du = AM dx + NV ay. (3)
Ou

 

Since JZ = ax? lf contains the derivatives of only those terms in

» which contain x, Integrating (1) with respect to x (y being con-
stant), we have

u = { Mdx + (3), (2)

where #() represents the terms in « which do not contain x.
To find @(y), differentiate (2) with respect to y.

ou =. CO 0g

ap _ a

9 P :
As was said, @ is independent of x and so also is = as is verified

Hence

by differentiating (3) with respect to +;

a a aN AM

ores a T d. = 7—- ~—-=0
Ox ' M oy pp a } Ox oy "
Integrate (3) with respect to_y.

ouy= fo [wx [veel ate.

Therefore the solution of (1) is

v= [aay [ \v—z [uelotese. (4)

In like manner, working first with JV instead of J/,

[ror f ra [ral wtene (5)

is also a solution of (1).
418 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIIL

302. Rule for Solving the Exact Equation.

{ MM dx contains all the terms of the primitive containing. Also,
a
since V — ap f Mae is independent of «x, apf Mae must contain

those terms in VV containing x. Therefore to obtain

[xg fore

integrate only those terms in JV which do not contain x. Hence the
tule. Integrate JZ dx as if y were constant; integrate those terms in
JV dy which do not contain »; equate the sum of these integrals to a
constant.

A like rule follows for effecting (5), § 301.

EXAMPLES.

1. Solve (32? — 4xy — 2y")dx + (39? — gay — 22°)dy =O.

Here OM _ oN
ge PS

[ara = WP aw aeety — aay fv ae

Therefore the solution is
we — 2x%y — axe tyre

 

2. Solve (x? + (x dx + y dy) + x dy —ydx=0.
2 2
Ans, ~ a + tant 2 = ‘
3. Solve (a? + 8xy — 2y?)dx + (2x—y)dy = 0.
Ans, a’x + $7 — 2a? + grty = c.
4. Solve (2zax + by + g)dx + (2cy + bx 4+ e)dy =O.

Ans. ax? by +o 4+ gxtea =k.
5. Solve (m dx + ndy) sin (mx + ny) = (n dx + mdy) cos (nx + my).
Ans. cos (mx + ny) + sin (wx + my) = ¢
6. Solve 2x(a + 2y)dx + (2x? — y2)\dy = 0,
Ans, x? + 30¢y — av =e.

303. Non-Exact Equations of the First Order and Degree.—We
have seen that when a primitive equation /(x, y) = 0 is differentiated
there results the exact differential equation d(x, y, y’) = 0, writing
y’ for the derivative of_y with respect to x.

If now between f= 0 and @ = o we eliminate any constant
occurring in fand @, we get another equation, 7:(x, y, _’) = 0, which
is a differential equation satisfied at every point on #= 0. Therefore
f= 0 is a primitive of 7» = 0. But » = o will not be an exact
ART. 303.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 419

differential of the primitive / = 0, although f= 0 is a solution of the
differential equation 7 = o.
To fix the ideas, consider the equation

ax+by+tevtk=o. (1)
The exact differential equation of (1) is
(a + oax + (b+ cx)dy = 0, (2)

_ When (2) is integrated the constant of integration restores the
parameter & of the family (1) and (1) is the solution of (2). That is
to say, the family of curves (1) obtained by varying the parameter 4
gives the solution of the exact differential equation (2),

The constant & was eliminated trom (2) by the operation of differ-
entiation and restored by the process of integration,
Eliminate @ between (1) and (2) by substituting

by +h
BT i
from (1) in (2). ‘There results the differential equation
ay ax
by LA MEF cay’ (3)
1dx ¢ ax

 

or
bay ax cdx

by -k x ex fo
Integrating and adding the arbitrary constant — log c’,
log (4v + &) + log(cx + 6) — log x — loge’ =o.
(y + (cx + 8) = e's,
or (kc — c’)x + By + be xy + kb =o.
Putting the arbitrary parameter in the form 4c — c’ = aé, this
equation becomes the original primitive

ax tb teytk=o.
This equation with the variable parameter a is the solution ot the

differential equation (3).
The differential equation (3), or

 

(by + Adz — x(5 + cx)dy = 0 (4)
is not an exact equation, for
wt k) = 3, (= bx —cx*) = — b — 2¢x.

But (1) is the primitive of (3) as well as of (2).
Again, if we eliminate first 6 and then ¢ between (1) and (2), we
420 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIII.

shall get two other differential equations, neither of which is exact,
but each of which has (1) for solution with variable parameters 4 and c¢
respectively.

Observe particularly that if (4) be multiplied by 1/x*, it becomes
an exact differential,

y te b+ cx ,

 

 

 

x —— ee yy =0, (5)
since
df/y +4 8 f b+ex\ 6
a ae) Oe a)

Integrating this exact equation (5) under the rule § 302, the
solution is

gx + by + cxy +h = 0,
the same equation as (1) with g for parameter.

304. Integrating Factors.—In the preceding article we have seen
that the same group of primitives can have a number of different
differential equations of the first order and degree. The form of any
particular differential equation depending on the manner in which an
arbitrary constant has been eliminated between the primitive and its
exact differential equation. In the example above, when the differential
equation was not exact, it was made exact by multiplying by 1/2°*.
Such a factor is called an zn/egrating factor of the differential equation
which it renders exact.

The number of integrating factors for any equation

Mdx+ Ndy=o (1)
is infinite. For, let be an integrating factor of (1). Then
M(M dx + N dy) is an exact differential, say du, and

MM dx + Nady) = du.

Multiply both sides of this equation by any integrable function of

u, say /(u),
wfe)(M dx + Nay) = flu)du, (2)
The second member of (2) is an exact differential, and therefore

also is the first. Hence, when y is an integrating factor of (1), so

also is u4/(w), where /(z) is any arbitrary integrable function of w..
In illustration consider the equation

ydx —xdy=o.
This is not exact, but when multiplied by either —, er =
yxy x
it becomes exact and has for solution

x
— = constant.
wv
ART. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 421

The general solution of the differential equation
Mdx+Ndy=o
consists in finding an integrating factor y such that
M( dx + Nady) =o
is an exact differential, then integrating by the method given as the

solution of the exact equation.

The integrating factor always exists, but there is no known method
by which it can be determined generally. The rules for determining
au integrating factor for a few important equations will now be given.

305. Rules for Integrating Factors.

I. By Lnspection.—While the process of finding an integrating
factor by inspection does not, strictly speaking, constitute a rule, in the
absence of a general law for finding the integrating factor it is an
important method of procedure. An equation should always be ex-
amined first with the view of being able to recognize a factor of inte-
gration. The process is best illustrated by examples.

EXAMPLES.
1. Solve ydx —xdyt+f(x)dx = 0.

The last term is exact; its product by any function of x is exact. Therefore
any function of x that will make y dx — x ay exact is an integrating factor, Such
a factor is obviously 1/x*,

oe pie oe BD ge = 0,
x x

or d (2) + an =0,

gives the solution
Shs fs eg.
x x

2. Solve ydx + log x dx = x dy.
Ans. ext+ty+logx+1=0.
3. Solve (1+ ay)y dx + (1 — xy)x dy = 0. (Factor 1/x%y?),
T

Ans. cx = yer.
4. Integrate «x*y8(ay dx + dx dy)= 0.
Obviously 2x?¢-'~4y44—" B is an integrating factor, where # is any number.
On multiplying by the factor we get
axka—tykb dx + bxkaykb—1 dy = rH xhaykb) = 0,
the solution of which is evident.

5. Integrate
xayB(ay dx + bx dy) + xaybr(a,y dx + bx dy) = 0.
422 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIIL.

The factors x4a—1—aykb—1-B, kya, -1-ayh\b,—1—-By
make the expressions
xayB(ay dx + bx dy) and xayBi(ay dx + b,x dy)
exact differentials respectively, whatever be the values of the arbitrary numbers

and &,.
Therefore, if 4 and £, be determined so as to satisfy

fa—'—-a=ha—1i1—aQ,
kb—-1—- ff =46,-1-— By,
the factors are identical and these values of £ and 4, furnish the integrating factor
of the equation proposed.
6. Solve (y3 — 2yx?)dx + (2xy? — x8)dy = 0.
Ans, xy yr — x)oe.
7. Solve the equation

Ly + A? + 9?)]dx = [x — f(x? + 9). (1)
This is the differential equation of the group or family of rotations. Put
at yar,

Rearranging (1),
yax —xdy + f(r*).(xdx+ yay) = 09,
2(y dx — x«dy) + f(7*)dr* = Oo.
This can be written
(y dx — x dy) — (x dy — y dx) + f(r*)dr? = 0,

 

 

or v a(=) — 2 a(2) + f7)drF = 0.
J x
An integrating factor is obviously 32 Sar . Whence
a= ay :
Z yo ee +20") dr? = 0.
eS y r
1+ Pp a

Integrating,
ap & J f(r")
tan-! , — tan — + [Pan i

Il. Whenever an integrating factor exists which is a function of x
only or of 9' only, tt can be found,

Making use of the fact that e* is always a factor of its derivative:

(2). Let 2 be a function of x.

In e(M dx + Ndy)=0,
put M’=¢M, N=eN.
aM’ 0M jan’ az aN
— = rel ioe a
Thea ay dy’? = Ax en ax a ax"
The condition that ¢* shall be an integrating factor is
OM dz ON

om _ aw
az = Yon,

or x.
N
ArT. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS, 423

If, therefore,

OAT AN
cag a
74 = P(x)

is a function of x only, then
o= f o(x)ax.

Hence a is an integrating factor of M@ dx +N dy =o
whenever
al = ON

y ae (2)
iV
is a function, @(x), of x only.
(6). In like manner, letting e* be a function of y only, we find

that ol TH eo integrating factor of
Mdx+ Ndy=o
when
ON AM
ax hy
a @)

is a function, #(_y), of y only.
(c). Whenever the expression (1), (2), or d(x), 2(y¥) is constant,
then ¢* or e’, respectively, is the integrating factor.

EXAMPLES.

1. One of the most important equations under this head is Leibnitz’s linear
equation, .

dy
+=, 5
where P and Q are functions of x or are constants.
This equation, (Py — Q)dx + dy = 0, is such that
6m aN
ye |
ye
‘ : ‘ [Pax
Therefore it has the integrating factor e¢
"Pa.
aid. pedded O ms: (2)
Pdx "Pa. "Pa.
Since sf dy + sy * Py ani eo =
on integrating (2), ;
ye elPt) fol? adc tel. (3)

This is the solution of the linear equation (1).
424 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXVIIL

2. Bernoulli’s Equation.—The equation known as Bernoulli’s
q
2+ By =O",

in which /, Q are functions of x or are constants, reduces to Leibnitz’s linear equa-
tion. For, multiply by (— 2-+ 1)/y%, and put v = y—*+1, Equation (1) becomes

ft — mh =r — m0,

which is linear in uv.
dq
3. Solve /’(y) = +Pf(y) = Q, where P, Q are functions of x.
Put v= /(y). The equation becomes
dv
ae + Pu = Q,
which is linear in v.

TI. When Mx + Ny 0, there are two cases in which the inte-
grating factor of Mdx + N dy = = 0 can be assigned,

(1). When JZ, WV are homogeneous and of the same degree, then
a ae is an integrating factor.
(2). When JZ, & are such functions as
M = y$(x X 9), N= x(x x ¥);

I : ‘ ‘
then Wa Wy is an integrating factor.
Proof: We have the identity
Madx + Nady
ine dx @
= fom (2) or = 9-8)

= 44 (Me + My)d log (xy) + (ALe — My) log (x/y)}.
(1). Divide by Adv + My.
Mdx+ Nady _ Mx Ny x
Meni —i1g log (xy) +i z He Ly 7'°8 (5),
= 4d log (xy) +r (= =) 410g (5),

if M, M are homogeneous functions in x, y and of the same degree.
Since x«/y= ges. this can be written
Mdx+Nd_, i x x
Waray a oe (108 *)a log (?),

= ddu + 1F(v)dv,
where uw = log (xy), v = log (x/y).
ArT. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 425

This case is otherwise solved by the substitution y = 2x, see
§ 297, II.
(2). Divide by Mav — Ny
Midx+Ndy_1 Mx+ MN
“Ma— Ny 2 We — Ny 7 8 OY) + 42 log (zp),
If M= yo(ay), NM = xp(2y), then
Mx + Ny _ dlxy) + ¥(2)
Mx — Ny — $(xy) — ooo
M dx + N dy
jay = a/(xv)@ log (xy) + 44 log (x/y),
= $F (log xy)d log xy + 4d log (x/y),
= 4F(u)du + 4do,
Writing as before, xy = el%*, uw log ay, v=log x/y.
(3). The cases in which J/v + My = 0 were solved in § 297, I,
Ex, 21.

EXAMPLES.

Solve by integrating factors the following equations:

1. y dx — x dy + logxdx =o. (I, Ex. 1.)
Ans. cx +y+lgx+1=—0.
2. a(x dy + 2y dx) = xy dy. Ans. alog #y =y +.
3. (x? + 2xy — y*)dx = (x? — axy — y)dy, Ans. Oya de+y).
a 4S 20(2-4). Ans, 2 —yptoay me
a =

5. (x4y? + xy) y dx + (x27? — I) x dy =o, Ans. y = cery,

6. (xty3 + 1) dy + y dx) + (xy? + ay) y de — x dy) =0, 5
Ans, xy — ao log cy’.
7. Bax +(322y + 2y%)dy = oO. Ans. tt atac xe py
8. (y ty Vay)dx — (x +x Yxy)dy = 0. Ans. y = cx.
9. (x? + 7% 4 2x)dx + 2y dy = 0. Ans. x* + y? = ce-*,
10. (327 — 9 dy = 2xy dx. Ans. x? = y? = oy
VW. 2xy dy = (x? + 9”)dx. Ans, xt? — 3? = cx.
12. (x¢y — 2xy2)dx = (x3 — 3x°y)dy. Ans. at log 3 =6
13. (32274 + 2xy)dx = (x? — 2xty")dy. Ans. ey3 + x? = oy.

14. (yt 4 2yjdx + (xy? + 2yt — andy = 0. Ans. ay py? + ax/y? =e
15. (2x2y — 3yt)dx + (32% + 2xy%)dy = 0. eee
Ans. ga ih y— 88 — ax y-t

16. (92 + 2x°v)dx + (2x8 — xy)dy = 0. Ans. 6 xy =ahy

wa

Se

=e
toe
426 INTEGRATION FOR MORE THAN

 

ONE VARIABLE. [CuH. XXXVIII.

 

 

ay x I
Wx ay axyn. HS: PS a
18. (1 + °)dy = (m + xy)dx. Ans. y= mete Wt +x.

dy tany _ . 3x pe
Wee Sage ee Ans. nS ea

wy
20. --.- =x —y. Ans. y =x —1+4 ce,
ax
ay 3, I
a. ty = Gi Het ay es
22 ay Bz ea exxn, Ans. y = x"(e* + o)
“dx x :
dy | l—2x y oh
23. as a =k Ans. Bal + cex.

 

306. Solution by Differentiation.—A number of equations can
be solved, by means of differentiation as equations of the first order
and degree.

EXAMPLES.

1. Let p = 2 . Let the differential equation be
r=f Pp).

Differentiating with respect to Z,
dx = f"(p) dp.

this gives the equation
dy =f" p) Pp &.
ya [Pee +e

The elimination of 2 between (1) and (2) gives the solution.
2. In like manner, if the differential equation is

dy = f"(p) @.
p dx =f" p) a,
Fe)
ax =
rs

= (Pape
The elimination of g between (1) and (2) is the general solution of (1).

3.«=p-+ logs.

(1)

Since dy =pdx,

(2)

(t)

on differentiation we have

or

(2)

Ans, xis 4 Yay tet log (—1 4 2y oe).
4, 2°77 —-1 + 7’.

Ans. ey + 2exey + 2 = 0.
5. ¥ = ap + &".

Ans. x+ Ya? + aby = alog(a + Wa F 4by) 4.
Art. 306,] INTEGRATION OF DIFFERENTIAL EQUATIONS. 427
EXERCISES.
1. 3e* tan y dx + (1 — e%) sec’y dy =O. Ans. tany = c(1 — e%)8.

2. (4v + 3x)dy + (y — 2x)dx = 0.
Ans. ¢(2y? + 2xy — x*)? vs

3. (2x —y + I)dx + (x + y — 2)dy = 0,

ay + (r+ 3)
ay ~ (1 — 73 )x

dns, log {2032 — 0! + Gy — 5} = Vz tan ¥2G2=9 4,

 

 

ayo 5
4. (x8ex — 2mxy?)dx 4+ 2mx*y dy = oO. Ans. x?ex 4+ my? = x.
5. y(2xy + ex)dx — e*dy = 0. Ans. x*y tex my.
6. dy + (y — e*)dx = 0. Ans. yee =x fe.
7. costx dy + (y — tan x)dx = 0. Ans. y — ce—tanx = tanx — I.
8. (« + Iady = ny dx + eX(x + 1)*tIdx. Ans. y = (e* + ox + 1%.
B si
9. dy = (by + asin x)dx. Ans. y = cehb® — a ee =
0. Fa (e+ ap Ans. 2y = (x + 1) + dx + 1)%
We. xady = ny dx 4 exxntidx, Ans. y = x"(ex + ¢).
12. dy = (y + 1)x dx. Ans. y = ce" — 1.
13. cos x dy + y sinx dx = dx. Ans. y = sin x + ¢cos x.
14, <(1 — a2 + (2%? — Iy = ast. Ans. y = ax + cx Yi — x’,
Le

15. (a + y)dy = @ dx.

16. (« — y)*dy = a ax. Ans. log =

x—y—a

17. x? dy + (y — 2x7 — x*)dx = 0. Ans. y = x? (; + ce*

ee Ans, cx =e
18. x ee a

19. (x? +9? — a%)x de + (x? —y? — By dy =O.

Ans. st a = tant te

a
Sea der er Ga oe

).

x
y

Ans. xt — y* 4 2x2? — 2a°x? — 20°? =o.

20. dy = (x*y? — 1)xy dx. Ans. p(xt-titce™) a.
21. 2xy dx + (7? — x*)dy = 0. Ans. ot x = oy.
22. (« ty)dy + (4 — pdx = 0. Ans. log ¥x? + y? + tant =G

23, (xi + x47? fay + ty dx + (ay — a? — ay + De dy =o.

Ans. x*y? — 2xy log gy = 1.

24, ay) Ay = ie Ans, xy = axe

@ x
6. y—2as(r+#Z). Ans. hyd) ee
CHAPTER XXXIX.

EXAMPLES OF EQUATIONS OF THE FIRST ORDER AND
SECOND DEGREE.

307. The equation of the First order and Second degree is a
dy

quadratic equation in a of the form
a\? Wp
(FZ) + 4h B=o, (x)
where A, Bare, in general, functions of x and _y.
d
We shall represent = by ~. Equation (1) can be written symboli-
x

cally :
I(*) Ip) = © (2)
308. There are three general methods which should be made use
of in solving (1):
(1). Solve for y; (2). Solve for x ; (3). Solve for p.

309. Equations Solvable for _y.—If (2) can be solved for y, the
equation becomes

J = Ex, p). (1)
Differentiate with respect to x.
oF  aF dp
Pa apn de (2)

This equation (2) is of the first order in a

The elimination of » between (1) and the solution of (2) furnishes
the solution of (1). The elimination of ¢ is frequently inconvenient
or impracticable. When this is the case, the expression of x and y in
terms of the third variable ~ is regarded as the solution.

EXAMPLES.
1. Solve p + 2xy = x? + y?. (1)
. - y=xt Vp.
Differentiating,
= 1 &@
bait apt Fa

428
ART. 309.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 429

= op
or ax = b
2pi(p me 1) (2)

 

 

ee
o x= Hog? =!
x eee?

I + e2x%—2e
or = a (3)

Eliminating 4, we have for the solution

ke ex
Dae ca Bb e2%
2. Solve « — yp = ap?. (1)
; ‘ x — ap
Differentiate y = a with respect to x, and put the result in the form
dx I ap

be pi—~)* ~7—p

Solving this linear equation,

 

x= ee (¢ + @sin—1f). (2)
Substituting in (1),
y= —ap +t + @ sin—! f), (3)

The values of x, y expressed in terms of the third variable in (2), (3) furnish
the solution of (1).

3. Clairaut’s Equation.—The important equation, known as Clairaut’s,

: Y=pxt+Kh) (1)
can be solved in this manner.
Differentiate with respect to x,

Eg es Sal nae
parte Ft sens

gq
or e+ rot =0. (2)
The equation (2) is satisfied by either
x+f"(p)=0, or

The solution of (1) is obtained by eliminating » between either of these equa-
tions and (1).
ap

+—=0 gives =c¢, constant.

Therefore one solution is

ap
ax

y=urtHO), (3)

which is the family of straight lines with parameter c.
The second solution is the result of eliminating ~ between

Y= pe +P) l
and o= r+//"(p).

The second of these equations is the derivative of the first with respect to p;
x and y being regarded as constants, f as a variable parameter. This result is

(4)
430 INTEGRATION FOR MORE THAN ONE VARIABLE, [CH. XXXIX,

clearly the envelope of the family of straight lines representing the first solution (3).
This envelope is called the szzgu/ar solution of (1).

Thus the general solution of Clairaut’s equation (1) is effected by substituting an
arbitrary constant for @ in the equation. The singular solution is the envelope of
the family of straight lines representing the general solution.

4. Lagrange’s Equation.—To integrate

Y= afb) + AP) (t)
Differentiating with respect to x and rearranging,
dx _ fd) Fup) _
B* Rae" + FH -7 ~~ ”

This is a linear equation in x and can be solved by § 305, II, Ex. 1.
Eliminating » between (1) and the solution of (2), the solution of (1) is obtained.
Otherwise x and y are obtained in terms of the third variable f.

5. Solve y=(Qitpje +7.
Differentiating, ‘ +x= — 2p.
Solving this linear equation,

x = 1 = p)-- ce;

we YS 2—pP + (rt per.

6. Solve 2«?(y — px) = yp?.
Put =u yoy

which is Clairaut’s form.
v=eu+e, Hence v2? = ex? +2.

310. Equations Solvable for +.—When this is the case
J, J; D) =o

* = FU, p). ()
Differentiate with respect to _y.
1. OF OF a&
po ay * op ay
This is of the first order in 2 The elimination of # between (1)
and the integral of (2), or the expression of x and y in terms of J,
furnishes the solution of (z).

 

becomes

(2)

 

EXAMPLES.
1. Solve «x =y-+ 2’.
I rs dp Ee pdp
pee ea. or Om 2s a
y=e— [P+ 2p + 2log(p —1)], «x = ¢— [2p + 2log(p — 1)}-
202 =y+ log p% Ans. y = ¢ — a log(p — 1), z= ef alog 2.

3. Solve py + 2px = y. Ans. yi s2xetea
ArT. 313.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 431

311. Equations Solvable for ¢.—The equation /(x, y, ~) = 0 is
a quadratic in p.

If this can be solved in a suitable form for integration for J, it

becomes
12 — O(*, »)} 1b — P(x, ”)} = 0.
Each of the equations

P= P(% ¥) and p= p(x, y)
is of the first order and degree in = and their solutions are solu-

tions of (1).
Such solutions have already been discussed.

EXAMPLES.
1. Solve 2? — (x + yp + a7 =0.
(P24 — 9) =0

q
gives dy —xdx=0, and 3 —dx=0.

2avox+e, and y = cer.
22—5/1+6=0. Ans. yroax+og yo3ete
312. In particular, if f(x, y, £) =o does uot contain x or does
not contain y, corresponding simplifications of the above processes
apply, see § 306.

312. Equations Homogeneous in + and y.—When the equation
I(x, ¥, p) = 0 is homogeneous in x and _y, it can be written

r(Z, 2) =o, (1)

(1). Solve, if possible, for and proceed as in § 297, II.
(2). Solve for y/x. Then the equation becomes

ya=xf(P). (2)
Differentiate (2) with respect to ~ and rearrange.
dx _ fp) @
x p—(/P)

EXAMPLES.
1. Solve x2? — 2yp + ax = 0. Ans. 2cy = Cx* + a.
2. Solve y = yp? + 2px. Ans. y? = 20x 4 ?.
3. x2f? — axyp — 377 = 0. Ans. yo x, sme.

: ORTHOGONAL TRAJECTORIES.

313. A curve which cuts a family of curves at a constant angle is
called a /rajectory of the family. We shall be concerned here only
with orthogonal trajectories. If each member of a family of curves
432 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XXXIX.

cuts each member ofa second family of curves at right angles, then
each family is said to be the orthogonal trajectories of the other family.

At any point x, y where two curves cross at right angles, the rela-
tion pp’ = — 1 exists between their slopes /, 7’.

314. To Find the Orthogonal Trajectories of a given Family of
Curves.

Let P(4, ¥, a) =0 (1)
be the equation of a family of curves having for arbitrary parameter a.
Let (x, ¥, p) =0 (2)

be the differential equation of the family (1), obtained by the elimina-
tion of the parameter a.
The differential equation

f (2% -3)= °,

or (x2, = =)= 9, (3)

is the differential equation of a family of curves, each member of
which cuts each member of (1) at right angles. Therefore the general

integral of (3),
c P(x, IY, 6) =o, (4)

is the equation of the family of orthogonal trajectories of (1).

EXAMPLES.

1, Find the orthogonal trajectories of the family of parabole y? = gax.
Differentiating and eliminating a, the differential equation of the family is

wey

dx ~ 2x
The differential equation of the orthogonal trajectories is
ax y
dy 2x
The integral of which is «? + 4y? = ¢?, a family of ellipses.
2. Find the orthogonal trajectories of the hyperbole «xy = a’.

The differential equation is y+ *f = 0. The differential equation of the
orthogonal trajectories is

giving the hyperbole 2? — y? =< for trajectories.
3. Find the orthogonal trajectories of y = mx.

4. Show that «? + y? — 2cy = 0 is orthogonal to the family
yp? = 2ax — x1,
Art. 316 ] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 433

2 2
5. Find the orthogonal system of = +5 == 1, in which 4 is the parameter.
Ans. x? +4 7? = a log x? +e.

6. Find the system of curves cutting x? + é%y? = a? at right angles, «
being the parameter of the family. Ans. yo = x”

THE SINGULAR SOLUTION,

314. We have seen in the case of Clairaut’s equation, § 309, Ex. 3,
that there may exist a solution of a differential equation which is not
included in the general solution. Such a solution, called the simgudar
solution, we now propose to notice more generally.

315. Singular Solution from the General Solution.

Let P(x, y, 6) = 0 (x)
be the general solution of the differential equation
Ix; I, p) = 0. (2)

A solution of the differential equation (2) has been defined to be
an equation (1) in x, y such that at any point x, y satisfying the

equation (1) the x, y, and p = = derived from this relation satisfies (2).

The general solution (1) being the integral of (2) satisfies the con-
dition for a solution. Also, however, the envelope of the system of
curves (1) isa curve such that at any point on it the x, y, p of the
envelope is the same as the x, y, f of a point on some one of the sys-
tem of curves (1), and must therefore satisfy (2). Consequently the
envelope of the family (1) is a solution of (2).

This is a simgular solution. It is not included in the general
solution, and cannot be derived from it by assigning a particular
value to the parameter c.

We may then find the singular solution of a differential equation
(2) by finding the envelope of the family (1) representing the general
solution of (2).

Thus the singular solution of (2) is contained in

p(%, ¥) =9,
which results from the elimination of c between
P(x, y¥,c) =o and G(x, 7, ¢) =o.

316. Singular Solution Directly from the Differential Equa-
tion.—It is not necessary to obtain the general solution of a differen-
tial equation in order to get the singular solution. The singular solu-
tion can be obtained directly from the differential equation without
any knowledge of the general solution.

Let the differential equation

Ke; 2; p) = 9 (1)

 
434 INTEGRATION FOR MORE THAN ONE VARIABLE. [CH. XXXIX.

be regarded as a family of curves having the variable parameter /.
Find the envelope :

X(%, ¥) = 0 (2)
of (1), as the result of eliminating between

A%; I; P)=0 and /j(%, 7, p) =o

d :
Since at any +, y satisfying (2) the +, y, < of (2) is the same as

@ f 5: .
the x, y, = of a point on (1), the equation (2) must contain a solu-

tion of (1).
EXAMPLES.

1. Find the general and singular solutions of 2? + xf = y,

This is Clairaut’s form, and the general solution can be written immediately by
putting 2 = const.

However, independently, we have on differentiation

ap

oO = (x + 2p) Te

a
dx
with respect to ¢c and eliminating ¢, we find the singular solution 4y + x? = o.

Integrating the other factor, x + 2f = 0, or eliminating ~ between this and the
differential equation, the same singular solution is found.

=o gives = c, and y = cx + ¢@ for the general solution. Differentiating «

2. Find the general and singular solutions of the equation y = px + a 4/1 +p
Ans, x+y? =a’,
3. Find the singular solution of x74? — 3xvp + 27? + 28 = 0,
Ans. x*(y? — 4x3) = 0.

317. The Discriminant Equation.—The discriminant of a func-
tion (x) is the simplest equation between the coefficients or constants
in #(«) which expresses the condition that / has a double root. If
has two equal roots, equal to a, then

F(x) = (vw — a) P(x),
where ¢ is some function which does not vanish when x = a. Hence,

differentiating and putting + = a, we have the conditions for a double
root at a,

 

F(a)=o0, F(a)=0, F(a) #0.

Eliminating @ between /(a) = 0, #’(a) = 0, or, what is the same
thing, eliminating x between #(x) = 0, F’(x) = 0, we obtain the
discriminant relation between the coefficients, the condition that
(x) shall have a double root.

318. c-discriminant and /-discriminant.

Let (x,y, c) = 0 be the general solution of the differential
equation /(x, y, 2) =o.
ART. 320.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 435

(1). The equation (x, v) = 0 which results from the elimination
of c between the equations

A(x, ¥,c)=0 and G(x, y,c)=0
is called the ¢-discriminant, and expresses the condition that the equa-
tion ¢ = o, in ¢, shall have equal roots.

(2). The equation y(%,_v) = 0 which results from the elimination
of ~ between the equations

Kx, p) =o and (4, y, p) =o

is called the f-discriminant., It expresses the condition that the equa-
tion f = 0, in J, shall have equal roots.

319. c-discriminant contains Envelope, Node-locus, Cusp-
locus.—The c-discriminant is the locus of the ultimate intersections
of consecutive curves of the family (+, y, c) = 0.

It has been previously shown that the envelope of the family is
part of this locus, and also that the envelope is tangent to each member
of the family. :

Suppose the curves of the family have a double point, node, or
cusp. Then, in case of a node, two neighboring curves of the family

 

Fic. 155.

intersect in two points in the neighborhood of the node, which con-
verge to the node-locus as the curves converge together. In the neigh-
borhood of the envelope two neighboring curves intersect in general
in but one point.

In the case of a cusp, two neighboring curves intersect, in general,
in three points in the neighborhood of the cusp-locus. Two of these
points may be imaginary.

We may expect to find the envelope occurring once, the node-locus
twice, the cusp-locus three times as factors in the ¢-discriminant.

320. p-discriminant contains Envelope, Cusp-Locus, Tac-Locus,
—If the curve family (x, y, 2) = 0 has a cusp, then for points along
the cusp-locus the equation vanishes for two equal values of p, as it

aq ,
does also for points along the envelope. But, in general, the ee of the

cusp-locus is not the same as the pf of the curve family and therefore
does not satisfy the differential equation.
436 INTEGRATION FOR MORE THAN ONE VARIABLE. [CuH, XXXIX.

Again, at a point at which non-consecutive members of the curve
family @ = o are tangent the x, y, p of the point satisfies the equa-
tion = 0. ‘The locus of such points is called the /ac-locus. The

d ‘
~ of the tac-locus is not the same as that of the curve family @ = 0,

and the tac-locus therefore is nota solution of f = o.

321. It has been shown by Professor Hill (Proc. Lond. Math. Soc.,
Vol. XIX, pp. 561) that, in general, the

the envelope oxce,
c-discriminant contains { the node-locus /wzce,
the cusp-locus ¢hree times,

the envelope once,
p-discriminant contains ~ the cusp-locus once,
the tac-locus /wice,

as a factor. ‘This serves to distinguish these loci. Of these, in
general, the envelope alone is a solution of the differential equation,
It may be that the node- or cusp-locus coincides with the envelope,
and thus appears as a singular solution.* The subject is altogether
too abstruse for analytical treatment here.

EXAMPLES.

1, xf? — (« — a)? = 0 has the general solution

y yte a gd — 2023,
eo or Ay + 6P = 44x — 3e)?.

fe The p-discriminant condition is x(x — a)? = 0, the
¢e-discriminant condition is.2(x — 3a)? = 0. «= 0 occurs
once in each, it also satisfies the ditferential equation and
is the singular solution or envelope. « = a occurs twice
in the f-discriminant and does not occur in the ¢ dis-
criminant. 2« =a is therefore the tac-locus. «+ = 3a
occurs twice in the c- and does not occur in the J-dis-

criminant. x = 3a is therefore a node locus.

ee

2. Show that (vy + c)? = xis the general solution of
4p" = gx, and «8 =o is a cusp-locus. There is no
singular solution.

3. Solve and investigate the discriminants in

 

 

 

 

O a a Poewpay.
ob General solution (243 4- 3xy + ¢)? = 4? + y)8. No
singular solution. Cusp-locus x? + y = 0.
Fic. 156. 4, In 8a? = 27y, show that the general solution is

ay* = (« —c), singular solution y = 0, cusp-locus y§ = 0.
5. Find the general and singular solution of y = xp — p?.
Ans. yoew— 0, «x= dp.

 

*Proc, Lond. Math. Soc., Vol. XXII, p. 216. Prof. M. J. M. Hill, +‘ Ox
node- and cusp-locit which are also envelopes.”
ART. 321.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 437

EXERCISES.
Find the general solutions of the following equations.
1 2? = ax. Ans. 25(y +c)? = 4gax,
2. 2 = ax. Ans. 343(¥ + 6) = 27ax7.

3. P(x + 2v) + 3px + y) + A(y + 2x) = 0. Factor and solve.
An.y=o xty=ey ytxettytae

4.2? —7p +12 =0. An. ya=qxto yo3rte
5. 2p? — 2yp + ax = 0, Ans, 207 = x? + a.
6. 9p? + 2xp = ¥. Ans, y* = 20x + c.
7. xp? - 2xyVp + ay? —-xt=0 Ans. sin-1 2 = log Cx.
x
By = pr — +5. Ans. y = x — 0) +.
9. xy? p? + 2) = 2py3 + x3. Ans. (x7 — y* + cy(x? — x? 4+ ext) = 0.
10. y + px = x'p’. Ans, xy =e 4 Cx.
W. ayf? + (2x + dp —y=Oo. Ans. act + eax — 6) —y* = 0,

12.7 —px = WI +p? f(x? +7”). Change to polar coordinates.

Ap’ )dp
Ans. 6+ ¢= SSS ees
fs Vp? — { Aip?)}?

13. (xp — 9 = a(t + pra? + 97)8,

Ans. tan 2+ ¢ = vers~t2a x? + y?,

14. (<p —yP =f? -2 “9 +1. Ans, sin-* z =sectxc,

15. 3f°y? — 2ayp + 47? — x? = 0, Put 27 — 3y? = 07,
Ans. 3(x7 + 9”) + 4ex +P =O.
16. (x7 + y?)(1 + A)? — Axe + NE + p~ +96) + (* + YP? = 0.
Ans. x? + y? — 2x +y) + 4A =0.

 

a =a. Ans. (y bePF tle —apP=t,
Vit?

18. y= px 4-p — #*. Ans. yamex+e—

19. y? — 2pxy — 1 = p(1 — +"). Ans. (y — ex =1+e.

20. y= 2px + yf. Put y= 2, Ans, y? = cx +30,

21. x(y — px) = Ip. Ans, y* = cx? ae

22. (px — yoy + +) = HB. Ans, y? — ext = — a

= BE atp?, Ans, y = ex + YR + ae’,

ema elds singular sdilion e/a + 7/P? = 1.
24. y = p(x — 4) + 2/f, singular solution, y? = 4a(x — 4).

25. (vy — xp)(mp — 2) = mnp. Ans, (y — cx)(me — n) = mnc,

singular solution, (x /m)* + (y/njt = 1.

26. — 2ayp tit y= Ans. (y — xP = 1 — A,

singular solution, y? — «* = 1.
438 INTEGRATION FOR MORE THAN ONE VARIABLE. (Cu. XXXIX.

27. 2° — gxyp + 8y? = 0. Ans. y = ¢(x — of,
singular solution, 27y = 4x3,

28. Find the orthogonal trajectories, A being the variable parameter, of the
following curve families:

(1) fF oa Ans, x* + y? = a? log x? +e,
* a2 A2
(2). 2? + my? = mA? Ans. y = cx.
2 2 2 2
(3). ect ee Ans. = - =
29. Find the orthogonal trajectories of the circles which pass through two fixed
points. a Ans. A system of circles.
30. Find the orthogonal braleetories of the parabolz of the th degree
an—ly = x", Ans. mPtvto=e,
31. Find the orthogonal trajectories of the confocal and coaxial parabolze
y? = 4a(x + A). Ans. Self-orthogonal.
32. Find the ortho-trajectories of the ellipses x?/a? + y?/2? = A2,

Ans. y? = cx.
33. Show that if

d,
ft ( ” 4, *) =o
is the differential equation of the family of polar curves @(p, 6, ¢) = ©, then
, 2
1(e 6, 0 Z)= °
is the differential equation of the orthogonal system.

34, Find the orthogonal trajectories of 9 = a(1 — cos 6).
Ans. p = c(i + cos 6).
35, Also the ortho-trajectories of —
(1). p% sin 7@ = a, Ans. p*cos x) = c#.
(2). p= log tan@ +a Ans. 2/p = sin’6 + ¢
CHAPTER XL.

EXAMPLES OF EQUATIONS OF THE SECOND ORDER AND
FIRST DEGREE.

322. The differential equation of the second order and first degree
is an equation in +, y, J, ¢,

IF: ty TD) = 9;

w ap ay
ae? 2= a dx?

in the first degree.

We shall attempt the solution of the equation for only a few of the
simplest cases.

We have seen that the general solution of the equation.of the first
order and degree gave rise to a singly infinite number of solutions, ,
represented by a family of curves having a single arbitrary parameter,
this parameter being the constant of integration.

In like manner, the general solution of the equation of the second
order and first degree, involving two successive integrations, requires at
each integration the introduction of an arbitrary constant. The
general solution, therefore, contains two arbitrary parameters, and is
correspondingly represented by a doubly infinite system of curves, or
two families, each having its variable parameter.

The process by which a differential equation of the second order is
derived from its primitive is as follows.

where p= and in the equation ¢ occurs only

Let P(X, Y, C1 ¢,) = 0 (1)

be an equation in x, y and two arbitrary constants c,, c,. Differ-
entiating (1) twice with respect to .v, there results

 

ae (Ow,
sty Yo, (2)

vd a i ay\? af dy
at * 7? Ox wast 5: ax? \dx +4 dy dx® aes (3)

Between these three equations can be eliminated the two arbitrary
parameters c,,¢,. The result is the differential equation of the

second order,

I(%, I Py 7) = %
439
440 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XL.

EXAMPLE.
The simplest equation of the second order is
aty
dx? ~
Here the integrations are immediately effected.
y
d(—)=2 ax,
a
q
> = kx + “1s

¢, being the first constant of integration. Integrating again, the general solution is
Y= ex + gx
The two arbitrary parameters c,, ¢, giving a doubly infinite system of parabola.
P yr &2 & g y P

323. The Five Degenerate Forms.—The ordinary processes of
integrating differential equations are of tentative character. We are
led to the solution of general forms through the consideration of the
simpler cases. Investigation of the general methods of treating this
subject is out of place in this text, and we shall consider here only a
few interesting and important equations of simple form.

A general method of solution can be proposed for the five degen-
erate forms of the general equation,

r. J(%, 9) = 03 2 J, 9) =0; 3 AP 2) = 93
4. K(*; P, 9) = 95 5 Nb 9) =o
324. Form /(x, 7) = o.—This being of the first degree in g,
d*’y
dx®
The differentials involved are exact, and it is only a question of
integrating twice. The solution is

Os
fe =f Fede + Cy.

ot gy = [ae f Fax + ¢x+ ¢,.

 

= F(x).

Ex. g = xe. Ans. y = (x — 2)e%§ + Ox + 0.
325. Form /(y, 7) = o.—Here
ey
ae Fy).
DQ _ dy _@_Wdp_

The equation becomes
pap = F(y) a.
ArT. 326.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 441

ee 2 fFo)4 +e

dy
12 fy =e ane

The integral of this gives the solution.

Integrating,

Bi OS

EXAMPLES.

2.
1. Solve a = ay.

2 f Ky)dx = ay’, Put «4 = ae
eS an
oY +4
Hence ax = log(y + Wy? +e) +e
Show that this can be transformed into
Y = o[e9* + clean,
Multiply the given differential equation by 2 a
dy dy _ d@ (dy\? 25,0 ot rae
ade ae (2) ¢. Og ">
Hence the first integral is, as before,
dy\ 2
(2) = ay? + k.
a
2. Solve ne + ay =o.

 

ax
Here 2 f FAy)dy = — a4, Put gq = ae’,
aax= a ‘
Ve—y~
ee
Hence ax + ¢, = sin“ os
or y = sin (ax 4- ¢,),

= 4, sin ax + &, cos ax.
Multiply the differential equation by 29 and obtain the first integral directly as

in Ex. 1. ; : .
Examples 1 and 2 are important in Mechanics.

3. Solve g ay =1. Ans. 3x = 2ah(y? — 24) t+ at +
326. Form /(?, 7) =o
dy af @ aD _
Here a ¥(2), oo F(?).
ap

oe

= FA) Cy.
442 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XL,

This is an equation of the first order, the solution of which is that

of the required equation.
EXAMPLES.

dry ay\?
1. Solve Ta + @(%) =0.
Integrating p-2d~ + adx, we have for the first integral
ax
ae axe
Paes 2 = log (ax +e) + ¢,

or P= 4K + Cy.
x
2. Solve a@ oS = = Ans, y = C674 ty
3g=f7/41. Ans. eY = ¢, cos (x + G).
4¢q+p7p?+1=0. Ans, y = log cos (x — 4) + 4.
327. Form /(x, ~, 7) =0.—Such equations are reduced to the
ap

first order in x and / by the substitution g =

dx*
feb) =f (2 #)=0

EXAMPLES.

is equivalent to

+ hee 7 4 ax=0

Bx =0.

ax
re hee
The first integral is
i ee ee yt * I+a
The second integration gives
y= — ax+elog (x + 1 + 2).
2 (r+ e+ P+1=0. Ans. y = Gx + (47 + 1) log ( — 4) +%
328. Form /(y, 2, 7) =0.
_ty Db _ wah ,@
oS aa ae die a a
Substituting for g, the equation is reduced to the first order in y
and /.
EXAMPLES.
2 y 4 = =0. Ans. ane = ealet0),

z
eae Ans. y8 = at Ge + &

3.79 — pf? = y* log y. Ans. log y = ce* + Ge-*.
ArT. 329.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 443
329. Solution of the Linear Equation
a@y dy
mero gee oe (1)
in which A, # are constants.

‘The solution of this equation is suggested by the solution of the
corresponding equation of the first order

 

a +a =o,
‘ . ay : boas
which Bie = — adx, the solution of which is y = ce-**,
If we try y = e”* in (1), we have
oF dO 4 Bee = obo dw Bye (2)
ax? ax ia :

I. Roots of the Auxthary Equation Real and Unequal.—The func-
tion (2) vanishes if m be one of the roots of the auxi/iary equation

m+ Am+ Bs (m—m,)(m—m,) = 0. (3)
Hence y = e”17is a solution. Also, y = ¢e”:* is a solution for
any arbitrary constant ¢. In like manner y= ¢,e2* is a solution.
The sum of these two,
i — cee + ce", (4)
is also a solution, and is the general solution of (1) since it contains
two independent arbitrary constants, ¢, and «¢,.

II. Roots of the Auxiliary Equation Real and Equal.—l{ m, = m,,
the solution (4) fails to give the general solution, since then

J = (¢, + €,)e"*,
and c,-++c, =c’ is only one arbitrary parameter.

The solution in this case is immediately discovered on differentiat-
ing (2) with respect to m. For then
eae : sa 4 Bem = (am 4+ Ayem* 4 (m?4 Am + B)xem.

If m= pt is the double root of (3), then (3) and its derivative
vanish when m — yw. Consequently y = xe“* is a solution, and
also is y = cve**. Hence the sum of the two solutions c’e#* and cxe*
is the general solution of (1) when y is a double root of (3), or

y= ote(0 + ox), (5)
III. Roots of the Auxiliary Equation Imaginary.—When the roots
of (3) are imaginary and of the forms

m,=a+i, m,=a— wd,
444 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cu. XL.

where? = 4/— 1, these roots may be used to find the solution. For
(4) becomes
y= c,e(at ibe + C60),
= e*(c,e* + c,e-#),
We have by Demoivre’s formula
e®* = cos dx + 7 sin bx,
e—%= — cos dx — 7@ sin bx.
Therefore the solution is
y = e*{(c, + ¢,) cos dx + (c, —c,) #sin dx},
= e*(k, cos 6x + &, sin bx), (6)
where &,=¢,+¢,, & = (¢,—c,)t. If the arbitrary constants

c,and c, be assumed conjugate imaginaries, the constants 4, and &, are
real.

By writing tana = 4,/k,, or cot 6 =4,/k,, the solution (6)
may be written respectively
y = c'e* sin(dx + a),
= ce cos(bx — £). (7)

EXAMPLES.
1. Solve g —p = 2.

The auxiliary equation is
m—m—2 = (m+ 1\m — 2) =0.
The general solution is therefore y = c,e-* + ¢,¢2%,
2.1f g—2p+y=0, (m—1P? =0, .. y = e%(e, + Gx).
3. Solve g +3f = 5y4y.
m* + 3m — §4 = (m — 6)(m + 9).
oe Y = 4% + Ge 9%,
4. Solve ¢+ 89+ 25y =0.
m+ 8m+25—=0 gives m=—443Y—L
‘. y¥ = €—4%(h, cos 3x + &, sin 3x).
330. Solution of the Equation
ee + Ax oo By =0, (z)
A, B being constants,
Put x =e, then z= log x. Also,
dy _dWde _idy dy _ 1 (dy wy
a dae ae gana (ae — a
On substitution, equation (1) becomes
os ia = 1) Oa By = ©;

which is the form solved in § 329.
ART. 331.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 445

Ex. Solve 229g — xp ty = 0.
The equation transforms into
ay gy
a _ a + y=O

Y= Ae, + O2) = x(q. + & log x).

331. Observations on the Solution of Differential Equations.
—The remarks made on the integration of functions are equally
applicable to the integration of differential equations. The process
is of tentative character, and skill in solving equations comes through
experience and familiarity with the known methods of solving the
integrable forms.

When the equation is not readily recognizable as one of the stand-
ard forms for solution, it can frequently ve transformed into a recog-
nizable form by substitution of a new variable.

Most of the processes given in this chapter for the solution of
certain forms of the equation of the second order and first degree are
immediately applicable to equations of higher orders. In the exer-
cises will be found certain simple equations of higher order than the
second, proposed for solution by the methods exposed in the text.

General methods of solving differential equations must be reserved
for monographs on the Theory of Differential Equations.

EXERCISES.

1 = sax + by. Put a@r+ey = 2, ete.

Ans. atx + by = c,eb* + ce,
2. o = ax — by, Ans. ax — by = ¢, sin dx + ¢, cos bx.
W289 + GF — ee v2
22 + P+ q a

or 2¢” = ¢? sec%(4e,x + ¢), according as the first constant of integration is

3.¢ =". Ans. ¢e1% =

+47, 0, or —4%

4.29 +p =0. Ans. yoologx +e.

5. 7 = xp. Ans. y= a fear + ey.

6. °¢ = ay. Putz = 2y/x% .. xy = Gr +e.

7.9 + 12y = 7p. Ans. y = 603% 4+ cet.

8. 3(¢ +7) = 10p. Ans, y = 46% + cet”.

9¢+ 4p =p. Ans. ye =¢,e* V5 4+ ce-* v5.
ax bx

10. ay +9) = (24+ &)p. Ans. y= Geo + Gee,

a*y ay

OP gk Ans. yy = ce2* + cge—2* 4 ey.
7 = 477 1 2

11.
446 INTEGRATION FOR MORE THAN ONE VARIABLE. (CH. XL.

12. 7 — 66+ 13y = 0. Ans. y = (ce, sin 2x + ©, cos 2x)e3*.
13. 7 — 2ap + by = 0. Ans. According as a > or < 4,
y = 0x (cer VaI=F8 4 ce—2Va2- 83), or c04(c, sin.x 4/0 — a? + c, C08 x VF — a).
14. ¢ — 4abp + (2 +B Yy = 0.
Ans. y = eabx{c, sin (a? — 8)x + c, cos(a? — 8x},
_ 15. g — plog.a? + [1 + (log a)]y =0. Ans. yrar(e, sin x + ¢, cos x).

16. 9 — 2ap + a = 0. Ans. = eax(C, + yx).
17. g =0. Ans. yout en.

ay
18. 7a =e Ans. y = cet + ey + Oye,
19. x*¢ — xp = 3y. Ans. xy = 4x4 + 6.

20. (2 + dx)¢ + da + dx\p 4 By =O,
Ans. y = ¢, sin log (2 + bx) + ¢, cos log (a + 4x).

 

21, x a = 2. Ans. y = 6, + Gx + oye? + x? log x.
By Fs
22. iam sin8x, Ans. y = G+ Gx + 6x + feos x — sycos x.
23. gv = a, Ans. (qx +) = ay? — a.
j x x

24. a7 =i1 +p Ans. 2y/a = cer + qe # +

25. a’¢? = (1 + 2”). Ans. (x +aqrP+(y+oraa.

26. (1 — x*)g — xp = 2. Ans. y = ¢, sin—tx + (sin-tx) +

27.99 +p? = 1. Ans. Pos x? taxt &.

28. (1 — logy) yg + (1 + logy)p?= 0. Ans. (qx + &)(logy — 1) = 1.

29. yg — 2? = 7" log y. Ans. log y = Ge* + ce.

30. (p — xg)’ = 1+ 9%. Ans. y= tari ta it ai+ e.

31. Find the curve in which the normal is equal and opposite to the radius of
curvature. [Catenary. ]

32. Find the curve in which the normal is equal to the radius of curvature and
in the same direction.

33. Find the curve in which the radius of curvature is twice the normal and
opposite to it. The parabola, «7 = 4e(y — ¢).

34. Determine the curve in which the normal is one half the radius of curva-
ture, and in the same direction.

The cycloid «+ ¢sin— < + yay —y =o.

35, Find the locus of the focus of the parabola y? = gax as the parabola rolls
on a straight line. (Catenary.]

36. Find the locus of a point on a circle as it rolls on a straight line.

37. Express the locus of the center of an ellipse as it rolls on a straight line in
terms of an elliptic integral.

38. The problem of curves of pursuit was first presented in the form: To find
the path described by a dog which runs to overtake its master.
ArT. 33!.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 447

The point 4 describes a straight line with uniform velocity; it is required to find
the curve described by the point 4, the motion of which is always directed toward
A and the velocity uniform.

Take the path of A for y-axis. The tangent intercept on the y-axis is y — x.
By hypothesis the change of this is proportional to the change of arc-length.

oo —xdpo=myi-+t pdx,
log x™ + log (2 + 71+ ~*) + log a =0,
2p = cyte — Gm,
axmeti , atm +1

Se a ee

 

lL m—i-

The curve is algebraic, except when m = 1, then we have to substitute log x
for — «—™+1/(m — 1).
APPENDIX.
SUPPLEMENTARY NOTES.

449
APPENDIX.

NOTE 1.

Supplementing § 30.

Weierstrass’s Example of a Continuous Function which ha
nowhere a Determinate Derivative.* ;

The function
J(*) = 2 8” cos (a%rx),

in which x is real, a an odd positive integer, 4 a positive constant less
than 1, isa continuous function which has for no value of x a deter-
minate derivative, if ad > 1 + 37,

Whatever assigned value x may have, we can always assign an
integer 44 corresponding to an arbitrarily chosen integer m, for which
—} <a" —pSth

Put Xp, = a"X — HM, and let
, H—I1 nH + I

 

 

= a
PO Ah ae
i+ ~ I— 4,
/ Aes m+ df m+r
SS oe. =
and at <p ol’,

The integer m can be chosen so great that x’ and x” shall differ
from .v by as small a number as we choose.

We have

K(x’) —/(*) op? cos (a"7rx') — cos (a*7x)

?
ee x Hx

n=O

cos (a"x') — cos (a"7x)

=) (aiy

a(x’ — .v)
uz=O
oo
weit cos (a"*" 7x") — cos (22th ae) .
go or eee ge eee, TN
m=0

 

* Taken from Harkness and Morley, Theory of Functions.

451
452 APPENDIX.

 

 

Since
sin | a” a= vn
1
cos (a" 2x’) — cos (a"ax) _ wf gke t 2
= — 2 sin( a*—__ 12 | — ‘—_ ,—_ —’,
a"(x’ — x) 2 at eee
Q'—— 14

2

and since the absolute value of the last factor on the right is less than
1, then the absolute value of the first part of (i) is less than

Sor = PD,

 

and therefore less than x(ei) ,»ifad > 1.
Also, since a is an odd integer,
cos (a”*"2x') = cos [a"(ué — 1)z7] = — (— 1)4,

cos (a"t" 7.47) = cos (aun + aX, 4,7) = ( — 1)" cos (2"%,,4, 72).
Therefore

gotn COS (a"*" mx’) — cos (a"*" 712)
x’ — x%
n=O
I cos (a"x,, 70
= (— 1)#(ad)™ on 1 605 (2 Races) (2° nt ).
I + Xintr
n=O

All the terms under the 2 on the right are positive, and the first
is not less than 2, since cos (#,,,,7) is not negative and 1 + x

 

lies between $ and 3. is
Consequently
Ae) — fe) _ sl, HN oe
ge ge eT a ae wD

where & is an absolute number > 1, and 7 lies between — 1 and 4-1.
In like manner
Nx") —/*) 2 a1 S
DAU 4 NT _ (— 1) (ab)” &'( =
a (—rmesme(E+ Fe), Git
where &’ is a positive number > 1, and 77’ lies between — 1 and + 1.
If ad be so chosen as to make
ab > 1-+ 32,
2 a
ab —1
the two difference-quotients have always opposite signs, and both are
infinitely great when m increases without limit. Hence /(x) has
neither a determinite finite nor determinate infinite derivative.

 

that is,

 
SUPPLEMENTARY NOTES. 453

Every point on such a line, if line it could be called, is a singular point.

Some idea of the character of the geometrical assemblage of points representing
such a function can be obtained by selecting
two particular fixed points 4, B of the as-
semblage. Between 4 and B, in progressive
order, select points P,, P,, . . . representing
the function corresponding to x, +, ...
Consider the polygonal line AP, P,.. . B.
Increase the number of interpolated points in-
definitely, and at the same time let the dif-
ference between each consecutive pair con-
verge to 0. Then, since the function /(x)
is continuous, each side, P, P,+,, of the broken
line converges to 0, But, instead of each
angle between consecutive pairs of sides of
this polygonal line converging to two right
angles, z, as their lengths diminish indefi-
nitely, as was the case when we defined a curve with definite direction at each point;
let now these angles converge alternately too and 27. The polygonal line folds
up in a sigsag. The point P converging to the neighborhood of a true curve 42.
But the difference-quotient at any point of the zigzag assemblage has no limit, it
becomes wholly indeterminate as the two values of the variable converge together.
It is also possible that the length representing the sum of the sides of the polygonal
between any two points of the assemblage at a finite distance apart (however small)
is infinite in the limit.

Such functions are but little understood and have been but little studied. It is
possible that they may have in the future far-reaching importance in the study of
molecular physics, wherein it becomes necessary to study vibrations of great velocity
and small oscillation.

 

Fic, 157.

NOTE 2.
Supplementary to § 42,
Geometrical Picture of a Function of a Function.

If z= /(y), where y = G(x), we can represent the function 2
geometrically as follows:

Draw through any fixed point O in space three straight lines Ox,
Oy, Oz mutually at right angles, so that
Ox, Oy are horizontal and Oz is vertical.
These lines fix three planes at right
angles to each other. .vOQy’is horizontal,
vOzs and 4’Oz are vertical.

The relation 1 = (x) can be repre-
sented by a curve P’Q’ in the plane
xvOy, Atany point ?’ on this curve we
can represent 3 by drawing Le = /();
up if ((¥”) is positive, down if /(y) is
negative. The relation 2=/(y) is
represented by the curve PQ” in yOz.

. =e (>), Fic, 158,
as a function of .v, is represented by the
curve P”Q” in «Oz. In other words, z as a function of + and y is

 
454 APPENDIX.

represented by a point in space having the corresponding values
z, ¥, x as coordinates with respect to the three planes. The assem-
blage of points representing z, y, « is a space curve PQ. The
orthogonal projections on the three coordinate planes of PQ represent
the functional relations

(P'Q), v= O(@)s (PPO), =O) (PO), 2 =S0).

The derivative D,y is represented by the slope of P’Q’ at P’ to
Ox. The derivative D,z is represented by the slope of the tangent
to P’’O"" at P’” to Oy; the derivative D,z by the slope to the
axis Ox of the tangent at P” to P’Q”.

The function of a function is represented by a curve in space.

NOTE 3.

Supplementary to § 56.

The zth Derivative of the Quotient of Two Functions.

Let vou/v. Then uw=yw. Applying Leibnitz’s formula to
this product, we have
“= vy,
nu’ vy! /
a ep
ul! vy’ vw’ yy yy"
a at 1! i! ai”
ul" y" yt yy y-2 yy” wy"
—— gD uv.
n} 2a Sey) — 1)! ri! @—a)lal a + n!
To find _y”, the th derivative of w/v, in terms of the derivatives of
pal
wandv. Eliminate gg ts Be icbag aaa from the 2 + 1 equa-
1! n— 1)!

tions. We get

I ppf@\. (— 1)"|4%# v oo...
a) ores
: eae

u!
= vo e
1! a!

ul! vl vy’ ;
a! af x!”

(n + 1) rows
SUPPLEMENTARY NOTES. 455

Also, in particular, if « = 1, we have

vi f
P| a Ee oe ee
n! v (v)"** ) a! :
vy’ v
— —v20
2} x!
gy vy! vy’
ee
| 7 YOWS
NOTE 4.

Supplementary to § 56.

To Find an Expression for the zth Derivative of a Function
of a Function.

Let z = /(y), where » = @(x). To find the zth derivative of
g with respect to x.
We have, by actual differentiation,
Se =SiIx
TE =e thet,
Jag = fis —3 i + fi yl
x ae Le
The law of formation of these first three derivatives of / with
respect to x shows that the th derivative must be of the form

An = Af + Ady Spite SP ASS (r)
where the coefficients, 4,, contain only derivatives of y with respect to .v
and are therefore independen/ of the form of the function ~ Conse-
quently, if we determine A, for any particular function /, we have
determined the coefficients whatever be the function % Let then
—6
fy) = 2
Then in (1) we have
te apo gts SBR. teh
ri “y — ) 2 (r— 1)! ak am r—I re

1!
Hence, when 4 = y, we have
I
= — {Dy — Br :
4 =a {Dual
which means that (vy — 4)” is to be differentiated ~ times with respect
to * and in the result y substituted for J.

(Zn = 52 | Dats — yr} (3)
456 APPENDIX.

This gives the mth derivative of f with respect to x in terms of the
derivatives of / with respect to y and those of _y with respect to x, and
is the generalization of the formula

qd _ 40) &

We can give another form to (3), as follows. Let_y= 4 when

xa. Then
— b= g(x) — (2) = (* — 2)p, (4)
where v stands fee the difference-quotient

P%) — Pla)

x—a
Apply Leibnitz’s Formula to (4), and we have
Diy — 6)" = Dix — a)’v",
= 2C,,,Drtu Dix — a).
p=o
But, Dé(x —a)’ =r(r—1)... (7 —p+1)(x — a)",

= 9 when p>y7,
=:9 when <7 and x=a,
= 7! when P=r and wx=a,

Therefore (3) becomes

(;)09= 3 eu tren(g) (EHY

Notes 3 and 4 give some idea of the complicated forms which the
higher derivatives of functions assume.

NOTE 5.
§ 64. Footnote.

If a function /() and its derivatives are continuous for all values
of x in (a, f) except for a particular value a of x at which f(a) =,
then all the derivatives of /(+) are infinite at a.

Let x, <v, <a. Then

A) —S(*) = (x, — x,)f"(8),
where & lies between x, and x, Let @— x, be a small but finite
number, and let x,(=)a. Then /(x,) is infinite, and /(x,) is finite.
(a—x)f"(E) =o.
Since a — x, isfinite, #’(&) = o ; and since /’(&) is finite ifa — §
is finite, we must have a — §&(=)o and

f(a) = 0.
SUPPLEMENTARY NOTES. 457

In like manner we show that /’"(a) = o , and so on.
Corollary. If /(2) = 0, then /’(a2) = ow, and also

J (*)
J\*)
becomes oo when + = a,
For, considering absolute values, if f (2) = 0, then also

log /(2) =. By the theorem established above, if log /(.x) is 0
when . = a, then
S'(*)

J()

 

D log f(x) =

also becomes oo when x = a.

 

NOTE 6.
Supplementary to Chapter VI

On the Expansion of Functions by Taylor’s Series.

1. This subject cannot be satisfactorily treated except by the
Theory of Functions of a Complex Variable. The present note is an
effort to present in an elementary manner by the methods of the
Differential Calculus a fundamental theorem regarding the elementary
functions.

An elementary function may be defined to be one which does
not become oO or oo an infinite number of times in any finite interval,
however small. Such functions are also called rational.

A function /(x) is said to be unlimitedly differentiable at + when
all the derivatives /”(x) of finite order are finite and determinate at
x, We consider only those functions which are such that neither
the function nor any of its derivatives become 0 or o an unlimited
number of times in the neighborhood of any value + considered.

2. In the same way that a function of the real variable . may be
o for an imaginary number p + 7g, such a function may be o fora
complex number p + 7g, where 7 = ¥—1. For example, the func-
tion
Fe) = 9)
F)
becomes oo at p + wif p+ 7 is a root of (x) and not of d(x). A
value of x at which #(x) is 0 or oo is called a roof or pole, respectively,
of the function. It being understood that there are not an indefinite
number of roots or poles in the same neighborhood. *

 

* A point in whose neighborhood there are an infinite number of poles is called
an essential singularity. An isolated pole is called a non-essential singularity,
458 APPENDIX.

The poles of a function, whether imaginary or real, enter into the
results which we shall obtain. Wherever we use the word function
in this note we mean a uniform function which has only roots and
poles, éu¢ no essential singularity, and which is unlimitedly differen-
tiable everywhere except at a pole.

3. Theorem I.—If /(x) is a one-valued, determinate, and unlim-
itedly differentiable function at x, then the series

S= Sere

is absolutely convergent for all values of_y less in absolute value than

R=+V(% —PP +,
where pf + 7g is the nearest pole of /(..), or any of its derivatives /”(.x),
to the number x; and the series S is « for any value of_y greater
than 2.

4. Represent x, y by the coordinates of a point in a plane «Oy.
Then (see § z5, Ex. 9, 10):
(1). At all points x, y at which
y_ f(a)
= < I
bin re pole

S is absolutely convergent, and also

YP) =o.
(2). At all points se at which
aI
LEG
S = o, and also =
- "(x)= 0.

5. It follows, therefore, that if

 

* Remembering that the modulus or absolute value of any number x + zy is

le+yla+ Vx? +7,
then of two numbers Z, + zg, and g, + zg, that one is nearest + for which we have
the difference
Pre]

least.
SUPPLEMENTARY NOTES. 459
has a finite limit different from 0, it is necessary that

fae:
a+r f"(xyo7 1

n=o

6. Since at all points of absolute convergence of
y
me = 0,

and at all points of infinite divergence of S

 

PC
= "(v) = o,

the boundary between absolute convergence and infinite divergence
of S is marked by the values of .., y which satisfy

wy " =
a f “(#) = 1.
7. The locus
IM” gatgy =
ma (x) = I,
for an arbitrary and great value of 2, will be a close approximation to

the boundary line we seek. Differentiating, this locus has the differ-
ential equation

 

ay _ ate
ae FRX
z r\ y f(x)
S(t lie fer
Which for # arbitrarily great gives, in the limit,
a
Wag Ve |r,

in virtue of

 

 

f J Pe ite
a+r f"(x)
on the boundary.

8. Therefore the absolute value of y is equal to the absolute value

of a linear function of x, of the form
e = |z — x| 2
for all values of x and_y on the boundary.

This is the equation of the family of boundary lines having the
parameter 4. These lines are fixed by the fact that whenever /(~) or
J’ (x), for any finite 7, is oo we have y =o.

9. If, therefore, x) = co when x = #, the corresponding bound-
ary lines for a real pole # are the two straight lines

— (p — Sa ae
or you—p and y= —x-+f.
460 APPENDIX.

If A(x) = «© when x = p + 7, then the corresponding boundary
lines for a complex pole p + 7g are the two branches of the rectangu-
lar hyperbola

od |p+ i — x),
=(p—«} + ¢,
= 2 2

5 Seat

having for asymptotes
yux—p and y= —x+.

10. Therefore for any function having real and compiex poles
the boundary lines consist of pairs of straight lines crossing Ox at 45°
at the real poles and of right hyperbole having as asymptotes similar
straight lines crossing Ox at the real part of the complex pole.

The vertices of the hyperbola corresponding to the pole p + zg
are p, + 9.

11. The region of absolute convergence of S is that portion of
the plane (shaded) such that from any point in it a perpendicular
can be drawn to Ox without crossing a boundary line. The nearest
boundary lines to Ox make up /he boundary of the region of converg-
ence of S. It consists of straight lines and hyperbolic arcs.

oN SS ZN rise OK

Fic. 159.

The boundary line of the region is symmetrical with respect to
Ox. The ordinate at any point of this boundary line of converg-
ence is the radius of convergence for the corresponding abscissa, and
is equal to the distance of its foot from the nearest pole point.

For any point on the boundary

G27 os
L nti /*(x) a

is less than 1 for any point inside, and greater than 1 for any point
outside, the region.
SUPPLEMENTARY NOTES. 461

12. Ifa function has two real poles a, £, and no pole between
a, f, the region of absolute convergence consists of a square between
aand #. If between @ and # there is an imaginary pole p+
such that lies between @ and , the imaginary pole has no influ-
ence on the region of convergence if

[2 —Ha+A)P+¢ > ta — fy.
If, however,
[2 —3(4+ AP +9 < }a— py,

the hyperbola y? = (¢ — x)? + ¢ cuts off a portion of the square of
convergence.

13. Theorem II. If /(x) is a one-valued, determinate, unlim-
itedly differentiable function (having only a finite number of roots
or poles in any finite interval), then

2

fetn= ) Fre) ()

for all values of « and_y for which the series is absolutely convergent.
That is, for all values of_y less in absolute value than the radius

R= oP V(x — p)* + q,
where p + 7g is the nearest pole of ((v) to x. Equation (1) is not
true for any value of_y such that [1] >.

Proof: The construction of the region of absolute convergence
shows that from any point P 7 this region can be drawn two straight
lines making angles of 45° with Ox to meet Ox without crossing or
touching the boundary of absolute convergence.

At any point #2, y in the region of absolute convergence the

series
i? rok
S’ = Bal S (x)

is absolutely convergent.

, oS as
But S => ax = ay"
Hence
as as
= — dv ——d
aS aa + ay yy

as as
=~ (dx — d
= 0, if x + _y is constant.

Therefore all along the line . + _¥ ==, 7 the region of absolute
convergence, S must be conséan/. This line passing through any
462 APPENDIX.

point P in this region meets Ox without touching the boundary. At
the point where . -+-_y = c meets Ox we have y = 0, x =, and
S= fc) =/(* +).
Consequently all along any such line passing through the region
of absolute convergence, and therefore at any point whatever in this
region, we have

ao

fetsy= ) Fre).

14. What is the same thing,
UPR yeas -
fay= SES py (x)

for all values of x andy which make the series absolutely convergent. *

If we make the investigation in the form (1), the regions consist
of parallelograms on the line_y = .v as diagonal, and having for sides
the straight lines

v= Af, x= 2v— p,
corresponding to a real pole f, and hyperbole
(ey? = P+ 9-9? = (2 P+ 2,

or eo say + py a= P+ A,
corresponding to a complex pole p -+ zg.

15. Observations.—In the preceding investigation the object has
been to point out as briefly as possible the salient points in the
establishment of the theorems proposed. Details have not been
entered upon. For example, we might discuss fully the behavior of
the approximate boundary line.

r “x)= I

n!\
at the zeros of /*(x). There the curve has vertical asymptotes, but
closes up on the asymptote as # increases. Also, /"(~) cannot have
the same zero point for an indefinite number of consecutive integers
nm unless the function is a polynomial.

Again, if at any assigned point x the derivatives are alternately
o, the radius of convergence is fixed by

J"(*)
fre -b 1) Fata | =
since for absolute convergence we must have!
De
L man Poy |<
*Tt being understood that y is at a finite distance from any value of the variable
at which the function is 0.

 
SUPPLEMENTARY NOTES. 463

This simply means that there are two poles that are equidistant
from the value x. If the poles of a function are all real, it is im-
possible for more than alternate derivatives to be zero continually.

If there are more than two poles equidistant from ., then at least
one must be complex.

If there be three equidistant poles from .v, then one must be real
and two imaginary, + 7g, and conjugate. Then the derivatives at
x are o alternately in pairs and the radius of convergence there is

ce |= | fm + 1)(” + 2) JE)

FPF ay
and so on.

_ Points , equally distant from several poles, are the singular
points on the boundary. Elsewhere, for three poles, we can always

write
w(2+1)(nL2 ST) = SJ) ite TUM *) | hes fe)
TODO ED pate = petay, EEE) porate) OP) paras?

 

the limit of which is 4%, and converges to the value #° at the singular
point as .v converges to the v of suchasingular point. ‘lhe generaliza-
tion of this is obvious.

EXAMPLES.

1. The region of absolute convergence and of equivalence of the Taylor’s series
of the functions tan x, cot a, sec x, csc x, consists of the squares whose diagonals
are the intervals between the roots of sin «, cos +, respectively.

2. In particular tan x is equivalent to its Maclaurin’s series for all values of x
in) — 4m, + 4.

Also for sec x in the same interval.

cot x, csc x are equal to their Taylor's series in the interval )o, 2(, the base of
the expansion being 42.

 

3. Expand — - by Maclaurin’s series.
oe
Put y equal to the function. Then

Yer —Y = Hx
Apply Leibnitz’s formula, and put x = 0 in the result. We have for deter-
mining the derivatives of y at 0,

n(n — 1)

 

my) $ Sb tT Io =O
Making # = I, 2, 3, . . . , we find these derivatives in succession, and there-
fore a ‘g
Be a a Bi es Bagh 8 oh es
Sn6= oii ar a
wherein B, = 2, B,=ay B= as Fy = eo B= + + are called

ree
Bernouilli’s numbers. They are of importance in connection with the expansion
of a number of functions.
Since e#?" = — 1, the poles + 7m are the nearest values of x« to o at which the
function becomes oo. The series is therefore convergent and equal to the function
forvin)— 2, + ™.
464 APPENDIX.

4. Show that for x in) — 4a, + oe

x x

wg Et SE et FE it...

either directly or from

 

ext ex — J ex — J]
5. Show ce from 4 that, for the same values of x,

 

 

 

ex — Byxt2t*— 1 Bx 26-1
—_ Boe ba a 3 Sake
ex ae I Stee a 4! 2 6! 2
6. Obtain the Maclaurin expansion
s : m m 1? — m*) m1? — m2)(32 — m2)
sin(# sin—1xv) = ge x] eB 1 bs aes

and find for what values of x the equation is true.
Put y = sin (m sin-1x).
(I — )y" — ay! + my = 0.
Apply Leibnitz’s theorem and deduce
(1 = a2) 74) — (20 + ne + (nt ny) = 0.

yr)
(— 2) — axe + DZ a — (08 — mt) ey =o.
. (ati)
Since f (2 + 4) aa aay IIA,
y (nti)
fe mt) = fe m3 ner a ylar2) |=] R,
we have

(I — +7) —-2x R-~-R=
or R| = |e +1.
Therefore if x is the base of a Taylor’s series for y, the function is equal to the
series in jx — 1,4-- 1(. If «= 0, the Maclaurin’s series is equal to the function

in )— 1, + th

When x = 0, the differential equation gives
yore) = (2? os m?) y(%),

which gives the coefficients in the series.

7. Treat in the same way cos (m sin—1x),

8. For what values of x is the Maclaurin’s series corresponding to the function
Fee (1 — 2°) y" — xy’ — ay =0
equal to the function?

Work as in 6. The function is e2 sin“,

9. In general, any function y satisfying a differential equation

(1 + ax®) yt2) 4 pain + 5) ytd + g(n — e\in — d)y) = 0,

where a, 4, ¢, d, f, g are any constants, is equal to its Taylor’s series (base x) in

the interval )x — R, x + A(, where R is the radius of absolute convergence, and
R is the absolute value of the least root of the quadratic

(1 + ax?) + prR+G9R? =0.,

A large class of functions can be treated in this way,
SUPPLEMENTARY NOTES. 465

10. If z is a function of x having only a finite number of roots in a finite interval,
find the region of equivalence of the function 1/z with its Taylor’s series.
Let y=1/u. Then ye =1, Differentiate 2 times by Leibnitz’s formula. Then
Yb egy YP A egy oy” + + ut =O.
Divide by y*, and make 2 = 00, Then, a being the radius of convergence,
we have, ifw = G(x),

2
Oe) +2 oe) + Pox) t...=0

But this series is nothing more than $(« + ,).
Therefore + + must be a root of

O* + p) = 9.

Consequently
x+p=h,
or prk—x,

where 2 is the nearest root of g(x) to x, the base of the expansion.

NOTE 7.
Supplementary to Note 6.

I. While, in this book, we are not interested in functions of a
complex variable z = x + 4, it is instructive and interesting to con-
sider the treatment of a function /(z) after the method of Note 6 for a
function of the complex variable z = x + z.

We assume that /(z) is one-valued, unlimitedly differentiable with
respect to 2 at all values of z in the finite portion of the plane except
at poles of /(z), which are, we assume, the only singularities the func-
tion has.

2, Let zax+t+y, C=x’+H’. The series

Cs}

sa) FO

°

is absolutely convergent (when the series of absolute values of its terms
is convergent) for all values of z and € which satisfy

fikropicm « fr aG ice

n=O no
The series S is co when these limits are greater than 1.
The boundary conditions are

f{frolicen fe Gorn

Therefore for arbitrarily great the boundary is arbitrarily near

Pipe, ee
jar" | =e gare ay

a and f being arbitrary constant real numbers.

 

 
466 APPENDIX.

From the first of these equations, we have

az 2 ft)

Ea PO
=e, when 2 = oo.
oe ae

k being an arbittary constant, But if ~ is a pole of ((z), then z = 0
when € =p.

Hence o=hk—e®s, or k= es,

Therefore, corresponding to any assigned €, the boundary corre-
sponding to the pole p is fixed by

a= e(p— 6),
which is a circle about the origin in the z-plane with radius |
k=|p-¢|,

since 3 is arbitrary.
3. If pis the nearest pole of (2) to €, then for all values of z for
which
le}< R=|p—¢|
the series is absolutely convergent, and is infinite for any value of z
if |z|> 2.
4. Put§ =z+. Then z= & —€.

The series

s= SES G—

is absolutely ‘conversant at all points
& inside the circle Cdescribed about

o| i.e #@ € asa center with radius
R=|E-Sl,

Fic, 160. p being the nearest pole of /(z) to €.
For any assigned value of & in this circle the series S is con-
stant with respect to & since

and this is o when 2 = oo.
Now we can always move € up to & along the straight line join-
ing them, the series .§ remaining constant in value. But when

€ = &, we have
ee (6 =)" eg) = AE).

iy

 

 
SUPPLEMENTARY NOTES. 407

Therefore * this equality is true for all values of &, € which make
the series absolutely convergent, i.e., at any point inside the cir-
cular boundary corresponding to any assigned € and the nearest pole
p of /(z), described about € as center with radius of absolute con-
vergence,

R=|p—6

 

NOTE 8.

Supplementary to Note 6.

Pringsheim’s Example of a Function for which the Maclaurin’s
Series is absolutely Convergent and yet the Function and
Series are different.

Let

= AY (— 1)"
Js ) ri para (1)
A and a being positive constants, @ > 1. This function is one-
valued, finite, continuous, and unlimitedly differentiable for all finite
values of the real variable 2. It has, however, infinitely many com-
plex poles

¥—1

ar

 

ae

an infinite number of which are in the neighborhood of = 0, which
is therefore an essentially singular point.

For the mth derivative of A(x) we find (¢ = + V— 1)

ae Ya Clay 1 1
Tt (y=sn 9 r\ | aay - @eF }

°

rot, 2,3,.-.

 

At v=o,
fie) Se,
Sto) = 9,
fo) = (— 1)"(2m)!e-e™,

Therefore the Maclaurin’s series is

S = (— yo (2)

This series is absolutely convergent for all finite real values of -v.

 

* This problem was first solved by Cauchy, by means of singular integrals.
See any text on the theory of functions of a complex variable.
468 APPENDIX.

Now let A <1, |4|<1.

 

fs I A I I
Wega I 7 ae > r+ 14 ax?
and S<er.
In particular, let x = a-t,
FEDS I 1 a—1I

r+ if ra oe

wo. f(a-*) > e-? > S when x = a+,
a—I e+1
er —1 >

when ape" ; or Oe eo

The function /(x) and the series S$
are different.

In the figure the solid line is the
curve y = /(x), the dotted line the curve
y = S, constructed with exaggerated
ordinates, for the values A = log 2,
a= 2.*

 

 

Fic. 161.

NOTE 9.
Supplementary to § 118.
Riemann’s Existence Theorem.

Any function /(x) that is one-valued and continuous throughout
an interval (a, 4) is integrable for that interval.

Let the numbers x,, x,,... , ,_, be interpolated in the inter-
val (a, 4) taken in order from a = x, tod = x,,.

We have to prove that the sum of the elements

Sus BAG %e = He) (1)

converges to a unique determinate limit, when each subinterval con-
verges to zero, whatever be the manner in which the numbers xv, are
interpolated in (a, 4).

I. The sum 5S, must remain finite for all values of 7, For /(x)
is finite, and if 47 and m are the greatest and least values of /(x) in

(a, 4),
m(6—a)<S, <M(b— a).

Also, since /(x) is continuous, there exists a value & in (a, 4) at

which
Sy = (8 — a/(6), (2)
A(&) being a value of /(x) between m and J

 

*For further information on this subject, see papers by Pringsheim, Jath.
Ann. Bd. XLII. p. 109. Math. Papers Columbian Exposition, p. 288.
SUPPLEMENTARY NOTES. 469

II. Interpolate in the rth subinterval of (1), in any manner,
m, — 1 values x/,..., X,,_, of. Then, asin I,

Sug = BAGH, = 2) = (% — HED 3)

where &’, is some number in the subinterval (x,, x,_,).

Form similar sums of elements for each of the # subintervals of
(1). Letp=ni+...-+n, Add the 2 sums of elements such
as (3).

Hence

Ss = Sw, + see + Sag?
p
a = f2,)(%, a Xypar)s

= 3 (e, — %,.)/(8). (4)

This is a new element sum containing p > elements, which is
to be regarded as a continuation of (1) by the interpolation of new
numbers in each subinterval of (1).

Subtracting (4) from (1), we have

So = 21K) BHD = es

Let 6 be the greatest absolute value of the difference between the
greatest and least values of /(x) in the subinterval (+, — +,_,),
r=i,...,%. Then, since /(z,) and /(&;) are values of /() in
(%,5 Xp),

Sx = Ss| < |o2 (*, i Xp_1)s

| <|d(4 — a), (5)
for all values of the integer f, however great. But when each sub-
interval converges to o, then d(=)o, since /(x) is continuous, and
at the same time 2 = o.

Therefore, by the definition of a limit, S, converges to a limit
when 2 =o.

III. To show that the limit of S, is wholly independent of the
manner in which the interval (a, 4) is subdivided:

Let there be an entirely different and arbitrary interpolation

x,/,..., «4_,. Consider the element-sum

1 ’
Sez a. (6)

Interpolate in (a, 4) the numbers
. + J
Hy ee x gp Bad Bag os 0 Mences

occurring in (1) and (6), thus dividing (a, 4) into m+ 2 intervals.
470 APPENDIX.

Interpolate in each of these m- intervals new numbers, thus
dividing (a, 6) into m-+»-+ > subintervals. Form the element-
sum 5S,, 4,4, Corresponding to these subintervals.

Then, by II, S, and -S,,4,4, converge to the same limit. In
like manner S,, and S,,4,4,% converge to the same limit. Therefore
S, and S, converge to a common limit. The uniqueness of the
limit of (1) under any subdivision whatever of (a, 4) is demonstrated.

This theorem gives the means of defining analytically the area and
length of a curve, and the volume and surface area of a solid.

NOTE 10.

Supplementary to § 135.

Formule for the Reduction of Binomial Differentials of the form
x(a + dx") dx.

Put y =at dx", Then
Dxyt = ax—y" + nybxrtr-ryr-t,
= aaxr— yi +. (aa + ny)bxetb-wi—s, (1)
= (a+ ny)xtyt — anya, (2)

In (1), put a=m—n+1, y=p+t, then
Dx" yt? = a(m — n+ i) "yt + (mp + m+ 1)dxmy’, (A)
In (2), put a=m+1, y=, then
Dat +¥yt = (np + m + 1)x%y? — anpxryh, (B)
In (1), put a=m+1, y=p+1, then
Dacrriyh*t = a(m + r)cry + (np + m+n + x)baryb,  (C)
In (2), put a=m+1, y=p+t1, then
Dat yh! = (np +m +n x)xmyht* — an(p-+ r)x%ys,  (D)
Integrating the formule (A), ..., (D), we have the formule of
reduction, where y = @ + bx”:

7 = gem tty ptt (m—n-+ 1)a sao
[vote ~ (pp tmtis ppm y ood * ane

 

 

SiS OO anp wigs
[eager pte ce e)

mag, cetyett (mp tm+nt3b (nin
[Pega Gee frrtin ©

 

i = emtiyptt np+m+n+i1 eee
[Pe =a ee frre ©
SUPPLEMENTARY NOTES. 471

NOTE 11.
Supplementary to § 165.

If_y = f(x) be represented by a curve, and y, Dy, Dy are uniform
and continuous, then we can always take two points P and P, on the
curve so near together that the curve lies wholly between the chord
and the tangents at P and P..

Let xv, y be the coordinates of P, and
X, F those of P’, any point on the curve
between P and P,.

The tangent at P has for its equation

F,=/(*) + (X — x) f'"(«).

At any point x, y of ordinary posi-
tion, not an inflexion, the difference
between the ordinate to the curve and
the tangent is

 

X — x)
fH= Ke SPS, @ Bae
where & is some number between x and X. We can always take X
so near tox that /’’(&) keeps its sign the same as that of f(x) for all
values of & in (x, X). Therefore the difference (1) keeps its sign
unchanged in (*, X) or the curve is on one side of the tangent, for
this interval.
The equation to the chord PP, is
F.=fN*) + (X — *)f(E,),
where /’(&,) is the slope of the chord PP,. The difference between
the ordinates of the curve and chord is
NA) — Fos (X — *) (8) — /(8))). (2)
Let x, be so near x that /’(x,), /’(&,) have the same sign as /’(.x).
Then this difference (2) keeps its sign unchanged for all values of X
in (x, x,). It can now be easily shown that (2)and (1) have opposite
signs, and there can always be assigned a number 1, so near x that the
curve PP, lies wholly in the triangle formed by the tangents at P, P,
and the chord P?,.

NOTE 12.
Supplementary to § 226, IV.
Proof of the Properties of Newton’s Analytical Polygon.

1. Let there be any polynomial in x and y, such as

f = Api?! +... Agi (1)
wherein the exponents a, ( of each term satisfy the linear relation
aa + bB =¢, (2)

c being taken a positive number.
472 APPENDIX.

Let / be arranged according to ascending powers of y, so that
B,<B,< ... Then

S= xyP( 4 + A, xtamayPa Bi, +... a

[ ah a |
= xt fa 4,+ 4, 4 Hae eo (3)
a :
&
where 4,,..., 4g, are the roots of the equation in / —

A APO Be APO See,

Therefore the locus of “= o consists of x = 0, y = o, and the
parabolic curves

yt = Bex? (FSHtyp se 2s y Pa — Be

&
2. In(3), lety = &v*, & being constant. Then

& &

PSAP OC AO ee pe
b &

a ted Pe ta Sag
= a (4p Aa 4 -

= Kx«*, X being constant.

3. Let /’ be a function A’x2'y8’, or the sum of a finite number of
such functions, such that the exponents a’, /’ of each term satisfy the
linear equation

aa’ + 6f’=c’'.

b
Then, asin 2, let y = £x*, and we have in the same way

cl

PEELS,
&’ being a constant.
4. Let a, 6 and ¢, c’ be positive numbers,
Then » goes
Gat
where x and y satisfy y* = hx’,

(1). If ce’ >, then
{ ° when x(=)o, H(=)o.
SUPPLEMENTARY NOTES. 473

(2). If ¢’ <c, then
fe
fous when x=0, yoo,
5. We are now prepared to prove § 226, IV, (1), (2).
Let L(x, vy) = SC xy =o.
(1). Let / represent that part of / which corresponds to a side of

the polygon as prescribed in § 226, IV, (1), and F’ re t th
remainder of #, Then ; wr eer ee ris

Laff’,
F fF’
or ee ae

Through each point corresponding to terms in /’ draw a line
parallel to the side corresponding to
Then by 3, (1), we have

f3 I, sinc a
—= ince s~=0
as £7 ;

when x(=)0, y(=)o.
Therefore in the neighborhood of the origin # = o and J=0
are the same.
But the form of f = o in the neighborhood of the origin is that of
a parabola
a kx,
Hence /’ = o goes through the origin in the same way as does
J = 0, whose form is that of a parabola of type y* = Ax’.
(2). Let / represent that part of /’ corresponding to a side of the
polygon as prescribed in § 226, IV, (2), and F’’ the remainder of Ff.
fF £’
Then == 1 =:
a;
Draw parallels to the side corresponding to_/, through all points
corresponding to terms in 7’.
Then by 3, (2), we have

fF since f = 0
po ff

when x = 0O,y= 0.
Therefore / = o and f= 0 pass off tooo inthesame way. Also,
J = 0 passes off to oo , as does a parabola of type yt = ha’.

 

Notre.—The same process can be extended to surfaces, using a polyhedron in
space. The part of the equation corresponding toa plane face such that there are
no points between that face and the origin gives the form of a sheet of the surface
at the origin, Likewise the part corresponding to a plane face such that no point
lies on the side opposite to the origin gives the form of a sheet at o. The plane
faces in each case cutting the positive parts of the axes,
INDEX.

[Zhe numbers refer to the pages.] -

Absolute number, 2
Anticlastic surface, 360 5
Appendix, 451 :
Archimedes,

spiral of, 117, 161

area of spiral, 234

length of spiral, 248
Areas of Plane Curves,

rectangular coordinates, 226, 396

polar coordinates, 233, 397
Asymptotes,

rectilinear, 121

to polar curves, 125
Auxiliary equation, 443
Axes, of a conic, 323, 325 '

of a central plane section of a coni-'

coid, 328 ,

Base of Expansion, 88

Bernoulli,
definite integral by series, 222
differential equation, 424

Binomial differentials, 193, 470

Binomial formula, 67

Binormal, 378, 379

Bonnet, 131

Boundary of region of convergence, 460

Cantor, definition of number, 5
Cardioid, 118, 163
area, 234; length, 248
surface of revolute, 261
volume of revolute, 401
orthogonal trajectory of, 438

 

Catenary,

normal-length, 116

radius of curvature, 134

center of curvature, 146

curve traced, 152

area, 228; length, 245

volume of revolute, 258

surface of revolute, 261

differential equation, 446
Cauchy,

theorem of mean value, 79, 87, 222

theorem on undetermined forms, 93

on expansion of functions, 467
Caustic by reflexion, 390
Circle,

area, 227, 234

length of perimeter, 246
Circle of curvature,

for plane curves, 100, 134

for space curves, 379
Cissoid,

tangent and normal, 115

subtangent, 116

curve traced, I51

area, 229
Clairaut’s equation, 429
Coordinates of center of curvature, 133
Computation of,

e, 84; logarithms, 86; 2, 89
Concavity and Convexity, 127
Concavo-convex, 128
Conchoid of Nicomedes, 160
Concomitant, 312
Convexo-concave, 128

475
476 INDEX.
Cone, Devil, 161
volume of, 257, 266 Difference,

equation of, 349
Conic, center of, 331
Conicoid of curvature, 360
Conjugate points. 334
Connectivity, law of, 19
Conoid, volume of, 266
Consecutive numbers, 7
Constant, 4
Contact,

of a curve and straight line, 127

of two curves, 130
Continuity,

theorem of, 23

of functions, 278
Continuum, 3
Convergency quotient, 14
Cubical parabola, 135, 150
Curvature, 130

radius of, 133

circle of, 134

of surfaces, 365

measure of, 370

spherical, 381
Curve tracing, 147, 340
Curves in space, 375
Cusp, 151, 155, 336, 337
Cusp-conjugate point, 335
Cusp-locus, 435
Cycloid,

tangent to, 114

curve traced, 163

area, 232

length, 252

surface of revolute, 261, 263

volume of revolute, 262, 263
Cylinder, equation of, 348

Decreasing function, 74

Definite integration, 215

Degenerate forms of differential equa-
tion of second order, 440

Degree of differential equation, 409

Descartes, 26

Developable surfaces, 374

 

of the variable, 35

of the function, 35

quotient, 36
Differential, 55, 63

quotient, 55, 64

coefficient, 55

relation to differences, 57

total, 294
Differentiation of,

logarithm, 41

power, 42

sum, 43

product, 44, 69

quotient, 45, 454

inverse function, 46

trigonometrical functions, 44, 45, 46

circular functions, 48

exponentials, 49

function of a function, 49, 70, 453,

455
implicit function, 66, 296
function of independent variables,
282, 306

under the integral sign, 391
Differential equations,

first order, 409 °

second order, 439
Discriminant equations, 434
Double points, 334
Double integration, 396
Dumb-bell, 160

Edge of envelope, 389
Elements of curve at point, 147
Elliott’s theorem, 236
Ellipse,

tangent and normal, 113

subnormal, 115

radius of curvature, 125

evolute, 144, 154

area, 228, 233

arc length, 246

length of evolute, 246, 251

norinal to, 331

orthogonal trajectory, 433
Ellipsoid, volume of, 268, 399
Elliptic functions, 208
Elliptic paraboloid, 268, 280
Envelopes,

of curves, 138, 343, 435

of surfaces, 385
Epicycloid, 164, 248
Equiangular spiral, 118, 162, 234
Equilateral hyperbola,

evolute of, 146

area of sector, 228
Euler’s theorem on curvature, 366
Eulerian integral, 217
Evolute of a curve, 144

length of, 250
Exact differential, 416
Exponential curve, 150

Family of curves, 138
Finite difference-quotient

fle) — fia)
x—a
nth derivative of, 138
Folium of Descartes,
tangent, 113; asymptote, 122
traced, 159; area, 232, 234
Fort, 18
Fresnel’s wave surface, 390
Function,
definition, 19, 273, 274
explicit, implicit, 19, 277
transcendental, 20
rational, irrational, 20
symbolism for, 21
uniform, one-valued, 21
continuity of, 22, 278
difference of, 35
derivative of, 36
difference-quotient of, 36
increasing, decreasing, 74
Function of a Function,
geometrical picture, 453
nth derivative of, 455

Gamma functions, 217

 

INDEX. 477

Gauss,
theorem on areas, 241
theorem on curvature, 372

Geodesic line, 384

Groin, volume of, 267
surface of, 408

Harkness, 451
Helix, 376, 384, 405
Holditch’s theorem, 238
Homogeneous,
coordinates, 339
differential equation, 414, 431
Horograph, 372
Hyperbola,
tangent, 113; asymptotes, 122
radius of curvature, 137
area, 228, 233
orthogonal trajectory, 432
Hyperbolatoid, volume of, 269
Hyperbolic sine, cosine, 29
Hyperbolic spiral,
traced, 162; subtangent, 117
area, 234; length, 248
Hyperbolic paraboloid, volume, 399
Hyperboloid of revolution, volume, 258
Hypocycloid,
tangent, 113; evolute, 146
traced, 154, 164; area, 229
length, 246
volume of revolute, 258
surface of revolute, 261

Increasing function, 74
Indicatrix of surface, 361
Infinite, infinitesimal, 2, 7
Inflexion, 128
Inflexional tangent, 332, 355
Integer, definition, 1
Integral, definition, 165
indefinite, 173; definite, 215
fundamental, 173
Integration, definition, 167
by transformation, 178
by rationalization, 182
by parts, 183
478

Integration by partial fractions, 185

under the /‘sign, 393
Integrating factor, 420
Interval of a variable, 4
Intrinsic equation of curve, 251

of the catenary, 252

of the involute of circle, 252

of the cycloid, 253
Illusory forms, 95
Inverse curves, 161
Involute of a curve, 144

Jacobi’s theorem on areas, 241

Lacroix, 335
Lagrange,

theorem of mean value, 78

differential equation of, 430

interpolation formula, 241
Leibnitz,

nth derivative of product, 69

symbol of integral, 170

linear differential equation, 423
Lemniscate, 99

traced, 156, 159; area, 234; length,

253; area revolute, 263

Lengths of curves,

plane curves, 243, 247

curves in space, 404
L’Hopital’s theorem, 94
Limit,

of a variable, 7

principles of, 7

theorems on, 8

of (1 + 1/2)*, 16
' of integration, 167
Limagon, area of, 239
Linear differentiation, 293, 301, 307
Linear differential equation, 443
Line of curvature, 384
Logarithmic curve,

traced, 150 ; length, 246
Logarithmic spiral,

area, 234; length, 248
Loxodrone, 384

Maclaurin’s series, 83, 467

 

INDFX.

Maximum and Minimum, 103
independent variables, 314
implicit functions, 321
conditional, 322

McMahon, 416

Mean Value,
theorem of, 76, 218 :
formula by integration, 220
for two variables, 309

Mean Curvature, 371

Meunier’s theorem, 366

Modulus of a number, 3,.458

Morley, 451

Neil, 245
Neighborhood, 7, 278
Newton,
binomial formula, 67
radius of curvature, 135
rule for areas, 240
analytical polygon, 340, 471
Nodal point and line, 362
Node, 156, 337
Node-locus, 435
Non-exact differential equation, 418
Normal,
to a curve, I14, II5, 330
to a surface, 358
Normal plane, 376

Oblate spheroid,

volume, 258; surface, 262
Omega, 2
Order of differential equation, 409
Ordinary point,

on curve, 329; on surface, 352
Orthogonal trajectories, 432
Osculation, 132
Osculating plane, 377

Parabola,
tangent, 113; subnormal, 116
radius of curvature, 134
evolute, 144; area, 228, 234
arc length, 245, 248
length of evolute, 251
orthogonal trajectory, 432
Paraboloid of revolution,
volume, 258; surface, 260
Parameter, 138
Partial derivatives, 282
Pedal curve, 239
Plane, equation to, 347
Planimeter, 239
Pole of a function, 457
Primitive of a function, 168
Principal,
sections of surface, 365
radii of curvature, 365, 368
normal, 378
Pringsheim, 87, 467
Probability curve,
traced, 151; area, 395
Prolate spheroid, volume, 257
Pseudo-sphere,
volume, 258; surface, 261
Pursuit, curve of, 446

Quotient of functions,
mth derivative, 454

Radius of convergence, 88
Radius of curvature,
plane curves, 100, 133, 250
at point of inflexion, 135
for surfaces, 365
for space curves, 370, 389
Real number, 3
Reciprocal spiral, 117, 126
Revolute, definition, 255
volume, 256 ; surface, 259
Riemann’s existence theorem, 468
Roche, 222
Rolle’s theorem, 75
Root of a function, 457

Saddle point, 336
Scarabeus, 160
Schlémilch, 222
Semi-cubical parabola, 245
Sequence, 14
Singular points,

on a curve, 158, 333

 

INDEX. 479

Singular points on a surface, 352, 362
Singular solutions of differential equa-
tions, 433
Singular tangent plane, 362
Singularity, essential and non-essential,
457
Solution of differential equations, gen-
eral, particular, complete, 410
by separation of variables, 410
when J and & are of first degree,
415
by differentiation, 426
when solvable for y, 428
when solvable for x, 430
when solvable for p, 431
Specific curvature, 371, 372
Sphere,
volume, 257, 400; surface, 260, 403
Spherical curvature, 381
Steradian, 371
Stewart, 238
Stirling, 83
Straight line, equations, 348
Subtangent, subnormal, 115, 116
Successive differentiation, 62
Surface, definition, 349
general equation, 349, 361
of solids, 255, 402
Synclastic surface, 360

Table of derivatives, 52
Table of integrals, 176
Tac-locus, 435
Tangent,
to plane curves, 112, 116, 330
length, 115
to space curves, 375
Tangent line to surface, 350
Tangent plane, 351
Taylor’s series, 82, 86, 87, 221, 457,
467
Tortuosity, 380, 382
Torse (developable surface), 374
Torus,
volume, 259; surface, 261
tangent plane to, 364
480

Total,
derivative, 291, 294
differentiation, 290
differential, 294
Tractrix,
tangent-length, 119; area, 235
arc length, 246
volume of revolute, 258
Trajectory, 431
Transcendental function, 83
Triple,
point, 337; integration, 398
Trochoids, 164

Umbilic, 361, 369
Undetermined forms, 92

 

INDEX.

Undetermined multipliers,
applied to maxima and minima, 323
applied to envelopes, 343, 388
Undulation, point of, 333

Variable, definition, 4
difference of, 35
Volumes of solids, 255, 398, 400, 40x

Weierstrass, 5
example of derivativeless function,
451
Witch of Agnesi,
tangent, normal, 115
traced, 152; area, 228
volume of revolute, 258

Zero, 2
MATHEMATICS

Evans’s Algebra for Schools.
By GeorGE W. Evans, Instructor in Mathematics in the English High

School, Boston, Mass. 433 pp.

I2mo.

$1.12.

Aside from a number of novelties, the book is distinguished

by two notable features:

(1) Practical problems form the point of departure at each

new turn of the subject.

From the first page the pupil is put

to work on familiar material and on operations within his
powers. Difficulties and novelties arise in a natural way and
in concrete form and are met one at atime, and he is led to see
the need for each operation and preserved from regarding
algebraic processes as a species of legerdemain.

(2) The book contains nearly 3,500 examples, none of which
are repeated from other books. The exercises are graduated
according to difficulty and are adapted in number to what ex-
perience has shown to be average class needs. Problems are
carefully classified with reference to the several types of
equations arising from them, and the pupil is specially drilled
upon typical forms (as, for example, ‘‘ the clock problem,” “the
cistern problem,” “day’s work problem,” etc.) and upon

generalized forms.

Paul H. Hanus, Professor in
Harvard University :—The author
has certainly been successful in
presenting the essentials of ele-
mentary algebra ina thoroughly
sensible way as to sequence of
topics and method of treat-
ment.

C. H. Pettee, Professor in the
N. H. College of Agriculture :—I
have actually become tired look-
ing over algebras, geometries, and
trigonometries that have no ex-
cuse for existence. Hence it is
with real pleasure that I have
examined Evans’s Algebra for
Schools. The author evidently
knows what a student needs and
how to teach it to him.

 

E. S. Loomis, Cleveland (Ohio)
West High School :—To_ pass
gradually from arithmetic to all
gebra, to bridge that intellectua-
chasm in the minds of many,
is no little thing to do. Evans
has done it more nearly than
any other author I have read.
I like his scheme of models,
but above all I like his coor-
dinating algebra and the other
sciences. I wish I could teach the
book, it is so full of good things.

Jas. E. Morrow, Principal Al-
legheny (Pa.) High School :—1I find
more to commend in this algebra
than in any book on the subject
since the publication of ’s in
1869.

 

32
Mathematics 33
Gillet’s Elementary Algebra.

By J. A. GILLET, Professor in the New York Normal College. xiv +412
pp. 12mo. Half leather, $1.10. With Part Il, xvi+512 pp. r12mo.

$1.35.

Distinguished from the other American text-books covering
substantially the same ground, (1) in the early introduction of
the equation and its constant employment in the solution of
problems ; (2) in the attention given to negative quantities and
to the formal laws of algebra, thus gaining in scientific rigor
without loss in simplicity; (3) in the fuller development of

factoring, and in its use in the solution of equations.

James L. Love, Professor in
Harvard University :—It is un-
usually good in its arrangement
and choice of material, as well as
in clearness of definition and ex-
planation.

J. B. Coit, Professor in Boston
University :—I am pleased to see
that the author has had the pur-
pose to introduce the student to
the reason for the methods of al-
gebra, and to avoid teaching that
which must be w2/earned,

F. F. Thwing, Manual Train-
ing High School, Louisville, Ky.:—
Two features strike me as being
very excellent and desirable ina
text-book, the prominence given
to the concrete problems and the
application of factoring to the so-
lution of quadratic equations.

J. G. Estill, Hotchkiss School,
Lakeville, Conn.:—The order in
which the subjects are taken up
is the most rational of any algebra
with which I am familiar.

Gillet’s Euclidean Geometry.

By J. A. GILLET, Professor in the New York Normal College.

1zmo. Half leather, $1.25.

436 pp.

This book is “ Euclidean” in that it reverts to purely geo-
metrical methods of proof, though it attempts no literal repro-

ductions of Euclid’s demonstrations or propositions.

Metrical

applications and illustrations of geometrical truths are inter-

spersed with

unusual freedom.

”

“Originals’’ are made an

integral part of the logical development of the subject.

Percy F. Smith, Professor in
Yale University :—The return of
the ‘‘spirit of Euclid’? should be
much appreciated, and it will be
interesting to watch the workings
in the classroom of the two alter-
native methods of Book V. Con-
sistency and rigor are carefully

maintained in both works, and I
shall take great pleasure in using
and recommending them.

E. L. Caldwell. organ Park
Academy, Ll/,:—1 find in them the
best results of modern research
conibined with rigid exactness in
definition and demonstration.
34

Mathematics

 

Keigwin’s Elements of Geometry.
By Henry W. KEIGwIn, Instructor in Mathematics, Norwich (Ct.) Free

Academy. iv+ 227 pp. 12mo.

$1.00.
This little book is a class-book, and not a treatise.

It cov-

ers the ground required for admission to college, and includes

in its syllabus the stock theorems of elementary geometry.

It

is, however, out of the common run of elementary geometries

in the following particulars :

1, The early propositions, and a few difficult and funda-
mental propositions later, are proved at length to furnish

models of demonstration.

2. The details of proof are gradually omitted, and a large
part of the work is developed from hints, diagrams, etc.

3. The problems of construction are introduced early, and
generally where they may soon be used in related propositions,

Oren Root, Professor in Hamil-
ton College, N. Y.:—I like the
book, especially in that it gives
‘‘inventional geometry” while
giving the fundamental propo-
sitions. Geometry is taught very
largely as if each proposition were
an independent ultimate end.
Pupils do not grasp the interlock-
ing relations which run on and on
and on unendingly. Mr. Keig-
win’s book, compelling pupils to
use what they have learned of re-
lations, must help to prevent this.

C.L. Gruber, Pa. Normal School,
Kutztown -—The method of the
book is an excellent one, since it
gradually leads the student to de-
pend in a measure upon himself
and consequently strengthens and
develops his reasoning powers in
a manner too often neglected by
teachers of the present day. It
gives neither too little nor too
much.

W. A. Hunt, High School, Den-
ver, Colo. :—It does not do for the
pupil what he should do for him-
self. With strong teaching, the
book is just what is needed in
preparatory schools.

 

Miss Emily F. Webster, State
Normal School, Oshkosh, Wise.:—
At the first I looked upon the
book as very small, but I now con-
sider it very large, for it is per-
fectly packed with suggestions
and queries which might easily
have extended the book to twice
its present size had the author
seen fit to elaborate, as so many
authors do; but in not doing so
lies one of the finest features of
the book, as much is thus left for
the student to search out for him-
self. The original exercises are
fine and in some cases quite un-
usual. The figures are clear and
the lettering is economical, some-
thing which is by no means com-
mon, and much valuable time is
wasted by repeating unnecessary
letters ina demonstration. Dem-
onstrations are made. general,
which is an advantage, for it is
often difficult to induce pupils to
do so when the author has failed
to set them the example.

George Buck, Dayton (0.) High
School.—I am highly pleased with
it and commend its general plan
most heartily.
Mathematics 35

 

Newcomb’s School Algebra.

By Simon NEwcomp, Professor of Mathematics in the Jehns Hopkins

posah ag x-+ 294 pp. I2m0o. 95 cents. (Key, 95 cents. Answers, 16

Newcomb'’s Algebra for Colleges.

By Simon NeEwcos, Frofessor in the Johns Hopkins University. evzsed.
xiv-+546 pp. t2mo, $1.30. (Key, $1.30. Answers, 10 cents.)

Newcomb’s Elements of Geometry.

By SIMON NeEwcoms, Professor of Mathematics in the Johns Hopkins
University. Revised. x+399pp. 12m0. $1.20.

Newcomb’s Elements of Plane and Spherical Trigo-
nometry. (WirH Five-pLace TaBLeEs.)

With Logarithmic and other Mathematical Tables and Examples of their
Use and Hints on the Art of Computation. By SIMON NEWCOMB, Pro-
fessor of Mathematics in the Johns Hopkins University. Revzsed. vi +
168 + vi+ 80+ 104 pp. 8vo. $1.60.

Elements of Trigonometry separate. vi+ 168 pp. $1.20.

Mathematical Tables, with Examples of their Use and Hints on the Art
of Computation. vi-+ 80+ 104 pp. $1.10,

The Tables, which are to five places of decimals, are regu-
larly supplied to the United States Military Academy and to
Princeton University and Yale University for the entrance
examinations.

Newcomb’s Essentials of Trigonometry.

Plane and Spherical. With Three- and Four-place Logarithmic and
Trigonometric Tables. By Stmon NeEwcoms, Professor of Mathematics
in the Johns Hopkins University. vi-+ 187 pp. 12mo. $1.00.

Much more elementary in treatment than the foregoing.

Newcomb’s Elements of Analytic Geometry.
By Simon NEwcomg, Professor of Mathematics in the Johns Hopkins
University. viii 357pp. 12mo0. $1.20.

Corresponds closely to the usual college course in plane
analytic geometry, but is so arranged that a practical course
may be made up by omitting certain sections and adding Part
II, which treats of geometry of three dimensions. The sec-
tions omitted in the practical course, together with Part III,
form an introduction to modern projective geometry.
36 Mathematics

 

Newcomb’s Elements of the Differential and Integral
Calculus.

By Simon NeEwcoms, Professor of Mathematics in the Johns Hopkins
University. xii+ 307 pp. 12mo. $1.50.

A complete outline of the first principles of the subject
without going into developments and applications further than
is necessary to illustrate the principles.

Nipher’s Introduction to Graphic Algebra.

For the use of High Schools. By FRaNcis E. NIPHER, Professor in Wash-
ington University. 1zmo. 66 pp. 60 cents.

Eighteen of the most elementary graphs illustrating the
‘solution of equations. It is thought that none of these graphs
is beyond the capacities of high-school pupils. By injecting
some such material here and there into the ordinary instruction
in algebra, new meaning can be given to mathematical opera-
tions and new interest to the whole subject.

Phillips and Beebe’s Graphic Algebra. Or Geometrical
Interpretations of the Theory of Equations of One Un-
known Quantity.

By A. W. PHILLIPS and W. BEEBE, Professors of Mathematics in Yale
College. Revised Edition. 156pp. 8vo. $1.60.