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'VUoxt..3oo9. OS. .a.;
P
SCIENCE CENTER LIBRARY
BOUGHT WITH THE INCOMK
FROM THE BBQUEBT IIP
PROF. JOHH FABBAK, U..B.
ELIZA FABBAB
AN ELEMENTARY TEXT-BOOK
ON THE
DIFFERENTIAL AND INTEGRAL
CALCULUS
WILLIAM H. |CHOLS
NEW YORK
HENRY HOLT AND COMPANY
1903
/WoLXi-v ic»o«^.o^.^\
\
\
I *- >'
C^/l*V^-^''
,,' ^''>^ ^
BT
HENRY HOLT & CO.
ROBERT DRUMMOlfD, PRINTER, KEW YORK.
PREFACE.
This 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 or more. 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.
• • •
lU
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-
ingy 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 finite numbers. 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 D/l 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 quanitiy 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 3 is that number whose
product by b is equal to a.
In preparing this text I 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 modem 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 w^orking out all of the exercises.
W. H. E.
University of Virginia, October, 1902.
CONTENTS .
INTRODUCTION.
FUNDAMENTAL ARITHMETICAL PRINCIPLES.
PAGB
Section I. On the Variable i
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 van-
able. The Principle of Limits. Fundamental theorems on the limit
Section IL Function of a Variable 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.
Chapter 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.
Chapter IL Rules for Elementary Differentiation 41
Derivatives of standard functions log x, jra, sin jr. 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.
Chapter III. On the Differential of a Function 55
Definition of differential. Differential-quotient. Relation to difference.
Relation to derivative.
Chapter IV. On Successive Differentiation 62
The second derivative. Successive derivatives. Successive differentials.
The differential- quotients, variable independent. Leibnitz's formula for
the »th derivative of the product of two functions. Successive derivatives
of a function of a linear function of the variable.
VI CONTENTS.
PAGE
Chapter V. On the Theorem of Mean Value 74
Increasin|? and decreasing functions, Rolle's theorem. Lagrange's
form of the Theorem of Mean Value, or Law of the Mean. Caucby's form
of the Law of the Mean.
Chapter VI. On the Expansion of Functions 82
The power-series. Taylor's series. Maclaurin's series. Expansion of
the sine, logarithm, and exponetitiaL Expansion of derivative and primi-
tive from that ot function.
Chapter VII. On Undetermined Forms 92
Cauchy*s theorem. Application to the illusory forms 0/0, 00 /oo ,
o X 00 I <» — « , o**! » S I*.
Chapter VIII. On Maximum and Minimum 103
Definition. Necessary condition. Sufficient condition. Study of a
function at a point at which derivative is a Conditions for maximum,
minimum, and inflexion.
PART II.
APPLICATIONS TO GEOMETRY.
Chapter IX. Tangent and Normal 112
Equation of tangent. Slope and direction of curve at a point. Equa-
tion of normal. Tangent-length, normal-length, subtangent and subnor-
mal. Rectangular and polar coordinates.
Chapter X. Reciilinear Asymptotes 121
Definitions. Three methods of finding asymptotes to curves. Asymp-
totes to polar curves.
Chapter XL Concavity, Convexity, and Inflexion 127
Contact of a curve and straight line. Concavity. Convexity. In-
flexion, concavo-convex, convexo-concave. Conditions for form of curve
near tangent.
Chapter XII. Contact and Curvature 130
Contact of two curves. Order of contact. Osculation. Osculating circle,
circle of curvature, radius, and center.
Chapter XIII. Envelopes 138
Definition of curve family. Arbitrary parameter. Enveloping curve
of a family. Envelope tangent to each curve.
Chapter XIV. Involute and Evolute 144
Definitions. Two methods of finding evolute.
Chapter XV. Examples of Curve Tracing 147
Curve elements. Explicit and implicit equations. Tracings of simple
curves.
CONTENTS. vii
PART III.
PRINCIPLES OF THE INTEGRAL CALCULUS.
PACB
Chapter XVI. On the Integral of a Function 165
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 Standard Integrals. Methods of
Integration 175
The irreducible form / du. The catechism of standard integrals. Prin-
ciples of integration. Methods of integration. Substitution (transforma*
tion, rationalization). Decomposition <parts, partial fractions).
Chaffer 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 215
Symbol of substitution. Interchange of limits. New limits for change
of variable. Decomposition of limits. A theorem of mean 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.
APPUCA TIONS OF INTEGRA TION
Chapter XX. On the Areas of Plane Curvf-s 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 243
Definition of ciu-ve-length. Leni^th of a curve, rectangular coordinates,
polar coordinates. Length of arc of evolute. Intrinsic equation of curve.
Chapter XXII. On the Volumes and Surfaces of Revolutes. 25^
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
PACK
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
Two Variables 282
On the partial derivatives. Successive partial differentiation. Theorem
of the independence of the order of partial differentiation.
Chaffer XXVI. Total Differentiation 290
Total derivative defined. Total derivative in terms of partial deriva-
tives. The linear derivative. Total differential. Differentiation of the
implicit function.
Chapter XXVII. Successive Total Differentiation 299
Second total derivative and differential of « = /(xt 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
Variables 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 of Sev-
eral Variables 314
Definition. Conditions for maxima and minima values of /{x, y) and
f{x, y, z). Maxima and minima values for the implicit function. Use of
I^grange's method of arbitrary- multipliers.
Chapter XXXI. Application to Plane Curves 329
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-con jugate 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 cur\'es with several
parameters subject to conditions. Use of arbitrary multipliers.
CONTENTS. ix
PART VI.
APPLICATION TO SURFACES,
PAGE
Chapter XXXII. Study of 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 Normal to a surfsice. Study of the 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 365
Normal sections. Radius of curvature. Principal radii of curvature.
Meunier's theorem. Umbilics. Measures of curvature of a surface.
Gauss' theorem.
Chapter XXXIV, Curves in Space 375
General equations of curve. Tangent to a curve at a point. Oscu-
lating plane. Equationsofthe principal normal. Thebinormal. Circle
of curvature. Tortuosity, measure of twist. Spherical curvature.
Chapter XXXV. Envelopes of Surfaces 385
Envelope of surface-family having one arbitrary parameter. The
characteristic line. Envelope of siuface-family having two independent
arbitrary parameters. Use of arbitrary multipliers.
PART VII.
INTEGRATION FOR MORE THAN ONE VARIABLE, MULTIPLE
INTEGRALS,
Chapi'er XXXVI. Differentiation and Integration of In-
tegrals 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 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 Degree 409
Rules for solution. Exact and non-exact equations. Integrating
fzictors.
Chapter XXXIX. Examples of Differential Equations of
the First Order and Second Degree 428
Rules for solution. Orthogonal trajectories. The singular solution.
c- and /-discriminant relations. Redundant factors not solutions. Node,
cusp, and tac- loci.
X CONTENTS.
PACB
Chapter XL. Examples of Differential Equations of the
Second Order and First Degree 439
The five degenerate forms. The linear equationi and homogeneous
linear equation having second member o.
APPENDIX.
Note I. Weierstrass's Example of a Continuous Function which has no deter-
minate derivative 451
Note 2. Geometrical Picture of a Function of a Function 453
Note 3. The »th Derivative of the Quotient of Two Functions. . • 454
Note 4. The nth Derivative of a Function of a Function •» 455
Note 5. The Derivatives of a Function are infinite at the same points at which
the Function is infinite 456
Note 6. On the Expansion of Functions by Taylor's Series • 457
Note 7. Supplement to Note 6. Complex Variable • 465
Note 8. Supplement 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 diiferent 467
Note 9. Riemann's Existence Theorem. Proof that a one- valued and con-
tinuous function is integrable 468
Note 10. Reduction formuls for integrating the binomial differential 470
Note II. Proof that a curve lies between the chord and tangent, when the
chord is taken short enough 471
Note 12. Proof of the properties of Newton's analytical polygon for curve-
tracing 47^
Index 475
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 Sjrmbols.
I, «, 3> 4, • . • (i)
are symbols used to represent the groups of marks which we call inle-
gers. Thus *
1=1,
2 = 1 + 1,
3 = 1 + 1 + 1,
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
• • • , i. i. i, I (ii)
constitute an extension of the table (i) to the left of the integer i,
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, *• w identicat with" or ** ij the same as."
I
2 INTRODUCTION TO THE CALCULUS. [Sec. L
and conversely. Those numbers in (ii) which are the reciprocals of
the infinite or unassignably great integers, are said to be infiniiesimaU
or unassignably small.*
5. The Absolute Number. The AbBolute-Number Continuum.
When in the table of numbers
• . . , i» ii I. 2, 3, . . . (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 conh'nuum. 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 absolute
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 infinite 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 infinitesimal or unassignably smalL
The absolute continuum is thus divided into two classes of num-
bers : the uniquely assigned or simply the assigned numbers, which we
call the finite numbers ; and the numbers which cannot be uniquely
assigned or transfinite numbers.
The transfinite numbers greater than i are called infinite, those less
than I 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 closed on the lefl by assigning in the
table a number zero whose symbol is o, which shall be less than any
number in the absolute system. Since now there is no number greater
than I to correspond to the reciprocal of this number o, we design
arbitrarily a number omega whose symbol is /2 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 £1 is the ultimate 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 CI is not an abso-
lute number, but is the superior boundary number of the absolute
system.
■ - • — — -
* The words * great * and * smaU * have in no sense whatever a magnitude mean-
ing when applied to numbers. They are mere conventional phrases and the woxds
' right * and * left 'or * in ' and * out,' might just as well be employed.
Art. 7.] ON THE VARIABLE. 3
7* The conventional symibol for the whole class of unassignably
great or infinite numbers is 00 . There has been adopted no conven-
tional symbol for the class of infinitesimals ; the symbol most com-
monly used is the Greek letter iota, t.
8. The Real-Nttmber S78tem.-^When in the algebraic system
of numbers
— i2, . . . , — 3, — 2, — I, o, + I, -f 2, + 3, . . . , -f /2,
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 real-
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 ariihmeiical or absolute
value. Thus, the modulus or absolute valye 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 boolLwe ^Ml be directly concerned only with real num-
bers and their absolutWralues. Hereafter when we speak of a number^
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.
— 1 , I , \ , \ , \, 1 -i-H -5- 1 — I ^^—^
D' O P B Jl O a B P C D
Fig. I.
Select any point C? on a straight line. Choose arbitrarily any unit length ; with
which coiistruct a scale of equal parts, Ay B^ 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
* + »>»
where x and y are any two real numbers, and i is a conventional symbol represent-
ing -4- ^ — I. 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 £1 is
conventionally a number common to all systems in the same way as is o.
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,
ibr 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 s is any complex number x + iy, its modulus or absolute value is
1,1 = 4. Vx*'-\-yK
4 INTRODUCTION TO THE CALCULUS. [Sec. L
toward the right, and A\ B*, C% . , . toward the left. Mark the points of divi-
sion, o at the origin O, and 4- it +2, etc., toward the right ; — i, — 2, etc.,
toward the left. Then, corresponding to any real number x there is a point P on
the line to the right oi 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 Pan the line there is a number
in the real-number system.
10. 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- ^c speak of such a 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 cons/an/ ac-
cording as it does or does not change its value^ring^n 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 «, v,
w, x,jf, z, etc., and constants by the initial letters j, b, c, etc., of
the alphabet. This is not always the case, however, as the context
will show.
11. Interyal of a Variable. — We shall sometimes confine our
attention to a portion of the number system. For example, we may
wish to consider only those numbers between a and 6. We shall employ
the symbol (a, b), a being less than b, to represent the numbers a, b
and all numbers between them. If we wish to exclude from this
system b only, we write (a, 3 ( ; if a only, we write ) a, b)\ when we
wish to exclude a and b and consider only those numbers greater than
a and less than ^, we repres,ent the system by )<i, ^(.
If jt is a general symbol representing any number in such a portion
of the number system, or interval, defined by a and 3, we have the
equivalent notations,
(fl, ^) = a ^x ^ b,
(at b(E:a ^ X < b,
){7, b)^a <ix ^ b,
ja, b{=a <:x <b.
A variable x is said to vary continuously through an interval (a, b),
when X starts with the value a and increases to b in such a manner as
to pass through the value of each and every number in (a, b). Or, x
Art. 12.] ON THE VARIABLE. 5
passes in like manner from d to a. The number x is said to increase
continuously from a to d, or decrease continuously from d to a.
o a X b
— I 1 1 I
Fig. 2.
In the geometrical picture, of § 9, illustrating the number system, if the points
A^ P^ Bf correspond to the numbers a, x, b^ the segment AB represents the inter-
val (a, b). As X varies continuously from a to b^ or through the interval la, b)^ the
point F corresponding to the number x generates the segment AB,
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 jt — a becomes and
remains less than any given assigned constant absolute number e what-
ever, we say that the number x has a for its limit.
In symbols, un^^e above conditions we write
which is read, '^^^^H of .^ is a. " The variable is said to converge
to its limit. ^Hi^
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 trraticnai number called a surd^ 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
rtj , flj , . . • , Hu , • . •
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 6 be given (however small), we can always assign an integer n for which
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 regular sequence of numbers is but the limit to
which converges the element Cr of that sequence, as r increases indefinitely. If a
be the symbol of the number thus defined, then in symbols we write,
«= ;f(«r).
It should be observed that irrational numbers having been thus defined, the
numbers ar in a regular sequence can be any numbers rational or irrational. The
regular sequence defines and assigns a number in its 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
I-:f« + i = (l — jr)(i+x + jf*+ . . . +jr«)
is established by multiplying the two factors on the right.
* Due to Cantor and Weierstrass.
INTRODUCTION TO THE CALCULUS.
[Sec. I.
Therelorei in compact symbolism, which we shall frequently employ,
I x«»+"
r-o
I — * I — Jf
If X is any number such that [Jri < I. we can make and keep ji^ + >, and there-
fore also the second term of the member on the right, less than any assigned
number €, by making n sufficiently great. Therefore, the limit of the sum of the
series on the left is i/(i — jc), or in symbols
£ 2 xr=i+x-\-x^-^ . . ,
I — JC
In this example the Tariable is
If now X is any assigned number in )o, i(, x is positive, and Sr continuously
increases as f* increases, llie variable Sr is always less than the limit. If jt is in
)— I, o(, it is negative, say Jr^ — « ; then
2 ( — !)''<»** = !—«+«' —
+ ( - lyan,
r-0
+
(— l)«fl»»+«
I -f-fl ' I -\-a
When H is even the variable St, is greater tha^^^^^Hi ; when n is odd
the variable Sn is less th'an its limit. Therefore, as ^m^l^es through integral
values, the variable converges to its limit, changing from greater than the limit to
less as n changes from even to odd and vice versa.
It is to be observed that if \x\ > i, the sum of the series and the equivalent
member on the right increase inacnnitcly with n, in absolute value, and can be
made greater than any assigned number and tlierefore become infinite. Under
these circumstances Ihe series has no limit ; its value becomes indeterminately
great.
Geometry furnishes numerous illustrations of the 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.
6. The evaluation of the irrational and transcendental number n 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 h- jt as x diminishes indefinitely
in absolute value/
Draw a circle with radius I. Draw MA =r
MB perpendicular to OT, Then
Area quadrilateral OA TB = tan jr.
Area triangle OAMB = sin x,
Area sector OANB = jr,
where x is, of course, the circular measure of
. Z AOT,
^ Then, obviously, from geometrical consider-
ations,
sin jc < X < tan x,
. X I
or I < - < .
smx
cos X
Fig. 3
• •
sinx
I > > cos X.
Art. 13.] ON THE VARIABLK 7
When jr diminishes indefinitely in absolute value, cos jt becomes more and more
nearly equal to i, and has the limit i as x converges to o. Consequently the quo-
tient (sin x)/jir converges to the limit i as j: converges to o. In our symbolism,
£m =
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, x is said
to be ififini/esimal.
When a variable becomes and remains greater, in absolute value,
than any arbitrarily great assigned number, the variable is said to be
infinile.
When a variable x is infinitesimal, we write* jf(=)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 a, or jQx = a, then by definition
£{x - a) = o.
When X — a\s infinitesimal, we write
X — tf( = )o.
This same relation we shall frequently express by the symbol
x{=)a,
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 Jf converges to a,** or ^^ x converges to a."
We shall frequently use the symbol € {epsilon) to represent an
arbitrarily small assigned absolute number. We then speak of the
interval (^ — 6, a -{- e) as the neighborhood of an assigned number
tf. The symbol x{=^)a means that "a: 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 j; = 00 . Such a
variable has no limit, it simply becomes indeterminately great. The
symbol Jtr = 00 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 witii 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.
|jf — a| = I 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. [Skc. L
It is impossible for a (one-valued) variable x to converge to two
unequal limits a and d. For, the differences |jkr — a| and | jt — 3| can-
not each be less than the assigned constant number i | ^ — ^ | 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 ^(a + ^)> where say a < 3.
Uxz=^{a + 6), .-. J»r-fl-i(3-a).
lfx>lla + 6), .'. X'-a>i{6-a),
Ux<i(a + 6), .-. 3 - ^ > i(d - a).
II. If two variables x andjf are always equal and eacA converges to
a limits then the limits are equal.
If J[^x = a, and J[^y = 3, and :i: =^ = «, then, by I, the variable
t cannot converge to two unequal limits simultaneously. Therefore
15. Theorems on the Limit.*
L If the limit of x is o, then also the limit of ex is o, where e is
finite and constant.
For, whatever be the assigned constant absolute number 6, we can
by definition of a limit make and keep |x| less than the constant
I e/c I , and therefore ex less than 6 in absolute value. Consequently,
by definition
£ {ex) = o = e£(x).
II. If each of k finite f number of variables ;C| , jc, , . . . , j;. , 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 n variables. Then
I :Cj + :c, + . . . 4" -*« I = «JP-
Since n is finite, the limit of this sum is o, by I.
III. If J[^Xy^ = «! , £,x^ =«,,..., J[^Xn = <»« • then when « is a
finite integer
£{^\ + ^1 + •••+*«) = £xy + £x^ + . . • + ;£*».
For, put JiFi = ^1 + ^1 > • • • » ^n = ^« + ^n- ^y definition, the
limits of or, , . . . , ^„ are o. Hence
*i + *« + •••+ •^» = K +•••+ ^«) + (^i +•••+ ''-)'
by II, gives
£{x^ + . . . + ^«) = <»i + • • • + ««•
Therefore the limit of the sum of a finite 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.
f If the number of variables is not 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 jc, which
have assigned limits a^ and a, , is equal to the product of their limits.
Let, as in III, jCj = ^ij + or^ , ^, = a, + a,.
By III, we have
But, jQa^ = o, jQa^ = o, and a fortiori J[^{oi^a^ = o. Therefore
Cor. The limit of the product of a finite number of variables having
assigned hmits, is equal to the product of their limits. In symbols *
£ n(x;i = n£{x;).
V. The limit of the quotient, x^/x^ , of two variables is equal to
the quotient of their limits, firoznded the limit of the denominator is
not o.
With the same symbolism as in IV,
^1 ^ ^1 + ^1 ^ ^1 . ^1 + ^. ^1
a a at — a ct
«,««K + «'.)"
By hypothesis, jQa^ = o, j£a^ = o, and a, :^ o. 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 I.f
VI. If X and y are two variables and a is a constant, such that^
always lies between x and a, then if j£x = a, also j£y = a.
* As the symbol 2 is used to indicate the sum, so i7 is used to indicate the
product of a set of numbers. Thus,
H
2Xr = Xi -|- JT, 4- . . . 4- ** f
X
M
X
The advantage of such symbolism is in compactness of the formulae.
f Notice particularly the provision that £xt =^ o. For, when £x^ = o and
£xj :^ o, the quotient xi/x^ inci eases beyond aU limit or becomes infinite as x,
and JT, converge to their limits. An infinite number cannot be a limit under the
definition.
Again, if /^x^ = o and also £xi = o, the quotient of the limits 0/0 is com-
pletely indeterminate, while the quotient Xt/x^ = q may or may not converge to
a determinate limit. The value of this quotient as x. and x% converge to o depends
on the law connecting the variables jt, and x^ as they converge to o. This case is
one of profound importance and is the foundation of the Differential Calculus.
lo INTRODUCTION TO THE CALCULUS. [Sec. I.
The truth of this is obvious, since |jif — <i| > !> — a|, and x —a
has the limit o.
In like manner, it follows that if x and z have the common limit
a, and ^^ is a third variable between x and s, then also must J[^ = a.
For, I J/ — a| must at all times be less than one or the other of the
differences |ji: — a| and |0 — a|, and each of these differences has
the limit o.
Vn. 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 a b? the common limit of x and y^ where x is always positive
and y is always negative. Then
+ U| = ii + a, and — |>'| = ii + /?,
where J[^a = o, £^fi = o. Subtracting,
\x\^\y\=a^p.
Since £^(a ^ /3) z=z o, . •. a -{- a = 2a = o, and a, the com-
mon limit of X and_y, is o.
VIII. If a 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 a: equal to or less than A.
(i). 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 (or 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 ^4 — a = A into 10 equal parts. The vari-
able X after attaining a must either attain a -f- -^A or remain always
less than a + -^h, l(x attains a -f ^A, it must either attain a + -^k
or remain always less than a + i\^. We continue to reason thus
until we find a digit /, such that x must attain a + — ^ and remain
* 10
^ + I
always less than a + '^ — ^' That is, x must enter and always
remain in one of the 10 intervals.
In like manner, divide the interval
(a + ^A, «+A±f>i)
\ 10 10 /
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 /,
such that
10 10 10 10* lO*
Art. 15.] ON THE VARIABLE. II
In like manner^ continue this process n times. Then
^+^y—r<^<<'+^y—r+—n'
' ^ IC^ ^ lO*^ ' ID*
I I
This process can be carried on indefinitely. Consequently the
construction leads to the constant number
CO
I
from which x can be made to differ by a number less than A/ 10* which
can be made and kept less than any given number e, for all values of
n greater than m, where ^/ic*" < €.
Therefore the constant a is the limit of x, 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,
IX. If there be two variables .r andj', such that > is always greater
than X, and if x continually increases and j' continually decreases, and
the difiference^ — x becomes less in absolute value than any assigned
absolute number e, then there is a constant number greater than x
and less than> which is the common limit of jt andj^.
By Theorem VIII, x has a superior limit a, and y has an inferior limit
d. For, any particular value^'^ 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, and y = d'\-/3f
we have
But, j£ {y ^ x) := o, j£{a -|- /5) = o; . •. a — 3 = o. This defines
the equality of a and d. Therefore x and y converge to a common
limit.
1 2 INTRODUCTION TO THE CALCULUS. [Sec. I,
EXERCISES.
1. The successive powers of any assigned number greater than i increase
indefiniteiy and become infinite as the exponent becomes infinite.
Let a be any absolute number, and m any integer.
Then (i -|- a)»» > I -f- ma, (i)
Infiact, (I -f-a)»=i-f 2a + a*> i -f 2a.
The formula (i) is true when m = 2. Assume it to be true when m = n. Then
(I + a)* > I -h na.
Multiply both sides by i -f ^>
.-. (I + a}«+i > I + (« + i)a + ua\
> i4-(if+i)a.
(I) is true also for » + i. But, being true for « = 2, it is also true foriw = 3,
and therefore for »« = 4, etc., and generally. Therefore, since ma and conse.
quently (i -|- cr)"* can be made greater than any assigned number, the proposition
is demonstrated.
2. The successive powers of any assigned absolute number less than I diminish
indefinitely and have o for limit.
Any number less than i can be written as the quotient 1/(1 -f- a). By Ex i
(1+ a)« i-^ma ma
This can be made less than any assigned number €, by sufficiently increasing m.
3. The successive roots of an absolute number greater than i continually
diminish ; those of an absolute number less than i continually increase ; and in
either case have the limit i.
Whatever be the absolute number a,
± («+i)-J / ' ) *+'
Therefore, by Exs. I, 2, whatever be the integer «,
I I
4f*"> a''+», iftf > I ;
a* < tf**"*"', is a < I.
I
If « > I, then fl" > I.
z
Let a =- I -J- cr, and tf * = i -|- y5 ;
I
then (I + or)" = I + fi,
or (I + a) = (I -I- /?)«>!+ nfi,
/Sf < a/«, and we have
I
^ < I + -,
n
1
Hence ;^ « *» = I.
Art. 15.] EXERCISES^ 1 3
Let a < I, say « = 1/(1 -f <r).
I I
Also, «*•<!, say «•« = 1/(1 -f fi).
Then, as before, fi < a/ftf and
1 I
I > tf« > ■- .--7-,
which shows again that
£ a'^ = I,
4. Show that when a is any assigned positive number,
£ "* = ».
whatever be the way in which x converges to o.
(I). Let w, ;#, /, ^ be any positive integers. Then
w > m p
t
If rt > I, then <»* > I.
.'. «*• *><»", and tf ^» V / < /7 *• .
Tlierefore rz* continually increases as x increases by rational numbers.
If ij < I, then a^ < i.
M > Ml
. •. a* I' < tf * , and
Therefore a* continually diminishes as x increases by rational numliers.
When |x| is rational and less than i, there can always be assigned two con-
secutive integers m and m -|- i such that
, < |jrl < ■ .
The above results show that whether a he greater or less than i, a" lies between
II II
rt** ^ ' and a*». When w = 00 . «*" "*^ * and <!"• converge to i, Ex. 3, and there-
fore also does a*; and £a* = i, when jf(=)o.
(2). When jr is irrational there can always be assigned two rational numbers a
and /} differing froiii each other as little as we choose, such that a < x < fi.
The number a* is defined by its lying between «« and afi. Since Jt=)o when
nr(=)y5( = )o, we have, as before, a* converging to I along with aa and afi,
5. Show that £a* :=z a^ z=i afi, if £x — ft.
We have «P — n-* = a^i — a* -/*).
Passing to limits, we have, by Ex. 4,
afi — £a* = o.
6. If it and fi are positive numbers, and £x = fi, show that
j^ loga ^ = loga £{x) = log^ fi,
H
We have log^ fi — loga x = log^ — .
X
The above exercises show that however x converges to fi, £ log^, (fi/x) = o.
Therefore
^^a fi - £ loga j: = o.
14 INTRODUCTION TO THE CALCULUS. [Sec. L
7. Utilize Ex. 6, to prove IV, V, from III, § 15.
8. Use Ex. 6, to show that
where ^ has a positive limit, and the limit oix is determinate.
9. A set of numbers a^^ a^^ . . . ^ ar y . . . ^ arranged in order is called a se-
quence. Any number of the sequence, ar » is called an element of the sequence ; the
number r is called the order of the element «r« Any sequence is said to be known
when each element is finite and known when its order is known.
\i a^, a^y . . . , Onf ... be a sequence of numbers such that Or is finite when
r is finite, then will ;fa«, when » = oo", be o or 00 according as
£\
On
is less or greater than i, respectively.
I^t, when « = 30 , £i^H + i/^n) — >&, and Jk > i. Then, by the definition of a
limit, we can always assign a number i^ such that !</''</', wht-nce corre-
sponding VoAfift can find an integer m for which we have, for all values of m.
By hypothesis, <j,„ is finite. Since we can make ^" greater than any assigned
number by sufficiently increasing «, we have £an = 00 .
In like manner, if £{an + i/^n) = /& < i,
which can be made less than any assigned number by increasing «, when as l)efore
I > >f < X*. . •. £aH = o, when « = 00 .
In order that the element an may have a finite-limit different from o, it is neces-
sary that *
/
fi±ii=ii.
The quotient, <7„ 4. ,/<?«, 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
^i + S +•••+<»«+•• • (0
is said to be absolutely convergent when the corresponding series of the absolute
values of the terms is convergent.
That is, when
^H - l^il + l^al + . . . H- \a„\
has a determinate limit when n =zqo ,
Show that (i) is absolutely convergent if
/
an
and if this limit is greater than I, the sum of the series is oo .
* When the symbols 1 = |, |> |, | <' are used, they mean that the equality or in-
equality of the absolute values of the two members of the equation is asserted.
Art. 15] EXERCISES. IS
Let the letters in (i) represent absolute numbers, and let
/
an
Then there can always be assigned an integer m corresponding to any number
k' such that >t < i' < I, for which
for all values of n. As in Ex. 9, we have
Hence the sum of the series after a^ is less than
^fl^-f . . . + ^«tf«+ . . . =M^+ • • • -f >^«+ . . . )•
= a
m
T^Tjp'
This is finite, since ^ =^ I. Therefore 5*00 must be finite. Also, by Ex. 9,
£a,K =0, when m = 00 . Consequently we can always assign an integer n such
that
for all values of m, where e is any assigned number. Hence Sn has a determi-
nate limit. Otherwise, the existence of the limit of Sn follows at once from VIII,
§15. For 5",. continually increases, but can never exceed
«i + «f + • • • 4- «« + «m j-^r^-
Again, if £{an-^i/an) > h say equal to ^. > i. Then, as before, we can assign
Jb' between k and i, and have the sum of the series after a^ greater than
am{^ + . . . +^*+ . . . ),
which is QO .
The number £iam+i/aM) is called the convergency quotient of the series.
11. The arithmetical average, or mean value of a sequence of n numbers,
flj, a^t . . . , Off ,
is one ifth of their sum, or
I *
n
I
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
where L and Af are the least and greatest elements respectively, the mean value
must remain finite. Also,
^■^^n+i «(» + /) ,
1 6 INTRODUCTION TO THE CALCULUS. [Skc L
But
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 n
that will make anJtp — ocn 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 Umit of
(-r)-
when z becomes infinite in any way whatever.
Divide both numerator and denominator in
m+i
where m is a positive integer, by jr** — i. Whence results
1. / J_\»i— I - — 1 I
I^-jC-.^- . . . '\\X^) ^-»+Jf« + . . . '\-X-m
i\\ If JT = I -I , then each of the m terms in the denominator of the frac-
tion on the right is less than I.
jp «« — I ,1 m-\-\
JP — I fn M
Hence "^ i
m
or ^i J '- — V > ( I-I-— 1 .
(.+,-irr>(-^i)'
Therefore, the value of the expression continually increases with «, and is
always greater than 2, by Ex. i.
i2\ If jr = I — ' , each of the m terms in the denominator of the same
fraction is greater than i.
I _^ -» I _ W+I
••• < * "r i; ~ — ir~'
I — X ^ ^
w + I
Hence i - jt * < -,
M + I
I
or * "* > ^ - 1:'
Art. 15.] EXERCISES. 17
or
(-^)->(--ir
TherefDre the expression continually diminishes as the positive integer m in-
creases.
^3). Whatever be the positive number Xj we have
Jfa > jc* — I.
X J^ 4- I
Hence (. - 1) '> (« + l)'.
whatever positive value x may have.
(4). Also,
= (■ - yh K- + yJ-
if we put jr s= >f + 1.
These results (i), . . . , (4), show that
(-4)'
continually increases as g increases by positive integers, and continually de-
creai>es 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
(I -h 1/6)6 = 2.521 . . . and (I — 1/6)- 6 = 2.985 . . .
We represent it, conventionally, by the symbol ^. More accurately computed,
its value is
^ = 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 is any positive number, rational or irrational, we can always find two con-
secutive integers m and m -\' i^ such that
m <i z <i m -\- If
"^ (' + ^r*(' + 7)'< (■+;)"*'•
" h^rh^r<{-^yr<hhTh^}
This shows that when m = 00 , then « = 00 , and
I + i-V= e.
« = 00
ij-
*Put (I — «-')-«* = amj [I -f (m - I)- »]-»-! = d^. Then assigning to
m the values i, 2, 3, . . . , we have two sequences of positive numbers. The
sequence o^ always diminishes, the sequence d^ always increases. The difference
^m — ^m is a positive number converging to o when » = oo . The two sequences
therefore define a common limit e.
x8 INTRODUCTION TO THE CALCULUS. [Sec. L
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 ;f (I -f x)* = ^.
*(-)o
14. Show that
£ logafl +4 1 = £ ^^'^^ + ^^' = loga-f,
«=oo \ •*/ x(»)o
and is i, iia = e. Use Ex. 6.
15. Show that ;f f i + - j = ;f (» + ^)^ = ^«
16. Show that £ (^ -^ i)A = ^^« **•
*(-)o
Hint Put a* = I + «.
17. If x» is a positive integer, show that
i
^ Wff** — '•
18. Show that Ex. 17 also holds true when m is a negative integer, also i£m
is any positive or negative rational number. JL i.
Hint. Put m -Si p/q. Divide the numerator and denominator by x<^ — a*,
to obtain the quotient in determinate form for evaluation.
19. Show that £ "" >y - sin a ^ ^^ ^ ^^ ^^ ^^ j ^^
«( = )« X — a
20i Let/,, represent some particular one of the digits o, i, . . . , 9, for a par-
ticular value of r. Show that the periodic decimal
a ./j . . . //// ^ ,...// ^ ,„// + X . • • // f « • • •
has for its limit the rational number
M+ 77 ,
' IO^(IO»»» — 1)'
where M^a-p^, . . // , and N = pipi ^. i . . .piJ^m, and // + ^ = /i + ^ + r»
^ being .any integer, and r any integer less than or equal to q.
*The evaluation here given is a modification of one due to Fort, 2^tschri/t fUr
Mathcmatik^ vii, p. 46 (1862}. See also Chrystal's Algebra^ Part II, p. 77.
SECTION II.
ON THE FUNCTION OF A VARIABLE.
16. Definition. — When two variables x zxsAy are so related that
corresponding to each value of one there is a value of the other they
are said to \iQ functions of each other.
If we fix the attention on ^ as the function, then x is called the
variable ; if on at as the function, then j' is called the variable.
Such functions as x and y defined above are not amenable to
mathematical analysis until the lanv of connectivity between them can
be expressed in mathematical language.
Classification of Functions.
Functions are classed as explicit 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,
^, sin X, \oga X,
and their inverse functions,
x~^^ sin~'jr, 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 ox transcendental ^.czoxA'-
ing as the number of operations (including only
addition, multiplication. involution,
subtraction, division, evolution,
by which the function is constructed from the variable), is finite or
infinite,
19
20 INTRODUCTION TO THE CALCULUS. [Sec. 1L
i8. The Explicit Rational Functions.
I. The Explicit Integral Rational Function.
The function of the variable x,
where the numbers a^ , . . . , a„ are independent of x, and n is definite
integer, is called an explicit integral rational function of x^ 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 of numbers. It can advantageously be expressed
by the compact symbolism
M
2;" a^ XT,
r=o
meaning the sum of terms of type a^ from r = o to r = «.
II. The Explicit Rational Function.
The quotient of two explicit integral rational functions of a vari-
able x^
K-vK^-{- . . . +^«^'
is called an explicit rational function of x^ or simply a rational function
of or.
Its place in the system of functions corresponds to that of the
rational or fractional number in the number system.
m. The Explicit Irrational Algebraic Function.
Any expression involving a variable x, or an integral or rational
function oi 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 oix.
Such a function in the function system- corresponds to those irra-
tional numbers in the number system called surds.
For example,
1 V^=^, a-^-bxi, VTfx/ Vi - or,
are irrational algebraic functions.
19. Explicit Transcendental Functions. — Any expression which
is constructed by an infinite (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 jr, ^*, log j:, tan - » jr, etc., 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 andj/, 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 implied 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^ 4" ^* — i" = o
defines :r as a function of y^ and, just as much so, ^ as a function of x. These
functions can be expressed explicitly by solving for x and y» Thus we have
= >('-^'' '"^ y-^~
-ax*
or X and^ are expressed as explicit irrational algebraic functions of each other.
In general, any algebraic polynomial in two variables x and^
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 F(x)^ 0(-^), ^'(•^), 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 s)rmbol
written y, F^ 0 or ^, etc.
We frequently employ the symbols j', «, «, v, etc., as functions of .r.
In like manner we write a function of two variables .^•, ^ as il>(x,y)
oxfix^y)^ etc., meaning a mathematical expression containing x and
y. The equation
implies, as said before, a functional relation between x and y^ and
defines J' as an implicit function of or, or jr as an implicit function of^'.
\if(pc) is a function of x^ and if a is any particular assigned value
of Xy we write /(tf) as the value of the function when j; = a, or, as we
say, the value oi/{x) at a.
For the present, when we use the word function we mean an
explicit function of one variable.
A function, f[pc)^ is said to be uniform or one-valued at a when the
function has one determinate value at a.
For example,
ax^ -\- bx A- Cy e»f sin x^
are one- valued functions for any value of jt.
li /\x) has two, three, etc., distinct values corresponding to a
2 2 INTRODUCTION TO THE CALCULUS. [Sec XL
value of the variable, it is said to be a two- , three- valued, etc., func-
tion.
For example, <ur*, |/<i* — jr*, are two- valued functions of jr.
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 ^(^ — jf* exists as a real function only in the inter-
val ( — a, -|- <*); the function represented by the series
I + JT -|- JC* + • • •
exists as a determinate finite function only in the interval ) ~ I, -|- i(.
22. Continuity of a Function.
Definition : f(pc) is said to be a continuous function of jrator = a»
when /(or) converges Xo f\a) as a limit, at the same time that x con-
verges continuously to a as a limit.
The definition and condition of continuity oi f{pc) at a are com-
pactly expressed in symbols by
£A^) ^A£^Y
The function /(;»:) is said to be continuous in an interval (o', ft)
when it is continuous for all values of ^ in (ar, p).
The definition of continuity of y*(^) at x asserts that whatever
absolute number S is assigned, we can always assign a corresponding
absolute number h such that for all values oix^ satisfying the inequality
we have
Since, by definition, the limit oi/[x^ \^/{pc)^ we can make and
keep
less than any assigned absolute number S for all values oi/{x^ sub-
sequent to an assigned value y(a:').
\{ X -\-h'\% the value of the variable corresponding X.of{x'^^ then
all the values of the function corresponding to the values of the vari-
able in (x, X -^ h) satisfy the inequality above.
An important corollary to the above and a principle which will
constantly be employed later is: \i/\a)^ the value oi /{pc) at <z, is
difi'erent from o and is finite, then we can always assign a finite num-
ber h such that for all values of x in the interval (a — ^, a -f- ^) the
function y^Ar) has the same sign as /*(«).
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
must be arbitrarily small in absolute value when the difference of the
corresponding values of the variable, x^— x^^ is arbitrarily small.
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 mce versa.
In symbols, wheny(jr) is continuous at a, we must have simulta-
neously
£{x - a) = o and ^[/(^) -/(a)] = o.
The definition and condition of continuity at a can be expressed in
the compact symbol
23. Fundamental Theorem of Continuity. — If /(jt) is a uniform
(§21) and continuous function of ^ in an interval \a, b), then what-
ever number iV^be assigned between the numbers _/(a) and/(^), there
is a value ^ of or in (a, b) such that at 5 we have
The proof of this theorem falls under two heads.
I. If a function /{x) is one-valued and continuous throughout an
interval (j, ^), and/* (a) and/* (3) have contrary signs, then there is a
number ^ in (a, b), at which we have
n$) = o.
Suppose /(a) is negative and/*(3) positive. T\\tTi/{x) cannot be
o or -f- arbitrarily near to j: = fl, nor can /"(or) be o or — arbitrarily
near io x = b, by definition of continuity oi/{x) at a and b.
Let b — a =.h. Divide this interval into 10 equal parts by the
numbers
^> ^ + tV^» • • • > ^ + A^> ^•
Either /*(j:) is o for x equal to one of these numbers, in which
case the theorem is proved, or it is not. In the latter case let a, be
the last of these numbers, proceeding from the left, at which /"(.r) is
negative, and b^ the first at which it is positive.
Proceed in exactly the same way, subdividing the interval {a^ , 3,)
into 10 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 b^ the first at
which it is positive.
Continuing this process n times, we find that either/* (jf) is o at
one of the interpolated numbers, or that /*(««) is negative and /(^«) is
positive (see § 15, VII, 3), and
n n
a,^a + h y^,, K^(^-\'h y^^ + A,
" ^ /^ 10'^ /^ ic 10"
I I
where each/r('' = i, 2, . . . , «) represents some one of the digits
o, I, . . . , 9. If /(at) is o for some one of the interpolated
numbers obtained by continually subdividing (<7, ^), the theorem is
proved; if not, then the two numbers a^ and 3„, the former always
24 INTRODUCTION TO THE CALCULUS. [Sec. II.
increasing, the latter always diminishing, converge to the common
limit
I
Meanwhile f{a^ and /{h^ converge to the common limit /'(5),
by the definition of continuity. The first of these y* (a.) is always
negative, the second y(^„) is always positive. Also, since b^ — a^ =
^/lo", we must have
£ \ /(*.) -/(«.) { = ^ ! I /(M I + 1 /(«.) I } .
But this limit is o, by definition of continuity.
. •. f(£) = o.
In like manner we prove the theorem when /(a) is positive and
f (b) is negative.
!!• The general theorem now follows immediately. For, what-
ever be the numbers /(a) and/(^), if Allies between them, then
f{x) - N
must have contrary signs when x :=i a, x -=.1. Therefore, by I, there
must be a number ^ in (a, 3) at which
The important fact demonstrated by this theorem is this: If a
function / \x) is uniform and continuous in an interval (a, b) of the
variable, then as the variable x varies continuously through the inter-
val (fl, ^), the function must vary continuously through the interval
determined by the numbersy(fl) and/"(^). That is, the function f(x)
must pass through every number between f[a) and f{b) 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 of a finite number of continuous functions is a con-
tinuous function throughout any common interval of continuity of
these functions.
\l f^{pc), fj^x), . . . ,y;,(j:), are continuous at jf, then
<f^(x) ^/^{x) + . . . +/^{x)
is a continuous function at x. For we have
^£<f>{^-') = /:iA(^')+ ■ ■ ' +/«(-^)],
= £A{^') + . • . £A{^').
=A{£^') + • • • A{£x'),
=/(.r) + . . . +A{x) = it>{x).
Art. 24.] ON THE FUNCTION OF A VARIABLE. 25
n. I'he product of a finite number of continuous functions is a
continuous function in any common interval of continuity of these
functions.
If 0Wh/(^) ./.W . . ./.W,
then £ 0(^) = £[/^(^') . . . /.(^O].
= £A{x') . . . £/n{x%
=/^(x) . . . Mx) = <t>{x).
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
flmction in their common interval of continuity, except at the values
of the variable for which the denominator is zero.
li /{x) = <l>(x)/^(x)^ then we can consider /(^) as the product of
€h{x) and i/tpix). The theorem is then true by the reasoning of the
preceding theorem.
Otherwise,
- i,(x) '
provided ^(x) ^ o. If ^(a:) = o and <p{x) ^ o, then/(jr) = 00 and
is not continuous at x. If tp(x) = o and also ((>(x) = o, then /{x)
may or may not have a determinate value as the limit oi /(x')y a case
which we shall investigate later.
IV. It has been shown, Exercises, Sec. I, Ex. 5, that
when a is positive. Therefore a-^'^ is continuous when /(x) is con-
tinuous.
V. In like manner, Ex. 6, Exercises, Sec. I,
£ log.yi^) = log, £r{x) = log^^^),
/[x) being positive. Therefore, log^or) is continuous.
VI. Again, if/\^) and <f>{x) are continuous andy(jt:) is positive,
we have
.-. log>' = <p{x) log/(:«:),
= £<t>(x).£\ogA^)'
• •• log ;^^ = <l>(£x) \OgA£^)'
= log [A£^)^*''^"■
.-. £{Ax)\*<'^ = \/{£x)-\*^'\
26
INTRODUCTION TO THE CALCULUS.
[Sec. 1L
and the function y is continuous when <t>(pc) is continuous and J\x) is
continuous and positive.
SpEaAL Theorems.
Since ^ = a: is a continuous function of x^ the product a^, where
a^ is independent of ^^ and r 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' -=. i\n X -\- 2 cos \(x' -}- x) sin \(pc^ — or),
and ;^ sin \{x' — ;»:)= o, when Ar'(=)a;,
we have ^ sin at' = sin jc = sin J[^x,
Therefore sin Jtr is everywhere continuous.
In like manner we show that cos x is everywhere continuous, and
by § 24, III, tan Xy cot x^ sec x and esc x are continuous fimctions
everywhere except at the roots of their denominators, cos x^ sin at.
The continuity of any algebraic function of
jc*, <z*, sin AT, log Xy
can now be easily determined.
35. Geometrical lilustration of Fimctions. — If we adopt the method of rep-
resenting the variable, in §§ 8, 1 1, by points on a straight line, such as Oxy then at
any point M on Ox corresponding to ;r = a we can represent the corresponding
K
■'79
V
K
J^-I_J_I_
P
^*
T
Pi
M,
Pj>
P^
MzM M,
-X
Fig. 4.
value of a uniform function /(jr) by a point Pin ^ plane xOy. The point P is con-
structed by laying off a perpendicular MP to Oxj such that the distance MP is
equal to the number /[a)^ and is measured upward if /{a) is positive, and down-
ward if /(a) is negative.
For each and every value of jt for which /(x) is a defined function, such as a.,
a^y . . . , we can construct corresponding points P^, P^, . . . , representing /(a\
f{a^\ . . .
This is the familiar method of Analytical Geometry, invented by Descartes.
If we put y = f[x)y then Oy perpendicular to Ox can be called the axis of the func-
tion, corresponding to Ox, the axis of the variable; and jr, y are the Cartesian coor-
dinates of the point P representing the functional form/(jr).
Art. 25.]
ON THE FUNCTION OF A VARIABLE.
27
If the function /(jr) is continuous in any interval (<7^, a^\ then corresponding to
any point M^ in M^M^ there is a point P^ in the plane xOy^ at a finite distance
from Oxy representing the function. Moreover, any two such points /*, P' cor-
responding to M'^ M" can be brought as near together as wc choose by bringing
M' and M** sufficiently near together. Can we say that the assemblage of all the
points, P, representing a continuous function in a given interval (a, b) of x^ is a
line?
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 length corresponding to any two arbitrarily
chosen points on the line, and also to have Erection at any point. This requires
a definition of direction and of lengthy 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 xOy representing a con-
tinuous function f\pc) 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 anal3rtical property that will define a determi-
nate length. These properties must be inherent in the function f[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 to a 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-
cemed 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 /\ , . . . , /m representing values of a continuous function in the interval
J/i M^ , nas the following properties :
3f,
Mr Mr^i
3f-
X
Fig. 5.
Join the consecutive points by straight lines. Consider the broken polygonal
line /*,/*,... Pn* Then, if Afj^ and Mn correspond to two fixed values a, d of x,
and we increase the number of points, J/, between M, and Afn indefinitely, in such
a manner that the distance between any two consecutive points Mr and Mr 4. i con.
verges to zero, we shall have :
First. The distance between the corresponding points Pr and Pr + 1 converges
to o. For, PrPr + j is the hypothenuse of a right-angled triangle, PrJ\/'Pr + u
whose sides Pr A^and AP^ + 1 =Mr Mr + x converge too together, when Mr{ = )Mr+if
since the function y(;ir) is continuous.*
* The point A^, not shown in the figure, is the point in which a straight line
through Pr + I parallel to Ox cuts Afr Pr*
28 INTRODUCTION TO THE CALCULUa [Sec. II.
Second. We assume that the angle /'r-i Pr^r-^x lietween any pair of con-
secutive sides of the polygonal line, such as /V — i Pr ^^ PrPr-\-\t converges to two
right angles as a limit
Third. We assume that the sum of the lengths of the sides of this polygonal
line /\ Pn converges to a determinate limit length.
The Hrst consideration secures continuity, the second determinate direction, and
tiie 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
EX£SCISES.
1. If /[x) = 2JC* — j:* — I2jr + I, show that the function has a root in
each of the intervals (o, i), (2, 3), (— 3i — 2).
2. If 4p{x) = (JT - i)/{x -\- I), show that
0 fl) - ip{d) __ g — ^
3. If iKO = ^ + e-', show that
^3/) = laoy - 3^0»
4. If J%x) = log "" ^, show that
5. What functions satisfy the functional equations
<K^) + <P(y) = 0(-^),
*(■') - *(JK) = ^^M
/{x - ;.) = F[x)/F{y).
6. If y(jr) H <*Jf* — ^* + ^» write /{sin jt).
7. If ^ = x'-|-jr — 5, write x as a function of^'.
X
8. Show that e* is discontinuous at x = o. Examine the behavior of this func-
tion as X increases through o.
9. If y = log {x -f- Vjt* -|- i), show that
X = \(gy — e-y).
This last function is called the hyperbolic sine of ^ and is written
sinh^ Er \(ey - e-y),
10. If y = log {x -f- 4/x' — i)t •^ is called the hyperbolic cosine of y and
written cosh y. Find this as a function of y,
11. Show that ey = sinh^ + cosh^.
12. Let X be any assigned real number. Consider the function
where n takes only positive integral values. Show that I{n) has the limit o when
« = 00 , whatever may be the finite value of x,
13. Show that
^, V _ «(« — 1) ... (a — r 4- I)
/('•) = -^^ —-r -^—^xr,
in which a and x are assigned real numbers, has the limit o when r = 00 , pro-
vided |jr| < I. What is the value of/(oo ) when |jr| > I ?
14. Investigate
/?■
for \x\ ^ I.
3© INTRODUCTION TO THE CALCULUS. [Sec. II.
15. The identity
- = m'- (^r
shows that the geometrical mean, ^ad, of two unequal numbers lies between them
and is less than their arithmetical mean l(a -f- d).
Finding the square root of any absolute number fi can be reduced to finding
the square root of a number between i and lOO. For, we can always assign an
integer n such that io"/fi? = or, where I < or < lOO; n being -}- or — according
as p is less or greater than i. Then
Consider any given number between I and lOO. Choose x^ from one of the
integers 2, . . . , 10, such that
(jT, - i)« < a < jcj«.
Then — < f'a < ^i ,
2.<Va<|(-x + ~) = -..
Show that if this construction be continued to jt^ y then
I
- Va <
« T »* -^ 2W« - I »
arfd therefore the sequence of numbers Xj , jr, , ... defines the square root of a,
and
£ x^ = f/a.
msoD
16. Apply 15 to show that 4/5, to six decimal places, is
2207 . ^o
^* "^ "^ ^ 2.2360689.
17. Show that the cubic function of x,
=A^)
a -x, h , g
h , b — Xy f
g > f y c-^x
always has three real roots.
Expanding with respect to the first row,
/(x) = (« - x)\{b - xtc - X) -/'] - [A\c -X)- ifgh +g»(6 - X)].
Let pt qy of which p is not less than 7, be the two roots of the quadratic
function
(I, _ x){c - jc) -/» = x» - (3 -f r)x 4 be -/».
Then p -\- q m b -{- c, 71x16. pq =1 be — p.
Therefore neither / nor q can be between b and c or equal to b or r, and / is
greater and q is less than either b or e. But
/(+oo) = — 00,
/{p) = + [A \^p-e + g i> ~ bf,
f{q) :=^[hVe^q^gVb^qY,
/(-oo) = +00.
Hence, by § 23, I, f{x) vanishes between + 00 and /, between p and q, also
between q and — ao , and the three roots are real. This exercise will be needed
in subsequent work.
Art. 25.] EXERCISES. 31
18. Dttermine the condition that the function
ajfi 4- 2bx -f- c
shall retain its sigrn unchanged for all values of the variable x.
The function can be written
_(ajc-f-^)«-|-(g^ — ^)
a
In order that this shall retain its sign unchanged for all values of jt, it is
necessary and sufficient that ac -' I^ shall be positive. This condition being satis-
fied, the function has the same sign as a iat all values of x.
19. Determine the condition that the function
shall retain its sign unchanged for all values of the variables x and^.
By completing the square, the function can be written
{ax -f hy)^ -{-y^jab — A*)
y
a
which, when ab — A* is positive, has the same sign as a for all values of x and^.
20. Determine the condition that the function
ax* -\- by* -\- cz* + 2fyt -\- 2gxz -f- ihxy
shall keep its sign unchanged for all values jt, y, jr.
By completing the square, the function can be written
^ ] (*^ 4-^ + W + («^ - ^^)y' + 2O - hg\yz + (ac - ^«)s« | .
The function will keep its sign unchanged and have the same sign as a what*
ever be the values of x, y^ x, provided the quadratic function
(ab — A»)y + 2(/a — hg)yz + (ac — ^*)««
is always positive. This will be the case, by Ex. 19, when
ab - h^ and (ab - h*\ac - g*) - {/a - A^y
are both positive. Or, what is the same thing, ab — A* and
a{abc -f 2/jgA — a/* — bg* — cA*) = a
must be positive.
Exercises 18, 19, 20 will be drawn on in the sequeL
a Ag
Abf
gfc
33
BOOK I.
FUNCTIONS OF ONE VARIABLE.
PART I.
PRINCIPLES OF THE DIFFERENTIAL CALCULUS.
CHAPTER I.
ON THE DERIVATIVE OF A FUNCTION.
26. The Difference of the Variable. — The difference of 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 arbitrarily
assigned value of the variable, say x^.
Or, in symbols, . .
x^ — X,
We use the characteristic letter A to represent the symbol of this
operation, and write
Ax = x^^ X.
This difference. Ax, is of course positive or negative according
as X is greater or less than x.
We sometimes for convenience write
Ax = x^ — or = A,
so that
x^ = x + A,
and call A 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 arbitrarily
chosen value of the variable, x^. In symbols
A^^ -AA
is the difference of the function y(j;) at x.
As in the case of the difference of the variable, we use the letter
A as the symbol of this operation, and write
4A*) H/(^.) -A^)-
35
36 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. L
28. The Difference-Quotient of a Function.
A difference of a function and a difference of tiie 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-quotient of the function at the particular value of the
variable.
Thus, in symbols,
^/(x) _Ax^ -fix)
^X X^ — X
= 9i
is the difference-^otient of the function y][Ar) 2X x.
For an assigned particular value x^ the number q^ depends on
the value assigned to the arbitrary number x^.
29. The Derivative of a Function.
Definition. — Whenever the function /"(jt) is such that when we
assign to the arbitrary value of the variable successive arbitrarily
chosen values
in such a manner that this sequence converges to the particular value
;i: as a limit, and the corresponding sequence of difference-quotients,
x^-^-x ~^" x^-x -y«"--' A, -Jtr -y----
has a determinate number as a limit, this limi/ is called the deriva-
iwe of the function /{x) at x.
In other words, the function yi[:r) is said to be differ entiable at x
when the difference-quotient
^x x' -^ X
converges to a determinate limit as x' converges to x as a limit in
any arbitrary manner whatever.
In symbols,
/{x') -Ax)
£
X^ — X
is called the derroaiive oi/(x) at x.
This derivative is, in general, a function of x^ and we shall
represent it, after Lagrange, by the symbol /'{x)y a convenient and
characteristic symbolism because it shows the association of the
derivative A{x) with the primitive function /(x) from which it has
been derived.
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 Z? 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
'/{x') -Ax)
m-x) = £■
X — X
But we have already agreed that this limit, the derivative, shall
be represented ^y f\x). Hence we have the equivalent symbolism
Or, the operation D performed on the function J\pc) results in
the derivative /"'(ji:).
This operation is called differentiation,
30. Observations on the Derivative. — We observe that in order
that a function /(a:) may be differentiable (have a derivative), it must
be continuous. For, unless we have
£ ^Ax) = £[A^) -Ax)] = o,
as is required by the definition of a continuous function, then, since
we do have
£ix' -x) = o.
xf(^)x
the value of the corresponding difference-quotient would be 00 , 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. [Ch. L
(the difference of the function, 4/^), and the limit of the denomi-
nator, Jjc, 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'ratio 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 constant is necessarily o by the definition.
For, the quotient of differences is constantly o and remains o for
Jjc(=)o.
EXAMPLES.
1. Differentiate the function x*.
We have the difference-quotient,
-J = y + jr.
XT '- X
The limit of this number when x'(=i)x is 2x.
.'. Dx^ = 2JC.
2. Differentiate the function jt*.
We have
the limit ot which is x~^/2 when jp'(=)jr.
3. If f(x) ~ sin x, show that Z? sin jc = cos x.
We have, by Trigonometry,
sin j/ — sin j: = 2 cos \{x^ + ^) sin \(xf — x).
sin jf* — sin JT , , ^ . x sin Ux* — x)
••• Zi = cos \{Jc^ + x) —-^3 —\
But, by § 12, Ex. 4, i "'"jy J^^ = I,
/sin |(y >
. •. f'{x) = cos X.
4. Show that the derivative of any constant is zero.
If ^ is any constant, it keeps its value unchanged whatever be the value of jr.
Therefore the difference-quotient is
A - A
jTj — j:
= o
for all values of x^ ^ x and when jrj(=r)jr. 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 a be constant and^ a function of x. Let^ take the value ^| when x takes
the value Xy The difference-quotient of ay is
aji - ^y _ ^ y\-y
jr, — jr
x^ — X
the limit of which is aDy.
Day = oDy,
31. Geometrical Picture. — We have seen that a difTerentiable
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.
^
/
V
/
I
^
jA
/^
^ry
u
0
Q^
j
V
Fig. 6.
In the figure, if/', P' represent /■(ji[:),y(jf'), then
Ax -=. x' — X •=. MM\
A/(x) NP' .,
v; • = -oTr = tan e\
Ax PN
where 6^ ^ /, NPP' is the angle which the secant PP^ makes with Ox,
By the definition of a tangent to a curve, the limiting position of the
secant PP^ as the point /" moves along the curve and converges to P
as a limit is the tangent PT to the curve at P, At the same time 6*
converges to 6^ as a limit, 6 being the angle which the tangent PT
makes with Ox. But tan 6 is the limit of tan 0\ and is therefore the
limit of the difference-quotient, or is the derivative oi /{x) at x.
Therefore we have
D/{x) ^/'(x) = tan 0.
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.
4© PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. I.
32. Ab Initio Differentiatioii. — 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 x and sin x by
the ad tm/io 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 initio
j)rocess 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.
EZSSCISES.
Differentiate the following functions :
1. 3jr* — tx, 6(jp — I).
2. 7JI:* — 13. T&jfl.
3. (X - i){2x -f- 3). 4Jf + I.
4. x-\ — x-a.
5. tfjr-3. — 3ajr^.
6. {X — a)/{x + a). 2a{x -f- <»)-».
7. x«. \j^.
a(^-2)». *(jr»-2)-».
9. 2{X + I)-*. - (* + l)~«.
10. **. kx-l,
11. jr". (if any finite integer.*) lu^-K
± p ±-x
12. x^. (/ and y positive integers.*) f-~x^ .
13. cos X, — sin x.
14. tanx. 8ec*jr.
15. log X, (See § 15, Ex. 11.) x-K
16. sec jr. sec jf tan x.
17. a'. (Use Ex. 15, § 15.) a* log, a.
18. Jp*. (x positive, a rationaL) ax^—^.
* Divide numerator and denominator of the diff.-quot. hy Xi — x in Ex. 1 1,
I 1
and by JTi^ — Jf* in Ex. 12.
CHAPTER II.
RULES FOR ELEMENTARY DIFFERENTIATION.
33. As was stated in Chapter I, when we have once differentiated
.r*, sin X, log x, by the ad initio process, we can differentiate directly
any elementary function of these functions by certain rules for
differentiation, without recourse to the ab 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 a:, are necessary, for x^ can be
differentiated by means of log x. Independent proofs, however, are
given in each case.
34* Derivative of logn x. — We have for the difference-quotient,
writing ^^ — jr = ^,
writing x/A = z. When ^(=)o, « = 00 .
...Z)log.^ = y*ilog.(i+i)',
s>ao
= ^log- ^(i + j)'. §15, Ex. 6.
SsOO
* As a matter of feet, the evaluation of only one of these functions, log jr, by
the ad initio process is necessary. That is, the differentiation of all functions can
te reduced to the evaluation of the single limit, (i -|- l/jr)«, when x=roo,
§ 15, Ex. 12. For, the differentiation of log x gives that of ^, and we have
sin X = —.(€^* — ^-"), where jE -h V'— '• We do not, however, recognize com-
plex numbers in this book directly, which necessitates an independent differentia-
tion of sin JT, and restricts us to a geometrical definition and differentiation of that
function.
41
42 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II,
The evaluation of this limit is effected in § 15, Ex. 12, and is
^he number
-\ — j = ^ = 2.7182 . . .
/('
s>aD
In particular, ii a = e, then log^ e = i, and
Dlogx = -.
According to common usage, when the base of the logarithm
employed is e we omit writing the base and put log x for log, x,
35. Derivative of x*. — Let a =p/gy where/ and ^ are positive
integers.
Dividing the numerator and denominator of the difference-
quotient
x^t — x<i
x^ — X
^y -^1* — j:?, the difference quotient-becomes
In the numerator there are / terms each of which has the limit
\X9] J and in the denominator there are g terms each of which
has the limit \xl] , when x^{=)x. Therefore the limit of the
difference-quotient is
= —xq .
If a = — p/q^ then the difference-quotient is
-t -^ A i.
x^ <i — X <i _^ — \ x^<i — Jf *
* x^<ix<t ^
the limit of which for jCj(=)ar is, by the above,
X<1 '
Art. 36.] RUI^S FOR ELEMENTARY DIFFERENTIATION. 43
Therefore, whatever be the rational number a,
Rule: Multiply by the exponent and diminish the exponent by i.
36. Derivative of sin z, cos z. — It has been shown in Chapter I,
§ 30, Ex. 3, that
/? sin a: = 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 or = — 2 sin ^(x' -\- x) sin ^{x' — x).
cos x^ — cos X . , / / . V sin Ux^ — x)
• • • 7 = — sin Ux' + x) ,/J, r-^
X' — X 2V I / 1^^ _ ^J
Hence, on passing to the limit,
D cos X =. — sin x.
Rules for Differentiation.
37. We proceed to establish rules for the derivative of the {1) suniy
(2) product, (3) quotient, (4) inverse function, and (5) lunction
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 y=u -\-v -\-w,
where «, v, w are differentiable functions of x. Let the differences
of these functions be 4>'> ^^9 ^^9 ^^> respectively, corresponding
to the difference Jx of the variable x. Then, if y, u, v, w take the
values ^j, «j, »j, Wj when x takes the value x^, we have
yx -'y = ^^9 . • . >'i =y + Ay,
and so for «, v, w,
y^ = u^ +^i + «^i,
yx -y = («i - «) + (^1 -v) + {^x - ^)9
or
Ay ■=^ Au \- Av -\- Aw,
Ay Au ^ Av , Aw
Ax "" Ax Ax Ax '
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 PRINaPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II.
Since the limit of a sum is equal to the sum of the limits, we have
for Jx{=)o, on passing to limits,
I>y = I}u'\'Dv + Dw, (I)
or D{u + v + w)=:I}u + Dv+ Bw,
Corollary. What has been proved for three functions here is
equally true for any finite number of functions «, , . . • f^n , and it can
be proved in the same way that
D2u^ = 2Du^ ;
I 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
corresi)onding theorem of limits, § 15, which justifies the equality.
EXAMPLES.
1 . The derivative of any polynomial in x,
«o 4- ^i^ + «r** + • • • + ^»**»
is
This can be expressed in the following rule :
Strike out every term independent of x, since its derivative is zero, and
multiply each remaining term by the exponent of that term and diminish that
exponent by i.
2. If ^ = 2x1 -f log X* — 3 sin X,
show that Dy = 5X* -f- 5 A — 3 cos x.
cx^ -I- dx •+• a
3. If /(x) EE -^—^ ^^^^, show that
/'(x) = r - ax-2.
4. Make use of the identity
sin (a -\- x) z= sin a cos x -f- cos a sin x,
to show that I> sin (a -f- x) = cos {a + x).
39. Derivative of a Product of Functions.
Let jf=uv.
Then, with notation as in § 40, we have
J>; = (« -f- ^u){v -|- Jv) — uv,
= V Ju -\- u /Jv -{- ^u • Jv, ,
^y _^ ^^ _y ^v Au-Av ...
Ax "^ Ax Ax ' 4^ ^ '
Since, by hypothesis,
Art. 40.] RULES FOR ELEMENTARY DIFFERENTIATION. 45
are finite, the last term on the right of (i) has the limit o when
j^x{=z)o ; for it can be written either
H^) °' (^)^"'
and Ju{^)o, ^v(=)o, when ^jr(=)o, the functions being con-
tinuous.
Therefore, in the limit, (i) becomes
B(uv) z= V Bu -{-^ u Ih. (II)
In particular, if v is constant, v = Oy then Da = o, and
I){au) = a Du.
Corollary. Show that
D(umi)) = uv Dw -f- tnv Dv + vu) 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.
1. Show that D{^ sin x) = «x*— « sin jc -f- •«* cos x,
2. I\xi^ log x) = je«-i (log X* -f i).
3. Show that £ [D log jc»ta « — cos j: log x) = i.
4. Show that D sin* x = sin 2x.
2
6. If y = (log x)*, show that Dy z= log (x)'.
6. If /{jc) = log jt*, then /'(x) = 2/x.
7. Show that Z> sin 2jr = 2 cos 2x,
8. Show that D cos 2jir = — 2 sin 2x.
Use cos 2x = (cos x -f- sin jir)(cas x — sin x).
9. Show that D (log jr*) = log * -f i.
40. Derivative of a Quotient.
Let V = — .
Then_y, it, v, become y + ^>'> ^ + ^^'j ^ + ^*'» when or becomes
^ -|- ^x, and we have
~ v{v + ^») *
4y Jx Ax
^j; v(^ -\- Av)
46 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II.
Since ^v(=z)o when /Jx{=:)o, we have, provided v i^ o, on
])assing to limits
^y = ^0 = "-^^^-^^ (III)
In particular, if u = a, any constant, then Du = o, and
EXAMPLES.
1. Show that D tan x = sec' x.
sin X
We have Z? tan x = D
cos jr
cos'jr 4- sin' x
"~ cos* X *
= sec' jr.
2. Show that Z> cot x = — esc* jc, using both
cot jf = cos jf/sin jc and cot jr = i/tan jc.
3. Show that D sec jt = sec x tan jr.
using both sec x = i/cos jr and sec x = tan x/sin jr.
4. Show that D esc x = — esc x cot x,
5. Show that D vers jc = sin x.
a A~ X b — a
6. Show that D -rp^ = ,-jr-r-4,-
0 -\- X {b-\- xy
7. Show that D f^-i^V= 4/j / "^ '^.
8. Show that D
I
log X log a *
log ox " (log axY
41. Derivative of the Inverse Function. — \iy is a continuous
function of x^ we must have 4y( = )o when Jjr(=)o, by the defini-
tion of continuity. Therefore for any particular value of x at which
J/ is a continuous function of x we can always make ^y converge to
o continuously in any manner we choose, such that simultaneously
we have Ax = o. Also, for corresponding differences Ay and Ax^
we have
Ay Ax _
llx Ay ~~
If we represent the derivative oiy with respect to .r, by D^^y, and
the derivative of x with respect to y, by D^^ then whenever j/ is a
differentiable function of x and D^y ^ o, we shall have x a differ-
entiable function of ^, and the relation
DjJ^'DyX = I
always exists.
Therefore, \iy and x 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 tlie first derivative.
If^ =:/[x)f then X = (f>{y), obtained by solving^' =/(x) for x^
is the inverse function o{/{x).
Geometrical Illustration.
/
If the curve AB represents the function y =/{x), and we con*
sider x as the function and y as the variable, we have ^= 0(>')
X
Fig. 7.
represented by the same curve, except that now Oy is the axis of
the variable and Ox the axis of the function. For a particular x,
the point X represents /"(ji:) and ^(j^), and we have
xX =^/{x) ; yX = 0(>).
Again, if 6, 0 are the angles made by the tangent to AB at X,
with OXf Oy respectively, measured according to the conventions of
Cartesian Geometry, we have
^^ =rW = tan e,
D^ = (t>\y) = tan 0.
But, since we always have tan 0 tan 0 = i,
. *. D^y-DyX = I.
EXAMPLES.
1. If ^ = jr» -f 2ax -f 6, then
D^y = 2{x -f a),
^ 2(jr + a)
If we solve for jc, we get the inverse function
X = — a ± Vfl* -{-y — ^t
a function which we do not yet know how to differentiate, but we know its deriva-
tive must be the value DyX obtained above.
2. If J' = Jri, find D^^r DyX^ and verify DyDx = i.
48 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II-
3. Differentiate sin— « jr.
J£y = sin-«jf, then x = sin^.
Hence DxV = — .
Vi -X*
We know from Trigonometry that the angle whose sine is x, sin— 'jr, is
multiple- valued and that
sin [HTt -f (— i)*^] = sin ©,
where n is any integer. In the derivative of sin-»jc above, the radical shows its
value is ambiguous as to sign. But if we agree to take sin— ijr to mean that angle
between — ^le and -j- \ft whose sine is x, there is no ambigruity, since then cos>' is
positive.
Then we have
D sin— »x = ==:=-.
4. Show in like manner that
^ I
D cos-»jc = -— -^— ,
where o < cos— »jir < jr.
This can also be shown immediately by differentiating the identity
sin— 'JT -j- cos-'JT = jjr.
5. Show that Z> tan-»j: = "V ' .
I -f Jt'
Put y = tan— >jc, then x = tan^, and
DyX = sec'^ = I -f x». Ex. I, § 41.
.-. Z? tan-ijc = -t-ljj ,
where we take tan— ^x to be that number such that
— i^ < tan-'j: < -f J*.
6. Show in like manner that
Z> cot-.. = ^.
where o < cot-»jr < 7C,
Also, by § 38, from
tan-»x -f cot-'jc = l^T.
7. Show that D sec-»x = - _ .
x^x'
H y =. sec-*x, then x = secj', and
DyX = sec^ tan^' = x ^x^ — i. Ex. 3, §41.
X ^X^ - I
8. Show, as in Ex. 7, that
D csc-'jr = ~~
X 4/x* — I
Also, by § 38, using the identity
csc-»x -f sec-*jr =: \ic.
Art. 42] RULES FOR ELEMENTARY DIFFERENTLVTION. 49
9. Difierentiate a*.
Put y — a^y then x =r log^^.
Therefore, by g 34, we have
y
.•. Djty = ~^— = a* log, a.
loga <r
In particular, \ia = ey then
/?a* = a* log a
becomes
Z><* = <!*,
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 a be a function of the variable y^ say z =/][_>'), and let ^^ be a
function of :r, say^y = <f>{x). We require the derivative of z with
respect to the variable x,
If«is a differentiable function of the variable^', and>'isadif-
ferentiable function of the variable x^ for corresponding values of z^
y and x, then we shall have
D^z^D^^D^y, (VI)
or /'Ay)^fy{y)'^^^
For, we have
Az _Az Ay
Ax^ Ay Ax^
and since by hypothesis D/s and D^jf are determinate limits, Djs is a
determinate limit equal to their product, and (VI) is true.
Corollary. If « is a function of v, v a function of w^w a, function
oiz, z 2L function of^^, and finally y a function of x, then the difference*
relation
Au _ Au Av Aw A 2 Ay
Ax ~~ Av Aw Az ^y Ax
leads to the derivative
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 jr«, when x is positive and a irrational.
Put^ = JK«, then taking the logarithm or, as we shall say, *< Iogarating,"f we
have
log^ = a log jr.
* For a geometrical picture of a function of a function, see Appendix, Note 2.
•f The term ** taking the logarithm " is the meaning of an operation so frequently
used that it seems to deserve a verb <*to logarate."
5© PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II.
Differentiate with respect to x. We have
y X
y
, •. Dy = a — = ajp«— I,
X
the same formula as when a is rational.
2. Differentiate {a -f- ^•«')*.
Put Jy^y ) = ,v«, where y ■= a -\- hx,
• •• /i(>') = ay^-^Dy^ and Dy z:z b,
. •. /?(a ^- ^jr)« = ^a(tf -\- bx) «— I.
3. To find D cos jr from D sinx =. cos x.
We have cos x = sin (^^ — x).
. •. /? cos jf = Z> sin (\i[ — x),
= cos (^jr — x) IWie — x),
= — sin X.
4. Deduce in like manner D cot x, D esc x, given the derivatives of tan x
and sec x.
5. If ^ = cos-ijT, then x = cos_^'.
Differentiate both sides with respect to x,
I = — sin^Zjy.
. •. D cos— »af, as before.
6. Find in like manner D cot-»x, D csc-»x, from D tan x, Z> sec x.
7. If ;^ = «*, then log^ = x log a.
Differentiating with respect to x, we have
DyXogy-Dxy = log a,
or - DxV = \og a,
y
.'. /^jr^' = a* log tf, as before
8. Differentiate Vrt" - jr». Put « = ^« - jc«.
-- X
Va* - or*'
9. As an example of the differentiation of a complicated function of functions,
differentiate
log sin tCiM{a-bx)*^
Let
y •= a — 6x. .-. Dj^y •= — b.
z =. {a — bxY =^', ^ .•. DyZ = 3^*.
u = cos {a — bxf = cos «, . • . Z>,« = — sin z.
w = sin ^ cos (a— 6jf) = sin r, . ■ . /?t,w = cos v.
Therefore the required derivative is the function
— r'^ sin z c<;s v.
w
which can be expressed as a function of x directly.
Art. 43] RULES FOR ELEMENTARY DIFFERENTIATION. 51
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
D(uv*^) Du Du
the upper signs going together and lower signs going together. Put^' =^ uv*^,
then taking the logarithm,
log ^^ = log f# ± log V,
Dy _Du Du
y u V
This expresses compactly the formulae for differentiating the product and the
quotient of two functions.
2. Show that if M| M, . . . un is the product of n functions of x, the derivative of
the product is given by
DU^jUr _ ST^Dur
I
3. Differentiate tP>^ where u and v are functions of x.
Put y rz w> and take the logarithm.
. •. log y =1 V log M.
Differentiating,
— ^ = z/v.loe u -|- V — .
y u
Dte> = «» [log uDv -\- '^Du\,
4. Differentiate log, u,
'?M\,y = log» M, then v' = «. Ix>garate this with respect to the base e, and
we have
y log V = log M.
Differentiating with respect to jt,
r^ . y ^ -^
log V Dy -| Dv = .
_ , (Du \ogu Dv\ I
,-. />log,«= — - - -5 j.^ .
\u log V V J log V
44. For general reference in differentiajtion a table or catechism of
the standard rules and elementary derivatives is compiled and should
be memorized.
In this table « or » is any differentiate function of a variable with
respect to which the differentiation is performed.
5^
PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IT.
I.
2.
3-
The Derivative Catechism.
D{cu) —cDu,
D{u + v)=I)u + Ih.
D{uv) —uDv-\-vDu.
lu \ _^v Du — uDv
5.
6.
7-
Dv
-(9 =-^
Du'
= au
Du.
Du ,
D log. « = - log. e.
8. Z? log tt =
Du
u
9. Z)fl» = fl» log <z /?«.
0. De"" = tf« Z?«.
1. Du"* = «• log « /?» -|- w""' Z?«.
2. Z> sin « = + cos u Du.
3. Z? cos tt = — sin « 2?tt.
4. J9 tan « = + sec' uDu,
$. D cot u = — esc* «2?«.
6. D sec tt = + sec u tan «Z?«.
7. Z? CSC « = — CSC 2/ cot u Du.
8. Z)sin-'«
9. Z? COS*"*!/ = —
Du
20.
21.
22.
Z? tan-'«
Z> cot-'tt
D sec~'«
= +
= +
VI - «*
Du
I + ««*
Du
I + «2'
Du
23. Z? csc~*« = —
Du
24.
Z? vers~'« =
Z?i/
V2« — ti^
Art. 44.] EXERCISES. 53
XZXRCISES.
1. Differentiate by the ab initio process, and check by the catechism, the follow,
ing functions :
(I), jr. (2), ex, (3), 2J^. (4), cx^^ (5), ^'. (6), ax'4. (7^ jr« - 2jr. (8),
54r« - 4jf -I- 7. (9), l/(ax H- ^). (10), Jf* - 3x - 2jr-«. (ix), (x - 1X3^? + 2).
(12), (jr - 3)/(jr -f SH. (13), x*. (14), xL (15), jr"*.
The solution in each case depends on the £ict that a" -- 6* is divisible by a — ^
when n is an integer.
(16), cos-. (17), sin ax, (18), tan ax. (19), esc ax.
2. Draw the curves = ^jfl and find the slope of the tangent where jt = 2.
3. Draw the curves = j(^ -f 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
1
of jH, x^f x~^, X « , and check the results by the rule for diflferentiating a function
of a function.
5. Show that the equation to the tangent to any curve y = /(x) is
the point representing/ (a) being the point of contact
6. Differentiate Vfl« - jc«, ^jfi - ««, Va* -f 2hx.
Ans. - :c(fl« - j:«)-*, :r(jc« - fl«)"^, %» -|- 23jr)-*.
(1) Aa + j:)* = r(j + x)*-'.
(2) A« + •«*/" = M^ 4- ■«•)*•
(3) />(^ 4. bj^f •= I2^jr«(^ + ^jr»)^.
(4) Z>(fljr* + ^or -f f)* = 5(fljr» + *x + €)\%ax + ^).
(5) /?(tfi - jr«)» = - ioLr(a* — x«)*.
(6) JXa^x 4- ^Jf*)' = 7(tf*x + bjflf{a^ + 2^x).
(7) D{b -j- rx")« = mfux-*-'{b -\- fx*")»-».
(8) Z?(i +flx*)"* = - axil + ajf*)"^.
(9) /)(„« - ^)* = _ jjc*(at _ jr»)~*.
(10) /> sin* ox = — /> cos' <ix-= a sin 2ax.
(11) D sin* ax ■= na sin»«-»«ix cos ax.
(12) D sin (sin x) = cos x cos (sin x).
7. Show that the equation to the tangent at x = a, >' = /9, for the curve
(1) X* -f ^* = tf* is xa -}- ^/5 = fl*.
^ . x* y , xa yfi .
(3) ^' = 4/-* » -^'Z' = 2/(jr + a).
jl. Given sin 3X = 3 sin x — 4 sin* x, find cos 3X.
9. Given cos 5X r= 16 cos^ x — 20 cos' x 4 5 cos x, find sin 5x.
10. Verify cos x = I — 2 sin' \x^ by differentiating.
11. Obtain new identities by differentiating
sin 3a 4 sii^ 2a — sin a = 4 sin a cos \a cos }tf,
sin b sin (^Jt — b) sin (|« -(- ^) = ^ sin 3^,
a and 3 being variables.
54 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IL
12. Differentiate the identity
cos* 2JC — 3 cos 2x = 4(cos* jr — sin* x).
13. Differentiate jc* sin x, jr* |/<i -^ ^jr, (<wr -f ^)*.
14. /?[(j: + i)*(2j: -!)»] = (idr + i)(jr -|- i)*(2jr - i)«.
15. />[(x« + i)(-r» - X)*] = ^'^";!'^"'.
16. Z>{(l - 2jr + 3JI:* - 4Jt«)(i + j:)»| = - 2a«»(i -f *).
17. D{{1 - 3Jf* -f 6r*)(i -f :r*)»} = 6a«*(i -f :r«)«.
18. Show that
\l-Jf (I - jr) Vi - jr«'
I — x
19. Show that
/? un-i — ==r- = — • Z)sin-»^-Z — -
20. Differentiate 8in-«(jr/fl), tan-»(ajr + ^), cos-i — , sec-»(a/x),
scc-'(jr + «*).
21. /> log sin X = cot x; 2> log cos x = ?
22. Differentiate ^, <-*, <r«*, ^^*, ^<«*.
23. Differentiate a'*, tf*»«*, ai<«* a*«*.
24. Differentiate log x*, log (a -\- x), log (ox + 6% x»e*, a^ex^ 2*,
r« log (x 4- a\ log (x + e»), e»/\og x, log (x#«), sin (^) log x, /«>•« log (cos x),
logatanx, 3»<w«, 5«^««, logj^a (cos ox).
26. /? sin [cos {ax -f ^)«] = — na(a -|- *x)»«-x sin (ax -f *)«• cos [cos(tfx + ^)«] .
26. If ^ = \{e* — <-*), show that
■^ = log (-K + i^rT7),
and that />x^ -^y^ = i*
27. In Ex. i| § 41, differentiate x as a function ofy 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 f(pc) represents any function of x^ and x^ — x any difference
of the variable, then
(^, - ^)/'(^)
is the differential oif{x) at x.
The value of the differential at a particular value x depends on
the value assigned to the arbitrary number at^.
We use, after Leibnitz, the characteristic letter d to represent the
differential, and write d/{pc) to represent the differential of the func-
tion y(^) at x. Thus
df{x) = {X, - x)/\x),
^/'{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, lety(jir)=^, ih^nf\x) = i, and
dx = Dx-Ax
= Ax,
Therefore we can write dx for Ax^ and have
df{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 always a finite
number as its value. In fact we cannot assign 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
♦ By some writers the derivative f'(x) is called the differential-coefficient of
the function /(jr), because of its relation to the differentials in the equation
df{x) ^ f\x) dx,
55
56
PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. III.
or, the differential-quotient 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
or
dx'
where J/ =/(jr). Also, yi\\tn/{x) is a complicated function we fre-
quently write
48. Geometrical Blnstratioii. — We have seen, § 31, that if
^ =:/{x) is represented by a curve PP^ , then the derivative /"'(jr) or
T
/
y
y
A
/
M
J
0
a
: a
h . ■
Fig. 8.
Dy is represented by tan 0, where d is the angle riiade by the tangent
FT to the curve at P with the axis Ox.
Assign any arbitrary number x^, and let P^ represent /"(jcJ, and
T the corresponding point on the tangent to the curve at P, Then
we have
PM = x^ — X := Ax = dx,
d/{x) = {x^ - x)r(x),
= PAftM MPT,
^MT,
^7* therefore represents the differential of the function /(:c) at .r
corresponding to x^. While
MP, =/[x,) -/{x) = J/{^).
((/"zTid A/sie more nearly equal when Ax or dx is a small number.
£
Art. 49.] ON THE DIFFERENTIAL OF A FUNCTION 57
Observe that for a particular x the differential-quotient
^> ^f(x) = tan <?
is constant for all values of x^,
49* Relation of Differentials to Differences. — Since the differ-
ence-quotient has the derivative for its limit, we can put
^ =/'W + -■
where flr(=)o, when -Jjr(=)o. Therefore
A/{x) —f(pc)Ax + a^x,
^=,f'(x) dx -\- a ^x.
Hence, wheny*'(;c) 7^ o, we have
dAx) - •
AxC=)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
formulae 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
d/{u) = DJ{u) dx.
For, by definition,
df{u) = D,f{u) du,
^DJ{u).Djudx,
= DJ(u) dx,
since D,J{u) =/"'(«) D^, and du = D^udx.
.-. DJ'{u)du^DJ{u)dx,
or the first differential of a function is the same whatever be the
variable.
More generally, let «, r, and w be functions of x. Distinguish-
ing differentials like derivatives by subscripts, we have
d^u) = DJiu) dv = DJ{u) D^ dv,
= D^u) du = dj[u).
In like manner, d^J\u) = d^/[u). Therefore
dj[u) = d^/{u),
58 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. 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 jr, y^ are the coordinates of a point on a curve, show that
or {Y --y)dx^(X ^ x)dy
is the equation of the tanfi^ent at or, y^ where AT, Y are the current coordinates
on the tangent. This equation can also be written
Y -y _ X ^x
dy '~' dx '
2. Show, with the above notation, that
{jr ^y)dy -\-(X^ x)dx^o
is the equation of the normal at x, y.
3. Show that d{j^ log jr) = jr«(log :fi -\- \)dx,
4. <^cos mx cos nx) = ^ m cos nx sin mx dx ^ n cos mx sin nx dx,
6. -r- sin* jp = » sin*—' x cos x.
dx
6. «/ sin (i 4- jc*) = 2 X cos (i + jc*) dx.
7. If >' = sin** X sin mjr, show that
dy
sin' jc -f- =s iw sin*»+«jf sin(m 4- i)x,
dx \ I /-
8. D{a sin* jt 4- ^ cos* or)* = if(fl — ^) sin zx (a sin* jc 4- ^ cos* x)*-».
9. </ sin(sin x) = cos x cos(sin x) dx.
10. /(jr) = sin-i(jr*), show/'(jc) = «jr*-«(i - x**)-*.
11. </sin-»(l - X*)* = - (I -j^r^dx.
^^ d ^ 4- fl cos X Vtf*~.ir^
12. -r cos-" — i— ^^ = — ;— T
dx a -^ b cos X a-f-^cosjp
13. d sec* X = n sec* x tan x dx.
^. , . 2</jr
14. d sec-'(j:») = ^-- .
x^x* — I
16. 4a* + X*)* = jt(fl« 4- j^-^dx.
16. </(«*- jr*)-*a x(a* - x^)-\dx,
17. ^x(:.* + a*)-*= ^1—
^ ^(x* -f fl*)» *
18. D^2ax - jr*)* = (fl - xXaajP — x*)-».
19. If fix) = ft — J sin 2JC, then /'(jc) = sin* x.
20. Show that d{^ -\- ^sin 2x) = cos* x dx.
-, d /cos«jf \ . ,
21. 2Z I cos jr J = sin*
dy
22. If ^ = sin jf — J sin* JT, then -j- = cos* jr.
23. d log cos X = ~ tan x <i[;r.
24. 7^ log sin x = cot x.
25. y = tan x — x. ^ =r tan* x dx.
X.
6o PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IIL
26. 0(0 = cot / + /. then 0'(^) = — cot* /.
27. If s = log tan ^y, show that
|. = c.c,= /?log^!.
— COS^
+ COS^
28. I>m log tan (^jr + ifn) s= sec m.
29. -V- log ^ / — ' — : = sec a.
aa A/ 1 — sin a
30. ^sin-i(3^ - 4^») = 3(1 - e«)-*</e.
31 ^ tan- ^<^ + ^ =, j/^r - ^«
36. — I 7 V**' - 7* 4- «' sin-« -^ [ = 2 V«' - »?».
^/ \ a - /? 2 V(a - /)(/ - fi)
37. -J- C08-I ■ . 7„ = ^ , . .
tfir a -{- ^ cos « A -|- ^ cos z
38. ^*(i - ;r*) = ^*(i - 3^ - **)^.
q-. ^ (sin mvy* __ fww (sin my)*-' cos {nv ~ i»y)
^t/ (cos wz')'* "" (cos m/)"«-M
._ </ sin* 0 sin*«-» B , , ^ . . , ^
*<»• W^^=l = SSSTFF^'" "••» + '•""' ")•
41. </r* = x«(i -f log x)<£r.
42. /?/** = e'^x*(i + log jf).
"•i(r"(c('+'-^-
d e* — r-* A.
' dx e* -^e-» (^ + ^~*)* '
45. </ log (^ + e-x) = ^ " ^* ^.
46. />^(«+*)' sin X = ^(«+«)' [2(tf -f x) sin x + cos x\
47 ^ * = <*(!- ») - I
' dzf ^ \ (^ _ i)2« •
48.
rf/ Vi + |/i 4- /«/ / |/i + /» \i + |/* + ^/
Art. 51.] EXERCISES. 6 1
49. If ^0 = fl^"-''> ~ , show that
^(/) = i a<»"-'')""* log a.
XI I
50. d tan « » = — a « u-' sec* a» logadu,
61. ^/ [6 + log cos (ijr - 0)] = 2(1 + tan e)-« dB,
52. A^ sin-»^) = sin-» 0 + ^(i - ^)"*.
53. Atan 0 tan-» 6) = sec« B tan-« ^ + (i + e«)-i tan 9.
54. i?^-*"-'* cos ^x = — /*'-'*(2fl«jc cos ^jr 4- ^ sin ^jt).
I
-a
55. ^Jr* = Jr* (I — log jc) <6f.
55. <//= /" e*dx.
57. /?x*^= x^ Jp«[4r-« + log j: + (log jr)«].
5a ihf*- /^JT-' (1 + Jf log jt) dx,
59. /?(i — tan Jf) cos Jf = — cos jc — sin x.
60. D log (log /) = I /log /'.
61. If 0(0 = ^' sin ^/, show that
0'(/) = ^' («« + ^)* sin (3/ + G),
where tan 0 = bja.
62. If sin^ = JTsin {a '\- y\ prove that
dy __ sin' (a -j- ^)
dx "~ sin fl
63. If x{\ + j^)* + ^'(i + *)* = o» show that
D:,y = - (I + j:)-» or I.
64. If y=^f{t) and x = /*(/), show that
65. If xy = ^-^, show that
dy log x
^ - (1 + logx)«'
66. </ (sin xY = (sin jc)* (log sin jc + jc cot x) dx.
- </ . ». log tan /
CHAPTER IV. .
ON SUCCESSIVE DIFFERENTIATION
52. The Second Deriyative. — ^The derivative/'(j:) of a function
/[x) is itself a function of Jtr, which is, in general, also differentiable.
The derivative of the derivativey"'(;i;) of a function /(jc) we call
the jeco«</ derivative ol/[x)y and write it/'^^{x).
Thus
*
For example, i{/{x) = x*'y the first derivative /"'(jc) is nx^-\ and
in the same way We find the second derivative
/'\x) = n{n - i)jc*-2.
Again, H/Ijc) = sin x, then
/\x) = cos X and f'\x) = — sin x.
If we use the symbol Df{x^ to represent the operation of differen-
tiation performed on J\x)i then two successive differentiations of
f\x), which result in the second derivative, are represented hy Ij^J\x),
.-. D\DAx)-\^D^A^)^r{xy
BZAMPLSS.
1. D{a -\-bx^C3^)-=.b \- 2fx,
/>»(« -h ^Jr 4- c^) = D{b-\- ^cx\
2. D cos ax = — a sin ajr,
/?* cos fljc = — aZ? sin aj: = — «• cos ax.
3. Z? log <7jr = a/x\ LP' log at = — a/x^,
4. /? |/fl» - .y* = - jr(a« - jc«)-*,
53. Successive Differentiation.— The second derivative like the
first is, in general, a ditferentiable function. Its derivative is called
the third derivative of the function, and written
/.»(,v, . ^/"W -/"(»)
62
Art. 54.] ON SUCCESSIVE DIFFERENTIATION. 63
In general, if the operation of differentiation be repeated n times
on a function yT[j;), we call the result the «th derivative of the func-
tion. We write the «th derivative in either of the equivalent symbols
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
The index of either D or / \n D^^f^ 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 oi/{pc) is
d^fix) = d\dAx)\
= d[/\x)dx],
= d[/'(x)].dx. (i)
since dx is constant. But, by the definition of the differential,
^[/'(x)l = 0\/\x)-\ dx,
=/"(x) dx. (ii)
Substituting in (i), we have for the second differential
d^Ax) =/"{x){dx)\
or the second differential of a function is equal to the product
of the second derivative into the square of the differential of the
variable.
It is customary to write the square of the differential of the
variable in the conventional form dx^ instead of (^)^ whenever there
is no danger of confounding
dx^ = {dx)^
with d{x)'^, the differential of the square of x. We shall write then
d^/(x) z=:f'\x)dx^.
In like manner for the third differential oiJ\x)
d\d-^J\x)-] = d\/'\x) d^\
= d[/-{x)ydx^,
since dx is constant ; and since by definition
d[/"{x)] = n[/"{x)-\ dx.
=/"'{x) dx.
64 PRINQPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV.
we have for the third diiferential
and so on.
In general, the nth differential of a function is equal to the
product of the «th derivative of the function into the »th power of
the differential of the independent variable. In symbols
where it is always to be remembered that d^ means {dxY^ and </*,
/* indicate the number of operations and order of the derivative
respectively.
EXAMPLES.
1. We have ^ sin jt = cos x dx^ and
</* sin jc = ^(cos X dx) = <^cos x)'dx = — sin x «£x*.
2. d^{a -f b3^) = d^zbxydx = %bdx\
X*'
3. d* log X ^ d(-\'dx =z ^
5$. The Differential-Quotients. — The »th differential-quotient
of a function is the quotient of the nth differential of the function by
tine «th power of the differential of the independent variable.
In symbols we have, from § 54,
This symbol is also written, for convenience, in the forms
X\\ of which notations are equivalent to either of
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, i.e.,
the theory of functions, we are required to deal with the general or
»th derivative of a function in order to know all the properties of the
function.
The formation of the «th derivative of a given function presents
no theoretical difficulty, but owin&r 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
nth derivative of the function whose properties we wish to investigate.
There are a few functions whose «th derivatives can be obtained
in simple form, as will be shown below.
We are aided in forming the nih derivatives of functions by the
following:
(i). The nth derivative of the sum of a finite number of functions
is equal to the sum of their ;fth derivatives.
(2). The «th 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 nth 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 (p{x, y) = o, we can of
course solve for^, when possible, and differentiate as before. It is
generally, however, better to differentiate (f}(x, y) with respect to x
and then solve for Dy. If we wish JD^y, we can either diSerentiate
Dy with respect to x, or differentiate <p{x,y) = o twice with respect
to X and solve the equations for D^,
In illustration,
2jr* — 3>^ — axy = o.
. • . 6jt* — ay — (gy* -j- tix)Dy •= o,
I2JC — aJ)y — (i8y JDy + a)I>y — (gy* + ax)L^ =: a
ThcrefDre
_ I2^Qy« A-fixf- 2a(6x^ - ay) ^« -f- ajc) - i8y(6af* - ay'f
^ (9y» ^ axf
Again, we frequently require the derivatives Dxy and Z?i^, when we have
given the polar equation (p(p, 6) = o, where jp = p cos 6, ^ =r p sin ©.
We have
^ sin Q D^o 4- /> cos 0
~ cos 0 D^p — p sin6* ^^'
66 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV.
Also.
- (^^ '
= p* + g(A/o)' -p/?;p
(cos S Z?^p — p sin 6)»'
In which D^p and Z>}p must be determined from the polar equation 0(pi 9) = a*
EXAMPLES.
1. The ifth derivative cix^j a being constant
(I). Let-<i= i» be a positive integer. Then
for all values of » < m. If m = m, then
Z>»jr*» = »i(w — i) . . . 3-2.1 =r m I
This being a constant, all higher derivatives are o.
... />*«+^x« = o
for all positive integers p.
Also, when x = o,
jy^x^ IS o, n <m,
(2). Let the constant a be not a positive integer. Then, as before,
]>x^ := a{a — 1) . . . (df — w -f i)jf«-«.
Whatever be the assigned constant a, we can continue the process until m > a,
when the exponent of jr will be negative and continue negative for all higher deriv-
atives.
Consequently, when j: = o,
D»x* = o,. M < a.
Z>»jc* = 00 , n > a,
* The differentiation of an implicit function 0(x, ^) = o is, properly speaking,
the differentiation of a function of two variables, and a simpler treatment will be
given in Book II.
It will be shown in Book II that the derivative of y with respect to jt, when
^x, y) = o, is
80
dy __ dx
dx" "" 30 *
where ^ means the derivative of 0(j:, y) with respect to jt, x being the ^^w^J/ vari-
ox
able ; ^^ means the derivative of <p with respect to^', y being the only variable.
oy
For example, if 0(jr, >') = 2je* — 3^ — ojry = o,
60 ^ . 90 ,
then ^ = 6jr* - oy ; ^ = - gy* - a*.
Therefore, as in the text,
dy _6jfi — ay
dx "" gy* -|- ax '
Art. 56.] ON SUCCESSIVE DIFFERENTIATION. 67
2. Deduce the binomial formula for (i -f- ^y^t when the exponent n is a posi-
tive integer.
We hare
(I + jfXl + ') = (I + ^)" = I + 2;c + x^.
By an easy induction we see that (x -f~ x)* must be a polynomial in x of degree
If. It is our object to find the numerical coefficients of the various powers of x in
this function. Let
(I + jr)« = ^0 + ^\X + ^r^ + • • • + <»«•**.
Differentiating this r times with respect to x^ we have
if(i»— I) . . . {n — r+iXi+Jf)"~*'=^l fl^ + . . . + nin—i) . . . (n^r-\'i)an
This equation is true for all assigned values of x and r, and when jt = o,
^ rX
r !(» — r) ! '
a number whidi it is customary to represent conventionally by either of the
symbols
Cn^ r or
P)-
This number is of frequent occurrence in analysis. In Algebra, when n is an
integer, it represents the number of combinations of n things taken r at a time.
Hence we have the binomial formula of Newton,
(l4-x)« = ic« ,xr. (I)
r - o
Corollary. If we wish the corresponding expression for (a + >')«, then
(a^yy^a^U^yy
n
} m
YxiXy/a for x in (i), and multiply both sides by tf*.
r ■ o
This can be written more symmetrically thus:
{a -f yY _ ^ fl»-^ Jf^
8. The ifth derivative of log JF. We have
Z> log jr = — =a j^x.
X
Therefore, by Ex. I,
Z?» log X = (- !)*-»(» - I) 1 ^.
4. The frth derivative of a*. We have
Da* = fl* log a.
.'. />•«•* = tf*^ (log a)*.
In particular, Z>^ = t^\ D^e* =r r*. This remarkable function is not changed
by differentiation.
68 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV.
5. The nth derivative of sin x and cos x.
We observe that
D sin X =1 -\- cos x\ D cos x •=. — sin x\
L^ sin X = — sin x; Z?* cos a: = — cos x\
Z)3 sin JT = — cos x\ Zfi cos j: = -|- sin x\
/>* sin jp = -|- sin x; Z>* cos jr = -f- cos jc.
Thus four differentiations reproduce the original functions and therefore the higher
derivatives repeat in the same order, so that
£)xtir-i sin x = (— i)*^' cos x\ /)»«»-x cos JT = (— i)« sin x\
JD^ sin jc = ( — i)« sin x\ D» cos jr = (— i)« cos x.
In virtue of the relations
cos JT = sin (\K -f- x\ sin x = — cos(-^]r -|- Jf),
these formulae can be expressed in the compact forms
Z>«sin X
/)« cos X
= sinfjr4-^3rj,
= cos f X -j — jrY
6. Given ff-|.:L=i, find D),y, Dly,
Differentiating with respect to x,
X y dy
Differentiating again,
dx^
^ , dy ^x
since -^ = = — ,
Diiierentiating again, we can find
d*y _ 3^x
^ "" « V '
7. If y* = 4flx, show that
<;^ 2tf ifly 4fl*
^ ^ 7 * ^ ~ ~ lr»"'
8. If ^' — 2x^ = fl', show that
dy y d*y a* d^y 3a»x
dx y - x' dx^ (y'^x)*' dx*" (^ — x)»*
9. From the relation x* -f- ^ — ^axy = o, show that
dy ^ x^ — av d^y
2e^xy
dx y* -- ax* djfi " (y^ - axf
10. If sec X cos^' = tf, show that
dy ^tanx d^ _ tan'j^ ^ tan'x
d!r tan^* dx* "~ tan'^
Art. 57.] ON SUCCESSIVE DIFFERENTIATION. 69
57* Leibnltz'8 Formula for the nth DeriyatiTe of the Product
of Two Functions. — Let u, v he any two functions of at. For sake
of brevity, let us represent the successive derivatives of u and v by
these letters with indices, thus :
r', v'\ 1/", . . . , l^, . . .
Then
D(uv) = u'v + r'«,
= «"r + 2«V + Iff;''.
In like manner, differentiating again this sum of products, we find
on simplification
Observing, when we use indices to indicate the derivatives, the
symbols JJ^u, /^{x), c^, mean that no differentiation has been
performed and the function itself is unchanged,
. • . iyu=tfi = u, and /»(jtr) =/{jc).
In the above successive derivatives of uv we observe that the
indices representing differentiation follow the law of the powers of
ti + V when 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 nth derivative and then differentiate again to see if it be true,
in consequence of that assumption, for n -|- i.
Assume that (see Ex. 2, § 56)
Differentiating this, we have
r»o
in virtue of the relation* C^,^ + C,f-i = C-n.r.
Therefore, when the law is true for any integer «, it is also true
for « + I. But, being true for » = 2, 3, it is true for any assigned
integer whatever.
r!(ii-r)l "^ (r-i)!(if-r+i)! (r-i)!(if-.r)! \r "^ if-r-f i^ "" r!(ii-|-i-r;
)l
70 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV.
We can express the nth derivative of the product uv symbolically,
thus:
D^{uv) = (« + »)«,
in which (u -f- v^ is to be expanded by the binomial formula, and
the powers of u and v in the expansion are taken to indicate the orders
of differentiation of these functions. Remembering that when the
index is a power we have t/^ = i, but when it means differentiation,
EXAMPLES.
1. To differentiate the product of a linear function by any function /(jc).
Let « = (fljc -f 6)f[x),
Then I>(ax -f ^) = a, jy*(ax + *) = o,
.-. I>u = (at -f d)/^x) -\- naf-^(x).
2. In like manner show that the »th derivative of the product of a quadratic
function of jt, say ^, by any other function/, is
3. Show, if *p(x) and ifjix) are differentiable functions of x,
l>[0ix)i^x)\ _ I, iP^jx) jf^x)
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'^d, and /(«) be any differentiable function of u. Then
= <(«).
= af':(u) Du,
and generally
1. Show that
2. I>e»* = a^e^.
EXAMPLES.
/>* sin ax "z^ aF sin (ax -j — ie\
D^ cos ax z= a* cos lax -| ft]
3. Show that Da (---^\ = , ^^-r-
* Appendix, Note 4.
AnT. 58.] EXERCISES. 7 1
Show that
XXSRCI8ES.
1. />'(^) = (- i)^
n(n 4- I) . . . (« -f r — i>
xn+r
3. />f— ^^l=:r!
2. Br^l) = (- lya
^^ — jry (^ — jc)»'+«
4. Z>-log(i+x) = (- ,)-.J^^^.
5. />(jf* log jr) = 6jr-«.
6. I^x^ -f- a sin 2x) =s 32a cos 2jr.
7. i?i(*») = JT -^« + r />■-»«, where m is any fdnction of x.
8. D^{a — jf)«i = (a — jt) I>u — rDr-iu.
9. /?^Jf* log jt) = — 4 ! j:-».
ia i>«(x log Jf) = (- !)«(» — 2)1 jr-«+^
11. D^x» := jp«(i 4- log Jf)* + jc*-«.
12. />■ log (sin jr) = 2 cos x esc* jc.
13. Z^(:r* log .vl) = 2*.T-».
14* D^af» = «« (log fl*)".
15 z>«^-±^-r- i)«-^— ?i±l-+ _fL=ii_l
Observe that by the method of partial fractions we can write
ax -\- b ^ I {aC'\'bac — b'^^
{X - ^x* + 0 " Tcxirri + irf7r
16. D- ^-^^ = />.^i_ f?Mii^ ?i±i\ ^ .
(x-ptx^q) p-q\x-p x-^qj
17. Make use of the method of partial fractions, to find the Mth derivative of
gj^ ^ bx -\- e _ I /at/« ^ pb J^ c aq* J^ bq '\- c\
{x-^p){x^q)'=J^\ ^37 x-^q . )+'^
18. Show that [-y] -5 , , ^ = — 6 ^ r— -.
19. If >^ = a(i 4- jc«)-», show that
(1 4- jr»)^(«) 4- 2#m:>'(«-0 4- «(« — i);^(«--a) = a
20. If /(x) = tf cos (log x) -\- b sin (log x), show that
x^/"(x) 4- jr/'(jr) 4.y(j:) = o.
21. Show in 20 that the following equation is true :
jy»+« 4- x{2n 4- i)/«+i 4- («« 4- i)/« = a
22. If y = /• ■*»""' J^, show that
(I — j^)d*y — xifydx^ a*y <&•,
thence find, as in 21, an equation in>«+a, >'*»+«, ^n.
72 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV.
23. If y = {x+ Vx* - i)* show that
(jr» — i)d*y -\- xdy dx — m^ <&• = a
24. If ^^ = sin(sin x)^ show that
d^y -|- tan x dy dx -^y cos'jr dx^ = o.
25. If y = A cos nx -^ B sin nx^ then
I^y -f n*y = o.
M
28. Z>»^** cos *JP = (tf* + ^*)^ tf«* co» {dx -\- n0), where tan 0 = d/a. Differ,
entiate once, twice, and observe that the law follows directly by induction.
27. Z>»tan-»jr-i == (_ i)«(,f «. i)! sin" (tan-«jr-«) sin «(tan-«jr-x).
Put y = tan-»jr->. Then x = coiyy and
Z),^ = - (I + x'Y'^ = - sinV-
Dj^y = — Z>j, sin'^ = ^ Dy sin* y Dy = sin*^ sin 2y
The rest follows by an easy induction.
28. If X •=. (p{t) and y = ^/), then>^ is a function of j:
Required Dxy^ ^ly*
We have
£^ = V^(/)*//, dx = 0'(/)d5f.
• ^ ^(0
A1<»A
■• ^^-^(/r
Also,
</«y _ d i>/ _ d if/ dt
dx*~ dx (pi dt 0t' ax '
i>i'4>i - i>i4>r . . dt _i
{4>if ' ' dx 0r
d*y dx dy d^x
"dfilt ^Jt~dF
(
i)'
I. If JT = sin 3/, ^ = cos 3/, show that
Dly^-y-t.
30. />. tan-x = (- i)«-.(« - 1)1 si" (^ tan-»^i)
(I + ^)*»
This follows immediately from Ex. 27, since
tan-'jr = \it — tan-'x-".
31. If y — tan-»jc, show that
(I + jt*)^(«+i) + 2«jry(«) + n{n - lljf(—i) = o.
32. />(a« -h j:*)-^ = (— i)»« ! ^-^a sin«+i0 sin (n -f- 1)0,
where tan 0 = j/jc. Hint. Use Ex. 30, and
D tkn-^{a/x) = - a(a* + x*)-t,
33. -O^A^fl* + jr*)-» = (— lYa-^-tn \ sin«+«0 cos (n -f i)0
where tan 0 = a/x. Use Ex. 32 and Leibnitz's Formula.
4- JT ^ ' (I -f Jf)»+1
Art. 58.] EXERCISES. 73
_ 2i<r — I .mtz
35. D^e'^^z 7=-^ '
36. If y(i + jt*) = (I - a: + jr«)», then
<6:* (i-f x«)*
37. If ^ = sin {m sin-»jc), prove
38. If ^ = sin-'jc, deduce
(I - *»)/' - xy' = o,
and • (I - JT^) " /■ - (2it + i)x ^ / - ii«3^ = o,
by applying Leibnitz's Formula to the above. The deduction of such difierential
equations is of fundamental importance for the expansion of functions in series.
39. Show that
Apply Leibnitz's Formula to the product/(jr)-(a ~ jp)-«.
40. Show that
where, in the differentiation indicated by ^ , jr is constant and y the variable.
The result follows at once when Leibnitz's Formula is applied to the product of the
two functions /(jr) —/(>') and {x — ^)~'.
This is one of the most important formulae 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. 40^
with respect to ^ (jt being considered constant during the operation), is
Hint. Di£ferentiating 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 increases as its variable increases, A function is said to
be a decreasing function when it decreases as its variable increases.
In symbols, /{x) is an increasing function at a: = a when
/[x) ^A^) (I)
changes from negative to posi/ive (less to greater) as x increases
through the neighborhood, (a — e, a + e), of a. In like manner
/[x) is a decreasing function at a when the difference (i) changes
from positioe to negative (greater to less) as x increases through the
neighborhood of a.
60. Theorem. — A function /{x) is an increasing or decreasing
function at a according as its derivative /\a) is positive or negative
respectively.
Proof: If f{pc) is an increasing function at a^ the difference-
quotient
f{x) ^/[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, '\i/(x) is decreasing at a, the difference-quotient
is negative for all values of x in the neighborhood of a and therefore
its limit cannot be positive. Hence, \i/\a) is a negative number,
the difference-quotient must be negative for x in the neighborhood
of a, and therefore /(^) is decreasing at a.
Geometrical Illustration.
\jtty =/(x) be represented by the curve A^Ay The function is increasing at
A^ and decreasing at Ay
We have
/'(a,) = tan e^ = +,
for Oi is acute, while
/'(«s) = tan 0, = -,
74
Art. 6 1.]
ON THE THEOREM OF MEAN VALUE.
75
since d, is obtuse. Remembering that, under the convention of Cartesian coordi-
nates, the angle which a tangent to a curve makes with the jr*axis is the angle
between that part of the tangent above Ox and the positive direction of Ox,
6i. Rolle'8 Theorem. — If a function /(x) is one- valued and
differentiable in (or, )5), and we have /{a) =zj\/!f), then there is a
value S oi X in {a, /5) at which we have
/'(S) = o,
provided y '(at) .is continuous in {a, /S).
li /(x) is constant in any subinterval of (a, /5), its derivative
there is o and the theorem is proved.
It/lx) is not constant in (or, /S), then at some value x^ in {a, fi)
we shall have/(^') ^A.^). ^i A^') > A") —A^)* ^^^ function
must increase between a and x^ and decrease between x^ and l3, in
order to pass from /(a) to the greater valuey(ji:'), and iromA^^) ^^
the lesser value y(4). Also, HAx") <A^*) =/(/^)i then the func-
tion must decrease in (a, J?) and increase in (x^, /3), for like
reasons. In either case the derivative /*'(^i) at some point x^ in
(flf, x^) must have contrary sign, § 60, to the derivative A{x^ at
some value x^ in (or', /?).
Since /\x^ and f'(x^ have opposite signs, and f'(pc) is, by
hypothesis, continuous in (aTj, a:,), then there is, § 23, I, a number
5 in (jTj , j:,), and therefore in (or, P)^ at which we have
/'(5) = o.
In particular, if/(«) = o and/()5) = o, then there is a number
5 between a and fi at which
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. — If A^) ^^ ^ o"^'
valued differentiable function having a continuous derivative in
(or, P)y and if a and h are any two values of x in (a, /?), then
Ab) -Aa) = {6- a)/'(S),
where 5 is some number in {a, 6),
76
PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V.
The truth of this theorem follows immediately from Rolle's
Theorem.
Let k represent the difference-quotient
Then
or
i-a
/(d) -A") = (* - «»)*.
/{5) - 66 =/{a) - ka. (i)
The function f{pc) — kx \% equal to the number on the left of
(i) when AT = ^, and to the number on the right when or = <z.
Therefore, by Rolle's Theorem, having equal values when x ziza and
when ^ = 3, its derivative must vanish f or Ji? = 5 between a and b.
Differentiating, f(pc) — kXy k being independent of Xy we have at S
which proves the theorem.
Another way of establishing the result is to observe that the
function
(a - b)/{x) -(X- 6)Aa) + {x - a)/{6)
vanishes when x = a, also when x =z d. Therefore its derivative
must vanish for some value of x, say S, between a and 3.
.-. (a-i)/'{S)-A''}+Ai) = o.
Geometrical Illustration.
Each of these processes admits of geometrical illustration,
(i). k is the trigonometrical tangent of the angle which the secant AB makes
with Ox, Draw OA'B' parallel to AB. Then
BB' -f(b) ^ kb- AA' ^f(a) - ka,
XX' —/(x) — kx is equal to A A* when x = a, and to BB* whenx = b. The
theorem asserts that there is a point E on the curve ^ '=A^) having abscissa § at
which/'(^) = k^ or the tangent at E is parallel to the chord AB,
(2). The function
(a - b)Ax) - (4r - b)f(a) ^ {x ^ a)/[b)
is nothing more than the determinant
Af>). h, I
Art. 63.] ON THE THEOREM OF MEAN VALUE. 77
which is the welUknown formula in Analytical Geometry for twice the area of the
triangle AXB^ in terms of the coordinates of its comers. This vanishes when X
coincides with A or B. It attains a maximum when the distance of X from the
base AB is greatest, or when X is at E^ where the tangent is parallel to the chord.
This theorem amounts to nothing more than RoUe'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,
\jcXj\x\ be a one-valued successively differentiable function of a: in
a given interval (a, /J). Let x represent any arbitrary value of the
variable, and y some particular value of the variable at which the
derivatives of/" are known.
(i). Consider the difference-quotient
X ^ y
If we hold X constant while we differentiate this n times with
respect to the variable y by Leibnitz's Formula, § 57, and then
multiply both sides by
(x — J')»+*
we shall obtain
A=c) -Ay) - (^ -jyV'iy) - ... - ^^^/^W
^ {X -jy)'+' / d Y tAx) -Ay) \
n\ \dyj \ X —y )'
For, we have
D;--(x -y)-^ = (« - r)\(x -^)-^-+^
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
X — j^
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^^. Thus
-Z'iy) = (^ -y)Q; - Q' («)
-/"(>') = {X -y)^; - 2Q'„ (3)
Multiply (2) by (x-y), (3) by ^ (x - y)*, . . . , and (« + i)
by —t{x —yy, and add the « + i equations. There results
A^)-AJ')-i^-j')/'(y)- • -^^>0')=^-^=^'ej-', (q)
the same formula as in (i).
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 § 6^, and consists in the evaluation of the «th derivative, Q^\ of
the difference-quotient 0 in a different form.
Consider the arbitrarily laid down function of z,
in which, as in § 63,
does not contain s and is constant with respect to s.
Observe that this function F{z) is o when z = x, because the first
two terms cancel and all the others vanish when z = x. Also, F{z)
is o when z =y, by reason of the identity (g).
Consequently, by Rolle's Theorem, § 61, the derivative I^\z)
must be o for some value S oi 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
F^{z) = - ilZl!):^«(,) + («+!) ^r^^Gi--'.
Hence, when » = 5, at which F\S) = o,
Art. 65.] ON THE THEOREM OF MEAN VALUE. 79
Substituting this value in (^), we have Lagrange's form of the
Theorem of Mean value,*
r-o
65. Theorem of Mean Value. Cauchy's Form. — Cauchy has
given another form to the evaluation of the difference
A')-J^
which for some purposes is more useful than that of Lagrange. Its
deduction is somewhat simpler.
Let X be constant and z a variable. Consider the function
^W =/l«) + (^ - «)/'(«) + . . . + ^^^>W. (i)
By the Theorem of Mean Value, § 62,
Fix) - F{a) = {x- a)F'{S), (ii)
where S is some number between x and a.
When s ^ X, we have from (i)
F{x) =/[x).
When z = a, then from (i)
F{a) =/(«) +{x- a)/'{a) + ... + ^^rL^/-(a).
Differentiating (i),
and
^•'w = ^^ «. '^ Vh«).
F'(S) = ^^ ^, ^^ V'(g).
Substituting in (ii), we have Cauchy' s form
(C)
r=o
* In order that this result shall be true, it is necessary that the function /[x)
and its first n-{- i derivatives shall be finite and determinate at x and at^, and
also for a// values of the variable between x and y. 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 00 at a given value of the
variable, then all its derivatives are 00 there, and also the quotient of the deriva-
tive by the function is 00 , see Appendix, Note 5.
8o PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V,
The numbers represented by £ in (C) and in (L) are not equal
numbers. AH 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 » = 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 n 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 n
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.
(i). It is customary to write R^ as a symbol of the difference
between the functions
/(*) and J^^±-^fr(a),
Art. 68.] ON THE THEOREM OF MEAN VALUE. 8 1
SO that
n
o
Or, more briefly,
/W = 5« + ^«,
where S^ represents the 2 function.
(2). In particular, if a = o, and /[x) is differentiable, « + i
times at o and in (o, x), we have
/{X) =/(0) + V'(0) + . . . + ^/"(O) + Jin ,
where, using Lagrange's form.
j^n+l
^« = (;r+7)T^"H5), ^ in (o, X),
or, using Cauchy's form,
i?. = x^-£^/.+i(/r), 5 in (o, X).
(3). If we write the difference x —y = A, so that x ^y + A,
/O + K) =/(>) + A/V) + . . . + ^/^w + ^..
(4). Again, since h is arbitrary we can put h = dy. Then
or
2! ^- • • • "^ «!
^/=^+-rf + ... + -f+^^
EZSSdSSS.
1, 1£/[x) = o when x = n, , . . . , jr = An > where
tf 1 < fl, < . . . < tf „ ,
and/(jr) and its first n derivatives are continuous in (a^, a«), show that
y(x) = (4f - tfj) . . . (Jf - <!«)-
#1!
where ^ is some number between the greatest and the least of the nambert
'» ^1 » • • • « ^«i»
2. In particular, if a| = a, = . . . = a« =: a, then
y(') = ^^^-ar^/'i^)'
where 4 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,
the derivatives /^(fl), r = i, 2, . . . , at a, are such that the series
^■-r'^^/^w.
r»o
has a finite limit when » = 00 , and we also have
flaOD
then for the values of x and a involved we have
y{x) =y(a) + (x- a)f\a) + <^lrLf)!/"(a) + . . . (T)
This is called Taylor* 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 R^ we choose in show-
ing £R^ = o.
71. Maclaurin's Series, — Under the same conditions as in § 70,
if a = o, •
Ax) =/(o) + x/'{o) + ^/"(o) + . . . (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,
EZAHPLES.
1. Any rational integral function or polynomial y^;r) can always be expressed as
f[a) + (X - a)f\a) + . . . + ^^^^"/"(a),
where n is the degree of the polynomial /(jr).
For, since /is of the nth degree, all derivatives of order higher than /^ are o.
Consequently the theorem of mean value gives
whatever values be assigned to x and a.
In particular, we may put a = o, and have
/[x) =y(o) + xfio) -h . . . +^/-(o),
TV.
and this must be the polynomial considered when arranged according to the
powers of x,
2. We may define as a transcemUnial integral function one such that all oi its
derivatives remain determinate and non-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
{x — a)*^^ ^
since /»+'(^) is finite for any $ between x and a, for all values of n. Also,
(x — af+^/(n 4- 1)! has the limit o when « = 00 (see § 15, Ex. 9).
Moreover, the series is absolutely convergent (Introd., § 15, Ex. 10), since
/
00
where ^ is a finite absolute number not less than the absolute value of any deriva-
tive of/ at a. The series on the right is absolutely convergent, since
HbOO
see g 15, Ex. 10.
* This formula is really due to Stirling.
84 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI.
Therefore, if /(x) be any transcendental integral function, we have for any
assigned value of j: or ^
Also,
A') =A') + (' - «)/'('') + ^^-^/"(fi) + . t •
jt»
A') =Ao) + '/"(o) + 7r^"(«>) + • • •
Such functions are sin x, cos jr, e*.
3. Show that ii/(x) is any transcendental integral function as defined in Ex. 29
then f(px 4- q) can be expanded in Taylor's series for any assigned values of/, ^,
X and a.
This follows immediately from 2, since
{^XfU>^ + (?) = /«/"(/■» + ^)-
4. To expand ^' by Maclaurin's formula.
We have D^e* = <« for all values of r. At o we have
D^e* = ^ = I. Also,
/
e^ = o.
(i»4-i)!
Hence, substituting in Maclaurin's series, we have
^='+-' + 2i + 3l+---
rl
r-o
In particular, when x = i,
which gives a simple and easy method of computing e to any degree of approxima-
tion we choose.
5. To compute sin ;r, given x, by Maclaurin's formula.
sin o =i o, /?»-« sin o = (— i)«-», and J>* sin 0 = 0,
by Ex. 5. § 56. Therefore
jr* X* jtT
6. To compute in the same way cos x, given x.
By Ex. 5, § 56, cos o = i, Z?»«-i cos o r= o, Z)»« cos o = (— i)«.
X* jr* X*
... cosx=i-^ + -,-^ + ...
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 Rn*
The limit of ^«, however, is very readily seen to beo, since we have respectively
^n = . " . \y. sin U + jar j, for sinx,
, — ; — -. cos ( 5 H- —je ], tor cos x.
l)!~*(^ + ¥4 ^
Art. 71.] ON THE EXPANSION OF FUNCTIONS. 85
7. The binomial formula for any real exponent.
Consider the expansion of (i 4~ ^Y ^7 Maclaurin's series, when a is any
assigned real number.
We have
Z>*(i + x)« = tf(tf - I) . . . (a - » -f i)(i + xY'*.
.-. [Z>»(i -f jr)-]^„ 0 = «(fl - I) . .. . (« - » + I).
Substituting in Maclaurin's series, we have
The quotient of convergency, § 15, Ex. 9, of this series is
/
\n -\- I
-X
= m- (I)
fliaao
Therefore the series is absolutely convergent when Ix| < i, or for all values of x
in ) — I, + i(. For U| > i, the series is 00 .
Also, by (C), § 65, or § 68, (2),
•« - •'"^Ti (I + €)«+'-» • ^ •
Whatever be the value of \ between x and o, so long as kl < 1 we have
a — n X — \
«<li- (3)
/:
«+ I I+$
flIaOO
For this limit is the same as
/
which is less than i when o<j:<i. Ifjr<o, put x = — jp* and € = ^ $'.
Then the limit is equal to
/
i-r
But :r' — $' < I — $', since o < x' < i and O < $' < x'.
Inequality {3) being true, £Rn = o, in (2). Therefore the series is equal to the
function for the same values of x for which the series is absolutely convergent.
.-. (I + x)^ = I + gx 4- ^ ^^ '^ + ^y V4-...
for all values of jr in )— i, -|- <( ' ^^'^ ^^ equality does not exist for any value of
X for which |jr| > I.
8. Expand log (i -f- •^) ^7 Maclaurin's series.
Let f[x) = log (I -I- x\
and /"(o) = (- !)*+«(» -1)1.
Substituting in Maclaurin's series, we get
jr - Jjr* + ijf» - . . .
The convergency quotient of this is
/I-
n
-X
n-\- I
Waco
= w.
86 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VL
The series is therefore absolutely convergent for |jr| < I, and is oo for ]j:| > I.
Al$o,. w« have, by (C), § 65,
Whatever may be ^ between x and o when |jr| < i, we have, as in Ex. 7,
£\
i + €
< I.
MbOO
Therefore £Rn ■=z o^ and
log (I + ^) = j: - iJT* + iJf* - i^f* + . . . (I)
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 (i -|- ;r}.
By changing the sign of x,
log (I - X) = - j: - fr« - ijc" - . . . (a)
By subtracting (2) from (i),
log \^ = 2(* + J^ + Ij^ 4- . . .) (3)
If n and m are any positive numbers, put
— then X =
I — X n 2n ■\- m
Substituting in (3),
(n + m\ _ / m I w* 1 m^ \
"^ \r^i~) " \^irT^ "*■ J(2« + mf + 5(2« + «»)» "+■ • • 7
a series which converges rapidly when n> tn, and gives the logarithm of m -f- ^
when log n is known.
The logarithms thus computed are of course calculated to the base e. To find
the logari&m to any other base, we have
log, fl
72. Observatioiis 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 S» The only information we have with regard
to S is that it is some number which lies between two given num-
bers. Moreover, we know that ^ is a function of n and in general
changes its value with «, . It is therefore necessary that we should
show that jQRn = o ^^^ ^^^ values of ^ between x and a, in order to
be sure that j£Rn 's o for the particular value £ involved in the law
of the mean whatever may be that number S between x and a. In
the deduction of the form R^ 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 S in the interval {a^ x)^ but it is also necessary.
The equality of the function and the series depends on R^ vanishing
for ail values of £ in (<z, x).*
It is desirable therefore, that we should have such general laws
with regard to the expansion of functions as will enable us, as kr as
it is possible, to avoid the formation of the nih derivative and the
investigation of the remainder term R^^ 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 /(at) and its derivative /'(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 /"(jf) and /"'(at) gives
/{x) ^/(d) + (^ - «)/■»+. . . H-^^^Z-W + ^«. (I)
/\x)=/\a) ^(x- a)f'\a) + . . . + ^^^l^f{a)+R;, (2)
where
R, = {X - <,)i^^/-+>(5), (3)
I. We observe that the quotients of convergency of (i) and (2),
as obtained by taking the limit of the quotient of the (» -^ i)th
term to the nth term, have the same value, for
«>aO NaOO
flaOO
* In the theorem of the mean, (I), § 70, the series
0 wi
may be absolutely convergent and yet not equal to the function f(x). For Prings-
heim's example, see Appendix, Note 8.
88 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI.
Therefore, if
= ^ (5)
/I
Aa) + (^ - «)/'(«) + ^T^/"(«) + • . .
is a finite determinate number, then the two series
2!
and
/'(«) + (* - <»)/"(«) + ^^^/'"w + . . .
are absolutely convergent in the common interval )<2 — i?, a + -^( >
and are both 00 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 R is called the radius
of convergence.
IL We observe that if, for all values of 5 between x and tf, we
have
'-*/"'«'<|,, (6)
/
ftaOO
« /"(5)
in which 5 has the same value wherever it occurs, then, § 15, Ex. 9,
must (3) and (4) be o when « = 00 whatever be the value of ^
between x and a in (3) or (4).
Consequently, if we have determined (5) for any function and
shown that (6) is true for values of jc 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
EXAMPLES.
1. Having proved that the requirements in g 73 arc satisfied for (i +x)*, and
this function is equal to its Maclaurin's series for all values of :i: between — i and
~\- I, and for no values of jr outside these limits, it follows immediately, in virtue
of § 73, that log (I -f- •^) is equal to its Maclaurin's series in the same interval,
since
Z> log (I + :r) = (I + x)-i.
2. The function tan-'jc is equal to its Maclaurin's series for j:* < i. For
Z)tan-»jir = -.
I -f x»*
and jc* < I is the interval of equivalence of (i -}- •*"*»"' with its Maclaurin's series.
Moreover, since
(I + jc*)-' = i-jc«-f:c*-jr<-f.,.,
and the primitive of (i -f j^)—^ is tan-'x, and -tan-K) = o, we have, by § 73,
tan-'* = jc — f«« + Jx* - |jc» + . . . ,
lor - I < jr < -f I.
Akt. 74.] ON THE EXPANSION OF FUNCTIONS. 89
We can verify this result directly, for
I> tan-»jf = (— 1)*-" ^ " .^ sin (if tan-'x-i).
^ ' (1+^*)*
.'. [Z>« tan-»jr]p = (- !)»-'(» — i)l sin(^«;r).
Also, sin (2m - j = o, sin {2m + i) - = (~ i)».
Therefore the Maclaurin's series for tan-'jc is
which has the interval of absolute convergence )— i, + i(.
For Hn » in Lagrange's form, we have
„ x» sin (» tan-»Jf-0
n (I + §*)*-
the limit of which, for » = 00 , is o when \x\ < i.
In particular, if jc — tan J* = 1/ i^ then
r= — TT' r^ — •755' • • • »
2 V^ 3 3 5 3' 73'
which can be used to compute the number n. A better method, however, is given
below
3. For all values of |x| < i we have shown that
(I - x^ r= I + Ijc* + i^ ;r* + 5-1^ jr« + . . .
\ / -r T- I 2.4 ' 2.4.6 '
But a primitive of (i — jfl)-^ is sin— 'jc, and since sin— »o = o, we have, by
§73,
sin-»;c =:jr+iljc»4.LL3i^^ . . .
for X in )— I, 4- !(•
In particular, since Jjt = sin— « -J, we have
6" 2 "^ 2.3 2*"^ 2.4-5 2*"^ • • ' »
from which ff' can be computed rapidly.
4. Determine the Maclaurin's series for cos-'x, cot-'jt, sec-'jr, esc— »x. In
each case determine the interval for which the function is equal to the series.
74, We can find the «th derivative of sin~*;r without difficulty,
but it would be difficult to evaluate the corresponding limit of i?„ by
the direct processes of Maclaurin's formula.
Observe that the coefficients in the power series for sin~'^ can be
determined from Ex. 38, § 58, where we have
(i — a:^)Z>"+^ sm-'x — (2« + i)aZ^+' sin-'^t — n^D^ s\n-'x = o.
. 2)1^2 sin-'o = «'2?"sin""'o.
When we have found D sin""*o, D^ 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 jQRn is a matter of considerable difficulty.
90 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. 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 n evaluated.
We add a few exercises in the application of the formula. The
cases in which the remainder term R^ 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 M
be the greatest and m the least absolute value of the {n -j- i)th deriv-
ative in the interval {a^ x)y the error committed in taking
A^) =2'i^>(«)
lies in absolute value between
m and ^-7 -^-rj-M,
{n+i)\ {n+i)l
by Lagrange's form of i?«. When we know the »th 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.
1. If r is the chord of a circular arc a, and 6 the chord of half the arc, show
that the error in taking
d
is less than -rs-. where a < radius of the arc.
7680'
2. VLd\% the distance between the middle points of the chord c and arc a^ in
Ex. I, show that the error in taking
8 d^
c = a
3 «
. t ^^ 32 </*
is less than ^ — =-.
3 ^
3. The series i4--^ + -^+ • • • is convergent for|x| < x. It is infinite
when X ^ I, and also 00 when j: < — i. Show that we can make x converge
to — I in such a way as to make the sum of the series equal to any assigned
number we choose.
Let X = — : I, where a is any assigned number. Then we have for
the sum of if -|- I terms of the series
Art. 74.] ON THE EXPANSION OF FUNCTIONS. 9I
I 4- ( — i)«| I ; — 1
1 — X
2 —
If If = 2i» or 2m, -f i» And m = 00 , this sum is respectively equal to
J(i 4- ^-) or 4(1 - ^),
one or the other of which can be made equal to any given number by properly
assigning a.
Show that
4. tan jc = jf -f Jj:* -f ^jfi + R^.
5. sec X = I + ijr» + Aor* + ^i^j^ + i?,.
6. log (I -f- sin X) = X - Jjc* + ^j:» — ^^jr* + ^,.
7. ^ sec Jf = I -f ^ + ^ + |^+ iJf* + A-^ + >P«.
8. Show that for |jc-| < I we have
£ jf» , 1-3 X*
2 3 '
Hint. /> log (or + f^l + x») = (I + •**)"*•
log (x 4- i^iT^ )=^^i+^?
2 3 2.4 5
2jr
9. Expand sin—* -3 and tan— » — , in powers of x, determin-
ing the intervals of equivalence, §§ 72, 75.
10. Expand xJ^x^ -|- a* -|- a* log (jc + -f^Jr* 4- «'), in powers of jp and deter-
mine the interval of equivalence.
Hint. The derivative is 2 -f/fl^-f-jf*.
11. Expand in like manner
4 4/2^ I — X yT-f- X* 2 f/J* I — *•
by using its derivative (i + •«*)"*•
12. Show that the «th derivative of (jr« + fix -f 8)~« at o is
Expand the function in integral powers of x and determine the interval of
equivalence.
13. Show by Maclaurin's formula that
Hint If ^ = (1 + jr)«, then \ogy = .^i£±f).
... y = ^(*), ^x) = I - ir + ix» - ij^ 4. . . . ,
and the first few derivatives can be found.
14. Compute the following numbers to six deamal places: e, itj log 2, log« 10,
sin 10°.
CHAPTER VII.
ON UNDETERMINED FORMS.
75. When u and v are functions of x, they are also functions of
each other. If, when a:(=)a, we have «(=)o and »(=)o, the quotient
u
V
will in general have a determinate limit when x{=)a. This limit
will depend on the law of connectivity between u and v. 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
A^) -A")
X — a
when /(x) — y(<j)( = )o and x — a(=)o.
The evaluation of the quotient u/v when x converges to the
common root a of u and v, is but a generalization of tKe idea
involved in the evaluation of the derivative. For, let <p{x) and tp{x)
be two functions which vanish when x = a, or, as we say, have a
common root a. Then
<p(a) = o and tp{a) = o.
We wish to evaluate the limit of the quotient
0(-y)
f/;{x)
when x{ = )a.
Since (p{a) = o, ^'(a) = o, we have
<P{^) _ 0(-^) - 0(^)
tp{x) ff:{x) - t/:{ay
X — a
tp{x) - il:{ay
X — a
Consequently if (p{x) and tl:(x) are differentiable functions at a,
Art. 76.] ON UNDETERMINED FORMS. 93
and the member on the left has a determinate limit when x{=)a, we
have
£
For example.
jr(-)fl
flogx
= I.
X — 1
It may happen that a is a common root of tp'Cx) and ^\x), then
0'(fl) = o and t/^^{a) = o. In this case we shall require a further
investigation in order to evaluate the quotient <f>/i/). For this pur-
pose we require the following theorems:
76. A Theorem due to Cauchy. — Let <f>{x) and t/){x) be two
functions which vanish at a, as also do their firet n derivatives, but
the {n -|- i)th derivatives of both <f>{x) and t/){x) do not vanish at a.
Then we shall have
where S is some number between x and a.
Let ;? be a variable in the interval determined by the two fixed
numbers x and a. Then the function
=: o when 2; = a, also when z=: x.
By the law of the mean, § 62, /\z) = o for some number z = £^
between x and a. But, in virtue of the fact that <f>^{a) = t/>'{a) = o,
we have /^{a) = o. Consequently /^\z) = o for some number S^
between S^ and a.
In like manner J"\z) = 0 for » = 5, between 5^, and a, and so
on until finally we have
y-+'(^) = 0-+'(^) ^{x) - r^\a) 4>{x) = o,
where £ is some number between x and a.
If ^*+^(«) is not o between x and a, we can divide by it. Hence
This theorem is of great generality and usefulness.
For example, the functions {x — a)«+«/(«-|- 1)! and
" ^jr — aV
J\x)mA')- i^— ;#-/'(«)
r" o
are such that they and their first n derivatives vanish at x = a, while the (n-\- i ,th
derivative of the first function is I. Therefore, by the theorem just proved, we
have
(x — «)«+>
-^^ ' (n-\- I)! -^ ^*^'
which is Lagrange*s formula for the law of the mean.
94 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. 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 PH6pital. — If (f>{x) and i/){x) are two func-
tions which vanish at a, as also do their first n derivatives, then we
shall have
/0(f) _ r 0*^'(g) ^ r 0*^'(-^)
X ^ w " X r^\^) X r^\^y
jn-)« M-v* «r-)«
For, by Cauchy's theorem, § 76,
f/^{x) r^\sy
where S lies between x and a. Hence, since S and x convei:g^e to
a together, we have for x(=)a
f 4>{x) _ <lf+\a)
1 t(^) ~r^"(«)'
Moreover, Cauchy's theorem shows that the quotients
;ff(xj' (r-i, 2,...,«)
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.
EXAMPL£S.
1. Evaluate, when x(zs)ij the quotient
•»* — a-*" 4- 2
j:* — 3Jf + 2 = o, when x z= 1.
Z?(jf* — 3JC + 2) = 2JC — 3, = — I, when * = i.
JT* — 1 = o, when x =z i.
D{x^ — I) = 2Jf, = 2, when jc = i.
'2jr — 3 I
2X 2'
2. Show that
J^x — I ^ I
X ^-i—s-
Art. 78.] ON UNDETERMINED FORMS. 95
3. Evaluate, when jr(=)o, the following:
/ sm Jf ^ j: — 2 sin jc
4. Show that for *(=)o, we have
jf — sin ;c "" ' ^ vers jr
5. Evaluate, when j:(=)o,
/jc — sin-»Jf _ I Ca* — ^ _ , « /"tan jr -
6. Find the limits, when x(=)o,
/jc — sin jc _ £ /* sin y _ _ 3 . /*
=s 3.
/tan jr — X
- = 2.
sin j:
cos mx m^
78. The Illusory Forms. — When u and » are two functions of x,
which are such that the functions
U/Vy UV, « — », «*,
tend to take any of the forms
0/0, 00 /oo , o X 00 , 00 — 00 , o^ 00 •, I*,
as X converges to a ; then when these functions have determinate
limits for x{=)a, the theorem of THdpital will evaluate these limits.
All these forms can be reduced to the evaluation of the first, 0/0,
as follows :
(i). 00/00 and o X 00 reduce directly to 0/0.
For, if 1/^ = 00 , »^ = 00 , then
*^« — !?. — ^^^' — °
«Vi "" 00 "" i/«a "~ o*
and we evaluate
If «^ = o, r^ = 00 , then
u„v^ =rr
«« o
'« « - /— ^ >
and we evaluate
2/
(2). In like manner, if «^ = 00 , tr^ = 00 , then
provided £i''^x/^x) = ^> otherwise this form has no determinate finite
limit and is 00 .
96 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIL
This illusory form can also be reduced to the evaluation of the
form o/o when x{ = )a, thus :
which takes the form o/o when x = a. Therefore, if jC^*^ = ^,
£{u — »)=(;, for x{=:)a.
(3). The last three forms, o®, 00®, i*, arise from the function
«', which can be reduced to 0/0, thus :
Since i/ = ^»<«».
In each of the cases o®, 00®, i*, the function v log u takes the
form o X 00 , which can be turned directly into 0/0 and evaluated as
in (i).
Examples of 00/00 ando/oo.
The eTaluation of u/v^ when « = 00 , t^ = 00 . for ;r = a, is carried out in the
same way as for 0/0. For we have
=/i
when jc(=)a. If now </>{x)/ilf{x) has a determinate limit A ^ o, when Jp(=)a,
then
X ^ (^)
Therefore, for Ji?(=)fl, when 0(x) = 00 , if{x) = 00 ,
if 4}'{a)/i/{a) is determinate.
/*tan X ^ r sec* 4: __ /'sec x
^ sec X ^ X sec JT tan X ■" ^ tan jit
//tan a:\«
tan JIT
Or immediately, by Trigonometry, = sin x»
2. Show that
i -y-n /*— f
= O.
^ x^
jC«oo
when If is a positive integer. Also when n is not an integer.
3. Show that £ x«(log jr)« = o.
A»T. 79.] ON UNDETERMINED FORMS. 97
4w Show that
A'
Zli^ = a
tanO
6. Evaluate, when ji:(=-)J^,
tanx log tan 2x ^ I — sin x -f cos jp ,
teifTsi ' log Un x • sin JP -f cos Jf - I*
log sin X secjc . ^an x
(* - 2x} * sec 3Jr ' tan 5j?'
6. Show that ;f (I - *) tan l(jrjf) = -.
Examples ofoo — oo.
7. £(9^cx — tan x) = o, for Jf(=)l^.
a ;^(x-> — cot x) = o, for j:(=)o.
9. jc tan X — J^ sec *(=) - i, when *(=)**•
,a fL:ij!15(=)i., when x(=)o.
11. {a» - I )/x (= ) log ^J, when jr(=)o.
Examples of«^.
1
12. (i + ji:)*(=y, 4=)o-
I
z
14w (^ + i)*'(=V' * = «•
IB. (cos 2jc>r*(=)r-a, 4=)«>-
I
16. x»"^(=y-s 4=)»-
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
^^_^yog.in,x^ when ^(=)i.
This takes the form o*^. To evaluate, equate the function to^and
take the logarithm,
log (x — i)
^ ^ log sin nx
I
r)^^-ji = rIEj-=L /fiEL!!^ sec ^*.
Jj log sin nx Jj nco&nx ^ X * "" '
&in nx
n.
.9.8 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIL
But £stcnx =:^ — 1, and
Hence ;^^ = e*, when jf(=)i.
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 Ar(=)o,
I
(I + xY- e
X
Differentiating numerator and denominator, we have
~^-(i + .y) log (I +^)
(I + X)'
x\i + X)
j£(i + xy = e, and the limit of the other factor is, by the ordinary
process, readily found to be — ^. Hence the limit is — e/2.
I
Otherwise, put for (i + at)' its value. Ex 13, Chap. VI,
and the result appears immediately without differentiation.
Geomktrical Illustrations.
(I). If/ (a) = Of <p(a) = o, /'(«) ^ o, <p'{a) y£. o, consider the curves rep-
resenting j/ = /(4p), y = <p{x).
y-/(«)
y-0(a)
Fig. II.
These curyes cross Ojc at x = <i at angles whose tangents are equal Xo f\a\
(p\a)y or
f*(a) =5 tan Oj, <p'{a) = tan 6,.
//(^) ^ // ^^x / ^-^» \ ^ tan e^
The limit of the quotient/ (x)/ip(x) is represented by the quotient of the slopes
of these curves at their common point of intersection with Ox.
Akt. 79.]
ON UNDETERMINED FORMS.
99
(0
(2). Consider the functions x and y in
Differentiate with respect to x and solve for Dy,
2Jf*^ 4- 2y» -f- tf ^ *
Zy takes the form 0/0 when jr = o, for then also ^ = q bj (i). To evaluate
-Dy =
this, difierentiate the numerator and denominator with respect to jr.
6jt* 4- 2^« -f \xy Dy ~- a*
- £^
-£
4:ty 4- (2Jf« + 6y« + a»yZ?y '
— tf '
- a^£Dy'
This means that the curv*- whose^^quation -is (i) in -Cartesian coordinates has
two branches passing* vthrough the origin jr = o, >> = o, which is a singular point.
There the slopes of Sie two branches to Ox are -\- i and — i. The curve is the
UmnisccUe^
Fig. 12.
We can find ZJj' at j: = o, ^ = o for the curve (i), without indetermination by
differentiating the equation (i) twice with respect to x. Thus
(2a«- i2jc»-4;'*) = \txy Z>/ + (4Jf«-|-l2;^«-f2tf»)(ZJ>^)«4- (4jr^-f 4;4 ^2a^yf)D^y^
which gives, as before, Z>y = ± i, when x = o, y z= o,
(3). We know, from trigonometry, that the radius p of the circle circumscrib-
ing a triangle ABC with sides a, b^ c having area Sy is
abc
Also, from Analytical Geometry, we have
25 =
i»
where x^ y\ x,, ^, ; jr„ ^,, are the coordinates of the comers 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 x^(=z)xy x^{'=r)Xy is
^ = — :^v~- -
• ;'We hare
** = ('I - *)• + (/i - ^J*.
f = (X, - ^ + {y, - y^,
a'=(x,-x,)'+iy,-y,y.
loo PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII.
Also,
X y 1
•^2 yt '
= ^1 - ^•'i + -yj(-^i - ^) - •^2(>'i - ;')•
Substitute tliese values in the expression for p. Observe, when x^{=)x, p is of
the form o/o. Divide the numerator and denominator by jt^ — x and let x^{=)x.
To evaluate
for x^(=)x, differentiate the numerator and denominator with respect toxj and then
let Xj(=:)r.
/
^ ^ = £(^^i-y) = xDy-y.
jTj — jr*
Therefore, when B(=)Af
I (Xj - xY
P =
l^+(g^)*}[^ + (^«
2 ^2 - J^ - (-^2 - jf)Z>y
The first factor takes the form o/o when jr, = jc. To evaluate it, differentiate
the numerator and denominator with respect to jr,, and we have
I 2(Xj — x)
T L>y^ - Dy'
this is again o/o when x, = jr. To find its limit when x^-=.)x^ differentiate the
numerator and denominator with respect to x^, and there results
I
which has the limit i/D^y when x^(=.)x.
Therefore when the points B and C converge to A along the curve, the circle
ABC converges to a fixed circle passing through A which has the radius
/"{x)
d\y
1^
This circle is called the circle of curvature of the curve ^ =/(•*") at the point
X, y^ and R is called the radius of curvature. Observe that when Xj(=)jc and
jTj 3^ X, the circle and curve have a common tangent at A^ or, as we say, are
tangent at A, When this is the case the curve and circle both lie on the same side
of the tangent at A, Also the circle lies on the same side of the curve in the neigh-
Art. 79.] ON UNDETERMINED FORMS. lot
borhood of A, But when also jr,(=)j: the circle crosses over the curve at A,
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.
BXSRaSBS.
1. Evaluate, when jr(=)o,
/e* — 2COS X -|- f~* __ , /* sin 2x ~|- 2 sin*jr — 2 sin jr __
JT sin x "~ * / ^'^ ^ "" ^^* ^ "~ ^
2. Also, for the same limit of x,
/sin 4jr cot x _ o. /*sin ^ cos 2j: __
vers XX cot* 2x'~ * j vers x cot x '~
3. Show, when x{z=)q,
msin X — sin mx m P tan «jr — « tan x
£
~)~ i' f
jr(cos X — cos mx) 3 T if sm x — sm «jc
= 2.
4. If x(=)o, then
/(^,2y^4-x-f 2_ I r
., fcx 2
— X) tan — = — .
' 2 %
6. Evaluate for x = 00 ,
^cos-J , ^cos--j . ^cos-) , (cos-J .
6. Find the limits, when x(=)o, of
(l\tanx /i\tln«
-j , f-j , (Sinx)-n* (Sinx)tan*
7. Find the radius of curvature of the parabola^' = 4^x at anj point x, y^ and
show that at the origin it is equal to 2/.
8. Evaluate
- fi«)* + (a - 0)* ^Ta
©8)* + (a - e)* I + « V's
9.-1 ; (=)fllogtf, when x(=)^)r.
log sm X ^
>« — 4 -|_ ^— * -|- 2 cos X __ I
-^ -6-
11. £ xe'^ = 00 .
*(-)o
I02 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIL
16. Evaluate, when jr = oo ,
i^x -{• a — t^x 4- ^, 4/jic» + ax — jr, a* sin (^/fl*).
16. Find where the quadratrix
^nx
y = j: cot —
•^ 2a
crosses the ^ axis.
17. Show that £[—. 1 W |.
*(Mo
CHAPTER VIII.
ON MAXIMUM AND MINIMUM.
go. Definition. — A function /"(x) is said to have a maximum
value 2Lt X = a when the value of the function, /{a), at a is greater
than the values of the function corresponding to all other values of
X in the neighborhood of a.
The function is said to have a minimum value at a when J\d) is
less thanyTJ^r) for all values of x in the neighborhood of a.
In symbols, f{pc) is a maximum or a minimum at a according as
Ax) -/(«)
is negative or positive^ respectively, for all values of x ^ a in
{a — e^ a-^- e) the neighborhood of a,
8i. 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 /*'(«) is o.
At a value a at which /(jt) is a maximum or a minimum, by defini-
tion the differences
A^)-A'^) and A^')-A'>), ■
where ji/ < a < x^\ have the same sign.
Consequently the difference-quotients
* x' — a ^ x" •— a
have opposite signs for all values of x' and x"' in the neighborhood
of a, since x' — a is negative and x" — a is positive. Therefore,
since q' and /' have the common limit f\c^ when Ar'(=)a and
j;"(=)a, we have
= =,|/'(«)|=o.
Hence /'(^) = <>•
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 x 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 /'(:»:)
changes irom positive to negative as x increases through a.
103
I04 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIIL
In like manner, at a minimum, x = a, the derivative, /'{a), is o,
2Ln6./'(x) changes from negative to posUwe 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 /{x) a jt« ~ 2:r -f- 3.
.-. f(x) = 2{x — i).
We have f\i) =0. Also for x < i, we have /^{x) negative, and for x > i,
f(x) positive.
Hence f{i) = 2 is a minimum value oi f{x).
2. Let f{x) = — 2jr» -f &r - 9.
.-. f{x) = 4(2 - X).
We have /(2) = o, f{2 - e) = +, f(2 -f e) = — .
•'• A?) = — I is a maximum.
82. The condition /*'(«) = o is necessary, but it is not sufficient,
in order that the function /(x) shall have a maximum or a minimum
value at a. For the derivative /"' (at) may not change sign as x
increases through a. It may continue positive, in which case /(a*)
continues to increase as x increases through a] ot /\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 f(x) is represented by the
slope of the tangent to the curve to the jr-axis. At a maximum or a minimum,
f{x) = o or the tangent to the curve is parallel to Ox, In the neighborhood of
Fig. 14.
a maximum point, such as A or Cy the curve lies below 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 Z>, the points B^ D are the lowest points
in their respective neighborhoods. At a point E the tangent is parallel to 0.r,
and f{x) = o, but the curve crosses over the tangent and is an increasing function
at Ey also the derivative f(x) is positive for all values of jr in the neighborhood.
It will frequently be impracticable to examine the signs of the
derivative in the neighborhood of a value of x at which /"'(a*) = o.
A more general and satisfactory investigation is required to discrimi-
nate as to maximum and minimum at such a point.
Art. 83.] ON MAXIMUM Ax\D MINIMUM. 105
83. Study of a Function at a Value of the Variable at which
the First n Derivatives are Zero.
(i). Let J\x) be a function such that /'(cl) 51^ o. Then by the
law of the mean, §§ 62, 64,
Ax) -y(a) = (^ - aY\S).
By hypothesis, f'{a) 7^ o is the limit of f\x) and of /'{S) as
x{=^)a, since S lies between x and a. Consequently we can always
take X so near a that throughout the neighborhood of a we have
/'{S) of the same sign as /'{a) for all values of x in that neighbor-
hood. Hence, as x increases through the neighborhood of a, the
difference y(.:r) —/(a) changes sign with x — a\ and by definition
/(x) is an increasing or decreasing function at a according as f'{<i)
is positive or negative respectively.
(2). Let/'(^) = o and f"(a) ^ o. Then
Throughout the neighborhood of a, /''(S) has the same sign as
its limit f"(ci) ^ o, and therefore does not change its sign as x
increases through a. But, as (x — ctf also does not change sign as
X passes through <2, we have the difference
Ax) -yia),
retaining the same sign for all values of x in the neighborhood of a,
and having the same sign as /'^{^y Consequently, by definition, the
functiony{:i;) has a maximum or a minimum y2A\x^ /{a) at a according
as /"(a) is negative ox positive respectively.
(3). Let /'(a) = o, f'\d) = o, f"\a) ^ o. Then
__ {x^ay
As before, in the neighborhood of a, f"\i^ has the same sign as
its limit f'^'{a) jL o. But {x — «)" changes its sign from — to +
as X increases- through a. Therefore the difference
must change sign as x increases through tf, and f{pc) is an increasing
or decreasing function at a according as f"\a) is positive or negative,
(4). Let/'(^) =/"(«) = . . . =/"(«) = o, but/-+i(a) ^ o.
Then, by the law of the mean,
Ax) -A<^) = V+T)r-^"(^)-
In the neighborhood of a, /^^^iS) has the same sign &s/'^'*'^{a).
If « + I is odd, then {x — a)"^* and therefore /l-^) ^/{^) change
sign as x increases through a\ and/^jr) is an increasing or decreasing
Ax) -y(«) = ^ ., ' /'"{S).
io6 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIIL
function at a according as /*^\a) is positive or negoHve* If, how-
ever, « -|- I is even^ then {x — a)""'"* does not change sign, nor does
the difference y(:r) —/{a), as x increases through a; consequently
/{x) is a maximum or a minimum at a according as /^^\o) is nega--
five or posiivoe. Hence the following
84. Rule for Hazimttm and Minimum, — ^To find the maxima
and minima values of a given function J\x)^ solve the equation
/\x) = 0. If a be a root of the equation y'(jir) = o, and the first
derivative of f(pc) which does not vanish at a is of even order, say
/^{a) ^ o, theny^fl) is a maximum ii/^{a) is negative, or a minimum
ii/^^(a) is positive.
EXAMPLES.
1. Find the max. and min. values, if any exist, of
^,r) = x^ — ^ + 24jr — 7.
We have 0'(x) = 3(jt« -- dr -f 8) = 3{x -. 2){x - 4).
.-. 0'(2) = o, ^'(4) = o.
Also, ip'ix) =6(x- 3).
... 0"(2) = -, ^'(4) = +.
^2) = -^ 13 is a maximum, 0(4) = 9 a minimum.
2. Investigate for msutima and minima values the function
^jr)9 ^* -|- ^~* -f- 2 cos X.
We have <p'(p) = 0"(o) = ^"'(o) = o, ^'▼(o) = 4.
. *. <p{p) = 4 is a minimum. Show that o is the only root of 0'(x)«
3. Investigate j* — 5jr* -f • 5jr» — i, at jc = I, jr = 3.
4. Investigate .«» — 3x« -|- 3jr + 7, at x = i.
6. Investigate for max. and min. the functions
Jf'* — 3^^ + ^ + 7i j:» - gjT -f 15* - 3.
3Jt* — I25j:» -f- 2l6our, jr» -f 3jr» -f dr — 15.
6. Show that (I — jr -f x*)/(i -f jc — **) is min. at jT = ).
7. It xjf(y — jr) = 2fl", show that^' has a minimum value when x =z a.
8. If 3aV + ■«?'• + 4**' = o» show that when x as 3^/2, then >^ = — 3tf
is a maximum. I^y being then —
9. If 2X* + 3<iy* — x^y* = o, then x = $'a makes ^ = 5*^ a minimum.
85. Observations on Maximum and Minimum.
(i). 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* -j- 5x -\- c»
Substitute^ — 6/2a for x. The function becomes
— 7^ ^ ^^'
Art. 85.] ON MAXIMUM AND MINIMUM. 107
which is evidently a maximum when y z=z o and a is negative, and a minimum
when y ■= o and a is positive.
(2). Labor is frequently saved by considering the behavior of the
first derivative in the neighborhood of its roots, instead of finding the
values of the higher derivatives there.
For example, see Ex. 6, § 85, and also
iP(x) = (X - ^f(x + 2)*.
Here <p^{x) = 3(zx - 2){x - 4)*(* + 2)».
<p' passes through o, changing from -f to — as ^ increases through — 2; there-
fore <p{— 2) is a maximum.
(f)' passes through o, but is always positive as x increases through 4 ; therefore
^4) is an increasing value of <p{x). Also 0' passes through o, changing from
— to 4- as jr 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 ofx which makes /"{x) a maximum or a minimum also
makes C/^ix) a maximum or a minimum when C is a positive constant,
and a minimum or a maximum when C is a negative constant.
/{x) and C -{-/'{x) 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, fi)y and have a maximum or a minimum value at a: = a in the
interval, while its derivative /'(x) is 00 at a, but continuous for all
values of (x) on either side of a.
In this case, to determine the character oi /{x) at a, we can use
(i) or (2) as a test. Otherwise we can consider the reciprocal
i//'(x)y which passes through o and must change sign as x passes
through a, for a maximum or a minimum oi/(x) at a,
BXAMPLSS.
1. Consider (f>{x) sb (x — 2)* -h i.
0 is a one>valued and continuous function and is always positive. It clearly
has a minimum at x ^ 2, where (p(x) = i. We have
^W = T
3 (*-,)!' »
and 072) = oo . Also, 0'(2 — A) is negative and
0'(2 -f- n) is positive.
2. In like manner ^^
ij{x) = I - (jr - 2)* "^
has a maximum at jr = 2. ^'C. 15.
Y
^
•X
lo8 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. Vm
3. Consider <p(x)s l + {x — 7)\
which is also uniform and continuous.
We have
<pf{x) = \ — '
* which is -f- oo when jr = 2. but is always -\- in the neigh-
Ytq j5 borhood of 2. Therefore, at x = 2» 0(x) is an increasing
function.
In like mareier l — (jc — 2)* is a decreasing function at jr = 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 two numbers, and let
X -\-y= c.
Then 4jcy = {X +yf ^ {X ^ y)^ ^ ^ - {X - yf,
which is evidently greatest when x •= y.
Let j: -f J' 4. « = f .
Then, as long as any two of jr, y^ 2 are unequal, we can increase the product
xyz without changing the third, by the above result. Therefore xyz is greatest
when X ^= y -=■ z» The method and result is general, whatever be the number
of variables.
EXERCISES.
1. Find the maximum and minimum values of ^, where
^ = (X - i)(jr - 2)«.
2. Find the max. and min. values of
(I). 2jf» - isjc* + 3dr -f- 6.
(2). (X - 2XX - 3)*-
(3). ^-3^-1-^+3.
(4). 3x* - 25Jf» 4- ear.
3. Show that(x* + x + !)/(•«• — X -\- I) has 3+« for max. and 3-"» for min.
4. Find the greatest and least values of
a^iii X -\- b cos x and a sin'jr -|- b cos'x.
5. Investigate (jc« + 2jr — l5)/(-^ - 5)i and also
x^ - 7jr -{- 6
jr — 10 '
for maximum and minimum values.
6. The derivative of a certain function is
{pc - i){x - 2)\x - 3)»(x - 4)*;
discuss the function at jt = i, 2, 3, 4.
7. Find the max. and min. values of
(a), (x - l){x - 2)(x -3), (4 jn(l - xXl - j:*),
{b). X^ - &*» -h 22JC» - 24^, (/). {X* - 1)/{X* + 3)»,
(<■}. {X — a)*{x — b), (^). sin x cos'x,
(d), {X - a)*{x - b)\ (A), (log x)/x.
Art. 85.] - ON MAXIMUM AND MINIMUM. 109
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 ^ 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 cva/ 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 comers 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
2(xr) a JTi -f- . . . -f jr« = r,
I
n
then n(jrr) a X| . . . x^
I
is greatest when Xj = jr, = . . . =s Jfn.
14. Apply 13 to show that if. -f*>' + ' = ^i ^^^ maximum value of xy*tfl is
15. Show that ifx -\-y -^ z = Cj the maximum value oix^y^t» is
16. Find the area of the greatest rectangle that can be inscribed in the
ellipse. (Use the method of Ex. 15.)
x^ y«
^ + -^, = I. [Ans. 2fl^.]
17. Find the greatest value of ^xyz^ if
^ + y. + 7i='-
fAns. 8 -^1
L 3 i^3 J
This is the volume of the greatest rectangular parallelopiped that can be
inscribed in the ellipsoid.
18. Show that the gjreatest 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 c from the centre (9 of a given circle, a secant is
drawn cutting the circle in A and B. Draw the secant when the area of the
triangle ^C>-5 is the greatest. [With C as a centre and radius equal to the diagonal
of the square on c, 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. \r = a /n, r = 00 .]
21. Find when a straight line through a fixed point P makes with two fixed
straight lines ACy A By a triangle of minimum area. [/* bisects that side.]
22. The product xy is constant; when is jc -f- ^ 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.
Iio PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIIL.
25. From a fixed point A on the circumierence of a circle of radius a, a perpen-
dicular AY v& drawn to the tangent at a point P ; show that the maximu'm area of
the triangle APYii. 3 vJayS. ^
26. Cut four equal squares from the comers 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. Constnict a cylindrical cup with given surfa.ce 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 fiill 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 time a 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 9 miles from the place to be reached.
33. The length of a straight line, passing through the point a, b^ included be-
tween the axes of rectangular coordinates is /. The axial intercepts of the line are
a, P^ and it makes the angle 9 with Ox, Show that
(a). / is least when tan B = (b/a)\
(*). a -I- /5 is least when tan 0 = {d/a)^.
(c), a/S is least when tan 0 = d/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 = 2ic{\ — ^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-» 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 1)Oth 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^' = ^x to a given point on the
axis.
40. The sum of the perimeters of a circle and a square is /. 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. 1 1 1
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 a has for its altitude (23 — Viy)a/i6 ; while that of the corresponding
right cylinder is {2 — 2/ V5)*<j.
46. Show that the altitude of the cone of least volume circumscribed about a
sphere of radius a is 4<j, 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 comer of a rectangle whose width is a is folded over to touch the
other side. Show that the area of the triangle folded over is least when fa 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 3^. The base of the triangle being parallel to
the major axis.
60. Find the least length of the tangent to the ellipse x^/a* -\-y*l^ = '» inter-
cepted between the axes. [Ans. a -^ b.\
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 imchanged. Show that
the inclination of the planes to the axis is sec~' V3 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 L feet long of diameters D and d at the ends has the volume
2 LL^
27 Z> — d'
53. A man in a boat off shore wishes to reach an inland station in the shortest
time. He can row u miles per hour and walk v miles per hour. Show that he
should land at a point on the straight shore at which
cos a : cos ft ■= u : v,
approaching the shore at an angle a and leaving it at an angle /3»
[This is the law of refraction.]
54. From a point O outside a circle of radius r and centre C, and at a distance
a from C a secant is drawn cutting the circumference at /^ and /^, The line OC
cuts the circle in A and B,
Show that the inscribed quadrilateral ARR!B is of maximum area when the
pn)jection of RI^ on AS 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 7/ is the height and R the radius of the base, then // = 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 j' =/(^), or
<l>{Xy y) = o, be the equation to any curve. The equation to the
p secant through the points or, y and
^ x^y y^ on the curve is
X - X x^^ x' ^'^
Xy J' 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
F'G- '7. PP^ when P,i=)P. But when
P^(=)P we have x^(=)x and j/j(=)y. The member on the right of
equation (i), being the difference-quotient of_>' with respect to x, has
for its limit the derivative of y with respect to x. At the same time
the arbitrary point X, Y on the secant becomes an arbitrary point
on the tangent. Therefore we have for the equation to the tangent
at P
X - X dx ^'
in terms of the coordinates .r, y of the point of contact.
(»)
112
AaT. 88.] TANGENT AND NORMAL. II3
The equation to the tangent (2) can be written
r-y=^X^.)^, (3)
or in differentials
{^F^y)dx--{X- x)dy = o, (4)
or in the symmetrical form
dx dy
SZAMPUBS.
(5)
1. Find the equation to the tangent to the circle j^ •\-y^ = <i*.
Differentiating, we have
2x -|- 2y Dy = o.
. •. Dy = — x/y^ and the tangent at x^yv&
y
or Yy -^ Xx ^ a\
2. Find the tangent at :r, >» to *«/«* -f y^/b'^ = I.
3. Find the tangent at x, y to jtV/z* — y*/b* = i.
4. Find the tangent at x, y to y^ = 4/jr.
5. Find the tangent at x, y to x^ -\- y^ -{- 2/y -{- 2gx -f- <^ = <X
6. Show that the equation to the tangent at x^ y to the conic
0(jr, y) = ax^ -i- fy^+ 2hxy + 2/r + 2gy +,</= o
is {ax 4- hy +/)-V+ [hx-\.by^- g)Y ^ (A +^ + ^) = a
7. Show that the equation to the tangent at ;r, ^ to the curve
— + 4- = I
^m ' i}m
IS h — , = I-
8. Find the tangent at jr, ^^ to jr^ = «'j'. [S'^A ~~ ^ ^/y = 30
9. The tangent at jr, ^ to jr' — 3^^ -^ y^ rrz o is
(;/» — ax)Y-{- {x'^ — <7>')A' = axy,
10. Find the equation to the tangent to the hypocycloid
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
^ = 0(/), J' = //•(/),
then, since dx =: <f>\f)d/, dy = ^''(/)<//, we have for the equation
to the tangent
{F-y)<f>'(/) = (X~x)f{/). (I)
114 APPLICATIONS rO GEOMETRY. [Ch. IX.
EXAMPLES.
1. If the coordinates of any point on a curve satisfy the cycloid
X = a{ii — sin 0), y = a(i — cos 6),
show that the tangent at x^ y makes an angle ^ with Oy^ and has for its equation
Y ^y z=z{,X - x) cot ^.
2. In like manner, if
jc = r sin 26(1 -j- cos 26), ^ = f 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.
If^ = 0(jf) and^ = ^(pc) are two curves, and these equations be
solved for x and^, we find the coordinates of the points of intersection.
If the curves intersect at an angle cj, then since 0'(a') and ^\x) are
the tangents of the angles which the tangents to the curves make with
Oxy we have
tan ai = ^* J' . (i)
The two lines cut at right angles when (p'xtl)^ = — i.
Ex. Show that je" + -V' = ^^^ ^^^ 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 fit and 0^ 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 ^;r and the other, we have tan ff^ tan 6^ = ^ i. Therefore
dx
tan ^» = — -— = — D^.
dy y^
Hence the equation to the normal at .r, ^ to a curve is
Y-yJ^(X-x)D^^o, (I)
or (F -)f)D^ ^X-x=.o, (2)
or in difTerentials
{Y-y)dy ^{X~ x)dx = o, (3)
where D^^y or DyX must be found from the equation to the curve.
EXAMPLES.
1, The equation to the normal at x, y to x*/a^ +^y^* = i is
— = a* — a\
X y
Art. 91.]
TANGENT AND NORMAL.
"S
2. The normal at x, ^^ to j^ = ax* is
fiyy-\' mxX =1 ny^ -f- mx*,
3. Show that the tangent and normal to the cissoid
y\2a - jf) = j;», at x = a, are,
at (tf, a), y ^ 2x — a, 2y -{- x =z ^a;
at (a, — a), y + 2X = a, 2y = x ^ 3a.
4. In the Witch of Agnesi, y(4a^ + x») = &?», the tangent and normal at
X = 2aj are
X -\- 2y =z 4a, y — 2x — ^a.
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 0(Xty) = o.
If d is the distance from a, /tf to a point -r, y on the curve, then
6^ = (a-xy^(/3-y)\
When this is a maximum or minimum,
</(5« = - 2(a - x)dx - 2(j3 -y)dy = o,
which is the equation (3), § 90, to the normal through a, /3.
91. Subtangent and Subnormal (Rectangular Coordinates).—
The portion of the tangent, FT, included between the point of
contact, P, and the Jir-axis, is called
the tangent4ength. The portion of ^1
the normal between the point of con-
tact and the ^r-axis is called the nor-
mal length. The projections yj/and
il/iVof the tangent- length and nor-
mal-length on the x-2Xi% respectively
are called the subtangent and subnor-
mal corresponding to the point P,
O
N
X
M
Fig. 18.
If /, «, *$•<, Sn represent the tangent-length, normal-length, sub-
tangent, and subnormal respectively, then we have directly from the
figure
dx'
.,=,/!• '=W-+G4)7
Sn=y
dx'
'*=^\^ +(
dxj'
St is measured from T to the right or left according as -S'^ is +
or — , and S^ is measured from i^to the right or left according as
S^ is -f or — .
EXAMPLES.
1. Show that the subnormal in the ellipse jt*/a« + ^V^' = l is
5; = - b'^x/aK
2. Show that 5< in_y = a* is constant.
Ii6 APPLICATIONS TO GEOMETRY. [Ch. IX.
3. In^' = 2mx^ show that ^^ = m is constant
/- --\
4. In the catenary^ = \a \e^ -|- e «/ , n z= y^/a,
5. Show that <p(x, y) ^ o must be a straight line if St/Sft is constant
6. Show, in the cissoid jc* = {2a — x)y^, that
St = (2ax — x*)/{za - jr).
7. Show that the circle x* -\- y* =z a^ has n constant
92. Tangent, Normal, Subtangent, Subnormal (Polar Coor-
dinates).— het/{p, 6) = o be the equation to any curve in polar coor-
dinates, tp the angle which the tangent at any point makes with the
radius vector, and <p the angle which the tangent makes with the initial
line. From the figure we have
Fig. 19.
n^n « psin ^8
tan MP.P = —r-i 7^,
^ p + Jp — p cos jJ0
sin J(f
_ Jd ^'^F~
' + ^j-p—jd—
When J^(=)o, we have, passing to limits,
tan^ = p^, (i)
since 4- = ^ sin jr = o, when ar(=)o.
Also, since (p =z 0 -\- ip, we have
tan 0 =
fjDf,0 -f tan 6
p -{- tan 0 D^p
D^o — fj tan ^'
(*)
Art. 92.] TANGENT AND NORMAL. 117
Observe that (2) is the same value as that obtained for D^ in
§56.
Draw a straight line through the origin perpendicular to the
radius vector, cutting the tangent in T and the normal in N, We
call /'iVand PT^ 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 normal-length and polar
tangent-length respectively ; and their projections, OiVand OT^ on this
perpendicular are called respectively the polar subnormal zxA subtangent.
We have directly from the figure
/ = p sec ^ = p j/i + f^(D,e)\ (3)
n — pcsctf) =i^p'^ + {I>$py. (4)
5"/ = p tan ^ = fi^Bf^O, Sn = p cot if? ^ D$p. (5)
When Dpd is positive (negative), Sg is to be measured from O to
the right (left) of an observer looking from O to P,
Putting p'^D0p, we have for the perpendicular from the origin
on the tangent
P= , ^ (6)
Vp' + p''
since // = pS^ This can be written
(7)
I
tt'-f
tau
\dB
if we
put p
^^^
i/u,
for then
dp^
dd"
du
de*
BXAMPLB8.
1. In the spiral of Archimedes p =£ <i9, show that tan ^ ^ 0, and Sn is
constant.
2. Show that St is constant in the reciprocal or hyptthcfdc spiral fA ^ a.
Il8 APPLICATIONS TO GEOMETRY. [Ch. IX.
3. In the equiangular spiral p = ae*<^^^, show that ^ t= a^ St = p tan a,
Sm = p cot a.
4. If p = «*i show that tan ^ = (log «)-«.
6. Show that the perpendicular from the foc\;is4o the tangent in the ellipse
(I — ^ cos G)/t> ^ a{i — ^)
18 p^ =z pa* ~ .
6. Determine the points in the curve p = a(i -|- cos 6), the cardioid, at which
the tangent is parallel tp the initial line.
7. If p = afi — OPS 0)i show that
^ = ^, / =3 '2a sin> ^, .S4 = 2a sin* ^ Un ^.
EXERCISES.
1. Show that in <p[Xf y) = o, the intercepts of the tangent at any point jt, ^ on
the axes are
Xi = x -^yDyX, Yi^y - xD^y.
2. The length of the perpendicular ^m the origin on the tangent is
p = xJ^y - y
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 Dxy byy, and the area by H. Then
y
Also
dn __ {y - xy'Xy + ^)y'
dx ~ /« '
where y" s5 ^J^'- For a maximum or a minimum D£l = o. The conditions
y 5KJ o, y 7i o, y -xy' ^o, y-\-xy=iO
show, by Ex. i, that Xi = zr, Yt z= 2y,
4. Find when the area of the triangle formed by the coordinate axes and the
tangent to the ellipse
jf' y*
a*^ ^
is a minimum.
6. Show that the tangent at the point (2, — i) of the curve
jr« -f 2x^ - ar* + 4^ -i-y -4 = 0
is &r -f- isy = I.
6. The line ex -^^ y = e{i -\- ic) is tangent to the curve
sin :r — cos jr = log^, at («', ^).
7. The line;' -f i = o is tangent at (-f- I, — i) to
X* — 2Jf»y« --3r*+4«y-f4Jf + 5)' + 3=0^
8. Determine the points at which the tangents to
•** +y = 3-*
are parallel to the coordinate axes. (x = o, y = o), {x = ± i, y =z ± f/i).
Art. 92.] TANGENT AND NORMAL. 119
9. At what point ofjr*-j-4y— 9 = ois the tangent parallel to jt — ^ = o ?
(j:= - 1,^ = 2.)
10. The tangents from the origin to
X* — y^ -{- yc*y -\- 2jcy* = o
are y = o, ^x — y = o, x -f- ^ = a
11. The perpendicular from the origin to the tangent at x, y of the curve
jci '\-y^ = a^ is / = ^axy,
12. Show that the slope of the curve x^y'^ = «*(jc+j') to the jr-axis is \jc
at o, o.
13. If jr, y are rectangular coordinates and p, 6 the polar coordinates of a point
on a curve, show geometrically that when Dxy = o we have D^fi = p tan 0,
and verify from the formulae in the text.
14. Show that the curves
-+^=l and _+ -=,
cut at right angle if a* — ^^ = a'* — ^'*.
15. In the parabola jc* -\- y^ = a*, show that at x, y the tangent is
Xy^ -f Kr* = (axy'^,
and that the sum of its intercepts is constant and equal to a,
16. The tangent at x, y to {x/df + {y/b} = i is
Xx/d^ -\-(Y+ 2y)/zb^y^ = i.
Also find the normal.
17. The tangent and normal to the ellipse
j^ -f~ 2y' — 2xy — jc = o
at X = I are,
at (I, o), 2r = X - I, y -\-2X=:2\
at (I, I), 2y = X + I, >^ -f 2x = 3.
18. In the curve ^(x — iX^f — 2)=:x — 3, show that the tangent is parallel
to the X-axis at x = 3 db 1^2.
19. In the curve {x/a^ -f {y/by = i, show that (see Ex. i.)
^2 -I- ^« - '•
20. Show that the tractrix
X + 4/r» — V» = - log
has a constant tangent-length.
21. In the curve^ =r a»«-ix, find the equation to the tangent; and determine
the value of n when the area included between the tangent and axes is constant.
22. In p(a^ -|- b^-^) = abf show that
5*^ = - ab/{ag« - br-9),
23. If p* cos 2$ = rt», show that sin ^ = <?Vp''
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 straight line passing through the two points as they converge to the
point of contact.
»20 APPLICATIONS TO GEOMETRY. [Ch. IX.
25. If Qf Py R are tlwec points on a ctirve, P the mid.point o* the arc QR, and
Kthe middle point of the chord QR, show that the normal at P is the limit of the
line /'Fas Q{=.)P, R{=)P^
26. Prove that the limit of any secant line through any two points R, ^ on a
curve is the tangent at a point P as R{=:)Pf Q(=i)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_^y 4-^-- ;c = o
and (K-jVilr'i + -^-^i=o,
where y, ^ represent jDm^ at or, j^ and ^yyy ^ the equations of a fixed normal at
X, y and a variable normal at x^, y^ EUmmating X^ we have
A/x -/) =J^iyi -yy + ^1 - -^i
M-/
jr, — j:
Also»
1 4-y*
= >' + — ^7r-» when ;ri(=)*.
y
Jr=^-yLi^', where /' = §.
This point is the center of curvature of the curve for x, y.
CHAPTER X.
RECTILINEAR ASYMPTOTES.
93. Definition. — An asymptote 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 Q,yxvi^f(Xyy) = o:
I. The equations to the tangent at jr, y and its axial intercepts
are
dy
^ dx
If we determine, for x ^y = oo ,
then the equation to the asymptote is
X y
a 0
Or, if we determine, for j; ^^^ = oo ,
and either a ox b za above, we have for the asymptote
y
y = mx -\-b or j; = ^^ + tf .
This method involves the evaluation of indeterminate forms,
which must be evaluated either by purely algebraic principles or by
aid of the method ol 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.
121
122 APPLICATIONS TO GEOMETRY. [Ch. X.
EXAMPLES.
1. Find the asymptotes to the hyperbola -= — 7= = l.
We have Xi = a^/x^ Vi = — l^/y. These are o when x = ^ = 00 .
Therefore the asymptotes pass through the origin. Also,
dy b'^x b I
dx d^ fl ^i — ayjt* *
the limits of which are ± b/a when jt = 00 . The equations to the two asymptotes
are ay = ± bx,
2. Find the asymptotes to the curves
(a), y z=.\ogx. x = o,
{b), y = ex. (Fig. 33.) y = o.
(0- ^ = ^*». (Fig. 34.) >' = a
(d), y^e*»=z j:« — I. >' = o.
X
W- I +^ = ^*. * = O, >' = O,
(/). ^ = tan ax^ y = cotaxy y =: sec ax,
3. Show that^' = X is ah asymptote of j:* = (jc* -f" S^'l^*
4. x +7 = 2 is an asymptote of>^ = 6jc* — jc*.
5. :r = 20 is an asymptote of Jf* = (2a — x)yK
6. j:* -|- ^' = fl* has >' + j: = o for an asymptote.
7i The asymptotes of {x — 2d)y^ = j^ — cfl are
X = 2a and «-[-'*= ± >'•
4r = 2a is readily seen to be an asymptote. For the others express ZJy in terms
of jr and make x = oo ; the result is i: i. Find the intercept in same way.
8. Find the asymptote of the Folium of Descartes
jc» +y = ^axy.
See Fig.' 49. The asymptote is x -{- y -{- a z= o. Vuty = mx in the equation
to determine slope and intercept.
II. We can sometimes find the as3nnptotes to curves by expansion
in a series of powers. Thus, if
then J/ = a^x + a^ is an asymptote. For, evaluating as in I, we
have m =2 a^, Fi = a^.
Observe also, if we have
then when ;i; = 00 the difference between the ordinate to this curve
and that of the curve y = (f>{x) continually decreases as x increases.
We say the two curves are asymptotic to each other.
Art. 94.] RECTILINEAR ASYMPTOTES. 123
EXAMPLBS.
9. In Ex. I, I| we have
As X increases indefinitely, the point Xy y converges to the straight-line asymptote
<y = ± bx,
10. Solve Ex. 3, I, by expansion.
11. Solve Ex. 6, I, by expansion. Here we find that the given curve and the
hyperbola
;^* = 4f« + ^ax 4- 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, /jjjf, y) = o, in j; and y, or
can be reduced to that form, we can always find its asymptotes as
follows :
Rule I. Equate to o the coefficients of the two highest powers
of X in
f{Xy mx + 3) = o.
These two equations solved for m and h furnish the asymptotes
oblique to the axes.
Rule 2. Equate to o the coefficients of the highest powers of x
and of ^ iny(ar, y) = o. The first furnishes all the asymptotes parallel
to the Jt-axis, the second thos^ parallel to thej'-axis.
Proof: (A). The straight line
y = mx '\- b (i)
cuts the curve
/{x,y) = o (2)
in points whose abscissse are the values of x obtained from the solu-
tion of the equation in at,
/{Xy mx + b) = o. (3)
If (2) is of the «th degree in x and ^, then (3) is of the «th
degree in x, and will furnish, in general, n values of x (real or
imaginary).
Let (3), when arranged according to powers of Xj be
A^+A,.^x^-^ + ...-\-A,x + A^ = o. (3)
If one of the points of section of (i) 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 x must be o, or A^ = o.
This is readily seen to be true by substituting i/z for x in (3),
and arranging according to powers of z. Then when 0 = 0, we have
jc = 00 , and A^ = o.
In like manner if a second point of intersection of (i) and (2)
moves off to an infinite distance on the curve, a second root of (3)
134 APPLICATIONS TO GEOMETRY. [Ch. X.
is infinite and we must have the coefficient of Jt*~* equal to o, or
^»-, = o.
When (i) and (2) intersect in two infinitely distant points, then
(i) is an asymptote of (2), and we have for determining the
asymptotes the two equations
A^ = o, A„_^ = o.
These two equations when solved for m and d give the slopes and
intercepts on thej'-axis of the oblique asymptotes of (2).
EXAMPLES.
12. Consider jc* = (jc* -j- 3<?*)y, see Ex. 3.
Here jc* — x^j^ — ^ay =r o
becomes (l — ^)^ — ^-^ — 3a*mx — ^M =^ O,
when mx + 3 is substituted fbr_y. Hence
I — «i = o and —6^0
give y z= X 2L% the oblique asymptote.
13. In x^ -]- y^ = ^axy^ Ex. 8, put y = mx -\- b,
.'. (I -f »i»)jr» -f "^mimb - a)x^ + . . . = o,
it Ixiinj/ unnecessary to write the other terms.
Hence »i= — i, b z= — a. Therefore the oblique asymptote is
y •= ^ x — a,
14. Show that y z=z x -\- \a is an asymptote of ^ = cufl -|- •**•
15. The asymptotes of y^ -^ x^ -\- zax^y = H^x
are ^ = * — |a and y -\- x -\- \a ■==: o,
16. ^ + ^x^y - Jfy* - ar'* + '* - 2-^ + a?'* + 4^ + 5 = o
has for asymptotes
(B). If the term A^_^af'-^ is missing in (3), or if the value of m
obtained from -4^ = o makes A^_^ vanish, then (3) has three infinite
roots when
^^ = o and A^_^ = o,
which equations give the values of m and b which furnish the
asymptotes. A^_^ will be of the second degree in b, furnishing two
3'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 00 .
If also the term A„_yXf^ is missing, or if A^_^ vanishes for the
value of m obtained from A^zn o, then the equations
A^ = o, ^H_3 = o
furnish three parallel oblique asymptotes, in general^ for each m,
EXAMPLES.
17. If {X -\- yY{x* -{^ y* ^ xy) z:z a*y* + €fi[x ^ y\
then A^^t - (i + fnf{t + » -f- nfi\
An-i — o.
An^^ zsb^ ^ a\
••• m =: — I, b sz ± a give asymptotes >' st — • jr ± a
Art. 95] RECTILINEAR ASYMPTOTES. 125
18. In Jt^(y 4- xy 4. 2tfr«(x -f ^) 4. Sa^xy + tf»)^ = o,
the asymptotes are ^ + jr = 2a, ^ -|~ -^ "j- 4^ = o.
19. Find the asymptotes to the airves:
(a). ^» - *»;/ = a V -h J^) + ^- ^ = 0, ^ s= o, jr =>
(3). y — jc* = <i«x. ^ = jr.
(<r). jf* — ^ •= a*jrK -|- ^>^. jr + ^ = o, x = y,
(C). For the asymptotes parallel to the coordinate axes, the fol-
lowing simple process determines them :
Arrange /{x, y) = o according to powers ofj', thus:
Ar-\^{Bx+L)y^^ + {Fx^ + Gx + ff)r-'+' . .=0. (4)
If the highest power ofy is, n, the degree of the curve, there will
be no asymptote parallel to Qy, since then A ^ o. If, however, the
term Ay* is missing, or ^ =0, then for any assigned x one root in
the equation (4) in >' will be 00 . If, now, Bx -{- C = o, a, second
root of (4), in_>', is 00 at ^ = — C/B, and this will be an asymptote
to the curve, since Z^j^y is 00 for the same value of x which makes
^ = 00 .
If the terms involving the two highest powers of y in (4) are
missing, then
Fx^ + Gx + B^= o
makes three roots of (4), in^, 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 ot'
the highest power of x.
Therefore the coefficients of the highest powers of x and^v 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 .r or y
they cannot be o and there are no asymptotes parallel to the axes.
EXAMPLES.
20. Find the asymptotes to the following curves:
(a). y*x — ay'^ = jr* -(- aj(^ -\- l^. x = a, y = x -j- af y -{- x -\- a = o.
(d). yix^ — ^bx -f 2^*) = x*— 3«jr* -\- c^, x = dj x = 2b^ ^ -f 3a = x -f 3/^
{c\ x^y^ = a*(x* + y ). X = i: a, y r=i ± a,
{d), x^y^ = fl'(jic* — y^). y -\- a = o, y — a ^^ o,
{e) y*a = y^x -\- J^. x := a,
if)' (x«-/)»-4y«-f>' = o.
Crt. x\x ^yf - a\x^ ■\.y^\ = o,
(f). jr^* = Jt^ 4- X 4- ^.
U). xv = (tf 4- yn^ - y^)'
(k), y(x - yf = y{x -. y) ^ 2,
95. Asymptotes to Polar Curves. — If y(p, ^) = o is the equa-
tion to a curve in polar coordinates, then, when it has an asymn-
126 APPLICATIONS TO GEOMETRY. [Ch, X.
tote, that asymptote must be parallel to the radius vector to the point
at oo 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, 6^) = o, determine
the values of 6 which make p = oo . These values of H give the
directions of the asymptotes.
If the equation can be written as a polynomial in />, the values of
0 are furnished by equating to o the coefficient of the highest power
of p.
To construct the asymptote when, & = a, the direction has been
found; evaluate for 0{=)a and p = oo the subtangent
»(-)o
where p« = i. The perpendicular on the asymptote is to be laid ofT
from the origin to the right or left of an observer at thS origin look-
ing toward the point of contact, according as / is -|- or — respec-
tively.
BXAMPL£S.
21. Let p = tf sec 5 + ^ tan B.
p = 00 when © = |jr; also,
,^ _ (a 4- ^ sin Q)'
^ dfi~ a sin 0 -f /J '
the limit of which \% a -\- b. The asymptote is then perpendicular to the initial
line at a distance a '\- bKo the right of O, Also, when 6 = \ny p = oo , and the
corresponding value of the subtangent gives a — b and another asymptote.
22. Show that p' sin (6 — ix) -f- ap sin (6 — 2cr) + a » = o has the asymp-
totes p sin (0 — a) = ± a sin a.
23. Find the asymptotes of p sin 0 = aO.
24. Find the straight asymptotes of p sin 4O = a sin 3O.
26. Show that p cos 0 = — a is an asymptote of p cos 0 = ^i cos 2O.
26. ^ = (p — a) sin 0 has p sin 0 = ^ fur asymptote.
27. Determine the asymptotes of p cos 26 = a.
Polar curves may have asymptotic circles or asymptotic points.
EXAHPLSS. '
28. Find the asymptotes of pO = a, for 6 = 0, 0 = 00 . Fig. 57.
29. Find the circular asymptotes of p(0 -|- a) = W, and of
_ //O* _ ^zG* _ g -f cos Q
^ ~ 0»~±T«' ^ " 6+ silTe* ^ ~ 6 + sin e*
CHAPTER XL
CONCAVITY, CONVEXITY, AND INFLEXION.
96. On the Contact [of a Curve and a Straight Line. — Let
y =/\x) be the equation to a curve. The equation to the tangent,
§ 87, at ;i; = a (J^ being the ordinate corresponding to x) is
y=A^) + (*• - «!/■'(«)•
The difference between the ordinates of the curve and tangent at
any point (by the theorem of mean value) is
/{x)-r=^{x-a)Y'\S).
If /^'{a) 5^ o, this difference will retain its sign unchanged for
all values of x in the neighborhood of a. Therefore throughout this
neighborhood the curve will lie wholly on one side of the tangent.
It will lie below the tangent when f'\ct) is — , and above it when
The curve y ^=.f(pc) is said to be concave at a when f"(a) is
negaiwe,.OT the curve lies below the tangent there; and is said to be
convex at a when f"{a) is positive^ or the curve lies above the tan-
gent there.
SXAMPLSS.
1. The curve ^ = e* is always convex, since Z>'^-« = e* is always positive.
2. The curve ^ = log x is always concave, since Z?* log y = —x~^ is always
negative.
3. The curve ^ -=1 3^ -\- ax \% convex when x is positive and concave when x
is negative, since D^y =. tx.
127
128
APPLICATIONS TO GEOMETRY.
[Ch. XL
PoiNTS OF Inflexion.
Fig. 22.
Suppose, at a: = tf, we have /"{a) = o, but /'"{a) ^ o. Then
the difference between the ordi nates of the curve and tangent at a is
Ax)-y=\{x-a)Y"'{S).
Since y'"(<2) 5^ o, then throughout the neiprhborhood of a, f'"{5)
keeps the same sign as its limit /"\ci). But {x — aY changes from
— to + as X increases through a. Consequently the corresponding
point PoTi the curve crosses over from one side of the tangent to the
other as P passes through A.
The curve is convex on one side of A and concave on the other.
The curve is said to have a point of inflexion at x, y when at this
point we have D^ y determinate and Z>y = o, D^y ^ o.
At a point of inflexion x = a ?l 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 x increases through a. See the points A and A^ in
Fig. 22.
SXAMPLES.
4. If y — 2(x — <j)' -j- 4Jt — I,
y = I2(jr — <7) = o, when x =: a,
and y" =12.
The curve has a concavo-convex inflexion at x = a.
5. Show that every cubic
/[x) s fljf* + ^Jt» -f- fjr -I- ^
has an inflexion a«id classify it.
Again, suppose at Jir = a we have
/^\a) = o, /'-(a) = o, /iv(^) ^ o.
Then
4-
In the neighborhood oia,/'^^{S) keeps its sign unchangea, as also
does {x — a)*. Consequently the curve lies wholly on one side of
the tangent, and is convex or concave according as/"'^(a) is + or — .
In general, if/''(^) = • • • • =/'"(^) = o,/'^^i(a) 5^ o, then
•^^ ^ (wz + i)! ^ ^ ^
Art. 96.] CONCAVITY, CONVEXITY, AND INFLEXION. 129
li m -\- I is even, the curve is concave or convex at a according as
ywri^^j is negative or positive.
If »i + I is odd, the curve has an inflexion atx = a, and is concavo-
convex {{/'^'^^(a) is -f-, and is convexo-concave if/*"*'*"»(fl) is — .
The tangent at a point of inflexion is sometimes called a station-
ary tangent, since D^0 = o there. For, 6 being the angle which the
tangent makes with Ox, we have tan 6 = Dj^y, etc.
The conditions for a point of inflexion given above, for/|[jt:, y) = o,
are exactly those which have been previously given for a maximum or a
minimum of i?x>'* ^^^ y = f(p^^ ^ * convexo-concave inflexion
whenever/"' (jr) is a maximum, and a concavo-convex inflexion whenever
/'(x) is a minimum. The investigation of^ =y*(jf) for points of
inflexion amounts to the same thing as investigating the maximum
and minimum values of^ = y'(jr).
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 jf* = (a* -f- ^)y ^^^ ^" inflexion at the origin. What kind of
inflexion ?
7. Show that a\v = bxy -\- cjc* -^^ dx^ inflects at o, o.
8. The origin is an inflexion on a^-^y = jr«», if m > 2 is an odd integer.
9. When is the origin an inflexion on >^ = kx^ ?
10. Find the point of inflexion on jr* — 3^jr* -f- ^V = o» and classify it.
[X = b,y z= 2^/a«.]
11. Show that the inflexions on/(p, 0) = o are to be determined from
See g 56. If we put p = i/m, this takes the simpler form
u + u'i = a
The polar curve is concave or convex with respect to the pole according as
u + u'i is + 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 p sin ^ = aQ,
13. In pO«» = a there is an inflexion when 0 = V^^ "- ^)'
14. Find the points of inflexion on the curves:
(fl). tan ax = y. (d), y = e-^^'
(b), y = sin ax, (^). 7 = (x - i)(jr - 2)(x - 3).
{c). y = cot ax, (/). p(e« - i) = «0«.
15. Show that the curve x{x^ ^ ay) = cfi has an inflexion where it cuts Ox,
Find the equation to the tangent there.
16. Show that jk* + >^ = <»• has inflexions on O^ and Oy.
17. The inflexions of :fiy = a\x — y) are at x = o, x=z ± a ^^,
18. X =r log (y/x) inflects at jc = — 2, y z=. — %e-K
19. pO = a has an inflexion at p ■=, a |/2.
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^ = (p{x) and y = ip{x) be two curves, the functions 0
and ^ having determinate derivatives at a.
If we solve y = (p{x) and y = ^(^) for x and y, we find the
points of intersection of the curves.
(i). If <p{a) = tf?{a) and 0'(«) 9^ ^'(<^), the curves cu/ at a, and
cross there. For, by the law of the mean applied to the function
IXx) = <f,{x) - f{x),
we have
<t>{x) - i^x) = (^ - a)l<l>\S) - nS)-]. (§ 62)
The derivatives <p\S), ^\S) are arbitrarily nearly equal to <p\a)
and V^'(«) for x in the neighborhood of a. Therefore, since
<l>\a) ^ t/}\a), the difference <p\S) — i>\S) keeps its sign unchanged
in the neighborhood of a, and x — a changes sign as x passes
through a.
(2). If we have <f>(a) = i/p{a), <p\a) = ip\a), but (p'\a) 9^ tp"{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
<l>{x) - fix) = \{x - ayi<p"{S) - i>"(S)]
shows that this difference does not change sign as x increases through
a, and therefore the curves do not cross at a,
(3). If 4>{a) = tp{a), <p\a) = ^'(a), 0-(a) = ^*», but 0'"(a)
T^ il)"\a)j then the curves have a contact of the second order at a,
and we have
<p(x) - fix) = i(* - aY[<p"'{S) - i>"'{S)l
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 (p(x) and t/}(x) and their first n derivatives at a
are equal, but their (« 4* ^)t^ derivatives are unequal, then the
130
Art. 97.] CONTACT AND CURVATURE. 13 1
curves are said to have an nth contact at a, or a contact of the nth
order. They do or do not cross at the point of contact according as
» -f- I is odd or even.
For we have, by the law of the mean,
which changes sign or does not according as » 4~ ' ^^ ^^ or even
when X increases through a.
Two functions are said to have a contact of order n at a value of
the variable when for that value of the variable the corresponding
values of the functions and their first n derivatives are equal.
II. The character of the contact of two curves is made clear by
the following theorem:
If two curves y = <f>{x) and y = ^{x) intersect in n distinct
points at tfj , a, , . . . , a« , then when these n points of intersection
converge to one point, the curves have a contact of order » — i.
To prove this the following lemma * will be established :
If F{x) vanishes at a, , a, , . . . ^ a,, , then
F{x) = (•^-^i)-'('^-0^^g^^
where ^ is some number between the greatest and least of the num-
bers AT, tf J , , , . , a„.
Consider the function of 2,
J{z) = (^ - a J . . . (;r - a^^F^z) - (« - a^) . . . (« - a:)F(x).
We havey(«) = o at the » + ^ values of z equal tox, a^, . . . , a^.
By RoUe's theorem, J\z) vanishes n times, once between each con-
secutive pair of these numbers. Also by the same theorem /"(«)
vanishes « — i times, once between each consecutive pair of numbers
at which /'(«) vanishes; and so on, until finally /*•(«) vanishes once
between the pair of values for which /"^^ (a) vanishes. This value,
say Sj at which /"(2) vanishes is certainly between the greatest and
least of X, tfj , , . . , fl,^ Hence
/«(^) = (^ - a,) ... (a; - a.)/^(f ) - n\ F{x) = o,
and the lemma is proved.
Now let F{x) s <l)[x) — tp{x). Then
This shows that when aj = a, = . . . = a^ = a, we have
where S lies between x and a.
* Due to Ossian Bonnet.
I
132 APPLICATIONS TO GEOMETRY. [Ch. XU.
This last equation shows that (f>{x) and ip{^:) and their first « — i
derivatives at a are equal, or the two curves have a contact at a of
order « ~ i. Therefore, when two curves have a contact of the nih
order, it means that they have « + i coincident points in common
at a ; or, as we sometimes say, they intersect in « + ^ consecutive
points. A curve which cuts another n times in the neighborhood of
a point, leaves that curve on the same side it approaches it when n
is even and leaves on the opposite side when n 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 = jfi and ^ = 3^:* — 3* -f I.
Solving the equations, we Bnd that jr=i,^ = iisa 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, Z>V» ^ire not equal
to each other. Therefore, at the point i, i the curves have a contact of the second
order.
2. Show that the straight line y ^=. x ^ i and the parabola \y •=z jc^ have a
first-order contact.
3. Find the order of contact of
gtv = x» — 3JI:* -f 27 and q;' -|- 3^; = 28. [Second.]
4. Find the orders of contact of the curves:
(rt). y = log (jc — i) and x^ — dx -\' 2y -\- % -=. o. [Second.]
\b), 4y = jf« — 4 and jk* -|- >/« — 2;/ = 3. [Third.]
[c), xy z= a^ and {x — 2^1)' + {y —2af = 2xy, [Third.]
5. Find the value of a in order that the hyperbola xv = 3Jr — i and parabola
^ = jc -|- I -|- a{x — I)* may have contact of the second order.
100. Osculation. — (i). We can always find a straight line which
has a contact of the first order with a given curve y = 0(.r) 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 4- ^ be the equation to a straight line, in which
m and b are arbitrary and are to be so determined that the straight
line shall have the closest possible contact with the curve y = <^x)
at a: = a.
Then we must have
ma -\-b ^z (p{a),
m = 0'(^)'
These two conditions completely determine m and b, and give
y^<P{a) + {x^a)<l>\a)
Art. ioo.] 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) = o, 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
(^-«)* + (r-/S)» = J?». (I)
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^ fi^ R^ the coordinates of the
centre and the radius.
Let us determine a, /?, and R^ so that the circle shall have the
closest possible contact with a given curve j/ = <t>{x) at a given point
Xy y of general position on the curve.
Differentiating (i) twice with respect to X^
X'-a + {r^/3)Dr=o, (2)
i + (r^/5)i}^r+{i)ry^o. (3)
The conditions for the contact are
y=jf, Dr= <p'{x), n'F= 4>"(x).
The values of a, /3, Ji determined from the three equations
(^x - «)« + {y~ fif = ie», (4)
x-a-\-{j>- P)<p'{x) = o, (5)
I + O' - fi)4>"ix) + [<t>"{x)-] = o, (6)
determine the circle of closest contact, of the second order, at at, y
on the curve. Solving these equations and writing >^, y^' for 0', 0",
we have for the coordinates of the centre of curvature
I 4- /' I -I- /*
fi=y + -^> a = x^y^^, (7)
and for the radius of curvature
ff _ (I +y»)»
(8)
Whenever the coordinates x, y are given, we can substitute in
these formulae and compute a, /?, and R^ 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, 111. (3), where we con-
sidered the circle which wa? the limiting position of a circle through
three points on the curve when these three points converge to x^ y
134 APPLICATIONS TO GEOMETRY. [Ch. 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 X, y, and R is called the radius of curvature, the point a, fi
is called the centre of curvature of the curve at x^y,
(3). In general, when the equation of a curve y = tl>(x) con-
tains a number, « -f '» ^^ arbitrary constants, we can determine the
values of these constants so that the curve shall have a contact of the
«th order with a given curve y = 0(:x^), at a given point of arbitrary
position and no higher contact. For, if we equate the values of the
function ^ and its first n derivatives to the corresponding values of
0 and its first n derivatives, we shall have n -\- 1 equations between
the » + I arbitrary constants in y;. These equations serve to deter-
mine the values of these constants which will make y = ^(x) have
an «th contact with y = (t>{x) at the point under consideration.
This is the highest contact such a curve ^ = ^ can have with a given
curve ;/ = 0 at a point of ordinary position. Then^y = ^ is said to
osculate the curve >^ = 0 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.
loi. 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 R as positive when j/" is positive, or when the curve is convex ;
and negative when y'^ is negative or the curve is concave. Conse-
quently the value of R is
R-
y
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 R oiy" is -[- or — .
EXAMPLES.
1. Find the radius of curvature at any point on the parabola jfi = 4^^.
Here 2my = Xy 2my" = I, I -f- /* = i -\-y/m ;
. 2{m + y)^
2. Find the radius of curvature in the catenary
Art. loi.] CONTACT AND CURVATURE. 135
Here y = i\e»— e~^J, y'=zy/a*; .-. p = -f ^/a.
Show that the radius of curvature is equal and opposite to the normal-length.
3. In the cubical parabola ^a^ = jc* we have
ay = jt«, ay* = 2x, (I 4- y*)i = («* + **)V^;
P =
2a*x
4. Newton's Rule for the Radius of Curvature. At any point /'on a given
curve draw a circle tangent to the curve and cutting it in a third point Q at dis-
tances / and a from the common normal and tangent respectively.
Let r be tne radius of the circle. Then, by elementary geometry, the products
of the segments of the secants are equal, and we have
/» = g(2r - g),
or r ^ - — U — .
2q ' 2
When Q{z=)Pf the circle becomes the circle of curvature at Pv^nA £r = P.
■•• « = £?'
when /( = )o, ^( = )o.
5. If Qi Pf P are three points on any curve, such that V is the middle point
of the chord QP, and P is the mid-point of the arc QPj show that
when Q{=)Pf P(=)P-
EZSRCISE8.
1. Find the parabola y = Ax^ -f -^^ H~ ^ which has the same curvature as a
given curve y = /(x) at a given point jr, y.
Y = /(x) + (X - x)/'{x) H- «^ - xfr{x).
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 is oo 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 P, and therefore does not cross the curve at such a point.
At a max. or min. value of P we have D^P^ = o. Differentiating (8), § 100, and
solving, we find for the curve
Computing D%y^ 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
^ ^ (g' sin' <» + *• cos' »)* ^ I'l , >'\ «„,^
ab \a* b^J
<P being the eccentric angle.
136 APPLICATIONS TO GEOMETRY. [Ch. XII.
7. jr -\- y* z^ i^ is satisfied by jr = « cos* 0, >' = a sin' 0. Show that
J? = — 3(flXK)^
8. Show that the radius of curvature of <^ = sec {x/a) is
^ = rt sec (x/a),
9. The coordinates of a point on a curve are
jf = r sin 2/(1 -|- cos 2/), y ■= c cos 2/(1 — cos 2/);
show that ^ = 4r cos 3/.
10. Find A' for jr* = ayK
11. Show that, yfh&Tiy = sin jr is a maximum, ^ = i.
12. Find the center and radius of curvature of xy = a\
a = (2^ ^- y^)/2x, /? = (x« + ay»)/2r, ^ = (jr« + ;/»)V2««.
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 jTj, y^ and -r, y are
The ordinate of tlieir intersection is
dx dx^
which takes the illusory form 0/0 for jTj = x.
\Vhen evaluated in the usual way, we have, when x^{=)x,
^ — y -r y,t »
which is the ordinate of the center of curvature.
Substitution of K — ^ in the equation of the normal gives X as the abscissa of
the center of curvature.
14. Find the radius of curvature at the origin for
2Jr» 4- 3xy — 4y* -f jr» — 6y = o.
Using Newton's method,
-*j 2y 2
15. Find the radius of curvature at the maximum ordinate of ^ = r-^"^**.
What is the order of contact of the circle of curvature ?
16. If fipy B) ■= o is the polar equation to any curve, show that at any point
/9, 6 the radius of curvature is
- p« + 2p-' - pp"'
where for brevity we write p' ^ ^tfp» p" ^ ^Jp«
This follows immediately from substituting for Dy and L^y, (i) and (2\ § 56,
in (8), § 100.
17. Show that if />» = i, «' = D^j «" = Z>Ji/, the value of the radius in
Ex. 16 becomes
■«* ^z —^ rrr-«
Art. ioi.] CONTACT AND CURVATURE. 137
18. Since at a point of inflexion y = o, we have there J^ = oa . Therefore the
inflexion condition for a polar curve is, as found before, u -^ t/* = o,
19. If p = tfO, show that i? = 0(1 + e«)V(2 + 6*).
20. K p = «•, then ^ = p[i -f (log «)«]*.
21. If p = 29 — II cos 2O, ^ r= 00 at cos 2O = ^.
22. Show that a = \a6, for p = a sin 6$, at the origin.
23. Find the radius of curvature for the hyperbola
jt«/a» -^y^ = I.
24. Find the radius of curvature of:
The circle p sr a sin 0; the lemniscate p* = a' cos 2O; the logarithmic
spiral p = g^; the trisectrix p = 2a cos 0 — a.
25. If /? is the radius of curvature of /{x, y) = o, show that
(^ + ^^.'j^
regardless of the independent variable.
Difiierentiate the equation of the circle of curvature,
J^^{x-ay^{y- bf,
. •. O = (jc — tf )d[r -f- (^ — ^yfyi
o = <i«« -f- (jc — tfy«jc -f <^2 -^. (^ — ^y^.
The elimination of j? — a and y — b gives the result
CHAPTER XIII.
ENVELOPES.
102. li/{x, ^) = o is the equation of a certain line contaiDing a
constant or, then we can implicitly indicate that the position of this
cuive depends on the value of a hy including it in the functional
symbol, thus:
A^.J', a) = o.
If we change a by substituting for it another number a^ , we get
another curve,
/{x, y, a,) = o,
which will, in general, intersect the fiist curve.
The arbitrary constant a in /[Xyj/, a) = o is called a parameter.
All the curves obtained by assigning different values to a are said to
belong to the same/amily of curves, of which a is the variable
parameter. Thus
/{x,j>,a)=o (i)
is the equation of a.^jwi^ of curves when we regard a as a variable, and
any curve obtained by assigning a particular value to tr is a particular
member of that family.
Thus, in the figure, let the curves i, a, 3, , . . be the particukr
curves of the family (i), obtained by assigning to a the particular
values a,, a , , . . taken in order.
Fig. 13.
Two curves of this &mily are said to be consecutive when they
correspond to consecutive values of a. The sequence of curves corre-
sponding to a,, a., . , . , as drawn in the figure, inteisect in points
A.S.C....
138
Art. 103.]
ENVELOPES.
139
Illustrations.
The arbitrary constant or parameter being a :
(a), y — ntx •\- a is the fomily of parallel straight lines sloped m to the axis
of jr. Consecutive members of this fsimily do not intersect in the finite part of the
plane.
{b), y = OCX -|- ^ is the family of straight lines passing through the point o, 6.
{c), X cos a -\- y %in a := p is the fa.mily of straight lines tangent to the circle
(d). y = ax 4- b/a is a family of straight lines tangent to a parabola^' = 4^jr,
and
^ = ajc — 2ba — bo^
is the fomily of normals to the same curve.
{e), {x — of + (/ — ^Y = ^* Js the family of circles with center a, b and
variable radii. The curves of i\i*t family do not intersect
(/"). jr* -J- ^» — iccx -f- ^ = o is the family of circles with radius r 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.
and /{x, y, a,) = o
—If
(0
(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'j(=)a all points on curve (2) converge
to corresponding points on (i), and in the limit curve (2) passes over
into curve (i) and they have an infinite number of points in common.
Therefore the attempt to determine the limiting position of the point
Xy y oi intersection of (i) and (2), by solving (i) and (2) for the
coordinates and then making a^{=)ay leads to indeterminate forms.
We shall proceed to find the limit to which converges the point
x,y of intersection of (i) and (2),
by finding a third line which also
passes through their intersection, and
which does not coincide with (i)
when a^(=)a.
Assign to X and y the numbers a
and d, the coordinates of the inter-
section of (i) and (2), and let a be
a variable number. Then /(a, d, a)
is a function of the single variable a,
and we have, by the law of the mean,
y
nr/
\
{iV
w
0
X
Fig. 24.
/{a, 5, a^) -/[a, h, a) = (a^ - «)/!(«, h M)f
where /i is some number between a^ and a.
But, a, b being on (i) and (2), we have
/{a, by a^ = o and /{a, b, a) = o.
Therefore
(3)
(4)
I40 APPLICATIONS TO GEOMETRY. [Ch. XIII.
For the particular value ^ assigned in (3) we havey^(^,x /*) = o
as the equation to some curve passing through the intersection of (i)
and (2), in virtue of equation (4). We do not know the number ^
in (3) and (4), since all we know about it is that it lies between a
and a^.
But, whatever be the number // satisfying (4), we know that the
curve
/l(-^» >', >w) = o (5)
passes through the intersection of (i) and (2). Now, when flrj(=)ar,
then /i(=)a. If, therefore, when arj(=)a, the two curves (i) and
(2) intersect in a point which converges to a fixed point as a limit,
then (5) becomes
/^{x, y, a) = o, (6)
the equation to a curve which passes through the limit of the inter-
section of (i) and (2) as (2) converges to (i). Moreover, (6), being
a curve distinct from (i), has in general a definite intersection with (i).
If, between the equations
/{x,y,a)=o, (I)
/:(x, y, a) = o^ (6)
the variable parameter a be eliminated, we obtain the locus
i:{x,y) = o (7)
of all points in which the consecutive curves of the ^mily /{Xy yy a)
= o intersect as a varies continuously.
The curve (7) is called the envelope'^ of the family (i).
• Illustration of the Envelope.
As the parameter a varies continuously, the curve /(x, ^, or) = o sweeps over
or generates a certain portion of the surface of the plane xOy^ 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 xOy.
104. The envelope, E(x, y) = o, is tangent to each member of
the family /i;^,^, a) = o which it envelops.
We are not prepared to give a rigorous proof of this statement
now. This prouf 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)y {B), (C) be three contiguous curves of the family, (A)
* Strictly speaking, the equation of the envelope is the equation gotten by
equating to o that factor of £(Xf y) which occurs only once in £(x, y). See
Chapter XXXIX.
Art. 105.]
ENVELOPES.
141
and (C) intersecting the fixed curve (B) in points P and Q re-
spectively. When (-<4) and (C) converge to coincidence with (-ff),
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 (B) and the envelope.
EXAMPLES.
The variable parameter being a, find the envelopes of the following curve
families:
1. ofcosa-h^sina— / = o= /(x, y, a).
/^ = — X sin or -j- A' cos a. Square and add. Hence
3^ -\- y^ -=. fiy a circle with radius /.
2. Envelope the family f m y — ax — b/a = o.
/|[ = — jf -f b/a\ .-. a = yb/x. Hence ^* = 4/bx.
3. Envelope the family /my — ax -|- 2ba + ba*.
/^ = — jf -|- 2^ -|- ^ba*. ,\ a^ = {x — 2b)/$b, Hence
2yyH = 4(jc — 2^)».
4. Find the envelope of (jt cos a)/ a -f- (y sin a)/b = i.
5. Find the envelopes of y =: ax -^ ^a*a* ± b*. [x^/a* ± y*/d^ = »•]
6. Envelope the family x* -^-y* — 2ax = r*.
105. Envelopes when there are Two Connected Parameters.
Let i/>{x, y, a, P) =: o (i)
be the equation to a curve, involving two arbitrary parameters a and
p which are related by the condition
^(«» P) = o- (2)
I. When we can solve (2) with respect to a or fi and substitute
in (i), we reduce that equation to that of a family with one parameter.
The envelope is then found as before.
i4a
APPLICATIONS TO GEOMETRY.
[Ch. XIIL
EXAMPLE.
Find the envelope of the ellipses
(0
when a -{- fi = c.
We have /5 = r — a.
x' y*
Therefore
a^ ~^ u — a)» ~
)'
= I.
Differentiating with respect to a, and
solving fur a.
ex
I
cz =
^ + r
and fi=
cy
i
which substituted in ( i) give
Fig. 26.
\
^y^ = c\
n. Otherwise, when it is inconvenient to solve (2), it is generally
simpler to proceed as follows :
Lctx,y be constant, and differentiate (i) and (2) with respect to
any one of the parameters, say /?. Eliminate a, /3 and a' = da/d/3,
between the four equations.
(pi'XyJ^, a, /3) = 0, (i)
<pfi{x, y, a, /3) = o, (2)
f/.ia, IS) = o, (3)
i^p{or> fi) = o. (4)
The result is the envelope E(x, y) = o.
For example, take the same question proposed in L
We hare for (i), (2), (3), (4),
:;-. + 4 = ».
a*
P"
(0
(2)
(3)
(4)
a' -f I = o.
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. L-^ +>' = ^ •]
2. Find the envelope of the ellipses
^+5i = *
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, [x* H-^* = A]
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/P = i when or* -|- fi^ = ^.
n « m "* n
6. Show that the envelope of xjl -\-y/m = i, where l/a -|- m[b = i is the
parabola (Jf/a)* + (.V/^)* = i-
7. From a point Pon the hypothenuse of a right-angled triangle, perpendiculars
/W, PN^x^ drawn to the sides; find the envelope of 3ie line MN.
8. Find the envelope of the circles on the central radii of an ellipse as diameters.
9. Find the envelope of^' = ^OLx ■\- a*. \}^ + 27** = 0.]
10. Find the envelope of the parabola y^ = oi{x — a). [4^* = jr*.]
11. Find the envelope of a series of circles whose centers are on Ox and radii
proportional to their distances from O,
12. The envelope of the lines x cos 30: + ^ sin 3a = tf(cos 2^)^ is the
lemniscate (jc" -j- .y')* = <i\^ — >'*)•
13. Find the envelope of the circles whose diameters are the double ordinates
of the parabola^* = ^x, [y^ = 4a(a -f- x).]
14. Find the envelope of the circles passing through the origin, whose centers
are on y^ = 4ax. [{x -\- 2a)y^ -|- jc* = o.J
15. Find the envelope of x^/o^ -f y^/p^ = i, when a* + /P = £^.
[{X ± yf = ^.]
16. Circles through O with centers on x^ — y* =. a^ are enveloped by the
lemniscate (jr* -f y^f = 4a\x^ — y*).
17. Show that the envelope of
La* 4- 2Ma -{- JV =z o,
in which Z, Aff N are functions of x and y, and a is a variable parameter, is
LN = M\
18. In Ex. 17 show that if L, M, JV^ltc linear functions of x and 7, the envelope
is a conic tangent to Z = o, iV = o and having Af z= o for chord of contact.
Differentiate LN^ — Af* = o with respect to jt,
.-. L'N-\- N'L-zMAi',
At the intersection of Z = o and J/ = o we have L'N = o; and since there
yv ptf o, we have L* = o. The Dxy from this is the slope of the tangent to the
envelope. Hence Z' = o is the tangent at the intersection of Z = ^ = o to the
envelope, etc.
CHAPTER XIV.
INVOLUTE AND EVOLUTE.
io6. Definition. — When the point of contact, /*, of the circle of
curvature of a given curve moves along the curve, the center of curva-
ture, C, describes a curve called the evoluie 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 involute of the evolute.
107. There are two common methods of finding the evolute of a
given curve.
I* If 0(jc, _y) = o is the equation of the given curve, and a, fi are
the coordinates of the center of curvature, then we have, § 100, (2),
If we eliminate x andj^ from these two equations and the equation
to the curve, 0(^, >») = o, we leave a and fi 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.
1. Find the evolute of the parabola y^ = \px.
We have / =/*jr"^; y = — ip^x"^. Hence
a - X = 2(j: + /). /J - J. = - 2(/"M + A*).
. % a = 2/ -f 3j:, 6 — — 2/"M.
Eliminating jr, we have for the equation to the evolute (g 112, Ex. 17, Fig. 44)
4(cr - 2/)« = 27/)^.
2. To find the evolute of the ellipse jc*/fl» -f ^V^' = '•
We can differentiate directly, or use the eccentric angle
X z= a cos 0, y ■=. b sin 0, and find
/ = - h^x/a^y, y = - ^v^y.
Hence (aa)' + {b^)^ = (a« - ^)*. (Fig. 43.)
M4
Art. io8.] INVOLUTE AND EVOLUTE. 145
U. The evolute of a given curve yT[ji^, >') = o is the envelope of
the normals to the curve.
The equation to the normal io/=. o at x,y\s
X-X + {F-j')y' = o. (i)
But J/ andy are functions of x, from the equation /= o to the
curve. Therefore x is a, parameter in (i), by varying which we
get the system or family of normals. Hence the required locus is to
be found by differentiating (i) with respect to x, keeping X, incon-
stant. Thus
^i + {F^yy^-y2 = o. (2)
Eliminating x between (i) and (2), we have
y-y="-^ and x-x=-y^--^/-,
in which A*" and -Fare' the coordinates of the center of curvature,
§ 100, (2). This proves the statement.
EXAMPLES.
1. Find the evolute of the parabola ^' =r 4/jr.
The equation to the normal is
y = ax — 2pa — pa*. (I)
. •. o = jt -- 2/ — Zpo^'
... «=(^L^^\«
which substituted in (i) gives as before in I, 4(x — 2/>)* = zjpy*,
2. Find the evolute of the ellipse jif*/a* -\- y*/i^ = i.
Taking the equation to the normal
ax s^Q a -^ by CSC a = a^ — ^',
. •. ax sec a tan a -\' by esc a cot a = o.
Hence tan a = — (by/axy, which leads to the same result as in I,
{ax)^ + {by)^ = («« - b^\
io8. The normal to a curve Is a tangent to the evolute.
Let (x-^ay+{y- fiy^I^ (i)
be the equation of the circle of curvature at Xy y. Then, letting x,y
vary on the circle, R remaining constant, we have, on differentiation
with respect to jt,
x-a+{y- fty-o, (2)
I +y' + (y - I^V = o. (3)
Now let x,y vary along the curve, R being variable. The num-
bers a and /i are also functions of ^. Differentiate (2), which is the
equation to the normal to the curve at x^ y, with respect to x.
.-. i+y* + o-/»K-^'-/?y = o, (4)
y
146 APPLICATIONS TO GEOMETRY. [Ch. XIV.
Subtract (4) from (3).
da dp dy __
d6 dx
da dy^
which proves the statement.
EXERCISES.
1. Find the centre of curvature of ^ = a*x.
These equations are the equations of the evolute, a and /9 being expressed in
tenns of ,v, a third variable.
2. Find the coordinates of the center of curvature ^f the catenary, Fig. 38,
a = * — — ^y^ — ai^^ fi — 2y,
a
3. Find the center of curvature and the evolute of the hypocycloid,
j^ -\.y^^ a*.
a=ix + zj^y\ p=y + :^Jy^', {a -j- /3)^ + (a - fi)^ = 2aK
4. In the equilateral hyperbola 2xy = a^,
(a -h /!0* - (a - p} = 2a\
5. In the parabola jr* -^y^ = a*, a + ft =• 3(jf +^).
CHAPTER XV.
EXAMPLES OF CURVE TRACING.
I09« 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.
1X0. 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 =y, D^y = v".
1. Equation of the tangent:
{F-jy)=.(_X-xy.
2. Equation of the normal :
{y-y)x'=-{x-x).
3. Subtangent and subnormal:
4. Tangent-length and normal -length :
/ =^vr+y=^, n =yVi +y\ •
5. Tangent intercepts on the axes:
Xi = X —yy'-K Vi =y - x/.
6. Perpendicular from origin on the tangent :
7. Radius of curvature :
8. Coordinates of center of curvature:
a ■- X -- y — yy — , p = y -{ -77—.
147
148 APPLICATIONS TO GEOMETRY. [Ch. XV.
n. Polar Coordinates. D$p = p', Dip = p". up = i,
1. Angle between the tangent and radius vector:
P
tan ^ = -4-.
P
2. Angle between the tangent and the initial line:
* ^ p + P' tan 6>
tan 0 = -7 ^-— 2j.
^ p' — p tan 6^
3. Subtangent and subnormal :
o* d$ dp
^'- f/ - du' ^'-^ - dr.
4. Tangent-length and normal-length:
' = 44//^ + /"'*' « = 4/p* + P^-
5. Perpendicular from the origin on- the tangent :
P = -7—- . -^ = «* -f «".
6. Radius of curvature :
p« -I- 2P'-* - pp'' '
III. 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 = <l>(x) ox X — t/.^{y)
as its equation, in which either <p(x) or ip(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
andy the curvature at the point.
A regular method of procedure for tracing a curve is:
I. Examine the equation for symmetry.
If the equation is unchanged when the sign of ^^ 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. m.]
EXAMPLES OF CURVE TRACING.
149
If the equation is unchanged when x and^ are interchanged, the
curve is symmetrical with respect to the line^^ = .r.
If the equation is unchanged when x and y are interchanged and
the signs of both x and y changed, the curve is symmetrica] with
respect to or -j-^' = o.
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 jc = o, ^ = o satisfy the equation, the curve passes through
the origin. Put x =z 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 = jr*. The curve is symmetrical with respect
to Oy, It passes through O and cuts neither axis else-
where. Since y •=. 2x is o at O^ Ox is tangent.
Also, y is positive as x continually increases from o, \ V
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 = 2jr' — 3x -f 4.
/ = 4Jf - 3. /' = 4.
The curve is always convex. It has a minimum
ordinate, ^ = 2}, at j: = }. Its slope db according as
X < ». It cuts Cy at ^ = -f 4, and neither axis else-
where. It is symmetrical with respect to the line x = 4.
mon parabola. (Fig. 28.)
0
Fig. 27.
The curve is the com-
FiG. 28.
Fio. 29.
3. Trace jr = 3 — 2j^ — 3y'. Here x is a one-valued function of y.
DyX = — 2 — 6>', Z?Jjr = — 6. X is a maximum at ^ = — i, when x = 3^.
The curve cuts Ox at ^ = o, x = 3, and Oy 2X y z=. -\- 0.78, ^ = — 1.44. It is
everywhere concave to Oy, x continually diminishes from its maximum value,
ISO
APPLICATIONS TO GEOMETRY.
[Ch. XV.
-»
and the curve has no asymptote at a finite distance. It is as before the common
parabola. (Fig. 29.)
4. Trace the cuHcal parabola y •=. 3^, Here
y = yfl, y = tx. 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 -f- when x is -f-» &n<l >' is — when x is — .
The curve lies in the first and thii^ quadrants. In the
first quadrant it is everywhere convex, in the third every-
where concave to Ox. It changes its curvature at the
origin where there is concavo-convex inflexion. There
are no asymptotes and the absolute value of ^^ is oo when
that of JT is 00 . (Fig. 30.) (Read foot-note, p. 164.)
5. Trace the semi.cubical parabola y* = ^r*. Ox is
an axis of symmetry. The origin is on the curve. When
j: is — , ^ is imaginary and the curve does not exist in the
IG. 30. plane to the left of Oy. x = ^i is a one- valued function
of ^. DyX = ly — i shows the slope 00 at C? with respect to Oy, and this slope is
± fory ± respectively. D^ = — fv—t is always negative, or the curve is con-
cave to Oy. There are no asymptotes and Xy y become 00 together. (Fig 31.)
Fig. 31,
Fig. ^2.
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 = o, positive and infinite for x = -f- 00 , and is o when x = i where it cuts the
axis Ox. The derivative y = i/x is infinite for x = o, which line is an asymp.
tote, y is always positive and decreases as x increases. The ordinate continually
increases, y r= — jr-» is always — , hence
the curve is everywhere concave and as in V
the figure. (Fig. 32.)
7. Trace the exponential curve y = e*.
y is always -|-, by convention e* is -\-.
^ = -f- 00 when x = -j- 00 ; y(=y^ when
X = — 00 . Also y •=. y" = e*. The curve
is always convex and increasing, and since
y(=)o when jp = — 00 , t?x is an asymptote.
When jf = o, >' = i, where the curve cuts
Oy. If we agree with some authors that y
has negative values for x = (2« -|- i)/2w,
m and n being integers, then there will be a corresponding series of points
representing the function lying below Ox on a curve represented by a dotted
Fig. 33.
Art. III.]
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
v=l + x + g + fl + . ..-*«.
The curve y ■= e* i^ identically the same as the curve in Ex. 6 if we inter-
change X and y, (Fig. 33.)
8. Trace the probability curve y — e-^. The ordinate is always -f- ; it has a
maximum at o, i ; it is o when jr is i: 00 . There is a cancavo-convex inflexion at
X ■=. -\- 1/V2 and a convexo-concave inflexion at jr = — i/i^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 cissoid of Diocles, (2^ — y^sfi = ^. The curve has Oy as an
axis of symmetry, and passes through O, and cuts the axes nowhere else. Since
Fig. 35.
^ + ^y = 2<M^, y cannot be negative if a is positive. We find that y z^2a\%
an asymptote in both directions, since x = £ 00 when^^ = 2a.
Again, corresponding to an assigned y^ there are only two equal and opposite
values of x. Therefore, lor an assigned jt, there is only one value of ^. Also,
is ± according as jr is ±, The curve is decreasing for x negative and increasing,
for X positive. To findy at the origin, the above value of y is indeterminate.
But we have directly from the equation to the curve
la ^ y
~y
= 00
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 cusp. By plotting a
sufficient number of points, we find the curve to have the form as drawn in the
figure. (Fig. 35.)
152 APPLICATIONS TO GEOMETRY. [Ch. XV.
10. Trace the witch of Agnesi, y = &i»/(jc* -}- 4j«). Tlie ordinate is always
Fig. 36.
-f-t &nd has a maximum ^ = 2^ , at j: = o. Oy is an axis of symmetry, and Ox
an asymptote. There are inflexions at j: = ± 2<i/ 1/3. (Fig. 36.)
11. Trace the cuHc a^y = ^x* — ax^ -\- 2tf", in which a is positive. There is
a maximum ^ = 2<j at ;r = o ; and a minimum >/ = }<2, at jr = 2<2. An inflexion
occurs at jr = a. For jt < <i the curve is concave, and for jt > a convex. There
are no asymptotes. The curve crosses Ox between jt = — a and x = — 20.
Also, ^ = ± 00 when jr = ± 00 . (Fig. 37,)
Fic. 37. Fig. 38.
12. Trace the catenary, y =^ \a. \^ ^ f'aj ^ in which a is a positive constant.
The curve is the form of a heavy flexible inextensible chain hung by its ends.
The ordinate ^ is a minimum and equal to a when x — o, and is -|- for all values
of jr. The curve is convex everywhere. >/ = -|- 00 when jc = ± 00 , and there are
no asymptotes. The slope continually increases with jr. (Fig. 38.)
13. Trace the cubical parabola jr* -= y\y — a)^ where a is positive.
Since jr = ± ^ 4/y — fl,
the point o, o is on the curve. But no other point
in the neighborhood of the origin is on the curve^
since for such values of y^ x is imaginary. The
origin is therefore a remarkable ptjint, it is an iso-
lated point of the curve, and such points are called
conjugate points. For each value of ^ greater than
a there are two equal and opposite values of jr.
The curve is symmetrical with respect to Oy. The
ordinate_>' is a minimum at jr =r o, where the tan-
o; gent is horizontal, y = o gives inflexions at
»» "^ — * .- tf', which for jr -f- is
convexo-concave and for jr — is concavo-convex. There is no asymptote, and
^=-1-00 for jr=±oo. (Fig. 39.)
y =
Art. III.]
EXAMPLES OF CURVE TRACING.
153
14. Trace y = {jfl — i)'.. The cu-ve lies above Ox and has Oy for 2ii axis
y
0
Fig. 40.
of symmetxy. y has a maximum at or = o, and minima at jr = ± i. There are
inflexions at x = ± 1/ 4/3^
The infinite branches have no asymptote. (Fig. 40.)
15. Trace the curve ^
'h'^
The ordinate has the limit e when
jr = ± 00 . This is the important
limit on which differentiation was
founded, y has the limit i when x = o
and continually increases with x. For
— I < j: < o the curve does not exist.
The point o, I is what is called a stop
point, the branch ending abruptly
there. For x < — I, and converging
to — I, y is greater than e and is 00 .
As X decreases to — so , the curve
decreases continually and becomes
asymptotic to^ = ^. (Fig. 41.)
EXERCISES.
1. Trace the ciirves ^ = sin x, y =. cos x.
2. Trace y = tan jr, ^ = cot jr.
3. Trace ^ = sec x, y =. esc x.
4. Trace y = vers x = I ~ cos x.
I
5. Trace y ^ e ■* .
6. Trace the curves
xy = I, (X - i){y - 2) = 3, y(x - l){x - 2)
7. Trace the curve y{x — i)(x — 2) = (x — 3)(x — 4).
8. Trace y{a^ + Jf*) = aHa — x).
9. Trace x»(y - a) = a* — xyK
10. Trace tf«x = y{x — af.
Tl. Trace ^ = x^(2a — x).
12. Trace (x* -|- 4) = yx*.
= I.
154
APPLICATIONS TO GEOMETRY.
[Ch. XV.
13. Trace yx{i — jcjy = i — 5*.
14. Trace the quadratrix y -=■ x tan ~ • .
2a
15. Trace the curve y = sin {n sin x).
16. Trace y = (ix — a)^{x — af.
112. Implicit Functions. — In general, when the equation to a
curve is given in the implicit form J\x, y) = o, 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 Book II. The ordi nates 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,
The curve is symmetrical with respect to O, Ox, and Oy, There are two
equal and opposite values ciy to each x, and two of jr to each^', for either variable
Vi
Fig. 42.
less than a. The curve does not exist for values ofxory greater than a. We have
in the first quadrant
/ = -
and the curve is tangent to Oj: at x = a, and to Oy Sit x = o, 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)* + (iy)* = (fl« - ^*
in the same way as above. (Fig. 43. )
Art. 112.]
EXAMPLES OF CURVE TRACING.
^SS
Show by inspection that four normals can be drawn to the ellipse from any point
inside the evolute.
From what points can i, 2, or 3 normals be drawn ? y
Fig. 44.
Fig. 43.
Fig. 45.
17. Trace the parabola ^^^ = 4/jr, and its evolute, 4(x — 2/)" = 27/^y'. 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 — jk«)» = jfi.
There are two branches,
The first continually increases as x increases from o. The second increases,
attains a maximum, and then descends indefinitely, crossing Ox a.tx := i. Both
branches are tangent to Ox 2it O since
y = 2x ± |jr
is o when * = o. The curve does not exist in the plane to the left of Oy. Ex-
amine for asymptotes. Find the inflexion and the maximum ordinate. The origin
is a sinf^lar point called a cusp of the second species, (Fig. 45.)
19. Trace in the same way the curve
x^ — 2ajfly — cucy^ -f a^y^ = o.
20. Trace the curve ^» rr: (jf -f- i)jc*.
^ is a two- valued function of ;r,
y z=z ± x^x-J^ I.
Ox is an axis of symmetry.
The curve passes through the origin in two branches, pj^,^ .^^
y = ^x i^x^ I, y =z --X 4/^-f- I-
The curve does not exist in the plane to the left of x = — I. Between — i
iS6
APPLICATIONS TO GEOMETRY.
[Ch. XV.
and o the ordinate is finite, having a maximum and a minimum. We have for the
slopes of the two branches passing through C? at jt = o,
-/-=± / V'x+i = ±i.
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 nvd^. In this curve the origin is a node.
21. Trace the curve (dx — cyf = (j: — <j)*.
Clearly, x = tf, y = ad/c,
is on the curve. But these values make the deriva-
tive y indeterminate. Differentiate the equation
twice.
... (^•_ rv')« - (^x - o')0'" - M^ -af = 0,
and at the point jt = a , y z= ah/c^
(b - cyy = o,
gives y z= b/c. Since y is imaginary when jr < a»
Fig. 47. J ^ . I ., rr
' and y -n^ -x ± — y{x — a)*,
C €
the curve is as in the figure. The point a, ab/Cy is a cusp oi the Jirsf species, (Fig. 47. )
22. Trace the curve 4v* = 4jc* -f- izx^ + gx.
Fig. 48.
23. Trace the lemniscate,
jr« -f ^'^ = -f fl 4/jr« — y^
shows that^' cannot be greater than x and only equal tojr when they are both o at
the origin. The curve is symmetrical with respect to (?, Ox^ Oy, Also,
+^-^|'-(5^
and since y ^x^ we have, when j: = o, _y = o,
/
1=..,
(See 111. (2), g 79.)
which are the slopes of the two branches of the curve passing through the origin.
Again,
^, ^x^a^-2(x^Jry^)
y a^ -h 2(jr> -\- y^)
i\'
Art. 113.] EXAMPLES OF CURVE TRACING. 157
In the first quadrant y is + from j: = o to the point determined by
where it changes sign, giving y a maximum, and y decreases until y' =z ao at
X = a, y rz o.
Being symmetrical with respect to the axes the curve is as in the figure. No
part of the curve exists for x > Oj since the equation is of the fourth degree and a
straight line cannot cut the curve in more than four points.
Put y = mxy and plot points on the curve by assigning different values to m.
Thus, in terms of the third variable m^ we have
x= ± a J— -, y=±am -^ -. (Fig. 48.)
I -f- »r I -f- ^
113. General Considerations In Tracing Algebraic Curves. —
The equation of any algebraic curve when rationalized is of the form
of a polynomial of the ;ith degree in x and y. It can always be
written
o = «o + «i + •••+««= ^. (0
where u^ is the constant term (independent of x and y), u^, 1/, ,
etc., are homogeneous functions or polynomials in x,j^ of respective
degrees i, 2, etc.
If u^ = o, the origin is a point on the curve,
(i). To find the tangent at the origin when u^ 7^ o.
When u^ = o, the line j/ = mx intersects the cur\e at O,
Substitute mx for y in the equation to the curve. Then, if
«/j = /j: -f- £v, the equation (i) becomes
{p + ^)x+T, + ... =0, (2)
where the terms 7*,, etc., contain higher powers of x than the first.
Divide the equation (2) by x, which factor accounts for one o root.
Then ]et x =z o, and (2) becomes
p -\- gm =z o, or « = — p/g.
This value of m is the slope of the curve at the origin, since now
the line^' = mx cuts the curve in two coincident points at the origin,
and
u^=px + gy=o
is the equation of the tangent at the origin.
If u^ = o, ttj = o, and u^ = rx^ + sxy -[- /j^.
Then, as before, put mx iorjf and the equation becomes
(r + j« + //„'^)^+7;+... =0, (3)
where the terms T',, etc., contain higher powers of x than 2.
Divide by x\ which accounts for two zero roots of (3) ; in the
result put X = o.
.*. /m^ -{- sm-\- r =1 o (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
u^ = rjfi ^- sxy -f- A^ = o.
158 APPLICATIONS TO GEOMETRY. [Ch. 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 i/, = o, the equation of the three tangents
at 6^is
«, = 0,
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 nih. degree in more
than n points. For, if we put mx -\- b for j/ in 6^ = o, we have an
equation of the «th degree in x for finding the abscissae of the points
of intersection oiy = nix + b and U z=z o.
If now Uy is the term of lowest degree in U, and we put mx for
ymUy then jc*" is a factor and represents r roots equal to o. The
line^' = mx cuts the curve U=. o, r times at the origin, and there-
fore cannot cut it in more than n ^ 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 singular point. Illustrations
have been given of nodes, cusps, and conjugate points.
At a singular point on a curve D^ is indeterminate. Points at
which D^ is determinate and unique are called points of ordinary'
position, or ordinary points.
To find a singular point on a curve <t>[Xyy) = o, differentiate
with respect to x. The result will be
M-^-Ny'-o, (i)
where Jll/and N are functions of x and y. At a singular pointy' is
indeterminate and M-=z o, iV= o. Any pair of values of x^ y satis-
fying the equations
0=0, J/=0, iV=:0
is a singular point. If (i) be differentiated again, we have
/> _|_ Q/ + Rfi + Ny" = o.
At the singular point iV=: o, leaving a quadratic in y' for deter-
mining the slopes of the curve, if the point is a double point. If a
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 or, /?, and
we transform the origin to tht singular point by writing x -{- a,
y + /3 for X and y in the equation to the curve, the construction of
the curve will be simplified as in (i), (2).
Art. 113.]
EXAMPLES OF CURVE TRACING.
159
EXAMPLES.
24. Trace the lemniscate^ Ex. 23.
Here «- = jr' — j/* = o is the equation to the tangents at o, or ^ = ± x, as
before in Ex. 23.
Put y — tnx in the equation and compute a number of points. Clearly m
cannot be greater than i.
25. Trace iht folium of Descartes,
^ -\-y^ — zaxy = a
The equation of the tangents at
origin is yxy = 0, or jf = o, _y = o.
find that
X '\' y -|- tf = o
is the only asymptote. Put y = mx^
y =
then
I -I- ^s* "^ I -|- nfi'
Xy y are finite for o < fw < + ^ •
Fig. 49.
Com-
pute a number of points corresponding to
assigned values of tn. Observe that if we
change m into i/m, x and y interchange
values. The curve is symmetrical with
respect to the line y =. x. In the first
quadrant there is a loop, the fEirthest point
from the origin being x =y ■= ^, Determine the maximum values of x and y for
this loop. For negative values of m we construct the infinite branches above the
asymptote, since ^^ = mx cuts the curve before it does the asymptote. (Fig. 49.)
26. Trace the curve (y — 2)\x — 2)jr = (x — i)*(jc* — 2x — 3).
Examining for singular points, we find
Therefore jt = i, ^ = 2 is a singular point. Transform the origin to this
point by writing x -\- I for x, ^ -|- ^ for >'.
Then the equation becomes
y\x^ — I) =t x\x'^ — 4).
Examining for asymptotes, we find the
asymptotes jc = ± i, y :=, ± x. The
equation to the tangents at O is y
.". y :=. ± 2x. When y =. o\ x
jf = o. The curve is symmetrical with
respect to Oxy Oy, and O, We neeii there-
fore trace it only in the first quadrant, in
X order to draw the whole curve.
The line y = mx cuts the curve in points
whose coordinates are
« = 4^,
= ± 2t
r
These increase continually as m increases
from o to I, and the branch approaches the
asymptote as drawn. The coordinates are
imaginary for i < »i < 2, and when m=: 2;
Fig. 50. X = Oy y = o. As m increases from 2 to
-j- 00 , X and y are real and increasing, and
m = 00 gives jp = ± i,^ = 00 , the curve approaches the asymptote as drawn.
The origin is an inflexional node. (Fig. 50.)
i6o
APPLICATIONS TO GEOMETRY.
[Ch. XV.
27. Trace the curve (x -f 3^^' = Jp(-* — i)(x — 2).
28. Trace the curve a^* = dx* -\- jfi.
29. Trace the dumbbell a*y* = a^x*- — jfi.
30. Show that x* 4-^ = 5<w*^* lias the form given in Fig. 51.
Fig. 51.
Fig. 52.
31. Trace x* = (jr* — y^.
The lowest terms are of third degree. The origin is a triple point. The
tangents there beings = o, ^ = ± jr. (9y is 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 = m{i — «*), y =r ««(i — w*).
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.)
y >. If
Fig. 53. Fig. 54.
32. Trace the curves
y* = ax* — jflf ^ = «• — X*, y*(x — a) = (x — ^)r».
33. Trace the conchoid of Nicomedes,
(x* -\-y*){b — 7)» = flV, when ^ =, <, > a.
34. Trace the curves
^ = (x — i){x ~ 2Xx — 3), a*x = y(b'^ + ^). x^ — y* + 2axy* = O.
35. Show that x*y^ + x* = tf'(x* — y*) consists of two loops and find the form
of the curve.
36. Show that the scarabeus
^x^ 4. v2 + 2tfx)«(x« + y'*) = b\x^ — y*f
has the form given m Fig. 53.
Art. 114.] EXAMPLES OF CURVE TRACING. 161
37. Show that the devil
y* — X* -\- ay* -{- bx^ = o, where a = — 24, ^ = 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 :
(i). Compute values of p corresponding to assigned values of ^^ 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 B which make
p = 00 for the directions of the asymptotes. Place the asymptote in
position by evaluating the limit of f^DJH = — Z?„^, 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 maximum and minimum values of /> and for points
of inflexion.
(6). Examine for periodicity.
115. Inverse Curves. — If /(/o, ^) = o is the polar equation to
any curve, then /{p~\ 6^) = o is the polar equation of the inverse
curve.* We have been accustomed to put p~' = «, so that/'{«, 0) =z o
is the equation of the inverse curve.
1. Show that if Xf jf are the rectangular coordinates of a point on a curve, the
equation to the inverse curve is obtained by substituting
for j: 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 ArchinudeSy p ■= afy. The distance from the pole is
proportional to the angle described by the radius vector, tan ^ = 0. The curve
is tangent to the initial line at O. The intercept PQ between two consecutive
revolutions is constant and equal to 2na, Therefore we need only construct one
turn directly. The dotted line shows the curve for negative values of d, which
» .^— ^.^^.^^__^__^_^_^___^_
* More generally two polar curves are the inverses of each other, when for the
same 6 their radii vectores are connected by p|p, z=. l^, k z=: constant.
l62
APPLICATIONS TO GEOMETRY.
[Ch. XV.
is the same as the heavy line revolved about a perpendicular to the initial line
through a (Fig. 55.)
Fig. 55. - Fig. 56. >
i
39. Trace the eqtUanguiar spiral p ziz ^, We can write the equation
e = /5 log p,
if we prefer, tan ^ = ^, or the angle between the radius and tangent is constant,
p = fl for 6 = o, and p increases as 0 increases. /o(=)o for 6 = — 00 .
The pole O is an asymptotic point. (Fig. 56.)
40, Trace the hyperbolic or reciprocal spiral
pQ = a. The pole O is an asymptotic point.
A line parallel to the initial line at a distance a
above it is an asymptote. For negative values
of G, rotate the curve through tc about a normal
to OA at O. (Fig. 57.)
41. Trace the lemniscate p^ = 2a' cos 2O.
Fig. 57.
42. Trace the conchoid p ■=. a sec 0 ± ^1
or {x^ -f-^'K* - fl)" = ^*-«*.
When a <b, there is a loop; when a = ^, a cusp; when « >^, there are two
points of inflexion. (Fig. 58.)
Fig. 58.
Fig. 59.
Art. 1 15. J
EXAMPLES OF CURVE TRACING.
163
43. Trace the cardioid /^ = tf(i -f- cos 0). The curve is finite and closed,
symmetrical with respect to Ox, p = 2^, a, o, for 0 = o^ ^jr, jr, and
diminishes continually as 0 increases from o to ^r. Also, tan ^ r= — cot 46. As
6(=))r, ^=)ir, or the curve is tangent to Ox at the pole, which point is a cusp.
The rectangular equation is
x» -f-y - flx = + a i^^ -\-y\ (Fig. 59.)
44. Trace the ihrei'Uaved clover p r=. a cos 39,
45. Trace the curves :
(l). p =7 0 sin 26, p=acos20.
(2). p = 0 sin 3O, p -= a sin 4O,
(3). p = a sec* 4^, p ^ a sec 6.
(4). p = a sin 6, p = a sin' |d.
46. Trace the curve p(©» — I) = aG*.
47. Trace p — a vers 0 and p = a(i — tan 6).
48. Trace the evolutes o£ 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.
Fig. 60.
(I). Let the radius of the rolling circle MPL be a, the point /'the generating
point, iff the point of contact with the fixed straight line Ox which is called the
base. Take AfO equal to the arc AfP; then O is the position of the generating
point when in contact with the base. Let O be the origin and jr, y the coordinates
of P, Z PCM = e.
Then we have
X z=i OM -^ NM =r. tf(0 - sin 0), ;/ = ^'A^ = fl(i - cos 0).
The coordinates are then given in terms of the angle 0 through which the rolling
circle has turned. OA = ^^ca is called the base of one arch of the cycloid. The
highest point V is called the vertex. Eliminating % we have the rectangular equation
X zs:. a cos
a — y f
1-1 :: — ^
2,ay
(Fig. 60.)
(2). To find the equations to the
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
X = aO -|- a sin 0, y ■= a — a cos 0.
Eliminating 0 for the rectangular
equation,
jr = <2 cos—* ^ -f- -f^2<jy — ^*.
(Fig. 61)
The cycloid is one of the most important curves.
164
APPLICATIONS OF GEOMETRY.
[Ch. XV.
50. The Trochoids, 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.
V
p:^
/
\
P
\
T
\
V
y A
r /r
M
Fig. 62.
Let CAf = a, CP = /, GP' = /', Z MCP = 0.
Then x = OJST = aO — / sin 0, y = PN = « — / sin fi,
for a point P on the hypotrochoid PV, Replacing / by /', 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
of the fixed circle. (Fig. 63.)
Let O be the center of the fixed circle of
radius a, and C the center of the rolling circle
of radius b^ and P the tracing point. Then
with the notations as figured, we have
arc AM = arc PM^ or aO = *^. Hence
X - ON z^ OL - NLy
= (tf 4- 3) cos 0 — ^ cos (0 + 0),
= (a -f- ^) cos 6 — ^ cos ^-T— 0;
y =z PN= CL^ CK=: {a -f- ^) sin 0 - ^ sin (6 -f- 0),
= (tf -f- ^) sin 0 — 3 sin — y-B,
for the coordinates of the epicycloid. For the hypocycloid change the sign of b.
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. Dly = + or
D^x = -f- means convexity with respect to O^ or Oy respectively. This is the
equivalent of viewing the curve from the end of the ordinate at — 00 > 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.
ii6. Definition. — The product of a difference of the variable
x^ — Aj into the value of the function /{x) taken anywhere irf the
interval (or, , x^) is called an element.
In symbols, if z is either of the numbers x^ or x^, or any assigned
number between x^ and x^, the product *
(^. - *.)/W
is the element corresponding to the interval {x^^ x^.
Geometrical Illustration.
If^ z=,/{pc) is represented by the curve AB in any interval {a, 6),
and jfj, jf, are any two values of x in
{a, d)y then the element corresponding
to (aTj , x^ is represented by the area
of any rectangle x^M^Mx^^ whose
base is the interval x^ — x^^ arid alti-
tude is the ordinate zZ to any point
on the curve segment PJ*^.
117. Definition. — The integral oi
a function f(pc) corresponding to an "o
assigned interval (a, U) of the variable
is defined as follows:
Divide (a, b) into n partial or sub-intervals (a, ^j), {x^^ :rj,
. . . , (^«_, , ^n-^i {p^n^x 9 ^)» by interpolating between a and d the
numbers .v, , . . . , x^_^ taken in order from a to 6. And for con-
tinuity of expression let x^ = a, a:^ s 3.
The integral of a function is the /liwiy of the sum of the elements
corresponding to the n sub-intervals, when the number of these sub-
intervals is increased indefinitely and at the same time eacli sub-
interval converges to zero.
165
i66
PRINCIPLES OF THE INTEGRAL CALCULUS. [Cii. XVL
In symbols, we have for the integral oi/{x) corresponding to the
interval {a, d),
•*r(=)-«'r-i r»*«
'r— I »
In which z^, is either x^ , x^^^ or some number between x^ and x,
or as we say, briefly, some number ^the interval {x^^, x^). At
the same time that « = oo we must have x^ — jtr^_j(=)o.
Geometrical Illustration.
If y •= /[x) is represented by a continuous and one-valued ordinate to a curve,
then the integral of /{x) for the interval (a, d) is represented by the area of the
surface bounded by the curve, the jr-axis, and the ordinates at a and d.
2fn B
Fig. 65.
For,, any elementary area, such as (x, — Jr,)/(«j), lies between the areas of the
rectangles x^PM^^ and xJV^P^^ constructed on the subinterval (jc, , jr.), or is
equal to one of them, according as «, = -^ji f ^s = -^s* Also, the corresponding area
x^P^P^^ bounded by the curve /^/j, Ux, and the ordinates at jc, and x^ lies
between the areas of the same rectangles, in virtue of the continuity of f(x\ when
Xj — x^ is made su6Eiciently small.
Hence the sum of the integral elements and the fixed area of the curve lie
between the sum of the rectangular areas
and
(0
(2)
Let RQ be not greater than the greatest of the subintervals into which («, //) is
divided. The difference between the areas (i) and (2) is not greater than the area
of the rectangle BDQRy whose base is RQ and whose altitude BR is equal to the
difference y(^) — JXa) and to the sum of the altitudes of N.M^ , N^^ » • • • » NnM^,
This rectangle BQ has the limit o, since each subinterval has the limit o; and so
also has RQ^ while its altitude is finite and constant, or does not change with n.
Consequently the areas (i) 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 off{x) for (tf, b) is equal to the area of the curve, as enun-
ciated.
Art. ii8.] ON THE INTEGRAL OF A FUNCTION. 167
ii8. Evaluation of tiie 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 this
interval (a, b) 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 limii defined as the integral, in
§ 117, is called integration.
In evaluating the limit
£^{x^- Ar^,l/(«,.), x^ - x^_\ = ) o.
we are said to integrate the function yfrom a = 3;^ to 3 = x^. The
numbers a and h are called the boundaries or limits of the integration
or integral. The lesser of the numbers a and h is called the inferior^
the greater the superior^ limit of the integration. f
In the differentiation of the elementary functions
j;*, ff*, log x^ sin jc,
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.
f The word limit as here employed does not in any sense have the technical
meaning limit of a variable as heretofore defined. It is an unfortunate use of th^
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.
1 68
PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI.
The systematizing of the artifices of integration is the object of
this part of the text.
119. Primitive and Derivatiye. — If we have two functions F(x)
2Lnd/[x), so related th2Lt/[x) is the derwa/we of F{x), then F{x) is
called a primiiive oi/(x). The indefinite article is used and F(x) is
called a primitive oiJ\x)y because if
DFIx) =/(x),
then also we have
D[F{x) + c] =yi;^),
where C is any assigned constant.
Any one of the functions
fix) + c
obtained by assigning the constant (7, is a primitive of /{x). The
primitive of /(x) is the family of functions containing the arbitrar}'
parameter C,
Geometrical Illustration.
The two curves
^ = /(x)+Ci. (I)
y^F{x)-\.C^, (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
Fig. 66.
stant. Each curve represents a primitive of /(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).
Lety^jf) be a continuous function in (a, A"), and let F{x) be a
primitive oi/{x).
Let x^ = fl, ^„ = X, Interpolate the numbers jr^, . . . , x\_^
between a and X in the interval (a, A"), in order from a to A', sub-
dividing the interval {a, X) into the n subintervals
V^*0» *^l/» \^\^ -^j/j • • • I (•^it— 2> '^n—i)* (-^fi— I » ^n)*
Art. I20.] ON THE INTEGRAL OF A FUNCTION. 169
We have the sum
n
2 {^r — ^r^i) = AT — a,
whatever be n.
Since y^Ji;) is the derivative of I%x),
F'{x) =/{x).
By the law of the mean value applied to each of the subintervals,
we have the n equations
F{X) - IXx^,) = (x, - A-._,)/(4-.),
J^x^,) - I\x^,) = {x^, - ^»_M*"«-,),
jr{x,) - f\x,) = (x, - x,)AS,).
I\x,) - F{a) = K - x,)Ae,).
Adding, we have
F\X) - F{a) = 2 (Av - x^.)A5.). (0
r-i
in which £^ is some particular number in the interval (j;^, ^^,).
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 «. The equation is true whatever be the integer «, and when
n = 00 . The sum on the right remains constant as we increase n,
and being finite when « = 00 ,
r-n
I\X) - F{a) = £2(x,- x,-:\A^r).
MaOO ral
Now let Zr be any number whatever ^the subinterval {x^, -^r-O*
fdr each subinterval. Then
where 6,.(=)o, when x^ — a:^,(=)o, by reason of the continuity of
Therefore
r«l
= 2{X^ - X^xVl^r) + 2{Xr - ^r-,)€r.
I X
Let 6 be the greatest, in absolute value, of the numbers
€,,...,€„. Then
n n
I I
the limit of which is o, when » = 00 ; provided each subinterval
x^ - x^_,{=)o
when « = 00 .
1 70 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI.
Therefore, when » = 00 , and at the same time each subinterval
x^ — Av_,(=)o, we have
z^ being any number of the interval (jr^, x^^); that is, b^ may be
x^, or a:^,, or any number we choose to assign between x^ and
The member on the right in (2) is, by definition, the integral of
/[x) from a to X, and we therefore have for that jntegral
£ S{X^ - X^.yiBr) = ^^ - /T[«),
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 X. The integral
is therefore a function of its limits,
1 21, In the preceding articles of this chapter we have fixed no
law by which the values a*,, . . . , ►r,,^, were interpolated between a
and X. 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 o
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 n equal parts, so that
x^ — AT^, = Jx^ = h = {X — a)/«,
and we shall take the value of the function to be integrated at x^^ ,
the lower end of each subinterval.
The integral oi/lx) from a to X \s then
Ax(«)o M
F{X) - F{a) = £ 2/(^;)Jx.
But observe that
Hence the integral of /{x) from x z= a to x = X 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, aftd 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. 1 71
more compact and serviceable symbol designed by Leibnitz. Thus^
in future we shall write in the suggestive symbolism
as the symbol for the integral o{/{x) from a to X.
The characteristic symbol j i^^ modification of the letter .9, the
initial of sum, and is taken to mean limii'Sum, or J =j£2. The
symbol /(x)dx represents the type of the elements whose sum is
taken.
If F{pc) is a primitive ofy(A:), then
F{X) - F{a) = f%) dx,
= j^Fix) dx,
X
= £dJ!\x).
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 dF(x), then F{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 quoiient 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 itmii 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 o, 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. XVL
Lemma. If the sum of » variables <^, , . . • , <^m ^^ ^ determinate
limit -4 when each converges to o for « = oo , so that
and there be any other n variables »j , . . . , »^ , such that each con-
verges to o for « = 00 , and at the same time
/^ /.T = -
then also
£iPx + . . . + f;«) = i4.
For, whatever be r,
~ = I + €^,
**r
where e^(=)o, when « = oo . Also,
If € is the greatest absolute value of e^ , . . . , e„ , then
the limit of which is o, and, § 15, III,
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 y = F{x) be represented by a curve, and let F\x) = f{x). Then J{x) is
tlie slope of the curve or of its tangent at x.
We have SQ equal to
l-iw;^, (I)
Also, the sum of the differentials of F
at a, jTj , . . . , is
q ^dF= M^ 7\ 4-iW,7;-h . . . -^-Mn Tn. (2)
The difference between this sum and
that in (I) is
SdF-- SJF=P^T^-\-F^T^-{' . . . -\-BTn.
W But we know that the limit of
JF _ MrPr_
dF " MrTr
is I when « = 00 and Jjr(=)o. Hence, by the lemma above, we have
f\,X) - F{a) =£2JF = £2dF,
= £2 F\x) dx,
=^£2A^)dx,
which is another illustration of the integral.
Xi Xt X3 X
Fig. 67.
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 F{pc) of a given function
J\x) is called the indefinite integral of /{x\ and we write conven-
tionally, omitting the limits,
j/{x) dx = F{x).
This, of course, becomes the definite integral
jy{x) dx = F{X) - F{a)
when the limits of integration a and X are assigned.
The indefinite symbol
JAx)dx
proposes the question : Find a function which differentiated results
\nj\x)\ or, find a primitive of/(jr).
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
X
C e^dx and f sin xdx
are called the fundamental integrals. They can be determined
directly by the ab initio process, and all other functions that can be
integrated in terms of the elementary functions can be reduced to
the standard form
/
du ■=. u
by means of these fundamental integrals.
I. We have, where (X — a)/n = h,
r e^dx ^ £ h\f + ^+* + . . . + ^•+<«^«)*],
Ja A(-)o
_ ^X
e"" — tf*.
174 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI.
2. AISO/
Jf sin xdx = ;£ A[sin a -f sin (a -{- ^) -f . . . -|- sin (<i+«— lA)],
a A( 3 )o
by a well-known trigonometrical summation.*
But the expression under the limit sign is equal to
(cos (a - ih) - cos [a + i(2« - i)A]\^^
= {cos (a - iA) - cos {X - *^)l-|p,
which, when ii( = )o, has the limit cos a — cos X,
/* sin A' ^ = — cos X + cos a.
* See Loney's Trigonometxy, Part I, § 241, p. 283.
CHAPTER XVII.
THE STANDARD INTEGRALS. METHODS OF INTEGRATION.
126. As Stated in the preceding chapter: ii/lx) is the derivative
of F{x), then I^{x) is a primitive of /(x), or an indefinite integral
of yl[j:). This and the next chapter will be devoted to finding primi-
tives of given functions.* This process is nothing more than the
inverse 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
I^/{x) means, find the derivative oi/[x) ; the symbol D~y[x) means,
find a primitive oi/{x).
It is usually preferable to work with differentials and employ the
symbol f/{x) dx to mean, find a primitive of /{x\ or simply,
integrate /][ar).
If u is any function of x^ then
•=/
du
and is the solution of the integral.
The solution of
//(x)^
invariably consists in transforming f(pc) dx into the differential du
of some function u of x, and when this is done the integral or primi-
tive u 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 \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
du.
/
* This is the starting-point of- the theoiy of differential equations, an extensive
branch of the Calculus.
175
176 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIL
The Integral Catechism.
t J cudx = c Ju dx.
2. J(u + v)dx = fudx + fv dx.
%. J u dv = uv — I V du.
4, J u^du z=
5. J-^ = log u,
6. / /» du = ^.
« 7^ = I.
>■/
7- J a» du =
sm otf
cot tfM
log a*
9, J sm au du = . I cos au du =
9. / sec» au du = . / esc* au du z= —
10. / sec u taai u du =: sec u, jcsc u cot u du =r — esc fi.
11. I — ■ = sm— ' — = — cos—' — .
J j^a^ — u^ a a
12. f-j^== = log (« + 4^ii^T^').
'*• / ■?-; — \ = - tan-» — , or cot-' — .
J 1^ -{-a^ a a a a
Ju^ — a* 2a ^u-\-a* ^^ 2a ^ a -\- u ^ J a* -- u*'
16. / tan u du z=: log sec u, jcot u du =z log sin u,
16. / sec udu = log tan (^< + ^jr). J esc m </« = log tan |ii.
17. / i^a^ - «« </« = ^ i^a*-u^ 4- ^» sin-»-.
18. y 4/«« ± a* du = ^tf i/tt» ± tf « ± |a* log (« + 4/«* ± ««).
19. / sin' « i/» = |ff — ^ sin 2m. J cos* » </i/ := ^ -f ^ sin 2«.
20. J ^og udu = «(log M — I).
--/•</« I « I fi
21. / ■ = — sec-i - = CSC-" — .
•^ u yu^ ^ a^ a a a a
du
J^2u — «»
22. / — r--==^ = vers-« « = — covery-' f^
COS ax
Art. 127.] METHODS OF INTEGRATION. 1 7 7
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 formulae, «, v, etc., are functions of x,
127. Principles of Integration. — ^The first two formulae in the
Catechism enunciate two fundamental principles of integration.
I. Since c du z=. d{cu), where c is any constant, we have
I cdu=zj d(cu) =. cu = cjduy
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 / to the
other without changing the integral.
EXAMPLES.
h fjc»dx=z ifjc^ dx = if(^)dx = if d{x*) = ijc*.
2. j sin axdx = I {— a sin ax)dx = / ^(cos ax) = —
II. Since d{u -^v + w) = du -\- dv -{- dw,
. •. f{du + dv + dw) =fd(u -{-v -{-w),
=zjdu +Jdv -\-Jdw.
It follows, therefore, that the integral of the sum oid^ finite number
of functions is equal to the sum of the integrals of the functions, and
conversely.
EXAMPLES.
1. f{ax + cj(^)dx z=jaxdx-\-Jcx^ dx,
z=ajxdx-\-cjjfi dx,
= njd{\x^) + cjd(lx^),
2. / (cos X — sin o-rWr = J cos xdx — I sin ax dxj
, I
= sm X A cos ax,
a
178 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. 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 Jdu by inspection, and
thus recognize u. 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 / du, or to the
recognized form of one of the standard tabulated functions in the
Catechism, are
I. Substitution.
(i) Transformation. (2) Rationalization,
n. Decomposition.
(3) Parts, (4) Partial Fractions.
129. While nearly all the standard integrals in the catechism are
immediately obvious by the inversion of corresponding familiar
formulae 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
J e'dx =: e*, J 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 {Substitution), — ^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
[x) dx,
fA-
and we put x = (f>{z), then dx = 0'(«) dz. The integral is trans-
formed after substitution into the new integral
A'
;0(«)]0'(«) dz.
This when integrated appears as a function of «, which is retrans-
formed into a function of x by solving x = (p{z) for z and substituting
this value z = tp{x). The final result is the proposed integral
f/{x) dx.
Art. 130.] METHODS OF INTEGRATION. 179
EXAMPLES.
1. Use a substitution to find f—.
Put u = ^f then du = e" dv.
• *• / — = J dv := V szlog u,
2. Make use of i^ du — e^ to find / «« dtu
Put «« = ^'. .'. au<^-^du = ^du. Hence
uf^du=z^dHidv=-^e '^"dv z=i — — e '^ dT'^ 'trV
<» fl tf + i v^; )^
?±!.
j u^ du = ^
a
««+»
fl+ I a-j- I
3. Integrate Tcos x dx, given /*sin « </« = - cos «.
We have cos xd^r = - sin {\ic - x)d(^ie - x).
Hence, if « = ^jr - x,
/ cos xdx = — I sin u du = cos « = sin x.
4. Integrate tan x dx. We have, by Ex. i,
/* - V /"sin JT , pdlcos x)
tanjf <^= / dx = - / 1 = - log cosx.
J cos X J COS X ^
5. Integrate icotxdx.
6. Show that
J sinaxdx = cos ax; J cosaxdx = ^ sin ax,
7. Show that / tan at </x = - log sec ox.
8. Integrate / ^ — = — .
•/ y I — x«
Substitute x = sin «. .-. i^r = cos * dg.
.•. / — 7== = I dz =z g = sin-'ji.
9. Integrate Au + ^x)^ </x.
Put a + dx =y, ... <& = ^/^.
•. f(a + dxVdx^ 1 /V^^v = J'i^ - (^ + ^^y^'
^, . ^. Put / = «tane. Then
d/ =z a sec* 0 ^. Hence
. , a = — /^ = — = — tan-' — .
a* -^ f a J a a a
l8o PRINCI1»LES OF THE INTEGRAL CALCULUS. [Ch. XVIL
11. / a» du, Put^ = a«. .•. dy ^ a* log a du.
^ log a .^ log a log tf
12. Integrate the functions
3', jr« - 2', tf -h ^* + ^«
13. Integate
. iV^ -^ »/•
JC-)-l' JC»-fI* x^-f-a"'
14. / — i — : — = / -^^ — ; — -. = log (I + sin x),
15. / sin' jr</jc = ^ I (I — cos zx^dx = ^x — Jsin 2jc.
/• sin JT //r /• - .
16. Find the integrals / , I cos*xdx.
^ J I — cos X ^
17. Given the definite integral
/d/
— = log /.
In the value of the definite integral, let c( — )— I, then (see § 75, 0/0 form),
^ — = /(•*'-^' log X - a-^+i 1<^ fl),
= log X — log a,
log a is the constant of integration and we have
. Put « = « sec B,
u 4/u^ - a*
19. / - . This can be written
•^ 4/2J — s*
/ds
|/l - (I - 'i/
Put I — X = cos 6. . *. ^j* = sin 0 ^, and the integral becomes
I dO = 0 = cos-i(i — j) = vers-»j.
-^ /• ^ . /» sin X , /• //(cos x)
20. / sec jr tan x dx = / — -— dx = — I —^-^, — -,
J J cos^ X J cos' X
= sec X.
21. / esc X cot X dx = ?
22. /* -^^ ■ Put l/x^ 4- a« = « — jr. .-. ^x = 1^—^dz.
... f /-^ . =f- = \ogz = \og(x 4- i/7^T^2).
Art. 130.] METHODS OF INTEGRATION. 18 1
23. Show by a like substitution that
dx
/
24. Integrate /
= iog(x-(-4/:;^-r7«).
sm X cos X
/dx /•sec* X dx r d (tan x) , ,^ .
% = / — = / — ^^ = log (tan jr).
sin X cos X J tan jr J tan x
25. f-T^ = (-^~T^-^—r- = log (tan ^jt), by Ex. 24.
J sin jr J sm ^jr cos ^ * ^ s /» y t-
/dx
, put X = iTt — z
cos JC *
• •• f£-x = -/t^ = - ^^ (^'^^ *'> = '^ (~* *'>'
= log[cot (iJf - ix)] = log tan (i* + i*).
These results can be identified with 16 in the table.
27. Observing that we can, by inspection, write
jr* — a' 2tf \x — a jr -j- fl / *
we have
/:
</r I jr — tf
x" — fl' 2a X -{- a
This process ii a particular case of the general method of decomposition into
partial fractions.
Integrate this case, using the substitution {x — a) = {x -{- a)z.
Also, integrate the more general integral
dx
f
{X - a)(x - dy
by means of the transformation x — a = (x — d)z.
28. We can make use of Ex. 27 to obtain the integrals in Exs. 25, 26. For
we have
/dx fcosxdx __ /• //(sin x) __ I fi -{- sin x \
cos X ~J cos'x "*"J I — sin'x "~ 2 ^ \i — sin x/ '
Show in like manner that
J sin X 2 ** \l -h cos x/
29. Integrate / — —^ . Put <?« = sec 9.
Then dx = tan 6 dB^ and the integral becomes
fdB = 0 = cos-i(r-«).
oA T * * r sin OdB
30. Integrate I r ^ .
** J a — b cos 9
We can complete the differential by inspection, for the integral becomes
I ^dia — /5 cos 0) I , / X flx
- / . V- = — log (a — 3 cos 9).
1 82 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVU.
Otherwise, put a - S cos B = z, . •. ^ sin 0 ^ = dz.
The integral is therefore
31./.
dx
Put JT* ± tf« = ««. Then x dx = z dz,
^x^ ± fl»
dx dz dx -{- dz d(x 4- z)
or — = — = — ; = -^ — ^ — - '
z X z -\' X x-f-«
/dx Cdx pdix -f «) , ^
= log (X + y jft ± fl»).
131. Rationalization (Suds/t/u/t'on). — 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.
EXAMPLES.
1. Integrate f{a + dx^)^x^ dx.
Put a-\- dx* = z^. bx^dx — z^ dz. On substitution the integral becomes
2. Integrate / = .
•^ (a + bx»f
Put a -f 3j;« = «». .-. xdx =. ys^dz/2b.
3
The integral is — ^^ ^ ^^ .
3. Put a '\' bx = ^, and show that
/— ^^^ = -kif^^ - Za){a + bx)\
4. Put «• — jc* = a', and show that
•^ (fl« - x^f 20
5. To integrate / —
^1+^
Art. 132.] METHODS OF INTEGRATION. 183
Put I + i/jt« = «». .-. <£r = - *«» dz.
The integral becomes
/I , ^ — I
(I - z^)dz = z - -a» = ^ |/i + x«.
/^
, ., . Put i/x^ — I = ««. .-. dx = —^ MdM.
After substitution the integral becomes
" -^ I -X*
The integral becomes
^Z^?^'— Z.-^- -{ f +^-+ '^<- ■'}■
since ^-4-, = »' + » + i+j^-
8. r |/a« — j[* dx. Put X = fl sin G. .-. ^ = a cos 0 ^.
.-. J j^a* ^x»dx = a»y COS* 0 dB = ^a* f{i -\- cos 2B)dB,
= ^a«(0 + |sin 2O),
= ^* sin-' 1- ^ J^a* — x*.
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,
d{uv) = u dv -{- V du,
. •. u dv =. d{uD) — V du.
Integrating, we have the formula for integration by parts,
J u dv =z uv — / V du.
EZAMPLSS.
1. Integrate IXogx dx.
Decompose the differential log x ^, so that
M = log X and dv = dx,
dx
,'. du z=i — and v =: x.
X
184 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII.
Hence
I log xdx =z X log X ^ I dx z= X log x ~ x.
2. Integrate Itain-^xdx.
Put u = tan— 'jc, dv z=. dx.
Then ^ = -, v = jr.
I -f jr«'
/ tan-"x dx = X tan— 'x — / — ; — ; ,
= X tan-»x — log Vl -f- X*,
3. Integrate j xe*dx.
Put « = X, dv z= €* dx.
Then ^m = <6r, v = ^*.
.'. fxe*dx = xt*—je*dx=ze»(X'-t).
4. Integrate I x^logxdx.
Put ti = log X, du = x^,
dx x«-»-«
X ' « -h 1
x«+« . r x» dx
/x«+« , r x» dx
x^logx dx = — ; log X — f — ; ,
x«+« , x«+«
logx —
fl 4- I «* (« -h i)»
6. Integrate j j^x* -f- a* dx.
Put « = 4/x« + a*, dv = <£r.
<fk# = — ^^- — ^, V = X.
Vx»-f ««
Hence
*«</x
J -f/x«4- a« ^ = X |/x*-f a* — /"-
But
jr»-f-tf« r a^dx , A x^dx
Adding, we have
.-. /* VxT^i ^/x = ^ i^x«-f a« + fi« log(x -f |^x*-f a*)»
by Ex. 22, § 130, or Ex. 31, § 137.
Art. 133.] METHODS OF INTEGRATION. 185
6. Show, in like manner^ that
f ^x^-d* dxz=\x Vjc»-a» - ^Mog {x + J^x^ ^a*).
7. We can frequently determine the value of an integral by repeating the process
of integrating by parts. Thus, integrate
/
g'* sin dx dx.
Put u = sin 6Xf dv = g^ dx.
I
du =z d a)s dx dx^ v =
a
.% / ^* sin bx dx ^ — e*^ sin bx \ e^ cos bx dx*
But, in the same way, we have
J g^ cos bx dx = — ^x C03 bx -}- - if^ sin bx dx.
Substituting and solving, we get the integrals
e*^ sin bx dx =: - , — -- (a sin ^x — ^ cos bx\
^* cos bx dx = ia cos *jc 4- 3 sin bx\
a» 4_ ^2 ^ • '
Put b/a = tan a, then these integrals can be written
— 1^== sin {bx — a) and — z — =^ cos (bx — a)
respectively.
8. Use Exs. 5, 6 to integrate
x^ dx . /• x* </jc
f ""^ and f
9. Show that / s\Ti—^xdx = x sin— »x -|- y'l — x* by putting u = sin— «jr,
dv = dx,
10. Use the method of Ex. 5 to show that
j 4/a«^^« dx = \x^a* - jr« + fi» sin-'—.
11. Use the work of Ex. 10 to get
/- ."^^ = - 4* V^rr:? + \at sin-^.
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 frnctions, each of which is
sim[)ler 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
^{x)_
1 86 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIL
and the degree of 0 is higher than that of ^, we can always divide
0 by ^y so as to get
in which the quotient y(;r) is a polynomial in jf and can be integrated
immediately. The remainder F(x)/il.'{x) is a rational function in which
F{x) is a polynomial of one lower degree than ^(or), the general
integration of which will be considered later.
SXAHPLBS.
■•/^.=/('-'+--7^)-'
= i-J^ - J4!» + JT - log (I + *).
J if - \ Jx'-^ zj Jr* — 4
= r'°«^-i-'"8<^-*)-
^ /•jr« — 3x4.1,
x« + 4 ^ + 4 ^^ + 4 ^ + 4
= l^ + itan-.^-Z.iog(^ + 4,.
/dx
{X — a){x — d)
We can always write
I = -!-(-i '-)
{x — a)(jc — ^) a — d\x ^ a X — b)
by inspection. Therefore
/dx __ I jf — a
(X - a){x - ^) " ^"^^ ^^T^TT*
134* Observations on Integration. — ^The processes of Substitution
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 a 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.
Inte^^rate Exs. i to 10 by the primary method of completing the differential
by inspection.
1. I jT* dx^ I ajT-^dx, I 2x-^dx,
2. f(x^ -f i)^xdx = {{x^ -f I )^
4. (iiofi — t-^)dt = 6^* -f i/-*.
6. A^-* -f x-'^)dx, ((s^ - i)ds/s, fv dv/{w* - I).
6- / r ■ ' <^" = log ♦/«■ + 2«.
J u* -\- 2U ^
7. f(fi - 2)»/~»^/.= 2/-* - 6/-« -^. J/» - log /•.
8. f(a* - x^f ^xdx, /(V^- i/*)*^, j{x + ifdx.
®- J ax^ + bx-]-r' jiT^^r^^^ J {ax^ + bx+cy"^'
C(i +:c«)-' . /»(! - :r«)-* . f dx
J tan-i;r ^•'^ J sm-^x "^'^ J l^fT''
10. Write immediately the integrals of
jp + 1 ' x + i' x» + I ' JK» + i' x« -f a«
cos' ^Xf cosV sin x, tan»x scc'jt.
1 88
PRINCIPLES OF THE INTEGRAI. CALCULUS. [Ch. XVII.
cos |/;
e» cos e*dx •='f
-dx = 2 sin -|/^.
3. /»JC»-» cos jr«^jr = ?
ycos(iogx)^^^
J I 4- X*
4
5
6
3^
dxz^l
- /* dx r dx rudv -\- vdu
9. / sin 3x ^, / sec' 40 dO, I cos ^0 d0.
22./
= log X -j- j:» -j- ^.
23
(X — 2) </jf
= 2V':r +
24. Jtan'' 0 r/0 = tan 0 - 0. /"cot* <f> d<f> z=i 'i
25. fsin 20 ^/^ = ? /"cos 20 ^ = ?
g^t, P J sin fw 4- «)j: sin (m — »)ji:
26. / cos mx cos nx dx z= 1— ' '^ J ^ '—.
J 2{m -j- n) ~ 2(m — n)
/, sin (m — n\x sin (m -4- h)x
sin mx sin nx dx — ^ 1- ^ ! C,
2 (ot — «) 2(w -j- n)
Use cos a cos y5 = i cos (a -}- iC^) -f i cos (or - /3), etc.
t%t f ' , cos (/» -I- n)x , cos (w — n\x
27. — / sin »Mr cos iMT d[af = / 4 ^ -^.
J 2{m -}- ») ' 2(/// — n)
28. / sin \x cos Ix dx =z } I cos ^ cos $xdx = 7
23. P~^ = ¥}oe xf.
«>• /^/^i-
- dx ■=. a sin-' 4/<j* — jf*.
X a
Put ;c :
Put e*
Put x«
Put log X
z\
= «.
:= Z.
= «.
Put X
% .
= «.
Put x» =
<.
2JC — a
Put lax — fl' = **.
ax —\
Art. 134.] METHODS OF INTEGRATION. 189
Multiply the numerator and denominator by ^a H- x,
31. fxi^T+r^ ^ = f (jr -f a)i - la(x + a)l Put x + a = z\
32. fx^e'dx = ^'(jr« - 2jr -f 2). Parts.
33. /':r»<* dx = ^(jc» - 3^:" -f 6ir - 6). Parts.
35. fcot-^x dx z=x cot-»jf + i ^o« (^ + •*')•
36. fx tan~>x die = i(x* + i) tan-«Jf - ^x.
37. /*:r» sin jc </r = 2 cos jr + 2Jf sin jr — jr« cos j:.
38. /"^r* cos .r <i> = *' sin jc 4- 2Jf cos j: - 2 sin x.
39. /cos jc log sin X ^Jf = sin x (log sin jr — I).
40. f^^* dx = e^
/» dx r dx f ^ ^
Hint. Complete the square.
42. /* — — "^""^ - — - = 2 log (f/^r=r^ + i-^^^^^).
J |/(j: - a)(jr - fi)
Put X — a = «', then </r = 2z dz,
/dx _ /• dz
V(x - a)(;r ^ fS) ~^J ^^+~a - /?
= 2 log (« + V^* + a - /J).
/dx . Ix — a
— =^==^ = 2 sin-»
|/(x - aX/^ - or) \/5-a
Put ^ — a = 2*, as above, and the integral becomes
2 f ^__.
J -f//5 - a - «»
44. f |/tf -f 2^j: -I- tjc« <£r.
_ r-i[(ra: -f /J)* — (^* — ac)].
Put ^x -f- ^ = «. . •. <ilr = dz/c^ and the integral becomes
J J j^^^^d^'^r^) dz,
the standard form 18, § 126.
19© PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIL
/dx
j-^f where m and n are positive integers,
OT m -\- H is 3t, positive integer greater than i.
Put JT — fl = (jc — d)Zt then
a ^ bz (a — d)z . ^ — ^ . a — 6 ,
I — z I — z I— « (I— a)*'
and the expression transforms into
(I — «)«+«-»</»
(a — fi)m+M-i gm'
Expand the numerator by the binomial formula and integrate directly.
46. Integrate / sin/jr cos^jt dx, whenever p -{- ^ is an even negative
integer.
Let / + ^ = — 2». Then
sin/jr cosffx = sin^jc oos-^»«jr = tan^jr sec'*jr,
= tan^x(l + tan*jr)»-« sec*jir.
Put tan jr = /. Then
fsin^x cos'fxdx = ff^i + /*)»-»<//.
Expand by the binomial formula and integrate directly.
47. Integrate sin^x cos9xdXf whenever / or ^ is an odd positive integer.
Let / = 2r 4- I, then
/ sin«''+»jr cos^xdx = — / (sin'x)'' cosVjr<ii[cos x),
= — / (I — cos'x)'' cos^j: d{cos x),
= — I {I — ^Yc9dc.
Expand by the binomial formula and integrate.
48. J sin«0 <ye = i co8»6 — cos B,
49. / cos"6 ^* = ? Check by putting ^ — x for jr.
50. /"cos^e dfl9 = sin e - I sin»e + \ sin»6.
51. /*sin«e cos'O d9 = ^^jCOS»«e - Jcos^.
52. fsin^x cos->4r rfx = 8ecjp + 2cosjp— ^ cos^x.
53. / i^sin x cos*jp <6r = f sin'x — f sin'x.
54. / cos'jr csc*jr dx = $ sin'x — f sin'x.
55. fcsc^x sec*jc <6r = f tan'x — 2 cot*jr.
Art. 134.] METHODS OF INTEGRATION. 191
56. / sin*jr sec«x dx = i tan'x -f- I tan»x.
58
59
61
62
63
67. / sin'x sec"x ^ = f tan' jr.
L / csc*x sec*A' </jp = 2 tan*x (I + ^ tan*;r).
I. Aan*e dB = /'tan«-ae (scc«6 — i)dB,
iSinn-iB /*^ ^ -
= — / tan«-a© <^.
» ~ I ,f
/cot**— »fl /•
cot«^ <* = — ^::^^ _ / oot«-39 dB.
. ytan<6 </9 = ^tan»e — tan ^ + 0.
:. fco^ ^ = - i cot«0 - log (sin 0).
1. fcoi*e dB = — icot«0 4- cot e + e.
64. fcoi*B ^ = - i cot*e + i cot«0 + log (sin 6).
65. / sin jr cos X (a* sin'jr -f ^ cos*jf)*<6r.
Note, </(«» sin'jr -f 3* cos**) = 2(a* — 3*) sin x cos x <^. Hence the integral
is
66. / -i 5 — . .^ ■ , • = -i tan-» (— tan x ) .
Divide the numerator and denominator by cos'x.
/dx
— ; --7 . Divide the numerator and denominator by ^a* 4- H^,
asiTix-\-o cos X J w \ 1
and put tan a = a/b. Then we have
1 r dx I
|/fla 4. ^2 J cos (jf - a) y'a* -f ^ T -r * /
68. /*— r^ . We have
J a -{- b cos X
fl -f- ^ cos jc = tf (sin' ^ + cos* ^jc) -f ^ (cos' ^x — sin' \x)
= (a -f ^) cos' ^jc -|- (tf — b) sin' |jr,
which reduces the integral to the form of Ex. 66.
Divide the numerator and denominator by cos' \xi and put m = tan \x. Then
the integral becomes
/dz
(« + ^) 4- (« - ^)«*'
which is standardized. Hence
— r—L = — tan-» \ ^ /^-^ tan — V , a > b-,
a-{-bco&x |/a2 _ ^a ^-^^^^ 2^' ^ '
1^^ l^^-f ^ -4- yy^^ tan jx
tog =r .^^ r f a < b.
|/^i — flS j^b-\-a — i^b — aidLii \x
tgi PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIL
69. f ^^". = i tan-. ^ "^ ^ ^" ^"
70. Integrate A-i+-"??A dx=-L i^_.
^ J (x -{- sin jc)' 2 (x 4- sin xf^
71. j X sin X iix =z sin X — X cos x.
72.yi^^^x = log(iH-x)«-x.
73. /* ^'^ _. _ i _JL_.
' -^ (^ + x»)' 3 (a3 ^ -pj)**
= log (tan-ix).
(I -t- x^) tan-«x ^ ^ '
__ /• ^/x Ix -{- I 2 — X
75. / — - = 2 sin-i^ I — ^ — =■ cos-i .
J VS-\- 4x^x' \ 6 3
76./"^^!?^-^-^^ = sin (log x;. Put x = log ..
__ /• ^/x I . tan 4x — 2
77. / ; = - log -^^
J 4 — 5 sin X 3 ^ 2 tan ^x — I
78. / = - tan-i (3 tan x).
^ 5 — 4 cos 2x 3 ^
CHAPTER XVIII.
GENERAL INTEGRALS.
General Forms Directly Integrable.
135. The Binomial Differentials. — Expressions of the type
x^{a -{- djc^)y dx, (A)
where a, fi^ y are any rational numbers, are called binomial differen-
iials.
This expression is directly integrable in two cases.
I» When — 3 — is a positive integer.
The substitution is a + hxP = ». Then
hence
•4-1
— T
x^{a + bxfi)y dx = i- ^-^^ — ^dz.
or -4- I
Consequently, when — ^ — is a positive integer, the transformed
expression can be expanded by the binomial formula and immediately
integrated.
II. When — ^ [- ^ is a negative integer.
T*he substitution is a + ^^ = ^■^•
For, if we substitute or = i/y in the differential x^{a + bx^ydxy
it becomes
Sv -\~ oc -f- I
which, by I, is integrable when — ^-^ — —r= — ■ — is a positive integer,
or, what is the same thing, when
flf + I
/5 + ^
193
194 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII.
is a negative integer. Also, the transformation a -\- bx^ = z becomes
Hence, under the transformation,
x^ia + bxf^)ydx = ^a'T^\b - zf^T^"^'^ z^dz.
P
In working examples it is better to make the transformations than
to use the transformed general formulae, which are too complicated
to be remembered.
When ay fi, y do not satisfy the conditions in I, II, the binomial
differential must be reduced by parts.*
EXAMPLES.
r x^dx j^ I a
4 f ^ Axks. -y \ _ ^^ )
6. / i. Ans. -'-^!-,.
J (a« 4- jf«)l 3(tf« + *«)«
•^ (I -f Jr)*
8t dx A ■*■
.' (I + :«•)* (I + ^)*
»• /^$:^- ^- - ^<^ + -*>*•
1. Ans. ;.
**(i -f ^ (I + ^)*
136. Integration Of ^^^-j-^^—-^^.
The substitution is a + co^ = a:*«*.
. • . c^jc z=. s? dx A- zx dz. or — = 5.
• xz c — v^
dx dz
•*• {A + Cr»)(tf + cj^\ " {Ac - Ca) - A^*
which is standardized, being 13 or 14 (§ 126) according as {Ac—Ca)/A
is negative or positive.
* For formulse of reduction see Appendix, Note 10.
(B)
Art. 137.] GENERAL INTEGRALS. 19$
If {Ac — Ca)/A = — , the integral is
' tan-^^^^2:
i/A{Ca -. Ac) j^A(a + cy?)
If (Ac — Ca)/A = -[-, the integral is
I i/-^(^ + ^•^) + x^Ac — Ctf
EXAMPLES
Ans. — =■ tan-»-
(I + jf»)(i - jr«)* 4/2 i/i - *»
^ I cx
Ans. — tan-«. ^
+ 4jf«)(4 - 3^:*)*" . * 5 |/J y £2 _ 9x
•'(3
8. f #-—;-. Ans. ± log «|3_+ 4«^);_+_S-
•' (4-3-**K3 + 4*')* ao ^ 2(3 + 4*')* - S'
137. Integration of _j^-+^^. (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 L represent the linear function / + ^.
Let Q represent the quadratic function a + ibx + cjnr*.
rdx
I. Consider / -y.
Completing the square in Q^ we have
/dx __ r cdx
a-\- 2bx + cj^ ~~J {ex + bf — (^ — ac)'
Put cj: 4" ^ = '• Then the integral becomes
dz
A
4j« -. (^ - ac)'
This is standardized, and depends on whether i^ ^ ac is positive
or negative. If negafwe, the roots of the denominator are imaginary
and the integral is an angle, the standard 1 3. If positive^ the roots of
the denominator are real and the integral is a logarithm, the standard
14 (§116),
If ac > *«,
If «: < ^,
dx I ex -\'b
— — , , tan ' . (i)
ijac - 3' i/ac-» ^ '
J Q"
/dx _ 1 ex + b — j/i^ ^ ac
Q "" 24/^ - ac °^ cx + b + \/W^^ac' ^^^
196 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIIL
II. Consider
Since the derivative, Q\ of Q is a linear function, we can alwajrs
determine two constants A and B, such that
Z = ^ + BQ\
or p -\-qx ^ A '\- 2bB + icBx.
Equating the constant terms and coefficients of x,
B = q/2c^ A ^=p — hq/c.
•• J ^- c J Q^2cJ -Q-
The first integral has been reduced in (i), (2), and the second
is log Q,
In working examples, carry out the process and do not substitute
in the general formula.
EXAMPLES.
■• J ^4.4x4- 5""^ I (^-f2)»-fi"^2^-f 4^4-5 ) '
= - 2 tan-»(x + 2) -I- i log (:r* 4- 4^ + 5).
2. [ ^^-^^ = - J—\j^-x'i±lj^l log (jfS + 2* + 3).
'• rf +^6^+10''' = " - log (** + e.* + 'o)' + " tan-C + 3).
'• fS+2l+2 = ' - '°g (^ + " + a)* + 3 tan-.(x + i).
- \.- dXy where F[x) is any polynomial in x, divide I\x)
by C until the remainder is of the form L/Q^ and integrate.
138. Integration of (/ + ^•^).^' — ^^ pj
Let, as in § 137, Z and Q represent the linear and quadratic
functions respectively.
I. Consider / --.
Complete the square in the quadratic, and then
/dx _ r r ^f
Q^^^J i^icx + bf - (^'^ - ac)'
Art. 138,] GENERAL INTEGRALS. 197
which is the standard 11 or 12 according as fi is greater or less than
ac. If a and c are both negative and ac > ^, the function is
imaginary.
We have, according as the roots of Q are real or imaginary,
I
-^ log [ex ^b-^ ^c(a -+- 2dx + cji^)],
I ex + b
-^ sm-' ==-,
^e yae + Ir
as the corresponding values of the integral.
Write, as in II, § 137, Z = -4 + ^Q\ and determine A and A
U. Consider / -^:^ ^.
Then
/l--/i+-/f-
The first integral on the right was reduced in I, the second is
dx
Z0*
HE. Consider / -
_ . qdx dz I — Aar
Put / + y^=,A. ...__=-_, x = -^^.
Substitute in the integral and it transforms into
dz
-/;
j^a' 4- 2b' z + c's»'
which can be integrated by I, then replace z by i/(/ -j- ^o:).
EXAMPLES.
1. r .-^^ = 2 log (vG?+ v:^"^^).
2. / — , = 2 sin-» I- = sin-i (— - i ) .
— ■ = 2 sin-V-=f — ^ = 8in->(2jr — 3).
4/3JP — jr« — 2
4. / --^-^ ^ = log (2X + 1 + 2i/i + X + :r«).
^' f\^^'^* = f/(x + «)(x+*)+ (fl-^)iog (1^^^=^ + vr+1).
*• /• ^ . 2j: 4- I
6. / = sin-' "L .
198 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVUL
JTl/jC* —
dx \ a
z = - cos-'—.
' J (I +x)|/i ^jc* \i H--^'
9. r-^^ = - i log ?^ti;^?i?.
10. / — = -^z. sin-« — -, — .
•^ (I 4- X)^l -\-2X - X^ 4/2 I + -^
11. f — ^1 =-Liog^^-^+3+j!g:
12. /-_g±iL^ = i/j^ + 2^ + 34.1og(x+i + ♦^^ + 2x4-3)*.
•' |/x« 4- 2JC -I- 3
Reduction by Parts.
139. Integration of Powers of Sine and Cosine.
J sin*A: dx =z j sin*~'ji; sin x dx.
Put u = sin*"*;*:, dv = sin xdx;
. •. </» = (»— i) sin*^:!: cos x dx, » = — cos or.
Hence, applying the formula for parts,
isin^xdx= — sin*~'Ar cos x -\- {n — i) /sin*~*.r cos* xdx,
= — sin^-'Ji: cos a: + (« — i)fsm*'-^x(i — sin' or) <£^,
= — sin*~'j:cos jr4-(«— i) J sin*~'Ardlr— («— i)Jsin''xdx^
. •. / sin"j: <£r = 1 J sin*^;*: dx. (i )
When » is a positive integer this reduces the exponent by 2, and
leads to Jdx or j sin x dx according as n is even or odd.
Since integration by parts depends only on the differential equa-
tion d{uv) =r udv -\- vdu, the formula is true when n is any positive
or negative rational number.
Change n into — » -4- 2 in (i), and we have
/dx _ — cos X n -- 2 r dx
sin*.r ~ (« — i) sin*~'Ar ' « — ly sin*~';c* ^ ^
In (i) and (2) change x into ^n — x, then
/^ , cos^-'jcsin^ , « — I /* « , • , ,
cos«j: dx = ^^ + — ^— / cos*-»Jf dx, (3)
/</ji: _ sin Jt n ^ 2 r dx
cos"^ ~" (« — i)cos""*:ir « — i »/ cos^^'jc ' ^
Art. 139.] GENERAL INTEGRALS. 199
These formulae are important. They reduce the integrals to stand-
ard forms whenever n is an integer.
Formulae (i), (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 u and dv is indicated in each
case in the formulae below.
/*«'"", xdx= f^^'^'Z xx^^x dx,
J COS* J COS*^' COS '
J CSC* / CSC*"' CSC*
In the part fvdu use sin*.r -f* cos*:ir = i, sec'jc = i -j- tan'or,
or csc'^.r = i -|- cot*^:, as the case requires.
/sin X cos X
sin*jf dx = 1- ^jf = |jf — J sin 2jr.
2. /d„», rf, = - t sin', cos, _ } cos , = I 00-, - CO. ,.
3. Tsin^jf </jir = — ^ cos jT sin jr (sin*jr -h I) + !•*•
4. jsivk*x </jp = — J sin*jf cos jr -|- } / sin*jf dx.
6. isin^x dx = — i cos jr (^ sin'jr + ^ sin*jr + | sin x) + ^x.
6. Find the corresponding values lor cos x, integrating by parts. Check the
result by putting | ir — jr for jr.
/dx
-, = log tan ^jr s log (esc x — cot jr).
sm x
8. \ -^- - CO'*-
J sin'jT
9. f . ■ = — ■ . , — Kt Jog tan —.
J sin'jt 2 sin*jc ^ 2 * 2
^m. t dx ICOSX 2^
10. / . . = :-T— cot jr.
J sin*jr 3 sm'j: 3
^^ t dx ICOSX 3cosj^.3,**
11. / -Tji — = — r ~-i 5" . , + s- log tan — .
J sin*x 4 sin*x 8 sm*x '8 * 2
^^ C dx ICOSX 4COSX 8^
12. I .-■=—- -7-T — — r-s -COtx.
^ sin"x 5 8in*x 15 sin'x 15
13. Deduce the corresponding integrals of cos x, and check the result by
putting l^r — X for x.
aoo PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIU.
140. Integration of fsin'^x cos'^x dx.
We have for all positive or negative rational values of m and n
d sin*"~'a: , . sin'"~*A: . ^ , sin^jc
— -- = («r — i) — -— — \- in — i) — -— .
ax cos^^'o: ^ ' cos*^a: ^ ' cos"*r
Therefore
/sin""jr , I sin"*~^;ir m — \ /*sin*"^:i: , , ,
— s— ^ = ==7- / — r-^- dx. (5)
cos*:ir « — I cos" *;«: n — \ J cos*~"'x ^''^
In particular, when m = »,
/tan«-'jr /* «_ .
tan*ji; dx = — — — — / tan*^ar dx. (6)
Put \n — jr, for x in (5) and (6). Then
/cos"»jr . — I cos"*~'jir m — 1 /*cos"""«:*: .
/cot*""*^ /*
cot"*:*; dx ^ ^ --—■ / cot*-«;r dx. (8)
The same results are obtained immediately by changing the signs
of m and n.
Change the sign of « in (5), then
/sin*^*;ir cos*+'ji: m — 1 /* . ^ ,
sin^xcos^xdx = ; 1 ; — / sin^~^x cos'^^'xdx.
n + i ^n + ij
But sin**^Jir cos*+*a: = sin"*"»j: cos*ji:(i — sin'jc),
= sin*"~*jr cos*:ir — sin"*;r cos*;c.
Substituting and solving, we have
/n ^ ^~^/*«., n J sin"'-'j;cos*+'^ , .
sin'^Ji; cos**a: dx = — ; — / sm*"~'Ji: cos*:^; dx ; . (o )
m + nj m + n ^""^
In like manner, change the sign of « in (7) and write i — cos'^j;
for sin^Jtr in the last integral. Then
/« • « ^ ^— ^ r «-« • « ^ I cos*"-'A:sin"+';r
cos'^.r sin*jc <ilr = / cos^ x sm^x dx-\ .(10)
n+mj m+ n ^ '
These formulae serve to integrate sin*ji: cos*"jir dx whatever be the
integers m and n.
It is well to be able to integrate the functions of this article in-
dependently. The forms below show the separation into the parts u
and dv which effect the integration directly when the trigonometrical
relations sin^ji; + cos^:ir = i , sec^jc = i + tan'jf, csc^^i: = i + cot^^r
are used in the integral jvdu.
Art. 141.] GENERAL INTEGRALS. 201
/ sin*"a: cos*xdx:= Jsin'^^x cos":rxsina:^= J sin^Ji; cos'*"'arxcos Jir^;c,
/tan* , rtan*-^ ^ tan'* ,
cot-^^^=y cot— ^ ^ cof-**^"*-
1. Jcos^x ain*xdx = isinxco8x(i sin*jf — ^ sin'j: — J) + t^jr.
2. / -;^ 5- = — f- log tan fr.
J sm jc cos'jc cos x ^^ ■
*• / "^-M r- = 1— 5-7- + - log tan — .
J sin^x cos'x cos X 2 sm^jr 2 * 3
4. / tan^jT ^ = ^ tan'x — tan x -\- x.
6. / COt^xdx = — i COt*JP -|- cot JT -|- 4f.
/dx I
- — r- = — — r— ; log (sin x).
tan>jf 2 tan^jf * ^ '
dx __ I
in*jf "~ 2 tai
dx — I I
J tan^jT 4 tan^x * 2 tan'jr ' * ^ '
Integration of Rational Functions.
141. General Statement. — Any rational function of x whose
numerator is a polynomial N and denominator a polynomial D can
by division be decomposed into
where 0 is a polynomial, and the degree of S is that of D less i.
We then have
fp.=fQ<Lc+fp..
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 n has n roots, real or imaginary, and can be written
If there is no second root equal to a^, then a^ is said to be a
single root. If, however, there is another root equal to a,, say
a^zzz a^, then the two factors can be written (x — a^)\ and we say
that a^ is a double root, or that the polynomial has two equal roots.
In like manner, if there are r equal roots equal to a, the correspond-
ing factor is (x — a)'', and we say that <z is a multiple root of order
r, or the polynomial has r equal roots of value a.
202 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. 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 imaginaiy root a-\- dV — i,
there must be another a ^-^ dV ^ i. Now the product of the factors
corresponding to these two roots is
(X --a — dV — i){x ^a + 6V - i) — {X'-'ay + P,
= jc* — lax + «* + ^,
which can be written = jc* -J- /jt -|- ^.
Moreover, if « + i3 (1 s 4^—1) is a multiple root of order r, so
also is <i — 1'^, and we have the corresponding factor in the
polynomial
{j^+px + gy.
Hence any pol3rnomial in x is composed of factors, linear and
quadratic, of the types
x^a, {x-^dy, ^+p^ + g, i^+P^ + gy-
If ^^
be a rational function, in which F{x) is of a d^;ree at least i lower
than that of Ax), we can always decompose the function into the
sum of partial fractions corresponding to the roots oi/[x), as follows :
For each single real root a there is a fraction
X — a'
for each multiple real root d of^order r there are r fractions
^^+7Air. + -..+ ^'
{X ^ 6) ' {x^dy ' " • • ^ (or - dy'
for each pair of conjugate imaginaiy roots there is a haction
C+ Dx
^+P^ + i'
for each pair of conjugate multiple imaginary roots of order s there
are s fractions of the types
» /^ _L_ ^^ I /?Vi I" • • • »■
xi ^ ax + /3 ' (x^+ ax + /Sy ' ' ' (^x^ ^ ax + py'
In these partial fractions the numbers A, B, C, D, E, F, etc.,
are constants. Since there are exactly as many of these constants as
there are roots of y(jr), they are n in number.
If now we equate F{x)/f(x) to the sum of the partial fractions
and multiply the equation through by /{x), we shall have F(x)
equal to a polynomial in x of degree « — i. 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 n linear equations in the constants A, B,
Cy etc., which serve to determine their values,*
The integral of the rational function then depends on
/dx r {i:+Fx)dx
The first of these can be integrated immediately, the second is
always of the type
r J^+J^£)dx_ _ r dz C zdz
J \sx-af^iPY~^^^ J K^^n^ J (f+v^r
wherein X'=. a-^-z. The last integral on the right is
(^ + l^Y "27 (?-+ ^r "" 2(r- 1) {z^ + i>r-''
To integrate the first integral on the right, f put 5 = 3 tan 6.
. •. dz — b sec«^ d0.
which can always be integrated by parts, § 139.
Hence the rational function can always be integrated.
EXAMPLES.
J x» — 4Jr
We have here single real roots; hence
x» -f 6^: - 8 _ x«-h6ir- 8 _A B C
X* — iMr "" x{x — 2)(jr -{- 2)^ X X — 2 X + 2*
Clearing of fractions,
jP« ^- 6jr - 8 = ^(x - 2Xx + 2) -I- Bx{x + 2) -f C{x - 2)x, (1)
=:(A + B+ C)*> -f 2{B - C)x — 4A.
Equating coefficients,
^ -f ^ + C = I, 2iB — C) = 6, — 4i4 = — 8.
.'. ^ = 2, B = 1, C = — 2.
Hence the integral is
J±^^—^dx = 2 log JT -h log (x - 2) - 2 log (JT -h 2),
If we assign particular values to or in (i), we can find ^, B^ C
more easily. Thus put jc = o, then — 4-4 = — 8; put x = 2, then
* Provided these » equations are independent, which they are.
f See also Ex. 88, at the end of the chapter.
204 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIIL
SB = 8; put jf = — 2, then 8C= — i6, which give the constants at
once. The general principle involved in this abbreviated process is :
when there are only single roots, put x equal to each root in turn, and
the constants are immediately determined.
^•/(T^lJFT^ = »'»«<'-3) + «•»«' (' + ')•
*• fx' + i- 3 = * '"8 (* + 3) + * 'og (^ - ')•
«• !J^Z-6'^ = log [(;r + Zf (X - 2)].
/jc* 4- I
^ <£r. Here there is one single root, o, and a triple root, x = I.
x{x — I)*
Hence
x(x - l)» ~ :t: "^ (:r - l)» "^ (jr - I)«"^ X - r
Clearing of fractions, we have
jr> + I = (^ + D)x^ 4- (C - 3i4 - 2Z>)jc« + (3^ -f- ^ - C+ Z));c - ^.
o = C — 3^ — 2A
o = 3^ + ^^C4.A
Whence A z= — i, ^ = 2, C = i, Z? = 2.
' ' jc(jc - I? "■ jr ^ (jr - l)» ^ (jr - i)« ^ X - I "
/x* -f- I I 1
— -=- dx =. — log Jr — , rz V- 2 log (x — l),
in /"^J^ — 2, I2x + 19 , , , , X
"•• y(.+2)«'''= (^7+^+ 3 log (*+*).
J jr* — 2jc» -I- X* ^ X* ^ jr(x— I)
/Jf Jx
/ — ; — rr-5— ; — :• Here there are a pair of imaginary roots.
(x+iX-r*+i)
X A Lx-^M
(x+i)(x«+i) I +x^ 14: JC«'
Art. 141.] GENERAL INTEGRALS. ^05
Clearing of fractions,
=:(A + M) + (L + M)x + (^ + ^)^-
Equating coefficients,
Z + ^ = o, L + Jif= 1, A + Af=o,
L = h M=h A = -i.
f _JL^i_ - i loe -i-i^ + - tan-«;r.
•*• J 0^+ i){^ -fi) "■ 4 ^ (I -f- ^f 2
'^'fj
dx
. We have i + jr» = (l + ^)(i - -r + '*)•
Clear the fracUons and put ^ = - i. Then ^ = i- Substituting this, we
/» dx i_ /'_j;__ , ' r (g - ■^)^
U /• ^ - ' loe i-±^^±^ + — tan- ^^'.
15. Tt \
J (x — I
Equating coefficients,
B ■\- Z = o.
^ + Z - 2iW = O.
... iI/=o, B = A-\, Z=-i.
Hence the integral is
-L_J_4.1log(x-i)-f log(x«H-i).
18./
19./
-j_l 1^4-3 __ *
I*^3T'^ = ^ + ^log-p- - V3 tan- ;j^-
■^ _ i log ^^^ i tan-x.
(;t« + 1)(JC» + X) - 4 ^ (' + »)'(•** + 0 2
2o6 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIIL
"^ T^'^ dx. Here there is a double pair of imaginary roots. Hence
we put
2x»-f jr-f 3 _Ax -i^ B Cjt -f />
(jr» -f i)» ~ (x* -I- I)* ■*" jr» -h I
.•■ 2Jt» H- * 4- 3 = C«» + Z>jc« -I- (^ + C>ar + ^ -f Z>.
.'. -rf = — I, iff = 3, C = 2, /> = o.
2Jf*4.x + 3_-x-f3 2jr
(jr» -h I)* ~ (jr» 4- i)« ' jr« + I .
, - , put jr = tan 0, then the integral becomes
/cos«e ^ = ie + isin2e=i tan-"* 4- , ' .
^ ^ * ^ ^ 2(jr» 4- I)
J (x« - S* -|-3)» 3(^* -3^ + 3)"^
26
r=- tan
2jr — 3
3 1^3 f'i *
22. /""^ J" f ~,- dx = f " -^ , + log (x« + 2)* L^. tan-.-4-.
J (jr«H-2)» 4(x« + 2)^^^ ^ ' 4^2 V2
142. Trigonometric Transformations. — On account of the simple
character of the reduction formulae in §§ 139, 140, it is often advan-
tageous to transform many algebraic integrals to these forms, and con-
versely many trigonometrical formulae can be transformed into useful
algebraic forms.*
SI «"■-"+ 'J sin"^ co8«-"«»-aO d$.
1. Put x = a tan 6, then
f* dx
+ x»)**
2. Put X = tf sin 0, then
(««-X*;
3. Put X = a sec 0, then
/x** dx /" cos"""""*© -
4. Put X = 211 sin>6, then
(2tfx - x*)** -^ cos«-'e
5. Make the same transformations in the above integrals when m or « is
negative.
*The reduction formulae for the binomial difierentials are given in the Ap-
pendix, Note 10.
Art. 144.] GENERAL INTEGRALS. 207
The general integral
/i
j^dx
(a + cofy
can always be transformed to the trigonometric integral when the signs
of a and c are known, whatever be the signs of m and «.
1 . Integrate by trigonometrical transformations
j ^a^-^ji* dx, f 4/jc»— «« dx, f V^^^T^* dx.
/dx P dx f
dx
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 x -=: sf*^ where n is the least common multiple of the
denominators of the several fractional powers.
For example,
+
Put X = s^. The transformed integral is
•«»(i + z) dz
./=v
Consequently the integral is
|j:* — 2x* — 4x* -f- 4 tan-«jr* — 2 log (i -j- jr*).
Again, any algebraic function containing integral powers of x along
with fractional powers of a linear function a •\' hx can be ration-
alized by the transformation a -|- ^jr = ^, in the same way as above.
1. /-^^= A (54^ + 6^ ^. &r + 16) Vx - I.
^ t X dx 2 2a 4- dx , , . ,
2- / r, = 7= — - _■■■ , by a 4- 3jr = «'.
Complete the differential, integrate and compare results.
• /* dx , / . / \ 2 ^ 2 ^x — I -f- I
•^ X ^x* 4- ajf« -|- I •' j^z* ^ a ± 2
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,
2o8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII.
/ |/v dXy when^ is a polynomial in x 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. Hy 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 ^a:x^ -\- bx -x- c can be
rationalized and integrated as follows :
Factor out the coefficient of ji::* and \^X.y = 4/±^ + /-^ + ^
The rational function F{x,y) is rationalized in x\
I. When the coefficient of .r^ in >' is positive, by th^ substitution
j^x^ -(- px -\- q -=. z — X,
z^—q ^J^pz^q 2(f-\-pz-^q)
Then x = — , z — x •=z — '-^ — ^—^ .dx=:] —^~ r^dz.
p + 2Z' p-j- 2Z ' (p + 2Z)^
n. When the coefficient of oi?' is negative and the roots of the
quadratic er, (i are real, then
— •^+/'^ + ^= (x — a){/3 — x).
The function F(Xfy) is rationalized by either of the substitutions
i^— x^ + px +q = ^(x — a) {/3 — x) = {x — a)z or (/? — x)z.
Then .=^1; ^. = "(^)^. (.-«> = i^f.
.-. J Fi.,y)d^ = 2{a - fi)J /-^-J:^, _p^ j_.^_.
When the roots of — x^ -{-Px -{- q are imaginary the radical is
imaginary.
145. Integration by Infinite Series.— We know that if a function
/{x) = a^^ a^x + a^x^+ ...
in an interval ) ^ H^ -\- H {y then also its primitive is equal to the
primitive of the series for this same interval (§ 72). Hence
[x)dx = ^0^ + \a^x^ + ^^ + . . .
/A
EXAMPLES.
1. /•-^— =^+^;+^^"+^^^;+...
J 4/1 _ jps I ' 2 6 ' 2.4 n ' 2.4-6 16 '
^ C dx ,-. / , I sin«jif i.3sin*:r , \
2. / — • = 2 4/sin JT ( I -f ^ . . . 1
J Vsinx V^2 5^2.4 9^ /
Art. 145.] GENERAL INTEGRALS. 209
Put sin jr = «. .*. dx •= dz/cos x^ and the integral is
/dz
J ^ ' ' \m q m -^ ft * 2 ! ^* «i + an ' /
For what values of x is this true ?
4. Show that
J 4/1-^x* I 25 ^2.49
5. Show that
_iiii.3i
* • 2 S** 2.4 9lx»^ ' ^
/.?:.•—' N('+-+f 4-'+?,'4^'+ •■■ !
Determine the values of x for which this is true.
Put d -j- X = M. .'. e^ =z r-^e^^ etc.
6. The eUiptic integral /" (i _ ^ sin*jr;*iZr, ^ < i, can always be ex.
panded by the binomial formula, and the general term f sin'*x dx integrated.
_ /• sin X , J /I I jr* \ x^ \
7. I —-= dx = 2x [ —J ■—...).
J \/x \3 7 3!^ "SI /
EXBRCISBS.
x^dx
^- f / ;u = * ^^''"'-^ - i-* Vi - ^ (3 + 2jf^
•^ (l — j:2)t
3
2. / =.Ilog ^ J-
J ^*^l-X^ ^ ^ X 2X*
/dx __ X jfi
(a* -f x^)^ " a*(d^ + Jt*)* ~ 3fl*(a« -j- jr»)i '
. C X^ dx — jfi .3/ . x\
^- fr^^-^^ir = - (2^ - ^)*(i* + 1^) + 3^' "^-\te
•^ (2tfx — j:«)* \2a
6. fx*e<^^ dx=—(x^--^x^-\- Ijc - IV
J a \ a a* cfi]
7. y^'* (log x)^ ^ = ix* [(log Jf)« - i log X + i].
8. / Jt* cos j«r </jr = JK* sin X 4- 3JP* cos x — dx sin x — 6 cos x.
9. / x* sin X </x = — X* cos X 4- 4** sin x + I2x* cos x — 24(x8inx -|- cos x).
10. / jn = 2 tan 4d — 0.
J (I -f- cos e)« ^
aio PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII.
11. Tcos^O sin 20 <il9 = — f cos^.
12. fsin^ cos»e <^ = i sin»e - i sm«e.
13. fsin^ cos^ <^ = - ^(cos 26 — I COS* 20 4- t cos»20).
14. fco3*x cscx dx = ^ cosfix + cos j: + log tan ^x.
15. fcos^x csc»;i: dx = (cos»x — | cos x) csc'jt — | log tan ^or.
16. f "^ ^^-l^iogi^^f-^CEZ.
17. f—^^^—, = (A^ + fx* 4- iWi + -»^r*.
•^ (I -^- JC')«
Show that f je*«[(fl' + Jf*)* -f j:]«<£r can be integrated by the same sub-
stitution when « is a positive integer.
J (I + JC*)* **
20. f4^ = ix^-U^(x^+i).
J jc*-f I 3 3
/dx 6 , , (jr* 4- 1)*
:tl 4- ;rl x« *
22. /•4-±-i'^=--,+ '^ + *^^--^^(** + »>-
J jr« + Jr« or* x»^
23. /'-i^-£=V + ^^^ irr^ + ***'''"**•
./ X* — X* 3 jr* + I
24. /•— ^= = log^"^
25.
_=-.* + 2jog^-
jr«<far 6jr* + ^ + 1
{^ 4- I)' ""
(4:f + I)* l2(iMf + i)«
^«^ 3.
26. r ^=:=|(x+0*-3(-^+i)* + 3log(i + fi + ^)-
27. /•.^^=i(;r*-2.)(:a + a)*.
i^
J 4/2JC* 4- I 30
29. . , ,
Art. 145.] GENERAL INTEGRALS. 2H
31
<^ I Vx" -f a« — tf
• / — - = — log
^^- /^:pf^^^ = '°« (^3"^^ + ')' + 1-8^ (i^3"=^ - 3)*.
33. /* ^ = -Llog i^^^JiM^^
34. r ^: = -L log V^j-^^+24-^- V?
•'*V-^-* + 2 ^^2 i/*» -X4-24-X+ 4/2*
38. /" ^ ^ _ V^»^ - 2x -f. 2
39. /'-^IZI^ flCr = log (X - 2) - il^ll.
J{x-2f "»^ *^ (*-2)«'
Put jr — 2
= «.
= e.
= B.
M,
^ y (* + i)« - ' Put '+I
*^ f^^ = »<'' - 3X'*+ I)*. Put *»+! = ,.
- ^ /• sin jr <6r
* J sin (;c -f «) = (' + ''> CO* « -sin tf log sin (jf + a). Put :p -(. «
^ /("i^T^ = '^^ -4X^+1)* Put ^+ I
46. yj:» log jr <& = J jt» (log jr - I).
47. y*Jr— I log jc ^ = i x« Aog jr - i\
4B. J xsinxdx = — x cos x -|- sin jr.
49. Jx log (:r -f. 2) flCr s= (jf« - 4) log VJ+2 - Jx* + *.
212 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVUL
4
50. fx tan-ijr dx = ^{x^ + l) ten-»jc — ^x.
61. Integrate
fx-*{a* - ;r»)-* dx, J (a* - *«)• dx, fx* (a* + x*)-^ dx, .
fx* ^a^ Zx'^ dxy fx* i^a^ + x*dx, f(a* - x*)^ dx.
52. / sin*x cos'x dx = ^ cos jr (^ sin'x — ^ sin'jp — | sin x) -|- -f^x*
53. / ; = - log ^ T— ^-— •
J 3 -h 5 COS jf 4 * tan ^jr — 2
dx I
/or I
= — tan~>(2 tan Ix)
5— 3 COS j: 2 ^ ■'
S6.r
Pat jn = I.
<£r 6 -{- ex
(a + 2^jr + cx*)^ (ac — S»)(a 4- 2dx + ex*}'
Complete the square and put ex -\- b :=z s. The integral reduces to 55I
57^ r (/ 4- y-y) ^-r ^ bp^aq^{ep^ bq)x
'J (a^ 2bx -\- ex^)l (ae — ^«)(fl -|- 2^jir -f- rjr«)*
For xz = I transforms
X dx . ^ — dg
into
(a -f 2^jr -f rx«)* («« 4- 2^a 4- 0* '
f X dx a ■}- bx
{a 4- 2bx -f <-x2)' (ae — ^){a 4- 2^jf 4- or')*
Combining with Ex. 56, the result follows at once.
(^-\-x)dx 7^-4
58. /"-ii
^ (I -
(I - 2JC+ 2jr«)* (I — 2jr 4- 2JIP«)*
go /• ^ ' lo^ (.r - tf)(x - ^)
J (or - a)(jr - /J)(2jc - a - ^) - (fl - ^)* ^ (2x- a - b)*'
gQ /• ^r _ I J (j: — a,*{x — b)
J (x- a){x - ^X3^ -2a-b)~ 2{a - bf ^ O^r - 2a - ^)»*
®^- y iS"^r6^«-fii;r-6 = 2 ^"^ (X - 2)* •
/■ x»^jr I ,_(.r-i)(x-3)"
'^2- y (X - i)(;r - 2Ax - Sy = * + ^*^« (X - 2Y*
63. f , ^" -^ = ilog^"~^)^^" + 3^
64
j:* ^jc I /•(3jr» - 7 4- 7)^-^
/■^ ^^ _ I r(ix* —
7^ + 6 ^ J X* —yx -\- 6
I . . m . ^^ . 7 1 (■«■ — 2Wjr 4- 3)
= - log (x» _ 7^ + 6) + ^logL_±L_t3;.
Art. 145] GENERAL INTEGRALS. 213
e$. / ; — == log(j:— i)H — log(*— 3)+-^ log(jr+4).
J j^— X3r-t-i2 2 10 *^ '^14^^ 35
^^ t X dx I.,jr — 2
68. / . r= r = h 2 log .
fifi C ^^ - g locr^"^ ' 2X~ll--3
'*''• J (x - «;«(a: - 3)« ~ (a - ^)»^ X- a {a - ^)« (jp - ^X-* - ^)"
-g. t dx I , I ^ X
J;** -4* 4- 3 6(4?-!)^"*^ (JC-I)* 184/T -/2
Notice JT* — 4*-f3z=(jr— i)«(jc* -f 2jr + 3),
^2. r ,.,**! "^ ; — : = ^=i +log J^ "" 'f + 2 Un-i(jr - 1).
J (x — iy(x* — 2Jr+2) X— I^^Jf* — 2Jf + 2^ ^ '
/<£r I X 1 X
r4. / ; iri = - log 5 H ; 3. Put JC* = «.
^ I . Jf* 1
(T+:^«(i + j^) 2
«fx* + 5jf» + 4 6 2
6. / ; . . . -, = - log . — : — T.-. — : — ST — - tan-»jr.
'• ; (i + ;r^i.|-4«») - r**" ' r+3^*
r9. lf/(jr) H (x — tfj) . . . (jT — tf „), and F{x) is a polynomial of degree lest
than n, show that
/5i-=im^"-^
80. Show that any algebraic function involving integral powers of x and frac-
tional powers of
a '\- bx
can be rationalized by putting ^ = s^, where tn is the least common multiple of
the denominators of the fractional powers. Apply to Exs. 81, 82.
a 14 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII.
83. If /(x) is a rational function of sin jt, cos x, then /[x) dx is rationalized
by the substitution tan ^ ■=z z.
rhen sm x = — ; — 5 , cos x = — ; — , , dx = — ; — 5.
In particular, when « — i, n — i, or m -f- n is even, say 2r, we get for these
respective cases
8in"»jif oos^xdx = — r'«(i — i^ydc,
= + j««(i — s*ydst
— ^^
■" (I -f- fly+i '
where j s sin jr, cm cos x, / s tan x.
/dx
-, where Q,, Q^ are any quadratic functions of x.
0,0,*
Write out Q{-^ in partial fractions. This reduces the integral to § 136, (B), or
to § 138, (D).
85. In general, ii f(x^ y) is any rational function of jt andJVf where
y^ ^ a + zhx^ ex* — c(x — d){x - /5),
then any one of the following substitutions will rationalize f{x^ y)^'*
^ = a* + x«,
= « + M
= t{x - ar)f7.
/■ dx __ f x-^dx _ Put u = jr-*»+^
*®' J ^fli _j. jfiyt "" J (I -j> a'x-«)*'*~ " ' dvfoT the other £i.ctor.
87. Given the signs of the constants a and ^, transform the Unomial differential
x*(a + bx^)y dx
into the trigonometrical differential
C sin«*jr co0*>x <i^»
determining the constants C, /» and n for each case.
CHAPTER XIX.
ON DEFINITE INTEGRATION.
146. The Symbol of Substitution. — We use the symbol
F{x)-]
x^b
or, in the abbreviated form when the variable is understood,
-'a
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 b is sub-
stituted for AT. Thus
F{x)\ s F(h) - F\a).
If F{pc) is a primitive of /"(at), then we have
/'/(/) dt = F\{)\= Fix) - /•(«).
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 f A^) dx^F\b)^ F{a\
.-. f^/(x)dx^^JlAx)dx.
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
f^A^)dx and f^A^)dx,
while they are equal in absolute value.
215
2i6 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX.
148. New Limits for Change of Variable. — If we transform
the integral
by the substitution of a new variable for x, then we have to find the
corresponding new limits.
Let the substitution be jc = 0(«), which solved for z gives
«= tl)(x). Then, when x = x^y we have z^ = tp{x^, and when x z= X,
Zz=:^{X). Also,
/{x)dx =/[</>{z)'\<p'{z) dz = F{z) dz.
'. f^/{x)dx^f^F{z)dz.
X
For example, put x = a tan 2. Whence z = tan—»— . When jr = o, then s = O;
a
when X = Of then g = ^x. Consequently
r dx I /•*' . I
I =1—1 cxM g az = =
since / cos z dz =. sin s.
149. DecompoBition of the Definite Integral Limits.
If
f/{x) dx = F{X) - F{x,),
then
f^JXx) dx = F{X) - F(a).
Whence, on addition,
r /[x) dx + f^/lx) dx = f'^/ix) 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 integrals:
Ji 3J1 3 3 3
2. /"— = log jrj = log / — log I = 1.
3. I sin X dx = I cos x dx ^ l.
Art. 149.] ON DEHNITE INTEGRATION. 2X7
10. /'*'sin«e </0 = i*. 11- /* 'cos 20dB = ^.
12. /*'«»•' sin jr i6r = J. 13. jf* (li^ - A '*)'* = * Vf- 5-
14. r -^ = -' «. r.--a:r = i.
Jo «* + * 2^ •'• ^
"*Vo co8*4r 3 JjL ~-
•'o
COS X 1/3"
6
dx 1 r* dx <f>
? ^ 2 Jo I
-f- 2jf COS 0 -f' ' 2 Jg I -j- 2jc COS 0 -|- jr* 2 sin 0'
19. jfr— sin »«^ = ^r-p^,. 20. jf;^ cos »«<& = ^-^
, ^L^ . ^^ = , when ac > ^.
22. Show, by putting jr •= I — «, that
This is called the First Euierian Integral. Integrating by parts,
f^\l - xr-' dx = **<' ~ ^^•'~' + 1^j:^x - ,)"-^<6r.
Use this to show that the value of the above integral is
when ^ is a positive integer, and therefore whenever / or ^ is a positive integer
the integral can be evaluated.
/•I .2* i»i , 2"
Jo ^ ^ 3-7"i3 Jo 5-7-9I3-I7
24>. The integral ie-'x^ dx is called the Second Euierian Integral or the
Gamma-function^ T{n -|- i).
We have, by parts,
r-^x*dx r= — ^*x« + n\ e-*x^*~^ dx.
Since /-«4r* = o when j: = o and when jr = 00 ,
r^*jr" dx zz n / r-*x*-» <^.
0 Jo
.-. r(n -\- \) = nr{n).
2i8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX.
A1so« when fs is an integer,
r(n+i)=:nl
The Eulerian Integ^ls are fundamental in the theory of definite integrals.
26. /^(log.).^ = Jr^i. = (-i).|^(.ogi)V
Hint. Put ^— « = jr in Ex. 24.
X"° It!
t-Mgn dz ^.—^^ Put ;r = tf« in Ex. 24.
cos-j: dx =z j sin«;r <&,
and that
•lo 2*4>o . . • 2m 2
Jo 3-5-7 . . . (2#f+ I)*
when m is a positive integer.
28. l^^fLlLJ^dxz=:e. Put
xs = I.
Put X — 2 = f*.
,10ff6
> 4/^ _ I
30. /* *"^^~' </jf = 4 - jr. Put ^ - I = ««.
•'o ^H-3
31. /' a + ^-^B = 7^i=^- W^*~ "^ > *•
'0
32. ' ' ' ^^'^ xdx =. 1,
150. A Theorem of Mean Value. — Since in
VJTq
^ keeps the same sign throughoat the summation,
mf dx< f /[x) dx<M C dx,
Jx^ Jx^ Jxo
where m and Msure the least and greatest values respectively of the func-
tion /{x) in (x^, X). Therefore the integral lies in value between
m{X - x^) and M{X - x^j,
S'\nce/{x) is continuous in the interval, there must be a value of or,
say S, in {x^, X), for which
rV(^) dx = {x^ x,)/{S),
Jx^
/{G) being a value of the function between m and M, its least and
greatest values.
Art. 150.]
ON DEFINITE INTEGRATION.
219
The value
A^ = x^ /^-^'^ ^
0*^0
is called the mean value of the function in {x^^ X).
If J^x) is a primitive o{/[x), then
F{X) - fix,) = {X - X,) /{S),
= {X-x,)F'{S),
since /''(jit) =/[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
^x ^{X -- x^)/n,
then
/{x)dx=£ 2JxA^r),
= (^-:.,)^-^"»>+-^"->+---
NaOO
If the limit of th^ arithmetical mean of the n values of the function
at the points of equal division of {x^, X) be indicated by/(^), the
result is the same as above indicated.
Geometrical Illustration.
If ^ =yTJc) is represented by the curve AB, then
i
ydxzrz area {x^ZBX),
This area lies between the rectangles
x^TX and x^SBXy constructed with x^
as base and the least and greatest ordinates
to the curve respectively as altitudes.
There is evidently a point ^ between x^
and X at which the ordinate |Z = /(^}
is the altitude of a rectangle x^RQXj inter-
mediate in area between the greatest and
least rectangles, whose area is equal to that
bounded by the curve.
BZAMPLES.
1. Find the mean value of the ordinate of a semi -circle, supposing the ordinates
taken at equidistant intervals along the diameter.
Let jc» -f y = fl' be the circle. Then
viz., the length of an arc of 45*.
220 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX.
2. In the same case, suppose the ordinates drawn through equidistant points
measured along the circumference. Then the arc length is the variablei and the
mean ordinate is
I /•»' 2
— I a sin $dB =: —a.
We shall see later that this is the ordinate of the centroid of the semi-
circumference.
3. A number n is divided at random into two parts; find the mean value of
their product.
— / x(n — x)ax = ^ IV,
n J^ 0
n
4. Find the mean value of cos x between — n and -\- n.
5. li M^*{y) is the mean value of^ =^A:X) in (jCj, 4:,), show that:
{a). M*(2x* + 3x - I) = 8J.
(c). m[{x + i)(^ + 2) = I2i.
(d), M^'{siTi 0) = 2/jr.
6. Find the mean distance of the points on the semi-circumference of a circle of
radius r, from one end of the semi-circumference, with respect to the angle.
i)f *'= - f^^'zr cosBdO=^.
By the mean value of n numbers is meant the »th part of their
sum. To estimate the mean value of a continuous variable between
assigned values, we take the mean of n 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. i and 2 above.
\iy is a function of /, then the mean value of^' with respect to /
for the interval (Z^, /,) is
I /•'»
151. An Extension of the Law of the Mean. — If 0(jr) and ^(x)
are two continuous functions of x, one of which, ^(at), has the same
sign for all values of x in (jf^, X)^ then we shall have
j^y{x)i^{x)dx = <p{S)iy{x)dx,
where S is some number between x^ and X.
For if m and M are the least and greatest values of <f>{x) in
(.v^, X), then the integral must lie between the numbers
ml if;{x)dx and Ml il;(x)dx,
since tf:(x)dx does not change sign in {x^, X), Therefore there
Art. 152.] ON DEFINITE INTEGRATION. 221
must be a number G in {x^y X) for which the integral has the value
proposed, since <t>{x) 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 0 be a variable in the fixed interval {a, x). Then
/{x) -Aa) = £/'{,) A = - jfVW d{x - B).
Put u z=:/\z)y dv = d(x — «), and integrate by parts.
•■ A^ -A<i) = - (* - «K'(»)]: + f^{x - zV"{z)dz,
= {x- ay'(a) - J\x - zY"(z) d{x - »).
Put « =y"(a), dv = (x — z)d(x — z), and integrate the
integral on the right by parts. Then
/(x)-Aa)={x-a)/'{d)+^-^^/''(a)-J^^'^^^/'''{'H^-z).
' Continue to integrate by parts in the same way, and there results
n
r-o
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 f{pc) and the series vanishes when
« = 00 , provided
/'
for all values of « in (a, x^ ; and moreover, if this limit is not zero
for any finite subinterval of (a, or), 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 (i) 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 5*00 = ^n + ^li , then we should write
The terminal term being R^ -f- Tn* In order that f{x) = S^ it is necessary that
lx)th ^Rn =^1 £^n =0. £Rn = o does not ensure £ T„ =0. See Appendix.
Notc^.
22 2 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX,
term given in the ditferential calculus. For, since (x — zy keeps
its sign unchanged for all values of b in {a, x), we have
where S is some number between a and a*. This result, (3), takes
Lagrange's form when p --=■ n^ and Cauchy's when / = o. The more
general form (3), where / is any integer, is due to Schlomilch and
Roche.
153. The Definite Integral Calculated by Series.—If /(?) can
be expressed in powers of (z — a) by Taylor's series, for all values
of z in (a, at), then also can the primitive of f{z)^ and the definite
integral of the function is equal to that of the series, taken term by
term, between a and x. Hence, integrating between a and x^
A') =/l«) + (« - <'V\<') + ^^^V"(«) + . . . ,
we have
jr>) dz = {x-a)A<i) + ^^/'(«) + ^^/"(i) + . . . (I)
In particular, put or = o, then we have
jf/i;.) -ft = \a<^) - ^/'(«) + ^/"(«) - . . . . (3)
a formula due to BemouUu
Knowledge of the derivatives at a serve therefore to compute the
integral. When a = o in (i), then
j'^Az) d. = xAo) + ^/'(o) + "-/"{o) + . . . . (3)
which is Maclaurin's form, and is more convenient, in general, for
computation than (2).
EXAMPLBS.
t. Deduce Bernoulli's formula (2), § 153, by using the formula for parts,
jfix) dx = xf{x) -jxf^x) dx.
X 3 52! 73J
4. yiog (tan 4>)d0 = -^1-^+^-^+...^.
154. Observations on Definite Integration. — In order that a
function may admit of definite integration in an interval (/y, fi) 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 any value of
the variable between the limits of integration, tiien 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
£/{x) dx
as the limit to which converges the integral
j^/{x) dx,
when AT = 00 , provided there be such a limit. The same remark
holds when one limit is — 00 and the other -\- 00 .
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 diSerentiable (see Appendix.
Note i).
The study of definite integrals will be taken up again in Book II.
BXSRaSES.
1. / ^ = 3 j/a, 2. / — =^= = ir.
3. P , = J*. 4. / sin-«jf dx z=z\tc ^X.
J. 2-1-0)8X3^3- J^ ^ '
7 /** ^ ~ * ft /*** <6f ^ 0
' J0 I -f- cos6 C06 jf ~ sin 6* J^ I 4- cos 6 00s jr "nnO*
' J^ a* sin*Jf -f- ^ cos'jr 2ad'
-n /**» dx jt(a* + ^
'"• J; (a» sinU -f. ^ co8«x)> " 4tf»^ •
«« /•^ dx ,- /•!» sin X dx , , ^ I
11. / — = ir. 12. / -— ; 1- = iir -f- tan-i — — .
13. Show that, when >(■ < I,
r -^ -^Ti M*^+ (i:-^)V4- (i:3:|)V + ..n-
This is an elliptic integraL
14. Show that
/(^
ffaoO
224 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX.
Put dx = i/«. The limit of the sum is then
'^ dx n
h I
15. Show that the limit of the sum
when « = 00 , is ^?r.
16. Show that
/ sin mx sin nxdx and j cos fnx cos nx dx
are zero when « and » are unequal integers, and are equal to 4jr if m and n are
equal integers.
sin'jc cos^jc dx = X.
io /•*' sin 0 -f- cos e ^
*• Jl 3 + sin2e~^ = ^^^3. Put sine-C08e=x
|0 />«+^»"" {x'-{.i)dx
'^' I, ^=; = logs. Put j: — 4r-« = *.
Oik C^ ^ 1C ^ /•-"
/ i o.. .^o ^ I ^« = "i i' 21. / xe* dx z= I,
Jo I — 2a cx)s X -f- « I — «* Jjj
22-/ FT^= logics: 23. jT
24
26
0 d'-i-^x^'^y ^^-Jl J^TT^?'"*"^^'^*-
J-i i/z-^x-x' Jo i^ + ^'i' ^ '
30. jf ' tan X ^/jT = log |/i; 31, /•*' se<^xdx = i.
3*- j[ ^(i - ^)*^ = 2 /'*'sin«e cos*0 fl^O =r J^«j.
36. r Jf»(i - ^)*^ = r^'sin^e cos*^ ^ = -.
•'o Jo 32
36. rx*(i~jr)*^x= 35 .
Jo 128
37. Putting e» -i =y\ show that
•log* >tl «2
^\;^^^dx^2r.y^^t
Jo Jo i-j-y*"
— ie
Art. 154.] ON DEFINITE INTEGRATION. 225
38. If Jf + I = r,
39. Putting X =za sin 6,
«a |/tf* - *• <& = «* f sin'0 cos«6 <* = -£5-
40. If jr = a tan e,
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 ^r-axis, and two ordinates corresponding to the values
x^ , jc, of Xf is represented by the definite integral
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 n
^ equal parts, each equal
to ^x. Let AB h^ the
curve representing the
equation y =:/{x), and
x^ABx^ the boundary of
the area required. Let
MN be one of the sub-
divisions of x^x^. Draw
ordinates to the curve at
each of the points of
X division, and construct
the n rectangles such as
^iG- ^9- MPqN, and also the n
rectangles such as MpQN. Since the curve is continuous, we can
always take Ax or MNso small that for each corresponding pair of
rectangles the cnrvtPQ lies inside the rectangle P/^^, and therefore
the area MPQNoi the curve lies between the areas of the rectangles
MPqN and MpQN, Hence the whole area x^ABx^ for the curve
226
Art. 155.]
ON THE AREAS OF PLANE CURVES.
227
lies between the sum of the rectangles represented by MPgN SiXid the
sum of those represented by MpQN, The difference between the
sums of these rectangles is the sum of n rectangles of type PpQq.
Which sum is equal to a rectangle represented by BRy whose base
BS is Ax and altitude ^.S* is y^ — >'o» where y^ = x^By y^ = x^A.
CVi> J'o) b^i^g t^^ greatest and least ordinates in the interval. When
the number of rectangles, n, 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 « = 00 , the area of
the curve, or
j£ 2yAx = f ^ydx.
If y = f(pc) is the equation of a curve, the area A included
between the curve, the ordinates ^^^^^jj/^ at x^^ j:, , and the A:-axis is
A = pA^) ^•
EXAMPLES.
1. Area of the circle.
Taking jr* -f ^' = «* as the equation of the circle,
.-. y - ± Vfl* - Jt».
«f
g
y""^^
■s
P
f
/
/
\
R
0
a
Q a
1? X
1
1 -:
■X
Fig. 70.
Taking the positive value of the radical, we have for the area XqP^^x^ ,
-f
'Xo
V
— Sin-' —
2 «J^o
If jr, a a, we get the area of the semi-segment x^Pf^A. If jr^ = o, and x^ = tf,
we have the area of the quadrant OBA equal to
r -/«» -x*dx = InaK
If Q is the angle POA, then >f = a sin 0, ;r = a cos 0.
. *. dx = — a smB dB. The area, A, of the circular quadrant is thei given by
A= rydx= -a* r sin«e dB = a* /**%in«© dQ,
= ^«^e - sin 0 cos 6)]*' = isra'.
The area of the entire circle is therefore jta*.
228
APPLICATIONS OF INTEGRATION.
[Ch. XX.
2. The area of the ellipse.
b ,
From the equation of the ellipse jfi/a^ -\- y^/l^ = i, we get^ = — y a* _ x*.
Consequently, as in Ex. I, the area of the elliptic quadrant is
which is b/a times the corresponding area of a circle
of radius a. Hence the area of the entire ellipse is
icab,
3. Area of the parabola.
Taking y* =zpx dis the equation of the curve,
and the positive value of the radical in^ = 4/^x, we
have the curve OP. The area OPNis then
Fig. 71.
= \xy.
But xy is the area of the rectangle ONPM. The area of the segment POP* of
the parabola cut off by a chord perpendicular to the
diameter is two thirds the rectangle MPPM*.
4. Area of the hyperbola.
Let -^y^' — y 1^^ = I be the equation to the
curve. Then the area of APN is
Fig. 72.
= — jr k/x^ — fl' log
2a ^ 2 *
ab X -\- ^x^ — tf «
= \xy ^ \ab log
(J+^)-
6. Area of the catenary.
The equation to the curve is
— X
Fig. 73.
y = \a\e* ^e «) .
The area OVPNis
= \a^ \e^— e''y = a f>» — <:
If NZ is perpendicular to the tangent at P^ show that the above area is twice
that of the triangle PLN. Observe that ten LNP = jDy, LN = ;/ cos LNP^ etc.
6. Show that the area of a sector of the equilateral hyperbola jr^ — ^' = <i*
included between the jr-axis and a diameter through the point x, y of the curve is
^t log ^.
7. Find the entire area between the witch of Agnesi and its asymptote.
The equation is (jt* -j- 4^*)^ = Stf*. Am. \iea*.
Art. 155.]
ON THE AREAS OF PLANE CURVES.
229
8. Find the area between the curve ^ = log jr and the x-axis, bounded by the
ordinates at jt = i and x. Ans, jr(log x — I) + i*
9. Find the area bounded by the coordinate axes and the parabolas^ -\-^ = tfi.
Ans, Jo*.
10. Find the entire area within the cmre | — j -)- | ^ j = i. Ans, \nab.
11. Find the entire area within the hypacyclaid j^ -f~ ^ = ^'*
Hint Put jc = tf 8in"6, y = a oos^9.
Ans, l^a*.
12. Find the entire area between the cissoid (2a — x)y* = j^^ and its asymp.
tote jr = 2fl. Ans, ^Ka*,
13. Find the area included between the paranoia x* = 4ay and the wiUA
y{x* + 4fl«) = &i». Ans. a\2n - 4).
The origin and the point of intersection of the curve give the limits of tne
integral.
14. Find the area of the loop of the curve
0^ = (jr - aX^r - bf.
Hint I>et x — « = «•. Ans.
8 ^
'5\ ^
Fig. 74.
15. Find the whole area of the curve a^y* = jc*(24i — x).
Ans. na^.
Fig. 75.
16. Find the area of the loop c. :he curve
ay = jc*(3 -h x).
The area of the loop is
2 /^ \2jfi
Put *-|-x = ««.
17. Show that if^ =/(-^) is the equation of a curve referred to oblique coordi-
nate axes inclined at an angle a>, then the area bounded by the curve, the 4r-axis,
and two ordinates at x^^ x^ is
A = sin (0 j y dx.
18. The equation to a parabola referred to a tangent and the diameter through
the point of contact is^^' = 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 = ^. If 09 is the angle between the asymptotes, show that the area between
the curve, x-axis, and two ordinates at jtq, jt^ is
^ sin 00 log |~ j .
20. liy zs ax* is the equation to a curve in rectangular coordinates, show that
the area from jr = o to x is
230
APPLICATIONS OF INTEGRATION.
[Ch. XX.
156. If the area bounded by a curve, the axis oiy^ and two abscis-
sae x^^ x^, corresponding to the ordinates^^, y^, is required^ then that
area is
= J xdy.
EXAMPLES.
1. Find the area of the curve y* = px between the curve and the >'-azis from
y zs o\oy = ^.
2. Find the area of the curves = €* between the curve, the^^-axis, and ab-
scissae at^ = i,^ = a. Check the result by finding the area between the curve
and the ;r-axis for corresponding limits.
Also find the area bounded by the curve, the ^^-axis, and the negative part of the
jT-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
axis of coordinates.
It is evident that if
the curve crosses the axis
<c between the limits of in-
tegration, then, y being
positive above the jc-axis
^'°' ^^* and negative below it,
those portions of the area above Ox are positive, those below are
negative. The integral
pytc
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.
EXAMPLE.
Find the area of ^ = sin x from jr = o to x =r \n.
We have
Jo
But j siux dx = 2,
'f.
sin X dx z= — cos x
f.
sin X dx =z — I.
Fig. 77.
.-. ^*' = ^'+ y^*' = 2 - I = 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, Or.
Art. i6o.]
ON THE AREAS OF PLANE CURVES.
231
If we have to find the area between two
curves V
and two ordinates at a and d, such as the
area LMNR in the figure, that area can be
computed by finding the area of each curve
separately. But if it is more convenient, thcQ
area is Fig. 78.
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 RL to NM.
EXAMPL£.
Find the area bounded by the curves
x{y — e*) =1 sin x and 2xy = 2 sin jr -|- jp*,
the^-axis, and the ordinate at jt = i.
1-55 +.
= fi^-i^)^ = ^-i =
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 MP to meet the
curve again in Q, and let MP =>',,
-xMQ = _y,. Let a and b be the abscissas
of the extreme tangents aA and bB,
Then the area of the curve is
a JIf b
Fig. 79.
This result also holds if the curve cuts the axis of :r.
EXAMPLE.
Find the whole area of the curve {y ^ mx)* = a* — j^.
Here
y =z mx ± |/<j> — X*.
.*. y^ = mx -|- i/«* — Jt*f
y^ = MX — J^a* — X**
/•+•
160. The area of any portion of the
curve
/
{^■i>
0)
Fig. 80.
232 APPUCATIONS OF INTEGRATION. [Ch. XX.
is ai times the area of the correspondiDg portion of the curve
/{x, y) = c. {2)
For (i) is transformed into (2) by putting x = ax'fy = ^ in (i);
and hence^' dx, from (i), becomes ad / dx*, and we have
ly dx = adfjf/ */.
KXAMPLS8.
1. The entire area of the circle x* 4- ^'^ = i is jr. Hence that of the elUpN
jfl/a* + y/i^ = I is a^n.
2. Find the whole area of the curve
(j)'^ {{)•- ^
In Ex. il| § 155, it is shown that the area of
j^+y^ = I
is {jr. Hence that of the proposed curve is fira^,
8. Check the result in Ex. 2 by putting x = a sin'0^ y zs b cqs*0l
Then ydx = ^ab sin*0 ca&^4> dtp,
.*. A •=. i2ab I sin'0 cos*0 dtp = fieab.
Jo
x6i« Sometimes the quadrature of a curve is to be obtained when
the coordinates are given in terms of a third variable, or is &cilitated
by expressing the coordinates in terms of a third variable. Thus if
^ = 0W» y = ^W'
the element of area is
ydx= fp{i)<l>\i)di.
BZAMPLBa
1, Find the area of the loop of ihitfoHum of Descartes^ ^
Put y zs tx\ then
••• '^ = (i 4. /sy 3'»*» and
yox-gaj ^, _^ ^^ -FTl* 2(i + /»)'
The limits for / are o and 00 . Hence A = |a'.
2. In the cyciHd,
X = a{g ^ sin /), ^ = tf(i — cos /),
.-. fy dx — a^ /*versV *// = 4a* f sin* ^i dt.
Taking / between o and fC^ we get 3ira* for the entire area between one arch of
Che cycloid and its base.
Art. i6a.] ON THE AREAS OF PLANE CURVES. 233
3. Find the area of the ellipse using j^ja^ ■\- y*/i^ = it where x -=, a cos 0,
^ = ^ sin 0.
4. Find the area of the hyperbola jc*/a* — y^/H^ = i, from x = a to x ^ x^
using jr = a sec 01 y^b tan 0.
i6a. Areas in Polar Coordinates. — Let p =/{0) 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
Fig. 82.
Let AB represent p =:/[0), 01 the initial line. Z^OA = a,
ZlOB = /3. Then- €iAB is the sector whose area is required.
Divide the angle A OB :=z fi — a into n equal parts each equal to
^ff, and draw the corresponding radii cutting AB in corresponding
points P, Q, etc. ; dividing the curve AB into n 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 =y(6'), we can
always take i/^ 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 t3rpe OPg and the sum of the circular sectors of
tjrpe OpQ. But the difference between these sums of circular sectors
is equal to the area
ALNM= ^0B» - OA^)Jd,
which has the limit o when J0(=)o, or when » = 00 . Therefore
the sum of either the external or internal circular sectors converges
to the area of the sector OAB as a limit when » = 00 .
Putting p^ = OA, p^ = OB, and Pr{r =1, 2, . . .), for the
radii to the points of division of AB, the area of the curvilinear
sector OAB is ^
w.jQ r-i •'•
234 APPLICATIONS OF INTEGRATION. [Ch. XX.
EXAMPLES.
1. Find the area swept out by the radius vector of the spiral of Archimedes^
p = aOy in one' revolution.
We have ^ =^ \ f p* d$ z= ^ T tf«e« dQ = 4ir»tf«.
2. Find the area described by the radius vector of the logarithmic spiral
p — ^•, from 6 = o to 0 = iir. Am, — (^'* — i).
3. Show that the area of the circle p = a sin 0 is ^flra*.
4. Find the area of one loop of p = a sin 2O. Ans, ^lea*.
5. Find the entire area of the cardioid p = a( i — cos 6). Ans. \fCa*,
6. The area of the parabola p ■= a sec* ^ from 0 = o to d = |^ is |a'.
7. Show that the area of the lemtdscate p* = a' cos 2O, is a*.
8. In the hyperbolic spiral ph •=. 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 «0. Ans, a^/n,
10. Find the area of the loop of \X\<t folium of Descartes^
jr* + >^ = 3flxy.
Transform to polar coordinates. Then
yi cos d sin 9
^ ~ sin«0 -f- cos»0 '
Therefore the area is
9a' z*^ sin'6 cos*© d^ _ 9^' /"* t*^ du _ - ,
T7o (sin»Q + cos»©)« " ~2 J^ (i -j- «*)*"**'' '
where u = tan 0.
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
*•+>»= (f)' and P' + e'= (I)'.
13. The area oi p = a cos 3O, from o to \iey is ^^KaK
14. Show that the area oi p ■=. a (sin 2O -|- cos 26), from o to 2^, is na^,
15. The area of p cos $ z:z a cos 2O, from o to iic, is ^(2 — ^iC)aK
163 • We come now to consider the area generated by a straight-
line segment which moves in a plane, under certain general conditions.
In rectangular coordinates we^have considered the area generated by
the moving ordinate to a 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 = /. 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 = /, P'Q' = /+Jf, PI- Si, P'l^ 6% and 6 be the
angle which the tangent at P makes with a fixed direction. Let A A
represent the area swept over by PQ in moving from PQ to P'Q' through
Fig. 83.
the angle Ad. Draw the chord PP' and the circular arcs QMy Q'M'
with /as a center. Then A A is equal to the area of the circular
sector QIMy plus a fraction of the area of the triangle PIP\ plus a
fraction of the area QM Q'M'. Or, in symbols,
AA = !(/ - difAO
+ Asi. d'i sin AS + -^[(Z + J/ + S'ty - (/ - 6/)^]A6,
where /", , /'^ are proper fractions. Observing that A/, S/, and tfV
converge to o when A0{=)o, divide by A 6 and let A0{=)o, Then
or dA = |/a </5.
Hence between the limits 8 = a, 0 = /5 the area swept over by
/is
A = ijT^/^^^.
When the law of change of /, the length of the tangent, is given
as a function of 0, the area can be evaluated. If / =z /[0) be this
relation, the curve / =z/[0), considering / as a radius vector and 0
the vectorial angle, is called the direcitng 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 itd^^ when the point of contact moves entirely
around the boundary of a closed plane curve.
2. The tradrix 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 PT = a
turning through the angle \ic as the point P moves from J along the curve JPS
asymptotic to Ox, Therefore the area in the first quadrant is ^iro'y and the whole
area bounded by the four infinite branches is iro'.
23<5
APPLICATIONS OF INTEGRATION.
[Ch. XX.
3. Check the aboye result by Cartesian coordinates and find the equation to the
tractrix.
We have directly from the fig^ure
-^ = - tan PTN= ^ ■
dx j^a* _ yt
. • . ydx = — f^tf ' — y^ dy.
Fig. 84.
Hence the element of area of the tractrix is the same as that of a circle of
radius a. It fallows directly that the whole area of the tractrix is nd^» This
gives an example of finding the area of a curve without knowing its equation. To
find the equation of the tractrix, we have
dx= ^^ ^dy.
Integrating, we get
. = - VSrirp + . log l±i^L=Jf ,
since x = o when^ = a. This is said to be the first curve whose area was found
by integration.
4. Show that the area bounded by a ciunre, its evolute, and two normab to the
curve is
•jC"''
</9t
where p is the radius of curvature of the curve, and 6 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
P^P^ moves in such a manner as to be always parallel and equal
to the radius vector of a known curve p =z/(^6) called the director
curve.
It is required to find the area of the closed curve described by a
point P on the line PJ^^ which divides the segment P^^ in constant
ratio.
Art. 164.]
ON THE AREAS OF PIJ^E CURVES.
237
Let (P), (-Pj), (iP,), (A) be the areas of the closed curves
described by the corresponding points as shown in the figure. Let
Fig. 85.
P^F^ and P^P^', Fig. 86, be two positions of the segment,
them to meet in ۥ
Produce
A
Fig. 86.
Let /o = Pji',, P^/PP^ = mJm^.
m.
' iWl + w.
^ A = ^lp»
/>/>. = — ^
* w, + «,
A^, = V.
where ^^ + ^i = i-
The element of area PJPJ^^P; is, § 163, if CP, = r,
^(/>.) - </(/>,) = i(p + r)2 ^^ - \f^ dS,
=zprdd + ^fJ^ dd. (i)
In like manner the element of area P^PP'P^ is
d(P) - d(P>i = K^,p + rf dd - ira ^<?,
^\(^de^\k^f?de. (3)
Multiply (i) by >ij and eliminate k^pr dO between (i) and (2),
remembering that k^-\- k^z=z i. Then
d{P) = i,d(P,) + i,d{P,) - i^i,d(A).
Integrating for a complete circuit of the points P^ and P^ about
the boundaries of the curves, we have
(P) = i,{p,) + i,{p,) - i^yi). (3)
where the area of the director curve is given by
{A)=:^fpl>de,
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 PJ^^ = p is constant and equal to a^ we have
Holdtich's theorem^
(P) = >J.(i',) + i,iP,) - ik,i,c^fdff.
If a chord of constant length a 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 : n is
if P is distant c^ and c^ from the ends of the chord.
SXAMPLES.
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 d from its ends is na6.
2. A chord of constant length c movea. aboi^t 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, \itf*.
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* = ^ax is
(jV* — 4<wr)(y -I- 4rt*) = aV.
Check the result in Ex. 2 by the direct integration
jz dy = Icht
fxomy =— ooto^ = -|-oo» * Wng 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 F and the given curve is equal to half
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 /9,t 6| and p,* ^s ^^^ ^1^^ polar coordinates of points /\ and /\ on a straight
line, then the radius vector p of a point on this straight line whicli divides the
segment /\/', = A. so that PP^ = k^X, PP^ = k^, is determined by
P^ = Vi' + v.' - Vt^'- (I)
This is Stewart's theorem in elementary geometry. If 0 is the angle which p
makes with P^P^t then
Pi» = p* 4- ^i''^* — 2^iAp cos 0,
p^ = /a' -j- k^X^ + 2^,Ap cos <p.
The elimination of cos <p gives (i) at once.
Multiply (i) through by \dB, then
iP« d^ = k, \p,^ dO -f ^1 ip,» dQ - k,i, iA» dBy (2)
or diP) = k^ d{P,) + k, d{P,) - V2 <A^)y
and Elliott's theorem follows immediately on integration.
Art. 164.J ON THE AREAS OF PLANE CURVES. 239
The geometrical interpretation of (2) is as follows : Let \ = /\/} be constant.
Construct the instantaneous center of rotation / of A as P^P^Mms through ^6.
Then P^P' PP\ P^P^ (Fig. 86) subtend the angle M at /. 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 Planinuter.
In Fig. 86, let /'.T'. = /be constant At P let there be a graduated wheel
attached to the bar P^Pn in such a manner that the axle of the wheel is rigidly
parallel to P^Py This wheel can record only the distance passed over by the luir at
right angles to the bar.
Let PyP = /j, PP^ = /,. Let CP = r.
Then with the symbolism of § 164 we have
d(Pit - d(Pi = \{r -f /,)« ^ - ir« fl«,
= rl^ d^ -I- W ^0.
d{P) - 4/>,) = ^r»d« - i(r - /J« ^,
= rl^dfi ^ \i^ dO.
Adding these two equations,
APt) - 4^1) = /•'- ^0 + iW - /i») d0.
But r dB =z dR 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
(^1) - (^1) = KRx - -f 1) + W - VK9. - fii).
0|, 62 being the initial and terminal angles which the bar makes with a fixed
direction, and ^. , R^ the initial and terminal records of the wh^l.
Notice that wnen the wheel is attached to the middle of the bar
(^.) - (^,) = KRx - ^i)-
The path of /\ is a circle in Amsler*s instrument.
EXERCISES.
1. Find the area of the l]ma9on p = a cos B -\- b^ when b > a.
Ans. (^ -I- fi«)jr.
2. Show that the area of a segment of a parabola cut off by any focal chord in
terms of/, the chord length, and/, the parameter, is \l^P^-
3. Show that the area of the curve x^y* = (a — x){x — 6) is jr(a* ~ ^*)*'
4. Show that the whole area between the curve ^(a* -!-*•} = mcfl and the
jT-axis is mita'.
6. Show that the whole area between the curve y\a* — jc*) = ^ and its
asymptotes is 29r^.
6. Show that the area between the curve and the axes in the first quadrant for
{x/a)k 4. (y/^)* = I is ad/20.
7. Show that the area of a loop of the curve y* — 2^^' -|- a*jc^ = o is 21^/3^.
8. The locus of the foot of the perpendicular drawn from the origin to the tan-
gent of a given curve is called the^da/oi the given curve.
(I). The pedal of the ellipse (x/a)* -f (^/3)» = i is
f^ =^a^ CO8*0 -f ^ sin*^.
240 APPLICATIONS OF INTEGRATION. [Ch. XX.
Show that its area is \ie{a* -)- ^')*
(2). The pedal of the hyperbola (x/af — (y/bf = X is
p« = fl« cos«e - *« sin«e.
Show that its area is ab -\- (a^ — ^) Uxi-^{a/b).
9. If ^i» ^^s) ^5 be three ordinates, y^ being midway between ^| and^,, o£ the
curve
y = ojr* -|- bx^ + ^x + </,
show that the area bounded by the curve, the jr>axis, and the ordinates y^^ and y^ is
If we transfer the origin to Xj , o, and put Xi = —■ k^ jr, = -|- ^^ the equatioa
of the curve can be written
jK = ax« + /?*» + r* + ^•
We have for the area
£
and iA(^| + ^t + 4^s) ^^^ ^^^ same value. This is called Newton's rule.
10. Show that the area of any parabola
y = ax^ -{- bx -\- c,
from xz= — hftox^-^Af can be expressed in terms of the coordinates Xi , y^
and x^ , y^ of any two points on the curve, whose abscissae satisfy x^x^ = — |A*.
Ans. ^ = 2k'^y* - ^.^''t.
The mean ordinate in the interval is
Xj-*,
I /•+*
•^^ "^ 2>i7.A -^"^ " **"** "*" '•
Let/ and ^ be two undetermined numbers. Then
The three equations in p, ^,
M +M =0. (2>
/ + ^ = I, G>
give determinate values of / and g, provided
*.'. i**
•*1 I •*! »
= 0,
1,1,1
or XjX^ = — ^*«.
Then ym=^i + m,
and the values of / and g from (2), (3) give the result
11. In Elliott's theorem, g 164, (3), show that the mean of the areas of the curves
described by all points on the segment /\-P, is i[{^i) 4- (-^t)] — 1(^)*
12. A given arc of a plane curve tarns, 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 r and the perpendicular
from the origin on the tangent/, prove that its area is given by
1 r pr dr
14^ Lagrange's Interpolation Formula.
We have seen, in the decomposition of rational fractions, that when
^Jf) s (jf - fljXjf — a,) ... (x - an\
and F{x) is a polynomial in j: of degree less than n,
See § 133, and Ex. 79, Chapter XVIII.
If F{x) is any differentiable function of x, then, since
vanishes at x = tf j , . . . , 0«g, and the second term is a polynomial of degree m — I,
we have, § 98, II, lemma,
where ^ is some number between the greatest and least of the numbers x^
The formula
is called Lagrange's interpolation formula. The member on the left computes the
value at x of an unknown function when its values at ^ j , . . . f an are known,
with an error which is represented by
U-a,)..^.(x..an) ^^j
15. Gauss' and Jacobi's theorem on areas.
li /\x) is any polynomial of degree 2» — i, then the exact area of the curve
y = J^x) between x -=. p^ x •=, q can be computed in terms of n properly assigned
ordinates.
Let
Ti^\ - V ^^) ^^^•'^
^ ^"f ^-«r ^K)'
where, as in Ex. 14, ^(jf) a (jr — tf|) . . . (jt — «<»).
Then J{x) s F{x) — L{x) is a polynomial of degree in — I, in which F[x)
is of degree 2ft — i^ L{x) of degree « — i. Also, y{x) vanishes when x = Oj,
. . . , <7m. Hence
J^x) -L(x) = A iP(x) ^^(x),
where A is some constant and 0(x) some polynomial of degree i» — I, since ^x)
is of degree n.
Integrating between / and q,
r^x) dx -rL(x)dx = Ar<p{x) ^x) dx.
242 APPLICATIONS OF INTEGRATION. [Ch. XX.
Jacob! has shown as £d11ow8 that we can always assign a^j . . . , aj, , so that
•9
£
^iffdxzsO,
Fori integrating by parts successively,
where tpi'') denotes the result of differentiating 0 r times, and ^r the result of
integrating ip r times, remembering that 0(*^O is a constant.
If we take, after Jacobi, for the values a^ , . . . , a^ > the n roots of the equa-
tion of the »th degree
(s)V-/X' -?)]- = <>.
then the integrals 0| , . . . , ^n between / and q are all o, since each contains
{x — /)(j: ~ ^) as a factor.
Therefore, for these values of a^ , . . . , a» , we have
or the proposition is established.*
If the degree of /^x) is 2ff, then the area can be expressed in terms of » -|- I
ordinates taken at the roots of
The area of ^ = ^jr) can be expressed in a singly infinite number of ways if
one more than the required number of ordinates bie used, in a doubly infinite
number of ways if two more than the required number be used, and so on.
16. Show that the area of
from — ^ to -f- ^* is equal to
where ^'j and >, are the ordinates at x = ± ^/VJ* Give a rule and compass
construction for placing these ordinates.
* See Boole's Finite Differences, p. 52.
CHAPTER XXI.
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 J' is a function of x such that j^, Ufy, JJ^y, are uniform and
continuous functions in an interval x = oTf x z= /3, then the assem-
blage of points representing
in {a, p) is called a curve.
We can demonstrate * that if P and P^ are any two points on this
curve, we can always take
P and P^ so near together
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 Qy R be any other
two points on the curve
between P and -P,, then,
however near together are ""q
Q and R, the same property ^^^ g
is true for Q and R. ' ''
If we divide the interval (a, b) into n subintervals and at the
points of division erect ordinates to the points A, L, . . , , 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 ATRSVB circumscribed, to the curve AB.
* Appendix, Note 11.
243
244
APPLICATIONS OF INTEGRATION.
[Cii. XXI.
Let c^ represent the length of the rth chord, and /^ that of the rth
side of the circumscribed line.
Clearly, whatever be the manner in which (a, b) is subdivided or
to what extent that subdivision be carried, we shall always have
2cr < 2t^ and
'r-o
^Cr ^ £ 2t =
o.
NBOO
If we interpolate more points of division in {a, b), then 2i
decreases while 2c increases. Consequently 2t and 2c converge
to a common limit. This limit we define to
be the length of the curve between A and B,
i66. Let P be a point x^y on z curve,
the length of which between A and P is s.
Let -Pj be a point on the curve having
coordinates x + ^x^ y + Jy, and let the
length of the curve between P and P^ be
Jj, the length of the chord PP^ be Ac.
ax "^ Draw the tangents at P and P^ Let the
angle which PT makes with Ox be d, and
FIG. »8. ^j^^ ^^gj^ between TP and TP^ be ^6.
Let TP = /, rPj = /,, then, by § 165,
Ac < As <t + /j.
But, from the triangle PTP^,
{Acf = /2 + /j« + 2//j cos AO,
= (/ + /,)'-4//iSin«-J-J^.
' 4//1
- (^J=-
(/ + /,)3«'"'^^''-
4^i/(^+A)' can never be greater than i, and when Ad{=z)o,
Jc:(=)o, / + /j(=)o, also sin^ \A6( = )o. Therefore when Ax{==)o,
we have
im- ■:
A*(-)o
Since, by definition, As lies between Ac and / + /^ , we also have
/Ac
A*(-)o
Now,
or
{Acf = ( J;t)« + {Ay)\
(^)'=(S)(0='+(^)"
Art. i66.] ON THE LENGTHS OF CURVES. 345
Therefore, in the limit, for Jx{^=)o,
(^)"='+(S)'
or
Hence the length of the arc of the curve from ^ to P is
s
In like manner, using 4y instead of Jx, we obtain
s
In differentials
=/'>l^ + {wh- <^>
Since dy/dx = tan 6^, B being the slope of the curve at x^ y, we
have
dx = cos (^ ds, dy = sin & ds,
dx dy
Therefore -=-, ^ are the direction cosines of the tangent to the
as as
curve at x, y.
EXAMPLES.
1. Rectify the semi-cubical parabola ay* = 3^,
the arc being measured from the vertex. This was the first curve whose length
was determined. The result was obtained by William Neil in i66o.
2. Rectify the ordinary parabola y* = 2ax,
We have DyX = y/a*
.-. ^=^f i/fl« 4- y* dy,
= 1, y^qiT^ + liog^liLi^^I?,
2« ^-^ ^ ^ 2 * a '
the arc being measured from the vertex.
3. Rectify the catenary y •=- \ci \/« -^ e */ .
We have Dy =^ \\e^ — e"^) .
.'. 1 = ^1" \e'^ + ^^ «^/ <£r = 4^ V^*" — e~^) .
246 APPLICATIONS OF INTEGRATION. [Ch. XXL
Show that s = PJL Fig. 73. Also, ATL = constant. The catenary is theie£>re
the evolute of the tractrix represented by the dotted line in the figure.
4w Rectify the evolute of the ellipse
(Ar)i -I- (by)\ = (tfi _ ^«)l.
Write the equation in the form
put x = a sin'0, y ^ fi cos'0.
. •• s = ^ J (a* sin«0 + fi^ cos* <p)kd{a^ sin»0 + jP cos^) •
Measuring the curve from x :=■ o, y = fi, we get
^ _ (a* sinV -I- /P cosV)i — yS»
^- a«-/3«
5. Find the length of the curve gay* = x{x — 3fl)*, from j: = o to x = 3<i.
6. Find the entire length of the kypocyclotd jc* •+ ^* = «♦. Ans. ta.
7. Find the length of the arc of the circle x^ -^ y* =z a\ from x = oto x =: d^
and the whole perimeter.
8. Find the length of the logarithmic curve y = ca^.
We have DyX = b/y^ where b = i/log <j.
9. Find the length of the tractrix (see § 163, Fig. 84)
= -«J y = - «log^ +
const.
y -' ■
Measured from the vertex, x = o, y = a,
.S z= a log (a/y).
X* y*
10. Length of an arc of the ellipse — -|- ^ = i.
Put X = a sin <pf y =s b cos <p. Then
5" = f{a* cosV + ^* sin V)* ^0,
= tf Ai — ^ sin* 0f dip,
where e 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 = o, 0 = \n. Since ^ sin'0 is always less than i, we can expand
the radical in a series of powers of sin 0, and integrate, obtaining the length of the
quadrant (see Ex. 27 § 149)
2 ( i\2/ I \2.4/ 3 \2.4.6/ 5 1
Art. 167.]
ON THE LENGTHS OF CURVES.
247
Polar Coordinates.
167. If p =/(^) is the equation to a curve, and p, B are the
coordinates oi P \ p + Jp, 5 + J^, those of -Pj, then, calling ^c
the chord PP^ , we have
{/ley = (p + ^P)' + P* - 2p{p + Jp)cos J6f,
= (Apf + 2p(p + Jp)(i - cos M).
Hence
But when J6^(=)o, we have Jp(=)o, Jc(=)o, and
/* I — cos
jr _^ I /*sin ^ __ I
Ji^-)0
- (^)'=(S)'+^.
Fig. 89.
ds dc . ds
-^ = ;^, since :t;= I, §166.
• • dd dd
ds^ =1 dfi^ + fy^ d(^,
dc
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 jc = p cos 0, y — p sin ft
. •. </j; = dp cos ^ — p sin 6 dd, dy =. dp sin 6^ -f p cos 6^ dO.
.-. ds^ = dx* + dy^ =z dfi^ + p^de^.
248
APPLICATIONS OF INTEGRATION.
[Ch. XXI.
1. Find the length of the cardioid p ■=. a{i -\- cos G).
D^p = — 47 sin 0, and therefore •
s — af[(i 4- cos 0)» + sin«e]» d$ = 2a f cos \B dB = ^ 9in ^ | *.
The entire length is &x.
2. Show that the length of the arc of the spiral of ArchitnedeSy p = oBj from
the pole to the end of the first revolution, is
a[n firpiSi" -f I log (2jr -f- f i + 41c* )].
3. Logarithmic spiral p ■=. a^. Put b = l/Iog a.
Then J = r^{i-\- ^»)* ^p = (I + ^•)*('-« - ''i)-
4. Show that the length of the arc of log p = aO, from the origin to (p, 0), is
a
5. Find the length of p + B* = aK
6. Find the length of p = a sin 0, from o to ^K.
7. Find the length of p •=. a sec 6, from o to |^.
8. Show that the entire length of p -=. a sin' ^0 is fira.
9. Show that the entire length of the epicycloid
4(p« — tft)» __ 2.^a^f^ sin'e,
which is traced by a point on a circle of radius \a rolling on a fixed circle of radius
0, is I2<7.
10. Find the entire length of the curve p = a sin 26.
11. Show that the length of the hyperbolic spiral p^ -=. a is
[
4/tf* -f- p* — « log
a 4- i/a«-|-p'
12. Show that the length of the parabola p — a sec* ^6, from 6 r= — ^«
to fl = -j- ^«, is 2fl(sec ijr 4- 1^ t^"^ 1^)-
x68. Geometrical Interpretations of the Differential Equations
ds^ =: di? ■\- dy^ and ds^ = </p« + p«</0«.
I. In Cartesian coordinates, if we take x
as the independent variable, then we have
PM = dx. Also, since
Dy = tan e = tan MPT,
.-. dy ^\3.iiB dx in MT,
. •. ds^ = PM^ -f- i«/r« = /^r*.
Hence ds = PT*, while tlie correspond-
ing difference of the arc is PP^* Also, we
have the relations
dx =. ds oy&By dy rz-ds %\nB.
It is easily shown geometrically that the
limit of the quotient oids •=. PT, by either
Fig. 90- As = PPy^ or Ac = PP^, is 1, when dxmAx
converges to a See definition of the length of a curve, § 165.
Art. i68.]
ON THE LENGTHS OF CURVES.
249
11. We can in the case of polar coordinates exliibit ds^ dp^ and p <^ as the
lengths of certain circular arcs as follows:
Let OA be the initial line and P a
point on the curve /(p, 6) = o, /'7'the
tangent at P. Draw OC perpendicular
to p = OPy cutting the normal at P
in C Then n =s PC is the normal
length, and S^ = OC is the subnormal. ^^
Let 0 be the independent variable, then
dB = JS is an arbitrarily chosen angle.
We have the difierentials dsy dp, p dB
proportional to the sides of the triangle
POC, or to If , Snt p, respectively. For
we have
(S)*= ©■+"■
But dp = Sn dBy by g 92, (5). Also, 5;> + p* = if*. Hence ds =:nde.
Draw i^' parallel and equal to OC, Strike the arcs PN, PL, and iW with centers
C radius 5*, C radius if, O radius p, having the common central angle ^6 = d^.
Then
ds = PL, dp^PN, pdB = PM,
It is interesting to notice that the rectilinear triangle PLNvi a right-angled tri-
angle similar to PCO\ the sides of which, PL, PN, NL, are equal to the chords
subtending the arcs PL, PN, and PM respectively.
Therefore, in the triangular figure PZA^ whose sides are
the circular arcs PL, PN, and an arc ZA^with radius p equal
to PM, ;we have the sides {PN) = dp, (PL) = ds, {LN) =pdB.
Also, the angles between the circular arcs are
(Z) = ijr-^, {P) = il,, (J\r) = iie+ de.
and ds* = dp* -f p'<**,
dp ss ds cos ^, p dQ = ds sin V*.
Fig. 92. ^" order to prove these statements, it is only necessary to
show that the rectilinear segment ZA^ is equal to the chord
subtending PM, Let jr, ^ be the chords subtending PL, PN, Then from the
rectilinear triangle PNL we have
LN* = 31* J^y* - 2^0^ cos LPN,
But I LPN = ^ -f i Je - \AB = ^.
Also, JT = 211 sin \AB, y = 2Sn sin \AB,
.', LN* z= 4(5',* -f «» - ^nSn cos ^) sin* \A^,
= 4p« sin« \AB = (chord MP)*,
The remainder follows easily. v p
Observe that if we draw PM perpendicular to OP, as ^» • *
in Fig. 93, and put PM = 5/, MP^ = 5p, then we have,
for J0(=)o,
/As _ r ^p ^ r^p _
Therefore the difference equation
J^ = 6p* •\- ^p»
leads at once to the differential equation
ds* z=^df^J^f^ dB*.
Fig. 93.
250
APPLICATIONS OF INTEGRATION.
[Ch. XXI,
169. Radius of Curyature and Length of Evolute.
li/{Xy ^)= o is the equation of a curve, then
dy
:f-=tan^,
ax
ebr dx
Fig. 94.
(1)
Hence if R is the radius of curvature at or, y^
^ y, -sec«/^^^^^,
since ds = sec 6 dx. Therefore ds = R dB.
The angle ^0 = dd is the angle between the tangents at P and P, ,
and is equal to the angle between the normals at P and /*,.
170, The length of the arc of the evo-
lute of a given curve is equal to the differ-
'*y ence of the corresponding radii of curvature
of the involute.
Let x,yht Si point on the involute cor-
responding to the point or, /3 on the evo-
lute.
Then we have for the radius of curvature
I^={a-xy+{/3^y)\
Differentiating, we get
PdP={a- x){da -'dx) + {/S ''y)(d/3 - dy),
= (a - x)da + {/3^y)d/S,
since {a — x)dx + {/3 —y)dy = o, this being the equation of R,
the normal to the involute, lid is the angle which the tangent to the
involute at jr, y makes with Ox, then, since R is tangent to the evolute,
R makes with Ox the angle 0 = -J^w + 6^, and we have
a — jc = — J? sin S = J? cos 0, fi — ^ =: RcosO = R sin 0.
Hence, on substitution in (i),
dR = da cos <f) -^ d/3 sin 0,
= da;
if c is the length of the arc of the evolute.
Integrating between a^, /3^ and a,, )5,, we have
^, - ^1 = ^. - ^v
This can be shown otherwise, for we have
{X - a)da + {y- /3)d/3 = o, (2)
the equation to the normal to the evolute at a, /3.
The perpendicular R, from x,y on the involute to the line (2), is
^^ {X - a)da+iy - /3)dfl
j^dd^ + dpi '
or {x - a)da -f (^^ - P)d§ = i? (/tr.
Equating with (i), we ge** dR ^=.do' 2& 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 nor a minimum value.
For when R passes through a maximum or a minimum value dR
changes sign. jdR would be zero when taken between two points
at which R 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 absolute 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.
SZAMPL£S.
1. Find the length of ay^y' = 4(^1: — 2^)*, the evolute of the parabola^ = ^ax.
We have for the coordinates y, y of the center of curvature and ^1 the radius
of curvature of the parabola at x, y,
x' = «« + K. / = -^. ^ = a<.(^)*.
Measuring the arc of the evolute from the cusp, x = 2a, j^ = o, to y, /^
we have
2. Find the length of
(tfx)I + (by)^ = (tf« - ^)I.
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 ^ of a curve with a fixed tangent, and the
angle 0 which the tangent at the end P of the arc makes with the
fixed tangent, are called the intrinsic coordinates of a point on the
curve.
The equation /![ J, 0) = o, which expresses the relation between s
and 0, is called the intrinsic equation of the curve.
To find the intrinsic equation of a curve y^ar,_>')=o ot/[p, 6)=o,
we have to find the length of the arc from a fixed point to an arbitrary
252 APPLICATIONS OF INTEGRATION. [Ch. XXI.
point on the curve, then the angle 0 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 catenary.
Take the vertex as the initial point, then
y^\a V^ + r -j .
Dy := \ V « — e~^) = tan 0.
Also, J = ^ (^^« — e ^) .
Eliminating x, we have s =a tan 0.
2. Find the intrinsic equation of the involute of the circle.
Let X* -{-y* ^ a* be the equation to the circle. Unwrap the arc beginning at
the point a^ o, and let the radius to the point of contact make the angle 0 with
Ox. Then 0 is also the angle which the tangent to the involute makes with Ox,
The radius of curvature is the unwrapped tangent length, or ^ = atp. But
ds = H d<p = a<p d(p, ,\ J = \a<fi^,
172. General Remark on Rectif cation. — 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.
EZSRCISBS.
1. Show that the length of Za^y = jc* -j- 6a'jr* measured from the origin is
2. Show that the whole length of 4(jr* -j- y^) — a* z= 3a V* is 6a.
3. Show that a^y^ = jr«+« is a curve whose length can be obtained in finite
terms when — or 1 is an integer.
4. Show that the intrinsic equation to ^ = ax^ is
s = -fifa (sec'0 — I).
5. Show that /o*" = a*" cos tn6 can be rectified when i/m is an integer.
6. If X = 0(/), y = ^/), then
(§)"=e)"+(D'=[^<o)-+w)p.
7. In the cycloid jc r= fl(9 — sin 0), y z= a vers G, show that
ds = 2a sin ^0 d%.
Hence the length of one arch is 8a.
Art. 172.] ON THE LENGTHS OF CURVES. 253
8. Show that the length of the cycloid
X = a cos—' ^ -f- V^^y — y*
from the origin to jr,^ is |^&7^.
9. Show that the intrinsic equation of the cycloid in Ex. 8 is
J = 40 sin <pf
the tangent at the origin being the initial tangent.
10. In the equiangular spiral p z=z ae»^ show that s = p sec ^, where tan 0
= ftf measuring s from the pole.
11. Find the length of the reciprocal spiral from 0 = 2ie toB =z 4X', the equa-
tion being pQ = a,
12. Show that the whole perimeter of the lemnisccU:
p* = a* cos 2$
\ ^2.5^2.4.9^2.4.6.13^ /
13. Show that the length of 7* = x* between x = o, jt = i is ^ (13' — 8).
14. Designating by Z*Zj (/) the length of the curve /(*, y) = o, from x = a
to X = ^, show that:
(fl). LVMx* - 6xy + 3) = «.
(d), L%:i(xl ^ yi ^ J) = 6a,
(0- J^'x'^iby - ■* Vi^"^^ - log(jr - i/^^~^i)'] =: fxdx^^
id), li'jj^x - i^di-^^+ a log ^ + ^^' -"'
= « log -.
(/). ^^"iC-K - V^^^TF* - sin-» y7) ^Tx-^dx = 2.
(^). Zj;:;(j»'» + 4^-2 log;^) = i(2 log 2 -f- 3).
15. Use the binomial theorem to evaluate
16. Show that
LllMy - sin jr) = «(i + i - A + ih - • • • )
17. ^f :f (P' - aopS + «') = r <3 + log 4).
P » 4
20. A hawk can fly v feet per second, a hare can run 1/ feet per second. The
hawk, when a feet vertically above the hare, gives chase and catches the hare when
the hare has run b feet. Find the length of the curve of pursuit.
254
APPLICATIONS OF INTEGRATION.
[Ch. XXL
Take O^ the starting-point of the hawk, as origin, the line O/fdnwn to the
starting-point of the hare /^as ^-axis, and a parallel Ox to the hare's path as jc-axis.
When the hawk has flown a distance s to /\ the hare will have run a distance or to
jP* in the tangent to the curve at P,
Fig. 95.
Let PP = /. We have d =t/T=z//C. 5 = v r, the length OPC. T being
the time of pursuit.
. •. -- = — = ^ =s constant.
S V
In like manner,
~~ 5S ^'~ — S K%
5 V
dcr = kds. Also,
/« = (o- - j:)» -h (fl - y)K
tdt = (<r — JcX^/o" — dx) — (fl -^ y)dy.
But o* — jr = / cos G, <z ~ ^ = / sin 0. Also, </(r ca& ^ \s k ds cos B or
>( ^x, and
dx cos 0 -|- ^ sin 6 = </^.
. *. dt sz k dx — ds.
Hence
S^kCdx-^fdiz=zkb-\~
Jo «/«
'\^^h'-
21. If ^ and rare the radii of the fixed and rolling circles in the epicycloid and
hypocycloid
R ±. r R + r
x = (R ± r)cos 0 1^ rcoB ^, y z= {R ± r) sin0 T r sin — =— ^,
show that the lengths of the curves from cusp to cusp are respectively
Sr{R ± r)/R,
22. In § 164 (Elliott's theorem), if ^i , x, s are the corresponding lengths of
the arcs described by the points /\, /",, ^respectively and a the corresponding
length of the director curve, show tnat
ds* = >t, ds^* + Jkj^ ds* - Jk^Ji^ da*.
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
hne.
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.
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.
Fig. 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 iJCthe 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 CD, the length of the generating
line.
^55
256
APPLICATIONS OF INTEGRATION.
[Ch. XXIL
It IS evident from the definition of a revolute that any section of
a revolute by a plane perpendicular to the axis AB 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
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-
^ tion of the line ACDB about the axis AB, then the
section AD' B is nothing more than one position of
the generating line ACB. Again, the revolute can
always be regarded as being generated by a circle
moving in such a manner that its center moves
along the axis to which its plane is perpendicular,
and its radius changes according to a given law.
Fig. 97.
.or
174. Volume of a Revolute. — Let y =yT[^) be a curve AB,
We require the volume of the solid generated by the figure aABb
revolving about Ox as axis
of revolution.
Divide (a, U) into n
subintervals, aiid pass
planes through the points
of division cutting the
solid into n parts, such as -^ J ^
the one generated by the >H a
revolution of xPP'x', ^
We can always take x' —
X = Ax so small that the
curve PP' will lie inside
the rectangle PMP'M\ if
f{pc) is continuous. Let
y be an increasing one- ^^^* 98-
valued function from x — a Xo x :=^ h. The volume, A F, of the
solid generated by xPP'x\ lies between the volumes of the cylinders
generated by the rectangles xPMx' and xMP'x', Hence for each
subdivision of the solid we have
ny^Ax < AV< ny^Ax.
(I)
The whole volume of the revolute, therefore, lies between the
sum of the n interior cylinders and that of the exterior cylinders, or
^ nfAx < V <2 Tty^Ax. (2)
t I
But if we interpolate more points of subdivision in {a, 5), 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
A4C<«)0
these sums have a common limit, which is K
I
Ax(-)o n
ffaOO I
= f^^y^ dx, (3)
Again, we have directly from the inequality (i)
Hence, for Jjir(=)o, we have
dV
= ^,
dx
since J[^y' = v, when Jjr(=)o.
.-. dV—nydx, (4)
and as before
y,= 7r£ydx. ' (5)
In like manner we show that if a: is a one-valued function of j^,
say X = 0(jk), then the volume generated by the revolution of the
curve about Qv as an axis, included between two planes perpendicular
to Qv at y = / and ^ = ^, is
V,:=nflx^dy. (6)
EXAMPLES.
1. Find the volume of the cone of revolution generated by the revolution of the
triangle formed by the lines jr = o,^ = o, x/a -\-y/b = i, about Ox as axis.
But a is the altitude and ^ 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
oi:^ -\' y^ = a* about Oy,
yy
=zT[f ""x^dy^nj ' (a« - v«) ^ = |jrfl».
3. The prolate spheroid is the revolute generated by the revolution of an ellipse
about its long axis, sometimes called the obiongum.
258 APPLICATIONS OF INTEGRATION [Ch. XXIL
Let a be the semi-major axis of x*/a^ -|- y*/6^ = I.
Then we have for the volume of the oblongum
Vjc = * J ' j,(fl* - J^)dx = |jrfl3«.
4. The ^6/a/^ spheroid or oblatum is the revolute obtained by revolving the
ellipse about its minor axis; show that its volume is \icaHy where b is the semi minor
axis.
5. Show that the volume of the revolute obtained by revolving the parabola
/• = 4Ar about Ox^ from jr = o to ;r = «, is 2;ra*.
This is the /arA^<7/(;i</ of revolution.
6. If the hyperbola -5 — tj = ' revolves about Oy^ the revolute is called the
hyperboloid of revolution of one sheet. Show that the volume ftx>m y znoXoy z=.y
If the curve revolves about Ox^ find the volume from x = a to j: = 2a. This
surface is called the hyperboloid of revolution of two sheets. ,
7. Find the entire volume generated by the revolution about Ox of the
hypocycloid jr* + ^* = a*. Am. ff^iea^-
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, rind its volume.
Here y* dx ^ ^ (a* '~y*)^y dy. Hence the volume from x z:iO\ox ^= x is
Vs^^ft f («* - y*)^ dy = \%{a^ - y )l.
The volume of the entire pseudo-sphere is f ir^, or one half that of a sphere
with radius a,
9. Find the volume generated by the revolution of the catenary
^=-|fl\^*-|-^ */ about Ox from o to jr. Ans, \ica(ys 4- ox\
10. The volume generated by revolving the witch (jr* -j- \ti^)y = &i* about its
asymptote is 4;r'a'.
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
MABCN and MADCK Hence the
volume of the solid ring generated by
ABCD revolving about Oy is
V,= nf{x*-x,*)dy,
where x^ = BD^ x^ = RB, and the
Fig. 99. limits of the integral are y = OM,
¥'=. ON. A corresponding integral gives the volume about Ox,
Art. 176.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 259
BXAMPLSS.
1. The solid ring generated by the revolution of a circle about an axis external
to it is called a torus. Show that the volume of the torus generated by the circle
(jr — af +/« = r»
(a ^ r) about Oy is %i^h*a.
We have jr, = a + 4/^ —y,
. •. Vy = jr J 4^ ^f* —y* dy = 23rV*tf.
Observe that the volume is eqtial 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
—55—+^ = '
{c > a) about Oy is 2j^abc.
176. The Area of the Surface of a Reyolttte.— We know, from
elementary geometry, that the carved 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
n(VD^AD'~ VC'BC).
Since BC/AD = VC/VD, we deduce for the
surfeice generated by the revolution of CD about Fig. loa
BA the area
2nMN'CD,
where ilfiV joins the middle points of AB and CD.
In the figure of § 1 74, Fig. 98, subdivide, as before, the interval
(a, h) into n parts; erect ordinates to the curve ^^ 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 AB
a polygonal line AB with n 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 Ac = PP' revolving about Ox has for its
area
^n^"^^ Ac = 2n(y + \Ay)Ac.
We define the surface generated by the revolution of the arc of
the curve AB about Ox to be the limit to which converges the isur-
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.
/
26o APPLICATIONS OF INTBGRATION. [Ch. XXII.
To evaluate this limit, we have for the area of the surface gen-
erated by the curve AB
MaOB I
Ax(-)0 M
= £ 2 27ryds.
«a« I
Since for each pair of corresponding elements of these two sums
we have
2nyds
Hence we have, by definition of an integral,
S^=27t yds.
In like manner, if AB revolves about Oy^ we have for the area of
the surface generated
o]y = 27r / xds,
EXAMPLES.
1. Find the surface of the sphere generated by the revolution of the circle
y* = a* — x^ about Ox.
dy X ds d
We have -r- = » "t- = — •
dx y dx y
.«. 5";, = 2itJ yds = 2;r(r, — x{}a.
Hence the area of the torn included between the two parallel planes is equal
to the circumference of a great circle into the altitude of the zone. If x, = 4~ ^i
jTj = — fl, we have the whole surface of the sphere 4,110*.
2. Show that the curved surface of the cone generated by the revolution of
y =x tan a about Ox, from x = o to j: =r A, is it A* tan a sec a. Verify the for-
mula deduced for the surface of a frustum in § 176.
3. Surface area of the paraboloid of revolution.
Let J* = zmx revolve about Ox. Then
^« = ^ ^y + '«')V </j' = g ] (^ + «.«)*-«•[ .
4. Let 2a be the major axis of fl^' + ^*-«^ = <'*^'» and e its eccentricity. Then
we have for the surface of the prolate spheroid
^ 2Kbe /•+« \a* ^ , . / , f- , sin-»A
Art. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 261
5. Show that the surface of ^t pseudo-sphere is
S^ = 2na i ify = 2ita(a — yy
Its entire surface is 2ira'.
6. In the catenary show that
S» = ie{ys + ax),
Sy = 23r(tf' + xs. — tfy),
from ;r = o to jr = X.
7. Show that the surface of the hypocydoid of revolution generated by
x^ -\.yi = rtl about Ox is J^*tf*.
8. A cycloid revolves around the tangent at the vertex. Show that the whole
surf&ce generated is ^f-jed*,
9. The cardioid /o = a(i -|- cosG) revolves about the initial line. Show that the
area of its surface is *{-iccfl.
17*7. If a plane closed curve having an axis of S3rmmetry revolves
about an axis of revolution parallel to
the axis of symmetry and at a distance ^
a from it, then we shall have for the «
volume and surface of the revolute gen-
erated, respectively,
V = 27raA, S = 27raL,
where A is the area and Z the length of
the generating curve. ^
Let X = ahe the axis of symmetry O
and Oy the axis of revolution. Then
for the volume
Xi a
Fig. ioi.
«»
But if CAf ■= CN = x\ .V, = fl + x\ jtr^ = a — j/,
^y = 27taj 2x' dy = 27raA.
For the surface
Sy = 2;r^ {x^ + x^) ds,
= 2;ra / 2ds r= 2naL,
The results obtained assume that the axis of revolution does not cut
the generating curve.
EZAMPLBS.
1. The volume and surface of the torus generated by the revolution of a circle
of radius a about an axis distant c from the center (c ^ a) are respectively in^a^c
and ^n^ac,
2. The volume generated by the revolution of an ellipse, having 2a, 2b as major
and minor axes, about a tangent at the end of the major axis is 2ie'^aH.
262 APPLICATIONS OF INTEGRATION. [Ch. XXII.
BZSRCISS3.
1. Show that the segment of the parabola y^ = 2/jr, made by the line x = a,
when rotated about Ox^ generates the volume
2,np I X dx =z Tc^aK
2. The figure in £z. i rotated about the ^-axis generates the volume
3. The volume generated by the closed curve x* — a*x^ -f- a*y* = o about the
jr-axis is
4. The curve x* -^ y^ =z i rotating about the ^'-axis generates a solid whose
volume is |;r.
5. The volumes generated hy y =i e* about Ox and Oy are respectively
jr / e^dx = iff. It I (log^)»d^ = zn.
6. The curve ;^ = sin x rotating about Ox and Oy^ respectively, for 4r = o,
jf = jr, generates the volumes
ir Ain'jr dx = ^jr*, 1C f (it* — 2itx) cos x dx =z 2it\
7. The volume generated by one arch of the cycloid
X =s a(B — sin 0), y z=z a{i — cos 9), rotating about Or, is
32jra» r^sin* ^ ^(^6) = sif^-
8. The same branch rotating about Oy gives the volume
4jr«fl' f'{it — e + sin 0) sin e ^ = 6jr«fl».
9. Show that the whole surface of an oblate spheroid is
e being the eccentricity and a the semi-major axis of the generating ellipse.
10. The curve y\x — 4a) = axix — 3a), from x = o to x = 3<i, revolving
around Ox generates the volume ^^^'(15 — 16 log 2).
11. The curve ^'( 2a — x) = x* revolves around its asymptote. Show that the
volume generated is 2ff*fl'.
12. The curve xy* = 4a*{2a — x) revolves around its asymptote. Show that the
volume generated is \i^a^,
13. Find the volume and the surfece generated by revolving >^' = \ax about Ovy
ftom jc = o to JT = fl. Am, V= \itc^. S = iffa«[6i/2— log(3 +21/2)] .
14. Show that the volume generated by revolving the part of the parabola
x^ -{- y^ = a^ between the points of contact with the axes about Ox or Oy is i^jro*.
Art. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 263
15. The surfkce generated by^ = jr*, from jr = o to jr = i, rotating about Ox,
IS
21c £ i^i + 9x*j*dx = ^ ( |/iooo- 1).
16. The surface generated by jr* — aV + Sa*y* = o, about Ox, firom x =: o
to X = a, is
4^' 1/0
17. If a circular arc of radius a and central angle 2a < fC revolves about its
chord, the volume and surfcice of the spindle generated are respectively
2jra'(} sin a -|- } sin tt cos'a — d: cos a), ^ica^sin a — a cos a),
18. The surface generated by jT* -}- 3 = 6xy turning about Oy, from j: = i to
X = 2j has for area it{^ -f- log 2), and f{;r when turned about Ox.
19. The surface generated by >^* -}- 4jr = 2 log ^, rotating about OXf fromjf = i
to^ = 2, is J^jr.
20. The area of the surface of revolution of
2y = X ^x* — I -f log (j: — ^x* — i),
about Oyy from j: = 2 to j: = 5, is 78?r.
21. The surface of the cycloid of revolution is ^fCa*, and its volume is 5)r*a',
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 ^na^ and ^a*.
23. When, in Ex. 21, the normal at the vertex is the axis of revolution the sur-
fiice and volume are respectively
24. Show that when the lemmscate p* = a' cos 2O is revolved about the polar
axis, the surface generated is
4jr<i' i sin 0 <fl9 = 2%a^(2 — ^^2),
26. Show that if the ciunre y* = ajt* -{- bx -{■ c be revolved about Ox^ the
volume generated between x^ » x^ is
where ^w is the ordinate at ^x, -|- -^i)*
This curve can be made any conic whose axis coincides with Ox, by properly
assigning the numbers a, by c. 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
ory = (X - d^b - X),
revolving about Ox as an axis, is
ir j(fl + *)log^ - 2(/^-fl)L
27. The volume of the heart-shaped solid generated by revolving p = a(i-f-C08 0)
about the initial line is \icefi,
28. Find the volume of the hour-glass generated by revolving the curve
y^ — 2ry 4- 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 H is the distance
between any two circular sections A^ and A^^ and A the area of the
circular section at a distance h from A^^ then the volume included
between the sections A^ and A^ is
=r
I
I
U
^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 -4 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 n sides which will generate a prism as the
curve moves in the manner described. If P is the
area of the polygon and H its altitude, then PH
is the volume of the prism. When « = 00 and
each side of the polygon converges to o, 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 AH^
179. Volume of a Solid. — ^Consider any solid bounded by a
surface. Select a point O and draw a straight line Ox in a fixed
direction. Cut the solid by two planes perpendicular to Ox at points
X^ , X^ distant X^ and X^ from O,
Whenever the area A of the section PM oi the solid by any plane
PM perpendicular to Ox, distant x from O, is a continuous function
264
Fig. 102.
Art. 179.]
ON THE VOLUMES OF SOLIDS.
265
/
/
/
/
of X, then the volume of the solid included between the parallel
planes at X^ and X^ is
To prove this, divide the interval between X^ and X^ into a large
number of parts, n. Draw planes through the points of division
perpendicular to Ox, thus dividing the solid into n thin slices, of
Fig. 103.
which MPP^M^ is a type. Let A be the area of the section PM, and
A^ that of section P^My at a distance x^ from O. \jtVAV be the
volume of the element of the solid included between the sections at
X and Xyy and x^^ x =. Ax the perpendicular distance between ^the
sections.
We can always take Ax 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 A V. Let
the area of the curve traced by this line on the plane PM be A',
Then the volume of this cylinder is
6V' = 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 x and x^ a cylinder
of which the element of volume of the solid is a part. Let this
straight line trace in the plane PM a curve whose area is A'\ The
volume of this external cylinder is A" Ax,
Hence we have
A'Ax < AF<A''Ax,
or
Also, necessarily, from the manner of construction of the lines
bounding the areas A' and A'\
A' <A < A'\
266
APPUCATIONS OF INTEGRATIOX.
[Ch. XXUL
If now the soiface of the solid is such that the bonndaiy of the
section P^^ at jr, converges to the bonndaiy of the section PJfzt
X, when jr^(=)jr, then also A\^)A, A''{=z)A, and we have
Therefore
When A is determined as a function of jt, say A = 4>{^)t ^^>^
the evaluation of F is a matter of integration, and we have
y = fl^<t>{^) <i*'
L If the parallel pUne sections of any solid have equal areas, then
Therefore, if a plane figure moves in any manner withoat 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-
tode is equal to the distance between the initial and terminal piisitions of the
generating plane.
2. The general definitka of a cone is as follows:
A straight line which passes through a fixed point and moves according to any
law generates a saAce called a £fffl^. In general, the cone u defined by a straight-
line generator passing through a fixed point, the vrrtexj and always intersecting a
given curve, called the directrix.
A cone is generated l>y a straight line passing through
a fixed point V, and always intersecting a closed plane
curve of area B. Find its volume.
Draw a perpendicular VM =z // to the plane of the
curve. Draw a plane parallel to B cutting the surface in a
curve of area A, at a distance fW = A fiom K Then we
shall have
A_ i*»
B-JP'
For, inscribe any polygon in the curve B and join the
comers to V. The edges of the pyramid thus formed intersect the parallel plane
containing A in the comers of a similar polygon inscribed in section A, HP and/
are the areas of these polygons, we have
from elementary geometry. But A and B are the respective limits of/ and P. The
volume of the cone is then
3. A catund 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
But
Fig. 105.
plane. If the generating line is always perpendicular to the fixed straight-line di-
rector and traces a curve in a 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
base.
Let A VC be the straight-line director at a dis-
tance VD = Iffxom the plane of the base.
Any plane VNM perpendicular to VC cuts out
of the surface a triangle of constant altitude H^ and
base MN •=. y. This triangle moving parallel to
itself generates the volume required. Hence
^ Tdx:=zf \Hydx,
0 Jo
where T = area MV//, x = OD = AK
t j ydx = B, the area of the base. Therefore
y = ^^B.
The volume of the conoid is therefore half that of a cylinder on the base B
having the same altitude //.
This is at once geometrically evident by constructing the rectangle on AfAT as
base with altitude M
4. On the ordinate of any plane curve, of area B^ as base a vertical triangle is
drawn with constant altitude //, Show that whatever be the curve traced by the
vertex Fin the plane parallel to the base, as the ordinate generates the area of the
base, the triangle generates a volume ^J/B.
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 A when at distance I/, show that the volume gen-
erated is ^AJ/.
6. The axes of two equal cylinders of revolution intersect at right angles. The
solid common to them both is called a groin. 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
sur&ces of the cylinders in the straight
lines AE and B£. The surfaces inter-
sect in CME. A plane DLMN parallel
Xa xOy cuts the cylinders and the vertical
planes xOC^ yOC in a square, which
moving parallel \o xOy generates one
eighth of the groin. Let x be a side of
this square, whose distance &x>m C? is ^
Then x' = «* — >l». Hence
^V = J\a^ - h^)dh ^ \c^.
The volume of the groin is ^'o*,
where a is the radius of the cylinders.
Knowing that any figiure drawn on a
cylinder rolls out into a plane figure, show that the entire surface of the groin
is i6a>.
7. Oxf Oy, Oz are three straight lines mutually at right angles to each other.
A cylinder cuts the plane xQv in an ellipse of semi-axes OA = tf, OB = b\ and
the plane xC7s in an ellipse with semi -axes OA = a, OC = c. The generating
268
APPLICATIONS OF INTEGRATION.
[Ch. XXIII.
lines of the cylinder are parallel to BC. Show that the volume of the cylinder
bounded by the three planes xOy^ yOs, zOxis ^idc,
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 |a* tan a.
9. On the double ordinates of the ellipse 3*x* -f- ^'V = ^*^* ^^^ ^^ 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 ^6* cot a.
10. Two wedge-shaped solids are cut from a right circular cylinder of radius a
and altitude A^ by passing two planes through a diameter of one base and touching
the other base. Show that the remaining volume is (jr ^ 1)^'^*
11. Two cylinders of equal altitude A have a 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-[^ie)/je times that of the sphere.
13. Two ellipses have a common axis and
their planes are at right angles. P^ind the
volume of thb 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 A OB represent quadrants of
the given ellipses.
OA = a, OB — b, OC = e.
Then LMN represents a quadrant of the
moving ellipse, having 'z and y as semi-axes.
Let X = OMht the distance of the plane LMN
from O. The area of the moving ellipse is nyz.
Also, c^x^ -I- «««• = flV and *«jc« -f a^ = <»'^'.
Hence we have for the volume
»+«
(icyz) dx = {itabc.
The surface is called the ellipsoid with three unequal axes.
14. Two parabolas 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 parabolse.
Let OM and ONXnt the two parabolse
whose equations referred to AOL^ BOL
as axes are jc' = oxi^z and^' = 2b^z,
MLN is the position of a quadrant .
of the generating ellipse at a distance •"
z = OL from O, The area of the
ellipse is icxy. The volume generated
from s = otos = ^is
r = r {rcxy)dz = nab^.
The surface generated is called the
elliptic paraboloid.
Fig. io8.
Art. 179.] ON THE VOLUMES OF SOLIDS. 269
15. Volume of the hyperbolatoid*
Given two parallel planes at a distance apart H, 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 hyperbolatoid.
If in one of the planes a fixed point P be taken, then a straight line through
/*, 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
,.(i±£..|).
where B-^^ Bj are the areas of the sections of the solid by the parallel planes dis-
tant apart if, and (7 is the area of the base of the director cone.
Hint. Ally plane parallel to the given planes cuts the generating line in seg-
ments that are in constant ratio. Therefore the area B of any such section is
B = kyB^ -f- ^l"^l ~" ^l^J^
(projecting on a plane parallel to the bases), by Elliott's theorem, § 164, (3).
k^ and k^ can be expressed in- terms of ^, the distance of the section B from
either base B. or B^, Then
V- \ Bdh,
where ^ is a quadratic function of h^ and the result follows directly.
Since ^ is a quadratic function of h^ the results of Exercises 10, 1 1, 16, Chapter
XX, apply also to the hyperbolatoid, when ordinates are read sectional areas.
An important general case is : If the generating straight line moves in such a
manner as to remain always parallel to a fixed plane, then C = o and
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* -f-^' = a', as a central diagonal, is con-
structed a regular polygon of n (even) sides, whose plane is perpendicular to that
of the circle. Show that the volume generated by the polygon is
. 2jr
sm —
n
and therefore the volume of the sphere is \7C€^.
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 R draw two circles whose planes are parallel and
distant RJ 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 out a solid between
the tangent planes whose volume is equal to that of the sphere.
BOOK II.
FUNCTIONS OF MORE THAN ONE
VARIABLE.
371
PART V.
PRINCIPLES AND THEORY OF DIFFERENTIATION.
CHAPTER XXIV.
THE FUNCTION OF TWO VARIABLES.
iSo. Definition. — When there is a variable z related to two other
variables x and y in such a manner that corresponding to each pair
of values of x, y there is a determinate value of z^ then z is said to
be a function of the variables x and j^.
We represent functions of two variables or, y by the symbols
f[x^ _y), 0(ji", y\ etc. , in the same sense that we employed the corre-
sponding symbols f{x)y <l>{x\ etc., to represent functions of one
variable x.
When it is so well understood that we are considering a function
/(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 f.
In like manner we frequently consider the single letter z as represent-
ing a function of the variables x and j', and write
x8x. Geometrical Representation. — Let « be a function of two variables x
and y. Let the value c oiz correspond to the values a of x and d oiy. Through
a point O in 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 xOy^ 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^, in which we represent
any pair of values of the variables x and y by a
point having these values as coordinates referred
respectively to Ox^ Oy as axes, as in plane ana-
lytical geometry.
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.
273
274 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIV.
To represent the value z=z c of the function corresponding to the values x = a,
y ■= d oi the variables : Construct the point JV^ in the plane, xOy, of the variables,
having for its coordinates OM = a, MN = b* The value c of the function z can
then always be represented by a point P^ which is constructed by drawing a per«
pendicular NP to the plane of the variables at N^ such that NP •=. c\% 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, b^ r, or in general jr, j, r, are the coor-
dinates of the point /'with respect to the orthogonal coordinate planes xOy^ yOz^
zOx,
We can then always represent any determinate function y(jr, ^)of two variables
by a poiot in space whose distance £rom a plane is the value of the function.
182. Function of Independent Variables. — ^Let z =/{x,yj be a
function of the two variables x and^'. When there is no connection
whatever between x and^, then z is said to be a function of the two
independent variables x and^.
This means that, within the limits for which ;? is a function oi x
and yy whatever be the arbitrarily assigned values of x and^ there
corresponds a value of z.
Geometrical Illustration.
Consider the function of two independent variables
+ V'a' - jc» -y\
This function has no real existence for values. of x and y such that Jf* +^* > <*'•
Also, for x^ -\- y^ •=. a^ the function is o, while for any arbitrarily assigned values
of jr and^' whatever, such thatx*+-y' < <**• ^^ func-
tion has a unique determinate positive value. Geo-
metrically speaking, the function exists for any point
on or inside the circumference of the circles* -\-y^ = a*
in the plane xOy^ and the point representing the
function for any such assigned pair of values of x, y^
is a point on the surface of the hemisphere
jpi _|_ ^« 4. ,1 -a^
which lies above xOy. The circle x^ -\- y* •= a^ is
called the boundary of the region of the variables for
which the function
« = -f |/a« — ;c* _ ;/«
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 surfa.ce of which z = /{Xy y) is the equation in Cartesian
orthogonal coordinates. The study of a function of two independent variables
corresponds, therefore, to the study of surfaces in geometr>% in the same sense that
the study of a function of om^ variable corresponds to the study of plane curves as
exhibited in Book I.
183. Function of Dependent Variables. — ^Let z = /{^yy) 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 «r is a defined function of .r andj'.
Art. 183.] THE FUNCTION OF TWO VARIABLES.
275
Fig. III.
I. In particular, we can hold j' fixed and let x alone vary. In
which case z is a, function of the single
variable x. For example, let_>' = ^ be
constant, then
is a function of the single variable x.
li 2 ^=,J\xyy) be represented by a sur-
face, then equation (i), which is nothing
more than the two simultaneous equa-
tions
is represented by a curve AB in a plane x'0'z\ parallel to and at a
distance h from the coordinate plane xOz. Or, is the curve of inter-
section of the surface z •=. J\x,y) and the vertical plane^ = 3, as
exhibited by the simultaneous equations (2). The equation z =/{Xj b)
of this curve is referred to axes O'x', O'z' of x and z respectively, in
its plane x'Oz\
II. In like manner, if we make x remain constant, say Jtr = a,
and let^ vary, then z ■=if{xyy) becomes
«=yi^,>'), (3)
a function of y only, and is represented
by a curve A B in 2l plane y'O'z', Fig.
112, parallel to and at a distance a from
^the coordinate plane yOz, Or, it is the
curve of intersection of the surface
z =i/(x,y) and the plane x z=z a, whose
equations are
z=A^yy)y \
X = a, \
(4)
Fig. 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,
X and y take the values of coordinates of points on the line x = a
or ^ = ^ in xOy, we can make them take the values of coordinates
of points on the straight line
X ^ a _^ y ^ d
COS a
sm a
(5)
which is a straight line through the point a, b in xOy and making
an angle a with the axis Ox,
Substituting
AT = a -f r cos a, y =z b -\-r sina
276 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIV.
in z =/[x, y) for x and y respectively, and observing that r is the
distance of or, y from a, b measured
on the line (5), we have
z =/(tf + r cos «', 3 4- r sin a). (6)
If a is constant, (6) is a function of
the single variable r, and is the equa-
tion of a curve APB cut out of the
surface z =/[x,y) by a vertical plane
through (5), and the curve has for its
equations
Fig. 113.
X — a y — d
(7)
cos a
sin a
The curve (6) is referred in its own plane, rO'z\ to O'r, O'z' as
coordinate axis. The coordinates of any point P on the curve being
r, z.
IV. In general, x and^ being independent, we can assume any
relation between them we choose.
For example, we may require the
point x^ y in xOy to lie on the curve
^(•^> y) = o.
Then, as in III, z = /{x, y) is a
function of the dependent variables x and
y which are connected by the functional
relation <t>(p^y y) = o. The geometrical
meaning of this is: The point F repre-
senting the function z must lie in the
vertical through the point P^ represent-
ing X, y on the curve (p(Xy y) = o. Or,
the function z of the dependent variables x, 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 0(jtr, y) = o, is the horizontal projection of the curve in space AP
representing the function.
Geometrically speaking, the function z = /"(a:, y) of two depend-
ent variables x and y, connected by the relation 0(ji', y^ =: o, is
represented by the space curve which is the intersection of the surface
z -j=.J\x^y) and the vertical cylinder (p^x^y) = o, whose equations
are
Fig. 114.
o = (f>{x,y). /
(8)
Art. 185.] THE FUNCTION OF TWO VARUBLES. 277
If we solve ^(x, ji) = o for y and get y = (t(Jr), then substitui-
ing iory \a/{x,y), we express « as a function of x only, thus :
z^/lx.^{x)-\. (9)
This equation (9) is the equation of the projection of the space
curve AP (8) on the plane xOx.
In like manner we can express s as a function of y only, and get
the equation of the orthogonal projection of (8) on the plane _>'0s.
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,
/{x,y) = o,
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,
of the two independent variables x and^, if we make » constant, say
2 = c, we have the implicit function in two variables
A',y) = '- (■)
Geometrically, this is nothing more than the equation to the
curve LMN, Fig. 1 1 5, cut out of the
surface » = /{x, y) by the horizontal
plane s = r, at a distance c from xOy.
Its equations are
The lines cut on a surface by a
series of horizontal planes are called
the contour lines of the surface. In
particular, if s = o, "CbRTi. /{x, y) = o ^'°- "S-
is the equation in the xOy plane- of the horizontal trace of the sur&ce
X = /(jT, y), or the curve ASC cut in the horizontal plane by the
surface.
In the same \\'ay that the implicit equation in two variables
defines either variable as a function of the other, the implicit function
/{x,y,t)=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, 3 as the coordinates of its points.
185. Observations on Functions of Several Variables. — The
general method of investigating a function of two independent
variables is to make one of the variables constant and then study the
278 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cii. 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 § 1 84, 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.
i86. Continuity of a Function of Two Independent Variables.
Definition. — The function z =/{x, y) is said to be continuous at
any pair of values x,y of the variables when corresponding to x^y
we havey(-r, y) determinate and
for jfj(=)ar, ^j(= jy, independent of the manner in which x^ and y^ are
made to converge to their respective limits x and^.
The definition also asserts that
£[A^xyyi)-A^yy)]=^^>
ioT x^{-)x, y^{ = )v.
In words : The function z = /(Xy y) is continuous at x, y when-
ever the number z^ = /{x^ , y^ converges to 0 as
a limit, when the variables x^ , y^ converge
simultaneously to the respective limits x, y in
an arbitrary manner.
Geometrically interpreted, the point P^,
representing x^ , y^, z^, must converge to P,
representing x,y, z, as a limit, at the same time
-a? that the point -A^, representing at, , y^y con-
verges to M, representing x,y; whatever be the
Fig, 116. P^^^ which N is made to trace in xOy as it
converges to its limit M.
A function y(:»;,^) is said to be continuous in a certain region A
in the plane xOy when it is continuous at every point x, y in the
region A.
An important corollary to the definition of continuity oi/{x,y)
at X, y is this; Whatever be the value oi/[Xy y) different from o, we
can always take x^ , y^ so near their respective limits x, y that we
shall hsLve /{x^ , y^) of the same sign 2Ls/{x,y),
187. The Functional Neighborhood. — A consequence of the
definition of continuity of z =^yi^x,y) is as follows:
Art. 187.] THE FUNCTION OF TWO VARIABLES.
279
^^/{•^»y) is continuous in a certain region containing a, 3, we can
always assign an absolute number e so small that corresponding to
6 there are two assigned absolute numbers k and k, such that for all
values of x 2Lndy for which
we have
The proof of this is the same as that given for a function of one
variable. For, let^ and a be fixed numbers, and let x vary. Then
whatever number ^e be assigned, we can always assign a correspond-
ing number A ^ o, such that for \x — a\ < ^ we have
sinc^A^fy) ^^ ^ continuous function of one variable x, and its limit
In like manner for [y — d\ <.k yre have
\A^>y)-A<'>^)\ <H
and on addition
\A^'.y)-A<'>^)\ <e
for all values of x, y, such that
\x-a\ <k, \y-b\ <k.
Geometrically speaking, whatever be the value c =/(<i, h), we
can always assign an arbitrarily
small number e, corresponding to
which there is a rectangle KLMN
in the plane xOy^ the coordinates
of whose comers are K^ {a — h,
^ + >&); Z, (fz + h, b+ky, M,
(a + A, 3-/i);^; (fl-A, b^k),
such that, whatever be the point
jf, y in the rectangle KLMN, the
corresponding point jt, y, z on the
surface z =y(A:,^) lies between the
parallel planes z = c— e, (STUV)
and » = c + e, ( WXVZ). The
point P representing a, b, c.
Such a region KLMN is called
the neighborhood of the point a, b. The point is called its center. In
like manner the corresponding parallelopiped STUV-WXVZ is
called the neighborhood of the point P in space.
The above results may be stated thus : When the variables x, y
are in the neighborhood of a, 3, then must the continuous function
A'Xy y) be in the neighborhood of /"(a, b).
An important consequence is this: HA^* ^) *s continuous in the
Fig. 117.
28o PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIV.
neighborhood oi/la, b) ^ o, then we can always assign a neighbor-
hood of a, b such that for all values of x^y in this neighborhood
the value y(ji;, y) of the function has the same sign asy^a, b),
EXERCISES.
1. Trace the surfEice representing the function
Put z = ^ — mx + b. When z = o, the surface cuts xOy in the straight line
V = mx — b. If X = Of we have for the section of the surface by the plane x = a
the straight line
« = ^ — ma -\- b.
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^ = mx ^ b in xOy^ and
therefore generates a plane.
In like manner it can be shown that the implicit function of the first degree in
X V St
/lx,y,s)mAx-\-By+Cz-{-D=zO,
is always represented by a plane.
2. Show that the function
^a* -X* ^ y*
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
a^ b* c* a b
by their plane sections.
4. Trace by sections the surfsice representing
(jt« - azY(a* - x^)-- xY -O.
5. Find the maximum value of the function
x^ V*
when the variables are subject to the condition x ■\- y ^ i.
Let z = /{x, y). Then z is immediately reduced to a function of one variable
by substituting i ^ xiory.
__ x* y*
••• ' - ' " ^" ^•
dz 2x , (i — x)
dx a* ' b^
gives x = «V(^ -f ^)' y = *V(«* 4- ^)» « = I - i/{a* + ^«), which is a
maximum value of z since Dlz is negative.
Consider the geometrical aspect of this problem. We have
j^ y*
' = '-:;«- Ai» (I)
the equation of the elliptic paraboloid whose vertex is o, o, 1, and which cuts xOy
in the ellipse x^/a^ + y^b* = i.
Art. 187.] THE JUNCTION OF TWO VARIABLES.
281
We wish the highest point on the curve cut out of the surface by the plane
x-^-y = I. Take (/r^ the horizontal trace of this plane, as the positive axis
of r, and ^s', its vertical trace on yOz, as axis
of s in the plane rC/t^, Then for the equation %
to the curve in the plane x -{-y ^ i, or
X — o y — I
Va — |/2
= r.
we substitute jc = r f^2, >» = i — r 4/2 in (i).
Hence the equation to the curve of section in its
own plane is
/ 1\ , 2 4^
-'(^
f*.
Fig. 118.
DrZ = o gives r = a^/(a* -|- ^) |/2, and
jyir = — . Hence the values o{Xfy,z as before.
The first method, in which we substitute for^^ in terms of x, is only possible
when we can solve the condition to which the variables are subject, with respect to
one of them. 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
in the 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.
i88. 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 z =^f{pCy y) be a function of the two independent variables
X and y. The simplest relation we can impose between x and j/ is
to make one of them remain constant while the other varies. We
then reduce the function « to a function of one variable, to which
we can apply all the methods of Book I for functions of one variable.
For example, let^ 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
Jb x^-^ X
This is called the partial derivative of the function z or / with
respect to x. To obtain the partial derivative of /(at, y) with respect
to x, make^ constant and differentiate with respect to x.
Correspondingly, the partial differential of /(x^ y) with respect to
X is the product of the partial derivative with respect to x, D^^ and
the differential of at or jf^ — jtr = Ax. If we represent the partial
differential of/ with respect to x by d^^ then we have
d,f=A{x,y)dx,
and the corresponding partial differential quotient is
It is customary to employ the peculiar symbolism designed by
282
Art. 190.]
THE FUNCTION Of TWO VARIABLES.
2«3
Jacobi for representing the partial differential quotient or derivative
of /{x, y) with respect to x. Thus the above will hereafter be
written (the svmbol d is called the round d)
dx dx '
The symbol d 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 cons/an/, then /{x, y) becomes a
function of one variable y, and has a determinate derivative with
respect to j/. This derivative we call the partial derivative ot/{x, y)
with respect to^', which is written and defined to be
x« const.
^y X yi-y
190. Geometrical Illustration of Partial Derivatiyes. — If
z •=./{x, y) is represented by the ordinate to a surface, then at any
point P{x^ y, z) on the surface
draw two planes PMQ and PMR
parallel respectively to the coor-
dinate planes xOz zxi^yOz, These
planes cut out of the surface the
two curves, PA!' and /y respectively,
passing through P.
z =/{x, y) {y constant)
is the equation of the curve PIC in
the plane PMQ.
z=/{x,y) (x constant) Fig. 119.
is the equation of the curve P/ in the plane PMR,
Draw the tangents /'7'and PS to the curves /'A!' and PJxn their
respective planes, and let them make angles 0 and ^ with their
horizontal axes, as in plane geometry. Then we have
bz ^ ^«
Therefore the partial derivatives of J\x, y) with respect to x and
y are represented by the slopes of the tangent lines to the surface
z =/[x, y), at the point x^yy z, to the horizontal plane xOy, These
tangents being drawn respectively parallel to the vertical coordinate
planes xOz^ yOz,
Also, draw /T parallel to MQ, and P^ parallel to MR. Then
we have
VT = (^, - x) tan 0, US = {y' -^) tan ^,
284 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXV-
if Q is x^, y, and -^ is x, y'. Or
represent the corresponding partial differentials of/" with respect to
X and J' at P{x, y, s).
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. — If z =:/{x,y) is a func-
tion of two independent variables x and y, then, in general, its
partial derivative with respect to x,
is also a function of x and^ as independent variables. This deriva-
tive can also be differentiated partially with respect to either x ovy^
as wasyi^AT, y). Thus, differentiating again with respect to x, y being
constant, we have the second partial derivative of/ with respect to x.
In symbols
In like manner /]^(jc, y) can be differentiated partially with respect
toy^ a: being constant. Thus we have for the second partial differen-
tial quotient of/" with respect first to x and then to^'
dydx "-^'y^^'-^f'
Similarly, differentiating /^(x, y) partially with respect to^, we
have
ay ■" dy —Jyy\^^y)^
and with respect to x we have
?!^, y) _ g/;(^> y) _ f,n^ ,.
dxdy~ "" dx -yy'^'^>y)'
Thus we see that the function z = /[x, y) has two firs/ partial
derivatives,
dz dz
dx' dy'
and four second partial derivatives,
d''^z dh dh d^z
dx^' dj^* dydx* dxdy
Each of these give rise t6 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
;fth order, of the forms
9*« 9"«
where p and g are any positive integers satisfying / -j- ^ = «. These
nth derivatives, however, are not all different, for we shall demon-
strate presently that dx^ and dy^ in the denominators are interchange-
able when the partial derivatives are continuous functions, and that
9*« _ 9*«
dx^dyt ^ dyfdx^'
or the order of e&cting the partial differentiations is indifferent
The number of partial derivatives oi/{x,y) of order n is then n -\- 1
EXAMPLES.
1. If * =x* + axy -j- cos jf sin y^
dz
.*. -T- = 2x -j- ay — sm X sin>,
^ = aar -f- cos x cos y.
8. In Ex. I,
d/ ^ 2x df _ 2y
e^z &^z
= a — sm j: cos^'zz
dy dx
d*z
r-5 = 2 — cos X sm V,
d»2
a>
^ = — cos j: siny.
4. In Ex. 2, show that
ay ov
dy dx dx dy
192. Theorem. — The partial derivatives are independent of the
order in which the operations are effected with respect to x and^.
In symbols, if a? =/[Xyjf), we have
dx dy~~ dy dx'
Consider the rectangle of the
four points
J/, {x,y); M^, {x,,y,)]
<?, {x^,y)\ ^, {x.y^).
The theorem of mean value applied
to a function of one variable x gives
Ax x^ — X '
=/i'{S,jy), (I)
where S is some number between or, and x, (See Book I, § 62.)
286 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXV.
Form the difference quotient of (i) with respect to>',
J J/ ^A^i^yi) -A^yy) -A^^yd +A^^y)
Ay Ax {^y^-y){x^^x) '
where 77 is some number between ^^ and>. The value (2) is therefore
equal to the second partial derivative of /J taken first with respect to
Xy then with respect to^, at a pair of values 5^, ;; ofx^y. Geomet-
rically, at a point S, v ^^ t^^e rectangle MQM^R,
In like manner, taking the difference-quotient ofy^ first with respect
to J', we have
4^ -y^ -M^y nh (3)
where rf is some number between JV^ and^.
Now taking the difference-quotient of (3) with respect to x, we
have
A A/ ^ A^^>yd -A^^y.) -A^.^y) ^A^^y)
Ax Ay {^x^^x){y^^y) '
»:. yiy V^v V ) '^Jti \^f V ) /-It ttct -,/\ /-\
— X ^x -^''» * ^^ ' ^ ^' ^^^
where ^' lies between x^ and x, r}' between y^ and^. The value of
(4) is then equal to the second partial derivative of yi taken first with
respect to y and then with respect to x at some point ^', rf ^ also
inside the rectangle MQM^R,
But (2) and (4) ar^ identically equal. Hence we have
^ I
This relation is true whatever be the values x^,yi*
If now the functions
''^ and 'y
dy dx dx dy
are continuous functions of x andy in the neighborhood of x, y, then
since ^', rj' and S, tf converge to the respective limits or, y when
x^{=z)x^ y^[=:)yy the two members of (5) converge toacommon limit
at the same time, and therefore
ay by
dy dx dx dy '
(6)
I
Art. 192.] THE FUNCTION OF TWO VARIABLES. 287
Incidentally, equations (2) and (4) show that the difference-
quotients
A Af _ jy A A/ _ Ay
Ay Ax ~ Ay Ax ' Ax Ay ~ zJor Ay
converge to a common limit whatever be the manner in which
Ax{=)o, Ay{=)o, and that common limit is
or
dydx dx dy'
Observe that in the symbols
dy bx-^ - ^'^
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
dx \dx by) ^ dxdydx~^ by dxdx ~ by 6jc* *
. 93/ _ by
•• bx^by bybx^'
and similarly for other cases. Hence, in general,
b^y _ b^^y
bx^byt "■ bybx^ '
in whatever order the differentiations be made.
EXERCISES.
1. If g ■= tan-» - , show that
y
dH d*z X* — y*
2. If *=^5^^, find />;,«, Dy\z.
3. Verify in the following functions the equation
ay _ ay
dx dy ~ dy dx'
jf sin >^ + ^ sin jt, log tan (y/x),
X ]ogy, {ay - dx)/(fy - ax),
xy, ^log(i-|-^).
4. If % — tan-» — ^ , show that
5. If M = x»y — 2jry* 4. 3Jcy , show that
X TT + J' :r- = 5«^
a 88 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXV.
7. If ^(- + ^) = <-*. then g +1 = (a + * _ ,),.
8. If ■ = e*^y + a tin x, «bow that
©'+(5)" = '- + '' + — "+'>
9. From « = Jify', show that
tanV:
(!)•+ (D-
Let P, {x, y s) be on the suriace, and
PRS (he laneent plane.
Draw MN = p perpendicular to RS.
Then y = PNM.
d£ _ PM_ ^f__PM
dx~ PM' dy~ SM'
Since PS-NAf = XAf-MS.
and PS* = RM* \ MS',
Fic. „,. ■■■ i^^)-' = i^^)-' + (*'^)'*.
and therefore
— .+ («)■+(!)•.
It In Fig. Ill, let ^ be nit)' point in the trace of the tangent plane with j-(?>'.
Let A'jl/' make an an^le 9 with Ox, and the tangent line A'/' to the sur&ice mate an
angle 0witli the horiionlal plane jOr. Then the triangle ^5j1/ la the Gum irfthe
triangles PMN, NMS, or
PM.SM= SM-NMcosH + RM-NM
PM _ PM
NM ~ PM '■'"'' T SM
y, c, whose vertical
dy
Art. 192.] THE FUNCTION OF TWO VARIABLES.
289
The values of sin B, cos 8 from this equation put in (i), Ex. ii, give for the
tangent line of steepest slope
/d/\ « . /dA «
Observe that this is the slope of the tangent plane in Ex. 10.
13. If <p{x, ^) = o is the equation of any plane curve, show that
dy _ dx
dx ~ 50(jr, y)'
—bjT
Let z = <p{x,y) be the equation of a sur-
face cutting the horizontal plane in the curve
0(x, y) = o.
Let Pj (Jf» v) and -/\, (xt, y^) be two
points on the curve 0(jr, y) = o. Draw the
vertical planes through P and P^ parallel
respectively to xOz and^Os, cutting the sur-
face in curves PQ, P^Q. Then ^ is a point
jTp ^, z on the surface. The derivative of^
with respect to x in <p(x, ^) = o is the limit of the difference-quotient
yi —y _ MP^ _ MQ cot MP^Q __ tan MPQ __ tan MPQ
" PM ~ ~MQ cot MPQ "" tan MP^Q ~~ ~ tan NP^Q'
JTi — jr
Also,
UnMJ'Q= ^^. t^NF,Q=^.^^,
4 being between x and jt^, ;; between y and y^ (by the theorem of the mean).
Therefore, when jr,(=)ji:, ^j(=)y,
d0
dx T. x^ — x dip'
dy
This usually saves much labor in computing the derivatives of implicit functions
in X and y.
The important results of Exs. 10, 1 1, 12, and 13 are deduced here geometrically
to serve as illustrations of the usefulness of partial differentiation. They will he
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* — y^/d^ —1=0, X siny — ^ sin Jf = o,
ax^y -f- dy^x — 4xy = 0, e» siny — log^ cos x =z o,
16. Show that the slope of the tangent at x, y on the conic
ax^ -f ^* + 2>4jrv -f 2i«f -f- 2vy -\- d = o
Is ^ _ _ ax-^hy -\-u
dx Ax -{• dy -\- V
CHAPTER XXVI.
TOTAL DIFFERENTIATION.
193. In the partial 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 oi/(Xy y) when both
X and y vary during the operation of evaluating the derivative. Such
derivatives are called total derivatives.
In order to make clear the nature of the total derivative of a
function
consider the simple case when there is a linear relation between x
^nd^v,
X '— x^ ^y —y __
where / = cos 6, m = sin d, and the differentiation is performed with
respect to r. Let x^, y ; /, »i ; be constant. Then r varies with x
SLudy, and
r^=(x-x'y+{y^y'y.
Also, X and^ are linear functions of r, and
jc = j/ -f. /r, y =y -|- mr.
Substituting these values of x and y in /{x, y), we reduce that
function to a function of the one variable r, and it becomes
/{j/ + lr,y + mr). (i)
The derivatives of this function with respect to r can now be
formed by the methods of Book I. Thus we get by the ordinary
process of differentiation
W ^JL ^ etc
for the successive derivatives of / with respect to r. These are
called the total derivatives oi/ with respect to r. Both variables x
zxidy V2ixy with r.
We can give a geometrical interpretation to this total derivative
as follows : The equation
x — x' _ y^y
— 7 — = = r (2)
I m ^ '
290
Art. 194.]
TOTAL DIFFERENTIATION.
291
is the equation of a straight line through x\ / in the horizontal
plane xOy, making an angle (^ with Ox. r being the distance
between the points oc'^ y' and at, y on the line. Let 0' be the point
x'^y. Draw (Xz' vertical. The vertical plane rO'z' through the
Fig. 123.
line (2) cuts the surface representing z r=,f{x^y) in a curve PP^^
whose equation in its plane, referred to O'r and O'z' as axes of
coordinates r and z, is
z^A^'^^r.y^mr). (3)
Let /', be a point on this curve whose coordinates in space are
x^, y, , «, and in rO'z' are r^ , z^. Let r^ — r z=. Ar. Then, by
definition, the derivative of z with respect to r at x^y is the limit
of the difference- quotient, when r^=.^r.
^l-^
^-^
Hence we have
dz
-r- = tan a?,
ar
where co is the angle which the tangent PM^' to the curve PP^ at
P, and therefore to the surface, makes with O^r, or the horizontal
plane xOy,
Observe that as x^ , y converge to x, y, the point M^ converges
to ilf along the line M^M.
By assigning different values to d 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/ = ;r or ^tc, 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. [Ch. XXVL
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 =z/{x, y), where x and ^ are connected
by any relation 0{x, y) ■= o. To find the derivative of s with respect
to /, where / is any differentiable function of x and^.
Let z take the value z^, and / become Z^, when x,y become
Let y be constant and x^ be a variable. Then the law of the
mean is applicable to the function /{x^^^y) of the one variable Jtr^,
and we have
A^i » y) -A^^ y) = (-^i - ^)q^AS^ y\ (O
where B is some number between x^ and x.
In like manner, let x be constant and^^ vary, then, by the law of
the mean,
a
A^i > yi) -A^i y >') = U -y)'^A^i ' v)* (a)
where ;; is some number between j/j and^'.
Adding (i) and (2), we have
A^i . y^-A^' y) = K - ^)Q^AS,y) + U -y)YJ{^x > v)- (3)
Therefore the difference-quotient with respect to / is
'j^t =AiS' y) § +/;(^. n)% (4)
/ix, Ay, AZy At converge to o together, and at the same time
x^{z=)x, 5{-)x,y^{-)y, 7?(=lv. Also,
ay^ ^^ ^A^
have the respective limits
¥i^^y) ^^^ ^A^>y)
dx dy
if these latter functions are continuous in the neighborhood of x, y.
Passing to the limit in (4), we have for the total derivative oi/[x, y)
with respect to /, at x, y^
di dx dt "^ by di' ^^'
The geometrical interpretation of (i) is this: In Fig. 123 we
have M, \x, y)-, M^ , {x^ , j^; Q, (x^ , y)-^ R^ (x, ,y,).
Also,
A^\>y) -A^>y) = Q'k = pq' tan q'pk.
Art. 195.] TOTAL DIFFERENTIATION. 293
But, since on the curve PK there must be a point JT, {fi, y, z) at
which the tangent is parallel to the chord,
In like manner for equation (3),
/l^i » ^1) -A^v y) ^LP, = - LK tan LKP^.
But, since there is a point JT, (;r, , T}y z) on the curve KP^ at which
the tangent is parallel to the chord, we have
-tanZJ5r/>, = ^^-/][.r,,v).
195. The Linear Derivative. — An important particular total
derivative is the case considered in § 193. Suppose there is a linear
relation between x and>^, such as
X — a y — b
:= — r,
I m
Then jf=:a-|-/r, >^ = 3 + 'W" To find the total derivative of
f(x, y) with respect to the variable r, we have
dx dy
- = /. -- = «.
/ = cos 0, m = sin B, being constant. Therefore
This is a much simpler way of evaluating this derivative than that
proposed in § 193.
As before (see Ex. 11, § 192, § 193),
tan c» = ^ = 1^ cos I? + 1^ sin ^ (2)
dr ox by ^ '
is the slope to the horizontal plane of a tangent line to the surface,
in a vertical plane making an angle B with xOz,
Again, suppose, as in § 194, that x and y are related by
0(jf , y) = o, and we wish the derivative of / with respect to j, the
lengtli of the curve <p{x, y) = o, measured from a fixed point to x, y.
Then, putting / = j in (5) § 194,
^_^^^ _^^
ds^ bxds^ by ds ^^'
dx (fy
But --p = cos 0, -=- = sin 0, where 6 is the angle which the tan-
ax as
gent to 0(jr, ^) = o at j:, ^^ makes with Ox. Hence we have the
same value of the derivative as in (i).
294 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI.
which is also the slope to the horizontal plane of the tangent line to
the surface.
196. The Total Differential of /[^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 <//, we have for the total differential of/ at x, y
Observe that
are the partial differentials oif. Hence
d/^bj^b^; (a)
or, the total differential of /at Xy 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 dx = MQ 2Lnd dy = MjR. Draw PR\ QM', Q''S
parallel to MR. Then
dj- = Q'Q" = M'S and dj = R'Rf' = SM'\
. • . d/- M[S + SM'' = M'M'';
or, the differential of the fui^ction is represented by the distance
from a point in the tangent plane to the surfeice at P from a hori-
zontal plane through P.
197. The Total Derivatiyes with respect to x and j^. — If, in the
total derivative
di '^ dx di "^ dy di'
we take / » at, then the total derivative of /with respect to jr is
dx~dx'^dydx' ^^
If we take / s^, then
dy ■" dxdy "^ by' ^ '
Equations (i) and (2) represent the total derivatives of /with
regard to x and^ respectively. These derivatives are quite distinct
and different from the partial derivatives, as is shown by the formulas,
and as is exhibited in their geometrical interpretations as follows:
The total derivative of z =/(^, y) with respect to x is the limit
of the difference-quotient
jfj — or'
Art. 197.1 TOTAL DIFFERENTIATION. 39$
X fmAj! varying as the coordinates of a point on some curve MHla
the horizontal plane.
Fio. 124.
If, P^ is x^, y^, s, , then, in Fig. 1 14,
a, - B = /P, = /'/>/, x^- x = N'M' = J'P'.
ds
Therefore — - = tan a is the total derivative of b with respect
to j;. That is, the total derivative ofywith respect to x is repre-
sented by the slope to Ox of the projection P'T' of the tangent PT
to the surface on the vertical plane xOz. The tangent PT being
in a vertical plane through P which makes with xOz the angle 0
dy
determined by ^ = tan 6, as determined from 0(jr, y) = o. That
dy
is, -J- is the slope to Ox of the horizontal projection MN ol the
tangent PT.
In like manner the total derivative ofywith respect to^ is equal
to tan 0, this being the slope to Oy of the projection of the same
tangent PToa the perpendicular p\ar\tyOs.
Equations (i) and (1) are immediately determined from the total
differential
'/='^^ + f*
by dividing through first by dx and then by dy.
In Fig. 134 we have
d/ =/T = fT' = J"T",
and equarions (i) and (z) can be verified by the differential qaotients
taken from the figure directly.
7g6 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI.
198. Differentiatioii of the Implicit Function y^ar,^') = o. — An
important and valuable corollary to the total differentiation of the
function 2 =/{x, y) is that which results in giving the derivative of
y with respect to x in the implicit function y(jr,j/) = o.
Since « = o in « ^/{x, y) gives /(or, y) = o, and in /[x, ^) = o
are admissible only those values of x and y which make z constantly
zero, the derivative of z with respect to any variable must be o.
Therefore, from (5), §194, or (i), § 196, § 197,
dy _^ dx
dx'^ ~"^
dy
This has been geometrically interpreted in Ex. 13, Chap. XXV.
In general, the plane z = c, c being any constant, cuts the surface
z =/'{Xy 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/(ji:, y) = c. In the same way as above.
bx dt "^ 'Sy'dt " dt " dt "
o.
dy ^ dt __^ dx
It ~dy
which corresponds to the slope of the tangent to the contour (at the
point X, y, c) to the vertical plane xOz.
BXERCISSS.
1. If jr« -f y — yucy = e, find D^y,
Here ^ = 3(x« - ay\ / = 3(y - ax)
dy _ x*^ —^
dx ^ ax ^ y^'
2. Find D^y in jr»«/«"« + y^/b^ = I.
a/ _ mx^-^ df my^-^ /dy\ (t\'* (f\
ajr — a« ' dy ~ ^m ' ' ' \dx/ "" \a / \y)
3. If x\ogy -ylogx^o, then ^=^ jog^*-^^
* -^ '^ * dx X \og xy ^ X
4. Let X = p cos 6. Find the total diflferential of x.
^^ ^ ^^ . /»
— = cos9, _=-ps,ne,
• •
dx = cos $ dp — p s\n B dQ.
Art. 198.J TOTAL DIFFERENTIATION. 297
5. Find the slope to the horizontal plane of the curve
1 = * +y.
^~- (i - ?) ^'-
6. Find the slope to xOy (the steepness) of the curve cut from the hyperbolic
paraboloid z = jfl/afl — v«/^ by the parabolic cylinder v* =r Apx.
We have
</f dt dx , dg dy
tan « = — = 1 —,
ds dx ds^ dy di
s being the length of the parabola y* = 4/jr. Here
5* _ 24r 6« _ 2y
ai "" ««"' ^ " "" ^"
tan 09 s
which is the declivity of the curve in space at jt, y^ s.
Find the points at which the tangent to this curve is horisontaL
7. If f# = tan-»(;^/x), du=z{xdy —y dx)/{x* +y*),
B, U s zs xff dM := yxT-^ <^ -f 07 log xdy.
9. Find the locus of all the tangent lines to a surface s r^ /(x, y) at a point
(fl, *, r), R
Through P draw a vertical plane. Fig. 123, rMP^ whose equation is
X — a y — b , .
Then the equation to the tangent line, PAf'\ to the sur&ce at P, in the plane
rAfPin terms of its slope at a, ^, c, is
s- <r 4^
s and r being the coordinates of any point on the tangent line. But at «, 3, r
^ - tan «, -*3?lil ^ 4. ?^^^
dr~ da dr db dr'
Therefore the eouation to the tangent line to the surface at a, b, c, whose hori-
zontal projection malces / 9 with Ox (where / = cos 6, m = sin 0), is
Eliminating r/ and rm between (i) and (2), we have
,-, = (,_«)g + (,-*)|, (3)
an equation of the first degree in x, y^ g, which is the locus in space of the tangent
lines at a, b, c on the su^ce. This locus is a plane, Exercise I, Chap. XXIV,
touching the surface at a, b, c, and is defined to be the tangent plane to the surface
at a, bf c.
298 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XX VL
10. Show that the equation to the tangent plane to the surface t z= ax^ -\- bj^
at any point x\ /, «' on the surface is
a -f- «* = 2{axx' -f hy')*
11. Use the equation to the tangent plane
to verify Ex. 12, § 192.
The direction cosines of the plane are proportional to ^, "51 • "" *' ^®^ce
if/,«.
n are these cosines,
/ m
a/ - a/ -
da bb
n
— I
I
'^1-
©■+ '
©■
Alsc
), 8ec*x = i/»S giving the
same result
: as Ex. 12.
12.
Show that when
•
ax"
0,
(0
the tangent plane to the sur£&ce is horizontal at values oix^y satisfying m = /{x^y)
and (I).
13. Show that the curve on the sur^Lce x =/lxt y) at all points of which the
tangent plane to the surfaice makes the angle 45* with xOy is the curve cut on the
sur&ce by the cylinder
I.
14. Apply Ex. 13 to show that the cylinder jfl -{-y* = 4^' cuts the sphere
x^ -{- y* -{- z* = 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 jp* -f-^* = a* represents a vertical cylinder of revolution whose
axis is Om and radius is a. Find the equations of the path of a point which starts
ztxzsafy=20,z = o and ascends the cylinder on a line of constant grade k»
This curve is the helix, a spiral on the cylinder, having for its equations
CHAPTER XXVIL
SUCCESSIVE TOTAL DIFFERENTIATION. «
199. Second Total Derivative and Differential of s =:/[x,y).
It has been shown in § 194, (5), that
dV dx dt "^ay dt ' ^ '
where x and_y are any differentiable functions of/.
If we differentiate again with respect to /, then
d(^
_dfbf_ dx\ d^(d£ _^\
dt\dx ' di)'^ di \dy ' dty
_dxd(d/\ B/d^x dy d /bf\ d/d»y
" ^ d/\dx)^ dx dfi ■*■ d/ dt \dyr ^y dt^'
(^)
Also,
Since ^ , v— are functions of a: and j' to which (i) is applicable,
in the same way we have
di\bx}'~ dx \dxj' di "^ by\dxy dt '
"" ajc» dt^ dy dx di ' ^^^
d/d/\_ ay dx ay dy.
dt \dy J^ dxdy dt"^ dy d/' ^^^
Substituting in (2) and remembering that
ay _ dy
dx dy'~ dy dx'
we have finally for the second total derivative of /{x^y) with respect
to/
d^/_dy/dxy dy dxdy ay/M« d/d^x a/^
df' "^ dx^\dt/ '^ dxdy dt dt'^dy^\dt) '^ dx dfi "^ dy dfi' ^5'
Multiplying through by <//*, we have the second total differential
299
300 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVII-
In (5), /is taken as the independent variable, and while d/ is per-
fectly arbitrary in (i) in actual value, we agree, as in Book I, that d/
shall be taken as having a constant value in the successive diiferen-
tiiuions.
Thus if we take x as the independent variable instead of /, then dx
d^x d (dx\
is taken constant, in which case -^-^ = — 1 — 1=0, and we have for
the second total derivative of /" with respect to x
dx^^ dj(^'^ hx dy dx "^ bf \dx) "^ by dx^' ^'^
In like manner taking j^ as the independent variable, changing / to^
ir^ (5)> we have dy constant, and the total derivative of/ with respect
to^' is
dy^^dx^\dy) '^^dxdydy'^dy'^dxdj/^' ^^
dh
^^' 1^' whenyi;;*:, y) = o.
The formulse of the preceding article furnish means of expressing
the second derivative of^ with respect to x in an explicit function
J\Xy y) = o, in terms of the partial derivatives oif(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 J' by
ff ft ffi ftt ftt g*^
and the first and second derivatives of^ with respect to x by >/', y".
Putting/* = « = o in (7), § 199, we have
But J'' = —fLIfi' Substituting this and solving for>/',
In like manner we get, by interchanging x and y^ the second
derivative D^x, Otherwise deduced from (8), § 199.
201. Higher Total Derivatives. — We shall not have occasion to
use the higher total derivatives of « = f(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 y with respect to / see Exercise 35 at
the end of this chapter.
Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 301
The higher total derivatives o(/[x, y) with respect to an arbitrary
function i oi 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/(ji:, 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. Sttccesslve Linear Total Derivatives.— To find the »th
derivative of /(x, y) with respect to r, when x and y are linearly
related by
X ^ a ^y — b _
a, h^ /, m being constants.
The first derivative is, as found before,
Differentiating again with respect to r, we have
Otherwise this follows immediately from (5), § 199, wherein
dx ^ dy d^x d^y
' = ''' ^-^' dr"""' :^*=5y^=°-
Differentiating (2) again with respect to r, and rearranging the
terms, we have
dr^ '^^ dx' ^ y dx^dy ^ ^'"^ dxdf + *" ay ^^^
We observe that (i), (2), (3) are formed according to a definite
law. The powers of /, m, and their coefficients follow the law of the
binomial formula.
a a
If we consider the s)rmbols ^ » "a~ ^ operators, on/) and write
conventionally
dx^ dyf "^ \dx) \dy) '
then we can write
^ = ('R+-|r)/. W
^ = (4 + -i)'-^' <s>
fly /, 9 3 \v
302 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVII.
in which the parentheses are to be expanded by the binomial formula
d d
and the indices of the powers of 5— and —— taken to mean the num-
ax oy
ber of times these operations are performed.
We can demonstrate that this law is general and that we shall
have
0 = ('s+-f)-/ «
as follows.
First, observe that
£¥ _±¥ d bf _ d' d/
'^ bx " bx br ' Jrliy "~ by "W
(8)
For
Also,
d_bf _dy dx by (fy
dr bx "" bx^ dr "*" ~bybx^'
bj^ ' bybx'
'^f -^ il ^f ^ni^f\
which proves the first equality in (8), and the second is proved in the
same way.
Now assume (7) to be true. Differentiating again with respect to r,
we have
d^^y _d / b b Y
Ad by d/
=('^+4)'('^+-i-y-
The memond iechnica (7) being true for n = 3, it is true for 4,
and so on generally.
Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 303
SZBSCISBS.
1. Given x* + j'* = a\ find Dy.
2. If Jf» -j- xy« - ay* =0, / = (3jc» 4- y*)/2y(a - x).
du i(a* — /« _ K«)
3. If (/> + «•)« = a-V - ««), -=ij_pi-^j.
♦.If .« = fL±_'. 1' =
-- Z^ dx 2SX — ftyX* — «»)*
5. If ««v« +fS^ - V^ = o, ^ = _. ^ .
6. If /(•^i >^) = o, is the equation to any curve, show that
are the equations to the tangent and normal at jt. y. The running coordinates
being a; Y.
7. Show by Ex. 6 that the equations of the tangent and normal to the eUipse
x^/a* -|- yyil^ = I are
— - 4- -^ = I and a« ^« — = <?» — *«.
a* r X y
8. Show that the second derivative oiy with respect to r, in/(jc, >^) = o, can
be expressed in the form
£V _ _ \bxdy by dx ) ■'
^" (!)■ ■
_ _ ay /a/
~ ajr« / dy'
9. Show that the ordinate of the curve y{jr, ^) = o is a maximum or a minimum
yrhen/^ = o, according as^]^ andy^ are like or unlike signed.
For a maximum value of ^ we must have
dx'^ dx / dy"^
or /i = o, yJJJ ^ o. When this is the case, by § 200,
djy
d^
which gives a maximum when fjx 2ind/y are like signed and a minimum when
unlike signed.
10. Show that the maximum and minimum ordinates of the conic
/ 3 OJf* + ^' + 2/lxy -f Ijrx -{. ?/y ^ d= O
are found by aid of
/i = ax-i- Ay -{-g = o.
If /y = 6y -^ Ax -\- /, is positive, the ordinate is a maximum; if negative, a
minimum.
11. Find the maximum ordinate in the folium of Descartes,
^•^ - ^axy + jr* = o sy^jr, y).
i/; =-ay-\- x\ ify^r.y*^ ax.
304 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVII.
Eliminating y between/ = o, yX = o, we have
jr« — aa'jr* = o.
. •. jr = o, jf = fl Y^' These values give y -=.0^ y •= a 4/47 For jr = o,
^ = o, we have^' = o, but for jr = a 4/2^ y r= a I/4, we have a maximum y if
a = -{-, since
</jc* ~~ d jr* / ^^^ ^ *
12. If two curves 0(jr, >^) = o, ^jc, y) = o intersect at a point x, y^ and if <»
be their angle of intersection, prove that
tanc« = -^i#ZL4!^.
4>lc i>y + 0; ^^'i
13. Show that two curves 0 = o, ^ = o cut at right angles if at their point
of intersection
H. Apply this to show that the ellipses
jr«/tf» + y^/b'^ = I, x»/a« + yV/^* = »
will cut at right angles if « ' — ^* = a' — /S*.
15. Show that the length of the perpendicular p from the origin on the tangent
to the curve 0(x, ^) = o at jr, ^^ is
V(0xf'-fm?'
16. Show that the radius of curvature of /(jr, >') = o at jr, ^^ is
~ /;;(/;)"- 2/;;/;/; +/j;(/;)« *
17. If /(jc, >^) = o, show that
dy dx*'^^ \dxdy'^ dy^ dx } dx^'^ y^x"^ ~dx lyj -^ "^
18. U y^ =z 2xy -f tf', show that
dy _ y d*y _ d* d^ ^ ^d*x d^x tf"
dx ~ y - X ' dJ^ "" (^ - a-)' * Z? ~ " (^/-jr)* ' ^ ~ " p*
Also, that X =: ± a are maximum and minimum values of x.
19. Investigate^ = sin {x -{- y) for maximum and minimum^.
dy ^ cos( JT + y) d*y _ — y
dx ~ I — cos (jr -f y) ' 7i" ~ [i — cos(jr-|- ;/)]* '
20. If « = jc*^' — 2xy* + 3xy, show that
dz dz
' ^:? + *^aT^y + ^V='°'•
22. If ,= ^-:i£, ?! = Zj^..
X — y dx {x — yy
Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 305
ds ds
23. If y — fu =/[x — *»«), then m—-\-H~-=i.
24. If « = y»y prove t/^ = y'-i(i + \ogy*) = u^^
25. If « = V^T7", prove (jr^^ +-^^) ' = °"
26. The curve jr* -j- ^ — 3^: = o has a maximum ordinate at the point
I, ^2, and a minimum ordinate at — I, — ^2.
27. The curve p(sin'6 -|- cos'O) = a sin 2O has a maximum radius vector at the
point a f^2, ItC.
28. The curve zx^y -\-y*-\-^ — 2=0 has no minimum ordinate, it has a
maximum ordinate at the point — ^, 2.
29. The curve x* -\-y* — 4X)^ — 2=0 has neither maximum nor minimum
ordinate.
30. Show that (o, 2) gives y a maximum, and ± ^ 1/3, » ^ a minimum, while
1 V3i } n^^l^cs X a maximum, and ^ } ^3, } gives x a minimiun in the cardioid
31. In jc* + 2fljt*^ — aj^ = o, ^ is a minimum at jr = i « .
32. In ^ay* -{- xy^ + 4ajc^ = o, ^ is a maximum for * = S^/2.
33. Investigate the conic ax* -j- 2Arx -f- ^^' r= i, for maximum and minimum
coordinates.
34. If -^ is the radius of curvature of /(x, ^) = o, and 9 the angle which the tan-
gent makes with a fixed line, show finom ds ^ R dB and 6 = tan— ^ dy/dx^ that
R ^ (' +/•)* ^ (^ + ^«)i
y </*^ dx — dy d*x '
The first when x is the independent variable, the second when the independent
variable is not specified and <£r, dy are variables.
35. The third total derivative of /(jr, y) with respect to any variable / is
"• \dt dx'^ dt dy) ^'^ dx IF '^ dy
dfi \dt dx ^ dt dy] -^ ^ bx dF ^ dy dt^
dij^dy^di^ dt ^
+ 31 :^^-^'dt'^ dxdy \dF di "^ dF
CHAPTER XXVIII.
DIFFERENTIATION OF A FUNCTION OF THREE VARIABLES.
203. We are particularly interested here in the differentiation of
a function
of three independent variables, for the reason that when w = o we
have
/{x, y, z) = o,
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 .v = a, >' = 3, z =z c, there
is a point in space which represents the three variables x, y^ 2 for
those particular values.
When, corresponding to any triplet x, y, 2, the function /(jt, y^ z)
has a determinate value or values it is defined as a function of
x,y, z.
The function y is a continuous function of a:, y, z at x, y, z when
for all values of x^,y ^ z^ in the neighborhood of x, y, z we have
the n umber y{jk:j,^j, z^ in the neighborhood oi/(x,y, z).
204. Differentiation of w =/{x, y^ z). — Let x^y, z and x^y^
z^ be represented by two points P, P^
in space. Complete the parallelopiped
PRQP^ with diagonal PP^, by drawing
parallels to the axes through P and P^.
Then in the figure we have the coordi-
nates of R, {x^.yy z), and of Q, (x^^
y^ , z). Let PP^ z= Jr, and let /, m,
•« n be the direction cosines of the angles
which PP^ makes w4th the axes Oxy
Oy, OZf respectively.
Fig. 125.
Then we have
x^ — X = lAr,
y^ —y ^mAr,
z^— z = nAr,
306
Art. 205.] FUNCTIONS OF THREE VARIABLES. 307
Applying the theorem of mean value for one variable, letting
z, y^ X in succession alone vary, we have
A^xyf^ -A^iyi^) = («i - ^l/K-^i^'iO,
A^xyi^) -A^v)^) - (yx- yV'rK^xV^\
A^xy^) - /(■^) = (-^1 - ^Vii^y^)*
where S, y, b; x^ , 17, z; x^ , j\, C, are points such as Z, Af, N,
respectively, on the segments PA, RQ, QP^ By addition, we have
w^^w =/^{Syz)Jx -\-/;{x^vz)^y + /i{x^y,Z)Az.
Now let / be any differentiable function of jc, y, z, such that
/ = /j , when Xy y, z become x^, y^j z^. Then for the difference-
quotient of w with respect to /,
v). ^ w ^,,' .Ax , -,, .Jy . _,, ^.Jz
'l^—f =Ai{^)'^^ +/iK';^)27 +/iK>'iQ j7-
dw dw div
If now the partial derivatives ^—-, -r— , -^r- are continuous func-
ox ay az
tions throughout the neighborhood of x, y, z, we have, on passing to
limits in the above equation, the total derivative of y with respect
to /,
di " dx dt '^ dy d/'^ dz di' ^'^
The process is obviously general for a function of any number of
variables, and if /^ is a function of n independent variables z^^ , . . . ,
v^ , then the derivative of F with respect to /, a function of these
variables, is
dF^ _ V^?^ ^^'•
I
Second Total Derivative of w =^J\x, y, z). — We can differ-
entiate (i) with respect to / and obtain in the same way
'dfi" \d/dJc'^d/d^'^d/d'z)'^'^dxW^ + d^dr^'^ dzdi^' ^^^
20$. Successive Linear Differentiation. — Of chief importance
are the successive linear total derivatives oi/lx, y, z) with respect to
r when
X — a y — h z — c
~r~ = ~ik IT ^ "*'
where a, 3, c, /, w, n are constants. Then
Jf=a + /r, y -=z b -{- mr, z = c -{- nr,
and
dx _ dy dz
— _/, -^=«, ^=«
are constants, their higher derivatives are o.
3o8 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cm. XXVUL
Equation (i), § 204, becomes
We can differentiate (i) again with respect to r and get
d^f /.a a a\«
or obtain the result directly from the equation (2) in § 204.
We can show, as for two variables, that the »th linear total
derivative can be expressed by
dv /,9 ^ a , a\« . .
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/*.
EZSRCISES.
bu du du
^ _, , , dz z(t — xey)
2. If x^> + logs-jK* = o, ^=A___J.
3. If « = log(:r» -\-y^ 4- «• - 3^*), «i 4- «; + «; = 3{Jf +^ + «)-'•
4. If w = log (tan x -f tan^' -f tan a), tirj^ sin 2jc + Wy sin 2y + wi sin 2« = 2.
6. If w = (jc« +^« + «•)■"♦, show that
6. If «; = ^», 5^-^^ = (I H- 3xy« + ^y««)^.
7. If 70z=x*g*-\' ^V -f ;r«^«a«, w/Jry. = ^^«* + ^^'•
dz
8. Show that ^ = oo at the point (3, 4, 2) on the surface
jr* 4- 3«' + -^y — 2y« — 3jf — 4« = o,
9. Show that ^-|- = o at the point (— 2, ~ i, o) on the surfiice
♦** + *■ — S^' + 4K» H-^' — 2« — IS = o.
10. 5 — 5— = — at the point (i, 2, — i) of the surface
^•* 3^ 343 *^ V »
jc* — >' 4- 2«* + 2xy — 4rii 4--^— >' + ' — 5 =®-
11. Show that the second total deriyatives of w = /(•*"» ^t «) with respect to x, y, z
are respectively
dj* ~ \dx^ dxdy^ dxbz)'''^dy dj^^ bz dx*'
dy* ~ \dyWx^ by^'^bz)-''^ dxdy^"^ dzdy^'
d*w _ [dxb^ ^I^IV/O- ^£— 4- ^^
CHAPTER XXIX.
EXTENSION OF THE LAW OF THE MEAN TO FUNCTIONS OF TWO
AND THREE VARIABLES.
2o6. Fttnctlons of Two Variables. — Let s :=/{x,y) be a function
of two independent variables.
When X = a,y = d,\ets become c = /(a, 6). Also, let
X ^ a j^ — d
Then » -A^^y) =/(« + ^^ ^ + ^) (2)
is a function of the one variable r, if a, by I, m are constants. This
function becomes c = /[a, d) when r = o. If this function of r and
its first « + I derivatives with respect to r are continuous for all
values of r from r = o to r = r, then, by the Law of the Mean for
functions of one variable,
^=^+ VA+ '-+nl [d^)o+U^Ti)\^^)rJ ^'^
Here ( t— ) means the /th derivative of « with respect to r
, j means the (« + i)th derivative of «
with respect to r taken at some value a oir between o and r.
Also, since these derivatives are linear derivatives of «, we have
\drl)r-drt^> when^ = ^.j.=3,
since / = (jr — fl)/r, « =(^ — 3)/r, from (i). Hence
309
SIO I-mSSCU-'LES AMf THECRY C¥ L^TELEXnATT-INL 'Ci
Is
im^i,
rts^^tcz.'itLj
<'^Kj
I I
d
a5
^»^-r•
tbeMemV
of ('4> acd 151 in iji.
to f^r4t;orK of two Tzri
bare the Lxv of
^**
\0l
207. Tbc gco-n€trical intcrpretaifoc of § 2 c6 is as foUr-ws :
Giycn tbc ordii:;arc to a sariact at a ^<ir::c-Lar po:-: j, i, and the
partijil derivatives of the ordi-
nate at that point. To nni the
ordinate at an artitiaiy p-c:nt
Let z = /T.r. I • be tbee:':ai-
tion to a surace oa which
A.iJj 3, 0 is the p-o:nt at w^-ich
the c«x>rd:ni:es and ronidl de-
rvatives of r are kno^n. Let
/* be the point on the surface
at which .r. 1 are given and 5 or
yf JT, » ) is req:iired.
Pis a vertical r'ane through
A and P, cutting^ the surface in the airve AP and the horizontal plane
in the straight line jBJ/, whose equation is
— a y — 6
— — = ' = r.
m
The equation of the curve AP cut out of the surface by this verti-
cal plane is
z =fya + Ir, 5+ mr),
referred to axes Br, Bz' and coordinates r, z, in its plane rBz' The
law of the mean is applied to this function of the variable r, resulting
in /3J. Then, since these derivatives are linear, they can be ex-
pressed in terms of the partial derivatives oi z at a, by and (3) is trans-
formed into (6).
2o8. Expansum of Functioiis of Two Variables. — ^Wlienever
the function (2;, § 206, of the one variable r can be expanded in
Ari. 2io] extension of THE LAW OF THE MEAN. 31 1
powers of r by Maclaurin's series as given in Book I, then we can
make n = 00 in (6), and we have
and the functiony(;ir,j') can be computed in terms of /[a, d) and the
partial derivatives at a, b.
209. Functions of Three Variables. — Following exactly the same
process as in § 206, for
we have the law of the mean for three variables,
where S, t]> C are the coordinates of some point on the straight-line
segment joining the points in space whose coordinates are x^y, z and
a, d, c.
Whenever the function of onp variable r,
/{a + /r, <5 + mr, c + «r),
can be expanded in an infinite series of powers of r by Maclaurin's
series, Book I, then we can make» = 00 in (i), and have
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 =./[x,y) is constant and o, then /[x^y) = o 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
In like manner, by (i), §209, the equation to any surface
/{x, y^z) = o can be written
n
312 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIX.
^1, being (5), § 206, for equation (i) above, and the correspond-
ing value in (i), § 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
AT— a = A, y -^ h =z ky 2 — c = I,
and in the result changing a, d, c into x^y^ z.
Thus for two variables
For three variables
/[x+h.y+k, z+1)^^±(a^+ i^^+l^^'A^.y. «). (II)
EXERCISES.
1. Show that the equation of any algebraic curve of degree n can be written as
either
0=^^^i.\(--'')ra + (y-')3jYA'^i). (I)
or
2. Show that any algebraic surface of »th degree can be written in either of the
equations
0 = 2^{(* - «)4 + (A- - *) l + (,-.)i f>. *. .). (,)
r=o ^
X— -i-y ^) A^> y) is called a concomitant o{/{x, y).
Find the concomitants of a homogeneous function /(jr, y) of degree «.
In (10), § 211, put h z=. gx^ k =. gy, then
A' + g', y -V gy) =^^ (* ai + -^ ^) ^'' ^>
Since /is homogeneous in x and j/ of degree if,
Art. an.] EXTENSION OF THE LAW OF THE MEAN. 31 3
This equation is true for all values of ^ including o. Therefore, equating like
powers oigy we have
In the same way, if/(jr, y^ z) is homogeneous of degree n, we find, by putting
h = gx^ k •=. gy, /=s^in(ii), §211, as above, the concomitants oif[x^ y, z)^
'a; +^ -^ + • Tzj '/=«('•-')•••('»-'•+ 0/.
for r =r I, 2, . . . , If.
The concomitant functions are important in the theory of curves and surfaces.
They are invariant 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 =/{Xj y) will be a maximum at
X = a^ y = b, when _/][«, b) is greater thanyT[;c, y) for all values of x
2Si6.y in the neighborhood of a, b.
In like mannery(<i, b) will be a minimum value oi/{x, y) when
/{a, b) is less th2in/{x,y) for a// values of x, y in the neighborhood
of a, b.
In symbols, we have /(a, b) a maximum or a minimum value of the
function y(Ar,^) when
is negative or positive, respectively, for all values of Xy y in the
neighborhood of a, b.
Geometrically interpreted, the point P, Fig. 115, on the surface
representing z =/\x,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 ^t P
accordingly.
Also, when jP is a maximum point, then any contour line ZMN,
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 =/[x, y), X and j' 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
x' -=. X '\- /r, y '=.y -|- mr,
314
Art. 213.] MAXIMA AND MINIMA VALUES. 315
where / = cos 6/, »i = sin 0. Then
is a function of the one variable r, if 0 is constant.
If s is a maximum or a minimum, we must have, by Book I,
^ = o, -r^ negative or positive,
respectively, /or all values of ft That is,
^ = cos^^ + sin«^=o.
This must be true for all values of B, But when 0 = o and
d = \n^ we have
blA^'^^ = o and a^-^-^'-^) = ° (^)
respectively. Equations (i) are necessary conditions in order that
x^ y which satisfy them may give z a maximum or a minimum. But
they are not sufficient, for we must in addition have
different from o and of the same sign for all values of ft When (2)
is negative for all values of ft then z at x^y is a maximum; and when
(2) is positive for all values of ft then 2 is a minimum.
The quadratic function in /, m (see Ex. 19, § 25),
AP + 2Hlm + Bm^, (3)
will keep its sign unchanged for all values of the variables /, w, pro-
vided
AB - m
is positive. Then the function (3) has the same sign as^.
(a). Therefore the function /"(a*, >') is a maximum or a minimum
at x^ y when
ao; " ' by" ' aa:2 ay \^aA- by) " "^' ^^^
9V* 9V
and is a maximum or a minimum according as either •:— or -^^ is
oxr ojr
negative or positive respectively.
(3). If AB — H^ = — , then will (2) have opposite signs when
m = o and /»// = — A /If; also when / = o and m/I = — /^/^. The
function cannot then be either a maximum or a minimum (see Ex.
i9> § 25).
3l6 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX.
(c). If AB — ff^ = o, and A, B, Zf are not all o, then the right
member of (2) becomes
(lA + mHy {mB + lHy
A "^ B '
and has the same sign as ^ or B for all values of 0, except when
;;;// = — A/If. Then (2) is o. This case requires further examina-
tion, involving higher derivatives than the second; as also does the
case when A, B, H zi^ all o.
To sum up the conditions, we have /{Xy y) a maximum or a mini-
mum at AT, y when
A = 0, /y — o,
ft' T- max. /•// ftt _i
J XX — -T min., Jxxjxy — "T-
or yj; = T sr."!; fi n
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), we
solve yi = o,/'y =z o, to find the values of x, 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 x, y
iny^AT, y), or by eliminating x,y between the three equations
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 x, y by vertical planes and
determining whether or not ai/ 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
A^,y) -A-,y) = W - ^)^^ + (y -J')^^-
For all values of x', y' in the neighborhood of x, y we have 5, 17 also
in the neighborhood of x^ y. If the valuesy^, /*, are different from
o, then the values y^, /!^ are in the neighborhoods of their limits and
have the same signs as those numbers for all values of ^, y in the
neighborhood of x, y. Therefore the difference on the left of the
equation changes sign when x^ = Xy Sisy' passes through^, ii/'y ^ o.
In like manner this difference changes sign when^^' =y, as x' passes
* 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 /^ ^ o. Hence it is impossible for /[Xf ^) to be a
maximum or a minimum unless /"^^ = o and/*^ = o.
When/jJ = o, y^ = o, we have
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 or a minimum at Xy y. But in this neighborhood the
sign of the member on the right is the same as that of its limit,
This gives the same conditions as in I, and leads to the same results.
EXAMPLES.
1. Find the maximum value of « = ^axy — jr* — ^.
This is a surface which cuts the horizontal plane in the folium of Descartes.
Here
9« . 3« _
^ = 3«).-3jr«, -=3flj:-3r», (i)
The equations (i) furnish
3tfy-3jf» = o, 3tfjr-3y» = o^ (3)
for finding the values of x, y at which a maximum or a minimum may occur.
Solving (3), we have
x = o, ^ = 0| and jr = a, ^ = a.
For jf = o, ^ = o,
aF« dp " \?xby) - ~ ^ '
and there can be neither maximum nor minimum at o, a
For jr = a, ^^ = « 1
and since -r— . = — ^» we have the conditions for a maximum yalue of s at a, a
fulfilled. Hence at a^ a the function has a maximum value a*.
2. Show that e^/vi is a maximum value of
(a- x)(tf-;.)(x+^ -a).
3. Find the maximum value of jt* -|- x^^ +>'• — at — ^.
Am, \{ab ~ a* ^ ^.
4. Show that sin x -)- sinj/ 4~ ^^^ (-^ '^^) is ^ minimum when jt =^ = |)r, a
maximum when x =^ = J^r.
6. Show that the maximum value of
(fljc + 4y + OV(J^ +-^* + 0 i» «* + *« + A
3l8 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX.
6. Find the greatest rectangular parallelopiped that can be inscribed in the
ellipsoid. That is, find the maximum value of Sjcyz subject to the condition
Let u = xyz. Substituting the value of z in this from (i), we reduce » to a
function of two variableS|
--"^•('-S-S)-
du* du*
From ^— = o, -^— = o, we find the only values which satisfy the con-
ditions X z= a/ 1/3, y ^= d/ 1/3. These give z z= c/ 4/3, and the volume
required is Sadc/^ ^3!
7. Show that the maximum value of x^y^z*^ when 2j: -j- 3^ + 4* = ^» *s (a/g)^.
8. Show that the surface of a rectangular parallelopiped of given volume is
least when the solid is a cube.
9. Desijjn a steel cylindrical standpipe of uniform 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 oi base.]
11. The function z = x^ -{- xy -\-y* — $x — ^ -^ i has a minimum for jc = 2,
y=u
12. Show that the maximum or minimum value of
« = flj:* -f ^j'« -f 2Ax}' -f- 2£-x -\. 2fy -\-c (l)
is
Z =:
We have
a hg
hb f
gf c
-r-\ a h
I h b
I dz 1 dz
-5^ =«-' + Ay + ^ = o» -~~ =Ax + by+/=o. (2)
Multiply the first by Xj the second by^, subtract their sum from (i), and we
get
«=i''^+^ + ^. (3)
Eliminating x and^' between (2), (3), the result follows.
The condition shows that when ab — A^ is positive, the above value of a; is a
maximum or minimum according as the sign of a is negative or positive. If
ab — A^ =: — , then z is neither maximum nor minimum. We recognize the
surface as a paraboloid, elliptic for ab — //* positive, and hyperbolic when
ab - >4« = -.
13. Investigate 2 = x* + 3l>'' — ^y -{- 3-^ — 7y + ' for maximum and min-
imum values oi z.
14. Investigate max. and min. of x* -|- >'* — x* -{- xy ^ yK
X = o^y = Oy max. ; x = y = ± i, min. ; x = — y = ± \ ^3, min.
15. The function (jr — j)' — 4y{x — 8) has neither maximum nor minimum.
16. The surface x* + 2j* — 4jr -}- 4>' -J- 35 -}- 15 = o has a maximum
^-ordinate at the point (2, — i, —3).
17. The function jc* -\-y* — 2x* -f- 4^>' — 2^* has neither maximum nor minimnim
for X = o, ^ = o; but is minimum at {-{- f^2, — ^2"), (—4/2, -J- ^2").
Art. 214.] MAXIMA AND MINIMA VALUES. 319
18. Show that cos x cos a -|- sin jit sin a cos (^ — /^ is a maximum when
X = a,y = fl.
19. Show that x* — txy* -\- cy* ^t o^ o is minimum if r > 9, and is neither
maximum nor minimum for other values of c. Hint Complete the square in x.
20. Show that (I 4" -^ ~l~ ^''VC' — ^*^ — ^^)has a maximum and a minimum
respectively at
£ _ ^ __ 1 ± ^i ^ a* -f- h^
a ~ b tf» -f b'^ •
21. Show that 3, 2 make x^y\6 — x — y) ^ maximum.
22. Show that a, b make (2ax — x*)(2by — y^) a maximum.
23. Show that 3 + 4 V^ i^ ^ maximum, — 6 — 4 |/2 a minimum, value of
y - 8y -f l8y - 8y + o^ - 3jr« - 3jc.
214. Maxima and Minima Values of a Function of Three
Independent Variables.
Let u =/{x,y, z),
x^ — X = Ir == /t, y^ —y = wr = ^, s^ — z =z nr = g.
As before, if « is a maximum or a minimum at .r,^', z, we must have
u=/{x + ir, _y + mr, s + nr),
a maximum or a minimum for all values of /, m, », or
du du du du
_^ x^ — X du y^ —-yduz^ — zdu
r dx r dy r dz'
must be o for all values of /, m, n or of x^, y^, z^ in the neighborhood
o{ x^yy z, or
Hence the necessary conditions
du du du , .
dx^""' 3^=°' a?=<'- (')
Now when the relations (i) hold, and for all values of x^,y^, z^ in
the neighborhood of x, y, z, we also have
or, what is the same thing,
Ah^ + Bk^ + Q^ + 2Fkg + 2Ghg + 2^/ii (2)
(wherein^ =/-, ^ = /;', C=/-, /-=/;', G=/-, H^f-)
negative (positive) for all values of h, k^ g, then will « 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 ^i kj g has been determined in Ex. 20, § 25, where it is shown that
when
A H
and A
A H G
H B
H B F
G F C
are both positive (2) has the same sign as A for all values of ^, k^g.
Therefore /{x^y, z) is a maximum or a minimum at x^y^ s,
determined from
^/ = o, /; = o, /; = o,
when we have
/i; = T
nmz.
min. 9
/ xxj yx
J xy J yy
= + ,
J XX Jyx J MX
t I P
/ XM y y« J MM
~" T" min.
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.
EXAMPLSS.
1.
f m J^ 4-^' -|- »* + jr — 2« — jry.
/; = 2jr-;/-f 1=0, y; = 2^-x = o, /; = 2«-2 = o.
.-. Jf=-|, ^ = -i, 5=1, give /=-!.
Also, /jjy, = 2| Jyy = 2, /]J, = 2, /)^ = — I, _^ = 0| fyg = O.
2 1
I 2
= 3,
2 I o
120
= 6.
002
Therefore — 4/3 is a minimum value of/.
2. Find the maximum and minimum values of
AT* -\- fy^ -{- CZ* -f- 2fyz -\- 2gXg -|- 2Ajiry + 2UX -|- 2V^ + 2WZ + <i
Here /,' = 2{ax + Ay + ^ + «) = o,
// = 2{hx + /^v -f /^ -h
. «) = o, )
t;)=o, V
w) = o. )
(I)
// = 2Cpx+j5. + « +
Multiply the first by x, the second by y^ the third by 2. Add together and
subtract the result from the function/.
,*, f=.ux-\-vy'\'Ws-\-d, (2)
Eliminating x, ^, z between (i) and (2), we have
/=
a k g u
-H
a h g
h b f V
h b f
gfcw
gf ^
u V w d
which is a maximum or a minimum according as
«= T ,
« ^ I = +»
k b
a kg
h b f
if c
= T
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 fix>m three
given points is a minimum.
Let -rii^i, '|y . . . x^i y^i *ii be the given points. Then
fy = 22(;/ -;/^) = o = 3^ - "Syr ,
/; = 22(« — «r) = O = 3« — 2a^
••• •» = JC-^i -h -^a + ^«)» ^^ = K.>'i +>'«+>'»)» « = i<*l + «l + *»)•
The point is therefore the centroid of the three given points.
/2, =/;;;=/;; = 6, /;; =/i; = Z^;; = o. show that the solution is a min-
imum.
Extend the problem to the case of n given points.
4. If ter = ojf* + ^^ + ^** + ^ + ''*^'» show that x = ^^ = a = O gives
neither a maximum nor a minimum.
315. Maximum and Mlnimiim for an Implicit Function of
Three Variables. — ^To find the maximum or minimum values of z in
A^^ y^ *) = o-
Since the total difTerentials of /are o, we have
Also, at a maximum or a minimum value of z we must have
ia =: — =- . ^ o
dz
for all values of dy and <i[r. It is therefore necessary that
¥ 9/ a/ , ^
Substituting these values in (2), we have at the values of ^tr,^' which
satisfy (3), and make dz = o,
- TT
In order that this shall retain its sign for all values of dy and dx^
we must have
/:^/p - (A'Y = +• (4)
322 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX.
Then the sign of d^z is that of/l/j^l^. (See Ex. 19, p. 31.)
Hence z will be a maximum (minimum) at x, y^ z, determined
from
/x =0, /J = 0, /=o,
wheny^^ is positive (negative), provided (4) is true.
EXAMPLES.
1. Find the maximum and minimum of z in
2JP* -f- 5^' + «* — 4Jry — 2jf — 4y — i = o.
/; = 4j:-4y-2 = o, fyz=z toy -4x -^ = 0,
give X = f ^ = 1, z = ± 2.
/; = 2« = ± 4, /j; = 4, /ij/;; - (/j^? = 24.
z is therefore a maximum and a minimum at }, i.
2. Show that z in «' -|- Jar* — 4xy + ^« = o has neither a maximum nor a
minimum at jr = - /r,;' = - ylfj * = - A-
216. Conditional Maximum and Minimum. — Consider the
determination of the maximum or minimum vahie of zz=/'(x,jf),
when X and j/ are subject to the condition 0{x,y) = o.
Geometrically illustrated, z = /{x,y) and (f>{x^y) = o are the
equations of the line of intersection of the surface z =/'and the ver-
tical cylinder 0 = o. We seek the highest and lowest points of this
curve.
Since, at a maximum or minimum value of z^
also ^dx + ^dy=zo, (2)
we have, eliminating dy^ dx, the equation
/*0;-^'0i = o (3)
to be satisfied by at, y at which a maximum or minimum occurs.
Ek^uation (3) together with 0 = o determines x and y for which a
maximum or minimum may occur.
I fsually the conditions of the problem serve to discriminate be-
tween a maximum, minimum, or inflexion at the critical values of a:, >'.
The test of the second derivative, however, can be applied as
follows: We have
which must keep its sign unchanged for all values of or, y satisfying
0 = o in the neighborhood of the x^y also satisfying (3). But we
also have
0;;^.r» + 20;;^^^ + 0;;.^ + 0;^^^ + 0;^^ = o. (5)
Art. 217.] MAXIMA AND MINIMA VALUES. 323
To eliminate the differentials from (4), (5), multiply (4) by 0^ , (5)
by/y, and subtract, having regard for (3). In the result substitute
ioxdy/dx from (2).
When this is negative (positive) we have a maximum (minimum)
value of z. The form of the test (6) is too complicated to be very use-
ful, and it is usually omitted.
EXAMPLES.
1. Find the minimum value of ^r* -\-y^ when x ziAy are subject to the condi-
tion ax -\- by -\- d ■=. o.
Condition (3) gives bx = ay. Therefore, at
we have
d}
which can be shown to be a minimum by (6). Otherwise we see at once from the
geometrical interpretations that this value of jr* 4~>'' must be a minimum.
First. ^^ 3^ -\-y* is the distance from the origin, of the point x^y which is on
the straight line ax -\- iy -^^ d = o, and this is least when it is the perpendicular
from the origin to the straight line, which was found above.
Second, g = x* -^-y* is the paraboloid of revolution. The vertical plane
ax -\' iy -\- d =z o cuts it in a parabola, whose vertex we have ^und above, and
which is the lowest point on the curve.
2. Determine the axes of the conic ax* + fy^ + ^Ary = i.
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 ;r*-f-^'} subject to the above condition of x^y being on
the conic.
Let « = 4f» -{-jA and 4> = ax* + by* -\- zhxy ~ i = o.
Condition (3) gives
X ax-\- hy
J^by -\-hx'
Multiply both sides by x/y and compound the proportion, and we get
(tf — «— «)x -}- A^ = o,
hx •\-{b ^ ir-«)y = o.
Eliminating x and y^ there results
for determining the maximum and minimum values of m.
217. The whole question of cre^^f^/r/ib/fa/ maximum and minimum is
most satisfactorily treated by the method of undetermined multinliprc
of I^grange.
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. [Ch. XXX
To find the maximum and minimum values of
when the variables jc, j^, z, w are subject to the conditions
ip{x,y, «, w) = o, (2)
f{x, y, z, w) = 0. (3)
Since, at a maximum or minimum value of u, we must have du = o,
the conditions furnish
/; dx+/;dy+/Jdz+/:,dw = o,)
0; dx +(f>;dy + <Pidz+i/>^dw=oA (4)
^; dx + tl:;dy + fi dz + i:^dw = o.)
Multiply the second of these by A, the third by fj, X and /< being
arbitrary numbers. Add the three equations.
(/; + Un + tJLf,)dz + (/: + A0: + ^ti::)dw = o. (5)
Since X and /i are perfectly arbitrary, we can assign to them
values which will make the coefficients of ^.vand dy vanish; ncioreover,
since equations (2) and (3) connect four variables, we can take two of
them, say z and w, independent, and therefore dz and dtv are arbitrary.
Consequently, in (5), after assigning X and // as above, we must have
the coefficients of dz and dw equal to o. Therefore
/; + X<t>', + ^r, = o,
/; + A^; + fx-n = o, f w
The six equations (2), (3), (6) enable us to determine or, _y, z, w,
A, //, which furnish the maxima and minima values of u.
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* -f- y* + '* when x, y, z are subject to
the condition
0=.ax-{'by-\-cz-{-dz=zO,
Here we have, as in equations (6),
^11^^ 111
dy ^ dy "^ ' *
- — t- A -^ = o = 2« I- Ar.
Art. 217.J MAXIMA AND MINIMA VALUES. 325
Multiply by a^ b^ c and add. Also, transpose and square. Then
2(ax -f 4y + cz) -f (a» + 3« -f. ia)A s - 2^ + (<i« + ^ -f i*)X = O,
^jc^ ^yt + ,») « (a« 4- ^« + /«)A.« a 4« - («" + ^ + ^)A« = o.
The problem is to find the perpendicular distance firom the origin to a plane.
2. Find when « = jr* + ^* + «* is a maximum or minimum, jr, y^ t being sub-
ject to the two conditions
js*/a^ -f ^'V^ H- *'A* = '» ilr -f »iy -f »« = o.
Geometrically interpreted: Find the axes of a central plane section of an ellip-
soid.
Equations (6) give
ay + A ^ + /«»» = o»
Multiply by jr, ^, « and add. We get A = — u. Therefore
" 2{u - ««) ' '"•^ "" 2(« - <^) ' *** "" 2{u - ^) '
Hence the required values of m are the roots of the quadratic
fl«/« ^«f* ^«« _
3. Find the maximum and minimum values of
« = a«jp* + ^y -f <«««,
o", y^ z being subject to the conditions
Jf* -f ^* + a* = I, /r 4- «wy -|- iMT = O.
The required values are the roots of the quadratic
it/{u — a«) + !»*/(« - ^«) 4- ««/(« — <•) = o.
4. Find the maximum and minimum values of
w = jr* 4- ^' 4" *'
when ji, ^, z are subject to the condition
€ufl 4" ^^' 4- ^** + 2/V* 4" ^^-^a 4- 2^jry «= I.
Geometrically interpreted: Find the axes of a central conicoid.
The conditions (6) give
X -\- {ax -\- ky + gz)X = O,
^4-(>b:4-/iy4./;rU = o,
* + igx '\-fy + «)A. = o.
Multiply by x^ y^ z and add. . •• X = — u.
Eliminating x, y, z from the above equations,
a — «-i, A , ^ = a
A , 3 — «-„ /
^ » / , c-u-i
The three real nx)ts of this cubic, see Ex. 17, g 25, furnish the squares of the
semi-axes of the conicoid.
324 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX.
To find the maximum and minimum values of
when the variables j:, y, z, w are subject to the conditions
<f>(x,y, 2, w) = o, (2)
^(x, y, z, w) = o. (3)
Since, at a maximum or minimum value of u, we must have du = o,
the conditions furnish
/; ^^ + /y4y+ /; dz + /^dw = o,)
0; dx+<f>;dy + (f>^dz + <f}^ dw=o,y (4)
tpj;dx+tl.^;dy + i/;idz + tl::,dw = o. )
Multiply the second of these by A, the third by //, A and ja being
arbitrary numbers. Add the three equations.
(/i + A0^ + ;i^>^ + (/; + '"^0; + Mtp;)dy +
(/; + A0; + Mf.^^ + {/: + A0^ + M^^:,)^ = o. (5)
Since X and /i 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 e and w, independent, and therefore dz and dw are arbitrary.
Consequently, in (5), after assigning A and // as above, we must have
the coefficients of dz and dw equal to o. Therefore
A + ^<Pi + P^'Px = o,
/; + ^0; + Mi^i = o,
/L + A0^ + pii\^i = o.
The six equations (2), (3), (6) enable us to determine x^ y^ z, tu,
\, /i, which furnish the maxima and minima values of u.
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 « = jc* -f ^* + ** when x, y, m are subject to
the condition
(f) ^=. ax -{- by -\- cz -\- d z= o.
Here we have, as in equations (6),
a« , _ 30 , _
^ + ^:^=° = *-^ + ^*'
5~ + A 3- = o = 2ar f Af.
oz dz
Art. 217.J MAXIMA AND MINIMA VALUES. 325
Multiply by a, b, c and add. Also, transpose and square. Then
2(ax 4- ^^ -f cz) -f (tf » -f ^ 4. <«)A a - 2</ + (a« -I- ^ H- <*)A = o,
4(^ ^yt + ,1) _ (tfl 4. ^ ^ ^A« H 4J# - (tf« + ^ + ^)A« = o.
. •• a/ii = = =r—
The problem is to find the perpendicular distance from the origin to a plane.
2. Find when u = x^ -\- y^ -\- z^ is z. maximum or minimum, x, y^ % being sub-
ject to the two conditions
jp«/fli -{•y^/H^ -f «V^ =1, /jT-f »»y -f if« = o.
Geometrically interpreted: Find the axes of a central plane section of an ellip-
soid.
Equations (6) give
2« -|- A -y -f- /III sr O.
Multiply by x, ^, « and add. We get A = — u. Therefore
^ - 2(11 - fl«) ' "^ - 2(« - ^) ' '^ - 2(f< - ^) •
Hence the required values of m are the roots of the quadratic
g«/« ^«i»f« <*«« _
3. Find the maximum and minimum values of
II = tf V -f- ^« -f ^f «,
jr, ^, « being subject to the conditions
x*4-^* -}"'*='» Ix -\- my -\' rut ^1 f^
The required values are the roots of the quadratic
/»/(« — fl«) 4- ««/(« - ^) -f- iiV(« — <*) = o.
4» Find the maximum and minimum values of
u = jt»-fy + ««
when x^ y^ % are subject to the condition
ax^ + ^>'* H- «* + 2/V* 4- ^gxz 4* ^Ary x= 1.
Geometrically interpreted: Find the axes of a central conicoid.
The conditions (6) give
X ■\-{ax-\- hy '\- gz)X = o^
^4.(Ar4.^4./«)A = o,
Multiply by x^ y, % and add. . *. A = — tf.
Eliminating x^ y^ z from the above equations,
a — if-i, h , ^
A , d ^M^, f
= a
The three real roots of this cubic, see Ex. 17, g 25, furnish the squares of the
semi-axes of the conicoid.
326 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX.
5. Show bow to determine the maxima and minima values of x* -^ ^ -{- s*
subject to the conditions
BXBRCISB8.
1. Show that the area of a quadrilateral of four given sides is g^atest 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 a into three parts x, y, z such that x'^tP may be a
maximum. ^ * y * ^
Ans. —=— = — =
m n p m-{-n-^p
5. Find the maximum value xy subject to the condition x*/a* 4->V^' = '•
This finds the greatest rectangle that can be inscribed in a given ellipse.
6. Find a maximum value of xy subject to iix-\- fy = c, and interpret the result
geometrieatty^
7. Divide a into three parts jr,^, *, such that xy/2 -j-xz/^ -|-^V4 ^^^^^ ^ *
maximum. Ans. x/21 = y/20 = t/6 s=: a/^j.
8. Find the maximum value of xyz subject to the condition
by the method of § 216.
9. Show that x-{-y -\-z subject to a/x + 6/y -\-c/z=zi is a minimum when
10. Find a point such that the sum of the squares of its distances from the
comers of a tetrahedron shall be least.
11. 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 in the plane of a triangle such that the sum of the squares
of its distances from the sides a, d, c is least A being the area of the triangle.
xyz 2A
a b c fl«4-^ + <«
13. Circular sectors are taken off the comers of a triangle. Show how to leave
the greatest area with a given perimeter. [The radii of the sectors are equal.]
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.
/,jr -f »f,j/ -I- »,« = /j , )
The equations of the planes being in the normal form.
Art. 217.]
MAXIMA AND MINIMA VALUES.
3^7
We have, if fi« = jr« +>^« + ««,
2JP+ A^+ l^ =0,
ay + m^ + ot,u = o,
Multiply these by x, ^, s in order and add. Multiply by /, , m^ , fi, in order
and add. Multiply by /, , i», , ir, in order and add. Whence the equations
2«« 4- p^X + PtM=o,
2/1 "I" A -f- cos 6// = o^
2/, 4- cos 6A. 4- M =0,
Since /j* -f- iHj' 4- *i' = V + *•!* + ''i* = '» A^j + iWjiw, 4" 'V'l = cos ^, where
0 is the angle between the normals to the planes. Eliminating A and /<, we have
«• A /, =0.
/i I cos 6
/, cos6 I
or «• sin*8 = /j* 4- /,» — 2/,/, cos 0,
which result is easily verified geometrically as being the perpendicular from the
origin to the straight line.
16. A given volume of metal, z^, 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 Xf y^ % are the external length, breadth, depth,
m — <jS
= -^ = "+-^1^5-: » = **•
17. Find a point such that the sum of the squares of its distances from the faces
of a tetrahedron shall be least. If V is the volume of the solid, x^ y^ s, w the per-
pendicular distances of the point from the fatces whose areas are A^ B, C, />, then
:S~5"" ^""5"" ^«4.^4- C«4-Z>«*
18. Of all the triangular pjrramids having a given triangle for base and a given
altitude above that base, find that one which has the least surface.
The surface is i<fl 4- ^ 4- r) ^t* -f A«, where a, b, c are the sides of the base, r
the radius of the circle inscribed in the base, h the given altitude.
19. Show that the maximum of (ajc 4- ^ 4- «y-**-»'-^'^"-l''«* is given by
20. Show that the highest and lowest points on a curve whose equations are
0(x, y, «) = o, ^jf, y,z)^o, (I)
are determined from these equations and
<f>x + A*,' = o, ^/ 4- Xi>; = o. (2)
21. Show that the maximum and minimum values of r* = jr* 4- ^ 4~ '*> where
Xf y, » are subject to the two conditions
fljr* -|_ ^y* 4- «« 4- 2fyz 4- 2/xs 4- 2hxy = l, ix + n^ -{- wi ^ o^
are given by the roots of the quadratic,
= a
" r-2 h g
I
k b^r-^ f
m
g f c^r^
n
i m n
0
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 uf max-
iraum and minimum conditions.
22. Investigate s = 2jc* — yc^^ + ^ = (y^ — ^){y* — **)•
At o, o we have «i =r i^ = jBr^J = «y'J = o, «jJi = 4. The conditions
g^ jpjj — («it)' = o makes the case uncertain. The function s vanishes along
each of the parabolse y^ = Xj y* = 2x. It is positive for all values x, y in the
plane 2 = 0, except between the two parabolse, where it is negative. The function
is neither a maximum nor a minimum at o, o, since it has positive and negative
values in the neighborhood of that point. In &ict s is negative all along y^ = 3Jt'/2
except at o, o.
23. « = a^y^ — 2ajc^y -)- jt* 4- ^*.
At o, o the case is uncertain. Put^ = mx^ then
z = jr«[(I + m*)x* — 2amx -f ««»!«].
When X ory is o the function is positive. For all values of m the quadratic
factor in the brace is positive.
Hence 2; is a minimum at o, o.
24. « •=y'* — xy^ — 2x^y -|- ■**•
As in 23, the condition is uncertain at o, o. Put^ = mx. Then
z = x\x^ — m(m -}- 2)x -\- w»].
The function is positive when x ory is o. For any value of m not arbitrarily
small z is positive for all arbitrarily small values of jr. But since
is negative for all arbitrarily small values of nty the quadratic function of x in the
brace has two small positive mots for each such value of m. Between each pair of
these arbitrarily small roots the quadratic factor, and therefore 2, is negative. The
function is neither a maximum nor a minimum at o, o. In fact along the curve
X* z=: y the function is « = — jc^.
25. Q>nsider the function z defined by the equation
or « = « — ^xap — p^y
wherein the positive value of the radical is taken and p* =z x* -i-y*. This is the
lower half of the surface generated by revolving the circle (x — a)* -{-y* = a* about
the ^'-axis.
Here
dz X a ^ p dz y a — p
^ "" "^ ^ J^2a^p * ^y " ~ ~^\ i^2a — p '
At all points satisfying .r* +>'* = p' = <»' these derivatives are o. Also at
such points
... »i, ,j5 - (^;;)« = o.
The function s is o at each point jr, y satisfying jr* 4- y^ = '''. 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 « = o is a singular tangent plane to the surface.
CHAPTER XXXI.
APPLICATION TO PLANE CURVES.
I. Ordinary Points.
2i8. We have seen that when the equation of a curve is given
in the explicit form y = f[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^' with respect to x^
or through the law of the mean, as given in Book I, for functions oif
one variable.
In the same way, when the equation of the curve is given in the
implicit form F(x^y) = o, 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 = F{x,y)^
whose intersection with the plane ;? = o is the curve we wish to
investigate.*
219. Ordinary Point. — If F{x^ ^) = o is the equation to a
curve, then any point x^y at which we do not have both
bF ^ dF
- — = o and -TT- = o
dx dy
is called a single point on the curve, or a point oi ordinary position ^ or
simply an ordinary point.
By the law of the mean,
hF hF
F(x,y) = /l[a, 3) + (;r - «) g^ + Cy - *) -£-.
If F(x, y) = o, and a, ^ is an ordinary point on this curve, then
Fia, 3) = o. Hence
From this we derive for ar( = )<J, j'(=)3,
dy _ ^dF /dF
dx ~' bx I by'
* For conyenience of notation we shall generally write the explicit equation to
a curve in the fsxmy =:/(jr), and the implicit equation as F[x^ y) = o.
329
330 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
Therefore the curve /l[a:, J') = o and the straight line
(._«)_ + (^_*)_ = o (I)
have a contact of the first order at a, 5, or (i) is the equation of the
tangent to the curve at a, b.
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 F{Xy ^) = o be the equation to a curve, then
is the equation to the curve in the form of the law of the mean. The
straight line
intersects this curve in points whose distances from Xy y are the roots
of the equation in r,
o = /X-.» + '-(4+«|)^+7(4 + «^)V. (4)
If the point x, y is on the curve, this is one point of intersection,
and one root of (4) is o, for F[Xy y) = o.
If in addition we have
,dF dF
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 Xf y.
Eliminating /, m between the condition of tangency (5) and the
equation to the straight line (3), we have the equation to the tangent
at X, y,
(^-*)a4 + (^->')f = 0. (6)
the current coordinates being X, Y.
The corresponding equation to the normal at x^y is
dF dF ' ^^^
dx dy
^ -- -f J^ -sr + «--i + 2«»-a + . . . 4- fWo = O.
Art. 219.] APPLICATION TO PLANE CURVES. 33 1
EXAMPLES.
1. Use Ex. 3, § 211, to show that if J^x^ }f) = cis the equation to a curve, in
which y^jT) y) is homogeneous of degree n^ then the length of the perpendicular
from the origin on the tangent is
2. If jF[xt y) m Un-h **i»-i -|- . • • + «i + «o = o >s the equation of a curve
of ifth degree, in which Uf. is the homogeneous part of degree r, show that the
equation of the tangent at x, ^ is
dx "^ dy
!£ X, K is a fixed point, this is a curve of the (n — i)th degree in jr, y which
intersects /(x, ^^ = o in ft{n — i) points, real or imaginary. These points of
intersection are the points of contact of the n(n — i) tangents which can be drawn
from any point -Y, K to a curve -^ = o of the nth degree.
3. If A^, y be a fixed point, the equation of the normal through JT, Kto /*= o
at jr, y is
dF dF
This is of the nth degree in jt, ^, which intersects /*= o in «• points, real or
imaginary, the normals at which to ^ = o all pass through X^ K There can
then, in general, be drawn n* normals to a given curve of the ffth degree from any
given point.
4. Show that the points on the ellipse x*/a* -f~ y*/^ = I at which the
normals pass through a given point a, /H are determined by the intersection of
the hyperbola
xy{a* - ^) = ad^ - /5^jf
with the ellipse.
5. If F{Xf ^) = o is a conic, show that its equation can always be written
(a). Show that the straight line whose equation is
-p=^=r, (4)
where / = cos 6, m = sin 0, cuts the curve in two points whose distances from <7, 6
are the roots of the quadratic
(^). Show that
('-'')S+(^-*)^=« (4)
is the equation of a secant of which a, d is the middle point of the chord.
(r). Show that the equations
dF dF
solved simultaneously, give the coordinates of the center of the conic.
(d). Show that
X — a y " b . ,dF . dF
— z — =< and / i 1- *» - - = o
/ m dx ^ dy
332 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
are the equations of a pair of conjugate diameters of the conic /* = o, whose center
is a, b,
6. K >&* < I, show that the tangent to jc*/tf* + y^/b^ = >&* cuts off a constant
area from x^/a^ -Vy^l^ = I-
7. In Ex. 5, show how to determine the axes and their directions in the conic
/* = o, by finding the maximum and minimum values of r in the quadratic (3)^ as
a function of 0, tlie center of the conic being a, b,
220. The Inflexional Tangent. — At an ordinary point x^ y on
the curve F{x^y) = o, the straight line
X ^x Y-v , .
-7-=~^=o (I)
cuts the curve in points whose distances from jr, y are the roots of the
equation in r,
o
If we have
the line (i) cuts the curve in two coincident points at x,y, and is tan-
gent to the curve there.
If, in addition to (3), / and m satisfy
then the line cuts the curve Z' = o in three coincident points at x, y,
provided
In this case the line (i) has a contact of the second order with
F =: o at Xf y, and this point is an ordinary point of inflexion. This
means that the value of I/m = tan 0 in (3) must be one of the roots of
the quadratic in i/m (4).
Eliminating / and m between (3) and (4), we have a condition that
X, y may be a point of inflexion,
F- Fp - 2F- Fl F; + F^ F;? = o. (6)
To find an ordinary point of inflexion ow F ^=^ o, solve (6) and
F =• oiox X and y. If the values of jr, y thus determined do not
make both Fl, and Fy vanish, and do satisfy (5), the point is an
ordinary point of inflexion.
The solution of equations (6) -and Z' = o is generally diflScult.
In general, if or, y is an ordinary point satisfying Z' = o, and
\by bx dx by ) ~ '
Art. 221.] APPLICATION TO PLANE CURVES. 333
r = 2, 3, ...,» — I, and
\dy dx dx by) ^ ^ ^'
then when n is odd we have a point of inflexion at which the tangent
cuts the curve in n coincident points at x, y. When n is even Xy y is
called a point of undulation and the curve there does not cross the
tangent but is concave or convex at the contact.
The conditions for concavity, convexity, or inflexion at an ordinary
point on /^= o can be determined as in Book I. For, differentiating
/^ = o with respect to a: as independent variable,
~~ bx dy dx'
-\dx "'' dx By) '^ "^ dyd^'
At an ordinary point d^ ^ o or b^ ^ o. Hence the curve is
convex, concave, or inflects at x, y according as
d*y \dx dx dy / \dy dx dx dy /
dx*~ dF /a/[\8
is positive, negative, or zero.
1. Show that the origin is a point of inflexion on
a^y = bxy -f- ^^ •\- dx*.
2. Show that x =. by y •=. 2^/rt* is an inflexion on
jr* — 3^jr* 4- d^y = O.
3. Show that the cubical parabola ^* = (x — tif{x — b\ has points of inflexion
determined by 3jc -|- « = 4^.
Hint. Solve the conditional equation for {x — a)/{y — b),
4. If ^* = /(-x*) be the equation to a curve, prove that the abscissae of its points
of inflexion satisfy
II. Singular Points.
221. If at any point at, ^^ on a curve F(x,y) = o
dF ^ dF
^=0 and ^ = 0,
the point jc, y is called a singular point.
Since 3^ = — a~ / a—* ^^ direction of a curve at a singular point
is indeterminate.
334 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXL
222. Double Point. — If at a singular point the second partial
derivatives of /'are not all o, we shall have
Divide through by {X — xY and let X{=)x, Then
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
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{Xy y) ==-o must satisfy the
equations
^-=0, /'; = o, f; = o. (i)
The slopes of the tangents there are the roots /^ and /, of the
quadratic
/»/--+ 2//-- + /';i=0. (2)
(A). Node'. If the roots of the quadratic (2) are real and different,
then
/•- F^ - F^^ -, (.5)
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 quadratic in / (2) are imaginary,
or
Fj; F- -F-^=+, (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
^\\^x-.)l^^(r-y)l-yF
For arbitrarily small values oi X '— 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 X^ K in the neighborhood of x,y,
(C). Cusp'Conjugaie. If the roots of (2) are equal, or
/--/-^'-/--a^o, (s)
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 F^^^ -^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 ;^« = (X - a){x - b){x - r).
(I)
y
where a, b, c are positive numbers, and a < b < c.
The curve is real, finite, two-valued, and sym-
metrical with respect to C?jc for « < x < b. It does
not exist for x < a or ^ < x < r; it is finite and sym- Q
metrical with respect to Ox for all finite values of
X > £' The ordinate is 00 when x = 00 . The curve
consists of a closed loop from a\o b^ and an infinite
branch from c on. The curve is shown' in Fig. 127.
y
^
Fig. 127.
Fig. 128.
In the limit we have
{b). Let € converge to b.
Hien the loop and open branch tend to come together,
and in the limit unite in
jK« = (x - a%x - b)\ (2)
giving at ^ a node. (See Fig. 128.)
(c). Let b converge Xo a in ( I). The closed oval
continually diminishes, shrinking to the [)oint a.
>^« = (X - «)«(x - c\
(3)
O
Fig. 129.
which consists of a single isolated or conjugate point
X = a, and an open branch for x > c (Fig. 129.)
(</). Let c and b both converge to a. The oval
shrinks to a, and the open branch elongates to a also,
resulting in
^« = (x - a)». (4)
which has a cusp at a, (Fig. 130.)
2. A clear idea of the meaning of singular points
on a curve is obtained when we consider the surface
s = /(x» 7), which for any constant value of « is a
curve cut out of the surface by a horizontal plane.
For example, using (i), Ex. I, we have the surface
z = {x^a){x-bXx-c)--y\
^y^' which is symmetrical with respect to the xOz plane,
and cuts the xOz plane in the cubic parabola
« = (X — fl)(x - b)(x - c\
and the horizontal plane in the curve
^« = (X - fl)(X - b){x - c).
336 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
A movinf^r horizontal plane cuts the surface in curves of the same family. For
example, DD is an open branched curve; BB is a curve with a node as in Fig. 128;
A A is a curve with a closed oval and one open branch as in Fig. 127; so also is CC.
As the horizontal cutting plane rises until it reaches a maximum point 7* on the
Fig. 131.
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 T, The
point of touch 7' is a conjugate of the curve TT and part of the intersection of the
surface by the plane. If the cutting plane be raised higher, to a position y, the
oval and conjugate point disappear altogether and the section is only the open
branch J.
Observe that the tangent plane at the node of BB 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
« = (x - a)» - y\
This cuts xOz in jr = (x — a)>, and the
horizontal plane in ^' = (jr — <i)'. All
planes parallel to yOz cut the surface in
• ..^ordinary parabolse. All sections of the
surface by horizontal planes are open
branched curves, none having cusps except
that one in xOy, All horizontal sections ftir
% negative have inflexions in the plane j:=a,
and their tangents there are parallel to Ox,
The horizontal sections above xOy have no
inflexions. As the plane of the horizontal
section below xOy rises, the inflexional tan-
gents unite in the unique double tangent at
the cusp in the plane xOy,
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 F(x^y) = o have a point of
this class, and let F\x^ ^) = o be the equation of the curve referred to the singular
point as origin and the tangent there as ;r-axis. The point is a cusp of the first
species if ^jr, o) changes sign as x passes tlirough o. If F\x^ o) does not change
sign as x passes through o, the point is either a conjugate or a cusp of the second
species. If in the neighborhood of such a point no real values of jr, y satisfy the
equation, the conjugate point is identified. Also, the conjugate points on i*^ = o
are the values of ;r, y which make z •=. Fz. maximum or a minimum.
* A cusp is of the first species 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 »
Fig. 132.
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.)
Node. Cusp, first species. Cusp, second species.
Fig. 133.
In fact, all other singular points of algebraic curves are but combi-
nations of these, together with inflexions.
SZAMPL£8.
1. Show that the origin is a node o{y*{a* -|- jc*) = x'{a* — :r*), and that the
tangents bisect the angles between the axes.
2. Show that the origin is a cusp in ay* = ;r*.
3. Find the singular point on^ = jfl(x -|- a). [Cusp.]
4. Investigate ^jc* -}-^') = •** *^ *he origin.
5. Investigate Jt^ — ^axy -^y* = o at the origin.
6. Find the double point of (^x — eyf = (x — of, aod draw the curve there.
[x = Cf y •=. ab/c. Cusp.]
7. The curve (^ — r )• = (jf — a^ix — b) has a cusp at a, r, if 0 ^ ^ ; conju-
gate if a < b,
8. Investigate ^' = x{x + a)* and jcl -f-^* = «* for singular points.
9. Investigate at the origin the curve
F^ ay* — 2xy* -f- $yx* — tfJiP*-f^-fjr«-f^ = o.
Here /^ = o. /J = o, /%* — ^^^ /j; = o, at the origin, and the third
partial derivatives are not all o. The origin is a point of the cusp-conjugate class,
and^'' = o is the double tangent
Since F(X, o) s — ojc* -|- jt* changes sign as x passes through o, the origin is
a cusp of the first kind.
223. Triple Point. — If jf, >' satisfy the equations
F= fj;=zF; = Fjii = f;; = Fi; = o, (i)
and do not make all the third partial derivatives of F vanish. Then
we have at any point X, IT on the curve
o = {(^--)^ + (K-.)|-}V.
Divide by {X — xY and make X{^)x, We have the cubic in /
for finding the three directions of the curve at or, y,
o = /-^'i; + ztFj^y + 3/^^ + fiF^. (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,
33^ PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
which is a triple point on the curve. The equation of the three tan
gents at or, y is
1
Some forms of triple points are shown in Fig. 134.
Fig. 134.
EXAMPLES.
1. Show that jc* = (j:* — y'^)y has a triple point at the origin.
2. Investigate at O the curve jr* — Z^xy^ -h 2ay' = o,
224. Higher Singularities. — In general, if
h^F _
dx^dyi "" °
for all values of p -^ g = r, and r = o, i, 2, . . . , « — i, then
the curve 1^=0 has an «-ple point at x,yt and in general passes
through the point n times.
The equation of the n tangents there is
Their slopes are the roots of
Examples of multiple points are shown in Fig. 135.
Q^^V
Fig. 135.
EXAMPLES.
1. Investigate «• + ^ = $ax*y\ at o, o,
2. Investigate (y — jt»)« = x*, at o, o.
3. In JT* + ^x* — t^y* = o, the origin is a double cusp.
4. Determine the tangents at the origin to
y^ = x\i — x^).
t* ± ^^ = o.]
Art. 225.J APPLICATION TO PLANE CURVES. 339
5. Show that jf* — 3rtxy + >'* = ^ touches the axes at the origin.
6. Investigate x* — ax^y -{- fy^ = o at o, o.
7. Show that o, o is a conjugate point on
ay^ — ji* -f~ ^^ = o
if a and d are like signed, and a node when not.
8. Show that the origin is a conjugate point on
/*(jf* — a*) = x*f and a cusp on (^^ — x*)* =r jr*.
9. Investigate {y — x^y = x** at o, o, for n ^ 4.
10. Investigate {x/a)^ + (^/^)' = '» where it cuts the axes.
11. Find the double points on
or* — 4ax^ -f 4a*x* — d*y* + 2£^y — a* — ^ = o.
12. Also on jr* — 2ax^ — axy* -f- ^'V* = ^■
13. Find and classify the singular points on
X* — 2ax^y — axy* -{" ^*y^ = ^
when tf = I, a > I, <? < I.
14. Show that no curve of the second or third degree in x and^ can have a cusp
of the second species.
Show that if I^x, _y) = o is any equation of the third degree, having a point of
the cusp-conjugate class at the origin and the jr-axis as tangent, the origin is a
cusp or conjugate point according as /^jr, o) does or does not change sign as x
passes through o, that is, according as the lowest power of x is oi/d or ^v^n.
15. If I*{x^ y) =z o is any curve of the fourth degree, having at the origin a
double point of the cusp-conjugate class, and the tangent there as ^r-axis, then the
origin is a cusp of the first species if the lowest power of x in /(x, o) is odd ;
otherwise, it is a cusp of the second kind or a conjugate point according as the
co-factor of j:* in J*[xy 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
jr* -|- ^ _ ay* — 2ax^y -|- axy* -\- a*y^ ^ O;
is a conjugate point in
x^ -{• y* — tfy* — oj^y -f- *^* + ^V = ^I
and a cusp of the first kind in
jT* 4- ^ — ^y* — bx^y -f- ^•^* + ^V = ^
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 « = o. If now we make the equa-
340 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
tion to any curve F{x,y) = o homogeneous in x,^, s, by writing the
equation
(I)
(r. i) =
and clearing of fractions, then we have the homogeneous equation in
three variables Xy jf, z,
^i(-^> ^> «) = o- (2)
F^ = o becomes F= o when we make z = 1,
But /'j = o being homogeneous in x, y, z, it is the equation of a
cone with vertex at the origin, and which cuts the horizontal plane
« = I in the curve /^ = o, which curve is the subject of investigation.
Consequently any investigation of /*, = o carried on for a homo-
geneous function in a, ^, z is applicable to the curve F = o when in
the results of that investigation we make z = 1.
III. Curve Tracing.
226. In the tracing of algebraic curves, the following remarks are
important.
(I). If the origin be taken on a curve of the «th degree, at an or*
dinary point, the straight line>' = mxc&n meet the curve in only n — i
other points.
If a curve has a singular point of multiplicity m, and this be taken
as origin, the line^ = mx can meet the curve in only n ^ m other
points.
Therefore, if any curve of the «th degree has at the origin a sin-
gularity of multiplicity « — 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
>'-axis parallel to this asymptote, we lower the degree of the equation
in ^' by i. If there be m parallel asymptotes, and we take the >'-axis
parallel to them, we lower the degree of the equation my by m. If
the degree of the equation inj/ can thus be made quadratic or linear
in >', then by assigning different values to x, the curve can be plotted
by points conveniently.
(III). In any algebraic equation of a curve /'= o, 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 to o 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 i.
(IV). The Analytical Polygon. — Newton designed the follow-
ing method of separating the branches of an algebraic curve at a
singular point, and tiacing 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 00 .
Let F(x, y) be any polynomial in x andjf which contains no con-
stant term. Then
J^{x, y)s 2 C>y = o
is the equation of a curve passing through the origin.
Corresponding to each term
C^-y*^ plot a point with reference
to axes Op^ Oq^ having as abscissa
and ordinate the exponents p and q
of X zxAy respectively. Thus lo-
cating points -^j, . . , , -<4j^, draw
thexf'm//^ polygon A^Aji^A^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 include them all.
The properties of the polygon are :*
(i). Any part of the equation /*= o, corresponding to terms
which are on a side of the polygon cutting the positive parts of the
axes Op, Oq, and such that no point of the polygon lies between that
side and the origin^ when equated to o is a curve passing through the
origin in the same way as does /*= o.
Thus, if we strike out of /*= o all terms except those correspond-
ing to terms on the side A^^y we have left a simple curve which
passes through the origin in the same way as /* = o. In like manner,
if we strike out all terms save those corresponding to points on the side
A^A^y 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 Opy Oq, 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 infinity in the same way as does /'= o.
Thus the part of /^ = o corresponding to the side A^^ gives
such a curve. Again, the part corresponding \xyA^^^ 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 ^ = o, and these are
more simply determined by equating to o the coefficients of the
highest powers of x and of^^ in /'= o. Such a side is A^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. [Ch. XXXI.
axes Oq, Op, dc&A^A^^, merely gives the points of intersection of
Z' = o with Ox or Oy accordingly.
(5). Any side of the polygon which is parallel to one of the axes
Opy Oq 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(i).
BZAMPLE8.
1. Trace x» -|- 2a^j^y — ^^ = o.
Numbering the terms in the order in which
they occur, we have A^, A^^A^, in the polygon
corresponding to the terms of the equation.
The curve passes through O in the same
way as does the curve
jr«+2a V^ s jf*(jc'-f 2a^) =0,
corresponding to A.A^ , or
as shown in Fig. 130.
AlsOf the curve passes
through O in the same way
as does
^(^^y — ^y* ss y{ria^:^ — by^ = o,
corresponding to A^^ , as shown in Fig. 139.
The curve passes off to 00 in the same way as
Fig. 137.
Fig. 138.
Fig. 139.
does the curve
jc* -i^y* = o,
or
J
= fy.
V
corresponding to A^A^ , Fig. 140.
The form of the curve is therefore as in Fig. 141.
y
Fig. 140.
Trace the following curves:
2. jr* — 2axy — axy^ -j- ^'j'* = °*
4. ay* — xy* — 2yji:* -f ax* — jr' = o
6. Jt* — a*xy -f- d^y* = o.
7. X* — ^axy* -\- 2ay* = O.
9. Jf* -j- a*xy — y = o.
11. a«(x»-f>/«)-2«(jf-;/)»-fA-*+y=0.
13. a(;^- Jr)»(JV-h*)=y^-^•
15. ax{y — xy =^*.
17. Trace x* — ax^y — axy* + d'y^
18. Jt* — a*xy* = ay^.
20. Jf* + ax^y = ay*.
22. x5 +^ = s^-^y-
24. {x — 2)y* = 4x.
26. {y — x){y — 4x)(y -f 2jc)= Sax*.
3. X* — ax*y + axy* -f a*y* = a
for a = If 0 > I, a < I.
B, y = x{x* — I).
8. JT* — 2a^x* -|- Sfl'jry — 2a V + ^ =0.
10. a^x' ^yt)^x* +^ = o.
12. a(y* - x^X;' - 2x) =y.
14. X* — fljry' -f-^* = o.
16. X* — a*jry + d*y* = o.
■= o, near the origin.
19. X* — a*xy = ay*.
21. x{y -xy = ^y.
23. (X - 31^' = (^ - I)**.
25. (X - iXx - 2y = x».
27. (^ - xY{y 4- x,Cy -f 2x) = i6a*.
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 fkmily. When x,y
moves along a curve of the family
F{xyj^, a) r= o, (i)
a is constant, and we have on differentiation
dF dF
But if X, y moves along the envelope, a is variable, and on dif-
ferentiation of (i)
dF . dF . dF ^ . .
dF dy
Also, on the envelope ^ — = o. Therefore j-, from (2) and (3),
are the same at a point x^y common to the curve and its envelope.
228. Again, let a, /3, y be variable parameters in the equation
F{x,y, a, fi, y) = o, (i)
where a, fi, y are connected by the two relations
0(«»Ay) = o> (2) i^(a, /S, y) = o. {3)
We require the envelope of the family of curves (i) when a, /5, y
vary. Obviously, if we could solve equations (2), (3) with respect to
two of the parameters and substitute in (i), or, what is the same thing,
eliminate two of the parameters between equations (i), (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 (i), (2), (3), the parameters being the variables,
dF. ^^F .. ^ dF.
^da+^dfi+-dy^o.
Multiply the second of these by A, the third by /^, and add. Deter-
mine \ and fJL so that the coefficients of da and dft are zero. Then
344 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
if we take d'j( as the independent variable parameter, the differentials
d0, dy are arbitrary and we can assign them so that the remainder of
the equation shall be zero. Then
dF , 90 dtb
dF dd> dib
The envelope is the result obtained by eliminating a^ fi, y, X, pi
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 n variable parameters connected by « — i
equations of condition.
229. We can get a concrete geometrical intuition of the relation
of curves of a family and their envelope, by letting « be a variable
parameter and considering
^{x, y,z)=:o
as the equation of a surface in space. Then the curves of the family
are the projections on the horizontal plane xOy of horizontal plane
sections of the surface, obtained by varying z = a,
EXAMPLES.
1. Find the envelope of a line of given length, /, whose ends move on two fixed
rectangular axes.
We have to find the envelope of
x/a -)- y/^ = I when «■ -[" ^' = ^''
.'. x/a* = Aa, y/i>* = Xb,
Hence X = a-^, and a = (/«x)*, b =z (Z^)*,
and the envelope is jc" -)- >* — ^ •
2. Find the envelope of concentric and coaxial ellipses of constant area.
Here x*/a^ +;'V<^' = 1 and ab = c.
. •. x^a^ = A^, jV<5» = Xa. . •. 2cX = I.
The required envelope is the equilateral hyperbola 2xy = c,
3. Find the envelope of the normals to the ellipse.
Here a*x/a - b*y//3 = a* — b* and a*/a* -f- /3^/b* = i.
. •. a*x/a* = Xa/a\ i^y//P = - Xfi/b^. . •. A = «« - bK
give the required envelope
(tfx)' 4- (b}^)i = {a* - 3«)5.
Art. 229.]
APPLICATION TO PLANE CURVES.
345
4. Show that the envelope oix/a + y/B = i, where a and b are connected by
+1.
5. Show that the envelope of x/I -|- y/"* = '» where the variable parameters
I, m are connected by the linear relation l/a -)- fn/b = i, is the parabola
{j)*+ (0'= ■■
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
of the squares of its distances from n fixed points x^.^ y^ is a constant >P| is the
locus
^Xr — ^, 2xryr * ^J^rt JT = O.
Sxryr , 2yl - it*, Syr. y
Sxr , 2yr , » , I
jf t y ,1,0
Let the line be Ix -^ my -^- p sz o. Then
i6» = /«2x; + m^2yj, + #f/« + 2iw/ Syr + 2/p Sxr + 2/m Sxryr,
= rt/* -f bm* -f <"^ + ^/'''P + ?f ^ -f 2>l/w.
Also, /* -)- ^' = If ^ ^^^ "* being direction cosines of the line.
Hence we have
a/ -f Aw -f ^ + A/ -f \pix = o,
hl+bm ^jp + Aw + i/iy = o,
gl^fm-^€p+ o -hi/* =0.
Multiply hy iy M^ p in order and add. . *. A = — >H.
Eliminating l^ m^ p^ ^ between the equations
A/-f (3 - i»)« 4.// + i^^ = o,
glj^ fm^cp^\li =0,
jr/ + y^ 4- / + o =0,
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 n points jr^, yr is equal to a constant k^ is a circle
whose center is the ccntroid of the fixed points and whose radius is one nth the
distance k.
Let /jc + iwgf+/ = o be the line. Then /* + ««= i, and
k = ISxr + mSyr + **/»
= tf/ -j- ^m -f- fP*
Here we have
fl + Ajt 4- 2>u/ = o,
b -^-Xy •\- 2fim = o,
c -{• X '\- o =0.
A = — r = — ff . Multiply these three equations by /, m, / in order and
add. Hence k -{^ 2fi=o,
346 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI.
The equations a — nx z= Jk/^ ^ ^ ny = Jtm, squared and added, give the
enTeiope
('-^)V(-"^)'=(^)--
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 tke envelope of the ellipses jc*/a^ +^/^ = ii when a* + ^ = >t*,
is a square whose side is A j^2,
12. Show that the envelope of line xa^ +>'^*" = <^+', when ««+ ^ = i/«, is
*^+y^ = (^) .
13. Find the envelope of the family of parabolse which pass through the origin,
have their axes parallel to Oy and their vertices on the ellipse x^/a* -\- y*/^ = x,
[A parabola,]
14. The ends of a straight line of constant length a describe respectively the
circles {^x ± cf -\' y* =. aK Show that the envelope of the curve described by the
mid-pomt of the line, c being a variable parameter, is
15. Find the envelope of a faunily 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 &mily is \iC€^ when
c > -la. Also, show that the entire area of the envelope is i^[|« — ^\
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 /(or, j') and /'(ar,>', s),
when abbreviated into^and F, to mean a function of (wo and ihree vari-
ables respectively.
The functions immediately under consideration are
z ^f(Xyy) and F(x,y, z) = o.
The first expresses e explicitly as a function of x and y, and is to
be regarded as the solution of the implicit function F=z o with respect
to 3.
It is to be observed generally that since
F^/^z,
results obtained from the investigation o( F = o are translated into
those for ^ =/* by
d^+^F a^+y dF d^^F _
231. In the present article we recall a few fundamental principles
of solid analytical geometry which will be needed subsequently.*
I. Hke Plane, The equation of the first degree in x, jf, z can
always be represented by a plane.
We repeat the proof of this from geometry as follows :
Let Ax + By -^ Cz + D = o, (i)
A, B, C,D being any constants. Assign to x and^ any values x^ ^y^
whatever, and compute «j, so that x^,y^^ z^ satisfy (i). In like man-
ner assign arbitrarily x^^y^^ and compute «, so that Jt,, y^y z^ also
satisfy (i).
Represent, as previously, x^y, ^ by a point in space with respect to
* For a more detailed discussion, see any solid analytical geometry.
347
348 APPLICATION TO SURFACES. [Ch. XXXIL
coordinate axes Ox, Oy, Oz. Then, whatever be the numbers m and «,
the point whose coordinates are
m -{- n ' m -{- n * m -^^ n *
is a point on the straight line through the points o:,,^ , b^ and ^,,^,, «,,
which divides the segment between these points in the ratio of m to ».
By varying m and « we can make x\y, z' the coordinates of any
point wliatever on this straight line. But the point x' , _y', z^ must
be on the surface (i), since, on substitution, these values satisfy (i).
Therefore, whatever be the two points whose coordinates satisfy (i),
the straight line through these points must lie wholly in the locus
representing (i). This is Euclid's definition of a plane surface.
The intercept of the plane on the axis Oz is -^D/C, Therefore,
when C = o, the intercept is oo , or the plane is parallel to Oz.
Hence (i) becomes
-4.r + J?y + Z> = o,
the equation of a plane parallel to Oz, cutting the plane xOy in the
straight hne whose equations are 2^ = o. Ax -\- By -^ D = o.
We use orthogonal coordinates unless otherwise specially men-
tioned. If /, m, n are the direction cosines of the perpendicular from
the origin on the plane and p is the length of that perpendicular, the
equation of the plane can be written in the useful form
Ix -\-my -{-nz — p =: o, (2)
where /^ 4- iw^ _|_ ,,2 -_ j
n. TTie Straight Line, Since the intersection of any two planes is
a straight line, the equations of a straight line are the simultaneous
equations
A^x-\.B^+C,zJrn,^o,
A^ + B^+Cj^ +
The equations (3) of a straight line can always be transformed
into the symmetrical form
x^a_y^b _z--c
where a, 3, c is a point on the line ; /, m, «, the direction cosines of
the line; and A is the distance between the points x, y, z and a, b, c,
III. 77ie 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 directrix of the cylinder.
With reference to space of three dimensions and rectangular coor-
dinates, any equation
Ax,y) = o (5)
is the equation of a cylinder generated by a straight line moving
2::::|
Art. 233.] STUDY OF THE FORM OF A SURFACE AT A POINT. 349
parallel to Oz and intersecting the plane xOy in the curve y^jTjj') = o.
Fot/'{x, ^) = o is nothing more than the equation
in three variables, in which the coefficients of s are zero, and which is
therefore satisfied by any x,y on the curve yj;ji:,_>') = o in xOy and
any finite value of z whatever.
In like manner /(^, 2) = o, /[x, 0) = o are cylinders parallel to
the Ox, Oy axes respectively,
IV. TAe Cone. A cone 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 base of the cone.
Any homogeneous equation of the nth degree in Xy y, z, such as
I\x, y, z) = o, (6)
is the equation of a cone having the origin as vertex.
Let (X, fiy y be any values of jc, _>', z satisfying (6). Then, since
(6) is homogeneous, ka^ kfi, ky will also satisfy (6), and we shall have
F(kx, iy, kz) = ^F^Xf >', z) = o
whatever be the assigned number k. The coordinates of any point
whatever on the straight line through the origin and aty fi, y can be
represented hy ka^ A(i, ky. Therefore all points of this straight line
satisfy (6). When the point a, ^, y describes any curve, the straight
line through O and a, y^, y generates a surface whose equation is (6),
and this is by definition a cone.
If we translate the axes to the new origin — u , — 3, — c, by writ-
ing X — a, y — b, z — c, for x^yj z in (6), we have
F{x ^ a,y -^ b, z — c) = o, (7)
a homogeneous equation in x ^ a, y — b, z — c, which is the equa-
tion of a cone whose vertex is a, 3, c.
232. General Definition of a Surface. — If F{x,y, z) is a continu-
ous function of the independent variables x, y, z, and is partially dif-
ferentiable with respect to these variables, we shall define the assemblage
of points whose coordinates x,y, » satisfy
F{x,y,z)=^o (i)
as a surface, and call (i) the equation of the surface.
233. The General Equation of a Surface. — Let F(x,y^z) = o
be the equation of any surface.
Then, by the law of the mean, we can write
F{x,yy z) = F[x\y'y z^
r»i
350 APPLICATION TO SURFACES. [Ch. XXXII-
in which the summation can be stopped at any term we choose, provided
we write S, 7, C instead oi x\y\ z' in the last term, where ^, ;;, C
is appoint on the straight line between x^y^ a and x\y\z' , We can
therefore always write the equation to any surface in the standard
form
r-\
This enables us to study the function as a rational integral function
of ;»;,_>', z.
If the equation of the surface be given in the explicit form
z =/{x^y)^ then in like manner, by the law of the mean, we have
for the equation to the surface
r-l
in which the summation stops at any term we choose, provided in the
last term we write £ for x^ and rj ior y'-y <f , t} being a point on the line
joining Xyy to x\y,
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 AB passing through a second point B
on the surface, when B converges to ^ as a limit along a curve on
the surface passing through ^ in a definite way.
To find the condition that the straight line
^=^i=y=izii:=A (I)
I m n
shall be tangent to the surface F{xj y, z) =0.
The equation of a surface in implicit form is, § 233, (i),
^(^',y, 2') + ix- *') l^ + (y -yf^ + (2 - «') If +^=°- (*)
Substitute /A, mX, nX for x — x\y — y, « — s\ from (i) in (2).
We have the equation in \,
= /'(.'.y,.') + (/a-5 + «.^ + «3^)A + ^.
(3)
for determining distances from x',y\ z' to the points in which (i)
intersects the surface (2). \i F{x\y' y ar') = o, or x' yy' y z' is on the
surface, one root of (3) is o. If in addition
,dF dF , dF
^ dx^ + "^ at' + ^ a? = ^' (^)
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 (i) cuts the
surface (2) coincide in Xyy\ z\ and the line will be tangent to the
surface at x\yy z\ and have the direction determined by /, «, n.
Observe that in conditions (4) and /* -j" '''* + ^* = ^ we have
only two relations to be satisfied by the three numbers I, m, n, and
therefore there are an indefinite number pf tangent lines to a surface
at a point x^y, z\
If the equation to the surface be in the explicit form z = /(-^k^)*
or
z = z' + {x-x')l^+{jy-y)^, + Jl, (5)
then, as before, the straight line (i) meets the surface (5) in x^^y, z'
when 2 = «' and other points whose distances from x'^y^ »' are the
roots of the equation in A,
=('i;+"^- ")*+'•
The condition of tangency is that a second point of intersection
shall coincide with ^',y, «', or
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^ ») = o at ^',y, z\
The straight line
X -— x' y — y' z — z'
is tangent to the surface F= o at x\y, z' when F{xf^y, z') = o and
dF dF dF
Tf now at x\ /, z' the derivatives
a/; a/; a/;
aP' a/' bz'
are not all o, we obtain the locus of the tangent lines to /*= o at
x'^y'y z' by eliminating /, »i, n between (i) and (2). Therefore this
locus is
(---')g+(>'-y)'^+('-«')^=o. (3)
Exjuation (3) is of the first degree in x^y^ ^, and therefore is a
plane tangent to /' = o at x\y^ z\
352 APPLICATION TO SURFACES. [Ch. XXXIL
Tangent plane to z z=z/(x,_y).
Eliminating /, m, n between (i) and
we have
as the tangent plane to z —fzX.oc'.y,
236. Definition of an Ordinary Point on a Surface.— We have
just seen that when at any point on a surface F=.o the first partial
derivatives,
dF dF dF
dx' ay"' dP
are not all zero, the surface has at that point a unique determinate
tangent plane. Such a point is called 2. point of ordinary position, or
simply an ordinary point.
On the contrary, if at x^y, & we have
dj" =0, byF= o, a^ = o,
the point is called a singular 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[xj y) shall be parallel to
xOy. Ans. b^f = 3 y/ r= o.
2. Find the conditions that the tangent plane to F{x^ y^ z) ■= o shall be hori-
zontal. Ans, bxF = byF = o, d^F ^ o.
3. Show that the tangent plane at x^, y , g' to the sphere x^ -\- y* -{- z* =z a* is
xx^ -f- yy + ^^' = «*•
jjji y' z^
4. Find the tangent plane to the central conicoid f- — H = 1.
a 0 €
xx" yy* zz'
a p c
5. Show that the tangent plane to the paraboloid ax^ -|- fy* = 2z at jc', y, z'
is axx' -f- iyy' = z -\- z*.
6. Show that the tangent plane to the cone F^x.y^ z) = o, having the origin as
vertex, is xdjc'F-^ydyF -\- zdz'F = o.
This follows directly from the fact that F is homogeneous, and therefore the
tangent plane is
dF , dF ^ dF . 9^ , , dF , , dF
•^ ay + ^ aZ + ^ ai^ = ^ a]7 + ^ V + "^ a?" = ^^^'' ^'' ^^ = ^'
where n 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
x^ ^y^ -^ z^ = a^f 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 jcyz = tf* is of constant volume.
9. Show that the equation of the tangent plane to the conicoid
ax^ 4" ^y* + ^'' 4" ^/y H" 2^-'^* + 2Ary -j- lux -f- ^^ + 2^* + ^ = ^^
at y , y, «', is
{ax' + My +^' + «>r -f (Ay + ^y +/«' + rir -f
{^^ -{-/y -{-cz* + w)z-\-ux' -^v/+w^ + d=o.
10. Show that
is the general equation of any conicoid, and that
df _dF _dF__
dx "^ dy ~ dz
are the equations of the center of the surface.
11. Show that the plane
,dF ^ dF bF
cuts the conicoid /* = o in a conic whose center is a, fi, y, and therefore this is
the tangent plane when a, fff y is on the surface.
12. Show that the locus of the points of contact of all tangent planes to the
surface /^ = o, which pass through a fixed point a, /^t Yt is the intersection of
F •=. o with the surface
bF hF bF
13. This surface is of degree « — i when /* = o is of degree if.
For, let Fz= £/»-{-... + 6^ + ^, where Ur is the homogeneous part of
degree r. Then, as in two variables, we have the concomitant
Therefore the tangent plane at jr, y^ z may be written
^^i + I'^ + 2^ = «^. + (» - 0^^-. + . . . + ^p or
^^+ *'^ + ^IF + '^"- + '"'-' + •••+(«- »)^i + "t; = o.
since U^-\- . , , -\- 6^^, = o.
14. Find the condition that the plane Ix -\- my -\- nz = o shall be tangent to
the cone
F = tfjr* -)- ^V' + ^^' + ^fy^ + 2^-^* + ^^^ = o
at x'y /, s'.
The equation to the tangent plane at x', y , s^ is, Ex. 6,
a/* . bF bF
354
APPLICATION TO, SURFACES.
[Ch. XXXII.
In order that this shall be identical with Ix -\- my -{• nz zs. o^ the coefficients
of X, yt z must be proportional.
dF / , bF i bF I
Hence
fly + >ft/ -h ^ = A,
/y 4" #i/+ ^^' = o-
In order that these equations shall be consistent we have
= o,
a
h
g
I
h
b
f
m
?
f
c
ft
I
m
n
o
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 c, the horizontal projection of the intersection of the plane and cone
is two straight lines
(an^ -\-cP — 2gin)x^ -f- 2(hn^ -\- dm — gmn ^fln)xy -f {bn^ -f cm^ — 2fmn)y^ = O.
These will be real and different, coincident or imaginary, according as
(a«» -f en — 2gln){bn'^ -f cm^ — 2fmn) — (hn^ -f dm — gmn — flnf,
which can be written as the determinant
A =
a
h
g
I
h
b
f
m
g
f
c
n
I
m
n
0
is negative, zero, or positive, respectively.
16. Show that the projections of the two lines in 15 can be written
dj
dJ
y* = 0,
db d/i -" ' da
with similar equations for the projections on the other two coordinate planes.
17. Show that J in Ex. 15 can be written
da ~ ^'^
db
dc
where
dD dD dD
+ /,,_ + n^n --^+im~ = ^ A,
D =
a h
g
h b
f
g f
c
2yj. Conventional Abbreviations for the Partial Derivatives. —
The elementary study of a surface 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 symlx)ls for the first and second partial deriv-
atives. These derivatives being the coefficients of the first and second
powers of x, y, z in the equation, it is customary to represent them by
Art. 238.] STUDY OF THE FORM OF A SURFACE AT A POINT. 35 S
•
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, z) = o,
by dz* dx dz* dx by
When z -zz.fyx^y^^
t^^^L a^^L r=?^ .= -^ / = ?Z
^"bx' ^^by' bx^' bxby' by^'
238. Inflexional Tangents at an Ordinary Point. — We have seen,
§§ 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
F = o are not all o, there are two of these tangent lines that are of
particular interest.
(i). Let s =y\Xyy) be the equation of a surface.
The straight line
I m n ^ '
cuts the surface z =/* in points whose distances from the point x, y, z
on the surface are the roots of the equation in X
we have seen that (i) is tangent to a =yat x,y, z.
If in addition we have /, m, n satisfying the condition
two roots of (2) are o, and the line (i) cuts the surface in three coin-
cident points at x, y, z.
The conditions
fi + m^ +n^ = I,
pi -i^ qm — « =0,
rP -f- 2sim -[- /w' = o,
356 APPLICATION TO SURFACES. [Ch. XXXII.
determine two straight lines, in the tangent plane, tangent to the
surfaces =/*at 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
in i/m, has real and different, double, or imaginary roots, or accord-
ing as
r,-^^'X'X-(JZ-Y (5)
dj^df
( ^V \
\dxdyj
is negative, zero, or positive.
Since any straight line, such as (i), cuts any surface of the «th
degree in n points, the straight lines in any plane cut the curve of sec-
tion of a surface of degree n in n points. Therefore a plane cuts a sur-
face of the «th degree in a plane curve of degree n.
The tangent plane to a surface of degree n cuts the surface in a
curve of degree n 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. I'his point is a node,
conjugate point, or cusp according to the value of condition (5). Cora-
pare singular points, plane curves.
Eliminating /, m, n between (i), (3), (4), we have for the equa-
tions of the inflexional tangents at x^y^ z
Z^z={X-x)p + {V^y)q,
{X - ^)V -f 2{X - x){y'-y)s + {F^yft=: 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
(i) cuts the surface in points whose distances from jc, j', z are the roots
of the equation in A,
„ ^/dF dF dF\ X^/ d d , 9\^^ „
If x,y, z is on the surface, or F{Xfy, z) = o, and
LI + Mm -{-Nn = o,
the line (i) is tangent at x,y, z. If in addition /, /«, n satisfy the
condition
Art. 239.] STUDY OF THE FORM OF A SURFACE AT A POINT. 357
the line (i) cuts the sur&ce in three coincident points at x^y^ 8. The
conditions
p + m^ + f^ = 1, (6)
LI + Mm + iV« = o, (7)
AP + Bm^ + Cn^ + 2Fmn + ^Gln + 2Hlm = o, (8)
determine the directions of the two inflexional tangents.
Eliminating /, w, n between (i), (7), (8), we have the equations of
the inflexional tangents at x^ y, z,
|(^— )-^ + (i'->)ay+(^-«)3^|/'=o, (9)
|(^--)3^+(^-^)|+(^-')|}V=o. (xo)
The first is the tangent plane, which cuts the second, a cone of the
second degree with vertex x^y, «, in the two inflexional tangents.
These tangents will be real and different, coincident, or imaginary,
according as the plane (9) cu/s 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)
A If G L
H B F M (11)
G F C N
L M No
is negative, zero, or positive.
239. Should the second partial derivatives also be separately o at
x^ y\ Zy and r the order of the first partial derivatives thereafter which
do not all vanish at x, y, 0, then there will be at x, y, z on the sur-
face r inflexional tangents, which are the r straight lines in which the
tangent plane at Xj y^ z cuts the r planes
or the cone of the rth degree,
{(^--)^+(^-»|; + (^-^)^,|> = o.
These r inflexional tangents to the surface are the r tangents to
the curve cut out of the surface by the tangent plane at the point of
contact, which point is an r-ple singular point on the curve of
section.
EXAMPLES.
1. Show that the inflexional tangents at any point y, y , ^ on the hyperboloid
jcJ/^a 4. yfijl^ — «Y^ = I, He wholly on the surface and are therefore the two
right-line generators passing through the point. Show that their equations are
X * - ^ ^ „ y -y _ z -^
35^ APPLICATION TO SURFACES. [Ch. XXXII.
2. Show that the inflexional tangents at a point jr, ^, s on the liyperbolic parab-
oloid jr'/a* — ^V^ = ^' ^'^ wholly on the surfisice, and that their equations
are
a " ±^ " ^:^ y
a S
the upper signs going together and the lower together.
3. Show that the inflexional tangents to the cone
AX* + 4y* + fs« -f 2/yz -f 2^xg -f 2kxy = o
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 a
maximum or a minimum the inflexional tangents are imaginary.
340. 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 norma/ to the surface at that point.
Since the equation to the tangent plane at x, y, z is
nr* nr» ar»
or Z-z = {X-x)^lj^(r-y)f-.,
the coefficients of ^Y, V, Z are proportional to the direction cosines
of the normal, and we have for the equation to the normal at jc, yy z
or
X X
dF ~
dx
r-y Z-Z
dF ~ dF '
dy dz
X-x
F-y _Z-,^
dx
a/ ^ -I
dy
EXAMPLES.
1. Show that the normal at jr, y, z to xyz = a* is
Xx - xi^= yy^y^ = Zz- zK
2. Find the equations of the normal to the central conicoid lufl + ^* + ^^' = '•
X -- X _ Y-y _ Z-j^
ax '~ by ~ cz '
8. Show that the normal to the paraboloid ax^ -\- by^ = 2z has for its equations
X-x y-y
ax . by
=i z - Z»
24X. 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 (i) with respect to the tangent plane ^ (2) with respect to
the conicoid 0/ curvature y (3) with respect to the plane sections parallel
Art. 242.] STUDY OF THE FORM OF A SURFACE AT A POINT. 359
to the tangent plane t (4) with respect to the plane sections through the
normal.
242. With respect to the tangent plane:
(i). Let z =./\x,y). Then the equation of the surface is
Let Xj Ky Z^ be a point in the tangent plane in the neighborhood
of the point of contact jc, y^ s. Then the difference between the
ordinate to the surface and the ordinate to the tangent plane is
This difference is positive for all values o( X, V in the neighbor-
hood of or, y when
are positive (Ex. 19, § 25). Then in the 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 — Zj is negative throughout the neighborhood
when rt — j* is positive and r is negative at the point of contact.
Then the surface in the neighborhood of the j)oint of contact lieis
wholly below the tangent plane and is said to be concave there.
(2). Let F(Xy y, z) = o. In the same way we have the equa-
tion of the surface,
(X-x)F'^+{y-y)F',+ {Z-z)F'. =
and for that of the tangent plane at Jtr, y^ z,
(X - x)FL+ {y-j')F;+ {z, - z)r, = o.
On subtraction,
{Z,-Z)F'.= \ { (X-x)^.^+(y-y)l-^+iZ-z)l-^ } V.
Therefore, at Xyy^ z, by Ex. 20, § 25, when
A H
and A
A H G
H B
H B F
G F C
are positive, the surface is convex when A and F^ are unlike signed,
concave when A and F'^zrt like signed.
360 APPLICATION TO SURFACES. [Ch. XXXII.
Observe that a surface is concave or convex at a point when the
inflexional tangents there are imaginary, and conversely. When a
surface is either concave or convex at a point, its fomi is said to be
synclasiic 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 in a
curve having a node at the point of contact and tangent to the inflex-
ional tangents. At such a point the form of the surface is said to be
anticlastiCy 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,
A H G L
H B F M
G F C N
L M N o
= + synclastic y
= — 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
of a circle about an external axis in its plane generates a torus. 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,
(i). The explicit equation z =/\x,y), or
z = .+i^-.) §^+iy-y) 1"+^ { (A'-.)4+(r-^) l}'/,
shows that in the neighborhood of x, y^ z the surface differs arbitra-
rily little from the paraboloid
z = .+(^-.) % +(1^-^)1^+ \ { ix-x)l^ +iy-y) I } /
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, z 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
rx^ + ^ + 2XJfy + 2/^ + 2^ — 2« + >J = o
is A? - {r + t)\ + {rt - j^) = o.
Art. 245.] STUDY OF THE FORM OF A SURFACE AT A POINT. 361
This gives the elliptic or hyperbolic form according as r/ — j' is
positive or negative.
(2). In the same way, the implicit equation F(x,_y, z) = o, or
dx
dz
shows that in the neighborhood of x,j^f s 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 x, y, 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 F.
The discrimination of the conicoid can be made through the
discriminating cubic (see Ex. 17, p. 30)
and the four determinants
A^X.Jf , G
H , B ^\, F
G , F , C-A
linants
A H G L
H B F M
G F C N
= o.
as in analytical geometry.*
244. The Indicatrlx of a Surface. — At an ordinary point x^y^ z
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 indicairix of the surface at or, y^ z.
Points on a surface are said to be circular (umbilic), elliptic^ para-
bolic^ or hyperbolic 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 3-plane and ^-axis. — If the equation is z =/{x^ y), then since
5 = o, / = o, ^ = o at the origin, the equation is
20 = rx^ + 2sxy + iv* + 2^.
The equation of the indicatrix at the origin is
z = rx^ + 2sxy + ^,
* See Frost's, Charles Smith's, or Salmon's Analytical Geometry.
(
362 APPLICATION TO SURFACES. [Ch. XXXII.
2 being an arbitrarily small constant. This is an ellipse or hyperbola
according as r/ ~ j^ 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 on bl.
surface /'rz o, we have independently
dF dF dF
the point is said to be a singular point.
If the second derivatives are not all zero, then all the straight lines
whose direction cosines I, m, n satisfy the relation
will cut the surface in three coincident points at x,y, z, and are called
tangent lines. Eliminating /, m, n by means of the equation to the
line and (2), we obtain the locus of the tangent lines at x^y, z,
I (A- - .-) /^: + (F-^) I + (Z - .) II V= o. (3)
This is the equation of a cone of the second degree, with vertex
■^> J'j ^9 which is tangent to the surface /^= o at the point x,y, z.
The form of the surface at x^ y, z 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 Jtr, y, z lie in one or the other of two planes.
The point is then called a nodal point. The condition for a nodal
point is that equation (3) shall break up into two linear factors, or
A N G =0. (4)
J/ B F
G F C
A line on the surface /'= o at all points of which (4) is satisfied
is called a nodal 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 in a 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 x,yy z of order r, and
a tangent cone there of the rth degree whose equation is
|(^--)aJ + (^-^')|; + (^-^)a^|>=o. (5)
247. A singular iangenl 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.
BZSRCISBS.
1. The tangent plane to yx^ = a^z at x^<, y\^ *\ is
Tjcxy^y^ -\- y x\ — a^t -=: zaU^.
Find the equation to the normal there.
2. The tangent plane to <(jf* -j-y*) = 2Axy at x^^ y^, «j is
2x(x^z^ - fy^) + 2y(y^z^ - ^x^) + z{xi -{-y*) - 2^^y^ = o.
The tangent plane meets the surface in a straight line, and an ellipse whose
projection on the xOy plane is the circle
{x* -^y^ixl - yl) + {x\ +yi){yy, - xx{i = o.
Show that the s-axis is a nodal line.
3. The tangent plane to a^y^ = x*(^ — **) at x^^ y^, z^ is
xjci(^ - z\) - d^y^ — zz^x\ + x\z\ = O.
At any point on Oz^ F,l =. Fy =. Fg = o, show that at any such point there
are two tangent planes
j=*/-^
4* Show that the tangent plane at x-^, y^, z^ to
j^ •{• y* -\- sfl — ^xyz = tf'
is x(x\ - yyz{i + y{y\ - x^z^) + z{z* - x^y^^ = a».
5. The tangent plane at jr,, ^j, Zj to x^y^zP •=. a is
M n p
6. Show that (2<z, 2^1, 24z) is a conical point on
xyz — a{x* + ^* + «•) -|- 4fl* = O,
and find the tangent cone at the point.
Afu, X* -\- y* -\- z* ^ 2yz -- 2zx — 2xy = o
7. Show that the surface
has two conical points.
The tangent cone at o, o, o is ^x^/a* + 3y*/^* + «V^ = ^*
8. Determine the nature of the surface
ay* + dz* + x(jc* +y^ + «») = o
at the origin.
The origin is a singular point, the tangent cone there is ay* -\- dz* =: o. Ha
and d are like signed, the origin is a cuspal point around the x-axis.
9. A surface is generated by the revolution of a parabola «' = 4mx about an
ordinate through the focus; hnd the nature of the points where it meets the axis of
revolution.
364 APPLICATION TO SURFACES. [Ch. XXXII.
Hint. The equation of the surf&ce can be written
i6w»(j:* -f >^») = («« - 4»l»)«.
The two right-angled circular cones x* -{-y* = (z ± znif are tangent to
the surface at the singular points.
10. If tangent planes are drawn at every point of the surface
a{yz+zx-^xy)=:xyzy
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 sur£ices of revolution having Oz ior axis
x^+y*=/lz).
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
(^ + J'* + «' + ^ - ^*? = 4^(*» +y).
Show that the tangent plane at j/, y , ^', is
(r - r)(^y -{- yy) + rzz' = r[a* + ^'^ - ^)],
where r* s xf^ -f-y*-
The tangent plane at any point on the circle jc* -|-y = (r — a)' cuts the sur£[ice
in a figure 8 curve whose form is given by the equation
(y^ -j- «*)* — \acy^ + 4^^^ — d)z^ = o.
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 j:* -f-.y* = ^ cuts the torus in two parts, one of which
is synclastiCi the other is anticlastic.
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 «-plane and ar-axis respectively.
Then the equation to the surface can be
written
z = i(rx' + 2sxy + /y*) + ^, (i)
since at the origin z = o, p= o, g = o.
Cut the surface by a plane passing
through Oz and making an angle 6 with
Ox, At every point of this plane let
or = /o cos ^, ^ = p sin ^. Fig. 142.
. •. z = \fl^{r cos«^ + 2J cos ^ sin /9 4. / sin^^) + T,
where T contains as a factor a higher power of p than p^.
The radius of curvature R of this normal section PO is, by New-
ton's method, §101, Ex. 4, given by
I _ r2Z
pf")o
(*)
= r cos'6' -j- 2J cos ^ sin ^ -j- / sin'tf,
= \(r + 0 + W - /) cos 2^ + J sin 26. (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
tan 26 ■=.
2J
(4)
If a is the least positive value of B satisfying (4), the general solu-
tion is \mc -[- tty 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 X\\t principal radii of curvature.
365
366
APPLICATION TO SURFACES.
[Ch. 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
4 = r cos2^ + / sin2^, (5)
since then j = o, by (4).
Let R^ and R^ be the radii of the principal sections.
Then when B — o, R-' =z r; 6 = ^rr, R-' = /, in (5).
I _ cos^6' sin^^
Also, if R^ is the radius of curvature of a normal section perpen-
dicular to that of R, then
I sin^^ cos^^
R
I
R
R,
R.
• ^ A' ^, R,
(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 xOz as the normal plane, and
let the oblique plane xPOQ make the
angle 0 with xOz.
jfp Then the equation of the surface is
22r = roi? -\- 2sxy + />'-
)>•
At any point P in the oblique sec-
1/ a . a
Fig. 143.
tion y = z tan <p.
2Z
X
= r + 2J- tan 0 + / . tan^0 +- .^ + - tan 0 -^
But since Ox is tangent to the curve OP at o,
-tan0-^j/.
/2 sec 0 /" ^
•■•/
20
-2
A- \dx^/^=,
as P converges to 6> along PO, Also, in the xOz section, if
MR = oTq, we have^ = o, and
2z _ /ay
/22: _
dx'^l
Art. 250.]
CURVATURE OF SURFACES.
367
-.=/&■
Let ^Q, -^ be the radii of the normal and oblique sections. Then,
for b( = )o,
i? = / — cos 0,
Hence R^z R^ cos 0'
This is Meunier's theorem. '
250. Observe, in the equation to the surface (i), § 248, the equa-
tion of the indicatrix is
2Z = rx^ -\- 2sxy -(- (v^. (i)
The principal sections of the surface at O pass through the axes
of the indicatrix conic, whose equation is
2S = rx^ -\- fy^ (2)
when xOz and^'^?^ are the principal planes.
Equation (i) shows that the radius of curvature of a normal section
varies as the square of the corresponding central radius vector of the
indicatrix. All the theorems in central conies 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
tenter 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 x,y^ z, and /, m, n the direction cosines of the tangent line
there. Let Q be another point A", V^ Z on the surface and in the
normal section.
Let QR be the perpendicular from Q on the tan-
gent line PR.
Then for R, the radius of curvature of the sec-
tion, we have
PR^ __ f p^(^Y___ r P(?
£
JL 20^'
^ ' ^QJ^" JL ^Q^\^Q.
The tangent plane at P is
(.Y - x)L + (r^y)M+ (Z - z)JV = o.
The distance of Q from this plane is
Fig. 144.
where
368
APPLICATION TO SURFACES.
[Ch. xxxm.
Also, Q being a point on the surface,
{X- x)L ^(r-y)M-^ {Z-z)N
:=AP + Brn^ + Cn^ + 2Fmn + iGln + 2Hlm,
since £T = o for e(=:)-P, and
(I)
= A.
/fi
»
is the equation of the tangent PI^, The derivatives Z, ^, etc. , of
course being taken at -P.
252. If the equation of the surface be /'(■^,^) — « = /*= o,
then since Z =^, /T/= ^, N= — 1, C = F = G = o, (i), §251,
becomes
I _ r/* + 2j/w -f- ^^'
253. To Find the Principal Radii at Any Point on a Surface.
— We have only to find the maximum and minimum values of ^ in
(i), §251, §252.
I. In (i), § 251, let /, m, n vary subject to the two conditions
IL + mM-\-nN =0, /« + w^ + «« = i.
Then, by the method of § 217,
^/ + J7;w + 6^« + AZ + /i/ = o,
HI -\- Bm -\- Fn + \M-\- ^m = o,
G/+ Fm + Cn -\-\N + pin =0.
Multiply by /, m, n, respectively and add. . *. ji = — k/R.
... (^ _ t</R)l-\- Hm-\- Gn -\-\L- o,
Hl-\- {B ^ K/R)m + Fn-\- \M— o,
G/ + Fm-\- {C-K/R)n -\-XN= o,
LI 4- Mm + Nn = o.
Eliminating /, zw, n, A, we get the quadratic
A -k/R, H , G , Z
H , B - k/R, F , M
G , F , C - k/R, N
M
N
= o,
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.
II. Ifs =/[x,y) be the equation to the sur&ce, then in (i),
§252, we have /, m, n subject to the two conditions
// + ^w — « = o, and /' + w* + «* = i,
which reduce to the single condition
(I + /)/« + 2pglm + (i + ^)«' = I.
Applying the general method for finding the maximum and
minimum values to (i), §252,
r/ + sm + A[(i +/•)/ + p^m-] = o,
x/ -f ^« + ^[/^^ + (i + ^')^0 = ®-
Multiply respectively by / and m and add. Whence A. = — k/H.
Eliminating / and m from
[r^ — (I +/')/f]/ + (s/^ — pgK)m = o,
(xje - /^a:)/ + [/^ - (i+f)K]m=o,
there results the quadratic
[r^ - (I +/^K-] [/^ - (I + ^)#f] - (xi? - pgK)^ = o,
or
{r/^s^)/^ - [^-(1 + ^) + /(I +/») - 2figs]Kj^ + /f* = o,
for finding the radii of principal curvature. In this equation
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 *
Aj(^ + B/ + C^ + 2Fyz + 2Gxz + 2Hxy = i,
made by the plane Lx + My -f iV^ = o.
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 (i), § 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 /j, »ij, o; /,, o, w^; o, Wj, «,, respectively. Then
equating the corresponding values of k/R in (i), § 251,
Al^-\'Bm^ + 2Hl^m^ = Al^ + Cn^ + 2G/,«, = Bm^-\- Cn^^+ 2/>w,«,.
Also, since these three tangent lines are parallel to the tangent
plane, the equations
Z/j + Mm^ = Z/, + Nn^ = Mm^ + A«, = o
give
.,_ M^ a_ Z»
370 APPLICATION TO SURFACES. [Ch. XXXIIL
and i^, m^ have opposite signs. The same equations give like values
for /,,»,, etc. On substitution we obtain the conditions which must
exist at an umbilic,
AM^ + BL^ - 2HLM_ AI^ + CL^ - iGLN
_ BN^ + CM^ - 2FMN
These two equations in x^y, s, together with the equation to the
surface, give the points at which umbilics occur.
If the equation of the surface isy^:»r,^) — « = o, results are cor-
respondingly simplified and the conditions which must exist at an
umbilic are immediately obtained from the fact that k/J^ is constant
for all values of /, m, n, satisfying the identical equations
-— = r/* -f- 2slm -[- ^',
I = (I +f)P + 2pqlm + (I + f)m\
Whence results, from proportionality of the constants,
255, Equations (2), §254, are very simply obtained by seeking
the point on the surface z ^i/{x,y) at which the sphere
<P{x,y, 0) = (:r - af+ {y - pf+{z - y)^ - /y^ = o
osculates the surface z =z/. The first and second partial derivatives
of a in 0 are the same as those for/ at the point of osculation. Dif-
ferentiating 0 = 0 partially with respect to x and^, we get
A" — « + («— y)P=^ o>
I + ^ + (« - r) ^ = o>
pqJ^(z^y)sz=o,
^ ^ ^ s r t
• •
Also, i? = — (» — y)^^ +/^ + f*» since the direction secant of
the normal with the «-axis is — (i -|-/* + f^'
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 in a 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 x,. Let 0^ and 0, 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 ^ ~ 0f This angle is also
the angle through which the normal has turned as a
ix>int 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 on a standard
circle of radius r, 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
s' — s'
0. - 01 = -Hr-^-
If the standard circle be taken with unit radius, the entire curvature
of P^P^ is measured by the arc j,' — x^' on the unit circle.
The mean curvature, or average curvature, of Pi^^ is the entire cur-
vature divided by the length of the curve P^P^,
or, is the quotient of the corresponding arc on the unit circle divided
by the length of curve PyP^-
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 0 with
respect to s. But since ds = PdKp^ where R is the radius of cur-
vature of the curve at a point, we have for the specific curvature
'ds "" :^'
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 go is defined to be the area of the
surface cut out of the sphere by the cone, divided by the square on
the radius, or
A
P?
The unit solid angle, called the steradian^ is that solid angle which
372
APPLICATION TO SURFACES.
[Ch. XXXIII.
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
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 korograph 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 sur£au:e.
If .S* be the area of the given portion and qd the entire curvature, the
mean curvature is ^
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 to o. In symbols, the curvature at
a point is doo
dS'
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
doo I
ds""
Let .S be any portion of a sur-
face containing a point Z'.
Draw the principal normal sec-
tions FAf = Jj„ FN = Jxj.
Jtr,, Jo", being the arcs of the
horograph corresponding to ^s^,
As^y on the surface.
AoD _ I
In the limit
doo I
dS
^i<
Art. 256.] CURVATURE OF SURFACES. 373
BXSRCISBS.
1. Find the principal radii of curvature at the origin for the 8ur£iice
2M = dx* — $xy — 6yK Ans, -f^, — -f^
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 i : 2.
3. Find the principal radii of curvature, at x, yt x, of the surface
y cos jr sin — = a Ans. ± —^ .
^ a a a
a — b
4. Show that at all points on the curve in which the planes % ■=. ± — -r~ cut
the hyperbolic paraboloid 2m = ojfi — hy^ the radii of principal curvature of the
latter surf&ce 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), g 248, that the mean curvature of all the normal sections of
a surjBice at a point is
i(^+i)-
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 a
9. Show that at t,sety point on the revolute generated by the tractrix revolving
about its asymptote, the specific curvature is constant and negative. This surface
xs called the pseudo-spliere.
10. If the plane curve given by the equations
x/a = cos 0 -{- log tan \ 9, yla = sin 0,
revolves about Ox^ the surface generated has its specific curvature constant
11. Ifi?,, i^, are the principal radii of curvature at any point of the ellipsoid
on the line of intersection with a given concentric sphere, prove that
12. Prove that the specific curvature at any point of the elliptic or hyperbolic
paraboloids*/^ + «*/<■ = x varies as (//«)*, p being the perpendicular from the
origin on the tangent plane.
13. In the helicoid y ^ x tan (*/a) show that the principal radii of curvature,
at tyery point at the intersection of the helicoid with a coaxial cylinder, are con-
stant and equal in magnitude, opposite in sign.
14. Prove that the specific curvature at every point of the elliptic paraboloid
2z =s x^/a '\-y^/by where it is cut by the cylinder ;f*/<i' -h^'V^ = I, is (4fl^)-«.
15. Prove that the principal curvatures are equal and opposite in the suifgice
x!H,y — «) -+■ ^' = o where it is met by the cone (x* -f" ^*if* = (/ — ')*•
16. The principal radii of curvature at the points of the surface
ii'jf* = «•» *•-!- y^ where x =.y ^ m^
are given by 2JP -f- 2 f^3 a^ — 9^' = a
374 APPLICATION TO SURFACES. [Ch XXXIII.
17. Prove that the radius ok curvature of the turfaice x» -4~>* -{• m» =: af* ^i
«•.— a
an umbilic is 3 •*» V(** ^ ^)*
^ _ y _ *
18b Show that — = 4- = — i* 9m umbilic on the sur&oe
a a c
19. Show that x =zy = m = {ahc)^ Is an umbilic on the sur&ce xyM =3 abc and
the curvature there is \{abc)~^,
X^ V* s*
20. Find the umbilici on the ellipsoid -z + 7= + -3 = <•
cr V* C^
a\a* — fi) c^Cfi {*\
Ans, The four real umbilics are x* = -^4 r— 1 , «* = -^ — ^'.
a^ — c^ «• — ^
21. At an ordinary point on a surfaice the locus of the centers of curvature of all
plane sections is a fixed surface, whose equation referred to the tangent plane as
s-plane and the principal planes as the x- and-^-planes, is
(*• + y + «') (^ + ^^) = »(*• + j^)-
22. Show that an umbilicus on the sur£i'ce
(V«)* + ky /*)* + (« A)* = I
,.,^^ £(J)-=-(5)-=l(l)-.
23. If /* =s o is the equation of a conicoid, show that the tangent cone to the
surface drawn from the vertex a, /tf, y touches a surface along a plane curve which
is the intersection of ^^ = o and the plane
bF ^ bF bF >
<' - ^^ a^--^^-^ - '^>a^- + (' -^> 17 +^^^' ^' ^) = °-
24. Find the quadratic equation for determining the principal radii of curva-
ture at any point of the surface
0(jr) H- *(y) + ;r(') = <>,
and find the condition that the priaiqipal curvatures may be equal and opposite.
26. Show that the cylinder
(a* 4. 4*)l^x* -f (^ + ^)flV = (a* -f ^)fl»^'
cuts the hyperboloid jf*/tf' -f y*/fi — t^/t^ = i in a curve at each point of which
the principal curvatures of the hyperbobid 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» +y^-{- M*) = 2a(x* 4. y).
27. In the surface in Ex. 24 show that the point which satisfies
is an umbilic.
28. Find the umbilici on the surface 2« = x^/a -{- y^/^.
Am. jr = o, >/ = — J^{ab - ^«;, 1 = ^a — ^), if 0 > 3.
29. Show that s = /{x, y) ib generated by a straight line if at all points
ay ay
a^
This is also the condition that the inflexional tangents at each point of the sur-
£icc shall be coincident. Such a surface is called a tone or developable surface.
by^- \bx by I '
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
<t>x(x,y, z) = o, 0,(:*r,^, g) = 0. (x)
■
If between these two equations we eliminate successively x^y, «, we
obtain the projecting cylinders of the curve on the coordinate planes,
respectively,
^i(^» «) = o» ^t(-^» *) = O' i\{^»y) = o.
Any two of these can be taken as the equations of the curve.
258. A curve in space is also determined when the coordinates of
any point on the curve are given as functions of some fourth variable,
such as f,
^=0(/), y^fif), «=a:(0. (2)
The elimination of / between these equations two and two give the
projecting cylinders of the curve.
259. Equations of the Tangent to a Curre at a Point. — If the
equations of the line are (i), the equations of the tangent line to (i) at
Xyyy B are the equations to the tangent planes to 4>\ = ^> 0s = <>,
taken simultaneously, or
(;r-.)g + (i'-,)^ + <z-.)»A = <..
(■)
Since the tangent line is perpendicular to the normals to these
planes, the direction cosines /, m, n of the tangent line are given by
/ __ m _ n __ I
where
Zj, iH/",, N^ being the first partial derivatives of 0^ at x^y, e, and
similarly Z,, M^, A\ are those of 0,.
375
(»)
(3)
376 APPLICATION TO SURFACES. [Ch. XXXIV.
260. If X is the length of a curve measured from a fixed point
to X, y, z, then the direction cosines of the tangent to the curve at
Xf yy z are
dx dy dz
'=5r' '"=^' "-5-'
and the equations of the tangent are
X--x_ _ r-y __ Z-z
dx "^ dy "" dz '
ds ds ds
If the equations to the curve be given by (2), § 258, then
dx di
-T- = 0'(O T" » ^*^'» *^^ ^^® equations (2) become
X--X _r-^ y _Z^z
In general the equations to the tangent are
X-x _^-^_^~g
dx " dy "^ dz *
without specifying the independent variable.
261. The Equation to the Normal Plane to a Curre at x,y, z is
^x-.)'^+ir-,)%+(Z-^)f^=o, (X)
the normal plane being defined as the plane Avhich is p>erpendicular
to the tangent at the point of contact.
Regardless of the independent variable, (i) becomes
{X - x)dx + {y^y)dy + (Z - z)dz = o. (a)
EXAMPLES.
1. Find the tangent line to the central plane section of an ellipsoid.
The equations of the curve are
jc* y* ««
The equations of the tangent at jt, y, g are
X- X _ Y—y _ Z- z
(4)
C^--^^ A^^C^ b-^^aL
b^ ^ ^ a^ a^ H^
2. Trace the curve (the helix)
jT = a cos /, ^ = a sin /, c = bt.
Show that the tangent makes a constant angle with the x^ y plane, and that the
curve 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 cunre of intersection of the
surfaces
2* = «*« -f by^. Ax -\- By •\- C% •\- D '=^ o,
from the fact that at these points the tangent to the cunre must be horizontal.
4. Show that at every point of a line of steepest slope on any surface /* s o
we must have
djr ^ by
5. Show that the lines of steepest slope on the right conoid x ^syf(t) are cot
out by the cylinders jr* -j- ^'^ = r*, r being an arbitrary radius.
262. Osculating Plane. — If P, Q, i? be three points on a curve,
these three points determine a plane. The limiting position of this
plane when Py Q, R converge to one point as a limit is called the
osculating plane of the curve at that point.
The coordinates x^y^ z of any point on a curve are functions of the
lengthy Sy of the curve measured from some fixed point to x^ y^ z.
Therefore, if j^ be the length to x^^^y^^ z^^
where a is the length to some point between x^y^ z and x^^y^^ Zy,
Put dJ s Ji — J, x' s D^, x" s D\x, etc. , then
ATj = .r + ds^x" + \6s^*x" + \^s^'X';\
Let P, Qy R htx.y, z; x^,y^, z^ ; o:,,^,, £,. Then
x^=x+ 6i'x'^, y, =y + ds-y'^, b, = 8 + ds-z"^. (i)
x^ = X + kSs'X' + ^I^Ss^'X^', ■)
y^ = y + kds.y + i^^s'-y;;^, [ (2)
The equation to the plane through P can be written
A{X - X) + B(y^y) + qz - s) = o. (3)
If this passes through Q and Py then
A(x^ - JT) + £{y, -y) + C(z^ - ^) = o, ) . .
^(•^, - ^) + ^U -^) + C{z, - ^) = o. f ^^^
Substitute the values of the coordinates from (i) and (2) in (4).
Divide by 6s, 6s^, and let 6s{=)o.
Ax" + B/ + Cz' = o,
Ax" + By" + Cz"
: :: } (=)
Eliminating A^ By C between (3) and (5), we have the equation to
the osculating plane at Xyy, Zy
X^Xy Y--yy Z^z
x' y z!
x" y z"
= o. (6)
373
APPLICATION TO SURFACES.
[Ch. XXXIV.
Or, regardless of the independent variable,
X^x, Y^y, Z^z =o. (7)
dx dy dz
d^x (Py d^z
263. To Find the Condition that a Curve may be a Plane
Curve. — If a curve is a plane curve, the coordinates of any point must
satisfy a linear relation
Ax + By-^-Cz + D—o,
where Ay B^ C, D are constants. Differentiating,
Adx + Bdy + Cdz =;o,
Ad^x + Bd^y + Cd^z = o,
Ad^x + Bd^y + Cd^z = ©•
Eliminating A^B, Cy we have the condition
dXy dy, dz = o,
d^Xy d^y, d^z
d^x, dy, d^z
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 /, My n be the direction cosines of the principal normal at
Xyy, z. Then, since this line lies in the normal and osculating planes,
y^'z!'
•■\- m
z" x'
z" x"
+ n
x'y
= 0,
/or' + my + ««' = o.
These conditions are satisfied by / = at", m =y'\ n = ar", since
^'VA+^'
z' x'
z"x"
+«"
x' y I
x"y'\-
x^y z!'
x" y z'
x"y' z!'
= o.
Also differentiating x^^ +y^ + *'* = i,
.-. x'x''+yy'+B's''^o.
Therefore the equations of the principal normal are
X-x r^y Z-z
jf
or
X
X-x
d^x
Jl
y
r-y
dy
.//
Z'
d^z
(I)
(2)
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,
/ z" - v" z' " z'x"' - «'' or' "" x^y - x'y* ^^^
Dividing through by d^^ 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^ fiy yht the coordinates
of the center, and p the radius of the circle of curvature at x^ y, z.
Then
(JT -«)« + (^ -/?)»+ (a -,.)» = p».
Let x^y^ z vary on the circle. Differentiate twice with respect to s.
Then .
(x - a)x" + \y - /jy + («-;/) 0'' + x'* +y 3 + «'« = o.
But ^*+y* + «''=!. Also, the line through x^y, z and
a^ fi^ Y is the principal normal, whose direction cosines, by (i),
§ 264, are
/ =
|/a/'"» +/'» + »''»'
with similar values for m and ». Since
X -^ a -=. Ipy V — >5 = wp, z — Y ^ *P>
The center of the circle is a = or — /p, etc
267* The direction cosines of the binormal are
/ = piy'z'' - «y' ), m = p{«V - x'z'% n = p(;ry' -/jt'O-
For, by (4), § 265,
v^z" - zy "" z'x" - xW' ■" xy ^yxf' • ^'^
Also differentiating^* -j-y*-|-«'*=i,
.•/ jT'^'+yy +«y' = o.
The sum of the squares of the denominators in (i) is
(x' » +y » + z! ^)(x" 2 +y' » + z" «)-(ar'a:" +y/' + s's'O s i/p».
Hence the results stated.
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 through the angle Jr as the point of
contact P moves to Q through the are Jx, the measure of torsion at /' is
ds ^ X ^^ *
when Js(=)o. The number c = DrS is sometimes called the radius
of torsion.
Let /j, m,, n^; / , «,, n^, be the direction cosines of two planes
including an angle 0. Then
sin^d' = («,«, -- «j«,)8 + («/, - A"a)' + (A'Wi - «/i)'-
Let i, m, n he the direction cosines of the osculating plane at P,
and / -f- J/, « + Jm, n -\- An those at Q,
Let Jr be the angle between these planes. Then
sin'Jr = {m^n - « J«)» + (« J/ - /Jii)» + (jAm - m^lf.
Divide by -Jj* and write
sinMr __ sin' Jr /^ A*
Let Jx(=) o. Then, in the limit,
/dr\^ I dn dm\* / dl Jn\^ ^ I ,dm dl\^ ,.
\li) = ("^- « ^) + («^ - Vx) + y-di -''-ds) • (^>
>• • , ,dl , dm ^ dn
Since /•4-»i'4-«'= I, .•. /^-4-»i-r-+«3- = o.
' ' ' ds ds * ds
Square this last equation and subtract from (i).
- (£)■= (!)'+ (^)'+ (^y- «
269. The measure of torsion can be expressed in another form, as
follows.
Let I, m, n he the direction cosines of the binormal, and
Z =yz'' — sy\ etc., as in § 267. Then
I m n
Whence Z» + M^ + iV« = i/p». (2)
Since the binormal is perpendicular to the tangent and principal
normal,
Zr' +my + nz' = o, (3)
Ix" 4- my + nz" =z o. (4)
Akt. 270.]
CURVES IN SPACE.
38"
DifTerentiating (3) and using (4),
/ V + my + n's' = o.
Differentiating, /^ + «« + ««= i.
. •. i/' + mm^ -f- »«' = o.
From (5) and (7) we get
/' __ m' _ n'
mz — ny
and each of these is equal to
(5)
(6)
(7)
(8)
my
.'-.'/
(9)
/Z+mAf+nN '
Differentiating (4),
Therefore (8) is equal to
ix"'^ 4, my + nz''' __ x'''L+y''M+z'''N
IL + mM+nN " L^ + M'^ + I^ •
Remembering that /,m,n; x^^y^ z' are the direction cosines of two
lines at right angles,
{mz' - nyy + {nx' - h^ + (// - mx^^ = sin» ^7C=zi.
Therefore, by (2), § 268, and (8),
'x'''L+y''M+z'''NY
m= (;
Z^ + M^-\-
or
L— — — n*
x" y z'
jc" y z"
(10)
by (2), and the determinant form of
x"'L-^y"M+z'''N.
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,
.-. (x-a)x' +(y _/?!/' +(,_y),' =0.
{X - .■r)x" +iy- /Sy + (* - r)s" = - t.
(X - a)x"' +iy- P)y"' + (« - y)*'" = o.
382
APPLICATION TO SURFACES.
[Ch. XXXIV.
Eliminating between the last three equations,
(^ - a) =
=z<rf^{yy -«>");
x' y b'
x^' yf «^'
j^// yfff gfff
y — /3 = (Tf^iyx"' - tf^'xT)', t - y = (Tffix'y" - x'y).
Squaring and adding,
j^ = a^p^Kxy - yx'y + (/»"' - syy + (^v - x's'^'y].
Clearly the circle of curvature lies on the sphere of curvature.
Let P, Qy Ry /be four points on a curve and in the same neighbor-
hood, R and p the radii of spherical and circular curvature.
Then, C being the center of the circle through /*, Q, R, and S
S that of the sphere through
P, Q, Rf /, we have directly
from the figure,
Jp
SC=
R^
-■=''+(1)"
\ds
dp
= /o»+<r'
'^y
5C = f- =
dr
ds
Fig. 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
d^x tPy €^z
the sum of the squares of the three determinants
{dyt^ — dztPy)^ + {dzc^x — dxd^z^ + {dxdfy — dy d^x)\
Then, regardless of the independent variable employed,
{dx^ + dy + dz^)^
P =
vl
X
dx dy dz
d^x d^y d^z
dx dy dz
d^x d^y d^z
c^x dy d^z
0)
dx dy dz
d^x d^ d^\j^
3
(»)
Art. 271.] CURVES IN SPACE. 3^3
(i) comes immediately from § 267, and (2) from putting the value
of /G^ from (i), §271 in (10), § 269.
BXSHaSBS.
1. Show that in a plane curve the torsion is o.
2. The equations of the tangent at x^ y, m \o the curve whose equations are
flJt* -|- ^ -f- f«* = 1, Sjfl -{- cy* -\- as* = I, are
x{X - X) _ y{Y-y) ^ »{Z - «)
ab — ^ be ^ a* ac -^ a^'
3. The equations of a line are
jc* + y -|- «« = 4fl* and j:* -f »* = 2ax,
Show that the equations of the tangent line and normal plane are
(x — a)X -\- zZ = aXf
yY-\- aX =r a(4/i
4. The equation of the normal plane to the intersection of
x^/a -\- yyb -^^ tsyc =i 1 and jr* + ^« + «« =s </«
is £ a{b - c) ^ jb{c ^ a) +^ c(a ^ i) = o.
5. Show that the curve m(x -f ^X-* — <») = «•»«(>'+ *)(^ — a) = «•, is a
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 surfiice x* +y* -^ ** r= ^ cuts the sphere
x*+y*-\^g*=: a*
in great circles.
8. Show that the equations of the tangent to the curve
^ = fljf — jf*, M* =z a* — tfjc,
- jr). y ' X y y x) \M y)'
are X^x= — 5:_(K-.^)= -?!(Z-«),
a — 2x^ a ^ '
9. Find the osculating plane at any point of the curve
X z= a cos /, J' = ^ sin /, « = ct,
Ans. c(Xy - Yx) + ab(Z — «) = a
10. Find the radius of circular curvature at any point of
x/h 4- y/it = I. jc« -I- «« = fl«.
aW J^A* 4- k»
11. Show that the curves of greatest slope to xOy on the surfaces xyx =r a* and
cz-=.xy are the lines in which these surfaices are cut by the cyliiider jr' — ^* = const.
12. Find the osculating plane at any point of the curve
JT = 0 008 6 4- ^ sin ^, ^^ = a sin 0 + ^ cos 0, t = r sin 26.
3^4 APPUCATION TO SURFACES. [Ch. XXXIV.
13. Find the principal nonnal at any point of
jfl -\- y* := a*^ fl« = j:* — ^.
Hint. Express x, y in terms of s as the independent variable.
14» Given the helix jr = a cos 6, ^ = a sin 0, 5 = 36) show that
(i). 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). i|jy sin 0 — ii Fcos 6 _ 3(s — M) = o,
^JT sin 6 >- ^Kcos 0 -f tf(s ~ M) = o.
(3). ^ = tani.
(4). p = fl(i + ^/^>
(5). o- = (fl« + l^yb.
15. Show that ^ rr o, /.' s= o are the equations of the line of contact of the
vertical enveloping cylinder of ^ = o, and that the horizontal projection of this
line is the envelope of the horizontal projections of parallel plane sections of ^= o.
16. Show that the equations of the level lines and lines of steepest slope on the
surface F z= o are
i^rzo, F*^dx -\- F; dy =: o and /•=©, F'^dy -- F*y dx — o
respectively, and that they cross each other at right angles.
17. Find the lines of steepest slope on the surfaces
ojc* + fy^ -|- ^a* = I and x = <ur* -j- /jy*.
18. A line of constant slope on a surface is called a Loxodrone. Find the loxo-
drone on the cone x^ -^ y^ — k(z — cf. Show' that its horizontal projection S& a
logarithmic spiral.
19. Find the loxodrone on the sphere jc* -f- >'* + *' ~ <**•
20. A line 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 sur&ce is a line whose osculating plane at any point
contains tlie 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 surfsice. Show that at every point on a geodesic on
the surface ^ s= o, we have
CHAPTER XXXV.
ENVELOPES OF SURFACES.
272. Envelope of a Surf ace - Family having One Variable
Parameter. — When /1(^, j', «) = o is the equation of a surface con-
taining an arbitrary parameter or, we can indicate the presence of this
arbitrary parameter a by writing the equation
F{x,y, Zy a) = o. (i)
The position of the surface (i) depends on the value assigned to a.
By assigning a continuous series of values to awe have a singly infinite
family of surfaces whose equation is (i).
If we assign to a a particular value a^ we have another position of
the surface (i) whose equation is
F{X, y, 2, OTj) = o. (2)
The two surfaces (1) and (2) will in general intersect in a curve.
When arj(=)a the surface (2) converges to coincidence with the sur-
face (i), and their line of intersection may converge to a definite posi-
tion on (i). At any point on the intersection of (i) and (2) the
values of Xyy, 3 are the same in both equations. By the law of the
mean^
F(x,y, z, a,) = F{x,y, z, a) -f (a, - a)Fi,(x,y, 8, a"),
a' being a number between a and a^.
At any point of intersection of (i) and (2)
F{x,y, Zy a) = F(x,y, z, a,) = o.
Therefore at any such point we have
F;.{x, y, z, a') = 0. (3)
If, when a^{=)a, the line of intersection of (i) and (2) converges
to a definite position on (i), then the coordinates of all points on this
line must satisfy, by (3), the equation
— F{x,y, z, a) = o, (4)
and the surface (4) passes through the limiting position of (i) and (2).
If from equations (i) and (4), i.e.,
F{x,y, z, a) = o, F:,{x,y, z, a) = o, (5)
385
386 APPLICATION TO SURFACES. [Ch. XXXV.
a be eliminated, the result is an equation <p{x^ y^ z) = o, which
is the surface generated by the line whose equations are (5), or 0 = o
is the locus of the ultimate intersections of consecutive surfaces of the
family (i). This locus is called the envelope of the family (i). 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, a) = o, at x,y, z, is
^dx + — dy + — dz==o. (I)
But in ^x,_y, s, a) = o, as x,_y, z vary along the envelope, or also
varies, and the equation to the tangent to the envelope is
— dx^^dy + —dz+^dcc=o. (2)
Since at any point x^y, z common to the surface /'= o and the
envelope, that is all along the characteristic, we have Fi = o, the
planes (i) and (2) coincide.
EXAMPLES.
1. Show that the envelope of a family of planes having a single parameter is a
iorse (developable surface).
Let « = x4>{a) + y^a) + ;t(a).
.-. ^ = 0(a), ^- = ^<x)\ ••• x<t>L +yK + ;r« = o-
Also,
Hence r/ — j* = o. See Ex. 29, § 256.
X A- V
2. Envelop — '-^ — \- za ■= 2,
Ans. Hj-perbolic cylinder, xz -\- yz =1 i,
3. Envelop x -\-y — 2az = a*.
Ans, Parabolic cylinder, x -f- >' + *' = o-
4. Generally if 0, V» X ^^^ linear functions of x, yj «, then the envelope of the
plane
0a^ + 2il;a + ;f = o
is iff* = <PXj * cone or cylinder having 0 = o, ;f =0 as tangent planes, and
0 = o is a plane through the lines of contact.
5. Find the envelope of the family of spheres whose centers lie on the parabola
jr* -f 4ay = 0, « = o, and which pass through the origin.
Ans. x* -\- y* -\- s^ = 2ax!^/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. jr* -J-^'* -|- ** = const
274. Envelope of a Surface-Family with Two Variable Param-
eters.
If F{ot, P) s F(x, y, 3, a,fi) = o (i)
is a surface of the family, then
^(^i» A) s F{x,y, z, a„ /?,) = o (2)
is a second surface of the family.
At any point x^yy z where (i) and (2) meet,
F{a„ A) = ^(«. /S) + («.-«) 37 + (A - fi)^n (3)
where a' is between a^ and nr, ^ between ^5^ and fi.
In virtue of (i) and (2), (3) gives
dF bF
This is the equation of a surface passing through the intersection
of (i) and (2). But for any fixed values x^y, 0, or, ft satisfying (i)
and (2) there are an indefinite number of surfaces (4) obtained by
varying or^, fi^^ all of which cut (i) in lines passing through ;i;, >^, z.
Consequently there are of these surfaces (4) two particular surfaces,
bF dF
which cut (i) in lines passing through x, y, z.
If now the point x^y, z has a determinate Umit when a^(=:)a,
/5,(=)/?, then the three surfaces
F{a, /J) = o, Fi{a, /3) = o, /?(«, /3) = o,
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 /S, we obtain the equa-
tion to the locus of intersections. This locus is a surface called the
envelope of the family (i).
275. The Envelope of the Family F(x, y, z,a, fi) ^ ois Tan-
gent to Each Member of iht Family. — The tangent plane to any
member of the family is
BF^ . ^F. dF. , .
3^8 APPLICATION TO SURFACES. [Ch. XXXV.
As the point x, y, z moves on the envelope, a and fi vary. The
plane tangent to the envelope is
Q-dx+-dy + ~dz + -da+^d^ = o. (2)
At a point x,y, a common to the envelope and one of the surfaces,
we have F^ = o, /J = o, and therefore the planes (i) and (2)
coincide. Since this point is the intersection of the line whose
equations are F^, =z o, Fp =0 with the surface F = o, the envelope
is tangent to the surface at a point, and not along a line.
276. Use of Arbitrary Multipliers. — l{F{x,y, 2, a, /3) = o,
where a, fi are two arbitrary parameters connected by the relation
0(£r, )5) = o, then Z' = o is a family of surfaces depending on a
single variable parameter. The equation of the envelope is found by
the elimination of a, )5, da, dfi between
F= o, 0 = 0, Fida+ F^d/S = 0, <pL^a + <f>^ d/3 = 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, fi, X from
F= O, 0 = 0, /'a + ^0a = 0, -^/J + ^<Pp = O.
The family of surfaces, represented by F = o^ containing n
parameters which are connected by « — i or « — 2 equations is
equivalent to a family containing one or two indepenc^ent parameters
respectively. Such a femily, 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. Foi 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 x^y, z could be eliminated, and a relation between the
parameters obtained, which is contrary to the hypothesis that they
are independent.
In general, if /*= o contains n arbitrary parameters aTj , . . . , a^
connected by the « — i equations of condition 0^ = o, . . . ,
0,^, = o, the equation of the envelope is found by eliminating the
2« — I numbers or^ , . . . , a„ , A^, . . . , A„_„ between the 2«
equations
F =z o, 0^ = o, . . . , 0^_, = o,
^'K) + K<PrM + . . . +^x0'--xK) = o,
^\0^n) + ^0/(««) + . . . + V,0Ui(««) = O.
Art. 276.] ENVELOPES OF SURFACES. 389
XX8KCI8ES.
1. Find the envelope of (6 being the variable parameter)
j: sin 0 — >^ cos 6 ^ oQ ^ cm.
Ans, X sin -—^ v cos ^—^ ^-^ = A/j^^yt .^i.
2. Find the envelope of a sphere of constant radius whose center lies on a circle
in the jt^^-plane.
Ans, If JT* + j^ = ^ is the circle, and the sphere has radius a, the envelope is
the torus j:* + ^ = [^ + («« — ««)*]«.
jr« v» «»
3. Find the envelope of the ellipsoids -^ + ii -|"7 = '» where a 4- ^ -)- ^ = >(.
<r It r
Am, x*+^»+»» =it».
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 jr.axis and whose
radii are proportional to the distance of the center from the origin.
Am, >« + «« z= «»(*« -f y ^- ,«).
6. Find the envelope of the plane ax -\' fy -\- y% ^ \ when the rectangle
under the perpendiculars from the points (a, o, o) and (— a, o, o)on the plane is
equal to 1^,
Am, a, ^positive. -j_p-^ + ■£— X__ - ,.
X y t
7. Find the envelope of the planes 1" f* H — = ' when a« -f ^« -|- ^ = i«.
Am, jr«+» +;'*+' + s*"*-' = i*+'.
X y 9
8. Spheres are described having their centers on -7 = ^ = — , 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^ n being direction cosines)
•** -\-y^ -f- «' = (/jf 4- wy + **' + ^)'' ^ = const.
9. Envelop the family of spheres having for diameters a series of parallel chords
of an ellipsoid.
10. If /(a) = o is the equation of a family of surfaces containing a single
arbitrar>' parameter or, then the equations of the characteristic line on the envelope
are F\a) = o, F'{ol) = 0. As a varies this line moves on the envelope; it will in
general have an enveloping line on that 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
Fyfl) = o, F{pC) = o, F'\a) = o.
11. Find the equations of the edge of the envelope of the plane
x sin 8 — ^ cos 6 = aO — cz,
Ans, jc* -f-^' = a', y := x tan — .
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. [Ch. XXXV.
Let Ix -\- my -\- ns z=: o be the plane; the parameters are connected by
/« -j. »,! + «« = I, /iflt -I- m^6» + »«^ = d*.
jc* y» «»
13. Spheres are described on chords of the circle x'-^-y* = 2ax^ £ = o which
pass through the origin, as diameters, show that they are enveloped by
(jr* +7»-|- «« - axY = a\x^ + j^«).
14. Show that the envelope of planes cutting off a constant volume from the
cone ax^ -{■ by^ -\- cz^ =z o is d. hyperboloid of which the cone is the asymptote.
15. Find the envelope of the plane /r -f- »?y +*** = ^» when /* + «■ -|- «* = i,
Ans. Fresnel's Wave-surface,
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 Une in which intersect
the sphere and cone
;c» -f-y -f «« = r», jc»/a«-f ^V^ -f «V^ =0.
17. Find the envelope of a plane which moves in such a manner that the sum of
the squares of its distances from the comers of a tetrahedron is constant.
18. Show that the envelope of a plane, the sum of whose distances iix>m n fixed
points in space is equal to the constant ky is a sphere whose center is the centroid
of the fixed points and whose radius is one nth of k,
19. Show that the envelope of a plane, the sum of the squares of whose distances
from n 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 frx>m a given surface, the
envelope of Sie reflected rays is called the caustic by reflexion.
Show that the caustic by reflexion of the sphere jk* -f- ^' 4- «* = r*, the radiant
point being h^ o, o, is
in which ff m 3^ "i"^' ~i~ "**
PART vn.
INTEGRATION FOR MORE THAN ONE VARIABLE
MULTIPLE INTEGRALS.
CHAPTER XXXVI.
DIFFERENTIATION AHID INTEGRATION OF INTEGRALS.
277. Differentiation under the Integral Sign. Indefinite In-
tegral.— Ijtl/^x, ^) be a function of two independent variables Xy y.
Let
^{^>y) = JA^>y)^>
the integration being performed for y constant. This integral is a
function of^ as well as of x. On differentiating with respect to x^
Again, differentiating this with respect to^
d^F ^ d/{x, y)
By dx By '
But
9^F
Consequently
_ B^F _ B/
BxBy " ByBx~ By'
By By
BF_ r
r-'
or
iA-)-=/^s^-
Therefore, to differentiate with respect to ;^ the integral taken with
respect to jc of a function of two independent variables jt, y, differ-
entiate the function under the integral sign.
39«
392 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVI.
In like manner we have
a* r X r^f
This process is useful in finding new integrals, from a known inte-
gral, of a function containing an arbitrary parameter.
EXAMPLES.
1. We hav« the known integral
a
Diflerentiating with respect to a, we find
/""•^ = T (' - j) •
And generally, diflerentiating n times,
2. Since I sin ax ax = ,
/x sin ax , cos ax
X cos ax ax z= = — .
a a*
3. From I (a 4- bx^dx = ■ / . ., show
/. , ^ inbx — a)(a 4- bx\»
x<^(/x = — ; — show that
tf -f I
f-^^o^'^=^^,{^ogx^^^.
278. Differentiation of Definite Integrals when the Limits are
Constants.
Let *^ ~ 7 A^' y)^f
where a and b are independent oi x. Then the result of § 277 holds
as before, and
that
On account of the importance of this an independent proof is
added. Let Ju denote the change in u due to the change Jy in j'.
Then, the limits remaining the same,
2Su _ /•* /{x,y + 4^) — {/x, y)
-f.
^y J a Ay
dx.
ART. 279.] DIFFERENTIATION OF INTEGRALS- 393
Hence, when 4y(=z)o, we have
and, geneially,
1. If r r^dx=i 1
be differentiated n times with respect to a, we get
/
2. From It iv \ - "~i »
J/** dx __ 1.3.5 . » • (2>» — I) )r
[^ (x« -f- «)*+' ~ 2.4.6 . . . 2» itf*"*"*'
The value of a definite integral can frequently be found by this method. Thus:
3. Let «= f'^^.^-^dx.
Jo log-^
Then -;- = / -1-'^=/ ''^^ = — T":*
••• •* =fjT^ = log (« + l)»
no constant being added since u = o when a = o,
4. Find /'^^ (I + tf cos e)<^.
ifw. « log (i + 4^1 — tf«).
279. Integration under the Integral Sign.
I. Indefinite IntegraL
Let F(x,j^)=f/{x,^)dx.
Then will
Let V =j/(x,y)dy.
Then ^ =^A^>y)-
Also, ^fvdx=J~dx=f/{x,y)dx=F{x,y),
.: Jvdx=Jl!\x,y)dy,
or / ) JAx>y¥y (''•*=/ 1 j A^>y)^ \ 4y-
394 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. 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
ff/dxdy^ff/dydx.
As in differentiation, the operation is to be performed first with
respect to the variable whose differential is written nearest the func-
tion, ur integral sign.
II. The same theorem is true for the definite double integral of a
function of two independent variables when ike limits are constants.
Let
JA^, y)dx = X(x, y), JAx, y)dy = F{x, y) ;
JfA^>y)dxdy =ff/{x,y)dydx = F(x,y).
Then
rv^^ = ^(x^yy) - x{^vy\
The last two values are the same. Hence
or
The integral sign with its appropriate limits and the corresponding
differential are written in the same relative position with respect to
the fiinction.
KZAMPLBS.
^/'
1. i x*-^dx = i . Hence
r rV-^ ^ =/••*= log ^
^ log* a.
Art. 28a] DIFFERENTIATION OF INTEGRALS. 395
Put X = e-*.
dz = log -i.
I I ^€-^'*sindxdadx= f ' , ,,,
Jo J a. J». ^' + ^'
sin 6x dx = tan-« -^ — tair-' -?i
If Oq s o^ Oj = 00 , then
J/»« sin ^x ,
8. ETalvate r*^""*'"^.
Put ^^ r '"**'^-
and r* ^-••(«+*")a ^. = ^^-••.
Also, jr%-(.+.%^=i._i_,
and *jf' np^ = * tan-'jf |*= ^ft = i".
.-. J^e"^dx=\yx,
e-a^x^dx = — |/)iF.
This glares the area of the probability curve.
280. If F[x^ y, b) is a function of three independent variables, the
same rules as for a function of two independent variables govern the
triple integral
JjJFdxdydz.
Examples of double and triple integrals will be given in the next
chapter.
CHAPTER XXXVir.
APPLICATIONS OF DOUBLE AND TRIPLE INTEGRALS.
Plane Areas. Double iN'recRATioN.
281. Rectangular Formola. — If ;r,^are the rectangular coor-
dinates of a point in a plane xOy, then
deo = dx ^y •= dx dy
is an element of area, being the area of the rectangle whose sides are
Ax and Jy.
Let the entire plane xOy be
divided into rectangular spaces
by parallijls to Ox and Oy, of
which Ax /^y \s ^ type. The
area of any closed boundary
drawn in the plane is the limit
of the sum of all the eniire rectan-
gular elements of type Ay Ax
included in the boundary, when
for each rectangle Ax{ = ')q,
Ay(=)o. For the area within
the closed boundary A is equal to
A = S Ay Ax
plus the sum of the fractional
rectangles which are cut by the
Fig- '48. 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 Ax{=)Ay{^)o.
The summation is effected by summing first the rectangles in a
vertical strip PQ and then summing all the vertical strips fr(Mn ^ to 7^
or, first sum the elements in a horizontal strip PL, then sum all the
horizontal strips in the boundary from S to C. These summations are
clearly represented by the double integrals
f-f*"j,j^, f'"r'""Wjj,.
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.
EZAKPISS.
1, Required the area between the parabola >■' = ax and the circle y* = aax — x*,
in (he first quadrant.
The curves meet at the origin and at x = a.
(I). ^ =£J^jj£'-''jydx=£lViax-x'- ^(^dx
~ 4 3 ■
(I). A = r^" r^''" d:, dy
'£\i
x) and inside the circle
3. Find the area common to the parabola jv' = 25X and 5^* = Cjy. Ani. j.
282. Polar CoordinateB.— 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 ladti includ-
ing the angle iJf, is
\\{p^^pY-(?\M
= p jp jff + i ^f^ ^e.
The entire area in any closed
boundary is the limit of the sum
of the entire checks in the'
boundary. Thesumof the par- *"'"■ '^5-
tialcheclcsonthe boundary being o when ^p(=)J#{=)o, as in § 381.
But, since
/pjp^0 + i/}p>je , ,/^p
pjpjff ~'+i2 P'
= I,
■when Jlp(=)J0(=)o, the area within any closed boundary is equal to
A = jCSpJp^e
when J/)(=)Jtf(=)o.
39^ INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII.
This summation can be effected in two ways :
(i). We can sum the checks along a radius vector ^^, keeping
^6 constant, then sum the tier of checks thus obtained from one value
of 6 to another.
(2). We can sum the checks along the ring UV, keeping p and
^p constant, then sum the rings from one value of p to another.
These operations clearly give the double integrals
J^i t^ps^W •'Pi J9=i
:A(p)
EXAMPLBS.
1. Find the area between the two circles /9 = acos9, p=^oosO, 6 > a.
b coctf
(1). A= f f fidpJB
(2). A= I f ^ dB pdp-^ I I p dBpdp,
J a •'0 •'0 •'co8-*i
which gives the same result as (i).
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, z 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 Ax, Ay, Az
be the edges of a typical elementary
parallelopiped. Then the volume
Ax Ay Az
is the elementary space volume.
The volume of any closed surface is
the limit of the sum of the entire elemen-
tary parallelopipeds included by the
surface when Ax{=^)Ay{=)Az{=)o.
Vz=£2 AxAyAz,
taken throughout the enclosed space.
(i). L,ctx,y, Ay, J;r be constant. Sum the elementary volumes
Fig. 150.
Art. 283.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 399
between the two values of b, obtaining the volume of a column AfS of
the solid. The result expresses 5 as a function of x and^ given by
the equation or equations of the boundary.
(2). Let X, Jx be constant. Sum the columns between two
values o(y for 4v(=)o. The result is the slice of the solid on the
cross-section x = constant^ having thickness Jx,
(3). Sum the slices between two values of jr for J;i:(=)o. The
result is the total sum of the elements, expressed by the integral
= r^r^edydx,
= /***-4, dx.
Jxi
'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
x^ V* ««
5+^ + ? = '-
d/7^* J, »• ^'
2. Find the volume bounded by the hyperbolic paraboloid xy s as, the xOy
plane, and the four planes or = Xi, x = x,, ^ ss y^, y = y^,
xf
V= C'^nrdzJydx,
Jxx Jy\ •/©
Jx, Jyx «
Jxx 2a
= ^ (j^i — *\iyt - viK-'i^i + ^%y% + •'i^. + ^.^A
400 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIL
The volume is therefore equal to the area of the rectangular base multiplied by
the average of the elevations of the comers. This is the engineer's rule for calcu-
lating earthwork volumes.
284. Polar Coordinates. — The polar coordinates of a point P in
space are p, the distance of the point from
the origin; ^, the angle which this radius
vector OP makes with the vertical Oz\ and 0,
the angle which the vertical plane POz makes
with the fixed plane xOz,
Through P draw a vertical circle PM
with radius p. Prolong OP to Ry PR = J p.
*« Draw the circle RQ in the plane POM with
radius p + Ap, If A A is the area PRQS,
then
//lA _
JppJd"^'
We may therefore take dA = pdp dO. This area revolving around
Oz generates a ring of volume
2 ;r p sin 6dA.
Therefore the volume generated by dA revolving through the arc
ds =. p sin (^ d<p 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 0 d(p dp d6.
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 limit of the sum of the
entire elementary solids included in the surface when
Jp(=)J0(=)J<>(=)o.
Or, the volume is equal to the triple integral
F =fffp^ sin 0d<pdp d0,
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,
w w
K= f ^' f ^^ r'''^p^dp'SinQdB'd(p,
t/^=o «/tf=o t/p=o
ir
= — I * / ' sine dB'd<p
3 •/♦=o Jb=q
w
=!/'-'<>=**<»•
Art. 285.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 40 ^
The first integration gives a pyramid with vertex O and spherical, base
a' sin 6 d^ //0. The next integration gives the volume of a wedge-shaped element
of a solid between two vertical planes determined by <p and 0 -{- J0. 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^
I^t a be the radius of the sphere, a the semi-vertical an^le of the cone. The
polar equation of the sphere with the vertex of the cone as origin is p = 2a cos 0.
Jo Jo Jo
3. The curve p = a(i -{- cos 6) revolves about the initial line; find the volume
of the solid generated.
y^ I I I p»dp.d<p.utiedB,
Jo Jo Jo
2Ktfi
J (I + COS 6)» sin 9 ^ = fjrtf*.
285. Mixed Coordinates. — Instead
of dividing a solid into columns stand-
ing on a rectangular elementary basis,
as in the method
y=ffzdxdy,
it is sometimes advantageous to divide
it into columns standing on the polar
element of area. Thus the elementary
column volume is
z p dd dp.
Therefore for the volume of a solid /y
we have
V= ffJds.pdOdp,
=zffzpdpd&,
taken between the proper limits.
Fig. 152.
1. Find the volume bounded by the surfaces s = o,
JK* -f" ^' = 4^' *^^ y* = 2^"* — -**•
Here z = pV4^ ^^^ ^^^ limits of p and B must be such as to extend the inte-
gration over the whole area of the circle jr* = 2cx — jc*. Let pi = 2^ cos 0;
then
a Jo 0422 oa
402 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIL
2. Find the volume of the solid bounded by the plane * = o and the surface
% =z a^
Here p» = *«+^. .«. F=affe "^ pdpd^.
^2W
j dB^ 2ft.
[See Todhunter, Int CaL p. i8i.]
Surfaces of Solids.
286. When the plane through any three points on a surface (the
points arbitrarily chosen) converges to a tangent plane as a 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 siurface we proceed as follows:
Let P be a point at,^, « on a surface,
and Q a point x -\- Jx,y + 4>'> * + ^^-
The prism MTNU on the rectangle
whose sides are Axy Ay cuts the surface
in an element of surface FRQS. Draw
the diagonal MN ^vA the two inscribed
triangles PRQ and PSQ, Let i>erpendic-
ulars to the planes of the triangles PRQy
PSQ at the point P make angles ^,, y^
with Oz respectively. The angles y^, y^
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 = MTNsec y,, PSQ = MUNstc y^.
Also, MTN= MUN,
... pRQj^psQ^^^^li±^^AyAx.
Fig. 153.
Art. 286.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 403
t.ien
By hypothesis, if A^S is the area of the element of surface PRQS^
£■
J«5
secy,+secy,^^^^
= I.
But when Q{=^)P the perpendiculars to the planes PRQ, PSQ
have the normal to the surface at i^ as a limit, since the planes PRQ^
PSQ converge to the tangent plane at Z' as a limit.
If y is the angle which the tangent plane at P makes with the
plane xOy^ then
/
sec y, 4- sec y,
i-s— i lJ = sec y,
=^i + (^) +
(I)'
and
dy dx
= sec y,
=//
sec y dy dx,
=#>/■+(£)'+ (I)*--
taken between the limits determined by the boundary of the portion
of the surface whose area is required.
1i Find the area of the sphere-surface x* -\- y* -\- tfi r:^ tfi.
dz X dM_^ y
Jx^o Jy^o ^a^ ^x*^yt
Vtf*-jr«
~ 2/1
dx s \ica\
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.
(I). Let the equations of the sphere and
cylinder be
x»+y + «* = ««,
x^-\-y^ = ax, y
as in the figure.
Fig. 154.
404 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIL
-ax dz dx
J^ax — jr*
-"IT
= ^a r !^[lZ^dx = 2a f\\^dx = 4^«.
Jo i/tfx-x« Jo \^
(2). Let J be the length of the arc of the base of the cylinder measured from
the origin. Then
S — ^ jzds,
taken over the semi-circumference. Let ^^ be the angle which the sphere radius to
P makes with Os, and 6 the angle which OM = p makes with Ox. Then
z :=z a cos ^ = a sin 6. J = <i6, ds -=. a d^*
.-. 5=4/* "*'«" sine </0 = 4<»'-
(3). Otherwise, immediately from the geometry of the figure,
as in (i).
8. Find the sur£a,ce of the sphere intercepted by the cylinder in Ex. 2.
From the figure,
(I). sec ;^ = ~ =
« 4/a« - x» - ^«
= j^ax-x* dy dx
o
Integrate directly, or put sin«e = x/{a + x) and integrate
Hence S = 2tf«(3r — 2).
(2). Again,
iS =i I /06 sec ;/- </p.
p = fl cos 0 = a sin ^. . •. ^ = -J* — 6
Gcos § <i9 = — fl* [6 sin e -f- cos 6]^ = - a\^ie — I)
Lengths of Curves in Space.
287. As in 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 cur\'e.
Art. 287.J APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 405
Since (^)*^^ = ^jc« + zjy» + /^s?,
. i=^-+(i)"+(S)'
ds ds
with similar valute for the derivatives -j-, -^,
with corresponding values for s when ^ or s is taken as the indepen-
dent variable.
If the coordinates of a point on the curve are given in terms of a
variable /| then
(§)•=(§)•+ (f)*+,(^)*.
and
EXAXPLBS.
1. Find the length of the helix
.' jr = acoSj, ^ = flsin-,
measured from s = o.
Take s as the independent yariable. Then
dx a . t dy a t
2. Find the length, measured from the origin, of the curve
2ay = jf*, 6aU = jr».
3. Show that the length, measured from the origin, of
^ = a sin X, 4* = tf"(jr + cos * sin 4f),
is jr 4~ '*
4. Find the length of _
measured from the origin. Ans, s ss x -{-y * s»
4«6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIL
6. Find the length, measured from the horizontal plane, of the oirve
288. Observations on Multiple Integrals. — The problem of
integration always reduces ultimately to the irreducible integral
d£ being the element of the subject to be integrated. Or this may
be taken ^ 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.
BZSSCIS£S.
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 sur&ices
X* -\- y* =z a*f « = o, « = jr tan a.
Ahs, ^ f I * J , d» dy dx rs ^e^ tzn a,
2. Find the volume bounded by the plane « = o, the cylindier
and the hyperbolic paraboloid xy = cz. Am, n — /?*.
3. Find the volume bounded by the sphere and cylinders
:t»+yt ^gt:sa*, jfi+y* = ^, /»« = «« cos«© + ^« sin«e.
Am. Ki6 - 33rK«» - ^)^
Art. 288.] APPUCATION OF DOUBLE AND TRIPLE INTEGRALS. 407
4b 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 sphei^ a. Find the
volume common to both suiiaces. Ans. {(jr — 1)0*.
5. Show that the volume included within the ntrhee
is a^c times the volume of the surface
^x, y, %) = o.
0. Show that the volume of the solid bounded by the sur£ioes
« = o, x« -|- ^« = 4^2, jf« -f ^* = 2f r, is |jr^/«.
7. Find the entire volume bounded by the positive sides of the three coordinate
planes and the surface
8. Find the volume bounded by the surface
jr* -f^t + «» = «♦. Ans. ^naK
9. Find the volume of the sur&ce
90
(7)*+(f)*+(7)*=«- -- A«*'-
10. Show that the volume included between the surfece 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
jt^/a* -\-y*/fi -f g*/c* = X. Am, \nabc.
12. Find the whole volume of the solid bounded by
(jr» -|- ^» -|- «»)> = 27a*jrys. Ans, Ja*.
13. Use § 285 to show that the volume of the torus
14. Find the volume of the solid bounded by the planes jr = o, ^^ = o, the sur-
face {x -\- y^ = 4tf'f and the tangent plane to the surface at any point/, gy h,
Ans, ^ak^.
15. Show that the surfaces ^> -f '* = 4'>^i ^^^ jr ~ s =: <i, include a volume
8jrfl».
16. Show that the volume included between the plane s = o, the cylinder
y^ = 2CX — jc', and a paraboloid ojt* -^ iy* = 2z is ^JT^Sfl-' + ^»).
17. Show that the whole volume of the surface whose equation is
(jT* -f-^* -+- z^f = cxyz is equal to ^/Z^o.
18. Show that the volume included between the planes^ = ± i and the sur&oe
fl»jr* -f ^«» = 2{ax -f bt)y^ is ^nl^/sab.
19. Find the form of the surface whose equation is
(*»/a» +y^/l^ + «V^)« = x^/a^ +y/** - «V<*»
and show that the volume is jt^abc/^ f^
4o8 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. 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. i6JP, R being the radius of the cylinders.
21. Find the area of the surface
«' -|- (or cos a 4" ^ sin a)' = a*
in the first octant. Ans, 2a^ esc 2a.
22. Find the volume of the solid in the first octant bounded hy xy = az and
X +y -f « = «. Am, (Jl — log 4)tf».
23. Find the sur&ce of the sphere jfl -{- y^ -{-»*:= a* in the first octant inter-
cepted between the planes or = o, ^ = o, jc = ^, ^ = ^.
Ans. a {20 sm— « — == — a sin— » -= =^ i
24to A curve is traced on a sphere so that its tangent makes always a constant
angle wiUi a fixed plane. Find its length from cusp to cusp.
CHAPTER XXXVin.
Iin^GRATION OF ORDINARY DIFFERENTIAL EQUATIONS.
289. Classiflcatloii. — A differential equation i$ an equation which
involves derivatives or difTerentials.
An ordinary differential equation is one in which the derivatives
are taken with respect to one 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 t)ie first order and first degree
when multiplied by dx^ are equivalent to the differential equations of
tbe first order and first degree
^ = cos X dXy 2xdy = (3^ — xy)dx, ao^j^dy •= 2xdy ^y dx.
In general, any linear flinction of ~^y
dy
1
in which 0 and ^' are constants, or functions of at or>^, or of .r and^,
is a derivative equation of the first degree and order. When multi-
plied by dx it become the general differential equation of the first
degree and order - - •
(p dy '\' tj) dx :=i o,
292. Examples of Equations of the First Order and Second
Degree. — ^The equations
(i)'=-. '{%)'-"{%)
+ AT =
are of the second degree and first order. Written differentially,
dj^ = ojfidj^f X dj^ -^ 2y dy dx-^ax dj^ = o,
409
4IO INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII.
In general, the type of an equation of the first order and second
degree is
where 0, tf:, x ^^^ functions of je:,^' or x and^'^ or constants.
293. Equations of the Second Order and First Degree. — Such
equations as
are of the second order and first degree.
294- Solution of a Differential Equation. — ^To solve a given
differential equation
dy
wherey s ^ , is to find the values x and j^ which satisfy the equa-
tion. Thus, if the values of x and j/ which satisfy the equation
fl>{x,y) = o.
satisfy a differential equation F = o, then 0 = o is a solution of
The solution of a given differential equation may be a particular
solution or it may be the ^^^o/ solution. The general solution in-
cludes all the particular solutions. Or the solution may be a singular
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.
Equations of the First Degree and First Order.
295. The simplest t3rpe of an ordinary differential equation of the
first order and degree is
dy^f{x)dx. (I)
Integrating, we obtain the solution
y^F\x)^ c, (2)
where F{x) is a primitive oi/(x) and c is an arbitrary constant. For
a particular assigned value of c^ (2) is a particular solution of (i),
Art. 295.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 4II
and is the equation of a particular curve in a definite position. At
each point of the curve (2),
is the slope, or direction of the curve (2). For different values of
c we have different curves. The ordi nates of any two such curves
differ by a constant. Equation (2) is then the equatiob 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 (i).
296. Every equation of the first order and first degree can be
written
Mdx + JVdy=:o, (i)
where, as has been said before, MsndJVajrt either constants, functions
of jr or^, or functions of x and^'.
297. Soltttioii by Separation of the Variables^ — ^This solution
consists in arranging the equation
Mdx + J\rdy = o, (i)
so that it takes the form
4>(x)dx 4- ip{y)dy = o. (2)
The process by which this is effected is called separaiton 0/ the
variables. When the variables have been thus separated the solution
is obtained by direct integration. Thus, integrating (2),
where c is an arbitrary constant, and is the parameter of the family
of curves representing the solution.
I. Variables Separated hy 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.
1. Find the curre whose slope to the jr-axis is — x/y^ and which passes through
the point 2, 3.
The geometrical conditions give rise to the differential equation
dy X
-;- = , or y dy -\- X dx -=: o,
dx y
The solution of which, obtained by integration, is the family of circles
*«+y = A
The particular curve of the family through 2, 3 is
x«+^= 13.
412 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIIL
2. Find the line whose slope is constant.
dy
^-•sz m gives the family of parallel straight lines ^ = mx 4- ^«
3. Find the curves whose differential equation is
X dy •\- y dx =z o,
llie variables when separated give
• i
dx , dy
X y
.-. logjr + log^' = r, or xy ^ k» '
Otherwise we may write the solution xy zn e'. This is a fumily of hyperbolae
having fqr asymptotes the coordinate axes.
If we observe that xdy -\- y dxis nothing more than d{xy\ the solution xy ■=. c
is obvious.
4. Find the curve whose slope at any point is equ^l to the ordinate at the point
Here -r- = y. .•, ~ = dx,
dx y
Hence 'og>' = x + f , or >» = ^+* =r e^e» = eu^^
which is the exponential fomily of curves.
6. Find the curve whose slope is proportional to the abscissa.
Ans. The family of parabolse^ = ax^ -\- c, in which Cf the constant of integra-
tion, is the parameter.
6. Find the curve whose slope at x, y is equal to xy, Ans, y = <<*•*" •
7. Find the curve whose subtangent is proportional to the abscissa of the point
of contact.
dx dx dy .
Here y -;- =r ax, .«. — = a - gives
^ dy X y ^
log X =. a\ogy -\- c^ or ^« = kx,
8. Find the curve whose subnormal is constant.
dy
y ~ =. a gives y* = 2ax -j- Cy the parabola.
' 9. Find the curve whose subtangent is constant. Af$s. y = ce^ *
10. Find the curve whose subnormal is proportional to the ifth power of the
ordinate. What is the curve when » is 2 ?
11. Find the curve whose normal-length is constant
1 Here the geometrical conditions give the differential equation
'-\^m- ■■■ ^'ww-
Integrating, x — e = — (a* — y*)^, or the fiimily of circles
(x - cY +yt = a\
with radius a, having their centers on the jr-axis.
12. Find the curve in which the perpendicular on the tangent drawn from the
foot of the ordinate of the point of contact is constant and equal to a.
Art. 297.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 4^3
The differential equation of condition is
hiW
The solution is therefore the fiimily of curves
r -f jp = fl log (^ + f^rzrjt ),
When r = o this is the catenary with Oy as axis.
13. Find the curve in which the subtangent is proportional to the subnormal.
14w Determine the curve in which the length of the arc measured £nom a fixed
point to any point P\% proportional to (i) the abscissa, (2) the square of the
abscissa, (3) the square r-xit <rf the abscissa of the point P*
(I). A straight line.
(2). The condition is / = — .
or
a dy sz l/jf* — a* dx.
The solution of this is
r 4- flv = ^x i/x* - a* - ^« log [x + f'x'-a*].
(3). The geometrical condition can be written j = 2 ^ax.
.-. </j=^-^jr. dx* + dy* = ds*=:^ gives
^Jf X •
^ = ^
— -— ajr.
Put X = M* and integrate. The result is the cycloid
tf 4- ^ = |/ar(tf "- x)-\-a sin-«
s
Ex. 14, really leads to a differential equation of the first order and second
degree, which furnishes two solutions which are tlie same.
15. Find the curve in which the polar subnormal is proportional to (i) the radius
vector, (2) to the sine of the vectorial angle, (i). p = cr^. (2). p'=c— a cos 0.
16. Find the curve in which the polar subtangent is proportional to the length
of the radius vector, and also that curve in which the polar subtangent and polar
sub-normal are in constant ratio. Arts, p •=■ cfn$,
17. Determine the curve in which the angle between the radius vector and the
tangent is one half the vectorial angle. Atu, p = c{i — cos 6).
18. Determine the curve such that the area bounded by the axes, the curve, and
any ordinate is proportional to that ordinate.
M
If /I is the area, £1 = ay. . •. dH ^ y dx = a dy, ,-. y = ce^,
19. Determine the curve si)ch that the area bounded by the x axis, the curve,
and two ordinates is proportional to the arc between two ordinates.
Jy
£1 =z as. ,\ y dx z= ads, dx = a
i^y* - a*
414 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIIL
This gives, on integration, the catenary
20. Find the curte 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 jt* + ^* = ^, or (x — y)* — 2c{x + ^) -f- <• = a
21. Solve M dx -\' N dy^ when Mx ± AJ' = o.
(I). Mx-^ Ny ^o gives M/N = — y/x.
Substituting in the equation, _ = — . . •. x z= cy,
X y
(2). Mx — AJ' = o gives M/N = y/x.
dx dy
SubstittttiDg in the equation, 1- -^ = o. . •, xy ^ c,
II. Solution when the EquaHon is homogeneous in x and y, — ^When
the equation
Mdx -{- Ndy = o
is such that M = <p(x, y), N = ^(^, y) are homogeneous functions
of X duAy and of the same degree, the solution can be obtained by
the substitution J/ = zx.
We have
Divide the numerator and denominator by j:*, n being the degree
of 0 or ^;.
••• 1=^+4=-^')-
Hence
dx dz _^
'x'^ z + F(z)^^'
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.
BXAMFLSS.
1, Solve the equation (2jr* —- y''')dy — zxy </x = O.
■D * dz ^z
Put y = «jf. . •. 5 4- or ^- — r-, or
dx 2 — «* .
-dz.
X z^
Integrating,
log JT = ^ 1 — log s.
2
Replacing z by ^/x, we have
X* =y(<: - log;/).
Aet. 298.1 INTEGRATION OF DIFFERENTIAL EQUATIONS. 41S
2. Determine the curve in which the perpendicular from the origin on the
tangent is equal to the abscissa of the point oi contact.
Ans, The circles *• -f- ^' = 2cx.
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 y^ z=. my, .•. (x — my)dx -^ y dy •=. o^ etc.
ax
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 + (jty -f j^^dy = a Ans, xy* = ^{x -f 2y).
7. Solve x^ydxz=t(:i^^y^)dy, Ans. log*?^ = ^.
298. Solution when M and iV are of the First Degree. — The
equation
(<»i-^ + ^^y +^1)^ = («r^ + Ky + C^dy (i)
can always be solved as follows :
Put a: = a:' + ^, y ■=, y -\- k^ where h and k are arbitrary
constants. Then (i) becomes
dy _ a^x' + Ky' + a,h + b^k + c,
dx' a^' + b^y' + a^h + bjc + c,*
!• If aj!>^ ^ ajf^t assign to h, k the values which satisfy
(»)
a,k + 3.i + '^i = o. ^ ^^j
a^h + V + f
Then (2) becomes
dx' ~ a^' + b^ ■
(4)
This is homogeneous and can be solved by § 297.
\ij\x\y) = o is the solution of (4), then /(a: — ^,>^ — ^) = o
is the solution of (i).
IL If a,3, = tfJ,, let -* = ^ = »i.
a 0
Then (i) becomes
dy_ ^ ^i-y + Ky + gi ..
dx m{a^x + bj) + c; ^^^
Put z = a^x + b^y. Then (5) becomes
in which the variables can be readily separated.
41 6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cii. XXXVIII.
EXAMPLES.
i. SoWe (ay - 7jc +7)^ + (7^ -pc+ ^)dy = o.
Ans, (y — x-\- i)\y -j- x — if = c.
2. Solve {zx +y +i)dx + (4:1: -|- 2y - i)dy = o.
Afu, X -{- zy -\- log{2x -}- y — i) =z c.
3. Solve. (7^ + Jp + 2)dx — (3JC -[- 5;/ -j- 6)</k = o.
Am. X + ^ -\- 2 =£{x —y + 2)*.
299. The Exact Differential Equation. — The differential equation
Mdx-\-Ndyz=zo
is said to be ^Xi exact differential equation ^vYitXi it is the immediate result
of differentiating an implicit function /(at^j/) =: o.
In ^t, if
«=/(■*,>') = 0,
then du = -^dx + -^dy = o
gives an exact differential equation.
300. Condition that Jf d::i; + iV^y = o be Exact. — Since J/ must
he the first partial derivative with respect to or, and JVthe first partial
derivative with respect \.oy of some function /(jc,^), then
^ dx' dy'
But since
dy __ ay
dy dx "^ dx 8/
we must have the relation
dM _dN
dy " dx ^ '
existing between J/ and .A^in order that Mdx+Ndy=:o shall be
exact. This condition is also sufficient, and when (i) is satisfied
Mdx -[- Ndy is an exact differential.
For,* let V= J Mdx.
•'• dx ~ ' dy dx " dy "" dx'
dN d'^V
dx dx dy*
• This is due to Professor James McMahon.
Art. 301.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 41?
where the constant of integration <t>'(y) is some function of j' or a
constant independent of x. Therefore
Mdx -\- Ndy = ^^dx + ^Zdy+<f>\y)dy,
an exact differential.
301. Solution of the Exact Equation.
If -5 — = 5— > there exists a function «, of or and^', such that
oy ox
du = Mdx-\'Ndy. (i)
bu
Since M z=z -— , iff contains the derivatives of only those terms in
ox
u which contain x. Integrating (i) with respect to x {y being con-
stant), we have
u=fMdx+<p(j), (2)
where (f>{y) represents the terms in u which do not contain x.
To find <f>{y)f differentiate (2) with respect to^'.
Hence
As was said, <p is independent of x and so also is -^^ as is verified
by differentiating (3) with respect to .a;;
{-1/-}=!^-
= 0.
bx ( by J ) bx by
Integrate (3) with respect toj'.
■ '■ • '^^^ " f 1 ^'hf'^^'^'' I '^ + «.
Therefore the solution of (i) is
u=jMdJc^ J \^N-tjMdx^dy + c = o. (4)
In like manner, working first with ^instead oi My
jNdy+J ^M-^^jNdy^dx + c^o (5)
is also a solution of (i).
4l8 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII.
302. Rule for Solving the Exact Equation.
j Mdx contains all the terms of the primitive containing jc. Also,
since iV— r— / Mdx is independent of x, ^— / j^^at must contain
those terms in JV containing x. Therefore to obtain
integrate only those terms in iV which do not contain x. Hence the
rule. Integrate Mdx as if^' were constant; integrate those terms in
JV dy which do not contain x; equate the sum of these integrals to a
constant.
A like rule follows for effecting (5), § 301.
EXAMPLES.
1. Solve (3Jr« — 4xy — 2y*)dx + (3;'* — 4xy — 2j(^)dy = o.
Here — - = — 4jc — ^y = ---,
dy dx
f^dx = JJ« — 2x^ — 2xy^\ JZy*dy =>^.
Therefore the solution is
j;* — 2j^y — 2xy* -{-y* = c,
2. Solve (jr* + y*){x dx -{-y dy) + x dy ^ y dx = o,
<*"' "4- y* V
Arts. j~^ 4- tan-« — = c,
2 • X •
3. Solve (fl« + Sjtv — 2y*)dx -f (2x—yydy = o.
Am, a^x -f- ^ — 2xy* -f 4^y = ^•
4. Solve (2ax -{- fy + ^)dx -f- (2rv -f 3jr -|- e)dy = o.
Ans. ax* -[- ^^ + CV' + ^ + C>' = '^*
5. Solve (wi dx 4- *'dy) sin {mx + ny) = (» </jr -f- « i/^) cos («r -|- my).
Ans, cos (otjc -f- ^y) + sin (iw: -f- ^) = ^'
6. Solve 2x{x -\- 2y)dx -|- (2x* — ^*)fl{K = o.
Ans, x' + 3^:*^^ — 2^* = ^.
303. Non-Exact Equations of the First Order and Degree. — We
have seen that when a primitive equation /(at, j^) = o is differentiated
there results the exact differential equation ^{x,y,y) = o, writing
y for the derivative of>' with respect to x.
If now between f =■ o and <t> •= o we eliminate any constant
occurring in /"and 0, we get another equation, ^-ix^y^y') = o, which
is a differential equation satisfied at every point ony= o. Therefore
/= o is a primitive of ^ = o. But tp = o will not be an exact
Art. 303.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 419
differential of the primitive/* = o, althoughy= o is a solution of the
differential equation ^ = o.
To fix the ideas, consider the equation
ax -{- by ^ cxy + >t = o. (i)
The exact differential equation of (i) is
(fl -}- cy)dx + (^ + cx)dy = o. (2)
When (2) is integrated the constant of integration restores the
parameter k of the family (i) and (i) is the solution of (2). That is
to say, the family of curves (i) obtained by varying the parameter >(
gives the solution of the exact differential equation (2).
The constant k was eliminated from (2) by the operation of differ-
entiation and restored by the process of integration.
Eliminate a between (i) and (2) by substituting
a + cy = !
X
from (i) in (2). There results the differential equation
dy dx
fy + k x{d + ex) '
1 dx c dx
(3)
or
b x bb + ex
bdy dx cdx
by + k "^ X ex + b'
Integrating and adding the arbitrary constant ^ log c',
log (by + ^) + \og(ex -f ^) — log jc — log c' = o.
•*• (fy + ^)(CX + ^) = ^'Xf
or (kc — c^x ^ l^y ^ be xy -+ kb =z o.
Putting the arbitrary parameter in the form kc ^ c^ = ab, this
equation becomes the original primitive
ax + by -\- exy -f i = o.
This equation with the variable parameter a is the solution ot the
differential equation (3).
The differential equation (3), or
(by + k)dx - x(b + ex)dy = o (4)
is not an exact equation, for
— (by'\-k) = b, _ ( — 3j: —co:^) = — 3 — 2ex.
But (i) is the primitive of (3) as well as of (2).
Again, if we eliminate first b and then e between (i) and (2), we
420 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII.
shall get two other differential equations, neither of which is exact,
but each of which has (i) for solution with variable parameters b and c
respectively.
Observe particularly that if (4) be multiplied by \/3^y it becomes
an exact differential,
=^—^ — dx = dy =0,
^ -^ , -^— . (5)
Since
^ /fy + k\ _ ^ / b + cx\ _ b^
dy\ j(^ )^bx\ X ;~^'
Integrating this exact equation (5) under the rule §302, the
solution is
qx '\-hy -\- cxy -|- ^ = o,
the same equation as (i) with q 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 \/x^.
Such a factor is called an integrating factor of the differential equation
which it renders exact.
The number of integrating Actors for any equation
Mdx-^Ndy:=^o (i)
is infinite. For, let /* be an integrating factor of (1). Then
fA{Mdx -\- Ndy) is an exact differential, say du, and
fjL{M dx '\- N dy) = du.
Multiply both sides of this equation by any integrable function of
«, say /][«),
fji/{u){Mdx + Ndy) =/[u)du. (2)
The second member tyf (2) is an exact differential, and therefore
also is the first. Hence, when ^ is an integrating factor of (i), so
also is /{/(«), where /(tt) is any arbitrary integrable function of u.
In illustration consider the equation
ydx^xdyzzz o.
This is not exact, but when multiplied by either — ^, — , or —,
it becomes exact and has for solution
X
— = constant.
y
Art. 305.] INTEGRATICW OF DIFFERENTIAL EQUATIONS. 421
The general solution of the diflferential equation
consists in finding an integrating factor pi such that
pi(MdX'^Ndy):=zo
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
ail integrating factor for a few important equations will now be given.
305. Rules for Integrating Factors.
I. By Inspection. — ^While the process of finding an integrating
factor by inspection does not, strictly speakings 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.
BXAMPLE8.
1. Solye y dx — xdy -f /(*)</* = a
The last term is exact; its product by any function of x is exact Therefore
any function of x that wiU make^ dx ^ x tfy exact is an integrating &ctor. Such
a foctor is obviously i/x*.
••• ^ + ^^^ = 0^
gives the solution
X
2. Solve y dx •{- lag X dx ^ X efy.
+f^'-='-
Ans* «r+^ + logjc+Ir=0.
3. Solve (I + xy)y <Zr -|. (i — xy)x <^ = o. (Factor i/x^*).
I
Ans. ex = ye*',
4. Integrate :^y^(ay dx -{- 6x dy)= a
Obviously jf*»-«-*y**-« fi is an integrating fitctor, where A is any number.
On multiplying by the ^cfor we get
axAa^xykb dx + bxkaykb~\ dy =r -^d(x^yM) = O,
the solution of which is evident.
5. Integrate
x*yfi{ay dx + 6x ify) + x^tyfii{a^y dx -f d^x dy) = O.
A.
I
422 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIIL
The factors jr*«-«-«^^— «-^, x*i«i-«-*i^*i^i-»-^i
make the expressions
x^yfi{ay dx -\- bx dy) and x^\y^\{ayy dx -f byX dy)
exact differentials respectively, whatever be the values of the arbitrary numbers
i and Jky
Therefore, if k and Ji^ be determined so as to satisfy
Jka ^ I — a = >!«! — I — OTi,
M -^ 1 - fl = k^d^- I - p^,
the factors are identical and these values of k and k^ furnish the integrating factor
of the equation proposed.
6. Solve {y^ — 2yx*)dx + (2J7» — ^)dy = o.
Am, x^\y* — . jt*) = r.
7. Solve the equation
[y + xA^ -^y^)'\dx =[x- yf{x^ + y^)yy, (l)
This is the differential equation of the group or family of rotations. Put
x*-^y* = r«.
Rearranging (i),
ydx-'xdy ■JirAf^)'{x^ + y<^y) = Oi
2(y dx^xdy) +/{r*)dr* = o.
This can be written
{ydx — xdy)—{xdy—ydx) +/(rV'^ = O^
or
An integrating factor is obviously ^ ^ . Whence
X -J" y
y — ^+^^^ = 0.
JT y^ r*
'+7. ' + ?
Integrating,
tan-« - - tan-i ^ + f^^dr* = c.
y X ^ J t*
II. Whenever an integrating /actor exists which is a /unction 0/ x
only or o/y only, it can be/ound.
Making use of the fact that ^ is always a factor of its derivative:
(a). Let » be a function of x.
In ^{Mdx + Ndy) = o,
put M' = e'M, j\r = ^j\r.
Then ?^-^?^. M:-^a^^ + ^?^
^^'^^ ay-^aj/^ aor "■^^^ + '^aJ'
The condition that e* shall be an integrating factor is
dy dx ^ dx*
dM _bN
by bx
or dz = -^ — r^ — tfr.
N
Art. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 423
If, therefore,
dAf dj\r
^y ^^ ^/ \
N
is a function of x only, then
z = j<t>(x)dx.
Hence f' ' * is an integrating factor oi M dx •\- N dy = 0
whenever
~dy' dx (i)
N
is a function, (f>{x), of .r only.
(d). In like manner, letting «* be a function of v only, we find
that e"^ is an integrating factor of
Afdx + J\rdy = o
when
dIV dM
^ ^ (2)
is a function, ^{v), ofy only.
(c). Whenever the expression (i), (2), or (f>{x), i^(y) is constant,
then e* or e'', respectively, is the integrating factor.
£XAHPL£S.
1. One of the most important equations under this head is Leibniti's linear
equation,
i + ^ = «' (,)
where P and Q are functions of x or are constants.
This equation, {fy — Q)dx -f <$^ = o, is such that
N
Therefore it has the integrating factor
J^^{dy -f /> ^) = J'^^Q dx. (2)
Since eJ cfy + e^ Py dx - d\yeJ y,
on integrating (2),
y = rS''*'\jJ''*'Qd.^c\ (3)
This is the solution of the linear equation (i).
434 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII.
2* Bemanlll't BqvAtioii. — The equation known as Bernoulli's
in which P, Q are functions of jt or are constants, reduces to Leibnits's linear equa-
tion. For, multipl/ by ( — if + i)/y", and put v =r .y-»+«. Equation (i) becomes
J + (i-«)/V=(i-ii)(2,
which is linear in v,
3. Solve f\y) -^ -^-PAy) = Ot where P, Q are functions of jr.
Put V = /(^). The equation becomes
A/
which is linear in v.
III. WAen Mx ± iVv ^ o, there are two cases in which the inte-
grating /acior of M dx -|- N dy = o can be assigned.
(i). When M^ iVare homogeneous and of the same degree, then
Mx + Ny ^^ ^^ integrating factor.
(2). When M^ N are such functions as
M = y<p{x xy), N— x^{x X y),
then -57 rr- is an integrating factor.
Mx -^ Ny
Proof: We have the identity
Mdx + Ndy
-\\{Mx-\. Ny)d log (xy) + {Mx - IVy)d log (x/y) ] .
(1). Divide by Mx -\- Ny.
^^^±|^=^log(^j^) + i^^Jlrflog 0).
= y log {xy) + i/ 0)'' log (jj,
if M^ N are homogeneous functions in Xy y and of the same degree.
X
lOff ~"
Since x/y = ^ ^ , this can be written
Mdx + NJy
= ¥ log {xy) + \F {log ^d\og l^y
Mx + Ny
= ^« + \F{v)dv,
where « = log (xy), v = log {x/y).
Art. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 4«S
This case is otherwise solved by the substitution y szzx, see
§ 297» n.
(2). Divide byMx — A/y
If ^ s y(t>{xy)y N = ^^(J^), then
Mx -{' Ny ^ <P(^y) + ^^C-^y)
Mx — iVy ~" 0(jpy) — ^(^)'
. •• ^Mxtliy = M^^)^ log {xy) + id log {x/y),
= i^log •^)<^ Jog :ry + ^ log {x/y),
Writing as before, xv = *'***', » = log .rv, » = logAr/v.
(3). The cases in which Mx ± Ay = o were solved in § 297, I,
Ex. 21.
Solve by integrating iacton the following equations:
^, y dx ^ X ify -^-logx dx = o, (I, Ex. I.)
•Am. ex -\- y + log X -\- i =sa
2. a(jr i^' + 2>' <ir) = xy dy, Ans, a\ogj^=y^c.
6. (jf^* + jcy);' <6f + (jtV* — I) jr <(y = o. -.^iw. y = ^i^.
Ans. *y - — = log fy*.
7. Jc* </jr Mlx^y + 2y*)<^ = o- ^^- *• -f 2y* = tf 4/jc« + ;^*.
Z. i^y -\- y V^)dx — (jr -f jr Vxy)dy = o. Ans, y = or.
9. (Jf* + 7* + 2jc)<& 4- 2^ ^' = a ^iw. jr* -f ^* = f/-*.
10. (3Jf* — y^]dy = 2jry <£r. -^iw. j:* — ^ = fv*.
11. 2XK ^(i' = (jr» -f y >/jf. ^«^. Jf« — y =s rx.
12. {j^y - 2jrK«)^ = (^ - 3Jf*^>». ^^^ |+ log^ = ^•
13. (3«V + 2jry)<£r = (oc* - 2jfy>^. ^iw. jr^ + J^* = <7-
14. (y^ + 2y)^ H- (^ + 2/* — 4^^)^ = o- -^'w- *r +>'' + ^/y* = ^•
15. (24:^ - 3r*)^ + (3Jf* + *-«r*)^ = o-
16. (y + 2jr>yjr + (2jr» - xy)dy = o. ^»x. 6 ^ = 4f"*^i + c.
426 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIIL
dy X I
17. x-i- — ay =z X 4- 1, Ans, y = f- cx^,
dx '^ \ — a a
18. (I -+- ^)dy = (« 4- xy)dx. Am, ^ = wur -|- ^ 4/1 -f x*.
19. ^H r^= {x — I) sec^. -^#ir. smy = , . , .
21. ^ + ^ = ^. ^«J. ^ = jr + i + fr»*.
22. ^ = » - + ^j:«. ^w. y = j:*(^ + ^).
306. Solution by Differentiatloii. — A number of equations can
be solved, by means of differentiation as equations of the first order
and degree.
EXAMPLES.
dy
1. Let p zzz -J- . Let the differential equation be
•*=/(/). (I)
Differentiating with respect to /,
Since dy '=.pdx^ this gives the equation
dy^f\P)pdp.
... y^Jf'(p)pdp + c. (2)
The elimination of/ between (i) and (2) gives the solution.
2. In like manner, if the differential equation is
y =An (I)
on differentiation we have
dy=/'{p)dp.
... p dx =z /'{p) dp,
ot dx^^Mdp.
... x^f-C^dp+c. (a)
The elimination of/ between (i) and (2) is the general solution of (i).
3. jf = / + log/.
Ans. x+i = ± V2y-\-c + log ( - i ± V^J^ + 0-
4. ^/* = 1+/*.
Ans, e^y + 2cxtf + ^ = o.
B. ;/ = «/ + ^/'.
i^«j. jr± V^' 4- 4^ = fl log (tf ± i^^*~+~45') -I- ^.
Art. 306.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 4^7
EXERCISES.
I. 3^ Unydx-^ (I - ^) sec*;/ ^y = o. Ans. tan;/ = i^i - ^)».
^>«J. log \2(Vc - i)« -f (3y - 5)'} = i^ tan-' ■ 3^ . ^ +^-
4. (;c»/* - 2/wxy«)ii:r + 2ifMrV dy = o. v4im. x«^ + my^ = ^a:».
5. y{2xy ^ex)dx-e-dy^o. Ans, x^ -{• e' = cy,
6. dy + iy-- ^')dx = o. Ans. yf = x + c,
7. cos'a:^;/ + C;' - UTix)dx = 0. ^fw. JK - ^^•"^^ = tan :r - I.
8. (x + i)dy =f^dx + e^x^ i)-+«^. Ans, y = (f' ^ c)(x + l)«.
^ sin j: + cos x
%. dy = {by^a sin x)i&. Ans. y ^ c^» - a ^-q— j,
II. jf ^;. = f»y ^ + ^^+'^. ^i«. ^^ = *"(^ + 0-
12. ^^^ = (;^ + i)^ ^. ^'"- ^ = '^'^ - '•
13. C08:r^j|/+>8inari^ir = ^. ^nj. ;r = sin Jf -f^ cos x.
14. ;r(l - ^)^ -h (2^ - i)y = «*•. ^«J. > = /MT + ^* l/nn?.
15. (X + y^^dy = «« ^. ^'«'- —r^^'' "^•
X — ;/ + a ^ -hy
16. (x ->')«*/ = «» ^■'- ^'"- '^ x-y-a = * "V"*
17. x« 4^ + (^^ - 2*y - ^)^ = o- ^'"- ;' = •«' \i +«'/ •
19. (*• + jv* - «V ^x + (*• - ^'^ - ^)>' ^y = o-
20. ^;' = (x!y» - 1)^ ^- ^'*'- •^'^** + ' + ^''*^ = *•
21. a^TK ^ + (JK* - ^)^;' = o. Am. >« + x« = CK-
22. (X +y)dy + (x ^y^dx = o. Am. log 4/4:* +y + tan-«- = c
23. (jcV + ^y + Jry + i)y ^ + (^ - •*^>'* - ^ + ^^^ ^^ = ^•
Am, j^V' — ^xj/ log ry = !•
24. ^4-->= -. ^"^^ J^ = «* + ^-
CHAPTER XXXIX.
EXAMPLES OF EQUATIONS OF THE FIRST ORDER AND
SECOND DEGREE.
307. The equation of the First order and Second degree is a
quadratic equation in -j- of the form
(l)v ^ %
+ B = o, (I)
where A, Bsxe, in general^ functions of jrand^^.
We shall represent ~ by p. Equation (i) can be written s3miboli-
ax
cally
308. There are three general methods which should be made use
of in solving (i):
(i). Solve for y \ {2). Solve /or x \ {j). Solve /or p.
309. Equations Solvable for j^. — If (2) can be solved for y, the
equation becomes
y^F\x,p). (I)
Differentiate with respect to x*
dF BF dp
dp
This equation (2) is of the first order in -j-.
The elimination of ^between (i) and the solution of (2) furnishes
the solution of (i). The elimination of/ is frequently inconvenient
or impracticable. When this is the case, the expression oix and^ in
terms of the third variable p is regarded as the solution.
EXAMPLES.
1. Solve/ + 2xy = *» +y. (i)
... >/ = 4r-h V?:
Differentiating,
428
Art. 309.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 4^9
or i6f=— — ^^ . (2)
or p^ = — -^- • (3)
Eliminating/, we have for the solution
2. Solve X —yp= apK (l)
Differentiate ^ = — -^ , with respect to jr, and put the result in the form
/
Jx I ap
dp Pil-P') ^l-p"
Solving this linear equation,
P
X ^
Substituting in (i),
{c -{- a sin— »/). (2)
y=^ap+ (r + tf sin-»/). (3)
The values of or, y expressed in terms of the third variable/ in (2), (3) furnish
the solution of (i).
3. Claintat's Equatioii. — The important equation, known as Clairaut's,
j^=/x +/(/), (I)
can be solved in this manner.
Differentiate with respect to jr.
or, [x +/'(/)] ^ = o. (2)
The equation (2) is satisfied by either
X +/'(/) = 0, or ^£=0.
The solution of (i) is obtained by eliminating p between either of these equa-
tions and (i).
Jp
~j- z= o gives / = f , constant.
ax
Therefore one solution is
y = cx +/(r), (3)
which is the family of straight lines with parameter c.
The second solution is the result of eliminating / between
y=px+/ip), )
and o = X -f /'(/). f ^^'
The second of these equations is the derivative of the first with respect to/;
X and y being regarded as constants, / as a variable paramc^ter. This result is
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 singular solution of (I).
Thus the general solution of Clairaut's equation (i) 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. LagnLn^e's Equation. — To integrate
y = xAP) + ^/)- (0
Differentiating with respect to x and rearranging,
^'^AW^'^ AP)-p-°- <^
This is a linear equation in x and can be solved by g 305, II, Ex. i.
Eliminating/ between (i) and the solution of (2), the solution of (i) is obtained.
Otherwise x and y are obtained in terms of the third variable /.
6. Solve ^ = (I -f /)jr + /«.
dx
Differentiating, -j — \- x z=. — 2/.
dp
Solving this linear equation,
jr = 2(1 — /) + ce~P\
6. Solve x*{y —/or) = yp^.
Put X* =s «, ^' = V'
dv . /dv\*
••• " = "55,+ U)'
which is Clairaut's form.
.'. V z=z eu -\' c\ Hence y^ = ex'* 4- ^.
310. Equations Solvable for x. — When this is the case
becomes
^ = ^y.py (0
Differentiate with respect toy,
1 dF QF dp , .
P dy ^ dp dy ^ '
This is of the first order in -f . The elimination ofp between (i)
ay
and the integral of (2), or the expression of J? and y in terms of/,
furnishes the solution of (i).
EXAMPLES.
1. Solve X =y -}-pK
— =1 + 2/ -7-, or dy = — ^'^ '^
p ' '^ dy' -^ p-i
.-. >^ = r- [/» + 2/ + 2log(/- I)], jf = f— [2/ + 2log(/- I)].
2. X — y -{- \og p^. Arts, y z=: c ^ a log(/ — i), jc = r -+- a log —^- — .
/ — I
3. Solve p'^y -f 2/jr = y. Ans, y* = 2cx -f c*-
Art. 313.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 43^
31 !• Equations Solvable for /• — ^The equation /\x^ y, p) =0 is
a quadratic in/.
If this can be solved in a suitable form for integration for/, it
becomes
Each of the equations
P = 0(^, y) and p = tl){x, y)
dy
is of the first order and degree in -f^, and their solutions are solu-
dx
lions of (i).
Such solutions have already been discussed.
EZAMPLSS.
1. Solve /» - {x-^-y^p + xy = o.
(/-^)(i>-->') = o
dy
gives dy — X dx :=zOy and i/r = o.
y •
2. /* — 5^ + 6 = o. Ans, y = 2x -^ c, y = yx •]- c.
312. In particular, if /(^, y, P) =0 does not contain x or does
not contain y, corresponding simplifications of the above processes
apply, see § 306.
312. Equations Homogeneous in x and y.— When the equation
/{x,y, /) = o is homogeneous in x andj', it can be written
(£' i) = "•
(0
(i). Solve, if possible, for/ and proceed as in § 297, IL
(2). Solve ioiy/x. Then the equation becomes
y^^Apy (2)
Differentiate (2) with respect to x and rearrange.
. dx_/\P)_^
" ^"p-i/py
EXAMPLES.
1. Solve xpi* — 2yp -f <jjr = o. Ans, ley = (*j^ -f- 0.
2. Solve y = ^'Z* + 2/jr. ^«j. ^« = 2cx -|- r*.
3. jf"/* — 2xyp — 3;'« = o. . <^«j. cy — ^^ xy — c.
Orthogonal Trajectories.
313. A curve which cuts a family of curves at a constant angle is
called a trajectory of the family. We shall be concerned here only
with orthogonal trajectories. If each member of a family of curves
or
432 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX.
cuts each member of a 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' = — I exists between their slopes /, /'.
314. To Find the Orthogonal Trajectories of a given Family of
Curves.
Let <l>(x,y,a)^o (i)
be the equation of a &mily of curves having for arbitrary parameter a.
Let /{x, y,p)—o (2)
be the differential equation of the family (i), obtained by the elimina-
tion of the parameter a.
The differential equation
is the differential equation of a family of curves, each member of
which cuts each member of (i) at right angles. Therefore the general
integral of (3),
H^^y^ ^) =o> (4)
is the equation of the family of orthogonal trajectories of (i).
BXAMPLBS.
1. Find the orthogonal trajectories of the family of parabolas y^ = ^x»
Differentiating and eliminating /z, the differential equation of the family is
dx zx'
The difTerential equation of the orthogonal trajectories is
dx _ y
dy ~ 2x'
The integral of which is jt* -j- ^» = ^, a family of ellipses.
2. Find the orthogonal trajectories of the hyperbolae xy = «■.
The differential equation is ^ + ;r/ = o. The differential equation of the
orthogonal trajectories is
dx
giving the hyperbolae j:* — >'• = <^ for trajectories.
3. Find the orthogonal trajectories of ^^ = mx,
4. Show that jr* -f ^' — ^^ = 0 is orthogonal to the family
^* = Zdx — jc*.
Art. 316 ] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 433
5. Find the orthogonal system of _ -l-*^ = i, in which b is the parameter.
Am, Jf* + ^' = a' log jf* -f r.
6. Find the system of curves cutting jr* -|- l^y* = ^«* at right angles, a
being the parameter of the faunily. Ans» yc = jr^'.
The Singular Solution.
314. We have seen in the rase 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 singular
solution, we now propose to notice more generally.
315. Singular Solution from the General Solution.
Let 4>{x,y, c) z= o (i)
be the general solution of the differential equation
/T^.>'>/) = o. (2)
A solution of the differential equation (i) has been defined to be
an equation (i) in x, j^ such that at any point x, y satisfying the
dy
equation (i) the x^y^ and/ = ^derived from this relation satisfies (2).
The general solution (i) being the integral of (2) satisfies the con-
dition for a solution. Also, however, the envelope of the system of
curves (i) is a curve such that at any point on it the x^ >', p of the
envelope is the same as the .r, ^, / of a point on some one of the sys-
tem of curves (i), and must therefore satisfy (2). Consequently the
envelope of the family (i) is a solution of (2).
This is a singular solution. It is not included in the general
solution, and cannot be derived from it by assigning a particular
value to the parameter <:.
We may then find the singular solution of a differential equation
(2) by finding the envelope of the family (i) representing the general
solution of (2).
Thus the singular solution of (2) is contained in
which results from the elimination of c between
0(jr, y^c) —o and 0^(^, y^ c) — 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
434 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX.
be regarded as a family of curves having the variable parameter /.
Find the envelope
X{^> y) = o (2)
of (i), as the result of eliminating p between
A^>J'yP) = ^ and /i(x, y, p) = o.
Since at any x,j^ satisfying (2) the x,y, -j-oi (2) is the same as
dy
the x^y, -T— of a point on (i), the equation (2) must contain a solu-
tion of (i).
1. Find the general and singular solutions of /' -j- x/ = y.
This is Clairaut's form, and the general solution can be written immediately by
putting/ = const.
However, independently, we have on differentiation
o = (;r + 2/) ^.
dp
J~ =: o gives p •=: Cf and y zn ex -\- ^ far the general solution. Differentiating
with respect to c and eliminating r, we find the singular solution \y -\- x* ^= o.
Integrating the other factor, x -|- 2/ = o, or eliminating/ between this and the
differential equation, the same singular solution is found.
2. Find the general and singular solutions of the equation y •=. px -\- a ^i -f-/*.
Ans, jT* -|- ^* = a*.
3. Find the singular solution of x^p^ — ycyp -|- ^y* -{- j^ z=i o.
Ans, x^(y* - 4jr*) = o.
317. The Discriminant Equation.— The discriminant of a func-
tion F(x) is the simplest equation between the coefficients or constants
in F(x) which expresses the condition that F has a double root. If
F has two equal roots, equal to a, then
F{x) = (.V - ayct>{x),
where 0 is some function which does not vanish when x =z a. Hence,
differentiating and putting x = a, we have the conditions for a double
root at <i,
F{a) = o, F'{a) = o, r\a) ^ o.
Eliminating a between F{a) = o, F\a) =1 o, or, what is the same
thing, eliminating x between F(x) = o, F'{x) = o, we obtain the
discriminant relation between the coefficients, the condition that
F{x) shall have a double root.
318. c-discriminant and /-discriminant.
Let (f>{x,y, c) = o be the general solution of the differential
equation /■(:<;, ^, p) = o.
Art. 3*0.] EQUATIONS OF nRST ORDER AND SECOND DECREE. 435
(i). Hie equation ^{x, y) = o which results from the elimination
of c between the equations
^C-J^i ^1 <^) = o ai"! 0((J^. ^. c) = o
is called the c-discriminant, and expresses the condition that the equa-
tion 0 = o, in c, shall have equal roots.
(2). The equation xt^/y) ~ ° which results from the elimination
o(p between the equations
A^' y,P)=o and /^{x, y,p) = o
is called the ^-discriminant. It expresses the condition that the equa-
tion/= o, in/, shall have equal roots.
319. odiBcriminant contains Envelope, Node-locus, Casp-
locus. — The c-discriminant is the locus of the ultimate intersections
of consecutive curves of the family <}>(x, y, c) = o.
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
Fig. 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-
liorhood 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 maybe imaginary.
We may expect lo find the envelope occurring once, the node-locus
twice, the cusp-locus three times as factors in the c- discriminant.
330. /^scriminant contains Envelope, Cusp-I^KUS, Tac-Locus.
— If the curve family y( a, j', /) — o has a cusp, then for |)oints along
the cusp-locus the equation vanishes for two equal values o( p, as it
does also for points along the envelope. But, in general, the ~- of the
cusp-locus is not the same as the p of the curve family and therefore
does not satisfy the dilferential equation.
43^ INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX.
Again, at a point at which non-consecutive members of the curve
family 0 = o are tangent the or, y^ p of the point satisfies the equa-
tion /■ = o. 'i*he locus of such points is called the tac-locus. The
dx
of the tac-locus is not the same as that of the curve family 0=0,
and the tac-locus therefore is not a solution oif = o.
321. It has been shown by Professor Hill (Proc. Lond. Math. Soc,
Vol. XIX, pp. 561) that, in general, the
C the envelope once^
c-discriminant contains \ the node-locus twice^
(the cusp -locus three limes ^
( the envelope once^
/-discriminant contains \ the cusp-locus once^
(the tac-locus twice^
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.
£XAMPLS8.
1. x^ — (x — <z)' = o has the general solution
y -\- c =. \x^ — 2a«*,
or 9(^ -h ^)' = \x{x - 3«)».
The /-discriminant condition is x{x — a)* = o, the
^-discriminant condition is x{^x — 3a)* =■ o. x = o occurs
once in each, it also satisfies the differential equation and
is the singular solution or envelope, x •=. a occurs twice
in the /-discriminant and does not occur in the c dis-
criminant. ;i: = a is therefore the tac-locus. jr = 3^
occurs twice in the c- and does not occur in the/ -dis-
criminant, j: = 3a is therefore a node locus.
2. Show that (y -\- cf = jr'is the general solution of
4^* = 9x, and i* = o is a cusp- locus. There is no
singular solution.
3. Solve and investigate the discriminants in
/« + 2j/ -y.
General solution (z** -|- 3xy + c)^ = 4(ji:' -f- y^. No
singular solution. Cusp-locus x^ -\-y = 0.
4. In Sfl/* = ^^y^ show that the general solution is
ay"^ = (jc — r)', singular solution ^ = o, cusp-locus^'* = o.
5. Find the general and sing^ular solution of ^ = x/ — /*.
Ans, y ■= ex — <^^ jc* = ^i^.
Fig. 156.
*Proc. Lond. Math. Soc., Vol. XXII, p. 216. Prof. M. J. M. Hill, ''On
node- and cusp-loci which are also envelopes.^*
Art. 321.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 437
BZSRCI8SS.
Find the general solutions of the folbwing equations.
1. /* = ax^, Ans, 2${y + ^)* = 4Ar».
2. /• = ax*. Am. 343(;' + ^f = 2jaxf.
3. /•(* + 2y) 4- 3^(jr +jf) + piy + 2jc) = a Factor and solve.
Ans.y^c, jf-|-^=r, xy '\- x* -{- y^ = c.
4. /• - 7/-f 12 = o. Ant. y sz4x-\' c. ,v = 34: -f r.
5. xp* — 2>'/ -|- ii4f = o, ^nx. 2cy = r*jr* -j- a.
6. ^/* -|- 2jp;^ = y. Ans. y^ = 2^ -f ^.
7. jry — 2jry/ -|. 2^ — 4f« =r a -rfiw. 8in-« — = log ex.
9. xy\^ + 2) = 2/r» + *». -rfiM. (j:» -^ + ^X^ -y 4- «*) = a
10. J' + /x = xy . -^^iw. xy = r + ^jf.
11. ay^ + (24r — ^)/ — >' = o. ^iw. fl^ -f ^2jr — ^) — ^« = o.
^»x. « + .= ' ^^P
12. ^ — /jr = 4/1 -f p^f(j^ + y). Change to polar coordinates.
u4#M. tan-' — -f f = Ten- *2a |/jr* -+- >'*.
14. (jf^ — J')* = /• — 2 ^ -f- I. ^iM. sin-« i = sec-ijf + c,
15. 2^y* - 2J9'/ -f 4y« - x« = o. Put Jt« — ay* =s »«.
^iw. 3(jr* + y') ± 4£x -|- <• = a
16. {^+y*K^ +/)• - 2(x +^Ki +/)(^ +>/) + (jf +-K/)« =0.
Ams. x* +y* — 2c{x 4- ^) -f ^^ = a
17. X H- ^ ^ = a. Ans. (y + o« -f (* - «)» = i.
18. >' = /Jr +/ — /". -^«J. J' = fx + ^ — A
19. >'* - 2pxy — I = /«(l — «^ -^iw. (^^ - rx)« = I + ^.
20. y = 2/JC 4. j^y. Put y s= f . Ans, y^ = cx-\- |^.
21. **(;^ - px) = ^/». ^«J. ^« = «« 4- A
22. (/Jf -^'K/^y + ■*) = >**/. ^~. y - r^ = - ~p^,
tZ. y =ixp + J^b* -f ««/». ^»J. >' = fx + V^ 4- fl»<«,
singular solution 3^/a* -f- ^/^ = !•
24. ^ = P{x — ^) + «//, singular solution, y^ = 4ii(x — S).
29. (^ — ^){p^P — «) = »wf/. ^w. (>' — cx)(me — ») = imir,
singular solution, (x/m)^ ± {y/n)^ z= i.
29. /» - 2xyp + (I + x>}^ = I. Ans. (y - rjr »« = i - <*,
singular solution, ^' -> jH = i.
43^ INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX.
27. /> - ^xyp + 8;/« = a Ans. y := c{x - c)\
singular solution, ^^y s 4:1:*.
28. Find the orthogonal trajectories, X being the variable parameter, of the
following curve families:
(I). ^+ ^ = '• -^'"- ^ +>* = ^' log^ + ^.
(2). jc^ + ff^y* = »»U«. Aus. y r= fjr**".
X^ yt yS yl
<5>- «« + x« + F ="• '*"'• :?^r3 - ^ = *•
29. Find the orthogonal trajectories of the circles which pass through two fixed
points. Ans, A system of circles.
30. Find the orthogonal trajectories of the parabolse of the Mth degree
an-xy — jpti, Ans, ny* -}- jc* = <•.
81. Find the orthogonal trajectories of the confocal and coaxial parabolse
y* sr 4X(x 4- A). Ans* Self-orthogonal.
32. Find the ortho-trajectories of the ellipses jc'/a' -{-y*/^ = XK
Ans, y^* = cx^\
33. Show that if
is the differential equation of the family of polar curves 0(p, 6^ c) = o, then
/(a9.-p'|) =
is the differential equation oi the orthogonal system.
34b Find the orthogonal trajectories of p s a(i — cos 0).
Ans, /9 IS ^ I 4- cos 0).
35. Also the ortho-trajectories of—
(I), pi sin n$ = a^, Ans, p** 00s n$ = c*,
(2). p = log tan $ -^•a. Ans, 2/p = sin'O 4~ ^*
V
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 \nx,y, p, g,
ify dp <Py
where / s -r- » ^ s 3-= -m , and in the equation g occurs only
ax ax 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 hdving 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.
Let 0(J^,^, <?i, 0 = 0 (0
be an equation in a:, ^ and two arbitrary constants {:,, c,. Differ-
entiating (1) twice with respect to jit, there results
a«0
bx^
BY dy &y (dyy d/d^ _
"^ dx dydx "*■ dx^ \dx) "^ by dx^'^ ^^'
Between these three equations can be eliminated the two arbitrary
parameters c^, c,. The result is the differential equation of the
second order,
A^f y^ A 9) = o.
439
440 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XL.
BXAMPLB.
The simplest equation of the second order is
Here the integrations are immediately effected.
^y t ,
^1 being the first constant of integration. Integrating again, the general solution is
y = i^** + CyK -f ^,.
The two arbitrary parameters c^^ c^ giving a doubly infinite system of parabolse.
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,
i-/l-^>^) = o; 2. /|;>, ^) = o; 3-/(A^) = o>
4. A^* A 2) = o; s- Ay^ py 9) = o-
324. Formyi[j;, g) = o. — TTiis being of the first degree in g,
The differentials involved are exact, and it is only a question of
integrating twice. The solution is
. •. y ^jdx jF{x)dx + c^x + c,.
Ex. q = xe*. Ans, ^ = (x — 2)e» -f- c^x + r,.
325. Form _/(>', q) = o. — Here
Put , = /. inen -j-^ = -— = v- -r- = ^ -r»
dx dx^ dx dx dy dy
The equation becomes
pdp^F\y)dy.
Art. 326.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 441
Integrating,
dy
dx =
The integral of this gives the solution.
1. Solve g = a^.
XXAMPUS.
2JJ\y)cix ss flV- Put ^1 = tf«A
dx =:
Hence ax = log(^ + f>* + c) + c^
Show that this can be transfbnned into
y = c^'e^ + f,'<-««.
dy
Multiply the g^ven differential equation by 2 -3-.
aJT
Hence the first integral is, as before,
2. Solve ^ + a^ = o.
Here ^jj\y)dy = - ay. Put ^j s <i»A
.'. aax = — ^
y
Hence <ur + f , = sin-« -,
or ^ = f sin (ax + <:,),
= ^j sin ox -f i, cos ojr.
Multiply the differential equation by 2/ and obtain the first integral directly as
in Ex. I.
Examples I and 2 are important in Mechanics.
3. Solve q |/fly = I. Afu, yc = 2tf*(^* — a^iX^* + O* + ^r
326. Form/(/, ^) = o.
H-e ^ =/•(!), or 1 = ^.).
... x=f
442 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XL.
This is an equation of the first order, the solution of which is that
of the required equation.
Integrating f^^dp -f- adx^ we have for the first integral
.-. ^ = log (<!* + <) + <',
or €r '=z c^ -\' c^,
2. Solve a ^ = ^. Am. y = c/^-^ r,. (
8. ^ = /* -^ X. Atis. r-J' = ^, cos (x + rj,
4. ^ -f /• -f I = o. Ans, ^ = log cos (x — ^i) + ^,.
327. Form /{Xy py ^) = o. — Such equations are reduced to the
first order in x and/ by the substitution ^ = ^.
.•• /(^>A^)s/(^,A^) =
BXAMPLS8.
1. (I + *^)^ +jr^-|-AiP = o is equivalent to
The first integral is
f/i + x*
The second integration gives
^ = f, — ojf -f ^1 log (x H- i/i + *•).
2. (I + jf«)^ -f/« + I = o. ^iM. jv = ^iJP+ (V + 0 log (* - ^1) +*^r
328. Form /{y, p, q) = o.
__d*y ^ dp ^ dy dp _^dp
~d[r*"" dx "^ dx dy ""^ 4^'
Substituting for ^, the equation is reduced to the first order in^^
and/.
tfjp* ax a '- y
^y%'^ (s)*= '• ^'"' y = x« + ^iJ: + ^,.
3. ^^ — /* = ^' log^. -<4#w. log^ = Cye» + f,r-«.
Art. 329.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 443
329. Solution of the Linear Equation
^+A^+By = o, (!)
in which A, B are constants.
'I'he solution of this equation is suggested by the solution of the
corresponding equation of the first order
5^ + ^-^ = °'
which gives — = — <idx, the solution of which is j/ = ce~**.
If we try_y = tf** in (i), we have
-^ + A -^ + Bi-" s {m* + Am + B)ir'. (2)
I. Boo/s of the Auxiliary Equation Real and Unequal, — The func-
tion (2) vanishes if m be one of the roots of the auxiliary equation
n^ + Am + -5 s (« — *«r,)(« — «,) r= o. (3)
Hence y = ^"•i-'is a solution. Also, y = c^e^^* is a solution for
any arbitrary constant c^ In like manner y = cj^^ is a solution.
The sum of these two,
y = c^e^x' + CJ^^, (4)
is also a solution, and is the general solution of (i) since it contains
two independent arbitrary constants, c^ and r,.
II. Roots of the Auxiliary Equation Real and Equal, — If m^ = m,,
the solution (4) fails to give the general solution, since then
J' = (^1 + ^t)^^
and ^1 + ^, = c' is only one arbitrary parameter.
The solution in this case is immediately discovered on differentiat-
ing (2) with respect to m. For then
d^x^"^* dxe"^*
-^^^ + ^--^+ i?^ef-«= (3« + ^)^ + («»+^«+ B)x^.
lfM = pi is the double root of (3), then (3) and its derivative
vanish when m = jj. Consequently y = xei^-^ is a solution, and
also is ^ = cxef^-*. Hence the sum of the two solutions c'e*^ and cxe*^
is the general solution of (i) when /< is a double root of (3), or
y = ^^(c' + cx). (5)
III. Roots of the Auxiliary Equation Imaginary. — When the roots
of (3) are imaginary and of the forms
m^= a-\- idy m^^ a -- id,
444 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XL.
where i s |/— i, these roots may be used to find the solution. For
(4) becomes
We have by Demoivre's formula
^* = cos bx + i sin hx^
er^* = cos bx — 1 sin bx.
Therefore the solution is
y = ^{(^1 + ^1) cos ^^ + (S^ - O » sin bx\,
= ^(^1 cos ^jr 4" ^1 sin bx), (6)
where ^^ = c^ + c,, ^, = (c^ — c,)i'. If the arbitrary constants
c^ and c, be assumed conjugate imaginaries, the constants k^ and k^ are
real.
By writing tan a = kjk^, or cot y^ = ^^i,, the solution (6)
may be written resp)ectively
y = c'tf" %m{bx + a),
= c''^ cos(^ar — ft). (7)
BZAMPLBS.
1. Solve ^ — / = 2y.
The auxiliary equation is
«• — iw — 2 a (»i -|- iX«w — 2) =s a
The general solution is therefore y = c^e-* -\- c/^,
2. If q - 2p -j-y = 0, (« - i)« = o, .-. ;^ = ^(fj + <'^).
3. Solve ^ + 3/ = 54y.
«' + a^w - 54 = (« - 6)(»f + 9).
4. Solve ^ + 8/ -f 25>^ = o.
#w* 4- 8»i -j- 25 = o gives iw = — 4 ± 3 f^— I.
. •. y =r t-A*{k^ cos 3JC 4- ^, sin 3jf ).
330. Solution of the Equation
^^+A.%+By=o, (I)
Ay B being constants.
Put j: = e*, then » = log x. Also,
<^ _dydz _!</>'. ^?V __ I /^ ^y\
dx ~' dzdx ~~ X dz ' dx^ "~ ;r^ \d^ dzj*
On substitution, equation (i) becomes
which is the form solved in § 329.
Art. 331.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE, 445
Ex. Solve jc^q — xp -{-y = o.
The equation transforms into
d^y dy ,
... y z= d^{c^ + c^) = x{€^ + c^ 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 oe 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.
BXSRCISES.
1. -£^ =z fl«jc + ^. Put fl«:r +^«y = «, etc.
Am, a^x -\- b^y = c^^» + c^».
d^y
2. -— = a^x — ^. Am. a^x — b^ = f, sin ^jt -|- r, cos bx.
V'2^y -+- C^ -I- f 1 C^ — X
or 2^ = c^ sec*(^ijr -|- <*), according as the first constant of integration is
+ c^, o, or — rj«.
4. xq +/ = O. Am, ^ = fj log jr -f Cy
B. ^ = xp. Am. y = r, / e^^^dx + c^
6. ^q — 2y. Put « = 2y/x^. . •. xy = c^ + ^i-
7. ^ 4- I2y = yp. Am. y = c^/i" + c^.
8. 3(^ + J') = ^9P' ^^' y = V^ + ^j^-
9. y -h 4^ = ^. Am. y^ :=^e^€* VI + c^-» Vf.
10. ab{y '\-q) = {a* + l^)p. Am. y = V* + V*-
11. ^ = 4^. Am. y = c,e2x J^ c^2x + Cy
446 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XL.
12. ^ — ^ + 13^ = a Ans, y ^r^ {c^wi 2x -\- c^ cos 2xys^,
13. ^ — 2ap + ^ = a Ans. According as a > or < ^,
y = /-»(^j4f**'i^^^ + ^^-♦'i*^:^), or^(<:i sinx 4^^ - a « -f- ^,cos x |/^ - ««)•
14. ^ - 4^/ + (<i« 4.3«)«y = a
-rfw. ^ = <*^{^i sin (tf» — ^)r -f r, cos(tf» — ^)jr j.
15. ^ — p log tf* -H [1 + (log ayiy =r o. -«4iw. ^ =s fl«(fj simr -f ^t co*-')*
16. ^ — 2tf/ -f fl^ = o. ^ifj . ^ = *»*(fj -f. r^).
17. ^ = o. ^«w. ^^ = rj + f^.
18. ^ = 4^. Ans. y =s ^i^ + ^, + V-
19. J^q — Jt^ = 3^. ^nx. xy = rjjc* + <■,.
20. (a + ^x)»^ + ^tf + bx)p + 3^ =r O.
^iM. y — Ci sin log (<i + ^Jp) + ^2 cos log (a -f" ^*)*
21. j: ^4- = 2. -«4iM. ^ = f J -j- f ^ -f f^ -|- JK* log jr.
22. -^^ r= sin'jc. A;is. y ^= c^-{- c^ -{- c^ 4- Jcos x — /^cos* jr.
23. i^^ = fl. ^«J. (^jX -\- c^f — c^'* — fl.
24. a^q^ = I +/'. Ans. 2y/a = ^,^ -f- c^-U « + r,,
25. aY = (I +/*)*. ^«J. (x + ^1)* + (^ + ^,)* = a*.
26. (I — Jr*)^ — jc^ = 2. ^»x. >' = ^1 sin-«jr -[- (sin-»jr)* + ^r
27. yq -{-p^ =z I. ^/w. ^* = jr» -|- r jjr -j- r,.
28. (I - log^)^^ 4- (I 4- log>')/>' = o. ^«j. {c^x 4- ^,)(log^ _ I) = I.
29. yq —p^ =y* \ogy. Ans. logy = c^e* + r,^*.
30. (/ - Jr^)' = I -f ^'. ^»J. ^ = i^iJ:' + Jr 4/1 + ^ + ^•
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, x* = 4c{y — c).
34. Determine the curve in which the normal is one half the radius of curva.
ture, and in the same direction.
The cycloid jr -|- ^ sin-« ^ + r ^ry — _y* = o.
35. Find the locus of the focus of the parabola^' = ^x 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 A describes a straight line with uniform velocity; it is required to find
the curve described by the point B^ the motion of which is always directed toward
A and the velocity uniform.
Take the path ciA for^-axis. The tangent intercept on the>^-axis is^ — xp.
By hypothesis the change of this is proportional to the change of arc-length.
log jc« + log (/ + |/i-h/*) -H log rj = o,
2y = r, — c. f.-» .
'^ * *»»-f-i *iw— I
The curve is algebraic, except when m = i, then we have to substitute log x
for — jr-««»+«/(»f — I),
APPENDIX.
SUPPLEMENTARY NOTES.
449
APPENDIX.
NOTE 1.
Supplementing § 30.
Weierstrass's Example of a Continuous Function which has
nowhere a Determinate Derivative.*
The function
/[x) = -2" 3* cos (a^TTx),
o
in which x is real, a an odd ]X)sitive integer, d a positive constant less
than I , is a continuous function which has for no value of a: a deter-
minate derivative, if o^ > i + |^.
Whatever assigned value x may have, we can always assign an
integer ja corresponding to an arbitrarily chosen integer m, for which
Put ^«+, = tf""-^ — f^9 and let
and x' <,x < x".
The integer m can be chosen so great that or' and x'^ shall differ
from A' by as small a number as we choose.
We have
A^) -A^)
to
X* — X
if=o
m — i
_ V^in^os (a"7rjr') — cos (c^nx)
_ ST^ . cos (<i";rA:') — cos {a^nx)
ar(x' — x)
n = o
00
V^^^^ cos (a"'+";r;c') — cos (<i'"+";rjf)
/ ^ •^ — •**
11=0
•Taken from Harkness and Morlcy, Theory of Functions.
451
45 2 APPENDIX.
Since
COS (a^TTx) •— cos (a*7rx) . ( x + J»r \ \ 2 /
a*{x — x) \ 2 J X — X
2
and since^the absolute value of the last factor on the right is less than
I y then the absolute value of the first part of (i) is less than
r— I
7t
r«— '5^.
o
and therefore less than -7^ — —9 i( ad > i.
ad — 1
Also, since a is an odd integer,
cos (a"+*;r;r') = cos [«*(/< — i);r] = — ( — i)**,
cos (a'"+";rj;) = cos (a'^jjTt +^"'^i«+i^) = ( — i)'*cos (a*^„^.,;r).
Therefore
00
r
iM+„ cos (a'^+^Ttx^) — cos (a'^^^TTx)
X* — X
00
»S=0
All the terms under the 2 on the right are positive, and the first
is not less than |, since cos {x^^^n) is not negative and i -|- ^'m^i
lies between \ and |.
Consequently
^!^^) = (-.,.W-j(i + ^,), (fi)
where S is an absolute number > i, and 7 lies between — i and + i.
In like manner
Ax") -A^) / X / zx ^,/2 . TCTf' \
(iii)
where £' is a positive number > i, and ;;' lies between — i and -\- i.
\{ abht so chosen as to make
2 n
that IS, — >
3 ab — i'
the two difference-quotients have always opposite signs, and both are
infinitely great when m increases without limit. Hence /"(a) has
neither a determinite finite nor determinate infinite derivative.
SUPPLEMENTARY NOTES.
453
r+i
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
p such a function can be obtained by selecting
i>j.t ^ i^Q particular fixed points A, B ci the as-
►"S" semblage. Between A and B^ in progressive
^ order, select points /\, P^^ . . , representing
the function corresponding to Xj, 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 o. Then, since the function /{x)
is continuous, each side, /V-^r-f-i* of the broken
line converges to o. But, instead of each
angle between consecutive pairs of sides of
this polygonal line converging to two right
angles, ie, as their lengths diminish indefi-
a x^x^
Fig. 157.
nitely, as was the case when we defined a curve with definite direction at each point;
let now these angles converge alternately to o and 29r. The polygonal line folds
up in a zigza^^. The point P converging to the neighborhood of a true curve AB.
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 ffir- reaching importance in the study of
molecular physics, wherein it becomes necessary to study vibrations of great velocity
and small oscillation.
NOTE 2.
Supplementary to § 42.
Geometrical Picture of a Function of a Function.
If z =/{y)y where ^ = 0(Ar), we can represent the function z
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. xOy is horizontal,
xOz znd yOz are vertical.
The relation^ = <f>{x) can be repre-
sented by a curve P Q' in the plane
xOv. At any point F' on this curve we
can represent z by drawing P'P =/iji),
up \i/(y) is positive, down '\i /[y) is
negative. The relation z =/{y) is
represented by the curve /^'''^'" inyOz.
as a function of x, is represented by the
curve P"Q" in xOz. In other words, ^ as a function of x and y is
Fig. 158.
454
APPENDIX.
represented by a point in space having the corresponding values
Zj y, X as coordinates with respect to the three planes. The assem-
blage of points representing 2, y, x is 2i space curve PQ, The
orthogonal projections on the three coordinate planes of PQ represent
the functional relations
(/"^O, ^ = 0W; {P"Q"), «=/i0(A)!; {P"'Q"')> 2 =/(.>•)•
The derivative D^jf is represented by the slope of P'Q' at P' to
Ox. The derivative D^ is represented by the slope of the tangent
to P"'Q"' at P'" to Oy\ the derivative D^ 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 n\\i Derivative of the Quotient of Two Functions.
Let y = u/v. Then « = ^. Applying Leibnitz's formula to
this product, we have
u
rt
= W,
u
2!
V
ft
v' /
2\^ + 1! i!
- ,. y
.//
■— V
2!
//
n\ ~ n\^ '^ (n — i)! il "^ (« - 2)! 2!
^rr+...+--^.
To find^", the «th derivative of u/v, in terms of the derivatives of
ftf and V. Eliminate y,"^ . . .
tions. We get
ff! \» / (»)"+■
' (» - I)!
from the « + i equa-
tt f 0 o .
V o
u V
?! il
{n '\- i) rows
SUPPLEMENTARY NOTES.
455
Also, in particular, if u = i, we have
_(-ir
v'
if
v"
2!
V 00...
v'
—rV 0 . . .
i!
3!
2! i!
n rows
NO IE 4.
Supplementary to
§56.
To Find an Expression for the nth Derivative of a Function
of a Function.
Let 3 =y(^), where >' = 0(^). To find the wth derivative of
s with respect to x.
We have, by actual differentiation,
A" =/;'%' -3/-'>4v;' +/>r.
The law of formation of these first three derivatives of /"with
respect to x shows that the «th derivative must be of the form
/r = Aj; + Aj;' + . . . + Aj^, (i)
where the coefficients, A^y contain only derivatives o(y with respect to x
and are therefore tndependen/ of the /or m of the function/*. Conse-
quently, if we determine A^ for any particular function /*, we have
determined the coefficients whatever be the function/! Let then
Then in (i) we have
- n:{y - by = \^—-l^A, + . . . +-^ A^, + A,. (2)
•Hence, when d =y, we have
A^ = l{l>:iy-,y]^^,
which means that (y — dy is to be differentiated n times with respect
to X and in the result y substituted for d»
r = 1
(3)
456 APPENDIX.
This gives the nth derivative of/* with respect to x in terms of the
derivatives of/* with respect to y and those of ^^ with respect to x, and
is the generalization of the formula
^jc-^yy) - dy dx'
We can give another form to {3), as follows. Let^' = h when
or = a. Then
^ - 3 = ^(x) — (p{a) = (x - d)v, (4)
where v stands for the difference-quotient
X — a
Apply Leibnitz's Formula to (4), and we have
= 2C,,^D:-^v^£>^,(x^aY;
^ B O
But, Dfi{x — a)" = r(r — i) . . . (r — / + i){x — a)*^,
= o when p '> r,
= o when / < r and x = a.
= r! when ' p =. r and x •=. a.
Therefore (3) becomes
r=i
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 y][.r) and its derivatives are continuous for all values
of A' in (ot^ P) except for a particular value <j of at at whichy][<z) = 00 ,
then all the derivatives oif^pc) are infinite at a.
Let x^ <^x^< a. Then
A^^ -Ax,) = (X, - x,)/'(S),
where S lies between x^ and x^ Let a — x^ht a small but finite
number, and let x^(=)a. Then /(;>;,) is infinite, andy^^fj) is finite.
Since t7 — x^ is finite, y"'(5^) = 00 ; and since /'(^) is finite if ^ — ^
is finite, we must have a — ^( = )o and
SUPPLEMENTARY NOTES. 457
In like manner we show that /"(a) = oo , and so on.
Corollary. l(/{a) = oo , then/\a) = oo , and also
/M
becomes oo when x = a.
For, considering absolute values, if/" (a) = oo , then also
\og/{a) = 00 . By the theorem established above, if log/[x) is oo
when X = a, then
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 o or oo an infinite number of times in any finite interval,
however small. Such functions are also called rational.
A function y(jc) is said to be unlimitedly differentiate at x when
all the derivativesy*'(A-) 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 o or oo an unlimited
number of times in the neighborhood of any value x considered.
2. Ill t}iO same way that a function of the real variable .v may be
o for an imaginary number p -|- 'V» such a function may be oo for a
complex number / + tg, where i = 4/— i. For example, the func-
tion
becomes oo at ^ -f li/ if / -|- ig is a root of il'(x) and not of (/>(x). A
value of jr at which ^{x) is o or oo is called a roof ox pole y 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.
45 S 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, but no essential singularity, and which is unlimitedly differen-
tiable everywhere except at a pole.
3. Theorem I. — \{f(x) is a one-valued, determinate, and unlim-
itedly differentiable function at x, then the series
00
o
is absolutely convergent for all values of y less in absolute value than
where / + '^ is the nearest pole of/j^.v), or any of its derivatives y*'' (a:),
to the number x) and the series 6" is 00 for any value ofy greater
than R,
4. Represent x^ y by the coordinates of a point in a plane xOy.
Then (see § 15, Ex. 9, 10):
(i). At all points x,y at which
« + 1 /"(*)
< I.
S is absolutely convergent, and also
«=00
(2). At all points x^y at which
y f-^\x)
£-.
n + I /-{x)
^' = 00 , and also
> I,
/
^/■w = «.
« = <»
5. It follows, therefore, that if
w=»
* Remembering that the modulus or absolute value of any number x -^ iy is
then of two numbers p^ -\- iq^ and p^ -\- iq^ that one is nearest x for which we have
the difference
\P + iq - x\
least.
SUPPLEMENTARY NOTES. 459
has a finite limit different from o, it is necessary that
/.-
I.
1
n + I f'{x)
6. Since at all points of absolute convergence of S
'^/\-) = o,
and at all points of infinite divergence of *$"
the boundary between absolute convergence and infinite divergence
of S is marked by the values of x, y which satisfy
/^/"(^•) = '•
7- The locus
for an arbitrary and great value of n^ will be a close approximation to
the boundary line we seek. Differentiating, this locus has the differ-
ential equation
dy _ y /"^'{x)
dx "^ n fyX)
Which for n arbitrarily great gives, in the limit,
dx
in virtue of
= I
£■•
y /-^^{x) , ^
« + I /«(A)
on the boundary.
8. Therefore the absolute value of j^ is equal to the absolute value
of a linear function of x, of the form
y^=\k^x\\
for all values oi x and^on the boundary.
This is the equation of the family of boundary lines having the
parameter k. These lines are fixed by the fact that whenever y(.^•) or
/''{x), for any finite r, is 00 we have_y = o.
9. If, therefore, y^A") = 00 when x =z p, the corresponding bound-
ary lines for a real pole p are the two straight lines
y = (/ - ^)\
or y z=, X — p and y — -^ x -^p.
46o APPENDIX.
If /{x) = eo when x = p -\- iq, then the corresponding boundary
lines for a complex pole/ -f- li? are the two branches of the rectangu-
lar hyperbola
= {p- xf + f,
«^ ^ +g> '■
having for asymptotes
y -^ X — p and J/ "= — ■*■+/■
10. Therefore for any function having real and complex poles
the boundary lines consist of pairs of straight lines crossing Ox at 45°
at the real poles and of right hyperbolx having as asymptotes similar
straight lines crossing Ox at the real |)art of the complex pole.
The vertices of the hyperbola corresponding to the pole/ + iq
o.Kp, ± g.
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 the boundary oi the region of convei^-
ence of .S, It consists of straight lines and hyperbolic arcs.
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
_ «+ " /V) '
is less than i for any point inside, and greater than i for any point
outside, the region.
/.
SUPPLEMENTARY NOTES. 461
12. If a function has two real poles or, /3, and no pole between
a, fiy the region of absolute convergence consists of a square between
a and fi. If between a and fi there is an imaginary pole p -\' tg
such that / lies between a and fi, the imaginary pole has no influ-
ence on the region of convergence if
0» - i(a + /S)? + ?* > i(« - /?)»•
If, however,
\P-hi« + fi)V-\-f<\{«-fi)\
the hyperbola j^ = (/ — .r)' + ^ cuts off a portion of the square of
convergence.
13. Theorem II. Ti /(x) is a one-valued, determinate, unlim-
itedly differentiable function (having only a finite number of roots
or poles in any finite interval), then
'30
o
for all values of x and y for which the series is absolutely convergent.
That is, for all values of ^ less in absolute value than the radius
where/ + iq is the nearest pole oi /[pc) to x. Equation (i) is not
true for any value of>' such that \y\'>R.
Proof: The construction of the region of absolute convergence
shows that from any point P in 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 x, y in the region of absolute convergence the
series
00
s' = Y;^j^r^\x)
o
is absolutely convergent.
But ^'= -=--.
ox oy
Hence
= o, if a: -f-^ is constant.
Therefore all along the line x -{- y z= c, in the region of absolute
convergence, S must be cons/an/. This line passing through any
462 APPENDIX.
point P in this region meets Ox without touching the boundary. At
the point where x -^-y = c meets Ox we have j^ = o, x ^=^ c^ and
Consequently all along any such line passing through the region
of absolute convergence, and therefore at any point whatever in this
region, we have
00
o
14. What is the same thing,
00
O
for all values of jcand^' which make the series absolutely convergent.*
If we make the investigation in the form (i), the regions consist
of parallelograms on the line^v = x as diagonal, and having for sides
the straight lines
x—p, x—iy—p,
corresponding to a real pole/, and hyperbolae
(X ^yf = 1;^ + iq -y\'^ = {p -y^ + f,
or x^ — 2xy + 2py = /^ _|. ^^
corresponding to a complex pole/ + iq.
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
at the zeros oif(pc). There the curve has vertical as3-mptotes, but
closes up on the asymptote as n increases. Also, /""(.r) cannot have
the same zero point for an indefinite number of consecutive integers
n 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
/
"C+O/.^^^NI^.
since for absolute convergence we must have '
/
«(«+0 /V)
< I.
* It being understood that^y is at a finite distance from any value of the variable
at which the function is 00 .
SUPPLEMENTARY NOTES. 4^3
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 x, then at least
one must be complex.
If there be three equidistant poles from jf, then one must be real
and two imaginary, p ± ig, and conjugate. Then the derivatives at
X arc o alternately in pairs and the radius of convergence there is
J^^\ = \£n{n+i){n+2)y'^^'^
y«+3(jt-)'
and so on.
Points A", equally distant from several poles, are the singular
points on the boundary. Elsewhere, for three poles, we can always
write
the limit of which is A^, and converges to the value ^^ at the singular
point as x converges to the x of such a singular point. The generaliza-
tion of this is obvious.
EXAMPLES.
t. The region of absolute convergence and of equivalence of the Taylor's series
of the functions tan x, cot x, sec x, esc x^ consists of the squares whose diagonals
are the intervals between the roots of sin jr, cos x, respectively.
2. In particular tan x is equivalent to its Maclaurin's series for all values of x
in ) —^it, -\- ^7r(.
Also for sec x in the same interval.
cot X. CSC X are equal to their Taylor's series in the interval )o, it{ , the base of
the expansion being ^it.
3. Expand by Maclaurin's series.
Put ^ equal to the function. Then
y^ — ^ = X.
Apply Leibnitz's formula, and put x :;^ o in the result We have for deter-
mining the derivatives of ^ at o,
Making » = i, 2, 3, . . . , we find these derivatives in succession, and there-
fore
<•« — I 22! 4 ! ' 6 !
wherein ^j = J[, -^j = uV? ^i = tV A = ■!rV» -^s = ¥*«» • • • *** called
Hernouilli's numbers. They are of importance in connection with the expansion
of a number of functions.
Since ^^'^ = — i» the poles ± tic are the nearest values of x to o at which the
function liecomes 00 . The series is therefore convergent and equal to the function
fr)r X in ) — TT, -|- TCy.
464 APPENDIX.
4. Show that for jr in ) — ^x, -f ^it(,
^ + I - 2 2 ! ^^ ^^ TT ^ '^ 6T (^ - 0+ . . *
either directly or from
X _^ X 2X
^ -|- I "" ^ — I ~~^a« _ I '
5. Show directly frond 4 that, for the same values of jr,
'^•= ^.X(2. - „- V?l^li.+ ^.-***- '
• • •
e»-{- I '' ' 4! 2 '61 2
6. Obtain the Maclaurin expansion
... »t m \^ — m^) mil* — «')(3* —«')».
sin(w sin-«^) ^-x^ j-^ 'j^ + -^^ -^ 'jfi+ , . . ,
and find for what values of jt the equation is true.
Put y = sin (m sin— »jr).
. •. (I — j^)y" — xy* -{- m^ = o.
Apply Leibnitz's theorem and deduce
(1 - x^)yi^-^^) - (2» + i)j9/(«+0 -f- (m« —f^)y(n) = o.
y(«+i) v(«)
(" + « JwT;rl=l^>
(»' - '"•)/75r = 2 ^" "^ '^;^^ ■ ^'' + '^ ^^=^ ' = ' ^'
we have
(I — jr*) — 2JC -^ — ^ = o,
or ^ I z= I ;c ± 1.
Therefore if jt is the base of a Taylor's series for y, the function is equal to the
series in )x — i, x -\- i(. If jc = o, the Maclaurin's series is equal to the function
in )- I, -h i(.
When jc = o, the differential equation gives
which gives the coefficients in the series.
7. Treat in the same way cos (iw sin— »jr).
8. For what values of x is the Maclaurin's series corresponding to the function
y in
(I - jr*)/' — x/ — a*y z=o
equal to the function?
Work as in 6. The function is ^ rtn~'jc.
9. In general, any function^ satisfying a differential equation
(I + ax^)yi^-^^)-{-pxin -f 6)yi^^t) ^ g(^n — <•)(» — d)yin) = o,
where a, 6, c, </, /, (/ are any constants, is equal to its Taylor's series (base x) in
the interval )r — /?, jr -(- /^{y where /? is the radius of absolute convergence, and
i? is the absolute value of the least root of the quadratic
(I -}- ax^) 4. /jr i^ -f- g/^ = o.
A large class of functions can be treated in this way.
SUPPLEMENTARY NOTES. 4^5
10. If f# is a function o£x having only a finite number of roots in a finite interval,
find the region of equivalence of the function i/u with its Taylor's series.
Let/ = i/u. Then yu=i. Differentiate n times by Leibnitz's formula. Then
Divide by/**, and make m = oo . Then, p being the radius of convergence,
we have, if ft = itJ{x\
<p{x)+^<p^(x) + ^ 0"(4r) + . . . = a
But this series is nothing more than <p(x -|- p).
Therefore x -\- p must be a root of
0(x -f. p) = a
Consequently
x-{- p = k,
or p rr i ~ jr,
where k is the nearest root of {p{x) to or, 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-\- ty, it is instructive and interesting to con-
sider the treatment of a function ^(0) after the method of Note 6 for a
function of the complex variable s = x -{- ^,
We assume that_/(«) is one-valued, unlimitedly differentiable with
respect to 0 at all values of z in the finite portion of the plane except
at poles o(/{e)f which are, we assume, the only singularities the func-
tion has.
2« Let s s X -{- ty, C s -^^ + jv'- The series
ee
o
is absolutely convergent (when the series of absolute values of its terms
is convergent) for all values of b and C which satisfy
/{^/-(C)}V,<|.. or y'-A-^Ki,
ffSOO lf=SOO
The series 5 is 00 when these limits are greater than i .
The boundary conditions are
Therefore for n arbitrarily great the boundary is arbitrarily near
a and /3 being arbitrary constant real numbers.
= I.
4^6 APPENDIX.
From the first of these equations, we have
^ "■ « /"(C) '
= — e'^y when » = oo.
k being an arbitrary constant. But if/ is a pole of/(«), then i? = o
when Z =p.
Hence o = >^ — /^/, or k = e'^p.
Therefore, corresponding to any assigned Z» the boundary corre-
sponding to the pole/ is fixed by
0 = /^(/-C),
which is a circle about the origin in the 0-plane with radius
Ji=\p-Z\,
since /3 is arbitrary.
3, If/ is the nearest pole of/{s) to Z, then for all values of z for
which
|«|<ie = |/-C|
the series is absolutely convergent, and is infinite for any value of z
if 1^1 >^.
4. Fut S = z + C. Then « = 5 - C
The series
00
p ^=Z^^^-^'^'^
is absolutely convergent at all points
^ inside the circle Cdescribed about
"* C as a center with radius
^ = |C-/|.
^^^- '^- / being the nearest pole oi/{z) to C-
For any assigned value of S in this circle the series S is con-
stant with respect to Ci since
dS^
^.^y ^±:^r^z) = (-^^/--(C),
dZ dZ/^ r\ -" ^"' n\
o
and this is o when « = 00 .
Now we can always move C up to 5^ along the straight line join-
ing them, the series S remaining constant in value. But when
Z = S, we have
00
y ^^ - P'/'(Q = AS)-
n I
SUPPLEMENTARY NOTES. 4^7
Therefore* this equality is true for all values of S, Z which make
the series absolutely convergent, i.e., at any point inside the cir-
cular boundary corresponding to any assigned C &nd the nearest pole
/ oi/{z), described about Q as center with radius of absolute con-
vergence,
^ = I/-C!.
NOTE 8.
Supplementary to Note 6.
Pringsheim'B Example of a Function for which the Maclaurin's
Series is absolutely Convergent and yet the Function and
Series are different.
Let
r-o
A. and a being positive constants, a > i. This function is one-
valued, finite, continuous, and unlimitedly differentiable for all finite
values of the real variable x. It has, however, infinitely many com-
plex poles
± —^-^ ^ = I, 2, 3, - . .
an infinite number of which are in the neighborhood of .r = o, which
is therefore an essentially singular point.
For the nth derivative oi/[x) we find (i* s + V -— i)
00
I ^ V^/_ xX^^^Y, i I I 1
At a: = o,
/2'« + i(o) = o.
Therefore the Maclaurin's series is
00
0)
This series is absolutely convergent for all finite real values of x,
* 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 ^ <i, |Jt| < I.
I A
A^)>
i+x» 1 + a^jfi ^ I + X* 1+ tf«jr»'
and ^ < ^-«.
In particular, let x = «-*.
Fig. i6i.
ff— I
.*. /(fl"*) > ^-* > ^ when X = a-*.
when — : — > g-i
or
^+1
The function /[x) and the series 49
are different.
In the figure the solid line is the
curve ^ =i/\x), the dotted line the curve
y =z Sf constructed with exaggerated
ftniiiuUes, for the values A, = log 2,
a = 2.*
NOTE 9.
Supplementary to § Ii8.
Riemann'8 Existence Theorem.
Any function /l[ji;) that is one-valued and continuous throughout
an interval {a, 6) is integrable for that interval.
Let the numbers x^, x^, . . . , x^^^he interpolated in the inter-
val {a, d) taken in order from a = x^ to 6 s x^*
e have to prove that the sum of the elements
5. « i/(«.)(^r - ^r-i)
(I)
converges to a unique determinate limit, when each subinterval con-
verges to zero, whatever be the manner in which the numbers Xr are
interpolated in (a, b).
I. The sum 5„ must remain finite for all values of «. For /{x)
is finite, and if M and m are the greatest and least values oi/{x) in
m{b ^a) <Sn < M{b — a).
Also, since /(a:) is continuous, there exists a value S in (a, b) at
which
S, = {b- a)/(S), (2)
/{S) being a value oi/\x) between m and M,
* For further information on this subject, see papers by Pringsheim, Maik.
Ann. Bd. XLII. p. 109. Math. Papers Columbian Ejcposition, p. 288.
SUPPLEMENTARY NOTES. 4^9
II. Interpolate in the rth subinterval of (i), in any manner,
n^ — I values arj j . . . , x^,,^ of x. Then, as in I,
I
where S'^ is some number in the subinterval {Xr, •^r-i)-
Form similar sums of elements for each of the n subintervals of
(i\ Let/ = n^ -{-... -^ »«. Add the n sums of elements such
as (3).
Hence
>
I
= 2 {X, - X,.,)/iS'r). (4)
s
This is a new element sum containing p > n elements, which is
to be regarded as a continuation of (i) by the interpolation of new
numbers in each subinterval of (i).
Subtracting (4) from (i), we have
5. - 5, = i \J{Zr) -AS'r)]iX, - X,,,).
I
Let d be the greatest absolute value of the difference between the
greatest and least values of /[x) in the subinterval {x^ — -^r-i)*
r = I, . . . , «. Then, since /{Sr) and /{Si) are values of /[x) in
i < i *('* - "). (5)
foi all values of the integer p, however great. But when each sub-
interval converges to o, then (f(=)o, since /(j;) is continuous, and
at the same time » = 00 .
Therefore, by the definition of a limit, S^ converges to a limit
when « = 00 .
III. To show that the limit of S^, is wholly independent of the
manner in which the interval (a, b) is subdivided :
Let there be an entirely di^erent and arbitrary interpolation
Jt/, . . . , JP^-,. Consider the element-sum
NV
SL m 2A''r)(xi - ^i...). (6)
I
Interpolate in (a, d) the numbers
X^t • • • f "^K—l f -^I > • • • > •^^■»— 1 f
occurring in (i) and (6), thus dividing (a, d) into m -f- ^ intervals.
470 APPENDIX.
Interpolate in each of these m -^ n intervals new numbers, thus
dividing (a, d) into m-f if-|-/ subintervals. Form the element-
sum S^^tt+^ corresponding to these subintervals.
Then, by II, 6« and .S^ ^ . ^_ « converge to the same limit. In
like manner S^ and 5'm+»+/ converge to the same limit. Therefore
Sn And S^ converge to a common limit. The uniqueness of the
Kmit of (i) under any subdivision whatever of (a, H) is demonstrated.
This theorem gives the means of defining analytically the area and
length of a curve, and the volume and surkce area of a solid.
NOTE 10.
Supplementary to § 135.
Farmute for the Rednctioii of Binomial Differentlalg of the form
Put J/ = a -f bjf. Then
= aax^-yy-* + (aa 4- ny)bx^^'-yr-^^ (i)
= (flf + ny)x^yf — anyx^-yv-^ (2)
In (i), put a = m — n+ 1, y=p + i, then
2?jc-»-^y+« = a(»i - » + i)jf^j^ ^(np+m + i)bafy. (A)
In (2), put a = « + I, y =p, then
Dxf^+y =:(np + m+i )x'y^ — anpxy^K (B)
In (i), put a = « + I, X = ^ 4. I, then
j9^+y4., », ^^^ ^ i)jc«y ^(np^m+n + i)^j;*+V. (C)
In (2), put a = « + i, y=p+i, then
Z)jr^y+' = (»> + »« + »+ i):«;*y^* — ««(/ + i)jfV- (D)
Integrating the formulae (A), . . . , (D), we have the formulae of
reduction, where^ s « + ^•«*:
y -^ {np + m + i)b (np + m+i)6J
fjry^dx = "^^'^ + !^^ . foTj^-'dx. (B)
Xy^dx=: .^-^ ,+ /^T ; / ^y^^dx. (D)
« 4
SUPPLEMENTARY NOTES.
471
AX)^r,=
-/"(^ (I)
NOTE 11.
Supplementary to § 165.
Ify =^/[x) be represented by a curve, and y, Ly^ IPy 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 x,ybt the coordinates of P, and
X, 1^ those of P\ any point on the curve
between P and Py
The tangent at P has for its equation
K =/W + (^ - ^)/'(-^).
At any point x,y of ordinary posi-
tion, not an inflexion, the difierence
between the ordinate to the curve and
the tangent is
2!
where S is some number between x and X. We can always take X
so near to jc that/'X^) keeps its sign the same as that oi/"{x) for all
values of S in {x, X), Therefore the difference (i) keeps its sign
unchanged in {x, X) or the curve is on one side of the tangent, for
this interval.
The equation to the chord PP^ is
where yX^i) '^ the slope of the chord PP^ The difference between
the ordinates of the curve and chord is
Let x^ be so near x thaty*'(^j), /\G^ have the same sign a&/\x).
Then this difference (2) keeps its sign unchanged for all values of -X"
in (at, jTj). It can now be easily shown that (2) and (i) have opposite
signs, and there can always be assigned a number ji:^ so near x that the
curve PP^ lies wholly in the triangle formed by the tangents at /*, P^
and the chord PP^*
NOTE 12.
Supplementary to § 236, IV.
Proof of the Properties of Newton's Analjrtical Polygon.
X. Let there be any polynomial in x and^, such as
/= ^,a;V> + . . . + ^^V'", (i)
wherein the exponents a, fi of each term satisfy the linear relation
tfor + 3/? = C, (2)
c being taken a positive number.
47* APPENDIX.
Let f be arranged according to ascending powers oiy, so that
>»,</?,<... Then
= j^ ^\_A, + A, \^x--^ ' '+ . . J, (3)
-= cf'^/^Kjix' - k) . . . \yx-'- i^.jA„ (4)
b
where J^^, . . . , ^^^-^, are the roots of the equation in / s^or " ,
^, + ^/•-^» + . . . +^/--^« = o.
Therefore the locus of /"= o consists of jr = o, ^^ = b, and the
parabolic curves
y = ^>*. (r = I, . . . , ^«, - /?J.
2. In (3), let J/ = >br*, ^ being constant. Then
= -AT JIT* , ^ being constant.
3. Let y ' be a function A'x^'y^\ or the sum of a finite number of
such functions, such that the exponents a\ p' of each term satisfy the
linear equation
h_
Then, as in 2, let j/ = ijr*, and we have in the same way
K' being a constant.
4. Let a, h and c, c' be positive numbers.
Then
where jc and> satisfy >^ = ^.
(i). If c' > c, then
y ^ = o, when :r( = )o, X=)0'
f'-tf
SUPPLEMENTARY NOTES. 473
(2). If c' < c, then
^ ^ = o, when x = 00 , ^^ = 00 .
$• We are now prepared to prove § 226, IV, (i), (2).
Let I{x, y) « ^C ^y = o.
(i). I^ty represent that part of /^ which corresponds to a side of
the polygon as prescribed in § 226, IV, (i), and F' represent the
remainder of F. Then
/'=/+^',
F , F'
Through each point corresponding to terms in F' draw a line
parallel to the side corresponding to/.
Then by 3, (i), we have
/F PF'
when x{ — ) o, X = )o.
Therefore in the neighborhood of the origin F = o and /= o
are the same.
But the form of/* = o in the neighborhood of the origin is that of
a parabola
Hence F = o goes through the origin in the same way as does
y = o, whose form is that of a parabola of typey* = ibc^.
(2). Let /" represent that part of /" corresponding to a side of the
polygon as prescribed in §226, IV, (2), and F' the remainder of F,
F F'
Then -= i + — .
Draw parallels to the side corresponding to/*, through all points
corresponding to terms in F',
Then by 3, (2), we have
/F PF'
7=^' """i/="'
when a: = 00 , >' = 00 .
Therefore F •=. o zn^/^r: o pass off to 00 in the same way. Also,
y = o passes off to 00 , as does a parabola of type>^ = kj(^.
Note. — The same process can be extended to surfsiceSf using a polyhedron in
space. The part of the equation corresponding to a 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 tlie origin gives the form of a sheet at ao . The plane
£&ces in each case cutting the positive parts of the axes.
INDEX.
[Tke numbers refer to the pages^
Absolute number, 2
Anticlastic surf&ce, 360
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
sur&ce 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, 151
area, 229
Clairaut's equation, 429
Coordinates of center of curvature, 133
Computation of^
^, 84; logarithms, 86; ir, 89
Concavity and Convexity, 127
Concavo-convex, 128
Conchoid of Nicomedes, 160
Concomitant, 312
Convexo-concave, 128
475
476
INDEX.
Cone,
volume of, 257, 366
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, 13b
radius of, 133
circle o^ 134
of sur&ces, 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
Devil, 161
Difierence,
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 theoi^m, 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, 25 1
normal to, 331
orthogonal trajectory, 433
INDEX.
477
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, X62, 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 o^ 250
Exact differential, 416
Exponential curve, 150
Family of curves, 138
Finite difierence-quotient
A') -A")
X — a
ifth 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
Gauss,
theorem on areas, 241
theorem on curvature, 372
Geodesic line, 384
Groin, volume of, 267
surfoce of^ 408
Harkness, 451
Helix, 376, 384, 405
Holditch's theorem, 238
Homogeneous,
coordinates, 339
differential equation, 4x4, 431
Horograph, 372
Hyperbola,
tangent, 113; asymptotes, X22
radius of ciurvature, 137
area, 228, 233
orthogonal trajectory, 432
Hyperbolatoid, voliune of, 269
Hyperbolic sine, cosine, 29
Hyperbolic spiral,
traced, 162; subtangent, 117
area, 234 ; length, 248
HyperboUc 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, I
Integral, definition, 165
indefinite, 173; definite, 21$
fundamental, 173
Integration, definition, 167
by transformation, 178
by rationalization, 182
by parts, 183
478
INDEX.
Integration by partial fractions, 185
under the /'sign, 393
Integrating £Etctor, 420
Interval of a variable, 4
Intrinsic equation of curve, 351
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, 7S
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 cxirves, 243, 247
curves in space, 404
L'Hopital's theorem, 94
Limit,
of a variable, 7
principles o^ 7
theorems on, 8
of (I + i/«)*, 16
of integration, 167
Lima9on, 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
Ix)xodrone, 384
Macliiurin's scries, 83, 467
Maximum and Minimiun, 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, 114, 115, 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
e volute, 144; area, 228, 234
arc length, 245, 248
length of evolute, 251
orthogonal trajectory, 432
INBEX.
479
Paraboloid of levolution;
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,
«lh derivative, 454
Radius of convergence, 88
Radius of curvature,
plane ciu^es, 100, 133, 250
at point of inflexion, 135
for surfaces, 365
for space curves, 379, 389
Real niunber, 3
Reciprocal spiral, 117, 126
Re volute, definition, 255
volume, 256 ; surface, 259
Riemann's existence theorem, 468
Roche, 222
Rollc's theorem, 75
Root of a function, 457
Saddle point, 336
Scarabeus, 160
Schl()milch, 222
Semi-cubical parabola, 245
Sequence, 14
Singular points,
on a curve, 158, 333
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 M and N are of first degree,
415
by differentiation, 426
when solvable for^, 428
when solvable for x, 430
when solvable for/, 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, 35 1
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
48o
INDEX.
Total,
derivative, 291, 294
differentiatioii, 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
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, 401
Weierstrass, 5
example of derivativeless function,
Witch of Agnesi,
tangent, normal, 115
traced, 152; area, 228
volume of revolute, 258
Zero^ 2
MATHEMATICS
Evans's Algebra for Schools.
B7 Georob W. Evans, Instructor in Mathematics in the English HIg:h
School, Boston, Mass. 433 pp. z2mo. $x.ia.
Aside from a number of novelties, the book is distinguished
by two notable features :
(i) 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 a time, 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 in a thoroughly
sensible way as to sequence of
topics and method of treat-
ment.
C. H. Pettee. Professor in the
N. H. College of Agriculture :—\
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 : — I find
more to commend in this algebra
than in any book on the subject
since the publication of '« in
1869.
32
Mathematics
ZZ
Gillet's Elementary Algebra.
By J. A. GiLLET, Profesaor in the New York Normal College. xiv + 4i9
pp. i9mo. Half leather, $i.io. With Part II, xvi + sxa pp. x2mo.
$1.35.
Distinguished from the other American tepct-books covering
substantially the same ground, (i) 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
chat 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 unlearned.
F. F. Thwiof^, Manual Train-
ing High School f Louisville , Ky, : —
Two features striice me as being
very excellent and desirable in a
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. 436 pp.
x2mo. Half leather, $1.25.
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
che "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. Morgan Park
Acadethy, Jll,:—\ find in them the
best results of modern research
combined with rigid exactness 10
definition and demonstration.
34
Mathematics
Keigwin's Elements of Geometry.
By Henry W. Keiowin, Instructor in Mathematics, Norwich (Ct ) Free
Academy, iv + asypp. lamo. $i.oo.
This little book is a class-book, and not a treatise. It rov-
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, K. ;— 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. Gtub^t^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, Wisc,:~-r
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 in a 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 {OJ)Hi,(ih
School . — I am highly pleased with
it and commend its general plan
most heartily.
Mathematics .^5
Newcomb's School Algebra.
B^ Simon Nbwcomb, Professor of Mathematics in the J*hns Hopkins
\j niversity. x -f- 294 pp. i2mo. 95 cents. (AVy, 95 cents. Answers^ 10
cents.)
Newcomb's Algebra for Colleges.
By Simon Newcomb, Professor in the Johns Hopkins University. Retnsed^
xiv -4-546 pp. xamo. $x.jo. (AVy, $i.jo. ^n^nv^rj, 10 cents.)
Newcomb's Elements of Geometry.
\j Simon Newcomb, Professor of Mathema
I niversity. Revised, x-f399pp. xsmo. $x.ao.
By Simon Newcomb, Professor of Mathematics in the Johns Hopkins
Un
Newcomb's Elements of Plane and Spherical Trigo-
nometry. (With Five-place Tables.)
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. Revised, vi -\r
168 + vi 4- 80 4- 104 pp. 8vo. $z.6o.
Elements of Trigonometry separate, vi + 168 pp. $i.aa
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 Simon Newcomb, Professor of Mathematics
in the Johns Hopkins University, vi + 187 pp. lamo. $1.00.
Much more elementary in treatment than the foregoing.
Newcomb's Elements of Analytic Geometry.
By Simon Newcomb, Professor of Mathematics in the Johns Hopkins
University, viii -f 357 pp. lamo. $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.
3^ Mathematics
Newcomb's Elements of the Differential and Integral
Calculus. ,
By Simon Newcomb, Professor of Mathematics in the Johns Hopkins
University, xii + 307 pp. lamo. $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. lamo. 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 Alg^ebra. Or Geometrical
Interpretations of the Theory of Equations of One Un-
known Quantity.
By A. W. Phillips and W. Beebe, Professors of Mathematics in Yala
College. Revised BdUwn. 156 pp. Svow $zj6a
.1 • ,
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