€ tit ntnj af

dVUenr

JlrrstuM bxi \l(»»,ii„^«M,,T ^, CV^

cr^aMm\

ill

x?%&5&& >UWGl.^

Wkt

AN ELEMENTARY TREATISE

ON

THE DIFFERENTIAL CALCULUS.

AN ELEMENTARY TREATISE

ON

THE INTEGRAL CALCULUS,

CONTAINING

APPLICATIONS TO PLANE CURVES AND SURFACES,

AND

A CHAPTER ON THE CALCULUS OF VARIATIONS.

BY

BENJAMIN WILLIAMSON, D.C.L., F.R.S. Crown 8vo, 10s. 6d.

AN INTRODUCTION TO THE THEORY

OF

STRESS AND STRAIN OF ELASTIC SOLIDS

BY

BENJAMIN WILLIAMSON, D.C.L., F.R.S. Crown 8yo, 5s.

AN ELEMENTARY TREATISE ON DYNAMICS,

CONTAINING

APPLICATIONS TO THERMODYNAMICS.

BY

BENJAMIN WILLIAMSON, D.C.L., F.R.S.,

AND

FRANCIS A. TARLETON, LL.D. Crown 8vo, 10s. 6d.

LONGMANS, GREEN, AND CO.,

LONDON, NEW YORK, BOMBAY, AND CALCUTTA,

AN ELEMENTARY TREATISE

ON

THE DIFFERENTIAL CALCULUS,

CONTAINING

THE THEORY OF PLANE CURVES,

WITH

NUMEROUS EXAMPLES.

BY

BENJAMIN WILLIAMSON, M.A., D.C.L., Sc.D.,

SENIOR FELLOW OF TRINITY COLLEGE, DUBLIN.

NEW IMPRESSION.

LONGMANS, GREEN, AND CO.,

39 PATERNOSTER ROW, LONDON;

NEW YORK, BOMBAY, AND CALCUTTA.

1912.

[all rights reserved.]

^s^e

U)

^TH

PREFACE.

In the following Treatise I have adopted the method of Limiting Ratios as my basis ; at the same time the co- ordinate method of Infinitesimals or Differentials has been largely employed. In this latter respect I have followed in the steps of all the great writers on the Calculus, from Newton and Leibnitz, its inventors, down to Bertrand, the author of the latest great treatise on the subject. An ex- clusive adherence to the method of Differential Coefficients is by no means necessary for clearness and simplicity ; and, indeed, I have found by experience that many fundamental investigations in Mechanics and Geometry are made more intelligible to beginners by the method of Differentials than by that of Differential Coefficients. While in the more ad- vanced applications of the Calculus, which we find in such works as the Mecanique Celeste of Laplace and the Mica- nique Analytique of Lagrange, the investigations are all conducted on the method of Infinitesimals. The principles on which this method is founded are given in a concise form in Arts. 38 and 39.

In the portion of the book devoted to the discussion of Curves I have not confined myself exclusively to the ap- plication of the Differential Calculus to the subject, but have availed mvself of the methods of Pure and Analytic

vi Preface.

Geometry whenever it appeared that simplicity would be gained thereby.

In the discussion of Multiple Points I have adopted the simple and general method given by Dr. Salmon in his Higher Plane Carves. It is hoped that by this means the present treatise will be found to be a useful introduction to the more complete investigations contained in that work.

As this book is principally intended for the use of begin- ners I have purposely omitted all metaphysical discussions, from a conviction that they are more calculated to perplex the beginner than to assist him in forming clear conceptions. The student of the Differential Calculus (or of any other branch of Mathematics) cannot expect to master at once all the difficulties which meet him at the outset ; indeed it is only after considerable acquaintance with the Science of Geometry that correct notions of angles, areas, and ratios are formed. Such notions in any science can be acquired only after practice in the application of its principles, and after patient study. •

The more advanced student may read with profit Carnot's Reflexions sur la Mitaphysique da Calcal Infinitesimal; in which, after giving a complete resume of the different points of view under which the principles of the Calculus may be regarded, he concludes as follows : —

" Le m^rite essentiel, le sublime, on peut le dire, de la methode infinitesimale, est de reunir la facilite des procedes ordinaires d'un simple calcul d'approximation a l'exactitude des resultats de l'analyse ordinaire. Cet avantage immense serait perdu, ou du moins fort diminue, si a cette methode pure et simple, telle que nous l'adonnee Leibnitz, on voulait, sous l'apparence d'une plus grande rigueur soutenue dans tout le cours de calcul, en substituer d'autres moins naturelles,

Preface. vii

moins commodes, moins conform es a la marehe probable des inventeurs. Si cette methode est exacte dans les re- sultats, comme personne n'en doute aujourd'hui, si c'est tou- jours a elle qu'il faut en revenir dans les questions difficiles, comme il parait encore que tout le monde en convient, pourquoi recourir a des moyens detournes et compliques pour la suppleer? Pourquoi so contenter de l'appuyer sur des inductions et sur la conformite de ses resultats avex ceux que fournissent les autres methodes, lorsqu'on peut la demontrer directement et generalement, plus facilement peut-etre qu'aucune de ces methodes elles-memes ? Les objections que Ton a faites contre elle portent toutes sur cette fausse suppo- sition que les erreurs commises dans le cours du calcul, en y negligeant les quantites infiniment petites, sont demeurees dans le resultat de ce calcul, quelque petites qu'on les sup- pose ; or c'est ce qui n'est point : Telimination les emporte toutes necessairement, et il est singulier qu'on n'ait pas apercu d'abord dans cette condition indispensable de Telimi- nation le veritable caractere des quantites infinitesimales et la reponse dirimante a toutes les objections."

Many important portions of the Calculus have been omitted, as being of -too advanced a character ; however, within the limits proposed, I have endeavoured to make the Work as complete as the nature of an elementary treatise would allow.

I have illustrated each principle throughout by copious examples, chiefly selected from the Papers set at the various Examinations in Trinity College.

In the Chapter on Eoulettes, in addition to the discussion of Cycloids and Epicycloids, I have given a tolerably com- plete treatment of the question of the Curvature of a Eoulette, as also that of the Envelope of any Curve carried by a rolling

viii Preface.

Curve. This discussion is based on the beautiful and general results known as Savary's Theorems, taken in conjunction with the properties of the Circle of Inflexions. I have introduced the application of these theorems to the general case of the motion of any plane area supposed to move on a fixed Plane.

I have also given short Chapters on Spherical Harmonic Analysis and on the System of Determinant Functions known as Jacobians, which now hold so fundamental a place in analysis.

Trinity College, October, 1899.

TABLE OF CONTENTS.

CHAPTER I.

FIRST PRINCIPLES. DIFFERENTIATION.

FAGH

Dependent and Independent Variables, I

Increments, Differentials, Limiting Ratios, Derived Functions, . . 3

Differential Coefficients, 5

Geometrical Illustration, ......... £

Navier, on the Fundamental Principles of the Differential Calculus, . 8

On Limits, ............ 10

Differentiation of a Product, ^ 13

Differentiation of a Quotient, . . . . • . . . • 15

Differentiation of a Power, . . . . . . . . .16

Differentiation of a Function of a Function, . . . . . 1 7

Differentiation of Circular Functions, 19

Geometrical Illustration of Differentiation of Circular Functions, . . 22

Differentiation of a Logarithm, 24

Differentiation of an Exponential, ........ 26

Logarithmic Differentiation, 27

Examples, 30

CHAPTEK II,

SUCCESSIVE DIFFERENTIATION,

Successive Differential Coefficients,

Infinitesimals,

Geometrical Illustrations of Infinitesimals, Fundamental Principle of the Infinitesimal Calculus, Subsidiary Principle, .....

Approximations, ......

Derived Functions of xm, ....

Differential Coefficients of an Exponential,

Differential Coefficients of tan-1 x> and tan"1 -, .

x

Theorem of Leibnitz,

Applications of Leibnitz's Theorem,

Examples, ... .

34 36

37 40

4i 42 46 48

50

5' 53 57

Table of Contents.

CHAPTER III.

DEVELOPMENT OF FUNCTIONS.

Taylor's Expansion, .

Binomial Theorem, ,

Logarithmic Series, .

Maclaurin's Theorem, ....

Exponential Series, ....

Expansions of sin x and cos x,

Huy gens' Approximation to Length of Circular

Expansions of tan"1 x and sin"1 #,

Euler's Expressions for sin x and cos xf .

John Bernoulli's Series,

Symbolic Form of Taylor's Series, .

Convergent and Divergent Series, .

Lagrange's Theorem on the Limits of Taylor's

Geometrical Illustration,

Second Form of the Remainder,

General Form of Maclaurin's Series,

Binomial Theorem for Fractional and Negative

Expansions by aid of Differential Equations,

Expansion of sin mz and cos mzy

Arbogast's Method of Derivation,

Examples, ......

Arc,

Series,

Indices,

PAGH

61 63 63 64

65 66 66 68 69 7o 70

n 76

78

79 81 82

85 87

S8 9*

CHAPTEK IV.

INDETERMINATE FORMS.

Examples of Evaluating Indeterminate Forms without the Differential , Calculus, ............ 96

Method of Differential Calculus, ........ 99

Form Oxoo, , . . . . 102

Form 22., ..,..103

Forms o°, 00 °, i±flD ., , * . .105

Examples, 109

CHAPTER V.

PARTIAL DIFFERENTIAL COEFFICIENTS.

Partial Differentiation, . .

Total Differentiation of a Function of Two Variables,

Total Differentiation of a Function of Three or more Variables,

Differentiation of a Function of Differences,

Implicit Functions, Differentiation of an Implicit Function,

Euler's Theorem of Homogeneous Functions, .

Examples in Plane Trigonometry, .....

Landen's Transformation,

Examples in Spherical Trigonometry, .... Legendre's Theorem on the Comparison of Elliptic Functions, Examples, .........

"3 "5 "7 119

120

123

130

137 14c

Table of Contents.

XI

CHAPTER VI.

SUCCESSIVE PARTIAL DIFFERENTIATION.

The Order of Differentiation is indifferent in Independent Variables, Condition that Pdx -f Qdy should be an exact Differentia], Euler's Theorem of Homogeneous Functions, Successive Differential Coefficients of <p (x -+- at, y -f fit), Examples, .........

PAGE

145 I46

I48

I48

'50

CHAPTER VII. lagrange's theorem.

Lagrange's Theorem, 151

Laplace's Theorem, ., . 154

Examples, 155

CHAPTER VIII. extension of taylor's theorem.

Expansion of <p {x + //, y + /•), . 156

Expansion of <f> (x + ht y -\ k, z + /), . . . . . , 159

Symbolic Forms, .160

Euler's Theorem, 162

CHAPTER IX.

maxima and minima for a single variable,

Geometrical Examples of Maxima and Minima,

Algebraic Examples, .....

Criterion for a Maximum or a Minimum,

Maxima and Minima occur alternately, .

Maxima or Minima of a Quadratic Fraction, .

Maximum or Minimum Section of a Right Cone,

Maxima or Minima of an Implicit Function, .

Maximum or Minimum of a Function of Two Dependent Variables,

Examples,

164

165

169 173 *77 181

185 186 188

CHAPTER X.

maxima and minima of functions of two or more variables.

Maxima and Minima for Two Variables,

Lagrange's Condition in the case of Two Independent Variables,

191 191

xii Table of Contents.

Page

Maximum or Minimum of a Quadratic Fraction, 194

Application to Surfaces of Second Degree, . . . . .196

Maxima and Minima for Three Variables, 198

Lagrange's Conditions in the case of Three Variables, . . . . 199 Maximum or Minimum of a Quadratic Function of Three Variables, . 200 Examples, ............ 203

CHAPTER XL

METHOD OF UNDETERMINED MULTIPLIERS APPLIED TO MAXIMA AND

MINIMA.

Method of Undetermined Multipliers, 204

Application to find the principal Radii of Curvature on a Surface, . . 208 Examples, 210

CHAPTER XII.

ON TANGENTS AND NORMALS TO CURVES.

Equation of Tangent, . . . . . . . . . .212

Equation of Normal, 215

Subtan gent and Subnormal, . . . . . . . . . 215

Number of Tangents from an External Point, 219

Number of Normals passing through a given Point, .... 220

Differential cf an Arc, 220

Angle between Tangent and Radius Vector, ...... 222

Polar Subtangent and Subnormal, ....... 223

Inverse Curves, . . . . . . . . . . .225

Pedal Curves, 227

Reciprocal Polars, 228

Pedal and Reciprocal Polai of rm = am cos m$, ..... 230

Intercept between point of Contact and foot of Perpendicular, . . 232

Direction of Tangent and Normal in Vectorial Coordinates, . . . 233

Symmetrical Curves, and Central Curves, 236

Examptea. 238

CHAPTER XIII.

ASYMPTOTES.

Points of Intersection of a Curve and a Right Line, .... 240 Method of Finding Asymptotes in Cartesian Coordinates, . . .242 Case where Asymptotes all pass through the Origin, .... 245

Asymptotes Parallel to Coordinate Axes, 245

Parabolic and Hyperbolic Branches, . . . . . . 246

Parallel Asymptotes, . . . . . . . . . -247

The Points in which a Cubic is cut by its Asymptotes lie in a Right Line, 249

Asymptotes in Polar Curves, 250

Asymptotic Circles, 252

Examples, ......•••••• 254

Table of Contents. xiii CHAPTEK XIV.

MULTIPLE POINTS ON CURVES.

Tage

Nodes, Cusps, Conjugate Points, 259

Method of Finding Double Points in general, 26 1

Parabolas of the Third Degree, 262

Double Points on a Cubic having three given lines for its Asymptotes, . 264

Multiple Points of higher Orders, 265

Cusps, in general, .......... 266

Multiple Points on Curves in Polar Coordinates, 267

Examples, •••• 268

CHAPTEK XV.

ENVELOPES.

Method of Envelopes, 270

Envelope of La2 -f 2 Ma + N = O, 271

Undetermined Multipliers applied to Envelopes, 273

Examples, # 276

CHAPTEE XVI.

CONVEXITY, CONCAVITY, POINTS OF INFLEXION.

Convexity and Concavity, . .278

Points of Inflexion, .......... 279

Harmonic Polar of a Point of Inflexion on a Cubic, . . . .281

Stationary Tangents, 282

Examples, 283

CHAPTEK XVII.

RADIUS OF CURVATURE, EVOLUTES, CONTACT,

Curvature, Angle of Contingence, 285

Radius of Curvature, . 286

Expressions for Eadius of Curvature, 287

Newton's Method of considering Curvature, • . . . . .291

Radii of Curvature of Inverse Curves, 295

Radius and Chord of Curvature in terms of r and py .... 295

Chord of Curvature through Origin, 296

E volutes and Involutes, 297

Evolute of Parabola, 298

Evolute of Ellipse, 299

Evolute of Equiangular Spiral, 300

XIV

Table of Contents.

Involute of a Circle, ,™

Radius of Curvature and Points of Inflexion in Polar Coordinates, . .301

Intrinsic Equation of a Curve, ^oi

Contact of Different Orders, [ <,Qt

Centre of Curvature of an Ellipse, * ?07

Osculating Curves, 9 oQg

Radii of Curvature at a Node, . . . . . . . .210

Radii of Curvature at a Cusp, . . . . . , , . 311

At a Cusp of the Second Species the two Radii of Curvature are equal, . 312 General Discussion of Cusps, . . . . . . . * t c

Points on Evolute corresponding to Cusp3 on Curve, . , . .316

Equation of Osculating Conic, . . . . , „ . .317

Examples, r m . ?io

CHAPTEE XVIII.

ON TRACING OF CURVES.

Tracing Algebraic Curves,

Cubic with three real Asymptotes,

Each Asymptote corresponds to two Infinite Branches,

Tracing Curves in Polar Coordinates,

On the Curves rm =«"» cos md,

The Limacon,

The Conchoid,

Examples, ....

322 323 325 328 328

3V 332 333

CHAPTEE XIX.

ROULETTES.

Roulettes, Cycloid, 33-

Tangent to Cycloid, ! ! 336

Radius of Curvature, Evolute, 337

Length of Cycloid, ] * 333

Trochoids, • * . 339

Epicycloids and Hypocycloids, 339

Radius of Curvature of Epicycloid, . 342

Double Generation of Epicycloids and Hypocycloids, .... 343

Evolute of Epicycloid, 344

Pedal of Epicycloid, m 346

Epitrochoid and Hypotrochoid, 347

Centre of Curvature of Epitrochoid, . . . . . '. . 351 Savary's Theorem on Centre of Curvature of a Roulette, . . . .352

Geometrical Construction for Centre of Curvature, 352

Circle of Inflexions, .......... 354

Envelope of a Carried Curve, 355

Centre of Curvature of the Envelope, .... ^57

Table of Contents,

xv

Radius of Curvature of Envelope of a Right Line,

On the Motion of a Plane Figure in its Plane,

Chasles' Method of Drawing Normals, .

Motion of a Plane Figure reduced to Roulettes,

Epicyclics, ....

Properties of Circle of Inflexions,

Theorem of Bobilier,

Centre of Curvature of Conchoid,

Spherical Roulettes,

Examples, • • • •

PAGE

35? 359 360 362 363 367 368

37o 370 372

CHAPTER XX.

ON THE CAKTESIAN OVAL.

Equation of Cartesian Oval, 375

Construction for Third Focus, 376

Equation, referred to each pair of Foci, . 377

Conjugate Ovals are Inverse Curves, 378

Construction for Tangent, . . . . • . . . 379

Confocal Curves cut Orthogonally, 381

Cartesian Oval as an Envelope, 382

Examples, 384

CHAPTER XXI.

ELIMINATION OF CONSTANTS AND FUNCTIONS.

Elimination of Constants, 384

Elimination of Transcendental Functions, 386

Elimination of Arbitrary Functions, 387

Condition that one expression should be a Function of another, . . 389

Elimination in the case of Arbitrary Functions of the same expression, . 393

Examples, 397

CHAPTER XXII.

CHANGE OF INDEPENDENT VAKIABLE,

Case of a Single Independent Variable, 399

Transformation from Rectangular to Polar Coordinates, .... 403

d2 V d2V Transformation of -— - + — — , 404

dx2 dyl

m . .. »&V d2V d2V

Transformation of — — - + — — + — -, 405

dx2 dy2 dz2

Geometrical Illustration of Partial Differentiation, 407

xvi Table of Contents.

PAGE

Linear Transformations for Three Variables, ...... 408

Case of Orthogonal Transformations, ....... 409

General Case of Transformation for Two Independent Variables, . .410

Functions unaltered by Linear Transformations, 411

Application to Geometry of Two Dimensions, 412

Application to Orthogonal Transformations, 414

Examples, 416

CHAPTER XXIII.

SOLID HAKMONIC ANALYSIS.

d?V d2V d2V

On the Equation— +—+— = 0, 418

Solid Harmonic Functions, 419

Complete Solid Harmonics, 421

Spherical and Zonal Harmonics, ........ 423

Complete Spherical Harmonics, . . . . . . . . 427

Laplace's Coefficients, 429

Examples, 432

CHAPTER XXIV.

JACOBIANS.

Jacobians, 433

Case in which Functions are not Independent, 435

Jacobian of Implicit Functions, .... .... 438

Case where / = o, . . . 44 1

Case where a Relation connects the Dependent Variables, . . . 442

Examples, 446

CHAPTER XXV.

GENERAL CONDITIONS FOR MAXIMA AND MINIMA.

Conditions for Four Independent Variables, 447

Conditions f or n Variables, 449

Orthogonal Transformation, ......... 452

Miscellaneous Examples, 454

Note on Failure of Taylor's Theorem, 467

The beginner is recommended to omit the following portions on the Jirst reading :— Arts. 49, 50, 51, 52, 67-85, 88, III, 114-116, 124, 125, Chap, vn., Chap. viii. ; Arts. 159-163, 249-254, 261-269, 296-301, Chaps, xxiii.,

XXIV., XXV.

DIFFEEENTIiL CALCULUS,

CHAPTER I.

FIRST PRINCIPLES DIFFERENTIATION .

i . Functions. — The student, from his previous acquaintance with Algebra and Trigonometry, is supposed to understand what is meant when one quantity is said to be a function of another. Thus, in trigonometry, the sine, cosine, tangent, &c, of an angle are said to be functions of the angle, having each a single value if the angle is given, and varying when the angle varies. In like manner any algebraic expression in x is said to be a function of x. Geometry also furnishes us with simple illustrations. For instance, the area of a square, or of any regular polygon of a given number of sides, is a function of its side ; and the volume of a sphere, of its radius.

In general, whenever two quantities are so related, that any change made in the one produces a corresponding variation in the other, then the latter is said to be a function of the former.

This relation between two quantities is usually represented by the letters F, /, 0, &c.

Thus the equations

U-F(9), *=/(#), I0«0(J?),

denote that u, v, w, are regarded as functions of x, whose values are determined for any particular value of x, when the form of the function is known.

2. Dependent and Independent Variables, Con- stants.— In each of the preceding expressions, x is said to be

6

2 First Principles — Differentiation.

the independent variable, to which any value may be assigned at pleasure ; and «, v, w, are called dependent variables, as their values depend on that of xy and are determined when it is known.

Thus, in the equations

V - 10*, y = x>, y = sm(P,

the value of y depends on that of #, and is in each case deter- mined when the value of x is given.

If we suppose any series of values, positive or negative, assigned to the independent variable x, then every function of x will assume a corresponding series of values. If a quan- tity retain the same value, whatever change be given to x, it is said to be a constant with respect to x. We usually denote constants by a, J, c, &c, the first letters of the alphabet ; variables by the last, viz., u9 v, w, x, y, z.

3. Algebraic and Transcendental Functions. — Functions which consist of a finite number of terms, involving integral and fractional powers of x, together with constants solely, are called algebraic functions — thus

are algebraio expressions.

Functions which do not admit of being represented as ordinary algebraic expressions in & finite number of terms are called transcendental : thus, sin x, cos x, tan a?, 0*, log x, &c, are transcendental functions ; for they cannot be expressed in terms of x except by a series containing an infinite number of terms.

Algebraic functions are ultimately reducible to the follow- ing elementary forms : (1). Sum, or difference (u + v, u - v).

(2). Product, and its inverse, quotient (uv> -J. Powers, and

their inverse, roots (wm, um).

The elementary transcendental functions are also ulti- mately reducible to : (1). The sine, and its inverse, (sin u, sin"1^). (2). The exponential, and its inverse, logarithm (V, log a). •

Limiting Ratios — Derived Functions. 3

4. Continuous Functions. — A function 0 (x) is said to be a continuous function of x, between the limits a and b, when, to each value of x, between these limits, corresponds a finite value of the function, and when an infinitely small change in the value of x produces only an infinitely small change in the function. If these conditions be not fulfilled the function is discontinuous. It is easily seen that all algebraic expressions, such as

atf? + diX71"1 4 . . . . a

ny

and all circular expressions, sin x, tan x, &c, are, in general, continuous functions, as also ex, log x.: &c. In such cases, accordingly, it follows that if x receive a very small change, the corresponding change in the function of x is also very small.

5. Increments* and Differentials. — In the Differen- tial Calculus we investigate the changes which any function undergoes when the variable on which it depends is made to pass through a series of different stages of magnitude.

If the variable x be supposed to receive any change, such change is called an increment ; this increment of x is usually represented by the notation Ax.

When the increment, or difference, is supposed infinitely small it is called a differential, and represented by dx, i.e. an infinitely small difference is called a differential.

In like manner, if u be a function of x, and x becomes x + Ax, the corresponding value of u is represented by u + Ait ; i. e. the increment of u is denoted by Au.

6. limiting Ratios, Derived Functions. — If w be a function of x, then for finite increments, it is obvious that the ratio of the increment of u to the corresponding increment of x has, in general, a finite value. Also when the increment of x is regarded as being infinitely small, we assume that the ratio above mentioned has still a definite limiting value. In the Differential Calculus we investigate the values of these limiting ratios for different forms of functions.

The ratio of the increment of u to that of x in the limit,

when both are infinitely small, is denoted by — . When

ax

B 2

4 First Principles — Differentiation.

u =/(#), this limiting ratio is denoted by /'(#), and is called the first derived function* off(x).

Thus ; let x become x + h, where h = A#, then u becomes

f{x + h), i. e. u 4 Aw =f(x + ti),

.\ Aw =f(x + A) -/(*),

Aw f{x + h)-f{x) A# h

The limiting value of this expression when h is infinitely small is called the first derived function of /(#), and represented

Again, since the ratio — has/' (a?) for its limiting value, if we assume

Ai =' {X) + "

a must become evanescent along with Air ; also — becomes

Ax

— at the same time ; hence we have dx

This result may be stated otherwise, thus : — If ux denote

the value of u when x becomes.^, then the value of the ratio

Ui ~~ u

, when xx - x is evanescent, is called ihe first derived

Xi— x

function of u, and denoted by — . ' J dx

* The method of derived functions was introduced by Lagrange, and the different derived functions off(x) were defined by him to be, the coefficients of the powers of h in the expansion of /(# + h) : that this definition of the first derived function agrees with that given in the text will be seen subsequently.

This agreement was also pointed out by Lagrange. See " Theorie des Fonctions Analytiques," N08. 3, 9.

Algebraic Illustration, 5

If Xi be greater than x, then ux is also greater than u, pro-

vided is positive ; and hence, in the limit, when xx - x

is evanescent, ux is greater or less than u according as — is

cix

positive or negative. Hence, if we suppose x to increase,

then any function of x increases or diminishes at the same

time, according as its derived function, taken with respect

to x, is positive or negative. This principle is of great

importance in tracing the different stages of a function of x,

corresponding to a series of values of x.

7. Differential, and Differential Coefficient, of

Let u =/(#) ; then since

we have du = d(f(x)) = f(x)dx,

where dx is regarded as being infinitely small. In this case dx is, as already stated, the differential of x, and du or f (x) dx, is called the corresponding differential of u. Also f'{x) is called the differential coefficient of /(#), being the coefficient of dx in the differential of f(x).

8. Algebraic Illustration. — That a fraction whose numerator and denominator are both evanescent, or in- finitely small, may have a finite determinate value, is

a na evident from algebra. For example, we have T = — 7 what- 0 0 no

ever n may be. If n be regarded as an infinitely small

number, the numerator and denominator of the fraction

both become infinitely small magnitudes, while their ratio

remains unaltered and equal to -r.

It will be observed that this agrees with our ordinary idea of a ratio ; for the value of a ratio depends on the relative, and not on the absolute magnitude of the terms which compose it.

Again, ff u

na + n2a'

nb + n2b" in which n is regarded as infinitely small, and a, b9 a' and V

6 First Principles — Differentiation.

represent finite magnitudes, the terms of the fraction are both infinitely small,

but their ratio is r>,

b + no

the limiting value of which, as n is diminished indefinitely, is j. Again, if we suppose n indefinitely increased, the limiting value of the fraction is j-r For

a + an a' ab' - ba

b + Vn b' V (b + Vn) '

It 7 /

but the fraction -rrn — ttt diminishes indefinitely as n

t V(b + Vn) J

increases indefinitely, and may be made less than any assignable magnitude, however small. Accordingly the

limiting value of the fraction in this case is Tr

o

9. Trigonometrical Illustration. — To find the values of - — g, and — tj— , when 0 is regarded as infinitely email.

Here - — - = cos 0, and when 9 = o, cos 6 = 1 . tanft

Hence, in the limit, when 0 = o,* we have

sin 6 _ tan# . .. ,.

: a = 1, and, .\ -; — s = 1, at the same time.

tan 0 sin 0

n

Again, to find the value of - — ^, when 6 is infinitely small.

From geometrical considerations it is evident that if 0 be the circular measure of an angle, we have

tan 0 > 6 > sin 0,

tan0 0 or -1 — £ > -r— 5 > 1 ;

sin v sin 0

* If a variable quantity be supposed to diminish gradually, tiUitbe less than anything finite which can be assigned, it is said in that state/wbe\ndefinitely small or evanescent; for abbreviation, such a quantity is often denoted oy,>cypher.

A discussion of infinitesimals, or infinitely small quantities of different orders, will be found in the next Chapter.

Geometrical Illustration.

but in the limit, i.e. when 9 is infinitely small,

tan0 SnU ~ *'

and therefore, at the same time, we have

6

sin0

= i

This shows that in a circle the ultimate ratio of an arc to its chord is unity, when they are both regarded as evanescent.

10. Geometrical Illustration. — Assuming that the relation y = f(x) may in all cases be represented by a curve, where

y = /(*)

expresses the equation connecting the co-ordinates (x, y) of each of its points ; then, if the axes be rectangular, and two points (x, y), (%l9 yx) be taken on the curve, it is obvious

that represents the tangent of the angle which the

X\ ~~ x

chord joining the points (#, y), (xly yx) makes with the axis of x.

If, now, we suppose the points taken infinitely near to each other, so that xY - x becomes evanescent, then the chord becomes the tangetit at the point (#, y), but

— — - becomes — - or f (x) in this case.

xx - x ax

Hence, f (x) represents the trigonometrical tangent of the angle tvhich the line touching the curve at the point (x, y) makes ivith the axis of x. We see, accordingly, that to draw the tangent at any point to the curve

V - /(*)

is the same as to find the derived function f(x) of y with respect to x. Hence, also, the equation of the tangent to the curve at a point (x, y) is evidently

y-Y = f{x){x-X), (2)

where X, Y are the current co-ordinates of any point on the

8

First Principles — Differentiation,

tangent. At the points for which the tangent is parallel to the axis of x, we have f (x) = o ; at the points where the tangent is perpendicular to the axis, f (x) = oo . For all other points f (x) has a determinate finite real value in general. This conclusion verifies the statement, that the ratio of the increment of the dependent variable to that of the independent variable has, in general, a finite determinate magnitude, when the increment becomes infinitely small.

This has been so admirably expressed, and its con- nexion with the fundamental principles of the Differential Calculus so well explained, by M. Navier, that I cannot for- bear introducing the following extract from his "Lecons d' Analyse": —

" Among the properties which the function y = f(x), or the line which represents it, possesses, the most remarkable — in fact that which is the principal object of the Differential Calculus, and which is constantly introduced in all practical applications of the Calculus — is the degree of rapidity with which the function /(#) varies when the in- dependent variable x is made to vary from any assigned value. This degree of rapidity of the increment of the function, when x is altered, may differ, not only from one function to another, but also in the same function, ac- cording to the value attributed to the variable. In order to form a precise notion on this point, let us attribute to x a deter- mined value represented by ON, to which will correspond an equally determined value of y, represented by PN. Let us now suppose, starting from this value, that x increases by any quantity denoted by Ax, and represented by NM, the function y will vary in consequence by a certain quantity, denoted by Ay, and we shall have

y + Ay = f(x + Ax), or Ay = f(x + Ax) -/(#).

The new value of y is represented in the figure by Q3f, and QL represents Ay, or the variation of the function.

Fig.

Geometrical Illustration. g

The ratio — of the increment of the function to that of

Ax

the independent variable, of which the expression is

/Qg+ Ax) -/(a?) Ax '

is represented by the trigonometrical tangent of the angle

QPL made by the secant PQ with the axis of x.

Ay 44 It is plain that this ratio — is the natural expression

of the property referred to, that is, of the degree of rapidity with which the function y increases when we increase the independent variable x ; for the greater the value of this ratio, the greater will be the increment Ay when x is in- creased by a given quantity Ax. But it is very important

Av to remark, that the value of — - (except in the case when

Ax

the line PQ becomes a right line) depends not only on the value attributed to x, that is to say, on the position of P on the curve, but also on the absolute value of the increment Ax. If we were to leave this increment arbitrary, it would be

impossible to assign to the ratio — any precise value, and

L±X

it is accordingly necessary to adopt a convention which shall

remove all uncertainty in this respect.

14 Suppose that after having given to Ax any value, to

which will correspond a certain value Ay and a certain

direction of the secant PQ, we diminish progressively the

value of Ax> so that the increment ends by becoming

evanescent ; the corresponding increment Ay will vary in

consequence, and will equally tend to become evanescent.

The point Q will tend to coincide with the point P, and the

secant PQ with the tangent PT drawn to the curve at the

Ay point P. The ratio — of the increments will equally

approach to a certain limit, represented by the trigonometrical tangent of the angle TPL made by the tangent with the axis of x.

"We accordingly observe that when the increment Ax}

io First Principles — Differentiation,

and consequently Ay, diminish progressively and tend to

A?/ vanish, the ratio — of these increments approaches in

general to a limit whose value is finite and determinate.

A?/ Hence the value of — corresponding to this limit must be

i\X

considered as giving the true and precise measure of the rapidity with which the function f (x) varies when the independent variable x is made to vary from an assigned value ; for there does not remain anything arbitrary in the expression of this value, as it no longer depends on the absolute values of the increments Ax and Ay, nor on the figure of the curve at any finite distance at either side of the point P. It depends solely on the direction of the curve at this point, that is, on the inclination of the tangent to the axis of x. The ratio just determined expresses what Newton called the fluxion of the ordinate. As to the mode of finding its value in each particular case, it is sufficient to consider the general expression Ay /{x + Ax) ./{x)

Ax Ax

and to see what is the limit to which this expression tends, as Ax takes smaller and smaller values and tends to vanish. This limit will be a certain function of the independent variable #, whose form depends on that of the given function

f(x) "We shall add one other remark ; which is, that

the differentials represented by dx and dy denote always quantities of the same nature as those denoted by the variables x and y. Thus in geometry, when x represents a line, an area, or a volume, the differential dx also represents a line, an area, or a volume. These differentials are always supposed to be less than any assigned magnitude, however small ; but this hypothesis does not alter the nature of these quantities : dx and dy are always homogeneous with x and y, that is to say, present always the same number of dimensions of the unit by means of which the values of these variables are expressed." 10a. Iiimit of a Variable Magnitude. — As the con- ception of a limit is fundamental in the Calculus, it may be well to add a few remarks in further elucidation of its meaning : —

Limit of a Variable Magnitude. 1 1

In general, when a variable magnitude tends continually to equality ivith a certain fixed magnitude, and approaches nearer to it than any assignable difference, hoicever small, this fixed magni- tude is called the limit of the variable magnitude. For example, if we inscribe, or circumscribe, a polygon to any closed curve, and afterwards conceive each side indefinitely diminished, and consequently their number indefinitely increased, then the closed curve is said to be the limit of either polygon. By this means the total length of the curve is the limit of the perimeter either of the inscribed or circumscribed polygon. In like manner, the area of the curve is the limit to the area of either polygon. For instance, since the area of any polygon circumscribed to a circle is obviously equal to the rectangle under the radius of the circle and the semi-perimeter of the polygon, it follows that the area of a circle is repre- sented by the product of its radius and its semi-circumfe- rence. Again, since the length of the side of a regular polygon inscribed in a circle bears to that of the correspond- ing arc the same ratio as the perimeter of the polygon to the circumference of the circle, it follows that the ultimate ratio of the chord to the arc is one of equality, as shown in Art. 9. The like result follows immediately for any curve.

The following principles concerning limits are of fre- quent application: — (1) The limit of the product of two quan- tities, which vary together, is the product of their limits; (2) The limit of the quotient of the quantities is the quotient of their limits.

For, let P and Q represent the two quantities, and p and q their respective limits ; then if