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AN ELEMENTARY TREATISE
ON
THE DIFFEEENTIAL CALCULUS.
AN ELEMENTARY TREATISE
ON
THE INTEGRAL CALCULUS,
CONTAINING
APPLICATIONS TO PLANE CURVES AND SURFACES,
AND
A CHAPTER ON THE CALCULUS OP VARIATIONS.
BY
BENJAMIN WILLIAMSON, D.C.L., F.R.S.
SEVENTH EDITION.
AN INTRODUCTION TO THE THEORY
OF
STRESS AND STRAIN OF ELASTIC SOLIDS.
BY
BENJAMIN WILLIAMSON, D.C.L., F.R.S.
AN ELEMENTARY TREATISE ON DYNAMICS.
CONTAINING
APPLICATIONS TO THERMODYNAMICS.
BY
BENJAMIN WILLIAMSON, D.C.L., F.R.S.,
AND
FRANCIS A. TARLETON, LL.D.
SECOND EDITION.
AN ELEMENTARY TREATISE
THE DIFFERENTIAL CAIOFLUS,
CONTAINING
THE THEORY OF PLANE CURVES,
NUMEROUS EXAMPLES.
BY
BENJAMIN WILLIAMSON, D.C.L., F.R.S.,
SENIOR FELLOW OF TRINITY COLLEGE, DUBLIN.
NINTH EDITION, REVISED.
LONGMANS, GREEN, AND CO.,
PATERNOSTER ROW, LONDON;
NEW YORK AND BOMBAY.
1899.
[all kights rkserved.]
c:Zi>l^C ^^^C-t^l^
Printed at the
By Ponsonby a Weldrick.
:^;y^a^^
PREFACE. l\^na^
In the following Treatise I have adopted the method of
Limiting Eatios 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 Greometry 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 M^canique 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 myself 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 Curves. 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 Metajohysique du Calcul 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 merite essentiel, le sublime, on pent le dire, de la
methode infinitesimale, est de r^unir la facilite des procedes
ordinaires d'un simple calcul d'approximation a Fexactitude
des resultats de 1' analyse ordinaire. Cet avantage immense
serait perdu, ou du moins fort diminue, si k cette methode
pure et simple, telle que nous I'a donnee Leibnitz, on voulait,
sous I'apparence d'une plus grande rigueur soutenue dans
tout le cours de calcul, en substituer d'autres moins naturelles,
Preface, vii
moins commodes, moins conformes k la.<,marclie probable
des inventeurs. Si cette methode est exacte dans les re-
Bultats, 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 detoumes et compliqucs pour
la suppleer? Pourquoi se contenter de I'appuyer but des
inductions et sur la conformite de ses resultats avex ceux que
foumissent les autres methodes, lorsqu'on pent la demontrer
directement et generalement, plus facilement peut-etre
qu'aucune de ces metbodes elles-memes ? Les objections que
I'on 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 n^cessairement, 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.
CHAPTEE I.
FIRST PEINCIPLES. DIFFERENTIATION.
Page
Dependent and Independent Variables, i
Increments, Differentials, Limiting Ratios, Derived Functions, . . 3
Differential CoeflBlcients, 5
Geometrical Illustration, 6
Navier, on the Fundamental Principles of tlie 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, 17
Differentiation of Circular Functions, 19
Geometrical Illustration of Differentiation of Circular Functions, . .22
Differentiation of a Logarithm, 24
Differentiation of an Exponential, 2b
Logarithmic Differentiation, 27
Examples, 3<^
CHAPTEE n.
SUCCESSIVE DIFFERENTIATION.
Successive Differential Coefficients, 34
Infinitesimals, . . ■ 3^
Geometrical Illustrations of Infinitesimals, 37
Fundamental Principle of the Infinitesimal Calculus, . . . .40
Subsidiary Principle, 4^
Approximations, 4^
Derived Functions of x^, 4^
Differential Coefficients of an Exponential, 4^
Differential Coefficients of tan-i a:, and tan-i-, 50
X
Theorem of Leibnitz, 5'
AppUcatious of Leibnitz's Theorem, 53
Examples, ... •• 57
X Table oj Contents,
CHAPTER III.
DEVELOPMENT OF FUNCTIONS,
PAGH
Taylor's Expansion, 6i
Binomial Theorem, 63
Logaritlimic Series, ..63
Maclaurin's Theorem, 64
Exponential Series, 65
Expansions of sin x and cos a?, 66
Huy gens' Approximation to Length of Circular Arc, . . . .66
Expansions of tan"^ a; and sin"^ a:, 68
Euler's Expressions for sin x and cos x, 69
John Bernoulli's Series, 7°
Symbolic Form of Taylor's Series, 70
Convergent and Divergent Series, 73
Lagrange's Theorem on the Limits of Taylor's Series, . . . .76
Geometrical Illustration, 7^
Second Form of the Eemainder, 79
General Form of Maclaurin's Series, 81
Binomial Theorem for Fractional and Negative Indices, . . . .82
Expansions by aid of Differential Equations, • ^5
Expansion of sin mz and cos mz, 87
Arbogast's Method of Derivation, 88
Examples, 91
CHAPTER IV.
INDETERMINATE FORMS.
Examples of Evaluating Indeterminate Forms without the Differential
Calculus, ; 96
Method of Differential Calculus, 99
Form o X 00 , 102
Form g-, 103
Formso% 00°, i*« 105
Examples, 109
CHAPTER V.
PARTIAL DIFFERENTIAL COEFFICIENTS.
Partial Differentiation, I13
Total Differentiation of a Function of Two Variables, . . . .115
Total Differentiation of a Function of Three or more Variables, . .117
Differentiation of a Function of Differences, 119
Implicit Functions, Differentiation of an Implicit Function, , , .120
Euler's Theorem of Homogeneous Functions, 123
Examples in Plane Trigonometry, 130
Landen's Transformation, 133
Examples in Spherical Trigonometry, • I33
Legendre's Theorem on the Comparison of Elliptic Functions, . . * 137
Examples, 140
Table of Contents, n
CHAPTER VI. ..
SUCCESSIVE PARTIAL DIFFERENTIATION,
The Order of Differentiation is indifferent in Independent Variables,
Condition that Fdx + Qdy should be an exact Differential,
Euler's Theorem of Homogeneous Functions,
Successive Differential Coefficients oi <p(x + at, y + fit).
Examples, ,
PAGE
146
148
148
CHAPTER VII.
Lagrange's theorem.
Lagrange's Theorem, •••.151
Laplace's Theorem, 154
Examples, 155
CHAPTER VIII.
EXTENSION OF TAYLOR 'S THEOREM.
Expansion of (^ (o; + /i, y + A-) , 156
Expansion oi (p (x + h, y + k, z + I), 159
Symbolic Forms, •...160
Euler's Theorem, 162
CHAPTER IX.
MAXIMA AND MINIMA FOR A SINGLE VARIABLE.
Geometrical Examples of Maxima and Minima, 164
Algebraic Examples, 165
Criterion for a Maximum or a Minimum, 169
Maxima and Minima occur alternately, 173
Maxima or Minima of a Quadratic Fraction, 177
Maximum or Minimum Section of a Right Cone, 181
Maxima or Minima of an Implicit Function, 185
Maximum or Minimum of a Function of Two Dependent Variables, . 186
Examples, 188
CHAPTER X.
MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE VARIABLES.
Maxima and Minima for Two Variables, . . . * • • '91
Lagrange's Condition in the case of Two Independent Variables, . .191
xii Table of Contents.
PAGB
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
CHAPTEE XI.
METHOD or UNDETERMINED MULTIPLIERS APPLIED TO MAXIMA AND
MINIMA.
Method of Undetermined Multipliers, 204
Application to find the principal Eadii of Curvature on a Surface, . . 208
Examples, 210
CHAPTER XII.
ON TANGENTS AND NORMALS TO CURVES.
Equation of Tangent, 212
Equation of Normal, 215
Subtaii 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
Reciproca'. Polars, • 228
Pedal and Reciprocal Polai of r*" = a"* cos md, ..... 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
Example 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
CHAPTEE XIV.
MULTIPLE POINTS ON CURVES,
Pagb
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 genera], 266
Multiple Points on Curves in Polar Coordinates, 267
Examples, ••••••• 268
CHAPTEB XV.
ENVELOPES.
Method of Envelopes, 270
Envelopeof Zo2+2Jfa + iV=o, 271
Undetermined Multipliers applied to Envelopes, . . . . .273
Examples, •• 276
CHAPTEE XVI.
CONVEXITY, OONOAVITY, 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
CHAPTEE XVII.
RADIUS OF CURVATURE, EVOLUTES, CONTACT,
Curvature, Angle of Contingence, 285
Eadius 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 p, , , • . 395
Chord of Curvature through Origin, 296
Evolutes and Involutes, 297
Evolute of Parabola, 298
Evolute of ElHpse, 299
Evolute of Equiangular Spiral, . 3^0
xiv Table of Contents,
Pagh
Involute of a Circle, • . 300
Eadius of Curvature and Points of Inflexion in Polar Coordinates, . .301
Intrinsic Equation of a Curve, 304
Contact of Different Orders, 304
Centre of Curvature of an Ellipse, 307
Osculating Curves, . 309
Radii of Curvature at a Node, 310
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, 315
Points on Evolute corresponding to Cusps on Curve, . . . .316
Equation of Osculating Conic, 317
Examples, 319
CHAPTER XVIII.
ON TRACING OF CURVES.
Tracing Algebraic Curves, 322
Cubic with three real Asymptotes, 323
Each Asymptote corresponds to two Infinite Branches, .... 325
Tracing Curves in Polar Coordinates, 328
On the Curves r*» =«»» cos »i0, 328
The Lima9on, 331
The Conchoid, • 332
Examples, 333
CHAPTER XIX.
ROULETTES.
Roulettes, Cycloid, • . 335
Tangent to Cycloid, 336
Radius of Curvature, Evolute, 337
Length of Cycloid, 338
Trochoids, 339
Epicycloids and Hypoeycloids, 339
Radius of Curvature of Epicycloid, ....... 342
Double Generation of Epicycloids and Hypoeycloids, .... 343
Evolute of Epicycloid, 344
Pedal of Epicycloid, 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
Pack
Radius of Curvature of Envelope of a Right Line, . .^ • . . 358
On the Motion of a Plane Figure in its Plane, . V* . . . 359
Chasles' Method of Drawing Normals, 360
Motion of a Plane Figure reduced to Roulettes, 362
Epicyelics, 363
Properties of Circle of Inflexions, 367
Theorem of Bobilier, 368
Centre of Curvature of Conchoid, 370
Spherical RoulQttes, 37°
Examples, 372
CHAPTEE XX.
ON THE CARTESIAN 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, •••• 3^4
CHAPTEK XXI.
ELIMINATION OF CONSTANTS AND FUNCTIONS.
Elimination of Constants, 3^4
Elimination of Transcendental Functions, 3^6
Elimination of Arbitrary Functions, . . _ 3^7
Condition that one expression should he a Function of another, . . 389
Elimination in the case of Arbitrary Functions of the same expression, . 393
Examples, . , 397
CHAPTER XXII.
CHANGE OF INDEPENDENT VARIABLE.
Case of a Single Independent Variable, 399
Transformation from Rectangular to Polar Coordinates, .... 403
cPV d^V
Transformation of —r^ + ——„ • • 4^4
dx^ dxp-
d'^V d^V d^V
Transformation of -— r- + — ^ + -— -, 4^5
dx^ dip- dv-
Geometrical Illustration of Partial Differentiation, 4^7
xvi Table of Contents.
PAGB
Linear Transformations for Tliree 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 HARMONIC ANALYSIS.
cPV d^V drV
OntheEguation^ + ^+^ = 0 ^li
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.
JACOBIA27S.
Jacobians, 433
Case in which Functions are not Independent, 435
Jacobian of Implicit Functions, . . .438
Case where /= o, 441
Case where a Relation connects the Dependent Variables, . . ' . 442
Examples, 446
CHAPTER XXV.
GENERAL CONDITIONS TOR MAXIMA AND MINIMA,
Conditions for Four Independent Variables, 447
Conditions for n Variables, 449
Orthogonal Transformation, 452
Miscellaneous Examples, . . 454
S"oTE ON Failure of Taylor's Theorem, 467
The beginner is recommended to omit the following portions on the^r*^
reading: — Arts. 49, 50, 51, 52, 67-85, 88, iii, 114-116, 124, 125, Chap, vii..
Chap. VIII. ; Arts. 159-163, 249-254, 261-269, 296-301, Chaps, xxm.,
XXIV., XXV.
DIFFERENTIAL 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{x), €>=/(«?), 'tr = 0(^),
denote that w, v, w, are regarded as functions of Xy whose
values are determined for any particular value of a?, 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
B
2 ^^ First Principles — Differentiation.
the independent variable, to which any value may be assigned
at pleasure ; and u, 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
y= lo*, y = 0^^ y = Qmx,
the value of y depends on that of a?, 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, h, c, &o., the first letters of the alphabet ;
variables by the last, viz., u, v, w, a?, y, z.
3. Algebraic and Transcendental Functions. —
Functions which consist of a finite number of terms, involving
integral and fractional powers of Xj together with constants
solely, are called algebraic functions — thus
are algebraic expressions.
Functions which do not admit of being represented as
ordinary algebraic expressions in a finite number of terms are
called transcendental : thus, sin Xj cos Xy tan x, e*, 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 : (i). Sum, or difference {u -^ VyU - v),
(2). Product, and its inverse, quotient (uvy - j. Powers, and
their inverse, roots (w"*, w"*).
The elementary transcendental functions are also ulti-
mately reducible to : (i). The sine, and its inverse, (sin w,
sin~*w). (2). The exponential, and its inverse, logarithm
Klogu).
Limiting Batios— ^Derived Functions. 3
^ 4. Continuous Functions. — A function 0 (x) is said to
be a continuous function of x, between the limits a and J,
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
Ooa^ + «iaj^* +....«„,
and all circular expressions, sin x, tan x, &c., are, in general,
continuous functions, as also 6*, 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 a; is usually
represented by the notation ^x.
"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 w be a function of a?, and x becomes
X + Ao?, the corresponding value of u\s> represented by w + Aw;
i. e. the increment of u is denoted by Aw.
y<' 6. liimiting 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
B 2
4 First Principles — Differentiation.
u =f[x)y this limiting ratio is denoted by /'(a?), and is called
the.^rs^ derived function* oif(x].
Thus ; let x become x + hj where h = Ax, then u becomes
f{x + h)f i. e. w + Au -f{x + A),
/. Aw =f{x + h) -f{x)y
Au _f{x + h)-f{x)
Ax h '
The limiting value of this expression when h is infinitely small
is called the first derived function of f[x), and represented
by/(^).
Again, since the ratio — has/' {x) for its limiting value,
if we assume
£ miist become evanescent along with Ax ; also — - becomes
° Ax
-r- at tlito same time ; hence we have
dx
This result may be stated otherwise, thus : — If Wi denote
the value of u when x becomes Xi, then the value of the ratio
fi. — or
, when Xi- X \% evanescent, is called ihe first derived
Xi- X
du
function of u, and denoted by -j-.
* The method of derived functions was introduced by Lagrange, and the
different derived functions of/ [x) were defined by him to be, the coefficients of
the powers of A in the expansion of/{a;+A): that this definition of the fijst
derived function agrees with that given in the text will be seen subsequently.
This agreement was also pointed out by Lagrange. See "Theorie dea
Fonctions Analytiques," N°*. 3, 9.
Algebraic Illustration. 5
If Xi be greater than a?, then Ui is also greater than u, pro-
vided — is positive ; and hence, in the limit, when Xx - x
Xi - X ^ ^w
is evanescent, Wi is greater or less than u according as — is
dx
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 Bifferential Coefficient, of
m-
Let u =f{x); 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 /' {x) dx, is called the corresponding differential of u.
Also /' (x) is called the differential coefficient of /(a?), 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
evident from algebra. For example, we have t = -7 what-
ever n may be. Jl 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 t-
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.
... - wflj + n^a'
Agam, :f w « -^ ^,
° nb + n^o
in which n is regarded as infinitely small, and a, h, d and V
6 First Principles — BifferentiaUon.
represent finite magnitudes, the terms of the fraction are
both infinitely small,
but their ratio is ■; 7^
b + nh
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-,. For
a + a'n a ah' - ha'
h + h'n V b'{h + h'n) '
but the fraction -^-r-Tz — 77-- diminishes indefinitely as n
^ h'(b + h'n) -^
increases indefinitely, and may be made less than any
assignable magnitude, however small. Accordingly the
limiting value of the fraction in this case is -,.
0
< 9. Trigonometrical Illustration. — To find the values
of 7 — ^, and — n~, when 0 is regarded as infinitely email.
Here - — j. = cos0, and when 9 = o, eosO = i.
tan 6
Hence, in the limit, when 6^ = o,* we have
sin 0 , tan 0 , ,, , .
7 — 7T = I, and, .*. -; — j: = I, at the same time,
tan 0 sm 0
a
Again, to find the value of -: — -x, when 0 is infinitely small.
sm 0
From geometrical considerations it is evident that if d be
the circular measure of an angle, we have
tan 0 > 0 > sin 0,
tan0 e
or -7— 5 > -7—^ > I ;
sm ff smu
* If a variable quantity be supposed to diminish gradually, till it be less than
anything Jinite which can be assigned, it is said in that state to be indefinitely
small or evanescent; for abbreviation, such a quantity is often denoted by cypher.
A discussion of infinitesimals, or infinitely small quantities of diflferent orders,
will be found in the next Chapter.
Geometrical Illustration,
but in the limit, i.e. when 0 is infinitely small,
tan0 __
sin e" ^'
and therefore, at the same time, we have
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.
lo. Oeometrical 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 (ic, y)
of each of its points ; then, if the axes be rectangular, and
two points {x, y), {xi, y^ be taken on the curve, it is obvious
that — represents the tangent of the angle which the
chord joining the points {x, y), {xi, y^ makes with the axis
of X.
If, now, we suppose the points taken infinitely near to
each other, so that Xi- x becomes evanescent, then the chord
becomes the tangent at the point (a?, y), but
t}—l. becomes -f ot f (x) in this case.
Xi- X ax
Hence, f {x) represents the trigonometrical tangent of the
angle which the line touching the curve at the point {x, tj) makes
with the axis of x. We see, accordingly, that to draw the
tangent at any point to the curve
y = /(^)
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
s
First Principles — Differentiation.
tangent. At the points for which the tangent is parallel to
the axis of x, we have /' (cc) = o ; at the points where the
tangent is perpendicular to the axis, f (a?) = oo . For all
other points /' (cc) 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 / {x) 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 a; a deter-
mined value represented by OiV", to which will correspond
an equally determined value of y, represented by PJ^. 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 = fix + Ax), or A^^ = /(a; + Ax) -fix).
The new value of y is represented in the figure by QM,
and QJj represents Ay, or the variation of the function.
Fig. I.
Geometrical Illustration, 9
The ratio — ^ of the increment of the function to that of
the independent variable^ of which the expression is
is represented by the trigonometrical tangent of the angle
QPL made by the secant PQ with the axis of x,
" It is plain that this ratio — - is the natural expression
i^X
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 ^y when x is in-
creased by a given quantity Aa;. But it is very important
Ay
to remark, that the value of — ^ (except in the case when
l^X
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.
" 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
point P. The ratio — of the increments will equally
Ax
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,
lo First Principles — Differentiation.
and consequently Ay, diminisli progressively and tend to
vanisli, the ratio — ^ of these increments approaches in
A^
general to a limit whose value is finite and determinate.
A'?/
Hence the value of — corresponding to this limit must be
considered as giving the true and precise measure of the
rapidity with which the function f {x) varies when the independent
variable x is ?nade 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 Arc and Ay, nor on the figure of the cui've 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 ^y /(^^A.)-/(^)
Acc Aa; '
and to see what is the limit to which this expression tends,
as ^x takes smaller and smaller values and tends to vanish.
This limit will be a certain function of the independent
variable x, 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."
loa, liimit 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 : —
Idmit of a Variable Magnitude.
In general, when a variable magnitude tends continualm to
equality ivith a certain fixed magnitude, and approaches nearer to
it than any assignable difference, however 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 eacli side indefinitely diminished,
and consequently tlieir 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: — (i) 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
P=p^a, Q = g + (B,
a and j3 denote quantities which diminish indefinitely as P
and Q approach their limits, and which become evanescent
in the limit.
Again, we have
FQ =pq +pf5 + qa + aj3.
Accordingly, in the Hmit, we have
PQ-pq-
12 First Principles — Differentiation,
.. P p + a p qa - pQ
Again, -pr =^ — 3 = - + ^7 — ^.
The mimerator of the last fraction becomes evanescent in
the limit, while the denominator becomes q^, and consequently
the limit of — is -.
Q q
1 1 . DiflTerentiation. — The process of finding the derived
function, or the differential coefficient of any expression, is
called differentiating the expression.
We proceed to explain this process by applying it to a
few elementary examples.
Examples.
I. y = x\
Substitute x + h for x, and denote the new value of y by yi, then
tn = {x+ hy = a;2 + 2xh + h^ ;
.*. — ; — or — ^ = 2a; + A.
h Ax
If h be taken an infinitely small quantity, we get in the limit
dy
~= 2X]
dx
or if /(«) = ^^1 we have/' {x) = 2X.
„ I
Here yi =
«+ A
I I
yi-y
X + h X X (x + h)*
y\-y or ^ = -
A ' Ax x{x +h)''
which equation, when h is evanescent, becomes
dy i_ \x/ i_
^" x^' ^ dx ~ x"
t
Differentiation of a Product. 1 3
12. Diirerentiation of the Algebraic Sum of a
Finite ^Rf umber of Functions. — Let "•
y = u + V - w -\- &Q,\
then, \ixi = x ■\- h, we get
yi = Wi + «?! - «ri + . . . ;
. yi - y _ 'Wi - ^ «^i - -^^ Wi^W
''' T" - "T~ "^ ~1 F" "*"•••'
which becomes in the limit, when h is infinitely small,
dy du dv dw
dx dx dx dx
Hence, if a function consist of several terms, its derived
function is the sum of the derived functions of its several parts,
taken with their proper signs.
It is evident that the differential of a constant is zero.
13. DifTerentiation of tbe Product of Two Func-
tions.— Let y = uv, where u, v, are both functions of x ; and
suppose Ay, Au, Av, to be the increments of y, u, v, corre-
sponding to the increment Ax in x. Then
Ay = (w + Au^{v + Av) - uv
= uAv + vAu + Au Av,
Ay Av , . Au .
or — = u — + (v + Av) — .
Aa; AiZJ Ax
Now suppose Ax to be infinitely small, then
Ay Av Au
Ax* Ax* Ax^
become in the limit
dy dv J ^^ .
dx* dx^ dx *
also, since Av vanishes at the same time, the last term dis-
appears from the equation, and thus we arrive at the result
dy dv du , .
/ = w — + «? — . (3)
dx dx dx ^
14 First Principles — Differentiation.
Hence, to difPerfentiate the product of two functions,
multiply each of the factors hy the differential coefficient of the
other, and add the products thus found.
Otherwise thus: let /(a;), 0 {x)j denote the functions, and
h the increment of x, then
Vi =/(a? + h) <^{x-\-h);
. yi-y y{^ + h)ip(x + h) -fix) 0 [x)
" h h
Now, in the limit,
f{x->rh)-f[x) ^,. , ,^ ,.
and (l){x + h) - <l> ix) ,. .
J = 0 W»
and, accordingly,
|=/(*)^'(!fc) + ^(<r)/(«,),
which agrees with the preceding result.
When y = au, where « is a constant with respect to x,
we have evidently
dy du
dx dx
14. Differentiation of the Product of any IVuniber
of Functions. — First let
y = uvw;
suppose VIC = Sf
then y = us,
and, by Art. 13, we have
dy dz du
dx ir dx
Differentiation of a Quotient. 15
but, by the same Article,
dz dv dw
hence
dx dx dx^
dy du dv dw
-f- = vw -rr + wu -rr -^ uv -—.
dx dx dx dx
This process of reasoning can be easily extended to any
number of functions.
The preceding result admits of being written in the form
16?^ I du I dv I dw
y dx u dx vdx w dx^
and in general, if y = yi . 2^a . 2^3 .... ^n,
it can be easily proved in like manner that
i ^ = i ^ + i ^^ + 1^ (a)
y dx yx dx y^dx' ' ' yn dx '
15. DiflTerentiation of a Cinotient — Let
y = -, then u = yv;
^1 i. r A_z ^^ dv dy
therefore, by Art. 13, ^ = 2^^^^^^^
dy du dv du udv
dx dx dx dx vdx
dx
dx
V
du
'Tx
" ■■ 9
dv
dy _ dx djc / (5)
* ' dx v^
This may be written in the following form, which is often
useful :
d (u\ 1 du u dv
dx\v] V dx v^ dx'
1 6 First Principles — Differentiation.
Hence, to differentiate a fraction, multiply the denominator
into the derived function of the numerator, and the numerator into
the derived function of the denominator ; take the latter product
from the former, and divide hy the square of the denominator.
In the particular case where w is a constant with respect
to X {a suppose), we obviously have
d fa\ a dv /^\
dx \vj v^ dx
Examples.
a — x ^ du
I. u = . An».
a + x dx {a + xy-
2. « = (a + ») (4 + a?). -— = a + b-\- 2x.
ax
V ' 1 6. Differentiation of an Integral Power. — Let
^ ' ' y = x^, where w is a positive integer.
Suppose yi to be the value of y, when x becomes a^i, then
■» = Xi + xxi + . . . + ar ^.
Xi- X Xi- X
Now, suppose Xi- X to be evanescent. In this case we
may write x for Xi in the right-hand side of the preceding
equation, when it becomes nx''^~^; but the left-hand side, in
the limit, is represented by ~ •
Hence -^ = mxf^K
dx
or — V— ^ = wa^ .
dx
This result follows also from Art. 14 ; for, making
2^1 = ^2 = ^3 = . . . = yn = w,
we evidently get from (4),
d (w") ^, du f.
-^ = nu"^'—. (7)
ax ax
This reduces to the preceding on making u ^ oj.
/
Differentiation of a Function of a Function. 1 7
17. nifferentiatioii of a Fractional Power. — Let
then jr = t*-, and-^i = -^^
hence, by (7),
^ dx dx'
d (u") dy _ mu^'^du m ^-idu . .
dx dx n y"~* dx n dx'
y = «""*, then 2^ = — , and by (6) we get
id
18. Differentiation of a ]¥egative Power. — Let
(6)
^(«-") ;^ '»«-™-'^- (9)
Combining the results established in (7), (8), and (9), we
find that
-\ ^ = mu'f*'^ —
dx dx
for all values of m, positive, negative, or fractional. When
applied to the differentiation of any power of x we get the
following rule : — Diminish the index by unity, and multiply the
power of x thus ohtained by the original index ; the result is the
required differential coefficient, with respect to x.
19. Differentiation of a Function of a Function. —
Let y = f{x) and u = <p(y), to find — . Suppose yi, Ui, to be
ax
the values of y and u corresponding to the value Xi for x ;
then if Ay, Aw, Ax, denote the corresponding increments,
we have evidently
Ui-u ^ Ux-u yi-y
Xx-x yi- y xi-x^
or
Au Au Ay
Ax Ay Ax'
1 8 First Principles — Differentiation.
As this relation holds for all corresponding increments,
however small, it must hold in the limit,* when AiP is
evanescent ; in which case it becomes
du _ dudy y v
dx dy dx
Hence the derived function with respect to x of u is the
product of its derived with respect to y ; and the derived of y
wi(h respect to x.
20. DiflTerentiation of an Inverse Function. — To
prove that
dx I
dy ~ dy^
dx
Suppose that from the equation
y-m («)
the equation
is deduced, and let jTi, yi, be corresponding values of x^ y,
which satisfy the equation (a), it is evident that they will
also satisfy the equation {b). But
yi-y ^ xi-x ^
Xx-x y,-y
As this equation holds for all finite increments, it must
hold when Xi - x and yi- y are infinitely small ; therefore
we have in the limit
^.^ = I. (II)
dx dy
The same result may also be arrived at from Art. 19,
as follows : —
When y = f{x), and u = <}>{y),
* The Student will observe that this is a case of the principle (Art. joa) that
the limit of the product of two quantities is equal to the product of their limits.
Differentiation of sin X, 19
we have, in all cases,
du du dy
dx dy dx
This result must still hold in the particular case when u = x,
in which case it becomes
dxdy
dydx
Examples.
1. « = (a«-a;2)ft.
Let a' - «2 _s y^ then « = y*,
- = 5S^,ana-.-«.
Hence — = - loa? (a* - a;*)*.
dx ^ '
2, u= {a + bx^)K Ans. ^ = izbx* {a + hi^)\
3. « = (i + af2)».
4. w = (i +a;»»)*. — = mwa:«-i(i+a:«)»»-K
^ ~ (i +z^)i'
du
dx
We next proceed to determine the derived functions of
the elementary trigonometrical and circular functions.
21. DifTerentiation of sin a;. — Let
y = sin iP, ^1 = sin {x + h)y
h
y\- y sin (a? + ^) - sin x _ " """' 2
~A h h
. h
sm-
But by Art. 9, the limit of — r— = i ; moreover, the limit of
... ( h\
2 sm - cos \x-\- -\
(a? + -Jis cos«.
cos
oa
2 Bin - sin [ a? + - j
zo First Principles — Differentiation,
Hence ^ = cos^g, (12)
22, DifTerentiation of cos a;.
y = cos Xf pi = cos (a; + A),
. h .
/ T \ A riin —
i/i - p _ coa (x + h) - ooB X 2
~~h h " h
Hence, in the limit,
c? cos a? . . .
-^- = -sm*. (.3)
This result might be deduced from the preceding, by substi-
tuting — s f or a;, and applying the principle of Art. 19.
It may be noted that (12) and (13) admit also of being
written in the following symmetrical form : —
^sina; . / ttN
«?cosa; / 7r\
— ; — = cos a? + - ).
dx \ 2)
23. Differentiatioii of tan x.
y = tan a?, yi = tan (a? + h)^
eiD.{x + h)
sin X
y\-y tan {x + h) - tan x cos {x + h) cos x
"X" h h
sin h
A cos a? cos (aj + h)*
which becomes — r- in the limit,
cos' X
Differentiation ofy =
= %in'^ X,
Hence
d (tan x) I
dx cos'' X
= seo'aJ.^
Otherwise thus,
d (tan x)
, sm a; d sm x
d . cos X — ; sm x
cos X dx
dm^x
dx
dx
dx ^
cos^a;
cos*aJ+ sin* a;
cos* a?
I
cos'' a;'
21
(14)
24. Differentiation of cot a?. — Proceed as in the last,
, , d (cot a;) I „ . .
and we get — ~ — - « - -r-r- = - cosec'^a;. (15)
° dx sm*a; ^ '
This result can also be derived from the preceding, by put-
ting — 2 for a;, as in Art. 22,
^
5. Differentiation of sec x.
I
y = see a? = ;
cos a?
dv sin a;
dx cos'* a;
= lan a; sec x.
Similarly
d cosec a?
- cot a; cosec x.
26.
Differentiation of y
= sin"*a?.
Htsr^
fl?«siny, .*.
3- = cosy.
dy
Hence,
by Art.
20, we get
c?y I
I
4- .
(16)
da <xi&y -/i -g»
22 First Principles. — Differentiation,
The ambiguity of the sign in this case arises from the ambi-
guity of the expression y = sin"^ x ; for if y satisfy this equa-
tion for a particular value of x, so also does ir - y; as also
2Tr + y, &o. If, however, we assign always to y its least value,
i. e. the acute angle whose sine is represented by x, then the
sign of the differential coeflficient is determinate, and is evi-
dently positive ; since an angle increases with its sine, so long
as it is acute. Accordingly, with the preceding limitation,
d . sin~^ XI , .
In like manner we find
d . cos"* X I
^ ^/in?
(i8)
with the same limitation.
This latter result can be at once deduced from the preced-
ing by aid of the elementary equation
sm * X + cos^ a? = -,
2
27. BiflRerentiation of tan"* a?.
y = tan'* a?, .*. x = tan y ;
hence
dx
«-, »
dy cos* y
ef.tan'*a? dy ^ \ , .
,,—- _ = oos'y.^-^. (.9)
28. CTeometrical Demonstration. — The results ar-
rived at in the preceding Articles admit also of easy demon-
Geometrical Demonstration,
23
stration by geometrical construction. We shall illustrate this
method by applying it to
the case of sin d.
Suppose XPQFtobe a
quadrant of a circle hav-
ing 0 as its centre, and
construct as in figure.
Let Q denote the angle
XOP expressed in circu-
lar measure ; then Fig. 2.
„ arc PX , - ^ ^ arc PQ
e = -^^, and A = A9 = -^^p-.
Accordingly,
8m(0 + A)-sm0 = — = ^.^ = COS PQi?.^;
N M
,\ ^^ T^ = cos PQR ^^r.
h arcPQ
But we have seen, in Art. 9, that the limiting value of
PQ
arcPQ
= I ; also PQIt = 9, at the same time ; hence — -r^ — = cos 0,
au
as before.
The student will find no difiiculty in applying the pre-
ceding construction to the differentiation of cos 0, sin~^ 0, and
cos"^ 6. The differential coefficients of tan 9, tan"^ 9, &c., can,
in like manner, be easily obtained by geometrical construction.
I. y = sin (nx + a).
Examples.
dy
dx
n cos {tm + a).
2. y = cos mx coswr.
3. y = sm^a:,
— = - [m cos nx sin mx + n cos mx sin nx),
dy
dx
= n 8in»"i X eos a?.
24 First Principles — Differentiation.
4. y = sin (i + »»). 'd'^'^^ ^°^ ^^ ^ ^^^*
5. Show that sin^ x — (sin»»a; sin mo;) = m sin^'+^a; sin (m + i) «.
^ uX
d
Here — (sin»»a; sin mx) = m sin^'-^x (cos x sin ma; + sin x cos wa;)
= m sin»»-i a: sin (m + l) a; : .-. &c.
6. y = {a sin* a; + J cos" a:)". •~ = n{a'~b) sin 2a; (a sin" x -vb cos' a;)"-'.
^ 'J. y = sin (sin a;).
Or y = sin «, where w = sin a:. -^ = cos a; cos (sin x),
o • 1 / ^\ ^y wa;«-i
8. y = sin-^ (a:»). "^
<fa; (I - a;2»)**
9. 2/ = sin-i (i - a;")i.
Here (i - a;")i = sin y ; .♦. a: = cos y.
l=-sin-^. . ^^
dx' " dx y'l _^2'
,* + «cosa; dy ./~^ li
10. y = cos-i ; . ■— = V ^ -^
a + icosa; dx a + bcosx
dy
1 1. y = sec» X, -f = n sec" a; tan x,
dx
12. y = sec"^ (a;«). -— - = / .
^ 29. DilTereiitiatioii of logoa;.
Let y = ^OgaX, Vx = loga {X + A),
^l0g^I+^^
yi~y ^ logg (a? + A) - logga;
h h h
Hence -j- is equal to the limiting value of
when h is infinitely small.
Again, let h = xu, then
I 1 / h\ I loga (i + w) I . . .J
Geometrical Demonstration. 25
.*. -^ = - multiplied by the value of loffo (i + w)** when u is
infinitely small.
To find the value of the latter expression, let - = 2, then
(i -^uY becomes ( i + - j , in which % is regarded as infinitely
great. Suppose the limiting value of this expression to he re-
presented hy the letter e, according to the usual notation. We
can then find the value of e as follows by the Binomial
Theorem : —
/ iV 15 I s (2 - i) I
\ 2/ 12 1.2 2*
■ . (-^) (-i)(-a ,
« I + - + .^ L + .^ LJ i + &o.
I 1.2 1.2.3
The limiting* value of which, when s = 00, is evidently
III I o
I + - + + + + &0.
I 1.2 1.2.3 1.2.3.4
By taking a sufficient number of terms of this series, we
can approximate to the value of e as nearly as we please.
The ultimate value can be shown to be an incommensurable
quantity, and is the base of the natural or Napierian system
of logarithms. When taken to nine decimal places, its value
is 2.718281828.
Again, since (i + uY = e when m = o, we get
d.loggx _ logge
X
dx ^ ■ ('")
Also, since the calculation of logarithms to any other
base starts from the logarithms of some numbers to the base e ;
* It -will be slio-wTi in Chapter 3, without assuming the Binomial expansion,
that e is the limit of the sum of the series
I + - H + + &c., ad infinitum,
I I . Z 1.2.3
26 First Principles — Differentiation.
and moreover, since the logarithms of all numbers are expressed
by their logarithms to the base e multiplied by the modulus
of transformation, the system whose base is e is fundamental
in analysis, and we shall denote it by the symbol log without
a sujffix. In this case, since log e = i, we have
|aog*)=i. (21)
Again,
d ,, . logioe M . .
^ (log,,*) = ^ = -, («)
where M or logio e is the modulus of Briggs' or the ordinary
tabulated system of logarithms. The value of this modulus,
when calculated to ten decimal places, is
0.4342944819.
On the method of its determination see Q-albraith's "Algebra,"
P- 379-
If a; be a large number, it is evident, from the preceding,
that the tabular difference (as given in Logarithmic Tables),
M
i. e. the difference between logio (-^ + i) and logioic, is — , ap-
proximately. The student can readily verify this result by
reference to the Tables.
30. Differentiation of cf.
Let y = a"y then log 1/ = x log a ;
. d{logy)
but
•• dx ^"^^'
djlogy) ^ d (log y) ^^}_dy_^
dx dy dx y dx*
d .a^ dy
dx dx
Also, since log e= i, we have
d.^ ^
dx
= ylog a = a' log a. (23)
Logarithmic Differentiation, %*j
EXOCPLES.
I. y = log (sina?).
Let sin a; = z, then y = log s.
dy dy dz
dx dz ' dx*
And since
dy cos a?
we get -r = - — = cot «.
dx sm X
, /i - cos a;
, I
/i - cos a; _ /
V I + cos a? /
2 sin*-
2 af
= tan - ;
2 cos'^ -
2
••. y = log tan -. Hence -^ = -: — •
2 aa; sina?
31. liosarithmic DifTerentiation. — When the func-
tion to be differentiated consists of products and quotients
of functions, it is in general useful to take the logarithm
of the function, and to differentiate it. This process is called
logarithmic differentiation.
Examples.
I. y = yi.y8.y8. . . yn, log2/ = logyi + logya + . . . + logyi..
Hence 1^ = i ^ + ifL% . . . +± ^».
y dx y\ dx yi dx yn dx
This furnishes another proof of formula {4), p. 15.
sin»»a; „ , , . ,
a. y = . Here, log y = w log sin a; - « log cos*;
co8»»a; y o^
I dy cos x sin x dy sin"*"^ x , , . • %
.'. --f = m + » ; .'. -f-= r- (»»cos*a? + nsin'a;).
y dx sin X cos x dx cos"-^* x
28 First Principles — Differentiation,
V
(a:-2)S(^-3)i-
Here logy = - log (a; - i) - - log (« - 2) - ^ log (« - 3) ;
243
hence ' ^^ - ^ i 3 x 7 i _ ix^ + ^ox - g-,
ydx 2x-\ 4 a? -2 3 a; - 3 12 . (a; - i) (a; - 2) (a; - 3) '
^ dy _ {x- i)i (7a;2 + 30a; - 97)
' ' dx \^.{x-^)l{x- 3)V
5. y = X*. Here log y = a; log «.
"^^ y^"^^"^^*"^')' .-. -^=x«(i+log«).
6. y = «**. Here log y = a;»,
I dy d.x* , . .
.-. ^ = ^*a;'(» +loga;).
7. y = «•, where m and v are both functions of *.
Here log y = v log «,
I dy , dv V du
••• - / = log «* 3- + - 3- ;
y da? dx udx
dy f. dv V du\ . dv .du
.: ~f-=:u* Mog « — + - — = M" log tt — + VM^l — .
dx \ ^ dx u dx) ° dx dx
32. The expression to be diJfferentiated frequently admits
of being transformed to a simpler shape. In such cases the
student will find it an advantage to reduce the expression to
its simplest form before proceeding to its differentiation.
Examples.
I. y = sin-
V'l+x^
\/i + X'
X x^
Here — ^ = sin y, or -„ = sin'' y ; hence z = tan y,
A / f a. -rZ I + x^
and we get — = cos* y —
dx I + «'
Here
Hence
Logarithmic Differentiation.
y = tan-i ^. 7=. ^
V I + »«- V I -a»
tauy=-5^-rr=r= -. -»
\/l +a;2-'s/i - «*
\/i + a;2 ^ tan y 4- I ^
^TT^-tany-i'
. ^^('+tany)g-(i-tany)^^ 2tany ^ ^.^
(i + tan y)2 + (i - tany)a i + tan'y
dy
-— cos 2y = *
^9
<^;r
Hence
Let a; =
cos ly Vi -»*'
y = iog r — — . = - log . — 7=-
V-v/i +»- -v/i -a; * '/i+ar-Vi-af
I , I 4-^/1 - «« I
-log
dy
ds
= -log(i + V^i -a;2)--
2a;
2a; Vi — a;2
^ , \/i +a;2 - I ^ ,
y=»tan-i-^- + tan-* .
a; I — a;'*
: tan z, and the student can easily prove tliat
5 , dy s \
y ra - z ; hence -—- 5.
'a <&; a I + a;2
30 First Principles — Differentiation,
Examples.
1 A ^y ^
I. y=sBeo-»*. Ana. -f- =
2. y = alogas. ^=I + log«.
3. y = log tan a;. -^ = — i- .
^d; sin 2^
4« y = logtaii-*jr.
5. y = a^/x.
<& (i 4 sfi) tan-» x'
dx ^*y X
c ' n \ ^y cos (log a?)
6. y = sin (log «). -f = — ^^—^—:,
dx X
^ A. ^ ^ dy \
7. y = tan-i ^
8. y = tan-
V X + V a <fy 1
9. y
I - -s/oi <^^ 2V^^ (i + a;).
Here y = tan-^ y^a; + tan-i \/a.
aj«»» <?«/ 2«ic2»-i
(I + ar^)*' rfa; (i + a;2)»*i*
). y = log - \ tan-» a;. -/ = -
VI -a;/ i^a; I
-««
, /\/i + a;2 + a; <fy 1
I. y = iog r — :^=-7=
. _, 3 + 2x dy I
, (i + a;2)J , , , dy x
^ 13- y = log 7 n- + ^ tan-i x, -f-= — — .
y "* "* ^ (i + a:)* ^ rfa; (i + «) (l + «»)
'^ a/TT^' ^ "(1+^)1-
Examples.
31
^ '^ X dx X sfl ^ '
16. y
I — tan X
dx X
dy
sec X dx
^/ 1 — x^ + X s/ i dy
17. y = log
/ ^/^.~x
- (cos aJ + sin «).
18. y
(I + x^)^ •
'^^ (v^i - a;2 + a;v^2) (i - a;2)
rfy _ (l + a^) ^ g« tan"^ «
^a; (i + x^)l *
20. y = log { (2a; - i) + 2\/a;3 - a; - i } .
rfa; (ar«-a?-i)i'
- il + X\/2 + X^ ^ ,X\/2 dy 2\/2
II. y = log ^ + tan-i -^. 3^ = --^..
\i -x^2 + x-^ ^-^' ^^ ' + *
t2. y = e** tan-> a;. r ~ ^'^ \ 2 "^ ^* **^"' ^^ (' + ^^S ^) ) ■
Ax y I + aj y
13. Being given that y t= a^s Ji-ajzWi 1 5 ^
dy ex'- + c'x* + d' x^
dx
x^W
(■-)'(■-?)
determine the values of c, c\ c*'. Ans. c = 3, c' = - 6, c" = f ,
24. y = log (log a;).
, 3 + 5 cos a:
25. y = cos-i ^— ^ .
5 + 3 cos a;
I - a;*
26. y = sin-^ r.
27. y = e"* sinw raf.
dx X log a;"
rfy 4
<^a; 5 + 3 cos*
rfv - 2
dip 1 + »«
/ 28. y = e^i
•—. = go* Bin»»-i ra; (« sin ra; + mr cos ra?).
aa;
^ = efl»*/a^ + r« sin (ra; + ^), ^ft —
where tan ^ s -.
32 - First Principles — Differentiation,
29. y = log {\/x-a + ^/x^b). Am. ^ = 7
Here inf = tan'^ ?^; ... a. = cos y; .-. ^ = L__.
31. y = a;*". ^ = a;*"+»»-i (» log x + i).
i , ^ . »» - 1
jJS 32. y = ( I + a;2) 2 sin ^^ tan"* a;). -^ = «» (i + a;2)~2~cos { (w- 1) taii-»;p} .
, ja cos a; - i sin a; <fy
33. y = log^/ . — = —
° \ a cos aj + d sin a; dx a^
ah
cos^ X — b"^ sin? x
34. Define the differential coeflS.cient of a function of a variable quantity,
with respect to that quantity, and show that it measures the rate of increase of
the function as compared with the rate of increase of the variable.
35. If y = -, prove the relation
dy dx
^/l + y*' -v/i + x*^
c rr ^ ^^ -V ax -V */ {x"^ -^ axf - bx ., . du .
36. If w = log --— T prove that — is of the form
a;2 + oa; - \/(a:3 + ax)'^ - bx «^
, and determine the values of ^ and B, Am. -4 = 3, JB = a.
Vix^ + ax)2 - bx
ABm^d + B8m^e+ C
J 37. Prove that j^ f sin 0 cos 0 ^/i - ^2 sin2 e )
and determine the values of A, B, G. Ana. A = 3^2, B = — 2 {i + c^), c = 1.
38. If w = a? + - — + ^-^ — + ' ^ ' i -,+ ,,. ad inf. ; find the sum
23 2.45 2. 4. 67'
of the series represented by — . Ana* (i — a?'^)"*.
uX
39. Reduce to its simplest form the expression
3^2 d a: (a;2 + 2a)i
Ana.
(a;2 + a)i {x^ + 2a)i dx ' {x"^ + a)% ' ' (x^ + a)i {x^ + 2a)i
40. If sin y = X sin la + y), prove that -^ = ? ^,
^ ^ "" " dx sma
Examples, 33
^ 41. If a:(r + y)» + y(l +a:)J = o,find j^. ^
In this case x^ (i + y) = y^ (r + ») ;
••. x^ -y^ = yx {y - x),
42. y = log (^ + V^' - «^) + sec-1 -. - = - ^^3^.
43. If a; and y are given as functions of t by the equations
x=f{t); y = F{t);
dij ^ d)^ F' {t)
find the value of — in terms of ^. — = jj-r^. •
^
X
44. y
1+ x^
I \x'
I + &c., ad infinitum.
«« dy X
Hence y=.j-^. 5- = ^=!
45. ar = fe i*
X
dy log ^
Hence y=—^—. efa;" (l + log.;)i'
( 34 )
CHAPTER 11.
SUCCESSIVE DIFFERENTIATION.
33. Successive Derived Functions. — In tlie preceding
chapter we have considered the process of finding the derived
functions of different forms of functions of a single variable.
If the primitive function be represented by/(ir), then, as
already stated, its first derived function is denoted hj f\x).
If this new function, f'(;x)j be treated in the same manner,
its derived function is called the second derived of the original
function /(a:), and is denoted by/''(^).
In like manner the derived function of f'\x) is the third
derived of /(a?), and represented by/'''(ir), &c.
In accordance with this notation, the successive derived
functions oifix) are represented by
/», f\x), f"'{x) /W(*),
each of which is the derived function of the preceding.
34. Successive DiflTerential Coefficients.
If y = /(a:)wehaveg=/».
Hence, differentiating both sides with regard to x, we get
dx\dx) dx-' ^ ' ■' *■ '
then J^ = '^»-
In like manner t"(t2 ) ^® represented by — , and so on ;
Successive Differentials, 35
hence . g =/" {x), &o. . • . g ^/("' {?>). (i)
The expressions
dy d'^y d^y d'^y
IJ 1^' d^' ' ' ' d?^
are called the first, second, third, . . . n'^ differential coef-
ficients of y regarded as a function of x.
These functions are sometimes represented by
/, f, r, . . . yW,
a notation which will often be found convenient in abbre-
viating the labour of forming the successive differential
coefficients of a given expression. From the mode of
arriving at them, the successive differential coefficients of a
function are evidently the same as its successive derived
functions considered in the preceding Article.
35. Successive Differentials. — The preceding result
admits of being considered also in connexion with differen-
tials ; for, since x is the independent variable, its increment,
dx, may be always taken of the same infinitely small value.
Hence, in the equation dy = f\x) dx (Art. 7), we may
regard dx as constant, and we shall have, on proceeding
to the next differentiation,
d {dy) =dxd [/ {x)-] = [dxyrix),
since ^[/W] =/' W dx.
Again, representing d {dy) by d'^y,
we have d'^y = f\x) {dxy ;
if we differentiate again, we get
d'y=r\x){d^)^
and in general
From this point of view we see the reason why/^"^ {x) is
called the n*^ differential coefficient oif(x).
d2
36 Successive Differentiation,
In tlie preceding results it may be observed tbat if dx
be regarded as an infinitely small quantity, or an infinitesimal
of the first order, {doay, being infinitely small in comparison
with dx, may be called an infinitely small quantity or an
infinitesimal of the second order ; as also d^y, if /" {x) be
finite. In general, d"y, being of the same order as (dx)"', is
called an infinitesimal of the n*^ order.
36. Infinitesimals. — We may premise that the expres-
sions great and small, as well as infinitely great and infinitely
small, are to. be understood as relative terms. Thus, a magni-
tude which is regarded as being infinitely great in comparison
with Q. finite magnitude is said to be infinitely great. Similarly,
a magnitude which is infinitely small in comparison with a
finite magnitude is said to be infinitely small. If any finite
magnitude be conceived to be divided into an infinitely great
number of equal parts, each part will be infinitely small with
regard to the finite magnitude ; and may be called an infini-
tesimal of the first order. Again, if one of these infinitesimals
be conceived to be divided into an infinite number of equal
parts, each of these parts is infinitely small in comparison
with the former infinitesimal, and may be regarded as an
infinitesimal of the second order, and so on.
Since, in general, the number by which any measurable
quantity is represented depends upon the unit with which
the quantity is compared, it follows that a finite magnitude
may be represented by a very great, or by a very small num-
ber, according to the unit to which it is referred. For ex-
ample, the diameter of the earth is very great in comparison
with the length of one foot, but very small in comparison
with the distance of the earth from the nearest fixed star, and
it would, accordingly, be represented by a very large, or a
very small number, according to which of these distances is
assumed as the unit of comparison. Again, with respect to
the latter distance taken as the unit, the diameter of the
earth may be regarded as a very small magnitude of the first
order, and the length of a foot as one of a higher order of
smallness in comparison. Similar remarks apply to other
magnitudes.
Again, in the comparison of numbers, if the fraction (one
million)*'^ or — ^, which is very small in comparison with
Geometrical Illustration,
37
unity, be regarded as a small quantity of^the first order, the
fraction — ^, being the same fractional part of — ^ that this
10
lO"
is of I, must be regarded as a small quantity of the second
order, and so on.
If now, instead of the series —5, f — -J , f — 5 j , . . .
we consider the series
in which n is
supposed to be increased without limit, then each term in the
series is infinitely small in comparison with the preceding
one, being derived from it by multiplying by the infinitely
small quantity -. Hence, if - be regarded as an infinitesimal
of the first order, — „, — ,
n^ n^
. — , may be regarded as infini-
tesimals of the second, third, . . , r*^ orders.
37. Creometrical Illustration of Infinitesimals. —
The following geometrical results will help to illustrate the
theory of infinitesimals, and also
will be found of importance in the
application of the Differential Cal-
culus to the theory of curves.
Suppose two points. A, B, taken
on the circumference of a circle ;
joiu B to E, the other extremity
of the diameter AE, and produce
EB to meet the tangent at A
in D. Then since the triangles
ADB and EAB are equiangular,
we have
Fig. 3.
AB BE BD AB
AD ~ AE' ^"""^ AD ~ AE'
Now suppose the point B to approach the point A and to
become indefinitely near to it, then BE becomes ultimately
AB
ecjual to AE, and, therefore, at the same time, -p^ = i.
38 Successive Differentiation,
Again, -j=r becomes infinitely small along with -j^,
i. e. BB becomes infinitely small in comparison with AB or
AB. Hence BB is an infinitesimal of the second order when
AB is taken as one of the first order.
Moreover, since BB - AE < BB, it follows that, when one
side of a right-angled triangle is regarded as an infinitely small
quantity of the first order, the difference between the hypothenuse
and the remaining side is an infinitely small quantity of the
second order.
Next, draw BN perpendicular to AB, and BF a tan-
gent at B', then, since AB > AN, we get AB - AB
<AD'-AN<BN',
AB-AB BN AB
BB ^ BB^ BB'
Consequently, ^yr — becomes infinitely small along with
AB; .'. AB - AB is an infinitesimal of the third order.
Moreover, as BF= FB, we have AB = AF + BF; .-. AF
+ BF- AB is an infinitely small quantity of the third order ;
but AF + FB is > arc AB, hence we infer that the difference
between the length of the arc AB and its chord is an infinitely
small quantity of the third order, when the arc is an infinitely
small quantity of the first. In like manner it can be seen
that BB - BN is an infinitesimal of the fourth order, and
so on.
Again, if AB represent an elementary portion of any
continuous* curve, to which AF and BF are tangents, since
the length of the arc AB is less than the sum of the tangents
AF and BF, we may extend the r^pult just arrived at to all
such curves.
♦ In this extension of the foregoing proof it is assumed that the ultimate
ratio of the tangents drawn to a continuous curve at two indefinitely near
points is, in general, a ratio of equality. This is easily shown in the case of
an ellipse, since the ratio of the tangents is the same as that of the parallel
diameters. Again, it can be seen without difficulty that an indefinite number
of ellipses can be drawn touching a curve at two points arbitrarily assumed on
the curve ; if now we suppose the points to approach one another indefinitely
along the curve, the property in question follows immediately for any con-
^uou3 curve.
Geometrical Illustration, 39
Hence, the difference between the length of an infinitely
small portion of any continuous curve and its chord is an infi-
nitely small quantity of the third order y i.e. the difference between
them is ultimately an infinitely small quantity of the second
order in comparison with the length of the chord.
The same results might have been established from the
expansions for sin a and cos a, when a is considered as infi-
nitely small.
If in the general case of any continuous curve we take
two points A, B, on the curve, join them, and draw BJS
perpendicular to AB, meeting in ^ the normal drawn to
the curve at the point A ; then all the results established
above for the circle still hold. When the point B is taken
infinitely near to A, the line AJE becomes the diameter of
the ci7xle of curvature belonging to the point A ; for, it is
evident that the circle which passes through A and B, and
has the same tangent at A as the given curve, has a contact
of the second order with it. See "Salmon's Conic Sections/'
Art. 239.
Examples.
1. In a triangle, if the vertical angle be very small in comparison with either
of the base angles, prove that the difference between the sides is very small in
comparison with either of them ; and hence, that these sides may be regarded as
ultimately equal.
2. In a triangle, if the external angle at the vertex be very small, show that
the difference between the sum of the sides and the base is a very small quantity
of the second order.
3. If the base of a triangle be an infinitesimal of the first order, as also its
base angles, show that the difference between the sum of its sides and its base
is an infinitesimal of the third order.
This furnishes an additional proof that the difference between the length of
an arc of a continuous curve and that of its chord is ultimately an infinitely
small quantity of the third order.
4. If a right line be displaced, through an infinitely small angle, prove that
the projections on it of the displacements of its extremities are equal.
5. If the side of a regular polygon inscribed in a circle be a very small
magnitude of the first order in comparison with the radius of the circle, show
that the difference between the circumference of the circle and the perimeter of
the polygon is a very small magnitude of the second order.
40 Successive Differentiation.
38. Fundamental Principle of the Infinitesimal
Calculus. — We shall now proceed to enunciate the funda-
mental principle of the Infinitesimal Calculus as conceived by
Leibnitz :* it may be stated as follows : —
If the difference between two quantities be infinitely
small in comparison with either of them, then the ratio of
the quantities becomes unity in the limit, and either of them
can be in general replaced by the other in any expression.
For let a, j3, represent the quantities, and suppose
a = /3 + z, or g = I + g.
Now the ratio -3 becomes evanescent whenever i is infinitely
small in comparison with /3. This may take place in three
different ways : (i) when j3 is finite, and i infinitely small :
(2) when i is finite, and j3 infinitely great ; (3) when j3 is
infinitely small, and i also infinitely small of a higher order :
thus, if i = /tj3^, then rr = A:i3, which becomes evanescent along
with /3.
* This principle is stated for finite magnitudes by Leibnitz, as follows : —
" Caeterum aequalia esse pnto, non tan turn quorum differentia est omnino nulla,
sed et quorum differentia est incomparabiliter parva." ..." Scilicet eas
tantum homogeneas quantitates comparabiles esse, cum Euc. Lib. 5, defin. 5,
censeo, quarum una numero sed finito multiplicata, alteram superare potest ; et
quae tali quautitate non differunt, aequalia esse statue, quod etiam Arcbimedes
sumsit, aliique post ipsum omnes." Leibnitii Opera, Tom. 3, p. 328.
The foregoing can be identified with the fundamental principle of Newton,
as laid down in his Prime and Ultimate Ratios, Lemma I. : "Quantitates, ut
et quantitatum rationes, quae ad sequalitatem tempore quovis finito constanter
tendunt, et ante finem temporis illius proprius ad invicem accedunt quam pro
data quavis differentia, fiunt ultimo aequales."
All applications of the infinitesimal method depend ultimately either on the
limiting ratios of infinitely small quantities, or on the limiting value of the
sum of an infinitely great number of infinitely small quantities; and it may
be observed that the difference between the method of infinitesimals and that of
limits (when exclusively adopted) is, that in the latter method it is usual to
retain evanescent quantities of higher orders until the end of the calculation,
and then to neglect them, on proceeding to the limit; while in the infinitesimal
method such quantities are neglected from the commencement, from the know-
ledge that they cannot affect ihejinal result, as they necessarily disappear in the
limit.
Principles of the Infinitesimal Calculus. 41
Accordingly, in any of tlie precediog cases, the fraction
-3 becomes unity in the limit, and we can, in general, substi-
tute a instead of j3 in any function containing them. Thus,
an infinitely small quantity is neglected in comparison with
a finite one, as their ratio is evanescent ; and similarly an
infinitesimal of any order may be neglected in comparison
with one of a lower order.
Again, two infinitesimals a, /3, are said to be of the same
order if the fraction — tends to a finite limit. If -^ tends
a or
to a finite limit, |3 is called an infinitesimal of the n*^ order
in comparison with a.
As an example of this method, let it be proposed to
determine the direction of the tangent at a point {x, y) on a
curve whose equation is given in rectangular co-ordinates.
Let X + a, y + j3, be the co-ordinates of a near point on
the curve, and, by Art. 10, the direction of the tangent
depends on the limiting value of -. To find this, we substi-
a
tute a; + a for x, and 2/ + j3 f or ^ in the equation, and neglect-
ing all powers of a and j3 beyond the first, we solve for -,
and thus obtain the required solution.
For example, let the equation of the curve 106x^ + 1/ = zaxy\
then, substituting as above, we get
a^ + sx^a + y^ + 3?/^j3 = ^axy + ^axl5 + saya :
hence, on subtracting the given equation, we get the
a ax - y,
39. Subsidiary Principle. — If ai + aj + 03 + . . . + a^
represent the sum of a number of infinitely small quantities,
which approaches to a finite limit when n is increased indefi-
nitely, and if j3i, jSa, . . • j^n be another system of infinitely
small quantities, such that
I + tn,
/3i ^ ft ^^
_3«
— = I + fi, — = I + £2, .
Ox Oa
* ' «»
42 Successive Differentiation.
where ci, fg, . . • Cnj ar© infinitely small quantities, then the
limit of the sum of ^i, jSa, . . • i3» is ultimately the same as
that of tti, as, . . . On.
For, from the preceding equations we have
i3i + jSg + . . . + j3» = ai + aa + . . . -f an + aiSi + azSi + . . . + Onen-
Now, if 17 be the greatest of the infinitely small quan-
tities, 61, f2, • . . f»> we have
j8i + ^3 + . • . + j3» - (oi + a2 -I- . .» . '. 4- On) < »; (ai + a, . . . + a„) ;
but the factor ai + as + . . • + a^ has a finite limit, by^ hypo-
thesis, and as ij is infinitely small, it follows that the limit of
i3i + /Ba + . . . + i3rt is the same as that of ai + az + . . . + a„.
This result can also be established otherwise as follows : —
The ratio §>±P,±s^^±h, -
oi + 02 + . . . + a»
by an elementary algebraic prraciple, lies between the greatest
and the least values of the fractions
|3i /3» /3,.
> i • • • >
oi 02 an
it accordingly has unity for its limit under the supposed con-
ditions : and hence the limiting value of /3i + /3a + . . . + j3» is
the same as that of oi + 02 + . . . + on.
^ 40. Approximations. — The principles of the Infini-
tesimal Calculus above established lead to rigid and accurate
results in the limit, and may be regarded as the fundamental
principles of the Calculus, the former of the Differential, and
the latter of the Integral. These principles are also of great
importance in practical calculations, in which approximate
results only are required. For instance, in calculating a
result to seven decimal places, if — ^ be regarded as a small
quantity o, then a^, a% &c., may in general be neglected.
Thus, for example, to find sin 30' and cos 30' to seven de-
cimal places. The circular measure of 3 o' is -^, or . 00 8 7 2 6 6 ;
360
Approximations, 43
denoting this by a, and employing the formulae,
a^ a*"
sm a = a — —, cos a = i ,
o 2
it is easily seen that to seven decimal places we have
— = .0000381, — = .0000001.
2 o
Hence sin 30' = .0087265 ; cos 30' = 9999619.
In this manner the sine and the cosine of any small angle
can be readily calculated.
Again, to find the error in the calculated value of the
sine of an angle arising from a small error in the observed
value of the angle. Denoting the angle by a, and the small
error by a, we have
sin (« + a) = sin « cos a + cos a sin a = sin « + a cos a,
neglecting higher powers of a. Hence the error is repre-
sented by a cos a, approximately.
In like manner we get to the same degree of approxima-
tion
tan {a + a) - tan a = — r-.
Again, to the same degree of approximation we have
a + a _a ba - af5
where a, /3 are supposed very small in comparison with a and h.
As another example, the method leads to an easy mode of
approximating to the roots of nearly square numbers ; thus
'/a^ + a = a + — ; ^/ar + a^ = a + — = a, whenever a^ may
2a 2a
be neglected.
Likewise, ^/a^ + a = a + — 5, &c.
If J = « + a, where a is very small in comparison with «,
we have ^/ah = y^a^ + aa = a + - = .
2 2
44 Successive Differentiation,
Again, in a plane triangle, we have the formula
C C
c^ = a^ + b^ - zah cos C = {a + by sin- — + {a - by cos'^ — .
Now if we suppose a and b nearly equal, and neglect {a - by
in comparison with {a + by, we have
e = {a + by sin* — + {a - by cos'* — = {a + b) sm — .
This furnishes a simple approximation for the length of
the base of a triangle when its sides are very nearly of equal
length.
Examples.
^ I. Find the value of (i + o) (i - 20^) (i + 30'), neglecting a* and higher
powers of o. ^ns. I + o - 2o^ + a^.
2. Find the value of sin (a + a) sin {b + 0), neglecting terms of 2nd order
in a and fi. Ans. sin a sin 6 + a cos a sin i + /3 sin a cos b.
3. liiri. = u - e sin w, e being very small, find the value of tan ^u.
Ans. ii-^e) tan — .
2
„ u m e . . t* . /fn \ , e .
Here - = — + - sm « ; tan - = tan ( — + a , where a = - sm « : .'. &c.
222 2 \2 / 2
4. In a right-angled spherical triangle we have the relation cos 0 = cos a cos b ;
determine the corresponding formula in plane trigonometry.
The circular measure of a is -, R being the radius of the sphere ; hence,
It
substituting I - -^ for cos «, &c., and afterwards making iJ = 00, we get
c2 = a2 + P.
5. If a parallelogram be slightly distorted, find the relation connecting the
changes of its diagonals.
Ans. dt^d + d' ti.A' = o, where d^ d' denote the diagonals, and Arf, Ac?' the
changes in their lengths. In the case of a rectangle the increments are equal,
and of opposite signs. .
6. Find the limiting value of
Aa^ + -gg^^^ + gg"*^^ + &C.
flg" + ig^'i + ca"-'- + &c.
when a becomes evanescent.
^g"* A
In this case the true value is that of ■ = — a"*^.
flo'» a
Hence the required value is zero, — , or infinity, according as m>, =, or < n.
Examples, 45
7. Find tte value of
I--V- + —
6 120
I + --
2 24
neglecting powers of x beyond the 4tli. Am. i + — + — .
y^ 8. Find the limiting values of - when y = o, a; and y being connected by
the equation yj = 2xy — sfi.
Here, dividing by y' we get
X
- 2 - = - y.
y^ y
If we solve for - we have
Hence, in the limit, when y = o, we have - = 2, or - = o.
y y
9. In fig. 3, Art. 37, if ^J5 be regarded as a side of a regular inscribed polygon
of a very great number of sides, show that, neglecting small quantities of the
4th order, the difference between the perimeter of the inscribed polygon and
that of the circumscribed polygon of the same number of sides is represented
by - BB.
2
Let n be the number of sides, then the difference in question is n {AD - AB) ;
, ^ irAE , ^^ ^^^ TT AH {AB - AB)
but n = — ; .-. n {AB - AB) = ^—— -'
arc^^ ^ ' AB
^wAE^^-^^r. '^{BE-AE) = 1bB, a. p.
This result shows how rapidly the perimeters of the circumscribed and in-
scribed polygons approximate to equality, as the number of sides becomes very
great.
10. Assuming the earth to be a sphere of 40,000,000 metres circumference,
show that the difference between its circumference and the perimeter of a regular
inscribed polygon of 1,000,000 sides is less than -j^th of a millimetre.
11. If one side 5 of a spherical triangle be small, find an expression for the
difference between the other sides, as far as terms of the second order in b.
Here cos e = cos a cos i + sin a sin J cos C.
Let z denote the difference in question ; i. e. <j = a — « ;
then cos a coaz + sin a sin z = cos a coab + sin a sin i cos C;
.*, sin a - sin 5 cos C = cot a (cos b — cos z).
46 Successive Differentiation,
Since z and b are both small, we get, to terms of the second order,
The first approximation gives z=b cos 0. If this he substituted for * m the
right-hand side, we get, for the second approximation,
^ 42 sin^C cot a
2
We now proceed to find the successive derived functions
in some elementary examples.
41. Derived Functions of of*.
Let 1/ = £i^f
then -/ = mx"''-^ -j^=m(m- 1) x*^-\
dx dx^ ^ ' '
and in general, -r-^ = m {m - i) (m - 2) . . . (w - w + i) a?'""".
GX
If m be a positive integer, we have
d'^iaf')
djf"
I • 2 . . . m.
and all the higher derived functions vanish.
If ni be a fractional, or a negative index, then none of the
successive derived functions can vanish.
Examples.
I. lfu = ax»-\^ 5a;«-i + cx^'"^ + &c., prove that
__ » n (« - 1) ax»-'^ + (» - I ) (n - 2) bx>*-^ + &c;
, ./ d»u , d"'-^u
also^ -7— = I . 2 . . . . « , «, and - — -, = <i-
a
, , ^ dy na d^y n{n ^ i\a
pi-ove thfa,t — = -, — ^ = ^ '
dx a;"-!' dx^ a;"+2 '
and ^^ = (- ,)m *»(^+0--.(" + "^-i)«^
Examples, 47
y 3. y=2a\/x\
dy a d^y a d^y i a
provethat ^^ = ;/-' ^ = -.11' ^ = 4^1'
- (- iV
<'**'^y _ / T\n 3 • 5 • 7 . . .{in-\)a
42. If y = a;^ log iP, to find ^ .
Here -^ = zx^ log a? + ic* ;
also —^ = 6.r log aj + 30? + 2:?^ = 6.r log a? + 5^,
O/X
It miglit have been observed that in this case all the
terms in the successive differentials which do not contain
log X will disappear from the final result — thus, by the last
Article, 3 ' = o, accordingly, that term may be neglected ;
cix
and similar reasoning applies to the other terms. The work
can therefore be simplified by neglecting such terms as we
proceed.
The student will find no difficulty in applying the same
mode of reasoning to the determination of the value of
d^y
■j^, where y = af^^ log x.
For, as in the last, we may neglect as we proceed all terms
which do not contain log a; as a factor, and thus we get in
this case,
d'^y _ (w - i) . . . 2 . I _ 1^- ^ ^
dx"" ~ X X
48 Successive Differentiation,
43. Derived Functions of sin mx.
Let y = sin mx^
then -— = m cos mx,
ax
-T- = - rrt sm mx,
dar
and, in general, -^-^ = (- i)"m^" sin mx^
—-— ^ = (- i)«w'«+' cos mx,
■I
(>:
It is easily seen that these may be combined in the single
equation (Art. 22),
d^ (sin mx) r - I ""
dxf
In like manner we have
d"^ cos mx
= m^ miimx + r-y {2]
= m^ cos [mx + r
dx" \
44. Derived Functions of e^.
Let y = e«*,
9
then ^ = flfe«^ ^ = «'c«*, . . . ^ = ««c'^^ (3)
This result may be written in the form
(IJ.^^^a'e-, (4)
where the symbol ( — ) denotes that the process of differentia-
tion is applied n times in succession to the function e"^.
Derived Functions of e** cos bx. 49
In general, adopting the same notation, we have
^^oa"^* + ^ia«-»e«* + ^^a^-V^ + &o.
= [^o«" + A^a""-' + A^a""-^ + &c. An'] c«*.
This result, if ^ (a?) denote the expression
A(^ + Ayof-^ + . . . ^„,
may be written in the form
in which 0 (a) is supposed to contain only positive integral
powers of a.
45 . To find the n*^ Derived Function of e^' cos hx, —
Let tj represent the proposed expression,
then -r = aef^" cos bx - be'^^ sin bx
ax
B ^* {a cos bx - b sin &a?) ;
b
If tan 0 = -, we have ^ = ^a'^ + 6^ sin 0, and a = y^a' + 6* cos ^.
Hence we get
50 Successive Differentiation,
Again,
^ = (a' + b')^ e«* [a cos (bx + cp) - h sin {hx + 0)]
= (a" + b^) e"' cos {bx + 2^).
By repeating this process it is easily seen that we have in
general, when n is any positive integer,
^ = (a= + b'f^'- cos {bx + n<l>). (6)
dx
46. To find the DeriTed Functions of tan "M -J,
and tan"^ x.
Let y = tan"^ f - J, or a? = cot y :
, , dy - I
then -r = ; = - sm' y.
«a? I + fl?^
« sin^ y — (sin'* y) = sin* y sin zy,
ay
« - sin^y— {^v^y sin 2^)
«= - I . 2 . sin' 2^ sin ly, {Ex. 5, Art. 28.)
Hence, also — = 1.2.3. sin*«/ sin 4y;
and in general, --^ = (_ i)« |7t -^i sin" y sin wy.
Theorem of Leibnitz, 5 1
^ • -1^
Again, since ioir^x = — tan~ -,
2 X
we have —^ — ■ = (" ^ ) h"^ ^m" y sm nt/, (7)
where y = cot'^ a?, as before.
This result can also be written in the form
sinfntan"^-)
c?" (tan"*ij?) / x« , , \ -/ /o\
^.^, = (- i)-^ In - I — ^ -^ . (8)
47. If y = sin {m sin"* a?), to prove tbat
Here
dy m cos (m sin"* x)
■~\ ^m^ cos' (m sin"* a;) « w* (i - y*).
Hence, differentiating a second time, and dividing by 2 -^,
we get the required result-.
48. Theorem of lieibnitz. — To find the n^^ differen
tial coefficient of the product of two functions of x. Le(
y ^uv\ then, adopting the notation of Art. 34, we write
, , , . dy du dv
and similarly, /', w", tf\ &o., for the second and higher
derived functions — thus,
'^ dx""' dx""'
F 2
52 Successive Differentiation.
Now, if we differentiate the equation y = uv, we have
xf = uv' -V vu\ by Art. 13.
The next differentiation gives
y" = uf + wV + «?V + V^' = w/' + 2«/t^ + WjI\
The third differentiation gives
f' = wt'"' + wV + 2wV' + 2«*V + t;V' + t;«*'"
= t^tj"' + 3wV' + 3wV + «^"',
in which the coefficients are the same as those in the expan-
sion of (a + Vf.
Suppose that the same law holds for the /»'* differential
coefficient, and that
1.2
then, differentiating again, we get
+ ?L(^Z_i) (^''^(n-i) + e^'''^(n-2)) + &0. . . . + tte***)^;
= ««;(«*^) + (fj + i) «*'«?(«) + 6Liil^«i"t.'(«-i) + &o. . . . ,
1.2
in which it can be easily seen that the coefficients follow the
law of the Binomial Expansion.
Accordingly, if this law hold for any integer value of w,
it holds for the next higher integer ; but we have shown that
it holds when w = 3 ; therefore it holds for ^ = 4, &c.
Hence it holds for all positive integer values of n.
In the ordinary notation the preceding result becomes
fl?» [m) d''v du ^"-1 V n(n- i) d'u d''-''v .
— ^^ — - s= u V n ■ + — ^^^ ■ V &c.
daf" ds^ dxdstf'-^ 1.2 dxUx''-''
Applications of Leibnitz^ a Theorem, 53
49. To prove that
(|)"(^«)=^(-0«. (n)
where w is a positive integer.
Let V = e"^ in the preceding theorem ; then, since
dv „^ d^v „ „^ d^v „ „^
■— = «e«^, -— = a'e«*, ... TIL = ^ ^ y
dx dx^ dx""
we have
which may be written in the form
(i)V")-|.-*~-i*"4:V-'""(i>*"(i)>.
« + — ) is supposed to be
developed by the Binomial Theorem, and — , — , . . .
dx' dx"' dsxf
dxj \dx) \dx
substituted for ( 3~)^> (j-J^j (t-) ^> iii ^^^ resulting ex-
pansion.
50. In general, if 0 (a) represent any expression in-
volving only positive integral powers of a, we shall have
For let <^\-f\ when expanded, be of the form
\dx)
M^ +^i(~) +• • • -^^n.
54 Recessive Differentiation,
then tlie preceding formula holds for each of the component
terms, and accordingly it holds for the sum of all the terms ;
.*. &o.
The result admits also of being written in the form
('^^i)-"
,-.^(^ )(,«.«).
This symbolic equation is of importance in the solution
of differential equations with constant coefficients. See
" Boole's Differential Equations," chap. xvi.
51. \f y = sin"^a?, to prove that
hence, by differentiation,
Again, by Leibnitz's Theorem, we have
fe) X
'^'^4!-£-f-2-
On subtracting the latter expression from the former, we
obtain the required result by (14).
If a; = o in formula (13), it becomes
Applications of Leibnitz's Tlieorem, 55
— j represents the value of -~^ when x becomes
cypher.
Also, since f -^ j = i, we get, when w is an odd integer,
V(&»*7o I • 3 • 5 • . . • « .
' 72 \
Again we have (v^j =0; consequently, when n
integer, we have [^) = o.
is an even
m
A 52. If y = (i +a;')^sin (mtan~*a?), to prove that
(i +^')^- 2(w - i)x-£-vm{m- i)y = o. (15)
Here
dy --1 --1
-J- = mx{i ■¥ a?y sin (m tan~^aj) + i»(i + a;')^ cos (mtan'^a;),
or
dy - -
(i + a?) -f = mx{i + a^) ^ sin (m tan'^a;) + m (i +a?) * cos w (tan"* a?)
m
= mx^ + m{i + Q^y cos (m tan~* x) ;
/ ox? / i , N I + a^dv
,\ (1 + sr) ^ cos (m tan"*ir) = ; — xv.
^ ^ ^ ' m dx
The required result is obtained by differentiating the last
equation, and eliminating cos {m tan"^ x) and sin (m tan'^a;) by
aid of the two former.
Again, applying Leibnitz's Theorem as in the last Article,
we get, in general —
56 Successive Differentiation,
Hence, when « = o, we have
Moreover, as when ar = o, we have y = o, and -j- = tn ; it
ax
follows from the preceding that
For a complete discussion of this, and other analogous
expressions, the student is referred to Bertrand, " Traits de
Calcul Differentiel," p. 144, &o.
Examples, 57
Examples.
^ I. y = «* log if, proYe that —=---..
2. y = x log ir,
3- y = ^,
^ 4. y = log (sin x\
^ , \/ I + X* - I , , 24?
5. y = tan-i • + tan'*
« I-*
6. y = «« log (a:!),
\ I - a: V ^
8. y = e^*Bmlx,
>»
_^=(_l)n
1
3'. X
>»
0=..(,+iog.r
' + «*-^
<f 3y 2 COS rr
»l
(^a:3 ~ sin3 x '
d^y 5^
/
>»
dx'^ ~ (1 + 0:2)2-
»»
iV 2«
d^" x'
it
d-iy 8^/2 . «s
dx^ ~ (i+x*)2'
»
rf«y ^♦•« sin {x + *?</>)
where tan ^ =
I
• — •
r
n-r / d\^
9. If y = e"' a:^, prove that
dx**
and (^)V^)=(;)""'(gV*')
>• 10. lfy = a cos (log a;) + b sin (log «),
prove that x^ ha;, +y = o.
^ dx^ dx
II. If y = ^sin-lx^
d^y dy
prove that ^' ~ ^^^ ^ ~ ^ ^ " ^'^'
58 Examples,
12 Froye that the equation
dx' dx
is satisfied by either of the following values of y :
y = cos (» sin"^ x), ox y = e^ "^ sta-)*,
13. Being given that y — [x -v ^ x'- — !)♦»,
d'^v dv
prove that (a;2 - i) --^ + a; -^ - m»y = o.
dx^ dx
14. If y = sin (sin a;),
prove that T^ + t" ^^^ ^ + y cos'a; =« o.
aa;^ air
15. In Fig. 3, Art. 37, HAB be regarded as a side of a regular polygon of an
indefinitely great number of sides, show that the difference between the circum-
ference of the circle and the perimeter of the polygon is represented by 2 SD^
6
to the second order of infinitesimals.
16. If y = -4 cos fw; + J5 sin «a?, prove that ( — + w^ I y = o.
//O Ti. I .1. .. ^"y / V I »» • sin"*' <p sin (n + i) A
vhere A = tan** -.
^ a;
This follows at once from Art. 46, since — f tan-»- J = ^ .. It can also be
proved otherwise, as follows :
_L_^_L__r_i i_l.
a^ + x* 2a(-i)iU -«(-!)* « + «(~i)U*
(- 1)« 1 . 2 . . . « r I I "I
Examples. 59
Again, since - = tan <^, ve hayo a = ^/'^^TV' sin </>^nd x = v/a^ + »2 cos ^p ;
ii+j
hence (« + a (- l)*)*^! = («» + «*) =* (cos ^ + (- i)» sin <^)»+i
= (a« + sfl) » {cos {« + I) <^ + (- I)* sin (» + i) ^},
and we get, finally.
^„y |n.8in(»+ i)<^.8in'»+i<^
i8. In like manner, if y = -r 5,
a* + a;*
^ny 1« • sin"+^ <p . cos (n + l) ^
prove that ^ = <" ^J" " ^Ji^ •
19. If « = a;y,
provethat d^n^" dTn + ^'d^i'
20. If w= (sin"^«)*,
provethat (' "* > ^ " ^^ == *'
a I. Proye, from the preceding, that
(l - a;2) (2n + i) a; - — - - «' 3-:
and
\^«+«/o~** \dx**/o
22. If y 5= ^* sin *a;, prove that -^ - 2» — + (a2 + A') y = a
aa; + i - , <^'H
23. Given y = ^j-^„ find—.
•^®^® ;c2-<;2 ~ 2<J X- e "^ 2C «: + <?*
rf«y (-i)-\n / ae+b ac - b \
( 6o )
CHAPTEE III.
DEVELOPMENT OF FUNCTIONS.
53. liemma. — If w be a function oi x + y which is finite
y^ and continuous for all values of a; + ?/, between the limits
a and h, then for all such values we shall have
du _ du
dx dy
For, let w =/(a; + ^), then if x become a? + ^,
dx h
when ^ is infinitely small.
Similarly, if y become y + A, we have
which is the same expression as before.
T-r du du
Hence -7- = -;-.
dx dy
Otherwise thus : — ^Let z = x + y, then u =f{z\
dz , dz
— = I, and :^ = i;
dx dy
du ^dudz _ ,. .
di~didx~'^^^^'
du dudz _ ,. . du
dy dz dy ^ dx'
Taylorh Expansion. 6i
54. If a continuous function fix + y) be supposed ex-
panded in a series of powers of y, the expansion can contain
no negative powers; for, suppose it contains a term of the
form My~^, where M is independent of y, this term would
become infinite, for all values of x^ when y = o\ but the given
function in that case reduces to f(x) ; and since / {x) cannot
be infinite for all values of x^ it follows that the expansion
of /(aj+ y) can contain only positive powers of y.
Again, if f{x) and its successive derived functions be
continuous, the expansion of /(a; + y) can contain no fractional
p
power of y. For, if it contain a term of the form Py'^%
where - is a proper fraction, then its {n + 1)*^ derived func-
tion with respect to y would contain y with a negative index,
and, accordingly, it would become infinite when y = o; but this
is impossible for the same reason as in the former case ; hence,
with the conditions expressed above, the expansion oif{x + y)
can contain only positive integral powers of y.
55. Taylor's Kxpansion otf{x + y).* — Assuming that
the iunoiion f {x + y) is capable of being expanded in powers of
y, then by the preceding this equation must be of the form
f{x + y) = Po + P^y + P,f + &C. + Pny"" + &C.,
in which Po, Pi, . . . Pn are supposed to be finite and con-
tinuous functions of x.
When y = o, this expansion reduces iof{x) = Po.
Again, l^i u=f{x + y)\ then by differentiation we have
du
Tx"
dP, dP, JP^
dx " dx ^ dx
du _
dy~
-- Pi + iP^y + aPay* + &
• The investigation in this Article is introduced for the purpose of showing
the beginner, in a simple manner, how Taylor's series can be arrived at. It is
based on the assumption that the function /(a; + y) is capable of being expanded
in a series of powers of y, and that it is also a continuous function. It demon-
strates that whenever the function represented by/(a; + y) is capable of being
expanded in a convergent series of positive ascending powers of y, the series
must necessarily coincide with the form given in (i). An investigation of the
conditions of convergency of the series, and of the applicability of the Theorem
in general, will be introduced in a subsequent part of the Chapter. The parti-
cular case of this Theorem when /(a;) is a rational algebraic expression of the n'*
degree in x is already familiar to the student who has read the Theory of Equations.
62 Development of Functions,
Now, in order that these series should be identical for all
values of y the coefficients of like powers must be equal.
Accordingly, we must have
dx dx -^ ^ ^'
' I. 2 dx ~ I . 2 dx" ~ I ,2^ ^ ^'
' Z dx 1.2.3 d^ " 1.2.3*^ ^^^'
and in general,
I . 2 . . . ;^ d^'' I . 2 . . .n
fW{x).
Accordingly, when f{x) and its successive derived func-
tions are finite and continuous we have
/(*+y) =/{<») + f/W + r^/"W + • • • + f/'"'W+- (0
This expansion is called Taylor's Theorem, having been first
published, in 17 15, by Dr. Brook Taylor in his Methodus
Incrementorum.
It may also be written in the form
•^ ^ i'/ ^ \ / J ^ 1^2 dx^ \n doif ^ ''
or, if u = f{x)f and Wi =f{x + y),
ydu y"^ d^u y^ d*^u - , .
To complete the preceding proof it will be necessary to
obtain an expression for the limit of the sum of the series
after n terms, in order to determine whether the series is
convergent or divergent. We postpone this discussion for
the present, and shall proceed to illustrate the Theorem by
The Logarithmic Series, 63
showing that the expansions usually giyon in elementary
treatises on Algebra and Trigonometry are particular cases
of it.
y 56. The Binomial Theorem. — Let w = (a? + y)** ;
here/(ij?) = x^, therefore, by Art. 41, " " '
f(iv) = niv'"-\ .../(*•) (a;) = w (w - i) ...(;*- r + i)af*^.
Hence the expansion becomes
-i ^ ^^ ^ ^«-^ y"-, I (4)
I . 2 . . . r ^^
If ?^ be a positive integer this consists of a finite number of
terms ; we shall subsequently examine the validity of the
expansion when applied to the case where n is negative
or fractional.
^ 57. The liOgarithmic Series. — To expandlog (a; + ?/).
Here /{x) = log (^),- r{x) = '-, f"{x) =-±
Accordingly
log(* + ^)=log. + l^-i?^+i^-l?^+&o.
If a? = I this series becomes
log(i+y)=|-J + ^-...(-,)-^..&o. (5)
When taken to the base a, we get, by Art. 29,
log, (i+2^) = Jlf('|-^ + ^-^ + &o.y (6)
64 Development of Functions,
58. To expand sin (a? + y).
Here f{x) =smx, f^{x) = coax,
f\x) = - sin X, f'\x) = - cos x, &o.
Hence
sin
(x^y)=mix{i -— + — &c. + p-. . . )
^ ^^ . V 1.2 1.2.3.4 \2n J
+ cos:p[^ ^— - + ^^ ...± ^
I 1.2.3 1.2.3.4.5
As the preceding series is supposed to hold for all values,
it must hold when a; = o, in which case it becomes
Qmy=.l 1-^^ 't &c. (8)
I 1.2.3 1.2.3.4.5
Similarly, ii x = —, we get
cos y = I - -^— + &o. (9)
I .2 I .2.3 .4
We thus arrive at the well-known expansions* for the sine
and cosine of an angle, in terms of its circular measure.
59. maelaurin's Theorem. — If we make a? = o, in
Taylor's Expansion, it becomes
/ (y) =/(o) + f/(o) + -^/'{o) + ... ^/W(o) + . . . , (10)
where /(o) . . .f^"^{o) represent the values which /(^r) and
its successive derived functions assume when x = o.
Substitute xior y in the preceding series and it becomes
/ W =/(o) + Y /'(o) + -^^ /"(o) + . . • + ^ /'"' (o) + &c.
* These expansions are due to Newton, and -were obtained by him by the
method of reversion of series from the expansion of the arc in terms of its sine.
This latter series he deduced from its derived function by a process analogous
to integration (called by Newton the method of quadratures). See Opuscula,
torn r., pp. 19, 21. Ed. Cast. Compare Art. 64, p. 68.
Exponential Series, 65
This result may be established otherwise thus ; adopting
the same limitation as in the case of Taylor's Theorem : —
Assume f{x) = A + Bx + Cx^ + Dx^ + Ex^ + &c.
then f {x) ^B-\-2Cx + sDx' + ^Ex' + &c.
f' (x) = 20 + 3 . 2Dx + 4 . ^Ex" + &c.
f''{x) = 3 . 22) + 4 . 3 . 2^iC + &c.
Hence, making a? = o in each of these equations, we get
f(o)^A, f'(o)=B, -^-^ = 0, fM.^ = D,&o.
whence we obtain the same series as before.
The preceding expansion is usually called Maclaurin's*
Theorem ; it was, however, previously given by Stirling, and
is, as is shown already, but a particular case of Taylor's series.
We proceed to illustrate it by a few examples.
60. Bxponential lieries. — Let y = cf.
Here f{x) = «*, hence /(o) =1,
f{x) = a'' log a, „ /(o) =logfl,
r{x)=a^{loga)\ „ r(o)=loga)\
/(«) {x) = a- (log a)% „ fW (o) = (log a)- ;
and the expansion is
(x loo; a) (x loff aV (x loff aY ^ , .
I 1.2 I . 2 . ..n
If e, the base of the Napierian system of Logarithms, be
substituted for a, the preceding expansion becomes
X x^ a^ , .
e**=i+- + + ...+ + ... (12)
I 1.2 I . 2 . , . n
♦ Maclaurin laid no claim to the theorem which is known by his name, for,
after proving it, he adds — "This theorem was given by Dr. Taylor, Method.
Increm." See Maclaurin's Fluxions, vol. ii., Art. 751.
F
66 Development of Functions,
If ^ = I this gives for e the same value as that adopted in
Art. 29, viz. :
III I
e = I + - + + + + . . •
I 1.2 1.2.3 1.2.3.4
61. £xiiaiisioii of sin x and cos x by llaclauriii's
Theorem. Let/(i?;) = sin a?, then
/(o)=o, /(o) = i, r(o) = o, /"(0)=-I,&0,
and we get
, X X 'Ju f.
sm ic = + &c
I 1.2.3 1.2.3.4.5
In like manner
cos X = \ -
2.3.4
the same expansions as already arrived at in Art. 58. .
Since sin (- a;) = - sin x^ we might have inferred at once
that the expansion for sin x in terms of x can only consist of
odd powers of x. Similarly, as cos (- x) = cos x, the expan-
sion of cos X can only contain even powers.
In general, if F{x) = F{- x), the development of F{x)
can only consist of even powers of x. li F{- x) = - F{x), the
expansion can contain odd powers of x only.
Thus, the expansions of tan x, sin'^^r, tan~^a;, &c., can con-
tain no even powers of x ; those of cos x, sec x, &c., no odd
powers.
62. Huy gens' Approximation to length of Circular
Arc.* — If A be the chord of any circular arc, and B that of
Q'P_A
half the arc ; then the length of the arc is equal to , q.p.
<j
For, let R be the radius of the circle, and L the length of
the arc : and we have
A .LB . L
♦ This important approximation is due to Huygens. The demonstration
given above is that of Newton, and is introduced by him as an application of
his expansion for tlie sine of an angle. Vid. " Epis. Prior ad Oldembnrgium."
Huygens' Approximation. 67
hence, by (8),
A = L z^^ + 7 — zr. - &0.
2.3.4. J?' 2 .3.4. 5 . i6.i2*
8B = 4L — ^+ -— 57 -&o.
2 . 3 . 4 . i22 2 . 3 . 4 . 5 . 64 . i2*
consequently, neglecting powers of ^ beyond the fourth, we
get
Hence, for an arc equal in length to the radius the error in
adopting Huygens' approximation in less than *^ part of
the whole arc ; for an arc of half the length of the radius
the proportionate error is one-sixteenth less ; and so on.
In practice the approximation* is used in the form
X = 2J5 + i (2^ - A).
3
This simple mode of finding approximately the length of
an arc of a circle is much employed in practice. It may also
be applied to find the approximate length of a portion of
any continuous curve, by dividing it into an even number of
suitable intervals, and regarding the intervals as approxi-
mately circular. See Rankine's Rules and Tables, Part I.,
Section 4.
* To show the accuracy of this approximation, let us apply it to find the
length of an arc of 30° in a circle whose radius is 100,000 feet.
Here B = 2li sin 7° 30', A = 2B sin 15" ;
but, from the Tables,
sin 7° 30' = .1305268, sin 15° = .2588190.
„ ^ 2B-A
Hence 2B + = 52359.71.
The true value, assuming v = 3.1415926, is 52359.88 ; whence the error is but
.17 of a foot, or about 2 inches.
F 2
^
68 Development of Functions,
yf 63. Expansion of ioxf^x. — Assume, according to Art.
61, the expansion of tan~^;z;to be
Ax + Bx" + ai?5 + Bx' + &c.,
where -4, B^ C, &c., are undetermined coefficients:
d \^x^^x
then -^ = -4 + iBx"- + sOc* + ^Bx^ + &c. ;
but -^—, = 1 = I - a;' + a?* - a;' + &o.,
ax \ ^■ x^
when a; lies between the limits ± i.
Comparing coefficients, we have
^=1, 5 = -i C = -, D = - i&o.
3 5 7
Hence
^ , x x"" x'' , . aj'"+i , ,
tan-^a; = + ... + (- i)" + . . . ; (14)
135 ^ ^ 2W + I » \ t/
when X is less than unity.
This expansion can be also deduced directly from Mac-
laurin's Theorem, by aid of the results given in Art. 46.
This is left as an exercise for the student.
64. Expansion of sin'^ic. — Assume, as before,
sin"^a; = Ax + Bx^ + Cx^ + &c. ;
then 7 TTT = A + ^Bx^ + $Cx^ ^ &o. ;
(I - x^p
but 7—^ — ^, = (1 -x')-^=i + lx'+L^x' +
(l -X)^ ^ ' 2 2.4
2.4... 2r
Hence, comparing coefficients, we get
-4=1, ^=-.-, C = i-^.-,&o.
23 2.45'
Finally,
. , X 10? 1.33?* i.3...2r-i ;r"*' , ,
sin"^ir = - + - .-+ — ^. — + .. . + — . +... (15)
123 2.4 5 2.4... 2r 2r+ I ^ '
Euler^s Expressions for Sine and Cosine, 69
Since we have assumed tliat mr'^x vanishes along with x we
must in this expansion regard but^x as oeing the circular
measure of the acute angle whose sine is x.
There is no difiSculty in determining the general formula
for other values of sin"^^, if requisite.
A direct proof of the preceding result can be deduced
from Maclaurin's expansion by aid of Art. 5 1 . "We leave
this as an exercise for the student.
From the preceding expansion the value of tt can be
exhibited in the following series :
TT I II I . 3 I p
6 2 2.38 2 .4.532
For, since sin 30° = -, we have - = sin"^ - ; .'. &c.
2 02
An approximate* value of w can be arrived at by the aid
of this formula ; at the same time it may be observed that
many other expansions are better adapted for this purpose.
65. £uler's Expressions for Sine and Cosine. — In
the exponential series (12), if x >y - i be substituted for x,
we get
e^-^ =1 + + &c. . . •
1.2 1.2.3.4
•■]
1.2.3
cos a; + v^ - I sin a; ; by Art. 59.
Similarly, e~^^"^ = cos a? - -y/ - i sin x.
Hence e^"^'^ + r^^-^ = 2 cos Xj
^^~i _ e-x-/'! ^ 2y/~^ sin X.
A more complete development of these formulae will be
found in treatises on Algebra and Trigonometry.
(16)
* The expansion for sin-^a?, and also this method of approximating to ir, were
given by Newton.
70 • Development of Functions,
66. John Bernoulli's §lcries. — If, in Taylor's Ex-
pansion (i) we make y = - x^ and transfer f{x) to the other
side of the equation, we get
f{:c) =/(o) + xf\x) -^J"(x) + 7^3 /'"(*) - &o. (.7)
This is equivalent to the series known as Bernoulli's,*
and published by him in Act. Lijos., 1694.
As an example of this expansion, let /(a;) = ^ ; then
/(o) = i, y» = ^, /»=^,&c.,
and we get
x^
e^ = 1 + xe" ef" + &c.,
1.2
Or, dividing by e*, and transposing,
e^ = I -x + &o.,
I '2
which agrees with Art. 60.
67. Symbolic Form of Taylor's Theorem. — The
expansion
may be written in the form
/<..„=|..,£.i(|.);....g|)\...|,,„,(.»)
in which the student will perceive that the terms within the
brackets proceed according to the law of the exponential
series (12) ; the equation may accordingly be written in the
shape
f{x^y)^e'rxf{x), (19)
* In his Reduc. Quad, ad long, curv., John Bernoulli introduces this theorem
again, adding — *' Quam eandum seriem postea Taylorus, inteijecto viginti
annorum intervallo, in librum quern edidit, a.d. 1715, demethodo incrementorum,
transferre dignatus est sub alio tantum characterum habitu." The great in-
justice of this statement need not be insisted on ; for while Taylor's Theorem is
one of the most important in the entire range of analysis, that of Bernoulli ia
comparatively of little use ; and is, as shown above, but a simple case of Taylor's
Expansion.
Symbolic Form of Taylor's Theorem. 7 1
where e ^ is supposed to be expanded as in the exponential
theorem, and ^- V^ written for r- ( — ) /(a;), &o.
^n dx"^ Y \dxj ^ ^ "
This form of Taylor's Theorem is of extensive application
in the Calculus of Finite Differences.
68. Other Forms derived from Taylor's Series. —
In the expansion (3), Art. 55, substitute h for y,
^, hdu h" d'u A" d''u ^
then u^ = u + - —■ + — + . . . z— + &o.
I dx I . 2 dx^ I , 2 . . ,n dx"-
If now h be diminished indefinitely, it may be represented
by dx, and the series becomes
du dx dhi dx^ d^hi daf*
Wl = w + + -^„ — + . . . + T-
dx I dx^ I . 2 dx^^ I . 2 . , ,n * '
or m-u = '^dx+'^^dx' + ^^^^dx' + &o., (20)
I 1.2 1.2.3 '\/
in which «i - w is the complete increment of u, corresponding
to the increment dx in x.
Again, since each term in this expansion is infinitely small
in comparison with the preceding one, if all the terms after
the first be neglected {hf Art. 38) as being infinitely small in
comparison with it, we get
du =f\x) dx,
the same result as given in Art. 7.
Another form of the preceding expansion is
du d^u d^u d^u « , ,
Wi - w = — + + + . . . + + &c. (21)
I 1.2 1.2.3 1.2... W ^ ^
69. Theorem. — If a function ofx become infinite for any
finite value of x then all its successive derived functions become
infinite at the same time.
If the function be algebraic, the only way that it can be-
come infinite for a finite value of x is by its containing a
term of the form --^, in which Q vanishes for one or more
7 2 Development of Functions,
values of x for whicli P remains finite. Accordingly, let
dPPclQ
P , , du clx Q dx ; this also becomes infinite when
Q = o.
d^i( d^ii
Similarly, — , — , &c., each become infinite when Q = o.
I
Again, certain transcendental functions, such as e"" ,
cosec {x - a), &c., become infinite when x = a; but it can be
easily shown, by differentiation, that their derived functions
also become infinite at the same time. Similar remarks apply
in all other cases.
The student who desires a more general investigation is
referred to De Morgan's Calculus, page 179.
70. Remarks on Taylor's Expansion. — In the pre-
ceding applications of Taylor's Theorem, the series arrived
at (Art. 56 excepted) each consisted of an infinite number of
terms ; and it has been assumed in our investigation that the
sum of these infinite series has, in each case, a, finite limiting
value, represented by the original function, /(a; + y), or f{x).
In other words, we have assumed that the remainder of the
series after n terms, in each case, becomes infinitely small
when n is taken sufficiently large — or, that the series is con-
vergent. The meaning of this term will be explained in the
next Article.
71. Convergent and Divergent Series. — A series,
Ui, U2, Uz, , , . Un, . . . consisting of an indefinite number of
terms, which succeed each other according to some fixed law,
is said to be convergent, when the sum of its first n terms
approaches nearer and nearer to a finite limiting value, accord-
ing as n is taken greater and greater ; and this limiting value
is called the sum of the series, from which it can be made to
differ by an amount less than any assigned quantity, on
taking a sufficient number of terms. It is evident that in the
case of a convergent series the terms become indefinitely
small when n is taken indefinitely great.
If the sum of the first n terms approximates to no finite
limit the series is said to be divergent.
Convergent and Divergent Series, 73
In general, a series consisting of real and positive terms
is convergent whenever the sum of its first n terms does not
increase indefinitely with n. For, if this sum do not become
indefinitely great as n increases, it cannot be greater than a
certain finite value^ to which it constantly approaches as n
is increased indefinitely.
yt^ 'J 2. Application to Oeometrical Progression. —
The preceding statements will be best understood by apply-
ing them to the case of the ordinary progression
I — »zj**
The sum of the first n terms of this series is in all cases.
I - X
(i). Letir< I ; then the terms become smaller and smaller
as n increases ; and if n be taken sufficiently great the value
of x^ can be made as small as we please.
Hence, the sum of the first n terms tends to the limiting
value ; also the remainder after n terms is represented
by — '■ — , which becomes smaller and smaller as n increases,
and may be regarded as vanishing ultimately.
(2). Let x> I. The series is in this case an increasing
one, and af^ becomes infinitely great along with w. Hence
the sum of n terms, or , as well as the remainder
I -X X - I
after n terms, becomes infinite along with n. Accordingly
the statement that the limit of the sum of the series
i+x-\-x^-\-,,,-i!-af^-\-,.,ad infini
is holds only when x is less than unity, i. e. when the
I — X
series is a convergent one.
In like manner the sum of n terms of the series
I - X + x^ - x^ ^- &c.
i-(-i)^a?"
13 ^^ .
I + a?
74 Development of Functions*
As before, when x<i. the limit of the sum is ; but
I + iC
when x> i, of* becomes infinitely great along with n, and the
limit of the sum of an even number of terms is - oo ; while
that of an odd number is + oo . Hence the series in this case
has no limit.
73. Theorem. — 1/^ in a series of positive terms repre-
sented by
Wi + W3 + . . . + «^n + &c.,
the ratio -^ he less than a certain limit smaller than unity ^ for
Un
all values of n beyond a certain number ^ the series is convergent,
and has a finite limit.
Suppose A; to be a fraction less than unity, and greater
than the greatest of the ratios ^^ . . . (beyond the number
Un
n), then we have
Un
/. Wn+1 < kUn.
"- < k,
Unn
.'. Wn+2 < k^Un.
«"- < k,
,'. UnH < k^Un.
Hence, the limit of the remainder of the series after w„ is
less than the sum of the series
kun + k^Un + . . . + IfUn . . ad infinitum ;
therefore, by Art. 72, less than
kUn . ,
r , since A- < I .
i-k
Hence, since Un decreases as n increases, and becomes infi-
nitely small ultimately, the remainder after n terms becomes
also infinitely small when n is taken sufficiently great ; and
consequently, the series is convergent, and has a finite limit.
Aj2:ain, if the ratio -^ be > i, for all values of n beyond
Un
Convergent and Divergent Series. 75
a certain number, the series is divergent^ and has no finite
limit. This can be established by a similar process; for,
assuming k > i, and less than the least of the fractions
-^, . . . then by Art. 72 the series
Un + kun + khcn + &c. ttd infinitum
has an infinite value ; but each term of the series
Un + Un+i + Un+2 + &C.
is greater than the corresponding term in the above geome-
trical progression ; hence, its sum must be also infinite, &c.
These results hold also if the terms of the series be alter-
nately positive and negative ; for in this case k becomes
negative, and the series will be convergent or divergent
according as - ^ is < or > i ; as can be readily seen.
In order to apply the preceding principles to Taylor's
Theorem it will be necessary to determine a general expres-
sion for the remainder after n terms in that expansion ; in
order to do so, we commence with the following : —
74. Xiemnia. — If a continuous function 0(aj) vanish token
X = a, and also when x = b^ then its derived function ^\x), if
also continuous, must vanish for some value of x between a
and b.
Suppose b greater than a\ then if <^'{x) do not vanish
between a and b, it must be either always positive or always
negative for all values of x between these limits; and
consequently, by Art. 6, (p{x) must constantly increase, or
constantly diminish, as x increases from a to b, which is
impossible, since (^{x) vanishes for both limits. Accordingly,
0'(a?) cannot be either always positive or always negative ;
and hence it must change its sign between the limits, and,
being a continuous function, it must vanish for some inter-
mediate value.
This result admits of being illustrated from geometry.
For, let y = (l){x) represent a continuous curve; then, since
<j){a) = o, and ^(5) = o, we have y = o, when x = a, and also
when x=b; therefore the curve outs the axis of x at distances
a and b from the origin ; and accordingly at some inter-
76 Development of Functions.
mediate point it must have its tangent parallel to the axis of
X. Hence, by Art. lo, we must liave 0'(^) = ^ ^^^' some
value of X between a and b.
75. I^agrange's Theorem on the liimits of Tay-
lor's Series. — Suppose jB„ to represent tlie remainder after
n terms in Taylor's expansion, then substituting X for x ^- y
in (i), we may write
/(Z) =/{.) + ^-^r [x] + -^^^V" w + . . .
in which /(jr),/' (a?) /" {x) are supposed finite and
continuous for all values of the variable between X and x.
If we now make a? = X, we get i^ = o ; accordingly it
contains X - ir as a factor ; hence we may write
Rn = [X-x)PP, {22)
where ph^ positive quantity, and P is a function of X and x.
Consequently we may write
f[X) - \f(x) + (-^/'W + . . . + ^^^/<"-' W
+ {X-xyp]^=o. (23)
Now, let % be substituted for x in every term in the pre-
ceding, with the exception of P, and let F (z) represent the
resulting expression : we shall have
F{z) =/(X) - j/(.) + i^--il/'(2) + . ..+ (X-2)?PJ, (24)
in which P has the same value as in {22).
Again, the right-hand side in this equation vanishes
wlienz = X; .-. F[X) =0.
Also, from (23), the right-hand side vanishes when z = x\
,'. F{x) = o.
Limits of Taylor* s Theorem. 77
Accordingly, since the function F[z) vanishes when 2 = X,
and also when z = x,it follows from Art. 74 that its derived
function F\z) vanishes for some value of z between the limits
X and X.
Proceeding to obtain F'{z) by differentiation from equa-
tion (24), it can be easily seen that the the terms destroy each
other in pairs, with the exception of the two last. Thus we
shall have
F\z) = - (^-^)"">(,) +^;(x - zy--P.
Consequently, for some value of z {z' suppose) between x
and X we must have
Again, if 0 be a positive quantity less than unity it is
easily seen that we may write
z' = x+ e(X-x),
since by assigning a suitable value to 0, x + 9{X - x) can be
made equal to any number intermediate between x and X.
If we substitute this value for z in the foregoing equation it
becomes
(I - or ^^I'f^y^i'^ + e{x- x)]={i - e)p[x - x)pp
^^^^-^ =(i-0)^i?„, by (22),
Hence we infer that p = n, and
iJ, = i^-^/(")(^ + 0(X-^)). (25)
L-
Making this substitution, equation {22) becomes
f{X) = fix) + i^^ fix) + ^-^fn^) + . . .
+ (ZjI^' /(n-l) (^) + (If ^ fin) {:, + e(X-x)}. (26)
The preceding demonstration is taken, with some modifi-
cations, from Bertrand's " Traite de Oalcul Differentiel" (273).
78 Development of Functions,
Again, if h be substituted for X - x, the series becomes
f{x^h)=f{x)+hf{x)+&(i.
■^p7/^'*"M^) + £/^^M^ + 0^). (27)
In this expression n may be any positive integer.
Jin = I the result becomes
f{x + h) =f{x) + hf {x + Oh). (28).
When n = 2,
fix + h) =f{x) + hf (x) + ^/" (x + Oh). (29)
The student should observe that 6 has in general different
values in each of these functions, but that they are all subject
to the same condition, viz., 6 > o and < i.
It will be a useful exercise on the preceding method for
the student to investigate the formulae (28) and (29) inde-
pendently, by aid of the Lemma of Art. 74.
The preceding investigation may be regarded as furnish-
ing a complete and rigorous proof of Taylor's Theorem^ and
formula (27) as representing its most general expression,
76. Geometrical Illustration. — The equation
/(X) =f{x) + {X-x)f [x^Q{X-x)]
admits of a simple geometrical verification; for, let y =f{x)
represent a curve referred to rectangular axes, and suppose
(X, F), {x, y) to be two points Pi, P2 on it : then
f{X)-f{x) Y-y
Y - y .
But 7= is the tangent of the angle which the chord Pi Pj
X — X
makes with the axis of x\ also, since the curve cuts the
chord in the points Pi, P2, it is obvious that, when the point on
the curve and the direction of the tangent alter continuously,
the tangent to the curve at some point between Pi and Pi must be
parallel to the chord Pi P2 ; but by Art. 10,/' (xx) is the tri-
gonometrical tangent of the angle which the tangent at the
Second Form of Remainder, 7^
point (iTi, f/i) makes with the axis of x. He;oce, for some value,
x^y between X and x, we must have
Y-y f{X)-f{x)
^(*'^ = Z^= X-x '
or, writing Xi in the form x + B {X - x)^
f{X) =/{x) + {X-x)flx+e{X-x)].
77. iSecond Form of Remainder. — The remainder
after n terms in Taylor's Series may also be written in the form
En = ^ . ^ AV(«) {X + Oh).
\n-i '
For it is evident that B.n may be written in the form
{X-x)F,^
.-./(X) =7(0.) + (X- ^) fix) + . . . 4 (^Z^V^**-^) (^)
■v{X-x) P,.
Substitute s for x, as before, in every term except Pi ; and the
same reasoning is applicable, word for word, as that employed
in Art. 75. The value of F' (s) becomes, however, in this
case
and, as F\z) must vanish for some value of 2 between x and
X, we must have, representing that value by a; + Q (X - x)y
p. = (^-'^)""('-g)'"'yW {^ + 0 (X - «)}, (30)
where Q, as before, is > o and < i.
If h be introduced instead of X - a?, the preceding result
becomes
JJ„ = (Luir' J^nfl^n) (^ , Q,0, (31)
which is of the required form.
8o Development of Functions.
Hence, Taylor's Theorem admits of being written in the
form
{i-e)^'f^-){x+Qh). (32)
The same remarks are applicable to this form* as were made
with respect to (27).
From these formulae we see that the essential conditions
for the application of Taylor's Theorem to the expansion of
any function in a series consisting of an infinite number of
terms are, that none of its derived functions shall become
infinite, and that the quantity
shall become infinitely small, when n is taken sufficiently
large ; as otherwise the series does not admit of a finite limit.
78. liimit of when n is indefinitely great.
1 , 2 ,.n
Let Un = , then -^ = ; .*. -^ becomes smaller
I . 2 . . w Un n+ I Un
and smaller as n increases ; hence, when n is taken sufficiently
great, the series Un+n ^n+s? • • . &c., diminishes rapidly, and
the terms become ultimately infinitely small. Consequently,
whenever the n^^ derived function f^'^'i {x) continues to he finite for
all values of w, however great, the remainder after n terms in
Taylor* s Expansion becomes infinitely small, and the series has
a finite limit.
* This second form is in some cases more advantageous than that in (27).
An example of this will be found in Art. 83.
y//^ Remainder in the Expansion of sin x. 8 j
r^yg- ©eneral Form of JUaclaurin's Sferies. — The
/expansion (27) becomes, on making x = o, and substituting
X afterwards instead of A,
A^) =/(o) +7/(0) --^^/"(o) + . . . + |J-i/("-" (o)
+ ^/'"'(9^)- (33\
Hence the remainder after n terms is represented by
where 0 is > o and < i .
This remainder becomes infinitely small for any functioi]
f{x) whenever ,— Z^**^ (Ox) becomes evanescent for infinitely
great values of n.
We shall now proceed to examine the remainders in the
different elementary expansions which were given in the
commencement of this chapter.
80. Reniainder in the Kxpansion of a^. — Our for-
mula gives for Bn in this case
^{loga)»a^.
Now, a^* is finite, being less than a'' ; and it has been proved
(x lo2r a^^
in Art. 78 that ^^ — becomes infinitely small for large
values of n. Hence the remainder in this case becomes
evanescent when n is taken sufficiently large. Accordingly
the series is a convergent one, and the expansion by Taylor's
Theorem is always applicable.
8 1 . Remainder in the Expansion of sin x. — In this
case
„ a?" . fmr ^ \
jB„ = r- sm — + (fx .
n \2 J
82 Development of Functions.
This value of Bn ultimately vanislies by Art. 78, and the
series is accordingly convergent.
The same remarks apply to the expansion of cos x.
Accordingly, both of these series hold for all values of x.
82, Remainder in tbe Expansion of log (i + x). —
The series
X x"^ a^ X*- o
+ + &c.,
1234
when a? is > i , is no longer convergent ; for the ratio of any
term to the preceding one tends to the limit - x ; conse-
quently the terms form an increasing series, and become
ultimately infinitely great. Hence the expansion is inappli-
cable in this case.
Again, since f^{x) = (- i)**"^ - — -, the remainder
(i + X)
Bn is denoted by - — —- 1 jr- ) ;; hence, if x be positive and
X
less than unity ^ -^ is a proper fraction, and the value of
Bn evidently tends to become infinitely small for large values
of n ; accordingly the series is convergent, and the expansion
holds in this case.
83. Binomial Theorem for Fractional and IVega-
tive Indices. — In the expansion
(i +aj)'» = I + — a; + — ^^ -x^ + ,...
^ ^ I 1.2
m(m- i) . . . im-n-\- i)x^ o
I . 2
if Un denote the n*^ term, we have
w„+i m-n+ I
the value of which, when n increases indefinitely, tends to
become -iz;; the series, accordingly, is convergent if a; < i,
but is not convergent ii x > i .
Binomial Theorem, 83
Accordingly, the Binomial Expansion .does not hold when
X is greater than unity.
Again, as
/(») {x) =m{m~ i) . . . (m - w + i) (i + a?)'"-",
the remainder, by formula (25), is
1 .2. . . w
or
m (m- i) . . . (m-w + i) a?**
I . 2 . . . yj (i + Oa;)"-"**
Now, suppose a; positive and less than unity ; then, when
n is very great, the expression
m{m- i) . . . (wi - w + i)
I . 2 . . . n
becomes indefinitely small ; also -. „ . _^ is less than unity;
hence, the expansion by the Binomial Theorem holds in this
case.
Again, suppose x negative and less than unity. We employ
the form for the remainder given in Art. 77, which becomes
in this case
(- i)"— ^^ \ r — ^ — (i - OY-' (i - 9x)*^^ ;
^ ^ 1 . 2 . . . (w- i) ^ ^ ^ '
or
U \«^(^^~ 0 . . .{m-n+ i) {1 - Q)'^^af^
^ ^^ 1.2 ... (/^-I)
1-0
I -Ox
1 — a
Also, since x< i, 6x< 9; .'. i -6x> i - 6 ; hence 7-
is a proper fraction ; .*. any integral power of it is less than
unity ; hence, by the preceding, the remainder, when n is
sufficiently great, tends ultimately to vanish.
G 2
84 Development of Functions,
In general {x + y)*^ may be written in either of the forms
x^[ I +-) or ^'"f I +-) :
now, if the index m be fractional or negative, and x> y, or
- a proper fraction, the Binomial Expansion holds for the
X
series
{x + y)"' = x'^[ I +-] =x"' + — x'^'-^y + -— ^^ (xf^'^y^ + &o.,
' \ Xj I 1.2
but does not hold for the series
(a; + y)"* = y"»f I -f - = ^T + - y"*~^a; + — ^^ <- 2/*""V + &c., '
since the former series is convergent and the latter divergent.
We conclude that in all cases one or other of the expan-
sions of the Binomial series holds ; but never both, except
when m is a positive integer, in which case the number of
terms is finite.
84. Remainder in the Expansion of tan^^ic. — The
series
X a^ s^
tan'^a; = + &c.,
135
is evidently convergent or divergent, according as a; < or > i . ,
To find an expression for the remainder when a?< i, we have,
^y (8)» P- 50—
, , „ |w - I . sin ( w n ioxr^x J
/(n)(^) = ( |. ] . tan-i-^, = (- i)-^ ^Z: IJ I .
\d^J (I + x'f.
Hence we have, in this case,
a;" sin jw n tan"^ (Ox) >
^'^"^ '^ n(i + 0V)5
which, when x lies between + i and - i , evidently becomes
infinitely small as n increases, and accordingly the series holds
for such values of x.
Expansion hy aid of Differential Equations, 85
85. Expansion of 8in"^a?. — Since the function sin'^a? is
impossible unless a; be < i, it is easily seen that the series
given in Art. 64 is always convergent ; for its terms are each
less than the corresponding terms in the geometrical pro-
gression
X -\- 3^ + x^ + &0.
Consequently, the limit of the series is always less than the
limit of the preceding progression.
A similar mode of demonstration is applicable to the
expansion of tan"^^; when x < i, as well as to other analogous
series.
In every case, the value of i^„, the remainder after n
terms, furnishes us with the degree of approximation in the
evaluation of an expansion on taking its first n terms for
its value.
86. Expansion by aid of DiflTerential Equations. —
In many cases we are enabled to find the relation between
the coefficients in the expansion of a function of x by aid of
differential* equations ; and thus to find the form of the
series.
For example, let y = e', then
dx~^~^'
Now suppose that we have
y = a^\ a^x -v a^st^ + . . . an^if* + . , . ,
then -;- = «i + za^ + . . . na^x**''^ + &o.
dx
Accordingly we have
* This method is indicated by Newton, and there can be little doubt that it
■was by aid of it he arrived at the expansion of sin {m sin'^ x), as well as other
series. — Vide Ep. posterior ad Oldemburfftum. It is worthy of observation that
Newton's letters to Oldemburg were written for the purpose of transmission to
Leibnitz.
86 Development of Functions,
hence, equating coefficients, we have
2 2' 3 2.3
Moreover, if we make a; = o, we get ajq = i,
.-. ^ = I + - + + + &c.,
I 1.2 1.2.3
the same series as before.
Again, let
y = sin (msin"*a?).
Here, by Art. 47, we have
Now, if we suppose y developed in the form
y = aQ + ayx-^ a^x"^ + . . . + UnP^ + &c.,
di/
then ^ " ^^ ■*" ^^^ "^ 3^3i2^ + . . . + nanx''-^ + &c.,
^ = 2«2 + 3 . 2«3aj + . . . + w (w - i) a„a^-* + &c.
Substituting and equating the coefficients of a^ we get
^Z ^2
«!.• (34)
'♦»+2 — 7 TT \ "■»•
{n + i) (w + 2)
Again, when a; = o we have y = o; .*. «o = o.
Hence we see that the series consists only of odd powers
of a; ; a result which might have been anticipated from Art.
6i.
To find tti. When a? = o, cos {m sin~^a?) = i, hence ( 7^ )= ^ ;
accordingly ai = m ;
m^ - I m{m^ - i)
•*• Q/z — Cli = — .
2.^ 1.2.3
m^ - g m (rr^ - i) {w^ - 9)
4.5 1.2.3.4.5
Expansion of sin mz and cos mz, 87
hence we get
• * / . _, \ ^ m(m^- i)
sin* im sin^a?) = - jz; ^ x^
I 1.2.3
mM - i) (w^ - 9) , p , ,.
+ — ^ '-^ ^ x' - &c. (35)
1.2.3.4.5
In the preceding, we have assumed that sin'^a; is an acute
angle, as otherwise both it, and also sin [m dic^x), would admit
of an indefinite number of values. — See Art. 26.
87. Expansion of sin mz and cosws. — If, in (35), she
substituted for sin"^ij?, the formula becomes
( I m^ - \ . .
sm mz = m sm z\ sm^s
(i 1.2.3
im^ - i) {m^ - q) . . _ ) ...
+ ^ LI ^sin*2 - &c. (36)
1.2.3.4.5 )
In a similar manner it can be proved that
m^ sin^z m^ (m"^ - 4) . . _ , .
cos mz = 1 + — ^ sirrz - &c. (37)
1.2 1.2.3.4
If m be an odd integer the expansion for sin mz consists
of a finite number of terms, while that for cos mz contains an
infinite number. If m be an even integer the number of
terms in the series for cos mz is finite, while that in sin mz is
infinite.
The preceding series hold equally when m is a fraction.
A more complete exposition of these important expansions
will be found in Bertrand's " Calcul Diiferentiel."
In general, in the expansion (36), the ratio of any term'
to that which precedes it is -. r-7 r sin^s, which, when
^ {n + i){n + 2)
n is very great, approaches to sin^s. Hence, since sin z is
less than unity, the series is convergent in all cases. Similar
observations apply to expansion (37).
* This expansion is erroneously attributed to Euler by M. Bertrand ; it wixa
originally given by Newton. See preceding note.
88 Development of Functions.
The expansion
I 1.2 1.2.3 1.2.3.4
can be easily arrived at by a similar process.
S^. Arbogast's Method of Berivations.
X X^ Q^
If f« = « + ^- + c + d + &c..
I 1.2 1.2.3
to find the coefficients in the expansion of 0 {u) in ascending
powers of x —
Let /W = 0 M,
B C
and suppose f{x) = A + —x + a^ + &o.
=/(o)+^/'(o)+:^/"(o) + &o.,
then we have evidently
^=/(o) = 0(a).
Also, writing u\ ic'\ u"\ &c. instead of
du d'^u dhi
di' d?' d?'^"^''
by successive differentiation of the equation /(a?) = 0 (««), we
obtain
. r {x) = 0' (u) . u%
f\x) = 0' ill) . ir + 3(/)" (i^) . u' . w" + ^"' (w) (w7,
/i^(;r) = 0' [u) . i^^ + ^" ill) {^u' tr + 3 (w'7] + 6f ''(w) . (w^. u"
Now, when a; = o, w, w', w", e*'", . . , obviously become
fl5, by c^ dy . . . respectively.
Arhogasfs Method of Derivations, 89
Accordingly,
j?= / (o) = ^' (a) . 6, / ;
B = f" (o) = ^' («) . ^ + 3^'' [o>) . &c + f' [a) . 6»,
^ = /iv (o) = ^' («) . e + ^" (fl^) (46^ + 3^') + 6^'" (a) . h'^c
+ ^^^ («) . 6*.
From the mode of formation of these terms, they are seen
to be each deduced from the preceding one by an analogous
law to that by which the derived functions are deduced one
from the other ; and, q.% f (x), f" {x) . . . are deduced from/(a;)
by successive differentiation, so in like manner, B, C, D^ . . .
are deduced from ^ {u) by successive derivation ; where, after
differentiation, a^ 6, c, &c., are substituted for
du d^u p
«'rf^' ^' •••*''•
If this process of derivation be denoted by the letter S, then
J5 = 8.^, C=^.B, B = ^.C,&G. (38)
From the preceding, we see that in forming the term
S . ^(a), we take the derived function (ji^a), and multiply it
by the next letter b, and similarly in other cases.
Thus 8 .b =c, S .c =d, .. .
S . J"* = mb'^'^c, S . c"* = mc'^-^d . . .
Also d .(l>'{a)b = ^' (a) c + f [a) b\
This gives the same value for C as that found before ; B
is derived from C in accordance with the same law ; and so
on.
The preceding method is due to Arbogast : for its com-
plete discussion the student is referred to his " Calcul des
Derivations." The Rules there arrived at for forming the
successive coefficients in the simplest manner are given in
" Galbraith's Algebra," page 342,
go Development of Functions.
As an illustration of this method, we shall apply it to find
a few terms in the expansion of
. f X a? ct^ o \
m[a + h - + c + d + &o. .
\ I 1.2 1.2.3 J
Sin
Here A = sma, JB = d .sina = b cos a,
C = S . boosa = c coBa - b^ BID. a,
D =S. C = d cosa - ^bc sin a - b^ cos «,
JE = d.D = eGOBa- {^bd + 3c') sin a - 6b^c cos a
+ ¥ sin a.
x^
If the series a-vbx + o + &c. consist of a finite num-
I . 2
ber of terms the derivative of the last letter is zero — thus, if
d be the last letter, S . c? = o, and d is regarded as a constant
with respect to the symbol of derivation §.
If the expansion of ^ (u) be required when u is of the
form
a + /3^ + 7a;2 + So?' + &c.,
the result can be attained from the preceding method by
substituting a, b, c, c?, &c. instead of a, j3, i . 2 7, i . 2 . 3 . S,
&c., and proceeding as before.
The student wiU. observe that in the expression for the
terms i), E, &c., the coefficients of the derived functions
^'(fl), (^' {a), &c., are Qom-^lQiolj independent of the form oi the
function ^, and are expressed in terms of the letters, b, c, d,
&c. solely ; so that, if calculated once for all, they can be applied
to the determination of the coefficients in every particular
case, by finding the different derived functions 0'(«), (^"{a),
&c., for that case, and multiplying by the respective coef-
ficients, determined as stated above.
Examples. 91
Examples.
1. If M = f(ax + bu), then --—=-—-. This furnishes the condition that
•^ ^ ' a dx b dy
a given function of x and y shouM be a function of ax + by.
2. Find, by Maclaurin's theorem, the first three terms in the expansion of
tan;p.
Ans. « + — + —
3 15
3. Find the first four terms in the expansion of sec x.
x^ 5«* 61 x^
Ans. I + — + ^— + .
2 24 720
4. Find, by Maclaurin's theorem, as far as «*, the expansion of log (i + sin x)
in ascending powers of x.
Let /(a;) = log (i + sin a;),
Ai. £>i \ cos a; I - sina:
then/ (a;) = : — = = sec a; - tan x.
I + sm a; cos a;
/"(a?) = sec a; tan a; - sec'^a; = - f'{x) sec x ;
f"\x) =-/"(a;) sec x -f\x) sec a? tan x = -f'[x) f'[x),
.'.f^^{x) =- {f"{x)Y -f{x)r'{x);
.•./(o) = o, /'{o)=l, /"(o) = -i, r'(o) = i, /-(o) = -2;
a;2 x^ x^
.'. log (i + sin a?) - a; + -7 + &c.
"^ ' 2 6 12
5. Find six terms of the development of in ascending powers of a;
cos X
^x^ x^ 2x^
Ans. i+a; + a;2 + — +- + :^— . . .
3 2 10
6. Apply the method of Art. 86, to find the expansions of sina; and cos a:.
7. Prove that
tan-i {x + h) = tan-i a: + A sin 2 ^^ - (A sinz)^ !^^_i? + (^ sinz)^ &c.,
where z = cot-^a:.
Here/(a;) = tan-^a; = — z; and by Art. 46, j— = (- i)" |«- » sin^zsin ««; .*. &q.
92 Examples,
8. Hence prove the expansion
TT sin z sin 2% , sin Xz .
- = 2 + cos z H cos'« + — C0S*2 + &c.
3 1 a 3
Let A = - cot a = - «, &c.
q. Prove that
» « sin a sin 2« sin 32
2 2 I 2 3
Let A sin z = - I in Example 7 : then A + x = — ; = - tan - ; .•. &o
Bin* a
10. Prove the expansion
TT sii
2 cos z ' 2 cos' 2 ' 3 cos'a
TT sin 2 I sm 22 I sm sz
Assume h = — : , then
sin 2 cos 2
X + h = - tana = tan (ir - a) ; .*. » - 2 = tan"* (z ■\- h), &c.
Substituting in Example 7, we get the result required.
The preceding expansions were first given by Euler.
11. Prove the equations
sin 9* = 9 sin a; - 120 sin'a; + 432 sin^a; - 576 sin'ar + 256 sin'as^
cos 6a; = 32 cos^x - 48 cos*a; +18 cos'a; - i.
These follow from the formulae of Article 87 .
12. If m = 2, Newton's formula, Art. 87, gives
( . sin^a? sin^a? „ )
< sm a; &c. 5 ;
verify this result by aid of the elementary equation sin 2a; = 2 sin x cos x.
13. If <^ (a; + A) + ^ (a; - A) = ^ (a;) ^ (A), for all values of x and h,
<l>"(x) 0'^(a;)
prove that ^—7-r = ■hrr-r = &c. = constant :
<t>{x) <i>"{x)
and also 4>'(o) = o, <p"'{o) = o, &c.
14. If, in the last, ^^ = ^^'^ prove that d>(x) = e^'-^-e-^.
d>" (x)
It . ■ = - a' ; prove that A (a;) = 2 cos (ox).
I
of
Examples, 93
5. Apply Arbogast'B method to find the first four terms in the expansion
(a + i« + cx^ + d3fi+ &c.)".
Am. «« + «««-! bx + (^ ^^~ ' ^2 ^ „a^ J ^-2 ;cj
+ n |(2jllli!L:ila«-3 ^8 + (n - i) an-zi^ + an-i^j a^ + &c.
e* + I
16. Prove that the expansion of . x can contain no odd powers of x.
For if the sign of x be changed, the function remains unaltered.
17. Hence, show that the expansion of contains no odd powers of x
beyond the first.
_r X x X e^ -It I .
Here + _ = - , ; .♦. &c
^ - I 2 2 e* - I
18. If t* = , prove that
n (d»-^u\ n{n-i) /d"^^u\ (du\ , ^
I \dx^--J(i 1.2 \dx"^/o \dxjQ
and hence calculate the coefficients of the first five terms in the expansion of «.
Here «*« = « + «, and by Art. 48, we have
(du n(n-i)d^u d»u\ d^u
dx 1.2 dx^ dx<^l dx^
x^ - , . .
19. If *
X ^ Bx ^ Bi
-r* 4-
[ 2 1.2 1.2.3
•4
ve that
.2. ..6
^1 = ^, ^2 = —, ^8 = —, ^4 = — , &C.
6 30' 42' 30'
These are called Bernoulli's numbers, and are of importance in connexion
with the expansion of a large number of functions.
20. Prove that
X X Bix^ , , , Btx^ , , , Bzi^
««+i 2 1.2^ ' 1.2.3.4^ ' 1.2. ..6^ '
94 Examples,
21. Hence, prove that
= BiX (2« - l) + 2* - I + -7 (»' - l) f &C
«*+i ^ ' 3-4 3.4.5.6' '
= + &c.
2 24 240
22. Prove that
2^-Bix^ 2*^23^ i^Bsmfi .
ar cot a; = I ■ 7 - &c.
1.2 1.2.3.4 1. 2. ..6
23. Also, tan - = Bix (2^ - i) + -^ (2* - i) + &c.
2 3.4
24. Prove that
X ^ X ^ a;2 ^ X*' Bzx^
- cot - = I - -Bi 1 B2-. — — ...
a
This follows immediately by substituting - for x in Ex. 22.
25. Given u{u— x) = i ; find the four first terms in the expansion of u in
terms of a;, by Maclaurin's Theorem.
26. If ar^^ + ^ + y = o,
dx^ dx '
expand y in powers of x by the method of indeterminate coefficients.
27. Show that the series
X «' a;' X*-
J III 2*** ^^ A***
is convergent when x< i, and divergent when a? > r, for all values of m.
28. Prove the expansion.
(x - «)»» <p {x) {x - «)« <p{a) {x - a)«-i ^a \<p{a) ]
I _ (dyif{a)\
"^ 1 . 2 . (a; - a)'«-2 U«/ U(«)) "^ °
29. Find, by Maclaurin's Theorem, the first four terms in the expansion of
(I + a)* in ascending powers of x.
Let f{x)={iArx)\
Examples, 95
But, by Art. 29, /(o) = «;
.•./•(o) = -;, /"(o)-^. /"'(o) = -y«-
Hence (i + a;)* = * + -^ a;3 + &c.
2 24 16
This result can be verified by direct development, as follows:
let u={i + xYj
I X x^ se^
then log M = - log ("i + a^) = I + + . . . ;
X 234
,'. u = e ^ 3'4"* = e.e"^ 3"4 •••
r fx x^ a? \ x^fi X x^ y x^ (i X \3 -|
[a; iia;'^ Tx^ "1
I-- + —." ' .
2 24 16 J
30. In Art. 76, if /(a;) and/'(x) be not both continuous between the points
Pi, Pz, show that there is not necessarily a tangent between those points, parallel
to the chord.
31. Find the development of : in ascending powers of ar, the coef-
sin X sm 2a;
ficients being expressed in Bemoullian numbers. " Camb. Math. Trip., 1878."
_. a; sin 3a; , . ,, . . ,- -l , ^
Since -; : = a: cot a; + a; cot 2a;, the expansion m question, by (22),
sin a; sin 2a; ' ^ ^ ' ^ '
is
3 22/?2a:2 2<^4ar< , , ^ 2^ B^x^ , ^ , „
( 96 )
CHAPTER IV.
INDETERMINATE FORMS.
89. Indeterminate Forms. — Algebraic expressions some-
times become indeterminate for particular values of the
variable on which they depend ; thus, if the same value a
when substituted for x makes both the numerator and the
denominator of the fraction *^-7-n vanish, then" -7-T becomes of
<l>(x) <p(a)
the form -, and its value is said to be indeterminate,
o
Similarly, the fraction becomes indeterminate iif{x) and
(p (x) both become infinite for a particular value of x. We
proceed to show how its true value is to be found in such
cases. By its true value we mean the limiting value which the
fraction assumes when x differs hy an infinitely small amount
from the particular value which renders the expression indeter-
minate.
It will be observed that the determination of the diffe-
rential coefficient of any expression /(i?;) may be regarded as a
case of finding an indeterminate form, for it reduces to the
. , . ^ fix + h] - fix) ,
determmation of — — =^^-^ when A = o.
h
In many cases the true values of indeterminate forms can
be best found by ordinary algebraical and trigonometrical
processes.
We shall illustrate this statement by a few examples.
Examples.
I. The fraction ~^ ?^ r-; becomes of the form - when a; = c; but since
hx^ - 2bcx + bc^ o
it can be written in the shape -rr L, its true value in all cases is -.
^ b{x -c)^ b
Examples. 97
2. The fraction — ir=- becomes - when * = o.
\/ a + X "■ V a -X °
To find its true value, multiply its numerator and denominator by the com-
plementary surd, ^ a + » + ^ a — x, and the fraction becomes
x{*/ a + a; + '\/ a - x) \/a + x+ \/a-x ^
■ or '■^-- ;
2X 2
the true value of which is ya when x = o.
*y a' + ax + x"^ - \/ d^ - ax + x"^
3. ^ , when * = o.
\/a + X — Ya-x
Multiply by the two complementary surd forms, and the fraction becomes
2ax {ya + a; + ^/^a - x)
2x {\/a'i + ax-\-x^ + \/a^-ax-\-x'^]
a ( V a + x-\- y a — x)
ts/ a^ ^ ax -V x^ ^ *y c^ — ax -V 3^
the true value of which evidently is \/a when a; = o. From the preceding
examples we infer that when an expression of a surd form becomes indeter-
minate, its true value can usually be determined by multiplying by the com-
plementary surd form or forms.
or
2x - \/ ^x^ — a} . , I
4. whfjn » = a. Ans. -.
X — y %a^ — a^ ^
a - y^a^ — x^ . . i
5. — when x = o. Ana. — .
x^ 2a
, a sin 0 - sin C0 , o .
6. becomes - when 0 = o.
0(cos Q - cos a9) o
To find its true value, substitute their expansions for the sines and cosines,
and the fraction becomes
a [9 + ...]- [ae + . . .
\ 1-2.3 / \ 1.2-3 /
e{ + ... + .. .}
^ (a3 _ a) + . . ,
m 53 .
-v«--i)-..
H
98 Indeterminate Forms.
Divide by 0"^ («- - i), and since all the terms after the first in the new numerator
a
3
and denominator vanish when 0 = o, the true value of the fraction is - in this
7. The fraction
aox'' + aiic"-! + . . . + an
becomes ^ when « = 00 :
its true value can, however, be easily determined, for it is evidently equal to
that of
^.+^+^+..
X^-n,
01 «2
X X^
Moreover, when a; •= 00, the fractions — , -r ...— ..., all vanish ; hence,
X X^ X
the true value of the given fraction is that of
Ao ^
xm-n — when x =00.
ao
The value of this expression depends on the sign of m — n.
(i.) If PI > n, «»""'» = 00 when a; = 00 ; or the fraction is infinite in this
case.
An
(2.) Itm = «, the true value is — .
ao
(3.) If m < M, then a;»»-» = o when^ = 00 ; and the true value of the frac-
tion is zero.
Accordingly, the proposed expression, when x = 00, is infinite, finite, or zero,
according as m is greater than, equal to, or less than n. Compare Art 39.
8. M = v^ic + a- Yx + b, when a; = 00.
a-b
\/x + a + *y X + b
9. \/a;2 ^ ax - x^ when a; = co. Ans. -.
10. « = a* sin ( — I , when a; = 00.
(i.) If « < I, a* = o when a; = 00, and therefore the true value of u is zero
in this case.
(2.) If a > I, then a* becomes infinite along with x ; but as — is infinitely
small at the same time, we have sin — ■= — . Hence, the true value
a' a"
of tt is c in this case,
Here w = — . , = o when a; = 00.
Method of the Differential Calculus. 99
r. « = \/a2 _ -j;3 cQt _ I is of the form o x oo when x = a.
2 \« + a;
Here m -
IT /« - «
tan - J
but, when a - a: is infinitely small,
tan
v la — X -K ja — X
2 y^T~x^ 2\«T«'
\/ a} — x^ a + X ±a ,
u = I ; = ' = — when x = a.
la- X Z *■
\7^~x 2
X sm (sm x) — sm'a;
12. w = -^ , when x — o.
Substitute the ordinary expansion for sin a;, neglecting powers beyond the sixth,
and u becomes
( . sin^a? sin'ar) / a? x^\^
X {sm X ; — + -; — [ - («-,— + — )
( |3 k ^ \ |3 |5/
i^ x'^ if a;3\3 a;5 / ajZ a;4\2
Hence we get, on dividing by a;', the true value of the fraction to be — when
lo
« = o.
(a sin> + )8 cos2<^)» - iS" , ■ ^ . ,
13. ' a^- fin ' ^^®^ " = ^- ^'**- ^"^^•
Similar processes may be applied to other cases ; there
are, however, many indeterminate forms in which such pro-
cesses would either fail altogether, or else be very laborious.
We now proceed to show how the Differential Calculus
furnishes us with a general method for evaluating indetermi-
nate forms.
90. — -method of the Differential Calculus. — Sup-
pose —^ to be a fraction which becomes of the form - when
(}>{x) o
X = a;
i. e,f{a) = o, and (ft (a) = o ;
H 2
lOO Indeterminate Forms.
substitute a + A for a; and the fraction becomes
f{a-^h) -f{a)
f±l3 or_L_.
^(a+ hY <p{a + h) - <p[a) '
but when h is infinitely small the numerator and denominator
in this expression become /'(«) and <^\a\ respectively; hence,
in this case,
f{a^h) f\a)
(^{a + h) ^\a)'
Accordingly, -tjA represents the limiting or true value of
the fraction -7-7.
0(a)
(i.) If /'(a) = o, and i^'{a) be not zero, the true value of
—7-^ is zero.
(2.) If / (a) be not zero, and <^\a) = o, the true value of
0(a)
(3.) If /'(a) = o, and 0'(a) = o, our new fraction ''-77^ is
0(aj
still of the indeterminate form -. Applying the
preceding process of reasoning to it, it follows that
its true value is that of'
■(«)'
If this fraction be also of the form -, we proceed to the
next derived functions.
In general, if the first derived functions which do not
vanish be/(**)(a) and 0(")(a), then the true value of '—j-r
is that of ^^^^jtH-
Examples, loi
Examples. '"*
tf
a? sin a; -
cos a;
2
when a: =
2'
/(^)
= a; sin a; -
2'
Here
^ {x) = cos a; ;
•*• /'(^) = a: cos a; + sinaf, /' f - J = i,
4)'(a;) = -sina?, <^' ^^j =- i.
Hence « = - i, when a? = -.
2
gmx _ gmo
2. M = r-, when a? - a.
(a? - a)*-
Here /(») = «»»* - «»»»,
^(a;) = (a; - «)•• ;
<p'{x) =r{x- «)♦"•*, ^'(») is o or 00, as r > or < i,
Hence the true value of w is 00 or o, according as r > or < i.
This result can also be arrived at by writing the fraction in the form
hence, expanding ^w*^, and making A = o, we evidently get the same result as
before.
3-
Here
X-
-sma;
a;3
when x = o.
f\x) = I - C03«,
/'(o) = o.
<?>'(«) == 3*S
^'(o) = o.
f"{x)=B>mx,
/"(o)=0.
<l>"{x)=6x,
^"(o)=o.
f"'{z) = coax,
/'"(O) = I.
<p"'{x) = 6,
<^"'(o) = 6.
I02 Indeterminate Forms,
Hence, the true value is -, as can also be immediately arriyed at by substituting
z— -2 + &c. instead of sin z.
o
4* when z = o, Ans. log a.
X
. ^/(x)-^/(a) /(^) +/»
5» T\ 7~» wnen a; = a. ,, —7— 77-r.
c'^(a;) - ^4)(fl) " <^(a)+ <^ (a)
It may be observed that each of these examples can be exhibited in the form
00 - 00 , that is, as the difference of two functions each of which becomes in-
finite for the particular value of a; in question.
91. Form o x cxs. — The expression /(a?) x <^{x) becomes
indeterminate for any value of x which makes one of its fac-
tors zero and the other infinite. The function in this case is
easily reducible to the form - ; for suppose /(a) = o, and 0 (a)
= 00, tlien the expression can be written — ^, which is of the
required form.
Examples.
irz
I. Find the value of (i — z) tan — when a? = i.
1 — z 2
This expression becomes — , the true value of which is - when a; = i.
cot —
2
2.
Sec z ( a: sin a; J ,
when a; = -,
2
xemz —
mi
bis becomes
2
a form
already discussed.
IJ
cos X '
3.
Tan {z - a)
.%(»-«),
when z = a.
Ans. o.
4. Cosec'^jSa; . log (cos ax), „ x = o.
Method of the Differential Calculus. 103
becomes indeterminate for the value x = a/ii
92. Form ^. As stated before, the Jraction ' ^73 also
f{a) = 00, and (^(os) =00
owev(
in the shape
It can, however, be reduced to the form - by writing it
I
I
The true value of the latter fraction, by Art. 90, is that of
{0W)' ^^m\fW
fix) 'VwUwr
fix)
Now, suppose A represents the limiting value of -j-^s^
when x = ttf then we have
f{a) <l>{a)
that is, the true value of the indeterminate form ^ is found
in the same manner as that of the form -.
o
In the preceding demonstration, in dividing both sides of
our equation by -4, we have assumed that A is neither zero
nor infinity ; so that the proof would fail in either of these
oases.
It can, however, be completed as follows : —
Suppose the real limit of -7-( to be zero, then that of
, . is kj where k may be any constant ; but as the
0W
I04 Indeterminate Forms.
latter fraction has a finite limit, its value by the preceding
method is
therefore ' ,, { = o ; i. e. when A is zero, '-7,-^ is also zero, and
vice versa.
fix)
Similarly, if the true value of ~j-\ ^® infinity when x = a,
then jj4- is really zero ; we have, therefore, 777^ = o, by what
■p ( n\
has been lust established ; /♦' ,, ; = cxs.
Accordingly, in all cases the value of '-77-f * determines
that of -)-{ for either of the indeterminate forms - or ^.
<l>{a) o °°
f'fx)
* On referring to Art. 69, the student will observe that "^7-^ is of the form
<f>{x)
^ whenever -j4 = ^, so that the process given above would not seem to assist
us towards determining the true value of the fraction in this case ; however, we
generally find a common factor, or else some simple transformation, by which
we are enabled to exhibit our expression, after differentiation, in the form -.
o
For example is of the form -^^— when jc =-: here /'(«)
^°s(^-i)
= aed^x, <i>'{x) = — ^, and the fraction "^^y-^ is still of the form — , but it can
a- _ 5 ^ \^)
2
V
X
be transformed into ■' , , which is of the form - : the true value of the latter
cos^a; o
fraction can be easily shown to be — 00 when x = -.
In some instances an expression becomes indeterminate from an infinite value
of X. The student can easily see, on substituting - for x, that our rules apply
y
equally to this case.
Indeterminate Expressions of the Form \f{ic))^^'). 105
93. Indeterminate expressions^ of the Form
(/(^)j*(*). Let u = {/(^))*(^), then log u = <p{x) logf(x).
This latter product is indeterminate whenever one of its fac-
tors becomes zero and the other infinite for the same value
of X.
(i.) Let ^(aj) = o, and log [f{x)] = ± 00 ; the latter re-
quires either /(a?) =co,orf{x)=o.
Hence, {/{x)]'^^'^^ becomes indeterminate when it is of the
form o", or oo<'.
(2.) Let (l){x) = + 00, and log {/{x)} = o, ot/{x) = i ; this
gives the indeterminate forms
I* and I"".
Hence, the indeterminate forms of this class are
o^ oo«, and i±-.
Examples.
I . (sin «;)**"« is of the form oo, -when a; = o.
log (sin x)
cota;
The true value of this fraction is that of
cot a?
= - cos a; sin x, or o when x = o.
- cosec^a;
Hence the value of (sin a;)**" * = «o = i at the same time.
v
2. (sin a;)*"" *, -vrhen x = -.
This is of the form i *, but its true value is easily found to be unity.
I
when a? = o.
/tan^\^2^
,0,(1^)
Here log u
, ^ tana; x^
but ss r + — + &c.
a^ 3
io6 Indeterminate Fortns.
, tan K , / aj2 \ a;2
.*. log = log f I + — + &c. J = — + &c.
Hence, the true value of log « is - when a? =o; and, accordingly, the value of m
is ei at the same time.
(-r-
when x = o,
log (i + az)
Let « = - then log u
z Z
.'. hy Art. 92, the true value of log u when 2 = 00 is that of , or is zero.
Hence, the value of « is r at the same time.
5. « = f I + - J , when a; = 00.
T A ^ .-, 1 log(i + az)
Let « = -, then log u = ^^ — ^',
the true value of which is a when z is zero.
Hence, the true value of w is ^ ; as also follows immediately from Art. 29.
(I\ Una;
(-9
when a; = o. An$. i.
** , when x = a, „ «''.
94. Compound Indeterminate Forms. — If an in-
determinate form be the product of two or more expressions,
each, of which becomes indeterminate for the same value of x,
its true value can be determined bj considering the limiting
value of each of the expressions separately ; also when the
value of any indeterminate form is known, that of any power
of it can be determined. These are evident principles : at
the same time the student will find them of importance in the
evaluation of indeterminate functions of complex form. We
will illustrate their use by a few elementary applications.
Examples.
I. Find the value of
x"^ (sin x) ta»» * ( ^T ^ ) , "When a; = -.
The value of a;*" is ( - J , and that of (sin x) *»»« is unity : see p. 105.
Examples. 107
Again, *^~ ^ becomes — -. on substituting — * for x : hence its true
'^ 2 sin 2a; 2 sin 2Z 2
value is - when 2 = 0.
2
IT IT***
Accordingly, the true value of the proposed expression when « = - is -^.
2. — when a? = 00,
This fraction can be written in the form / -r- \ • The true value of ~, by the
method of Art. 92, is that of — ^ ; but the value of the latter fraction is zero
-en
n
when a; = 00 ; hence the true value of the proposed fraction is also zero at the
same time.
3. u = x» (log a;)»», when x = o, and m and n are positive.
n
Here t* = («"* log a;)*»i
-^ is of the form ~ when « = O;
its true value is that of
I
X
X »»
Hence, the true value of the given expression is zero.
This form is immediately reducible to the preceding, by assuming x" = rv.
««»
4. M = — when a? = 00.
Here
but if J > I, and n > m, J*""*" = oo when x = oo. Consequently the vtJue of m is
of the form o* , or is zero in this case.
Again, if w > w, fr^"''" = o when a; = oo, and the true value of u is oo.
lo8 indeterminate Fortns,
a * .
5. « = — - wnen a; = o.
Let aj = -, and this fraction is immediately reducible to the form discussed in
z
the previous Example.
(i - cos ir)»{log (I +«;))♦» , . i
6. ^ ' ^ ^ ^^ , when x = o. Ant. — .
7. « = , when x = 0.
From Art. 29, this is of the form - ; to find its true yalue, proceed by the
o
method of Art. 90, and it becomes
(' + ^^ 1 ^(Tm — J-
Again, substituting for (i + «)* its limiting value «, we get
fa;-(i +a;)log(i+a;))^
the true value of which is readily found to be — when x = o. Compare Ex, 29,
p. 94.
I m* - I I C« sin a; - sin ax) »
8. — < -; : > , when x^o,
I sm a; I (x(co8a;- cosaa;))
w* — I
The true value of — : , when a; = o, is log m ;
sma; 7 o »
a sin a; — sin aa; ,
and that of -. r, whena^so,
a;(cosa;-cosaa;) ^
a
has been found in Example 6, Art 89, to be - ; hence the true value of the given
3
expression when x = o, is ( - j log m.
6.
«+ I '
(«*
- a:2)i + (a
-a:)l
9'
(«3.
- x^)\ + («
-;r)i'
lO.
rri tan »
(.«■
-i)^
^8ma:_ ^j
log
sin*'
12.
n
X
-©.
13.
a;2 -f. 2 COS a; -
2
a:*
'~>
14.
f a; + sin 2a; -
6sm?^
2
(.
+ cos a; -
5 cos-
Examples. log
Examples.
»= -
sin a; + cos « ir
x= -.
» = o.
« = o.
«♦».
(sin waj\ «•»
— )•
cos xd - cos «0
^* (a;2-w2)r *
\/ a + a; — ^/2x
4- y 7^
-v/ a + 3a; - 2-/ {»
2
„ tana: -sin* I
8. r-s 1 iCssO.
2
I +
aV l
a; = - . a log a.
a? = c, o.
(?)•
no Examples,
v/2 + co8 2a;- sina; , » ^ "v/io
15. : , when oj = -. Ans. ,
a; sin 2a; + as cos X 2 3-ir
- x<* sin na — w« sin xa
16. , X = n.
tan na - tan a;a
n<^^ (« cos Ma - sin na) co&hia.
17.
x^ tan na: - M tan a;
I — cos mx ' n sin x — sin nx '
18.
(2 sin a? - sin 2x)^
(sec a: -cos 2a;) 3 '
x = o.
8
t25
[9. a;^'*, a: = I.
-»'(y)}-2(/)(x) + 2»(y)
(a:-y)» " ' ^-«'- 6
,n (^ - y){<ft'(^) + <P'{y)} - 2(p{x) + 2»(y) «^'"(y)
a;log(i + a;)
22. X . e''. a? = o.
23. , ; r, X = 0.
•* log(n-a;)'
ira; — I
log (tan 2x)
•^ log (tan x)
- e* + log(i -a;) - I . i
26. 2J i a; = o. - -.
tana; - x 2
2»» ,*/t ^ I + m
27. 7 , tan-i Y I -m ^^g ^ _ ^os A, w = i.
cos 3<^
3
log(i+a; + a;2)+log(i-a; + a;')
25. — — -« a? = O. It
sec a; — cos a;
t
"• ( ^ )
aiUi . , , On*
Examples, 1 1 1
'log x\x
/log x\x
31.
32.
33. a;2 f I + - j - «2r3 log / 1 + j ,
I -a; + loga;
I - a; + log a;'
cos X - log (r +a;) +sma; - i
e" + sm a; - I
^ • log{i+a;)^*
e* — e"* — 2a;
^^' "tan a; - iP~»
d lax^ + 5a; + c\
fi?a; \ aia; + ^»i / '
a — X
-«log(?)
when a? = 00. ^«*. i.
i *a;
(I + a;)*-e +-
^ ' 2 we
? » '=°- ^-
sin a; - log (^* cos a;) i
34- / ,»
I - V 2a; - a;-
x^ — X
35. ^ , a? = I.
^^ I - a; + log x'
x'^-x
36.
2
«i
41. y -„ ^ * = «• -I.
a — \/ 2«a; — a;*
tan (a + a;) - tan (« - a;) i + a'
42. ^^ ^ — ~ a; = o. .
tan-i (« + a;) - tan"^ (a — xy coa^a
a;^ — 30? 4- 2
1 1 2 Examples.
44. (sina;)"»», when a; = o. Ans, i,
45. (seca;)»»»ec«, x = o. i.
tar. » ^ ^
46. (sin a;) , as = -. i«
47. Find the value of
{x -i/)a» + (y - a) a?" + (g - a?)y** w
{x - y) {t/ - a) {a - x) » 1.2^
when X = y = a.
Substitute a+ h for a;, and a + k for y, and after some easy transformations we
get the answer, on making h= o, and k = o.
X + tan X — tan 2x ^7
4S. , z- cs Am. -^,
a: + sin a; — sin zx — 7
40. — ^ , x=-o. — .
^ 2a; + tan a; - tan 3a;' 52
50-
\/^ - s/a + \/a; - a
2 . I ^
a; sin a; — tan x
5.- ^ . ^ ■
»=sa.
» - o.
v/a;2-a2 -v/za
- I
20'
( 113 )
CHAPTEE Y.
PARTIAL DIFFERENTIAL COEFFICIENTS AND DIFFERENTIATION
OF FUNCTIONS OF TWO OR MORE VARIABLES.
95. Partial Differentiation. — In the preceding chap-
ters we have regarded the functions under consideration as
depending on one variable solely ; thus, such expressions as
e***, sin hx, x^, &c.,
have been treated as functions oi x only ; the quantities
a, b, m, . , . being regarded as constants. We may, however,
conceive these quantities as also capable of change, and as
receiving small increments ; then, if we regard x as constant,
we can, by the methods already established, find the differen-
tial coefficients of these expressions with regard to the quan-
tities, a, b, m, &c., considered as variable.
In this point of view, e"* is regarded as a function of a as
well as of X, and its differential coefficient with regard to a
is represented by -^-, or x e""^ i-^ Art. 30 ; in the derivation
cia
of which X is regarded as a cons^iant.
In like manner, sin {ax + by) may be considered as a
function of the four quantities, c, y, a, b, and we can find its
differential coefficient with respect to any one of them, the
others being regarded as constants. Let these derived functions
be denoted by
du du du du
Tx 'dkf la' dh'
respectively, where u stands for the expression under con-
eideration, and we have
du , . . du . , , .
— = « cos (ax + by)^ — = b cos (ax + by),
du . . . du / I \
Y = ^ Gos (ax+ by), — = y cos (ax + by).
114 Partial Differentiation.
These expressions are called the partial differential coef-
ficients of u with respect to x, y, a, b, respectively.
More generally, if
denotes a function of three variables, x, p, z, its differential
coefficient, when x alone is supposed to change, is called the
partial differential coefficient of tlie function with respect to x;
and similarly for the other variables y and s. If the function
be represented by u, its partial differential coefficients are
denoted by
du du du
^' di/' dz'
and from the preceding it follows that the partial derived
functions of any expression are formed by the same rules as
the derived functions in the case of a single variable.
Examples.
I. M = (aa?2 4. byi J. C22)«.
Here -y = 2««a; (ax^ + hy^ + cz')«-^,
dx
du
■■ 2nby {ax^ + hy^ + C2*)»-^
du
dz"
: 2nez {ax^ + hy^ + e^Y'K
u =
. , X
y
du
I
du -X
dx~
Vt-
T^-i dy y^^-
^'
u =
■.XV,
dn , du
■.XV
log*.
u = xi(p {xy).
— = 2X<p {xy) + x'^y<f> {xy).
Differentiation of a Function of Two Variables, 115
96. DiflTerentiation of a Function^ of Two Vari-
ables.— Let u- (jt {x,y), and suppose x and y to receive the
increments h, k, respectively, and let Aw denote the corre-
sponding increment of w, then
Aw = 0 (x + h, p ■{ k) - (p {x, y)
=-<!> {x+ h,y + k) -~ (ji {x,y + k)+(j>{x,y + k)-^{Xfy)
(f) {x + h,y + k) -(f> (x,y + k) (p {x, y + k) - (j) {x, y) ^
h k
If now h and k be supposed to become infinitely small,
by Art. 6 we have
<fi (x + hf y + k) - (j) (x,y -^ k) _ d , (ft {x, y+ k)
h ^ *
and 0 (^> y + ^^) - 0 (^^ y)_^ d ' ^ (^> y)^
k dy
In the limit, when k is infinitely small, (jt {x, y + k)
becomes 0 (a?, y), and
. li±%JLli^ becomes ^4^;
cfa; dx
hence we get- neglecting infinitely small quantit^^ of the second
order,
dr ^ du ,
du ^ -— h ^ -T- ki
djf dy
where h and k are infinitely small.
If dXf dy, be substituted for h and k, the preceding
becomes
- dii . du _ , .
du = -r dx + -— du. (i)
dx dy ^ ^ ^
In this equation du is called the total differential of w,
where both x and y are supposed to vary.
The student should carefully observe the different mean-
ings given to the infinitely small quantity du in this equation.
Thus, in the expression — dx, du stands for the infinitely
ax
I 2
ii6 Partial Differentiation,
small change in u arising from the increment dxmcc,y being
du
regarded as constant. Similarly, in — dy, du stands for the
infinitely small change arising from the increment dymy,x
being regarded as constant. If these partial increments be
represented by dxU, dyU, the preceding result may be written
in the form '
du = dxU + dyU.
That is, the total increment in a function of two variables is
found by adding its partial increments, arising from the
differentials of each of the variables taken separately.
EXAMPLES.
I . Let a; = r COS 0, in which r and d are considered variables, to find the
total differential of x.
(Ix dx
Here ■— = cos 0, — - = - r sin 9*
dr ' de
Hence dz = cos 0 rfr - r sin 0 d6.
Here ^ = ^, ^t =.%.
dx «»' dy ^*
.\du^-dx-^.-^dy.
(- I . Let - = 2, then m = f («),
yj y ' . Tv/»
(;)
du du dz
dx dzdx ~ y '
_ ,/?\
du du dz _ \y)
dy dz dy~ y'- *
ydx — xdy
^« = ^'(?)
y
Again, multiplying the former of the two preceding equations by x, and the
latter by y, and adding, we get
du du
dx dy
Differentiation of a Function of Three or more Variables. 1 1 7
97. Differ entiation of a Functioji of Three or
more ITariables. — Suppose
u = (f, {x, y, s),
and let h, k, I represent infinitely small increments in x, y, z,
respectively; then
Au = (l>(x + h,t/ + kjZ + I) -(l>{x,y,z)
^(l) {x + h, y + k, z + I) - ((> (x, y + Ic, z + I) ,
h
^{x,y + k,z+l)-<f){x,y,z+l)j^ (^{x,y,z+l)-(l>{x,y,z) ^
+ ^ k+ ^ /,
wHch becomes in tkv limit, ^y the same argument as before,
when dXj dy, dz, are substituted for h, k, I,
du ^ du ^ du ^ , .
du = — dx + — dy + -r dz. (z)
dx dy dz
Or, the infinitely small increment in u is the sum of its
infinitely small increments arising from the variation of each
variable considered separately.
A similar process of reasoning can be easily extended to
a function of any number of variables ; hence, in general, if
w be a function of n variables, a?!, a?2, iCs, . . . a?«,
du ^ du ^ dv ^ / .
az* = -T- aa?i + — cwrg + . . . + -t— dxn, (3)
(a/X\ aXi uXn
98. If
u =/(^, w),
where v, w, are both functions of x ; then, from Art. 96, it is
easily seen that
du _ df{v, w) dv df{v, tv) dw
dx dv dx dw dx'
This result is usually written in the form
du du dv du dw , .
dx dv dx dw dx'
In general, if
1 1 8 Partial Differentiation,
'Sphere yi, 2/2, .. . y«> are eacli functions of ir, we have
du du dxfY du dy^ du dyn . .
dx dyi dx dy-i dx ' ' ' dyn dx'
Also, if yi, yzf &o., y„, be at the same time functions of
another variable s, we have
du du dyx du dy^ p du dy^
— + -; 1- OCC. + -T—
d% dxjx dz dy% dz ' dyn dz '
and so on.
^•^JAMPLES.
r. Let u^ (p{X, F),
where X = ax + by, T = a'x + J'y ;
, du _ du dX du dY
®° dx~ dXlx^dflx*
du __ du dX du^ dY ^
dy~ dXly ^ dY dy*
, ^ dX dX . dY , dY .,
but __ = <» — - = J, -— = a , — - = 6 »
dx dy dx dy
Hence
dX
'' Ty
= J.
dY
dx
du
^dX^'
4.
"dX^
^fr
du
dx
du
dy
2. More generally, let
where Z = aa; + iy + ca,
r = a'a? + Vy + e\
Z=a"x^-h"y-k-i^'z.
When these substitutions axe made, u becomes a function of «, y, z, and we
haye
du du ,du ,,du
^ = *^ + ''^'^'^ ^'
du .du du du
Tr^dx^ df^ ^'
du du ,du „du
Tz^-'dX-^'df^' dz'
Differentiation of a Function of Differences,
119
98*. DiflTerentiation of a Function of Diflfe-
rences. — If w be a function of the differences of the vari-
ables, a, j3, 7 : to prove that
du du du
da dp dy
Let a-/3 = ir, j3-7 = y, 7-a = s; then, w is a function
of Xy y, z\ and, accordingly, we may write
Eence
Similarly,
u = (i>(x, y, 2).
du _ du dx du dy du d% du du
da dxda dy dn dzda dx dz
du du vuu
dj3 dy w;'
du du du
da dfi dy
du _ du
dy dz
du
This result admits of obvious extension to a function of
the differences of any number of variables.
EXAMPUSS,
I. If
dA dA
If
dA dA dA dA _
da"" Is'^ 'Jy'' IE ~ ^
I,
I,
I,
h
0,
3,
7»
s,
o\
62.
7S
5^
a\
a-',
7^
8»,
dA
dy
dA
'IE
= 0.
I,
h
h
I,
0,
B,
7>
5,
«%
&\
7^
5\
«s
/3*,
7S
5S
I,
I,
I,
I>
0,
fi,
7»
5,
0-,
i8^
7^
s^
«^
i8^
y,
5^
, prove that
prove
that
120 Partial Differentiation.
yA
99. Definition of an Implicit Function. — Suppose
that y, instead of being given explicitly as a function of x, is
determined by an equation of the form
/(^, y) = o,
then y is said to be an implicit function of x ; for its value, or
values, are given implicitly when that of x is known.
100. DifTerentiation of an Implicit Function. —
Let Iz denote the increment of y corresponding to the incre-
ment h in X, and denote /(^, y) by u.
Then, since the equation f[x^ ?/) = o is supposed to hold
for all values of x and the corresponding values of y, we
must have
f{x ■vh,y-¥k)= o.
Hence du = O) and accordingly, by Art. 96, we have,
when h and k are infinitely small,
du ^ du ^
-rh+-rk=oi
ax dy
du
hence m the limit h^ IT ^'IT ^^
dy
This result enables us to detc^iine the differential
coefficient of y with respect to x whenever the form of the
equation /(a;, y) = oib, given.
In the case of implicit functions we may regard x as
being a function of «/, or 3/ a function of x, whichever we
please — in the foimer case y is treated as the indejyendent
variable, and, in the latter, x : when y is taken as the inde-
pendent variable, we have
du
dx dy I
dy du dy
dx dx
This is the extension of the result given in Art. 20, and
might have been established in a similar manner.
Differentiation of an Implicit Function. 1 2 1
EXAJIPLES.
I. If »3 + y' - Zaxy = e, to find — ,
ax
Here
iS^tf Art. 38.
2. If
dy x^-
dx ax — y^
«»» g>^ ' dx
du mx^'^ du my^'^ ^ ^ dy fx\**~^fby
dx a^ * dy b^ * ' ' dx
^y loi. If w = 0 (iT, I/), where x and 2/ axe connected by the
equation f(x, y) = o, to find the total differential of u with
respect to a? ; 2/ being regarded as a function of x.
Here, by Art. 90, we "get
Also
du d<^ dtp dy
dx dx dy dx
dx dy dx
/
Hence, eliminating —-, we get
d^d£ df^d^
du _ dx dy dx dy /-\
dx df
dy
22
Partial Differentiation,
This result can also be written in the following deter-
minant form :
d<lt dcji
dx' dy
d£ ^
dx* dy
du
dx
dy
More generally, let u =- ^(r, y, s), where x^ y^ s, are con-
nected by two equations,
/i(aj, ^. z) = o, /,(<r, y, s) = o ;
then, as in the preceding case, we have
and also
du d(l> dcf) dy dift dz
dx dx dy dx dz dx*
dx dy dx dz dx *
dx
dfidy dfidz _
dy dx dz dx
= 0.
Hence, we get
dcj) d<p d(f>
di' d^' di
df: dA dA
dx' dy' dz
du
dA dA df,
dx' dy' dz
dx
df, df,
dy' dz
dA dA
dy' dz
This result easily admit
s of generalizat]
Lon.
(8)
Enter's Theorem of Homogeneous Functions, 123
102. Euler's Tbeorem of Homogeneous Func-
tions.— If
u^AsxPy^^ Bo^ 1/ -f Co^' y^" + &o.,
where
p -^ q - p' ^- ^ = p" ^ f][^ = &(i. = n,
to prove that
du du , .
Here ^ :r " ^^^ y^ "^ -^^^ ^^^' ^'^' "^ ^°- »
y — = ^3'^ %A-^B(i ^ y'i' + &o. ;
.'. «^ + 2^ ^ = ^0 -+ (/)^ 2^^ + B{p' ^^aP'y^' ^- &c.
= w^aj^ y^ + wjB^jP' 2^*' + &c. = nu.
Hence, if u be any homogeneous expression of the w**
degree in x and y, not involving fractions, we have
du du
dx dy
Again, suppose u to be a homogeneous function of a
fractional form, represented by — ; where 0i, 62, are homo-
geneous expressions of the n*^ and m^^ degrees, respectively,
in X and y ; then, from the equation
we have
and
0.
du
rf0i rf^.
dx~~
' W'' '
du
dy
^01 , C?02
1 24 Partial Differentiation,
accordingly we get
du du H^^^^r^'v^^^i^
dx ^ dy {<p,Y
but, by the preceding, »
, du du ndii Q»2 - m6x (t>2
hence ^zr + ^:r= /^\8
= (w - w) — = (w - m) w ;
which proves the theorem for homogeneous expressions of a
fractional form.
This result admits of being established in a more general
manner, as follows :
It is easily seen that a homogeneous expression of the n^^
degree in x and y, since the sum of the indices of x and of y
in each term is w, is capable of being represented in the
general form of
a;"0
(I)
Accordingly, let
u^x-<^\^ = x-v.
where
Then
d^^ „, „ dv
— = nx''-H + a?" — ,
dx dx'
- du ^dv
and -rr = x^-r-i
dy dy
multiply the former equation by x, and the latter by 3/, and
add; then
du du „ ^f dv dv\
Euler^s Theorem of Homogeneous Functions. 125
but (by Ex. 3, Art. 96),
dv dv
du du „
hence a; — + y — = naf'v = nuy
dx dy
whicli proves the tbeorem in general.
In the case of three variables, x, y, z,
suppose u = AxP y^ s**,
then we have
du . ^ ^ du J „ „ ^ du J „ n .,■
X— =ApxP y'i z^, t'-j- = Aq x^ y^ z^, z— =ArxP y^ z' ;
du du du ., s „ „ ^ / N
,\ X — + y — + z — = A{p + q + r) xP y^ z*^ = (p + q + r) u ;
uiX dy uz
and the same method of proof can be extended to any homo-
geneous function of three or more variables.
Hence, if w be a homogeneous function of the n*^ degree
in Xf y, z, we have
du du du , V
It may be observed thai the preceding result holds also
if n be a fractional or nrfgative number, as can be easily seen.
This result can also be proved in general, by the same
method as in the case of two variables, from the considera-
tion that a homogeneous function of the n^^ degree in x, y, z
admits of being written in the general form
« = ^"^l|^)»
or in the form
u = x*^ (j) {Vf w)y where v = -, and to = -.
X X
Proceeding, as in the former case, the student can show,
126 Partial Differentiation,
without difficulty, that we shall have
du du du
X -T- + y —- + z -T- = nu.
dx dy dz
Another proof will be found in a subsequent chapter, along
with the extension of the theorem to differentiations of a
nigher order.
Examples.
Verify Euler's Theorem in the following cases by direct difEerentiation : —
a;3 4- «/3 ^^ ^^ cu
{x + yyi " dx dy 2
a^ + ax'^y -^ by^ du du
'a;* + :
y2 ' " '^dx
. , a;2 _ V* du du
3- "=«'""?T?' " ^^^nr"-
(x^-y^\ du «
du
103. Theorem. — If C= Wo + «*i + «^ • • . + W«,
where Wo is a constant, snd «^i, Wj, • . • ««» are homogeneous
functions of x, y, z, &c., nf the ist^ 2nd, . . . n*^' degrees,
respectively, then
dU dU dU , ,
'"^'■^^^'■^''•••"^^^'^^■'^^■'■•••^^'''- (")
For, by Euler's Theorem, we b^ve
dUr dUr dUr «
aj4.^ +2 +&C.= rWr»
dx dy dz
since w^ is homogeneous of the r*^ degree in the variables.
Cor. liU=Oy then
dU dU dU , » / X
dx dy dz
This follows on subtracting
we^o + w^i + . • . + VlUn - o
from the preceding result.
ous
Remarks on JEuler's Theorem. 1 2 7
104. Remarks on Ruler's Theorem. — In the appli-
cation of Euler's Theorem the student should be careful to
see that the functions to which it is applied are really
homogeneous expressions. For instance, at first sight the
expression sin'^ I -r — ^ ) might appear to be a homogene
function in x and y ; but if the function be expanded, it is
easily seen that the terms thus obtained are of different
degrees, and, consequently, Euler's Theorem cannot be
directly applied to it. However, if the equation be written
X -b i/
in the form -r — ^ = sin u, we have, by Euler's formula,
d sin u d sin u _ sin u
''~d^'^^~di~^~r'
or cos w 1 a? -T- + 2/
uli
du dii\ sin u
dx dyj 2
. du du ^m u I X + y
hence x-=--^ y -r ^ = ,
d^ dy 2 ^^/{i^^ylY-{x^■y)''
When, however, the degrees in th-i numerator and the
denominator are the same, the function is of the degree zero,
and in all such cases w^ hs-yfi
du <^^
dx dy
T-i . (^ + y^ X + y —
For example, sm'* ( ^ _ "^ ^ j, tan"^ -, e^, &c., may be
treated as homogeneous expressions, whose degree of homo-
geneity is zero. The same remark applies to all expressions
which are reducible to the form ^ ( - ) ; as already shown in
Ex. 3, Art. 96.
105. If X = r COS 6y y = r sin 0,
to prove that xdy - ydx = r^dO. (13)
128 Partial Differentiation,
In Ex. I, Art. 96, we found
dx = cos Qdr - r sin QdQ ;
similarly dy = mi ddr -1- r cos Odd.
Hence scdy = r cos 0 sin Odr + r^ cos^ OdO^
ydx = r Gos 0 sin Odr - r'^ s\v^ OdO ;
.'. xdy - ydx = r^ dO.
106. If X and y have the same values as in the last, to
prove that
{dxY + {dyY = {dry + r' {dO)\ (14)
Square and **,dd the expressions for dx, dy, found above,
and the required result follows immediately.
The two preceding formulae are of importance in the
theory of plane curves, and admit of being easily established
from geometrical considerations.
107. If u= ax^ + hy"^ + cz^ + zfyz + igzx + ihxy,
to find the condition among the constants that the same values of
X, y, z should satisfy the three equations
du
dx~
du
dy '
du
Here — = 2ax + 2% + 2gz - 0,
— = zhx^ 2hy + 2/s = 0,
du
— = 2gx + 2fy + 2CZ = 0.
Hence, eliminating x, y, z between these
the required condition is
abc - af - Ig"" - ch^ + 2fgh = 0;
or, in the determinant form.
a h g
h h f
= 0.
g f c
Remarks on Eulcr^s Theorem. 129
The preceding determinant is called the (Mscviminant of the
quadratic expression, and is an invariant of the function ; it
also expresses the condition that the conic represented by
the equation u = o should break up into two right lines.
(Sahnon's Conic Sections, Art. 76.)
The foregoing result can be verified easily from the latter
point of view ; for, suppose the quadratic expression, u, to be
the product of two linear factors, X and Y ;
or w = XY,
where X = Ix + my + nz, Y = Vx -1- m'y + n'% ;
., du ^dY ^dX .,^ -„
then — = X-r- + Y ^r = l^ + ^Y,
dx dx dx
du ^dY -^dX ,^ ^
-r=X-r + F-— = m'X + mY,
dy dy dy
du dY ^.dX ,^ ^
— =X— - +Y -r = nX + nY.
dz dz dz
Here the expressions at the right-hand side become zero for
the values of x, y, s, which satisfy the equations X = o, 1^= o,
or Ix + my + nz = o, l^x + ?ny + nz = o.
Hence in this case the equations
du du du _
dx ^ dy ' dz
are also satisfied simultaneously by the same values.
We shall next proceed to illustrate the principles of
partial differentiation by applying them to a few elementary
questions in plane and spherical triangles. In such cases we
may regard any three* of the parts, «, 6, c, A^ B, (7, as being
* The case of the three angles of a plane triangle is excepted, as they are
ft^uivalent to only two independent data.
K
1 30 Partial Differentiation,
independent variables, and each of the others as a function of
the three so chosen.
108. Equation connecting the Yariatious of the
three iSides and one Angle. — If two sides, a^ b, and the
contained angle, C, in a plane triangle, receive indefinitely
small increments, to find the corresponding increment in the
third side c, we have
c^ = a^ + b"^ - 2ab cos (7;
.*. cdo = {a - b cos C) da + (b - a cos C) db + ab sin CdC\
but « = ^ cos (7 + c cos ^, h = a Qo^ C + c cos A.
Hence, dividing by c, and substituting c sin B for b sin C,
we get
dc = cos Bda + qo^ Adb -v a sin B dC. (15)
Otherwise thus, geometrically.
By equation (2), Art. 97, we have
dc dc „ dc ^^
dc = -r da + -r: db + —^ dC,
da db dC
dc
Now, in the determination of -— we must regard b and C as
da °
constants ; accordingly, let us sup-
pose the side CB, or «, to receive a
small increment, BB^ or A a, as in
the figure. Join AB^^ and draw B'D
perpendicular to AB^ produced if
necessary; then, by Art. 37, AB^ -Op^
= AB when BB' is infinitely small, y~
neglecting infinitely small quanti-
ties of the second order. _.
Hence ^'^- ''
Ac = AB' -AB^AD-AB = BD;
dc .. .. .Ac BD p
.•. -r- = limit 01 = -— =r7 = COS B,
da An BB
Examples in Plane Trigonometry, 131
Similarly, — = cos -4 ; which results agree with those arrived
U/O
at before by differentiation.
do
Again, to find -r^. Suppose the angle C to receive a
dU
small increment A (7, represented by
BCB^ in the accompanying figure;
take CB' = CB, join A&, and draw
BB perpendicular to AB^.
Then
Ac = -4^ - ^5 = B^D (in the limit)
= BF cos AB^B = BF sin ^^^(q.p.). Fig. 5.
Also, in the limit, BB' ---- BG sin BCB^ -- a b.G,
Hence -r-^ = limiting value of — ^ = « sin 5 ;
the same result as that arrived at by differentiation.
In the investigation in Eig. 5 it has been assumed that
AB - AT) is infinitely small in comparison with BB ; or that
AB - AD
the fraction ^yr — vanishes in the limit. For the proof
of this the student is referred to Art. 37.
When the base of a plane triangle is calculated from the
observed lengths of its sides and the magnitude of its vertical
angle, the result in (15) shows how the error in the computed
value of the base can be approximately found in terms of the
small errors in observation of the sides and of the contained
angle.
dC
1 00. To find -r-. wben a and 0 are considered
dA
Constant. — In the preceding figure, BAB^ represents the
change in the angle A arising from the change A (7 in (7;
moreover, as the angle A is diminished in this case, we must
denote BAB^ by - A^, and we have
^^ ABAA ABAA c^A
sin AB'B cos B cosB'
K 2
132 Partial Differentiation,
Also, BB =a^C',
c?^ A-^4 ^ ^ a cos ^ ^ '
This result admits of another easy proof by differentiation.
For « sin jB = 5 sin ^ ;
hence, when a and h are constants, we have
a Qo^ B dB = h cos A dA ;
also, since ^ + J? + (7 = tt, we have
dA + dB + dC= o.
Substitute for c?^ in the former its value deduced from the
latter equation, and we get
(a cosB + b cos A) dA = - a cos B dC;
or c dA = - a Gos B dO, as before.
no. Equation connecting the Variations of tivo
Slides and the opposite Angles. — In general, if we take
the logarithmic differential of the equation
a sm B = b sin. A,
regarding a, b, A, B, as variables, we get
da . dB db dA
a tan B b tan A
(17)
III. lianden's Transformation. — The result in equa-
tion (16) admits of being transformed into
dA dC
aoosB c '
but
c = va^ + J^ - lab cos (7, and a cos B = \/a* - b^ sin^^;
hence we get
dA dO_
*/a^ - ¥ sin ^A \/ a^ + b"^ - zab cos C
Examples in Spherical Trigonometry, 133-
If C be denoted by 180° - 2^,, tbe ag^le at A by ^, and
- by h, the preceding equation becomes
d6 2d(j)i idifii
2 ^*/>i
(i + k) \/i - kx sin^^i'
2v^
(18)
wbere kx-
i +k
Also, tbe equation a ^mB = h sin A becomes
sin (2^1 - <!>) =h sin ^.
The result just established furnishes a proof of Landen's*
transformation in Elliptic Functions.
We shall next investigate some analogous formulae in
Spherical Trigonometry.
112. Relation connecting the Variations of Tiirec
Slides and One Angle. — Differentiating the well-known
relation
cos c = cos « cos J + sin a sin ft cos (7,
regarding a and h as constants, we get
dc sin a sin h sin C
dO sin c
sm a sin B,
dc
Again, the value of — , when h and C are constants, can
cia ,
be easily determined geometrically as follows : —
* This transformation is often attributed to Lagrange ; it had, however, been
previously arrived at by Landen. (See Philosophical Transactions, I'j'ji and
I77S-)
r34
Partial Differentiation.
In the spherical triangle ABC, making a construction
similar to that of Fig. 4, Art. 108, we have
BB =Aa; .-. — = limit of — - = -=r^
da Aa BB
(in the limit) = cos B,
Similarly, when a and G are con'
stantSf — =cosA.
db
Hence, finally, ^^- ^'
dc = cos Bda -^ cos Adh -^ sin a sin B dC, (19)
This result can also be obtained by a process of diffe-
rentiation. This method is left as an exercise for the
student.
As, in the corresponding case of plane triangles, we
have assumed that AB" = AD in the limit ; i. e., that
AB' - AD
-7— — is infinitely small in comparison with ^D in the
limit ; this assumption may be stated otherwise, thus : —
If the angle ^ of a right-angled spherical triangle be
c — b
very small, then the ratio — — becomes very small at the
same time, where c and b have their usual significations.
This result is easily established, for by Napier's rules we
have
tan b BID. b cos c
00s ^ = 7 = r—. — ;
tan c cos b sm c
I - cos A _ sin c cos b - oosc smb _ sin {c - b) ^
* * I + cos ^ sin c cos b + coa c sin b sin (c + b)^
or
sin {c - b) = tan^ — sin {c + b) ; .-. -=-^ — j^= sin {c + b) tan—.
tan —
2
But the right-hand side of this equation becomes very small
along with A, and consequently c - b becomes at the same
time very small in comparison with that angle.
Examples in Spherical Tngonometry. 135
The formula ( 1 9) can also be written in the form
,^ dc da ^h , .
dC = ; 5 - 1 5 - • 7 X A' (20)
sm a sm B sm a tan B sm 0 tan A
The corresponding formulae for the differentials of A and B
are obtained by an interchange of letters.
Again, from any equation in Spherical Trigonometry
another can be derived by aid of the polar triangle.
Thus, by this transformation, formula (19) becomes
dC = - cosb dA - cos adB -\- sin A sin h dc, (2 1 )
These, and the analogous formulae, are of importance in
Astronomy in determining the errors in a computed angular
distance arising from small errors in observation. They also
enable us to determine the most favourable positions for
making certain observations ; viz., those in which small errors
in observation produce the least error in the required result.
113. Remarks on Partial Differentials. — The be-
ginner must be careful to attach their proper significations to
the expressions — , -7-^, &c., in eacli case. Thus when a and
dc
0 are constants^ we have -7-^ = sin ^ sin ^ ; but when A and a
are constants, we have -^yz = 7^ ; these are quite different
dC tan (7 ^
quantities represented by the same expression — ^.
The reason is, that in the former case we investigate the
ultimate ratio of the simultaneous increments of a side and
its opposite angle, when the other two sides are considered as
constant ; while in the latter we investigate the similar ratio
when one side and its opposite angle are constant.
Similar remarks apply in all cases of partial differentia-
tion.
When our f ormulse are applied to the case of small errors
in the sides and angles of a triangle, it is usual to designate
these errors by Aa, A&, Ac, AA, AB, AC; and when these
expressions are substituted for da, db, &c., in our formulae,
they give approximate results.
136 Partial Differentiation.
For instance (19) becomes in this case
Ac = Aa cos 5 + A5 cos ^ + A(7 sin « sin B ; {22)
and similarly in otlier cases.
It is easily seen that the error arising in the application of
these formulae to such cases is a small quantity of the second
order ; that is, it involves the squares and products of the
small quantities A^, A^, Ac, &c. This will also appear more
fully from the results arrived at in a subsequent chapter.
1 14. Theorem. — If the base c, and the vertical angle C,
of a spherical triangle bo constant, formula (19) becomes
da db
= o.
cos A cos B
Now, writing ^ instead of a, \p instead of b, and k for
-; — , this equation becomes
sm c
sin A sin B
since k
sin a sin b
dip
(23)
a/i - k^ sm^<l> a/i - F sin'i//
where 0 and \p are connected by the following* relation : —
cos c = cos 0 cos 1// + sin 0 sin i// cos (7,
or cos c = cos 0 cos t/* + sin ^ sin x^ ^ \ - k^ sin^ c,
115. In a ISpherical Triangle, to prove that
da db dc , ,
1 + ^ + 7i = o> (24)
cos A cos -B cos (7 ^ '
sin C .
when -; is constant.
sm c
* This mode of establishing the connexion between Elliptic Functions by
aid of Spheiical Trigonometry is due to Lagrange.
Examjjles in Spherical Trigonometry. 13'
Let sin C = Z: sin c, and we get
,^ Z; cose , sin^ cos'tJ ,
c?C = 77 dc=- ^dci
cos C sm a cos C
substitute this value for c?(7 in (19), and it becomes
. „ _. , cos c sin ^ sin .B ,
dc = cos Ado + cos B da + ^ dc ;
cos C
■(■
. ,, T* , I COS c sm ^ sm ^\ .
or cos Ado -VGOBBda = [ i 7; ) dc
cos C
cos^ cos -6 ,
cos C
since sin A sin ^ cos c = cos (7 + cos ^ cos B.
_. ^<x c?& dc
Hence r + :^ +
cos ^ cos B cos (7
Again, since cos -4 = y^i - sin^^ = \/i - Ic^ sin^cf, &c.,
the preceding result may be written in the form
da dh dc
-v/i - k^ ^m^a v^i - F sin'^^ \/i - W sin^a ' '
where «, J, c, are connected by the equation
cos c = cos fl5 cos J + sin « sin h v i - F sin^c.
116. Theorem of I^egendre. — "We get from (24)
cos B cos Cda + cos A cos (7c?i + cos B cos ^c?c = o,
or (cos ^ - sin ^ sin C cos a) da + (cos 5 - sin ^ sin 0 cos h) dh
+ (cos (7 - sin ^ sin B cos c) cfc = o ;
/. cos Ada + cos ^cfJ + cos Cdc
= sinB sin Cd (ain a) + sin A sin Cd (sin ft) + sin A sin 5c? (sin c)
= A' (sin 6 sin cd (sin cr) + sin a sin cc? (sin b) + sin a sin bd (sin c) j
= k'^d (sin a sin i sin c) ;
138 Partial Differentiation.
or -v/i - F sin'^ ada-\- y^i - k"^ sin^ hdh -^^i-k^ sin'^ c dc
= Fc? (sin a sin 5 sin c). (26)
This furnishes a proof of Legendre's formula for the compa-
rison of Elliptic Functions of the second species.
The most important application of these results has place
when one of the angles, C suppose, is obtuse ; in this case
cos 0 is negative, and formula (25) becomes
da dh dc
V^i - F sin'' a v^i - F sin^6 ^/i - ¥ sin'^c'
where the relation connecting a, b, c is
cos c = cos a cos J - sin fl sin & ^/l - ¥ sin'^c.
In like manner, equation (26) becomes, in this case,
v/i - W m^a da + y^i - F sin'^ h dh
= yi - ¥ sin^ cdc + ¥d (sin a sin h sin c).
117. If w = ^(a; + atf y -{■ f5t), where a?, y, a, j3, are in-
depeudent of t, and of each other, to prove that
du du ^du , .
Let of = X + at, y' = y + fit]
then ^ = ^{^9^)9
, dx' dif daf di/
Also, since / is independent of x, we have
du du dx' _du j ^^ ^^
dx dx' dx dx'* dy d^/'
_. du du dx du dy' du ^ du
Hence 3r = 3-7-:7r+T^-^ = «:r + PT"-
dt dx dt dy dt dx ' dy
Partial Diferentiation. 139
In like manner, if x\ y\ 2', be substituted for a; + a^, y + /3^,
2 + 7^, in the equation ^
w = ^ (a; + a^, y + (5t, 2 + 7^),
it becomes u = (p {x\ y\ 2') ;
du du dx du dy' du dz ^
^^®^ 'd^^l^'d^^'d^d^'^d^'d^'
dx' d^ dz'
but -7- = i> "3" = o? T~ = o>
^ du du 1 t^^ _ ^^ ^^ _ ^^
dx dx'^ dy dy'^ dz dz''
. du _du dx' du dy' du dz'
^^^'° 'dt"d^'~M^li/~db^'d^'dV
dx' d\/ dz'
T-r du du f^ d'^ du , „.
This result can be easily extended to any number of variables.
140 Examples,
Examples.
dx
(x\ fv\ ^^ ^'.
- I + sin-i ( - J , prove that dii = . + — 7=d
If w = a;y<|) f-j, », *-T- + y3- = 2W.
; \ y
dx dy
3. Find the conditions that «, a function of a?, y, z, should he a functipn of
X \ y \ z.
. du du du
dx dy d*
4. If f{ax + *y) = tr, find ^.
5. If /(m) = ^(v), where m and t; are each functions of x and y, prove that
(fw dv dv du
dx dy dx dy'
t. Find the values oi x -z- ^ y -r, when
dx dy
7. If w = sin «« + sin iy + tan"^ f - J , prove that
zdy — ydt
du = a cos axdx+ b cos Jy ^y H r — ^.
y* + «'
„ x/. 1 n n^^ 3^^ . du I (fW - log X
8. If w = logycc, find — and —. Ana. — = — , — = -— — .
dx dy dx a; logy dy y {log yy
Q. If 0 = tan-1 -, prove that
(x^ + y^) dd = ydx - xdy.
10. If w = y'y prove that
du = y*»-^ {xzdy + yz log ytfa; + a;y log ydz).
Examples, 141
II. If a + \/a' - y^ =ye « , prove that w«
dy -y
dX ^^2 _ y%'
12. In a spherical triangle, when o, h are constant, prove that
dA tan ^ , <?C sin G
-, and -— =
dB tan^' <^i? sin ^ cos^
13. In a plane triangle, if the angles and sides receive small variations,
prove that
cLB + h cos -4 AC = o ; a^h heing constant,
cos CLh + cos Bd^c = o ; a, .^4 heing constant,
tan-4A5 — bAG; a, B heing constant.
14. The hase c of a spherical triangle is measured, and the two adjacent
hase angles A, B are found hy observation. Suppose that small errors dA, dB
are committed in the observations of A and B ; show that the corresponding
error in the computed value of G is
- cos adB - cos bdA.
15. If the hase c and the area of a spherical triangle be given, prove that
a . b
sin' -dB + sin^ - dA = o.
2 2
16. Given the base and the vertical angle of a spherical triangle, prove that
the variation of the perpendicular p is connected with the variations of the sides
by the relation
sin Gdp — sin i'da + sin sdb^
s and s' being the segments into which the perpendicular divides the vertical
angle.
17. In a plane triangle, if the sides a, b be constant, prove that the variations
of its base angles are connected by the eq^uation
dA dB
a/u^ -b^ ain^A ^Z h^ - a^ ern^B'
18. Prove the following relation between the small increments in two sides
and the opposite angles of a spherical triangle,
da
+
dB
dA
+
db
tana
tan^
tan A
tan 3*
19. In a right-angled spherical triangle, prove that, if A be invariable
sin 2cdb = sin zbdc ; and if c be invariable, tan ada + tan bdb = o.
142 Examples,
20. If a be one of the equal sides of an isosceles spherical triangle, whose
vertical angle is very small, and represented by d<a, prove that the quantity by
which either base angle falls short of a right angle is - cos a dot.
21. In a spherical triangle, if one angle C be given, as well as the sum of
the other angles, prove that
da db __
sin a sin 5 *
22. If all the parts of a spherical triangle vary, then will
cos Ada + cos Bdb + cos Cdc = kd{k sin a sin i sin e) ;
sin ^ sin J5 _ sin C
sin a sin i sin c *
*i ^^ db dc . ^ ^ /i\
Also - — ^ + — - + -^, = tan^ tan5 tan Gd[-\
cos A cos B cos C
These theorems can be transformed by aid of the polar triangle? — M^Cullagh^
Fellowship Examination, 1837.
These are more general than the theorems contained in Arts. 1 15 and 116,
and can be deduced by the same method without difficulty.
23. If z = ^ (a;2 - y2), prove that
dz dz
d» dy
24. If z = -/ ( - J , prove that
dz dz
x—-\y-r-\-f=o,
dx dy
dy dz
25. Find -j- and — when x, y, z are connected by two equations of the
form
/(«,y, 2)=0, <^ (a;, y, «) = o.
df_d,\> dfd^
dy _ dx dz dz dx
' Jx^WdT^Wd?
dz dy dy dz
d£d^ _ d£d<p
dz _dy dx dx dy
" 'dx~ dfd<p df d<f>'
dz dy dy dz
Examples.
26. Prove that any root of the following equation in y,
143
y^-\-xy
satisfies the differential equation
dx^
dy^
27. How can we ascertain whether an expression such as
admits of being reduced to the form
Ans.
dtp d^l/
dx dy
d^
28. lilX^-mY+nZ, l'X + m'T+ n'Z, l"X+ m"Y ^ n"Z, be substituted
for X, y, z, in the quadratic expression of Art. 107 ; and if a', b', </, c^, ^,/', be
the respective coefficients in the new expression, prove that
of f ^
f V d'
(f d' c'
Ia f e
f b d
e d c
29. If the transformation be orthogonal, i. e. if x"- -{■ y"^ -\- z^ = X* ■{■ Y^ + Z-,
prove that the preceding determinants are equal to one another.
30. If M be a function of |, rj, (, and | = y + ", ri = z + -, ^= x + -,
z X y
shoM' that
du du
X— +
dx
du du du du .du / du du du\
( 144 )
CHAPTER YI.
SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE
VARIABLES.
1x8. Successive Partial Differentiation. — We have in
the preceding chapter considered the manner of determining
the partial differential coefficients of the first order in a func-
tion of any number of variables.
If w be a function of x, y, s, &c., the expression
du du du -
Tx' Ty' ^' ""•'
being also functions a?, y, 2, &c., admit of being differen-
tiated in the same manner as the original function ; and the
partial differential coefficient of — -, when x alone varies, is
ax
denoted by
d fdy\ d^u
d^\^}^^~d^'
as in the case of a single variable.
du
Similarly, the partial differential coefficient of — , when y
CtiX
alone varies, is represented by
d fdu\ d^u
dy \dx/ dydx*
d^***u
and, in general, denotes that the function u is first
ciy ax
differentiated n times in succession, supposing x alone to
vary, and the resulting function afterwards differentiated m
times in succession, where y alone is supposed to vary ; and
similarly in all other cases.
The Order of Differentiation is Indifferent. 145
We now proceed to show that the values of these partial
derived functions are independent of the oj^der in which the
variables are supposed to change.
119. If t^ be a Function of x and ?/, to prove that
d fdu\ d fdu\ d'^u dhi
dy\dx) dx\dyj dydx dxdy
where x and y are independent of each other.
Let u = <p{x, y), then — represents the limiting value of
uX
0 (a; + h, y) -(f> {x, y)
h
when h is infinitely small.
This expression being regarded as a function of y, let y
become y + k, x remaining constant ; then -ri-r] is the
limiting value of
((>{x + hyy + k)-(l>{x,y + Jc)-(t>{x + h, y) +0(a?, y)
Ilk
when both h and k become infinitely small, or evanescent.
In like manner -7- is the limiting value of
dy
(t>{x,y + k)-(l> {x, y)
k
when k is infinitely small ; hence -ri-r]^^ the limiting value
of "^
((>(x + h,y + k) -(i>{x+h,y) - (f>{x,y + k) + <l>{x,y)
hk
when both h and k are infinitely small.
Since this function is the same as the px-eceding for all
L
146
Successive I*artial Differentiation.
finite values of h and k, it will continue to be so in the limit ;
hence we have
d (du^
dx
fdu\ _ d fdu\
\dy) ~ dy \dxj'
d'u
d'u
dx^dy dyda?*
d^u _ d^u ^
dxdy dydx '
d'^u
d d
du
d d
du
dydx
dx' dy
Jx
dy ' dx
~dx
In like manner
for by the preceding
d r d^u\ d
dx [dxdy J dx
similarly in all other cases. Hence, in general,
dP'^'iu __ dP'-Hi
dxPdy^ ~ dy^dxP'
Again, in the case of functions of three or more variables,
by similar reasoning it can be proved that
d^u dhc
dzdxdy dxdyd£
Hence we infer that the order of differentiation is in all cases
indifferent, provided the variables are independent of each
other.
verify that -7-^- = ^—7-.
dydx dxdy
Examples foe Verification"
3. If « = sin {ax'^ + by^)^
d^u
dy^-dx
d*u
dxdy*'
diu
" dx^dy^' dy^dx*
120. Condition that Pdx + Qdy shall be a total
DiflTerential. — This implies that P dx + Qdy should be the
exact differential of some function of x and y. Denoting this
function by w, then
du = P dx + Q dy,
(2]
Condition for a Total Differential, 147
and, by (i), Art. 95, we must have
dP_£u_ dQ_d^
dy dydx^ dx dxdy
Hence tlie required condition is
dF^dQ,
dy dx'
121. If z^ be any Function of x and y^ to prove that
|(^<-)S)=s(^«|> »
where a; and y are independent variables.
Here each side, on differentiation, becomes
122. More generally, to prove tbat
d I dv\ d ( dv\ , .
where u and v are both functions of s, and s is a function of
X and y.
d I dv\ dudv d'^v
For -T- tt-r ] = -T--^ + u
but
dy \ dx) dy dx dydx^
du _ du dz dv _ dvdz ^
dy dz dy* dx dz dx '
d ( dv\ du dv dz dz d^v
dy\ dx) dz dz dx dy dydx '
and Tx'^t) ^^^ evidently the same value.
L 2
148 Successive Partial Diferentiation,
123. £uler's Theorem of Homogeneous Func-
tions.— In Art. 102 it has been shown that
du du
where w is a homogeneous function of the w'* degree in
a? and y.
Moreover, as — and — are homogeneous functions of the
degree n- i, we have, by the same theorem,
d fdu\ d (du\ , . du
dx \dxj dy \dxj ' dx^
d fdu\ d fdu\
du ^
dx \dy) ' ^ dy \dy) ^"' '^' dy'
multiplying the former of these equations by x^ and the
latter by y, we get, after addition,
, d^u d^u ^d^u , .( du du\
= (w-i)ww. (5)
This result can be readUy extended to homogeneous
functions of any number of independent variables.
A more complete investigation of Euler's Theorems will
be found in Chapter YIII.
124. To find the iSuccessive DilTerential CoeflS-
eients iivith respect to t^ of the Function
(^{x +at, t/ + (5t)y
where x, y, a, /3, are independent of t, and of each other.
By Ait. 1 1 7 we have in this case, where <j) stands for the
expression (p{x + at, y + (it),
d^ d(^ ^ dcj)
dt dx dy
Differentiation of(^{x-v at, y -^ ^t), 149
^0 t^ (d^\ d fd<f\
~''dx\dtj'"^ dy\dt)
d ( d(b ^d(}>) c d \ d^ d(l>\
o?;i; ( (i^iz; ^ f/?/ ) "^ dy { dx '^ dy)
This result can also be written in the form
in which ( « T" + P T" ) is supposed to be developed in the
dx ^ dy^
usual manner, and --^, &c., substituted for f — j 0, &o.
Again, to find — .
f^ _d f^\ ^dfd_ d\
df ~ dt\dt'J ~ dtydx ""^ dy)"^
( d ^ dVdcp f d ^ dVf d<f> „4\
By induction from the preceding it can be readily shown
that
f/»0 / d ^dy
H^^K^dx^^dy)'^'
This expression, when expanded by the Binomial Theorem,
gives the n^^ differential coefficient of the function in terms of
its partial differential coefficients of the n*'^ order in x and y.
150 Examples,
ExA2kIPLES.
1. If « = sm ix^y), verify the equation — -— = — — -,
2. If « = sin (y + ax) + (y ~ «a?)*, prove that
d'^u _ ^dru
dx^ dy^'
3. In general, \iu=f{y + ax)-^<p{y- ax), prove that
d^u „ d^u
J-^ = <^ 3-1-
dx' dy^
4. If M = y*, prove that
— = y-i(i+.logy) = _.
5. If w = — V"! ) fi°<l the values of
ax -V by ■\- cz
d^u d^u dr-u
^' ^^' ^^ ^*
6. If M = {x"^ + y2)J^ prove that
.d^u d^u ^d^u
^Si + ^^^^ + S-V^"-
7. If M = (a:^ ^ yS.^\^ prove that
aa;2 aa;«?y 6?y- 4
8. If F = ^y3 + iBy^x + 3Cya;2 + jj^rS^ prove that
d2VdV^_ fV^dVdV d^Vdr^_
dx^ dyi ^ dxdy Ix dy '^ If 1^ ~ ^^
A, B, 0
B, C, D
and show that the left-hand side of this equation vanishes when F is a perfect
cube.
d^u d^u d^u
1 H = O.
dx^ df ^ dz^
( 151 )
CHAPTER VII.
Lagrange's theorem.
125. IJagrange's Theorem. — Suppose that we are given
the equation
. z = x + y(li{z), (i)
in which x and y are independent variables, and it is required
to expand any function of z in ascending powers of ij.
Let the function be denoted by F{z), or by Uy and, by
Maclaurin's theorem, we have
I \dylQ I . 2 V^y Vo 1.2... n\dy''Jo ^ '
where Wo» ( -r 1 » &c., represent the values of w, -r, &c., when
VWo . . . dy
zero is substituted for y after differentiation.
It is evident that Uq = F{x).
To find the other terms, we get by differentiating (i) with
respect to a?, and also with respect to y,
dz ,, s dz dz , . ,, ,dz
Tx^'^y'^^'^dx^ dy^'^^^^y'^^'^dy^
1 dz , .dz
Also, since w is a function of 2, we have
du du dz du du dz
dx dz dx* dy dz dy'*
152 Lagrange's Theorem,
hence we obtain
Jy = *(^)^- (3)
Again, denoting ^(s) by Z, we have by Art. 121, since
Z is a function of u^
d I rrd'^^\ ^ f rT^^\ d^^U „ , .
T.[^ dp) = d^A^ T.) - df' ^'""^ ^'^'
dhi d f r7.du\ i \
df^A^di} '"^
Hence also -=-. = 3-3- ( Z^-^\
dy^ dxdy \ dxj
since x and y are independent variables ;
" |(>^S)-7.(^|)-£(^-S)V(3).
dxdy \ dxj \dxj \ dx) '
, d'u (d\(du\ ,.
^^^.^^ -df=Wj[^'dx} (51
To prove tliat the law here indicated is general, suppose
then.sinee ^(z»^"] = ^f Z«^] 4(^«« J)
— fz^-V-f'^^-
f/y \ dx) dx \ dy ) dx
Lagrange's Theorem. 153
This shows that if the proposed law hold for any integer
n, it holds for the integer n+i-, but it has-been found to hold
f or w = 2 and w = 3 ; accordingly it holds for all integral values
of n,
dti d u
It remains to find the values of -n -7-0 » &c. when we
dtj df
make y = o. Since on this hypothesis Z or 0(s) becomes
0(a?), and -7- becomes J^ ^ or F^{x), it is evident from (3),
(4), (5), (6), that the values of
du d^u dH d^*^u
dy' dy^' df " ' ~^'
become at the same time
....^[{0(.))'."2P'(.)].
Consequently formula (2) becomes
F{?) = F{x) + \ i>{x)r{x) + ^ I [ (0 WP J'(<^)] + &c
. +
This expansion is called Lagrange's Theorem.
If it be merely required to expand z, we get, on making
F(z) = s,
7/** d^-^
^ ^^[a>)]^ + &o, (8)
I . 2 , , . n do^'^
154 Laplace^ s Theorem.
126. Ijaplace's Theorem. — More generally, suppose
that we are given
«=/{a? + ^(^(z)), (9)
and that it is required to expand any function F{z) in ascend-
ing powers of y.
Let t = X ^- y<p{z), then z =f{t), and we have
^ = ^ + 2/^(/(0)- (10)
Also F{z) = F[f(t)] ; and the question reduces to the
expansion of the function F[f{t)] in ascending powers oiy
by aid of (10) ; accordingly, formula (7) becomes in this case
F{z) = F (/(<)) = F{f[x)} + I ,s,{f{x)} r [fix)] + &o.
^" ** '[0i/(^))]»-2;"[/W]I+&o. (>
I .2 . . . (fi + l) 6/12;"
This formula is called Laplace's Theorem, and is, as we
have seen, an immediate deduction from the Theorem of
Lagrange. These theorems evidently only hold when the
expansions are convergent series.
JExamples. 155
Examples. ^
1. Expand z, being given the equation
z = a + bz\
Here x = a, i/ = b, ^(z) = sfi,
and we get, from formula (8),
z = a + ba^ + Z^^a5+ iiPa' + &c.
Lagrange has shown that this expansion represents the least root of the pro-
posed cubic, and that a similar principle holds in like cases.
2. Given z = a + bz», find the expansion of z.
b^ *»
Ans. z = a + a^b + 2«a2"-i 1- 3w(3w - i) a3»»-2 + &c.
1.2 ^ 1.2.3
3. Given z = x + ye", find the expansion of z.
Ans. z = x + ve* + \f-^ + -^— z^'-^ + — - — \^el^ + &c.
1.2 1.2.3
4. z = a + « sin z, expand (i) «, (2) sin %.
^ d ^ f d\^
(l). Ana. 2 = « + « sin a + — (sin^a) + . ( — - j (sin%) + &c.
^ I . 2 da^ '1.2.3 \dal
^ d
(2). ,, sin z = sin « + e sin a cos a + -j- (sin'a cos a) + &c.
I . 2 aa
5. If « = a + - (z* — i), prove that
2
^ I . 2 . . . » \da) \ 2 / ^
6. Hence prove that
J . 3 . . . « \daj \ 2 )
( 156 )
CHAPTER YIII.
OB MORE VARIABLES.
127. Expansion of (l)(x + h,i/ + k). Suppose w to be a func-
tion of two variables x and y, represented, by the equation
^« = 0(a?, y) ; then substituting x + hior Xy we get, by Taylor's
Theorem,
0(^ + ^,2^) = 0(ar, y) + h— {(t>{x,y)] +j^^, {^(^, !/)) + &c.
Again, let y become y + ky and we get
(p{x + h,y + k) = (l>(xy y + k) + h— {^(ar, y + k)]
But
d T^ d^
<l>{x,y+k) = ^{x,y)+k — [^{Xy 2/)) +7y^^,(0(^,y)) + &c.
, c?w /c'* cPu
= w + /fe— + — + &0.
ay 1.2 dy^
Also
^ T" 0(^> y + ^0 ) = h— + hk -—r + 3-7-. + &o.,
and
h'' d- .. ,.. h^ dH Wk d'u .
1 . 2(^;z;=' ^^^ '^ ^^ i.2afa;' i.idx'dy
Extension of Taylor's Theorem* 157
Substituting these values in (i), we get
1 ,2 dx^ dxdy i, zdy"^ ' ^ '
128. This expansion can also be arrived at otherwise as
follows : — Substitute x + at and y + f5t for x andy, respectively,
in the expression ^ {x, y), then the new function
<t>{x + at, y + l^t),
in which x, y, a, (5, are constants with respect to t, may be
regarded as a function of tj and represented by F{t) ; thus
(j,{x + at,y + pt) =F(t).
The latter function F{t)f when expanded by Maclaurin's
Theorem, becomes, by Art. 79,
F{t) =i^(o) + ^F'{o) + ~ r\o) + . . .
+ |^i^(«)(0O, (3)
where F'[o) is the value of i^(^) when ^ = 0, i.e. F{o) = 0 (ar, y)
= u\ also F\o)y F'\o), &c. are the values of
^ ^^- &c
dt' df ^""'^
when ^ = o ; wher^v ^^ stands for (l>{x + at, y + j5t).
Moreover, by Ait. 1 1 7, we have
dd) d(b ,, dd)
dt dx dy
* Since it is indifferent whether we first change x into a; + A, and afterwards
change y into y -^ k, or vice versa ; the expansion given ahove furnishes an in-
dependent proof of the results arrived at in Art. 1 19.
158 Extension of Taylor^ s Theorem.
but, when ^ = o, ^(a; + a^, 3/ + ^t) becomes u, or F{p), and -j
at
becomes a -7- + /3 ^- at the same time.
ax ay
Hence i?^'(o) = a ^' + j3 ^.
' dx ay
Also, by the same Article,
dp "■ dx^^^ "^dxdy ^ ^ df (
which, when ^ = o, reduces to \
F (o)=„'- + 2„/3^H-^'^„ (4)
&C. &C. &0.
These equations may also be written in the symbolic
form
"(°) = (-i^-^|J«'
^»=(-|-'^|J«'
Again, f a — J w = a*" — , &o., since a, /3, are independent
of X and y : and hence the general term in the expansion of
F(t) can be at once written down by aid of the Binomial
Theorem.
Extension of Taylor'' s Theorem. 159
Finally, we have, on substituting h for at, and k for fit,
, , ,. ^ du ^ du h^^d^u „ d'^u
'Tr^W^-^^^KTx^^Jy] 't'i-^6h,,.ek). (5)
129. Kxpansion of (l>{cc + h, 1/ + k, z + I). — A function
of three variables, x, y, z, admits of being treated in a similar
manner, and accordingly the expression
(}){x + at, y + fit, z + jt),
when u is substituted for (p[x, y, z), becomes
e f d ^d dv .
or
(h(x + h, y + k, z + I) = u + ih—- + k-rr + l-r)^*
^^ »^ ' ^ \ dx dy dz)
I (, d ^ d d\ ^
+ h-— + k-r + 1-t\u + &c.
I .2\ dx dy dzj
,du ,du Au 1^ d^u ¥ dht P d^u
dx dy dz i . 2 dx^ i ,2 dy^ 1.2 dz^
,, d^u „ dhi .. d^u p ...
+ hk——r + Ih ^— + kl---- + &o. (6)
dxdy dzdx dydz
The general term in this expansion, and also the re-
mainder after n terms, can be easily written down.
1 60 Extension of Taylor's Theorem,
These results admit of obvious generalization for any
number of variables.
Also, by making x, y, s eacb cypher in (6), we have
.*(^'*'^)=M»^KSVKSV'
0
1 . 2 ydx^ja
where ( -r 1 » ( -r ) » • • • denote the values which the functions
\dxjQ \dyJo
du du , . ,
-7-, -r? . • . assume on making x = 0, y = o, and 2 = 0.
dx dy
This result may be regarded as the extension of
Maclaurin's Theorem.
130. Symbolic Expression for preceding Results. —
Since
^-*k- (^d , d\ I f^d , dy
"" \ dx dy) I . 2 \ dx dy)
1 f^d , dY .
equation (5) may be written in the shape
J. *
e*'^**^^^ {x, y)=<l>{a; + h,y + k). (7)
This is analogous to the form given for Taylor s Theorem
in Art. 67, and may be deduced from it as follows : —
We have seen that the operation represented by e^'^
when applied to any function is equivalent to changiag x
into X + h throughout in the function.
d
Accordingly, e '^(j>{x, y) = <^(x + h^y), since y is indepen-
dent of X.
Extension of Taylor* 8 Theorem, i6i
In like manner, the operation ^^^, wheji applied to any
function, changes y into y + k;
/. /"" . A^0 {x, y) = /""^ (a; + ^, y) = 0 (;^; + A, y + A;),
or e^y^^^'^i^ {x, y) = <l>{x + h^y + k),
assuming that the symbols k — and h -r- are combined ac-
cLy ux
cording to the same laws* as ordinary algebraic expressions.
In an analogous manner we obtain the symbolic formula
^Kx^W^' ^[x, y,z) = <p{x-¥h,y + k,z + l). (8)
131. If in the development (2), dxhe substituted for ^,
and dy for k, it becomes
(j){x + dx, y + dy) = (j) + -p- dx -i- -^ dy
ux ay
If the sum of all the terms of the degree n in dx and dy
be denoted by ^^"0, the preceding result may be written in
the form
, , , . d(b d^d) d^<h
<^{x + dx^y + dy) = (b + -'^ ^ —^ 4- 1— + . . .
^^ ' ^ I 1.2 1.2.3
+ -7-^ + &o.
Since t^, dy, are infinitely small quantities of the first
dru cPu
* That this is the case appears immediately from the equations — — - = ^-- ,
dxdy dydx
d^u d^u
-, &c.
dp dx'
1 62 Extension of Taylor* s Theorem,
order, each term in the preceding expansion is infinitely small
in comparison with the preceding one.
Hence, since c?^^ is infinitely small in comparison with
d<^y if infinitely small quantities of the second and higher
orders he neglected in comparison with those of the first, in
accordance with Art. 38, we get
d(p = ^(^7 + dx,y ■{- dy) - (p{x, y) ^-^dx-i- -^ dy,
which agrees with the result in Art. 97.
132. £uler's Theorems of llomogeneous Func-
tionjs. — ^We now proceed to give another proof of Euler's
Theorems in addition to those contained in Arts. 102 and 123.
If we suhstitute gx for A and gy for k in the expansion (5),
it becomes
, . / du du\
<p{x + gx,y-^ g?/):=u + g[x— + y ~\
g^ f „ d^u d^u
1.2 V dx^ -^ dxdy ^
where u stands for ^(a;, y).
But (^{x + gx, y -^^ gy) = ^[{i -\- g)x, (i + g)y} ;
and, if 0 [x, y) be a homogeneous function of the w** degree
in x and y, it is evident that the result of substituting (i + g)x
for ir, and (i + g)y iov y in it, is equivalent to multiplying it
by (i + gY* Hence, we have for homogeneous functions,
i^[x + gx,y + gy) = (i + g)"" (^{x, y) = (i + gYu,
, . [ du du\
or (i+^)«„ = « + j,(^^_ + 2,_J
1 . 2 \ dx^
d'u J'u\ .
where w is a homogeneous function of the n*^ degree in x
and y.
Euler^s Theorems, 163
Since the preceding equation holds for all values of ^, if
we expand and equate like powers of g, we* obtain
du du
,d^u d^u ,d^u , V
.d^u „ d^u . d^u .d^u . . , .
&0. &0. &0.
The foregoing method of demonstration admits of beiiii/
easily extended to the case of a homogeneous function of three
or more variables.
Thus, substituting gx for h, gy for k, gz for /, in formula
(6) Art. 1 2 9, and proceeding as before, we get
du du du
dx dy dz
„d^u ^d^u ^d^u d'u d^u
d^u f .
+ 2yz ;t-7-= ^(^ - i) ^>
&a &o. &o.
These formulae are due to Euler, and are of importance
in the general theory of curves and surfaces, as well as in
other applications of analysis.
The preceding method of proof is taken from Lagrange's
M^canique Analytique.
M 2
( i64 )
CHAPTEE IX.
MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE.
133. Definition of a Maximum or a Minimum. — If any
function increase continuously as the variable on which it de-
pends increases up to a certain value, and diminish for higher
values of the variable, then, in passing from its increasing to its
decreasing stage, the function attains what is called a maximum
value.
In like manner, if the function decrease as the variable
increases up to a certain value, and increase for higher values
of the variable, the function passes through a minimum stage.
Many cases of maxima and minima can be best determined
without the aid of the Differential Calculus ; we shall com-
mence with a few geometrical and algebraic examples of this
class.
134. Creometrical Example. — To find the area of the
greatest triangle which can he inscribed in a given ellipse. Sup-
pose the ellipse projected orthogonally into a circle ; then any
triangle inscribed in the ellipse is projected into a triangle
inscribed in the circle, and the areas of the triangles are to
one another in the ratio of the area of the ellipse to that of
the circle (Salmon's Conies, Art. 368). Hence the triangle in
the ellipse is a maximum when that in the circle is a maxi-
mum ; but in the latter case the maximum triangle is evidently
equilateral, and it is easily seen that its area is to that of the
circle as -v/iy to 47r. Hence the area of the greatest triangle
inscribed in the ellipse is
where a, h are the semiaxes.
Moreover, the centre of the ellipse is evidently the poiiit
of intersection of the bisectors of the sides of the triangle.
Algebraic Examples of Maxima and Minima. 165
Examples,
1. Prove that the area of the greatest ellipse inscribed in a given triangle is
X
, — (area of the triangle).
2. Find the area of the least ellipse circumscribed to a given triangle.
3. Place a chord of a given length in an ellipse, so that its distance from the
centre shall be a maximum.
The lines joining its extremities to the centre must be conjugate diameters.
4. Show that the preceding construction is impossible when the length of
the given chord is >a\/ z or <b*yz ; where a and h are the semiaxes of the
ellipse. Prove in this case that if the distance of the chord from the centre be
a maximum or a minimum the chord is parallel to an axis of the curve.
5. A chord of an ellipse passes through a given point, find when the triangle
formed by joining its extremities to the centre is a maximum.
6. Prove that the area of the maximum polygon of « sides, inscribed in a
given ellipse, is represented by - ab sin — .
135. Algebraic Examples of Maxima and minima.
— Many cases of maxima and minima can be solved by ordi-
nary algebra. We shall confine our attention to one simple
class of examples.
"Lei fix) represent the function whose maximum or mini-
mum values are required, and suppose u =f{x), and solve
for X ; then the values of u for which x changes from real to
imaginary^ are the solutions of the problem. TJiis method is,
in general, inapplicable when tEe equatioii in x is beyond the
second degree. We shall illustrate the process by a few ex-
amples : —
Examples.
r . To divide a number into two parts such that their product shall be a
maximum.
Let a denote the number, x one of the parts, then x (a — x) is to be a maxi-
mum, by hypothesis.
Here u = x(o - x)y or x"^ ~ ax ^^ u = o ;
solving for x we get
i±
>/i-"^
accordingly, the maximum value of Mis —, since greater values would make x
imaginary.
i66 Maxima and Minima of Functions of a Single Variable.
1. To find the maximum and minimum values of the fraction .
a;2+i
TT X ^ X 1 '\X( I — 2w) ( r + 2«)
Here u = , or «« + f = - ; ••. a; = — + ^— ^^ i-i— - — '-
X^+l U 2U~ 2M
In this case we infer that the maximum and minimum values of u are - and
2
; and the proposed fraction accordingly lies between the limits - and
for all real values of x.
These results can be also easily established, as follows. We have in all cases
{x + yf = {x- yf + ^y.
Accordingly, if a; + y be given, xp is greatest when x-y= o,or when x = y.
Conversely, if xy be given, the least value oix ^-y ia when x — y.
Hence, denoting xy by a^, the minimum value of a: + — is 2 a, for positive
values of x.
Again, it is evident that when a function attains a maximum value, its in-
verse becomes a minimum ; and vice versd.
Accordingly, the max. value of -5 j is — , under the same condition.
x^ + a* 2a
3. Find the greatest value of
(a + «) (i + x)
(a + x) (b + x) . ^ , . . ab . . /—■
Here is to be a mmimum, or h a? is a mm. ; .*.«=%/ «*»
and the max. value in question is
{x + «)(« + h)
4* *
af + c
Let x + e = Zy and the fraction becomes — -»
z
In order that this should have a real min. value, {a — c){b — e) must be posi-
tive ; i. e. the value of e must not lie between those of a and b, &c.
5. Find the least value of a tan 6 + bcot d. Ans. i\/ ab,
6. Prove that the expression ; will always lie between two fixed
*^ x^ + bx + c^
finite limits if a> + c' > o5 and b"^ <4.c^', that there will be two limits between
which it cannot lie if cfi + <^> ab and i' > 4 <j3 : and that it will be capable of all
values if a* + c* < ab.
136. Xo find the Maximniii and minimum values
of
act^ + zhxy + c?/'
e/ir* + il/xy + cV*
Algebraic Examples of Maxima and Minima, 167
X
Let u denote the proposed fraction, and substitute a for -;
y
then we get "^
az^ + 2hz + c , .
aV + 2b'z+ c' ^
or (a - a^ujz^ + 2{b- b'u)z + c - c'w = o.
Solving for z, this gives
[a - a'u)% +b-b'u=± ^{b - b'uy - {a- a'u) (c - cu) . (2)
There are three cases, according as the roots of the equation
{b'' - aV) u^ + {ad + ca'-2 bb') u+b''-ac = o (3)
are real and unequal, real and equal, or imaginary.
(i). Let the roots be real and unequal, and denoted by
a and j3 (of which /3 is the greater) ; then, if 6'* - a'c' > o, we
shall have
{a -a'u)z + b- b'u = + y{b'^ - dd) (u-a) (u-fS),
Here, so long as u is not greater than a, s is real ; but
when u > a and < j3, z becomes imaginary ; consequently, the
lesser* root (a) is a maximum value of u. In like manner, it
can be easily seen that the greater root (j3) is a minimum.
Accordingly, when the roots of the denominator, «V + 2b^x
+ d = o, are real and unequal, the fraction admits of all pos-
sible, positive, or negative values, with the exception of those
which lie between a and j3.
If either a' = o, oi d = o, the radical becomes
b'y(u-a)(u-(5),
and, as before, the greater root is a minimum, and the lesser
a maximum, value of u.
* In general, in seeking the maximum or minimum values of 1/ from the
equation, 1/ = tp (x), if for all values of y between the limits a and ^8, the corre-
sponding values of x are imaginary, while x is real when y = a, or y = )8 ; then
it is evident that the lesser of the quantities, o, )8, is a maximum, and the greater
a minimum, value of y. This result also admits of a simple geometrical proof,
by considering the curve whose equation is y = <^ {x).
1 68 Maxima and Minima of Functions of a Single Variable.
(2.) When a = j3, the expression under the radical sign is
positive for all values of w, and consequently u does not admit
of either a maximum or a minimum value.
(3.) When the roots a and j3 are imaginary, the expres-
sion under the radical sign is necessarily positive, and u in
this case also does not admit of either a maximum or a mini-
mum value.
Hence, in the two latter cases, the fraction admits of all
possible values between + 00 and - 00 .
In the preceding, the roots of the denominator are sup-
posed real ; if they be imaginary, i.e. if b'"^ - dc < o, we have
{a-du^z'^b-b'u = ±^{dd ~b"^) (ti-a) {f5-u).
It is easily seen that z is imaginary for all values of u
except those lying between a and j3. Accordingly, the greater
root is a maximum, and the lesser a minimum, value of u.
Hence, in this case, the fraction represented by u lies be-
tween the limits a and /3 for all real values of x and y.
137. Quadratic for determining 2. — Again, the value
of 2, corresponding to a maximum or a minimum value of w,
must satisfy the equation
{a - a'u)% + b - b'u = o.
Substituting for u in (i) its value derived from this latter
equation, we obtain the following quadratic in z :
(ab' - bay + z iac' - cd) + bd - cV = o. (4)
This equation determines the values of z which correspond
to the maximum and minimum values of u. It can be easily
seen that if the roots of equation (3) are real so also are those
of (4) ; and vice versa.
The student will observe in the preceding investigation
that when u attains a maximum or a minimum value, the
corresponding equation in z, obtained from (2), has equal
roots. This is, as will be seen more fully in the next Article,
the essential criterion of a maximum or a minimum value, in
general.
Condition for a Maximum or Minimum. 169
Find the maximum or minimum values of u in the follow-
ing cases : —
Examples.
«* + 2a; + 1 1 . 5 .
I. u = — . ^n*. M = 2, a max., « = ^ a nun.
a2 + 4a; 4- 10 6
X^ - X ■'r I 2 - 2X
I +
X^ ->r X — I X' + X — I
\ — X . . . X^ -h X — I .
-T is a max. or a min. according as ia a min. or a max., i. e.
x^ -i X- I ° 1 - X
I . . .
as « IS a maximum or a minimum.
I — X
.'. a;= o, or a; = 2 ; the former gives a maximum, the latter a minimum solution.
We now proceed to a general investigation of the condi-
tions for a maximum and minimum, by aid of the principles
of the Diif erential Calculus.
138. Condition for a Afaximnni or Minimum. — If
the increment of a variable, a?, be positive, then the corre-
sponding increment of any function, f{x), has the same sign
as that of /'(ip), by Art. 6 ; hence, as x increases,/ (37) increases
or diminishes according Q.^f\x) is positive or negative.
Consequently, when f[x) changes from an increasing to a
decreasing state, or vice versa, its derived function f\x) must
change its sign. Let oj be a value of x corresponding to a
maximum or a minimum value oif{x) ; then, in the case of
a maximum we must have for small values of h,
f{a) >f(a + h)y and/(a) >f{a-'h) ;
and, for a minimum,
f{a) <f{a + h), and/(a^) <f{a-h).
Accordingly, in either case the expressions
f{a + h) -f(a), and/(« - h) -f{a),
have both the same sign.
1 70 Maxima and Minima of Functions of a Single Variable,
Again, by formulae* (29), Art. 75, we have
f{a + h) -f{a) = hfia) + -^/'(a + 6h),
f{a - h) -f{a) = - hf{a) + -^/'(« - ^i*).
Now, when h is very small, and /"(«) finite, the second
term in the right-hand side in each of these equations is very
small in comparison with the first, and hence f {a + h) -f{a)
and f{a- h) - f{a) cannot have the same sign unless
/(«) = o.
Hence, the valines of x which render f{x) a maximum or a
minimum are in general roots of the derived equation f {x) = o.
This result can also be arrived at from geometrical
considerations ; for, let y = f{x) be the equation of a curve,
then, at a point from which the ordinate y attains a maximum
or a minimum value, the tangent to the curve is evidently
parallel to the axis of x ; and, consequently f{x) = o, by
Art. 10.
Moreover, if x be eliminated between the equations
f{x) = u and/'(a7) = o, the roots of the resulting equation in
u are, in general, the maximum and minimum values oif{x).
This is the extension of the principle arrived at in
Art. 134.
Again, since /'(«)•= o, we have
f{a-h)-f{a)=^J'{a-e,h)
(5)
* In the investigation of maxima and minima given above, Lagrange's form
of Taylor's Theorem has been employed. For students who are unacquainted
■with this form of the Theorem, it may be observed that the conditions for a
maximum or minimum can be readily established from the form of Taylor's
Series given in Art. 54, viz.,
f{a + h) -f{a) = hf{a) + ^^f'ia) + ^-^ /'"{a) + &c. ;
for "when h is very small and the coefficients /(a),/" (a), &c. finite, it is evident
that the sign of the series at the right-hand side depends on that of its first
term, and hence all the results arrived at in the above and the subsequent
Articles can be readily established.
Condition for a Maximum or a Minimum. 171
But the expressions at the left-hand side in these equations
are both positive for small values of h wb^n/"(«) is positive;
and negative, when f^\a) is negative ; therefore f{a) is a
maximum or a minimum according as f\a) is negative or
positive.
If, however, /"(flf) vanish along with /'(«), we have, by
Art. 75,
/(a + h) -f(a) =_|--/-(«) + rT^^/n^ + Oh),
/{a - h) -f{a) = T^r W •+ TTI^/^^C^ - OJ^)'
Hence it follows that in this case, f{a) is neither a
maximum nor a minimum unless f'\a) also vanish; but if
f"{a) = o, then /(a) is a maximum when /^^(«) is negative,
and a minimum when/^'^(<?) is positive.
In general, let/(")(a) be the first derived function that
does not vanish ; then, if n be odd, f{a) is neither a maximum
nor a minimum ; if n be even, f{a) is a maximum or a mini-
mum according as /(") [a] is negative or positive.
The student who is acquainted with the elements of the
theory of plane curves will find no difficulty in giving the
geometrical interpretation of the results arrived at in this
and the subsequent Articles.
Examples.
1. «= a sin* + J cosa?.
Here the mazimum and minimuin values are giyen by the equation
du a
-7- = fl cos a: - i sm iP = o, or tan a; = 7.
dx ' b
Hence, the max. value of u is -v/^^T^and the min. is - ^/a^ + b"^. This is
also evident independently, since u may be written in the form
\/a* + ^2 Bin i^x +0),
b
where tan o as — .
a
2. « = a; - sin «.
r ^, . du cPn dh*
In this case — = i - cos «, — = sm x, — ^ = cos «
dx ax'' dx^
172 Maxima and Minima 0/ Functions of a Single Variahk.
Accordingly, if 7- = o, we have ^-s = O1 ^^^^ -— = r.
Consequently, the function x - B\nx does not admit of either a maximum or a
minimum value.
This result can also be easily seen from geometrical considerations.
3. M = a cos a; + J cos 2x, a and b being both positive.
„ du . , .
Here -— = - a smrr - 20 sm 22?,
dx
d^u
— = - a cos T - 40 cos 2X.
The maximum and minimum values are given by the equation a sin* + 2b
sin 2a; = o :
— a
.'. we have, (i), sin a; = o ; or (2), cos x = —7-.
4*
The simplest solution of (r) is a; = o, in which case
w*: a + b, -— = — a — 40;
ax'^
consequently this gives a maximum solution.
Again, let a; = tt, and we have u = b - a, —^ = a - ^b ; consequently this
gives a maximum or a minimum solution, according as a is < or > ^b.
If a = 45, we get when a; = ir, -— = o.
ax^
On proceeding to the next differentiation we have
dhi
^ = a (sin « + a sin 2x), = o when a? *= «.
d^u
Again, ■— = a (cos a; + 4 cos 2x) = 3a. Consequently the solution is a
minimum in this case.
Again, the solution (2) is impossible unless a be less than 4J. In this case,
L e. when a < ^b, we easily find — ^ positive, and accordingly this gives a min.
value of «, viz. — -^ — h,
ob
4. Find the value of x for which sec « — a: is a maximum or a minimum.
Alls, smx = .
Application to Rational Algebraic Expressions. 173
139. Application to Rational Algebraic Expres-
sions.— Suppose fix) a rational function containing no
fractional power of x, and let the real roots of f{x) = o,
arranged in order of magnitude, be a, j3, 7, &c. ; no two of
which are supposed equal.
Then /{x) = {x - a) {x - (5) {x - y) . . .
and r(a)=(a-/3)(a-y) . . .
But by hypothesis, a - j3, a - y, &c. are all positive ; hence
/"(a) is also positive, and consequently a corresponds to a
minimum value oif{x).
Again, /'(/3) = 0-a)0-7)
here j3 - a is negative, and the remaining factors are positive ;
hence /"(j3) is negative, and/(|3) a maximum.
Similarly, /(y) is a minimum, &c.
140. Maxima and Minima lvalues occur alter-
nately.— We have seen that this principle holds in the case
just considered.
A general proof can easily be given as follows : — Suppose
f{x) a maximum when a; = a, and also when x = b^ where b is
the greater ; then when x = a + h, the function is decreasing,
and when x = b - hjitis increasing (where A is a small incre-
ment) ; but in passing from a decreasing to an increasing
state it must pass through a minimum value ; hence between
two maxima one minimum at least must exist.
In like manner it can be shown that between two minima
one maximum must exist.
141. Case of £qual Roots. — ^Again, if the equation
f{x) = o has two roots each equal to a, it must be of the form
In this case /"(a) = o,f"{a) = 21// (a), and accordingly,
from Art. 138, a corresponds to neither a maximum nor a
minimum value of the function /(a;).
In general, if /'(^) have n roots equal to a, then
Here, when n is even, /(a) is neither a maximum nor a
minimum solution : and when n is odd, / (a) is a maximum or
a minimum according as ;//(a) is negative or positive.
174 Maxima and Minima of Functions of a Single Variable.
142. Case where f'{x) = 00. The investigation in
Art. 138 shows that a function in general changes its sign in
passing through zero.
In like manner it can be shown that a function changes
its sign, in general, in passing through an infinite value ; i.e. if
<l){a) = 00, (j){a - h) and ^{a+h) have in genexal opposite signs,
for small values of h.
For, if u and - represent any function and its reciprocal,
they have necessarily the same sign ; because if u be positive,
- is positive, and if negative, negative.
u
Suppose Ui, U2, Usy three successive values of w, and
— , — , — , the corresponding reciprocals.
Then, if Wa = o, by Art. 138, Ui and u^ have in general
opposite signs.
Hence, if — = 00 , — and — have also opposite signs ; and
tt-i U\ W3
we infer that the values of x which satisfy the equation f\x)
= 00 may furnish maxima and minima values oif(x),
143. We now return to the equation
/{x) = (x-a)-^p{x),
in which n is supposed to have any real value, positive, nega-
tive, integral, or fractional.
In this case, when x = a,f^(x) is zero or infinity according
as n is positive or negative.
To determine whether the corresponding value of /(a?) is
a real maximum or minimum, we shall investigate whether
f^{x) changes its sign or not as x passes through a.
When it = a + hy f(a + h) = A" ;// (a + A),
„ X = a- h, f{a -h) = (-h)**\P(a-h):
inow, when h is infinitely small, -(pla+h) and \p(a-h) become
each ultimately equal to ip {a) : and therefore /'(a + h) and
f\a - h) have the same or opposite signs according as ( - i)"
is positive or negative.
Examples. 175
(i). If w be an even «n%er, positive or negative, /'(a;) does
not change sign in passing through a, and. accordingly a cor-
responds to neither a maximum nor a minimum solution.
(2). linhQ an odd integer, positive or negative, f{a + h)
and /'(a - h) have opposite signs, and a corresponds to a real
maximum or minimum.
(3). If w be a fraction of the form ± — , then ( - i)
r
= 1 ^ = I , and a corresponds to neither a maximum nor a
minimum.
ar.fl £
(4). If n be of the form ± i^LHl^ then ( - i) " = (- 1) '';
this is imaginary Up be even, but has a real value ( - i) when
p is odd. In the former case, f'(a - h) becomes imaginary ; in
the latter,/'(flJ + y^) and/'(«-A) have opposite signs, and/(fl)
is a real maximum or minimum.
Thus in all cases of real maximum and minimum values
the index n must be the quotient of two odd numbers.
Examples.
I, f{x) = ax^ + 2bx + c.
b
Her© f{x)=2{ax + b)=iO\ hence a: = ,
fix) = 2a,
j^ji^ — Z — is a maximum or a minimum value of ax^ + zJx + c, according
a
as a is negative or positive.
a. /(a?) = 2a:» - iSa;" + 36a; + 10.
Here f {x) = 6{a;2 _ 5a; + 6) = 6{a; -2){x- 3).
(i.) Let a; = 2 ; then /"(a;) is negative ; hence /(2) or 38 is a maximum
(2.) Let a; = 3 ; then/ "(a?) is positive; hence / (3) or 37 is a minimum.
1 7 6 Maxima and Minima of Functions of a Single Variable.
It is evident that neither of these values is an absolute maximum or mini-
mum ; for when a: = oo , f{x) = so , and when x = - o^ , f{x) = — go , accord-
ingly, the proposed function admits of all possible values, positive or negative.
Again, neither + oo nor — oo is a proper maximum or minimum value, because
for large values of a:, f{x) constantly increases in one case, and constantly dimi-
nishes in the other.
It is easily seen that as x increases from — oo to + 2, f{x) increases from —00
to 38 ; as ic increases from 2 to 3,f(x) diminishes from 38 to 37 ; and as x in-
creases from 3tos<,f{x) increases from 37 to 00. When considered geome-
trically, the preceding investigation shows that in the curve represented by the
equation
y = 2x^ - i^x^ + 36a; + 10,
the tangent is parallel to the axis of x at the points a; = 2, y = 38 ; and a; = 3,
y = 37 ; and that the ordinate is a maximum in the former, and a minimum in
the latter case, &c.
3. /(ar) = a + b{x - c)i. Ans. x = e. Neither a max. nor a min.
4. f{x) = b + e{x- a)^ + d{x - «)i
Substitute a + A for x, and the equation becomes
/(a + A) = i + c0 + d?^;
also f{a - A) = J + cA^ + dh^ ;
but when A is very small h^ is very small in comparison with Ai, and accordingly b
is a minimum or a maximum value of /(a-) according as c is positive or negative.
5- /(^) = S^^ + »2a;5 - 15a;* - 4oa;3 + i^x"^ + 60a; + 17.
Ans. X = ± I gives neither a max. nor a min. ; x =— 2 gives a min.
(x - i)(x- 6)
6. . Let a; — 10 = z, and the fraction becomes
X- 10
(g + 9)(g+4) ,,,^,,^36
, or z + 1 3 + — .
z z
The maximum and minimum values are given by the equation i - ^=0;
,-. « = + 6, and hence a; = 16 or 4 ; the former gives a minimum, the latter
a maximimi value of the fraction.
'• ^^^^-{x-^if'
Hence /'(^) = |^^ (^ + 5).
If a; = T,/(ir) is neither a maximum nor a minimum; if a? = - S,f(x) is a
maximum.
, _,. ^ ax" + 2hxy + cy^ 177
Max. and Mm. of rr- ri •
ax"^ + 20xy+ cy^
Again, the reciprocal function j is evidently a max. when z = ~ 1;
for if we substitute for x, — i + h, and - i - A, successively, the resulting
values are both negative ; and consequently the proposed function is a minimum
in this case.
This furnishes an example of a solution corresponding to /'(a;) =00. See
Art. 142.
144. We shall now return to the fraction
ax^ + ihxy + cy"^
aV + 2b' xy + cy*
the maximum and minimum values of which have been already
considered in Art. 136.
Write as before the equation in the form
z^{a - a'u) + 2z{h - Vu) + (c - c'u) = o,
where 2 = -.
y
Differentiate with respect to 2, and, as — = o for a maxi-
uZ
mum or a minimum, we have
z{a - du) + (5 - h'u) = o.
Multiply this latter equation by s, and subtract from the
former, when we get
z{b - h'u) + (c - c'u) = o.
Hence, eliminating z between these equations, we obtain
{a - du) (c - c'u) ={b- b'u)\
or u^ia'd - b"^) ~ u{ac + ca' - 2bb') + {ac - b"-) =0; (3)
the same equation (3) as before.
The quadratic for 2,
z'^{ab' - ba) + z{ad - ca') + bd - cV = o, (4)
is obtained by eliminating u from the two preceding linear
equations.
178 Maxima and Minima of Functions of a Single Variable.
This equation can also be written in a determinant form,
as follows : —
I
"Z
2'
a
h
c
a'
y
d
= o.
It may be observed that the coefficients in (3) are in-
variants of the quadratic expressions in the numerator and
denominator of the proposed fraction, as is evident from the
principle that its maximum and minimum values cannot be
altered by linear transformations.
This result can also be proved as follows : —
Let
^~ dX-" -\-2b'XF + c'Y''
where X, Y denote any functions of x and y ; then in seeking
the maximum and minimum values of w we may substitute
z for — , when it becomes
az^ + 2bz + c
dz^ + ib'z^- c"
and we obviously get the same maximum and minimum values
for u, whether we regard it as determined from the original
fraction or from the equivalent fraction in 2.
Again, let X, Y be linear functions of x and y, i. e.
X = Ix -v my, Y= Tx + m'y^
then u becomes of the form
Ax- + iBxy + Ctf
2V + iB^xy + Oy
where A, B, C, A\ B\ C\ denote the coefficients in the trans-
formed expressions ; hence, since the quadratics which deter-
mine the maximum and minimum values of tc must have the
same roots in both cases, we have
AC-B' = X{ac - b'), AC + CA' - 2BB' = \{ad + ca' - zbb"),
AC - B" = A(«V - b''). Q.JE.n.
Application to Surfaces, 179
It can be seen without difficulty that
A = {Im' - mr)\
We shall illustrate the use of the equations (3) and (4) by
applying them to the following question, which occurs in the
determination of the principal radii of curvature at any point
on a curved surface.
145. To find the Maxima and Minima Values of
r cos*a + 2s cos a cos jS + ^ cos'jS,
where cos a and cos j3 are connected by the equation
(i +p^) cos'a + 2pq cos a cos /3 + (i + q^) cos'^/B = i,
and p, q, r, s, t are independent of a and /3.
Denoting the proposed expression by w, and substituting
» cos a ,
z lor — p., we ffet
cos/3 °
rZ^ + 28Z + t
(i +p^)z^ + ipqz + (i + ^y
The maximum and minimum values of this fraction, by
the preceding Article, are given by the quadratic
u^i ■¥p' + q^)-u[[i+q'')r- zpqs-v (i +/)^) +r^ -«* = o; (6)
while the corresponding values of z or • ^ are given by
z^[{i +p^)s -pqr] + s((i +p^)t - (i + q^)r]
+ {pqt-{i+q')s} =0* (7)
The student will observe that the roots of the denominator
in the proposed fraction are imaginary, and, consequently, the
values of the fraction lie between the roots of the quadratic
(6), in accordance with Art. 136.
* Lacroix, Dif. Cal, pp. 575, 576.
N 2
i8o Maxima and Minima of Functions of a Singh Variable,
146. To find tbe Maxiniuiii and Mininram Radius
Vector of tlie Ellipse
ax^ + ilxy + cy* = i.
(i). Suppose the axes rectangular ; then
r^ = x^ -\- y^ iB> to be a maximum or a minimum,
X
Let - = s, and we get
as* + ibz + c
Hence the quadratic which determines the maximum and
minimum distances from the centre is
r* {ac - i') - r' (^ + c) + I = o.
The other quadratic, viz.
ha? - {a- c) xy - 6y* = o,
gives the directions of the axes of the curve.
(2.) If the axes of co-ordinates be inclined at an angle w,
then
r* = ar* + ^ + ixy cos (u
z^ + 2s cos w + I
az^ + 2bz + c '
and the quadratic becomes in this case
r* {ac - b^) -r^ {a + c ~ 2b cos w) + sin*w = o,
the coefficients in which are the invariants of the quadratic
expTfessions forming the numerator and denominator in the
oxprtffision for r*.
The equation which determines the directions of the axes
A the conic can also be easily written down in this case.
Maximu7n and Minimum Section of a Right Cone. 1 8 1
147. To investigate tlie maximmu and Jfinimi
Values of u«
Substituting z for -, and denoting the fraction by w, we have
az^ + 3/^2- + 3CS + fl?
~ a>z^ + 35's'^ + 3c'2 + flf*
Proceeding, as in Art. 144, we find that the values of u and 2
are given by aid of the two quadratics
az^ + 2bz + c = (az^ + 2b'z + c^u,
bz^ + 2CZ + d= (b'z^ + idz + d')u.
Eliminating u between these equations, we get the following
biquadratic in z : —
z\aV - ba!) + zz^ac' - cd) ^■z^[ad' - a'd + i{bd - cb')]
+ 2z{^d' - db') + led' - cd) = o. (8)
Eliminating z between the same equations, we obtain a
biquadratic in u, whose roots are the maxima and minima
values of the proposed fraction. Again, as in Art. 144, it
can easily be shown that the coefficients in the equation in u
are invariants of the cubics in the numerator and denominator
of the fraction.
148. To cut tlie maximum and minimum Ellipse
from a Right Cone wbich stands on a given circular
base. — Let AD represent the axis of
the cone, and suppose BP to be the
axis major of the required section; 0
its centre ; a^ b, its semi-axes. Through
0 and P draw LM and PR parallel to
BC. Then BP = 2a, b^ = LO . OM
(Euclid, Book in., Pr. 35) ; but LO
= — ,0M = — ;.'. P=-.BC. PR.
224
Hence BP^' . PR is to be a maximum
or a minimum. ^^* ''
1 82 Maxima and Minima of Functions of a Single Variable.
Let Z BAD - a, PBC = Q, BO = c.
Then BP^BC''''^^^ ''''''
FB = BP
sin ^P(7 cos {d - a)'
sin PBB c cos (0 + a)
sin Pi25 cos (6/ - a) '
cos (0 + a) .
— TTTi ( IS a maximum or a mmimum.
COS^ (0-a)
du _ sin 20 - 2 sin 2a
dQ cos* (0 - a)
TT uu 8m2(7- 2 sm 2a . ^
iience -777 = -r-r?, ^ — = o ; .*. sm20 = 2 sm 2a.
The solution becomes impossible when 2 sin 2a > i ; i.e. if
the vertical angle of the cone be > 30°.
The problem admits of two solutions when a is less than
15°* ^or, if Qi be the least value of 0 derived from the
equation sin 20 *= 2 sin 2a ; then the value - - 0i evidently
gives a second solution.
Again, by differentiation, we get
d^U 2COS20 , , . . . .
3215 = — TTTx r (when sm20 = 2 sm 2a). »
dd^ cos* (0 - a) ^ '
This is positive or negative according as cos 20 is positive or
negative. Iience the greater value of 0 corresponds to a
maximum section, and the lesser to a minimum.
^ In the limiting case, when a = 15°, the two solutions
coincide. However, it is easily shown that the corresponding
section gives neither a maximimi nor a minimum solution of
the problem. For, we have in this case 0 = 45° ; which value
d^u
gives -r^^ = o. On proceeding to the next differentiation, we
find, when 0 = 45°,
d^u _ - 4 _ ^4
dO' ~ cos*(45°-a) " ~ "9"
Hence the solution is neither a maximum nor a minimum.
When a > 15°, both solutions are impossible.
Oeometrical Examples, 183
149. The principle, that when a function is a maximum
or a minimum its reciprocal is at the same time a minimum
or a maximum, is of frequent use in finding such solutions.
There are other considerations by which the determina-
tion of maxima and minima values is often facilitated.
Thus, whenever w is a maximum or a minimum, so also
is log (w), unless u vanishes along with — .
Again, any constant may be added or subtracted, i.e. if
f(x) be a maximum, so also is /(a?) ±c.
Also, if any function, w, be a maximum, so will be any
positive power of w, in general.
150. Again, if z = f{u), then dz = f(u)du^ and conse-
quently s is a maximum or a minimum; either (i) when
du = o, i.e. when u is a maximum or a minimum ; or (2) when
/(«) = o.
In many questions the values of u are restricted, by the
conditions of the problem,* to lie between given limits;
accordingly, in such cases, any root of f\u) = o does not
furnish a real maximum or minimum solution imless it lies
between the given limiting values of u.
We shall illustrate this by one or two geometrical
examples.
(i). In an ellipse, to find when the rectangle under a pair of
conjugate diameters is a maximum or a minimum. Let r be any
semi-diameter of the ellipse, then the square of the conjugate
semi-diameter is represented by a' + i^ - r^, and we have
u = r^ (a* + h"^ - r^) a maximum or a minimum.
Here -r- = 2(0^ + b^ - zr^) r,
dr ^ '
Accordingly the maximum and minimum values are,
(i) those for which r is a maximum or a minimum ; i.e. r-a,
or r = 6 ; and, (2) those given by the equation
r(c^ + 6^ - 2r*) = o;
• See Cambridge Mathematical Journal, vol. iii. p. 237.
1 84 Maxima and Minima of Functions of a Single Variable,
or r = o, and r
'JH
The solution r = o is inadmissible, since r must lie between
the limits a and b : the other solution corresponds to the
equiconjugate diameters. It is easily seen that the solution
in (2) is the maximum, and that in (i) the minimum value
of the rectangle in question.
151. As another example, we shall consider the following
problem* : —
Given in a plane triangle two sides (a, b) to find the
maximum and minimum values of
I A
- . cos — ,
e 2
where A and c have the usual significations.
Squaring the expression in question, and substituting x
for c, we easily find for the quantity whose maximum and
minimum values are required the following expression :
I2b_ a" -y"
X x^ ar* '
neglecting a constant multiplier.
Accordingly, the solutions of the problem are — (i) the
maximum and minimum values of a;, i.e. a + 6 and a -b.
du
(2) the solutions of the equation — , i.e. of
I Ab 3 (^^ - b^)
x^ x^ x^
or x^ + Ofbx - ^{a^ - b^) = o;
whence we get x = ^/^a"^ + b^ - 2b,
neglecting the negative root, which is inadmissible.
Again, if J > a, \/3a'^ + b'^ - 2b is negative, and accord-
ingly in this case the solution given by (2 ) is inadmissible.
* This problem occurs in Astronomy, in finding when a planet appears
biightest, the orbits being supposed circular.
Maxima and Minima Values of an Implicit Function. 185
If a > 6, it remains to see whether y^sa' + 6* - ih lies
between the limits a ■\- h and a -h. It* is easily seen that
-v/3a* + 6*-2Jis>«-6: the remaining condition requires
a ^ b > v^3rt^ + h^ - zby
or « + 36 > 'v/s^'* + ^S
or a^ + tab + 96^ > 3«^ + 6%
i.e. 4 J' + 3^5- a^ > o,
or (46 - flj) (6 + a) > o ;
or, finally, ft > -.
We see accordingly that this gives no real solution unless
the lesser of the given sides exceeds one-fourth of the
greater.
When this condition is fulfilled, it is easily seen that the
corresponding solution is a maximum, and that the solutions
corresponding to x = a + b, and x = a - b, are both minima
solutions.
152. Haxima and Minima ITalaes of an Implicit
Function. — Suppose it bo required to find the maxima or
minima values of ^ from the equation
/(x, y) = o.
Differentiating, we get
du du dy
dx dy dx
where u represents f{x, y). But the maxima and minima
dy
values of y must satisfy the equation -^ = o : accordingly the
ax
1 86 Maxima and Minima of Functions of a Single Variable,
maximum and Trn'm'-mnm values are got by combining* the
equations -r = o, and u = o,
153. maximum and JVUnimum in case of a Func-
tion of two dependent Variables. — To determine the
maximum or minimum values of a function of two variables,
X and y, which are connected by a relation of the form
f{x, y) = o.
Let the proposed function, 0 (a?, y) be represented by u ;
then, by Art. loi, we have
d^df d(f) df
du _ dx dy dy dx
dx df
dy
But the maxima and minima values of u satisfy the
equation ;t- = o, hence the values of x and y derived from
dx
the equations /(a?, y) = o, and
d^ df d^ df
dx dy dy dx '
furnish the solutions required. To determine whether the
solution so determined is a maximum or a minimum, it
d''^u
is necessary to investigate the sign of -r-^. We add an
example for illustration.
154. Gimn the four sides of a quadrilateral, to find when its
area is a maximum.
Let a, by c, d be the lengths of the sides, ^ the angle
between a and b, yfj that between c and d. Then ab sin 0
+ cd sin ;// is a maximum ; also
a^ ■{- b^ - 2ab cos 0 = c* + gP - zed cos ;/»
being each equal to the square of the diagonal.
• This result is evident also from geometrical considerations.
Maximum Quadrilateral of Given Sides* 187
Hence ab cos 6 + cd Goa xL-r- = o
^ ^ d(l> -4
for a maximum or a minimum ; also,
^ . i#
ab sm 6 = ed sm \p-j- ;
•'. tan <;^ + tan rp = o, OT<j> + \p=i 80®.
Hence the quadrilateral is inscribable in a circle.
That the solution arrived at is a maximum is evident
from geometrical considerations; it can also he proved to be
so by aid of the preceding principles.
For, substitute —r—. — -, instead of -7^, and we get
cd sin \p dip °
du ab sin (^ + \li)
d(p sin ^ '
TT d*u ab cos (6 + \IA f d\L\ , , . ,
Hence -r^ = —-. — — i + -;^ ] + a term which
d(j>^ &ui\p \ d(l)J
vanishes when ^ + ^ = 180°; and the value of — ^ becomes
in this case
ab
(-s>
sin ip
which being negative, the solution is a maximum.
I S8 Examples,
Examples.
1. Prove that a sec 6 + 3 coseo 0 is a minimum when tan fl = I/*.
2. Find when 4a;' - 153:2 + 12a; - i is a maximum or a minimum.
Ans. af = 1^, a max. ; oj = 2, a min.
3. If a and * be such that /(a) = /(i), show that f{x) has, in general, a
maximum or a minimum value for some value of x between a and b.
4. Find the value of x which makes
sin X . cos X
cos2(6o» -ar)
•5- I* 7t\ — ttI ^e a maximum, show immediately that -^^is aminimum.
f{x) - <p{x) ^ (p{x)
6. Find the value of cos x when - is a maximum.
V 5 - 4 cos a;
Ans. cos a?:
7. Find when ■ is a maximum. „ as ss -•.,
\/4 + S*'* ^
8. Apply the method of Ex. 5 to the expression
5-a/i3
6
x^ - ax + b
9. "What are the values of x which make the expression
2x^ - 2ix- + 36a; - 20
a maximum or a minimum ? and (2) what are the maximum and minimum
values of the expression ? Ans. a: = i, a max. ; x= 6, & min.
*o. u = X^ui — xY. Ant. X = , a maximiu-
^ m-\- n
1 1. Given the angle C of a triangle ; prove that sin^^ + sin'^ is a maximum,
and cos*-4 + cos*^ a minimum, when A = B.
12. Find the least value of ««*■» + le'^'. Ana. z's/ab
{a + x){b + x) .—.
■* {a- x){b - :r) »i _ v
Examples, 1 89
14. Show that h -^ e {x - a)?, when ar = a, is a minimum or a maximum
according as e is positive or negative. ^
15. M = a; cos X. Ans. x = cot x.
16. Prove that x" is a maximum when x = e.
17. Tan^a; . tan" (a — a:) is a maximum when tan (a — ix) = tan a ?
n + m
18. Prove that ; is a minimum when x = e.
log a;
19. Given the vertical angle of a triangle and its area, find when its base is
a minimum.
20. Given one angle A of a. right-angled spherical triangle, find when the
difference betweeen the sides which contain it is a maximum.
do
Here tan ccoaA = tan b ; and since e — b is a. maximum, -77 = i.
do
Hence we find tan b = v cos A>
This question admits of another easy solution ; for, as in Art. 112, we have
Sin («J + i) 2
consequently sin (e — b) becomes a maximum along with sin {c + b), since A is
constant ; and hence c — b 13 & msiximum when c + b = gO°.
This problem occurs in Astronomy, in finding when the part of the equation
of time which arises from the obliquity of the ecliptic is a maximum.
21. Prove that the problem, to describe a circle with its centre on the
circumference of a given circle, so that the length of the arc intercepted within
the given circle shall be a maximum, is reducible to the solution of the equation
$ - cot e.
22. A perpendicular is let fall from the centre on a tangent to an ellipse,
find when ther intercept between the point of contact and the foot of the perpen-
dicular is a maximum. Prove that p = \/ ab, and intercept = a -b.
23. A semicircle is described on the axis-major of an ellipse ; draw a line from
one extremity of the axis so that the portion intercepted between the circle and
the ellipse shall be a maximum.
24. Draw two conjugate diameters of an ellipse, so that the sum of the
perpendiculars from their extremities on the axis-major shall be a maximum.
25. Through a point 0 on the produced diameter AB of a semicircle draw a
secant ORE', so that the quadrilateral ABRR' inscribed in the semicircle shall
be a maximum.
Prove that, in this case, the projection of RB^ on AB is equal in length to
the radius of the circle.
26. If ^m<\) = k sin »|/, and i^ + »|/ = a, where a and k are constants, prove
that cos 1^' cos ^ is a maximum when tan^^ = tan y^ tan ^'.
190 Examples,
27. Find the area of the ellipse
ax^ 4- 2hxy + by^ = e
in terms of the coefficients in its equation, by the method of Art. 146.
(l) for rectangular axes. Ans. ■.
s/ab - A2
,.-,,. vc sin Q)
, (2) for oblique. „ — ■-.
^/ab - A2
28. A triangle inscribed in a given circle has its base parallel to a given line,
and its vertex at a given point ; find an expression for the cosine of its vertical
angle when the area is a maximum.
29. FLnd when the base of a triangle is a minimum, being given the ver-
tical angle and the ratio of one side to the diflference between the other and a
fixed line.
30. Of all spherical triangles of equal area, that of the least perimeter is
equilateral ?
31. Let u^ -\- x^— "^axu = o; determine whether the value x = o gives u a
maximum or minimum. Ans. Neither.
32. Show that the maximum and minimum values of the cubic expression
ax^ + ibx^ ^-^cx^-d
are the roots of the quadratic
aV _ 2Gz\ A = 0;
where G = aH- labc + ih^, and A = cC-cP- + d^a^ + 6^db^ - ib\^ - dabed.
33. Through a fixed point within a given angle draw a line so that the
triangle formed shall be a minimum.
The line is bisected in the given point.
34. Prove in general that the chord drawn through a given point so as to
cut off the minimum area from a given curve is bisected at that point.
35. If the portion, AB^ of the tangent to a given curve intercepted by two
fixed lines OA, OB, be a minimum, prove that FA = NB, where F is the point
of contact of the tangent, and N the foot of the perpendicular let fall on the
tangent from 0.
36. The portion of the tangent to an ellipse intercepted between the axes is
a minimum : find its length. Ans. a + b.
37. Prove that the maximum and minimum values of the expression, Art. 147,
are roots of the biquadratic
{a- my {d - ud'f + 4 (a - ud) {c - ue'^ + 4 (^ - ud') {b - ub'f
- 3 (6 - iib'Y {c — uc'Y — 6 (a — ud) {b — ub') {c — uc) (d ~ ud') = o.
( 191 )
CHAPTER X.
MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE IN-
DEPENDENT VARIABLES.
155. Maxima and Minima for Two Tariables. — In
accordance with the principles established in the preceding
chapter, if 0 {cc, y) be a maximum for the particular values
Xq and 2/0, of the independent variables x and y, then for all
small positive or negative values of h and ^, 0 (jto, y^ must
be greater than ^ {^Cq -\-h,yQ-\- k) ; and for a minimum it must
be less.
Again, since x and y are independent, wo may suppose
either of them to vary, the other remaining constant;
accordingly, as in Art. 138, it is necessary for a maximum
or minimum value that
du . du
- = o, and- = o; (,)
omitting the case where either of these functions becomes
infinite.
156. liagrange's Condition. — "We now proceed to
consider whether the values found by this process correspond
to real maxima or minima, or not.
Suppose aJo, yo to be values of x and y which satisfy the
equations
du J du
^ = 0, and-=o,
and let A. B. C be the values which -r^, -r—r, -t-„ assume
dx^ dxdy dy^
when Xq and 2^0 are substituted for x and y ; then we shall
have
i>{xQ+h,y^^li) - (p{xo,yo)= {Ah''+ 2Bhk+ Ck'') +&c. (2)
1 92 Max. and Min.for two or more Independent Variables,
But when h and k are very small, the remainder of the
expansion becomes in general very small in comparison with
the quantity Ah^ + 2Bhk + 0^^ . accordingly the sign of
<P{xq + h, 7/0 + k) - <p{xo, po) depends on that of
AT,2 7,1.7 ni^ ' s^ Uh + Bkf + ¥{AC-B')
Ah^ + zBhk + OF, i.e. of -^ '—2 — ^^ •
.jOL
Now, in order that this expression should be either always
positive or always negative for all small values of h and k,
it is necessary that AC - B^- should not be negative ; as, if
it be negative, the numerator in the preceding expression
would be positive when A; =0, and negative when Ah + Bk = o.
Hence, the condition for a real maximum or minimum is
that AC should not be less than B"^, or
d^di^ ^^^\dx(fy)'
and, when this condition is satisfied, the solution is a maxi-
mum or a minimum value of the function according as the
sign of A is negative or positive.
If j5^ be > -2 (7 the solution is neither a maximum nor a
minimum.
The necessity of the preceding condition was first estab-
lished by Lagrange ;* by whom also the corresponding con-
ditions in the case of a function of any number of variables
were first discussed.
Again, ii A = o, B = o^ C = o, then for a real maximum
or minimum it is necessary that all the terms of the third
degree in h and k in expansion (2) should vanish at the same
time, while the quantity of the fourth degree in h and k
should preserve the same sign for all values of these quan-
tities. See Art. 138.
The spirit of the method, as well as the processes em-
ployed in its application, will be illustrated by the following
examples.
157. To find the position of the point the sum of the
squares of whose distances from n given points situated in
the same plane shall be a minimum.
' Thiorie des Fonctions. Deuxieme Partie. Ch. onzieme.
Maxima and Minima for Two or more Variahles. 193
Let the co-ordinates of the given points referred to
rectangular axes be
(«i, Ji), («2, ^2), (^3, h) . , , {an, bn)y respectively;
(x, y) those of the point required ; then we have
w = (ic - «i)* + (s^ - hiY + (a? - a.^"" + {y - hf + . . .
+ (^ - anY +(y- bnY
a minimum ;
du f .
dx
du
-T-=y-bi^y-h\-...-^y-hn=ny-[hi+h+,..-^bn)^o.
-_r «i 4- «2 f . . . + «„ Si + Ja + . . . + fi«
Hence a; , y = ;
and the point required is the centre of mean position of the
n given points.
From the nature of the problem it is evident that this
result corresponds to a minimum.
This can also be established by aid of Lagrange's con-
dition, for we have
_ ^^^ _ D _ ^^^^ _ rr _ ^^^ _
~ dx^~ * " dxdy ~ ' ~ dy'^~ '
In this case AC - B^ \s> positive, and A also positive;
and accordingly the result is a minimum.
158. To find the maximum or Minimum ITalue
of tlie expression
ax^ + hf + iJixy + 2gx + 2fy + c,
* Denoting the expression by u, we have
\ du ^ ,
I ^^ 7,7 ^
2dy ^ ^
194 Maxima and Minima for Two or more Variables.
Multiplying the first equation by x, the second by y, and
subtracting their sum from the given expression, we get
u = gx^-fy + c\
whence, eliminating x and y between the three equations,
we obtain
a h g
u{ah-h')^ h h f . (3)
9 f <i
This result may also be written in the form
d^ ^
where A denotes the discriminant of the proposed expression.
. . d'^u d'^u . d^u _
Asram, -—; = 2a. -r^ = 20. -z—r = 2h.
^ ' dx' ' dy"" ' dxdy
Hence, if ah - h^ be positive, the foregoing value of t* is a
maximum or a minimum according as the sign of a is negative
or positive.
If h^ > ah, the solution is neither a maximum nor a
minimum.
The geometrical interpretation of the preceding result is
evident ; viz., if the co-ordinates of the centre be substituted
for X and y in the equation of a conic, w =? o, the resulting
value of u is either a maximum or a minimum if the curve
be an ellipse, but is neither a maximum nor a minimum for
a hyperbola ; as is also evident from other considerations.
159. To find the maxima and !9Iininia Yalues
of the Fraction
aa^ + hy'^ + ihxy + igx + zfy + 0
«V+ b'y^+ 2h'xy+ 2g'x+2f^y+(f'
Let the numerator and denominator be represented by
^1 and 02 ; then, denoting the fraction by w, we get
01 = W02. (a)
Examples for Tico Variables, 195
Differentiate with respect to x and y separately, then
d(^i du
dx dx dx '
2 d(px du "* G?02,
dy dy dy * V
but for a maximum or a minimum we must have
du
du
dy
= o
hence, the required solutions are given by the equations
ax -{■ hy -v g = u(a'x + Ky + /),
hx -:- hy +/ = u{Kx + Vy +/).
Multiplying the former by x^ the latter by y^ and subtracting
the sum from the equation (a), we get
These equations may be written
(a - a'u)x + {h - h'u)y + ^ - /w = o,
{h - h\()x + {b- b'u)y +f -fu = o,
{g - ^u)x + {f-fu)y + c - c'^f = o.
Eliminating x and y, we get the determinant
a - du h - h'u g - g^u
h-Ku b~ b'u f -fu = o. (4)
9 -^'i^ f-fu c -du
The roots of this cubic equation in u are the maxima and
minima required.
This cubic is the same as that which gives the three
systems of right lines that pass through the points of
intersection of the conies ^1 = 0,^2= o.*
* Salmon's Conic Sections, Art. 370.
02
196 Maxima and Minima for Two or more Variables.
The cubic is written by Dr. Salmon in the form
AV + 0 V + 0«^ + A = o, (5)
where A, A' denote the discriminants of the expressions 0i and
02, and 0, 0' are their two other invariants.
On the proof of the property that the coefficients are in-
variants compare Art. 144.
The cubic reduces to a quadratic if either the numerator
or the denominator be resolvable into linear factors ; for in
this case either A = o, or A' = o.
If both the numerator and denominator be resolvable into
factors, the cubic reduces to the linear equation
0'w + 0 = O,
and has but one solution, as is evident also geometrically.
160. To find the Maxima or Minima lvalues of
^^ + y' + z^i where
oa;* + J/ + cs* + ihxy H- 2gxz + 2f%y = i.
X XI
Let w = a:* + y* + «* ; substitute x' and y' for - and -, and
z z
we have
aJ^ + j^'* + I
(6)
aa?'* + ly"^ + c + 2hx']f + 2gx + 2//*
Accordingly the cubic of formula (4) becomes in this case
a-u~^ h g
h b'vr^ f
g f c-ir'
This is the well-known cubic* for determining the axes of
a surface of the second degree in terms of the coefficients in
its equation : when expanded it becomes
u-^-{a + b + c)ur^ + {ab + be + ac -p - g" - h^)w^
+ («/' + bg^ + cA' - abc - 2fgh) = o.
♦ See Salmon's Geometry of Three Dimensions, 3rd ed., Art. 82.
Application of Lagrange's Theorem. 197
161. Application of Itagrange's Condition. — In
applying this condition to the general "case of Art. 159, we
write the equation in the form
01 = W02>
from which we get, on making — = o, and — = o,
dxdy dxdy dxdy^
c?'0i _ c?202 d:'u
Hence
^'|S?-(S^)l = 4K«-«'«X*-*'«)-(^-^'«)')-
Accordingly, the sign of AC - JB^ is the same as that of
the quadratic expression
{ab - hT) - {ah' + h(/ - zhK) u + (dV - K^)u\ (7)
where w is a root of the cubic (4) or (5).
If A2 represent the determinant in (4), the preceding
quadratic expression may he written in the form —^,
ac
Again, Wi, u^y Uz representing the roots of the cubic (4) ;
a, /3, those of the quadratic (7) ; if Ui be a real maximum or
minimum value of u, we must have {ui - a){ui - j3)(a'6'- h'^)
a positive quantity.
Accordingly, if a'l> - K"^ be positive, Ux must not lie be-
tween the values a and /3. Similarly for the other roots.
198 Maxima and Minima for Tivo or more Variables.
If all the roots of the cubic lie outside the limits a and j3,
they correspond to real maxima or minima, but any root
which lies between a and ]3 gives no maximum or minimum.
In the particular case discussed in Art. 1 60 the roots of
the cubic (6) are all real, and those of the quadratic
a - ir\ h
= o are interposed between the roots of the
h, h-u-'-
cubic. (See Salmon's Higher Algebra^ Art. 44). Accord-
ingly, in this case the two extreme roots furnish real maxima
and mininia solutions, while the intermediate root gives
neither. This agrees with what might have been anticipated
from the properties of the Ellipsoid ; viz., the axes a and c
are real maximum and minimum distances from the centre to
the surface, while the mean axis h is neither.
It would be unsuited to the elementary nature of this
treatise to enter into further details on the subject here.
162. maxima or Minima of Functions of three
Tariables. — Next, let u = <^{x, y, z), and suppose iTo, yo, 2o
to be values of Xj y, z, which render u a maximum or a mini-
mum ; then, if sr, y, z be independent of each other, by the
same reasoning as before, it is obvious that a?o, yo , 2© must
satisfy the three equations
du du du
dx ^ dy ' dz
omitting the case of infinite values.
Accordingly we must have
(p{xo + hf yo + kyZo + l) - (pixc, yoi Zo) ^^—^ "^""^ i~2 "^ ^~2
+ Fkl + Ghl + Ehk + &c.
where A, B, (7, F, G, H, are the values that ^
d^u d^u d^u d^u d^u d^u
dx^' df' cfe^' dydz' dxdz' dxdy
respectively assume when iTo, yo> 2o are substituted for x^ y, 2
in them.
Maxima or Minima for Three Variables. 199
Now, in this, as in the case of two independent variables,
it is necessary for a real maximum or ^jninimum value that
the preceding quadratic function should be either always
positive or always negative for all small real values of hy k,
and /.
Substituting al for A, and /3/ for k, and suppressing the
positive factor l'\ the expression becomes
Aa' + JBj^'+0+ 2F[5 + 2Ga + 2lIaP, (8)
A[a' + 2a (^±^)1 + Bi3' + 2F13 + a
or
Completing the square in the first term, and multiplying by
A, we get
{Aa + ^/3 + Gy+ {AB-S')^' + 2{AF- QE)^ + [AC- G').
Moreover, since the first term is a perfect square, in order
that the expression should preserve the same sign, it is neces-
sary that the quadratic
{AB - JT^)j3^ + 2{AF- CH)^ + AO-G'
should be positive for all values of j3 : hence we must have
ab-e:'> o, (9)
and {AB - E'){AC - G') > {AF- GH)\
or A{ABC + 2FGE- AF' - BG' - OR') > o, (10)
i.e. A and A must have the same sign, A denoting the diS'
criminant of the quadratic expression (8), as before.
Accordingly, the conditions (9) and (10) are necessary
that Xo, ^0, Zo should correspond to a real maximum or mini-
mum value of the function u.
When these conditions are fulfilled, if the sign of A be
positive, the function in (8) is also positive, and the solution
is a minimum ; if A be negative, the solution is a maximum.
163. 9Iaxinia and Minima for any number of
¥ariables. — The preceding theory admits of easy extension
200 Maxima and Minima for Two or more Variables.
to functions of any number of independent variables. The
values which give maxima and minima in that case are got
by equating to zero the partial derived functions for each
variable separately, and the quadratic function in the ex-
pansion must preserve the same sign for all values ; i.e. it
mnst be equivalent to a number of squares, multiplied by
constant coefficients, having each the same sign.
The number of independent conditions to be fulfilled in the
case of n independent variables is simply w - i , and not 2" - i ,
as stated by some writers on the Differential Calculus. A
simple and general investigation of these conditions will be
given in a note at the end of the Book.
164. To investigate the 9Iaxiiuuiii or minimaiii
Yalue of tbe Expression
am? + hy^ + cz^ + ihxy + zgzx + ifyz + ipx + zqy + irz + d.
Let u denote the function in question, then for its maxi-
mum or minimum value we have
du
dx
du
dy
du
= 2(ax + hy + gz +^) = o,
= 2{hx + hy +/« + j) « o,
= 2{gx+fy ^cz^ r) «o;
hence, adopting the method of Art. 158, we get
Eliminating x, y, z between these four equations, we obtain
a h g p
h h f q
g f 0 r
p q r d
= u
a h g
h h f
g f c
. . . d'^u
Again, smce —
2a,
d^
df
2h, &e.
Maxima or Minima for two or more Variables. 201
the result is neither a maximum nor a minimum unless
h h
is positive, and
a
h
9
h
b
f
g
f
c
has the same sign as a.
The student who is acquainted with the theory of surfaces
of the second degree will find no difficulty in giving the
geometrical interpretation of the preceding result.
165. To find a point such that the sum of the
squares of its distances from n given points shall he
a ^Minimum. — Let {a, b, c), (a\ b\ c), &c., be the co-ordi™
nates of the given points referred to rectangular axes ; Xy y^ a,
the co-ordinates of the required point ; then
(x - aY + (y - hY + (z - cY
is equal to the square of the distance between the points
(a, b, c), and (x, y, 2).
Hence
w= (a^-a)* + (y-5)»+ (z-c)«+ (ar-a'/+ (y-5')' + (2-^r
+ &c. = 2(aJ - aY + S(y - b^ + S(2 - c)*,
where the summation is extended to each of the n points.
For the maximum or minimum value, we have
2S(aj - a) = 2nx - 2S« = o,
du
dx
— = 2S(y - J) = 2ny - 2S5 = o,
dy
du ^ . „
T- = 22 s - c) = 2n% - 2^c = o ;
dz '
^a 26 2c
a?o = — > yo= — ) 2o = —
n ^ n n
i.e. a^o, yo> 2o are the co-ordinates of the centre of mean posi-
202 Maxima and Minima of Independent Variables.
tiou of the given points. This is an extension of the result
established in Art. 157.
. . d^u d^u d^u d^u _
Agam ^ = ^«. Of = '"' ^ = -'"' S^ = °' *°-
The expressions (10) and (11) are both positive in this case,
and hence the solution is a minimum.
It may be observed with reference to examples of maxima
and minima, that in most cases the circumstances of the prob-
lem indicate whether the solution is a maximum, a minimum,
or neither, and accordingly enable us to dispense with the
labour of investigating Lagrange's conditions-
Examples. 203
Examples. ^
Find the maximum and minimum values, if any such exist, of
ax + by + c . c + \/ c^ + i' + c^
C. -:; 5 . Am. -= -.
«- + y2 + I X
ax + bt/ + 0
(a), a; = o, y = o, a maximum.
(i8). a; = y = + -, a minimum.
(7). a; = - f/ = + - — , a mmimum.
2
4. ax^ + bxy + dz^ + ?a;z + myz.
X = y = z = o, neither a maximum nor a minimum.
5. If « = ax^y^ - X^y* - a^y^j prove that a; = -, y = - makes u a maximum.
2 3
6. Prove that the value of the minimum found in Art. 165 is the -th part of
n
the sum of the squares of the mutual distances between the n points, taken two
and two.
7. Find the maximum value of
(axi-by + cz)e . ^«,. ^- ^- + - + -|,
8. Find the values of x and y for which the expression
{aix + b\y + CiY + [a^x + ^2^ + cj)' + . . . + {c^x + i»y + <s,}'
b'^comes a minimum.
( 204 )
CHAPTER XI.
METHOD OF UNDETERMINED MULTIPLIERS APPLIED TO THE
INVESTIGATION OF MAXIMA AND MINIMA IN IMPLIJIT
FUNCTIONS.
1 66. Jflethod of Cndetermined Multipliers. — In many
cases of maxima and minima the variables which enter into
the function are not independent of one another, but are con-
nected by certain equations of condition.
The most convenient process to adopt in such cases is
what is styled the method of undetermined* multipliers. We
shall illustrate this process by considering the case of a func-
tion of four variables which are connected by two equations
of condition.
Thus, let w = 0(a?„ iPaj ^8, a^i),
where iCi, Xz, Xz, Xi are connected by the equations
Fi{xi^ Xiy Xs, Xi) = o, F^ixi, x^f X3, xi) =0. (i)
The condition for ajmaximum or a minimum value of w
evidently requires the equation
dd) ^ dd) . dd) . d(b .
-T- dxi + -^ dxz -\--^ dxz +— dXi - o.
dxi dxt dx% axi
Moreover, the differentials are also connected by the rela-
tions
dF, , dF, . dF, . dF, . ^
-J— dxi + -:r- dx2 + -r- dx3+^- dxi « o,
dxi dxz dxz dXi
dFz ^ dFz ^ dFz ^ dF^ ^
-7- dxi-^-r- dxz + -7— dXi + -7— dxt, = o.
dxi dxi dxz dXi
Multiplying the first of the two latter equations by the arbitrary
* This mctliod is also due to Lagrange. See Mec. Anal., tome i., p. .74.
Method of Undetermined Multipliers. 205
quantity Xi, the other by X2, and adding their sum to the pre-
ceding equation, we get ^
c?0 , dF, . dF,\. fdcf> . dF,. dFA^
dxi dxi dXiJ \dXi aXi ax%j
fd<b ^ dF, ^ dF,\^ fdci> . dF, . dF,\.
\dxz dxz dxzj \dXi ax^ dx^j
As Ai, A2 are completely at our disposal, we may suppose
them determined so as to make the coefficients of dx^ and dxi
vanish. Then we shall have
di^ . dFx . dF2\, fd((> . dF, . dF,\ ,
dx^ dxz dxzJ \dxi dx^ dxj
Again, since we may regard X3, x^ as independent variables,
and Xi, X2 as dependent on them in consequence of the equa-
tions (i), it follows that the coefficients of dxs and dXi in the
last equation must be separately zero, for a maximum or a
minimum ; consequently, we must have
d(b ^ dFi . dF2
-^ + Ai -T- + A2 ;^ = o,
dXi dXi dxz
d6 . dFi . dFi
-^ + Ai ;7- + X2 -T- = o.
dXi dXi, dXi
These, along with equations (i) and
d(b . dFi . dF2
~- + Ai -7- + A2 -r- = o,
dxi dxi dxx
d6 . dFi . dFi
-^ + Ai -7- + A2 -^ = o,
dx2 dx2 dx2
are theoretically sufficient to determine the six unknown
quantities, i^i, iC2, a?3, x^, Ai, Aj ; and thus to furnish a solution
of the problem in general.
This method is especially applicable when the functions
J^i, J^2, &c., are homogeneous ; for if we multiply the preceding
2o6 Method of Undetermined Multipliers.
differential equations by fl?i, x^, Xz, x^, respectively, and add,
we can often find the result with facility by aid of Euler's
Theorem of Art. 103.
There is no difficulty in extending the method of undeter-
mined multipliers to a function of n variables, Xi, x^, x^, . . .
Xn, the variables being connected by m equations of condition.
i^i = O, i^2 = O, i^3 = O, . . . Fm = O,
m being less than n ; for if we differentiate as before, and
multiply the differentials of the equations of condition by the
arbitrary multipliers, Ai, X2, . . . X;„ respectively; by the same
method of reasoning as that given above, we shall have the n
following equations,
d(f> dF^ dFm
-z h Ai — \- . . , + Am —} —
axi dxi dxi
. dF, . dFm
+ Ai -J- + . . . + A,» — — = O,
dXi dXi dx',
dXn dXn dXn
These, combined with the m equations of condition, are
theoretically sufficient for the determination of the m -i- n
unknown quantities
a?i, Xzf . . , Xn, Ai, A2, . . . A»|.
Examples.
I. To find the triangle of maximum area inscribed in a given circle.
Let R denote the radius of the circle, A, £, C, the angles of an inscribed
triangle, u its area ; then
u = -— = zH^ sin A sin B sin C.
Also, A + £ +0=1^0°; .\ dA + d£ +dC^.Oi
and, taking logarithmic differentials, we get
cot AdA + cot BdB + cot CdC = o,
Examples. 207
and consequently
tan ^ = tan i? = tan C; hence A = JB =0 = 6d* ;
and therefore the triangle is equilateral.
2. Find a point such that the sum of the squares of the perpendiculars
drawn from it to the sides of a given triaogle shall be a minimum.
Let a, y, z denote the perpendiculars : a, J, c the sides of the triangle ; then
« = a;* + 3/2 4. z3 is to be a minimum ;
olso ax ■\- by + cz — double the area of a triangle = 2A (suppose) ;
. '. xdx + ydy 4- zdz =■ o, adx + bdy + cdz — o ,
.'. a; = \a, y =:\b, z = \c : multiplying these equations by a, b, c, respectively,
and adding, we obtain
2A
ax + by + cz:=\ (a* + b^ + c^), or A. =
X —
a3 + ^3 + c*'
■z£ia 2Ab 2A0
,2 + ^2 + c2» y a3 + ^2 + ^2' ^i + J2 + c«'
which determine the position of the point. The minimum sum is obviously
fl2 + i2 + c2
3. Similarly, to find a point such that the sum of the squares of its distances
from four given planes shall be a minimum. Suppose A, B, C, D to represent
the areas of the faces of the tetrahedron formed by the four planes ; x, y, z, w,
the perpendiculars on these faces respectively; then, as in the preceding
example, we have
Ax-v By\Cz^ Bw = three times the volume of the tetrahedron = 3 F (suppose),
and M = a;2 + y'-* + 2^ + w^^ a minimum;
.*. xdx + ydy + zdz + xcdw = o,
Adx + Bdy + Cdz + Bdw = o ;
hence X = \A, y = ?J3, z = \C, w = \D;
and proceeding as before, we get m= .. ^ ^^ ^3.
4. To prove that of all rectangular paralldepipeils of the same volume the cube
has the least sm'face.
Let X, y, 2 represent the lengths of the edges of the parallelepiped ; then^ if
A denote the given volume, we have
xyz = A, and xy + xz + yz a minimum ;
.'. ysdx + xzdy + xydz = o,
(1/ + z) dx + {x + z) dy + {x + y) dx=o;
hence yz = \{y + z), xz = \{x-\- z), xy = \ {x + y) :
from which it appears immediately that x — y = z.
208
Method of Undetermined Multipliers.
167. To find the Maximum and Jflinimum
¥alues of
ax^ + hy^ + cz^ + zhxy + igzx + 2fyz,
where the variables are connected by the equations
Lx + My + iVs = o, and x^ + y"^ + z^ = i.
In this case we get the following equations :
ax -\-hy + gz + \iL + Asiz; = o,
hx + hy +/s + \xM + Xzy = o,
gx +fy + cs + XiiV + A22 = o.
Multiply the first by x, the second by y, the third by z, and
add; then
u + A2 = o, or Xi
u.
Hence
[a- u) X + hy + gz + XiL = o,
hx + {b -u) y +fz + XiM = o,
gx +fy + (c-u) z + XiN = o,
Lx + My + Nz = o :
eliminating x, y, z and Ai, we get the determinant equation
a — u, h, g, L
h, h - Uy /, M
g, /, c-u, N
L, M, jsr, o
= o.
(2)
The roots of this quadratic determine the maximum and
minimum values of u.
The preceding result enables us to determine the principal
radii of curvature at a given point on a surface whose equa-
tion is given in rectangular co-ordinates.
Application to Surfaces.
209
Again, the term independent of u in this determinant is
evidently
a,
h.
g.
L
h,
b,
f,
M
9,
f,
e,
N
L,
M,
N.
0
and the coefficient of u^ is L' + M^ + N^, Accordingly, the
product of the roots of the quadratic (2) is equal to the frac-
tion whose numerator is the latter determinant, and denomi-
nator L^ + M^ + iV\ From this can he immediately deduced
an expression for the measure of curvature* at any point on a
surface.
• Salmon's Geometry of Three Dimensions. Art. 2 95.
2IO Examples,
EXAMPLEB.
1. Find the minimum value of
^1 + ^J + ^^ + • • • + ^'^
where x\, Xiy . . . Xn are subject to the condition
a\x\ + a^2 f . . . + Uf^n = k, Ans.
2. Find the maximum value of
xP y9 2^,
where the variables are subject to the condition
ax ■{■ by -^^ cz = p -^ g + r.
a; + a;...+<
Am.
mrc^)
6 <b
3. If tan - tan - = m, find when p^"r P - ♦»? sin * is a maximum.
4. Find the maximum value of (« + i) (y + i) (z + i) where a' bv c^ = A.
{log {Aabc)}^
Ans.
27 log a . log b . log c
5. Find the volume of the greatest rectangular parallelepiped inscribed iu
the ellipsoid whose equation is
^2 y9 gi
Ans.
6. Find the maximum or the minimum values of «, being given that
« = a2a;2 + ^2^2 + c^z-, a;2 ^ y2 ^ 2? = I, and Ix + my + nz = o.
Proceeding by the method of Art. 167, we get
a^x + \X + fil= O, b^y + Ay + fini = o. c-z -^ \z -t /xn ~ o.
Again, multiplying by x, y, s, respectively, and adding, we get A. = - u.
.'. (m - a^) x = fxly (u - b^) y = fim, (u - c^) z = /*».
Hence, the required values of u are the roots of the quadratic
l^ m* «2
+ f =t o.
« - a' u - If^ « - a
Examples. 2 1 1
7. Giren -5 + i^ + -^ =* i, and Ix + fny + «» = o, find when «* + y' + z* is a
a* 0* c* .4
maximum or minimum. Proceeding, as in the last example, we get the quadratic
aH'^ h^m^ c^n^
+ r:; + -. = o.
w - «« u - b^ u - c^
This question can be at once reduced to the last by substituting in our equations
rta:, Jy, and («, instead of a?, y, «.
8. Of all triangular pyramids having a given triangle for base, and a given
altitude above that base, find that whose surface is least.
Ans. Value of minimum surface is \/ r^ + p^^ where a, 5, c repre-
2
sent the sides of the triangular base ; >*, the radius of its inscribed circle ; and p^
the given altitude.
9. Divide the quadrant of a circle into three parts, such that the sum of the
products of the sines of every two shall be a maximum or a minimum ; and
determine which it is.
10. Of all polygons of a given number of sides circumscribed to a circle, the
regular polygon is of minimum area ? For, let ^i, ^2, • . . ^n be the external
angles of the polygon, then the area can be easily seen to be in general
r« (tan ^ + tan ^ + . . . + tan — J ,
where ^1 + ^2 . . . + 0n = 2t.
Hence, for a minimum, ^1 = ^2 = ^s = . . . = ^».
11. Of all polygons of a given number of sides circumscribed to any closed
oval cujTve which has no singular points, that which has the minimum area
touches the curve at the middle point of each of the sides.
12. Given the ratio sin ^ : sin i|/, and the angle 0, find when the ratio
sin (0 + 0) : sin (^^ + &) is a maximum or a minimum. Ans. <p + rp — 0.
1 3. Required the dimensions of an open cylindrical vessel of given capacity,
so that the smallest possible quantity of material shall be employed in its con-
struction, the thickness of the base and sides being given.
Ans. Its altitude must be equal to the radius of its base.
14. Show how to determine the maximum and minimum values of flp'+y*+«'
subject to the conditions
te + wy + «z = o.
P 2
{ 212 )
CHAPTER XII.
TANGENTS AND NORMALS TO CURVES.
1 68. Equation or the Tangent. — If (a?, y), (a?i, ^/i), he the
co-ordinates of any two points, P, Q, taken on a curve, and
if (X, Y) be any point on the y
line which joins P and Q ; then
the equation of the line PQ is
Y-y = {X-iv)
yi-y
^ M
in which X and Y represent the ^
current co-ordinates. ^^s- ^•
If now the point Q be taken infinitely near to P, the line
PQ becomes the tangent at the point P, and, as in Art. lo,
we have for its equation
y
(I)
where X, Y are the co-ordinates of any point on the line,
and ic, y those of its point of contact.
For example, to find the equation of the tangent to the
curve
x^y^ = a^^*^.
Taking the logarithmic differentials of both sides, we get
ny
n mdy
X y dx
&y_
dx
mx
and the equation of the tangent becomes
nX mY
X y
m + n.
Tangents Parallel to a Given Line, 2 1 3
If we make X* o, and F= o, separately, we get -^ y
fn 4" n
and X for the lengths of the intercepts made by the
tangent on the axes of x and y, respectively. This result
furnishes an easy geometrical method of drawing the tangent
at any point on a curve of this class.
li m = I, n = I, the preceding equation represents a
hj^erbola ; ii m = 2, and n = - i, it represents a parabola.
169. If the equation of the curve be of the form
/{x, y) = o, and iS/(x, y) be denoted by w, we have from
Art. 100,
du
dx du^
dy
and hence the equation of the tangent becomes •
.— .du ,-. .du
The points on the curve at which the tangents are
du
parallel to the axis of x must satisfy the equation -— = o ;
they are accordingly given by the intersection of the curve,
w = o, with the curve whose equation is -7- = o. The y co-
ax
ordinates at such points are evidently in general either
maxima or minima.
Similar remarks apply to the points at which the tangents
are parallel to the axis of y.
To find the tangents parallel to the line y = mx + n. The
points of contact must evidently satisfy
du du
-I- + m— =0.
dx dy
The points of intersection of the curve represented by
214 Tangents and Normals to Curves,
this equation with the given curve are the points of contact
of the system of parallel tangents in question.
The results in this and the preceding Article evidently
apply to oblique as well as to rectangular axes.
Examples.
1. To find the equation of the tangent to the ellipse
du _2x du _^ 2y
and the required equation is
2. Find the equation of the tangent at any point on the curve
— + J- = 1. Ans. + -f — = I.
3. If two curves, whose equations are denoted by w = o, m' = o, intersect in
a point (a?, y), and if « be their angle of intersection, prove that
du dt/ di4f du
. dx dif dx dy
tan w =
du dvc du du
dx dx dy dy
4. Hence, if the ciures intersect at right angles, we must have
du du' du du'
dx dx dy dy "
5. Apply this to find the condition that the curves
a;2 yi ^2 yi
^ + i» = '» ^+i^'°'
should intersect at right angles. Ana. a* - Ja = a'* - h'*.
Equation of Normal,
215
1 70. Equation of ^^orinal. — Since the normal at any
point on a curve is perpendicular to the tangent, its equation,
when the co-ordinate axes are rectangular, is
or
(3)
The points at which normals are parallel to the line
y = mx + n are given by aid of the equation of the curve u = o
along with the equation
du
dy
-7- = m
d'i&
d»
Examples.
I. Find the equation of the normal at any point (ar, y) on the ellipse
Ans.
X y
= 0;^-^,
3. Find the equation of the normal at any point on the curve
yf^ = ax». Ans. nYy + mXx -ny^ -i- mx^.
171. ^ubtangent and l§lubiiornial. — In the accom-
panying figure, let PT repre- y
sent the tangent at the point P,
PiV the normal; OM, Pilf the
co-ordinates at P ; then the
lines TM and MlSf are called
the subtangent and subnormal 0'
corresponding to the point P. Fig. 9.
To find the expressions for their lengths, let ^ = Z FT My
then
PM
TM
MN , dy
TM=^,
dy'
MN^y
dx
dy
dx
2i6 Tangents and Normals to Curves.
The lengths of FT and PN are sometimes called the
lengths of the tangent and the normal at P : it is easily
seen that
-4^&
dp
dx
Examples.
1. To find the length of the subnormal in the ellipse
a2 + ^2 " '•
Here y— = - — x\
dx a^
the negative sign signifies that MN is measured from M in the negative
direction along the axis of x, i.e. the point N lies between M and the centre 0 ;
as is also evident from the shape of the curve.
2. Prove that the subtangent in the logarithmic curve, y = «*, is of constant
length.
3. Prove that the subnormal in the parabola, y^ = imx, is equal to m.
4. Find the length of the part of the normal to the catenary
. X X.
^3
intercepted by the axis of a?. Am. — .
5. Find at what point the subtangent to the curve whose equation is
ay2 = a^ (a - x)
IS a maximum. Ans. a? = -, y = a.
172. Perpendicular on Tangent. — Let p be the length
of the perpendicular from the origin on the tangent at any
point on the curve
F{x, y) = c,
Length of Perpe^idkular on Tangent, 217
then the equation of the tangent may be written
X cos a> + J' sin w = p^
where w is the angle which the perpendicular makes with
the axis of x.
Denoting F (x, y) by w, and comparing this form of the
equation with that in (2), and representing the common value
of the fraction by X,
we get
Hence
du du du du
dx dy dx dy
cos a> sin a> 'O
(du\^ fdu\
du du
J dx ^ dy / V
and 'P = ,, -^ — • (4)
^du^"^
m*
CoR. If F{x^ y) be a homogeneous expression of the n*^
degree in x and 2/, then by Euler's formula, Art. 102, we have
du du
*&.+^^ = »"=""'
and the expression for the length of the perpendicular
becomes in this case
nc
UduV fduV
173.
In the
curve
to
prove that
m
p""-^ ^ (a cos w)"*-^ + {h sin a.)'""*. (5)
2iS Tangents and Normals to Curves.
By Ex. 2, Art. 169, the equation of the tangent is
= I ;
comparing this with the form
X cos w + ]r sin (0 = j»,
we get
or
cos w _ a;'^^ sin a> _ y*"~^
la cos <.>N"^'_ OS fb sin w\'^'_ y
XT') "«' \^^j °V
Hence, substituting in the equation of the curve, we obtain
the result required.
174. liocns of Foot of Perpendicular for the same
Curve. — Let X, Fbe the co-ordinates of the point in ques-
X T
tion, and we have, evidently, cos w «= — , sin a> = — : substi-
tuting these values for cos w and sin a> in (5), it becomes
(X^ + F^)*^^ = (aX)"*-^ 4 {h Y)"'-\
since j9* = X^ + T^.
175. Another Form of the Fquation to a Tan-
gent.— If the equation of a curve of the n^^ degree be
written in the form
<p{x, y) =Un + W„-i + W„.a + . . . f tt, + Wi + Wo = O,
where Un denotes the homogeneous part of the n^^ degree in
the equation, w„_i that of the {n - i)'^, &o.; then, by Cor.
Art. 103, we have
x-r + y-r- - ^ f«„.; -f 2Un^ + &0. . . + nuo\.
ax ay
Number of Tangents from an External Point, 219
Hence the equation of the tangent in Art. 169 becomes
X-r + F -i^ + w^i + 2Wn-2 + . . . + wwo = o ; (6)
ax ay
an equation of the (w - i)^'^ degree in x and y.
176. Xumber of Tangents from an External
Point. — To find the number of tangents which can be
drawn to a curve of the n*^ degree from a point (a, j3), we sub-
stitute a for X, and /3 for Y in (6) , and it becomes
a^+ j3^ + Ur^i + 2W,*_2 + . . . + wwo = o. (7)
This represents a curve of the {n - lY^ degree in x and y,
and the points of its intersection with the given curve are the
points of contact of all the tangents which can be drawn
from the point (a, j3) to the curve. Moreover, as two curves
of the degrees n and n - i intersect in general inn {n - i)
points, real or imaginary (Salmon's Conic Sections, Art. 238),
it follows that there can in general be n{n - i ) real or
imaginary tangents drawn from an external point to a curve
of the n^^ degree.
If the curve be of the second degree, equation (7) be-
comes
a J- + j3 ^ + Wi + 2Wo = O,
an equation of the first degree, which evidently represents
the polar of (a, /3) with respect to the conic.
In the curve of the third degree
U3 + Ui + Ui + Uo = o,
equation (7) becomes
d6 ^ d(j>
a— + ^-^ + U2 + 2Ui + sUo = Oy
ax ay
which represents a conic that passes through the points of
contact of the tangents to the curve from the point (a, /3).
This conic is called the polar conic of the point. For the
origin it becomes
Ui + 2Ui + 3We = o.
220 Tangents and Normals to Curves,
177. KTamber of UTormals which pass through a
Ctiven Point. — If a normal pass through the point (a, j3),
we must have from (3),
.du du
This represents a curve of the n^^ degree, which intersects the
given curve in general in n^ points, real or imaginary, the
normals at which all pass through the point (a, j3).
For example, the points on the ellipse
at which the normals pass through a given point (a, j3),
are determined by the intersection of lie ellipse with the
hyperbola
xp (a* - b^) = c^ay - V^x,
For the modification in the results of this and the pre-
ceding article arising from the existence of singular points on
the curve, the student is referred to Salmon's Higher Plane
Curves, Arts. 66, 67, iii.
178. DifTerential of the Arc of a Plane Curve.
Direction of the Tangent. — If the length of the arc of a
curve, measured from a fixed point A on it, be denoted by s,
then an infinitely small portion of it is represented by ds.
Again, if ^' represent the angle QPL (fig. 8), we have
, PL ^ . , QL
but in the limit, PL = dx, QL = dy, and PQ = ds* and also
0' becomes PTX, or (p (fig. 9).
* In Art. 37 it has been proved that the difference between the length of an
infinitely small arc and its chord is an infinitely small quantity of the second
■, . . . , , , , » , , ■■ . arc FQ - FQ . . „ . ,
order in comparison with the length of the chord; i.e. — is infinitely
small of the second order, and therefore this fraction vanishes in the limit.
Hence J~p?) ~ '' ultimately.
Differential of the Arc of a Curve,
221
Hence
dx
dy
cos^ = ^, sin0 = ^
squaring and adding, we get
tm- '
(8)
(9)
Hence, also, we have
and therefore
= dx" + dy\
\
(10)
On account of the importance of these results, we shall
give another proof, as follows : —
Let, as before, FR he the tangent to the curve at the
point P,
MN=PL = ^x, QL = Ay.
LPTX=(l),a>TGPQ= As,
Then, if the curvature of
the elementary portion PQ
of the curve be continuous,
we have evidently the line
PQ<qxqPQ<PR+QR', 0 T M N
Fig. 10.
or vAa?' + /^y"^ < As < Ax sec (^ + Ay - Ax tan ^ ;
H
.— ^ < — <sec0 + — ^- tan 0.
\Axj Ax Ax
Again, in the limit ir- = -f- = tan (^, and J i + (
Ay_
Ax
222 Tangents and Normals to Curves,
becomes ji + (^^j or sec0 j accordingly each of the pre-
ceding expressions converges to the same limiting value, and
we have j~ = Ji+fj^- 5 which establishes the required
result.
179. Polar Co-ordinates. — The position of any point
in a plane is determined when its distance from a fixed point
called a pole, and the angle which that distance makes with a
fixed line, are known ; these are called the polar co-ordinates
of the point, and are usually denoted by the letters r and 0.
The fixed line is called the prime vector, and r is called the
radius vector of the point.
The equation of a curve referred to polar co-ordinates is
generally written in one or other of the forms,
r=/(0), ori^(r,0)=o,
according as r is given explicitly or implicitly in terms of 0.
Also, if 6/ be positive when measured above the prime vector,
it must be regarded as negative when measured below it.
1 80. Angle betTFeen Tangent and Radius Tector.
Let 0 be the pole, P and Q two near
points on the curve, PMo. perpendicular
on OQ, OP = r, POX = 0, and ^p the
angle between the tangent and radius
vector. Then
cos OQP = -77^ : but in the limit when
QP Fig. II.
Q and P coincide, the angle OQP
becomes equal to ;//, and*
QM dr PM rdO , ,^
^57^ = Tf -7T7S = -7- > at the same time ;
PQ ds PQ ds '
dr r , rdO , , rdO , .
or cosi// = -T-, sm^=-7-, tan;//=— r-. (n)
^ ds ■ ds dr ■
• These residts can be easily established from Art. 37.
^olar Suhtangent and Subnormal,
21$
Also,
\ds J '^ \ds) ~
(12)
Hence, also, we can determine an expression for the
differential of an arc in polar co-ordinates ; for, since
we get, on proceeding to the limit,
ds__ j 7d¥
dr \ dr
or
-4
.2 '
dr'
dr.
(13)
These results are of importance in the general theory of
curves.
i8i. Application to the liOgarithmic Spiral. —
The curve whose equation is r = a^ is called the logarithmic
spiral. In this curve we have
^ . rdO I
tan w - —r- = :; .
dr log a
Accordingly, the angle between the radius vector and the
tangent is constant. On account of this property the curve
is also called the equiangular sjnral.
182. Polar liubtangent and Subnormal. — Through
the origin 0 let 8T be drawn perpendi- g
cular to OF, meeting the tangent in T,
and the normal in S. The lines OT and
08 are called the polar subtangent and.
subnormal, for the point P. To find
their values, we have
r'^dO
0T= OF tan OFT=r tan ^ = -^.
08 = OF tan 0F8
Also, if
H =
224 Tangents and Nmvnals to Curves,
Again, if ON be drawn perpendicular to PT, we have
PiV^=OPcosi/. = r^. (15)
183. Expression for Perpendicular on Tangent.- —
As before, let p = ON, then
^ as
The equations in polar co-ordinates of the tangent and
the normal at any point on a curve can be found without
difficulty : they have, however, been omitted here, as they
are of little or no practical advantage.
Examples.
1. To find the length of the perpendicular from a focus on the tangent to an
ellipse.
The focal equation of the curve is
ail - e^ I - e cosfl
r = '. or M= —. -r- ;
I - « cos 6' a (I - e^)
du « sin 0
dd a(i-c*)'
**' ^ ^ a« (I - e2)vs - a*{i -e^) \r )'
2. Prove that the polar subnormal is constant, in the curve r~a9; and thft
Dolar subtangent, in the curvft rrt = a.
Inverse Curves.
225
184. Inverse Curves. — If on any radius vector OP,
drawn from a fixed origin 0, a point P' ^e taken such that
the rectangle OF . OF" is constant, the point P' is called the
inverse of the point P ; and if P describe any curve, P'
describes another curve called the inverse of the former.
The polar equation of the inverse is obtained immediately
from that of the original curve by
F .
substituting - instead of r in its
equation ; where k'^ is equal to the
constant OP . OP",
Again, let P, Q be two points,
and jP', Q' the inverse points ; then
since OP . OF = OQ . 0Q\ the
four points P, Q, Q\ P", lie on a
circle, and hence the triangles
OQP and OP^Q' are equiangular ;
Fig. 13.
PQ
fq:
OP
oq:
OP. OQ
OQ . OQ'
OP. OQ
(17)
Again, if P, Q be infinitely near points, denoting the
lengths of the corresponding elements of the curve and of its
inverse by ds and ds, the preceding result becomes
<fe=J
(18)
185. Direction of the Tangent to an Inverse
Curve. — Let the points P, Q belong to one curve, and F, Q!
to its inverse ; then when P and Q coincide, the lines PQ,
FQ' become the tangents at the inverse points P and F:
again, since the angle SPP' = the angle SQ'Q, it follows that
the tangents at P and F form an isosceles triangle with the
Hne PF.
By aid of this property the tangent at any point on a
curve can be drawn, whenever that at the corresponding
point of the inverse curve is known.
It follows immediately from the preceding result, that if
two curves intersegt at any angle, their inverse curves intersect at
the same angle. ^
2 26 Tangents and Normals to Curves,
1 86. fquation to the Inverse of a Criven Curve. —
Suppose the curve referred to rectangular axes drawn through
the pole 0, and that x and y are the co-ordinates of a point P
on the curve, X and Y those of the inverse point P' ; then
X~ 0F~ OF' ~ X'+Y'' ^^^^^^^^^ Y~ X^+Y'''
hence the equation of the inverse is got by substituting
and
x^ + y"^ x^ +
mstead of x and y in the equation of the original curve
Again, let the equation of the original curve, as in Art.
174, be
tin + Un-\ + W»-2 + . . . + ^2 + Wi + Wo = O.
When-r and , , are substituted for x and v, '^
x^ + y^ x' ■¥ y^
becomes evidently t^ ^ .
(:z;^ + ?/^)^
Accordingly, the equation of the inverse curve is
A-'"w„ + /v^"-2?/„_i (aj2 + y"") + ^'"-%n-2(^' + y')' + . . .
+ WoC^'^ + 2^^^)" = o. (19)
For instance, the equation of any right line is of the form
^^l + ^^o = o ;
hence that of its inverse with respect to the origin is
hhh + Uq {x^ + y"^) = o.
This represents a circle passing through the pole, as is
well known, except when Uq = o ; i.e. when the Hne passes
through the pole 0.
Again, the equation of the inverse of the circle
x^ + y^ ^ Ui -»- W'o = o,
with respect to the origin, is
(k' + ¥u, + u, {x' + f)) (x" + y') = o,
which represents another circle, along with the two imaginary
right lines x' + y^ = o.
Pedal Curves, ii'j
Again, tlie general equation of a conic is of the form
hence that of its inverse with respect to the origin is
k^u^ + Wu^ix^ + \f) + Uq{x^ -^ iff =' o,
which represents a curve of the fourth degree of the class
called '* bicircular quartics."
If the origin be on the conic the absolute term Uq vanishes,
and the inverse is the curve of the third degree represented
by
Fw2 + Wi (3? + y^) = o.
This curve is called a " circular cubic."
If the focus be the origin of inversion, the inverse is a
curve called the Limac'on of Pascal. The form of this curve
will be given in a subsequent Chapter.
187. Pedal Curves. — If from any point as origin a per-
pendicular be drawn to the tangent to a given curve, the locus
of the foot of the perpendicular is called the ^e^«/ of the curve
with respect to the assumed origin.
In like manner, if perpendiculars be drawn to the tan-
gents to the pedal, we get a new curve called the second pedal
of the original, and so on. With respect to its pedal, the
original curve is styled the first negative pedal, &c.
188. Tangent at any Point to the Pedal of a
given Curve. — Let OJS', ON'
be the perpendiculars from the
origin 0 on the tangents drawn
at two points P and Q on the
given curve, and T the intersec-
tion of these tangents ; join iViV^';
then since the angles ONT and
ON'T are right angles, the qua-
drilateral ONN'T is inscribable
in a circle,
.-. lON'N^^lOTN,
In the limit when P and Q coincide, L OTN = L OPN,
and NN' becomes the tangent to the locus of JV; hence the
Q 2
228 Tangents and Normals to Curves.
latter tangent makes the same angle with ON that the
tangent at P makes with OP. This property enables us
to draw the tangent at any point N on the pedal locus in
question.
Again, if p' represent the perpendicular on the tangent at
N to the first pedal, from similar triangles we evidently have
r-X
P . . . . ^
Hence, if the equation of a curve be given in the form
p^
r =f{p), that of its first pedal is of the form ^ =/(??), in
which p and p' are respectively analogous to r and p in the
original curve. In like manner the equation of the next
pedal can be determined, and so on.
189. Reciprocal Polars. — If on the perpendicular ON
a point P' be taken, such that OP". ON is constant {k^ sup-
pose), the point P' is evidently the pole of the line PiV" with
respect to the circle of radius k and centre 0 ; and if all the
tangents to the curve be taken, the locus of their poles is a
new curve. We shall denote these curves by the letters A
and B, respectively. Again, by elementary geometry, the
point .of intersection of any two lines is the pole of the line
joining the poles of the lines,* Now, if the lines betaken as
two infinitely near tangents to the curve A, the line joining
their poles becomes a tangent to B ; accordingly, the tangent
to the curve B has its pole on the curve A. Hence A is the
locus of the poles of the tangents to B.
In consequence of this reciprocal relation, the curves A and
B are called reciprocal polars of each other with respect to the
circle whose radius is k.
Since to every tangent to a curve corresponds a point on
its reciprocal polar, it follows that to a number of points in
directum on one curve correspond a number of tangents to its
reciprocal polar, which pass through a common point.
Again, it is evident that the reciprocal p)olar'to any curve
is the inverse to its pedal with respect to the origin.
We have seen in Art. 1 76. that the greatest number of tan-
geats from a point to a curve of the n^^ degree is w (» - i) ;
♦ Townsend's Modern Geometry y vol. i., p. 219.
Reciprocal Polara. 229
hence the greatest number of points in which its reciprocal
polar can be cut by a line is n{n - i), qj? the degree of the
reciprocal polar is n{n- / ) . For the modification in this
result, arising from singular points in the original curve, as
well as for the complete discussion of reciprocal polars, the
student is referred to Salmon's Higher Plane Curves.
As an example of reciprocal polars we shall take the curve
considered in Art. 173.
If r denote the radius vector of the reciprocal polar cor-
responding to the perpendicular p in the proposed curve, we
have
^ = 7-
r
Substituting this value for j9 in equation (5), we get
m m
1 -m-3 . tn-l
= (a cos (u) -»- {0 Sin w) ,
n< m
\m-!\
or A;"*-* = {ax)"^^ + (hy)'
which is the equation of the reciprocal polar of the curve re-
presented by the equation
In the particular case of the ellipse,
a- ^ h' ''
the reciprocal polar has for its equation
¥ = aV + by.
The theory of reciprocal polars indicated above admits of
easy generalization. Thus, if we take the poles with respect
to any conic section ( U) of aU the tangents to a given curve
A, we shall get a new curve B ; and it can be easily seen, a£
before, that the poles of the tangents to B are situated on the
curve A. Hence the curves are said to be reciprocal polars
with respect to the conic Z7.
It may be added, that if two curves have a common poiD^
230 Tangents and Kormals to Curves.
their reciprocal polars have a common tangent; and if the
curves touch, their reciprocal polars also touch.
For illustrations of the great importance of this " principle
of duality," and of reciprocal polars as a method of investi-
gation, the student is referred to Salmon's Conies, ch. xv.
We next proceed to illustrate the preceding by discussing
a few elementary properties of the curves which are comprised
under the equation r"* = a"* cos md.
190. Pedal and Reciprocal Polar of r*" = a^QO^mB.
"We shall commence by finding the
angle between the radius vector and
the perpendicular on the tangent.
In the accompanying figure we
(It
have tan PON = cot OPN = - -^.
rm
Pig. 15-
But m log r = m log a + log (cos mO) ;
hence -m = - tan mQ,
rdd
and accordingly, L PON - mO. (20)
Again, p = 0N= r cosmO
or r""^^ = a'^p. (21)
The equation of the pedal, with respect to 0, can be im-
mediately found.
For, let lAON = w, and we have
«•»'
Also, from (21), (-)=(-
Hence, the equation of the pedal is
pmn = ^m+l QQ^ \ (22)
On the Curve r"' = a"* cos md. 231
Consequently, the equation of the pedal is got by substi-
tiitinsr instead of m in the eqiiatioir*of the curve.
By a like substitution the equation of the second pedal is
easily seen to be
^2nt+i _ /72fn+l
rzrrr nid
cos ;
zm + I
and that of the n*^ pedal
-^ -— - m8
^n.n*i ^ fjmn+. ^OS . (23)
mn +1 ^ '
Again, from Art. 184, it is plain that the inverse to the
curve r^ = a^^ cos w^0, with respect to a circle of radius a, is
the curve r^ cos mO = «*".
Again, the reciprocal polar of the proposed, with respect
to the same circle, being the inverse of its pedal, is the curve
tn /I w*
— mu —
r"**^ cos = a"***. (24)
m+ 1 ^ ^^
It may be observed that this equation is got by substitut-
ing for m in the original equation.
Accordingly we see that the pedals, inverse curves, and
reciprocal polars of the proposed, are all curves whose equa-
tions are of the same form as that of the proposed.
In a subsequent chapter the student will find an additional
discussion of this class of curves, along with illustrations of
their shape for a few particular values of m.
Examples.
t. The equation of a parabola referred to its focus as pole is
r (i + cosO) = za,
to find the relation between r and p.
a
Here ri cos - = aJ, and consequently p* = «y,
a well-known elementary property of the curve.
232 Tangents and JSTormals to Curves.
2. The equation r' cos 2e = d2 represents an equilateral hyperbola; prore
that ji?r = a'.
3. The equation r^ = a^ cos 2^ represents a Lemniscate of Bernoulli ; find
the equation connecting p and r in tnis case. Ans. r^ = c^p,
4. Find the equation connecting the radius vector and the perpendicular on
the tangent in the Cardioid whose equation is
r = a{x + cos e). Ans. r' = zap^.
It is evident that the Cardioid is the inverse of a parabola with respect to
its focus ; and the Lemniscate that of an equilateral hyperbola -with respect to
its centre. Accordingly, we can easily draw the tangents at any point on either
of these curves by aid of the Theorem of Art. 1 85.
5. Show, by the method of Art. 188, that the pedal of the parabola, p* = aty
with respect to its focus, is the right line p = a.
6. Show that the pedal of the equilateral hyperbola pr = a^ is a Lemniscate.
7. Find the pedal of the circle r^ = 2ap. Ans. A Cardioid, 1^ = 2ap\
igi. lExpres«iioii for PiV. — To find the value of the
intercept between the point of
contact P and the foot JV of
the perpendicular from the
origin on the tangent at P.
Let p = OJSr^cj =L NO A, >
PN= t) then L NTN'==lNON'
= Aw, also SN'=TS sin STN';
SN'
.-. TS = — iTfTTi^,; but in the
limit, when PQ is infinitely small,
and TS becomes FN or t :
d(t)
Also OP' = ON' + PN^ ;
^=^'.(iy. (36)
192. To prove thai
ds dt i m\
dd) du)
Vectorial Co-ordinates, 233
On reference to the last figure we have
ds .. ., .PT^TQ dt .. ,''^.QN'-FN
-J- = limit of , -— = limit of ;
but FT+ TQ - QN' + PN = TN - TN';
hence ~ -7- = limit of = limit of = ON=p ;
aw au) All) Ao>
ds dt
att) aui
This result, which is due to Legendre, is of importance in
the Integral Calculus, in connexion with the rectification of
curves.
dp
If — be substituted f or ^, the preceding formula becomes
ds d^p , ^,
aw ^ dw^ ^ '
This shape of the result is of use in connexion with curva-
ture, as will be seen in a subsequent chapter.
193. Birection of Hiormal in tectorial Co-ordi-
nates.— In some cases the equation of a curve can be
expressed in terms of the distances from two or more fixed
points or foci. Such distances are called vectorial co-ordi-
nates. For instance, if ri, r-i denote the distances from two
fixed points, the equation ri + ra = const, represents an ellipse,
and ri - ^2 = const., a hyperbola.
Again, the equation
ri + mri = const,
represents a curve called a Cartesian* oval.
Also, the equation
ri r2 = const,
represents an oval of Cassini, and so on.
The direction of the normal at any point of a curve, in
such cases, can be readily obtained by a geometrical con-
struction.
* A discussion ot the principal properties of Cartesian ovals will be found
in Chapter XX.
234 Tangents and Normals to Curves.
For, let
-F(ri, rz) = const.
be the equation of the curve, where
then we have
dFdn dFdr^ _
dvx ds drz ds
Now, if PTbe the tangent at P, then, by Art. 1 80, we have
— ^ = cos ;//!, -~ = cos ;/»2, where ■^x = L TFFi, xp2 = I. TFF2.
TT dF , dF . . .
±lence — cos \pi+ — cos \pz = o. (29)
ari ar%
Again, from any point R on the normal draw RL and
RM respectively parallel to F^P and FiP, and we have
PL : LR = sin RPM : sin RPL = cos 1/^2 : - cos ;//i
_(^^ dF
Accordingly, if we measure on PFi and PF2 lengths
PL and PM. which are in the proportion of -r- to -;— , then
dri drz
the diagonal of the parallelogram thus formed is the normal
required.
This result admits of the following generalization :
Let the equation of the curve* be represented by
•P(n, n, ^3, . . . rn) = const.,
* The theorem given above is taken from Poinsot's Elements de Statigue,
Neuvieme Edition, p. 435. The principle on which it was founded was, how-
ever, given by Leibnitz {Journal des Savana, 1693), and was deduced from
mechanical considerations. The terra resultant is borrowed from Mechanics,
and is obtained by the same construction as that for the resultant of a number
of forces acting at the same point. Thus, to find the resultant of a number of
lines Pa, Fb, Fc, Fd, . . . issuing from a point F, we draw through a a. right
line a£, equal and parallel to Fb, and in the same direction ; through B, a right
line BO, equal and parallel to Fs, and so on, whatever be the number of lines:
then the line FH, which closes the polygon, is the resultant in question.
Normals in Vectorial Co-ordinates. 235
where rj, ^2, . . . r^ denote the distances from n fixed points.
To draw the normal at any point, we connect the point with
the n fixed points, and on the joining lines measure off
lengths proportional to
dF dF dF dF ,. .
^' d^.; d7r-d^:'''^''^'^'^^'
then the direction of the normal is the resultant of the lines
thus determined.
For, as hefore, we have
dFdj\ dFd_n ^^_
dri ds dr2 ds ' ' ' drn ds
TT dF , dF , dF ^ . ,
Hence -7- cos t//i + — cos U/2 + . . , -r- cos ip« = o. (30)
(tr\ arz ^^»
dF , dF , dF ,
Now, -7- cos t//i, — cos t//2, . . . TT COS U/„,
ar\ ar^ ■ w/ j^
are evidently proportional to the projections on the tangent
of the segments measured off in our construction. Moreover,
in any polygon, the projection of one side on any right line
is manifestly equal to the sum of the projections of all the
other sides on the same line, taken with their proper signs.
Consequently, from (30), the projection of the resultant on
the tangent is zero ; and, accordingly, the resultant is normal
to the curve, which establishes the theorem.
It can be shown without difficulty that the normal at any
point of a surface whose equation is given in terms of the
distances from fixed points can be determined by the same
construction.
Examples.
T. A Cartesian oval is the locus of a point, P, sucli that its distances, Pi/,
PM', from the circumfereuces of two given circles are to each other in a constant
ratio ; prove geometrically that the tangents to the oval at P, and to the circles
at M and M'j meet in the same point.
2. The equation of an ellipse oi Cassini is r/ = ab, where r and / are the
distances of any point P on the curve, from two fixed points, A and B. If 0
be the middle point of ^P, and PiVthe normal at P, prove that L AFO= I BPN.
3. In the curve represented by the equation ri^ + ra' = «', prove that the
normal divides the distance between the foci in the ratio of r% to ri.
236 Tangents and Normals to Curves.
194. In like manner, if the equation of a curve be given
in terms of the angles 0i, 02, • • . ^n> which the vectors drawn
to fixed points make respectively with a fixed right line, the
direction of the tangent at any point is obtained by an analo-
gous construction.
For, let the equation be represented by
F{Q,, 02, . . . 0«) = const
Then, by differentiation, we have
dF^dd, dFdOi dFddn_
dOi ds dOz ds ' ' ' dOn ds
Hence, as before, from Art. 180, we get
I dF . , I dF . , I dF , , .
Accordingly, if we measure on the lines drawn to the fixed
points segments proportional to
i_dF i_dF i_dF
r.ddy r.dO,' ' " rndOn*
and construct the resultant line as before, then this line will
be the tangent required. The proof is identical with that of
last Article.
195. Curves Symmetrical Yrith respect to a liine,
and Centres of Curves. — It may be observed here, that
if the equation of a curve be unaltered when 1/ is changed
into - y, then to every value of x correspond equal and oppo-
site values of «/; and, when the co-ordinate axes are rect-
angular, the curve is symmetrical with respect to the axis of a;.
In like manner, a curve is symmetrical with respect to
the axis of y, if its equation remains unaltered when the sign
of x is changed.
Again, if, when we change x and y into - x and - y, re-
spectively, the equation of a curve remains unaltered, then
every right line drawn through the origin and terminated by
the curve is divided into equal parts at the origin. This
takes place for a curve of an even degree when the sum of
Symmetrical Curves and Centres, 237
the indices of x and y in each term is even ; and for a curve
of an odd degree when the like sum is odd.- Such a point is
called the centre* of the curve. For instance, in conies, when
the equation is of the form
flwj* + zhxy + hy'^ = c,
the origin is a centre. Also, if the equation of a cubicf be
reducible to the form
Wa + Wi = o,
the origin is a centre, and every line drawn through it is bi-
sected at that point.
Thus we see that when a cubic has a centre, that point
lies on the curve. This property holds for all curves of an
odd degree.
It should be observed that curves of higher degrees than
the second cannot generally have a centre, for it is evidently
impossible by transformation of co-ordinates to eliminate the
requisite number of terms from the equation of the curve.
For instance, to seek whether a cubic has a centre, we substi-
tute X + a for a?, and F + j3 f or y, in its equation, and equate
to zero the coefficients of X^ XFand F^ as well as the abso-
lute term, in the new equation : as we have but two arbitrary
constants (a and /3) to satisfy four equations, there will be
two equations of condition among its constants in order that
the cubic should have a centre. The number of conditions is
obviously greater for curves of higher degrees.
* For a general meaning of the word " centre," as applied to curves of
higher degrees, see Chasles's Apercu Historique, p. 233, note.
t This name has been given to curves of the third degree by Dr. Salmon,
in his Higher Plane CurveSj and has been generally adopted by subsequent
writers on the subject.
238 Exmnples.
Examples.
I. Find the lengtlis of the subtangent ana subnonnei at any point of the
curve
yn = ^n-la;.
Ans, nx. — .
nx
;. Find the subtangent to the curve
X>'*yn — «»»•«•,
nx
Am, .
m
;. Find the equation of the tangent to the
curve
x^ = o3yS.
Ans.
5^ 2r_
X y ^'
4. Show that the points of contact of tangents from a point (o, ^8) to the
curve
are situated on the hyperbola (m + n)zi/ = nfix + may,
5. In the same curve prove that the portion of the tangent intercepted be-
tween the axes is divided at its point of contact into segments which are to each
other in a constant ratio.
6. Find the equation of the tangent at any point to the hypocycloid, xt + 1^
= al ; and prove that the portion of the tangent intercepted between the axes is
of constant length.
7. In the curve a;'» + 2/" = a**, find the length of the perpendicular drawn
from the origin to the tangent at any point, and find also the intercept made by
the axes on the tangent.
an tf2n
Ans. p = — — ; intercept = -. r.
8. If the co-ordinates of every point on a curve satisfy the equations
a; = c sin 20(i + cos 20), y = c cos 20(1 — cos 20),
prove that the tangent at any point makes the angle B with the axis of x.
9. The co-ordinates of any point in the cycloid satisfy the equations
X = a{Q -sme), 2/ = a(i -CO8 0):
prove that the angle which the tangent at the point makes with the axis of y
. e
18 -.
2
Examples, 239
Here -^ = -- = cof -.
ax ax 2
10. Prove that the locus of the foot of the perpendicular from the pole ou
the tangent to an equiangular spiral is the same curve turned through an angle
11. Prove that the reciprocal polar, with respect to the origin, ofanequi-
angular spiral is another spiral equal to the original one.
13. An equiangular spiral touches two given lines at two given points ; prove
that the locus of its pole is a circle.
13. Find the equation of the reciprocal polar of the curve
y* cos - = a*.
3
d
with respect to a circle with radius a. Ans. The Cardioid H = ai cos -.
2
14. Find the equation of the inverse of a conic, the focus heing the pole of
inversion.
15. Apply Art. 184, to prove that the equation of the inverse of an ellipse
with respect to any origin 0 is of the form
lap = OFi . pi + OF2 . p2,
where F\ and F-2, are the foci, and p, pi, 02 represent the distances of any point
on the curve from the points 0, /i and /2, respectively ; f\ and /g being the
points inverse to the foci, F\ and Fz.
16. The equation of a Cartesian oval is of the form
r + ^r' = a,
where r and r' are the distances of any point on the curve from two fixed points,
and a, k are constants. Prove that the equation of its inverse, with respect to
any origin, is of the form
opi + 3o2 -f 7J3 = o,
where pi, pz, pa are the distances of any point on the curve from three fixed
points, and o, )3, 7 are constants.
17. In general prove that the inverse of the curve
api + i8p2 + 7P3 = o,
with respect to any origin, is another curve whose equation is of similar form.
18. If tlie radius vector, OP, drawn from the origin to any point Pon a
240 Examples.
curve be produced to Pi, until PPi be a constant length ; prove that the normal
at Pi to the locus of Pi, the normal at P to the original curve, and the perpen-
dicular at the origin to the line OP, all pass through the same point.
This follows immediately from the value of the polar subnormal given in
Art. 182.
19. If a constant length measured from the curve be taken on the normals
along a given curve, prove that these lines are also normals to the new cixrve
which is the locus of their extremities.
20. In the ellipse -^ -t — = i, if a? = a sin a,
prove that
ds
s= ay/ I - ^2 Bin2«^.
21. If i« be the element of the arc of the inyerse of an ellipse with respect
to its centre, prove that
,- « a/'i - «^ sin*0 , - «*- i'
d»=Tfi-^ , , ^ dip, where n = —75—.
22. If w be the angle which the normal at any point on the ellipse
— + J- = I makes with the axis-major, prove that
, *« doo
d9= -
a {i — tf^sin^w)^
23. Express the differential of an elliptic arc in terms of the semi-axis major,
fi, of the confocal hyperbola which passes through the point.
^ a\^ — y?
24. In the curve r»» = o»» cos m0, prove that
ds — ,
-— =a a sec "» mQ.
de
25. If F{x, y) = o be the equation to any plane curve, and <p the angle be-
tween the perpendicular from the origin on the tangent and the radius vector to
the point of contact, prove that
dF dF
y X —
dx dy
'™* = -7f — dF'
( 241
CHAPTEE XIII.
ASYMPTOTES.
196. Intersection of a Curve and a Right One.—
Before entering on tlie subject of this chapter it will be ne-
cessary to consider briefly the general question of the inter-
section of a right line with a curve of the n^^ degree.
Let the equation of the right line hQ y = fix -\- v, and sub-
stitute fix + V instead of y in the equation of the curve ; then
the roots of the resulting equation in x represent the abscissae
of the points of section of the line and curve.
Moreover, as this equation is always of the n*^ degree, it
follows that every right line meets a curve of the n^^ degree in n
points^ real or imaginary^ and cannot meet it in more.
If two roots in the resulting equation be equal, two of the
points of section become coincident, and the line becomes a
tangent to the curve.
Again, suppose the equation of the curve written in the
form of Art. 175, viz. :
Un + lln-\ + W«-2 + . . . •Wa + Wi + «^o = O ;
then, since % is a homogeneous function of the n^^ degree in
X and yy it can be written in the form x^fA-\ similarly
And accordingly, the equation of the curve may be written,
*"/.(f)+-"-y.(f)-"-y.g)+&o. = o. (r}
Substituting ^ + - for - in this, it becomes
X X
242 Asymptotes.
Or, expanding by Taylor's Theorem,
+ &c. = o. (2)
The roots of this equation determine the points of section in
question.
We add a few obvious conclusions from the results arrived
at above :—
1°. Every right line must intersect a curve of an odd de-
gree in at least one real point ; for every equation of an odd
degree has one real root.
2°. A tangent to a curve of the n^^ degree cannot meet it
in more than n - 2 points besides its points of contact.
3°. Every tangent to a curve of an odd degree must meet
it in one other real point besides its point of contact.
4°. Every tangent to a curve of the third degree meets
the curve in one other real point.
197. Definition of an Asymptote. — An asymptote is
a tangent to a curve in the limiting position when its point
of contact is situated at an infinite distance.
1". No asymptote to a curve of the n*^ degree can meet it
in more than n - 2 points distinct from that at infinity.
2°. Each asymptote to a curve of the third degree inter-
sects the curve in one point besides that at infinity.
198. Method of finding the Asymptotes to a Curve
of the n*^ Degree. — If one of the points of section of the
line y = fxx + V with the curve be at an infinite distance, one
root of equation (2) must be infinite, and accordingly we
have in that* case
/.W - o. (3)
Again, if two of the roots be infinite, we have in addition
v/o'M +/.W = O. (4)
* This can be easily established by aid of the reciprocal equation ; for if we
substitute - for x in equation (2), the resulting equation in z will have one root
z
zero wnen its absolute terra vanishes, i.e., when/o(/i) = o ; it has two roots
zero when we have in addition yfo'ifi) +fi(ji) = o ; and so on.
Method of finding Asymptotes in Cartesian Co-ordinates. 243
Accordingly, when the values of fi and v are determined
so as to satisfy the two preceding equations, the correspond-
ing line
y = fix + V
meets the curve in- two points in infinity, and consequently is
an asymptote. (Salmon's Conic Sections, Art. 154.)
Hence, if /wj be a root of the equation /©(/x) = o, the line
is in general an asymptote to the curve.
If /i(/i) = o and/o(ju) = o have a common root (jUi suppose),
the corresponding asymptote in general passes through the
origin, and is represented by the equation
y = fxix.
In this case % and w^-i evidently have a common factor.
The exceptional case when fo{fx) vanishes at the same
time will be oonaidered in a subsequent Article.
To each root of /o(ju) = o corresponds an asymptote, and
accordingly,* every curve of the n^^ degree has in general n
asymptotes, real f/r imaginary.
From the preceding it follows that every line parallel
to an asymptote meets the curve in one point at infinity.
This also is immediately apparent from the geometrical
property that a system of parallel lines may be considered
as meeting in the same point at infinity — a principle intro-
duced by Desargues in the beginning of the seventeenth
century, and which must be regarded as one of the first
important steps in the progress of modern geometry.
Cor. No line parallel to an asjrmptote can meet a curve
of the fi!'^ degree in more than (^ - i) points besides that
at infinity.
Since every equation of an odd degree has one real
root, it follows that a curve of an odd degree has one real
* Since /o(^) is of the «''• degree in /i, unless its highest coefficient vanishes,
in which case, as -we shall see, there is an additional asymptote parallel to the axis
of ft
R 2
244 Asymptotes.
asymptote, at least, and has accordingly an infinite brancli
or branches. Hence, no curve of an odd degree can he a closed
curve.
For instance, no curve of the third degree can be a finite
or closed curve.
The equation fifi) = o, when multiplied by j»**, becomes
w„ = o ; consequently the n right lines, real or imaginary,
represented by this equation, are, in general, parallel to the
asymptotes of the curve under consideration.
In the preceding investigation we have not considered
the case in which a root of /o(/u) = o either vanishes or is
infinite ; i.e^, where the asymptotes are parallel to either
co-ordinate axis. This case will be treated of separately in a
subsequent Article.
If all the roots of /o(iu) = o be imaginary the curve
has no real asymptote, and consists of one or more closed
branches.
Examples.
To find the asymptotes to the following curves :—
1. y' = ax^ + a^.
Substituting fxx -k- v for y, and equating to zero the coefficients of sfi and a;*,
separately, in the resulting equation, we obtain
^8 — I = o, and Tff^v = a ;
a
hence the curve has but one real asymptote, viz.,
a
y = X+-.
3
2. y* — «* + 2ax^y = S^sfi.
Here the equations for determining the asymptotes are
/t* - I = o, and 4jtt3j/ + 2o/t =» o j
accordingly, the two real asymptotes are
a ^ a
y = a: — -, and y + a; + - = o.
3. «» + 3a:2y _ xf - 3y^ + x^ - 2Xt/ + 3y3 + 4a; + 5 = o.
a^ 3 I 3
Ans. j/ + -+- = o, y = ic + y y -\- z = -.
3 4 4 »
Asymptotes Parallel to Co-ordinate Axes, 245
199. Case in ii'hich t^„ = o repres^jnts the n Asymp-
totes.— If the equation of the curve contain no terms of
the {n - lY^ degree, that is, if it be of the form
Un + UnJi + Wn-8 + &C. . . . + Wi + Wq = O,
the equations for determining the asymptotes become
/o(/i) = o, and i/o'(ju) = o.
The latter equation gives v = o, unless /o'(iu) vanishes along
with/o(ju), i.e., unless /o(iu) has equal roots.
Hence, in curves whose equations are of the above form,
the n right lines represented by the equation Un = o are the
n asymptotes, unless two of these lines are coincident.
This exceptional case will be considered in Art. 203.
The simplest example of the preceding is that of the
hyperbola
ax^ + ihxy + hy"^ = c,
in which the terms of the second degree represent the asymp-
totes (Salmon's Conic Sections, Art. 195).
Examples,
Find the real asymptotes to the curves
1. xy* - x^y — <^{x + y) + i*. Ans, a; = o, y = o, » —y = 0.
2. y^ — 3fi = c^x. „ y - x = o.
3. «* - y* = a^xy + d*y*. „ x + y = OjX-y = o.
200. Asymptotes parallel to the Co-ordinate
Axes. — Suppose the equation of the curve arranged accqrd-
ing to powers oix, thus
Oo^" + («iy + b)x*'-^ + &o. = o ;
then, if ^0 = o and aiy + b = o, or y = , two of the roots
of the equation in x become infinite ; and consequently the
line ttiy + 6 = o is an asymptote.
246 Asymptotes.
In other words, whenever the highest power of x is
wanting in the equation of a curve, the coefficient of the
next highest power equated to zero represents an asymptote
parallel to the axis of x.
If Uq = o, and 6 = 0, the axis of x is itself an asymptote.
If x^ and x^~^ be both wanting, the coefficient of x^~'^ re-
presents a pair of asymptotes, real or imaginary, parallel to
the axis of x ; and so on.
In like manner, the asymptotes parallel to the axis of y
can be determined.
Examples.
Find the real asymptotes in the following curves : —
1 . y^x - ay2 = a;^ + ax^ + b^. Ans. x = a, y = x + a, j/ + « + a = o.
2. y(a;2 - ^bx + 2b'^) =a? - ^aafi + a^. x = b, x = 2b, y + ^a = x + ^b.
3. ic'ys = «2 (a;2 + y'i), x = ±a,y = ±a.
4. a;2y« = 0^(2:^ - y«). y + a = o, y -a^o,
5. y'^a-y^x = x^, x = a.
201. Parabolic Branches. — Suppose the equation
/o(/x) = o has equal roots, then/o'(/xi) vanishes along with/o(jLt),
and the corresponding value of v found from (5) becomes in-
finite, unless /i(/u) vanish at the same time.
Accordingly, the corresponding asymptote is, in general,
situated altogether at infinity.
The ordinary parabola, whose equation is of the form
{ax + j5yy = Ix + my + n,
furnishes the simplest example of this case, having the
line at infinity for an asymptote. (Salmon's Co7iic Sections,
Art. 254.) ^ ^ ■
Branches of this latter class belonging to a curve are
called parabolic, while branches having a finite asymptote are
called hyperbolic.
202. From the preceding investigation it appears that
the asymptotes to a curve of the n^^ degree depend, in
general, only on the terms of the n^^ and the {n - i)'^ degrees
Parallel Asymptotes. 247
in its equation. Consequently, all curves which have the
same terms of the two highest degrees have generally the sami
asymptotes.
There are, however, exceptions to this rule, one of whicli
will be considered in the next Article.
203. Parallel Asymptotes. — We shall now consider
the case where /o(/z) - o has a pair of equal roots, each repre-
sented by jui, and where /i(/ui) = o, at the same time.
In this case the coefficients of x^ and af^~^ in (2) both
vanish independently of v, when /x = yui ; we accordingly
infer that all lines parallel to the line y = fxix meet the curve
in two points at infinity, and consequently are, in a certain
sense, asymptotes. There are, however, two lines which are
more properly called by that name ; for, substituting /xi for ju
in (2), the two first terms vanish, as abeady stated, and the
coefficient of x^''~^ becomes
2
■fo'ini) + vfiifXi) +MfMl).
I . 2
Hence, if vi and v-z be the roots of the quadratic
Y^/«"W + '//(i«.)+/^W=o), (6)
the lines y = fxix + vi, and y = fiiX + V2,
are a pair of parallel asymptotes, meeting the curve in three
points at infinity.
If the roots of the quadratic be imaginary, the corre-
sponding asymptotes are also imaginary.
Again, if the term tCn-i be wanting in the equation, and
if /o(^) = o have equal roots, the corresponding asymptotes
are given by the quadratic
I . 2
fo'ifx,) +M^;) = o.
In order that these asymptotes should be real, it is
necessary thQ,tf2{ini) and fo'dmi) should have opposite signs.
There is no difficulty in extending the preceding investi-
gation to the case where fo(jm) = o has three or more equal
roots.
24S Asymptotes.
Examples.
I. (a; + y)' {x^ + y^ + xy) = a^f + a^(x- y).
Here /o(/t) = (i + /x)2(i + /* + /i^), /iCu) = o, f^iji) ^- a^tj?-,
.♦. /*i = - I, fa" (/ti) = 2, /2 Otti) = - a=« ;
accordingly yi = «, j/g = - o,
and the corresponding asymptotes are
y ■{. X - « = o, and y + » + a = o.
Tbe other asymptotes are evidently imaginary.
a. x^ {x + y)2 + 2ay'^{x -[■ y) + Za^xy + a^y = o.
Here /,(/i) = (i + /i)2, /lOu) = 2«/*«(i + ^), f 2{,j,) = Sa^f, ,
.•• /*i = - I, /o"(/t) = 2, /i'(mi) = 2«, /2(mi) = - 8«»,
jtad the corresponding asymptotes are
y + a? - 2a = o, and y + « + 4a = o.
204. If the equation to a curve of the n*^ degree be of
the form
(p + ax + I5)(l>i + 02 = o,
where the highest terms containing x and 2^ in ^i are of the
degree n - i, and those in 03 are of the degree w - 2 at most,
the line
y + ax + 15 = o
is an asymptote to the curve.
For, on substituting - ax- (5 instead of y in the equation,
it is evident that the coefficients of of' and x^~^ both vanish ;
hence, by Art. 198, the line 3/ + a^ + j3 = o is an asymptote.
Conversely, it can be readily seen that ii y + ax + /3 be an
asymptote to a curve of the n*^ degree its equation admits of
being thrown into the preceding form.
In general, if the equation to a curve of the n*^ degree
be of the form
{y + aiX + Pi){y + a2X + /Bz) . . . 0/ + anX + j3«) + 02 = o, (7)
Asymptotes to a Cubic. 249
where <pi contains no term higher than the degree n - 2y the
lines
j^ + a,a; + /3i = o, y + a-iX -{■ fi^ = O^ . . . ^ + a«3? + j8« = o
are the n asymptotes of the curve.
This follows at once as in the case considered at the com-
mencement of this Article.
For example, the asymptotes to the curve
xy {x + y + a^{x + y -^ a^ + byx + bzy = o
are evidently the four lines
x = o, y = OjX + y + ai = o, x + y + a2 = o
If the curve be of the third degree, ^2 is of the first, and
accordingly the equation of such a curve, having three real
asymptotes, may be written in the form
{y + aix + j3i)(y+ a2X+ P>2){y + aa^ + jSa) + Ix ^ my + n = o, (8)
Hence we infer that the three points in which the asymp-
totes to a cubic meet the curve lie in the same right line, viz.,
Ix + my + w = o.
The student will find a short discussion of a cubic with
three real asymptotes in Chapter xviii.
205. To prove that, in general, the distance of a point
in any branch of a curve from the corresponding asymptote
diminishes indefinitely as its distance from the origin increases
indefinitely.
If y + aa; + j3 = o be the equation of an asymptote, then,
as in the preceding Article, the equation of the curve may be
written in the form
{y + ax + /3) 01 = 02,
where 02 is at least one degree lower than 0i in rr and y.
250 Asymptotes,
Hence y + ax-vfi = -^
and the perpendicular distance of any point {xo, t/o) on thf
curve from the line 1/ + ax + (5 = o is
yo + aXo + /3 I f(j)2
. -, or . , , ,
where the suffix denotes that a;© and y^ are substituted for ^
and y in the functions 0i and 02.
Now, when Xq and y^ are taken infinitely great, the value
of the preceding fraction depends, in general, on the terms
of the highest degree (in x and 2/) in ^i and ^2 ; and since the
degree of ^2 is one lower than that of ^i, it can be easily
seen by the method of Ex. 7, Art. 89, that the fraction —
01
becomes, in general, infinitely small when x and y become
infinitely great. Hence, the distance of the line y + ax + (5
from the curve becomes infinitely small at the same time.
It is not considered necessary to go more fully into thia
discussion here.
The subject of parabolic and other curvilinear asymptotes
is omitted as being unsuited to an elementary treatise.
Moreover, their discussion, unless in some elementary cases,
is both indefinite and unsatisfactory, since it can be easily
seen that if a curve has parabolic branches, the number of its
parabolic asymptotes is generally infinite. The reader who
desires full information on this point, as well as the discussion
of the particular parabolas called osculating, is referred to a
paper by M. Pliicker, in Liouville's Journal, vol. i., p. 229.
206. Asymptotes in Polar Co-ordinates. — If a
curve be referred to polar co-ordinates, the directions of its
points at an infinite distance from the origin can be in gene-
ral determined by making r = 00, or m = o, in its equation,
and solving the resulting equation in 9. The position of the
asymptote corresponding to any such value of 9 is obtained
by finding the length of the corresponding polar subtangent,
dfi
i.e., by finding the value of — corresponding to u = o.
Asymptotes in Polar Co-ordinates. 251
It should be observed that when — is positive, the asymp-
tote lies above the corresponding radius vector, and when
negative, below it ; as is easily seen from Art. 182.
If we suppose the equation of the curve, when arranged
in powers of r, to be
r^MQ) + ^"-yi(0) + . . . + rUm +/«(6>) = o,
the transformed equation in u is
u%{Q) + u^-'U^iQ) + . . . + uf,{Q) +/o((9) = o : (9)
consequently, the directions of the asymptotes are given by
the equation
/.(0) = o. (10)
Again, if we differentiate (9) with respect to d, it is easily
seen that the values of -r^ corresponding to u = o are given
by the equation
/.(e)^+/«'(e) = o, (I,)
provided that none of the functions
/.(0), MO), • • • /«W
become infinite for the values of 0 which satisfy equation (10).
Consequently, if a be a root of the equation /o(0) = o, the
curve has an asymptote making the angle a with the prime
vector, and whose perpendicular distance from the origin is
, - , /i(a)
represented by - 777-^.
Jo \a)
It is readily seen that the equation of the corresponding
asymptote is
r sm(a - Q) --TTTs = o-
This method will be best explained by applying it to one
or two elementary Examples.
252 Asymptotes.
Examples.
I. Let the curve be represented by the equation
r = a sec 0 + J tan fi.
cosd
Here
a + i sini
When 8 = -, we have « = o, and — = r.
Accordingly, the corresponding polar subtatigent is a + i, and hence the line
perpendicular to the prime vector at the distance a + b from the origin is an
asymptote to the curve.
Again, « vanishes also when 0 = — , and the corresponding value of the
2
polar Bubtangent is a - i ; thus giving another asymptote.
2. r = a sec wO + 4 tan mQ,
Here
a + J sin i
V du — tn
"When d = — , we have «♦ = o, and -— = : ,
2m ' dd a + *'
whence we get one asymptote.
. • 1. /» 3^ du m
Again, when 0 = — , w = o, and — = -,
which gives a second asymptote.
On making d = — , we get a third asymptote, and so on.
2m
It maybe remarked, that the first, third, . . . asymptotes all touch one
fixed circle; and the second, fourth, &c., touch another.
3. Find the equations to the two real asymptotes to the curve
r28in(0 -a) + ar sin (0 - 20) + a^ = o.
uins. r sin (0 - tt) = + a sin o.
207. Asymptotic Circles. — In some curves referred to
polar co-ordinates, when 0 is infinitely great the value of r
tends to a fixed limiting value, and accordinglj the curve
Asymptotic Circles. 253
approaches more and more nearly to the circular form ^t the
same time : in such a case the curve is said to have a circular
asymptote.
For example, in the curve
aO
so long as B is positive r is less than a^ a being supposed
positive; but as B increases with each revolution, r con-
tinually increases, and tends, after a large number of revo-
lutions, to the limit a ; hence the circle described with the
origin as centre, and radius a, is asymptotic to the curve,
which always lies inside the circle for positive values of 0.
Again, if we assign negative values to 0, similar remarks are
applicable, and it is easily seen that the same (jircle is asymp-
totic to the corresponding branch of the curve ; with this
difference, that the asymptotic circle lies within the curve in
the latter case, but outside it in the former. The student
will find no difficulty in applying this method to other
curves, such as
^ aO _ aO' _ a{e + Goae)
** ~ d + sin 0' ^~e' + a'' '*" e + Bmd~'
254 Examples,
Examples.
Find the equations of the real asymptotes to the following curves : —
1. y{a^ - x^) = P{2x + e). Ans. i/ = o, x + a = o, x-a = o.
2. X^ - ap-y^ + fl[2^« + i* = o. x + i/ = o, X — i/=o, x = o.
3. x^ — x^t/^ +x^ + y* — a^ = o. x— i=OyX+i=o,x — p = o, x + t/ = o.
4. (a + a;)2(32-a;2) = a;^y2. x = o.
5. {a + a;)2(52 + a:^) = x^y^. z = o, y = x + a, y + x + a = o.
6. arV - 2a;2f/2 + try^ = a^ajS 4. b'iy'i, x = o, y=o, x — y = ± v o* + b"^.
7. a' — 4iry2 _ ^x^ 4. i2a;y - izy^ + Sx + 2y + 4. = o.
Ans. X + 3 = o, X — ^y ■= o, x + 2y = 6
?. a;2y' — «a;(a; + y)2 — 2a^y^ — a* = o. x — 2a = o^ x + a = o.
9. If the equation to a curve of the third degree he of the form
W3 + Ml + Wo = o,
the lines represented hy W3 = o are its asymptotes.
10. If the asymptotes of a cuhic be denoted by 0 = 0, 3 = 0, 7 = 0, the
equation of the curve may be written in the form
ojSy = Aa'r£$ + Cy.
1 1. In the logarithmic curve
prove that the negative side of the axis of x is an asymptote.
12. Find the asymptotes to the curve
r cos nd = a.
13. Find the asymptotes to
r cos md = a cos n9.
14. Show that the curve represented by
x^ + aby — axy = o
has a parabolic asymptote, x^ + bx + b^ = ay.
Examples. 255
15. Find the circalar asymptote to the curve
16. Find the condition that the three asymptotes of a cuhic should pass
through a common point.
Let the equation of the curve he written in the form
«o + Zhx + 3*1^ + zcqx"^ + ^c\xy + ZCzy^ + dox^ + zdix'^y + zd^xy^ + dzy^ = o,
then the condition is
da, di, d2f
diy «?2, ds,
Co, Ci, C2,
This result can he easily arrived at hy substitutint; x -V a and y + j8 instead
of X and y in the equation of the cuhic, and finding the condition that the part
of the second degree in the resulting equation should vanish. See Art. 204.
17. When the preceding condition is satisfied show that the co-ordinates,
a and $, of the point of intersection of the three asymptotes, are given by the
equations
e\di — Cod^ _ Cod\ — c\do
dodz — di^ d^d^ — d\^
18. If from any point, 0, a right line be drawn meeting a curve of the n*^
degree in i2i, E%, . . . Rn, and its asymptotes in ri, r2, . . . rn, prove that
ORi + Oi?2 + . . . ORn = On + Orz + . . . 0/-„.
N.B. — The terms of the w<^ and {n — i)'* degrees are the same for a curve
and its asymptotes.
19. If a right line be drawn through the point {a, o) parallel to the asymptote
of the cubic {x - a)^ - x'^j/ = o, prove that the portion of the line intercepted by
the axes is bisected by the curve.
20. If from the origin a right line be drawn parallel to any of the asymptotes
of the cubic
y{ax^ + ihxy + ly"^ + igx + 2/y 4- c) - a^ = O,
show that the portion of this line intercepted between the origin and the line
gx -V fy ■\- c = o \& bisected by the curve.
21. If tangents be drawn to the curve a:^ + y3 _ ^3 ixova. any point on the
line y = X, prove that their points of contact lie on a circle.
32. Show that the asymptotes to the cubic
ax'^y + bxy"^ + a'x^ + b'y^ + a'x + b"y = o
are always real, and find their equations.
Ans. bx + b' = o, ay + a' = o,
ab{ax + by) - a^b' - a'b^ = o.
( 256 )
CHAPTER Xiy.
MULTIPLE POINTS ON CURVES.
208. In the following elementary discussion of multiple
points of curves the method given by Dr. Salmon in his
Higher Plane Curves has been followed, as being the simplest,
and at the same time the most comprehensive method for
their investigation. The discussion here is to be regarded as
merely introductory to the more general investigation in that
treatise, to which the student is referred for fuller information
on this as well as on the entire theory of curves.
"We commence with the general equation of a curve of the
^th degree, which we shall write in the form
+ Joa? + hy
+ c^"^ + Cixy + Cay'
+ &0. + &0.
+ 4a;" + haf^-^y + &o. + In't/^ =» o,
where the terms are arranged according to their degrees in
ascending order.
When written in the abbreviated form of Art. 175, the
preceding equation becomes
Wo + Wi + «^ + . . . + w^i + «^« = o.
We commence with the equation in its expanded shape,
and suppose the axes rectangular. Transforming to polar
Multiple Points. 257
co-ordinates, by substituting r cos H and r sin 6/ instead of
X and yy we get "*
«o + (^0 cos d + hi sin &) r
+ (co cos'^O + Ci cos 0 sin 0 + Cg sin^O) r* + . . .
+ (/ocos"0 + /iCOs"-^0 sin0 + . . . + /„sin"0)r'» = o. (i)
If 0 be considered a constant, the n roots of this equation
in r represent the distances from the origin of the n points
of intersection of the radius vector with the curve.
If Gq = o, one of these roots is zero for all values of 0 ; as
is also evident since the origin lies on the curve in this case.
A second root will vanish, if, besides a^ = o, we have
&o cos 0 + 5i sin 0 = o. The radius vector in this case meets
the curve in two consecutive points* at the origin, and is
consequently the tangent at that point.
The direction of this tangent is determined by the
equation
Jo cos 0 + 5i sin 0 = o ;
accordingly, the equation of the tangent at the origin is
b(fio + hxy = o.
Hence we conclude that if the absolute term be wanting
in the equation of a curve, it passes through the origin, and
the linear part (u^ in its equation represents the tangent at
that point.
If Bq = o, the axis of a; is a tangent ; if ^1 = o, the axis
of 2/ is a tangent.
The preceding, as also the subsequent discussion, equally
applies to oblique as to rectangular axes, provided we sub-
stitute mr and nr for iv and y ; where
sin (u)-6) - sin 0
m = ^:^ -, andw=-^ ;
sm (0 sm CO
u) being the angle between the axes of co-ordinates.
From the preceding, we infer at once that the equation of
the tangent at the origin to the curve
iv''{x' -vy^) =a{x-y)
* Two points which are infinitely close to each other on the same branch of
a curve are said to be consecutive points on the curve.
258 Multiple Points on Curves,
is iP - 2/ = o, a line bisecting the internal angle between the
co-ordinate axes. In like manner, the tangent at the origin
can in all cases be immediately determined.
209. Equation of Tangent at any Point. — By aid
of the preceding method the equation of the tangent at any
point on a curve whose equation is algebraic and rational
can be at once found. For, transferring the origin to that
point, the linear part of the resulting equation represents the
tangent in question.
Thus, iif{x, ?/) = o be the equation of the curve, we sub-
stitute X + Xi for aj, and F + yi for y, where {xi, y^) is a
point on the curve, and the equation becomes
/(X + ^x, F+yO=o.
Hence the equation of the tangent referred to the new axes is
-f-) =0.
On substituting x - Xi, and y - 2^1, instead of X and F, we
obtain the equation of the tangent referred to the original
axes, viz.
This agrees with the result arrived at in Art. 169.
210. Double Points. — If in the general equation of a
curve we have % = 0, b^ = o, hi = o, the coefficient of r is
zero for all values of 0, and it follows that all lines drawn
through the origin meet the curve in two points, coincident
with the origin.
The origin in this case is called a double point.
Moreover, if Q be such as to satisfy the equation
<?o cos*0 + Cicos 0 sin 0 + c^ sin^0 = o, (2)
the coefficient of r* will also disappear, and three roots of
equation (i) will vanish.
As there are two values of tan Q satisfying equation (2), it
follows that through a double point two lines can be drawn,
each meeting the curve in three coincident points.
Double Points. 259
The equation (2), when multiplied by-^r', becomes
CqOi^ + Cixy + C2?/^ = o.
Hence we infer that the lines represented by this equa-
tion connect the double point with consecutive points on the
curve, and are, consequently, tangents to the two branches of
the curve passing through the double point.
Accordingly, when the lowest terms in the equation of a
curve are of the second degree (wo), the origin is a double
point, and the equation u% = o represents the pair of tangents at
that point.
For example, let us consider the Lemniscate, whose equa-
tion is
(x^ + y''y = (C^{x^ -y").
On transforming to polar co-ordinates its equation becomes
r* = ah'"^ {cos^O - sin'^^), or, r' = «^ cos 2O.
Now, when 9 = o, r = ±a;
and, if we confine our atten-
tion to the positive values of
r, we see that as 0 increases
from o to -, r diminishes
4
from a to zero. When 0 >-
and< — , r is imaginary, &c., Fig- '8.
and it is evident that the figure of the curve is as annexed,
having two branches intersecting at the origin, and that the
tangents at that point bisect the angles between the axes.
The equations of these tangents are
X + y = o, and x - y = o,
results which agree with the preceding theory.
211. Socles, Cusps, and Conjugate Points.* — The
pair of lines represented by ^^2 = o will be real and distinct,
coincident, or imaginary, according as the roots of equa-
tion (2) are real and unequal, real and equal, or imaginary.
* These have been respectively styled crunodes, spinodes, and acnodes, by
Professor Cayley. See Salmon's Higher Plane Curves^ Art 38.
S 2
26o Multiple Points on Curves.
r^^-J^
Hence we conclude that there may be one of three kinds
of singular point on a curve so far as the vanishing of Uq and Wi
is concerned.
(i). For real and unequal roots, the
tangents at the double point are real
and distinct, and the point is called a
node; arising from the intersection of V^___^^j'^'^\n^
two real branches of the curve, as in y"^ NX
the annexed figure. \ ^
(2). If the roots be equal, i.e. if U2 j,-
be a perfect square, the tangents coin-
cide, and the point is called a cusp ; the
two branches of the curve touching each ^^^
other at the point, as in figure 20.
(3). If the roots of iti be imaginary,
the tangents are imaginary, and the
double point is called a conjugate or '^* *°'
isolated point ; the co-ordinates of the point satisfy the equation
of the curve, but the curve has no real points consecutive to
this point, which lies altogether outside the curve itself.
It should be observed also that in some cases of singularities
of a higher order, the origin is a conjugate point even when u^
is a perfect square, as will be more fully explained in a sub-
sequent chapter.
We add a few elementary examples of these different
classes for illustration.
Examples.
1. y2 («2 + a;2) = ^2 (^2 _ a.2).
Here the origin is a node, the tangents bisecting the angles between the axes of
co-ordinates.
2. ay^ = x^.
In this case the origin is a cusp. Again, solving for y ve get
Hence, if a be positive, p beeomes imaginary for negative values of x ; and,
accordingly, no portion of the curve extends to the negative side of the axis of a:.
Moreover, for positive values of a:, the corresponding values of y have opposite
signs. This curve is called the semi-cubical parabola. The form of the curve
near the origin is exhibited in Fig. 20.
Double Points in General. 261
3. y* = x^ {x + a).
Am. The origin is a cusp.
4. • h{x'i\y^)=x\
Ans. The origin is a conjugate point.
5. «* - 3«^y + 2/3 = o.
Ans. The two branches at the origin touch the co-ordinate axes.
212. Double Points in Ceneral. — In order to seek
the double points on any algebraic curve, we transform the
origin to a point {xx^ y\) on the curve ; then, if we can deter-
mine values of Xi, y^ for which the linear part disappears from
the resulting equation, the new origin (a;i, y^ is a double point
on the cuTve.
From Art. 209 it is evident that the preceding conditions
give
moreover, since the point (iPi, y^ is situated on the curve,
we must have
/(^i, y^ = o.
As we have but two variables, a^i, ^i, in order that they
should satisfy these three equations simultaneously, a con-
dition must evidently exist between the constants " in the
equation of the curve, viz., the condition arising from the
elimination of x^^ y^ between the three preceding equations.
Again, when the curve has a double point (i^i, yi), if the
origin be transferred to it, the part of the second degree in
the resulting equation is evidently
'^'(SV^^(S).^^^(S,
Accordingly, the lines represented by this quadratic are
the tangents at the double point.
^ The point consequently is a node, a cusp, or a conjugate
point, according as
(dhi\. ((Pu
\dxdy)\ \dx^
A fd''u\
'JAdfj:
262
Multiple Points on Curves.
It may be remarked here tliat no cubic can have more
than one double point ; for if it have two, the line joining
them must be regarded as cutting the curve in four points,
which is impossible.
Again, every line passing through a double point on a cubic
must meet the curve in one, and but one, other point ; ex-
cept the line be a tangent to either branch of the cubic at
the double point, in which case it cannot meet the curve else-
where; the points of section being two consecutive on one
branch, and one on the other branch.
In many cases the existence of double points can be seen
immediately from the equation of the curve. The following
are some easy instances : —
Examples.
To find the position and nature of the double points in the following
curves : —
I. {bx- cyY = {x- o)5.
The point a; = a, y = — , is evidently a cusp,
at -which bx - cy = o\& the tangent, as in the
accompanying figure
2.
{^-cY = {x-aY{x-b).
The point x = a, y = e, is &. cusp if « > i, or 0
if a = i ; but is a conjugate point ii a<b.
Fig. 21.
3. y'^ = x{x + a)2.
The point y = o, a; = -aisa conjugate point.
4. id + 2/* = «'.
The points x=o^y = ±a; and y = o, a; = + a, are easily seen to be cusps.
213. Parabolas of the Third Degree. — The follow-
ing example* will assist the student towards seeing the dis-
tinction, as well as the connexion, between the different kinds
of double points.
Let y^ = {x - a) [x -h) {x - c)
be the equation of a curve, where a<h < c.
♦ Lacroix, Cal. Bif.^ pp. 395-7. Salmon's Higher Plane Curves, Art. 39.
Parabolas of the Third Degree.
263
Here y vanishes when x = a^ OTx=b, ot x=c] accordingly,
if distances OA = a, OB = b, OC = c, be taken on the axis of
X, the curve passes through the points A, B, and C.
Moreover, when x < a, y^ is negative, and therefore
y is imaginary.
„ x> a^ and < J, y"^ is positive, and therefore
y is real.
„ x>b^ and < c, y"^ is negative, and therefore
y is imaginary.
„ x> Cy y"' is positive, and therefore
increases indefinitely along with x.
Hence, since the curve is sym-
metrical with respect to the axis of
X, it evidently consists of an oval
lying between A and B, and an
infinite branch passing through
(7, as in the annexed figure. It
is easily shown that the oval is
not symmetrical with respect to
the perpendicular to AB at its
middle point. Again, \ib = c, the
equation becomes
y^ = {x^ a){x - by.
In this case the point B co-
incides with (7, the oval has ,
joined the infinite branch, and o
B has become a double point,
as in the annexed figure.
On the other hand, let b
y^= {x-ay{x-c)]
in this case the oval has shrunk
into the point -4, and the curve
is of the annexed form, having o
A for a conjugate point.
Next, \Qt a = b = c, and the
equation becomes
2/^= {x-aY\
y
is real ; and
Fig. 22.
A
Fig. 23.
^, and the equation becomes
Fig. 14.
264 Multiple Points on Curves.
here the points A, JB, C, have
come together, and the curve
has a cusp at the points, as in
the annexed figure.
The curves considered in
this Article are called parabolas Fig. 25.
of the third degree.
As an additional example, we shall investigate the fol-
lowing problem : —
214. Given the three aBymptotes of a cubic, to find its equa-
tion, if it have a double point.
Taking two of its asj^mptotes as axes of co-ordinates, and
supposing the equation of the third iohQ ax + by + c = o, the
equation of the cubic, by Art. 204, is of the form
xy [ax + by + c) = Ix + my + n.
Again, the co-ordinates of the double point must satisfy
the equations
du _ du _
dx ^ dy '
or [lax + by -^ c) y = I, {ax + ihy + c) x = m;
from which I and m can be determined when the co-ordinates
of the double point are given.
To find w, we multiply the former equation by x, and the
latter by y, and subtract the sum from three times the equa-
tion of the curve, and thus we get
cxy = zlx + zmy + t^u ;
from which n can be found.
In the particular case where the double point is a cusp,*
its co-ordinates must satisfy the additional condition
d^(Pu _ f d^u Y
dx^ dy"^ \dxdy) *
or {2ax + 2by + cy = ^abxy,
and consequently the cusp must lie on the conic represented
by this equation.
* It is essential to notice that the existence of a cusp involves one more
relation among the coefficients of the equation of a curve than in the case of an
ordinary double point or node.
Double Points m a Cubic. 265
It can be easily seen that this conic* touches at their
middle points the sides of the triangle formed by the asymp-
totes.
The preceding theorem is due to Plucker,t and is stated
by him as follows : —
" The locus of the cusps of a system of curves of the third
degree, which have three given lines for asymptotes, is the
maximum ellipse inscribed in the triangle formed by the
given asymptotes."
It can be easily seen that the double point is a node or a
conjugate point, according as it lies outside or inside the
above-mentioned ellipse.
215. Multiple Points of Higher Curves. — By follow-
ing out the method of Art. 208, the conditions for the existence
of multiple points of higher orders can be readily determined.
Thus, if the lowest terms in the equation of a curve be of
the third degree, the origin is a triple point, and the tangents
to the three branches of the curve at the origin are given by
the equation Uz = o.
The different kinds of triple points are distinguished,
according as the lines represented by Uz = o are real and
distinct, coincident, or one real and two imaginary.
In general, if the lowest terms in the equation of a curve
be of the m*^ degree, the origin is a multiple point of the m^^
order, &c.
Again, a point is a triple point on a curve provided that
when the origin is transferred to it the terms below the third
degree disappear from the equation. The co-ordinates of a
triple point consequently must satisfy the equations
du du d^u d/^u d^u
"=°' ;& = °' ^=°' ^=°' ;&^=°' ^=°-
Hence in general, for the existence of a triple point on a
curve, its coefficients must satisfy four conditions.
The complete investigation of multiple points is effected
♦ From the form of the equation we see that the lines a; = o, 1/ = o are
tangents to the conic, and that 2ax + zb?/ + c = o represents the line joining the
points of contact ; but this line is parallel to the third asymptote ax + bj/ + c = o,
and evidently passes through the middle points of the intercepts made by this
asymptote on the two others.
t Liouville's Journal^ vol. ii. p. 14.
266 Multiple P/tnts on Curves.
more satisfactorily by introducing the method of trilinear co-
ordinates. The discussion of curves from this point of view is
beyond the limits proposed in this elementary Treatise.
215 {a). Cusps, in Greneral. — Thus far singular points
have been considered with reference to the cases in which
they occur most simply. In proceeding to curves of higher
degrees they may admit of many complications ; for instance
ordinary cusps, such as represented in Fig. 20, may be called
cusps of the first species, the tangent
lying between both branches : the cases in
which both branches lie on the same side,
as exhibited in the accompanying figure,
may be called cusps of the second species. p. ^5
Professor Cayley has shown how this is
to be considered as consisting of several singularities happen-
ing at a point (Salmon's Higher Plane Curves, Art. 58).
Again, both of these classes may be called single cusps,
as distinguished from double cusps extending on both sides of
the point of contact. Double cusps are styled tacnodes by
Professor Cayley. These points are sometimes called points
of osculation ; however, as the two branches do not in general
osculate each other, this nomenclature is objectionable. It
should be observed that whenever we use the word cusp with-
out limitation, we refer to the ordinary cusp of the first species.
Cusps are calledpom^s de rehroussement by French writers,
and Eiickkehrpunkte by Q-ermans, both expressing the turning
backwards of the point which is supposed to trace out the
curve; an idea which has its English equivalent in their
name of stationary points. A fuller discussion of the different
classes of cusps will be given in a subsequent place. We
shall conclude this chapter with a few remarks on the multiple
points of curves whose equations are given in polar co-ordi-
nates.
Examples.
1. (y - ir»)2 = a;5.
Here the origin is a cusp ; also
y = a;2 + a;5 ;
hence, when a; is less than unity, both values of y are positive, and consequently
the cusp is of the second species.
2. Show that the origin is a double cusp in the curve
re* + 4x* — a^y^ = o.
Multiple Points with Polar Co-ordinates, 267
216. multiple Points of Curves in Polar Co-ordi-
nates.— If a curve referred to polar co-ofSinates pass through
the origin, it is evident that the direction of the tangent at
that point is found by making r = o in its equation ; in this
case, if the equation of the curve reduce to f{d) = o, the
resulting value of d gives the direction of the tangent in
question.
If the equation /(0) = o has two real roots in Q, less than tt,
the origin is a double point, the tangents being determined
by these values of d.
If these values of Q were equal, the origin would be a cusp ;
and so on.
In fact, it will be observed that the multiple points on
algebraic curves have been discussed by reducing them to
polar equations, so that the theory already given must apply
to curves referred to polar, as well as to algebraic co-ordi-
nates.
It may be remarked, however, that the order of a multiple
point cannot, generally, be determined unless with reference
to Cartesian co-ordinates, in like manner as the degree of a
curve in general is determined only by a similar reference.
For example, in the equation
r = a coa^O - b ahi^O,
the tangents at the origin are determined by the equation
tan 6 = ± Jt, and the origin would seem to be only a double
point ; however, on transforming the equation to rectangular
axes, it becomes
{x' + fy={ax'-bfy;
from which it appears that the origin is a multiple point of the
fourth order, and the curve of the sixth degree. In fact,
what is meant by the degree of a curve, or the multiplicity of
a point, is the number of intersections of the curve with any
right line, or the number of intersections which coincide for
every line through such a point, and neither of these are at
once evident unless the equation be expressed by line co-ordi-
nates, such as Cartesian, or trilinear co-ordinates; whereas
in polar co-ordinates one of the variables is a circular co-
ordinate.
263 Examples.
Examples.
1. Determme the tangents at the origin to the curve
j/"^ = a;2 (i — x^). Ans. a; + y = o, ar — y = o.
2. Show that the curve
a;* - S"^P + y* = o
touches the axes of co-ordinates at the origin.
3. Find the nature of the origin on the curve
a;* - az^i/ + bi/^ = o,
4. Show that the origin is a conjugate point on the curve
ay^ — 0:^ + 5x^ = 0
when a and b have the same sign ; and a node, when they have opposite signs.
5. Show that the origin is a conjugate point on the curve
6. Prove that the origin is a cusp on the curve
(y - a;2)2 = a^.
7. In the curve
(y-a;2)2 = a:«,
show that the origin is a cusp of the first or second species, according as n is
< or > 4.
8. Find the numher and the nature of the singular points on the curve
X* + 4ax^ -2ai/^ + 4a^x^ - "^c^y^ + 4a* = o,
9. Show that the points of intersection of the curve
0'* ©'
with the axes are cusps.
10. Find the douhle points on the curve
x^ - 4aa;^ + \c?x^ — h'^y'^ + ^h'^y — «* — A* = O.
Examples. 26g
11. Prove that the four tangents from the origin to the curve
wi + wa + «3 = o
are represented by the equation 4W1 us = w".
12. Show that to a double point on any curve corresponds another double
point, of the same kind, on the inverse curve with respect to any origin.
13. Prove that the origin in the curve
x* - zax^y - axy^ + a'^y^ = o
is a cusp of the second species.
14. Show that the cardioid
r = a (i + cos d)
has a cusp at the origin.
15. If the origin be situated on a curve, prove that its first pedal passes
through the origin, and has a cusp at that point.
16. Find the nature of the origin in the following curves: —
r^ = a^ sin 30, r^ = a** sin nd, r = — .
bu + c
17. Show that the origin is a conjugate point on the curve
X* - ax^y + axy"^ + a^y"^ = o.
18. If the inverse of a conic be taken, show that the origin is a double point
on the inverse curve; also that the point is a conjugate point for an ellipse, a
cusp for a parabola, and a node for a hyperbola.
19. Show that the condition that the cubic
xy"^ + ax^ + bx'^ + CX+ d + 2ey = o
may have a double point is the same as the condition that the equation
aa^ + hx^ + cx^ + dx- 6^ = 0
may have equal roots.
20. In the inverse of a curve of the n*^ degree, show that the origin is a
multiple point of the w^'» order, and that the n tangents at that point are parallel
to the asymptotes to the original curve.
( 270 )
CHAPTEE XY.
ENVELOPES.
217. method of Envelopes. — If we suppose a series of
iifferent values given to a in the equation
/(^, y, a) = o, (i)
the for each value we get k distinct curve, and the ahove
equation may be regarded as representing an indefinite
number of curves, each of which is determined when the
corresponding value of a is known, and varies as a varies.
The quantity a is called a variable parameter, and the
equation /(;r, y, a) = o is said to represent a. family of curves;
a single determinate curve corresponding to each distinct
value of a ; provided a enters into the equation in a rational
form only.
If now we regard a as varying continuously, and suppose
the two curves
/(^, y, a) = O, f(x, 2/, a + Aa) = O
taken, then the co-ordinates of their points of intersection
satisfy each of these equations, and therefore also satisfy the
equation
f{x, y, g + Aa) -/(a?, y, a)
Aa
Now, in the limit, when Aa is infinitely small, the latter
equation becomes
*%l^ = o; (2)
da
and accordingly the points oi" intersection of two infinitely
near curves of the system satisfy each of the equations (i)
and (2).
Envelopes. 271
The locus of the points of ultimate intersection for the
entire system of curves represented by /(a?, y, a) = o, is ob-
tained by eliminating a between the equations (i) and (2).
This locus is called the envelope^ of the system, and it can Joe
^asily_ seen. that it is touched ty e very" cur veoFthe systejn*.
For, if we consider three consecutive curves, and suppose
Pi to be the point of intersection of the first and second, and
P2 that of the second and third, the line Pi P2 joins two infi-
nitely near points on the envelope as well as on the inter-
mediate of the three curves ; and hence is a tangent to each
of these curves.
This result appears also from analytical considerations,
thus : — the direction of the tangent at the point x, y, to the
curve /(ii;, 7/, a) = o, is given by the equation
dx dy dx
in which a is considered a constant.
Again, if the point x, y be on the envelope, since then a
is given in terms of x and y by equation (2), the direction of
the tangent to the envelope is given by the equation
df df dy df fda da dy\ _
dx dy dx da\dx dy dx) '
df df dy
or -f-^^T-^^'
dx dy dx
7/*
since -^ = o for the point on the envelope.
da
Consequently, the values of -^ are the same for the two
(ZX
curves at theircomm^^ppint, and hence they have a common
tangent aFiEat pointr^""^
One or two elementary examples will help to illustrate
this theory.
The equation x eoQa + y sina = p, in which a is a variable
parameter, represents a system of lines situated at the same
272 Envelopes.
perpendicular distance p from the origin, and consequently
all touching a circle.
This result also follows from the preceding theory ; for
we have
/(^> ?/, a) = a; cos a + y sin a - ^ = O,
df(x. y, a) .
-^ — • = - xsiaa + y cos a = O,
da
and, on eliminating a between these equations, we get
x^ + y''=p\
which agrees with the result stated above.
Again, to find the envelope of the line
m
y = aaj + — ,
.a
where a is a variable parameter.
Here f{x, yy a) = y - ax = 0,
a
df(x, y, a) ^^^ ^. . /^
= -«! + — = 0; ,\a= /— .
da a' \ X
Substituting this value for a, we get for the envelope
which represents a parabola.
" 21 8. Envelope of La^ + 2Ma + N= o. Suppose Z, M, iV,
to be known functions of x and y, and a a parameter, then
f{x, y, a) = Lv? + iMa + iV= O,
-7- = iLa + 2JZ = o;
da
accordingly, the envelope of the curve represented by the
preceding expression is the curve
Undetermined Multipliers applied to Envelopes. 273
Hence, when L, M, N are Knear functions in x and y,
this envelope is a conic touching the lines L, N, and having
M for the chord of contact.
Conversely, the equation to any tangent to the conic
LN - M"^ can be written in the form
La^ + 2Ma + ]Sr=0*
where a is an arbitrary parameter.
219. ITndeternimecl multipliers aiiplied to Enve-
lopes.— In many cases of envelopes the equation of the
moving curve is given in the form
f{x,i/, a, ^) =c„ (3)
where the parameters a, /3 are connected by an equation of
the form '
^ (a, (5) = c„ (4)
In this case we may regard /3 in (3) as a function of a by
reason of equation (4) ; hence, differentiating both equations,
the points of intersection of two consecutive curves must
satisfy the two following equations :
df df d[5 , d(b d(h d3
da dp da da dp da
df_ d£
Consequently g = ^.
da df5
If each of these fractions be equated to the undetermined
quantity X, we get
^ = X^
da da
rf/3 dfi
* Salmon's Conies, Art. 270.
T
274 Envelopes.
and the required envelope is obtained by eliminating a, /3, and
X between these and the two given equations.
The advantage of this method is especially found when
the given equations are homogeneous functions in a and /3 ;
for suppose them to be of the forms
/(a?, y, o, P) = Ci, 0 (a, (5) = c,,
where the former is homogeneous of the w'* degree, and the
latter of the m*^, in a and /3. Multiply the former equation
in (5) by a, and the latter by /3, and add ; then, by Euler's
theorem of Art. 102, we shall have
nci = mCiX. or A = — ^ , (6)
mcz
by means of which value we can generally eliminate a and /3
from our equations.
Examples.
I. To find the envelope of a line of given length (a) whose extremities move
along two fixed rectangular axes.
Taking the given lines for axes of co-ordinates, we have the equations
et p
Hence
^,=K., | = ^A
from which we get
I
.-. a =(«%)!, fl=(<>V)',
and the required locus is represented hy ^^^
CEZEE^L^
2. To find the envelope of a system of concentric and coaxal ellipses of con-
slant area.
Here ~: + •z^= i, a)8 = c ;
o» )8"-
hence — , = A/S, -— = Ao ; .•. r\e = 1
and the required envelope is the equilateral hyperbola
2x1/ = c.
Examples. • 275
3. To find the envelope of all the normals to an ellipse.
Here we have the equations
where a and B are the co-ordinates of any point on the ellipse.
Hence, -_ = A-, -_ = _x--
fl2' i82 i2
consequently A. = «^ — b\
and we get a*x = (a« - J2) a', b^y = - (a^ - i2)j8';
•'*</ \a'i-bi)' b~ \a-i-by'
Substituting in the equation of the ellipse, we get for the required envelope,
(aa;)l + (%)l = {a^ - *2)l.
This equation represents the evolute of the ellipse.
X y
4. Find the envelope of the line - + — = i» where a and j8 are connected by
a B
the equation
m M m
a»» 4. jgm = c»». Am. a;*"*^ + y»»»+^ = <;*»+^
2 20. The preceding method can be readily extended to the
general case in which the equation of the enveloping curve
contains any number, w, of variable parameters, which are
connected by w - i independent equations. The method of
procedure is the same as that already considered in Chapter
XI. on maxima and minima, and does not require a separate
investigation here.
T 2
2^6 Examples,
Examples.
I. Prove that the envelope of the system of lines - + — = i, where I and m
I m
I tn
are connected by the equation - + — = i, is the parabola
a
©'* (I)'-
2. One angle of a triangle is fixed in position, find the envelope of the
opposite side when the area is given. Jlns. A hyperbola.
3. Find tbe envelope of a right line when the sum of the squares of the
perpendiculars on it from two given points is constant.
4. Find the envelope of a right line, when the rectangle under the perpen-
diculars from two given points is constant.
Ans, A conic having the two points as foci.
5. From a point P on the hypothenuse of a right-angled triangle, perpen-
diculars FM, PJV" are drawn to the sides ; find the envelope of the line MK.
6. Find the envelope of the system of circles whose diameters are the chords
drawn parallel to the axis-minor of a given ellipse.
7. Find the envelope of the circle
xi + i/i- 2aex + o' - J2 = o,
where a is an arbitrary parameter ; and find when the contact between the
circle and the envelope is real, and when imaginary.
(a). Show from this example that the focus of an ellipse may be regarded as
an infinitely small circle having double contact with the ellipse, the directrix
being the chord joining the points of contact.
8. Show that the envelope of the system of conies
a a — h
where a is a variable parameter, is represented by the equation
(X ± ^hf + 2/2 = o.
Hence show that a system of conies having the same foci may be regarded
as inscribed in the same imaginary quadrilateral.
9. Find the envelope of the line
iro'» + yj3"» = a«+^
where the parameters a and /5 are connected by the equation
Am. «»-»» +
Examples, 277
10. On any radius vector of a curve as diameter a circle is described : prove
geometrically that the envelope of all such circles itf the first pedal of the curve
with respect to the origin.
11. If circles be described on the focal radii vectores of a conic as diameters,
prove that their envelope is the circle described on the axis major of the conic as
diameter.
12. Prove that the envelope of the circles described on the central radii of an
ellipse as diameters is a Lemniscate.
13. Find the envelope of semicircles described on the radii of the curve
r»» = a** cos nd
as diameters.
14. If perpendiculars be drawn at each point on a curve to the radii vectores
drawn from a given point, prove that their envelope is the reciprocal polar of
the inverse of the given curve, with respect to the given point.
15. Find the envelope of a circle whose centre moves along the circum-
ference of a fixed circle, and wbich touches a given right line.
16. Ellipses are described with coincident centre and axes, and having the
sum of their semiaxes constant ; find their envelope.'
17. Find the equation of the envelope of the line Aa; + ^y + v = o, where
the parameters are connected by the equation
a\« + b/x^ + cv^ + ^ffiv + 2gvK + zhKfi = O.
a, h, ff, X
h, b, /, y
Ans.
18. At any point of a parabola a line is drawn making with the tangent an
angle equal to the angle between the tangent and the ordinate at the point ;
prove that the envelope of the line is the first negative pedal, with regard to the
focus, of the parabola ; and hence that its equation is ri cos -0 = ai, the focus
being pole.
N.B. — This cTirve is the caustio by reflexion for rays perpendicular to the
axis of the parabola.
19. Join the centre, 0, of an equilateral hyperbola to any point, P, on the
curve, and at P draw a line, PQ, making with the tangent an angle equal to the
angle between OP and the tangent. Show that the envelope of PQ is tlie first
negative pedal of the curve
r^ = 2a* sin -d sin - Q.
3 3
the centre being pole, and axis vamox prime vector.
N.B. — This gives the caustic by reflexion of the equilateral hyperbola, the
centre being the radiant point.
20. A right line revolves with a uniform angular velocity, while one of its
points moves uniformly along a fixed right line ; find its envelope.
4w*. A cycloid,
( 278 )
CHAPTER XVI.
CONVEXITY AND CONCAVITY. POINTS OF INFLEXION.
221. Convexity and Concavity. — If the tangent be
drawn at any point on a curve, the neighbouring portion of
the curve generally lies altogether on one side of the tangent,
and is convex with respect to all points lying at the opposite
side of that line, and concave for points at the same side.
Thus, in the accompanying figure, the portion QPQf~w^^
convex towards all points
lying below the tangent, and
concave for points above.
If the curve be referred
to the co-ordinate axes OX
and OY, then whenever the
ordinates of points near to
P on the curve are greater
than those of the points on
the tangent corresponding to ^^' ^^'
the same abscissae, the curve is said to be concave towards
the positive direction of Y.
Now, suppose 2/ = 0 (^) ^^ ^^ ^^Q equation of the curve,
then that of the tangent at a point x, y, by Art. i68, is
Y-y=(X-x)
dx
Let P be the point x, y, and MN =h = MN\ QN = j/i,
TN = Fi, and we have
2/1 = 0(aj + h) = <l>(x) + h(l>\x) + 4,'\x) + — - — <^'"{x) +&o.
Fi*= y + h<^\x) = ^{x) + h(^\x) ;
/. y. - F. = ^^ ^'\x) + Y^/\^) + &o. (I)
Points of Inflexion,
279
When h is very small, the sign of the right-hand side of
this equation is the same in general as that of its first term,
and accordingly the sign of 3/1 - Zi, or of QT, is the same as
/) fchatof_£(;r).
' ' * Hence, for a point abjave th e jixis^f av the curxe ia.coiurex
to;w-arJs~that axis when (p'^{x) is positive, and concave when
nQggJjigie- *"
We accordingly see that the convexity or concavity at any
point depends on the sign of ^"(ic) or -7^, at the point.
-i
222. Points of Inflexion.-
the point P, we shall have
-If, however, ^''(a?) = o at
2-5 .4
0^^(a;) + &c. {2]
Fig. 28.
Such points on a curve are called points of
Now, provided ((/^\x) be not zero, 1/1 - Yx changes its sign
with h, i.e. if MN' = MN= h,
and if Q lies above T, the
corresponding point Q[ lies
below T\ and the portions of
the curve near to P lie at
opposite sides of the tangent,
as in the figure.
Consequently, the tangent
at such a point cuts the curve,
as well as touches it, at its
point of contact,
inflexion.
Again, if <^"\x) as well as <^\x) vanish at the point P, we
shall have
A*
1.2.3.4
and, provided <^^'^{sc) be not zero at the point, yx - *Yx does not
change sign with ^, and accordingly the tangent does not
intersect the curve at its point of contact.
Generally, the tangent does or does not cut the curve at
its point of contact, according as the first derived function
which does not vanish is of an odd, or of an even order ; as
can be easily seen by the preceding method.
28o Points of Inflexion.
From the foregoing discussion it follows that at a point
of inflexion the curve changes from convex to concave with
respect to the axis of x, or conversely.
On this account such points are called points of contrary
flexure or of inflexion.
22 s The subject of inflexion admits also of being treated
by the method of Art. 196, as follows : — The points of in-
tersection of the line y = /ux + v with the curve y = ^{x) are
evidently determined by the equation
^{x) =fix + v. {3)
Suppose A,B, C, 2), &c., to represent the points of section in
question, andleta^i, x^, .. . Xn
be the roots of equation (3) ; —^
then the line becomes a f-^
tangent, if two of these
roots are equal, i.e., if ^ig- ^g-
^'(^1) = jM, where Xi denotes the value of x belonging to the
point of contact.
Again, three of the roots become equal if we have in
addition (l>\xi) = o ; in this case the tangent meets the curve
in three consecutive points, and evidently cuts the curve at its
point of contact ; for in our figure the portions PA and CD
of the curve lie at opposite sides of the cutting line, but
when the points Aj B, G become coincident, the portions A£
and BC become evanescent, and the curve is evidently cut as
well as touched by the line.
In like manner, if (l>"\xi) also vanish, the tangent must
be regarded as cutting the curve in four consecutive points :
such a point is called o, point of undulation.
It may be observed, that if a right line cut a continuous
branch of a curve in three points. A, B, (7, as in our figure,
the curve must change from convex to concave, or conversely,
between t]ie extreme points A and (7, and consequently it
must have a point of inflexion between these points ; and so
on for additional points of section.
Again, the tangent to a curve of the n^^ degree at a point of
inflexion cannot intersect the curve in more than w - 3 other
points: for the point of inflexion counts for three among
the points of section. For example, the tangent to a curve
Harmonic Polar of a Point of Inflexion on a Cubic, 281
of the third degree at a point of inflexion cannot meet the
curve in any other point. Consequently^ if a point of in-
flexion on a cubic be taken as origin, and the tangent at it
as axis of x, the equation of the curve must be of the form
a-^ + ?/0 = o,
where 0 represents an expression of the second and lower
degrees in x and y. For, when y = o, the three roots of the
resulting equation in x must be each zero, as the axis of x
meets the curve in three points coincident with the origin.
The preceding equation is of the form
or, when written in full,
x^ + y {ax^ + ihxy + hy"^) + y {zgx + ify + c) «= o. (4)
Now, supposing tangents drawn from the origin to the
curve, their points of contact, by Art. 176, lie on the curve
U-i + 2Ui = o,
i.e. on the curve
(g^+fy + c)y = 0,
The factor y = o corresponds to the tangent at the point
of inflexion, and the other factor gx + Jy + c = o passes
through the points of contact of the three other tangents to
the curve.
Hence, we infer that from a point of inflexion on a cubic
but three tangents can be drawn to the curve y and their three
2mnts of contact lie in a right line.
It can be shown that this right line cuts harmonically
every radius vector of the curve which passes through the
point of inflexion.
For, transforming equation (4) to polar co-ordinates, and
dividing by r, it becomes of the form
Ar + JBr+ C = o.
If /, /' be the roots of this quadratic, we have
r r V
282 Points of Inflexion,
Now, if p be the harmonic mean between / and r", this
gives
2 I I 5 _ 2/7 cos 0 + if sin d
Hence the equation of the locus of the extremities of the
harmonic means is
gx + fy -^ c = o, Q.E.D.
This theorem is due to Maclaurin [De Lin. Geom. Prop.
Gen.y Sec. iii. Prop. 9).
From this property the line is called the harmonic polar of
the point of inflexion. This line holds a fundamental place
in the general theory of cubics.*
224. /Stationary Tangents. — Since the tangent at a
point of inflexion may be regarded as meeting the curve in
three consecutive points, it follows that at such a point the
tangent does not alter its position as its point of contact
passes to the consecutive point, and hence the tangent in this
case is called a stationary tangent.
The equation -7-^ = 0 follows immediately from the last
ax
consideration ; for when the tangent is stationary we must
fIfK
have -J- = o, where (f>, as in Art. 171, denotes the angle
ax
dt
which the tangent makes with the axis of x ; but tan (jt = —^
d'^y
hence -p^ = o, which is the same condition for a point of
ax
inflexion as that before arrived at.
* Chasles, Aper^u His tongue, note xx. ; Salmon's Higher Plane Curves.
Art. 179.
Examples, 283
Examples.
1. Show that the origin is a point of inflexion on the curve
a^y = bxy + cx^ + dx*,
2. The origin is a point of inflexion on the cubic M3 + mi = o ?
3. In the curve a^'^y = a?'",
prove that the origin is a point of inflexion if m be greater than 2.
4. In the system of curves
find under what circumstances the origin is (a) a point of inflexion, (b) a cusp.
5. Find the co-ordinates of the point of inflexion on the curve
«' - 3^0:2 4. a^y = o. Ans. x = b, y = — .
6. If a curve of an odd degree has a centre, prove that it is a point of
inflexion on the curve.
7. Prove that the origin is a point of undulation on the curve
Ui + Ui + U5 + &C., + M» = O.
8. Show that the points of inflexion on curves refeired to polar co-ordinates
are determined by aid of the equation
d^u ^ I
^ + -7;:^ = °i where « = -,
de^ r
9. In the curve rd»* = a, prove that there is a point of inflexion when
d =\^m (i - m).
10. In the curve y = c sin -, prove that the points in which the curve
a
meets the axis of x are all points of inflexion.
11. Show geometrically that to a node on any curve corresponds a Hue
touching its reciprocal polar in two distinct points ; and to a cusp corresponds a
. point of inflexion.
284 Examples.
12. If the origin be a point of inflexion on the curre
«l + «» + M8 + . ..+•«■ = o,
proye that «2 must contain t/i as a factor.
13. Show that the points of inflexion of the cubical parabola
y»=(*-a)«(*-i)
lie on the line
3x + a = 4* :
and hence proTe that if the cubic has a node, it has no real point of inflexion ;
but if it has a conjugate point, it has two real points of inflexion, besides that
at infinity.
14. Prove that the points of inflexion on the curve y^ = x'(aE* + ^px + y)
are determined by the equation zx* 4 6px^ + 3 {p- -^ q) x ■\- 2pq = o.
15. If y* = /(x) be the equation of a curve, prove that the abscissae of its
points of inflexion satisfy the equation
16. Show that the maximum and minimum ordinates of the cnrye
y=2/(x)r(x) -{/(*)}»
correspond to the points of intersection of the curve y* =/(x) with the axis
ofx.
17. "When y* =/(x) represents a cubic, prove that the biquadratic in x
which determines its points of inflexion has one^ and but one, pair of real roots.
Prove also that the lesser of these roots corresponds to no real point of inflexion,
while the greater corresponds, in general, to two.
18. Prove that the point of inflexion of the cubic
fly* + 3ixy- -h :«x2y + dj^ + 3«b* = o
lies in the right line ay + bx = o, and has for its co-ordinates
2a-e , %abe
, = -— ,andy=— ,
where O is the same as in Example 32, p. 190.
19. Find the nature of the double point of the curve
y= = (x - 2)' {X - 5),
and show that the curve has two real points of inflexion, and that they subtend
a right angle at the double point.
20. The co-ordinates of a point on a curve are given in terms of an angle •
by the equations
« = sec3e, y = tan0Eec^fl;
prove that there are two finite points of inflexion on the curve, and find the
values of B at these point*.
{ 285 )
CHAPTER XVn.
RADIUS OF CURVATURE. E VOLUTES. CONTACT. RADII OF
CURVATURE AT A DOUBLE POINT.
225. Corvatore. Angle of Contingence. — Every con-
tinuous curve is regarded as having a determinate curvature
at each point, this curvature being greater or less according
as the curve deviates more or less rapidly from the tangent at
the point.
The total curvature of an arc of a plane curve is measured
by the angle through which it is bent between its extremities —
that is, by the external angle between the tangents at these
points, assuming that the arc in question has no point of in-
flexion on it. This angle is called the angle of contingence of
the arc.
The curvature of a circle is evidently the same at each of
its points.
To compare the curvatures of
different circles, let the arcs AB
and ah of two circles be of equal
length, then the total curvatures
of these arcs are measured by the
angles between their tangents, or
by the angles ACB and ach at
their centres : but °' ^ '
^^^ , aroAB src ab i i
lACB'. Lacb = — — --: = -rps*— •
AG ac AC ac
Consequently, the curvatures of the two circles are to each
other inversely as their radii; or the curvature of a circle
varies inversely as its radius.
Also if As represent any arc of a circle of radius r, and
A<p the angle between the tangents at its extremities, we have
As
r = — .
2 86 Radius of Curvature.
The curvature of a curve at any point is found by deter-
mining the circle which has the same curvature as that of an
indefinitely small elementary arc of the curve taken at the
point.
226. Radius of Curvature. — Let ds denote an infi-
nitely small element of a curve at a point, d^ the corresponding
ds
angle of contingence expressed in circular measure, then —
evidently represents the radius of the circle which has the
same curvature as that of the given curve at the point.
This radius is called the radius of curvature for the point,
and is usually denoted by the letter p.
To find an expression for p, let the curve be referred to
rectangular axes, and suppose x and y to be the co-ordinates
of the point in question ; then if ^ denote" the angle which the
tangent makes with the axis of x, we have
dy d . tan 0 d^y
^'"I'^di' ■'•~dr =;&"
, . dd) dd) dx d(f> , cPy
Hence
dy\y
sec»(^_V \dx) J (j)
d(p d'^y d^y
ds d^ dx*
At a point of inflexion —^ = o : accordingly the radius of
ax
curvature at such a point is infinite : this is otherwise evident
since the tangent in this case meets the curve in three conse-
cutive points. (Art. 222.)
Again, as the expression f i + f^J 1 has always two
values, the one positive and the other negative, while the
Expressions for Radius of Curvature. 287
curve can have in general but one definite circle of curvature
at any point, it is necessary to agree upon-Avhicli sign is to be
taken. "We shall adopt the positive sign, and regard p as
being positive when -7^ is positive ; i. e. when the curve is
(ix
convex at the point with respect to the axis of x.
227. Other ^Expressions for p. — It is easy to obtain
other forms of expression for the radius of curvature ; thus
by Art. 178 we have
dx . dy
cos rf> = -7-, sm 0 = -^.
^ ds ^ ds
Hence, if the arc be regarded as the independent variable, we
get
d6 d^x dd) d^y
- sm 0 -r = -T-, cos d) -7- = -r^,
^ ds ds^' ^ ds ds"'
from which, if we square and add, we obtain
p* \ds) \dsy {ds'J'
Again, the equations dx = cos ^ds, dy = sin ^ds,
ds
give by differentiation (substituting — for c?^),
(dsY (dsY
d^x = cos (pd^s - sin 0-^—^, d^p = sin ^d^s + cos ^ -^^. (3)
Whence, squaring and adding, we obtain
(d'xy + {(Py)' = id's)' + i^*,
P
(2;
'"' '' ^/(d'xy + {d'yy - (d'sf ^^'
288 Radius of Curvature.
Again, if the former equation in (3) be multiplied by
sin ^, and the latter by cos ^, we obtain on subtraction,
ds^ ds^
cos (p d}y - sin 0 df a; = — , or dxd'^y - dyd^x = — .
P P
S®^^® P = T^Ti T^' (5)
dxd^y - dyd^x
The independent variable is undetermined in formulae (4)
and (5), and may be any quantity of which both x and y are
functions.
For example, in the motion of a particle along a curve,
when the time is taken as the independent variable, we get
from (4) an important result in Mechanics.
Examples.
I. To find the radius of curvature at any point on the parabola x"^ = ^my.
Here 2m/ = a:, 2m-^=i, i + f^) = i + -^ = i + i^;
2 (m + y)g
'•'^'=— ^^i
3. Find the radius of curvature in the catenary
y-\{f^^-'\
Here
dx 2\ 1' dx' a'-' '
-f-
a
Hence the radius of curvature is equal to the part of the normal intercepted
by the axis of a:, but measured in the opposite direction (Ex. 4, Art. 171).
3. In the cubical parabola 30^^ = 3?^ we have
dx dx' ( \dxl ) ,e ' ^a*x
General Expremonfor Radius of Curvature 289
4. To find the radius of curvature in the ellipse -;; J- r- 53 I.
Let a; = a cos ^, then y = 5 sin <p, and we have
dx = — a sin <pd(p, d^x = - a cos (pdc^"^ — a sin ^^fz^^
dy ^h 00a <j>d(]), d^i/=:-b sin ^<?<^2 + j ^qs ^^20.
Hence by formula (5) we obtain
(a*sin2^ + 3*0082^)5
^ ^ •
5. In the hypocycloid a:t + yi = ai, let a; = a cos'^, then y = a sin^^, and re-
garding (f) as the independent variable, we have
dx = -Za cos'^ sin <pd<p, d^x = S(t cos (f> dcp- (2 sin^ (|) - cos^ ^),
<?y = 3a sin^^ cos ^ ^^, d^i/ = 3« sin 4) (?^2 (2 cos^ ^ - sin^ ^),
whence
{dx^ + dy^)i = 3a sin^ cos <pd(i>, and «?a;c?2_j/ _ dyd'^x = - gd^ wa.^<f> cos*^c?^3,
from which we obtain
p = - 3 {axi/y.
6. Find the radius of curvature at any point of the curve
«« = secf-j. Ans. p = a Beo I- U
228. Creneral Sxpression for Radius of Curva-
ture.— The value of p becomes usually diJficult of determi-
nation from formula ( i ) whenever y is not given explicitly in
terms of a;, that is, when the equation of the curve is of the
form
"We proceed to show how the equation for p is to be trans-
formed in this case. Suppose
du du _ d^u _ d^u _ d'^u __ ^
Tx'-^' dy~^' ^"^' d^y'-^' df"^'
then, by Art. 100, we have
dx
290 Radius of Curvature.
Again, differentiating this equation with respect to x,
regarding y as a function of a; in consequence of the given
equation, and observing that
d . dL dLdy ^ /iir\ _ ^^ dM dy
dx^ ' dx dy dx^ dx^ ' dx dy dx^
we obtain
dL dLdy_ (dM dMdy\dy^ d^y
dx dy dx \ dx dy dx) dx dm^
A^2Bf.C%^M'^ = o; (6)
dx dx^ dx^ ' ^ '
whence, on substituting - -^ for -f, we obtain
JzL (XX
(Py _ AM^-zBLM+CD
dx'^ ~ M'
Consequently
^ ^ AM' - 2BLM -^ CU' ^^^
Or, on replacing X, M, A, B, C, by their values,
m
,..- *wi
d'^u fduV d'u du du d'u fduV
dx' \dy) dxdy dx dy dy' \dx)
The result in (6) enables us to determine the second diffe-
rential coefficient of an implicit function in general; a process
which is sometimes required in analysis.
22q. The Centre of Curvature is the point of
intersection of two Consecutive IVormals. — We shall
next proceed to consider the subject from a geometrical
point of view.
As a circle which passes through two infinitely near
points on a curve is said to have contact of the first order with
Newton^ s Method of Investigating Curvature. 291
the curve, so the circle which passes through three infinitely
near points on a curve is said to have contact of the second
order with it, and is called the circle of curvature, or the
osculating circle at the point.
Again, the centre of the circle which passes through
three points, P, Q, jK, is the intersection of the perpendicu-
lars drawn at the middle points of PQ and QR ; but when
P, Q, R become infinitely near points on a curve, the per-
pendiculars become normals, and the centre of the circle
becomes the limiting position of the intersection of two infinitely
near normals to the curve. (Compare Art. 37, note.)
ds
From this it is easily seen that we obtain — for the length
a^
of the radius of the circle in the limit, as before.
230. IVewton's Method of investigating Radii of
Curvature. — When the equation of the curve is algebraic
and rational it is easy to obtain an
expression for its radius of curvature*
at any point.
For, take the origin 0 at the
point, and the tangent and normal
for co-ordinate axes; let P be a
point on the curve near to 0, and
describe a circle through P and 0
touching the axis of x\ draw PN
perpendicular to OX and produce
it to meet the circle in Q ; then we have
ON' = PN.NQ.
Hence, if x and y be the co-ordinates of P, we get
*
But when P is infinitely near to 0, NQ becomes OB, the
* This method of finding the radius of curvature is indicated by Newton
(Principia, lib. I., Sect, i., Lemma xi.), and has been adopted in a more or le^
modified form by many subsequent writers.
V 2
292 Radius of Curvature.
diameter of the circle of curvature, and if p be its radius, we
have
. . a?*
2p = limit of — when x is infinitely small.
Again, since the axis of x is the tangent at the origin,
the equation of the curve, by Art. 208, is of the form
hy = CqX^ + 2Cixy + Ci'if' + terms of the third and higher degrees
= C(fio^ + 2Cixy + (?2y' + W3 + W4 + &o. (9)
On dividing by y we obtain
Oi = Ca— + 2CiX + C2I/ +— + &0.
' y y
Again, when x is infinitely small, — becomes 2p, and
each* of the other terms at the right-hand side becomes infi-
nitely small ; hence
Thus, for example, the radius of curvature at the origin in
the curve
ty - 2x^ + ^xy - 4y* + «•
is -, the axes being rectangular.
Ms Hi
♦ We have assumed above" that the tenns — , — , &c., become evanescent
y y
along with x ; this can be readily established as follows : —
Let «3 = oa^ + ^x^y + fxy* + Jy*,
then - = o- +i8x'» + 7ary + 5y»;
each of the terms after the first vanishes with a;, while the first becomes a — «,
or 2apx, which also vanishes with a?, when p is finite.
Similar reasoning is applicable to the terms, — , &c.
Case of Oblique Axes. 293
From the preceding it follows that wien the axis of x is
a tangent at the origin, the length of the radius of curvature
at that point is independent of all the coefficients except
those of y and x^.
231. Case of Oblique Axes. — If the co-ordinate axes
be oblique, and intersect at an angle w, then PQ no longer
passes through the centre of the circle in the limit, but be-
comes the chord of the circle of curvature which makes the
angle w with the tangent ; accordingly, we have in this case
2p sin w = -:fr^ = — , in the limit.
PN y^
Hence, in the case of oblique axes, we have
hi . .
p Bin 01 = — . (10)
If &i and Co have opposite signs, p is negative ; this
indicates that the centre of curvature lies below the axis of x,
towards the negative side of the axis of y.
The preceding results show that the radius of curvature
at the origin is the same as that of the parabola, h^y = CqX^, at
the same point ; and also that the system of curves obtained
by varying all the coefficients in (9), except those of y and
;r', have the same osculating circle, in oblique as well as in
rectangular co-ordinates.
Again, as in Art. 223, the osculating circle, since it meets
the curve in three consecutive points, cuts the curve at the
point, in general, as well as touches it.
If Co = o in the equation of the curve, and h^ be not zero,
the radius of curvature becomes infinite, and the origin is a
point of inflexion. This is also evident from the form of the
equation, since the axis of x meets the curve in this case in
ihree consecutive points.
232. In general, the equation of a curve referred to any
rectangular axes, when the origin is on the curve, may be
written in the form
ih^x + ibxy = CqX^ + 2CiXy + c^y"^ + t^g + &o.
294 Radius of Curvature.
Here h^x -\- hy = o is the equation of the tangent at the
origin ; and the length of the perpendicular PN from the
point {x, y) on this tangent is
haX + hiy
Also, OF^ = x^ + y\ and OF^ = 2p . PiVin the limit.
Accordingly, we have, when x and y are infinitely small,
I zPN 2bQX + zhiy
P OP' (x' + y')ybo' + bi'
\ ^ CqX^ + 2Cixy + C2y^ Ws «
~{^' + f)yK^' {x' + f)^b^Tb?'^
(since the point x, y is on the curve).
Affain, the terms contained in , ^ „ &o., become
cent in the limit, as before {see note. Art. 230).
Hence we have
I c^x^ + iCixy + c^y^ x \x
' ' evanes-
9 {x"
^^^)^*»^*'"(-g)Vv^'
But for points infinitely near the origin we have
X 0\
Substituting this value instead of - in the preceding equation,
X
it becomes
I cj)y^ - 2b^bxCi + Cib*
P {bo' + b.
(II)
The student will find no difficulty in showing the identity
of this result with that given in (7).
Radius of Curvature in terms of r and p.
295
233. Radii of Cnrvatiire of Inverse Curves. — ^It
may be convenient to state here that if tWo curves be inverse
to each other with respect to any origin, their osculating circles
at two inverse points are also inverse to each other with respect
to the same origin.
This property is evident geometrically from the con-
sideration that a circle is determined when three points on
it are given.
Again, since the centres of the two inverse circles are
in directum with the origin, we can construct the centre of
curvature at any point on a curve, when that for the cor-
responding point on the inverse curve is known.
Also, if the osculating circle at any point on a curve
pass through the origin, the corresponding point is a point of
inflexion on the inverse curve. .
We shall next proceed to establish another expression for
the radius of curvature, which is of extensive application in
curves referred to polar co-ordinates.
234. Radius of Curvature in terms of r and p. —
Let PN and P(7 be the tangent
and normal at any point P on a
curve, P'iV' and F'C those at
the infinitely near point P', then
C is the centre of curvature cor-
responding to the point P. Let
0 be the origin.
Join 0(7, and let OC = 8,
OP = r, or = /, ojsr = p,
ON'=p\ CP=CP'= p; then
we have
or
Fig. 32,
OC =OP'+CP'^ 2OP.CP. ooB OPO,
^^r^ + p^" 2pp.
In like manner we have
r * + p^ - 2pp
Subtracting, we get
r^ -r^ = 2p {p' - Jt)), or —
/-
2/>
p -p r -v r
296 Radius of Curvature.
Hence we have
dr p dr . , ^
— = -, or /o = r— . (12)
dp r ^ dp ^ ^
This formTila can also be deduced immediately from Ait.
191 : thus
I T^ TIT dp ; dp ds dp dp dr . dp
d(t) ds do) ^ ds drds ^ ^ dr
dp dr
235. Chord of Curvature through the Origin. —
Let 7 denote half the intercept made on the line OF by the
circle of curvature, and we evidently have
y = psinOPJr=pf=^|. (,3)
This and the preceding formula are of importance when-
ever we can express the equation of the curve in terms of the
lines represented by r and p.
Their use will be illustrated by the following elementary
examples : —
ExOtPLES.
1. To find the radius of curvature at any point on a parabola.
Taking the focus as pole, the equation of the curve in terms of r and p
evidently is p^ = zmr.
— dr pr /ar^Xi dr p^
Hence /> = -^ = -= (-) i «!=». 7 =/'^ = - = ^r.
2. To find the radius of curvature in an ellipse.
Taking the centre as origin, the equation of the curve is
P
_ dr _ a^lP-
''' ^~^d^~~p^'
3. To find the radius of curvature in the Lemniscate.
Here, by Ex. 3, Art. 190, we have r^ = a^p;
' *• 3**^ T = ^^'f lience p = — ; also, 7 = - .
dp Sr 3
Evolutes and Involutes.
297
4. To find the chord of curvature which passes through the origin in the
Cardioid ^
r = a(r +cos0).
In this case, we have r* = 2ap'^.
Hence 7 = r- — - = - r.
dp z
5. To find the radius of curvature at any point on the curve r^=:a^ cos m6.
Here r»»*i = a*»p, by Art. 190.
Hence p = r z = r- ; also, 7 = .
(w -t i^r*""^ \m-v \)p w + I
This result furnishes a simple geometrical method of finding the centre of cur-
vature in all curves included under this equation.
236. To prove that p =p + — ^. If p and w have the
same signification as in Art. 192, the formula of that Art.
becomes
ds d^p
^ d(^ ^'^dw'''
Examples.
In a central ellipse prove that
(>4)
p = *y d^ cos^ ctf 4- i^ sin'oj,
and hence deduce an expression for the radius of curvature at any point on the
curve.
2. In a parabola referred to its focus as pole, prove that p = m sec «, and
hence show that p = ^m sec^w.
237. Evolutes and Involutes. — If the centre of cur-
vature for each point on a curve be p^ p^
taken, we get a new curve called the ~^^~~' -^
^volute of the original one. Also, the
original curve, when considered with
respect to its evolute, is called an in-
volute.
To investigate the connexion be-
tween these curves, let Pi, P2, P3, &c.,
represent a series of infinitely near
points on a curve; (7i, (72, Os, &c., the
corresponding centres of curvature,
then the lines PiC,, TzCi, PgCs, &c.,
are normals to the curve, and the lines
OiCz, (72(73, (73(74, &c., may be regardedin
Fig. 33.
the limit as consecutive elements of th© evolute; also, since
298 Radius of Curvature,
eacli of the normals PiCi, PzC'zjPaCaj&c, passes through two
consecutive points on the evolute, they are tangents to that
curve in the limit.
Again, if pi, /02, JO3, /04, &c., denote the lengths of the radii
of curvature at the points Pi, P2, P3, P4, &c., we have
^1 = PiCi, p2 = P2C2, /03 = PzCzy Pi = Pid, &c. ;
•*• f>l "~ ^2 = PlOi — X2^2 = P2^1 ~ P2C2 = C1C2J
also p2- p3= CiCz, pz- pi= GzCi, . . . |On-i - /On = 0,»-i(7n ;
hence by addition we have
pi- Pn= C1C2 + C2C3 + Czd + . . . + Cn-1 On.
This result still holds when the numher n is increased
indefinitely, and we infer that the length of any arc of the
evolute is equal, in general, to the difference between the radii of
curvature at its extremities.
It is evident that the curve may be generated from its
evolute by the motion of the extremity of a stretched thread,
supposed to be wound round the evolute and afterwards
unrolled ; in this case each point on the string will describe
a different involute of the curve.
The names evolute and involute are given in consequence
of the preceding property.
It follows, also, that while a curve has but one evolute, it
can have an infinite number of involutes ; for we may regard
each point on the stretched string as generating a separate
involute.
The curves described by two different points on the
moving line are said to be parallel; each being got from the
other by cutting off a constant length on its normal measured
from the curve.
238. £volates regarded as EiiTelopes. — From the
preceding it also follows that the determination of the evolute
of a curve is the same as the finding the envelope of all its
normals. We have already, in Ex. 3, Art. 219, investigated
the equation of the evolute of an ellipse from this point of
view.
239. Evolute of a Parabola. — ^We proceed to deter-
mine the evolute of the parabola in the same manner.
Evolute of Ellipse.
299
Let the equation of the curve be y"^ = 2ma;, then that of
its normal at a point («, y) \&
m
or
if
y* + 2my{m - X) - 2m^Y = o.
The envelope of this line, where y is regarded as an arbi-
trary parameter, is got by eliminating y between this equa-
tion and its derived equation
^y^ + 2m(m - X) = o.
Accordingly, the equation of the
required envelope is obtained by
substituting = instead
° 2 m - X
in the latter equation.
Hence, we get for the required
evolute, the semi-cubical parabola
27mY^ = 8 (Z - m)\
The form of this evolute is exhi-
bited in the annexed figure, where
VN=m = 2VF. If P, F, repre-
sent the points of intersection of the
evolute with the curve, it is easily seen that
240. Evolute of an Ellipse. — The form of the evolute of
an ellipse, when e is greater
than yv 2, is exhibited in
the accompanying figure ;
the points M, N, M\ N\ are
evidently cusps on the curve,
and are the centres of cur-
vature corresponding to the
four vertices of the ellipse.
In general, if a curve be
symmetrical at both sides
of a point on it, the oscu-
lating circle cannot intersect Fig. j^.
Fig. 34.
300 Radius of Curvature.
the curve at the point ; accordingly, the radius of curvature
is a maximum or a minimum at such a point, and the corre-
sponding point on the evolute is a cusp.
It can be easily seen geometrically that through any point
four real normals, or only two, can be drawn to an ellipse,
according as the point is inside or outside the evolute.
It may be here observed that to a point of inflexion on
any curve corresponds plainly an asymptote to its evolute.
241. Evolute of an Equiangular Spiral. — We shall
next consider the equiangular or logarithmic spiral, r = a^.
Let P and Q be two points
on the curve, 0 its pole, P(7,
Q(7the normals at P and Q ; join
OC. Then by the fundamental
property of the curve (Art. 181),
the angles OPC and OQO are
equal, and consequently the four
points, 0, P, Q, (7, lie on a circle :
hence L QOC = z QPC; but in
the limit when P and Q are coin- p. ^
cident, the angle QPC becomes
a right angle, and C becomes the centre of curvature belong-
ing to the point P ; hence POC also becomes a right angle,
and the point C is immediately determined.
Again, lOCP = L OQP 1 but, in the limit, the angle
OQP is constant; .*. L OCP is also constant ; and since the
line CP is a tangent to the evolute at C, it follows that the
tangent makes a constant angle with the radius vector OC.
From this property it follows that the evolute in question is
another logarithmic spiral. Again, as the constant angle is
the same for the curve and for its evolute, it follows that the
latter curve is the same spiral turned round through a known
angle (whose circular measure is — logoi/).
241 {a). Involute of a Circle. — As an example of
involutes, suppose APQ to represent a portion of an involute
of the circle BAC, whose centre is 0. Let
OC=a, LCOA = ip,
and CA the length of the string unrolled; then
CP=CA = a6.
Points of Inflexion in Polar Co-ordinates,
301
Draw ON perpendicular to the tangent at P, and lei
ON = Pf then we have "^
p = atp.
Hence, since
lBON=aCOA = (P,
the pedal of the curve APQ is a
spiral of Archimedes.
Also, since
OP' = OC + CP\
we have
r'^ =p'^ + a^,
Fig- 37.
which gives the equation to the involute of a circle in terms
of the co-ordinates r and p.
Again, if AP = s, we have
from which it is easily seen that
242. Radius of Curvature, and Points of In-
flexion, in Polar Co-ordinates. — We shall first find an
expression for p in terms of u (the reciprocal of the radius
vector) and d.
By Article 183- we have
I , (du\
hence
Also
I dp^
p*du
.■r- = U +
p >= r
dr
d^
dd'"
I du
dp u^dp^
302 Radius of Curvature
, , / <Pu\ I ( f du
consequently ^ ^« + - j = _ = j i + (^_
... I , c?w I <?r
Affam, since t/ = -, we nave -77; = - -^ -^
!\!
(15)
!■
(16)
This result can also be established in another manner, as
follows : —
On reference to the figure of Art. 1 80, it is obvious that
^ = 0 + ^// ; where (/> is the angle the tangent at P makes with
the prime vector OX.
.^ d<h d\L dd> ds d\L
s^^^^ ^ = ^^^' ''d^do^'-'Je'
dxL
I _ ^ dd^^
' * p ds ds^
dS
df d r
Again, denoting ^ and -^^ by r and r, we have
r
tan ^ = - ; and hence
r
di> ^ 'r"^ - rr r* - rr
-5 = cos^^ — — — = -^ rr ;
dd ^ r^ r^ + r^
d\L r^ - rr + zr^ . ds ., ,
dQ r^ + r' a^
Hence, we get
Intrinsic Equation of a Curve, 3^3
Or, replacing r and r by their values,
-nil'
Again, since /o = oo at a point of inflexion, we infer that
the points of intersection of the curve represented by the
equation
O'r fdr\
with the original curve, determine in general its points of
inflexion.
In some cases the points of inflexion can be easier found
by aid of (15), which gives, when p = 00,
d'u
Examples.
1. Find the radius of curvature at any point in the spiral of Archimedes,
(1+0^)1
r = a0, Ans. a-^-^-^.
2. Fiad the radius of curvature of the logarithmic spiral r = a ,
Ans. r(i + (loga)2)l.
3. Find the points of inflexion on the curve
9
r = 29 - 1 1 cos 20. Ans, cos 2d = — .
II
4. Prove that the circle r = 10 intersects the curve
r = II — 3 COS50
in its points of inflexion.
5. Prove that the curve
r = « + * COS «e
has no real points of inflexion unless a is > J and <(i + «') J. When a lies he-
tween these limits, prove that all the points of inflexion lie on a circle ; and show
how to determine the radius of the circle.
304 Radius of Curvature.
242 {a). Intrinsic Equation of a Curve. — In many
cases the equation of a curve is most simply expressed in
terms of the length, 5, of the curve, measured from a fixed
point on it, and the angle, 0, through which it is bent,
i. e. the angle of deviation of the tangent at any point from
the tangent at the fixed point, taken as origin. These are
styled the intrinsic elements of the curve by Dr. Whewell,*
to whom this method of discussing curves is due.
The relation between the length s and the deviation 0 for
any curve is called its intrinsic equation.
If this relation be represented by the equation
then if /> be the radius of cvirvature at any point, we have
Also, if 5i denote the length of the evolute, from Art. 237
it is easily seen that the equation of the evolute is of the form
Si =/'(0) + const.
From this it follows that the series of successive evolutes
are in this case easily determined by successive differentiation.
The simplest case of an intrinsic equation is that of the
circle, in which case we have
5 = a<^.
Again, from Art. 24i(<?), the intrinsic equation of the
involute of a circle is reducible to the form
a(b^
2
We shall meet with further examples of intrinsic equa-
tions subsequently.
243. Contact of Different Orders. — As already
stated, the tangent to a curve has a contact of the first order
with the curve at its point of contact, and the osculating
circle a contact of the second order. We now proceed to
distinguish more fully the different orders of contact between
two curves.
Cambridge Philosophical Transactions, Vols. viii. and rx.
Contact of Different Orders, 305
Suppose the curves to be represented by the equations
y=f{x)y and 2/ = ^ (a?),
and that Xi is the abscissa of a point common to both curves,
then we have
/(a?i)=0(^i).
Again, substituting Xy + h, instead of a; in both equations,
and supposing yi and yi the corresponding ordinates of the
two curves, we have
Vx =/(^i + h) ^/{xi) + hf{x,) + j-^/"(^i) + &c..
Subtracting, we get
y.-y.=A{/(*,)-^'WI + j^, (/>0-'^"W)+&c. (17)
Now, suppose f^{xi) = (p'ixi), or that the curves have a
common tangent at the point, then
t/.-p> = ^^ {/'(*.) - fi'^d ) + Y^ {f"M - rC'^O )+ &o.
In this case the curves have a contact of the first order ;
and when h is small, the difference between the ordinates is
a small quantity of the second order, and as yi - y-i does not
change sign with A, the curves do not cross each other at the
point.
If, in addition
then j^i - 2/2 = j^^ {/"(^i) - ^'"(^0 } + &o.
In this case the difference between the ordinates is an in-
finitely small magnitude of the third order when h is taken
an infinitely small magnitude of the first; the curves are
then said to have a contact of the second order ^ and approach
infinitely nearer to each other at the point of contact than in
3o6 Radius of Curmture.
the former case. Moreover, since y^ - y^ ^anges its sign
with h, the curves cut each other at the point as well as touch.
If we have in addition f'\x^ = 0"'(a?i), the curves are
said to have a contact of the third order: and, in general, if
all the derived functions, up to the n^^ inclusive, he the same
for both curves when x = a?i, the curves have a contact of the
n^^ order, and we have
y. -y. =1^^ {/(""'(«.) - *(""' («.)) + &o. (i8)
Also, if the contact be of an even order, w + i is odd, and
consequently If'^^ changes its sign with A, and hence the curves
cut eac other at their point of contact ; for whichever is the
lower at one side of the point becomes the upper at the
other side.
If the curves have a contact of an odd order, they do not
cut each other at their point of contact.
From the preceding discussion the following results are
immediately deduced : —
(i). If two curves have a contact of the n^^ order, no curve
having with either of them a contact of a lower order can
fall between the curves near their point of contact.
(2). Two curves which have a contact of the n*^ order at
a point are infinitely closer to one another near that point
than two curves having a contact of an order lower than
the n^^.
(3). If any number of curves have a contact of the second
order at a point, they have the same osculating circle at the
point.
244. Application to Circle. — It can be easily verified
that the circle which has a contact of the second order with a
curve at a point is the same as the osculating circle determined
by the former method.
For, let {X-fxf^{Y-^J^B^
be the equation of a circle having contact of the second order
at the point {x^ y) with a given curve ; then, by the preceding,
dti d ii
the values of -r- and tt must be the same for the circle and
dx dx^
for the curve at the point in question.
Application to Circle, 307
Differentiating the equation of the circle twice, and sub-
stituting X and y for X and Y, we get
a;-a+(y-/3)^ = o, (19)
Hence y-^ ^5— ' * - « Sj^"^' (^')
This agrees with the expression for the radius of curvature
found in Art. 226.
The co-ordinates a, j3 of the centre of curvature can
be found by aid of equations (21) ; and the equation of the
e olute by the elimination of x and y between these equa-
tions and that of the curve.
In practice, the following equations are often more useful :
thus, by differentiation with respect to x, we get from (19),
In like manner, from the equation
we obtain
(y-/3)+(^-a)^ = o,
d^x d f dx\ , .
245. Centre of Curvature, and £volute of Ellipse.
— As an illustration, we shall apply these equations to de-
X 2
3o8 Radius of Curvature.
termine the co-ordinates of the centre of curvature, and the
equation of the evolute of the ellipse
a^ If
TT dy 6* dx a*
" dx\ dx) a^' dy\dy) b''
^ d'y h" (dy\ y" ¥ x"
In like manner, we have
d^x a*
Substituting in {22) and (23), we obtain for the co-ordinates
of the centre of curvature
/3 j^ , « ^ • (24)
Again, substituting the values of x and y given by these
equations, in the equation -^ + •77= i, we get for the equation
of the evolute
{aa)t + (j3&)l = {a' - h')h
246. It may be noticed that the osculating circle cuts the
curve in general, as tcell as touches it. This follows from
Article 243, since the circle has a contact of the second order
at the point.
At the points of maximum and minimum curvature the
Osculating Curves, 309
osculating circle has a contact of the third order with the
curve ; for example, at any of the four vertices of an ellipse
the osculating circle has a contact of the third order, and does
not cut the curve at its point of contact (Art. 240).
247. Osculating Curves. — When the equation of a
curve contains a numher, w, of arbitrary coefficients, we can
in general determine their values so that the curve shall have
a contact of the (n - ly^ order with a given curve at a given
point ; for the n arbitrary constants can be determined so
that the n quantities
dy d^ dr^y
^' dx' da?' ' ' 'dx^-''
shall be the same at the ♦point in the proposed as in the
given curve, and thus the curves will have a contact of the
{n - lY^ order.
The curve thus determined, which has with a given curve
a contact of the highest possible order, is called an osculating
curve, as having a closer contact than any other curve of the
same species at the point.
For instance, as the equation of a circle contains but
three arbitrary constants, the osculating circle has a contact
of the second order, and cannot, in general, have contact of a
higher order ; similarly, the osculating parabola has a contact
of the third order ; and, since the general equation of a conic
contains five arbitrary constants, the general osculating conic
has a contact of ihe fourth order. In general, if the greatest
number of constants which determine a curve of a given
species be w, the osculating curve of that species has a contact
of the (n - lY^ order.
248. Geometrical Metliocl. — The subject of contact
admits also of being considered in a geometrical point of view ;
thus two curves have a contact of the first order, when they
intersect in tivo consecutive points ; of the second, if they inter-
sect in three; of the n^\ if in w + i. For a simple investi-
gation of the subject in this point of view the student is
referred to Salmon's Conic Sections, Art. 239.
249. Curvature at a ©ouble Poiut. — We now pro-
ceed to consider the method of finding the radii of curvature
of the two branches of a curve at a double point.
3 1 o Radius of Curvature,
In this case the ordinary formula (8) becomes indetermi-
nate, since
du , du
-7- = o, and — - = o
dx dy
at a double point. The question admits, however, of being
treated in a manner analogous to that already employed in
Art. 230 : we commence with the case of a node.
250. Radii of Curvature at a IVode. — Suppose the
origin transferred to the node, and the tangents to the two
branches of the curve taken as co-ordinate axes, w represent-
ing the angle between them.
By Art. 210, the equation of the curve is in this case of
the form
zhxy = ax^ + ^x^y + '^xy'^ + ^y^ + W4 + &o. :
dividing by xy we obtain
2h = a- + 3x + yy + 6— + — + &o.
y ' X xy
Now, let pi and pa be the radii of curvature at the origin
for the branches of the curve which touch the axes of x and y,
respectively; then, by Art. 231, we have
2p\ sin (!>=—, and 202 sm w = — , in the limit.
y X
Again, it can be readily seen, as in the note to Art. 230,
that the terms in — , &o., become evanescent along with x
xy
and y, and accordingly the limiting values of — and — can
be separately found, as in the Article referred to.
Hence we obtain
h h , ,
Also, if a = o, we get pi = 00, and the corresponding
branch of the curve has a point of inflexion at the origin.
Similarly, if § = o, pa = 00.
Radii of Curvature at a Cusp, 311
If a = o, and S = o, the origin is a point of inflexion on
both branches. This appears also immediately from the
consideration that in this case Uz contains u^ as a factor.
If the equation of a curve when the origin is at a node
contain no terms of the third degree, the origin is a point of
inflexion on both branches. An example of this is seen in
the Lemniscate, Art. 210.
Examples.
1. Find the radii of curvature at the origin of the two branches of the curve
ax^ - 2bxy + ey^ = «* + y*,
b h
the axes being rectangular. Ans, - and -.
2. Find the radii of curvature at the origin in the curve
a [y"^ - x^) = a?.
Transforming the equation to the internal and external bisectors of the angle
between the axes, it becomes
^axy^^ = {x-yf'y
hence the radii of curvature are 2a -y^ 2 and - T-a^/ 2, respectively.
251. Radii of Curvature at a Cusp. — The preceding
method fails when applied to a cusp, because the angle w
vanishes in that case. It is easy, however, to supply an in-
dependent investigation : for, if we take the tangent and
normal at the cusp for the axes of x and y, respectively, the
equation of the curve, by the method of Art. 210, maybe
written in the form
y^ = ax^ + ^x^y + yony'^ + hf + e^4 + &c. (26)
Now in this, as in every case, the curvature at the origin
depends on the form of the portion of the curve indefinitely
near to that point ; consequently, in investigating this form
we may neglect ?/V, ?/^, &c., in comparison with y- ; and x*^
3?y^ &c., in comparison with 7?.
312 Radius of Curvature.
Accordingly, the curvature at tlie origin is the same, in
general, as that of the cubic
y^ = aic^ + ^x^y. (27)
Dividing by x^, we get
«/"
Hence, in immediate proximity to the origin, - be-
X
comes very small, i. e. y is very small in comparison with x.
Accordingly, the form of the curve near the origin is repre-
sented by the equation
y"^ = ax^.
From this we infer that the form of any algebraic curve
near a cusp is, in general, a semi-cubical parabola {see'Ex. 2,
Art. 211).
Again, since
we have, by Art. 230,
a^ X
from which we see that p vanishes along with x, and accord-
ingly the radii of curvature are zero for both branches at the
origin.
This result can also be arrived at by differentiation, by
aid of formula (i).
252. Case where the Coefficient of a;^ is wanting. —
Next, suppose that the term containing x^ disappears, or
a = o, then the equation of the curve is of the form
y'' = /3^3/ + 'yxy'' + ^y^ + aV + &c. ;
and proceeding as before, the curvature at the origin is the
same as in the curve
if = p^y + aV, (28)
Radii of Curvature at a Cusp. 313
The two branches of this curve are determined by the
equation
P. 3.^'
y = - ic- ± - y /3' + 4a'. (29)
2 2
The nature of the origin depends on the sign of j3^ + 4a', and
the discussion involves three cases.
(i). If j3^ + 4a' be positive^ it is evident that the curve
extends at both sides of the origin, and that point is a double
cusp (Art. 215(a)).
On dividing equation (28) by y"^, and substituting zp iot
— , we get I = 2]3|0 + 4a'/o'. (30)
The roots of this quadratic determine the radii of curva-
ture of the two branches at the cusp.
These branches evidently lie at the same, or at opposite
sides of the axis of x, according as the radii of curvature
have the same or opposite signs : i. e. according as a has a
negative or positive sign.
These results also appear immediately from the circum-
stance, that in this case the form of the curve very near the
origin becomes that of the two parabolas represented by
equation (29).
(2). If jy^ + 4a' be negative, y becomes imaginary, and the
origin is a conjugate point.
(3). If j3^ + 4a' = o, the equation (30) becomes a perfect
square : we proceed to prove that in this case the origin is a
cusp of the second species.
To investigate the form of the curve near the origin, it is
necessary in this case to take into account the terms of the
fifth degree in x (y being regarded as of the second) : this gives
{y - ^xy = yxf + jSV?/ + aV = x{yy' + jd'x'y + aV). (31)
It will be observed that the right-hand side changes its
sign with x ; accordingly the origin is a cusp. Also, the cusp
is of the second species, for the two roots of the equation in y
plainly have the same sign, viz., that of |3 ; and consequently
both branches of the curve at the origin lie at the same side
of the axis of x.
314 Radius of Curvature,
Moreover, as equation (30) has equal roots in this case,
the radii of curvature of the two branches are equal, and the
branches have a contact of the second order.
We conclude that when the term involving x^ in equation
(28) disappears, the origin is a double cusp, a cusp of the second
species, or a conjugate point, according as (5"^ + 4a' > = or < o.
Moreover, if a = o, one root of the quadratic (30) is in-)
finite, and the other is —5. The origin in this case is a double
cusp, and is also a point of inflexion on one branch. Such a
point is called a point of oscul-inflexion by Cramer.
If j3 = o in addition to a = o, the origin is a cusp of the
first species, but having the radii of curvature infinite for both
branches.
It is easy to see from other considerations that the radii
of curvature at a cusp of the first species are always either
zero or infinite.
For, since the two branches of the curve in this case
d^y
turn their convexities in opposite directions, —- must have
opposite signs at both sides of the cusp, and consequently it
must change its sign at that point ; but this can happen only
in its passage through zero, or through infinity.
It should be observed that the preceding discussion applies
to the case of a curve referred to oblique axes of co-ordinates,
provided that we substitute 7 instead of p ; where y is half
the chord intercepted on the axis of y by the osculating circle
at the origin.
253. Recapitulation. — The conclusions arrived at in the
two preceding Articles may be briefly stated as follows : —
(i). Whenever the equation of a curve can be transformed
into the shape y^ = ax^ + terms of the third and higher degrees,
the origin is a cusp of the first species ; both radii of curva-
ture being zero at the point.
(2). When the coefiicient of x^ vanishes,* the origin is
* In this case, if Vi be the equation of the tangent at the cusp, the equation
of the curye is of the form
Vi^ + ViVi i- V4 + &c. = o.
This is also evident from geometrical considerations.
General Investigation of Cusps* 315
generally either a double cusp, a conjugate point, or a cusp
of the second species. In the latter case the two branches
of the curve have the same centre of curvature, and conse-
q^uently have a contact of the second order with each other.
(3). If the lowest term in x (independent of y) be of the
5*^ degree, the origin is a point of oscul-inflexion.
If, however, the coefficient of a^y also vanish, the origin
is not only a cusp of the first species, but also a point of
inflexion on both branches of the curve.
254. <jreiieral Investigation of Cusps. — The pre-
ceding results admit of being established in a somewhat more
general manner as follows : —
By the method already given, the equation which deter-
mines the form of an algebraic curve near to a cusp may be
written in the following general shape :
y"^ = 2Aafy + Bx^ + Cx% (32)
where lAx'^ is the lowest term in the coefficient of y, and
Bx^, Cx^* are the lowest terms independent of y.
By hypothesis, a, b, c are positive integers, and a > i, b> 2,
(J > 3 ; now, solving for y, we obtain
1/
= Ax^ ± yA'x'"' + Bx^ + Caf,
which represents two parabolasf osculating the two branches
at the origin.
The discussion of the preceding form for y resolves itself
into three cases, according as 2 a is >= or < 6.
(i). Let 2a = h ■{■ h^ then
6 + A 6
y = Ax^± x^yB + A^a^ + Ca^'^.
h
(a). If h be odd, 0^ becomes imaginary for negative values
of X, and accordingly the origin is a cusp of the
first species in this case.
♦ This term is retained, as it is necessary in the case of a cusp of the second
species.
t The word parabola is here employed in its more extensive signification.
^,5 Radius of Curvature.
(j3). If h be even, and B positive, y is real for all values
of X near the origin ; accordingly that point is a
double cusp.
(y). If b be even, and B negative, the origin is a conjugate
point.
(2). If 2a = b, we have
y = Ax^± X'* ^/{A^ + B) + CxP-*.
In this case, the origin is either a double cusp, or a conju-
gate point, according b.8 A^ + B is positive or negative.
Again, ii A^ + B = o, we have
e-b
y = x^(A + x ^ yC)-
(a). Ifc-b be an odd number, the origin is a cusp of the
second species.
(j3). li c-b be even, the origin is a double cusp or a con-
jugate point according as C is positive or negative.
(3), 2a < b, OT b = 2a + h.
Here p = Ax" ± x"* ^/A^ + Bx^ + Cx'-^'*,
and the curve evidently extends at both sides of the origin,
which accordingly is a double cusp.
This method of investigating curvature is capable of being
modified so as to apply to the case of multiple points of a
higher order ; the discussion, however, is neither sufficiently
elementary, nor sufficiently important, to be introduced here.
255. Points on E volute corresponding to Cusps on
Curve. — In connexion with evolutes and involutes, the pre-
ceding results lead to a few interesting conclusions.
(i). If a curve has a cusp of the first species, its evolute
in general passes through the cusp. However, if in addition
the cusp be a point of inflexion, the normal at it is an asymp-
tote to the evolute.
(2). To a cusp of the second species corresponds in general
a point of inflexion on the evolute : in some cases the point
of inflexion lies altogether at infinity.
(3). To a double cusp corresponds a double tangent to the
evolute.
Equation of Osculating Conic. 317
256. Equation of tbe Osculating "Conic. — As an
additional illustration of the principles involved in the pre-
ceding investigation, it is proposed to discuss the question of
the conic which osculates an algebraic curve at a given point.
Transferring the origin to the point, and taking the tangent
as axis of Xj the equation of the curve may be written in the
form
ay = aj* + aixy + ttip^ + hoO^ + hio^y + biXy^ + hy^
+ CqX^ + CiO^y + &c. + doo^ + &c. (33)
In considering the form of the curve near the origin, as a
first approximation we may, as in Art. 251, neglect xy, if^ &c.,
in comparison with y ; and x^^ x^, &c., in comparison with x^ ;
thus the equation reduces to the form
ay = a?*. (34)
Hence the form to which every curve of finite curvature
approximates in the limit is that of the common parabola, as
already seen in Art. 231.
To proceed to the next approximation, we retain terms of
the third order (remembering that when ic is a very small
quantity of the first order, y is one of the second), and the
equation becomes
ay = x^ + ttixy + haO?,
On substituting ay instead of x^ in the term hoO^, the pre-
ceding equation becomes
ay = iP' + {a^ + h^a) xy, (35)
This represents a conic having contact of the third order
with the proposed curve at the origin. When ai + h^a = o, the
parabola ay = x^ has a contact of the third order at the origin,
and accordingly so also has the osculating circle.
^ In proceeding to the next and final approximation, we re-
tain terms of the fourth order, and we get
ay-a^ + aixy + azy^ + boX^ + hix'y + Coic*. (36)
3i8 Radius of Curvature.
Moreover, from the preceding approximation we have
haojxy = ha^ + hoO^y («i + ah^.
Hence, we get for the equation of the conic having a
contact of the closest kind with the given curve
ay = a^+ {ai + ha)xy-\-[at + a{h-aihQ) +a«(<;o - Jo*)]s/'. (37)
This conic, since it has the closest contact possible with
the given curve at the origin, is the osculating conic (Art. 246)
for that point.
In like manner the parabola
«y = :r' + (ff, + Ma^ + ^-^^^^^V, (38)
since it has the closest contact possible for a parabola, is the
osculating parabola at the point.
Examples. 319
Examples,
1. Prove that the radius of curvature at the vertex of a parabola is equal to
its semi-latus rectum.
2. Find the length of the radius of curvature at the origin in the curve
y* + a^ + a (ir2 + y2) = a^y. Ans. -.
3. Find the radius of curvature at the origin in the curve
a^y = bii^ ■{■ cx^y. Ans. oo.
4. Prove that the locus of the centre of a conic having contact of the third
order with a given curve at a common point is a right line.
5. Prove that the locus of the centres of equilateral hyperbolas, which have
contact of the second order with a given curve at a fixed point, is a circle, whose
radius is half that of the circle of curvature at the point.
6. Prove geometrically that the centre of curvature at any point on an ellipse
is the pole of the tangent at the point, with respect to the confocal hyperbola
which passes through that point.
7. The locus of the centres of ellipses whose axes have a given direction, and
which have a contact of the second order with a given curve at a common point,
is an equilateral hyperbola passing through the point ?
8. Prove that the locus of the focus of a parabola, which has a contact of
the second order with a given curve at a given point, is a circle.
9. Prove that the radius of curvature of the curve rtf""^ y = a;*» at the origin is
zero, -, or infinity, according as w is < = or > 2 : m being assumed to be greater
than unity.
10. Two plane closed curves have the same e volute : what is the difference
between their perimeters ?
Ans. 2trd, where d is the distance between the curves.
11. Find the radius of curvature at the origin in the curve
3y = 44; - i5a;2 - %si^ :
find also at what points the radius of curvature is infinite.
12. Apply the principles of investigating maxima and minima to find the
greatest and least distances of a point from a given curve ; and show that the
problem is solved by drawing the normals to the curve from the given point.
(fl). Prove that the distance is a minimum, if the given point be nearer to
the curve than the corresponding centre of curvature, and a maximum if it be
further.
320 Examples,
(J)). If the given point be on the evolute, show that the solution arrived at
is neither a maximum nor a minimum ; and hence show that the circle of curva-
ture cuts as well as touches the curve at its point of contact.
13. Find an expression for the whole length of the evolute of an ellipse.
«3 - ^3
Ans. 4 J—.
ab
14. Find the radii of curvature at the origin of the two branches of the curve
«*— - ax^y — axy"^ + a^y"^ = o. Ans. a and -.
2 4
15. Prove that the evolute of the hypocycloid
jrl + ^ = al
is the hypocycloid
(a + )8)i + (a - ^)i = 2«l.
16. Find the radius of curvature at any point on the curve
y + V ^ ( ' - ^) = sin-i ^x.
17. If the angle between the radius vector and the normal to a curve has a
maximum or a minimum value, prove that y = r; where 7 is the semi-chord of
curvature which passes through the origin.
18. If the co-ordinates of a point on a curve be given by the equations
a; = c sin 20 (i + cos 20), y = c cos 20 (i — cos 20),
find the radius of curvature at the point. Ans. 40 cos 30
19. Show that the evolute of the curve
r* - a2 _ ffipi
has for its equation
f2 _ (i - w) a2 = m.p\
30. If a and )8 be the co-ordinates of the point on the evolute corresponding
to the point {x, y) on a curve, prove that
dy da
Tx ^ + ' = °-
21. If p be the radius of curvature at any point on a curve, prove that the
radius of curvature at the corresponding point in the evolute is — : where o»
da
is the angle the radius of curvature makes with a fixed line.
22. In a curve, prove that
1 = 1. l^\
p dx \dsj
Examples, 321
23. Find the equation of the evolute of an ellipse -by means of the eccentric
angle.
24. Prove that the determination of the equation of the evolute of the
curve y = kx^ reduces to the elimination of x between the equations
w - 2 Jc'^n- „ , . ^ 2n - I . I
a;2n-ij and fi = kxn-\.
n - I n — 1 n— I kn{n— i)a;"-«
25. In figure, Art. 239, if the tangent to the evolute at P meet the parabola
in a point H, prove that EN is perpendicular to the axis of the parabola.
26. If on the tangent at each point on a curve a constant length measured
from the point of contact be taken, prove that the normal to the locus of the
points so found passes through the centre of curvature of the proposed curve.
27. In general, if through each point of a curve a line of given length be
drawn making a constant angle with the normal, the normal to the curve locus
of the extremities of this line passes through the centre of curvature of the pro-
posed. (Bertrand, Cal. Dif., p. 573.)
This and the preceding theorem can be immediately established from geome-
trical considerations.
28. If from the points of a curve perpendiculars be drawn to one of its tan-
gents, and through the foot of each a line be drawn in a fixed direction, pro-
portional to the length of the corresponding perpendicular ; the locus of the
extremity of this line is a curve touching the proposed at their common point.
Find the ratio of the radii of curvature of the curves at this point.
29. Find an expression for the radius of curvature in the curve p =
p being the perpendicular on the tangent.
30. Being given any curve and its osculating circle at a point, prove that
the portion of a parallel to their common tangent intercepted between the two
curves is a small quantity of the second order, when the distances of the point
of contact from the two points of intersection are of the first order.
Prove that, under the same circimistances, the intercept on a line drawn
parallel to the common normal is a small quantity of the third order.
31. In a curve referred to polar co-ordinates, if the origin be taken on the
curve, with the tangent at the origin as prime vector, prove that the radius of
y
3urvature at the origin is equal to one-half the value of - in the limit.
32. Hence find the length of the radius of curvature at the origin in the
curve y = a sin nd. Ans. p = — .
'^ 2
33. Find the co-ordinates of the centre of curvature of the catenary; and
show that the radius of curvature is equal, but opposite, to the normal.
34. If p, p be the radii of curvature of a cujve and of its pedal at corre-
sponding points, show that
p{2r'''-pp)=t^.
Ind. Civ. Ser. Exam.y 1878.
y
( 322 )
CHAPTER XYIII.
ON TRACING OF CURVES.
257. Tracing Algebraic Curves. — Before concluding the
discussion of curves, it seems desirable to give a brief state-
ment of the mode of tracing curves from their equations.
The usual method in the case of algebraic curves consists
in assigning a series of different values to one of the co-ordi-
nates, and calculating the corresponding series of values of
the other ; thus determining a definite number of points on
the curve. By drawing a curve or curves of continuous cur-
vature through these points, we are enabled to form a tolerably
accurate idea of the shape of the curve under discussion.
In curves of degrees beyond the second, the preceding
process generally involves the solution of equations beyond
the second degree : in such cases we can determine the series
of points only approximately.
258. The following are the principal circumstances to be
attended to : —
(i). Observe whether from its equation the curve is sym-
metrical with respect to either axis; or whether it can be
made so by a transformation of axes. (2). Find the points
in which the curve is met by the co-ordinate axes. (3). De-
termine the positions of the asymptotes, if any, and at which
side of an asymptote the corresponding branches lie. (4) . De-
termine the double points, or multiple points of higher orders,
if any belong to the curve, and find the tangents at such
points by the method of Art. 212. (5). The existence of
ovals can be often found by determining for what values of
either co-ordinate the other becomes imaginary. (6). If the
curve has a multiple point, its tracing is usually simplified by
taking that point as origin, and transforming to polar co-or-
dinates : by assigning a series of values to 6 we can usually
determine the corresponding values of r, &c. (7). The points
On Tracing of Curves, 323
where the y ordinate is a maximum or a minimum are found
from the equation j- = o: by this means the limits of the
curve can be often assigned. (8). Determine when possible
the points of inflexion on the curve.
259. To trace the Curve y^ ^x^ {x-a)\ a being sup-
posed positive.
In this case the origin is
a conjugate point, and the
curve cuts the axis of ^z? at a
distance OA = a. Again,
when X is less than «, y is
imaginary, consequently no
portion of the curve lies to
the left-hand side of A.
The points of inflexion, I '
and r, are easily determined
^2y Fig. 38.
from the equation -r-f = o ; the
dx
corresponding value of a? is — ; accordingly AN = .
Again, if TI be the tangent at the point of inflexion /, it
can readily be seen that TA = - = .
This curve has been already considered in Art. 213, and
is a cubical parabola having a conjugate point.
260. Cubic with three Asymptotes. — We shall next
consider the curve*
y^x + ey = ax^ + hx^ + ex + d, (i)
where a is supposed positive.
The axis of y is an asymptote to the curve (Art. 200), and
the directions of the two other asymptotes are given by the
equation
y^ - ax^ = Of or y = ± X ^/a.
* This investigation is principally taken from Newton's Enumeratio Li-
nearum Tertii Ordinis.
Y 2
324 On Tracing of Curves,
If the term hx^ be wanting, these lines are asymptotes ; if h
be not zero, we get for the equation of the asymptotes
/- h /- *
On multiplying the equations of the three asymptotes
together, and subtracting the product from the equation of
the curve, we get
ey = [e ]x + a:
this is the equation of the right line which passes through the
three points in which the cubic meets its asymptotes. (Art.
204.)
Again, if we multiply the proposed equation by x^ and
solve for xy, we get
\ax^ + 5a;^ + ca?' + t^a? + - : (2)
from which a series of points can be determined on the curve
corresponding to any assigned series of values for x.
It also follows that aU chords drawn parallel to the axis
of y are bisected by the hyperbola xy-v- = o\ hence we infer
that the middle points of all chords drawn parallel to an
asymptote of the cubic lie on a hyperbola.
The form of the curve depends on the roots of the bi-
quadratic under the radical sign. (i). Suppose these roots
to be all real, and denoted by a, j3, 7, §, arranged in order of
increasing magnitude, and we have
xy = --±^a{x-a){x- (5){x - y){x - g).
Now when x is < a, y is real ; when x > a and < /3, ^ is
imaginary ; when x> $ and < 7, y is real ; when x> y and
<d,y is imaginary ; when x > d, y is real.
Asymptotes.
325
We infer that the curve consists of three branches, extending
to infinity, together
with an oval lying
between the values
/3 and 7 for x.
The accompany-
ing figure* repre-
sents such a curve.
Again, if either
the two greatest
roots or the two
least roots become
equal, the corres-
ponding point be-
comes a node.
If the interme-
diate roots become ^ig- 39-
equal, the oval shrinks into a conjugate point on the curve.
If three roots be equal, the corresponding point is a cusp.
If two of the roots be impossible and the other two im-
equal, the curve can have neither an oval nor a double point.
If the sign of a be negative, the curve has but one real
asymptote.
261. Asymptotes. — In the preceding figure the student
will observe that to each asymptote correspond two infinite
branches ; this is a general property of algebraic curves, of
which we have a familiar instance in the common hyperbola.
By the student who is acquainted with the elementary
principles of conical projection the preceding will be readily
apprehended ; for if we suppose any line drawn cutting a
closed oval curve in two points at which tangents are drawn,
and if the figure be so projected that the intersecting line is
sent to infinity, then the tangents will be projected into
asymptotes, and the oval becomes a curve in two portions,
each having two infinite branches, a pair for each asymptote,
as in the hyperbola.
The figure is a tracing of the curve
9a?y2 + io8y = (iP - 5) (a? - II) (a; - 12).
326
On Tracing of Curves.
It should also be observed that the points of contact at
infinity on the asymptote in the opposite directions along it
must be regarded as being one and the same point, since they
are the projection of the same point. That the points at
infinity in the two opposite directions on any line must be
regarded as a single point is also evident from the considera-
tion that a right line is the limiting state of a circle of in-
finite radius.
The property admits also of an analytical proof ; for if
the asymptote be taken as the axis of x, the equation of the
curve (Art. 204) is of the form
2/01 + 02 = o,
^^ ^ = " t"'
where 02 is at least one degree lower than (pi in x and y.
Now, when x is infinitely great, the fraction — becomes in
general infinitely small, whether x be positive or negative ;
and consequently the axis is asymptotic to the curve in both
directions.
262. To trace the Curve
ay = bx^ + x^,
where a and b are both positive.
Here ya^ = ± x"^ {x + b)^.
The curve is symmetrical with respect
to the axis of x^ and has two infinite
branches ; the origin is a double cusp.
The shape of the curve is exhibited in the
figure annexed.
If b were negative, we should have
ya^ = ±x^ {x - b)k
Here y becomes imaginary for values of x less than b ;
accordingly, the origin is a conjugate point in this case : the
curve has two infinite branches as in the former case
263. Xo trace the Curve
a^y^ = zabx^y + x^.
Fig. 40.
Form of Curve near a Double Point.
327
From the form of its equation we sea that the origin is
a point of oscwi-inflexion (Art. 251'
Solving for y, we can easily
determine any number of points
on the curve we please. It has
two infinite branches at opposite
sides of the axis of a;, and a loop
at the negative side of that axis,
as exhibited in the figure.
264. To trace the Curve
x*" + x^y"^ +y^ = x [ax^ - hy"^).
(i). Let a and h have the
same sign, then the origin is
a triple point, having for its
tangents the lines
a + y vb
X
o,
Fig. 4a.
and x^ya-y \/h=o.
Moreover, since the curve
has no real asymptote, it is
a finite or closed curve with
three loops passing through the
origin ; and it is easily seen that its shape
is that represented in the accompanying
figure.
(2). \ia and h have opposite signs, the
lines represented by a^ - hf = o become
imaginary. The curve in this case consists
of a single oval as in the figure.
This and the preceding figure were
traced for the case where b = 3a: if the
value of - be altered, the shape of the curve Fig. 43.
a
will alter at the same time. If a be greater than h, the
curve (2) will lie inside the tangent at the point X
265. Form of Curve near a Double Point.— When-
ever the curve has a node or a cusp, by transforming the
origin to that point, the shape of the curve for the branches
328 On Tracing of Curves,
passing through the point admits of being investigated by tlie
method explained in Arts. 250, 251. It is unnecessary to
enter into detail on this subject here, as it has been already
discussed in the articles referred to.
266. In connexion with the tracing and the discussion of
curves there is an elementary general principle which may
be introduced here.
If the equation of a curve be of the form
Lr - MM' = o,
where Z, Jf, L\ M' are each functions of the co-ordinates x
and y, the curve evidently passes through all the points
of intersection of the curves represented by the equations
L = o and Jf = o ; similarly it passes through the intersec-
tions oi L = o and if' = o ; and also those of Jf = o and
L = o\ and of L' =0 and M' = o. Moreover, if L and L'
become identical, the points of intersection coincide in
pairs, and the equation of the curve becomes of the form
X^ - MM' = o ; which represents a curve touching the curves
Jf = o, M' = o, at their points of intersection with the curve
L = o.
This principle admits of easy extension ; but as the subject
belongs properly to the method of trilinear co-ordinates, it is
not considered necessary to enter more fully into it here.
267. On Tracing Curves given in Polar Co-ordi-
nates.—The mode of procedure in this case does not differ
essentially from that for Cartesian co-ordinates. We have
already, in Arts. 206 and 207, considered the method of
finding the asymptotes and asymptotic circles in such cases.
It need scarcely be observed that the number and variety of
curves whose discussion more properly comes under the
method of polar co-ordinates are indefinite. We propose to
confine our attention to a few varieties of the class of curves
represented by the equation
r^ = «"» cos mQ.
268. On the Curves r"* = a^ cos mB. — In this case,
since the equation is unaltered when B is changed into - 0,
the curve is symmetrical with respect to the prime vector :
again, when 0 = o, we have r = a ; and as B increases from zero
Curves of the Form r'"^ = a'" cos md.
329
to — , r diminislies from a to zero. Wlien m is a positive in-
2 m
teger, it is easily seen that the curve consists of m similar loops.
There are many familiar curves included under this
equation. Thus, when m = i , we have r = a cos 0, which
represents a circle : again, if m = - i , the equation gives
r cos 0 = a, which represents a right line. Also, if w= 2, we
have r^ = a^ cos 20, a Lemniscate (Art. 210). If m = - 2, we
get r^ cos 20 = a^j an equilateral hyperhola.
If m = - we get r^ = ah cos -, whence r =- (\ + cos 0), a
1 ... , 0
-, it IS rs cos -
2 2
cardioid (Ex. 4, p. 232) ; with m
parabola (Ex. i, p. 231) ; and so on. As already observed,
if we change m into - m we get a new curve, inverse of
the original. Also, the reciprocal polar is obtained by sub-
stitutinsr - instead of m.
° m + I
The tangent and normal can be immediately drawn at
any point on a curve of this class by aid of the results arrived
at in Art. 190. The radius of curvature at any point has
been determined in Ex. 5, Art. 235. The method of finding
the equations of the successive pedals, both positive and
negative, has been also already explained.
A few examples in the case of fractional indices are here
added.
Example i.
rl = rt* cos -.
3
Here when 0 = o, we have r = «,
and the curve cuts the prime vector
at a distance OA equal to a : again,
when 0 = — , r = — - — : also when
2 8
0 = 7r, r
a a
-3, or 05= g.
Fig. 44.
The shape of the curve is given in the accompanying
figure. This curve is the inverse of the caustic considered in
Example 18, p. 277.
330
On Tracing of Curves.
Ex. 2.
Ex.3.
Ex. 4.
rl = «t cos- d. r^ = a^ cos - d. r^ = a^ cos - d.
4 5 3
In Ex. 2, as 0 increases from zero to 120^, r diminishes
from a to zero : when Q increases
from 120° to 240°, r increases from
zero to a : when d increases from
240° to 360°, r diminishes from a
to zero. By assigning negative
values to d, the remaining part of
the curve is seen to be symmetrical
with that traced as above. The
same result plainly follows by con-
tinuing the values for Q from 360°
up to 720°. The form of the curve
is exMbited in the annexed figure. j?ig. 45.
In Ex. 3, according as cos - 0 is positive or negative, we
get equal and opposite real values, or maginary values, for r.
Hence it is easily seen that for values of B between ± ^ tt the
o
radius vector traces out two symmetrical portions of the
IS 2 s
curve : again, between — tt and -^ tt we get two other
o o
Fig. 46.
Fig. 47.
symmetrical portions. The shape is that given in the former
of the two accompanying figures.
The Limagon.
zy
The latter figure represents the curve is-Ex. 4 ; it consists
oifive symmetrical portions ranged round the origin.
The results above stated admit of generalization, and it
can be shown, without difficulty, that in general the curve
r^ = a'i cos — consists of p similar portions arranged about
the origin; and that the entire curve is included within a
circle of radius a when p is positive, but lies altogether
outside it when p is negative.
Many curves can be best traced by aid of some simple
geometrical property. We shall terminate the Chapter with
one or two examples of such curves.
269. The liimai^oii. — The inverse of a conic section
with respect to a focus is called a Limacon. From the polar
equation of a conic, its focus being origin, it is evident that
the equation of its inverse may be written in the form
r = a cos 0 + ft,
where a and h are constants.
It is easily seen that - is the eccentricity of the conic.
The curve can be readily traced by drawing from a fixed
point on a circle any number of chords, and taking off a
constant length on each of these lines, measured from the
circumference of the circle.
If a be less than ft, the curve is the inverse of an ellipse,
and lies altogether outside the circle.
If a be greater than ft, the
curve is the inverse of a hy-
perbola, and its form can be
easily seen to be that exhibited
in the .annexed figure, where
OD = a -hy and the point 0 is a
node on the curve.
If ft = a, the curve becomes
the inverse of the parabola,
and is called a cardioid. The
inner loop disappears in this
case, and the origin is a cusp ~
on the curve. Fig. 48.
332
On Tracing of Curves,
When a= 2b, the Limacon is called the Trisectrix; a
curve by aid of which any given angle can be readily
trisected.
270. The Conchoid of IVicomedes. — If through any
fixed point A a secant PiAP be
drawn meeting a fixed right line LM
in JS, and BF and BPi be taken
each of the same constant length;
then the locus of P and Pi is called
the conchoid.
This curve is easily traced from
the foregoing geometrical property,
and it consists of two branches,
having the right line LM for a
common asymptote. Moreover, if
the perpendicular distance AB oi
A from the fixed line be less than
BP, the curve has a loop with a
node at A, as in the annexed figure.
It is easily seen that when
AB = PP, the point ^ is a cusp
on the curve ; and when AB is
greater than PP, ^ is a conjugate
point.
The form of the curve in the
latter case is represented by the dotted lines in the figure.
If AB = a, BP = h, the polar equation of the curve is
(r ± h) cos 0 = flj.
When transformed to rectangular co-ordinates, this
equation becomes
The method of drawing the normal, and finding the
centre of curvature, at any point, will be exhibited in the
next Chapter.
Examples. 333
Examples.
I. Trace the curve y = [x - i) {x - i){x - 3), and find the position of its
point of inflexion.
a. Trace the curve y^ - ^axy +x^ = o, drawing its asymptote.
This curve is called the Folium of Descartes.
3. Tra the curve a^x = y {b^ + x^), and find its points of inflexion, and
points of greatest and least distance from the axis of x,
4. If an asymptote to a curve meets it in a real finite point, show that the
corresponding branch of the curve must have a point of inflexion on it.
5. Find the position of the asymptotes and the form of the curve
«* — y* + 2axy'^ = o.
6. Show that the curve r = a cos ie consists of four loops, while the curve
»• = a cos 39 consists of but three. Prove generally that the curve r = a cos nd
has n or 2n loops according as n is an odd or even integer.
7. Trace the curve
y^ (x - a){x - b) = e^x + a)ix + b).
8. Show that the curve x^y^ + x^ = a^{x^ - y^) consists of two loops passing
through the origin, and find the form of the curve.
9. Trace the curve y{x + a)^ = b^z(x + c)% showing the positions of its
asymptotes and infinite branches.
ro. Trace the curve whose polar equation is
r = a cose + b co3 2df
und show that it consists of four loops passing through the origin.
11. Gi7en the base and the rectangle under the sides of a triangle, fijid the
equation of the locus of the vertex (an oval of Cassini). Exhibit the different
forms of the curve obtained by varying the constants, and find in what case the
curve becomes a Lemniscate.
12. Trace the curve y^ = ax^ + ^bx^ + zcx + d, and find its points of greatest
and least distance from the axis of x.
Show that two of these points become imaginary when the roots of the cubic
in X are all real.
13. Given the base and area of a triangle, prove that the equation of the
locus of the centre of a circle touching its three sides is of the form
x^y - a{x^ + y^) - b- {y - a) = o.
334 Examples.
14. Prove that all curves of the third degree are reducible to one or other of
the forms
(i). a?«/2 ^ey = ax^ + bx^ ■{■ ex + d.
(2). xy = ax^ + bx"^ + ex + d.
(3). y2 = aa^-\- bx'^ + ex + d,
(4). y = ax^ + bx^ + ex + d.
Nosvton, Enum. Linear. Ter. Ordinis.
15. Prove that all curves of the third degree can be obtained by projection
from the parabolas contained in class (3) in the preceding division. [Newton.]
For every cubic has at least one real point of inflexion : accordingly, if the
curve be projected so that the tangent at the point of inflexion is projected to
infinity, the harmonic polar (Art. 223) will bisect the system of parallel chords
passing through this point at infinity. Hence the projected curve is of the
class (3). [This proof is taken from Chasles, Ristoire de la Geometi'ie, note xx.]
16. Trace the curve r = —z , and show that it has a point of inflexion
B^ - I
when 0^ = 3 ; find also its asymptotes and asymptotic circle.
17. Trace the curve y = a sin-, and show how to draw its tangent at any
point. (This is called the curve of sines.)
18. The base of a triangle is fixed in position ; find the equation of the locus
of its vertex, when the vertical angle is double one of the base angles.
Trace the locus in question, finding the position of its asymptote.
19. Show geometrically that the first pedal of a circle with respect to a
point on its circumference is a cardioid.
20. Show in like manner that the Lima9on is the first pedal of a circle with
respect to any point.
21. Trace the curve
y* + ^axy'^ = aa^ + a;*,
and find the equations of its asymptotes, and of the tangents at the origin.
Ind. Civ. Ser. Ex., 1876.
( 335 )
CHAPTEE XIX.
KOULETTES.
271. Roulettes. — When one curve rolls without sliding
upon another, any point invariably connected with the rolling
curve describes another curve, called a roulette.
The curve which rolls is called the generating curve, the
fixed curve on which it rolls is called the directing curve, or
the base^ and the point which describes the roulette, the tracing
point. We shall commence with the simplest example of a
roulette : viz., the cycloid.
272. The Cycloid. — This curve is the path described by
a point on the circumference of a circle, which is supposed to
roll upon a fixed right line.
The cycloid is the most important of transcendental
curves, as well from the elegance of its properties as from its
numerous applications in Mechanics.
We shall proceed to investigate some of the most
elementary properties of the curve.
Let LPO be any position of the rolling circle, P the
generating point, 0 the point of jg-' l b
contact of the circle with the fixed
line. Take the length AO equal
to the arc PO, then, from the
mode of generation of the curve, ■ ■■ — ,- r?
A is the position of the generating
point when in contact with the I'ig. S©*
fixed line ; also, if AA^ be equal to the circumference of the
circle, A^ will be the position of the point at the end of one
complete revolution of the circle. Bisect AA' in D, and
draw DB perpendicular to it and equal to the diameter of
the circle, then JB is evidently the highest point in the
cycloid. Draw PiV perpendicular to AA\ and let PI^ = y,
AN =x,L PCO = e, OG=a, and we get
x = AO-]SfO = a{d-&me), y = PN = a{i -cos0). (i)
336 Boulettes*
The position of any point on the cycloid is determined by
these equations when the angle 0 is known, i. e. the angle
through which the circle has rolled, starting from the position
for which the generating point is upon the directing fine.
273. Cycloid referred to its Vertex. — It is often
convenient to refer the cycloid to its vertex as origin, and to
the tangent and normal at that point as axes of co-ordinates.
In the preceding figure let
then we have
x = £]Sr' = a{e' + 8me'), y = PN' = a{i--co8e'), (2)
274. Tangent and IVormal to Cycloid. — It can be
easily seen that the line PO is normal at P to the cycloid ;
for the motion of each point on the circle at the instant is one
of rotation about the point 0, i. e. each point may be regarded
as describing at the instant an infinitely small circular* arc
whose centre is at 0 : and hence PO is normal to the curve.
This result can also be established from the values of x
and y in (i) : for
dx , n^ dy . ^ , .
^ = «(i - cos 0), ^ = « sm 0 : . (3)
, dy sin0 6 , „^^
"-y- = n = cot - = cot PLO ;
dx I - CO80 2
and, accordingly, PL is the tangent, and PO the normal to
the curve at P.
dx d)j
Again, if we square and add the values of -^ and -7^, we
du do
obtain
fds\
. = aM (i - cos ey + sin'^ 0] = ^a" sin'' ^ 9 ;
* This method of finding the normal to a cycloid is due to Descartes, and
evidently applies equally to all roulettes.
The Cycloid.
J37
hence
--: = 2fl5SlIl- = PO. -*
dd 2
(4)
275. Radius of Curvature and £volute of Cycloid.
— Let p denote the radius of curvature at the point P, and
lPOA=<p = -;
then
dcj)
2— = 4fl^sin- = 2PO;
or the radius of curvature is douhle the normal
value of p the evolute of the curve
can be easily determined. For,
produce PO until OP' = OP, then
P' is the centre of curvature be-
longing to the point P. Again,
produce LO until 00' = OL, and
describe a circle through 0, P' and
(/ ; this circle evidently touches
AA\ and is equal to the generating
circle LPO.
Also, the arc OP" = arc OP = ^0 ;
/. arc ap" = aro -F0 = AD-A0=0D
(5)
From this
Fig. 51
P'O'.
Hence the locus of P' is the cycloid got by the rolling of
this new circle along the line
P'O'; and accordingly the evo-
lute of a cycloid is another
cycloid. It is evident that the
evolute of the cycloid ABA'
is made up of the two semi-
cycloids, AF and B^A\ as in
figure 51. Conversely, the
cycloid ABA' is an involute of
the cycloid AB'A'.
The position of the centre of
curvature for a point P on a
cycloid can also be readily de-
termined geometrically, as fol-
lows : —
Suppose 0, a point on the
lircle infinitely near to 0, and take 00%
z
UiF
3^8 Roulettes,
be the centre of curvature required, and draw POi and P'Oz.
Now suppose the circle to roll until Ox and O2 coincide, then
COi becomes perpendicular to AD, and POi and FOt will
lie in directum (since P' is the point of intersection of two
consecutive normals to the cycloid). Hence
z OCOx = z PO,Q = z OPO, + z OP'Oi,
since each side of the equation represents the angle through
which the circle has turned.
But L OCOi = 2lOP Oi. (Euclid, III. 20.)
Hence L OPO^ = z OFOx\
.-. P0i = P'0r,
and consequently in the limit we have
PO = P'O,
as before.
We shall subsequently see that a similar method enables
us to determine the centre of curvature for a point in any
roulette.
276. liength of Arc of Cycloid. — Since AFB^ (Fig. 51)
is the evolute of the cycloid ^PP, it follows, from Art. 23 7, that
the arc AP' of the cycloid is equal in length to the line PP",
or to twice P'O ; hence, as A is the highest point in the
cycloid AP'B^y it follows that the arc AP' measured from the
highest point of a cycloid is double the intercept P'O, made
on the tangent at the point by the tangent at the highest
point of the curve.
Hence, denoting the length of the arc AP^ by s, we have
s = 4^^ sin P'OD = \a sin 0. (6)
This gives the intrinsic equation of the cycloid {see Art.
242 (a)). Hence, also, the whole arc AB^ is four times the
radius of the generating circle : and accordingly the entire
length ABA' of a cycloid is eight times the radius of its
generating circle.
Again, if the distance of P' from AA' be represented by
y, we shall have
FO^ =00' xy= lay.
Hence s" - 4P'0- = ^ay, fj)
Epicycloids and JSypocycloids.
330
Fig. 53.
Their forms are exhibited in
This relation is of importance in the applications of the
cycloid in Mechanics.
Again, since ^0 = arc OP', if we represent AOhy v, we
have*
V = 2a<p. (8)
277. Trochoids. — In general, if a circle roll on a
right line, any point in the.
plane of the circle carried round*
with it describes a curve. Such
curves are usually styled tro-
choids. When the tracing
point is inside the circle, the
locus is called a prolate tro-
choid ; when outside, an oblate,
the accompanying figure.
Their equations are easily determined; for, let x, y be
the co-ordinates of a tracing point P, referred to the axes
AD, and AI {A being the position for which the moving
radius CP is perpendicular to the fixed line).
Then, if CO = a, CP ^ d, L OCP = 0, we have
x^^AN^AO-ON^aQ-dmi 0,
y = PiV" = « - (? COS 0.
278. Epicycloidsf and Hypocycloids. — The investi-
♦ This is called, by Professor Casey, the tangential equation of the cycloid,
and by aid of it he has arrived at some remarkable properties of the curve ("On
a New Form of Tangential Equation," Fhilosophical Transactions, 1877). "In
general, if a variable line, in any of its positions, make an intercept v on the axis
of ar, and an angle ^ with it; then the equation of the line is
ar + ^cot^-y = o;
and V, <p, the quantities which determine the position of the line may be called
its co-ordinates. From this it follows that any relation between y and tp, such
as
will be the tangential equation of a curve, which is the envelope of the line.'*
For applications, the reader is referred to Professor Casey's Memoir. See also
Dub. Exam. Papers^ Graves, Lloyd Exhibition, 1847.
t I have in this edition adopted the correct definition of these curves as
given by Mr. Proctor in his Geometry of Cycloids. I have thus avoided the
anomaly existing in the ordinary definition, according to which every epicycloid
Z 2
(9)
340
Roulettes.
ffation of the properties of the cycloid naturally gave nse to
the discussion of the more general case of a circle rolling on a
fixed circle. In this case the curve generated by any point
on the circumference of the rolling circle is called .an epicycloid,
or a hypocycloid, according as the rolling circle touches the outside,
or the inside of the circumference of the fixed circle. We shaU
commence with the former case.
Let P he the position of the generating point at any in-
stant, A its position when
on the fixed circle ; then
the arc OA = arc OP.
Again, let (7 and Che
the centres of the circles,
a and b their radii,
LACo = e,/.oc'P = e';
then, since arc OA = arc
OP, we have aO = hO.
Now, suppose C taken
as the origin of rectangu-
lar co-ordinates, and CA
as the axis of a?; draw PN
and C'L perpendicular,
and PJf parallel, to CA, and we have
Fig. 54.
x=^CN=CL-NL = {a+b) cosO -b cos {0 + Q'),
y = PN^CL- C'M= {a + J) sin 0 - J sin (0 + ^);
or, suhstituting - d for d\
X = {a + b) cosB - b cos —7— 0,
. a + b
y = {a + b) sinO - b sin —7— 0.
i
(10)
is a hypocycloid, but only some hypocycloids are epicycloids. While according
to the correct definition no epicycloid is a hypocycloid, though each can be gene-
rated in two ways, as will be proved in Art. 280.
Epicycloids and Hypocycloids, 341
When the radius of the rolling circle is a submultiple of
that of the fixed circle, the tracing point, after the circle
has rolled once round the circumference of the fixed circle,
evidently returns to the same position, and will trace the
same curve in the next revolution. More generally, if the
radii of the circles have a commensurable ratio, the tracing
point, after a certain number of revolutions, will return to its
original position : but if the ratio be incommensurable, the
point will never return to the same position, but will describe
an infinite series of distinct arcs. As, however, the suc-
cessive portions of the curve are in every respect equal to
each other, the path described by the tracing point, from
the position in which it leaves the fixed circle until it returns
to it again, is often taken instead of the complete epicycloid,
and the middle point of this path is called the vertex of the
curve.
In the case of the hypocycloid, the generating circle rolls
on the interior of the fixed circle, and it can be easily seen
that the expressions for x and y are derived from those in (10)
by changing the sign of h ; hence we have
« = (a - 6) cos 0 + 6 cos —7— Oi
a-b • ^"^
y - {a -h) Bm.0 -h sin —7— 0,
The properties of these curves are best investigated by
aid of the simultaneous equations contained in formulas (10)
and (11).
It should be observed that the point A, in Fig. 54, is a
cusp on the epicycloid ; and, generally, every point in which
the tracing point P meets the fixed circle is a cusp on the
roulette. From this it follows that if the radius of the rolling
circle be the n^^ part of that of the fixed, the corresponding epi-
or hypo-cycloid has n cusps : such curves are, accordingly,
designated by the number of their cusps : such as the three-
cusped, four-cusped, &c. epi- or hypo-cycloids.
Again, as in the case of the cycloid, it is evident from
Descartes' principle that the instantaneous path of the point P
is an elementary portion of a circle having 0 as centre ; ac-
342 Roulettes.
oordingly, the tangent to tlie path at P is perpendicular to
the line PO, and that line is the normal to the curve at P.
These results can also be deduced, as in the case of the
cycloid, by differentiation from the expressions for x and y.
We leave this as an exercise for the student.
To find an expression for an element ds of the curve at
the point P; take 0\ 0'', two points infinitely near to 0 on
the circles, and such that 00' = OC/'', and suppose the gene-
rating circle to roll until these' points coincide :* then the
lines CO' and CO" will lie in directum, and the circle wiU
have turned through an angle equal to the sum of the angles
OCO' and OC'O''; hence, denoting these angles hjdO and dd",
respectively, we have
ds = OP (dO + d&) = OP^i + f)dd\ (i2)
since d& =^ - dd.
0
279. Radius of Curvature of an Epicycloid. —
Suppose w to be the angle OSN between the normal at P and
the fixed line GA, then
i^ = aos-o'C8 = ----Q) .', daj = -de[i+-^\.
22' { 2b)
Hence, if p be the radius of curvature corresponding to
the point P, we get
p= * = Opi(^). (,3)
dii) a + 2b ^ ^
Accordingly, the radius of curvature in an epicycloid is
in a constant ratio to the chord OP, joining the generating
point to the point of contact of the circles.
♦ It may be observed that O'O" is infinitely small in comparison with OC/ ;
hence the space through which the point 0 moves during a small displacement
18 infinitely small in comparison with the space through which Pmoves. It is
in consequence of this property that 0 may be regarded as being at rest for the
instant, and every point connected with the rolling circle as having a circular
motion around it
Double Generation of Epicycloids and Hypocycloida^ 343
Fig. 55«
280. Double feneration of Epicycloids and Hypo-
cycloids. — In an Epicycloid, it can be easily sliown that
the curve can be generated in a second manner. For,
suppose the rolling circle in-
closes the fixed circle, and join
P, any position of the tracing
point, to 0, the correspond-
ing point of contact of the two
circles; draw the diameter O-SD,
and join C/E and PD ; connect
C, the centre of the fixed circle,
to (/, and produce CO to meet
DP produced in D\ and describe
a circle round the triangle (yPD'\
this circle plainly touches the
fixed circle ; also the segments
standing on OP, O'P, and OiJ are obviously similar ; hence,
since OP = OC/ + UP, we have
arc OP = arc 00' + arc (/P.
If the arc 00' A be taken equal to the arc OP^ we have
arc OA = arc QfP ; accordingly, the point P describes the same
curve, whether we regard it as on the circumference of the
circle OPD rolling on the circle OO'E^ or on the circumference
of O'PD' rolling on the same circle ; provided the circles each
start from the position in which the generating point coincides
with the point A, Moreover, it is evident that the radius oj
the latter circle is the difference
between the radii of the other two.
Next, for the Hypocycloid,
suppose the circle OPD to roll
inside the circumference of OO'E,
and let C be the centre of the
fixed circle ; join OP, and pro-
duce it to meet the circum-
ference of the fixed circle in 0' ;
draw O'E and PD, join CO',
intersecting PD in D', and de-
scribe a circle round the triangle
PD'Qf, It is evident, as be-
fore, that this circle touches the Fig. 56.
344
Roulettes,
larger circle, and that its radius is equal to the difference be-
tween the radii of the two given circles. Also, for the same
reason as in the former case, we have
arc 00' = arc OP + arc O'P.
If the arc OA be taken equal to OP, we get arc UP
= arc O^A ; consequently, the point P will describe the same
hypocycloid on whichever circle we suppose it to be situated,
provided the circles each set out from the position for which
P coincides with A.
The particular case, when the radius of the rolling circle is
half that of the fixed circle, may be noticed. In this case the
point B coincides with C, and P becomes the middle point of
0(/, and A that of the arc 00^. From this it follows im-
mediately that the hypocycloid described by P becomes the
diameter CA of the fixed circle. This result will be proved
otherwise in Art. 285.
The important results of this Article were given by Euler
[Acta.Petrop.y 1781). By aid of them all epicycloids can be
generated by the rolling of a circle outside another circle ;
and all hypocycloids by the rolling of a circle whose radius
is less than half that of the fixed circle.
281. E volute of an Epicycloid. — The evolute of an
epicycloid can be easily
seen to be a similar epi-
cycloid.
For, let P be the trac-
ing point in any position,
A its position when on the
fixed circle ; join P to 0,
the point of contact of the
circles, and produce PO
untU Pi" = OP ?^*,
then P is the centre of
curvature by (13) ; hence
or^op
a + lb' Fig. 5-7.
Next, draw P'C/ perpendicular to P'O ; circumscribe the
Evolute of Epicycloid, 345
triangle OP' (7 by a circle ; and describe a circle with C as
centre, and CC/ as radius : it evidently touches the circle OF'Qf.
Then Oa : 0E= OF : OP = a:a + zh-^COiCE',
.-. CO-Oa:CE-OE = CO:CE,
or Ca:CO=CO:CE;
that is, the lines CE, CO, and 0(7 are in geometrical pro-
portion.
Again, join C to B", the vertex of the epicycloid ; let CF
meet the inner circle in D, and we have
arc an : arc OJ? = CO' : C0= CO : CE^aO.EO
But arc OB = arc OQ ; /. arc CD = arc FO^.
Accordingly, the path described by F is that generated by a
point on the circumference of the circle OFO^ rolling on the
inner circle, and starting when P' is in contact at D. Hence
the evolute of the original epicycloid is another epicycloid.
The form of the evolute is exhibited in the figure.
Again, since CO : OE = CO^ : (/O, the ratio of the radii
of the fixed and generating circles is the same for both epicy-
cloids, and consequently the evolute is a similar epicycloid.
Also, from the theory of evolutes (Art. 237), the line
PF i:> equal in length to the arc P'A of the interior epicy-
cloid ; or the length of FA^ the arc measured from the
vertex A of the curve, is equal to
a CO ca
Hence, the length* of any portion of the curve measured from
its vertex is to the corresponding chord of the generating circle as
twice the sum of the radii of the circles to the radius of the fixed
circle.
* The length of the arc of an epicycloid, as also the investigation of its
evolute, were given hy Newton {Principia, Lib. i., Props. 49, 50).
346
Roulettes.
With reference to the outer epicycloid in Fig. 57, this
gives
aroP^=2P^.^. (,4)
The corresponding results for the hypocjcloid can be
found by changing the sign of the radius b of the rolling
circle in the preceding formulae.
The investigation of the properties of these curves is of
importance in connexion with the proper form of toothed
wheels in machinery.
282. Pedal of Epicycloid. — The equation of the pedal,
with respect to the centre of the
fixed circle, admits of a very
simple expression. For let P be
the generating point, and, as be-
fore, take arc OA = arc OP, and
make AB = 90°. Join CA, CB,
CP, and draw CN perpendicular
to LP. Let lPDO = <p,l BCN
= io,/.ACO = e, CN = p.
Then since AO = PO, we have
aO = 2b(i); .-. 9 = —(j).
Again, w = go° - ACW = 0 + 6
I + —
a
Fig. 58.
hence
Also
0 =
ow
a + 2b
CN= ODsin^;
.*. p= {a + 2b) sin
au)
a + 26'
(15)
(16)
which is the equation of the required pedal.
283. Equation of Epicycloid in terms of r andj!?. —
Again, draw OL parallel to DiV^, and let CP = i\ and we have
r^ -/ = PN^ = OL' = OC - CD
a + 2b
f
Epitrochoids and Hypotrochoids, 347
^-'^'^'^^'■^ c^)
Also, from (16) it is plain that the equation of DiV", the tan-
gent to the epicycloid (referred to GB and CA as axes of x
and y respectively), is
izjcos (u + ?/ smo) = (« + 20) sm 7. (18)
The corresponding formulae for the hjrpocycloid are
obtained by changing the sign of J in the preceding equa-
tions.
Again, it is plain that the envelope of the right line re-
presented by equation (18) is an epicycloid. And, in general,
the envelope of the right line
«? cos fc> + y sin (u = A; sin ma>,
regarding w as an arbitrary parameter, is an epicycloid, or a
hypocycloid, according as m is less or greater than unity. For
examples of this method of determining the equations of epi-
and hypo-cycloids the student is referred to Salmon's Higher
Plane Curves, Art. 310.
284. £pitroclioids and Hypotrocboids. — In general,
when one circle rolls on another, every point connected with
the rolling circle describes a distinct curve. These curves are
called epitrochoids or hypotrochoids, according as the rolling
circle touches the exterior or the interior of the fixed circle.
If d be the constant distance of the generating point from
the centre of the rolling circle, there is no difficulty in
proving, as in Art. 278, that we have in the epitrochoid the
equations
X = {a ■\-h) QO^O - d cos — — 6,
« + i > (-9)
y = {a + b) sinO - d sin —7— B.
348
Roulettes.
In the case of the hypotrochoid, changing the signs of h
and dy we ohtain
a-h
X = {a - b) cos 6 + d
cos
b
a - b
e,
1
^ = (a-b) sin 9 - d sin "^-r-^ B.
(20)
In the particular case in which a = iby i.e. when a circle
rolls inside another of double its diameter, equations (20)
become
x = [b +d) cos Q, y = (b - d) sin 0 ;
and accordingly the equation of the roulette is
^' ^ f
(b+dy {b-dy *
which represents an ellipse whose semi-axes are the sum and
the difference of b and d.
This result can also be established geometrically in the
following manner : —
285. Circle rolling inside another of double its
Diameter. — Join Ci and 0 to any
point L on the circumference of the
rolling circle, and let CiL meet the
fixed circumference in A ; then since
L OCL = zOCA, and OG, = 2 00, we
have arc OA = arc OL ; and, accord-
ingly, as the inner circle rolls on the
outer the point L moves along CiA,
In like manner any other point on
the circumference of the rolling circle
describes, during the motion, a dia-
meter of the fixed circle.
Again, any point P, invariably connected with the rolling
circle, describes an ellipse. For, if L and M be the points in
which OP cuts the rolling circle, by what has been just
shown, these points move along two fixed right lines C^A
and CiBy at right angles to each other. Accordingly, by a
Epitrochoids and Hypotrochoids. 349
well-known property of the ellipse, any other point in the
line LM describes an ellipse.
The case in which the outer circle rolls on the inner is
also worthy of separate consideration.
286. Circle rolling on another inside it and of
half its Diameter. — In this case, any diameter of the rolling
circle always passes through a fixed point, which lies on the
circumference of the inner circle.
For, let CiL and C^L be any two positions of the moving
diameter, (7i and d being the corresponding positions of the
centre of the rolling circle : 0 and O2 the corresponding posi-
tions of the point of contact of the circles. Now, if the outer
circle roll from the former to the latter position, the right
lines C1O2 and CO2 will coincide in
direction, and accordingly the outer
circle will have turned through the
angle C2O2C1 ; consequently, the mov-
ing diameter will have turned
through the same angle ; and hence
Z C2LC1 = lCiO^Cx", therefore the
point L lies on the fixed circle, and
the diameter always passes through
the same point on this circle.
Again, any right line connected Fie" 60
with the rolling circle will always touch
a fixed circle.
For, let DE be the moving line in any position, and draw
the parallel diameter AB\ let fall Cii^ and ZIT perpendicular
to DE. Then, by the preceding, AB always passes through
a fixed point L ; also LM= CiF= constant ; hence D-E" always
touches a circle having its centre at L.
Again, to find the roulette described by any carried point
Pi. The right line Pi(7i, as has been shown, always passes
through a fixed point L ; consequently, since CiPi is a con-
stant length, the locus of Pi is a Limagon (Art. 269). In like
manner, any other point invariably connected with the outer
circle describes a Limacon ; unless the point be situated on
the circumference of the rolling circle, in which case the
locus becomes a cardioid.
^eo Examples — Roulettes.
1. "When the radii of the fixed and the rolling circles become equal, prove
geometrically that the epicycloid becomes a cardioid, and the epitrochoid a
Liraa^on (Art. 269).
2. Prove that the equation of the reciprocal polar of an epicycloid, with
respect to the fixed circle, is of the form
r sin ma = const.
3. Prove that the radius of curvature of an epicycloid varies as the perpen-
dicular on the tangent from the centre of the fixed circle.
4. If a = 4J, prove that the equation of the hypocycloid becomes
5. Find the equation, in terms of r &ndp, of the three-cusped hypocycloid ;
«. e. when a = zb. Ans. r^^a^ - %f.
6. Find the equation of the pedal in the same curve.
Ans. jt? = 5 sin 3w.
7. In the case of a curve rolling on another which is equal to it in every
respect, corresponding points being in contact, prove that the determination of
the roulette of any point P is immediately reduced to finding the pedal of the
rolling curve with respect to the point P.
8. Hence, if the curves be equal parabolas, show that the path of the focus
is a right line, and that of the vertex a cissoid.
9. In like manner, if the curves be equal ellipses, show that the path of the
focus is a circle, and that of any point is a bicircular quartic.
10. In Art. 285, prove that the locus of the foci of the ellipses described by
the different points on any right line is an equilateral hyperbola.
ir. -4 is a fixed point on the circumference of a circle ; the points L and M
are taken such that arc AL — m arc AM, where m is a constant ; prove that the
envelope of LM is an epicycloid or a hypocycloid, according as the arcs AL and
AM are measured in the same or opposite directions from the point A.
12. Prove that ZM, in the case of an epicycloid, is divided internally in the
ratio w : I, at its point of contact with the envelope ; and, in the hypocycloid,
externally in the same ratio.
13. Show also that the given circle is circumscribed to, or inscribed in, the
envelope, according as it is an epicycloid or hypocycloid.
14. Prove, from equation (14), that the intrinsic equation of an epicycloid is
4J (a + b) . a<b
8 = — i sm — r.
where s is measured from the vertex of the curve.
15. Hence the equation « = ^ sin «<^ represents an epicycloid or a hypo-
cycloid, according as » is less or greater than unity.
Centre of Curvature of an Epitrochoid or Sypotrochoid. 35 1
16. In an epitrochoid, if the distance, d, of the moving point from the centre
of the rolling circle be equal to the distance between the centres of the circles,
prove that the polar equation of the locus becomes
r = 2 (a + h) cos
+ 2l
17. Hence show that the curve
r = a dnrnQ
is an epitrochoid when w < i, and a hypotrochoid when m > i.
This class of curves was elaborately treated of by the Abbe Grandi in the
Philosophical Transactions for 1723. He gave them the name of " Rhodonese,"
from a fancied resemblance to the petals of roses. See also Gregory's Examples
on the Differential and Integral Calculus^ p. 183.
For illustrations of the beauty and variety of form of these curves, as well as
of epitrochoids and hypotrochoids in general, the student is referred to the admi-
rable figures in Mr. Proctor's Geometry of Cycloids.
287. Centre of Curvature of an Epitrochoid or
Hypotrocboid. — The position of the centre of curvature for
any point of an epitrochoid can be easily
found from geometrical considerations. For,
let Cx and G^ be the centres of the rolling
and the fixed circles, Pi the centre of cur-
vature of the roulette described by Pi ; and,
as before, let Oi and O2 be two points on the
circles, infinitely near to 0, such that OOx
= OO2. Now, suppose the circle to roll until
Ox and Oz coincide; then the lines CxOi
and C2O2 will lie in directum, as also the
lines PxOx and P2O2 (since P2 is the point Fig. 61.
of intersection of two consecutive normals to
the roulette).
Hence z OCxOx + L OC^Oi = L OPxOx + L OP^O^,
since each of these sums represents the angle through which
the circle has turned.
Again, let L CxOPx = 0, OOx = 00^ = ds ;
then
L OCxOx
OCx' "-^^'^'-oci'
,OPxOx='-^, ^OPM = '-^
352 Roulettes,
consequently we have
Or, if OPi = n, OP2 = i\,
-^ok='''Ko?/ok) ^''^
II ,M I
- + - = cos d) — + - |.
From this, equation n, and consequently the radius of curva-
ture of the roulette, can be obtained for any position of the
generating point Pi.
If we suppose Pi to be on the circumference of the rolling
OP
circle, we get cos ^ = —7777 ; whence it follows that
2 C/Oi
OP, = — ^ OPi,
which agrees with the result arrived at in Art. 279.
288. Centre of Curvature of any Roulette. — The
preceding formula can be readily extended to any roulette : for
if (7i and C, be respectively the centres of curvature of the
rolling and fixed curves, corresponding to the point of contact 0,
we may regard OOi and 00, as elementary arcs of the circles
of curvature, and the preceding demonstration will still
hold.
Hence, denoting the radii of curvature 00 i and 00, by
Pi and Pi, we shall have
— + i = cosd)|- ^■ -\ (22)
Pi /02 ^Vi rij
It can be easily seen, without drawing a separate figure,
that we must change the sign of p, in this formula when the
centres of curvature lie at the same side of 0.
It may be noted that Pi is the centre of curvature of the
roulette described by the point P2, if the lower curve be sup-
posed to roll on the upper regarded as fixed.
289. Creometrical Construction* for the Centre of
♦ This beautiful construction, and also the formula (22) on which it is based,
■were given by M. Savary, in his Leqona des Machines a VEcole Foly technique.
See also Leroy's Qiometrie Descriptive, Quatrilme Edition, p. 347.
Construction for Centre of Curvattcre.
353
Curvature of a Roulette. — The formula {22) leads to
a simple and elegant construction for the centre of curva-
ture Pa.
We commence with the case when the base is a right
line, as represented in the accom-
panying figure.
Join Pi to (7i, the centre of curva-
ture of the rolling curve, and draw
ON perpendicular to OPi, meeting
FyCi in N; through N draw JVM
parallel to 0(7i, and the point P2 in
which it meets OPi is the centre of
curvature required.
For, equation {22) becomes in
this case
cos ^ I TT^^ +
Fig. 62.
oc.
OF, OF..
k)
V^hence we get
Pi Pa
OPi . OP2
NF,
NC, . OF, '
00, sin C, ON NO, sin C,NO
. i^iPa ^ NF, ^
" OF2 NC,'
and, accordingly, the line NF2 is parallel to Od, Q. E. D,
The construction in the general case is as follows : —
Determine the point N as in the former
case, and join it to Cg, the centre of curva-
ture of the fixed curve, then the point of
intersection of NCz and F,0 is the required
centre of curvature.
This is readily established ; for, from
the equation
CC^ _ cos ^PiPa^
OC.OC OF,. OF J
CiCa OF2 O(7iCOS0
I
OC,
we get
oa'F^F,
OF,
2 A
Fig. 63.
354
Roulettes,
But, as before,
OGi cos ^ =
CN.OPx . OCiCos0
NPr ' " OP,
NO,
NP,
hence
C,C, OP2 ^NCi
OCP^P," NPx
Consequently, by the well-known property of a transversal
cutting the sides of a triangle, the points d, P2, and N are
in directum.
The modification in the construction when the rolling
curve is a right line can be readily supplied by the student.
290. Circle of Inflexions. — The following geometrical
construction is in many cases more Cj
useful than the preceding.
On the line OCi take ODi such
that
I
OBy
00, OO2'
and on OD, as diameter describe a
circle. Let JE, be its point of inter-
section with OPiy then we have
and formula {22) becomes
I I
+
(23)
OPx OP2 OAcos^ OEi'
Hence, if the tracing point Pi lie on the circle* OEiD,,
♦ This theorem is due to La Hire, who showed that the element of the
roulette traced by any point is convex or concave with respect to the point of
contact, 0, according as the tracing point is inside or outside this circle. (See
Envelope of a Carried Curve, 355
the corresponding value of OP2 is infinite," and consequently
Pi is a point of inflexion on the roulette.
In consequence of this property, the circle in question is
called the circle of inflexions, as each point on it is a point of
inflexion on the roulette which it describes.
Again, it can be shown that the lines Pi P2, PiQ-^^ Pi Ei
are in continued proportion ; as also CiCz, CiOy anJ CxBi.
For, from (23) we have
P1P2 I
OPi.OF, OE,
Hence P1P2 :F,0=OP^: OEr,
.-. PiPa : PiO = P1P2 -OP,:P,0- OE, = P,0: P^E,. (24)
In the same manner it can be shown that
CiA:CiO=(7iO:(7ii)i. (25)
In the particular case where the base is a right line, the
circle of inflexions becomes the circle described on the radius
of curvature of the rolling curve as diameter.
Again, if we take OD2 = ODiy we shall have, by describing
a circle on ODg as diameter,
C2C/1 '. O2C/ = {y-iiJ '. L/2-L^2)
and also P2P1 : P2O = P2O : P^E^. (26)
The importance of these results will be shown further on.
291. Envelope of a Carried Curve. — We shall next
consider the envelope of a curve invariably connected with the
rolling curve, and carried with it in its motion.
Since the moving curve touches its envelope in each of its
Memoires de V Academie des Sciences, 1706.) It is strange that this remarkable
result remained almost unnoticed until recent years, when it was found to
contain a key to the theory of curvature for roulettes, as well as for the
envelopes of any carried curves. How little it is even as yet appreciated in
this country v/ill be apparent to any one who studies the most recent produc-
tions on roulettes, even by distinguished British Mathematicians.
2 A 2
356 Roulettes.
positions, the path of its point of contact at any instant must
be tangential to the envelope; hence the normal at their
common point must pass through 0, the point of contact of
the fixed and rolling curves.
In the particular case in which the carried curve is a
right line, its point of contact v^ith
its envelope is found by dropping a
perpendicular on it from the point of
contact 0.
For example, suppose a circle to
roll on any curve : to find the envelope*
of any diameter PQ : —
From 0 draw ON perpendicular
to PQ, then N, by the preceding, is Yis. 6$.
a point on the envelope.
On 0(7 describe a semicircle; it will pass through iV",
and, as in Art. 286, the arc ON = arc OP = OA, if A be
the point in which P was originally in contact with the
fixed curve. Consequently, the envelope in question is the
roulette traced by a point on the circumference of a circle
of half the radius of the rolling circle, having the fixed curve
AO for its base.
For instance, if a circle roll on a right line, the envelope of
any diameter is a cycloid, the radius of whose generating circle
is half that of the rolling circle.
Again, if a circle roll on another, the envelope of any
diameter of the rolling circle is an epicycloid, or a hypocycloid.
Moreover, it is obvious that if two carried right lines be
parallel, their envelopes will be parallel curves. For ex-
ample, the envelope of any right line, carried by a circle
which rolls on a right line, is a parallel to a cycloid, i.e. the
involute of a cycloid.
These results admit of being stated in a somewhat different
form, as follows :
If one point. A, in a plane area move uniformly along a
right line, while the area turns uniformly in its own plane,
then the envelope of any carried right line is an involute to a
cycloid. If the carried line passes through the moving point
* The theorems of this Article are, I believe, due to Chasles : see his Histoire
de La Geometrie, p. 6q.
Centre of Curvature of the Envelope of a Carried Curve. 357
A, its envelope is a cycloid. Again, if tB*e point A move
uniformly on the circumference of a fixed circle, while the
area revolves uniformly, the envelope of any carried right
line is an involute to either an epi- or hypo-cycloid. If the
carried right line passes through A, its envelope is either an
epi- or hypo- cycloid.
292. Centre of Curvature of the Envelope of a
Carried Curve. — Let aj)i represent a
portion of the carried curve, to which Dm
is normal at the point m ; then, by the
preceding, m is the point of contact of aihx
with its envelope.
Now, suppose fl'2^2 to represent a por-
tion of the envelope, and let Pi be the
centre of curvature of «i6i, for the point m,
and P2 the corresponding centre of cur-
vature of ^2^2-
As before, take d and O2 such that
00, = OO2, and join PiOi and P2O2.
Again, suppose the curve to roll until
Oi and O2 coincide; then the lines PiOx
and P2O2 will come in directum, as also
the lines dCi and O^Cz ; and, as in Art.
288, we shall have
Fig. 66.
zOi + z:C2 = zPi + zPa;
and consequently
ok "" ^r '"' H^. ■" ^)
(27)
From this equation the centre of curvature of the enve-
lope, for any position, can be found. Moreover, it is obvious
that the geometrical constructions of Arts. 289, 290, equally
apply in this case. It may be remarked that these construc-
tions hold in all cases, whatever be the directions of curvature
of the curves.
The case where the moving curve aihi is a right line is
worthy of especial notice.
358
Roulettes
In this case the normal Om is perpendicular to the moving
line ; and, since the point Pi is ini&nitely
distant, we have
COS(^
= 7777 (^^^- 290);
OP, 00, 00, OD
whence, F, is situated on the lower circle of
inflexions. Hence we infer that the dif-
ferent centres of curvature of the curves en-
veloped hy all carried right lines, at any pj g^;
instant, lie on the circumference of a circle.
As an example, suppose the right line OM to roll on a
fixed circle, whose centre is C,, to
find the envelope of any carried right
line, LM.
In this case the centre of cur-
vature, P2, of the envelope of LM,
lies, by the preceding, on the circle
described on 00 as diameter; and,
accordingly, OP, is perpendicular
to the normal Pi P,.
Hence, since L OLPi remains
constant during the motion, the line
OPi is of constant length; and, if
we describe a circle with 0 as centre, pig. 68.
and OP1 as radius, the envelope of
the moving line LM will, in all positions, be an involute of a
circle. The same reasoning applies to any other moving
right line.
We shall conclude with the statement of one or two other
important particular cases of the general principle of this
Article.
(i). If the envelope a^h, of the moving curve aihx he a right
line, the centre of curvature Pi lies on the corresponding circle of
inflexions.
(2). If the moving right line alicays passes through a fixed
point, that point lies on the circle OB^E,.
292 (a). Ei:presision for Radius of Curvature of
Envelope of a Right liine. — The following expression
for the radius of curvature of the envelope of a moving right
On the Motion of a Plane Figure in its Plane, 359
line is sometimes useful. Let p be tlie perpendicular distance
of the moving line, in any position, from a fixed point in the
plane, and w the angle that this perpendicular makes with a
fixed line in the plane, and p the radius of curvature of the
envelope at the point of contact; then, by Art. 206, we have
^^P / ON
Whenever the conditions of the problem givejo in terms of
w (the angle through which the figure has turned), the value
of p can be found from this equation. For example, the re-
sult established in last Article {see Fig. 68) can be easily
deduced from {2^). This is left as an exercise for the student.
293. On the Motion of a Plane Figure in its Plane.
— "We shall now proceed to the consideration of a general
method, due to Chasles, which is of fundamental importance
in the treatment of roulettes, as also in the general investi-
gation of the motion of a rigid body.
We shall commence with the following theorem : —
When an invariable plane figure moves in its plane, it can
he brought from any one position to any other by a single rotation
round a fixed point in its plane.
For, let A and B be two points of the figure in its first
position, and Ai, Bi their new
positions after a displacement.
Join A Ax and BBi, and sup-
pose the perpendiculars drawn
at the middle points of AAi
and BBi to intersect at 0 ;
then we have AO = AiO, and
BO = BiO. Also, since the
triangles AOB and AiOBi
have their sides respectively ^^' ^'
equal, we have Z.AOB = lA.OB^ ; .•. z AOA^ = lBOBx.
Accordingly, AB will be brought to the position A^Bx by
a rotation through the angle AOAx round 0. Consequently,
any point C in the plane, which is rigidly connected with AB,
will be brought from its original to its new position, Cx, by
the same rotation.
This latter result can also be proved otherwise thus : — Join
OC and OCx ; then the triangles OAO and OAiCx are equal,
360 Roulettes.
because OA = OAi, AC = A,Ci, and the angle OAC, being
the difference between OAB and BAC, is equal to OA.d,
the difference between OA^Bx and BiAiCi ; therefore OC
= OCi, and lAOC = lA.OC^ ; and hence z ^O^i = / COC^.
Consequently the point G is brought to Ci by a rotation
round 0 through the same angle AOAx. The sarae reasoning
applies to any other point invariably connected with A and B.
The preceding construction re-
quires modification when the lines
AAi and BBi are parallel. In this
case the point, 0, of intersection of the
lines BA and BiAi is easily seen to be
the point of instantaneous rotation.
For, since AB = A^Bx, and AAi, ^1^^ ^o^
BBi, are parallel, we have OA = OAi,
and OB = OBi. Hence, the figure will be brought from its
old to its new position by a rotation around 0 through the
angle AOAy.
Next, let AAij and BBi be both equal and parallel. In
this case the point 0 is at an infinite distance ; but it is
obvious that each point in the plane moves through the same
distance, equal and parallel to AA^ ; and the motion is one of
simple translation, without any rotation.
In general if we suppose the two positions of the moving
figure to be indefinitely near each other, then the line AAi,
joining two infinitely near positions of the same point of the
figure, becomes an element of the curve described by that point,
and the line OA becomes at the same time a normal to the curve.
Hence, the normals to the paths described by all the points of the
moving figure pass through 0, which point is called the instan-
taneous centre of rotation.
The position of 0 is determined whenever the directions of
motion of any two points of the moving figure are known ; for it
is the intersection of the normals to the curves described by
those points.
This furnishes a geometrical method of drawing tangents
to many curves, as was observed by Chasles.*
• This method is given by Chasles as a generalization of the method of Des-
cartes (Art. 273, note). It is itself a particular case of a more general principle
concerning homologous figures. See Chasles, Eistoirc de la Geometrie, pp. 548-9 :
9\wi Bulletin Universel dea Sciences, 1830.
Chasles' Method of drawing NormaU. 301
■f
The following case is deserving of special consideration : —
A right line always passes through a fixed
point, while one of its points moves along a
fixed line : to find the instantaneous centre of
rotation. Let A be the fixed point, and AB
any position of the moving line, and take
B' A = BA ; then the centre of rotation, 0, is
found as before, and is such that OA = 0A\
and OB = OB". Accordingly, in the limit the ^ ^
centre of instantaneous rotation is the inter- j.-
section oi BO drawn perpendicular to the fixed
line, and A 0 drawn perpendicular to the moving line at the
fixed point.
In general, if ABhe any moving curve, and Zilf any fixed
curve, the instantaneous centre of rotation is the point of inter-
section of the normals to the fixed and to the moving curves j for
any position.
Also the normal to the curve described by any point in-
variably connected with AB is obtained by joining the point
to 0, the instantaneous centre.
More generally, if a moving curve always touches a fixed
curve -4, while one point on the moving curve moves along a
second fixed curve B, the instantaneous centre is the point of
intersection of the normals to A and B at the corresponding
points; and the line joining this centre to any describing
point is normal to the path which it describes.
We shall illustrate this method of drawing tangents by
applying it to the conchoid and the limacon.
294. Application to Curves. — In the Conchoid (Fig. 49,
ps-g® 332), regarding AP as a moving right line, the
instantaneous centre 0 is the point of intersection oi AO
drawn perpendicular to AP, with RO drawn perpendicular to
LM] and consequently, OP and OPi are the normals at P
and Pi, respectively.
For the same reason, the normal to the Limacon (Fig. 48,
page 331) at any point Pis got by drawing OQ perpendicular
to OP to meet the circle in Q, and joining PQ.
362 Roulettes.
Examples.
1. If the radius vector, OP, drawn from the origin to any point P on a curve,
be produced to Pi, until PPi be a constant length ; prove that the normal at P\
to the locus of Pi, the normal at P to the original curve, and the perpendicular
at the origin to the line OP, all pass through the same point.
2. If a constant length measured from the curve be taken on the normals
along a given curve, prove that these lines are also normals to the new curve
which is the locus of their extremities.
3. An angle of constant magnitude moves in such a manner that its sides
constantly touch a given plane curve ; prove that the normal to the curve de-
scribed by its vertex, P, is got by joining Pto the centre of the circle passing
through P and the points in which the sides of the moveable angle touch the
given curve.
4. If on the tangent at each point on a curve a constant length measured
from the point of contact be taken, prove that the normal to the locus of the
points so found passes through the centre of curvature of the proposed curve.
5. In general, if through each point of a curve a line of given length be
drawn making a constant angle with the normal, the normal to the curve locus
of the extremities of this line passes through the centre of curvature of the pro-
295. Motion of any Plane Figure reduced to
Roulettes. — ^Again, the most general motion of any figure
in its plane may be regarded as consisting of a number of
infinitely small rotations about the different instantaneous
centres taken in succession.
Let 0, 0\ 0'\ 0"\ &c., represent the successive centres of
rotation, and consider the instant when i,r^
the figure turns through the angle OxOO q /'' ' ^
round the point 0. This rotation will q Zl-' t^
bring a certain point Ox of the figure to _^i,^^
coincide with the next centre (7. The next '^^^^^" "^^
rotation takes place around 0'; and suppose o\;--..
the point O2 brought to coincide with the (fV./
centre of rotation 0". In like manner, by "V^^ T'
a third rotation the point O3 is brought to ^ 1\
coincide with (/", and so on. By this ^ \ ,.
means the motion of the moveable figure ^i
is equivalent to the rolling of the polygon ^^' ^^'
OOiOiOz . . . invariably connected with the figure, on the
polygon Oaa'O'" . . . fixed in the plane. In the limit, the
polygons change into curves, of which one rolls, without
Epicyclics. 363
sliding, on the other ; and hence we conclude that the general
movement of any plane figure in its own plane is equivalent to the
rolling of one curve on another fixed curve.
These curves are called by Eeuleaux* the "centrodes" of
the moving figures.
For example, suppose two points A and B of the moving
figure to slide along two fixed right
lines CX and CY; then the instan-
taneous centre 0 is the point of inter-
section of ^0 and BO, drawn perpen-
dicular to the fixed lines. Moreover,
as AB is a constant length, and the
angle ACB is fixed, the length CO is
constant ; consequently the locus of
the instantaneous centre is the circle
described with C as centre, and CO as ^^* ^^'
radius. Again, if we describe a circle round CBOA, this
circle is invariably connected with the line AB, and moves
with it. Hence the motion of any figure invariably connected
with AB is equivalent to the rolling of a circle inside another
of double its radius {see Art. 285).
Again, if we consider the angle XCY to move so that its
legs pass through the fixed points A and B, respectively ; then
the instantaneous centre 0 is determined as before. More-
over, the circle BCA becomes a, fixed circle, along which the
instantaneous centre 0 moves. Also, since CO is of constant
length, the outer circle becomes in this case the rolling curve.
Hence the motion of any figure invariably connected with the
moving lines CX and CY is equivalent to the rolling of the
outer circle on the inner (compare Art. 286).
295 (a). Epicyclics. — As a further example, suppose one
point in a plane area to move uniformly along the circum-
ference of a fixed circle, while the area revolves with a uniform
angular motion around the point, to find the position of the
"centrodes."
The directions of motion are indicated by the arrow
heads. Let C be the centre of the fixed circle, P the position
* See Kennedy's translation of Eeuleaux' s Kinematics of Machinery,
pp. 65, &c.
364
Roulettes.
of the moving point at any instant, Q a point in the moving
figure such that CP = FQ.
Now, to find the position of
the instantaneous centre of
rotations it is necessary to
get the direction of motion of
the point Q.
Let Pi represent a con-
secutive position of P, then
the simultaneous position of Q
is got by first supposing it to
move through the infinitely
small length QR, equal and
parallel to PPi, and then to
turn round Pi, through the
angle RPiQi, which the area
turns through while P moves
to Pi. Moreover, by hypo-
thesis, the angles PC^Pi and RPxQi are in a
Fig. 74.
constant ratio :
if this ratio be denoted by m, we have (since PQ = PC)
RQr = mPPi = mQR.
Join Q and Qi, then QQ, represents the direction of mo-
tion of Q. Hence the right line QO, drawn perpendicular
to QQi, intersects CP in the instantaneous centre of rotation.
Again, since the directions of PO, PQ, and QO are, re-
spectively, perpendicular to QR, RQi, and QQi, the triangles
QPO and QiRQ are similar ;
.'. PQ = mPO, i.e. CP = mPO.
Accordingly, the instantaneous centre of rotation is got
by cutting off
m
(29)
Hence, if we describe two circles, one with centre C and
radius CO, the other with centre P and radius PO; these
circles are the required centrodes; and the motion is equivalent
to the rolling of the outer circle on the inner.
Epicyclics.
365
Accordingly, any point on the cirdlimference of the outer
circle describes an epicycloid, and any point not on this cir-
cumference describes an epitrochoid. When the angular
motion of PQ is less than that of CP, i.e. when m < i,
the point 0 lies in PC produced. Accordingly, in this
case, the fixed circle lies inside the rolling circle; and the
curves traced by any point are still either epitrochoids or epi-
cycloids.
In the preceding we have supposed that the angular
rotations take place in the same direction. If we suppose them
to be in opposite directions, the construction has to be modified,
as in the accompanying figure.
In this case, the angle ^!%Pf%
RPiQi must be measured in
an opposite direction to that
of PCPx ; and, proceeding as
in the former case, the direc-
tion of motion of Q is repre-
sented by QQi', accordingly,
the perpendicular QO will in-
tersect CP produced, and, as
before, we have
PO
PO
m
Fig.75.
Hence the motion is equi-
valent to the rolling of a circle
of radius PO on the inside of a fixed circle, whose radius is
CO. Accordingly, in this case, the path described by any
point in the moving area not on the circumference of the
rolling circle is a hypotrochoid.
Also, from Art. 291, it is plain that the envelope of any
right line which passes through the point P in the moving
area is an epicycloid in the former case, and a hypocycloid
in the latter.
Again, if we suppose the point P, instead of moving in a
circle, to move uniformly in a right line, the path of any
point in the moving area becomes either a trochoid or a
cycloid.
Curves traced as above, that is, by a point which moves
366 Boulettes,
uniformly round the circumference of a circle^ whose centre moves
uniformly on the circumference of a fixed circle in the same
plane, are called epicyclics, and were invented by Ptolemy
(about A.D. 140) for the purpose of explaining the planetary
motions. In this system* the fixed circle is called the deferent,
and that in which the tracing point moves is called the
epicycle. The motion in the fixed circle may be supposed in
all cases to take place in the same direction around 0, that
indicated by the arrows in our figures. Such motion is called
direct. The case for which the motion in the epicycle is direct
is exhibited in Fig. 74.
Angular motion in the reverse direction is called retro-
grade. This case is exhibited in Fig. 75. The corresponding
epicyclics are called by Ptolemy direct and retrograde epicy-
clics.
The preceding investigation shows that every direct epi-
cyclic is an epitrochoid, and every retrograde epicyclio a
hypotrochoid.
It is obvious that the greatest distance in an epicyclio
from the centre C is equal to the sum of the radii of the circles,
and the least to their difference. Such points on the epicyclio
are called apocentres and pericentres, respectively.
Again, if a represent the radius of the fixed circle or
deferent, and /3 the radius of the revolving circle or epicycle ;
then, if the curve be referred to rectangular axes, that of x
passing through an apocentre, it is easily seen that we have
for a direct epicyclic
iP = a cos 0 + j3 cos mB, \
p = asmd + j5 sm mO. )
* The importance of the epicyclic method of Ptolemy, in representing ap-
proximately the planetary paths relative to the earth at rest, has recently heen
brought prominently forward by Mr. Proctor, to whose work on the Geometry of
Cycloids the student is referred for fuller information on the subject.
We owe also_ to Mr. Proctor the remark that the invention of cycloids, epi-
cycloids, and epitrochoids, is properly attributable to Ptolemy and the ancient
astronomers, who, in their treatment of epicyclics, first investigated some of
the properties of such curves. It may, however, be doubted if Ptolemy had
any idea of the shape of an epicyclic, as no trace oi" such is to be found in the entire
of his great work, The Almagest.
Example on the Construction of Circle of Inflexions. 367
The formulae for a retrograde epicyclio are obtained by
changing the sign of m (compare Art. 284).
It is easily seen that every epicyclic admits of a twofold
generation.
For, if we make md = <{), equation (30) may be written
(h
a? = 3 cos d> + a cos — ,
^ . .6
y = j3 sm d) + a sm — ,
which is equivalent to an interchange of the radii of the
deferent circle and of the epicycle, and an alteration of m
into — . This result can also be seen immediately geometri-
m
cally.
It may be remarked that this contains Euler's theorem
(Art. 280) under it as a particular case.
296. Properties of the Circle of Inflexions. — It
should be especially observed that the results established in
Art. 290, relative to the circle of inflexions, hold in all cases
of the motion of a figure in its plane, and hence we infer
that the distances of any moving point from the centre of curva-
ture of its path, from the instantaneous centre of rotation, and
from the circle of inflexions, are in continued proportion.
Again, from Art. 292, we infer that if a moveable curve
slide on a fixed curve, the distances of the centre of curvature of
the moving, from that of the flxed curve, from the centre of in-
stantaneous rotation, and from the circle of inflexions, are in
continued proportion.
The particular cases mentioned in these Articles obviously
hold also in this case, and admit of similar enunciations.
These principles are the key to the theory of the curvature
of the paths of points carried by moving curves, as also to the
curvature of the envelopes of carried curves.
We shall illustrate this statement by a few applications.
297. Sxample on tlie Construction of Circle of
Inflexions. — Suppose two curves aj)i and Cidi, invariably con^
nected with a moving plane figure, always to touch two fixed
curves a^h and c^dz^ to find the centre of curvature of the roulette
described by any point Rx of the moving figure.
J 68 Roulettes.
The instantaneous circle of inflexions is easily constructed
in the following manner : — Let
Pi and P3 be the centres of cur-
vature for the point of contact
m for the curves aih and a^h^,
respectively : and let Qi, Qz, be
the corresponding points for
the curves c^d^ and c^di. Take
then, by Art. 290, the^ points ' p-g ^5
Ex and Fi lie on the circle of
inflexions. Accordingly, the circle which passes through 0,
Ei and Pi, is the circle of inflexions.
Hence, if PiO meet this circle in Gi, and we take
R1R2 = ^^-pTi ^^® point P2 (by the same theorem) is the
centre of curvature of the roulette described hy Pi.
In the same case, by a like construction, the centre of cur-
vature of the envelope of any carried curve can be found.
The modifications when any of the curves «i6i, a-ih^ &c.,
becomes a right line, or reduces to a single point, can also be
readily seen by aid of the principles already established for
such cases.
298. Theorem of Bobillier.* — If two sides of amoving
triangle always touch two fixed circleSy the third side also always
touches a fixed circle.
Let ABC be the moving triangle ; the side AB touching
at <? a fixed circle whose centre is 7, and AC touching at J a
circle with centre /3. Then the instantaneous centre 0 is the
point' of intersection of 5/3 and cy.
Again, the angle jSOy, being the supplement of the con-
stant angle BAC, is given; and consequently the instanta-
neous centre 0 always lies on a fixed circle.
* Coiirs de g dome trie pour les ecoles des arts et metiers. See also Collignon,
Traite de Mecanique Cinematique, p. 306.
This theorem admits of a simple proof by elementary geometry. The in-
vestigation above has however the advantage of connecting it with the general
theory given in the preceding Articles, as well as of leading to the more general
theorem stated at the end of this Article.
Analytical Demonstration,
369
Fig. 77.
Also if Oa be drawn perpendicular to the third side BCy
a is the point in which the side
touches its envelope (Art. 291).
Produce aO to meet the circle
in a ; and since the angle a 0/3
is equal to the angle ACB, it
is constant ; and consequently
the point a is a fixed point on the
circle. Again, by (4) Art. 292,
the circle jSOy passes through
the centre of curvature of the
envelope of any carried right
line; and accordingly a is the
centre of curvature of the enve-
lope oi JBC; but a has already
been proved to be a fixed point ;
consequently BCin all positions touches a fixed circle whose
centre is a. (Compare Art. 286.)
This result can be readily extended to the case where the
sides^jB and AG slide on any curves ; for we can, for an in-
finitely small motion, substitute for the curves the osculating
circles at the points b and c, and the construction for the point
a will give the centre of curvature of the envelope of the
third side BC.
298 (a). Analytical Demonstration. — The result of the
preceding Article can also be established analytically, as was
shown by Mr. Ferrers, in the following manner : —
Let ay h, c represent the lengths of the sides of the moving
triangle, and pi, p^^ pz the perpendiculars from any point
on the sides a^ h, c,. respectively ; then, by elementary
geometry, we have
api + hjh -^ cpz= 2 (area of triangle) = 2A.
Again, if jOi, /Q2, /03 be the radii of curvature of the enve-
lopes of the three sides, and w the angle through which each
of the perpendiculars has turned, we have by {2d>)y
api + hp2 + cpz = 2A.
(31)
Hence, if two of the radii of curvature be given the third
can be determined.
2 B
370
Roulettes.
We next proceed to consider the conchoid of Nicomedes.
299. Centre of Curvature for a Concboid. — Let A
be the pole, and LM the directrix of a conchoid. Construct
the instantaneous centre 0, as before : and produce AO until
OA, = AO,
It is easily seen that the circle circumscribing AiOEi is
the instantaneous circle of inflexions : for the instantaneous
centre 0 always lies on this circle ; also jRi lies on the circle
by Art. 290, since it moves along a right line : again, A lies
on the lower circle of inflexions of same Article, and conse-
quently A: lies on the circle of inflexions.
Hence, to find the centre of curvature of the conchoid
described by the moving point Pi, produce PiO to meet the
circle of inflexions in Fi, and take
PiPa = -^^ ; then, by
PiPi
22
P. is
the centre of curvature belonging to
the point Pj on the conchoid.
In the same case, the centre of
curvature ot the curve described by
any other point Qi, which is inva-
riably connected with the moving
line, can be found, ll'or, if we
produce QiO to meet the circle of
inflexions in JSr, and take Q1Q2
= Tp^ ; then, by the same theorem,
Q2 is the centre of curvature re-
quired.
A similar construction holds in all other cases.
300. Spherical liouiettes. — The method of reasoning
adopted respecting the motion of a plane figure in its plane
is applicable identically to the motion of a curve on the sur-
face of a sphere, and leads to the following results, amongst
others : —
(i). A spherical curve can be brought from any one
position on a sphere to any other by means of a single
rotation around a diameter of the sphere.
(2). The elementary motion of a moveable figure on a
sphere may be regarded as an infinitely small rotation
Fig. 78.
Motion of a Rigid Body about a Fixed Point. 371
around a certain diameter of the sphere." This diameter is
called the instantaneous axis of rotation, and its points of
intersection with the sphere are called the poles of rotation.
(3). The great circles drawn, for any position, from the
pole to each of the points of the moving curve are normals to
the curves described by these points.
(4) . When the instantaneous paths of any two points are
given, the instantaneous poles are the points of intersection
of the great circles drawn normal to the paths.
(5). The continuous movement of a figure on a sphere
may be reduced to the rolling of a curve fixed relatively to
the moving figure on another curve fixed on the sphere.
By aid of these principles the properties of spherical roulettes*
can be discussed.
301. Motion of a Rigid Body about a Fixed
Point. — We shall next consider the motion of any rigid
body around a fixed point. Suppose a sphere described
having its centre at the fixed point ; its surface will intersect
the rigid body in a spherical curve A, which will be carried
with the body during its motion. The elementary motion of
this curve, by the preceding Article, is an infinitely small
rotation around a diameter of the sphere ; and hence the
motion of the solid consists in a rotation around an instan-
taneous axis passing through the fixed point.
Again, the continuous motion of A on the sphere by (5)
(preceding Article) is reducible to the rolling of a curve
i, connected with the figure ^, on a curve X, traced on the
sphere. J3ut the rolling of i on A is equivalent to the
rolling of the cone with vertex 0 standing on X, on the cone
with the same vertex standing on A. Hence the most general
motion of a rigid body having a fixed point is equivalent to
the rolling of a conical surface, having the fixed point for its-
summit, and appertaining to the solid, on a cone fixed in
space, having the same vertex.
These results are of fundamental importance in the gene-
ral theory of rotation.
* On the Curvature of Spherical Epicycloids, see Eeaal; Journal de lEeoU
JPolytechnique, 1858, pp. 335, &c.
2 B 2
372 Examples,
Examples.
1. K the radius of the generating circle be one-fourth that of the fixed,
prove immediately that the hypocycloid becomes the envelope of a right line of
constant length whose extremities move on two rectangular lines.
2. Prove that the evolute of a cardioid is another cardioid in which the
radius of the generating circle is one-third of that for the original circle.
3. Prove that the entire length of the cardioid is eight times the diameter of
its generating circle.
4. Show that the points of inflexion in the trochoid are given by the
equation cos 0 + - = o ; hence find when they are real and when imaginary.
5. One leg of a right angle passes through a fixed point, whilst its vertex
slides along a given curve ; show that the problem of finding the envelope of
the other leg of the right angle is reducible to the investigation of a locus.
6. Show that the equation of the pedal of an epicycloid with respect to any
origin is of the form
r = (<? + 2h) cos - c cos (6 + o).
a -\- 2b ^ '
7. In figure 57, Art. 281, show that the points C, F and Q are in directum.
8. Prove that the locus of the vertex of an angle of given magnitude, whose
sides touch two given circles, is composed of two lima9ons.
. 9. The legs of a given angle slide on two given circles : show that the
locus of any carried point is a limacjon, and the envelope of any carried right
line is a circle.
10. Find the equation to the tangent to the hypocycloid when the radius of
the fixed circle is three times that of the rolling.
Ans. X cos CO + y rnxoi = b &m 3«.
This is called the three-cusped hypocycloid. See Ex. 5, Art. 286.
11. Apply the method of envelopes to deduce the equation of the three-
cusped hypocycloid.
Substituting for sin ia> its value, and making t = cot w, the equation of the
tangent becomes
xt^ + {y- lh)fi + xt+h^-y = o,
in which t is an arbitrary parameter. If t be eliminated between this and its
derived equation taken with respect to t, we shall get for the equation of thft
hypocycloid,
(«» + 3^2)2 + i8J2 (a;» ^ j/S) ^ 24ia;2y _ Uy^ = 27 J4.
Examples, 373
12. If two tangents to a cycloid intersect at a constant angle, prove that the
length of the portion which they intercept on the tangent at the vertex of the
cycloid is constant.
13. If two tangents to a hypocycloid intersect at a constant angle, prove
that the arc which they intercept on the circle inscribed in the hypocycloid is of
constant length.
14. The vertex of a right angle moves along a right line, and one of its legs
passes through a fixed point : show geometrically that the other leg envelopes a
parabola, having the fixed point for focus.
15. One angle of a given triangle moves along a fixed curve, while the
opposite side passes through a fixed point : find, for any position, the centre of
curvature of the envelope of either of the other sides, and also that of the curve
described by any carried point.
16. If a right line move in any manner in a plane, prove that the locus of
the centres of curvature of the paths of the different points on the line, at any
instant, is a conic. — (Resal, Journal de I' £cole Foly technique, 1858, p. 112).
This, as well as the following, can be proved without difficulty from equa-
tion (22), p. 352.
17. When a conic rolls on any curve, the locus of the centres of curvature
of the elements described simultaneously by all the points on the conic is a new
conic, touching the other at the instantaneous centre of rotation. — (Mannheim,
same Journal, p. 179.)
18. An ellipse rolls on a right line : prove that p, the radius of curvature of
the path described by either focus, is given by the equation - = ; where
par
r is the distance of the focus from the point of contact, and a ia the semi-axis
major. — (Mannheim, Ibid.)
1 9. The extremities of a right line of given length move along two fixed
right lines : give a geometrical construction for the centre of curvature of the
envelope in any position.
20. Prove that the locus of the intersection of tangents to a cycloid which
intersect at a constant angle is a prolate trochoid (La Hire, Mem. de I* Acad, des
Sciences, 1704).
21. More generally, prove that the corresponding locus for an epicycloid is
an epitrochoid, and for a hypocycloid is a hypotrochoid. (Chasles, Sist. de la
Giom., p. 125).
22. If a variable circle touch a given cycloid, and also touch the tangent at
the vertex, the locus of its centre is a cycloid. (Professor Casey, Thil. Trans.,
1877.)
23. Being given three fixed tangents to a variable cycloid, the envelope of
the tangent at its vertex is a parabola. {Ibid.)
24. If two tangents to a cycloid cut at a constant angle, the locus of the
centre of the circle described about the triangle, formed by the two tangents and
the chord of contact, is a right line. {Ibid.)
25. If a curve {A) be such that the radius of curvature at each point is n
times the normal intercepted between tlie point and a fixed straight line (B),
374
Examples,
then when the curve roUs along another straight line, (5) will envelope a curve
in which the radius of curvature is « + i times the normal.
Thus, when « = - 2, {A) is a parabola, and {B) the directrix ; and when
the parabola rolls along a straight Una, its directrix envelopes a catenary (for
which w = - i), to which the straight line is directrix.
When the catenary rolls along a straight Hue, its directrix passes through a
fixed point, for which w = o.
When the point moves along a straight line, the straight line which it car-
ries with it envelopes a circle (w = i), and {B) is a diameter.
When the circle rolls along a straight line, its diameter envelopes a cycloid
(n = 2), to which {B) is the base. When the cycloid rolls along a straight line
its base envelopes a curve which is the involute of the four-cusped hypocycloid,
passing through two cusps, and is in figure like an ellipse whose major axis is
twice the minor. (Professor Wolstenholme.)
The fundamental theorem given above follows immediately from equation
(27), P- 357-
26. Prove the following extension of Bobillier's theorem : — If two sides of a
moving triangle always touch the involutes to two circles, the third side will
always touch the involute to a circle.
27. Investigate the conditions of equilibrium of a heavy body which rests on
a fixed rough surface.
In this case it is plain that, in the position of equilibrium, the centre of
gravity G of the body must be vertically over the point of contact of the body
with the fixed surface.
Again, if we suppose the body to receive a slight displacement by rolling on
the fixed surface, the equilibrium will be stable or unstable, froni elementary
mechanical considerations, according as the new position of G is higher or
lower than its former position, i. e. according as G is situated inside or outside
ths circle of inflexions (Art. 290).
Hence, if pi and f% be the radii of curvature for the corresponding fixed and
rolling curv 3, and h the distance of G from the point of contact of the surfaces,
the equilibrium is stable or unstable according as h is < or > — . See Walton' s
P1 + P2
Problems, p. 190 ; also, for a complete investigation of the case where h =
pi + p2
Minchin's Statics, pp. 320-2, 2nd Edition.
28. Apply the method of Art. 285 to prove the following construction for
the axes of an ellipse, being given a pair of its conjugate semi-diameters OP, OQ,
in magnitude and position. From P draw a perpendicular to OQ, and on it take
PB = PQ ; join P to the centre of the circle described on 01) as diameter by a
right line, and let it cut the circumference in the points H and F ; then the right
lines OE and OF are the axes of the ellipse, in position, and the segments P^
and PF are the lengths of its semi-axes (Mannheim, Nouv. An. de Math. 1857,
p. 188).
29. An involute to a circle rolls on a right line : prove that its centre describes
30. A cycloid rolls on an equal cycloid, corresponding points being in con-
tact : show that the locus of the centre of curvature of the rolling curve at the
point of contact is a trochoid, whose generating circle is equal to that pf either
cycloid,
( 375 )
CHAPTEE XX.
ON THE CARTESIAN OVAL.
302. £quatioii of Cartesian Oval. — In this Chapter*
it is proposed to give a short discussion of the principal pro-
perties of the Cartesian Oval, treated geometrically.
We commence by writing the equation of the curve in its
usual form, viz.,
n ± fivz = a,
where n and r^ represent the distances of any point on the
curve from two fixed points, or foci, F^ and Fz, while fx and
a are constants, of which we may assume that (i is less than
unity. We also assume that a is greater than FiFz, the dis-
tance between the fixed points.
It is easily seen that the curve consists of two ovals, one
lying inside the other ; the former corresponding to the
equation n + fir^ = a, and
the latter to n - nn = a.
Now, with Fi as centre,
and a as radius, describe a
circle. Through F2 draw
any chord DE, join FxB
and FxE; then, if P be
the point in which FxD
meets the inner oval, we
have
PD = a-n = llr^ = liPF^.
From this relation the
point P can be readily Fig» 79.
found.
* This Chapter is taken, with slight modifications, from a Paper published
by me in Mermathena, No. iv., p. 509.
376 On the Cartesian Oval,
Again, let Q be the corresponding point for the outer
oval Ti - jLLVi^ a; and we have, in like manner, DQ = /utFiQ ;
.-. F,Q : F,P =QD:DP;
consequently, F2D bisects the angle PF2Q.
Produce QF2 and PF2 to intersect FiFj and let Pi and Qi
be the points of intersection.
Then, since the triangles PF2D and P1F2E are equiangular,
we have PiF = juPiJPl ; and consequently the point Pi lies on
the inner oval. In like manner it is plain that Qi lies on
the outer.
Again, by an elementary theorem in geometry, we have
F2P . F2Q = P1).DQ + FJ)' ;
/. (i - fx') F2P . F2Q = F,D\
Also, by similar triangles, we get
F2P : F2P1 =^ F2I) : F^E ;
consequently
(i - ^^) FoQ . P2P1 = F2D . F2E = const. (2)
Therefore the rectangle under F2Q and FzPi is constant; a
theorem due to M. Uuetelet.
303. Construction for Third Focus. — Next, draw
QFz, making /.F2QF3 = LF2F1P1 ; then, since the points Pi,
Pi, Q, Fs lie on the circumference of a circle, we get
P1P2 . P2P3 = F2Q . P2P1 = const. (3)
Hence the point P^ is determined.
We proceed to show that P3 possesses the same properties
relative to the curve as Pi and P2 ; in other words, that P3 is
a third focus.*
For this purpose it is convenient to write the equation of
the curve in the form
mri ± Ir^ = ncz, (4)
in which c^ represents PiP'2, and /, m, n are constants.
It may be observed that in this case we have n > m > I.
* Thia fundamental properly of the curve was discovered by Chasles. See
Histoire de la Ge'ome'trie, note xxi., p. 352.
Construction for Third Focus. 377
Now, since L FiF^Q = 1F1P1F2 = Z F.PFi, the triangles
F1PF2 and F1F3Q are equiangular ; but, by (4), we have
mFiP+ lF^P = nFiFi;
accordingly we have
mFiF^ + IFsQ = nF,Q,
or nFxQ -lF,Q = mF,F,;
i. e. denoting the distance from F^ by r^ and F^F^ by c^,
nr^ - Irz = mc%.
This shows that the distances of any point on the outer oval
from F^ and F^ are connected by an equation similar in form
to (4) ; and, consequently, F^ is a third focus of the curve,
304. Klquations of Curve, relative to eacb pair of
Foci. — In like manner, since the triangles FiQF-i and FiF^P
are equiangular, the equation
mF,Q - IF^Q = nF^F^
gives
mF,F^ - IF J* = nF,P.
Hence, for the inner oval, we have
nri + Ir^ = mc^.
This, combined with the preceding result, shows that the con-
jugate ovals of a Cartesian, referred to its two extreme foci,
are represented by the equation
nr^ ± Irz = mcg. (5)
In like manner, it is easily seen that the conjugate ovals re-
ferred to the foci Fi and F^ are comprised under the equation
nri - mrz = ± Ici, (6)
where
c, = F,F,.
305. Relation betiveen the Constants. — The equa-
tion connecting the constants /, m, n va. o, Cartesian, which
has three points F,, F^y F^ for its foci, can be readily found.
378 On the Cartesian Oval,
For, if we substitute in (3), c^ for F^F^, &c., the equation
is easily reduced to the form
Pc^ + n^Cz = m%,
or PF^z + rn'F^, + n'F.F^ = o, (7)
in which the lengths F^Fz, &c., are taken with their proper
signs, viz., FzF^ = - F^Fz, &c.
306. Conjugate Ovals are Inverse Curves. — Next,
since the four points -F2, -P, Q, -^3, lie in a circle, we have
F,F . F,Q = F,F^ .F\Fz = const. (8)
Consequently the two conjugate ovals are inverse to each other
with respect to a circle* whose centre is jPi, and whose radius
is a mean proportional between F^Fi and Fx F^.
It follows immediately from this, siuce F^ lies inside both
ovals, that Fz lies outside both. It hence may be called the
external focus. This is on the supposition that the constants!
are connected by the relations n > m > I,
Also we have
L PF,F, = L PQF, = L F,Q,P, = L F^FzP, ;
hence the lines F^P and F^Pi are equally inclined to the
axis FiFz, Consequently, if Pz be the second point in which
the line FiP meets the inner oval, it follows, from the sym-
metry of the curve, that the points P^ and Pi are the
♦ It is easily seen that when ? = o the Cartesian whose foci are Fi, F2, jPs,
reduces to this circle. Again, ifw = o, the Cartesian becomes another circle,
whose centre is F^y and which, as shall be presently seen, cuts orthogonally the
system of Cartesians which have Fi, F2, Fi for their foci. These circles are
called by Prof. Crofton {Transactions, London Mathematical Society ^ 1866), the
Confocal Circles of the Cartesian system.
t From the above discussion it will appear, that if the general equation of
a Cartesian be written \r + fxr' = vc, where c represents the distance between
the foci; then (i) if, of the constants, A, /*, v, the greatest be v, the curve is
referred to its two internal foci ; (2) if v be intermediate between A and yu, the
curve is referred to the two extreme foci ; (3) if v be the least of the three, the
curve is referred to the external and middle focus ; (4) if \ = jx, the curve is a
conic ; (5) iiv = K, or v - jx, the curve is a lima9on ; (6) if one of the constants
\, IX, V vanish, the curve is a circle.
Construction for Tangent at any Point 379
reflexions of each other with respect to thb axis F1F2, and the
triangles F^PzF^ and F1P1F2 are equal in every respect.
Again, since
z F^PFs = L F^QF, = L F^xPi = z. F^FxP^,
the four points P, P2, F^ and F^ lie on the circumference of a
circle.
From this we have
P3P . F^P^ = F^Fi . FzF^ = constant.
Hence, the rectangle under the segments, made by the inner oval,
on any transversal from the external focus, is constant.
In like manner it can be shown that the same property
holds for the segments made by the outer oval.
If we suppose P and P2 to coincide, the line FzP becomes
a tangent to the oval, and the length of this tangent becomes
constant, being a mean proportional between F^F^ and F^o.
Accordingly, the tangents drawn from the external focus
to a system of triconfocai Cartesians are of equal length.
This result may be otherwise stated, as follows : — A system
of triconfocai Cartesians is cut orthogonally hy the confocal circle
whose centre is the external focus of the system (Prof. Orof ton) .
This theorem is a particular case of another — also due, I
believe, to Prof. Crof ton — which shall be proved subsequently,
viz., that if two triconfocai Cartesians intersect, they cut each
other orthogonally.
307. Construction for Tangent at any Point. —
We next proceed to give a geometrical method of drawiog
the tangent and the normal at any point on a Cartesian.
Eetaining the same notation as before, let R be the point
in which the line F^D meets the circle which passes through
the points P, Fi, F^, Q ; then it can be shown that the lines
PR and RQ are the normals at P and Q to the Cartesian
oval which has F^ and F^ for its internal foci, and F^ for its
external. For, from equation (4), we have for the outer oval
dr\ dr-i
m - - /-;- = o.
i(s ds
38o
On the Cartesian Oval.
Hence, if wi and w% be the angles which the normal at Q
makes with QJFx and QF^ respectively, we have
m sin iM)\ = I sin wz ; or sin w\ : sin cuz = / : m, (9)
Q
Fig. 80.
Again, we have seen at the commencement that
also, by similar triangles,
EQ:BF, = I)Q:F,Q = l:m;
BQ : BF, = sin i^QP : sin i^Qi?; ;
BiaBQF^ : sin BQF^ = l:m.
but
hence
(10)
Consequently, by (9), the line jRQ is the normal at Q to the
outer oval. In like manner it follows immediately that PB
is normal to the inner oval.
This theorem is given by Prof. Crofton in the following
form : — The arc of a Cartesian oval makes equal angles with the
right line drawn from the point to any focus and the circular arc
drawn from it through the two other foci.
This result furnishes an easy method of drawing the
tangent at any point on a Cartesian whose three foci are
given.
Confocal Cartesians intersect Orthogonally. 381
Tlie construction may be exhibited in the following
form : —
Let i^i, F2, Fz be the tbree foci, and P the point in question.
Describe a circle through P and two foci Fz and Fz, and let
Q be the second point in which F\P meets this circle ; then
the line joining P to R, the middle point of the arc cut off
by PQ, is the normal.
308. Confocal Cartesians intersect Orthogonally.
— It is plain, for the same reason, that the line drawn from
P to Riy the middle point of the other segment standing on
PQ, is normal to a second Cartesian passing through P, and
having the same three points as foci — F2 and Fz for its in-
ternal foci, and Fi for its external.
Hence it follows that through any point two Cartesian ovals
can he drawn having three given points — which are in directum —
for foci.
Also the two curves so described cut orthogonally.
Again, if JK(7 be drawn touching the circle PRQ, it is
parallel to PQ, and hence
F^C'.F.C = F,R : RB = F2E' : F,R.RD;
but F^R . RD = RP' ;
.-. F,C : F,C = F,R' : PR' = m' : P, (i i)
Hence the point C is fixed.
Again
CR:F,D==RF,:DF, = m'':nf-P;
.*. OR = — -, (12)
which determines the length of CR.
Next, since RP =RQ, if with R as centre and RP as
radius a circle be described, it will touch each of the ovals,
from what has been shown above.
Also, since (7 is a fixed point by (i i), and CR a constant
length by (12), it follows that the locus of the centre of a circle
which touches both branches of a Cartesian is a circle (Quetelet,
Nouv. M^m. de VAcad. Roy. de Brux. 1827).
382
On the Cartesian Oval,
This construction is shown in the following figure, in
which the form of two conjugate
ovals, having the points i<\, Fz,
Fzf for foci, is exhibited.
Again, since the ratio of
FiR to JRP is constant, we get
the following theorem, which
is also due to M. Quetelet : —
A Cartesian oval is the
envelope of a circle, whose
centre moves on the circum-
ference of a given circle, while
its radius is in a constant ratio
to the distance of its centre
from a given point.
310. Cartesian Oval as an Envelope. — This con-
struction has been given in a different form by Professor
Casey, Transactions Royal Irish Academy, 1869.
If a circle cut a given circle orthogonally, while its centre
moves along another given* circle, its envelope is a Cartesian
oval.
This follows immediately ; for the rectangle under FiP
and FiQ is constant (8), and therefore the length of the tan-
gent from Fi to the circle is constant.
This result is given by Prof. Casey as a particular case of
a general and elegant property of bicircular quartics, viz. : if
in the preceding construction the centre of the moving circle
describe any conic, instead of a circle, its envelope is a bicir-
cular quartic.
* It is easily seen that the three foci of the Cartesian oval are : the centre
of the orthogonal centre, and the limiting points of this and the other fixed
circlo.
Examples, 383
Examples.
1. Find the polar equation of a Cartesian oval referred to a focus as pole.
If the focus F\ be taken as pole, and the line FiFz as prime vector, we easily
obtain, for the polar equation of the curve,
(m* - P)r^ - 2cz {mn - l^ cos e)r + c-^ {n^ - P) = o.
The equations with respect to the other foci, taken as poles, are obtained by
a change of letters.
2. Hence any equation of the form
r^ - 2{a + b cose)r + e^ = o
represents a Cartesian oval.
3. Hence deduce Quetelet's theorem of Art. 302.
4. If any chord meet a Cartesian in four points, the sum of their distances
from any focus is constant ?
For, if we eliminate 6 between the equation of the curve and the equation of
an arbitrary line, we get a biquadratic in r, of which — 4a is the coefficient of
the second term.
5. Show that the equation of a Cartesian may in general be brought to the
form
S^ = PI,
where S represents a circle, and L a right line, and A; is a constant.
6. Hence show that the curve is the envelope of the variable circle
\^kL 4 2\S + F = o.
Compare Art. 309.
7. From this show that the curve has three foci; i.e. three evanescent
circles having double contact with the curve.
8. The base angles of a variable triangle move on two fixed circles, while
the two sides pass through the centres of the circles, and the base passes through
a fixed point on the line joining the centres ; prove that the locus of the vertex
is a Cartesian.
9. Prove that the inverse of a Cartesian with respect to any point is a bi-
circular quartic. {See Salmon, Higher Flane Curves, Arts. 280, 281.)
10. Prove that the Cartesian
r-2 _ 2 (a + i cos 0) r + 0^ = o
has three real foci, or only one according as
a - 4 is > or < «,
( 384 )
CHAPTEE XXI.
ELIMINATION OF CONSTANTS AND FUNCTIONS.
311. Elimination of Constants. — The process of dif-
ferentiation is often applied for the elimination of constants
and functions from an equation, so as to form differential
equations independent of the particular constants and func-
tions employed.
We commence with the simple example y"^ = ax + b. By
dv
differentiation we get 2y-j- = a, 2i. result independent of h,
A second differentiation gives
(S)
<py /
a differential equation containing neither a nor b, and which
accordingly is satisfied by each of the individual equations
which result from giving all possible values to a and b in the
proposed.
, In general, let the proposed equation be of the form
f{xy y, a) = o. By differentiation with respect to a?, we get
dx dydx *
The elimination of a between this and the equation/(ir,y,a) = o
leads to a differential equation involving x, y and -^, which
(XX
holds for all the equations got by varying a in the proposed.
Again, if the given equation in x and y contain two
constants, a and b ; by two differentiations with respect to x,
we obtain two differential equations; between which and the*
Examples. 385
original, when tlie constants a and h are eliminated, we get a
differential equation containing x, y, — and -7^.
In general, for an equation containing n constants, the
resulting differential equation contains a?, y, -7-, -rr . . . — ;
^ ^ ^ dx dx^ dx""
arising from the elimination of the n constants between the
given equation and the n equations derived from it by suc-
cessive differentiation.
Examples,
^ I. Eliminate a from the equation
y'^-iay + x'^a'. Am. (a;2-2y2) (^V- 4a;y^-a;«=0.
ya. Eliminate a and ^ from the equation
<.-«)»=p(.-.). ^»...(|)V/J = o.
s/3. Eliminate the constants a and )8 from the equation
y = a cos nx + )8 sin nx, Ans. — - + tv^y = o
dx^
J 4. Eliminate a and b from the equation
{x-af+{y-bf = c\ Ans. c'^l -A^^f — .
(d-yy
\dxy
This agrees with the formula for the radius of curvature in Art. 226.
5. Eliminate a and /8 from the equation
/n ^\ ^ d^y n^y
y=axcoa - + )8 . Ans. —^ + -f- = o.
\a / dx^ x^
6. Eliminate the constants uq, ai, , , . «« from the equation
d^+^ij
y = f {x) + aox» + aix"-^ +...««. ^ns» --^^ = 4>('»+i) {x).
y/ 7. Eliminate o and fi from the equation
JS. Eliminate a and J from the equation
d-y dy
xy = ae'' + be-». Ans. x--^ + 2-^-xy=o.
dx^ dx
9. Eliminate e and e' from the equation
,^y
y = o'ice + c'are * • Ans. a;* 3-5 = y.
2G
386 Elimination of Constants and Functions,
312. Elimination of Transcendental Functions. —
The process of differentiation can also be employed for the
elimination of transcendental functions from equations
of given form ; for example, the logarithmio__f unction
can be eliminated by differentiation from the equation
fj/7/ (h ( Jy)
y = log(l>{x)y which gives -j- = . We have met several
instances of this process of elimination already ; thus, in
Art. 86, we found that the elimination of the symbolic
functions, sin and sm~\ from the equation y = sin {m sin'^a?)
leads to the differential equation
. ^.d^'y dp „
The principles involved in this process of elimination are
of great importance in connexion with the converse problem —
viz., the procedure from the differential equation to the
primitive from which it is derived. This part of the subject
belongs to the Integral Calculus in connexion with the solu-
tion of differential equations.
Examples,
I. </ = tan-'a;. Ans. -f = 5.
dx 1 + x^
3. Eliminate the exponential and logarithmic functions from the equation
4. Eliminate the circular and exponential functions from 3/ = ^ siu x.
Here
"'if
— = «« Sin a; + g* COS a; = y + ^
cos a; ;
therefore
d^i/ dy
-^ = -^ -1- ^ COS a; -
dx^ dx
-^*6ina; =
4-
5.
-•S='-^-
6.
y = sin (log a;).
, rf2y du
Ans.x^-+x- + t/ = 0.
Elimination of ArUtrary Functions, 387
In the preceding examples we have only considered the
case of a siiyle independent variable : the differential equa-
tions arrived at in such cases are called ordinary differential
equations.
When our equations are of such a nature as to admit of
two or more independent variables the equations derived from
them by differentiation are called partial differential equa-
tions. We proceed to consider some cases of elimination
which introduce differential equations of this class.
y3i3. £liiiiinatioii of Arbitrary Functions. — The
equations hitherto considered contained only two variables ;
we now proceed to the more general case of an equation in-
volving three variables, two of which accordingly can be
regarded as independent. We shall denote the independent
variables by the letters x and .y, and the dependent variable
by 2. It will also be found convenient to adopt the usual
notation, and to represent the partial differential coefficients
dz dz d^z dh - dh
dx* dy^ dx^^ dxdy dy"^
by the letters p, q^ r, s and t, respectively.
We proceed to show that in this case we are enabled by
differentiation to eliminate functions whose forms are alto-
gether arbitrary. In fact we have already met with examples
of this process : for instance, if s = x^ij) f - ], we have seen, in
Art. 102, that in all cases we have
dz dz , .
*&"^«, = "^' (•)
whatever be the form of the function ^ : this function accord-
ingly may be regarded as completely arbitrary in its form,
and the preceding differential equation holds whatever form
is assigned to it. This can also be shown immediately by
differentiation. Conversely, it can be established without
difficulty that ic'^^f-l is the most general form of z which
satisfies the preceding partial differential equation, and con-
sequently z = x^(pl-^ J is said to be the solution of equation (i),
2 G2
388
Elimination of Constants and Fundions,
where the function 0 is perfectly arbitrary. This latter
process, as in the case of ordinary differential equations,
comes under the province of the Integral Calculus, and is
mentioned here for the purpose of showing the connexion
between the integration of differential equations and the
formation of such equations by the method of elimination.
As another simple example, let it be proposed to eliminate
the arbitrary function from the equation z =f{x^ + y"^).
Here p = j^^ixfix^ -^f), q = -^ = 2yf(x'-vy'')\
hence we get yp- xq = o\
an equation which holds for all values of 2 whatever the form
of the function (/) may be.
1. « = <p{ax + hy).
2. y — bz = <p{x — az).
(y — B\
4. (pix** + y»') = z^.
5. z'^ = xy+ <p (h.
Examples.
Ans. aq — hp.
, ap-\-hq=\, J
, (a;-o)i?+(y-^)? = a-7,
, wa;"-^ q = my*'*^^p.
, xzp + yzq = xy. J
z =px + qy + nvx^ + y^ + 2*.
n//3I4. Condition that one Expression should be a
Function of another. — Let z = ^(t^), where v ia o, known
function of x and y. y '^
Here
therefore
dx
dz dv
dx dy
, . V .vc dz ,r .dv
dx'
dzdh
dv
dy dx ^ dy ^ dx
This furnishes the condition that z should be a function of
the quantity represented by v. Also, denoting z by F, and
supposing V and v to be tw^i^iven explicit functions of x and
y, the condition that Vis a function of v is that the equation
^(^_dVd^_ |\ 2
dx dy dy dx \\ ' ^ ^
Condition that one JExpression is a Function of another. 389
shall hold for all values of x and y, i. e. shall be identically
satisfied. For instance, if
F= a/i-^'-Zi-^'^ and V = V^^' + y/^^,
cc + y
dVdv dVdv ., .. n
we get —r-, ] — r = o> identically ;
° dx dy dy dx
hence F is a function of v in this case.
This can also be independently verified : for, \ix=- sin 0,
and y = sin 0, we get
^ cos (^ - cos 0 , 0+0
F= -^—R ^ = - tan ;
sm t/ - sm 0 2
«? = sin 0 cos 0 + cos 0 sin 0 = sin (0 + 0) :
this establishes the result required.
We have here assumed that whenever equation (2) is satis-
fied identically, V is expressible as a function of v : this can
be easily shown as follows : —
Since V and v are supposed to be given functions of x and
y^ if one of these variables, ^, be eliminated between them we
can represent F as a function of v and x.
Accordingly, let
y-Ax, v) ;
dV _df_ d£dv_ d_V_4fdi^
dx dx dv dx^ dy dv dy '
therefore
dVdv__dVdv__^dv^ I
dx dy dy dx dxdy
Hence, since the left-hand side is zero by hypothesis, we must
have 3^ = o ; i.e, the function /(a;, v) or F reduces to a funo-
dx
tion of e? solely ; which establishes the proposition.
390 Elimination of Constants and Functions,
315. More generally, let it be proposed to eliminate the
arbitrary function (p from the equation
where V and v are given functions of three variables, x, p,
and z.
Regarding x and y as independent variables, we get by
differentiation
dV
dtj
eliminating ^\v) we obtain
dVdv^ _dVdv
dx dy dy dx
dV ,, Jdv dv\
fdV(h^ dvdV\
\dz dy dz dy J
fdVdv dvdV\
^^[d^Iz-Txl^r'''^ (3)
a result independent of the arbitrary function 0.
This equation can also be established as follows : —
Differentiating the equation V = 0(«?), considering x, y, z
as all variables, we get
Then, since the form of ^(t?) is perfectly arbitrary, this equa-
tion must hold whatever be the form of the function <p\v)y
and hence we must have
-7- dx + -—dy + ~— r/2 = o,
dx dy dz
\ (a)
dv ^ dv , dv ^ ' ^ '
■r- dx + y- dy + — dz^'O
dx dy dz
Condition that one Expression is a Function of another, 391
Moreover, introducing the oondition^that z depends on x
and y, we have
dz = pdx + qdy ;
consequently, eliminating dxy dy, dz between this and the
equations in (4), we get
dV dV^ dV^
dx* dp ' dz
dv dv dv a o. (5)
dx* dy* dz
This agrees with the result in (3).
Examples.
Eliminate the arbitrary functions in the following cases : —
I. x = <p[amix-\-b sin y).
2. J5 = ««^(a?— y).
A' --- = (}> I---]
^ z X \y XI
S. ^^■
y^<p [y) + X
■X(p{y)
6. 2 = aV^2+y2 + <^fn.
7. z={x + yy*<l>{x'i-y^).
J 8. «2 + y2 + i52 = 0(a;p+ hy-\- cz).
. . dz dz
Ans. bcosy- — « cos a: — = o.
dx dy
dz dz z
" dx dy a
dz dz xy
dx dy
dz dz /—
dz dz
*» y— + x — = nz.
" dx dy
dz dz
Ans. {bz— ey) -- + (<?a: — az) — = ay- bx.
392 Elimination of Constants and Functions*
316. Next, let it be required to eliminate the arbitrary
function 0 from the equation
where w is a given explicit function of x, y and z.
Eegarding x and y as the independent variables, we may
differentiate the equation with respect to x, and also with
respect to y\ then, since s is a function of x and y, we have
d , (hiu) ,, Jdu du \
_ d,(b{u) ., Jdu du \
hence we obtain two partial differential equations involving
^» Vi 2, p, q, (j){u) and ^'(«<). Accordingly, if ^(w) and ^'(w) be
eliminated between these and the original equation, we shall
have a resulting equation containing only x, y, z, p and q.
317. Case of Two or more Arbitrary Fauctions. —
If the given equation contain more than one arbitrary func-
tion, we must proceed to partial differentiations of a higher
degree in order to eliminate the functions : thus, in the case
of two arbitrary functions, ^(w), and \p{v), the first differen-
tiations with respect to x and y introduce the functions (p'(u)
and \p\v). It is plainly impossible, in general, to eliminate
the four arbitrary functions between three equations; we
accordingly must proceed to form the three partial differen-
tials of the second order, introducing two new arbitrary
functions <p''{u) and \fj''(v). Here, again, it is in general
impossible to eliminate the six functions between six equa-
tions, so that it is necessary to proceed to differentials of the
third order : in doing so we obtain four new equations, con-
taining two additional functions, 0'''(^«) and \p''\v). After
the elimination of the eight arbitrary functions there would
remain, in general, two resulting partial differential equa-
tions of the third order.
318. There is one case, however, in which we can always
obtain a resulting partial differential equation of the second
order — viz., where the arbitrary functions are functions of
the same quantity, u.
Case of Two or more Arbitrary Functions. 393
Thus, suppose the given equation of the form
F[x,y,%,^[u),-^[u))=Oy (6)
where t« is a known function of a?, y and s.
By differentiation we get
dF dF dFfdu du\_
dx dz du\dx dz) '
dF dF dFfdu du
dy dz du \dy t '
. . . dF
Eliminating — between these equations, we obtain
dFdu _ dFdu fdFdu _ dFdu\
dx dy dy dx \dz dy dy dz)
(dFdu dFdu\
^^[d^dz'd^Txr''' (7)
This equation contains only the original functions ^{u),
4i{u), along with ^, y, z, p and q. Again, if we apply the
same method to it, we can form a new partial differential
equation, involving the same functions (p{u) and ^(w), along
Vfithx.y, z,p, q,r, s, t.
The elimination of the unknown functions, (p(u) and {(/{u),
between this last equation and equations (6) and (7), leads to
the required partial differential equation of the second order.
The result in (7) admits also of being arrived at by the
method adopted in the second proof of Art. 315. For, re-
garding Xj y, z, as aU variables, we get from (6), on differen-
tiation,
wj? , dF , dFfdu , du ^ du ^ \ ,^,
dx + -^ ay + -— dz + -ri -r dx -h -r dy + — dz ]= o. (8)
dz du\dx dy dz J ^ '
dF ^ dF
-j-dx-\--—
dx dy
T> i. dF dF „ . dF „, ,
du d(p{u) ^^ ' d\p{u) ^ ^ '
394 Elimination of Constants and Functions.
and accordingly, since (8) must hold for all values of (^\u)
and -^'{u), we have
and
--dx^-j'dy + -T'dz = o,
dx dy ^ dz \
du , du , du \
-r dx + —dy + -ydz^ o J
(9)
Eliminating between these equations and
dz = pdx + qdy,
we get the following determinant :
dF dF dF
dx^ dy* dz
du du du = o
dx* dy^ dz
(10)
which, plainly, is identical with (7).
This admits also of the following statement: substitute c
instead of u in the proposed equation ; then regarding c as con-
stant, diiferentiate the resulting equation, as also the equation
u = G {on the same hypothesis) : on combining the resulting
equations with
dz = pdx + qdy,
we get another equation connecting ^ (c) and \p (c) ; and
applying the same method to it, we obtain the result, on
eliminating the arbitrary functions <j)(c) and \p{c) between
the original equation and the two others thus arrived at.
These methods will be illustrated in the following ex-
amples : —
Examples, 395
Examples.
1. 9^xip{z) + y^{z).
Here p = </)(z) + {x(t>'{z) + y^l>'{z)}py
q = rp{z)+{x<t>{z) + y^l,'{z)]q.
Hence f " ^ "•^'^' ^^PP^'®'
Applying the principle of Art. 314, we have
d (p\ d (p\
or q^r - 2pqs + pH = o.
Otherwise thus : let 2 = e, and we get dz = o, and <}> {c) dx + \l/{c) dy = o,
also pdx + qdi/ = o ;
therefore ^ = ^.
Differentiating again, we have
qdp - pdq = O,
or q\rdx + sdy^ — p{sdx + tdy) = O,
which, combined with pdx -f qdy = o,
leads to the same result as above.
2. z = x<p{ax + Jy) + y^{ax + by).
Here p = <l>(«!a; + iy) + a{x(p {ax + by) + yyp {ax + by)},
q = ip{ax + by) + b{x<p'{ax + by) + y\l/'{ax + by)];
therefore bp — aq— b(p{ax + by) — a\p{ax + by) ;
hence br- as = a{b({>'{ax + by) - a\l/'{ax + by)];
bs-at= b{b<i>'{ax + by) - a\p'{az + by)];
therefore b^r - 2abs + aH = o.
396 Elimination of Constants and Functions.
Otherwise thus : let ax+ bj/ = c, then adx + bd^ = o; also,
dz = <j>{c) dx + yp{c) dy^ and dz = pdx + qdy ;
hence
hp-aq= b<p{c) - a\l>{e).
Differentiating again, we get
bdp - adq = o, or b{rdx + sdy) - a{sdx + tdy) = Ow
Combining this with the equation adx + bdy = o, we get
b'^r - 2abs + aH = o,
as before.
319. Case of n Arbitrary Functions of same
Function. — It can be readily seen that the preceding
method is capable of extension to the elimination of any
number n of arbitrary functions from an equation, provided
that they are all functions of the same quantity u.
For the equation (7) plainly holds in this case, and, pro-
ceeding as in the last Article, we obtain a series of equations
(the last being of the n*^ order of differentiation), each con-
taining the n arbitrary functions along with the variables and
their derived functions. If the n functions be eliminated
between the n differential equations and the original equation,
we obtain a differential equation of the n*^ order which is
independent of the arbitrary functions in question.
Examples. 397
Examples,
1. Given y = ^*(C + C'x), prove that
-— - 2a — + «V = O-
dx^ dx
2. Eliminate the constants from the equation
d^y dy
y = (71^2* cos 3a; + (72«2ar g^ 3^. ^WS. ^ " 4 j" + 13^ = 0-
3. Eliminate (7 and C from the equations —
. . cosmic ^ ^, .
(a) V = -i. :: + C'cos nx + G sin«a;,
^ ' ^ n^ - m^
(J) y = a;sinwa; + Ccos wa; + C'sinwa:.
Ans. {a) -^^ + n'^y = eosmx. (b) -— y+n^y = 2n cob nx.
4. Eliminate the arbitrary functions from the equation
z = -/- -\- <b{y ->!■ ax) + \l(y - ax), Ans. r-a^t- xy.
0
,y^. Eliminate the functions from the equation
y = Aco8 {a sin-^- + b)» Ans. (c^-x-) — j - a; — + «V = o»
6. Eliminate A and a from
y = ^cos(wcosa; + o). Ans. ^ — cot a: -r- + w V sin^a; = o.
7. li z = COB ax <p I -] + Bin ax\^ r-] f prove that
rx^ + 2sxy + ty^ + a^x'^z = o.
8. If «!, a2, fl!3 be the roots of the equation
^gS Examples.
prove that the result of eliminating the exponentials from the equation
d^y dP-y dy
9. Find the result of the elimination of the arbitrary functions from
z = ^{x + ay) + ^{x — ay). Ans. d^r — t = O.
/ 10. If z =/ ( - J + <^{xy\ prove that
x^r — yH ■\^ xp - yq = o.
11. If ae-y + he-v = ee* + de-'', prove that
\d^ idyy ^ir/^V-i1=^^ /— ^V
Idx^ "*" \dx) " ^^J L \dx) J "^ rfa; \rfa;3 j •
12. z=a;«^(-J +a;»«»|/(-j.
Ana. x^r + 2a;ys + j/'^^ - (w + w — i)(j3a; + jy) + mnz = o.
/ 13. Eliminate the arbitrary functions from the equation
z = <p{x +f{y)}. Ans. ps - qr = O.
14. Prove that y = Ae"' satisfies the differential equation with constant
coefficients
d»y d**-^y dy
provided a is a root of the equation
2" + piz»-i + . . . + jon-ia + ;?n = o.
15. Show that
y = Aie^i" + A2e'*2'> + . . . + AnS^*'^
is the general solution of the equation in Ex. 14, M'here «i, ^2 . . • «n are the
w roots of the equation in z, and Ai, A2, . . . An are arbitrary constants.
16. Eliminate the constants from the equation
ax^ + 2bxy + cy2 + zdx + 2ey +/= o.
Ana. 40r3 _ 45yr2« + ^qH = o, where i? = ^, q = ^. r = ^, &c.
dx dx^ da^
( 399 )
CHAPTEE XXII.
CHANGE OF THE INDEPENDENT VARIABLE.
320. Case of a Single Independent Variable. — We
have already pointed out the distinction between indepen-
dent and dependent variables in the formation of differen-
tial coefficients.
In applications of the Differential Calculus it is sometimes
necessary to make our differential equations depend on new
independent variables instead of those which had been origi-
nally selected.
To show how this transformation is effected we commence
with the case of one independent variable, and suppose V to
7 72
represent any function of x, y, — , — , &c. "We proceed to
show how the expressions for —, — ^, &o., are transformed,
(tX ClOG
when, instead of x, any function of x is taken as the indepen-
dent variable.
Let this new function be denoted by t, and suppose that
dx d X
— , — , &c., are represented by x, id, then, in all cases
dt (to''
we have
du du dx du
dt dxdt dx''
where u is any function of a? ;
Hence j! = 4|; (.)
dx xdt ^ '
400 Change of the Independent Variable,
d'y_d_fdy\_ d_(ldy\^i_d_(i_dy\^
^^^ 1^'' dx\dx)~ dx\xdtj xdt\xdt)' ^^^
hence
. d'^y .. dy
d'y ^ ""df^'^di (3)
. gy d{ df dt\ I dl dt' dt
^^^^^ d^'~ dx\ [xf J~xdt\ x'
dp ''^''dt' ^ dt^^^''^ ^^^
\x)
(4)
and so on for differentiations of higher degrees.
If y be taken as the independent variable, we obtain the
corresponding values by makiog
dy
Tt"
' '. \l = 0. &0-
^x
dy
dx'
I tfy <Py
dy \dy)
\df) dydf
d?y
(&»
/dx\' »
\dy)
Hence //^ ./^» ^^2 /^^\3» v5;
(6)
and so on.
The preceding results can also be arrived at otherwise,
as follows: — The essential distinction of an independent
variable is, that its differential is regarded as constant ; ac-
cordingly, in differentiating -f, when x is the independent
dx
Case of a Single Independent Variable. 401
variable we have ^( t- ) = -f-' However,-'w]ien x is no longer
regarded as the independent variable we must consider the
dtj
numerator and the denominator of the fraction -f- as both
ax
variables, and, by Art. 15, we get
ri(^y\ _ ^*^ ^^^ ~ ^y ^"^^ ^ l^y\ dxd'^y - dy d^x
\dxj dx^ ' dxKdx) dx^
Differentiating again on the same hypothesis, we get
d (d^y\ dx^ d^y - dx dy d^y - zdx d^x d^y + 3 (d^x) ^ dy
dx\dx^J dx^ *
These results are perfectly general whatever function of x
be taken as the independent variable. Their identity with
the equations previously arrived at is manifest.
Examples.
1. Being given that x = a{9 — sin d) and y = a{i - cos 0), find the value of
dx" « ( I - cos oy
2. Hence deduce the expression for the radius of curvature in a cycloid.
3. li X = {a + b) cos 0 - i cos — 7— Q and y = {a-\-b) sin d—b sin —7— 0, find
dry
the value of — -.
dx^
a + b
cos 0 - cos —r— 6
Hero /= y =iml—^+ i]d,
dx .a + b . ^ \2b J
sm -— — 0 - sm 0
b
d^y a-V2b
4i(a + J)sin-cos3 (^^+1 j^
d^t/
4. Change the independent variable from a; to 0 in the expression -7-^, sup*
UX"
posing :c = sin 0.
dy
^^® dx cos Odd* " dx^ cos d dd V cos 6 dd) cos^ d dd"^ r-*
^ cos30
2 D
402 Change of the Independent Variable,
5. Transform the equation
»^^Aax-f^hy = o
dx^ dx €
7i =.- e
into another in which d is the independent variable, being given a; = ^5,
Here dj_^dyd_x^^dj_^
dd dx dd dx '
hence U-) = ^~i--), ov^^ = x^^^ ^ x%
dd\ddl dx\ dxj' d&^ dx^^ dx'
therefore ^^^^^^^J.
dx-' de^ dd*
and the transformed equation is
6. Transform the equation
into another where z is the independent variable, being given x=-.
z
It is evident that in this case x^^-z"^- hence
dx dz '
d
as
d ldy\ d I dy\
dx\^Tx)-'dzVliy
dx^^'^dx-'' dz' +'^J
therefore x'^^A-zx^^- .» ^'^
"" dx^^^'^Tx-'d^^*
and the transformed equation is
_+«2y^0.
7. Change the independent variable from ^ to . in the equation
d'-y
dx'
^* drr-i + ^^y = O' where x^-.
^■Z--/i^'^y-^.
Ttco Independent Variables, 403
321. Two Independent Variables. — We will next
consider the process of transformation for two independent
variables, and commence with the transformations intro-
duced by changing from rectangular to polar coordinates
in analytic geometry. In this case we have
sv = r Gosd, ^ = r sinO; (7)
and therefore r^ = x^ -h y^, tan 0 = -. (8)
Accordingly, any function V oi x and y may be regarded
as a function of r and d, and by Art. 98 we have
dV_dVdx dVdjT)
dQ ~~dxJQ^'dy di) I
> • (9)
dV _dVdx d_Vdy | ^^^
dr dx dr dy dr }
But, from (7),
dx ^ dx . . dy . ^ dy , ,
hence we obtain
dV dV dV , ,
o?F <^F c?r , ,
^^=^^^^^- ^''^
These transformations are useful in the Planetary Theory
Again, we have
dV^dVdr d_Vde-)
dx dr dx dO dx I
>• (13)
dV_dJFdr dVdO |
dy dr dy dO dy J
2D2
404 Change of the Independent Variable.
But from (8) we have
therefore
dr X n ^^ ' Q
_- = _ = cos 0, -- = sm d,
dx r ' dy
d9 ,^v sm0 dd
GOSO
r
dV ^dV sinOdV
dx dr r dd'
dV . ^dV cos OdV
dy dr r dd
(H)
(15)
(i6)
(17)
The two latter equations can also be derived from equa-
tions (i i) and (i 2) by solving for — and — .
d^V d^V
322. Transformation of — -r- and -— -. — Since for-
dx^ dy^
mula (i6) holds, whatever be the form of the function F,
we have
d , . n ^ r \ sin 0 c? , .
dx
dr
dQ
where (p stands for any function of x and y. On substituting
dV
— instead of (p, this equation becomes
d^rdV
dx\dx
cos0^-
dr
^dV sin BdV
dr r dO
sinO d
~V7e
^dV sin OdV
dr r dU
^^d^V cos Q sin Q d'V cos B sin BdV
cos a -7^; — ^ — — +
dr"
drdB
dB
sinOr ^dW . .
cos B-r-n\ - sm B
r [_ drdB
dVl
dr }
sinJrcos^^ smBd'jn
"^ r I r dB '^ r dB" f
Transformation of j-^ ^^^^;tt • 4^5
or -z-z- = QOB^B -rT- +
dx^ dr^ r
idV d'Vl
_rde drdOJ
singed V sin'Od^V
r dr r^ db^ '
In like manner we get
d^ _ . d'^V 2sm0cos0ri^r 6fF1
dy'^ dr^ r \_r dd drdB]'
QOs'^OdV GOB^Od^V
"^ ~V dr "^ r' dO' '
The latter result can also be readily deduced from the
preceding by substituting in it — 0 for 0.
If these equations be added we have
dx" "^ dy^ ~ dr"" ^ r dr '^ r' d6' ' ^^ ^
^23. Transformation of -— - + ~-^ + --^ to Polar
^ dx '^'" '^"
Coordinates.
dW d'V d^
dx^ dy^ dz^
Let the polar transformation be represented by the equa-
tions
a; = r sin 0 cos 0, y-r sin Q sin (p, z = r cos 6 ;
also, assume p = r sin 0, and we have
x = p cosfji, y = pe,m<}>;
T. / n^ d'y dW d'^V idV I d'V
hence, by (18), -r-r- + -r^ = -7-^- + - -r~ + -^ ttT'
' *^ ^ ^' dx^ dy^ dp^ pdp p^ d<l>'
4o6 Change of the Independent Variable.
Again, from the equations
p = r %mO, % = r cos d,
we have in like manner
(TV ^_(PV idV I d'V
d^"^ dz' ~ dr" "" rdr '^ p' dO^ '
Accordingly,
dT d'V d^_(PV i^dV i_d2V idV j_(PV
'd^'^df^d^'d^'^pdp '^ p'dcfi' '^ rdr ^ r' dd' '
But by (17) we have
dV . ^dV cos OdV
dp dr r du
^^ ^ idV idV GotddV
therefore - 3- = - 3— + —tt 3zr«
pdp r dr ir du
Hence we get finally
d'V d'V d^_d^ ■ I <fF
dx'' ^df"^l^~ dr" "^ r'sin^e di^^^
I d'V 2dV Got OdV
■*" r' dS^ '^ r dr '^ r' dd
lid f JV\ I d ( . ^dV\ I ^F) , .
=^|7A'Vj^-i^^dr^^^j^s^0^!- ('9)
324. Remarks on Partial Differentials. — As already
stated in Art. 113, the student must be careful to attach the
correct meaning to the partial differential coefficients in each
case.
dec
Thus in finding — in (10) we regard a? as a function of r
(xr
and 0, and differentiate on the supposition that d is constant;
dr
in like manner the value of — in (14) is found on the suppo-
sition that y is constant.
Geometrical Illustration.
407
The beginner, accordingly, must not fall into tlie con-
ur uos
fusion of supposing that in this case we have — x -r= i*
OiX ar
This caution is necessary, as some mathematical writers,
from not paying proper attention to the meaning of partial
derived functions, have fallen into a similar error.
325. Crcometrical Illustration. — The following geo-
dr dx
metrical method of determining the proper values of — and —
O/JO 0/1
under the preceding hypotheses may assist the beginner
towards forming correct ideas on this important subject.
Let P be the point whose coordinates are x and y ; then
OM = Xy FM = y, OP = r,
POX = Q. Now, in finding
dx
— regarding B as constant,
we take on the radius vector
OP produced a portion PQ
= Ar, and draw QN perpen-
dicular to OX; then Aa?, the
corresponding increment in x,
is represented by MN or PL ;
Fig. 82.
therefore
^. PL ^ dx
T- = ^7; = cos C^, or —
Ar PQ dr
dr
A«
Ar
COS0.
Again, to find ^ on the supposition that y is constant :
ax
let MN be i^x, the increment in x, and draw the parallelo-
gram PLMJY, and join OZ, meeting in 7 a circle described
with radius r and centre 0 ; then LI represents the corre-
sponding increment in r, and we have
— = limit of -- = limit of =ry: = cosO;
dx Ax PL
dr dx
so that in this case the values of -r and — are each equal to
dx dr
cos 0 or -, as before.
4o8 Change of the Tndependent Variable.
d/T
The values of 3^, &e. can also be readily represented
geometrically in a similar manner.
326. lilaear Transformations. — If we are given
x = aX^-hT^cZ, 1/ = a'X + b'Y+c'Z, z =a'X+b''Y+ c''Z, (20)
then any function F, of x, y and s, is transformed into a
function X, Y^ Z\ and, as in Ex. 2, Art. 98, we have
dV ^ d_r ,dV ,,d_V
dX dx dy dz '
dY dx dy dz '
dV_ dV ,dV „dV
dZ dx dy dz '
Again, proceeding to second differentiation, we get
d'V d( dV ,dV „dV\ .dfdV ,dV „dV\
-7^ = «— U— - +a —-+a—- ] + a —[a—- +a — - + a -7-
dX^ dx\ dx dy dz J dy\ dx dy dz J
„d_f dV ,dV ,,dV\
dz\ dx dy dz J
= r/.2 .
d'V , d'V „ d^V „ , d^Y
= «'— ^ + zaa -——^ laa -r—r + 2a a -r— 7-
dx^ dxdy dxdz dzdy
(TF „,d'}
dy'' "''' dz'
Similarly we have
dY* dx' dy' dz' dxdy
+ 2bb''-—-+ 2V'b'——\
dxdz dzdy
Orthogonal Transformation, 409
(PV_ ,d'V ,,(PV „J'V ,_d'V
dZ''' d^'^'' df ^' d^^^^'dxdy
„d'V „, d'V
+ 2CC -—- + 2C C -T—r-
dxdz dzdy
327. Orthogonal Transformations. — If the transfor-
mation be such that
we have
fl^ + a2 + «"2=i, h'+h"'-^h''^=i, o^ + c'2 + c"2=i. (21)
ah^db' + a"b''=o, ac-¥a'G + a"c'=o, hc + h'c' + h"c'=o. {22)
Again, multiplying the first of equations (20) by a, the
second by a\ and the third by a'\ we get on addition, by aid
of (21) and (22),
X = ax + a'y-\- a"z,
In like manner, if the equations (20) be respectively
multiplied by ^, h\ V\ we get
Y=hx^Vy^h"z\
similarly,
Z= cx^ c'y + c'z.
If these equations be squared and added, we obtain
a" -vh^ + & = I, a"" + V + c'2 = I, d"' + h"^ + c"" = i. (23)
ad-\- h})-\- cc = o, ad'+ hh"-\- cc"= o, dd'^- Vh"^- c'c'= o. (24)
Hence in this case, if the equations of the last Article be
added, we shall have
(PV_ d'V d'V d'V d^V d'V
dx" ■*" df ^ dz' ' dX' "^ dY' ^ dZ'' ^^^^
410 Change of the Independent Variable,
The transformations in this and the preceding Article
are necessary when the axes of co-ordinates are changed in
Analytic Geometry of three dimensions ; and equation (25)
shows that, in transforming from one rectangular system to
d^V d^V d^ V
another, the value of the function ~-r + tt + -rr is un-
dur dy^ dz^
altered.
328. Creneral Case of Transformation for Two
Independent Variables. — Suppose that we are given the
equations
ic = <l>{r,e), p = xp{r,e), (26)
then any function of x and y may be regarded as a function
of r and 6, and we have, from (9),
dV_dVd^ dVdy
dd ~ dx dd'^ dy W
dV dVdx dVdy
dr dx dr dy dr
where the values of -^, ~, -^, -/ can be determined from
dO dO dr dr
equations (26).
Whenever these equations can be solved for r and 0,
separately, we can determine, by direct differentiation, the
values of — , — , -, — , and hence, by substituting in (13),
we can obtain the values of — and — •
dx dy
When, however, this process is impracticable, we can ob-
i. • XI. 1 . dr dr „ , , . „ dV ^ dV
tain the values of — , -, &c., by solving for -7- and -7-
dx d/ ' '^ ° dx dy
in the preceding equations.
Thus we obtain
dVdy dVdy
3— dQ dr dr dS, /,-\
dx =-7—; T-r5 ^^^
ax ay dx dy
Tddi^'drle
Transformation for Two Independent Variables* 411
(28)
dVdx _dVdx
dV_Wdr~l^de
dy dx dy dx dy '
drde'dOdr
The values of -t-t , tt"? &c., can be deduced from these :
dx" dy^
but the general formulse are too complicated to be of much
interest or utility.
329. Concomitant Functions. — We add one or two
results in connexion with linear transformations, commencing
with the case of two variables. We suppose x and y changed
into a'X. ■\- hY and a''K + h' Y, respectively, so that any func-
tion 0 {x, y) is transformed into a function of X and Y : let
the latter be denoted by «^i(X, F), and we have
<p{x,y) = <p,[X, Y),
Again, let x' and y' be transformed by the same substitu-
tions, i. e.f
x'^aT + bY\ t/ = a'X'+bT;
then since x + kx = a {X + kX) +b{Y+kY),
and y+ky' = d(X + kX) + b\ Y+kY),
it is evident that
ip[x + kx\ y-vkij) = (l^,[X+kX\ Y^-kY").
Hence, expanding by the theorem of Art. 127, and
equating like powers of k, we get
,d(li ,d(li ^,d<pi ^,dcl>i
" d^ ^^''y dxdy^y df ^ dX'^^^ ^ dXdY^^ dY"
&e, &o. (30)
412 Change of the Independent Variable.
Accordingly, if u represent any function of a? and y, the
expressions denoted by
f ,d ,d\ ,d ,dV
\ dx ^ dy ' \ dx ^ dyj
are unaltered by linear transformation.
Similar results obviously hold for linear transformations
whatever be the number of variables (Salmon's Higher Algebra^
Art. 125).
Functions, such as the above, whose relations to a quantio
are unaltered by linear transformation, have been called con-
comitants by Professor Sylvester.
330. Transformation of Coordinate Axes. — When
applied to transformation from one system of coordinate
axes to another, the preceding leads to some important
results, by applying Boole's method* (Salmon's Conies,
Art. 159).
For in the case of two dimensions, when the origin is
unaltered we have
x"" + 2xY COS io + y^=^ X^ + 2Z'F' cos Q + Y'\ (31)
where w and Q, denote the angle between the original axes
and that between the transformed axes, respectively.
Multiply (31) by X, and add to (30): then denoting
0(a;, y) by u, and ^i(-X^, T) by U we get
Now, suppose A assumed so as to make the first side of
this equation a perfect square, it is obvious that the other
side will be a perfect square also. The former condition
gives
(d^u , \ (dhi . \ / d^u . \*
IH .\(dhi .\ f d'u
i^^^^lW^^J^KMy
M — - — > — +Acosa>
* I am indebted to Prof. Bumside for the suggestion that the equations of
this Article are immediately obtained by Boole's method.
Transformation of Coordinate Axes, 413
or X'Bm^. + A^-.^-2^^^co8a,j
d^(fu f cPu Y_
dx^ dy^ \dxdyj
Accordingly, we must have at the same time
A^sm^i2 + A^^,+ ^,-2^^^cosl2j
d:'Ud'U f d'U \^
Hence, comparing coefficients, we get
(fu dhi _ f dhi Y d^Ud^ _ / d'^U V
dx' dy" \dxdy) dX'^dY' \dXdY) ^ z.^)
sin^w sin^Q
and
(fw d'u d'u d'U d'U d^'U
dx^ dy^ dxdy _dX^ dY^ dXdY r \
sin'^faj sin^Q
Consequently, if u he any function of the coordinates of
a point, the expressions
d^u d^u f d^u Y d\ du d^u
dx^ dy^ \dxdyj , dx^ dy"^ dxdy
sin^w sin^w
cos (U
are unaltered when the axes of coordinates are changed in any
manner^ the origin remaining the same.
In the particular case of rectangular axes, it follows that
d^u d^u d^u d^ii f d\ Y
dx"^ dy^ dic^ dy"^ \dxdy)
preserve the same values when the axes are turned round
through any angle.
414
Change of the Independent Variable,
331. ilpplication to Orthogonal Transformation.' —
When the transformations are orthogonal it is easy to extend
the preceding results to three or more variables (Art. 327).
Thus, in the case of three variables, we have
Multiplying this by A, and adding the result to the equation
that corresponds to (30), it follows that the expression
+ 2ZX
d'u
dzdx
+ 2xy
,, (Pu
dxdy
is unaltered by orthogonal transformation.
Next, suppose that X is such that the quadratic function
in x\ y and 2' is the product of two linear factors ; then, by
Art. 107, we have
d'u
dxdy*
dxdz
d'u
d^u
dxdy* dy"^ ' dydz
d'u
d'u
dxdz' dydz' dz^
+ X
o.
(34)
But, as the transformed expression must also be the product
of two linear factors, we have
d^u . dH d^u
dxdy' dxdz
d'u
d^u
dydx
dxdz dydz dz^
dydz
Ux
d'U . d'^U
2 + X,
d'U
dX'
dXdY' dXdZ
d'U d'U
dXdY' dY'
» :7tF2 + X>
dU
dYdZ
U
d'U d'U ,
r + X
dXdZr dYdZ!" dz-
(35)
Orthogonal Transformation,
415
Equating tlie coefficients of like powers of A^we see that the
expressions
d^u (Pu d\
(Pu (Pu / dhi Y dhi^d^ f d\i Y d^u d^u /d'uV
dx" dy" \dxdy) ^ dx" dz" \dxdz) ~df dz^ ~ [d^zj '
and
d^u d^u
d'u
dx^^ dxdif dxdz
dhi d^u dhc
dxdy* dif* dzdy
d^u d^u d^u
dxdz dydz^ dz^
are unaltered by orthogonal transformation.
The first of these results has been already arrived at by
direct substitution (Art. 327).
These results readily admit of generalization.
41 6 Change of the Independent Variable,
Examples.
1. Being given y =f{u) and u = <f>(x), find — •
2. liy = F{t), t =f{u), u = (l>{x), find the value of -^ •
Ans. r{t)f'{u)<!>"{x) + {<l>'{x)}Hr{u) F'{t) + {f{u))^F"it)}.
3. Change the independent variable from a; to « in the equation
«* — ^ — 2nx^ -^ + a^y = o, where x = -,
dx^ dx z
d'^y 2[n^i)dy
^«*- ^^-^— ^+^'^ = °-
d^y dy
4. Transform {i - x"^) j-^- x — ^-a^yn^o^ being given x = sin z,
d^y
Ans. _ + «-y = o.
5. If ip = r sin 0 cos <^, y = r sin 0 sin 0, z = r cos 0, prove that — = — -,
dr dx
where 0 and ^ are regarded as constants in finding — ; while y and z are re
dr
dr
garded as constants in finding —- , •
dx
6. If 2 be a function of two independent variables, x and y, which are
connected with two other variables, u and v, by the equations
fi{x^ y, u, v) = o, f2{Xf y,u,v) = o;
ehow how to express — - and -— in terms of — and — •
dx dy du dv
7. Transform the equation
d'^y , 2x dy _y
+ x^dx (i + a;2)
"^ ^ JL ^2 dr. (t -4- *r2\2 ~ '
into another in which d is the independent variable, supposing x = tan9.
Examples.
417
8. If 2 be a function of x and ^, and u = px-^qy-z, prove that when
p and q are taken as independent variables, we have
du
du
d^u
dht
dpdq
d^u r
?* dq^ ^ rT^
rt — s'- dp dq rt ■
where i?, q, r, s, t denote the partial differential coefficients of z, as in Art. 304.
9. If the equation
d^*v (?'«"^v dv
ilS^-T^+ AiX>^-^ j-^ + . . . + An-lX / + A = O
dx>*
dx'^-^
dx
be transformed to depend on 0, where x = e9, prove that the coefficients in the
transformed differential equation are all constants.
K). Given x = ^, y = ^^^ prove that
^ f -^(0 r
dx'^ \F{t)ip'{t)-'<p{t)F'{t^
F{t), F'{t), F"{t)
<p{t), <^'{t), 4>"{t)
3B
( 4i8 )
CHAPTEE XXIII.
SPHERICAL HAHMONIC ANALYSIS.
332. It is proposed in tHs chapter to give a brief discussion
of the differential equation
d'V d'V (PV
an equation which occurs so frequently in physical investi-
gations. We shall denote the symbolic operator
d" d^ d' ,
^-'dy^-'d^^^^^'
Adopting this notation, we readily see that
_o/ X o » fdu dv du dv du dv\ , ^
\Uj<i> U/tt/ {Aiy Uy Ct-is UflA J
Again, since
~ (r*") = mxr^-\
we have -— (r"*)
= mr"*-'^ -v m{m- 2)x^ r"*"*,
and we readily get
V^
(r'") = w(m+ i)r»»-».
Hence, from (2),
we have
V^(r"»F) = r'»y'
'F+m(m+ i)r"»-'F
(3)
„,/ rfF rfF rfF\ , ,
Moreover, if F be a homogeneous function of the n^^ de-
gree in X, y, 2, we get, by Euler*s theorem of Art. 98,
y2 (^m Y>^ = ^my2 F+ m (w + 2w + i) r»»-« r. (5)
8oMd Sarmonic Functions. 419
.'•
333. Solid Harmonic Fanctions. — Any homogeneous
function in ic, y, z which 'Satisfies equation (i) is called a solid
spherical harmonic function, and frequently a solid harmonic.
We shall denote a solid harmonic of the n^^ degree hy
Vn, in which the degree n may be positive or negative, in-
teger or fractional, real or imaginary.
It is evident that any constant multiple of an harmonic is
also an harmonic of the same order.
From (5) it follows that a solid harmonic of the w^'^ degree
satisfies the equation
V' {r"" Vn) =m{m+ 2n+ i) r"^'^ F„. (6)
Y
Hence we see that if Fn be a solid harmonic, -r-?- is also
a solid harmonic, whose degree is - (w + i).
Again, from (3) we see that - is a solid harmonic of the
r
degree - i. Also it can be readily shown that - is the only
function of r that satisfies equation (i). For by (19), Art.
323, we can transform that equation into
d ( ,dV\ I d ( . .dV\ I ^F , ,
Hence, if F be a function of r solely, we must have
— ( r^ — J = o. This gives F in the form - + J.
In like manner, if F be a function of the angle ^ solely,
d^V
it must satisfy the equation — — - = o : this leads to F= «<^ + h.
Hence we observe that tan"^ I - 1 is a solid harmonic of the
degree zero.
Again, if F be a function of Q solely, we have
|(sme^=o.
2 B 2
420 Spherical Harmonic Analysis,
Hence we see that log ( tan -j satisfies the equation
V2p'= o, and we infer that log f —— is a solid harmonic.
r + 2
r- z
r ■¥ X r -V V
In like manner, I02: and log are also solid har-
' ^ r -X r - y
monies.
It is readily seen that <f> log tan ( - J satisfies equation (7) ;
hence we see that
V . r + 2
tan-^-log
X ^ r - z
is a solid harmonic, of zero degree.
If V satisfies* equation (i) it is seen immediately that
— , - — , and -r- also satisfy it, as also the general expres-
dx dy d% ^ ^ ^
dP+q+ry
sion , in which p, q, r are any positive integers.
Hence, from any solid harmonic, a numher of others can
be immediately deduced by differentiation.
Again, since -7-^ is a harmonic of degree w - i, it follows
uX
from (6) that -^^^ -z-^ is also a solid harmonic, whose degree
is - w : and so on.
For example, any expression of the form . ^^ ^ f - j
is a solid harmonic, whose degree is - (y + ^ + / + i).
Examples.
I. Find the condition that
ax^ + by* + ez^ + dxff + 0ses + fffz
should be a solid harmonic. Ana, a + b + cso.
Complete Solid Sarmonica, 4^1
2. Prove that
OK nr. y T •{■ Z
z tan-^ -, and z log 2r,
r+ z* r{r+ z)* x* r - z
are solid harmonics.
3. If Fo be a solid harmonic of degree zero, prove that r^"-* -—■ is also a
solid harmonic.
4. Hence prove that ~ — ^^ is a harmonic function.
For, let Vo = tan'^ ( ) > then, since y = x tan ^, it can be shown, as in
Art. 46, that
^3n~l gin fi(h
Hence ^ is a solid harmonic, as also any function derived from
{^^ + y'}l
it by differentiation.
5. Prove that u = — < is a solution of the differential equation
2(2^ + 3)
334. Complete iSolid Harmonics. — A solid harmonic
that is finite and single valued for all finite values of the co-
ordinates is said to be a complete harmonic. It can be proved,
by aid of the Integral Calculus, that every complete solid
harmonic is either a rational integral function of the coordi-
nates, or is reducible to one by multiplication by some power
of r. Assuming this, it follows that the number of indepen-
dent complete harmonics of degree n is 2n + i, when n is
positive.
For it is readily seen that the number of terms in F«, a
rational homogeneous function of the n*^ degree in x. y, s, is
(fi + 2^ (fi + i^ __
-; and also the number of terms in V*Fn is
; hence, since v' Fn ^ o identically, we must have
-^ linear equations connecting the coefficients in F« ;
K)nsequently, the number of independent constants is
{n + 2) {n + i) n(n - 1
2 2
422 Spherical Harmonic Analysis.
It can now be shown that every complete harmonic can be
deduced from - by differentiation.
r -^
For the solid harmonic
^+A+/ I
dxJdy^dz^ (a^ + ^' + z^)*'
when the differentiations are performed, is readily seen to be
a fraction of which the numerator is a homogeneous func-
tion of the degree n, and whose denominator is -^;^, where
Vn
n = k +J + I. If this function be represented by -j~, the
numerator Vn, by (6), is also a solid harmonic.
We can now show that the number of independent har-
monics of degree n that can be thus derived is zn + i.
For, since
dz' \rj ~ [dx' "^ dfj \r^
we see that
dz^^ \rj ^ ^ \dx' dfj \r/
. ,. ,f d^ d^Y
m which ( 7~j + j-2 ) ^^^ ^® expanded by the binomial theorem
as if — and — were algebraic quantities, and the resulting
differentiations of - taken.
r
Hence, if I be even, we have
i dJ*^ fd" _^Y/'A
dx^ dy^ dz^ \rj ^ ^ dx^ dy^ \da^ dfj \rj
and, if I be odd,
i-\ i-\
dxfdy^dz\r) ^ ^^ dx^ dy^\d^ '^ dy^ Jz^r}
Spherical and Zonal Harmonics, 423
Aeoordingly, in the former case, we get aliumber of terms
each of the form , , , „ - , where p + q = n\ and in the
dxPdi/\rf ^ ^ '
latter, terms of the form ^ „ , „ J— ( - )} , in which p + q^n-i,
dx^dy^ \dz\rjY ^ ^
Now there are p + q+i, ot n+ i terms in the former case,
and n in the latter. Hence there are 27^+1 independent
forms, as was to be proved.
33 5 • Spherical and Zonal Harmonics. — If a solid
. harmonic Vn be divided by r", the quotient may be regarded
as a function of the two angular coordinates, or spherical
surface coordinates, Q and 0. Such a function is called a
spherical surface harmonic of the degree n.
Hence, if F« = r**F„, then F„ is a spherical harmonic of
the n^^ degree.
It is obvious that the general spherical harmonic of the first
degree is of the form aao^O + h sin 6 cos (j> + c sin 9 sin <^, where
a, bf c are arbitrary constants. Also, the general expression
for Z2 can be written down readily (see Ex. i, Art. 333).
Again, by {19), Art. 323, we see that Yn satisfies the
differential equation
sm %d9\ dQ J sm^0 d<j)^ ^ ^
This equation admits of a useful transformation : for, let
fjL = cos 0, then, since
we get
d ( , ,, dYn) I d'Yn , .^ / X
Again, if a spherical harmonic be a function of 9 solely, it
is called a zonal harmonic. Hence, if P« be a zonal harmonio
of the n^^ order, it must satisfy the equation
i-j(.-,»)fJH-M-x)P.= 0. (.0)
424 Spherical Harmonic Analysis.
When w is a positive integer, the value of Pn can be readily-
represented by a finite series. For since, by hypothesis, P«
is a function of the n*^ degree in /u, we may assume
m=»
P„ = S Kiu"»).
m=o
dP
Hence -^^ = S {mamfj^-^) ;
.-. — (i -ju'^) — ^ = 2m(m - i)a^ju*"-'- Sm(m + i)flf^ju"».
Substituting in (lo), and equating the coefficient of ^*" to
zero (since the result must vanish identically), viQ get
[m + i) (m + 2) a^+a = - (w - m) (m + w + i) am-
Hence, observing that the highest power of ju is n, we
have
w(w-i) ^
2 (2W - I)
and we may write
r 2(2^i-l)'^ 2.4(2W-l)(2W-3)^ )'
where fif« is an arbitrary constant.
This is the general form of a zonal harmonic of integer
positive degree ; and we see that two zonal harmonics of the
same degree can only differ by a constant multiplier.
It can be shown independently of the above that
KIT I ^^^ ~ ^^** s^^isfies the equation (10).
In order to prove this we shall assume u = im^ - i , and
write the symbol 2) for — ; then we have to prove that
D {wi)«^^ (w»)) - w (w + i) D» (w*») = o.
Zonal Sarmonic8» 425
du '*
Now, observing that — = Zfx^ we get, by Leibnitz's
ClfX
theorem of Art. 48,
Again, since
D («*«*!) = 2 (;^+ i);i«t",
we have
J)n+i (^^n+i) = 2 (w + l) D»^ OuW")
= 2 (?i + i) juZ)" (w") + 2w (/^ + i) 2)"-^ (e**).
Equating these values of D''^'^ («*"^^), we get
wZ)"-^^ (w") = w (w + i) 2)«-^ (t***) ;
hence D {wD«+^ («««)} - w (w + i)2)" «) = o. (12)
Consequently -D" (w**) satisfies the equation in question.
Hence we infer that
■•=Ki)v-
i)«
The student can verify, by direct differentiation, that
this expression differs only by a constant factor from the
value of Pn found in (11).
It is usual to assume that Fn is that value of the pre-
ceding expression which becomes unity when /i« i.
To find this value, we have
(|)V-.)»-«(|;J"'{MM'-.r)
by Leibnitz's theorem.
426 Spherical Harmonic Analysis.
Now it is readily seen that f — ) (jn* - i )**~^ = o when ju = i ;
hence, when ju = i, we have
(|^)V-.)"=-(|f(/-o-
= 2^n {n - i) (^ \fji' - i)^^ &c.
Consequently, when ju= i,
The foregoing result can be readily shown in another
manner. For 2 [w f — J (ju'' - i)" is the coefficient of h" in the
expansion of (i - 2/^^ + h^)-^ (see Ex. 6, p. 155).
Again
( (x - ay + y' + z")'^ = {r^ - zarjj. + a')"*
I f
= - (i - hu, + A*)-i, where h = -,
a^ ^ ^ a
and we have
a-'^ 2-[n\dfxJ ^^ ^ '
in which we suppose a> r.
But V^iix-ay + y^ + z^y^^o;
hence
\d~) ^'^^ ~ ^^" satisfies equation (10), &o.
The functions Pi, . . . Pn are usually called Legendre's
Coefficients.
Complete Spherical Marmonics, 427
Examples.
1. If ^ = I, prove that P,, = 1 for all values of «.
2. If /A = - I, prove that P„ = (- i)».
3. If /* < I, show that the series
Pii- Pa + . . . + P„ + . . .
is convergent.
4. Prove the relations
P4= g , P6 = g
5. Prove the equations
|-(P«.i-Pn-i) = (2»f i)P„,
(» + I) P,M - (2W + I )/iP» + WP„_1 = O.
336. Complete Spherical Harmonies. —
From Art. 334 it follows that a complete spherical har-
monic Yn of the n^^ order, when n is an integer, contains
2n + I arbitrary constants. Its value can be expressed by
aid of the corresponding zonal h^monic P«, as we proceed
to show.
Since F„ is in this case a rational integer function of
sin Q cos ^, sin Q sin ^ and cos d, we may suppose it expressed
in a series of sines and cosines of multiples of <^, whose coef-
ficients are functions of 0, or of fx. We accordingly assume
that Tn consists of a number of terms each of the form
il/, coss<^; then, substituting in equation (8), and observing
that ■ = - 8^ cos 5^, we obtain, on equating to zero the
(X(p
coefficient of cos 5^,
If, as before, we write u f or u* - i, and B for -7-, this
dfi
uB[uDMs] - sWs -n{n+i) uMs = o. (15)
428 Spherical Harmonic Analysis.
s
Now, let Ms'^u^v) then
« <
uDMs = u^ Dv + SjuLU^v;
therefore « ' I i
uD {uDMs) = u'^^D'v + 2 (s + i)fxi^'l)v + su^% + syu\.
Substitute in {15), and divide by u^ ; then
uD^v-¥ 2(5+ i)iLiDv+ {s(s+ i)-n{n+ i))«? = o. (16)
It is readily seen that this equation is satisfied by assum-
ing V = D^Fml foi" substituting this value for v in (16), it
becomes
ujy^'Fm + 2 (S + l)tlB''^Pm + S (s + l) J^Pm
- n{n + I ) D'Pm = o ;
but by Leibnitz's theorem the first three terms are equivalent
to i)**^ [uDPm) ; whence the equation becomes
i?" [uDPn,] -n[n-\-i) D'Pm = o.
But this equation follows immediately from (10) by
differentiating it s times with respect to ju.
^.coordingly, the expression
satisfies equation (14), and hence
coss^(^^-i/(|-Jp„
satisfies (8).
T Ti ^(sinsd>) „ . I,
In like manner, as \ ^ = - s' sms0, the expression
rin«^0..-i)^(|-)'.
also satisfies the same equation.
Accordingly, equation (8) is satisfied by the expression
(^.cos s^ + B, sin s^) (n' - i)^^AY(p„), (i 7)
in which A, and £, are arbitrary constants.
Laplace's Coefficients, 429
This expression is called a Tesseral Surfade Harmonic, and
is said to be of the degree n and order s.
If we give all integer values to s from i to n, the com- ^
plete spherical harmonic F» can be written in terms of '
Tesseral harmonics as follows: —
s=n td'Pn
Yn = AoPn + ^ (^,coss^ + j5,sins<^)(/x^-i)=^-^, (18)
f d\^
in which (-7-) (fx^- i)" may be substituted for Pn if neces-
\dfij
sary.
This equation contains the proper number 2n+ i of arbi-
trary constants, and consequently may be regarded as a
general expression for a complete spherical harmonic of in-
teger positive degree.
There is no difficulty in showing by differentiation that
— ) (juL^- lY differs only by a constant from
{n-s){n-s-i)
^ 2{2n-i) ^
{n - s){n ^ s - i){n - s - 2) (n - 8 - s)
2.4. (zn- i)'{2n- 3)
Hence that part of Yn which depends on the angle stp may
be written
This agrees with the general expression given by Laplace
{M^canique Celeste, tome iii., chap, ii., p. 46).
337. Ijaplace's Coefficients. — It is immediately seen
that the expression 77 ttz — 7 jr^ — , ttth satisfies the
general equation (i), as also the corresponding equation
<^F O'V d'V _
daf' "^ di/' ^ dz'' ~ ^'
430 Spherical Harmonic Analysis,
Transferring to polar coordinates, the preceding expression
may be written {r^ - 2\r/ + r^) , where
af = / sin & cos 0', y' = / sin Q' sin ^\ s' = / ooa 0' ;
and X = cos 0 cos Q' + sin Q sin 0' cos («^ - 0') .
If P and P' be the points whose coordinates are ccyz and
x'y'z\ respectively, then
Accordingly, if
(i - 2\h + A'»}-i= i + Lih+ Lih^ + . . . + Z„A» + . . .,
we have
I I Xi/ Za/* Zn/«
and
I I LiV LzT^ LnT^
= h 1 + . . . + —
Hence, since VN -pp?) = o, we must have v*(-;^) = o,
and also V"" [Lnr") = o.
From this we see that Z„ is a spherical harmonic of the
degree n, and that it satisfies the equation
The functions Xi, Z2, . . . Ln are called Laplaceh Coef-
ficients,^ after Laplace, to whom their introduction into
analysis is due.
The value of Ln may be deduced from that of Pn in (11)
or (13), by substituting juju' + -v/i - ju'' -v/i - ju'* cos (0 - 0') in
place of ft, where ju = cos 0, and [i = cos 0\ Hence it is a
function of the n*^ degree in ju, y^ i - ju^ cos 0 and y^ i - /i'* sin 0 ;
as also in ju', \/i - ju''^ cos 0' and y^i - /^ sin 0'.
Laplaceh Coefficients. 431
Moreover, since Ln is a spherical harmonic of the n*^
degree, and symmetric in fi and ju', as also in <p and 0', it
must, by (18), be of the form
in which the coeflScients a^, «!,...<?«... are constants, the
values of which remain to be determined.
It is immediately seen that ao= i: for ii fx = i, we have
P„'= I, andX« = P„.
In the Integral Calculus, Art. 233, it is shown that
2I ^ ~ s
as = (- i)* , ' Assuming this result, we have
?^ + s
«=»lw — s « *<f*P d^P»^
Some further applications of spherical harmonics will be
found in the Integral Calculus, Arts. 230-5, but for a more
complete treatment of the subject, which involves the applica-
tion of Multiple Integrals as well as the solution of Differential
Equations, the student is referred to Thomson and Tait's
Treatise on Natural Philosophy ; to Ferrers' Spherical Har-
monics ; or to Todhunter's Treatise on Laplace* s, Lam^^s, and
BesseVs Functions,
432 Spherical Harmonic Analysis,
Examples,
1. If « be a solution of the differential equation
du du du ... . . , ,
prove that x^ + P^ +e-T ^'^ ^so be a solution of it.
^ dx dy dz
2. Show that each of the quantities
--fj?, (I -A*') cos 20, (i-)u2)8in20, /tVi-/ii«cosa, /t\^i-/*'«sini
is a surface harmonic of the second degree.
3. Prove that the expressions
, r+z z r+z z , r + 2 2rxz
zlog 2r, -^log -, arlog + —
^ r-z r'' ° r- z r^' ° r - z x^ + y^
are solid harmonic functions.
4. If the polar variables be replaced by « and v, where
cot-e^ = u, tan-^^* = v, and i = V - i,
prove that any surface harmonic of the order n satisfies the equation
d^V n{n+i)F
— o.
dudv (m + v)*
5. If />', pi', pa' be the roots of the equation in \,
i = ^+ y' . «'
and if
A A. - A2 X
K"
V(p2 - A2) (p2 - k2)' «i - V(A2-pi2)(pi2-kY
prove that
transforms into
V(p22-A2)(p22-k2)'
<f2u flf^U <;2w
(^^-^.')S+(''-''^')|r.+(^''-p')^.
( 433 )
OHAPTEE XXIV-
JACOBIANS.
S^8. JTacobians. — The results obtained in Articles 330
and 331 are particular cases of a class of general theorems
in determinants which were first developed by Jacobi (Crelle's
Journal, 1841).
Thus, if u, V, w be functions of x, y, z, the determinant
J^
du
du
^'
dy'
^v~
~'dv
dx'
dy'
dw
dw
^'
Ty'
dz
dz
(I)
was styled by Jacobi a functional determinant. Such a
determinant is now usually represented by the notation
d{u, t). w)
d{x,y,zf
and is called the Jacobian of the system w, v, w with respect
to the variables x, y, z.
In the particular case where u, v, w are the partial diffe-
rential coefficients of the same function of the variables x, y, s,
their Jacobian becomes of the form given in Art. 331, and
is called the Hessian of the primitive function. Thus the
determinant in Art. 331 is called the Hessian of u, after
Hesse, who first introduced such functions into analysis, and
pointed out their importance in the general theory of curves
and surfaces.
2F
434
Jacobians.
More generally, if yi, y^, 2/3 ... ^n t© functions of oji, x^^ x^
Xm the determinant
dyi
dXx'
dyi
dx,"
dy.
" dXn
dyi
dx,'
dyz
dx^''
dy%
' * dXn
dyn
• •
dyn
dyn
dx,'
dx,''
dXn
is called the Jacobian of the system of functions yi, 2/2? .. . yn
with respect to the variables a?i, x^, . , . Xn\ and is denoted by
J^
d{yi,y2,'"y^
d(Xi, X2y... Xn)
(2)
Again, if yi, ^2, • • . yn be differential coefficients of the
same function the Jacobian is styled, as above, the Hessian
of the function.
A Jacobian is frequently represented by the notation
J{yi, ^2, . . . yn),
the variables Xi, Xzy . . . x^ being understood.
If the equations for ^1, ?/2, . . . y» be of the following form,
yi =f2{i^i, a;^),
2/3 =/3(aJl, Xiy X»)y
yn =/n(a?i, Xz, , , , Xn)f
Examples, 455
it is obvious that their Jacobian reduces to -its leading term,
VIZ.
J « -7 — . — ; — . . . -z — • 13;
dxi aXi axn
This is a case of a more general theorem which will bo
given subsequently (Art. 343).
Examples.
1. Find the Jacobian of yi, yz, • • . yn, being given
yi = I ~ xi, yz = x\{i- Xi)y yz = xixz {1 - xz) . , .
yn = i»ia;2 . . . Xn.i (I - Xn). Ans. 7 = (- ly^Xi'^-'^Xi^-'^ . . . x„-i.
2. Find the Jacobian of xi, xz, . . , a?« with respect to 0i, 62, . . . 9n, being
given
Xi = cos 61, xz = sin ^i cos 62, X3 = sin Oi sin 62 cos 63, . . •
Xn = sin 01 sin 62 sin 03 • • • sin 6n-i cos dn.
Ans. ^(^1' ^2> • • • ^») = (_ i)„ sin„0j . ginu-i^j . . . sin0„.
339. Case wbere tbe Functions are not Indepen-
dent.— If 2/1, ^2 • . . 2/n be connected by a relation, it is
easily seen that their Jacobian is always zero.
For, suppose the equation of connexion to be represented
by
F[yi,y2, . . .2/n) =0;
then, differentiating with respect to the variables a?!, Xz,,.. a?«,
we get the following system of equations : —
dF dxjx dF dyz dF dy^
d/yx dxi dy^ dxi " ' dy^ dxi '
2F3
436 Jacohians,
dF dj/i dFdpt dF dpn _
dyx dXi dt/idxt ' " dyn dsh '
dFdyx dFdpi dF dpn _
dyxdxn dy^dxn "' dy^dx^
,. . ^. dF dF dF
whence, eliminatma: -7-, -7- • • • -;— > w© ?>^^
^ dyx dyt dyn ^
d {Xx, Xi,,,.Xn)
The converse of this result will be established in Art. 344 ;
and we infer that whenever the Jacobian of a system of
functions vanishes identically the functions are not indepen-
dent. This is an extension of the result arrived at in Art.
314.
340. Case of Functions of Functions. — If we sup-
pose ih, tizj W3 to be functions of yi, y^^ y^, where yi, yz, yz are
functions of Xi, icz, (^3 ; we have
dui _ dui dyx dUx dy^ dih dy^
dxx dyx dxx dy% dxx dy^ dxx '
dUx dui dyx dUx dy^ dUx dyz
dxz dyx dXi dyx dxz dy^ dx%^
dux _ dtix dyx dux dyz dux dy^
dxz dyx dxs dy^ dx^ dyz dxz'
• ••••• &o.
Case o/Funcftons of Functions,
437
Hence, by the ordinary rule for the multiplication of
determinants, we get
ayz ay2 ay^ / x
dui dui dux
du.
dui dth
dyx
dyx
dyx
dxx dx^^ dxz
dyx
dyi dyz
dx^
dx^'
dxz
dUi du-i dUi
dxx dXi^ dxz
=
dUi
dyx'
dut duz
dy% dyz
•
dyz
dxx'
dy2
dx^'
dy^
dXz
dus dUi duz
dus
duz duz
dyz
dy%
dy%
dx^ dXi dxz
dy:
dyi dy
dxx'
dx,'
dxz
or
d{uxy Ujy Uj) djuiy u^y th) d{yx, ya, yz)
d{xxy X2y Xz) d{y ly yzy yz) * d{xxy x^y Xz)
It follows as a particular case, that
djyxy yi, yz) d{xx, a?2, xz) ^
d(xxyX2yXz) d{yx,y2yyz) '
(6)
These results are readily generalized, and it can be shown
by the method given above that
d{uxy Ujy... Un) ^ d{uxy u^y . , . u^) d[yxy yz, . . . yn)
d{Xxy iTa, . . . Xn) d{yxy ^2, . . yn) ' d{Xx, X2,... XnY
(7)
This is a generalization of the elementary theorem of
Art. 1 9, viz.
du du dy
dx dy dx
Again,
d{yx, yz, « . » yn) d{xi, x^,,,, a?„)
d{xxy iCz, . . . iP«) d{yx, yz, . . ^ yn)
This may be regarded as a generalization of the result
dx ^ dy
dy ' dx
(8)
438
Jacobians,
341. The Jacob! an is an Invariant. — In the parti-
cular case of linear transformations we have a system of
equations as follows: —
yi = hxXx + hxi + bf^n,
In this case
, + InXn^
d{yx, y2 yn)
d{Xi, Xi Xn)
hi bi . . . bn
h k
k
This determinant is a constant, and is called the modulus
of the transformation.
Accordingly, in linear transformations the transformed
Jacobian is equal to the original Jacobian multiplied by the
modulus of the transformation.
In the case of orthogonal transformation (see Art. 327)
the modulus of the transformation is unity, and accordingly
the Jacobian is unaltered by such a transformation.
342. Jacobian of Implicit Functions. — Next, if
Uy V, w, instead of being given explicitly in terms of x, y, z,
be connected with them by equations such as
Fi{x,y,z,u,v,w) = Q, Fi{x,yfZ,u,v,w) = o, Fi[x,y,z,u,v,w) = o,
then w, V, w may be regarded as implicit functions of a?, y^ «.
In this case we have, by differentiation,
dFx dFx du
dx du dx
dFi dv dFi dio _
dv dx dw dx
dFi dFi du dFi dv dFi dw _
dy du dy dv dy dw dy '
iFz dFi du dFi dv dF^ dw
dx du dx dv dx dw dx '
Jacohian of Implicit Functions,
m
Hence we observe, from the ordinary rule for multiplica-
tion of determinants, that
dF^ dF, dF,
du dv dw
dFi dFx dF,
du^ dv^ dw
dx dx' dx
dx' dy' dz
dF^ dF, dF^
du ' dv^ dw
•
du dv dw
dy^ dxf dy
= -
dF^ dF^ dF^
dx' dy' dz
dF^ dFs dFs
du^ dv' dw
du dv dw
Tz' 1%' d^
dF, dFz dF,
dx' dy' dz
UJJ 2 , M-tt U/U UW U/JCz CI/J.' 2 U/JL' 2 . »
~d^ ' Ty' dy' ~- = " ~^' "^^ ~^ ' ^^^
dFz du dv
dw dz' dz'
This result may be written
d{F„F2,F,) d[u,v,w) ^ d{Fr,F„F,)
d{u, V, w) ' d{xy yy z) " d[x, y, z)
The preceding can be generalized, and it can be readily
shown by a like demonstration that if yi, 2/2, ^3, • • . yn
are connected with a?i, a?2, 0% . . . a;» by /* equations of the
form
i^i(a?i, aJa . . . Xn, yi, ^2 . . . 2^n) = O,
F2[xiy ajj, . . . iTn, 2^1, y2 . . . y») = o.
(10)
Fn[Xi, aJa . . . Xn, yi, y2*>.yn)=' o,
we shall have the following relation between the Jacobians
d[F,,F2,,..Fn) ^ d{y,, ^2, . . . Vn) ^ f .^ d[F,,F2,.,.Fn)^
'^^^'ij 2^3, .• • Vn) ' d{Xif X2,,,, Xn] d{Xiy X2, . . . Xn)'
Accordingly
djyu ^2, . . » yn)
d{xi, Xi, ,. . Xn)
(-1)'
d{F^, F2,.., Fn)
d{Xi, X2, . , . Xn)
d{F,,F2,.,.Fn)'
d[yi, 2^2, .. . yn)
(II)
440
Jacohians.
343. Again, the equations connecting tlie variables are
always capable by elimination of being transformed into the
following shape : —
(l>l[Xx, iPz, . . . Xn, yi) = 0,
02 fe, xz, ., . Xn, Vii y%) = o,
(pz[Xz, X^y .,. Xn, Vi, Vi, yz) = o,
In this case the Jacobian determinant
(^{yi, yiy " ' ynY
as in Art. 338, reduces to its leading term
d<l>i d(i)2 difiz d(f)n
dyi dyi dy^'" dy„
In like manner,
^(<^ij 02, . . . ^n)
d(Xi, Xiy ... Xn)
reduces to
d<j)\ d(p2 d(f>n
dXi dXz" ' dXn
Accordingly, in this case, the Jacobian
d(pi d(p2
d[yx,y2,.,.yn) . .„dx, dx, "
d{xi, ajj, . . . Xn) ^ ^ d(j>i dtp^
d(Pn
' dXn
dcpn
dyx dy-i "
' dyn
(12)
(■3)
344. €ase where JTacobian vanishes. — We can now
prove that if the Jacobian vanishes, the functions yi, ^2, . . . y»
axe not independent of one another.
Case where Jacohian Vanishes, 441
For, as in the previous Article, the equations connecting
the variables are always capable of being transformed into
the shape given in ( 1 2), and accordingly, if J {yi ^2, . . . y„) = o,
we must have
d'Xi dx-i ' ' ' dxn '
that is, we have -^ s o for some value of i between i and n,
dxi
Hence, for that particular value of i the function (fn must
not contain Xi ; and accordingly the corresponding equation
is of the form
(f>i (a?f+i, . . . iTn, Vx, ^8, . . . Vi) = o.
Consequently, between this and the remaining equations,
0H1 = o, </>f+2 = 0, . . . ^„ = o,
the variables xui Xi+2, . . . a?„ can be eliminated so as to give
a final equation between ^1, ^2, • • • yn alone. This establishes
our theorem.
Also, it follows that if the Jacobian does not vanish,
the functions are independent.
345. In the particular case where
yi = Fi{xi, xo, . . . a?J,
t/2 = F^iyi, X2, . . . Xn),
we have
t/n = Fn{yi,y2, . . .yn.i,!^n),
d (yi> ^2, . . ♦ yn) ^dyj_ dy^, dy
d [Xiy iTs, . . . Xn) dXi ' dx2 ' ' ' dXi
(14)
It may be observed that the theory of Jacobians is of
fundamental importance in the transformation of Multiple
Integrals (see Int, Calc, Art. 225).
^^^ jacobians.
Examples.
I. Find the Jacobian of yi, j/2, . . . y« with respect to r, 0i, 62, . . . 0«-i,
being given the system of equations —
yi — r cos di, 1/2 = r sin Oi cos 62, ys = »* sin ^i sin 02 cos Os, • • •
yn = r sin 0i sin 02 . . • sin 0n-i.
If we square and add we get
yi'^ + y8« + . . . yn' » r».
Assuming this instead of the last of the given equations, we readily find
7= r»-i sin»»-'0i sin«-^02 ... sin 0„.a,
2. Find the Jacobian of yi, yz . . . yn, being given
yi = «! (i - a;2), y2 = ariar* (I - a^s) . . •
y»-i = aJia^2 . . . sun-i (I — a^)»
yn = X1X2 . . . Xn>
Here, yi + y2 + . • • y» = J»?i, and we get
a [Xif X2f . . , Xn)
346. Case where a Relation connects the Depen-
dent ITariables. — If yi, 2/2 .. . y^ which are given func-
tions of the n variables iCi, a^, . . . Xn, be connected by an
independent relation,
J^(2/i» ya, . . . y«) = o, (15)
we may, in virtue of this relation, regard one of the variables,
ii?„ suppose, as a function of the remaining variables, and thus
consider 2/1, ^2, • . • Vn-i as functions of a?i, arg, . . . a?n_i. In
this case it can be shown that
dF
d[yi, ^2, . . . Vn-i) _ dyn d (yi, ya . . . yn) .
d[Xx,X2^ . . . Xn-i) dF_d(xi,X2, . . . aj„)
dXn
Case where a Belation connects the Dependent Variables. 443
For, if we regard Xn as a function of cPi, xz, &c., we have
^ , . _ ^1 dyx dXn _^( \_^ ^^2 dXn »
^^^^^"^1 "*" ^^' dxy^'^'~dxx dxndxx
Also, from equation (15),
dF dF dxn dF dF dXn «
aa?i dxndxi dx^ dxndx%
dF
let Ax = ^,
dF dF
. dx^ . c?ir„-i ^
^a?n ^ dXn
dx," ^'' dx.
dx^
— ~~ A2, . . . - , An-l»
then
TT d , . dijx ^ dyx d . . dyx ^ dyx -
Hence ^— (yi) = -T--Ai3— , -3— (2/i) = tt--A2 t~, &o.
rfiPi ^^^ dxx dXn dxi ' dx^ dxn
&o.
Accordingly, substituting in the Jacobian
dixxyXiy .. . n;„.i)'
it becomes
dyx ^ ^yi ^-x J^ ^^-\ i^
dxx dxn dx2 ^ dx^ " * dxn.\ **" c?a?»
dXx dXn ' dXi "^ dXn * * * <^iJ?n-l **" dXn
dyn-x . #«-i e?yn_i . dyn.\ dy^^x ^ %i»-i
— ; Ai — ; — « — ; A2 — ; — j • • • "7 ^n^\ ~3
dXi dXn dX2 dXn axn.x axn
If this determinant be bordered by introducing an addi-
tional column as in the following determinant, the other
444
Jacohiam.
terms of the additional row being cyphers, its value is readily
seen to be
dyi
dx:
dyi
dx,' • • •
dyx
dXn
dy%
dx,'
dy2
dx,' ' ' '
dyz
dx^
»
dyn-i
dyn-i
dyn-x
•
dx,'
dx^' ' ' '
dXn
K
A», • . .
I
or
dyi
dx^
dyi
dx,' ' •
dyi
dXn
I
dyi
dx,^
dyz
dx^' ' * '
dtjz
dXn
dF
•
.
.
•
dXn
dyn-x
dyn-i
dyn-x
dXx'
dx,' ' ' '
dXn
dF
dF
dF
dXx'
dx,' ' ' '
dXn
Again, we have
dF dF dy^
dxi dyi dxi
dF dy^
dyi dxi
^ dF dy^
dyn dxi '
dF
dx.
dF dyx
dyx dXi
^ dF dy, ^
dy^ dxz
dl
"^Ty
ndx,'
Substituting these values in the last row of the preceding
Case where a Relation connects the Dependent Variables, 445
the theorem is established, since we readily find that the
determinant is reducible to
dy,
dx^'
dyi
dx^' * •
dyi
'dXn
dF
dyn
dF
dyi
dxC
dy^
dx^ ' '
dyi
' dXn
dxn
dyn
dx:
dyn
dyn
' dXn
(16)
It may be well to guard the student from the supposition
that this latter determinant is zero, as in Arts. 339 and 344.
The distinction is, that in the former cases the equation
F{yu yi ' ' ' yn) = o, connecting the y functions, is deduced
by the elimination of the variables Xi, X2, . . - Xn from the
equations of connexion ; whereas in the case here considered
it is an additional and independent relation.
446
I. Being given
Jacohians,
Examples.
yi = r BinOi sin dz, ys = r sin 0i cos 0a,
yz=r cos Qi sin dz, y^ = r cos di cos Os,
find the value of the Jacobian
^(r, 01, 02, 03) '
^««.|»^sin0icos0i.
a. Find the Jacobian ■ , 5' /, being given
» = r cos 0 cos ^, y = rsind V I— M^sin^^, a = rBin</)\/i — w'^sin^e,
"vrhere w* + M* = i.
r2{M2cOSV + «2cos2a)
^M9.
Vl-m'sin^^ Vi - nHxn^Q
3. Being given
X%Xz X\X% X\X%
yi=— -, y2 = -— , ^3=-—,
0^1 0:3 it^S
find the value of the Jacobian of yi, yz, ya.
4. In the Jacobian
Ans. 4.
d[y\,y%. ..yn)
d{^Xi,Xi,...Xn)'
if we make
«1 «2
yi = — , 2/2 = —,... yn =
prove that it becomes
«, Wl, «2, ... «»
^M efe^l <f?^ <fMn
dx\ dx\ dx\ ' ' ' dxi
I
du du\ du% dun
«n+l
dXi dxz dxi* ' ' ' dxz
Un
du du\ dui dun
dXn dXn dXn ' ' ' dXn
This determinant is represented by the notation K{Uf ui, , . . Un).
5. If a homogeneous relation exists between u, «i, . . . w», prove that
Jr(w, «i, . . . «„) = o.
6. In the same case if yi, yg, . . . yn possess a common factor, so that
yi = UiU, &c., prove that
•^(y, ya, . . . y») = 2m«/(wi, «a, ...«») - ««-1jK'(«, «!,«»,... 1%).
( 447 )
OHAPTEE XXV.
GENERAL CONDITIONS FOR MAXIMA OR MINIMA.
347. Conditions for a maximum or Minimum for
Four ITariablesi. — The conditions for a maximum or a
minimum in the case of two or of three variables have been
given in Chapter X.
It can be readily seen that the mode of investigation, and
the form of the conditions there given, admit of extension
to the case of any number of independent variables.
We shall commence with the case of four independent
variables. Proceeding as in Art. 162, it is obvious that the
problem reduces to the consideration of a quadratic expres-
sion in four variables which shall preserve the same sign for
all real values of the variable.
Let the quadratic be written in the form
u = anXx + av^^ + a^z + a^^x^
+ 2a'iiX2Xi + ittziX^i^ (i)
in which «ii, 012, ^22, &c., represent the respective second
differential coefficients of the function, as in Art. 162.
We shall first investigate the conditions that this ex-
pression shall be always a positive quantity. In this case
«ii, 022, «33, &o., evidently are necessarily positive : again,
multiplying by o^n, the expression may be written in the
following form : —
«iiW = [aixXi + ttnXi + fl5i3a?3 + «i4a?*)^ + («i i«22 - dn) Xz + («n<i?a3 - ^is'^) ^i
+ («11«Z44 - «U ) X^ + 2 («5iia23 - «12«13)iP2^
+ 2 [cLwCLu — (ln(l\i\ X^^ + 2 {anasi - dndu) X^i. (2)
448
General Conditions for Maxima or Minima,
Also, in order that the part of this expression after the
first term shall be always positive, we must have, by the
Article referred to, the following conditions : —
and
^11^22 — ^12 > 0>
(«u«22 - (in) («ii«33 - Cln) - («n«23 - aiidizY > O,
^11^22 "■ ^12 ^11^23 ~ ^12^13, ^11^24 ~ ^12^14
^11<^23 ~ ^12^13) <5fll^33 — ^13 > <^11^34 ~" ^13^14
^11<5524 — ^12<?14> ^11^34 — <^13^14j ^11^44 " ^14
> o.
(3)
(4)
(5)
To express the determinant (5) in a simpler form, we
write it as follows: —
I
an
aiif «i2> (^I3y ^14
O, fl5]1^22 ~ ^12 > ^11^23 ~ ^12^13> ^11^24 "" ^12^14
O, <Zufl!23 — ^12<^13) ^11^33 ~ ^13 ) ^11<?34 ~ <?13<^14
O, fl5ll<3!24 ~ ^12^14> <^11<5534 — a23fl!l4> ^11^44 — ^14
(6)
Next, to form a new determinant, multiply the first row
by 0^12, aisi au, successively, and add the resulting terms to
the 2nd, 3rd, and 4th rows, respectively; then, since each
term in the rows after the first contains an as a factor, the
determinant is evidently equivalent to
«ii
(7)
^11> ^12) ^13) ^14
^12) ^22) ^23> ^24
^13> ^23) <3533, au
«i4, a2i, auf an
In like manner the relation in (4) is at once reducible to
the form
f^Uf ^12) ^13
an «12, a22f ^23 > O'
^13) ^23> ^33
Maxima and Minima,
449
Hence we conclude that whenever the following condi-
tions are fulfilled, viz.
any ^12) ^13) au
an > o,
aui ^12
^12, fl^za
> O,
<^12) ^22) <^23
^13) ^23j ^33
>0,
<^12j ^22j ^23j a^i
^13) ^23) ^33> ^34
^/l4, ^24> <^345 ^44
> o.
(8)
the quadratic expression in (i) is positive for all real values
of Xi, Xi^ Xz^ X4^.
Accordingly, the conditions are the same as in the case
(Art. 162) of three variables, with the addition that the deter-
minant (7) shall be also positive.
In like manner it can be readily seen that if the second
and fourth of the preceding determinants be positive, and
the twp others negative, the quadratic expression (i) is
negative for all values of the variables.
The last determinant in (8) is called the discriminant of
the quadratic function, and the preceding determinant is
derived from it by omitting the extreme row and column,
and the others are derived in like manner.
When the discriminant vanishes, it can be seen without
difiSculty that the expression (i) is reducible to the sum of
three squares.
It can now be easily proved by induction that the preceding
principle holds in general, and that in the case of n variables
the conditions can be deduced from the discriminant in the
manner indicated above.
348. Conditions for n l^ariables. — If the notation
already adopted be generalized, the coefficient of Xr^ is
denoted by am and that of XrXm by larm- In this case the
discriminant of the quadratic function in n variables is
^iij ^12) a
13>
an,
az2,
«'23, ...
«2«
an,
«23,
«33, . . .
a^
•
•
•
•
am,
a%m
«3n, . . .
2G
ann
(9)
450
General Conditions for Maxima or Minima*
and the conditions that the quadratic expression shall be
always positive are, that the determinant (q) and the series
of determinants derived in succession by erasing the outside
row and column shall be all positive.
To establish this result, we multiply the quadratic func-
tion by «ii, and it is evident that
aiiU = [a^^Xx + anXi + . . . ai«a?„/ + (au^za - an']x^ + ... .
+ (^iiflfnn - ax^)Xr^ + 2(an«23 - ax-iflxz^X^X^ +
+ 2 {axxarn ~" ^ir'^in) XrXn + &C.
In order that this should be always positive it is necessary
that the part after the first term should always be positive.
This is a quadratic function of the n- \ variables Xx, a^a, . . . Xn>
Accordingly, assuming that the conditions in question hold
for it, its discriminant must be positive, as also the series of
determinants derived from it. But the discriminant is
^H^22 "" <^12 ) ^11^23 ~ ^12^13> • • • <'5ll^2n "" ^is^in
^11^23 ~ ^12^13j ^11^33 "~ ^13 > ... axx^^in ~ ^13^1»
^11^34 "~ ^i2<55u> (i'wa^ — ax^ax^^ . . . ^ufl^in "~ (^\^fi\n
^ii^2n— ^i2^m) ^ii^3» ~ ^i3^in>
fluffy
(lO)
Writing this as in (6), and proceeding as in Art. 347, it
is easily seen that it becomes
(")
«.e. the discriminant of the function multiplied by ^u""''.
Hence we infer, that if the principle in question hold for
n - I variables it holds for n. But it has been shown to hold
in the eases of 3 and 4 variables, consequently it holds for
any number.
«11,
^12,
«13,
. . . axn
^12,
fl'22,
«23,
... ain
«5l3,
«23,
«33,
... azn
•
•
•
.
«in,
a%m
«3n,
. . . ttnn
Maxima and Minima.
451
"We conclude finally that the quadratic expression in n
variables is always positive whenever the series of determi-
nants
^n,
^iij a\z
a\2y a%-i
ail, «i2, «5i3
an, «22, ^23
J • • •
^13, «23, «33
an, aiz, .
^12) ^22j •
^ln> a2nj
«2^
(.2)
are all positive.
According as the number of rows in a determinant is
even or odd, the determinant is said to be one of an even or
of an odd order.
If the determinants of an even order be all positive, and
if those of an odd order, commencing with ^fn, be all negative,
the quadratic expression is negative for all real values of the
variables.
Hence we infer that the number of independent conditions
for a maximum or a minimum in the case of n variables is
n - I, as stated in Art. 163.
It is scarcely necessary to state that similar results hold
if we interchange any two of the suffix numbers ; i. e. if any
of the coefficients, ^22, «33, • • • ann, be taken instead of an as
the leading term in the series of determinants.
If the determinants in ( 1 2 j be denoted by Ai, Aa, A3, . . . A«,
it can be seen without difficulty that whenever no one of these
determinants vanishes the quadratic expression under con-
sideration may be written in the form
Ai A2
Un^
(13)
Hence, in general, when the quadratic is transformed into a
sum of squares, the number of positive squares in the sum
depends on the number of continuations of signs in the series
of determinants in (12).
It is easy to see independently/ that the series of conditions
in (12) are necessary in order that the quadratic function
under consideration should be always positive ; the preceding
2G2
452 General Conditio)) fi for Maxima or Minima.
investigation proves, however, that they are not only necessary,
but that they are sufficient.
Again, since these results hold if any two or more of the
sujfix numbers be interchanged, we get the following theorem
in the theory of numbers : that if the series of determinants
given in (12) be all positive, then every determinant obtained
from them by an interchange of the suffix numbers is also
necessarily positive.
Also, since when a quadratic expression is reduced to a
'sum of squares the number of positive and negative squares
in the sum is fixed (Salmon's Higher Algebra, Art. 162), we
infer that the number of variations of sign in any series of
determinants obtained from (12) by altering the suffix
numbers is the same as the number of variations of sign
in the series.
349. Orthogonal Transformation. — As already stated,
a quadratic expression can be transformed in an infinite
number of ways by linear transformations into the sum of
a number of squares multiplied by constant coefficients ;
there is, however, one mode that is unique, viz. what is
styled the orthogonal transformation (see Art. 341).
In this case, if Xi, X2, X3, . . . Xn denote the new linear
functions, we have
x^^-vxi + .,.^ xn^ = Xi^ + X2' + &c. f X„2 = F;
also, denoting the coefficients of the squares in the transformed
expression by ^1, <?2, • • . «n,
17= ayxXi + a-iiX^ + . . . + a^n^n + . . . + lax^X^X^ + ZaxrlCxXr + . . .
= axXx + ^2X2^ + «f„X„^
Hence, equating the discriminants of U - XV ior the two
systems, we get
^11 ~ A, ai2y . . . ttin
^i2j ^22 ~ A, ... a2n
<^13) ^23> • • • azn
^ A
^i«> ^2n>
= [ai-\){a2-X),.,{an-\).
(14)
Conditions of Maxima or Minima in General. 453
Accordingly, the coeflBcients ai, az, . . . an are the roots of
the determinant A.
Moreover, in order that the function U should be always
positive or always negative for all real values of the variables
(^1,(^2, ... Xny the coefficients «i, a-i . , . a„ must be all positive
in the former case, and all negative in the latter ; and con-
sequently, in either case, the roots of the determinants in (14)
must all have the same sign.
For a general proof that the roots of the determinant A
are always real, and also for the case when it has equal
roots, the student is referred to Williamson and Tarleton's
Dynamics^ Second Edition, Chapter XIII.
454
Miscellaneous Examples,
Miscellaneous Examples,
I. If a, )8, 7 be the roots of the cubic
x^+px'^ + qx + r = o,
show that
dp dq dr
da da da
dp dq dr
dk* dfi* dh
dp dq dr
dy* dy* dy
(7-i8)()8-a)(a-7).
2. Being given the three simultaneous equations
<p\{Xu SC2, Xz, Xi) = O, (f>2{Xi, X2, X%, xC) - O, ^z{Xi^ X^, Xz, Xi) = o,
determine the values of -— , — — , — — .
dx\ dxi dx\
d-ti
d^u
S. If X and y be not independent, prove that the equation - — r- — ^ ^
does not hold good, in general.
4. Prove that the points of intersection of a cui-ve of the fourth degree with
its asymptotes lie on a conic ; and in general for a curve of the degree n they
lie on a curve of the degree n — z.
5. Prove that every curve of the third degree is capable of being projected
into a central curve, (Chasles).
For if the harmonic polar of a point of inflexion he projected to infinity, the
point of inflexion will be projected into a centre of the projected curve {see p. 282).
6. Two ellipses having the same foci are described infinitely near one
another, how does the interval between them vary ?
{a) How will the interval vary if the ellipses be concentric, similar, and
similarly placed?
7. Eliminate the arbitrary functions from the equation z = <p{x) ^{y).
8. Show that in order to eliminate n arbitrary functions from an equation
containing two independent variables, it is, in general, requisite to proceed to
differentials of the order 2n — i. How many resulting equations would be ob-
tained in this case ?
9. In the Lemniscate r^ = a^ cos 20, show that the angle between the tan-
gent and radius vector is - + 2$.
Miscellaneous Examples,
10. If the detenninant of the n*^ order
455
X, a,
a, a
a, a
X, a
a. X
be denoted by An, prove that —— = wA„-i.
11. Prove that the ellipses
ahj^ + i%2 = ^2^2 (I) ^ ^2^2 sec* <p + bhj^ cosec* <p = a*e* (2),
are so related that the envelope of (2) for different values of 0 is the evolute of
(i) ; and the point of contact of (2) with its envelope is the centre of curvature
at the point of (i) whose excentric angle is <p.
12. Being given the equations
bx = \fiy by= V(A2-i2)(J2-;i2),
prove that
-^2 ' J2_
13. If i—y- ay^ = o, develop y^ in terms of a by Lagrange's Theorem.
14. Being given a; = r cos d, y = r sin d, transform
h(l)T
rfa:2
into a function of r and 0, where Q is taken as the independent variable.
15. Apply the method of infinitesimals to find a point such that the sum of
its distances from three given points shall be a minimum.
If p\ , p2, p3 denote the three distances, we have dpi + dpz + dp3 = o: suppose
dpi = o, then d{p2 + pz) = o, and it is easily seen that pi bisects the angle be-
tween p% and p3, and similarly for the others ; therefore &c.
16. Eliminate the circular and exponential functions from the equation
y = gsra"^*,
17. One leg of a right angle passes through a fixed point whilst its vertex
slides along a given curve, show that the problem of finding the envelope of the
other leg of the right angle may be reduced to the investigation of a locus.
45 6 Miscellaneous Examples.
1 8. If two pairs of conjugates, in a system of lines in involution, be given
by the equations
u = ax'^-it 2bxy + cy"^ = O, u' = a'x^ + 2h'xy + c'y"^ = o,
show that the double lines are given by the equation
dudu' dudu' ,„ . , n - a _i. , - >
=0. (Salmon s Comes, Art. 342 ).
dx dy dy dx
. X\ X-i Xn-\
19. If Wl = — , W2 = — , Wn-1 = ,
Xn Xn Xn
where Xi, X2, . . * Xn are connected by the relation
Xi"^ ^xz'^ + xs'^ + . . .+ Xr? = I,
prove that the Jacobian
d{Ui, »2, • • • t^n-l) _ I
d{xi, X2, . . . Xn-l) Xn'*-^'
20. If the variables yi, yz, ... y» are related to xij X2, ... Xn by the
equations
yi = aiXi+ azxz + . . . + anXn,
f/i = biXi + biXi + . . • + bnXnt
yn= hXi +hX2 + . . ,-^ InXn,
and if we have also
X,^+X2^ + ...+ Xn^ = I,
yi2 + ya^ + . . . + y«2 = i,
prove that the Jacobian
^(a;i, a;2, . . . a;n-i) a^n'
21. Prove that the equation
ry"^ — 2sxy + tx"^ =px + qy - z
d^z
may be reduced to the form — + z = o by putting x = ucoa v, y = w sin v.
22. Investigate the nature of the singular point which occurs at the origin
of coordinates in the curve
X* — 2ax^y - axy^ •{■ a-y^ = 0.
Miscellaneous Examples, 457
23. Investigate the form of the curve represented hy the equation t/ = e'"'
24. How would you ascertain whether an expression, V, involving ar, y, and z,
is a function of two linear functions of these same variables ?
Ans. The given function must be homogeneous ; and the equations
dV_ dV_ dV_
dx * dy ^ dz '
must be capable of being satisfied by the same values of x,y^z\ i. e. the result
of the elimination of a;, y, and z between these equations must vanish identi-
cally.
25. Iiy= <p (a;2), prove that
dy
dx**
~ = (2x)«^(«) (a;2) + w(w - I) (2a;)«-2 ,^(«-i) {x^
I • 2
26. If a; + ty = (a + i^S)", where * = V - i, prove that
df_+_dy^ _ 2 <fa^ + d$^
27. If tan 0 tan ,// = 7^^, prove that ^ + JLlfj^ = o.
VI - C2' ^ d^L \l^C^ Billed,
28. If a; = — , prove that
ky
dx
transforms into
V(i-a;2)(i-F-a;2) V(i -y2)(i -/fcV)'
Prove that — Ucu) = (i +a;— Jw.
dx ' ' \ dx)
29. Hence prove that
I d\( d \ ^dr-u
\ dx) \ dx ) dx*
^ I d\ I du\ I ^ d\ du du ^ d^u
458 Miscellaneous Examples,
30. Pro7e that
(4)('l-)K-0'-S-
By the preceding example we have
therefore ^^_ . . j^._ = ^ _.
31. Prove, in general, that
H)(^|-')("s-^)---(4-"+')''=^
d»u
This can he easily arrived at from the preceding hy the method of mathematica'
induction : that is, assuming that the theorem holds for any positive integer n
prove that it holds for the next higher integer (« + i), &c.
32. Find - + — „ I - J in terms of r. when r^-a^ cos 2d. Ana. —r •
"^ r dd^ \r/ f*
33. If w = {x^ + y^ + z^)i, prove that
rf*w d^u d*u d^u d^u <^*^ _^
34. If a = , ^ ,, and d> = tan-M - ) , prove that
d**z _ I . 2 . 3 . . . n. cos(w + i)^.co8»^^<ft
dx^ ~ ^~ ^ a;"+^ '
(f2n2 I . 2 . 3 . . . 2«.C0S(2W + l)</) .COS^"-^^)
d2tM-l« 1 . 2.3 . . .(2W+l)sin(2W+2)(ft.COs2>'^2<^
Miscellaneous Examples,
459
35. If w be a homogeneous function of the n<^ degree in x, y, z, and if ui, W2, M3,
denote its differential coefficients with regard to x, 1/, z, respectively, while
«ii, wi2, &c., in like manner denote its second differential coefficients, prove that
Wn, W]2, «13, «i
M2I, W22, «23> «2
W3I, W32, W33> «3
Ml, W2, W3, O
«11, «12, «13
«21, «22, «33
«31, «32, «33
36. If M be a homogeneous function of the «*'» degree in x, y, z, w, show
that for all values of the variables which satisfy the equation m = o we have
«11, M12, M13, Wj
«21, «22, W23, «2
«31, «32, W33, «3
«1, «2, «3, O
(n-i)^
Wn, «12, «13, Wl4
«21, «22, «23, «24
«31, W33» M33, M34
«41, «42, «43, «44
37. If « + A be substituted for a; in the quantic
, w (n — i)
aoa;" + naix>*-^ + — ^ azX**'^ + &c. + ««,
and if «'o, «'i, .... aV • •
quantic, prove that
denote the corresponding coefficients in the new
da'r
It is easily seen that in this case we have
r(r-i)
a'r = ar + rar.\\ + -^ <?r-2A.'+ &c. . . . + aoV; .*. &c.
1 . 2
38. If ^ be any function of the differences of the roots of the quantic in the
preceding example, prove that
fflO
d d d
:5- + 2<^^ -— + 3^2 :r- +
da\ daz das
+ nan-\
dan)
<p = 0.
This result follows immediately, since any function of the differences of tbe
roots remains unaltered when a; + A is substituted for Xy and accordingly
d(p
d\
o in this case.
39. Being given
u = «y + Vi - «« - y2 + a;2y2^ » = « Vi - y^ + y Vi - «2,
460 Miscellaneous Examples,
prove that
du dv dv du
— — sa Q
dx dy dx dy
and explain the meaning of the result,
40. Find the minimum value of
sin ^ sin ^ sin C , ^ , ■« „ « «
+ . ^ . -■ + -. — ■ . p, where ^ + i? + (7= l8o\
sin ^ sin C sin G &\n.A sin A siuB
41. Prove that
where <p {x) is a rational function of x.
42. Show that the reciprocal polar to the evolute of the ellipse
-^ + 17 = I»
with respect to the circle descrihed on the line joining the foci as diameter, has
for its equation
43. If the second term be removed from the quantio
(ao, a\, 02, . , , a,) {x, y)«
by the substitution of a: y instead of x, and if the new quantic be denoted
by {Ao, o, A2, A3, . . . An){x, y); show that the successive coefficients
A2, Az . . . An are obtained by the substitution of ai for x and — oo f or y in
the series of quantics
(rto, «i, az) {x, y\ (ao, «!, «2, ^3) (a?, y), • . . (ao, «i, . . . «n) («, y).
44. Distinguish the maxima and minima values of
I + 2a: tan-^ x
I +a;2
^, a'a;2 + 2h'x + c'
45- If y = — , prove that
^"^ ax^ + 2bx -Y c ^ ^
i^^ {ac - b^) y2 + (^ac' ^ g'c _ 2bb') y + a'g' - b"^
2 <?« (a^') a;2 - i^ca) x + (Ac') '
where {ab') = ab' - ba', &c
Miscellaneous Examples*
461
46. If IX + mY+ nZ, I'X + m'T+n'Z, l"X + m"r+ n"Z, be substituted
for X, y, z, in the quadratic expression ax^ + bip- + cz^ + 2dyz + 2<?'a; + 2/a;y ;
and if a', b', c', d', e\ f be the respective coeflS.cient3 in the new expression ;
prove that
= o whenever
/:
, d\ c\
47. If the transformation be orthogonal, i.e. if
a;3 + y2 + 22 == ^2 + j-2 + ^3^
prove that the preceding determinants are equal to one another,
48. Prove that the maximum and minimum values of the expression
ax^ + /[hx^ - dcx^ + ^dx + e
are the roots of the cubic
aV - 3 {a^I - 32-2)23 + 3 (aZ2 - i8B-/) 2 - A = O,
where E=ac-h\ I = ae- ^bd + ^d^.
fl, b, c
by Of d
c, d, e
, and A = 73 - 27/2.
By Art. 138 it is evident that the equation in z is obtained by substituting
e - z instead of e in the discriminant of the biquadratic ; accordingly, since the
discriminant of the biquadratic is
73 _ 27/2 = o,
we have for the resulting equation
(/-«»)» = 27(7- 25")2.
In general, the equation in z whose roots are the n — t maximum and mini-
mum values of a given function of n dimensions in x can be got from the dis-
criminant of the function, by substituting in it, instead of the absolute term,
the absolute term minus z.
It is evident that the discriminant of the function in a; is, in all cases, the
absolute term in the equation in z.
49. If A be the product of the squares of the differences of the roots of
afi - px^ + qx - r = 0,
462 Miscellaneous Examples,
find an expression for — - by solving from three equations of tlie form
dr
dA dA dp dA dq dA dr
da dp da dq da dr da*
Ans. 2()8 + 7-2o)(7+a-2j8) (a+/3-27).
50. If X + F V- I be a function of a; + y V- i, prove that X and Y satisfy
the equations
d^X d^X ^ d^Y d^Y
1 = o. and \- = o,
dx'' ^ dt/^ ' dx-i ^ df-
51. If the three sides of a triangle are «, a + o, a + )8, where o and j3 are
infinitesimals, find the three angles, expressed in circular measure.
TT a + )8 IT 2o-)8 TT 2)8 -a
Ans. — , -+ — , - + — .
3 as/i 3 a>/3 3 «>v/3
52. If y = a; + aa;', where a is an infinitesimal, find the order of the error in
taking x = y - ay^.
53. The sides a, h, c, of a right-angled triangle become a + o, 3 + )3, c + 7,
where o, ;8, 7 are infinitesimals ; find the change in the right angle.
. cy - aa - bfi
^. — ^ — .
54. If a cui've be given by the equations
2x = ^yW^t + v^i2 _ 2t^
2y = *yW^t - ^i^^ - 2t,
find the radius of curvature in terms of t.
55. In the curve whose equation is y = r*", determine all the cases where
the tangent is parallel to the axis of x.
If 6 be the greatest angle which any of its tangents makes with the axis of a,
prove that tanfl = J-.
56. In a curve traced on a sphere, prove the following formula for the
radius of curvature at any point :
sin rdr
tan p = — .
cos pdp
57. Apply this form to show that in a spherical ellipse we have
sin^ sinj9' = const.,
where p and p' are the perpendiculars from the foci on any great circle whii.h
touches the ellipse.
Miscellaneous Examples, 463
58. Prove the following relation between (p, p'), the radii of curvature at
corresponding points of two reciprocal polar curves :
^^ ~C0S3,|/' \
where t// is the angle between the radius vector and normal.
59. If AB, BG, CD, ... be the sides of an equilateral polygon inscribed in
any curve, and if AJD be produced to meet BCinF, prove that, when the sides
p2
of the polygon are diminished indefinitely, BF = Z—,, where p and p' are the
radii of curvature at B and at the corresponding point of the evolute.
00. If /7=V(i-^)(i+y+y') + ^(i-y)(i+^ + ^'),
dnd r = I —
'-y
i\2
dnd the value of
dx dy dx dy*
61. If r«ir" + — , and 2 = a; + -,
prove that
62. Determine h and h so that the curve
(a;2 + ip-) (a; cos o + y sin a - a) = F (a; cos )8 + y sin j3 - J)
may have a cusp ; o, )8, and a being given, and the coordinates being rectan-
gular.
Prove that in this case the cuspidal tangent makes equal angles with the
asymptote and with the line drawn from the cusp to the origin.
63. Find the coordinates of the two real finite points of inflexion on the
curve y^ = {x — 2)- {x - 5), and show that they subtend a right angle at the
double point.
64. If X, y, z, be given in terms of three new variables, u, t>, w, by the fol-
lowing equations : x = Fu, y={F-b)v, z= {F -c)w, where
_ I + Jv^ 4. cw"^ ^
"^ t<2 + t;2 + «J2 *
464 Miscellaneous Examples,
it is required to prove that dx^ + dy"^ + dz^ = LHu^ + M^dv^ + N^dto'^^ and to
determine the actual values of L, M, N.
6$. li X -{■ y = X, y = XY^ prove that
dP'U dP-u du ^d-u d^u du
'^dx^'^^dzdu'^dx^ dT'~ dXdY'^dX'
bb. Beinsr given x = u^ - ^uv^, y = xu^v — v^. find what — ; '—— becomes
\x.i^xm^oiu,v,du,dv, xdz-^ydy
udv — vdu
Ans. — -.
udu + vdv
a
67. If the polar equation of a curve be r = a sec^ -, find an expression for
its radius of curvature at any point.
dx
68. Show that the differential — — is transformed into
^x^ - sx^ + 3
Idy
-/(I + ynan2 A) (I + 2/2 eot2 A)'
by assuming x = */r — -, and find the value of \.
Ans. A. = 7* 30'.
vj. If y^ + xy= I, prove that
„ d^y dy^ dv"^
dx^ dx^ dx^
70. The pair of curves represented by the equation
,•2 _ 2rF (a>) + c- = O
may be regarded as the envelope of a series of circles whose centres lie on a
certain curve, and which cut orthogonally the circle whose radius is c, and
whose centre is the origin (Mannheim, Journal de Math., 1862).
71. A chord FQ cuts off a constant area from a given oval curve ; show that
the radius of curvature of its envelope will be ^FQ (cot 0 + cot ^), d and <p being
the angles at which FQ cuts the curve.
72. In the polar equations of two curves,
i?'(r, ft,) = o, /(r, «)=0,
if J2*»» be substituted for r, and nO. for w, prove that the curves represented by
the transformed equations intersect at the same angle as the original curves.
W. Eoberts, Liou^ille's Journal, Tome 13, p. 209.
Miscellaneous Exatnples, , 465
This result follows immediately from the property that — — is unaltered by
ar
the transformation in question.
73. A system of concentric and similarly situated equilateral hyperbolas is
cut by another such system having the same centre, under a constant angle,
which is double that under which the axes of the two systems intersect.
Ibid., p. 210.
74. In a triangle formed by three arcs of equilateral hyperbolas, having the
same centre (or by parabolas having the same focus), the sum of the angles is
equal to two right angles. Ibid., p. 210.
75. Being given two hyperbolic tangents to a conic, the arc of any third
hyperbolic tangent, which is intercepted by the two first, subtends a constant
angle at the focus. Ibid., p. 212.
An equilateral hyperbola which touches a conic, and is concentric with it, is
called a hyperbolic tangent to the conic.
76. A system of confocal cassinoids is cut orthogonally by a system of equi-
lateral hyperbolas passing through the foci and concentric with the cassinoids.
Ibid., p. 214.
The student will find a number of other remarkable theorems, deduced by
the same general method, in Mr. Roberts' Memoir. This method is an exten-
sion of the method of inversion.
77. If on each point on a curve a right line be drawn making a constant
angle with the radius vector drawn to a fixed point, prove that the envelope of
the line so drawn is a curve Avhich is similar to the negative pedal of the given
curve, taken with respect to the fixed point as pole.
78. If 2U=ax--\ 2bxy + cy"^, 2 F s a'x" + zh'xy + c'y\
^AU^+2BUV+ Cr\ find A, B, V.
and
dU dU
dx* dy
d_V dV
dx* dy
79. Prove that the values of the diameters of curvature of the curve y^=f{x)
at the points wheie it meets the axis of x are/'(o),/'(j8), .... if a, /8, ... be
the roots of /(a;) = o.
Hence find the radii of curvature of y- = [x^ - m^) {x-a) at such points.
80. A constant length FQ is measured along the tangent at any point F on
a curve ; give, by aid of Art. 290, a geometrical construction for the centre of
curvature of the locus of the point Q.
81. In same case, if FQ' be measured equal to FQ, in the opposite direction
along the tangent, prove that the point F, and the centres of curvature of the
loci of Q and Q' lie in directum.
82. A framework is formed by four rods jointed together at their extremities
prove that the distance between the middle points of either pair of opposite aides
2 H
466 Miscellaneous Examples,
is a maximum or a minimum when the other rods are parallel ; being a maximum
when the rods are uncrossed, and a minimum when they cross.
83. At each point of a closed curve are formed the rectangular hyperbola,
and the parabola, of closest contact ; show that the arc of the curve described by
the centre of the hyperbola will exceed the arc of the oval by twice the arc of
the curve described by the focus of the parabola ; provided that no parabola has
five-pointic contact with the curve. {Camh. Math. Trip., 1875.)
84. A curve rolls on a straight line : determine the nature of the motion of
one of its involutes. {Frof. Crofton.)
85. Prove the following properties of the three- cusped hypocycloid : —
(i). The segment intercepted by any two of the three branches on any
tangent to the third is of constant length. (2). The locus of the middle point
of the segment is a circle. (3). The tangents to these iM-anches at its extremities
intersect at right angles on the inscribed circle. (4). The normals corresponding
to the three tangents intersect in a common point, which lies on the circum-
scribed circle.
Definition. — The right line joining the feet of the perpendiculars drawn to
the sides of a triangle, from any point on its circumscribed cii'cle, is called the
pedal line of the triangle relative to the point.
86. Prove that the envelope of the pedal line of a triangle is a three-cusped
hypocycloid, having its centre at the centre of the nine-point circle of the
triangle. (Steiner, Ueber cine bcsondere curve dritter Masse und vierten grades,
Crelle, 1857.)
This is called Steiner^s Envelope, and the theorem can be demonstiated,
geometrically, as follows : —
Let Pbe any point on the cin umscribcd circle of a triangle ABC, of which D
is the intersection of the perpendiculars ; then it can be shown without difficulty
that the pedal line corresponding to I' passes through the middle point of JDF.
Let Q denote this middle point, then Q lies on the nine-point circle of the
triangle ABC. If 0 be the centre of the nine-point cii'cle, it is easily seen that,
as Q moves round the circle, the angular motion of the pedal line is half that of
OQ, and takes place in the opposite direction. Let R be the other point in
which the pedal line cuts the nine-point circle, and, by drawing a consecutive
position of the moving line, it can be seen immediately that the corresponding
point T on the envelope is obtained by taking QT= QR. Hence it can be
readily shown that the locus of T is a three-cusped hypocycloid.
This can also be easily proved otherwise by the method of Art. 295 (a).
87. The envelope of the tangent at the vertex of a parabola which touches
three given lines is a three-cusped hypocycloid.
88. The envelope of the parabola is the same hypocycloid.
For fuller information on Steiner's envelope, and the general properties of
the three-cusped hypocycloid, the student is referred, amongst other memoirs, to
Cremona, Crelle, 1865. Town send, Educ. Times. Reprint. 1866. Ferrers,
Qaar. Jour, of Math., 1866. Serret, Nouv. Ann., 1870. Painvin, ibid.^ 1870
Cahen, ibid., I075.
On the Failure of Taylor^ s Tfieorem. 467
On the Fatlitre op Taylor's Theorem.
As no mention has been made in Chapter III. of the cases when Taylor's
Series becomes inapplicable, or what is usually called the failure of Taylor's
Theorem, the following extract from M. Navier's Zegons d' Analyse is intro-
duced for the purpose of elucidating this case : —
On the Case Trhen^ for certain particular Values of the
Variable, Taylor's Series does not give the DeTelopment of
the Function. — The existence of Taylor's Series supposes that the function
f{x) and its differential coeflScients/'(a;),/"(a:), &c. do not become infinite for
the value of x from which the increment h is counted. If the contrary takes
place the series will be inapplicable.
Suppose, for example, that /(a;) is of the form - — ^— ^-j w being any positive
number, and F{x) a function of x which does not become either zero or infinite
when X = a.
If, conformably to our rules, - — ^^-r — be developed in a series of posi-
tive powers of h, all the terms would become infinite when we make x = a. At
the same time the function has then a determinate value, viz. : — ^ ; but,
A»»
as the development of this value according to powers of h must necessarily con-
tain negative powers of h, it cannot be given by Taylor's Series.
Taylor's Series naturally gives indeterminate results when, the proposed
function f{x) containing radicals, the particular value attributed to x causes
these radicals to disappear in the function and in its differential coeflScients.
In order to understand the reason, we remark that a radical of the form
p_
{x — a)?, p and g denoting whole numbers, which forms part of a function /(a:),
gives to this function q different values, real or imaginary. As this same radical
is reproduced in the differential coefficients of the function, these coefficients also
present a number, q, of values. But, if the particular value a be attributed to x,
the radical will disappear from all the terms of the series, while it remains
p
always in the function, where it becomes M. Therefore the series no longer re-
presents the function, because the latter has many values, while the series can
have but one. The analysis solves this contradiction by giving infinite values
to the terms of the series, which consequently does not any longer represent a
determined result.
The development oif{x) ought, in the case with which we are occupied, to
p
contain terms of the form h^. We should obtain the development by making
a; = a + A in the proposed f unction.
468 On the Failure of Taylor's Theorem.
Fractional powers of h would appear in the latter development : for example,
suppose
f{x) = 2ax - 3;2 + a'/x^ - a»;
tliis giyes
ax
f{^) = 2{a-x)+ ■;
y'a;2 - a-
On making a; = «, we have /(a;) = a^y and all the differential coefficients
become infinite. This circumstance indicates that the development oif{x + h)
ought to contain fractional powers of h when a; = a : in fact the function be-
comes then
f{a + A) = «« _ A2 + a\/2ah + A%
of which the development according to powers of h would contain A^, A^, A^, &c.
It should be remarked that a radical contained in the f unction/(a;) may-
disappear in two different ways when a particular value is attributed to the
variable x ; that is, i°, when the quantity contained under the radical vanishes :
2°, when a factor with which the radical may be affected vanishes.
In the foi-mer case the development according to Taylor's Theorem can never
agree with the function /(a; + A) for the particular value of a; in question, for
the reason already indicated.
But it is not the same in the latter case, because the factor with which the
radical is affected, and which becomes zero in the function, may cease to affect
the radical in the differential coefficients of higher orders ; in fact it may not
disappear at aU, and the series may in consequence present the necessary number
of values.
For example, let the proposed function be
/(a;)as(a;-a)»»Va;-^
m being a positive integer.
Here we have
in{x-~a)''^
f{x) = m{x - o)*"-! -^x-h + 2^^— y'
r{x) = m(m - i)(a:-. a)-2 v/J3j + ^i^^T'^, _ {x^a)M^
•slx-b 4(:c-J)i
Each differentiation causes one of the factors of {x — a)"» to disappear in the
first term. After m differentiations these factors would entirely disappear ; and
consequently the supposition x — a, in causing the first m-derived functions to
vanish, will leave the radical Vx - d to remain in all the others.
INDEX
ACNODE, 259.
Approximations, 42.
further trigonometrical applica-
tions of, 130-8.
Arbogast's method of derivations, 88.
Arc of plane curve, differential ex-
pressions for, 220, 223.
Archimedes, spiral of, 301, 303.
Asymptotes, definition of, 242, 249.
method of finding, 242, 245.
number of, 243.
parallel, 247,
of cubic, 249, 325.
in polar coordinates, 250.
circular, 252.
Bernoulli's numbers, 93.
series, 70.
Bertrand, on limits of Taylor's series,
77.
Bobillier's theorem, 368, 374.
Boole, on transformation of coordi-
nates, 412.
Brigg's logarithmic system, 26.
Bumside, on covariants, 412.
Cardioid, 297, 372.
Cartesian oval, or Cartesian, 233, 375.
third focus, 376.
tangent to, 379.
confocals intersect orthogonally,
381.
Casey, on new form of tangential
equation, 339.
on cycloid, 373.
on Cartesians, 382.
Cassini, oval of, 233, 333.
Catenary, 288, 321.
Cayley, 259, 266.
Centre of curve, 237.
Centrode, 363.
Change of single independent variable,
399.
Change of two independent variables,
403, 410.
Chasles, on envelope of a carried right
line, 356.
construction for centre of instan-
taneous rotation, 359.
generalization of method of draw-
ing normals to a roulette, 360.
on epicycloids, 373.
on Cartesian oval, 376.
on cubics, 454.
Circle of inflexions in motion of a plane
area, 354, 358, 367, 374-
Complete Solid Harmonics, 42 1 .
Conchoid of Nicomedes, 332, 361.
centre of curvature of, 370.
Concomitant functions, 411.
Condition that Pdx + Qdy should be a
total differential, 146.
Conjugate points, 259.
Contact, different orders of, 304.
Convexity and concavity, 278.
Crofton on Cartesian oval,^378, 379,
380.
Crunode, 259.
Cubics, 262, 281, 323, 334.
Curvature, radius of, 286, 287, 295,
297, 301.
chord of, 296.
at a double point, 310.
at a cusp, 311, 313.
measure of, on a surface, 209.
Cusps, 259, 266, 315.
curvature at, 311.
Cycloid, 335, 356.
equation of, 335, 336.
radius of curvature, and evolute,
length of arc, 338.
Descartes, on normal to a roulette, 336.
ovals of, 375.
Differential coefficients, definition, 5.
successive, 34.
470
Index.
Diflferentiation of, a product, 13, 14.
a quotient, 15.
a power, 16, 17.
a function of a function, 17.
an inverse function, 18.
trigonometrical functions, 19, 20.
circular functions, 21, 22.
logarithm, 25.
exponential functions, 26.
functions of two variables, 115.
three or more variables, 117.
an implicit function, 120.
partial, 113, 406.
of a function of two variables,
"5-
of three or more variables,
115.
applications in plane trigono-
metry, 130.
in spherical trigonometry,
133-
successive, 144.
of </) (a; + at, y + ^t) with respect
to t, 148.
Discriminant of a ternary quadratic
expression, 129, 194, 196.
of any quadric, 449.
Double points, 258, 261.
Elimination, of constants, 384.
of transcendental functions, 386.
of arbitrary functions, 387, 396.
Envelope, 270.
of Lo? + iMa + JV = o, 272.
of a system of confocal conies,
Ex. 8, p. 276.
of a earned curve, 355.
centre of curvature of, 357.
Epicyclics, 363.
are epi- or hypo-trochoids, 366.
Epicycloids and hypocycloids, 339,
356, 466.
radius of curvature of, 342.
cusps in, 341.
double generation of, 343.
evolute of, 344.
length of arc, 345.
pedal, 346, 372.
regarded as envelope, 347.
Epitrochoids and hypotrochoids, 347.
ellipse as a case of, 348, 363.
centre of curvature of, 351.
double generation of, 367.
Equation of, tangent to a plane curve,
212, 218.
normal, 215.
Errors in trigonometrical observation,
135.
Euler, formulae for sin x and cos x, 69.
theorem on homogeneous func-
tions, 123, 127, 148, 162.
on double generation of epi- and
hypo-cycloids, 344.
Evolute, 297.
of parabola, 298-
of ellipse, 299, 308 ; as an enve-
lope, 297.
of equiangular spiral, 300.
Expansion of a function, by Taylor's
series, 6r.
by Arbogast's method, 88.
oi (l>[x + h, y 4-^), 156.
oi (p{x-^ h, y^k, z + l), 159.
Family of curves, 270.
Ferrers, on Bobillier's theorem, 369.
on Steiner's envelope, 466.
Folium of Descartes, 333.
Functions, elementary forms of, 2.
continuous, 3.
derived, 3.
successive, 34.
examples of, 46.
partial derived, 113.
elliptic, illustrations of, 136, 138.
Graves, on a new form of tangential
equation, 339.
Harmonic polar of point of inflexion
on a cubic, 281.
Huygens, approximation to length of
circular arc, 66.
Hyperbolic branches of a curve, 246.
Hypocycloid, see Epicycloid.
Hypotrochoid, see Epitrochoid.
Indeterminate forms, 96.
treated algebraically, 96-9.
treated by the calculus, 99, etseq.
Infinitesimals, orders of, 36.
geometrical illustration, 57.
Inflexion, points of, 279, 281.
in polar coordinates, 303.
Intrinsic equation of a curve, 304.
of a cycloid, 338.
of an epicycloid, 350.
of the involute of a circle, 301.
Index.
471
Inverse curves, 225.
tangent to, 225.
radius of curvatui-e, 295.
conjugate Cartesians, as, 378.
Involute, 297.
of circle, 300, 358, 374.
of cycloid, 356.
of epicycloid, 357.
Jacobians, 433-45-
;"agrange, on derived functions, ^,note.
on limits of Taylor's series, 76.
on addition of elliptic integrals,
136.
theorem on expansion in series,
151-
on Euler's theorem, 163.
condition for maxima and minima,
191, 197, I99» 202.
La Hire, circle of inflexions, 354.
on cycloid, 373.
Landen's transformation in elliptic
functions, 133.
Laplace's theorem on expansion in
series, 154.
coefficients, 429.
Legendre, on elliptic functions, 137.
on rectification of curves, 233.
coefficients of, 426.
Leibnitz, on the fundamental principle
of the calculus, 40.
. theorem on the n^^ derived func-
tion of a product, 5 1 .
on tangents to curves in vectorial
coordinates, 234.
Lemniscate, 259, 277, 296, 329, 333.
Lima9on, is inverse to a conic, 227,
331, 334> 349, 361, 372.
Limiting ratios, algebraic illustration
of, S.
trigonometrical illustration, 7.
Limits, fundamental principles, 11.
Maclaurin, series, 65, 81.
on harmonic polar for a cubic, 282.
Manheim, construction for axes of an
ellipse,^ 374.
Maxima or minima, 164.
geometrical examples, 164, L83.
" " 3, 166. X
algebraic examples,
ax^ + zlxy + cy^
01 — ; ,
a a;2 + 2b' xy + c'y"^
condition for, 169, 174
166, 177.
problem on area of section of a
right cone, 181^
for implicit functions, 185.
quadrilateral of given sides, 186.
for two variables, 191 ; Lagrange's
condition, 191, 197.
for functions of three variables,
198.
of n variables, 199, 449.
application to surfaces, 200.
undetermined multipliers applied
to, 204.
M\iltiple points on curves, 256, 265,
. 367.
Multipliers, method of imdetermined,
204.
Napier, logarithmic system, 25.
Navier, geometrical illustration of
fundamental principles of the
calculus, 8.
on Taylor's theorem, 467.
Xewton, definition of fluxion, 10.
prime and ultimate ratios, 40.
expansions of sin Xy cos a;, sin-^,
&c., 64, 69.
by differential equations, 85.
method of investigating radius of
curvature, 291.
on evolute of epicycloid, 345.
Nicomedes, conchoid of, 332.
Node, 259.
Normal, equation of, 215.
number passing through a given
point, 220,
in vectorial coordinates, 233.
Orthogonal transformations, 409, 414,
452.
Osc-node, 259,
Osculating curves, 309.
circle, 291, 306.
conic, 317.
Oscul-inflexion, point of, 314, 317.
Parabola, of the third degree, 262, 288.
osculating, 318.
Parabolic branches of a curve, 246.
Parameter, 270.
Partial differentiation, 113, 406.
Pascal, lima^on of, 227.
Pedal, 227.
tangent to, 227.
examples of, 230.
negative, 227.
472
Index.
Pliicker, on locus of cusps of cubics
having given asymptotes, 265.
Points, de rebroussement, 266.
of inflexion, 279.
Polar conic of a point, 219.
Proctor, definition of epi- and hypo-
cycloids, 399-
epicyclics, 366.
Ptolemy, epicyclics, 366.
Quetelet, on Cartesian oval, 376, 381-
Radius of^curvature,'286.
in Cartesian coordinates, 287, 289.
in r, j3 coordinates, 295.
in polar coordinates, 301.
at singular points, 310.
of envelope of a moving right
line, 358.
Reauleaux, on centrodes ot moving
areas, 363.
Reciprocal polars, 228, 230.
Remainder in series, Taylor's, 76, 79.
Maclaurin's, 81.
Resultant of concurrent lines, 234.
Roberts, W., extension of method of
inversion, 464.
Rotation, of a plane area, 359.
centre of instantanous, 360, 364.
of a rigid body, 371.
Roulettes, 335.
normal to, 336.
centre of cui-vature, 352; Sa-
vary's construction, 352.
circle of inflexions of, 354.
motion of a plane figure reduced
to, 362.
spherical, 370.
Savary's construction for centre of
* curvature of roulette, 353.
Series, Taylor's, 61, 70, 76.
binomial, 63, 82.
logarithmic, 63, 82.
for sin x and cos x, 64, 66, 81.
Maclaurin's, 64, 81.
exponential, 65, 8r.
Bernoulli's, 70.
convergent and divergent, 72, 75.
for sin-' a;, 68, 85.
for tan-^a;, 68, 84.
for sin mx and cos inx, 87.
Arbogast's, 88.
Lagi-ange's, 151.
Solid Harmonic functions, 419.
Spherical Harmonics, 423.
Spinode, 259.
Stationary, points, 266.
tangents, 282.
Subtangent and subnormal, 215.
polar, 223.
Symbols, separation of, 53.
representation of Taylor's theo-
rem by, 70, 160.
Tacnode, 266.
Tangent to curve, 212, 218, 258.
number through a point, 219.
expression for perpendicular on,
217, 224.
expression for intercept on, 232.
Taylor's series, 61.
symbolic form of, 70,
Lagrange on limits of, 76.
extension to two variables, 156.
to three variables, 159.
symbolic form of, 160.
on inapplicability of, 467.
Tesseral Surface Harmonics, 429.
Three-cusped hypocycloid, 350, 372,
466.
Tracing of curves, 322, 328.
Transformations, linear, 408.
orthogonal, 409, 452.
Trisectrix, 332.
Trochoids, 339.
Ultimate intersection, locus of, 271.
for consecutive normals, 290.
Undetermined multipliers, application
to maxima and minima, 204.
applied to envelope, 273.
Undulation, points of, 280.
Variables, dependent and indepen-
dent, I .
Variations of elements of a triangle,
plane, 130; spherical, 133.
Vectorial coordinates, 233.
"Whewell, on intrinsic equation, 304.
Zonal Harmonics, 423.
THE END.
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to find a ready welcome among
schools which decide for themselves
how much geometry they shall teach.
I believe it covers entirely sufficient
ground for the average student of
geometry. I especially commend
the sound sense which does not
hesitate to put things in the form in
which they are found elsewhere,
instead of straining after new (and
luorse) methods."
Prof, Arthur E. Haynes, Uni-
versity of Minnesota: — " The book
is one of much merit, and presents
the subject to the beginner in a very
interesting way."
L. L. Jackson, State Normal
School, Ypsilanti, Mich. : — "I believe
it to be the best geometry for the
07'dinary high school that has yet
been brought out. I base this on
the quantity of matter, its import-
ance, the simple and yet sufficiently
rigid treatment of the same."
W. T. Reid, Belmont School,
Belmont, Cal. : — "His idea of making
the book exactly coincides with mine.
In fact, only a few days ago I took
the geometry we are using and
checked over a certain number of
propositions that were to be mas-
tered, following in my checking
exactly the theory that Mr. Gore has
been following in making his book.
I have again and again felt that we
were wasting a good deal of time in
useless detail. Certainly the essen-
tial thing is to get the main princi-
ples of geometry thoroughly well
fixed in the mind, and so clearly
before the boy that he can make
use of them in his after work."
Longmans, Green, &- Go's Publications.
Nichols— Elementary and Constructional Geometry.
By Edgar H. Nichols, A.B., of the Browne & Nichols School, Cam-
bridge, Mass. i2mo. 150 pages. $0.75*
*^*This book is prepared mainly with reference to the reconn-
mendations of the National Committee of Ten, and is designed for
Pupils beginning Geometry at the age of twelve, or even younger.
It is based upon the author's class-room experience with young
boys during the last twelve years.
A pupil who has acquired a familiar knowledge of the principles
developed in this book should be able to take up the study of
Theoretical Geometry, both plane and solid, in a text-book where
no complete proofs, but suggestions only, are given to aid in the
solution of the more difficult problems and theorems.
Journal of Pedagogy, Bingham-
ton, N. Y.: — " It is superior to any
other book that we have seen de-
signed for the same grade of
pupils."
Prof. B. F. Chase, Central High
School, Kansas City, Mo,: — " I am
much pleased with Prof. Nichols'
book. His treatment of Moulding
of Areas, p. 79, appears to be espe-
cially strong." [Adopted later for
use in the Manual Training High
School of Kansas City.]
Emily P. Wolcott, Girls' Latin
School, Baltimore: — "It contains
among its many good features, some
things that none of the other books
of its kind possess."
W. A. Francis, Phillips (Exeter)
Academy: — " I am well pleased with
M. Nichols' book. It is the best
that I have seen."
School Review, Chicago, 111. : —
"New ideas are revealed, not by
mere statement, but by judicious
questioning. Most of the principles
the student is led to formulate for
himself. Many of the points which
are difficult for the child to under-
stand— as the subjects of angles, of
equivalent figures, and of areas — are
explained with more than usual sim-
plicity and clearness."
Casey — A Treatise of the Analytical Geometry of the
Point, Line, Circle, and Conic Sections.
Containing an Account of its Most Recent Extensions. By John
Casey, LL.D., F.R.S., Fellow of the Royal University of Ireland,
etc. (Dublin University Press Series.) i2mo. $3.50*
Casey — The First Six Books of the Elements of Euclid,
and Propositions I.-XXI. of Book XL, and an Appendix of the
Cylinder, Sphere, Cone, etc. With Copious Annotations and Numerous
Exercises. By John Casey, LL.D., F.R.S. Sixth edition, revised
and enlarged. i6mo. 332 pages. $1.40*
Casey — A Sequel to the First Six Books of the Elements
of Euclid.
Containing an easy Introduction to Modern Geometry. With Numerous
Examples. By John Casey, LL.D., F.R.S. Seventh edition, revised
and enlarged. Parti. Crown 8vo. 168 pages. $1.10*
Longmans, Green, & Go's Publications. 7
Low — Text-Book on Practical, Solid, and Descriptive
Geometry.
By David Allan Low (Whitworth Scholar), Principal of the People's
Palace Technical School, London. Author of " An Introduction to
Machine Drawing and Design."
Part I. Crown 8vo. With 114 Figures. 118 pages. $0.60*
Part II. Crown Bvo, With 64 Figures. 140 pages. $0.90*
Farif I. Treats of Projection of Points and Lines, Simple Solids
in Simple Positions, Changing the Planes of Projection, Additional
Problems on Lines, Planes other than the Co-ordinate Planes, Prob-
lems on the Straight Line and Plane, Sections of Solids, Projection
of Plane Figures, etc.
Part II. Additional Problems on the Straight Line and Plane,
Projection of Solids, Isometric Projection, Horizontal Projection,
Curved Surfaces and Tangent Planes, Developments and Projections
of Screw Threads, Intersection of Surfaces, Projection of Shadows,
etc.
Morris — Practical Plane and Solid Geometry.
By I. Hammono Morris, South Kensington Art Department. Fully
Illustrated with Drawings done specially for the Book by the Author.
(LoNGMi^Ns'' Elementary Science Manuals.) i2mo. 264 pages.
$0.80*
The Volume treats of: i. Construction and Use of Plain Scales
and Scales of Chords. 2. Proportional Division of Lines. 3. Mean
and 4th Proportional. 4. Lines and Circles required in drawing
out Geometrical Patterns. 5. Reduction and Enlargement of plane
Figures. 6. Polygons on Lines and in Circles. 7. Irregular Poly-
gons. 8. Irregular Figures : the Ellipse, etc. 9. Plan, Elevation,
and Section of Cube, Pyramid, Prism, Cylinder, Cone, and Sphere
in Simple Positions.
Solid Geometry. The Principles of Projection. Definition of
Terms, etc., etc. Simple Problems relating to Lines and Planes.
Plan and Elevation of Simple Solids resting on the Horizontal Plane,
and also when Inclination of Two Sides or of Plane and One Side
are given. Sections of such Solids by Vertical and Horizontal
Planes.
Graphic Arithmetic. The Representation of Numbers by Lines.
The Multiplication of Numbers by Construction. The Division of
Numbers by Construction. The Determination of the Square Root
of Numbers by Construction.
Morris — Geometrical Drawing for Art Students.
Embracing Plane Geometry and its Application, the Use of Scales, and
the Plans and Elevations of Solids. With nearly 600 Figures. By
I. Hammond Morris. Crown 8vo. 192 pages. $0.60
8 Longmans, • Green, &- Go's Publications.
Henrici— Elementary Geometry.
Congruent Figures. By Olaus Henrici, Ph.D., F.R.S., Professor
of Pure Mathematics in University College, London. Second edition.
141 Diagrams. (London Science Class Books.) i6mo. 210 pages.
$0.50*
Taylor — An Introduction to the Differential and Integral
Calculus and Differential Equations.
By F. Glanville Taylor, M.A., B.Sc, Mathematical Lecturer at
University College, Nottingham. Crown 8vo. 592 pages. $3.00*
Salmon — A Treatise on Conic Sections.
Containing an Account of some of the Most Important Modern Alge-
braic and Geometric Methods. By G. Salmon, D.D., F.R.S. 8vo.
416 pages. $3.75*
Smith — Geometrical Conic Sections.
By J. Hamblin Smith, M.A., of Gonville and Caius College, Cam-
bridge. i2mo. 172 pages. $1.10*
Sutherland — Primer of Geometry.
By James Sutherland, M. A, Crown 8vo. 116 pages. $0.75*
This little book is an attempt to give a First Course in Geometry,
Mensuration, and Measurements that shall be of real value as an
Educational Training, and that shall be the foundation for more
advanced work in the higher parts of the school. It gives practical
applications of the early Propositions of Euclid's First Book to
actual measurements, and endeavors to make an interesting study
of what is generally looked upon by the schoolboy as the least
interesting part of his work. The exercises are such as can easily
be worked in any school having a playground, or available piece of
land, and the necessary apparatus is of the simplest description, in
fact, most of it can be made by the pupils themselves.
Watson— ^Plane and Solid Geometry.
By the Rev. H. W. Watson, M.A., formerly Fellow of Trinity College,
Cambridge, (Text-Books of Science.) i2mo. 308 pages. $1.25
Contents : Book I. Triangles, Angles, Parallel Straight Lines,
Polygons, and Loci. Book II. On the Circle. Book III. Problems
of Construction connected with the Straight Line and Circle. Book
IV. On Areas. Book V. Ratio, Proportion, etc. Book VI. Ap-
plication of Proportion to Geometry. Book VII. On Planes, and
Lines in Space.
Wilson — Geometrical Drawing.
For the Use of Candidates for Army Examinations, and as an Intro-
duction to Mechanical Drawing. By W. N. WiLSON, M.A., Master at
Rugby School, Crown 8vo. 160 pages. $1.35*
Taylor — An Introduction to the Practical Use of Log-
arithms, with Examples in Mensuration.
By F. Glanville Taylor, M.A., B.Sc, Mathematical Lecturer at
University College, Nottingham. Crown 8vo. $0.50*
14 DAY USE
)RROWED
STATISTICS USkARY
This book is due on the last date stamped below, or
on the date to which renewed.
Renewed books aj;)»«if5jeh; to immediate recall.
SEI^ONILL
APR 2 6 2001
U. C. BERKELEY
LD 21-40m-10,'65
(F7763sl0)476
General Library
University of California
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