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LIBRARY
UNIVERSITY OF CALIFORNIA.
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AN ELEMENTAEY TEEATISE ON THE
DIFFERENTIAL AND INTEGRAL CALCULUS.
CambrftfQc :
Prtnteti at t|je SSntfacrsitg ^rcss.
FOR MACMILLAN AND CO.
aonOon: GEORGE BELL.
IBublin: HODGES AND SMITH.
©Hinbutgf) : EDMONSTON AND DOUGLAS.
©lasfloto: JAMES MACLEHOSE.
©ifOtB: J. H. PARKER.
AN ELEMENTARY TREATISE ON
THE DIFFERENTIAL AND
INTEGRAL CALCULUS,
FOR THE USE OF COLLEGES AND SCHOOLS.
BY G. W. HEMMING, M.A.,
FELLOW OF ST. JOHN's COLLEGE, CAMBKIDQE.
SECOND EDITION,
WITH CORRECTIONS AND ADDITIONS.
CambritiQe :
MACMILLAN & CO.
1852.
>* Of THB
wUHIVBRSITTj
*^g/"^-^ zing
library
5T7/3
PREFACE TO THE SECOND EDITION.
Many corrections and additions have been made in
the present Edition, chiefly with the view of fitting it for
younger students by bringing out the leading principles
of the subject with greater clearness. Some additional
illustrative examples are given, but for purposes of practice
the student is still referred to Gregory's Collection of
Examples.
Lately Published, hy the same Author, 12mo. cloth, 45. Qd.
First Book on Plane Trigonometry, comprising Geometrical Tri-
gonometry, and its Application to Surveying, with numerous
Examples for the use of Schools.
ijITIVEESIT
CONTENTS.
CHAPTER I.
Variables and Functions.
Article Page
1 — 7. Definitions of variables and functions . . . . 1
8. Equations of one or more independent variables . . 7
CHAPTER H.
Limiting Values of Functions.
9. Definition and explanation of a limiting value . , .8
10. Axioms respecting limiting values . . . ii
arc PQ
''' ^'^^"cho^dPQ^^ 11
12. /^.=o^" = l 12
X
x" — 1
13. lt^=o- — T=« ....... 12
X - I
1
14. lt,=^{l + xf=e 13
^^' '-^ X loga 1^
16. /^^o— r— = logfl ...... 14
CHAPTER in.
Differentiation.
17. Definition of differentials and differential coefficients . .15
18. Differentials proportional to the rates of increase of the variables 17
1 9. Geometrical illustration of the definition . . .17
20 — 44. Differentiation of elementar}' functions . . . 19
45 — 51. Differentiation of a function of a function . . .27
52. Examples ....... 30
53. Summary of results . . . . . .31
Vlll CONTENTS.
CHAPTER IV.
Integration.
Article Page
54. Definition of integration . . . . .34
56, 57. Addition of a constant in integration ... 35
58 — 65. Integration of elementary functions . . .36
66. Summary of results ..... 40
67. Integration by algebraical transformations . . .41
68, 69. Integration by parts ..... 42
70 — 76. Integration of rational fractions . . . .43
78. Rationalization ...... 49
79,80. Criteria of integrability of a;" (a + ii;")^ c?a; . . .49
81 — 87. Integration by redviction . . . . 51
88. Exceptional cases . . . . . .56
89. Integration of sin"' x cos" xdx in particvilar cases . 57
90, 91. Integration of the same fimction by reduction • . 57
CHAPTER V.
Successive Differentiation.
92. Definition of an independent variable . . .60
93. Geometrical illustration . . . . . 61
94, 95. Successive diflferentiation . . . . .62
96 — 98. Relations between successive differentials and differential coeffi-
cients, when the independent ^'ariable is general . 63
99. Form of the above relations when the quantity (a?), under the
functional sign, is independent variable . - 64
100. Examples ..... 65
101 — 104. Formation of differential equations . . 67
105. Homogeneity of differential equations ... 70
106. Change of independent variable . . . . 70
107. To pass from an equation among differentials, with x as inde-
pendent variable, to one among differential coefficients
with respect to x, and the converse ... 70
108, 109. To pass from a general independent variable to a', and the converse 7]
110. To pass from a general independent variable to any function of a? 71
111 — 113. To change the independent variable from x to any function ofx 72
114.. To change the independent variable from x to y . 74
115, 116. Illustrations and examples ..... 74
CHAPTER VI.
Differentiation of Functions of several Variables and
Implicit Functions.
117, 118. The total differential of a function of two connected variables
equal to the sum of the partial differentials . . 78
CONTENTS.
IX
Article Page
119—121. Examples ....... 80
122. Extension to functions of any number of variables . 82
123 — 125. Implicit functions of one independent vaiiable . . 82
126 — 128. Relations between corresponding implicit and explicit functions 84
129. Examples . . . , . . .86
130 — 132. Definition of the total differential of a function of two inde-
pendent variables . . . . . 87
133, 134. Another form of the definition . . . .92
135. Geometrical illustration ..... 94
136. Successive total and partial differentiation . . .95
137. Order of successive partial differentiation immaterial . 96
138. Notation for successive differentials and differential coefficients 97
139 — 142. Relations between successive total and partial differentials
and differential coefficients, with general independent variables 99
143. The same relations, with particular independent variables 101
144 — 146. Change of independent variable in equations involving partial
differential coefficients ..... 101
CHAPTER VII.
Development of Functions.
147 — 149. Taylor's theorem. Limits of the remainder. Examples 104
150, 151. Stirling's theorem. Examples . . . .107
152, 153. Failure of Taylor's theorem .... 108
154. Other methods of expansion . . . .110
155. Extension of Taylor's theorem to functions of two variables 111
15fi. d''f(^x,y) _ d'^f{x,y) ^^^
dx'^'dy' dy'dx^~'
157. Limits of remainder of the above series . . . 112
158,159. Other forms of Taylor's theorem . . . 113
160. Lagrange's theorem . . . . .113
161. Laplace's theorem . . . . . 115
162. Laplace's theorem deduced from Lagrange's . . 116
Maxima and Minima
CHAPTER VIII.
Limiting values of Indeterminate Functions,
values of Functions.
163. Fractions of the form J; . . . . 117
164,165. Fractions of the form -§■ . . . . .117
166. Other indeteraiinate forms . . , . 119
167 — 1 71 . Maxima and Minima of Functions of one independent variable.
Geometrical illustration . . . . .119
172 — 177. Maxima and Minima of Functions of two or more independent
variables. Geometrical illustration . . . 123
178. Examples . . . . . . .129
CONTENTS.
CHAPTER IX.
Tangents and Asymptotes.
Article
Page
179.
Definition of a tangent ....
133
BO, 181.
Equations to tangent and normal, dx^ + dy'^ = ds^
133
182.
Length of tangent and normal
135
183.
Concavity and convexity of cm^^es
135
184.
Points of inflexion .....
136
185.
Asymptotes ......
136
186.
Examples ......
137
187.
Inclination of tangent to radius vector, dr- + r'dd' = ds^ .
140
188.
Values of ^T and SF .....
141
189.
Asymptotes of polar curves ....
143
190.
Concavity and convexity of polar curves. Points of inflexion
143
CHAPTER X.
Contact of Curves.
191, 192. Definition of contact. Curves whose contact is of a higher
order, indefinitely closer than those whose contact is of a lower
order ...... 144
193. Curves whose contact is of an even order cut, otherwise not 145
194. Conditions of contact modified . . . 145
195. Curve of closest contact . . . . .146
196 Circle of curvature . . . . . 147
197. Radius of curvature and coordinates of centre of curvature 147
198,199. Various forms of these expressions . . . 149
200. Circle of curvature has a contact of third order when the radius
is a maximum or minimum . . , .151
201. Equation to evolute . . . . . 152
202,203. Properties of evolute . . . . .153
204. Evolute of polar curves . . . . 156
CHAPTER XI
Singidar Points.
205.
Definitions
.
206—208.
209.
210.
211.
212.
Multiple points
Cusps . . . .
Conjugate points
Other singular points
Summary. Example .
157
157
161
162
164
164
CONTENTS.
XI
Article
213, 214.
215.
216.
Explicit Curves
Implicit Curves
Polar Curves
CHAPTER XII.
Tracing of Curves.
Page
Kit)
167
170
CHAPTER XIII.
Envelopes — Differentials of Areas, S)-e.
217,218. Envelope of a group of curves .
219. Ultimate intersection of contiguous normals
dr
2^-0- ^ = 'Tp
221. Differential of an area
222. Differential of a sectorial area
223. Differential of the volume of a solid of revolution
224. Differential of the surface of a solid of revolution
172
175
175
176
176
177
177
CHAPTER XIV.
Integration between Limits. Successive Integration.
225. Integral the limit of the sum of a series
226. Area of a curve ....
227- Length of a cun^e
228. Sectorial area of a curve
229. Volmne and surface of a solid of revolution
230. The same results differently obtained
231. Further application of Art. 225
232 — 238. Successive integration. Examples
239. Integral expressed by a series .
240. Remainder of Taylor's series as a definite integral
178
180
181
182
182
182
184
185
192
ral . 192
DIFFERENTIAL AND INTEGRAL CALCULUS.
CHAPTER L
VARIABLES AND FUNCTIONS.
1. In all analytical inquiries the symbols with which we are
concerned are divisible into two classes — those which by the con-
ditions of the problem admit of variation, and those which are in-
capable of it. The former are called variables, the latter constants.
Thus, the co-ordinates of any point of a given curve are variables,
being capable of a change of value in passing from one point of the
curve to another, while those quantities which enter into the equa-
tion to determine the particular curve of which we treat, as the
latus rectum of a parabola, the semi-axes of an ellipse, the radius of
a circle, &c., are constants so long as the equation is restricted to
one particular curve.
It should however be borne in mind, that the same quantities
may appear in one problem as variables and in another as constants.
Thus, if X and 7/ represent the co-ordinates of any point in a given
circle, they are connected by the equation
where x and 1/ are variables and r constant. But if we have to
determine the locus of the intersection of pairs of equal circles,
having their centres in two fixed points whose distance is 2b, the
equations to any pair of circles whose radii equal r may be put in
the form
(x - by + ?/' = r^
(x + by + y^ = r'.
The equation to the required locus will then be a relation between
X and r/, which holds for every possible value of r and will be found
from the above equations by treating r as a variable. The distance
2i, between the centres is the same for every pair of circles. In this
problem therefore x, y and r are variables and h constant. By
H. D. c. \
Z VARIABLES AND F€NCTIO.NS.
eliminating r between them we find the equation to the locus
required, viz. j: = 0.
Hence we obtain the following definitions.
A variable is a quantity which, by the conditions of the problem
that we are investigating, may assume a number of different values.
A constant is a quantity which the conditions of the problem
suppose to be invariable.
Quantities, which in some problems appear as variables and in
others as constants, as the radius of the circle in the above example,
are sometimes termed parameters.
2. A continuous variable is one which is capable of receiving
any value whatever ; a discontinuous variable, one which only ad-
mits of a particular series of values separated by finite intervals.
Thus, the co-ordinates of an indefinite straight line are continuous
variables, while a symbol which represents any integer is a discon-
tinuous variable.
In the ordinary operations of Algebra and in the present subject,
all the variables employed are supposed to be continuous.
Results, however, obtained on the supposition of continuity are
applicable to variables which are continuous within certain finite
limits, and become discontinuous beyond them. Thus, the general
equation to a straight line is applicable to the side of a triangle,
provided we restrict the values of the co-ordinates to the limits
imposed by the figure.
Variables are usually represented by the final letters of the
alphabet, as u, w, x, y, z, &c., and constants by the initial letters a,
h, c, &c. ; a notation purely arbitrary, and adopted for convenience.
3. A function of any variable is a quantity whose value depends
upon and changes with that of the variable, so that when an arbi-
trary value is given to the vai-iable the function receives a definite
corresponding value.
Thus, the distance which a man walks in a given time is a func-
tion of his rate of walking ; and the algebraical expressions x"", a",
a + bx", sin x, &c., are all functions of x, and x a function of each of
them. So also, the co-ordinates of a curve are mutually functions
of each other, since when an arbitrary value is given to one of them,
the other receives a definite corresponding value.
VARIABLKrf AND i-UiXCTIOXS. 3
Functions of the variables x, y, &c. are writteny(j:), f{y), (p (t),
When the same functional sign is prefixed to different variables,
it denotes that the function is obtained from the variables in the
same way. Thus, if /(x) = sin x, f{z), if it occurs in the same in-
vestigation, must = sin 2.
4. A function of a continuous variable is said to be continuous
when its values are obtained from those of the variable hy a Jixed
law, such that a gradual change in the variable produces a corre-
sponding gradual change in the function, otherwise it is discon-
tinuous. The meaning of these two conditions of continuity becomes
more apparent by the aid of a geometrical illustration.
Suppose a curve to be traced whose equation is y =zf(x). Such
a curve must always exist whatever be the form ofy, since we can
fix as many points of it as we please by giving a series of arbitrary
values to x, and determining from the equation the corresponding
values o^ y. In order that the ordinate of this curve may be a con-
tinuous function of the abscissa, according to the above definition,
two conditions must be satisfied :
(1) The change of j/ consequent on a gradual change of x, must
be gradual.
(2) It must follow a fixed law.
The former of these conditions excludes the case of a series of
disconnected points, or of lines terminating abruptly, and restricts
us to curves which may be traced without raising the pen from the
paper. Where however y has several values corresponding to each
value oi X, the condition is satisfied if each branch may be thus con-
tinuously drawn. The latter condition excludes from the class of
continuous curves, figures composed partly of one curve and partly
of another, as for instance a triangle, the co-ordinates of whose
several sides are related by three distinct laws or equations. It is
even possible to trace curves which satisfy the geometrical idea of
continuity whose ordinates are nevertheless discontinuous functions
of the abscissae in the sense of our analytical definition.
A curve is considered continuous in geometry where it can be
traced by the motion of a point, without any abrupt changes of
direction (Newt. Prin. Bk. i. Lem. 5). A triangle is therefore geo-
1—2
'4 VARIABLES AND FUNCTIONS.
metrically discontinuous, but the egg-shaped curve formed by con-
necting the extremities of a semi-circle with those of a semi-ellipse
of equal diameter satisfies the geometrical definition. The equation
to the latter figure however does not give ?/ as a continuous function
of X, since for some values of the abscissa it follows the law of the
circle and for others that of the elhpse. But the equations to the
distinct parts of such composite figures are continuous, and results
obtained by treating them as such may be applied to the geometrical
figure if the values given to the variables are confined within the
limits appropriate to the particular equation employed. Thus in
the last example the results deduced from the continuous equation
to the circle are true of the semi-circular portion of the figure.
In the above instances of discontinuity the change in value of
the function was gradual, but the law was not fixed. In other cases
the law remains fixed, but there is an abrupt change in value at
particular points. Discontinuity of the latter kind occurs most fre-
quently in the form of a sudden change from + co to -co. Thus
for example, when x passes through the value - , sec x changes
abruptly from + co to - co , the law by which the value of the func-
tion is ascertained remaining the same on both sides of the point
of discontinuity. Geometrical examples are found in many curves
having asymptotes parallel to the axes of co-ordinates. Thus the
equation to the hyperbola referred to its asymptotes is xi/ = c*, and
when X passes through zero, 1/ suddenly changes from -co to + co .
In such cases results obtained from the equation common to both
portions of the curve, on the hypothesis of continuity are applicable
to all points (however near to those where the discontinuity occurs),
except the actual points of discontinuity themselves. After reading
the following chapter, the student will be able to see that such
general results may be said to have an application in the limit to
these points of isolated discontinuity themselves.
There are other forms of discontinuity, but the ordinary alge-
braical functions as ax"', sin a, log a;, are continuous except for par-
ticular values of the variables. In all the propositions which follow,
the continuity of the functions treated of will be assumed, except
when the contrary is expressly stated. Our results, therefore, cannot
be applied, without a separate investigation, to discontinuous func-
tions except within their continuous limits.
loff I iv . sin I
VARIABLES AND FUNCTIONS. 5
5. A function of two ov more variables is a quantity whose
value depends upon and changes with the values of those variables,
so that when arbitrary values are given to the variables, the func-
tion receives a definite corresponding value. Thus, the distance
•which a man walks is a function of his rate of walking, and of the
time in which he performs the distance, and the algebraical ex-
pressions {x+yY,a^, sin (-j are functions of x and y, and
-©}
a function of the three variables x, jj, and z.
Functions of two or more variables are represented by the sym-
bols F(x, y), (p(x, y, z), f{x, y, z, u), &c. Thus the equation
z = F(x, y) signifies that s is a function of the two variables x and y.
Those who are familiar with geometry of three dimensions, will
recognize an illustration of such a function in the ordinate of a
surface.
6. In the equation F {x, y) = 0, x and y are not functions of
two variables, but each is simply a function of the other, since on
giving an arbitrary value to either, the other receives a definite cor-
responding value. Thus if x and y are co-ordinates of a circle, y is
a function of x alone, whether the equation be written in the form
y = Ja^ — x^ or x" + y' = a'.
And in general if any two variables x and y are connected in any
manner, so that the value of either depends on that of the other (as
is the case with the co-ordinates of a curve), this relation may be
expressed in either of the following forms,
(1) F{x,y)=0, (2) y^f{x), (3) x = 4>{y),
■where the forms of the functions F,f, <p, depend upon the nature of
the law connecting the variables.
When this connexion is expressed by an equation of the form
(1), each variable is said to be given as an hnplicii function oi the
other; in equation (2), y is said to be given as an explicit function
of a:, and in equation (3), x as an explicit function ofy.
So also if we have an equation between three variables, of the
form
F (x, y, z) - 0,
b. VARIABLES AND FUNCTIONS.
each variable is given as an implicit function of the other two, the
corresponding explicit functions being of the form
and similar remarks apply to functions of any number of variables.
It is evident that the forms of the functions^J (p, yp-, depend on
that of F in any particular case^ and in many instances the methods
of Algebra enable us to determine them from it. This process of
expressing one variable as an explicit function of the others with
which it is connected by any equation, is called solving the equation
with respect to that variable. Thus, for instance, if ^, j/ are the
co-ordinates of a point in an ellipse referred to the centre as origin,
they are connected by the equation
from which we derive the equivalent equations
giving y and x respectively as explicit functions of the other.
7. Although it follows from our definitions that a constant
cannot be a function of a variable, yet it will be found subsequently
that many of the propositions which will be established with respect
to functions, will apply to constants as a particular case, in a manner
exactly analogous to that in which the properties of symbols of mag-
nitude are extended to zero, although the symbol of no magnitude
at all.
8. If in any equation between two variables, as (1), (2), or
(3), we give to either variable an arbitrary value, the other assumes
a definite value dependent on the former. For this reason such an
equation is said to be an equation of one independent variable, since
we are at liberty to give an independent or arbitrary value to one
variable only.
If three variables are connected by a single equation, and we
give an arbitrary value to one of them, there exists an indefinite
number of pairs of values of the other two which, together with the
arbitrary value of the first, satisfy the equation ; but if a second
variable is arbitrarily determined, the third assumes a definite value
dependent on the arbitrary values of the other two. Such an equa-
VARIABLES AND FUNCTIONS. 7
tion is therefore called an equation of two independent variables ;
and generally, if any number (??) of variables are connected by a
single equation, there are (« — 1) independent variables.
If three variables are connected by two equations, as f{x, y, z)
— 0, and <b (^, y, z) = 0, and an arbitrary value is given to one of
thera, the values of the other two become definite, and there is
therefore only one independent variable ; and generally, if n varia-
bles are connected by r equations, and we give arbitrary values to
(re — r) of them, we obtain r equations among the remaining r varia-
bles, which are therefore determinate. In this case, therefore, the
number of independent variables is («— r).
CHAPTER 11.
LIMITING VALUES OF FUNCTIONS.
9. Let f{x) be any function of a variable x, a any particular
value which we may give to the variable.
Then if by giving to x a value sufficiently near to a, we can
make /{x) differ from some other value A hy a, quantity less than
any assignable quantity, A is called the limiting value of f{x)
when X approaches a, or, as it may be concisely written,
A ^ It ^ fix).
It will easily be seen^ that if /(«) is the actual value of /(.r)
whenx = a, this limiting value A will equal /(a) ; for /(a-) ~/(«)
may clearly be made as small as we please by giving to a? a value
sufficiently near to a. Hence it follows, that in general the defini-
tion above given of the limiting value of a function, when the
variable approaches a particular value, is nothing else than the
actual value of the function when the variable already equals that
value.
Thus, for example, let f{x) be the number of miles which a
person, walking at the rate of 4 miles an hour, can walk in x
hours, whence we have/(,r) -4j'. When the time is taken equal to
2 hours suppose, the distance will evidently be 8 miles. This then
is the actual value of /(a-) when x = 2. But this is also the limiting
value of /(a;) when x approaches 2; for the more nearly the time
approaches 2 hours, the more nearly does the distance approach
8 miles ; and we may, by diminishing the former difference, dimi-
nish the latter as much as we please. Hence, by our definition, 8
miles is the limiting value of /(.r) when x approaches 2 hours.
The question then naturally arises — What is the use of the some-
what complex definition of a limiting value of a function, when it is
in no respect different from the actual value of which the idea is so
much more obvious ? Why speak of the value that f{x) approaches
when X approaches a, when we might so much more simply speak of
the value which /(x) has when x has the value a} The answer
to this is, that there frequently are pai'ticular values of the variable
LIMITING VALUES OP FUNCTIONS. 9
for which we are unable to determine the actual corresponding value
of the function, and yet can determine its corresponding limiting
value. And it is for such values, and for no others, that the idea of
a limiting value is resorted to.
In the example given above, there is evidently no value of x
for which we cannot assign the corresponding value of /(x). If
the time is given, we can at once determine the actual distance
walked. In this and similar cases, therefore, although the idea of
a limiting value is as admissible as in any- other, we shall always
confine ourselves to the simpler idea of an actual value.
The following example will illustrate the necessity and use of
the idea of a limit.
Let /(x) = ^^^ (1).
If we give to x any value diiFerent from a, as 6, we obtain
the actual value of the above function, viz.
/(^) = T^ = ^-^'^ -■(^^'
If, however, we give to x the value a, and endeavour to obtain
the value of /(a), we find /(«)=-, an expression which has no
meaning whatever, and is perfectly indeterminate ; /(x), therefore,
has no actual value when x=^a. Nor can the difficulty be obviated
by saying, that because
•^ ^ ■' X — a
.'. f(x) = x + a,
and therefore, when x = a, /(a) = 2a;
for in the above process we have first divided by (x—a), and then
put the divisor =0, a proceeding which is repugnant to the first
principles of Algebra, and by which any fallacy might be esta-
blished. Thus we might on the same grounds assert that, because
x' — a^ = X - a, when x —a,
.'. x + a-\, when x^a,
.-. 2a = 1, whatever a is,
a manifest absurdity, arising from our having divided by a quan-
tity which is the representative of zero. The above function, there-
fore, has no actual value when x = a.
10
LIMITING VALUES OF FUNCTIONS.
But since equation (2) is evidently true, however near h is to
a, provided it does not actually equal it, it follows that we may
make f{x) differ as little as we please from 2a, by making x
sufficiently near to a. Za is therefore the limiting value of/(x)
when X approaches a.
It is so important that the reader should have a perfectly clear
idea of a limiting value, that we shall add the following geome-
trical illustration.
Let PQ, be any curve, PN, NQ lines drawn parallel to two
rectangular axes ; then, if the nature of the
curve is known, NQ and PN will be func-
tions one of another.
Let then QN= <f> {PN),
.•.t.nNPQ = t(m,
We can therefore in general determine
the value of tan NPQ for any assigned value
of PN ; but if PN equals zero, QN or
(p (PN) also vanishes ; and tan NPQ being
the ratio of two lines, each of which is equal to zero, has no deter-
minate actual value.
Since however, when (p is known, we can obtain an actual value
of tan NPQ for any assigned value of PN different from zero, how-
ever small, we are able to determine the value to which tan NPQ
approaches when PN approaches zero, i. e. the limiting value of the
tangent.
Let r be a point in any line MT which is drawn perpendi-
cular to PN produced, so taken that
TM
^ = Zfp^., (tan iVPQ);
then PT is evidently the line to which the chord PQ continually
approaches as Q approaches P, or, in other words, PT is the limit-
ing position of the secant PQ when Q approaches P. This is
called the tangent to the curve at the point P ; and it will be seen
that the investigation of the inclination of tangents is one of the most
important applications of the Differential Calculus.
LIMITING VALUES OF FUNCTIONS. 11
10. From the preceding explanations the truth of the following
axioms becomes evident.
(1) If a function f(^x) has an actual determinate value when
X = a,
lt^f{.x)=f{a).
(2) If an equation is true for all values of the variable up to
the limiting values, it is true in the limit ; that is, if the equation
f{x) = <p{x) is true for all values of j:, however near to a.
If one of the functions, as (p{x), has a determinate actual value
when x=-a, this equation becomes, by axiom (1),
and if both/(j:) and (p{x) have actual values when x = a, it reduces
itself to the self-evident form
/(«) = </>(«)•
We shall now proceed to determine the limiting values of certain
quantities which will be required hereafter.
11. If PQ is a curve of continued curvature,
arc PQ _
"^•^o chord pa "
A curve of continued curvature is one in which the exterior
angle between two tangents at P and Q gradually diminishes, and
ultimately vanishes when Q moves up to and ultimately coincides
with P.
Let then tangents at the points P, Q, of
such a curve intersect in T, and draw TN
perpendicular to the chord PQ.. Then -^
chord PQ = PT. cos NPT+ QT . cos NQT.
The angles NPT, NQT are always less than the exterior angle
between the tangents ; and therefore when Q moves up to P, each
of these angles vanishes, and each of the above cosines becomes
equal to unity;
PT+QT J PT . cos NPT+ QT. cos NQ T
•'• "''<^=» chord PQ " ^«=» chord PQ
= 1 , by tlie above equation.
12
LIMITING VALUES OF FUNCTIONS.
But the chord PQ, the arc PQ, and the broken line PT+QT,
are in order of magnitude always^ and therefore in the limit ;
. arc PQ
" "^^=» chord PQ"-*'
12. IL
\, where x is the circular measure of the angle.
Let PQ be the arc of a circle whose centre is C ; draw CNA
perpendicular to the chord PCl, and let the angle PCQ ■— 2x : whence
we have
PN=^ PQ, AP = ^ arc PQ, and ACP = x,
r
sin X _ NP chord PQ
~V ^~AP^ arc PQ '
chord PQ
'"'=' arc PQ
It ^^^ = It
Hence
33. //.
1
= 1.
, tan X J sin x
X X
= 1,
sm X - , x ,
^^0 = 1> a"d cos (0) = 1.
x-1
Whether ?i is integral or fractional, positive or negative, it may
be expressed in the form 3 ^ where p, q, and r are positive
integers ;
x"-l a; " -1
x — 1 X — 1
z^"-' - 1
if s' — j:
zi z'-l
LIMITING VALUES OF FUNCTIONS. 13
Therefore, by dividing numerator and denominator by -s-1, we
obtain
x"-l _ 1 (z''-' + 2^-^+ ... + 1) - (zi-' + z''-" + ... + 1)
l^'I^ fJ-' + z'-^+ ... + 1
Now, when x=l, 2 = 1, and the actual vahie of the right-hand
side of the above equation, and therefore the limiting value of the
other side, is - — ^ or «.
r
IL
x~i
14. //f^o (1 + ^y = ^> where e is the base of the Napierian system
of logarithms.
Assume - = 7i. Then
X
lt^,{\ + x'f=K^^(i+l^\
Also by the binomial theorem, we have
1 n.n-1 1
1 + 71 . - + ^ ^ . -5 + &C.
n 1.2 ir
When 71 becomes = <x) , the part of this series which involves
negative powers of n vanishes, and the series becomes
1 + 1 + — — + — - — + &c.
1.2 1.2.3
1
.-. ll^^{\ + xy = e.
15 n log,(l+:r) 1
li>. U^, ^ _log,e-
loff a
For ^-^?^^^=\og^{l^x)%
X
= log„e by (14)
1
r^ =
log a
14 LIMITING VALUES OF FUNCTIONS.
16. K^^=\op;a.
Let a'' -1 = 3,
.: x = \oga{l + z)
and when x=0, z also vanishes,
a'- 1
.-. lu, =K
= log«, by (15).
The reader will observe, that none of the above quantities have
determinate actual values for the particular values assigned to x.
CHAPTER III.
DIFFERENTIATION.
17- Let X and y be two variables connected by the equation
F{x,7/) = (1),
and let the equivalent explicit equations be
y=fi.^) (2), ^ = 0(y) (3).
Let X and y, .r + Aa: and y + Aj/, be any two pairs of values of
the variables which satisfy the above equations. The quantities A,r,
Ay are called respectively the increments of x and y. Then, em-
ploying equation (2), we have
y + ^y =/(^ + Ao:) ;
■whence Ai/=f(x + Ax)-f(x) (4),
If we suppose x and y to remain unaltered, while Ax and Ay
vary, equation (4) gives Ay as an explicit function of Ax.
In the same manner we might . have obtained, from equation
(3), the equation
Ax = (p{y + Ay)-(p(y) (5),
giving Ax as an explicit function of A^, an equation which is
obviously equivalent to (4). From the equations (4) or (5) we can
. Ay
express the ratio -^- as a function of Ax or Ay. This ratio will
generally change in value as Ax and Ay diminish, and since Aa-
and Ay vanish together, it will ultimately assume the indetermi-
nate form - . Since, therefore, the above ratio ceases in this case
to have a determinate actual value, we must have recourse to the
idea of a limiting value, as explained in the previous Chapter. Let
then (/x, dy be any two quantities whose ratio is equal to the limit-
1 6 DIFFERENTIATION.
ing ratio of the increments Ax, Ay, when these increments approach
zero, so that
■^ = lt ~~ when A.r and A^ approach zero,
dx and dy are called the differentials of x and y.
From equation (4) we have
, A?/ - . , ^ , „ (f(x + Ax)-f(x)]*
It —^ when Ax and Ay approach zero = «a=o \ jr- — ^^^^ \ •
This limit must be a function of x dependent on the form of (/), and
is therefore written /'(a:). Being the coefficient by which the diff'er-
ential of x must be multiplied to give that of y, it is termed the
differential coefficient of y with respect to x. Hence we have
dy=f'{x)dx, and/(^) = Z/Ayi^^-^\
From equation (5) we might have obtained, in a similar manner,
(p'{ij) being called the differential coefficient of x with respect to y, and
being obviously the reciprocal off'(x).
The object of the differential calculus is to investigate the ratio
of the differentials, or, what is the same thing, the value of the differ-
ential coefficients of variables connected by equations of different
forms. From the preceding explanations the following definitions
become intelligible.
Def. When two variables x and y, connected by any equation,
receive corresponding increments, any two quantities whose ratio
equals the limiting ratio of these increments when they approach
zero are called the differentials of the variables ; and the coefficient
by which the differential of x must be multiplied to give that of^, is
called the differential coefficient of y with respect to x ; and its reci-
procal, by which the diff'erential ofy must be multiplied to give that
of X, is called the differential coefficient of x with respect toy.
As differentials are defined merely by their ratio to one another,
their actual magnitude is perfectly arbitrary; this, however, does
* By?<A=o is meant the limit when all the increments approach zero. In the above
case since Ax and Ay necessarily vanish together, it is the same thing as ltA;,=o^ or
DIFFERENTIATION.
17
not render an equation involving differentials indeterminate, since
their relative magnitude is definite, and since, from the nature of the
definition, it is impossible for a differential to appear as a factor on
one side of an equation without another connected with it appearing
on the other.
18. It is sometimes convenient to view the differential coefficient
under the following aspect.
If X and 1/ were so related that Ax and Ay were always in a
constant ratio (as is the case when x and y are the co-ordinates of
a straight line), the ratio -^ would be the measure of the relative
rates of increase of the variables. When, however. Ax and Aij do
not preserve a constant ratio, the relative rates of increase of the
variables will be properly measured by the limiting ratio of these in-
crements. Hence the differentials of the variables are proportional
to their respective rates of increase, and the differential coefficient of
y with respect to x measures the relative rates of increase of y
and x.
19. As has been already observed, every equation between two
variables may be regarded as the equation to a curve, and every
proposition founded on such an equation admits of a geometrical in-
terpretation. The following geometrical illustration will aid the
reader's conception of the definitions of Art. 17.
Let PQiB be the curve corres- y
ponding to the equations (l) (2) ^
and (3) of Art. 17.
Since X, y and x + Ax, y 4 Ay
are taken to be values of the va-
riables which satisfy the equations,
they must be the co-ordinates of
two points in the curve. Let these
be P and Q so that we have
OM = x, ON = x + Ax, MN^Ax,
Om = MP=^y, On = NQ = y + Ay, mn = QU=Ay.
Let Pr be the tangent at P, T any point in it.
Now suppose Q to move up towards P. As it does so, the ratio
Ay Q.U . , TV
'Ax °'' PIJ "^*^°™®s more and more nearly equal to the trigono-
metrical tangent of the angle which PT makes with the axis of.r.
H. D. C. ^
T/
y
ii
p
/"
a
/
II
M
] 8 DIFFERENTIATION.
QU
When Q reaches P, QU and PU both vanish, and the ratio ^^
ceases to have any actual value^ after having approached indefinitely
TF
near to pT>- • Hence
TV and PV therefore satisfy the definitions of dy and dx.
Since this is equally true at whatever point of the tangent T is
taken, TV and PV may have any magnitude we choose to assign to
them, the only restriction imposed by the figure being that they shall
be lines parallel to the axes and bounded by a point in the tangent
to the curve.
The differentials therefore are not indefinitely small (as they are
sometimes represented) but of a purely arbitrary magnitude, their
relative magnitude only being fixed.
Their ratios, that is, the respective differential coefficients are
evidently given by the equations,
fix) = ^= tan TPV= cot TPv,
PV
<P'(y) = ^ = cot TPV= tan TPv,
that is,f'(x), <p\y) are the tangents of the angles which the tangent
of the curve makes with the axes of x and y respectively.
The geometrical meaning of Art. 18, is equally clear. The
assertion that dx and dy are proportional to the respective rates of
increase of x and y merely amounts to this, that TTand PV measure
those rates of increase at the point P, the increase of y being pro-
portionally greater as the inclination of the tangent to the axis of x,
increases, and vice versa. The truth of this is apparent from the
figure.
In the case we have taken, x and y are increasing together at the
point P, and all the quantities
A* {PU) Ay (QU) dx (PV) and dy (TV),
and the tangent of the inclination of PT to
the axis of x are positive. If an increase of
X had corresponded to a decrease of y, the
figure would have been as below in which dx
and dy have opposite signs, and the tangent
of the inclination of PT to the axis of x is negative.
DIFFERENTIATION. 1 9
Hence, fix) is positive or negative according as an increase of x
produces an increase or decrease of ^.
20. The remainder of this Chapter will be devoted to the deter-
mination of the ratio of the differentials of variables connected by
equations which can be reduced to an explicit form. The treatment
of equations in an implicit form will be deferred to a subsequent
Chapter.
It may be remarked here, that the process of finding the differ-
ential or differential coefficient of any function is called differentiating
the function.
21. Let y = Ui + U2+ +?/„,
Ui, tiz ?/„ being all functions of x.
Let X receive an increment Ax, and let the corresponding incre-
ments of J/, u^ w„, be A?/, Aif, A?/„. Then
i/ + At/ = Ui + Aui + 7<j + Awg +n„ + Au„,
.'. At/ =Aui + Au2+ Au„;
Ay _ Au^ Au^ Au„
" Ax~ AF "^ Ax "^ A^'
which, being always true, is true in the limit.
di/ dti, dito die,,
dx dx dx dx '
or dy = dui + du^ + . . . dii„ ;
that is, the differential and differential coefficient of the sum of a
number of functions are equal to the sum of the differentials and
differential coefficients respectively of the several functions.
22. Let y =f(x) + C, where C is independent of x and y.
Then, if Ay, A/'(x) be the increments of y and f(x), corresponding
to an increment Ax of x,
since the change of x into x + Ax does not alter the value of C,
. Ay ^ Af{x) ^
"Ax Ax '
therefore, in the limit,
dji _ d/(x)
dx dx '
or dy^dfix).
2—2
20 DIFFERENTIATION.
Since we should have had by (21), if C had been a function o£ x,
dy = df{x) + dC,
this result may be expressed by saying that the differential of a con-
stant is zero.
23. Let y = Ui. ti2,
M, and Mg being functions of x. Then, with the same notation as
before,
y + Ay = («, + Am,) (w^ + Am^),
. •, Ay = w, A?/2 + U2 Aui + Am, Amj,
Am Am, Am, Am, .
Ax 'Ax * Ax Ax *
When we approach the limit by making the increments approach
zero, the last term becomes -r^.ll.Auz', which (since ->- is gene-
rally finite, and UAu^^ 0) equals zero,
dti duo du,
' ' dx ' rfx ^ dx '
or dy = til du^ + u^ dtii .
24. This rule may be extended to the case •where y is the pro-
duct of any number of functions, of x. Thus, let
y = Ui.U3. . .ii„.i.u„.
Then, by (23), dy = dui.{u2. . .u„} + 7iid{TC2. . .u„},
d {m, . . . M„} dui rf {mj . . . «„}
M, . . . M„ «, «2 • • • "« '
£? {Mj . . . M„} JMj J {Wg . . . M„}
U3. . .U„ M2 ?<3 . . . M„
By substituting this value in the former equation, and continually
repeating the process, we obtain
d (m, . . . u,,} du, diia dit„
M,. . .M„ M, Mg *'«
a theorem of which the result of the preceding article is a par-
ticular case.
25. If ?/ = Cf{x)^ where C is independent of x and y,
Ay = CAf{x),
DIFFERENTIATION. 21
' ' Ax Ax '
which being true in the limit, we have
dx dx '
or dy = Cdf(x) ;
that is, if a function of a; is multiplied by a constant, its differential
and differential coefficient are multiplied by the same constant.
26. Letv = -'.
With the usual notation, we shall have
. 7/j + Am,
.'. Ay =
Us + Alls
>
til + Am,
Us + Aus
u,
>
Us
MbAm, -
UiAu.2
U2 (Ms +
Aus) '
Am,
""^Ax-
A«2
- ''' A:.
Ax Us {us + Amj)
Therefore, in the limit, we have
dui diis
dy ^ dx ' dx
dx u^
Usdui — Uidus
dy =
u„.
27. Let y = j:", n being any number whatever.
Then Ay = {x + Ax)" - x",
Ay _(x + Ax)" — x"
' ' Ax~ Ax
_ n-l \ X )
22 DIFFERENTIATION.
Let 1 + — =2. Then when Ax approaches 0, 2 approaches 1 ;
dx ^~' z-\
The value of this Hmit has been found in (13), and =n,
dy n-X
••• ^="^ '
or dy = n x"~^ dx.
From the results of this and former articles, we have, if
7/ = a + bx + ex- + px",
dy — [b + 2cx + + n^o;""'} dx.
28. Let y = log^ x.
Then Ay = log„ {x + Ax) - log^ x ;
. x + Ax
Ax Ax
1 log„{l +z} .„ Ax
X Z X
And when Ax approaches zero, z approaches zero;
. ^y -^ It log« (^ + ^)
dx X z
This limit has been found in (15), and =
log a '
%_ 1
dx xloga'
1 dx
dy
log; a X
Hence, if «/ = log x, this becomes
, dx
log X meaning the Napierian logarithm of x.
29. Let y = a".
Then A^ = a'+^"-fl';
Ay , a^^ - I
' Ax Ax '
a' It
d.v~ '^^ Ax
A=o
DIFFERENTIATION. 23
The value of this limit has been found in (l6), and = log a.
.-. -j^=loffa.rt',
or dy = log a . a'dx.
Hence if y=^^>
dy = e'dx.
30. Let y - sin x.
Then A?/ = sin {x + Aa;) - sin x
= 2 cos {x + ^ Aa;) sin 5 Aj; ;
... ^=cos(x+iA^).'^^|^,
• ^-^^^ cosfx+^Ax) '^"^^''
and //a^o cos (x + | Ax) = cos x, and /<a=o ^T^^ = 1» ^y (12),
dy
.'. -r = cos X,
ax
or «?^ = cos X rfx.
31. Let J/ = cos X.
Then A_y = cos (x + Ax) - cos x
= - 2 sin (x + 5 Ax) sin 5 Ax ;
... ^ = -sin(x+^Ax)^^||^,
dy „ • / 1 A ^ sin jAx
•*• ^=-^^A=oSin(x + 5Ax)-j^^,
sm /\,y
and //a=o sin (x + | Ax) = sin x, and Z^a^o ^^ = 1» by (1 2),
dy
.'. -r- = — sin X,
ax
or dy = — sin ardx.
32. Let ^ = sec x.
Then A_y = sec (x + Ax) - sec x
1 1_
~ cos (x + Ax) cos X *
2 sin (x 4- ^ Ax) sin ^ Ax ^
~ cos X . cos (x + Ax)
24 DIFFERENTIATION.
A?/ sin(a; + ^A;r) sin | At
Ax cos X . cos {x + Ax) ' 5 Ax '
»\n(x + ^Ax) sin J? , ,, sinAAo: , , ^
and //a=o i -2 J—- = , and U^_o -^^ ^1, by (12),
cos or . cos (or + Ao;) cos^ a; ~° ^Ax ' J ^ J>
dy sin x
dx cos^ X '
y sin X
or dy = ^— . dx.
"^ cos X
33. Let y = cosec x.
Then Ay — cosec (,r + Ax) — cosec ^r
sin {x + Aj) sin X
_ 2 cos (j; + |Ar) sin ^Ax
sin j: . sin {x + Ax) '
Ay _ _^ cos (x + ^Ax) sin 5 Ax
Ax sin X . sin (x + Ax) ' 5 Ax '
, ,, cos (r + 5 Ax) cos X , , sin ^ Ax
and ^^A=o -^ \ , .\ = -^T- , and Ia_. — 5-?-^= 1 by (12),
sm X sin (x + Ax) sin* x -" lAx "J K'^^J>
dy cos X
•*• rfx"~ihP^'
J cos X
sill X
34. Let y = tan x.
Then Ay = tan (x + Ax) - tan x
_ sin (x + Ax) sin x
cos (x + Ax) cos X
sin Ax
cos X . cos (x + Ax) '
Ay 1 sin Ax
Ax cos X . cos (x + Ax) * Ax '
1 ij 1 1 17, sin Ax
^"^ ^''='> cos(x-fAx) = c3^' ^"^ ^'^=^-A^=^^ ^y (^2)'
dy_
dx
= sec X,
dy = sec* x dx.
DIFFERENTIATION. 25
35. Let y = cot x.
Then Ay = cot (x + ^x) - cot x
_ cos {x + Aa;) cos a;
sin {x + Ax) sin x
sin Ax
sin X . sin (x + Ax) '
Ay _ 1 sin Ax
Ax sin ^ sin (x + Ax) Ax '
and U^^—L-^^ =^J_ and lt^_^%^= 1, by (12),
sin (x + Ax) sin x Ax •> j \ j'
• . jT = — cosec" X,
or dy= — cosec^ x dx.
36, Let y = sin~' ^.
Then x = sin _y,
/. dx = cos y cify, by (30),
.•. dy = ■ dx;
"^ cos 7/
or, expressing the differential coefficient in terms of x,
37. If it is required to find the differential of the inverse sine
without assuming that of the direct sine, we must proceed as
follows :
y = sin~' X,
.'. sin y=-x,
.: sin (y + Ay) - sin y = Ax,
.-. 2 cos {y + ^Ay) sin ^Ay = Ax,
, I . . sin \Ay Ax
.'. cos (y + 5 Aw)— ,^ >'- = —-.
and Itt^ cos (y + \Ay) = cos y, and It^^^}^^^ = 1, by (12),
dx
26. DIFFERENTIATION.
Since the steps in the determination of the differential of the
inverse function are precisely the same as for the direct function, it
is needless to repeat the independent proof for cos"' or, sec-'x, &c.,
as the reader will be able to supply it without diflBculty when re-
quired. We shall therefore deduce the differentials of these func-
tions from those already found.
38. Let y = cos~* x.
Then x = cos y,
.'. dx=-s\y\ydy^ by (31),
.*. dti = — -. — dx,
"^ sm y
*=~7a^)
dx.
39. Let y = sec~^ x.
Then x = sec y,
.'. dx = ^^dy, by (32),
, cos^ V , 1 7
.-. dy = ■—. — - dx = dx,
^ sm y tan y sec y
40. Let y = cosec"^ x.
Then x = cosec y,
sin^y , 1 J
.-. du = dx= 7 -— dx,
' ^ cos y cot y cosec y
41. 'Lety = tan~^x.
Then x = tan y,
.'. dx = sec'y dy, by (34),
.'. dy = ■ — 5— dx,
^ sec* y
or
% = Y
1
+ x'
dx.
DIFFERENTIATION. 27
42. Let I) = cor' X.
Then x — cot y^
.'. dx = — cosec^^ di/, by {35),
.'. dy = s— dx,
cosec y
43. Let 3/ = versin"' x.
Then a: = versin y = 1 — cos y,
.-. dx = smydy, by (31),
.*. (/?/ = -; dx,
"^ sm y
and sin^ = ^(l - cos-» = ^(1 - (1 - xj] - J{9.x - x%
•'. dy = — r— ^ dx.
44. Let y = suversin"' x.
Then a; — suversin j/ = 1 + cos r/,
.-. dx= -siny dy, by (28),
.•. dij = : dx,
and sin y = Ji} - cos^ y) = J{i-{x- 1)'] = J(2x - x'),
.'. dy = 7— rr dx.
^ J{2x - x^)
45. We have now shewn how to differentiate all the simple
functions which ordinarily occur in analysis, as well as any func-
tion formed by the addition or multiplication of any number of
them. It remains to differentiate functions compounded of these
simple functions, such as log sin x, d' '°s ^j"^ (.qs (a + bx)", &c. All such
functions are included in the general forms /{«/> (a;)}, /Q0 {\j/^ (•»)}]>
&c., where/, <p, yp-, &c. denote simple functions already differentiated.
The following theorem will enable us to differentiate such compound
functions.
46. Let !/=f{^(^)}-
Denote (x) by w, so that the above equation gives
u = (p {x),
28 DIFFERENTIATION.
Let Ax, Au, Ay be corresponding increments of te, u, y, all of
■which evidently vanish together.
If <Zm is the differential of?/ corresponding to the differential dx
of X, it is determined by the equation
du = 0' {pc) dx ;
and if % is the differential of y corresponding to du, it is determined
by the equation
dy =f (m) du.
And, substituting the above value of du, we have
dy=f{u)(}>'{x)dx,
which gives the required relation between dy and dx,/'(ii) and ^'(x)
being determinable by our previous methods.
47. The correctness of the above result may not appear quite
obvious, because we have in effect defined dy, not by the funda-
mental definition
dx^^^^-'A-x ^^^
but by compounding the two definitions
£-"-^:' <^>
^•^tr''--'^ ('>
This is however immaterial since the last three equations are not
independent, any one of them being deducible from the other two.
Thus from (2) and (3) we can obtain the same value of dy as that
given by (1).
Ay _/^ Am .
^""^ Ax~ A21 Ax'
•• ^^^='>'^x~^'='Au^='Ax-
By equations (2) and (3) this becomes
Ay _ dy du _ dy^
which is identical with equation (1).
DIFFERENTIATION. 29
48. The following proof of the above proposition is free from
the apparent difficulty. As before, let
u = <p{x), (1)
and .-. 1/ =f{u) (2)
Let Ajr, ^u, Aj/ be corresponding increments of x, u, y, which
all vanish together.
Then il = U.._,^ = U,_,'^y^.
dx "Ax "A?( Ax
But from (l) and (2),
^^^^^'^u "^' ^")' ^"^ ^^^=» ti = '^' ("'^ '
.■.f^=fiu)<p'{x),
and dy=f'{u)(p'{x)dx.
49. The last article gives the proposition in a form which, consi-
dered as a mere proof of the theorem, is preferable to that of Art. 46.
The former proof has been retained for the purpose of shewing more
clearly the connexion between the equations (1), (2), {3) of Art. 46.
The student will perceive that such equations as
dt/ dy du
dx du dx''
may be used without fear of error.
This is self-evident, provided the differentials which enter twice
have the same value in both places, which (as appears by Art. 47)
will be the case, even though they are defined on the one side by
equations (2) and (3), and on the other by equation (1).
50. The introduction of the new symbol (u) for (x) may be
dispensed with, when the process of differentiation will take the
following shape,
.-. dy = df{<p{x)},
=f'{<P{x)}d<t>{x)
=f'{<P{x)}cp'{x)dx,
/'{0(x)} being the same as /'(«). that is, the same function of 0(x)
which /'(x) is of x^ and therefore determinable by the rules above
investigated.
30 DIFFERENTIATION.
51. In like manner we may differentiate z, where
z = Flf{4>{oc)}2.
For putting u = (j> (x),
and ^ =/(«),
we have z = F(i/) ;
___^ .-. dz = F'{y)Mtf
dy =/ («) du,
du = (p' (x) dx ;
.-. dz = F'{y)f'{u)<i>'{x)dx,
or without introducing any new symbols,
dz = F' [/{0 (x)}]/ {</) (x)}.^' (x) rfx,
and the theorem may evidently be extended to expressions com-
pounded of any number of functions.
52. The following examples may serve to illustrate the method
of differentiating any explicit function of one variable.
(1) Let 3^ =/ (x) = ha" + ca-'.
Then dy = bda" + cda-^, by (25),
and da'' = log a . a'^dx, by (29),
and da"^ = da", if u = — x
= log a .a'' du ) , ^ , , , ^,
1 *= -X, , X h by (29) and (46 ,
= log a.rt (-f/x) J .' V /
.•. dy = log a {6a' — cw^} dx,
and /' (x) = log a {ba" — ca~'}.
The actual introduction of a new symbol is useless where the
function of x, which it represents, is not very complicated. Thus
(2) Let t/ = log sinx=/(x).
Then dy = d log sin x
d sin X
sinx
by (28),
.*. du = S^!j^-f = cot X dx, by (30),
•^ smx
and /'(x) = cotx;
where the process is evidently the same as if we had substituted
u for sin x.
DIFFERENTIATION. 31
(3) Let^ = a('"'^')"=/(x).
Then dy = log a . aC''»s»)" d {{x log a;)"} by (29),
= log a . a("°g^)" {w (a: log j;)"-' d{x log jc)} by (27 j,
= 71 log a . aC^'os')" a;"-! (log x)""' {dx Xogx^xd log x} by (23),
= « log a . aC^'os-)" X"-' (log x)"-' (1 + log x) dx, (by (28),
and/'(x) = « log a . aC^'os')" a;""' (log a:)""' (1 + logx).
(4) Let y = tan (cot"' x), whence y = -.
We will find dy from each form of the expression as an illus-
tration of Art. 46.
From the second form of the equation we obtain immediately,
since
y = x-\
dii = — x~^ dx = — 5- .
^ x^
Also from the other expression we have
y = tan u, if u = cot~' w,
~— dx
.'. dy = sec* u du, and du = ^ ,
••• d^
-
=
1+ !.
cot M
1 + X*
dx
=
^.dx
1+x'
= . as before.
X-
For examples on this and other parts of the subject, the reader is
referred to Gregory's Examples in the Differential Calculus.
53. The results of this Chapter are here collected, and should be
carefully remembered. ^
lfy=/(:c)=f,(x)+Mx).../,,(x),
dy = d/,(x) + dflx) + ... dfXx), ovf'{x) =f:{x) +f./{x) + ...f,:(x).
32 DIFFERENTIATION.
dy =/,{x) df,(x) +/.(x) dflx), or fix) =/,(^)// W +/.(^)//(^).
= CA(x), dy = Cdflx), f{x) = C//(x),
= x", dy - nx"-' rfx, /' (x) = «a:""\
= loff„ X, dy = . • /' (x) = , ,
° ' "^ \oga.x J \ / \oga.x
= 0", dy = log aM"" dx, f {x) = log a-a",
= sin X, rfy = cos x dx, f (x) = cos x,
~- = cos X, dy = — sin x dx, fi^) ~ — siii ■^>
, sin X , .... sin x
= sec X, dy = 5— ax, / (x) = 5— >
•^ COS^X ^ V v cos^x
, COS X J .... cos X
= cosec X, aw = — ^-s— dx, / (x) = — r-^ — ,
"^ sin'^x ./ V y sin^x
= tan X, dy — sec* x c?x, /X^) = ^ec^ x,
= cot X, dy = - cosec* x dx, f (x) = — cosec' x,
• 1 , dx /.// X 1
= cos-x, dy=-j0^^, /(x) = _^_li_,
= sec-' X, dy = — -y^r-;^ — — , /(x) =
xj{x'-iy -> ^^~ xj{_x'-\y
= cosec-' X, dy = -~j-^ — -^, /'W =
= tan-x, dy = ^,, /'(^) = rT^'
= cot-x, ^i'^f^, /(^)=Ili'
DIFFERENTIATION. 33
(j^j» 1
y =f{x) = F{u) and m = (x), <^y = F' {ii) cp'(x) dx, f (x) - F'(«) </>'W-
These results are expressed both in the notation of differentials and
of differential coefficients, in order to familiarise the reader with
both. Both notations express the same fact, namely, that the ratio
^ is equal to a certain function in each case, and, so far as we have
carried the subject at present, may be used with equal convenience.
H. D. c.
CHAPTER IV.
INTEGRATION.
54. In the preceding Chapter we have shewn how to differen-
tiate any explicit function of a single variable, that is, having any
equation of the form y =/(A we have shewn how to determine
the value of /'(a;) in the equation dy =f\x) dx.
Integration is the converse of differentiation, that is, it consists
in finding from any differential equation dy -f'{x) dx the integral
equation ^=/ (a), from which it has been derived.
The symbol (/) by Avhich this operation is represented is the
converse of the symbol {d) which represents differentiation, and is
therefore defined by the equation
jdy=y.
Hence, i^/Xx)dx is the differential of /(^),
|/'Gr)^.r=/(x),
/(x) is called the integral of /'(^) dx^ or sometimes, though not very
correctly, the integral o?f'{x).
55. From the nature of the rules which define a differential,
the form of /'(j:) can always be determined where that of/(j:') is
given. We have only to find — ^— j and determine its limiting
value, and f'(x) is known. No such general method of integration
can be found: for since the integral of any diffei'ential is defined to
be the function from which it may be obtained by differentiation, we
can integrate those functions only to which the differentiation of
other functions has chanced to lead us. Thus, if we are required to
integrate any function (p(x)dx, we cannot be sure that the process is
even possible, that is, we cannot say that there is atiy function whose
differential is cp {x) dx, unless we have observed that this quantity, or
some other to which it is equivalent, has been obtained by the dif-
ferentiation of some known function. It will, however, appear here-
after that all functions of the form (p (x) dx can be integrated in the
form of infinite series, although there are very many whose integrals
cannot be expressed in finite terms by any of the ordinary symbols
INTEGRATION. 35
of analysis. The method to be pursued in integration will therefore
be, first to collect a number of integrals by examining the results of
the differentiation of some simple functions, and then by various
artifices to make the integrals of other functions depend upon those
so determined by inspection. In this way large classes of functions
have been integrated in finite terms, although many are incapable of
such reduction.
56. Since (Art. 22) /(x) + C and /(x) have the same differential
f'{x)dx, it follows that we may take as the integral o£ /' (x) dx
either of the above quantities, giving to C any value we please in-
dependent of X. Hence it appears that while there is only a single
differential of a given function, there is an indefinite number of in-
tegrals of a given differential, all of which are included in the general
form/(x) + C, where C may have any value independent of a; which
we choose to assign to it; C is called an arbitrary constant, and must
be added to every integral in order to express it in its most general
form. Where, however, the form of (/) is the only object of the in-
vestigation, the particular integral f{x) is often spoken of intead of
the general form f{x) + C.
57- By considering the differentials of /(^) and x as measures
of the rates of increase of /(x) and x respectively (as explained in
Art. 18), the problem of differentiation may be said to be — Having
given a function of x, to compare its rate of iiicrease with that of x;
and that of integration. Having given the ratio of the rates of in-
crease of the function atid the variable, to determine the function.
Now it is evident that, for a given function, the rates of increase
can have but one definite ratio for a given value of the variable,
whereas an indefinite number of functions, differing only by constant
quantities, will have the same value of this ratio.
Thus we may draw as many parallel straight lines as we please,
in all of which the rates of increase of the co-ordinates have the
same ratio, since they depend only on the inclination of the line and
not at all upon its distance from the origin. If y = mx is the equa-
tion to one of these lines, they are all included in the general form
y = mx + C, where C is perfectly arbitrary, and all have the same
value of the differential coefficient. The equation to each may there-
fore be obtained by integrating the same differential equation
dy — mdx.
3—2
36 INTEGRATION.
To take a more general case. Referring to the figure of Art. 19-, it
is clear that an infinite number of curves may be drawn having
their tangents at the extremities of a common ordinate, equally in-
clined to the axis of x\ The equation which includes the whole class
of curves is i/ =f(x) + C\ where C is arbitrary, since the addition of
a constant to every ordinate merely changes the distance of the curve
from the axis of x, without altering its form or the inclination of its
tangent. In all these the differentials therefore have the same ratio
for the same value of x, and any one of the equations may be ob-
tained by integrating the differential equation dy =f'{,x) dx which is
common to them all. These observations shew the meaning of the
proposition established in the last article.
We shall now proceed in the empirical manner indicated in
Art. 55., to determine the integrals of the differentials given in
Art. 53., and some others immediately deducible from them.
58. Since
d{A{^) +fM ... +/„(.^)} = f(/. W + 4/'.G^) + ••• dM^)'
.'. /]{x) +Mx) +M^) = j{dMx) 4 dMx) + ... d/jx)},
or f{f/{x) +/,Xx) + . . ./,Xx)\ dx = j/\'{x) dx 4 j/,'(x) dx...+ [//(a-) dx,
that is, the integral of the sum of any number of functions equals
the sum of the integrals of the several functions.
59. Since d Cf{x) = Cdf(x),
.'. C/{x) = jCd/ix) = {Cf{x) dx,
or [Cf'{x)dx^C if'{x)dx;
that is, if the quantity to be integrated is multiplied by a constant
factor, the constant may be placed outside of the sign of integration,
60. ^mce d(x") = nx"~'^dx, where n is any number, integral or
fractional, positive or negative, different from zero.
'dx = -+C,
n
dx:^ , +C,
11+ \
where n is any number different from - 1 ; that is, to integrate
any power of x except - we must increase the index by unity and
INTEGRATION. 37
divide by the index so increased, an arbitrary constant being added
to obtain the general integral.
61. Since dloi^x = — .
I
dx , -, ^
— = log .r + const. == log C x,
if we write the arbitrary constant in the form log C. This deter-
mines the integral of x" in the case excepted from the previous
article. Hence
d e p
f
ax" + bx"~^ . . .+ c A h-5 + ...-^
X X X
= a + 6 — . . . + cj? + ri log a: ... - -, - , „, ■ , + C.
n + \ n ° X {7)1 - 1 ) X
f dx f dx
62. From I — we can find / .. „ „, as follows.
J X J J{x'' ± a-)
Let «* ± a^ = U-,
.', xdx = udu,
dx dii dx + die
' ' n X X + 71 '
"' J Ji.^^ =^a^)~ 111 ~ j x + u '
= log {x + li) + const.,
r= log C^*^ ^^ ,
^ a
C
writing the constant in the form log — to give the integral a more
convenient form.
63. From the same integral we can also deduce the values of
dx , f dx
[ dx . f dx
j?^'""'' j^^^'-
a — a; 2a [a + x a - a: j
and d (a — a) = — dx, and d(a + x) = dx ;
f dx If dx 1 j' dx
J a' — x" 2a J a + X 2a J a — x '
1 fd(a + x) 1 fd(a-x)
2a J a + X 2a J a - x
h
= — loff (a + x) log (a - x) + const. ;
dx 1 , ^,n + X
7, o ~ — log 6 .
' - 1^ 2a ^ a-x
38 INTEGRATION.
f dx _ J_ [ dx _ J_ /• dx
•'■ }x^-a''"^]x-a 2ajx + a'
1 Cd{x-a) 1 /• (/ (x + a)
~ 2a J x-a 2a J x + a *
= — log (x -a)-x~ log {x + a) + const. ;
' ■ j x' - a'' 2a ^ x + a'
f dx
64. From Art. 62, we can deduce the value of / — . 2^_y2\
For let x = -, .'. log a; = log a - log u ;
dx du
' ' X ?< '
1 1
C dx _ f du
ail
du
_ 1 /• du
therefore, by Art. 62,
dx
f dx 1 X
j X Jl^a'^x') ~ a^"^^^ a + J{a' ^ar')'
65. Recurring to Art. 53, we find
d (a^) = log a . a'^dx,
.'. I a'^dx = , . a" + C.
J log«
H a = e, this becomes, / e^dx = e^ + O.
INTEGRATION. 39
Again, d sin x = cos x dx,
cos X dx ^ sin a: + C.
/'
And since d sin 7nx = cos mx d(inx) = in cos mx dx,
f
cos mx dx = — sin mx + €'.
m
Similarly, d cos inx = - m sin mx dx.
' sin mx dx = — cos mx + C-
m
Also, d tan mx = m sec^ 7nx dx ;
I
sec^ mx dx = — tan ?«x + C
m
And d cot mx = — m cosec^ mx dx ;
}
cosec^ mx dx = cot mx + C.
m
d(-\
/
■■= sin"' - + C.
^(a^ - x^) a
o 1 . J -ix \aj dx
bo also, since d cos - = •
/;
// ^ 2\ = - COS ' - + (7.
J{ar-x') a
Here we have obtained a second form of the same integral, but
since
. , X . X IT
sin~ - + COS" - = - = const.
a a 2
the two expressions include precisely the same system of values
when all possible constant values are given to the arbitrary con-
stants.
. . 7 _i X ^ \a) a dx
Again, 6?sec -= 77-5- — 7 = — 77-5 ^;
a X fx' \ xj{x^-a-)'
h
dx 1 iX ^
-^ = - see" - + C.
X J{x^ - a') a a
40 INTEGRATION.
As in the preceding case, this integral might have been expressed in
the form
— cosec"' - + C,
a a
which may be shewn as before to be identical with that already
given,
_^x \a) adx
bmce o tan - -.
" 1 +
/ ^ 7, = - tan - + C.
J a' + X a a
As before, we might have obtained the equivalent form
- -cot-'-+C.
a a
„. J . _, a: V«/ dx
bince aversm
f dx . , X _,
•'• 1^177, j7 = versm-' - + C
; J{2ax - X-) a
or = — suversin"^ - + C.
a
66. Collecting the results of the preceding Articles, we have
I x"dx = h C unless w = - 1, and then
./ 71 + 1
\-J = log ^ + C,
-nr— -2 = - tan-' - + C,
J a^ + x^
a a
dx 1 - ^a + X
= — loff 6 ,
2a ® a-x'
f dx
I a' - x""
f dx 1 , ^x-a
~5 s =7— iogC; ,
] X- - a' 2a ° X + a
I —rr^ 2\ = sm-' - + C, or = - cos"' - + (7,
J-/r^A4xJ<rn>< C{/dL
I INTBGRATION. 41
= - sec ' - + C,
X J(x^ - a') a a
dx ■ I ^ ^
-= versin"'- + C,
J(2ax - x^) a
a'dx = ; 1- C,
log a
sin mx dx = cos mx + C,
m
cos mx dx = — sin mx + (7,
m
sec^ mj: <?.r = - tan mx + C,
m
cosec'mx dx = cot mx + C.
m,
The above are termedj'undamenlal integrals, and should be care-
fully remembered.
All other integrals are obtained by reducing them to some one
of the above forms.
It may be observed that where the function under the integral
sign is homogeneous in x and a, the integral is also homogeneous
and one dimension higher in terms of x and a; or of the same
dimensions, if we consider dx as of the same dimensions as x.
The remainder of this Chapter will be occupied with various
methods of reducing the integrals of several classes of differentials
to some of the fundamental forms.
Integration hy Algebraical Transformations.
67. An integral may often be reduced to the form of one of
the fundamental integrals, or to the sum of two or more by simple
algebraical transformations.
As an example, we may take the following :
{ '^^*'' _ { ^^^
Jl+a; + x'~ Jf + i^ + xy
da4-x)
-\u
2 , 2x- + 1 .,
42 INTEGRATION.
The student will find numerous examples of the application of
this method in Gregory's Examples^ p. 246, et seq., with which he
should make himself familiar.
Integration by Parts,
68. The following theorem is often of great use in reducing
integrals to the required forms.
Since d{uv) = udv + v du,
u and V being any functions of a variable x,
.'. uv = ludv+ Ivdu,
.'. ludv = uv — ivdu (!)•
This is called the formula of integration by parts, and enables
us to integrate any function tidv if the function vdu is in an inte-
grable shape.
69. As an example of its application, we may take
\x log xdx.
f'
Assume log x = u, and x dx = dv,
whence — = du, and -pr = 'v,
X 2
the arbitrary constant being omitted in the value of v for the sake
of simplicity, since the formula holds when v is any quantity whose
diiferential is dv. The constant would, in fact, disappear from the
result if it had been introduced.
Hence \x log xdx= lie dv,
= uv — Ivdu by (1 ),
x^ , fx^ dx
= 2^*'^^-j2-T'
x^ x^
= _]ogx-- + C,
x^
= - {log x-i} + C.
The student is again referred for examples to Gregory's Exatn-
ples, which he is recommended carefully to study in all parts of
the subject.
INTEGRATION. 43
Rational Fractions*.
70. A rational fraction is a fraction of the form
x" + 51^;""' ... + q^„
-where the indices of x are all positive integers and the coefficients
constant.
If the numerator contains powers of x as high or higher than any
in the denominator, the fraction can always be reduced by division
to the sum of a rational integral function of x, and a rational frac-
tion, in which the largest index which occurs in the numerator is
less by unity than the largest in the denominator : the first part of
this can be integrated by inspection, and therefore we need only
consider those rational fractions in which the dimensions of the
numerator are less than those of the denominator.
Rational fractions are integrated by resolving them into the
sum of a number of fractions with simpler denominators, called
partial fractions.
71. This can always be done as follows. Let the rational
fraction be
U _ poX'" + PiX""-^ + ... pn
V x" + qiX"~^ + ... q„
where m is not greater than n -1.
Let the equation V =0 have one real root equal to a, r real roots
equal to b, one pair of impossible roots equal to a ± /3 ^(— 1 ), and s
pairs of impossible roots equal to a'±/3'^(— 1), which comprise all
the forms in which the roots of any rational equation can occur.
U A Br Br-, B, M+Nx
Assume -7> = h -; rrr + r ttt-i • • • + r +
F x-a {x-bj {x-by-'"' x-b x'-2ax + a''+f3'
K, + L,x Ki + LiX
(x'-2a'x+a"+/3'y x'-2a'x + a''+(S"
Then real values of A, Br, ^._,. . .7?,, M, JV, K,. . .K„ L,.. .L„
can always be found to satisfy this equation. For by multiplying
both sides of the equation by V, we obtain an equation
U-/(x) = (2),
" Those readers who are not familiar with the Theory of Equations are recom-
mended to omit the remainder of this chapter. •
44< INTEGRATION.
where £7" is a rational integral function o£ x of not more than (fi — 1)
dimensions, and /(a:) a rational integral function of (« — 1) dimen-
sions, in which the coefficients of the several powers of x are linear
functions of the indeterminate quantities A, Br, &c. in equation (1).
In order that the two sides of equation (l) may be identical, the
coefficients of the several powers of x in equation (2) must vanish.
This condition will give us n linear equations to determine A, Br,
&c. Now the number of these coefficients is evidently equal to the
number of roots of V = 0, that is, to 7i, and therefore the ?i linear
equations will give real values of them which satisfy equation (l).
Hence -„ may always be resolved into partial fractions of the forms
assumed in equation (1).
If the dimensions of U had been greater than n - 1, this method
would have failed, because the coefficients of the higher powers of ^r
would have contained none of the indeterminate quantities, and
could not therefore have been made to vanish by assigning particular
values to these indeterminate coefficients.
If the equation V—0 contains more than one of each of the four
classes of roots, we must add partial fractions of the forms above
given for each of these roots, and it will appear, as above, that real
values may be found for the indeterminate coefficients which will
satisfy the assumption. By this method / -p dx, where -p is a
rational fraction, may always be reduced to the sum of a number of
integrals of the forms
jAdx f Bdx r {M+Nx)dx f (K+Lx)dx
The two first of these are integrable by inspection, the third can
always be integrated by the algebraical artifices before considered,
and the last may be reduced to the third form by the method of
reduction which will be investigated below.
The general method of determining the partial fractions above
given is often very laborious, and may be simplified in practice in
the manner indicated by the following examples.
72. Let V ~0 have neither equal nor impossible roots. We may
then proceed in every case as in the following example.
Let
INTEGRATION. 45
Udx c ^dx
rudx _ r xdx
j ~V' ~ j (x - a) {x -b){x- c)
X ABC
1 hen assume , r-~, j^r? r = 1 r h ,
{x - a){x — o){x - c) X — a x — b x — c
.'. x = A{x-h){x-c)+B{x-a){x-c) + C{x-a){x-b). (i)
Since this must hold for all values of x,
let x — a, .'. a = A (a — h)(a — c),
x = 6, .-. b = B{b-a)ib-c),
X = c, .'. c = C (c — a) {c — i),
which equations determine A, B, and C; and then
( xdx _ fAdx CBdx f Cdx
j(x-a)(x-b){x-c)~ J x-a J x-b J x-c'
= A\og(x-a)-{-B log {x -b) + C log (x - c).
73. If we had pursued the method of Art. 71, we should have
determined the values of ^, B and C, by equating the coefficients
of x^ and x and the constant terms of equation (1). This would
have given us the equations
= A+B + C,
l=-A{b + c)-B(a + c)-C{a + b),
= A be + B ac + C ab,
which give A, B, and C though less readily than the equations of
the last article.
74. Let V =0 contain equal i-oots but no impossible ones^ as in
(x' + x) dx
. f (x^ + X) dx
the example j y-^ — r^rr t\ ■
^ J(x- ay (x - b)
x^ + x As Ai B
Assume , r„ r jv = 7 xa "■ •" r >
{x—a)-{x — b) {x — ay x — a x-b
.-. x^ + X = A2 (x - b) + A^ {x - a) (x - b) + B (x - a)- . . . (a),
Let x = a, .'. a- + a=Ar^(a-b) (l),
therefore subtracting,
{x^-a^)-¥{x-a)=A„_{x-a) + A,{x-a) {x - b) + B (x ~ a)',
and dividing by x-a,
X + a + I =^ A2 + Ai {x - b) + B (x - a) ((3).
Let x = a, .: 2a + l=A^ + Ai{a-b) (2).*
* This step is admissible although we have previously divided by x — a, because
(a) and {ft) are not equations for the determination of <^', but identical equations.
Thus when the proper values are given to A2, Ai and B both sides of equation {ft)
become x + a+l, and must therefore have the same value when ,r is put equal to a.
46 INTEGRATION.
B may be determined by putting x = b either in (a) or {ft) ; the
former is preferable, as it gives B independently of ^2 by the
equation
b'+b.= B(b-ay (3).
Equations (1), (2), and (3) give A2, ^,, and B, and
r (a;^ + x)dx _ f -^2 dx fA, dx CB dx
J(x — ay {x— b) J (x — ay j x — a j x — b
= ^ +A,\og{x-a) + B log {x-b) + a
The same method applies when there are more than two roots equal
to a, by repeating the substitutions and subtractions.
75, Let V = contain unequal pairs of impossible roots, as in
, , /■ (x — c)dx
the example j ^^,^^,^^^,^^^^^,^ .
x-c Ax + B Cx + D
Assume t-z i^tt-^ — ; tt: = —^ ir +
{x' + a'){x' + bx + b') x' + a^ x^+bx + b"
.-. x-c = {Ax + B) (x' + bx + b') + (Cx + D)(x'+a') ... (a).
Let x^ = — a^; an assumption which we are at liberty to make,
since the above equation, being identical, must be true even when
impossible values are given to x;
.-. X -c = (Ax + B) {bx + b' - a^),
= Abx' + {Aib'- a') + Bb} x+B(b'- a'),
■^{A(b'- a') + Bb}x-Aba' + B {b'- a:),
which cannot hold unless
i=A{b'-(e) + Bb (1),
and -c = -Aba'+B (//- u') (2).
Equations (l) and (2) determine A and B ; C and J) may be
determined in exactly the same manner.
Thus, in equation (a), let x^ = — {bx + b'),
.'. x-c = (Cx + D) (a- -bx- b'),
= -Cbx'+{C (a* - b') -Db}x+D (a' - b'),
= [C (b' + a' - b') - Bh] x + Cbb'+D (fl^ - b'),
which cannot hold unless
l = C{b' + a'-b')-Db (3),
-c=Cbb'+D((r-b') (4).
INTEGRATION. 47
Equations (3) and (4) determine C and T> ; and
\Cx + D) dx
r (x-c)dx _ f {Ax + B)dx f{C
J{x' + a')(x' + bx + b')~J x' + a"" "^ j "^
+ bx + b'
These integrals can be found for all values of a, b, b' by alge-
braical methods.
When only one partial fraction remains to be determined, as in
this example after A and B have been found, it may be done by
substituting for the other indeterminate quantities in (a) their values,
and dividing by the coefficient of {^Cx + D). This is generally a
simpler method of determining the last partial fraction correspond-
ing to a pair of impossible roots ; we shall employ in the following
example,
C x^dx
\{x-\y{x'+i)'
. x^ A^ A. Cx+ D
Assume ; ,.3. , rr- = 7 ~T5 + + — „ , ,
.: x^ = A, (x'+ 1) + A^(x-l){x'+l) + (Cx + D)(x-iy .., (a).
Let x = l, .-. 1 = ^2.2, .-. A.
1.
•2 '
therefore subtracting,
x^~\=A^ {x^ -1) + A,{x- 1) {x' + 1) -t- (C'x + D){x- ly,
.-. x^ + x + l^A^(x+l)+A^(x' + l) + {Cx + D) (x - 1).
3 — '^A
Let x = l, .'. 3 = ^2.2 + >4,.2, .-. ^, = ^^1—^=1;
substituting these values in (a), it becomes
|'-x + l = (Ca:+Z))(x-iy,
.-. Cx + D = ^, or C-0, Z) = A,
r x^dx _ I f ^^ f ^^ 1 ( ^
■'■ J{x- ly {x'+i)~ ^Jj^'^TYy '^jTZ^i + ^jxTl
+ log (jc - 1) + i tan~'^ + C
X - 1
When there are more than two pairs of impossible roots, the
partial fractions may be determined in succession by the method
given in the first example of this article, the last fraction being most
conveniently found by substitution, as in the last example.
48 INTEGRATION.
76. Let V = contain equal pairs of impossible roots, as in
(2x^ — x) dx
f (^x-'-xJdx
the example li-^ — ^^g, ^ — rr
^ J {x- + 2y {x^ + 1)
^x'-x Ax + B A'x + B' Cx+ D
'"^ (x' + 2y (x' + l) '(^^f^ x' + 2 "^ o;^ + 1 '
.-. 2x''-x
= (Ax + B)(x'+l) + (A'x + B') (x^ + 2)(x' + 1) + (Cx + D) (x' + 2)- . . . («)•
Let a:' = -2, .-. - 4 - x = (^x + 5) (- 1) ... (/3')>
which cannot hold unless
-1 = -A, or ^ = 1,
- 4 = - 5, B = 4>.
Subtracting (/3) from (a) we have
2j'+4-(^x + ^)(^' + 2) + (^'x + J5')(*^'+^)(-^'+l) + (^'*^ + -^)(-^' + 2)';
dividing by x^ + 2,
2 = (Ax + B) + (A'x + B') (x^ + 1) + (Cx + D) (x' 4 2) (7).
Let x^ = - 2, .-. 2 = (^x + B) - (A'x + B') (S),
.-. 0=A-A', .'. A'^1,
2 = B- B', B' = 2.
The remaining numerator, Cx + Z), may be determined by put-
ting a;* = - 1, as in Art. 75, or, being the only remaining numerator,
more conveniently by substituting for A, B, A', B', their values in
(a ). If there had been three or more equal pairs of impossible roots,
we must have subtracted (3) from (7), divided by (x^ + 2), and then
put x^ = —2, and repeated the process till all were determined.
The partial fractions being determined, we shall have
C (2x'-x)dx _ f(A x+B) dx f (A'x + B')dx f(C^+D)dx
J(x' + 2y(x'+l)~J (x' + 2y~'^j x' + 2 ^] x' + l '
The first of these integrals may be determined by the method of
reduction, and the others by known methods.
77- We have now considered every case except where roots of
all the different kinds occur together, in which case the different
methods must be applied in succession, as it is easily seen, by observ-
ing the preceding examples, that the determination of any partial
fraction is not at all affected by the number or nature of the rest.
INTEGRATION. 49"
Rationalisalion.
78. Many functions, which are not in the form of rational frac-
tions, may be reduced to a rational form, or, as it is termed, ration-
alized by different transformations. Rules for the rationalization of
several classes of functions have been investigated, but the student
will find that practice will enable him to discover the appropriate
assumptions in most of the cases which ordinarily occur, without
burdening his memory with a variety of rules for the purpose.
, f x^dx
pie, -i i
Take as an exam
Assume x = z^,
1 2
.r^ = ^,
and dx = 6z^dz,
f z' dz
and the integral becomes 6 I — — —^ , which is in a rational form, and
J z^+ a^
may be integrated by the methods already investigated.
Integ7-ation of x'" (a + bx"ydx.
79. Functions of the above form are of very frequent occur-
rence. We shall first shew how to integrate them when the indices
7«, n, and p satisfy certain conditions, and then investigate a general
method of integration applicable to all cases.
, . ,-„ 7rt + 1 . ... 11-
(1) Where is a positive integer, m, n, and p being posi-
tive or negative, integral or fractional.
T ...
Let p — - , where r and q are positive or negative integers.
Assume {a + bx") = z'',
z'' - a
b"
.-. x"'dx=^-^{rJ-a)" dr.,
nh "
H. D. C. 4
50 INTEGRATION.
and multiplying by z", we have
f n f »i+i
n
h"
Now when the above condition is satisfied 1 inust either
n
equal zero or a positive integer, and the function under the integral
sign can be expressed in a finite series of integral powers of ~, and
can therefore be integrated in finite terms.
If had been negative or fractional, the expanded binomial
would have contained an infinite number of terms, and the method
would have failed to give us the integral in a finite form.
The condition = a positive integer, is called the First
Criterion,
/r^\ iTTL tn + 1 , .
(2) VViiere — + p = a negative mteger.
Then x'" (a + bx'y - ar'"^"^ {6 + ax-'Y.
The function in this latter form may be integrated by the
, , .„ 7n + up + 1 . . . , ..„?« + I
previous method if ' = a positive integer ; that is, if
+ /) = a negative integer.
The assumption to be made in this case is therefore
b + ax~" = z'', putting p = - as before.
The condition + p = a negative integer, is called the
>Seco7id Criterion.
80.
The
following
are
examples
of
the
application
of
these
methods.
(0
f x^^dx
J (ah
+ x^'
Here
VI +
n
1 i+ 1
3,
a positive
integer
; therefore
the
first
criterion is satisfied.
Since p = —\t the proper assumption is
fl5 -f x-i = Z-.
INTEGKATION. 51
-T Tri = ^\{^-a^dz,
= 4 [{z'-2a^z'' + a)dz,
= f (rt^ + o;')^ - f a^ («^ + x--f- + 4«(a^ + o:^)^ + C.
Here = , and the first criterion is not satisfied.
n 2
But + p = — i + 5 = — 2, and the second criterion holds.
.{a'-x')Ux ( {a'x-'-\)^dx
Then I ^^ « = | ^^3
/• («^- j:^)'c/jr _ f{
and we must assume ax - — 1 = 2 ,
rt
_ _ -z{z'-^\)dz _
(a^j;-^- l)i («^j:-^-l)t
5a* 3a*
n^x" \ 5
} + C,
(a'-x"-) H3 a' + 2x-) ^ ^
15aV
81. When neither of the criteria is satisfied, we must employ
the following method, which is applicable to all cases. It consists
in making /j;'"(a + ia;")''(/j; depend on another integral of the same
form but with smaller values of 7h or ;; ; by repeating which process,
4 — 2
52 INTEGRATION.
we arrive at last at an integral which can be determined by methods
already given. This is called the method of reduction.
As the treatment of the integral depends on the signs of m and ;?,
we shall use these letters to represent positive quantities only, which
will give the following cases.
(1) To reduce m in f "^ ^^„. - or jx'"{a + bx"y dx.
. r dx {(a + bxy J
(2) To reduce m in \^,^^-^. or j ^^^^ dx.
(3) To reduce p in j x'" (a + bx"y dx or j^ — -^ dx.
, ^ _ -, . f x'^dx f dx
(4) To reduce p m -. j—^ or | ^ — , , „xp-
^ >' ' J {a + bxy J x^{a + bx"y
The mode of treatment is exactly the same in each pair of integrals
as above arranged, that is, the method of reducing m or p depends
only on the sign of the index to be reduced.
n is always supposed to be positive, as the function can always
be reduced to that form by altering the value of m. The follow-
ing are the methods to be employed in the four cases respectively.
••''•"" j -^ (« + bxy ~J \nb (p - 1) (a + bx"y-'j '
therefore, integrating by parts,
71 b [p
71 + 1 f x'"~"\a
p-l)J (04
_ „,_„^i 1 7)1-11 + 1 f x'" "dx
'"""''*' 7ib{p-l){a + bx"y-' ^ lib (p - 1 ) J {a + bx"y-' '"^ ^'
j,...-n+i jfi -^ ji + I f ,r'"~" [rt + bx"] dr
P,n = 7-. T^^ . „ ■ , +
7ib Q)-l){a + bx''jP-' nb {p - 1) J (a + bx")''
p _ J^'""'"^' 7)1 -71+1 f , ,
•'• "'~~ 7ib(p- 1) {a + bxy-' ^ 7ib{p-l) ^" '"-" "^ '"^ '
,, a"'~"+' ?« — ?j + 1 „
. •. j[^ = -)- a I _ '
b {)ip -7)1—1) (a + bx^y-^ b {lip -7)1-1) '" " '
by which formula P,„ is made to depend on P,„_„, that is,
f x'"dx C x"'~"dx
J (a + bx"y °" J (fl 4 bx"y '
and by repeating the process, P,„_„ may be made to depend on
^
INTEGRATION.
P,n-i,n and generally P„, may be made to depend on P„,_r„ where
r is any integer.
By employing the form (A) we make the integral depend on
another of the same form, in which m is diminished by n, and p by
unity, by the repetition of which m may be reduced by any multiple
of n and p by the same multiple of unity.
I x"'(a + bx"y is treated in exactly the same manner, and we obtain
formulae by which yn alone may be reduced by any multiple of?/,
or m reduced by a multiple of 71, and p increased by the same
multiple of unity.
dx
83. Let P„
J X"" (rt
+ bxy '
, \ a h
then -—-, ,—^-, = -—, ,— tni; +
and
dx
x'" (a + /jjc")P-' .1"' (« + bxy x'"-' (fl + bx"y '
j a;'" (a + 6a;"/-' ' "'
1 w6(p — 1) /* dx
-1) r rfj
1 j^"'-"(a-
(/« - 1 ) a;"^i (a + 6x")'-' w - 1 j a;"'"" (a + bxy
by integrating by parts :
. «p I n{p-l) + m-l ,
(^ _ ] ) j;-' (a + bxy-' m - 1
by which formula P„ is made to depend on P,„_„, and therefore
P„ may be made to depend on an integral of the same form, in
which m is reduced by any multiple of n. The other integral in (2)
is treated in the same manner.
84. Let P^
.-. P„
= I x"' {a + bx"ydx ;
; - a [ «"• (a + bxy-' dx + b ( x""-" (a + bx"y-' dx ;
and j x-^" (a + bx^-' dx = j x^^' d {^^^iT ~ } '
a'"^' (a + bx'J' OT + 1 r , ,
= ^^- ^ , — x'" (a + bx^yUx,
nbp nop J ^
by integrating by parts :
. p ^p a-'""' (a + bx-y m + 1
•• ^r = "Pr-r+ ^ ^r,
54! INTEGRATION.
P - 5^ — + « Fp
'' np + ?« + 1 np + in + \
by which formula P^ is iTiade to depend on P^_,, and therefore
P may be made to depend on another integral of the same form in
which p is reduced by any integer. The other integral in (3) is
treated in the same manner.
85. Let P
then
and
''^1 {a + hxy '
x"" _ ax"' 6x"'+"
(fT+^jx")^-' ^ {a + hx"Y "*" (« + bxy '
j (a + hxy J Xnb (p - 1) (o + bxy-'j
7ib{p-i)(a+bx"y-' "^ «6(p-l) j (a + bxy-' '
by integrating by parts :
•'• ''^" ^n{p-l)ia + bxy-' ~ \?i (p - 1) j ^-•'
by which formula Pp is made to depend on P^_,, and therefore
P may be made to depend on an integral of the same form in
which p is reduced by any integer. The other integral in (4) is
treated in the same manner.
86. Of these four forms the first two reduce 7n alone by 7i at
each step, and the last two reduce p alone by 1 at each step, while
the form (A) which occurs in Art. 82 reduces both in and p at once,
and a similar form reduces m and increases p in the other integral
of Art. 81, (1). On account of the increase of p, this latter form is
never more advantageous, and often much less so, than the final
form of Art. 82. Where however it is equally applicable, it has the
advantage of being more readily obtained. The mode of proceeding
with an integral of the general form under consideration, will be to
reduce m and p together by (A), where that is possible, until either
171 is less than 7i, or p equal to unity. If the function is not then
integrable, we must reduce p in the former and ?h in the latter case,
until we arrive at an integrable form. Where formula (A) is not
applicable we must begin by reducing w or p, choosing that one by
INTEGRATION. 55
the reduction of which we arrive most easily at an integrable form.
That we shall always obtain at length an integrable form is evident,
since we may always reduce p to unity, either in the numerator or
denominator, when the function is integrable at once in the first
case, and in the second is either a rational fraction or admits of im-
mediate rationalization.
The reduction of ?« is often preferable to that of p, because it
proceeds by 7t instead of unity at each step, and in many cases,
as where w^ = n^-l, leads to a function which is integrable by
inspection.
The methods employed in the four distinct cases, and Jiot the
resttlling formulae, should be remembered and applied in any par-
ticular case.
/x dx
(a* + x^
x'dx
Here we can first reduce 7 by 2 and f by 1, by means of for-
mula (A), and then reduce the index of x alone by 2, twice, which
will bring us to I j which can be immediately integrated.
J {a^ + x^Y
Employing therefore the method of Art. 82, we have
/■ x''dx _ (x^.xdx
x^ „ [ x^dx
= i + 6 i .
(a" + xy J («' + xy
As we have now to reduce the index of x twice, we will put it
equal to m, that the same process may serve for both reductions.
■r 1 T^ [ ^'"^-^•^
Let then P,„ = — -j ;
J (a- + xy
.: (as in Art. 82) P,„ = a:'-' (a' + .r)' - (m - 1) jx'"-' (a' + x')' dx
= x'"-' {a' + x-'f - (m - 1) {a' P„._„ + P,„},
... P„ = i .r"- (a' + x')'' - "^ a' P,„_, ;
711 ^ ' VI
.'. P, = ix'(a' + x')^^-U'Pz,
an d P, = i x' (a' 4 x') ' - § a' P, ,
56 INTEGRATION.
and P, = I — ^— ^-^ = (rt- + x^)^ + const.
J (a^ + X-)-
{a^ + x")
Hence
r x'dx _ x"
J (a' + x-)i ~ {a- + x-f^
+ 6 lix' (a- + a--)' - ia^ {^x" (a' + .r )* - f «' (a^ + x^)^} + const.]
88. There are some particular values of the indices p, m, and 7i,
in which the above methods become inapplicable on account of the
coefficients which occur becoming equal to infinity. This happens
in the following cases :
(1) Where the index of x is (-1) the reduction of m is im-
possible, as appears from Art. 8.3. The integral is then of the
C n Y C fl Y
form — ; ; — r- or — (« + bx")^. The most general form of p
J x{a + bx")'' } X ^ '
T ...
IS -, r and q being positive integers.
Assume then
a + hx" = z'',
(z^ - af
.-. X- , ;
b"
dx q z^-'dz
" X n {z''-a)'
and the integrals become
91 J z'^-a ?i J z^-a
both of which are integrable as rational fractions.
m + 1 . f x"'dx C(a + bx"Ydx
(2) When p = in ^-— - or ^ — —^ ,
the reductions of m and p respectively become impossible, as appears
from Articles 82 and 84.
r x"'dx _ f. x"''"''dx
^^"' J (a + bxy " I {ax- + by
f dx
= I —7 r^ by the above condition.
J x{ax-+by ^
INTEGRATION. 67
This is the same form as that just integrated, and may therefore
r
be solved by the assumption ax~" + J = 2' if p = - .
(3) When the index of (« + bx") is (- 1) the reduction of p
becomes impossible, as appears from Art. 85. In this case the in-
tegral is / -. — or —, i—r^ , both of which are rational
^ J a + Ox" J x"' {a + 6j: )
fractions.
Integration of sixfx co%"xdx.
89. This function may be integrated by methods similar to
those applied to x"'(rt + bx'y. We will first consider some particular
cases in Avhich the integral may be immediately obtained, and then
proceed to the method of reduction.
(1) Let one of the indices (as m) be an odd positive integer,
then m =: 2r + 1, where r is a positive integer ;
• •. I sin"'a: coa^Xilx = I cos"a; (1 — cos'o;)'" sin xdx.
By expanding the binomial, the integral is reduced to a finite num-
ber of integrals, each of which may be determined by inspection.
If the index of cos x is an odd positive integer, and equal to
2r + 1, we have
isivTx cos^'xdx = i sin"'a: (1 - sin'x)'' cos xdx,
which is integrable as before.
(2) Let {m + n) be an even negative integer =— 2r.
Then I sin'"a: cos"a: dx = j tan'"j: cos"'+"x dx
tan^j: (1 + tan-j;)''"' ^ec'x dx.
i
By expanding the binomial, the integral is reduced to the sum of
a finite number of integrals, each of which may be determined by
inspection. •
90. When neither of these conditions is satisfied we must pro-
ceed by I'eduction.
For this there are only two distinct modes of proceeding, that
for the reduction of a positive index, and that for the reduction of
a negative one.
58 INTEGRATION.
(1) To reduce m in \s.m!" x co^" x dx,
being positive, and n either positive or negative, let
Then P^ = \ sin"'~' x cos" x sin x dx,
sin"^'xcos"+'x m-l f . , . „^, , ...
= ; + , /sin"'-='xcos"+'xrfx (A),
n + 1 n + 1 J ^ '^
sin'^'xcos"+'x »«-l,/"- ^2 „ 7 Tj 1
= + { I sni'^^x cos"x dx - P„],
fi + 1 7^ + 1 V
„ sin"^' X cos"+' X »/ - 1 „
^•^ p ^5 ^. jP
' ' " m +n in +n '""^ *
If the index to be reduced had been that of cosx, the method would
have been exactly analogous. By formula (A) both indices are
altered ; that of sin x reduced, and that of cos x increased or reduced
according as it is positive or negative.
, ^ r^ 1 . {&m"'xdx
(2) To reduce n in I — ,
^ ^ J cos^»
n being positive and m either positive or negative,
„ rsin'"j;(/x
let P„ = „ — ;
} cos X
/" sin*"x (cos^x + sin^x)c?a:
J cos X
r, f • .„xi ^mxdx
= Pn-2 + sm'"+'x -— - — ,
) COS X
p sin"'''"*a: 7n+l fsm'"xdx
"~^ (?J — 1 ) cos""' X 71 — 1 J COS""^X '
sin^+'x ( w + n „
= ? 7^ ^TT- + ^ I r r "r-2-
(m - 1 ) COS X ( M — 1 J
If the index to be reduced had been that of sin .r, the method
* would have been exactly analogous.
It will be seen that in every case the reduction is by 2 at
each step.
91. Take as an exam
, [ dx
pie —5-.
J cos* j;
INTEGRATION. 59
LetP„=f-^^-,
J cos a;
(cos^^ + sin^x)dx
cos'',r
= -r„_a + / sin a: ,
J cos x
_ p s in j; 1 f dx
"~^ {n — \) cos""' X n — I J cos"~^ j; '
.. p sin a: , " - - p
" (/J— l)cos"~'a; ?« — 1 "
• *~4cos^a;"^4^'
_ sin.r 1
'~2cos^r 2 ''
and P. = f = lot? C cot d tt - la:) ;
jcoso; ° ^* ^ '^'
/■ (Ix sin j: 3 sin j;- 3 , ^ ^ / , , s
— 5^ =1 — + o — 2" + ologCcot(V- 5^')-
CHAPTER V.
SUCCESSIVE DIFFERENTIATION.
Theory of the Independent Variable.
92. In Chapter III. we have shewn how from any equation of
the form
y =f{^\ ox x = (p{y) (1),
to determine the value oi f'{x) or ^'{y) in the corresponding dif-
ferential equation
dy=f'{x)dx, or dx = (p'(y)dy (2).
It may now be asked^ are the differentials c?x, dy to be con-
sidered as functions of the variables x and y ? The answer obviously
is, that they may have any values consistent with equations (2), that
is, any values whose ratio equals the differential coefficient. Now
these values may evidently be such as to make them both functions
of the variables; but we are at liberty also, if we please, to give to
either of them any arbitrary constant value, when the other will
receive such a corresponding value as to satisfy equations (2), and
will therefore be a function of the variables.
We have already said that a single equation between two variables
is called an equation of one independent variable, without applying
the term independent variable to one rather than the other ; we will
now add the following definition.
Def. In an equation between two variables, that one whose
differential is assumed to be constant is called the independent
variable.
When no such assumption is made and the differentials are left
in their original generality, the equation is said to have a general
independent variable.
When y is given explicitly in terms of cT, it is generally most
convenient to make x independent variable, and vice versa, as
thereby equations (2) are presented in a more advantageous form.
It is sometimes, however, better to keep the independent variable
general for the sake of symmetry in equations derived from equa-
tions (2) in a manner which will presently be explained.
SUCCESSIVE DIFFERENTIATION.
61
isr L,
93. The following geometrical illustration will make the last
article more intelligible.
Let P, Q be any two points of
the curve corresponding to equa-
tions (1); T, S, any points arbitra-
rily chosen in the tangents at P
and Q. Then, the remaining lines
of the figure being drawn parallel
to the axes respectively, PF and
TF may be taken to represent the
values of the differentials at P, and
QX and SX to represent their values at Q.
The positions of T and S being perfectly ai'bitrary and inde-
pendent of one another, the magnitudes of the differentials at Q are
quite independent of their magnitudes at P. But we may if we
will so fix the positions of T and S that PF and QX shall be con-
nected by any law we please.
Thus we may impose the restriction that one of the differentials,
as da;, shall at every point of the curve be some determinate function
of the corresponding abscissa, or ordinate, when dy will also take a
variable value dependent on the assumed value of dx and on the
form of the curve. Suppose for instance we determine that dx
x^
shall always have the value — , i. e. that T, S, &c. shall be so
chosen that
and that the same law may prevail throughout the curve.
The corresponding values of dy will then be given by the
equations
TF = -^ . tan TP F, SX = -^^- tan SQX.
By this assumption both dx and dy are made functions of the
variables, their analytical values being
dx = —
^i/ = -/W-
In like manner we may determine the positions of the points T^
S, &c., throughout the curve, by any other rule, or we may leave
them if we please, perfectly arbitrary and unfettered by any rule at
62 SUCCESSIVE DIFFERENTIATION.
all, since the equations (2) are equally true whatever rule is adopted
and whether any rule be imposed or not.
One of the simplest rules for fixing the values of the differentials
is to give PV the same value at all points of the curve, so that
QX= PV wherever P and Q may be. In this case TF will not
preserve a constant value, since TV = PV. tan TPF and the angle
TPV varies from one point of the curve to another. In fact we shall
have with the above rule
dx=C, dy = Cf'{x).
Another equally simple rule is to make TV constant throughout
the curve, that is, to assume
dy = C and thence dx = C(p'{y).
When PV retains a constant value x is (as above stated) called the
independent variable, and when TV is constant y is called the inde-
pendent variable.
Another form of restriction sometimes imposed upon the differen-
tials is to take some function of one of the variables as >// (x) and
make its differential ^|/'(x) dx constant, that is, to determine dx and
dy by the equations
, C , Cf{x)
\}/-{x)' ^ ^^ (•■»■)
When this is done, \//(a;) is said to be the independent variable.
When no restriction is imposed on the values of PFand TV, the
independent variable is said to be general. In this case therefore
the differentials must be treated as arbitrary functions of the vari-
ables, to which we can at any time assign such definite forms as may
be found convenient.
94. Since dx, dy are in general (before any variable is made the
independent variable) functions of the variables, they, like all other
functions, may have differentials, and we shall meet with such quan-
tities as d{dx), d{dy), d{d{dx)], d{d{di/)). Sec, which are written for
consciseness
d^x, d'y, d\v, d^y . . . d''x, d"y,
and are called the second, third ... ?«'*' differentials of x and y re-
spectively.
Again, f'{x) being a function of x must have a differential co-
efficient, and this a third, and so on. These are written
f"{x),f"'{x)...f'%x),
SUCCESSIVE DIFFERENTIATION. 63'
and are called the second, third . . . n^^ differential coefficients of
/(.r) or 1/ with respect to x. So also we have
the second, third ... m"* differential coefficients of (p{i/) or x with
respect toy.
95. It should be observed that the idea of a particular inde-
pendent variable applies only to differentials and not to differential
coefficients. The ratio -~- ov f\x) is the same whether dx or dy be
variable or constant and is in no way affected by the assumption of
a particular independent variable. So from the figure of Art. QS,
it is evident that tan TPV is independent of the position of T.
df'(x)
Again, /"{x) v/hich = - ' . has the same value whatever be the inde-
pendent variable, and the same is evidently true of all the successive
differential coefficients. An equation among differential coefficients
therefore, if it involves no differentials, does not imply the assumption
of any particular independent variable and is equally true irrespective
of any such assumption. It will be seen hereafter that an equation
among differential coefficients with respect to x, is immediately con-
vertible into a corresponding equation among differentials with x for
indepe?ident variable. In consequence of this, x is sometimes incor-
rectly called the independent variable in the former as well as in the
latter equation. It is important that the student should avoid this
error*, as some of the following propositions wholly rest upon the
fact that an equation among differential coefficients is unaltered by
a change of the independent variable.
96. When /'(x) or ^'(y) is known, the ratio o^ dy to dx is also
known. So when /'(a;) and/" (a;) or (p'{r) and (p'\x) are known, we
can find the relation between dx, d"x, dy and d'y ; and generally
when the first w differential coefficients of one variable with respect
to the other are known, we can find the corresponding I'elation among
the differentials of the variables of the orders up to the «"'. These
relations we shall now determine.
" The phraseology referred to is called errorieous, with reference only to the defini-
tions before given. Totally different definitions are sometimes employed wliich
would justify that application of the term ' independent variable.'
64 SUCCESSIVE DIFFERENTIATION.
97. To find the relations between the successive differentials
and differential coefficients when the independent variable is general.
Let y =/(x),
then dy=f'{x)dx (l);
differentiating both sides of this equation, we obtain
d'y^d{f'{x)dx}
= df'{x) dx +/{x) d'x. Art. 23.
.-. d'y ^f"{x) dx' +f(x) cPx (2).
By differentiating this equation, we obtain (Art. 21 and 23),
d^y = df"{x) da? + d (dx')/"{x) + df'{x) d'x + d^xf'{x),
.'. d^y =f'\x) dx^ + 3f"{x) dx d'x +f{x) d^x (3),
and similar equations may be found connecting the differentials and
differential coefficients of higher orders.
These equations give the successive differentials of y explicitly
in terms of the differentials of x and the differential coefficients of y
with respect to x. The differential coefficients may be found expli-
citly in terms of the differentials from the above equations, but more
readily as follows.
98. To express the differential coefficients in terms of the dif-
ferentials when the independent variable is general.
If y^fi'^).
/'«4: ^*)'
... (An..6)/'W = i<^{g} = 5?^^?^^/ (5).
Again,/-(:.) =-^ = d^^XTx"^ \tJ]
dx (dxd^y - dy d^x) - 3d'x (dxd^y - dyd^x) ^
= d^' ~" ^^^'
and similar equations may be found for the differential coefficients of
higher orders, the equations (4), (5), (6), &c. being equivalent to
(1)' (2)5 i.^)' ^^' ^^^^ deducible from them.
99. To find the relations between the successive differentials
and differential coefficients when x is independent variable.
SUCCESSIVE DIFFERENTIATION. 65
When X is made independent variable dx is constant, and therefore
d^jc, d^x, &c., all equal to zero. Hence we obtain by differentiating
the equation,
dy =/'(.^) doe,
.'. d^y =f"(x)dx', since dx is constant,
d'y=f"{x)dx',
equations which might have been obtained from those of Arts.
97 and 98 by equating d^x, d^x, &c. to zero.
100. Examples of successive differentiation.
1. To find the successive differential coefficients of 1/ where
2/ =/(/) = «'^"-
We have by repeated applications of the rule of Art. 27,
/'(x) = 11 . aa;""^
f"{x) = n.n-l. cx"-^
f"\x) = n. 71-1... 2.1 .a,
• From the above equations we see that when x is independent variable the
successive difFerential coefficients may be written as above — — ^...---^. These
dx dx^ dx"
expressions are sometimes used instead of /' (a-), f"{x), &c., when x is not inde-
pendent variable, but in that case it is clear that the numerators will no longer
represent the successive diff'erentials of y, and that these expressions in fact cease to
be fractions, and become mere symbols equivalent to fix), f"{x)...f'-"l{x). Thus,
if this practice were adopted, equation (2) of Art. 97 would become
the d^y on the left of the equation, meaning the second differential of y, while in the
numerator of -r-^ it has no such meaning. It is better to avoid such a use of the
notation, but as it is often met with, it must be remembered that -r^, &c. are then
mere symbols and not fractions.
II. D. c. 5
66 SUCCESSIVE DIFFERENTIATION'.
and all the difFerential coefficients after the Ji*-^ vanish, since /"(x)
is constant.
The successive differentials when x is independent variable follow
at once from the differential coefficients: as appears generally in
Art. 99.
Thus we have
di/ =f'{j'c) dx = n . ax"~Ulx,
d^y =f"{x) dx^ = n.n-l. ax^'-dx".
d^y =f"\x) dx" = n.n-l ...2.l.a dx''.
If it is required to find the values of dy, d^i/, &c. when the inde-
pendent variable is general, we must proceed exactly as in the
general case of Art. 97- Thus, by differentiating repeatedly and
considering dx variable, we obtain
y = ax"",
dy = n. ax"'^ dx,
d'y = n.n-l. ax^-^dx" + n . ax""' d%
d^i/ = n.n-l.n-2. ax^'^dx^ + n.n-l. ax''-\2dxd'x)
+ n .n-1. ax"~- dxd^x + n . ax"~^ d^x,
= n.n-l .n-2. ax"-^dx^ + 3n.n-l . ax^-'dxd^x + n . ax"-^d%
and by pursuing the same method the higher differentials may be
found.
2. Let t/ =/{x) = e\
Then proceeding as before, /'C-^) = ^^»
f\x) = e\
f'Xx) = e-,
and when x is independent variable,
di/ = e"dx,
d^y = e'dx^,
d^y = e^Jx".
When the independent variable is general, we have
dy = e^dx^
d^y = e'dx' + e^d^
SUCCESSIVE DIFFERENTIATION. 67
d^y = e'dx^ + e' . 2dxd'x + e^'dxcPx + efd^x,
= e^dx^ + Se'dxd'x + e'd^x,
Sec. = &c.
Further examples of successive difFerentiation will be found in
Gregory's Examples, Chap. ii. Sect. 1. The quantities there obtained
are successive differential coefficients expressed by the notation men-
tioned in the note to Art, QQ ; or if x is considered to be the inde-
pendent variable </y, d^t/...8cc. properly represent the successive
differentials. The student is recommended to work a few of Gre-
gory's examples also with a general independent variable.
101. We have already employed the term differential equation
in speaking of equations involving differentials. For convenience
sake, equations involving differentials or differential coefficients up to
the 1st, 2nd. . .n*^^ respectively, are called differential equations of the
1st, 2nd. ..n'^ order. Thus the equations (1), (2), (3), of Art. 97 are
differential equations of the 1st, 2nd, and 3rd orders respectively.
Equations which (like those last mentioned) are obtained imme-
diately by successive differentiation, are also sometimes called the
1st, 2nd, 3rd, &c. derived equations, and the original equation (as
y=f(x) in Art. 97) is then styled the primitive. Thus in the 2nd
example of the previous article the last equation is the third derived
equation of the primitive y - e^, and is a differential equation of the
3rd order.
Besides the derived equations which are the immediate results of
differentiation, an infinite variety of differential equations of every
order may be formed by combining together the primitive and its
derived equations in any manner we please.
Every differential equation, again, whether an immediate derived
equation or compounded out of several, may be presented in a
number of different forms by adopting a general or different particu-
lar independent variables. Thus the derived equations of Art. 97
assume the simple forms of Art. 99, when x is made independent
variable. By altering the independent variable we vary only the
form and not the substance of a differential equation. The form
corresponding to a general independent variable includes all the
others, and we shall shortly prove that any one of the particular
forms may be deduced from any other. Before investigating the
5—2
'68 SUCCESSIVE DIFFERENTIATION.
methods of effecting such transformations, we will give some exam-
ples of the formation of differential equations by combinations of
various primitives with their derived equations.
102. Let the primitive be
y = ax + hx^ (1).
The first two derived equations (when the independent variable is
general) are
dj/ = adx+ 2bxdx (2),
d'i/ = 2bdx' + {a + 2bx)d'x (3).
1 . Let it be required from these equations to obtain a differential
equation of the first order, in which x shall not appear.
To do this we have only to eliminate x between (1 and (2).
Thus, from (2) we have
d^- = (a- + 4fabx + 4iV)^x^
= (a'' + ^by)dx' by (1),
which is the required equation.
2. From the same primitive let it be required to find a differen-
tial equation from which a and b shall disappear.
For this purpose we must employ equations (1), (2), and (3), as
there are two quantities, namely a and b, to be eliminated. Thus,
eliminating a between (l) and (2), we find
xdt/ — ydx = bx^dx ;
and between (I) and (3),
xd^y —yd^x = 2bxdx^ + bx^d^x,
and eliminating b between the two last equations we obtain the
required differential equation
x^d'^y dx - 2x^dydx^ - x^dyd^x + %vydx^ = (4).
If we had worked with x for independent variable we should have
had, instead of equations (1), (2), (3),
y — ax + bx^,
dy = adx + 2bxdx,
d'y = ^bdx'.
By eliminating a and b between these, we find
x^d^i/ - 2xdydx + 2ydx^ = (5),
which differs from equation (4) only by the omission of the terra
involving d^x, which vanishes when x is independent variable.
SUCCESSIVE DIFFERENTIATION. 69
If we divide the last equation by dx^ it becomes
dx^ dx ^ '
which by Art. Q[) is equivalent to
j;y"(^)-2^/(x) + 2^ = (6),
(4), {p), {6), are three different forms of the required equation.
103. Equations formed as in the last example by eliminating
constants are of frequent occurrence. In general, the order of the
resulting equation must be equal to the number of constants elimi-
nated. For to eliminate n constants we require in general 7^ + 1
equations. The primitive and its first n derived equations must
therefore be used, and the equation thence obtained will involve an
«"' differential or differential coefficient, that is, it will be of the Ji"*
order. In particular cases, however, it may happen that the steps
necessary to eliminate n constants cause others also to disappear
when the order of the differential equation will be lower than that
given by the general rule.
104. In some cases also we can eliminate functions of the
variables as well as constants.
Thus, let the primitive be
y = a sin (a; + 6) (1).
From this we may eliminate the circular functions, as well as the
constants a and h.
The first two derived equations are (making x independent
variable)
dy = a cos Qc + b)dx (2),
d'y =- a sin (j; + b) dx^ (3).
Between (l) and (3) we may eliminate a sin (x + b) and obtain
d^y+ydx''=0.
If we had expressed this in the notation of differential coefficients
it would have been
and if we had kept the independent variable general,
d^y dx - dyd^x + ydx^ = 0,
as may easily be verified.
70 SUCCESSIVE DIFFERENTIATION.
Examples of the formation of dilFerential equations by the elimi-
nation of constants and functions will be found in Gregory's Ex-
amjjles, Chap. iv. The results with a general independent variable are
not there given, but if obtained, their accuracy may be tested by
making d^x, (Px, &c. vanish when they ought to reduce themselves
to the given forms.
105. If we consider the ?j'^ power of a first differential as
homogeneous with an n"^ differential, it will be seen that all the
differential equations we have yet met with are homogeneous in the
differentials. Thus, in this sense, every term of equation (4) in Art.
102, is of three dimensions, and every term of equations (5) and (6)
of two dimensions and of no dimensions, respectively.
This homogeneity is not accidental, but must be found in every
differential equation. For all such equations are either derived
equations or are formed out of them. Now all derived equations are
included in the general forms of Art. 97, which are homogeneous in
the above sense: the same property must therefore belong to all
particular derived equations and to all equations deduced from them.
106. It has been before stated (Art. 101), that the different
forms assumed by a differential equation when expressed in terms
of differentials with various independent variables or of differential
coefficients may be deduced one from another, so that when any one
form is given all the others may be obtained. This we shall now
effect in the several cases.
107. Having given an equation among the differential coefficients
of J/ with respect to x, to obtain the equivalent equation among the
differentials of x and y when x is independent variable.
To do this we have only to substitute for f'{x\ f"(x) . . . f"'\x)
in the given equation their values -^ , (S'"^dx" ^^°^ ^^^' ^^' ^^^
if we wish to get rid of fractions, multiply by the highest power of
dx which appears in the denominators. The converse transition is
equally easy. For an equation among the differentials when x is
independent variable can only contain the differentials dx, di/,dy...d"i/.
By the equations dt/ =f(x)dx, d'y =f"{x) dx\..d"y =f'\x) dx" of Art.
99, we can eliminate the differentials of y ; when dx will be the only
differential remaining, and since the equation must be homogeneous
SUCCESSIVE DIFFERENTIATION. 71
(Art. 105) dx must enter to the same power in every term and may
be divided out, leaving differential coefficients only.
108. Having given an equation among the differentials of x
and y, with a general independent variable, to find the equivalent
equation where x is independent variable.
To do this we have only to make dx constant, and therefore
dPx, d^x d'^x equal to zero, when the equation is immediately
reduced to the required form.
109. Having given an equation among the differentials of x and
y, in which x is independent variable, to find the equivalent equa-
tion when the independent variable is general.
The given equation may be transformed into that among the
differential coefficients of y with respect to x, by dividing by some
power of dx. (Art. 107.)
This may then be transformed into the equation among the
differentials when the independent variable is general, by the fol-
lowing equations, obtained (as in Art. 98) by the successive dif-
ferentiation of the equation
f{x)=y, viz.
1 ,ri JdtfW dxjdxd^i - dyd^x) - Sd' xjdx d'y - dyd'x)
rx^^=dAdx'Kdi)\-'^--'^ — dx^ '
&c. = &c.
dx having been considered variable in the diflPerentiations because
the independent variable is general.
The expanded forms after that for /'(.r) are difficult to remem-
ber, and should be worked out from the first or unexpanded forms
in each particular case by performing the operations indicated, re-
membering that both dx and dy are to be considered as variables.
110. Having given an equation among the differentials of x
and y when the independent variable is general, to find the equi-
valent equation when any function of .r or y is independent variable.
In this case, since neither dx nor dy is to be made constant, the
equation is still true in its original shape ; but it may be simplified
72 SUCCESSIVE DIFFERENTIATION.
by eliminating one of the variables, and so obtaining a new equation
between the differentials of the other and of the new independent
variable. This may be done as follows.
Let z be the new independent variable, and let x = (z), where
(p is known. Then differentiating this equation, remembering that
z is independent variable, we obtain
dx = <p'{z')dz,
d'x = ((>" {z)dz%
d"x = (p^"\z)d2" ;
by substituting which values in the original equation we obtain the
required equation between the differentials of ^ and z.
By dividing by some power of dz, this may be at once trans-
formed into the equation among the differential coefficients of ^ with
respect to z. (Art. 107).
111. In the preceding cases, one of the two forms of the equa-
tion, either that to which or that from which we have had to pass,
has had a general independent variable. In those that follow, both
equations have determinate independent variables. Such transfor-
mations are of much more frequent occurrence than the former ones.
112. Having given an equation among the differentials of x and
y when X is independent variable, to find the equivalent equation
among those of i/ and z when z is independent variable, z being a
function of one of them as x, or as it is commonly expressed, to change
the independent variable J'rotn x to z.
By dividing by some power of dx, the original equation may be
transformed into the equation among the differential coefficients of _y
with respect to x. (Art. 107).
Then by differentiating /(x) = i/, remembering that, when z is
independent variable, dx is variable, we obtain
w/.^N _1 ^/M _ d'i/dx~d'xdi/
J ^''^~ d'x'^Kdx) ~ dx'
^"'( \ ^ rl\^ .(dy\[ _dx{dxd^y- dyd^x)-3d^x{dxd^y-dyd^x)
J ^''^^di'^Ui \tx)]- ' d^' '' '
&c. = &c.
Also, if the given relation between x and z is put in the form
x = (p{z\
SUCCESSIVE DIFFERENTIATION". 73
we have dx = (f)'{z)d3.
d"x = (}>^"\z) dz",
when z is independent variable.
By substituting these values of x and its differentials in the
former equations, /' (x), /"(x), &c. may be expressed in terms of z,
dz and the differentials of ?/, and the required transformation effected
by the substitution of these values in the original equation.
By dividing by some power of dz, this equation is transformed
into the equation among the differential coefficients of ^ with respect
to z. (Art. 107).
113. Instead of first expressing the differential coefficients at
length in terras of the successive differentials of x and i/, and sub-
stituting for those of x their values derived from the equation ^ = (z),
it will be found more convenient in practice to invert the order of
these operations, and first substitute for dx its value in the unex-
panded forms off(x), f"{x), &c., and then perform the differenti-
ations, remembering that dz is constant. As an example, we will
change the independent variable from x to z, where x = e' in the
equation
x^ d^y + 9.xdy dx +y dx"^ = 0.
This is equivalent to
"" dx'^"^ dx^^ '
or to ^y (^) + 2x/(:r) +/(x) = 0,
since x is independent variable.-
Then, since x = e% dx = e''dz,
"J ^^~dx~ e'dz'
-^W- ,1^ -eUlz'^Vdz) e'^dz' '
since z is independent variable, and therefore dz constant.
Hence the equation becomes
d-y + dy dz + y dz^ = 0,
74 SUCCESSIVE DIFFERENTIATION.
where -j- , j4 ^^"^ ^^ differential coefficients of y with respect to z.
114. Having given an equation among the differentials of a; and
y when x is independent variable, to find the equivalent equation
when y is independent variable, or to change the independent variable
from X to y.
By dividing by some power of dx, the given equation may be
transformed into that among the differential coefficients of 3/ with
respect to x. (Art. 107)«
Then, differentiating successively, remembering that y is inde-
pendent variable, we obtain from the equation
, Sid-xY-d^xdx
= ^•^^-^1? '
&c. = &c. ;
and by substituting these values in the original equation the required
transformation is effected.
The resulting equation may be transformed into that among the
differential coefficients of x with respect to y, by dividing by some
power of c?^. (Art. 107)-
115. The methods of effecting these transformations will be easily
remembered, if it is observed that
(1) An equation with x as independent variable is immediately
transformed into an equation involving only differential coefficients
with respect to x, by division by some pov/er of dx.
(2) The expressions for the differential coefficients when any
variable is made independent, are obtained from the expressions in
which the independent variable is general, viz.
/
'W=l-/"W=i<l)./"'W=i;Mij''(©}'^'=-
SUCCESSIVE DIFFEHENTIATION. 75
by making the differential of the variable which is to be indepen-
dent, constant, and performing the operations indicated, on that
supposition.
(3) If the variable selected for independent variable is neither
X nor 1/, but a given function of x, dx must be expressed in terms of
the differential of this quantity, and substituted in the above ex-
pressions, and the opei'ations performed on the supposition that this
differential is constant. In like manner, if the new independent
variable is given as a function of y, dy must be eliminated from
the above expressions before performing the differentiations.
116. We will take as an illustration the equation
y — e' cosx=f{x),
and deduce from it equations among the differentials in various
forms, and then apply the foregoing methods to obtain them one
from another. By differentiating
y = e^ cos X,
we have, when the independent variable is general,
dy — e" (cos x — sin x) dx,
d^y = — Se" sin xdx^ + e" (cos x — sin x) d^x.
From these equations different relations among the differentials may
be found: we will take that one from which cosj: and sinx dis-
appear. Then, eliminating sin x and cos x, we obtain
d'ydx - dy (2dx' + d'x) + 2ydx^ = (1).
If we had differentiated with x as independent variable, and then
eliminated sin x and cos x, we should have found
d'y-2dydx + 2ydx''=^0 (2).
Again, the successive differential coefficients of ^ are
/' (x) - e" (cos X - sin x),
f"{x) = — 2e''s'mx,
whence we have, by eliminating sin x and cos x,
/"{x)-2/(x) + 2fix) = (3).
Suppose now x = log s, the original equation becomes
y = z cos (log 2),
whence, if z is independent variable,
dy = {cos (log ~) - sin (log z)} dz,
d'^y = {sin (log 2) + cos (log z)] dz^ ;
76 SUCCESSIVE DIFFERENTIATION.
whence we obtain
z'dPy-sdi)ch-^-2ydz' = (4).
Lastly, differentiating the original equation with y for inde-
pendent variable, we obtain
dy = e' (cos x — sin x) dx,
= — 2e" sin xdx"^ + e" (cos x — sin x^ d'x;
whence, by eliminating sin x and cos x, we have
d^xdy + Qdx'dy-2ydx^ = (5).
These five equations are all equivalent and may be deduced one
from another by our previous rules.
Thus (2) is obtained immediately from (1) by putting d^x=0,
and (3) from (2) by dividing by dx^. (Arts. 107, 108).
(l) is deduced from (2) by the method of Art. IO9. Thus, first
change (2) into (3), and then we have
^"(r\- ^ ^('^A d'ydx-d'xdy
J ^^^-dx'^KTxJ- d? '
Substituting these values in (3), we have
d'^ydx — d^xdu ^di/ ^
.: d^ydx - dy (2dx^ + d^x) + 2ydx^ = 0,
which is equation (l).
Again, (1) may be transformed to (5) by making d'y = 0, and to
(4) by the method of Art. 110.
To interchange (2) and (4), that is, to change the independent
variable from x to 2 when x = log;?, we must proceed as in Art. 112
or 113.
By dividing by dx^, (2) becomes identical with (3), and when z
is independent variable, we have
and since x = log 2, dx = -^,
"J^'^)- dz '
SUCCESSIVE DIFFERENTIATION. 77
= -r^d(2dy) since dz is constant,
_z{zd^y + dzdif)
then, substituting in (3), we obtain
.*. z^^y — zdzdy + 2y(/s^ = 0,
which is equation (4).
In the same way (2) may be obtained from (4).
To interchange (2) and {S), or to change the independent variable
from X to y, we must proceed as in Art. 114.
First reduce (2) to (3) as above ; then
f"{x) = X. ^ ( J/ ) ' ^y being now constant,
Then, substituting in (3), we have
_dylx dj
dx^ ^dx^-y ^'
or ^xdy + 2dydx^-2yda^ = 0,
which is equation (5).
In the same way (2) may be obtained from (5).
The student should work out those transformations which are
not given at length in the text.
For examples in the change of independent variables, see Gre-
gory's Examples, Chap. iii. Sect. 1.
CHAPTER VI.
DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
AND IMPLICIT FUNCTIONS.
117. The quantities we have hitherto differentiated, have always
been given as explicit functions of a single variable. A function of
one independent variable may, however, be expressed by an equa-
tion of the form u = F{x, y), where x and y are themselves connected
by some other equation (Art. 8). In such a case the differential of
u may be found by expressing one of the variables x and 3/ as a
function of the other by means of the given relation between them,
whereby u will be reduced to the form of an explicit function of a
single variable and may be differentiated by our former methods.
Thus suppose we have
u = Jx^ + f,
where x and y are connected by the additional equation
y = inx + c.
By substituting this value of j/ in the former equation we obtain
u = Jx^ + {inx + cf,
which may be differentiated by the rules of Chap. in.
The following theorem will enable us to differentiate such func-
tions more readily, without the necessity of any previous elimination.
118. Let u = F(x,y), (1),
where x and y are connected by the equation
l/=/(.^) (2)-
When X receives an increment Ax, y and ii will receive definite
corresponding increments Ay, Am which vanish with Ax, Au is
given by the equation
Au='F{x + Ax, y + Ay) - F (x, y)
= {Fix + Ax, y + Ay) -F{x + Ax, y)} + {F(x + Ax, y) -F{x, y)}.
The latter part of this expression is the increment which F (x, y)
would have received, if x alone had been variable and y constant;
it may be written A^F{x, y).
The former part is the increment which F (x + Ax, y) would
have received if y alone had been variable and x constant, and may
in like manner be written AyF(x + Ax, y).
DIFFERENTIATION, &C. 79
Hence the above equation becomes, on dividing by one of the
increments, as Ax,
Am _ A, F (x, y) \jF{x^-Ax,y)
Ax ~ Aa; Aj? '
which being true always, is true in the limit when the increments
approach zero,
A« A,F{x,y) AyF(x,y)
X being written for x + A.r in the second term because x and x + Ax
become identical in the limit.
A?< da
A,F{x,y) d,F(x,y)
Also //a.o ^^ = ^^— = ^ W>
where d^F(x,y) and i^'(x) signify the differential and differential
coefficient respectively of F (x, y) obtained on the hypothesis that y
is constant and F (x, y) a function of x alone. Similarly
// ^yFJ.^, y) _ dyF(x, y) _
"^-^ A^^ ~ dy -^ ^-^^
where dyF(x,y) and F'{y) are the differential and differential co-
efficient respectively of F(x, y) obtained on the hypothesis that x is
constant and F(x, y) a function of j/ alone;
. // ^yF(x,y) _ F(x,y) Ay
•• '^^-» A^ «A=o ^^ •"^=OAx
_ (Zy F (.r, ?/) dy^p,, .di
dy dx ^"^^ dx
_ d,jF (x, J/) ^
Substituting these values in the above equation and multiplying
by c?x, we obtain
du = d,F(x,y) + d,F(x,y) (3),
or du = F' (x) dx + F' (y) dy (4).
F'(x), F'{y), being differential coefficients obtained by treating
F(x, y) as a function of one variable only, may be found by the
methods of Chap, iii., and equation (4) will then give one relation
between the three differentials du, dx and dy.
80 DIFFERENTIATION OF
By differentiating equation (2) we obtain
cly=f{x)dx,
by which equation together with (4), du may be expressed in terms
either of dx or dy.
d:,F(x,y) and d^ F(x, y) are sometimes written for conciseness
dji dyU*, and are called the partial differentials of u with respect to
X and 3/ respectively; F'{x), F'(ij) are called the partial differential
coefficients of u with respect to x and y respectively ; du is what we
have already defined (in Art. 17), as the differential of «: to dis-
tinguish it from the partial differentials, it may be termed the total
differential of u. It is evidently the same quantity which we should
have obtained by eliminating one of the variables between equations
(l) and (2) and differentiating the result.
The theorem proved above may therefore be stated thus :
The total differential of a function of two variables, which are
themselves functions of one another, equals the sum of the partial
differentials with respect to the separate variables, or (with the ab-
breviated notation)
du = d^u + dytt.
119. We will apply the theorem to the example of Art. 117-
u = Jx' + f,
where j/ = mx + c.
* In using this notation, it must be borne in mind that u is merely an abbrevia-
tion for the function F (.r, t/). The pair of equations (1) (2) may be transformed into
a number of different pairs, each of which would give the same value of u in terms of
either of the other variables, but all leading to different values of dxU dyU. Thus
the pair of equations u = xy^ and j^ = c' is equivalent to the pair m = wye' and y = ^' %
but the former give d^u = y'^dx = e^'dx, and dyU — Ixydy = 2xe'dy, while the latter
give dxU={\-vx)ye'dx ={\ + x)e^'dx and dyU = xe'dy. Hence when dxU and dyu
occur, u must be understood to stand for the same function of x and y in both.
Another variation of notation must also be noticed. The partial differential co-
d 'it d It
efficients F'(a') F'(y) which with the notation of the text equal -^, -~- respectively,
dll du , , . , . • 1 fni 1
are sometimes written — , — , the subscripts being omitted. Ihese latter quantities
then cease to be ratios, and become mere symbols equivalent to F'{x), F'{y), and the
partial differentials are then written -r- • dx, -7- . dy. This is the notation used in
*^ dx dy
Gregory's examples. We shall recur to the general subject of notation in Art. 138,
infra.
FUNCTIONS OF SEVERAL VARIABLES. 81
Here cim = , _ , a,.u =
J^' + f' " J^' + f'
, , , X dx + 11 dif
.: an = dji + dji = — , ' .
du
If we wish to obtain -7- as a function of x we must employ the
second equation to eliminate y and dy. Thus differentiating, we have
dy = m dx, .
and substituting in the value of du for y and dy their values, we find
, X dx + hnx + c)m dx
du = . / '^ - ,
Jx^ + (rnx + cy
du (I + 7)1^') X + mc
or -^ = ^ == ,
"^ sjx'^ + {mx + cf
the same result which we should have obtained by pursuing the
method indicated in Art. 11 7.
In like manner -— may be obtained.
dy ^
120. The relation between x and ^ was expressed in Art. 118.,
for simplicity's sake, by the explicit equation y =f{x) ; but the
theorem du = dji + dyti, is independent of that assumption (which
nowhere enters into the proof) and is equally true whatever be the
form of the relation between x and y provided they are in fact
functions of one another. Thus the relation may be expressed by
the implicit equation /(x, y) = 0, or by the pair of equations
^ = 0(2)^ y = ^{^)>
or more generally still by a system of n equations involving j:,^and
n — \ other variables, since by solution or elimination any of these
forms will give ^ as a function of x.
121, As an example, let it be required to express du in terms
of dz from the equations
Here dju = cos
H. D. c.
82 DIFFERENTIATION OF
dx -
= e" (/~,
dy-
= 22 dz,
du =
= dji + dyif,
-.
--©{"'/■
vdt/)
--(i){—
- 2e'z dz
z*
_
= r-3 cos (-\ (z-2
)dz.
}'
If we had first substituted for x and y their values, we should
have had
= sin f pj
by differentiating which equation we may obtain the same result
as before.
In like manner, du may be expressed in terms o£ dx or dy.
122. If there are more than two variables under the functional
sign, it may be proved in a precisely similar manner that the total
differential equals the sum of the partial differentials with respect to
these variables. Thus, if u =f{pc, y, z . . . v) where x, y, z ... v are
connected by a sufficient number of equations to make u a function
of one independent variable only*,
du = d^u + dyU + d^^u ... + d^u.
Drfferenliation of Implicit Funciiojis of one Independent Variable.
123. We are now able to find -^ where x and y are connected
dx ^
by an implicit equation, without solving it with respect to either of
them. Let x and y be connected by the equation
F{x,y) = 0.
Then if Ai-, Aj/ be corresponding increments of x and y, the
equation must be satisfied by x + Aj; and y + Ay,
.-. F(x + Ax, y + Ay) = 0,
* This condition is essential. The very iirst step in Art. 118 depends upon it,
since without it, an increment An' of x would not produce corresponding increments
of y and u.
FUNCTIONS OF SEVERAL VAKIADLES, p3
^ ^F{x,y) _ F{x + ^x,y + ^y)-F{x,y) _^^
"Ax Ax
which being true always is true in the limit;
... ^^j^'-y) = 0, or dF {x, y) = 0.
But dF(x, y) is the sum of the partial differentials of F(x, y),
.'. F'(x)dx + F'(y)dy = 0,
whence -r--- t^,, i •
dx F'(^)
124. In like manner, if
F{x,y,z) = 0,
where x, y and 2 are connected by a second equation so as to make
them functions of one independent variable the variables will have
corresponding increments Ax, Ay, Az, all of which vanish together.
Then
F(x + Ax, y + Ay, s + Az) = 0,
AF(x, y, z) _ F{x + Ax, y + A^, z + Az) - F{x,y, g) _ ^
A^ Ax ~ '
.-. iE^^^lil) = 0, and dF{x, y,z) = 0;
.-. F'(x) dx + F'{y) dy + F'{z) dz = 0,
and generally if
lil F(x,y,z...v) = 0,
where the variables are connected by a sufficient number of equa-
tions to make them functions of one independent variable,
dF{x,y...v) = 0,
and therefore (Art. 122)
F'(x) dx + r(y) dy+... F'(v) dv = 0.
125. Hence if n variables x„ X2.. .x„ are connected by the « - 1
equations
i^i (j:„ ^2 . . . a;,,) = 0,
F^i^i, Xs.. .x„) = 0,
F„.i(xi, irj...x„) = 0,
in which case they are functions of one independent variable only,
we shall have
F/ (x.) rfx, + F/ (r,) dx,...+ F,' (xj dx, = 0,
6—2
hi DIFFERENTIATION OF
F/ (x.) dxi + F/ (j-g) dxa...+ F,' (x„) dx„ = 0,
F'„_i{x,)dx^ + F'„_,(x,)dx,... + F'„_,{x„)dx„ = 0.
From these w — 1 equations, any n-2 of the diiferentials may be
eliminated, leaving an equation to determine the ratio of the remain-
ing two, that is, the differential coefficient of one of the corresponding
variables with respect to the other.
126. Let the explicit equations corresponding to
F(x,i/) = (1),
be 7/=/{x), x = <p{y) (2).
Then since the forms of /and depend on that of F, those of/' and
(p' must be connected with those of F'{x) and F'{y). The relations
between them may be obtained as follows.
The ratio of the differentials dx and c?y must be the same whether
obtained from (1) or (2). But from (1) we have (Art. 123)
F'{x)dx + F'(i/)dy = 0.
Also from (2) dy =f' {x) dx,
and dxr=(p'{y)dy.
These equations must be identical ;
" F\y) J ^'"^ cl>\yy
127. Again, let three variables be connected by the two
equations ^tf
Fix,y,u) = (1),
F,{x,y,v) = (2),
and let the explicit equations obtained by the solution of (l) be
u=f{x,y), x = <t>{u,y), y^y^{x,u) (S).
Then the ratios of the differentials may be determined by combining
the equation dFi = with the differential of either of the above
forms of equation (1), that is, by combining dF^ = with either of
the following equations :
F'{x)dx + F'{y)dy + F'(n)du = (4),
du=fix)dx+/(y)dy....^ (5),
dx = (p'(y)dy + tp'(ii)du (6),
dy = \lrXx)dx + \l/'(ti)du (7),
and since the ratios of the differentials must be the same from which
ever form they are derived, the above equations must be identical.
FUNCTIONS OF SEVERAL VARIABLES. t5
Hence by dividing (4) by F' (?<) and comparing the coefficients
with those of (5), we obtain
So, from (4) and (()) we find in like manner.
And from (4) and (7)
^^^^ = -F^y ^(") = -7^y
^W = -T^)- ^(«) = -p(7)-
128. The same results may be obtained somewhat differently,
as follows.
If we differentiate equation (1) on the hypothesis that _y is con-
stant, we obtain an equation between the differentials of x and ti,
which may be written in either of the forms
P (x) dx + F' (u) dji = 0,
or F'(x) d„x + F' (m) du ^ 0,
the former expressing the fact that y is constant, by assuming it to
be expressed as a function of x and y, and differentiated with respect
to X alone, the latter by assuming x to be expressed as a function of
u and y, and differentiated with respect to ii alone. The one form
tacitly refers to the first, the other to the second of equations (3).
Again, by differentiating equations (3) partially, we have
d^u =/' (x) dx, d^x — (p'{ii) du,
■which being substituted in the above give
F'{x) + F'{it)f(x)^0,
r{x)4>'(jc)+F\u) = 0;
•• F'{u) ^ ^^~(p'{u)'
Similarly, by differentiating equation (1) on the hypotheses that
£ is constant and that ii is constant respectively, we should have
obtained
F'{y) ^^' <P'{yy
the same relations as those found in the last article.
B6 DIFFERENTIATION OP
129. Examples of the differentiation of implicit functions,
CiV
1. To find -— from the equation
(x + yf = x^ + my^.
The equation may be put into the form
7t — (x + yY- x^— my' = ;
.% d^u = 2 (x + y) dx — 2 X dx = 2y dx,
and dyii = 2 (x + y) dy — 2 my dy = 2{x + {l— m)y} dy ;
.'. = dji + dyU = 2y dx + 2{x+ (1- m)y] dy,
... ^ = _ y .
" dx x+{\ —m)y'
The same result may be obtained by solving the equation with
respect to y and differentiating the result, -when the equations of
Art. 126, will be verified.
In differentiating an implicit function it is not necessary to re-
duce it to the form tc = 0. For if we have
Aa;,y) = <i>{x,y),
we obtain by equating the differentials of each side,
f{x)dx+f{y)dy = cp'{x)dx + 4>'{y)dy,
the same equation which results from the differentation of
« =/('^, y)-<P (^^ p) = 0,
since dji = {f'{x) — (p'{x)] dx,
dyn^{f'{y)-cp'{y)]dy.
Thus, if we had differentiated the equation
{x + yy = x^ + 7ny-,
as it stands, we should have got the equation
2{x + y) (dx + dy) = 2xdx + 2my dy,
which is identical with that found above.
2. Let the three variables x, y, z be connected by the equations
x^ if z^ ^
^-^p-^? = ^' (1)^
x" +y +2' = r', (2).
Then from (1)
2x dx 2y du 2z dz
A — =i— ^ -I =
FUNCTIONS OF SEVERAL VARIABLES.
and from (2) 2x dx + 9,ij dy + Zz dz = ;
.'. eliminating dz,
dy a^ c^ X
Similarly^ ^-^— J.
87
The same results may be obtained by first eliminating one of the
variables between (l) and (2) and the relations of Art. 127, may be
verified by solving equation (1) with respect to the several varia&les,
and comparing the results of differentiating equation (1) and the
explicit equations so obtained.
130. In Art. 118, and in the other propositions of this chapter
deduced from it, the quantities differentiated are functions of one
independent variable only, for the number of variables in no case
exceeds the number of equations by more than one. Thus, in the
general case of Art. 125, we might have eliminated from the given
equations, all but two of the variables, leaving an equation of the
form
from which, by the fundamental definition we could have obtained
dxi =f'{x^dXi.
The differential equations of Art. 125, merely afford a different
and generally a readier method of determining the ratio of the same
differentials.
If however the number of equations between the variables is
not sufficient to make them functions of a single independent variable,
the reasoning of Art. 118, not merely fails, but the result becomes
unintelligible, because the definition on which it is based ceases to
have any meaning.
88 DIFFERENTIATION OF
Thus let u, X, y be connected by a single equation whose implicit
and explicit forms are
F{ti,x,y) = 0, (1),
^^=fi.^^y)> (2)>
x = (p{tc, y), (3),
i/ = \}^(u, x), (4).
Then it will be found impossible to attach any meaning to the
differentials du, dx, dy.
For the only definition yet given of a differential is that of
Art. 1 1, which is in terms restricted to the case of functions of one
independent variable^ and may be stated as follows.
If X and y are functions of one another and an increment i\x be
given to x, y will receive a corresponding increment Ay bearing a
definite ratio to Aa; and vanishing with it. Such increments will
have a limiting ratio^ and this limiting ratio is denoted by the
ratio -f- .
dx
For the sake of perspicuity we have, in Art. 118, prefixed the
term total to the differential of u, but we added nothing to the
definition and the -j- of that article means nothing more than the
A?<
limiting value of -r— which might have been obtained by first elimi-
nating y between the given equations. So the partial differentials
of Art. 118 are merely the differentials of u, obtained from the
fundamental definition on a particular hypothesis, and involve no
extension of the definition.
Now if we endeavour to apply the definition to an equation such
as (1) or (2), we find that it gives no meaning to the term differ-
ential (J. e. total differential) of n, because the hypothesis on which
the definition rests, viz., that ?< is a function of some one other variable
is no longer true.
Thus, if
X + /\x, y + Aj/, n + An,
be values of the variables which satisfy equation (2) we have
Au=fix + Ax, y + Ay)-f{x,y),
and no other relation exist between the increments. When an arbi-
FUNCTIONS OF SEVERAL VARIABLES. B9
trary value is given to Ax, Au does not assume a definite value,
until Ay also has been arbitrarily determined. Hence Au does not
bear a definite ratio either to Ax or Ay but depends on both of them,
and no such thing exists as a definite limiting value of either of the
ratios -.— , -r— , on which our definition of a differential was founded.
Ax Ay
In short, the definition of Art. 17, was framed exclusively with re-
ference to functions of one independent variable and has no applica-
tion to the case we are now considering.
It is otherwise however with respect to the partial differentials
and diffei-ential coefficients. The values of /'(*), /'(i/) rnay l^e ob-
tained as before, since the supposition of only one variable receiving
an increment, while the other remains constant, reduces the function,
so far as these operations are concerned, to a function of one variable
only, and brings it within the scope of the original definition. Thus,
if ,r receives an increment Ax while y remains constant, u will re-
ceive a corresponding increment A^u, bearing a definite ratio to Ax.
A II.
—^ will therefore have a limiting value when Ax approaches zero.
Ax
and we may define the differentials d^u and dx to be any quantities
whose ratio equals that limiting ratio, whence we have (as before),
dM ^ , Am ^, . , I'M! dji . A„?/ .,, ^
W = -^^A^o ^=/(^-), and similarly -^ = //,,=„ ^^ =f{y).
Hence, partial differentials of functions of two independent vari-
ables have the same meaning as when the variables are connected.
Total differentials of such functions have no meaning whatever.
If then, the term total differential is to be applied at all to func-
tions of two independent variables, it must be by virtue of some new
definition. As yet, it is undefined, and we are at liberty either to
leave it undefined and unused, or to define it in any way which may
be found convenient. In selecting a definition, we shall seek one,
which, while it applies to functions of two independent variables, in-
cludes within it the definition before given of a total differential of
a function of one independent variable only. The advantage of sucli
a definition is obvious as it will enable us to work with differentials,
without enquiring in each case into the number of independent vari-
" We cannot now use ll^_^ instead of ltjy,_^ becaitse Ac and i\y do not necessarily
vanish toijetlier.
90 DIFFERENTIATION OF
ables. Indeed, unless we attached to the term when extended to
functions of two independent variables, a meaning analogous to that
which it bears when there is only one, it would be better to leave it
undefined and to confine its use exclusively to the latter case.
In order to maintain this analogy we define the total differential
as follows.
Def. The total differential of a function of two independent
variables is the sum of the partial differentials.
When expressed in analytical language (with reference to
equation 2,), the definition is as follows. The total differential, dit,
of ?< is a quantity which satisfies the equation
du = d^u + dyii,
or du^f(x)dx+f'{y)dy (5).
So {with reference to equations (3) and (4)} the total differentials
of X and_y are defined by the equations
dx = (p'(u) du + (p'{y) di/, (6),
and dy=^-^'{u)du + y\/'{y) dy, (7),
or still more generally with reference to the implicit equation (l),
the definition assumes the following shape.
If dx, dy, du are any three quantities which satisfy the equation
F'{x)dx + F'{y)dy + F'{z)dz = 0, (8),
dx, dy and du are called the total differentials of x, y and u.
The several equations (5) (6) (7) and (8) are all identical, by
virtue of the necessary relations which exist between the forms of
F,f,(p and y^/, as appears fi-om Art. 127, so that the differentials on
the right hand side of equations (5) (6) and (7) as well as those on
the left, represent total differentials.
When u is a function of more than two independent variables,
the total differential is defined in like manner as the sum of the
partial differentials. Thus, if
F{x, y ... ti, v] = 0,
one of the explicit equations being
the total differentials of the variables are defined by the equation
F'{x) dx + F'{y) dy ... +F' {ic) du + F'(v) dv = 0,
FUNCTIONS OF SEVERAL VARIABLES.
91
or by any of the equivalent equations, as
du =f'{x) dx +f'{y) dy+... f'{v) dv.
These equations, therefore, having been deduced from the original
definition of Art. 17, in the only case where that definition applies,
and being themselves taken as the definition in all other cases, are
universally true whatever may be the number of independent
variables.
131. Hence, generally, if n variables are connected by r equa-
tions,
i^,(xi,A-2 ... j:„) = 0,
F^ (j?! , Xa . . . X,^ = 0,
the relations among the total differentials will in all cases be given by
the equations
F\(x^) dxi + F\{x;) dx^...+ F\{x„) dx„ = 0,
F'g{xi) dxi + F\{x^ dx^ ...+ F's,(x„)dx„ = 0,
F\{xi) dx^ + F'X^,) dx^...+ F',(x^) dx„ = 0.
132. Whatever be the number of equations between the varia-
bles, there will be the same number between their differentials.
Hence, when the variables are functions of one independent variable,
the number of equations between the differentials will be one less
than the number of differentials, and one differential may have an
arbitrary value given to it when the others will take definite corre-
sponding values. When there are two independent variables, arbi-
trary values may be given to any two of the differentials, and in
general as many differentials may be arbitrarily determined as there
are independent variables in the original equations.
Thus when the only equation between the variables is
F{x, y, u) = 0,
dx, dy, du are by the definition any quantities which satisfy the
equation
F'(x)dx + F'(y)dy+F'(u)du = 0.
To any two of them, therefore, we may give what values we
please, when the above equation determines the value of the other.
So in all cases where there is but one equation given between the
0% . DIFFERENTIATION OF
variables, all but one of the differentials may have arbitrary values
assigned to them,
]33. The definitions of Art. 130 will not at once enable the
student to attach any other idea to differentials of functions of several
independent variables than that of quantities which satisfy a par-
ticular equation. There is nothing in this notion analogous to the
fundamental idea of differentials of a function of one independent
variable, viz. quantities whose ratio equals the limiting ratio of the
increments. Such an idea is however involved in the definition of
Art. ISOj and it is important to consider it, on account of its value in
geometrical and other applications of the subject.
An alteration in the form of Art. 17 will help to make this clear.
Let y —fix) be any equation connecting x and y ; Ax, Ay, corre-
sponding increments of x and y.
Conceive two quantities Ix, ly which do not vanish with Ax and
A_y but which (as Ax and Ay vary) so change that ly always bears
the same ratio to Zx which Ay bears to Ax. Then, instead of the
definition of Art. 17, we might have defined the differentials to be
the limiting values of Ix and ly. For this would have given us
|! = ''^l?- =;,,_. J = ft.., ^ , as before.
dx It^^o'l^ ^^ Ax
The definition so modified may readily be extended to functions
of any number of independent variables when it will be expressed
as follows.
Def. The total differentials of variables connected by any num-
ber of equations, are the limiting values of any quantities bearing the
same proportion to one another as the increments, but not vanishing
Avith them.
It remains to shew that this is identical with the analytical defini-
tion of Art. 130, which may be done as follows.
Let u =f{x, y).
Let Ax, Ay and A?e be any increments consistent with the above
equation. Then
Ah =/(x + Ax, y + Ay) -/(x, y)
= {/(x + Ax, y + Ay) -/(x + Ax, y)} + {/(x + A.v, y) -/(x, y)\
= Aj,/(x + Ax,y) + A,/(x,y)
= P.Ax+Q.Ay,
FUNCTIONS OF SEVERAL VARIABLES.
93
where P= ^^— , Q- ^^,
Therefore if hi, Sx, By be any quantities bearing the same pro-
portion to one another as Am, Ax, Ay, we have
lu = P.lx + a.hj.
In the limits when all the increments approach zero, hi, Ix and
ly become, according to the above definition, du, dx and dy, and we
have
du = It A=oP ' dx + It^i^Q . dy.
Also as in Art. 118,
and/^.=.Q=.Z/.=„M(-^^.lI./(y);
.-. du=f{x)dx+fiy)dy,
which is the equation by which the differentials were defined in Art.
130. A similar proof may be applied to functions of any number of
variables*.
* The mode in which the definition of a differential is extended in the text to
cases not contemplated by the original definition may occasion some little difficulty.
The principle however is not peculiar to this subject and is precisely the same as that
adopted with reference to the fundamental definitions of Algebra. One instance will
be familiar to most readers : «"" is defined when m is a positive integer as the product
obtained by multiplying a by itself m - 1 times. Hence result the equations
a'".a'' = «"■+",
(a™)" = a"'-".
The above definition is obviously unmeaning when m is negative or fractional. We
can attach no idea for instance to the multiplication of a by itself half a time. A new
definition becomes necessary, and accordingly we define a" (when m is other than a
positive integer) to be such a quantity as satisfies the above equations, which then
hold universally. This is precisely analogous to what we did in Art 130 of the text.
Having so defined a" in all cases, the next question is whether we caji attach any
meaning to it beyond that of a quantity satisfying particular equations, and the
result obtained is that a-"" means — and a™ means "Ja. The analogy between this
explication of the definition of a" and the process of Art. 133 will be readily seen.
The primary rules of algebra require to be extended beyond their original
meaning in the same way as the definition of a", although such an extension is com-
monly omitted in elementary works for the sake of evading a difficulty at the
94
DIFFEUENTIATION OP
134. It may be observed, that to any two of the increments
Am, A.r, Ay, in the last Article, we may give arbitrary values.
Hence two of the quantities Sm, ^x, hi/, which are in the same pro-
portion, will also be arbitrary, and the same must be true of their
limiting values du, dx, dy^ a conclusion somewhat differently arrived
at in Art. 132.
135. The foUowing figure will be intelligible to those who are
acquainted with Analytical Geometry of three dimensions, and will
shew the geometrical meaning of the differentials of functions of two
independent variables.
PQRS represents a portion of the surface whose equation is
2=f(x,7/), the axis of a; being parallel to Pw?, that of _y parallel toPn,
and that of s downwards (which makes the figure clearer than if
s were measured, as usual, upwards).
Let P be the point x, y, z ;
PM = Ax, PN=Ay.
Then the increment of z corresponding to an increment of x alone
threshold of the subject, and the rules are in fact proved only for particular cases and
then tacitly assumed to be universally true. In strictness they can only be gene-
ralized by a method similar to that of the text and affording a useful illustration of it.
On this subject the reader is referred to Ohm's Mathematical Analysis, trans-
lated by Ellis, liondon, 1843.
FUNCTIONS OF SEVERAL VARIABLES. 95
is MQ, that corresponding to an increment of y alone is NR, and
that corresponding to increments of both variables is LS ;
Also RF and QU being drawn parallel to x and ^,
LU=/l,z, LV=A^z,
VS = A,/(x, y + Aj/), US = Aj{x + Ax, y\
since VS is what MQ, becomes when y + Ay is put in the place of
y, and US the value of NR when x + Ax is put for x.
If Pm, Pii are lines of any length parallel to x and y respectively,
Pqsr the tangent plane at P, and therefore Pq, Pr the tangents to
the sections PQ, PR, we may put
Pm = c?x, P?J = dy ;
whence we have (as in Art. 19)
mq = d^z = lu, nr = d^z = Iv.
Also since qs is parallel to Pr, us = nr;
.: Is = III + us = lu + nr = d^z + d^z^
.: ls = dz.
Hence the significance of the definition of Art. 133 appears. For
as PM and PiV diminish, the surface becomes more and more nearly
identical with its tangent plane ; and since the increments are the
co-ordinates from P of any point whatever {S) of the surface, the
differentials (or the limiting values of quantities in the same pro-
portion as the increments) must be the co-ordinates from P of any
point whatever (s) of the tangent plane at P ; which agrees with
the above construction.
136. Exactly as in the case of equations of one independent
variable, so when there are more, those variables whose differentials
are assumed to be constant are called the independent variables, and
when none of the differentials are considered constant, the inde-
pendent variables are said to be general.
Since the partial differentials and differential coefficients are gene-
rally variable, we may have second, third... n"" partial differential
coefficients and partial differentials, as has already been seen with
respect to total differentials of functions of one variable. The same
will also be true of the total differentials of functions of several
variables, these being generally variable.
96 DIFFERENTIATION OF
Again, if u is a function of two variables x and y, the partial
differential or differential coefficient of ti with respect to x will
generally be a function of ^ as well as x, and may therefore be dif-
ferentiated partially with respect to y. Hence Ave may have such
quantities as
dAd.{u)}, dMij-% -^y-, -^'
These latter may be represented in the functional notation by the
symbols f"{y, x), f"(x, y), these symbols being understood to mean
that we are to differentiate u twice, first partially with respect to x,
and then partially with respect to y, and in the reverse order re-
spectively. We shall now shew that the order in which such dif-
ferentiations are performed is immaterial.
'^(f)_'^(g)
137. To shew that ,- j-
dx ay
By the definition of partial differentials, we have, if ti — /(.r, y'),
dy^ 7, J /(.r, ;/ + Ay) -f(x, y) \
dy ^ ^'^-^ I A^ / •
Now when x becomes ■r + A^, the above expression becomes
i f(x + Ax,y + A y ) -f(x + Ax, y) \
and A- -.''— is the difference between these two values :
dy
<w
, , which by the definition = ll^r-o — 7—^
dx '' Ax
[;, / /(^+Ax,.y+A;/)-/(x+Ax,^) | f /(x,j/+A?/)-/(x,.y) )]
- «A^o [ ^J- -J
_ / /(x + Ax. ^ + A^) -/(x + A.r, ;/) -/(x, y + Ax) +/(x, y) \
~"^1 AxA^ j'
writing U\^ because it appears by the previous step that the limits
are to be taken by making both Ax and Ay approach zero,
^dji^
dx)
The value of . may be found in the same way, and will
FUNCTIONS OF SEVERAL VARIABLES. 97^
be precisely the same as that of -r-^ since the above expression
(being perfectly symmetrical) remains unaltered when x and y are
interchanged.
' dx dy
and the symbol, /"(-y, !/) ™ay be used to denote either of them.
CoR. 1. If X and y are independent variables, we obtain from
the above equation, since dx, dtj are constant,
d.^dyU _ dydji
dxdy dxdy'
In this case therefore, dj,dyU = dydji.
It must be remembered that this equation does not hold, unless
X and y are independent variables.
, . "^(f ) <f^ , . ,
Cor. 2. From the equation — -r-L- = — j^ it is easily seen
that when a function is to be differentiated m times partially with
respect to one variable, and n times with respect to another, the
order in which the differentiations are performed is immaterial; for
by repeatedly interchanging any two contiguous symbols (by virtue
of the above theorem) the order may evidently be changed in any
manner we please.
138. On the notations employed to represent successive partial
differentials and differential coefficients.
Having employed dji, d^u to represent the partial differentials
of u with respect to x and y, the successive partial differentials must,
consistently with our notation in the case of functions of one inde-
pendent variable, be represented by
d,u, dju dj^hi, d,ji, dyhi d,j"Ui, dj^u d:"d;u (1),
and representing the partial differential coefficients of i< or F{x,y) by
F'{x) and F'{y), the successive partial differential coefficients with
respect to x alone and y alone must be represented by
F\x-), F"{x) F^"\x), F'(y), F"(y) F">(j/) ••• (2).
When we first find the partial differential coefficient of ic with
respect to x, and then with respect to y, the result has been repre-
H. D. V. '
98 DIFFERENTIATION OF
sented by F''{x, ?/) ; a notation which cannot conveniently be ex-
tended to the case where the function is differentiated m times with
respect to x, and n times with respect to y. This might be expressed
by the symbol i^ <'"+"' (,r,„, ?/„), but such a notation would be very in-
convenient in practice.
Since, as far as partial differentiation is concerned, u may be con-
sidered as a function of one variable only, the values of the differ-
ential coefficients in terms of the differentials will be, (Art. 98), when
the independent variables are general,
dji \dx J f. d^u ^\dyj \dy J
d7' —37^'^'' tj' —dT'^^' -TT-'^'---^'^'
which differ from the expressions in the case of one independent
variable, only by the use of the subscripts, to denote the variable
with respect to which the differentiations are performed. When x
and y are both independent variables, these become (Art. 99).
dji d/u d"ii d^u dfu d,"n d^dyU d/'dy"u . .
~d^' ~d^"'li^' liy' 'df-""tf' dxdy'" dx"'df"'^ ^'
These notations (2), (3), (4-) are precise, but the functional nota-
tion (2) cannot be conveniently extended to the general case ; (3) if
used merely as a notation to represent differential coefficients, is
excessively cumbersome, and (4) holds only when x and y are made
independent variables. To obviate these objections a less expressive
but more convenient notation is generally employed.
In the first place the expressions (4) are used for the differential
coefficients, not only when x and y are independent variables, but in
all cases. When this is done, the numerators cease to represent the
successive differentials as heretofore ; but this will occasion no error
if we regard those expressions merely as symbols, and not as frac-
tions. This inseparability of numerator and denominator is indi-
cated by enclosing them in brackets ; and since the variable with
respect to which the differentiations are performed is then sufficiently
indicated by the inseparable denominator, the subscripts also are
omitted.
The partial differential coefficients are then represented by
fdu\ f(fn\ fdUi\ fda\ f(fu\ (dJ^juS
W' \dx"-)--W)' \dy)' \dfj-\dfj'
( dhi \ ( d-^-u \
\dxdy)'"\dx"'dy"J"'^ '''
FUNCTIONS OF SEVERAL VARIABLES. 99
Where the natui-e of the problem precludes any probability of
error, even the brackets are frequently omitted. .
The notation (2) is recommended in preference, where no differ-
ential coefficients beyond the second occur ; in other cases (5) should
be employed. The latter however is more generally adopted in all
cases. For the sake of familiai-ising the reader with it, we shall
employ it in the following propositions.
139. To express the successive total differentials of ?< or F(x, tj)
in terms of those of x and y, and the partial differential coefficients of
«, neither x nor y being independent variables.
We have
du = dji + dyU (1),
•■•■''-''{(£)''4-4(|)M'
Substituting these values, remembering that (j^J = (x"^) ' '^^
have
(2)
Equations (l) and (2) give du and d'u, and by proceeding in the
same way the expressions for the differentials of higher orders may
be obtained, but they increase rapidly in complexity as we advance,
when X and y are not independent variables.
140. The expressions for the successive total differentials in
terms of the successive partial differentials of u are easily obtained
Thus
dti = dxU + djjii,
d'u = d, {dji + dyii] + dy {dji + d^u},
.-. d^u = d/u + d^ d,ji + dy dji + dyU ;
7—2
100 DIFFERENTIATION OF
and similar equations for the differentials of higher orders. It will
be observed that d^ d^u and dy d^u are not equal because x and y are
not independent variables.
Equations in the above form are of little use, but we may deduce
from them the equations of the preceding Article, which we will
now do as an illustration of the principles above explained.
141. From the equation
d^u = d/ii + d^ dyii + dy dji + dy^ii,
to express d^u in terms of the partial differential coefficients of m.
similarly
d.dyu = d^ { ©^-^l = (.Sy)^"'^^^Q^^^^'
chdji ^ dy I (I;) dx } = Q^;;^ d. drj + g) dydx.
Adding these quantities, we obtain
since f/j, f/>r + dydx = d'x and c?^ f/y + dy dy — d'y.
142. The expressions which we have found for the differentials
of II, hold not only when x and y are connected by some further
equation, when one at least of the differentials dx, dy must be vari-
able, but also when there is no other relation between the variables
than the equation u = F(x, y), so long as the independent variables
ai'e left general. On account of the complexity of those expressions
this is seldom done, but the variables under the functional sign are
almost always made the independent variables.
FUNCTIONS OF SEVERAL VARIABLES. 101
143. To express the successive total differentials of u or F {x, y)
in terms of the partial differential coefficients when x and y are
independent variables.
We have du = dji + dyU,
Also ^M = djiu + dydu,
since dx and dy are constant.
By differentiating d'hi, we obtain, in like manner,
In these equations it may be observed that the coefficients and
the indices of the operations -y-, -j- and of the differentials dx, dy
are the same as those of the binomia theorem. By assuming this
for d'li, it may be proved for d"^hi, by differentiation, and being true
for <fM, dhi, is therefore generally true. This theorem may be
written in an abbreviated form, thus,
with the convention that wherever in the expansion we find \-jA dx'',
we are to write (J^^ dx", and where we find (^^j \^^j dx^dx'', to
The theorem may easily be extended to functions of any number
of variables. Thus if
ti = F(xi, X2...ir,„),
a;,, X3...x^ being independent variables.
144. To change the independent variables in an equation in-
volving partial differential coefficients of a function of two inde-
pendent variables, from one system to another.
Let n = F(x,y),
and let it be required to change the independent variables from
102 DIFFERENTIATION OP
X and y to r and 6, where these latter are connected with the former
by the equations
r = cp (x, y), 6 = ^!^ (x, ij).
By substituting the values of x and y obtained from these equations,
ti becomes a function of r and 9. Also
(du\ _ fdu\ fdr\ fdu\ fdd\
\dJ ~ \drj \dxj "^ \ddj \dx) '
(y-j, (7-) being the differential coefficients of r and 6 on the sup-
position that y is constant, and therefore equal to (p\x), ^|^'(•^)> which
may be found from the above equations.
In like manner, we have
/^\ _ fdti\ fdr\ fdu\ fdd\
\dy)-\dr)\dy)^\de)\Ty)'
by substituting which values in the given equation the transforma-
tion is effected.
The expressions for the differential coefficients of higher orders
are obtained in a similar manner, but the general expressions are
somewhat complicated, and all that is necessary is to pursue the
same method in any particular case.
145. When the equations connecting r and Q with x and y can-
not be reduced to such a form as to give r and Q explicitly in terms
of X and y, we must proceed as follows.
Let x = (p (r, 0), y = ylr (r, 0).
(S) = (s)(S)-(|)©'
(l') = (S)(l)-(S)(|)-
Eliminating successively [-/-) and ( t- ) , we obtain
^du\ (dy\ /du\ /dy\
\drj \ddj~\de) [drj
© =
^dx\ /dy\ /dx'K ^di/y
\dr) \ddj~\dd) W
^dic\ ^dx\ /dH\ fdx\
fdu\ fdx\ /du\ /dx\
/chA _ \dd) \dJ^) ~ \d?) \ddj
\dy) ~ (dx\ fdy\ _ fdx\ (dj/\ "
\dr) \dd) \dd) \dr)
FUNCTIONS OF SEVERAL VARIABLES. 103
The partial differential coefficients of x and y are immediately de-
ducible from the equations connecting x and y with r and 0, and the
transformation is effected by substituting in the given equation the
above values of (-7- j and (;7-).
146. We will take as an example the expression
fdR\ /dR\
the new independent variables being determined by the equations
r- = x^ + if. tan = ^ .
^ ' X
T,, (dR\ (dR\ X (dR\ -y
^^^" \-d^)=Kd?)-r^\le)-^l^e'
= (f)-s-(f)
'dR\ sin
Ai fdR\ /dR\ y A/i?\ 1
^^^° {-dt,)-{-dF)r'-{-de)jl^e'
fdR\ . . fdR\ cos d
/dR\ /dR\ z'dRs , ^ . ^,
••• ^ U) "-^ W; = Kdi) ^^ ''''^'■^ «"^ ^J
f—\ \y cos - X sin 01 _ fdR\
\dd)\ r \-'\d^)-
We will treat the same function by the method of Art. 145, em-
ploying the equations
x = r cos Q, y = r sin Q.
-» (|)=(§);--(f)-«.
•••(S)=(f)---(S)^.
(f)Kf)--(f)^.
whence, as before, the expression is reduced to the form
Examples will be found in Gregory's Examples, Chap. i. 11.
and III.
CHAPTER VII.
DEVELOPMENT OF FUNCTIONS.
147. To expand f(x + Ji) in ascending powers of h.
Lemma, If (p (x) be any function of x, such that cp (a) = and
(p'(x) is positive for all values of a; from a to a + h inclusive, then
(p (.r) will (between the same limits) have the same sign as h ; and
if cp' (x) be negative for all values of x within those limits, then <p {x)
will have a different sign from /^.
This follows at once from the consideration, that the differential
coefficient of (p (x) with respect to x equals the ratio of the rates of
increase of (p {x) and x *.
Let now F(x) be a function of x such that F (a) = 0, and let A
F'{x)
and B be the greatest and least values of ,
° n {x — a)
from a up to « + h. Then between these limits,
-^^ when X varies
A-
F'{x)
and
F'{x)
-B
n {x — ay ' M (x — a)""'
are constantly positive,
.-. An(x- a)"-' - F'(x) and F' {x) - Bn {x - a)'^\
are both constantly positive or constantly negative, since (x - «)""'
cannot change its sign within the given limits ;
.-. if <p, (x) = A{x- ay - F (x) and <^, (x) = F (x) - 5 (x - a)»,
* The Lemma becomes more obvious from the following construction. Let OAB
be the axis of x, OA being equal to a, and OB to a+h, h being in this case positive.
Fig. 1. Fig. 2.
Then the curve p=(t> (.r) must by the assumed conditions pass through A, and (if the
differential coefficient is positive from A to B) its form must be that of figure 1, and
y will be positive between A and B. So if A is negative, and therefore OB' less than
OA, y -will be negative. If </>' is negative, the curve will be asin figure 2.
DEVELOPMENT OF FUNCTIONS. 105
the above quantities -will be the values of (p/ (x) and cp^'ix), and
(pi (x) and (p2 (x) will satisfy the conditions of the lemma, and will
thex'efore both be positive or both negative within the given limits,
. F{x) . F(x) ^
.: A - 7 — M- and 7 — Mr„ - B,
{x — a) {x — a)
have the same sign ; wherefore, giving to x one of the admissible
values, viz. a + h, ■ — ^~ — '- lies between A and B.
This may be expressed by the equation
where 0, has some unknown value between and 1, since the
quantity on the right of the equation may, by varying B-^ from to 1,
be made to assume all values from B to A, some one of which must
therefore satisfy the above equation.
If not only F {a) but also F'(ci) equals zero, we may prove in the
same way that
F'{a + e,h) _ F"{a + dji)
where 6^ has some unknown value between and 0, , and therefore
a fortiori between and 1 .
If all the quantities F(a), F'{a) .. . F'"-"(a), equal zero, we may
obtain a series of equations similar to the above, of which the last is
F^{a + P„,i/0 _ J^w (a + eh)
ie„_,hy 1
where the value of 6 must lie between and 1 .
Collecting these equations, Ave have (with this last set of con-
ditions),
F(a + h) _ J-f"' ( g + e/Q
A" \n ^^^'
where Q has some unknown value between and 1.
We will now give a particular value to F(.i;), viz.
By differentiating F {£) successively, we obtain.
106 DEVELOPMENT OF FUNCTIONS.
F"ix) =r W- |/"(a) +/-(«) (^-«) + - +/""'(«) ^^^TT- } '
j«(x) =/t^) - { fK^) +/^^"(«) (X - a) + . . . +/"-"(«) ^^^V } '
2r.-i)(^)=/-)(x) -{/■-' (a)}, ,
i^(x)=/"'(x).
All the terms -within the brackets in the above expressions, ex-
cept the first, are multiplied by some power of (x-a), and will
therefore vanish when x is made equal to a, unless some of their
coefficients are infinite.
If, therefore, f(x) be such a function that none of the differential
coefficients /(a) /"(a) .../""^(a) are infinite, we shall have
F (a) = 0, FXa) = ... F^%a) = ... F"-"(a) = 0,
and F^^x) =f'(x),
F(x) therefore satisfies the conditions necessary to equation (1) and
by substituting the values of F(x) and F'"\x) in that equation it
becomes
/(a + ^)=/(«)+/'(«)AV''(«)-|^••V''^''(«)J|^+/"(« + 0/O^,
~ (2).
In this expression we may give to n any value we please (if by
so doing we introduce no infinite differential coefficients) ; and if
none of the differential coefficients o^ f{x) become infinite when
x = a, n may be taken indefinitely large, and we have
f{a-i.h)=f(a)^f'{a)h+f"{a)^ + &iC.ininf. (3).
The series in this form is called Taylor's theorem.
The last term in expression (2) is evidently the difference between
this infinite series and its first (w) terms. Its value cannot be exactly
determined, since 6 is unknown, but we can determine the greatest
and least values which it can assume while Q lies between and 1.
Since the remainder of the series after («) terms must lie between
these values, they are called the limits of the remainder of Taylor's
series.
148. The above series is true for all values of h, and therefore
among others for the value dx. If u =f{x) and h = dx, the general
DEVELOPMENT OP FUNCTIONS. 107
term of the series becomes •^^^—^'^-^, or (if x is taken for indepen-
dent variable) -7— .
In this case, therefore, we have
/(a: + (/a,) =« + <?«* + -|^ +.-.+ -|^- • •«« ««/•
149. As an example of the application of Taylor's theorem, let
f{x) = \og X, and let it be required to expand log (1 + x) in powers
of X.
We have
f{x) = \o^x, .-. /(l) = log(l) = 0.
/(-'(x) = (- 1)-' 1= , /""(O = (- 1)"-' \!l^'
.-. log(l + a:)=x-|" + [2^... + (-ir'i^:ii^.. .«■«"?/:
= x +— + (-1) ^n mf.
2 3 ^ ^ n •"
The limits of the remainder after n terms will be the greatest
and least values of -^-^-^^-^a;" from = to = 1, and
\n
\n n {l + Oxf
Hence the limits are (- l)""' - and (- 1)"-' nil+x)" '
150. To expand f(x) in ascending powers of x.
If in the expansion of /(a + h) given by Taylor's theorem, we put
a = 0, ^ = a;, we have,
Ax) =/(0) +/(0) X 4-r(0) I . . . +/->(0) ^^ +/"'(0x) ^;
or /(^) =/(0) +/(0)x +/'(0) ^+ . . .. m inf.
108 DEVELOPMENT OF FUNCTIONS.
This is called Stirling's or Maclaurin's theorem, and is, as we
see, only a particular case of Taylor's theorem.
The condition that none of the quantities /(O), f'(0), &c. should
be infinite is necessary to its truth in the second form, the first re-
quiring only that those up to the h"* order should be finite.
151. As an example of the application of Stirling's theorem, let
it be required to expand sin x in powers of x.
We have f(x) = sin x, .: /(O) = 0,
f(x) = cosx, .-. /(0) = 1,
/'Xx) = -smx, .•./"(0) = 0,
fXx) = sin fx + « l) , .-. /'"'(O) = (- 1)''""" if n is odd,
= if 71 is even.
x^ x^ ^r'"*""'*
••• ^^^^ = ^--[3^ [5 +(-^^""'[2^31 i^inf.
The limits of the remainder after the term involving a:""-' will be the
greatest and least values of /'^'"'(0a-) -t^— ,
and f^-'Hex) -r— = sin {dx + mir) . — ,
= (- irsinr^x) -^.
Hence the limits are and (- 1)"' sin x t^— •
152. To determine the nature of the expansion of /(a + li) when
Taylor's theorem fails.
This will happen when any of the functions f{a),f'{a),f"{a), &'c.
become infinite.
(1) Suppose /(a) = CO .
Then f{x) can generally be expressed in the form
where m has such a value that ^ (a) is finite. Hence Taylor's
theorem holds with respect to (a + A),
... f(a + h) = ^^-^ = h-'" {<p{a) + cp' (a) h + cp'Xa) i h' + &c.} ;
DEVELOPMENT OF FUNCTIONS. 109
i.e. the expansion o£/(n + h) contains negative powers of h when
/(«) = 00 .
(2) Suppose that the first of the series of functions /(o^), /'(x),
f"{x), &c. which becomes infinite when x = a^ is /^"^(x).
Let F{x) =/(x) - {/(«) +f{a){x -a)... +f'-^\a) ^^Ll^ \ .
LTZ— -'
Then, since all the quantities introduced in the above expression are
finite, we shall have, as before, (Art. 147),
F{a) = 0, F'{a) = 0, F'-\a) = 0, F^^a) =f%a) = o:> .
Hence we have, as in the proof of Taylor's theorem,
F{a + h) _ F^'-\a + Oh)
F(a + h) _ F^Xa + dh)
h- ~ [l '
which equations being always true, are true in the limit j
. F(a + /0_ F^-\a)
' ' "h=o ITZi \ 7— = y>
and .^.£&i±5=^,co.
Now there will generally be some value of m such that
V ( n ■X-h\
^fk=o — j^ — shall be neither zero nor infinite; and in order that
the above equations may be true, m must lie between r — 1 and r.
We shall therefore have
where C is independent of h and P a quantity which vanishes with
h, and which, when arranged in powers of h, must be of the forni
C'A" + C"h^ + &c.
.'. F{a + h) = C/r + CVr-^+p' + (7'7«''»+*' + &c.
or, putting for F{a + Ji) its value,
/(« + h) =f{a) +f{a)h +/'•-"(«) r^ + CA'" + 07^+" + &c.
i. e. the expansion follows the law of Taylor's theorem so long as that
gives finite coefficients, after which a series of fractional powers of h
commences, the first of which lies between the last integral index
and the integer next greater than it.
110 DEVELOPMENT OF FUNCTIONS.
153. It will be observed that the term generally has been used
in these demonstrations. The fact is, that our assumption that /(a:)
may be expi-essed in the form <p{x){x — 0)'"", where (p{x) neither
equals zero nor infinity when x = a,\s not universally true ; and in
these cases it is impossible to expand the function f{x) in ascending
powers of the required quantity h. Thus, for example, \i f{x)
= log (x - a), f{a) = log = - CO, and U,^f{x) {x - «)"' = 0, for all
values of m, however small*; i. e. (p(a')=0 for all finite values of
m, and = co when ?w = : there is therefore, in this particular case,
no function such as we have supposed (p (x) to be, which shall
=f{jX) {x — a)"*, and yet be neither zero nor infinity when x = a. In
this case the expansion of /(a + h), i.e. of log h in powers of h,
cannot be found.
154. Taylor's and Stirling's theorems enable us to expand all
functions of one variable whose successive differential coefficients
can be found.
Expansions may sometimes be found more readily by the fol-
lowing method.
Let/(a:) be the function to be expanded in powers of jr.
Assume f{x) = A + Bx + Cx^ +
then f(,x)= B + 2Cx+
/"(j')= 2C+&C.
Now if any of these differential coefficients are quantities which can
be readily expanded by algebraical processes, or if any relation can
be found between the quantities f(x), /'(x), &c., it is evident that,
by substituting for them the above values, we shall obtain an equa-
tion for determining the indeterminate coefficients A, B, C, &c. by
equating coefficients of equal powers of x.
* This may be proved as follows:
assume log(a' — 0)= — y ;
.'. (.r -«)"' = e-^j
.•. y = os when x = a ;
ltx = a(^v- ay log {<r-a) = -lfy=^ e-'"yy = - Ity^^
m.' „
Li-
»M + jY 2' + —
however small m may be.
DEVELOPMENT OF FUNCTIONS. Ill
As an example of this method, let/(j:) = sin a:. Assume
sina: = J.0 +AiX + Az't^'" . + A„x'' + . . .
or, as it may be written,
sin X = '2.A„af.
Differentiating this equation twice, we obtain
— sin X = 1.71 (71 — 1) yl„a;"~%
.-. = lA^x" + 1.71 (n- 1) ^„J;"-^
The coefficient of a;"~^ in the above is A^2 + n{n — l) J.„,
.'. J[„-2 + 71 (71 - 1) ^„ = 0.
. 1 . 7 sin ^ A 1
Also, since If^a = '^> A^^O, ^, = 1,
and making « in the above equation successively 2, 3, 4 . . . 2m,
2w + 1, we obtain
^, + 2.1.^2 = 0,
^,4 3.2.^3 = 0,
^2 + 4.3.^4 = 0,
A,
= 0,
A,
1
~ 2.3'
A,
= 0,
-A 2m
= 0,
A 2m-2 + 2?«.27W - 1 ^2m = 0,
^2m-l + 2?M + 1.27W ^2m+l = 0, .•- ^2,„+i = (- l)"
\2m + l '
„2m+l
155. To expand /(a: + h, 7/ + k) in powers of k and k.
If we consider f(x, 7/) as a function ofy alone, and differentiate
partially with respect to j/, we shall have, by Taylor's theorem,
/(x, 7/ + k)
If in the above we write x + h for x, we obtain
f(x + h,7/ + k) =f{x + h, y)
^ df{x + h, 7/) j^ ^ (Pf{x + h, y) F +^7(^±]bjlK +
dy dy' [2*" df \j'_ '"
" The brackets are omitted from the partial differential coefEcients for convenience,
as no error can arise from so doing.
y f i,^ i-'^j - /fr ^ ^ t/^J
112 DEVELOPMENT OF FUNCTIONS.
Expanding, in like manner, each term of the above series,
d/ix + h^ ^ df{x, y) ^ dy(x, y) j^ _^dy{x,y) J^ ^ __
dy dy dxdy dx'"-"^ dy \n-\
d'f(x + h,y) _ ^Ax, y) , drf{x,y) h"-' ^ _
flfy dy" '" dx"~'dy' \ n-2
drf{x^-h,y) _ ^ dy{x,y) A- ^
dy" dx" ''dy" I n — r
Therefore, substituting these values in the above equation, we obtain
^/ 7 7^ ^/ X . ( df(x, y) df{x, y) , \
+
L I mx^ ;^. ^ , dY(x, y) ^^^. ^ dYJx, y) ^, I
2 \ </x^ <^^t(y dy- J
12
[«\ dx" dx"-'dy \n-r dx'^Ulf j
+
It will be observed that the coefficients and the indices of h
and k, in the quantities within the brackets, are those given by the
binomial theorem.
156. If in the above proof we had changed x into x + h before
changing y into y + k, the coefficient of h"~''k'' would have been
changed from
1 drf{x, y) ^^ _J_ dy{x, y)
\n — r dx"~''dy'' [ n — r dy'^dx"'"
Hence ^j"/ (x, 3/)_ ^7 (x, .y)
Hence dx'^-df dfdx""' '
or the order of partial difflsrentiation is immaterial, a property which
we have before proved. (Art. 1 37-)
157. The limits of the remainder of the above series, after
terms of the (ji- 1)"* degi'ee in h and k, will be given by putting
X + Bh, y + dk for x and y in all the terms of the yi"* degree, and
determining the greatest and least values of each term when 6 varies
from to 1.
DEVELOPMENT OF FUNCTIONS. 113
This series, like that on which it depends, fails when any differ-
ential coefficients become infinite for the particular value assigned
to X.
158. If we put h = dx, k = dy, and u =/(a-, y\ and consider
X and y, independent variables, Taylor's series for two variables,
. d^ii d^u . . ,.
as appears by Art. 1 43.
159, With the same conventions as those employed in Art. 143,
Taylor's theorem for one and for two variables may be written
f{x+h) = e^^f{x\
fix + h,y + k)= e^"'"^'"^^ fix, y).
In like manner it may be proved that
fix + h,y + k,z + l...) = e^'^^ '" "~^"'' f{x, y, z...).
If the variables under the functional sign are made independent
variables, all the above forms are included in the general expression
f{x+dx, y + dy, ...)^ e'f^x, y ...)
d being the total differential.
160. Lagrange's theorem. From the equations
y = z + x(p{y) (1),
u=f{y) (2).
to expand u in ascending powers of x.
By Stirling's theorem,
From(l) ^£=cp(y) + xcl>'{y)f^,
•• di-\^x\i>'{yy'^^^^ dz'
and therefore, if C7be any function of ?/,
dU ^ , ^ dU (Q\
H. D. C. °
114 DEVELOPMENT OF FUNCTIONS.
Let U=u, therefore -r- = (p(y)-r'
cix ^ ^^ ^ dz
Also, since £=/(^)|,
0(^)^ = (/;/(i/)|.
dV
If therefore V be such a function of _?/ that -p = ^ {y)f'{y)>
^^y^d^.^Tz'
dhc d dV d dV d {, ^dV\ , , ,
Now suppose 57.=7;,^|0(j/)"jJ>
dx"^^ dx di
d" fdV\
d^ilK
dz" dx
d" i , sdF„) , , ,
I. e. if the law of formation above assumed holds for -;— , it holds
dx"
for -^j ; and, as it has been proved for -j-^, it holds generally.
Hence we have ?/„ =/(^),
DEVELOPMENT OF FUNCTIONS. 115
and the expansion becomes
161. Laplace's theorem. From the equations
ij = F{z + xcp{y)} (1),
^«=/(^) (2),
to expand ti in ascending powers of x.
By Stirling's theorem,
fdu\ /d"u\ x" . . „
dx ^^^^ dz'
dU ^, .dU
From
where U is any function of y,
L
of i^;
Let £/■=?/, and assume (p{y) — = —- where F is some function
^ dN,_ d fdV\ _
dx* dx \dz)
dz dx '
By the same process as in Lagrange's theorem, we may shew
from this equation and equation (3) that the successive differential
coefficients are given by the same law,
d"u d"-' ( , .du\
••• -d^^=d^^\^^y^dz\'
Hence u^=f{F{z)),
+
116 DEVELOPMENT OF FUNCTIONS.
dz^-' Y^^^-'J^ dz j \n'
162. Laplace's theorem may be deduced from Lagrange's as
follows.
Let z+x(p{i/) = v, .'. y = F{y\
V = z + x(p {F(v)}.
These equations are now in the form required by Lagrange's
theorem, in which we have only to change/ and (p into /jP and (pP,
when we obtain the expansion of u as given by Laplace's theorem.
CHAPTER VIII.
LIMITING VALUES OF INDETERMINATE FUNCTIONS-
MAXIMA AND MINIMA VALUES OF FUNCTIONS.
163. To determine the limiting value of a fraction which
assumes the form ^ for a particular value of the variable.
Let -^-^ be a fraction whose numerator and denominator both
vanish when x = a.
By the definition of a differential coefficient,
•' ^^=°(plx + Ax)-cp{a:) cpXx)
In the above equation put x = a,
f(a + Ax) _f'(al
" "^='(p{a + /^x) cp\ay
'^- <p{x)- cp'iaY
I^-CSl^ is of the form o, this equation is indeterminate, but in that
case we have, since equation (l) is true in the limit,
It f^-lt f'^'^.
^^-cpixr''-<p'(x)
Let now the first pair of differential coefficients of the same
order o£ /{x) and (p^x), whose ratio is determinate when x = a, be
/(")(rt), <^'"'(a); then, by applying the above result to the differential
coefficients in succession, we have
164. To determine the limiting value of a function which as-
sumes the form § for a particular value of the variable.
118 LIMITING VALUES OF INDETERMINATE FUNCTIONS.
fix)
Let the fraction be ) i where /(o) = co, (p (a) — co. Then
1
Ix [(pix) }
7(7) ^ i 70^ I
-cpixy/ix)'
the same expression as that found when /(a) and (p (a) equal zero.
If/' (a) and (p' (a) have a determinate ratio, this becomes
,/ /M = fM.
-<p{x) cp\a)'
but if f^"\a), ^'"'(a) is the first pair of differential coefficients whose
ratio is actual, we have, as before,
;/ /(^)^ /">(«)
-cp(x) r^^y
165. If all the differential coefficients of/(jr) and (p{x) have
indeterminate ratios when x = a, this method fails to determine the
limiting value of the fraction. In this case we can only obtain it by
algebraical methods, as in the examples of Chapter II. We may
proceed as follows :
"'="«/) (a:) """=>(« + 70 ■
Let the expansions off(a + h), (p(a +k) in ascending powers of A,
obtained algebraically (since, when the differential coefficients become
infinite, Taylor's theorem fails) be
/(a + h) ■= A/r + J5/i" + . . .
fp(a + Ii)^A'k""+B'k"'+...
where m and m' must be negative or fractional. Then
// /(■^) ^ // ^A"' + Bh"+ ...
Ah"^"" + Bh"-""+ ...
- iff.=o — ^, :^B'h"'-"" +777 '
= if m is > m',
A .„
= -77it m = m'
A
= CO if m < ?«'.
LIMITING VALUES OF INDETERMINATE FUNCTIONS. 119
If we had employed this method when neither f(a), (p (fl) nor any
of the differential coefficients are infinite, we should have obtained
the results of Art. l63, as is evident from the fact that the expan-
sions will be those given by Taylor's theorem.
166. There are some other indeterminate forms which a func-
tion may assume, the limiting values of which may be made to de-
pend on those of functions which assume the form -^. Such forms
are 0", co", 1=^=.
(1) Let ?< =/(x)<fW where /(«) = 0, (a) = ; so that when
X = a, n is of the form of 0°. Then
and when x = a, (p (x) \ogf(x) assumes the form O.co or -^, the limit
of which can therefore be determined, and thence that of u.
(2) Lety(a) = co, (p{(i) — 0; then u assumes the form co" when
x = a\ but ^(x)log/(j;) assumes the form O.co or 0, the limit of
which, and thence that of v, can be determined.
(3) Let /(c) = 1, ^(^fl)=±co. Then u assumes the form 1'*''°
when x = a, and (p{x)\ogf{x) assumes the form co.O or §^, the limit
of which, and thence that of u, can be determined.
Maxima and Minima.
167. To find the maxima and minima values of functions of one
variable.
Def. If, when x receives a continuous increase, f{x^ increases
until X has a certain value, a, and afterwards decreases,/(a) is called
a maximum value of/(x); and if/(^) decreases until x attains the
value a and afterwards increases, f{a) is called a minimum value
of/(x).
Since f\x) equals the ratio of the rates of increase of f{x) and x,
it follows that when an increase of x produces an increase of /(x),
f'{x) must be positive, and when an increase of x produces a de-
crease of fix), fipc) must be negative. Hence when x, in the
course of its increase, passes through a value, a, which makes f(x')
a maximum or minimum, f'{x) must change its sign from positive to
negative in the former, and from negative to positive in the latter
case.
120
MAXIMA AND MINIMA VALUES OF FUNCTIONS.
Hence at such a point f{a) must either equal zero or infinity.
The converse of this however^ viz. that all values of x which
make f\x) either zero or infinite correspond to maxima or minima
values of/(x), is not true; because a function, as f'{x), may pass
through these values without changing its sign, which is the criterion
of /(a;) having a maximum or minimum value.
To determine, therefore, all the values of x which make f{x) a
maximum or minimum, we must first find the values of x which
satisfy either of the equations
1
f'{x)=0,
A^)
= 0.
(1%
and for each of these values inquire whether /'(•^) changes its sign
in passing through it. If not, this value of x does not make f(x)
either a maximum or a minimum : if there is a change of sign from
positive to negative, /(x) has a maxirmim, and if from negative to
positive, a minimum value.
The existence and nature of the change of sign of /'(x) for any
value, a, of x which satisfies either of equations (1) is easily deter-
mined by observing the signs of /'(a + h) and /'( a - h), when h is
made so small that there shall be no change of sign in f(x) between
X— a and x = a + h, or between x = a and x = a — h.
168. The last proposition becomes self-evident when stated in a
geometrical form.
Let the figure represent a curve whose equation is y =/(x)
/ ^
T L M" IV
having a maximum ordinate at A and a minimum one at B. Then
if P be any point x, y, tan FTL =/(^). From P to ^, the value of
this tangent is evidently positive ; at ^ it passes through zero to a
negative value, which it retains until at B it again passes through
zero and becomes negative. The change from positive to negative
therefore corresponds to the maximum, and that from negative to
positive to the minimum value.
The case where /'(x) passes through infinity, will be considered
geometrically in a subsequent chapter. (See Art. 209).
MAXIMA AND MINIiMA VALUES OF FUNCTIONS. 121
169. The criterion by whicli we distinguish between maxima
and minima is sometimes more convenient in the following form.
If any value of x makes /{■x) a maximum, /'{x) changes from
positive to negative in passing through that value ; hence, in this
case, an increase of x makes f'(x) decrease at the point in question ;
and similarly, if f(x) is a minimum, an increase of x makes f\x)
increase.
The differential coefficient of f'(x) must therefore be negative in
the former, and positive in the latter case; i.e. if a be any value of
X which satisfies either of equations (1),
/(a) is a maximum if/"(a) is negative,
minimum positive.
This criterion fails when /"('^) is either zero or infinite : in the latter
case we must proceed as in the previous article, in the former we
may either employ that method or the following.
Let the first differential coefficient of /(x) which does not vanish
when X =a,he /"(«). Then, by Taylor's theorem,
y(a + /,)_/(rt)^^ v_ if^u^
and /(« - h) -/•(«) = -^'1^ ^^'^ (- hy.
But if f(a) is a maximum, both these quantities must be negative;
and if a minimum, both must be positive, if h is taken sufficiently
small.
Also, when h is sufficiently diminished, both /'"'(a + 0A) and
/'"'(« -S/i) assume the same sign as /'"'(a). Unless therefore n is
even, the above expressions will have different signs, and f{a) will
be neither a maximum nor a minimum value; if n is even, the signs
of both will be the same as that of /'"'(a);
•*• /(('') is a maximum if /'"'(a) is negative,
minimum positive.
Hence the rule. If the first differential coefficient which does not
vanish is of an even order and negative, there is a maximum, if of
an even order and positive, a minimum value; if of an odd order
there is neither a maximum nor a minimum.
170. The above rule is equally true if we read differential ftfr
122 MAXIMA AND MINIMA VALUES OP FUNCTIONS.
differential coefficient, and this, whatever may be the independent
variable. For, from the expressions of Art. 98, it appears that if
y =/('^) a"d dy = 0, fix) will also vanish ; and if all the differentials
of _y, up to rf""'j/ inclusive, vanish, the differential coefficients up to
the (w — 1)* will also vanish. Again, if dJ'y is the first differential
which does not vanish, for a particular value of x, as a, d"y must
(when x = a) become equal to /<"' (a) Jx", as well when the indepen-
dent variable is general as when x is independent variable ; for by-
observing the equations of Art. 97^ it is obvious that the first term
in the expression for c?"y will be /'"'(a)(/a:", and that all the other
terms will be multiplied by lower differential coefficients which (as
we have seen) vanish with the corresponding differentials. Hence,
when n is even, the sign of d^y (for the value a of x) will be the
same as that of /'"'(o), since dx" must then be positive. The condi-
tions dy = 0,dy=0...cr~\y) = 0, and d"(y) a positive or negative
eve7i differential are therefore identical with those before obtained,
viz,/'(rt) = 0, /"(a) = 0.. ./'""''(a) = and /'"'(a) a positive or nega-
tive even differential coefficient.
171. To determine the maxima and minima values of u from
the equations
^=fi^>y) (1)'
= F{x,y) (2).
This may be done by substituting in the former equation for y its
value in terms of x, when u becomes an explicit function of x and
may be treated by our former methods. Those maxima and minima
values corresponding to the condition du = may however be more
conveniently obtained as follows.
When M is a maximum or minimum, du — 0;
.:f{x)dx+f'(y)dy^O,
also F' (x) dx + F' (y) dy = 0.
By eliminating dy and dx between these equations, we obtain an
equation which, together with (2), determines the values of x and y
which make u a maximum or minimum. The sign of d^u will distin-
guish them.
If the same value of x which satisfies the above equations also
makes d^ vanish, we must find the first differential, d^u, of u which
does not vanish for that value of x, and observe whether n is even or
MAXIMA AND MINIMA VALUES OF FUNCTIONS. 123
odd, and in the former case whether dTu is positive or negative. In
doing this it will always be convenient, though not necessary, to
make either x or y independent variable.
172. To find the maxima and minima values of a function of
two independent variables.
Def, If any values a, b, of the independent variables in the
function /(x, y) make /(«, V) always greater than /(a + ^, h-vh\
whatever be the relative magnitudes and signs of h and h, provided
they are taken sufficiently small, /(a, 6) is called a maximum value
of/(x, y); and if/(«, b) is always less than /(a + 7^, 6 + A), it is
called a minimum value of /(x, y).
lif{a,h) is a maximum or minimum, the above conditions, which
hold for all relative values of h and k, must hold for that particular
system of values of h and k which satisfy the condition k = mh or
y-h = m{x-a) ; and conversely, if they hold for all such systems
of values, when m is varied in all possible ways, they must hold for
the general values, since there is no pair of values of A and k which
cannot be found in some one of the particular systems. But the
supposition y-h =m{x-a) reduces /(x, y) to a function of one
variable, and our definition to that of a maximum or minimum value
of such a function.
Let therefore ^ be a function of one independent variable deter-
mined by the equations,
^=f{^,v) 0)'
y — b = 7n(^x — a) (2).
The conditions that z shall be a maximum or minimum are (by
the previous articles).
th=f'{x)dx+f'{y)dy = (3),
and dy = m dx,
whence dz = dx{f'{x) + mf{y)\ = (4).
In order that f{x, y) may have a maximum or minimum value,
this equation must hold for all values of ?«, which cannot happen
unless both the equations,
n^)-0,f{y) = 0,
are satisfied. Hence if /(«, b) is a maximum or minimum value,
a and b must be a pair of roots of these equations.
124 MAXIMA AND MINIMA VALUES OP FUNCTIONS.
z will be a maximum or minimum according as the sign of rf^s is
negative or positive for the above values of x and y. If we differen-
tiate equations (1) and (2) a second time with x as independent
variable *, we obtain from equation (2) (Fij = and then from
equation (1),
d'z^f"{x)dx^ + 2f"{x,y)dxdy+f"{y)df (5)
= rf^''{/"(^) + 2/'(x,3/).«+/"(i/)"^'} (6).
When x = o^ y = ^) the value of drz becomes,
dx'' {/"(«) + 2/"(a, h) m +f"{b) m%
U f{a, b) is a maximum the above quantity must be negative,
and if a minimum it must be positive, for all assignable values of ??e.
In both cases therefore, it must be incapable of changing its sign for
any change in the value of ?«, which will be the case if the equation
/" {a) + 2/" (fl, b) m +f" (b) m^ =0 (7),
has impossible roots, that is, if
r{a)f"(b) is > {/"(«, b)Y.
This is called Lagrange's condition.
If it is satisfied, /(o, b) will be either a maximum or minimum:
to discriminate between them, we observe that the sign of the above
expression, being always the same, must be the same as that which
it has when m = Q that is the same as that of /"(a), which again
must be the same as that of f "(b), since otherwise Lagrange's con-
dition would be impossible. Hence /(a, b) will be a maximum or
minimum, according as f"{a) and f"{b) are negative or positive.
If Lagrange's condition is not Satisfied the equation (7) will have
i'eal roots and the value of d'S, when x = a and y = b, will in general
be negative for some values of m and positive for others.
Hence, some values of m will make z a maximum and others a
minimum, in which case f{a, b) will be neither a maximum nor a
minimum value of f(x,y).
To determine therefore, the maxima and minima values off(x,y)y
we must find all the roots of the equations
/'(^) = 0, and/'(i/) = 0,
* .r is made independent variable for simplicity's sake, but we might have left the
independent variable general, when two additional terms would have been added to
(5) which would have vanished when a. b were substituted for .i- and y, leaving the
subsequent steps the same as in the text.
MAXIMA AND MINIMA VALUES OF FUNCTIONS. 125
which satisfy Lagrange's condition and then distinguish the maxima
from the minima values by observingthe sign either of/"(«) or of/"(6).
173. There is one case in which f{x, y) may have a maximum
or minimum value even though Lagrange's condition fails, viz. where
equation (?) has equal roots, and therefore
f"{a)f"{b) = {f"{a,h)Y.
For if the roots be equal to ?«, , (Pz will retain a constant sign for
all values of ?«, except my, and for that value cPz will equal zero.
For the value /«,, of m the condition that z may be a maximum
or minimum will be that the first differential, d"z, which does not
vanish shall be of an even order, and then z will be a maximum or
a minimum according as the sign of (Pz is negative or positive.
If therefore, 71 is even and the sign o^ d^z when m = m^^ the same
as that of d^z for all other values of m, s will be a maximum for all
values of ?n including m, or a minimum for all such values. In this
case therefore /(or, b) will be a maximum or minimum accordingly.
But if n is odd, z will be neither maximum nor minimum when
m = mi, and if « is even and the sign of d"z different from the sign
of d^z for other values of m, z will have a maximum or minimum
value when m - ??«, corresponding to minima or maxima values re-
spectively for all other values of m. In neither of the latter cases
therefore will /(a, b) be a maximum or minimum.
Again, if d^z vanishes for all values of ?w, which will be the case
if /"(a) = 0, /"(a, 6) =0, and /'(&) = 0, the condition that /(«, b)
shall be a maximum or minimum value, will be that the first actual
differential d"z shall be of an even order and retain a constant sign
for all values of 7n. The condition which will then take the place
of Lagrange's will be that the equation of n dimensions in m,
d"z = 0,
shall have no real roots, with further conditions in the case of pairs
of equal roots analogous to those above found. Such conditions are
too complicated to be of much practical value, and in the majority
of cases, Lagrange's condition will be found to apply.
174. The hypothesis introduced in Art. 172, that m is an arbi-
trai'y constant in the expressions (4) and (6), is evidently the same as
the assumption that dx and dy are arbitrary constants in (3) and (5),
in which case those expressions -become the first and second total
126 MAXIMA AND MINIMA VALUES OF FUNCTIONS.
differentials o^ f(x, y) considered as a function of the two independ-
ent variables x and y. The higher differentials of z are in like
manner converted by the same hypotheses into the total differentials
of/(x, y) with X and y as independent variables. The conditions
for determining the maxima and minima values of f(^x, y) may
therefore be stated as follows :
(1) d/(x, y) = 0, which involves f(x) = 0, f {y) = 0.
(2) d'f{x^ y) must retain a constant sign for all values of the
differentials, the negative sign corresponding to maxima and the
positive to minima values.
(3) If d^ f{x, y) vanishes for the same values of x and y as
d/(xy), the first actual differential must be of an even order, and
must itself retain a constant sign for all values of dx and dy ; and
if dy(x, y) vanishes only for particular relative values of dx and dy,
retaining a constant sign for all others, the first actual differential
for those particular values must be of an even order, and must have
the same sign which dy(x, y) has for other values of dx and dy.
(4) In every case a negative sign corresponds to a maximum
and a positive to a minimum value.
By a process similar to that of Art. 172, these conditions may be
extended to functions of any number of independent variables.
175. The geometrical significance of Art. 172 appears thus.
Let a surface be constructed whose equation is 2 =f(x, y)^ and
let/(a, b) be a maximum or minimum ordinate of the surface. Let
a plane be drawn through this ordinate, making an angle tan~'7«
with the plane of xz. Its equation will be equation (2) of Art. 172,
and it will intersect the surface in a curve determined by the pair of
equations (l) and (2). By making the intersecting plane revolve
round the ordinate f(a, h), i. e. by giving to m all possible values,
•we obtain a system of curve sections in some one or other of which
every point of the surface is included. An ordinate which is greater
than any of the surrounding ordinates of the surface, must be greater
than the neighbouring ordinates in any of the curve sections, and
conversely if it exceeds the neighbouring ordinates in all the sec-
tions, it must exceed all the surrounding ordinates of the surface ;
that is, if /(a, b) is a maximum in all the sections it will be a maxi-
mum also in the surface, and similarly, if it is a minimum in all the
sections it will be one also in the surface ; but if the ordinate is a
MAXIMA AND MINIMA VALUES OF FUNCTIONS. 127
maximum in some sections and a minimum in others^ it will be
neither a maximum nor a minimum ordinate of the surface, since
some of the surrounding ordinates will be greater and others less
than it. Lagrange's condition is the analytical expression of these
facts. When it is satisfied the ordinate will be a maximum or mini-
mum in the surface, if it is so in any one of the sections. The most
convenient sections to examine are those parallel to the co-ordinate
planes, and this is what we have done, in employing the sign off" (a)
or f '(b) as the test for the purpose.
176. In the preceding articles we have considered only those
maxima and minima values of functions of several variables, which
make the differentials vanish. There are others which make them
infinite, but they are of comparatively little interest except with
reference to the singular points of surfaces, which is a subject quite
beyond the scope of this work.
177. To determine the maxima and minima values of a function
of n variables connected by r equations.
Let M = i^(j;,, x^ ... x,^ = xa2LJi. or min. ,
where the variables are connected by the equations
=y 2 C-^i > X2 •■■ X„j,
(0.
o=Mxi, 01:2 '"^„)'
Then, since u is a maximum or minimum,
du = F'(Xi) dxi + F'(x2) dx, ... + F^x,,) dx„ = ^
also df/i =fi{xi) dxi +fi'{x^ dxn ... +fi'{x„) dx„ = I
df, ^fr'ipc,) dx, +fj{x,) dx,... +//(x„) dx„ =
From these (r + 1) equations we may eliminate r of the differ-
entials, leaving a linear equation involving the remaining (n — r)
differentials. This elimination may be conveniently effected by the
method of indeterminate multipliers.
From the system of equations (1), we obtain
du + \,rf/"i + X^dfi ... + A,(//, = 0,
which holds for any values of Aj, Aj .., A,.
128 MAXIMA AND MINIMA VALUES OF FUNCTIONS.
If we give to du, df^ ... df\, their values from equations (1) and
collect the coefficients of each differential, this equation becomes
M, dxi + M2dxo ... + M„dx„ = 0,
where each quantity M is of the form
F'(^) + X,//W + \2/Aa:) ... + K//{a:).
The multipliers Aj, A.^ ... A,, being indeterminate, we will give them
such values as to satisfy the r equations
3f, = 0, ]\L = ... M, = (2),
by which assumption the above equation is reduced to the form
M^i dx^+i + M^+j tZx^+2 ... + M„ dx„ = 0.
Now in the given system of equations there are (n — r) indepen-
dent variables, and therefore the (« — r) differentials in this equation
are perfectly arbitrary, and the equation cannot be satisfied unless
M,^^ = 0, M,+8 = . . . M„ = 0.
By substituting in these equations the values of A,, A, ... A,, obtained
from (2), we have (« — r) equations, which, together with the (r)
given relations among the variables, are sufficient to determine the
values of x,, X2 ••• x„, which make u a maximum or minimum.
Hence, whatever be the number of dependent and independent
variables, we have the (« + r) equations
/.(or,, x.,... x„) = 0,
/^(x,, x,...x„) = 0,
fr{Xi, Xo ...x„)=0,
F'(x,) 4 A,//(x.) + Kf,'{x,) ... + Kf/(x,) = 0,
F\x.;) + A,//(x,) + A,//(j:,) ... + A,//(x,) = 0,
F'{x„) + A,//(x„) + A,/,'(x„) ... + Kf/(x„) = 0,
involving the (n + r) quantities x,, Xa ... .?•„, A,, A, ... A,, from which
we may determine the values of j-,, x„ ... x„, which make u a maxi-
mum or minimum.
The introduction of interminate multipliers, makes this investi-
gation rather complex. The principle involved is however very
simple. It is (as may be seen) merely this. First to apply the test
of a maximum or minimum value, viz., du = 0, and then after elimi-
MAXIMA AND MINIMA VALUES OF FUNCTIONS. 129
nating as many differentials as possible to make the coefficients of
the remaining independent differentials separately vanish. In most
examples the use of indeterminate multipliers will be found much
simpler than any other mode of elimination.
The investigation of the condition which in this case supplies the
place of Lagrange's condition, and the application of the test to dis-
tinguish maxima from minima values, would be extremely trouble-
some. In most problems, however, it is easy to see from « priori
considerations whether maxima or minima values exist, and to dis-
criminate them from one another, the exact values of the variables
only remaining to be determined. In such cases the investigations
alluded to become superfluous.
178. Ex. 1. Find the maxima and minima values of x^(a-j;)'.
71 = A® (a — x)",
— = x^ (a - x) {3a — 5x] = or co .
The only values of a; which satisfy this condition are
x = (}, x = rt, andx = f«,
except X =00 , which renders u also infinite, and need not be consi-
dered.
When X passes through the value 0,
— changes from ¥{0, + h) (3a + 5h)
to h\a-h) (3a- 5h),
both of which have the same sign when h is sufficiently small.
Hence when x-0, u is neither a maximum nor a minimum.
When X passes through the value a,
-— changes from - (a- lifh (2a — Sh)
ax
to (a + hy h (2a + 5h),
the former of which is negative and the latter positive, when h
is sufficiently small. Hence when x = a, u has a minimum value
equal to zero.
When X passes through the value \a,
-J- changes from (fa - hf (|« + h) h
io-(\a+hf(la-h)h,
H. D. C. 9
130 MAXIMA AND MINIMA VALUES OF FUNCTIONS.
the foi'mei* of which is positive and the latter negative when h is
sufficiently small. Hence, when .r=f«, 71 has a maximum value
lOS
equal to (f«y(|ayor^^a\
The same results may be obtained by the method of Art. I69.
Since
■J- = x^(a — x) (Sa — 5x),
dx'
= ^^(a-.)(3«-5x)|^-^-^^},
g = 2(«-x)(3a-5x) + P,
where P is a quantity which vanishes with x;
Therefore when x = Q, -^-^ = 0, ^-3 = ba ;
dx' dx
that is, the first differential coefficient which does not vanish is of an
odd order, and therefore u is neither a maximum nor a minimum.
d^u
When X = a, -r-j = 2a^, which is positive, therefore ti has a mini-
mum value.
When X = f a, y-^ = - 2 (f )'' a', which is negative, therefore u has
a maximum value.
Ex. 2. To find the maxima and minima ordinates of the lem-
niscate.
The equation is
(^x' + yy=a''{x'-r) 0), ^^ ^'''
and its form is that of the figure.
By differentiating (1) we obtain,
2 (x- + y-) (x dx + 1/ dy) = a^ {x dx - y dy),
, dii a^ — 2.r^-2w* J?
whence -y- = -^ — —^ — r^ - •
dx a^+ 2x^ + 2y y
The equation -f- = ^ will be satisfied when
MAXIMA AND MINIMA VALUES OF FUNCTIONS. 131
The values of x and 1/ obtained from this equation together with
(1) are
which may therefore correspond to maxima or minima values. To
distinguish them we will apply the criterion of Art. I69.
By differentiating -^ , we obtain
dx
+ P,
dx" y {o^ + 2a;^ + 9.y-\
when P includes only terms which vanish when equation (2) i?
satisfied.
Hence when j? — ± */ ^ a-, and y = + a/ - «,
dx^ J^a '
which corresponds to a maximum value. \
If therefore OM=OM' = ^| a, and MP = M'P' = ^^ a, '
MP and M'P' are maxima ordinates,
/3 1 A
when x=^ ./ -a, and y=-x/ 5 «,
^^'^^ ~ x/2a '
or y has minima values at the points Q and Q,'.
Since a maximum value of (- y) corresponds to a minimum value
of y, the geometrical ordinates MQ, M'Q! are maxima.
The values x = ± a and y = will make ^ infinite, but they do
not correspond to maxima or minima values, since x cannot exceed
a without rendering the equation (1) impossible.
In the following example we shall employ the method of indeter-
minate coefficients.
9—2
132 MAXIMA AND MINIMA VALUES OF FUNCTIONS.
Ex. 3. To find the rectangular parallelopiped which shall con-
tain a given volume under the least surface.
Let the edges be x, y, z, c? the given volume. Then
u ■= xy -v xz -vyz — min.
xyz = a^^
.'. (^ -V z) dx -v {x -¥ z) dy + {x + y) dz = 0,
yz dx + xz dt/ + xy dz = 0,
.'. (y + z + Xyz) dx + (x + z + Xxz) dy + (x + y + Xxy) dz = 0.
.-. y + z + Xyz = (1),
x + z + Xxz = (2),
X +y + Xxy = (3).
(l);c-(2)^gives z{x-y) = 0,
similarly x {jj — £;) = 0,
and y (z - x) = 0^
These, together with xyz = a^, are satisfied by x = y = z = a; that
is, the parallelopiped must be a cube.
It is obvious that this form makes the surface a minimum^ since,
by diminishing one side, we may increase the surface as much as we
please.
Examples will be found in Gregory's Examples, Chap. vii.
CHAPTER IX.
TANGENTS AND ASYMPTOTES.
179*. Def. Let APQ be any curve, T'PQ a straight line
cutting it in the points P and Q. y
Suppose Q to move along the curve
towards P, then the direction of
T'PQ will continually change, and
it will approach a certain limiting
position as Q approaches P. Let
TP he this limiting position ; then
TP is called the tangent to the curve APQ. at the point P.
A line PG drawn through P perpendicular to the tangent at P,
is called the normal to the curve at the point P.
180. To determine the inclination of the tangent to the co-
ordinate axes.
Let the equation to the curve referred to rectangular axes Ox
Oy, be y =f{x),
OM = .r, MP = y, co-ordinates of P,
ON =x + Ax, NQ=y + Ay, co-ordinates of Q.
Draw PR parallel to the axis of x; then
PR = Ax, RQ==Ay,
and since PT is the limiting position of QPT' when Q approaches
P, that is, when Ax and Ay approach zero,
■■ " A-o •
tanPra:-Z^A=otanPrx
'Ax'
tanPrx =
di
dx'
^ is known from the equation to the curve, and the inclination of
dx
the tangent to the axis of x is determined.
• The substance of this and the following article has already been given by way
of illustration. It is repeated here in a more condensed shape, as it properly belongs
to the subject of this chapter.
134 TANGENTS AND ASYMPTOTES.
Hence^ if x' , y' are the current co-ordinates of the tangent^ its
equation is
y'-y = %'^^'~^^
v' —y x' — X
or - — — = ■ .
dy dx
Since the normal is perpendicular to the tangent,
tan PGx = — cot P jTj; = — T- ;
dy
therefore the equation to the normal is
y-y=-Ty^''-'^^'
or "^^1^ = 0.
dy dx
In the figure of Art. 179, ^T, MG are called the suhlangerd
and subnormal.
Now MT= MP cot PTx, MG = MP tan PTx,
.'. subtangent = ?/ — ,
subnormal =y —-.
181. The sine and cosine of the angle PTx may be conveniently
expi'essed as follows.
Let the arc ^P = *, arcPQ = A*, chord PQ = c; then
RQ
sin PTx = Z/a=o sin PT'x — U^^-^ ^^
_ Ay As
Now //a=o ■7r-= -t- and ll^^ — = 1, by Art. 11,
L\s as c
.'. sm PTx = ^;
ds
PR
also, cos PTx = lt/:^-a cos PT'x — It^-o^^^
Ax As
A* c '
.*. cosPij: = -r-;
ds
TANGENTS AND ASYMPTOTES. 135
hence (^J + (^J = cos^ PTx + sin^ PTx^l,
or dx^ + dif = ds^.
182, The lengths of the portions of the tangent and normal
between the point of contact and the axis of x, are given by the
equations
Pr= MP cosec pro;
ds Jjdx'+df) j{i+/ixy}
-^dy-^ dy y fix) '
PG= MP sec PTx
The expressions for MT, MG, PT, PG, may also be immediately
obtained from the equations to the tangent and normal.
Thus, putting y'-O in the equations to the tangent and normal,
we obtain in the two cases
, dx
x-x =y-j-,
dy
, dy
Hence, in order that our expressions for the subtangent and
subnormal may hold in all cases, we must reckon MT positive when
T lies on the negative side of i\f, and MG positive when G lies on
the positive side of M. In the figure of Art. 179 these quantities
are positive.
183. To determine the concavity or convexity of a curve at any
point.
A curve is convex to the axis of x, so long as the inclination of
the tangent to the axis of x increases with x, and concave when it
decreases with an increase of x.
In the former case, therefore, the differential coefficient of
tan PTx must be positive, and in the latter negative.
Now, tan P 7^0;= •'^ when j/ is positive,
= — —- when y is negative ;
dx "^ ^
136 TANGENTS AND ASYMPTOTES.
therefore the curve is convex or concave to the axis of x according as
-T^ and y have the same or different signs.
184. To determine the position of points of inflexion.
Def. a poinl of inflexion is a point where the direction of the
curvature changes from convex to concave, or from concave to
convex.
At such a point the inclination of the tangent to the axis of x, after
increasing up to the point of inflexion, begins to decrease, or vice
versa. The inclination therefore, and consequently its tangent or
fly
-J- , must be a maximum or a minimum at a point of inflexion, the
analytical condition of which is
^-^^Ooroo.
If not only -—^ but also -— vanishes, we must have the addi-
tional condition, that the first differential coefficient which does not
vanish shall be of an odd order.
The values of x which make -yz infinite, will not necessarily cor-
respond to points of inflexion, since this condition may be satisfied
without the inclination having a maximum value.
When the tangent is parallel to the axis oi y,-~ , and therefore
generally -r^, are infinite, Avhether the point is a point of inflexion
d^x
or not. In this case we may employ the condition ~r^i = or co , in-
stead of the above, as it might have been obtained in a precisely
similar manner, and is free from the ambiguity attaching to our
former criterion.
185. To find the equation to the asymptote of any curve.
Def. The asymptote of a given curve is a straight line or curve
which the given curve continually approaches but never meets. If
the equation to the given curve is algebraical, the asymptotes may in
general be determined as follows.
TANGENTS AND ASYMPTOTES 1S7
Let the equation to a curve be /{x, y) - 0; and let this be
reduced to the form
3/ = Jx"+J5a;'"...+ G^+iyx-''+&c (1),
the indices being arranged in descending order. Let the equation
to another curve referred to the same axes be
y' = Ax" + Bx'^...+ G (2),
then {y-y') = Hx-^ + 8^c.,
i. e. the distance between the two curves is generally finite when x
is so, but approaches zero when x approaches infinity.
The curve (2) is therefore the asymptote to the curve (t).
If 71 = 1, m=0, the equation (2) becomes
y = Ax + B,
and the asymptote is rectilinear.
If in the equation to the curve, y is given explicitly in terms of
X, the expanded form may be obtained by the ordinary algebraical
methods. If t/ is only given as an implicit function of x, the ex-
pansion must be found by means of indeterminate coefficients.
The most important cases are those in which the asymptotes are
rectilinear.
It is evident that a rectilinear asymptote coincides with the limit-
ing position of the tangent when one or both of the co-ordinates
approach infinity.
This is sometimes given as the definition of an asymptote, and
in many cases the asymptotes may be readily determined from this
consideration.
Whenever there is a rectilinear asymptote, the perpendicular upon
it from the origin, and therefore one at least of its intercepts on the
co-ordinate axes, must be finite. Hence the test of the existence of
an asymptote is, that the intercepts of the tangent shall not both
become infinite when the co-ordinates of the point of contact are put
equal to infinity.
Asymptotes parallel to either of the co-ordinate axes may be at
once discovered, by observing whether any finite value of one of the
co-ordinates renders the other infinite*.
186. Examples on the preceding articles.
(1) Let the equation to the curve be
" Symmetrical investigations of points of inflexion and asymptotes will be found
in the Muthemaliail Joxirnal for February 1(541 and November 1843,
138 TANGENTS AND ASYMPTOTES.
• ^ xdx _ydy dy _h^ x
a dx a' y
and the equation to the tangent becomes
x(x'-x) _ y{y'-y)
a" ¥ '
^-f- = lby(i).
The equation to the normal is
or a'yx' + h^xy' = (o^ + h^) xy.
The equation to the asymptote may be found from that to the
tangent by making x and y approach infinity. The equation to the
tangent may be put into the form
y = —i — x: .
Now when x and y approach infinity — approaches zero, and by (l),
Avhich approaches ± j when y approaches infinity. Therefore the
equation to the asymptotes is
7/ =^-x'.-
^ a
This may also be found b)'^ expanding y in descending powers of
X. Thus
=44'-^--}'
Hence, by our definition, the equation to the asymptotes is
y' -^ti. - x'.
a
Since y-y'=^^~ +
TANGENTS AND ASYMPTOTES. 139
the sign of (y-j/'), when x is positive and sufficiently large, is nega-
tive for the asymptote whose equation is ^' = + - x', and positive for
that whose equation is ^' = x' ; that is, in the former case the
curve lies below the asymptote when x is positive, and in the latter
above it. Since the sign o? y—y' changes with x, the reverse will
be the case when x is negative.
This is easily seen to agree with the known form of the hyperbola.
(2) To find the asymptotes of the curve
{y' + bf-){x-a)-x'^0 (1).
When X approaches «, the value of y^ + hy^, and therefore that of
y, approaches infinity, and there is therefore an asymptote parallel to
the axis of ^ at a distance a from it.
To find any other asymptotes that may exist, we must expand y
in a series of descending powers of x. Assume, therefore
C
y = Ax + B +— +&C.,
.-. y^ = A'x^ + 3A'Bx^+8iC.,
... (^3:-a)y^ = A^x* + (3A'B-aA')x^ + &c.,
b(x-a)y'= bA'x'+8cc.,
-x'^-x\
Adding these quantities, we have, by (l),
= x*(A'-1)+x'' (3A-B - aA' + hA') + &c (2),
which must hold for all values of x;
.-. ^'-1=0, .-. A = l,
3A'B-aA'+bA' = 0, ■.-. B = "^.
o
Therefore the expanded form of the equation to the curve is
a-b G .
j/ = x ^ — ~ — I 1- &c. . . .
3 X
and consequently that to the asymptote,
/ r f^-b
^ 3
If it had been required to determine whether the curve lies
above or below the asymptote, we must have collected the coefficient
140
TANGENTS AND ASYMPTOTES.
of one more term in the expansion (2), by equating which to zero
the value of C might have been determined.
The relative position, however, of the curve and asymptote in
such cases may generally be more easily determined by indirect con-
siderations, as will be seen when we consider the methods of tracing
curves.
(3) To find the points of inflexion of the curve i/ = - — >,-^ .
!/ =
(x-«r
Then
dy 3 (x — a)'
which vanishes when x = a ; and as the next differential coefficient
does not vanish when x — a, this point is a point of inflexion. Since
y and -^ also vanish when x = a, the tangent at the point of inflexion
is the axis of J, and the form of the curve is that of the figure
y
Curves referred to polar co-ordhiates.
187. To determine the angle between the tangent and the
radius vector of any curve.
Let QPT' be a secant through two
points P, Q of a curve APQ. FT the
tangent at P; r, d co-ordinates of P,
r + Ar, + A0 co-ordinates of Q, refer-
red to a pole aS", and prime radius Sx.
Draw PR at right angles to SQ..
Then, by the definition of a tangent,
tan SPT=ltpQ=o tan SQT,
PR
TANGENTS AND ASYMPTOTES. 141
r sin A0
= ^^^=° r + Ar - r cos Ad '
rA0 s[n A0 1
= «A=o -^^ -^^g ~ln7iA0 '
1-^2^ Ar
- rA0 dd
And "-^=0^,:-^^'
sin Afl
, sill tJ" ,
2smHA0_. siniA0A0 ^^^
//a=o ^^^ "-^=9 iA(? Ar ^
If we call AP, s, and therefore PQ, A.v, and the chord PQ, c,
we have
cos SPT = lfr(i=o —7" 5
dr
"ds'
^ [ ^ sin*iAa\
A* c
since
Also sin SP T = /<pq=o
c
Hence,
rA0 sin A0 A?
" '^^='' a7 "aF" T '
dB
ds '
(ST -'©'->
or rfr^ + r^ja- = ds\
188. If 5r, 5r are drawn at right angles to SP, Prrespec
tively, we have
ST = r tan SPT,
-'■'t-
ST is called the j^olar suhlangenl.
142 TANGENTS AND ASYMPTOTES.
Also, if
SY=
= P,
P =
= ?• sin
SPT,
=
ds
This quantity may be conveniently expressed in terms of the
reciprocal of r. If this be u, we have
du
dr^
UUf
.-. ds' =
du'' d&^
1
,ds^
=
„ fdu\
It may
also be expressed
in terms of the rectangular
co-
ordinates
as
follows.
If -Sis
the
origin and Sx the axis of x,
we have
e
= tan->-^;
x'
.'. de
xdy — ydx
~~ 1 . .2
p
xdy — ydx
r^dQ X dif —ydx
ds ds
The expression for ST has been found from a figure in which
dr .....
r, 0, -ja have all positive signs, but it applies equally to all cases, if
we interpret properly the sign with which it is affected. By con-
structing the figui'es in the cases where any of these quantities are
negative, it is easily seen that the direction of ST is always deter-
mined by the following rule, when the positive direction of Q is
taken to the left of the prime radius. Look along the positive direc-
tion of the radius vector, and draw ST to the right or left, according
as it is positive or negative.
It may also be observed that the expression for tan SPT is, in
all cases, positive when the tangent falls behind the radius vector,
and negative when on the other side of it.
TANGENTS AND ASYMPTOTES. 143
189. To determine the asymptotes of polar curves.
When any value of renders r infinite, there may be a recti-
linear asymptote. This will be the case if the polar subtangent re-
mains finite when r becomes infinite. Since in this case the angle
SPT vanishes, the asymptote will be parallel to the infinite radius,
and at a perpendicular distance from it equal to the corresponding
value of ST, and measured in the direction indicated by its sign.
When the radius vector approaches a constant limiting value, as
6 approaches infinity, the curve will approximate to a circle which is
called an exterior asymptotic circle when r increases in magnitude up
to the limiting value, and an ifiterior asymptotic circle when r de-
creases towards it, as in the former case the circle is without, and in
the latter within the curve.
190. To determine the concavity or convexity of curves referred
to polar co-ordinates and the position of points of inflexion.
It is easily seen that when a curve is concave to the pole, p
increases with r, and when convex p decreases with an increase of r.
The curve is therefore convex or concave according as -j- is nega-
tive or positive.
At a point of inflexion, the curve changes from convex to concave
or vice versa, and therefore -j- changes its sign. At such a point
therefore we must have
-f- = 0, or CO .
ar
Those values of r which satisfy this condition, and also produce a
change in the sign of -,- , will correspond to points of inflexion.
Examples will be found in Gregory's Examples, Chap. ix.
CHAPTER X.
CONTACT OF CURVES.
191. Let y =f{x), y = F{x) be the equations to two curves ;
then if, when a certain value, o, is given to x, the values of the
ordinates of the two curves are the same ; that is, if
f{a) = T{a),
the two curves have a common point whose co-ordinates are a, f{a),
and are said to intersect. If at the point of intersection we have also
f{a) = F'{a),
the curves have a common tangent, and are then said to have a con-
tact of the first order.
And generally, if we have
f{a)=^F{a), f{a) = F'{a), f"{a) = F"{a) ...f'^\a) = F^^a),
the curves are said to have a contact of the h"* order at the point in
question.
The reason of this definition appears from the following pro-
position.
192. If any curve has a contact of the nf^ order with a second
curve, and a contact of the n*'^ order with a third at the same point,
where m is greater than w, and an ordinate is drawn cutting the
three curves, the portion intercepted between the curves whose con-
tact is of the m}^ order bears to that intercepted between those whose
contact is of the n^'^ order, a ratio which approaches zero as the ordi-
nate approaches that through the point of contact ; or, as it may be
expressed, the curves whose contact is of the higher order are, in the
limit, infinitely closer than those whose contact is of the lower order.
For, let the curve y =f(j>c) have a contact of the m*^ order, and
y = (p(x) a contact of the n**" order, with y = F{x) at a point whose
abscissa is a ; let an ordinate be drawn to the three curves whose
abscissa is a + ^, and let A„,, A„ be the portions intercepted between
y = F{x) and the two other curves.
Then A,„ =/(« + h)-F{a + h\
and /(a + h) =f{a) +f(a)k ... +/-'(«) j^^ +/'-'(« + Ok) p^^^.
CONTACT OF CURVES. 145
[m ^ '' [in + I *
The first (?m + 1) terms of these series are equal by the conditions
of contact of the ?«'*' order,
7m+l
similarly, A„ = ——^ {0'"+"(« + ^^0 - ^<"+"(« + 0/«)},
... ^ = /r "• j^^^ (^'"+')(a + 0A) - F'"^\a + Oh) '
When h approaches zero, the coefficient of /i'""" approaches a finite
value, since otherwise the contacts would be of higher orders than
those supposed ; and if m is greater than n, the limit of /t"*"" is
zero.
A_„
A"
Ith^o -r^' - 0.
193. Curves which have a contact of an even order cut, and
those which have a contact of an odd order touch without cutting at
the point of contact.
For if A„ is the excess of the ordinate of i/ =f(x) over that of
y = F{x) with which it has a contact of the m^^ order,
^- = tItt ^^"""^'' "■ ^^'^ - ^'"'^'^ + ^^'^^-
When h is made sufficiently small, the sign of the coefficient is
that of/''"+^'(a) - jP>"'+'*(«) whether h is positive or negative, and the
sign of /^'"+' changes with that of h^ when m is even and is unaltered
by a change of sign in h when m is odd. In the former case
therefore the ordinate of one curve is in excess on one side of the
point of contact, and that of the other on the other side, while
in the latter, the ordinate of the same curve is in excess on both
sides.
Hence, when the contact is of an even order the curves cut, and
when of an odd order, touch without cutting.
194. If we suppose the same quantity to be independent vari-
able in two curves which have a contact of the m"' order, the con-
ditions are that at the point of contact the values of
X, ;/, dx, dy (f "x, dJ'^y
shall be the same in the two curves.
H. D. c. 10
146 CONTACT OP CUEVES.
For if any function of x is independent variable, the values
of dx, d^x . . . d"'x must be the same in both curves when the same
value is given to x, and dj/, d^y . . . d"y can be expressed in terms
of these quantities and of the m first differential coefficients, which
are also the same in the two curves by the conditions of contact of
the ?«"" order.
When any of the differential coefficients have infinite values,
the proof in Art. I9I, which depends on Taylor's theorem, fails.
If the infinite differential coefficients are of the second or higher
orders, the curves are of peculiar forms (as will be subsequently
shewn) at the point in question, such that the notion of orders of
contact is inapplicable. If however the first differential coefficients
are infinite, there may be no peculiarity except that of the tangent
being parallel to the direction which has been chosen for the axis
of ^. The difficulty is at once obviated by equating the differential
coefficients of x with respect to y in the two curves, instead of those
of ?/ with respect to x, since these conditions might have been ob-
tained exactly as the others.
When the differentials are equated, no difficulty can arise, since
the conditions of contact in this form might have been obtained from
either of the other systems.
For this reason, and also for the sake of symmetry, it is often
better to employ the differentials than the differential coefficients.
195. To determine the particular curve of a given group which
has the closest possible contact with a given curve.
Let j/ =/(x) be the equation to any curve
y = F{x, ai, a^ ... a„),
that to a group of curves having 71 parameters Oj Oj. . .«„, by assign-
ing particular values to which, any curve of the group may be
determined.
Let now such values be given to a^ a„ . . . «„, as to satisfy the
n equations,
y'-y ^-^ d^^_i1r^y
y ~y' dx~ dx dx-' ~ daf-' ^ ^'
when a particular value, a, is given to x. This can evidently be done,
since o, a2,...a„ enter into the values of y' and its differential co-
efficients ; the n parameters will then be absolutely determined by
CONTACT OF CURVES. 147
the assumed equations and cannot be made to satisfy any additional
conditions. When these values ai'e given to the parameters, the par-
ticular curve thus selected from the group will have a contact of the
(n - 1)"" order with the curve 1/ =f{x) at the point whose abscissa is a.
Hence the order of the closest contact which a curve of a given
species can be made to have with a given curve at any point of it, is
equal to the number of parameters in the general equation to curves
of that species, diminished by unity.
It may happen in particular cases, that the same values of the
parameters which satisfy the 71 equations (1) will also make some of
the higher differential coefficients equal in the two curves, and in
that case the contact will be of a higher order; but in general the
order of closest contact is less by unity than the number of parameters.
The general equation to the circle contains thi'ee parameters ; a
circle may therefore always be found which shall have a contact of
the second order at any point of any given curve.
196. The curvature of a curve at any point may be conveniently
measured by determining the radius of the circle which has the
closest possible contact with the curve at that point. Hence the fol-
lowing definition.
The circle which has a contact of the second order with a curve
at any point, is called the circle of curvature of that point, and its
radius the radius of curvature.
197- To determine the radius of curvature and the co-ordinates
of the centre of the circle of curvature at a point (x, y^ of a given
curve.
Let p be the radius, a, /?, the co-ordinates of the centre. Then
the equation to the circle is
(^-«r+(^-/5y=/ (1),
■whence {x — a)dx+ {y — ft)dy = (2),
(x - a) d^x + {y-ft) dhj = - dx" -dy'- = - ds^ (3) ;
where dx, dy, d^x, d^y, ds are the differentials of the co-ordinates of
the circle.
We have now to give to o, /:?, and p such values, that the circle
and curve may have a contact of the second order at the point (x, y'),
that is, such values that x, y, dx, dy, d-x, d'y may have the same
values in the circle as in the curve. , If therefore we give to the
10—2
148 CONTACT OF CURVES.
differentials in equations (l), (2), (3) their values derived from the
equation to the curve, those three equations will give the required
values of a, /3, and p. To find these explicitly, we have from (2)
^ X — a y — ft ^(x - a) d^x + {y — ft) cPy
dy — dx dy d^x — dx d^y
dyd'x-dxd'y' ^ ^^^
:c-a_ y-ft _ J{{x-ay + {y-ftr]
dy - dx J{dx^ + dy^)
= -£.by(0.
Equating these values, we obtain
dyd?x — dxd^y
dy ds
P rl,.rl^ ^ ^ rl-^rfi,,'
X +
ft=y-
dyd^x — dxd^y
dxds^
dy^x — dxd^y '
The magnitude of p without regard to sign being the object of
investigation, we must employ the upper or lower sign in any par-
ticular curve according as the former or the latter renders the above
expression positive. The direction in which p is to be drawn will
be determined by the values and signs of the expressions for a
and ft.
" The artifice of elimination here used, depends on the following algebraical
proposition. If
a _ c
t hen -7 = -. = — ; jt tor all values of «i and n,
o d nib + nd'
" >J(,b^+d')'
For ma+nc = jlmb +71 - c\=-{mb + nd),
and VK + 0=| \/ [f^' + ^.c^yiy/Cb^ + d^).
whence the above equations follow.
By giving convenient values to m and n, as in the instance in the text, this often
affords a ready method of elimination.
COXTACT OF CURVES. 149
198. If we denote by t the angle which the tangent to the
curve makes with the axis of x, we have
T - tan"' -; :
ax
<ftldx — d-X(h/
ay + ax
(It dijd^x — dx(Fy
*'• r77" d? '
or using the lower sign in the value of p,
ds_
dT'
p =
199. In the expressions above found for p, a, /3, the independent
variable is general ; they may be simplified in particular cases by
assuming different independent variables. The following are the
most important forms and should be remembered. In the first three
eases the upper sign of the expression for ^ in Art. 197 is used, in
the fourth the lower. These are the signs commonly used in the
respective cases, but for the reason above given, it is perfectly im-
material which sign we adopt.
(1) X independent variable.
Putting d^x = 0, the expressions become
ds- l^^'VJi
' dxd'y d'y
d?
14-f^
dti ds^ dy \dx
a = X- / ,„ =x- -^
dxd^y dx d'y
dx^
ds^ '"-
^ ^ Vdx J
dx^
Exactly analogous equations may be obtained when y is inde-
pendent variable.
(2) $ independent variable.
Since the expressions do not involve d^s they will still be correct
in their general form, but it is often convenient to modify the ex-
pression for p as follows.
150 CONTACT OF CURVES.
We have (dj/d^x - dxd^t/Y = -3- (1) ;
also dx^ + df = ds';
.'. dx(fx + dy(fy = Of since ds is constant.
Adding the square of this equation to (1), we have
{<r-xy {dx- + df) + {dhjy (dx- + df) = ^4 ;
ds' 1
j{{d^xy + (d-,/y} /I /./^y _^ /<yy i '
The expression for p may also be put in the following forms,
when s is independent variable.
As before, dx d'x + di/ d'^y = ;
" '^'^~ ^~dr '
ds'
H-
df d'y
dx
- dx d-y
dx
ds dx
ds
dhj
dhf
ds-
In like manner, by eliminating d^y we obtain
dy
_ ds dy ds
P^ d^T^d^'
ds-
(3) 6 independent variable, where
x — r cos 6, y =f sin 6.
Then dix — dr cos 6 — ddr sin 6,
d-x = d-r cos - 2 dr dd sin B-dQ-r cos 0,
dy = dr sin 6 + ddr cos 6,
d^y = d-r sin 6 + 2drd6 cosO- dd-r sin 6 ;
.-. dy d-x -dx^y = rd-r d0 - 2dr- dd - r-dd\
and ds'={dr- + r'-dd-}^',
CONTACT OF CURVES. 151
{d7^+r^df}^
''• ^ ~ rcPr d0-2dr-dd - rUW^ '
dd' ^ \dd
(4) It is often convenient to express p in tei*ms of p, the per-
pendicular upon the tangent. From Art. 188, we have,
p =
X dy — y dx
Is '
.'. dp = -j—j {(x dry — y d"x) ds" - ds d^s (x dy —y dx)} ;
.'. dp ds^ — {{x d-y —y ^x) (dx^+ dy^) + (ydx — x dy) {dx d^x + dy dy)}
= X {dx'^ d^y — dy dx d'x} + y {dx dy d^y — d^x dy^}
= {x dx + y dy) [dx d^y — dy d^x} ;
— ds^ xdx + y dy
dy d^x — dx d'y dp '
rdr „
or p = -^, if x- + y'^r\
the negative sign of the expression for p in Art. 197, being used in
this instance in order to make the expression in terms of p and
r positive.
Cor. a chord of the circle of curvature drawn through the
point of contact is called a chord of curvature. Hence the chord of
curvature through the pole of a curve referred to polar co-ordinates,
equals the diameter of curvature multiplied by the cosine of the
angle between the radius of curvature and the radius vector;
.*. Choi'd of curvature through pole = 2r -r- - = 2/) -r- .
dp r ^ dp
200. Since the circle of curvature has a contact of an even order
with the curve, it will generally cut the curve. If however the
values of a, ft, and p, which satisfy the conditions of contact of the
second order, happen to be such as to make the third differential co-
efficients identical in the curve and circle, the contact will be of the
third order, and the circle will touch without cutting the curve.
This will be the case whenever the radius of curvature attains a
152
CONTACT OF CURVES.
maximum or minimum value. For at such a point, making x inde-
pendent variable, y =f{x) being the equation to the curve,
dp {1 +777^121 "
••• = 3/' (a:) {r{^-)Y-f"{x) {1 +f{x)Y] (1).
Now if we differentiate three times the equation to the circle,
with X as independent variable, we have
{x-ay + (y-(iy=p^,
1 +
or
:i)vg(.-/5)=o,
<Py dx \dxy
by the conditions of contact of the second order ;
••• 'B=/"W^ by(l);
or when the radius of curvature is a maximum or minimum, the con-
tact is of the third order, and the circle does not cut the curve.
This will always be the case at the vertex of a curve.
201. To find the equation to the evolute of a given curve.
The co-ordinates (a, /3) of the centre of the circle of curvature at
any point {x, y) of a given curve have been found, in Art. I97, as
functions of x and y. As the point of contact varies, the position of
the centre of curvature will also vary, and if x, y vary continuously,
the centre of curvature will trace out a curve. This curve, for
reasons which will be explained below, is called the evolute, the
given curve receiving the correlative title of i?woh/te.
CONTACT OF CURVES. 153
The equation to the evolute will be found by eliminating x and
y between the equation to the involute and those which give the
values of a and /3.
The elimination is generally difficult, and, with a few exceptions,
impracticable.
202. To prove the properties of the evolute from which it
derives its name.
In determining the values of a, /3, p, in Art. 197, we differen-
tiated the equation to the circle of curvature twice, and substituted
in the result, for x, y and their differentials of the first and second
orders, their values derived from the equation to the given curve,
the equality of these differentials being the condition of contact of
the second order between the curve and the circle.
In what follows, x and y will be considered as co-ordinates of any
point of the involute, and a, /3 those of the corresponding point of
the evolute ; so that p, a, /? are functions of x and j/, and no longer
constants, as when .r and y were considered as current co-ordinates
of the circle and not of the curve. With this understanding we may
proceed as follows to determine the properties of the evolute.
The values of/), a, ft are given by the equations (Art. 197-)
(x-«y + (^-/3)^ = P^ (1),
(x-a)dx + {y-ft)dy = (2),
{x-a)d'x + (y-ft)dy = -ds' (3).
Differentiating (1) and (2), considering a, ft, p as variables, we
obtain
(x - a)dx + {y- ft) dy - {x - a)da- {x- ft) dft = p dp,
(x - a) d-x + (y- ft) d-y - dx da-dy dft= - ds'',
from which equations, together with (2) and (3), we have
(x - a)da + (y - ft) dft = - p dp (4),
dxda + dydft=^0 (5).
Equation (2) is the equation to the normal to the involute, a, ft
being the current co-ordinates. Hence any point of the evolute is
in the normal to the corresponding point of the involute.
Again, from equations (2) and (5), we obtain
f^=-^^ (6).
da dft ^ ^
154
CONTACT OF CURVES.
This is the equation to the tangent to the evolute at a point {a, ft),
X, y being the current co-ordinates.
Hence any point of the involute is in the tangent to the cor-
responding point of the evolute. The radius of curvature, therefore,
which is the line joining corresponding points of the involute and
evolute, is at the same time a normal to the former and a tangent to
the latter.
If we denote by o- the arc of the evolute, we have, from equa-
tion (6),
X — a. y — ft
_{x — a)da-V {y — ft)dB — pdp
da'+dft' da"
J{{x-ay + (y-fty} _^ p
J{du" + dft') d<T
by (4),
by (1).
Equating these values, we have
dp ^d(T = 0.
If o- is so measured that it increases as p decreases, we must take
the upper sign, whence
p + <r = C;
that is, the arc of the evolute added to the length of the radius of
curvature is invariable.
From these properties it is easily seen that the involute must be
a curve traced by the end of a string unwound from the evolute,
and it is from this property that the name is derived.
Thus, let a point P of the string pP in unwinding from the
curve pqA trace out the curve PQ,
and let j)P, qQ he any two posi-
tions of the string; pP and qQ
must evidently be tangents to Aqp
at p and q.
Again, the portion of PQ in
the immediate neighbourhood of
P, may ultimately be considered
as traced by the revolution of pP
about p, and therefore coincides
with a circle whose centre is p
and radius pP.
CONTACT OF CURVES.
155
Pp is therefore a normal at P and p is the centre of curvature
of PQ at P. So Qq is a tangent to Aq at q and a normal to PQ at
Q, and g is the centre of curvature of PQ at Q. The curve pq is
therefore the locus of the centres of curvature of PQ. Again, if A
be a fixed point in Aqp, Qq evidently exceeds Pp by the length of
string unwound from qp ;
.-. Aq +qQ = Ap+pP,
which is the property expressed by the equation
p + <y = C.
This latter property is a necessary consequence of the former one,
viz. that the tangent to the one curve is a normal to the other, and
was in fact deduced from it in the last article.
If P'Q' be a curve traced out by any other point, P', of the string,
the same properties will belong to P'Q' as to PQ, and both of them
will be involutes of Apq.
Hence it follows, that while for a given involute, there can be
but one evolute, an indefinite number of involutes can be obtained
from a given evolute, by varying the length of the string unwound.
203. To find the radius of curvature at a point of an evolute
corresponding to a given point of any curve or involute.
Let p be the radius of curvature at any point of the given curve ;
p' that at the corresponding point of the evolute. Then if t, t', be
the angles made with the axis of x by tangents at the corresponding
points of the involute and evolute,
since the tangent to the evolute is a normal to the involute.
Also (Art. 198) P^'Jr'
And similarly, p' = y-,
dp
since dr' = dr and da- = dp ;
, dp ds _ dp
' ' " ds' dT " ds'
If a second evolute be drawn to the former as involute, and p" be
156
CONTACT OF CURVES.
the radius of curvature at the correspondhig point, we shall have
in like manner
= P
dp
And if/)'", /o". . ./)'"* be the radii of curvature at corresponding points
of a succession of evolutes, we shall have in like manner
P -de"
P P Jp^n-2, ,
by which equations the radii of curvature may be successively deter-
mined.
204. To find the equation to the evolute of a polar curve.
Let PO be the radius of curvature of the curve AP ; SP = r ;
p
SY=p, aS'F' = ^/, perpendiculars to the tangent and normal at P,
SO - ?•'. Then
.: p' + p" = r (1),
„ , o /drY ^ dr
•■"■^'-{dj,)''-i'rp w-
From the equation to the curve in terms of p and r, and these
two equations, i- and p may be eliminated, leaving an equation be-
tween jt>' and r', which is the equation to the evolute.
For examples on the present Chapter, see Gregory's Examples,
Chapter xii.
CHAPTER XI.
SINGULAR POINTS.
205. Certain points of curves present peculiarities of different
kinds inherent in the curves themselves, and not dependent on the
position of the co-ordinate axes. Such points are called singular
points: the most important are miultiple points, conjugate points,
cusps, and points of inflexion.
Multiple points are those through which several branches of a
curve pass.
Conjugate points are isolated points through which no continuous
branch passes.
Cusps are points at which the curve suddenly stops and returns
in the opposite direction. At a cusp, therefore, the curve has two
branches with a common tangent.
Cusps are of two kinds :
(1) When the two branches are on opposite sides of the tangent.
(2) When they are on the same side of it.
The two kinds are represented in the following figures.
Points of inflexion have already been defined in Art. 184.
In the following investigations of the analytical properties of
singular points, the equations to the curves are supposed to be alge-
braical and not transcendental or circular.
206. If a straight line is drawn intersecting any curve, it will
meet it in a number of points which
may equal but cannot exceed the order
of the curve. (Hymers' Conic Sections,
Sect. 238).
If we suppose the line to move
parallel to itself, the number of points
in which it meets the curve may in the
course of its motion undergo vai'ious
changes. Thus in the accompanying ^
158 SINGULAR POINTS.
figure the line MPp meets the curve in two points. When by-
moving parallel to itself it reaches either the point A where it be-
comes a tangent or the multiple point B, the two points of intersection
coincide. Beyond B again, the double intersection is restored.
So if there had been more than two branches passing through B a
greater number of intersections would have been reduced to a single
one.
This property Avill furnish an analytical test of the existence
of a multiple point. For convenience, suppose the intersecting
line to be parallel to the axis of i/. The ordinates MP^ Mp of
the points of intersection for any assigned value of x will then be
the roots of the equation to the curve, when solved with respect to y.
Whenever two of these roots become equal the corresponding point
must either be a multiple point as at B or must have its tangent
parallel to the axis of j/ as at A.
To distinguish multiple points from such points of contact, sup-
pose another set of intersecting lines to be drawn parallel to the axis
of X. Then the points of intersection for any assigned value of y
will be given by the roots of the equation when solved with respect
to x; and as before, when any two of these become equal there
must either be a multiple point or a point whose tangent is parallel
to the axis of x.
If therefore for any values of x and y two or more roots of the
equation become equal both when solved with respect to x and with
respect to y, the corresponding point must be a multiple point,
since the same tangent cannot be parallel to both axes at once.
Hence the following proposition.
207. To determine the analytical properties of a multiple point,
and to find the number and direction of the branches which pass
through it.
'Let f\x,il)-0 (1),
be the equation to a curve cleared of radicals; ^,, y^ co-ordinates
of a multiple point.
If in (1) we put ;r = x,, the equation must have two or more
roots each equal to y^; and if we put y=^y^, it must have two
or more roots each equal to x^. Since the equation is in a rational
SINGULAR. POINTS. 159
form, the condition of its having equal roots is that the derived
equation shall have a root of the same value*. The equations,
/W = 0, /(y) = 0,
■will therefore be satisfied as well as equation (1) by those values
of X and _y which correspond to a multiple point.
To determine the values of -^ at the point in question, we have,
by differentiating (1),
/'0^)+/(J/)|^=0 (2),
which gives only an indeterminate value of j^ when the conditions
of multiplicity are satisfied.
Differentiating (2), we have
/"W+2/"(.,y)g+/'W(|y+/'(^)g=0 (3),
and putting x = x^^ y ^y^'
/'(.,) +2/"(x..,,)|u^"(y,)(|y=o (4).
If any of the coefficients in this equation are finite, it will give
two values of -i~: if these are real, two branches of the curve pass
through the point {x^ , _?/,) at the inclinations determined by the two
values of — : if they are impossible, no branch passes through the
rtXj
point, which is therefore a conjugate point, since its co-ordinates
satisfy equation (1).
If all the coefficients in (4) vanish, the values of the differential
coefficient remain indeterminate.
To determine them we must differentiate (3), and in the result
make x and y equal to x, and j/, : to simplify the process we may
observe that (4) might have been obtained from (2) by differentiating
7 72
it as if -T- were constant, since the term involving -j4 must be mul-
dx ax
tiplied by a coefficient which vanishes when .r, y are put equal to
• See Hymers' Theory of Equations, Sect. IV. It will be observed that the
proof of the property employed in the text depends upon the equation being cleared
of radicals.
160 SINGULAR POINTS.
Xi, 3/,; the same will be the case with the equation derived from
(3), which will be a cubic equation giving three values of —^ .
If these are all real, three branches pass through the point at
inclinations determined by these values ; if only one root of the
equation is real, there is a single branch of the curve through the
point (x,,3/,).
If this equation becomes indeterminate by its coefficients vanish-
ing when X, y equal ar, , y^, we must proceed as before, differen-
tiating as if -J- were constant, until we arrive at an equation which
has finite coefficients. The number of real roots will indicate the
degree of multiplicity of the point.
If the equation for determining -j- has equal roots, two or more
branches will touch. Points where this occurs are sometimes called
points of osculation.
The equation obtained after n differentiations is generally of the
«'^ degree in -^ , but it may happen that the coefficients of some
number (r) of the highest powers of ->— vanish while other coeffi-
cients remain finite, thus reducing the degree of the equation to
dx, .
n — r. If, however, we had sought the values of -j- instead of those
of ^, these would have been the coefficients of the r lowest instead
ax,
of the r highest powers, and there would have been r roots of the
equation each equal to zero. In the former case, therefore, there
must be r roots each equal to infinity, and r branches of the curve
pai-allel to the axis of y, besides those determined by the other real
roots of the equation.
208. In this method of determining the values of — it has
been assumed that -j— vanishes when multiplied by a coefficient
equal to zero; this may not be true, however, when the second
differential coefficient becomes infinite at the point in question, a
condition which, as will be seen, generally indicates some other
SINGULAR POINTS. 161
peculiarity besides multiplicity, such as the existence of a point of
inflexion or a cusp. Any error arising from this source may be
obviated by transferring the origin to the point which we are ex-
amining, and finding the limiting value of ^ when x and j/ approach
zero, which will then be identical with the differential coefficient
at that point.
This is more laborious than the preceding method, except when
the multiple point is originally at the origin, and as the error
alluded to is of rare occurrence the former method is generally
preferable.
209. To determine the analytical properties of cusps.
Since a cusp is in fact a species of multiple point, it will
appear, by the same reasoning as before, that the co-ordinates of
such a point in a curve whose equation is /(or, ij) = must in general
satisfy the equations
That which distinguishes a cusp from a multiple point, is that the
two branches of the curve lie only on one side of the point instead of
passing through it.
If, therefore, a, b are the co-ordinates of a cusp, and we give to
X in the equation to the curve the values a + h, a - h, one of these
must give two real values of y while the other renders y impossible,
h being taken sufficiently small.
This can only be the case, in algebraical curves, if the equation,
when solved with respect to y, is of the form
y = F(x) + (p{x) (x ~ a)",
where w is odd and 7i even, since the radical has two real values
when (x ~ a) is positive and is impossible when this quantity is
negative.
By repeatedly differentiating the above equation, we shall at
length arrive at one of the differential coefficients of y which con-
tains a term involving a negative power of (x ~ a) and which there-
fore becomes infinite when x = a. Hence the distinguishing pro-
perty of cusps is that their co-ordinates render some of the differ-
ential coefficients of ^ infinite.
H. D. c. 11
162 SINGULAR POINTS.
There is one case to which the above reasoning does not apply,
viz. that where the cusp is of the first kind and the tangent paral-
lel to the axis of 3/, since in this case the values a-h, a + h of x
will each give a single real value of y.
Here, however, the first differential coefficient is infinite, and our
general criterion that some of the differential coefficients must become
infinite still holds, the other necessai-y conditions being those of 7/
having a maximum or minimum value at the point in question,
which have been already investigated.
If the tangent is parallel to the axis of 1/ and the cusp of the
second kind, the substitution of a + h and a-h for x in the equa-
tion to the curve will give respectively two and no real values of ^.
To distinguish between the two kinds of cusps when the tan-
gent is not parallel to the axis of 1/, we must observe whether the
difference of the ordinates of the curve and the tangent at the cusp,
corresponding' to an abscissa a ^ h, has the same sign for both the
values of y, or whether it is affected by a double sign. This differ-
ence =/(fl ± A) -{/(«) =^/(«) ^'}-
If we give to h that sign which corresponds to the real values
of 1/, and expand the above expression in ascending powers of h,
the development must contain an even root of Ii, since a change in
the sign of h renders y impossible.
If this root appears in the first term of the series the cusp is
of the first kind, otherwise it is of the second kind ; for in the
former case the two values of the series have opposite signs, and
in the latter the same sign, when h is taken so small that the sign
of the first term is that of the whole series.
If /"(a) is finite, the test assumes a simpler form; for then the
difference becomes = -^/"(« ^ Gh), which ultimately has the same
sign or signs as /"(a). If, therefore, /"{a) has a double sign the
cusp is of the first kind, otherwise of the second.
210. To determine the analytical properties of conjugate points.
Whenever the co-ordinates of a point of a curve render the first
differential coefficient impossible, no branch of the curve can pass
through the point, since at every point of a continuous branch the
SINGULAR POINTS. 163
tangent of the angle which its tangent makes with the axis of x must
have a real value.
If the equation to the curve, f(x, y) = 0, is in a rational form,
the value of -p obtained from the equation
cannot be impossible for any values of x and y, since both /'(x)
and fXy) niust be free from radicals. If, therefore, -^ have an
impossible value, this equation must become indeterminate, or the
co-ordinates of the conjugate point satisfy the equations
f{x,i/) = 0, f(x) = Q, f{y) = 0.
There may, however, be conjugate points in a curve without
-J- becoming impossible, in which case the above equations are not
satisfied.
For the point (o, b) will be a conjugate point if y is real when
x^a but impossible when x = a + h or a — h, h being taken suffi-
ciently small. This can only occur, in algebraical curves, when the
equation solved with respect to _?/ is of the form
y = F(x) + <p {x) {x - aj {x - c)",
where m is odd, n even, and a less than c ; for when x = a, y = F (a)
and when x is put equal to a ± h,
y = F{a ± /«) + ^ (a ± k) (± h)''(a - c ± kf,
which is impossible when h is sufficiently small, whether we take
the upper or lower sign, since a — c^k may in either case be made
negative by giving a sufficiently small value to h.
If r is equal to unity, -p will contain a term involving (x— c)",
whose coefficient remains finite at the point in question, and will
therefore have an impossible value. In this case, therefore, the
above conditions are satisfied by the equation in the rational form.
But if r is greater than unity, the terms in -^ , involving the radix
cals, will all disappear when x = a, and -j^ , as well as y, will have
a real value at the conjugate point.
11—2
16-i SINGULAR POINTS.
211. There are some other peculiarities which can occur only in
curves whose equations involve circular or transcendental functions.
Thus y may become impossible in such curves, on one side of a cer-
tain point, without having double values on the other, since the
impossibility may arise from other causes than the existence of an
even radical in the value of ?/. These are called points d' arret.
The curve y = x\ogx will be found to have a point d'arret at the
origin.
Again, the differential coefficient may change its value abruptly,
so that two branches of the curve meet in a point at a finite angle.
Such points are called points saillants. The curve y = x tan~^ - has
a point saillant at the origin.
212. The principal results of this chapter may be summed up
as follows.
If /(«, y) = is an algebraical equation in a rational form, those
values of x and y which satisfy the equations
f(x,2/)=0, /(x) = 0, f(,y) = 0,
correspond to points which are either multiple points, cusps, or
conjugate points. They may in general be distinguished thus.
At a multiple point -^ has two or more real values.
At a cusp some of the differential coefficients of i/ become in-
finite, and the ordinate is generally impossible on one side of the
point.
At a conjugate point -j- is generally impossible, and j/ is impos-
sible on both sides of the point.
Ex. To find the nature of the curve
y^ — axi/' + x'^ —
at the origin. Differentiating we have
dx
There is therefore a singular point at the origin. Differentiating
istant, we obtain
as if -^ were constant, we obtain
dx
SINGULAR POINTS. 165
all the coefficients of which vanish when x and y are put equal to
zero. Differentiating again as if -.- were constant^ we have
.4.-6«(|)-..4,(|y=0.
which gives when x = 0, y = 0,
There are therefore at the origin two branches parallel to the axis
of a; and one parallel to the axis of j/, indicated by the disappearance
of the coefficient of (-^j .
If we solve the above equation with respect to y, we obtain
which gives four real values of y when x is positive and sufficiently
small, and no real value when x is negative.
The branches parallel to the axis of x must therefore form a
cusp.
Since the tangent at the cusp is the axis of x and its apex at the
origin, the difference between the ordinates of the curve and tan-
gent is simply the value of _y, and since the corresponding values of
y are of opposite signs, the cusp is of the first kind.
Additional examples will be found in Gregory's Examples,
Chap. X.
CHAPTER XII.
TRACING OF CURVES.
213. When the equation to any curve is given, we can, by-
giving a series of different values to one of the co-ordinates, deter-
mine as many points of the curve as we please.
By determining the values of -^ or ^ t- at these points, we can
find the direction in which the branch of the curve passes through
them; and by the methods already investigated we can find the
position of their asymptotes, and the nature and situation of their
singular points.
Having done this, we can trace the form of the curve.
214. When the equation to the curve is of the form y -fix)^ it
is generally sufficient to determine the position of the asymptotes,
if any, and the values of y and -~ for a few points where they can
most easily be found, when the positions of points of inflexion, maxi-
ma or minima values of the co-ordinates, &c., readily suggest them-
selves. When it is required to determine the position of such points
more exactly, our previous methods must be resorted to.
The following example will shew the method to be pursued :
a being greater than h.
^=-(--«)y(^)
Since the curve is symmetrical with respect to the axis of x, we
may trace it with the upper sign only, and add similar branches on
the opposite side of the axis of x^
... j/ = (^-a)y(^),
.^A-u-ds /C^U— +i-i---l
(l) To find the asymptotes not parallel to the axes.
TRACING OF CURVES.
167
( ^/l 1^ 1^' \
=-(«+|)4G-i)^-^
Therefore the equation to the asymptote is
and the curve lies above or below the asymptote, according as x
is positive or negative.
(2) Positive values of x.
Let x = Q, .•. ^ = CO J or the axis of j/ is an asymptote,
X >Q<h, 3/ is impossible.
x = h,
dx
CO
x>b <.a, ^ is — ,
y =
dy_ _ /(ct-h\
dx Sj \ a )'
x> a, y \% -V,
X =TO
y = co,
dx
(3) Negative values of ^:
- x = 0, J/ = 00 ,
r/ = co,
dy
~ X = (X> ,
dx
= 1.
Hence the form of the curve must be that of the figure.
The curve may be verified and more exactly determined by
seeking the positions of the maxima and minima ordinates, and the
points of intersection of the curve and the asymptotes.
215. When the equation to the curve gives y only in an im-
plicit form, we may often trace the curve by obtaining simple ap-
proximate forms of the equation which hold when the co-ordinates
approach zero and infinity respectively.
Let, for example, the equation to the curve be
y^ - 20" x^y + x^ = 0.
168
TRACING OF CURVES.
(1) To find the asymptote, assume
(J
i/=Ax + B + — + &c.,
.-. 9/'=A'x' + 5A*Bx* + 5A3(AC + 2B')x^ + &c. "
- Sa" x'lf = - 20," Ax^ + &c.,
x^ = x^,
.: = (^^+1)0;''+ 5A'Bx' + A {5A'C + 10A'B'-2a')x'+ &c.,
.-. ^'+1=0, .-. ^=-1,
5A'B = 0, .-. B= 0,
5A^C+ 10 A'B'-Oa'=0, .'. C = -la\
Therefore the expanded form of the equation to the curve is
2 a^
y = — x — - — + &c.
•^ 5 X
The equation to the asymptote is therefore
and the curve lies below or above it according as x is positive or
negative.
(2) Omit the term x^ in the equation, and consider whether the
equation thus obtained is for any values of the co-ordinates an ap-
proximate representation of the given curve. It is
y^ = 2a Vy,
or y* = 2a'x^
When this equation holds, the dimensions in x of each of the
terms retained are t, while those of the neglected term are 5 ; if,
therefore, we suppose x to be small, the term omitted is small com-
pared with those retained; and the equation so obtained is there-
fore, Avlien x is small, an approximate form of the original equation.
Hence the given curve has a branch which approximates, when
near the origin, to the form of the curve
y^ = ^j2ax,
which is represented in the accompany- y
ing figure.
(3) Omit the term y\
Then the equation becomes
2fl V^ = x\
" ^ 2a-'
TRACING OF CURVES.
169
Here the dimensions in x of the terms
retained are 5, and those of the term neg-
lected 15. The above equation is therefore
approximately true when x is small. Its
form is that of the figure.
By omitting 2a^x^y, we should only obtain the equation to the
asymptote, which we have already found. This would give the
asymptote accurately in the present example; but in general, when
the asymptote does not pass through the origin, it would give only
an approximate form of it, namely, a line through the origin parallel
to it ; the former method of determining the asymptote is therefore
in general necessary.
From the above data we can construct the curve, which must be
of the form represented.
This may be verified by seeking the position of the points of in-*
flexion and the maxima and minima values of the ordinates.
It may be further verified by examining the nature of the mul-
tiple point at the origin, which we will now do.
y" - 2aV_y + x' = (1),
.-. (5x^-4a^^j/) + (5y-2«y)^ = (2).
There will be a multiple point when, together with equation (1),
the equations
5x* — A:a^xy = 0,
5/ - 2a'x' = 0,
are satisfied. The only values of x and y which satisfy these three
equations are x = 0, y = o.
Differentiating (2) as if -.- were constant,
170
ITUCING OP CURVES.
(20a:« - A.a'y) - %a^x ^ + 9.0 f (^Y = 0,
which becomes indeterminate when x and j/ are put equal to zero.
Differentiating again, after dividing by 4, we obtain
which becomes, when x = Q, y-0,
f- = 0.
ax
There is therefore one branch parallel to the axis of x, and two
parallel to that of y, corresponding to the two infinite roots due to
the disappearance of the two highest powers of the cubic. This
result accords with the figure.
216. When the equation to the curve is expressed in polar
co-ordinates, the direction of the curve at any point is determined
by the inclination (</>) of the tangent to the radius vector where
tan ^ = r-7-. It is also necessary to find the polar subtangent in
order to determine the position of the asymptotes.
Let, for example, the equation be
^ "TTTcosT' ^ ^^^"^ greater than unity;
differentiating the logarithm of r, we have
dr e sin 1 + e cos Q
JA = -k J •'• tan rf> = ; ;; ,
r dQ 1 + e cos ^ e sm
/ST = r tan = — %—^ .
Let = 0^ 1+e'
( tan ^ = CO ,
> < a, r is +,
a being the smallest value of cos~^ f — 1 , and therefore between
and TT ;
( r = 0D,
e sin a -H ^(e* - 1)
TRACING OF CURVES.
171
> a < TT, 7" IS — ,
e + 1'
tan (p =(X) .
When = 6' ± 2w TT, the value of r is the same as when 6 = 6'.
Hence all portions of the curve corresponding to values of 6, either
negative or greater than ^ir, only repeat those already found between
the limits and Stt.
By observing the rule with respect to the sign of ST, the asymp-
totes corresponding to = a, and = 27r — a respectively, will be
Ppy Qq, and the curve will be represented by the figure.
A being the point where 6=0 and a that where 6 = Stt. The
asymptotes evidently intersect at C the bisection of Aa^ since
sT=sr=
and
ST
Additional examples will be found in Gregory's Examples,
Chapter xi.
'-r ?jyi^
/> . OS*
CHAPTER XIII.
ENVELOPES, DIFFERENTIALS OF AREAS, ETC.
217. To find the equation to the envelope of a group of curves.
Let f{xy,a) = (1),
be the equation to the curve whose parameter is (a),
f{x,y,a + Aa) = (2),
that to another curve of the same group, whose parameter is
(a + Aa).
These curves will intersect in the points whose co-ordinates are
given in terms of a and Ac, by equations (1) and (2). As Aa varies,
this point of intersection changes its position, and the limiting posi-
tion to which it approaches when Aa approaches zero is called the
point of ultimate intersection of the curve (1), and the contiguous
curve of the group. At the point of intersection of (1) and (2) we
shall have
/(x, y,a + Aa) -/(x, t/, a) _ ^ .
Aa ""'
f(x, t/,a + Aa) -f{x, y,a) _f.
and, therefore, at the point of ultimate intersection,
+ Aa) -/(j
Aa
... <%^ = (3).
da '
Equations (l) and (3) therefore determine the point.
By eliminating a between (1) and (3), we obtain an equation
between x and y, which holds at the point of ultimate intersection,
and which is independent of a. This is therefore the equation to
the locus of the points of ultimate intersection of curves of the group
represented by equation (1).
Let the equation to this locus be
<^(^,^) = (4).
Then it may be shewn that the curve (4) touches each of the
curves of the group (l), and that every point of (4) is touched by
some curve of the group. For this purpose it is only necessary to
ENVELOPES DIFFERENTIALS OF AREAS, ETC. 173
bear in mind that any one of the equations (I) (3) and (4) is deriv-
able from the other two.
Thus, let a have a particular value in (1) so that that equation
may represent some one curve of the group. Then at the point of
intersection of (l) and (4.) equations (1) and (4) hold together, and
therefore at that point equation (3) also holds.
Now the value of -^ at the point of intersection, in the curve (l)
ax
is given by the equation
/W+/Wj = (5),
and that in the curve (4) by the pair of equations
/'W+/w|-/'«| = ol (g),
which are identical with (5), and therefore the curve (4) touches the
particular curve (1).
So also, if we take any point of (4) whose co-ordinates are
(x,, y^), it will meet that curve of the group whose pai'ameter is
given by equation (1) when x,, y^ are put for x,y.
The parameter thus obtained will therefore also satisfy equation
(iV
(3)with the same values of x, y ; and the values of ^- at the point of
contact will be determined by equations (5) and (6) respectively,
which as before are identical. Hence, any point x,, y^ of the curve
is touched by one of the curves of the group (1),
For these reasons the locus of ultimate intersections of a group
of curves is also called the envelope of the group.
218. If the equation to the group contains two or more parame-
ters connected by equations, so that one only is independent, it may
be reduced to the assumed form by eliminating all the parameters
except one, by means of the equations connecting them.
It is however often more convenient to differentiate the equation
before such elimination, and afterwards eliminate the differential
coefficients of the dependent parameters by means of the given
equations between them. The operations in this case become more
174 ENVELOPES DIFFERENTIALS OP AREAS, ETC.
symmetrical by the use of difterentials instead of differential coeffi-
cients, when the elimination may generally be most readily effected
by the method of indeterminate multipliers, explained in Art 177-
As an example, we will find the envelope of the system of lines
of a constant length whose extremities lie in two lines at right
angles to each other.
Let the two lines be taken for the axes of x and 1/ ; c the constant
length of each line of the group.
The general equation to any line of the group is
M- (-)•
where a, h are parameters connected by the equation
a' + b'^^c^ (2).
The equation to the envelope will be given by the elimination of
a and b, between (1), (2), and the equations
xda ydb _
ada + bdb = 0,
which together are equivalent to equation (3) of Art. 217.
Using an indeterminate multiplier X we have
i^-Xa\da + \^^-Xb\db = Oy
whence we may obtain
X
—,-Xa-O,
a
and p — a6 = 0,
X y
X y _ a b I
•'• ^ " P " 6^ ~ a' + b^ " ? '
a WxJ '
b=\7if'
therefore substituting in (1), the required equation is
2. z z
For examples see Gregory's Examples, Chapter xiii.
ENVELOPES DIFFERENTIALS OF AREAS, ETC. 175
219. To find the point of ultimate intersection of a normal to a
given curve with the contiguous normal, and the locus of all such
pointe.
Let the equation to the curve be
y=K^) (!)•
Let j:', y' be the co-ordinates of the point of ultimate intersection
of the normal at (x, 7/) with the contiguous normal, and let the dis-
tance between {x, y) and (x', y') be denoted by p. Then
{x'-xj^^y'-yy^p' (2).
The equation to the normal is
{x'-x)clx + {y'-y)dy = Q (3).
And at the point of ultimate intersection the differential of this equa-
tion with respect to the parameters x, y, must also vanish ;
.-. {x'-x^d^x + {y'-y)d?y = dx^ + dxf (4).
Equations (2), (3) and (4) will give x', y' and p in terms of x
and y and their differentials, y and the differentials may then be
eliminated by means of equation (l) and its derived equation, and
j;', y' and p obtained in terms of a: alone. By eliminating x between
the values of x' and y' we shall obtain the locus required.
The equations for determining x\ y' and p are the same as those
found in Art. 197 for the determination of a, /3 and p.
The point of ultimate intersection of contiguous normals is there-
fore the centre of curvature, and the locus of all such points the
evolute of the given curve, a result which might have been foreseen
from geometrical considerations.
220. This property furnishes a more convenient method of ex-
pressing p in terms of ^ and r than the one given in Art. 199-
Let P be any point of a curve,
SP = r, SY = p, \^>
the centre of curvature at P, and therefore , , . ,
PO = p. o/
Then
SO' = SP' + P0'-2 SP. PO cos SPO
= r^+p'-2pp. S
Since is the point of ultimate intersection of contiguous nor-
mals, a change in the position of P will in the limit produce no
176
ENVELOPES DIFFERENTIALS OF AREAS, ETC.
change in the position of or the length of OP. The above equa-
tion may therefore be differentiated with respect to r and p, con-
sidering SO and p as constant. Hence
0^2rdr-2pdp;
dr
''■P^'Tp-
221. To find the differential of the area included by a curve
and two ordinates.
Let Pd be two points of a curve whose co-ordinates are AM = x,
MP = y, and AN=x + Ax, NQ. = y+Ay;
and let the area between AM, MP, the
curve CP, and some fixed ordinate equal ^.
Then the area MPQN will with the
same notation equal AA, AA being the
increment of A corresponding to an in-
crement Ax of X,
This lies between the areas of the pa-
rallelograms MR, MQ, i. e. between i/Ax, and (y + A?/) Ax ; and
since the limiting ratio of these areas when Ax approaches zero is
unity, that of AA and either of them must equal unity,
AA
't/A.
or dA =ydx.
It
■AzzO
= h
222. To find the differential of the area included by a curve
and two radii vectores.
Let P, Q be points of a curve whose polar co-ordinates are ?; 6,
and r + Ar, 6 + A0, and let A be
the area between the curve CP,
the radius SP, and some other
fixed radius. Then will SPQ equal
AA, the increment of A corre-
sponding to an increment Ad of d.
AA lies between the areas of the
circular sectors whose radii are SP, SQ respectively, and vertical
angle equal to PSQ, i. e. between
r
-A0
2
and ^SZ^rlAe:
2
ENVELOPES DIFFERENTIALS OF AREAS, ETC.
177
and since the limiting ratio of these sectors when Ad approaches
zero is unity, that of AJ. and either of them must equal unity ;
AA
lis=,-
= h
Ad
.-. dA = -dQ.
Q
223. To find the differential of the volume included by a sur-
face of revolution and two planes perpendicular to its axis.
Let V be the volume pi'oduced by the revolution round AN
of the area bounded by MP, the curve CP and a fixed ordinate,
(see fig. Art. 221). Then AV will be the volume produced by the
revolution of TQ,, AV being the increment of V corresponding to an
increment Ax of x; AV lies between the cylinders produced by the
revolution of PR, QS respectively, i. e. between
TTT/^Ax and Tr(i/ + Ayf Ax :
and since the limiting ratio of these volumes when Ax approaches
zero is unity, that of AFand either of them must also equal unity;
AV
f(s=o
TxfAx
: dV
= 1,
= ir'lfdx.
224. To find the differential of the surface of a solid of revolu-
tion bounded by two planes perpendicvilar to its axis.
Let S be the surface produced by the revolution of a curve CP
about the axis of x, A M. Draw Pn, -m q
Qm parallel to the axis of x and each p^
equal to PQ. Then A^", the incre-
ment of S corresponding to an incre- ^
ment Ax of x, will be the surface pro-
duced by the revolution of PQ, and ^ ~ '^ ip
PQ = As if arc CP = s ; AS lies betwen the surfaces produced by the
revolution of Qm and Pit, i. e. between IttjjAs and 27r(y + Ay) As.
Since the limiting ratio of these surfaces when A^ approaches zero is
unity, that of AS and either of them must equal unity ;
A^
'i.TryAs
.'. dS =Q.-7r)/df,
= S'/ry J(dx- + df).
II. D c. 12
//v-
= 1,
CHAPTER XIV.
INTEGRATION BETWEEN LIMITS AND SUCCESSIVE
INTEGRATION.
225. To express an integral as the limit of the sum of a series.
Let/'(a:) be any function of x, a any constant^ and let the interval
X- ahe divided into n parts each equal to Ax, so that
Ao; = , and .•. x = a -vn Ax.
n
Let (j) (x) =f(a) Ax +f'{a + Ax) Ax +f(a + 2 Ax) Ax
+f'{a + (w - 1) Ax} Ax.
The value of (p (x) depends on the number of terms (re) of this
series, and the consequent magnitude of Ar. Now when « ap-
proaches infinity. Ax and therefore each term of the above series
approaches zero, and (p (x), whose actual value then becomes indeter-
minate, will approach a certain limiting value ; let this be \}/- (x).
To determine the value of xjy (x) cliange x into x + Ax in the
above series, which is equivalent to adding to it the term
/ (a + n Ax) Ax, or f'{x) Ax.
Hence, (p (x + Ax) - (p (x) =f'{x) Ax ;
(p (x + Ax) — (p (x)
Ax
-fix).
In this equation let Ax approach zero; then (pix) becomes in
the limit V^(^),
.:lt^.i^^^^^^=^^^^A/(x),
.:^'{x)=f{x);
therefore \\/ (x) must be one of the values contained in the general
integral, /(x) + C, of/'(x). To determine the value to be given to C,
we have
y}.{x)=fix) + C,
.: x|,(fl)-/(a) + (7=0.
INTEGRATION BETWEEN HM1T8. l79
since (p (a) and therefore \lr {£) evidently vanishes when x is made
equal to a ;
••• C = -/(a),
and x/.(*)=/(a)-/(a) (1).
The general form of the integral of /'(x), viz. /(x) + C, is
written l/Q'')dx, the indeterminate constant being included in the
integral sign. When such a value is given to C that the integral
shall vanish when x equals any particular value a, the integral
f{x) — /(«) is written I /'(x)dx ; and if in this we give to x any
Ja
particular value b, the resulting value of the integral, viz.
/(&)—/(«) is written \f'{x')dx,
\f'{x)dx is called the indefinite integral of f\x),
\f'{x)dx the corrected integral of /'(x),
Ja
j f'(x)dx the dejinite integral off'(x), between the
limits a and b.
With this notation equation (l) becomes, giving \|/(x) its value,
and writing f(x) instead off'(x),
^y(x) dx = ftA=o {/(«) ^^ +/(« + -^0 ^x ... +f(x - Ax) Ax} ;
: j/(x) dx = Iti^ {/(a) Ax +f(a + Ax) Ax ... +f(b - Ax) Ax}.
i-
The indefinite integral must therefore equal the limit of the
sum of any number of terms of a series formed like the above,
commencing with any term whatever of the series, and termi-
nating with that in which the quantity under the functional sign
is X — Ax.
The above equations may be written for the sake of brevity as
follows :
^f{x)dx = //a=o 2/-^'{/(x)Aa:},
^J{x)dx = U^^, ^t^' {fix) Ax}.
12 2
180
INTEGRATION BETWEEN LIMITS.
226. To determine the area included by a curve and two given
ordinates.
In Art. 221 it was proved that if A is the area included by a
curve, a fixed ordinate, and an ordinate whose abscissa is x,
clA = y c?x,
= \ydx,
= ^ (x) + C suppose.
The area and the integral are alike indeterminate, the former on
account of the indeterminate position of the fixed ordinate, and the
latter on account of the arbitrary constant which enters into it.
If the fixed ordinate is made definite and taken to be that
whose abscissa is a, the arbitrary constant must have such a value
given to it that the area shall vanish when x is made equal to a ;
.-. (a) + C = 0,
and A—f^ (x) — <p (a).
= ydx.
(1).
By giving different values to x, we obtain the values of the areas
bounded by the ordinate whose abscissa is a and any other. Thus
the area included between ordinates whose abscissae are a and b
= j^ydx
.(2).
It may be observed that as soon as a constant value is given to
the abscissa of the second ordinate, the form of the function (p
disappears from the expression, so that while from (l) we can
obtain the value of any one of a system of such areas, (2) will
determine only the particular area to which it is equal.
For example, let the curve be a circle.
J?
Taking C for origin, we have
2/ = J{(r - x-),
INTEGRATION BETWEEN LIMITS. 181
.-. area CBNP = f''j(a'-x')dx
Jo
= i^ V(«' - ^') + 5 «' sin-' - + C.
Since the area vanishes when x = 0, C must also equal zero,
.-. area CBNP = iocJ{a- - a?) + ^a" sin"' - .
Let now x = a,
.'. quadrant ACB = ^a^ 2 "' = 4'^ ''^
.'. whole area of circle = ira^.
From this expression the form of the general function from which
it is derived has disappeared.
227. To determine the length of the arc of a given curve.
If s is the arc of a curve measured from any fixed point to the
point whose abscissa is x,
ds = J{dx- + dy^),
If the abscissa of the fixed point is a,
And the arc between two points whose abscissae are a and b,
=iv{-(iy}-
As an example we will find the arc of the cycloid from the vertex
to a point whose abscissa is x.
In the cycloid fx = J^~^)'
a being the radius of the generating circle, and the vertex the
origin ;
= 2 J{2ax) + C,
182 INTEGRATION BETAYEEN LIMITS.
and C =s 0^ since the arc vanishes when x = 0,
.'. arc = 2 J{2ax).
If we put X = 2a, we obtain the arc of a semi-cycloid which there-
fore = 4«.
228. To determine the area included by a curve and two given
radii vectores.
If A be the sectorial area included between a curve, a fixed
radius and a radius inclined at an angle 6 to the prime radius
.-. A^Ur'de.
If the fixed radius is inclined at an angle, a, to the prime radius
A = ^jydd,
and the area bounded by radii inclined at angles a, ft to the prime
radius
=*r
■de.
229. In like manner if V, S be the volume and surface of a
solid of revolution bounded by planes whose abscissae measured
along the axis of revolution are a and x.
F=7r I y-dx, Art. 223,
If the two abscissae are a and b,
V — IT I y^dx.
--i>y{-©>-
230. These expressions for areas, arcs, &c., may be found at
once from the considerations of Art. 225.
Thus, to find the area ABPN. Let 0A = a, ON=b, Om = x,
mp = 1/. Divide AN into a number of parts each equal to Ax, and
INTEGRATION BETWEEN LIMITS.
183
A m n jsr
on these as bases construct parallelograms as in the figure.
Then, assuming- that the area is equal to the limit of the sura of
the parallelograms when Aa- approaches zero, we have
area AB^P = It^ 'EJ'-^^yAx,
since i/Ax is evidently the area of any of the parallelograms, as tip,
.-. area ABNP= j 1/dx, Art. 225.
The parallelograms or other portions into which the whole quantity
is divided are called elements.
By taking as our element the cylinder generated by the revolution
of one of the pai'allelograms, and assuming that the volume of the
solid of revolution is the limit of the sum of the cylinders when
Ax approaches zero, we have
= TT / y-dx.
By taking the chord pq as our element and assuming that the
arc BP equals the limit of the sum of the chords, when Ax
approaches zero, we have
s = lts^i:j'-''^J(Ax' + Af),
=iV{-(g)'}-
By taking as our element the surface generated by the revo-
184 INTEGRATION BETWEEN LIMITS.
lution of the chord pg about the axis of .r, and assuming that the sur-
face of revolution is the limit of the sum of these elements when Ax
approaches zero, we have
S = U^^^ 2/-^- 27rijJ{Ax' + Aif),
=-/>y{-(S)v^-
By taking as our element the circular sector whose radius
p
is Sp and vertical angle A0 and assuming that the sectorial area
BSP between radii, for which the values of 6 are a and j3, is
the limit of the sum of such elements when A0 approaches zero,
we have
area BSP - It^^, ^^'-^^^r'AO,
<
rcld.
It will be observed that the assumptions made in each case are
equivalent to the equations giving the differentials of the areas,
arcs, &c., in Arts. 221 —224.
In fact it is the same thing to say that the whole quantity
equals the limit of the sum of the elements, as that the increment of
the whole and the element have ultimately a ratio of equality, which
is what those equations amount to when expressed in words instead
of symbols.
231. This method of dividing a quantity into elements may
be applied to other functions besides areas, volumes, &c.
Thus let it be required to find the limit of the sum of every
portion of a circular area, multiplied by the square of its distance
from the centre, when the magnitude of each portion is indefi-
nitely diminished. Let the radius of the circle be a. Take as an
element the annulus whose inner radius is r and breadth Ar.
INTEGRATION BETWEEN LIMITS. 185
The required function for this annulus
= (area of annulus) r*,
neglecting the difference of the values of r for different parts of the
annulus, since when the limits are taken any error from this source
vanishes.
Therefore the required quantity
= 27r / r^ dr
= 2t-
4
- (area of circle) — .
2
Successive Integration.
232. Hitherto we have considered integration only as the con-
verse of the simple differentiation of a function of one variable. It
may however be extended in the same way as differentiation, and the
converse of the operations of successive and partial differentiation
will be called successive and partial integration.
Such operations are denoted by merely prefixing the integral
sign ( () to the differential, of whatever kind it may be, as many
times as there are operations to be performed.
Thus if u —f{x)
we have, when x is independent variable,
du =/' (x) dx,
d'u=f"{x)dx\
the converse of which equations will be written,
tl = If (j;) dx,
u^jjr(x)dx^
So if u =/(x, 2/),
djc =/' {x) dx, d^u =f" (or) dx\ dj,u = f" (.»;, y) dx dy.
186 INTEGRATION BETWEEN LIMITS.
X and y being the independent variables^ whence we have
n = j/ {x) dx = fjf" (x) dx- = J|r (^, y) d^ dy.
The last integral is more often written \dx \A\)f''{x,y), a form
which will be found convenient when the integration is between
definite limits.
An integral involving two or more signs of integration is called
a double or multiple integral.
233. We have seen that in order to obtain the general value
of \f'(x)dx it is necessary to add an arbitrary constant, because
such a quantity would disappear in differentiation. The same must
be done at each stage of the successive integration of a function of
one variable.
Thus ff"(x)dx=f(x) + C;
••• f ff"(^) d^' = f(/'^ + O) dx =f(x) + Cx+ C,
where C and C' are arbitrary constants.
So if we integrate n times in succession with respect to x we
must add,
since such a quantity would disappear in the corresponding differ-
entiation.
For the same reason, when we integrate a partial differential
djf{x, if) we must add the most general quantity which would have
disappeared in obtaining c?^/(a:, _?/). This may be any function ofy
or a constant, both of which are considered as included in the general
form (p {y) where <p is perfectly arbitrary.
If therefore u =f{x, y),
jf'{x)dx=f{x,y) + <p{y).
INTEGUATIOX BETWEEN LIMITS. 1 S7
Similarly, /"/(y) dy =f(x, y)+^ (x),
when 0, yp- are perfectly arbitrary functions.
So if we have to integrate twice partially with respect to x, we
must add an arbitrary function of _?/ at each stage ;
//^
jf'ix) dx' =f{x, y) + cp,(y) x + 4>,{y),
(pi and (p2 being arbitrary functions.
Again, if we have to integrate first partially with respect to one
variable and then partially with respect to the other we have
jdyjdxf" ix, y) =jdy {/{y) + <p {y)}
where >// {y) is the integral of the arbitrary function <p (y) and there-
fore itself a perfectly arbitrary function of y, and ;^ (.r) a perfectly
arbitrary function of x.
In like manner the general value of any multiple integral will
be given by adding to the particular integral, such functions of the
variables as would have disappeared in the corresponding differen-
tiation.
234. The arbitrary functions introduced into the value of par-
tial and successive integrals may be determined, and the definite
integral obtained between given limits exactly as in the case of
simple integration.
The case of most frequent occurrence is that of successive inte-
gration with respect to several different independent variables, as
for instance where
Here
n= idx \dyf"{x,y).
\dyf"{x,y)J-l^ + <p{xl
where <p is arbitrary.
If the integral vanishes when 1/ equals some function, X„ of x,
we have
dfix. X.) . .
188 INTEGRATION BETWEEN LIMITS.
which determines (p and then
If both limits of y are given and equal X, and Xj, we have
= F' (x) suppose,
where F'(x) will be a function of x whose form depends on that of/
and on the values of X, and Xj.
/ ^dijf"{x, y) is thus expressed as a definite function of a; in the
same way as / dxf'{x) was in Art. 225 expressed as a definite
constant. Substituting the above value in the general integral,
we have
jdx j\ f" (x, y) = ^dx F' (x)
= F(x) + a
If this integral is to vanish when x = a,
= F(a) + C;
.-. jjxj'\fXx, y) = F {x) - F(a).
And if two limits a and b are given,
f/xf%r{x, y) = F{b) -F(a).
235. The interpretation of multiple integrals as the limiting
sums of series, is similar to that of simple integrals. Thus in the
above instance f' 'dyf"(x, y) is the limiting value of the series
Jxi
whose general term is f"(x,y)Ay, the first and last terms being
those in which y = X, and X^- A^ respectively. The limit of the
sum so found is what we have denoted by F'{x). This is a quantity
varying with x, and a series may therefore be formed whose general
term is F'{x) Ax, and the first and last terms those in which x equals
INTEGRATION BETWEEN LIMITS.
J 89
a and b — Aj; respectively. The limit of the sum of this series will
then be I F'{x)dx. Hence
\ldxj2dyf"{x, y) = It^ 2/-^' Ao: {Sj"^^ A^/"(x, .y)},
that is, if an element be formed whose value is f"(x, 7/)AxAj/, the
limit of the sum of all such elements will be given by the mul-
236. We will apply this method to find the area included be-
tween the two curves APB, AQB, whose equations are respectively
y=f{x) and t/ = (p(x).
V
/
P
P
^
r \
X
/
A.
\
J
I
J
li"
^ J
^ X
Let X, y be the co-ordinates of any point p within the area ;
X + Ax, y + l\y those of any neighbouring point q ; x and y will there-
fore be independent of one another. Then the area of the paral-
lelogram pq whose sides are Ax and Ay ~ Ax Ay.
Also the area of a parallelogram of which one side is PQ and
the other Ax is the sum of all the smaller parallelograms obtained
by giving to y all values between MP{/(x)} and MQ {(p (x)] ;
.*. parallelogram PR = j dy Ax
= {(p(x)-/(x)}Ax,
and the whole area between the curves equals the limit of the sum
of all parallelograms formed in the same way as PR, and comprised
between the ordinates HA, KB, that is, between the limits a and b
ofx,i£OH=a, OK = h.
190 INTEGRATION BETWEEN LIMITS.
.-. Area ^ \ dx \ dy=\dx{(p [x) -/{x)}
= I ^(x)dx- j f{x) dx.
We might have obtained the same result by the formula of Art.
226, which would have given.
Area HAQJBK= i cp(x)dx,
Area HAPBK= ('/(x) dx ;
.'. Area between the curves = I (p (x) dx ~ j f(x) dx.
237. As another example, let it be required to find the centre
of gravity of a quadrantal area.
Let 5, y be the co-ordinates of the centre of gravity G; x,y those
of any point p within the area. Then the moment cv
of G about OB must equal the limit of the sum of
the moments of all the elements of which the area is
made up.
"^ O M A E
Or if we take an element Ax Ay as before, we have,
area BOC x AG = 111. MpAx Ay,
also area BOC = IfE Ax Ay ;
_ W^ yAxAy
' ' y " ICL Ax Ay
jdxjydy
jdx jdy
the limits being first from 7/ = to ^ = MP = J a" - x', and then from
a- = to ;k = OB = a, a being the radius of the circle ;
ra rVo^I^
\ dx \ xdy
- _ Jo Jo
fa /"^a^—x^
I dx j dy
Similarly
INTEGRATION BETWEEN LIMITS. ]91
Now I ''"^'ydy^'^+C; .'. = C;
Jo
/•>/XP r-Jd^-j:^
Also I dy = sja--x^, and j^ xdy = x Jcc" - x- ;
Also j^ Jx j^ dy - j^ Jx ^a--x^ = \x Ja^- x- + 1 «' sin-^ Ji'^ ^'
=!;«■•
"3'
3 4m
y ~ IT „ ~ 3^
"7 «
4
o
nd -=— ^= —
4
238. The method of converting the limiting sums of series into
multiple integrals, is equally applicable whatever may be the number
of independent variables. In every case, the increments are changed
into the corresponding differentials and the sign of summation into
that of integration, the limits of the integration being the same as
those of the series. The order of integration is very material. For
in general the limits at each integration are functions of all the
variables with respect to which the integration remains to be per-
formed; their values therefore depend on the order of integration,
192 INTEGRATION BETWEEN LIMITS.
and the latter cannot be varied after the limits are once fixed. The
only exception to this, is when all the limits are constant, as some-
times happens, in which case we may differentiate in any order we
please. A careful consideration of the examples above given, will
make the truth of these observations apparent. In working ex-
amples great care is requisite to avoid errors in fixing the limits,
which can only be done with accuracy by keeping constantly in view
the series of which the integrals are the limiting values.
Examples of the determination of areas, surfaces, &c. will be
found in Gregory's Exainples, Part ii. chap. ix. Applications of in-
tegration to summation, are met with in all physical subjects, and
form the principal use of the Integral Calculus.
239. To express an integral by an infinite series of finite terms.
By integrating by parts, we obtain
\f{x) dx =/(*) X - \f'{x) X dx,
jf(x)xdx=^fix)'^-ljr(x)x^dx,
ff'ix) x^ dx =f" {x) I' - 1 jr'{x) x^ dx,
&c. = &c.
.-. fAx)dx^f(x)x-f{x)^...^f-^x)^^l^jf"^ix)x''dx.
If in this series we make ?t infinite, we have
f/(x)dx=fix)x-fix)~ +/"(x)-^-&c hi inf.
The term t— | /"*"' (j?) x" dx is therefore the difference between
\ny
the infinite series and its first n terms. An arbitrary constant must
be added to give the general value of the integral. In the first form
of the expression the constant is included in the last term.
240. From the formula of integration by parts we may obtain
a proof of Taylor's Theorem, which presents the remainder after n
terms in the form of a definite integral.
INTEGRATION BETWEEN LIMITS. 193
let x = a + k — z;
.'. dx = — dz;
and when x = a, z = h,
X = a + k, z = ;
.'. f{a + h) -/{a) = JV(« + k-z) dz.
Now by repeatedly integrating by parts, as in Art. 239,
jf(a + h-z)dz=f(a+h-z)z+/"(a + h-z)^......
+f"-\a + h-z)^^+^jf^(a+h-2)z''-^dz.
Therefore taking the integral between the limits and h,
/(« + /^) =/(«) +/(«.) h+f"(a)^ +/"-)(«) -^
+ -. f V"' (a + h- 2) z"-' dz.
The value of the remainder after 7i terms is, therefore,
j^ />(»+/, -.).-&.
This is easily shewn to lie between the limits of the remainder
before found.
For
[ "/"" (a + h- z) z"-' dz = It^^ So"-A-~ {/(") {a + h- z) z"-' Az).
The particular value ofy'"'(a + h - z), by which any term of this
series is multiplied, must lie between the greatest and least values
which that function can assume while z varies from to h.
There must, therefore, be some intermediate value of z which
gives to the function a value, which multiplied into every term,
gives the same result as when each term is multiplied by the par-
ticular value belonging to it. This vaUie of rr may be represented
by (1 — 0) h, where lies between and 1. By substituting this, the
series becomes
f''{a + eh)lt^^,^:-'^-'z''-'dz,
H. D. C. 13
194 INTEGRATION BETWEEN LIMITS.
••• (/*"' (a + h- z) s"-' dz =/<"> (« + eh) f z"-' dz
Jo . Jo
= i>O(a-f0/O.
And the value of the remainder is, therefore,
^f'Ka + eh),
the expression before obtained.
The expression in the form of a definite integral is of course
preferable, when the integi'ation can be effected, as it determines
the exact value of the remainder and not merely the limits between
which it lies.
THE END.
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