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ELEMENTS
OF THE
DIFFERENTIAL AND INTEGRAL
CALCULUS
BY CHARLES DAVIES,
AUTHOR OF MENTAL AND PRACTICAL ARITHMETIC, FIRST LESSONS IN
ALGEDRA, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIP-
TIVE GEOMETRY, ELEMENTS OF ANALYTICAL GEOMETRY,
AND SHADES SHADOWS AND PERSPECTIVE.
SECOND EDITION.
REVISED AND CORRECTED.
PUBLISHED BY
A. S. BARNES & Co, Hartford.— WILEY & PUTNAM; COLLINS,
KEESE & Co, New York.— THOMAS, COWPERTIIVVAIT &
Co, Philadelphia.— CUSHING & SONS, Baltimore.—
TRUMAN & SMITH, Cincinnati.
183 P.
Entered according to the Act of Congress, in the year one thousand
eight hundred and thirty-six, by Charles Davies, in the Clerk's Office
of the District Court of the United States, for the Southern District of New
York.
HENRY W. REES, STKREOTTPKK,
45 QOID STREET, HEW YORU
44 i)\) 1.--
■X
PREFACE.
The Differential and Integral Calculus is justly con-
sidered the most difficult branch of the pure Mathematics.
The methods of investigation are, in general, not as
obvious, nor the connection between the reasoning and
the results so clear and striking, as in Geometry, or in
the elementary branches of analysis.
It has been the intention, however, to render the sub-
ject as plain as the nature of it would admit, but still,
it cannot be mastered without patient and severe study.
This work is what its title imports, an Elementary
Treatise on the Differential and Integral Calculus. It
might have been much enlarged, but being intended for
a text-book, it was not thought best to extend it beyond
its present limits.
4 PREFACE.
The works of Boucharlat and Lacroix have been
freely used, although the general method of arranging
the subjects is quite different from that adopted by
either of those distinguished authors.
Military Academy,
West Point, October, 183&
CONTENTS
CHAPTER I.
Constants and variables, ...... 9
Functions defined, ....... 9
Increasing and decreasing functions, ... 10
Implicit and explicit functions, . . . . 11
Differential coefficient defined, .... 16
Differential coefficient independent of increment, . 20
Differential Calculus defined, ..... 23
Equal functions have equal differentials, . . 23
Reverse not true, ....... 23
CHAPTER II.
Algebraic functions defined, .... 25
Differential of a function composed of several terms, . 26
" " the product of two functions, . . 27
" " " any number of functions, 28
" " a fraction, 29
Decreasing function and its differential coefficient, . 30
Differential of any power of a function, ... 30
" of a radical of the second degree, . . 31
" coefficient of a function of a function, . 33
Examples in the differentiation of algebraic functions, 34
Successive differentials — second differential coefficient, 39
Taylor's Theorem, .43
Differential coefficient of the sum of two variables, 43
Development of the function u = (a -f a;)", . . 46
" second state of any function, . 47
Sign of the limit of a series, .... - 47
6
CONTENTS.
Page,
Cases to which Taylor's Theorem does not apply, . 48
Maclaurin's Theorem, ...... 50
Cases to which Maclaurin's Theorem does not apply, 53
Examples in the development of algebraic functions, . 54
CHAPTER III.
Transcendental functions — logarithmic and circular, . 55
Differential of the function u = a*, . . . 55
" " logarithm of a quantity, ... 58
Logarithmic series, , . . . . . 59
Examples in the differentiation of logarithmic functions, 62
Differentials of complicated exponential functions, . 64
" " circular functions, .... 66
" <£ the arc in terms of its functions, . 70
Development of the functions of the arc in terms of the arc, 73
Development of the arc in terms of its functions, , 75
CHAPTER IV.
Partial differentials and partial differential coefficients
defined, 79
Development of any function of two variables, . . 80
Differential of a function of two or more variables, . 82
Examples in the differentiation of functions of two va-
riables, 85
Successive differentials of a function of two variables, 86
Differentials of implicit functions, .... 89
Differential equations of curves, .... 93
Manner of freeing an equation of constants, . . 96
•« " the terms of an equation from ex-
ponents, 97
Vanishing fractions, ....... 98
CHAPTER V.
Maxima and minima defined, 105
General rule for maxima and minima, . . , .108
Examples in maxima and minima, . . . . 109
Rule for finding second differential coefficients, . . 112
CONTENTS. 7
CHAPTER VI.
Expressions for tangents and normals, . . . 116
Equations of tangents and normals, . . . .118
Asymptotes of curves, ...... 122
Differential of an arc, 125
" " the area of a segment, .' . . 127
Signification of the differential coefficients, . . 128
Singular points defined, 132
Point of inflexion, . . . . . . .133
Discussion of the equation y = bzizc(x — a) m , . 134
Condition for maximum and minimum not given by Tay-
lor's Theorem — Cusp's, . . . . . 139
Multiple point, 143
Conjugate or isolated point, . . . . . 144
CHAPTER VII.
Conditions which determine the tendency of curves to
coincide, ........ 147
Osculatrix defined, . . . . . . 150
Osculatrix of an even order intersected by the curve, . 152
Differential formula for the radius of curvature, . 154
Variation of the curvature at different points, . . 155
Radius of curvature for lines of the second order, . 156
Involute and e volute curves defined, . . . .158
Normal to the involute is tangent to the evolute, . 160
Difference between two radii of curvature equal to the
intercepted arc of the evolute, . . . 162
Equation of the evolute, 163
Evolute of the common parabola, . . . . 164
CHAPTER VIII.
Transcendental curves defined — Logarithmic curve, . 166
The cycloid, 169
Expressions for the tangent, normal, &c, to the cycloid, 171
Evolute of the cycloid, 173
Spirals defined, 175
CONTENTS.
INTEGRAL CALCULUS.
Page.
Integral calculus defined, 189
Integration of monomials, . . . . , 190
Integral of the product of a differential by a constant, 192
Arbitrary constant, . . . . . . 194
Integration — when a logarithm, . . . . .194
Integration of particular binomials, . . . 195
Integration — when a logarithm, . . . . .195
Integral of the differential of an arc in terms of its
sine, 196
Integral of the differential of an arc in terms of its
cosine, . . . . . . . .198
Integral of the differential of an arc in terms of its
tangent, . . . . . . . .199
Integral of the differential of an arc in terms of its
versed-sine, ....... 200
Integration by series, ...... 201
" of differential binomials, .... 207
Formula for diminishing the exponent of the parenthesis, 212
Formulas for diminishing exponents when negative, . 213
Particular formula for integrating the expression
=-, ....... 214
V2 ax — oc 2
Integration of rational fractions when the roots of the
denominator are real and equal,
the roots are
Integration of rational fractions when
equal, .....
Integration of rational fractions when the denominator
contains imaginary factors,
Integration of irrational fractions,
Rectification of plane curves,
Quadrature of curves,
" of curved surfaces.
Cubature of solids, .
I>ouble integrals,
216
221
226
233
243
250
261
269
274
DIFFERENTIAL CALCULUS
CHAPTER I.
Definitions and Introductory Remarks.
1. There are two kinds of quantities which enter into
the Differential Calculus : variables and constants.
The variable quantities are generally designated by the
final letters of the alphabet, x, y, z, &c. ; and any values
may be attributed to them which will satisfy the equations
in which they enter.
The constant quantities are designated by the first
letters of the alphabet, a, b, c, &c. ; and these preserve
the same value throughout the same investigation, what-
ever values may be attributed to the variables with which
they are connected.
2. If two variable quantities are so connected together
that any change in the value of the one will necessarily
produce a change in the value of the other, they are said
to be functions of each other.
Thus, in the equation of a given straight line
y = ax -f b,
10 ELEMENTS OP THE
if we change the value of the ordinate y, the value of x
will also undergo a change : hence, y is a function of x,
or a? a function of y.
This general relation, which merely implies a depen-
dence of value, is expressed by
y = F(x), or, x = F{y);
and the equations are read, y a function of x, and x a
function of y. This dependence of value may also be
expressed by the equation
F(x, y) = 0,
which is read, function of x, y, equal to 0, and merely
implies, that x depends for its value on y, or y on x.
3. The letter which is placed in the first member of the
equation is called the function, and the one in the second
member is called the variable. In the equation
y = F(x),
y is the function and x the variable, and in the equation
x = F{y),
x is the function and y the variable.
4. In the equation of the straight line
y = ax -f b,
it is plain that if the value of x is increased the value of
y will also increase, or if x be diminished the value of y
will diminish : hence, y and x increase together, or de-
crease together, and y is then said to be an increasing
function of the variable x.
DIFFERENTIAL CALCULUS 11
In the equation of the circle
a? + y 2 = R 2 , or y 2 = R 2 -x 2 ,
the value of y increases when x is diminished, and de-
creases when x is augmented : when this relation subsists
between y and x, y is said to be a decreasing function,
of the variable x.
5. If in any equation of the form
y = F(x),
the value of y is expressed in terms of x and con-
stants, as for example, if
y — ax 5 , or y = 3 x 2 + bx 3 , &c,
y is then said to be an explicit function of x.
But if the value of the function is not directly expressed
in terms of the variable on which it depends, as in the
equation
y 4 — 3axy + x 5 = 0;
or if the dependence is expressed by means of an inter-
mediate variable, as in the equations
y = F(u), u = F(x),
y is then said to be an implicit or implied function of x.
The roots of an equation, for example, are implicit func-
tions of the coefficients. •
6 In every equation of the form
y = F(x),
either the function y, or the variable x, may be made to
12 ELEMENTS OF THE
change its value according to any law whatever, and the
corresponding change which takes place in the other, will
be determined by resolving the equation. Thus, in the
equation of the circle
x 2 +if = R 2 ,
if we change the value of either x or y by a quantity
± h, the corresponding value of the other variable may be
determined from the equation, and the difference between
it and the primitive value, will express the change of
value.
The law of change is generally imposed on the variable
a?, and as this law is arbitrary, x is called. -an independent
variable.
It simplifies the operations of the calculus, to increase or
diminish the variable x uniformly; that is, to change it
from one state of value to another by the addition or sub-
traction of a constant quantity; and since the law of
change is arbitrary, this supposition does not render the
calculus less general.
7. Although the values of the variable quantities may be
changed at pleasure without affecting the values of the
constants with which they are connected, there is, never-
theless, a relation between them which it is important to
consider.
If in the equation
y = F(x\
a particular value be attributed either to x or y, the other
will be expressed in terms of this value and the constant
quantities which enter into the primitive equation. Thus,
DIFFERENTIAL CALCULUS. 13
in the equation of the straight line
y = ax + 6,
if a particular value be attributed to a?, the corresponding
value of y will depend on a and b ; or if a particular
value be attributed to y, the corresponding value of x will
likewise depend on a and b. The same will evidently be
the case in the equation of the circle
x 2 +y 2 = R 2 ,
or in any equation of the form
y = F(x).
Hence, we see that, although the changes which take
place in the values of the variables are entirely indepen-
dent of the constants witli which the variables are con-
nected, yet the absolute values are dependant on the
constants.
8. Since the relations between the variables and con-
stants are not affected by the changes of value which the
variables may experience, it follows that, if the constants
be determined for particular values of the variables, they
will be known for all others.
Thus, in the equation of the circle
x*+tf = R\
if we make x = 0, we have
y=±R;
or if we make y = 0, we have
x= ±R,
2
14 ELEMENTS 0? THE
and the value of R will be equal to the distance from the
origin to the point in which the circumference cuts the co-
ordinate axes, whatever be the value of x or y.
9. The function y, and the variable x, may be so re-
lated to each other as to reduce to at the same time.
Thus, in the equation of the parabola
y 2 =2px,
which may be placed under the general form
F(x,y) = 0, or y = F(x) t
if we make x — 0, we have y = 0, or if we make
y = 0, we shall have x = 0.
10. We have thus far supposed the function to depend
on a single variable ; it may however depend on several.
Let us suppose for example, that u depends for its value
on x, y, and z, we express this dependence by
u = F(x, y, z,)
If we make x — 0, we have
u = F(y,z);
if we also make y = 0, we have
u = F(z);
and if in addition, we make z = 0, we have
u = a constant,
which constant, may itself be equal to 0.
11. Let us now examine the change which takes place
DIFFERENTIAL CALCULUS- 15
in the function for any change that may be made in the
value of the variable on which it depends.
Let us take, as a first example,
u = ao? t - •
and suppose x to be increased by any quantity h. De-
signate by v! the new value which u assumes, under this
supposition, and we shall have
v! =a(x + A) 2 ,
or by developing
v! — ax 2 -\- 2axh -f ah 2 .
If we subtract the first equation from the last, we shall
have
u' —u — 2 axh -f- ah 2 ;
hence, if the variable x be increased by h, the function
will be increased by 2 axh -{-ah 2 .
If both members of the last equation be divided by h,
we shall have
— - — = 2ax-\-ah, (1)
which expresses the ratio of the increment of the function
to that of the variable.
12. The value of the ratio of the increment of the func-
tion to that of the variable is composed of two parts, 2 ax
and ah. If now, w r e suppose h to diminish continually, the
value of the ratio will approach to that of 2 ax, to which
it will become equal when h = 0. The part 2 ax, which
is independent of h, is therefore the limit of the ratio of
IS ELEMENTS OF THE
the increment of the function to that of the variable. The
term, limit of the ratio designates the ratio at the time
h becomes equal to 0. This limit is called the differen-
tial coefficient of u regarded as a function of x.
We have now to introduce a notation by which this
limit may be expressed. For this purpose we represent
by dx the last value of li, that is, the value of h wliich
cannot be diminished according to the law of change to
which h is sidjjectcdy without becoming ; and let us
also represent by du the corresponding value of u : we
then have
du
-r- = 2ax. (2)
dx v
The letter d is used merely as a characteristic, and the
expressions du, dx, are read, differential of u, diffe-
rential of x.
It may be difficult to understand why the value which k
assumes in passing, from equation (1) to equation (2), is
represented by dx in the first member, and made equal
to in the second. We have represented b}r dx the
last value of h, and this value forms no appreciable part
of h or x. For, if it did, it might be diminished without
becoming 0, and therefore would not be the last value of h.
By designating this last value by dx, we preserve a trace
of the letter x, and express at the same time the last
change which takes place in h, as it becomes equal to 0.
13. Let us take as a second example,
u —. ax 3 .
If we give to x an increment h, we shall have
uf = a (x + h) 3 — ax 3 -f- 3 ahx 2 + 3 ah 2 x + ah 9 .
hence, u r — u = 3 ahx 2 -f 3 ah 2 x -f ah 3 ,
DIFFERENTIAL CALCULUS.
17
and the ratio of the increments will be
— 3ax^ + Sahx + ah 2 ,
h
and the limit of the ratio, or differential coefficient,
^ = 3aa*.
ax
In the function
nx*, we have
du
dx
47207 3 .
14. We have seen, in the preceding examples, that the
differential coefficient, or limit of the ratio of the increment
of the function to that of the variable, is entirely indepen-
dent of the increment attributed to the variable.
Indeed, when the ratio is obtained as in the examples
already given, and the increment made equal to in the
second member, it is plain that the first member can no
longer depend upon the increment. As this, however, is
an important principle, we shall add another proof of it ; in
the course of which we shall discover the value of the dif-
ferential coefficient under particular suppositions, and also
the form under which the new value of the function u may
be expressed.
Every relation between a function u and a variable x,
expressed by the equation
u =± F{x),
will subsist between the ordinate and abscissa of a curve.
For, let A be the origin of the rectangular axes, AX, A Y.
In the equation y
u = F(x),
make x — 0, which will
give
u = a constant :
18
ELEMENTS OF THE
lay off AB equal to
this constant. Then
attribute values to x,
from to any limit,
-as well negative as
positive, and find from
the equation
u - F{x),
the corresponding values of u. Conceive the values of a?
to be laid off on the axis of abscissas, and the values
of u on the corresponding ordinates. The curve described
through the extremities of the ordinates will have for its
equation-
u = F{x). (1)
15. Let x represent any abscissa, AH for example,
and u the corresponding ordinate HP.
If now we give to x any arbitrary increment 7^, and
make HF—h, the value of u will become equal to FC y
which we will designate by v! . We shall then have
u'=zF(x + h).
But F(x + h) = HP+CD, and HP
F(x).
Now, for a given value of h, CD will vary if P be
moved along the curve : hence, CD will depend for its
value on x and h } and we shall have
CD = F f (x } h):
the notation, F f , F", &c, designating new, or different
functions of x.
DIFFERENTIAL CALCULUS. 19
But since CD becomes 0, when h = 0, h must be a
factor of the second member of the equation, and we may
therefore write,
CD = F"(x, h)h.
Hence, vl = F{x + h) = F(x) + F" (a?, k) h, (2)
and transposing F(x) — u, we have
u'-u = F // (x,h)h. (3)
But since v! ' —u= CD = tang CPD. h, we have
u r -u= tang CPD.li = F f, {x, h)h ;
hence, ^^- = Uxi S CPD = F / (x i h). (4)
If now we suppose h to diminish continually, the point
C will approach the point P, the angle CPD will be-
come nearer and nearer equal to the angle PTH, which
the tangent line forms with the axis of abscissas. If we
pass to the limit of the ratio, we shall have
i£- = tang PTH=F"'{x); (5)
ax
and it remains to show that, this differential coefficient is
independent of h.
To prove this, we will observe, that whatever value may-
be attributed to h, a secant line, APC, can always be
drawn through P and the extremity of the corresponding
ordinate. The ratio of the increments of the ordinate and
abscissa may then be expressed by the tangent of the an-
gle CPD ; and since any secant will become the tangent
PT, when we pass to the limit, it follows that, the limit
20 ELEMENTS OF THE
of the ratio which is represented by the tangent of the
angle PTH, is independent of the increment h.
When, therefore, we pass from equation (4) which
expresses the ratio of the increment of the function to that
of the variable, to equation (5) which expresses the limit of
trfat ratio, the second member of equation (4) must be
made independent of h, which is done by making h = ;
and since the second member itself does not become 0, it
follows that there is at least one term in F" (x, h) which
does not contain h.
If then, we divide the second member of equation (4)
into two parts, one independent of h, and the other con-
taining h as a factor, it may be written under the form
F"(x, h) = F ff \x) -f F 1Y (x, h)h.
Substituting this value of F f/ (x, h) in equation (2),
we obtain
u'= F(x) + F ,,f {x)h -V F 1Y (x, h)h 2 ,
or, u! == u + F m {x) h + F l \x, h) h\
or by omitting a part of the accents,
u'=u + F(x)h + F"(x,h)W. (6)
Hence, also,
jt^- = F'(x) + F"(x,h)h, (7)
and by passing to the limit
16. Let us now resume the discussion of equation (6),
uf=u + F(x)h + F" (x, h) h\ (9)
DIFFERENTIAL CALCULUS. 21
This equation expresses the relation which exists be-
tween the primitive function u, and the new value v!
which it assumes, when an increment h is attributed to
the variable x. We see that the new value of the func-
tion is composed of three parts.
1st. The primitive function u.
2d. A function of x multiplied by the first power of the
increment h.
3d. A function of x and h multiplied by the second
power of the increment h.
We may also remark that, the coefficient of h in the
second term, is the differential coefficient of the function
u, and that the third term will vanish when ive pass to the
limit or make h = 0.
In order to render the form of the equation as simple as
possible, let us make
F' O) = P, and F" (x, h) = P f ;
the equation will then become
u^u + Ph + P'K 2 , (10)
or, vf -u = Ph + P f h 2 .
The coefficient P is in general a function of x, yet the
relation between u and x may be such as to make P a
constant quantity, in which case P' will be 0, or the
relation may be such as to render P' constant. These
cases will be illustrated by the examples.
17. If we take equation (8), which is
22 ELEMENTS OF THE
and multiply both members by dx, we have
du — Pdx:
hence, the differential of a function is equal to its dif-
ferential coefficient multiplied by the differential of the
variable.
18. The differential of the function may also be ex-
pressed under another form. For, if we multiply both
members of the equation
by dx, and omit to cancel dx in the first member, we shall
have
— - -dx = Pdx,
dx
in which either member expresses the differential of the
function u.
19. We may conclude from the preceding remarks that
the differential of a variable function, is the difference be-
tween two of its consecutive values, by which term we
mean to designate that difference which cannot be dimin-
ished according to the law of change to ivhich the function
has been subjected, without becoming 0.
20. We also see that, the Differential Calculus is that
branch of mathematics, in which the properties of quan-
tities are determined by means of the changes which take
place when the quantities pass from one state of value to
another.
21 . If two variable functions u and v, are so connected
DIFFERENTIAL CALCULUS. 23
together as to be always equal to each other, whatever
value may be attributed to either of them, their differ-
entials will also be equal.
For, suppose both of them to be functions of an inde-
pendent variable x. We shall then have (Art. 16),
u> _ u = Ph + P7/ 2 ,
v f -v=Qh+Q'h 2 .
But, since u! and v' are, by hypothesis, equal to each
other, as well as u and v, we have
Ph + P'h 2 = Qh + Qh\
or by dividing by h and passing to the limit
P=Q,
hence,
du dv
dx dx'
and,
du j dv 7
— — dx = -y— dx,
dx dx
that is, the differential of u is equal to the differential of v
(Art. 18).
22. The reverse of the above proposition is not gene-
rally true : that is, if two differentials are equal to each
other we are not at liberty to conclude that the functions
from which they were derived are also equal.
For, if we have the function
bu + a = F(x),
the values of a and b will not be affected by attributing
24 ' ELEMENTS OF THE
an increment h to x : we shall therefore have (Art. 16),
bu' + a = bit + a + Ph -f- P'h 2 ,
7 (u' — u) „ , „,,
or, b v — - — '- = P -f- P7?,
&
or by passing to the limit
b -j— ±= P, hence, Mw = P<f a?.
Now, bdu is the differential of the function bu as
well as of the function bu-\- a: and hence we may
conclude
1st. That every constant quantity connected with a
variable by the sign -plus or minus will disappear in the
differentiation.
2d. That the differential of the product of a, variable
quantity by a .constant, is equal to the differential of the
variable multiplied by the constant.
3d. That the differential of a constant quantity is
equal to 0.
DIFFERENTIAL CALCULUS. 25
CHAPTER II.
Differentiation of Algebraic Functions — Succes-
sive Differentials — Taylor's and Maclaurirts
Theorems.
23. Algebraic functions are those which involve the sum
or difference, the product or quotient, the roots or powers,
of the variables. They may be divided into two classes,
real and imaginary.
24. Let it be required to find the differential of the
function.
u = ax.
If we give to x an increment h, and designate the
second state of the function by u f , we shall have
v! — ax + ah — u -f- ah,
h
= a
du
hence, du = adx, or -j-dx = a ^»
25. As a second example, let us take the function
u = ax 4
3
26 ELEMENTS OF THE
If we give to x an increment h, we have
u / =ax z +2ahx-\-ah' i f
u f —u
— r — = 2 ax -f ah :
h
hence, du = 2axdx.
26. For a third example, take the function
u — ax 3 :
giving to x an increment h, we have
— - — =: 3 ax 2 + 3 axh -f ah 2 ,
or passing to the limit
— — - =3 3 ax 2 ; hence, du = 3 ax 2 dx.
ax
27. Let us now suppose the function u to be composed
of several variable terms : that is, of the form
u = y -\- z — iu = F (a?),
in which y, z, and iv, are functions of x.
If we give to x an increment h } we shall have
v! —u — (y r — y) + {z f — z) — (io r — w) :
hence, (Art. 16),
«'_ » = (Pft + PW) + ( Q& + <W - (Lk + 2/A 2 ),
or, *^ = (P + P'h) + (Q + Q>h)-(L + I/h),
or by passing to the limit
DIFFERENTIAL CALCULUS. 27
and multiplying both members by dx, we have
—r-dx = Pdx -j- Qdx — L dx.
ax
But since P, Q, and L, are the differential coefficients
of y, z, and to, regarded as functions of x, it follows (Art.
17) that, the differential of the sum, or difference of any
number of functions, dependent on the same variable, is
equal to the sum or difference of their differentials taken
separately.
28. Let us now determine the differential of the product
of two variable functions.
If we designate the functions by u and v , and suppose
them to depend on a variable x, we shall have
v/ = u + Ph -f P'h\
v' = v + Qh + Q'h 2 ,
and by multiplying
u'v' = ( u f Ph + Fh 2 ) (v + Qh + Q'h 2 ) ;
if we perform the multiplication, and omit the terms which
contain Jl 2 , which we may do, since these terms will vanish,
when we pass to the limit, there will result,
j = vP-\-uQ + $c.
or passing to the limit,
d(uv) „
therefore, d(uv) = vPdx + uQdx = vdu -f- ndv.
Hence, the differential of the product of two functions
dependent on the same variable, is equal to the sum of the
23 ELEMENTS OF THE
products which arise by multiplying each by the differ-
ential of the other.
29. If the differential of the product be divided by the
product itself, we shall have
d(uv) du dv
uv u v
that is, equal to the sum of the quotients lohich arise by
dividing each differential by its function.
We can easily determine, from the last formula, the
differential of the product of any number of functions.
For this purpose, put v = ts, then
dv _ d(ts) _ dt i ds
v ts t s '
and by substituting for v in the last equation, we have
d(uts) __ du dt ds m
- — |- _ ( £
uts u t &
and in a similar manner, we should find
djutsr . . . .) _ du dt ds dr »
utsr .... u t s r
If in the equation
d(uts)_du dt ds
" — — H I r >
uts uts
we multiply by the denominator of the first member, we
shall have
d(uts) = tsdu -+- usdt + utds ;
and hence, the differential of the product of any number
of functions, is equal to the sum of the products which
DIFFERENTIAL CALCULUS. 29
arise by multiplying the differential of each function by
the product of the others. ^,
30. To obtain the differential of a fraction of the form
— we make
v
— = t, and hence u as tv.
v
Differentiating both members, we find
du = vdt + tdv ;
find]
ing
the value
of dt, and
substituting
for
t
its
value
u
>
V
we obtain
V
udv
v 2 '
or by reducing to a common denominator
7 ^ vdu — udv
at = ;
hence, the differential of a fraction is equal to the deno-
minator into the differential of the numerator, minus the
numerator into the differential of the denominator, divided
by the square of the denominator.
31. If the numerator u is constant in the fraction t = — ,
v
its differential will be (Art. 22), and we shall have
, udv dt u
dt= 5-, or -?-=— -o • '
v l dv v 2
When u is constant, t is a decreasing function of v (Art.
4), and the differential coefficient of t is negative.
This is only a particular case of a general proposition
80 ELEMENTS OF THE
For, let u be a decreasing function of x. Then, if we
give to x any increment, as + h, we have
u'^u + Ph + P'h 2 ,
or, u'-u = Ph-\-P f h\
But by hypothesis u > u r ;. hence, the second member
is essentially negative - y and passing to the limit,
du _ p
dx
hence, a decreasing function and its differential coefficient
will be affected with contrary signs.
32. To find the differential of any power of a function,,
let us first take the function w* in which n is a positive
and whole number. This function may be considered as
composed of n factors each equal to u.. Hence, (Art. 29)*
d(u n ) __ d{uwuu .. ) _du du du du
u {uuuu . . . . ; u u u u
But since there are n equal factors in the first member,,
there will be n equal terms in the second ; hence,
d(u n ) _ ndu
u n ~~ u *
therefore, d(u n ) = nu n ~ l du.
r
If n is fractional, represent it by — , and make
s
v = u'y whence, if = v"' r
and since r and s are supposed to represent entire num-
bers, we shall have
ru T " l du = sv*~ l dv ;
- DIFFERENTIAL CALCULUS. 31
from which we find
r 1 r i
, ru , ru 7
av = — — T du = — dUy
su"
or by reducing
dv = — u" du;
s
which is of the same form as the function
d{u n ) — nu n ~ x d,u,
T
by substituting the exponent — for n.
Finally, if n is negative, we shall have
-n 1
from which we have (Art. 31),
1 \ — d(u n ) __ — nu n ~ x du
hence, by reducing
d(u~ n )= -nu~ n - l du.
Hence, generally, the differential of any power of a
function, is equal to the exponent multiplied by the func-
tion with its primitive exponent minus unity, into the
differential of the function.
33. Having frequent occasion to differentiate radicals of
the second degree, we will give a specific rule for this
class of functions.
Let v = -y/u^ or v — u* ;
then, dv=z—u % du = —u * du
2Vu~'
32 ELEMENTS OF THE
that is, the differential of a radical of the second degree,
is equal to the differential of the quantity under the sign,
divided by twice the radical.
34. It has been remarked (Art. 2), that in an equation
of the form
u = F(x),
we may regard u as the function, and x as the variable,
or a? as the function, and u as the variable. We will
now* show that, the differential coefficient which is obtained
by regarding u as a function of x, is the reciprocal of
that which is obtained by regarding x as a function of u.
If we consider u as the function, the ratio of the in-
crements will be represented by
u' — u 1
x 1 — X x 1
(1)
u' — u
or since
xf-
-x = h,
we have (Art.
16),
v! — u
1
1
h ~
h
1
Ph + P'h 2
P + P'h
or by passing
to the limit
du 1
dx 1
P
But when we pass to the limit, the denominator of the
dx
second member of equation (1) becomes -7—; hence,
dx _ 1 _ 1
du P / du\
\~dx~)
DIFFERENTIAL CALCULUS.
To illustrate this by an example, let
i
u = x 3 f whence x — y/u =u 3
Now, — = 3x? = 3u 3 ;
but regarding x as the function
dx 1-1 1
du 3 £
Su 3
35. If we have three variables u, y, and x, which are
mutually dependant on each other, the relations between
them may be expressed by the equations
u — F(y), and y = F f (x).
If now we attribute to x an increment h, and designate
by k, the change which takes place in y, we shall have
(Art. 16),
uf=u + Pk + P'h\ y'=y+Qh+ Q'h 2 ,
and ^ZJL^PJrP% ?Lz]L = Q+q] h
k a
If we multiply these equations together, member by
member, we shall have
u ^ x ^Zl = (P^Pk)(Q + Qh);
k II
but k = y r — y ; hence, by dividing and passing to the
limit, we have
du _ du dy
dx dy dx
aid hence, if three quantities are mutually dependant on
34 ELEMENTS OP THE
each other, the differential coefficient of the first regarded
as a function of the third, will be equal to the differential
coefficient of the first regarded as a function of the second^
multiplied by the differential coefficient of the second re~
garded as a function of the third.
36. Let us take as an example
we find
v = bu 3 , u = ax*,
^L = 3bu*, ^ = 2acc.
du dx
But, — =— — x —r- = Sbu 2 x 2ax=6abu 2 x;
ax du ax
and by substituting for u*, its value a
2 x\
—— = 6a?bx 5 , and dv = 6a 3 bx 5 dx.
dx
EXAMPLES.
1. Find the differential of u in the expression
u — V# 2 — oc 2 .
Put a 2 — x 2 = y, then •w = y 2 , and the dependence be-
tween u and x, is expressed by means of y, and u i'a
an implicit function of x. Differentiating, we find
^=+ly -* = .!(<,»_*»)-*, and ^=-S*;
dy 2 J 2 v ' dx
DIFFERENTIAL CALCULUS. 35
by multiplying the coefficients together we obtain
-j— = (a 2 — x 2 ) 2 2x =
dx 2 V Va 2 -x 2 '
hence,
— xdx
2. Find the differential of the function
m
u — {a-\- bx") .
Place a -+- bx n == y : then u — tf; and
~— = my m ~ l = 7n{a -\- bx n )
hence,
ax
du , m ~ l „_,
— -- = mnb(a + ox ) x
ax
du = ?nnb(a-{- bx n ) a" ' dx.
3. Find the differential of the function
u = x{a 2 + x 2 ) Va 2 — x 2 ,
du = {{a 2 + x 2 ) Va 2 ^?) dx + x ^Tf^x 2 d{a 2 + a*),
+ x(a 2 + x 2 )d^d i -x 2 ,
in which the operations in the last, two terms are only
indicated. If we perform them, we find
d(a 2 + x 2 ) = d(x 2 ) = 2xdx,
d( — x 2 ) — xdx
dVa 2 -x 2
2^a 2 -x 2 Va'-x 2 '
36 ELEMENTS OF THE
Substituting these values, we find
= [(*
du = \ (a 2 + a?)Va 2 -a? + 2a?Va 2 -a?-
V a 2 —oc 2
dx;
or, reducing to a common denominator and cancelling the
like terms,
V^—x 2
4. Find the differential of the function
u =
a 4 + a V + x'
, _ (a 4 + a¥+ x*)d(a 2 - x 2 ) - {a 2 - x 2 ) d{a*+ aV+ x 4 )
dU ~ (a 4 +aV+* 4 ) 2
from which we find
, _ — 2x(2a i + 2a 2 x 2 — x*)dx
5. Find the differential of the function
Make y — — — , z = -/(c 2 — or 2 ) 2 ,
ya?
then we shall have
u = iJ/(« - y + z) 3 = (a - y -f z) A ;
DIFFERENTIAL CALCULUS. 37
we therefore have (Art. 32),
3 --i
du = —(a - y + z)' d(a-y + z),
3 --
_ — 3dy + 3dz
4;Va — y + z
But from the equations above, we find
, _ / b \ d -\fx~~ — bdx
dy = d(— 7 =:)= —b = =•
W^ * 2*W
dz = d(c 2 - x 2 ) 1 = ^{c 2 - x 2 f~ l d{c 2 - x 2 ),
'2 -- — 4xdx
= -^-(c 2 — x 2 ) 3 x — 2xdx = • 3/
Substituting these values of cly and dz, in the ex-
pression for du, we find
36
dw= <^
2a?Va: ^c 2 -^
1
i 4/ b
1
» v X J
6. w = — ,
0?
du
1
7. w = — =,
a? n '
du
y dx*
dx
x 2
— ndx
38
ELEMENTS OF THE
8. u == V2ax-\-x 2 ,
<i?/
(« + x) dx
V2ax+x 2 '
10. u = a 6 +' 3 «V + 3aV + # 6 , Jm = 6 (a 2 + tf 2 ) 2 ^.
xdx
11. M =
Vl-a?
du
12.
a?
+ VI
e?w
dx
13. w= (a-f V^),
T=^(^+ yT=^) 2
3 (a-}- V^) c?a?
c£w =
2V'c
14. m = [a + yl
15. u = x 2 y 2
4cf
^ 3 '
3
<ia?
dw = 2x 2 ydy + 2y 2 xdx.
16. w= va+^X Vb 2 +y\ du = - ^ %
y Va 2 +* 2 V& 2 +*/ 2
— o+*r
18.
1+a 2
19.
x-\-y
dw
d«
nx n x dx
4 07^07
, _z(dx-\-dy)— (x-hy)Sdz
DIFFERENTIAL CALCULUS. 39
20 . .._vr+^+yi^ ,,_ .tod+vrr?)
Vi4-^_Vl-a? ^Vl-a? 2
21. Find the differential coefficient of
F(x)=z8x i -3x 3 -5x
Ans. 32# 3 — 9.Z 2 — 5. .
22. Find the differential coefficient of
F(x) = (x 3 + a)(3x 2 + b)
Ans. 15a? 4 -f 3ar^-f6ao?.
23. Find the differential coefficient of
F(x) = (ax + x 2 ) 2 ,
Ans. 2 (ax + x 2 ) (a + 2 a?).
24. Find the differential coefficient of
F(x)
x+ Vl — x 2
< . 1
Ans.
Vl-x 2 (l-]-2xVl-x 2 )
Of Successive Differentials.
37. It has been remarked (Art. 16), that the differ-
ential coefficient is generally a function of x. It may
therefore be differentiated, and x may be regarded as the
independent variable, A new differential coefficient may
thus be obtained, which is called the second differential
coefficient.
40 ELEMENTS OF THE
38. In passing from the function u to the first differ-
ential coefficient, the exponent of x in every terra in
which x enters, will be changed; and hence, the rela-
tion which exists between the primitive function u and
the variable x, is different from that which will exist
between the first differential coefficient and x. Hence,
the same change in x will occasion different degrees of
change in the primitive function and in the first differential
coefficient.
The second differential coefficient will, in general, be
a function of x i hence, a new differential coefficient
may be formed from it, which will also be a function
of x ; and so on, for succeeding differential coefficients.
If we designate the successive differential coefficients
by
p, q, r, s, &c,
we shall have
du _ dp- _ * dq _ *
dx dx dx
But the differential of p is obtained by differentiating
du
its value — , regarding the denominator dx as con-
dx
stant. ;
we therefore have
<©=*.
or,
d 2 u
dx
== dp,
and by
substituting for dp its value,
we ]
have
d 2 u
dtf
= q.
DIFFERENTIAL CALCULUS. 41
The notation d 2 u, indicates that the function u has
been differentiated twice, and is read, second differential
of u. The denominator dec 1 expresses the square of the
differential of x, and not the differential of x 2 . It is
read, differential square of x, or differential of x squared.
If we differentiate the value of q, we have
j/d 2 u\ , d 3 u 7
d {d?) = d *> or ' ih?= dq;
, d 3 u j.
hence, -— - = r, &c,
doer
and in the same manner we may find
d 3 u
The third differential coefficient -7-^, is read, third
ax 6
differential of u divided by dx cubed; and the differ-
ential coefficients which succeed it, are read in a similar
manner.
Hence, the successive differential coefficients are
du
d 2 U
d 3 U
d*u
dx =P >
dx 2 ~ q '
dx*- r >
dx*
&c
from which we see, that each differential coefficient is
deduced from the one which precedes it, in the same
way that the first is deduced from the primitive function.
39. If we take a function of the form
u = ax n ,
4*
42 ELEMENTS OF THE
we shall have for the first differential coefficient,
du n _ l
— - = nax .
dx
If we now consider n, «, and dx, as constant, we
shall have for the second differential coefficient
d 2 u
and for the third,
2^ = n{?i-l)ax n ~\
d 3 u
■^ = n(n-l){n-2)ax n -*,
and for the fourth,
d*u
dx*
n(n — l)(n — 2)(n — 3)ax n ~\
It is plain, that when n is a positive whole number, the
function
u = ax n ,
will have n differential coefficients. For, when n dif-
ferentiations shall have been made, the exponent of x in
the second member will be ; hence, the nth differential
coefficient will be constant, and the succeeding ones will
be equal to 0. Thus,
<^ = n(n-l)(n-2)(n-.3) a.l,
and, 7 w .. = 0.
dx n + l
DIFFERENTIAL CALCULUS. 43
Taylor's Theorem.
40. Taylor's Theorem explains the method of de-
veloping into a series any function of the sum or difference
of two variables that are independent of each other.
41. Before giving the demonstration of this theorem,
it will be necessary to prove a principle on which it de-
pends, viz : if we have a function of the sum or dijference
of two variables of the form
u = F(x ± y),
the diffe?'ential coefficient will be the same if we suppose x
to vary and y to remain constant, as when we suppose y
to vary and x to remain constant.
For, make x ± y = x' :
we shall then have
u = F(a/)
, du
and 17 =?■
If we suppose y to remain constant and x to vary,
we have
da/ = dx,
and if we suppose x to remain constant and y to vary,
we have
aW — dy.
But since the differential coefficient p is independent
of da/ (Art. 15), it will have the same value whether,
da/ = dx } or, dx/ = dy.
44 ELEMENTS OF THE
To illustrate this principle by a particular example, iet
us take
u = (x + y)\
If we suppose x to vary and y to remain constant,
we find
and if we suppose y to vary and x to remain constant,
we find
du f . \« i
the same as under the first supposition.
42. It is evident that the
F(x + y),
must be expressed in terms of the two variables x and y,
and of the constants which enter into the function.
Let us then assume
F(x + y) = A + By a + Cy h + Dy e +, &c,
in which the terms are arranged according to the ascend-
ing powers of y, and in which A, B, C, D, &c, are inde-
pendent of y, but functions of x, and dependant on all
the constants which enter the primitive function. It is
now required to find such values for the exponents a, 6, c,
&c, and the coefficients A, B, C, D, &c, as shall ren-
der the development true for all possible values which
may be attributed to x and y.
DIFFERENTIAL CALCULUS. 45
In the first place, there can be no negative exponents.
For, if any term were of the form
By-
it may be written
B
and making y = 0, this term would become infinite, and
we should have
F(x)=cc,
which is absurd, since function of x, which is independent
of y, does not necessarily become infinite when y = 0.
The first term A, of the development, is the value
which the primitive function assumes when we make
y — 0. If we designate this value by u, we shall have
F(x) = u.
If we make
F(x + y) = u',
and differentiate, under the supposition that x varies and y
remains constant, we shall have
<M__dA_ dB_ a .d£ h ,dD lC , &
dx dx dx dx ' dx
and if we differentiate, regarding y as a variable and x
as constant, we shall find
^i=aBif- x + bCy h ~ l + cDy e ~ l +, &c. :
dy
But these differential coefficients are equal to each other
( Art. 41); hence, the second members of the equations
46 ELEMENTS OP THE
are equal, and since the coefficients of the series are
independent of y, and the equality exists whatever be the
value of y, it follows that the corresponding terms in each
series will contain like powers of y, and that the coef-
ficients of y in these terms will be equal (Alg. Art. 244).
Hence,
a — 1 = 0, b — l = a, c — 1 = &, &c.,
and consequently
a = 1 , b = 2, c = 3, &c. ;
and comparing the coefficients, we find
" dA C-J-— — D- 1 dC
dx 2 dx 3 dx
And since we have made
F(x) = A=u, and F{x-\-y) = u' i
we shall have
„ du - (Pu ^ d?u
A = u, B = -^~, C= _ , „ ^ D
dx' '■■ 1.2 dx 2 1.2.3 dx 3 '
and consequently,
, , du cPu y 2 , d 3 u y 3 , p
B - M + ^ y+ -^tV + ^T±3 + ' &c -
43. This theorem gives the following development for
the function
u f =(x + y) n ,
du __, d 2 u f , \ „_i p
u = x% —~=znx n \ -j^-=n(n-\)x n \ &c:
DIFFERENTIAL CALCULUS. 47
hence,
w' = (x + y)' 1 = x n + nx n ~ l y + ??(n ~ 1) ^~ 2 y 2 ,
' n(n—l)(n — 2) n _ 3 3 , o
1.2.3 y '
44. The theorem of Taylor may also be applied to the
development of the second state of any function of the
form
u = F(x),
when x receives an arbitrary increment h, and becomes
x 4- h. For, if we substitute h for y, we have
. , du 1 , dhi h 2 . d?u li 3 , „
dx dx 2 !.* dx 3 1.2.3 ' '
hence, the difference between the two states of the func-
tion is
, du 1 , c^w 7<? , d?u h 3
in which the difference is expressed in terms of the
differential coefficients and the ascending powers of the
increment.
If we now suppose h to diminish continually, the sign of
the limit of the series will depend on that of the first term
-— h, or if h is positive, on that of the coefficient — .
dx dx
For, by dividing by h, we have
u' — u _du d?u h d?u h 2 -
48 ELEMENTS OF THE
and by passing to the limit
, du 7
au — —~xax:
ax
hence, when a series is expressed m the powers of a
variable which we suppose to be continually diminished,
the sign of the limit of the series will depend on the sign
of the term which contains the loioest power of the variable.
45. Remark. The theorem of Taylor has been demon-
strated under the supposition, that the form of the function
u'=F(x + y),
is independent of the particular values which may be
attributed to either of the variables x or y. Hence, when
we make y = 0, and obtain
F(x) — u ;
this function of x ought to preserve the same form as
F(x + y) ; else there would be, values of x in one of the
functions,
u r =F{x + y), u = F(x),
which would not be found in the other, and consequently
some of the values of x would be made to disappear when
a particular value is assigned to y, which is entirely con-
trary to the supposition.
If the function be of the form
u 1 =b-\- Va — x-\-y,
we shall have
u = b+ Va — x.
DIFFERENTIAL CALCULUS. 49
If we row make is — a, we shall have
v! = b + VyT and u = b,
hi which we see, that v! and u are expressed under dif-
ferent forms ; and hence, the particular value of y =
changes the form of the function, which is contrary to the
hypothesis of Taylor's theorem. When, therefore, the
function
v! — F(x 4" y),
shall change its form by attributing particular values to
x or y, the development cannot be made by Taylor's
theorem.
46. The particular supposition which changes the form
of the function will, in general, render the differential
coefficients in the development equal to infinity.
If we have
then,
k' =
--c+^f+x-
~!A
u =
da
1
dx ~
d 2 u
2(/W
1
dx 2 ~
dht
2x2(/ +
1 . 3
3
x)z
dx s ~
&c.
2x 2x2(/ -f
&c.
5
in which all the coefficients will become equal to infinity
when we make — x —f. ^
50 ELEMENTS OF THE
47. If we have a function of the form
v! — b -f Va — x + y,
in which n is a whole number, all the differential coeffi-
cients of u, for x— a will become infinite. For, we have
hence,
u -
= b + y a
— X -
~b-{-(a-x) n 7
da
dx~~
1
n
1
n-1 >
! — ^) "
d 2 u
(1-
a) 1
dx 2
?i 2
2b-1»
(a — #) "
&c.
&c.
all of which become infinite when we make x = «.
MaclaurirCs Theorem.
48. Maclaurin's Theorem explains the method of
developing into a series any function of a single variable.
Let us suppose the function to be of the form
u = F{x).
It is plain that the value of F(x) must be expressed in
terms of x, and of the constants which enter into F(x).
Let us therefore assume
u = A -f Bx a + Cx b + Dx e + , &c.,
in which the terms are arranged according to the ascend-
ing powers of oc, and in which A, B 7 C, D, &c, are
DIFFERENTIAL CALCULUS 51
independent of x, and dependent on the constants which
enter into F(x).
It is now required to find such values for the exponents
a, b, c, &c, and the coefficients A, B, C, D, &c, as
shall render the development true for all possible values
which may be attributed to x.
If Ave make a? = 0, u takes that value which the F(x)
assumes under this supposition, and if we designate that
value by U we shall have
U~A.
The first differential coefficient is
d ^.^aBx a ~ l + bCx b ~ l + cDx~ x + &c,
ax
and since this does not necessarily become when we
make x = 0, it follows that there must be one term in the
second member of the form x° : hence,
a — 1 = 0, or a = 1 ;
and making x = 0, we have
du __ rt
dx
The second differential coefficient is
C ^ = b(b- l)Cx b ~ 2 + c(c- l)Dx- 2 + &c. ;
but since the second differential coefficient does not neces-
sarily become 0, when x = 0, we have
b - 2 = 0, or b = 2:
52 ELEMENTS OF THE
hence, by making x = 0, we have
2C, or £*4*X
dx 2 ' da? 2
We may prove in a similar manner that
dhi 1
c = 3 and D = -^-^ , &c.
dx 3 1.2.3
If then we designate by U what the function becomes
when we make x = 0, and by U, V" , V 1 ", &c, what
the successive differential coefficients become under the
same supposition, we shall have
Fix) =U+irx+U /f — + V" jt~ -f &c.
k ■ 1.2 1.2.3
49. The theorem of Maclaurin may be deduced imme-
diately from that of Taylor.
In the development
, , du d 2 u y 2 , d 3 u if , e
the coefficients u, -r—, -~-p &c,
are functions of x, and also dependent on the constants
which enter into F(x + y)»
If we make x = 0, the F(x -+- y) becomes F(y), and
each of the differential coefficients being thus made inde-
pendent of x, will depend only on the constants which
enter into F(x-\-y\ and which also enter into F(y}*
Hence, if we designate by
U, U f , V", U f \ V""\ &c. f
DIFFERENTIAL CALCULUS. 53
the values which the coefficients assume under this
hypothesis, we shall have
50. If we take a function of the form
u = (a + x) n ,
we shall have
Tx = n{a + ocy-\
&c. == &c.
which become, when we make x = 0,
U = a", V = na n ~\ V" = n(n-l) a n ~\ &c. ;
hence,
{a + x) n = <f + na n ~ l x + n<<n ~ l \ n - 2 x 2 + &c.
1 • 10
51. Remark 1. The theorem of Maclaurin has been
demonstrated under the supposition that the F(x) reduces
to a finite quantity when we make x = 0. The case,
therefore, is excluded in which x = renders the function
infinite. Thus, if we have
u = cot x, u = cosec x, or u = log x,
and make x = 0, we find u = oo ; hence, neither of these
functions can be developed by the theorem of Maclaurin.
5*
54 ELEMENTS OF THE
Remark 2. We have already seen (Art. 45.), that the
theorem of Taylor does not apply to those cases in
which the form of the function is changed by attributing
a particular value to one of the variables : the theorem
therefore fails for -particular values, but is true for all
others, and hence, the general development never fails.
In the theorem of Maclaurin the failure arises from the
form of the function : hence, it is the general development
which fails, and with it, all the particular cases.
EXAMPLES.
1. Develop into a series the function
u= Va 2 ~T^ = a(l+^y.
2. Develop into a series the function
3. Develop into a series the function
a + oc \ as
4. Develop into a series the function
u= 4y - — a (l-H-i) .
DIFFERENTIAL CALCULUS. 55
CHAPTER III.
Of Transcendental Functions.
52. If we have an equation of the form
u = a x ,
in which a is constant, it is plain that u will be a function
of x ; and if a be made the base of a system of logarithms,
x will be the logarithm of the number u (Alg. Art, 257).
When the variable and function are thus related to each
other, u is said to be an exponential or logarithmic func-
tion of x.
53. The functions expressed by the equations
u = sin x, u— cos x, u — tang x, u = cot x, &c.,
are called circular functions.
The logarithmic and circular functions are generally
called transcendental functions, because the relation be-
tween the function and variable is not determined by the
ordinary operations of Algebra.
Differentiation of Logarithmic Functions.
54. Let us resume the function
u = a*.
56 ELEMENTS OF THE
If we give to x an increment h, we have
and v!- u = a** h - a x = a x (a h -l).
In order to develop a h , let us make <2 = l+6, we shall
then have
rf = (i + 6y = i + Aft + ^- 1 )y + ^" 1 X^-gy +&c .
V ; 1 1.2 1.2.3
hence,
1 1.3 1.3.3
\ X 1 2 1.2 3 /
from which we see, that the coefficients of the first power
of h will be
Vl 2 3 / '
replacing b by its value a — 1, and passing to the limit,
we obtain
du_da*__ ,/a-l (a-lf j («-l) 3
C?07 da?
or if we make
VI 3 3 /*
«-l (a-iy («-l) 3 .
k-— — + 8 - «c,
— — = ka*, or da* == ka'dx ;
ax
in which k is dependent on a.
DIFFERENTIAL CALCULUS. 57
The successive differential coefficients are readily found.
For we have
da x
a x k>
d (^j-) = da*k = a x k 2 dx ;
dx
da 1
hence, -———(fl?,
d " a * -a x k 3
&c. &c.
d n a x , „
dx n
55. It is now proposed to find the relation which exists
between a and k. For this purpose, let us employ the
formula of Maclaurin,
u = F{x) = U+ U— + V"— + V'"— — + &c.
v ; 1 1.2 1.2.3
If in the function
and the successive differential coefficients before found,
we make x = 0, we have
17 = 1, ir==k t U"=k 2 , U"=k\ &c;
hence,
_ * , kx t k 2 x 2 | /cV , B
a = 1 H h &c.
If we now make a? = — , Ave shall have
k
a "= 1+ T + T<r + Tih- t - &c - :
58 ELEMENTS OF THE
designating by e the second member of the equation, and
employing twelve terms of the series, we shall find
e = 2.7182818;
hence, a k —e, therefore a = e k .
But, 2.7182818 is the base of the Naperian system of
logarithms (Alg. Art. 272) ; hence, the constant quantity
k is the Naperian logarithm of a.
By resuming the result obtained in Art. 54,
da x — a x Jc dx,
we see that the differential of a quantity obtained by
raising a constant to a power denoted by a variable ex-
ponent, is equal to the quantity itself into the Naperian
logarithm of the constant, into the differential of the
exponent.
56. If now we take the logarithms, in any system, of
both members of the equation
e* =
a.
we shall have
kle =
: la, or
k =
la
: 7?
whence,
da x =
= ka T dx =
la
■Te a
x dx ;
or by recollecting
that
u -
=v,
we have
du
dx
la ,
-Te a;
DIFFERENTIAL CALCULUS. 59
or, if we regard x as the function, and u as the variable,
we have (Art. 34),
dx _ Ze 1
du la a''
Let us now suppose a to be the base of a system of
logarithms. We shall then have x= the logarithm of
w, Za=l, and le = the modulus of the system (Alg.
Art. 272); and the equation will become
d{lu) = le — ,
that is, the differential of the logarithm of a quantity is
equal to the modulus of the system into the differential of
the quantity divided by the quantity itself.
57. If we suppose a = e the base of the Naperian
system, and employ the usual characteristic V to desig-
nate the Naperian logarithm, we shall have
d(l' u ) = — ;
u
that is, the differential oj the Naperian logarithm of a
quantity is equal to the differential of the quantity divided
by the quantity itself.
The last property might have been deduced from the
preceding article by observing that the modulus of the
Naperian system is equal to unity.
58. The theorem of Maclaurin affords an easy method
of finding a logarithmic series from which a table of
logarithms may be computed. If we have a function of
the form,
u — F(x) = Ix,
60 ELEMENTS OF THE
we have already seen that the development cannot be
made, since F(x) becomes infinite when x = (Art. 51.)
But if we make
u = F(x) = l{l-\-x\
the function will not become infinite when x = ; and
hence the development may be made.
The theorem of Maclaurin gives
v ' 1 1.2 1.2.3
If we designate the modulus of the system of the loga-
rithms by A, we shall have
du
dx
A- 1
1 +
- = A(l
X
+ x)~\
dhi
- A i
1
= -a(i +
x)-\
1 -f xf
d 3 U
dx*~
-ZA (i
1
+ oof ~
2A{l+x)-
-3
If we
now make t
v = 0,
we have
U =
0,
U f = A,
&'*
= 'Za,
V" = 24,
&c. :
hence,
1(1+sd) = A(x- — + - — + - &c.^
v; V 2 3 45 )
This series is not sufficiently converging, except in
the case when x is a very small fraction. To render the
series more converging, substitute — x for x : we then have
z(i-,)=^(_,-4-4-4.4_ & c.)
DIFFERENTIAL CALCULUS. 61
and by subtracting the last series from the first, we obtain
^+*)-^-J=KS)= 84 (t + t + t + &c
If we make
1 -fa? t , z , z
1 -\ , we have x —
1 — a? n* 2n-\- z y
and by observing that
we have
l{n + z)-ln = 2A[-?—+ -V-^- V+— (-^— Y+ &c,l
from which we can find the logarithm of n + z when the
logarithm of n is known. This series is similar to that
found in Algebra, Art. 270.
If we make n — 1, and z = l, we have 71 = 0, and
If we make the modulus A = 1, the logarithm will be
taken in the Naperian system, and we shall have
Fa = 0.693147180,
2l f 2 = Z'4= 1.386294360;
and by making z = 4, and n = 1, we have
Z'5 = 1.609437913,
and l f 2 + Z'5 = HO = 2.302585093.
6
62 ELEMENTS OF THE
If we now suppose the first logarithms to have been
taken in the common system, of which the base is 10, we
shall have, by recollecting, that the logarithms of the same
number taken in two different systems are' to each other
as their moduli (Alg. Art. 267),
HO : ^10 : : A : 1,
or, 1 : 2.302585093 : : A. : 1 ;
whence, A = = 0.434284482.
2.30258509
Remark. To avoid the inconvenience of writing the
modulus at each differentiation (Art. 56), the Naperian
logarithms are generally used in the calculus, and when
we wish to pass to the common system, we have merely
to multiply by the modulus of the common system. We
may then omit the accent, and designate the Naperian
logarithm by I.
59. Let us now apply these principles in differentiating
logarithmic functions.
1. Let us take the function u = l( . — ).
\Va 2 +a?J
Make
and we shall have
a?dx
dx ya 2 + oc 2
Va 2 +x 2 aHx
but dz=
a 2 -f x 2
(a 2 +a?)%'
whence, du —
DIFFERENTIAL CALCULUS 63
a 2 dx
x(a 2 +a?)
2. Take the function u — l\ , — , ;
Lvi-h^— VI — x J
and make yT+#+ -y/l— x—y, -y/T+x— ^\/l—x=z i
which gives
u = I (—) — ly — lz, and du = -± .
\ z J J v z
z/< y z
But we have
dx dx — dx
T
zdx
, dx dx — dx / /— /- \
dy — t - = -. ( -y/l+X— Vl— X),
2-Vl + x 2^1-* 2Vl-^ V V /'
2Vl — a*'
2V1+J? 2V1-07 2Vl-a^ /
Whence,
y *" 2yVl-ar 2*Vl-# 2 '
— (y 2 -}-* 2 )^
2y2:Vl — a? 2
and observing that ?/ 2 -}-2; 2 =:4 and yz = 2x,
•L 7 da?
we have dw = .
^Vl-^ 2
64
ELEMENTS OF THE
3. U = l(c 0+ -y/ 1 + a ^,
5. u
TV a + x + <\/a — x
0. u = / - / / •
L V a + a? — Va — #_
dx
Vl + x 2
c?w
dx
Vl-x 2
f/Wr
dx
Vl + x 2
^M =
adx
xVa 2 -*?
60. Let us suppose that we have a function of the form
u == (lx) n .
Make Za? = z 9 and we have
w == z n , du == rcz* ~ ! cfe,
and substituting for z and dfe their values,
d(lxf= n V x)n ~ l dx.
x
61. Let us suppose that we have
u = l(lx).
Make Ix = z, and we shall have,
7 j dz t dx
u = lz, du = — , dz — — ;
z x
hence,
du
dx
xlx
DIFFERENTIAL CALCULUS. 65
62. The rules for the differentiation of logarithmic func-
tions are advantageously applied in the differentiation of
complicated exponential functions.
1 . Let us suppose that we have a function of the form
in which z and y are both variables.
If we take the logarithms of both members, we have
hi = ylz ;
iience, — — dylz+y — ■ ;
u z
or, du = ulzdy + uy — t
or by substituting for u its value
du = dz y = zHzdy + yz y ~~ 1 dz.
Hence, the differential of a function which is equal to
a variable root raised to .a poiv-er denoted by a variable
■exponent, is equal to the sum of the differentials ivhich
•arise, by differentiating, first under the supposition
that the root remains constant, and then under the sup-
position that the exponent reiqmns constant (Arts. 55,
and 32).
2. Let the function be of the form
U F= of.
Make, b* = y, and we shall then have (Art. 55),
w = o^, du = a v lady; but dy~b*lbdx 7
X
tience, du = a h b*lalbdx..
6*
66 ELEMENTS OF THE
3. Let us take as a last example
u = z*,
in which z, t, and s, are variables.
Make, f = y, we shall then have
u = z y , du = z y Izdy + yz y " 1 dz.
But dy = fltds + sf ~ l dt ;
hence, du = z f lz(t s ltds + sf~ l dt) + Fz^dz,
du^/fQtlzds + ^+^y
Differentiation of Circular Functions.
63. Let us first find the differential of the sine of an
arc. For this purpose we will assume the formulas (Trig,
Art. XIX),
sin a cos b -f sin b cos a
sin (a + b) =
sin {a — b) =
R
sin a cos b — sin b cos a
If we subtract the second equation from the first,
2 sin b cos a
sin (a + b) — sin (a — b)
R
and if we make a-\-b = x + h, and a — b = x, we shall
have
2 sin — A cos (ff + -^-^)
sin (a? + A) — sma? = 5 — ,
" DIFFERENTIAL CALCULUS. 67
and dividing both members by A,
2sin — hcos(x-\ h)
x + h)-smx 2 \ 2 /
sin (x + A)
h hR
sin — A cosfaH h)
2 \ 2 /
1a * '
2
If we now pass to the limit, the second factor of the
COS X
second member of the equation will become
R
• 1 i
sin — a
2
In relation to the first factor — z its limit will be unity,
-n Rs'ma , sin« cos«
ror, tanffa = , whence == ;
cos a tang a R
Now, since an arc is greater than its sine and less than
its tangent*
sin a ■ , , sin a * sin a
<1, and >
a tang a
* The arc DB is greater than a straight line
drawn from D to B, and consequently greater
than the sine DE drawn perpendicular to JIB.
The area of the sector ABD is equal to
~ABX BD, and the area of the triangle ABC
2
is equal to —AB XBC. But the sector is less A E B
&
than the triangle being contained within it : hence,
^ABXBD^ABXBC,
consequently, BD < BC.
DO ELEMENTS OF THE
hence, the ratio of the sine divided by the arc is nearer
unity than thcLt of the sine divided by the tangent. But
when we pass to the limit, by making the arc equal to 0,
the sine divided by the tangent being equal to the cosine
divided by the radius, is equal to unity : hence the limit
of the ratio of the sine and arc, is unity.
When therefore we pass to the limit by making h == 0,
we find.
d since __ cos a? >
dx R
cosxdx
hence, dsinx =
R
64. Having found the differential of the sine, the diffe-
rentials of the other functions of the arc are readily de-
duced from it.
cos# = sin (90° —a?), dcosx = cZsin(90° — x\
and by the last article,
dsin(90° - x) = -^-cos(90° - x)d{§0° - x),
R
= ——cos (90° — w)dx:
R
, 7 smxdx
hence, dcosx = - — ;
R
the differential of the cosine in terms of the arc being
negative, as it should be, since the cosine and arc are
decreasing functions of each other (Art. 31.)
DIFFERENTIAL CALCULUS.
65. Since the versed sine of an arc is equal to radius
minus the cosine, we have
7 . 1/T . s sin xdx
a ver-sin x = a(R — cos x) =
R
R sin x
66. Since tang x = , we have (Art. 30),
cos X
7 R cosxd sina? — R s'mxd cosa?
a tang x =
(cos 2 j?+ sin 2 x)dx ._
2 ~~ '
COS^*?
but cos 2 a? + sin 2 07 = R 2 :
RHx
hence, d tango? =
R 2
67. Since cot# = , we have
tang a?
7 R 2 d tang x R^dx
a cot# = ^— 5= 5 5—;
tang^a? tangpc cos^a?
, # , jR 2 sin'
but, tang^a? = —
hence, J cot a? = — —
which is negative, as it should be, since the cotangent is a
decreasing function of the arc.
70 ELEMENTS OF THE
R 2
68. Since sec^ = , we have
cos x
7 R 2 d cosa? R sinxdx
a sec x = = . -
cos^r cos w a?
. R sin x , R 2
but, = tang x, and =sec^;
cos x cos x
, 7 sec a? tanga?c£r
hence, a sec a? = =2 .
R 2
R 2
69. Since cosec # = — : , we have
sin a?
* R 2 d sin 5? 7? cos 5? da?
a cosec a? = — ■ — = — - — ■ — ;
, , cosec a? cotxdx
hence, d cosec x —
R 2
70. If we make R-\, Arts. .63, 64, 65, 66, 67,
will give,
d sina?= cosxdx (1),
dcosx = — sinxdx (2),
d ver sina?= sinxdx (3),
<* tanga;= i (4)>
cZcotO? = r-r- (5).
simr
The differential values of the secant and cosecant are
omitted, being of little practical use.
71. In treating the circular functions, it is found to be
most convenient to regard the arc as the function, and the
DIFFERENTIAL CALCULUS. 71
sine, cosine, versed-sine, tangent, or cotangent, as the
variable. If we designate the variable by u, we shall
have in (Art. 63) sin x = u, and
7 Rdu Rdu
ax
cos x Vli 2 —u 2
If we make cosx = u, we have (Art. 64),
_ Rdu Rdu
sin a? \/W^u*
If we make ver-sina? = u, we have (Art. 65),
, Rdu
ax = — : .
But, s'mx—-\/R 2 —cos' i x, and cosx=R — u,
therefore, cos 2 a? = R 2 — 2 Ru + u 2 ,
hence, sin a? == ^2Ru — u 2 ,
Rdu
and consequently, dx =
V2Ru — u 2 >
If we make tang x — u, we have (Art. 66)
7 cos 2 x du
ax
2 >
R
. cos x R , cos 2 a7 R 2 R
but — =r— = , hence
J? sec a? ' J? 2 sec 2 a? J^+tang 2 ^'
# 2 dw
hence, - da?
R 2 + u 2 '
7£ ELEMENTS OF THE
Now, if we make R—l, the four last formulas
become
, du , du
dx = — . dx =
Vl-U 2 ' Vl-U 2 '
7 du , du
doc= =, dx — ;
V211-11 2 l + u
and these formulas being of frequent use, should be care-
fully committed to memory.
72. The following notation has recently been introduced
into the differential calculus, and it enables us to designate
an arc by means of its functions.
s'm~ l u — the arc of which u is the sine,
cos -1 it= the arc of which u is the cosine,
tang -1 w — the arc of which u is the tangent,
&c. &c. &c.
If, for example, we have
du
x — sin l u, then, dx
Vl-u 2
73. We shall now add a few examples.
1 , Let us take a function of the form
Make cos x — z, and sin x = y ;
then, u = z y , and (Art. 62);
du = z v Izdy + yz y ~ x dz:
DIFFBRENTIAL CALCULUS. 73
also, dz=—smxdx, and dy = cosxdx:
hence, du = z v flz dy + -¥-dz\
= cosrf iax (lcosxcosx )dx,
\ coso? /
2. Differentiate the function
. _, 7 mdu
x — sm mu, ax =
Vl — m 2 u 2
3. Differentiate the function
#? = cos l (uV 1 — u 2 )
d - (—1 + 2u 2 )du
' V(I37+m 4 )(1-w 2 )'
4. Differentiate the function
_ x u j 2du
x = tans — , 007 = -
5 2' 4 + w 2
5. Differentiate the function
a? = sin 1 (2w Vl — u 2 \ dx = —
\ / Vl-u
6. Differentiate the function
, x * ydx — xdv
u = tang" — , du = Z — —Z-.
V y 2 + a?
74. We are enabled by means of Maclaurin's theorem
and the differentials of the circular functions, to find the
7
74 ELEMENTS OF THE
value of the principal functions of an arc in terms of the
arc itself.
Let u = F(x) = sin a?: then,
du d 2 u . d 3 u
— = cosa?, — — .— — smo?, _ .. = -— cosa?,
dx dxr dx 6
d*u . d 5 u
— — = sin x, ——■ — + cos x.
dx* dx
If we now render the differential coefficients independent
of x, by making x — 0, we have (Art. 49),
17=0, £* = !,
u>' = o, £/";=- i,
£/"" =0,
17""= + 1:
hence, sina? =
* 3 | * &c.
1 1.2.3 1.2.3.4.5
75. To develop the cosine in terms of the arc, make
u — F(x) = cos x ; then,
du . cPu d 3 u
— =— sma?, -— = — cos a?, -rr = sma?,-
aa? aar oar
— —- = COSa?, — — r=— SUIS?,
d# 4 dx 5
and rendering the coefficients independent of a?, we have
£7=1, 17=0, 17"= -1, £/"'=0,
£/"""= 1, 17""'= 0:
hence, cos* = 1 - ^ + j-^-^ - &c.
DIFFERENTIAL CALCULUS. 75
The last two formulas are very convenient in calculating
the trigonometrical tables, and when the arc is small the
series will converge rapidly. Having found the sine and
cosine, the other functions of the arc may readily be
calculated from them.
76. In the two last series we have found the values of
the functions, sine and cosine, in terms of the arc. We
may, if we please, find the value of the arc in terms of
any of its functions.
77. The differential coefficient of the arc in terms of
its sine, is (Art. 71),
du Vl-u 2
developing by the binomial theorem, we find
dx i , 1 2 , i-3,,4 , 1.3.5 6 ; -
— = H u H u H u 6 4- (fee.
du 2 2.4 2.4.6
In passing from the function to the differential coeffi-
cient, the exponent of the variable in each term which
contains it, is diminished by unity ; and hence, the series
which expresses the value of x in terms of u, will contain
the uneven powers of u, or will be of the form
x = Au + Bu 3 + Cu 5 + Du 7 + &c;
and the differential coefficient is
^ = A + 3Bu 2 + 5Cu* + 7Du*+6cc
du
76 ELEMENTS OF THE
But since the differential coefficients are equal to each
other, we find, by comparing the series,
A = l, **4 C = -i#-, D h35
2.3' ~~2.4.5 ~2.4.6.7'
hence,
u , 1 u 3 , 1.3u 5 , 1.3.5 7 , .
3C=sm~ 1 u= H — w+ &c.
1 2 3 2.4.5 2.4.6.7
If we take the arc of 30°, of which the sine is -—-
(Trig. Art. XV), we shall have
and by multiplying both members of the equation by 6,
we obtain the length of the semi-circumference to the
radius unity.
78. To express the arc in terms of its tangent, we have
(Art. 71),
doc 1
du 1 + w 2
which gives
= {l + U>) ,
du
hence the function x must be of the form
x = Au + Bu z + Cu 5 + Du\
and consequently
$• = A + ZBu 2 + 5 Cw 4 + 7Dw 6 ^
DIFFERENTIAL CALCULUS. 77
and by comparing the series, and substituting for A, B, C,
&c. their values, we find
_! u u 3 , u 5 u 1 ■ „
a? = tanff w = h &c.
* 1 3 5 7
If we make x — 45°, u will be equal to 1 ; hence,
arc45°=l- — + — - — + &c.
3 5 7
But this series is not sufficiently convergent to be used
for computing the value of the arc. To find the value
of the arc in a more converging series, we employ the
following property of two arcs, viz. :
Four times the arc whose tangent is — , exceeds the
5
arc of 45° by the arc whose tangent is — ( — *.
* Let a represent the arc whose tangent is — . Then (Trig. Art.
XXVI),
: 2 tans a 5
tang 2 a ■.
tang 4a =
1 — tang 2 a 12 ■
2 tang 2 a 120
1— tana 2 2a 119
The last number being greater than unity, shows that the arc 4 a ex-
ceeds 45°. Making
4c=w?, 45 Q = B,
the difference, 4 a — 45° = A — B = b, will have for its tangent
, . _ _ % tang A — tang B 1
tang 6= tang (Jl—B) a ■■ . ° — ^—=r = 7^r ;
s sv } 1 -f tang A tang B 239*
hence, four times the arc whose tangent is -r-, exceeds the arc of 45° by an
1 ' ■
arc whose tangent is ■—- -.
78
But
tang
ELEMENTS OF THE
♦ -1 1 l
tang 1 - r =
1 1
+
.i 1
5 5 3.5 3 5.5 5 7.5 7
1 1.1 1
f &C.„
239 239 3(239) 3 5(239) 5 7(239)'
7-f &c;
hence,
arc 45° = J
A 5 3.£
1 1
5 3.5 3 5.5 5 7.7 1
(± !_ +_J L_ + >>
\239 3(239) 3 5(239) 5 7 (239V /
(239) 3 ' 5(239) 5 7(239) 7
Multiplying by 4, we find the semi-circumferenee
= 3.141592653.
DIFFERENTIAL CALCULUS. 79
CHAPTER IV.
Development of any Function of two Variables
— Differential of a Function of any number
of Variables — Implicit Functions — Differential
Equations of Curves — Of Vanishing Fractions.
79. We have explained in Taylor's theorem the method
of developing into a series any function of the sum or dif-
ference of two variables.
We now propose to give a general theorem of which
that is a particular case, viz :
To develop into a series any function of tivo or more
variables, and find the differential of the function.
80. Before making the development it will be necessary
to explain a notation which has not yet been used.
If we have a function of two variables, as
u = F(x,y),
we may suppose one to remain constant and differentiate
the function with respect to the other.
Thus, if we suppose y to remain constant, and x to
vary, the differential coefficient will be
SO ELEMENTS OF THE
and if we suppose x to remain constant and y to vary,
the differential coefficient will be
d ^ = F'{x,y). (2).
The differential coefficients which are obtained under
these suppositions, are called partial differential coef-
ficients. The first is the partial differential coefficient
with respect to x, and the second with respect to y.
81. If we multiply both members of equation (1) by
dx, and both members of equation (2) by dy, we obtain
— - dx = F f (x, y) dx, and —^dy = F" (x, y) dy.
The expressions,
du 7 du 7
T* dx ' %**•
are called 'partial differentials ; the first a partial diffe-
rential with respect to x, and the second a partial diffe-
rential with respect to y ; hence,
A partial differential coefficient is the differential co-
efficient of a function of two or more variables , under
the supposition that only one of them has changed its
value : and,
A partial differential is the differential of a function
of two or more variables, under the supposition that only
one of them has changed its value.
82. If we differentiate equation (1) under the suppo-
sition that x remains constant and y varies, we shall have
dy
DIFFERENTIAL CALCULUS. 81
and since x and dx are constant
j(du\ _ d{du)
\dx) dx
which we designate by
d*u t
dx *
hence, -^ — =- = F"' (x, y).
dxdy
The first member of this equation expresses that the
function u has been differentiated twice, once with respect
to x, and once with respect to y.
If we differentiate again, regarding x as the variable,
we obtain
which expresses that the function has been differentiated
twice with respect to x and once with respect to y. And
generally
d n+m u
dx n dy m '
indicates that the function u has been differentiated n-\-m
times, n times with respect to x, and m times with respect
to y.
83. Resuming the function
u = F(x,y),
if we suppose y to remain constant, and give to x an arbi-
trary increment h, we shall have from the theorem of Taylor,
r,, , , N duh , d 2 u h 2 , d 3 u h 3 , e
82
ELEMENTS OF THE
■, . , du d 2 u d 3 u
mwluch ' "■ s; as? rf?'
are functions of a?- and ?/, and dependent on the constants
which enter the F(x,y).
If we now attribute to y an increment k, the function
M, which depends on y, will become
die
and the function — will become
aa?
du , d?u k , d 3 u k 2 , d 4 u k 3
(fa? dxdy 1 7 dxdy 2 1.2^ dxdy 3 1.2.3 '
d?u
and the function — r-^-, will become
(far
c^w , d 3 u k , d% k 2 , d 5 w /c 3 . p
L Arc
da* (far% 1 <fa*% 2 1>8"T cfa 2 ^ 3 1.2.3 *
d 3 u
and the function -j-j-, will become
d 3 u d l u k_ d 5 u k 2 d 6 u k 3 „
~dx J + dx T dyT + dx 3 dy 2 1.2 + <fa 3 <fy 3 1.2.3 /
&c. &c. &c. &c.
Substituting these values in the development of
F(x + h, y),
DIFFERENTIAL CALCULUS. 83
and arranging the terms, we have
r/ . , . 7N du k . dht k 2 d 3 u k 3 s
F(oc+h,y+k)=u+- — — h— \- &c,
dull <Pu lik * d?u hk 2 o
dx 1 dxdy l.l dxdy 2 1.2
da? 2 1.2 (fo^y 1.2
(ir 3 1.2.3
+ -r^T-^r+&c.;
which is the general development of a function of two
variables, when each has received an increment, in terms
of the increments and differential coefficients.
84. If we now transpose u = F(x, y) into the first
member, and pass to the limit, Ave find
d[F (x, y)] = du = — dx-{- j-dy.
The differential of F (x, y) = du, which is obtained under
the supposition that both the variables have changed their
values, is called the total differential of the function.
85. If we have a function of three variables, as
u = F{x,y,z),
and suppose one of them, as z, to remain constant, and
increments h and k to be attributed to the other two, the
development of F (x -f h, y -f k, z) will be of the same
form as the development of F(x + h, y + k) ; but u and
all the differential coefficients will be functions of z.
84 ELEMENTS OP THE
If then an increment I be attributed to z, there will be
four terms of the development of the form
du , du 7 du 7
u i T h -> ~T k * ~T L
ax ay dz
If u were a function of four variables, as
u — F (x, y, z, s),
there would be five terms of the form
du 1 du 1 du 7 du
u, — h, —k, —I, —on;
ax ay dz as
and a new variable introduced into the function, would
introduce a term containing the first power of its increment
into the development.
If we transpose u into the first member, and pass to
the limit, we shall have
d[F{x >y' z)] =TJ x+ Zy d!/+ ^ d *'
and
, r x du 7 du 7 du , du ,
d[F{x,y,z,s)} = - [x dx + -dy + Tz az + -d S ,
from which we may conclude that, the total differential
of a function of any number of variables is equal to the
sum of the partial differentials.
86. The rule demonstrated in the last article is alone
sufficient for the differentiation of every algebraic function.
1. Let u = x 2 -{- y 3 — z; then
du
—dx=2xdx. 1st partial differential ;
ax
DIFFERENTIAL CALCULUS. 85
du
— dy — 3y 2 dy, 2d partial differential J
dy
^dz=-dz, 3d
dz
hence, du — 2x dx + 3y 2 dy — dz.
2. Let u = xy ; then,
du 7 7
—-ax — y ax 3
dx
--du = xdy :
dy J
hence, du — ydx 4- xdij.
3. Let u = x m y n ; then,
— dx = ?nx m ~ l y" dx,
d>
— dy == m/ n ~ l x m dy : hence,
du = mx m ~ l y n dx + ny n ~ x x m dy — x m - x y n ~\mydx + nxdy\
x
4. Let u — — , then,
y
dx = —,
ax y
du , xdy
— dy=: f-
dy y 2
, 7 ydx — xdy
hence, du = 2 £„
f
s
86 ELEMENTS OF THE
5. Let u — - .. ■' = ay (x 2 -\-y 2 ) 2 ; then,
Vx 2 + if
du 1 ai/xdx
-dx = 1 |f
(x 2 +y 2 )
du_j_ ady arfdy
y (x 2 + y 2 Y (x 2j ry 2 Y
, 7 ayxdx — ax 2 dy
hence, du = - — -.
6. Let u — xyzt ; then,
du = yztdx + xztdy + xytdz -f xyzdt.
7. Let u — z y ; then,
pdy = z y lzdy (Art. 55),
pdx = yz y ~ l dz (Art. 32):
hence, du = z y lzdy -\- y z v ~ x dz.
Remark. In chapter II, the functions were supposed
to depend on a common variable, and the differentials were
obtained under this supposition. We now see that the dif-
ferentials are obtained in the same manner, when the func-
tions are independent of each other, and unconnected with
a common variable.
87. We have seen (Art. 39), that a function of a single
variable has but one differential coefficient of the first
order, one of the second, one of the third, &c. ; while a
DIFFERENTIAL CALCTTI/l T S. 87
function of two variables has two differential coefficients
of the first order, a function of three variables, three ; a
function of four variables, four ; &c.
It is now proposed to find the successive differentials
of a function of two variables, and also the successive
differential coefficients.
We have already found
7 du , du ,
du — — ax -\- — dy.
dx dy
Since — and — are functions of x and y. the
dx dy
du du
differentials ~rdx, ~r-dy, must each be differentiated
dx dy J
with respect to both of the variables; dx and dy being
supposed constant: hence,
d(-dx\-—dx 2 A- dhl dxdv
C \dx dX )-dar dX +d^dy dXdy >
j/du 7 \ dhi 7 9 , dhi , 7
and since the second differential of the function is but the
differential of the first differential, we have
d u d hi d "u
dhi = —^ dx 2 + 2 -= — — dx d\j -f -— dy 2 .
dxr dxdy dy 1
If we differentiate again, we have
d 3 ii, 3 dhi dx 2 dy,
d( ( ^Ldx 2 \- U dx* + —
d \dx 2dX )~dx^ dX + dx*
d{<lp-dxdy\ = 2-^-dxhly + 2-^- 2 dxdy\
v dxdy J dxrdy dxdy z
00 ELEMENTS OF THE
and consequently,
It is very easy to find the subsequent differentials, by
observing the analogy between the partial differentials and
the terms of the development of a binomial.
We also see that, a function of two variables has tivo
partial differential coefficients of the first order T three of
the second, four of the third, &c.
88. There are several important results which may be
deduced from the general development of the function of
two variables (Art. 83).
1st. If we make x = 0, and y — 0, u and each of
the differential coefficients will become constant, and we
shall have
t-i/7 7n I rdu 1 , du 1 \
+T^w + 2 ^V df k )
+ &c,
which is the development of any function of two variables
in terms of their ascending powers, and coefficients which
are dependent on the constants that enter the primitive
function.
2d. If, in the general development, we make y = 0, and
k = 0, we shall have
DIFFERENTIAL CALCULUS. 89
_L. 7X , du h , d 2 u h 2 , d 3 u h? , s
r(;r+ , )=M+ _ T+ __ + „___ +&c .,
which is the theorem of Taylor.
3d. If we make y — 0, k = 0, and x — 0, we have
„,-.. du h , d 2 u h 2 , d?u h 3 , s
which is the theorem of Maclaurin.
Lnplicit Functions.
89. When the relation between a function and its
variable is expressed by an equation of the form
y = F{x)
in which y is entirely disengaged from x, y has been
called an explicit, or expressed function of x (Art. 5).
When y and x are connected together by an equation of
the form
F(x,y) = 0,
y has been called an implicit, or implied function of x
(Art. 5.)
It is plain, that in every equation of the form
F(x,y) = 0,
y must be a function of x, and x of y. For, if the
equation were resolved with respect to either of them, the
value found would be expressed in terms of the other
variable and constant quantities.
90 ELEMENTS OF THE
90. If in the equation
u = F(x,y)-0 K
we suppose the variables x and y to change their values
in succession, any change either in x or y, will produce a
change in u : hence, % is a function of x and y when
they vary in succession. The value, however, which u
assumes, when x or y varies, will reduce to when
such a value is attributed, to the other variable as will:
satisfy the equation
F(x,y) = 0.:
Now the partial differential
du -.
-Y-ax^
ax
represents the limit of the change which takes place in the-
function u under the supposition that x varies (Art.- 81);
and the partial differential
du ,
ts the limit of the change which takes place' in the function
u under the supposition that y varies. But the change
which takes place in u when x and y both vary is :.
iience,, -^dx + —dy = &.
dx ay
91. In discussing the equation
F{x y y)^O h
DIFFERENTIAL CALCULUS. 91
it is often necessary to find the differential coefficients of
one of the variables regarded as a function of the other,
and this may be done without resolving the equation. For,
from the last article,
du j , du ,
— dx-\- — dy = 0:
dx dy
or,
hence,
du du dy _
dx dy dx
du
dy _ dx
dx du
dy
Hence, the differential coefficient of y regarded as a
function of x, is equal to the ratio of the partial differen-
tial coefficients of u regarded as a function of x, and u
regarded as a function of y, taken with a contrary sign.
Let us take, as an example, the equation of the circle
F(x,y) = a? + y 2 -R 2 = u = 0;
a. du , du
then, r— =. 2x 3 and — - = 2y :
dx dy J
hence,
dy _ x
dx y
Although the differential coefficient of the first order is
generally expressed in terms of x and y, yet y may be
eliminated by means of the equation F(x,y) = 0, and the
coefficient treated as a function of x alone. In the circle,
we have
y= VlV-x 2 ,
92 ELEMENTS OF THE
dy _ X
hence,
dx Vj^ 2
92. If it be required to find the second differential
coefficient, we have merely to differentiate the first diffe-
rential coefficient, regarded as a function of x, and divide
the result by dx. Thus, if we designate the first diffe-
rential coefficient by p, the second by q, the third by
r, &c, we shall have
dp dq .
z.
93. To find the second differential coefficient in the
circle, we have
dy x
dx y ■
j (dy\ — ydx -f- xdy t
Kdx) ~ y 2
, dy
d?y " V ^ X Tx i
hence, -^ == - »
dx 2 y 2
and by substituting for —■ its value -
X
, we have
y
d 2 y x 2 -f y 2
dx 2 ~ y 3
1. Find the first differential coefficient of y y in the
equation
y 2 — 2mxy -f- x 2 — a 2 — u = 0,
du , rt du
—- = — 2my + 2x< — — 2v — 2mx:
dx y dy y
DIFFERENTIAL CALCULUS. 93
1 fy _ r—2my-\-2x~\_ my — a?
dx L 2 y — 2 ??za? J y — ma?
2. Find the first differential coefficient of y in the
equation
y 2 + 2077/ + a? - a 2 = 0.
dx
3. Find the first and second differential coefficients
of
y>
in the equation
y 3 — 3 axy -f x 3 = 0,
du
dx
= Sx 2
o du n
- 3 ax,
hence,
<&2
dx
3 x 2 — 3ay ay — a?
3y 2 —3ax y 2 — ax
For the second differential coefficient, we have
, d v oJ\ f Mi ^f^Jy
dhj
(y 2 — ax) (a -^ — 2a? J — (ay — x 2 ) (%y C -f- — a\
or, by substituting for -f- its value, and reducing,
dx 2 (y 2 — axf
dy
dx
d?y 2 a?y 4 — 6 ax 2 y 2 + 2 yaf_ + 2 a 3 a?y
dx 2 ~ (y 2 - ax) 3 '
2xy ( y 3 — 3 tfajy + a? 3 ) + 2a 3 a?y _
(y 2 - ax} 3
but from the given equation
y 3 — 3axy -f a? 3 = 0.
, d 2 v 2a 3 xy
hence, -±4 = - — — ?—
aar (y — axy
\i
94 ELEMENTS OE THE
Differential Equations of Curves.
94. The Differential Calculus enables us to free an
equation of its constants, and to find a new equation which
shall only involve the variables and their differentials.
If, for example, we take the equation of a straight line
y = ax-\- b,
and differentiate it, we find
dy _
dx
and by differentiating again,
The last equation is entirely independent of the values
of a and b, and hence, is equally applicable to every
straight line which can be drawn in the plane of the co-
ordinate axes. It is called, the differential equation of
lines of the first order.
95. If we take the equation of the circle
x 2 + y 2 = R 2 ,
and differentiate it, we find
ocdx + ydy ~ 0.
This equation is independent of the value of the radius
R, and hence it belongs equally to every circle whose
centre is at the origin of co-ordinates.
f=
2Rx
-x?;
2R
_f
+ x\
J
X
DIFFERENTIAL CALCULUS. 05
96. If the origin of co-ordinates be taken in the circum-
ference, the equation of the circle (An. Geom. Bk. Ill,
Prop. I, Sch. 3) is
from which we find
and by differentiating,
x ( 2 ydy -f- 2 xdx ) — ( y 2 + x 2 ) dx
or by reducing
(x 2 — y 2 ) dx-\-2xydy = 0,
which is the differential equation of the circle when the
origin of co-ordinates is in the circumference.
The last equation may be found in another manner.
If we differentiate the equation of the circle,
y 2 = 2 Rx — x 2 ,
we have, after dividing by 2
ydy = Rdx — xdx ;
. „ ydy + xdx
hence, R — y J 7 .
dx
If this value of R be substituted in the equation of the
circle, we have
(x 2 — y 2 )dx + 2xydy == ;
the same differential equation as found by the first method.
96 ELEMENTS OF THE
97. If we take the general equation of lines of the
second order (A.n. Geom. Bk. VI. Prop. XII, Sch. 3),
y 2 — mx + nx 2 ,
and differentiate it, we find
2ydy = mdx + 2nxdx ;
differentiating again, regarding dx as constant, we have,
after dividing by 2,
dy 2 + yd 2 y = ndx 2 .
Eliminating m and n from the three equations, we obtain
y 2 dx 2 + x 2 dy 2 — 2xy dxdy + yx 2 d 2 y = 0,
which is the general differential equation of lines of the
second order.
98. In order to free an equation of its constants, it will
be necessary to differentiate it as many times as there are
constants to be eliminated. For, two equations are neces-
sary to eliminate a single constant, three to eliminate two
constants, four to eliminate three constants, &c. : hence,
one constant may be eliminated from the given equation
and the first differential equation ; two from the given equa-
tion and the first and second differential equations, &c.
99. The differential equation which is obtained after the
constants are eliminated, belongs to a sjiecies or order of
lines, of which the given equation represents one of the
species.
Thus, the differential equation (Art. 94),
DIFFERENTIAL CALCULUS. 97
belongs to an order or species of lines of which the
equation
y zzzo.x -r b,
represents a single one, for given values of a and b.
The equation of a parabola is
if = 2px,
and the differential equation of the species is
_ 2 xdy — ydx — 0, or dy 2 -f yd-y = 0.
100. The differential equation of a species, expresses
the law by which the variable co-ordinates change their
values; and this equation ought, therefore, to be indepen-
dent of the constants which determine the 77iag7iiiv.de, and
not the nature of the curve.
101. The terms of an equation may be freed from their
exponents, by differentiating the equation and then com-
bining the differential and given equations.
Suppose, for example,
P and Q being any functions of x and y.
By differentiating, we obtain
nP n - l dP=dQ:
by multiplying both members by P, we have
nP n dP = PdQ,
and b}' substituting for P" its value,
7iQdP = PdQ.
9
98 ELEMENTS OF THE
The same result might also have been obtained by
taking the logarithms of both members of the equation
For, we have
and (Art. 57).
P n =Q.
nlP = lQ,
dP dQ
hence, nQdP = PdQ.
Of Vanishing Fractions, or those which take the
form — .
J
102. It has been shown in (Alg. Art. Ill), that — is
sometimes an undetermined symbol, and that its value
may be 0, a finite quantity, or infinite.
This symbol arises from the presence of a common
factor in the numerator and denominator, which, becoming
for a particular value of the variable, reduces the fraction
to the form — .
If we have, for example, a fraction of the form
P{x-a) m
Q(x-a) n '
in which P and Q are finite quantities, and make x = Oy
we shall have
P{x-a) m _ Q
Q{x-a) n ~ 0'
DIFFERENTIAL CALCULUS. 99
The value of this fraction will, however, be 0, finite or
infinite, according as
m>n, m = n, ?n<n,
for under these suppositions, respectively, it takes the form
P(x-a) m - n P P
~Q J Q' Q{x-a)"- M '
Let the numerator of the proposed fraction be desig-
nated by X, and the denominator by X', and let us sup-
pose an arbitrary increment h to be given to x. The
numerator and denominator will then become a function
of a? -f h, and we shall have from the theorem of Taylor
Y . dx h ; d 2 x h 2 , (Px K 3 ; .
, dX h , d 2 X' h 2 d*X h 3
+ dx 1 ' dx 2 l/2 + dx 3 1.2.3 +&C '
If the value of x = a, reduces to the differential
coefficients in the numerator as far as the ?nth order, and
those of the denominator as far as the nth. order, the value
of the fraction will become,
d m X h"
dx m 1.2.3.4..
h &c,
. .171
d n X If
+ &C.
If we make h = 0, the value of the fraction will be-
come 0, finite, or infinite according as
m>n, ?n=zn, m < n,
and hence, if the value x = a, reduces to the same
number of differential coefficients in the numerator and
100 ELEMENTS OF THE
denominator, the value of the fraction will be finite
and equal to the ratio of the first differential coefficients
which do not reduce to 0.
103. Let us now illustrate this theory by examples.
1. If in the fraction
l-x n
1-x'
we make x— 1, we have — . But
dX n l dX
= -nx n \ -— = -
ax ax
-i;
in which, if we make x= 1, we have
dX , dX
— — = — n 9 and —r— =
ax ax
~h
, dX
hence, — , — ■
dx
— n.
dX
dx 7
therefore, the value of the fraction when x = 1, is +ft.
2. Find the value of the fraction
ax 2 — 2 acx -f- ac 2 ,
— r-r-. when x — c,
bx 2 — 2bcx-\-bc 2
4^ = 2ax-2ac, ^- = 2bx-2bc,
dx dx
both of which become 0, when x = c. Differentiating
again, we have
d?X <M' ,
-aW= 2a > -^r =2h;
hence, the true value of the fraction when x = c is ~.
DIFFERENTIAL CALCULUS. 101
3. Find the value of the fraction
x 3 — ax 2 — a 2 x + « 3 i
, when x —
Ans. 0.
-, when oo = a.
or — a 2
4. Find the value of the fraction
ax — x 2 ,
— 5 g -, when x=za.
a i —2a 3 x + 2ax 3 — x*
Ans. oo.
5. Find the value of
, when x = 0.
x
Ans. la — lb.
6. What is the value of the fraction
1 — sin# + cos.a? , nnn
- ., when 37 = 90°.
sin^-h cos x — I
Ans, 1.
7. What is the value of the fraction
a — x — ala -f- alx ,
-j — , wnen x = a.
a— V2ax — x 2
Ans. — 1.
8. What is the value of the fraction
*-, when a? =1,
\ — x + lx
Ans. — 2.
9. What is the value of tke fraction
Q n /£«
^ =-, when,a? = a.
ia — Iz
Ans. nc? e
9*
102 ELEMENTS OF THE
104. It has been remarked (Art. 47), that the theorem
of Taylor does not apply to the case in which a particular
value attributed to x renders every coefficient either or
infinite. Such functions are of the form
(a? -a 8 )-
(x - a ) n '
in which m and n are fractional.
In functions of this form we substitute for x, a-\-h 9
which gives a second state of the function. We then
divide the numerator and denominator by h raised to a
power denoted by the smallest exponent of h, after which
we make h — 0, and find the ratio of the terms of the
fraction.
When we place a -f- h for x, we have in arranging
according to the ascending powers of h,
F{a + h) _ Alt + Bh h + Ch* + &c,
F\a + h)~ A'lt' + Bh b ' + Ch* + &c.
Now there are three cases, viz. : when
a > a\ a = a', a < a' .
In the first case the value of the fraction will be ; in
the second, a finite quantity ; and in the third it will be
infinite.
105. In substituting a-\-h for x, in the fraction
i. '
(x-ay
DIFFERENTIAL CALCULUS. 103
, (2ah + h 2 y ,_ , ,J.
we have \ — J — = (2a-\-h) 2 ,
and by making h = 0. which renders x = a, the value of
the fraction becomes
(2a)\
2. Required the value of the fraction
a
(a?-3ax+2a 2 y .
; — - — when x = a.
(x'-cfy
Substituting a + h for x, we have
.1 1 1 1
/** (3a 2 + Sah + S 2 ) 8 (3a 2 + 3a/* + h 2 )*
which is equal to 0, when h—0.
106. Remark. The last method of finding the value of
a vanishing fraction, may frequently be employed advan-
tageously, even when the value can be found by the
theorem of Taylor.
107. There are several forms of indetermination under
which a function may appear, but they can all be reduced
to the form — .
1st. Suppose the numerator and denominator of the
fraction
X
T'
to become infinite by the supposition of x = a. The
fraction can be placed under the form
104 ELEMENTS OF THE
X
X
which reduces to — , when X and X f are infinite.
2d. We may have the product of two factors, one of
which becomes and the other infinite, when a particular
value is given to the variable.
In the product PQ, let us suppose that x = a, makes
P = and Q = oo . We would then write the product
under the form,
P
PQ=:
which becomes — when x = a.
108. Let us take, as an example, the function
(l-tf)tang~K-a?;
in which t designates 180°.
If we make x=l, the first factor becomes 0, and the
second infinite. But
1 1
tang — 7tx =
2 ♦ 1
COt 7TX
2
hence, (1 — a?) tang — ttx = ,
cot — a-a?
2
2
the value of which is — when # = 1.
it
DIFFERENTIAL CALCULUS. 105
CHAPTER V
Of the Maxima and Minima of a Function of a
Single Variable.
109. If we have
u = F(x),
the value of the function u may be changed in two ways :
first, by increasing the variable x ; and secondly, by dimin-
ishing it.
If we designate by u! the first value which u assumes
when x is increased, and by u" the first value which u
assumes when x is diminished, we shall have three con-
secutive values of the function
u f , u, u".
Now, when u is greater than both u f and u f/ , u is said
to be a maximum : and when u is less than both v! and
u" y it is said to be a minimum.
Hence, the maximum value of a variable function is
greater than the value which immediately precedes, or the
value that immediately follows : and the, minimum value
of a variable function is less than the value which imme-
diately precedes or the value that immediately follows.
110. Let us now determine the analytical conditions
which characterize the maximum and minimum values of
a variable function.
106 ELEMENTS OF THE
If in the function
u = F(x),
the variable x be first increased by h, and then diminished
by h, we shall have (Art. 44),
r,r ''. x\ (lull d 2 U ll 2 , d 3 U h 3 , s
v ; dx 1 dx 2 1.2 dx 3 1.2.3
,, ^, 7X du h , c/ 2 ^ It" dhi li 3 , r,
u"= Fix — li) — u — - h —7-7- -r-r — : h &c;
v ; dx I dx 2 1.2 cte 3 1.2.3
and consequently,
du h d?u It? j_ d 3 w A 3 - o
du h d 2 u h 2 d 3 u h 3 o
dxT d~aFT^2 ~~ ^ 3 1.2.3
Now, if w has a maximum value, the limits of v! '—u
and u" — u, will both be negative ; and if u is a minimum,
the limits of u'—u and u"—u will both be positive.
Hence, in order that a may have a maximum or minimum
value, the signs of the limits of the two developments must
be both minus or both plus.
But since the terms involving the first power of h, in
the two developments, have contrary signs, it follows that
the limits of the developments will have contrary signs
(Art. 44) ; hence, the function u can neither have a maxi-
mum or a minimum unless
^ = 0;
dx
DIFFERENTIAL CALCULUS. 107
and the roots of this equation will give all the values of
x which can render the function u either a maximum or
a minimum.
Having made the first differential coefficient equal to 0,
the signs of the limits of the developments will depend on
the sign of tl.e second differential coefficient.
But since the signs of these limits are both negative
when u is a maximum, arid both positive when u is a
minimum, it follows that the second differential coefficient
will be negative when the function is a maximum, and
positive when it is a minimum. Hence, the roots of the
equation
'- = o,
being substituted in the second differential coefficient, will
render it negative in case of a maximum, and positive in
case of a minimum ; and since there may be more than
one value of x which will satisfy these conditions, it fol-
lows that there may be more than one maximum or one
minimum.
But if the roota of the equation
ax
reduce the second differential coefficient to 0, the signs of
the limits of the developments will depend on the signs
of the terms which involve the third differential coefficient ;
and these signs being different, there can neither be a
maximum or a minimum, unless the values of x also reduce
the third differential coefficient to 0. When this is the
case, substitute the roots of the equation
I OS ELEMENTS OF TIU
7 = o,
ax
in the fourth differential coefficient ; if it becomes negative
there will be a maximum, if positive a minimum. If the
values of x reduce the fourth differential coefficient to 0,
the following differential coefficient must be examined.
Hence, in order to find the values of x which will render
the proposed function a maximum or a minimum.
1st. Find the roots of the equation
.^ = 0. ■
dx
2d. Substitute these roots in the succeeding differential
coefficients, until one is found which does not reduce to 0.
Then, if the differential coefficient so found be of an odd
order, the values of x will not render the function either
a maximum or a minimum. But if it be of an even
order, and negative, the function will be a maximum ; if
positive, a minimum .
111. Remark. Before applying the preceding rules to
particular examples, it may be well to remark, that if a
variable function is multiplied or divided by a constant
quantity, the same values of the variable which render the
function a maximum or a minimum, will also render the
product or quotient a maximum or a minimum, and hence
the constant may be neglected.
2. Any value of the variable which will render the func-
tion a maximum or a minimum, will also render any root
or power a maximum or a minimum ; and hence, if a func-
tion is under a radical, the radical may be omitted.
DIFFERENTIAL CALCULUS. 109
EXAMPLES.
I . To find the value of x which will render y a maxi-
mum or a minimum in the equation of the circle
f + x 2 = R\
dy __ x
y
dx
x
making — 0, gives x — 0.
The second differential coefficient is
d?y _ x 2 -;- y 2
dx 2 ~ f '
and since making x = 0, gives y = R, we have
dx 2 ~ R
which being negative, the value of x — renders y a
maximum.
2. Find the values of x which will render y a maximum
or a minimum in the equation,
y — a — bx -f x%
differentiating, we find
g=-6 + 2 ,, and 5 = 2,
making, — b + 2 a? = 0, gives w = — ;
A
and since the second differential coefficient is positive, this
value of x ™;il render y a mir> ; ^um. The minimum
10
110 ELEMENTS OF THE
value of y is found by substituting the value of x t in the
primitive equation. It is
3. Find the value of x which will render the function
u — a* + b*x — c 2 x 2 ,
a maximum or a minimum,
du ,. n ., , IP
— = b J — 2c x. hence x = — g ;
dx 2c 2 '
and, --— = — 2c 2 :
ciatr
hence, the function is a maximum, and the maximum
value is
4. Let us take the function
u = 3 a 2 x 3 — ¥x -\- c 5 ,
du b 2
we find -Y- — 9 a V — Z> 4 , and # = ± — .
ax da
The second differential coefficient is
d 2 u 10 2
V-5 = 18 era?
a a?*
Substituting the plus root of a?, we have
LTFFERENTIAL CALCULUS. Ill
which gives a minimum, and substituting the negative
root, we have
which gives a maximum.
The minimum value of the function is,
1 2b"
and the maximum value
9a'
9a
112. Remark. It frequently happens that the value
of the first differential coefficient may be decomposed into
two factors, X and X\ each containing x, and one of
them, X for example, reducing to for that value of x,
which renders the function a maximum or a minimum.
When the differential coefficient of the first order takes
this form, the general method of finding the second diffe-
rential coefficient may be much simpliiied. For, if
— = A4 ,
dx
we shall have
dhi _ XdX XdX'
dor dx dx
But by hypothesis A" reduces to for that value of x
which renders the function u a maximum or a minimum :
d 2 u X'dX
benC6 ' M = -dx~ ;
112 ELEMENTS OP THE
from which we obtain the following rule for finding the
second differential coefficient.
Differentiate that factor of the first differential coef-
ficient which reduces to 0, multiply it by the other factor,
and divide the product by dx.
5. To divide a quantity into two such parts that themth
power of one of the parts multiplied by the nih power of
the other shall be a maximum or a minimum.
Designate the given quantity by a and one of the parts
by x, then will a — x represent the other part. Let the
product of their powers be designated by u ; we shall then
have
u = x m (a — x) n y
whence, -^ = mx m ~ x (a - x) n - nx m (a - a?)- 1 ,
= (ma — mx — nx}x m ~ l (a — a?) B_ V
and by placing each of the factors equal to 0, we have
ma .
x — , x — 0, x — a.
m-\-n
The second differential coefficient corresponding to the
first of these values, found by the method just explained,, is
°^-=-{m + n)x m ~ l {a - x) n ~ x ;
ax
and substituting for x its value, it becomes
i+n-2
(m-\- n) 1
ifn- 3
hence, this value of x renders the product a maximums
The two other values of x satisfy the equation of the
DIFFERENTIAL CALCULUS. 113
problem, but do not satisfy the enunciation, since they are
not parts of the given quantity a.
Remark. If m and n are each equal to unity, the quan-
tity will be divided into equal parts.
6. To determine the conditions which will render y a
maximum or a minimum in the equation
yi _ 2vixy + x 2 — a 2 — 0.
The first differential coefficient is
dy _ my — x <
dx y — mx '
x
hence, my — x = 0, or y = — .
m
Substituting this value of y in the given equation, we
find
ma
VT3
,2
and the value of y corresponding to this value of x is
a
Vl — m l
To determine whether y is a maximum or a minimum,
let us pass to the second differential coefficient. We have
-f- — {iny — x)(y — mx)~ x ;
ax
d?y
(-2-0
hence,
dx 2 y — mx
10
114 ELEMENTS OF THE
and since -f- = y we have
ax
dx 2 y — mx
and by substituting for y and x their values, we have
*y = i .
hence, y is a maximum.
7. To find the maximum rectangle which can be in-
scribed in a given triangle.
Let b denote the base of the triangle, h the altitude,
y the base of the rectangle, and x the altitude. Then,
u = xy = the area of the rectangle.
But b : h : : y : h — x:
_, bh — bx
lience, y — — T
and consequently,.
bhx — bx 2 b ,, ™
u — - = —(kx - Of) o
/i a
and omitting the constant factor,
d u -l o ^
— = h — 2x. or #=: — :
<fc 2
hence, the altitude of the rectangle is equal to half the
altitude of the triangle : and since
do?
the area is a maximum.
2,
DIFFERENTIAL CALCULUS. 115
8. What is the altitude of a cylinder inscribed in a given
right cone, when the solidity of the cylinder is a maximum ?
Ans. One third the altitude of the cone.
9. What are the sides of the maximum rectangle in-
scribed in a given circle ?
Ans. Each equal to R V2.
10. A cylindrical vessel is to contain a given quantity
of water. Required the relation between the diameter of
the base and the altitude in order that the interior surface
may be a minimum.
Ans. Altitude = radius of base.
11. To find the altitude of a cone inscribed in a given
sphere, which shall render the convex surface of the cone
a maximum. .
Ans. Altitude = — R.
3
12. To find the maximum right-angled triangle which
can be described on a given "rrae.
Ans. When the two sides are equal.
13. What is the length of the axis of the maximum
parabola that can be cut from a given right cone ?
Ans. Three-fourths the side of the cone.
14. To find the least triangle which can be formed by
the radii produced, and a tangent line to the quadrant of a
given circle.
Ans. When the point of contact is at the middle of the
arc.
15. What is the altitude of the maximum cylinder which
can be inscribed in a given paraboloid ?
Ans. Half the axis of the paraboloid.
116
ELEMENTS OP THE
CHAPTER VI
Application of the Differential Calculus to the
Theory of Curves.
113. It has been shown in (Art. 13), that every relation
between a function and a single variable on which it
depends, may subsist between the ordinate and abscissa of
a curve. Hence, if we represent the ordinate of a curve
by a function ?/,• the abscissa may be represented by
the variable x.
Of Tangents and Normals.
114. We have seen (Art.
15), that if y represents
the ordinate and x the ab-
scissa of any curve as CP,
the tangent of the angle
PTA, which the tangent
forms with the axis of ab-
scissas will be represented
R N
by
dy
dx~'
dy and dx being the differentials of the ordinate and ab-
scissa of the point of contact P.
DIFFERENTIAL CALCULUS. 117
But wc have (Trig. Th. II),
1 : TR :: tangT : RP;
that is, 1 : TR :: &■ : y:
hence. T.R — y—- — sub-tangent.
115. The tangent TP is equal to the square root of
the sum of the squares of TR and RP ; hence,
\/l
TP = y\/ 1+^= tangent.
116. From the similar triangles TPR, RPN, we have
TR : PR : : PR : RN,
dx
hence, ^ t— : y : : y : RN,
dij
consequently, RN = y~ = sub-normal.
117. The normal PN is equal to the square root of the
sum of the squares of PR and RN ; hence,
PN = y V 1 + -^1 = normal.
118. Let it be now required to apply these formulas to
lines of the second order, of which the general equation
(An. Geom. Bk. VI, Prop. XII, Sch. 3), is,
y 2 — mx + nx 2 .
Differentiating, we have
dy _m-\-2 nx _ m -\- 2 nx
dx 2y 2Vmx + nx 2
118
ELEMENTS OF THE
substituting this value, we find
, „, r> dx 2(mx + nx 2 )
sub-tangent TR = y— = -± - '
° y dy m + 2nx
| mx
TP = y\/l + ~~=\/mx + nx 2 +
v ay* y \_m + 2nx
sub-normal RN = y-f = ■:
J dx 2
PN = y\/l + ^=\/:
mx + nx 2 + — (m + 2nx) 2 .
4
By attributing proper values to m and n, the above
formulas will become applicable to each of the conic
sections. In the case of the parabola, n = 0, and we have
TR = 2x, TP=V mx -f- 4 x 2 ,
RN =f,
PN
w
mx -\-
119. It is often necessary to represent the tangent and
normal lines by their equations. To determine these, in
a general manner, it will be necessary first to consider the
analytical conditions which render any two curves tangent
to each other.
Let the two curves, PDC,
PEC, intersect each other at
P and C.
Designate the co-ordinates of
the first curve by x and y, and
the co-ordinates of the second by
x' ', y' . Then, for the common
point P, we shall have
x = x f , y = y'.
DIFFERENTIAL CALCULUS. 119
If we represent BG, the increment of the abscissa, by
h, we shall have, from the theorem of Taylor (Art. 44),
CG-PB=CF=^-^A + d -ft-^-+ &c,
dx 1 dor 1.2 dx 3 1.2.3
CG-PB=CF=%± + %^%-^- + &c;
do/ 1 da/ 2 1.2 3Pl>2.3
hence, by placing the two members equal to each othgr,
and dividing by h, we have
dy d 2 i/ Ji_ '• dy r d?y f h -
<fc + ^1.2 ' '~"Z7' + "^" 1.2 +
If we now pass to the limit, by making h = 0, we shall
have
cAr da/
in which case the point C will become consecutive with P,
and the curve PEC tangent to the curve PDC. Hence,
two lines will be tangent to each other at a common point,
when the co-ordinates and first differential coefficient of
the one, are equal to the co-ordinates and, first differential
coefficient of the other.
120. The equation of a straight line is of the form
y — ax + b,
hence, -f- — a.
dx
But the equation of a straight line passing through a
given point, of which the co-ordinates are a/' \ y /f , is (An.
Geom. Bk. II, Prop. IV),
y-y"=a(x-a/'),
120 ELEMENTS OF THE
or by substituting for a its value, we have, for the equation
of a straight line passing through a given point,
y-y"=||(*-*").
This line may be made tangent to a curve at any point
of which the co-ordinates are x ;/ , y" , by substituting for
~% the first differential coefficient found from the equation
ax
of the curve, and making x'\ y l/ , equal to a/ ,: \ y" of the
curve.
121. Let it be required, for example, to make the line
tangent to a circle at a ooint of which the co-ordinates are
a/', y" . Since the co-ordinates of this point will satisfy
the equation of the curve, we have
x" 2 + y" 2
= R 2
and
by
differentiating,
dx" "
x"
and by substituting this value in the equation of the line,
and recollecting that x' n -\-y ff2 ~ R 2 , we have
yy"+xx"=R\
which is the equation of a tangent line to a circle.
122. A normal line is perpendicular to the tangent at
the point of contact, and since the equation of the tangent
is of the form
DIFFERENTIAL CALCULLS. 121
the equation of the normal will be of the form (An. Geom.
Bk. II, Prop. VII, Sch. 2),
and this line will become normal to a curve at a point of
dx"
which the co-ordinates are x' f , y", if the value of — — be
ay"
found from the equation of the curve, and substituted for
— , and the co-ordinates a/ f f y" of the straight line be
ay
made equal to x /f , y n of the curve.
The equation of the normal in the circle will take the
form,
y»
123. To find the equation of a tangent line to an ellipse
at a point of which the co-ordinates are a#, y ff , we have,
Ay /2 + ^V /2 = A 2 5 2 .
By differentiating* we have
dy" _ B 2 x ,f m
do/' ~~~AY~ ;
hence, we have
which becomes, after reducing,
A 2 yy ff -\-B 2 xx"=A 2 B 2 .
The equation of the normal is
ii
122
ELEMENTS OF THE
124. To find the equation of a tangent to lines of the
second order, of which the equation for a particular point
(An. Geom. Bk. VI, Prop. XII, Sch. 3) is
y rr2 z=z mx" + nx" 2 .
By differentiating, we have
m + 2 nx"
dx"
2y'
hence, the equation of the tangent to a line of the second
order is
t//
y-y = — ^77 — (*--*"),
and the equation of the normal
2y"
y-y"=-
m + 2 nx 1 '
-,{x-x"\
Of Asymptotes of Curves.
125. An asymptote of a curve is a line which continually
approaches the curve, and becomes tangent to it at an
infinite distance from the origin of co-ordinates.
Let AX and AY be E
the co-ordinate axes, and
y-y'^%^-^'
the equation of any tan-
gent line, as TP.
DIFFERENTIAL CALCULUS. 123
If in the equation of the tangent, we make in succes-
sion y = 0, x=0, we shall find
x-Al-x y dy//i y-AJJ-y -oC d ^.
If the curve CPB has an asymptote RE, it is plain
that the tangent PT will approach the asymptote RE,
when the point of contact P, is moved along the curve
from the origin of co-ordinates, and T and D will also
approach the points it and Y, and will coincide with
them when the co-ordinates of the point of tangency are
infinite.
In order, therefore, to determine if a curve have asymp-
totes, we make, in succession, x = oo and y = oo in the
values of A T, AD. If either of these become finite, the
curve will have an asymptote.
If both the values are finite, the asymptote will be in-
clined to both the co-ordinate axes : if one of the distances
becomes finite and the other infinite, the asymptote will
be parallel to one of the co-ordinate axes ; and if they both
become 0, the asymptote will pass through the origin of
co-ordinates. In the last case, we shall know but one
point of the asymptote, but its direction may be deter-
di/
mined by finding the value of -^-, under the supposition
that the co-ordinates are infinite.
126. Let us now examine the equation
y 2 = moc -\- nx 2 ,
124 ELEMENTS OF THE
of lines of the second order, and see if these lines have
asymptotes. We find
A rr 2if — mx
AT—x - = ,
m-\-2nx m-\-2nx
. r. mx 4- 2 nx 2 mx
AD = y-
2y 2 Vmx + nx 2
which may be put under the forms
AT = ~ m , Al)±
+ 2n 2\J- + n
and making x = qo , we have
AR=-^-, and i^=-^=.
2n ^ 2 Vn
If now we make n — 0, the curve becomes a parabola,
and both the limits, AR, AE, become infinite : hence,
the parabola has no rectilinear asymptote.
If we make n negative, the curve becomes an ellipse,
and AE becomes imaginary : hence, the ellipse has no
asymptote.
But if we make n positive, the equation becomes that
of the hyperbola, and both the values, AR, AE, become
finite. If we substitute for n its value — , we shall have
A z
AR=-A, and AE=±B.
DIFFERENTIAL CALCULUS.
125
Differentials of the Arcs and Areas of Segments
of Curves.
127. It is plain, that the chord and arc of a curve will
approach each other continually as the arc is diminished,
and hence, we might conclude that the limit of their ratio
is unity. But as several propositions depend on this rela-
tion between the arc and chord, we shall demonstrate it
rigorously.
128. If we suppose the ordi-
nate PR of the curve, POM to
be a function of the abscissa, we
shall have (Art. 16),
N
and
in which
PQ = h,
MQ = (P+P / h)h;
dx'
R
Hence, PM= '\f^+{P-\-P'hfK 2 =h-y/l^{P-{-P , h) 2 .
We also have NQ = Ph ;
heiice, PN = Vh 2 + P 2 h 2 = h Vl + P 2 ,
NM= NQ -MQ=- P f fc
hence, we have
PN+MN hVl+PZ-P'h 2 Vl + P 2 -P'h
PM
Wl+(P + P'h) 2 Vl + (P -f P'hf '
126 ELEMENTS OF THE
of which the limit, by making h — 0, is
VT+W = 1
VI -f P 2
But the arc POM can never be less than the chord PM,
nor greater than the broken line PNM which contains it ;
hence, the limit of the ratio
POM
PM ~ '
and consequently, the differential of the arc is equal to the
differential of the chord. But when we pass to the limit
of the arc and chord, PM becomes the differential of the
chord, and PQ and QM, become the differentials of x
and y ; hence, if we represent the arc by z, we shall have
dz = -y/dx 2 + df~:
that is, the differential of the arc of a curve, at any point,
is equal to the square root of the sum of the squares of
the differentials of the co-ordinates.
129. To determine the differential of the arc of a circle
of which the equation is
x 2 + y 2 = R 2 ,
xdx
we have xdx -f- ydy = 0, or dy — ;
/ x d Or dx ———^——
-whence, dz=y dx 2 + — y~ — — Vx 2 + y 2 ,
Rdx _ Rdx
Vi^ 2 -ar ^,
DIFFERENTIAL CALCULUS.
127
the same as determined in (Art 71). The plus sign is to
be used when the abscissa x and the arc are increasing
functions of each other, and the minus sign when they
are decreasing functions (Art. 31).
G M
130. Let B CD be any segment
of a curve, and let it be required
to find the differential of its area.
The two rectangles DCFE,
DGME, having the same base
DE, are to each other as DC to
EM ; and hence, the limit of their D E
ratio is equal to the limit of the ratio of DC to EM,
which is equal to unity.
But the curvelinear area DCME is less than the rect-
angle DGME, and greater than the rectangle DCFE :
hence, the limit of its ratio to either of them will be
unity. But,
DCME DCME DEFC_ DCME _j^n
DE ~ DE X DEFC ~ DL * DEFC y '
X
or by representing the area of the segment by s and the
ordinate DC by y, and passing to the limit, we have
ds
dx
or ds = ydx ;
hence, the differential of the area of a segment of any
curve, is equal to the ordinate into the differential of the
abscissa.
\
128 ELEMENTS OF THE
131 To find the differential of the area of a circular
segment, we have
o? + f = R\ and y = VR'-x 2 ;
hence, ds — dx yR z — x 2 .
The differential of the segment of an ellipse, is
B
ds = -—dx^/A 2 —x 2 ,
A.
and of the segment of a parabola
ds = dxV2px.
Signification of the Differential Coefficients.
132. It has already been shown that, if the ordinate of
a curve be regarded as a function of the abscissa, the first
differential coefficient will be equal to the tangent of the
angle which the tangent line forms with the axis of abscis-
sas (Art. 15). We now propose to show the relation
between a curve and the second differential coefficient,
the ordinate being regarded as a function of the abscissa.
Let AP be the abscissa
and PM the ordinate of a
curve. From P lay off
on the axis of abscissas
PP ; = h, and PP // = 2h.
Draw the ordinates PM,
P'M, P"M" ' ; also the lines
MMN, MM" ; and lastly,
MQ, M'Q!, parallel to the
DIFFERENTIAL CALCULUS.
129
axis of abscissas. Then will M'Q = NQ r , and we shall
have
PM = y,
dy h d 2 y h 2
P"M"=y-\
y +T x T
dor 1.2
d»/2h ' d 2 y ilt
dx\ dx 2 1.2
+ &c,
+ &c,
P'M- PM = M'Q =^A+i^JL+ &c .,
dx 1 dx 2 1.2
■p"M !, -P r M' = M"Q= ^h + 4^— + &c.
ax dor 1 . 2
M 1 ' Q' - M> Q = + JIT^iV - -^r # + &c.
ptar
Now, since the sign of the first member of the equation
is essentially positive, the sign of the second member will
also be positive (Alg. Art. 85). But if we pass to the
limit, by diminishing h, the sign of the second member
will depend on that of the second differential coefficient
(Art. 44) : hence, the second differential coefficient is
positive.
If the curve is below
the axis of abscissas,
the ordinate s will be
negative, and it is easily
seen that Ave shall then
fiave
P P'
I
Ml
A
Q
M
r 5 **^^
M>
Q'
N
M"
\
M ,r Q f -MQ
dx 2
130
ELEMENTS OF THE
Now, since the first member is negative, the second
member will be negative : hence we conclude that, if a
curve is convex towards the axis of abscissas, the ordi-
nate and second differential coefficient will have like signs.
133. Let us now con-
sider the curve CMM'M",
which is concave towards
the axis of abscissas. We
shall have,
N
/
M"
M
y-
^
Q
m/
/
C
Q
A
P 1
3/
J
u
runir? , dy h , d 2 y h 2 , s
dx 1
dx 2 1.2
tv/ tit// , dy2h , d?y 4/* 2 , p
* ■ dx I dx 2 1.2
P ff M ,f -P'M f = M fr Q f = C ^-h-{-~l-~ + &c,
dx dx- 1 . 2
M'Q-MQ = - NM"=-^X-h 2 + &c.
dor
But since the first member of the equation is negative,
the essential sign of the second member will also be
negative : hence, the second differential coefficient will
be negative.
DIFFERENTIAL CALCULUS.
131
If the curve is below the
axis of abscissas, the ordi-
nate will be negative, and it
is easily seen that we should
then have
M"Q-M>Q
+ NM // =^K i -\- &c;
a-ar
hence we conclude that, if a curve is concave towards the
axis of abscissas, the ordinate and second differential
coefficient will have contrary signs.
The ordinate will be considered as positive, unless the
contrary is mentioned.
134. Remark 1. The co-ordinates x and y, determine
a single point of a curve, as M. The first differential of
y is the limit of the difference between the ordinates PM,
P f M f , or the difference between two consecutive ordinates.
The second differential of y is the limit of M~ f N, and
is derived from M''Q or dy, in the same way that dy is
derived from the primitive function. The abscissa x being
supposed to increase uniformly, the difference, and conse-
quently the limit of the difference between PP' and P'P !f
is : therefore its second differential is 0. The co-ordi-
nates x and y, and the first and second differentials deter-
mine three points, M, M, M" , consecutive with each other,
135. Remark 2. When the curve is convex towards
the axis of abscissa, the first differential coefficient, which
132 ELEMENTS OF THE
represents the tangent of the angle formed by the tangent
line with the axis of abscissas, is an increasing function of
the abscissa : hence, its differential coefficient, that is, the
second differential coefficient of the function, ought to be
positive (Art. 31).
When the curve is concave, the first differential coeffi-
cient is a decreasing function of the abscissa ; hence, the
second differential coefficient should be negative (Art. 31).
Examination of the Singular Points of Curves.
136. A singular point of a curve is one which is distin-
guished by some particular property not enjoyed by the
points of the curve in general : such as, the point at which
the tangent is parallel, or perpendicular to, the axis of
abscissas.
137. Since the first differential coefficient expresses the
value of the tangent of the angle which the tangent line
forms with the axis of abscissas, and since the tangent is
0, when the angle is 0, and infinite when the angle is 90°,
it follows that the roots of the equation
ax
will give the abscissas of all the points at which the tan-
gent is parallel to the axis of abscissas, and the roots of
the equation
dy _ doc _
dx dy
DIFFERENTIAL CALCULUS. r33
will give lire abscissas of all the points at which the tan-
gent is perpendicular to the axis of abscissas.
13S. If a curve from being convex towards the axis of
abscissas becomes concave, or from being concave becomes
convex, the point at which the change of curvature takes
place is called a point of inflexion.
Since the ordinate and differential coefficient of the
second order have the same sign when the curve is convex
towards the axis of abscissas, and contrary signs when it
is concave, it. follows that at the point of inflexion, the
second differential coefficient will change its sign. There-
fore between the positive and negative values there will be
one value of x which will reduce the second differential
coefficient to or infinity (Alg. Art. 310) : hence the roots
of the equations
d 2 y rPi/
dx> = °> 0T -dJ = ™
will give the abscissas of the points of inflexion.
139. Let us now apply these principles in discussing
the equation of the circle
x 2 + y 2 = R 2 ,
vt e nave, u
y u
J S>
and placing
dy
dx;
X
>
y
-
X
y
=o,
we
\ haye x =
0.
Substituting this value in
the equation
of the
curve,
we
have
y =
±R;
12
134 ELEMENTS OF THE
hence, the tangent is parallel to the axis of abscissas at
the two points where the axis of orclinates intersects the
circumference.
If we make
dx
X
y ~
00,
or
_.y,=
X
0,
we have y = :
; subs
3titutino[ this value in
the
equation,
we find
x —
±R,
and hence, the tangent is perpendicular to the axis of
abscissas at the points where the axis intersects the cir-
cumference.
The second differential coefficient is equal to
?\
which will be negative when y is positive, and positive
when y is negative. Hence, the circumference of the
circle is concave towards the axis of abscissas.
If we apply a similar analysis to the equation of the
ellipse, we shall find the tangents parallel to the axis of
abscissas at the extremities of one axis, and perpendicular
to it at the extremities of the other, and the curve concave
towards its axes.
140. Let us now discuss a class of curves, which may
be represented by the equation
y = b±c(x — a) m ,
in which we suppose c to be positive or negative, and
different values to be attributed to the exponent m.
DIFFERENTIAL CALCULUS.
135
1st. When c is positive, and m entire and even.
By differentiating, we have
dy
dx
J-
</__
dar
??w(x — a) T
m(?n — l)c(x — a) m 2 .
dy
If we place the value — = 0, we find x = a, and sub-
dx
stituting this value in the equation of the curve, we find
y = b :
hence, x = a, y = b, are the co-ordinates of the point
at which the tangent line is parallel to the axis of
abscissas.
Since m is even, m — 2 will
also be even, and hence the second
differential coefficient will be posi-
tive for all values of cc. The curve
will therefore be convex towards
the axis of A", and there will be
no point of inflexion.
The value of x = a renders the ordinate y a minimum,
since after m differentiations a differential coefficient of an
even order becomes constant and positive (Art. 110).
The curve does not intersect, the axis of A", but cuts the
axis of Y at a distance from the origin expressed by
y = b + ca m .
136
ELEMENTS OF THE
141. 2d. When c is negative, and m entire and even.
We shall have, by differentiating,
( ^.= - 7 nc(x-a) m - 1
doc
and
^ = _ W j( wl -l) c ( a? -fl)»- •
dx
The discussion is the same as
before, excepting that the second
differential coefficient being nega-
tive for all values of x, the curve
is concave towards the axis of:
abscissas, and the value of x = a,
fenders the ordinate y a maxi-
mum (Art. 110).
142. 3d. When c is plus or minus, and m entire and
uneven.
We shall have, by differentiating-,
^-=±mc{x-a) m -\
dx
<Py
and
— J = ±m{m — \) c(x — a) T '
The first differential coefficient will be 0, when x~a;
hence, the tangent will be parallel to the axis of abscissas,
at the point of which the co-ordinates are x = a, y = b.
DIFFERENTIAL CALCULUS.
137
Since the exponent m — 2 is
uneven, the factor (x— a) m ~' 2 will
be negative when x < a, and
positive when x > a ; hence, this
factor changes its sign at the
point of the curve of which the
abscissa is x = a.
If c is positive, the second differential coefficient will be
negative for x < a, and positive for x > a : hence there will
be an inflexion when x — a. If c were negative, the curve
would be first convex and then concave towards the axis
of abscissas, but there would still be an inflexion at the
point x — a. At this point the tangent line separates the
two branches of the curve.
There will, in this case, be neither a maximum nor a
minimum, since after m differentiations a differential coef-
ficient of an odd order, will become equal to a constant
quantity (Art. 110).
143. 4th. When c is positive or negative, and m a
2
fraction having an even numerator, as m = — .
By differentiating, and supposing c positive, we have
2c
dy 2 y
~- = — c\x — a)
dx 3 v ;
3(x — ay
d?l_
dx 2 ~
2c
a »
9(x-ay
If we make x = a, the first differential coefficient will
become infinite ; and the tangent will be perpendicular to
12*
138
ELEMENTS OF THE
trie axis of abscissas, at the point of which the co-ordinates
are x = a, y = b»
In regard to the second differen-
tial coefficient, it will become infi-
nite for x = a, and negative for
every other value of x t since the
factor (x — a) of the denominator
is raised to a power denoted by an
even exponent. Hence, the curve
will be concave towards the axis of
abscissas.
If we take the equation of the curve
y=zb + c (x — a) 3 ,
and make x = a -\- h t and x = a — h, we shall have, in
either case,
y — b-\- ch 3 ;
and hence, y will be less fov_x = a, than for any other
value of x, either greater or less than a. Hence, the
value x = a, renders y a minimum.
If c were negative, the equation would be of the form
y — b — c(x — a) 3 ;
and we should have, by differentiating,
dy 2c
and
3{x — a) 3
&y__ 2c
aar 9{x-ay
DIFFERENTIAL CALCULUS,
139
The first and second differen-
tial coefficients will be infinite for
x = a, and the second differential
coefficient will be positive for all
values of x greater or less than a ;
and hence, the curve will be con-
vex towards the axis of abscissas.
If, in the equation of the curve
y — b — c(x — a) 3 ,
we make x — a -f li, and x — a — h, we shall have, in
either case,
y = b — ch 3 ;
and hence, y will be greater for x = a, than for any other
value of x either greater or less than a. Hence, the
value x — a, renders y a maximum.
144. Remark. The conditions of a maximum or a
minimum deduced in Art. 110, were established by means
of the theorem of Taylor. Now, the case in which the
function changes its form by a particular value attri
buted to x, was excluded in the demonstration of that
theorem (Art. 45). Hence, the conditions of minimum
and maximum deduced in the two last cases, ought
not to have appeared among the general conditions of
Art. 110.
We therefore see that there are two species of maxima
and minima, the one characterized by
g=°>
the other by
dy
dx
00 .
140 ELEMENTS OF THE
In the first, we determine whether the function is a
maximum or a minimum by examining the subsequent
differential coefficients ; and in the second, by examining
the value of the function before and after that value of x
which renders the first differential coefficient infinite.
The branches DE, ME, which are both represented by
the eolation
y — b ± c{x — a) 3 ,
are not considered as parts of a continuous curve. For,
the general relations between y and x which determine
each of the parts DE, ME, is entirely broken at the
point , M, where x — a. The two parts are therefore
regarded as separate branches which unite at M. The
point of union is called a cusp, or a cusp point.
145. 5th. When c is vositive or negative and m a
3
fraction having an even denominator, as m = — .
Under this supposition the equation of the curve will
become
3
y = b± c(x — a) 4 ,
and by differentiating, we have
dy _ 3 c
dx 4(a?-a)V
- &y 3c
a - 4.40r-a)7
DIFFERENTIAL CALCULUS.
141
M
The curve represented by this
equation will have two branches :
the one corresponding to the plus
sign will be concave towards the
axis of abscissas, and the one cor-
responding to the minus sign will be l\~
convex. Every value of x less than
a will render y imaginary. The co-ordinates of the point
My are x — a, y==b.
146. 6th. When c is positive or negative and m a
fraction having an uneven numerator and an uneven de-
3^
5
Under this supposition the equation will become
y--
— b±c(x
-ay,
and
by
differ
entiating,
we have
dx
3
c
50-
% 9
-a) 5
d?y
dot?'
3
2c
7
5.5(,t — a)'
from which we see that if we use the superior sign of the
first equation, the curve will be convex towards the axis
of abscissas for x < a, that there will be a point of inflexion
for a? = a, and that the curve will be concave for x > a.
Ef the lower sign be employed, the first branch will become
concave, and the other convex.
147. The cusps, which have been considered, were
formed by the union of two curves that were convex to-
142
ELEMENTS OF THE
wards each other, and such are called, cusps of the first
order.
It frequently happens, however, that the curves which
unite, embrace each other. The equation
(y — x 2 ) 2 = x 5 , -
furnishes an example of this kind. By extracting the
square root of both members and transposing, we have
d 2 ij
dx 2
2 2
and by differentiating
-£ = 2x± — x 2 .
dx 2
We see by examining
the equations, that the curve
has two branches, both of
which pass through the
origin of co-ordinates. The
upper branch, which corres-
ponds to the plus sign, is constantly convex towards the
axis of abscissas, while the lower branch is convex for
„<!!*-, and concave for x>-^- and ,<l. At
the last point the curve passes below the axis of abscissas
and becomes convex towards it. If Ave make the first dif-
ferential coefficient equal to 0, we shall find x = 0, and
substituting this value in the equation of the curve, gives
y — ; and hence, the axis of abscissas is tangent to both
branches of the curve at the origin of co-ordinates. At
this point the differential coefficient of the second order
is positive for both branches of the curve, hence they
DIFFERENTIAL CALCULUS.
143
are both convex towards the axis. When the cusp is-
formed by the union of two curves which, at the point
of contact, lie on the same side of the common tangent, it
is called a cusp of the second order.
148. Let us, as another example, discuss the curve
whose equation is
y z=b±(x — a) Vx — c.
By differentiating, we obtain
d?/ , x — a
dx 2^x-c
We see, from the equa-
tion of the curve, that y will
be imaginary for all values
of x less than c.
For x — c, we have y~h ;
and for x > c, we have two
values of y and conse-
quently two branches of
the curve, until x = a when they unite at the point M\
For x>a there will be two real values of y and conse-
quently two branches of the curve. The point M, at
which the branches intersect each other, is called a mul-
tiple point, and differs from a cusp by being a point
of intersection instead of a point of tangency. At the
multiple point M there are two tangents, one to each
branch of the curve. The one makes an angle with the
axis of abscissas, whose tangent is
4- Va-
144 ELEMENTS OF THE
the other, an angle whose tangent is
— yfl — c '
149. Besides the cusps and multiple points which have
already been discussed, there are sometimes other points
lying entirely without the curve, and having no connexion
with it, excepting that their co-ordinates will satisfy the
equation of the curve.
For example, the equation
Gif — x 3 -\- bx 2 = 0,
will be satisfied for the values
oc= ± 0, y = ±0 ; and hence,
the origin of co-ordinates A,
satisfies the equation of the
curve, and enjoys the property
of a multiple point, since it is
the point of union of two values
of a?, and two values of y.
If we resolve the equation with respect to y, we find
and hence, y will be imaginary for all negative values of
x, and for all positive values between the limits x = and
oc = b. For all positive values of x greater than b } the
values of y will be real.
The first differential coefficient is
dy _ x(Sx—2b)
<te~2Vax 2 (x-b) '
DIFFERENTIAL CALCULUS. 145
or by dividing by the common factor x,
dy _ Sx-2b
df ~ 2 Va{z — b)
and making x = 0, there results
dy_ 26
dx 2 V—ab '
which is imaginary, as it should be, since there is no point
of the curve which is consecutive with the isolated or con-
jugate point. The differential coefficients of the higher
orders are also imaginary at the conjugate points.
150. We may draw the following conclusions from the
preceding discussion.
1st. The equation ^ = 0, determines the points at
ax
which the tangents are parallel to the axis of abscissas.
2d. The equation -~ == qd , determines the points of
the curve at which the tangents are perpendicular to the
axis of abscissas. The two last equations also determine
the cusps, if there are any, in all cases where the
tangent at the cusps is parallel or perpendicular to the
axis of abscissas.
3d. The equation -5-^ = 0, or -^-=00 determines
n dx 2 dx 2
the points of inflexion.
4th. The equation -f- = an imaginary constant, in-
dicates a conjugate point.
13
146
ELEMENTS OF THE
CHAPTER VII.
Of Osculatory Curves — Of Evolutes.
151. Let PT be tangent to the curve ABP at the point
P, and PN a normal at the same point : then will PT
be tangent to the circumference of every circle passing
through P, and having its centre in the normal PN.
It is plain that the cen-
tre of a circle may be
taken at some point C,
so near to P, that the cir-
cumference shall fall with-
in the curve APB, and
then every circumference
described with a less ra-
dius, will fall entirely
within the curve. It is
also apparent, that the centre may be taken at some point
Oy so remote from P, that the circumference shall fall
between the curve APB and the tangent PT, and then
every circumference described with a greater radius will
fall without the curve. Hence, there are two classes of
tangent circles which may be described; the one lying
within the curve, and the other without it.
DIFFERENTIAL CALCULUS.
147
ACE
152. Let there be
three curves, APB,
CPD, EPF, which
have a common tan-
gent TP, and a com-
mon normal PN ; then
will they be tangent to
each other at the point
P. It does not follow,
however, from this cir-
cumstance, that each curve will have an equal tendency to
coincide with the tangent TP, nor does it follow that any
two of the curves CPD, EPF, will have an equal ten-
dency to coincide with the first curve APB.
It is now proposed to establish the general analytical
conditions which determine the tendency of curves to
coincide with each other, or with a common tangent.
Designate the co-ordinates of the first curve APB by
x and y, the co-ordinates of the second CPD by a?, y' ,
and the co-ordinates of the third EPF by x' f , y" . If we
designate the common ordinate PR by y, if, y" , we shall
then have
>/ '__ , dy h d 2 y If
9 R' = y +
+
dhj
do: 1 da?1.2 dx 3 1.2.3
4- &c.,
sR*
dy' h &y' h 2 <Pi/ h 3
da/T da? 2 1.2 dx r3 1.2.3
&c;
nR!
y V dx" 1 *
d?y" h 2 _^ d 3 y" h?
dx' 2 1.2 da?" 3 1.2.3
&c.
But since the curves are tangent to each other at the
point P, we have (Art. 119),
148 ELEMENTS OF THE
y=y'=y"> and %=%=%■■ hence >
A-. a -^( $"y <Py'\ h 2 ( d 3 y dy \ h 3 „
q \dx 2 da/ 2 ) 1.2^ \dx 3 (fa/Vl.2.3 + '
* U 2 ^Vli 1 ^^ dx"V\.2.2>^
Now, in order that the first curve APB shall approach
more nearly to the second CPD than to the third EPF,
we must have
d<d',
and consequently,
h 2 h 3 l? h 3
A— + B-^— + &c, < A'— + B'-^— + &c,
1.2 1.2.3^ '-> i:sT 1.2.3 '.
in which we have represented the coefficients in the first
seiies by A, B, C, &c.-, and the coefiicients in the second
by A', B' 9 'o\ &c.
Now, the limit of the first member of the inequality will
always be less than the limit of the second, when its first
term involves a higher power of h than the first term of
the second. For, if A = 0, the first member will involve
the highest power of h, and we shall have
h 3 h 2 h 3
6—^— + &c, < A' — + B r -±— n + &c,
1.2.3 1.2 1.2.3
and by dividing by h\
B_^_-j-&c.,<A' — + B f -^—- +&c,
1.2.3 ' 1.2 1.2.3
and by passing to the limit
DIFFERENTIAL CALCULUS. 149
But when A = Q, we have
dy _ jy
da? ~ do/ 2 '
and hence, when three curves have a common ordinate, the
first will approach nearer to the second than to the third,
if the number of equal differential coefficients between the
first and second is greater than that between the first and
third. And consequently, if the first and second curves
have m + 1 differential -coefficients which are equal to
each other, and the first and third curves only m equal dif-
rential coefficients, the first curve will approach more
nearly to the second than to the third. Hence it appears,
that the order of contact of two curves will depend on
the number of corresponding differential coefficients which
are equal to each other.
The contact which results from an equality between the
-co-ordinates and the first differential coefficients, is called
a contact of the first order, or a simple tangency (Art. 119).
If the second differential coefficients are also equal to each
other, it is called a contact of the second order. If the first
three differential coefficients are respectively equal to each
other, it is a contact of the third order; and if there are m
differential coefficients respectively equal to each other, it
is a contact of the mth order.
153. Let us now suppose that the second line is only
given in species, and that values may be attributed at
pleasure to the constants which enter its equation. We
13*
150 ELEMENTS OF THE
shall then be able to establish between the first and second
lines as many conditions as there are constants in the
equation of the second line. If, for example, the equation
of the second line contains two constants, two conditions
can be established, viz. : an equality between the co-
ordinates, and an equality between the first differential
coefficients ; this will give a contact of the first order.
If the equation of the second curve contains three con-
stants, three conditions may be established, viz. : an. equality
between the co-ordinates, and an equality between the first
and second differential coefficients. This will give a con-
tact of the second order. If there are four constants, we
can obtain a contact of the third order ; and if there are
m + 1 constants, a contact of the mth order.
It is plain, that in each of the foregoing cases the highest
order of contact is determined.
The line which has a higher order of contact with a
given curve than can be found for any other line of the
same species, is called an osculatrix.
Let it be required, for example, to find a straight line
which shall be osculatory to a curve, at a given point of
which the co-ordinates are a/ r , y" .
The equation of the right line is of the form
y = ax + b,
and it is required to find such values for the constants a
and b as to cause the line to fulfil the conditions,
x = a/>, y = y", and % = %■
DIFFERENTIAL CALCULUS. 151
By differentiating the equation of the line, we have
ax
and since the line passes through the point of osculation
y — y" — -j-(oc — xf f ).
Substituting for ~ its value -tit, we have
for the equation of the osculatrix.
In the equation of the circle
a/ f2 + y" 2 = R\
dy" cc"
we find ±T=-yr
hence, the equation of the osculatrix of the first order, to
the circle, is
x"
y-y ! '=--rM--x"\
or by reducing yy" -f xx" = R 2 .
154. If et, and £ represent the co-ordinates of the centre
of a circle, its equation will be of the form
( X -ccf-\-(y-^f = R 2 .
If this equation be twice differentiated, we shall have,
(x — «)dx + (y — /3) dy = 0,
dx 2 -i-dy 2 + (y-^(Ty = 0;
152
ELEMENTS OF THE
and by combining the three equations we obtair
dx 2 -f dy 2
y-P =
d 2 y '
dy /dx 2 4 dy c<
dx \ dhj
R=±
{dx 2 4- dy 2 ) ]
dxd 2 y
If it be now required to make this circle osculatory to
a given curve, at a point of which the co-ordinates are a/',
y" f we have only to substitute in the three last equations,
the values of
dy
dx
dy^
dx"
d 2 y _ ffy"
"dx 2 ~dx 772
deduced from the equation of the curve, and to suppose, at
the same time, the co-ordinates x and y in the curve to
become equal to those of x and y in the circle.
If we suppose x ;/ , y", to be general co-ordinates of the
curve, the circle will move around the curve and become
osculatory to it, at each of its points in succession.
155. If the circle CD
be osculatory to the curve
EF, at the point P, we
shall have
qsz=Cx +
W
1.2.3
for h positive ; and
h 3
q's'
Cx -
1.2,3
&c.
&c,
DIFFERENTIAL CALCULUS. 153
for h negative: hence, the two lines qs, c/s', have contrary
signs. The curve, therefore, lies above the oscillatory cir-
cle on one side of the point P, and below it on the other,
and consequently, divides the oscillatory circle at the point
of osculation. Hence, also, the oscillatory circle separates
the tangent circles which lie without the curve from those
which lie within it (Art. 151).
In every osculatrix of an even order the first term in the
values of qs, qW, will, in general, contain an uneven power
of h ; and hence the signs of the limits of their values will
depend on that of It. The curve will therefore lie above
the osculatrix on one side of the point P and below it on
the other ; and hence, evert/ osculatrix of an even order
will, in general, he divided by the curve at the point of
osculation.
156. The first differential equation of Article 154,
(x — a)dx + (y — f)dy —
may be placed under the form
dx
If we make the circle oscillatory to the curve we have
a? — x f/ , y = ij" , and
dx dx" .
-r — -r^j\ hence,
dy dy n
dx"
which is the equation of a normal at the point whose co-
ordinates are x" y" (Art. 122). But this normal passes
through the point whose co-ordinates are * and /3. Hence,
the normal drawn through the point of osculation, ivill
contain the centre of the osculatory circle.
157. It was shown in (Art. 155) that the osculatory cir-
cle is, in general, divided by the curve at the point of oscu-
154 ELEMENTS OF THE
lation.. The position of the curves with respect to each
other indicates this result.
For, the osculatory circle is always symmetrical with
respect to the normal, while the curve is, in general, not
symmetrical with respect to this line. If, however, the
curve is symmetrical with respect to the normal, as is the
case in lines of the second order when the normal coincides
with an axis, the curve will not divide the osculatory circle
at the point of osculation ; and the condition which renders
the second differential coefficients in the curve and circle
equal to each other, will also render the third differential
coefficients equal, and the contact will then be of the third
order.
158. The radius of the osculatory circle
dxd?y
is affected with the sign plus or minus, and it may be well
to determine the circumstances under which each sign is
to be used.
If we suppose the ordinate to be positive, we shall have
(Art. 133)
— ~, and consequently d 2 y
negative when the curve is concave towards the axis of
abscissas, and positive when it is convex. If then, we
wish the radius of the osculatory circle to be positive for
curves which are concave towards the axis of abscissas, we
must employ the minus sign, in which case the radius will
be negative for curves which are convex.
DIFFERENTIAL CALCULUS. 155
159. If the circumferences of two circles be described
with different radii, and a tangent line be drawn to each, it
is plain that the circumference which has the less radius
will depart more rapidly from its tangent than the circum-
ference which is described with the greater radius ; and
hence we say, that its curvature is greater. And gener-
ally, the curvature of any curve is said to be greater or less
than that of another curve, according as its tendency to
depart from its tangent is greater or less than that of the
curve with which it is compared.
160. The curvature is the same at all the points of the
same circumference, and also in all circumferences described
with equal radii, since the tendency to depart from the tan-
gent is the same. In different circumferences, the curva-
ture is measured by the angle formed by two radii drawn
through the extremities of an arc of a given length.
Let r and r' designate the radii of two circles, a the
length of a given arc measured on the circumference of
each ; c the angle formed by the two radii drawn through
the extremities of the arc in the first circle, and d the
angle formed by the corresponding radii of the second.
We shall then have
2*r : a . : : 360° : c, hence, c
also,
2*r 7
2*r / : a :: 360° : d, hence, c' = ?^:
2*-/ '
and consequently
156
ELEMENTS OF THE
that is, the curvature in different circumferences varies
inversely as the radii.
161. The curvature
of plane curves is meas-
ured by means of the
osculatory circle.
If we assume two
points P and P 7 , either
on the same or on dif-
ferent curves, and find
the radii r and r' of the circles which are osculatory at
these points, then
curvature at P : curvature at F : : — : —7;
that is, the curvature at different points varies inversely
as the radius of the osculatory circle.
The radius of the osculatory circle is called the radius
of curvature.
162. Let us now determine the value of the radius of
curvature for lines of the second order.
The general equation of these lines (An. Geom. Bk. VI,
Prop. XII, Sch. 3), is
y 2 = moo + noo 2 ,
which gives,
, |_ (m + 2nx)dx ^ , 2 = [4y 2 +(m + 2^) 2 ]^
2y ' J 4z/ 2
J2 _ 2n y d% 2 —(m J \-2noo)dxdy_[Ani/ 2 —(m-\-2 nxf] doo 2
ay ~ Ty 2 ~— ^3 •
DIFFERENTIAL CALCULUS. 157
Substituting these values in the equation
. R= {dj+df)
dxd l y
.we obtain
3
21
„ _ [4(?nx 4- nor) -+- (hi + 2«a;) 2 ] a
which is the general value of the radius of curvature in
lines of the second order, for any abscissa x. i
163. If we make x = 0, we have
K = — m = — - ;
2 A
that is, in lines of the second order, ^c radius of curva-
ture at the vertex of the transverse axis is equal to half
the parameter of ilia t axis.
If be required to find the value of the radius of curva-
ture at the extremity of the conjugate axis of an ellipse,
we make (An. Geom. Bk. VIII, Prop. XXI, Sch. 3),
2J5 2 B 2
m == — — , n— -5 , and x = A,
A A A
which gives, after reducing,
hence, the radius of curvature at the vertex of the conju-
gate axis of an ellipse is equal to half the parameter of
that axis.
In the case of the parabola, in which n = 0, the general
value of the radius of curvature becomes
14
158
ELEMENTS OF THE
p __(m 2 4- 4 ma?) 2
~ 2 m 2
164. If we compare the value of the radius of curvature
with that of the normal line found in (Art. 118), we shall
have
„ (normal) 3
that is, the radius of curvature at any point is equal to
the cube of the normal divided by half the 'parameter
squared : and hence, the radii of curvature at different
points of the same curve are to each other as the cubes oj
the corresponding normals.
Of the Evolutes of Curves.
165. If we suppose an os-
culatory circle to be drawn at
each of the points of the
curve APP'B, and then a
curve ACQ ' C !l to be drawn
through the centres of these
circles, this latter curve is
called the evolute curve, and
the curve APP f B the invo-
lute.
166. The co-ordinates of the centre of the osculatory
circle, which have been represented by «> and /?, are con
stant for given values of the co-ordinates x and y of the
DIFFERENTIAL CALCULUS. 159
involute curve, but they become variable when we pass
from one point of the involute curve to another.
167. We have already seen that the osculatory circle is
characterized by the equations (Art. 154)
(x-«Y + (y-Pf = R 2 , (1)
(x-*)dx + (y-P)dy = 0, (2)
da?+dif-\-(ij-Q)d 2 i/ = 0. (3)
If it be required to find the relations between the co-
ordinates of the involute and the co-ordinates of the
evolute curves, we must differentiate equations (1) and (2)
under the supposition that * and /3, as well as x and y,
are variables. We shall then have
(x - *)dx + (y-p)dy-(x- a)d* - (y - p)dp = RdR,
dx 2 -j- dy 2 -f (y — /3) (Py — dadx — dp dy = 0.
Combining these with equations (2) and (3), we obtain
- {y - p)dp - {x - «)dx = RdR } (4)
— dccdx — dp dy = 0.
The last equation gives
dp _ dx f s
T,-~~dy [D)
But equation (2) may be placed under the form
which represents a normal to the involute (Art. 1 22), and
which becomes, by substituting for — -^ its value -7-,
dy da
160 ELEMENTS OF THE
y-*=s;(*-«)> ( 6 )
or ,8 - y = -£(« - x) (Art. 120).
dec
This last is the equation of a straight line passing
through a point whose co-ordinates are x and y, and tan-
gent to the curve whose general co-ordinates are # and /3 ;
hence, a normal line to the involute curve is tangent to
the evolute.
168. It is now proposed to show, that the radius of cur-
vature and the evolute curve have equal differentials.
Combining equations (2) and (5) we obtain
( a; -«) = (y-/ 3 )J ) (7)
or by squaring both members,
combining this last with equation (1) we have
mM^t?-^ (8 )
Combining equations (4) and (7), we have
-(y-fi)dfi-(y-fi)^- = RdR,
or -^ — — >-(y-P) = RdR;
DIFFERENTIAL CALCULUS. 161
or by squaring both members
Dividing this last by equation (8), member by member,
we have
(dR) 2 = d« 2 + dfi 2
or dR = VdJ + dp.
But if s represents the arc of the evolute curve, of which
the co-ordinates are a and /3, we shall have (Art. 128),
ds= Vd«? + d!i 2 ;
hence, dR = ds;
that is, the differential of the radius of curvature is equal
to the differential of the arc of the -evolute.
169. It does not follow, however, from the last equation,
that the radius of curvature is equal to the arc of the evolute
curve, but only that one of them is equal to the other plus
or minus a constant (Art. 22). Hence,
R = s -\- a
is the form of the equation which expresses the relation
between them.
14'
162
ELEMENTS OF THE
If we determine the radii
of curvature at two points of
the involute, as P and P f ,
we shall have, for the first,
R = s + a,
and for the second
B! = s f + a ;
hence,
R'-R = s'- s =C / C // ;
and hence, the difference betiveen the radii of curvature at
any two points of the involute is equal to the part of the
evolute curve intercepted betiveen them.
170. The value of the constant a will depend on the
position of the point from which the arc of the evolute
curve is estimated.
If, for example, we take the radius of curvature for lines
of the second order, and estimate the arc of the evolute
curve from the point at which it meets the axis, the value
of s will be when R = — m (Art. 163): hence we
shall have
— m = 0-\-a
or
m
and for any other point of the curve
R = s + —m.
DIFFERENTIAL CALCULUS.
163
Either of the evolutes, FE,
FE', F'E', or F'E, corres-
ponding to one quarter of the
ellipse, is equal to (Art. 169)
B
A
171 . The e volute curve takes
its name from the connexion which it has with the corres-
ponding involute.
Let CC'C" be an evolute
curve. At C draw a tan-
gent AC, and make it equal
to the constant a in the equa-
tion
R = s + a.
Wrap a thread ACQ ' C"
around the curve, and fasten
it at any point, as C".
Then, if we begin at A,
and unwrap or evolve the
thread, it will take the positions PC f , P'C" , &c, and the
point A will describe the involute APP f : for
PC'-AC = CO and P'C" -AC= CC'C", &c
172. The equation of the evolute may be readily found
by combining the equations
do? + dy 2
x
with the equation of the involute curve
_ dy{dx?+dy 2 )
dxd?y
164 ELEMENTS OF THE
1st. Find, from the equation of the involute, the values of
— and d?y,
dx
and substitute them in the two last equations, and there
will be obtained two new equations involving «s, /3, x and y.
2d. Combine these equations with the equation of the
involute, and eliminate x and y : the resulting equation
will contain *, /3, and constants, and will be the equation
of the evolute curve.
173. Let us take, as an example, the common parabola
of which the equation is
y 2
— mx.
We shall then have
dx
2?
<?,,=
and hence
m 2 dx 2
„ _ a - 4 yy4y 2 + ?n 2 x _ 4y 3 + m 2 y _ 4£
y P ~ m 2 \ 4z/ 2 )~~ w? ~ w?^ y
and by observing that the value of x — * is equal to that
of y — £ multiplied by — j--> we nave
4z/ 2 -f- m 2 m
u> — «* -—
hence we have,
2m
m 4y 3 , 2iP m
— j8 = -i- and a: — <* = -—
m 2 m 2
DIFFERENTIAL CALCULUS 165
substituting for y its value in the equation of the involute
y = m' 2 x~,
we obtain
3.
If!
m 2
= — 2x
m
and by eliminating x, we have
27 ??i\ 2 J'
which is the equation of the evolute.
If we make ^ = 0, we have
1
2 '
and hence, the evolute meets the /
axis of abscissas at a distance from
the origin equal to half the param-
eter. If the origin of co-ordinates
be transferred from A to this
point, we shall have
1
and consequently
2 '
21m
The equation of the curve shows that it is symmetrical
with respect to the axis of abscissas, and that it does not
extend in the direction of the negative values of *'. The
evolute CC corresponds to the part AP of the involute,
and CO' to the part AP'.
166
ELEMENTS OF THE
CHAPTER VIII.
Of Transcendental Curves. — Of Tangent Planes
and Normal Lines to Surfaces.
174. Curves may be divided into two general classes :
1st. Those whose equations are purely algebraic ; and
2dly. Those whose equations involve transcendental
quantities.
The first class are called algebraic curves, and the
second, transcendental curves.
The properties of the first class having been already
examined, it only remains to discuss the properties of the
transcendental curves.
Of the Logarithmic Curve.
175. The logarithmic curve takes its name from the
property that, when referred to rectangular axes, one of
the co-ordinates is equal to the logarithm of the other.
If we suppose the logarithms to be estimated in paral-
lels to the axis of Y, and the corresponding numbers to
be laid off on the axis of abscissas, the equation of the
curve will be
y — ljc.
DIFFERENTIAL CALCULUS.
167
176. If we designate the
base of a system of loga-
rithms by a, we shall have,
(Alg. Art. 241)
and if we change the value
of the base a to a r , we shall
have
It is plain, that the same value of x, in the two equations,
will give different values of y, and hence, every system of
logarithms will give a different logarithmic curve.
If we make y = 0, we shall have (Alg. Art. 257)
x == 1 ; and this relation being independent of the base of
the system of logarithms, it follows, that every logarithmic
curve will intersect the axis of numbers at a distance from
the origin equal to unity.
The equation
a y — x,
will enable us to describe the curve by points, even with-
out the aid of a table of logarithms. For, if we make
y
; &C,
we shall find, for the corresponding values of x,
x = 1, x = -\fa, x — a -y/a, x = tfa &c.
177. If we suppose the base of the system of logarithms
to be greater than unity, the logarithms of all numbers less
168 ELEMENTS OF THE
than unity will be negative (Alg. Art. 256) ; and therefore,
the values of y corresponding to the abscissas, between the
limits x~0 and oc = AE = 1, will be negative. Hence,
these ordinates are laid off below the axis of abscissas.
When x = 0, y will be infinite and negative (Alg. Art.
264). If we make x negative, the conditions of the equa-
tion cannot be fulfilled ; and hence, the curve does not
extend on the side of the negative abscissas.
178. Let us resume the equation of the curve
y = Ix.
If we represent the modulus of the system of logarithms
by A, and differentiate, we obtain (Art. 56),
dy =
. dx
X
dy
dx
_A
X
or
But r-«£ represents the tangent of the angle which the
ax
tangent line forms with the axis of abscissas : hence, the
tangent will be parallel to the axis of abscissas when
x = oo , and perpendicular to it when x '■=. 0.
But when x = 0, y = — oo ; hence, the axis of ordinates
is an asymptote 1o the curve. The tangent which is
parallel to the axis of X is not an asymptote: for when
x— co , we also have y = oo .
179. The most remarkable property of this curve be-
longs to its sub-tangent 7 Y i? / , estimated on the axis of
logarithms. We have found, for the sub-tangent, on .he
axis of X (Art. 114),
DIFFERENTIAL CALCULUS. 1C9
™=g*
stnd by simply changing the axes, we have
T'R'^cc = A:
dec
hence, the sub-tangent is equal to the modulus of the
system of logarithms from ichich the curve is constructed.
In the Naperian system AT = 1, and hence the sub-tangent
will be equal to 1 = AE.
Of tlie Cycloid.
G b
\
A N L
ISO. If a circle NPG be rolled along a straight line
AL, any point of the circumference will describe a curve,
which is called a cycloid. The circle NPG is called the
generating circle, and P the generating point.
It is plain, that in each revolution of the generating circle
an equal curve will be described ; and hence, it will only
be necessary to examine the properties of the curve
APBL, described in one revolution of the generating circle.
We shall therefore refer enly to this part when speaking
of the cycloid.
181. If we suppose the point P to be on the line AL
at A, it will be found at some point, as L, after all the
15
170
ELEMENTS OF THE
points of the circumference shall have been brought in
contact with the line AL. The line AL will be equal to
the circumference of the generating circle, and Is called
the base of the cycloid. The line BM, drawn perpen-
dicular to the base at the middle point, is equal to the
diameter of the generating circle, and is called the axis of
the cycloid.
182. To find the equation of the cycloid, let us assume
the point A as the origin of co-ordinates, and let us sup-
pose that the generating point has described the arc A P.
If N designates the point at which the generating circle
touches the base, AN will be equal to the arc NP.
Through N draw the diameter NG, which will be
perpendicular to the base. Through P draw PR perpen-
dicular to the base, and PQ parallel to it. Then, PR = NQ
will be the versed-sine, and PQ the sine of the arc NP.
Let us make
ON^r, AR = x, PR = NQ=y,
we shall then have
PQ = 4%ry^y\ x = AN-RN= arc NP - PQ ■
hence, the transcendental equation is
x = ver- sin~ l y — y2ry — y 2 .
DIFFERENTIAL CALCULUS. 171
1 63. The properties of the cycloid are, however, most
^^asily deduced from its differential equation, which is
readily found by differentiating both members of the trans-
scendental equation.
We have (Art. 71),
rdy
J(ver-sin \y) —
d( — V% ry — y 2 )
hence,
V ' 2ry — y
rdy-ydy .
V2ry—y 2
dx
~dy rdy — ydy
or dx
V2 ry — y 2 V
y d y
2ry- y 2
V2ry — if
which is the differential equation of the cycloid.
184. If we substitute in the general equations of (Arts.
114, 115, 116, 117), the values of dx, dy, deduced from
the differential equation of the cycloid, we shall obtain the
values of the normal, sub-normal, tangent, and sub-tangent.
They are,
normal PN = v2ry, sub-normal RN = \2ry — y 2 ,
tangent P T = l ^DL— ^ sub-tangent TR -
V2ry — y 2 \2ry — y 2
These values are easily constructed, in consequence of
their connexion with the parts of the generating circle.
The sub-normal RN, for example, is equal to PQ of
the generating circle, since each is equal to -y/2ry — y 2 :
hence, the normal PN and the diameter GN intersect
the base of the cycloid at the same point.
1 72 ELEMENTS OF THE"
Now, since the tangent to the cycloid at the point P is.
perpendicular to the normal, it must coincide with the
chord PG of the generating circle.
If, therefore, it be required to draw a normal or a tan-
gent to the cycloid, at any point as P, draw any line, as
ng, perpendicular to the base AL, and make it equal to
the diameter of the generating circle. On ng describe a
semi-circumference, and through P draw a parallel to the
base of the cycloid. Through p y where the parallel cuts
the semi-circumference, draw the supplementary chords
pn, pg, and then draw through P the parallels PN, PG r
and PN will be a normal, and PG a tangent to the cycloid
at the point P*
185. Let
us resume the differential
equation
of
the
cycloid
a y* ...
V%ry — y 2
which may 1
be put
under the form
dy _
dx
■y/2ry — y 2 a /2r
y v y
-i.
If we make y, — 0, we shall have
dx
and if we make y = 2r, we shall have
ax
DIFFERENTIAL CALCULUS. 173
hence, the tangent lines drawn to the cycloid at the points
where the curve meets the base, are perpendicular to the
base ; and the tangent drawn through the extremity of the
greatest ordinate, is parallel to the base.
186. If we differentiate the equation
dx
_ y d y
-y/2 ry — y 2
regarding dx as constant, we obtain
o=^ + dy)vw^-- ydy{ ;' dv - yd / ) -,
V2ry — y 2
or by reducing and dividing by y,
= {2ry-y 2 )d 2 y + rdy\
whence we obtain
J 2ry — if
and hence the cycloid is concave towards the axis of
abscissas (Art. 133).
187. To find the evolute of the cycloid, let us first sub-
stitute in the general value of
R ^ (da? + dy 2 f
dx d 2 y
the value of cPy found in the last article : we shall then
have
R = 2 2 (ry) 2 = 2i/2ry:
hence, the radius of curvature corresponding to the ex-
tremity of any ordinate y, is equal to double the normal.
15*
174
ELEMENTS OF THE
The radius of curvature is when y — 0, and equal to
twice the diameter of the generating circle for y = 2r:
hence, the length of the evolute curve from A to A' is-
equal to twice the diameter of the generating circle.
Substituting the value of d 2 y in the values of y— p f
as — *■ (Art. 172), we obtain
y — p=:2y, x — «>——2^2ry — y 2 ;
hence we have
y=— j3, x — *— 2V—2rp-~p 2 .
Substituting these values of y and x in the transcen-
dental equation of the cycloid, we have
ver-sm
/*+ V-2r/3-
which is the transcendental equation of the evolute, re-
ferred to the primitive origin and the primitive axes.
Let us now trans-
fer the origin of co-
ordinates to the point
A f , and change at
the same time the_2y
direction of the posi-
tive abscissas : that
is, instead of estima-
ting them from the X'
left to the right, we will estimate them from the right
to the left. Let us designate the co-ordinates of the
evolute, referred to the new axes A! M,. A f X f , by *' and fif+
DIFFERENTIAL CALCULUS. 175
Since A'X' — AM — the semi-circumference of the gene-
rating circle, which is equal to rr, we shall have, for the
abscissa A r R r of any point P',
A!K! —^ — rTT — x, hence, <*. ■=. m — *! \
and for the ordinate, we shall have
R'P'= #~ RE - P'E = 2r- (- /3) = 2r + 0,
hence, P = — 2r + p f , or — fl = 2r — p'.
Substituting these values of * and £ in the transcen~
dental equation of the evolute, we obtain
rw — *'= ver-sin- 1 (2r — p') -f- V2r/3' — /3 /2 ,
or
(/== rv — ver-sin" 1 (2 r - £') - V2rp'—& 2 .
But the arc whose versed-sine is 2r — ,$', is the supple-
ment of the arc whose versed-sine is jS', hence
«' = ver-sin l £' — V 2 r& — /3 /2 ,
which is the equation of the evolute referred to the new
origin and new axes.
But this equation is of the same form, and involves the
same constants as that of the involute : hence, the evolute
and involute are equal curves.
Of Spirals.
188. A spiral is a curve described by a point which
moves along a right line, according to any law whatever,
the line having at the same time a uniform angular motion.
176
ELEMENTS OF THE
Let ABC be a straight
line which is to be turned
uniformly around the
point A. When the
motion of the line be-
gins, let us suppose a
point to move from A
along the line in the
direction ABC. When
the line takes the posi-
tion ADE the point will
have moved along it to some point as D, and will have
described the arc AaD of the spiral. When the line
takes the position AD'E f the point will have described
the curve AaDD\ and when the line shall have comple-
ted an entire revolution the point will have described the
curve AaDD'B.
The point A, about which the right line moves, is
called the pole ; the distances AD, AD f , AB } are called
radius-vectors, and if the revolutions of the radius-vector
are continued, the generating point will describe an in-
definite spiral. The parts AaDD'B, BFF'C, described in
each revolution, are called spires.
189. If with the pole as a centre, and AB, the distance
passed over by the generating point in the direction of the
radius-vector during the first revolution, as a radius, we
describe the circumference BEE ', the angular motion of
the radius-vector about the pole A, may be measured by
the arcs of this circle, estimated from B.
If we designate the radius-vector by u, and the measur-
ing arc, estimated from B, by t, the relation between u
DIFFERENTIAL CALCUXUS. 177
and t, may in general be expressed by the equation
u = at n ,
in which n depends on the laiv according to which the
generating point moves along the radius-vector, and a on
the relation which exists between a given value of u and
the corresponding value of t.
190. When n is positive the spirals represented by the
equation
u — at n ,
will pass through the pole A. For, if we make t = 0, we
shall have u = 0.
But if n is negative, the equation will become
u — at'",
or
u ~r>
in which we shall have
U — 00
for
t = o,
and u =
for
t = co:
hence, in this class of spirals, the first position of the
generating point is at an infinite distance from the pole :
the point will then approach the pole as the radius-vector
revolves, and will only reach it after an infinite number of
revolutions. x
191. If we make n = 1, the equation of the spiral be-
comes
u — at.
If we designate two different radius-vectors by v! and
u", and the corresponding arcs by t' and t' f , we shall have
v! — at' and u" = at''
178
ELEMENTS OF THE
and consequently
if
to the measur-
This spiral is
that is, the radius-vectors are proportioned
ing arcs, estimated from the point B.
called, the spiral of Archimedes.
192. If we represent by unity the distance which the
generating point moves along the radius-vector, during one
revolution, the equation
u~at,
will become
1 = at
or
lx±
a
t.
2t, and consequently
a =
But since t is the circumference of a circle whose
radius is unity, we shall have
1
a
193. If the axis BD, of
a semi-parabola BCD, be
wrapped around the circum-
ference of a circle of a
given radius r, any abscissa,
as Bb, will coincide with
an equal arc BV , and any
ordinate as ha, will take the
direction of the normal Ab'a' . \.. _..-«-''
The curve Bale' , described
through the extremities of the ordinates of the parabola, is
called the parabolic spiral.
The equation of this spiral is readily found, by observing
that the squares of the lines b'a f , c c' , &c., are propor-
tional to the abscissas or arcs BU, Be .
DIFFERENTIAL CALCULUS. 179
If we designate the distances, estimated from the pole
A, by u, we shall have Va! — u — r: hence,
(u — r) 2 = 2pt,
is the equation of the parabolic spiral.
If we suppose r = 0, the equation becomes
u 2 — 2pt.
If we make n— — 1, the general equation of spirals
becomes
u = at~ l , or ut — a.
This spiral is called the hyperbolic spiral, because of the
analogy which its equation bears to that of the hyperbola,
when referred to its asymptotes.
194. The relation between u and t is entirely arbitrary,
and besides the relations expressed by the equation
u = at?,
we may, if we please, make
t = loga.
The spiral described by the extremity of the radius-vec-
tor when this relation subsists, is called the logarithmic
spiral.
195. If in the equation of the hyperbolic spiral, we
make successively,
we shall have the corresponding values,
u = a, u = 2a, u = 3a, u = 4a,&c.
180
ELEMENTS OF THE
Through the
pole A draw AD
perpendicular to
AB, and make
it equal to a :
then through D
draw a parallel
to AB. From
any point of the
spiral as P draw PM perpendicular to AB, we shall
then have
PM = u sin MAP - u sin t.
If we substitute for u its value
Pilf =
smJ?
we shall have
Now as the arc t diminishes, the ratio of - — - will ap-
proach to unity, and the value of the ordinate PM will
approach to a or CM: hence, the line DC approaches
the curve and becomes tangent to it when t = 0. But
when t = 0, u = oc ; hence, the line DC is an asymptote
of the curve.
196. The arc which measures the angular motion of the
radius-vector has been estimated from the right to the left,
and the value of t regarded as positive. If we revolve
the radius-vector in a contrary direction, the measuring
arc will be estimated from left to right, the sign of t will
be changed to negative and a similar spiral will be de-
scribed. The line DC is an asymptote to the hyperbolic
spiral, corresponding to the negative value of t.
DIFFERENTIAL CALCULUS.
181
197. Let us now find a general value for the subtangent
of the spirals. The subtangent is the projection of the
tangent on a line drawn through the pole and perpendicular
to the radius-vector passing through the point of contact.
The equation of the spirals may be written under the
form
u = F{t\
in which we may suppose t the independent variable, and
its first differential constant.
Let AO =1 be the radius of
the measuring circle, PTa tan-
gent to the spiral at the point P,
and A T drawn perpendicular to
the radius- vector AP, the sub- /
tangent.
Take any other point of the
spiral as P', and draw AP' .
About the centre A describe the
arc PQ, and draw the chord PQ.
Draw also the secant PP' and
prolong it until it meets AT,
drawn parallel to QP, at T. T T
From the similar triangles QPP' t A T'P\ we have
PQ : QP' :: AT' : AP' ;
hence,
QP'
PQ
AP f
AT'
But when we pass to the limit, by supposing the point
P' to coincide with P, the secant VPP' will become the
tangent PT, and AT will become the subtangent AT.
16
182
ELEMENTS OF THE
> <£
But under this supposition
the arc NN f will become equal
to dt, the arc PQ to the chord
PQ (Art. 128), AP' to u, and
the line QP f to du.
To find the value of the arc
PQ> we have
1 : NN' : : AP . : arc PQ;
hence,
1 : c?i : : u : arc PQ,
and PQ = udt.
Substituting these values, and passing to the limit, we
have
du ■. u
~u~di~~lT : ,
hence, we have the subtangent
u 2 dt
AT
du
198. If we find the value of u 2 and du from the gen-
eral equation of the spirals
u = af",
we shall have
AT=—r+ l ,
n
DIFFERENTIAL CALCULUS. 1S3
In the spiral of Archimedes, we have
n = 1, and a= — ;
2*"
i 2
hence, AT= — .
2*-
If now we make t = 2x-= circumference of the mea-
suring circle, we shall have
AT — 2*- = circumference of measuring circle.
After m revolutions, we shall have
t~ 2«wr,
and consequently,
A T = 2 ?n 2 7r = m.2 m-* ;
that is, the subtangent, after "m revolutions, is equal to
m times the circumference of the circle described with
the radius-vector. This property was discovered by
Archimedes.
199. In the hyperbolic spiral n = — 1, and the value of
the subtangent becomes
AT=-a;
that is, the subtangent is constant in the hyperbolic spiral.
200. It may be remarked, that
AT _udt
~AP~~dhl
expresses the tangent of the angle which the tangent makes
with the radius-vector.
184 ELEMENTS OF THE
In the 'logarithmic spiral, of which the equation is
t=]ogu,
we have dt — A — ;
u
. A T udt .
hence, ——=——— A;
AP du
that is, in the logarithmic spiral, the angle formed by the
tangent and the radius-vector passing through the point of
contact, is constant; and the tangent of the angle is equal
to the modulus of the system of logarithms. If -f is the
Naperian logarithm of u, the angle will be equal to 45°.
201. The value of the tangent in the spirals is
202. To find the differential of the arc, which we will
represent by z, we have
PP^^qF /2 +QP 2 ;
or, by substituting for QP f and PQ their values, and
passing to the limit, we have
<k = Vdu 2 + ic 2 dt 2 .
DIFFERENTIAL CALCULUS.
185
203. The differential of the
area ADP when referred to the
polar co-ordinates, is not an ele-
mentary rectangle as when re-
ferred to rectangular axes, but /
is the elementary sector APP 1 . \
The limit of the ratio of the
sector APP' with the arc NN' t
will be the same as that of
either of the sectors APQ,
AP"P' between which it is
contained, with the same arc
NN'. Hence, if we designate
the area by s, and pass to the limit, we shall have
u 2 dt
ds
di
APxPQ _u 2
VNN' ~ 2 ° r
ds
which is the differential of the area of any segment of a
spiral.
Of Tangent Planes and Normal Lines to Surfaces.
204. Let u = F(x,y,z) = Q t
be the equation of a surface.
If through any point of the surface two planes be passed
intersecting the surface in two curves, and two straight
lines be drawn respectively tangent to each of the curves,
at their common point, the plane of these tangents will be
tangent to the surface.
205. Let us designate the co-ordinates of the point at
which the plane is to be tangent by a/ r , y", z" .
16*
186 ELEMENTS OF THE
Through this point let a plane be passed parallel to the
co-ordinate plane YZ. This plane will intersect the
surface in a curve. The equations of a straight line tan-
gent to this curve, at the point whose co-ordinates are
a/', y" , z ff , are
* = *"=«", y-y" = ^(*-*");
dz
the first equation represents the projection of the tangent
on the co-ordinate plane ZX, and the second its projec-
tion; on the co-ordinate plane YZ (An. Geom. Bk. IX.
Art. 70).
Through the same point let a plane be passed parallel to
the co-ordinate plane ZX, and we shall have for the
equations of a tangent to the curve
y = y" = h", cc - a/' '== ^(z - z")
The coefficient -j- represents the tangent of the angle
which the projection of the first tangent on the co-ordinate
plane YZ makes with the axis of Z ; and the coefficient
doc
— represents the tangent of the angle which the projection
CbZ
of the second tangent on the plane ZX makes with the
axis of Z (An. Geom. Bk. VIII, Prop. II).
. But these coefficients may be expressed in functions of
the surface and the co-ordinates of its points. For, we
have
u = F(x,y,z)=0,
and if we suppose x constant, we shall have (Art. 87)
du — -r- dy + — dz = ;
dy J dz
DIFFERENTIAL CALCULUS. 187
du
. dy dz
hence, -~- = — — :
dz du
dy
and if we suppose y constant, we shall find, in a similar
manner,
du
dx _ dz
dz du
dx
hence, the equation of the projection of the first tangent on
the plane of YZ becomes
du
dz
y-?/' =
■ g* *">;
dy
and the equation of the
projection of the second tangent
on the plane of ZX is
du
x-x" =
• Tz (z z")
■ du {Z Z h
dx
The equation of a plane passing through the point whose
co-ordinates are a/' ', y", z u is of the form
A(x -x") + B(y - y") + C(z - *") = 0,
C
in which— —will represent the tangent of the angle which
the trace on the co-ordinate plane YZ makes with the
Q
axis of Z, and — T" tne tangent of the angle which the
A.
trace on the plane of ZX makes with the axis of Z.
1S8 ELEMENTS OF THE
But since the tangents are respectively parallel to the
co-ordinate planes YZ, ZX, their projections will be
parallel to the traces of the tangent plane : therefore,
du
du
€
dz
B~
du'
dy
du
C
dz
A~
~~du~'
dx
hence, — B =
% n.
~~du~ '
dz
du
hence, — A =
doc c
du
dz
Substituting these values of B and A in the equation
of the plane, and reducing, we find
which is the equation of a tangent plane to a surface at a
point of which the co-ordinates are a/', y !l , z" .
206. A normal line to the surface being perpendicular
to the tangent plane at the point of contact, its equations
will be of the form
du du
x -x" = ~{z - z"\ y-y = -^L(z- z"\
du du
dz dz
ELEMENTS
INTEGRAL CALCULUS
Integration of Differential Monomials.
207. The Differential Calculus explains the method of
finding the differential of a given function. The Integral
Calculus is the reverse of this. It explains the method
of finding the function which corresponds to a given
differential.
The rules for the differentiation of functions are explicit
and direct. Those for determining the integral, or func-
tion, from the differential expression, are less direct and
are deduced by reversing the process by which we pass
from the function to the differential.
208. Let it be required, as a first example, to integrate
the expression.
x m dx.
We have found (Art. 32), that
d{x m + l )=(m + l)x m dx,
dx m+l ,/tf OT+1 \
whence, xdx — — - = a ( — — - ),
m + \ \wi + l/
190 ELEMENTS OF THE
x m+l
and consequently — ,
1 J m + l'
is the function of which the differential is x n dx.
The integration is indicated by placing the character /
before the differential which is to be integrated. Thus,
we write
fx m dx
m + 1
from which we deduce the following rule.
To integrate a monomial of the form x m dx, augment
the exponent of the variable by unity, and divide by the
exponent so increased and by the differential of the
vaiiable.
209, The characteristic / signifies integral or sum.
The word sum, was employed by those who first used the
differential and integral calculus, and who regarded the
integral of
x m dx
as the sum of all the products which arise by multiplying
the mth power of x, for all values of x, by the con-
stant dx.
dx
210. Let it be required to integrate the expression — .
x
We have, from the last rule,
f—'^fdxx- 3
J x 3 J
x~ 3+l
x~ 2 _
i
-3 + 1
-2~
2^*
a similar manner,
we
find
2
fdx\/oc 2 = fx~*
dx :
2
x1 + l
5
x's
6
3x3
f«
~5~*
INTEGRAL CALCULUS. 191
211. It has been shown (Art. 22), that the differential
of the product of a variable multiplied by a constant, is
equal to the constant multiplied by the differential of the
variable. Hence, we may conclude that, the integral of
the product of a differential by a constant, is equal to the
constant multiplied by the integral of the differential :
that is,
x m + x
fax m dx — afx m dx — a
Wi + 1
Hence, if the expression to be integrated have one or
more constant factors, they may be placed as factors with-
out the sign of the integral.
212. It has also been shown (Art. 22), that every con-
stant quantity connected with the variable by the sign
plus or minus, will disappear in the differentiation ; and
hence, the differential of a -j- x m , is the same as that of
x m ; viz. mx m ~ x dx. Consequently, the same differential
may answer to several integral functions differing from
each other in the value of the constant term.
In passing, therefore, from the differential to the integral
or function, we must annex to the first integral obtained,
a constant term, and then find such a value for this term
as will characterize the particular integral sought.
For example (Art. 94),
-f. = a, or dy = adx.
ax
is the differential equation of every straight line which
makes with the axis of abscissas an angle whose tangent
is a. Integrating this expression, we have
192
ELEMENTS OF THE
fdy-afdx,
or
y = ax,
or finally,
y = ax -\-C.
If now, the required line is to pass through the origin
of co-ordinates, we shall have, for
x = 0, y = 0, and consequently, C — 0.
But if it be required that the line shall intersect the axis
of Y at a distance from the origin equal to + b, we shall
have, for
x = 0, y = -f- b, and consequently, C = -f b ;
and the true integral will be
y = ax -f 6.
If, on the contrary, it were required that the right line
should intersect the axis of ordinates below the origin, we
should have, for
x = 0, y — —b, and consequently, C = —b ;
and the true integral would be
y = ax — b.
213. It has been shown (Art. 95), that
xdx + ydy ==
is the differential equation of the circumference of a circle.
By taking the integral, we have,
/ xdx-\- fydy = 0, or x 2 -\-y 2 = 0,
or finally, x 2 -f y 2 + C = 0.
INTEGRAL CALCULUS. 193
If it be required that this integral shall represent a given
circumference, of which the radius is R, we shall have,
by making
x = 0, y 2 =-C = R\
and hence, C= — R 2 ;
and consequently the true integral is
ar> 4. f -R 2 = 0, or Z* + f = R\
The constant C, which is annexed to the first integral
that is obtained, is called an arbitrary constant, because
such a value is to be attributed to it as will cause the
required integral to fulfil given conditions, which may be
imposed on it at pleasure.
The value of the constant must be such, as to render
the equation true for every value which can be attributed
to tJic variables.
214. There is one case to which the formula of Art. 208
does not apply. It is that in which in = — 1 . Under this
supposition,
r m1 x m +' x~ l + l x° 1
/ x ax = = = — a= — = oo.
J m-fl -1 + 1
But when ?n = — 1,
.dx
fx m dx = fx~ l dx = f
x
and f— = logx+C. (Art. 57).
./ x
215. Since the differential of a function composed oi
several terms, is equal to the sum or difference of the diffe-
rentials (Art. 27), it follows that the integral of a differen-
17
194 ELEMENTS OF THE
tial expression, composed of several terms, is equal to the
sum or difference of the integrals taken separately. For
example, if
bdx
du-~
- adx
X
— -\-x-\/x dx,
we
have
fdU:
—J (adx —
bdx
-f- x <\/x~dx),
ani
U —
■x*+C.
216. Every polynomial of the form
{a -f- bx -}- ex 2 4- &cc.) n dx,
in which n is a positive and whole number, may be inte-
grated by the rule for monomials, by first raising the poly-
nomial to the power indicated by the exponent, and then
multiplying each term by dx.
If, for example, we make n — 2, and employ but two
terms, we have
f(a + bxfdx —f(a 2 dx + 2abxdx + b 2 x 2 dx),
7,2 3
= a 2 x + abx 2 + ?^-+C.
Integration of Particular Binomials.
217. If we have a binomial of the form
du — {a-\- bx n ) m x n ~ l dx ;
that is, in which the exponent of the variable without the
parenthesis is less by unity than the exponent of the vari-
able within, we may make
INTEGRAL CALCULUS. 195
a + bx n = 2, which gives
dz
nbx n l dx = cfc, or a?" ' c&r = — T ;
no
dz z m + l
whence du — z m ^-, or u = — -r- ;
no {m-\-\)no
and consequently
u . Ja + borr^ c
(m-j- l)nb
Hence, the integral of the above form, is equal to the bino-
mial factor with its exponent augmented by unity, divided
by the exponent so increased, into the exponent of the vari-
able within the parenthesis into the coefficient of the
variable.
For example,
f(a + 3a: 2 ) 3 xdx = <1±1^1 4. Q; and
- m 1
f(a + bx 2 ) 2 mxdx =z—j-(a-{- bx 2 ) 2 + C.
218. A transformation similar to that of the last article
will enable us to integrate certain differentials correspond-
ing to logarithmic functions. If we have an expression of
the form
adx
du
bx
dz
make c-\-bx = z, which gives dx = —, and by sub-
stituting, we have
/adx Cadz a C dz a , ~
196 ELEMENTS OP THE
and by substituting for z its value
In a similar manner, we should find
C at
J c—
4.iog(fr-foo+c,
bx b
in which the integral is negative, since d( — x) = — dx.
We can find, in a similar manner, the integral of e very-
fraction of which the numerator is equal to the differential
of the denominator, or equal to that differential multiplied
by a constant.
If, for example, we have
, _ {b + %cx) mdx #
a-\-bx -\- ex 2
make a-\-bx-\- ex 2 = z, which gives, bdx + 2cxdx = dz,
and hence,
mdz
du = , or u — mlogz,
z ' ° '
and by substituting for z its value
u == mlog(<2 -h ia? -f ca, 12 ).
Of Differentials whose Integrals are expressed by
the Circular Functions.
219. We have seen, Art. 71, that if x designates an arc
and u the sine, to the radius unity, we shall have
du
INTEGRAL CALCULUS. 197
du
hence, / , ~x-\- C ;
J Vl-U 2
or adopting the notation of Art. 72,
du
/
Vl-u 2
= sin u-\- C
If the arc expressed in the second member of the equa-
tion be estimated from the beginning of the first quadrant,
the sine will be 0, when the arc is ' 0, and we shall have,
for u = Q
du
/■
■0, and consequently C = 0,
and under this supposition, the entire integral is
/
du
Vl-u 2
To give an example, showing the use of the arbitrary
constant, let us suppose thatgthe arc which is to be ex-
pressed by the second member of the equation, is to be
estimated from the beginning of the second quadrant. This
supposition will render
du
/
VT^u~ 2
= for u = 1 .
But when u=l, sin 1 w = — ?r ; hence,
-L*+C=0, or C=——ir:
2 ' 2
and we have, for the entire integral, under this supposition,
/
du . , 1
= SWT l U -*\
Vl-u 2 2
17*
198
ELEMENTS OF THE
220. It frequently happens that we have expressions to
integrate of the form
dz
Let us suppose, for a moment, that a is the radius of a
circle, and z the sine of any arc of the circle ; and that u
is the sine of an arc containing an equal number of degrees
in a circle whose radius is unity : we shall then have,
u :
a : z;
hence,
u =
and consequently,
du
/du /•
and
r dz
a
, dz
du — — ;
a
v
J Va
dz
hence.
/du f ^ dz . _j z
the arc being still taken in a circle whose radius is unity.
221. We have seen (Art. 71), that if x designates an
arc, and u the cosine, to the radius unity, we shall have
du
dx =
Vi
hence,
or adopting the notation of Art. 72,
t du -l , ^
/ , = cos l u+C.
INTEGRAL CALCULUS. 199
If the arc be estimated from the beginning of the first
quadrant, it will be equal to — tt for u — 0; hence, the
2 1
first member of the equation becomes equal to — n when
I 2
u = 0. But under this supposition, cos -1 u— — *■ : hence,
C = 0, and the entire integral is
du
f
Vl-u 2
222. By a method analogous to that of Art. 220, we
should find
r dz _, z
J 7 — COS ,
J Va 2 -z 2 o
the arc being estimated to the radius unity.
223. We have seen (Art. 71), that if x represents an
arc, and u its tangent, to the radius unity, we have
7 du
ax = ;
1 + u 2 '
hence, / r = x + C :
J l + u 2
or, adopting the notation of Art. 72,
du
/
\ + u 2
tang 1 u-\-C.
If the arc is estimated from the beginning of the first
quadrant, we shall have
/* du
tang _1 = 0, when /- 5 = 0; hence, C = 0,
J 1 + w
200 ELEMENTS OF THE
and the entire integral is
/
du
tang" u.
l + u 2
224. To integrate expressions of the form
dz
let us suppose for a moment that a is the radius of a circle,
and z . the tangent of any arc, and that u is the tangent
of an arc containing an equal number of degrees in a circle
whose radius is unity : we shall then have
1 : u : : a : z ;
z z dz
hence, u = — , u 2 = — -, and du = « — .,
a a 1 a
and consequently,
/du r dz U z
hence, by dividing by a,
/,
dz 1 , z
= — tang — .
d 2 + z 2 a "a
the arc being estimated to the radius unity-
225. We have seen (Art. 71), that if x represents an
arc, and u the versed-sine, to the radius of unity, we have
7 du
ax = . — ;
V2u — u 2
hence, / . = x = ver-sin" 1 u + C :
J V2u — u 2
INTEGRAL CALCULUS. 201
and if the arc is estimated from the beginning of the first
quadrant, C = 0, and we shall have
/du
V2u — u
226. To integrate an expression of the form
dz
V~2
az — z
Suppose, as before, a to be the radius of a circle, and
we shall have (Art. 224),
dz
a
a
and consequently,
du r dz
/du r
V2u-u* ~ J V2*
\az
to the radius unity.
Integration by Series.
227. Every expression of the form
Xdx,
in which X is such a function of x, that it can be developed
in the powers of x, may be integrated by series.
For, let us suppose
X == Ax a + Bx h + Cx e + Dx d -f &c, then,
Xdx = Ax a dx + Bx h dx -f Cx c dx + Dx d dx + &c.,
fXdx=^—x a + l -\--^-x b + l + — ^ + 1 -f-5-^+ 1 + &c.
J fl+1 b + l c+l ^d+i ^
202 ELEMENTS OF THE
Hence, the integration by scries is effected by develop-
ing the function X in the powers of x, multiplying the
series by dx, and then integrating the terms separately.
(J /Y>
Let us take, as a first example, — ,
a-{- x
— = dx x == dx(a + a?) -1 ,
a -\- x a + x
a a 4 a 6 ar
and consequently,
ydx /V 1 7 xdx x 2 dx x 3 dx , D \
— — =± I (—dx 2-H 3 — T -4-&c.):
a 4~ x J \ a a a ar J
and integrating each term separately, we obtain
/aX X XX X , p i /^r
/dx
— — log (a + x) (Art. 218),
a ~r~ X
we have
X u X
X X 2
log (a + x) = — - " + -^o - f4 + &c + C.
sv ; a 2a 2 3a 3 4a 4
To determine the value of the constant, make x= 0,
which gives
log a = 4- C, or C = log a ; hence,
log( a + .) = loga + |-£- 2 + g-£ ; +&c. )
log(«+«) - log«=log (l + £) = | - ^ + ^- &c,
a result which agrees with the development in Art. 58.
INTEGRAL CALCULUS. 203
dx
l + x 2
228. Let us take, for a second example
We have, dx _ =dx{l + x 2 )- 1 ;
l^j-ir
and by developing and integrating,
/* dx x 3 , x 5 x 1 , - , ^
/ — x h &c + C.
J 1 + x 2 3 5 7
When we make a? = 0, the arc is ; hence,
tang #=:a? -j h&c.:
6 3 5 7
a result which corresponds with that of Art. 78.
229. If, in the expression — | — - , we place x 2 in the
1 — p" x
first term of the binomial, and then develop the binomial
x 2 + 1 , we obtain
rj^ = /74-^ + i-i.+&c.)<fo;
J .'<r + 1 J \xr or x b x b J
and by integrating, we have
tang l x— -f —3 — — ^ + &c. + C.
x 3x° 5x 5
To find the value of the constant C, let us make the
arc = 90° = -«■. This supposition will render the tan-
gent x infinite, and consequently every term of the series
will become 0, and the equation will give
— *• = <)+ C, or C = — sr.
2 ' 2
204 ELEMENTS OF THE
t
Making this substitution, we have, for the true integral,
/-.
dx . _! 1 1.1 1
tang X x = —-7T f- _ _j- & c .
and 2 i i '> are > as tne y sh 011 ^ be, essentially the same.
a? 2 +l to 2 a* 3cc 3 5x 5
230. The two series, found from the expressions — ■ — -
dx
x 2 -\-l
For, the tangent of an arc multiplied by its cotangent,
is equal to radius square or unity (Trig. Art. XVIII).
Hence, if we substitute for x, in the first series, -, we
x
shall have, for the complemental arc,
tana; 1 -— = ^4-
x x 6 x
Q..3 5a fi >
and subtracting both members from -v,
1 ,1,1 11 1
— 7F — tang — = tang x~--7r .4- ___ -__j_ & c .
2 b x ° 2 a? 3^ 3 5a? 5
231. We have found (Art. 71),
dx
sin
'^/vCT^ 1 -^'-
and by developing, we find
(l-a*)~* = 14-— ** + — .— x* 4-i-.JL.JL ajp+dcc;
v ' 224 246
multiplying by c&r, and integrating, we obtain,
. . , 1 x 3 1 3 x 5 , 1 3 5 a? 7 , &
sm a? = a? 4 • • — • h &c,
232 452 4 6 7
INTEGRAL CALCULUS. 205
the constant being when the arc is estimated from the
beginning of the first quadrant.
If we take the arc of 30°, the sine of which is equal
to half the radius (Trig. Art. XIV), we shall have
• -lonn 1,111,13 1113 5 11.
sin * 30° =- + -.-,-5^ — .-.-. — + -.-.-. -.-r-4- &c.;
223 2 3 2 45 2 5 2467 2 7
hence,
n . _, onn n( \ 1.1.1 1.3.1.1 1.3.5.1.1 , - \
w = 6sin *30° = 6(-H ,-f -A =+&c.J,
V2^ 2.3.2 3 ^2.4.5.2 5 ' 2.4.6.7.2' )
and by taking the first ten terms of the series, we find
*■ = 3. 1415962,
which is true to the last decimal figure, which should be 5.
232. We will add a few more examples.
1. To integrate the expression - —
^ Vx-x 2
By making s/~x~ = u, we have
dx dx 2 du
V x — x 2 VxVl—x V 1 — u 2
But from the last series
r 2du / lv? , 1 3m 5 , 1 3 5u 7 \
J Vl-u 2 \ 2 3 245 2467 / T '
hence
JV^? V 23 24 5 T 246,7 ;vt
2. da?-*/ 2a# - a? 2 = {2a)*x*dx(l - — ) ¥ .
18
206 ELEMENTS OF THE
But
V 2a)
} n'1 1 x 1 1 x 1.1.3 a! 3 .
2 — 1 ^-g <
2aJ 2 2a 2*4 4a 2 2.4.6 8a 3
£ 7
|__1_ j?_^___J_ i_ 2 a? 8
2 ' 5 2a 2 '4 ' 7 4a 2
hence
Jdxy 2ax — x 2 — ( — x
1 1 3 2 tf 2 \
. — . — . r-— &c. ) *J2a-\- C:
2 4 6 9 8a 3 ) y J
and consequently
J_ _1_ 3_ J_jr 3
2 ' 4"6"9 8(
g
^- &c.)2a?</2aff-h C,
If the radius of a circle be represented by a, and the
origin of co-ordinates be placed in the circumference, the
equation will be (An. Geom. Bk. Ill, Prop. I, Sch. 3),
y 2 — 2ax — x 2 \ hence y—yllax — x 2 ,
and consequently (Art. 130)
dx V 2 ax — x 2 = ydx
is the differential of a circular segment.
If we estimate the area from the origin, where x =
we shall have C — 0. If then we make x = a, the series
will give the area of one quarter of the circle, if we make
x = 2a, of the semicircle.
J V1 +
lx 3 1.3a? 5 1.3.5a? 7 , s
^^-23- + 2^y-2^6Y +&C - + a
INTEGRAL CALCULUS. 207
f d * - !x l 13 L3 - 5 fee I C
J V^^lT 2.2^ 2.4.4a; 4 2.4.6.6* 6 ° 6C - + °'
Integration of Differential Binomials.
234. Differential binomials may be represented under
the general form
v_
x m ~ x dx(a + bx n )%
in which, without affecting the generality of the expres-
sion, m and n may be regarded as entire numbers, and n
as positive.
For, if m and n were fractional, and the binomial of
the form
JL 1 L
x 3 dx{a + bx*y
make x = z 6 , that is, make the exponent of z the least
common multiple of the denominators of the exponents
of x, and we shall then have
x' 6 dx(a + bx' 2 ) q — 6z 7 dz(a + bz 3 )* t
in which the exponents of the variable are entire.
If n were negative, we should have,
x m ~ l dx(a 4- bx~ n y,
and by making x — — f we should obtain
p_
z- m - l dz(a + bz n y,
the same form as before.
208 ELEMENTS OF THE
Furthermore, the binomial
p
x m ~ l dx(ax r + bx n y
may be reduced to the form
x m * dx(a + bx n ~ r y,
by dividing the binomial within the parenthesis by x r , ant ;
pr
multiplying the factor without by x q .
235. Let us now determine the cases in which the
binomial x ni ~ l dx{a -f hx n y has aai exact integral.
Make a -f bx n ~ z q ; we shall then have*
m
n z q — a . ; , £ „ m (z*—a\ n
x n = —j-— f (a + baT)* =:z p , x m = (——) t
and by differentiating,
no
lience
x m ~ l dx = rZ-^lZ-Z) dz ;
nb \ b J
l dx(a -f bx n y = ±s?+^dz^-j^j ,
which will have- an exact integral in algebraic terms when
— is a whole number and positive (Art. 216). If — is
n n
negative see Art. 260.
Hence, every differential binomial has an exact inte-
gral, when the exponent of the variable without the paren-
thesis augmented by unity, is exactly divisible by the
exponent of the variable within.
Thus, for example, the expression
p_
x 5 dx{a+bx 2 y
INTEGRAL CALCULUS. 209
has an exact integral. For, by comparing it with the
general binomial, we find
171
m~Q, ?i — 2, and consequently, — = 3,
and the transformed binomial becomes
26 \ b J
236. There is yet another case in which the binomial
zf n - l doc(a-\-bx n )i has an exact integral.
If we multiply and divide the quantity within the paren-
thesis by x n , we have
2. L
ac m ~ l dx(a + bx n Y = x m - l dx[(ax- n -f &)#"]«
p_ np
=zx m ~ l dx{ax~ n -[-bYx^
= a? « dx{ax~ n -\- b)" y
Now, if we add unity to the exponent of x without the
parenthesis, and divide by — n, the quotient will be
— ( (- — )> and the expression will have an exact
integral when this quotient is a whole number (Art. 235).
Hence, every differential binomial has an exact integral,
when the exponent of the variable without the parenthesis
augmented by unity and divided by the exponent of the
variable within the parenthesis, plus the exponent of the
parenthesis, is an entire number.
237. The integration of differential binomials is effected
by resolving them into two parts, of which one at least has
a known integral.
We have seen (Art. 28) that
d{uv) = udv + vdu,
2J0 ELEMENTS OF THE
whence, by integrating,
uv =fudv -\-fvdu,
and, consequently,.
fudv — uv —fvdu.
Hence, if we have a differential of the form Xdx, in
which the function X may be decomposed into two factors
P and Q, of which one of them, Qdx, can be integrated,,
we shall have, by making / Qdx — v and P = u y
JPQdx = Pv-fvdP,
in which it is only required to integrate the term fvdP*
238. To abridge the results, let us write p for — , in
which case p will represent a fraction, and the differential
binomial will take the form
dx(a+bx n )
If now, we multiply by the two factors x n and. x \ the
value will not be affected, and Ave obtain
x m - n x n - l dx(a + bx n y.
Now, the factor x n ~ 1 dx(a-\- bx n ) p is integrable, whatever
be the value of p (Art. 217) ;, and representing this factor
by dv, we have
V -~T , i \ i> > and u=zcc >
(p+l)nb
and, consequently,
oc m - n (a-\-bx n Y* 1 m-n
{p+l)nb (p+l)nb
fx m - l dx{a + bx n ) p =
■fx m - n - l dx{a + bx n y+ l .
INTEGRAL CALCULUS. 211
But, fcc m - n - l dx(a + bx n ) p+l =
Jx m - n - l dx{a + bx n ) p (a + bx n } =
afx m ~ n - l dx(a + bx n ) p + bfx m ~ l dx(a + for")* ;
substituting this last value in the preceding equation, and
collecting the terms containing the integral
Jx m ~ x dx(a + bx n y,
we have
x m - n (a + bx n ) p+l - a(m - n) f x m ~ n ~ x dx(a + bx) p m
{p+l)nb '
whence,
formula (A.) fx m ~ 1 dx(a + bx n ) p =z J\.
x m - n (a + bx n ) p+l -a{m- n) J x m ~ n ~ x dx(a -f bx n y
b(pn-\-m)
This formula reduces the differential binomial
fx m - l dx(a + bx n ) p to that of fx m - n - l dx(a + ta n )* ;
and by a similar process we should find
fx m ~ n - 1 dx(a-\-bx n y to depend on fx m -* n ~ x dx(a + bx n y ;
and consequently, each process diminishes the exponent
of the variable without the parenthesis by the exponent
of the variable within.
After the second integration, the factor m — n, of the
second term, will become m — 2n; and after the third,
m — 3n, &c. If m is a multiple of n, the factor m — ?i,
m — 2n, m — Sn, &c, will finally become equal to 0, and
then the differential into which it is multiplied will disap-
21 2 ELEMENTS OF THE
pear, and the given differential will have an exact integral,
which corresponds with the result of Art. 235.
239. Let us now determine a formula for diminishing
the exponent of the parenthesis.
We have
fx m - x dx(a + bx n ) p = foc m - l dx{a + bx n ) p ~\a + bx n ) =
afx m - l dx(a + bx n ) p ~ l -f bfx m+n ~ l dx{a + bx n ) p ~\
Applying formula (A) to the second term, by placing
m + n for m, and p — 1 for p, we have
fx^ n - l dx{a + bx n y- l =:
x m (a + bx n Y - amfx m - l dx(a + bx n y~ l
b(pn-\-m)
Substituting this value in the last equation, we hare
H formula (B) , . .fx m ~ l dx(a + bx n ) p =
x m (a + bx n y+pnafx^- x dx{a + bx n y~ l
pn + m
which, diminishes the exponent of the parenthesis by unity
for each integration,
240. By means of formulas (A) and (B), we reduce
fx m - l dx(a + bx n ) p to fx m ~ rn - l dx(a-{-bx n y- 8 ;
rn being the greatest multiple of n which can be taken
from m—1, and s the greatest whole number which can
be subtracted from p.
For example, fx 7 dx(a -f bx' 3 y is reduced, by formula
(A), to
fx*dx(a-{-bx 3 y, and then to / xdx (a -f bx 3 ) 2 :
INTEGRAL CALCULUS. 213
and by formula (B) fxdx(a-\- bx 3 ) 2 , reduces to
fxdx(a -f- bx 3 ) 2 , and finally to fxdx(a + bx 3 )
241. It is evident that formulas (A) and (B) will only
diminish the exponents m — 1 and p, when m and p arc
positive. We will now determine two formulas for dimin-
ishing these exponents when they are negative.
We find from formula (A)
fcc m - r - l dx(a + bx n ) p =
x m ~"{a + bx n ) p+l — b{m -f np)fx m - x dx(a + b af 1 ) 7 * t
a(m — n)
and placing for m, — m-\-n, we have
formula (C) fx- m ~ l dx(a-\-bx n ) p = O
x~ m {a + bx n ) p+1 + b(m — n — np)fx- m + n - l d^(a + fo n ) p
— awi
in which formula, it should be remembered that the nega-
tive sign has been attributed to the exponent m.
242. To find the formula for diminishing the exponent
of the parenthesis when it is negative.
We find, from formula (B),
fx m - l dx(a + bxy- l =
x m (a + bx n ) p — (m + np)Jx n ~ x dx{a + bx n ) p
pna '
writing for p, — p + 1, we have
formula (D) fx m - l dx(a -f &c*)- p = '1)
x m (a + bx n )- p + l - (m + n - np)fx m ~ 1 dx(a + bx n )~ p+l
(p— \)na
214 ELEMENTS OF THE
This formula does not apply to the case in which p = 1.
Under this supposition, the second member becomes infi-
nite, and the differential becomes that of a transcendental
function.
243. It is sometimes convenient to leave the variable in
both terms of the binomial. We shall therefore determine
a particular formula for integrating the binomial
x q dx
x q {2ax — x 2 ) ^dx~
V2i
„2
i CIX — X
This binomial may be placed under the form
fx 1 *dx(2a-x)~ 2 ,
and if we apply formula (A), after making
™ = q+— , ri-=h JP=-— , a = 2a, Z>=-.1,
we shall have
fx ~dx(2a—x) 2 =
q q J y > '
and if we observe that
i .i 3 i
q — — q-l -i- ? — — 7 — 1 — —
x 2 =x X 2 X * —X X 2 ,
and pass the fractional powers of x within the parentheses,
we shall have
E formula (E) f— r * — —
x q ~W2ax-x 2 (2q-l )a C x q ~ l dx
q q J V2UX-X 2 '
INTEGRAL CALCULUS. 215
which diminishes the exponent of the variable without the
parenthesis by unity. If q is a positive and entire num-
ber, we shall have, after q reductions
/dx x
/n 2 =s ver-sin -1 — . (Art. 226).
V2ax — x 2 a
Integration of Rational Fractions.
244. Every rational fraction may be written under the
form
Px n ~ l + Qx n ~ 2 + Rx + S ,
Fx n + QV- 1 +R'x-%S' '
in which the exponent of the highest power of the varia-
ble in the numerator, is less by unity than in the denomi-
nator. For, if the greatest exponent in the numerator was
equal to or exceeded the greatest exponent in the denomi-
nator, the division might be made, giving cue or more
entire terms for a quotient and a remainder, in which the
exponent of the leading letter would be less by at least
unity, than the exponent of the leading letter in the divisor.
The entire terms could then be integrated, and there
would remain the fraction under the above form.
Place the denominator of the fraction equal to : that
is, make
Fx n +Q'x n - 1 R'x+S' = 0,
and let us also suppose that we have found the n binomial
factors into which it may be resolved (Alg. Art. 264).
These factors will be of the form x — a, a? — &,#—- c,
x — d, &c. Now there are three cases :
216 ELEMENTS OF THE
1st. When the roots of the equation are real and
unequal.
2d. When they are real and equal.
3d. When there are imaginary factors.
We will consider these cases in succession.
1st. When the roots are real and unequal.
adx
245. Let us take, as a first example,
Hi — CO
By decomposing the denominator into its factors, we
have
adx adx
x 2 — a 2
(x — a)(x -\- a)'
and we may make
adx
(x — a) (x + <
»)
= ( A + B
■ \x — a x -\- a
dx,
in which A and B are constants, whose values are yet to
be determined. In order to determine these constants,
•let us reduce the terms of the second member of the
equation to a common denominator ; we shall then have
adx {Ax + Aa + Bx — Ba) dx
(x — a){x J r a) (x — a)(x + a)
In comparing the two members of the equation, we find
a — Ax -\~ Aa + Bx — Ba ;
or, by arranging with reference to x,
(A + B)x + (A-B-l)a = 0.
But, since this equation is true for all values of x, the
INTEGRAL CALCULUS. 217
coefficients must be separately equal to (Alg. Art. 208) :
hence
A + B = 0, and (A-B-l)a = 0,
which gives
A ~2' ■- 2'
Substituting these values for A and B, we obtain
adx -odx ~_dx
x 2 — a 2 x — a x-\-a*
and integrating, we find (Art. 218)
I A = 1 los(x - a) - ¥°° {x + a) + c '
and, consequently,
T ft3 H - bX 2
246. Let us take, as a second example, — ; ~dx.
arx — or
The factors of the denominator are x and a 2 — x 2 ; but
a 2 — x 2 = (a + x) (a — x) :
hence, the given fraction becomes
a?(a — a?) (a -{-a?)
Let us now make
a 3 4-ba? A . B C
x(a — x)(a + x) x a — x a-\-x
19
218 ELEMENTS OF THE
reducing the terms of the second member to a common
denominator, we have
a 3 + bx 2 Aa 2 - Aa?+ Bax + Bx 2 + Cax -Cx 2
x(a — x) (a + x) x(a — x){a J r-x)
and, comparing the like powers of x (Alg. Art. 208),
B-A-C^b, Ba+Ca = 0, Aa 2 = a 3 .
From these equations, we find
A = a, B=«±±, C=- a -±*,
2 ' 2 '
and substituting these values, we obtain
a 3 + bx 2 7 d x , a + b 7 a -f b 7
— ~^rdx~a 1 - -ax - -ax ;
a z x — x 6 x 2{a — x) 2(a-\-x)
and integrating (Art. 218),
Ca 3 + bx 2 , , a + b, (
-^log(a + X )+C
= a log a; — ^— — [log(a - x) + log(a + x)] + C
= alogx -log(a — x) {a + x) + C
= alogx log(a 2 — x 2 ) + C
— alogx — (a + 2>)log <\/a 2 — x 2 +C.
247. Let us take, for a third example, -5 — ~- -dx.
INTEGRAL CALCULUS, 2K*
Resolving the denominator into the two binomial factors
(Alg. Art. 142), (a? — 2), (a?— 4), we have
3^-5 A , B .
H -, nence
x 2 — 6x + 8 x — 2 ^-.4'
3x-5 _ Ax-4A + Bx-2B
x 2 - 6a7+8 ~~ # 2 -6^ + 8 '
and by comparing the coefficients of x, we have
-5= -4 A- 2B, 3 = A + B,
which gives
B = — , A = — -,
2' 2'
and substituting these values, we have
dx
J x 2 — 6x4-8 ' 2 J x — -2^~ 2 J x —
4
= ^log(x-±)-±-log(x-2)+C.
"248. Let us take, as a last example,
xdx
aP + ^ax — b*'
Resolving the equation
a? + 4«a? — #=«,
■we find
x=—2a4- V4a 2 + £ 2 , #= — 2a- V4a 2 + 6 2 ,
and consequently, for the product of the factors,
(x+2a+ V4a 2 +b 2 )(x4-2a- ^±a 2 4-b 2 )=x 2 4-4:ax-fi l .
220 ELEMENTS OF THE
To simplify the work, represent the roots by — K and
— L, and the factors will then be
cc + K, x + L,
and we shall have
1 — : hence
a? + 4 ax — b 2 x + K x + L
x Ax + AL + Bx + BK
3? + Aax — b 2 ~~ x 2 + 4:ax—l? ~~
whence,
AL + BK = 0, A + B=l,
and, consequently,
K B= L
K-L 7 K-L
hence,
/
K log(a.+/0--A-log(a ? +L)+C.
x 2 +±ax-b 2 K-L &v ' ' K-L
249. In general, to integrate a rational fraction of the
form
Par-'+Qx™-* +Rx+S
x m +QV- 1 .. .. -fi^+S' °°'
1st. Resolve the fraction into m partial fractions, of
which the numerators shall be constants, and the denomi-
nators factors of the denominator of the given fraction.
2d. Find the values of the numerators of the partial
fractions, and multiply each by dx.
INTEGRAL CALCULUS. 221
3d. Integrate each partial fraction separately, and the
sum of the integrals thus found will be the integral
sought.
250. The method which has just been explained, will
require some modification when any of the roots of the
denominator are equal to each other. When the roots are
unequal, the fraction may be placed under the form
Px' + Qx 3 -\- Rx 2 -{- Sx -]- T
(x — a) (x — b) (x — c) (x — d) (x — e)
A + * +- c -+^-, + E
x — a x — b x — c x — d x — e '
if several of these roots are equal, as for example,
a = b — c, the last equation will become
Px*+Qx 3 +$ce. A + 5+C D E
(x — af (x — d) (x — e) x—a x — d x — e*
in which A + B + C may be represented by a single con-
stant A! .
Now, in reducing the second member of the equation to
a common denominator with the first, and comparing the
coefficients of the like powers of x, we shall have five
equations of condition between three arbitrary constants,
A', D, and E : hence, these equations will be incompati-
ble with each other (Alg, Art. 103).
If, however, instead of adding the three partial fractions
x — a x — b'
x — c
which have the same denominator, we go through the
19*
222 ELEMENTS OF THE
process of reducing them to one, their sum may be placed
under the form
A! + B'x+Ox 2
(x-af '
or, by omitting the accents,
A + Bx+Cx 2
(x-af '
251. Let us now make
x — a = z, and consequently, x = z -+- a ;
we shall then have
A + Bx + Cx 2 A + Ba+ Ca 2 + Bz+2 Caz + Cz 2
(x-a]
A + Ba+Ca 2 , B + 2Ca , C
substituting for z its value, and representing the numera-
tors by single constants, we have
A + Bx+Cx 2 A' \ B' C
+ 7- IV2 +
(x — af (x — af (x — af x — a" 1
the form under which the fraction may be written.
Since the same reasoning will apply to the case in
which there are m equal factors, we conclude that
Px m - l +Qx m -* +Rx+S _
(x-af
A A! A! 1 . A"../
(x-a) m ^(x-a)^- l ^(x-a) m ~ 2 ' x-a
252. In order, therefore, to integrate the fraction
Px*+Q f a? + &c.
(x — af (x — d)(x — e)
dx,
INTEGRAL CALCULUS. 223
place it equal to
A . A' A!' D . E
(a? — af (x — df
then, reducing to a common denominator, and comparing
the coefficients of the like powers of x, we find the values
of the numerators of the partial fractions. Multiplying
each by dx, and the given fraction may be written under
the form
a A f A" D E
■.dx -f —dx + dx -\ dx -\ dx.
(x—a) 3 (x—a) 2 (x—a) x—d x — e
The first two fractions may be integrated by the method
of Art. 217, and the three last by logarithms. Hence, finally,
/
P x*+Qx 3 + Rx 2 +Sx+T dx _ A A!
(x — a) 3 (x — d)(x — e) 2(x — a) 2 x — <
-f A"\og(x —a) + Dlog(x - d) + Slog (a? - e) + C.
253. Let it be required to integrate the fraction
2 ax ,
7 ^dx.
{oc + a) 2
We have
2 ax A A!
(x + a) 2 (x + a) 2 x + a
reducing the fractions of the second member to a common
denominator, and comparing the coefficients of x in the
two members, we have
2a — A! and A + A'a — 0:
hence,
A = — 2a 2 , and A! = 2a ;
224 ELEMENTS OP THE
and, consequently,
2axdx 2a 2 dx 2adx
(g-hd)> (* + «) 2 0* + a) '
hence, (Arts. 217 & 218),
/2axdx 2 a 2 , ~ , , N
(?T^ = ^+7 +2alog(iBH - a) -
254. Let us find the integral of
aPdx
x s — ax 2 — a 2 x -+- <r
By placing the denominator equal to 0, we see that, by
making x = a, the terms will destroy each other :. hence, a
is a root of the equation, and x—an factor. Dividing by
x — a, the quotient is x 2 — a 2 : hence, the fraction may be
placed under the form
a?dx a?dx
(a? — a 2 ) (x -
-a)
(x + a) (x — a)(x -
a?dx
-a)
Let us now make
x 2
~ (x — a) 2 (x + a)'
B
(x — a) 2 (x-\-a) (x — a) 2 (x — a) x-\-a
Reducing the terms of the second member to a common
denominator, we have
x 2 _ A(x + a) + A'(x 2 - a 2 ) + B(x- a)
(x — a) 2 (x-\-a)~~ (x — a) 2 (x-\-a)
and developing, and comparing the coefficients of the like
INTEGRAL CALCULUS. 225
powers of x, we obtain the equations
A' + B=l, A-2Ba = 0, Aa- A! a 2 -f Ba 2 = 0.
Multiplying the first equation by a 2 , and adding it to the
third, we have
Aa + 2Ba 2 = a 2 ;
then multiplying the second by a, and adding it to the last,
we have
a 2 = 2Aa, and consequently, A = — a ;
substituting this value of A, we find
B = — and A' = — .
4 4
Substituting these values of A, A f , and B, we have
aPdx adx Sdx dx
(x — a) 2 (x-\-a) 2(x — af 4(x — a) 4(a? + a)'
and consequently,
/a?dx _ a 3 , . .
a?-ax 2 -a 2 x + a*-~~2(x-a) + T i0g[X - a)
+ -^hg(x + a)+C.
255. We can integrate, in a similar manner, when the
denominator contains sets of equal roots. Let us take, as
an example,
adx adx
( a? 2_l)2-( a7 _ 1 )2 (a7+l) 2-
226 ELEMENTS OF THE
Make
, ^ , B , 5'
(^-1) 2 (^+1) 2 (a?— I) 2 ' oc-1 ' (^+1) 2 J?+l'
reducing the second member to a common denominator,
we find the numerator equal to
A(^+l) 2 +A / (a7-l)(a?+l) 2 +5(^-l) 2 +5 / ( J r+l)(a?-l) 2 ;
and comparing the coefficients with those of the numera-
tor of the first member, we have the following equations :
A' + B' = 0,
A +A f + B-B f = 0,
2A -A'-2B-B' = 0,
A ^A'+ B + B ; = a.
Combining the first and third equations, we find A = B;
and combining the second and fourth, gives 2 A + 2B == a:
hence, we have
a — R — ^L A'——— B'— — '
consequently, the given differential becomes
1 r dx dx dx dx ~)
T a L(a?-l) a + (^+l) 2 ~^=T + ^+TJ ,
and by integrating,
Jx^rf^i-^i- ^-iog(.-i)+iog(, +1 )] + a
256. If an equation of the second degree has imaginary
roots, the quantity under the radical sign will be essential!}/
INTEGRAL CALCULUS. 227
negative (Alg. Art. 144), and the roots will be of the form
x = =f <2 + h -y/— 1, a?==Fa — h->J — 1,
and the two binomial factors corresponding to the roots
will be
(a? ± a -6 y/^X) (# ± a + h V — 1) = «? ± 2a# + a 2 -f- 6 2 .
Hence, for each set of imaginary roots which arise from
placing the denominator of the fraction equal to 0, there
will be a factor of the second degree of the form
x 2 ± 2ax + a 2 + b 2 .
257. If the imaginary roots are equal, we shall have,
a = 0, x= -\-b-\/ — 1 , x= — b V — 1,
and the factor will become x 2 + 6 2 .
In the equation,
a?-6cx+\Qc 2 = y
the roots are,
x = 3c -f c y r —T # = 3c — c V— 1 ;
comparing these values of a? with the general form, we
have
a = — 3c & == c,
and the given equation takes the form
x 2 — 6c;r-f 9c 2 +c 2 = 0.
Comparing the roots of the equation,
a^-H4a7+12 = 0,
with the values of x in the general form, we have
a = 2,. b=</8,
228 ELEMENTS OF THE
and the equation may be written under the form
a? + 4a? + 4 + 8 = 0.
258. Let us first consider the case in which the deno
minator of the fraction to be integrated contains but one
set of imaginary roots. The fraction will then be of the
form,
P+Qx + Ra?+Sx 3 + &c.
(a? — a) (a? — b) .... (a? — h){x 2 -\- 2ax + a 2 -f- b 2 )
which may be placed under the form
dx y
Adx Bdx Hdx Mx + N ,
+" Z h ' ' ' ' + Z I + i'L^i „2 i S^'
x — a x — b x — h x 2 -f 2aa? + ft2 + W
The first three fractions may be integrated by the methods
already explained: it therefore only remains to integrate
the last, which may be written under the form
Mx + N .,
,dx.
{x + a) 2 + b z
If we make x -f a — z, the expression becomes
Mz + N—Mci j
¥+b 2 dZ '
and making N — Ma = P, it reduces to
Mz+P,
z l + b l
which may be divided into the parts,
Mzdz . Pdz
z * + b 2 ' z 2 + b 2 '
which may be integrated separately.
INTEGRAL CALCULUS. 229
To integrate the first term, we have
zdz M f 2zdz_
b 2
fMzdz _ f zdz _M f 2zd
J z 2 + b 2 ~ J z 2 +b 2 ~ 2 J z 2 +
in which the numerator, 2zdz, is equal to the differential
of the denominator: hence (Art. 218),
fMzdz M, , ., , , 2 .
z 2 + b 2
or by substituting for z its value, x + cl,
fMzdz M, r/ . x2 , 7 21
M
= ~log(x 2 + 2ax-\-a 2 + b 2 )
= M log i/a* + 2ax + a 2 + & 2 .
Integrating the second term by Art 224, gives
f Pdz P ,/z\
J^Th 2 = T l ^'(-b}
or by substituting for z its value, x -f a, and for P,
iV" — Ma, we have
/' Pcfe N—Ma _,/a? + a\
ft^— — tan « (-r) ;
and finally,
*/* 2 + 2aaH-a 2 + & 2
Mlog V^+2^ + « 2 + o 2 -f- ^— tang- 1 (^p ).
259. Let us take, as an example, the fraction
20
230 ELEMENTS OP THE
in which, if + 1 be substituted for x, the denominator
will reduce to : hence, x — 1 is a factor of the denomi-
nator. Dividing by this factor, the fraction may be put
under the form
c+fx
(x-l){x 2 + x+l)
dx,
m which x 2 + x + 1 is the product of the imaginary
factors. Placing this product equal to 0, finding the roots
of the equation, and comparing them with the general
values in the form
a?+2ax + a 2 + b 2 =0,
we find
a 2 ~V 4*
We may place the given fraction under the form
c+fx _ A Mx + N m
(x— 1) (x 2 + x + 1)~~ x— 1 x 2 + x + 1 '
reducing the second member to a common denominator,
and comparing the coefficients of x in the numerator with
those of x in the numerator of the first member, we obtain
Substituting these values of M and N, as also those of a
and 6, in the general formula of Art. 258, and recollecting
that
/' Adx c+f r dx c+f, . .
INTEGRAL CALCULUS. 231
we find
j
'^^d X = ^±l\og{x-l)-^±^h g V^+7+Y
+ /^ ta „ g -r^+r|+c.
VI
UVI
260. The equation which arises from placing the de-
nominator of the fraction equal to 0. may give several
pairs of imaginary roots respectively equal to each other
In this case, the factor x 2 zt 2ax + a 2 -\- b 2 will enter
several times into the denominator, or will take the form
2 +2^ + a 2 -f6 2 ) p ;
and hence, that part of the fraction which contains the
pairs of equal and imaginary roots, must be placed under
the form (Art. 251)
H+ Kx H' + K!x
+
(x 2 + 2ax + a 2 -{-b 2 y {x 2 + 2ax + d 2 + b 2 ) p - 1
II" + K"x H n + K n x
(x 2 +2ax + d 2 + b 2 ) p - i x 2 + 2 ax + a 2 + b 2
Now, reducing to a common denominator, and comparing
the coefficients, we find the values of the constants
H, K, H\ K', iF, K" H\ K n . . .
after which, multiply each term by dx, and then integrate
the terms separately.
Since all the terms are of the same general form, it will
only be necessary to integrate the first term, which may
be written under the form
H+Kx
[{x + af + b 2 y
dx ;
232 ELEMENTS OF THE
which, if we make x + a = z, will reduce to
H-Ka + Kz A
and making M—H— Ka> it will become
M+Kz , Kzdz v , Mdz
■ dz = -— — dz '
(b 2 + z 2 ) p (b 2 + z 2 Y ' {b 2 + z 2 f
The first term of the second member may be placed under
the form
Kf(b 2 + z 2 )-»zdz,
and integrating by the formula of Art. 217, we have
Kzdz IK 1
I;
5=i+C-
(b 2 +z 2 y 2 (i- P )(b 2 + z 2 y
It then only remains to integrate the second term
Mdz
By comparing the second member of this equation with
formula (D), Art. 242, we see that it will become identical
with the first member of that formula, by supposing
m=l, a = b 2 , b = l, and n~2\
and hence, by means of that formula, the exponent — p
may be successively diminished by unity until it becomes
— 1, when the integration of the term will depend on
that of
dz
W+z 2 '
But we have already found (Art. 224),
f dz 2 1 - x fz\
INTEGRAL CALCULUS. 233
and hence the fraction may be considered as entirely in-
tegrated.
261. It follows, from the preceding discussion, that the
integration of all rational fractions depends on the follow-
ing forms :
x m+l
1st. fx m dx — .
• m+1
== ± loff (a ± x).
a±x 5V ;
/dx 1 , / x \
3d.
Integration of Irrational Fractions.
262. The method of integrating rational fractions having
been explained, we may consider an irrational fraction as
integrated when it is reduced to a rational form.
263. Every irrational fraction in which the radical
quantities are monomials, may be reduced to a rational
form.
Let us take, as an example,
i_
^ x — \a x % — \a
ax, or — [ r .
%f~x— Voc
Having found the least common multiple of the indices
of the roots, (whicii indices are the denominators of the
fractional exponents,) substitute for x a new variable, z,
with this common multiple for an exponent, and the frac-
tion will then become rational in terms of z.
234 ELEMENTS OF THE
In the example given, the least common multiple is 6;
hence we have
x = z 6 and tJx = z' 3 , ^/x — z 2 , dx = 6z 5 dz ;
and substituting these values, we obtain
an expression which may be integrated by rational frac-
tions ; after which we may substitute for z its value, -\/x.
264. If the quantity under the radical sign is a polyno-
mial, the fraction cannot, in general, be reduced to a
rational form. We can, however, reduce to a rational
form every expression of the form
X( VA + Bx± Cx 2 ) dx,
in which X is supposed to be a rational function of x.
If we write a denominator 1, and then multiply the
numerator and denominator by VA -f Bx =b Cx 2 , the
expression will take the form
X!dx
VA + Bx± Cx 2 '
in which X r is a rational function of x : hence the two
forms are essentially the same.
If now, we can find rational values- for V A+ BxzLCx*
and for dx, in terms of a new variable, the expression will
take a rational form.
There are two cases to be considered: 1st., when the
coefficient of x 2 is positive ; and, 2d, when it is negative..
INTEGRAL CALCULUS. 235
Let us consider them separately. First, make
VA + Bx+Gx*= Vcy/A + ^v + a*
= V~C Va + bx -f x 2 ,
in which a = —-, b = —.
G C
In order to find rational values for dx and Va + bx+x 2 ,
place
Va -f 6a; + x 2 = oc + 2:, (1)
from which, by squaring both members, we find
a + fo?=:2^ + ;2: 2 , (2)
and hence,
and substituting this value in equation (1),
V a + bx + a? 2 = T — — - -f z ;
b — 2z
and by reducing to the same denominator,
V a + b X + 3 ? =- Z °- bz + a . (4)
— fiZ
Let us now find the value of dx in terms of z. For this
purpose we will differentiate equation (2), we then find
bdx = 2xdz + 2zdx + 2zdz ;
whence we have
{b — 2z)dx = 2(x + z)dz ;
236 ELEMENTS OF THE
and by subtracting equations (1) and (4), and substituting
for x + z the value thus found, we have
f-L o \j 2{z 2 — bz -\- a) ,
(b — 2z)dx = i— - —^dz 9
b — 2z
, , 2(z 2 -bz + a) , ,_.
8nd **=- \b-2z? ^ (5)
265. Let us take, as an example,
dx
xVa + Bx+Cx 2
which may be written under the form
dx
V~C x x Va + bx + x 2 '
and substituting the values of Va -f bx + x 2 and dx } from
equations (4) and (5), we have
dx ■■' 2dz
Va + bx + x 2 ~b-2z'
and multiplying the denominator by the value of x, in
equation (3),
dx __ 2dz
xVa + bx + a?~ * 2 -a'
and then by Vc, we have
dx dx 2dz
or
VC X x Va + to -f a? a? V-A+BaH- C* 2 (z 2 - a) VC
which is a rational form, and may be integrated by the
methods already explained.
INTEGRAL CALCULUS. 237
266. Let us take, as a second example,
dx
Vh + c 2 x 2
which may be placed under the form
dx
v&*
c
c
and comparing this with the form of Art. 264, gives
c = -yJC, b = 0, — = a.
&
Hence,
dx 1 r dx
/dx 1 r d
V7H- c 2 x 2 ~~~ J V^
Having placed
i/iT+x 2 = z-{-x,
we found, Art. 264, equations (5) and (4),
hence
dx r dz
/ax C — 1
Va~+~x^~ J ~T~~ ° g
z.
Substituting for z its value, and multiplying by — , we
c
have
and substituting for a its value, ~, we have
238
ELEMENTS OF THE
/
dx
Vh + c 2 a?
~ log
— ( Vh -r c 2 a? — ex)
l_ c
+ c
— log log( Vh + cV — ex) + C.
But since the difference of the squares of the two terms
within the parenthesis is equal to h, it follows that if h
be divided by the difference of the terms, the quotient will
be their sum (Alg. Art. 59). But the division may be
effected by subtracting their logarithms. Let us, then,
add to, and subtract from, the second member of the equa-
tion, — \ogh. We shall then have,
/
1, 1 1,
-log loffA-| — log h loe( VA+cV— cx)4- C:
Vh+<?a* c c c c c
or by representing the three constants — log log Zf,
c c c
and C, by a single letter C, we have
-log( VT+cV + ex) + C.
/
dx
Vh+c^x*'
267. Let us take, as a third example,
dxVm 2 + x 2 .
Comparing this with the general form, we find
a = m 2 and 6 = 0;
hence (Art. 264),
2% I 772
VnF+x 2 = and dx
6Z
2z 2
INTEGRAL CALCULUS. 239
and consequently,
7 r^TT-^i (z 2 + m 2 ) 2 7
dx V m z + x z — — - ,. ' dz,
which is rational in z ; and, having found the integral in z,
substitute the value of z in terms of x.
268. Let us now consider the case in which the coeffi-
cient of x 2 is negative. We have
<y/A + Bx- Cx 2 = V^\/^- + ~--<
= Vc" -yja -\-bx-x 2 .
If now, we make as before,
Va + hoc — x 2 = x -+■ z t
and square both members, the second powers of x in each
member will not cancel, as before ; and therefore, x can-
not be expressed rationally in terms of z. We must,
therefore, place the value of the radical under another
form. We will remark, in the first place, that the bino-
mial a + bx — x 2 , may be decomposed into two rational
factors of the first degree. For, if we make
x 2 — bx — a = 0,
and designate the roots of the equation by a and <*', we
have (Alg. Art. 142)
(x 2 — bx — a) = (x — et) (x — *'),
and consequently, by changing the signs,
(a + bx — x 2 ) = — (x — *) (x — <x!) = (x — *) {«! — x) ;
240 ELEMENTS OF THE
and placing the second member under the radical, we
may make
V{x — *) (*' — x) = (x — cc)z\ (1)
squaring both members
{x _ u ) [a! — x) = (x— ec) 2 Z 2 j
and by suppressing the common factor x — #,
«! — x = {x — *)z 2 , (2)
whence,
X-
ct' + ctZ 2
~ 1+Z 2 '
and
X-
— C6 r
ct! + az 2
l+z 2
or by reducing,
,r —
- C6 —
a! — a,
fh
l+z"'
which, being substituted in the second member of equa-
tion (1), gives
V(» -.)(*_ *) = ■£=£*; (4)
and by differentiating equation (3), we obtain
2 («' — «) , ...
dx =-{T+W ()
269. To apply this method to a particular example of
the form
dx
Va + bx — oe*
INTEGRAL CALCULUS. 241
substitute the values of V a -f- bx — x 2 and dx, found m
equations (4} and (5) : we find
dx 2(x! — u)z , _ 2dz
Va + bx-x 2 (l | z 2 ) 2 z ^'~ u) l + z2)
\-\-z 2
hence
dx
f—= =-2t<mg- l z+C;
J yci + bx — x 2
or, by substituting for z its value from equation (1),
/' dx =r,-^-i( V ^-«^«[-^ )
J Va-i-bx-x 2 ^ («-«) J
= C-2tang- 1 A / /?^.
V x — cc
270. If, in the last formula, we make
a = 1 and b = 0,
the trinomial under the radical will become 1 — x 2 , and
the roots of the equation x 2 — 1 = are
« = — 1 and ce! = 1 .
Substituting these values, and the general formula becomes
and if we suppose the integral to be when x = 0, we
enan have
0=C-2tang- x (l)
= C - 2(450) (Trig. Art. VIII)
= C-90°: hence C = — .
2
21
242 ELEMENTS OF THE
Substituting this value, and we have
/• dx * _ x /l—x
I — — 2tang l \/ ,
271. We have already seen (Art. 219) that
/dx
and hence,
— 2tan£ -1 \/- —
° V 1 +
should also represent the arc of which x is the sine
To prove this, we have (Trig. Art. XXV)
„ , 2tangA
tang2A = - . 8 . .
6 1 — tang 2 A
/l — x
Substituting for tang A, \J , and reducing, we have
Ung2A = y^ER;
/l — cc
that is, twice the arc whose tangent is i/ — — is equal
» 1 ~~J~ x
/— — rj
to the arc whose tangent is Z
x
I- g
But the arc whose tangent is 1 — # ? j s t jj e com .
a?
plement of the arc whose tangent is . , (Trig.
Vl — x 2
Art. XVIII) ; and this arc has x for its sin*. Hence,
either member of the equation
INTEGRAL CALCULUS. 243
r dx a- _ /i_ x
1—7== = 2 tang \/ ,
J Vl-x 2 2 ° V l + ar'
represents the arc whose sign is a?.
272. Let us take, as a last example, the differential
dxy 2 ax — x 2 .
In comparing this with the general form, we find (Art
268)
cc — and ec ; = 2a;
and Art. 268, equations (4) and (5), give
2az j _ 4<2.2r
i + ?' (T+?) :
^/a:(2a — x) = t "~ 2 , dx = — ^ri^z.
Substituting these values, we have
8a 2 z 2 dz
dxy2ax — x 2
(i+* 2 ) :
which may be integrated by the method of rational
fractions.
Rectification of Plane Curves.
273. The rectification of a curve is the expression of
its length. When this expression can be found in a finite
number of algebraic terms, the curve is said to be rectijiable,
and its length may be represented by a straight line.
274. The differential of the arc of a curve, referred to
rectangular co-ordinates, is (Art. 128)
dz= V d ^ + dy 2 -
244 ELEMENTS OF THE
Hence, if it be required to rectify a curve, given by its
equation,
1st. Differentiate the equation of the curve.
2d. Combine the differential equation thus found with
the given equation, and find the value of dx 2 or dy 2 in
terms of the other variable.
3d. Substitute the value thus found in the differential
of the arc, which will then involve but one variable and
its differential. Then, by integrating, we shall find the
length of the arc, estimated from a given point, in term;
of one of its co-ordinates.
21 h. Let us take, as a first example, the common para
bola, of which the equation is
y 2 = 2px.
Differentiating, and dividing by 2, we have
ydy = pdx,
and consequently,
dx 2 = — „ dy 2 ;
p 4
substituting this value in the differential of the arc. we
have
= —dyVp 2 + y 2 ;
which, being integrated by formula (B) Art. 239, gives,
by supposing m = 1, a =p 2 , b=l, n = 2, p = -r-
INTEGRAL CALCULUS. 245
f , / • = iog( Vp 2 + y 2 + y) ;
and integrating the second term by the formula of Art.
266, we have, after maKing h =p 2 , c 2 = 1,
dy
p 2 -\-y
and consequently,
If we estimate the arc from the vertex of the parabola,
we shall have
y — for s = : hence
= -^\ogp+C or C=--|-logp;
and consequently,
z _ yVp 2 + y 2 + p lo r Vp 2 + y 2 + y \ .
and hence, the value of the arc, for a given ordinate y, can
only be found approximatively.
276. The curves represented by the equation
y n =px m ,
are called parabolas. This equation may be placed under
the form
1 m
y^p n x T ' f
or by placing p n ==p', and -~ = n f , we have
y=p'x n '' y
21*
246 ELEMENTS OF THE
or finally, by omitting the accents, the form becomes
y=px\
By differentiating, we have
dy = ripx n ~ l dx,
and by substituting this value of dy in the differential of
the arc, we have
i
z=f(\+nyx 2n - 2 fdx.
The integral of this expression will be expressed in a
finite number of algebraic terms when is a whole
to 2n — 2
number and positive (Art. 235). If we designate such
whole and positive number by i, we have for the condition
of an exact integral in algebraic terms,
1 2i+I
— i, or n=
2n-2 7 2i '
and substituting for n, we have
2H-1.
y—px 2i > or y 2i —p^x %i+l y
which expresses the relation between x and y when the
length of the arc can be found in finite algebraic terms.
There is yet another case in which the integral will be ex-
pressed in rinite and algebraic terms, viz. when o +1T
is a positive whole number (Art. 236 and 235.)
3
277. If we make i = 1, we have n = — , and
At
y 2 =p 2 a?,
which is the equation of the cubic parabola.
INTEGRAL CALCULUS. 247
Under this supposition, the arc becomes (Art. 217)
z=f(l +nYa>-^ = JL.(i + lf x f 4- C;
and hence, the cubic parabola is rectifiable (Art. 273).
If we estimate the arc from the vertex of the curve, we
have x = 0, for z = : hence
and consequently,
278. If the origin of co-ordinates is at the centre of *he
circle, the equation of the circumference is
R 2 = x 2 + y 2 ,
and the value of the arc,
r dx
If the origin be placed on the curve
y 2 = 2Rx - x 2 ,
dx
and z = R
i
V2Rx-a?'
both of which expressions may be integiated by holies,
and the length of the arc found approximative^.
279. It remains to rectify the transcendental cur"- s.
The differential equation of the cycloid is (Art. 182)
V 2ry — y 2
248
which gives
ELEMENTS OF THE
dx 2
_ y 2 dy'
2ry-y>*
Substituting this value of dx 2 in the differential of the
arc, we obtain
& =v^S= rf V
2ry
2ry — y*
2ry-y l
= d y V^7~ = ( 2r f( 2r - y)~^ d y>
But (Art, 217)
f(2r-y)-hy=-2(2r-yy + C;
and hence,
r
z= -(2r) 2 2-v/2r-y+ C= — 2^/ r 2r(2r — y) + C.
If now, we estimate
the arc z from B, the
point at which y = 2 r,
we shall have, for z — 0,
y = 2r; hence ^
= + C, or C = 0,
and consequently, the true integral will be
z= _^2V2r(2r — y) ;
the second member being negative, since the arc is a
decreasing function of the ordinate y (Art. 31).
If now, we suppose y to decrease until it becomes
equal to any ordinate, as DF = ME, DB will be repre-
sented by z, or by 2 -y/2r(2r — y\ and jB£ = 2r — y.
But JBG =BMxBE: hence
.££ = V2r(2r — y),
JNTEGRAL CALCULUS. 249
and consequently,
BD = 2BG;
or the arc of the cycloid, estimated from the vertex of the
axis, is equal to twice the corresponding chord of the
generating circle : hence, the arc BDA is equal to twice
the diameter BM ; and the curve ADBL is equal to four
times the diameter of the generating circle.
280. The differential of the arc of a spiral, referred to
polar co-ordinates, is (Art. 202)
dz = Vdu 2 + ii z dt 2 .
Taking the general equation of the spirals
u = at",
we have du 2 = n 2 aH Zn ~*dl? ;
and substituting for du 2 and u 2 their values, we obtain
dz = at n - l dtVn 2 + f.
If we make n = 1, we have the spiral of Archimedes,
(Art. 191), and the equation becomes
dz = adtVl + ^;
which is of the same form as that of the arc of the com-
mon parabola (Art. 275).
281. In the logarithmic spiral, we have t — hgu, and
the differential of the arc becomes
dz-duV2+ C;
and if we estimate the arc from the pole,
250 ELEMENTS OF THE \
Consequently, the length of the arc estimated from the
pole to any point of the curve, is equal to the diagonal of
a square described on the radius-vector, although the
number of revolutions of the radius-vector between these
two points is infinite.
Of the Quadrature of Curves.
282. The quadrature of a curve is the expression of its
area. When this expression can be found in finite alge-
braic terms, the curve is said to be quadrdble, and may be
represented by an equivalent square.
283. If s represents the area of the segment of a curve,
and x and y the co-ordinates of any point, we have seen
(Art. 130), that
ds = ydx.
To apply this formula to a given curve :
1st. Find from the equation of the curve the value of y
in terms of x, or the value of dx in terms of y, ivhick
values will he expressed under the forms
y = F(x), or dx — F(y)dy.
2d. Substitute the value of y, or the value of dx, in the
differential of the area : we shall have
ds = F(x) dx, or ds = F(y) dy :
the integral of the first form will give the area of the
curve in terms of the abscissa, and the integral of the
second will give the area in terms of the ordinate.
INTEGRAL CALCULUS. 251
284. Let us take, as a first example, the family of para-
bolas of which the equation is
y n =px m :
we shall then have
and
i. J?
y = p n x n ,
— — 7VO n m + n 72
fF(x)dx=fp n x n dx=—±- — x n = ■ — xy+C\
J v ' J 1 m + n m + n J
1 m
by substituting y for its value, p n x n .
If, instead of substituting the value of y in the differential
of the area
ydx,
we find the value of dx from the equation
y n =px m >
dx = j — dy,
we have
and consequently,
•J ^ " ~ m J ~ " ~~ m + n - m + n * '
p m p m
n
by substituting x for its value, — , which is the same re-
p m
suit as before found.
Hence, the area of any portion of a parabola is equal
to the rectangle described on the abscissa and ordinate
252 ELEMENTS OF THE
multiplied by the ratio . The parabolas are ihere-
^ m + n r
fore quadrable.
In the common parabola, n = 2, m=l, and we
have
fF(x)dx = — xy y
that is, the area of a segment is equal to two thirds of
the area of the rectangle described on the abscissa and
ordinate.
285. If, in the equation
y n =px m ,
we make n = 1, and m — 1, it will represent a straight
line passing through the origin of co-ordinates, and we
shall have
JF(x)dx = —xy,
which proves that the area of a triangle is equal to half
the product of the base and perpendicular.
286. It is frequently necessary to find the integral or
function, between certain limits of the variable on which
it depends.
A particular notation has been adopted to express such
integrals.
Resuming the equation of the common parabola
y 2 = 2px,
and substituting in the equation ydx the value of dx -= ^-^
we have
P d P
INTEGRAL CALCULUS.
253
or, if the area be estimated from the
vertex A, we have C = 0, and
fydx — — — .
Jy Sp
P
F f
If now, we wish the area to terminate "
at any ordinate PM = b, we shall then
take the integral between the limits of y = and y = b ;
and, to express that in the differential equation, we write
1 rb o 7 b 3
which is read, integral of y 2 dy between the limits y =
and y = b.
If we wish the area between the ordinates MP = b,
MP' = c, we must integrate between the limits y = 6,
y — c. We first integrate between and each limit, viz. :
AMP
\i\ yHy
3p'
AM
we then have
PMM
y P' = —f C y 2 dy = — :
pj y y Sp
'P === AMM'P r - AMP = — fl tfdy
__J_ b 3 _ 1
287. Let us now determine the area of any portion of
the space included between the asymptotes and curve of
an hyperbola.
22
254
ELEMENTS OF THE
The equation of the hyperbola referred to its asymp-
totes (An. Geom. Bk. VI, Prop. IX,) is
xy
M.
In the differential of the area of a curve ydx, x and y
are estimated in parallels to co-ordinate axes, at right an-
gles to each other.
The differential of the
area BCMP, referred to
the oblique axes AX,
A Y, is the parallelogram
PMM'P', of which
PM=y and PP f = dx.
If we designate the
angle YAX=MPX by
|8, we shall have
area PMM'P = ydx sin i
and substituting for y its value — ,
x
the area BCMP by s, we have
7 nr • d°c
as = MsmQ — ,
and representing
C&x
and 5 — M sin/3 / — = M sin/3 logx + C.
J x
If AC is the semi-transverse axis of the hyperbola, and we
make AB=1, and estimate the area s from BC, we shall
have, for x = 1 , 5 = 0, and consequently C = ; and the
true integral will be
s = Msinp\ogx.
INTEGRAL CALCULUS. 255
But, since ABCD is a rhombus, and M—AB x BC (An.
Geom. Bk. VI, Prop. IX, Sch. 2), and since AB — 1, we
have M= 1, and consequently,
s = sin /3 logo?.
Now, since s, which represents the space BCMP for any
abscissa x, is equal to the Naperian logarithm of x multi-
plied by the constant sin/3, s may be regarded as the loga-
rithm of x taken in a svsiem of which sin (3 is the modu-
lus (Alg. Art. 268). Therefore, any hyperbolic space
BCjIP is the logarithm of the corresponding abscissa
AP, taken in the system ichose ?nodulus is the sine of the
angle included between the asymptotes.
If we would make the spaces the Naperian logarithms
of the corresponding abscissas, we make sin/3 = 1, which
corresponds to the equilateral hyperbola. If we would
make the spaces the common logarithms of the abscissas,
make siiiS =: 0.43429945, (Alg. Art. 272).
2S3. The equation of the circle, when the origin of co-
ordinates is placed on the circumference, is
y 2 — 2rx — x 2 , or y — V 2rx — x 2 ,
and hence, the differential of the area is
dx y2rx — x 2 ;
mid this will become, by making x — r — u,
\_
-fdu(i 2 -u 2 y.
If we integrate this expression by formula (B) Art. 239,
256 ELEMENTS OF THE
we have
- fdu{r> - u 2 f =-\u{r z - u 2 f - ±-r 2 fdu(?*-u 2 f*
du
— — u-y/ r 2 — u 2 -\ f 2 i
2 2 J ■
But we have (Art. 253)
/—du _ _ i /u\
and placing for u its value
fdxV 2rx — x 2 =
(r — x) V 2rx — x 2 -\ r- 2 cos _1 ( ) 4- C ;
2 v ; 2 \ r J ?
and taking this integral between the limits x — and
x = 2r, we shall have the area of a semicircle.
For x — 0, the area which is expressed in the first
member becomes 0, the first term in the second member
becomes 0, and the second term also becomes 0, since
the arc whose cosine is 1, is 0: hence the constant
C = 0.
If we now make x == 2r, the term
— (r — x) -\/ 2rx — x*
reduces to 0, and the second term to
-l^cos-X- l) = —r 2 *r (Trig. Art. XIV),
z z
and consequently, the entire area is equal to r 2 ^, which
INTEGRAL CALCULUS. 257
corresponds with a known result (Geom. Bk. V, Prop. XII,
Coi. 2).
The equation of the ellipse, the origin of co-ordi-
nates being at the vertex of the transverse axis (An. Geom.
Bk. IV, Prop. I. Sch. 8), gives
r>
y = —V^Ax—x 2 ,
A.
and consequently, the area of the semi-ellipse will be
equal to
fydx = — I dxyj 2Ax — x 2 .
A J
Integrating, as in the last example, between the limits
* = 0, and x = 2A, and multiplying by 2, we find ABtt
for the entire area. This corresponds with a known result
(An. Geom. Bk. IV, Prop. XIII).
*- 289. The differential equation of the cycloid (Art. 183) is
dx =
ydy
V2ry - y 2 '
whence
y 2 dy
fydx= j-
■y/2ry — y 2
and applying formula E, (Art. 243) twice, it will reduce to
f . dy - ; and (Art. 226)
«/ V2ry-y 2
/V^== = ver-sin-<m.
J V 2ry -y 2 v r /
But we may determine the area, of the cycloid in a more
simple manner by introducing the exterior segment AFKH,
22*
258
ELEMENTS OF TUB
Regarding FB as a
line of abscissas, and de-
signating any ordinate as
KH, by z — 2r — y, we
shall have
F K
B
\
B
/
f
&"J >>.
A
M L
But
whence
zdx
d(AFKH) =
(2r-y)ydy
V27
-y-y
ZC&T.
dy^2ry~-y\
AFKH=fdyV2ry—y 2 -f a
But this integral expresses the area of the segment of a
circle, of which the abscissa is y and radius r (x4rt. 288):.
that is, of the segment MIGE. If now, we estimate the
area of the segment from M, where y = 0, and the area
AFKH from AF, in which case the area AFKH= for
y — 0, we shall have
AFKH = MIGE;
and taking the integral between the limits y = and
y = 2r, we have
AFB = semicircle MIGB t
and consequently,
area AHBM = ,4 \F£M - i¥IGJ5.
But the base of the rectangle AFBM is equal to the semi-
circumference of the generating circle, and the altitude is
equal to the diameter, hence its area is equal to four times
the area of the semicircle MIGB ; therefore,
axQa.AHBM=3MXGB r
INTEGRAL CALCULUS. 259
and consequently, the area AHBL is equal to three times
the area of the generating circle.
290. It now remains to determine the area of the spirals.
If we represent by s the area described by the radius-vec-
tor, we have (Art. 203)
7 u 2 dt
as == :
2
and placing for u its value at (Art. 189)
7 a 2 t 2 "dt , a 2 f n+1 ; n
as — and s = h C,
2 4n + 2
and if n is positive C — 0, since the area is when t = 0.
After one revolution of the radius-vector, f = 2?r, and we
have
_ a 2 (2*Y n+l
S ~ 4:71 + 2 '
which is the area included within the first spire.
291. In the spiral of Archimedes (Art. 192)
a — — and n = 1 ;
2*
hence, for this spiral we have
f
s
24^ 2?
which becomes — , after one revolution of the radius-
3
vector ; the unit of the number — being a square whose
o
side is unity. Hence, the area included by the first spire,
is equal to one third the area of the circle whose radius is
equal to the radius-vector after the first revolution.
In the second revolution, the radius-vector describes a
260 ELEMENTS OF THE
second time the area described in the first revolution ; and
in any revolution, it will pass over, or redescribe, all the
area before generated. Hence, to find the area at the end
of the with revolution, we must integrate between the limits
t = (m— 1)2*- and t = m.2?r,
which gives
m 3 — (?n — 1 ) 3
1 —7T.
3
If it be required to find the area between any two spires ?
as between the with and the (m + 1 )th, we have for the
whole area to the (m -f l)th spire equal to
(m + 1 ) 3 — mf
3 " ;
and subtracting the area to the mth spire, gives
0+l) 3 -2m 3 -{-0- l) 3
—7F — 2mir,
3
for the area between the with and (wi + l)th spires.
If we make m =± 1, we shall' have the area between the
first and second spires equal to 2t: hence, the area he-
tiveen the mth and (m + 1 )th spires, is equal to m time's
the area between the first and second.
292. In the hyperbolic spiral n = — 1, and we have
ds =c dt and s = .
2 2t
The area s will be infinite when t — 0, but we can find
the area included between any two radius-vectors b and c
by integrating between the limits t = b, t = c, which will
give
a 2 / 1 1 \
S =-2KT-V)'
INTEGRAL CALCULUS.
293. In the logarithmic spiral t — \ogu : hence, dt
261
du
l dt udu
hence,
2 2
udu u 2
/udu ir n
and by considering the area s — when u = 0, we have
C = and
Determination of the Area of Surfaces of
Revolution.
M
M
B
294. If any curve BMM f , be re-
volved about an axis AX, it will de-
scribe a surface of revolution, and
every plane passing through the axis
AX will intersect the surface in a me-
ridian curve. It is required to find the
differential of this surface. For this A P P' X
purpose, make AP == x, PM == y, and PP' — h : we shall
then have
PM = F(x) = i/,
P'M'
*(.+»)=, + £* + $£+ to.
262
ELEMENTS OF THE
In the revolution of the curve BMM ,
the extremities M and M! of the ordi-
nates MP, M'P f , will describe the cir-
cumferences of two circles, and the
chord MM! will describe the curved
surface of the frustum of a cone. The
surface of this frustum is equal to
(Geom : Bk. VIII, Prop. IV.)
(circ.MP -J- circ.MP')
2
(2 7 rMP+2vM / P>)
(■ x chord MM
P P' X
that is, to
X chord MM' =v (MP +MP') X chord MM' ';
and by substituting for MP, MP' their values, the expres-
sion for the area becomes
dy
d 2 u h*
If now we pass to the limit, by making h — 0, the chord
MM' will become equal to the arc MM (Art. 128), and the
surface of the frustum of the cone will coincide with that
of the surface described by the curve at the point M. If we
represent the surface by s and the arc of the curve by z,
we have, after passing to the limit,
ds — 27rydz,
and by substituting for dz its value (Art. 128), we have
ds — 27ry ydx 2 + dy 2 :
whence, the differential of a surface of revolution is equal
to the circumference of a circle perpendicular to the axis,
into the differential of the arc of the meridian curve.
INTEGRAL CALCULUS. 263
Remark. It should be observed that X is the axis about
which the curve is revolved. If it were revolved about
the axis Y, it would be necessary to change x into y and
y into x.
295. If a right angled triangle CAB be revolved about
the perpendicular CA, the hypothenuse CB will describe
the surface of a right cone. If we represent the base BA
of the triangle by b. the altitude CA by h, and suppose
the origin of co-ordinates at the vertex of the angle C, we
shall have
x : y : : h : b: hence
y — -—x and dy = —-dx.
h h
Substituting these values of y and dy, in the general for-
mula, we have
bx , /tt, — nr bar
f27ry^dx 2 +dij 2 =f27r — dxVJ^+b 2 = 7 r--V}i 2 +b 2 -i-C,
and integrating between the limits x = and x — h, we
obtain
surface of the cone = ?r b -\Jli- -j- li z = 2nb x
2
• AJ, CB
= circ.AB x .
296. If a rectangle ABCD be revolved around the side
AD, we can readily find the surface of the right cylinder
which will be described by the side BC.
Let us suppose the axis AD — h, and AB — b : the
equation of the line DC will be y = b : hence, dy = 0.
Substituting these values in the general expression of the
differential of the surface, we have
f27ryVdx z +dy 2 = f27rbdx = 27rbx+C;
264 ELEMENTS OF THE
and taking the integral between the limits x = x = h,
we have
surface = 2vbh = circ.AB x AD.
297. To find the surface of a sphere, let us take the
equation of the meridian curve, referred to the centre as
an origin : it is ,
x 2 + y 2 = R 2 ,
and by differentiating, we have
xdx + ydy = ;
hence
, XO/X -, 79 X OjX
and ay'
y y 2
Substituting for dy its value, in the differential of the
surface
ds — 27ry Vdx 2 -j~ dy 2
we obtain
s = f2*y y dx 2 -+- ~dx 2 =/2irRdx = 2rrR X + C.
If we estimate the surface from the plane passing through
the centre, and perpendicular to the axis of X, we shall
have
5 = for x — 0, and consequently C = 0.
Now, to find the entire surface of the sphere, we must
integrate between the limits x = + R and x ■=. — R, and
then take the sum of the integrals without reference to
their algebraic signs, for these signs only indicate the po-
sition of the parts of the surface with respect to the plane
passing through the centre of the sphere.
INTEGRAL CALCULUS. 265
Integrating between the limits
x = and x — -f- R,
we find
s = 2*R 2 ;
and integrating between the limits x = and x = — R,
there results
s=-2*-# 2 ;
hence,
surface = 4*-Pc 2 = 2*Rx2R;
that is, equal to four great circles, or equal to the curved
surface of the circumscribing cylinder.
298. The two equal integrals
s = 2~R~ and s=-2*R 2
indicate that the surface is symmetrical with respect to the
plane passing through the centre.
299. To find the surface of the paraboloid of revolution,
take the equation of the meridian curve
which being differentiated, gives
dx= y^y and ^m.
p p
Substituting this value of dx in the differential of the sur-
face, it reduces to
**&/(£££)*? = jydy vV+TT
23
266 ELEMENTS OF THE
But we have found (Art. 217)
fjydyV¥+f = ^(y 2 +P 2 f + C:
hence,
and if we estimate the surface from the vertex at which
point y — 0, we shall have,
~ 2*P 2 „ %*P 2
. = -£-+C, whence, C= - -f- f
and integrating between the limits
y=0, y = b,
we have
s = f p [(V+P % T-y*\
300. To find the surface of an ellipsoid described by
revolving an ellipse about the transverse axis.
The equation of the meridian curve is
A 2 y 2 + B 2 x 2 = A 2 B\
whence
T B 2 xdx B ocdx
A " y A Va 2 -* 2
substituting the square of this value in the differential of
the surface and for y its value
^VA 2 -x?
we have
: 2 7T- dx v/A 4 - ( A 2 - B V*
INTEGRAL CALCULUS. 267
and «'="*« || -fA^&fdx \J-^p - °? ;
and if we represent the part without the sign of the inte-
gral by D, and make
A 4
A 2 -B 2
we shall have
R\
s = DJdxjW-o?.
But the integral of das yR 2 — x 2 is a circular segment
of which the abscissa is x, the radius of the circle being
R. If, then, we estimate the surface of the ellipsoid from
the plane passing through the centre, and also estimate the
area of the circular segment from the same point, any
portion of the surface of the ellipsoid will be equal to the
corresponding portion of the circle multiplied by the con-
stant D. Hence, if we integrate the expression
s =fdx -y/R 2 — x 2
between the limits x = and at = A, and designate
by D' the corresponding portion of the circle whose
radius is R, we shall have
— surface ellipsoid = D x D f ;
hence, surface ellipsoid = 2D x D f .
301. To find the surface described by the revolution of
the cycloid about its base.
The differential equation of the cycloid is
dx.
ydy
V2ry —
,2
268 ELEMENTS OF THE
Substituting this value of dx in the differential equation
of the surface, it becomes
, _ 2^^2ry * dy
-\/2ry — y 2
Applying formula (E), Ajt. 243, we have
But,
/v©=/^ = ^ (2! - yr ^- 2(2) - y)l;
hence,
l~ 2 — 8 - -1
s = 2^/2r —~y\/2ry — y 2 r{2r — y) 2 \+ C.
_ 3 3 _J
If we estimate the surface from the plane passing through
the centre, we have C — 0, since at this point 5 =
and y = 2r. If we then integrate between the limits
y == 2r and y = 0, we have
s = — surface = ^r 2 ; hence,
2 3
s— surface = Trr 2 ,
3
that is, the surface described by the cycloid, when it is
revolved around the base, is equal to 64 thirds of the
generating circle.
The minus sign should appear before the integral, since
the surface is a decreasing function of the variable y
(Art. 31).
INTEGRAL CALCULUS.
269
Of the Cubature of Solids of Revolution.
302. The cubature of a solid is the expression of its
volume or content.
303. Let u represent the volume or
solidity generated by the area ABMP,
when revolved around the axis AX. If
we make AP = a?, PP' = h, we have
MP'— F(x-Yh). Now, the solid gene-
rated by the area ABMMP', will ex-
ceed the solid described by ABMP, by
the solid described by the area PMM'P' .
The solid described by the area ABMP is a function of
co, and the solid described by the area ABMM'P' is a simi-
lar function of (x-\-h). If we designate this last by v!,
we have
dhi h 3
du 1 , d?u lr
u-{-—-h + -—
dx dx i 1.2
+
dx* 1.2.3
hence, the solid described by PMM , P / is
h 2 . d?u h 3
, du , <Pu
U r —U = — h + t-5-
dx dx 2 1.2
dx 3 1.2.3
+ &c;
+ &c.
Let us now compare the cylinder described by the rectan-
gle P'M with that described by the rectangle P'C. The
equation of the curve gives
MP = y = F(x) M'P'=F(x + h);
hence, since PP' = h,
cylinder described by P'M— v [F(x)fh f
cylinder described by P'C = * [F(x + k)fh;
270 ELEMENTS OF THE
and the ratio of the cylinders is
[F(x + h)] 2
[F{x)] 2 ■
the limit of which, when h — 0, is unity .
But the solid described by the area PMM / P / is less
than one of the cylinders and greater than the other ;
hence, the limit of the ratio, when compared with either
of them, is unity. Hence,
du , , d 2 u h 2 , o du , d 2 u h c
- T - h + -— - h&c. —r--\--rr + &c.
dx dx 2 1.2 (fo? ^ 2 1 .2
^[FO)] 2 /* "■[J , («)] !
the limit of which, when h — 0, is
du
dx
I,
whence
and finally
<2w = Try 2 dx ;
the differential of the solidity iry 2 dx being a cylinder whose
base is ^y 2 and altitude dx.
304. Remark. The differential of a solid, generated by
revolving a curve around the axis of Y, is
nx?dy.
305. Let it be required to find the solidity of a right
cylinder with a circular base, the radius of the base being
INTEGRAL CALCULUS. 271
r and the altitude h. We have for the differential of the
solidity
ny 2 dx,
and since y — r, it becomes
nr^dx ;
and taking the integral between the limits x=0 and x = h,
we have
which expresses the solidity.
306. To iind the solidity of a right cone with a circular
base, let us represent the altitude by h and the radius of
the base by r, and let us also suppose the origin of co-or-
dinates at the vertex. We shall then have
r
y = —x ana y
J h y h
XI
and substituting, the differential of the solidity becomes
.2
and by taking the integral between the limits x = and
x = h, we obtain
— r 2 -h = ^r 2 x — ;
3 3
that is, the area of the base into one third of the altitude.
307. Let it be required to find the solidity of a prolate
spheroid, (An: Geom : Bk. IX, Art. 33).
The equation of a meridian section is
272 ELEMENTS OF THE
which gives
hence the differential of the solidity is
B 2
du — n-— 2 (A 2 — x 2 )dx,
and by integrating
u = ^( A 2 x--) + C
If we estimate the solidity from the plane passing through
the centre, we have for'a? = 0, u = 0, and consequently
C = 0; and taking the integral between the limits x — Q
and x = Aj we have
1 2
— solidity = —"B 2 x A ;
and consequently
2
solidity ±= -— - nB 2 x 2 A.
o
But a-jB 2 expresses the area of a circle described on the
conjugate axis, and 2A is the transverse axis : hence,
the solidity is equal to two-thirds of the circumscribing
cylinder.
308. If an ellipse be revolved around the conjugate axis,
it will describe an oblate spheroid, and the differential of
the solidity would be
du = TcxPdy :
INTEGRAL CALCULUS. 273
and substituting for x 2 , and integrating, we should find
2
solidity = —n A 2 x 2 B :
o
that is, two-thirds of the circumscribing cylinder.
309. If we compare the two solids together, we find
oblate spheroid : prolate spheroid : : A : B.
310. If we make A = B, we obtain the solidity of the
sphere, which is equal to two-thirds of the circumscribing
cylinder, or equal to
4 1
R 3 = ^
3 6
311. Let it be required to find the solidity of a para-
boloid. The equation of a meridian section is
if = 2px,
and hence the differential of the solidity is
du = 2 TTjyxdx ; hence
u = TV par + C ;
and estimating the solidity from the vertex, and taking the
integral between the limits x = and x == h, and designa-
ting by b the ordinate corresponding to the abscissa x = h,
we have
u = nph 2 z=z nb 2 x — ;
r 2
that is, equal to half the cylinder having an equal base
and altitude.
312. Let it be required, as a last example, to determine
274
ELEMENTS OF THE
the solidity of the solid generated by the revolution of the
cycloid about its base.
The differential equation of the cycloid is
y d y
hence we have
dx~
du
V^ry — y 2
Kifdy
V2nj-if
which, being integrated by formula (E) Art. 243, and then
by Art. 226, we find the solidity equal to five-eighths of
the circumscribing cylinder.
Of Double Integrals.
313. Let us, in the first place, consider a solid limited
by the three co-ordinate planes, and by a curved surface
which is intersected by the co-ordinate planes in the curves
CB, BD, DC.
Through any point of
the surface, as M, pass
two planes HQF and
EPG respectively paral-
lel to the co-ordinate planes
ZX, YZ, and intersect-
ing the surface in the
curves IIMF and EMG.
The co-ordinates of the
point M are
AP=x, PM=y, MM'=z
INTEGRAL CALCULUS. 275
It is now evident that the solid whose base on the co-ordi-
nate plane YX is the rectangle A QM'P, may be extended
indefinitely in the direction of the axis of X without chang-
ing the value of y, or indefinitely in the direction of Y
without changing x. Hence, a? and y maybe regarded
as independent; variables.
If, for example, we suppose y to remain constant, and x
to receive an increment Pp = h, the solid whose base is
the rectangle AQM'P, will be increased by the solid
whose base is the rectangle M'm'pP ; and if we suppose
x to remain constant, and y to receive an increment
Qq — /c, the first solid will be increased by the solid whose
base is the rectangle Qqn f M f .
But if we suppose x and y to receive their increments
at the same time, the new solid will still be bounded by
the parallel planes epg, hqf, and w T ill differ from the prim-
itive solid not only by the two solids before named, but
also by the solid whose base is the rectangle n'M'mJW .
This last solid is the increment of .the solid whose base is
the rectangle M'Ppm!, when we suppose y to vary; or
the increment of the solid whose base is the rectangle
Qqn'M', when we suppose x to vary.
Let us represent by u the solid whose base is the rect-
angle A QM'P ; u will then be a function of x and y, and
the difference between the values of the increments of u,
under the supposition that x and y vary separately ; and
under the supposition that they vary together, will be equal
to the solid whose base is the rectangle n'M'm'N'. By
taking this difference (Art. S3) we have
276 ELEMENTS OF THE
hence,
solid n'N'm'M...M d?u 1 d 3 u , 1 d?u
hk dxdy ' 2 cfo 2 dz/ 2 cfody 2
and passing to the limit, by making h = and h=0, the
second member becomes -j — 7-.
dxdy
As regards the first member, the rectangle
n'N'm'M 7 = hxh
and the altitude of the solid becomes equal to M!M—z
when we pass to the limit : hence
d 2 u
dxdy ~
314. Although the differential coefficient
&u
dxdy ~
has been determined by regarding u as a function of two
variables, we can nevertheless return to the function u by
the methods which have been explained for integrating a
function of a single variable.
For we have
d?u ~\dxJ
«©
dxdy dy
hence
and integrating under the supposition that x remains con-
INTEGRAL CALCULUS. 277
slant, and y varies, we have
du
whence
**/*+*•
— dx = dx fzdy + JT c?07 ;
aa?
and if we integrate this last expression under the supposi-
tion of x being the variable, and make j X! dx = X\
u = fdxfzdy + X + Y.
It is plain that the constant, which is added to complete
the first integral, may contain x in any manner whatever;
and that which is added in the second integral, may contain
y : the first luill disappear ivhen we differentiate with
respect to y, and the second when ive differentiate with
respect to x.
The order of integration is not material. If we first
integrate with respect to x, we can write
d?u _ \dy) .
d,xdy dx (
and by integrating, we find
-——fzdxy u=fdyfzdx:
hence we may write
u — J fzdy dx, or u = ffzdxdy,
which indicates that there are two integrations to be per-
formed, one with respect to x, and the other with respect
to y.
24
278 ELEMENTS OF THE
315. If we consider the differentials as the indefinitely
small increments of the variables on which they depend,
we may regard the prism whose base is the rectangle
n'N'm'M', as composed of an indefinite number of small
prisms, having equal bases, and a common altitude dz.
Each one of these prisms will be expressed by dxdydz,
and we shall obtain their sum by integrating with respect
to z between the limits z — and z = MM', which
will give
/ dx dy dz — zdx dy.
316. It is plain that zdx is the differential of the area
of the section made by the plane HQF parallel to the
co-ordinate plane ZX ; and consequently
f z dx = area of the section HQF.
Hence, (Jzdx)dy is equal to the elementary solid in-
cluded between the parallel planes HQF, hqf t or
f(fzdx)dy=ffzdxdy
is equal to the solid which is limited by the surface and
the three co-ordinate planes. If we consider a section
of the solid parallel to the co-ordinate plane YZ, we have
fzdy = area of the section EPG, and ffzdxdy '■==. solidity
of the solid.
317. Let us suppose, as a first example, that
_ 1
z ~ tf + y 2 '
we shall then have
JJaZ + y 2, J J a? + y 2 J J J ap + y 1
INTEGRAL CALCULUS. 279
Let us now integrate under the supposition that x is con-
stant ; we then have
/*_^_ = .i tang -^ + jr,
J x l -\- y x x
in which X! represents an arbitrary function of x. If we
now make fX / dx = X, and integrate again under the
supposition that x is a variable, we have
fdx P\ 9 dy "l =Jdx[— tang- 1 -^- + X'l
J J x z + y z J Lx ° x J
= fe ang -y +x
J X X
dx u
The integral of — -tang -1 — is obtained in a series by
xx
substituting the value of (Art. 228),
x x 3x 6 5x 5 7x 7
and since, in integrating with respect to x, we must add
an arbitrary function of y, which we will represent by Y,
we shall obtain
r rdxdij _ x , Y _y_ + _£ l/Lljl. &c
We shall obtain the same result by integrating in the in-
verse order, viz., by first supposing y to be constant.
Under this supposition
r^^ = -tang- , ^+F )
J a? + y 2 y ° y
280 ELEMENTS OF THE
then integrating with respect to x,
'*/^=^[7 , '"< r 7 +I '']
-/
y y
y y
But by observing that (Trig. Art. XVIII),
tang 1 _ = __tang l -±-,
we shall have, after the second integration, and the addi-
tion of an arbitrary function of x,
and as we can include the term — logy in the arbitrary
function Y, this result may be placed under the form
ffj^y_ X+Y _fdy tJLt
J J xr -\-y z J y x
which is the same as the result before obtained, as may be
shown by placing for tang -1 — its value, multiplying each
term by — , and integrating.
318. When we consider
ffzdxdy
as expressing the solidity of a solid, it is necessary to con-
sider the limits between which each integral is taken, and
these limits will depend on the nature of the solid whose
cubature is to be determined. Let it be required, for ex-
INTEGRAL CALCULUS. 281
ample, to find the solidity of a sphere, of which the centre
is at the origin of co-ordinates. Designating the radius
by R, we have
a? + y* + z 2 =zR 2 ,
and consequently,
ffzdxdy =ffdxdy VR 2 -x*-y 2 .
If now, wc suppose y constant, and make R 2 — y 2 = R /2 ,
and then integrate with respect to x, we have
/ dx VR" - x' z -if=fdx VR /2 - x 2 ,
and integrating this last expression, first by formula (B)
Art. 239, and then by Art. 220, we have
fdxVR /2 -cc 2 = — ^n^^+ — R f2 sm- l ~ + Y;
and substituting for R' 2 its value, we obtain
It should be remarked, that fzdx expresses the area of
a section of the sphere parallel to the co-ordinate plane
ZX t for an v ordinate y — AQ, and to obtain this area we
must integrate between the limits x — and x — QF.
But since the point F is in the co-ordinate plane YX t
we have for this point z — 0, and the equation of the sur-
face gives
QF=:x=VrT^ 2 ;
therefore, for every value of y the integral fzdx must be
taken between the limits x = and x — VR 2 — y 2 . Inte-
23*
282 ELEMENTS OF THE
grating between these limits we have
fdxVR 2 -x 2 -f = —{R 2 -y 2 )sm" l {l)
since, sin -1 (l) = — :
hence,
fdyfzdx = ^-fdy(R> - f) = ^y-t> ) + X,
and taking this last integral between the limits y = and
y = AC = R, we obtain
6 '
which represents that part of the sphere that is contained
in the first angle of the co-ordinate planes, or one-eighth
of the entire solidity. Hence,
4 1
solidity of the sphere = — R?^~ — D 3 n.
We might at once find the solidity of the hemisphere
which is above the horizontal plane YX, by integrating
between the limits
x = — VR 2 — y 2 and x = + V R 2 — y 2 .
Taking the integral between the limits
x = and x = — V R 2 — y 2 , •
we have fzdx = — — (R 2 — y 2 ) ;
and between the limits
x = and x = -f VR 2 — y 2 >
INTEGRAL CALCULUS. 283
we have fzdx =— (R 2 — y 2 ) ;
hence, between the extreme limits, we have
fzdx = ?L(R*-tf).
Then taking the integral
Sdyfzdz = ^fdy(R*-f) p
between the limits
y=—R and y = 4- R,
we find the solidity to be
I**
or the solidity of the entire sphere is,
3
THE END
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