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Full text of "Elements of the differential and integral calculus"

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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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