ri
DEPARTMENTS OF MtCHANtCAL
& £LECTR)CAL E-NOiNEERING
OCT 1930
Universi'y of California
INTEGRAL CALCULUS.
A TREATISE ON THE
INTEGRAL CALCULUS
AND ITS
APPLICATIOKS
WITH NUMEROUS EXAMPLES.
By I. TODHUNTEK, M.A., F.RS,,
HOirOEAET FELLOW OF ST JOHN'S COLLEGE,
CAMBEIDGE.
SIXTH EDITION.
Hontron :
MACMILLAN AND CO.
1880
[The right 0/ translation and reproduction is reserved.]
library
CTambriligc :
PKINTED BY C. J. CLAY, M.A.
AT THE UKIVEESITY PEESS.
PEEFACE.
Ix writing the present treatise on the INTEGRAL Calculus
the object has been to produce a work at once elementary and
complete — adapted for the use of beginners, and sufficient
for the wants of advanced students. In the selection of the
propositions, and in the mode of establishing them, I have
endeavoured to exhibit fully and clearly the principles of the
subject, and to illustrate all their most important results.
The process of summation has been repeatedly brought for-
ward, with the view of securing the attention of the student
to the notions which form the true foundation of the Integral
Calculus itself, as well as of its most valuable applications.
Considerable space has been devoted to the investigations of
the lensrths and areas of curves and of the volumes of solids,
and an attempt has been made to explain those difficulties
which usually perplex beginners — especially with reference
to the limits of integrations.
The transformation of multiple integrals is one of the
most interesting parts of the Integral Calculus, and the ex-
perience of teachers shews that the usual modes of treating
it are not free from obscurity. I have therefore adopted a
method different from those of previous elementary writers,
VI TREFACE.
and have endeavoured to render it easily intelligible by full
detail, and by the solution of several problems.
The Calculus of Variations seems to claim a place in the
pi'esent treatise with the same propriety as the ordinary
theory of maxima and minima values is included in the
Differential Calculus. Accordingly a chapter of the treatise
is devoted to this subject ; and it is hoped that the theory
and illustrations there given will be found, with respect to
simplicity and comprehensiveness, adapted to the w^ants of
students.
In order that the student may find in the volume all that
he requires, a large collection of examples for exercise has
been appended to the several chapters. These examples
have been selected from the College and University Exami-
nation Papers, and have been verified, so that it is believed
that few errors will be found among them.
The work has been carefully revised since its first ap-
pearance, and additions made to it with the hope of increas-
ing its utility for the purposes of instruction, and of render-
ing it still more worthy of the favour wdth which it has been
received. An Elementary Treatise on Laplace's Functions,
Lamp's Functions, and BesseVs Functions has been published
as a sequel to the Treatises on the Differential Calculus and
the Integral Calculus.
T. TODHUNTER.
CaMBEIDCtE,
September, 1878.
CONTENTS.
CHAP. PAGE
I. Meaning of Integration. Examples 1
II. Eatioual Fractions 23
III. FormulaB of Keduction ........ 42
IV. Miscellaneous Eemarks ....... 52
V. Double Integration 77
TI. Lengths of Curves 87
VII. Areas of Plane Curves and of Surfaces .... 121
Vm. Volumes of Solids 166
IX. Differentiation of an Integral with respect to any quantity
which it may involve . 191
X. Elliptic Integrals 201
XI. Change of the Variables in a Multiple Integral . . . 210
Xn, Definite Integrals 242
Xni. Expansion of Functions in Trigonometrical Series . . 295
XTV. Application of the Integral Calculus to Questions of Mean
Value and Probability 323
XV. Calculus of Variations 338
XVI. Miscellaneous Propositions 395
INTEGEAL CALCULUS.
3 1 > } 4 J
CHAPTER I.
MEANING OF INTEGRATION. EXAMPLES.
1. In the Differential Calculus we have a system of
rules by means of which we deduce from any given function
a second function called the differential coefiticient of the
former ; in the Integral Calculus we have to return from the
differential coefficient to the function from which it was
deduced. We do not say that this is the object of the
Integral Calculus, for the fundamental problem of the subject
is to effect the summation of a certain infinite series of in-
definitely small terms; but for the solution of this problem we
must generally know the function of which a given function is
the differential coefficient. This we now proceed to shew.
2. Let (/) (x) denote any function of x which remains
continuous for all values of x comprised between two fixed
values a and b : where continuous has the meaning defined in
Art. 90 of the Differential Calculus. Let x^, x^,...x^_^ be a
series of values between a and b, so that a, x^, x^,...x^_^, b
are in order of magnitude ascending or descending. We
propose then to find the limit of the series
(x^- a) (j)(a) + (x^- x^ (ji (^J + {x^-x^) (f> (a?J +
when X, — a, x„ — x,,...b — x , are all diminished without
1 '2 1 ' ^ n—1 , , . .
limit, and consequently ?i increased without limit.
Put x^-a = h^,x^-x^=h2,...h- x„_^= \; thus the series
may be Avritten
\c^ (a) + K4> [x^ . . . + K-, <P i^n.,)+K ^ (^„-i)'
and may be denoted by 2A0 (x), for it is the sum of a number
T. I. c. 1
2 MEANING OF INTEGRATION.
of terms of which /i<^(a;) may be taken as the tjrpe. Since
each of the terms of which h is the type may be considered
as the difference between two vahies successively ascribed to
the variable x, we may also use the symbol {x) Aa:; as the
type of the terms to be summed, and 20 {x) Aa; for the sum.
We may shew at once that %^ {x) Aa? can never exceed a
certain unite quantity. For let A denote the numerically
greatest value which ^ (cc) can have when x lies between a and
h ; then X'^ (a?) l\x. is numerically less than (]i^ + h^-\-... + h,^A,
that is numerically less than ip — a)A.
We now proceed to determine the limit of S(^ {x) Aa;, Let
-y^r (x) be such a function of x that (x) is the differential
coefficient of it with respect to x. Then we know that the
limit of ^^^^^^ Y — when h is indefinitely diminished
is (j) {x). Hence we may put
'f {^.) - -^ (^i) = K [4^ (^i) + fii
where p^,p^,...p^ ultimately vanish. From these equations
we have by addition
'^{h)-f (fl) = 2(^ (x) Ax + thp.
Now SAp is numerically less than (b — a) p where p denotes
the greatest of the quantities p^, p^,...p^; hence %hp ulti-
mately vanishes, and we obtain this result, the limit of'Z^(x)Ax
when each of the quantities of which Ax is the type diminishes
indefinitely is y^r (b) —-^jr (a).
3. The notation used to express the preceding result is
I (j) (x) dx—'\lr{h)—'\lr (a) ;
J a
the symbol / is an abbreviation of the word " sum," and duj
represents the Ace of 2 ^ (x) Ax.
MEANING OF INTEGRATION. 3
4. Suppose that h^, h^,...h„ are all equal; then each of
them is equal to , and x^ is equal to a + - (& — a).
Hence I <^ [x) dx is equivalent to the following direction :
J a
"divide h — a into n equal parts, each part being h; in <^ (x)
substitute for x successively a, « + h, a + 2h,...a + {n — l)h;
add these values together, multiply the sum by h and then
diminish h without limit." If these operations are performed
we shall have as the result t/t (6) — ^/^ (a), where i/r (x) is the
function of which ^ {x) is the differential coefficient with
respect to x.
The student then must carefully observe that for the
foundation of the Integi'al Calculus we have a certain theorem
and a corresponding notation. The theorem is the following :
let -v/r {x) be any function of x, and [x) its differential co-
efficient with respect to a;; let w be a positive integer and
'nh = h — a, and suppose <^ {x) finite and continuous for all
values of x between a and h) then the limit when n is inde-
finitely increased of
A M (a) + (j!) (a + /i) + ^ (a + 2A) + . . . + <^ (6 - A) [
Is >/r (J) - -«/r [a).
The notation is that this limit is denoted by I j> (x) dx,
J a
SO that 1 (j){x)dx = '\^ {b) — y^ (a).
J a
As a particular case we may suppose a to be zero ; then
nh = b, and the limit Avhen n is indefinitely increased of
hU (0) + 4> (h) + cj, {2h)+...+ (b {b-Ji)\
is denoted by I ^ (x) dx, and is equal to t^t (6) — ^Ir (0).
5. A single term such as <^ (x) Ax is frequently called an
element. It may be observed that the limit of 20 (x) Ax will
not be altered in value if we omit a fiiiite number of its
elements, or add a finite number of similar elements; for
1—2
4 APPLICATION OF INTEGRATION.
in the limit each element is indefinitely small, and a finite
number of indefinitely small quantities ultimately vanishes.
6. The above process is called Integration; the quantity
rb
i (j> {x) dx is called a definite integral, and a and h are called
the limits of the integral. Since the value of this definite
integral is -^(h) —-^ (a) we must, when a function ^ {x) is to
be integrated between assigned limits, first ascertain _ the
function i/r {x) of which ^ [x) is the differential coefficient.
To express the connexion between ^ {x) and '^ {x) we have
and this is also denoted by the equation ,' •
\ (^ [x] dx = ■'Ir {x).
!'
In such an equation as the last, where we have no limits
assigned, we merely assert that '\lr{x) is the function from
which (ji {x) can be obtained by differentiation; y^r {x) is here
called the indefinite integral of <f) (x).
7, The problem of finding the areas of curves was one
of those which gave rise to the Integral Calculus, and fur-
nishes an illustration of the preceding Articles.
Let DPE be a curve of which the equation is i/ = (f) (x),
and suppose it required to find the area included between this
curve, the axis of x, and the ordinates corresponding to the
abscissae a and h. Let OA = a, OB = h; divide the space
AB into n equal intervals, and draw ordinates at the points
APPLICATION OF INTEGRATION", 5
of division. Suppose OM = a + (r - 1) h, then the area of
the parallelogram PMNp is
hj> [a + (r - 1) h].
The sum found by assigning to r in this expression all values
from 1 to n differs from the required area of the curve by
the sum of all the portions similar to the triangle PQp, and
as this last sum is obviously less than the greatest of the
lig-ures of which PMNQ is one, we can, by sufficiently
diminishing h, obtain a result differing as little as we please
from the requfred area. Therefore the area of the curve is
the limit of the series
h |(/) (a) + <^ (a + /i) + (^ (a + 2A) + + (/>(&- li)]- ,
and is equal to '«|r(6) — '\|r(a).
8. If yjr^x) be the function from which ^{x) springs by
differentiation, we denote this by the equation
/■
^ (x) dx = -^ {x),
and we now proceed to methods of finding A/r {x) when ^ [x) is
given. We have shewn, in Art. 102 of the Differential
Calculus, that if two functions have the same differential co-
efficient with respect to a variable they can only differ by
some constant quantity ; hence if -^ [x) be a function having
4>{x) for its differential coefficient with respect to x, then
if (x) + C, where G is any quantity independent of x, is the
only form that can have the same differential coefficient.
Hence, hereafter, when we assert that any function is the
integral of a proposed function, we may if we please add to
such integral any constant quantity.
Integration then will for some time appear to be merely
the inverse of differentiation, and we might have so defined
it ; we have however preferred to introduce at the beginning
the notion of summation because it occurs in many of the
most important applications of the subject.
We may observe that if (^^[x) and j)^{x) are any func-
tions of x,
j[cf>^{x) + <f>,{x)] dx =^j>,{x) dx +j<p,{x) dx\
6 INTEGRATION" BY SUBSTITUTION.
or at least the two expressions which we assert to be equal
can only differ by a constant, for if we differentiate both we
arrive at the same result, namely, (f>^(x) + (f)^{x).
Also, if c be any constant quantity
\c<f)(a;) dx=-c \4>[x) dx;
or at least the two expressions can only differ by a constant.
9. Immediate integration.
When a function is recognized to be the differential coeffi-
cient of another function we know of course the integral of the
first. The following list gives the integrals of the different
simple functions ;
lx'"dx = — — : , / — = loga7,
J m + 1 J X * '
I a'^dx = -, , I e'dx = e",
J log^a J
jsinx dx = — cosx, jcosx dx = smx,
f dx , f dx
I — ^7— = tan X, I . „ = — cot X,
J cos a; Jsurx
/:
= sm - or = — cos
/;
^{a^ - ic') a a'
= - tan - or = — cot - .
a^ + x^ a " a a " a'
10. Integration hy substitution.
The process of integration is sometimes facilitated by sub-
stituting for the variable some function of a new variable.
Suppose (f) (x) the function to be integrated, and a and b the
limits of the integral. It is evident that we may suppose
a; to be a function of a new variable z, provided that the
function chosen is capable of assuming all the values of x
required in the integration. Put then x=f{z), and let a' and
b' be the values of z, which make f{z) or x equal to a and h
respectively ; thus a =f{a) and b =f{b'). Now suppose that
INTEGRATION BY SUBSTITUTION. 7
i/r(.c) is the function of which (f){x) is the differential co-
efficient, that is suppose (p{x) =-^ — > then
/,
b
But by the principles of the Differential Calculus,
^-i^^' = * 1/(^)1/ W;
therefore t (/(*')! "t (/(<'')1 = i' ^ 1/(^)1/W d^
rb rb' fj[x
hence <^{x) dx= \ j> {x) ^ dz.
This result we may write simply thus
j<^(a;) dx=j(f){x) ^^dz,
provided we remember that when the former integral is taken
between certain limits a and b, the latter integral must be
taken between corresponding limits a and b'.
h
11. As an example of the preceding Article suppose that
is required. Assume x = a — z, then -^ = —1,
and ^ax — x" = a^ — z^. Thus
/,
dx r 1 Ax 1 f dz
f 1 dx ^
sji^ax - x^) j V(2a.r - x^) dz J VK " ^)
_,z _,a — X -,x
= cos - = cos = vers - .
a a a
a
X
Again, let | — t?^ — ^ be required. Assume «/ — . ,
*= ' J X Aj{2ax - a"") ^ 1-^
thus
dx _ a , r dx _ r 1 dx ,
d^~ {l-zf I X ^(2ax - a') ~~ J ^(2aa; - a'} dz ^
8 INTEGRATION BY PARTS.
dz 1 f dz
-I
aV{2(l-^)-(l-^r} aJv(l-^^)
1.-1 1 . _i a; — a
= - sill s = - sm .
a ax
Here we have found the proposed integrals by substituting
for X in the manner indicated in the preceding Article. This
process will often simplify a proposed integral, but no rules
can be given to guide the student as to the best assumption
to make ; this point must be left to observation and practice.
12. Integration hy parts.
From the equation
d (uv) _ dv du
dx dx dx
we deduce by integrating both members,
f dv J f du ,
uv = u-y- dx+ V -r dx,
J ax J ax
therefore ju -r dx = uv — \v -r- dx.
J ax J dx
The use of this formula is called " integration by parts.*'
For a particular case suppose v = x; then we obtain
ju dx = ux ^ j^ T ^^'
For example, consider I x cos ax dx. Since
1 d sin ax
cos ax = ^ ,
a ax
we may write the proposed expression in the fonn
fx d sin ax
J a dx
dx,
and this, by the formula, supposing u = - and v = sin ax,
X sin ax fsin ax ,
= dx
a J a
INTEGRATION BY PARTS.
X Sin ax cos ax
1 8 — .
a a
Again, ix^ cos axdx= I —
x^ d sin ax
dx
dx
•^ sin ax f 2a;
_ X sm ax f
~ a J
a
sin ax dx
0^ sin ax C2xd cos ax
_ x^ sin ax f
~ a J
a^ dx
dx
0? sin ax 2,x cos ax [2 cos ax ,
H 5 r, — dx
J a"
a a
x^ sin ax 2x cos ax 2 sm aa;
. . r ^, . - /"sin aa; de"' ,
Again, I e sm axax= j -r— ax
s'max ,, fae" cos ax ,
e — — dx
c J G
sin ace ,, ffl cos asc cZe'
c
e
-/'
,« ( --~^^~ -^^^_ ^^
c" da;
sm ax _ « cos ax ,, /a" sin aa?
,^ .. .„^^™^ ^„ _ j a o.^^^ ^,^ ^^^
e"^' 2 — e^^ - r
c c J c
By transposing,
(-?)/
e' V . a
e'"" sin axdx = — i sin aa: — - cos axj ,
f . , 6°* (c sin ax — a cos aa;)
therefore e"' sm axdx = r-; — 5 •
j a' + c'
Similarly we may shew that
e" (c cos ax + a sin ax)
I'
e" cos ax dx = 2 , a
a + c
10 EXAMPLES OF INTEGRATION.
13. The differential coefficient of any function can always
be found by the use of the rules given in the Differential
Calculus, but it is not so with the integral of any assigned
function. We know, for example, that if m be any num-
^n.+l
ber, positive or negative, except —1, then lx"^dx= ,
but when m = — 1 this is not true ; in this case we have
fdx
I — = log X. If however we had not previously defined the
term logarithm, and investigated the properties of a logarithm,
we should have been unable to state what function would
give - as its differential coefficient. Thus we may find our-
selves limited in our powers of integration from our not
having given a name to every particular function and investi-
gated its properties.
In order to effect any proposed integration, it will often
be necessary to use artifices which can only be suggested
by practice.
14f. We add a few miscellaneous examples.
Ex. (1). !^{a^-x')dx.
l^/{a^— X') dx = x\J{a^ — x^)+\ -jt-^ ^^ , by Art. 12, sup-
posing u=- \f{a? — x^) and v = x.
A 1 r // 2 2N 7 [ a^ — x^ ^ [ a^dx f x'dx
And \iJ\a —x)dx= -77-2 2\ dx = -tj—^ ir — -77-2 j. ;
therefore, by addition,
2 \\/{a^ — x") dx = X \l{<r — x^) + a" (-77-2 2\ »
therefore fv(a' " ^1 dx = ^^^"^"^"^ + ^ sin"^ - . Art. 9.
EXAJVrPLES OF INTEGRATION. 11
Assume \J{x' + a^) = z -x, therefore c^ = z'^~2zx,
dx _z— X
dz z
„ f dx r 1 dx J [dz ,
Ex. (3). [ ., f "" ., .
As in Ex, (2), we may sliew that the result is
log[x + ^/{x''-a%
Ex. (4). js/{x' + a')dx.
fv(^' + a') dx = x sf{x^ + a?) -J"/?^^^ ^J ^'t. 12.
Also Jv(^^+a^cZ^=/^P^c?..=|^^^^
therefore, by addition,
2 J V(«^' + a') cZ^ = ^ v'(^' + a') + Q^'j // 2^^-2x ;
therefore [v(-^' + «') f?^ = ^^^'^^+'''^ + 1" log (a; + V(^'+ a^)}.
Similarly I V(^" — «") c^^'^ = t, ■ — -^ log [x + \/(^^— a^) ]•
r (Za; — ^ { ^^
J Jia+bx + cx^) JcJ
' >s/(a+bx + cx^) \/cj I fa bx ^
\JC c
)
'JcJ'
dx
^Jc] /{( ,6V, ^ac-b'
v^{("+s;y
4o*
1*2 EXAMPLES OF INTEGRATION.
Putting oc-i ^ = z, our integral becomes, by (2) and (3),
— log {2cx + 6 + 2 Vc \/(« + hx + cx')],
where we omit the constant quantity —f- log 2c.
In a similar manner, by assuming s = ^ + — • we may make
\\J{a-\-hx-{- cx^) dx depend upon Ex. (4).
C dx _ J^ r dx
J \/{a + hx — cx^) ^cj I la bx ^ A
Put h^ for — t-t; — and z for x—^r, then tbe integn-al
4c 2c °
1 f dz . 1 . _ z
becomes —- -777^ irr , wliicb gives — - sin ^ 7- , that
^1 c J 1^ [h^ — z') ^ sJg h
IS
1 . _, 2cx — h
sm
^~2o
In a similar manner, by assuming z = x — -^ we may make
I V(« + t^ — c^'^) c?a; depend upon Ex. (1).
Put :. = i,tken|j^^^|^^=J^-^^i-^|(;y
EXAMPLES OF INTEGRATION. 13
f du If chi 1 . _,
1 . _!«
= — Sin - .
a X
-I a -I a 17
Since sin"* - + cos ^ - = -^ , a constant, we may also write
our last result thus,
f dec 1 _i a
77-2 T\ = - COS - .
]x\J[x —a) a X
By putting cc = - , as in Ex. (7), we deduce for the
required result
1 , X
iOOf
dx , C dx
Ex. (9). [7-^, and f- .
^ ^ J{x-ay' Jx-a
r dx _ _ 1 1
]{x- a)'" ~ m-1 {x - a)'"-' '
These are obvious if we differentiate the right-hand
members.
dx
Ex.
f dx _ 1^ r/_i i_\ ,
jx" — <^ 2aJ\x — a x + aj
_ 1 f dx 1 C dx
~2ajx — a 2ajx + a
14 EXAMPLES OF INTEGRATION.
1 , x — a
2a ° x + a
a .
,— positive: ..
x+a ^ x+a
This supposes — -- positive; if ~ — - be negative, we
must write
a — X
r dx 1
Jx'-d'^Ic
lo
or
2a ^ a + x
Ex. (11). [- ^^
+ bx + cx^ '
dx 1 [ dx
r dx 1 r
ja + bx + cx^ ~ c J
(. + ,
^Y 4ac-6'''
2cJ ' 4c'
If — ^-^— be negative, we obtain the integral by Ex. (10),
namely
1 1 ^ 2cx + h — Aj{b' — 4ac)
V(6'-4ac) ^^2cx + b + ^{b'-4^ac)'
If — T--2 — be positive, then by Art. 9, the integral is
2 ^ _, 2cx + b
rjrtan
V(4ac - b') V(4ac - b')
Ex. (12). f-^pL^^dx.
Ja + bx + cx
Ax + ~~ + B-
]a + bx + ex' "^^ - j a + bx + ex' "^^
Af '2cx + h ,/D_^^^ dx
2c] a
has been found in Ex. (11).
+ bx + cx' ^\ 2c J J a + bx + cx''
A
2c
The former integral is — log {a + hx + cx'), and the latter
Ex. (13). I
EXAMPLES OF INTEGRATION. 15
dx
COS a;
f dx fcosxdx f dz .„
= 2 — = 1 2 , n z= sm X,
j cos a; J cos cc Jl — z'
, , 1 + sin a; , , /tt x\
= h loo- ^— = log cot -r — s •
- ° 1 - sm cc * \4: 2/
r cIt^ CO
Similarly -.-^- = locj tan - .
'' jsinic ° 2
Ex. (14). f— 4^ , and f ^f . .
^ j a + 6 cos 05 j a + 6 sm a;
c?a; r ^aj
[ dx _ C dx
J a + b cos X J f . r.x ^ " ^A , 7 / 2
a ( sm' - + cos" k J + ^ ( cos'
2 - ''" 2
= /
sec*^ ^ dx
a + b + (a — h) tan^ ^
= 2 I -, — ^ — iT-2 > if ^ = tan - .
Ja + b+{a-b)z" 2
Hence, if a be greater than b, the integral is
2 ^ ...V(a-;.) 2 . /(«-^)fa°|
rj-tan —77^ tt^ or — r-^ — n{^^^ ~'
and if a be less than 6, the integral is
1 , z^/(b-a) + ^/(b + a)
loff-
^/{U' - a') ° ^ V(^ - «) - V(^ + «) '
X
,^(b — a)tim~ + \/{h + a)
that is -j-p 5- log —
16 EXAJVIPLES OF INTEGRATION.
r 7
To find I — —J—. — assume a; = - + ^ ; thus the iuteOTal
becomes — —j- , which has iust been found. Or we may
proceed thus,
dx
f dx _ r
J a + b sinx J
-I
a f sm 2 + cos' ^ 1 + 2b sm - cos ^
sec* X ^a;
a{\^ tan' |j + 2h tan |
=2/;^
, II = tan - ,
Put v = s + - , and the integral becomes
■^ a ft
9,
(Zy
and this can be found as before.
Ex, (15). Let i/r {x) denote any function of x, and let
•^"^ [x) denote the inverse function, so that -\/r [i/r"^ [x)'\ = u; : if
the integral of "^{x) can be found so can the integral of
■^'^ (a). For consider I ^~^ (a?) dx ; jDut i/r"^ (cc) = 2;, then
x = -^{z) : thus
I -v/r"^ (a;) dx= jz-r- dz = zx— \xdz =zx— \^ [z) dz.
In any of these examples, since we have found the in-
definite integral, we can immediately ascertain the definite
integral between any assigned limits. For example, since
therefore
"" dx
/
J a
EXAMPLES OF INTEGRATION. 17
= log [2a + ^/[{2ay + a']] - log [a + ^{a' + a')]
, 2 + V5
= log:.
p
15. The integral /.«'""' (ct + hx^y dx can be found imme-
diately if -^ is a positive integer, for (a + hx^f can then be
expanded by the Binomial Theorem in a finite series of powers
of X, and each term of the product of this series by a;'""^ will
be immediately integrable. There are also two other cases in
Avhicli the integral can be found immediately.
For assume a + hx"^ = f ;
i 1.1
therefore -=f-^j , 21 = 1^ [-T^
Hence L'""^ (a + hx^dx = L"^"' (a + hx'')
'P'
Q /,p+j-i /^ '^
-1
li — he a positive integer we can expand (f^—a)" in
a finite series of powers of f, and each term of the product
of this series by t^'"^'^ Avill be immediately integrable.
Again, J a;"'-' (a + Ix"") 'dx= x' (ax-" + h) ' dx ;
and by the former case, if we put ax~" + h = t\ this is im-
mediately integrable if
n
1 ... tn t)
be a positive integer; that is, if - + - &5 a negative integer.
T. I. c. 2
18 EXAMPLES OF INTEGRATION.
In the first case, if — were a negative integer the integral
might still be found, as we shall see in the Third Chapter, and
similarly, in the second case, if — f- - were a 'positive integer:
but as in these cases some further reductions are necessary, we
do not say that the expressions are immediatehj integrable.
Ex. (1). \x^ [a + xf dx.
Here — = 3 : assume a + x = f; the integral becomes
n
2 [(f - ay fdt or 2 [(f - 2at* + aY) dt,
which gives
it' 2af a'f]
thus jx'{a + x)'dx = 2{a + x)H^^-~{a + x)+^
r dx
Ex. (2)
x' (1 + xy
Here ??i = - 1, n = 2, ^- = — -; therefore — + " = - 1.
' ' q 2 n q
Assume x'^+l^ f : therefore x^ = js — 7 > ^i = 1
f-l'dt (f-if
dx
Also f ^" , = [ ^L^.dt
Jx'{i+xy~ J x'{x-'+iY'
Substitute for x and -^ their values, and this becomes —J dt,
which = — ^ or — -!^-!^ -.
X
EXAMPLES, 19
Here m = l, n = 2, ^ = --, therefore —+^ = -1.
Assume a^a;"^ + ! = «*, therefore «" =
dx
3 5
r dx _ r dt j._ 1 r^^_ 1
EXAMPLES.
r dx _ . _i 3 + 2a;
• Jv(l-3.'c-a;'0~^''^ "VIS"'
2. 1 log xdx = x (log ic — 1).
3. a;" log a; Ja; = r-^loga; irL
J *= ?i + 1 [ ° n + 1 j
4. f^sin^J^ = -^cos^ + sm^.
5. [^^^ = tan-i(0.
G. I / [ ^^^^ — ^j dx = sj{mx + x^) + m log [Jx + V(wi + a;)}.
This may be found by putting x = z^,
f - 1 + x^ _
7. a; tan ^ a3 cZx = — ^-^ tan ^ a; — ■!»;.
o ffi \2 7 Sa; „ . , sin 2a;
8. I (1 — cos X) dx = -^ — 2 sm x H r— •
2-2
20 EXAMPLES.
!-.
{i-xy 1 - cc 2 (1 - xy
C afdx _ J.
10. -. «=7r^l02f
a' + x'
a^'^^a'-x''
6? . _, a; — a
11. \\J i^ax - x") dx = ^-^ ^/{2ax - a;') + ^ sin
12, 1^7^ -^ = -J(2ax-x'') + avers~^-.
JJ{2ax-x-) ^^ ' a
14, -. — dx = log (x + sin x).
J x + smx
, ., fiC-rsiiiu; -, .<^
lo. I :^-r ax = x tan - .
ic + sm a? , , X
ax = x tan -
+ cos X z
16 ■ ^^
j a; (logic)" (n - 1) (log^a;)""' •
-j^y^ / og I Qg ^j ^^ = log a; . log (log x) — log x.
J X
r . ^ e"'' a sm( m + n) x-{m+n) cos {m+7i)
20. e'"'sin??2cccoswa;aa;=-7^; 2~r"7 — rz^^i
e"" a sin (m — n)x — (m — n) cos {m — n) x
21. [e"'' cos^ .r (?a; = i fe"'' (cos Sic + 3 cos a;) cZx
= J- (3 sin Sx - cos 8a:;) + -^ (sin x - cos x).
40^ ^ 8
EXAMPLES. 21
2
22
2a
23. rV(2a^'-a'')tZ^ = ^-
•'o '^
r" _ £c
24. vers ^ - c^x = 7ra.
Jo «
Proceed thus : let vers"^ - = 0, therefore x = a (1 — cos 6),
and the integral becomes 1 a9 sin 6 dd.
J
25. X vers"* -dx = — -t— .
Jo « 4
2G. ^ vers - cZic = — ^7— .
27. rW6'cos'^(Z^ = :^.
28. f-^^^ = 1 logtanf|+^'
J sm ic + cos X i\J2. ° \2 8
29.
dx
X f^{a + bx-\- cx') '
1
y
Put x = - and this becomes a known form.
30. fV(l^sm..<;.=-!^i-\^^-^-!^^
j X dX bx o
This may be obtained by putting sin"* x = 0.
SI. ; £7.r = ^ tan ^ + log cos 6, where sin 6 = x.
J{l-x')^
32. 1 ^ — 5 = —. (cot 6-{ — ^r— ) , where ic = a cos 6.
C sin^xdx _/« + &y, _i \/a tan .t a;
ja + 6 cos' a; ~ V ab' J ^^ \/(« + ^) ^'
22 EXAMPLES,
[ dx _(2a;''-l)_V(ljf^
]x^
35
3G
41,
42
43
^/{l+x^) 3^'
[tan''" ede = 2^^ - ftan^"-^ 9 d9
+ -{-lyx+i-iye,
^^n-l ^2n-3
2n - 1 271-3
X being = tan 6.
37. Shew tliat I sin mx sin wa; cZic and 1 cos wa; cos wa; dx ar
Jo •'o
zero if m and « are unequal integers, and = -^ i
m and w are equal integers.
.38. /{log ©I'd-- {log (^)}'- 3. {log^}V6.1og^-6.
39. I -IT?:; 5r dx = -^i — 6 tan ^ — log cos 6, where cot ^ = a
J X- (1 + ic^) 2 o '
40. [^^i±^./f^^.. = V(a»-a;^)-?^;^^
] a^-x y \a^rx) ^ ' ^/{a + x)
/ vers ^ - -2
f/v/(2aa; — ic^)
I t; = -rrq sr cos"V, if c is less than 1.
Jo 1 + ccosa; V(l — c)
f " e-^ cos' ede=fQ (ei" + e"*'').
J -Jn-
44. I — },^ . ^ g jr- . Assume 2; = a; + - .
_ [(a + Jr«")^ (7a; . . 7 n 4
4o. ^^ ^^ . Assume a + ox = .s .
J a;
23
CHAPTER II.
RATIONAL FRACTIONS.
16. We proceed to the integration of such expressions as
A' + B'a;+C'x\..+M'.
x""
A + Bx+Cx\..+Nx''
where A, B,...A\ B',... are constants, so that both numerator
and denominator are finite rational functions of x. If m be
equal to n, or greater than n, we may by division reduce the
preceding to the form of an integral function of x, and a
fraction in which the numerator is of lower dimensions in x
than the denominator. As the integral function of x can be
integrated immediately, we may confine ourselves to the case
of a fraction having its numerator at least one dimension
lower than its denominator. In order to effect the integration
we decompose the fraction into a series of more simple frac-
tions called partial fractions, the possibility of which we j)ro-
ceed to demonstrate.
Let -p. be a rational fraction in its lowest terms which is
to be decomposed into a series of partial fractions; suppose V
a function of x of the w**' degree, and U a function of x of
the (?i — l)"" degree at most; we may without loss of gene-
rality take the coefiicient of a;" in Fto be luiity. Suppose
F= {x -a){x- by (x' - 22X + d' + /3=) {x^ - 2^x + 7^ + h')',
so that the equation F= has
(1) one real root = a,
(2) r equal real roots, each = h,
(.3) a pair of imaginary roots a + /8 \/(— !)>
(4) s pairs of imaginary roots, each being 7 + S \/(— 1),
24 RATIONAL FRACTIONS.
By the theory of equations V must be the product of factors
of the form we have supposed, the factors being more or fewer
in number. Since V is of the rf" decnree we have
Assume
l+r + 2 + 2s = n.
^ A ■ s.^+ ^. +, ^. + ^-
Cx + D
+
a^-2oix + d' + l3'
E^x + F, E,x + F, . E^x + F ^
where A, B^, B^,...C, D, E^,... are constants which, in order
to justify our assumption, we must shew can be so determined
as to make the second member of the above equation identi-
cally equal to the first. If we bring all the partial fi'actions
to a common denominator and add them together, we have V
for that common denominator, and for the numerator a func-
tion of X of the (n — 1)* degree. If we equate the coefficients
of the different powers of x in this numerator with the cor-
responding coefficients in U, we shall have n equations of the
first degree to determine the n quantities A, B^, B^,... and with
these values of ^, B^, B^,... the second member of the above
equation becomes identically equal to the first, and thus -^
is decomposed into a series of partial fractions.
If V involves other single factors hke x — a, each such
A
factor wiU give rise to a fraction Hke ; and any repeated
factor like {x — by will give rise to a series of partial fractions
B Ti
of the form ^-—., |-^=i,.... In like manner other
factors of the form x" - 2oix + cr -^ ^"^ or [x" -2r^x+ rf + h'')'
will give rise to a fraction or a series of fractions resjpectively
of the forms indicated above.
17. The demonstration given in Art. 16 is not very satis-
factory, since we have not proved that the n equations of the
RATIONAL FRACTIONS. 25
rst degree which we use to determine A, B^, B.^,... are in-
ependent aud consistent.
A method of greater rigour has been given in a treatise on
be Integral Calculus by Mr Homersham Cox, which we will
ere briefly indicate. Suppose F {x) to contain the factor
: — a repeated n times; wo have, if
^{x) <}>{x) ^^^^ -^{ay^^ , ^(^0
Fix) {x-aY-y\r{x) {x-a)"f{x) [x-ay'
Now 6 (x) — r-^ "^ {x) vanishes when x = a, and is there-
^ ^ '' yjr [a) '
Dre divisible by x — a; suppose the quotient denoted by
; {x), then
F [x) {x - ay-' ylrix) f{a){x-ay'
'V (cc)
The process may now be repeated on _ ^. „-i , , . , and
„ (b (x)
hus by successive operations the decomposition ot ry-^
ompletely effected. In this proof a may be either a real
oot or an imaginary root of the equation F (x) = -, if
; = a + /3v'(— 1), then a — /3V(— 1), will also be a root of
<" (x) = 0; let b denote this root, then if we add together
he two partial fractions
y}r [a) {x - ay -f (6j {x - by '
<fe shall obtain a result free fi-om \/(— !)•
18. With respect to the integration of these partial
ractions we refer to Examples (9) and (12) of Art. 14 for all
he forms except -r-r, — ^ ^ — ^^^ , and this will be given
^ (x' - 2yx + y^ + 8'y *
lereafter. See Ai-t. 32.
Having proved that a rational fraction can be decomposed
n the manner assumed in Art. 16, we may make use of
26 RATIONAL FEACTIONS,
different alg^ebraical artifices in order to diminish the labour
of determining the constants A, B^, B^, — The most useful
consideration is, that since the numerator of the proposed
fraction is identically equal to the numerator formed by
adding together the partial fractions, if we assign any value
to the variable x the equality still subsists.
19. To determine the j^df'tial fraction corresponding to a
single factor of the first degree.
Suppose %r7-\ represents a fraction to be decomposed,
and let F{x) contain the factor x — a once; assume
where -4 is a constant, and . , , represents the sum of all
the partial fractions exclusive of , and F{x) = {x—a)-<^{x).
From (1)
4>{x)=A^{x) + {x-a)x{^) (2).
In (2), which holds for any value of x, make x=-a, then
</) (a) = Ay^r (a),
therefore A = %-t-\ •
'>/r(a)
Since F' (x) = '\lr{x) + (x— a) t/^' (x), we have
F' (a) = t (a),
therefore A = ^, , \ .
F (a)
20. To determine the partial fractions corresponding to a
factor of the first degree which is repeated.
Suppose F (x) contains a factor x — a repeated n timeSj
and let F (x) = {x — a)" -v/r (x). Assume
RATIONAL FRACTIOXS. 27
^(■^•)^ A , A , A , A , x(^2
F{.v) {x - a)" {x- a)"-' (x - a)""' x-a^y{r{x)'
where , ; , - denotes the sum of the partial fractions arising
ylr (x) ^ °
from the other factors of F{x). Multiply both sides of the
equation by {x - ay and put f{x) for -^rhl {x - a)" ; thus
J^ {x)
f{x)=A,+A^{x-a)+A^{x-ay...+A„{x-ay-'+^^fx-a)\
Y (ic)
Differentiate successively both members of this identity
and put x = a after differentiation ; then
/'(a) = 1.2^3,
r-'{a)^\n-lA,,.
Thus A^, A^, ... A„ are determined.
21. To determine the partial fractions corresponding to
a pair of imaginary roots which do not recur.
(b(x)
Let - U. ( denote the fraction to be decomposed; and
o' ± /3 \/(— 1) a pair of imaginary roots ; then if we denote
these roots by a and b and proceed as in Art. 19, we have
for the partial fractions
F'{a)x-a F'{b)x-b'
Suppose P ' = A—B v'(- 1) ; then since w^jri may be
obtained from ^, . by changing the sign of /^{—l), we
must have ^, . = A +B \/(— !)• Hence the fractions ai'o
28 RATIONAL FEACTIONS.
ic-a-/3V(-l) a;-a + /3V(-l)'
and tlieir sum is
2 A {x-a) + 2^/3
{x-ay + /3' '
22. Or we may proceed thus. Suppose x' — px+q to
denote the quadratic factor which gives rise to the pair of
imaginary roots a±^ V(— 1) ; then assume
(ji(x) _ Lx + M x(^)
F{x) x'-px+q^ ■^f{x)'
so that F{x) = {x^-px+q) ■\lr(x). Multiply by F{x) ; thus
(P {x) = {Lx+ M) ^lr{x) + {x'-px + q) xi^) (1).
Now ascribe to x either of the values which make
x^ —px + q vanish ; then (1) reduces to
<fi{x) = {Lx + M)f{x) (2).
By the repeated substitution of px— q for x^ in both
members of (2), Ave shall at last have x occurring in the first
power only, so that the equation takes the form
Px+Q = Fx + Q\
Put for X its value a + ^ a/(— 1) ^^^ equate the coeffi-
cients of the impossible parts ; thus
P = P' and therefore also Q = Q'.
Here P and Q are known quantities, and P' and Q' in-
volve the unknown quantities L and 31 to the first power
only, so that we have two equations of the first degree for
finding L and Jf.
23. To determine the jmrtial fractions corresponding to
a pair of imaginary roots which is repeated.
"We may proceed as in Art. 20. Or we may adopt the
following method. Suppose x^—px + q to be the quadratic
factor which occurs r times: assume
RATIONAL FRACTIONS. 29
F(x) (cc'-px + qY {x''-2)x + qY^ x'-JJX+q i/^C^) '
SO that F{x) = {x' -px + qY ^|r {x). [Multiply by F(x) ; thus
«/, (x) = {L,x + 2Q ^ {x) + {L^_,x + J/,_,) {x" -px + q)f (x)
+ + (x'-p^ + qYx{-^) (!)•
Now ascribe to x either of the values Avhich make
x^—px + q vanish; then (1) reduces to
<f> (x) = {L,x + il/J f (.r).
Proceed as in Art. 22, and thus find L^ and M^. Then
from (1) by transposition we have
<}> {x)-{L^x^M^)^{x) = {L^_^x+M^_^ {x'-px+q)y\r{x) +...
The right-hand member has x' —px + q for a factor of
every term : hence as the two members are identical we can
di^dde by this factor. Let <^j {x) indicate the quotient ob-
tained on the left-hand side ; then
01 {^-) = (^r-l ^ + K-^) t (^) + i^r-,^ + K-2) i^"- 1^-^ + 5) t {^)
+ + {^''-p^+qrx{^) (2).
From (2) we find L^^ and M^_^ as before ; then by trans-
position and division
<^,{x)HL,..^'rM^_„)^^{x) + {L,_,x^M,_^){x'-px+q)y^{x)+...
and so on until all the quantities are determined.
^2 _ 3^ _ 2
Take for example ^-^ _.,„ . '^ — zrr:-,. Assume it equal
^ {x- + x+lf{x + \y ^
to
L^x + M^ , L,x+M^ X¥)__.
{x' + x + lf^ x- + x^-l (x + iy
then a;" - 3^ - 2 = {L^x -f J/,) {x + 1)'
+ {L^x + il/J {x'+ x + l){x + lY-\- {x^ + x + iyx{^)"- (3).
30 EATIONAL FRACTIONS.
Suppose x^ -}■ x + 1 = 0; thus the equation reduces to
= {L^x + ]\Q(x'' + 2x + r).
Put —x — 1 for «^ ; thus
- 4^ - 3 = {L^x + M^ X = L^x^ + M^x
= -i:,(a; + l)+ilf,^;
therefore — 4 = — i^^ + ^v ^^^ — 3 = — Xj ;
thus L^ = 3, and M^ = - 1.
From (3) by transposition
x^ -2x-2-{2x- 1) (a; + l)**
= {L^x+M;){x'+x + i){x + iy+{x''+x + iyx{x).
The left-hand member is — Sx^ — 4<x^ — 4a; — 1 ; divide by
a)'^ + x + 1; thus
-{2x-t-l) = iL^x + MX^ + iy+{^'+^ + 'i-)x(.^) (4).
Again, suppose x^ + x +1 = ; thus
- 3^ - 1 = (i,«; + JlQ («;' + 2j; + 1) = (i.a; + il/J ic
therefore —S = —L^ + M^, and — 1 = — i^;
thus ij = 1 and M^^ = - 2.
Thus the joartial fractions corresponding to the quadratic
factor are found. The partial fractions corresponding to the
factor (x + iy may then be found by Art. 20. Or we may
from (4) by transposition and division by x"^ + x + l obtain
-{x-l)=x i^-)'
Thus
XJx) _ _ x-1 ^ _ x + 1 2 ^ 1_ 2 .
{^iy~ {x + iy {x+iy'^ix+iy x+i'^ {x+iy'
therefore
x''-Sx-2 3^-1 , x-2 , 2 1
~" + .2 .... 1 + TZmTs ~
{x'-^x+iyix+iy (x'+x+\y^ x'+x+i ix+iy x+i
RATIONAL FRACTIONS. 31
5x^ + 1
24. Examples. Kequired the integral of .^_» ^ .
By division we have
5a;' + 1 ^ , T . , S5« - 29
x^-Sx + 2 "--'"' af-Sx-\-2
85aj-29 A , B
Assume -k — ^i^ — —r^ = r +
a;'-3^ + 2 a;-la;-2'
therefore 35a; - 29 = -4 (a; - 2) + 5 (a; - 1).
Make x successively equal to 1 and 2 ; then
35 - 29 = - JL, or A =-6,
70 - 29 = B, ox B= 41 ;
, , 5a;' + 1 . ,. G 41
therefore — — ^ ^ = oa; + Id +
a;^-3a; + 2 ' a;-l^a;-2'
therefore I -J^-^ xfZa; = -^ + 15a;-61og(a;-l) + 411og(a;-2).
J X — ox "T" ^ ^
_ .,,,., T ^ 9a;'^ + 9a;-128
Eequured the integral of ^3_g^2^3^^c ) •
Since a;' — 5a;'' + 8a; + 9 = (a; — 3)" (a; + 1), we assume
9a;" + 9a;-128 _ A ^ B^ i ^' •
a;' - 5a;' + 3a; + 9 a; + l (a; -3)' a;- 3'
therefore 9a;'+9a;-128=^(a;-3)'+5j(a; + l)+5,(a;+l)(a;-3).
Make a; = 3 and — 1 successively, and we find
B^=-5, A = -8.
Also by equating the coefficients of x^, we have
9 = A+B„
therefore ^2~ ^^ }
therefore
/OTTiifs cZ. = - 8 log (. + 1) + ^ + 17 log (. - 3).
32 RATIONAL FRACTIONS,
x^ + 1
Required the integral of
Assume
x' + l
(x - 1)* {x' + 1)
_ A 1 -^. , ^ , A , ^ , c^ + D .
~ {x-1)* {x-iy {x-iy^ic-i^ x + i^x'-x + i'
therefore x' + l = {A^+A^{x-l)+A^(x-iy+A^{x-iy]{x'+l)
+ {B{x'-x + l) + {Cx + I)){x + l)](x-iy...{l).
Put£c = l, then 2 = 2A^ (2);
therefore ^^ = 1.
From (1) and (2) we have by subtraction,
x''-l=A^(x'-l) + [A^+A^{x-l)+A^{x-iy]{x-l){x'+l]
+ {B{x'-x + l) + {Cx + D)(x + iy{x-iy.
Divide hj x — 1, then
x + l = A^{x' + x + l) + {A^ + A^{x-l)+A^{x-iy}ix' + l]
+ {B{x'-x + l) + {Cx + D) (x + 1)} {x - l)^..(3).
Put a;=l, then 2 = SA^ + 2A^ ....(4);
therefore A^ = — ^.
From (3) and (4), by subtraction,
x-l=A^{x'+x-2)+A^{x'-l) + {A^-\-A,{x-iy{x-V){x'+V
+ [B [x' -x^\) + {Cx + D) {x + l)]{x-iy
Divide by a? — 1, then
l = A^{x + 2) + A,^{x'-\-x + l)-\-[A^ + A^{x-l)]{x'+l)
+ [B{x'-x + \) + {Cx + D){x^l)]{x-iy...[o).
RATIONAL FRACTIONS. S3
'P\itx = l, then 1-3.4^ + 3^+2^3 (G);
tliorefore A^ = — l.
From (5) and (6), by subtraction,
+ {B(x'-x+l) + [Cx + D)[x + 1)1 (a; - 1)\
Divide by a; — 1, tlien
= A^ + A^{x-V2) + A^{x' -^-x+l) -\- A^{x' ^1)
■\-[B{x'-x+l)-v{Cx-^D){(c + l)]{x-l) (7).
Put x=\, then 0=^, + 3^2 + 3-^3 + 2^^ (S);
therefore A^
5
From (7) and (8), by subtraction,
= ^ , (^ - 1 ) + ^3 (a^'' + a; - 2) + ^, (a;' - 1 )
+ {B{x'-x-\-l)^-{Cx + D) (aj+l)}(a;-l).
Divide by a; — 1, then
= ^,+ ^3(a; + 2) + yl,(a;^ + a;+l)
+ B{x''-x + l) + {Cx + D){x+l) (9).
Put a; = — 1, then
0=^2 + ^3 + ^, + 3j5 (10);
therefore B = ■^^.
From (9) and (10), by subtraction,
= ^ (a; + 1) + ^, (a;' + a;) + ^ (a;'' - a; - 2) + (Ca; + D) (a; + 1).
Divide by a; + 1, then
= ^3 + J^a! + ^(a;-2)+(7aj + X> (11).
Put a; = 0, then
A,-2B + D = (12);
therefore J^ = h
T. I. c. 8
34! RATIONAL FRACTIONS.
From (11) and (12), by subtraction,
therefore C' = — f ;
therefore
{x-iy{x'+i) {x-iy 2{x-iy 4^{x-iy
5 1 2a; - 1
+
8(a;-l) ' 24(a; + l) 3{x'-x + l)
Jix-
{x' + l)dx _ 111
tnereiore I (^_ i)*(^3+ i)- S{x-1Y'^ ^ix-lf^ 4{x-l)
5 11
+ g log (^ - 1) + ^ log (a; + 1 ) - - log (x' - a; + 1) .
25. We will give as additional examples the integration
x"'~^
of —„ — Y , supposing m and qi positive integers, and 7n — 1
less than n.
Required the integral of -j^ — zr when n is supposed even.
The real roots of cc" — 1 = are 1 and — 1, and the imagi-
nary roots are found from the expression cos rd±t,J{—l) sin rO,
where 6 = — , and r takes in succession the values 2, 4, ... up
n
to n — 2; see Flane Trigonometry, Chapter xxiil. Now by
<h(x)
Art. 19 if %TF-x he the fraction to be decomposed, the partial
±{x)
fraction corresponding to the root a is !^, , \ . In the
^ ^ F (a) x-a
present case
m(a) a a a . n i
■ Z,, \ = — — : = — -= — , smce a = 1.
F (a) wa" ' ?ia" n '
Hence corresponding to the root 1 we have the partial
fraction — -, :;- , and corresponding to the root — 1 we have
n {x — 1)
the partial fraction — --r . And corresponding to the pair
n (x "v i-)
of roots cos 7'd + \/(- 1) sin 7'd we have the pair of partial
fractions
RATIONAL FRACTIONS. 35
{ cosr^ + V(-l)s i nr6'r ( cosr^ - V (-l) sin rOY"
n [x — cos r0 — v'(— 1) sin rd] n [x — cos rd + v'(— 1) sin t6] '
that is
cos 7nr9 + \/(— 1) sin mrd cos ??i?'^ — a/(— 1) sin mrO
n[x~ cos rd — \J{—1) sin r^} w [a; — cos rO + V(— 1) sin rO] '
, , , . 2 cos mrd (x — cos r6) — 2 sin 77zr^ sin rO
that is ^-^, — ^ ^ — .
n {x — Zx cos rd + 1)
2 ^ cos mrO {x — cos r^) — sin mr^ sin rO
n ix — cos i-df + sin" r^ '
where S indicates a sum to be formed by giving to r all the
even integral values from 2 to 7i — 2 inclusive. Hence
/
x^'-'dx 1, , ,, (-1)"*, , ,,
^j;,— ^ = - log (a; - 1) + ^^ log (a; + 1)
+ - ^cos m?-^log(^^—2^cos 7-^+1) — " ^Ssin mr9 tan~^ — -. r- .
„m— 1
X
26. Bequired the integral of —^ — r- ivhen n is supposed
odd.
The real root of a;" — 1 = is 1, and the imaginary roots
are found from the expression cos rd ± V(— 1) sin rd, where
^ = — , and r takes in succession the values 2, 4, ... up to
w — 1. Hence as before we shall find
/•^"'-i dx 1 1
I -^ — =- = - log (x — l)+-^ cos ??zr^ log (x" — 2x cos r^ + 1)
J X —1 , n ^ n ^ '
2 ^ . - ^ _, a; — cos r^
— 2, sm mrd tan — ^ rr—
n sm rt7
3—2
SG RATIONAL FRACTIONS.
27. Required the integral of -„ — r when n is supposeo
Ob "T~ X.
even.
The equation a;"+ 1 = has now no real root; the imaginary
roots are found from the expression cosr^ + V(— 1) sinr^,
where 6 = — , and r takes in succession the vaUies 1, 3,... ui;
n
to w — 1. And if a be a root of x" + 1 — 0, we have
(f>{a) _ g"'"^ _^__^.
F' (a) " ^-' " wa" ~ ¥ '
thus the sum of the two fi-actions corresponding to a pair ol
imaginary roots is
2 cos mrd {x — cos rd) — sin mr6 sin rd
n {x — cos rO)' + sin'^ rd
Hence
/
—;, — =- = — S COS mrO log (a;^ — 2x cos 7-^ + 1)
, 2 ^ . o , -1 ^ - cos r^
+ - 2, sm ??i?'t^ tan — -. ^ — ,
n sm rU
where ^ indicates a sum to be formed by giving to r all the
odd integral values from 1 to w — 1 inclusive.
28. Required the integral of — ^^ — ~ when n is supposeo
odd.
The real root of 5?"+ 1 = is in this case — 1, and the imagi-
nary roots are found from the expression cos r6±\/{— 1) sin r9
IT
where d = — , and r takes in succession the values 1, 8, . . . ur
to n — 2. Hence we shall obtain
/
af-^dx (-1)*"', , ,,
— 2, cos Wi?'^log (x'—^x cos r^+ 1) + - S sm mrd tan — -. — ~y- .
RATIONAL FRACTIONS. 87
29. AVc will finish tlic Chapter with some miscellaneous
remarks on the decomposition of rational fractions.
6 (x) .
I. Suppose we have to decompose the fraction -p— - into
partial fractions where (f) (x) is not of a lower dimension than
F(x). Divide (f) (x) by F{x) ; let (}>^ (x) denote the quotient,
and ^2 {x) the remainder; then
<l>(x) = <f>,{x)Fix)+4>,{x);
therefore |^ = '/'i (^) + #(^ "
Accordingly we have now to decompose ^", '. into partial
fractions. It should be observed that we shall obtain the
same values for the partial fractions whether we apply the
methods of Arts. 19, 20, 21, 22, and 2.3 to %Pc or to-%-p( .
Take, for example, the case of Art. 19: since, by hypothesis,
F(a) = 0, and cj>{x) = <^^ {a:) F {x) + (j).^{x) , we have <{)(a)^(f)^(a).
II. From considering the values of ^j, A^, ... in Art. 20
we see that the foUoAving result holds: let r stand for any
integer from 1 to n both inclusive, then A^ is equal to
the coefficient of Jif"^ in the expansion oif{a + h) in powers
of h. Accordingly we may obtain A^ by ordinary algebraical
processes. For example, suppose we have to decompose
, ir-, rrs into partial fractions. Denote the required
{x — a) {x — by '■
partial fractions by
— '^^ h^. + + — ^
{x-ay {x-af x-a
'^ {x-by^ (^x-by-''^ x-b'
Here/(cc) = (a;-6)~^; therefore ^, is equal to the coeffi-
cient of A'"' in the expansion of (a-J + A)"^ in powers of //.
38 RATIONAL FRACTIONS.
The expansion can be effected by the Binomial Theorem;
thus we obtain
A =
p(p + l)...{p+r-2) J-jr'
r
| r-l ' {a-b)
p-tr-l '
Similarly if s stand for any integer from 1 to p, both
inclusive, then B^ is equal to the coefficient of h'~^ in the ex-
pansion of (6 — a + li)'" in powers of h.
III. Suppose that
and F{.) = [l-^)\l-^,) ^1-^.J;
here 6 {x) and F {x) are of the same dimensions. By decom-
posing: 1, ' { we obtain the term ^^ together with a series of
^ '^ F{x) h
partial fractions, a pair of which may be denoted "by
x — p x + p'
where p stands for -j- .
Then, by Art. 19,
^^-F'{py ^^ F'{-py
Let h be less than 1c, and suppose n to increase indefinitely;
then the term j^^ vanishes. And, by Plane Trigonometry,
Chapter xxiii. we have
, , , sin/^aj r,. > sinA;a;
RATIONAL FRACTIONS. 39
therefore ^ (p) = , , and since sin kp = 0, we have
a; + /3 /i
2r7r sin
a; - p a; + /3 A cos kp \x — p x + p
hrir
7 7 ( 1 ^"'^'
A/j cos rir \x jj-
Hence finally, if h be less than k,
. hrtr
. , r sm — ,—
&VQ.hx ^ ^ ^*
= 27r2
sm kx cos rTT [k x —rir )
where ^ denotes a summation with respect to r from r = 1
to r = oo .
The method of this example may be applied in other
similar cases.
IV. Some additional information on the theory of the
decomposition of rational fractions will be found in the first
volume of Serret's Cours cTAlg^bre Saperieure, 1866. Sug-
gestions which are intended to diminish the numerical labour
involved in the process of decomposition will be found in the
Cambridge and Dublin Mathematical Journal, Vol. iii., in the
Mathematician, Vol. ill., and in the Quarterly Journal of Ma-
thematics, Vol. V.
40 EXAMPLES.
EXAMPLES.
, f dx 1, (x-iy 1 ^ _i2a;+l
„ Cx'-l , , /a;-2\i
.2 K ^1 ^_^
r2a;'-3a' , 5 ^ ^,x 1 , a^-c
— 7 ^4- ao; = 77- tan 7- ^og — — -
J x*-a* 2a a 4:a °x + c
j{x' + l){x' + x + l) 2 * x' + l ^ ^/S V3
a; + a.
^- '' ■^ + l)(a;''' + a; + l) 2^ x' + l ^"^
^ f x^dx 1 , X — 1J2 _i X
7- j^q:^^ = 6^^«^+i + -F*^" V2'
8. . . . . . dx = - log -
/:
x' + x'+l 2 ='a;' + ^ + l'
9- -7 Tv^ ^dx = x + \og
{x-l){x-2) ^ ^iC-1
— - tan ^ a;.
EXAMPLES. 41
, . r a;c?a; 2 1 1 , /i . n
^*- j(l+a:)(l+i>^)''(l+aO~5 14-2^ 2^og^-L+^V
/■ x'dx _ 1 , a;'^ — ic \/2 + 1
J Z+1 ~ 4V2 ° ^N^^2T 1
+ ^ [tan-^ {x V2 + 1) + tan"^ {x v/2 - 1)}.
+ ^ {tan"' (2a; - V3) - tan"' (2^ + V3)}.
^ Y "
17. L,,-, "^ o. . Assume 1 — ?/' = vV.
18. It- c-oTTi is ?r-^ • Assume y = -, .
J (1 + rr) ^(1 + 3^ H- 3«;-) -^ 1 + «
42
CHAPTER III.
FORMULAE OF REDUCTION.
•SO. Let a + hx"" be denoted by X; by integration by
parts we have
L~-^X^ dx = ^^ - f^ pX'-' ~ dx
J m J 771 ax
^^^-^Mh'^^n-^XP-^dx (1).
m m J
The equation (1) is called &, formida of reduction; by
means of it we make the integral of ic™'' X'' depend on that
of a;'""^""^jr''~\ In the same way the latter integral can be
made to depend on that of a;'""^^""^X'^^; and thus, if p be an
integer we may proceed until we arrive at a;"'*"*"^X-^~^, that
is a;™'^"''"\ which is immediately integrable.
From (1), by transposition, ♦
r m+„-i^p-i ^^ ^ <^ _ ^ L--^x' dx.
J bnp onpj
Change m into 7n — n and p into ^ + 1 ; thus
{x'^-X'dx = f^^^'l _ '^~\, U-^-X^^' dx...{2).
J bn{p + l) hn{p + l)J
This formula may be used when we wish to make the
integral of cc"'Z^ depend on another in which the exponent
of X is diminished and that of X increased. For example,
if m = 3, w = 2, and ^ = — |, we have
r x^dx _ X 1 r dx
FORMULA OF EEDUCTION. 43
The latter integral has already been determined, and thus
the proposed integration is accomplished.
Since L^-^Y" dx = fa;"'-'X^-' (a + hx'') dx
= a lx'"-'X'-' dx + 6 [*•'"+"-' X"-^ dx,
we have by (1)
^^' - ^ [a;"'"^"-' J''-' dx==a \ x^'-'X^' dx + h [x^'^-'X^'dx,
m m J J J
therefore ( x^'-'X^'' dx = ^^ - ^ ^'"^ + ""^^ f a;™-^8-» X'-' dx.
J am am J
Change p into ^:) + 1, and we have
(x-^X^ dx = ^^^" - ^ ^^^^ + ^^ + ^^ f o;-"-^ Z^ ^^ (3).
Change m into m — n and transpose, then
f.2;"'-^X^£?a;- ■^""'""^''' (m-n)a f„.-n-^j, ^^^^u),
J b [m + np) b [m + np) J • • • v >•
We have already obtained from (1) by transposition
J bnp bnpj
also [a;"'-^Z^ dx = a f cc'-'X-^"' <Za; + 6 L-"'+"-^X^-^ cZo; ;
therefore [x'^-'X^dx = a (x^'-'X^' dx + ^-^ - — \x"'-'X''dx:
J J np npj
therefore fx'^-'X'dx = ^^ + -^^^^ f a^^-'X""^ cZoj (5).
J m + np m + npj ^ ^
Change p into p -\-\ and transpose ; thus
[."-X'& = - -g'-Jg^ + "' + "^ + " L-X'" <fe (6).
44 FORMULA OF REDUCTION.
31. If an example is proposed to wliicli one of tlic pre-
ceding formulaj is applicable, we may either quote that
particular formula or may obtain the required result inde-
r x^dx
pendently. Thus, suppose we require I - ,, ^_ .f. ; we have
J V(c -a;") J ax
= _ ^(c^ _ x') x'^-' + {m - 1) [^""'-^ V(c' - x') dx
= - sJic^ - x^) x-^ + (m - 1) /^'-^^^^ •
By transposition.
therefore
/
_jt;;^d^_ ^ _ X--' ^{c' - x') [m - 1) c^ r ar-' dx
V(c' - x') " m m J V(c' -of) ^ ^'
This result agrees with the equation (4) of the preceding
Article if we make a = c\ h = - I, n = 2, j) = - ^, and change
m into in + l.
f dx
Again, suppose we require j ^,^ // 2^^gN • We have
By transposition,
, , „ f (^.r _ V(a ' + x') [ dx
FORMULAE OF REDUCTION. 45
and by changing vi into m — 2 we obtain
dx _ ^/{a^ + x-) 071 — 2 f dx
/.
x'" V(a' + x') {m - 1) d'af"-' {m - 1) a^ ^"'~' V(a' + ^')
(2).
f x^dx
Another example is furnished by I —-^ 2\ > which may
r x"*~^^dx
be written I -r-^ r ; if in equation (4) of the jueceding
J Y yZa -^ Xj
Article we make h = — 1, n = 1, p = — \, and change a and m
into 2a and ??i + \ respectively, we have
/■ x^dx x^~^s/{1ax-x') a {2m -1) f x'^'^dx
J \j{2ax - a") "" m "^ 7?i J7(2^^^^^
(3),
which of course may be found independently.
32. In equation (6) of Art. 30 put a = c\ m = 1, n = 2,
5 = 1, and p = — r; thus
[ dx _ X 2r — Srdx
J{x'T^~2{r-l)c'{af + c'r''^2{r-l)c'J{x' + cT''
This formula Avill serve to reduce the form
{Ax + B) dx
k
32\r >
{^x'-2ojc + a^ + ^y
wdiich occurs in Art. 18 ; for this last expression may be
written thus,
^ A {x — a) dx . . _ /* dx
that is
_ A 1 .. p, r dx
~2{r-l)[{x- af^^Y' ^ 1 {(^ -°-' + ^')Y '
"By putting cc - a = cc', we have
r dx _ f dx'
and thus the above formula becomes applicable.
46 FORMULA OF REDUCTION.
83. These formulae of reduction are most useful when th(
integral has to be taken between certain limits. Suppos(
<}){x), x{^)> '^{^)> functions of ic, such that
i(f){x) dx='x_{x) + j-^ (x) dx,
then I cf){x) dx = x {b) —%(«)+/ '^ (^) dx,
J a J a
as is obvious from Art. 3.
For example, it may be shewn that
n
({c' - x'fdx = ^^^'-fy + J!^ [(c^ _ ^2) f-^ dx ;
ti . . . -
suppose -x a j^ositive quantity, then x (c^ — x"^) ^ vanishes both
when x = and when x = c. Hence
{c' - x') ' dx = -^ (c' - xy dx.
The following is a similar example. By integration by
j)arts
jx^' (1 - xy-' dx = - ^^^" or' + ^ L"-^ (1 - xfdx.
Hence f a'"' (1 - x^ dx = ^^ I x"^' {1 - cK)"tZa:.
Thus if r be an integer we may reduce the integral to
r^ 1
(1 — xY^'~^ dx, that is to ^ ; hence
Jo ?i + r-l'
34. The integration of trigonometrical functions is faci-
litated by formula; of reduction. Let </> (sin x, cos x) denote
FORMULA OF REDUCTION. 47
any function of sin x and cos x ; tlicn if wo put sin x = z, we
have
c C clx
(f) (sin X, cos x)dx=\(f) [z, V(l - z^)] ^ ^^
For example, let ^ (sin a;, cos x) = sin" a; cos' a; ; then
[sin" X cos' xdx = U" (1 - z'^^^-'^dz (2) .
If in the six foraiulce of Ai't. 30 we put a = 1, h = -l,
n = 2, j3 = i {q — 1), we have
/"s-'-i (1 _ z'y^^-'^dz
m m J
4+1 2 + 1 J ^
^ .-(1 -/)'"" ^ m±£±l r «« (1 _ ,=,.„_„^,
??z + 2 — 1 m + q — lj
2 + 1 g + 1 J ^ ^
If we put m=p + l, and ^^ = sin x, the first of the above
equations becomes
f • V n 1 sin^^^ X cos'"^ a; , 0' — 1 r . ,^,, 0-2 j
sm" a cos^ a;c?a: = ^ h ~ ^- sm''+- a; cos" xdx^
J _?J + 1 ^ + lj
and similarly the other five equations may be expressed.
48 FORMULA OF REDUCTION.
35. The following is a very important case :
r • [ d cos oc
/sin" xdx = — I — -T — sin""^ xdx
= — cos X sin""^ x-\- {n—l)\ cos'' x sin""^ xdx
= — cos X sin""^ a; + (n — 1) I (1 — sin^ a;) sin"~^ xdx.
Transposing, we have
n I sin" xdx = — cos x sin"~^ a:; + (n — 1) Isin""'^ xdx ;
,, „ f ■ n 7 cos o; sin'^'V-r ?i-ir. „_2 ^
therefore sm xdx = 1 1 sm xdx.
n n
From the last equation we deduce, if n be positive and
greater than unity,
L
sin" xdx = 1 sin" ^ xdx.
?i ^0
Similarly, if n be positive and greater than 3,
/,
in- _ w — 3 r^"" .
sin" ^ icc?^; = sin""* ircZ^.
n — 2i
'' -^.'
Proceeding thus, if n be an even positive integer we shall
arrive at I dx or ^tt ; if n be an ocZcZ positive integer we
•'
shall arrive at sin xdx, w^hich is unity. Hence, if n be a
-'0
positive integer,
I sin" xdx = — — y^ / .. »' Z (n even),
Jo n{n-2){n-4!) 3
These two results hold if we change sin x into cos x, as
will be found on investigation.
36. From the preceding results we may deduce an im-
portant theorem, called Wallis's Formula.
FORMULAE OF REDUCTION. 49
Suppose n an even positive integer ; then
rsin"^^^ = '^ 71^3 n-5 8 1 TT
* • n-i J n-2 11-4 n-6 2 .
sm xdx = r-. — ;, . — — -. (2).
1 w — Iw — 3?i — 3
/,
Now it is obvious that I sin""' xdx is less than
Jo
sin" ^ xdx and greater than I sin" icc?^ ; because each
Jo Jo
element of the first integral is less than the corresponding
element of the second integral and greater than the corre-
sponding element of the third integral. And it has been
shewn that
sin" xdx -
^^ - 1
sin"~^ xdx
n
I sin" xdx -,
Therefore ~ is less than 1 and greater than .
I sm" xdx
Jo
Hence the ratio of the right-hand member of (1) to the
right-hand member of (2) is less than unity and greater than
: thus
n
TT. • , ,, 2.2.4.4.6.6 (n-2)(w-2)
— is greater than - — - — - — = — ^— ,= -. ^rz-, ^\ .
2 *= 1.3.3.5.5.7 (?i-3)(w-l)
2.2.4.4.6.6 {n-2){n-2) n
and less than - — „ „ ., — ^-^ > 1-..~, , v t ■
1.3.3.0.0.7 (w - 3) (n - 1) ?i — 1
EXAMPLES.
n
1. (d' + xydx=-^ r-^ + -v-r (« + ^ ) f^-^-
J ^ ' 71 + 1 n + l J
T. I. c. 4
50 EXAMPLES.
r ,, , X-" ' V'lax -
m + 2 j ^^ ^
3. \x sf(2ax -x'')dx = -^ (2ax - x'f- + a j^/{2ax - x^) dx.
4. a; i^[zax — a; } aa; = -^ .
Jo ^
5. [ xV(2aa; - a;^) ^^ = - 1 {2ax-ar)^ + ^-^jxj(2ax - x') dx
6. x^ J{2ax — x") dx =
JO
r2a 77ra*
7. I ic' ^(2aa; — x^)dx = — q- .
8. fa;" (log xr dx = "^"^ ^^^^/''^"' - ^ f^" (log a.)'""^ dx.
J ^ ° 71+1 W + lj
r x"'^^ ( 2 2
9. J^" (log o.)^ c^o. = ^^-^ |(log xY - ^^ log ^ + ^^^:^
5«V
10. i^\ec'ede = i
^^- Jo v(«+^) U v"-
1 2. [sin' ^ cos' 9de = -l cos" ^ + i cos' ^.
13. [ f = 3 (tan 61 - cot 6') + i (tan' ^ - cot' 0).
J sm ^ cos a
fsm''ed9_ sine ^. 1 - sm ^
^ • J "^s' ^ ~ 2 cos'-' '^ ^ '° 1 + sin 6* •
EXAMPLES. 51
15. I ' (cos 2^)^ cos ede= ^"L^J^ .
Assume J {2) sin 6 = sin </>.
IG. p(a-^^)cos-^^x=(l + ^y^.
17. fvers"^ -J (Za; = (tt- - 4<) a.
/••'' sin'a;c/^ c'^-l, ,, , v 2-c
18. V = — s- log (1 + c) + -s-r •
Jo 1 + ccosa; c* ® ^ 2c'
19. If ^ (/O = I (1 + c cos a;)"" (fe, shew that
{n - 1) (1 - c') </) (/i) = - c sin a; (1 + c cos a;)""'''
+ (2;i -S)<p{n-l)-{n-2)(l) (n-2).
20. sji^ax — X') vers"^ -dx = —r— .
Jo OK' TC
21. ic v(2ow; — a;") vers - cte = -^ + -y- .
22. (tan x)' dx=j%-i log 2.
Jo
c being less than unity.
24.. Let P =Ax'' + Bx" + Ca;= + . . ., V^_ „= j x"'P''dx,
a = vi + l + na, (3 = m + l+7ih, <y = m + 1 + nc,. . .
Then
V„, „ = A F„,,, „_, + B F:,, „_, + C F„,,, „_, + ...
{Cambridge and Dublin Mathematical Journal, Vol. III.
page 242.)
4—2
52
CHAPTER IV.
MISCELLANEOUS REMARKS.
37. We have at the beginning of this book defined the
integral of ^ {x) between assigned limits a and h as the limit
of a certain sum 2^ {x) Aa?, and have denoted this limit by
/ ^ {x) dx. We have shewn that this limit is known as soon
•< a
as we knoAV the function -^ (x) of which <^ (x) is the differen-
tial coefficient. In the pages immediately following we gave
methods for finding '\jr (x) in different cases. We shall now
add some miscellaneous remarks and theorems, mainly in
order to recall the attention of the student to the process of
summation which we placed at the foundation of the subject.
88. Suppose we wish to find the integral of sin ic between
limits a and h immediately from the definition. By Art. 4 we
have to find the limit when n is infinite of
h [sin a +sin (a + h) + sin (a + 2/i) + sin [a + {n — 1) li]\
where h = - (h — a).
It is known from Trigonometry that this series
- . / n — \.\.nh 1 . ( h — a h\
A sm I a H ^— li\ sm -^ /i sm ( a -1 ^ o J
. h . h
sm 2 sm 2
MISCELLANEOUS KEMARKS. 53
The limit of r when n is infinite and therefore h zero
. h
sin 2
is 2; hence the required integral
^ . h + a . b — a ,
= 2 sm — ^r— sm — ^— = cos a — cos o.
39. Kequired the limit when w is made infinite of the
series
n n n n n
This series may be written
"^^'%'^''% "^'''
putting ^ for - , Ave obtain
jl 1 1 1 ]
|l "^ 1 +/i^ "^ 1 + i-lhj "^1 + {n-lfU] '
Comparing this with Art. 4 we see that the required limit is
i nor f ' fi 0(j
what we denote by I = 2* Now (r^ ~2 = tan~^«; hence
J is the required limit.
40. We define I ^ [x) dx as the limit when n is infi-
J a
nite of
\(^ (a) + h^(^ [x^ + h„(^ (^„_ J.
Now let A and B be the greatest and least values which
^ [x) takes between the limits a and h ; then the series is
less than
(/^ + /'2+ +^0^i.
54 MISCELLANEOUS REMARKS,
and is greater than
{K'rh + + K)B;
that is, the series lies between
{h — a)A and {h - a) B.
The limit must therefore be equal to (h — a) G, where G is
some quantity lying between A and B ; but since (^ [x) is
supposed continuous, it must, while x ranges from a to h,
pass through every value between A and B, and must there-
fore be equal to G when x has some value between a and h.
Thus G = ^[a + 6 [h — a)], where 6 is some proper fraction,
and
•b
4> (x) dx={b- a) (j){a + d{b- a)].
J a
Similarly if yjr (x) retains the same sign while x lies be-
tween a and b, we may prove that
f
J a
^ (x) -^Ir (x) dx = (f) [a + 6 {b — a)] j i|r (x) dx.
•I a
41. The truth of the equation
rb re rb
I (ji{x)dx=j <p{x)dx+j (f){x)dx (1)
will appear immediately; for suppose ^fr (x) to be the integral
of (j) (x), then we have on the left-hand side
'^{b)-ir (rt),
and on the right-hand side
i/r (c) — -v/r (a) 4- -x^ (J) — i/r (c).
In like manner the equation
[ (f>{x)dx = -r <f){x)dx (2)
J a J b
is obviously true. We may shew also that
I <}>{x)dx=l j){a — x)dx (3).
MISCELLANEOUS REMARKS. 55
For putting a — a: = z wc have
j (f)(a — x)dx = — j (J) (z) dz,
therefore I (f){a — x)dx—— cf) (z) dz
J J a
= ["(/, (.) dz, by (2).
ra fa
Of course 1 (f>(z) dz— j if) (x) dx, since it is indifferent whe-
•' ''0
ther we use the symbol x or z in obtaining a result which
does not involve x or z.
We have from (1)
I (j) (x) dx= \ (f){x) dx+ I cf) (x) dx.
J J -la
The second integral on the right-hand side, by changing
X into 2a — x , will be found equal to
1 (2(X — o:) dx or (2a — x) dx.
•'a J a
Hence
I <^ (a;) cZa; = I {^ {x) + ^ (2a — x)] dx.
•J ■Jo
Hence, i£ (f) (x) = cf) {2a — x) for all values of x comprised
between and a, we have
[ (l){x)dx='2l <f>(x)dx (4),
•'0 -^
and if (f> (2a — x) = — <^ (x), we have
/•2a
<}>{x)dx = (5).
For example,
sill' d d9= 2 [^\m' Odd by (4)
n
•'
56 MISCELLANEOUS REMARKS.
rir
and I cos'edd = by (5).
•'0
42. Such equations as those just given should receive
careful attention from the student, and he should not leave
them until he recognises their obvious and self-evident truth.
/,
cos' 6 dd is by definition the limit when n is infinite of the
series
h [cos' h + cos' 2A + cos' U + cos' (w - 1) h],
where nh = tt. Now
cos' h = — cos' (/I - 1) h, cos' 2/« = — cos' {n — 2)h, ;
thus the positive terms of the series just balance the negative
terms and leave zero as the result.
In the same way the truth of sin' dd6 = 2 sin' 6 d9
follows immediately from the definition of integration, and the
fact that the sine of an angle is equal to the sine of the sup-
plemental angle.
Suppose h greater than a, and <^ {x) always positive be-
tween the limits a and h oi x; then every term in the series
2^ {x) Aa; is positive, and hence the limit 1 <^ [x) dx must
he a positive quantity.
43. All the statements which have been made suppose
that the function which is to be integrated is always finite
between the limits of integration; for it must be remem-
bered that this condition is included in the word continuous
of the fundamental proposition, Art. 2. If therefore the func-
tion to be integrated becomes infinite between the limits of
integration, the rules of integration cannot be applied; at
least the case must be specially examined.
r dx ^ . .
Consider I ' — r ; the value of this integral is
2 — 2^/(1 — «). Here the function to be integrated becomes
infinite when ^=1; but the expression 2 — 2^/(1— a) is
finite when a = 1. Hence in this case we may write
MISCELLANEOUS REMARKS.
57
•1 a
clx
V(l - ^r)
= 2, provided that we regard this as an abbrevia-
te j?
tion of the following statement: " , ,,^ ,
*= 'oV(l-^)
is always finite
if a be any quantity less than unity, and by taking a suffi-
ciently near to unity, we can make the value of the integral
differ as little as we please from 2."
Next take
dx
; the value of this integral is - log (1— a),
which increases indefinitely as a approaches to unity. Hence
in this case we may write \ ^ = go provided that we
X
reo-ard this as an abbreviation of the following statement :
fd fly
" I increases indefinitely as a approaches to unity, and
by taking a sufficiently near to unity we can make the inte-
gral greater than any assigned quantity."
Next consider
7- rr,; the integral here is :;
[l—xY ° l — x
If
without remarking that the function to be integrated be-
comes infinite when x = 1, we propose to find the value of the
integral between the limits and 2, we obtain —1 — 1, that is
— 2. But this is obviously false, for in this case every term
of the series indicated by S (x) Ax is positive, and therefore
the limit cannot be negative. In fact . ' ,. and .-. _ .^
are both infinite. This example shews that the ordinary
rules for integrating between assigned limits cannot be used
when the function to be integrated becomes infinite between
those limits.
44. In the fundamental investigation in Ai't. 2, of the
rb
value of j) (x) dx, the limits a and b are supposed to be
finite as well as the function ^ [x). But we shall often find it
convenient to suppose one or both of the limits i7ifinite, as we
will now indicate by examples.
58 MISCELLANEOUS REMARKS.
Consider I .; 5 ; the inte;]jral is tan"*j?. Hence (
Jl + aJ ^ Jol+i
= tan"^ a ; the larger a becomes, the nearer tan~^a approach(
IT
to - , and by taking a sufficiently large, we can make tan~^
TV
differ as little as we please from - ; hence we may wri1
as an abbreviation of this statement
d.
X IT
l^x' 2 •
Similarly I — - = log (1 + a) ; and by taking a laro
enough we can make log (1 + a) greater than any assigne
quantity. Hence for abbreviation Ave may \vrite
/.
dx
z =CO.
l + x
45. Suppose the function ^ (x) to become infinite om
between the limits a and b, namely, when x— c. We cannc
then apply the ordinary rules of integration to 1 (p [x] dx; bi
J a
we may apply those rules to
"C-;u. rb
<j> (x) dx+ i (j) (x) dx
for any assigned value of fjb however small. The limit of tl:
last expression when yu, is diminished indefinitely is called I
Cauchy the principal value of the integral I (x) dx.
J a
For example, let cf) {x) =
c — x
,, , '1^ dx , c — a
then = log - — ,
a c-x "^ /J,
, I - dx [^ dx , h —c
and I = - - — ~ = - log
MISCELLANEOUS EEMARKS. 59
ence the principal value is log log , that is
c — a
f dx X
46. The value of 7^^ ^r is sin"' - ; hence
jJia—x) a
I
dx
itudents are sometimes doubtful respecting the value which
5 to be assigned to sin"' (1) and to sin"' (-1) in such a result
s the above. Suppose we assume ^=asin^; thus the integral
lecomes \ dO or 6. Now x increases from — a to a, hence
he limits assigned to 6 must be such as correspond to this
ange of values of x. When x = -a then 6 may have any
alue contained in the formula (4/i -1) ^ , where n is any
Qteger. Suppose we take the value (4w — 1) ^ , where n is
ome definite integer, then corresponding to the value x = a
ve must take 6 = (4:n - 1) "^ + ir ; this will be obvious on
ixamination, because x is to change from — a to + a, so that
t continualbj increases and only once passes through the value
ero.
Hence -77-2 T\ = '^^
As this point is frequently found to be difficult by begin-
lers we will consider another example.
Suppose we require]^ ^2-^;^,-^.
^'^ f sec'ddO 1^ _i/tan^\
We have - ^ , . 2 a = - tan -— I ;
J a + tan U a \ a /
md as the integral is to be taken between the limits and tt,
60 MISCELLANEOUS REMARKS.
we must determine the values of tan"' ( ) in these eases,
, /tan ^\
\ a /
Suppose 0, 6^, 0^, 0^, ...$„, TT, to be a series of quantities in
order of magnitude. By the nature of integration
I tidd = I ' ud6 + / ' udO + I \idd +... + ( ^icW.
Jo Jo J di J 3j J e„
Now each of the integrals on the right-hand side can be
made as small as we please by increasing 7i and making two
consecutive quantities as 0^ and ^,.^^ to differ as little as we
j^lease. Hence we see that the symbol tan"^ |-^- — J must be
so taken that tan~^ (— — ^^j — tan"' ( ^j shall diminish
indefinitely when 6^_^_^ — 6^ does so.
/tan ^\
Therefore tan"^ [ — — 1 must increase continuously with 0, ;
and it can only pass once through an odd multiple of ^ while
6 passes from to tt. If then we take mir for the value of
tan"' ( ] when ^ = 0, we must take {m + l)7r for the value
when ^ = TT ; and thus the value of the integral between the '
assiojned limits is - .
A common mistake with beginners is to take the second
value the same as the first, instead of taking the second value
to exceed the first by tt; thus the value of the proposed inte-
gral is made to be zero, which contradicts the last paragi-aph
of Art. 42.
. . . f (a - c cos 6) d9
Agam, suppose we require J^ ^. ^. ,. _ 2^^ cos ^ '
r (a-c cos 6) dO _ ^ [L . a^ - c" \ .g
ja^ + c'-2accos^~2cJ| "^ a' + c' - 2ac cos ^j '
TT a^— & /■"■
Thus the required integi'al is ^; — | — ^ — —
^ ^ 2a 2a Jo a
dd
+ c""— 2accos^ '
\
S^ow
r di
MISCELLANEOUS REIMARKS. 61
de
ac cos 6
[ sec^edO __2 -ifa±c. ^ .\
2 TT
When taken between the assigned limits this gives —^ 2-^
2 TT .
if a is greater than c, and 5 2 „- if a is less than c.
Hence the value of the proposed integral is - if a is greater
than c, and zero if a is less than c.
47. The Integral Calculus furnishes simple demonstra-
tions of some important theorems relating to the convergence
and divergence of series.
If (j) (x) continually diminish as x increases tvithout limit
from the value a, then the infinite series
</) (a) + (^ (a + 1) + <^ (a + 2) +
and the integral \ (f) {x) dx are hath finite or both infinite.
J a
rn+1
For since cf) (x) continually diminishes 1 (f) {x) dx is less
J a
than (f) (a) dx, and is greater than I (fj{a-i-l)dx; that is
•'a J a
(ji {x) dx is less than (f) (a) and is greater than (}){a + 1).
J a
ra+2
Similarly I ^ (x) dx is less than </> (a + 1) and is greater
than ^ (a + !2). Proceeding in this way we can shew that
the integral 1 <^ [x) dx is less than
<^(a) + </)(a + l)+</)(a + 2) +
G2 MISCELLANEOUS EEMARKS.
but is greater than
</)(«+ !) + </) (a + 2) + (/)(« + 8) +
Hence the series and the integral are both finite or both
infinite.
48. Now let log x be denoted by \ {x), let log (log x) be
denoted by 'h? (x), and so on. Then we shall demonstrate the
following theorem :
The series of which the general term is the reciprocal of
n\ (n) \' (w) X'{n) {\'^' {n)Y,
is convergent if p he greater than unity, and divergent if p he
less than unity.
^®* ^ ^^'^ " xX{x)\\x) X{x){X^\x)Y '
then I ^ {x) dx = ^ — ■* — , if jp be not unity, and = X*"^ (x)
if p be unity.
f "" \X^^ (a) '^"^
Hence (f){x) dx = — - — — ? if 1> be greater than
unity, and is infinite ifp be equal to unity or less than unity.
Hence the theorem follows by Art, 47.
49. We now proceed to investigate rules for determining
whether a proposed infinite series is convergent or divergent.
Let there be an infinite series
1
denote the sreneral term by . , , . It is obvious that the
series is certainly divergent unless -^ (x) increases indefinitely
with X : we will suppose that -v/^ {x) increases indefinitely
with X.
MISCELLANEOUS EEMARKS. G3
I. Suppose, as x increases indefinitely from a certain
value a, that , . is always less than -j, where G and 'p
are constants, ja being greater than unity; then the proposed
series is less than a certain series which is known to be con-
vergent by Art. 47: therefore the proposed series is con-
verirent.
1 G
If -r-,—s^ is less than — . , then x^ is less than G^ ioc); and,
y {x) x^
loo- (3\|r (x)
taking logarithms, we find that » is less than — —^ — ^
*= ° ^ logic
The last expression assumes the form — when x is infinite;
by the ordinary rules for evaluating such an expression we
iCvIr ix^
obtain — p- — - as its equivalent. Therefore if the limit of
^ {^)
tc^ (X)
, , , , when X is infinite, is greater than unity, we can find a
^{x) ' ' '=' -''
quantity p, greater than unity, such that x^ is always less '
than Cyjr [x). Hence the proposed series is convergent.
In a similar manner it may be shewn that if the limit of
: , / , when X is infinite, is less than unity, we can find a
ylr{x)' ' /'
quantity p, less than unity, such that x'' is always greater
than Cyjr (x). Hence the proposed series is greater than a
certain divergent series, and is therefore itself divergent.
xyl/* (x)
II. Thus if the limit of , /, -, when x is infinite, is
Y[x)
either greater than unity or less than unity, the nature of the
series is determined : but if this limit is unity, further investi-
gation is required.
Suppose, as x increases indefinitely from a certain value a,
1 C
that , , , is always less than — . , ,,„ , where G and n are
■^{x) '' x\K{x)Y
constants, p being greater than unity; then the j)roposed
series is less than a certain series which is known to b'^
convergent by Art. 48 : therefore the proposed series is con-
vergent.
C4 MISCELLANEOUS EEMAKKS.
1 . G
If , , . is less than — ,^ , ,,„ , then lX(x)\'' is less
y [x) X \k{oc)Y ^ '
C-dr (x) . .
than — L-\J- ^ and, taking logarithms, we find that p is less
than — ■,.,,, — , that is, » is less than — ^ — \.\ , . ^ .
X {x) ' ' ^ X^ (x)
The limit of this expression when x is infinite is the same
as the limit of X(x)\ ],/ — Ih. Hence if the limit of
\ [ir{x) J _
this last expression is greater than unity the proposed series
is convergent.
In a similar manner it may be shewn that if the limit
of the last expression is less than unity the proposed series
is divergent.
{X'U/^ (xj I
, ■ , — Ir , when x is in-
finite, is also unity, further investigation is required : the
general term of the proposed series may then be compared
•tl 1
xX{x){X'{x)]^'
Proceeding in this way we obtain the following result :
^^* ^^ = ^^^' ^"* ^^ = ^ ^"'^ (^» - ^)' ^'^ ^^ = ^' ^^^ ^^^ ~ ^^'
and generally let P^ = X"' (x) (P^_i — 1) ; and supj)0se that P^
is the first of the terms P^,P^,P„,... which has its limit, when
X is infinite, different from unity: then the proposed series
is convergent or divercjent accordincj as the limit of P is
greater than unity or less than unity.
We have supposed the general term of the series to be
denoted by ; if it be denoted by % (^) we have to
put r instead of i/r (x) in the preceding result : hence
A. V /
MISCELLANEOUS REMARKS. 65
we find that Po = - ^^. "^ , and that this is the only modifi-
X W
cation required.
50. Another form may be given to the result. "We know
by the Differential Calculus that % (a? + 1) - % (a;) = %' (.t; + 6),
where 6 is some proper fraction. Hence
therefore the limit, when x is infinite, of '^,\ is equal
X i^)
to the limit of x\l ^ , i\ r • Thus we may put
I X{^ + ^))
p =,^\. Xi^) _ i\ in the result of Art. 49.
The theorems in Arts. 47, 48, and 49 have been derived
from De Morgan's Differential and Integral Calculus ; there
is a valuable memoir on the subject of convergence by
Bertrand in the seventh volume of the first series of Liou-
ville's Journal de MatMmatiques. An elementary demon-
stration of the theorem of Art. 48 will also be found in the
Algehra, Chapter LVI.
51. Required I logsina;^?^.
J
By equation (3) of Art. 41,
j log sin xdx = [ ' log sin \^-x\dx=^\ log cos xdx.
Hence, putting y for the required integral,
2y = (log sin X + log cos x) dx
= log (sin X cos x) dx
J
T. L C.
6G MISCELLANEOUS REMAEKS.
P'f , sin 2x 7
= I log —2— «•*
=: I (log sin 2a3 — log 2] dx
-'
/•if . 1
= I log sin Ixdx — ^tt log 2.
But putting 2x = a;', we have
log sin 2xdx = ^ j log sin a;' c?a;'
- ''
= logsina;d!.r, by equation (4) of Ai't. 41 ;
TT
therefore 2_5/ = ?/ — ^ log 2,
therefore 2^ ~ 9 ^^S 9 •
Again, I ^logsin^cZ^= (tt- ^)^logsin ^fZ<?, by equa-
tion (3) of Art. 41 ; therefore
0= r(7r"-27r^)logsin(9fZ^,
•'
therefore 6 log sin 6 dB = - \ logsin ^fZ^ = -^ log ^,
Required I —^ ^ — -~- dx. Put x = tan 1/, and the integi\al
becomes I log (1 + tan 3/) d^; but by equation (3) of Art. 41
•'
IT TT
j "" log (1 + tan y) dy=y log |l + tan ^ - 3/ j| d,j,
T -. , /7^ \ -■ 1 — tan w 2
and 1 + tan 7 - V = 1 -h ,-—-— ^ = t-— ; ;
\4 '^y l+tany 1+tan?/
MISCELLANEOUS REBLVIIKS. 07
IF
therefore 2 I log (1 + tan y) dy = -^ log 2 ;
therefore I -^ — ^ dx = -, log 2.
Mog(l+^)
8
See Cambridge Mathematical Journal, Vol. iiL page IGS.
52. The remainder after ?i + l teraas of the expansion
of (f){a + h) in powers of h, maybe expressed by a definite
inteofral. For let
n
Differentiate with respect to z, then
i^»=-p(/,"^'(a;-.s).
in
Integrate both members of this equation between the
limits and h ; thus
that is
(li{x-h)+Ji(ji'(x-h)+%(ji"ix-h) + ^f-cj>''{x-h)-<p{a;)
=-^rz"cp"-'\x-z)dz.
\nJo
Put cH- A for a; and transpose, then
</>(a + /0 = </>(a)+7.f(«) + ~<^"(«) + (^ </>"(«)
Iji IIL
+
Thus the excess of (f>(a + h) over the sum of the first n + 1
terms of its expansion by Taylor's Theorem is expressed by
the definite integral
}- ( z''6''^Ha + h-z)dz.
5—2
6cS MISCELLANEOUS EEMARKS.
By means of the first result in Art. 40, we may put for
this definite integral
<}>"^'{a + h-eh),
where ^ is a proper fraction.
By means of the second result in Art. 40, we may put for
this definite integral
or
where ^^ is also a proper fraction.
53. Bernoulli s Series. By integration by parts we have
I ^ [x) dx = X(f) (x) — I X(j>' {x) dx,
jxcji' (x) dx=^(l>'{x)-j'^ I/)" {x) dx,
\x^(f)"{x)dx = -^(j)"{x) — I -^^"'{x)dx,
Thus \(f){x)dx = X(}){x) - =— Q (f)'(x)+j^<j>"{x)
Therefore,
[a ff fj3
j^(j){x)dx=a(f>{d)-^—^^'{a')+^f{a)
hi \n Jo
mSCELLANEOUS RKAIAEKS. 69
This series on the right hand is called Bernoulli's series. In
»me cases this process might be of use in obtaining I cf) (x) dx ;
Jo
r example, if <^ (.r) be any rational algebraical function of
le (?i — 1)'" degree, </>" {x) is zero ; or it might happen that
k''(/)" (x) dx could be found more easily than 1 ^ [x) dx. Or
;-ain, we may require only an approximate value of
I ra
(f) (x) dx and the integral / x"(p" {x) dx might be small
lough to be neglected.
54. By adopting different methods of integrating a func-
3n, we may apparently sometimes arrive at different results,
ut we know {Differential Calculus, Art. 102) that two func-
ms which have the same differential coefficient can differ
dy by a constant, so that any two results which we obtain
ust either be identical or differ by a constant. Take for
ample
I {a^ + h) {a'x + h') dx ;
tegrate by parts, thus we obtain
{ax + hY . , , fa' , j.^ -
at is
{ax-^hYJa'x^- h') a' {ax + b)'
2a Qa'
If we integrate by parts in another way, we can obtain
{a'x + by{ax + b) a{a'x + b'Y
2a 6a" '
lerefore
(ax + by [Sa (a'x + b') - a {ax + b) ]
. (a'x + b'Y [Za' (ax + b)-a (a'x + b')}
6a"
70 MISCELLANEOUS REIVrARKS.
can differ only by a constant. Hence multiplying by 6aV^
we have
a' [ax + hf [Sa {ax + h') - a (ax + h)]
- a^ {ax + by {na{ax +b)-a {ax + b'}]= 0,
where C is some constant. This might of course be verified
by common reduction. We may easily determine the value
of C; for since it is independent of x we may suppose
ax + b = 0, that is, x = ; then the left-hand member
a
becomes (ab' — a'b)^, which is consequently the value of C.
Similarly from
I {ax +b) dx+ I (ax + b') dx = {{a + a) x + b + b'] dx
we infer
{ax + b)\ {a'x + by {(a + a')x + b + b'Y- , ,
^-— — ^ + ^ — -^-^ = ^-^ ^ + constant.
2a za 2 {a + a)
Multiply by 2aa' {a + a) and then determine the constant by
supposing x = 0', thus we obtain the identity
a {a + a) {ax + b)~ + a {a + a') {ax + b'f
= aa! [{a + a)x + b + b'Y + {ba' - b'af.
If we integrate a function between assigned limits the
result must be the same by whatever method we proceed ;
and in this manner Ave may obtain various algebraical
identities.
Take for example I x'"{l — x)"dx, where w is a positive
integer. We have, by integrating by parts,
fx'^ (1 - xY dx = ^"'"'(^-^)" + _J^ [^-+1 (1 _ ^y-^ dx ;
J ^ ' m + 1 m + lj ^ ^
therefore [ a;"' (1 - xj dx = -^^ f x'^^' (1 - a;)""' dx.
MISCELLANEOUS REiLiRKS. 71
Proceeding in this way wo obtain
J, ^ ^ {in + l){}n + 2) ...{m + n + 1) ^^
Again fa;"' (1 - x)" dx = f^'" jl - wa; + — ^^-^^ a;* - ... 1 dx
_ 1 n 1 n{ii-l) 1 iNn 1 /9N
m-\-l 1 ' 7?i + 2 1.2 m + 3 ^ m+w+1
Therefore the expressions on the right-hand side of (1)
and (2) are equal if n be any positive integer.
55. By l(/) {x) dx we indicate the function of which (^{x)
is the differential coefficient; suppose this to be 'v^(.r). Then
we may require the function of which '^{x) is the differential
coefficient, which we denote by \-^{x)dx, or by \\ (j){x) dxdx,
and so on. For example, the integral of e'^ is j e'" + C^,
where (7^ is a constant ; the integral of this is
j-^e'' + C,x + C^;
the integ-ral of this is
'O
c .
where -^ being still a constant may be denoted for simplicity
by B if we please. Proceeding thus we should find as the
result of integrating e^' successively for n times
^ + ^,x"-^ + ^,^"-^+ +A^_,x + A^,
where A^, A^, A^ are constants.
u.
u„
72 MISCELLANEOUS REMARKS.
It is easy to express a repeated integral in terms of
simple integrals. For let u be any function of x ; let
t/j = ludx; let 21^= lic^dx] let Wg = \u/lx;
and so on.
By integration by parts we have
= I u^dx — xu^ — \ X -^ dx = X ludx — Ixudx ;
= I u^dx = l\x { udx — I xudx)- dx ;
therefore by integration by parts,
11^ = -^ \ udx — I -^ udx — X \ xudx + Ix'udx
= -^ I udx — X jxu dx+ ix Ix'udx.
The general formula is
\n w,^^j = x^ I udx — nx^~^ Ixudx -{ — ^j — ~ ^""^ Ix^udx —
+(-ir "^"-^^--^ ^"-±^^-fruc^^+
+ {-iyjx"udx.
The truth of this formula may be easily established by
induction ; for if we differentiate both sides we obtain a
similar formula with 7i — 1 in place of n.
MISCELLA^^EOUS EXA3IPLES. 73
2a
>
MISCELLANEOUS EXAMPLES.
I r x'-dx 57ra^ .
r [a? - e'x^) dx _'rr^(. _e\
dx
''+x'){h'+x^) 2ab(a + b)'
). If (f){x) = (f>{a + x), and n is a positive integer, shew that
rna ra
I <^ (a-) cZa; = ?i I {x) dx.
Jo J
!. Shew that I (^ (a;) cZ« = -^ 1 <^(— i,-^ + -^a;J (Za;.
r ci 4.1 J. r^siniccZa; tt^ zr^i
'. Shew that -— — = — , (Change x into tt - x .)
J Q -L "T" COS OG T
i. Shew that r (2ax-x')^Yers-'-dx = ?^.
Jo a 16
(Change x into 2a — x.)
I. Find the limit when n is infinite of
1 . 1 , 1 1
n ^{n'-l) ' VK-2^} "^VK-(n-in-
Result ^.
74 MISCELLANEOUS EXAMPLES.
10. Find the limit when n is infinite of
i- -i + i- + TT-i + \7^ + 7r-\ +...tow terms
Result.
i-<r
11. Find the limit when n is infinite of j ^ r .
Result. - . (Take the logarithm of the expression
12. Shew that | log tan ^ cZa; = 0.
IT
13. Shew that 1 sin a; log sin a; cZa; = log 2 — 1.
''
14. If /'(.^) be jDOsitive and finite from a; = a to x = a + (
shew how to find the limit of
{/(«)/(<'+3 /(r+'^f
when 71 is infinite ; and prove that the limit in ques
1 ra+o
tion is less than - ( f(x) dx, assuming that the gee
1 /-a+o
- f{x) dx,
^ J a
metric mean of a finite number of positive quantitie
which are not all equal is less than the arithmetic.
Hence prove that e " is less than I e'^dx, unless
J
be constant from x = io x = \.
MISCELLANEOUS EXAMPLES. 75
5. The Vcolue of the definite integral I log {1 + n cos^ 6) cW
may be found whatever positive value is given to n
from the formula
/;
TT
log (1 + n cos^^) d0=~ log {(1 + n)(l + « j' (1 + n,)*. . .]
where 77, TZj, TZj) ^^^ quantities connected by the
equation
n =: I
.6. Shew that
/" «. 7 e''' cos (ax — 6), ^ ^
le cos ax ax = ^ j—^ + a- constant,
J {a' + c')'
where tan 6 = - . Hence shew that if e'' cos ax be
^ c
integrated 7i times successively the result is
e^cosiax-ncly) _^ ^ ^ ^^^ ^ ^^^, ^ C„_y-\
(a^ + c^)^
1
L7. Shew that the series of which the n^ term is a" — 1 is
divergent.
^ a+-
L8. Shew that the series of which the n^ term is (-) is
convergent if a is greater than unity, and divergent
if a is not greater than unity,
[9. Shew that the series of which the n^ term is
p(p + a)(p + 2a) (p+ no)
q{q + d){q-)r'2.a) {q + na)
is convergent if ^ is greater than p-\-a, and divergent
if q is not greater than p + a. See Art, 50.
76 MISCELLANEOUS EXAMPLES.
20. Suppose that the ratio of the (n + l)"" term of a series
to the n^ is equal to
where j9 is a positive integer, and A, B, ... a, h, ... are
constants : shew that the series is convergent if a is
greater than A + 1, and divergent if a is not greater
than ^ + 1.
21. Let A= ju'^dx, B=\uvda;, C=\v'dx, and suppose
the hmits of the integration the same in the three in-
tegrals ; then shew that A G is never less than B^.
[Consider each integral as the limit of a certain
summation ; then the Example depends on the known
algebraical theorem, that
K'+«/+ + 0(c/+c/ + + o
is never less than
77
CHAPTER V.
DOUBLE INTEGRATIOX.
56. Let </> (x) denote any function of x ; then we have
;en that the integral of cf) (x) is a quantity ii such that
- = <f> {cc). The integral may also be regarded as the limit
- a certain sum (see Arts. 2..,G), and hence is derived the
rmbol / (f) (x) dx by which the integral is denoted. We
ow proceed to extend these conceptions of an integral to
ises where we have more than one independent variable.
57. Suppose we have to find the value of ^t which satis-
es the equation , , = <^ {x, y), where ^ [x, y) is a function
[ the independent variables x and y. The equation may
e written
dy\
dv
dy
' v= -j-. Thus V must be a function such that if we differ-
dx
ntiate it with respect to y, considering x as constant, the
3sult will be (}> (x, y). We may therefore put
latis _|=|^(a.^y)f;y.
78 DOUBLE INTEGRATION.
Hence u must be sucli a function that if we differentiate it
with respect to x, considering 7/ constant, the result will be
the function denoted by I (f> {x, y) dy. Hence
u
=\\j<^{^>y)dy\dx.
The method of obtaining u may be described by saying
that we first integrate ^ {x, y) with respect to y, and then
integrate the result with respect to x.
The above expression for u may be more concisely written
thus,
\\ j> {x, y) dy dx, or \\ cji (x, y) dx dy. ■
On this point of notation writers are not quite uniform ; we
shall in the present work adopt the latter form, that is, of the
two symbols dx and dy we shall put dy to the right, when we
consider the integration with respect to y i3erformed before the
integration with respect to a?, and vice versa.
58. We might find u by integi'ating first with respect to
X and then with res]3ect to y ; this process would be indicated
by the equation
^= jj (1^ (.^> y) dy dx.
59. Since we have thus two methods of finding w from the
d\i
equation t—t" = </> (^) y)y it "^^ill be desirable to investigate if
more than one result can be obtained. Suppose then that u^
and u^ are two functions either of which when put for u satis-
fies the given equation, so that
^ = ^{^,j) and ^ = *(»., 2,).
We have, by subtraction,
d'ii^ _ cPti^ __ ^
dxdy dxdy
that is, -J- ( ^- ) = 0, where v = u, — il.
dx \dyj ^ ^
DOUBLE INTEGRATION. 79
Now from an equation -7- = we infer that w must be a
constant, that is, must be a comtant so far as relates to x ; in
other words, w cannot be a function of x, but ina7/ be a func-
tion of any other variable which occurs in the question we are
consiJerinu:.
o
Thus from the equation -r-|-7-l = we infer that -r-
dx \ajjj dij
cannot be a function of x, but may be any arbitrary function
of y. Thus we may put
By integration we deduce
V = if Of) dy + constant.
Here the constant, as we call it, must not contain y, but
may contain x\ we may denote it by %(^r). And \f{y)dy
we will denote by -v/r (?/) ; thus finally
^ = 'f (.y) + %(^)-
Therefore two values of u which satisfy the equation
, — T- = {x, y) can only differ by the sum of two arbitrary
functions, one of x only and the other of y only,
60. We shall now shew the connexion between double
integration and summation. Let <^ [x, y) be a function of x
and y, which remains continuous so long as x lies between
the fixed values a and h, and y between the fixed values a
and /3. Let a, x^, x^, x„_^, 6 be a series of quantities in
order of magnitude; also let a, y^, y.^, y,„_^, /3 be another
series of quantities in order of magnitude.
Let x^-a^h^, x^-x^ = h.^, ^-^„-i = ^^i;
also let y,-ci=l\, y._-y,^k, (3- y„,_^ = k„,.
80 DOUBLE INTEGRATION.
We propose to find the limit of the sum of a certain
series in which every term is of the form
hjc^^ {x,_„ ?/^_J,
where r takes all integral values between 1 and n inclusive,
and s takes all integral values between 1 and m inclusive ; and
ultimately m and n are to be supposed infinite ; also x^ and
T/o are to be considered equivalent to a and a respectively.
Thus we may take hkcf) (sc, y) as the type of the terms we
wish to sum, or we may take AccA;/^ {x, y) as a still more
expressive symbol. The series then is
\[k^4>{a, a) + k,cf){a, y^) + k,(ji {a, y^ + ^-„. </>(«, Vm-d]
+ h^ {k^(f> {x^, a) + k,^{x„ y,) + k^(f> {x^, y^ + \<^ {x^, ?/„,_J}
+ h^ {k^cf) {x^_^, a) + k^(f> {x„_„ y,) + + ^'> {x„.„ y „,.,)].
Consider one of the horizontal rows of terms, which we
may write
h^^^ {k^cji {x,, a) + k^(f> {x^, y,) + kj>{x,, y,) + k„,(f> {x,, O}.
The limit of the series within the brackets when k^, k^,.. .k
are indefinitely diminished is, by Art. 3,
m
/'
J a
^{«^r>y)dy.
Since this is the limit of the series, we may suppose the
series itself equal to
/•|3
J a
where p^^^ ultimately vanishes.
Let I <p{x^, y) dy be denoted by •>^{x^ ; then add all the
horizontal rows and we obtain a result which we may de-
note by
Sli "^ (x) + %hp.
dx.
DOUBLE INTEGRATION. 81
Now diminish indefinitely each term of wliicli h Is the type,
then S/i/3 vanishes, and we have finally
rb
I -^^ (x) dx ;
J a
that is, J U j>{x,7j)dij[
This is more concisely written
b f?
(f) (x, y) dxdy,
dy being placed to the right of dx because the integration is
performed first with respect to y.
61. We may again remind the student that writers are
not all agreed as to the notation for double integrals. Thus
we use I \ ^ {x, y) dxdy to imply the following order of
J a J a
operations : integrate ^ {x, y) with respect to y between the
limits a and ^ ; then integrate the result with respect to x
between the limits a and h. Some writers would denote the
same order of operations by I I ^ {x, y) dydx.
J a •! a
G2. We might have found the limit of the sum in Art. GO
by first taking all the terms in one vertical column, and then
taking all the columns. In this way we should obtain as the
r/S fb
sum I I (f) (x, y) dydx; and consequently
J a J a
'•p rb rh ,-^
I </> (^> y) dydx =\ 1 4>{x,y) dxdy.
J a J a J a J a
The identity of these two expressions may also be esta-
bhshed by the aid of Art. 59, as we will now shew.
Let F{x, y) denote the integral of (j) (x, y) with respect to
y, supposing x constant ; and let / (x, y) denote the integral
of F (x, y) with respect to x supposing y constant. Then
T. I. c. G
82 DOUBLE INTEGRATION,
h rP rb
(f) {w, y) dxdy = \ [F (x, ^) — F{x, a)] dx
J a. J a
= { F{x, /3)dx- [ F{x, a)dx
J a J a
=f{^> /3) -/(«, ^) -f{h, 0) +/(a, a) (1).
Now let us first integrate </> {x, y) with respect to x, sup-
posing y constant, and then integrate the result with respect
to y, supposing x constant; \Qif\{x, y) denote the final result.
Then we obtain
' \\[x, y)dydx=f^{b, ^)-f,{h, o)-f,{a, /3) +f,{a, a). ..(2).
a. J a
But, by Ai-t. 59,
f^{^> y) =fix' 2/) +f (y) + %(^) (S),
where "^ (y) is some function of y without x, and -^^ (x) is
some function of x without y. By making use of (3) we
shall find that the right-hand member of (2) reduces to the
right-hand member of (1).
The function ^(x, y) is assumed to be fijiite through the
range of the integration : for that is involved in the notion
of continuity : see Arts. 2 and 43.
63. Hitherto we have integrated both with respect to x
and y between constant limits ; in applications of double
integration, however, the limits in the fust integration are
often functions of the other variable. Thus, for example, the
rb r^ix)
symbol I I <})(x, y) dxdy will denote the following opera-
JaJxix)
tions : first integrate with respect to y considering x con-
stant ; suppose F {x, y) to be the integral ; then by taking
the intecrral between the assigned limits we have the result
F[x,^{x)]-F[x,x{x)}.
We have finally to obtain the integral indicated by
\f[x,^{x)]-F[x,x{x)]'\cIx.
The only difference which is required in the summatory
process of Art. GO is, that the quantities a, y^, y^, ...3/„,_i will
DOUBLE INTEGRATION. 83
not have the same meaning in eacli horizontal I'ow. In the
{r + 1)^ row, for example, that is, in
we must consider a as standing for x{^r)> ^^<^ 2/i> 2/2' ^^
a series of quantities, such that x{^'r)> V^^IU, 2/m-i. '^ (^r),
are in order of magnitude, and that the difference between
any consecutive two ultimately vanishes. Hence, proceeding
as before, we get ^ ix^, y) dy for the limit of the sum of
the terms within the brackets in the {r + 1)"* row.
64. It is not necessary to suppose the same number of
terms in all the horizontal rows ; for m is ultimately made
indefinitely great, so that we obtain the same expression for
the limit of the (r + 1)"' row whatever may be the number
of terms with which we start.
65. "When the limits in the first integration are functions
of the other variable we cannot perform the integrations in a
different order, as in Art. Q'2, without special investigation to
determine what the limits wdll then be. This question will
be considered in Chapter XI.
QQ. From the definition of double integration, it follows
that when the limits of both integi'ations are constant,
\\ <f> (^) f (y) dxdy =j(f){a;)dxx j^lr{y) dy,
supposing that the limits in j-^ (y) dy are the same as in the
integration with respect to 1/ in the left-hand member, and the
limits in I ^ (x) dx the same as in the integration with respect
to X in the left-hand member. For the left-hand member is
the limit of the sum of a series of terms, such as
and the right-hand member is the limit of the product of
h,<f>{x,) +kcf>{x;) +h^c}i{x^) + h^c}> (a;„J,
and Jc.yjr (t/J + k^f (y^) + k^f (yj + k^yjr (y^ J.
6—2
84 DOUBLE INTEGRATION.
G7. The reader will now be able to extend the processes
given in this Chapter to triple integrals and to inultiple
integrals generally. The symbol
I j (p (x, y, z) dxdydz
will indicate that the following series of operations must be
performed : integrate {x, y, z) with respect to z between the
limits ^0 and ^^ considering x and y constant ; next integi-ate
the result with respect to y between the limits tj^ and t]^ con-
sidering X constant ; lastly integrate this result with respect
to cc between the limits ^^ and l^j. Here ^^ ^^^ ?i ^'^^J be
functions of both x and y ; and rj^ and rj^ may be functions
of w. This triple integral is the limit of a certain series
which may be denoted by %(fi (x, y, z) Aa; A^/ As;.
MISCELLANEOUS EXAMPLES.
Obtain the following eight integrals.
/
^lia'-x')
dx. (Put y = x".)
Result. -X sin"^ —
2. \
J ix — a
a-
x"^
"^dx
{x — a){x — h) {x — c) '
{a—h){a — c) {b—a){b—c) {c—a){c — h) '
I i2cn.xdx -r> 7, log {co5^ X + m^ sm^ x)
3. z r> ^— . Result
'•/r
+ ni^ tan^ x' ' 2 (??r — 1)
Result. — ^ log -^
na a + Y (ot -rx )
MISCELLANEOUS EXAMPLES. 85
5. I sec a; sec 2xdx.
T. ,, 1 , 1 + V2sina; 1, l + sin^.
lies lilt. -r=r lOOr ——. — - log -. .
V2 ° 1 - V2 sm a; 2 ° 1 - sin a;
,. ,"tan a — tana; ,
0. I ^ ; dec.
tan a + tan a;
Result, sin 2(X log sin (a +x) — x cos 2a.
t7a
r dx
J x^ + a V" + a
-r, , 1 1 x^ + ax+a^ 1 , _i a;a\/3
Result -r-s log -^s — ■ — ^ + ^^ 3 /o t^J^ -^ i •
(a — hx^) dx /-p + ^
«■ J^^^f^i- (P-!-^-^)
Result, cos ^ -77 — -■■ ■ ,,
/v/(c + 4a6)
9. Find the limit when n is infinite of
1
f . TT . 27r . Stt . wtt - tt " „ ,, 1
-^sm- sm — sm — sm —> . Mesult. -^
{ n n n n } z
10. Shew that
fx (tan-^ xf dx = '^('^-l)+ log ^2.
11. Shew that
8
•' 0-' 0-'
/ e'''^dxdyd2 = -^--^ + e''-
12. Let ^ = 1 1 ?t- cZa; cZy, B= ijuvdxdy, G=ljv^dxdi/,
and suppose the limits of the integrations the same in the
three integrals ; then shew that A C is never less than B^.
(See Example 21 at the end of Chapter IV.)
86 MISCELLANEOUS EXAMPLES,
13. If I 4> {z) dz is equal to unity, and j> (z) is always
positive, shew that
(\(j)(z) cos czdzj +( (f) [z) sin cz dzj is less than unity.
(See History of...Prohahility, page 564.)
14. If ( (^ (z) dz is equal to unity, and (/> (2) is always
positive, shew that
j 2'^^(z)d2-l\ z(p {z) dz\ is positive.
(See History of... Probability, page 566.)
87
CHAPTER VI.
LENGTHS OF CURVES.
Plane Curves. Rectangular co-ordinates.
68. Let P be any point on the curve APQ, and let x, y
be its co-ordinates ; let s denote the length of the arc AP
measured from a fixed point A up to P;
O 'jP
jtr
JV' •»
then {Differential Calculus, Art. 307)
-(I)}-
Hence
ds
dx
s =
i+'l)V-
dy
From the equation to the curve we may express -^^ in
terms of x, and thus by integration s becomes known.
69. The process of finding the length of a curve is called
the rectification of the curve, because we may suppose the
question to be this : find a right line equal in length to any
assigned portion of the curve.
88 LENGTHS OF CURVES.
In the preceding Article we have shewn that the length of
an arc of a curve will be known if a certain integral can be
obtained. It may happen in many cases that this integral
cannot be obtained. Whenever the 'length of an arc of a
curve can be expressed in terms of one or both of the co-
ordinates of the variable extremity of the arc, the curve is
said to be rectijiahle.
70. Application to the Parabola.
The equation to the parabola is ?/ = A/(4ax) ; hence
dy _ /a ds _ I Ix + a\ _
dx" y x' 'dx~ y \ X ] '
thus s = \\/\- ) dx (See Example 6, page 19.)
= >,/{ax + x^) + a log yx + \/(« + ^)] + 0.
Here (7 denotes some constant quantity, that is, some quan-
tity which does not depend upon x ; its value will depend
upon the position of the fixed point from which the arc s is
measured. If we measure from the vertex, then s vanishes
with X ; hence to determine C we have
a log \fa+ (7=0;
and thus s = i\J{ax + x^) + a log [^/x + ^J [a + x)] — a log /y/a
,/ . 2\ , 1 \/x + ^J{a + x)
= iJ(ax + ic^) 4- « log -r .
\/a
If then we require the length of the curve measured from
the vertex to the point which has any assigned abscissa, we
have only to put that assigned abscissa for x in the last
expression. Thus, for example, for an extremity of the
latus rectum x = a', hence the length of the arc between
the vertex and one extremity of the latus rectum is
71. In tlie preceding Article we have found the value of
the constant G, but in applying the formula to ascertain the
lengths of assigned portions of curves this is not necessary.
LENGTHS OF CURVES. 89
For suppose it Is required to find the length of the arc of a
curve measured from the point whose abscissa is ^, up to the
point Avhose abscissa is x.^. Let ■xjr {x) denote the integral of
a/ \^ + [-7 ) \ > and let s^ and s^ be the lengths of arcs of the
curve measured from any fixed point up to the points whose
abscissa) are a?j and x^ respectively, so that s^ — s^ is the
required length ; then
hence s^=-^ (x) + C; s^^^jr (x^) + C ;
therefore s.^ ~^i—'^ (■^2) "~ "^ (^i)*
Hence to find the required length we have to put x^ and x^
Buccessively for a; in i/r (x) and subtract the first result from
the second. Thus we need not take any notice of the constant
C; in fact our result may be written
-».= rA/n+(t)V^.
xi
dxj
72. Application to the Ci/cloid.
In the cycloid, if the origin be at the vertex and the axis
of ?/ the tangent at that point, we have {Differential Calculus,
Art. 858)
ds^_ //2a\
dx~\/ \^) '
therefore s = »J{8ax) + C.
The constant will be zero if we measure the arc s from the
vertex.
Conversely if 5 = ^/(Sax) + (7 we infer that the curve is a
cycloid. And more generally if we have
s + A= ^{B + C^x + C„jj),
where A, B, C^, and C^ are constants, we infer that the curve
90 LENGTHS OF CURVES.
is a cycloid. For by suitable changes in tbe origin and
axes the last equation can be put in the form
s = s/{8ax) + C.
73. Application to the Catenary.
c - --
The equation to the catenary is y = ^ (e" + e ") ; hence
thus s = \ Ue^ + e~^) dx = ^ (e' - e~') + C.
The constant will be zero if we measure the are s from the
point for which a) = 0.
7-i. Application to the Curve given by the equation
z
x3 +y3 z= aK
Here ^ = _^ ^ = Ctulf ^^•
dx x^' dx \ x^ / x^'
,1 i [da Sa'\v^ „
thus s = a^ -^ = — ^ h C.
J X'i ^
The constant will be zero if we measure the arc from the
point for which a; = 0. The curve is an hypocycloid in which
the radius of the revolving circle is one-fouith of the radius of
the fixed circle. (See Differential Calculus, Art. 362.)
75. In the same way as the result in Art. 68 is obtained
we may shew that
VI- (l)}^^-
LENGTHS OF CURVES, 91
Or we may derive this result from the former thus ;
From the equation to the curve we may express -j- in
terms of y, and thus by integi-ation s becomes known. In
some cases this formula may be more convenient than that in
Art. 68.
76. Application to the Logarithmic Curve.
X
The equation to this curve is ?/ = ha", or i/ = he'' if we
y
b
suppose a=e°; thus a;=clog^,
therefore ^==^, ^^'JSlllfl
dy y dy y '
and s-i'^-^^^^dv-l-^y +{ y^y
s-j ^ rf^_J^^^^,_^+J__^.
The latter integral is \/(c^ + y^) ; the former is
77. If sc and y are each functions of a tliird variable t,
we have {Differential Calculus, Art. 807)
ds _ /ifdxV /dy
Jt~V [VdiJ '^\di
^=''/^'¥+©J^'-
92 LENGTHS OF CURVES.
78. Application to the Ellipse.
The equation to the ellipse is -^ + t^ = 1. We may there-
fore assume a; = a sin ^, y = h cos (/>, so that <^ is the com-
plement of the excentric angle [Plane Co-ordinate Geometry,
Art. 168). Therefore, by the j)receding Article,
ds
'_,— = V(a^ cos^^ + 6^ sin^^),
and s = I /^/(a^ cos^ (f) + b^ sin^^) d(f) = a j V(l — e^ sin^ ^) d(p.
The exact integral cannot be obtained ; we may however
expand VCl — e^ sin" <^) in a series, so that
s = al[l —\e^ sin^^ — ^-^ e^ sin*^ ~ ^' a' p 6^sin^(/) ]d^,
and each term can be integrated separately. To obtain the
length of the elliptic qiiadrant we must integrate between the
limits and — .
Plane Curves. Polar Co-ordinates.
79. Let r, 6 be the polar co-ordinates of any point of
a curve, and 5 the length of the arc measured from any fixed
point up to this point; then {Differential Calculus, Art. 311)
lence s =
Hi^V'-
80. Application to the Spiral of Archimedes.
In this curve r = ad, thus -v^ = ci :
dd
hence s= j^/{r'-\- a')de = a j^/{l + 6"-) d0
The constant will be zero if we measure the arc s from the
pole, that is, from the point where ^ = 0.
LENGTHS OF CURVES. . 93
81. Application to the Cardioid.
The equation to this curve is r = a (1 + cos 6) ; thus
= [V [a? (1 + cos Of- + a' sia' 6] dd = a f V(2 + 2 cos 6) dO
= 2./
cos ^ dO = 4a siu ^- + C.
The constant will be zero if we measure the arc s from the
point for which ^ = 0, that is, from the point where the curve
crosses the initial line.
The length of that part of the cuive which is comprised
between the initial line and a line through the pole at right
angles to the initial line is 4 a sin ^ • The length of half the
perimeter of the curve is 4a sin - , that is, 4a.
82. Suppose we require the length of the complete peri-
meter of the cardioid ; we might at first suppose that it
riiT
would be equal to 2a I cos - dO; but this would give zero as
Jo ^
the result, which is obviously inadmissible. The reason of
this may be easily seen ; we have in fact shewn that
^| = aV(2 + 2cos^),
a a
and this ought not to be put equal to 2a cos ^ but to + 2a cos - ,
and the proper sign should be determined in any application
of the formula. Now by s we understand a positive quantity,
and we may measure s so that it increases with 0, and thus
~ is positive. Therefore when cos - is positive, we take the
dU ^
upper sign and put -^^ = 2a cos ^ ; when cos ^ is negative, we
take the lower sign and put -^=- 2a cos 2. Hence the
9-i , LENGTHS OF CURVES.
r2^ Q
length of the complete perimeter is not 2a I cos - d9, bi
2a cos ^d9- 2a I cos ^ cZ^, that is, 8a. This result migl
have been anticipated, for it will be obvious from the sjtc
metry of the figure that the length of the complete perimet
is double the length of the part which is situated on one sic
of the initial line, and this was sheAvn to be 4a in the prece(
ing Article.
83. It may sometimes be more convenient to find tl
length of a curve from the formula
which follows immediately from that in Art. 79.
84. AjjplicatioJi to the Logarithmic Spiral.
The equation to this curve is r — ha^, or r = he'' if we su
- r dd e
pose a = 6"; thus ^ = c log ^ ; therefore j- = ~ and
s =1 V(l + c') dr = V(l + c') r + C.
Thus the length of the portion of the curve which has
and r^ for the radii vectores of its extreme points is
r V(l + c') dr, that is, V(l + C) (r, - r J.
The angle between the radius vector and the correspondii
tangent at any point of this curve is constant {Differenti
Calcidus, Art. 354) ; and if that angle be denoted by
ds
we have c = tan a ; thus /^(l + c^) = sec a ; therefore -v; = sec
and s = r sec a + C. Hence {r^ — rj sec a is the length of tl
portion mentioned above.
LENGTHS OF CURVES. 95
Formulce involving the radius vector and perpendicular.
85. Let (f) be tho angle between the radius vector r of
any point of a curve and tho tangent at that point ; then
cos^ = -T- (Differential Calculus, Art. 310). Let p be the
perpendicular from the j)ole on the same tangent ; then
sin (p = - , therefore cos </> = — — ;
dr^ ^{r'-f) ^
ds r '
thus
ds __ r J — f '"'^^
therefore -t- = —rrr, w . and
86. Application to the Epicycloid.
With the notation and figure in the Differential Calculus,
Art. 360, it may be shewn that the equation to the tangent
to the epicycloid at P is
COS a — cos — ^ — o
sm d — sm ■ — 5 — t7
where x and y are the co-ordinates of P, and x and y the
variable co-ordinates. Hence it will be found that the per-
pendicular p from the origin on the tangent at P is given by
p = [a + 2h) sm ^ ;
also r = a^ + 45 (a + h) sin" ^7 ;
thus p' = ^-A Y^ > where c =- a + 2J.
^ c — a
Hence, by Art. 85,
9G LENGTHS OF CUIIVES.
At a cusp 7' = a, and at a vertex r = c ; thus the length of
the portion of the curve between a cusp and the adjacent
vertex is
-^ -Tr^> 2T ■, ttia-t IS , that IS — ^ .
Hence the length of the portion between two consecutive cusps
. 8h(a + h)
a
87. A remark may be made here similar to that in
Art. 82. If we apply the formula
a ^ ^
to find the length between two consecutive cusps, we arrive
at the result zero, since r = ft at both limits. The reason is
that we have used the formula
ds _ V(c^ — (f) r
dr a \J{c^ — r")
while the true formula is
ds \/(c^ — <^") ^
dr ~ a ^/{c^ — r') '
Since s may be taken to increase continually, it follows that
ds . . .
■J- is positive when r is increasing, and negative when r is
diminishing. Now in passing along the curve from a cusp to
the adjacent vertex r increases, thus -j- is positive, and wc
should take the upper sign in the formula for y- ; then in
passing from the vertex to the next cusp r diminishes, thus
ds . .
~r- is negative, and the lower sign must be taken. Hence the
dr ^ ' *=
length from one cusp to the next cusp
LENGTHS OF CURVES. 97
_ V(c'-a') p rdr ^/(c^-a') f^ rdr
_ 2 V(c^ - g") p rdr ^ 86 (a + h)
88. From what is stated in the preceding Article, it ap-
pears that if the arc s begin at a vertex the proper formula is
ds \/{c^ — a^) r
therefore s = -^-^l^ \ ^^^^Jl^^^[o^ -r^).
No constant is required since we begin to measure at the
point for which r = c; the formula holds for values of s less
a
It may be observed that thus
a
89. Similarly for the hypocycloid we may shew that
p = — 4 2-^ > where c = a — 26.
^ a^ — c^
Suppose (? less than a' ; then we may shew that
ds _ \/(a^ — c^) r
dr a '^{r^ — c")'
and thus s may be found. The length of the curve between
two adjacent cusps is «''("-*)
a
Next suppose c" greater than a^ ; then we should -vviite
(is
the value of -7- thus,
dr
ds _ a/(c^ — <^^) ^
dr ^ a s/{c'-r')
T. I. C.
,»\ »
98
LENGTHS OF CURVES.
in this case h is greater than a, and we shall find the length
of the curve between two adjacent cusps to be — ^^ .
When a = 2b we have c = and p = 0; in this case the
hypocycloid becomes a straight line coinciding with a dia-
meter of the fixed circle.
If a = J we have c^ = a" ; in this case the denominator in
the value of p^ vanishes ; it will be found that the hypocycloid
is then reduced to a point, and r = a.
It may be shewn as in Art. 88, that if s be measured from
a vertex to a point not beyond the adjacent cusp, we have
s = +
a'
J{r^-f),
the upper or lower sign being taken according as c is greater
or less than a.
Formulce involving the Perpendicular and its Inclination.
90. Another method of expressing the length of a curve
is worthy of notice.
Let P be a point in a curve ; x, y its co-ordinates. Let s
be the length of the arc measured from a fixed point A up
to P. Draw F a perpendicular fi'om the origin on the
tangent at P, suppose OY=p, PY = u, Y0x = 6; then
LENGTHS OF CURVES. 99
p = x COS ^ + 2/ sin 0,
u = x&inO — y cos 0,
dy . a ^s a
~= —coiO, -T-= — cosec ;
ax ax
therefore
dp . yy „ ^dx . ^dy
j^ = — a: sm ^ + 7 cos ^ + cos p -jt; + sin -y^ = — M,
do da da
d^p du „ . ^ . ^dx -, Jy
-f^ = — -Tn = — xcos0—ysm0 — sin0-j^ + cos9-f^
d(f d0 ^ d0 d0
ndx ds
therefore, by integration,
^ = -lpd0 + s.
d0
this may also be written
s + u= I pd0.
Suppose 5j and Mj the values of s and u when 9 has the
value 6^ , and s^ and u^ their values when ^ has the value 0^, then
therefore « = -th +
5,^ -s^ + u.
re,
— u^=\ pd0.
We have measured u in the direction of revolution from P
and have taken it as positive in this case ; when u is negative
it will indicate that Y is on the other side of P.
The preceding results may be used for different purposes,
among wdiich two may be noticed.
(1) To determine the length of any portion of a curve
when the equation to the curve is given ; for from that equa-
tion together with -^ = — cot ^ we can find x and y in terms
of 0, and therefore j) which is equal to x cos -{y sin ; then
s may be found from the equation
= |4./....
7—2
100
LENGTHS OF CURVES.
(2) To find a curve such that by means of its arc a pro-
posed integral may be represented ; for if the proposed inte-
gral bo jpdd, where p is a function of 9, the required curve is
found by eliminating 6 between the equations
X =]) cos 6 -~sm0, y = psin6 + -^ cos 6
and then the integral may be represented ^J ^-^q-
This Article has been derived from Hymers's Integral
Calculus, Art. 136.
91. The results of the preceding Article may be obtained
in another way. Let p denote the radius of curvature of the
curve at F ; let OP = r, and let s, u, and have the same
meaning as before, then from the Differential Calculus we
have
, = r|, therefore f^=r^' .
dr
ds J
d9
Also
Pr=r cos OP F=-r^;
LENGTHS OF CURVES. 101
therefore ~:. = — PY= — u.
do
Let PC be the radius of curvature at P; draw OQ perpen-
dicular to FC. The locus of C is the evolute of the curve
AP; and QG is with respect to this locus what PF is with
respect to the locus of P. Let 6', p be the polar co-ordi-
nates of Q, and let QC= u; then
0' = 6 — -^ and p = u.
Ana OC-u'--^- dp'__du_d:'p .
Also p=PQ + QG=p+u'=p + j^,',
but P~W *^®^6fore s = -L + jpdd.
From the value of PY we can obtain an easy proof of a
theorem of some interest in the Differential Calculus (Differ-
ential Calcidus, Art. 329). Let p^ denote the perpendicular
from on the tangent at Y to the locus of Y; then [Differ-
ential Calculus, Art. 284)
1 1 1
prp^'^p'
(dpV
\dd)'
since ^ is
the radius
vector of Y.
Thus
1
P.'
_1 ^r_2)'
f P'
therefore
•
A particular case
; of the formula
^2
-s^ + u^-u^
= pdd
should be noticed. Suppose we take a complete oval curve
without singular points; then 6^=6^ + 27r, and u^=u^', thus
the complete perimeter of the curve is I ^;rf^.
J 01
102 LENGTHS OF CURVES.
92. Application to the Ellipse.
• ' c A
Let APB be a quadrant of an ellipse, CFthe perpendicu-
lar on the tangent at P; let ACY=e. Then {Plane Co-
ordinate Geometry, Art. 196) CY=^ a ^J{1 - e' sin' 6) ;
therefore AP + FY =^ aL{l -e" sin= 6) dO,
the constant to be added to the integral is supposed to be so
taken that the integral may vanish with 6. If 7i be a point
such that its excentric angle is ^ — ^, we have, by Ai't. 78,
thus
And
PY
BP = aj >^{l-e' sin' 6) d9;
AP + PY=BP
_ dp _ ae^ sin 6 cos 6
.(!)•
dd V(l - e' sin' 6) '
Let X be the abscissa of P ; then by Art. 90,
a: = p cos — -, sin 6
dd
,,-, s • 2/1N n ae'sin^^cos^
= a \/{l — e' sm'^) cos 6 +
a cos
V(l-e'sin'^) V(l-e'sin'^)'
Thus PY= e'x sin ; and if x' be the abscissa of P we have
-^ — ^ j so that PY= . Thus (1) may be written
BB-AP = -xx ...
a
this result is called Faemani's Theorem.
.(2);
LENGTHS OF CURVES. 103
From the ascertained values of x and x we have
2 a' — a' sin*
"^ 1-e'^sin-^^
a'
- x'^
1-
(TX'^
therefore eV^-'* — a* {x^ + x'^) + a* =0.
Thus the equation which connects x and x involves these
quantities symmetrically ; hence from (2) we can infer that
BP—AR = — XX . This is also obvious from the figure.
a
The length of FY is also equal to the length of the
corresponding straight line at R.
We may observe that the value of FY may be obtained
more simply by means of a known property of the ellipse.
For suppose the normal at F to be drawn meeting GA at O ;
and through F di-aw a straight line parallel to GA meeting GY
at Q. Then FQ, = GG — e^x, by the nature of the ellipse ; and
FY=FQ^me = e^xdnd.
93. Application to the Hyperbola.
Let G be the centre and A the vertex of an hyperbola,
CFthe perpendicular on the tangent at P. Let AGY=6,
and CY = p ; then it may be proved that
PY-AF= a [ V(l - e' shi' 6) dO.
104 LENGTHS OF CURVES.
This may be proved in the same manner as the corresponding
result of the preceding Article ; we may either make the
requisite changes of sign in the formulae of Art. 90, which
are produced by difference of figure ; or we may begin from the
becfinninof asfain in the manner of that Article. The constant
to be added to the integral is supposed to be so taken that
the intes^ral may vanish with 6.
Suppose a the greatest value which 6 can have, then
PF has its least inclination to the axis GA, and {Plane Co-
ordinate Geometry, Art. 257) cot a — V(e^ — !)• When P moves
off to an infinite distance PY— AP becomes the excess of
the length of the infinite asymptote from C over the length
of the infinite hyperbolic arc from A. Thus this excess
', [V(l - e' sin^ 6) de.
IS a
Inverse questions on the lengths of Curves.
94. In the preceding Articles we have shewn how the
length of an arc of a known curve is to be found in terms
of the abscissa of its variable extremity; we will now briefly
notice the inverse problem, to find a curve such that the arc
shall be a given function of the abscissa of its variable ex-
tremity.
Suppose <^ {x) the given function ; then s = ^{x);
therefore ^ (^) =£= V {l+ (^
thus f^= [{<!>' {x)Y-l]\
and 2/=/[(^'(^)r-l]^^^-
95. As an example of the preceding method, suppose
(f){x) = ^/{'icx) ; thus <f>'ix) = jJ^; therefore
LENGTHS OF CURVES. 105
Ih
- — x]dx . ,
ax
J iJicx — x') 2 J I
sjicx — X') 2 J /^{cx — x^)
c _, 2x
= \/{cx —x^)+^ vers ^ — • + C.
We may ^vrite y for y — C and thus we find that the
curve is a cycloid. {Differential Calculus, Art. 858.)
f(^)
9G. For another example suppose (f)(x) = a log x ; thus
A"
' (a"^ — x^) dx
Here 2/=/y(^- l) ^^ = /l^(^._^.^
Involutes and Evolutes.
97. We may express the length of an arc of a curve with-
out integration when we know the equation to the involute
of the curve. Suppose s' to represent the length of an arc of
a curve, p the radius of curvature at that point of the involute
which corresponds to the variable extremity of s, then {Dif-
ferential Calculus, Art. 331) s ±p = l, where I is a constant.
If the equation to the involute is known, p can be found in
terms of the co-ordinates of the point in the involute ; then
these co-ordinates can be expressed in terms of the co-ordi-
nates of the corresponding point of the evolute, and thus s
is known. By this method we have to perform the pro-
cesses of differentiation and algebraical reduction instead of
integration.
106 LENGTHS OF CURVES.
98. Application to the Evolute of the Parabola.
Take for the involute the parabola which has for its equa-
tion y^ = ^ax\ let x, y be the co-ordinates of the point of
the evolute which corresponds to the point [x, y) on the
parabola. Then by the ordinary methods {Differential Cal-
culus, Art. 330} we have
«' = 2a + ^x, y' = -^2,
and p = 1ai j .
Thus we shall obtain for the equation to the evolute
and /, = 2a(^);
therefore s + 2a \—~-^ ] = I.
3a J ~
Suppose we measure s from the point for which x = 2a,
that is, from the point which corresponds to the vertex of the
parabola; then we see that s increases with x, so that we
must take the lower sign in the last equation ; also by sup-
posing X = 2a and s' = we find l= — 2a; thus
This value of s may also be obtained by the application of
the ordinary method of integration.
99. When the length of the arc of a curve is known in
terms of the co-ordinates of its variable extremity, the equa-
tion to the involute can be found by the ordinary processes
of elimination.
For we have (Differential Calculus, Art. 331)
dx'
dx _ Ids
x —X ~ p dx'
LENGTHS OF CURVES. 107
where the accented letters refer to a point in a curve, and
the unaccented letters to the corresponding point in the in-
volute. Thus
, — ax y- «
^ = ^"+^^' ^^)-
Similarly y = y'T p^ (2).
If then s is known in terms of x, or of y', or of both, by
means of this relation and the known equation to the curve
(IX civ
we may find -tt and -p ; and p is known from the equation
s + p = I. It only remains then to eliminate x and y from
(1) and (2) and the known equation to the curve; we obtain
thus an equation between x and y, which is the required
equation to the involute.
100. Application to the Catenary.
The equation to the catenary is
y
=1^
+ e
XT
s'
=l(^
— e
xf
and
supposing s' measured from the point for which a;' = and
y' =c', we shall now find the equation to that involute to
the catenary which begins at the point of the curve just
specified.
We have then
dy _ s ds _y\
dx c ' dx c '
, dy _ s' dx' _ c
^^""^ di~y" dl~^'
and p = s , no constant being required, because by supposition
p vanishes with s\
108 LENGTHS OF CURVES.
Hence equations (1) and (2) of the preceding Article become
_ ' ^'^ .
CO — vC ■"" ■ J J
y
y=y - — = ■ — ; — = -0
y y y
And s = V(3/" - c^) = ^/(p -0^) = '- V(c' - y) ;
therefore i^^^lil^lf),
y c
thus x = x' — \J{c^ — y^) ; therefore x = \/{c^ — y^) + x.
We have then to substitute these values of x' and y in
the equation to the catenary, and thus obtain the required
relation between x and y. The substitution may be con-
veniently performed in the following manner :
c - --'
y=-^{^'+e')',
therefore ^J[y'^ _ c^) = £ (/ _ e" ^) ;
therefore y + \/{y'^ — c^) = ce°,
thereiore x = c lo? ^-^ .
c
Thus finally, x + V(c' -y') = c log "^ "*" ^^''""''^'^ .
This curve is called the tractor^y ; on account of the ra-
dical, there are two values of x for every value of y less than
c, these two values being numerically equal, but of opposite
signs. There is a cusp at the point for which x = and
y = c\ and the axis of x is an asymptote.
101. The polar formulae may also be used in like manner
to determine the involute when the length of an arc of the
evolute can be expressed in terms of the polar co-ordinates
of its variable extremity. We have {Diferoitial Calculus,
Art. 832)
r" = p' + r'-2pp (1),
p'' = r'-p'' (2).
LENGTHS OF CURVES. 109
Here, as before, the accented letters belong to the known
curve, that is, to the cvolute, and the unaccented letters to the
required involute ; hence since the evolute is known, there is a
kno\\'n relation between 'p and r. And s' + p = /, so that if
s can be expressed in terms of j'j' and r we may eliminate
I) and r by means of (1), (2), and the kuo-vvn relation between
'p and ?•'. Thus we obtain an equation connecting 'p and r,
which serves to determine the involute.
102. Application to the Logarithmic Spiral.
In this curve p' = r sin a, where a is the constant angle of
the' spiral. If we suppose the involute to begin from the
pole of the spiral, and s to be measured from that point, we
have /3 = s' = ?•' sec a (Art, 84). Thus (1) of the preceding
Article becomes
r'^ = r'^ sec' a + 7"' — ^r'p sec a
= r"^ sec^ a + r'* sin* a +p^ — 2r'p sec a, by (2).
From this quadratic for p we obtain
p^r sec a = + r' cos a.
V (1 ~f" cos'' Ot)
If we take the upper sign we find » = — ^^ , and
^^ ° ^ cos a
then from (2) we find r' = '—^ r'*. But this solution
^ ' cos a
must be rejected, because from it we should find p or
r^r — ■ — ■ — 7^ 2-T f', which is inconsistent with the
dp cos a (1 + cos a)
equation p = r sec a.
/ • 9
r SNi ct
If we take the lower sign we find « = , and then
^ ^ cosa
'2 „• 2
from (2) we find r" = ^ — : thus » = r sin a. Hence the
^ ' cos a -^
involute is an equiangular spiral with the same constant
angle as the evolute has. ,
110 LENGTHS OP CURVES.
Intrinsic Equation to a Curve.
103. Let s denote the length of an arc of a curve measured
from some fixed point, (f> the inclination of the tangent at the
variable extremity to the tangent at some fixed point of the
curve; then the equation which determines the relation
between s and <j> is called the intrinsic equation to the curve.
In some investigations, especially those relating to involutes
and evolutes, this method of determining a curve is simpler
than the ordinary method of referring the curve to rectangular
axes which are extrinsic lines.
104. We will first shew how the intrinsic equation may
be obtained from the ordinary equation.
Suppose y =f{x) the equation to a curve, the origin
being a point on the curve, and the axis of y a tangent at
that point; from the given equation we have
S = -^'(^)=tiZ^^y^yP"^^''^^'
thus X is known in terms of tan (f), say x = F (tan 0) ; then
-j-i=F' (tan 0) sec'^ j> ;
also J- = cosec <^ ;
ds
therefore 'Tl~^' ^^^ ^ ^^^ ^ cosec ^ ;
from this equation s may be found in terms of (f) by integra-
tion. A similar result will be obtained if at the origin the
axis of X be the axis which we suppose to coincide with a
tangent.
105. Application to the Cycloid.
By the Differential Calculus, Art. 358, we have
^= //2a -^N 1 .
dx V \ tc / tan ^ '
LENGTHS OF CUEVES. Ill
therefore — = . „ , , cc = 2a sin* 6,
X sin'' </) ^
-7-T = 4a sin cos ^,
c/s , dx . .
-y7 = cosec 9 -77 = 4a cos 9 ;
09 a(p
therefore 5 = 4a sin <^ + G.
The constant will be zero if we suppose s measured from
the fixed point where the first tangent is drawn, that is, from
the vertex of the curve.
106. Having given the intrinsic equation to deduce the
ordinary equation.
We have -7- = sin ^ ;
therefore x= Ids sin </>.
Similarly y = Ids cos (p.
Now s is by supposition known in terms of ; thus by
integration we may find cc and 7/ in terms of <^, and then by
eliminating (f> we obtain the ordinary equation to the curve
in terms of a;*"and y.
107. Application to the Cycloid.
Here s = 4tt sin <^ ;
thus a; = I c?s sin ^ = 4a I sm<j} cos tp dcji =C — a cos 2(f),
y— ids cos (}) = 4'a j cos' ^ dcf) =C'+ 2a^ + a sin 2(f).
Hence by eliminating (f> we can obtain the ordinary equa-
tion ; if the origin of the rectangular axes is the vertex of
the curve, we shall have C = a and C" = 0.
112 LENGTHS OF CURVES.
108. "We shall now give some miscellaneous examples of
intrinsic equations.
The intrinsic equation to the circle is obviously s = a(p.
109. The equation to the catenary is
the origin being on the curve. Hence
thus if ^ be the angle which the tangent at any point makes
with the tangent at the origin,
s = c tan (j).
110. We have seen in Art. 86, that for the epicycloid
, cos 6 — cos — J— a
-^ = =; = tan 6 suppose,
ax . a + ^ . ^
sm — -J — £^ — sm c/
o
thus </) = 26 ^•
Again, from the same Article,
a
4<b(a + h) aO , ^
= cos TTT + C
a 2b
if we suppose s measured from the point for which ^ = 0.
Thus .'"—r-^i^-^^^T^i)-
We may simplify this result by putting
^ 7r(a + 25) ., , 4<h{a + h) , ,.
LENGTHS OF CURVES. 113
this amounts to measuring tlie arc from a vertex instead of
from a cusp. Thus
, 45(a + J) . axb'
s = — ^^ sm — ^ ,
where the accent may now be dropiDcd.
Ill, Similarly the intrinsic equation to the hypocycloid
may be written
46 (a-h) . a<h
s = — ^ sm ^
a a — lh'
112. It appears from the last two Ai'ticles that s = ? sin n^
represents an epicycloid or hyiJocycloid, according as n is less
or greater than unity. For examjjle, if
s = ls,ni-, s = ism^, s=lsm^, s = lsm^ , ...
^ O 4! o
we have epicycloids in which - = ^ , 1, ^, 2, ...
Ob Ji A
If s = l sin 2<^, s = l&m. Scj), s = l sin 4;(f), s = l sin ocf), ...
6 113 2
we have hypocycloids in which - = --,, -^ , - , -, ...
ct 4 b 8 5
113. If p be the radius of curvature of the curve at the
point determined by s and cp, we have (Differential Calculus,
Ai-t. 324)
ds
In the logarithmic spiral we know that p varies as s if the
arc be measured from the pole ; thus
J ds
1 ds
therefore Jc = - -r-r , and therefore by integration
s d(fi JO
k^ + constant = log s ;
therefore s = ae^^^,
T. I. C. 8
114! LENGTHS OF CURVES.
■wliere a is a constant. If we put s = s' + a we have
s' = a (e'^* - 1),
and now s is measured from the point for which <^ = 0.
114. If the intrinsic equation to a curve be known, that
to the evoKite can be found.
Let AP be a curve, B Q the evolute ; let s be the length of
an arc oi AP measured from some fixed point up to P ; s the
length of an arc of BQ measured from some fixed point up
to Q. It is evident that ^ is the same both for s and s, if in
BQ we measure cf) from BA, which is perpendicular to the
straight line from which (j) is measured in AP.
ds
In the left-hand figure s'==p — C = -^j^ — G.
ds
In the right-hand figure s =C— p= ^~'ll.'
Thus if s be known in terms of (f), we can find s in terms
of ^'. The constant G is equal to the value of p at the
point corresponding to that for which s' = 0.
115 For example, in the cycloid 5 = 4a sin ; thus
s = C- 4a cos 4).
LENGTHS OF CURVES. 115
Put <p = '^ + -i^ and s' = a + C; thus
cr = -ia siu i|r.
This shews that the evokitc is an equal cycloid.
116. Similarly if the intrinsic equation to a curve be
known, that to the involute may be found. For by Art. 114
therefore s = I {C ± s) d(f).
Thus if s' be known in terms of cj), we can find s in terms
of 0.
117. For example, in the circle s =a(f). Thus
s = [( a ± «</)) dcp =Ccf)± '4^ + C.
If we suppose s to begin where ^ = we have C — 0, and
further, if we suppose s to begin where the involute meets
J 2
the circle we have C=0; thus s = -~ . [^qq Differential
Calculus, Art. 333.)
118. It is obvious that by the methods of Arts. 114 and
116 we may find the evolute of the evolute of a curve, or the
involute of the involute of a curve, and so on.
119. The student may exercise himself in tracing curves
from their intrinsic equations ; he will find it useful to take
such a curve as the cycloid, the form of which is well known,
and ascertain that the intrinsic equation does lead to that
form ; he may then take some of the epicycloids or hypocy-
cloids given in Art. 112. For further information on this
subject, and for illustrative figures, the student is referred to
two memoirs by Dr Whewell, published in the Cambridge
Philosophical Transactions, Vol. viii. page 659, and Vol. ix.
page 150.
8—2
116 LENGTHS OF CURVES.
Curves of double Curvature.
120. Let X, y, z be the co-ordinates of a point on a curve
in space ; x + Aa?, y + A?y, 2; + As the co-ordinates of an
adjacent point on the curve. Then it is known by the prin-
ciples of soUd geometry, that the length of the chord joining
these two points is \/{(^^)^+ (Av)' + (^•^)'}- ^^^ ^ ^^ the
length of the arc of the curve measured from some fixed point
up to {x, y,z); and let s4-As be the length of the arc measured
from the same fixed point up to (x + Aa:, y + Ay, z + Az).
We shall assume that As bears to the chord joining the adja-
cent points a ratio which is ultimately equal to unity when
the second point moves along the curve up to the first point.
Thus the limit of
As
As A/r?
is unity. Hence
therefore ^ = V{l +(£/+(£)} ^--
uii dz
From the equations to the curve ~ and -7- may be ex-
pressed in terms of x, and then by integration s is kno\^^a in
terms of x.
121. With respect to the assumption in the preceding
Article, the student is referred to Differential Calculus,
Arts. 307, 308 ; he may also hereafter consult De Morgan's
Differential and Integral Calcidus, j)age 444, and Homer-
sham Cox's Integral Calculus, page 95.
122. Suppose, for example, that the curve is determined
by the equations
y^ = 4<ax (1),
z = \/{^cx — x~) -t-c vers"^- (2),
c
LENGTHS OF CURVES. 117
SO that the curve is formed by the intersection of two cylin-
ders, namely a cylinder which has its generating lines parallel
to the axis of z, and which stands on the parabola in the
plane of {x, y) given by (1), and a cylinder which has its
generating lines parallel to the axis of y, and which stands
on the cycloid in the plane of {x, z) given by (2). Then
dy _ / fa\ dz _ // 2c — x \
di~W[x)' dx~\/[~l^)'
therefore s = i^{2c + a) \^- = 2\/(2c + a) \/x.
No constant is required if we measure the arc from the origin
of co-ordinates,
123. The formula given in Art. 120 maybe changed into
=/y
-f|y-^($yi^
and in some cases these forms may be more convenient than
that in Art. 120.
124. Sometimes a Curve in space is determined by three
equations, which express x, y, z respectively in terms of an
auxiliary variable; then by eliminating this variable, we may,
if necessary, obtain two equations connecting cc, y, and z, and
thus determine the curve in the ordinary way. Suppose then
Xy y, z each a known function of t ; therefore
dy dz
dt . dz _dt
dx' dx dx
It dt
118 LENGTHS OF CURVES.
=M(iy-(iy-(S)]-
125, Application to the Helix.
This curve may be determined by the equations
a; = a cos i, 3/ = a sin ty z = ct\
tlius 5 = V (a' + c') [f/^ = t sj{a: + c") + G.
126. Wlien polar co-ordinates are used to determine tlie
position of a point in space, we have the following equations
connecting the rectangular and polar co-ordinates of any
point,
x — r sin 6 cos ^, 2/ = ^^ sin ^ sin ^, z = r cos 6.
And as a curve in space is determined by two equations
between x, y, and z, it may also be determined by two equa-
tions between r, 6, and ^. Thus we may conceive r and
(fi to be known functions of 6, and therefore x, y, and z
become known functions of 6.
Hence
dx . ^ , dr • a • id(h , n J
-vTi = sm y cos <i -77i — r sm ^ sm 6 ^ + r cos a cos *,
dd ad ad
-ii = sm6 sin 6 — .-, + r sin^ cos 6-]-: + r cos 6 sin <f),
dd do dO
dz ^dr . ^
^7^ = cos P -^ - r sm 0.
do do
"^- ^-$h^-^-'^'Q
2
LENGTHS OF CURVES. 119
This may be transformed into
ciuto .= /y [■■'gy +(!)'+.' sin' «}#.
127. If J5 be the perpendicular from the origin on the
tangent to a curve in space, then the equation
ds _ r
which was proved for a plane curve in Art. 85, will still
hold. For each member of the equation expresses the secant
of the angle which the tangent maizes with the radius vector
at the point of contact.
r vdr
Therefore s = , - .>■ — j; ^
EXAMPLES,
1. For what values of m and n are the curves a"'?/" = a-"'"^"
rectifiable? (See Art. 15.)
Result. If TT- or ^^ — 1- ^ is an integer.
'2.111 2m 2 °
2. Shew that the length of the arc of a tractory measured
from the cusp is determined by s = c log - .
3. Shew that the cissoid is rectifiable.
4. Shew that the whole length of the curve whose equation
is 4 (a;' + ?/") - a* = Sa^y^ is equal to fia.
It may be shewn that i-r] = — k — i i^
120 EXAMPLES.
5. The length of the arc of the curve
"between the limits {x^, y^ and [x, y) is
2^ {(^ + yf +{x- yff - 2^ {(x, + y^^ + {x, - y^f.
6. If s = ae% find the relation between x and y.
7. Shew that the intrinsic equation to the parabola is
ds 2a a , 1 + sin <f> , a sin rf>
-TT = TT or S = 75 log 3 -. 2 + ^1 ^2-r •
d<^ cos' (j) 2°l-sin(/) 1- sm' ^
8. The intrinsic equation to the curve y^ = aa^ is
9. Shew that the length of the arc of the evolute of a
parabola from the cusp to the point at which the
evolute meets the parabola is 2a{S/^3 — 1) ; where 4a
is the latus rectum of the parabola.
10. The evolute of an epicycloid is an epicycloid, the radius
2
of the fixed circle being - — ^ and the radius of the
° a + zo
generating circle t^. (Arts. 110 and 114.)
11. Shew that if the equation to a curve be found by
eliminating 6 between the equations
and y = cos dy^r' (6) — sin 6^|/' {6),
then s = yir{e)-\-ylr"{e).
12. Shew that the length of the curve SaJ'y = x* + Ga'x'^
measured from the origin is ^-3 (x^ + 4a") .
121
CHAPTER VII.
AREAS OF PLANE CURVES AND OF SURFACES.
Plane Areas. Rectangular Formulce. Single Integration.
128, Let DPE be a curve, of which the equation is
7/ = (x), and suppose x, y to be the co-ordinates of a point
P, Let A denote the area inchided between the cui-ve, the
axis of X, the ordinate PM, and some fixed ordinate DB, such
that OB is algebraically less than x ; then {Differential Cal-
culus, Art. 48)
dA
bence -^ = I <^ (*) dsc.
Let '^{x)-\-C be the integral of ^ (cc) ; thus
A=^\r{x) + a
Let J. J denote the area when the variable ordinate is at a
iistance x^ from the axis of y, and let A^ denote the area when
122 AREAS OF PLANE CURVES AND OF SURFACES.
the variable ordinate is at a distance x^ from the axis of y\
then
therefore A^ — A^ = '>Jr (x^) — yjr (x^) = (^{x) dx.
J Xi
129. Application to the Circle.
The equation to the circle referred to its centre as origin
is i/^ = a^ — x^ ; here (f> (x) = ^{a^ — x^) ; thus
A = U (x) dx = L{a'- x') dx = '^^^^^ + ^ sin' - + a
The constant G vanishes if we suppose the jixed ordinate
to coincide with the axis of y. It will be seen by draAving a
figure, that the area comprised between the axis of x, the axis
of y, the circle, and the ordinate at the distance x from the
axis of y, may be divided into a triangle and a sector, the
values of which are given by the first and second terms in the
above expression for A. This remark may serve to assist the
student in remembering the important integral
JV(«' - ^') dx = - ^^^^ ~ ^'^ 4- 1' sin-^ ^ .
130. Application to the Ellipse.
Suppose it required to find the whole area of the ellij)se.
The equation to the ellij)se may be written y^ = —^{a'^—x').
Co
Hence the area of one quadrant of the ellipse
= / J v(a' - .V- :-/„ ^/(«= -') ^- 1 ^ =
hence the area of the ellipse is irah.
131. Application to the Parabola.
The equation to the parabola is y"^ = 4<ax ; here then
^ (.^•) = '\/{4!ax),
AREAS OF PLANE CURVES AND OF SURFACES, 123
and I /^{^ax) dx = —^- x- + C;
tlius with the notation of Art. 128
^2-^1=1 'J{-^ax) dx = -^ (x^ - x^).
If x^ = we have for the area — ^ x^-, that is, two-thirds
of the product of the abscissa x^ and the ordinate i^{\ax.^.
132. Application to the Cycloid.
The integration required by the formula 1 ydx becomes
sometimes more easy if we express x and y in terms of a new
variable. Thus, for example, in the cycloid we can put
{Differential Calculus, Art. 358)
a; = a (1 — cos ^), y = a [0 ■\- am 6) ',
therefore I ydx = a" 1 {d 4- sin 6) sin 6 dd
= a' je sin Odd + tfO-- cos -^) ^^'
,, , o / /, /I . /IN «* //I sin 2^\
this gives «" (— ^ cos 6/ + sm t7j + -^ ( a ^ — I .
If we take this between the limits and tt for 6, we obtain
the area of half a cycloid ; the result is — ^ . Hence the
area of the whole cycloid is equal to three times that of the
generating circle.
133. The equations to the companion to the cycloid are
x = a (1 — cos 6), y = ad;
hence it may be shewn that the area of the whole curve is
twice that of the generating circle.
134. If a curve be determined by the equation x = ^(y),
then the area contained between the curve, the axis of y, and
124 AREAS OF PLANE CURVES AND OF SURFACES.
straiglit lines drawn parallel to the axis of x at distances
respectively equal to y^ and y^'^'s> \ <p [y) dy. This is ob-
•'3/1
vious after the proof of the similar proposition in Art. 128.
185. The formulse in Arts. 128 and 134 furnish one of
the most simple and important examples of the application of
the Integral Calculus. As we have already remarked, the
problem of determining the areas of curves was one of those
which gave rise to the Integral Calculus, and the symbols
used are very expressive of the process necessary for solving
the problem. In the figure to Art, 7, the student will see
that the rectangle PpNM may be appropriately denoted by
?/A.T, and the process of finding the area of ADEB amounts
to this ; we first effect the addition denoted by l.yAx, and
then diminish Ax indefinitely.
136. Suppose we require the area contained between the
curve y = c sin - , the axis of x, and ordinates at the distances
x^ and x^ respectively from the axis of y. We have
P» . X
! I sm-
dx = ca{ cos — — cos — ) .
Suppose then x^ = () and x^^air; the area is 2ca. Next
suppose a;^ = and x^ == 2a7r ; the result
ca ( cos — — cos —
a a
becomes zero in this case, which is obviously inadmissible,
since the area must be some positive quantity. In fact sin —
is negative from x = aTr to a; = lair, but in the proof that the
area is equal to I ydx, it is supposed that y is 2>ositive. If
y be really negative the area will be I (— y) dx.
Thus in the present example the area will not be
c I sin - dx but c\ %\n- dx + c\ ( — sin - dx.
AREAS OF PLANE CURVES AND OF SURFACES.
125
X
2aTT
X
that Is, c I sin - dx — c \ sin - dx ;
Jo a JaT «
this will give 2ca + 2ca, that is, 4ca.
Plane Areas. Rectangular Formula. Double Integration.
137. In Art. 128 we have sui>plied a formula for finding
the area of a curve ; that formula supposes the area to be the
limit of a number of elemental areas, each element beinc;' a
quantity of which T/Aa; is the type. We shall now proceed to
explain another mode of decomposing the required area into
elemental areas.
^
y^
/
^
tJ
r
_j^
1
,i'
■^
■^
6
u
?
i
y
1
i J
r
-B
Suppose we require the area included between the curves
BPQE and hpqe, and the straight lines Bh and Ee. Let a
series of straight lines be drawn i^arallel to the axis of y, and
another series parallel to the axis of x. Let st represent one
of the rectangles thus formed, and suppose x and y to be the
co-ordinates of s, and x + Aa? and y -\- Ay the co-ordinates of
t; then the area of the rectangle st is Aa:;A_y. Hence the
required area may be found by summing up all the values
of AxAy, and then proceeding to the limit obtained by sup-
posing Ax and Ay to diminish indefinitely.
We effect the required summation of such terms as AxAy
in the following way : we first collect all the rectangles
12G AREAS OF PLANE CURVES AND OF SURFACES.
similar to st which are contained in the strip PQqp, and
we thus obtain the area of this strip ; then we sum up ail
the strips similar to this strip which lie between Bh and
Ee. The error we may make by neglecting the element of
area which lies at the top and bottom of each strip, and
which is not a complete rectangle, will disappear in the limit
when Ace and Ay are indefinitely diminished.
Let y = 4> {x) be the equation to the upper curve, and
y—-y\r{x) the equation to the lower curve; let OC'=cand
0H= h, then if A denote the required area, we have
rh r4>{x)
A= I I dxdy,
J cJ >l'{x)
for the symbolical expression here given denotes the process
which we have just stated in words.
r r4> (X)
Now \dy = y, therefore dy = <h {x) —-^(x); thus we
J J 4>(j:)
have
rh
A= {(f){x)-ylr (x)] dx.
J c
In this form we can at once see the truth of the expression,
for (f>{x)—^lr (x) =PL—j)L = Pp\ thus [(^{x)—'^ {x) ] Aa? may
be taken for the area of the strip PQ^p, and the formula asserts
that A is equal to the limit of the sum of such strips.
The straight lines in the figure are not necessarily equi-
distant : that is, the elements of which AxAy is the type
are not necessarily all of the same area.
138. The result of the preceding Article is, that the area
A is found from the equation
A = r{(P(x)-ylr{x)]dx.
■J c
This result may be obtained in a very simple manner as
shewn in the latter part of the preceding Article, so that it
was not absolutely necessary to introduce the formula of
double integration. We have however drawn attention to
the formula
rhr<j>{x)
A =^ I i dxdy
J cJ <p ix)
AREAS OF PLANE CURVES AND OF SURFACES. 127
because of tlie illustration ^Yllicll is here given of the process
of double integration ; the student may thus find it easier to
apply the processes of double integration to those cases where
it is absolutely necessary, of which examples will occur here-
after.
139. If the area which is to be evaluated is bounded
by the curves x = y\r{y), and x = j>{y), and straight lines
parallel to the axis of x at distances respectively equal to c
and /(, we have in a similar manner
dydx=\ {<^i:y)-f{y)]chj.
Some examples of the formulas of Arts. 137 and 139 will
now be considered ; we shall see that either of these formuhe
may be used in an example, though generally one will be
more simple than the other.
140. Kequired the area included between the parabola
if=ax and the circle y" = 2ax — x^.
The curves pass through the origin and meet at the point
for which x = a; thus if we take only that ai'ea which lies
Dn the positive side of the axis of x, we have
A = j {'^ {2ax — x^) — aJ (ax)] dx = -J ^.
The whole area will therefore be 2 (— ^^j .
Suppose that we wish in this example to integrate with
•espect to X first. From the equation y'^=2ax — x^ we deduce
: = a ± /\/ (a" — 1/') , and it will appear at once from a figure
hat we must take the lower sign in the jsresent question.
<?
['bus let x^ stand for a — \J {a' — y") , and x,^ for — , then
2 2 2 O 2
a , ira nra la , „
128
AREAS OF PLANE CURVES AND OF SURFACES.
The reader should draw the figure and pay close attention
to the limits of the integrations.
141. In the accompanying figure S is the centre of a
circle BLD, and S is also the focus of a parabola ALC; we
shall indicate the integrations that should be performed in
order to obtain the areas ALB and LDG. This example is
introduced for the purpose of illustrating the processes of
double integration, and not for any interest in, the results:
the areas can be easily ascertained by means of formulae
already given ; thus ALB is the difference of the parabolic
area ALS and the quadrant SLB; and similarly LDG is
known.
Take 8 for origin. In finding the area ALB it will be
convenient to suppose the positive direction of the axis of x
to be that towards the left hand ; thus if 4a be the latus
rectum of the parabola, and therefore 2a the radius of the
circle, the equation to the j)arabola is y^ = 4ia(a — x). and the
equation to the circle is y^ = 4a^ — x\
Suppose we integrate with respect to x first, then
where
area ALB
_ _y_
^ 4a
r2a rx^
= I dydx,
Jo «' Xi
For here {x^—x^)^7/ represents a strip included between the
two curves and two straight lines parallel to the axis of x; and
AEEAS OF PLANE CURVES AND OF SURFACES. 129
strips are situated at distances from the axis of x ranging
between and 2a, so that the integration with respect to y is
taken between the Hmits and 2a.
Suppose we integrate with respect to ?/ first; we shall then
have to divide the area into two parts by the straight line AF,
parallel to 8Y. Let
2/i = V(-ia' - 4aa;), ?/, = V (-ia" - x"") ;
dxdy =\ {y^ — y J dx ;
- ji Jo
area AFB = \ dxdy= \ y^dx ;
J a J J a
the sum of these two parts expresses the area ALB.
Next take the area LDG; suppose now the positive direc-
tion of the axis of x to be that towards the right hand, then
the equation to the parabola is / = 4a (a + x), and the equa-
tion to the circle is y^ = 4a^— x^.
Suppose we integrate with respect to y first ; let
2/i = \/(4a''-^') and 2/^= \/(4a' + 4a^),
/•2a ryi
then areajDZC= dxdy.
J J y,
Suppose we integrate with respect to x first; we shall then
have to divide the area into two parts by the straight line LK,
parallel to SD: Let
^^ = V(4a*-/), x^ = ^-a;
then we shall find that 1)0= 2a^JS = h suppose; thus
ria r2a
&ve£iDLK= dydx,
rb r2a
area(7ZA^= dydx;
J 2a J X,
the sum of these two parts expresses the area LDG.
T. I. c.
130 AREAS OF PLANE CURVES AND OF SURFACES.
142. One case in whicli the formuloe of Arts. 137 and 139
are useful is that in which the bounding curves are different
branches of the same curve. Suppose the equation to a curve
to be {y — mx — cf = a^ — x' ; thus
y = mx + c + \l{a? — x^).
Here we may put
•\|r (x) = mx + c — \/{a^ — x^),
(}) {x) = mx + c + \/{a- — x^) \
thus <^{x) —■^{x) = 2s/ia' — x"), and the complete area of
the curve is
1 2sJ{c^ — x^)dx, that is, ira^.
J -a
143. We have hitherto supposed the axes rectangular,
but if they are oblique and inclined at an angle co, the for-
mula in Art. 3.28 becomes
A = sin to I (p (x) dx,
and a similar change is made in all the other formulae. It is
obvious that such elements of area as are denoted by y^x
and A^Aj? when the axes are rectangular will be denoted by
sin w yAx and sin co Ay/lx when the axes are inclined at an
ancdc (o.
"O"
For example, the equation to the parabola is y^= 4a'^ when
the axes are the oblique system formed by a diameter and
the tangent at its extremity; hence the area included be-
tween the curve, the axis of x, and an ordinate at the point
for which x = c, is
f / / i r s 7 4 sin CO \/ac-
sin 0) v (4a x) ax = ,
Jo ^
that is, two thirds of the parallelogram which has the abscissa
c and the ordinate at its extremity for adjacent sides.
AREAS OF PLANE CURVES AND OF SURFACES. 131
Plane Areas. Polar Formula}. Single Integration.
144. Let CPQ be a curve, of which the polar equation is
r = {6), and suppose r, 6 to be the co-ordinates of a point P.
Let A denote the area inchidcd between the curve, the radius
vector SP, and the radius vector SG drawn to some fixed
point C, such that the angle CSx is algebraically less than 6 ;
then {Differential Calculus, Art. 313)
dA_ [4>{6)Y
cW 2 •
Hence A=h [\4>(0)Yd0,
=hj[4>m
Let fid) be the integral of i^^ , then
A = 'f{e) + c.
Let ^j denote the area when the variable radius vector is
at an angular distance 6^ from the initial straight line, and
let A^ denote the area when the variable radius vector is at
an angular distance 6„ from the initial straight line ; then
A = ^(^J + a A,=^{r{e,) + C,
therefore A^- A^= f[d^) - f (^J = h l^'^W)]' dd.
J Oi
145. Ajoplication to the Logarithmic Spiral.
In this curve r = he'' ; thus
2« 6^c ^'
A = \\l/e^dd = ^-fe''-{-G,
9—2
132 AREAS OF PLANE 'CURVES AND OF SURFACES.
and A,-A,= ^jye"^ cZ^ = '^^ (e « - e ^ ) = | (r/ - r/),
where r, and r„ are the extreme radii vectores of the area
considered.
146. Application to the Parabola.
Let the focus be the pole, then
COS 2 J COS -
= — f 1 + tan^ ^ J sec^ ^ "^ = « tan - + -^ tan^ 2 + ^•
Hence ^,- ^, = «'/tan | - tan |) + | (tan^ | - tan' |) .
TT
Suppose that 6^ = and ^2 = 9' ''^^'^'^ "^^ obtain for the
. 4a'
area a^ -^ -^ , that is, ^; this agrees with Art. 131.
o o
For another example we will suppose the parabola refer-
red to the intersection of the directrix and the axis as pole,
the axis being the initial straight line. Here
^ cos^-V(cos26')
^ = -^ sh^^^ '
. c 2 fcos'^ + COS 2^-2 cos 6 V(cos 26) , .
thus A = 2a I r-^-fl "t/
J sm t/
„ , r2 cos"^ - sin'^ , . . , [cos d V(cos 20) ,.
= 2a — — ^-Ta- d6- 4a- t-^^ "^•
J siyrd J sm 6
AREAS OF PLANE CURVES AND OF SURFACES. 133
Now [ ^cos'^^-sm"^ ^^ ^ f ^^^,^ _ ^^^^^, ^^^^
J sm d J ^
= -§ cot'^ + cot^.
, /• COS ^ V(cos 2e)dd _ fJ(l-2 sin' ^)Jsm^
assume sin 6 = ~ , then the intesral becomes
t' °
- h{t- - 2) tdt, that is, - 1 (f - 2)1
Hence, adding the constant, we have
^ = ^ (cosec' ^ - 2) t - ^' cot^ ^ + 2a' cot 6' + C
4a' (cos2g)^-c os^ ^
3 sin' d
= 2a cot ^ + -5- -'^ J 3 ^ 1- C.
The constant will be zero if A commences from the initial
straight line ; for it will be found on investigation that
„ , . 4 (cos 2^)^ - cos'' 6 . , , ^ ^
2 cot o + 7i f^-TTi vanishes when £7 = 0.
3 sm
147. Application to the curve r = a (^ + sin 6). Here
A = I" [(^ + sin ey dd = ^j(e^' + 20 sin e + sin' ^) dd ;
and I ^ sin 6d9 = — 6 cos ^ + sin 0,
[sin' ^ cZ^ = 1 f (1 _ cos 2^) c^^ = I -
^v ^ «M^' ozi n ^ ■ n ^ sin 2^] ^
thus ^= 2]3-2^cos^+ 2sm(9 + 2 ^f+^-
Suppose we require the area of the smallest portion which
is bounded by the curve and by a radius vector which is
si n 2^
4 '
134
AREAS OF PLANE CURVES AND OF SURFACES.
inclined to the initial straight line at a right angle ;
have and hrrr as the limits of the integration,
required area is
then Ave
Thus the
a'
2
1 1-2
24 ^ 4 ^
Plane Curves. Polar Formulce. Double Integration.
148. In Art. 144 we have obtained a fonnula for finding
the area of a curve ; that formula supposes the area to be the
limit of a number of elemental areas, each element being a
quantity of which ^r"^ A^ is the type. We shall now proceed
to explain another mode of decomposing the required area
into elemental areas.
x/
Suppose we require the area included between the curves
BPQE and hpqe, and the straight lines Bh and Ee. Let a
series of radii vectores be drawn from 0, and a series of circles
Avith as centre ; thus the plane area is divided into a series
of curvilinear quadrilaterals. Let st represent one of these
elements, and suppose r and 6 to be the polar co-ordinates of
s, and r + Ar and 6 -f A^ the polar co-ordinates of t ; then the
area of the element st will be ultimatelv rA6^Ar. Hence the
required ai'ea is to be found by summing up all the values of
rA^A?-, and then proceeding to the limit obtained by sup-
posing A^ and Ar to diminish indefinitely.
AREAS OF PLANE CURVES AND OF SURFACES. 135
We effect the required summation of such terms as ?-A^Ar
in the following way: we first collect all the elements similar
to St which are contained in the strip PQqp, and thus obtain
the area of the strip ; then we sum up all the strips similar
to this strip which lie between Bh and Ee.
Let r = (^ (6) be the equation to the curve BPQE and
r = -^{6) the equation to the curve hpqe, let a and ^ be the
angles which OB and OE make respectively with Ox ; and
let A denote the required area, then
A= rcld dr ;
J a J ^(0)
for the symbolical expression here given denotes the process
which we have just stated in words.
Now lrc?r = -^, therefore
I'
J li
thus we have
J a
In this form we can see at once the truth of the expres-
sion, for 0P= (6) and Oj) = -^ {&), and thus
may be taken for the area of the strip PQqp, and the formula
asserts that the area A is equal to the limit of the sum of
such strips.
149. The remark made in Art. 138 may be repeated
here ; we have introduced the process in the former part of
the preceding Article, not because double integration is
absolutely necessary for finding the area of a curve, but
because the process of finding the area of a curve illustrates
double integration.
150. If the area which is to be evaluated is bounded by
the curves whose equations are d=(p{r), 6=-^{r) respectively,
13G AREAS OF PLANE CURVES AND OF SURFACES.
and by the circles whose equations are r — a and r = h re-
spectively, it will be convenient to integrate with respect to
6 first. In this case, instead of first summing up all the
elements like st, which form the strip PQqp, we first sum up
all the elements similar to st which are included between the
two circles which bound st and the curves determined by
6= <^ (it') and 6 = ylr[r). Thus we have
rb r^{r)
A = \ \ rdr dO.
J aJ <p(r)
Some examples of the formulee in Arts. 148 and 150 will
now be considered ; we shall see that either of these formulee
may be used in an example, although one may be more con-
venient than the other.
151. We will apply the formulte to find the area between
the two semicircles OPB and Opb and the straight line bB.
Let Oh = c, OB = h, then the equation to OPB is r = h cos 0,
and the equation to Opb is r = c cos 6. Thus the area
IT
r2 rhcosd
= I I rdddr.
Jo J ccosd
rhcosB
Now rdr=h{li'-c')coii'e;
J c COS 9
ir
therefore the area = i {Ji' - c') cos'' dde - J {h'' - c').
Jo o
Suppose we wish to integrate with respect to 6 fii'st ; we
shall then have to divide the area into two parts by describing
an arc of a circle from as centre, with radius Ob. The
AREAS OF PLANE CURVES AND OF SURFACES. 137
area bounded by this arc, the straight line Bh, and the larger
semicircle is
cos-i^-
rdr do.
The area bounded by the aforesaid arc, the semicircle Oph,
and the larger semicircle is
c C h
/:/
rdr dd.
cos-i-
c
The sum of these two parts expresses the required area.
152. Let us apply polar formulae to the example in
Art. 141. With S as ]3ole, the polar equation to the parabola
is r (1 + cos 6) = 2a or r cos^ g ^ ^' where 6 is measured from
SB ; and the polar equation to the circle is r = 2a. Hence,
if we integrate with respect to r first,
area ALB = ^ rdO dr.
J J at
If we integrate with respect to 6 first, we shall have if
^ _i 2a - r
t'j = cos ^
r2a rSi
area ALB = I rdr dd.
J a J
Next consider the area L)LG. The equation to 1)0 is
r cos ^ = - 2a ; the length of SO is 4a, and the angle BSO
IS—. Let ^, = cos' , ^2 = cosM J. Then if
w^e integrate with respect to 6 first.
area
/•4a rPo
DLC= rdrdO.
J 2a J 01
If we integrate with respect to r first, we shall have to
divide the area into two parts, by the straight line joining *S'
138 AREAS OF PLANE CURVES AND OF SURFACES.
with C. The area of the portion wliich has LC for one of its
boundaries is
2n- „9
.— - asec^-
o f 2
/7
J n J 2
rdO dr.
Tr_ J 2a
2
The area of the remaining portion is
m r—2asec8
J 277 J 2
nW dr.
2nJ 2a
3
The sum of these two parts expresses the required area.
153. A good example is supphed by the problem of find-
ing the' area included between two radii vectores and two
different branches of the same polar curve.
Suppose BPph, CQqo to be two different arcs of a spiral,
and that the area is to be evaluated which is bounded 'by
these arcs and the straight lines BC and he ; then the area is
jir,'-r:)dd,
AREAS OF TLAXE CURVES AND OF SURFACES. 139
where r, denotes any radius vector of the exterior arc, as SQ,
and ?'j the corresponding radius vector SP of the interior arc.
The limits of 6 will be given by the angles which SB and
Sb respectively make Avith the initial straight line.
Take for example the spiral of Archimedes ; let 6 be the
whole angle wliich the radius vector has revolved through
from the initial straight line until it takes the position SF ;
so that 6 may be an angle of any magnitude. From the
nature of the curve we have SP or r = a6, where a is some
constant. If then CQ is the next branch to BP, and 6 still
corresponds to SP, w^e shall have SQ = a {6 + 2??). Suppose
6^ and 6^ the values of 6 for SB and Sb respectively; thus
the area BbcC
2 rfi
a '■
^ J 01
d'
154. The student will remark a certain difference be-
tween the formulse \\dxdy and \\rd9dr, which exjaress the
area of a plane figure. The former supposes the area decom-
posed into a number of rectangles and AicA^/ represents the
true area of one rectangle. Hence in taking the aggregate of
these rectangles to represent the required area the only error
that can arise is owing to the neglect of the in-egular ele-
ments which occur at the top and bottom of each strip ; as
we have already remarked in Art. 137. But in the second
case rA^A?' is not the accurate value of the area of one of
the elements, so that an error is made in the case of every
element. It is therefore important to shew formally that the
error disappears in the limit, which may be done as follows.
The element st in the figure of Art. 148 is the difference of
two circular sectors, and its exact area is
Kr+Ar7A^-ir'A6',
that is rArA^ + ^(A?f A^.
140 AREAS OF PLANE CURVES AND OF SURFACES.
In taking the former term to represent the area we neglect
■J (A?')'A^. Hence the ratio of the term neglected to the
term retained
_ ^(Ar)°Ag _Ar
- 7'ArA^ ~2r*
By taking Ar small enough this ratio may be made as small
as we please. Hence we may infer that the sum of the
neglected terms will ultimately vanish in comparison with
the sum of the terms retained, that is, all error disappears in
the limit.
Other Polar Formulce.
155. Let s be the length of the arc of a curve measured
from some fixed point up to the point whose co-ordinates are
r and 6; let p be the perpendicular from the origin on the
tangent at the latter point: then the sine of the angle between
this tangent and the corresponding radius vector is ^^ ^ {Dif-
ferential Calculus, Art. 810) ; also ^ is another expression for
this sine ; hence, r V=-. Let A denote the area between
' as r
the curve and certain limiting radii vectores ; then
A = i J7- dd = ip ^ cZs = 1 Jr '^ds = ^,jpds ;
the limits of s in the latter integral must be such_ as corre-
spond to the limiting radii vectores of the area considered.
The result can be illustrated geometrically ; suppose P, Q
adjacent points on a curve, S the pole, j^' the perpendicular
from S on the chord PQ ; then, the area of the triangle PQS
= i/x chord PQ.
Now suppose Q to approach indefinitely near to P, then
2) =p, and the limit of the ratio of the chord PQ to the
arc PQ is unity.
AREAS OF PLAXE CURVES AND OF SURFACES. 141
Since /^ci.=/^**-=/^^^(Art.85),
, ^ 1 r V'>'dr
wo nave A = ^\ —-f-r, ttt .
156. Application to the Epicycloid.
Here «^ = — -, r— : thus
^ c'-a' '
A - i[^ \/(?'" — «^) rdr _ c f \/{r'^ — a^ rdr
'J a^ic'-r') ~2^Jv{c'-a'''-(/''-a-')}
= TT -jT^i -i T\ y where r = r - a .
Now
& — (T . ^^ z z \j{(? — a" — 2;^)
c' - a^ . _, V(r' - a') V(^-' - a') V(c' - r')
= — ;; — sm / _ V N / V
V(c'-a')
Taking this between the limits r = a and r = c, we get
2 2
) — a TT
■, that is, b (a + h) ir. Hence the area is ^-h {a+ h) tt,
that is, " ^ ^ — . By doubling this result we obtain
the area between the curve and the radii vectores drawn to
, ,. -u- 1 • ^-u f (a + 2?;) i (a + &) TT
two consecutive cusps, which is thereiore ^
a
The area of the circular sector which forms part of this area
is irah ; subtract the latter and we obtain the area between
an arc of the epicycloid extending from one cusp to the next
142 AREAS OF PLANE CURVES AND OF SURFACES,
cusp and the fixed circle on which the generating circle rolls ;
the result is
— (3a +26).
Similarly in the hypocycloid the area between the fixed
circle and the part of the curve which extends between two
consecutive cusps may be found. If a is greater than h the
result is
— (3a -26).
Area between a Curve and its Evolute.
157. In the figures to Art. 114, if we suppose the string
or straight line PQ to move through a small angle A0, the
figure between the two positions of the straight line and the
curve AP may be considered ultimately as a sector of a circle ;-
its area will therefore be -|p^A0, where p = PQ. Thus if ^
denote the whole area bounded by the curve, its evolute, and
two radii of curvature corresponding to the values 0^ and cf)^
of </), we have
A
r<p2
Hi
Since -^^ = - , we may also "write this
ds p -^
A
= ^jpds,
the limits of s being properly taken so as to correspond with
the known limits of <f). Or we may write the formula thus,
A =^lp'^dx.
ds
dx
158. Application to the Catenary.
Here s=c tan 0, Art. 109 ;
therefore /3 = csec^^, -^ = il c*sec*^cZ^;
AREAS OF PLANE CURVES AND OF SURFACES. 143
auJ l sec'* (j)d(l> = tan (f) + I t-An^<f) + C;
thus A is known.
Area of a Pedal Curve.
159. Suppose that perpendiculars are drawn from one
and the same point in the phxne of a curve on all the tangents
to the curve ; the locus of the feet of the perpendiculars is
called a pedal curve, the point from which the perpendiculars
are drawn is called a j^edal origin, and the curve from which
the pedal curve is derived is called the immitive curve.
"We have already had occasion in Arts. 90... 93. to notice
some relations between the primitive curve and a pedal
curve : we shall now give a proposition respecting the areas
of the various pedal curves which can be formed from the
same primitive curve by varying the pedal origin.
By the area of a pedal curve is meant the area described
by the perpendicular as the point of contact describes a given
arc of the primitive curve.
IGO. The origins of i^edals of a given area lie on a conic
section; and the conic section has the same centre whatever he
the given area.
Let A denote the area corresponding to a certain pedal
origin ; let A' denote the area corresponding to another
pedal origin 0' ; let r and 6 be the polar co-ordinates of 0'
witli respect to 0. Let j} denote the length of the perpendi-
cular from on any tangent to the primitive curve ; let p'
denote the length of the perpendicular from 0' on the same
tangent. Let (f> be the angle between these perpendiculars
and the fixed initial line. Then, as in Art. 157,
A = ^jpUl<p, A'^l^jp'^dcf^i
the integTations are to be taken between fixed limits.
144 AREAS OF PLANE CURVES AND OF SURFACES.
Now p =p-r cos ((f) — 6); therefore
A'=A- [pr cos ((^ - ^) # + ijr' cos^ (</> - 6') cZc/) (1).
Let a; = r cos 9, y = r sin 6 ; then
A'=A- {hx + hj) + M + 2ma;?/ + nf (2),
where h, Jc, I, m, n are certain quantities which remain con-
stant for every position of 0'.
Now (2) shews that the locus of [x, y) for a given yahie of
A is a conic section ; and that the conic sections obtained by
assijrnino: different values to A' are concentric.
The conic section is in general an ellipse. For, by putting
for I, m, n their values, we have
4 (m' - In) = ] [sin cf) cos </) (7</)[ - ] jcos'^cf) d^i x |jsin'</) dc})]- ,
and it may be shewn that the expression on the right-hand
side is negative ; see Example 21, at the end of Chapter IV.
Hence by Chapter xiii. of the Fla^ie Co-ordinate Geometry,
the conic section is an ellipse.
If the conic section were referred to its centre as origin,
the terms of the first degree in x and y would disappear from
the equation (2) ; thus we see indirectly that there must be
some pedal origin for which A = and ^ = 0. Suppose this
origin taken for 0, then we have from (1),
A'=A + ^jr'cos'{(j>-e)dcl);
as the second term on the right-hand side is positive. A' is
necessarily greater than A, so that the origin is that which
makes the pedal area least.
In the particular case in which the primitive curve is a
closed curve the conic section becomes a circle. For the
limits of (/) may then be supposed to be and 27r ; and thus
we have l = 7i and m = 0.
AREAS OF PLANE CURVES AND OF SURFACES. 145
We may just advert to the effect of the existence of
singular points on the primitive curve. In this case it may
liappen that (f) does not always increase from tlie lower limit
jf the integrations to the upper limit, but sometimes increases
md sometimes decreases. Suppose now, for example, that
irst increases from Q to -it, then diminishes from -tt to - tt
o 3 4'
md then increases from - tt to - tt. The values of h, k, I, m,
I Avill then be the same as if ^ had always increased from
) to - TT. The area of that part of the pedal curve traced out
-s ^ decreases from -^-ir to - tt will count as a negative
[uantity.
A memoir by Professor Hirst on the Volumes of Pedal
hirfaces will be found in the Philosophical Transactions for
8G3.
Area of Surfaces of Pevolution. Pectangidar Formidce.
161. Let ^ be a fixed point in the curve APQ ; let x, y
e the co-ordinates of any point P, and s the length of the
re ylP. Suppose the curve to revolve round the axis of cc,
ad let S denote the area of the surface formed by the revolu-
on of ^P; then {Differential Calculus, Art. 315)
dS .
T. I. C.
10
14G AREAS OF PLANE CURVES AND OF SURFACES,
therefore >S' = I I'lryds (1) ;
thus S = I 27ri/ -J- dx (2),
f ds
and S = \ 2^-7/ -J- dy (3).
Of these three forms we can choose in any particular ex-
ample that which is most convenient. If y can be easily
ds
expressed in terms of s we may use (1) ; if -7- can be easily
expressed in terms of 7/ we may use (3) ; generally however
ds
it will be most convenient to express y and -j- in terms of x
and use (2).
In each case the area of the surface generated by the arc
of the curve which lies between assigned points will be found
by integrating between appropriate limits.
162. Application to the Cylinder.
Suppose a straight line parallel to the axis of x to revolve
round the axis oi x, thus generating a right circular cylinder:
let a be the distance of the revolving straight line from the
axis of X ;
then 2/ = ^. ^^^ ^ ~^>
thus by equation (2) of Art. 161,
/S = 27r I adx = 27rax + C.
Suppose the abscissae of the extreme jDoints of the portion
of the straight line which revolves to be x, and x^: then the
surface generated
= 2-!ra I dx — 2'Tra {x^ — x^.
J Xi
AHEAS OF TLANE CURVES AND OF SURFACES. 147
1G3. Application to the Gone.
Let a straight line whicli passes tlirougli tlie origin and is
inclined to the axis of x at an angle a revolve round the axis
of X, and thus generate a conical surface. Then
y = x tan a, and -r-; = sec a ;
thus by equation (2) of Art. 161,
S=27r I tan a sec a xdx = 'ir tan a sec ax^+ C,
Hence the surface of the frustum of a cone cut off by planes
perpendicular to its axis at distances x^, x^ respectively from
the vertex is
TT tan a sec a (x^ — x^).
Suppose x^ = 0, and let r be the radius of the section made
by the plane at the distance x^., then r = x^isoa.a, and the
area is
TT cosec ar*.
1G4. Application to the Sphere,
Let the circle given by the equation y^=a^ — x^ revolve
round the axis of x ; here
dy _ X
dx y '
a?\ a
1+-J =
y^ y
Hence by equation (2) of Art. IGl,
dx = lirax + G.
*S = 27r (y- dx = 2TTa
r y
Thus the surface included between the planes determined by
x — x^ and x = x^ is 27ra (^2 — rrj.
Hence the area of a zone of a sphere depends only on the
height of the zone and the radius of the sphere, and is equal
10—2
148 AREAS OF PLANE CURVES AND OF SURFACES.
to the area wliicli the planes that bound it would cut off from
a cylinder having its axis perpendicular to the planes and
circumscribing the sphere; and thus the surface of the whole
sphere is 4!7rd\ These results are very important.
165. Application to the Prolate Spheroid.
Let the ellipse given by a^ + Z'V= a'h'^ revolve round the
axis of X which is supposed to coincide with the major axis
of the ellipse ; here
dif _ Ifx
dx (ly '
dx V \ a^if) ay
Hence by equation (2) of Art. 161,
The surface generated by the revolution of a quadrant of
the ellipse will be obtained by taking and a as the limits of
X in the integration. This gives
166. For another exami3le suppose the catenary
to revolve round the axis of x. Here s = ^[6" - e " j , by
Art. 73, if we measure from the point for which x = 0. Thus
we see that y- = s^+c'^. In this case we shall find that we
can use any of the three formulae in Art. 161; but (2) will
be the most convenient.
167. Suppose one curve to have for its equation y = (^),
and another curve to have for its equation y = -^{^), ^nd let
AREAS OF PLANE CURVES AND OF SURFACES. 149
both curves revolve round the axis of x. Let s^ and s^ denote
the lengths of arcs measured from fixed points in the two
curves up to tlie point whose abscissa is x. Let & denote the
sum of the areas of both surfaces intercepted between two
planes perpendicular to the axis of x at the distances x^
and x„ respectively from the origin. Then, by Art. 161,
^=277
£{^wg + f(.)grf..
For a simple case suppose that there is a curve which is
bisected by the straight line y = a, so that we may put
■y = C5 -f ^ (a;) for the upper branch and y = a — xi-^) ^^r the
lower branch. Hence
ds^ _ ds^
dx dx'
p2 ds f
and S = 47ra I -j-^dx = iira I ds^ ,
the limits for s^ being taken so as to correspond with the
assigned limits of x.
Hence, if there be any complete curve which is bisected
by a straight line and made to revolve round an axis which is
parallel to this straight line at a distance a from it and which
does not cut the curve, the area of the whole surface gene-
rated is equal to the length of the curve multiplied by '"lira.
For example, take the circle given by the equation
[x-hy+{y-kY-c''=0.
Here the area of the whole surface generated by the revolu-
tion of the circle round the axis of x will be ^irk x lire.
There is no difficulty in this example in obtaining sepa-
rately the two portions of the surface. For the part above
the straight line y = k, -we have 27r I yds, that is,
27rjllc + ^{c'-(x-hf}]ds,
that is, 2'rrjkds + lir f V [c' -{x- lif] ds.
150 AREAS OF PLANE CURVES AND OF SURFACES.
The former of these integrals is ^-nhs ; the latter is equal to
which will reduce to 27r I cdx, that is, ^ircx. Hence the sur-
face required is found by taking the expression '^itlis + ^ircx
between proper limits.
Area of Surfaces of Revolution. Polar Formulw.
168. It may be sometimes convenient to use polar co-
ordinates ; thus from Art. IGl we deduce
S = hiryds =kiry %de = J2'7rr sin ^ dO,
ds
where ^
169. Application to the Cardioid.
Here r = a (1 + cos ^) ; thus
^ = a V[(l + cos 6')^+ sin^6'} = a V(2 + 2 cos 6) = 2a cos ^ ;
do
therefore
S = 47ra' (1 + cos 0) cos - sin 0d9= IGTra'Jcos* ^ ^^^^ ^^
= 5- cos 2+ C.
The surface formed by the revolution of the complete curve
about the initial straight line will be obtained by takmg
„, . . 327ra
and TT as the limits of in the integral. This gives —^ .
AREAS OF PLANE CURVES AND OF SURFACES. 151
Any Surface. Double Integration.
170. Let X, y, z be the co-ordinates of any point 2> of a
surface ; x + A.r, y + A^, ^ + A^ the co-ordinates of an ad-
x*^
/y
jacent point q. Through -p draw a plane parallel to that
of ix, z), and a plane parallel to that of (y, z) ; also through
q draw a plane parallel to that of {x, z) and a plane parallel
to that of (7/, z). These planes will intercept an element ^5-
of the curved surface, and the projection of this element on
the plane of {x, y) will be the rectangle PQ. Suppose the
tangent plane to the surface at p to be inclined to the plane
of {x, y) at an angle 7, then it is known from solid geometry
that
sec 7
^ + '1,
+
chV
dy)
where ^- and -^ must be found from the known equation to
ax dy
the surface. Now the area of PQ is AxAy, hence by solid
geometry the area of the element of the tangent plane at p of
which FQ is the projection is AxAysecy. We shall assume
that the limit of the sum of such terms as Ax Ay sec 7 for all
152 , AREAS OF PLANE CURVES AND OF SURFACES.
values of x and y comprised between assigned limits is the
area of the surface corresponding to those limits. Let then >S^
denote this surface ; thus
^-(£hm>'y
dy)
the limits of the integrations being dependent upon the
portion of the surface considered.
171. With respect to the assumption in the preceding
Article, the reader is referred to the remarks on a similar
point in the Differential Calculus, Art. 308 ; he may also here-
after consult De Morgan's Differential and Integral Calculus,
page 444, and Homersham Cox's Integral Calculus, page 96.
172. Application to the Sphere.
Let it be required to find the area of the eighth part of
the surface of the sphere giveu by the equation
a^' + y + s' = a^
,T clz X dz V
Here -r- = — . -i- = — -:
a'
thus 8 =
dx s ' dy
^^{d^-x^-f)
Now in the figure we suppose OL = x\ put ?/j for 12,
then 3/j = VC^* — ^0' ^^^ ^^ value of y^ is obtained from the
equation to the surface by supposing 2 = 0. If we integrate
with respect to y between the limits and y^, we sum up all
the elements comprised in a strip of which LMml is the pro-
jection on the plane of (x, y). Now
pi dy^ _ r
V(a^-^^-y) U^[y{-f) 2
y^- dy _ ir _
thus S= ~^idx.
If we integrate with respect to x from to a, we sum up
all the strips comprised in the surface of which OAB is the
AREAS OF PLANE CURVES AND OF SURFACES. 153
ira
projection. Thus --^ is the required result ; and therefore
the whole surface of the sphere is 4>7ra^.
If we integrate with respect to x first, we shall have
r fi. adr/dx
ioio ^{d'-x'-f)'
where x^ = \l{a^ — if).
As another example let it be required to find the area of
that part of the surface given by the equation
z^ + (x cos a + 3/ sin a)""* — a^ = 0,
which is situated in the positive compartment of co-ordinates.
This surface is a right circular cylinder, having for its axis
the straight line determined by z = 0, xcosa +?/ sin a = 0,
and a is the radius of a circular section of it. Here
dz _ cos a {x cos a + ;/ sin a)
dx z
dz _ sin a [x cos a + y sin a)
dy~ z
thus &= [[^^^ = (I , . , "^"^^ ■ ,., .
J J z jj Vl«— (^' COS a + 3/ sm a)}
The co-ordinate plane of [x, y) cuts the surface in the
straight lines a = + (a; cos a + ?/ sin a), and if the upper sign
be taken, we have a straight line lying in the positive quad-
rant of the plane of [x, y).
To obtain the value of >S' we integrate first with respect to
y between the limits y = and y={a — x cos a) cosec a ; now
r dy 1 . _.x cos a + V sin a
J i^[d^ — [x cos a -1- y sin a)"} sin a a '
take this between the assigned limits, and we obtain
TT . _, rr cos a
-X — sm
sma V2 a ' '
154 AREAS OF PLANE CURVES AND OF SURFACES.
Gj C I tt . 00 cos ol)
therefore >S'= -. — \ \^ — sin"^ -[ dx,
sm a j ( ti a )
and the limits of the integration are and . Hence we
cos a
shall find
sin a cos a
173. It is worthy of notice that two different surfaces
may have their corresponding elements of area equal. Take
■ for example the surfaces determined by 2a2 = ac^ + y^, and
by az=-xij\ in each case
dz^ (dz_^ _ x^-^y-
\dxj
\dy) a'
Euler has discussed this matter in a Memoir entitled
Evolutio insignis paradoxi circa cequalitatem super Jicier am.
Novi Comm. Acad. Petrop. Tom. xvi. Pars prior. He calls
two such surfaces superficies congruentes.
The following surfaces are congruent :
the cone {z - c)'' = {(a; - aY +(y- hf] tan' 7,
and the plane x cos ci + y cos ^ + z cos y= p.
Again, the surfaces determined by the following equations
are congruent:
2az = «" + 1/^,
2az = {x' -7f)c + 2xy V(l - c'),
2a2 = [{x' + yj - Uxy + 2c {x' - y') + J' + c^}^,
2az = {x^ - 2/') cos e+2xysmd-[<f> (6) dd,
where ^ (6) is any function of 6, and ^ is a function of x aud
y determined by
2xy cos 6 — {x' — y') suxO = ^ {6).
174. Instead of taking the element of the tangent plane
at any point of a surface, so that its projection shall be the
AREAS OF PLANE CURVES AND OF SURFACES. 155
rectangle Aa;A//, it may be in some cases more convenient to
take it so that its projectiou shall be the jjolar element rAOAr.
Thus we shall have
S= 1 1 secy rdO dr.
For example, suppose we require the area of the surface
cvij = az, which is cut off by the surface x~ + if = c" ; here
sec7 = y^ (1 +-3 + ^.j = ^-^ smce x- + f = r\
Thus S=rr^^^^^^rdedr = '^{{c^+a^f-a^].
J Jo ct. ou
175. Suppose a; = r sin ^ cos (j),y = rsmO sin (f),z = r cos 9,
so that r, 6, ^ are the usual polar co-ordinates of a point in
space; then Ave shall shew hereafter that the equation
may be transformed into
An independent geometrical proof will be found in the
Cambridge and Dublin Mathematical Journal, Vol. IX., and
also in CarmichaeFs Treatise on the Calculus of Operations.
It will be remembered that in this formula 7' = V(d:;^-l- 3/^ + 2''),
while in Art. 174 we denote V(-^^ + 2/'^) ^7 *'•
Approximate Values of Integrals.
176. Suppose y a function of x, and that we require
ydx. If the indefinite integral \ydx is known we can at
once ascertain the required definite integral. If the inde-
finite integral is unknown, we may still determine approxi-
mately the value of the definite integral. This process of
/,
15G AREAS OF PLANE CURVES AND OF SURFACES,
approximation is best illustrated by supposing y to be an
ordinate of a curve so that 1 ydx represents a certain area.
Divide c— a into n parts each equal to li and draw n — \
ordinates at equal distances between the initial and final
ordinates; then the ordinates may be denoted by y^, y^,
Vn^ Vn^v Hence we may take
^^(2/1 + 2/2 + + :!/n)
as an approximate value of the required area. Or we may
take
as an approximate value.
We may obtain another approximation thus ; suppose the
extremities of the r"" and r + lj"" ordinates joined; thus we
have a trapezoid, the area of which is (yr + ^/r+Jo* ^^^
sum of all such trapezoids gives as an approximate value of
the area
^{l+y.+ys +y.+'^}-
This result is in fact half the sum of the two former
results. It is obvious we may make the approximation as
close as we please by sufficiently increasing n.
The following is another method of approximation. Let
a parabola be drawn having its axis parallel to that of y ; let
Vv V"-' 2/3 represent three equidistant ordinates, h the distance
between y^ and 3/^, and therefore also between y^ and y^.
Then it may be proved that the area contained between the
parabola, the axis of x, and the two extreme ordinates is
3 (2/1 + %2 + 2/3)-
This will be easily shewn by a figure, as the area consists of
a trapezoid and a parabolic segment, and the area of the
latter is known by Art. 143.
AEEAS OF PLANE CURVES AND OF SURFACES. 157
Let us now suppose that n is even, so tbat the whole area
we have to estimate is divided into an even number of pieces.
Then assume that the area of the first two pieces is
3 {y, + 4y, + 2/3).
that the area of the third and fourth j)ieces is
. 3(^3 + 43/4 +yJ>
and so on. Thus we shall have finally as an approximate result
0(^1 + 2(2/3 + ^5+ 2/«-i)+Z/«+i + 4(y.. + 2/4 +yj]-
3
Hence we have the following rule : add together the first
ordinate, the last ordinate, twice the sum of all the other odd
ordinates, and four times the sum of all the even ordinates ;
then multiply the result by one-third the common distance
of the ordinates. This rule is called Simpson's Rule: see
Simpson's Mathematical Dissertations 1743, page 109.
Simpson however merely made the obvious extension of
supposing n to be any even number ; the case of « = 2 really
involves the whole principle, and this had been given before :
see Cotes De Methodo Differentiali, page 32.
As an example of Simpson's rule let it be required to find
n dx
the value of :j 5 . Suppose n = 10; then we have
-1 1 __]_ ^ '
1 + -01' '^^ 1 + -04'*" '^^ 1 + 1"
If the calculation be carried to six places of decimals it will
be found that the approximate value of the definite integTal
is equal to •785398.
In this case the exact value is known, namely - ; and
this agrees to six places of decimals with the approximate
value.
158 AREAS OF PLANE CURVES AND OF SURFACES.
177. Instead of referring to Art. 143 in the preceding
investigation we might have used the following method.
Assume for the equation to the curve y =A + Bx + Cx^,
where A, B, and G are constants; and let y^, y.^, y^ denote
the values of y corresponding to the values 0, h, 2h of x
respectively. Then
y^=A, y^ = A+Bh+Ch\ y^ = A + 2Bh+4.Ch';
and from these equations we can express A, Bh, and Cli' in
terms of y^, y^, and y^ The area contained between the
curve, the axis of x, and the two extreme ordinates
= ydx = 2Ah + 2Bh^ +
JO
substitute the values of u4, Bh, and Ch"^, and this expression
becomes
If the first of the three equidistant ordinates had been
drawn at any point x = a, instead of the point x=0, we
should have obtained the same result. For put x = a + x in
the equation to the curve ; the equation will become
y = P + Qx + Bx'y
where P, Q, and R are constants; and y^, y.^, y^ will now
denote the values of y corresponding to the values 0, h, 2h
of x\ so that the process and result will be as before.
If we take y = A-\-Bx-\- Cx" + Dx^ for the equation to
the curve, then as we have only three equations connecting
the four quantities A, Bh, Ch'', and Dli? with ;/, , y^, and y^ we
cannot determine these four quantities ; it is however worthy
of notice that the area will still be expressed by the formula
just given. For we have
o o
and this is equal to
/,
2h
{A + Bx+ Cx"" + Dx') dx.
AREAS OF PLANE CURVES AND OF SURFACES. 159
Let us now investigate an analogous expression f(^r tho
case in which four equidistant ordinates are known. Assume
foi: the equation to the curve y = A+Bx+ Cx" + Dx^, and let
Vv 2/2' Vv y* denote the values of y corresponding to the
Values 0, h, 2/i, 3A of x respectively. Then
y^ = A + Bh + Ch' + Dh\
y^ = A + 2Bh + 4C/i' + SDh\
y, = A + 2Bh + 9 Ch' + 27 Dh' ;
and from these equations we can obtain A, Bh, Ch^, and D?i^
ill terms of y^, y^, y^ and y^. The area contained between
the curve, the axis of x, and the two extreme ordinates
r3'» , ^^^ dBh' ^^,3 , 8Wh*
= ydx = SAh+—^ + 9Ch^+~-^ ;
substitute the values of J., Bh, (7/l^ and DA", and this expres-
sion becomes -r;-''(y,+ 8^^+ Si/s + ^/J- This result Avas given
by Newton ; see the end of his Methodus Differentialis.
Then proceeding as in the latter part of Art. 176 we ob-
tain the following approximate rule, the whole area being
supposed divided into a number of pieces which is some
multiple of three : add together the first ordinate, the last
ordinate, twice the sum of every third ordinate, excluding
the first and the last, and three times the sum of all the
other ordinates ; then multiply the result by three-eighths of
the common distance of the ordinates.
In the methods of finding approximate values of areas of
curves which we have explained, we have supposed the
successive ordinates to be drawn at equal distances. Another
method of approximation has been proposed by Gauss in
which the successive ordinates are drawn, not at equal dis-
tances, but at intervals which the method shews will ensure
the most advantageous results. For an account of this
method the student may consult the tenth Chapter of the
Elementary Treatise on Lajilace's Functions, Lamp's Functions
and Bessel's Functions.
IGO EXAMPLES.
EXAMPLES.
1. If A denote the area contained between the catenary,
the axis of x, the axis of y, and an ordinate at the
extremity of the arc s, shew that A = cs. The arc s
begins at the lowest point of the curve.
2. The whole area of the curve {-) +[f)"^'^ ^^ i'^^^-
(The integration may be effected by assuming
a; = a cos^ <^.)
3. The area of the curve y {x^ + a^) =c^{a- x) from x =
to a; = a is c'^ (1^ — 2 log 2 J .
4. Find the whole area between the curve y'^x = 4a' (2a — x)
and its asymptote. Result. 47ral
5. Find the whole area between the curve y^ (x^-\r a^) = aV
and its asymptotes. Result. 4a^
/jQ [ft _1_ n(y\
G. Find the area of the loop of the curve y"^ = .
Result. 2a' (l - ^) •
x^ (a, -\- x')
7. Find the area bounded by the curve y'^ = — ^ and
the asymptote x = a, excluding the loop
4
Result. 2aM 1 + "^
8. Find the whole area between the curve y' (2a — x)= x^
and its asymptote. Result. oTra".
9. Find the whole area of the curve (y - xf = a" — x".
Result. 7^a^
10. Fiad the area included between the curves
2/'-4aa; = 0, x'-4ay = 0. Result. -^^
11. Find the whole area of the cur^^e a*y^ + tV = a'b'xi
Result, ^ab.
EXAMPLES, IGl
12. Find the area of a loop of the curve a'y* = x*(a' — x^).
Result, —zr- .
5
.13. The area between the tractory, the axis of y, and the
asymptote is -t~ . (See Ai't. 100, and Ai't. 134.)
1-i. Find the area of a loop of the curve
i/~ (a* + cc") = a;- (a^ — x^). Result -^ (tt - 2).
15. Find the area of the loop of the curve
16aV = Z>V (a^ — 2ax). • Result.
^ ^ ^ SO
IG. Find the area of the loop of the curve
2f {a' + x') = {a' - xy-.
Result, a" [S V2 log (1 + V2) - 2}.
17. Find the whole area of the curve
2if {a? + x") - ^ay (a' - x") + (a' - xj = 0.
Result. aV •] 4 ^— Y .
18. Find the area of the curve
y = c sin - . lofj sin -
from ic — to a; = air. Result. 2ac (1 — log 2).
19. Find the area of the curve -= (-) between x = a and
c V«/
X = /3, and /ro??i tJi5 result deduce the area of the
hyperbola xy = a* betvyeen the same limits.
20. Find the area of the ellipse whose equation is
ax~ + 2hxy + cy" = 1. Result.
TT
»J{ac — y^) '
21. Find the area of a loop of the curve r^ = a^ cos 29.
Result. -^.
T. I. C. 11
1G2 EXAMPLES.
22. Find the area contained by all the loops of the curve
r = a sin nd.
2 2
7ra Tra
Result. -4- or —r according as n is odd or even.
4 2
23. Find the area between the curves r = a cos nO and r = a.
24. Find the area of a loop of the curve r^ cos 9 = oj' sin 3^.
O 2 2
Result -r- — IT log 2.
4 2
25. Find the whole area of the curve r=a (cos 2(9 + sin 2^).
^ ■ Result. Tra'.
26. Find the area of a loop of the curve (x^ + y'^f = 4aV7/^
Result. -5- .
o
27. Find the whole area of the curve
(af + yy = 4!a'x^ + Wf. Result. 277 (a" + b^.
28. Find the whole area of the curve
-^ + Tz = ^2\—2 + i^] ' Result. ^riV^+^J-
29. Find the area of the loop of the curve
2/' - oaary + x^ = 0. Result. — .
30. Find the area of the loop of the curve
r cos 6 =a cos 20. Result. [ 2 — ^ j a\
31. Supposing a greater than h find the area of the curve
V(a— 6cos"d^) \l(a—b) 2
32. In a logarithmic spiral find the area between the curve
and two radii vectores drawn from the pole.
EXAMPLES. 103
33. Find the area between the conchoid r = a + h cosec
and two radii vectores drawn from the polo.
3-i. In an ellipse find the area between the curve and two
radii vectores drawn from the centre.
35. In a parabola find the area between the curve and two
radii vectores drawn from the vertex.
36. Find the area between the curve r = a. (sec ^ + tan ^)
and its asymptote r cos = 2a. Result. ( ^ + 2 ) a\
37. Tlie whole area of the curve r = a (2 cos ^ + 1) is
/ 3 /3\
a- f 2-77 H — ^— j , and the area of the inner loop is
a'
( 3 \/o\
38. Find the whole area of the curve r = a cos ^ + J, where
a is greater than h. Also find the area of the inner
loop.
39. If X and y be the co-ordinates of any point of an equi-
lateral h}T)erbola x''-y-=a^, and u the area inter-
cepted between the curve, the central radius vector
drawn to the point {x, y), and the axis, shew that
40. Find the whole area of the curve which is the locus of
the intersection of two normals to an ellipse at right
angles. Besult. ir [a -%)~.
It may be shewn that the equation to the curve is
, ^ ( a''-&-)'(a''sin^^-6^cos^^ )''
^' (a' -F ¥) {a' sin^ O+h- cos''' Bf '
(See Plane Co-ordinate Geometry, Example 53, Chap-
ter XIV.)
11—2
164 EXAMPLES.
41. Find the area included within any arc traced by the
extremity of the radius vector of a spiral in a com-
plete revolution, and the straight line joining the ex-
tremities of the arc. If, for example, the equation to
^}> prove that the area corre-
sponding to any value of 6 greater than 27r is
ira'
2 f ' fi \ '^"■^'^ / ^ -"'''^
.)^ ) 19^-^
2w + l [V^tt/ XLtt
42. Find the area contained between a parabola, its evolute,
and two radii of curvature of the parabola. (Art. 157.)
43. Find the area contained between a cycloid, its evolute,
and two radii of curvature of the cycloid.
44. Find the area of the surface generated by the revolution
round the axis of x of the curve xy — k^.
X
45. Also of the curve y = ae''.
46. Find the area of the surface generated by the revolution
c - --
of the catenary y = ^ (e° + e ") round the axis oi y.
47. Shew that the whole surface of an oblate spheroid is
48. A cycloid revolves round the tangent at the vertex :
shew that the whole surface generated is -^ ira'.
49. A cycloid revolves round its base : shew that the whole
G4
surface generated is -7,- ira^.
50. A cycloid revolves round its axis : shew that the whole
surface generated is 87ra" (tt — f ).
EXAMPLES, 1G5
51. The whole surface generated by the revolution of the
tractory round the axis of x is iirc".
62. A sphere is pierced perpendicularly to the plane of one
of its gi-eat circles by two right cylinders, of which
the diameters are equal to the radius of the sphere
and the axes pass through the middle points of two
radii that compose a diameter of this great circle.
Find the surface of that portion of the sphere not
included within the cylinders.
Result. Twice the square of the diameter of the
sphere.
53. Find the surface generated by the portion of the curve
cc
7/ = a 4- a log - between the limits x = a and x = ae.
o - o ^
Result, ^-na- |l + V(l + O - V2 + log .^ _^Y(/+,-J "
54. Find I — , where d^ represents an element of surface,
and 'p the perpendicular from the origin upon the
tangent plane of the element, the integral being ex-
tended over the whole of the ellipsoid -7,-\-^,-\- —, = \.
^ a b c
Result s^(«'^' + ^'c'^ + cV).
166
CHAPTER VIII.
VOLUMES OF SOLIDS.
FormulcG involving Single Integration. Solid of
Revolution.
178. Let ^ be a fixed point on a curve APQ, and P any
other point on the curve whose co-ordinates are x and y ; and
suppose X algebraically greater than the abscissa of A. Let
the curve revolve round the axis of x, and let V denote the
volume of the solid bounded by the surface generated by the
curve and by two planes perpendicular to the axis of x, one
through A and the other through Pj then {Differential
Calculus, Art. 314)
dV
Tx=''y
therefore
V
= liry'dx.
From the equation to the curve 3/ is a knoAvn function
of X \ suppose i/r {x) to be the integral of iry"^ ; then
v=^{x)^-a
VOLUMES OF SOLIDS. 1G7
Let Fj denote the volume when the point P has x^ for its
abscissa, and V^ the vokime when the point P has x^ for its
abscissa; thus
therefore V^— T"i = "^/^ (-^'o) — "^ (^i) = tt I y^dx.
179. Application to the Bight Circular Cone.
Let a straight line pass through the origin and make an
angle a with the axis of x ; then this straight line will gene-
rate a right circular cone by revolving round the axis of x.
Here y = x tan a ; thus
V= jirtan^ax'^dx = — ~ — x^ + C,
■TT- TT- "^ tan Ct , g g.
Suppose iCj = 0, and let r = x^ tan a ; thus the volume
becomes — — ^ — - , that is, — ^ . Hence the volume of
3 3
a right circular cone is one-third the product of the area of
the base into the altitude.
180. Application to the Sphere.
Here taking the origin at the centre of the sphere we
have y^= c^ — x^ ; thus
\'iry^dx = 'JTi<yx — -^\-\-G,
The volume of a hemisphere = I 'Ky^dx = — ^— ,
181. Application to the Paraboloid.
Here the generating curve is the parabola, so that
y^ = 4aa;,
16"S VOLUMES OF SOLIDS.
Thus V.^- F^ = TT I 4!ax dx = 2a7r {x^ — x^) .
Suppose cCj = 0, then the volume becomes 'iairx^, that is
yny^x^, where y^ = 4ax^ ; thus the volume is half that of a
cylinder which has the same height, namely rr^, and the same
base, namely a circle of which y,^ is the radius.
182. For another example we will take the solid gene-
rated by a cycloid which revolves round its axis ; here
(Differential Calculus, Art. 358)
y = \/(2ax — x^) + aYers'^-.
a
The integration is best effected by putting for x and y their
values in terms of 6 {Differential Calculus, Art. 358). Thus
TT hfdx = Tra' [(6 + sin 6)' sin 6 d9.
To obtain the volume generated by a semi-cycloid the
limits for x would be and 2a ; thus the corresponding limits
for 6 are and tt.
Now Id'' smed9 = -e'cose + 2ld COS e de
= - ^* cos (9 + 2^ sin ^ + 2 cos 9,
therefore I ^" sin ^ c^^ = tt" — 4 ;
o f /J • ^a.m fan on. m ^' 9sm29 cos 2^
2 9sin^9d9 = l9 (1 — cos 2^) d9 = -^ ^ ,
therefore 2 9 sin' 9 d9 = '^.
IT
And I ''sm'9d9 = 2 f sm'9d9 = 2 . | . (Art. 85.)
Thus the required volume
, f , , TT* 4] 3 /Stt* 8
= TraUTT^ - 4< + — + ^y =7ra''
VOLUMES OF SOLIDS. 1G9
183. This formula for the vokime of a solid of revolution,
F= I Try'dx, like others which we have noticed, is one, the
truth of which is obvious, as soon as the notation of the
Integral Calculus is understood. In the figure to Art. 7, if
PM be y and MN be denoted by A:?;, then tti/Ax is the
volume of the soUd generated by the revolution of MNjiP
about the axis of x. Thus Xiry'^Ax wall differ from the volume
generated by the revolution of A DEB by the sum of such
volumes as are generated by PpQ ; and the latter sum will
vanish in the limit. Therefore the volume generated by the
revolution of ADEB is equal to the limit of Xtt^'Ax, that is,
to liry'dx.
184. Similarly, if V denote the volume bounded by the
surface formed by a curve which revolves round the axis of y,
and by planes perpendicular to the axis of y, we shall have
F= l^irx'^dy.
And, as in Art. 178, we shall have
V,-t,= \'\x^dy.
J Vi
185. Suppose two curves to revolve round the axis of x,
and thus to generate two surfaces, and that we require the
difference of two volumes, one bounded by the first surface
and by planes perpendicular to the axis of x, and the other
bounded by the second surface and by the planes already
assigned. Let y = ^ {x) be the equation to the first curve,
and 2/ = -v/r (.t) that to the second. Then if F denote the
required difference, we have
V = [tt [c/. {x)Y dx -L [>/r {x)Y dx
-'^\\:['i>{x)Y-{<lr{x)Y]dx.
170 VOLUMES OF SOLIDS.
If the planes which bound the required volume are de-
termined by a; = a;^ and x = w^, we must integrate between
the limits x^ and x^ for x.
For a simple case suppose that a closed curve is such that
the straight line y = a bisects every ordinate parallel to the
axis of y ; then we have (^) = a + % (x) and -xlr (x) = a — x {^) ,
where ;^ (x) denotes some function of x. Thus
{cf>{x)Y-{ir{x)Y = 4ax{x),
and F= tt j-iiax (x) dx.
Suppose the abscissge of the extreme points of the curve
are x^ and x^, then the volume generated by the revolution
of the closed curve round the axis of x is 4(Z7r 1 ^ (^) ^^•
And 2 1 % (a;) dx is the area of the closed curve, so that the
volume is equal to the product of 2a7r into the area. This
demonstration supposes that the generating curve lies en-
tirely on one side of the axis of x.
If the generating curve be the circle given by
{x-hr+{y-hy = c\
we have ttc^ for its area, and therefore 2/ircV^ for the volume
generated by the revolution of it round the axis of x.
186. In a similar way if the curves x= q> (y), x = -^ (y),
revolve round the axis of y we obtain for the volume bounded
by these surfaces and by planes jDerpendicular to the axis of y
V='jrj[[cf>iy)Y-{f{y)Y]dy.
187. The method given in Art. 178 for finding the volume
of a solid of revolution may be adapted to any solid. The
method may be described thus : conceive the solid cut up into
thin slices by a series of parallel planes, estimate approxi-
mately the volume of each slice and add these volumes ; the
limit of this sum when each slice becomes indefinitely thin is
the volume of the solid required. Suppose that a solid is cut
VOLUMES OF SOLIDS. 171
up into slices by planes perpendicular to the axis of cc; let
(j) (.^') be the area of a section of the solid made by a plane
which is at a distance x from the origin, and let £c + Ax be
the distance of the next plane from the origin; thus these
two planes intercept a slice of which the thickness is Ax, and
of which the volume may be represented by <^ (x) Ax. The
volume of the solid Avill therefore be the limit of ^0 {x) Ax,
that is, it will be (f> (x) dx; the limits of the integration will
depend upon the particular solid or portion of a solid under
consideration.
For example take a prism as defined in Euclid, Book XL
Cut up the prism into slices by planes which are parallel to
the two equal and similar ends; take the axis of x perpen-
dicular to the two ends. Thus ^{x) is a constant, say A ; the
volume of the i^v'ism = A dx ^ Ah, where h is the perpen-
dicular distance between the two equal and similar ends.
188. Application to an Ellipsoid.
The equation to the ellipsoicj is
if a section be made by a plane perpendicular to the axis of x
at a distance x from the origin, the boundary of the section
is an ellipse, of which the semiaxes are h ,/ (1 A and
c / f 1 jj ; hence the area of this ellipse is irhc ( 1 — — ^ ) ;
this is therefore the value of [x). Hence the volume of
the eUipsoid
r 7 ft ^\ 7 ^-rrdbc
= I iroc 1 7,\dx= — ^ — .
■/>(i-?)
189. Application to a Pyramid.
Let there be a pyramid, the base of which is any recti-
linear figure ; let A be the area of the base and h the height.
172 VOLUMES OF SOLIDS.
Take the origin of co-ordinates at the vertex of the pyramid,
and the axis of x perpendicular to the base of the pyramid,
then the volume of the pyramid
rh
= (f> {x) dx.
J
Now the section of the pyramid made by any plane pa-
rallel to the base is a rectilinear figure similar to the base,
and the areas of similar figures are as the squares of their
homologous sides; and x and h are proportional to homo-
logous sides ; hence we infer that
x^
4> {x) = p A.
Thus the volume of the pyramid
_A [^ 27 _^
fl J o
This investigation also holds for a cone, the base of which is
any closed curve.
190. For another example we will find the volume lying
between an hyperboloid of one sheet, its asymptotic cone,
and two planes perpendicular to their common axis.
Let the equation to the hyperboloid be
„2 , .2 _2
and that to the cone
2 2 2
a' b' c'
If a section of the former surface be made by a plane
perpendicular to the axis of x and at a distance x from
the origin, the boundary is an ellipse of which the area is
irhc ( — 2 + 1 ) ; the section of the second surface made by
the same plane also has an ellipse for its boundary, and its
VOLUMES OF SOLIDS. 173
•nhcx^
area is — s— . Therefore the difference of the areas is irhc.
a'
Hence the required volume, supposing it bounded by the
planes x = x^ and x = x„, is
I rrhcdx, that is, ':rlc{x^ — x^.
101. Sometimes it may be convenient to make sections
by parallel jDlanes not perpendicular to the axis of x. If a
be the inchnation of the axis of x to the parallel planes, then
4> {x) sin a^x may be taken as the volume of a slice and
the integration performed as before.
192. The remarks made in Arts. 176 and 177 have an
apjjlication to the subject of the j^resent Chapter.
Let there be a solid such that the area of a section made
by a plane jDarallel to a fixed plane and at a distance x from
it is always equal to P + Qx + Rx^ + Sx^, where P, Q, B, S
are constants. Let three equidistant sections of the solid be
made by planes parallel to the fixed plane, 2/i being the dis-
tance between the two extreme sections. Let the area of the
sections, taken in order, be denoted by A^, A^, A^. Then
the volume of the portion of the solid contained between the
two extreme sections is equal to
{A^ + 4^A^ + A^).
3
lifour equidistant sections be made, oh being the distance
between the extreme sections, and the area of the sections
taken in order be denoted hy A^,A^, A^,A^, then the volume
of the portion of the solid contained between the two extreme
sections is equal to
^(a^ + sa^+sa^+a;).
Hence we may obtain rules for estimating approximately
the volume of any solid. Make equidistant parallel sections
of the solid ; the areas of these sections must then take the
place of the ordinates which occur in tlie Eules given in
Arts. 176 and 177.
174
VOLUMES OF solids;
Formulce involving Double Integration.
193. We will first give a formula for the volume of a solid
of revolution. In the figure, let x, y be the co-ordinates of s,
and X + Ax, ;/ + A^/ those of t Suppose the whole figure to
revolve round the axis of x, then the element st will generate
a ring, the volume of which will be ultimately 2713/ A^ A?/:
this folloAvs from the consideration that Aa;A?/ is the area of
st and 27r_?/ the perimeter of the circle described by s. Hence
the volume generated by the figure BEeh, or by any portion
of it, will be the limit of the svmi of such terms as Stt?/ Aa; Ay.
Let F denote the required volume, then
V— 27r \\ydxdy \
the limits of the integration being so taken as to include all
the elements of the required volume.
194. Suppose that the volume required is that which is
obtained by the revolution of all the figure BEeh; let y=j){x)
be the equation to the upper curve, y=-y\r{x) that to the lower
curve, and let OG=x^, 011= x^. We should then integrate
first with respect to y between the limits y = -\^{x) and
y — (f)(x); we thus sum up all the elements like 27ryAxAy
which are contained in the solid formed by the revolution of
VOLUMES OF SOLIDS. I7u
the strip PQqp ; then we integrate with respect to x be-
tween the limits x^ and x^. Tims to express the operation
symbolically
rx., r^(x)
F= 27r y dx dij
J x^J ^ (x)
= 'rr\^\[j>{x)Y-[f{x)Y]dx.
The second expression is obtained by effecting the inte-
gration with respect to y between the assigned limits, and it
coincides with that already obtained in Art. 185.
195. Thus in the preceding Article we divide the solid
into elementary rings, of which liryb^xl^y is the type; in
the first integration we collect a number of these rings, so as
to form a figure which is the difference of two concentric
circular slices ; in the second integration we collect all these
figures and thus obtain the volume of the required solid.
The truth of the formulee of the preceding Article is obvious
as soon as the notation of the Integral Calculus is under-
stood.
196. Suppose the figure which revolves round the axis
of X to be bounded by the curves x = (f) (y) and x = ylr (?/), and
by the straight lines y =y^ and y = y^ ', then in applying the
formula for V it will be convenient to integrate first with
respect to x ; thus
rvi ri'iy)
V=27r ydydx.
■J Vi J ^ {y)
In this case in the integration with respect to x we collect
all the elements like 27ry^yAx which have the same radius
y, so that the sum of the elements is a thin cylindrical shell,
of which Ay is the thickness, y is the radius, and (p{y) — "^{y)
the height. Thus
o
F=27r [c^{y)-'^{y)]ydy.
197. As an example of the preceding formulte, let it be
required to find the volume of the solid generated by the re-
17G VOLUMES OF SOLIDS.
volution of the area ALB round the axis of x in the figure
already given in Art. 141. This volume is the excess of the
hemisphere generated by the revolution of SLB over the pa-
raboloid generated by the revolution of ABL ; the result is
therefore known, and we propose the example, not for the
sake of the result, but for illustration of the formulae of double
integration.
Let S be the origin. Suppose the positive direction of the
axis of X to the left, then the equation to AL is ?/*= 4a (a — a;)
and that to BL is if= 4(1^— x". Let V be the required volume,
then
/•2a rV(4a--2/^
V= I ^iry dij dx.
Jo J jgl-p-i
ia,
If we wish to integrate with respect to ;/ first, we must, as
in Art. 141, suppose the figure ALB divided into two parts;
thus
V=\ I 27rydxdi/+j lirydxdy.
•! J ^{ia'-iax) J a J
Again, let it be required to find the volume generated by
the revolution of LDG about the axis of x. Let the positive
direction of the axis of x be now to the right, then the equa-
tion to LC is 2/^ = 4a (a + x-) and that to LD is i/^ = 4<a^ — x".
Let Fbe the required volume, then
V= I / 27ry dx dy.
J J s/Ua^-x"-)
If we wish to integrate with respect to x first, we must, as
in Art. 141, suppose the figure LBG divided into two parts;
thus
r2a r2a ria\fz rza
F = 27ry dy dx+ Stt?/ dy dx.
Jo J sjiiu'-y-) J ia J y'^-4a*
ia
198. Similarly, if a solid is formed by the revolution of
a curve round the axis of y, we have
V — il 27rx dy dx.
VOLUMES OF SOLIDS.
199. We now proceed to consider any solid.
177
573;
Let X, y, z be the co-ordinates of any point ^j of a
surface, x + ^x, y + Ay, z -\- ^z the co-ordinates of an ad-
jacent point q. Through j) draw planes parallel to the co-
ordinate planes of {x, z) and (?/, z) ; through q also draw
planes parallel to the same co-ordinate planes. These four
planes will include between them a column, of which PQ is
the base and Pp the height. The volume of this column will
be ultimately zb,x^y, and the volume between an assigned
portion of the given surface and the plane of {x, y) will be
fomid by taking the limit of the sum of a series of terms
like ^-AcT A?/. Let V denote this volume, then
'= I jzdxdy.
The equation to the surface gives 2; as a function of x and
y ; the limits of the integration must be taken so as to in-
clude all the elements of the proposed solid.
T. L c. 12
178 VOLUMES OF SOLIDS.
If we integrate first with respect to y, we sum up the
columns which form a slice comprised between two planes
perpendicular to the axis of x ; thus the limits of the inte-
gration with respect to y may be functions of x, and we shall
obtain.
jzdy=f(x),
where f{x) is in fact the area of the section of the solid con-
sidered made by a plane perpendicular to the axis of x at
a distance x from the origin. Then finally
V=\f{x)dx-
this coincides with the formula already given in Art. 187.
200. Application to the Ellipsoid.
Let it be required to find the volume of the eighth part of
the ellipsoid determined by the equation
«> O t>
-^ + r2 + ^ = 1-
a c
Here we have to find
First integrate with respect to y, then the limits of y are
and LI, that is, and b ./ (l — -rA ; we thus obtain the sirna
of all the columns which form the slice between the planes
Zpl and 3Icpn. Now between the assigned limits
1-"
,2 .fi^ „J
d'
l)^^ = ?(-J)^
thus V=\j^lc(l 7,)dx.
The limits of x are and a ; we thus obtain the sum of
VOLUMES OF SOLIDS. 170
all the slices which are comprised in the solid OABG. Hence
■,,_7rt/6c
201. Suppose the given surface to be determined by
xy = az, and we require the volume bounded by the jjlaue
of (x, y), by the given surface, and by the four planes x = x^,
so = x^, y = y^, y = y„. Here the volume is given by
= 4 - (^2 - ^'i) {1/2 - 2/1) l-^iyi + ^,y, + ^i3/2 + ^,i/.x]
where z^, z^, z^, z^ are the ordinates of the four corner points
of the selected j)ortion.
202. Find the volume comprised between the plane
2; = and the surfaces xy = az and {x — hy + {y — kj = c\
Here we have to integrate li—dx dy between limits de-
termined by {x — Kf +(y— k)" = cl
Now lydy = ^, and the limits of y are
^' - Vic' - {x - h'f] and /.• + ^{c' -{x- JiY].
Thus we obtain
2k Vic' ~{x- hy].
Hence finally the required volume
= ~ja^^{c'-(x-hr}dx,
where the limits of x are h — c and h-]-c.
12—2
180 VOLUMES OF SOLIDS.
And
jx v[c' -(x- ny] dx =j{x - h) v[c' -(x- hy] dx
+ hj>^[c'-{x-hy]dx.
Put x — h = t; thus we obtain
jt V(c' - f) dt + a[v(c' - f) dt
The limits of t are — c and + c ; therefore the result is
AcV , ,, • J 1 • ^^'cV
— ^r— ; and the required volume is .
^ Co
This result however assumes that ^;/ is positive throughout
the limits of the integration ; that is, the circle determined by
(^x — hy+{y—ky = c^ is supposed to lie entirely in the first
quadrant or entirely in the third quadrant. If this condition
be not fulfilled our result does not give the arithmetical value
of the volume, but the balance arising from estimating some
part of the volume as positive and some part as negative; for
example, if h and k vanish our result vanishes.
Similarly in the result of the preceding Article, it is
assumed that xy is ^jositive throughout the limits of the in-
tegration.
203. Instead of dividing a solid into columns standing
on rectangular bases, so that zl^xi^y is the vokime of the
column, we may divide it into columns standing on the
polar element of area; hence 2;?'A^Ar is the volume of the
column. Therefore for the volume F of a solid we have the
formula
=//
zrdO dr.
From the equation to the surface z must be expressed as a
function of r and 6.
For example, required the volume comprised between the
plane z = 0, and the surfaces x^ -^if= iaz and y"^ = 2cx - x\
Here z = z-\ and the limits of r and 6 must be such as to
4a'
VOLUMES OF SOLIDS. 181
extend the inteQfration over the whole area of the circle
y' = 2cx — x^. Let )\ = 2c cos 6 ; then the required volume
\~dedv = ^\ cos*ede==— cos*dd0
204, Required the volume of the solid comprised between
the plane of {x, y) and the surface whose equation is
z = ae ~c^ .
Here, since x^ + if = r"^, we have V= a \\e ''' rdO dr.
The surface extends to an infinite distance from the origin
in every direction ; thus the limits of 6 are and Itt, and
those of r are and oo .
>
Now je '^^ rdr = ^ c^\
thus
Jo
rdr =
2'
And
r2rr
dO
Jo
= 27r.
Hence
the
required volume is
Trad'.
There is a point involved in this Example which deserves
notice ; it relates to the limits of the integral. It is plain
that in general corresponding to the limits ± c for ^ and y
it would not be sutficient to integrate between the limits
and 27r for 0, combined with the limits and c for r ; the
integration in the latter case instead of extending over a
certain square would extend only over the inscribed circle.
In like manner the limits + go for x and y do not certainly
con-espond to the limits and 27r for 6, combined with and
CO for r. But in the present Example it is easy to see that
182 VOLUMES OF SOLIDS.
no error arises; tlie part of the integral which depends as it were
on the difference between the square and the circle vanishes
in comparison with the rest of the integral. The subject has
been noticed by mathematicians: see the Melanges Math4-
onatiques et Astronomiques, St Petersbourg, 1859, Vol. 2, page
Go, and a paper by Professor Cayley in the Messenger of
Mathematics, 1874.
Formulce involving Triple Integration.
205". In the figure to Art. 199, suppose we draw a series
of planes perpendicular to the axis of z; \et z be the distance
of one plane from the origin and s+As the distance of the
next. These planes intercept from the column jjqPQ an
elementary rectangular parallelepiped, the volume of which is
A^A_?/A2. The whole solid may be considered as the Hmit
of the sum. of such, elements. Hence if V denote its volume,
V = \\\ dxdydz.
206. Eequired the volume of a portion of the cylinder
determined by the equation
x^ + y""- 2ax = 0,
wdiich is intercepted between the planes
z = x tan a and z = x tan /3.
Here if y, stand for ^{2ax-x-), we have
Y—jj dxdydz
Jo J -y J X tun a
(I fill
\ (tan/3 — tana) iTcZicay
^ -2/1
= 2 (tan /3 - tan a) x »J{2ax - x"") dx
Jo
= 2 (tan /3 — tan a) -^ .
VOLUMES OF SOLIDS. 183
207. The polar element of plane area is, as we have seen
in previous Articles, rA0/\r. Suppose this were to revolve
round the initial line through an angle 27r, then a solid ring
would be generated, of which the volume is 27rrsin ^rA^A?-,
since 27r?- sin 6 is the circumference of the circle described by
the point whose polar co-ordinates are r and 0. Let (f) denote
the angle which the plane of the element in any position
makes with the initial position of the plane, (f>+A(f) the angle
which the plane in a consecutive position makes with the
initial plane ; then the part of the solid ring which is inter-
cepted between the revolving plane in these two positions is
to the whole ring in the same proportion as A^ is to 27r.
Hence the volume of this intercepted part is
r^sm0A(f)MAr.
This is therefore an expression in polar co-ordinates for an
element of any solid. Hence the volume of the whole solid
may be found by taking the limit of the sum of such ele-
ments ; that is, if V denote the required volume,
V=jjlr'sm0d(f>d0ch
The limits of the integration must be so taken as to in-
elude in the integration all the elements of the proposed solid.
The student will remember that r denotes the distance of any
point from the origin, the angle which this distance makes
with some fixed straight line through the origin, and (f> the
angle which the plane passing through this distance and the
fixed straight line makes with some fixed plane passing
through the fixed straight line.
208. Suppose, for example, that we apply the formula ta
find the volume of the eighth part of a sphere. Integrate
with respect to ?' first ; we have
/
r^ dr = - .
Suppose a the radius of the sphere, then the limits of r are
and a ; thus
V = jj'^s,med<pd0.
IS-i VOLUMES OF SOLIDS.
In thus integrating with respect to r, we collect all the
elements like r'^sin ^A0 A^Ar which compose a pyramidal
solid, having its vertex at the centre of the sphere, and for its
base the curvilinear element of spherical surface, which is
denoted by a'^sin OAcpAd.
Integrate next with respect to ^ ; we have
sin 6 dd = — co&0;
1^
IT
the limits of 9 are and - ; thus
r=jp^
In thus integrating with respect to 0, we collect all the
pyramids similar to ^sin^A^A^ which form a wedge-
o
shaped slice of the solid contained between the two planes
through the fixed straight line corresponding to (j) and (f> + /\(p.
TT
Lastly, integrate with respect to (f) from to ;^ ; thus
In this example the integrations may be performed in any
order, and the student should examine and illustrate them.
209. A right cone has its vertex on the surface of a
sphere, and its axis coincident with the diameter of the
sphere passing through that point : find the volume com-
mon to the cone and the sphere.
Let a be the radius of the sphere ; a the semi-vertical
angle of the cone, V the required volume, then the polar
equation to the sphere with the vertex of the cone as origin
is r = 2a cos 6. Therefore
'2w fa r2acoa
r2w fa f2aco3 9
F= r'' s'm d d<p dd dr.
Jo JoJo
EXAMPLES. 18.5
210. The curve r = a(l + COS ^) revolves round the ini-
tial straight Hue, tind the volume of the solid generated.
Here the required volume
'tt fin ra{l+cos0)
r^ sin 6 cW dcf) dr
.' .'0
' [1 + COS ey sin dd9.
q ,3
It will be found that this = — ^ ,
o
EXAMPLES.
1. If the curve y^ (x — 4:a) = ax (x — Sa) revolve round the
axis of X, the volume generated from x—0 to ic = 3a
is ^'(15 -16 log 2).
2. A cycloid revolves round the tangent at the vertex :
shew that the volume generated by the curve is ttW.
3. A cycloid revolves round its base : shew that the
volume generated by the curve is SttW.
4. The curve y"^ (2a — x) = o? revolves round its asymp-
tote : shew that the volume generated is 27^'^a^
5. The curve xy^ = 4a^ (2a — x) revolves round its asymp-
tote : shew that the volume generated is 47^'''a^
G. Find the volume of the closed portion of the solid
generated by the revolution of the curve {y^ — Vf = (£x
round the axis of?/.
Result. „i ~ -^r-
31 o a
18G EXAMPLES.
7. Express the volume of a frustum of a sphere in terms of
its heicrht and the radii of its ends.
Result. '^[h' + ^{r,'-\-r^)]
8. If the curve y"^ = 1mx-\-nx^ revolve round the axis of x,
find the volume of any frustum ; and shew that it
may be expressed either by
-^ ( J^ + c^ - i^^F) or by irh {r^ + ~) ,
where h is the altitude of the frustum and h, c, r are
the radii of its two ends and middle section. Deduce
expressions for the volume of a cone and spheroid.
9. Find by integration the volume included between a
right cone whose vertical angle is CO", and a sphere
of given radius touching it along a circle.
Ftesidt —^ .
o
10. If a paraboloid have its vertex in the base, and axis in
the surface of a cylinder, the cylinder will be divided
into parts which are as 3 to 5 by the surface of the
paraboloid ; the altitude and diameter of the base of
the cylinder and the iatus rectum of the paraboloid
being all equal.
11. A paraboloid of revolution and a right cone have the
same base, axis, and vertex, and a sphere is described
upon this axis as diameter : shew that the volume in-
tercepted between the paraboloid and cone bears the
same ratio to the volume of the sphere that the Iatus
rectum of the parabola bears to the diameter of the
sphere.
12. Find the whole volume of the solid bounded by the
surface of which the equation is
o? y-" z'
!-•—-; —
jResult. — '— — .
EXAMPLES. 187
13. Find the whole volume of the solid bounded by the
surface of which the equation is
9
Besult. K «'•
2
1-i. Find the volume of the solid formed by the revolution
of the curve {x^ + y^ = aV+ ^7/ round the axis of x,
supposing a greater than h. Shew what the result
becomes when a=h.
S.SUU. I (2a' + m a + „^^'_y.^ log ""^^f"'''' •
15. Determine the volume of the solid generated by the re-
volution of the curve {x'^-\-y^f = a'x^ + by^ round the
axis of y, supposing a greater than h. Shew what
the result becomes when a = b.
Result 77 (2Z)-+ ocC) h + ^ ,, , — 7^ sm -^ -^ .
IG. Find the volume of the solid formed by the revolution
of the curve {y^ + x^f = a^ {x' — y") round the axis of x.
EcsuU. !^{±l„g(l + V2)-i}
17. A paraboloid of revolution has its axis coincident with
a diameter of a sphere, and its vertex outside the
sphere : find the volume of the portion of the sphere
outside the paraboloid.
Besult. -77- . where h is the distance of the two
o
planes in which the curves of intersection of the sur-
faces are situated.
18. Find the volume cut off from the surface
c
by a plane parallel to that of (y, z) at a distance a
from it. Ilesult. ttoj^ '^{hc).
188 EXAMPLES.
19. A quadrant of an ellipse revolves round a tangent at
the end of the minor axis of the ellipse : shew that
the volume included by the surface formed by the I
curve IS
•jra
(10 - Stt).
20. Find the volume enclosed by the surfaces defined by
the equations
x"^ + y^ = cz, x^ + y^ = ax, ^ = 0,
illustrating by figures the progress of the summation.
Result. - ,, .
o2c
21. If /S be a closed surface, dS an element of S about a
point P at a distance r from a fixed point 0, and
^ the angle which the normal at P drawn inwards
makes with the radius vector OP, shew that the
volume contained by the surface
= -^0]''^ *^os ^ dS,
the summation being extended over the whole sur-
face.
Taking the centre of an ellipsoid as the point 0,
apply this formula to find its volume, interpreting geo-
metrically the steps of the integration.
22. Find the value oi \\\x^ dx dy dz over the volume of an
ellipsoid. Result. — ^.,- .
23. Determine the limits of intefrration in order to obtain
the volume contained between the plane of {x, y) and
the surface whose equation is
Ax''-\-Bxy-^Cy''-Dz-F=0.
EXAMPLES. 189
24. State the limits of the integration to be used in apply-
ing the formula 1 1 i dx d>/ dz to find the volume of a
closed surface of the second order whose equation is
ax^ + bi/ + cz' + a'yz + h'xz + c'xy = 1.
25, State between what limits the integrations in
dx dy dz
must be performed, in order to obtain the volume
contained between the conical surface whose equa-
tion is z = a — i\J(x^ -{-y"^), and the planes whose equa-
tions are x = z and x = Q; and find the volume by
2a^
this or by any other method. Residt. -^ .
26. State between what limits the integrations must be
taken in order to find the volume of the solid con-
tained between the two surfaces cz = mx^ + ny^ and
z= ax+ hii : and shew that the volume is - — when
m = 11 = a = h = 1.
27. A cavity is just large enough to allow of the complete
revolution of a circular disc of radius c, whose centre
describes a circle of the same radius c, while the plane
of the disc is constantly parallel to a fixed plane, and
perpendicular to that of the circle in which its centre
moves. Shew that the volume of the cavity is
f (.Stt + 8).
28. Find the volume of the cono-cuneus determined by
2 2
X
which is contained between the planes x = nnd
X = a. Result. -^—^ .
190 EXxiMPLES.
29. The axis of a right cone coincides with a generating
line of a cylinder ; the diameter of both cone and
cylinder is equal to the common altitude : find the
surface and volume of each part into which the cone
is divided by the cylinder.
Mesidts.
„ „ 47r Vo - 3 V15 2 J 27r -v/5 + S^/lo ^
Surfaces, ^ a and — — a ;
87r + 27V3-64 3 , 64-27V3-27r 3
Volumes, ^ a and ^, ■ — a^ ;
where a is the radius of the base of the cone or
cylinder.
30. A conoid is generated by a straight line which passes
through the axis of z and is perpendicular to it. Two
sections are made by parallel planes, both planes
being parallel to the axis of z. Shew that the
volume of the conoid included between the planes is
equal to the product of the distance of the planes into
half the sum of the areas of the sections made by the
planes.
191
CHAPTER IX.
DIFFERENTIATION OF AN INTEGRAL WITH RESPECT TO ANT
QUANTITY WHICH IT MAY INVOLVE.
211. It is sometimes necessary to differentiate an inte-
gral with respect to some quantity wliicli it involves ; this
question we shall now consider.
Required the differential coefficient of I (f> {x) dx with
J a
respect to h, supposing (^ {x) not to contain h, and a to be
indej^endent of b.
Let 10= \ (f) (cr) dx ;
J a
suppose b changed into b + Ab, in consequence of which
w becomes u + Au ; thus
rb + Ab
u + A?( =1 ^ (•^) dx 5
J a
rb + Ab rb
therefore Au= j </> (x) dx — I (j) {x) dx
J a J a
b + Ab
b
(f>{x) dx.
Now, by Art. 40,
rb + Ab
' j <f>{x)dx = Abc}){b + eAb),
where 6 is some proper fraction ; thus
^^ = c^ (6 + dAb).
192 DIFFERENTIATION OF AN INTEGRAL
Let Ab and Au diminish without limit ; thus
212. Similarly, if we differentiate u with respect to a,
supposing (f) {x) not to contain a, and b to be inde23endent
of a, we obtain
213. Suppose (p (x) to contain a quantity c, and let it
be required to find the differential coefiicient of I cf) (x) dx
J a
with respect to c, supposing a and h independent of c.
Instead of ^ (x) it will be convenient to write ^ {x, c),
so that the presence of the quantity c may be more clearly
indicated ; denote the integral by u, thus
•b
u= i (f> (x, c) dx.
J a
Suppose c changed into c + Ac, in consequence of which
u becomes u + Au ; thus
ti+ Au= i 4) {x, c + Ac) dx ;
J a '
therefore Aw = (f){x,c + Ac) dx — j (f) {x, c) dx
J a •'a
= 1 {<f} (x, c + Ac).— (f) {x, c)] dx;
J a
,, Au [^ (f) (x, c + Ac) - (^ (x, c) J
thus -A = — ^ X "•^^
Ac Ja Ac
Now by the nature of a differential coefficient we have
<j){x, c + Ac) — (f) (x, c) _ cl(f) (x, c)
Ac dc
+ P>
WITH RESPECT TO AXY QUANTITY. 193
where p is a quantity which diminishes without limit when
Ac does so. Thus we have
Ac J a etc J a
When Ac is diminished indefinitely, the second integi'al
vanishes ; for it is not gx'oater than {b — a) p, where p' is
the greatest value p can have, and p ultimately vanishes.
Hence proceeding to the limit, Ave have
du f^dd) (x, c) ,
-J- = — ^-1 dx.
dc J a dc
214. It should be noticed that the preceding Article sup-
poses that neither a nor b is infinite ; if, for exam2:)le, b were
infinite, we could not assert that (6 — a) p' would necessarily
vanish in the limit.
215. We have shewn then in Art. 213 that
We will point out a useful application of this equation.
Suppose that -v/r [x, c) is the function of which <p [x, c) is
the differential coefficient with respect to x, and that ;^ [x, c)
d(h (x c)
is the function of which , is the differential coefficient
dc
with respect to x ; thus (1) may be written
'^-'^^ = X{''.C)-X('>,c) (2),
let us suppose that b does not occur in (f) (x, c), and that
a is also independent of b ; then (2) may be written
^?^+C = %(i.o) (S).
dc
where C denotes terms which are independent of b, that
is, are constant with respect to b. Hence as b may have
T. I. c. 13
lO-i DIFFERENTIATION OF AN INTEGRAL
any value we jolease in (3), we may replace h by x, and
write
%(-.c) = '^%^ + c w.
dc
This equation may be applied to find ;^ {x, c) ; as the
constant may be introduced if required, we may dispense
with writing it, and put (4) in the form
I -?-^-^—^ dx = -j-\(p{x, c) dx.
For example, let ^ {x, c) = , ., .^ 5 then
dx 1 , _i
= - tan ex,
I 6 (x, c) dx = \ ^ .-, 2 —
thus -1- ( - tan"^ ex] = I -^ ( -, r-j dx
ac\c J J dc \l + c'x J
-I
2cx''
dx.
(1+cV/
dx
Thus from knowing the value of = s— „ we are able to
^ J 1 + ex
deduce by differentiation the value of the more complex
intes^ral ( r-- — ^-^r-„ dx.
* j (1 + cV)"
216. Eequired the differential coefficient cf 1 (^{x, c) dx
J a
Avitli respect to c when both b and a are functions of c.
Denote the integral by u ; then -j- consists of three terms,
one arising from the fact that ^ (x, c) contains c, one from
the fact that h contains c, and one from the fact that a
contains c.
WITH KESPECT TO ANY QUANTITY. 195
Hence by the preceding Ai-ticles,
du _ /"* d^ (x, c) , du db du da
dc J a dc db do da dc
J a dc
C^d(b (x, c) J , ,, .dh . , .
217. With the suppositions of the preceding Article we
may proceed to find -^ . By differentiating with respect to c
the term ( -^-^-^ dx we obtain
J a dc
r^ c?'^ {x, c) 7 ,d(f) {h, c) db d(f) (a, c) da
J a dc^ dc dc dc dc
From the other terms in -r- we obtain by differentiation
^^ ' ^ dc' db \dcj dc dc
,, s d'^a d(}> (a, c) /daV d(f> (a, c) da
Thus ^!=r^(^'^^.
dc J a dc
^ ^ ^ dc db \dcj dc dc
__, f ^ d^a d<f) (a, c) fdaV ^ d(f)(a, c) da
-<?>l«,cj^^, d^[d^) ~^ dc Tc-
dhi
Similarly -j-^ may be found and higher differential co-
efficients of u if required.
13—2
19G
DIFFERENTIATION OF AN INTEGRAL
218. The following geometrical illustration may be given
of Art. 216.
M M'
uV N'
Let y = (f)(x, c) be the equation to the curve APQ, and
y = cf>(x, c + Ac) the equation to the curve A'P'Q'.
Let
0M= a,
MM' = Aa,
NN' = M.
Then u denotes the area FMNQ, and u + Au denotes the
aveaP'iM'N'Q'. Hence
and
Au = P'pq Q + qNN'q - PMM'p,
Au _ P'pqQ' QNN'q PMM'p
Ac Ac ' Ac Ac
It may easily be seen that the limit of the first term is
the limit of f ^ ^(^>g + ^c)-</>(^'»g) ^^ that the limit of the
L Ac
second term is the limit of ^ (h, c) -r- , and that the limit
of the third term ia the limit of ^ {a, c) -^ . This gives the
result of Art. 216.
WITH RESPECT TO ANY QUANTITY. 107
219. Example. Find a curve such that the area between
the curve, the axis of x, and any ordinate, sliall bear a con-
stant ratio to the rectangle contained by that ordinate and
the corresponding abscissa.
Suppose ^ {x) the ordinate of the curve to the abscissa x;
then I <^ (x) dx expresses the area between the curve, the
J
axis of X, and the ordinate (c) : hence by sujoposition we
must have
Jo n
where n is some constant. This is to hold for all values of c;
hence we may ditferentiate with respect to c ; thus
^ n n
therefore c0' (c) = {n— 1) cf) (c),
, 6' (c) n — 1
</)(c) c
By integration log (j> (c) = {n — 1) log c + constant ;
thus (}) (c) = ^c""S
where A is some constant ; thus we have finally
(f> {x) = Ax''~\
which determines the required curve.
220. Find the form of ^ {x), so that for all values of c
x[(^ {x)Y dx
I'
Jo
c
n
{(}i{x)Ydx
By the supposition
fx{cf>ix)Ydx = ir{cf^{x)Ydx.
JO itJo
198 DIFFERENTIATION OF AN INTEGRAL
Differentiate with respect to c ; thus
thus o{l-^{i>{c)r = lj\<f>{x)Yda,.
Differentiate again with respect to c ;
thus (l - i) [c/, (c)r + 2c (l - ^^) c/> (c) f (0) = ^-^^
hence (l " ^) ^ (c) + 2c (l - ^^) </>' (c) = ;
. ^ <^'(C) 2-71 1
therefore , , . = zr? iT ~ •
(p (c) 2 (n — 1) c
Integrate ; thus
2—7?.
log (/) (c) = 2-(,7:riy log c + constant ;
2-n
therefore <j) (c) = ^c^t^-D,
where J. is some constant ; thus we have finally
2-w
<f) {x) = Ax^^"-'^\
This is the solution of a problem in Analytical Statics,
which may be enunciated thus. The distance of the centre
of gravity of a segment of a solid of revolution from the
vertex is always - th part of the height of the segment ; find
the generating curve, The required equation is ?/ = ^ (x).
re (k ^/^^ qTj.
221. Find the form of 6 {x) so that the integral / —jj ^
may be independent of c, supposing that ^ {x) is independent
of c.
Denote the integral by «, and suppose x = oz; thus
'" (f) {x) dx _ p Vc(^ {cz) dz
o\/{c-x) Jo \/(l-^)
Jo
WITH RESPECT TO ANi' QUANTITY. 199
Since w is to be iiulcpcndent of c, tlic differential coeffi-
cient of u with respect to c must vanish. Now
do Jo ^(1--) Jo 2c^/{c — x)
This last intcfjral then must vanish whatever c mav be.
If (f) {x) were not independent of c, this would not necessarily
require that (p (x) + 2xj) [x] should always vanish ; for such
re /p^
an integral as 1 cos — - dx vanishes whatever c may be. But
Jo ^
(f) (x) + 2.r0' (x) must vanish since (x) is supposed inde-
pendent of c. For suppose that (j) (x) + 2x(f)' (x) is not always
zero ; then as x increases from the sign of <^ (x) + 2x<p' {x)
will remain unchanged through some interval, which does
not depend on c, say until x = a. Thus the integral
''(b{x) + 2x(b' (x) ,
la sjia — x)
cannot vanish, since every element is of the same sign.
Hence we see that j> (x) + 2x(f)' (x) must be zero.
Therefore £^ = -1-.
9 {x) zx
therefore log (^ (a:) = — ^ log x + constant,
therefore (h (x) =—- ,
\/x
where A is some constant.
This is the solution of a problem in Dynamics, which may
be enunciated thus. Find a curve, such that the time of
falling down an arc of the curve from any point to the lowest
point may be the same. If s denote the arc of the curve
measured from the lowest point, x the vertical abscissa of
the extremity of s, then we have
ds
-V- = (a;) and s=2A >Jx;
30 that the curve is a cycloid (Art. 72).
200 MISCELLANEOUS EXAMPLES.
MISCELLANEOUS EXAMPLES.
1. If the straight line SP^P^P^ meet three successive revo-
lutions of an equiangular spiral, whose equation is
r= a^, at the points P^, P^, P^, find the area included
between P^P^, P^Pz^ ^"^ ^^^^ *wo curve lines P^P^,P^P^.
Result. ~rj^ — (P,-PJ'.
2. Find the area of the curve y'^ — axy^ + x^ = 0.
7ra"\/2
Result.
16
3. Find the area of the curve a'" + 2/^" = a" {xyY ^, where n
is a positive integer.
2
Result. If n is an even integer -^r- ; if n is an odd
. , aV
integer .
4. A string the length of which is equal to the perimeter
of an oval is wound completel}'' round the oval, and
an involute is formed by unwinding the string, begin-
ning at any point : shew that when the length of the
involute is a maximum or a minimum the length of
the string is equal to the perimeter of the circle of
curvature at the jooint from which the unwinding
beojins.
5. Find the portion of the cylinder oc^ + y^ — rx= inter-
cepted between the planes
ax + by + cz = and ax + h2/ + cz — 0.
TT (a' — a) ?-'
Result.
8c
6. Find the volume of the solid bounded by the para-
boloid y'^+ z^= 4a(a;+a) and the sphere x^ + y^-{- z'^ = c\
supposing c greater than a.
Result. 2'jra ( c^ — -^j- 1 .
201
CHAPTER X.
ELLIPTIC INTEGRALS.
222. The integrals [ „, ^f . .^, , fv(l - c' siu'^) cW,
\ (IB
and /-, . ., ,,, — -r-, o . „ -, , are called elliptic fanc-
j(l + asm-^) \/(l -c'sm^^; ' ^ -^
tions or elliptic integrals of the first, second, and third order
respectively; the first is denoted by F{c, 6), the second by
E (c, 6), and the third by II (c, a, 6). The integi'als are all
supposed to be taken between the limits and 6, so that they
vanish when 6 vanishes. 9 is called the amjilitude of the
function. The constant c is supposed less than unity; it is
called the modulus of the function. The constant a, which
occurs in the function of the third order, is called the para-
meter. When the integrals are taken between the limits
and — , they are called coinplete functions; that is, the ampli-
tude of a comjDlete function is ^ .
223. The second elliptic integral expresses the length of
a portion of the arc of an ellipse measured from the end of
the minor axis, the excentricity of the ellij)se being the
modulus of the function. JFrom this circumstance, and from
the fact that the three integrals are connected by remark-
able properties, the name elliptic integrals has been de-
rived.
22-i. The theory of elliptic integrals and the investiga-
tions to which it has led constitute a part of the Integral
Calculus of great extent and importance, to which much
attention has been recently devoted. We shall merely give
a few of the simpler results. For further information the
student is referred to the elementary treatise on the subject
by Professor Cayley.
202 ELLIPTIC INTEGRALS,
225. If 6 and cf) are connected by the equation
F(c,e) + F{c,c}>) = Fic,f^),
where /m is a constant ; then will
cos 6 cos ^ — sin ^ sin ^ ^(1 — c^ sin^ fi) = cos /x.
Consider $ and <j) as functions of a new variable t, and
differentiate the given equation; thus
1 ^\ 1 # = ...(1)
V(l - c' sin' d) dt >^{l- & sin' </)) dt
Now as i is a new arbitrary variable, we are at liberty to
assume
de
^ = V(l-c^sin'^),
thus from the equation (1)
#=_V(l-c^sin'<^).
Square these two equations and differentiate; thus
-y^ = — c sm 6 cos u, -T-J- = — c sm (p cos <f> ;
therefore df "" ~ 2 ^^^"^ - ^^^ '^^'
Let ^ + ^ = -v//- and ^ — ^ = ^j^ ; thus
-^ = -c-sm'»/rcosx, -^ = -c sm^cosi/r.
therefore , , , = cot y, ■ , , , = cot ^Ir ;
d±dx d^dx
at dt dt dt
ELLIPTIC IXTEGRALS. 203
therefore
therefore log -^ = log sin x + constant, _
clylr , .
therefore W^ ^^^"^ ^
■(2),
and similarly -jj; = B sin ^p'
where A and B are constants.
Hence -^ sin % -^'^ = jK sin -v/r — ,
therefore A cos % = -B cos -v/r + C (3).
Now from the original given equation we see that if ^ =
F{c,6) = F{c,,.);
therefore then 6 = (x and % = ■^/^ = /^ ;
thus from (3) {A- B)cos[Jb= C;
thus A cos {e-(p)=B cos (^ + <^) + {A - B) cos ^l ;
therefore
(A - B) cos ^ cos (^ + (^ + -B) sin ^ sin <^ = (^ - B) cos /a.. .(4).
In (2) put for -^ its value ^{1 - c^ sin^ 0) - ^(1 - c" siir 0),
and for ^ its value V(l - c' sin' 6) + V(l - c' sin'' </>), and then
suppose ^ = ; thus
V(l — c" sin' /x) — 1 = ^ sin //,,
and V(l - c' sin' fi) + 1 = B sin fx.
Substitute for ^ - ^ and ^ + J5 in (4) ;
thus cos ^ cos ^ - sin ^ sin ^ \f(l - c' sin' jjl) = cos fi.
204 ELLIPTIC INTEGRALS.
226. The relation just found may be put in a different
form. Clear the equation of radicals ; thus
(cos 6 cos (j) — cos /a)" = (1 — c" sin" /x) sin" 6 sin'^ <f) ;
therefore
cos" + cos" (f> + cos^ ^ — 2 cos d cos (f) cos /i
= 1 — c^ sin^ fjb sin^ 6 sin^ </>.
Add cos* (f) COS" yu. to both sides and transpose ; thus
(cos 6 — cos (f) cos nY
= 1 - cos" ^ — cos" fj, + cos* ^ cos* /x — c* sin* yu. sin* ^ sin* ^
= sin* (f) sin* /z (1 — c* sin* 6) ;
therefore cos ^ = cos cf) cos /i + sin (j) sin /x. a/(1 — c* sin* ^).
The positive sign of the radical is taken, because when
^ = 0, we must have ^ = /i.
227. We shall now shew how an elliptic function of the
first order may be connected with another having a different
modulus.
Let F[c,6) denote the function; assume
c + cos 2(^
therefore
1 de 2 (1 + c cos 2(^)
cos' dd(f3 {c + cos'2cf)y '
therefore
dd _ 2 (1 + c cos 2(f))
dcf) 1 + 2ccos2</) + c* *
And
1 ,^.in^^-l c*sin*2</.
1 c.m^-1 i + 2ccos20 + c*
l + 2ccos2<^+c*cos*20
l + 2ccos2(p + c
ELLIPTIC INTEGRALS. 205
therefore
[ dd ^ /• 2 (1 + c cos 24>) V(l-|-2ccos2(^+c'-') ,
J J{l-c^sm^0) il + 2ccos20 + c«* l + ccos2<j!, "^
V(H-2ccos20 + c=) 1+c /[, 4c .,J-
No constant is added, because (p vanishes with 6. Thus
1 "T C
2 4c J . /) sin 20
c. = -n rs and tan U = ^
' [l + cY c+cos2(/)
The last relation may be written thus,
c sin ^ = sin (20 — 6).
"We may notice that c^ is greater than c, for
c,^_ 4
\2 >
c' c (1 + c)'
and since c is less than unity, 4 is gi-eater than c (1 + c)^
IT
If (f)= - , then 6 = 7r ; thus
jf^l'(.„|)=f(c,.) = 2i.(o,|).
228. We will give one more proposition in this subject,
by establishing a relation among Elliptic Functions of the
second order, analogous to that proved in Art. 225 for func-
tions of the first order.
If cos Q cos — sin ^ sin ^^/(l -~ c^ sin*^/*) = cos /x,
then will
E (C; 6) +E (<?, <f))~E {c, fj,) = c^ sin ^ sin sin yw,.
20G ELLIPTIC INTEGRALS.
By virtue of tlie given equation connecting the amplitudes,
^ is a function of 6 ; thus we may assume
Differentiate; thus
/'(^)=V(l-c^sin'-'^)+V(l-c^sin^</))^
_ cos d — cos (f) cos fi COS (f) — COS 6 cos fx d(f)
sin ^ sin /u, sin sin /a c?^
(by Art. 22G),
_ J{sin''^ + sin^(^ + 2cos^cos(^cos/A} 1
dd 2 sin ^ sin (p sin /^ '
But sin^ + sin^ ^ + 2 cos 6 cos ^ cos /j,
= 1 + cos^ yu, + c** sin^ 6 sin* <^ sin'' yct ;
thus f ' (6) = 0'. in i^i^pii.
Therefore, by integration
f (^) = c^ sin 6 sin ^ sin fi.
No constant is added, because f{6) obviously vanishes
with 0.
If /i = ^ the present result coincides with Fagnani's Theo-
rem, demonstrated in Art. 92 ; this will be easily seen by the
aid of some developments which we will now give.
In Art. 92 we have the relation
1 a cos , a cos ff
where x = -77:; ., . ., ^, , x =
V (1 - e'-' sin' 0)' V (1 - &' ^^^' &)
MISCELLANEOUS EXAMPLES. 207
hence we obtain
e^ cos' e cos" 6' - cos' 6) (1 - e' sin' &) - cos' e'{l - e' sin' 6)
+ (1 - e' sin' 6) (1 - e' sin' ^') = ;
that is e* sin' 6 sin' ^' + e' (1 - sin' ^ - sin' 6' - sin' d sin' ^')
+ sin'6' + sin'-^'-l = 0,
that is
e' (e' - 1) sin' 6 sin' ^' + (e' - 1) (1 - sin^ ^ - sin' &) = 0,
that is e' sin' ^ sin' 6'' + 1 - sin' 6 - sin' (9' = 0.
This relation may be put in the following forms :
(1 - e') sin'^ sin' d' = cos' cos' e\
cos' 6'
sin'^ =
sm £/ =
1-e'sin'^"
cos'^
1 - e' sin' ^ •
MISCELLANEOUS EXAMPLES.
1. Find the whole volume of the solid bounded by the
surface of which the equation is
s'
TTO?
Result, -jr ', supposing the radical restricted to the posi-
tive sign.
2. Eind the whole volume of the solid bounded by the sur-
face of which the equation is
ij'nf/n"cr=i-
Result -—rz —
So
208 MISCELLANEOUS EXAMPLES.
3. Prove that the volume of that portion of the solid
bounded by the surface whose equation is
which lies on the positive side of the plane of xy is
21 •
4. Find the value of 1-^, where dS denotes the element
of the surface of a sphere, and r the distance of this
element from a fixed point without the sphere ; the
integration being extended over the whole surfiice of
the sphei'e.
27ra f 1 1 ]
Result. — 77T -^7 rszo —7 ^^2( ; where a is the
c(w-2) ((c-tt)" ' (c + a; J
radius of the sphere, and c the distance of the fixed
point from the centre of the sj)here.
5. A cylinder is constructed on a single loop of the curve
r = a cos n 6 having its generating lines peipendicular
to the plane of this curve ; determine the area of the
portion of the surface of the sphere a? •\- 'if + z^ = d'^
which the cylinder intercepts ; determine also the
volume of the cylinder wdiich the sphere intercepts.
Results. Area = — f :^ — 1 1 ; volume =
n \:L J' 3?iV2 3
6. Find the volume of the solid generated by the revolu-
tion of the closed part of the curve x^ — Saxy + y' =
round the straight line x-\-i/ = 0.
Resnlt.
o ^6
7. If the axes of two equal circular cylinders of radius a
intersect at an angle j3, the volume common to both is
-——. — >,: and the surface of each intercepted by the
3 sm/3
other IS
sin/3'
MISCELLANEOUS EXAMPLES. 209
8. The centre of a variable circle moves along the arc of
a fixed circle ; its })lane is normal to the fixed circle,
and its radius equal to the distance of its centre from
a fixed diameter : find the volume generated ; and if
the solid so formed revolve round the fixed diameter,
shew that the volume swept through is to the volume
of the solid as 5 is to 2.
9. The centre of a regular hexagon moves along a diameter
of a given circle of radius a, the plane of the hexagon
being perpendicular to this diameter, and its magni-
tude varying in such a manner that one of its diago-
nals always coincides with a chord of the circle : shew
that the volume of the solid generated is 2V3a\
Shew also that the surface of the solid is
a^ (27r + 3 v/3).
10. Prove that
f ^ dx ^ / ^\
where c = ^.
11. Shew that the length of an arc of the lemniscate
T^ = a^ cos 26 measured from the vertex can be ex-
pressed as an elliptic integral of the first kind.
12. P and Q are any two points on a lemniscate of which A
is the vertex, and is the pole. Find the relation
between the vectorial angles of P and Q in order that
the arcs ^Pand QO may be equal.
nesult. Cos xi OP cos AOQ = ^,.
V2
T. L C. 14
210
CHAPTER XI.
CHANGE OF THE VARIABLES IN A MULTIPLE INTEGRAL.
229.
We have seen in Art. G2 that the double integral
rb rp ^ rp rb
I \ (}> {x, y) dx dtf is equal to I I <^ {x, y) dy dx when the
J a J a. J a J a
limits are constant, that is, a change in the order of integra-
tion produces no change in the hmits for the two integrations.
But when the limits of the first integration are functions of
the other variable, this statement no longer holds, as we have
seen in several examples in the seventh and eighth Chapters.
We' give here a few additional examples.
230.
Change the order of integration in
••a r-^/(a^-x!>)
<^{x, y)dxdy.
The limits of the integration with respect to y here are
,y = and y = ^f{a^-x''); that is, we may consider the
'integral extending from the axis of x to the boundary of a
CHANGE OF THE VARIABLES.
211
circle, having its centre at the origin, and radius equal to a.
Then the integration with resjDect to cc extends from the axis
of 1/ to the extreme point A of the quadrant. Thus if we
consider z = (f) (.r, y) as the equation to a surface, the above
double integral represents the volume of that solid which is
contained between the surface, the plane of {x, y), and a
straight line moving perpendicularly to this plane round the
boundary OAPBO.
It is then obvious from the figure that if the integration
with respect to x is performed first, the limits Avill be ic =
and x = \/(a^ — y"), and then the limits for y will he y=0
and y = a. Thus the transformed integral is
a r^J(a*-y-)
CO
</> {^y y) dy dx.
231. Change the order of integration in
2 rSacosfl
/7'
JO ^0
(/)(r, e)rd9dr.
Let OA = 2a, and describe a semicircle on OA as dia-
meter. Let POX= e, then 0P= 2a cos 6. Thus the double
integral may be considered as the limit of a summation of
values of ^(r, d)rM^r over all the area of the semicircle.
Hence when the order of integration is chancjed we must
integrate for 6 from to cos"' ~ , and for r from to 2a.
14--2
212
CHAXGE OF THE VARIABLES
Thus the transformed integral is
r2a rcos-'gr
(f>{r,e)rdrd9.
JO ^0
Jo
232. Change the order of integration in
"2a rZa-x
(j) (x, y) dx dy.
X
The integration for y is taken from 3/ = t- to ?/ = 3a — ar.
The equation y = j- belongs to a parabola OLD, and the
equation y=2>a~x to a straight line BLC, which passes
through L, the extremity of the latus rectum of the parabola.
Thus the integration may be considered as extending over
the area OLBSO. Now let the order of integi'ation be
changed ; we shall have to consider separately the spaces
OLS and BLS. For the sj^ace OLS we must integrate
from x = to x=2 \/{ay), and then from y = to y = a\
and for the space BLii we must integrate from a; = to
x = 3a — y, and then from y = a to y = 3a. Thus the trans-
formed integral is
ra r2^(,ay) rSa rZa-ij
cj) {x, y) dy dx + \ </> {x, y) dy dx,
J J J a J
IN A MULTIPLE INTEGRAL. 213
233. Change the order of integration in
/:/
1 /•x(2-ar)
(f> (x, y) dx dy.
Here the integration with respect to y is taken from y = x
to y = x {^— x). The equation y = x represents a straight
line, and the equation y = x(^ —x) represents a parabola.
The reader will find on exainin>ng a figure, that the trans-
formed integral is
•1 ry
/:/
(^ {x, y) dy dx.
QJ i-Vd-y)
234. Change the order of integration in
a rx+2a
<^{x,y)dxdy.
Here the integration with respect to y is taken from
y = /^{a^ — x") to y = x-i-2a. The equation y = sj{a? — x^)
represents a circle, and the equation 2/ = a? + 2a represents a
straight line. The reader will find on examining a figure,
that when the integration with respect to x is performed
first, the integral must be separated into three portions; the
transformed integral is
ra ro r2a ra
(f){x,y)dydx+ j>{x,y)dydx
J J -Jia^-y-) J a J
rZa ra
<l>{x,y)dydx.
J 2a J y-2a
235. Change the order of integration in
+
_b
'a rb+x
ra rb+x
(f>(x,y)dxdy.
J J
Here the integration with respect to y is taken from y =
to y = r • The equation y = j represents an hyper-
bola ; let BDE be this hyperbola, and let OA = a. Then
the integration may be considered as extending over the
2U
CHANGE OF THE VARIABLES
space OBDA. Let the order of the integration be changed;
we shall then have to consider separately the spaces OADG
and CDB. For the space OADG we must integrate from
h
a; = to x = a, and then from -?/ = to y = j—_ — . For the
' '^ ^ 6 + a
space CDB we must integrate from a; = to x — -.,
and then from ?/ = t to y = 1. Thus the transformed in-
tegral is
fjO-y)
rb+a fa f^ f ^
I ^{x,y)dydx+ 4> (x, y) dy dx.
JQ Jo J b Jo
b+a
236. Chancce the order of integration in
where h =
y
rh cc-ixx
<^{x,y)dxdy,
J •/ Ax
. The transformed integral is
c-y
rxh TA re r iJ.
4>{x,y)dydx+ <}> (x,y) dy dx.
J J J Kh J
IN A MULTIPLE INTEGRAL. 215
237. Change the order of integration in
I I j>{x,y,z)dxdydz.
J J J
The integration here may be considered to be extended
throughout a pyramid, the bounding planes of which are
given by the equations
z = 0, z = 1/, y = x, x = a.
The integral may be transformed in different ways, and
thus we obtain
or
or
or
or
fa ra ry
(^{x,y, z) dy dx dz,
Jo J y J
ra ry ra
<f> {x, y, z) dy dz dx,
J J J y
ra ra ra
<^{x,y,z)dzdydx,
Jo J s J y
fa rx fx
\ <^{x,y, z) dx dz dy,
J J J z
fa fa fx
I j (f) {x,y, z) dz dx dy.
J J S J 2
These transformations may be verified by putting for
(j) (x, y, z) some simple function, so that the integi'als can
be actually obtained ; for example, if we replace <p {x, y, z)
by unity, we find — as the value of any one of the six
forms.
238. These examples will sufficiently illustrate the sub-
ject ; it is impossible to lay down any simple rules for the
discovery of the limits of the transformed integral. It is not
absolutely necessary to draw figures as we have done, for the
figures convey no information which could not be obtained by
reflection on the different values Avhich the variables must
have, in order to make the integration extend over the range
indicated by the given limits. But the figiu-es materially
assist in arriving speedily and correctly at the result.
216 CHANGE OF THE VARIABLES
"We now proceed to the problem which is the main object
of the presenfChapter, namely, the change of the variables in
a multiple integral. We begin with the case of a double
integral.
239. The problem to be solved is the following. Required
to transform the double integral 1 1 Vdx dy, where F is a
function of x and y, into another double integral in which
the variables are u and v, the old and new variables being
connected by the equations
<^i {x, y, u, v) = 0, (f).^ (x,y,u,v) = (1).
We suppose that the original integral is to be taken be-
tween known limits of y and x; as we integrate with respect
to y first, the limits of y may be functions of cc. Of course
while ibtegrating with resjDect to y we regard w as constant.
We first transform the integral with respect to y into an
integral with respect to v. This is theoretically very simple;
from equations (1) eliminate u and obtain ^^ as a function of
X and V, say
y = ^(j,C,V) (2),
from which we get
dy = yjr' (x, v) dv,
where y\r {x, v) means the differential coefficient of ^jr (x, v)
with respect to v.
Substitute then for y and dy in IVdy, and we obtain
I ^i^' {^> ^) ^^> where V^ is what V becomes when we put
for y its value in V. Hence the original double integral
becomes
1 1 ^i^' C-^' ^) ^^ ^^•
Thus we have removed y and taken v instead. As the
limiting values of y between which we had originally to
IN A MULTIPLE INTEGRAL. 217
integrate are kno^v^l, we shall from (2) know the limiting
values of v, between which we ought to integrate. It will be
observed, that in finding -— from (2), we supposed x constant ;
this we do because, as already remarked, when we integrate
the proposed expression with respect to y wo must consider x
constant.
The next step is to change the ordei^ of the above integra-
tions with respect to x and v, that is, to perform the integra-
tion with respect to x first. This is a subject which we have
already examined ; all we have to do is to determine the new
limits properly. Thus, supposing this point settled, we have
changed the original expression into
I / ^li^' {^> ^) ^^ ^^•
It remains to remove x from this expression and replace it
by w. We proceed precisely as before. From equations (1)
eliminate y, and obtain a; as a function of v and u, say
a- = %(v, u) (3),
from which we get
dx=')l [v, ti) du,
where ^ (v, u) means the differential coefficient of ;!^ (y, m)
with respect to u.
Substitute then for x and dx, and the double integral be-
comes
\\ V'yjr' (x, v) X {v, w) dv du,
where F' is what F, becomes when we put for x its value in
Fj. Thus the double integral now contains only u and v,
since for the x which occurs in yjr' (x, v) we suppose its value
substituted, namely, ^ (y> '^)- Moreover since the limits
between which the integration with respect to x was to be
taken have been already settled, we know the limits between
which the integration with respect to u must be taken.
218 CHANGE OF THE VAEIABLES
We have thus given the complete theoretical solution of
the problem ; it only remains to add a j^^ctd^cci^ method for
determining ^^r' (x, v) and ^ (^'; ^0 • ^^ this we proceed.
We observe that -v|r' {x, v) or ■— is to be found from equa-
tions (1) by eliminating u, considering x constant ; the fol-
lowing is exactly equivalent : from (1) we have
dy dv du dv dv ' dij dv da dv dv
d^dy^d^^ d^dy^d^
_,.. . , du ,, dy dv dv dy dv dv
Llimmate -y- : thus , , = -^ n >
dv d^ tt9j
du du
therefore
d<^^ cZ(^2 d<^i d'^2
dy _ dv du du dv
dv d(f)^ d(f)^ c/(/)j d(f>,^ '
du dy dy du
This then is an equivalent for ^jr' (x, v), supposing that after
the differentiations are performed we put for y and u their
values in terms of x and v from (1).
Again, x {^y ^0 or -r- is to be found from equations (1) by
cttt
eliminating y, regarding v as constant ; the following is
exactly equivalent : from (1) we have
d^ dx d^ dy_ d^^ _ d(f>„ dx d^ dy d^ _ „
dx du dy du du ' dx du dy du du
. . • du
From these equations by eliminating -.- we find
d4id(f>^_d^d^
dx _ du dy dy du
du d(f)^ d(f>^ d(f)^ d(f),^ '
dy dx dx dy
IN A MULTIPLE INTEGRAL. 219
This then is an equivalent for ^ {y, xi).
, . - , , . dv dii. da dv
Thus y {x, V) X [v, u) = . , , , , , .. .
dy dx dx dy
Hence the conckision is that
//K.W,.//K ^_gJg^ ...,. (*),
dy dx dx dy
where after the differentiations have been performed, we must
substitute for x and y their values in terms of w and v to be
found from (1); also the values of x and y mvist be substituted
in V.
An important particular case is that in which x and y are
given explicitli/ as functions of u and v; the equations (1)
then take the form
oc-f,{u,v) = 0, y-fAu,v)=0 (5).
Here ^^ = 1 ^^ = ^^ = ^^=1
dx ' dy ' dx ' dy '
and the transformed integral becomes
J J \dudv dvduJ
where we must substitute for x and y their values from
(5) in F.
Thus we may write
\\^'^^HKtt-r/l>^'^ («)•
Again ; suppose that u and v are given explicitly as func-
tions of X and y ; the equations (1) then take the form
u-F,{x,y) = Q>, v-F^{x,y) = (7). .
220 CHANGE OF THE VARIABLES
Hence we obtain
Vdv du
j!vdxd>/ =
\dF^ dh\ dF^ dF^ '
jjvdxdy
dx dy dx dy
wliere we must substitute for x and y their values to be
obtained from (7).
Thus we may write
V dv du
\du dv du dv ^ ''
' dx dy dy dx
The formula in (4), (6), and (8) are those which are
usually given ; they contain a simjile solution of the proposed
problem in tliose cases where the limits of the new integra-
tions are obvious. But in some examples the difficulty of
determining the limits of the new integrations would be very
great, and to ensure a correct result it would be necessary
instead of using these formulse, to carry on the process pre-
cisely in the manner indicated in the theory, by removing
one of the old variables at a time.
240. The following is an example.
Kequired to transform Vdx dy, having given
Jo ^0
y + X = u, y = 2iv.
From the given equations we have x = ic (1 —v), y = uv;
- dx _ dx dy dy
thus '^ = l—v, -j- = — u, -T-=v, -r- = u;
da dv du dv
- , dx dy dx dy ._ ,
therefore -^ -f- —-j- ~ = u (I —v) + uv = u.
du dv dv au
Hence by equation (6) of Art. 239, we have
I 1 Vdx dy= jl Vu dv du ;
but we have not determined the limits of the integrations with
respect to u and v, so that the result is of little value. We
IN A MULTIPLE INTEGRAL. 221
will now solve this example by following the steps indicated
in the theory given above.
From the given equations connecting the old and new
variables we eliminate u ; thus we have
y = ~ — ; therefore - —
X
1-v' dv {l-vf
to the limits y = Q and y = h, correspond respectively v =
and V = 1 ; thus
b + x'
h
ra fb fa rb+x
Vdxdy=\ J\x{l -vpdxdv.
J J J J
We have now to change the order of integration in
nb+x
i\x{i -vrdxdv.
This question has been solved in Art. 235 ; hence we obtain
b
fa fb fa rb+x
b b{\-v)
fb+afa fl f V
= j^ j^V^x{l-vrdvd.x+j j V^x{l-v)-'dvdx.
b+a
We have now to change x for ii where
6 a b
fb + a fl-v f\ fv
thus we obtain 1 V'udvdu+ 1 V'udvdu,
Jo Jo J b Jo
a + b
since to the limits and a for x correspond respectively and
Y^^ for u, and to the limits and — ^^ ^ for x correspond
respectively and - for w.
222 CHANGE OF THE VARIABLES
1{ a = h the transformed integral becomes
1 a a_
V'udvdu+ V'udvdu.
D Jl Jo
a
If a is made infinite, these two terms combine into the
single expression
nV'u do du.
241. Second Example. Required to transform
nc-x
Vdxdy,
having given y -\-x = u, y = uv.
Perform the whole operation as before ; so that we put
vx , dy X
y = ~ and -~ =
1 — v dv {1 —v)^'
When y = we have v = 0, and when y = c — x we have
V = . Thus the integral is transformed into
c
r [ " V^x{l-v)-^d^dv.
Jo Jo
Now change the order of integration ; thus we obtain
/•I rc{i-v)
J\x{l-v)-'dvdx.
Jo J
dx
Now put x = u(l — v) and -7- =1 — v; the limits of
u
will be and c. Hence we have finally for the transformed
integral
n
J J (
1 re
V'u dv du.
QJO
IN A MULTIPLE INTEGRAL. 223
2-t2. Third Example. Transform \\V dxdy to a double
integral with the variables r and 6, siq^posing
X = r cos 6, y = r sin 6.
"We may put 6 for v and r for u in the general formulae ;
thus
dx dij dx dif J ^ , . 2 -
du av dv du
and the transformed intec^ral is
V'rdedr.
This is a transformation with which the student is pro-
bably already familiar ; the limits must of course be so taken
that every element which enters into the original integral
shall also occur in the transformed integral.
A particular case of this example may be noticed. Sup-
pose the integral to be
1 1 <}) (cix + hy) dx dy ;
by the present transformation this becomes
^ [hr cos {6 — a)] rdO dr,
■11^
where Jc cos a = a and k sin a = h. Now put — a = 6', so
that the integral becomes
I (f> {kr cos 6') rdd'dr ;
then suppose r cos d' = x and r sin 6' = y and the integral
may be again changed to
1 1 ^ (kx) dx dy.
224 CHANGE OF THE VARIABLES
Thus suppressing the accents we may write
1 1 (j) (ax +hy) dxdi/= jj(f> [hx) dx dy,
where k = '^{a^+h^). The limits will generally be different
in the two integrals ; those on the right-hand side must be
determined by special examination, corresponding to given
limits on the left-hand side.
243. Fourth Example. Transform I 1 Vdx dy, having
JO io
given
x=au-\-hv, y = hu + av, a being greater than h.
Eliminate u, thus ay —lx= [a? — ¥) v, and the first trans-
formation gives
X
«2 _ 52 re r^^
I V.dxdv,
a JoJ__^_ ^
hx a' — ¥
where V, is what V becomes when we put [ v for
^ ^ a a
y. Next change the order of integration ; this gives
c
V^dvdx + - '-\ \ V^dvdx.
^ Jo J {a + b)v ^ J _ be J jC'-h"-
a-^-b-i b '"
We have now to change from x to u by means of the
equation x = au -\-hv, which gives -^ = a ; the limits of ^^,
corresponding to the known limits of x are easily ascer-
tained.
Thus we have finally for the transformed integral
e c-bv c—bv
ra+b r~~a fO f a~
(tt^-60 V'dvdu-\-{d'-V)\ V'dvdu.
J J V J be J av
a- — b'^ b
The correctness of the transformation may be verified by
supposing V to be some simple function of x and y ; for
\
IN A MULTIPLE INTEGRAL. 225
example, if V be unity, the value of the original or of the
transformed integral is - .
244. Fifth Example. The area of a surface is given by
the integral
//<i.,iyy{l+ (*)'+(!)] (Art. 170);
required to transform it into an integral with respect to 6 and
(^, having given
z — r cos 6, x = r&in.d cos ^, y = r&\n6 sin jy.
From the known equation to the surface z is given in
terms of x and y ; hence by substituting we have an equation
which gives r in terms of 6 and ^.
We will first find the transformation for dx dy :
dx dr . . , n ,
^a = -jh siQ " cos 9 + r cos V cos ^,
dx dr . ' a ' ,
-T7 = ;t7 sm 6/ cos ^ — r sm 6' sm ^,
dy dr . ^ . , n • ,
-7^ = -^ sm t/ sm 9 + r cos v sm 9,
du dr . . . , . n
-f, = -T7 sm 6/ sm 9 + r sm 6/ cos 9.
T-r dx dy dx du • nf ^ dr . ^
Hence -771-77 — r; -771 = ^ sm ^ r cos ^ + -^ sm ^ :
da d(p dip dd \ do /
thus dx dy will be replaced by
r sin ^ f r cos6+ jy. sin j dj) dO.
We have next to transform
^/l-(IT^(|)]•
T. I. c. 15
22G CHANGE OF THE VARIABLES
__ , dz dz dx dz dij
^^^^^^ dd^^dxTd-^TydO'
dz dz dx dz dy
d(f) dx defy dy d^ '
dz dr ^ . ^
^'"^ de^dd''''^-'^''''^'
dz _ dr ^
d(p d^
iz . . .
Thus -7- is a fraction of whicli the numerator is
dx
dz dy dz dy
ddi<f~d^^r
that is, (-^ cos ^ - r sin ^ j (~^ sin ^ sin ^ + r sin 6 cos ^J
(I'V f dv • \
— — cos ^ ( -7^ sin ^ sin ^ + r cos 6 sm ^1 ,
that is,
. dr . n n , dr „ . o /, ,
""'''^dy
- / OiU. 1/ V^UO I.
1 \j\ja u/ ^ >. 1 Kjiii »
/ v>
\j Kj ^
r'
and the denominator is
dx dy
ded<p
dx dy
dcfidO'
the value of which
was found before ; thus
dz
r sin cos (
9cosc^^^
■ , dr 2-2
r sm (i -77 — r sm
^ d(f>
'^
cos
10
dx
f ■ dr\
r sin Q ( r cos 6 + sin 6 -,^
Similar
dz
■ly
r cos d) ^-r
dxp
+ r sin Q cos
, ^ sm (/> ^ - r sm
e
sin
</>
dy
A* on v» fi \ fv* n,
net fl _l_ cin fl
IN A MULTIPLE INTEGRAL. 2 1^7
therefore
/^Y fdz\^ \d4)) V
\dx) \du/ 1 ■ 'ia( /J . • a^^^'\
^ ^^ 7-^sm^ (rcos^ + sm^-^j
and finally the transformed integral is
• \ '•* '
245. There will be no difficulty now in the transformation
of a triple integral. Suppose that F is a function of x, y, z,
and that \\\V dxdy dz is to be transformed into a triple
integral with respect to three new variables u, v, w, which are
connected with .-r, ;/, z by three equations. From the investi-
gation of Art. 239, we may anticipate that the result will
take its simplest form when the old variables are given ex-
plicitly in terms of the new. Suppose then
^ =/i ("» ^'' ^^')y y =/2 ("^ ■^^ '^^)' ^ =fz i^'' '^'' '^^) (!)•
AYe first transform the integral with respect to z into an
integral with respect to ?«. During the integration for z we
regard x and y as constants; theoretically then we should
from (1) express ^ as a function of x, y, and w, by eliminating
XI and V ; we should then find the differential coefficient of z
with respect to w regarding x and y as constants. But we
may obtain the required result by differentiating equations (1)
as they stand;
ji df. du df. dv df,
ail aw dv aw aw
df^ fZw .df^dv^df^^
die dw dv dw dw '
df^ du dfg dv df^_ dz
du div ' dv dw dw diu'
15—2
228 CHANGE OF THE VARIABLES
Eliminate -y- and -^— ; thus we find
dw aw
dz N
dit dv du dv
where N = ^-^^ (if,iL Jf^if^ j.'^L (¥Af, _if^¥^^
dw \du dv du dv ' dw \du dv du dv
df,fdf,df,_df,df,
dw \du dv die dv
Hence the integral is transformed into
lll'^^wwr^m'"''''''
du dv du dv
where V^ indicates what V becomes when for z its value in
terms of x, y and lo is substituted. We must also determine
the limits of w from the known limits of 2. Next we may-
change the order of integration for y and w, and then pro-
ceed as before to remove y and introduce v. Then again we
should change the order of integration for tv and x and then
for V and x, and finally remove x and introduce w. And in ex-
amples it might be advisable to go through the process step by
step, in order to obtain the limits of the transformed integral.
We may however more simply ascertain the final formula
thus. Transform the integral with respect to 2 into an inte-
gral with respect to lo as alaove; then twice change the order
of integration, so that we have
du dv du dv
Now we have to transform the double integral with respect
to X and y into a double integral with respect to u and v by
means of the first two of equations (1). Hence we know
by Art. 239 that the symbol dxdy will be replaced by
('
du dv du dvj
IN A MULTIPLE INTEGRAL. 229
and the integral is finally transformed into
V'N divdvdi',
where V is what V becomes when for x, i/, and z, their values
in terms of n, v, and w are substituted.
The student will now have no difficulty in investigating
the more complex case, in which the old and new variables
are connected by equations of the form
<^i (^, y, ^, "■, V, w) -
(}).^{x, y,z, u, V, iv) = \ (2).
</>3 ('^. y> ^> "> V, w) =
Here it will be found that
dz_^N^ dl_N^ dx ^ N^
dw ~J),' dv~l)^' da ~ I)^ '
also that N^ = D^, and N^ = D,^.
Thus 1 1 1 Vdx dy dz = 1 1 1 V -j^ du dv dw, where
^ ^ dj>^ ^d^ d^ _ d^^ d^A ^ f% /d^ d^ _ dc^^ d^\
* dw \du dv du dv J dw \du dv du dv J
+ ^^3 /^ #2 _ % #^^
dw \d2i dv du dv J '
and — i)j is equal to a similar expression with x, y, z instead
of u, V, w respectively.
It may happen that equations (2) will impose some restric-
tion as to the way in which the transformations arc to be
effected. For example suppose we have
x + y-\-z — u = 0, x + y — uv=0, y — uviu = 0.
From these equations we cannot express z in terms of w and
X and y, and therefore we cannot begin by transforming from
z to IV. We may however begin by transforming from ;; to ti
or from ^ to v ; or we may begin by transforming from x or y
to M or V or IV.
280
CHANGE OF THE VARIABLES
246. It may be instructive to illustrate these transforma-
tions geometrically. We begin with the double integral.
-bB'
Let 1 1 Vdx d)/ be a double integral, which is to be taken
for all the vahies of x and ?/ comprised witliin the boundary
A BCD. Suppose the variables x and y connected with two
new variables u and v by the equations
3/=/.(«,^) (!)•
u and V be found in terms of
X
■■ft ("> ^).
From these equations let
sc and y, so that we may write
u = F^ (x, y),
v=F,{x,y) (2).
Now by ascribing any constant value to u the first equa-
tion of (2) may be considered as representing a curve, and by
giving in succession different constant values to w, we have a
series of such curves. Let then APQGho. a curve, at every
point of which F^ {x, y) has a certain constant value u; and
let A' SRC be a curve, at every point of which F^ (.r, y) has
a certain constant value u + hu. Similarly let BPSD be a
curve, at every point of which F.^ {x, y) has a certain constant
value V \ and let B'QRD' be a curve, at every point of which
IN A MULTIPLE INTEGRAL. 231
F„ (.r, y) has a certain constant value v + Sv. Let x, y nov/
denote tlic co-ordinates of P ; we shall proceed to express
the co-ordinates of Q, 8, and li.
The co-ordinates of Q are found from those of P, by chang-
ing V into v + hv; hence by (1) they arc ultimately, when Zv
is indefinitely small, x -t- -,- Si' and y + -r ^v.
•' dv ^ dv
Similarly the co-ordinates of 8 are found from those of P
by changing u into u + hw, hence by (1) when hic is indefinitely
dx dii
small they are ultimately a; -f -y- hit, and y + -f- ^u-
The co-ordinates of P are found from those of P by
changing both u into u + Bu and v into v + Bv; hence by (1)
they are ultimately x + -j- Bu -{■ -j- Bv, and y + -f^ Bti +
ctu civ au
av
These results shew that P, Q, P, 8 are ultimately situated
at the angular points of a parallelogram. The area of this
parallelogram may be taken without error in the limit for the
area of the curviliuear figure PQB8. The expression for the
area of the triangle PQB in terms of the co-ordinates of its
angular points is known (see Plane Co-ordinate Geometry,
Art. 11), and the area of the parallelogram is double that of
the triangle. Hence we have ultimately for the area of
PQRS the expression
fdx dy dx dy\ ^ ^
~ \du dv dv duj
Thus it is obvious that the integral 1 1 Vdx dy may be
replaced by ± JjF'g | - g |) du dv ;
the ambiguity of sign would disappear in an example in
which the limits of integration were known. In finding the
value of the transformed integral, we may suppose that we
first integrate with respect to v, so that u is kept constant ;
this amounts to taking all the elements such as PQRS, which
232
CHANGE OF THE VARIABLES
form a strip such as A A' CO. Then the integration with
respect to u amounts to taking all such strips as AA'C'C
which are contained within the assigned boundary ABCD.
247. We proceed to illustrate geometrically the trans-
formation of a triple integi-al.
::>£
,->^
Let 1 1 1 Vdxdy dz be a triple integral, which is to be taken
for all values of x, y, and z comprised between certain as-
signed limits. Suppose the variables x, y, and z connected
with three new variables u, v, w by the equations
^ =/i (w, V, to), 2/ =/, {u, v,w), z =/3 {ii, v,'vc) (1).
From these equations let xi, v, and iv be found in terms of
X, y, and z, so that we may write
u = F^ (x, y, z), '0 = F^ [x, y, z), w = F^ (x, y, z) (2).
Now by ascribing any constant value to u, the first equa-
tion of (2) may be considered as representing a surface, and
by giving in succession different constant values to u we
IN A MULTIPLE INTEGRAL, 233
have a series of such surfaces. Suppose there to be a surface
at every point of which F^ {x, ?/, z) has the constant vahie u,
and let the four points P, B, D, be in that surtice ; also
suppose there to be a surface at every point of which
•^i.(^. y> ^) has the constant value u + Zu, and let the four
points A, F, G, E be in that surface. Similarly suppose
P, A, E, G to be in a surface at every point of Avhich
-^2 {^' y. ^) li3,s the constant value v, and B, 1), G, F to be in
a surface at every point of which F^ {ic, y, z) has the constant
value v + Sy. Lastly suppose P, A, F, B to be in a surface
at every point of which F^ {x, y, z) has the constant value w,
and C, D, G, E to be in a surface at every point of which
F^ {x, y, z) has the constant value w + hw.
Let X, y, z now denote the co-ordinates of P; we shall
proceed to express the co-ordinates of the other points. The
co-ordinates of A are found from those of P by chano-ino- u
into iL + hu; hence by (1) they are ultimately when SiT is
indefinitely small,
, dx ^ di/ dz ^
du •" da du
The co-ordinates of B are found from those of P by chang-
ing V into V -\-hv\ hence by (1) they are ultimately
dv ^ do dv
Similarly the co-ordinates of are ultimately
dx ^ dy ^ dz ^
diu ^ dio dw
The co-ordinates of D are found from those of P by chang-
ing V into v + 8v, and w into w + Stu; hence by (1) they are
ultimately
Similarly the co-ordinates of ^, P and G may be found.
These results shew that P, A, B, G, D, E, F, G are ulti-
mately situated at the angular points of a parallelepiped ; and
the volume of this parallelepiped may bo taken without error
234 CHANGE OF THE VARIABLES
ia the limit for the volume of the solid bounded by the six
surfaces which we have referred to. Now b)'^ a known theo-
rem the volume of a tetrahedron can be expressed in terms
of the co-ordinates of its angular points, and the volume of
the parallelei^iped PQ is six times that of the tetrahedron
ABPG. Hence finally we have for the volume of the paral-
lelepiped
+
{dx fchi dz dy dz\ dy fdz dx_ _dz^ dx\
\diL \dv dw dw dvj du \dv dw diu dvj
dz dx dii dx dy\\ ^ ^ ^ , at 5. ^ 5.
du\dv dw dwdvj)
Hence the triple integral is transformed into
±{{[v'Ndudvdw\
the ambiguity in sign would disappear in an example where
the limits of intecfration were kno\vn.
248. We have now given the theory of the transforma-
tion of double and triple integrals ; the essential point in our
investigation is, that we have shewn how to remove the old
variables and rejalace them by the new variables one at a
time. We recommend the student to pay attention to this
2)oint, as we conceive that the theory of the subject is thus
made clear and simple, and at the same time the limits of the
transformed integral can be more easily ascertained. We do
not lay any stress on the geometrical illustrations in the two
preceding Articles ; they require much more development
before they can be accepted as rigid demonstrations.
249. Before leaving the subject we will briefly indi-
cate the method formerly used in solving the problem. This
method w^e have not brought prominently forward, partly
because it gives no assistance in determining the new limits,
and partly on account of its obscurity ; the latter defect has
been frequently noticed by writers on the subject.
Suppose 1 1 Vdxdy is to be transformed into an integral
with respect to two new variables u and v of which the old
variables are known functions.
IX A MULTIPLE INTEGRAL. 235
Let the variables undergo infinitesimal changes : tlius
dx = -, du + -j- dv (1),
du dv
du = -/du + -rdv (2).
'^ du do ^ ^
Now in the original expression Vdx dy in forming dx we
suppose y constant, that is, dy = ; hence (2) becomes
= ^du + '$dv (3),
du dv
find dv from this and substitute it in (1) ; therefore
dx dy dx dy
, du dv dv du ^
Tbj ^''••- ('^)'
dv
Again, in forming dy in Vdxdy we suppose x constant,
that is, dx = 0; hence by (4) we must suppose du = 0; there-
fore from (2)
^y=£^^ (^)-
From (4) and (5)
, , [dx dy dx dy\ , ,
dx dy = — - -/ — - / du dv :
■^ \du du dv du) '
and 1 1 Ydx dy becomes
[[r[^^^-^^'f\dudv.
J J \au dv dv an)
With respect to the limits of integration we can only-
give the general direction, that the new limits must be so
taken as to include every element which was included by the
old limits.
23G CHANGE OF THE VARIABLES.
250. Similarly in transforming a triple integral
Vdx dy dz
///'
the process was as follows. Let tlie new variables be ?i, v, w ;
in forming dz we must suppose x and y constant ; thus we
have
7 dz , dz , dz ^
dz = -y- du + ^r dv + -j— dw,
du dv dw
^ dx , dx , dx J
= -7- ait + -7- rfy + -,— aw,
du dv dw
= -/ du + -^ dv+ -~ dw,
du dv dw
therefore dz= , , , , (l),
dxcly dxdy ^ ^
du dv dv du
where N has the same value as in Art. 247.
Next in forming dy we have to regard x and z as constant;
hence by (1) we must regard w as constant ; thus we have
= f,
du
du + '^ dv;
dv '
fdy dx
\dv du
dy dx\
du dv)
dx
du
therefore dy — 1 (2)
And lastly in forming dx we suppose y and z constant,
that is, by (1) and (2) we suppose w and v constant; therefore
7 dx , , ,
dx = -T- du (3).
From (1), (2), and (3)
dx dy dz = Ndu dv dw.
EXAMPLES. 237
251. The student who wishes to investigate the history
of the subject of the present Cliapter may be assisted by the
following references. Lacroix, Calcid Dif. et Integral, Vol. ii.
p. 208 ; also the references to the older authorities will be
found in page XI. of the table prefixetl to this volume. De
Morgan, J) If. and Integral Calculus ^ p. 392. Moigno, Calcul
lyif. et Integral, Vol. Ii. p. 214; Ostrogradsky, Memoires de
V Academie de St Petershourg, Sixieme S(^rie, 1838, p. 401.
Catalan, Memoii'es Couronnes par I' Academie... de Bruxelles,
Vol. XIV. p. 1. A memoir by Haedenkamp in Crelle's Journal,
Vol. XXII. 1841. Boole, Cambridge Mathematical Journal,
Vol. IV. p. 20. Cauchy, Exercices d' Analyse et de Physique
Mathematique, Vol. IV. p. 128. Svauberg, Nova Acta Regice
Societatis Scientiarum, Upsaliensis, Vol. xiii. 1847, p. 1. De
Morgan, Transactions of the Camhridg.e Phil. Society, Vol. ix.
p. [133]. Winckler, Denhschriften der Kaiserlichen Akad.
Math....Classe, Vol. xx. Vienna 1862, p. 97. A memoir by
Holmgren was communicated to the Stockholm Academy
in 1864, and published in Vol. V. of the Transactions.
EXAMPLES.
1. Shew that if ic = asin^sin^ and y~h cos 6 sin (}), the
double integral Ijdxdy is transformed into
+ 1 1 a5 sin ^ cos ^ d^ dO.
If X = u sin a. -\- V cos a and y — u cos a — v sin a, prove
that
//■^(^' s-) jiM^^:f) =///■ ("' ")
dudv
^^{l-io'-v')'
3. In the problem of Art. 239, supposing the limits of x
and y are both constants, shew how the limits of
u and V are to be found, in each of the three parts of
which the transformed integral will in general be
composed,
238 EXAMPLES.
4. Prove that
(•OO -00 »x>
I ^ (aV + 6^7/') c?a; f/y = -"^y \ 6 (x) dx.
J J 'iUDJ
5, Transform ilVdx di/, where y = xu and x= — ~ .
If the limits of y be and x and the h'mits of x be
and a, find the limits in the transformed integral.
ri ra(\+u)
Besult V'v[l+u)-\Jadv.
J oJ
6. Transform lje~^^'''^-'^^'^''^'''^^J'^dxdi/ from rectangular to
polar co-ordinates, and thence shew that if the limits
both of x and y be zero and infinity, the value of the
integral will be
*= 2 sin a ■
7. Transform 1 i (f) {x, y) dx dy to polar co-ordinates, and
J J ^
indicate the limits for each order in the transformed
integi'al.
Shew that
p p dx dy 1 ^^^_, ab
Jo Jo (c' + x' + y'f c c V(a' 4- i"" + c') '
8. Apply the transformation from rectangular to polar co-
ordinates in double inteofrals to shew that
+ 00 r +00
iZI
a dx dy 27r
-«> ^a;' + y- + a'f {x' -\- y" + a'^ « + «
1). Transform the double integral \jf(x, y) dxdy into one
EXiVJtfPLES. 239
in -svliich r and 6 shall be the independent variables,
having given
x = r cos ^ + a sin ^, y = r sin 6 + a cos 6.
Result.
Uf{r cosd + a sin 6, r sin ^ + a cos 6) (a sin 29 - r) dO dr.
10. Transform U e' ''''-'■'' dxdy into a double integral where
r and t are the independent variables, where - = t and
r^ = x"^ + if ; and if the limits of x and y be each
and CO , find the limits of r and t
Result. I -, — ^2 — .
Jo h !+«
11. If X and y are given as functions of r and 9, transform
the integral \\\dxdydz into another where r, 9 and
z are the variables ; and if a? = r cos 9 and y = r sin 9,
find the volume included by the four surfaces whose
equations are r = a, s = 0, ^ = 0, and z = mr cos 9.
Result. The volume = | \ r^m cos 9d9dr = -^ .
J
12. If ax = yz, fiy = zx, ^z = xy, shew that
jjjf{ru,^,ry)doid^dy = 4>jjjf(^^, J, ^)dxdydz.
13. Transform jjjjvdx^dx.^dx^dx^ to r, ^, (^ and -v/r where
iTj = r sin 9 cos ^, ^3 = '^ cos 9 cos A/r,
cc^ = r sin ^ sin ^, x^ = r cos ^ sin ilr.
i?eswZ^ jjjjy'r' sin ^ cos ^ cfr cZ^ # cZ^/^.
240 EXAMPLES.
14. Find the elementary area included between the curves
(f> {x, y)=u, yjr (x, y) = v, and the curves obtained by
giving to the parameters u and v indefinitely small
increments.
Find the area included between a parabola and the
tangents at the extremities of the iatus rectum by
dividing the area by a series of parabolas which touch
these tangents and by a series of straight lines drawn
from the intersection of the tangents.
15. Transform the triple integral I i j f{.v, y, z) dx dy dz into
one in which r, y, z are the independent variables,
having given -^ {x, y, z, r) = ; and change the vari-
ables in the above integral from x, y, z to r, 6, (f),
having given
f (^, y, ^, r) = 0, -f 1 (?/, ^, r, 6) = 0, f^ (z, r, 0, <^) = 0.
df ^, djr^
dx dy dz
16. Transform the double integral
in which x, y, z are connected by the equation
a;^ + ?/^ + 2" = 1, to an integral in terms of Q and <^,
having these relations,
X = sin ^ /^(l — m" sin^ 0), y = cos 6 cos (f),
z = sind a/CI — n' sin'^ (f)), m^ + n^ = 1.
Hence prove that
ff JT
'^'■2 on^ cos~ 6 + n^ cos^ (f) -,„.. ir
11'
•' •
V(l - m' &m' 6) V(l - n' sin'^ </>) ^^^'^ 2
EXAMPLES. 241
17. Transform the integral Indxdijdz to r, 6, (J), where
a; = r sin ^ ^/(l — n"^ cos" 6), y = r cos (p sin 0,
z = r cos 6 \/(cos^ ^ + n^ sin' <^).
p , [/•/ >•' {(?^' - 1) cos'-' (/) - n'^ sin' 6] dr d6 d<^
18. Transform the expression 1 1 - sin ^ J^ d(f) for a volume,
to rectansfular co-ordinates.
Besult. ^ jj(z—px — qy)dxdi/; this should be in-
terpreted geometrically.
19. lfx+7/ + z = u, x-i-7/ = uv, 1/ = uvw, -pvove ihat
Vdxdydz=\ Vu^vdudvdio.
J J oJ
CO ;. 00 /• 00
J ^
20. If x^ = rcose^,
x^ = r sin 6^ cos ^j,
iTg = r sin 6^ sin ^^ cos ^j^,
^„-i = ?' sin ^j^ sin $^. . .sin ^^^ cos 0„_^,
x^ = r sin ^^ sin 0^. . .sin ^^.^ sin 6'„_j',
shew that \\\ Vdx^ dx^. . . dx^
= ±jlj y V'-'Edr d9^ dd^ dd^_„
where V is any function oi x^, x^,...x^, and V what
this function becomes when the variables are changed,
and // stands for
(sin ^J"-» (sin ^,)"-' sin^„_,.
T. I. c. 16
242
CHAPTER XII.
DEFINITE INTEGRALS,
252. When the indefinite integral of a function is known,
we can immediately obtain the value of the definite integral
corresponding to any assigned limits of the variable. Some-
times however we are able by special methods to assign the
value of a definite integral when we cannot express the
indefinite integral in a finite form; sometimes without actually
findino- the value of a definite integral we can shew that it
possesses important properties. In some cases in which the
indefinite integral of a function can be found, the definite
integral between certain limits may have a value which is
worthy of notice, on account of the simple form in which it
may be expressed. "We shall in the present Chapter give
examples of these general statements.
We may observe that a collection of the known results
with respect to Definite Integrals has been published in a
quarto volume at Amsterdam, by D. Bierens de Haan, under
the title of Tables d'Integrales Definies.
253. Suppose f{x) and F{x) rational algebraical functions
of cc, and f (x) of 'lower dimensions than F (x), and suppose
the equation F{x) = to have no real roots ; it is required to
find the value of
It will be seen that under the above suppositions, the
expression to be integrated never becomes infinite for real
values of x.
Let a + /3 v'(- 1) and oc - ^ V(- I) represent a pair of the
imaginary roots o^F{x) = ; then the corresponding quadratic
DEFINITE INTEGRALS. 243
fix)
fraction of the series into which "v, can be decomposed,
may be represented by
the constants A and B being found from the equation
A-B ^/{- 1) - ^r^^q:^,3-j^j (Art. -1).
Now f '^^./"... = 2i?tan--^-"
therefore I ,— ^^' — r^i — '—^, = 'UBir,
and hence it mioht be said in a certain sense that if the
integ-ral be taken between the limits — cc and + co the
result will be zero. This however is not satisfactory, for the
positive part of the integral and the negative part are both
aumerically infinite, so that it is not safe to assume that they
balance. But \if{x) is at least two dimensions lower than
F(.r), we shall find that the sum of the terms of the type
which we are considering is finite for each part of the
integral, and then the positive part may be safely taken to
balance the negative part. For suppose we require the
integral between the limits and h. Let A^, A^,...A^^ denote
the constants of which we have taken A as the type ; and let
1 similar notation hold with respect to a and ^. Then we
bave for the integral the expression
...+ J„log
16—2
244 DEFINITE INTEGRALS.
This may be put in the form
2[A^ + A^+...+A,]logh
M\A^ A «.V,/^./
, 4 ^ \ h) ]i* ■ , \^ hj ' /r ,
••• + Alog ^ . ...
Now since /(a;) is at least two dimensions lower than F{x)
we have A^ + A„^ ... + ^^ = 0. Thus the above expression
reduces to the second part, which is iinite when h is infinite.
Hence when the limits are — cc and + co the sum of the
terms we are considering: vanishes.
If then we suppose F{x) to be of 2;i dimensions, and
Jj^,B^, B^ to be the n constants of which we have taken
B as the type, we have when / {x) is at least two dimensions
lower than F(x)
f_^-l^^-^dx = 2^{B, + B, + +B„].
254. As an example of the preceding Article we take
a?-"' dx
i
1+x""
where m and n are positive integers, and m less than 7i. Here
1
A-B^{-1) = ^^^ ^^^^^^_ -^^|.„-.in^,
and it is known that the values of a 4-/3 \/(— 1) are obtained
from the expression
(2r + l)7r^ ., ^- . C2r + l)7r
cos • ^-^- + V(- 1) sm ^ ,
by giving to r successively the values 0, 1, 2, up to
w — 1 : see Plane Trigonometry, Chapter xxiii.
Thus, by De Moivre's theorem,
DEFINITE INTEGRALS. 24'.')
(a + /? V(- l)!^"-^-"-^ = cos (/. + V(- 1) sin 4>,
where
so that
cos cf) + v'(- 1) sin (^ = - cos (2r + J)d + V(- 1) sin (2r + 1) ^,
where = — -; tt.
2n
Hence
2n - cos (2r + 1) ^ + V(- 1) sin (2r + 1) ^
cos(2r + l)^ + \/(-l)sin(2r+l)(9
2?i '
therefore i? = ^^— -— .
zn
Hence
^-^'^^ = '^Jsin^ + sin3^ + sin5^+... + sin(2/i-l)^l .
j _ oc 1 + iC Jl ( ' )
The sum of the series of sines may be shewn to be
sin'^ nO
— — — ; see Plane Trigonometry, Chapter xxil. ; and in the
present case nO = — - — tt, so that sin^ n6 = 1. Therefore
-^l + x^" . 2m +1
n Sm t: TT
2n
It is obvious that
that is,
•" x^"" dx
^ ^ is half of the above result
1+a;'
r x'"" dx _
Jo l + x'"'
TT
„ . 2m +1 •
zn sin — ^ ■ TT
246 DEFINITE INTEGRALS.
255. In the last formula of the preceding Article put
2m + 1
a;^" = y, and suppose —^ = h ; thus we obtain
/.
1 + 2/ ^i^ ^'^'^
.(1).
This result holds when k has any value comprised between
and 1. For the only restriction on the positive integers m
and n is that m must be less than n, and therefore by pro-
2/?i + 1
perly choosing m and n we may make — ^ — — equal to any
assigned proper fraction which has an even denominator when
2m + 1
in its lowest terms. And althousjh we cannot make ^^- — —
* 2'7J
exactly equal to any fraction which has an odd denominator
when in its lowest terms, yet we can make it differ from
such a fraction by as small a quantity as we please, and thus
deduce the required result.
In the last result put x" for y, where r is any positive
quantity ; thus
' rx'^" X"^' dx IT , . rx'^-'dx TT
that IS,
1 + x'' sin /cTT ' ' J 1 + ^''" ^' sm Ictt
Let Jcr = s ; thus I
J I
X ax 77
___ ___
r sm - TT
r
The only restriction on the positive quantities r and s is
that s must be less than r.
The student will probably find no serious difficulty in the
method we have indicated for proving the truth of equation
(1) when A; is a fraction which has an odd denominator when
in its lowest terms ; nevertheless a few remarks may be made
which will establish the proposition decisively, and which
will also serve as useful exercises in the subject of the jDre-
sent Chapter.
Let «=rp'^; then«=rC^+r^^;
h 1 + y .'o 1+y Ji i+y '
DEFINITE INTEGRALS. 247
and by putting - for y we find that
ii l+y Joi + 2 Jo 1 + y
^'--f- t<W^-'-'^-'^"^ ^^-
Equation (2) shews that -^u ^^ negative if ?/*"'-?/'* is con-
stantly positive, and positive if if~^ - y'" is constantly nega-
tive, between the limits and 1 for y. Hence -^r is negative
or positive according as h is less or greater than ^ • Thus u
1
diminishes as h increases from to - , and u increases as k
increases from - to 1.
Now let -7^ denote any fraction in its lowest terms, in
which ^ is an odd integer ; and let p be any even mteger.
Let ^^ = ^^^~ , and k,=^ ^^ ^ , and let h„ denote -^ . Let
^ pl3 PP _ P
ti,, v„, w, denote the values of I K. when for /jwe sub-
stitute ?c^, Jc^, Jc^ respectively. Then by equation (1)
u. = —. — ; — and u„ = —
^ smAr^TT ^ smA-gTr
Now we may take p so large that ^■^ and \ shall be both
greater or both less than - ; and then by the inferences drawn
from equation (2) it follows that ^l.^ must lie numerically be-
tween Mj and Wg. Thus u.^ cannot differ from ?/, or u^ by so
much as the ditference of u^ and u^ ; and therefore afortioH
u, cannot differ from -r^, — by so much as the difference of
^ sm A- TT
248 DEFINITE INTEGEALS.
u^ and u^. Hence as i^ ^i^-y be indefinitely increased we
TT
have finally u„= - — -. — .
"^ ^ sin k.^ir
Eulerian Integrals.
2oG. The definite integral f x''' (1 - a;)'""' dx is called
y
the first Eulerian integral; we shall denote it by the symbol
B (/, m). This integral is sometimes called the Beta function.
The definite integral I e'" x""^ dx is called the second
Jo
Eulerian integral; it is denoted by the symbol V [n). This
integral is sometimes called the Gamma function.
We shall now give some of the properties of these inte-
grals ; the constants in these integrals, which we have denoted
by /, m, n, are supposed loositive in all that follows.
257. In the first Eulerian integral put x = l — z\
thus \\'~'{l-xy-'dx=\\'''-'{l-zY-'dz; ' .
Jo Jo
this shews that the constants I and m may be interchanged
without altering the value of the integral ; that is,
B{l,m)=B{m, I)..
Again in the first Eulerian integral put x = :r— : thus
J/ ^^ -^^ "^^ Jo (1+2/)"""
In the same integral put x = ^ ; thus
j^x'-'{i-xY-'dx = ^^ (T+^-
258. Let e' =y, so that x = log - ; then we have
[ e-^x'^-'dx^l (log-) dy,
which consequently gives another form of V (n).
DEFINITE INTEGRALS. 249
259. We have by integration by parts
j e' x''dx = - e~-^ x'' + n\ e' a;""' dx ;
and e"* x" vanishes when x = 0, and also when a; = co . (See
Differential Calculus, Art. 153) ; thus
I e~^ ^" dx — n I e'' x"'^ dx ;
Jo ^0
that is, T{n + l)=nV(u) (1).
Since I e"' dx = — e~~ we have I e""^ cZo; = 1 ; that is,
r(i) = i (2).
From (1) and (2) we see that if n be an integer
T{n-\-\)=[n.
When n is not an integer we may by repeated use of
equation (1) make the value of V (n) where n is greater than
unity depend on that of V (ni) where m is less than unity.
260. By assuming kx = z we have
/.
e-^ «"-^ dx = hl e-' z^-' dz = ^ ^''^
1^ J
Jc'Jo k' '
2G1. We shall now prove an important equation which
connects the two Eulerian intec^rals.
.00 .00
Integrate the double integral f [ x'^"'-'y"'-^e~^'^'-'^' dij dx
Jo Jo
first with respect to a; ; we thus obtain, by Art. 2G0,
T{1 + m)
[i+y)
l+m •
Again, integrate the same double integral first with respect
to y; we thus obtain
r("0| ^^ - dx,
250 DEFINITE INTEGRALS.
that is r (??i) I e'' x'~^ dx,
Jo
that is r (m) T (l).
* . .m— 1
Hence
rj
Jo (1
y'^-'dj/ T(l)T(m)
Hence, by Art. 257,
^ ^ r (^ + m)
2G2. In the result of the preceding Article, suj^pose
l-{-m = l', thus, if m is less than unity.
Jo 1+y ^ ^ ^
since T (1) = 1. Hence, by Art. 255, if m is less than unity,
r(m)r(l-7n)=-^^^^.
^ ^ ^ ' sin??i7r
263. Put on = 1^ in the last result ; then
rQ)r(i) = 7r,
therefore T (h) = V"^-
Or, without using Art. 255, we have
^^^'^^ -Jo l+.V ~^io l+a;^~^''2-'''
therefore F (i) = ^Jir.
We will give another proof of the last result.
Let u = I e'^'Va; ; then it is obvious that u also
Jo
DEB'INITE INTEGRALS. 2-31
thus 11^=1 e~''' dx X I e'^' dij
Jo Jo
/* 00 ,*00
e''""''"'dxdi/ (Art. 66).
JO Jo
This double integral is shewn in Art. 204 to be
1- e-^'rdedr
Jo JO
4'
therefore
Vtt
«= 9
/-CO
Now r (i) = e'^'^'-tZx-; jDut x^tf,
Jo
thus r (JL) = 2 I e'""' d>/=2u = sjir.
Jo
264. We shall now give an expression for F (n) that will
afford another proof of the result in Art. 262. We know that
the limit of — r — when h is indefinitely diminished is log x ;
hence
(logl) = limit of (^-") ';
so we may write
where ?/ is a quantity that diminishes without limit when h
does so.
Put h = -; , then, by Art. 258,
r (w) = r"-' [ (1 - x'-y-' dx+{ y dx.
J •'0
In the first integi'al put x = z' ) thus
r (n) -fydx = 7-™ [' z"--' (1 - zy-' dz.
252 DEFINITE INTEGRALS.
Wc have it in our power to suppose r an integer; tlicn
the integral on the right-hand side, by Art, 33, is
1.2.3 r „_,
?i(?i+l) (w + r — 1)
Let r increase indefinitely, then y vanishes and we have
1 2 3 ... r
r (n) = limit of —. -^-^ — '^^ ^ , r" \
^ ' n{n+\) {n + r-V)
2G5. From the result of the preceding Article we have
A particular case of this is obtained by suj)posing n = l;
thus
V{l-m)T{l + m)^\} ~ rj (^ ~ F; V ~ 3V '
the expression on the right-hand side is known to be equal to
; see Plane Trigonometry, Chapter xxili. : thus
rtnr
r (1 -^ m) r (1 + m) =
sm viir
therefore T im) Til- m) = . '^ (Art. 259).
266. We shall now establish the following equation, n
being an integer,
11- V^
then reversing the order of the factors we have
x = r(i-?)rfi--) rfl
\ nj \ nj \ii
DEFINITE INTEGRALS. 253
Multiply, and use Art. 2G2 : thus
—n-l
. IT . 'lir . (?J — ijTT
sm - siu — sm
n n n
n
The denominator is equal to ^7^1 : see Plane Trirjonometry,
Chapter XXIII. Thus the result is established.
267. A still more general formula is
r(.)r(. + l)r(.+ 5) t{.+'^)
n-l
= r {nx) {27r) ''n^^'',
■which we shall now prove. Let <f) {x) denote
n-l
nV (iix)
we have then to shew that <p (x) = (27r) * /i~V
We have
n"'^"T(x + l)v(x + l + '^y..r (x + l+'^^
•5^^^ + ^) = nTjnJ+ n) ^^
7l''x(x + -](x+ -] (x+^
= ^ — ' ^ ^ (b(x) = (b (x).
(7ix + 7i-l}{nx+n-'2) nx ^^ ^ ^^ '
Similarly (}> {x + 2) = (j) {x + 1) = (J3 (x) ; and by proceeding
thus we have (f) (x) = (f> (x + m), where on may be as great as
we please. Hence {x) is equal to the limit of (J3 (/x) Avhen
fj, is infinite ; thus ^ (x) must he independent of x, that is,
must have the same value whatever x may be ; hence (f) (x)
must have the same value as it has when x = -: thus the
n
theorem follows by the preceding Article. This theorem is
254 DEFINITE INTEGRALS.
ascribed to Gauss ; a more rigid proof is given in Legend re's
Exercices de Calcid Integral, Vol. ii. p. 23 ; see also the
Journal de VEcole Poly ted inique, Vol. xvi. p. 212.
268. Take the logarithms of both sides of the formula
established in the preceding Article, and differentiate with
respect to x; thus we obtain
rv ^ TV ^ r'fa. + -) r{x + '^—^]
nV {nx) ^ r_Or) V nj \ n )
r(„.)-r(.) J,/ l^ + ^'TlI^^
n
("+^)
+ n\ogn (1),
where r'(^) stands for , .
Differentiate again; then, putting z for nx, we obtain
|.iogr(.)
d' log r {x + ^) d' log r (x + "^"j '
^ 1 fZ^grjx) _^ ^^
if [ dx^ dx' I • • • 1 ^^2
If n be made infinite the right-hand side vanishes, for it
becomes ultimately
1 p-+icZMogr(,r)
nJr dx
thati, iF'ogr(.+2)_.nogr(aj)
n ( a.c ax j
Hence we see that if z be infinite ? ,— ^- vanishes.
^^ x x{x+l) a-(ic+l)(a; + 2) '
take the logarithms and differentiate twice with respect to x ;
thus ^'^°g^^^^ = i + -J:^- + ^ 4- ad inf (^)
DEFINITE INTEGRALS. 255
The series just given is convergent for every positive
value of X.
Integrate between the limits 1 and x ; thus
d lo<T r (x)
■where — C stands for the value of — --, — ^^ when x = l.
ax
.1 1 .
The series whose if^ term is — ^ is convergent
n n-\- X — 1
for every positive value of x, as we may infer from the fact
that it is obtaiued by integrating between finite limits a con-
verging series in which all the terras have the same sign ; or
we may infer the convergence of the series from the fact that
x — 1
the general term, being —, =-, , is numerically less than
° ■ ° n{n-{- X — 1)
-, :, SO that the series is numerically less than another
(n-l)-'
which is known to be convergent.
The quantity C is called Elder s constant; it may be
r'Yi)
presented under various forms. It appears above as — -p , .,
that is as— r'(l). Nowr(?i) = l e~\c""' cZ.f ; tlierefore we
have V'{n)= f e"" a;""Mog .r fZ.c, and r'(l) = I e-'logxdx.
Again sujjpose x—1 in (1) ; thus
1 r'(l) \ n V "
-<Trh< + ^ d-+ + — ) T
"|r(i) ' r(i+l) r(,
n
256 DEFINITE INTEGRALS.
Increase n indefinitely; then the right-hand side be-
comes a certain integral, namely I -p log V {x) dx, that is
log r (2) - log r(l), that is zero.
Hence the limit of p-7~\ — log ^^> "when n is tnade infinite,
is zero.
In (3) suppose x infinite : hence, with the aid of the result
just obtained, we see that G is equal to the limit when n is
infinite of
-, 111 1 1
1 + 2 + 3 + 4+ +,-l°g^^-
It is easy to shew by elementary considerations that this
limit is finite. See Algebra, Chapter LV, Examj^le 12.
The value of C to 10 places of decimals is '5772156G49 ;
the calculation has been carried to 263 places of decimals:
see a paper by Professor J. C. Adams in the Proceedings of
the Royal Society, Vol. xxvii. page 88.
269. In equation (2) of the preceding Article change x
into a; + 1 ; thus
<^''logr(l+a-) _ 1 1 1
dx"" ~(aj+l)' + (^+2)'' + (a;+8)^+"'*'
differentiate w — 2 times ; thus
d"l0gr(l + ^-)_, _l(_l)nf 1 . 1
dx'' 1 ^ ' \{x + iy (a; + 2)"
+ (a; + 3)"+"-J •
Let S^ denote the infinite series 1 + .^ + -57, + . . . ; then,
if n be not less than 2, the value of ° , „ , when
ax
a: = 0, is |w-l(-l)">S^„.
DEFINITE INTEGRALS. 257
Also the value of — ^^^— y^ -, when x = 0, is — C\
ax
and log r (1 + .r) = when ic = 0. Hence, by Maclaurin's
Theorem,
hgT{l + x) = -Cx+^ 3' + "t~~-"
The series is convergent as long as x is numerically less than
unity. Now by the property of Art. 2G2, combined with that
contained in equation (1) of Art. 259, it follows that F (x) is
known for all positive values of x if it be known for all
values of x between and - , or for all values between ^
and 1, or for all values between 1 and 1^, and so on. And
the series just given will enable ns to determine the value of
log r (x), and thence of F (x), for all values of x between 1
and H ; so that we may consider that F (x) can be calcu-
lated for any positive value of x.
Legendre has constructed a table of the values of log F {x) ;
and an abbreviation of this table is given in De Morgan's
Differential and Integral Calculus, pages 587... 590. We may
also refer to an article by H. M. Jeffery on the Derivatives of
the Gamma-Function in the sixth volume of the Quarterly
Journal of Mathematics.
270. A higher degree of convergence may be given to
the series obtained for log T {1 ■\- x) thus :
^ x^ S v^
logF(l+^)=-Ca. + '-|---^- + ...,
logF(l-^)= (7^ + 3^+*-- + ...;
now F (1 + a;) . F (1 - a;) = xF {x) T {I - x)
= -.^^,byArt. 2G2;
sin CCTT "^
T. I. c. 17
258 DEFINITE INTEGRALS.
(fjir 1 1
therefore log -. = Sjc' + -r ^.x^ + ;^ S>a^ + • • • ,
and logr(l+a;) = ^log-^ Cx--f i--....
The result may also be written thus :
1 XTT \ 1 1+x
iogr(i + ^0 = ^iog^-ii^-2ios-i3^
the series in the last line converges rapidly when x is numeri-
cally less than - .
271. From equation (2) of Art. 26S we see that
^- is always positive, and is finite if x be positive :
hence —j — ^ increases algebraically as x increases from
to infinity, and therefore cannot vanish more than once.
Thus r (x) cannot have any maximum Avithin this range of
values of x, nor can it have more than one minimum. It is
easy to see that F (x) has one minimum, between x = l and
ic = 2 ; for F (2) = F (1).
To determine the minimum of F (1 + x) we differentiate
one of the series found for log F (1 +x), and equate the result
to zero. This gives an equation from which it is found by
trial that l+x = 1-4G16:321....
272. Many definite integrals may be expressed in terms
of the Gamma-function; we shall give some examples.
The integral I e~"'^^ dx becomes by putting y for a~x^
Jo
I
^ — .-^ , that IS, jr- F (i), or -:r- .
2a Vy 2a ^-^ 2a
DEFINITE INTEGRALS. 259
Again, in — - — '—^ — put — — = .r-^ ; thus we
7o ix + ar"" ^ x + a 1 + a
i-^ + a)
obtain
I f V- (1 - V)-- d>, that is 1 r (0 r jm)
«"• (1 + ay Jo ^ ^' ^^ '^^' ^^'""^ ''' a'" (1 + ay r{l + m) '
Again, in I x'~^ (1 — ic')""' tZiz; put x' = y; thus we obtain
,r i-. r(|)r(„o
ij »" (1 - y)- rfy, that is, — i^^ ..
2r(|+,„)
IT
Thus [^ sin" 6 cos' ddd=\ x^{l- x^y' dx
•' .'o
r r^ + ^^ r /^^ + ^
. . . r x'-'il-xr-\Ix ^ ly
we obtain
-1- [ V-i fi _ y\-^-^du that is ^ ^^) ^ (^^)
a'6"'Jo^ ^-^ 2/} rfy, thatis,^,^„.j,^^_^^^^.
273. In 1 ic'"^ (« — a:;)"*"^ c?x put x = ay; thus we obtain
a--- [ ' 3,'- (1 - yy~^ dy, that is, a^-- ^J:^^'^ '
Jo i {(> + VI)
ST-t. It is required to find the value of the multii^le in •
teoral
{\L . .a;'-^ y""-^ «"-^ ...dxdy dz.
17—2
2G0 DEFINITE INTEGRALS.
the integral being so taken as to give to the variables all
positive values consistent with the condition that x + y + z-^...
is not greater than unity.
We will suppose that there are three variables, and conse-
quently that the integral is a triple integTal ; the method
adopted will be seen to be applicable for any number of
variables.
We must first integrate for one of the variables, suppose z;
the limits then will be and 1-x-y; thus between these
limits
r.-i,, a-x-yY _ v{n)
Next integrate with respect to one of the remaining varia-
bles, suppose y ; the limits will be and \ — x\ and between
these limits, by Art. 273,
r , , , (1 - xT^'^ V (m) r (n + 1)
Lastly integrate with respect to x between the limits
and 1 ; thus between these limits
J ^ ^ r (Z4-W-1-W + 1)
Hence the final result is
r Qn) r (m) r (n + 1) r(or(>/i + n + i)
r (n + 1) r (to + ?i + 1) r (^ + m + ?i + 1) '
V{1) V (TO) r (n)
"'''^'^' r(^-i- TO 4-71 + 1)
275. It is required to find the value of the multiple
the integral being so taken as to give to the variables all
integral
DEFINITE INTEGRALS. 261
positive values consistent with the condition that
is not greater than unity.
Assume a. = ^ , ^=Q, ^=(^^
Then the integral becomes
with the condition that x + i/ + z+ ... is not greater than
unity. The value of the integral is, therefore, by the pre-
ceding Article,
, fff i_i «-i :?-! , , ,
- j...x^ y"^ z'' ...dxdydz
a'yS^v"..
\p) \qj \rj
pqr ...
T-, fl m n ^\
V - + - + - + ... +1
\p q r J
This theorem is due to Lejeune Dirichlet ; we shall give
Liouville's extension of it in Arts. 277 and 278.
276. As a simple case of the preceding Article we may
suppose p, q, r, ... to be each unity, and a, /3, 7, ... each equal
to a const-ant h; thus the condition is that ^+t] + ^+... is
not to be greater than h. Therefore the value of the integral
jjj...rv"'-'r'-d^dvdc...
ig ;j.»........ r (r)r(m)r(n)...
r{i+Qu + ti+ ... + 1) '
which we may denote by
Similarly if the integral is to be taken so that the sum of
the variables shall not exceed h + Ah, we obtain for the result
Hence we conclude that the value of the integral extended
over all such positive values of the variables as make tlie
2G2 DEFINITE INTEGRALS.
sum of the variables lie between h and h + Ah is
and when Ah is indefinitely diminished, this becomes
N{l + m + n+..:)h'^'''^''^--'Ah,
rfflr(m)r(n).
tiiatis, Til + m + n + ...)
277. It is required to transform to a single integral the
multiple integral
the integral being so taken as to give to the variables all
positive values consistent with the condition that a; + 3/ + ^ +. . .
is not greater than c.
We will suppose for simplicity that there are three
variables. By the preceding Article if / (.x- + 3/ + 2) were
replaced by unity that part of the integral which arises from
supposing the sum of the variables to lie between h and
h + All would be ultimately
r (i -1- VI + n)
And if the sum of the variables lies between h and h^- Ah
the value of f{x^y-¥z) can only differ from /(A) by a
small quantity of the same order as Ah. Hence, neglecting
the square of Ah, that part of the integral_ which arises from
supposing the sum of the variables to lie between h and
h + Ah is ultimately
r(?)r(m)r(» ) „.^.-,^^^^
r (/ + m + ?o -^ ^ ^
Hence the whole integral is
r(/ + m + w) Jo^ ^
This process may be applied to the case of any number of
variables.
DEFINITE INTEGRALS. 2G3
278. Similarly the triple integral
///r.-'ry{(^)'+(|)V©]'^f<'^''?
for all positive values of the variables, such that
is not greater than c, is equal to
Pi"- yI- + - + -] -'
\p q rj
This process may be applied to the case of any number of
variables.
279. It is required to transform to a single integral the
double integral
11 {u + ax + hyT" '
where the integral is to be taken for all positive values of
X and y such that ^ + y is not greater than k ; the quantities
p, q, u, a, and h being all positive constants.
Suppose that a is not less than h. We have
11 + ax + hy = u + a {x + y) - {a - h) y = U - 7),
where U stands for u-^a{x-\- y), and 77 for (a - h) y. Thus
(w + ax + hyy^^
the series here given being convergent.
The proposed double integral may now be transformed by
applying the method of Art. 277 to every term. Thus the
double integral
264 DEFINITE INTEGRALS.
^ [' [ r(j.)r(g ) jF^ T(p)T{q + i) ( a - h) r^
h\V{p + q) {u + atr'''^ Fip + q + l) ^^^^^ (u + at)'^'^'
T{p) T(q + 2)(p + q)(p + g + l)(a-hrr^'' )
■^ r{p + q + 2) 1.2 {u + aty^'^^''^'--r^
= r(v) (' '^"^ f ^^^^^ {p + q)Tiq + l)ia-h)t
^^' Jo {ii + city' [r {p + (/) "^ r{2) + q + 1) u + a^
, {p + q){p + q^l)V{q+^^) {a-hYf \
r(p + 2 + 2) 1.2(M + aO'' J
_ r ( y) r (y) p r^-^ f, , q{a-h)t
r(i> + ?) io ((i + a^/'-^^l ^ u + at
q{q + l) {a-hfe \
"^ 1.2 ' {u + atj +"-|^^
r(i) + ?) Jo (^i + aO^H ^* + a^J
In a similar manner we may transform to a single integral
the triple integral
a,P-l yi-i ^r-l ^^ ^^ ^^
(it + a-c + Z'j/ + c^)
p+3+r »
"where the integral is to be taken for all positive values of x,
y, and z such that x -\-y + z is not greater than k ; the quan-
tities p>, q, r, u, a, b, and c being all positive constants.
Suppose that a is not less than b or c. We have
u + ax + bi/ + cz — u + a {x + z) + by — {a — c) z.
Proceeding as before we find that the proposed triple inte-
gi-al can bo transformed into a series, each term being of the
form represented by the product of
DEFINITE INTEGRALS. 2G5
(j) + q + 7-){p + q + r+l)...{2y + q + r + p-l)
and the triple iiite<Tral •- , , ,"f ,.^
y+a+T+p •
Then, as before, we can shew that the triple integral just
expressed can be transformed to
r (p) r (q) r (r + p) [" t^^^'-' at
r {i) + q + r + p) j (« + aO^'*' ('* t 6^' "
Hence finally the proposed triple integral is seen to be
equal to
r ip) V (g) r (r) r^ t^-^^^ r _ (a-cytr-
T{2i + q + r) Jo (" + (^ty^ (" + ^ty \ ^ + "^ I
that is, to
T{p + q + r) Jo (m + ««)" {u + i«j« (i^ + cty '
This process may be applied to the case of any number
of variables; and it may receive extensions similar to those
which Arts. 277 and 278 supply of the process in Art. 275.
280. It is required to transform to a single integral the
multiple integral
///.../(<,... + »...+ + «...) ci..d.,...d..,
the integral being so taken as to give to the variables all
values consistent with the condition that x^' + x^ ... +x^ is
not greater than unity.
By successive applications of a transformation for a double
integral given in Art. 242, the multiple integral may be
reduced to
]]]•••/ (^^^i) ^^1 ^-^a • • • ^'^n^
where k = V(«/ + c,/ + . . . + a J) ;
2GG DEFINITE INTEGRALS.
and these transformations do not affect tlie condition that the
sum of the squares of the variables is not to be greater than
unity.
We have first then to find the value of the multijale integral
1 1 1 ...dx^dx^ ... dx^, the variables being supposed to have all
values consistent with the condition that x,^ + x^ + ... + x,^
is not greater than 1 — x^. If the variables are to have only
positive values then we obtain the value of the integral by
supposing in Art. 275, that each of the quantities, I, m, ... is
unity, that each of the quantities i'>, q, ... is equal to 2, and
that each of the quantities a, y3, , . . is equal to ^J{1 — x^).
Thus the result is
im]r^(i_^,)
n-l
On-ip/!^!^^
But if the variables may have negative as well as positive
values, this result must be multiplied by 2""'. Thus we get
m-l "-1
^ • (1 - <) '
Hence, finally, since the limits of x^ will be — 1 and 1, the
multiple integral is equal to
n-l
2 n n-l
TV
r('!i^+i
f /(fc.)(i-^.Vrf^..
This agrees with the result given by Professor Boole in
the Cambridge Mathematical Journal, Vol. in. p. 280, as it
may be found by integrating his equation (15) by parts.
281. It is required to transform to a single integral the
multiple integral
• " \r\ T-^^-. -^ dx. dx„... dx.
DEFINITE INTEGRALS. 2C7
the integral being so taken as to give to the variables all
values consistent with the condition that x^'' + x,^^ + ... + xj^
is not greater than unity.
As in the preceding Article the integral may be trans-
formed into
ii ... —-Z AA T. dx^ dx^ ...dx^.
First integrate with respect to the variables a;,, a*3,,.,a'„,
the limits being given by the condition that x^^ + x^' ...+x^
is not greater than 1 —x^. If the variables are to have only
positive values then the integral
dx^ dx^ . . . dx^^
by Art. 278 would be equal to
1 (r f-^Vf""' r^"'^'' 1 "--1
r
that is, to
l,Jm^a_..>)^■!:£IG), (A,.t.273).
that is, to 2^1 • — ^^ (1 - x{) " .
r
(1
But if the variables may have negative as well as positive
values, this result must be multiplied by 2"'\ Thus we get
2G8 DEFINITE INTEGRALS.
Hence finally, since the limits of x^ are —1 and 1, the
multiple integral is equal to
TT r .,-, V ,, o, „--i
282. Many methods have heen used for exhibiting in
simple terms an approximate value of V (^n + 1) when n is
very large : we give one of them.
The product e~' x" vanishes when x = and when x = cc;
and it may be shewn that it has only one maximum value,
namely when x = n. We may therefore assume
e X =e 11 e (1),
where t is a variable which must lie between the limits — co
and + CO .
Thus
[ e-' x" dx = e'" tf I e-^'^dt (2).
Jo J -cc dt
Take the logarithms of both members of (1) ; thus
x — n loga:; = n — n\og n + f (.3) ;
put x = n + u) thus
n — n log (n + w) =f— n log n (4).
But by Taylor's Theorem
log {n + u) = log n +
u u^
n 2 (n + Oaf '
where ^ is a proper fraction ; thus (4) becomes
therefore V (n) u
V(2} [n-i-eu) ^ ''
DEFINITE INTEGRALS. 2G9
therefore u = ,. " ^^ ,,, (G).
But from (3) ^ = ^^^ - = 2< + —
= V(2^0 + 2(i-^)^ l)y(C).
Hence (2) becomes
/•CO (* GO
Jo j - 00
and I e~^'' dt = \l{'Tr); thus
But since 1 — ^ is positive and less than unity, the nume-
rieal value of e~*' {1 — 0)tdt is less than I e'^'tdt, that
j -00 Jo
is, less than ^. Hence we conclude from (7) that as n is
increased indefinitely, the ratio of F (?i + 1) to e''' if ^ {'Imr)
approaches unity as its limit.
We may observe that in the original equation (1) we
have f and not t itself; hence the sign of i is in our power,
and we accoi'dingly take it so that equation (5) may hold,
sapf)osing \/n and ^2 both jDOsitive.
(See Liouville's Journal de Mathematiques, Yol. x. p. 464,
and Vol. xvii, p. 448.)
Definite Integrals ohtained hy differentiating or integrating
with respect to constants.
283. We shall now give some examples in which definite
integrals are obtained by means of differentiation Avith respect
to a constant. (See Art. 213.)
270 DEFINITE INTEGRALS.
To find the value of I e"'*'-^' cos 2rxdx.
Jo
Call the definite integral u ; then
du
J = — 2 xe'"-''^' sin 2?-^ dx.
dr Jo
Integrate the right-hand term by parts ; thus we find
.
du 2ru
dr~ a' '
therefore
d log u 2?*
dr d' '
therefore
log zt = - -, +
0/
therefore
u = Ae a"- ,
where yl is a quantity which is constant with respect to r.
that is, it does not contain r. To determine A we may suppose
?' = 0; thus u becomes / e-"'-*^" (^^, that is, 7^, (Ai't. 272)
Hence -4 = -77— , and e'^''''^" cos2rxdx = '^r~&~"'^'
2a Jo 2a
284. We have stated in Art. 214, that when one of the
limits of integration is infinite the process of differentiation
with respect to a constant may be unsafe ; in the present case
however it is easy to justify it; we have to shew that
Q-a"x- p^i^ vanishes where p is ultimately indefinitely small;
it is obvious that this quantity is numerically less than
/< 00
Pj I e~"'-^V,c where p^ is the greatest value of p, that is,
J Q
\/ IT
less than — p^ ; but this vanishes since p^ does. Similar
considerations apply to the succeeding cases.
DEFINITE INTEGRALS. 271
sin rx dx
,— kx
285. To find the value of e
Jo "^
Denote it by u, then
du _ r*
dr~Jo
But e "^-^ cos rx dx = e *-^
e ** cos rx dx.
r sin j-a; — Ic cos r.r
therefore e '''^ cos rxdx — j^ ^ ,,
. du k
thus
h
dr k' + r'
therefore u = tan ' , •
No constant is required because u vanishes with r. This
result holds for any positive value of A; ; if we suppose k to
diminish without limit, we obtain
r°° sin rx , tt
J ^ '^
TT
if r be positive ; if r be negative the result should be — ^ .
"We can now determine the definite integral
/,
^ sin rx cos sx ,
— ax;
^
for it is equivalent to
'Jo ^ ".'o ^
and the value of each of these two definite integrals can be
assicmed. Thus if r + s and r-s are both positive the result
is ^ ; if they are both negative it is - ^ ; if they are of con-
trary signs it is zero.
272 DEFINITE INTEGRALS.
286. To find the value of f e~V^^-7dx.
Jo
Denote it by u, then
du
du _ [ -(x-+-)djc
:j- = — 2a e \ «v — r ;
da j *■'
da
assume x = - , then the Hmits of z are <x and ; and we
2
obtain
therefore
therefore
therefore
To determine A we may suppose a = ; then i^ = -^ ;
du
da
-2u;
d log ?i
da
— _ 9 .
log w
= — 2a + constant-,
u
= ^e-''".
therefore A = -^r- ; thus
/,
00
e
287. We may also apply the principle of integration with
respect to a constant in order to determine some definite in-
tegrals ; the principle may be established thus.
rb
Let u= \ (p [x, c) dx,
J a
rp rp rb
then lidc =1 I </> (^, c) dc dx
J a. J a J a
a J a
b fP
a J a
(f) {x, c) dx dc ;
since when the limits are constant, the order of integration is
indifferent (Art. 02). We sliall now give some examples of
this method.
DEFINITE INTEGRAI-S. 273
r 1
288. We know that e"*' dx = j .
Jo f^ .
Integrate both sides with respect to k between the limits
a and b ; thus
dx = lo£f -
j ^ ^ a
f" e'""" dx ["^ e~"' dx
It should be noticed that I ■ — • and I are both
Jo X Jo X
. . .^ . {"e-^'dx . , „ -oa P^-c 1 ['dx
mnnite : lor is greater than e \ — , and I —
is infinite. But this is not inconsistent with the assertion
that dx is finite, and without findinsr the value
Jo ^ . . .
of this integral it is easy to shew that it must be finite. For
., . 1 , x-u c f''4^(^)dx J f'°(f)(x)dx ,
it IS equal to the sum ot — and I - ^ — where
J ^ J c «-'
(f){x) = e "■^ — e""^ ; the second of these integrals is finite, for
it is less than - I ^ (a?) c?^, that is, less than - ( 5- .
cJc c\a J
[c Ay r^\
"We have then only to examine I ^-A_^ ^.j,^
Jo •^
Now by Maclaurin's Theorem
<P{x) = {b-a)x + ^<f^"ixe),
where 6 is some fraction ; thus ^^ ^ is less than 6 — a 4 -,
ic 2
where -4 is the greatest value which (/>" (a;) can assume for
values of x less than c. Hence
/.
■ax IS less than {b — a)c + —r- ,
'0 ^ 4
and is therefore finite.
T.i.c. 18
274 DEFINITE INTEGRALS.
289. We know that
h
I e"^"" cos rx dx= jy
Jo 1^
F + r '
Integrate both sides with respect to Ic between the limits
a and b ; thus
pe-^^-e-"- 6^ + r^
• cos rx ax = -h log —r. r, .
Jo X " a + r
r^ sin o^x f cos ?'tX'
290. Let I cZij; be denoted by A, and ^ cZ
J X J L -T X
by J5 ; we shall now determine the values of ^ and B ; the
former has already been determined by another method in
Art. 28.5.
In the integral A put y for rx ; thus
A =
sin y dy _
'o y
this shews that A is independent of ?\
dB r°° a; sin ra; dx
V\ e have
1 f^ 71 7 r"" sin r^ cZ^
and ijar = - ., , — jx ;
Jo Jo a; (1+^0
r„, ^S f"'l+3^'sinr.r , ,
thus Bdr p-= T^— — ^dx = A;
Jo dr Jo X \-\-oi?
hence j^Bdr-'^-A=0 (1).
Multiply by e"' and integrate ; we obtain since A is con-
stant with respect to r
e"' 1 1 Bdr + B-AI= constant.
Now whatever be the value of r, it is obvious that the
integrals represented by ^, J5, and 1 Bdr, are finite ; hence
DEFINITE INTEGRALS. 275
the constant in the last equation must be zero, for the left-
liand member vanishes when r is infinite.
Thus
I* Bdr -{■ B - A = {) (2).
J
From (1) and (2) ^=-^;
therefore B=Ce^,
where C is some constant. And from (2)
therefore B = Ae~^ (3),
Now when r is indefinitely diminished, B becomes
r (JQ(^ ITT
T-, — 2 ) that is 7i ; hence from (3)
Jo l+a:"" 2 ' ^ ^
^=^ and 5=^6-'.
2 2
We have supposed r positive ; it is obvious that if r be
negative, B has the same value as if r were positive, and
A had its sign changed ; that is, if r be negative B = -^ e^
TT
and A= — -. {Transactions of the Royal Irish Academy,
Vol. XIX. p. 277.)
f COS VCC doc 77"
From I ^; 5— = r: e'^ , we obtain by differentiation
Jo 1+a; ^
with respect to r,
I
X sin rx dx tt
= 77 C
l + x' 2
And from the same integral by integrating with respect
to r between the limits and c, we have
/■" sin ex dx _'ir , _^.
Jo <rT^~2^^~' ^'
IS— 2
276 DEFINITE IXTEGRALS.
291. The preceding Article contains a rigorous investi-
gation of the values of the integrals A and B; another
method has been sometimes given for finding the value of
B which is more simple but far less satisfactory. We will
however now give this method, as it will lead us to notice a
point of importance.
Let £=1 ^ idx,
Jo l + a?
, clB r'" a;sinr^ ,
then -7- = — -^i—, — 2 "-^j
dr Jo l+x""
d?B r x^'co^rx ,
and -y-o- = — —5 2" "^
dr" Jo l+«
['-^ , r*cos7'a; ,
= — cos rxax+ I ^ 5 dx
Jo Jo l+x^
i-OO
= ~io
cos rx dx + B.
Now we will assume on grounds presently to be examined,
/•OO
that I cos rxdx = Q; thus
Jo
d'B
dr'
= B,
and we have to find B from this equation. Multiply both
r77?
sides by 2 -y-; and integrate with respect to r ; hence
where ^ is a constant, that is, h is independent of r. Thus
dB
dr
therefore
dB V(/i + B')
DEFINITE INTEGRALS. 277
by integrating we have
where k is another constant.
Thus e^* = i^ + V(;i + B').
By transposing, squaring, and reducing we finally obtain
where C^ and C^ are constants. We must now determine the
values of these constants. Since B cannot increase indefi-
nitely with r we must have C^ = 0; and then since -S = - when
IT
?■ = we have C„ = ^ . Therefore
^ 2
We now proceed to consider the assumption involved in
the preceding method.
o- r -ax • 7 _„- a sin ra; + r cos ra;
Smce \e ?,m.rxdx = — e r. n ,
a~ + r'
1 r -/M 7 -ni »* sm rx — a cos rx
and I e cos rxax = e —
a^ + r
r
we have I e""'^ sin rx dx
J (
2 r 2 '
a +r^
and I e "' cos ra; c?a; =
_______ 2 , 2 ?
a + r
if a be a positive quantity/.
If it were allowable to suppose a = we should obtain
/•CO 1 I* CO
I sin rxdx = - , and 1 cos rx dx = 0.
Jo r Jo
278 DEFINITE INTEGRALS.
sm rx
r cos TJC f
Since I sin rx dx = ^^ , and cos rx dx =
we
r
are thus apparently led to the conclusion that the sine and co-
sine of an infinite angle are both zero. The same conclusion
seems to be suggested in other cases, so that it has been
stated, that " the indeterminate symbols sin go and cos od
are found in numberless cases to represent each of them,
0, the mean value of both sin x and cos x."
On this point however diversity of opinion exists among
mathematicians, and the discussion of it would be unsuitable
to an elementary work ; the student may hereafter consult
three memoirs in the eighth volume of the Cambridge Philo-
sophical Transactions, numbered XV, xix, and xxxii.
Definite Integrals obtained by Expansion.
292. If we expand log {1-ae^^'-^*} and log [l-ae"'^^'-^'}
and add, we obtain
a^ „ a'
log (1 — 2a cos X + a^)
= — 2 (a cos ic + -^ cos '2x +-K-cos3a;+ ),
the series being convergent if a is less than unity. Integrate
both sides with respect to x between the limits and tt;
thus
( "log (1 - 2a cos X + a') dx = 0,a being less than 1.
Jo
If a is greater than 1, since
/ 2 1>
log (1 - 2a cos a; + a') = log a^ + log f 1 - ^ cos a; + ^j ,
we have
I log (1 — 2a cos x + a^dx = 'ir log a^ = 27r log a.
JO
If a = 1 it may be shewn by Art. 51 that the definite in-
tegral is zero.
DEFINITE INTEGRALS. 279
We may put the result in the following form ;
I log (a" — 2ac cos x + &) civ = ir log k^,
Jo
Avhere P is the greater of the two quantities a~ and c^ and
is equal to either of them if they are equal.
By differentiating this result with respect to a we arrive
at the result which constitutes the last Example of Art. 46.
293. By integration by parts we have
log (1 — 2a cos X + a^) dx
Hence, if a be less than 1,
^ J = x- log (1 + a)^ that IS, - log (1 + a) ;
Jol-2acosx+a' 2a °^ ' 'a *^
if a be greater than 1, the result is
- log (1 + a) — log a, that is, — log ( 1 + - ) .
294. In like manner we have, if r be an integer,
I cos rx log (1 — 2a cos x-\-a^')dx = — a^ or — cC,
Jo r r
according as a is less or greater than unity.
295. Integrate by parts the integral in the preceding
Article ; thus we find
"■ sin a? sin ra: ^ic _7r „ , ir
1 — 2a cos x + a' 2
= ^a'*-i OY^a-^r+i)
according as a is less or greater than unitv.
280 DEFINITE INTEGRALS,
29 G. Similarly from the known expansion
1 — 2a cos X + d^
= 1 + 2a cos ic + 2a^ cos '2x + -a? cos 3x +
where a is less than 1, we may deduce some definite integrals;
thus if r is an integer
/.
cos rx dx TTCi
ji. J
' 1 — 2a cos x-\- c^ 1 — a^
for every term that we have to integrate vanishes with the
assigned limits, except 2a'' / cos^ rx dx.
Jo
dx
f 1
297. To find the value of t— ^ = —
Jo J- + X 1 —
2a cos ex + a
2
The term ^^ k may be expanded as in the
1 — 2a cos ex + a "^ ^
preceding Article ; then each term may be integrated by
Art. 290, and the results summed. Thus we shall obtain
TT 11 + ae~'
2 ■ 1 - a- 1 - ae-" '
Similarly
r°° dx
log (1 — 2a cos ex + a^) ^ -, = it log (1 — ae ').
Jo i + X'
298. It is also known from Trigonometry that
= sin ex + a sin 2cx + a' sin Sex + . .
1 — 2a cos ex + a^
a being less than 1. Hence by Art. 290, we obtain
X sin cxdx _ "^
(1 + x"") (1 - 2a cos ex + a") ~ 2(6" -a) "
This also follows from the last formula of Art. 297, by differ-
entiating with respect to c.
DEFINITE INTEGRALS. 281
299. To find f^^dx.
Jo l—X
By expanding (1 — x) \ we find for the integral a series
of which the type is
1
x" los: X dx.
o
By integi-ation by parts this is seen to be equal to
1
Hence the result is
(! + «)"
'l 4-1 1 1 1
that is, by a kno'^Ti formula, — — ,
oOO. Let V denote e■^^^^~l\ that is, cos ic + \/(— 1) since ;
then if/ denote any function, we have by Taylor's Theorem,
/(a + v) +/(a + O
= 2 |/(a) +/' (a) cos x +-^^ cos 2x + | .
And
1 — 2c cos x + c
Therefore
= 1 + 2c cos £c + 2c" cos 2x + 2c^ cos '^x +
j; i-2ccosUc- ^"=i:r^-|/^-H^/("Hi:^/'(^)+.--|
2-77
1-c
In this result it must be remembered that c is to be less
than unity, and the functions /(a + v) and /(a + v~^) must be
such that Taylor's Theorem holds for their expansions, and
gives convergent series
282 DEFINITE INTEGRALS.
In a similar way it may be shewn that
and r , ^-^^Q^^ ^ (/(a + r)+f{a + Ol dx
j„ 1 — 2ccosa' + c" ^-^ ^ '^ "^ ^ '^'
= 7r[/(a + c)+/(a)}.
Substitution of impossible values for Constants.
301. Definite integrals are sometimes deduced from
known integrals by substituting impossible values for some
of the constants which occur. This process cannot be con-
sidered demonstrative, but wdll serve at least to suggest the
forms which can be examined, and perhaps verified by other
methods (see De Morgan's Differential and Integral Calculus,
page 63 0). We will give some examples of it.
We have e-'"" x""-' dx=p-''V (ji).
Jo
For p put a + h V(- 1), and suppose r = »J(a' + h^) and
tan ^ = - , so that p = r [cos 6 + VC" 1) sin 0} ; thus
a
.00
Jo
Thus by separating the possible and impossible parts we
have
/.
n tan"^ -
e "" X" - cos Ox ax = n
{a-' + by
•J
rOi)sin^?itan-'-
1 e~" x"'^ sin bx dx = „ •
-" {a' + by
For modes of verification see De Morgan, page 630.
DEFINITE INTEGRALS. 283
302. In the formula
Jo
■^' dx =
Za
, — a-I' -7— »
. ^ l + V(-l) ,1
change a mto j^ — ■ c ; thus
Jo 2c V2'
therefore I jcos c'x" — \f{— 1) sin cV - cZ^ = ^^^ r^ ;
therefore I cos c"a;^ dx = z^ — — , and | sin c^a;^ dx = - — ~ .
Jo 2c V2 Jo '^cx^'2
If we write y for c^x^, these become
r°° sin y dy _ /"° cos _?/ <7?/ _ lir
Jo Vi/~~Jo "^"""V 2'
These results may be verified in the following manner.
By Alt. 272 Ave have
1 2 r"
\JX f^TTJo "'
f,
Jo
therefore I — ; — dx = — 1 cos x dx I e~ -'-^ cZ^
V^ V*"" ./
2 '•=°
, dz I e~-'^ cos a; cZj;
\/7r Jo Jo
2 f " /-^, , by Art. 285,
V 2
TT
9*
Similarly we can shew that
Jo
sm a; , _ /tt
284 DEFINITE INTEGRALS.
803, In the integral 1 e~\ """^V^cZa;, suppose y = a: \/^;
Jo
thus the integral becomes —pr \ e \^ 'f / dy, which is
known by Art. 286. Thus
g-2aft
Now put cos Q + \/(— 1) sin ^ for Z;; thus the right-hand
member becomes
1
a a '
cos - + V(- 1) sin ^
^'"" «-2a{cos0+V(-l)sinfl}
2 ^
that is,
VTTf
„„„^o.„:„^, ^\ //
.^.:^f^..:^n0\]
Thus [" e-("'+S)<=°^^os K^;^ + ^) sin 6'| cZ^
= '^e-2«cosecos C2a sin 6' + 1) ,
and I e V •<-■'/ sm ^ a; + — 2 ) sm ^ ^ aa;
/ 6
dn f 2a sin ^ + ^
_ V^g-2acose gi^ /o™ „:„ ^ , "
Euler's Theorem.
304. We will now give a theorem which connects inte-
gration with the summation of a finite number of terms, and
which is sometimes employed for the approximate calculation
of the value of definite integrals ; the theorem is usually
called Euler's, though more strictly due to Maclaurin : see
History of the Mathematical Theory of Probability, page 192.
DEFINITE INTEGRALS. 285
By Taylor's Theorem we have
/(a + A) -f{a) = //' ((0 + %f" («) + %,/'" («) + •••;
change a successively iuto a-{-h, a + 2k, a+oh, ... a-^ ('i — 1) ^h
and add ; then if we put a; for a + 7ih we obtain the following
result :
h' /^
3
where S/' (x) denotes /' (a) +/' (a + A) i- ... + / (a; - A),
and S/" (cc), 2/'" {x),... have similar meanings.
For/' {x) put ^ («) ; thus
j </, (x) dx = h^cl> (x) +|2</)' {x) + ~^4>" (^) + ... ,
and, by transposition,
S0(^) = ^j"^"%(^)^^-|2f(^)-|s</>"(a^)- (1).
In the same way we have
(2).
(3),
20'" (^) = I {f {^)-<t>" («)} -^20"" (0.) -^Sc/, (a:) - ...
W,
and so on.
oc;5 DEFINITE INTEGRALS.
Now from the series in (1) we may eliminate "Ecj)' (x),
2<^" (x),... by the aid of (2), (3) — The elimination may be
effected thus: multiply (2) by AJi, multiply (3) by AJi^,
multiply (4) by AJi^, and so on ; then add the results, and
determine A^, A^, A^,... by the equations
A, A. I ^
Hence we obtain
1 ra+nh ( ]
20(^)=^j^ cf>{x)dx+A,)^(f>{x)-cf>(a)^^
+ A^ if {x) - f {a)| h^A^ If {x) - <p" (a)\ h' + ..
Having thus she^vn that 2(^ (x) can be put in this form,
where A^, -4j, A^,... are numerical quantities, which are in-
dependent of the variable x and of the function denoted by
(f) (x), we may adopt an indirect method of determining these
numerical quantities. Let <f> (x) = e" ; then
a + nh _ a
Thus
e» _ e« e" - e"^
+
so that
+ A^ (e^ - 0+ ^/* (e' - O + AJi^ (e* - e")
e"
Therefore ^„^ is the coefficient of 7i"' in the expansion of
in ascending powers oih. The expansion is effected
iu the Differential Calculus, Art. 123; it is there shewn that
DEFINITE INTEGRALS. 287
1 1 1 Bh BJo' BJi'
= - r. + H . V +
+ (-ir
L2n ' •••'
/)j, B^,... are called Beruouilli's J\"fi??i6e?'5 ; their values are, as
far as B^,
' 6' ^~30' "■ 42' '"80' '~m'
with respect to the values of the Numbers beyond B^ inform-
ation and references will be found in a paper by Mr Glaisher
in the Cambridge Philosophical Transactions, Vol. xii.
Thus it follows that of the quantities A^, A^, A^,... those
in which the suffix is an even number are zero, excejat A^
which is —I, and those in which the suffix is an odd number
are determined by
^^"-^"^ ^^ I2n'
We have then the following result:
By the aid of this we may calculate apj)roximately the
value of the definite integi'al 1 cj) (x) dx.
J a
The result may be put for abbreviation in the form
Xcp{x)=C + jJcf>{x)dx-^cf>{x) + ^(}i'{x)-^^
1 h^
where C represents a series of terms independent of x.
28S DEFINITE INTEGRALS.
The series thus obtained for 2<^ (x) will be in general an
infinite series, and as we cannot ensure that the series is con-
vergent the preceding investigation is not rigorous: we shall
return to the subject in Art. 332.
As an example of the last formula take <j> (x) = - , and
h = l. Thus we get by adding - to both sides
111 1/7,1 .1 1 ,
1+2 + 3 + +^=^ + ^"S^ + 2:^-12^^ + -
Hence by making x infinite we infer that in this Example G
is Elders Constant : see Ai't. 268.
EXAMPLES.
1. Evaluate I \ . ,„ . . ,. . Result. — .^,3 ,» " •
Jo X +0X +0 ^0 wo
2. Evaluate cos (a tan x) dx. Residt. -^ e ".
Jo
8. Evaluate x""-'^ e''" dx. Result. -.
Jo ^i
• J {a' cos'^ x + b' sin^ xf 4 \ab' ^ a'bj
IT
5. Prove f V(tan <^) # = -^ [^ + log {V(2) - 1]
Jo V -^ L-'
6. ProveJV(cot(/>)dc^=;^ | + log {V(2) + 1}
7. Find the limiting value of a:e~*' I e^' c7a; when ic = co .
Jo
Result. ^ .
EXAMPLES. 289
^, ^, , f "" COS ax — COS hx , , 7>
8, Shew that ax = losf - .
Jq X ° a
9. If F (x, -j be any symmetrical function of x and - ,
then
9
dx ^ fl f/.P
"-^(--;) ^"-^(--;)
10. If i^(^) be an algebraical polynomial of less than n
dimensions
-:
11. Prove that e-^^^^cos (sin ^) cZ^ = 27r.
Jo
12. Prove that !^— -~,^ =———— when c is indefinitely
Jo 1 - ccos"^ V(2/t) -^
nearly equal to unity, n being a positive quantity.
13. Evaluate f (a cos ^ + ^^ sin 6) log (a cos' O + h sin' ^) (^^.
Jo
Result. 2b {log a - 2 + -,v -^-, , cos"^ ^^ 1 ,
supposing a greater than b.
14. Shew that
f "" , 1 + 2?i cos a.r + if dx
Jo * 1 + 2u cos bx + )i' ' X
is equal to log [1 + n) log — , or log ( 1 + - J log -^ ,
according as n is less or greater than unity
T. I. c. 19
290 EXAMPLES.
15. Find the value of
/,
^
"where a and & are positive, but a and /3 positive or
negative ; and shew that it is wholly real when
a b
IG. Prove that cot^{l — x+x'^)dx = ^— log 2.
r°° dx f i\
17. Prove that ^-^ log a; + - = tt log 2.
18. From the value of dx deduce that of
Jo ^
Result. The two integrals are equal.
19. Prove that J^ ^ j dx = log j ^^^^aia+fc) •
20. Shew that [ " ^f'-'^^f ^« = tt.
Jo (1 + if)
21. Shew that (e~^' - e"-^'') dx=(b- a) sJtt.
Jo
(Sohithns of Senate- House Problems, by O'Brien and
Ellis, page 44.)
/• » e^ + l TT^
22. Shew that / log-^^ — ^ dx = -r .
Jo °e"-l 4
23. Prove that —, . — = log — , and reconcile with
Jo log a; X n
rl x""^ dx
this equation the result of transforming I y-^^ — by
making ic'' = y.
EXMIPLES. 291
24. Shew tliat f 'sin"^ cld = '^. ^ ^ ' .
Jo 2 ^/n + -2\
25. Shew that f'^qi^^P^ = miM
(O + cx)"'' r(; + m) 6"(i-t c)'
20. Shew that f'.SS-" f ^'f :' ^f ^ T (Q r (,„) ^
Jo (a cos- + b sm'^j'^'" 2r (Z + 7?i) a' 6
2/. fcShew that ^ — , . ., ^ =- , — ,
Joacos-0 + hsm-d 2cosA«7r ^-^ ^~ '
n being less than unity.
28. Shew that ^^ ^"^""'^^^ -^^^ ^""^
o(a+^cos^)" r(,0 ^^._^.^.
29. Shew that f" ^"'"' "^^^ =
Jo
( 1 — xr n sm — •
^ 7i
SO. Shew that ' ^ ^^-^ '^
(1 + ex) (1 - xy (1 + c)" sin UTT '
OT oi ,1 , f" sinaa^sin-ra; , tt tt
ol. Shew that dx = Q, or + - , or + - .
^
accordino- to the values of a and c.
S2. Trace the locus of the equation
CO
, shi^cos 6x ,.
y=\ — y — dd.
19—2
292 EXAMPLES.
83. Trace the locus of the equation
I = ["log [1 - 2e"" cos + e-^] d9,
Jo
where u = sin - .
a
34. Trace the locus of the equation
2 cc cos 6 dd
J
2
in which the sign of the square root is always taken so
as to make the quantity in the denominator positive.
35. Shew that
It 77
r2 Ci ... TT^ TT
I I sin ic sin~^ (sin x sin y) dxdy = -j^— -x^
36. Compare the results obtained from
sin ax e'^ dx dy,
by performing the integrations in different orders.
37. Find the value of e "' -*" dx, and hence shew that
Jo
---— 7 oa \/7r „ / fx a\ -— .-> 7
e «- •^- dx = — tV- =0 -2 - ^ e "- ■^- (7a-.
4e" J Va ic /
38. Shew that
JJV(l + ^^ + 2/^) -^ 4U V'
the integral being extended over all the positive
values of x and y which make x^ + y" not greater than
unity.
r /x a
Jo W ^^
EXAMPLES. 293
39. Shew that
n+l
■i
dx chj dz ... IT
9,
the number of variables being n, and the integration
being extended over all positive values which make
«* + 2/' + -"' + not greater than unity.
40. If A,-^A^x^A^x^^ ^F{x),
and a^ + a^c + a^x^ + = / (x),
prove that A^fi^ + A^a^a? + A^a.^x^ +
= ^/> (^0 + ^(^01 {/(") +/W1 ^^ - ^o«o.
where u = xe^^'^~'^^ and v = xe~^'^^~'^K
41. If the sum of the series a^ + a^x + a,jK^ + can be
expressed in a finite form, then the sum of the series
a/ + ttjV + a^x*' + can be expressed by a definite
integral. Prove this, and hence shew that the sum of
the squares of the coefficients of the terms of the expan-
sion of (1 + xy when n is a positive whole number,
may be expressed by
Q2n+2 r —
'" i'^'ecos'nede-i.
«r5-
^ cos-"<
Jo
42. Shew that
cos ex dx TT f e'
Jo
Fc +
'o 1 + cc^' 2 [1 + 0-' ' l + C-
Shew that
/,
(ji (sin 2x) cos xdx=\ (p (cos'^a;) cos x dx.
Jo
(Liouville's Journal de 2Iathe'niatiques, Vol. XVili.
page IGS.)
294< EXAMPLES.
2 ■•
X X
2 n
TT.
44. Shew that 1 — ^5 + -^i —
cos [x sin y) dy.
"JO
45. Shew that
r^m-i Q-x^dx [ /-"'-' e-^" dy = — ^
4G. Shew that
r (,2cos29+2^,s>n20)COS f , ^.^^ ^^ + i^^ COS 2^ dx
J -co sm ( 2x J
7U7r
?i sin —
n
1 „cos ,^ .
= 77-6-" . ^ + a);
sm
TT
^ being comprised "between the hmits + -j .
rx+i
47. Shew from Art. 2G7 that I log F (.r) c/o; is equal to the
J X
limit when w is infinite of - log -^F {nx) (27r) 2 7?^'-
48. Hence by the aid of Ai't. 282 shew that
rx+l 1
log T (x) dx = xlog X — X + ^ log 27r.
295
CHAPTER XIII.
EXPANSION OF FUNCTIONS IN TEIGONOMETEICAL SERIES.
305. The subject ^xe are about to introduce is one of
the most remarkable applications of the Integral Calculus,
and although in an elementary work like the present, only
an outline of the subject can be given, yet on account of the
novelty of the methods, and the importance of the results,
even such an outline may be of service to the student. For
fuller information we may refer to the Differential and Integral
Calculus of Professor De Morgan, and to Fourier"s Theurie...
de la Chaleur. The subject is also frequently considered in
the "vvi'itings of Poisson, for example, in his Traite de Meca-
nique,\ol. i. pp. 64-3. ..653; in his Theorie...de la Chaleur; and
in different 5lemoirs in the Journal de V Ecule Pohjtechnique.
The student may also consult a Memoir by Professor Stokes,
in the 8th Vol. of the Camhridge Philosophical 'Transactions,
a Memoir by Sir W. E. Hamilton, in the 19th Vol of the
Transactions of the Roijal Irish Academy, and a Memoir by
Professor Boole, in the 21st Vol. of the same Transactions.
80G. It is required to find the values of the m constants
A^, A.^, A^,...A^^, so that the expression
Aj^ sin X + A^sm2x + A^sin Sx + + -^m ^^^ ^^
may coincide in value with an assigned function of x when x
77"
has the values 6, 26, Sd,...m9, where d = - — — ^ .
m + 1
20G EXPANSION OF FUNCTIONS
Lety(j:) denote the assigned function of x, then we have
by hypothesis the following m equations from which the
constants are to be determined,
f{e) = A^ sin e + ^2 sin 16 + A^ sin 3^ + +^„, sin mO,
f (2^) = A^ sin 2^ + A^ sin 4^ + ^3 sin 6^ + + A,, sin 2m^,
f{rii&) = J.,sinm^+^2sin2??z^ +^3sin 3?w^+ +^,,^sin mmO.
Multiply the first of these equations by sin rQ, the second
by sin 2r^, , the last by sin 7???'^; then add the results.
The coefficient of A^ on the second side will then be
sin rQ sin sO 4- sin %'d sin IsO + + sin mrd sin msO ;
we shall now shew that this coefficient is zero if s be different
from r, and equal to \ {m + 1) Avhen s is equal to r.
First suppose s different from r.
Now twice the above coefficient is equal to the series
cos (r — 5) ^ + cos 2 (r — s) ^ + ■\- qq^ m {r — s) 6 ,
diminished by the series
cos (r + 6^) ^ + cos 2 (r + s) ^ + + cos m {r + s) 6.
The sum of the first series is known from Trigonometry
to be equal to
sm [zm + 1) ^ — ^ sm -- — ^—
9 ■ {r-s)d '
2 sm-^ — j^— -
{r-s)e\ . (r-s)d
sm -({i — s) TT — - — ^ — Y — sm
^ I 2
that is, to ^ / N /I — — ^ — .
. (r -s)0
2 sm -^^ — A-^ —
This expression vanishes when r — s is an odd number,
and is equal to — 1 when r — 5 is an even number.
The sum of the second series can be deduced from that of
the first by changing the sign of s; hence this sum vanishes
IN TRIGONOMETRICAL SERIES. 207
"vvhen r + 5 is an odd munber, and is equal to — 1 wlicn r+s
is an even number.
Thus Avheu 5 is different from r, the coefficient of Ag is
zero.
When s is equal to r, the coefficient becomes
sin^ r6 + sin" 2rd + + sin^ mrd,
that is, 9 — 9 ] cos 2r6 + cos 4r^ + + cos 2mr6 1 .
And by the method already used it will be seen that the
sum of the series of cosines is — 1 ; therefore the coefficient
of ^1,. is h {m+1).
A =
Hence we obtain
2
m +
- sin ref{6) + sin 2r6'/(26^)+ + sin mref{md) ,
and thus by giving to r in succession the different integral
values from 1 to m, the constants are determined.
Now suppose m to increase indefinitely, then we have
ultimately
2 f'^
A^ = - \ s'mrvf{v) dv.
TT J Q
TTJo
And as / (cc) now coincides in value with the expression
A^ sin x+ A^ sin 2x+
for an infinite number of equidistant values of x between
and TT, we may write the result thus
f(x) = - Zf sin nx I sin nvf (y) dv,
where the symbol S" indicates a summation to be obtained
by giving to n every positive integral value.
307. The theorem and demonstration of the preceding
Ailicle are due to Lagrange ; we have given this demonstra-
298 EXPANSION OF FUNCTIONS
tion partly because of its historical interest, and partly because
it affords an instructive view of the subject. We shall how-
ever not stop to examine the demonstration closely, but pro-
ceed at once to the mode of investigation adopted by Poisson.
308. The following expansion may be obtained by ordi-
nary Trigonometrical methods :
= 1 + 2A cos ^ ^
■(1),
^ ; IT iV — X) T ^ I
1 - 2/i cos —^ — ^ + /i*
V
+ 2/r cos — — + 2/i^cos — —, -+ . .
h being less than unity, so that the series is convergent.
Multiply both sides of (1) by (f>{v), and integrate with re-
spect to V between the limits — I and I; also make h approach
to unity as its limit. On the right-hand side the dilferent
powers of h become ultimately unity. The numerator of the
fraction on the left-hand side will ultimately vanish, and thus
the integral would vanish if the denominato?' of the fraction
were never zero. But if x lies between — I and I, the term
cos — ~ will become equal to unity during the integra-
tion, and thus the denominator of the fraction will be (1 — /i)^,
and will tend towards zero as h approaches unity. Hence the
integral will not necessarily vanish; we proceed to ascertain
its value. Let v — x = z and h=l —g, then
{l-h^)4>{i^dv Cg{l + h)4>{x + z)dz
^7 TriV — x) ,,, J „ ... „1TZ
1 - 2/i COS — ^-v — '- + h' J g" ^ 4/i sm' ^
Now the only part of the integral which has any sensible
value, is that which arises from very small positive or nega-
tive values of z\ thus we may put
. irz _iTZ
and (j){x +z) = (ji {x);
IN TRIGONOMETRICAL SERIES. • 299
and the intesfral becomes
"O'
9{i+J0<f>{^) — ~p=2^</.(a;:
i;'
2l(b (x) , _, irz
tan —r
IT gl
Suppose a and — /3 to be the limits oi z', we thus get
?^^|tan-^^tan-!^l.
TT I gi gl]
Hence, finally, when g is supposed to vanish, we have
2/0 ix). Therefore if x lies between — I and /,
4> W = lif <i> W dv + '^j Sr|' (^ {v) cos '''' ^''~ '''^ dv ...(2).
If however x = I or — /, then the integral on the left-
hand side has its sensible part when v is indefinitely near to
I and —I; we should then have to perform the above process
in both cases, but the integral with rosiject to z would only
extend in the former case from — /S to 0, and in the latter
from to a. Hence instead of 2l(J3 {1} on the left-hand side,
we should have 1(f) [l) + I(f> (— I) ; and instead of (f) [x) on the
1 1
left-hand side of (2) we should have ^ ^ (^) + ^^ (— 0- Thus
we have determined the value of the right-hand member
when X hes between I and —I, both inclusive; its value in
other cases can be determined by the method which will be
explained hereafter in Ai't. 321.
809. In the same way as the result in Art. 308 is found,
we have, if we integrate between and I,
^ (^^ = \i\\^ ^'^ ^' + 7 ^^ \y ^'^ ^"^ """ V ""^ ^' ^^^'
this holds if x has any value between and I; but when
a; = the left-hand member must be ^ </>(0), and when x = I
the left-hand member must be h (f>{l). Thus we have deter-
mined the value of the right-hand member when x lies
800 EXPANSION OF FUNCTIONS
between and I, both inclusive ; its value in other cases
can be determined by the method which will be exjalained
hereafter in Art. 821.
Similarly
Jl] ^(^)^^"+I^"j </'Wcos ^^ ' dv (2);
this holds for any value of x between and I, but when
.T = the left-hand member must be ^ (0), and when x = 1
the left-hand member must be | ^ (/).
From (1) and (2) by addition
<l>{x) = \ Ccf> {v) dv + j S;° cos^ f^cos ^<^(t;)(^v...(3).
L J b b J Q b
This holds for any value of x between and I, both in-
clusive.
From (1) and (2) by subtraction
'^ 7? TT T* r Tinr })
(*') = 7 ^r s^^ ~/~ s^^ "~r" 'Pi'v) dv (4).
This holds for any value of x between and I both exclu-
sive ; and when x=0 or I, the left-hand member should be
zero.
Equation (4) coincides with Lagrange's Formula.
We may observe that either of the formulae (3) and (4)
may be deduced from the other. Suppose we take (8) aud
write sin —j- (f) [x] instead of {x). Thus
TTX X r . TTV
sin -Y <f>{x) = -j j sin -y ^ (v) dv
2.^00 nirx [^ nirv . ttw . . , ,
-H -J 2j cos -y- I cos — ,- sm -j (p [v] dv.
■^.-r nirv .TTV 1 . (n + l)'rrv 1 . (n—l)'rrv
JN ow cos — , sm -T- = ^ sm -^ j-^ ^ sm j ;
IN TRIGONOJEETRICAL SERIES. 301
and therefore it will be found that the result may be exhibited
thus,
sin —j-<p[x) =
I
1 V=o f (n — VjTTX {n + l)T7x\ (^
1 sin — ,- ^ [v) dv ;
, (n—l^TTX (n-]rV\'jrx _, . mrx . irx
also cos , cos-^ J- — = I sm -y- sm -y ;
ITX
and then by division by sin—,- we obtain the formula (4).
For another investigation of the fundamental theorems
we may refer to Chapter xviii. of the Treatise on Laplace s
Functions. We will now give some examples.
310. Expand cc in a series of sines. Take formula (4) of
Art. 809, and suppose Z = 7r; then
/
, V cos nv sm nv
V sm nvav = — H
n n^ '
therefore I v sin nv dv = - if ?i be odd, and if ?i be even,
j W 71
Thus
x = 2 [sin x — ^ sin 2x + 1 sin 3^ — ^ sin 4a; + ].
This holds for values of x between and tt, and as both
sides vanish with x it holds when x = 0; and it is obvious
that if it holds for any positive value of x it holds for the
corresponding negative value; hence it holds for values of x
between — tt and tt, exclusive of these limiting values.
311. Expand coscc in a series of sines. Take formula
(4) of Art. 309 and suppose i = tt; then
I cos V sin nv dv = ^ j [sin (?i -{^1) v + sin (n — 1) v] dv
1 [cos (n + 1) V cos (n — l)v
?H- 1 71—1
802 EXPANSION OF FUNCTIONS
therefore I cos v sin nv dv =0 if n is odd,
Jo
2n
if n is even:
?r — 1
therefore
2 (4 . „ 8 . , l + (-l)" • • 1
cos a; = - -", - sm 2x + v^ sin 4« + . . . H ~ — -~ w sm w^ -f . . . ^ .
TT [3 lo 7r — 1 j
This holds from x = to cc = tt, exclusive of these limit-
ing values.
812, Suppose "we endeavour to expand a constant quan-
tity in a series of sines. Denote the constant by c ; then
]3utting c for ^ (v) in formula (4) of Art. 309, and supposing
I = IT, we obtain
4c f . 1 . 1 . 1
c = — \ sin X + Ti sin Sx + ■:; sin 5.r +•••-■ .
TT ( d o j
Hence dividing by c we obtain
TT . 1 . „ 1 . ,
-;- = sm 0) + -: sm 6x + ~ sm oj? +. . .
4 3 o
This holds from ^ = to a; = tt, both exclusive.
If we put ■^ — y for cc, we obtain the following formula
which holds from y = — ^toy=^, both exclusive,
TT 11^
2" = cost/ — ^ cos 3y + ^ cosoy — ...
813. Expand a: in a series of cosines.
Take formula (3) of Art. 809, and suppose Z = tt; then
/
- V sin nv cos nv
VC03 nvav= 1 o — ',
n n
r . 2 .
therefore I v cos nv dv = if n be even, and s if w be
Jo n-
IN TRIGONOMETRICAL SERIES. 303
.2
odd; and I v dv = -^
thus x = -r [cos a; + ;^ COS 3a; + -rij COS 5a; + ].
This holds from a; = to .r = tt, both inclusive.
If we put a; = ^^ — y, we obtain the following formula,
which holds for any value of y between — - and -^ , both
inclusive,
4 1 1
2/ = - {sin y - g^ sin 3y + ^ sin 5y - . . . ] .
314. Expand e""" in a series of sines.
We shall obtain
e = - 2.. —. 5 (1 — cos nire'^n sm nx.
This holds from a; = to a; = tt, both exclusive.
315. Expand e'^^ in a series of cosines.
We shall obtain
gftTT _ 1 2a ^« cos WTre*'" - 1
avr TT a" + n •
This holds from a; = to ic = tt, both inclusive.
316. Expand sinao; in a series of sines, a not being an
intesjer.
We shall obtain
TT sin ax sin x 2 sin 2x 3 sin 8,«
+ —
2 sin aTT 1'^ — a" 2^ — a' o' — a'
This holds from x = to a; = tt, the former inclusive, the
latter exclusive.
304 EXPANSION OF FUNCTIONS
317. Expand cos ax in a series of cosines, a not being
an integer.
We shall obtain
TT COS ax 1 a cos x a cos 2x
This holds from a; = to a; = tt, both inclusive.
318. Expand e^^ — e"*^ in a series of sines.
Here (e«" - e""'') sm ?iv ctv = ^-^ 3 — -
cos nTT.
„, „ TT e'^^ — e~"^ sm cc 2 sm 2x 3 sm 3;^;
ihereiore ^ — = jtt, 5 — ^r^ 5- + -7T2 2 ~
319. Expand e<'('^-^' + g-at'^--*-) in a series of cosines.
Here I (e'^''^"''' + e-^'""^''] cos ?2vcZy =
io
a + n
pirn __ p—aiT
fir gan- _
and |ea(7r-r) ^ g-a(:r-i-)| ^y ^
Jo o,
^, „ IT e^t"^-^) + e-«(''-^) 1 cos 5; cos 2x
Thereiore ^ v- „„ = ^^^ + ,2""; — ; + 1^2~, — 2 +
2(X e«T_ e-a'T 2a 1 4- a 2^ + a
320. It may be observed that from the formulae -svhich
have been given others may be deduced by integration ; and
in general the series thus obtained are more raj)idly conver-
gent than those from which they were deduced.
For example, take the formula for cos^ in a series of
sines given in Art. 811; integrate, thus
TT . cos 2x cos 4a; cos Gx
-■ sin X ■— constant z. — ^ ^ — = =— ^^ . . .
4 1.3 3.0 0.7
By putting x=0, we find that the constant is \. The result
agrees with what we should obtain by expanding sin a; in a
series of cosines.
IN TRIGONOMETRICAL SERIES. 305
As another example wo may take the last result of
Art. 313, aud integrate both sides with respect to y. The
IT
constant may be determined by putting — for y : thus
^ = g — - j cos y - -^^ cos oy + ^ cos 0?/ - . . . ,^ .
821. We have shewn that the formula (3) of Art. 309
holds for any value of x between and I both inclusive;
it is easy to determine what the right-hand member is equal
to when x lies beyond these limits. Suppose os positive, and
between I and 21 ; put x = 21 — x' so that x is less than /,
then
HTTX /- mrx'X niTX
cos —y- = cos I 'Lmr ^ — 1 = cos — ,~ ,
therefore the value of the right-hand member is (j) (x). Next
suppose X gTeater tlian 21; and suppose it equal to 2ml + x',
where x' is less than 21; then
HTTX niTX
cos —r- = COS
SO that the value is the same as it would be if x were put
instead of x; that is, the value is {x) if x be less than I,
and (f) (21 — x) if x be greater than I.
It is obvious that for any negative value of x the value is
the same as for the corresponding positive value.
Similarly we may shew that if x is positive and =2ml + x,
the value of the right-hand side of equation (4) of Art. 309 is
the same as if x were put instead of x, and is <^ (x) if x' be
less than I, and —<^{2l—x) if x be greater than I. And for
negative values of x the value is the same numerically as
for the corresponding positive value, but with an opposite
sign.
322. It maybe observed that in the fundamental demon-
stration of Art. 308, we suppose that when A aj)j)roaches unity
as a limit, the expression
k (p {v) cos —J ^ dv
T. I. c. . 20
306 EXPANSION OF FUNCTIONS
may be replaced by
f 7?7r (v oo)
I ^ {v) COS —J — — dv,
however large n may be. We may shew that no error arises
from this supposition, by proving that the latter integral
vanishes when n is increased indefinitely. We have
/* , , , mr (v — x) ^ J(J}(v) . nir (v — x)
6 (v) cos — ^— ^ do — ^ sm ^^^
J i irn I
I [ ,, , . . niriv — x)
which shews that the intecfral on the left-hand side will vanish
when n is infinite, at least if (f)' (v) be not infinite.
323. We have not yet alluded to one of the most re-
markable points in connexion with the formulae (3) and (4) of
Art. 309. lu these formulge (f> (x) need not be a continuous
function; for example, from x = to x = a we might have
(f> {x) —f\ (x), then from x = a to x = b we might have
[x) =f^ (x), then from x = b to x = c we might have
(f) (x) =f^ (x), then from a? = c to x = l we might have
^ {x) =f\ (x). The formula (3) for instance would still be
true for all values of x between and I inclusive, as is evident
from the mode of demonstration, except for the values where
the discontinuity occuis. When for example x = a, then the
value of the right-hand member would not be f^ (a) or f^ (a)
but i {./^i W +/2 (^)}- If therefore for x^a we have
fi (^) —f-i i^)> ^^® formula holds also when x = a.
Some writers adopt a mode of expression for such a
formula as (3) of Art. 309 which draws attention to the pos-
sible discontinuity. Instead of (f) (x) on the left-hand side
they put ^ {cf) {x + e) + (j) (x — e)], where e represents an inde-
finitely small positive quantity. Thus when there is no dis-
continuity the limit of <f)(x + e) is </> (x), and so also is the
limit of <p {x — e). But suppose that when x = a we have the
discontinuity just indicated ; then the limit of (p (a + e) is
Jl (a), and the limit of cf) (a — e) isf (a).
324. Find an expression which shall be equal to c when
X lies between and a, and equal to zero when x lies between
a and I.
IN TRIGONOMETRICAL SERIES. 307
Take formula (8) of Art. 300. Here </> (r) = c from v =
to v=a, aud then from v = a to v = 1 it is zero ; therefore
cos -^- 6 (v) av becomes c I cos -^- dv that is — sm — j-
Jo t Jo I nir i
therefore the required expression is
ca 2c ( . ira irx 1 . ^ira ^ttx
-— H — -^sm -y- cos -,- + - sm — r- cos — y-
l IT [ L I z L L
1 . 37ra ^TTX 1
H-gsm-^cos-^- +...';
this will give \c when x = a.
Or we may use formula (-i) of Art. 309. Then
f"' . nvTT , d /, na7r\
c sm —7- dv = — 1 — cos — J— ,
Jo i n7r\ I J'
and we have for the required expression
2c f 7ra . TT^ 1 27ra . 2ttx
— \ vers -y- sm -^ + - vers —r- sm -r-
TT ( t I 'I L L
1 Svra . Zirx )
+ ^ vers — ;— sm
3 '^'^ I ^^" I ^ I'
this gives when x=0, and -|c when x=a.
325. Find an expression which shall be equal to kx from
a; = to ^ = ^ , and equal to k (l — x) from x = - to x = 1.
u 2
Here
I
9 \v) COS — j- dv = ^y cos -y- ay + ^- (i — ?;) cos -7— dv
Jo t J o J I I
2
nrr
k]^(l . nir 1 WTT 1 ] H' , .
= ^ -^ s- sm -^ + -2- cos -?i 5- \ -\ sm mr — sm
TT [271 2 wV 2 wVj ?i7r
TT
T T COS -r-
1 . 1 . ?i7r cos7i7r 2
- sm nir — ;r- sm — - + — r, n—
kF { nir ^
20—2
808
EXPANSION OF FUNCTIONS
This is ^„ wheu n
TT n
other case, and
is of the form 4r + 2, and in every
j){v) dv = h \ V dv + h
{l-v)dv=~',
thus the required expression is
TT -I
1 "llTX 1
1, COS ~~i 1- ^ COS
I
Qttx
+ .
If we denote this by y, then from x = to x = \l both in-
chisive y = kx, and from x—l^l to x = l both inclusive
y =k{l — x); for values of x greater than I the values of y
recur as she'wn in Art. 321. Thus the value of y is the
ordinate of the figure formed by measuring from the origin
equal lengths along the axis of x to the right and left, and
drawing on each base thus obtained the same isosceles triangle.
As another example we may propose the following :
find a function ^ {x) in terms of sines which shall be equal
to X from ic = to x= a, then be equal to a from a; = a to
x = 7r— a, and then be equal to tt — a; from x = '7r — u. to x^tt.
The result is
4 f . 1 . 1 .
(b(x) = - -^sin a sin x + -xr7, sin 3a sin Sx+ ^, sin 5a sin ox +
TT { 6' o
this is true from a; = to ^^ = tt both inclusive.
We may give the following y
geometrical interpretation of this ^j q
result :
Let OACB be a square, such
that OA = TT, and OB = ir. Take
for the origin, OA for the o A
axis of X, and OB for the axis of
y, and let the axis of z be at right angles to the axes of x
and y. Let a pyramid be formed having OA CB for its base,
TT TT TT
and its vertex at the point x==-^, y = -^ , z= -r: then the fol-
^ Zi Id
IN TRIGONOMETRICAL SERIES. 309
lowing equation represents the four faces of the pyramid
which meet at the vertex,
z= - {sin a; sin ?/ + ^ sin 3^ sin 3^ + ^^sin 5a; sin 5?/+ ...},
By the mode of obtaining the result it apphes to that j)art
of the surface for which y is less than - ; and then by in-
spection we see it applies to that part of the surface for
TT
which y is between ^ and tt, "We may conveniently put
I + 2 for X, and ?; + - for y.
The student may verify the following examples.
If X be numerically less than a the exjDression
tn + 1
is equal to a — a: if ic be positive, and a + a? if ^ be negative.
Prove that for values of x between — vr and ir inclusive
x^ _ TT^ COS 1x cos 3x
__--_cosa; + -2^ 3^"^
This may be obtained from Art. 310 by integration ; or
from equation (3) of Art. 309. Integrate this result : thus
x^ TT^x _ . sin 2a; sin 3a;
Find an expression in terms of sines which shall be equal
to sin — from a; = to x = a, and equal to from a; = a to
cc = TT. The result is
fsin a sin x sin 2a sin 2a; sin 3a sin 3a;
TT'^-a'^ "^ 7r''-2V ^ 7r'-3V
Find an expression in terms of cosines which shall be
equal to -j — x^ from a; = to x= ^, and equal to from
310 EXPANSION OF FUNCTIONS
X
"2
to X
= 77.
The result is
TT"
12
4
■f -■
TT
COS X-
cos 3^
-1-
cos5j?
f .. ..
fcos 2x
■ z ■< = —
COS 4ic cos 6x
826. Other formulae may be given analogous to those in
Art. 809 ; we will here investigate some. We have by Art. 809
<j,ix) = ^^j' <P (v) dv + '^j^rf ^ {v) cos '"'^ ^Y ^^ dv. ..{!).
This holds when x has any value between and I ; but
when x = the left-hand member must he^cfi (0), and when
x = l the left-hand member must be ^(}>(l)- In the same
manner as this result was obtained we may also prove that
-^ (^•) = ^ijl ^ (^) ^^^ + 7 ^^ !l ^ (^) ^°^ ^^^^^-^—^ dv... (2).
This holds when x has any value between and Z; but
when a; = the left-hand member must be <^ (0), and when
x = l the left-hand member must be ^ (l).
Subtract (1) from (2); thus
(}){x)=-j2^ I (f){v)cos^ 2^ dv (.3).
This holds when x has any value between and I; but
when x = the left-hand member must be ^(/) (0), and when
x= I the left-hand member must be |(^ (I).
Now in the same manner as (8) was obtained, we may
obtain the following result, starting with v + x instead of
V — X,
A lv<»/■^/^ {'^n-l)'7r(v + x) . ,,.
^ = -^^Tj </)Wcos^ — "IT ^^'
This holds when x has any value between and I ; but
when x = the left-hand member must be -^^ (0), and when
X = I the left-hand member must he —^(f) (l).
IN TIUGONOMETRICAL SEMES. 311
From (3) and (4) by addition and subtraction we obtain
^W = ? Sr oos(?^il^/VwcosS!i^^„...(5),
* W = I Sr sin ^?^^/ V {") - e^i:i/)^" c;„.. .(G).
These hold when x has any vakie between and I in-
clusive, except that when a; = the left-hand member of (6)
must be 0, and w'hen x = l the left-hand member of (5) must
be 0.
As an example of (6) we have
TTX . X 1 . 3a' 1 . 5w
_ = sm--g,sm-^ + ^sm^-...;
this coincides with the last result of Art. 313.
827. We shall apply the formula (5) of the preceding
Article to establish a remarkable theorem first given by J ohn
Bernoulli. Let there be any curve AB the tangents of which
at A and B are at right angles ; let the involute of this curve
be formed beginning at A, and denote it by AC; let the
involute of AC be formed beginning at C ; and so on con-
tinually: then the ultimate figure obtained will be a cycloid.
Let s be the length of the arc of the original curve mea-
sured from A to any point P; let p be the radius of curvature
at P, and 6 the inclination of the tangent at P to the tangent
at A. Let p^ be the radius of curvature at the corresponding
point of the first involute, p^ that of the second involute,
Pg that of the third involute; and so on. Then 6 expresses
the inclination of p, p^, p^... to the normal of the original
curve at A; and 6 also expresses the inclination of p^, p^,
p,,... to the nori:pal of the original curve at B. Moreover
p,, ^3, Ps,... vanish when 6 = 0; and p^, p^, Pg,... vanish
when ^ = 9 .
ds [^
Now p= ^> ^^^ Pi=^> thus p^= i p dd.
312 EXPANSION OF FUNCTIONS
TT
f2
Similar! V, p.,= p^d6,
' Jo
P3= I PM
Jo
P.- [%/i0
and so on.
Now in formula (5) of tlie preceding Article suppose
i= ^ ', then since p is some function of 6, we have
p = A^cos9 + A^cos39 A- A^cos5$ + ...
where A^, A^, A^,... are certain constants determined by that
formula (5).
Thus
p^ = A^ sin ^ + - ^3 sin 3^ + - A^ sin 56 +
o o
P^= A^ cos 6 + — J3 cos 3^ + -2 --^5 cos 06 +
o O
/03 = ^4jSin^+-r^J[3sin3^ + Ti-4^sin 06 +
o
Proceeding thus we obtain, when n is indefinitely large,
p^ = A^smd, or p„ = A^cos6;
and these equations represent a cycloid; see Ai't, 105.
It should be observed that the formula which we have
used for p assumes that p vanishes when 6=^ '• see Art. 326.
But this does not really affect the demonstration ; for by the
nature of the problem p^ really does vanish when ^= -, and
therefore a formula for p, like that given for it will hold,
IN TRIGONOMETRICAL SERIES. 813
and the process can then be continued by which p^, p.,,... are
successively obtained. In Art. 102 it is shewn that the in-
vohite of an Equiangular Spiral beginning from the pole is an
Equiangular Spiral ; but close to the pole this curve forms an
infinite number of coils, and this singularity renders our pre-
sent investigation inapplicable : thus the apparent contradic-
tion between the result obtained in Ai't. 102 and the theorem
here investigated is explained.
We may next examine the nature of the result when the
tanjrents at the extremities of the orioinal curve are not
inclined at a right angle. Suppose these tangents to be in-
clined at an angle a; and j^ut a for I in the formula (5) of
the preceding Article. Then we have
p = A^cos ~+A^cos-^- + A^cos-^ + ;
and by proceeding in the same way as before we arrive at
the result
Pn = ^ cos — , or p^ = k sm ~ ,
where k = A
2a ■ - Z.X
'- TT
If k were a finite quantity, we should thus obtain an
epicycloid if a is greater than — , and a hypocycloid in which
the diameter of the revolvinfj circle is less than the radius of
TT
the fixed circle if a is less than - ; see Arts. 110 and 111;
and this is the usual statement of the results. But it will
be observed that h becomes indefinitely great in the former
case and indefinitely small in the latter case ; so that in the
former case the radii of the fixed and revolving circles must
be supposed to increase indefinitely, and in the latter case to
diminish indefinitely.
328. Suppose a, h, and J — a to be positive quantities.
Consider the double integral I 1 cos ux (p(v) cos uvdudv.
814 EXPANSION OF FUNCTIONS
By integration by parts we have
r , , . , <p (v) sinwy fcb' (v) sin uv ,
(v) cos uv dv = — I ay;
J ^ u J u
therefore
f * , , . , <^ (&) sin w5 <i (a) sin ^/a
I (/) [v) cos MV at; = ^—^^ ^-^-^
it u
r^ 4>' (v) sm uv ^^
Ja U
Thus tlie proposed double integral becomes
, ,,. r°° cos tta; sin w6 , , , , /■°° cos Ma:; sin wa ,
i> (}) (^^- </> (a) du
J U J u
cos ux (f)' (v) sin uv , ,
—^ du dv.
Ja U
The first and second terms may be easily found by
Ai't. 285. In the third term we can chan^je the order of
integration, and apply Art. 285 to fiud the result of integra-
tion with respect to u. We shall then obtain the following
results, assuming cc to be positive :
I. Let X be greater than b. Then each of the three in-
tegrals vanishes.
II. Let X be between a and b. Then the first term is
TT
equal to - ^ (h) ; the second term is zero. And, by Ai't. 285,
r°° cosM>rsin»<u , . , , 7r „ , „ , . ,
du IS equal to - tor values oi v which are
Jo u \ 2
greater than x, and zero for other values of w ; so that when
we multiply this expression by </>' {v) and integrate witli
TT 77
respect to v, we obtain - cf) (b) — - (f) (x). Thus on the whole
we have
|c^(6)-||0(&)-^(^(^)|,
TT
tliat is ;^ (oo), as the value of the original double integral.
IN TRIGOXOMETRICAL SERIES. 815
III. Let X be less than a. Then the first term is ^ <^ v^).
the second term is | </> (a), and the third term is ^ {<^ (&) - 9 («) ] •
Thus on the whole we have
l4>{h)-l<^{a)-l{^[h)-cl>[a)],
that is zero, as the value of the original double integral.
TT
Hence finally the double integral is equal to or to ^ ^ (^),
according as x lies beyond or within the limits a and h.
TT
It may be conjectured that ii x=a the value is 7 ^ (cf),
and \i x = h the value is ~ ^(6) ; and this conjecture is easily
verified.
If X is negative tlie value of the double integral is tlie
same as for the corresponding positive value of x; since
cos (— ux) = cos ux.
329. In like manner supposing a, h, h — a, and x to be
positive we can shew that s,mux<^{v)&\nuvdudv has
the same value as the former double integral. If x is nega-
tive the value is numerically the same as for the correspond-
ing positive value of x, but of contrary sign ; since
sin (— ux) = — sin ux.
330. By combining the results in Arts. 328 and 329 we
obtain the following. If a, b, b — a, and x are positive
r" rb
I (f){v) cos u[x— v) ill dv
J J a
is equal to or to tt^ (x) according as x lies beyond or with-
316 EXPANSION OF FUNCTIONS
"TT . , . , TT
ill the limits a and h ; and is equal to ;- </> (a) and ^ ^ (6)
respectivelj at the limits.
This result admits of extension. The limitation that x Is
to be iwsitive may be removed: for, by virtue of the remarks
at the ends of Arts. 328 and 329, If x is negative, so that it
is beyond the limits a and h, the double integral vanishes.
Again, suppose that a and h are negative quantities : put
a = — h, and h = — k; also put v = —v , and x = — x. Tlieu
I cos u {x — v)dv = — \ cos u {x — v) dv = I cos u {x — v) dv,
J a J h Jk
where h — k is positive.
331. In this way we find that, \i p — q be positive,
\ <f)(v) cos u {x — v) du dv
JO J q
is equal to or to ircj) (x), according as x lies beyond or
TT
2
^ ^ (q) respectively at the limits.
The result just enunciated may be called Fourier s Theo-
rem ; this name however is usually applied to that case of
the general formula in which we suppose q = — oo, and ^^ = oo ;
we have then for any finite value of x
(J3 (x) = ~ I 1 ^ (y) cos u (v — x) du dv.
TTJo J -Qo
Poisson has given a demonstration of the last result, which
we will now reproduce. Take the formula
4> i^) = ^J_^ ^ (v) dv+] ^-f ^cos "^ZA ^ (^) dv •
within the limits ^j and q; and is equal to g^(j>) and
TT
TT , 7177
put ~j = h, -J- = nil = u; thus we have
1 r^ . , . , 1 ^ f r^
</>W=2|| (^(v)fZy+- S jl Go&u[v-x)j>{v)dv[h,
IN TRIGONOMETRICAL SERIES. 317
u being a mnltiple of h, and the summation denoted by 2
extending for all values of u from /i to co , But if I becomes
indefinitely great the difference h of successive values of u
becomes indefinitely small, and the sum denoted by S be-
comes an integral taken with respect to u from u = () to
w = 00 . Thus if we make I— x , and put du for h, and the
sign of integration instead of 2, and suppose (v) is such
I ri .1
that ^, 6 [v] dv vanishes with -j , we have
"1 /• 00 /«00
(b(x) = - \ I cos u (v — x)<i> [v) du dv.
332. We shall now return to the subject introduced in
Art. 804, and shall give another demonstration, due to
Poisson. of the formula there obtained.
In Art. 308 we have obtained the following result:
\[<^{j)-\-c^{-i)]=~^^[^4>{v)dv^\^^'j{v)cos'^:^^
where the summation denoted by 2 applies to the positive
integer r, and extends from 1 to oo .
Change I into ^ : thus
-I /•2 2 r^
cos
2 2
Change ^ [v) into </> {Id + v) ; then the result becomes
l\ . /„ l_
2
y I <^ [U + v) dv + jX \ (j) {Id + v) cos 2 dv.
' ' 2r7r(v-^
818 EXPANSION OF FUNCTIONS
Put z tor kl + v; then the right-hand side becomes
-I (f) {z) dz + J '^ j (}) (^) cos — J- — '^^^•
ki--^ ki- 2
13 5 2n — 1
Now put for k in succession the values ^ , ^ , ^ . . • — ^ — ,
and add the results ; thus observing that cos— —
I
reduces to cos — r— , we have
-.-0(0) + <^(/) + </)(20+... + <^(7!Z-0 + ^>(»iO
nl Tm^ 9-)
1 f"' 2 p" 27'7r^
= -y (f)(z)dz+j'Z ^{z) COS ^^-j^dz,
tj o J I'
therefore
(j){0) + (f>{l) +0(20 + ... +(f>(^nl-l)
= ]j%(z)dz-l{<p{nl)-<}>{0)] + ^^^ f%{z)cos^-^dz.
It will be seen that this result resembles that of Art. 30-i ;
we shall now compare them more closely.
By integi-ation by parts
■ I 0(0) cos mz dz = — 4){z) sin mz 10' (z) sin mz dz
= — cf) fs) sin mz -\ :, 6' (z) cos mz — -^A 6" (z) cos mz dz.
CVintinue this process, and then take the integral between
n 7 n 2r7r ^,
the limits and )U; put qu tor— - : thus
|%(.) cos ^dz = 1 {0' {nl) - 0' (0)1
_ 1, {0'" Od) - 0'" (0)1 + . . . + ^'^^ (0^-^ {nl) - 0^-1 (0)1
(— i\« fill
+ ^ —„^- 02* (z) cos mz dz.
m-^ Jo
IN TRIGONOMETRICAL SERIES. 319
Now effect the summation with respect to r, and denote
bv S the infinite series 1 +-:ri + -^ + -rz + ... Thus
= ]jy(z)dz-l{<f>(nl)-<f>(0)]
+ El^'("O-f(O)}-S[f"O^O-0"'(O)} + ...
+ 2^.-1^2. - ,-:2. j^ ^'''(^) COS -j-dz.
The fact that this result, up to the last term exclusive,
agrees with that in Art. 30i depends upon the property of
Bernoulli s Numbers involved in the known formula
Cf _ " -"2r-l
The last term in the result just obtained gives us an equiva-
lent in the form of a definite integral for the remainder after
a certain number of terms of the series in Art. 30-i.
The property of Bernoidlis Numbers may be established
thus. Use the formula for sin 6 which is given in Plane
Trifjonometry, Art. 322, take the logarithms, and differentiate
with respect to 6; thus we obtain an exiaression for Scot 9
2 S*
in which the coefficient of 6"'' is ^r- Again we have
TT
COt^ =
26'\/-l
Thus ^ cot ^ = ^ V- 1 + - ;—
the last term can be expanded in powers of 6 by Art. 123 of
i\i& Differential Calculus; and by comparing the coefficient of
6-" with that already given we obtain the required formula.
320 MISCELLANEOUS EXAIVIPLES.
Let V be any integer less than n. The sum of the series
(f){i'l) + ^{vl + l) + ... +(p{nl-l)
may be obtained by subtracting the value of
(l>iO)+(fi{l)+...+cf){vl-l),
from that of <^ (0) + ^ (Z) + . . . + {nl - 1).
MISCELLANEOUS EXAMPLES.
Change the order of integration in the expression
n
(fi {a;, y) dx dy.
2a
Change the order of integration in the expression
'2a rV(4a.f)
<^ {x, y) dx dy.
J \/{2ax-x^)
re rb.v
3. Transform I ^{x,y) dxdy into an integral with
J J ax
respect to u and v, having given u = y + x, y = tiv;
and determine the limits of the new inteo-ral,
4. Transform I \ 4>{x, y) dx dy into an integral with
respect to u and v, having given y -\-cx = u, y = uv ;
and determine the limits of the new integral.
5. Transform j j (x— y) (y — z) {z - x) dx dy dz into an in-
tegral in which u, v, lu are the independent variables,
where u^ = xiiz, -=-^ 1- - , w^ = x~ -\- ir -\- z^.
"^ V x y 2 ^
MISCELLANEOUS EXAMPLES. 321
G. Prove that
U ) --Jo *^ n + af"Y
where t = x" and t = i".
(See Arts. 203 and 66 ; and transform as in Art. 242.)
/T 7J-2
7. Prove that tan ^ log: cot ^ cZ^ = -r^ -
Jo 48
8. Prove by transforming the expression from rectangular
to polar co-ordinates that the value of the definite
integral g-U*+2xVcosa+2/^) ^/.^ ^/^ -g Qonal to
Jo Jo
|- V^ri^ f sin - j , where i^fsin-j denotes a complete
elliptic function of the first order of which sin ^ is the
modulus.
9. Prove that
I e-x->ncot2^ sin (,,^.2 ^ ,^) ^^ ^ g-^ (.^ ^ ^^ //7rsm2/3^^ ^
10. Shew that
TT
f ' tan-^ {« V(l - tan^ x) ]dx = '^ tan"^ " V^ - ? cot"^ 'l(l±^^
' - 2 w
sin (^f tan - j
11- If ./ (B = L ,; n dO, determine the geometrical
J sin u °
meaning of the equation y ='xf{sinx).
12. A curve of double curvature revolves round the axis
of;r; shew that the surface generated
= 27r|v'{(yc/y + ^chy + (/ + z"-) {dxy}.
T. L C. 21
822 MISCELLANEOUS EXAMPLES.
13. Shew that
/.
dx
TT
a' + bx' + x* 2aV(& + 2a)'
, /■ afdx _
^"""^ Jo d' + hx' + x*~2
TT
V(^ + 2a)'
assirming that the denominator of the expression
under the integral sign does not vanish for any real
value of the variable.
14. Find an expression in terms of sines which shall be
TT
9
TT TT
equal to x when x lies between — ^ and — , and shall
77"
be zero when x lies between — tt and — — , or between
^ and TT.
1 ( 1 1 . '
Result. ^ -^ sin 2iC — ^ sin 4a; + ^ sm 6a; — . . . |-
2 f . 1 . „ 1 . .
+ - ^sm a; — - sm Ja; + ^rzr sm ox — .
823
CHAPTER XIV.
APrLICATIOX OF THE INTEGRAL CALCULUS TO QUESTIONS
OF MEAN VALUE AND PROBABILITY.
833. We will here give a few simple examples of the
application of the Integral Calculus to questions relating to
mean value and to probabiliti/.
Let <^ (x) denote any function of x, and suppose x succes-
sively equal to a, a + h, a + 2h, ... a + {ii — 1) h. Then
4> {a) + 4> {a + h) + <^ {a + ^h) + . .. + <f> [a + (n - 1) h}
n
may be said to be the mean or average of the w values
which (f) (x) receives corresponding to the n values of x. Let
h — a — nh, then the above mean value may be written thus,
[ (f> (a) + (fi{a + h)+4>{a + 2h} + ... -f [g + ( y^ - 1) /,}] h
b — a
Suppose a and h to remain fixed and n to increase inde-
finitely ; then the limit of the above expression is
^ (x) dx
a
This may accordingly be defined to bo the mean value of
^ (x) when x varies continuously between a and h.
21—2 '
o24 APPLICATION OF THE INTEGRAL CALCULUS
334. As an example we may take the following ques-
tion ; find the mean distance of all points within a circle
from a fixed point on the circumference. By this enunciation
we intend the following process to be performed. Let the
area of a circle be divided into a large number n of equal
small areas; form a fraction of which the numerator is the
sum of the distances of these small areas from a fixed point
on the circumference, and the denominator is n ; then fintl
the limit of this fraction when n is infinite.
Suppose i\, r„, ... r^ to denote the respective distances of
the small areas ; then the fraction required is
1 r 1
n - "'
Multiply both numerator and denominator by rAOAi^ which
represents the area of a small element (Art. 148), thus the
fraction becomes
{7\ + 7\^+ ...+7\,]rA0Ar
nrAOAr
The limit of the denominator will represent the area of the
circle, that is, ttc", if c be the radius of the circle. The limit
of the numerator will be, by the definitions of the IntegTal
Calculus, llr^cWdr, the integration being so effected as to
include all the elements of area within the boundary of the
circle. Thus the result is
IT
■ 1c COS 9
r' (Id dr
^■^ J a
This will be found to srive
82c
^ '^ 9
TT
335. The equation to a curve is r = c sin 6 cos 6, find the
mean lengtli of all the radii vectores drawn at equal angular
intervals in the first quadrant.
TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 825
It easily follows, as iii tlie last Article, that the required
vican lenqth is
I c sin 6 cos 9 dd
c
, that is, — .
TT vr
2
Again, suppose the portion of this curve which lies in the
first quadrant to revolve round the initial line, and thus to
fjenerate a surface. Let radii vectores be drawn from the ori-
gin to different points of the surface equably in all directions:
it is required to find the mean length of the radii vectores.
The only difficulty in this question lies in apprehending
clearly what is meant by the words in Italics. Conceive a
spherical surfiice having the origin as centre ; then by equable
angular distribution of the radii vectores, we mean that they
are to be so drawn that the number of them which fiill on
any portion of the spherical surface must be proportional to
the area of that portion. Now the area of any portion of a
sphere of radius a is found by integrating a^ iismddcpdO
within proper limits: see Art. 175. Hence a'^ sin 6 Acf) A9
may be taken to denote an element of a spherical surface,
and 27ra^ is the area of half the surface of a sphere. Thus we
shall have as the required result
d'c sin 6 cos 6 sin 6 dcj) dd
'lira:'
the integration being extended over the entire surface con-
sidered.
Hence we obtain
IT
2,1 /-2
['[%sin''^cos^f/</)(Z^
, that is, ~ .
■' ,, , . c
ZTT
826 APPLICATION- OF THE INTEGRAL CALCULUS
836. An indefinitely large plane area is ruled with
parallel equidistant straight lines; a thin rod, the length of
which is less than the distance between two consecutive lines,
is thrown at hazard on the area: find the chance that the rod
will fall across one of the straight lines.
Let 2a be the distance between two consecutive lines
and 2c the length of the rod. It is easily seen that we do
not alter the problem by supposing the centre of the rod
constrained to fall on a straight line drawn between two
consecutive lines of the given system and meeting them at
right angles, for the proportion of the favourable cases to
the whole number of cases remains the same after this limit-
ation as before.
Let the centre of the rod be at a distance x from the nearer
of the two selected parallels ; then sui^pose the rod to revolve
round its centre, and it is obvious that in this position of its
centre the chance that it crosses the straight line is -^ , where
ZTT
A, "27
c cos c}) = X. And we may denote by —^ the chance that
the centre of the rod falls between the distances x and x + Ax
from the nearer of the two parallels. Thus the chance re-
quired will be denoted by the limit of the sum of such quan-
2(6 Ax . . . 2 f X
titles as ^ — that is, it will be — 6 dx, where cos <f) = - .
TT a rrraj^ ^ c
The limits of x are and c ; hence the result
= - — (p smd) ad) = — .
TraJo ' ' Tva
This problem was first proposed by the celebrated naturalist
Buffon, and was afterwards discussed by Laplace: &qq History
of the Mathematical Theory of Probability, Art. 1020.
337. An indefinitely large plane area is ruled with
j>arallel equidistant straight lines, the distance between two
consecutive lines being b ; a closed curve having no sin-
gular points, whose greatest diameter is less than b, is
thrown down on the area : then the chance that the curve
TO QUESTIONS OF MEAN VALUE AND PROBABILITV. 327
will fiill on one of the straight lines is -. , where I is the peri-
meter of the curve.
Let AA be the lonsfest diameter of the closed curve, and
assume that the curve is symmetrical with respect to AA'.
It is easily seen that we do not alter the problem by sup-
posing the point A constrained to fall on a straight line
drawn between two consecutive lines of the given system,
and meeting them at right angles, for the jDroportion of the
favourable cases to the whole number of cases remains the
same after the limitation as before. Take two such con-
secutive straight lines, and consider one of them, which we
will denote by MN; we shall estimate the chance that the
closed curve will cross MN, and by doubling the result we get
the chance that the closed curve will cross the system.
Let A be at the distance x from MN; draw A Y per-
pendicular to MN, so that AY = x. Suppose the curve to
revolve around A, and it is obvious that in this position of A
26
the chance that the curve crosses MN is -^ , where 6 is the
ZTT
ang-le between AA' and A Y when the closed curve touches
Ax
MN; and we may denote by — the chance that A falls be-
tween the distances x and x + Ax from MN: thus, as in
Art. 336, we obtain finally — r l(f)dx for the required chance.
Now \(pdx = X(f) — Ix defy;
when a; = we have ^ = tt, and when x = AA' we have
= 0; the limits of x are and AA' ; thus
l(f)dx = — j xd(j)= \ x d(}>.
[' 1
xd<l> = ^lhj Art. 91 ; thus the chance of crossing MN is
Jo ■"
-T—T : and doubling this we obtain for the required chance — r .
S28 APPLICATION OF THE INTEGRAL CALCULUS
We assumed that AA' divides the curve symmetrically;
but the result will be the same if this restriction be removed.
Instead of the expression >,- we shall now have — — ^-
where (f)^ denotes the angle between A A' and A Y when the
closed curve touches MN at a point on one side of AA', and
</>„ denotes the corresponding angle Avhen the closed curve
touches MN at a point on the other side of AA' . Then finally
1 f'^ 1 P If"'
instead of — r xd6 we shall have ^— , cc deb, + ---, x deb ■
TrbJo 27r67o ^' 27!*6Jo
and the sum of these is — -^ as before.
ztto
This problem was given as an Example for the particular
case of an ellipse in the first edition of the present work ; in
the second edition the problem was put in the general form
here discussed : a verification by simple reasoning mav be
seen in Bertrand's Calcid Integral, page 484. This problem
includes that of Art. 33G ; for a rod of length 2c may be
regarded as a very slender oval curve of perimeter 4c ; t^hus
^-7- becomes -^ , that is -^ — , that is ^^ .
TTO TTO ItTO, IT a
338. A very curious theorem in the Integral Calculus
was obtained by Professor Crofton, by the aid of the Theory
of Probability, and published in the Plidosophical Trans-
actions for 1868; this w^e will now give. The method of the
discoverer of the Theorem well deserves the attention of the
student, on account of its novelty ; we will however here
mainly follow that adopted by Bertrand in his Galcid Inte-
gral, which involves nothing but the ordinary principles of the
Theory of Probability.
339. An indefinitely large plane area is ruled with
parallel equidistant straight lines ; suppose two closed curves
fixed in one plane, each completely outside the other, and let
them be thrown down on the area; suppose also that the
distance between two consecutive parallel straight lines is
such that the two curves cannot cross more than one straight
line at a time : required the chance that one of the straight
lines shall cross both of the curves.
TO QUESTIONS OF MEAN VALUE AND TJIOBABILITY. 321)
Imagine a string drawn tightly round the two curves, so
as to enclose them both, and to form two common tangents
ivhicli do not cross ; let l^ be the length of this string. Again,
imagine a second string drawn tightly round the two curves,
so as to enclose them both, and to form two common tansfents
which cross ; let l.^ be the length of this string. Then the
required chance is -^ — j~^ , where b is the distance between two
consecutive parallel straight lines.
For it is seen on investigation that — ^ ex23resses the chance
of the boundary formed by the second string being crossed by
a straight line; but this includes the cases in which the
common tangents are crossed, and not any part of the peri-
meter of the two curves : and moreover cases in which both
perimeters are crossed are counted twice over. The cases not
required constitute the aggregate corresponding to -~ ; and
TTO
thus by subtraction we obtain the result -"— r^ .
TTO
SiO. We now apply the general result of the precedino-
Article to a particular case ; w^e supi^ose one of the two
curves to become an infinitesimal straight line, that is a
curve in w^hich the longest diameter is infinitesimal, and
the shortest is infinitesimal compared with the longest. Let
PQ denote this infinitesimal straight line, and suiDjDose its
situation such that PQ produced would intersect the closed
curve associated w^ith PQ : w^e proceed to estimate l^ — l^.
Of the two ends, P and Q, let P be the more remote from
the closed curve. Let PA and PB be the tangents from P to
the curve ; let QC and QD be the tangents from Q, so that G
is very near A , and D is very near B. Then
4 - /, = AC+ CQ + 2PQ + QD-\- DB-{AP+ PB)
^2PQ + AC+ CQ-AP+BD + BQ-BP
= 2PQ - PQ cos a-PQ cos /3,
where QPA = a, and QPB = /3.
830 APPLICATIOX OF THE INTEGRAL CALCULUS
Therefore in this case the required chance
= — ^ (2 — cos a — cos (S).
TTO
341. Our object is now to solve this problem : tioo
straight lines are drawn at random across a plane closed
curve : it is required to find the chance that they ivill in-
tersect within the curve. But this will require some develop-
ment ; and in the first place we must explain the sense in
which we use the phrase a random straight line drawn across
a plane curve.
Suppose a plane curve thrown doAvn on such a system of
parallel straight lines as we have considered in the problems
of Arts. 336... 339; and let this process be repeated until
a straight line crosses the curve: the straight line which
thus first crosses the plane curve is called a random straight
line drawn across the plane curve, or briefly a random line.
It follows from this definition that unless the curve be a
circle random lines will not occur Avith equal facility in all
directions with respect to the curve ; for instance, if the curve
be an ellipse of great eccentricity random lines will occur
parallel to the minor axis with much greater facility than
parallel to the major axis. Let us determine the chance that
a chord of a curve drawn at random should lie between two
assigned directions including an infinitesimal angle d9 ; this
may for brevity be described less accurately as the chance
that a chord drawn at random should have an assigned direc-
tion 6. Let p denote the breadth of the curve measured at
risrht angles to the assicjned direction, that is the distance
between the two tangents to the curve which are parallel to
that direction ; then the required chance is obviously propor-
tional to p dO, and so may be denoted by Cp dO, where G is
some constant. We may determine C from the circumstance
that the sum of the chances corresponding to all directions is,
unity, as the chord must have some direction. Thus
J Q
TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 331
but bv the aid of Art. 01 we see that this becomes Cl=\
1 '
where I denotes the perimeter of the curve ; therefore G = y •
342. One chord drawn at random is parallel to a given
direction : find the chance that it will be intersected by
another chord drawn at random.
The chance that the first chord should cross an assigned
breadth j:) of the curve, which is at right angles to the given
direction, within an assigned space dp of p, and fall within
the angular distance dO from the given direction is f >
that is -^, — . Suppose such a chord denoted by il/iVin a
6
diagram ; and let z denote the chance that it will be inter-
sected by a second chord drawn at random.
If we throw the curve on the system of parallel straight
lines we have, as in Art. 339, the expression — j— for the
chance that the chord MN is intersected. This may be con-
sidered as the chance of a compound event, namely, the
chance that the curve is intersected, and that it is inter-
sected along il/i\\ Thus
therefore z = ■ — -, — .
Hence the chance that the first random chord is MN, and
that this chord is intersected by a second random chord, is
dpdd niN
that IS — - — do dp.
343. We can now return to the problem proposed at the
befrinnincT of Art. 341. If Ave sum all the values of the ex-
IT
pression just given we obtain the chance that two chords
o32 APPLICATION OF THE INTEGRAL CALCULUS
drawn at random u'ill intersect within the curve: this chance
then is
j,jJ3fNd0dp.
But iMN'diy, between the proper limits, is equal to the
area of the closed curve, which we will denote by H ; and
IdO between the limits is equal to tt. Thus finally we have
for the required chance — „ - .
344. We now proceed to find the chance that two
random chords produced will intersect ivitliout the closed
curve ; and we begin by finding the chance that the inter-
section takes place Avithin a certain infinitesimal area which
occupies an assigned position. We may naturally expect that
this chance will be proi^ortional to the magnitude of the in-
finitesimal area, and independent of its form ; but we will
not assume this : the reader may draw the infinitesimal area
of any form, as circular or rectangular.
Consider first the direction which makes an ano-le 6 with
a fixed straight line; let r denote the breadth of the infini-
tesimal area, and p tlie breadth of the closed curve, both
measured at right angles to the specified direction. The
chance that a random chord should have this direction is
^-—j— ; and the chance that with this direction it should cross
T
the infinitesimal area is - ; the chance of the compound event
rdd ,
is — — . The chance that this intersection occurs within an as-
signed portion dr of r is y- , that is • — - — . Let a sti'aight
line in the specified direction be denoted by 3INQP, cutting
the closed curve at M and N, and the infinitesimal area at Q
and P.
The chance that a second random choi'd intersects the
TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 33o
first within tho infinitesimal area is the same as the cliance
that it intersects the straight line FQ. Let z denote this
chance ; then, by Art. 340, and as in Art. 842,
— p {'1 — cos a — cos p) = —J- X z,
iro ITU
PQ
therefore z = — ,- (2 — cos oc — cos /3).
Hence tlie chance that there will bo intersection, and
that one of the chords will be PQ
drcW PQ ,_ _. , ,/,
= — T— z = —jf- (2 — cos a — cos p) dr do.
Therefore the whole chance of intersection within the as-
signed infinitesimal area is
~i JJPQ (2 - cos a - cos /3) dr dO.
Now \PQ dr between the proper limits is the infinitesimal
area, which we will denote by a ; thus tlie expression becomes
J. I (2 — cos a — cos /3) d9.
Let -v/^ be the angle which the closed curve subtends at
any point of a ; then ^ + oi = -^, so that jS ^-yjr — a and we
may put the expression in the form
y„ {2 — cosa- COS (->/r-ct)] fZa ;
f-' .
and this will be found equal to
^ (i^ — sin -vlr).
345. Til us the whole chance of intersection without the
closed curve is
2 f
J. Ida) {'^ — sin -v^)
where dco is put for a, and denotes an element of area ; the
integration is to extend over the whole area outside the
closed curve. The sum of this chance, and of that found in
Art. 343, must obviously be unity ; thus
334- EXAMPLES.
therefore I fZw (-v/^ — sin ■\/r) = - — ttO.
Here /I represents the perimeter of any closed curve, D.
the area, -v//^ the angle which the closed curve subtends at any-
external point, d(o an element of area there ; and the integral
is to extend over all the area outside the closed curve. This
formula in the Integral Calculus constitutes the theorem dis-
covered by Professor Crofton.
346. A large number of very interesting problems rela-
ting to the subject of the present Chapter will be found in
the volumes entitled Mathematical Questions, with their solu-
tions. From the Educational Times....
EXAMPLES.
If r=f{9) and y=f[-) he the equations to two curves,
f(9) being a function which vanishes for the values
6^, 6,,, and is positive for all values between these
limits, and if A be the area of the former between the
limits 6 = 6^^ and d—0„, and M the arithmetical mean
of all the ti-ansverse sections of the solid generated by
the revolution about the axis of x of the portion of the
latter curve between the limits x = a0^ and x = ad^,
27J-
shew that J/ = ^ tt ^> suiwosiug &„ greater than 0^.
A ball is fired at random from a gun which is equally
likely to be presented in any direction in space above
the horizon : shew that the chance of its reaching
more than — th of its greatest range is . / 1 1 ] .
m ^ V \ 111/
EXAMPLES. 335
!. From a point in the circumference of a circular field a
projectile is thrown at random with a given velocity,
which is such that the diameter of the field is equal to
the greatest range of the projectile : find the chance of
its falling within the field. ,, , 1 2
° Iiesult. -. (V2 — 1),
\ On a table a series of straight lines at equal distances
from one another is drawn, and a cube is thrown at
random on the table. Supposing the diagonal of the
cube less than the distance between consecutive
straififht lines, find the chance that the cube will rest
without covering any part of the lines.
Besidt. 1 , where a is the edge of the cube and c
CTT
the distance between consecutive straight lines.
i. Prove that the mean of all the radius-vectors of an
ellipse, the focus being the origin, is equal to half the
minor axis, when the straight lines are drawn at equal
angular intervals ; and is equal to half the major axis
when the straisrht lines are drawn so that the abscissae
of their extremities increase uniformly.
). An indefinite number of equidistant parallel straight
lines are drawn on a plane, and a rod whose length is
equal to r times the jierpendicular distance between
two consecutive lines is thrown at random on the
plane : find the chance of its falling upon n of the
2
straight lines. If n = ?• = 1, shew that the chance is - .
7. Two arrows are sticking;' in a circular tarfjet : shew that
the chance that their distance is greater than the
radius of the targ^et is *-, - .
° 47r
8. Supposing the orbits of comets to be equally distributed
tlirough space, prove that their mean inclination to
the plane of the ecliptic is the angle subtended by an
arc equal to the radius.
Result.
ooQ EXAMPLES.
9. A certain territory is bounded by two meridian circles
and by two parallels of latitude which differ in longi-
tude and latitude respectively by one degi-ee, and is
known to lie within certain limits of latitude : find
the mean superficial area.
10. A straight line is taken of given length a, and two other
straioht lines are taken each less than the first straight
line and laid down in it at hazard, any one position
of either being as likely as any other. The lengths of
these straight lines are b and b' ; it is required to find
the probability that they shall not have a part ex-
ceedmff c in common.
(^a-h-b' + cY
(a-b){a-b')'
{Canib. Fhil. Transactions, Vol. viil. page 386.)
11. From any point within a closed curve straight lines are
drawn at equal angular intervals to the circumference:
shew that the mean value of the squares on these
straight lines is the product of — into the area of
the curve.
12. A messenger M starts from A towards B (distance a) at
a rate of v miles per hour, but before he arrives at JS a
shower of rain commences at A and at all j^laces occu-
pying a certain distance z towards, but not reaching
beyond, B, and moves at the rate of w miles an hour
towards A ; if M be caught in this shower he will be
obliged to stop until it is over ; he is also to receive
for his errand a number of shillings inversely propor-
tional to the time occupied in it, at the rate of ?i shil-
lings for one hour. Supposing the distance z to be
unknown, as also the time at which the shower com-
menced, but all events to be equally probable, shev/
that the value of il/'s expectation is, in shillings,
nv (1 u u (u ■\- v) , u-{-v\
a 2 y V w ]
EXAMPLES. 3o7
13. A large plane area is ruled with parallel equidistant
straight lines, and also with a second set of parallel
equidistant straight lines at right angles to the former
set ; a thin rod is thrown at hazard on the area : find
the chance that the rod will fall across a line.
(See History of...ProhahiUty, page 347.)
14. Suppose a cube thrown on the system of lines described
in the preceding Example : find the chance that the
cube will fall across a line.
(See History of... Probability, page 348.)
15. Let there be a number n of points ranged in a straight
line, and let ordinates be drawn at these points ; the
sum of these ordinates is to be equal to s ; moreover
the first ordinate is not to be greater than the second,
the second not greater than the third, and so on :
shew that the mean value of the ?-*^ ordinate is
5 fl 1 1 1 )
n [n n — 1 n — 2 ?i — ?• + Ij
(See History of ...Probability, page 545.)
IG. Verifv the formula in Art. 345 by direct intefrration in
the case where the closed curve is a circle.
T. I. c. 22
SS8
CHAPTER XV.
CALCULUS OF VARIATIOXS.
Maxima and Minima of integrals involving one dependent
variable with fixed limits.
347. The theory of maxima and minima values of given
functions is fully considered in works on the Differential
Calculus. If, for example, y denotes any given function of an
independent variable x, then we can find the value or values
of X which make y a maximum or minimum, or we can shew
that there are no such values in some cases.
We are now however about to consider a new class of
maxima and minima problems. Let ;/ denote a function of x
which is at present undetermined ; and let V denote a given
7 y-7-
function of a*, y, -^, -~,... Suppose we wish to find the
relation which must hold between x and y in order that the
integral I Vdx, taken between given limits, may have a maxi-
mum or minimum value. We cannot here efifect the integra-
tion, because y is not known as a function of o:, and therefore
V is not known as a function of x; thus the ordinary methods
of solving maxima and minima problems do not apply. We
require then a neAv method, which we shall now proceed to
explain.
348. The department of analysis to which we are about
to- introduce the student is called the Calculus of Variations ;
its object is to find the maxima or minima values of inte-
gral exj)ressions, the exj^ressions being supposed to vary by
CAL',ULTTS OF VARIATIONS. 339
assigning d'l^evont forms to the functions denoted hj the de-
pendent variables. It will be seen, as we proceed, that the
method of finding these maxima or minima values is ana-
logous to that of finding ordinary maxima or minima values
by the Differential Calculus. »
340. It will be useful to recur to the method given in
the Differential Calculus. The student will remember that
the terms maximuin and minimum are technical terms, which
are defined and illustrated in treatises on the Differential
Calculus ; and they are used in mathematics in the sense
there assigned to them. Mistakes are frequently made by
confounding a maximum, value in the technical sense of the
word maximum, with the greatest value in the ordinary sense
of the word greatest.
Suppose y a given function of an independent variable x;
then if an indefinitely small change is given to x, in general
an indefinitely small change is consequently given to y, which
is comparable in magnitude with that given to x. The pro-
cess of finding a maximum or minimum value of 7/ may be
said to consist of two parts. First we determine such a value
of x that an indefinitely small change in it does not produce
in y a comparable indefinitely small change, but a change
which is indefinitely small compared with that of x. In the
second place, we examine the sign of this indefinitely small
change which is produced in y by the change of x; and for
a maximum this sign is to be necessarily negative, and for
a minimum positive.
We may therefore describe this process briefly thus; we
make the terras of the first order in the change of the depend-
ent variable vanish, and we examine the sign of the terms
of the second order. We shall pursue a similar method
with the problem wliich we have now to discuss ; we confine
ourselves, however, at present entirely to the first part of the
process, and shall hereafter recur to the second part.
S'yO. We have first to explain the notation which will
be used. Let x denote an independent variable, ?/ any func-
tion of a-, and ^^, -A,-., the differential coefficients of y
340 CALCULUS OF VARIATIONS.
with respect to x. We shall use hj to denote an indefinitely
small quantity which may be any function of x; and if u
denote any quantity whatever which depends on y we shall
denote by da the increment which 2t receives when y is changed
into y + hj. Thus, for example, consider the ditt'erential co-
efficient -^ ; when y receives the increment S_y this differen-
tial coefficient receives tlie increment -y- - , so that by S —
we mean — ^ • It is often convenient to use the symbol p
ax
for -,- : and so also hp is a convenient symbol for -y^ .
dx' -^ dx
Acfain, consider the second differential coefficient -fr, : when
*= ' dx
y receives the increment Zy this second differential coefficient
receives the increment -7-25 ^'^^ ^^ the second differential
coefficient is often denoted by q we may conveniently use S^
for — r-^ . Similarly r and s may be used for the third and
CLjC
fourth differential coefficients of y respectively, and hr and Ss
for -y-v and ~r4 respectively : and so on.
ax dx
The differential coefficients are also often denoted by
y'y y"' y"'---5 ^^^"I ^^^^^ %' V' W'>--- "^^y ^e used as equi-
valent to hp, hq, h',... resj)ectively.
351, The introduction of the symbol S is due to La-
grange. The student will see that this symbol resembles in
meaning the symbol d, which is used in the Differential Cal-
culus. Both dy and hy express indefinitely small increments;
dy however is generally used to denote the change in value of
a given function consequent upon a change in the value of the
dependent variable, hy is used to denote the change made by
ascribing an arbitrary change to the form of a function. The
quantity denoted by dy is called the variation of y.
CALCULUS OF VARIATIONS. 341
7 72
352. Let V denote a given function of a;,y,~ , -^-^ , ... ;
and let U= 1 Vdx, -wliere x^^ and x^ are supposed to denote
fdven limits. The value of Z7 cannot' be found so lonf^ as we
do not know what particular function y is of a; ; but without
knowing this we are able to obtain an expression for the
increment made in U by ascribing the arbitrary increment 8i/
to 1/, from which important inferences can be drawn.
Suppose V=(f) {x, y, y, y", y", ...);
then by definition
gF= </) {x, y + By, y + hy\ y" + hj", y'" + By'", ...)
The first term may be expanded by the ordinary exten-
sion of Taylor's theorem ; thus
where -r- is the partial differential coefficient of V with
ay
dV
respect to y, also -^ is the partial differential coefficient of V
with respect to y' ; and so on.
In the above expression for BV we have only expressed
terms of the first order, that is, we have omitted the terms of
the second and higher orders with respect to the small quan-
tities By, By', .... This we shall continue to do throughout the
remainder of the investigation.
Then
BU=rBVdx
f^^{dV. dV dV dV ^ ,^, ]j
i^o [dy -^ dy '^ dy -^ dy '' J
342 CALCULUS OF VARIATIONS.
We shall now transform this expression by integration by
parts, Eor shortness put
^-AT ^-P 'K-n ^-P
Then fpSy' dx =jp^^dx = PSjj - l-^ hj dx ;
therefore [" P V dx = {PSi/)^ - (PSy), - ["" "^ g^ dx.
Here (P3^)i is used to denote the value of TSi/ Avhen x^
is put for a^, and (P^y)^ is used to denote the value of PSt/
when x^ is put for x ; a similar notation will be used through-
dP
out. It is to be carefully observed that -^ means the coiii-
jilete differential coefficient of P with respect to x, that is to
dP
say, in forming -t-_ we are to remember that ?/ and its dif-
ferential coefficients all involve x implicitly.
Again
therefore
Similarly
/ „ (^'S?/ t^2? (75y d'E .\ f^d'E^ J
CALCULUS OF VARIATIONS. 843
This process may be continued until all the symbols
Si/, Bi/", Si/'", Si/"", ... are brought from under the integral
sign. It is to be observed that all the differential coeffi-
. , dQ d'Q dR d'R d'R ,, ,.^. ,. ,
cients ', , -T.2 > -J-;, ~T~J' j •■> ^^® complete diiiorential
coefficients.
Hence finally
+ S>l^[R~...\-Sq,[R-...],
+
, {^^[.^ dP d'Q d'R , \. ,
Here we have adopted some obvious simplifications of nota-
tion ; thus we use Si/^ for {Si/)^, and Sp^ for (-p) , a-nd
so on.
353. The value oi SU may be denoted thus,
SU=H^-E^+ r KSy dr,
J Xq
where H. denotes a certain afjonreorate of terms in which x, is
put for X, and H^ a similar aggregate of terms in which x^ is
put for X ; these aggregates do not involve any integrations.
Also
dx dj? dx^
344 CALCULUS OF VARIATIONS.
Since JTj — H^ involves only the values of the variables at
the limits, we shall sometimes speak of H^ — H^ as the terms
at the limits.
354. We can now determine the conditions which must
hold in order that U may have a maximum or minimum
value. For, in order that U may have a maximum or mini-
mum value, h U must vanish, whatever 5?/ may be, provided
only that it is an indefinitely small quantity. This requires
that
7^=0 and 11^-11^ = 0.
For if K is not always zero, it will be in our power to give
such a value to Sy as will make B U positive or negative at
our pleasure, and not zero. Suppose, for example, that the
highest differential coefficient of Zy which occurs in H^ — H^ is
the w*^ Put hy = a{x — oc^Y" {x — x^Y", where a is a function
of X which is indefinitely small, and is at present undeter-
mined. Then this value of 8y makes H^ — H^ vanish, so that
oZ7 reduces to I KZy dx. Now take a such that it is always
^ .To _
positive when K is positive, and negative when K is nega-
tive ; then SC/" is necessarily positive. And if the sign of a
be changed, SZ7 is necessarily negative. Thus if K is not
always zero, it is in our power so to take hy as to make h U
positive or negative at our pleasure.
Hence for a maximum or minimum value of U we must
have /v = 0; and then I KSydx vanishes, and therefore
also TTj — H^ must = 0.
355. The student has now become acquainted with the
essential features of the Calculus of Variations; these are
(1) the reduction of BU to the form H^ — H^+ \ Khydx,
J Xq
(2) the principle that K must vanish in order that U may
be a maximum or minimum. Although the subject admits
of considerable development, by various extensions of the
problem we have considered, still the two results we have
already obtained are the chief results.
and therefore in -rr the differential coefficient -y^ will
CALCULUS OF VARIATIONS. 845
S5G. We now proceed to examine more closely the
nature of the two conditions
K= and 11^ - IT^ = 0.
The equation K = is Avhat is called a diferential equa-
tion. Suppose that ~^ is tlie highest differential coefficient
which occurs in V; then this will in general occur in R also,
—-^ the differential coefficient -7^
dx ax'
occur, and this will be the highest differential coefficient which
occurs in K, so that the differential equation /i=0 will be of
the sixth order. And in general the order of the differential
equation is twice the order of the highest differential coeffi-
cient which occurs in V.
It is shewn in treatises on Differential Equations that the
solution of a differential equation involves as many arbitrary
constants as the number which expresses the order of the dif-
ferential equation. We must now shew how the arbitrary
constants which arise from the solution of the equation K=
are to be determined, so that a definite result may be ob-
tained. The condition H^ — H^=0 serves for this purpose.
Two cases may arise.
(1) Suppose that no conditions are imposed by the pro-
blem on the values of y and its differential coefficients at the
limits of the integi-ation ; then Sy^, Sy^, Sp^, Sp^,... are all arbi-
trary quantities, that is, we have it in our power to suppose
any indefinitely small values we please for these quantities ;
for example, we may suppose that as many of them as we
please are zero. Since %, , 8y^, Bp^, Sp^,... are thus all arbi-
trary, in order that H^ — IT^ may certainly vanish, the coeffi-
cient of each of the arbitrary quantities must vanish. This
furnishes for determining the constants as many equations as
there are constants.
(2) Suppose that conditions are imposed by the problem
upon the values of 7/ and its differential coefficients at the
limits of the integration; then Bi/^, By^, Bp^, Bp^,... are not all
arbitrary, for some of them can be expressed in terms of the
346 CALCULUS OF VARIATIONS.
rest by means of the given conditions. Let as many as pos-
sible of the quantities 87/ ^, Sy,,, Bj)^, Bp^,... be eliminated from
H^—IIf^, and then the coefficients of those which remain must
be equated to zero. The equations thus obtained, together
with those which express the given conditions, will form a
system equal in number to the number of constants, and
therefore will serve to determine those constants.
357. The principal difficulty in examples consists in the
solution of the difterential equation K = 0, and this difficulty
is frequently insuperable.
We will now shew that when V does not explicitly con-
tain the independent variable, one step in the solution of the
differential equation can always be taken. It will be suf-
ficient for practical purposes to confine ourselves to the case in
which V involves no differential coefficient of y higher than
the third.
Since V is supposed not to involve x explicitly, we have
for the complete differential coefficient of V
dx dx dx dx dx
And by supposition
dx dx^ dx^
.(1).
Thus
dV _ dP dy p dp d^Q dy ^ dq d^R dy j^dr
dx dx dx dx dj? dx dx dx^ dx dx
Now
dP dy jy dp _ d p dif
dx dx dx dx dx *
d^Q dy r\dq_d (dQ dy „ d^y]^
dx* dx dx dx \dx dx dx^\ *
d'R dy Tfdr_d ((£R dy_dRd^y ^ d'y)
dx^ dx dx dx\dx^ dx dx dx^ dx^)
CALCULUS OF VARIATIONS. 347
Hence, by integ-ration,
dii d Q di) „ dij d'R d>i dU d\ij d'y ^
dx dx dx dx dx dx dx dx ax
where C is an arbitrary constant. The highest differential
.. . , • /.x • d'li ,. , . d'R
coefficient that can occur m (2) is ~ which occurs in -^ ;
thus (2) is a differential equation of the fifth order, which is
a first integral of the equation (1) which is of the sixth order.
Particular cases may be obtained by supposing R or Q or P
to bo zero. For example, the most useful case is that in
which V involves only y and ^- ; so that (1) becomes
dx
and (2) becomes
358. The differential equation /r=0 is also susceptible
of one integration when V does not contain the dependent
variable. For then N= 0, and the equation becomes
dP_d;Qd^_ ^^
dx dx' dx^
and therefore
dQ^d'R ^
dx dx
r xx r (J^^
359. We know that I Vdx= 1 V -j- dy, supposing the
limits- of the integration with respect to y taken to corre-
spond to those of the integration with respect to x. And the
differential coefficients of y with respect to x may be expressed
in terms of the differential coefficients of*- with respect to y.
f dx
Thus in V J- dy we may regard y as the independent vari-
able, and X as the dependent variable, and proceed to find
the maximum or minimum value of the integi-al in this new
348 CALCULUS OF VARIATIONS.
form. We may feel a j)viori certain, as the problem is really
not changed by this change of the independent variable, that
we shall obtain the same result as if we had kept the original
independent variable.
Hence the cases considered in Arts. 357 and 358 may be
seen to coincide.
SCO. Again, let us suppose that V involves only p and
q. Then the differential equation K=0 reduces to
dx dx^ '
therefore, by integration,
^ = f+^.-
Also ^^P^ + Q'k
dx ax dx
dx dx dx dx
therefore, by integration,
Here (7, and C^ are arbitrary constants. In this case the
differential equation K=0 is of ih.Q fourth order, and the
result we have obtained is a differential equation of the second
order ; so that we have effected two steps in the integration
of the differential equation K=0.
861. We shall now proceed to consider some examples ;
as we have already intimated we confine ourselves entirely to
the first part of the process for finding maxima and minima
values ; see Art. 349.
302. To find the shortest line between two points.
This example is introduced merely for the purpose of
illustrating the formulae, as it is obvious that the result must
be the straight line joining the two points.
CALCULUS OF VARIATIONS. 34-9
Here F=V(1+/) and U= T^/il +p'') dx.
J Xq
Thus F involves onlyp, auJ the equation K=0 reduces to
— = ; hence P must be a constant, that is, . ,/, — jr must
dx vCj-+p;
be a constant. This shews that p must be a constant, and
therefore the required curve must be a straight line.
In this case E^ - H, =^r^^ - vIT+pT) •
If now tlie two points are fixed points, we have Sij^ =- and
gy^ = ; thus //j - H^ vanishes. Then the value of p must
be" found from the condition that the straight line must pass
through the two fixed points.
Suppose however that the ordinates of the two points are
not fixed ; the ahscissce are fixed because x^ and x^ are taken
to be invariable. In this case %, and S?/^ are arbitrary ; and
therefore i/^— ZT^will not necessarily vanish unless the coeffi-
cients of Sy\ and 3?/, vanish. This requires that p, and p^
should vanish, and as p is a constant by supposition this con-
stant must be zero. Thus our formulae are consistent with
the obvious fact, that when two straight lines are parallel the
shortest distance betw^een them is obtained by drawing a
straight Line perpendicular to them both.
363. To find the curve of quickest descent from one
given point to another.
The following is a fuller statement of the meaning of this
problem. Suppose an indefinitely thin smooth tube con-
necting the two points, and a heavy particle to slide down
this tube ; we require to know the form of the tube in order
that the time of descent may be a minimum. The problem
is known by the name of the hrachistochrone ; it was first
proposed by John Bernoulli in 1696, and gave rise to the
Calculus of Variations.
We shall assume that the required curve lies in the ver-
tical plane which contains the two given points. Let the axis
of y be measured vertically do^vnwards, and take the axis of
350
CALCULUS OF VARIATIONS.
X to pass through the upper given point. The particle is
supposed to start from rest, and then by the principles of
mechanics the velocity at the depth ?/ is i^{2gy). Thus the
time of descent is / — ,,^i-\ dx. We may then take
J 'To \/(2/7.5/) -^
Here V involves only y and jj ; so that, by Art. 357, for
a minimum we must have
that is,
therefore
V=Pp+C,
p
V(l+/) _
= G.
+ ^;
V[2/(l+/)}
Hence ?/(l +^/) = a constant = 2a suppose ;
y .
therefore
therefore
/ =
2a-'
y
dx
dij
y
V
a-yj ^{2ay-fj'
therefore x = a vers" - — \J{2ay — y^) + h, where h is another
a
constant.
This shews that the required curve is a cycloid with its
base horizontal, its vertex downwards, and a cusp at the
upper point. We may suppose the origin at the ujDper point
so that x^ = 0, and then 5 = 0.
Ilerell^-H,=
phf
1
phf
V(2a)
{(jP%)i-(i^^.'/)ol'
As we suppose both the extreme points fixed hy^ and Zy^
vanish, and therefore H^ — H^ vanishes.
\
CALCULUS OF VARIATIONS. 351
The constant a must be determined by the condition that
the cycloid shall jDass through the lower given point.
Suppose however that only the abscissa of the lower point
is given, and not the ordinate. Then, as before, //„ vanishes,
and II, = - -,,/[ -. Now S//, is arbitrary, so tbat in order that
' V(-«)
//j may vanish, we must have j)^ = ; tlius the tangent to
the cycloid at the lower limiting point niust be horizontal.
This condition must be used in tliis case to determine the
constant a.
3G4. We may modify the preceding problem by sup-
posing that the particle does not start from rest, but starts
with an assigned velocity. In this case we will suppose that
the axis of x is not drawn through the upper point, but is so
taken that the velocity at starting is that which would be
gained in falling from the axis of x to the upper fixed point.
The solution remains as before ; the cusp of the cycloid is
however no longer at the upper fixed point, but in the axis
of x. This might have been anticipated. For let ACB be
an arc of a cycloid, having its cusp at A; then this is the
curve of quickest descent from rest at A to B, and there-
fore CB must be the curve of quickest descent from C to B,
starting with the velocity at C.
8G5. To find the curve connecting two fixed points such
that the area between the curve, its evolute, and the radii of
■';urvature at its extremities may be a minimum.
By Art. 157 the expression which is to be made a mini-
mum may be taken to be
Here V involves only j) and q ; and therefore, by Art. SCO,
for a minimum we must have F= Qq + Cj) + C.,,
that is, (1±£l = Jl±fl + c,p + C, ;
1 9.
therefore ffl^±^^)i^^>
\\
S52 CALCULUS or VARIATIOXS.
By integration
G,i^u^p^~f^ = ^x+C,.. (1).
Also _^^__,j/-^=9^;
therefore by integration, C^tan"^/) ^^ ~ = ^^y + constant
add Cjj to both sides of this equation, and we have
0,tan-j)4-^i^^^ = 42/+a, (2).
Eliminate tj},n~'jj from (1) and (2); thus
therefore ^/[l +/) = ^rT77^~ ^'
Avhere B is such that 4<B = C.^G^— G^G^.
Let s denote the lens^th of the arc of the curve measured
from a fixed point ; then, by integrating the last equation,
we have
s+G=^/{G^y-G^x + B).
This shews that the required curve is a cycloid ; see Art. 72.
G'jj — G^x + B = is the equation to the tangent at the ver-
tex of the cycloid.
We must now examine the expression H^— 11^; we have
^-^^•(^-T3/^i'««»
As the extreme points are sujDposed fixed, Sj/^ and Zij^
vanish ; thus
\:
CALCULUS OF VARLVTIONS. 353
Suppose we impose the condition that the tangents to the
required curve are to have fixed directions at the extreme
points ; then Sp^ and 8p^ vanish, and B^ — H^ vanishes. In
this case the cycloid must be determined from the conditions
that it is to pass through two given points, and its tangents
are to have fixed directions at these points.
If, however, no condition is imposed on the values of ^ at
the Umits, Ave must have Qj = and Q^ = 0, in order that
H^ — H^ may vanish. Now Q = — / — ; and the radius of
3.
curvature = -^^ ^--^. Thus the radius of curvature must
vanish at the extreme points, that is, the cycloid must have
cusps at those points.
oQQ. To find the form of a solid of revolution, that the
resistance on movinc? throuQ-h a fluid in the direction of its
axis may be a minimum, adopting the usual theory of re-
sistance.
Take the axis of x as the axis of revolution. Then adopt-
ing the theory of resistance which is explained in works on
Hydrodynamics, the expression which is to be a minimum is
I
yp
dx.
I I i^'
1+/
Here V involves only y and p, and therefore by Art. 857,
for a minimum we must have
V=Pp+C,
that is, JIlL^,=.y2l±t^C;
therefore tt-'^'^W +0=0.
(1 +i>-Y
This is a differential equation for determining the required
curve.
T. L c. 23
Vv
So-i CALCULUS OF VAEIATIONS.
Integrals with limits subject to variation.
367. We have now sufficiently explained and illustrated
the method of finding the maximum or minimum value of an
integral expression involving one independent variable, when
the limits of the integration are supposed invariable. We
shall proceed to some extensions of the problem ; and we
begin by considering the modification which arises from sup-
posing the limits of the integration variable.
Suppose, for example, that we have two given curves in
one vertical plane, and that we wish to find the curve of
quickest descent from one of these curves to the other, the
particle starting with the velocity obtained in falling from a
given horizontal straight line. Here we have to find the
point at which the particle is to leave the upper curve, and
the point of the lower curve towards which it is to proceed, as
well as the path which it is to describe. We have therefore
to effect more than in the examples hitherto considered, and
we shall now explain how we may proceed.
We know, from what has been already given, that the
curve must be a cycloid with its base horizontal and a cusp
on the given horizontal straight line. For suppose any other
curve drawn from any point in the upper curve to any point in
the lower ; this curve cannot be that of minimum time, for we
know that, without changing the extreme points, we can find
a curve of less time of descent than this curve, namel}^ a
cycloid with its base horizontal, and a cusp on the given horizon-
tal line. Since then we know that the required curve must be
such a cycloid, the part of the problem which depends on the
Calculus of Variations may be considered solved; and we
may investigate, by the ordinary rules for maxima and minima,
the position of the particular cycloid for which the time is a
minimum. In fact, taking any arbitrary initial and final
points, we may find the equation to the cycloid passing
through these points ; then the time of descent will become
a known function of the co-ordinates of the initial and final
points, and we may determine for what values of these co-
ordinates the time is a minimum.
CALCULUS OF VARIATIONS. 355
S6S. We have shewn in the preceding Article that it is
not absolutely necessary to make an}'- moditication in our for-
muhc in order to include the case in which the limits of the
integration are supposed to be susceptible of change ; for the
process already given, combined with the ordinary rules of
the Differential Calculus, would enable us to solve any ex-
ample. It is however convenient to bring together all that is
wanted for solving such examples, and accordingly we shall
now supply the requisite modification of our original for-
mulce. As before, let
rxi
U= Vdx.
•_ P
J Xa
Suppose that in addition to the change of ;/ into y + ^y
the limits x^ and x^ are changed into x^-\-dx^ and x^-\-dx^
respectively. In consequence of this change of limits f/" re-
ceives the increment
rx(t+dxa
Vdx- Vdx,
J Tn
rxi+dxi fxo+dxo
J Xi •' Xo
that is, neglecting squares and higher powers of dxj^ and dx^,
U receives the increment
V^dx^ — V^dx^,
If we annex this to the expression already given for BIT, we
shall obtain the complete change in U consequent upon the
variation of y, and the change of the limits.
369. If no condition is imposed on the limiting values of
the co-ordinates, the additional terms just obtained,
V^dx^ - V^dx^,
can only be made to vanish necessarily by supposing V^ =
and Fp = 0. We thus introduce two new equations in ad-
dition to those which are obtained from H^— H^ = {)\ and at
the same time we have two new quantities to determine,
namely, x^ and x^. However, a more common case is that in
which the limiting values have to satisfy given equations.
Such a case we have already indicated in Art. 367, where a
23—2
356 CALCULUS OF VARIATIONS.
curve is required, the extreme points of which are to lie on
given curves.
We will consider that limit of the integration for which
the quantities are distinguished by the subscript 1. Let
Y=y+hy,
then if there had been no change of the limit, the extreme
values of the variables would have been x^ and y^ before
variation, and x^ and Y^ after variation. If however x^ is
changed into x^-\-dx^, we have Y^ changed into
that is, neglecting squares and higher powers of dx^ we have
Y^ changed into Y^-\-{-j-\ dx^, that is, neglecting the product
^■p^dx^y'mio y^-^hj^-\-i^ \ dx^. Supposing then that the
\CtX/ ^
given relation which is to be satisfied by the extreme
values is
we must have 3/i = ^ (^i)'
and also
to the first order. Thus
8y.={t' (-)-!}/-.
This gives a relation between hj^ and dx^, so that we can
eliminate one of them from the complete value of hll.
Similarly, the relation can be found between hj^ and dx^.
In geometrical problems (^J is the tangent of the inch-
nation to the axis of x of the straight line which touches the
required curve at the limiting point ; and y^' [x^ is the tan-
CALCULUS OF VARIATIONS.
357
gent of the inclination to the axis of x of the straight hnc
■which touches the givox curve at that point.
A particular case may be noticed which is sometimes
useful. Suppose the comjilete change of y^ is to be zero ;
this gives 8y^ + (—) dx^ = ; similarly if the complete change
of 2/0 is to be zero, hj^ + (J^^ dx^ = 0.
870. We may illustrate the preceding Article by a figure.
Let AB represent the required curve, and MEN the given
curve on which the extremity B of the required curve is to
lie. Let A'B' represent the curve derived from AB hy
ascribing the variation hj to each ordinate y. Draw BG and
B'C'D parallel to the axis of y, and BD parallel to the axis
of .r. Then ultimately
BC=By^, BD = dx^, B'D = f'{x;)dx^, C'D^T^ dx^.
Hence B'C = l-yjr' (x^) — {-~\ y dx^. Thus the geometrical
interpretation of our process is that if we reject quantities
of a higher order than those we retain, we have B'C = BO
ultimately.
358
CALCULUS OF VARIATIONS.
371. Let us now consider the case of the brachistochrone
problem which has been enuuciated in Art. 367.
Let the notation be as in Art. 363. Then
+
P %
Lvij/(i+/)}.
phj
X LV{3/(i+i/)i.
\y-'^>^'-
dP
As before from the equation N— -r- = we deduce
thus
V{2/(l+/)}=V(2a);
Let us suppose that the equation to the fixed curve from
which the particle is to start is F=%(2!), and that the
equation to the fixed curve at which the particle is to arrive
is Y= yjr (X). Then by the j)reccding Article we have
%i = {^'G'^) -P]i dx„ Sy„ = [x{x) - p]o dx,.
Thus the value of SU can be put in the form
BJJ = \dx^ — \dxg ;
wbere
and similarly
V(2a)
1
^0 V(2a)
{l+p,^'{x;)},
{l+PaX'Wl-
Since dx^ and dx^ are arbitrary, Sfywill not necessarily
vanish unless X^ = and \ = 0, Thus
l+2^i^'('^i)=0 and l+i)o%Vo) = 0; .
CALCULUS OF VARIATIONS. 359
and these shew that the cycloid must cut each of the two
fixed curves at right angles.
372. We have hitherto tacitl}'' assumed that the function
V does not involve the limiting values of the variables or of
the differential coefficients. Suppose now however that V
does involve x^, x^, t/^, y^,2'>o,l\>---
(1) Suppose that x^ and x^ are not susceptible of any
change. When y is changed into y + hy, besides the varia-
tion we have already investigated, V will receive an addi-
tional variation arising from the change in yo,yi,--. which
occur explicitly in V. These additional terms in 6 F are
and consequently the following additional terms occur in
Now Sj/g, By^, Bp^, S;:»,, ... ate not functions of the variable
X, but only of the limiting values of x; we may therefore
bring these quantities outside the integral sign and write the
additional terms thus,
ha T- (^^+ hi -j-dx + dpj -j-dx
+
Thus the occurrence of these additional terms will not
affect the reasoning by which it is shewn in Art. 354 that we
must have K = in order that U may be a maximum or
minimum. These additional terms must be annexed to the
expression H^—H^, and the whole then made to vanish.
Since the relation between x and y is supposed to be found
from the equation K=0, the expressions under the integral
signs in these additional terms become definite functions of x,
so that the integrations which are indicated can be effected,
at least theoretically.
360
CALCULUS OF VAKIATIONS.
(2) Suppose that x^ and x^ are also changed, and let
them become x^ + dx^ and x^ + dx^ respectively. Then V
receives the additional increment
where
dV'
dx^
and
' dV
dx^j
'dV'
_dx^_
dx^ +
dV
dx
1-1
dx^,
indicate complete differential coeffi-
cients ; that is to say, we are to remember that x^ occurs
implicitly in y^, p^, ..., and similarly for x^.
Thus besides the additional terms we have already given
SZ7 receives the increment
dx,
J Xa
d_r
dx + dx^
Xi
^0
dV'
d
X.
dx,
and this expression must be annexed to the aggregate formed
of H^ — Hq and the additional terms already given.
373. For an example we will take another modification
of the brachistochrone problem. Suppose two given curves
in the same vertical plane, and let it be required to find the
curve of quickest descent from one of these to the other, the
motion commencing at the first curve.
Let the axis of y be measured vertically downwards;
let 3/0 be the ordinate of the starting point, then when the
ordinate is y the velocity is ^J[2g {y—y^].
Thus we may take
We have then to change y into y —y^ in the solution of
Art. 371, and to add to the expression there given for SU
the terms found in Article 372.
Here V= ^^t ^ ; so that y„ is the only limiting value
CALCULUS OF VARIATIONS.
361
which occurs in V. We are therefore to add to the former
value oi BU
dV'
Lf^^^oj
dx;
and
dV
dx,
OJ
dVfdji
Hence by Art. 371, after putting A' = 0, we have
SU= \dx^ - X^dx^ + \Bj/o+ (-£ ) ^'^0
where \ and \ have the values assigned in Art. 371
Now in the present case
dP
dx'
^'^dV
dx,
dt/o dy~
therefore ^^dx = P,-P, =%^ ;
and %o + ( / ) dx^ = -^^ {x^ dx^, as in Art. 371.
Thus W=^ \dx^ - \dx, + ^^ {p, -r- p^ dx^
V(2a)
-V(2^^^+M'(ac?^o
Then by equating to zero the coeflficients of dx^ and dx^
we have
1 +p,f (x^) = and 1 -\-2:>jc (xj = 0,
so that X (^q) = ^' (^i)-
Thus the cycloid cuts the lower fixed curve at right
angles, and the tangent to the upper fixed curve at the
initial point is parallel to the tangent to the lower fixed curve
at the final point.
362 CALCULUS OF VARIATIONS.
Integrals with tiuo dependent variables.
874. We have hitherto supposed that F is a function
with only one dependent variable ; let us now suppose that V
is a function of two dependent variables.
Let F be a function of x, y, z, and the differential co-
efficients of y and z with respect to .r ; let
U=\^'Vdx,
and let us investigate the variation in the value of ?7when y
and z receive variations.
By proceeding as in Art. 352 we shall obtain the follow-
ing result,
hU=^H^-H, + J,-J,+ \''\Khj + Uz)dx,
where the symbols have the following meanings :
hj, as before, denotes an arbitrary variation given to y, that is,
hy is an indefinitely small arbitrary function of x ;
K, as before, denotes
dJ[_ddV d^d]^_
dy dx dy dx dy"
-where ^— , i-r, -i-„,..- are partial diffei'ential coefficients,
dy ay ay
and -^ -,-, , -T-T.-m,--' are complete differential coefficients
dx dy dx dy
relative to x ;
hz is an arbitrary variation given to z, that is, Zz is an in-
definitely small arbitrary function of x ;
L is relatively to z the same as K relatively to y, that is,
~ dz dx dz (?^ dz"
//, - H^ has the meaning already given, and J^ - J^ is rela-
tively to z the same as H^ - H^ relatively to y.
CALCULUS OF VARIATIONS.
3G3
875.
"We now proceed to find a maximum or minimum
value of U on the suppositions of the preceding Article.
(1) If 7/ and z are independent, in order that hU may
certainly vanish we must have
K=Q and L = 0;
and also
ir,-iT, + J,-Jo^o.
The values of y and z in terms of x must be found by
solving the differential equations K=0, L = 0; and the
arbitrary constants which occur in these solutions must be
determined by equating to zero the coefficients of the arbitrary
quantities %„, Bij^, (S ^^] , ... Bz^, Bz^, (B -j^j , ... which occur
(2) Suppose however that y and z are not independent,
but that they are connected by the relation <^ (x, y, z) — 0,
which is always to hold. Since this relation is supposed to
hold alwavs, we have also
(ji^x, y + By, z + B2) = 0;
and therefore ultimately
i^By+^Bz^O.
ay ^ dz
rxi
Thus the integral I {I^By + LBz) dx becomes
r
d(f)
- By dx,
I.
dz .
and in order that this i
dition
may vanish
K L .
d(f> d<fi'
dy dz
we have the single con-
364 CALCULUS OF VARIATIONS.
and from this differential equation combined with <^{x,y,z)=0,
we must find y and z.
As before, we must also have
876. For an example we take the following problem : to
determine a line of minimum length on a given curved surface,
between two given points.
Here we have
thus A = — J TTj— — 75— — -^ > -^ — 7 77T~, — 'J I „'2\ >
let (f) [x, y,z)=0 be the equation to the surface on which the
line lies. Then by the preceding Article we have, as the con-
dition for a minimum,
d y' d z'
d^ d^
dy dz
Let s represent the length of the arc of the curve ; then
y _ dy , / dz
Thus the above equation may be written
d\j d^z
ds^ _ ds^ /^x
d</) (^0
dy dz
From this we may conjecture by symmetry that each of
these fractions is equal to
d£^
d^'
dx
CALCULUS OF VARIATIONS, 3G5
and tliis we can demonstrate; for from (1) each of the frac-
tious by a known theorem of algebra is equal to
d\) cZ*?/ dz d'-s
ds ds' dsds'
d}i d(f) dz d(f)'
ds dy ds dz
ami since the equation (f)(x,7/,z) = holds for every point of
the curve, we have
d(f) dx dcf) dij d(f) dz
dx ds dy ds dz ds '
also by a known theorem
dx d^x dy d'y dz d^z _
ds ds^ ds ds^ ds ds'
Hence a line of minimum length is determined by the
symmetrical equations
,(2).
d'x
d^l
d'z
ds'
dr
ds'
d(f) "
dx
dy
~ d(ji
dz
It is proved in works on Geometry of Three Dimensions
that the equations (2) shew that the osculating plane at any
point of the curve contains the normal to the surface at that
point. Such a curve is called a geodesic curve.
877. Let us suppose that instead of being drawn be-
tween two fixed points, as in the preceding Article, the curve
is to be drawn between a fixed point and a fixed curve. Let
x„ correspond to the fixed point, and x^ to the fixed curve.
We have to consider the terms denoted by H^ + J^. As in
Art. 371, we find that these are
'v..-^(|).%.4.(l),^^.
86 G CALCULUS OF VARIATIONS.
Now since the extremity of the required curve is to lie on
a given curve we may suppose that at this extremity there
are two relations to be satisfied, which we may denote by
Then, as in Art. 369, we shall find that
Substitute in 11^ + J"^, and by reduction we obtain
and in order that this may vanish we must have
and this shews that the required curve must cut the fixed
curve at right angles.
Suppose that from a fixed point on a given surface geo-
desic curves of a given length are drawn in every direction,
then the other ends of these geodesic curves will form a
locus such that every one of the geodesic curves cuts it at
right angles. For the locus may be taken as the fixed curve
of the preceding investigation, and so by that investigation
any geodesic curve cuts the locus at right angles.
Relative Maxima and Minima.
378. A class of problems still remains to be considered,
called problems of relative maxima and minima values. Sup-
pose we require that a certain integral U shall have a maxi-
mum or minimum value while another integral W, involving
the same variables, has a constant value ; for example, we
may require a curve which shall include a minimum area
tmder a given perimeter. Here we do not require that 8 U
shall always vanish, but only that it shall vanish for such re-
lations among the variables as give a definite constant value
CALCULUS OF VARIATIONS. SG7
to W; that is in fact, wo require that SIT shall vanish for
all such relations among the variables as make 5 IF vanish.
The problem is solved by finding a maximum or minimum
value of t/+ air, where a denotes a constant; for in this
solution we ensure that BJJ+aSW necessarily vanishes, and
therefore 8 f/ must vanish whenever Sir does. The constant
a occurs in the solution, and its value must be determined
by making the integral W have the constant value which is
supposed given.
If we require that W shall be a maximum or minimum
while U remains constant, we shall in the same way proceed
to find the maximum or minimum of W+bll, where 6 is a
constant ; and if we suppose h—-, we obtain the expression
-iU+aW). Thus the same solution will bo obtained for
a ^
this problem as for that in which U is to be a maximum or
minimum while W is constant.
We now proceed to some examples.
379. It is required to find a curve of given length join-
ing two fixed jDoints, so that the area bounded by the curve,
the axis of x, and ordinates at the fixed points may be a
maximum.
Here U= ^ydx, W = rV(l +/) dx ;
•) Xo J Xq
let V=y + a/^(l ■]- p°), then we have to investigate a raaxi-
mum value of I Vdx. Under the integral sign we have
J Xq
only y and p ; hence for a maximum, by Art. 357, we must
have
V=Pp + C^,
that is, 3/ + rt V(l + p') = ^(JXf ) "^ ^^ '
that is, y + __^=C,
368 CALCULUS OF VARIATIONS.
Thus 1+/
(^
therefore ( ^ ) = — = -^ — \-7=r^ — ^ ;
therefore x-^ C^ = ^J[a^ - {C^- yf].
This shews that the required curve is a circular arc.
Since the extreme points are supposed fixed, the part of
SFAvhich depends on the limits vanishes.
The constants C^, C^, a must be determined by making
the circular arc pass through the given fixed points and have
the given length between them.
880. Given the length of a curve, find its form so that
the depth of the centre of gravity may be a maximum.
Take the axis of x horizontal, and the axis of y vertically
downwards. Let h denote the length of the curve ; then the
1 pi
depth of the centre of gravity is ^ I 2/ V(l +!>') dx, and the
length is V(l +i^^) d^-
J Xq
Xo
Let F= ^ y V(l +/) + a V(l H-/),
then we require a maximum value of Vdx.
Here by Art. 357 we must have
V=Pp+C„
that is,
therefore -^^ = &(7,;
theretore 1 + p = ■ ' .,, .,.j ,
and therefore
CALCULUS OF VARIATIONS. 8G9
dx 10
hence x = A\og[y + B+ sj[{y + Bf - A"]] + C,,
where C^ is a new constant, and A = bC^ and B = ah.
This equation shews that the required curve is a catenary.
If the ends of tlie required curve are supposed fixed, the terms
depending on the limits vanish, and the constants A, B, C^
must be determined by making the catenary pass through the
fixed points and have a given length between them. Suppose
however that instead of being fixed the ends are only con-
strained to lie on fixed curves. By proceeding as in Art. 871
we obtain the following limiting terms :
V,dx^ - V,dx^ + P,%i - PoS^o.
Consider the terms with the suffix 1 ; we have V^dx^+P^Si/^,
that is, (I + .) V(l + i^n d., + (f ■ + a) -0^^ .
Now supposing y = -\^{x) the equation to the fixed curve,
we have hy^ = [^'{x^) —p^ dx^, so that the term reduces to
^^ + "^ [i-\-v,^'{:x,)]dx,.
To make this vanish we must have 1 + ^i'»/^'(^,) = 0, for
7/^-\-ab cannot vanish, as then x^ would be impossible. A
similar result holds at the other limit ; and thus it appears
that the catenary must cut the fixed curves at right angles.
881. Given the surface of a solid of revolution, to find its
nature that the solid content may be a maximum.
Take the axis of x as the axis of revolution. Then the
surface is Stt 2/V(1 +i^^) ^-^i ^^^ ^^^ volume is tt I y dx.
Let V=y" + ay>s/(l +j)^) ; then we have to find a maxi-
mum value of 1 Vdx. Here by Art. 357 we must have
J Xq
T. J. c. 24
o70 CALCULUS OF VARIATIONS,
that is, y^ + ay V(l +/) = ,,ff ... + C,
therefore ^ + -j^^ = C.
This is a differential equation to the curve which Avould by-
revolution generate the required surface. Supposing that the
ends of the generating curve are required to pass through
fixed points, the terms at the limits vanish.
If either of the fixed points is on the axis of revolution, the
value 1/ = is to satisfy the equation to the curve ; thus (7 = 0.
Then the general equation reduces to
this gives a circular arc as the generating curve.
382. Given the mass of a solid of revolution of uniform
density, required its form so that its attraction upon a point in
its axis may be a maximum.
Let the axis of x be taken as that of revolution, and the
position of the attracted point as the origin.
Let the solid be divided into indefinitely thin slices by
planes perpendicular to the axis of x. If y represent the
radius of a slice, x its distance from the attracted point, k its
thickness and p its density, the attraction is (see Statics,
Chapter Xiii.)
27roK -fl - ""
ZTTpK
{ ^{x' + f
Therefore the whole attraction of the solid is
and the mass of the solid is
X,
irp j y-dx.
CALCULUS OF VARIATIONS. 871
Thus let V= 1 — yj-^ 57 + ail"; then we have to invcsti-
gate a maximum value of Vax.
Xa
dP
The condition N — 7- + = reduces here to N = 0,
ax
that is, 2ay -\ ^ — -, = ;
{x^ + y^f
therefore 2a (a;' + y'f- + x = 0.
If we suppose the limits a;, and x^ susceptible of change
we have the limiting terms V^dx^ — Vjioc^ ; and to make these
vanish we must have Fj = and Vq = 0; this leads toy^ = and
1/^ = 0. Thus the solid must be formed by the revolution
round the axis of x of the whole closed curve determined by
the equation 2a {x^ + y^ ^- a; = ; the value of a must be
found from the condition that the mass, and therefore the
volume, is given.
Douhle Integrals.
382. We shall now consider the problem of finding a
maximum or minimum value of a douhle integral; and we be-
gin by finding the variation of a double integral.
Let z he a function of the independent variables x and y at
dz
present unknoAvn ; let F be a given function of x, y, z, -j-
/7f C^L fill
and -T^; let Z7= I I Vdxdy; the integration is supposed
effected with respect to y first, and the limits y^ and y^ are
supposed given functions of x. It is required to determine
what function z must be of x and y in order that U may
have a maximum or minimum value.
Let hz denote an indefinitely small arbitrary function of x
and y ; let SF denote the variation made in F when z receives
the variation hz, and let 8 U denote the variation in U; then
we have first to obtain an expression for h U.
24—2
372 CALCULUS OF VARIATIONS.
Let L denote the partial differential coefficient of V with
respect to z, M the partial differential coefficient of V with
dz
respect to 7- , and N the partial differential coefficient of V
dz
with respect to -j- ; then we have
dx dy '
where, as heretofore, we confine ourselves to the first power
of the indefinitely small quantities. Hence
The value of S F may be written thus ;
and therefore
The differential coefficients with respect to x and y which
are here indicated are complete differential coefficients.
Also [''' f "' ^ {Mz) dxdy = [''' [{Mz)^ - {mz\] dx,
where (NSz)^ denotes the value of NSz when y^ is put for y,
and {Ndz)^ denotes the value of N8z when y^ is put for y.
And by Art. 216,
where (MBz)^ denotes the value of MSz when y^ is put for y,
and {M8z)^ denotes the value of MBz when y,, is put for y.
CALCULUS OF VARIATIONS. 873
Therefore ^ ["' ^ {^ISz) dx dy
aro -' 2/0
^"^ Mhzd}j) -(r'lIBzdj/)
2/0 /x^Xi \' l/o J x-=-x„
Therefore S L = 1 [^ ~ 'dx ~ dTj '^
+ fj' \{mz), - {Nhz)}^ dx
+ {\''Mhzd^ -(T'lmzdu)
-[^,^mz)^^-i^Mlz)M^x.
If the limits y^ and y^ are constants, the terms in the last
line vanish.
Of the four terms which compose SfT" it will be seen that
the second is similar in character to the third, and might be
expressed in a similar manner.
We have supposed that the limits of the integrations are
not susceptible of change ; if they are it is easy to see that
we must add to the expression for 3^7 the terms
dx f ''' Vdy] - (dx r Vdij)
+ r{V,dy,-VJy,)dx.
J Xo
In geometrical applications the limits of the integrations
with respect to x and y will frequently be determined by the
perimeter of a closed curve ; in this case y^ = y^ both when
x = x. and when x = x ; and therefore MBz dy and
/•ji
dx " Vdy vanish when x = x^, and also when x = x^,
- 2/0
S74 CALCULUS OF VARIATIONS.
884. In the value of 5C/ found in the preceding Article,
there is one term which is a double integral involving Sz
under the integral signs, and various single integrals de-
pending upon the limiting values of Bz. By the method
already used in Art. 354, it will follow that 8U wall not
certainly vanish unless the coefficient of Bs under the^ double
integral sign vanishes; thus for a maximum or minimum
value of U we have as a necessary condition
J- cUI dN _^
dx dy
This is a partial differential equation for finding z in
terms of x and y ; and we may say that the arbitrary func-
tions which occur in its solution must be determined so
that the remaining terms in Sf/" may vanish. But the dif-
ficulty of integrating the partial differential equation in
general prevents any practical examination of these terms
at the limits.
385. As an example, let it be required to determine a
surface of minimum area bounded by a given curve.
Here by Art. 170,
-/:7:/i-(£)*Hi)}-^--
let us put as usual
dz dz d'z_ ^l__^ drz^
dx=^'' d^r^' d^''"^' d-^dy-'' df '•
The condition for a minimum reduces to
dM . dN
dx dy
= 0,
^1, ^ • + d p , d q _r.
that IS, to -;: -7:^- o— ^ + -j If-, ,.-i, ^2> ~ "'
that is, to
r (1 +/ + 2') - {pr -\-qs)p^t{\ +/ + 2') - {ps + qt)q=^ 0,
that is, to (1 + q") r - '2pqs +{l+p")t = 0.
CALCULUS OF VARIATIONS, 375
It is shewn in works on Geometry of Three Dimensions
tliat this equation indicates that the required surface is such
that at every point the two principal radii of curvature are
equal in magnitude and of contrary signs.
Since we suppose the boundary of the required surface
to be a fixed curve Bz vanishes all round this boundary ; thus
the terms relative to the limits in BU all vanish.
Discrimination of Maxima and Minima values.
S8G. We shall now give some examples which illustrate
the second part of the investigation of raaxirna and minima
values of integrals ; see Art. 849.
Consider the example of finding the shortest line between
two given points. Here
F=V(1+/), U=rVdcc,
Suppose y changed into ?/ + S?/, and consequently p into
p + Bp; put p + Bp instead of jj in V and expand ; thus V
becomes
^(^+^^ + V(iT?) + 2(i+/)^ ■■••
where the terms which are not expressed are of the third and
higher orders in Bp. Thus we obtain
The first of these terms is what we formerly denoted by
BU, and the investigation of the minimum value of U so far
as it has hitherto been carried, consists in making this term
vanish. Supposing then that this term vanishes, and neg-
lectino- terms of the third and higher orders, we have
2.U(l+/f
376 CALCULUS OF VARIATIONS.
If Xj — Xq Is positive, every element of this integral is
positive ; thus 8 U is j)ositive, and therefore a minimum value
of U has been obtained.
S87. Again, take the case of the brachistochrone, when
the extreme points are fixed. Here
Change y ihto y + Sy, and j) into ^ + S^ ; and expand
the new value of V. Thus F becomes
V(l+p') ^ ^i}.^f)8y ^ pZp ^
^ 8 (1 +/)^ (Sy)^ ^ j ^^g;^ ^ ^ (¥)"
8/ 2/(1+/)^ 23/^(l+FJ^
and from this we can obtain h U.
Now by the process of Art. 363 the terms of the first
order in 8 fare made to vanish; then, neglecting terms of
the third and higher orders, we have
,^^p|3(i+rf%)-_ P^yh^ ^^ m u
ixol 8/ 2/(1+/)^ 2/(1+/)*)
We have now to investigate the sign of this expression
when the relation between x and y is that which is deter-
mined in Art. 363 ; and we shall shew by some transforma-
tions that hU is> positive.
Since 3/^ (1 +/)' = (2^)*,
we have 3,. r--(3(2a)^ (%)^iLM^
CALCULUS OF VARIATIONS. 377
and as the extreme points ai-e supposed fixed Sij vanishes at
the limits ; therefore
Now -^f^-U- ?^ ^-^'' = -- --^-^-^^^^
dx\y/ y^ chj if f y' f
Therefore f '* ^-^^^ da: = \ f '' {ly^ ^^ dx ;
and w^-i-rm^'^ma..
Thus 8 Z7 is positive, and therefore a minimum value of U
has been obtained.
The discussion is much simplified by taking the axis of x
vertically downwards, keeping x as the independent variable,
888. The preceding Article shews that it may be possible
to change the expression of the second order to which SCT is
reduced by our previous investigations, from a form in which
the sio-n is uncertain to a form in which the sign is obvious.
A general theory with respect to suitable transformations of
Ruch terms of the second order has been given by Jacobi ;
for this we refer to the works named at the end of the present
Chapter.
It may be observed that many of the problems discussed
in the Calculus of Variations a're of a kind in which we may
infer, with more or less certainty, the character of the result
from the nature of the particular problem. Thus, for instance,
we may perhaps see in a particular case that a least value
must exist ; so that if a solution presents itself, and only one,
which may be a maximum or a minimum, we infer that it
must correspond to the least value.
889. In the problem discussed in Art. 385 it is easy to
shew that the result really gives a minimum. Here
378 CALCULUS OF VARLiTIONS.
V= V(l + F + q'), U= r r V(l +/ + ql dxdy.
•1 XaJ Va
Suppose z changed into z + hz, in consequence of which p
becomes p + hp and q becomes q + hq. Thus V becomes
W pq^p^q J (1+
2(l+p^ + 2^)i (l+/ + 2^)t 2(1+/ + 2^)^
(1 + r) {SpY _ pqSpSq (1 +/ ) (S^)^
Then supposing the terms of the first order made to
vanish, and neglecting terpis of the third and higher orders,
we have
^ 1 p. fy^ {SpY+{Sqy + (qSp-pSqy
Thus the term under the integral signs is necessarily
positive ; so that a minimum value of U has been obtained.
Condition of InUgrahility.
890. In Art. 854 we have found that /f = is a neces-
sary condition for the existence of a maximum or minimum
value of the integral there considered. It may however
happen that in certain cases the relation K = is satisfied
identically ; this case we proceed to exemplify and interpret.
Suppose we are seeking a maximum or minimum value of
K\!/ y yi
CALCULUS OF VARIATIONS. 379
/ '2 "
Here F= ^ - -4" + -^ ,
y y y
^ dV V 2xf
^~ dy- f^ f
xy"
dV 1 2xj/
dy'-y / '
^ dy" V'
^^ dx'^ dx'~ / • / y'
On collectinsr the terms it will be found that
dx dx'
vanishes. Thus the relation /t = is an identity in this
example, and we cannot obtain from it any value of y.
In this example we shall find that
/
Fc7^ = ^,
y
that is, the integral Vdx can be pbtained without assigning
the value of y in terms of x. Thus if we wish to find a
maximum or minin^um value of Vdx, we must mvestigate
a maximum or minimum value of ( -— ) — ( -^ . We are
V J/ A \y K_
therefore not concerned with the maximum or minimum of
an undetermined integral expression of the kind hitherto
880 CALCULUS OF VARIATIONS.
considered, but with the maximum or minimum of an expres-
sion free from the integral sign.
This species of maximum and minimi:m problem is con-
sidered in some of the exhaustive treatises on the Calculus
of Variations ; as it does not present much intei'est we will
refer the student to such works.
391. We shall now prove universally that the necessary
and sufficient condition in order that V may be integrable
without assigning the specific value of y in terms of x, is that
K = should be identically true. An expression which is
integrable without assigning the specific value of the depend-
ent variable in terms of the independent variable is sometimes
said to be integrable per se, and is sometimes said to be ira-
mediately integrable.
392. We first prove that the condition is necessary.
Suppose that V involves x, y and the differential coefficients
of y with respect to x up to -7-^ inclusive.
If the function V is immediately integrable the integral
Vdx can be expressed in the form
Xa
whore the form of the function denoted by <^ remains un-
changed whatever may be the value of y in terms of x. Now
suppose that y receives such a variation as leaves the values
of y and its differential coefficients at the limits unaltered ;
then from the value of I Vdx it follows that
J Xo
B Vdx = ;
f
J X
L
CALCULUS OF VARIATIONS. 381
thus by Art. 352
r=^'. (dV d dV d' dV I , ^
But this cannot be true whatever Sy may be, unless
dV__^dV _^d/F_
dy dx dy' dx^ dy" * " '
and unless this is identically true it determines yas a function
of X. Thus if Fis immediately integrable the relation K=
must be identically true.
Next we shall shew conversely that if this condition
holds V is immediately integrable. It is usually considered
sufficient to say, that if this condition holds the variation of
Vdx depends solely on the liviiting values of x, y, and
the differential coefficients of y ; and therefore I Vdx must
itself depend solely on these limiting values, that is, V must
be immediately integrable. We shall however reproduce a
more satisfactory demonstration which has been given of the
proposition.
Suppose F= (/) {x, y, y',y",...).
Let u and v denote two functions of x at present unde-
termined ; let a denote a quantity which we shall vary inde-
pendently of x. Let -^ {ol) denote what F becomes when we
put u + av instead oi y, and u +av' instead of y', and u" + av"
instead of y", and so on ; thus
^Ir (y.) = (f) [x, u + av, XL + av , w" + av", . . .).
Differentiate both sides with respect to a, so that we have
a result which we may denote thus,
^^'^ du'^^du'^'^du"'' +
382 CALCULUS OF VARIATIONS.
Integrate both sides, from a = to a = 1 ; thus
that is, we have the following identically true,
(j) {x, u + V, It + v, u" + v", ...)
= (p {x, u, u', III' , ...)
[du clu aib J
Integrate both sides with respect to x ; thus
I ^ {x, u + V, it' + v', il" + v" , ...) dx
= I ^ (a"y M, u', u", ...) dx
+ I du.
\~v + -j~,v' + -j^,v" + ...[dx
[ait die du
where in the last term the order of the independent integra-
tions has been changed.
By integration by parts
d(b , , d(b f d d(b ,
-j-, V dx = V -r^,— Iv -r~ -^h dx,
du du J ax du
[d(f) „ , , dd> d d(b f d'^ dd) ^
j^^^^=^ d^'-'rxd^'-^rd?d^"^''^
and so on.
Thus
CALCULUS OF VARIATIONS.
I (f) (x, u 4- V, II + v, u" + v", ...) dx
383
= l(/>(a7, u, u, ib",...)dx
'^]o"\du' dxdu"^dx'du"'
fi , fd(b d dcj) \ ,
V {Yy,--j--j-777+ •'• Ida
doL
+
+
+ 1 da
d(f) d d(f) d^ d(b
du dx du dx^ du"
d^dj)
dx^ dii"
dx
Now by supposition the relation K= is satisfied identi-
cally whatever may be the value of ?/ ; so it is satisfied if
u + av be put for y. Hence
d<^ d d(f> d" d(p
du dx du dx^ du"
0.
The functions u and v are at present in our power; put
y — u for V and we have
\^{^> 2/> V' y",-")dx
= l(f>(x, u, u, u", ...) dx
^y '},\du' dxdu'^dx'du ;
■^^y-''^?JS-'-Tx^'^'-)^''
+
38-i CALCULUS OF VARIATIONS.
Thus I Vdx is here actually exhibited as an expression
consisting of terms, one involving only ordinary integration
with respect to cc, and the others ordinary integration with
respect to a. The function u is still in our power; it should
be chosen so that none of the quantities which occur become
infinite or indeterminate; it may happen that consistently
with this limitation we may put u = 0.
393. It will now be easy to give the necessary and suffi-
cient conditions for ensuring that a function shall be integrable
per se more than once.
Let Fhave the same meaning as before.
We have, whatever V may be,
I \ I VdxY dos = sc j Vdx — I x Vdx.
In order then that V may be integrable per se twice, the
condition must of course be satisfied which ensures that it is
integrable per se once ; and then the only additional condition
is that a; F must also be integrable ])sr se once. Thus in order
that V may be integrable ^jer se twice, the necessary and
sufficient conditions are that the following relations must be
identically true,
dV_d_dV ^dV_
dy dx d})' dx^ dy" ^ ''
dVx d dVx d" dVx
dy dx dy' dx^ dy" ^ ''
We may modify the form of (2). For
dVx_ drr dVx_ dV dVx_ dV
dy '""dy' dy '"^ dy" dy"~''df' '
£dVx^ d_dV dV
dx dy' dx dy' dy '
^dVx^ ^dV_ ^ d_dV
dx' dy" ' dx^ dy" " dx dy" '
CALCULUS OF VARIATIONS. 3Si
dx' dy'" ~ "^ dx' dy'" ^ dx' dy" '
Substitute in (2) and omit the terms which are zero by (1) ;
then we obtain
dy dx dy" dx^ dy'" " ^
Thus (1) and (2) may be replaced by (1) and (3).
By a formula given in Art. 55 the 'n}^ integral of any pro-
posed expression is exhibited in terms of n single inte-
grals. From this formula we infer that in order that V may
be integrable per se n times, it is necessary and sufficient that
each of the following expressions should be integrable per se
once: V, xV, x'V, x^'-'V.
For example, in order that V may be integrable ^jer se
three times, besides the conditions (1) and (2) or (1) and (3;,
the following must be identically true,
dV^_±dy^^dV^_ ^^ '
dy dx dy dx^ dy" '" ^
We may modify the form of (4). For
d^dVx^ _ ^d_dV dV
dx ay dx ay ay
^ il^ ^ar^^^^ + ^x~^ 2 ^— ^
dx^ dy" dx^ dy" dx dy" dy" *
d' dVx'^ ^ d? dV , ^ d" dV . ^ d dV
— iC 7 3 7 '/' ~r VX -. 2 7 /// "r t)
dx^ dy"' dx^ dy" dx^dy" dx dy'" '
Substitute in (4) and omit the terms which are zero by (1)
and (3) ; then we obtain
dtj" \.ld,xdy" '^l.^dx'dy ~ ^'^''•
Thus (5) may be taken instead of (4), in conjunction with
(1) and (2) or (1) and (3).
T. I. c. 25
o
86 CALCULUS OF VARIATIONS.
Addition on the Variability of the Limits.
394. In the method we have adopted of treating problems
involving changes of the limits we have followed the example
given in two most elaborate works on the subject, those of
Strauch and Jellett ; and we decidedly recommend this
method as the best. We do not ascribe any variation to the
independent variable, but only to the dependent variable.
Another method however has been frequently adopted, and it
should be explained in order that the student may understand
any reference to it which may occur in his reading. In this
method a variation is ascribed both to the dependent and
independent variables.
Let X become x-\-hx and let y become y + Sy, ^x and hy
beino- indefinitely small arbitrary functions of x) it is required
to find the variations or -j- , -j~^ , ...
w?/ dti
"We denote the variation in -i^ by 8 -^ ; therefore
^dy ^ d(y + B y) _ dy
dx d{x-^ hx) dx
dy_^dB^
dx dx dy
dSx dx
dx
_ dy d Sy dy d Sx dy
dx dx dx dx dx '
neglecting small quantities of the second order.
Thus adopting the usual notation for a differential co-
efficient, we have
^ , d'Su , d6x d(Sy — y'Sx) , ,,->
dx "^ dx dx
dx
^ , „. d(8y-y'Sx)
or ^y — y ^^^ = ■■ - - ■ ■_ — .
CALCULUS OF VARIATIONS.
la this result change y into y ; thus
hy"-y"hx = '^^^y^
dx
Similarly Sy -y 6x= ^/ »
and so on.
Put ft) for hy — y'hjc ; thus
p. // tt ft) ,/, ~,
ts til (t (O rr>r<^
387
Now let F be any function of x, y, and the differential
coefficients of y with respect to x; and let t^= I Fi^a;.
J rro
Let it be required to express the variation of Z7 which arises
from the variations Bx and By in x and y respectively. Let
8 F denote the change made in V; then
Xl
d (x + Bx)
Xo
dx
BU=j {V+B]^'^'-^^^^^dx
= rv^4^dx+rBVdx,
J g;^ ax J X(,
neglecting a term of the second order.
Vdx
Now
[v~dx=VBx-
dx
'dr
dx
Bx dx,
therefore I V'^ dx= {VBx)^-{VBx\-^ -~ Bxdx,
25—2
388
where
CALCULUS OF VARIATIONS.
dx
denotes the complete differential coefEcient of
V with respect to x.
dV'
dx
hx[ dx.
. . »„ dV^ , dV^ dl ^ . dV ^ „
And oV=-rox + ~j~6y+-j-,by+~j-^,hy + ... ,
dx dy "^ dy ^ dy
-dr] dV dV . dV ,. dV ,„ ^
-^dx^dyy^dy'y ^di'y +•••'
dx
thus
and finally
'dV
dx
- dV dV . dV „
CX= -J- C0 + -^-7 Oi + -j^-, CO +
dy dy dy
SU={VSx\-{VBx),+j^(^'La,+
dV ..., ,.
d.i
dV
"We need not proceed further as we have arrived at a
result equivalent to that in Art. 368; we have here co instead
of the By which occurs there, and Bx^ and Sx^ for dx^^ and dx^
respectively.
In geometrical applications it will be observed that x and
y become by variation x + Bx and y+ By respectively. Thus
^1 + ^^1 ^^^^ correspond to the x^^ + dx^ of Art. 369, and
y^ + By^ will correspond to the iY+-7~dx\ of Art.
369.
Discontinuous Solutions.
395. Some problems in the Calculus of Variations admit
of discontinuous solutions, and as the subject has attracted
much attention in recent times a few words may be here
conveniently devoted to it.
Let there be an integral I cj) dx which is required to be
a maximum or a minimum, where <^ is a given function of
X and y and the differential coefficients of y with respect
CALCULUS OF VAEIATIONS. 889
to X. Change y into y+By; then in the usual way we
obtain for the variation of the integral to the first order an
expression of the form L + I MZydx, where L depends on
the values of the variables and the differential coefficients
at the limits of the integration. Now if hy may have either
sign we must have Jl/ = as an indispensable condition for
the existence of a maximum or a minimum.
Suppose however that owing to some conditions in the
problem w& cannot always give to hy either sign : for ex-
ample suppose that throughout the whole range of the inte-
gration hy is essentially positive, then it is no longer necessary
that M should vanish. If M is positive througli the whole
range of the integration we are sure of a minimum ; and if
M is neo^ative through the whole range of the integration
we are sure of a maximum. We assume here that we are
able to satisfy the condition Z = ; or to ensure that L shall
be positive in the former case and negative in the latter case.
Next suppose that hy may have either sign through part
of the range of the integration, but that it is essentially
positive through the remainder of the integration. Then if
M vanishes through the former part and is positive through
the latter part of the range we are sure of a minimum ; and
if M vanishes through the former part and is negative through
the latter part of the range we are sure of a maximum. We
assume as before that the condition relative to L can be
satisfied.
For illustration we may take the problem which first sug-
gested these remarks. Required to determine the greatest
solid of revolution the surface of which is given, and which
cuts the axis of revolution at two fixed points.
With the usual notation we have to make tt i y^ dec a,
maximum while Stt | y V(1 + p"^) dx is given. Let a be a
constant at present undetermined ; then we have by the well
known theory to make u a maximum, where a denotes
^[if + Uy^{\+f)]dx,
390 CALCULUS OF VARIATIONS.
"We obtain Bu = L + j M By dx,
where M stands for 2y +2a^/{l +})') - 2 ^ J(T+P^) '
By the known principles of the subject we put 31=0,
and this leads in the usual way to ,-, "j^ 2x = & - 2/^ where
b is another constant, which is introduced by the integration.
Since the curve is to meet the axis of x at given points
we have y = at those points ; hence 6 = 0, and the equation
reduces to
^ + /=0.thatis,{^ + y
= 0.
Take — ——ir. + w = ; this leads to a circle which has
V(l+/) .
its centre on the axis of x and its radius equal to - 2a.
Let A and B denote the given points on the axis of x.
If the given surface is exactly equal to that of a sphere on
AB as diameter such a sphere fulfils all the conditions of
the problem.
But if the given surface be not equal to that of a sphere
on AB as a diameter, suppose C and I) points on the axis
such that the given surface is equal to that of a sphere on
CD as diameter. Then we obtain a discontinuous solution
by taking for the generating curve the part of the axis of x
between A and C, the semicircle on CD as diameter, and
the part of the axis of x between I) and B. This solution
was first suggested by observing that the fundamental equa-
tion obtained above splits into the two factors y = and
2a „
We shall see on examination that if vanishes for the
semicircle on CD as diameter ; and for the parts of the axis
of X which enter into the solution M reduces to 2a. Thus
EXAMPLES. 391
wlicn L is made to vanish Zu reduces to 1 2a hj dx^ for limits
corresponding to ^ C and DB. Now Sy is essentially posi-
tive for all this range, and 2a is negative as we see from its
geometrical meaning. Thus hit is a negative quantity, indi-
cating the existence of a maximum.
On this subject the student is referred to the Researclies
in the Calculus of Vacations, principally on the theorij of Dis-
continuous Solutio7is, by the present writer.
89C. For further information on the Calculus of Varia-
tions the student may consult Professor Jellett's treatise, and
the History of the Progress of the Calculus of Variations
during the Nineteenth Century, by the present writer.
The most interesting examples in this subject are those
which are connected with physical science, as the problem
of the brachistochrone ; accordingly we shall include some
more applications of this kind in the following selection for
exercise.
EXAMPLES.
1. A curve of given length has its extremities on two
given intersecting straight lines : determine its form
when the area included between the curve and its
chord is a maximum.
2. Determine a plane closed curve of given perimeter which
shall include a maximum area.
(See History... y page 68.)
3. Required to connect two fixed points by a curve of
given length so that the area bounded by the curve,
the ordinates of the fixed points, and the axis of
abscissae shall be a maximum, supposing the given
length greater than is consistent with the solution ob-
tained in Art. 379.
(See History..., page 427.)
4. A rectangular dish is to be fitted with a tin cover of
given height having the ends vertical : determine the
form so that the amount of material used may be the
least possible.
892 EXAMPLES.
5. A mountain is in the shape of a portion of a sphere,
and the velocity of a man walking upon it varies
as the height above the horizontal great circle of the
complete sphere : shew that if he wishes to pass from
one point to another in the shortest possible time, he
must walk in the vertical plane which contains the two
points.
G. When a curved surface can be divided by a plane into
two symmetrical portions the intersection of the plane
and surface, when an intersection exists, is in general
a line of minimum length on the surface.
(See History..., page 365.)
7. Find the minimum value of
fl(^g)'sin. + < y + ".-^"'">' U.
J [ \dxj sm a; J •
(See Philosophical Magazine for December, 1861, and July
1862.)
8. Required the minimum value of I |-^] dx under the
following conditions : 2/0 ~ -^> I ~ dx = — l.
(See ZTtstory..., page 432.)
9. Required the variation of 1 Vdx, where F is a function
of X, y, ~- , -j~, ... and v, where v =■ \ Vdx, and V is
also a function of x, y,-j- , -y-^ , . . .
(See History..., page 21.)
10. Let s denote I v'(l +P^) dx, and let j> (s) be any function
JO
of s ; then the relation between x and y is required
which makes I (s) dx a maximum or a minimum
Jo
while I ^(1 +P^) d^ ^ss ^ given value, a being a con-
J
stant. For a particular case suppose (/> (s) = s.
(See History. . ., page 453.)
EXAMPLES.
S93
11. Required the curve at every point of wliicli
is a maximum or a minimum.
(See Ilistory..., page 1.)
„ , . 1 dy .
12. Required the curve at every pomt ot wIiicH 2/^ ^^
maximum or a minimum, the variations of y and
being so taken that at any point yx — y
a
dy
dx
da; "J
shall undergo no change by variation.
(See History..., page 41-i.)
13. Apply Art. 375 to prove the point assumed in Art. 363,
namely, that the required curve in thebrachistochrone
problem lies in the vertical plane which contains the
two given points.
14. The form of a homogeneous solid of revolution of
given superficial area, and described upon an axis of
given length, is such that its moment of inertia about
the axis is a maximum : prove that the normal at any
point of the generating curve is three times as long as
the radius of curvature,
15. A given volume of a given substance is to be formed
into a solid of revolution, such that the time of a,
small oscillation about a horizontal axis perpendi-
cular to the axis of figure may be a minimum : de-
termine the form of the solid.
(See History..., page 891.)
IG. A vessel of given capacity in the form of a surface of
revolution with two circular ends, is just filled with
inelastic fluid which revolves about the axis of the
vessel, and is supposed to be free from the action of
gravity. Investigate the form of the vessel that the
whole pressure which the fluid exerts upon it may be
the least possible, the magnitudes of the circular ends
being given.
Result. The generating curve is a catenary.
^O O'
S94 EXAMPLES.
17. Find the equation given by the Calcuhis of Variations
for the transverse section of a straight and uniform
canal, when one of the three quantities, the surface, the
capacity, and the normal hydrostatic pressure, is either
a maximum or a minimum, and the other two are
given, the terminal surfaces and pressures not being
taken into account.
Shew also that when the surface is a minimum and the
capacity only is given, the section is circular; and
when the normal pressure is a minimum the section
is a catenary or two sti-aight lines, according as the
surface or tlie capacity is given,
18. If there are two curves with their concavities down-
wards and terminated in the same extremities, a par-
ticle moving under the action of gravity will take a
longer time to describe the upper curve than the lower
curve, the initial velocity being supposed the same in
the two cases.
(See History..., page 848.)
19. Assuming that a ship's rate of sailing is a function
of the angle which the direction of its course makes
with the direction of the wind, shew that the bra-
chistochronous course between two given positions is
rectihnear, and that unless it be in the straight line
joining the positions it is in two directions always
making the same angle with the direction of the
wind.
(See Philosophical Magazine for September, 1862.)
20. A solid of revolution is to be formed on a given base
with a given volume so as to experience a minimum
resistance when it moves through a fluid in the di-
rection of its axis : determine the figure of the solid.
(See i^esearc/tes... Chapter X.)
395
CHAPTER XVI.
MISCELLANEOUS PEOPOSITIONS.
397. In the present Chapter we shall investigate a few
miscellaneous propositions of interest.
398. It is required to transform the series
1^ _ 1 X 1 x^ _ _1 x'
VI m+ll-x'^ m + 2{l-xy m+ 3 (1 -;r)'"^ "*
into a series arranged according to powers of a;; it being
supposed that ^ is less than unity.
JL ^~ X
X ?/ • •
Put -, = ?/, so that X = -, -— . The given series
1-a; '^' 1+// °
y Jo
1 {y-ir^dy
Jo 1 + z/ io-^ \y y+y ''
^f:_rrdy
111 Jo .V+ 1 *
396 MISCELLANEOUS PROPOSITIONS.
Then by repeated integration by parts we have
y+1 (m + l)(2/+l) «2 + lJo(_y+l/
1 p
■ v"^^ ^ _ y
{in+V){y + V) (m + l)(m + 2)(y+l)'
2 fy ^v^'^^ dxj
and so on.
Thus we see that
1 [yy-'dy
f'U 1+2/
1 f X a? 2x'
' / ... , t \ r ... . ON <
m [m + l (m+l)(m4-2) (w+l)(m + 2)(7/i + 3)
2.3^* ]
'^ (m + l) {m + 2) {m + 3) {m + 4) ■^' " * " j
Hence the required transformation is effected
1
2
For example put m = -x, and divide both sides of the
equation by 2 : thus
, 1 a; 1 ic'^ 1 x^
l-r.-. +
3 1-a; ' 5 {\-xf 7 (1-^)'
\x 2^2 2 Ax
.3
~^ |3"''3.5"^3.5.7'^*
If we put sin'^^ for x this gives a known transformation
a
for ;r : see Differential Calculus, Art. 374.
tan d ""
399. In Art. 62 it is shewn that if we integrate a
function of two independent variables, with respect to both
variables, between fixed limits we obtain the same result
when we adopt either order of integration, provided the
function remain finite between the assigned limits. Con-
versely if by changing the order of integration we change
MISCELLANEOUS PROPOSITIONS. 897
the result it follows that the function must have been in-
finite within the range of the integrations. This principle
has been applied by Gauss to shew that every rational inte-
gral equation has a root real or imaginary.
Consider the expression
Put r (cos 6 + J— 1 sin 6) for x ; then the proposed expres-
sion takes the form P +QJ— 1, where
P =zr" cos nO + py~^ cos {n — \)6 +. . .+p„_ir cos 9 + /?„,
Q = r" sin nO -{-p^i-'^'^ sin (n — 1) ^+...+^„_irsin d.
_ p
Let F= tan * ^^ ; then
dV
^ dd
pdQ
dd
dd
dV
P'^
pdQ
'^dr
dr P' + Q'
Hence Y~Ja iii"^*ol^Gs (P^+ Q")^ in the denominator; and
d'^V
if we can shew that -5 — tt, becomes infinite "within a certain
drdo
range of values for 6 and r, it follows that P and Q must
simultaneously vanish.
We shall take and Itt for limits of 6, and and a
for limits of r, where a is large but finite ; and we shall
d^V . . cZ^F
integrate -, — t-, between these limits. Integrate , — 7-, first
"^ drdd ° drdO
dV
with respect to 6 ; thus we obtain -y— : now take this be-
tween the limits and lit, then the result is zero, for P and
Q and their ditferential coefficients have the same value
when 6 = '2.77 as when (9=0. Hence by adopting this order
398 MISCELLANEOUS PROPOSITIONS.
of integration we obtain zero as the result of the first inte-
gration, and therefore zero also as the final result.
Now adopt the other order. Integrate —j — j-3 first with
drew
d V dP dO
respect to r ; thus we obtain -^-r . Now Q ^n and P -7^
ad dO dU
dV
both vanish when r = 0, so that -j-r vanishes when r = 0.
do
When r = a we have for the value of Q -j-r — P ,^ a series
do dO
proceeding according to descending powers of a ; the first
term of which is — 7ia'^" (cos** nd + sin^ n6), that is — ^m^"* : and
a may be taken so large as to render all the other terms
insignificant in value compared with this. In like manner
P^ + (^ may also be made to differ as little as we please
from its first term, that is from a^".
Hence I ^ — rpi dr = — n,
Jo drdO
that is we have a result differing as little as we please from
this by taking a large enough. Then, integrating with resjject
to 6 between and 27r, we obtain — 2?i7r.
Thus by performing the integrations in different orders
we obtain two different results ; and therefore the function
must become infinite within the range of the integrations :
and therefore P and Q must simultaneously vanish within
that range. Bertrand's Calcid Integral, page 188.
400. It is shewn in Art. 177 that if a curve havins; the
equation y = A+Bx+Gx^ + Dx^ be made to pass through
three given points the ordinates of which are equidistant,
the area bounded by the curve, the extreme ordinates, and
the axis of x is equal to ^ [y^ + 4y,^ + y^ ; where y^, y^, and y^
are the ordinates and h the distance between two consecutive
ordinates. It will be observed that an infinite number of
such curves can be drawn, since there are four coefficients
A, B, C, D at our disposal, and only three conditions to de-
MISCELLANEOUS PROPOSITIONS. S0.9
termine them : thus wc might make the curve pass througli
any fourth point we please. Nevertheless the area men-
tioned remains always the same. This result admits of gene-
ralisation into the following theorem :
Let a curve having the equation
y = A^ + A^x + Ajf + ... + A^^_^x'"~^
be made to pass through 2/i — 1 given points, of which the
ordinates are equidistant, then the area bounded by the
curve, the extreme ordinates, and the axis of x is always
the same.
The demonstration is of pi-ecisely the same kind what-
ever may be the positive integral value of n ; we will suppose
for simplicity that n = 3.
Let 2/j, 7/2, y^, y^, y^ denote the ordinates of the given
points ; and let h be the distance between two consecutive
ordinates. Suppose the first ordinate to correspond to the
abscissa x=0. Then from the elements of the Theory of
Finite Differences we have
X . x{x — Ji) . „ X (x — h) (x — 2h) . ,
^ = ^' + A ^y- + Ani~ ^^- + h' \3 — ^ ^-
x{x-h)(!c-2k)(!c-3h) ,,
+ - ^Mi ""'^
X (x - h) (x - 2/0 (x - 370 (x - 4/0
where ^y,=y,-y„ ^"y, = y,-y,-{v,-y^ = y,-^u, + y„
and so on. Thus the value of A*_y^ involves y, , y„, ... up to
y^; and the value of A^?/j involves y^, y^,--- np to y^, where y^
is the ordinate of any arbitrary sixth point, corresponding to
an abscissa oh.
Now the area which we require = / ydx, so that the
Jo
term which involves A^y^ is
^\' cc {x - h) (x - 2A) {x - 3/0 (x - 4/0 dx,
a |£.'o
400 MISCELLANEOUS PROPOSITIONS.
In the integral put f + 2A for x, then it becomes
rth
-2/1
2fe
[■•ih
that is, (I' _ 47,2^ (^_ 7,^)1^^.
J -2fe
and this vanishes by first principles : see Art. 42. Hence
I ydx does not involve A'^/f ^^^ o^lj ^2/1' ^^I/i> '•• "P to
A^y^; and so when expressed in terms of y^, y„,... involves
these ordinates up to y^ inclusive. This establishes the re-
quired result.
It is scarcely possible that a result so general and so
simple has not been already given ; but the writer has not
met with it.
401. From Wallis's Formula we may deduce in an ele-
mentary way the formula for the approximate value of
1.2.3 ... X, when x is very large. Professor De Morgan
seems to have first noticed this in his Differential and Inte-
gral Calculus, page 293 ; and the process has been put in a
very simple form by Serret : see his Cours de Calcul Dif-
ferentiel et Integral, Vol. Ii. page 206.
According to "Wallis's Formula, as given in Art. 36, we
have
7r^ 2.2.4.4...f2a;-2)(2.r-2) 2x
2 ~ 1. 3. 3. 5.. .(2^-3) (2a;- 1) 2a;- 1 ^ ''
when X is infinite.
1 . 2 3 ... .a;
Now let dt (x) stand for -1^^ ; then it will be
e"'' x'' sj2Trx
found that (1) gives, when x is infinite,
1 2 ~ ^ I
{i>(2x)y
MISCELLANEOUS PROPOSITIONS. 401
and therefore by extracting the square root we have, when x
is infinite,
From the form of ^ {£) we obtain
■ -^- = 1(1 + 1)-^ p);
therefore log ^^ = " 1 + (^ + g) '"S (l + J.)
In this series the terms are alternately positive and
negative. The numerical value of the ratio of the terra
rt" 1
which involves a;""*"^ to the preceding- term is -^. r . - ,
which is certainly less than unity Avhen n is greater than 2,
provided x is not less than unity. Hence the value of the
series is less than t-— ^ , and therefore
12a;-
° </) (a; + 1) x' ^ "
where 6^ is some positive fraction less than — .
From (5) by successive changes we obtain
^°° fl^^TT) + ^"- f{x-^ + • • • + ^"S -^ ^2:«) "
~-x^^ [x + if^'"^{:ix~if '^^■>'
where 6^, 6„, d^,--. are all positive fractions less than -- .
T. I. c. 2Q
402 MISCELLANEOUS PROPOSITIONS.
Hence the sum of the terms on the right-hand side of (G)
is less than z _-„ x ^> that is less than ^-^t— .
(b Lv) . 1
Therefore log , ,\\ is less than :r^— , and therefore when
cc is infinite
= 1 (T).
From (2) and (7) we have when x is infinite
and therefore
r= = 1 + /5»
e "x"' J'Ittx
where ^ vanishes when x is infinite.
Thus the required formula is established.
402. We proceed to some further developments which
are due mainly to Serret.
A limit closer than that assigned by (4) may be found
for log -J-. — —^.
» </) (a; + 1)
For we have
cf>{x)
(1 + x) log
<^(a;+l)
^ ^^\l2x' 12a,''^40x-^ '"^ 2?i(?z + l)^" ^•••j
J ^^ , (n-2)(-ir ■
12.^- 120a,'' "• 2«(n + l)i?i + 2)a;"
In this series the terms are alternately positive and
negative. The numerical value of the ratio of the terra
which involves ^"'^'^ to the preceding term is
n{n — V) 1
n{n-l) + ''i{n-:i)'~x'
MISCELLANEOUS PROPOSITIONS. 408
wliich is certainly less than unity if n is greater than 2, pro-
vided X be greater than unity. Hence
(1 + x) log -,--— — 3-r is less than q-^r- ,
^ ' °0(.c+l) 12j7
and therefore
log -~ — r^ IS less than
4>{x + l) l^x{x + l)'
403. "We have identically
loo- 6 U) = loo- _<K:^ + loo- ii^±l) +
, (h (x ■{- vi) ■, . ,
^ (a;+ m + 1) =" ^ ^ '^
Let (^ (a;) have the form assigned in Art. 401 ; and suppose
m infinite. Then
log (^{x + m + 1) = log 1 = ;
and we obtain
loq- (p{x) = Z lo2[ , , ~,
where 2 indicates a summation with respect to n from
w = to n = CO .
But as in (4) we have
^(}){x + n+l) V ^/ °\x + nj
therefore
log</,(^0 = s|(x + 7^ + 2)log(l+;^J-lj....(8).
404. From the definition of </> [x] we have when a; is a
positive inter/er
log r (^ + 1) =-Iog 27r — a; + (^+9) log a;
+ log^(.r) (0);
404 MISCELLANEOUS PROPOSITIONS.
therefore by (8), when x is any positive integer,
1 /I'
log r (x + 1) = - log 27r — a; + f X + ~ I log x
+ 2|(. + »+l)log(l + --l-)-l| (10).
Eut this equation can now be shewn to hold when x has
any positive value.
For denote by ■>^ (x) the expression on the right-hand
side of (10) ; then it may be shewn by differentiating twice
that
^"(^) = ^ Or + .!+!)-
therefore by equation (2) of Art. 268
Hence, b}'' integration,
log V {x + 1) = -^^ {x) + Ax + B,
where A and B are arbitrarv constants.
But we know that for all positive integral values of x
we have log T (x + 1) = ->\r (x) ; hence A and B must be zero,
and therefore equation (10) must hold for all positive values
of X.
405. By Art. 403 we see that log {x) is equal to the
sum of a series of quantities, which are all positive by
equation (5). Hence log [x) is positive. Hence by equa-
tion (9) it follows that
log r {x + 1) is greater than ^ log "^tt — x ■\- ix + ~\ log x ;
and therefore V {x -\-V) is greater than e~''x'' J'lirx.
We shall now find an opposite limit for F (^ + 1).
By Arts. 402 and 403 we see that
log (h (x) is less than i^^ 2 , r-/ -, \ »
MISCELLANEOUS PROPOSITIONS. 405
that is lofr (b (ic) is less than ^^ ^ •] -I ;
°^^ ^ 12 \x + n x + n+1}'
therefore loir cb (cv) is less than r— — .
Hence by equation (9) it follows that
log r (x + 1) is less than - log 27r — x + (x + -j log x + —^ ,
-J \ «-/ i^x
and therefore
r (.r + 1) is less than e"'^'^ x"" J 'lirx.
40C. We proceed to an investigation of Stirling's Theorem,
which amounts to an expansion of logr(a;+l) in a series
proceeding according to inverse powers of x.
From equation (8) we obtain by differentiating twice
d-'\og<^{x) _ 1 1^ ^ ^ 1
dx^ X 2x' "* (^ + ny '
But for any positive value of z we have
-= e~'^~da, —.= I e~°-ad'X.
z Jo z Jo
Therefore, if x is positive.
But ^e-"°= _ _^ , so that
Integrate twice with respect to x, observing that log ^ (x)
and -J- log (j) (x) both vanish when x is infinite. Thus
CLOD
iog^w=/3(r^.-|-i)''-"'«-
406 IVUSCELLANEOUS PROPOSITIONS.
Therefore by (9) we have
logr (a; + 1) = ^\og2'rr- X + (x + ~j log x
+r?(?^-'^'»'^'"'=' '"^-
Now suppose — — r expanded in powers of a. By Arts. 05
and 123 of the Differential Calculus the result is
^"2^+|2°^"g'°^ + |6 ^— + |27T2~^ •
here B^, B^, ... are the Numbers of Bernoulli, as in Art. 304,
and their values are
■^^^6' ^^^30' ^^^42' ^'"SO' •^«"6U'""'
and /-"^ (^a) denotes that -^;^ — =- is to be differentiated 2r + 2
times with respect to a, and then 6x put for a, where ^ is a
positive proper fraction.
Now, observing that by Arts. 259 and 260,
Jo x"'^'
we have finally
logr (a; + 1) = 2 log 1'Tr-x + (x+^ \ogx
'^2x '6Ax'^ --^ {±r-\)2rx^'-'
+ 2^r^""^'^^^'^'"^^^"^^'-
This formula includes Stirling s Theorem; for that amounts
in fact to removing the definite integral at the end of the
expression just given, and allowing the series to continue
indefinitely.
MISCELLANEOUS PROPOSITIONS. 407
With respect to the early history of Stirling's Theorem
see the Ilistorij of...Prubabiliti/, page 188.
407. As an example of the formula obtained in the pre-
ceding Article suppose r = () ; thus
log r (^ + 1) = ^ log 277 - .c + Ix + -] log X
^J
a
., e« (1 - 7) - 1
_ e"l(a-2)e- + a + 2}
,,„ , , e" [(3-a) e-'^-4xe°-a-3}
/ W = i^^r^Ty •
It is easy to shoAv, by expanding the numerator of /""'(a)
in powers of a, that /'"{''■>■) is always negative so long as a. is
positive. Hence /" (a) continually diminishes as a increases
from to 00 ; and we can sheAV that _/"' (a) is positive so long
as a is : hence the greatest value of /"(a) for positive values
of a is when a = 0. By evaluation we find that f"{'J-) is
- when a = 0. Therefore
log r (^ + 1) = ^ log 27r - a; + (^^ + - j log « + — ,
where X is some positive proper fraction.
This result includes the two limits obtained in Art. 405.
408. Differentiate equation (11) ; thus
- 1 + |j e-'^da
00
'°'^"-'„ a 1.-^:^1- 1'^""''^-
408 MISCELLANEOUS PROPOSITIONS.
But, by Art. 288,
log x= - (e~" — e-°^) dz ;
Jo CO
therefore ^-logr(.+ l)= j^ Cf-S)''^ (I-)-
Therefore by putting .r = we obtain
r (1) Jo Va e"-l/ Jo \oL l-e"",
Hence, by Art. 268, we have another form for Eiders
constant, namely
L •-- e-»</a.
Jo Vl-e"" ocj
409. Integrate (12) and determine the constant so that
the expression on the left hand shall vanish when a; = ;
thus
log r (^ + 1) = f "^ I^^V ""T'lM ^^
Jo i a a (e" — l)j
.'o a ( l-e-"^) '
this presents log F (^ + 1) compactly as one definite integral,
but the form given in (11) may be in general more useful.
THE END.
CUrr.KIDGE: printed by C. J. clay, M.A. at the UNIVEKSITY PKEsS.
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