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I. F. Quinby Horatio Nelson Robinson.

# A new treatise on the elements of the differential and integral calculus

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therefore

that is;

Putting - for x in the first member of this last equation,

we have

r« zp^ (g — 2 )g-i rf^ __ r( p)r(q)

or r 2^-i(a - 2)^-id« = a^+^-i . -^(^)^(g ) .

j36i?. The last formula in the preceding article is a particu-
lar case of a more general formula by which may be expressed,
in terms of F functions, the multiple integral

fff...xP-hj^-^z''-K..{a-~x — y - z...)'-^dxd2/dz...

extended to all positive values of cc, y, z..., which satisfy the
condition a; + y -|- 2. . . <^ a.

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456 INTEGRAL CALCULUS.

Limiting ourselves to three variables, let

Now, by the last article.

Multiplying this by y^^^dy, and integrating with respect to y
from y==Otoy = a — aj, the result is

(a - xi^+r+.-i nq)nr+B) rir)r(s)

^ ' nq + r + 8)'

and finally, multiplying this last by x^^^dx, and integrating
with respect to x from a? = to a? = a, we have

In this, making a = 1, » = 1, wo have

the limits of integration being any positive values of a?, y, z,
which satisfy the inequality x-^-y ■\'Z <^\.

Assume (^y=„ (tj =„ (t^' =i;

then A = J J J t"'' ¥ 4 '' d^rf^rfj,

subject to the condition that ft + <; + 2; < 1 : therefore

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EULERIAN INTEGRALS. 457

270* By means of Formula 2 of the last article, we can
find the volume bounded by the co-ordinate planes, and the
surface having for its equation

(0"+(f/+e7=

Example. When a = |^ = 7 = 2, and p = g = r = 1, the
surface is that of an ellipsoid of which 2a, 26, 2c, are the axes.
Then, by the formula, the volume V oi \ of this ellipsoid
will be

afte r (2)!

and r (-j = \/^7t; for, let u=:J e-^' dx, then also

J

•'0 •^ "^

Now, / / e"^* ~^* dxdy is obviously \ of the volume the

equation of whose surface is z = e-^*"*'*. In terms of polar
co-ordinates, the expression for the same part of the volume is

r^ r zrdddr= H f e-"-' rdOdr.

But

J e-'^* rdr = - ^e-"* ; .' . T e-"* rdr = ^:

68

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468 INTEGRAL CALCULUS.

and JdOt=d} .-. 2^**^0 = |«;
.*. w = j e~*■*rfa? = HV'''•

Now, r/- j = r e-'x'* dx by defimtion,

— ^f e-^ dif=i2te=^ \/« by patting aBr^sy*:

„ abc \ ^\2)> dbc
therefore, V= -g- —,3 ^A ^ T*

J?7J. Differentiatioti under the sign J enables ns to find
new integrals from known definite integrals. Thus,

dx

n

Example 1. \ —f^ — ='-^a""*.

Difierentiating each member of this equation n times with
respect to a, we get

^» 1.2. 3. ..71 ^ 13 5 271 — 1 1 7t

(a?2 -J- a

)«+! 2 2 2 2 a»-^*2'

/.• c?:t? _ 1.3.5...(2 7t — 1) ;g

whence J ^ ^^, ^ ^^„_^, - 2T476~^^i 2^^^ *

/I
a

After 71 —^ 1 differentiations of the two mdmbers of this
equation, it becomes

g-ax^n-i^-,! 2.3...(7* — l)a^»;

that is J* e-«'aj'»-i cfr = €^ (Art 266),

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DIFFERENTIATION UNDER THE SIGN /. 459

The last formula holds good when a is replaced by the
imaginary quantity a + 6 V — 1, in which a is positive ; for

a+ftV—l

+ G

a + bV^^ +G(Art.73).

therefore

Jo a + bV-l'

and, by differentiating this equality n — 1 times with respect
to a, we get

r . ^../-^, 1^ 1.2.3...(n-l)

j?7;?. The formula just found leads to other integrals by
the separation of real from imaginary quantities.

Assume a + ^V— l=p (cos. 0-^-^—1 sin.^?), in which

Then

•/

= r e"*^(cos.6a; — V — 1 sin. 6a;) a?**"^ da?
J

and

1.2.3...n — 1 _ r{n) 1

(a + 6\/ — 1)** P** cos.n^ + V — Isin.n^

= -~-^ (cos.n^ — \/ — 1 sin. ti^) :

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460 INTEGRAL CALCULUS.

= — ^ (cos.n^ -— \/ — 1 siiL7i(?) ;
an equation which may be separated into the two,

r 1.7. ^(^) .

r(«)

cos.w^.

P'*

j?7\$. Making 71 = 1 in the last formula of the preceding
article, it becomes

a^ 4- i^

therefore, denoting by c a constant less than a, we have

But

j^dafy-.o..hxdx^f]^,.

da I e~'^ cos.bxdx = i dx \ e~^ cos.lzda

c •^ J J c

= / cos. toaoj.

J a X

Again :

/a ada _^ 1 a^ + 6^

/ cos.oxax = - I .J , i.» '

Jo ^ 2 c^ + 6^

Making 6 = in the last equation, it becomes

/• e-*'* — e-*" , , a
dxz=l-'j
X c

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J

DIFFERENTIATION UNDER THE SIGN /. 4G1

a result that may also be obtained by multiplying both mem-
bers of the equation

a

by da, and integrating the result between the limits a and c.
274:. In like manner, from the formula

/• b

a^ + 6^

e^*^%ixi.hxdx=.
J

we get

a X 1 c

=z tan.""^ =- — tan.""^ ^ .
6

But

// e^^mi.hxdx^:] dx \ Bm.hxc^da
c •/ J 9 J c

sin. bxdx ;

x

f Bin,iax?a; = tan,~^ =- — tan,~^=-«

J Q X

In this formula, making a = oo , c = 0, it reduces to

/:

-dx = ^

* m,bx 7 «

a? 2

whenJ^;^; whenJ^L.,, the second member becomes — ^:
from which it is seen that the integral / — ^ — dx changes

abruptly from - to — ^, when 6, in passing through zero,
changes from positive to negative.

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462 INTEGRAL CALCULUS,

275. The integral C e-"^ dx=z\y/n (Ex. Art. 270) leads
Jo A

to r e~^dx=z j^Tt; for

J —00 J —00 •/

Now, if we change x into — x, we have

r e-'^'dx^f e-''^dx = ^V^'

f e-"^^ dx z=z j^Ti,

And generally, if /(a?) is a function of the even powers of a?,
that is, such a function that /(x) zzi/^-^ x), then

r /{x)dx=2 r f{x)dx;
for J* /{x)dx=zf f{x)dx+r f{x)dx.

But J f{x)dx=rf{—x)dx=J /{x)dx:

C /{x)dx=2 f^/{x)dx.

In like manner, it may be shown that if/{x) is a function
of the odd powers of x, that is, if /(— x) = — -/(aj), we should
have

r /{x)dxz=0.

e-^ dx=j^7iy putting x^a for
ir, we have

which, by n diflferentiations with respect to a, becomes

J -00 ^ 2*

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DEFINITE INTEGRALS. 463

In this, making a = 1, we have

J —to Ji^

277m Changing a? into ^ + a in the formula
of the preceding article, we get

that is, ^""'/* e-**"2«r^a? = v'^:

But r e-'^-^dx= r c-**-2«;rfa;+ r^e-^^-^^'da?,

J —00 J —00 V

and J e-^^-^'^^cfo^z r"e-**+2aar^a?
by changing a? into — x:

whence

In this equation, replace a by aV— 1 ; then, since
6'^ + e-2-* = e-2ax^^ J. e2a''^~i = 2 cos. 2cca5 (Art. 73),
we have

/* i 1

e""* co&.2axdx=z-e^<^*^/7t.
2 ^

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464 INTEGRAL CALCULUS.

' This example is another instance in which the value of a
definite integral is found by passing from real to imaginary
quantities.

278. Another process by which i e"** cos. 2aoodx may be

found consists in differentiating with respect to a and subse-
quent integration : thus, put

w = I e~'* cos. 2axdx ;
du

then

-— = — I sin. 2aa;e""* 2xd3s = i sin. 2aa?, d, e"*^.
da J J Q

Integrating by parts, and observing^ that, at the limits,
sin. 2aa^""* is zero, we have

— z=z ^f e^'* COS. 2ax2KXjdx — — 2tm:
da J

da

u
But, regarding i^ as a function of a, we have
du

da u da.

Integrating with respect to a, we get

by making e" = C. To determine G, make a = ; then

u = fe-^'dx = l^7t= G:
therefore

/e""'* COS. .200^03 = - f^^j^n.
2

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/e ^0
, —^ = n(c, a, d\

SECTION X.

ELLIPTIC FUNCTIONS.

279. JElUptic Functions or Elliptic Integrals is

the name given to the following integrals : —

— = F(c, 0).

dd^

» (I +asin.^(9) Vl^^^sin:^^ '

The constant c is called the modulus of the fiinction, and is

supposed less than unity ; the constant a, which appears in the

third function, is cflJled the parameter ; and the variable 6 is

called the amplitude of the function. The function is said to

be complete when the limits of the amplitude are and -.

z

The integral of the second order expresses the length of the
arc of an ellipse estimated from the vertex of the conjugate
axis {Art. 240) ; the semi-transverse axis being unity, and the
eccentricity of the ellipse the modulus of the integral. From
this fact, and from the relations whicli exist between the sev-
eral functions, the term elliptic functions has been derived.
Our limits permit us to investigate but a few propositions
relating to such functions.

280. Putting X for sin.t^, the integral of the first order
becomes c' dx

•^0 4/1 — ;

69 466

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466 INTEGRAL CALCULUS.

In like manner, for another value of x denoted by x^j we

have

dxi

•/

Vl-oj'Vl-^c^rcJ

Now assume the relation

^^ + ,. ^^,L„. =0 (1).

Multiply through by the product of the denominators, divide
by 1 — c" a; J , and integrate ; then

J l-c-'x'x' ^J 1-c^xh

1

= constant.
Integrating the first term by parts, we get

J l-c'x'xl '^~ l-c'x^xl

J ' {i-c'x'x\y ■ VT^^,Vi-c*xi

«/ 1 — c^x^x^
In this result, interchanging x and x^ we have the second
term. Adding results, observing that by (1) the terms of the
sum which are under the sign f reduce to zero, we find

a; Vl —xWl- c'^x' + rci Vl - x'' Vl — c'x'

'- ^-^, = const. (2).

Eq. 1 expresses the condition that the variables x and Xi are
so related that the sum of the integrals

dx /•*! dxi

r^ dx /•*! dxi

;

1

shall be constant.

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ELLIPTIC FUNCTIONS. 467

Put r , /i - =T=^ = a, a? = 5^(oc),

Vr^^^ = C7(a), Vl-c^a;^ = i?(a) ;
also

Then, by Eq. 1, we have

* da + d^ = 0:

a -|- ^ = constant = y.

It is also seen from (1) that the constant y is the value of
a?! when a; = ; and, further, when a = 0, we have

therefore, by making the proper substitutions in (2), it be-
comes

^^« + ^)- l-c^j6'(«)}^{^(^)r '
which is the fundamental formula as given by Euler in the
theory of elliptic functions.

281. Suppose the variables <?, di, to be connected by the
equation

r^ do . r<^i ddi p dd

J Vl — c"^8in>^ •^o Vl — c'-'sin.^^i"^'^ o Vl — c^sin.'^^

(1),
or F{c,d) + F{c,d,) = F{c,fx),

in which ^ is a constant. If (?, ^i, be regarded as functions of
a third variable t, and (1) be differentiated with respect to the
latter variable, we have

do do I

Vl -c''8in.2(?^Vl — c2sin.2<?i ^ ^'

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468 IXTEORAL CALCULUS.

Sinco the new variable t is arbitrary, let us assume
do

di

whence, from (2),

1J=V(l-c^8in.'d) (3);

^J = -V(l-c'8in.«<?0 (4).
Squaring (3) and (4), and differentiating, we get

.•. ^?^~372^~ — c'(8m.(?co8.^±sin.(?iC08.^i),

or ^ (<? ifc <?i) = - ^ (sin.2(? ± 8in.2(?i) (5).
Put <? + (?i = qp, and «? — (?i = ip ; then

sin. 2/0 = sin. <)p cos. \ff -\- cos. 9 sin. xpy

sin. 2^1 = sin. <p cos. 1^ — cos. 9 sin. if; :

therefore, from (5), we have

d^cp , . d^xp « .

-jp^ = — c'sin. 9 cos. V, ^r^ = — c* sm. xff cos. 9.

We also have

d^ dip _ /cfe Y _ /^iV — a \ 1 — COS. 20^ 1 — cos. 2/A
dt~dt^\dt) \dt)'^^ ) 2 2 y

„/cos.2^ cos.2<?A . . .

= c^ ( ) = — c^ sin. op sm. yp :

\ 2 2 / ^ r

d^ d^xp

, - =cot.V;, , ^ =cot.9.
agp at/; aqp dip

di di dili

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ELLIPTIC FUNCTIONS, 469

dhp

_ ^ d , . . dcf d jdxp W

But ^^Bm.(y=:cot.(y-^^-, ^^i^- = -^:

dt

d /, d(f\ d d / dxp\ d

— ( 6 — ^ ) = — 6 sm. \p, ~{l—-]=z - l sm. op.
. dt\ dtj dt ^' rf^V rf^/ rf^ ^

Whence

I ^ z=il sin. t/^ + ^> ^ ^ ~ ^ ^^^* ^ + ^i5

or by putting G^IA, Gi = lAij and passing from logarithms
to numbers,

g = ^sin.v^, 5 = ^i8in.g, (6):

. . dip dcp

. • . -d sm. -I/; -7-/ = Ai sm.op ^- ;
dt dt

Acos.\p = Aicos.qi -]- (7).
From Eq. 1 wo see that F{Cjd)=F{c,ii) when <?i = 0:
therefore we then have =:fji= cp =:\pj and (7) then becomes
(A — Ai) COS. fi = 0; and therefore

A COS. (^ — ^1) — Ai COS. (<? + di) = (^ — ^1) COS. fi ;
'whence, by developing cos. (^-—^j), cos. (^ + <?i), and re-
ducing,
{A — Ai) COS. 6 COS. 6i'\-{A-{' Ai) sin. ^ sin. di

= {A - Ai) C03. fi (8).
Now,

t = l + §' = V(l-«"-'»)-V(l-c'™..,,),

-^ = V(l - <^^ sin.2 ^) + y^(l — c2 sin.2 <?i).

Substitute these values in (6), and make 6i=z0; then
\/{l — c^ sin.^ f*) — 1 = ^ sin. ^,
y^(l — c^sin.V) + 1 =-4isin.jM.

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470 INTEGRAL CALCULUS.

From th^se equations, getting the values ofA + Ai,A — Ai,
and substituting in (8), we get, finally,

COS. COS. Oi — sin. sin. O^j^il — c^ sin.^ft) = cos. fi (9).

This relation, by an easy transformation, may be made to
take the form

COS. ^ = cos^iC08.jM -j- sin.^i8in.jMy^(l — c^sin.^^) (10).

Eqs. (9) and (10) express the connection which exists be-
tween the variables in two elliptic functions of the first order
which have a common modulus.

282* Let F{c, 0), F{c, d^), be two elliptic functions in which
c, Cj, and 0, d^ are connected by the equations

2 4c /i\ X /, sin. 2/?i /^.

It is proposed to prove that

Difierentiate Eq. 2, regarding di as the independent variable;

then

1 dd _ 2(1 +c COS. 2^0
cos.'-'^:^ d^i"~ (c + cos.2^i)2

From (2) we also get

cos^^= - (^ + ^0^-2^0^ :
1^2ccos.2^i + c2

dd __ 2(l+cco8. 2^i)
d^ 1 "^ 1 + 2c COS. 2^ 1 -f c2 '

Also, from the same equation, we get

c2sin.2 2^i

1 — c2sin.2(? = l

1 +2ccos.2^i-|-c2

l + 2ccos.2^i + c^cos.^2<?i ,
l + 2ccos.2(?i + c2

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• /

ELLIPTIC FUNCTIONS. ATI

dd

V(l -c^sin.^^)

_ r 2(1 -fc COS. 2^1) y^(l + 2ccos.2^i -f c^) ,
"~J l + 2ccos.2di + c^ l + ccos.2^1 *

+ 2c COS. 2^1 + c^ 1 + c COS. 20 1
ddi

_ 2 r dOj

^'"'Ji-(T^— ■)■

Bnt the last integral, when ■ = Cj , becomes

If we suppose di = -^ 6 = 7t, then

ffc4'i)=^<"""=^4i)

j?55. Having shown (Art. 281) that there exists, between
the variables of two elliptic functions of the first order having
a common modulus, the relation

COS. d COS. di -^ sin. 6 sin. di ^{1 — c^ sin. (a) = cos. i^ (1),

then, between the corresponding functions of the second
order, there exists the relation

£(c, 0) + E{Cf d{) — F{c, fi) = c' sin. sin. ^^ sin. f*.
From the equation between the amplitudes 6, 0^ di, may
be considered as a function of 6 ; that is, we may assume
Eic, d) + E{c, d,) - E{c,(,) =/(<?),

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472 INTEGRAL CALCULUS.

and diflFerentiate, thug getting

By Eq. 10, Art. 281, the first member of this equation may
be put under the form

COS. — COS. di COS. ft COS. 6 1 — cos. d COS. fi ddi
sin. di sin.^ '" sin. d sin. ii dd

_^d{'^m?d -\-%m?di-\' 2cos.(?cos.^iCos.ft 1

do 2sin.^8in.^i8in.^'

But putting Eq. 1 under the form

COS. d COS. di — COS. jM =: ^(1 — c^ siu.^fi) sin*, d sin. di ,
and squaring, we get
cos.^^ -|- cos.^^i + cos.^/i — 2 COS. 6 cos. (?i cos. ^

= (1 — c2sin.V)8in.'^sin.2(?i.
Adding cos.^^j cos.^ft to both sides of this equation, transpos-
ing, and reducing by the relation cos.^ = 1 — sin.^, we find

sin.^ d -j- sin.^ ^i + 2 60s. cos. d^ cos. ^

= 1 + cos.^ II -|- c^^in.' 6 mi? di sin.^f*,

-y- (1 + 008.^^1^ + c^ sin.'^ Bin} 6 1 bvh? fi)

2 sin. <?sin. <?isin.jM

^ . c?(sin. ^sin. ^,)
= cHm.i. ^^— ^:

and therefore, by integration,

/(^d) = c^ sin. ^ sin. di sin. ^

8356

3^

r\

B

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