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Composition of two uniform circular motions.

Fourier's
Theorem.

This is intimately connected with the explanation of two sets of important phenomena,—the rotation of the plane of polarization of light, by quartz and certain fluids on the one hand, and by transparent bodies under magnetic forces on the other. It is a case of the hypotrochoid, and its corresponding mode of description will be described in a future section. It will also appear in kinetics as the path of a pendulum-bob which contains a gyroscope in rapid rotation.

75. Before leaving for a time the subject of the composition of harmonic motions, we must, as promised in § 62, devote some pages to the consideration of Fourier's Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth's crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance. The following seems to be the most intelligible form in which it can be presented to the general reader :

THEOREM.—A complex harmonic function, with a constant term

added, is the proper expression, in mathematical language, for any arbitrary periodic function; and consequently can express any function whatever between definite values of

the variable.

76. Any arbitrary periodic function whatever being given, the amplitudes and epochs of the terms of a complex harmonic function which shall be equal to it for every value of the independent variable, may be investigated by the "method of indeterminate coefficients."

Assume equation (14) below. Multiply both members first

2ἰπξ

by cos dέ and integrate from 0 to p: then multiply by

P

dέ and integrate between same limits.

QiπTÈ

sin

p
you find (13).

Thus instantly

Theorem.

This investigation is sufficient as a solution of the problem, Fourier's -to find a complex harmonic function expressing a given arbitrary periodic function,-when once we are assured that the problem is possible; and when we have this assurance, it proves that the resolution is determinate; that is to say, that no other complex harmonic function than the one we have found can satisfy the conditions.

For description of an integrating machine by which the coefficients A, B, in the Fourier expression (14) for any given arbitrary function may be obtained with exceedingly little labour, and with all the accuracy practically needed for the harmonic analysis of tidal and meteorological observations, see Proceedings of the Royal Society, Feb. 1876, or Chap. v. below.

77. The full theory of the expression investigated in § 76 will be made more intelligible by an investigation from a different point of view.

Let F(x) be any periodic function, of period p. That is to say, let F(x) be any function fulfilling the condition

F(x+ip) = F(x)

(1),

where i denotes any positive or negative integer. Consider the
integral

and therefore

and

do

F(x) [

and greater than F(*) [4x2

C'

where a, c, c' denote any three given quantities. Its value is
dx
less than F
if z
a2 + x2,
α2 +
and z' denote the values of x, either equal to or intermediate
between the limits c and c', for which F(x) is greatest and least
respectively. But

.

dx

La

[= 1 (tan-tan-1)

a2 + x2

c'

[° F (x) ada

a2 + x2

F(x) dx
a2 + x2

2

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29

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(2),

(3)

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Hence if ▲ be the greatest of all the values of F(x), and B the

least,

and

Also, similarly,

and

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But, by (1),

F(x) ada _ F(2) (tan ̄1

a2 + x2

F(x) adx
a2 + x2

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c' F(x)adx
a2 + x2

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T

2apv

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с

a

1

= ∞

= =
a2 + (x + ip)2,

(öö

i

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p

F(x)dx

[_ady - [F(x) d x (

2

a2 + x2

.00

Now if we denote √−1 by v,

1

T

2apv

ap

с

and > F(2) (tan~1 C - tan-1o) + B (~

a

απα

p

< A

> B

{cot I

που

p

tan-1

sin

< A ( tan

Adding the first members of (3), (4), and (5), and comparing with the corresponding sums of the second members, we find

π

π

2πα

p

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1 (1

2av x

p

2παυ

P

Cos2

- 2 cos

[ocr errors]

1

> B (tan- / + 1).

a 2

tan

tan

1

av)

TX

p

a

1

x + av

Σπα

p

2πα P

[blocks in formation]

1

1

1

a2 + (x+ip)2 ̄ 2av \x + ip – av x + up + av

and therefore, taking the terms corresponding to positive and equal negative values of i together, and the terms for i=0 separately, we have

+ €

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с

a

COS

π - tan

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cot

π

1

{ == (a

(= (x + ip})}

2

a2 +

[ocr errors]

Σπα

p

tan

+ 2Σ.

αυ

{2--2- 2Σ.
αυ 'i=1 ¿ 3μ3 — (x — av)2

2

1

-1

Ωπαυ

p

[blocks in formation]

π (x + αv)

p

T 2παυ
sin

apv

η

+tan

[ocr errors]

un-10).

a

COS

X

-∞

x + av

'i=1 ¿ 3p2 — (x + av}3 {

2

[ocr errors]
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2πχ

P

(4)

(5)

(e)

(7).

Hence,

and

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[* F(x) dx

a2 + x2

[ocr errors]

απα

p €

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2πXV

Next, denoting temporarily, for brevity, e

p

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and therefore

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α

Σπα

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Σπα

p

Hence, by (6), we infer that

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Hence, according to (8) and (9),

e

Σπα

P

e

2πX

Απα

== (1 + 26 cos 2 + 26° cos 40

2e COS

2e2

1

e2

Call m

P

p

ap F(x)dx(1 + 2e cos

b

[subsumed][ocr errors]

πα

e

2

2

= _ _—_ p { 1 + e (5 + 5 − '1) + e2 (¿a + 5 ̃2) + e3 (53 + ¿ ̃3) + etc. }

e2

1

p – 2 cos

[ocr errors]

1

1 - -1 - 1)

F(x)dx

2πα

p

F(≈) (tan ̄1 1-12 ) + 1 (π - tax

tan

(*.

a

F(2)

[ocr errors]

2πα

P

σπα

cos + 25 cos + etc.)

2e3

P

+ 2e2 cos

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c = C, and x = έ' - έ,

¿'

πα
20

Απα

p

с

(tan-tan)+(-tan- + tan)<

B

V

a

[F(x) dx (1+2 008 2+ etc.)

cos

p

by, and putting

F(x) = (x + §) = $(§′),

F (≈) = $(§ + ≈),

F' (≈') = $(§ + ≈').

C

1

12 + tan-10) >

a

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(8).

+ etc.)... (10).

(9),

Now let

έ' being a variable, and έ constant, so far as the integration is
concerned; and let

Fourier s Theorem.

Fourier's
Theorem.

The preceding pair of inequalities becomes.

P($+ z). 2 tan

1(F

and

C

$($+z'). 2 tan-1 +

a

If, for cos

COS

A.

1

A;

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Bi

a

[ocr errors]

COS

=

+ A

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B(T

P

where & denotes any periodic function whatever, of period p.

Now let c be a very small fraction of p. In the limit, where c is infinitely small, the greatest and least values of (') for values of έ between έ+c and έ − c will be infinitely nearly equal to one another and to (§); that is to say,

$(§ + z) = $(§ + ≈′ ) = $(§).

Next, let a be an infinitely small fraction of c. In the limit

π

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and

e = €

Hence the comparison (11) becomes in the limit an equation which, if we divide both members by π, gives

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1

=X

• (6) = = { [s] + (€) d€ + 2 = = ï [][] + (E') de' cos

2iπ (έ'-))
p

...(12).

i=1

P This is the celebrated theorem discovered by Fourier* for the development of an arbitrary periodic function in a series of simple harmonic terms. A formula included in it as a particular case had been given previously by Lagranget.

2iπ (§' — §) P 2inέ'

p

2inέ
P

2iπέ'
+ sin

p

and introduce the following notation :

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2 tan-1

2 tan

p = 1.

Þ (§) d§,

-1 C

n)<

α

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π

2'

cos

p

2 [(6) sin

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we take its value

,

sin

virk dk;

P

2ἰπξ

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(11)

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P

* Théorie analytique de la Chaleur. Paris, 1822.

+ Anciens Mémoires de l'Académie de Turin.

(13)

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