The applications of elliptic functions

Edward Bright

Mathematics Dept

THE APPLICATIONS OF ELLIPTIC FUNCTIONS.

THE APPLICATIONS OF

ELLIPTIC FUNCTIONS

BY

ALFRED GEORGE GREENHILL M.A., F.R.S.,

PROFESSOR OK MATHEMATICS

IN"

THE ARTILLERY COLLEGE, WOOLWICH

MACMILLAN AND AND XETT YORK 1892 [All riyhlft reserved]

CO.

AY*

CONTENTS. PAGE

INTRODUCTION,

vii

CHAPTER THE ELLIPTIC FUNCTIONS,

-

-

I.

-

CHAPTER

1

II.

THE ELLIPTIC INTEGRALS,

-

CHAPTER GEOMETRICAL AND MECHANICAL FUNCTIONS,

-

III.

ILLUSTRATIONS

THE

OF

ELLIPTIC

-

66

CHAPTER THE ADDITION THEOREM FOR

IV.

ELLIPTIC FUNCTIONS,

CHAPTER THE ALGEBRAICAL FORM

IN

-

-

112

V. 142

VI. -

-

175

-

200

-

254

VII.

GENERAL AND THEIR APPLICATIONS,

CHAPTER THE DOUBLE PERIODICITY

-

OF THE SECOND AND THIRD KIND,

CHAPTER THE ELLIPTIC INTEGRALS

-

OF THE ADDITION THEOREM,

CHAPTER THE ELLIPTIC INTEGRALS

30

VIII.

OF THE ELLIPTIC FUNCTIONS,

781468

-

-

CONTENTS.

vi

CHAPTER

IX. PAGE

THE RESOLUTION

OF THE

INTO FACTORS AND

ELLIPTIC FUNCTIONS

...

SERIES,

CHAPTER

X.

THE TRANSFORMATION OF ELLIPTIC FUNCTIONS,

APPENDIX, INDEX,

277

-

-

-

305

-

-

340

353

INTRODUCTION.

"L

ETUDE approfondie de

la

nature est

la

source

la plus

feconde des decouvertes mathematiques.

Non

seulement cette etude, en offrant aux recherches un but

determine a Favantage d exclure ,

calculs sans issue 1

;

elle est

Analyse elle-meme,

importe

le

et

questions vagues et les

les

encore un

moyen

d en decouvrir

les

assure de former

elements qu

il

nous

plus de connaitre et que cette science doit toujours

conserves

Ces ele ments fondamentaux sont ceux qui se reproduisent dans tous les effets naturels." (Fourier.)

These words of Fourier are taken as the text of the present treatise, which is addressed principally to the student of ApjDlied Mathematics,

who

will in general acquire his

matical equipment as he wants definite actual

problem

;

and

it

it is

mathe

for the solution of

some

in the interest of such

students that the following Applications of Elliptic Functions

have been brought together, to enable them to see how the purely analytical formulas may be considered to arise in the discussion of definite physical questions.

The Theory of Elliptic Functions, as developed by Abel and Jacobi, beginning about 1826, although now nearly *-~~^^*~ seventy years old, has scarcely yet made its way into the "

THE APPLICATIONS OF ELLIPTIC FUNCTIONS.

viii

ordinary curriculum of mathematical study in this country

and

is still

;

considered too advanced to be introduced to the

student in elementary text-books. In consequence of this omission,

many

of the

most interest

ing problems in Dynamics are left unfinished, because the

complete solution requires the use of the Elliptic Functions these could not be introduced without a long digression, unless a considerable knowledge is presupposed of a course

;

of Pure Mathematics in this subject.

But by developing the Analysis

as

it is

required for some

particular problem in hand, the student of Applied

Mathe

matics will obtain a working knowledge of the subject of Elliptic Functions, such as

he would probably never acquire

from a study of a treatise like Jacobi s Fundamenta Nova, where the formulas are established and the subject is in

developed

Mathematical application

they

of

originate

strictly

order

logical

without

Analysis,

the formulas,

any

Pure

of

on

digression

the

on the manner in which

or

independently,

branch

a

as

the

as

some

of

expression

physical law.

we

In introducing these applications

are following, to

some

extent, the

plan of Durege s excellent treatise on Elliptic Functions (Leipsic, Teubner); and also of Halphen s Traite

des fonctions

elliptiqucs

et

de

leurs

applications

(Paris,

1886-1891).

But while volume

I.

of

Halphen s

treatise

to the establishment of the formulas

of the functions,

volume

II.

;

is

devoted entirely

and analytical properties

and the applications are not discussed

in the following pages

it

is

till

proposed to develop

immediately from some definite physical or and the reader who wishes to follow geometrical problem the formulas

;

up the purely analytical development of the subject to such treatises as

Abel

s

(Euvres, Jacobi s

is

referred

Fundamenta Nova,

INTRODUCTION.

i

x

already mentioned, or the Treatises on Elliptic Functions of Cayley, Enneper,

Kb nigsberger,

The following works

also

H. Weber,

etc.

be mentioned as having been

may

consulted in the preparation of this

work

:

Legendre: Theorie des fauctions elliptiques ; 1825. Thomas Abriss einer Theorie der complexen Functionen :

und der Tketafunctionen einer Verdnderlichen 1873. Schwarz: Formeln und Lehrsdtze zum Gebrauche der ;

elliptischen Functionen.

Klein (Morrice)

:

Lectures on the Icosahedron

;

1888.

Klein und Fricke; Vorlesungen uber die Theorie der ellip

Modalfunctionen ; 1890. Darboux: Cours de niecanique ; 1886. Despeyrous tischen

et

R A. Roberts

:

Integral Calculus

1887.

;

Bjerknes: Niels Hendrik Abel; tableau de sa vie action scientifique

We

shall begin

;

by the discussion ,

of the

and

an idea

and importance.

Pendulum could only be

circular functions, small,

Problem of the

to give the student

Previously to the introduction of the the Circular

de son

as the problem best calculated to

define the Elliptic Functions, of their nature

et

1885.

by considering the

and by assimilating

its

Elliptic

treated

Functions,

by means

of the

oscillations as indefinitely

motion to that of Huygens

Cycloidal Pendulum, of 1673.

But now the employment

of the Elliptic Functions renders

the ordinary discussion of the Cycloidal

and of mere

historical interest,

such expressions as "reducible

"

Pendulum antiquated

and banishes from our

an integral which cannot be

to a matter of quadrature" in describing

integral, expressions

found,"

an

which aroused the indignation of

Mathematical Papers,

p. 562).

treatises

or

elliptic

Clifford

THE APPLICATIONS OF ELLIPTIC FUNCTIONS.

x

According to the

new

in

May

regulations for the

come

Tripos at Cambridge, to

1893, the schedule

II.

of Part

I.

includes

"

Elementary

excluding the Theta Functions and the

Elliptic Functions,

"

theory of Transformation

to be

hoped that this Functions into the ordinary mathe

reintroduction of Elliptic

;

so it

is

receive

more

Applications

have

matical curriculum will cause the subject

general

Mathematical

into force in the examination

and

attention

These

study.

to

been put together with the idea of covering this ground by exhibiting their practical importance in Applied Mathematics,

and of securing the

interest of the student, so that he

may

if

he wishes follow with interest the analytical treatises already mentioned.

We

begin with Abel

elliptic integral of

the

idea of the inversion of Legendre s

s

first

kind, and employ Jacobi

s

notation,

with Gudermann s abbreviation, for a considerable extent at the outset.

The more modern notation

of Weierstrass

is

introduced

subsequently, and used in conjunction with the preceding notation,

and not

to its exclusion

;

as

it

will be

sometimes one notation and sometimes the other

found that is

the more

suitable for the problem in hand.

At the same time explanation

is

given of the methods by

which a change from the one to the other notation can be speedily carried out. It has

been considered

sufficient in

many places,

in the reduction of the Integrals in Chapter

for instance

II.,

down the results without introducing the intermediate as the trained mathematical student to

whom

this

to write

analysis

book

;

is

addressed will have no difficulty in supplying the connecting steps,

and

this

work

will at the

exercises in the subject

students,

many

;

and

same time provide instructive

further, in the interest of such

important problems have been introduced in

INTRODUCTION.

XI

the text, forming immediate applications of theorems already

developed previously. I have to thank Mr. A. G. Hadcock for his assistance in preparing the diagrams, and in drawing them carefully to scale.

ERRATA. Page

6.

Line 9 from bottom, read Huygeiis. 1

*/ -. V x-y

42.

Line

48.

Line 5 from bottom, read -

64.

Line

99.

The diagram must be replaced by the one given below. The Xodoid in fig. 12, p. 99, was described by a point which was not a focus of the rolling hyperbola.

107.

138. 158.

205.

213.

227. 282.

328.

6,

read

19,

sin"

read Fonctions

4tt-(9e-

-r

4/r)

J .

elliptiques.

Line 2 from bottom, delete minus sign before radical. 2 ctf/D. Equation (7), read (r., Line 12, read 3QX(x, y). + v). Line 6 from bottom, read $(u - v) Line 7 from bottom, read G + Lx - X(yz - y ~] with the corresponding subsequent corrections. Line 7, read P s/-Y i + Q\ -Y 2 = $>(u

;

-

Line 5 from top, for rectangle read ribbon. Line 12 from bottom, read Pw. L. M. .?., IX.

ABBREVIATIONS. Quarterly Journal of Mathematics. S., Proceedings of the London Mathematical Society. Proc. G. P. $., Proceedings of the Cambridge Philosophical Society. Q. J. M.,

Proc. L.

Am.

J.

M.

M.,

F. E.,

Math. Ann., Phil.

Mag.,

American Journal

of

Mathematics.

Fonctions elliptiques (Legendre and Halphen). Mathematische Annalen.

Phil. Trail*.

Philosophical Magazine. Philosophical Transactions of the Royal Society of London.

Berlin Sitz.,

Sitzungsberichte der Berliner Akademie.

CHAPTEK

I.

THE ELLIPTIC FUNCTIONS. 1.

The Pendulum;

introducing Elliptic Functions into

Dynamics.

When a pendulum OP swings through a finite angle about a horizontal axis 0, the determination of the motion introduces the Elliptic Functions in such an elementary and straight forward manner, that

we may

take the

elliptic functions as

by pendulum motion, and begin the investigation of their use and theory by their application to this problem. Denote by the weight in Ib. of the pendulum, and let = OG h (feet), where G is the centre of gravity let Wk 2 denote defined

W

;

the

moment

of inertia of the

pendulum about the horizontal

+k )

axis through G, so that W(h* about the parallel axis through

2

is

the

moment

of inertia

(fig. 1).

OG makes

with the vertical OA an angle 6 radians at the time t seconds, reckoned from an instant at which the pendulum was vertical and if we employ the absolute unit

Then

if

;

of force, the

poundal, and denote by g (32

celoes,

roughly)

the acceleration of gravity, the equation of motion obtained

by taking moments about

is

since the impressed force of gravity is G so that ; vertically through

or,

on putting

G.E.F.

h + k*/h = I,

Wg

poundals, acting

THE ELLIPTIC FUNCTIONS.

to

THE ELLIPTIC FUNCTIONS.

3

If the gravitation unit of force, the force of

employed, then the equation of motion

w fr/TO 2

(A.

J lft

+&2 )-p = 7

.

is

a pound,

is

written

TTTf sm 0, Wh

O\*-^ v/

/\

"

reducing to (1) as before. 2.

Producing

OG

OP =

to P, so that

GP = k /h, 2

l,

P is

the point

called the centre of oscillation (or of percussion) and is called the length of the simple equivalent pendulum, because ;

AP

P

oscillates on the circle the point manner as a small plummet suspended

2); as along the arc

is

(fig.

the

same

by a fine thread from

seen immediately by resolving tangentially = s = l9 when the equation of motion of

AP

)

=

is

plummet

in exactly the

2

or

2

I(d 0/dt

-g sin#; ........................ (1) = C-gversO ......................... (2)

)= 2

and

g sin#=

integrating, U(dO/dt) These theorems are explained in treatises on Analytical Mechanics, such as Kouth s Rigid Dynamics, or Bartholomew Price s Infinitesimal Calculus, vol. IV., and might have been assumed here but now we proceed further, to the complete ;

integration of equation 3.

First suppose the

oscillation

BO A +AOB

of oscillation

is

Navez

Ballistic

church

bell,

oscillations,

books

;

in

(2).

pendulum

to oscillate, the angle of

being denoted by 2a

purposely

made

(fig. 2);

the angle

large, as in early clocks, in the

Pendulum, in a swing, or as in ringing a

so as to emphasize the difference from small the only case usually considered in the text

fig.

2 the angle of oscillation is made 300. when 6 = a, so that in equation (2)

Then dO/dt =

(7=# versa

and now denoting

g/l

2 by n

,

so that

;

n

is

what

Sir

W. Thomson

calls the

speed (angular) of the pendulum, 2 = n 2 ( vers a vers 0) ^(dO/dt)

= 2?i2 (sin2 Ja-sin ^), .............................. (3) vers = 2 sin 2 J0 2 dO/dt = 27i x/(sin ia -sin i0), 2

since

;

_ 2

and

nt-f

^

-..

.

..(4)

THE ELLIPTIC FUNCTIONS.

4,

and (4) is called by Legendre an elliptic integral of kind ; it is not expressible by any of the algebraical,

the first

circular,

or hyperbolic functions of elementary mathematics. 4.

To reduce

this elliptic integral to the

standard form con

by Legendre, we put

sidered

sinA0 = sinJa sin 0, by equivalent geometrically to denoting the angle as diameter, touching is the circle on (fig. 2), where in Q. BBf in D, and cutting the horizontal line

ADQ

AD

AQD

For, in the circle and, in the circle

PN

AP,

AQ,

AN= \AD vers 2$ = AD sin =I

Now

sin2 \ a

vers a sin

2

2

= 21 sin2 |a sin2 0.

= sin2 |a cos 0, ~ J$ = sin (sin Ja sin 0), 2

sin 2 J$

1

and

dle=

so that

*ft,2 v .^i*.* snrja sm 0)

^/(l

nt =

-

r~r ^r-, Jfr^/(1-siu iasm

and therefore

>

2

2

^,)

which is now an elliptic integral of the standard form employed by Legendre.

(Fonctions Elliptiques, 5.

In Legendre 2

ic sin

t. I.,

chap VI.)

replaced by ic; the quantity or A(0, K) and the integral

s notation, sin-Jet is

is. ,

K) for

Table IX.)

Legendre spent a long

life in investigating the properties of the elliptic integral of the first kind but the subject was revolutionised by the single remark of Abel (in

the function

Fc/>,

;

THE ELLIPTIC FUNCTIONS. 1823), that

is

of the nature of

.5

an inverse function

F(j>

we put u

then

;

and that

we should study

the properties of 0, the amplitude, as a function of u, and not of u as a function of as carried out by Legendre in his Fonctions Elliptiques. if

F(f>,

,

= am u, or am(it, /c) when Jacobi proposed the notation to be the modulus K is required put in evidence and now, 6.

;

we have

considered as functions of u, cos

Jacobi

= cos am u, sin = sin am u,

s

= A am u,


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THE ELLIPTIC FUNCTIONS.

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