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No. 5.

HARMONIC FUNCTIONS.

WILLIAM E. BYERLY,

PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY.

FOURTH EDITION, ENLARGED. FIRST THOUSAND.

NEW YORK:

JOHN WILEY & SONS.

LONDON: CHAPMAN & HALL, LIMITED.

1906.

COPYRIGHT, 1896,

BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD

UNDER THE TITLE

HIGHER MATHEMATICS.

First Edition, September, 1896. Second Edition, January, 1898. Third Edition, August, 1900. Fourth Edition, January, 1906.

Engineering &

Mathematical

i *~

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EDITORS' PREFACE.

THE volume called Higher Mathematics, the first edition of which was published in 1896, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume is now discontinued and the chapters are issued in separate form. In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature.

It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theory of num- bers, the group theory, the calculus of variations, and non- Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this form of publication may tend to promote mathematical study and research over a wider field than that which the former volume has occupied.

December, 1905.

444680

AUTHOR'S PREFACE.

THIS brief sketch of .the Harmonic Functions and their use in Mathematical Physics was written as a chapter of Merriman and Woodward's Higher Mathematics. It was intended to give enough in the way of introduction and illustration to serve as a useful part of the equipment of the general mathematical student, and at the same time to point out to one specially inter- ested in the subject the way to carry on his study and reading toward a broad and detailed knowledge of its more difficult portions.

Fourier's Series, Zonal Harmonics, and Bessel's Functions of the order zero are treated at considerable length, with the inten- tion of enabling the reader to use them in actual work in physical problems, and to this end several valuable numerical tables are included in the text.

CAMBRIDGE, MASS., December, 1905

CONTENTS.

ART. i. HISTORY AND DESCRIPTION Page 7

2. HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 10

3. PROBLEM IN TRIGONOMETRIC SERIES 12

4. PROBLEM IN ZONAL HARMONICS . . . 15

5. PROBLEM IN BESSEL'S FUNCTIONS 21

6. THE SINE SERIES 26

7. THE COSINE SERIES 30

8. FOURIER'S SERIES ' 32

9. EXTENSION OF FOURIER'S SERIES 34

10. DIRICHLET'S CONDITIONS 36

11. APPLICATIONS OF TRIGONOMETRIC SERIES 38

12. PROPERTIES OF ZONAL HARMONICS 40

13. PROBLEMS IN ZONAL HARMONICS 43

14. ADDITIONAL FORMS 45

15. DEVELOPMENT IN TERMS OF ZONAL HARMONICS 46

1 6. FORMULAS FOR DEVELOPMENT 47

17. FORMULAS IN ZONAL HARMONICS 50

18. SPHERICAL HARMONICS 51

19. BESSEL'S FUNCTIONS. PROPERTIES 52

20. APPLICATIONS OF BESSEL'S FUNCTIONS - 53

21. DEVELOPMENT IN TERMS OF BESSEL'S FUNCTIONS 55

22. PROBLEMS IN BESSEL'S FUNCTIONS 58

23. BESSEL'S FUNCTIONS OF HIGHER ORDER 59

24. LAME''S FUNCTIONS 59

TABLE I. SURFACE ZONAL HARMONICS 60

II. BESSEL'S FUNCTIONS 62

III. ROOTS OF BESSEL'S FUNCTIONS 63

IV. VALUES OF J0(xf) 63

INDEX 65

HARMONIC FUNCTIONS.

ART. 1. HISTORY AND DESCRIPTION.

What is known as the Harmonic Analysis owed its origin and development to the study of concrete problems in various branches of Mathematical Physics, which however all involved the treatment of partial differential equations of the same general form.

The use of Trigonometric Series was first suggested by Daniel Bernouilli in 1753 in his researches on the musical vibrations of stretched elastic strings, although Bessel's Func- tions had been already (1732) employed by him and by Euler in dealing with the vibrations of a heavy string suspended from one end; and Zonal and Spherical Harmonics were introduced by Legendre and Laplace in 1782 in dealing with the attrac- tion of solids of revolution.

The analysis was greatly advanced by Fourier in 1812-1824 in his remarkable work on the Conduction of Heat, and im- portant additions have been made by Lame" (1839) ar"d by a host of modern investigators.

The differential equations treated in the problems which have just been enumerated are

8 HARMONIC FUNCTIONS.

for the transverse vibrations of a musical string :

o *

for small transverse vibrations of a uniform heavy string sus- pended from one end ;

which is Laplace's equation ; and

for the conduction of heat in a homogeneous solid.

Of these Laplace's equation (3), and (4) of which (3) is a special case, are by far the most important, and we shall con- cern ourselves mainly with them in this chapter. As to their interest to engineers and physicists we quote from an article in The Electrician of Jan. 26, 1894, by Professor John Perry:

" There is a well-known partial differential equation, which is the same in problems on heat-conduction, motion of fluids, the establishment of electrostatic or electromagnetic potential, certain motions of viscous fluid, certain kinds of strain and stress, currents in a conductor, vibrations of elastic solids, vibrations of flexible strings or elastic membranes, and innumerable other phenomena. The equation has always to be solved subject to certain boundary or limiting conditions, sometimes as to space and time, sometimes as to space alone, and we know that if we obtain any solution of a particular problem, then that is the true and only solution. Further- more, if a solution, say, of a heat-conduction problem is obtained by any person, that answer is at once applicable to analogous prob- lems in all the other departments of physics. Thus, if Lord Kel- vin draws for us the lines of flow in a simple vortex, he has drawn for us the lines of magnetic force about a circular current; if Lord Rayleigh calculates for us the resistance of the mouth of an organ-pipe, he has also determined the end effect of a bar of iron which is magnetized; when Mr. Oliver Heaviside shows his match-

HISTORY AND DESCRIPTION, I)

less skill and familiarity with Bessel's functions in solving electro- magnetic problems, he is solving problems in heat-conductivity or the strains in prismatic shafts. How difficult it is to express exactly the distribution of strain in a twisted square shaft, for example, and yet how easy it is to understand thoroughly when one knows the perfect-fluid analogy! How easy, again, it is to imagine the electric current density everywhere in a conductor when transmitting alter- nating currents when we know Mr. Heaviside's viscous-fluid analogy, or even the heat-conduction analogy!

" Much has been written about the correlation of the physical sciences; but when we observe how a young man who has worked almost altogether at heat problems suddenly shows himself ac- quainted with the most difficult investigations in other departments -of physi€S, we may say that the true correlation of the physical sciences lies in the equation of continuity

dt=a \a*8 + ay a*'/

In the Theory of the Potential Function in the Attraction •of Gravitation, and in Electrostatics and Electrodynamics,* V \r\ Laplace's equation (3) is the value of the Potential Func- tion, at any external point (x, y, 2), due to any distribution of matter or of electricity; in the theory of the Conduction of Heat in a homogeneous solid f V is the temperature at any point in the solid after the stationary temperatures have been •established, and in the theory of the irrotational flow of an incompressible fluid \ V is the Velocity Potential Function and (3) is known as the equation of continuity.

If we use spherical coordinates, (3) takes the form

=0-

* See Peirce's Newtonian Potential Function. Boston. f See Fourier's Analytic Theory of Heat. London and New York, 1878 •or Riemann's Partielle Differentialgleichungen. Brunswick.

\ See Lamb's Hydrodynamics. London and New York, 1895.

10 HARMONIC FUNCTIONS.

and if we use cylindrical coordinates, the form

~~

In the theory of the Conduction of Heat in a homogene^ ous solid,* u in equation (4) is the temperature of any point (x, y, z) of the solid at any time /, and c? is a constant deter- mined by experiment and depending on the conductivity and the thermal capacity of the solid.

ART. 2. HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS.

The general solution of a differential equation is the equa- tion expressing the most general relation between the primi- tive variables which is consistent with the given differential equation and which does not involve differentials or derivatives- A general solution will always contain arbitrary (i.e., undeter- mined) constants or arbitrary functions.

A particular solution of a differential equation is a relation between the primitive variables which is consistent with the given differential equation, but which is less general than the general solution, although included in it.

Theoretically, every particular solution can be obtained: from the general solution by substituting in the general solu- tion particular values for the arbitrary constants or particular functions for the arbitrary functions ; but in practice it is often easy to obtain particular solutions directly from the differential equation when it would be difficult or impossible to obtain the general solution.

(a) If a problem requiring for its solution the solving of a differential equation is determinate, there must always be given in addition to the differential equation enough outside condi- tions for the determination of all the arbitrary constants or arbitrary functions that enter into the general solution of the equation ; and in dealing with such a problem, if the differen- tial equation can be readily solved the natural method of pro-

HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS. 11

cedure is to obtain its general solution, and then to determine the constants or functions by the aid of the given conditions.

It often happens, however, that the general solution of the differential equation in question cannot be obtained, and then, since the problem, if determinate, will be solved, if by any means a solution of the equation can be found which will also satisfy the given outside conditions, it is worth while to try to get particular solutions and so to combine them as to form a result which shall satisfy the given conditions without ceasing to satisfy the differential equation.

(b) A differential equation is linear when it would be of the first degree if the dependent variable and all its derivatives were regarded as algebraic unknown quantities. If it is linear and contains no term which does not involve the dependent variable or one of its derivatives, it is said to be linear and homogeneous.

All the differential equations given in Art. I are linear and homogeneous.

(c) If a value of the dependent variable has been found which satisfies a given homogeneous, linear, differential equa- tion, the product formed by multiplying this value by any constant will also be a value of the dependent variable which will satisfy the equation.

For if all the terms of the given equation are transposed to the first member, the substitution of the first-named value must reduce that member to zero ; substituting the second value is equivalent to multiplying each term of the result of the first substitution by the same constant factor, which there- fore may be taken out as a factor of the whole first member. The remaining factor being zero, the product is zero and the equation is satisfied.

(d) If several values of the dependent variable have been found each of which satisfies the given differential equation, their sum will satisfy the equation ; for if the sum of the values in question is substituted in the equation, each term of the sum

12 HARMONIC FUNCTIONS.

will give rise to a set of terms which must be equal to zero, and therefore the sum of these sets must be zero.

(e) It is generally possible to get by some simple device particular solutions of such differential equations as those we have collected in Art. i. The object of this chapter is to find methods of so combining these particular solutions as to satisfy any given conditions which are consistent with the nature of Jthe problem in question.

This often requires us to be able to develop any given func- tion of the variables which enter into the expression of these conditions in terms of normal forms suited to the problem with which we happen to be dealing, and suggested by the form of particular solution that we are able to obtain for the differential equation.

These normal forms are frequently sines and cosines, but they are often much more complicated functions known as Legendre's Coefficients, or Zonal Harmonics; Laplace's Coef- ficients, or Spherical Harmonics; Bessel's Functions, or Cylin- drical Harmonics; Lame's Functions, or Ellipsoidal Har- monics; etc.

ART. 3. PROBLEM IN TRIGONOMETRIC SERIES.

As an illustration let us consider the following problem : A large iron plate n centimeters thick is heated throughout to a uniform temperature of 100 degrees centigrade; its faces are then suddenly cooled to the temperature zero and are kept at that temperature for 5 seconds. What will be the tempera- ture of a point in the middle of the plate at the end of that time? Given a3 =0.185 in C.G.S. units.

Take the origin of coordinates in one face of the plate and the axis of X perpendicular to that face, and let ti be the temperature of any point in the plate t seconds after the cool- ing begins.

We shall suppose the flow of heat to be directly across the plate so that at any given time all points in any plane parallel

PROBLEM IN TRIGONOMETRIC SERIES. 1<>

to the faces of the plate will have the same temperature. Then u depends upon a single space-coordinate x ; ^ - = o and

- = o, and (4), Art. I, reduces to

dz

* = <& (i)

3/ - a*1

Obviously, u = 100° when t = o, (2)

u = o when x = o, (3)

i

and u = o when x = rr ; (4)

and we need to find a solution of (i) which satisfies the con- ditions (2), (3), and (4).

We shall begin by getting a particular solution of (i), and we shall use a device which always succeeds when the equa- tion is linear and homogeneous and has constant coefficients.

Assume* u = e&x+yt, where ft and y are constants; substi- tute in (i) and divide through by <?£*+?' and we get y = c? ft* ; and if this condition is satisfied, u = e^x+y( is a solution of (i).

u = gP*+**P* is then a solution of (i) no matter what the value of ft.

We can modify the form of this solution with advantage. Let ft — /i/,f then u = ^-"VV4** is a solution of (i), as is also u — e-We'1^.

By (d), Art. 2,

(V*» 4- e ~ i***} u _ ,-•*»<£ — Xf 1 = e-'W cos IAX (5)

2

is a solution, as is also

(e\*-xi e~v-xt\

« = *-» Vifl _ 1 / = #— ^"sin/wr; (6)

22 .

and /* is entirely arbitrary.

* This assumption must be regarded as purely tentative. It must be tested by substituting in the equation, and is justified if it leads to a solution, f The letter i will be used to represent 4/ — i.

14 HARMONIC FUNCTIONS.

By giving different values to jj. we get different particular solutions of (i) ; let us try to so combine them as to satisfy our conditions while continuing to satisfy equation (i).

u = ^>-aV sin }juc is zero when x = O for all values of // ; it is zero when x = n if yu is a whole number. If, then, we write u equal to a sum of terms of the form Ae'"3"1** sin mx, where m is an integer, we shall have a solution of (i) (see (d), Art. 2) which satisfies (3) and (4).

Let this solution be u = A^-**' sin x-\-A,e-*ayt sin 2x -f A^-^1 sin 3* + ..., (7)

.Alt At, At, . . . being undetermined constants. When / = o, (7) reduces to

u — At sin x -j- At sin 2.x -j- A3 sin ^x -f- • • . • (8)

If now it is possible to develop unity into a series of the form (8) we have only to substitute the coefficients" of that series each multiplied by 100 for Al , At, A3 . . . in (7) to have a solution satisfying (i) and all the equations of condition (2),

'(3), and. (4>

We shall prove later (see Art. 6) that

I — ~ sin x + - sin \x -4- — sin tx 4- . . .

TtL 3 5 J

for all values of x between o and n. Hence our solution is u = - -L--"" sin x + -<T9a" sin 3* + Le-&0t sin 5 x _j_ _ _ ^

To get the answer of the numerical problem we have only to compute the value of u when x = — and / = 5 seconds. As

there is no object in going beyond tenths of a degree, four- place tables will more than suffice, and no term of (g) beyond

the first will affect the result. Since sin - = i, we have to

fi

• compute the numerical value of

PROBLEM IN ZONAL HARMONICS. 15

A OO

- — e-"* where a1 = 0.185 and t = 5.

71

log a? = 9.2672 — 10 log 400 = 2.6021

log / = 0.6990 colog n = 9.5059 — 10

log a*t — 9.9662 — 10 colog tf** = 9.5982 — 10

log log e = 9.6378 — 10

log u = 1.7062

log log ^ = 9.6040 — 10

log ean = 0.4018^ u — 5Q°-8.

If the breadth of the plate had been c centimeters instead of it centimeters it is easy to see that we should have needed the development of unity in a series of the form

TIX 2.71X . iTtX

A. sin — -4- A. sm 4- A, sin + . . . .

c c c

Prob. i. An iron slab 50 centimeters thick is heated to the tem- perature 100 degrees Centigrade throughout. The faces are then sud- denly cooled to zero degrees, and are kept at that temperature for 10 minutes. Find the temperature of a point in the middle of the slab, and of a point 10 centimeters from a face at the end of that time. Assume that

4 f • nx , i •? nx .1.5 Tfx , \ ,

i — — sin — sin f- - sin * \- . . . from x = o to x = c.

x\ c '3 '5 f I

Ans. 84°.o; 49°, 4. ->

ART. 4. PROBLEM IN ZONAL HARMONICS.

As a second example let us consider the following problem : Two equal thin hemispherical shells of radius unity placed together to form a spherical surface are separated by a thin layer of air. A charge of statical electricity is placed upon one hemisphere and the other hemisphere is connected with the ground, the first hemisphere is then found to be at poten- tial i, the other hemisphere being of course at potential zero. At what potential is any point in the " field of force" due to the charge?

We shall use spherical coordinates and shall let Fbe the potential required. Then F" must satisfy equation (5), Art. i.

16 HARMONIC FUNCTIONS.

But since from the symmetry of the problem V is obviously independent of 0, if we take the diameter perpendicular to the

tfV . plane separating the two conductors as our polar axis, — — -^ is

zero, and our equation reduces to

9r sin 0 30

V\s given on the surface of our sphere, hence

V = f(ff) when r=i, (2)

where f(tf) = I if o < B < -, and /(0) = o if - < 0 < n.

2 £

Equation (2) and the implied conditions that V is zero at an infinite distance and is nowhere infinite are our conditions.

To find particular solutions of (i) we shall use a method which is generally effective. Assume* that V = RQ where/? is a function of r but not of 0, and & is a function of 0 but not of r. Substitute in (i) and reduce, and we get

„, Dx —jn , .

i ra*(rK) _ i dOJ. (3)

R dr* ~ @ sin 0 dO

Since the first member of (3) does not contain 0 and the second does not contain r and the two members are identically equal, each must be equal to a constant. Let us call this constant, which is wholly undetermined, m(m-\- i) ; then

d&

whence r ^ — m(m -f \}R = o, (4)

and — — -- — -j^ •}- m(m -\- i)& = o. (5)

* See the first foot-note on page 175.

X »*?

PROBLEM IN ZONAL HARMONICS.

Equation (4) can be expanded into

d*R dR r'-^-j + 2r-^r - m(m + i)R = o,

and can be solved by elementary methods. Its complete solution is

R-Arm + Br-m~\ (6)

Equation (5) can be simplified by changing the independ- ent variable to x where x = cos 0. It becomes

an equation which has been much studied and which is known as Legendre's Equation.

We shall restrict m, which is wholly undetermined, to posi- tive whole values, and we can then get particular solutions of (7) by the following device :

Assume* that © can be expressed as a sum or a series of terms involving whole powers of x multiplied by constant coefficients.

Let & = 2anxH and substitute in (7). We get

2[n(n — i)anx"-* — n(n -f- i)anxn -\- m(m + i)anxn~] = o, (8)

where the symbol 2 indicates that we are to form all the terms we can by taking successive whole numbers for n.

Since (8) must be true no matter what the value of x, the coefficient of any given power of x, as for instance x*, must vanish. Hence

(k + 2)(k + iX+2 — k(k + i)ak + m(m + i}ak = o,

m(m-\- i) — k(k -f- i) a=

If now any set of coefficients satisfying the relation (9) be taken, © = ^a^ will be a solution of (7).

If k = m, then at+, = o, ak+t = o, etc.

* See the first foot-note on page-*75^ \

18 HARMONIC FUNCTIONS.

Since it will answer our purpose if we pick out the simplest set of coefficients that will obey the condition (9), we can take a set including am.

Let us rewrite (9) in the form

(m-.k)(m+k- I)'

We get from (10), beginning with k = m — 2, m(m — i)

"~-= -"

_ m(m — \}(m — 2)(m — 3) 2. 4. (2m- i)(2;//-3)

m(m — i}(m — 2)(m — 3)(;« — 4)(m — 5)

"-• ~ 2.4.6. (2m - i)(2/« - 3)( 2/« - 5) a"" G

If m is even we see that the set will end with a0; if m is odd, with «,.

2,(2m—l)

m(m- i)(m-2)(m-

m(m- i)(m-2)(m-3)^m_ 2.4.(2;« — i)(2;«— 3)

where «m is entirely arbitrary, is, then, a solution of (7). It is found convenient to take am equal to

(2m — \)(2m — 3) ... I ~^TT '

and it will be shown later that with this value of am , @ — i when x = I.

Q is a function of x and contains no higher powers of x than xm. It is usual to write it as Pm(x\

We proceed to write out a few values of Pm(x) from the formula

= (M, - Qfr* - 3) - - - 1 r . _ >»(>>> - •) „-. w! L 2. (2m— i)

w(« - i)(w - 2)0;/ - 3) n

.*• — ... 2 .4.(2;« — i)(2m — 3) J

PROBLEM IN ZONAL HARMONICS. 19

We have:

x) - i or /'.(cos 0) = I,

x) = x or /^(cos 0) = cos 0,

*) = £(3-^ — i) or /^(cos 0) = £(3 cos20 — i),

-v\ — if r *-3 ? f\ f\r P (m<z H\ — i( C r* r>c3 H 2 rr>c fi\

x) — 2\ 5-* — 3-*^ or ^sv1-015 pj — 1^5 C0b c 3 cos f7;) < , •

— 3Ox* -f- 3) or

/'.(cos 0) = i(35 cos40 - 30 cos'0 + 3),

— JQX* + l Sx) or

/'.(cos 0) — 1(63 cos5 0 — 70 cos30 + 15 cos 0). J

We have obtained & = Pm(x) as a particular solution of (7), and 6) = Pm(cos 0) as a particular solution of (5). /^U'j or Pm(cos 0) is a new function, known as a Legendre's Coefficient, or as a Surface Zonal Harmonic, and occurs as a normal form in many important problems.

j7__ rmpm(Cos 0) is a particular solution of (i), and rmPm(cos 0) is sometimes called a Solid Zonal Harmonic.

V = A,P,(cos 0) + AsPfros 6} + A,r2P,(cos 0)

+ ^,r>/>,(cos60+... (13)

satisfies (i), is not infinite at any point within the sphere, and reduces to V = AJ>.(cos 0) + ^(cos 0) + AtPt(cos 0)

+ .43/>3(cos0)+... (14) when r = i.

r/_yJ0/>0(cos0) , A^cosff) , A,Pt(cos0) -7- -75- -p-

satisfies (i), is not infinite at any point without the sphere, is equal to zero when r = oo , and reduces to (14) when r = i.

If then we can develop f(ff) [see eq. (2)] into a series of the form (14), we have only to put the coefficients of this series in place of the A0, Alt At, ... in (13) to get the value of Ffor a point within the sphere, and in (15) to get the value of Fat a point without the sphere.

20 • HARMONIC FUNCTIONS.

We shall see later (Art. 16, Prob. 22} that if /(#) = I for o < 0 < — and/(0) = o for -'- < 0 < n,

- • ' jD-(cos

Hence our required solution is V= l + 3rP>(c°S *> ~ ' ' r3/3'(G

-f — '— 3r5/>6(cOS0)-., (17)

12 2-4

at an internal point ; and

_|_ii..Lll />(COS ^) _ . . .

1 12 2.47- v

?t an external point.

If r = - and 6 = 0, (17) reduces to

Tr I , 3 I 7 I I . II 1.3 I

L^ — — u ±1. . -- :_ . _ . -- -- . — ^ . - since P (I) — i

— /> I ^ ,, O o ^3 I TO O > ^B * • ' » S111*-^ -1 fKV1/ - »"

2 44 024 12 2. 4 4

To two decimal places F= 0.68, and the point r = -, 0 = o is at potential 0.68.

If r = 5 and 0 = — , (18) and Table I, at the end of this 4

chapter, give

and the point r = 5, # = - is at potential 0.12.

4

If the radius of the conductor is a instead of unity, we

f

have only to replace r by — in (17) and (18).

a

PROBLEM IN BESSEL'S FUNCTIONS. ;: 1

Prob. 2. One half the surface of a solid sphere 12 inches in di- ameter is kept at the temperature zero and the other half at 100 de- grees centigrade until there is no longer any change of temperature at any point within the sphere. Required the temperature of the center; of any point in the diametral plane separating the hot and cold hemispheres ; of points 2 inches from the center and in the axis of symmetry ; and of points 3 inches from the center in a di- ameter inclined at an angle of 45° tp the axis of synimetry.

Ans. 50°; 50°; 73°-9 ; 26°.! ; 77°.! ; 220-9.

** '* ^ MO * T- v~-

ART. 5. PROBLEM IN BESSEL'S FUNCTIONS.

As a last example we shall take the following problem : The base and convex surface of a cylinder 2 feet in diameter and 2 feet high are kept at the temperature zero, and the upper base at 100 degrees centigrade. Find the temperature of a point in the axis one foot from the base, and of a point 6 inches from the axis and one foot from the base, after the permanent state of temperatures has been set up.

If we use cylindrical coordinates and take the origin in the base we shall have to solve equation (6), Art. I ; or, represent- ing the temperature by u and observing that from the sym- metry of the problem u is independent of 0,

tfu I du tfu

s7' + r^ + ^'=0' (I)

subject to the conditions

u = o when z = o, (2)

u = o " r = I, (3)

U = IOO " 2 = 2. (4)

Assume u = RZ where R is a function of r only and Z of z only; substitute in (i) and reduce.

i d*R . i dR i d*Z

We get _f = (5)

R dr rR dr Z dz

The first member of (5) does not contain z\ therefore the second member cannot. The second member of (5) does not

ZZ HARMONIC FUNCTIONS.

contain r ; therefore the first member cannot. Hence each member of (5) is a constant, and we can write (5)

l^?_i-_L^- L^- R~dS~^^R~dr~ ~Z~d?~- ***

when yua is entirely undetermined.

Hence ^-^Z=o, (7)

cTR . idR .

and ^+7*+"* = a (8)

Equation (7) is easily solved, and its general solution is

Z = Ae** -\-Be~ **', or the equivalent form

Z = C cosh (us) -f- D sinh (//£). (9)

We can reduce (8) slightly by letting /-<r = x, and it becomes

d*R , i dR .

-4-^ = 0. (10)

dx1 ' x dx n

Assume, as in Art. 4, that 7? can be expressed in terms of whole powers of x. Let R = ~2anxn and substitute in (10). We get

2[n(n — \]anxn ~ ' + nanxn - 3 + ajc*\ = o,

an equation which must be true, no matter what the value of x. The coefficient of any given power of x, as xk~*, must, then, vanish, and

k(k — i)ak + kak -f- ak_ , = o, or ^X + «A-2 = 0,

whence we obtain at-* = — ^at (n)

as the only relation that need be satisfied by the coefficients in order that R = ^a/^ shall be a solution of (10).

If £ = o, ak_t = o, ak_t = o, etc.

We can, then, begin with k = o as the lowest subscript.

PROBLEM IN BESSEL S FUNCTIONS.

From (I i) at= —

Then «. = - *. = ;., «. = - 5, etc. = «.[,- i + - - -,-f^ + ...],

Hence

where at may be taken at pleasure, is a solution of (10), pro- vided the series is convergent.

Take a0 = I, and then ^ =Jo(x) where

T ' \ I I / \

J v\XJ T« I 0» -a Os .a zrz I -2 ,,2 /;» o» * ' * \12/

2 2.4 2.4«O 2.4*0.0

is a solution of (10).

Ja(x) is easily shown to be convergent for all values real or imaginary of x, it is a new and important form, and is called a Bessel's Function of the zero order, or a Cylindrical Har- monic.

Equation (10) was obtained from (8) by the substitution of x = JJLT ; therefore

is a solution of (8), no matter what the value of jn ; and u =J0(^r) sinh (fjiz) and u =Jn(fA.r) cosh (//#) are solutions of (i). «=yo(jur) sinh (jjz) satisfies condition (2) whatever the value of /*. In order that it should satisfy condition (3) JA must be so taken that

/.(/<) = o; T,0*p (13)

that is, // must be a root of the transcendental equation (13).

It was shown by Fourier that ./„(//) = o has an infinite num- ber of real positive roots, any one of which can be obtained to any required degree of approximation without serious diffi- culty. Let //,, /*„, //,,... be these roots ; then

u = A,J9(ns) sinh (pjs) + AJ.(^r) sinh (/i^)

+ AJ0(^r) sinh (^2) + . . . (14) is a solution of (i) which satisfies (2) and (3).

24 HARMONIC FUNCTIONS.

If now we can develop unity into a series of the form

+ BJJwr) + ^./.(/V) + . • • , . sinh ( u.s) T . . J5, sinh ( u.

~l

>+- • J

satisfies (i) and the conditions (2), (3), and (4). We shall see later (Art. 21) that if//*) = -

dx

I — 9\ -/o''r~'' / I -'QV"«' / _ I ^ ov^-s- / / .r-v

I, /-/M\~M 7^y\^,, /•/'»\T^<>* V10^

for values of r < i. Hence

— 200l J °^1' ' S'"h ^'^ -I- /.(^r) sinh (^fg) . / 7x

^s«JU| /•/..\_:.-i-/'^...\~t .. 7" / .. \ _:_u /~ .. \ I ' * * I \ 7 )

lifi(^i) sinh (2/1,) A*«/i(^t) sinh(; is our required solution.

At the point r = o, £ = I (17) reduces to sinh /*, . sinh yu2

sn u. sn u. -\

u — 200 — -I — • —I-

L/'./aOO sinh (2/^J ^,/1(//t) sinh (2yw2) ^ ' J

i 1

= IOO —

./X/O cosh ^ ^7,00 cosh since /0(o) = i and sinh (2*) = 2 sinh ^r cosh ;r.

If we use a table of Hyperbolic functions* and Tables II and III, at the end of this chapter, the computation of the value of u is easy. We have

/i, = 2.405 yua= 5.520

/,(/*.) = o-5 i?p /,W = - 0.3402

colog //, = 9.6189 — 10 colog //, = 9.2581 — 10 " JM = 0.2848 " 7,W= o.4683«

" cosh^,= 9.2530 — 10 " cosh^4=: 7.9037 — 10

9.1567 — 10 7.63oi« — 10

* See Chapter IV, pp. 162, 163, for a four-place table on hyperbolic func- tions.

PROBLEM IN BESSEL S FUNCTIONS.

»/i(/0 cosh /O~' = 0.1434

~' = - 0-0058

0.1376; « = i3 At the point r = £, z — I, (17), reduces to

.

/',/, W cosh /i, /*,/,(/0 cosh;*,

j = 0.6698

- 10 ,(/!,) cosh //, = 9. 1 567 - 10

8.9826 — 10; /.(*/«,) = - 0-1678

log /0(tM = 9.2248W - 10

colog /*j/,(/0 cosh/7, = 7.6301;* — 10

6.8549 — 10;

= 0.0961

,) cosh //,

0.0007 m

cosh ^ 0-°968 ' « = 9°-7

If the radius of the cylinder is a and the altitude b, we have only to replace // by j*a in (13) ; 2/1, , 2//,, ... in the denomi- nators of (15) and (17) by pj, pj), . . . ; and //,, //.,, ^us, . . . in the denominators of (16) and (17) by /*,#, //.,#, //s«, ....

Prob. 3. One base and the convex surface of a cylinder 20 cen- timeters in diameter and 30 centimeters high are kept at zero tem- perature and the other base at 100 degrees Centigrade. Find the temperature of a point in the axis and 20 centimeters from the cold base, and of a point 5 centimeters from the axis and 20 centimeters from the cold base after the temperatures have ceased to change.

Ans. 1 3°. 9; 9°.6.

26 HARMONIC FUNCTIONS.

ART. 6. THE SINE SERIES.

As we have seen in Art. 3, it is sometimes important to be able to express a given function of a variable, x, in terms of sines of multiples of x. The problem in its general form was first solved by Fourier in his " Theorie Analytique de la Chaleur" (1822), and its solution plays an important part in most branches of Mathematical Physics.

Let us endeavor to so develop a given function of x,f(x\ in terms of sin x, sin 2.x, sin $x, etc., that the function and the series shall be equal for all values of x between o and n.

We can of course determine the coefficients at, at, a3, . . . an so that the equation

f(x) = tf, sin x -f- tf2 sin 2x -f- a3 sin $x -}-... -f- an sin nx (i)

shall hold good for any n arbitrarily chosen values of x between O and n\ for we have only to substitute those values in turn in (i) to get n equations of the first degree, in which the n co- efficients are the only unknown quantities.

For instance, we can take the n equidistant values Ax,

71

^Ax, . . . nAx, where Ax = - , and substitute them for x in

n -\- i

(i). We get

f[Ax) = at sin Ax -f- a^ sin 2 Ax -\- aa sin 3 Ax -f- . .

-f- an sin nAx,

j\2.Ax) = #, sin 2Ax -f- #a sin ^Ax -(- a3 sin 6 Ax -{- . .

-f- an sin 2,nAx,

\ (2) f[$Ax) — at sin 3 Ax -j- «a sin 6 Ax -f- «3 sin <^Ax 4- . .

+ an sin 3;? Ax,

J\nAx} = #, sin n Ax -f- a, sin 2nAx -j- #3 sin

-f- an sin n*Ax, n equations of the first degree, to determine the n coefficients

<*,-» a*> af •••&*•

Not only can equations (2) be solved in theory, but they can be actually solved in any given case by a very simple and.

THE SINE SERIES. 27

ingenious method due to Lagrange,* and any coefficient am can be expressed in the form

-^ AfA*) sin (K*nA*\ (3)

K = l

If now n is indefinitely increased the values of x for which (i) holds good will come nearer and nearer to forming a con- tinuous set ; and the limiting value approached by am will probably be the corresponding coefficient in the series required to represent /(.z) for all values of x between zero and n.

Remembering that (» + \)Ax = n, the limiting value in question is easily seen to be

IT

am = - Cf(x) sin mxdx. (4)

7Tt/ 0

This value can be obtained from equations (2) by the fol- lowing device without first solving the equations :

Let us multiply each equation in (2) by the product of Ax and the coefficient of am in the equation in question, add the equations, and find the limiting form of the resulting equation as n increases indefinitely.

The coefficient of any a, aK in the resulting equation is

sin KAx sin mAx . Ax -\- sin 2,KAx sin 2,mAx . Ax -}- , . . -f- sin nKAx sin nmAx . Ax.

Its limiting value, since (n-\- \)Ax = TC, is

ir

/ sin KX sin mx.dx\ but

w w

I sin KX sin mx . dx = \ I [cos (m — K)X — cos(m -}- K)x~\dx—Q

0 0

if m and K are not equal.

* See Riemann's Partielle Differcntialgleichungen, or Byerly's Fourier's Series and Spherical Harmonic?.

28 HARMONIC FUNCTIONS.

The coefficient of am is

//;tr(sin2 mAx -\-s\rf 2m Ax -j- sin2 ynAx -f- . . . -f- sin2 nmAx\ Its limiting value is

IT

71

y» . s

sm w;r .x = —.

2 o

The first member is

/( J^r) sin 7«z/;r . J^r -\-f(2.Ax) sin 2mAx . Ax -f- . . .

-j-/(«^-^) sin mnAx . Ax, and its limiting value is

/ f(x) sin mx , dx.

0

Hence the limiting form approached by the final equation as n is increased is

»r

/ J\x] sin mx . dx = — am.

0

Whence

2 /'

«» = -J f(*} sin «»•<& (5)

7T0

as before.

This method is practically the same as multiplying the equation

f(x) = #, sin x -j- a3 sin 2^r -f- «s sin $x -}- . . . (6)

by sin mx . dx and integrating both members from zero to ft.

It is important to realize that the considerations given in this article are in no sense a demonstration, but merely estab- lish a probability.

An elaborate investigation * into the validity of the develop- ment, for which we have not space, entirely confirms the results formulated above, provided that between x = o and x = n the

* See Art. to for a discussion of this question.

THE SINE SERIES. Z'J

function is finite and single-valued, and has not an infinite num- ber of discontinuities or of maxima or minima.

It is to be noted that the curve represented by y = f(x) need not follow the same mathematical law throughout its length, but may be made up of portions of entirely different curves. For example, a broken line or a locus consisting of finite parts of several different and disconnected straight lines can be represented perfectly well by y = a. sine series.

As an example of the application of formula (5) let us take the development of unity.

Here f(x) = I.

«• am = — / sin mx . dx ;

7t i/

/ si

cos mx sin mx . dx = -- .

m

V

/I I

sin mx.dx = — (i — cos mrr] = — [i — ( — iY*l m m

0

= o if m is even

= — if m is odd. m

4 /sin x . sin \x . sin ^x . sin Jx . \ Hence , = i(— + _J- + _J- +-J-. + ...). (7)

It is to be noticed that (7) gives at once a sine development for any constant c. It is,

c = 4c(s\nx sin 3* sin 5* \

n \ i 3 5 "/'

Prob. 4. Show that for values of x between zero and it

x sn zx . sn 30: sn

fL\ // \ _ 4 T8^11 x s^n 3^ s'n S1^ s'n 7-^ j^

= "" "~ ~~ ~~

"30 HARMONIC FUNCTIONS.

if /(x) = x for o < x < — , and f(x) = n — x for — < x < n.

to /(*) =

2 Fsin x . 2 sin 2x , sin T.X . sin zx , 2 sin 6x . sin nx ,

--

if /(.#) — i for o < x < — , and /(x) = o for — < a: < TT.

(d} sinh jc =

2 sinh 7t Pi . 2 .

nh 7t Pi . 2 . i 3 • 4 •

— sin x -- sin 2X -\- — sin T.X -- sin AX -+- . . . . n [_2 5 rio 17 J

(e) x" =

2f/zr2 4\ . T? . In* 4\ . n* .

--- -, sin x -- sin 2x -\-\ --- a sin -ix -- sin 4x4- . . 7rL\i i / 2 \3 3/ 4 J

ART. 7. THE COSINE SERIES.

Let us now try to develop a given function of x in a series of cosines, using the method suggested by the last article. Assume

f(x) = bt -J- bl cos x -\-b^ cos 2.x -j- /^s cos 3^ -f- . . . ( i )

To determine any coefficient ^m multiply (i) by cos mx .dx and integrate each term from o to TT.

cos mx .x = o.

0

j

0

IT

/ bk cos kx cos w^r . dx=.Q, if »z and y^ are not equal. o

7T

/7T bm cos" w^r ^ = — bm, if w is not zero. 2 .

o

7T 2 /'

Hence £w = — / f(x) cos mx .dx, (2)

0

jf /« is not zero.

THE COSINE SERIES. 31

To get b0 multiply (i) by dx and integrate from zero to n.

•a Jb.dx = bjt,

0

IT

/ bk cos kx . dx = o.

0

w

Hence b0 = —J*f(x}dx, (3)

0

which is just half the value that would be given by formula (2) if zero were substituted for m.

To save a separate formula (i) is usually written f(x) = ££0 + b, cos x + £, cos 2x + £s cos 3* + . . ., (4)

and then the formula (2) will give bn as well as the other coef- ficients.

Prob. 5. Show that for values of x between o and n

_TT 4 /cos* cos 3* cos 5* \

-2~n\~^~ ~7~ ~7~ "J5

7t 8 /COS 2* COS 6.X , COS 10

\

- - -j,

if /(^c) = * for o < x < — , and f(x) = TT — x for — < A, x TT;

W

if /(*) = i for o < x < — , and f(x) = o for — < x < TTV

2 2

21 I I

(</) sinh x = - -(cosh 7f — i) -- (cosh n -j- i) cos *

7T|_ 2 2

-| — (cosh it — i) cos 2^: -- (cosh n -j- i) cos 30: + . . . ;

COS 2X COS t* COS

32 HARMONIC FUNCTIONS.

ART. 8. FOURIER'S SERIES.

Since a sine series is an odd function of x the development of an odd function of x in such a series must hold good from x = — it to x = TT, except perhaps for the value x = o, where it is easily seen that the series is necessarily zero, no matter what the value of the function. In like manner we see that if f(x) is an even function of x its development in a cosine series must be valid from x = — n to x = n.

Any function of x can be developed into a Trigonometric series to which it is equal for all values of x between — n and n.

Let/(;r) be the given function of x. It can be expressed as the sum of an even function of x and an odd function of x by the following device :

*) A*) -A-*)

identically ; but ' ~'-/^ -- 1 is not changed by reversing

£

the sign of x and is therefore an even function of x\ and when

f(x\ — f(— x\ we reverse the sign of x, -- - is affected only to the

2

extent of having its sign reversed, and is consequently an odd function of x.

Therefore for all values of x between — it and n

f(x\ -4- /[ — x) i , , ,

yv ; ' • — ' = -£0 -(- 1>, cos x -f- £, cos 2x -|- £3 cos 3* -f- . . .

2 2

2 rA*) +A— *) ;

where bm — — I yv ' —*• cos w^r . dx ;

7T t/ 2

ffx\ _ ft _ x\

and -l-^—L — ^ - '- = al sin ^ + ^, sin 2x -f- ^3 sin ^x -f- . . .

2 /y[;r) — /(— ^r) . where «„ = - / :iA— - — - - sin mx . dx.

7t »/ 2

FOURIER'S SERIES. 33

bm and am can be simplified a little.

2 //(*) + /(-•*) £» = --/ yv '^ ^ -cos mx. d^

71 U 2

o

It IT

= — jf(*) cos mx . dx+Jf(—x) cos mx . dx\; o o -1

but if we replace x by — x, we get

it — » 0

J /(•— #) cos nix . dx=—J f(x) cos mx.dx = J f(x)cos mx.dx,

0 -IT

»r

and we have ^w = — I /(*) cos w^r . dx.

—•a

In the same way we can reduce the value of am to

ir

— / f(x^ sin mx . dx.

71 t/

— it Hence

f(x) =• - 60 -\- l>l cos x -}- bt cos 2.x -f- <^s cos 3* -(-...

M

-|- #, sin JT -}" «, sin 2^r -|- «3 sin 3^r -}-..., (2)

JT

where #„, = — / f(x) cos wjr . ^, (3)

— n •n and am = — I f(x} sm mx . dx, (4)

7T t/

— it

and this development holds for all values of x between — n and TI.

The second member of (2) is known as a Fourier's Series.

The developments of Arts. 6 and 7 are special cases of development in Fourier's Series.

Prob. 6. Show that for all values of x from — n to n 2 sinh TrFi i i i i ~|

f* =• COS* H COS2X COS $X-\ COS4.X-H...

7T [_2 2 '5 10 17

54 HARMONIC FUNCTIONS.

2 sinh 7t |~i . 2 . 3 . 4 .

H — sin x sin 2X -4- — sin 3^ sin AX + . . . .

7i L_2 5 10 17 J

Prob. 7. Show that formula (2), Art. 8, can be written

f(x) = -f0 COS/?0 -j- £, COS (x — /?j) + ^a COS (2^ — fi^)

+ ^3 COS (3* — /?,) -j- . . . ,

where cm — (a^ + £,„")» and fim = tan"1 -r^-

Prob. 8. Show that formula (2), Art. 8, can be written

T

f(x) =-c, sin ft. + fl sin (^ + A) + ^ sin (2;c + A)

2

+ ^3 sin (3* + /?,) + . . . ,

b where cm — (am* + bm )* and pw = tan"1 — .

ART. 9. EXTENSION OF FOURIER'S SERIES.

In developing a function of x into a Trigonometric Series it is often inconvenient to be held within the narrow boundaries x = — rt and x = n. Let us see if we cannot widen them.

Let it be required to develop a function of x into a Trigonometric Series which shall be equal to f(x] for all values of x between x = — c and x = c.

Introduce a new variable

which is equal to — n when x = — c, and to n when x = c.

f(x) = /( — z\ can be developed in terms of z by Art. 8, (2), (3), and (4). We have

/(.*"•*) = 2 *° ^ ^ C°S * ~^ ^ C°S 2Z + £• cos 3* + . . .

-f- tft sin ^ -(- a, sin 2£ -J- ^3 sin 3.2 -j- . . . , (i)

where bm = — //( — *} cos *»# . afe, (2)

71 U \ 71 I

EXTENSION OF FOURIER'S SERIES,

and am = —Jf\—z\ sin mz . dzy (3)

— IT

and (i) holds good from z = — n to z = n.

Replace z by its value in terms of x and (i) becomes

i nx

/(•*•) = -£. + £,cos

£

a sin

C> c-

•nx 2,nx

^nx + tf3 sin— -+...; (4)

6-

and (2) and (3) can be transformed into

c , I /* ./v \ WIT-*" 7

bm = —J f(x) cos —^-dx, (5)

i r ft \ - mnx j

am = — J f(x) sin —^—dxt (6)

— c

and (4) holds good from x = — c to x •= c.

In the formulas just obtained c may have as great a value as we please so that we can obtain a Trigonometric Series for f(x] that will be equal to the given function through as great an interval as we may choose to take.

It can be shown that if this interval c is increased indefi- nitely the series will approach as its limiting form the double

00 00

integral -- / f(\)d\ I cos a(h — x}da, which is known as a

— oo 0

Fourier's Integral. So that

+ 00 ao

f(x) = -W" /(A>A f cos a(\ - x}da (7)

-09 0

for all values of x.

For the treatment of Fourier's Integral and for examples of its use in Mathematical Physics the student is referred to Riemann's Partielle Differentialgleichungen, to Schlomilch's Hohere Analysis, and to Byerly's Fourier's Series and Spherical Harmonics.

36 HARMONIC FUNCTIONS.

Prob. 9. Show that formula (4), Art. 9, can be written

xt \ l (7tX o \ , l27tX \

f(x) = -cQ cos /?„ + c, cos (—- Pi) + f* cos (— -- fl*j

- A +...,

where c.... = (<",,.," \- bm^ and ftm = tan"1 ~,

bm

Prob. 10. Show that formula (4), Art. 9, can be written /(*) = ^0 sin ^o + ^i sin (-^ + A,J + c, sin f^— ^ +

where ^», = awa + £«** and

ART. 10. DIRICHLET'S CONDITIONS.

In determining the coefficients of the Fourier's Series rep- resenting f(x) we have virtually assumed, first, that a series of the required form and equal to f(x] exists; and second, that it is uniformly convergent ; and consequently we must regard the results obtained as only provisionally established.

It is, however, possible to prove rigorously that if f(x) is finite and single-valued from x = — n \.Q x =. n and has not an infinite number of (finite) discontinuities, or of maxima or minima between x = — n and x — rry the Fourier's Series of (2), Art. 8, and that Fourier's Series only, is equal to f(x} for all values of x between — n and TT, excepting the values of x corresponding to the discontinuities of f(x\ and the values it and — TT; and that if c is a value of x corresponding to a .discontinuity of f(x), the value of the series when x = c is

«)]; and that when * = n or

x = — Tt the value of the series is ^[/(TT) -}-/(— TT)].

This proof was first given by Dirichlet in 1829, and may be found in readable form in Riemann's Partielle Differential- gleichungen and in Picard's Traite d'Analyse, Vol. I.

DIRICHLET'S CONDITIONS.

3?

A good deal of light is thrown on the peculiarities of trigo- nometric series by the attempt to construct approximately the curves corresponding to them.

If we construct y — al sin x and y — at sin 2x and add the ordinates of the points having the same abscissas, we shall ob- tain points on the curve

y = al sin x + ay sin 2x.

If now we construct y — as sin 3* and add the ordinates to those of y — at sin x + at sin 2.x we shall get the curve

y — a, sin x + at sin 2x + a3 sin 3^.

By continuing this process we get successive approximations to y — a, sin x + *, sin 2x + a, sin 3* + a, sin 4* + ...

O/S

II

X

y

o

CE3

y

Tt III

!

IY

Let us apply this method to the series y - sin x + \ sin 3* + | sin *>x + (i) (See (7), Art. 6.)

i/ = o when x — o, — from * = o to x = TT, and o when x — TT.

4

It must be borne in mind that our curve is periodic, hav- ing the period 2?r, and is symmetrical with respect to the origin.

The preceding figures represent the first four approxima-

444680

38 HARMONIC FUNCTIONS.

tion to this curve. In each figure the curve y = the series, and the approximations in question are drawn in continuous lines, and the preceding approximation and the curve corre- sponding to the term to be added are drawn in dotted lines.

Prob. 11. Construct successive approximations to the series given in the examples at the end of Art. 6.

Prob. 1 2. Construct successive approximations to the Maclaurin's

x3 x* Series for sinh x, namely x -\ -\ j- + • • •

O * D *

ART. 11. APPLICATIONS OF TRIGONOMETRIC SERIES.,

(a) Three edges of a rectangular plate of tinfoil are kept at potential zero, and the fourth at potential I. At what po- tential is any point in the plate ?

Here we have to solve Laplace's Equation (3), Art. I, which, since the problem is two-dimensional, reduces to

subject to the conditions V = o when x = o, (2)

V = o " x = a, (3)

V = o " y = o, (4)

V = i " j' = 3. (5)

Working as in Art. 3, we readily get sinh fiy sin /ir, sinh /?/ cos fix, cosh /ty sin fix, and cosh /?j cos fix as particu- lar values of V satisfying (i).

. , mny . tmtx . r , \ , \ , \ i / \ V = sinh — - sin - - satisfies (i), (2), (3), and (4).

• • i_ ny • 1 ^y

sinh - sinh-i-^

V = ±\ ^sin^+l- _• *,«=i+... (6)

is the required solution, for it reduces to i when y = b. See (7), Art. 6.

APPLICATIONS OF TRIGONOMETRIC SERIES. 39

(b] A harp-string is initially distorted into a given plane curve and then released ; find its motion.

The differential equation for the small transverse vibrations of a stretched elastic string is

9>- fl.9^

2 ' * * '

as stated in Art. i. Our conditions if we take one end of the string as origin are

y = o when x = o, (2)

y = o " x = /, (3)

3J

-a/ = ° * = °> (4)

y = fx « t - o. (5)

Using the method of Art. 3, we easily get as particular solutions of (I)

y = sin fix sin afit, y = sin fix cos afit,

y — cos fix sin «/?/, and y = cos /?.*• cos a fit.

. mnx mnat ,

y = sm — j— cos— -. — satisfies (i), (2), (3), and (4).

. mnx mrcat /A\

am sm — j— cos — ; — , V°>

, /» WTT^r , /_\

where am = j f(x) sin — j— . «*

0

is our required solution ; for it reduces to/(^) when/ = o. See

Art. 9.

Prob. 13. Three edges of a square sheet of tinfoil are kept at potential zero, and the fourth at potential unity ; at what potential is the centre of the sheet ? Ans. 0.25.

Prob. 14. Two opposite edges of a square sheet of tinfoil are kept at potential zero, and the other two at potential unity ; at what potential is the centre of the sheet ? Ans. 0.5.

Prob. 15. Two adjacent edges of a square sheet of tinfoil are

40 HARMONIC FUNCTIONS.

kept at potential zero, and the other two at potential unity. At what potential is the centre of the sheet ? Ans. 0.5.

Prob. 16. Show that if a point whose distance from the end of a

harp-string is -th the length of the string is drawn aside by the n

player's finger to a distance b from its position of equilibrium and then released, the form of the vibrating string at any instant is given by the equation

mnat

y — 7 — — r~i> ~~t sin — sin — T cos

J ^-

7 — — r~i t — — T — ~r '

(n— \\n ^- \m n I I

^ ' m=l '

Show from this that all the harmonics of the fundamental note of the string which correspond to forms of vibration having nodes at the point drawn aside by the finger will be wanting in the complex note actually sounded.

Prob. 17.* An iron slab 10 centimeters thick is placed between and in contact with two other iron slabs each 10 centimeters thick. The temperature of the middle slab is at first 100 degrees Centigrade throughout, and of the outside slabs zero throughout. The outer faces of the outside slabs are kept at the temperature zero. Re- quired the temperature of a point in the middle of the middle slab fifteen minutes after the slabs have been placed in contact. Given a3 = 0.185 in C.G.S. units. Ans. io°-3.

Prob. 18.* Two iron slabs each 20 centimeters thick, one of which is at the temperature zero and the other at 100 degrees Centigrade throughout, are placed together face to face, and their outer faces are kept at the temperature zero. Find the temperature of a point in their common face and of points 10 centimeters from the com- mon face fifteen minutes after the slabs have been put together.

Ans. 22°.8 ; 15°. i ; i7°.2.

ART. 12. f PROPERTIES OF ZONAL HARMONICS.

In Art. 4, z = Pm(x) was obtained as a particular solution of Legendre's Equation [(7), Art. 4] by the device of assuming that z could be expressed as a sum or a series of terms of Ihe form anx* and then determining the coefficients. We

* See Art. 3.

f The student should review Art. 4 before beginning this article.

PROPERTIES OF ZONAL HARMONICS. 41

can, however, obtain a particular solution of Legendre's equa- tion by an entirely different method.

The potential function for any point (x, y, z) due to a unit of mass concentrated at a given point (x^y^ £,) is

= '

and this must be a particular solution of Laplace's Equation £(3), Art. i], as is easily verified by direct substitution. If we transform (i) to spherical coordinates we get

V= - l _ = (2)

yr1 — 2rr1[cos 0 cos 0, -j- sin 0 sin Ol cos (0—0,)] -f- r,2

as a solution of Laplace's Equation in Spherical Coordinates [(5), Art. i].

If the given point (x^y^ #,) is taken on the axis of X, as it must be in order that (2) may be independent of 0, 0, = o, and

J7— _ _ ^_ _ * ^3)

Vr* — 2rr, cos 0 -{- r,8

is a solution of equation (i), Art. 4. Equation (3) can be written

(4)

/ r ra\-*

and if r is less than rl (1—2— cos 0 -f- — J can be developed

'"i '"i

^ra

into a convergent power series. Let 5"/>OT — be this series,

rm

pm being of course a function of 0. Then F= — ^pm—^ is a

i i

solution of (i), Art. 4.

Substituting this value of V in the equation, and remem- bering that the result must be identically true, we get after a

slight reduction

42 HARMONIC FUNCTIONS.

but, as we have seen, the substitution of x = cos 0 reduces this- to Legendre's equation [(7), Art. 4]. Hence we infer that the coefficient of the mth power of z in the development of (i — 2xz-\-z*)~* i^ a function of x that will satisfy Legendre's equation.

(! _ 2** + *")-* = [I -Z(2X -*)]-*,

and can be developed by the Binomial Theorem ; the coefficient of zm is easily picked out, and proves to be precisely the func- tion of x which in Art. 4 we have represented by Pm(x\ and have called a Surface Zonal Harmonic. We have, then,

if the absolute value of z is less than I. If x = i, (5) reduces to

but (i — 2,3r + *')'* = (i — *)"' = i+<s + ^ + <s3 + - • •;

hence ^«(0 = i- (6)

Any Surface Zonal Harmonic may be obtained from the two of next lower orders by the aid of the formula

(n + !)/>, + ,(*) - (2» + i)^^) + w/7,,. ,(^) = o, (7>

which is easily obtained, and is convenient when the numerical value of x is given.

Differentiate (5) with respect to z, and we get

(I - whence

or by (5)

(i - 2Mt + **)(/>,(*) + 2^f(«) .*+ S/'.W •*' ' • ->

+ P1(X}.z + P^.^+ .-0=0. (8)

PROBLEMS IN ZONAL HARMONICS. 43

Now (8) is identically true, hence the coefficient of each power of z must vanish. Picking out the coefficient of z11 and writing it equal to zero, we have formula (7) above.

By the aid of (7) a table of Zonal Harmonics is easily com- puted since we have P0(x) = i, and P^x) = x. Such a table for x = cos 0 is given at the end of this chapter.

ART. 13. PROBLEMS IN ZONAL HARMONICS.

In any problem on Potential if Fis independent of 0 so that we can use the form of Laplace's Equation employed in Art. 4, and if the value of Fon the axis of X is known, and

•can be expressed as 2amrm or as ^> -^qij, we can write out •our required solution as

F=2amrmPm(cos0) or V = ^ ^">/?">^S ^ ;

r

jf or since Pm(i) = I each of these forms reduces to the proper value on the axis ; and as we have seen in Art. 4 each of them .satisfies the reduced form of Laplace's Equation.

As an example, let us suppose a statical charge of M units <of electricity placed on a conductor in the form of a thin circu- lar disk, and let it be required to find the value of the Poten- tial Function at any point in the " field of force " due to the charge.

The surface density at a point of the plate at a distance r

from its centre is

M

(T =

Vd' - S

•and all points of the conductor are at potential — . See Pierce's Newtonian Potential Function (§ 61).

The value of the potential function at a point in the axis ot the plate at the distance x from the plate can be obtained without difficulty by a simple integration, and proves to be

M x* - a"

V — — cos-1-— — . (i)

2a x -- a

44 HARMONIC FUNCTIONS.

The second member of (i) is easily developed into a power series.

M , x* - a*

— cos

- 1

2a x* -j- «'

MVn x x3 x* x'

"I

•J

.

lfjr>*

Hence

y = Mr* _ ^/> (cos 0) + - # l_2 « 3

...

is our required solution if r < # and # < -, as is

,7 M\~a i a* n , N i a1 „ ,

F = -[- - _. -P9 (cos 60 + - -- P4(cos 0)

(5)

The series in (4) and (5) are convergent, since they may be obtained from the convergent series (2) and (3) by multiplying the terms by a set of quantities no one of which exceeds one in absolute value. For it will be shown in the next article that Pm (cos 6) always lies between i and — i.

Prob. 19. Find the value of the Potential Function due to the attraction of a material circular ring of small cross-section.

The value on the axis of the ring can be obtained by a simple

M

integration, and is . , if M is the mass and c the radius of the

v £ -|- r

ring. At any point in space, if r < c

V = y [/>0(cos 0) - I £ and if r > c

ADDITIONAL FORMS. 45

= *L\£-p o(COS 0) - - -(COS 0) + Il£ C

* Lf 2 r 2 . 4 r

ART. 14. ADDITIONAL FORMS.

(a) We have seen in Art. 12 that Pm(x) is the coefficient of in the development of (i — 2xz-\- z*}~^ in a power series.

l - 2X2 2* - 1 = I - ***'' * - •' ^-i

If we develop (l — #**')-* and (i — #*-*)-* by the Bi- nomial Theorem their product will give a development for ( i — 2xz -j- z*} - *. The coefficient of zm is easily picked out and reduced, and we get

/>«(cos 0) =

1.3.5... (2m —

2. 4. 6. ..a»

i 3.^-1)

I . 2 .(2W— l)(2W — 3)

If m is odd the parenthesis in(i) ends with the term con- taining cos 0 ; if m is even, with the term containing cos o, but in the latter case the term in question will not be multiplied by the factor 2, which is common to all the other terms.

Since all the coefficients in the second member of (i) are positive, Pm(cos 0) has its maximum value when 6 = o, and its value then has already been shown in Art. 12 to be unity. Obviously, then, its minimum value cannot be less than — i.

(b) If we integrate the value of Pm(x) given in (11), Art. 4, m times in succession with respect to x, the result will be

1.3.5.. '(2m — i), , .„

lound to differ from — ±-^—, — Ti - (x — l) by terms m-

(2iri) \

volving lower powers of x than the mih.

Hence />„« = JL f - ,)«. (2)

46 HARMONIC FUNCTIONS.

(c) Other forms for Pm(x), which we give without demon- stration, are

- (- 0" Q" * . /x

P .A -

m

— I . COS

nj ' o

0 L ™J

(4) and (5) can be verified without difficulty by expanding •and integrating.

ART. 15. DEVELOPMENT IN TERMS OF ZONAL HARMONICS.

Whenever, as in Art. 4, we have the value of the Potential Function given on the surface of a sphere, and this value de- pends only on the distance from the extremity of a diameter, it becomes necessary to develop a function of 6 into a series of the form

AnPa(cos ff) + A/3, (cos 0) + A,PJcos (f) + .

0 u\ / I * 1> / I • *\ / I

or, what amounts to the same thing, to develop a function of x into a series of the form

The problem is entirely analogous to that of development in sine-series treated at length in Art. 6, and may be solved by the same method.

Assume f(x) = A.PJx) + Afl*) + AtPfc) + . . . (i)

for — i < x < i. Multiply (i) by Pn(x)dx and integrate from — I to i. We get

M C

V

FORMULAS FOR DEVELOPMENT. 47

We shall show in the next article that

i I Pm(x)Pn(x}dx — o, unless m = n,

and that

-1

?}}1 I I />

Hence Am = —^~ J f(x}Pm(x}dx. (3)

— i

It is important to notice here; as in Art. 6, that the method we have used in obtaining Am amounts essentially to deter- mining Am, so that the equation

A*) = A.P.(x) + ASM + ASM + • • • + ASM

shall hold good for n -f- I equidistant values of x between — i and i, and taking its limiting value as n is indefinitely in- creased.

ART. 16. FORMULAS FOR DEVELOPMENT. We have seen in Art. 4 that z = Pm(x] is a solution of

Legendre's Equation -j-\ (i — ^2) ~ -f- m(m -f- 1)3 =• o. (i) ctx L- dx _J

dPm(x\~\

~*9)~ir J +w(w+ o^w = o, (2) and -(l-^-

Multiply (2) by Pn(x) and (3) by Pm(x), subtract, transpose. and integrate. We have

[«(«+ i) - n(n +

48 HARMONIC FUNCTIONS.

by integration by parts,

= o.

Hence fpm(x)P«(x)dx = O, (6)

-i unless m = «.

If in (4) we integrate from x to i instead of from — i to I, we get an important formula.

y* n y n y

P (x]P (r}dx = L -* (7)

m(m-\- i) i — n(n-\- i) ' v/'

X

and as a special case, since P0(x) — i.

(8)

- -, - : — - , m(m -f- i)

unless m = o. i To get flPJ^xftdx is not particularly difficult. By (2),

-i Art. 14,

By successive integrations by parts, noting that

Jm - K

-T- —^(x1 — i)m contains (x? — i)K as a factor if K < m, and

FORMULAS FOR DEVELOPiMENT. 40

^"'"(jtT1 _ \\m

that — i— -5 - '— = (2m}\ we get

- \}mdx = j\x - \y(x

m + i

'

Hence f\PJ(xNdx = -t . (u)

/ l_ '« \ / J ^ ^2 __[_ T

-1

1 •Prob. 20. Show that / Pm(x}dx = o if m is even and is not zero

0 m—l j - - _ ^

= (— J) z -~r~ — \ • ;. — -T if m is odd.

m(m+ i) 2.4.6 ... (m—i)

y» i

iPJMr^f = -- L~~- Note that 2m -i- i

0

jc" is an even function of x.

Prob. 22. Show that if f(x) = o from x = — i to x = o, and ) = i from .r =• o to x = i,

Prob. 23. Show that /7(0) = 2 S^^cos 0) where

>«=o

B] Sm0d0.

.30 HARMONIC FUNCTIONS.

Prob. 24. Show that

0= -^fi + sfi)'/', (cos 60 + 9(^)X(ccs H) + . . .1.

2 I \2 / \2 . 4' I

i, Art. 14.

esc See (i), Art. 14.

Prob. 25. Show that

+ (.. - ,)(" + -)("-•) />._(,) + .

2.4 J

1 1

TSTote that / xnPm(x)dx = — - — I xn— ]—dx, and use the

«/ 2mm \ v dxm

-i -i

method of integration by parts freely.

Prob. 26. Show that if Fis the value of the Potential Function at any point in a field of force, not imbedded in attracting or repel- ling matter; and if F = /(0) when r = a,

V=2Am~Pm(cos6}ifr<a

Ur

am+1 and V = 2^m-lPm(cos 0)i(r> a,

where ^4« = I f(6)Pm(cos 0) sin 0^.

2 t/

0

Prob. 27. Show that if

ca .,

y •= c when r = a ; K = rur<a, and r = — if r > «.

r

ART. 17. FORMULAS IN- ZONAL HARMONICS.

The following formulas which we give without demonstra- tion may be found useful for reference :

_—„_,

SPHERICAL HARMONICS. 51

ART. 18. SPHERICAL HARMONICS.

In problems in Potential where the value of V is given on the surface of a sphere, but is not independent of the angle 0. we have to solve Laplace's Equation in the form (5), Art. I, and by a treatment analogous to that given in Art. 4 it can be

proved that

d*Pm(u) d'TJi-n

V = r'" cos nd) sin" (i— • and V = r'" sin n<h sin" 6 —

dvn <//<"

where // = cos 0, are particular solutions of (5), Art. i.

The factors multiplied by r"1 in these values are known as Tesseral Harmonics. They are functions of 0 and 0, and they play nearly the same part in unsymmetrical problems that the Zonal Harmonics play in those independent of 0.

FM,O, 0) = AtPm(p) +n2l(Ancos n<fy + Bn sin «0)sin" d*-?4£

m=i «/<"

is known as a Surface Spherical Harmonic of the wth degree, and F=r'"Fm(,u, 0) and V = -^ Ym(,.<, 0)

satisfy Laplace's Equation, (5), Art. I.

The Tesseral and the Zonal Harmonics are special cases of the Spherical Harmonic, as is also a form Pm(cos y] known as a Laplace's Coefficient or a Laplacian ; y standing for the angle between r and the radius vector rl of some fixed point.

For the properties and uses of Spherical Harmonics we refer the student to more extended treatises, namely, to Ferrer's Spherical Harmonics, to Heine's Kugelfunctionen, or to Byerly's Fourier's Series and Spherical Harmonics.

ART. 19.* BESSEL'S FUNCTIONS. PROPERTIES. We have seen in Art. 5 that z =^J^(x] where

* The student should review Art. 5 before reading this article.

52 HARMONIC FUNCTIONS.

is a solution of the equation

cfz . I dz .

^> + --r + 2 = 0'' (2)

ax x dx

and we have called Ja(x] a Bessel's Function or Cylindrical Harmonic of the zero order.

_ dj,(x] _ xV x* x» x6 -1

~^7~ ~2l_ 2T4+2.42.6 2.42.6<.8~i ••J<2

is called a Bessel's Function of the first order, and

*'=/,(*) is a solution of the equation

which is the result of differentiating (2) with respect to x.

A table giving values of J*(x) and /,(-*') W1'U De found at the end of this chapter.

If we write J9(x) for x in equation (2), then multiply through by xdx and integrate from zero to x, simplifying the resulting equation by integration by parts, we get

dx

or, since /,(*) = — -^ — ,

J'xJ0(x}dx = xj,(x\ (5)

0

If we write Jt(x) for z in equation (2), then multiply through 'by x*— j- -, and integrate from zero to x, simplifying by inte- gration by parts, we get

APPLICATIONS OF BESSEL'S FUNCTIONS.

If we replace x by jjix in (2) it becomes

(Fz . i dz .

^ + ^ + /J2 = °

(See (8), Art. 5). Hence z — /9(f*x) is a solution of (7).

If we" substitute in turn in (7)J<>(vKx) and /0(/vr) for -, mul- tiply the first equation by xj^x), the second by xJ^Kx\ subtract the second from the first, simplify by integration by parts, and reduce, we get

(8)

Hence if //K and //t are different roots of ./„(//#) = O, or of — °» or of wAd"*) — A/0(y"«) = o,

= o. (9)

0

We give without demonstration the following formulas, which are sometimes useful :

I n

^(x) = - I C0s(^r COS 0)^0. (lO)

7ft/ 0

»r Jf /*

t(x\ = - I sin" 0 cos (x cos 0)^/0. (i i)

TTe/

They can be confirmed by developing cos (x cos 0), inte- grating, and comparing with (i) and (3).

ART. 20. APPLICATIONS OF BESSEL'S FUNCTIONS.

(a) The problem of Art. 5 is a special case of the following : The convex surface and one base of a cylinder of radius a and length b are kept at the constant temperature zero, the temperature at each point of the other base is a given function of the distance of the point from the center of the base ; re-

HARMONIC FUNCTIONS.

quired the temperature of any point of the cylinder after the permanent temperatures. have been established.

Here we have to solve Laplace's Equation in the form

(see Art. 5), subject to the conditions

u = o when z = o,

u = o " r = a,

u = /(r) " z = b. Starting with the particular solution of (i),

u = sinh(/^r)/0OO, (2)

and proceeding as in Art. 5, we get, if //,,//,, /*„, . . . are roots of Jt(t*a) = o, (3)

and f(r) = AJ^r] + AJ^r} + AMM + • - - > (4)

(b) If instead of keeping the convex surface of the cylinder at temperature zero we surround it by a jacket impervious to heat the equation of condition, u = o when r = a, will be re-

placed by — = o when r = a, or if u = sinh (jjufyjfyir) by

= o when r = a,

dr that is, by —

or /,(/«*) = o. (6)

If now in (4) and (5) ;/,, ;*,, jw,, . . . are roots of (6), (5) will be the solution of our new problem.

(c] If instead of keeping the convex surface of the cylinder at the temperature zero we allow it to cool in air which is at the temperature zero, the condition u = o when r = a will be

replaced by ~ '• -\- hu = o when r = a, h being the coefficient

or

of surface conductivity.

DEVELOPMENT IN TERMS OF BESSEL's FUNCTIONS. 55

If u = sinh (jjiz]J ^(JJLT'] this condition becomes

— Pj&r) + hJt(nr) = o when r — a, or /"*/,(/"*) — ahj^d] — o. (7)

If now in (4) and (5) //,,//,, //3 , . . . are roots of (7), (5) will be the solution of our present problem.

It can be shown that

SM = o, (8)

/.(•*) = o, (9)

and */,(*) - A/«U') = 0 (10)

have each an infinite number of real positive roots.* The earlier roots of these equations can be obtained without serious difficulty from the table iorj^x) and J^x) at the end of this chapter.

ART. 21. DEVELOPMENT IN TERMS OF BESSEL'S FUNCTIONS.

We shall now obtain the developments called for in the last article.

Let Ar) = AJJM-+AtfJM + AJ.(jis) + ... (0 /I,,;*,, //, , etc., being roots of /„(//«) = O, or of /,(yw«) = o, or of

To determine any coefficient Ak multiply (i) by rJQ(f.ikr}dr and interate from zero to a. The first member will become

Every term of the second member will vanish by (9), Art. 19, except the term

o

o o

by (6), Art. 19.

* See Riemann's Partielle Differentialgleichungen, § 97.

.50 HARMONIC FUNCTIONS.

Hence Ak = - / rf(f)J <$J*ip)dr. (2)

The development (i) holds good from r = o to r = # (see Arts. 6 and 15).

If yu, , yua , yus , etc., are roots oij^a) = o, (2) reduces to

> /<» i Ms > etc-> are roots of /,(/*#) = o, (2) reduces to

If //lt ^s, /i,, etc., are roots of jjaj^a) — A/0(/^) = o, (2) reduces to

For the important case where f(r) = i

a a /J.f.a

frf(ryt>(nkr}(tr= frj,(fif)etr=^

r o

by (5), Art. 19; and (3) reduces to

(4) reduces to

^* = o, (8)

except for k = i, when /^ = o, and we have

A, = i ; (9)

(5) reduces to ^ = 7—— , - r. (io'\

Prob. 28. A cylinder of radius one meter and altitude one meter has its upper surface kept at the temperature 100°, and its base and convex surface at the temperature 15°, until the stationary temper- atures are established. Find the temperature at points on the axis 25, 50, and 75 centimeters from the base, and also at a point 25 .centimeters from the base and 50 centimeters from the axis.

* \ O y O/* O Or>

Ans. 29 .6; 47 .6; 71 .2; 25 .8

DEVELOPMENT IN TERMS OF BESSEL S FUNCTIONS. ,j ,

Prob. 29. An iron cylinder one meter long and 20 centimeters in diameter has its convex surface covered with a so-called non-con- •ducting cement one centimeter thick. One end and the convex surface of the cylinder thus coated are kept at the temperature zero, • the other end at the temperature of 100 degrees. Given that the con- ductivity of iron is 0.185 an<^ °f cement 0.000162 in C. G. S. units.

Find to the nearest tenth of a degree the temperature of the mid- dle point of the axis, and of the points of the axis 20 centimeters from each end after the temperatures have ceased to change.

Find also the temperature of a point on the surface midway be- tween the ends, and of points of the surface 20 centimeters from •each end. Find the temperatures of the three points of the axis, supposing the coating a perfect non-conductor, and again, suppos- ing the coating absent. Neglect the curvature of the coating. Ans. iS°.4; 4°°. 85 ; 72°.8; 15°. 3; 40°.? ', 72°-5 5 °°-° 5 °°-° ; i°-3-

Prob. 30. If the temperature at any point in an infinitely long cylinder of radius c is initially a function of the distance of the point from the axis, the temperature at any time must satisfy the

3« , /3*« . i 3«\ , x ...

equation — = a \^r\ H -- ~— 1 (see Art. i), since it is clearly in-

dependent of z and 0. Show that

•where, if the surface of the cylinder is kept at the temperature .zero, /*, , jua , //s , . . . are roots of Ja(nc) = o and Ak is the value :given in (3) with c written in place of a ; if the surface of. the cylin- der is adiabatic ju,, //,, yw3, . . . are roots of J^c] = o and Ak is ob- tained from (4); and if heat escapes at the surface into air at the tem- perature zero yw,, /*„, A/,, ...are roots of HcJ^yc} — Ayo(/v) = o, .and Ak is obtained from (5).

Prob. 31. If the cylinder described in problem 29 is very long and is initially at the temperature 100° throughout, and the con- vex surface is kept at the temperature o°, find the temperature of a point 5 centimeters from the axis 15 minutes after cooling has begun ; .first when the cylinder is coated, and second, when the coating is -absent. Ans. 97°. 2 ; o°.oi.

Prob. 32. A circular drumhead of radius a is initially slightly -distorted into a given form which is a surface of revolution about .the axis of the drum, and is then allowed to vibrate, and z is the •ordinate of any point of the membrane at any time. Assuming that

58 HARMONIC FUNCTIONS.

z must satisfy the equation „— = f{~~. + - r \> subject to the con- ditions z = o when r = a, = o when / = o. and - = f(r] when

ot

( = o, show that z =- A i_/0(A<1^) cos ^^ct -f- AJJ^Hj'} cos /./// -(- . . . where /<,, yw,, yU8, , . . are roots of Ja(na) — o and Ak has the value given in (3).

Prob. 33. Show that if a drumhead be initially distorted as in problem 32 it will not in general give a musical note ; that it may be initially distorted so as to give a musical note ; that in this case the vibration will be a steady vibration ; that the periods of the various musical notes that can be given are proportional to the roots of Jn(x) — o, and that the possible nodal lines for such vibrations are concentric circles whose radii are proportional to the roots of /.(•*) = o.

ART. 22. PROBLEMS IN BESSEL'S FUNCTIONS.

If in a problem on the stationary temperatures of a cylinder u = o when s = o, 21 = O when z = b, and u = f(z) when r = a, the problem is easily solved. If in (2), Art. 20, and in the cor- responding solution 2 = cosh (^J^r] we replace // by //z, we can readily obtain z — sin (yu^)/0(//r/') and 3 = cos (l*z)J <,(}*? i) as particular solutions of (i), Art. 20 ; and

x"1 x* x*

Jn(xi) = i +— + ^-r, + y , & 4- ... (i)

and is real.

!^° . • k*iz f(z\ — ^ AK sin —r J v ' frr *

^ /> J? -ft <?

where ' Ak —-j- I f(z) sin ,- dz i2\

o by Art. 9.

Hence u = Ah sin

l*

u T{*

b ) is the required solution.

*.i - , / ^M \

J\-

LAME'S FUNCTIONS. 5'.i

A table giving the values of Jo(xi) will be found at the end of this chapter.

Prob. 34. A cylinder two feet long and two feet in diameter has its bases kept at the temperature zero and its convex surface at 100 degrees Centigrade until the internal temperatures have ceased to change. Find the temperature of a point on the axis halt way between the bases, and of a point six inches from the axis, half way between the bases. Ans. 72. °i; 8o°.i.

ART. 23. BESSEL'S FUNCTIONS OF HIGHER ORDER.

If we are dealing with Laplace's Equation in Cylindrical Coordinates and the problem is not symmetrical about an axis, functions of the form

rn i-2 „«

T(x\— I — - - A

2"7 X« + I ) L 2\H + I ) ~ 24. 2 !(« + I )(W -f- 2 )

play very much the same part as that played by J0(.r) in the preceding articles. They are known as Bessel's Functions of the «th order. In problems concerning hollow cylinders much more complicated functions enter, known as Bessel's Functions of the second kind.

For a very brief discussion of these functions the reader is referred to Byerly's Fourier's Series and Spherical Harmonics ; for a much more complete treatment to Gray and Matthews' admirable treatise on Bessel's Functions.

ART. 24. LAME'S FUNCTIONS.

Complicated problems in Potential and in allied subjects are usually handled by the aid of various forms of curvilinear co- ordinates, and each form has its appropriate Harmonic Func- • tions, which are usually extremely complicated. For instance, Lame's Functions or Ellipsoidal Harmonics are used when solutions of Laplace's Equation in Ellipsoidal coordinates are required ; Toroidal Harmonics when solutions of Laplace's Equation in Toroidal coordinates are needed.

For a brief introduction to the theory of these functions see Byerly's Fourier's Series and Spherical Harmonics.

60

HARMONIC FUNCTIONS.

TABLE I. SURFACE ZONAL HARMONICS.

e

P, (cos 0)

P3 (cos 0)

P3 (cos 0)

P4 (cos 0)

PS, (cos 0)

P6 (COS 0)

I P^ (cos 0)

0°

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1

.9998

.9995

.9991

.9985

.9977

.9967

.9955

2

.9994

.9982

.9963

.9939

.9909

.9872

.982!>

3

.9986

.9959

.9918

.9863

.97i)5

.9713

.9617

4

9976

.9927

.9854

.9758

.9638

.9495

.9329*

5

.9962

.9886

.9773

.9623

.9437

.9216

.8961

6

.9945

.9836

.9674

.9459

.9194

.8881

.8522

7

.9925

.9777

.9557

.9267

.8911

.8476

.7986

8

.9903

.9709

.9423

.9048

.8589

.8053

.7448.

9

.9877

.9633

.9273

.8803

.8232

.7571

.6831

10

.9848

.9548

.9106

.8532

.7840

.7045

.6164

11

.9816

.9454

.8923

.H238

.7417

.6483

.5461

12

.9781

.9352

.8724

.7920

.6966

.5892

.4732"

13

.9744

.9241

.8511

.7582

.6489

.5273

.3940

14

.9703

.9122

.8283

.7224

.5990

.4635

.3219'

15

.9659

.8995

.8042

.6847

.5471

.3982

.2454

16

.9613

.8860

.7787

.6454

.4937

.3322

.1699-

17

.9563

.8718

.7519

.6046

.4391

.2660

.0961

18

.9511

.8568

.7240

.5624

.3836

.2002

.0289

19

.9455

.8410

.6950

.5192

.3276

.1347

-.0443-

20

.9397

.8245

.6649

.4750

.2715

.0719

-.1072

21

.9336

.8074

.6338

.4300

.2156

.0107

-.1662

22

.9272

.7895

.6019

.3845

.1602

-.0481

-.2201

23

.9205

.7710

.5692

.3386

.1057

-.1038

-.2681

24

.9135

.7518

.5357

.2926

.0525

-.1559

-.3095

25

.9063

.7321

.5016

.2465

.0009

-.2053

-.346*

26

.8988

.7117

.4670

.2007

-.0489

-.2478

-.3717

27

.8910

.6908

.4319

.1553

-.0964

-.2869

—.3921

28

.8829

.6694

.3964

.1105

-.1415

-.8211

-.4052"

29

.8746

.6474

.3607

.0665

-.1839

-.3503

-.4114

30

.8660

.6250

.3248

.0234

-.2233

-.3740

-.4101

31

.8572

.6021

.2887

-.0185

-.2595

-.3924

-.4022

32

.8480

.5788

.2527

-.0591

-.2923

— .4052

-.3876

33

.8387

.5551

.2167

-.0982

-.3216

-.4126

-.3670

34

.8290

.5310

.1809

-.1357

-.3473

-.4148

-.3409

35

.8192

.5065

.1454

-.1714

-.3691

-.4115

-.3096

36

.8090

.4818

.1102

-.2052

-.3871

-.4031

— .2738

37

.7986

.4567

.0755

-.2370

-.4011

-.3898

-.2343

38

.7880

.4314

.0413

-.2666

-.4112

-.3719

—.1918

39

.7771

.4059

.0077

-.2940

-.4174

-.3497

-.146&

40

.7660

.3802

-.0252

-.3190

-.4197

-.3234

-.10031:

41

.7547

.3544

-.0574

-.3416

-.4181

-.2938

-.0534

42

.7431

.3284

-.0887

-.3616

-.4128

-.2611

— .0065

43

.7314

.3023

-.1191

-.3791

-.4038

-.2255

.039*

44

.7193

.2762

-.1485

-.3940

-.3914

-.1878

.0846

45°

.7071

.2500

-.1768

-.4062

-.3757

-.1485

.1270

TABLES.

01-

TABLE I. SURFACE ZONAL HARMONICS.

e

Pj (COS 0)

P2 cos 6)

Ps (cos 9)

P4 (cos 9)

P5 (cos 9)

P6 (cos 9)

P7 (cos 9)

45=

.7071

.2500

-.1768

-.4062

-.3757

-.1485

.1270

46

.6947

.2238

-.2040

-.4158

-.3568

-.1079

.1666

47

.6820

.1977

-.2300

-.4252

-.3350

-.0645

.2054

48

.6691

.1716

-.2547

-.4270

-.3105

-.0251

.2349

49

.6561

.1456

-.2781

-.4286

-.2836

.0161

.2627

50

.6428

.1198

-.3002

-.4275

-.2545

.0563

.2854

51

.6293

.0941

-.3209 -.4239

-.2235

.0954

.3031

52

.6157

.0686

-.3401 -.4178

-.1910

.1326

.3153

53

.6018

.0438

-.3578

-.4093

-.1571

.1677

.3221

54

.5878

.0182

-.3740

-.3984

-.1223

.2002

.3234

55

.5736

-.0065

-.3886

-.3852

-.0868

.2297

.3191

56

.5592

-.0310

-.4016

-.3698

-.0510

.2559

.3095

57

.5446

-.0551

-.4131

-.3524

-.0150

.2787

.294i>

58

.5299

-.07-8

-.4229

-.3331

.0206

.2976

.2752

59

.5150

-.1021

-.4310

-.3119

.0557

.3125

.2511

60

.5000

-.1250

-.4375

-.2891

.0898

.3232

.2231

61

.4848

-.1474

-.4423

-.2647

.1229

.3298

.1916

62

.4695

-.1694

-.4455

-.2390

.1545

.3321

.1571

63

.4540

-.1908 -.4471

-.2121

.1844

.3302

.1203

64

.4384

-.2117 -.4470

-.1841

.2123

.3240

.0818

65

'.4226

-.2321 -.4452

-.1552

.2381

.3138

.0422

66

.4067

-.2518 -.4419

-.1256

.2615

.2996

.0021

67

.3907

-.2710 -.4370

— .0955

.2824

.2819

-.0375

68

.3746

-.2896 -.4305

-.0650

.3005

.2605

-.0763

69

.3584

-.3074

-.4225

-.0344

.3158

.2361

-.1135

70

.3420

-.3245

-.4130

-.0038

.3281

.2089

-.1485

71

.3256

-.3410

-.4021

.0267

.3373

.1786

-.1811

72

.3090

-.3568

-.3898

.0568

.3434

.1472

-.2099

73

.2924

-.3718

-.3761

.0864

.3463

.1144

-.2347

74

.2756

-.3860

-.3611

.1153

.3461

.OV85

-.2559

75

.2588

-.3995

-.3449

.1434

.3427

.0431

-.2730

76

.2419

-.4112

-.3275

.1705

.3362

.0076

-.2848

77

.2250

-.4241

-.3090

.1964

.3267

-.0284

-.2919

78

.2079

-.4352

-.2894

.2211

.3143

-.0644

-.294:1

79

.1908

-.4454

-.2688

.2443

.2990

- .0989

-.2913

80

.1736

-.4548

-.2474

.2659

.2810

-.1321

-.2805

81

.1564

-.4633

-.2251

.2859

.2606

-.1635

-.2709

82

.1392

-.4709

-.2020

.3040

.2378

-.1926

-.2536

83

.1219

-.4777

-.1783

.3203

.2129

-.2193

-.2321

84

.1045

-.4836

-.1539

.3345

.1861

-.2431

-.2067

85

.0872

-.4886

-.129"!

.3468

.1577

-.2638

—.1779

86

.0698

-.4927

-.1038

.3569

.1278

-.2811

-.1460

87

.0523

-.4959

-.0781

.3648

.0969

-.2947

-.1117

88

.0349

-.498-3

-.0522

.3704

.0651

-.3045

-.073.>

89

.0175

-.4995

-.0262

.3739

.0327

-.3105

-.0381

90°

.0000

-.5000

.0000

.3750

.0000

-.3125

.0000

62

HARMONIC FUNCTIONS.

TABLE II. BESSEL'S FUNCTIONS.

X

Jt(x)

Jl(X)

X

JttX)

J\(X)

X

Jo(x)

Ji(x)

0.0

1.0000

0.0000

5.0

-.1776

-.3276

10.0

-.2459

.0435

0.1

.9975

.0499

5.1

-.1443

-.3371

10.1

-.2490

.0184

0.2

.9900

.0995

5.2

-.1103

-.3432

10.2

-.2496

.0066

0.3

.9776

.1483

5.3

-.0758

-.3460

10.3

-.2477

-.0313

0.4

.9604

.1960

5.4

-.0412

-.3453

10.4

-.2434

-.0555

0.5

.9385

.2423

5.5

-.0068

-.3414

10.5

-.2366

-.0789

0.6

.9120

.2867

5.6

.0270

-.3343

10.6

-.2276

-.1012

0.7

.8812

.3290

5.7

.0599

-.3241

10.7

-.2164

-.1224

0.8

.8463

.3688

5.8

.0917

-.3110

10.8

-.2032

-.1422

0.9

.8075

.4060

5.9

.1220

-.2951

10.9

-.1881

-.1604

1.0

.7652

.4401

6.0

.1506

-.2767

11.0

-.1712

-.1768

1.1

.7196

.4709

8.1

.1773

-.2559

11.1

-.1528

-.1913

1.2

.6711

.4983

6.2

.2017

-.2329

11.2

-.1330

-.2039

1.3

.6201

.5220

6.3

.2238

-.2081

11.3

-.1121

-.2143

1.4

.5669

.5419

6.4

.2433

—.1816

11.4

-.0902

-.2225

1.5

.5118

.5579

6.5

.2601

-.1538

11.5

-.0677

-.2284

1.6

.4554

5699

6.6

.2740

-.1250

11.6

-.0440

-.2320

1.7

.3980

.5778

6.7

.2851

-.0953

11.7

-.021H

-.2883

1.8

.3400

.5815

6.8

.2931

-.0652

11.8

.0020

-.2323

1.9

.2818

.5812

69

.2981

-.0349

11.9

.0250

-.2290

JJ.O

.2239

.5767

7.0

.3001

-.0047

12.0

.0477

-.',284

2.1

.1666

.5683

7.1

.2991

.0252

12.1

.0697

-.2157

2.2

.1104

.5560

7.2

' .2951

.0543

12.2

.0908

-.2(60

2.3

.0555

.5399

7.3

.2882

.0826

123

.1108

-.1943

2.4

.0025

.5202

7.4

.2786

.1096

12.4

.1296

-.1807

2.5

-.0484

.4971

7.5

.2663

.1352

12.5

.1469

-.1655

2.6

-.0968

.4708

7.6

.2516

.1592

12.6

.1626

-.1487

2.7

-.1424

.4416

7.7

.2346

.1813

12.7

.1766

-.1307

2.8

-.1850

.4097

7.8

.2154

.2014

12.8

.1887

-.1114

2.9

-.2243

.3754

7.9

.1944

.2192

12.9

.1988

-.0912

3.0

-.2601

.3391

8.0

.1717

.2346

13.0

.2069

-.0703

3.1

-.2921

.3009

8.1

.1475

.2476

131

.'2129

-.0489

3.2

-.3202

.2613

82

.1222

.2580

13.2

.2167

-.0271

3.3

-.3443

.2207

8.3

.0960

.2657

13.3

.2183

-.0052

3.4

-.3643

.1792

8.4

.0692

.2708

13.4

.2177

.0166

35

-.3801

.1374

8.5

.0419

.2731

135

.2150

.0380

3.6

-.3918

.0955

8.6

.0146

.2728

13.6

.2101

.0590

3.7

-.3992

.0538

8.7

-.0125

.2697

13.7

.2032

.0791

3.8

-.4026

.0128

8.8

-.0392

.2641

13.8

.1943

.0984

3.9

-.4018

-.0272

8.9

-.0653

.2559

13.9

.1836

.1166

4.0

-.3972

-.0660

9.0

-.0903

.2453

14.0

.1711

.1334

4.1

-.3887

-.1033

9.1

-.1142

.2324

14.1

.1570

.1488

4.2

-.3766

-.1386

92

-.1367

.2174

14.2

.1414

.1626

4.3

-.3610

-.1719

9.3

-.1577

.2004

14.3

.1245

.1747

4.4

-.3423

-.2028

9.4

-.1768

.1816

14.4

.1065

.1850

4.5

-.3205

-.2311

9.5

-.1939

.1613

14.5

.0875

.1934

4.6

-.2961

-.2566

9.6

- .2090

.1395

14.6

.0679

.1999

4.7

-.2693

-.2791

9.7

-.2218

.1166

14.7

.0476

.2043

4.8

-.2404

-.2985

9.8

-.2323

.0928

14.8

.0271

.2066

4.9

-.2097

-.3147

9.9

-.2403

.0684

14.9

.0064

.20«9

5.0

-.1776

-.3276

10.0

-.2459

.0435

15.0

-.0142

.2051

TABLES.

TABLE III. — ROOTS OF BESSEL'S FUNCTIONS.

63

n

xn for Jt(xn) = 0

xn for J\(xn) = 0

71

xn for J0(xn) = 0

xn for «/iGrB) = 0

1

2.4048

38317

6

18.0711

19.6159

2

5.5201

7.0156

7

21.2116

22.7601

3

8.6537

10.1735

8

24.3525

25 9037

4

11.7915

13.3237

9

27.4935

29.0468

5

14.9309

16.4706

10

30.6346

32.1897

TABLE IV.— VALUES OF Ja(xi).

X

J<t(xi)

X

J0(xi)

X

J0(xi)

0.0

1.0000

2.0

2 2796

4.0

11.3019

0.1

1.0025

2.1

2.4463

4.1

12.3236

0.2

1.0100

2.2

2.6291

4.2

13.4425

0.3

1.0226

2.3

2.8296

4.3

14.6680

0 4

1.0404

2.4

3.0493

4.4

16.0104

0.5

1.0635

2.5

3.2898

4.5

17.4812

0.6

1.0920

2.6

3.5533

4.6

19.0926

0 7

1.1263

2.7

3.8417

4.7

20.8585

0.8

1.1665

2.8

4 1573

4.8

22.7937

0.9

1.2130

2.9

4.5027

4.9

24.9148

1.0

1.2661

3.0

4.8808

5.0

27.2399

1.1

1.3262

3.1

5.2945

5.1

29.7889

1.2

1.3937

3.2

5.7472

5.2

32.5836

1.3

1.4963

3.3

6.2426

5.3

35.6481

1 4

1.5534

3.4

6.7848

5.4

39.0088

1.5

1.6467

3.5

7.3782

5.5

42.6946

1.6

1.7500

3.6

8.0277

5.6

46.7376

1.7

1.8640

3.7

8.7386

5.7

51.1725

1.8

1.9896

3.8

9.5169

5.8

56.0381

1.9

2.1277

3.9

10.3690

5.9

61.3766

INDEX.

Bernoulli!, Daniel, 7. Bessel's Functions:

applications to physical problems,

53-55-

development in terms of, 55-56. first used, 7.

introductory problem, 21. of the order zero, 23. of higher order, 59. problems, 25, 56-59. properties, 51-53- series for unity, 24, 56. tables, 62-63.

Conduction of heat, 7.

differential equations for, 8, 9, 10, 13, 21, 54, 57-

problems, 12-15, 2I~25» 4°, 56» 57- Continuity, equation of, 9. Cosine Series, 30.

determination of the coefficients, 30.

problems in development, 31. Cylindrical harmonics, 52.

Differential equations, 10.

arbitrary constants and arbitrary functions, 10.

linear, 10.

linear and homogeneous, 10.

general solution, 10.

particular solution, 10. Dirichlet's conditions, 36. Drumhead, vibrations of, 57, 58.

Electrical potential problems, 15, 39,

4°, 43- Ellipsoidal harmonics, 59.

Fourier, 7.

Fourier's integral, 35. Fourier's series, 32-36.

applications to problems in physics, 38-40.

Dirichlet's conditions of developa- bility, 36.

extension of the range, 34-35.

graphical representation, 37.

problems in development, 33, 34.

Harmonic analysis, 7. Harmonics:

cylindrical, 12, 21, 25, 51-59, 62-

63-

ellipsoidal, 55.

spherical, 7, 12, 51.

tesseral, 51.

toroidal, 59.

zonal, 12, 15-21, 40, 50, 60-61. Heat v. Conduction of heat, 7 Historical introduction, 7.

Introduction, historical and descriptive, 7, 8, 9-

Lame, 7.

Lame's functions, 12, 59.

Laplace, 7.

66 INDEX.

Laplace's coefficients, 12, 51. Laplace's equation, 17, 41, 43, 51.

in cylindrical coordinates, 10, 21.

in spherical coordinates, 9, 12. Laplacian, 51. Legendre, 7.

Legendre's coefficients, 19. Legendre's equation, 17, 40, 41, 47.

Musical strings, 7.

differential equation for small vibra- tions, 7. problems, 39, 40.

Perry, John, 8.

Potential function in attraction: problems, 44, 51.

Sine series, 26.

determination of the coefficients, 26-

28.

examples, 29. for unity, 12, 29.

Spherical harmonics, 7, 12, 51. Stationary temperatures: problems, 21, 25, 56, 57, 59.

Tesseral harmonics, 51. Toroidal harmonics, 59. Tables, 60-63.

Vibrations :

of a circular elastic membrane, 57, 58-. of a heavy hanging string, 7. of a stretched elastic string, 7, 39, 40.

Zonal harmonics:

development in terms of, 46-49.

first used, 7.

introductory problem, 15.

problems, 21, 43, 44, 49, 50

properties, 40, 43.

short table, 19.

special formulas, 50.

surface and solid, 19

tables, 60-6 1.

various forms, 45-46.

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