LIBRARY AN ELEMENTAKY TREATISE FOURIER'S SERIES SPHERICAL, CYLII^DHIOAL, AND ELLIPSOIDAL HARMONICS, APPLICATIONS TO PROBLEMS IN MATHEMATICAL PHYSICS. WILLIAM ELWOOD BYERLY, Ph.D., PROFE.SSOK OF MATHEMATICS IN HARVARD UNIVERSITY. BOSTON, U.S.A.: GINN & COMPANY, PUBLISHERS, 1895. Copyright, 1893, Bv WU.IJAIVt JELWOOD BTERLY. 'all rights KESEKVED. A-C-^X*'^-^''^ itr.H- PREFACE About ten years ago I gave a course of lectures on Trigonometric Series, following closely the treatment of that subject in Riemann's "Partielle Differentialgleichungen," to accompany a short course on The Potential Function, given by Professor B. 0. Peirce. My course has been gradually modified and extended until it has become an introduction to Spherical Harmonics and Bessel's and Lame's Functions. Two years ago my lecture notes were lithographed by my class for their own use and were found so convenient that I have prepared them for publication, hoping that they may prove useful to others as well as to my own students. ]Meanwhile, Professor Peirce has published his lectures on " The Xewtonian Potential Function " (Boston, Ginn & Co.), and the two sets of lectures form a course (jNIath. 10) given regularly at Harvard, and intended as a partial introduction to modern Mathematical Physics. Students taking this course are supposed to be familiar with so much of the infinitesimal calculus as is contained in my " Differential Calculus " (Boston, Ginn & Co.) and my " Integral Calculus " (second edition, same publishers), to which I refer in the present book as <' Dif. Cal." and " Int. Cal." Here, as in the " Calculus," I speak of a " derivative " rather than a " differential coefficient," and use the notation D^ instead of — for " partial derivative with respect to cc." The course was at first, as I have said, an exposition of Riemann's "Partielle Differentialgleichungen." In extending it, I drew lai'gely from Ferrer's " Spherical Harmonics " and Heine's " Kugelfunctionen," and was somewhat indebted to Todhunter ("Functions of Laplace, Bessel, and Lame"), Lord Rayleigh ("Theory of Sound"), and Forsyth ("Differential Equations "). In preparing the notes for publication, I have been greatly aided by the criticisms and suggestions of my colleagues. Professor B. 0. Peirce and Dr. Maxime Bocher, and the latter has kindly contributed the brief historical sketch contained in Chapter IX. W. E. BYERLY. Cambridge, Mass., Sept. 1893. 981582 ANALYTICAL TABLE OF CONTENTS. CHAPTER I. PAGES Intkoimctiox 1-29 Airr. 1. List of some important homogeneous linear partial differential equations of Physics. — Auts. 2-4. Distinction between the (jeneral solution and a particular solution of a differential equation. Need of additional data to make the solution of a differential equation determinate. Definition of linear and of linear and homogeneous. — Arts. 5-(3. Particttlar solutions of homogeneotis linear differential equations may be combined into a more general solution. Need of development in terms of normal forms. —Art. 7. Problem: Permanent state of temperatures in a thin rectangular plate. Need of a development in sine series. Example. — Art. 8. Problem: Transverse vibrations of a stretched elastic string. A develop- ment in sine ser/es stiggested. — Art. 9. Problem: Potential function due to the attraction of a circular ring of small cross-section. Surface Zonal Harmonics (Legendre's Coefficients). Example. — Art. 10. Problem: Permanent state of temperatures in a solid sphere. Development in terms of Surface Zonal Har- monics suggested. — Arts. 11-12. Problem: Vibrations of a circular drumhead. Cylindrical Harmonics (Be.ssers Functions). Recapitidation. — Art. 13. Method of making the solution of a linear partial differential equation depend upon solving a set of ordinary differential eqtiations by assttming the dependent variable equal to a product of factors each of which involves but one of the independent variables. Arts. 14-1-5. Method of solving ordinary homogeneotts linear differential equa- tions by develoiDment in power series. Applications. — Art. Ki. Application to Legendre's Equation. Several forms of general solution obtained. Zonal Harmonics of the second kind. — Art. 17. Application to Bessel's Equation. General soltttion obtained for the case where ?;; is not an integer, and for the case where m is zero. Bessel's Function of the second kind and zeroth order. — Art. 18. Method of obtaining the general sohttion of an ordinary linear differential equation of the second order from a given particular soltttion. Application to the equations considered in Arts. 14-17. CHAPTER IL Development in Trigonometric Series .'30-54 Arts. 19-22. Determination of the coefficients of n terms of a sine series so that the sum of the terms shall be equal to a given fttnction of x for n given values of X. Numerical example. — Art. 23. Problem of development in sine series treated as a limiting case of the problem jtist solved. — Arts. 24-25. Shorter method of solving the problem of development in series involving sines of whole mitltiples of the variable. Working rttle deduced. Recapitulation. — Art. 26. A VI TABLE OF CONTENTS. PACKS few important sine developments obtained. Examples. — Abt«. 27-28. Develop- ment in cosine series. Examples. — Art. 29. Sine series an odd function of the variable, cosine series an even function, and both series periodic functions. — Art. 30. Development in .series involving both sines and cosines of whole multiples of the variable. Fourier's series. Examples. — Art. .31. Extension of the range witliin which the function and the series are equal. Examples. — Art. 32. Fourier's Integral obtained. CHAPTER III. Convergence or Fourier's Series -55-68 Arts. 3.3-36. The question of the convergence of the sine series for unity con- sidered at length. — Arts. 37-38. Statement of the conditions which are sutiicient to warrant the development of a function into a Fourier's series. Hi.storical note. Art. 39. Graphical representation of successive approximations to a sine series. Properties of a Fourier's series infen-ed from the constructions. — Arts. 40-42. Investigation of the conditions under whicli a Fourier's series can be differentiated term by term. — Art. 43. Conditions under which a function can be expressed as a Fourier's Integral. CHAPTER IV. Solution of Problems in Physics i$v the Aid of Fourier's Integrals and Fourier's Series 69-134 Arts. 44-48. Logarithmic Potential. Flow of electricity in an infinite plane, where the value of the Potential Function is given along an infinite straight line; along two mutually perpendicular straight lines; along two parallel straight lines. Examples. Use of Conjugate Functions. Sources and Sinks. Equipotential lines and lines of Flow. Examples. — Arts. 49-52. One-dimensiunal flow of heat. Flow of heat in an infinite solid; in a solid with one plane face at the temperature zero; in a solid with one plane face whose temperature is a function of the time (Riemann's solution); in a bar of small cro.ss section from whose surface heat escapes into air at temperature zero. Limiting state approached when the tem- perature of the origin is a periodic function of the time. Examples. — Arts. 53- 54. Temperatures due to instantaneous and to permanent heat sources and sinks, and to heat doublets. Examples. Application to the case where there is leakage. — Arts. 55-56. Transmission of a disturbance along an infinite stretched elastic string. Examples. — Arts. 57-58. Stationary temperatures in a long rectangular plate. Temperature of the base unity. Summation of a Trigono- metric series. Isothermal lines and lines of flow. Examples. — Art. -59. Potential Function given along the perimeter of a rectangle. Examples. — Arts. 60-63. One-dimensional flow of heat in a slab with parallel plane faces. Both faces at temperature zero. Both faces adiathermanous. Temperature of one face a function of the time. Examples. — Art. 64. INIotion of a stretched ela.stic string fastened at the ends. Steady vibration. Nodes. Examples. — Art. 65. Motion of a string in a resisting medium. — Art. 66. Flow of heat in a .sphere whose surface is kept at a constant temperature. — Arts. 67-68. Cooling of a sphere in air. Surface condition given by a differential equation. Development in a Trigo- nometric series of which Fourier's Sine Series is a special case. Examples. — TABLE OF CONTENTS. ■AGES Arts. 69-70. Flow of heat in an infinite solid witli one plane face which is exposed to air whose teiuperatuie is a function of the time. Solution for an instantaneous heat source when the temperature of the air is zero. Examples. — Arts. 71-73. Vibration of a rectangular drumhead. Development of a function of two variables in a double Fourier's Series. Examples. Nodal lines in a rectangular drumhead. Nodal lines in a square drumhead. Miscellaneous Prohlems 135-148 I. Logarithmic Potential. Polar Coordinates. — II. Potential Function in Space. III. Conduction of heat in a plane. — IV. Conduction of heat in Space. CHAPTER V. Zonal Harmonics 144-194 Art. 74. Recapitulation. Surface Zonal Harmonics (Legendrians). Zonal Har- monics of the second kind. — Arts. 75-70. Legendrians as coefficients in a Power Series. Special values. — Art. 77. Summary of the properties of a Legendrian. List of the first eight Legendrians. Relation connecting any three successive Legendrians. — Arts. 78-81. Problems in Potential. Potential Function due to the attraction of a material circular ring of small cross section. Potential Function due to a charge of electricity placed on a thin circular disc. Examples: Spheroidal conductors. Potential Function due to the attraction of a material homogeneous circular disc. Examples : Homogeneous hemisphere ; Heterogeneous sphere ; Homogeneous spheroids. Generalisation. — Art. 82. Legendrian as a sum of cosines. — Arts. 83-84. Legendrian as the ??ith derivative of the vith power of a;'^ — 1. — Art. 85. Equations derivable from Legendre's Equation. — Art. 86. Legendrian as a Partial Derivative. — Art. 87. Legendrian as a Definite Integral. Arts. 88-90. Development in Zonal Harmonic Series. Integral of the product of two Legendrians of different degrees. Integral of the square of a Legendrian. Formulas for the coefficients of the series. — Arts. 91-92. Integral of the product of two Legendrians obtained by the aid of Legendre's Equation; by the aid of Green's Theorem. Additional formulas for integration. Examples. — Arts. 93- 94. Problems in Potential where the value of the Potential Function is given on a spherical surface and has circular symmetry about a diameter. Examples. — Art. 95. Development of a power of x in Zonal Harmonic Series. — Art. 96. Useful formulas. — Art. 97. Development of sin n6 and cos nd in Zonal Harmonic Series. Examples. Graphical representation of the first seven Surface Zonal Harmonics. Construction of successive approximations to Zonal Harmonic Series. Arts. 98-99. Method of dealing with problems in Potential when the density is given. Examples. — Art. 100. Surface Zonal Harmonics of the second kind. Examples: Conal Harmonics. CHAPTER VL Spherical Harmonics 195-218 Arts. 101-102. Particular Solutions of Laplace's Equation obtained. Associated Functions. Tesseral Harmonics. Surface Spherical Harmonics. Solid Spherical Harmonics. Table of Associated Functions. Examples. — Arts. 103-108. De- velopment in Spherical Harmonic Series. The integral of the product of two Viii TABLK OF CONTENTS. PAGES Surface Spherical Harmonics of different degrees taken over the surface of the unit sphere is zero. Examples. The integral of the product of two Associated Functions of the same order. Formulas for the coefficients of the series. Illustra- tive example. Examples. — Arts. 109-110. Any homogeneous rational integral Algebraic function of x, ?/, and z which satisfies Laplace's Equation is a Solid Spherical Harmonic. Examples. — Art. 111. A transformation of axes to a new set having the same origin will change a Surface Spherical Harmonic into another of the same degree. — Arts. 112-114. Lcq^laclans. Integral of the product of a Surface Spherical Harmonic by a Laplacian of the same degree. Development in Spherical Harmonic Series by the aid of Laplacians. Table of Laplacians. Ex- ample. — Art. 115. Solution of problems in Potential by direct integration. Examples. — Arts. 110-118. Differentiation along an axis. Axes of a Spherical Harmonic. — Art. 119. Roots of a Zonal Harmonic. Roots of a Tesseral Har- monic. Nomenclature justified. CHAPTER VII. Cylindrical Harmonics (Bessel's Functions) 219-237 Art. 120. Recapitulation. Cylindrical Harmonics (Bessel's Functions) of the zeroth order; of the uth order; of the second kind. General solution of Bessel's Equation. — Art. 121. Bessel's Functions as definite integrals. Examples. — Art. 122. Properties of Bessel's Functions. Semi-convergent series for a Bessel's Function. Examples. — Art. 123. Problem: Stationary temperatures in a cylinder (o) when the temperature of the convex stirface is zero; {b) when the convex surface is adiathermanous; {(■) when the convex surface is exposed to air at the temperature zero. — Art. 121. Roots of Bessel's functions. — Art. 125. The integral of r times the product of two Cylindrical Harmonics of the zeroth order. Example. — Art. 12(). Development in Cylindrical Harmonic Series. Formulas for the coefficients. Examples. — Art. 127. Problem: Stationary temperatures in a cylindrical shell. Bessel's Functions of the second kind employed. Example: Vibration of a ring membrane. — Art. 128. Problem: Stationary temperatures in a cylinder when the temperature of the convex surface varies with the distance from the base. Bessel's Functions of a complex variable. Examples. — Art. 129. Problem: Stationary temperatures in a cylinder when the temperatures of the base are unsyuimetrical. Bessel's Functions of the nth order employed. INIiscellaneotis examples. Bessel's Functions of fractional order. CHAPTER VIII. Laplace's Equation in Curvilinear Coordinates. Ellipsoidal Harmonics 238-2(30 Arts. 130-131. Orthogonal Curvilinear Coordinates in general. Laplace's Equa- tion expressed in terms of orthogonal curvilinear coordinates by the aid of Green's theorem. — Arts. 132-135. Spheroidal Coordinates. Laplace's Equation in spheroidal coordinates, in normal spheroidal coordinates. Examples. Condition that a set of curvilinear coordinates should be normal. Thermometric Parameters. Particular solutions of Laplace's Equation in spheroidal coordinates. Spheroidal Harmonics. Examples. The Potential Function due to the attraction of an oblate spheroid. Solution for an external point. Examples. —Arts. 130-141. TABLE OF CONTENTS. ix PAGES Ellipsoidal Coordinates. Laplace's Equation in ellipsoidal coordinates. Normal ellipsoidal coordinates expressed as Elliptic Integrals. Particular solutions of Laplace's Equation. Lame's Equation. Ellipsoidal Harmonics (Lame's Func- tions). Tables of Ellipsoidal Harmonics of the degrees 1, 2, and 3. Lamp's Functions of the second kind. Examples. Development in Ellipsoidal Harmonic series. Value of the Potential Function at any point in space when its value is given at all points on the surface of an ellipsoid. — Akt. 142. Conical Coordinates. The product of two Ellipsoidal Harmonics a Spherical Harmonic. — Art. 143. Toroidal Coordinates. Laplace's Equation in toroidal coordinates. Particular solutions. Toroidal Harmonics. Potential Function for an anchor rina;. CHAPTER IX. Historical Summary 207-275 APPENDIX. Tables Table I. Surface Zonal Harmonics. Argument 6. Table II. Surface Zonal Harmonics. Argument x. Table III. Hyperbolic Functions Table IV. Roots of Bessel's Functions Table V. Roots of Bessel's Functions Table VI. Bessel's Functions CHAPTER I. INTRODUCTION. 1. In many important problems in mathematical physic;^ ,;we: fire ol>lig.exl, to deal with 'partial differential equations of a comparatively simple form. For example, in the Analytical Theory of Heat we have for the change of temperature of any solid dne to the flow of heat within the solid, the equation D,u = a\D'^u + D^u + D;n)* [i] where u represents the temperatiire at any point of the solid and t the time. In the simplest case, that of a slab of infinite extent with parallel plane faces, where the temperature can be regarded as a function of one coordinate, [i] reduces to D,n = a^I);u, [ii] a form of considerable importance in the consideration of the problem of the cooling of the earth's crust. In the problem of the permanent state of temperatures in a thin rectangular plate, the equation [i] becomes D^u-{- D-^u=^0. [Ill] In polar or spherical coordinates [i] is less simple, it is A n = '^. [i>.0--^I>,. ") + ^^ A (Sin A ^0 + ^^ I>^ ^'.]. [IV] In the case where the solid in question is a sphere and the temperature at any point depends merely on the distance of the point from the centre fivl reduces to n / \ o -r,o, \ ,- -, •- -' Dt(ru) = a-I)^(ru) . [vj In cylindrical coordinates [i] becomes I)tn = a^ [D; n + - I),, v + -, Z>| n + Z*/ iq . [ vi] In considering the flow of heat in a cylinder when the temperature at any point depends merely on the distance r of the point from the axis fvil becomes i I),u = a'\I)^-u-\--I),.u). [vii] * For the sake of brevity we shall often use the symbol V^ for the operation Da'- +D,/- +!),'-; and with this notation equation [i] would be written DfU = a'^'V'^ u. 2 INTRODUCTION. [Art. 1. In Acoustics in several problems Ave have the equation Bly = (rD;i/; [viii] for instance, in considering the transverse or the longitudinal vibrations of a stretched elastic string, or the transmission of plane sound waves through ,^,th,e air. ^ . ^ . . "I '' J'f in, e.dn§i^^{ng the transverse vibrations of a stretched string Ave take . , .accpJint ,a£ .the j-esistance of the air [a'iii] is replaced by ^'v:j-';>;'.^'::. .:.••'. .•* i)!>/ -^ 2ki),i/ = criJ^>/. [ix] In dealing Avith the vibrations of a stretched elastic membrane, Ave have the equation ^ , _ D^z = c^(I)^z + I)^z), [X] or in cylindrical coordinates Dfz = <^{p^z + J D^^ + 1 Dlz). [XI] In the theory of Potential Ave constantly meet Laplace's Equation D-^ V + Dj V + I)-; V = [xii] or V'V^O Avhich in spherical coordinates becomes 7. [>■!>; 0-r) + ^^ Z>, (sin 6 AT) + ^ I>,j;r] - 0, [xiii] and in cyUndrical coordinates B:: V+ll)J'+ -, I>lV-\- D! V = 0. [XIV] In curvilinear coordinates it is /'./-«. [i'„(o;^,.r)+/',(^^„'-) + i>.,()r^i>.J-)] = o. [xvi Avhere /i {x,y,z) = pi , /:, {xjj.z) = p. , /; (.r,//,^) = pg represent a set of surfaces Avhich cut one another at right angles, no matter Avhat values are given to pi, pn, and pg; and Avhere h,' = (i),p,r + {i),^p,f + {iKp.y h^ = {D,p,y + ipyp.:)^ + (Ap2)^ ll^ = (D^PsY + (^yPsY + (Aps)S and, of course, must be expressed in terms of pi , p., , and p. . If it happens that V-pi = 0, V-p.. = 0, and V^ps — 0, then Laplace's Equation [xv] assumes the very simple form Chap. L] PARTICULAR SOLUTIONS. 3 2. A differential equation is an equation containing derivatives or differen- tials with or without the primitive variables from which they are derived. The general solution of a differential equation is the equation expressing the most general relation between the primitive 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 solution particular values for the arbi- trary constants or particular functions for the arbitrary functions; but in practice it is often easy to obtain particular solutions directly from the differ- ential equation when it would be difficult or impossible to obtain the general solution. 3. If a problem requiring for its solution the solving of a differential equa- tion is determinate, there must always be given in addition to the differential equation enough outside conditions 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 differential equation can be readily solved the natural method of procedure 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 equa- tion 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 jjarticidar solutions and so to combine them as to form a result which shall satisfy the given conditions without ceasing to satisfy the differential equation. 4. 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 collected in Art. 1 are linear and homogeneous. 5. If ci value of the dependent variable has been found ivhich satisfies a given homogeneous, linear, differential equation, the product formed by muliipdy- ing this value by any constant will also be a value , of the dependent variable which icill satisfy the equation. 4 INTRODUCTION. [Art. t>. For if all the terms of the given equation are transposed to the first mem- ber, the substitution of the lii-st-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. If several values of the dependent variable have been found each of tvhich satisfies the given differential equation^ their sum ivill satisfy the equation ; for if the sum of the values in question is substituted in the equation each term of the sum will give rise to a set of terms which must be equal to zero, and therefore the sum of these sets must be zero. 6. It is generally possible to get by some simple device ^yarticular solutions of such differential equations as those- we have collected in Art. 1. The object of the branch of mathematics with which we are about to deal is to find methods of so combining these particular solutions as to satisfy any given conditions which are consistent with the nature of the problem in question. This often requires us to be able to develop any given function of the varia- bles 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 sug- gested 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 Coefficients, or Spherical Harmonics ; BesseVs Funefions, or Cylindrical Harmonics ; Lame's Functions, or Ellipsoidal Harmonics, &c. 7. As an illustration, let us take Fourier's problem of the permanent state of temperatures in a thin rectangular plate of breadth ir and of infinite length whose faces are impervious to heat. We shall suppose that the two long edges of the plate are kept at the constant temperature zero, that one of the short edges, which we shall call the base of the plate, is kept at the tempera- ture unity, and that the temperatures of points in the plate decrease indefi- nitely as we recede from the base; we shall attempt to find the temperature at any point of the plate. Let us take the base as the axis of X and one end of the base as the origin. Then to solve the problem we are to find the temperature u of any point from the equation r^o i 7.0 n r -1 * ^ i ■■• D- u -f 1)^1 It = [ill] Art. 1 subject to the conditions u = when .r = (1) u = " ,r = TT , (2) u = " y = 00 (3) n = l " // = 0. (4) Chap. I.] KECTANGULAR PLATE. 5 We shall begin by getting a particular solution of [m], and we shall use a device which always succeeds when the equation is liiiear and homor/eneous and has constant coefficients. Assume* u = e°-y'^^^, where a and ft are constants, substitute in [iii] and divide by e'^y + ^-'\ and we have a- + /3' = 0. If, then, this condition is satis- fied ^l = e'^y + ^•'' is a solution. Hence u = e'^^ - "•'' t is a solution of [m], no matter what value may be given to a. This form is objectionable, since it involves an imaginary. We can, how- ever, readily improve it. Take -u ^ e'^y e'^", a solution of [m], and zi = e'^^'e"'^', another solution of [ill]; add these values of u and divide the sum by 2 and Ave have e'^y cos ax. (v. Int. Cal. Art. 35, [1].) Therefore by Art. 5 u = e'^y cos ax (p) is a solution of [m]. Take n = e'^y e'^' and u = e°-y e~'^\ subtract the second value of u from the first and divide by 21 and we have e°-y sin ax. (v. Int. Cal. Art. 35, [2]). Therefore by Art. 5 u = e°-y sin ax (Q) is a solution of [m]. Let us now see if out of these particular solutions we can build up a solu- tion which will satisfy the conditions (1), (2), (3), and (4). Consider u = e°-y sin ax . (6) It is zero when x = for all values of a. It is zero when a- = tt if a is a whole number. It is zero when 3/ = oo if a is negative. If, then, we write u equal to a sum of terms of the form Ae~"''' sin mx, where f/i is a positive integer, we shall have a solution of [iii] which satisfies conditions (1), (2) and (3). Let this solution be II = Aie~" sin x + A^e---' sin 2x + Jge-"" sin 3x -f- J^e"'*^ sin 4:X -{- - • ■ (7) Ai, A2, As, Ai, &c., being undetermined constants. When 1/ = (7) reduces to II — Ai sin X + A„ sin 2x -{- A^ sin ox -\- A^ sin 4.r + • * ' • (8) If now it is possible to develop unity into a series of the form (8), our problem is solved; we have only to substitute the coefficients of that series for Ai, A2, A3, &c. in (7). * This assumption must be regarded as purely tentative. It must be tested by substi- tuting in the equation, and is justified if it le ads to a solution, t We shall regularly use the symbol i f or v^ — 1 . 6 INTRODUCTION. [Akt. 8. It will be proved later that 1 = -| sin .r + o sin 3x + ir sin 5x + f sin T.r + • • • ) for all values of x between and tt; hence our required solution is « = - e~'' sin .r + o" e^^" sin 3x -\- y e~'^'' sin 5x + =r e~'^ sin 7x -{- ■ ■ • (9) for this satisfies the differential equation and all the given conditions. If the given temperature of the base of the plate instead of being unity is a function of x, we can solve the problem as before if we can express the given function of cc . as a sum of terms of the form A sin m x, where m is a whole number. The problem of finding the value of the 2yote7itial function at any point of a long, thin, rectangular conducting sheet, of breadth tt, through which an electric current is flowing, when the two long edges are kept at potential zero, and one short edge at potential unity, is mathematically identical with the problem we have just solved. Example. Taking the temperature of the base of the plate described above as 100° centigrade, and that of the sides of the plate as 0°, compute the temperatures of the points (-)(ia); (^)(i,2); (c)(|,3), correct to the nearest degree. Ans. (a) 25°; (b) 15°; (c) 6°. 8. As another illustration, we shall take the problem of the transverse vibrations of a stretched string fastened at the ends, initially distorted into some given curve and then allowed to swing. Let the length of the string be /. Take the position of equilibrium of the string as the axis of X, and one of the ends as the origin, and suppose the string initially distorted into a curve whose equation y =f{x) is given. We have then to find an expression for ij which will be a solution of the equation Bt^l/ = a^D;y [viii] Art. 1, while satisfying the conditions l, = when .r = (1) ,/^0 " x^l (2) v/=/(,r) '' f = (3) X»,y = " t = 0, (4) the last condition meaning merely that the string starts from rest. Chap. I.] VIBRATING STRING. 7 As in the last problem let * y =^ e'^ + ^* and substitute in [viii]. Divide by e«^ + ^« and we have yS^= a^d^ as the condition that our assumed value of y shall satisfy the equation. _ ^^^at ,r\ U — '^ (o; is, then, a solution of (viii) whatever the value of a. It is more convenient to have a trigonometric than an exponential form to deal with, and we can readily obtain one.by using an imaginary value for a in (5). Replace a by ai and (5) becomes ?/ = Qi.^^o.t)ax^ ^ solution of [viii]. Replace a by — ai and (5) becomes _?y :^ e-C'^^nOa*^ another solution of [viii]. Add these values of y and divide by 2 and we have cos a{x ± a£). Subtract the second value of y from the first and divide by 2i and we have sin a{x ± at). y = cos a(.r + at) y = cos a(.r — at) y = sin a(x -{- at) y = sin a(x — at) are, then, solutions of [vm]. Writing y successively equal to half the sum of the first pair of values, half their difference, half the sum of the last pair of values, and half their difference, we get the very convenient particular solutions of [viii]. y = cos ax cos aat y = sin ax sin aat y z= sin ax cos aat y = cos ax sin aat . If we take the third form y = sin ax cos aat it will satisfy conditions (1) and (4), no matter what value may be given to a, and it will satisfy (2) if a := -r- where m is an integer. If then we take . . TTX irat , . . 2'Trx lirat , , . Zirx Sirat , y:=A-i^ sm — cos — — \- Ao sm —j— cos — 1- A3 sm — r— cos — 1 (6) where Aj , Ao , A3 • • • are undetermined constants, we shall have a solution of [viii] which satisfies (1), (2), and (4). When t = it reduces to y = Ai sm —-{- A2 sm -— + As sm -y- + • • • (7) If now it is possible to develop f(x) into a series of the form (7), we can solve our problem completely. We have only to take the coefficients of this series as values of Ai, A2, A3 . . . in (6), and we shall have a solution of [viii] which satisfies all our given conditions. * See note on page 5. 8 INTRODUCTION. [Art. 9. In each of the preceding problems the normal function, in terms of which a given function has to be expressed, is the sine of a simple multiple of the variable. It would be easy to modify the problem so that the normal form should be a cosine. -We shall now take a couple of problems which are much more complicated and where the normal function is an unfamiliar one. 9. Let it be required to find the potential function due to a circular wire ring of small cross section and of given radius c, supposing the matter of the ring to attract according to the law of nature. We can readily find, by direct integration, the value of the potential function at any point of the axis of the ring. We get for it where M is the mass of the ring, and x the distance of the point from the centre of the ring. Let us use spherical coordinates, taking the centre of the ring as origin and the axis of the ring as the polar axis. To obtain the value of the potential function at any point in space, we must satisfy the equation vD'; (r V) + J^^ Deism OBe V) + ^^ Z>,^ V= 0, [xiii] Art. 1, subject to the condition V=j;^^^, ^vhen ^ = 0. (1) From the symmetry of the ring, it is clear that the value of the potential function must be independent of ^, so that [xiii] Avill reduce to rl):: (r V) + ^^^ A (sin 6 A V) = . (2) We must now try to get particular solutions of (2), and as the coefficients are not constant, we are driven to a new device. Let * V:= r'" P, Avhere P is a function of 6 only, and m is a positive integer, and substitute in (2), which becomes . (m + 1);-'"P + ^— „ De (sin $DgP)=0. * See note on page 5 . Chap. I.] POTENTIAL DUE TO WIRE KING. 9 Divide by r'" and use the notation of ordinary derivatives since P depends upon 6 only, and we have the equation -(- + 1) P + ^^ -^-^^ = , (3) from which to obtain P. Equation (3) can be simplified by changing the independent variable. Let a- = cos and (3) becomes I [ f ] + "'("' + i'^="- w Assume *. now that F can be expressed as a sum or as a series of terms involving whole powers of x multiplied by constant coefficients. Let P = 2 ((„ a;" and substitute this value of P in (4). We get 2 [h (n — 1) ((„ X" -'^—11 {)i + 1) a„ X" + m, {in + 1) a„ cc"] = , (5) where the symbol 2 indicates that we are to form all the terms we can by taking successive whole numbers for n. As (5) must be true no matter what the value of .r, the coefficient of any given power of x, as for instance x'', must vanish. Hence (A' + 2) (/.■ + 1) a, ^ , - /.■ (7.- + 1 ) a, + >n (m + 1) a, = (6) and a^+o = (k-\-l) (k +2) "'^- * (^^ If now any set of coefficients satisfying the relation (7) be taken, P = S '(k^^ will be a solution of (4). If l-^m, r,,^, = 0, ..,,,=: 0, &c. Since it will answer our purpose if we pick out the simplest set of coefficients that will obey the condition (7), we can take a set including a„^. Let us rewrite (7) in the form (/.■ + 2)(/.'+l) ^_, "' ~ (m - k) (m + A- + 1) ''*■ + 2 • (»; We get from (8), beginning with k = m — 2, ID (ill 1) ""'-- ^ ~ 2. (2/// — 1) ■'"' _ III (ill — 1) (ill — 2) (m — 3) «™_4 — 2A.(2iii —i)(2iii — 3) "'« _ m (m — 1) (H^ — 2) (in — 3) (m — 4) (m — 5) «^ _ e — 2. 4. 6. (21)1 - 1) {2vi — 3) (2m — 5) ^*'» ' *^^' * See note on page 5. 10 INTRODUCTION. [Art. 9. If m is even we see that the set will end with ^o , if ^ is odd, with «! . r VI (m — 1) „ , m (m. — 1) (m — 2) (m — 3) ~| r — a^ ^x 2.(2;», - 1) •'' ^ 2.4.(2w - 1) (2m - ^) '' J where a„^ is entirely arbitrary, is, then, a solution of (4). It is found con- venient to take «„j equal to {2m — 1) (2vi - 3) • • • 1 m\ and it can be shown that with this value of <•/,„ P ^ 1 when x ^=\. P is a function of x and contains no higher powers of x than a-'". It is usual to write it as P„, {x). We proceed to compute a few values of P„, (x) from the formula (2m — 1) (2m^ — 3) • • • 1 r m Cm. — 1) ^^ ' ml L 2. (27)1 — 1) m(m-l)(m-2)(m-3) _ n "^ 2. 4. (2m - 1) (2m — 3) ^ J • C^^ We have: Po(:r) = 1 or Po(cos 6) = 1 p^(x) ^x " Pi (cos 0) = GOSO p^(x) = i (3x2 _ 1) a p^(cos e) = i(3 cos2 ^ - 1) P^(x) = i (5a;3 — 3.x) " P3(cos 6) = i (5 cos^^ — 3 cos 6) p}J) = X (35^4 - 30^2 + 3) or p^(cos 0) = i(35 cos"^ — 30 cos'^^ + 3) P^(x) = i (63^5 — 70^3 + 15.X-) or P,(cose) = i(63 cos^^ — 70 cos^^ + 15 cos^) . We have obtained F = P,^(x) as a particular solution of (4) and P = P^ (cos 6) as a particular solution of (3). P,„ (.r) or P,„ (cos 6) is a new function, known as a Lef/endre's Coefficient, or as a Surface Zonal Har- monic, and -occurs as a normal form in many important problems. V = r'^'P^^ (cos $) is a particular solution of (2) and r'»P,„ (cos 0) is some- times called a Solid Zonal Harmonic. We can now proceed to the solution of our original problem. r= AroPo(cos ^) + .liV-Pi(cos ^) + J2»-'A(cos 6) +A^r^P^(cos 6) -\ (11) where Aq, A^, A^, &c., are entirely arbitrary, is a solution of (2) (v. Art. 5). When ^ = (11) reduces to r=A^, + A,r+A,r'-\-A,r'+---, since, as we have said, P„, (x) = 1 when x = 1, or P„j (cos $) = 1 when ^ =: 0. By our condition (1) J^^ when ^ = 0. I -t-' ) (10) Chap. I.] ZONAL HARMONICS. 11 By the Binomial Theorem provided >• < <•. Hence is our required solution if y < r ; for it is a solution of equation (2) and satis- fies condition (1). Example. Taking the mass of the ring as one pound and the radius of the ring as one foot, compute to two decimal places the value of the potential function due to the ring at the points (,,) (,. = .2,^ = 0); (d) (r = .Q,0 = O); (/) ^, = .6, ^ = 0; (h) (. = .2,^ = ^); (.) (. = .r,^ = ~); (r/) (r = .6,0^1); (r) (r = 2, ^ = "); ^^»'- ('0 "^S; ('') -99; (r) 1.01; (rf) .86; V -/ (,) .90; (/) 1.00; (,y) 1.10. The unit used is the potential due to a pound of mass concentrated at a point and attracting a second pound of mass concentrated at a point, the two points being a foot apart. 10. A slightly different problem calling for, development in terms of Zonal Harmonics is the following: Required the permanent temperatures within a solid sphere of radius 1, one half of the surface being kept at the constant temperature zero, and the other half at the constant temperature unity. Let us take the diameter perpendicular to the plane separating the unequally heated surfaces as our axis and let us use spherical coordinates. As in the last problem, we must solve the equation ''^'^''"^ + ^mO ^' ('^'' ^ ^' "^ + ^^hTO ^* " ^ ^- t""'"^ ^''^- ^ which as before reduces to rD;\ru) + ^^ A(sin 6 A ^0 = (1) from the consideration that the temperatures must be independent of <^. Our equation of condition is It = 1 from ^ = to = ^ and u = from ^ = J to ^ = tt , (2) when /■ ^ 1. 12 INTRODUCTION. [Art. 10. As we have seen u = r"'P,„(cos 0) is a particular solution of (1), m being any positive whole number, and i( = Ao roPo (cos 0)+A, rPi (cos $) + A. r'Fo^ (cos 0) + A^ y^P^ (cos 6) -\ (3) where Aq, A-^, A, A^ ■ ■ • are undetermined constants, is a solution of (1). When r ^1 (3) reduces to u =^ AoPo (cos 6) + ^1 Pi (cos 0) + APo (cos 0) + ^.Pg (cos 6) H (4) If then we can develop our function of 6 which enters into equation (2) in a series of the form (4), we have only to take the coefficients of that series as the values of Aq, A^, A^, &c., in (3) and we shall have our required solution. 11. As a last example we shall take the problem of the vibration of a stretched circular membrane fastened at the circumference, that is, of an ordinary drum- head. We shall suppose the membrane initially distorted into any given form which has circular symmetry * about an axis through the centre perpendicular to the plane of the boundary, and then allowed to vibrate. Here we have to solve PA. = .^ (P,r'^ + I D,.z + \, D^^ [XI] Art, 1 subject to the conditions z =/(!■) when ^ = (1; D,z = '' / = ' (2) ^ = - . = ... (3) From the symmetry of the supposed initial distortion z must be independ- ent of ^, therefore [xi] reduces to P,2,- = r2 (n^rz + - D,.z\ (4) and this is the equation for which we wish to find a particular solution. We shall employ a device not unlike that used in Art. 9. Assume f z =^ R.T where P is a function of r alone and T is a function of t alone. Substitute this value of z in (4) and we get bd;'t = c^T (D,m + -d^r\ 1 (PT _ 1 uPB 1 dR '^T'df' ~E \d7 + r 7/7 ; The second member of (5) does not involve t, therefore its equal the first member must be independent of f. The first member of (5) does not involve * A function of the coordinates of a point has circular symmetry about an axis when its value is not affected by rotating the point through any angle about the axis. A surface has circular symmetry about an axis when it is a surface of revolution about the axis. t See note on page 5. Chap. I.] VIBRATING DRUMHEAD. 13 r, and consequently since it contains neither t nor r, it must be constant. Let it equal — /ot^ where fx of course is an undetermined constant. Tlien (5) breaks up into the two differential equations -, + ^^<'T=i) (6) d'^R. 1 (IE ^^+-^ + /.-^A> = (). (7) (6) can be solved by familiar methods, and we get T = cos ixct and T = sin /xd as simple particular solutions (v. Int. Cal. p. 319, § 21). To solve (7) is not so easy. We shall first simplify it by a change of inde- pendent variable. Let r = - ■ (7) becomes cPB 1 dR (kr^ X dx ' * ^ Assume, as in Art. 9, that R can be expressed in terms of whole powers of X. Let R = ^a,^x" and substitute in (8). We get 2 [«('« — l)a„x^'~^ -\- 7ia„x"~'^ -j- aj,x"^ = , an equation which must be true no matter what the value of x. The coeffi- cient of any given power of x, as .t*'-^, must, then, vanish, and k(k — l)a^. -f ka^. + r?^._, = or A--(^^. -f (f,_, = whence we obtain %._2 = — k'^a^. (9) as the only relation that need be satisfied by the coefficients in order that i? = 2 a^.x'^ shall be a solution of (8). If A- = 0, a^._,_=0, a,._,= 0. &c. We can then begin with ^ ^ as our lowest subscript. From (9) a, = -'^- Then a^ = — -^^ 9'-' .4^.6^ &c. Hence R = a^ fl - |o + gfj. - 22.42,(3-2 + ' ' 1 where «o may be taken at pleasure, is a solution of (8), provided the series is convergent. 14 IXTEODUCTION. [Art. 11. Take ao=^l, and then li = Jq(x) where Jq (a') = 1 — 22 + 2^ ~ 2-.4-.6- "^ 2-. 4-. 6^. 8- ~ ' ' * (10) is a solution of (8). t/o (^) is easily shown to be convergent for all values real or imaginary of x. since the series made up of the moduli of the terms of Jo (a;) (v. Int. Cal. Art. 30) 1 ~r 2^ "T 2^^ I 2- -4"- 6^ i ' ' ' ' where r is the modulus of x, is convergent for all values of r. For the ratio of the n + 1 st term of this series to the 7i th term is and approaches 4«- zero as its limit as n is indefinitely increased, no matter what the value of ;■. Therefore Jq (x) is absolutely convergent. Jq (x) is a new and important form. It is called a BesseVs Function of the zero th order, or a Cylindrical Harmon ic. Equation (8) was obtained from (7) by the substitution of x = fir, therefore J? r/ ^ 1 (/^'■■)' (M^ (f'^-y , B = J,(ixr) = 1 - -2^ + 2U"^ - -22;42;go -f • • • is a solution of (7), no matter what the value of /*, and z '= Jq {fir) cos fict or z = Jo(f^r) sin fict is a solution of (4). ~ ^ Jo (/^'") cos fict satisfies condition (2) whatever the value of /x. In order that it should also satisfy condition (3) fi must be so taken that Jo(A^^0 = O; (11) that is, fi must be a root of (11) regarded as an equation in fx. It can be shown that Jq (x) = has an infinite number of real jjositive roots, any one of which can be obtained to any required degree of approxima- tion without serious difficulty. Let Xy, X2, a-g, • • • be these roots. Then if ,-(■1 .r, a-3 z = Ai Jo (/^i r) cos /^i rf + A.yJo (fjL2 r) cos fi. ct + A^ Jo (^3 »■) cos /Ug c^ + • • ' » (12) where A^, A^, A^, &c., are any constants, is a solution of (4) which satisfies conditions (2) and (3). When t = (12) reduces to z = A, Jo (fi, r) + A, Jo (fi, r) + A, J, (f^s r) + • • • . (13) If then /(?•) can be expressed as a series of the form just given, the solution of our problem can be obtained by substituting the coefficients of that series for .li, A^, As. &c., in (12). Chap. I.] DISCUSSION OF METHODS. 15 Example. The temperature of a long cylinder is at first unity throughout. The convex surface is then kept at the constant temperature zero. Show that the tem- perature of any point in the cylinder at the expiration of the time t is where /Xi, fx^, &c., are the roots of Jii{fJic) = 0, and where 1 =^A,J,{fx,r) + A,J,{iJ.,r) + A,J,{fjL,r) + • • •, c being the radius of the cylinder. 12. Each of the five problems which we have taken up forces upon us the consideration of the development of a given function in terms of some normal form, and in two of them the normal form suggested is an unfamiliar function. It is clear, then, that a complete treatment of our subject will require the inves- tigation of the properties and relations of certain uqw and important functions, as well as the consideration of methods of developing in terms of them. 13. In each of the problems just taken up we have to deal with a homo- geneous linear partial differential equation involving two independent vari- ables, and we are content if we can obtain particular solutions. In each case the assumption made in the last problem, that there exists a solution of the equation in which the dependent variable is the product of two factors each of which involves but one of the independent variables, will reduce the question to solving two ordinary differential equations which can be treated separately. If these equations are familiar ones their solutions can be written down at once; if unfamiliar, the device iised in problems 3 and 5 is often serviceable, namely, that of assuming that the dependent variable can be expressed as a sum or series of terms involving whole powers of the independent variable, and then determining the coefficients. Let us consider again the equations used in the first, second and third problems. {a) D^u -f D^u = (1) Assume u = X. Y where X involves x but not ij, and Y involves // but not x. Substitute m (1) , FD^^ x -|- A'X»/ 1' = 0, or, since we are now dealing with functions of a single variable, 1 d'X . 1 d-'Y (2) Xd^^-^ Ydf'-''' 1 d'Y 1 d'X Y df - X dx^ ■ 16 INTRODUCTION. [Akt. 13. Since the first member of (2) does not contain a?, and the second member does not contain y, and the two members must be identically equal, neither of them can contain either x or y, and each must be equal to a constant, say d\ Then ^^^Y=0 (3) and ^T + a-U'=0; (4) and if (3) and (4) can be solved, Ave can solve (1). They have for their com- plete solutions p ^ ^^^y -\- Be-'^y and A' = C sin a:r + I) cos ax . (v. Int. Cal. p. 319, § 21.) Hence F= e"^' and F= e~"^ are particular solutions of (3), A^= sin ace and A'= cos ax are particular solutions of (4), and consequently u = f'°-v sin ax , ti ^^ e'^^ cos ax , u = e~°-y sin ax , and ?f ^= e~'^y cos ax are particular solutions of (4). These agree with the results of Art. 7. (l>) D-^,1 = a'Dly (1) Assume // = T.X where T is a function of t only and X a function of x only; substitute in (1) and divide by (i^TX. We get 1 d'^T _ 1 d}X a^ 'df' ~X~di^ ' (^) 1 d^X hence as in the last case -y-f^ is a constant; call it — a^, and (2) breaks up into d'^X , , ,, ^ ^ ^^ + a^A'=0 (3) d'^T _^ + «V^=0. (4) The complete solutions of (3) and (4) are X^ A sin ax + B cos ax and T = C sin ar/?- + D cos aai, (v. Int. Cal. p. 319, § 21). 2/ = sin aa'cos ao^ _?/ ^ sin a^c sin ao^?, ?/ =: cos a.r cos art/, // = cos aa; sin aa# are particular solutions of (1), and agree with the results of Art. 8. (c) rD:-{r V) + -^q A (sin 6 DeV) = . (1) Assume V= R.® where B involves r alone, and © involves 6 alone; sub- stitute in (1), divide by B.®, and transpose; we get / d®\ r cP(rB) _ _ ^_ 'T^'^W B dr'^ ~ 0sin^ dO ' (^) Chap. I.] SOLUTION AS A PRODUCT. 17 Since by the reasoning used in («) and (b) each member of (2) must be a con- stant, say a^, we have d-'(rE) ' d^^ = «'^ (3) and 1 H''''^d0) sin^ dO +«^® = 0- (4) (3) can be expanded into dm dB ,._ + 2r^-a-^A' = 0. (5) (5) can be solved (v. Int. Cab p. 321, § 23), and has for its complete solution B = Ar'" + B?-" , where m ^ — -J- + v' a'^ + i and ?i = — ^ — V «"^ + i • Hence n = — m — 1, and a^ may be written ))i(^m, -\- 1), m being wholly arbitrary; and B = Ar'" + Br-'"-^ . B^V", and B^-f^-^ are, then, particular solutions of dm dB r' -^ + 2r -^- - m(7,i +1)B = 0. (6) With the new value of a^ (4) becomes / d®\ SrT ^-irO + ^K- + 1)0 = . (7) which has been treated in Art. 9 for the case where m is a positive integer, and the particular solution = P,„ (cos 0) has been obtained. Hence V = r'"P„^ (cos 0) and r=-^-,P,„(Gose), m being a positive integer, are particular solutions of (1). The first of these was obtained in Art. 9, but the second is new and exceedingly important. 14. The method of obtaining a particular solution of an ordinary linear differential equation, which we have used in Articles 9 and 11, is of very extensive application, and often leads to the general solution of the equation in question. 18 INTKODUCTiO^'. [Akt. U As a very simple example, let us take the equation Art. 13 (a) (4), which we shall write £; + "- = ". (i> Assume that there is a solution Avhich can be expressed in terms of powers of x; that is, let .i = 2«„-v", Avhere the coefficients are to be determined. Substitute this lvalue for z in (1) and we get 2 [n(i/ — T)a„x"-- + a-a„x"'] = . Since this equation must be true from its form, without reference to the value of X, that is, since it must be an identical equation, the coefficient of each power of X must equal zero, and we have (» + !)(» + 2) whence "„ = o «„4.2 is the only relation that need hold between the coefficients in order that 2: = 2 a^x" should be a solution of (1). If « + 2 = or « + 1 = , "„ will be zero and «„_.., rt„_4, &c., will be zero. In the first case the series will begin with Uq, in the second with Oj . If we begin with a^ we have and a* 4-, «o , fte = a" - ^; «o .. , &c., . : + 4: 6! + ...) (2> — «o COS ax (3> is a particular solution of (1). If we begin witii «! we have a' a" «7 = =^. rtl ,!"l &C., . . . (a-x" wx" a"!- \ Chap. I.] SOLUTION IN POWER SERIES. 19 is a solution of (1) ; ai can be taken at pleasure. Let «i = a, (4) becomes aV a^x^ a^.r'' , = a.r - .- + ^-j- _ -^ + . . . or z = sin ax which, then, is a particular solution of (1). z = A sin ax + B cos ax (5) is, then, a solution of (1), and since it contains two arbitrary constants it is the general solution. 15. As another example we will take the equation cPz dz ^' ,7-2 + ^-^^- >"("' + 1)^ = , (1) which is in effect equation (6), Art. 13 (c), and let m be a positive integer. Assume s == 2 «„;>■" and substitute in (1). We get 2 [n(;>i + 1) — vi(m + 1)] a„x" = . This is an identical equation, therefore [«(« + !)- w(;^^ + l)]r.„ = 0. Hence a„ = for all values of u except those which make n(n + 1) — m(m. + 1) = , that is, for all values of n except n =: »>, and n- = — m — 1 . Then z=Ax'" -{- Bx-'"-'^ (2) is the general solution of (1) and z = X'" and z = --^-, are particular solutions. If m, is not a positive integer this method will not lead to a result, and we are driven back to that employed in Art. 13 (c). 16. Let us now take the equation Txl(^--"^dx]+^''(^"+^^' = ^ (1) which is in effect equation (4), Art. 9, and is known as Legendre's Equation. (1) may be written (^ - ^') £ ~ -^ S + "'('"' + ^^'^ = ^ • (2) 20 INTRODUCTION. [Art. 16. Assume .-: := % a^x" and substitute in (2). We get 2 {n^^a — l)a„ .r"-2 + [w(»; + 1) — «(" + l)]^'„a'"} = . Hence {n + 1) (m + 2)a„ ^ , + [_m{m + 1) - «(m + 1)] «„ = , (.+!)(» +2) _ .^3^ "^ "« - /»(»^ + l)- ?;(/; +1) "" + 2- If o^^ = 0, then (/„ _ 2 = 0, a,^ _ ^ = 0, &c. ; but ^^, = if n = — 2 or n = — l. For the first case we have the sequence of coefficients m{in — 2) {m + 1) (m + 3) fH = jj ^^0 m(m - 2) { //^ - 4) {111 + 1) (»^ + 3) {m + 5) Let us take (Iq, which is arbitrary, as 1. Then z = 2^m (-0 '^vhere r- m(m+l) ., w(//^ - 2) (/>/ + 1) ("^ + 3) ^ ~| /ix i>;« (^0 = |_1 2! '" + ~ II .Y - • • • J (4) is a solution of Legenclre's Equation if ^j„, (a-) is a finite sum or a convergent series. For the second case Ave have the sequence of coefficients (»^-l)(//^+2) (ni - 1) (m - 3) (m + 2) (m + 4) a, = 5i a, (m - 1) (//^ - 3) (m - 5) (>n + 2) (^^^ + 4) (m + 6) _ ^^ 7 - - I Let us take a^ , which is arbitrary, as 1. Then z = q,„(x) where ^ r- (m-l)(m + 2) ^ , (m-1) (n> -S) (m +2) (>n +i) ^ n q,n(^) = |_a- 3! ^- + 5l •'• J (^> is a solution of Legendre's Equation if i — 3) {m — 4) (??i — 5) _ 2.4.6.(2 m — 1) (2 7m — 3) (2 m — 5) ^"^ ^"^ as a particular solution of Legendre's Equation. If, however, we begin with w = — m — 3, we have {m + l){m + 2) ^-m-3— 2(2 VI + 3) *- '" - 1 {m + l)(m + 2)(m+3){m + 4:) ^-m-6— 2A.(2m + 3) {2m + 5) ^-™-i (»^ + 1) {m + 2) (^m + 3) (»^ + 4) {m + 5) (». + 6) a_^_7 — 2.4.6. (2 /M + 3) (2m + 5) (2»^ + 7) «'-m-i » «c. «_m_i niay be taken at pleasure, and is usually taken as i q k . . . /9. , _l i y and z = Q,n{x) where ^ ,,_ !^i! r 1 {m + l){m + 2) 1 fj^m C-^; — ^2m + 1) (2//^ — 1) • • • 1 L a-'" + ^ "^ 2. (2m + 3) a;'" + ^ • (». + l)(m + 2)(m+3)(m+4) 1 n "*" 2.4.(27?^ + 3) (2m + 5) a;- + ^ ^ J ^^'' is a second particular solution of Legendre's Equation, provided the series is convergent. Qm{x) is called a Surface Zonal Harmonic of the second kind. Chap. I.] ZONAL HARMONICS. 23 It is easily seen to be convergent if x- < — 1 or a- > 1, and divergent if -1< a- < 1. Hence if m is a positive integer, z=AP,„{.v) + BQ,„{x) (10) is the general solution of Legendre's Equation if x < — 1 or x > 1. We have seen that for — 1 < ,r < 1 -" r (m + 1) 2"'[r(,>i)] if m is an eveu integer, and ?i^' r ill, + 1) if rn is an odd integer. If now we define (),„ (.r) as follows when — 1 < ./• < 1 "±=1 r ( w + 1) Q^ (x) = (- 1) -^ —--4- ^, Pm (•'•) (13) if ?« IS an odd integer, and 2'"[r(f + 1)]' - . r (m -f- 1) ..-.[r(^)J if m is an even integer, then (10) will be the general solution of Legendre's Equation if m is a positive integer when — 1 < a; < 1, as well as when x < — 1 or X > 1. 17. Let us last consider the equation (r-z 1 dz / m-\ l^^xdx+(}--¥)^ = ' (1> which is known as Bessel's Equation, and which reduces to (8) Art. 11, that is, to d^z 1 dz _ dx~ X dx "^ ■when ?n = 0;* (1) can be simplified by a change of the dependent variable. * This equation was first studied by Fourier in considering tlie cooling of a cylinder. \Ve shall designate it as "Fourier's Equation." 24 INTKODUCTION. [Art. 17. Let z =^ cc"' V and we get cPv 2m + 1 dv s? + —r- &: + " = » (2) to determine v. Assume <; = 2 a^x", and substitute in (2). We get 2 [?i(2 m -\- n)a^x^^~'-^ + «„a'"] ^ 0; whence o„_2 = — n(2 m + n)a„. If we begin with n = 0, then a„_2 = 0, a„_^ = 0, &c., and we have the set of values ^2 = 2(2;// +2)- 2%m + l) '* ~ 2A(2m + 2)(2m + 4) ~ 2\2\{m + \){m + 2) 2.4.6(2?M + 2)(2;m + 4)(27>i + 6) 2^3 !(m + l)(»i + 2)(m + 3) whence z = a,x-^ [l - r^^;^!^^ + 2*.2!(m +%0" + 2) + 2«3!(w +1)(»/ + 2)(;m + 3) ] (3) is a solution of Bessel's Equation, a^ is usually taken as -^^ — ^ if m is a pos- itive integer, or as <^,„ -p , — aTT^ i^ ^'^ is unrestricted in value, and the second member of (3) is represented by t/^(a;) and is called a Bessel's Function of the With order, or a Cylindrical Harmo7iic of the ?«th order. If m = , Jm{x) becomes Jo{x) and is the value of z obtained in Art. 11 as the solution of equation (8) of that article. If in equation (1) we substitute a--"'/? in place of x"\v for z, we get in place of (2) the equation dh^ 1 —2m f/r dx'^ x dx' and in place of (3) r x V 2^2!(1— m)(2 — m) 2«.3!(1 — m)C2 — m){3- n>) + ' ' ' J (^^ Chap. T.] BESSEL's EQUATION. 25 If Oq is taken equal to the second member of (4) is the same U " '" r (1 — m) ^ function of — in and x that J,,, (.r) is of + m and x and may be written Therefore z = AJ,„ (,r) + BJ^,^ (.r) (5) is the general solution of (1) unless '/,„(»') and J_,,^(^x) should prove not to be independent. It is easily seen that Avhen m = , J_ „, (x) and J,„ (x) become identical and (5) reduces to r: = (A+I-r)J,(x) and contains but a single arbitrary constant and is not the general solution of rourier's Equation (8) Art. (11). It can be shown that (/_„j(.r) = (— l)'"cr,„(.T) whenever m is an integer, and consequently that the solution (5) is general only when m if real is frac- tional or incommensurable. The general solution for the important case where m == is, however, easily obtained. Let F(m,x) be the value which the second member of (3) assumes when rto = 1 ; then the value which the second member of (4) assumes when ((q = 1 will be F( — w,a-), and it has been shown that z = F(7n,x) and z = F( — vi,x) are solutions of Bessel's Equation; z = F(in,x) — F(^ — m,x) is, then, a solution, as is also F{m.,x) — F{— ni,x) z = -^ ■ > (O; 1)11 ^ but the limiting value which — ^ ■ ' ^ — ^^ — — -— approaches as m approaches zero is [A/i^("^v'')]/»=ii and consequently z = lD„,F{in,x)l,^, (7) is a solution of the equation d:'z 1 dz d^'r-j^ + ^^O, (8) and the general solution of (8) is z = A J,(x) + B [A„ F (m, .t)],,^„ . i^0.,..) = .-^-"{i-2^T) + 2\m + 1) ' 2\2l(m + l)(w + 2) ^ 2V3l(m.+l)(Ni-{-2)(m +3) + ' ' 26 INTRODUCTION. [Art. 17. D,„F{m., ,x) = ,r"' log .. [l - ..^ J+ ^^ + 2^2 !(».. + !)(.. + 2) ] + ■'^"' A„, [_1 - 2%i^-+T) + 2''.2!(;m + 1)(w +2) ^ J • The general term of tlie last parenthesis can be written (- 1)^- '2-''.kl{m + l){m. + 2) ■ ■ • (m + k) and its partial derivative with respect to m is 1 (- 1)' o:«-7-. B 2*. /.•! ■^"> (m + 1)(^»^ + 2) • • • (;m -+- A-) 1 log ^j— ^— j-^ = — flog (la + 1) + log (III +2) + • • • (ill -f- l)(i7i + 2) . . . (?»- -\- k) L & V I / I » V I / I + log (m + ;.■) J . Take the D,„ of both members and we have 1 ^"^ (w. + 1) {iii + 2) • • • (ill + A") ~ (III + l)(«i +2) • • • (III + k) \_iii + 1 ^ »^ + 2 ^ //; + /.■ J ^'" |_-^ ~ 2-^(»^ + 1) + 2*.2!(w + l)(w + 2} ~ 2-^.3 !(;;/ +l)(w+ 2)(»? +3) -1 _ r^ 1 j^ 1 r 1 in + ■ ■ ■ J ~ 2^ (»^ + 1/ " iK2! (/M + 1)(/// +2) L^^^Tl ^ ^^H^' J .T« 1 fill 1 ~| and Ave have x^ 1 ,t4 /I 1\ .r^ /I 1 1\ [X>^i^(m, a;)] „,^ = ^o(-^') log .r + 2^!)2 1 ~ 2V2 ij^ Vl ^ 2/ + 2W!p U + 2 + 3/ x' n 1 1 i\ ~ 2«(4 !7 U + 2 + 3 + 4/ "* ' and z=A Jo (x) + B A'o (a-) , (9) X, 5C* /I 1\ x^ /I 1 1\ where Ko (x) = J^ (x) log x + 02 " 2V2 !? U "^ 2/ + 28(3 !)- U "^ 2 + 3/ .T^ /I 1 1 1\ -2^J404l+2 + 3+4) + --- (^^^ is the general solution of Fourier's Equation (8). Kq(x) is known as a BesseVs Function of the Second Kind. Chap. I.] GENERAL SOLUTION. 27 18. It is worth while to conlirni the results of the last few articles by- getting the general solutions of the equations in question by a different and familiar method. The general solution of any ordinary linear differential equation of the second order can be obtained when a particular solution of the equation has been found [v. Int. Cal. p. 321, § 24 (r?,)]. The most general form of a homogeneous (uxlinary linear differential equa- tion of the second order is where P and Q are functions of x. Suppose that // = " (2) is a particular solution of (1). Substitute y := nz in (1) and we get (]:'z / dv \ dz dz Call -j~ = z'. Then (3) becomes a differential equation of the first order in which the variables can be sepa- rated. Multiply by dx and divide by vz' and (4) reduces to dz' dr -^ + 2 --- + Pdx = . Integrate and we have log z' + log v' + fPdx = C or z'r' = e^'- f^''^ = Be-f^'^^ , Iz ^ e-J''"- fPdx dx and ^^A+BJ--^- dx ; ,j = >, (a + Bf'^\lx^ (5) is the general solution of (1), the only arbitrary constants in the second mem- ber of (5) being those explicitly written, namely, A and B. (a) Apply this formula to (1) Art. 14, £ + «'^ = 0; (1) 28 INTEODUCTIOX. [Akt. 18. given : x = cos ax, as a particular solution. Substituting in (5) we have since F = z =^ cos ax \ J cos^ a.r/ / B \ I -4 H tan ax I = A cos ax + 7>i sin ax , (2) as the general solution of (1), and this agrees perfectly with (5) Art. 14. (b) Take equation (1) Art. 15. d-'z dz ^r7^ + 2^^-"K'- + l> = 0; '(1) given: z = .r™, as a particular solution. 2 r -rpdx 1 Here P = - , j Pdx = 2 log x = log x'- , and e ' =-.,. Hence by (5) that is z — Ax"^ + rr^^i (2) is the general solution of (1), and agrees with (2) Art. 15. (c) Take Legendre's Equation, (2) Art. 16. <^^ ~ ^'^ rfe^2 - 2a; ^ + 7/^(m. + 1)". = ; (1) given : z = P,,^ (.r) , as a particidar solution. Here P = j^z:'"^ , J Pdx = log (1 - x^) , and e-/^^'- = j^— a* Hence by (5) . = P,„(.) (.4 + p J^-^-_|^-^^.^) (2) is the general solution of (1) and must agree Avith (11) Art. 16, if m is an integer, and therefore where C is as yet undetermined, and no constant term is to be understood with the integral in the second member. (d) Take Bessel's Equation, (1) Art. 17. d'^z Idz / m\ ^^ + ^^^ + (l-^>^ = ^' (1) given : z ^ J^(x) , as a particular solution. Chap. I.] GENERAL SOLUTION. 29 1 /> /> 1 Here ^ = ^ > I ^^^^ = log *' » ^^^^ e"^ ^''^ = - . Hence by (5) .,=^„.(.)(..i+/./-^,) (2) is the general solution of Bessel's Equation. If m = (2) becomes ^ = M-)(-i+^f-^;^:) (3) and must agree with (9) Art. 17. Therefore if.(x) = C^.(.)Jjp^^, (4) where C is at present undetermined, and no constant term is to be taken with the integral. The first considerable subject suggested by the problems which we have taken up in this introductory chapter is that of development in Trigonometric Series (v. Arts. 7 and 8). CHAPTER II. DEVELOPMENT IN TRIGONOMETRIC SERIES. 19. We have seen in Chapter I. that it is sometimes important to be able to express a given function of a variable cc, in terms of the sines or of the cosines of multiples of x. The problem in its general form was first solved by Fourier in his "Analytic Theory of Heat" (1822), and its solution plays a very important part in most branches of modern Physics. Series involving only sines and cosines of whole multiples of x, that is series of the form />o + hi cos X -\- 1)2 cos 2x -\- ■ • • -\- (ii sin x + «2 sin 2x-\- • • • are generally known as Fourier's series. Let us endeavor to develop a given function of x in terms of sin x, sin 2x, sin 'dx, &c., in such a way that the function and the series shall be equal for all values of x between cc = and ic = tt. To fix our ideas let us suppose that Ave have a curve, given, and that we wish to form the equation, ij = (ii sin X -\- a^ sin 2x + a^ sin 3x -\- • ■ • , of a curve which shall coincide Avith so much of the given curve as lies between the points corresponding to a- = and a- = tt. It is clear that in the equation y = r^i sin X (1) a I may be determined so that the curve represented shall pass through any given point. For if we substitute in (1) the coordinates of the point in ques- tion we shall liave an equation of the first degree in which a^ is the only unknown quantity and which will therefore give us one and only one value for a-i . In like manner the curve y =. a I sin X + (to, sin 2a! may be made to pass through any two arbitrarily chosen points whose abscissas lie between and tt provided that the abscissas are not equal; and ij =. Ki sin X + Go sin 2x + (h sin 3x -\- • • ■ -\- a„ sin nx may be made to pass through any w arbitrarily chosen points whose abscissas lie between and ir provided as before that their abscissas are all different. If, then, the given function f(x) is of such a character that for each value of x between x = and cc = tt it has one and only one value, and if between a; =: and x = ir it is finite and continuous, or if discontinuous has only finite discontinuities (v. Int. Cal. Art. 83, p. 78), the coefficients in y = a I sin X + a^ sin 2x -\- a^ sin 3a; ■{-•■• + ''« sin nx (2) PRELIMINARY STUDY OF A FINITE SUM. 81 can be determined so tliat the curve represented by (2) will pass tlirougli an}- II arbitrarily chosen points of the curve 'I/ = f(^) (3) whose abscissas lie between and tt and are all different, and these coefficients will have but one set of values. For the sake of simplicity suppose that the n points are so chosen that their projections on the axis of A' are equidistant. TT Call ,— iT'T = '^•^" ; then the coordinates of the n points will be [A,x,/(zia;)], [2A.x',/(2Aa")], [3Ax,/(3Aa-)], • • • [MAcc,/(?^Aa:)]. Substitute them in (2) and we have /(Aa;) = (ii sin Ax -\- a^, sin 2A.r -\- a., sin 3A;r + ' ' ' + "« sin 7iAx'^ /(2Aa-) = «i sin 2Aa; + (1.2 sin 4A.i' + rtg sin 6Ax + h "„ sin 2uXr \ /(SAa-) = ('i sin 3\x + >i., sin 6A.r + x.^ sin 9A.r -\ \- >i,, sin onAx [ '' /(« A.x) = a-^ sin iiAx -\- a^ sin 2n\x + a^ sin 3»A./; -[-•••+ "« sin ii^Ax, j »j equations of the first degree to determine the n coefficients a-^ , "^o,, (tz, • • - a,, . Xot only can equations (4) be solved in theory, but they can be actually solved in any given case by a very simple and ingenious method due to Lagrange. Let us take as an example the simple problem to determine the coefficients cti, (u, «3, cti, and rtg, so that // = r/i sin X -\- a.2 sin 2x + «3 sin 3x -\- u^ sin 4, ,— ? -,— ? .1- ' '^^it*- -?r > 'f here being A.-r . b () b b 6 b ® We must now solve the equations TT . TT . 27r . StT . 47r OTT — := c/i sin TT + (lo sin -rr- + «3 sin -jr + ''/4 sin — ;- -j- «5 sin -rr- 27r 27r 47r Gtt Stt 10,t — - =: <^i sin -7,- + r?3 sin -7 1- ^^3 sin -^ -\- a^ sin -- + a^ sin —.- 37r Stt Qtt . Ott 127r Iott > (61 -- = (^j sin -TT + r^.o sin -77- + ^3 sm -r,- + 11^ sm — , [- ^5 sm -— .- 47r . 47r . Stt . 127r . IOtt . 207r -~ = a^ sin —p~ + *■" -g- «■" T &C. Hence the coefficient of (u becomes TT 27r Stt 47r Stt cos TT + cos -77- + cos -g- + cos -g — h cos Stt Gtt dir VItt 6 157r "6" (7) and this may be reduced by the aid of an important Trigonometric formula which we proceed to establish. 20. Lemma. cos ^ + cos 2 ^ + cos 3 ^ + • • • + cos n6 ^ ^ sin(2« + 1) 2 919 \ ^ (1) For let *S' = cos ^ + cos IB + cos 3^ + • • ° + cos nQ and multiply by 2 cos ^ . 2»S cos ^ = 2 cos^ ^ + 2 cos ^ cos 2^ + 2 cos ^ cos 3^ H H 2 cos cos nO = 1 + cos ^ + cos 2^ H h cos (m — 1) ^ + cos 20 + cos 3^ + cos 4^ H \- cos (ii + 1)0 = 2>S' + 1 + cos {11 + 1) ^ — cos Hence S = + cos nO — cos {n + 1) ^ 2(1— cos ^) ^ ^ sin (2« + 1)2 '^ "^ ~ 2 "^ 2 ~e sin 1^ Chap. II.] NUMERICAL EXAMPLE. 33 21. Applying (1) Art. 20 to (7) Art. 19 the coefficient of a^ reduces to ^37r l2 IItt . 337r —^-7v sin- 2 sm j2 2 sm j-^- IItt 12 -'^- TT . 337r 37r ■ j2 ' ^^^cl ^1^ = 37r — ^2 refore sin(7r-f,) ™(s^-f|) J J and a.2 vanishes. In like manner it may be shown that the coefficients of a^, a^, and a^ vanish. The coefficient of ^i is 2 sin^ 5+2 sin-^ -J + 2 sin'^ ^r + 2 sin^ ^ o b (> () =1+1+1+1 + 1 27r 47r Gtt Stt — cos cos — cos cos -~ 6 6 6 6 IOtt cos g • IIt^ • /.. TTX sm --- sm ( 27r — - J 2 Sin - 2 sm - 6 6 = 6. The first member of the final equation is 2tt . TT 27r . 2'ir OTT . 37r 47r . 47r , ^ Stt . 57r ^ -sm-+2--sm-- + 2--sm-- + 2-sm-, + 2--sm--. Hence k=o 2 -KT^ kir . k.TT TT - 2^ — sm — = - (2 + ^3) = 2 approximately. If we multiply the first equation of (6) Art. 19 by 2 sin -^ , the second by 2 sin ^ , the third by 2 sin -^ , the fourth by 2 sin ^ , the fifth IOtt by 2 sin -— , add and reduce as before we shall find t = 5 2 ^ /.-TT . 2A-7r TT ,., , ^ i=i 84 DEVELOPMENT IN TKIGONOMETllIC SEKIES. [Akt. 22. and in like ananner we get /I- =5 2 .r-V A'TT . 3A'7r IT ^ _ «. = iXT-T = f(2-^«) = «-l Therefore ^ = 2 sin X — 0.9 sin 2.r + 0.5 sin 3.r — 0.3 sin 4a? + 0.1 sin 5x (1) TT 27r 37r cuts the curve >/ = oc. at the live points whose abscissas are tt , -^ , -tt > 47r , Stt 22. The equations (4) Art. 19 can be solved by exactly the same device. To find any coefiicient «^ multiply the first equation by 2 sin m^x . the second by 2 sin 27nAx, the third by 2 sin 3m^x, &c. and add. The coefiicient of any other a as o^. in the resulting equation will be 2 sin kAx sin m\x + 2 sin 2kL^x sin 2/HA.r + 2 sin 3/l'Acc sin 3mAx -{-■•■ -\- 2 sin nk\x sin nm^x = cos(/ft — A:) Ax + cos 2 (//i — A-)A.T + cos3(;yi — A')A.ifH \-cosn(m — k)Ax — cos(v» + 7.") Ax— cos 2(m + k)\x — cos 3(?m + A-) Ax cos /i(Hi + '''■) Ax by (1) Art. 20. n -{- 1 — - and (/i + l)Ax = tt . ^Hence the coefficient of a^. may be written sin (»i - k)7r - -^-^ sm (m + /(> - ^ — -^ — 2 sm ^^ o'' ^ sni ^ ^ — 2)1 4-1 sin ^y — (/ft — A') Ax 2n 4- 1 sin — (m + /.■)Ax 2 sm ^ ^ 2« + 1 _ ^ . (>ft + A-)Ax 2 sm ^ 2 , _L 1 o^irl /» _1_ 1\ but this is equal to ^ — ;^ or — - + - according as m — k is odd and so is zero in either case. or even Chap. II. ] DETERMINATION OF COEFFICIENTS. 35 The coefficient of a„, will be 2 sin"^ )n.\x + 2 sin^ 2?/iAx + 2 sin^ 3m^x + • • • + 2 sin^ nm^x 1 + 1 + 1 +•••+ 1 — cos 2wAu? — cos 4mA.x — cos 6m/\x — ■ • • — cos 2)im/lx = ■„ + I - ^'"f'! + ^>'"^" , by (1) Art. 20. ' 2 2 sill mi\x -^ ^ ^ But (2n + l)mA« = 2m(??, + l)Aa; — mA.'K = 2m7r — mAx, ^ p ^ sin (2n-\- l)))iAx sin (2m 7r — 7ftAa;) 1 2 sin /mA.t 2 sin 7nAx 2 ' and the coefficient of a^ is n -\- 1 . The first member of our final equation will be 2 "^/(kiXx) sin km/^x . 2 '^" *™ ^ ^7+1 2y /('^^^) Sill '^'"^'^^ ; (1) = «i sin X + f/2 sin 2x -\- • • • -^ a„ sin ?ia; , (2) Hence and the curve where the coefficients are given by (1) will pass through the n points of the TT curve ?/ =f(x) whose abscissas are Ace, 2Aaj, SAx, ■ • • nAx. Ace being —p- . It should be noted that since the n equations (4) Art. 19 are all of the first degree there will exist only one set of values for the n quantities «i, a,, 03, • • • a„ that can satisfy these equations. Consequently the solution which we have obtained is the only solution possible. 23. The result just obtained obviously holds good no matter how great a value of n may be taken. If now we suppose n indefinitely increased the two curves (2) Art. 22 and y = f(x) will come nearer and nearer to coinciding throughout the whole of their portions between .r = and cc = tt , and consequently the limiting form that equation (2) Art. 22 approaches as n is indefinitely increased will represent a curve absolutely coinciding between the values of as in question with y=f(x). 36 DEVELOPMENT IN TRIGONOMETltlC SERIES. [Aht. •2i. Let us see what limiting value (t,,^ approaches as n is indefinitely increased. a^.^^ = — ^ y /(A-Aic) sin kmAx (1) Art. 22. 2A.T -r-^ ^ '- 2_, /(A'-^-^O sin Jcm/lx 1 = 1 = - [/(Aa-) sin 7rtA.-r.A.r + /(2Aa:;) sin 2»^Aa:'.A./'H {-/(iiAa') sin 7im\x.Xr _ 2 r f ( A.r) sin mXr. A.r + /'(2 A.r) sin 2 /». A,i'. A.r H -j ^|_ +/('"' — A.r)sin»;(7r — A.'').A./'J since A.r = — ; — - • n + 1 As n is increased indefinitely Aa. approaches zero as a limit. Hence the limiting value of «,„ as n increases indefinitely is 2 limit r/(Ax) sin mAx.Ax +/(2A./-) sin 2«iA.r. A.r -\ ~| ^ TT Aa' = oL +fi'^ — -^•'•) sin ini-ir — A./-).A,i J ^a-) sin mx.dx . [v. Int. Cal. Arts. SO, 81.] ^/'^^ Hence /(cc) = «i sin .r + a^ sin 2x + ^^3 sin 3.r + • • •, (2) Avhere any coefficient a,„ is given by the formula a„^ = - j /(^O sin mx.dx , (3) is a true development of f(x) for all values of x between x = and .*• = tt provided that the series (2) is converr/ent, for it is in that case only that we can assume that the limiting value of the second member of (2) Art. 22 can be ob- tained by adding the limiting values of the several terms. When X = and when x = it every term in the second member of (2) is zero, and the second member is zero and will not be equal to f(x) unless f(x) is itself zero when x = and x = tt ; but even when f(x) is not zero for X ^ and a; = tt the development given above holds good for any value of x between zero and tt no matter how near it may be taken to either of these values. 24. Instead of actually performing the elimination in e.juations (4) Art. 19 and getting a formula for «,„ in terms of n, and then letting n increase indefinitely, we might have saved labor by the following method. * We shall vise the sign == for approaches. Ax = is read Ax approaches zero. Chap. II.J ABRIDGED METHOD. 37 Return to equations (4) Art. 19 and multiply the first by Aic sin inb>.Xy the second by Aa; sin 2mA.r, and so on, that is multiply each equation by ^x times the coefficient of a^ in that equation, and then add the equations. We get as the coefficient of % sin k\x sin m^x. Aj:- + sin 2^-Ax- sin 'ImH^^x. A.r -j -|- sin nk^x sin nniL^x. Ax . Let us find its limiting value as n is indefinitely increased. It may be written, since (/i + 1) A»' = tt , limit rsinA;Aa:;sin7/iAa:'. A;j" + sin2A^A,Tsin2w.Aa;. Aa--] ~1 L.X' = L + sin ^ (tt — Aa;) sin m (.tt — \x) . Ax J = I sin kx sin vix. dx ; but I sin kx sin mx.dx = i | [cos (m — k^x — cos {^n + k>)x\dx (I = if w, and k are not equal. The coefficient of a,„ is Aa;(sin- m!^x + sin"^ 2?MAa- + sin'-^ ?>mAx + • • • + sin'^ v^mA,-;c) Its limiting value limit Ax _ ,. sin'^ y«,Aa:;.A;r + sin- 2mAa:;.A.r + ■ • • + sin2?«,(7r — A,r)A.:f sin'^ mx.dx = — • The first member is /"(Aa')sin 7/?A./'.A.r H-/(2Aa-) sin 2mAx.Ax + ■ ■ ■ +/(wAa;) sin mnAx.Ax and its limiting value is I f(oii) sin mx.dx . Hence the limiting form approached by the final equation as n is increased is ^ f{x) sii sin mx.dx = - a. //(.X-) Whence a,,, ^ - | f(x) sin mx.dx as before. This method is practically the same as multiplying the equation f(x) = cii sin X -\- ^2 sin 2x + a^ sin 3x + ' ' ' (1) by sin mx. dx and integrating both members from zero to tt . 38 DEVELOPMENT IN TRIGONOMETRIC SERIES. [Art. -2.5. It is exceedingly important to realize that the short method of determining any coefficient cf„^ of the series (1) which has just been described in the itali- cized paragraph, is essentially the same as that of obtaining «,„ by actual elimination from the equations (4) Art. 19, and then supposing 7i to increase indefinitely, thus making the curves (3) Art. 19 and (2) Art. 19 absolutely coincide between the values of x which are taken as the limits of the definite integration. 25. We see, then, that any function of x Avhich is single-valued, finite, and continuous between x = and x = tt, or if discontinuous has only finite discontinuities each of which is preceded and succeeded by continuous por- tions, can probably be developed into a series of the form f(x) = a^ sin x -\- 0.2 sin 2x + r/., sin 3.t + • • • (1) where «„, = - | f(x) sin mx.dx — - i f(a) sin tna.da ; (2) and the series and the function will be identical for all values of x between X = and cr = tt, not including the values .t = and x = tt unless the given function is equal to zero for those values. An elaborate investigation of the question of the convergence of the series (1), for which we have not space, entirely confirms the result formulated above * and shows in addition that at a point of finite discontinuity the series has a value equal to half the sum of the two values which the function approaches as we approach the point in question from opposite sides. The investigation which we have made in the preceding sections establishes the fact that the curve represented by ?/ — 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 // = a sine series. 26. Let us obtain a few sine developments. (a) Let fi^')^'^. (1) We have x = a^ sin x + f'2 sin 2.r + '''3 sin 3a:; + • • • (2) 'here TT j X sin mx.dx (3) * Provided the function has not an infinite number of maxima and minima in tlie neioh- borhood of a point, v. Arts. 37-38. Chap. IL] EXAMPLES OF SINE SERIES. I X sin m.x.dx = -- (sin mx — mx cos ?»,r), (-IV'TT and {h) Let '<-. /x sin mx. dx III sin X sin 2x , sin 3.r sin 4,7- + m 9 a^ ^ — I sin mx.dx ; (4) (1) (2) / /^^ sin mx.dx = — 1 cos VIX III sin ma;. (7a- = - (l — eos iinr) [l-(-l)-] if m is even = — if VI is odd. Hence 4 /sin X , sin 3.r , sin 5a' , sin 7x (3) It is to be noticed that (3) gives at once a sine development for any constant c. It is, 4c /sin X sin 3a' sin 5x \ (4) If we substitute a- = — in (4) (rr) or (3) (b) we get a familiar result, namely (5) 4 13 5 7, a formula usually derived by substituting x = 1 in the power series for tan- la', (v. Dif. Cal. Art. 135.) (4) («) does not hold good when x = tt, and (3) (fj) fails when x = and when X = TT, for in all these cases the series reduces to zero. (c) Let f(x) = X from x = to .'■ ■ and f(x) from a- to .r = TT • That is, let ^ = f(x) represent the broken line in the figure. As the mathematical expression for f(x) is different in the two halves of the ^ curve we must break up 40 DEVELOPMENT IN TRIGONOMETillC SERIES. [Art. 26. j /(.t) sin itix.dx into j f{x) sin mx.dx + | /(a-) sin mx.dx. „, cos'^ mx.d.v = — &„j , if 7n is not zero. 2 /• 2 /^ Hence Z>„, = — j f(x) cos mx.dx = — j /(a) cos ma.da , (2) n if m is not sero. Chap. II.] EXAMPLES OF COSINE SEIllES. 43 To get Ji^y multiply (1) by d.r and integrate from zero to tt. I ^^. cos kxjJ.r = 0. Hence h,= - Cf(x)dx = - f{a)da, (3) TT J TT which is just half the value that would be given Ijy formula (2) if zero were substituted for m. To save a separate formula (1) is usually written f(x) = i^o ~t~ ^^1 cos X -\- 1)2, cos 2x + ^3 cos 3,r + • • • (4) and tlien the formula 2 r 2 /i . J^ = — I f(x) cos mx.dx = — j f{a) cos via.da (2) will give (iq as well as the other coefficients. It is important to see clearly that what we have just done in deter- mining the coefficients of (1) is equivalent to taking n -\- 1 terms of (4), substituting in 1/ = 4-/^5 + I'l cos X + h. cos 2,r + • • • + ^„t cos nx (5) in turn the coordinates of the n + 1 points of the curve whose projections on the axis of A' are equidistant, determining h, bi, b^, • • • /*„ by elimination from the n + 1 resulting equations, and then taking the limit- ing values they approach as n is indefinitely increased, (v. Art. 24.) TT If Ax ^= — q— : the abscissas of the n + 1 points used are 0, Ax, 2 Ax-, 3A.r , • • • uAx, so that we should expect our cosine development to hold for x ^ as well as for values of x between zero and tt. 28. Let us take one or two examples : (a) Let /'(.t) = X . (1) TT J TT J 2 r 2 2 J„ = - 1 X COS mx.dx — --r (cos vitt — 1 ) = -^ [(— 1) '" — 11. "i-i DEVELOPMENT IN TIUGONOMETRIC SERIES. [Art. 28. Hence .r ■^ -1 / cos o.r , COS 5.r , cos 7.1 ■^ *i / I cos o.r cos t).r , cos i x , \ (2) holds good not only for values of x between zero and tt but for x = and X = TT as well, since for these values we have - which are true by Art. 26 ('.')(3). (b) Let f(x) = .7- sin .r . (1) 2 r 2 ^U = - I ./• sin X.l/X = - TT =: 2 , TT J TT and __-, 4/. , 1 , 1 , 1 *■ = - TT j ./■ sin a- cos x.f/x = - r,/' sin 2x.dx = — - , r.r sin .r cos mx.dx = - C [_x sin {in + l).r — x sin (/// — l)^](Za; 2 if ^y; is odd {ni - l){m + 1) 2 ^ — z — r^rr if III is even. (/;/ — !)(/» + 1) Hence . . cos X 2 cos 2x , 2 cos 3.r . 2 cos 4.'" , X sin ,/■ = 1 — — \- • ' • (T\ 2 1.3 ^ 2.4 3.5 ^ ^"'' Jf ^' = o we have l=\ + h-h + h--- (^) EXAMPLES. Obtain the following developments: (1) f{^) _ TT _ 2 F cos 2x cos 6a? cos lOx cos 14.r n ~ 4 TT L P "^ 3^ "^ ^5^~ + ~T^ ^ ■ ■ ■ J if f{x) = a; from cf = to a- = - and /(a;) = tt — .r from a- = ^ to ,r = tt . Chap. II.] EXAMPLES. 45 (2) /(.) = 1 + 2 piip _ ^:^ -+ '^ _ '^ + ^ .-| . ^ TT \_ L O O i _J if f(x) = 1 from .r = (» to .r = - and ,/'(./■) = () from x = ^ to ./■ =: tt. ,„. „ TT^ . rcos ,/• cos U.r , cos o,i' cos 4.r , n (3) a- = --4|_^^ ^ + -S' 4^+-J- (4) a'^ =. — --- |^(^— - pj cos .r - ^ cos 2x + ^^ - .Tij cos 3x - ^ cos 4,r +(?;;- ^^,) cos 5.r J . (5) /(.r) = I + I [^^^ - l) cos ,r - I cos 2,r - | ( y + l) cos 3.r + -^ / -7; 1 ) cos 5,z- — -^^ cos 6j- — • • , TT if f{x) = ,r from .r =: to .r = - and /(.r) = from :r ^ -;^ t( 2 ri ) cos 2./; (6) «' = ^ [2 (<" - 1 ) - iTT' (<•■ + 1) ™^ ■'■ + r^, ("- - 1 : ,^, , 2 sinh TT rl 1 , 1 o 1 (/) cosh .j^- = o — o cos a? + - cos 2.r — —- cos ox TT l_J J O 10 + -p. cos 4,T — ■ ■ • 2 rl 1 (8) sinh a; = — - (cosh tt — 1) — - (cosh tt + 1 ) cos x + r (cosh TT — 1) cos 2,r — T— (COsh IT -\- I) cos o.r + • ■ • . .„. 2/x sin /LtTT r 1 cos .r , cos 2x cos .S.r (9) cos ^lx = -^= ^ — ., ^ + - ., .-, - 7r \_2/x- jx- — 1^ fi- — 2" /u- — .V cos 4.r ~1 + /I^^^T^ - • • • J ' if /i is a fraction. 29. Although any function can be expressed both as a sine series and as a cosine series, and the function and either series will be equal for all values of X between zero and tt, there is a decided difference in the two series for other values of x. Both series are periodic functions of ,/■ liaving the period 27r. If then we let // equal the series in question and construct the portion of the correspond- 40 DEVELOPMENT IN TRIGONOMETKIC SERIES. [Art. 29. iug curve Avhich lies between the values x = — tt and x = it the whole curve will consist of repetitions of this portion. Since sin mx = — sin (— mx) the ordinate corresponding to any value of X between — tt and zero in the sine curve will be the negative of the ordinate corresponding to the same value of x with the positive sign. In other words the curve y = cii sin X -\- a^ sin 2x -\- «3 sin 3x -{- • • • (1) is symmetrical with respect to the origin. Since cos mx = cos ( — mx) the ordinate corresponding to any value of x between — tt and zero in the cosine curve will be the same as the ordinate belonging to the corresponding positive value of x . In other words the curve 1/ = ^bo + bi cos X -\- b2 cos 2x + b^ cos 3x -\- • • • (2) is symmetrical with respect to the axis of Y. If then f(x) =^ — /( — x) , that is if /(a?) is an odd function the sine series corresponding to it will be equal to it for all values of x between — tt and tt, except perhaps for the value x = for which the series will necessarily be zero. If f(x) =f(—x), that is if f(x) is an even function the cosine series cor- responding to it will be equal to it for all values of x between x = — tt and X = TT, not excepting the value a' = 0. As an example of the difference between the sine and cosine developments of the same function let us take the series for. x . sin 2x , sin ox y = 2 [sii TT 4 r , cos 3.r , COS O.C // = t; — - COS X H -, h ?in 4. + 3^ ■ 5^ [v. Art. 26(rt) and Art. 2S(«)]. (3) represents the curve (3) (4) / and (4) the curve Chap. II.] FOUMEK's SERIES. 47 Both coincide with y = x from x = to x = tt, (3) coincides witli y =z X from x = — ir to .i; = tt , and neither coincides with y z=. x for values of x less than — tt or greater than tt. Moreover (3), in addition to the continuous portions of the locus represented in the figure, gives the iso- lated points (— 7r,0) (7r,0) (37r,0) &c. 30. We have seen that if f(x) is an odd function its development in silfe series holds for all values of x from — tt to tt , as does the development of f(x) in cosine series if f{x) is an even function. Thus the developments of Art. 26(^0> ^i"t- 26 Exs. '(2), (4), (6); Art. 28 (^<) Art. 28 Exs. (3), (7), (9) are valid for all values of x between — ir and tt. Any function of x can be developed into a Trigonometric series to which it is equal for all values of x between — tt and tt. Let fix) 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. identically ; but '-^^-^ — -^ — '— is not changed by reversing the sign of x and is therefore an even function of x\ and when we reverse the sign of x, "-^^ — ~ is aifected only to the extent of having its sign reversed and is consequently an odd function of x . Therefore for all values of x between — ir and tt /(■^■) -\-f{- ■^) ^ 1 ^^^ _^ ^^^ ^^g ^, ^ ^^ ^Qg 2,r + /,3 cos 3.r + • • • where h,,^ = - j ^-^^^ — ^ — --' cos mx.dx ; and u -'^^ f^ — = (I I sin X -\- (u sin 2x -\- a^ sin 3.r + • • • 2 n f( ^') — f( — x) where a... = - | '^^^ — -f^ — ^ sin mx.dx . -rrj 2 ^,„ and o„j can be simplified a little. t,. = Jf /W +/(-''■) eos«xA ttJ 2 = - I /(a-) cos mx.dx -\- i f{— x) cos mx.dx , 48 DEVELOPMENT IN TRIGONOMETRIC SERIES. [Art. 30. but if we replace x by — x, we get , 77 —IT I /'( — x) COS mx.dx = — I f(x) cos mx.dx = | /(.r) cos mx.dx , and we have h„^ = — | /(»•) cos mx.dx . In the same way we can reduce the vahie of a,^ to — I j\x) sin mx.dx . Hence f(x) = - bo + &i cos ^ + ^2 COS 2x + ^.^s cos 3cc + (2) where and + a I sin X + "2 sin 2x + f^3 sin 3x + • 6,„ = — j f(x) cos mx.dx = — j /(a) cos ma. da. (3) 1 /^ . 1 /^ . a,„ ^ — I /(cr) sin mx.dx = — j /(a) sin ma.da . (4) and this development holds for all values of x between — tt and tt. ^ The second member of (2) is known as a Fourier's Series. EXAMPLES. 1. (,)btain the following developments, all of which are valid from x = — tt to j^' = 7r : — (1) e-^ = ^^- _ cos a^ + p cos 2,r- — — cos 8x + — cos 4.r H 7r| -, Z o lU li I , 2sinh7rrl . 2 . ,, , 3 . ., \ -| H sm X — - sm 2x -\- — - sm ox — — ; sin 4.7^ + • • • . TT |_J O 10 1< _| TT 2 r cos 3.7- COS 5.r cos 7x ~\ sin ,r sin 2x sin 3.r sin 4.r "^^^ 2~+~3 r~+'"' where /(ic) = from .r = — tt to x = and /(.r) = x from j' = to x = tt. Chap. II.] EXTENSION aF FOURIEIl's SERIES. 49 (3) /(^) ~ " T? + ~ T-2 cos a; + - cos 2x + - cos 3x + - cos 5x + ^^ cos Qx + • • • J -TT Sin where f(x) = x from a; — — tt to .^- = , f(x) = from x = to .i? = -^ > TT TT and /(.r) = a' — — from .r = ^^ to x = tt . 2. Show that formula (2) Art. 30 can be written .%) = 9«o COS /3o + ('i COS (X ~ /8i) + r, COS (2.r — (3.2) + 6-3 COS (3x — iSg) H where (•„ = («„; + />„-;) "' a^iid /3„j = tan" ^ ~ ■ 3. Show that formula (2) Art. 30 can be written /(a) = 1 6-0 Sin ^0 + r, sin (,/■ + ^0 + r, sin (2x + (3,) + ^-3 sin (3.^ + P,) + ■ ■ • where c,„ = (a,- + bjj^ and I3„, = tan~^ -^ • 31. In developing a function of x into a Trigonometric series it is often inconvenient to be held within the narrow boundaries .r = — tt and x = tt . 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 /(.z-) for all values of x between x = — c and x = c . Introduce a new variable TT Z =: — X , which is equal to — tt when x = — c and to tt when x = c. /(.r) =,//- -) can be developed in terms of z by Art. 30 (2), (3), and (4). We have f{z ') = 9 ^^" + ^ ^os z + b, cos 2z + b, cos 3,- + • • • ) ^^^ -\- «i sin z + (u sin 2,? + ffg sin 3z -{-■■■ ) where 6„, = - j /(- ■s ) cos /msj.c?^; . (2) 50 DEVELOPMENT IN TKIGONOMETRIC SERIES. [Art. 31. and ^™ ~ ^ f'^TT / ^^^^ ^nz.dz . (3) and (1) holds good from s = — tt to z ^ ir . Replace z by its value in terms of x and (1) becomes /.. N 1 7 1 7 ''''•^17 27ra- . - Sttx . f(x) = -hQ-\-b^ cos — + Ih cos — h h cos — 1- • • • . 7r.r , . 2irx , . .STT.r , ' ^^ + «i sm ■ 1- a.2 sm f- a^ sm + • c c c The coefficients in (4) are the same as in (1), and (4) holds good from X = — (■ to X = c . Formulas (2) and (3) can be put into more convenient shape. or 5,„ = - \ f(^) cos — f^a:; = - j /(A) cos c?A . (5) — c — c In like manner we can transform (3) into «m = - J /(a-) sm -^ dx = -J f(X) sm -^ rfA . (6) By treating in like fashion formulas (1) and (2) Art. 25 and formulas (4) and (2) Art. 27 we get ^_ , . TTX , . 2'Trx . . Sttx . ^„^ f(x) = a, sm \- (u sm 1- cu sm \- ' ' ' (1) ^ c c c ^ where «« — "7 1 /(^') sin dx=-i /(A) sin — ^ rZA . (8) tl and f(x) := - ho -{- hi cos — + ^»2 cos 1- b^ cos — ^ r • ' • (9) where ^»„, = - I f(x) cos dx = - i /(A) cos dX . (10) and (7) and (9) hold good from .r = to x = c . Chap. II.] EXTENSION OF FOURIER's SERIES. 51 EXAMPLES. 1. Obtain the folloAving developments: (1) 1 =-• - sm \- - sm + p sin h " • ' from .T ^ to .r = c . 2c r . 7r,r 1 . 2'jr.T , 1 . Stt.t 1 . 47r.r , "1 (2) X ■= — sm sm 1- - sm sin h * " * ^^ IT \_ r 2 c 6 c 4: c J from a- ^ — c to x =. e . c 4'' r 7r.r , 1 37ra; , 1 Bttx 1 Ttt./- , "I .T = ^, cos \- -, cos h — cos 1" - cos h • ■ • 2 IT- \_ e 3^ c D^ c (- c J from X = to x = c „, , 2r-^r/7r2 4\ . TTX 7r2 . 2'7rx , /tt^ 4\ . 37r.r (3) ^■■' = ^LVT - 1^) ^^^^ T - Y ^^^^ -T + (t - ,^^) ^^" — TT- . iTTX , (IT- 4\ . 7)17 X . ~\ __3ni — + (^--;sm-^+-.-J from X = to j: =: c . fA 4^-2 p ^^, -^ 27rx , 1 Sttx 1 47r./; •T^ = TT ~ \ COS cos h 02 COS — y^ COS 3 TT- L '■ -^ c 3^ c 4^ c from ./• = — r to X = (' . r. ri + t'" 7r.r , 2(1—^^) . 27rx (4) e^ = 27r .,7" -2 s"^ ^ ) , , .; sm , 3 (1 + e'-) . Sttx , A(l — ef) . Attx , "] + 1 a - s^" ^ o\ -, .. o sm h • • • > c- -\- 97r- c t- + IbTT- (' J ^ _ ri e/ - 1 e'^ + 1 7r.x , ^' - 1 2'rrx e^ = 2c\ - — — cos h .2 , . ., cos L2 c^ c^ + TT^ c c^ + 47r'' c cos '-^ + • • • J from X = to ',r = c . c" -\- 1 oTT.r from .'P = to X = c , c c where f(x) = a; from x = to a; = - and f(x) = c — x from .a^ = :^ to 52 DEVELOPMENT IN TRIGONOMETRIC SERIES. [Akt. ■^2. 2. Show that formula (4) Art. .'31 can be written f(x) = - r, COS ^0 + ''1 COS {^— - (3,) + Co COS (^-— - (3o) 4- 03 COS /^ — M + ■ • • where ^-^ =(«,' + ^„l)^ and ^„, = tan-i^. 3. Show that formula (4) Art. .31 can be written 1 /TT.r \ /27r.r \ /(«') = r, '0 sin ^0 + '-1 sin (^— + (3^) + 'v sm (-^ + /i^j + .3 sin (•^" + a) + • • where r-„, = («J + ^>,f)* and /3„ = tan-^ ^ . 32. In the formulas of Art. 31 c may have as great a value as we please, so that we can obtain a Trigonometric Series for f(x) that will represent the given function through as great an interval as we may choose to take. If, then, we can obtain the limiting form approached by the series (4) Art. 31 as c is indefinitely increased the expression in question ought to be equal to the given function of x for all values of x. Equation (4) Art. 31 can be written as follows if we replace ^o? ^u ^25 " * ' Ui, a^, • • • by their values given in Art. 31 (o) and (6). -H I /(A) cos -y cos — ''A + I /(A) cos — — cos c/A + • • • + Cf(X) sin — sin — r/A + Cf(X) sin ^^^ sin ^^^ (ZA + • • • | 1 /* , N 7. rl , ''■A TT.l- . . ttA . TT.r T I /(^) ^^^ \ V ~'~ *-**^^ ~r *^*"^ ~T "^ '^^^^ ~T ^^^^ ~T 27rA 27ra" , . 27rA . 27r.i -j- cos — — COS \- sm sin Chap. II.] FOURIER'S INTEGRAL. 53 /■(.r) - i ff(X)dX [^ + cos ^ (A - .r) + cos ^ (A - .>■) + • • -1 = ^_ ff(K)dX fl + cos ^ (A - .'■) + cos y (A - .'■) + • • • + cos (- ^) (A - *•) + cos (- =^) (A - ./■) + • • -1 since cos ( — <^) = cos <^ . .m = ^ ffi^yi^ [• • • + 7 «o« (- v) ^^ - "') + 7 •^'^^ (- 7) (^ -''^^ — c H cos -^ (A — .r) H — COS — (A — X) + _ COS — (A - ./■) + • • • J (1) As c is indefinitely increased the limiting value approached by the parenthesis in (1) is I cos a(A — x).(la. Hence the limiting form approached by (1) is f(x) = ^ J f(\)d\y cos a (A - .r).da , (2) and the second member of (2) mnst be equal to f(^x) for all values of x . The double integral in (2) is known as Fou/'ier's Integral, and since it is a limiting form of Fourier's Series it is subject to the same limitations as the series. That is, in order that (2) should be true f(x) must be finite, continuous, and single valued for all values of x, or if discontinuous, must have only finite discontinuities.* (2) is sometimes given in a slightly different form. X 3: Since | cos a (A — x).da = | cos a (A — x).da -\- j cos a (A — x).da and II (I I cos a(\ — x).da = j cos (— a) (A — x).d(— a) = — | cos a(A — x).da i cos a(A — x).da = 2 | cos a (A — x).da * See note on pajxe P>S. 54 DEVELOPMENT IN TRIGONOMETRIC SERIES. and (2) may be written /(.r) = - Cf(\)dX fcos a(A — x).da . (3) If f(x) is an even function or an odd function (3) can be still further simpli- fied. Let /(.^.) =-/(-.,;). Since the limits of integration in (3) do not contain a or X the integrations may be performed in whichever order we choose. That is ff(X)dX fcos a(X — x^.da = Cda Cf(X) cos a(X — a:).dX . — o= —=c Now Cf(X) cos a(X — x).dX = Cf{X) cos a{X — x).dX + Cf(X) cos a(X — x).dX . Cf(X) cos a(X — x).dX = Cf{— X) cos a(— X — x).d{— X) = — Cf{X) cos a{X + x).dX and (3) becomes f(x) = - Cda Cf(X) [cos a(X — x) — cos a(X + x)^.dX = — (da I /(A) sin aX sin ax.dX or f(x) = - Cf(X)dX I sin aX sin ax.da . (4) If f(x) = /( — x) (3) can be reduced in like manner to 1 - /-^ f(x) = - Cf(X)dX j cos aX cos aa^.rfa . (5) Although (4) holds for all values of x only in case f(x) is an odd function, and (5) only in case f(x) is an even function, both (4) and (5) hold for all positive values of x in the case of any* function. EXAMPLE. (1) Obtain formulas (4) and (o) directly from (7) and (9) Art. 31. CHAPTER TIL COXVERGENCE OF FOURIER's SERIES. 33. The question of the convergejire of a Fourier's Series is altogether too large to be completely handled in an elementary treatise. We will, however, consider at some length one of the most important of the series we have obtained, namely 4 r . , sin 3.r , sin hx , sin 7.r , ~\ - I sm X + --g \ ^ \ h • • • . [v. (3) Art. L^6(//).] and prove that for all values of x between zero and it its sum is absolutely equal to unity; that is, that the limit approached by the sum of n terms of the series - sin X I sin a.da + sin 2x j sin la.da + sin ox Ain oa.da + • • • , as n is indefinitely increased, is 1, provided that x. lies between zero and tt. Let '^w = ~ sin X I sin a.da + sin 'Ix | sin la.da + sin '.\x | sin oa.da + • • • + sin nx j sin na.da . (1) Then S„= ~ I [sin a sin x -\- sin 2a sin 2x -\- sin 3a sin ox + • • • + sin /i.a sin nx'\")]^^« • 56 CONVERGENCE OF FOUEIER'S SERIES. [Art. 84. Therefore by Art. 20 (1) 'rrJ\_J 1 . a — X J sm — - — u sm ^— ^ ^sin(2. + l;^ ^ ^sin(2. + l)^ '^"^2^j — -7-^^r — ^^"-2^J — r^+:r— ^"' () sm — - — sm — - — In the first integral substitute ^ for — — ^ , and in the second integral sub- stitute /3 for — -y^ . We get ttJ sm B ^ ttJ sm B ^ ^ It remains to find the limit approached by /S^ as n is indefinitely increased. 34. E pin(2„ + l)/i ^^. (1) ./ sm yS 2 Fo ^''Vsin"^ ^^^ = i + cos 2/3 + cos 4/3 + ■ • • + cos 2?z^ , by Art. 20. and rcos2A-/3.(//3 = 0. Let us construct the curve sin (2?? + l).r 1/ = ^-^ — . sm X We have only to draw the curve // = sin (2ti + l)x and then to divide the length of each ordinate by the value of the sine of the corresponding abscissa. In 1/ =1 sin (2n -\- l)x the successive arches into which the curve is divided by the axis of A' are equal, and consequently their areas are equal. SU31MATI0X OF A SINE SEKIES. 57 Each arch ]ms for its altitude unity and for its base and is symmetrical witli respect to the ordi- 2.7, + 1 _ nate of its highest or lowest point. sin (2». + l).r sin X ■ If now we form the curve // ^ ^-^ — ' — '— from the curve i/ = sin (2ii -\- l)x , it is clear that, since TT sm X increases as x increases from to — , the ordi- nate of any point of the new curve Avill be shorter than the ordinate of the corresponding point in the preceding arch, and that consequently the area of each arch of // = ' — '- will be less than sin x that of the arch before it. If fto? «ij ^h, ' ■ ■ '^n-i ai'6 the areas of the suc- cessive arches and ((i^ that of the incomplete arch termi- nated by the ordinate corresponding to x = — sin (2n + l)x dx — r/y — (i^ + ^'2 — "3 + • • • • in {2n + l).x- , r sin {2n, + 1)^ tt / sin /? dli = i, by(l). . + ''4) if a is even, and 5 = "0 + (- "1 + "2) + (- "3 + ''4) + (- "5 + «c) + ••• + (-"«-. + <-'n - 1) + (- ''«) if n is odd. 68 rONVERGENCE OF FOUEIER's SERIES. [Art. 35. In either case each parenthesis is a negative quantity since tto > «i > «2 ^ "3 ^ (f'nf and it folloAvs that r/o is greater than — • Again TT - = ((0 — a I + (r/2 — rts) + (a4 — r/5) -i [- (r/„ _2 — «„ _i) + % if ?? is even and TT - = r,„ — ,,j -f (r/2 — r/3) + («4 — r/5) + • • • + (rt„ _j — aj if 7i is odd. In either case each parenthesis is j)Ositive and it follows that a^ — «i is less than — . Since tto and (Iq — (I I differ from — by less than they differ from each other, that is, by less than a^ . In like manner we can show that «o — ^1 and a,-, — c/i + "^s differ from TT — by less than a.y, and in general that r/,, — "1 + '/o — <^3 + ' ' " i '^t differs ■^ TT from - by less than fij.; or even that ^'0 — ''1 + «2 — f'3 + ■ • • ± -' differs from — by less than r/^. no matter what the value of p , provided p is greater than unity. 35. Prom what has been proved in the last article it follows that sin (2)1 -\- 1)3 f^""~- * sm X TT , TT where b is some value between 7; — r— - and — , differs from — by less than 2/^ + 1 2 ' 2 "^ sill (^)i "4" l^x the area of the arch in which the ordinate of u = ''"^- — correspond- sm X ing to X = I) falls if this ordinate divides an arch, or by less than the area of the arch next beyond the point (b , 0) if the curve crosses the axis of A' at that point. Chap. III.] SUMMATION OF A SINE SERIES. 59 TT The area of the arch in question is less than - — — — - , its base, miTltiplied by 1 a value greater than the length of its longest ordinate. h r sin (2« + l)x , iheretore I ^-^ ^ aa J sm X differs from — by less than ,:.^ — —:, 2 -^ III + 1 sm lb— - — — I V 2n + 1/ TT 1 If now n is indefinitely increased - — -— approaches ^ 1,1+1 . / TT \ zero as its limit, and we get the very important result h limit r r sin (2ii -\- l)x n _ tt ,^^ ^= 2o |_J sin ic "^ J ~ 2 if ()\ COS x + hn cos 2x + />3 cos 3.T + • • ■ 2 + f/j sin X + (^'o sin 2,r + «3 sin 3a" -j- • • • Chap. HI.] DIKICHLET's CONDITIONS. 61 TT where a^-=- \ f{a) siu ma.da and &,„ =: - j f(a) cos ma.da , and that Fourier's Series only is equal to f(x) for all values of x between a; = — TT and x = ir , excepting the values of x corresponding to the discon- tinuities of f(x), and the values tt and — tt if /(tt) is not equal to /(— 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 1 limit ^ ,,^ , , . . . -, and that if /(tt) is not equal to /(— tt) the value of the series when x = — ir and when a- =: tt is ' If f(x) while satisfying the conditions named in the preceding paragraph except for a finite number of values of x, becomes infinite for those values, the series is equal to the function except for the values of x in question provided that ('fi-'c)dx is finite and determinate, (v. Int. Cal. Arts. 83 and 84.) 38. The question of tlie convergency of a Fourier's Series and the condi- tions under which a function may be developed in such a series was first attacked successfully by Dirichlet in 1829, and his conclusions have been criticised and extended by later mathematicians, notably by Riemann, Heine, Lipschitz, and du Bois Reymond. It may be noted that the criticisms relate not to the sufficiency but to the necessity of Dirichlet's conditions. An excellent resume of the literature of the subject is given by Arnold Sachse in a short dissertation published by Gauthier-Villars, Paris, 1880, entitled "Essai Historique sur la Representation d'une Fonction Arbitraire d'une seule variable par una Serie Trigonometrique." 89. A good deal of light is thrown on the peculiarities of trigonometric series by the attempt to construct approximately the curves corresponding to them. If we construct y = a^ sin x and y = a.2 sin 2x and add the ordinates of the points having the same abscissas we shall obtain points on the curve t)2 CONVEKGENCE OF FOUKIEK'S SERIES. [Art. 39. ij = c/j sin X -\- a^ sin 2x . If now we construct y = (h sin ?>x and add the ordinates to those of // r= 11^ sin X -\- «2 sin 2x avb shall get the curve // = «! sin ./• -|- do sin 2.x + a^ sin 3.t. By continuing this process we get successive approximations to // = a-^ sin X + no sin 2a:' + r/3 sin ox -\- a^ sin \x 4- • • • Let us apply this method to a few of the series which we have obtained in Chapter II. Take ?/ ^ sin a? + - sin '?,x + - sin 5.t + • • • (1) IT = when x ^ , - from x = to .1; = tt , and when x^ir , 4 V. Art. 26 [/>](3). // = 2 ^sin X — \ sin 2,r + ~ sin 3,/- — J sin 4.r + • • ■) (2) = ./■ from .r = to .r=:7r, and when .r = 7r, Art. 26[a.](4). // = - r-o sin X — - sin 3.r + p^ sin o.r — -,_ sin 7.7- + • • • (3) = .r from x = to a- = — , and ir — x from x = - to .r = tt . Art. 2G [r](2). 12 112 1 y = - sin X + - sin 2x -\- - sin 3.'" + - sin 5./ — - sin 6x + - sin 7x-\- ■■ ■ (4) 1 w o o O ( TT TT TT ^0 when .c = 0, — from .r = to .'• = — , and from ''' = -^ to x = 7r. v. Art. 26 [./](2). It must be borne in mind that each of these curves is periodic having the period 27r, and is symmetrical with respect to the origin. The following figures I, II, III, and IV represent the first four approxima- tions to each of these curves. In each figure the curve i/ = the series, and the approximation in question are drawn in continuous lines, and the preceding approximation and the curve corresponding to the term to be added are drawn in dotted lines. Chai'. 111.] SUCCE ^CESSIVE APPilOXlMATlOlS'S TO A SINE SEKIES. 63 Y / \y "7 ^ "v" Y /■ — ^<' "~>^^ — "\ / y -A /•' "^N / -^\ X / V.^'' TC \ Y 7\-^. — .c— jc^ ''7" ,-'"~^, .'-'"N -> X t V.y 'v../' "• '' 71 I (54 CONVEllGENCE OF FOUKIER's SERIES. [Art. .39. Chap. TII.l PROPERTIES OF FOURIER S SERIES. 6o Figs. I, II, III, and IV immediately suggest the following facts : (a) The curve representing each approximation is continuous even when the curve representing the series is discontinuous. (b) When the curve representing the series is discontinuous the portion of each successive approximate curve in the neighborhood of the point whose abscissa is a value of x for which the series curve is discontinuous approaches more and more nearly a straight line perpendicular to the axis of X and con- necting the separate portions of the series curve. ((•) The curves representing successive approximations do not necessarily tend to lose their wavy character, since each is obtained from the preceding one by superposing upon it a wave line whose waves are shorter each time but do not necessarily lose their sharpness of pitch. This is the case in Figures I, II, and IV. In Fig. Ill the waves of the superposed curves grow rapidly flatter. It folloAvs from this that in such cases as those represented in Figures I, II, and IV the direction of the approximate curve at a point having a given abscissa does not in general approach the direction of the series curve at the corresponding point, or indeed, approach any limiting value, as the approxima- tion is made closer and closer; and that the lengj;h of any portion of the approximate curve will not in general approach the length of the correspond- ing portion of the series curve. Analytically this amounts to saying that the derivative of a function of x cannot in general be obtained by differentiating term by term the Fourier's Series which represents the function. (d) The area bounded by a given ordinate, the approximate curve, the axis of X, and any second ordinate will approach as its limit the corresponding area of the series curve if the series curve is continuous between the ordinates in question; and will approach the area bounded by the given ordinate, the series curve, the axis of X, any second ordinate, and a line perpendicular to the axis of A', and joining the separate portions of the series curve if the latter has a discontinuity between the ordinates in question. Analytically this amounts to saying that the Fourier's Series corresponding to any given function can be integrated term by term and the resulting series will represent the integral of the function even when the function is discontinuous (v. Int. Cal. Art. 83). We may note in passing that if the function curve is continuous a curve representing the integral of the function will be continuous and will not change its direction abruptly at any point; while if the function curve is dis- continuous the curve representing the integral will still be continuous but will change its direction abruptly at points corresponding to the discontinuities of the aiven function. 66 CONVERGENCE OF FOUKIER's SEKIES. [Art. 40. 40. The facts that the derivative of a Fourier's Series cannot in general be obtained by differentiating the series term by term and that its integral can be obtained by integrating the series term by term are so important that it is worth while to look at the matter a little more closely. Let us consider the differentiation of the series represented in Art. 39 Figure I. Let Sn = sin a- + - sin 3^" + - sin ox + • • • + ^ — j—r sin (2n + l)ic . Then ~ = cos x + cos 3a- + cos 5x -\- ■ ■ • -{- cos (2m + l)x. ax u . = 1 f"=0 ax and the curve is parallel to the axis of X for x = - no matter what the value of ii . If a.- = or X = IT ^" = 1 + 1 + 1 + 1 + • • ■ + 1 = v/ + 1 ax and the curve // = S^ becomes more nearly perpendicular to the axis of X at the origin and for x =z tt as we increase n. If . = 1 dx 2 '^22 2 That is — r^' = - if n = or n = ^k dx = — - •• /i = 1 " n = 3/t + 1 = '' w = 2 -' n = Sk + 2. Consequently when x = — ~' does not approach any limiting value as n is indefinitely increased. Indeed, in the successive approximations the point whose abscissa is — is successively on the rear, on the front, and on the crest o or in the trough of a wave, and although the waves are getting smaller they do not lose their sharpness of pitch. If X has any other value between and tt — " will change abruptly as n is changed and will not approach any limiting value as n is increased. Chap. III.] DIFFERENTIATION OF FOURIEIi's SERIES. 67 41. In general if we differentiate a Fourier's Series 'S' = 9 <^o + ^'\ COS :i: + ho, cos 2.r + !>. cos ox + • • • + c/j sin .r + (U2 sin 2,r + r/3 sin o^ + ' * * we get — hi sin X — 2Z'2 sin 2x. — Sh^ sin 3.« — • • • + <(i cos x + 2a2 cos 2.i' + 3a s cos o.« + • • • . Differentiate again and we get — hi cos x — 2'\ cos 2x — 3%s cos 3,> — • • • — «! sin X — 2^(f2 sin 2.t — oVj^ sin 3j' — • • ■ . We see that each time we differentiate we multiply the coefficient of sin 7cx and of cos kx by k while the term still involves cos kx or sin kx . Since the series cos X + COS 2x -\- cos ox + • • • + sin X + sin 2x + sin 3x + • • • is not convergent, and a Fourier's Series converges only because its coefficients decrease as we advance in the series, the differentiation of a Fourier's Series must make its convergence less rapid if it does not actually destroy it, and repetitions of the process will usually eventually make the derived series diverge. It is to be observed that the derived series are Fourier's Series, but of some- what special form, that is they lack the constant term. (v. Art. 30.) If now we integrate a Fourier's Series - h„ -f hi cos X + ho cos 2x + hs cos 3x + • • • + a I sin X -\- a^ sin 2x + ag sin 3.^ + • • • we get ^ "^ 9 ^■'o'^" ^~ ''^i ^^^^ ^ ~^ ~ ^'" ^^^^ ^^' + ~ ^3 ^^^^ '^•■*^ -\- ■ " 1 O 1 Q — r/i cos a — - «2 cos 2x — 7- a^ cos ox — • • • , 2 o a Trigonometric Series which converges more rapidly than the given series. It is to be observed that the series obtained by integrating a Fourier's Series is not in general a Fourier's Series owing to the presence of the term ^box. (V. Art. 30.) 42. We are now ready to consider the conditions under which a function of X can be developed into a Fourier's Series whose term by term derivative shall be equal to the derivative of the function. 68 CONVERGENCE OF FOURIEU'S SERIES. Let the function f(x) satisfy the conditions stated in Art. 37. Then there is one Fourier's Series and but one which is equal to it. Call this series *S'. Let the derivative /'(«)* of the given function also satisfy the conditions stated in Art. 37. Then f'{x) can be expressed as a Fourier's Series. By Art. 39 (d) the integral of this latter series will be equal to the integral of f\x)^ that is to f(x) plus a constant, and one integral will be equal to f(x) . If this integral which is necessarily a Trigonometric Series is a Fourier's Series it must be identical with S. It will be a Fourier's Series only iu case the Fourier's Series for /'(.r) lacks the constant term ^h^. But ^jy'(,x)dx by (3) Art. 30. Therefore h^ = - [/(tt) — /{— tt) ] ; and will be zero if /\7r) =/'(— tt). In order that f'(x) shall satisfy the conditions stated in Art. 37 f(x) while satisfying the same conditions must in addition be finite and continuous between x = — tt and cc = tt. If, then, /(.r) is single-valued, finite, and continuous, and has only a finite number of maxima and minima, het^eew .r = — tt and .r = tt, (the values a: = — TT and x =^ ir being included), and if /(tt) =/( — tt) f(x) can be developed into a Fourier's Series whose term by term derivative will be equal to the derivative of the function. It will be observed tKat in this case the periodic curve y = S is continuous throughout its whole extent. 43. Since a Fourier's Integral is a limiting case of a Fourier's Series the conclusions stated in this chapter hold, mutatis mtitandis for a Fourier's Integral. For example if a function of x is finite and single-valued for all values of x and has not an infinite number of discontinuities or of maxima and minima in the neighborhood of any value of x it Avill be equal to the Fourier's Integral Ifiajm COS a (A — x).d\ and to that Fourier's Integral only, and the integral with respect to x of this Fourier's Integral will be equal to Cf(x)dx. If in addition /(.r) is finite and continuous for all values of x the derivative of the Fourier's Integral with respect to x will be equal to ^y7 • * We shall regularly use the notation f'{x) for '—j^ . v. Dif. Cal. Art. 124. CHAPTER IV. SOLUTION OF PROBLEMS IN PHYSICS BY THE AID OF FOURIER'S INTEGRALS AND FOURIER's SERIES. 44. In Art. 7 we have already considered at some length a problem in Heat Conduction which required the use of a Fourier's Series. We shall begin the present chapter with a problem closely analogous in its treatment to that of Art. 7, but calling for the use of a Fourier's Integral. Suppose that electricity is flowing in a thin plane sheet of infinite extent and that the value of the potential function is given for every point in some straight line in the sheet, required the value of the potential function at any point of the sheet. Let us take the line as the axis of X and consider at first only those points for which y is positive: We have, then, to satisfy the equation BIV^DIV=^ (1) subject to the conditions F=0 when >j=^cr> (2) F=/(.x-) " ^ = (3) where f{x) is a given function, and we are not concerned with negative values of //. As in Art. f we have e~"" sin ax and e~"^ cos ax as particular values of F which satisfy (1) and (2). We must multiply them by constant coefficients and so combine them as to satisfy condition (3). By (3) Art. 32 f(x) = - Cda Cf(X) cos a (A — x) .dX . A) We wish to build up a value of F which will reduce to (4) when // = (). This requires a little care but not much ingenuity. 70 SOLUTION OF PROBLEMS IN PHYSICS. [Airr. 4i. Take e^^^cosaa.- and e~'^^sinax and multiply the first by cos aA, and the second hj sin aA ; they are still values of V which satisfy (1). Add these and we get e-«^cosa(A — .t)j still a value of V which satisfies (1), no matter what the values of a and A. Multiply by f(\)dX and we have e~''yf(X) cos a(A — x).dk (5) as a value of V which satisfies (1). V= Ce-'^i' f(\) cos a(\ — .r).dX (6) is still a solution of (1) since it is the limit of the sum of terms covered by the form (5); and finally V= - Cda Ce- <'yf{\) cos a(A — x).dX (7) is a solution of (1) as it is — multiplied by the limit of the sum of terms TT formed by multiplying the second member of (6) by da and giving different values to a. But (7) must be our required solution since while it satisfies (1) and (2), it reduces to (4) when ?/^0 and therefore satisfies condition (3). If f(x) is an even function we can reduce (7) to the form F= - j da r^""^/(A) cos ax cos aA.c^A (8) and if f(x) is an odd function to the form 2 r r r=- I da I e-'^yfik) sin ax sin aX.dX. (9) (7), (8), and (9) are valid only for positive values of ?/, but as the problem is obviously symmetrical with respect to the axis of A'. (7), (8), and (9) enable us to get the value of the potential function at any point of the plane. EXAMPLES. 1. Obtain forms (8) and (9) directly by the aid of (5) and (4) Art. 32. 2. State a problem in statical electricity of which the solution given in Art. 44 is the solution. Chap. IV.] FLOW OF ELECTRICITY IN AN INFINITE PLANE. 71 45. As a special case under Art. 44 let us consider the problem : — To find the value of the potential function at any point of a thin plane sheet of infinite extent where all points of a given line which lie to the left of the origin are kept at potential zero, and all points which lie to the right of the origin are kept at potential unity. Here f(oc) = if a-<0 and /(.r) = 1 if x>0. (7) Art. 44 gives us the required solution. It is T'= - Cda Ce- -^^ cos a(A — .iS).dX ; (1) but this can be much simplified. We have F=- j flk I e~"2'cosa(A — x).da. Now i e~"^ cos vix.dx =^ -^—, -, J a- -\- »r if ^/ >0. (Int. Cal. Art. 82, Ex. 8.) Hence /ir'^y cos a (A — x).da = -^ — ^ = 1 C y^ = - (''^ + tan- ^ '^Y tan I — — tan-^ - ) = ctn ( tan" ^ - ) ^ - : V2 y/ \ y/ x' and consequently F=-(? + tan-2:) = l_itan-^. (2) TT \2 I// TT X ^ Since log z = log (.« + ijl) =^ - log (x^ + //-) + * tan- ^ - , [Int. Cal. Art. 33 (2)], ; - - log ^ = /• - - log (X + yi) = - — log (x' + y-') + / (l - - tan" ' ^) If -1.'/ 1 1 — tan ^ - and — Cal. Arts. 209 and 210.) Hence and 1 tan ^ - and — -^ log (x- + y'^) are conjugate functions, (v. Int. ri = -^log(.r2 + 2/2) (3) is a solution of the equation DlV,ArD-^V, = ^', (4) 72 SOLUTION OF PKOBLEMS IN PHYSICS. [Akt. 45. and the curves - (- + tan-' -) = a (5) and - — log(.r' + ,f)=b (6) cut each other at right angles. If we construct the curves obtained by giving different values to a in (5) we get a set of equipotential lines for the conducting sheet described at the begin- ning of this article, and the curves obtained by giving different values to (h) in (6) will be the lines offloiv. Moreover since Vi= — — log {x^ + y/2) (3) is a solution of Laplace's Equation (4), the lines of flow just mentioned will be equipotential lines for a certain distribution of jiotential, for which the equi- "potential lines above mentioned will be lines of flow. F=a, that is ^(J + tan-'^) (5) reduces to U'^~ ^ ^^^^ '^'^ ■ (^) If now we give to a values differing by a constant amount we get a set of straight lines radiating from the origin and at equal angular intervals. V-^ = b, that is -iTT b, (6) reduces to x^-^,/^e-'^\ (8) If we give to i a set of values differing by a constant amount we get a set of circles whose centres are at the origin and whose radii form a geometrical progression. They are the equipotential lines for a thin plane sheet of infinite extent where the potential function is kept equal to given different constant values on the circumferences of tAvo given concentric circles or where we have a source as the origin; and for this i system the lines (7) are lines of flow, and (3) is the complete solution. The figure gives the equipotential lines and lines of flow for either sys- tem, but only for positive values of y. The complete figure has the axis of X as an axis of symmetry. Chap! IV.] FLOW OF ELECTRICITY IN AN INFINITE PLANE. 73 EXAMPLES. 1. Solve the problem of Art. 44 for the case where /(.x)= — 1 if x<0 and /(■*■) = 1 if .t>0. 2 :r A71S., V^ - tau"^ ^ - . TT // 2. Solve the problem of Art. 44 for the case where f(x) = a if X < and f(x) = h if .;• > . Ans., V= \ (l. V=- ftan- 1 ^^^^ + tan- ' ^^^^ . (1) Here Kow - log [(1 - z)ir\ = - log [(1 - X - ///)/] = - log [//+ (1 - ./■)*] l-log[(l-..)^ + /] + ^tan-^. and - ;^ log [(- 1 - ;^) /] = - _^ log [(-!-./•- yO C = - ^ log [y - (1+ x)q ^-~ log [(1 + .ry + //] + ^ tan- ' ^^' . - log — = ~- log Vn , • + - tan- ^ h tan ^ . 74 SOLUTION OF PEOBLEMS IN PHYSICS. [Art. 4(j. Hence vrV // // / 'Jtt ° (1 -\- X)- -\- i/' are covjuf/ate functions; * and - (tan- 1 ^-i£ + tan- ' -^ -) = « (2) is any equipotential line, and 1 , (1 - Xf + / , ,ON any line of flow for the system described at the beginning of this article; and is the solution of a new problem for which (3) represents any equipotential line and (2) any line of flow. The function conjugate to 1 tan-i h tan-i \_ y 'J J miglit have been found as follo\v.s. If 4> is the required function and \p the given function we have by Int. Cal. Arts. 208, 209, and 210 the relations Dx(p = D,f\f/ and D,j4> = — D^-^p . If now we integrate Dyf with respect to x treating ?/ as a constant and add an arbitrary function of y we shall have . So that .^ = - :;^ { log [(1 + x)-^ + y^] - log [(1 - xf + y'] { +Ay) . _ 1 r y y ~| I '^ ^•""^ ~ TT |_ ( 1 + x)-^ + 7/ (1 - x)--^ + 7/-^ J "^ " 1%) Comparing this with its equal — D^V' above we find -^r^ = and f(y) — C a constant therefore ^ log ^^'""tl'^'^ + ^ ' 27r '' (1 + X)- + ?/- where C may be taken at pleasure, is our required conjugate function. CiiAP. IV.] SOURCE AiSD SINK IN AN INFINITE PLANE. (2) reduces to ——. — '— == tan (ITT x' + y' — i or x^ + (// — etn (nry~ = csc-«7r ; and (3) to ^■^+//^+2 e^'^+l + 1 = 75 (5) (.r + ctnh ^vtt)-^ + y- = csq.Ii'-Ik (6) (5) and (6) are circles. The circles (5) have their centres in the axis of Y, and pass through the points ( — 1 , 0) and (1 , 0) ; and the circles (6) have their centres in the axis of A'. (4) is the complete solution, (6) is any equipotential line and (o) any line of flow for a plane sheet in which the points in the circumferences of two given circles whose centres are further apart than the sum of their radii are kept at different constant potentials, or where a source and a sink of equal intensity are placed at the points ( — 1,0) and (1, 0). An important practical ex- ample is where two wires connected with the poles of a battery are placed with their free ends in contact with a thin plane sheet of conducting material. The figure shows the equipotential lines and lines of floAV of either system. The complete figure would have the axis of A' for an axis of symmetry. v=o If^„ v=i v,-^ v=o EXAMPLES. 1. Show that if /(.;•) = (?i when :f b , r=^^+^ -\^(a,-a,) -ith tan-^ — ' 1- (c/2 — ''3) tan-^ - 76 SOLUTION OF riiOBLEMS IX PHYSICS. [Akt. 47. 2. Show that if f(x)=() if ,v<0, f(x) = a, if () < .r < ^>i, /(.r) = ./., if bi < X < h.^, f(x) = as if /', < .r < bs , &C., i[ 1 1 tan~^ - + (('i — ((.->) tan~^ '- + (>/., — a,) tan~^ ^ .'/ U .'/ + (''3 — c-i) tan-^ — + 1/ 3. Show that if f(x) = - 1 if x < — 1. f(x) =x if — 1 < ./• < 1 . f(x) = l if x>l. r=i[(l+.)tan-.l±^-(l-.)ta„-.l^ + |log|l^^]. 4. Show that if f(x) = - 1 if .r < - 1 , f(x) =0 if - 1< ,/■ < 1 , /(.T) = l if a->l, T- ir ,l±x ^ , l-.r ~| ttL // ?/ J Show that the equipotential lines are equilateral hyperbolas passing through the points ( — 1, 0) and (1, 0), and that the lines of flow are Cassinian ovals having (— 1, 0) and (1, 0) as foci. The lines of flow are equipotential lines and the equipotential lines are lines of flow for the case where the points ( — 1, 0) and (1, 0) are kept at the same infinite potential, or where very small ovals surrounding these points are kept at the same finite potential. The case is approximately that of a pair of wires connected with the same pole of a battery whose other pole is grounded, and then placed with their ends in con- tact with a thin plane conducting sheet. 5. Show that if f(x) = if x < , f(x) = — 1 if ()<' ,/■ < a, f(x) = if a b, ,. ir-TT jA The conjugate function -■'■f + ir^(J->-x^+>f] is the solution for the case where a sink and two sources of equal intensity lie on the axis of A', the sink at the origin and the sources at the distances a and b to the right of the origin. One of the lines of flow is easily seen to be the circle x^ -\- if- = ab . 47. If the plane conducting sheet has two straight edges at right angles with each other and one is kept at potential zero while the value of the poteu- Chap. IV.] EXAMPLES. TT tial function is given at each point of the second, that is if r=() when x = () and V^ /(■'') when y = 0, tlie sohition is readily obtained. It is 00 cc F= - Cda fe-^'-^/XX) sin ax sin aX.dX . (1) V. (9) Art. 44. This reduces to V. Ex. 3 Art. 45. EXAMPLES. 1. If V=0 when ?/ = and V^F(//) when a; = show that r= - j (la i f'-"-^ F(\) sin a// sin aA.rfA = -fF(X)dX \__^,^^^_^y - ^.+ (l + ^)j • 2. If 7'=/(;r) when y := and F=^F(//) when ,t = <) show that 3. If i^(//) = ^* the result of Ex. 2 reduces to 2/^ TT tan- '^ + i f/(A)rfA T^^^ -, - , , ^ , 1 4. If F(!/) = 1 for < ,y < 1 and FQ/) = for // > 1 while f(x) = 1 for 0 1 F= iftan- ^-^ - tan- ^-±^ + 2 tan- "^ + tan- 1^^ - tan- ^-^^ + 2 tan- '^l • X X //J 78 SOLUTION OF PKOBLEMS IN PHYSICS. [Art. 48. 5. If one edge of the conducting sheet treated in Art. 47 is insulated, so that D^.V=0 if .r = and V=f(x) Avhen y = V=- ( '^a i e-'^yf^X) cos ax cos aX.dX 48. If the conducting sheet is a long strip with parallel edges one of which is at potential zero while the value of the potential function is given at all points of the other, that is if F=0 when y = (i and V^=F(x) when y^h the problem is not a very difficult one. Since we are no longer concerned with the value of V when y ^= cc V= e'^y sin ax and F= pJ^^ cos ax are available as particular solutions of the equation i),2F+i>/r=0 (1) as Avell as F= e~'^y sin ax and F^ e~'^y cos ax . ,.a;/ _j_ ^- ay Consequently — sin ax = cosh ay sin ax [Int. Cal. Art. 43 (2)] and — sin ax = sinh ay sin ax [Int. Cal. Art. 43 (1)] and cosh ay cos ax and sinh ay cos ax are now available values of F and can be used precisely as e~ °-y cos ax and e~'^y sin ax are used in Art. 44. Following the same course as in Art. 44 we get as a solution of (1) which will reduce to F= F(x) Avhen // = 6 and to F=0 when // = 0, since sinh = — - — =0, and (2) is therefore our required solution. If F is to be equal to zero when y = h and to f(x) when ?/ = we have only to replace ^ by h — y and F(x) by f{x) in (2). We get CllAl'. IV.] FLOW IN A LONG STKIP WITH PAKALLEL EDGES. ( 9 If V^f(a--) when y = and V=F(x) when y = /> then y^l P„ r sinha(^-^) ^^^ _ ttJ J sinh ah " ^ ^ —00 This can be considerably simplified by the aid of the formula . pTT ^ . , sm — /smh px , TT (I . ■■ — cos rx.dx = -— • smh ox 1(1 j)ir . . rir <| cos h cosh — if 7^^< '/'-. [Bierens de Haan, Tables of Def. Int. (7) 265] and becomes 1 . TT „ . /^.„_ d\ F=5^ain^(«-,„)j:/-(A)— ^^^^ cos , ' 1- cosh - (A — x) ^ ^ 2b b J ^ _ TT// d\ cos -j- -\- cosh T (A — .'•) 'cosh — (A — x) — cos -f- cosh — (A — x) 4- b b b ^ cos , EXAMPLES. 1. Given the formula — r~7~ — r~ = , tan- 1 (a/ / , tanh J ) if 6 > ff , « + 6cosha' sjb'^ — a- \^b-\-a 2/ ' show that if F=l when ^=^0 and V=^Q when [/=^b V=j(b — i/). 2. Show that if V=^0 when i/=^b, V^ — 1 when t/ = and X TT L , TT// J tan- The solution for the conjugate system, that is, for a strip having a source at (0, 0) and an infinitely distant sink is log cosh^ — ^ — cos^ -77- ■ TT L ^b zb _i 80 SOLUTION OF PROBLEMS IX PHYSICS. [Art. 48. 3. Show that if 7'= — ! when ^ = and .r<.(). r=l when // = and .r > . V^ — 1 when y=^h and .r < , and V^l Avhen y — h and ./'>0. 2 / TT 7r.''\ 2 / Ti" 7r>"\ 1'= - tan-i ( tan ^ (/' — y) tanh -77 I H — tan"^ I tan — // tanh -— I TT \ Jib Zb / IT \ lb ' 2b/ . TT.r 2 ,r^^^ ^1 = - tan-M ■ TT L . TTI/ J sm —r- b The solution for the conjugate system, that is, for a strip having a source and a sink at the points (0, 0) and (0, b) is TT.T Try -cosh — - + cos ^ . r -■ ■ * '■ TT 1- Tra' 7r//J 4. If 7'=0 Avhen .r = 0, 7' = /(.r) when //=:() and ,r>0, and V={) when // = Ji and a- > , V=^~ Cda p"iha(/;-//) ^^.^^ ^^^ _ ^.^ _ ^^^ ^^^ ^ .r)]/(X)^X TT.' ;;^ sinh aJ> = h -" t/T ^ ^ 1/W''^ 2// '' r L , TT^^ ^ 73-// 1 TT .. I X TT'/ cosh —(>^ — .'■) — cos -i- cosh —(AH- .<■) — cos -r- b b ^ b for positive values of x and for values of 1/ betweeen and b. 0. If J'l^O when .t = 0, Vi = F(x) when y =b and ./•>0, and Ti =: when ij = and .r > r, = J_ sin -^ f r ^ : 1 ~]F(X)dX . cosh — (A — y) + cos — ^ cosh — (A + x) + cos -j- for positive values of .r and values of .y between and b. 6. If Vo = Avhen ;r = , F2 = /(•'^) when y = and .*■ > , and F2 = -Z^(a:-) when // = b and .-:?;> 7; =7^+ 7^1 for .X- > and ^ ''^ ^ ^ Chap. IV.] FLOW OF HEAT IN ONE DIRECTION. 81 By the aid of the formula / , rir pTT cosh — cos -— COSh/JCC TT J,q Jy COS rx.dx = '^ =^ if P < 2? ; ITT 7 cosh qx ' q lyir . . rir ^ cos \- cosh — [Bierens de Haaii, Def. Int. Tables (6) 265] , reduce this to 1 , . ./Wco.h|(A-.) V= 7 sin -y j ^ dX. ' _^ cosh T (■^ — ■^j — cos -y- 8. If F=0 when ^ = or Z* and .r< — «, F=l when y = or b and — o<.r and x> a . 7r(a—.r) . 7r(r/ + ./•) smh — ^— smh — ^— F= - tan-i h tan-i -' ■ • TT L Try . TT// J sm -J- sm — ^ 9. If r=0 when ?/ = or /> and x'< — a, F=l when // = and — a < ,r < « , F= when // = or b and x > a, and F= — 1 when y = b and — ^/ < .r < a V- tanh^^^^ tanh^^^^^ ^ [tan- ^- + tan- ^—1 TT \_ ^ Try ^ TT?/ J tan -y- tan -j- 10. A system conjugate to that of Ex. 9 is F^ + oo when // = or h and .r = — (^? , V= — cc when ?/ = or b and a' = a . In this case . ., TT// . 7r(^f — .r) sm- -; — \- smh — ^— 2^ sin^:^^ + sinh-^^^^^> 49. Let us take now a problem in the flow of heat. Suppose we have an infinite solid in which heat floAvs only in one direction, and that at the start the temperature of each point of the solid is given. Let it be required to find the temperature of any point of the solid at the end of the time t. Here we have to solve the equation D,u = a'I)^^u (1) [v. Art. 1 (ii)] subject to the condition uz=f(x) when ^^ = 0. (2) 82 SOLUTIOX OF PKOIiLEMS IN PHYSICS. [Art. 49. As the equation (1) is linear with constant coefficients we can get a particu- lar solution by the device used in Arts. 7 and S. Let M = eP' + <" and substitute in (1). We get 13 = a-a^ as the only relation which need hold between /? and a. Hence u = e"-^ + ""-^"-' = e""-"" e'^' (3) is a solution of (1) no matter what value is given to a. To get a trigonometric form replace a by al. Then u = e-"'"" e"^. If in (3) we replace a by — at we get As in Arts. 7 and 8 Ave get from these values H =. e-"-"^-' sin ax and ii = e~"^°--* cos ax as particular solutions of (1), a being Avholly unrestricted. From these values we wish to build up a value of u which shall reduce to /(a-) when t = and shall still be a solution of (1). We have /(.r) = 1 Cda Cf{X) cos a (A — x).d\ (4) V. Art. 32 (3), and by proceeding as in Art. 44 we get u = l Cda Ce-"'''" f(X) cos a(X — x).dX (5) as our required value of u. This can be considerably simplified. Changing the order of integration u = 1 ff(X)dX fe-""'" cos a(X — x).da . (6) fe-"-"-' cos a(X — x).da = — ^- . e~ H^ (7) by the formula 00 J Ce-"'^' cos bx.dx = —■ e~Si [Int. Cal. Art. 94 (2)] Hence ic = p^|/(A)e T^dX. (8) Chap. TV.] EXAMPLES. 83 X — .r Let now B = p- , 2a\f then X = ,r + 2asJ}.^ and u = ^Cf{x-{-2a^t.(3)e-^'d/3. (9) EXAMPLES. 1. Let the solid be of infinite extent and let the temperature be equal to a constant c at the time ^ = . Then it = ^ Ce- ^' dfi = ~ A-- ^ y//? = . Int. Cal. Art. 92 (2). 2. Let u = X when t = . Then ^^ = ^ fc^' + 2asJll3)e-^'d/3 = x. 3. Let ?< = x'^ when t — :(). Then ?^^ = ■■x^ + 4. Let W : = if x<- -l>, U = 1 if hen t = 0. Then h /> •^avt n=^Ce-^^,B = UJi ^3 + 3,,. ,5+10.3.^ + 5^ _^_^^^ V^J "^ \/7rL2r/v/^ 3(2aV^^' 5.2!(2«v/^)5 J h + x 5. Let « = if .:*;<0 and u = l if a- > when ^ = 0. Then ~2 ^^L2a^t 'd.(2a^ty 5.2\(2a\Jtf 7.31(2(^0' J' 6. An iron slab 10 c. m. thick is placed between and in contact with two very thick iron slabs. The initial temperature of the middle slab is 100°, and of each of the outer slabs 0°. Required the temperature of a point in the middle of the inner slab fifteen minutes after the slabs have been put together. Given «^ = 0.185 in C.G.S. units. Arts., 21°. Q>. 84 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 50. 7. Two very thick iron slabs one of which is at the temperature 0° and the other at the temperature 100° throughout are placed together face to face. Find the temperature of each slab 10 c. m. from their common face fifteen minutes after they have been placed together. Ans., 70°.8, 29°.2. 8. Find a particular solution of D,u^ a'^D^u on the assumption that it is of the form n = T.X where T is a function of t alone and X is a function of X alone. 50. If our solid has one plane face which is kept at the constant tem- perature zero, and we start \vith any given distribution of heat, the problem is somewhat modified. Take the origin of coordinates in the plane face. Then we have as before the equation D, u = (v^l)^: a , (1) but our conditions are u = [) when x = (2) u=f(x) " ^ = (3) and we are concerned only with positive values of x. We may then use the form (4) Art. 32 f(x) = - (da I /(A) sin ax sin aX.dk , (4) and proceeding as in the last section w^e get u^=— ( da i e'"''^-'f(X) sin ax sin aX.dX (6) I-' as our reqiiired solution. This may be reduced considerably. II = - if{X)dX I e-"-°--' [cos a{X — x) — cos a{X + x)'\da, n = — ^ Cf{X) (e- ^i;;^' — e- ^-^^)dX (6> bhis may be reduced to the form 2a\/'7Tt;; by (7) Art. 49, and this may be reduced to the form 2av'« 2) ^ fe" ^^ dp . vV 7. If the earth has been cooling for 200,000,000 years from a uniform tem- perature, prove that the rate of cooling is greatest at a depth of about 76 miles, and that at a depth of about 130 miles the rate of cooling has reached its maximum value for all time. Let a^ = 400. 8. Show that if the plane face of the solid considered in Art. 50 instead of being kept at temperature zero is impervious to heat u = -^-= Cf(X) (e """-' + e" ~^^^ )cl\ . V. (6) Art. 50. 51. If the temperature of the plane face of the solid described in Art. 50 is a given function of the time and the initial temperature is zero, the solution of the problem can be obtained by a very ingenious method due to Riemann. Here we have to solve the equation I),u = a^D^u (1) subject to the conditions u = F(t) when x=^() ■> I (2) We know that X is a solution of (1), v. Ex. 1 Art. 50. It is easily shown that u = ^ Ce-^'dfi, (3) VttJ where c is any constant, is a solution of (1). For i» '» Vtt 2a>^t — c 2 1 ^r — r ~2 ' sItt 2a^f — c 4«'(^ — c) Of^ir and . D,v =^a^D:u. €hai'. IV.] TEMPERATURE OF FACE A FUNCTION OF THE TIME. 8< Let cji(_jc, t) be a function ot x and / which shall be ec|uai tu zero if t is negative and shall be equal to '-hf'-""' if t is equal to or greater than zero; so that if a' = () cp(,r, f) ^1 and if We shall now attack the following problem, to solve equation (1) subject to the conditions u = if f = n = F{{)) '' X = and < / < r u = F(kT)'' x = " A;t(x, t); if x = u = if ;' > (/.■ + 1)t and it = F(kT) if kT t. If, then, 71t is the greatest whole multiple of T not exceeding f. f: = « n = ^ F(kT)[^(x,f - kr ) - c^(.r, f - (k + 1 )t)] . (6) If now we decrease r indefinitely the limiting form of (G) will be the solu- tion of the problem stated at the beginning of this article. (6) may be written = XF(kr) ^ (■*^^-A^r)-<^(..,^-(^-+l)r) ^ (') 88 SOLUTION OF PIIOBLEMS IN PHYSICS. [Art. 51. and if r is indefinitely decreased the limiting form of (7) is u = - CF(\)D, (^ - A)". ; , 2«V7r and (8) may be written t ,(2 u = -^ CF(X)e~ '"'"-'' (f - X)~ I dX , (9) or if we let 2a\t — X EXAMPLES. 1. If /( = nt when ./■ = and u = when t = 2. A thick iron slab is at the temperature zero throughout, one of its plane faces is then kept at the temperature 100° Centigrade for 5 minutes, then at the temperature zero for the next 5 minutes, then at the temperature 100° for the next 5 minutes, and then at the temperature zero. Required the tem- perature of a point in the slab 5 cm. from the face at the expiration of 18 minutes. Given; cf^=.185. .1/^s'., 20°.l. 3. If It =^ F(t) Avhen ,r = and ii ^f(^v) when ^ = , then V. (6) Art. 50. Chap. IV.] TEMPERATURE A PERIODIC FUNCTION OF THE TIME. 89 4. If in Art. (51) F{t) is a periodic function of the time of period T it can be expressed by a Fourier's series of the form F(t) = - />„ + ^ [«,„ sin mat -\- b,j^ cos mat'] , where a = ^ , ,H= 1 or F(t) = ^ bo + X Pm s"i (">^f + Ki) , where p„^ cos A,„ =: a,,^ and p,,^ sin A„j = i,„. v. Art. 31 Ex. 3. Show that with this value of F(t) (10) Art 51 becomes H = ^ bofe-^^//3 + ^ X P,n [sin (mat + A,J J«-^= cos ^J r/yS — cos (m,at + A,„) | f~^^ sin "^^^ (7y8"l and that as t increases u approaches the value Given that Ce—' sin -, (7.^; = ^ e-'^^^-gin ^ 72 ; fe- '^ cos -, (/,/• = — e-*-^^" cos b V2. J .r'' 2 J x'^ 2 f V. Riem^ann, Lin. par. dif. gl. § 54. 0. If we are dealing with a bar of small cross-section where the heat not only flows along the bar but at the same time escapes at the surface of the bar into air at the temperature zero we have to solve the differential equation DfU = (I'W^ii — b~u . v. Fourier, Heat § 105. SlioAv that for this case a = e-^^- + "'""'>' sin ax and ;^ = g- "*' + "-^- " cos ax are particidar solutions, and that if k =_/'(,/•) when t = u = '-^ re~-^f(\)dX = ^' P'-^=/(.f + 2asft.^)dfi . -'^sJirtJ sIttJ cf. (8) and (9) Art. 49. 90 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 5L If u^O when .r = ajid u =f(-f) when ^ = u = ~p [_f''-^\f(^ + 2a\/f.f3)df3 -fe-P\fi- .r + 2a^t.f3)d(3~j . cf. (7) Art. 50. If n^= — e'~77 when f^O and )f=^0 when .<■ = and if u = l when .r = and u = when f == we have only to add e~f to the second member of the last equation, since u^e"^ satisfies the equation D, 11 = (iW^ u — h- II . If ft =^ F(t) when .« = and ii = when f = we can employ the method of Art. ol. 2aVt 3 ZhStt and u =-^7= r(^ - X)~2e-^^('-A)-,-;;i^^ i'\A)f/A , 2a VW cf. (9) Art. 51, 2aV^« cf. (10) Art. 51. If F(t) is periodic and has the value taken in Ex. 4, show that the value approached by u as t increases is 1 , '>.'■ "'^ xv/2„ . / , ■'>-s/2 , ^ \ " = ^ ^) '' « + 2^ Pm f'-'ir '' sm I mat— -^ y + A, J , where p = (//- + V'''' + »r-a-)^ and -/ = (— //^ + V/'* + '»"^a^)^ • Chap. IV.] ANGSTilOMS METHOD. 91 Given and where /^- - ^" 7 ^■^ -■',■ O 7 t;~'-~ ,2 COS — f/.r =^ -T- « cos Zrt , ll (^,,2 _|_ v/,,4 ^ /,4^i ^j,^i ,7 =, V2 ^_ ^^.. ^ ^^^^ ^ ^^^^1 Angstrom's method of determining the conductivity of a metal is based on the result just given (v. Phil. Mag. Feb. 1863), and is described by Sir Wm. Thomson (Encyc. Brit. Article "Heat'') as by far the best that has yet been devised. 52. If II is a periodic function of the time when ,/; ^ as in Art. 51 Ex. 4 and we are concerned with the limiting value approached by u as t increases we can avoid evaluating a complicated definite integral if we take the following course. , Since as we have seen in Art. 49 u = e^' + °-'' is a solution of J), II = ci^I); u (1) provided only that /3 = d-a- we have as a solution. Replacing /3 by ± /3I tins becomes or ir = e^^''^-^f^'^'^ v// = ±^n/2(1 + o. and V— / = ± 7^ V2 (^ T ''^ ■ Hence ., = e-t^^ sin (l3t - I yjl) , „ = ^^\ COS {pt - ^ ^1) ' (2) « = .:^lsin(/3* + ;^^f), „ = „;;V|„„,(^, + i^|), (3) are particular solutions of (1). (^) 92 SOLUTION OF PROBLEMS IN PHYSICS. [Akt From these we get readily xJma . / X \ma , \ " = /3„,^-'-„ *T sill {^waf — -\^r- + K,) as a solution. (4) reduces to n = p„, sin (maf + A,„) Avhen .r =: and to n = p„, g-f ^T sin ( A„, — - a / — ) when ?^ = . If we add a term which satisfies (1) and which is equal to zero Avhen x- = and to — /D,„^"o 2 sin ( A,„ — -\/— ;- j when f — ( v. Art. 50) we shall have a solution of (1) which is zero when t = and whicli is p,„ sin (mat + A,„) when x — O. The term in question approaches zero as f increases [v. (7) Art. 50] and we have at once the solution given in Art. 51 Ex. 4, as our required result. EXAMPLE. Show that t/^(^^t + '^-^ is a solution of J),v=z(rlJ;u — 1/n if /3='>-a- — f/^, and hence that ?<:^e~^siii where (f^f ± -^-) , and » = e^ 3l cos (tSt ± ^) , p = [v/y3^ + lA + ^>^]i and y = [v//3^ + />^ - ^--^ji , are solutions. Hence "-=/>V'-;Ssin(^^--^ + A„,) is a solution. If (3 = ma this last result reduces to ?« = p^ sin (waf + A^) when x = and by the reasoning of Art. 52 it must be the value tt approaches as t increases if we have the same conditions as in the last part of Art. 51 Ex. 5. 53. The whole problem of the flow of heat is treated by Sir William Thom- son (v. Math, and Phys. Papers, Vol. II), and other recent writers from a dif- ferent and decidedly interesting point of view, which we shall briefly sketch in connection with the problem of Linear Flow. Suppose we are dealing with a bar having a small cross-section and an adia- thermanous surface, and take as our unit of heat the amount required to raise by a unit the temperature of a unit of length of the bar. If at a point of the bar a Chap. IV.] INSTANTANEOUS HEAT SOURCES. 93 quantity Q of heat is suddenly generated the point is called an instanfrmeoiis heat source of strength Q. If the heat instead of being suddenly generated is generated gradually and at a rate that would give Q units of heat per unit of time the point is called a jyermanent heat source of strength Q. The temperature at any point of the bar at any time due to an instantaneous source of strength Q at the point x = X is easily found by the aid of formula (8) Art. 49 as follows: — If a quantity of heat Q is suddenly generated along the portion of the bar from .r = A to .r = x -(- AA, where AA is any arbitrary length, the tem- perature of that portion will be suddenly raised to -^ , and we shall have by (8) Art. 49 u=^^J>'^dX (1) as the temperature of any point of the bar at any time t thereafter. If now we write u equal to the limiting value approached by the second member of (1) as AA is made to approach zero we get n = e icr-t (2) as the solution for the case where we have an instantaneous source at the point ./• = A . It is to be observed that in (2) v = when ^ ^ and n = ' 2nsJ'Trf when X = A and fX). If we have several sources we have only to ad 1 the temperatures due to the separate sources. Formula (8) Art. 49 may now be regarded as the solution for the case where we start with an instantaneous heat source of strength f(X.)dk in every element of length of the bar. A source of strength — Q is called a sink of strength Q; and (6) Art. 50 may be regarded as the solution for the case where we have at the start an instantaneous source of strength /(A)cZA in every element of the bar whose dis- tance to the right of the origin is A , and an instantaneous sink of strength /(X)dX in every element of the bar whose distance to the left of the origin is A. If we have an instantaneous source at the origin (2) reduces to ('' _Zl /ON 2iiSTt 94 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 54. For a permanent source of constant strength Q at the origin (3) gives u = -^ ( e- i^^iTT^) (f - r)- 2 dr (4) 2a\7rJ and for a permanent source of variable strength f(f) u = —-= Ce-^^^,^ (t — T)-if(T)dT . (5> 2a\7rJ In (4) and (5) u obviously reduces to zero when t = and ./■ > , but its value when a- = is not easily determined. We can avoid the difficulty by introducing the conception of a doublet. 54. If a source and a sink of equal strength Q are made to approach each other while Q multiplied by their distance apart is kept equal to a constant P the limiting state of things is said to be due to a doublet of strength F whose axis is tangent to the line of approach and points from sink to source. A doublet of strength — P differs from a doublet of strength P only in that its axis has the opposite direction. Let us find the temperature due to an instantaneous doublet of strength P placed at the origin. For a source of strength Q at .v — rj and an equal sink at X = — f) we have (J (7)-J-)2 (r? + x)2 (6 4a2< 6 4a2< ) or if 2r]Q = P, 2a>f7ri P _ (r)2 + x2) rfX n^ j=e 4n2« (e2a2t — e 2a".l) ■ia-qSTTt P (t)2 + t2) . -nx -= e i^ST" smh -^ . 2arj>^'Trt ^a^ If 7} is made to approach zero "[^'"''II]=l4' limit -V and u = p= e-^, (1) is the solution for the temperature at any time and place due to an instantane- ous doublet of strength P placed at the origin. For a doublet at any other point a- = X we have P(X — X) (.r-A)2 ^. 4«V7r^^ if .r > , and Chap. IV.] PERMANENT DOUBLET. 95 For a penuanent doublet of constant strength P placed at the origin we have and for a permanent doublet of variable strength f(t) ^' = 7^7= re-i:fc) {i - ryKf(r)dr , (4) a x< 0, if we let /3 = y^^ • From (5) and (6) Ave see readily that u = Avhen ^ = and that M ='77^ when *■ = if we approach the origin from the right and that u = —^TT^ when x = if we approach the origin from the left. If the point a- = is kept at the constant temperature b and we are con- cerned only with positive values of x we can get from (5) the solution given in Art. 50 Ex. 4 by supposing a permanent doublet of strength 2a^b placed at the origin. To solve the problem treated in Art. 51 we have only to suppose a permanent doublet of strength 2a^F(t) placed at x = and from (5) we get at once (10) Art. 51. EXAMPLE. Show that if D^u = aW^u — li-u and an instantaneous source of strength Q is placed at x ^ \ Q h->, (•^ - ^)- . -^ 1^ ^ u=^ -,:=- e "■' 4„2( V. Art. ol, F,x. o. 2a\7rt Show that if an instantaneous doublet of strength P is placed at the point _ Px _,,_^ 96 SOLUTION OF FliOBLEMS IN PHYSICS. [Art. 55. If a permanent doublet of strength f(t) is placed at x^=0 -Jj-^f'-^'i'-^)"'' 2a>/'t f(f) whence t( = when ^=^0 and .r > or ,r < and ii^=--±'-~ when 'Ji/- x = 0. Hence if we place at ;?" = a permanent doublet of strength 2a'^F(f) we get the solution given in Art. 51 Ex. 5 for the case where u = F(t) when .r = () and t/ = when ^ =; provided we are concerned only with positive values of x . If F(t) = c this reduces to 2r, /^ b2x^ u = -= I e-^-~i^2 (113 . \7rJ 2aV'7 55. As another example of the use of Fourier's Integral Ave shall consider the transmission of a disturbance along a stretched elastic string. Suppose we have a stretched elastic string so long that we need not consider what happens at its ends, that is so long that we may treat its length as infinite. Let the string be initially distorted into some given form and then released ; to investigate its subsequent motion. Let us take the position of equilibrium of the string as the axis of X and any given point as origin. We have, then, to solve the differential ecjuation I)f!jz=a^D^l/ (1) [v. (vin) Art. 1] subject to the conditions 1/ =f{x) when t = (2) I),>/ = " t = 0. (3) As in Art. 8 we find 1/ = cos a(x ± at) and y = sin «('^' i at) as particular solutions of (1). From these we must build up a value that will reduce to f(x) = - Cda Cf(X) cos a(X — x).dX (4) Chap. IV.] INFINITE STEP^TCHED ELASTIC STRING. 97 when f^O and will at the same time satisfy (3). // = cos aX cos a(.r -\- at) -{- sin a\ sin a(.r -\- at) or // = cos a(X — .>• — at) is fi solution of (1). Hence // = - Cda Cf(k) cos a(\ — .r — at).d\ (5) is also a solution of (1). (5) reduces to // =/(.t) when t = but it gives i)^7/ = — Tr/a CflX.) sin a (A — ,r).d\ when t = and consequently does not satisfy equation (3). If in forming (5) we use cos a(x — at) and sin a(.r — at) instead of cos a(.r + at) and sin a(.r -f- at) we get ?/ == - Cda C/IX) cos a (A — ,r + at).dX (6) which is a solution of (1), and reduces to //=,/'(>) when t = , l)ut it gives Dtl/ = — — Cda Cfl\) sin a(A — .r).d\ w^ren t^O and does not satisfy (3). If, however, we take one-half the sum of the values of // in (5) and (6) we get /O.^/A + - Cda Cf(\) cos a (A — ,r + ^^O-r/A | . (") a solution of (1) which satisfies both (2) and (3), and is, therefore, our required solution. This result can be very much simplified. If we substitute ,-; = x -{-at - Cda ("/(A) cos a(\—x — at).d\ = - Cda Cf(\) cos a(X - z).dX =./U) =/(.r + .^0 ; TT./ »/ 98 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 5(5. and in like manner we can show that - Cda Cf(X) cos a(A — x + at).dX =f(x — at) . —00 Hence our solution becomes >/=lLf(:'-+"0+f{-'--'^t):\- (8) This result is of great importance in the theory of elastic strings and it shows that the initial disturbance splits into two equal waves which run along the string, one to the right and the other to the left, with a uniform velocity a, and that there is nothing like a periodic motion or vibration of any sort unless the ends of the string produce some effect. 56. If the string is not initially distorted but starts from its position of eqiylibrium with a given initial velocity impressed upon each point we have to solve the equation U^>/ = aW^l/ (1) subject to the conditions // = () when t = (2) D,y = Fi^x)- f-0. (3) We get by the process used in Art. 55 = :^Jf(X),JX j^pin_a^:X-. + ^^0 _ ^n. a(X-.r - nt) y^ , but *^ a '^ . a it, anc if X — at <.X<. X + at, and is equal to zero for all other values of A; since "^•^7.,._ - if "'>0 = if m = 0. V. Int. Cal. Art. 92 (3). x + at Hence y = ^CF{X)dX (4) X — at is our required solution. Chap. IV.] LONG RECTANGULAR PLATE. 99 EXAMPLES. 1. If the string is initially distorted and starts with initial velocity so that y=_/'(,r) and I),j/=F(.r) when ^ = v = \ UV + "^) +/(•'■ - "0] + 1; J^X^)^^ • 2. If the initial disturbance is caused by a blow, as from the hammer in a piano, which impresses upon all the points in a portion of the string of length c an equal transverse velocity b show that the front of the wave which will be seen to run to the left along the string will be a straight line having a slope equal to — and a length equal to — v'4''/- + Ifi . Of course a wave having a front of the same length with a slope equal to — — will be seen to run to the right along the string, and the effect of the two waves will be to lift the string bodily and permanently to a distance — above its original position. 57. We shall now take up a few examples of the use of Fourier's Series. In the problem of Art. 7 let the temperature of the base of the plate be a given function of x, the other conditions remaining unchanged. Since f(x) = ^ (a„^ sin m x) »i= 1 a =: — I f(a) sm ma.da IT J ?f = — ^ ^^~"''' sin mx I f{a) sin ma.da . (1) If the breadth of the plate is a instead of ir "=J,X\j " ''" "7" J • W sin -7^ f^^ J • (2) 58. If the temperature of the base is unity and the breadth of the plate is TT the solution is, as we have seen in Art. 7, ?f = - e"-' sin x-\- - e~ •''■" sin 3.r -|- - er'^" sin 5x + • • • . (1) 7r L o 5 A This series can be summed without difficulty. We have the development log(l + .) = ^-f + |-^+--- if the modulus of z is less than 1. Int. Cal. Art. 221 (4). where we have 100 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 58. Hence log (1 — -) = — -, -,2 ^3 .A if mod. .- < 1 . 1 and if mod. .i < 1 . But [log (1 +^) -log (1 -^j] =1 +:^ +^ + ... (2> log (1 + z) = log [1 + >'(eos <^ + ; sin <^)] = I log [(1 + ;• cos <^)-^ + (>■ sin = I log (1 + 2r cos ck + >) + / tan- 1 . ' '"' ^ ^ , J ^ 1 -j- /■ cos cfi and log (1 — --■) = - log (1 — 2r cos + >'-) — / tan-^ ^'^!" ' [Int. Cal. Art. 33 (2)], and (2) becomes 1 ri - 1 + 2y cos cf> + r' . . 2r sin ^H 2 L2 ^^^ l-2>-cosc^ + .- + ' ^'-^'^ T^J ^cos <^ + / sin <^) >-^(cos 3<^ + >' sin 3<^ ) i *" 3 (3) From (3) we get two equations 1 , 1 + 2r cos <^ + /•- >' cos cf> r^ cos 3<^ /-^ cos 5c^ . . 4 ^^ 1 - 2r cos <^ + r' ~ ~^ ^ 3 ^ 5 ^ ^^ 1 , ,2/' sin d) r sin by .r in log [1 + /'(cos <^ + / sin <^)] it becomes - log [1 + e"'' cos x + ' ''""" sin .r] or log [1 + cos z + i sin z] V. Int. Cal. Art. 35 (3) and (4) a function of « as a whole; and log [1 — 7'(cos (f) + I sin <^)] becomes log (1 — cos .-; — i sin ,-) ; hence by Int. Cal. Arts. 209 and 210, 1 , 1 + 2e-'-' cos cc + e--^ , 1 , %r " sin x T h)g :; -; j T and - tan~^ — -r- 4 ^ 1 — 2e-^ cos £c + e--^ 2 1 — e--" 1 , cosh 1/ + cos X . 1 ^ , sin a; or - log z-^ and - tan ^ -^-j — - 4 cosh t/ — cos X 2 smh y are conjugate functions, and 1 , cosh ?/ + cos X .^. «i = -log ^-^^ (0 TT cosh ^ — cos X is the solution for the problem where the isothermal lines are the lines of 'flow of the present problem and the lines of flow are the isothermal lines of the present problem. * For our problem, then, the isothermal lines are given by the eqviation o tan sni X IT sinli or -^-j — == tan -— («> smli // 2 and the lines of flow by 1 , cosh >/ + cos X — log 7-^ := b , IT cosh // — COS X cosh If + cos X r-^ = e'^6 . cosh u — cos X EXAMPLES. (9) 1. If D;n -\- D^ii = , and » = 1 when [/ = , and u = when = and when x = a , 4 r wj, . TTX , 1 ■ATT,, . OTT.J' , 1 ,577,, . 57rX ~\ a^— \ e a sm r 7^ ^ » sm \- i^ & « sm [- ■ • TT L (to a o a J 2 - tan~^ vr . , Try smh -^ l-K'-l SOLUTION OF PHOBLEMS IX PHYSICS. [Akt. 59. 1'. Ji' f -- ^'" ~r sm I <^(A) sm dX + i sin !^- fr 1 ^ 1/-(A)rfX 2(1 (I J \_ , TT TT.*- , IT . 77./; ^ (I cosh - (A — If) — COS — cosh - (A. + //) — cos — -" + r^ -" - rr — ' '- >(«"^ ■ TT "-cosh - (A — v) + COS — cosh - (A + y) + cos — V. Art. 48. Exs. 4, o, and 6. 59. If three sides of a plane rectangular sheet of conducting material be kept at potential zero and the value of the potential function at every point of the fourth side be given; to find the value of this potential function at any point of the sheet. To formulate: — D;r+i)/r=o. (i) V=i) when ./■ = (). (2) 7'=0 " :r = a. (3) 7'=0 - y = h. (4) y=f{^) " 2/ = <'- (5) Working as in Art. 48 we get sinh — - (/) — //) 'I . mirx , sm . , imrli a smh a as a value of F which satisfies equations (1), (2), (3), and (4) if m is an integer. Therefore F= - y -— sm I /(A) sm dk (6} „, = 1 smh " a is our required solution. Chap. IV.] RECTANGULAR PLATE. 108 EXAMPLES. 1. If f{x) = 1 Eq. (6) Art. 59 reduces to sinh - (A — ij) sinh — (J> — //) 4r «. . TT.r ,1 (( . Sttx I —-\ -, sm ho T—, sm TT L . 1 TTO a 6 . . OTTO (l siiin — smn « a smh — {(> — >/) + i ^^ sm 1 2. If r=0 when x = 0, V=0 when cc = a, V=0 when ?/ = 0, and F = F(x) when // = b , then m=co smh a F== - V r sm ( FCk) sm d\ • a -^ I • 1 ^" ''■'^ « J ^ a J 7n=i smh a 3. If i^(cc) = I the answer of Ex. 2 reduces to . , irii . , Stt?/ . , Stt?/ smh -^ smh — ^ ^ smh — - „ _ _^ 4 r ^/ . TTX ,1 O . OTT.X , 1 a . OTTX , \ ' = - 7 sm ho 5-7 sm h i Tl sm • "sinh — sinh ■ sinh a a a 4. If r=0 when x=^0, F=0 when x=za, V^f(x) when y = 0, and V^F(x) when y^h, then m = Qo gml^ /^ _ ^A a F= - >, sm { I f(\) sm dX m = 1 smh t> smn " V V -. sinh 5. If f(x) = F{x) the answer of Ex. 4 reduces to .cosh 7i^ / A _ \ ^ ii V2 ^/ . niTTX r^^. . mirX , H = - >, sm I f(X) sm dX a''~i\_ . vnrb a J' ^ ' a J cosh — - — 2a 104 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 59. 6. If /(.r) =: F[.r) = 1 the answer of Ex. 5 reduces to V=z- \ sill h .. :r—, ^"^ TT L , -TTh a 3 , S-rrb « cosh -— cosh — — cosh^- (^ — I/) . _, + ^ ^^ sm . O , OTTO a J cosh — — 2a 7. If r=/(a') wlien // = 0, V=F(x) when // = /^ r=/) when x=za, then smh — - (b — If) r=-X sm ( ^ )/(A)sin- m=\ smh " a . . miry smh " ^ _, sinh a smh — — (ft — x) sinh , h . , m.iTX smh + -7^>)^^"T'"^)] sinh , b 8. If /(.t) = <^(^) = and i'^(a') = x(//) = 1 the answer of Ex. 7 may be reduced to sinli 7 (7; — .^' ) . cosh ^ (- — .*• ) V=-\ TTT- sm ^ + o :;^ sm -t^ TT L 26 . , TTff 6 2 , lira b sinh— cosh 2^ 3 . , 37ra 6 4 . 4,7ra smh^_ cosh — 47r// "1 b J Chap. IV.] FLOW OF HEAT IN A SLAB. 105 9. Find the temperature of the middle point of a thin sqviare plate whose faces are impervious to heat; 1st, when three edges are kept at the tem- perature 0° and the fourth edge at the temperature 100°; 2d, when two opposite edges are kept at the temperature 0° and the other two at the tem- perature 100°; 3d, when two adjacent edges are kept at the temperature 0° and the other edges at the temperature 100°. See examples 3, 6, and 8. A71S., (1) 25°; (2) 50°; (3) 50°. 60. Let us pass on to the consideration of the flow of heat in one dimension. Suppose that we have an intinite solid with two parallel plane faces whose distance apart is c. Take the origin in one face and the axis of X perpendicular to the faces. Let the initial temperature be any given function of x and let the tAvo faces be kept at the constant temperature zero; to find the temperature at any point of the slab at any time. We have to solve the equation I),n = r>^D^u (1) subject to the conditions ;, = when ,r = (2) ;^ = " ./• = « (3) u =f(x) '•' ;- = . (4) In Art. 49 we have found u = 6~ """■' sin ax and i( = e-"-'^-' cos ax as particular solutions of (1). {(=6-"^'^'' sin ax satisfies (2) Avhatever value is given to a. It satisfies (3) if a = provided m is an integer. Let us try to build a value of ii out of terms of the form Ae- " "U sin — ^ whieli shall satisfy (4). We have /(.,.) = fX [sin ^^//'(X) sin =-\/x]. (S) »i = 1 U ?< = - 2^ e -r- sm —-^ J /"(A) sm —— <1X \, (6) n, = 1 reduces to (5) when t ^Q and is our required solution. 106 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 61. EXAMPLES. 1. If f{X) —I>. a constant, (6) Art. 60 reduces to 4:b r (r-n"-! . ir.r 1 >M"-n''-t . Sttx . 1 257^^1 . Bttx , ~\ u = — e" c2 sm [- 7; c~ c2 sm \- ~ i~ ^^ sm h ' " ' • TT L c 6 CO (■ J 2. An iron slab 10 cm. thick is placed between and in contact with two other iron slabs each 10 cm. thick. The temperature of the middle slab is at first 100° throughout, and of the outside slabs 0° throughout. The outer faces of the outside slabs are kept at the temperature 0°. Eequired the temperature of a point in the middle of the middle slab fifteen minutes after the slabs have been placed in contact. Given ^2 = 0.185 in C.G.S. units. Ayis., 10°.3. 3. Two iron slabs each 20 cm. thick one of which is at the temperature 0° and the other at the temperature 100° throughout, are placed together face to face, and their outer faces are kept at the temperature 0°. Find the tem- perature of a point in their common face and of points 10 cm. from the com- mon face fifteen minutes after the slabs have been put together. Arts., 22°.8; 15°. 1; 17°.2. 4. One face of an iron slab 40 cm. thick is kept at the temperature 0° and the other face at the temperature 100° until the permanent state of tem- peratures is set up. Each face is then kept at the temperature 0°. Required the temperature of a point in the middle of the slab, and of points 10 cm. from the faces fifteen minutes after the cooling has begun. Ans., 22°.8; 15°.6; 16°.T. 61. If the faces of the slab treated in Art. 60 instead of being kept at the temperature zero are rendered impervious to heat, the solution of the problem is easy. In this case we have to solve the equation subject to the conditions l)^u = {) when .r = y, =/(.r) " ^ = 0. We have only to use the particular solution as we used u = e "-°--' sin ax in Art. 60. We get u = I [^pWrfX +X(.-'^ COS '^pm cos '^ " = , S(' "^^ '"' ^ 2, j /(^) Suggestion: Assume u=^() when x = 2 c niir ^ mir 108 SOLUTION OF PEOBLEMS IN PHYSICS. [Art. 63. and ?/., = -^yl ^ ^ e ^^ siii ) • (o) TT ■^ \ m c / Hence ?;j = y _ -j- - > / :^^- — ^ g- ^2 sm — — ) • (' ) If the left-hand face of the slab considered in Art. 60 is to be kept at a constant temperature jS and the right-hand face at the temperature zero we can get the term n^ which must be added to the second member of (6) Art. 60 by replacing y hy /3 and x l)y c — a: in (7). We then have r— ■--'"f(i.-^si„=^-)i L C TT -^^ \iH C / _| (8) EXAMPLES. 1. Show that -if n ^ (3 when .7- ^ . v =^ y when x = c, and i'=f(x) when t=^0 + f X(.-=^' .sin '^fim -ffl sin '^ ,/a). m = \ n 2. ShoM' that if if = fi when x = i) , u = {) when f = 0, and D^^h^O when X ^= c = /? 1 {(- 4.-2 sm —- -j--e- ^^ sm h t: «~ 4c2 sm — [-•••) • 63. If the temperature of the right-hand face of the slab just considered is a function of the time instead of a constant and the temperature of the left- hand face is zero the problem can be solved by a method nearly identical with tliat of Art. 51. Chap. IV.] TEMPEHATUllE OF 0:NE FACE VARIABLE. 109 Let (ti(x,t) be a function of x and t which shall be zero if f is less than zero and shall be equal to fi—l)'" vfl,r-irh . mirx^ — e — ';r~ sin [v. (7) Art. 62] if t is equal to or greater than zero. So that <^(.r,0 = if t < <^(.T,^)=0 " ^ = unless x = r. ^(x.f) — 1 " f = and ,r = c <}>(x,f) = 1 " ,r = r <^(.r,0 = '' ;r = . Precisely as in Art. 51 we get A- = (I *- as the required solution of our problem, n being as in Art. 51 the largest integer in - where t is any given value of the time. On our hypothesis the last term of (1), that is, — F(nT)(^[jx,t — (n + 1)t] = 0; the next to the last term F(^nr)c})(x,f — vt) has for its limiting value while as in Art. 51 the limiting value of the rest of the sum is -CF(\)D;,(x,f-X),/K. A*(.,, t-x) = ^-^"X[(- 1)°'"-- ^'"-" -■• T] ■ Henci "=-»[r+;X(^'-T)] -^ Zi\i— 1)'"'"^ sm -^J F{k) e ^ <'-^> dk\ , 110 SOLUTIOX OF PROBLEMS IN PHYSICS. [Art. 63. ^ T, ^ , -x-^r( — 1)'" • "^"'•^ / 7-1^ N If we substitute f3 = -^ — (f — A) we get « = f ^(0 + S^" -' "^ ('■"> -f'-' i' - i:^')'"^)] • <^> EXAMPLES. 1. If the temperature of the left-haud face is a function of t and the tem- perature of the right-hand face is zero and the initial temperature is zero " = (i-f>(0-g[isi.T(-(0-/-'-('-^»]- 2. If the temperature of the left-hand face is a function of f, the initial temperature is zero, and the right-hand face is impervious to heat ^ , 4.^-\r 1 . ('2in 4- l)7r.v / ^ . 3. If in Arts. 60-63 we are dealing with a bar of small cross-section and of length c and heat is radiating from the surfaoe of the bar into air at the tem- perature zero so that I>,u ^a^D;v — IJ^u, show that: (a) the second mem- bers of (6) Art. 60 and (1) Art. 61 must be multiplied by e~^'^ \ (h) equation (7) Art. 62 becomes . , bx ismh— m=i ' ) Chap. IV.] VIBRATION OF A STRING FASTENED AT THE E:SDS. Ill (/') equation (2) Art. 63 becomes siuli ~ Fit) + 3„VX { ,4+,i,,v' =" — L^» sinh a F^/«-'*^"f^'"->nA),/A]} 64. The problem of the motion of a finite stretched elastic string of length I fastened at the ends and distorted at first into some given curve j/^=f[.r) , and then allowed to swing, has been treated and partially solved in Art. 8. The complete solution is easily seen to be l/ = j2^sin —j- cos —j—^ ./(A) sm —j- dX . (1) in=l The second member of (1) is a periodic function of i having the period 21 —. The motion, then, unlike that in the case of an infinite string (Art. 55) is * 2/ a true vibration, a perio(Jic motion. The period ~ is the time it takes a dis- turbance to travel twice the length of the string (v. Art. 55). A careful examination of (1) will show that the actual motion is a good deal like that in the case considered in Art. 55. The original disturbance breaks up into two waves one of which runs to the right until it reaches the ^nd of the string and is then reflected, and runs back to the left or the under side of the string, while the other wave runs to the left and is reflected at the left- hand end of the string and runs back to the right under the string and is again reflected, runs back to the left over the string and so on indefinitely. If the curve into which the string is distorted at the start is of the form y=^b sin — y^ the solution is , . w^TT.T imrat ,_, // = h sm --— cos — - — • (2) Xo matter what value t may have the curve is always of the form . . viirx y^A sm —J- ; that is, for different values of t we have a set of sine curves differing only in the amplitude and not at all in the period of the curve. In this case either the whole string if m = l, or each mth of the string if m is not equal to one, rises and falls, and there is no apparent onward motion. When this is the case we are said to have a stead// vibration. 112 SOLUTION OF PROBLEMS IN PHYSICS. [Aim. Hi. If m = 1 we get steady motion of the string as a Avhole and if the vibration is rapid enough to give a musical note the note is said to be the pure funda- mental note of the string. If vi = 2 the vibration is twice as rapid as when w = I , the middle point of the string does not move and is called a node, the two halves of the string are in opposite phases of vibration at any instant, and the note given is an octave higher than the fundamental note and is called its pure first harmonic. If ,11 = 3 the vibration is three times as rapid as in the first ease, there are 1 21 . tAvo nodes a.' = - and .r = — , and the note is the pure second harmonic of o o the fundamental note. For any value of m the vibration is m times as rapid as when m =1, there are ut — 1 nodes at the points .r = — , .r ^ — , • ■ • x^ /, and we get the jii ni m m — 1st harmonic of the fundamental note. It is clear from (1) that no matter what the original form of the string the resulting vibration can be regarded as a combination of steady vibrations each of which alone would give the fundamental note of the string or one of its harmonics, and that the complex note resulting is really a concord of the fun- damental note and some of its harmonics. A finely trained ear can often recognize in a complex note the fundamental note of the string and some of its harmonics and is capable of analyzing a complex note into its component pure notes precisely as Fourier's Theorem enables us to analyze the complex function representing the initial form of the string into the simpler sine-functions which must be combined to form it. EXAMPLES. 1. 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 player's finger to a distance 11 b from its position of equilibrium and then released, the form of the vibrating string at any instant is given by the equation 2hn' -^ / 1 . mTT . mirx m,'jrat\ TT~^ Zr ( ~ s^^^ — s^^^ ~r~ c^s — 1~) n — IjTT- -^^ \/»' n I I / 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. Chap. IV.] VIBRATION OF A STRING IN A RESISTING MEDIUM. 113 2. If a stretched string starts from its position of equilibrium, each of its points having a given initial velocity, so that we have 1/ = when ;' = D,;/ = F(x) " f = y = " .r = I , the solution of the problem of its vibration is easy and gives »( = x / 2^/1 . viiTX . m7r((t f ^^. . viirX , \ y = ~X{-, «"^ — «^^^ -j-j ^(^) ^"^ -T ""V ■ 3. Write down the solution for the case where the string is initially dis- torted and each point has a given initial velocity. 65. If we do not neglect the resistance of the air in the problem of the vibration of a stretched string the differential equation is rather more compli- cated and the solution is not so easily obtained. The equation is given as (ix) Art. 1. Let us solve the problem for the case where there is no initial velocity. Here we have Df>/ + 2kD,u = l6 c2 sin I \f{\) sm dX I m—l (I EXAMPLES. 1. If /(;■) = b (8) Art. 66 reduces to n^b and there is no change of temperature. 2. If the initial temperature is constant and equal to /3 1 I -'' .o 7N r --% • -^^ 1 ^-'^% • 27r>- + 3e-^'sm^; J. Chap. IV.] COOLING OF A SPHERE IN AIR, 117 3. An iron sphere 40 cm. in diameter is heated to the temperature 100^ centigrade throughout; its surface is then kept at the constant temperature 0°. Find the temperature of a point 10 cm. from the centre, and find the tem- perature of the centre, 15 minutes after cooling has begun. Given (r = 0.1Sr} in C.G.S. units. Ans., 2°.l; ;r.3. 67. If instead of having the temperature of the surface of the sphere constant, the sphere is placed in air which is kept at the constant tem- perature zero, the problem is much more complicated. For in this case the surface temperature can no longer be simply expressed but is given by a ucav differential equation D^ ic -\- h II. = when r = c , (1) where h is an experimental constant depending upon what is called the sur- face conductivity of the sphere. Our equations, then, are D,{ru) = a^DXru) (2) ^t'=f{r) when ^ = (3) D/ii-\- hu^Q when r=^c. (4) As in Art. 66 let /; := rii ; then we have I),v = a-D;v (5) V = rj\)') when ;■ = (6) v = " r = (7) D,r + (h — ^) " = when r = c. (8) y? = g-"^"^' cos ar and /? = e~"'""' sin ar have already been found as par- ticular solutions of (5) (see Art. 60). V ^ e~ "'"■'' iiiw ar (9) satisfies (7) for all values of a. Substitute this value of v in (8) and we have ac cos ac + (lie — 1) sin ac = 0. (10) If Uj. is a value of a which is a root of the transcendental equation (10) v^e- "'v' sin a^r (11) will satisfy (5), (7), and (8). It remains to see whether out of terms of the form given in (11) we can build up a value of v which will satisfy (6). 118 SOLUTION OF PROBLEMS IX PHYSICS. [Art. 67. "When / = the second member of (11) reduces to sina^.r. If then we can express \/(^") as a sum of terms of the form 1/f.ii'ui a^.r where a^. is a root of (10) *' = 5) ^'i-- ''~ "'''^' '^^^^ "-k '' (1- .^ will satisfy all of the equations (5), (6), (7), and (8), and will be the required solution. Here, then, we have a new problem analogous to that of developing in a Fourier's Series, but rather more complicated, namely, to develop any function of .T in a series of the form ^o„j sin a„j.-y where a„j is a root of the equation (11); or if Ave call ar = and he — 1 =7^ Avhere a,„ = — ^, <^,„ being a root of the equation <^ cos 4> -\- p sin <^ = (13) or more simply of <^+yy tan = (); (14) remembering that the series and the function must be equal for all values of x between zero and c. If <^^ is a root of (14) — <^,„ is also a root. Since sin —^ x = — sin | '" .r ) the terms of the required development which correspond to negative roots may be combined with those corresponding to positive roots, and therefore we need consider only positive roots. ^ = is a root of (14) but as sin ^= there will be no corresponding term in the development. If we construct the curve and the curve y=^--x (15) y = tan X (16) the abscissas of their points of intersection are values of x which satisfy - + tana- = 0, that is, are roots of equation (14). It is easy to see that there will always be an infinite number of real positive roots, one for each of the branches of the periodic curve i/ = tan x which lie to the right of the origin. The numerical values of these roots can be obtained by an easy com- putation. The construction suggested above shows that as in increases <^^ will rapidly approach the value (2»i — 1) — if ]) is positive or if 2^ is negative and numerically less than unity, and (2?» -\- 1) — if j^ is negative and numer- ically greater than unity. Chap. IV.] COOLING OF A SPHERE LN AIR. 119 There exist, then, an infinite number of positive real roots of +i> tan <^ = and consequently of ac cos ar + (//'• — 1) sin ac = . 68. The development called for in the last article can be obtained very easily from a simpler one which we shall now consider, namely, to develop f(x) into a series of the form f(x) = (1-^ sin <^i,r + .,, <^a • • • are roots of the equation ^ cos <^-{- j> sin /.x sin 0,„.r.r/.e , J"sin^>,„, x.dx 120 SOLUTION OF PKOBLEMS IN PHYSICS. [Art. 68. I sin <^^.ic sin (f)^x.dx = 7, i [cos (<^^. — <^,„)a' — cos (<^^. + (li„^)x^dx = 1 F sin (<^, — c^,„) _ sin(<^,. + «^J ~1 ^ k COS 4>k Si^^ ^m — ^,n Sm «^^. COS (ji^ But 4>k COS <^^. -{-p sin <^^. = and <^„, cos <^„, +p sin ,„ = by (2). Hence the numerator of the second member of (4) is zero, and the coefficient of a^. vanishes if k is not equal to ))i. fsin^ ci>,„x.dx = ^ [,„ - sin <^,„ cos <^ J = ^ fl - '^^^1 ' (5) 2 sin 2<^,, 2 r Therefore «„, = ^^ — ^r—- I f(x) sin c{>„^x.dx . (6) 1 The coefficient of the integral in (6) can be transformed as follows so as not to involve trigonometric functions. ^ni cos <^,„ + J) sin <^,„ = , by (2) <^,„ cos'^ <^„, + tJ sin 2<^,„ = , sin 2c^,„ cos^2gin2^^^ (<^,„2+^2)cos2<^^=_p2, (7) (8) 7^ „r+/>0> + l) and «. = ^l^p^^;^ sin ^„^a.da . (9) Therefore our required development is Chap. IV.] COOLING OF A SPHERE IN AIR. 121 From (10) it easily follows that for values of x between and c /(■'') ^^ ''i si^ ^1*' + ^'2 sin a^x + ('3 sin a^x + • • • (11) '^here "« = f ,,"+j;f+l) PW -n -..MIX, (13) and a,„ is a root of the equation ac cos ar + p sin ac = . (13) It is to be observed that if p is infinite (13) reduces to sina6'=:0, a„j becomes —^ and (11) and (12) give our regulation Fourier sine series (v. Art. 31), and therefore the ordinary Fourier development in sine series is merely a special case of the problem just solved. Moreover since the Fourier method of .determining the coefficients of such a series requires that I sin a„^x sin a,iX.dx ^= , that IS that sin (a, - Of _ sin (a + a> _ ^ a« — a« «,« + «» T . ^, , a„,r cos a..,c a„r, cos a„c or recuicmg, that -^. — = -^ ~ , sm a„/! sm a„e or that a„j and a„ should be roots of the equation ac cos ac —. =p sm ac Avhere ^j is some constant, it follows that we have obtained in (11; the most general sine development that can be obtained by Fourier's method. EXAMPLES. 1. Show that the solution of the problem of Art. 67 is •u = ^^'',«e~"^V' sin a,„r , T , 2 a,ff2+(Ac— 1)2 r . . . ''''''' '- = c- aP+^(A.^ J^^">"""-'-^ and a„ is a root of ac cos ac + (Ac — 1) sin ar = . 122 SOLUTION OF PROBLEMS IN PHYSICS. [Art. (J8. 2. If the initial temperature of the sphere is constant and equal to /3 ru = X ^'m '' "'"'"' si^^ <^m '■ wliere b„ = 2/3A- .,"' .\^ Vx ^ ^ 2/3Ar [a,^r--' + (he - l)-^]7 3. If the temperature of the air is a constant y instead of zero the surface equation of condition is Z>,, u -{- h (u — y) ^ when r = e . The substitution of u^ = v — y, however, brings the problem under Ex. 2 and we get where i"~y)=X b„,e~"''^m sin a^r '- = r „TfT*ji„-z:ij/^[/W - V] -" "..M^- 4. An iron sphere 40 cm. in diameter is heated to the temperature 100° centigrade throughout; it is then allowed to cool in air which is kept at the constant temperature 0°. Find the temperature at the centre; at a point 10 cm. from the centre; and at the surface; 15 minutes after cooling has begun. Given «2 = 0.185 and ^' = ^ in C.G.S. units, (v. Ex. 3, Art. 66.) Ans., 96°.46; 96°.16; 95°.26. 5. Show that if in the slab considered in Art. 60 one face is exposed to air at the temperature zero, so that we have DtU = a^D^u, « = when x = 0, n=f(x) when t = 0, and D^.)/ -\- /n/ =^0 when x^c, then w = ^a^e" "'">«' sin a„^x m = l where «m = 2 —„ — r^i-Ti — r^r I tW sin a,„ X.dX, a-c-i- Jl(/tr-\-l)J' a,„ being a root of ac cos ae + /^c sin ae = . Chap. IV.] TEMPERATURE OF AIR VARIABLE. 123 <■>. If in the problem of Art. 57 heat escapes from one side of the plate into air at the temperature zero so that we have I); ii -\- />,; u = , ?f = when x=^0, ti^f{.c) when // = , and D_j.ii -\- hu ^0 when ,r = a, then u == ^''m^^"'"'' sin a„,.« where '>n = 2 -^-^^^_ j:/'(A) sm a„Ml^ , a„j being a root of aa cos aa + ha sin a« ^ . 7. If in the problem of Art. 59 there is leakage at one side of the sheet so that we have D;V -\- Dp^=(), F = Avhen x = {), F = when t/^h, V^=f(x) when i/ = 0, and Dj,V -\- hV— when x = a, then sinh a,,^^ where «,„ has the valne given in Ex. 6. 69. If we have an infinite solid with one plane face which is exposed to air at the temperatures U = F(t) and heat can flow only at right angles to this face, we can solve the problem readily for the case where the initial tem- peratures are zero. We have I),v = aWlu subject to the conditions u = when ;' = and Dj.v-\- ]t(^U — ii) = Q when a- = 0. Let i Then v will satisfy the equation Let v = u — J D^u. (1) and we shall also have v = U when x = , Since U=FCt) '- ^/^"^^^(^ - J^^^/^ ' (2) by Art. 51 (10). Hence V. Int. Cal. § 4, page 314, D_^u — hii = — hv by (1). ue-''-' = — h fe-''-^ vdx + C ; 124 SOLUTION OB^ PROBLEMS IX PHYSICS. [Akt. 70. Determining C by the fact that ue~''-'' = when .r = cc we have u = he''''' Ce-'^'vdx . (3) Substituting the vahie of r from (2) we have as our required solution. For an extension of this method to the flow of heat in two and three dimen- sions and for the interpretation of the results by the aid of the theory of Images, see E, W. Hobson, Proc. Lond. Math. Soc, Vol. XIX. EXAMPLES. 1. If the temperature of the air is a periodic function of the time, say p,„ sin (uiaf + A.„,) and we care only for the limiting value of u as t incre show that this value is ^ip,»'''V^ r/, ,1 jma\ . I X \ma , \ 1 Ima / X \ma ^ \~\ Art. 52 and Art. 51 Ex. 4. C . , , e'"'(aQU].hx — bcoshx) that I '"" sm hx.ax = ^ ., . ,., J «- + b- s hx + a' + Ir , r , , €"''((( cos hx 4- b sin bx) and I e""" cos bx.dx = — ^^ V. Int. Cal. Table of Int. (235) and (23t3). 2. If D^.V-^D;jV=Q>, V=0 when ?/ = and D^^V-\- hlF{y) — V^ = when a? = show that V= — f e-"-^ dx CF(>.)dX f-rn-^ y, - o , ;^ , J ; TT J ^ L.r-+(A — ^)^ a-+(A + //)-J V. Art. 47 Ex. 1. 70. The solution for an instantaneous heat source of strength Q at the point X ^ A if heat escapes at the origin into air at the temperature zero, so that Brii — hu=^0 when a- = , can be obtained by the aid of Art. 53. Chap. IV.] TEMPERATURE OF AIR ZERO. 125 Let u = Ui + »2 where u-^ is the temperature that woukl be due to the given source if we had no boundary at the origin, so that ?^i = — j=- G- ^-^. [Art. 53 (2)] 2a\7rt D^.u — Jilt = Dj.Ui — livi -\- D,.i(-2 — hi(.2 = when x* = . Therefore £>, u. — h u. = — (Z/,. u^ — hu^) (1) when a' = . But , _ sfrrf when X = 0. This is easily seen to be the vahie to which reduces when a' = , and this last expression is 2aS/7rt and therefore satisfies the equation D,u = a^D^-tc; (2) Q (\ + X)2 since - j=. e ia-.t is the temperature due to a source at x = — X. If, then, we determine u^ from the condition that T^ . 7 X Q /X-\- X , \ (k + xfl !>,(,. _/„,,) = --i^^(^- 7.),-^ (3) taking care not to introduce any arbitraiy constant or arbitrary function of t in our integration, lu will satisfy equation (2) and condition (1). Integrating (3) [v. Int. Cal. § 4, page 314] and determining the constants of integration suitably we get Q r (A + xy „, . /* . (A + xyi -1 "' ^ ^^t L^""^- 2^^^"y ^-'-^^ dx^ . (4) Therefore the solution of our problem is n = -^^[e-^^-{- e-'-^- - 2he'- f.--"^^^ dxl . (5) 2aV7r^L J J 126 SOLUTION OF PROBLEMS IN PHYSICS. [Art. 7L If we replace Q by f(X)dX and integrate from to co we get as the solution for the case where u=f(x) when ;; = and a- > , and Dj.u — hu^=0 when aj = u = ^^^Cf(\)dX p- ^^ + e- '-iSf^ - 2he"-Je~ "—'-^ dx~] (6) For an interpretation of this result by the theory of Images and tlie extension of the method to the conduction of heat in n dimensions see G. H. Bryan, Proc. Lond. Math. Soc, Vol. XXII. EXAMPLE. Show that if u=f(x) when t = and D^u + h[F(t) — u] = when .!■ = we must take w equal to the sum of the second members of (6) Art. 70 and of (4) Art. 69. 71. As another problem requiring a slight extension of Fourier's Theorem let us consider the vibration of a rectangular stretched elastic membrane fastened at the edges, that is of a rectangular drumhead. If two of the sides are taken as axes and the plane of equilibrium of the membrane as the plane of XY the equation for the motion of the membrane is r^z = c%i)^z + i>,fz) (1) see [x] Art. 1. Let the membrane be distorted at the start into some given form ,- =f(oc, y) and then allowed to swing. Our equations of conditions are then (2) (3) (4) (5) (6) (7) We can get a particular solution of (1) by our usual device. Assume and substitute in (1) . We get y- = e-(a"-^ + /3^) as the only relation that need hold between a, yS, and y, in order that ;s =:r e »-^ + ^^ + v' may be a solution. This gives Therefore ^ _ g a. + ^^ ± c^ V^IT^ is a solution of (1) no matter what values are given to a and /3. ,t = when x = i) z = X = a z = y = o z = y = b z=f{x, .2/)" t = ,.v = t = Chap. IV.] VIBRATION OF A KECTAKGULAK DKUMHEAD. l^-^ Replace a cind (3 by at and /3i and we have as a solution, and from this we get z = sin (a.r + |8.v ± r'f Va'^ + /3^) and cos (aa- + j8y ± '-'^ Va- + /S") (8) .(9) as particular solutions of (1), a and /8 being unrestricted. From (8) and (9) we can get solutions of the foUov/ing forms = sin ax sin ygy sin cf \a-^ + (i^ = sin ax sin y8^ cos ct \a^ + ^- = sin ax cos /Sy sin ct \a'^ -\- (3'^ sin ax cos /3^ cos c;^ \a^ + yS* s; = cos ax sin ;8y sin ct \a^ -\- ^" z = cos ax sin /?^ cos ct ^a^ + ^- s: = COS aa; cos /Si/ sin c^^ Va- + yS" (10) s; = COS ax cos /3// cos e?^ Va" + /3"^ each of which will satisfy equation (1). The second of these will satisfy also (2), (4) and (7) whatever values be taken for a and ft. It will satisfy (3) and 7ii7r mr (5) if a and ft are equal and — respectively. If, then, we can so combine terms of the form . imrx . mrii sm • sm —r- cos cirt a h V: as to satisfy (6) our problem will be completely solved. This can be done if we can express f(x, y) as a sum of terms of the form ^ sin sin— —=-, the sum and the function being equal when .'■ lies between and ct and y between and h. f(x, y) can be expressed in terms of sin — regard y as constant. We have by Fourier's Theorem ii \\ - A^, y) = % '^m sill "^ (11) 128 SOLUTION OF PEOBLEMS IN PHYSICS. [Akt. 71. where «m = - ff(^^ V) sin ?^ dk . (12) f(X, y) in (12) is a function of y and may be developed by Fourier's Theorem. We have f{\, y)=^ b„ sin ^ (13) h where K = j ff(>^, /*) sin '-—^ dfx, . (14) Substituting for /(A, y) in (12) the value just obtained we have «« = -J a(J ^^^J /(^' f") sm ^^ sm -^ df^j sm -^ m=l and Hence '^ = XX ( ^"''^ ^^^ ~V ^^^ "^ "^^^ ^'^'^ \ -^T + ^ ) ' ^^''^ n!=lH=l a h where ^,„_„ = — | f/A | /(A, /i) sin - — sin ——- dfx . (17) is our required solution. EXAMPLES. 1. Show that if the membrane starts from its position of equilibrium but Avith a given initial velocity impressed upon each point so that z^O when /» = and DfZ = F(x,y) when f = the solution is z = —y\ y,{A,„ „ , sm sm —r^ sm cirt J!!L^L\ + b^ -here ^-" = ^/'^/^(^' ^> ^^^^ "^ '''' T '''^ Chap. IV.] RECTANGULAR DRUMHEAD. 129 2. If there is both initial distortion and initial velocity ^ — -jZj Zj Sill sm -— A,„ „ cos rirt V — + jr + ^',n » sm <( if x = - , or — , or — • • III III III (hi — 1)>I ■ or III a 2a the lines .r = — , .<■ = — , • III III z =^ for all values of f, and i-emain at rest during the Avhole vibration and are nodes. The same thing is true of the lines h 2h ?>h (II — l)l> y = -, 1jr= — ,,/ = — ,■■■ ,j= . a II n. II 73. If the membrane is square it may have much more complicated nodes than if the length and breadth are unequal, as in this case the period of any term of the general solution reduces to o„ (1) and there will in general be two terms having the same period, and a musical note of the pitch corresponding to that period may be produced by initial cir- cumstances that bring in both terms. Thus . imrx . niri/r ^'^^ ,/~^t -^ i r> ■ '"^'' »/~^rn :,"! z ^= sm sm — - J,„_„ cos \iir -[- ir -f- /),„_„ sm - — • S/nr -\- ir CC (X \ it Ct J , . IITTX . mTTI/ r . CTTt / — 3— .^ , „ . ^'TT^ /^ r,~l + sm sill ~ A„ „, cos ■ — \iir -j- ir + J^,, „. sm — Snr -\- ir a a L « « J Chap. IV.] NODES OF A SQUARE DEUMHEAD. 181 is a form of vibration that will give a musical note. Let us write this '"^^ J~~^^ -iV A ■ '^'''^■'' ■ """'/ 1 7, ■ ^'''■•^' • W'TTV"! z = cos — \ III- -]- i)^\ A sm sm — - -\- B sm sin 'I L (( a a a J I • '^"^^ J~^r~\ — ?r ^i ■ ^Ji'^-x^ ■ nirii . ^^ . 717TX . i)i7ri/~\ + sm V"' + n C sm sm — '- + D sm • sm (2) (I L (' (( 11 'I A and in studying the forms of musical vibration of which the membrane is capable we may take A, B, C, and D at pleasure. Consider the simple case where ^4 = C and B = D; then (2) reduces to / , . 7mrx . mry . ^, . nirx . miriiX/ cirt ,— r-; , z^\A sm • sm y B sin sm )| cos — Vw + ;r \ a a a a f\ a , . CTTf I p- \ + sm — S III- + II- J . (3) Values of X and y that will reduce the first parenthesis in (3) to zero will cor- respond to points of the membrane remaining motionless during the vibration. Let us consider a few cases at length. (ft) If VI = 1 and n^l , the first parenthesis in (3) becomes (A + B) sin — sin '^ , ^ a a which is equal to zero only Avhen x = i) or y^O, or .r = <■/ or ?/=»'/, that is. for the four edges of the membrane. If, then, the membrane is sound- ing its fundamental note it has no nodes. {I)) If ni = 1 and /^ = 2 , we have . TTX . Iirij . 2'Trx . irii A sm — sm + P> sm sm ^ = a (I a a to give the nodes. TTX 27r// ri Let i? = 0, then sin — sin — ^ = 0, which is satisfied by >/=-: and ft a ^^ -^ '' 2 in addition to the edges the line y = -, is at rest and is a node. If J = a- = - is a node. li A = B ^ 2'Try . . 2'jrx . iri/ — - + sm sm — ^ = TTX . Try TV , o • '"■•^' TT'^' • TT'/ /^ 2 sm — sm -^ cos -^ + 2 sm — cos — sm — ^ = ft a a a a a • ■"■// / "T^// 1 '^^\ nn — ^ ( cos -^ + cos — I =^ . ft \ ft ft / . TTX sm — sm 132 SOLUTION OF PEOBLEMS IN THYSICS. [Art. 73. The first factor gives the four edges of the membrane. The second Avritten equal to zero gives 'Try TTX / ■7r.i\ — = — cos — = cos I vr I a a V a / TTIj TTX a a ;r + ^ = a, which is a diagonal of the square. If B = -A . TTX . 27n/ sm — sm — - a a sm sm — ^ = which is the other diagonal of the square. Other relations between A and B will give Trigonometric curves of the form Try B TTX cos ^^ ^ -. COS — (I A a which are easily constructed and Avhich obviously all agree in passing through the middle point of the square. We give the figures for a few of the cases Chap. IV.] NODES OF A SQUARE DRUMHEAD. 133 (c) If m = 11 = 2 we have (J + B) sm sm — ^ = to give the nodes, which are merely the lines a a X — - , and 11 = 7,- This form gives the octave of the fundamental note. {d') If la = 1 and u = 3 Ave have , . irX . StTI/ . „ . OTTX . TTII A sm — sm — =- + B sm — — sm — ^ a (I, a a to give the nodes. If A = we get If B = Ave get If A = — B Ave get a , 2a = - and x=:—- o o // = 3 and y = -- . . TTX . StTI/ . OTTX . TT?/ sm — sm — '■ sm sm — ^ = (1) (2) rx . 77)/F — sm — ^ ^ a a \_ a J TT// irx ,^ cos^ — cos^ — =: a (I ( TT// '^^\( Try , TT.rX ( COS -^ COS I ( COS -^ + COS — ■ ) = V (/ a J \ a a / If A=z B we get .? — // =^ and ./• -\- ij=i a . TT// TT.r 1 cos-^ 1- COS^ ^ - (3) COS '- -\- COS = — 1 , (4) a Trigonometric curve easily constructed. For other relations between A and B we get more complicated Trigonometric curves coming under the general form A cos — - + B cos = — A + B (5) 134 SOLUTION OF PROBLEMS IN PHYSICS. which all agree iu containing the points (a (i\ (a 2a\ /2a a\ , /2a. 2a\ MISCELLANEOUS PROBLEMS. I. Lofjarithmic Potential. Polar Coordinates. 1. Show that D^V-{- D;jV=() becomes if we transform to Polar Coordinates. 2. If in we let V^ R.<^ we get ^^ A cos a(^-\- B sin a<^ whence (1) ^ = A^ cos (a log /■) + Bi sin (a log r) V^ — cos a^ r= — sin a< V= e"-'^ cos (a log r) J'= e«'f> sin (a log /•) r= e-"* cos (a log r) r= e-""^ sin (a log ?■) J^= cosh ac^ cos (a log ;•) Y = cosh a) when r = a V= - l\, + X(~)"V^« *^°^ ""^ + -''» ^^^^ "'^) ^^^' '' *^ "■ ni=l and ^— T7^'o + X(~) (^'m cos 7«<^ + a,„ sin ?m^) for r>a, where ^>^ = - Cf(4>) cos 7n(f>.d(f) and o„j = - r/'(<;f)) sin ??;^.(/<^ 136 MISCELLANEOUS PROBLEMS. 4. Show that if F satisfies (1) Ex. 2 and V—f(r) when ^ = and r>0 V = ^ ^ cosh - (A - log r) = - sin ^ I /-(eM — —^ d\ TT 2 J • ^ ^ cosh (X — log /•) — cos 1/^/./ >N 7. /^cosh a(7r — d)) 1 . 5. If F=l when (^ = and <;■1 ( sm^ ) 2v//-.sin^ -" 6. If r=/(r) when cf> = and r=0 when cf> = (3 V=- ff(e^)clX n^^^\(l^-J ^ cos a(A - log .)•-) when <^ = and r = -^ 2 n — r- ~I r= - tan-' ,-— — :, ctn • TT Li + r- J TT 10. If V—0 when r = 1 , F= 1 when = . T'= 1 when = tj bau-i — r-^ — — • \_2r- sm 2(i J TT L2r- sm J(^. 11. If V=f() when r=^^, r=0 when <^ = 0, and V=0 when 7 = 2^a,,, l-J n sm — — - i± r > <: where «,„ = - Cf\) sin ^^^ (/<^ and < < ;8. 12. If V=^ /() when >■ = «., F= when r = h, r'=0 when ' and < (/> < ^ m=l rt p — b P where a„, = -J /((^) sm — - f/<^ . 13. If V^F((I>) when )' = b, r=() when r = a, V^O when (^ = 0, and V=0 when <^ = i8, then if a < >• < ^> and 0< <^ < ^ ^=X{ ^ — = lU"- 17.) ' J "» ^■" -;8- \ 3 where «„j ~ ^ I ^(^) '''^^^ — o~ '^'^ ■ P»y P 14. If V=xQ') ^''l^en <^ = 0, F=0 when c/)=r^, r=0 when r=a, and r= when /• = ^», then if a — loar a . mirilo^ r — log a) '} '^ — Zji «m ^^ ^-- sm — T^ — —^ \ ^ i. . , mir log b — log a ) log Zy — log a 138 ]VnSCELLANEOUS PKOBLEMS. log- where «,„ = . 7 -, | x(^'<^ ) sm , • d.r . log /> — log a J '^^ ' log h — log a 15. If V=if/()-) when <^ = y8, 7^=0 when <^ = 0. 7'=0 when ;-=:r/, and F= when r = b, then if a <)- < /3 r=X snih ^ log i — log a . «i7r(log > — log a) | sm 1 '" . , VITT log /y — log a S smh log /> — log a log 2 /* DITTOC where a™ = :; : ^ I ib(ae^) sin , ; ^ dx. log/> — log r/J ^^ log/> — logrf II. Potential Function in Space. 1. Show that /(■-^, y) = -\ fda Cd(3 Cdk Cf(\. fi) cos a (A — .r) cos ^(/x — j/).d/x, for all values of x and //. 2. Find particular solutions of D^V-\- D,JV-{- D^J^^=0 in the forms T^= e * .-^a= + P^ cos (ax ± jBi/) r= e * = ''«' + ^"- sin (ax ± (Sij) V= sinh z SI a- + p-. sin (a.r ± /?//) r= cosh s: Va' + /?-. sin (aa* ± y8//) &c. 3. Given D;V-\- D^'^ -\- D;V=0 , and V=f{x,y) when ,t':=0, solve for positive values of z. Remit. 1 Cdx C ^/"(^^z^)^^ . 4. Confirm the result of the last example by showing that if f(x, y) is inde- pendent of y Y^ 1 r zf{\,ti)dX .,. Ex. 3 Art. 45). irjz^+(k-xy ^ POTENTIAL FUNCTION IN SPACE. 139 5. If D^ F + />,; r + I); / ' = , and V = 1 when ,~ = for all points within the rectangle bounded by the lines a- = «, x = — a, >j^=b, and yz= — h; and F=0 when ,v = for all points outside of this rectangle, then ., ,._ b-U f TT 1 (a - xy\h - yf - z^j,, - aif + (b - yf + z^^ -'^ '' - s/(^r^^'^ 1 2 "^ 2 ''^ {a - xy(h - yf + ,.^[(« - x)' -^ (h ~ y^ + z^-] 1 ._, (a + -rfih - yy- - z^a + .-)^ + (^ - !/Y + .-^] -t- 2 (,, + ,rf(J^ - yy + .^[(« + ;,.)2 + (^, _ y)2 + .V^] . _1±J^ t TT 1 0^, - ■t)'^(^> + y)-^ - z\{a - a-y + ( b + yy + zT^ ^ ^j:fyy 12-^2' (a - xy(b + yy + zX(a - xy +(b + yy + z^] , 1 ., {a + ^ryU, + yy - .^[(. + .r)^ + (^ + Z/)^ + ^^] 1 ^ 2 ■ (a + .'■)^(^ + uT + ^TC' + .'f -^{b + yy + ^^] ^ if — ^; < X < rt, and ■iTT >- — ^= "1 sin -^-7 ^-j — T— — , — — — ; — — yj(b - yy I (« - -^yi^ - II)- + ■"[(« - ''•;- + i!> - yf + -'] - .in-> (« + ^y{b - yy - ^{a + ^0-^ + {b - yy + ^^] (« + '^n^ - yy + ^T(« + -^o' + c^' - yy + •^'] '^iw+w '^ ('' - ''y^^' + •'^)' + '^'[(^ - '-y' + (^' + '^y + '^'^ (.. + xy^b + 7/)^ + .-T(« + '^O"^ + (^ + y)-^ + ^^jr if a; < — « or x > « . 6. If the value of the jDotential function V is given at every point of the base of an infinite rectangular prism and if the sides of the prism are at potential zero the value of V at any point within the prism is Tr "i -^ X-*' „- \J^^"- + "^ • m'rrx . miry r .^ f,^^^ , . mirX . mr/x , ,« = 1 ,( = 1 If V=l on the base of the prism this reduces to . (2m + l)7ra; . (2n + l)7r?/ m= .. „= X sin ^ '- — sm ^^ TT-' ^ ^ "' "- (2w^ + 1) (2» + 1) »i = n = ^ 7. If the value of the potential function on five faces of a rectangular parallelopiped, whose length, breadth, and height are a, b, and c, is zero, and 140 MISCELLANEOUS PROBLEMS. if the value of V is given for every point of the • sixth face, then for anj- point within the parallelopiped smh 7r(c — z) \— + - V= > \,Ann — sm sm — - where J,„ „ ^— \dk \f{\, y) sin -^- — sin —j^ ^A*- 8. If the value of the potential function is given on two opposite faces of a rectangular parallelopiped and is zero on the four remaining faces, then within the parallelopiped n^^n^ sinh7r(c-,.)^-+- . viTTX . ntni sm sm - ■ . , \m- , n- (^ b smh 7rc\\^ -\- -- ^ ri- I- ^here and m = :r.n = ^ Smh 'TTZ V— + — Xx-v T^ * « '> . niTTX . nirii S ""'•' ~, — W^' '" ^ "" ^ m= 1 H= 1 smh ire \ — -\- — ' a- Ir a b An,n = ^ J ((^ I f(K HO sm —^ sm —^ rf/^ ^w,H = —, I ^^-^ I ^(^j h) sm sm —-^ dfi . 9. If the value of the potential function is given at every point on the surface of a rectangular parallelopiped, what is its value at any point within the parallelopiped? III. Conduction of Heat in a Plane. 1. Find particular solutions of I)fU = a-(I)^u -\- I),ju) of the forms u = e- "'("' + ^"->' sin (ax ± j3i/) u = e~ "'''"' "^ ^'^' cos (ax ± (Si/) . 2. Given the initial temperature of every point in a thin plane plate, find the temperature of any point at any time. CONDUCTION OF HEAT IN A PLANE. 141 3. For au instantaneous source of strength 4^ at (A, /a) '^ = -. r e 4n2, V. Art. 5-1 47ra-f For an instantaneous dovMct of strength P at (0, /a) with its axis perpen- dicular to the axis of Y PX .r^+(M-.v) ^ . ^ ^^ f ia2t V. Art. 54. 87ra4^2 For a permanent doublet of strength P at (0, /x) with its axis perpendicular to the axis of Y — JL --^ _.d+Oi^iO! If the strength of the doublet were Pdjx and the heat were uniformly generated and absorbed along the element dfx of the axis of Y beginning at (0, ix) we should have P _ a-2 + (|^-y)2 xdjJL P ^^(,u.-vf , ^ , t^— If « = 7) — "2 ^ 4a2« o I , To = ri — -o e i^^s;^ f? tan-' , and since d tan"' — ^ is the angle ARA\ where A and A' are the points (0, (x) and (0, /u, + r/yu.) and P. is the point (■*',//), ?^ == when .ri= unless P fjL <. 1/ < fi -{- dfi, in which case « = ^^Tj if x approaches zero from the positive side ; and u = when t = except in the element d/x. If then ti^O when t = and n =f(y) when x = we have only to suppose a doublet of strength 2a\f(x)dx placed in each element of the axis of Y and then to integrate; we get ttJ X- -\- (/x —>/)- For a permanent doublet of strength F(f) at (0, /x) we have II ^ 1 r xF(0) ^2+(^_,,^2 A xF'(t) .^+(^-,f- -| 142 MISCELLANEOUS PROBLEMS. From the reasoning above this must be zero when t=^0 except at the point (0, fi), must be 2a^F(t) at the point (0, fi), and at every other point of the axis of Y when t is not zero. Hence if m = when t = and u = F(i/, t) Avhen ./• = ttJ :,- + (/* - >/)- "^ ttJ O ./■- + (/i - i/Y "" ^' '' For an extension of this solution by the method of images to the case where there are other rectilinear boundaries and for its application to the correspond- ing problems in the flow of heat in three dimensions see E. W. Hobson in Vol. XIX Proc. Lond. Math. Soc. 4. If the perimeter of a thin plane rectangular plate is kept at the tem- perature zero and the initial temperatures of all points of the plate are given, then for any point of the plate n = j-^2^ 2_i. 0<,?' sin (ax ± (3>/ ± y.^) u = e-a-(a^- + P' + y'>t cos (ax ± (Sy ± yz) . 4 Given the initial temperature of every point in an infinite homogeneous solid find the temperature of any point at any time. = -3 Ce-^' d/S fe- y' dy Ce- ^'/(x + 2>>\ft.f^ , // + 2a>Jt.y , z + 2as[f.^)dB . 5. If the surface of a rectangular parallelopiped is kept at the temperature zero and the initial temperatures of all points of the parallelopiped are given, then for any point of the parallelopiped U=2^2^ A-'^m,n,p'i "'"'(t where ^,„,„,p = r^ j f/A I d^x j /(A, /w, v) sin — — - sin ^^ sin-^ sm -—— sm — - sm-'-— - bed plTV dv. 6. An iron cube 40 cm. on an edge is heated to the uniform temperature of 100° Centigrade and then tightly enclosed in a large iron mass which is at the uniform temperature of 0°. Find the temperature of the centre of the cube fifteen minutes afterwards. An>i., '.\^°A. 7. An iron cube 40 cm. on an edge is heated to the uniform temperature of 100° and then its surface is kept for fifteen minutes at the temperature 0°. Required the temperature of its centre. Ans., 9°. 5. CHAPTER V. ZONAL HARMONICS. 74. In Art. 16 we obtained z = Ap^(x) + Bq^(a-) (1) [v. (6) Art. 16] as the general solution of Legendre's Equation (1 - •^') X7. - 2^^ £ + M'"'^ + !)- = <> (2) dx'^ ' d,x m, being wholly unrestricted in value and x lying between — 1 and 1; where ^„(,, = 1 _ "-C'^+l) ,» + ».0«-2)(m +!)(« + 3) ^. m(m - 2)(,» - 4)(«, + !)(<« + 3)(»,. + 5) , 3: -t" (^o; 6! and . . _ ^. (m - l)(m + 2) ., (». - \){m - 3)(m + 2)(;» + 4) ,' and we found 3! ' 5 (m - l){m - 3)(w — ^){m + 2)(m + 4)(/;^ + 6) . 7! ^' V= r'^q^,^ (COS 6) + •••; (4) (P) 'J m being unrestricted in value, as particular solutions of the special form assumed by Laplace's Equation in spherical coordinates when V is independ- ent of ^; that is, of the equation rD::{r V) + ^^ D, (sin 6 De V) = . (6) * Before reading this chapter the student is advised to re-read carefully articles 9, 10, 13(r), 15, 10. and 18(r-). SURFACE ZONAL HARMONICS. 145 For the important case where m is a positive integer we fonnd . = .4P,,(:.)+iH>,„(.'') (7) [v. (10) Art. 16] as tlie general solution of Legenclre's Equation (2), whence F=r'"P,„(cos^) ■ V= -\-, P,„ (cos 0) V =>'"%)„, (COS 0) V= -^ Qm (cos 6) are particular solutions of (6) if m. is a positive integer. p i,.^ - (2^^^-l)(2>M-3)-- X V _ JM^ (8; 1) !(2m-l) m{in — 1) (rii — 2) (m — 3) ^^^ "^ 2.4.(2m — 1) (2?/i — 3) ■^' •] (9) [v. (8) Art. 16] and is a finite sum terminating with the term which involves X if m is odd and with the term involving .r° if mi is even. It is called a Surface Zonal Harmnnic, or a Legendre's Coefficient, or more briefly a Legendrian. ^™^' ^ (2>N-\-l)(2m-l)---lLx"'-'^ 2.(2»;, + 3) x'" + ^ (>»+l)(m-^2)(m + 3)(m + 4) 1 "^ 2.4.(2///. + 3)(2;» + 5) a."' + 5-^ if .r< — 1 or a->l. [v. (9) Art. 16.] It is called a Surface Zonal Harmonic of the second, kind. •] (10) Q,n(:r) = (-!) — .d and U^) = (-i)' r(»^ + 1) 2»[r(| + i)] ^ 2 2.4.6. • • • (/// — -2 P>n(^) 1) 3.5.7. •• • /// [v. (13) Art. 16] if ///. is odd and — 1< .r < 1 . r(/// + l) Pr»(.^) (11) K^')! [v. (14) Art. 16] if m is even and — 1 < r < 1 . (12) 146 ZONAL HARMONICS. [Art. 75. In most of the work that immediately iollows we shall regard x in P,Jx) as equal to cos ^ and therefore as lying between — 1 and 1.* 75. In Article 9 the undetermined coefficient (% of x"^ in Pn(x) was arbitrarily written in the form ~ ^" , '-^ for reasons which shall now be given. In Articles 9 and 16 z = t',n{^) "^^^^ obtained as a particular solution of Legendre's Equation ^^ ~ ""'^ S ~ ^"^ !l + "' ^"^ + 1) .^ = (1) by the device of assuming that z could be expressed as a sum or a series of terms of the form a^x'^ and then determining the coefficients. We can, how- ever, obtain a particular solution of Legendre's Equation by an entirely differ- ent method. The potential function due to a unit of mass concentrated at a given point (^i> Vi, «i) is V= , ^ (2) N/(.r - x,f + (^ - VxY + (.^ - .n)^ and this must be a particular solution of Laplace's Equation Dl V + Z>; V + D/ r = , (3) as is easily verified by direct substitution. If we transform (2) to spherical coordinates using the formulas of transformation X = /■ cos B y:=r sin $ cos <^ s = r sin 6 sin ^ we get V= , ^ (4) y/,.2 _ 2/Ti[cos e cos $1 + sin 6 sin 0^ cos(<^ — <^i)] + 'f as a solution of Laplace's Equation in Spherical Coordinates rD';{r V) + ^l~ A(sin 6 A V) + ^^ Dl V= [xiii] Art. 1. If the given point (xi, y^, z^,) is taken on the axis of X, as it must be that (4) may be independent of <^, ^i = , and V = —=2== (5) V'"^ — 2/Ti cos 9 -f rf * English writers on Spherical Harmonics generally use n in place of x for cos d. We shall follow them, however, only when we should thereby avoid confusion. Chap. V.] PARTICULAR SOLUTION OF LAPLACE's EQUATION. 147 is a solution of rI);'(rV) + ^.^- De(sm 9 D, V) = . (6) Equation (5) may be written .. 1 (J) («) V 1 — -.V cos 6 + .V-' is finite and continuous for all values real or complex of It is double-valued but the two branches of the function are distinct except for the values of z which make 1 — 2," cos 6-\-z- = namely z = cos 9 + / sin 9 and ,-.' = cos 9 — / sin 9, both of which have the modulus unity and which are' crifirt/l values. , is finite and continuous except for the values of V/I - 2z cos 9-\-z' z =z 009. 9 — i sin 9 and ,v = cos ^ + / sin ^ for which it becomes infinite; it is double-valued but has as critical values only these values of .-.-. It is then holomorjyjdc within a circle described Avith the origin as centre and the radius unity, and can be developed into a power series which will be convergent for all values of z having moduli less than one. (Int. Cal. Arts. 207, 212, 214, 220.) If then r > i\ — = can be developed into a convergent series '^'" cos^ + ^; v-^ involving whole powers of — . Let z^Pm ~7i ^^6 this series, ^„j, of course, being a function of cos 9. Then [v. (7)] is a solution of (6). Substitute this value of Fin (6) and we get As this must hold whatever the value of r provided r > )\ the coefficient of each power of r must be zero, and hence the equation must l)e true. iA/^&"^' 148 ZONAL HARMONICS. [Akt. 75. But as we have seen in Art. 9 the substitution of .r = cos in (9) reduces it to ^^ - "'^ S™ - -^' ^ + "'^"' + 1) ^- == 0' and therefore ^^J'„i is a solution of Legendre's Equation (1). If /• < /'i — ^ can be developed into a convergent series a/i — — cos ^ + -, \ )\ r{ involving whole powers of — • Xj.»i ''i j9„j — be this series. Tlien (v. 8) is a solution of (6) ; substituting in (6) we get whence it follows as before that ^ = Pm is a solution of Legendre's Equation. But 7_>,„ is the coefficient of the mth power of — in the development of (1 — 2 — cos 6 -\ — -„ I 2 according to powers of — , or of the mth ijower of — in the development of ( 1 — 2 - cos ^ + -yj 2 according to powers of -^ , or more briefly it is the coefficient of the mth power of z in the development of (1 — 2xz + z^)- 2 according to powers of z, x standing for cos 6. (1 - 2,r.. + z^- h = \l- z(2x - z)y h and can be developed by the Binomial Theorem ; the coefficient of z^ is easily , picked out and is (2m-l)(2m-3)---l r _ ,,^ _ m(m-l) ml [_' 2 (2 III — 1) + 2)0»-3) ^.,„_, . 2.4.(2//; -1)(2;«- 3) J But this is precisely -P,„(a"). [v. Art. 74 (9)] Hence P^ix) is equal to the coefficient of the mth power of .- in the development of [1 — 2xz + «^]" i into a power series, the modulus of z being less than unity. Chap. V.] PROPERTIES OF SURFACE ZONAL HARMONICS. 149 76. If x = l P„,(,x) = l. For if x = l (l-2xz + z^)-h reduces to (1 — 2z -\- z^)- 2 that is to (l—z)-^, which develops into H-^ + ^^+^3 + ^^ + ---, and the coefficient of each power of z is unity. Therefore ^„.(i) = i- a) We have seen that if m is even P„^(oc) contains only even powers of .r and terminates with the term involving cc°, that is with the constant term. The value of this constant term can be picked out from the formula for P„^{x) [v. Art. 74 (9)]. It is (— l)^ ' t' a c^^'.^ 7 5 or it can be found as follows: — It is clearly the value P„iQic) assumes when a" = 0; it is, then, the coefficient of z"^ in the development of (1 + .~-)~ 2^ ; but and the coefficient of «'", vi being an even number, is ( — 1)2 -^-^- ^— ^ . ^ 2.4.6 • • • ni If m is odd Pj,^(x) contains only odd powers of x and terminates with the term involving x to the first power. The coefficient of this term can be picked out from (9) Art. 74 and is (—1) ^ ' '-—; or it can be 2.4.0. ■ ■ ■ (i)i — Ij found as folloAvs : — It is clearly the value assumed by — "'^' when x = . It is, then, the coefficient of 5;'" in the development of 3 . (1 + z )v ^ = .. - .. I 3-^ ... 3-5-T ., , (l + «^)l ■' 2" ^2.4''' 2.4.6'' ^ m-I 3 5 7 • • ■ III and the coefficient of «'" in this development is (— 1) 2 ' ' — III being an odd number. 77. To recapitulate: „ , 1.3.5 • • • (2???. — 1) r m(7n — 1) m(m-l)(m-2)(;M-3) "^ 2.4.(2?» — l)(2w — 3) ■ m(iii — 1)(7» — 2)(//^ — ?-,)(m — A)(m — 5) ,„_,;, ' 2.4.6. (2m — l)(2m — 3) (2?;^— 5) ''"''" " ^ . (1) (3) 150 ZONAL HARMONICS. [Akt. 77. m being a positive integer, is a Siirface Zonal Harmonic or Leyendrian of the mth order. It is a finite sum terminating with the first power of .r if m is odd, and with the zeroth power of x if m is even. P,„(.r) is the coefficient of the mth power of z in the development of (1 — 2xz + z^)~h into a power series. Hence if s < 1 (1 - 2.r.t + .-^)-i = Po{x) + P,{x).z + P,(x).z' + P3(.r)..t3 + P,{x).z^ + P,(.r).,.^ + ■ • • + P,.(.r)..^'" + ■ • • . (2) Whence -^====i===^ = 7 r A(cos 6) + ^; P,(cos ^) + 5 P.3(cos ^) + ■ • • V'" — 2r}\ cos ^ + ' 1 •- ' ' + '^'p,„(cos^>+- --I if /■> 1 r r r^ = - Po(cos 0)+- Pi (cos 6) + — P,(cos 0) + • + ^' P„,(cos ^) + • • •] if r<, ^ = P„,(.r) is a solution of Legendre's Equation '^ -■'>£- --"I + "'('" + !)* = « when in is a positive integer. r= y'»P„^(cOS e) and F=^^;^^P,„(cos^) are solutions of the form of Laplace's Equation in Spherical Coordinates which is independent of , namely rD;'(rV) + ^^I)e(»iriODgV) = 0. (4) P>na) = l- (5) A,«(--'')=^2«(^-)- (6) P.n,^(0)=0. (8) P.(0)^(-r)- ^-^;f-,;::^:-^> . (9) Chap. V.] TABLE OF SURFACE ZONAL HARMONICS. \- dP....(^^ l ^ . .yn 3.5.7. •• •(2m + 1) L d-r J, = o 2.4.6. •••2m 151 (10) For convenience of reference we write out a few Zonal Harmonics. They are obtained by substituting successive integers for vi in formula (1). T\.(.r} = 1 P,(.T)=i(3x^-]; ^(•^■)-=o(35.r^-30,r^ + 3) F^(x) = - (63.rs - 70.^3 + 15a;) P,(,r) = ^ (231.T« - 315.T'' + 105.r2 - 5) • F,(.r) = ^ (429.1-" — 693x^ + SlS.r^ — 35.x) Pg(a-) := ^ (6435a;« - 12012,r6 + 6930,r'* - 1260,r-^ + 35). Any Surface Zonal Harmonic may be obtained from the two of next lower orders by the aid of the formula (n -f 1)P„^ 1 (x) - (2u + l)xP, (x) + nP„_,(x) = (12) which is easily obtained and is convenient when the numerical value of x is given. Differentiate (2) with respect to z and we get (11; whence (l-2xz + z')^ {^-'■^) P,(x) + 2P,(^).z + 3P,(x).z^ + Hence by (2) (1 - 2xz + z')(P,(x) + 2A(a-).^ + 3Ps(x).z'+ ■ ■ •) -\-(z- x){P,(x) + Pi(.xO-- + P-2{x)-^-' + •••)=() (13) 152 ZONAL HARMONICS. [Art. 78. (13) is identically true, hence the coefficient of each power of z must vanish. Picking out the coefficient of s" and writing it equal to zero we have formula (12) above.* 78. We are now able to solve completely the problem considered in Art. 9. We were to find a solution of the differential equation rD,^(r V) + -^^ De(sm 0DeV)=O (1) subject to the condition We know (v. Art. 77) that 31 F=— — — -1 when ^ = 0. (2) (c^ + H)^ ^ ^ V=r"'F„,(cos6) and F=-^,P,„(cos^) are solutions of (1). Por values of r < c M Mr lv2 1.3 r* (6-2 + ;-^)i ~ e\_ 2 c2 + 2.4 c" 1.3.5 /•' 2.4.6 c' Therefore for values of r < c •]■ (3) ^=f[^o(cos^)-^^>,(cos^) + 11 Jl^4(cos ^) -|||r_;p,(cos ^) + • • •] (4) is our required solution; because each term satisfies equation (1), and there- fore the whole value satisfies (1), and when ^ = P,,^(cos^) = P,„(l) = l [v. (5) Art. 77], and hence (4) reduces to (3) and (2) is satisfied. For values of r > c (c2 _|_ ^2^)1 ,. |_1 V ,.2 + 2.4 T^ 2.4.6 r'^ J ^^^ \_r 2 r^ ^ 2.4 7'^ 2.4.6 i-' ^ J * For tables of Surface Zonal Harmonics v. Appendix Tables I and II. Chap. V.] PROBLEMS IN POTENTIAL. 153 Therefore for values of r > c F= - I - Po (cos 6)-- -3 Po (cos 6) + H 75 ^*(cos ^) - m ^. P. (cos ^) + • • •] (6) is our required solutiou. For it satisfies (1) and reduces to (2) when ^ = . 79. As another example let us suppose a conductor in the form of a thin circular disc charged with electricity, and let it be required to find the value of the potential function at any point in space. If the magnitude of the charge is M and the radius of the plate is a the surface density at a point of the plate at a distance r from the centre is M and all points of the conductor are at the potential -^r— • (v. Peirce's New- tonian Potential Function, § 61.) The value of the potential function at a point in the axis of the plate at the distance x from the plate is easily seen to be Mr rclr M , a--^ — «2 2a x' + a' d_/M _, .r^ — (?A _ 3f dx \2a ^°^ x" + aV ~ a' + x" _ _ Jf r _ a-" a^* _ .T® ~j if a* < a, ^_£ri_^+^_^+ ...1 a:^L a-^^x-* x'^ J if x>a. Integrating and then determining the arbitrary constant we have M TT- COS a- MFtt X X x^ _ * I •^■' ~1 x-^ + a^ a L2 if x a . 164 ZONAL HARMONICS. [Ai We have, then, to solve the equation r A'('" n + gi^ A (sin Bo V) = subject to the conditions when ^ = and r < a Avhen ^ := and r^ a . The required solution is easily seen to be ,,3 1 ,.5 r if r < a and ^ < -L^--P,(cos^) + 3-3P3(cos^)-.-,P,(cos^) + -J nd ^a. EXAMPLES. 1. Given that if a charge M of electricity is placed on an ellipsoidal con- ductor the sxirface density at any point P of the conductor is equal to -; — ^ ? 4:7rabc where p is the distance from the centre of the conductor to the tangent plane at P (v. Peirce, New. Pot. Punc. § 61) ; find the value of the potential function at any external point when the conductor is the oblate spheroid generated by the rotation of the ellipse — „ + 7^ ^ 1 about its minor axis. ^ a^ b^ Ans. (1) If the point is on the axis of revolution If r= 2\Jir ■ [sin- (J^L±£^£\ _ sin- (J!^1^£A£V\ X being the distance from the centre. (2) If the point is on the surface of the spheroid r= '' 2\la^ — V" [f— (^^]=,-^.[f— (v-pb)] Chap. V.] . EXAMPLES. 155 (3) If the distance r of the point from the. centre is less than v'^- — h- and V= --i= r? - X -2 '' z- Ti A (cos 6) (4) If the distance r of the point from the centre is greater tlian \J .... .^ (1^!-^)^ or' P4 (cos ^) - ^^^^^^- Pe (COS ^; + • • -1 2. If the conductor is the prolate spheroid generated by the rotation of the x^ ir lipse — , + p, = 1 abont its major axis, show that if t point and is on the axis at a distance x from the centre, X 11 ellipse — , + p, = 1 abont its major axis, show that if the point is an external M . x + \Ja^-b^ V= — , - log , 2n/«- — fr X — >Ja' — b^ If the point is not on the axis and r > \a- — b ..^|I-^= + fc^^P.(cos, 80. As a third example we will find the value of the potential function due to a thin homogeneous circular disc, of density p, thickness k, and radius a. The value of F at a point in . the axis of the disc at a distance x from its centre is readily found and proves to be To = 2irpk(sjx^ + a' — x) = '^ \sjx^ + a" — x']. If x>a ^•^ +^'^T+:^v ^•a^ + 2:^^~2:4;?*+2X6.T« ~2iA8^«+''' T ,, 231 ri a 1.1^^3 , 1.1.3 a^ 1.1.3.5 r,^ , "1 156 ZONAL HARMONICS. [Art. 80. It X - — 771 -3 ^2 (cos ^) . 1.1.3 a^ „ , ,, 1.1.3.5 r." „ , ., , if r > a, and V = — ri--Pi(cos6) 1 r^ 1 1 r* 1.1.3 r^ ~| + ^-,A(cos6)- — -,P,(cos^)+2;^-,Pe(cos^)---J 73- if r < ffi and ^ < ^ ■ EXAMPLES. 1. The potential function due to a homogeneous hemisphere whose axis is taken as the polar axis, is if r > a, and is + |if>.(cos«)-|iil>.(cos «) + ...] TT if 7- < « and 6> - ■ 2. The potential function due to a solid sphere whose density is propor- tional to the distance from a diametral plane is, at an external point, ^^ 8 3fr5.3a . 5.3.1 a^ „ , ., 5.3.1.1 a' ^ ,, , 5.3.1.1.3 a' ^ ^ ^^ -] Chap. V.] ZONAL HARMONICS. 157 3. The potential function clue to the homogeneous oblate spheroid generated by the rotation of — 2 + ttj ^^ ^ about its minor axis is, at an external point, 3 M r ,:^^a^-l^ / 0^^-^^ + M „Va.-2 _|_ ,,2 _ 1,2) J if the point is on the axis of the spheroid at a distance x from its centre. y = -r^-, 7^1 3-5 ^^ — TT^ ~ 1 — - P2 (cos 6) if r> {fr--l>')^, and if r<(n-'-h'')^ and ^ (a^-h^)i. + ^^^^-^'F,(eos&) + --^ 81. The method employed in the last three articles may be stated in general as follows: — Whenever in a problem involving the solving of the special form of Laplace's Equation rD,^ (/• V) + -7^ Be (sin ^ Z», F) = , ^ sm $ ^ the value of V is given or can be found for all points on the axis of X and this value can be expressed as a sum or a series involving only whole powers positive or negative of the radius vector of the point, the solution for a point 158 ZONAL HARMONICS. [Akt. 82. not on the axis can be obtained by multiplying each term by the appropriate Zonal Harmonic, subject only to the condition that the result if a series must be convergent. It Avill be shown in the next article that P„j (cos 6) is never greater than one nor less than minus one. Hence the series in question will be convergent for all values of r for which the original series was absolutehj convergent. 82. In addition to the form given in (1) Art. 77 for P,„ (.r) other forms are often useful. It ought to be possible to develop P,„(cos 6), which may be regarded as a function of 0, into a Fourier's Series, and such a development may be obtained, though with much labor, by the methods of Chapter II. The development in terms of cosines of multiples of may be obtained much more easily by the following device. We have seen in Art. 75 that P^ (cos 6) is the coefficient of the mth power of z in the development of (1 — 2z cos 6 -\- z^)~i; in a power series, and that if mod « < 1 (1 — 2z cos 6 + s^)"^ can be developed into such a series. We know by the Theory of Functions that only one such series exists, so that the method by which we may choose to obtain the development will not affect the result. (1 — 2z cos e + «2)-i = (1 — z{e.^' + «-»') + .t-)-7 ^(l-ze^Y^(l-ze-^Y^. (1 — 5;e^')-i may be developed into an absolutely convergent series if mod ,t < 1 , by the Binomial Theorem. We have 2.4 ' 2.4.6 ' 2.4.6.8 1 „. , 1.3 „ „„. , 1.3.5 , „.. , 1.3.5 l-ze-^>)-\=l^^ze-^-^ + ^z'e-^~^-^^^^z zei + 2 4.6. z^e The product of these series will give a development for (1 — 2z cos + z^)- i in power series. The coefficient of z^ is easily picked out, and must be equal to P^(cos^). We thus get ^„.{cosU) 2.4.6 2m L ^ ^2 2^-1*^ ^ ^ ^2.4 (2//i-l)(2/M-3)<^ ^^ ^^ J Chap. V.] ZONAL SUEFACE HARMONIC AS A SUM OF COSINES. 159 P" (-^ ") = -Iji^S^^ b '"' '"" + - ui^, -^C" - 2)0 , ^ 1.3 'm(m — 1) + ^ 1.2(2.M-l)(2m-3J ^"^("^ - ^'^ , .,1.3.5 m(m — l)(m — 2) , "1 + ^ 1X3 (2..-l)(2.H---3)(2m-5) '°^^'" - *^^^^ + ■ • • J • ^^^ If TO is odd the development runs down to cos 6; if m is even to cos (0), but in that case the coefficient of cos (0), that is, the constant terra, will not contain the factor 2 which is common to all the other terms, but will be simply r i.3.5---(m-l) ~1^ L 2.4.6. •••?« J' We write out the values of P,„ (cos 0) for a few values of m Pq (cos &) = ! Pj (cos 6) = cos e p,(cos^)=i(3cos2^ + l) Ps (cos e) = -(5 cos 3(9 + 3 cos 0) P, (cos 0)=^ (35 cos 40 + 20 cos 20 + 9) 1 K2) Ps (cos ^) = TTTg [63 cos 50 + 35 cos 3^ + 30 cos ^] P, (cos 0) = — [231 cos m + 126 cos 4^ + 105 cos 20 + 50] P, (cos ^) = J— ^ [429 cos 7^ + 231 cos 50 + 189 cos 3^ + 175 cos 0] P, (cos 0) = ^^T^g^ [6435 cos 80 + 3432 cos 60 + 2772 cos 40 + 2520 cos 20 + 1225] . Since all the coefficients in the second member of (1) are positive, and since each cosine has unity for its maximum value it is clear that P,„ (cos 0) has its maximum value when ^ = 0; but we have shown in Art. 76 that P„, (1) = 1. Therefore P„j (cos 0) is never greater than unity if is real. It is also easily seen from (1) that P„j(cos 0) can never be less than — 1. 160 ZONAL HARMONICS. [Art. 83. 83. P,„ (.-r) can be very simply expressed as a derivative. We have rJtu — 1) rJin. — ,3) • • • 1 r m(m — 1) "'^^~ ml L 2.(2;/i-l) m(m-l)(m-2)(m-3) H ^ 2.4.(2?H — l)(2m — 3) J J ^,„ (X jc^:r - ^^^^ _^ ^^ , ^x 2. (2». - 1) ^ ^' 2.4.(2w — l) (2m — 3) J a' a; X f'P,,, (»-)rfa-2 = Cdx Cp„^ (x)dx ^ (2m-l)(2m-3)---i r ,„^., (», + 2)(m4-l) , (;m + 2)! L' -'- ^^ ^ (m + 2)(m + l)m(».-l) ^.,_, H 2.(2/« — 1) I -\- T)m(m - 2A.(2m — l(2m — 3') 2m(2in — 1 (2w)! L" '" 2(2m — 1) 3m — 1) (2m — 2) (2. 2.4.(2m — l)(2m — .3) - 3)---l (2m) r-p rr).Z^'" - (2m-l)(2m-3)---l F ,„ _ 2.K2m - 1) ^.,_, J P„,(.r)r/^ - L-. 2(2;>.-l) (2m-l)(2;.-.3)---l r ,„ _ ^^^^,„_, ^_ n^Q.-l) ^,_^ m(m-l)(m-2 )_^.,„_,, , "I The quantity in brackets obviously differs from (x^ — l)"' by terms involving lower powers of x than the mth. Hence P (x) - 1-3-5 • • • (2;. - 1) ^ _ „ Hence P,„(.^) - ^gm)! f/x- ^^ "^^ ' or P„, (.r) = tJ-t :^, (^' - 1) "' • (1) This important formula is entirely general and holds not merely when a- = cos 6 , but for all values of x. Chap. V.] EQUATIONS DERIVED FKOM LEGENDRE's EQUATION. 161 84. The last result is so important that it is worth while to confirm it by obtaining it directly from Legendre's Equation (^ - ^'^ £ " ^^ I + '"<"' + ^^' = ^ ^^^ V. (1) Art. 75. Let us differentiate (1) with respect to a; a few times representing ^^hyz',^,hjz",-hjz"',&e. We get (1 - .T-^) ^' - 2.3x '^' + [m(»i + 1) - 2 (1 + 2)],t" = , (^ ~ -^"'^ S" ~ ^-^^ '^' + t"^^''^ + 1) - 2 (1 + 2 + 3)].'" = , and in general (1 - .T--^) ^ - 2(7. + l)a^ -^ + [m(m + l)-2(l+2 + 3 + --- + »)]^("> = or (1 - .r^) ^' - 2(n + l)x '^ + lm{m + 1) - •«(«. + 1)],."" = . (2) Following the analogy of these steps it is easy to write equations that will differentiate into (1). Let '^ = z, ~: = z, '^ = z,&c. Then dx dx- dx^ {^-^^')'^.+H^ + ^>^ = '^^ will differentiate into (1), (^ - "^ & + "-^-^ £ + t^'^^" + ^^ - --^^"^ ^ ^ if differentiated twice will give (1), if differentiated three times will give (1), and in general (1 - x^ '^ + 2(71 - l)x ^' + [m(m + D - n(n - 1)] z„ ^ (3) if differentiated n times with respect to x will give (1). If ?i = m + 1 (3) reduces to (l-^'^ + 2»..'i^=0, (4) 162 ZONAL HARMONICS. [Art. 8.5. and the (m + l)st derivative with respect to x of any function of x which satisfies (4) will be a solution of (1). (4) can be written (1— '^f-' + -'"'-« = and can be readily solved by separating the variables and integrating, v. Int. Cal. (1) page 314. It gives z„^= C(.r- — 1)'«. Hence z = ^-f = C ^ , ,„ (o) dx"^ ax'" is a solution of Legendre's Equation (1) and agrees with the value of Pin{x) obtained in Art. 83. 85. The equations obtained in Art. 84 are so curious and so simply related that it is worth while to consider them a little more full3^ We have seen that differentiates into that if we differentiate (2) m times we get Legendre's Equation (1 - x') ^, - 2x ^, + m (m + l).t = ; (3) that if Ave differentiate (2) 2m times we get (l-^^)£-2(- + 1)^1 = 0; (4) that if we differentiate (2) m — n times we have (1 - .r'O 0,+ 20^ - 1).>-- 1. + i>»{>u + 1) - ni^n - l)].t = 0; {o) and that if we differentiate (2) m. + a times we have (1 - x^ '^ - 2(u + l)x f^ + [m(>n + 1) - n(u + 1)]-^ = 0. ((3) By the aid of (1) we found in the last article a particular solution of (2), namely Chap. V.] GENERAL SOLUTIONS OF THE DERIVED EQUATIONS. 163 If we substitute in (2) z = u(x^ — 1)"' following the method illustrated fully in Art. 18, we get as the general solution of (2) , ^ ^(,.. _ 1). + B(x^ - 1)"'Jp^,;;T: ' a) A and B being arbitrary constants. /dx — — ^^^^ is easily written out [v. formula (42) page 6. Table of Inte- (x 1)'" grals. Int. Cal. Appendix]. If x < 1 it vanishes when a; = 0. If a; > 1 it vanishes when .r z= oo . If then x + B(x-' — 1)'" C ^ (x^-ir-^ and if x > 1 and in these forms unnecessary arbitrary constants are avoided. From (7) we can get the general solutions of (3), (4), (5), and (6). (/"(/>•- — IV" d'^ r r dr ~1 dx>" ^ dxj»L^ ^ J (x'^ — iy+^J ^ ' is the general solution of (3). is the general solution of (4). ^ .^-"(..--1):^ + ^ i!:z!. r.,. _ ^y. f__^_i n 2) is the general solution of (5). is the general solution of (6). In each of these forms A and B are arbitrary constants and the integral is to be taken from to a; if ic < 1 and from a; to oo if a? > 1. Of course (10) must be identical with the forms already obtained in Arts. 16 and 18 as general solutions of Legendre's Equation. Equation (4) is so simple that it can be solved directly, and we get its solution in the form ^=^'' + «-/(;^^^. ("' which must be equivalent to (11). (13) 164 ZONAL HARMONICS. [Art. 85. Comparing (14) with (7), the solution of (2), we see that every solution of (4) can be obtained from a solution of (2) by dividing the latter by (x^ — 1)'", or in other words that if we write (2) {l~x"Y^^-^2im-l)xj^_-\-2mz = 0, (2) and (4) as (1 -x^)'^_- 2{m + l)^' ^J = (4) z = z-^(x'^ — 1)'" ; and the substitution of this value in (2) will give (4), and the substitiTtion of z-, = ^ ., ~ ^ ^ in (4) will give ('2). (x^ — iy ^ ^ ^ ^ ^ We have, then, two ways of obtaining (4) from (2) ; we may differentiate (2) 2m times with respect to x, or we may replace z in (2) by z^ix^ — 1)'". If we use the first method we have seen that Legendre's Equation (3) is midway between (2) and (4). That is if we differentiate (2) m times we get (3) and if we then differentiate (3) m times we get (4). Let us see if the half-way equation in our second process is Legendre's Equation. If z = y{x^-iyi and y = z^^x'^ — l)f z=z,(x-^-ir. So that if in (2) we replace z by y(.r"^ — l)f and then repeat the . operation on the resulting equation we shall get (4). Making the first substitution we find, ^1 - "'> S ~ ^^ I + [^^^^'^ + ^> - r^J ^ = ^' (^^> not Legendre's Equation but a somewhat more general form. Of course its solution is y = Aix^ - l)f + Bix-^ - l)f J^-^_ . (16) (2) and (4) are special forms of (5) and (6). Let us try the experiment of substituting in (5) z = y(l — x^y\ and in (6) z = ^ " • We find that (l — xf both substitutions give the same equation Chap. V.] ZONAL HARMONIC AS A PARTIAL DERIVATIVE. 165 The solution of (17) can be obtained from either (12) or (13) and is _ 1 f d"'-"{x" ^-'(l^^iV dx>-- or ^A^'^^^^^^^^bI^^I^ (IS) y = a- r-^f I A. rZ"»-^>'(.r^-l)'« , i;!:^ [ , _ ^. ,„ C ^Ix 1 ^ which of course must be equivalent. 86. In addition to the value of P^i^) given in (1) Art. 83 there is another important derivative form which we shall proceed to obtain. It is ^..(cos^) = ^%-"-'i>/(i). (1) We have seen in Art. 75 that - — zr can be developed into ^l_2^^cos^ + ^' a convergent series if i\ V^ sjx'^fj^^-2xr,-^r^ Regarding this as a function of (x — r-^) and developing according to powers of i\ by Taylor's Theorem we get as the (m + l)st term Hence £.l55ii) = (^ z)". (1) ■ 87. We have now obtained four different forms for our zonal harmonic^ a polynomial in x, an expression involving cosines of multiples of 6, a form involving an ordinary mth derivative with respect to x, and a form involving a partial ?/ith derivative with respect to x. We shall now get a form due to Laplace, involving a definite integral. f '!± = !L__^ (1) J (( — h cos (a^ — b'^y^ if a^ > h-" [v. Int. Cal. page 68]. 166 ZONAL HARMONICS. [Art. 87. — ; ; — --] can be expressed in the form —-5 — -1 by taking i(=l — zx (1 — 'J.rs: + z% {a — b')^ and l) = z \Jx- — 1 and no matter what value x may have z can be taken so small that (/- Avill be greater than Ir. Theii by (1) 1 _\_ r dcj> _i^ r d^ (1 — 2XZ + Z^)l ~ TtJ 1 _ -,,. _ ;. \/x-—l. COS cj> "^^ 1 — Z(X + sJx-'—l. COS c^) =: - r[l + (.r + \/x' — 1. COS <^),~ + (X + V.^^^. COS ) V + (•'• + y/^-'^^l. COS cj^yz^ + • • -yicf, if z is taken so small that the modulus of z(x + '^x- — 1. cos <^) is less than 1. But by Art. 77 (2) P,„(x) is the coefficient of z'" in the development of , ^ "• (1 - 2.7- + Z^)\ hence FJx) = - fl^^' + V^^' - 1- cos <^]'».7c^ . (2) By replacing c^ by tt — (f> in (2) we get n.(-^-) = ^ /"[■'■ - V-^^^^l. e(.s <^]"v7c^. (3) and if mod - < 1 or in other words if mod .. .^ 1 ^ ^^j^ l^g developed into a convergent series involv- ing powers of -, and the coefficient of (-) will be FjJ-'^); but this will be z \z/ -^ the coefficient of z-"'~^ in the development of — >, _ _i_ -771 according to descending powers of z, mod z being greater than 1. If now we let a =zzx — 1 and b = z \x^ — 1, a'^ — //- = 1 — 2xz + s:'^ and z may be taken so great that a^ — b^X). Then by (1) 1 _ 1 ^ d4> (1 — 2./-.V + «2)4- "~ ttJ ....-^. — 1 — z ^x-' — 1. cos <^ -^/- r/c^ ^(a; - V,r- - 1. cos c/)) fl , "^ 1 L z(x — >^x^ — l.coscf>)-i =1 f ^i r.-+ =L ,.- V C^' — V^.^-' — 1- cos <^) L (x — ^x^ — 1. cos <^) 1 (.X — ^x' — 1. cos y + ',^^. r. -' + • • •] f^<^ Chap. V.] DEVELOPMENT IN ZONAL HARMONIC SERIES, 167 and the coefficient of z"'"-'^ is — ( ttJ r [x — V^"^ — 1. cos c^]" Replace ^ by tt — <^ and we get Hence - ,„, y . , [x — \x" — 1. cos ?? If, then, ??i is not equal to n j'p„,(x)P,(x)dx = 0. (4) — 1 1 If vi=zn we have to find ^[^P,J^(x)Yd^• -1 by (3), = (— l)"'(2/«) ! C{x^ — lYdx. 170 ZONAL HARMONICS. [Art. 90 1 1 ^ C(x' - i)"'dx = C(x - i)'"(,r + ly^dx = - £^f(x - iy'-\x + ly^+hix -(-i)"' (.. + ,)e^:+,). r7^(-- + i)-^ = (-ir 2"' + hn\ (m + l)(m + 2)---(2m-^l) Hence ClP„,(;x)Ydx = ,^,„,/ ,,., / !\:: liL !^ i in — I 90. The solution of the problem in Art. 88 is now readily obtained, and we have f(x) = APo(*) + A,P,(x) + A,P,(x) + ■■■ (1) where A,^ = ^-^fmP.(^')dx. (2) The function and the series are equal for all values of x from x^ — 1 to x = l, and f(x) is subject to no conditions save those which would enable us to develop it in a Fourier's Series, [v. Chapter III.] Of course (1) can be written /(cos 0) = A,F,(cos 0) + ^iPi(cos 6) + ^2Po(cos 6)^ where A,, = ^ J /(cos 0)P,„ (cos 6)^Z(cos 6) or if /(cos 6) = F(0) F(d) = AoPo(cos 6) + ^1 Pi (cos 6) + A ^2 (cos 0) -\ (3) rhere 2 m + 1 A^ CF(e)P, „(cos 0) sin e.de (4) and the development holds good from ^ = to = 7r. If f(x) is an even function, that is, if /(— a')=/('^) (1) ^^^ (2) can be somewhat simplified. For in that case it can be easily shoAvn (v. Art. 77) that pXx)P,,(x)dx = 2pXx)P,,(x)dx, Chap. V.] DEVELOPMENT IN ZONAL HARMONIC SERIES. 171 an d that | f(x) P^^. + j (x) dx = 0; -1 so that if /(— x) =f(x) fix) = A,Po(x) + A,P,(x) + A,P,(x) + J,P,(x) + ■" (5) where J,, = (4k + 1 )ff(x)P,,(x)dx . (6) If f(x) is an odd function, that is, if /(— x) = — f(x) it can be shown in like manner that f(x) = A,P,{x) + A,P,i.r) + A,P,i.r) + A,P,{x) + • • • (7) where A,,^, = (4k + 3)jf(x)P,,^,(x)dx. (8) If it is only necessary that the development should hold for < a; < 1 any function may be expressed in form (5) or (7) at pleasure. a more gen- 91. We can establish the fact that Cp„^(x)P,^(x)dx = by eral method than that used in Art. 89. Let A'„j be any sokition of Legendre's Equation jx[_^^~ -'"^ £] + '"^'" + ^>' = ^ [^^- ^^> ^'^'- ^^^' which with its first derivative with respect to x is finite, continuous, and single-valued for values of x between — 1 and 1, — 1 and 1 being included. '"'™ £[(i-->'-f']+'»("'+i)-^-=« (1) Multiply (1) by A'^ and (2) by X„^ and subtract and integrate and we get lm(m + 1) - n(u + l)-]fx^XJx =fx,, £ [(1 - ^') '^'^x — 1 172 ZONAL HARMONlCSo [Art. 91. Integrate by parts, lm(m + 1) - »(« + l)]j'x„X,cb = [j.i;.(l -='')^- X.(l - »•») !^-]" '_ ^ (^ic dx Whence Cx,„X„dx = (4) — 1 unless m = n. (3) gives at once the important formula r,,, a-^-)[-v/i°'-x„f-] from which come as special cases (i-.^[p.(.)^-p„(.)^] J P„(.x)A(.T),fa = ,„(„,_m_„(„ + i) (6) and since Po(ic) = 1 unless m = . EXAMPLES. 1. Show that I P^,(x)dx ^0 if vi is even and is not zero. (-1) - ...u..A_^^OAa ....^_ix if J^ 3.5. <■ •••m m(m + 1) 2.4.6. • • • (m — 1) odd. V. Art. 91 (7) and Art. 77 (10). 2. Show that Cp^{x)P„{x)dx = Q if m and ??. are both even or both odd. (-1)' , t ./. I 2'"-^"-y(n^ - n)(vi + n + l)^!) (^l) if m + ?i is odd. v. Art. 91 (6) and Art. 77 (8), (9), and (10). cf. J. W. Strutt (Lord Kayleigh) Lond. Phil. Trans. 1870, page 579. 3. Show that ClP^^{x)Jdx = I v. Art. 89 (5) TT 2w + 1 Chap. V.] DEVELOPMENT IN ZONAL HARMONIC SERIES. 173 92. Formula (4) Art. 91 can be obtained directly from Laplace's Equation by the aid of Green'' s Theorem (v. Peirce's Newt. Pot. Punc. § 48). Take the special form of Greeiv's Theorem [(148) § 48 Peirce's Newt. Pot. Func] fff(^ V' V-VV^ U) dxchjdz =J( UI)„ V - VD„ U)ds (1 ) where V^ stands for {D^ + D^ + D^), D^ is the partial derivative along the external normal, and the left-hand member is the space-integral through the space bounded by any closed surface, and the right-hand member is the surface integral taken over the same surface, (v. Int. Cal. Chapter XIV.) If ?7and F are solutions of Laplace's Equation V^F=V'^f^=0 and (1) reduces to ^{ UI),^ V - VD, U) ds = . (2) / Now r"'X„^ and r"Z„ are solutions of Laplace's Equation if .r = cos $ (V. Art. 16). If the unit sphere is taken as the bounding surface and U = r"'A',„ and F= r"X^ (1) and (2) will hold good. Z)„ U = D,.(r"XJ = w.r'" - 'X^ , D„V=nr"-KY„, ds=z sin 0.dedcf>, and (2) becomes Cdcf, C(nX„^X„ — mX^^^XJ sin e.dO = or 27r(w — m) rA''„,A;, sin d.de = . (3) Since x = cos 6 , sin 6.d0 ^= — dx and (3) reduces to 'x^X^dx^O* (4) P unless m=:n. 93. We can now solve completely the problem of Art. 10 which was in that article carried to the point where it was only necessary to develop a certain function of in the form AoPo(cos 6) + .4iPi(cos 6) + A.P^icos e)-\ * It should be noted that this proof is no more general than that of the last article, for, in order that Green's Theorem should apply to r"'X„;, this function and its first derivatives must be finite continuous and single-valued within and on the surface of the luiit sphere, (v. Peirce, Newt. Pot. Func. § 48.) 174 ZONAL HARMONICS. [Art. 94- given that f(6) = 1 from ^ = to 6 = '^ and f(0) = fi'om ^ = ^ to O^tt. This amonnts to the same thing as developing F(x) into the series F(x) = A,P,(x) + A,P,(x') + A,P,(x) + AsP,(x) + ■■• where F(x) = from x ^ — 1 to .r = and F(x) = l from ■<■ = to .r = l. By Art. 90 (1) and (2) 2m + 1 / and any coefficient -•i/n = — 7) — " I ^mi^')'-^^' By Art. 91, Ex. 1 I Pj^(x)dx =: if 7n is even ^ ^'^ 1 3.5.7. •'•m •£ ■ jj ^ ^ //i(??i + 1) 2.4.6. • • • (ill — 1) Hence A,,^ = if 7n is even ^ '^2>/^+l 1.3.5. • • • (m - 2) .„ ... = (-1)^ 2^+2- 2.4.6. •••(m-1) ^f--«°dd. Then i.(..) = 1 + ^ P,(.r) - 1. 1 AC^O + ^- H Ps(r) -■■■ (1) and «-= ^ + ^ rP,(cos ^) - ^ • ^ /•^P3(cos ^) + ^ • ^ r^PsCcos 6) + ■ ■ ■ (2) for any point Avithin the sphere. 94. If in a problem on the Potential Function the value of V is given at every point of a spherical surface and has circular symmetry* about a diameter of that surface the value of V at any point in space can be obtained. We have to solve Laplace's Equation in the form rD,\i- V) + gA-^ A(sin ei)eV)=0 (1) * See note on page 12. Chap. V.] POTENTIAL GIVEN ON A SPHERICAL SURFACE. 175 subject to the conditions V=f{6) when rz=a F=0 *' ;-=oo. We have f{d) = AoPo{cos 0) + -4iPi(cos 0) + A.P.(%^ +i2-2:4;r^^^^^°' for an internal point V= ■■ ;. + (^ - «') [I 7. A(cos 0)-l.l'^^ P3(cos 0) for an external point. 176 ZONAL HAinrONICS. [Art. 94. 3. If Fi =/(cos 6) when r = a and V^ = when r = h show that for a , and the outer one of radius b is at potential q. Find V for any point in space. V^2^ if r 1 or a; < — 1. 178 ZONAL HARMONICS. [Art. 9.5. Let us obtain this development. By Art. 9(» (1) and (2) X" = A,P,(x) + A,P,(x) + A,P,(x) + • • • (1) where A^ = — ^ \ x" P „^{x)dx . (2) — 1 A„ = '-^ ^J.> '^^iBLr^ ^ by (1) A.. 83. — 1 By integration by parts we get r^" ^ "7 .7 dx = n(n — l)(w — 2) • • • (n — »i + 1) fa;"- «(1 — x^'dx , (3) —1 —1 if VI u. By integration by x^arts we readily obtain the reduction formula 1 ^ 1 ra;^(l - xydx = -^^ C^" ^ '(1 - ^^y ~ ^dx whence Ln-mn - ^yndx = i^wi! r . ,„ , r 2 . . I »■" + "'cZa; = — ; r—. if n + ?/i is even , J n-\-m-\-l == if n + wi is odd. Hence A, Therefore (2m + l)n{n — 1) fa — 2) • • • (w — m + 1) (m — m -\- 1) (?i — 7)1 -\- 3) (?i — m -\- 5) • • • (11 -\- m + 1 if »i /« or if m -\- n is odd. "" = 1.3.0 ■■■p.t + l) [(2" + l>^"('"> + (2» - 3) ^^^^ ^.-,(-») + (2,.-7) <^"+^f-^^ P._.(.) ending witn tne term — n- the second member ending with the term — r— , Pq(x) if n is even and with o ?i + 1 ^ ^ 3 the term — r— r Pxix) if /i is odd Chap. V.] USEFUL FORMULAS. 179 For convenience of reference we write out a few powers of x. X = P^(x) 1 ,2^ -A(.t) + -Po(x) ^' = q:^Ps(x)+^Ps(x) + ^P,(x) 16 24 10 1 ^' = 231 ^«^^'^ + 77 ^^("""^ + 21 ^'^''^ + ^ ^°^^^ (5) ^^ = Mk ^«(-^) + H ^«(^-> + ll ^^(^-^ + 9-9 ^^('^^ + -9 ^ o^-J If a given function of x can be expressed as a terminating power series it can be developed into a Zonal Harmonic Series by the aid of (4). Given that /(,r) =ao + ((iX + a.x- + (laX^ -\ . let f(x) = B, + 7AP,(.r) + P,P,(x) + B,P,(x) + • • • ; then picking out carefully the coefficient of P,„(^) we have (m 4- 1^ «. + .+ ••• • (6) '" 1.3.5. • • • (2m — 1) (w + l)(m + 2)(m + 3)(m + 4) 2.4.(2m + 3)(2m + 5) 96. The development of — y-^ is useful and is easily obtained. Let '^l^ = A,P,{x) + A,P,{x) + A,P,{x) + • • • Then ^,„ = ^1JP,„^,.)^,. (1) by Art. 90 (2) ; 180 ZONAL HARMONICS. [Art. 97. [P,„(.r)P„(*)f ~ = if VI + n is even = 2 if m + n is odd. (IP (x) Since P„(x) is an algebraic polynomial of the nth. degree in x, — y^-^ is an algebraic polynomial of tlie n — 1st degree in .r. Therefore in (1) m is less dP (x) than n; consequently — p"-^ is an algebraic polynomial iu x of lower degree than n and i Cp„{x) '^^f^ dx = by Art. 95 (3). — 1 "We get then J,„ = 2 in + 1 if ni + n is odd and m < n, ==: if ni -\- n is even or m '> n — 1 ; and ^^^ = (2n - l)P„_i {x) + (2;. - o)P„_,{x) + (2m - 9)P„_,(^) + • • • (3) the second member ending with tlie term ^Pi{x) if n is even and with the term Pq{x) if n is odd. From (3) a number of simple formulas are readily obtained. For example jP.{:c)dx = 2^ IP, _,(.,■) - P„ , ,(»•)] . (6) X [v. (4) and Article 77 (12)]. r7P /^^ [v. (5) and Article 91 (7). 97. By the aid of the formulas of Art. 96 a number of valuable develop- ments can be obtained. Let us get cos nO and sin nO n being any positive real. z = cos 7i6 and z = sin nO are solutions of the equation , ^^^ _L 2 A — + n2^ = Chap. V.] ADDITIONAL DEVELOPMENTS. 181 or if we let x = cos 6, of the equation Let aoA(*0 + ') H be the required development of cos n6 or of sin 7i0. Then X «» [(1 - -') ^^^ - - ^^ + -^P.(-)] = by (1). = F„^(x) is a solution of Legendre's Equation (v. Art. 77). Hence ,, (PF,Joc) dP„,(x) dF,Jx) , , , ^ , dx^ dx dx and (1) becomes (2) Formulas (4) and (6) of Art. 96 enable us to throw (2) into the form ^"'\_2?n-i-l dx 2/« + l dx J~^- ^'^^ dP (x) (3) must be identically true. Therefore the coefficient of — "'T must equal zero, and we have ^"l' I ^ I" III' , , N 2m -{- 5 91^ — )/i If we are developing cos n6 ao = - jcos nO sin e.cW by Art. 90 (4), = - ffsin (w + 1)^ — sin (n — l)$2de , 11+ cos nTT and 2 n'- — 1 3 (5) «! = - j COS ?«.^ cos (9 sin d.dO by Art. 90 (4), (6) 3 1 — COS mr 182 ZONAL HARMONICS. [Art. 97. (4). (5), and (6) give us COS nO = - —^^7^ T- Po(cos ^) + 5 -. ;-, PJcos 6) 2[n- — 1) L n^ — o- ^ ^^ [3P,(eos 6) + 7 '^, P.(cos 6) 1 — COS mr 2(n^ — 2^) r or 1 — COS ', ^„(cos^). For If % is a whole number 1 + cos 7i7r or 1 — cos mr will vanish and the series will end with the term involving P„(cos 6). For this case (7) maybe rewritten + (2»-3)5^r{~ly>.-.(«»^«) ^ ^ [)r — (n — 2)^]['/r — (>i ~ Vj J If we are developing sin nO ao = - I sin n6 sin O.c 1 sin HIT J n- — 1 op. ^ /,./,■,„ o sin nir a, = ^Csin nO cos 6 sin ^..7^ = - ■ ^^^, and 1 mil 7>7r I 7/ sin «^ = - ;^. V^ ^o(cos ^) + 5 -^^ -, P^fcos 6) , 1 sin??7r ToTi / m 1 -"^"~1^ 75 / m + 2- ;7:r2"2 [_3^i(«os 6) + * :^^r:z-^2 ^3(cos ^) If 11 is a whole number sin //tt = , and all the terms of (9) vanish except those involving P„_i(cos 6), P„_^i(cos 6), P„ + 3(cos 6) &c., which become inde- terminate. For this case it is necessary to compute a„_^ independently. Chap. V.] EXAMPLES. 183 We have 2/? — ir «„_! = — - — I sin n$F„ _j^(cos 6) sin d.dO 2n — 1 - r[cos {n — 1)^ — cos (w + l)^]P„_,(cos Q)^ le. TT 2?i — 1 1.3.5. ••• ('2«, — 3) r- A i- oo /i\-i Hence «„_ , = -^- . 0.4.6. ■■ .(2, - 2 ) '^ ^''^ ^''- '^^ ^^^^' and sm ^^ 1.3.->(2.-3)r 4 2.4.--'(2»-2)L^ ^)^n-A^ „2 /., iNa +(2H-T) g:|;;;^;;ir;;::(;;+^);i ^„<.os.)H----].(io) EXAMPLES. 1. Show that csc^ = J [1 + r> (^)>.(eos 0) 4- 9 (^D'^^Ccos ^) + 13 (HI)" A(cos ^) + ■ •] whence 1 TT v/f f [1 + 5 @ V.(..) + 9 (II) A(.) + 13 {^P.i^) + ■ • •] [v. Art. 90 (4) and Art. 82]. 2. Show that ctn^ whence = f [3 (i)i>,(co.s .) + 7 (f)(^)p.(co. .) + 11 (|)(^)V,(cos ., + . ] Vl = = I [^ a) A(..) + ^ (Da)''^.(^) + n (|)0'a(.) + ■ ■] [v. Art. 90 (4) and Art. 82]. 3. By integrating the resnlt of Ex. 1 and simplifying by the aid of Art. 96 (5), obtain the development sin- ^ = f [3 (^) V,(..) + 7 Q^JP,(^) 184 ZONAL HARMONICS. [Akt. 97. whence ^ = J rPo(cos ^) - 3 0-) Pi (cos O) — ! (~yP^(cos 6) 4. By integrating the result of Ex. 2 and simplifying by the aid of Art. 96 (5) obtain whence sin e = I [i P,(cos ff) - 5 (^)(i) P,(cos 0) - 9 (?)(ij)V.(eos «)-•■•]. To make clearer the analogy of development in Zonal Harmonic Series with development in Fourier's Series we give on page 185 a cut representing the first seven Surface Zonal Harmonics Pi(cos 6), P2(cos 6), • • •P7(cos 0), which are of course somewhat complicated Trigonometric curves resembling roughly cos^, cos 2^, •• -cos 7^; and on page 186, the first four successive approxi- mations to the Zonal Harmonic Series 1 + ^ P,(cos 6)-l.l P3(cos ^) + j|- II J'sCcos 6) [I] [v. (1) Art. 93], and I [Po(cos 0) - 3 Q Vi(cos 0) - 7 (^)'^3(cos 0) -iK2sy^^(^°^^)--]M (v. Ex. 3 Art. 97). 7j- 7j- [i] is equal to 1 from ^ = to = jj, and to from 6 = -^ to = 7r; and [ii] is equal to 6 from 6 = to 6---^=7r. The figures on page 186 are constructed on precisely the same principle as those on pages 63 and 64, with which they should be carefully compared. 98. By applying Gauss's Theorem (B. 0. Peirce, Newt. Pot. Func. § 31) or the special Form of Green'' s Theorem. ( ( f^^ '^dx dij dz = Cd,^ Vds = — 4.'7rCC Cpdx dy dz , Chap. V.] ZONAL HARMONIC CURVES, 186 ZONAL HARMONICS. [Art. 9/ '-] "^ ^^ ^ ~--,~' \ ^\ ,'— \-. » / '>" <: / ■^^ V .^. It __,. v_ V V. page 184. Chap. V.] THIN SPHERICAL SHELL. 187 [Peirce, N. P. F. § 49 (149)] to a box cut from an intiuitely thin shell of attracting matter by a tube of force whose end is an element of the surface of the shell we readily obtain the important result 47rpK = I)J\-I),^r,. (1) Avhere p is the density and k the thickness of the shell, Fi the value of the potential function due to the shell at an internal point and V^ its value at an external point, and where D„ is the partial derivative along the external normal to the outer surface of the shell. If we have to deal with a surface distribution of matter we have only to replace pK in (1) by O" where o" is the surface density, whence 47rcr = I)„ Fi — 1)„ V^ ' (2) (v. Peirce, N. P. F. §§ 45, 46, and 47). Formulas (1) and (2) enable us to solve problems in attraction when we know the density of the attracting mass, and problems in Statical Electricity when we know the distribution of the charge, by methods analogous to that of Art. 94. For example let us find the value of the potential function due to a thin material spherical shell of density p and radius a. Since V must be a solution of Laplace's Equation and must be finite both when r = and r = oo we have Fi =2^^„r'«P,„(cos 6) Fi and Fg must approach the same limiting values as r approaches a. Hence or B„, = A,^/i:'"' + K D„ V, = D, V, = -^{m + 1) -^— P„,(cos 6) . Therefore by (1) 47r/3K: =^(2m + 1) J,X'~^^m(cos 0) if K is the thickness of the shell. 188 ZONAL HARMONICS. [Art. 99. Let p =/(cos 0) =2) C,„PJeos 6) 1 C„, = ^-^^^ jf{^)Pn.(x)d:r by Art. 90 (2). where and and Then ^ttkC,,^ = (2))i + l)A,„a"'-' , and ^^-^4-«'^X2;7+i^"^'"^^'"^^^' ^^^ Fe = ^-rraK^^;;^^ "^ P.(cos ^) . (4) 99. We can noAv get the value of the potential function due to a spherical shell of finite thickness, provided that its density can be expressed as a sum of terms of the form C>-^P„;(cos 6). Let a be the radius of the outer surface and b be the radius of the inner surface of the shell. 1st. — Let p ^ C?'^'P„,(cos 0). Then for the shell of radius s and thickness ds J\ = 4:7rsds ^ ^'\_ ^1 P„,(cos 0) by (3) Art. 98, Cs and Fs = 4:7rsds ' ^— -j P,„(cos 6) by (4) Art. 98. Then if )• < h J - (2;« + l) (A- + »^+3; r'" + i ^^ /> < r < a J -^J ' 2»i + lL(A; + m + 3)?-"' + i if ?• > a ,jt- — m + 2 2d, — If p =^ C„,?'^'P,„(cos ^) the solutions will consist of sums of terms of the forms given in (1), (2), and (3). Chap. V.] ZONAL HARMONICS OF THE SECOND -KIND. 189 EXAMPLES. 1. If the shell is homogeneous V^27rp(a'-b') if ra, O V I' r 2h^ r-~l V=27rp\a'-j^--\ if /ja, and F= a constant if r a. Compare Ex. 2 Art. 80. 100. We have seen in Art. 18 {e) (3) that ^"(•-) = "'^"(-)/ (l-.')tf„.(.r)]- ' (1) no constant term being understood with \ — ^^^ , ^■,, . 1 J (l-^')[-P.«(^)]' oN,--r. . N-io is a rational fraction and becomes infinite only for a; = 1, (1 — cc^)[P,„(a;)]^ '' cc = — 1, and for the roots of P,„(.r)=0, all of which are real and lie between — 1 and 1, as can be proved by the aid of the relation . 1 n^^-1)- If .r- > 1 I — — - is finite and determinate and contains no J (l-.r-)[P,„(x-)]^ constant term. Hence if a;^>l for the constant factor of (>,„(a:') has been chosen so that C = — 1 . 190 ZONAL HARMONICS. [Art. 100. If x^i. From Art. 85 (10) it follows that C can be determined and is equal to ^^ . ^ ' if a-^ < 1 , and is equal (-l)m2»^m ! .„ ,^, ^^"'^• (2m)! Hence ,„(.) . (^ig^' ^, [(.- 1)-/^,^,] (T) if x'i_ (7) and (8) give us for Qo(x) and Qi(x) the values already written in (3), (4), (5), and (6). By the repeated application of the formula (m + 1)^, + i(a-) - (2m + l)xQ„,(x) + mQ^_,(x) =0, (9) which may be obtained for the case where x^ 1 from Art. 16 (9), any Surface Zonal Harmonic of the Second Kind can be obtained from ^o(^) and Qi(x) as given in (3), (4), (5), and (6). Chai'. v.] examples. 191 Analogous formulas for 2)m{^) ^^^^ 1m(^) can be obtained without difficulty from Art. 16 (4) and (5). They are (m + l)^- 4- -^A r ^^».fcos ^)P,„(cos 6) - F„,(co8 /3)()„,(cos ^) -1 - Z^{^ 'n ^r'"-i;L^«(cosa)^7,„(cos^)-P„Xcos;8)(>,„(cosa)J " 4. Find V for points corresponding to values of between a and /3 when V can be given in terms of whole powers of r for 6 = a and for = /3. 5. Find by the method of Art. 16 solutions of Legendre's Equation of the form i(m + l) ,^ _ -, , ^ (m-l)m(m-\-l)(m + 2 ) 22/2!)- . = ,pj.) = 1 + ;^^iv-^ (X - 1) + '"■ ^""v; ";""'• ^"^ (- - 1)-^ (,„ - 2)(m - l)m(m + 1)(,» + 2)(w + 3) ,^ _ -, , , , 2^(31)'^ ^ ^ ' . = .,P„(.) = 1 - V^&^ (, + !) + (>"■ - 1)"'(- + 1)"" + 2) (. + 1). (M - 2)(,« -!)«(»» + l)(w + 2)(m+ 3) 2^^3 !)^ ^'^' """ ^ I" If m is a whole number, ^P,„{x) = P^^(x) and _iP,„(a;) = (— 1)'«P„,(^). N,^ matter what the value of m, iF,„(x) is absolutely convergent for — 1 < »■ < 3 , and _lP„^(x) is absolutely convergent for — 3 < x < 1. 19: ZONAL HARMONICS. [Art. 100. 6. By the aid of (7) Art. 16 show that J'= -^ sin (n log r)k „(cos 6) , Sir F^ -= cos (ii log r)A:„(cos 6) , are solutions of Laplace's Equation F= -r. sin (ii log ?')Z„(cos 0) , Sr r'= -7; cos (11 log /•)Z„(cos 6) , V'r /•i>;(;- TO + ^^Q A(sin ^ D, F) = , A'„(a^) and ^«(^)=-'Z-i + m(^)==-^- + -+(i)\,[-+©i-+an ,r« + + — X' + ^„(.r) and /„(a;) are convergent if 0:- < 1, but are divergent if x^=l. 7. Show by the aid of Example that F^ -p sin (?i log ?')A''„(cos 6) . \r V= -p cos (n log r)ir„(cos 6) . F= -^ sin (n log r)K„(— cos ^) , Sr F= -^ cos (w log r)K„(— cos ^) , are solutions of if K„(x) = ,P_^,„,(x) = 1 ^ (.r - 1) , ["-+a)i-'^©i ^^, ['"+©l"-+(Dl-'+@'] 23(3 !)2 (^_1)3_^... Chap. V.] EXAMPLES. 193 and K„(- x) = _,P_^_, „,(.t) = 1 + ^ i^x + 1) , [-+©;][;:H-©l , ,.,, -ff'„(cos ^) is convergent except for = '7r, and 7t„(— cos^) is convergent except for 6^=0. ^n(^)) 4(^0 J -^'wC-'^)) and 7i'„(— .>■) are sometimes called Conal Harmonics. They are particular values of z which satisfy Legendre's Equation written in the form 0. For an elaborate treatment of them see E. W. Hobson on "A Class of Spherical Harmonics of Complex Degree." Trans. Camb. Phil. Soc, Vol. XIV. 8. If V=S\r) when e = j3, 9. If V=f{r) when 6 = jB and r^) ^."^^'^^ f^ sin aX sin fa log -)da ; if ^ < /3 . TT ^?'J J ^ ^/i„(C0S/3) V "^ O/ ' ^ — « 10. If V = /(>') when ^ = /3 and a<)' fi COS 6 must be replaced by (— cos 6) in examples 8, 9, and 10. 12. If r =/(>•) when 6 = ^, and r=0 when e = y, -,. 1 /» 7. r ^ y./ i\ ^"afCOS 6)IJC0S y) — /^„(COS 7)4(C0S ^) r /\ 1 n-i 7 ttV/^J J -^ ^ ^ A;a(cos /3)4(cos y) — A;„(cos y)4(cos ^) '- ^ s yj , — ^ if /3<^, J^' -{- hV=0 when /• = i , '"=" /!„ (cos^) . ^,. ^=S^m ^^'^'t m sin ( a,„ log ) , where ^ /f,,^^ (cos /3) V '' / log - ^^ "" a,r:(iog b - log «) V/i^'L^Hiog ^ - log '0 + 1]/"'^-^^"'"^ '''' '"'"''•'^^ and a,„ is a root of the equation a cos (a log A + hb sin (^a log -) = v. Art. 68 Ex. 5. CHAPTER YI. SPHERICAL HARMONICS. 101. When we are dealing with problems in finding the potential function due to forces which have not circular symmetry * about an axis and are using Spherical Coordinates, we have to solve Laplace's Equation in the form rD:-{r V) + J-, i»e(sin DeV) + -Ar^ I>1 ? ' = < » (1) [v. (xiii) Art. 1]. To get a particular solution of (1) Ave shall assume as usual that V is a product of functions each of which involves but a single variable. Let V-^= i?.©.$; where B, involves r only, © involves Q only, and O ^ only. Substitute in (1) and we get r = . (7) sm B (IB ' [_ V ' / gjj-,2 ^ J V / (6) was solved in Art. 13(c) and gives R = A,r"' + B,r-"^-\ (8) If in (7) Ave replace cos ^ by /* we get ib'- "'^ |] + ["•('" + 1) - r^G® = » ' w the equivalent of (1 - ^') S - -■'^ I + [•»('» + 1) - r^J ■' = " (i»> [v. (17) Art. 85], which was solved in Art. 85 for the case where m and n are positive integers and w < m + 1. v. (18) and (19) Art. 85. From (19) Art. 85 we get as a particular solution of (9) if we restrict ourselves to whole positive values of m and n, as we shall do hereafter unless the contrary is explicitly stated, and suppose m not less than n. A second but less useful particular solution of (9) is Combining our results we have as important particular solutions of (1) V= r"'(A cos ncl>-\-B sin »<^) sin" ^^^^^ ' (12) and V= ^ (A cos n^ + B sin nxf>) sin" ^^"^'"[^^ ' (13) where m and n are positive integers and n < /n + 1- Chap. VI.] TESSERAL HARMONICS. 197 (P^P (u) d"P (a) 102. sin" 6 r^— or (1 — ix% r^— is a new function of a, that is d/x" ^ dfi" of cos $, and we shall represent it by F„''(ijI') * and shall call it an associated /miction of the nth. order and mth degree. It is a value of ® satisfying equation (9) Art 101. By differentiating the value of P„,(x) given in (9) Art. 74 we get the formula Pm (/^) - 2. ^, , (,,, _ ,,) ! L^ 2. (2m - 1) ^ (»^ - 70(»^- - n - l)(m - n - 2)(m-?^-3) _, "I .;l) ■^ 2.4.(2/«-1)(2;m-3) ^ J^^ the expression in the parenthesis ending with the term involving x^ if m — n is even and with the term involving x if vi — n is odd. Eor convenience of reference we give on the next page a table from which P,;^(fi) can be readily obtained for values of m and n from 1 to 8. cos ?i<^Pj(/i) and sin n(f> Pj}(fi), that is, . ^d"PJa) , . . ^d"Pjfi) cos ncf> sm" 6 — r^ — ■ and sm neb sm" t* r-— are called Tesseml Harmonics of the mih degree and nth order, and are values of V which satisfy the equation m,Qu + 1) r+ ^^ Deisms A V) + ^ A,^ V= (2) or its equivalent m(m + 1) r+ zv[(i - ^^)i), r] + ^^, i>|F= . (3) There are obviously 2m + 1 Tesseral Harmonics of the mth degree, namely PM, cos<^sin^^i^\ sin <^ sin ^^^^^ cos2c^sin^/^^^, sin2?.th degree, and is a solution of equations (2) and (3). We shall represent it by Y„^(fl, ) or by Y„^(0, (t>). * Most of the English writers represent this function by T',;^^). 198 SPHERICAL HARMONICS. [Art, 102. Table 'for - m 1 2 3 4 5 6 7 8 n=.l. ,1 = 2. 1 n = 3. 1 3/. 3 |(5m-'-1) 15m ! 15 |(7.-3.) f(7M-l) 105m ^ (21m^ - IV- + 1) 105 ,., 3 , f(..-.) "^ (33m5 - 30m^^ + 5m) ^ (33m* - IBm^ + 1) 315 - (42V — 495m-' + 135m2 — 5) '-^ (143m5 - 110m« + 15m) o ^ (143m* - 06m2 + 3) ^ (715m' - IOOIm^ + 385m3 - 35m) ^(mM^-HSM-i + SSM^-l) ^ (39m5 - 26m-'' + 3m) r '"!",„ (/Lt, <^) and — —^ l^m(y"'j ^) ^^'S called *SoZiVZ Spherical Harmonics of the mtli degree, and are solutions of Laplace's Equation (1) Art. 101. To formulate: — r„,(M, «/>) ^Xf'^'^" ''''^ ""^ '^'''" ^ '^""^"'f^^ + B„ sin «<^ sin" ^ '^""^'"f^H (4) n = or r,,(/i, <^) = A,P,,{ix) +^IA, cos n^P^lfi) + B„ sin 7^<^P;,;(/^)] (5) is a Surface Sijlierical Harmonic of the mth degree. A Tesseral Harmonic is a special case of a Surface Spherical Harmonic, and a Zonal Harmonic a special case of a Tesseral Harmonic; -P„j(/i) being the Tesseral Harmonic of the zeroth order and the mth degree; it might be written P,^(/i). EXAMPLES. 1. Show that reduces to (1 - X') g - 2(M + 1)^ ^ + Vn(m + 1) - n(n + 1)]// = if we substitute (1 — x-)\y for z, even when m and n are unrestricted. Chap. VI.] TABLE FOE ASSOCIATED FUNCTIONS. 199 csc«^P„» = ^^>. n = 4. 11 — 5.' n = 0. 71 = 7. n = 8. 105 94.5m 945 f (11.-1) 10395/x 10395 ^(13^3-3,) 'T<''^ » 1.35135/x 1.35135 l-f^\65.4_ 26.^ + 1) ''Ti'^' -) 'Td^^ 1) 2027025m 2027025 2. Show that if in the second equation of Ex. 1 we let y = 2 a^x'' we get (»?, — 71 — k)Om + n-\-l + k) . . . ^ ^. whence z=2yZ(a:) and z — qZ(^) are sohitions of the first equation of Ex. 1, no matter what the values of m and n, if i.;;W = (1 - .■^)S [i - ('—")('- + - + r> ,, and "T- 41 .T •■•J (v/A — w — 1) (m — ?^ — 3) (m + 7?, + 2) (m + ». + 4) g_ .-| ^ ^ 5! -^^ "J- If m. — w is a positive integer, 2j>",(x) or 2',"(a;) will terminate with the term involving x"'~"; and in that case (ill — n)(iii — n — 1) „j_^ _ 2.(2/«-l) ^"" , {m — 11) (in — 11 — 1) (m — n — 2) (>/<- — ?<, — 3) "*" 2.4. (2»i — 1) (2m — 3) ^,„_„_4 "I ^ liUU SPHERICAL HARMONICS. [Art. 103. the parenthesis ending with a term involving x° if ui — n is even and x if /n — n is odd, is a solution of the first equation of Ex. 1. If m and ti are integers this value or z is -^^ — -; — — -r'[(^) . (2m) I ^ 103. We have seen in the last chapter that in many problems it is import- ant to be able to express a given function of cos 0, that is of /x, in terms of Zonal Harmonics of fi. So it is often desirable to express a given function of fi and being independent variables, and that we wish to express it in terms of Surface Spherical Harmonics. Assume that f(H; ^) ^X[_A,nPM +X(^i.,„ cos ncl>P:(fji) + 5„,,„ sin »c^P„X/i))] • (1) Let us consider first a finite case, and attempt to determine the coetticients so that />, ) =X[^°''«^"'^^^ +X(-^".'« ««s ''P:>M + ^n,n. sin ?i<^P:(;u)) J (2) shall hold good at as many points of the sphere as possible. The expression in brackets in the second member of (2) is a Surface Spherical Harmonic of the mth degree and contains 2/?? + 1 constant coefficients. The whole number of coefficients to be determined is then the sum of an Arithmetical Progression of i^ + 1 terms the first term of which is 1 and the last is 2^? + 1 , and is therefore equal to (p + 1)1 Let the interval from yu,^ — ltofi=:lbe divided into ]) -{-2 parts each of which is Afi so that (j) + 2)A/i = 2, and let the interval from <^ i= to <^ = 27r be divided into p-\-2 parts each of which is A<^ so that (p + 2)A^ == 27r. Chap. VI.] DEVELOPMENT IN SPHERICAL HAIIMONIC SERIES. 201 Then if we substitute in equation (2) in turn the values ( — 1 -|- A//., ^4>), (- 1 + 2A/X, A<^), • • • [- 1 + (/> + l)Afi, Ac^]; (- 1 + A/i, 2Acjf>), (- 1 + 2\fi, 2AcA), •••[-! + iP + 1 ) Ay^, 2A<^]; • . • [- 1 + A/., (p + 1)A<^], [- 1 + 2A/ti, (p + l)Ac/>), • • • [- 1 + (^> + 1)A/^, (p + 1)A<^]; since the first member in each case will be known we shall have Qj -\- 1)^ equations of the first degree containing no unknown except the (^v + 1)"^ coefficients, and from them the coefficients can be determined. When they are substituted in equa- tion (2) it will hold good at the (p + 1)^ points of the unit sphere where p-{-l circles of latitude whose planes are equidistant intersect ^> + 1 meridians which divide the equator into equal arcs. If now p is indefinitely increased the limiting values of the coefficients will be the coefficients in equation (1), and (1) will hold good all over the surface of the unit sphere. To determine any particular constant we multiply each of our (/; + 1)^ equations by A/tt A<^ times the coefficient of the constant in question in that equation and add the equations and then investigate the limiting form approached by the resulting equation as /^ is indefinitely increased. As p is indefinitely increased the summation in question will approach an integration; and since d/jbd4>^= — sin O.dO d(f> is the element of surface of the unit sphere, and as the limits — 1 and 1 of yu, correspond to tt and of the integration is a sm^face integration over the surface of the unit sphere. In determining any coefficient as A^„^ in (1) the first member of the limiting form of our resulting equation will be p7) COS n(^ P;^,(/x)d/j,. In the second member we shall come across terms of the forms Cdcf) Tsin Icj) cos ncji P„[(iii)P,][(fi)dfx,, Cd(f> Tcos left cos ncl> P,l{^^)Pl{p)dfjL, 0-1 0-1 2t 1 2T 1 Cd(i> Tsin «<^ cos ?? [P,;;(/A)]'^r//ti, p/) and F,(At, <^) are Surface Spherical Harmonics of different degrees. If we are determining a coefficient B„,„ the only difference is that sin n(f> and cos n(j) will be interchanged in the forms just specified. 202 SPHERICAL HARMONICS. [Akt. 105. 105. The integral over the surface of the unit sphere of the product of two Surface Spherical Harmonics of different degrees is zero. That is lfdj Y,{fi, ) Y„,{fji, 4>)dp, = . (1) -1 For as we have seen V'^=)^Yj(/j,, ) and F:= ?-'"y,„(yLi, (f>) are solutions of Laplace's Equation. Hence by Greenes Theorem C{UD„ V— VD„ Zr)ds = V. Art. 92. J}^^ U =D,.U= Ir' - 1 Yi(fi, ) ; UD„ V- VD,, U= („> - ly-^'"-^ Y,(f., H-l)Y,(fx,ct>)Y.„^(fi,ct>) on the surface of the unit sphere; and (m - l)j^Y,(fi, (/>) !;„(/., )ds = (m ~ Ij^dcfyJ Y,(fx, <^) Y^(fM, )dfi, = —1 277 1 Jrf<^J'F,(/x,)Y-,„(/.,c^)f7/. Hence unless / = ?h 277 1 EXAMPLES. 1. Obtain (1) Art. 105 directly from the equation m(m + 1) Y,^(fi, ) = V. (3) Art. 102, and Art. 91. 2. Show that the integral over the surface of the unit sphere of the product of two Tesseral Harmonics of the same degree but of different orders is zero. Suggestion : 27r 2t 2ir j sin k

= j sin l-cf> sin lcf>.d(f) = j cos kef} cos Iffi.dcj) := . 106. CFp(fi)FZ(luL)dfi = unless I = (m±nll .^ ^_ 2m + 1 (m. — n) ! Chap. VI.] DEVELOPMENT IN SPHERICAL HARMONIC SERIES. 203 For by integration by parts. Replacing n by n — 1 in equation (2) Art. 84 and remembering that — ^^"'j' is a possible value of s<«-i' we get (1 - .') '^9;i#-^ - -"«. ^ + [«(». + 1) - «(« - 1)] '^^ = 0, or if we multiply by (1 — /x-)"~'^ + ^;,, + n)^ni - „ + 1)(1 - /i^)«-^ '^" /f!'{^^ = 0, (111 or Hence follows the reduction for mala — 1 Using this formula n times we get -i • -1 =■- unless / ^ /». 2 (w+».)! .„ , = 7, ]— T 7 t"i ^^ l = tn 2m -\-l {m — m)! V. Art. 89 (4) and (5). 204 SPHERICAL HARMONICS. [Art. 107. 107. We are now able to complete the solution of the problem in Art. 104 and since j cos^ nt^t-dc^ = j sin^ n4>.d = tt and | d(^ = 27r we get as the coefficients in (1) Art. 104 An, = ^-^^P^ffifl, corresponding to points on the unit sphere, provided only that the given function satisfies the conditions that would have to be satisfied if it were to be developed into a Fourier's Series. If we use fXi and c^i in place of fi and cf> in (1), (2), and (3), we can write (4) in the form +X(^^!/'^^^/^^'^^ ' <^i)^."(/^)^".(/^0 COS n(cf> - cl>,)df,,'j . (5) Formulas (1), (2), (3), and (4) are convenient for actual work; (5) is rather more compactly written. 108. As an example let us express sin^ cos^ sin ) = - ,x\l — fx^) sin 2 . _2jn + l A-- Sir jli\l - fi')P,,(fi)dfiPm 2^.d4> = , Chap. VI.] ILLUSTRATIVE EXAMPLE. 205 A,,n = 2w. -\- 1 (vi — n) ! 47r im + n)\j ^'^^ ~ l^')Pl{l^)dfxj sin 2<^ cos n.d = ., 5„„„ = —^ ■ ^^^; J /^-^(l - A^^)P;,;(/.)r//. Jsin 2<^ sin .4, 1 = 720 C{iJi,' - lydfi = '^'^^^ if /M = 4 , -^ ^ d _ J_ 9 2J 4096 _ 1 ^^ '•■'~2*4!'4'6!' 7 ~ 105 ' By a like process we find ^23 = and B.o^—- Hence 42 sin^ ^ cos^ sin <^ cos c^ = — P.|(/Lt) sin 2<^ + -— P|(/ti) sin 2<^ , (1) 4_ lOo = — sin- ^ sin 2c^ + :q sin'^ ^ (7yLi2 — l) sin 2<^ . (3) The required expression might have been obtained without using the formulas of Art. 107, by a very simple device, as follows: sin^ 6 cos^ 6 sin cos <^ = - /i- sin- d sin 2 2. Show that cos r 5 9 2' 134' "1 2c/> = 2 cos 2c^ [- Piif.) + -^- P|(/.) + -gy-- P|(/x) + •••]• 3. If in a problem on the Potential Function F=/(/i, <^) when ?■ = «, we shall obviously have ^=X^* [A,«^™(/^) +X(A,n COS .^<^ + P„,^ sin ncf>)P;:,(,u,)~] m ^ n = 1 -" at an internal point and ^^i)^' \^ArnP,M +X(A,,n COS ^C^ + P„,„, Sin ?^«^)P»(/x)] at an external point, where Aq,„, A„„^, and B„„^ have the values given in (1), (2), and (3) Art. 107. 4. Solve problems (3), (4), and (5) of Art. 94 for the case where Fis not symmetrical with respect to an axis. 109. Any Solid Spherical Harmonic r'"I'„(yu., ^) being a value of F that satisfies Laplace's Equation in Spherical Coordinates will transform into a function of x, y, and z satisfying V^F^O if we change to a set of rectangular Chap. VI.] ANOTHER DEFINITION OF A SPHERICAL HARMONIC. 207 axes having the same origin and the same axis of X as the polar system. Moreover the new function will be a homogeneous rational integral Algebraic function of a;, y, z, of the mth degree. For each term of r"* cos w^P;/,(/x) is of the form Qj,m cos»--^<^ sin-^'<^ sin" 6 cos'"--'-" where 2k ) is a homogeneous rational integral Algebraic func- tion of the mth degree in x, y, and ,~. 110. Any homogeneous rational integral Algebraic function S.^(x, y, z) of the mth degree in x, y, and 2;, which is a value of V satisfying V^F=0 con- tains 2))i-\-l arbitrary constant coefficients. For S„^(x, y, z) will in general consist of ^ '-^ ■ — ^ terms and will therefore contain ^ -^— — coefficients. V^'?,„(.r, y, z) will be homogeneous of the (m. — 2)d degree and will contain ^ ~ coefficients, Avhich, of course, Avill be functions of the coefficients in S,„(x, y, z). Since V-'S'„,(.7', y, z) = independently of the numerical values of X, y, and z the .-, coefficients in \7'^S,,^(x., y, z) must be separately zero, and that fact will give us — —^ equations of condition between the ^^ -^~ — '-^ original coefficients and will leave ^^ ^ — '^^—^ ~ — '--_ '- or 2m. -}- 1 of them undetermined. S.„^(x, y, z) contains, then, the same number of arbitrary coefficients as r"^Y„^(iJ,,<}>). We can then choose the coefficients in r'" F,„(/Lt, ) so that it will transform into any given Sjj^(x, y, z). Consequently a Solid Spherical Harmonic of the m\X\ degree might be defined as a homogeneous rational mtegral Algebraic function of x, y, and z, S,^(x, y, z), of the mth degree satisfying the equation \^'^S„^(x, y, z) =0; and a Surface Spherical Harmonic of the mth degree as such a function divided by (a-2 + y2 _^ z')'^, that is by r"K 208 SPHERICAL HARMONICS. [Art. 111. EXAMPLES. 1. Show that if S„^(.^•, y, z) is a Solid Spherical Harmonic of the mth degree V^[r".%„(.r, 7/, -t)] = n(2m + n + 1)/--^S;„(^, y, ,t;). Suggestion : V'S„, = 0. VV = ^- D,S^ = '"f^- (I),ry+(D^ry+(B,7f = l. 2. Show that if f„(x, y, z) is a rational integral homogeneous function of x, y. and z of the ?;th degree it can be expressed in the form f„{x, y, z) = S„{x, y, z) + r\S,,_,(x, y, z) + 7-'S„_,{x, y, z) + ■ ■ ■ , (1) terminating with r"-^S'i(.T, y, z) if n is odd, and with 7^''So(x, y, z) if n is even. Suggestion: If a term rS,^_-^ were present in the second member of (1), and we were to operate with V^ on both members we should by Ex. 1 have a term — '?„_! which would be irrational when all the other terms of the resulting r equation were rational. No such term, then, could occur. In the same way it may be shown by operating twice on (1) with V^ that there can be no term »'^*Sv,_3 in (1); and thus step by step we can reach the result formulated in (1). 3. Express x-yz in the form Si + r^S.2 + ?'*'%• Suggestion: tet x^yz — Si-\- r^S. + >'^So and take V^ of both members we get 2yz = US, + 20r^So. Operate again with V^. = 120>S'„. Whence S,^(), S, = \yz, and S, = )^{Qx-' -f - z')yz. 4. Express sin^ B cos- sin ^ cos <^ in terms of Surface Spherical Harmonics. Suggestion : sin- 6 cos^ 6 sin <^ cos <^ = — ^ • Eor result v. Art. 108 (3). 111. A transformation of coordinates to a new set of axes having the same origin as the old set will change a given Surface Spherical Harmonic into another of the same degree. Eor such a transformation does riot change the form of Laplace's Equation V^F=0 if both sets of axes are rectangular, and it is effected by replacing x, y, and z in the Solid Harmonic correspond- ing to the given Surface Harmonic by x cos a^ -\- y cos a^ + z cos a^, X cos (3i-{-y cos jSa -f- s: cos jSs , and x cos yi + y cos yo + « cos yg respectively, where the cosines are the direction cosines of the new axes, and it will leave Chap. VI.] LAPLACIANS. 209 the function a homogeneous function of the mth degree in the new variables, and on dividing this by the vith power of the unchanged radius vector we shall have a Surface Spherical Harmonic of the mth degree. 112. We have seen in Art. 75 that if (x^, y^, «i) are the coordinates of a given point V = , ^ -^ (1) is a. solution of Laplace's Equation V'^r^O, and transforming to spherical coordinates that V= . - ^ (2) V?-2 — 2?Vi[cos 6 cos 6 1 + sin B sin di cos (<^ — <^i)] + *f is a solution of rD:^{r V) + ^^ A(sin D^ V) + ^^ D^ V^ . (3) If y is the angle between the radii vectores /■ and r^ of the points (x, y, z) and (j-i, ?/i, «i) (1) can be written V=^ , ^ (4) \)'- — 2r)\ cos y + i'l which must be equivalent to (2), and hence cos y = cos 6 cos di + sin 6 sin Ox cos (^ — <^i) . (4) which is a solution of (3) is of the same form as (5) Art. 75 and by developing it as we developed (5) Art. 75 we find that r=P,„(cosy) is a solution of the equation m(m + 1) F + ^ A(sin d A V) + ^-^ D^ V= (5) and that F= r"'P,„(cos y) and r= ^,;^ P^(cos y) are solutions of (3). If we transform our coordinates keeping the origin unchanged and taking as our new polar axis the radius vector of (xi, y^-, %). y becomes our new Q and P„j(cos y) reduces to P„j(cos &) , a Surface Zonal Harmonic, or a Leyendrian* of the mth. degree. It is then a Legendrian having for its axis not the original polar axis but the radius vector of (x^, y^, z-^. Since a Legendrian is a Sur- face Spherical Harmonic, P^(cos y) = P,„[cos 6 cos 6i + sin 6 sin 6i cos (<^ — <^i)J is a Surface Spherical Harmonic of the mth degree. « V. Art. 74. 210 SPHERICAL HAltMOXICS. [Art. 113. It is, however, of yery special form, since being a determinate function of fi, , /Ml, and (f>i it contains but two arbitrary constants if we regard it as a function of fx and ^, instead of containing 27)i + 1. It is known as a Lajdace's Coefficient, or briefly as a Laplacian, of tlie ?/ith. degree. We shall soon express it in the regulation form of a Surface Spherical Harmonic. The radius vector of (x^, y^, z^) is called the axis of the Laplacian and the point where the axis cuts the surface of the unit sphere is the'^^oZem _the Laplacian. We shall represent the Laplacian P,„(cos y) by L„^(/j,, ^, /^i. <^i). Of course Ljjjx. , 1 , <^i) ^ Pm(f^) ^= -Pm(cos 0), and is really independent of <^. 113. If the product of a Surface Spherical Harmonic of the mth degree by a Laplacian of the same degree is integrated over the surface of the unit sphere, the 47r result is equal to - — — — multiplied hy the value of the Spjherical Harmonic at the pjole of the Laplacian. That is, pz.^ Jr,,(^, <^:)L^{iM, ct>, fMi, ^,)d/ji = ^^i^ F„//xi, <^o . (1) Ti -1 Transform to the axis of the Laplacian as a new polar axis, and let -^^(At, <^) be the transformed Spherical Harmonic. L^^^/x, ^, fXi, <^i) will become P^(/a), and (1) will be proved if we can show that f.li.fz„(t^, «)P„W,V = ^^ ZJl, 0) . (2) —1 . Z„,{/,, )P,M = AlP„,{/x)J +X^A, cos ncj. + B, sin n<^)P,;;(/x)P„.(/.) (V. (5) Art. 102). Jz,„ (/x, <}>)Prn(f^)d<}> = 27rAlF^(,x)Y and fd/.yz„Xf^, cf>)PU,x)dct> = 2^^ A (V. (5) Art. 89) But Z„,(l, 0) = Ao, since P„,(l) = 1 and P;;,(l) contains (1 — 1)1 as a factor and is equal to zero. Hence (2) is proved. « Chap. VI.] LAPLACIANS. 211 114. We can now express a Laplacian in the regulation form as a Spherical Harmonic, by the formulas of Art. 107. X,„(/A, (j), fii, <^i) = P„,(cos y) = P„,[cos e cos 0^ + sin 8 sin Oi cos (<^ — ^i)J where 2m -\- 1 r* /* = ^^ 2;;rTi ^-^^'^ = ^-^^^^ ^^ ^^^ ^''- ^^^' -^^^^ (m + ^oJ J ^^^' '^' ^" '^'^ "^^^ nP:Xf^)dfx, 2(ni — ??.) ! 2m-ML (mj— w)_! ^4-^0!. ^,„ + ,,) I sin n4,F,:(fx^) by (1) Art. 113, and J,„,t- = -'^H.A- = A,i- = by Art. 105 unless k^m . Hence i,„(/x, c/>, ;.!, -- 1 X ^^^ ■©^ ct' 1 ^^ + s 3 "-^ CC s. o o" 1 1 § rH a. CJ g ^ "s. 1 t<-. w "s. o — I't' ^ITJH i t*^ § -IS -. — N M »:! k:: ►-; k;; 4 CilAP. VI.] SOLUTION BY DIRECT INTEGRATION. 218 EXAMPLE. Work the problems of Art. 108 and Art. 108 Exs. 1 and 2 by the aid of (3) Art. 114. 115. Such problems as we have handled in Arts. 98 and 99, and also prob- lems differing from them in not having circular symmetry about an axis, can now be solved by direct integration. For instance let it be required to find the value at an external point of the potential function due to the attraction of a solid sphere whose density at any point is proportional to the product of any power of the radius vector by a Surface Spherical Harmonic. Let p = Cv^ Y,n{l^ii 4>i) • Then using our ordinary notation we have « 27r 1 r r - 1 y>-'^ — 2r>-i c( cos y + ?f But by (3) Art. 77 1 1 r r, , = - Po(cos y) + - Pi (cos y) V/r'^ - 2rr, cos y + rf r L °^ ^^ ^ >' '^ ^' + ^; P,(cos y) + • ■ • + ^" P,„(cos y) + • • •] if r>r,. Consequently since iTt 1 - 1 V reduces to the single term a 277 1 (I 0-1 47r(7 a"' + ^ + 3 Y„,(fi,cl>) 2m + 1 m-\- k + 3 + 1 214 SPHERICAL HARMONICS. [Art. 115. EXAMPLES. 1. Solve by direct integration the problems worked in Arts. 98 and 99 and Examples 1, 2, 3, and 4 of Art. 99. 2. The density of a solid sphere is proportional to the product of the squares of the distances from two mutually perpendicular diametral planes; find the value of the potential function at an external point. Ans. p = A-/f cos"^ 6i sin- 6i cos"^ ^i kr \' [^ ^o(y^i) + ^ A(i^O + ^ cos 2^>- Vo = AoPo(cos ^) + J 1 - Pi (cos 0) + ^2 - Po(cos 0) + --- if r a. When r = a V^ + 1^ = . Hence m ma ma^ Ao — — —, A^———, A2— TT'"* Chap. VI.] AXES OF A SPHERICAL HARMONIC. 215 and F2=-^rPo(cos^)+^Pi(cos^) + pP2(cos^) + ---] ii' ra. Hence the effect of the induced charge is precisely the same at an external point as if the sphere were replaced by -^ units of negative electricity con- centrated at the point /• = — , ^ = . v. Peirce, Newt. Pot. Punc, § 66. 116. If the two points P and P' are taken on the line OH whose direction cosines are \, fx, and v, and if u and u' are the values at P and P' of any con- tinuous function of the space coordinates, then . is called the partial derivative of ti along the line OH and will be represented by D/,u. Let X, y, z be the coordinates of P and x + \x, y -\- Ay, z-\- ^z the coordinates of P'; then w' — u = D^u.i\x + DyU.Ay + Z>, u.Az -f- e where e is an infinitesimal of higher order than the first if Ax, Ay, and As are infinitesimal (v. Dif. Cal. Art. 198). 2i' — u _ A.i- , ^^ A?/ , ^ Az , e Hence -^^ = P*^ u. ^^ + D^ u. ^^ + P, u. ^^ + -^^ • T> , Acc ^ Ay T A« But := A , — ^ = u , and = v . PP' ^' pp' -"' ''"'' PP' Therefore D,^ u = XD^ n + /xP^ « + vP, « . (1) If V^i/^0, D'^D^D^u is a solution of Laplace's Equation. Por \7'XPl'P>^D;uj = P^'P;P;( V ?0 = • Hence if V^w = P^?< is a solution of Laplace's Equation, and if OHi, OH2, OHs, • • • are a set of lines through the origin P^^P^^P^^- --u is a solution of Laplace's Equation. 117. If H/^. is a rational integral homogeneous Algebraic function of x, y, and z of the A;th degree IxH, g,_,^ IxH, ,-'g,._, yi+2 r" ^.i j.i + 2 "T ,^.i + 3 » rr and is of the form -^^ . 216 SPHERICAL HARMONICS. [Art. 118. The same thing can be proved of Dy (-7 ) and D^ \~t) and therefore liolds good of A (f'). If w is a homogeneous function of x, y, and z of the degree — m — 1 and V2?< = then V^(>-'" + ^w)=0. \j2^^sm + 1 „) ^ ^2m + 1) (2m + 2)/-2'»- ^ u + 2(2m+ l)r2'»-i(a;D^?t + //D^^w + zD^u) + /•2'» + iV^it = 0, since xDj.u + ?/-0j^?^ + si)^?/ = — (m + l)?^ by Euler's Theorem (v. Dif. Cal. Art. 220). 118. — = =^=i is a solution of Laplace's Equation (v. Art. 75) '• S/x' + t/ + z' TT and is of the form — - • r an D^^JD^^D^^'--D^^i^) is then a solution of Laplace's Equation by Art. 116; is of the form degree — in — 1 , TT it is of the form ^J'^ ^ by Art. 117 and is a homogeneous function of the Therefor ■e ?" ^I),,^D/^^B^^---D^^l—) is a solution of Laplace's Equation, and is a rational integral homogeneous Algebraic function of x, y, and z of the mth degree, and is consequently a Solid Spherical Harmonic of the ???th degree (v. Art. 110) ; and 7-"' + ^AiA.Ag- ' " A,„ (^) is a Surface Spherical Har- monic of the ??ith degree. Moreover since the direction of each of the lines OHi, OH^, • ■ • 0H„^ depends upon two angles which may be taken at*pleasure, these angles and M are 27n + 1 arbitrary constants and may be so chosen that r"'-^^D^ D^_^- • ■ I)^^^ I — j may be any given Surface Spherical Harmonic. Consequently any given Surface Spherical Harmonic may be regarded as formed by differentiating ^ successively along m determinate lines OH^, OH^ • • • OH^, and is given except for the undetermined factor ilf when these lines are given. The lines OH^, OH^, OH^, • • • Olfj^ are called the axes of the Harmonic, and the points where they meet the surface of the unit sphere the j^oles of the Harmonic. The m axes of a Zonal Harmonic coincide with the axis of coordi- nates (v. Art. 86) and consequently the m axes of a Laplacian coincide with what we have called the axis of the Laplacian (v. Art. 112). CiiAP. VI.] ROOTS OF ZONAL AND TESSERAL HARMONICS. 217 119. Any Surface Zonal Harmonic PrtiitJ) is equal to zero for m real and distinct values of /x which lie between — 1 and 1 ; and any Associated Func- tion Pliii) is equal to zero for m — n real and distinct values of y. which lie between — 1 and 1 . —^ — : — — contains (fx-— Xy-'' as a factor, v. Art. 89. From Eolle's Theorem, " If f{x) is continuous and single-valued and is equal df(x) to zero for the real values a and h of x, -"^-^ is equal to zero for at least one real value of x between a and i," (v. Dif. Cal. Art. 126) it follows that since d(a- — 1)'" (/x- — 1)"' = when /^ = — 1 and when /x = 1 ^ =0 for at least one value of u between — 1 and 1. -^'^—-, — cannot be equal to zero for '^ dy more than one value of \x between — 1 and 1, for it contains {y^ — 1)'"~^ as a factor and is a rational Algebraic polynomial of the 2m — 1st degree. In like manner we can show that ^ , ^ — — = has m — 2 roots equal to d/x'' — 1, 111 — 2 roots equal to 1 and two real roots between —1 and 1 which separate the three distinct roots of ^ — =0; and in general if k < ?/^ + 1 that — ^^-, ^0 has m — k roots equal to— 1, m — k roots equal to 1, '^-^ rf^-Yu--l)"' and k real roots separating the k + 1 distinct roots of jlT^i ~ ^ • \ clm(fj;i — Y)in ^^ Hence P„Yu) = or ^ , — =0 has in real and distinct roots '"^^ 2'"»i! rt/u,'" between — L and 1, and it has no more since it is of the int\\ degree. The same reasoning shows that ; — = has m — n distinct real dfx'"-^" roots between — 1 and 1, and therefore that P,'',(/U,) is equal to zero for m — )i distinct real values of /x between — 1 and 1. Since P,;;(/u) contains sin" 6 as a factor it is also equal to zero when /x = — 1 and Avhen fi^l . cos n is equal to zero for 2?i equidistant values of <^, and sin n is equal to zero for 2>i values of <^. Hence any Tesseral Harmojiic sin nP;,',(iJi) or cos Ncl>P^[(fx) is equal to zero for 2n equidistant values of <^, for /x ^= 1, for /x = — 1, and for i7i — n real and different values of /x between — 1 and 1. It follows that the value of any Surface Zonal Harmonic -P,„(/x) at a point on the surface of the unit sphere will have the same sign so long as the point remains on one of the zones into which the surface of the sphere is divided by 218 SPHERICAL HARMONICS. the m circles of latitude corresponding to the m roots of Pm(H') = ^? ^^^ "^^^^^ change sign whenever the point passes from one of these zones into an adjoin- ing one; and that the value of any Tesseral Harmonic sin 7i(f>P,l(fM) at a point on the surface of the unit sphere will have the same sign so long as the point remains on any one of the tesserae into which the surface of the sphere is divided by the m — n circles of latitude corresponding to the roots of P/,|(//.) = and the 2n meridians corresponding to the roots of sin n + l) + 2''.2!(/i + l)(7i + 2) ~ 2«.3!(w + l)(« + 2)(7i + 3) "" J ^^^ and is called a Cylindrical Harmonic or BesseVs Functio7i of the wth order ; and that unless n is an integer z = AJ„(x) + BJ_„(x) is the general solution of Bessel's E(]uation. *The student should re-read carefully Arts. 11, 17, and 18{cl) before beginning this chapter. 220 CYLINDRICAL HARMONICS. [Art. 121. If n is an integer it can be shown that J-„(aO = (-l)"JL„(x), (v. Forsyth's Diff. Eq. Art. 102), and then is the general solution of Bessel's Equation and {A-„(.)} = U.) log . - 1 (g'^k^JL^ @^'- V. M. Bocher, Ann. Math. Vol. VI, No. 4. 121. A useful expression for Jj,{x) as a definite integral can be obtained without difficulty from Bessel's Equation [(5) Art. 120] by a slight modifica- tion of the method given by Forsyth (Diff. Eq. Art. 136). It was shown in Art. 17 that z = x"v is a solution of Bessel's Equation if V satisfies the equation dx^ X dx Assume = CTQQ^{xt)dt (2) where x and t are independent, T is an unknown function of t, and a and h are at present undetermined. Then ^ = - p T sin {xt)dt h d-v r and -^., = -J t-T cos {x£)dt . Substituting in (1) after multiplying through by x, we have h h ("(1 — f)Tx cos {xi)dt — ({'In + \)tT sin {xt)dt = . (3) Chap. YII.] BESSEL'S FUNCTIONS AS DEFINITE INTEGRALS. 221 By integration by jparts we find that h 6 ((\ — f)Tx cos {xt)dt = r(l — f-)T?iin (xt)~\ a (, ^ ~/[^^ ~ ^'^ Tt~ ^^^] '"' ^^^'^^^^ ' and (3) reduces to r (1 - f) T sin {xt) 1 - j"|^ (1 - t'i^ + (2m - l)i^r1 sin {xt)dt = 0. (4) If we determine T so that a-f')^+(^>^-'^)fT=o, (5) and a and ^» so that (1 — t^)T sin (xt)\ = (6) (4) will be satisfied and our problem will be solved. (5) gives T=C(l-t^r-i, (7) and (6) will obviously be satisfied if re = — 1 and b = l. ^, r(l — f^Y cos (xt)df . , . , ,, Hence v= C \ ~ ; — ^ — is a solution of (1), I and z=C X" r(l - tr cos (xt)dt 3. is a solution of Bessel's Equation. If we let t =^ cos ^ in (8) we get C x" I sin'^" fji cos (x cos cf>)d. Expand cos (x cos ^) into a series involving powers of x cos <^, integrate term by term by the aid of the formulas f sin" x.dx = ^- ^ - , [Int. Cal. (1) Art. 99], ^ ' rg + i) 222 CYLINDRICAL HARMONICS. [Art. 122. I sin" X cos'" x.dx = 2r(^'+i) (Int. Cal. Art. 99 Ex. 2), and compare with (6) Art. 120, and we get J^(x) = — I sin^" — - I sin-" <^ cos (^ sin (x cos <^)f/c^ = ^ — -j^ j sin-" + - ^ cos (ic cos (f>)d(j>. II 3. Prove by integration by jmrts that if n"^ - I sin-" ^ cos , if n> ~\ ^C^=^ = —'...(.) (3) (2) can be written Jx"J,, _ , (x)dx = x"J, (x) (4) if 71 > ~ (2) and (3) can be written and X- " ^^ - nx- " - './„ (.r) = - .7- "J,, ^ ^ (x), ^-^-.(^)-^^(^) (5) whence 2 ?^ = ,/„ _,(..) - ,/„ ^ .(.r) and (J) Jn{^)=jn-.{X)+'1.^,{X). (8) The repeated use of formula (8) will enable us to get from J^ix) and Ji(x) any of Bessel's Functions whose order is a positive integer. For example, we have -^3(-r) = (|-l)^i(^-)-^^o(^). 22-i CYLIXDillCAL HAKMONICS. [Art. 122. From a table giving the values of Jo(x) and Ji(x), then, tables lor the functions of higher order are readily constructed. Such a table taken from Eayleigh's Sound (Vol. I., page 265) will be found in the Appendix (Table VI.). By the aid of (5) and (6) any derivative of Jn(x) can be expressed in terms of J„(x) and J„ + i(x). For example If we write Jo(x) for z in Fourier's Equation [(2) Art. 120], then multiply through by xdx and integrate from zero to x, simplifying the resulting equa- tion by integration by payts, we get x'^^^^ ^xJ,(x)dx = ^; whence by (1) ( xj,,{xyx = xJ^(x) . (9) If we write Jq(x) for z in Fourier's Equation, then multiply through by ^2 — ^LJ ^x and integrate from zero to x, simplifying by integration by parts we get whence by (1) ^xi,T,{x)Ydx = |' [(Jo(^-))'^ + ('A(-^))^]. (10) In like manner we can get from Bessel's Equation [(o) Art. 120] the formula Jx(J^(x)ydx = \ [.r^ (^^) + {^' - nWn{^)y~] (11) which (6) enables us to reduce to the form J;r07„(a;))2fZ^ = |'[(J-„(:r))^+ (.7„^i(x))-^] -..a-J„(^K^i(a;). (12) Formulas (9), (10), (11), and (12) will prove useful when we attempt to develop in terms of Cylindrical Harmonics. Chap. VII.J PROPERTIES OF BESSELS FUNCTIONS. 225 Values of J„(oc) for Larger values of x than those given in Table III., Appendix, may be computed very easily from the formula 2 ! (8.')--' _ +^ — 4178^0^ — rx-i-"v 3!73^)-3^ + ■ ■ • J"K" -4 - ''V- (1'^ > v. Lommel, Studien fiber die Bessel'schen Functionen, page 59. The series terminates if 2n is an odd integer, but otherwise it is divergent. It can be proved, however, that in any case the sum of 7n terms differs from J„(x) by less than the last term included, and consequently the formula can safely be used for numerical computation. EXAMPLES. 1. Confirm (1), (2), and (3), Art. 122, by obtaining them from (3) and (6), Art. 120. 2. Confirm (1), Art. 122, by showing that Fourier's Equation will differ- entiate into the special form assumed by Bessel's Equation when w = 1. 3. Show that (9), Art. 122, is a special case of (4), Art. 122. 4. Show that the limit approached by J„(x) as n increases indefinitely is zero, and by the aid of this fact and of (8), Art. 122, prove that ^-i(-^-) =;[^^^«(-^-) - (:n + 2)J„^,(x) + (,^ + A)J,^,(x) + •••], 2 X 5. Prove that dJ„(x) 2 dx .ii^^^"('^"> - ('^ + 2).^„,,(a') + (M + 4).7-„,,(x-) -•..]. 6. Show that the substitution of ( 1 — —,) for x in Legendre's Equation will reduce it to the form \ //-/ (h/- \// n-f an \ nf and that the limiting form approached by this equation as n is indefinitely increased is Fourier's Equation, and hence that Jq{x) can be regarded as some constant factor multiplied by the limiting value approached by -P„(l— — j" as n is indefinitely increased. 226 CYLINDRICAL HARMONICS. ' [Art. 123. To complete the solution of the drumhead problem taken up in Art. fl, ,\" found that it would be necessary to develop a given function of r in the form /'(/•) = Ayl,(fx,r) + .LJ,(fi,r) + A,J,{fi,r) + • • • where /Xj, /Ug, /A3, &c., are the roots of the transcendental equation jQ^^fxa) = ; and in Art. 11, Ex. the development of unity in a series of precisely the same form was needed. (o) Let us consider another problem. The convex surface and one base of a cylinder of radius a and length h 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 centre of the base ; required the temperature of any point of the cylinder after the permanent temperatures have been established. Here we have to solve Laplace's Eqiuation in Cylindrical Coordinates ([xiv] Art. 1). ir-u + \ J>,.n + ]:, l>ln + l:n = (1) subject to the conditions ?( = when ,-; = K = '' /• = a »=/(;•) - z = h, and from the symmetry of the problem we know that D^u = 0. Assuming as usual u = R.Z we break (1) up into the equations ■^ + 7-^ + ^'^'='^' Avhence u = sinh (^fMz)J^(/jt.r) (2) and 11 = cosh (/xz)J^(/xr) (3) are particular solutions of (1). If fi^. is a root of f^oif^a) = (4) n. = sinh (fx,^.z)J,,(fj.,,i-) satisfies (1) and two of the three equations of condition. If then /(;■) = JA>(/^,r) + A„J,(fi,r) + A,J,(tM,r) + • • • (5) fi^, fXo, /Xg, &c., being roots of (4), sinh (/ii.~) X I r sinh (fj...t) sinh (^3,-:) satisfies (1) and all of the equations of condition, and is the required solution. Chap. VIL] FLOW OF HEAT IN A CVLINDEK. ^27 (I)) If instead of keeping the convex surface of the cylinder at thr tempera- ture zero we surround it by a jacket impervious to heat, the ("quation of condition » = when r = «. will be replaced by D,.u = when r = a, or if u = sinh (fxz)Jo(fir), by — -ALL^ — when r = a, that is by ^^^,'(^,,) = * or (v. (1) Art. 122) by ^i(/ur/)=0. (7) If now in (5) and (6) /Wj, [x^, /j.^, &c., are roots of (7), (6) will be the solu- tion of our new problem. (c) If instead of keeping the convex surface of the cylinder at the tempera- ture zero we allow it to cool in air at the temperature zero, the condition u = when r = a will be replaced by D,.n -\- //// = when r = <(■, or if n = siuli (/>i,?)t/y(/xr) by /xJo(/xr) + /^/o(/x?-) = when r = a "that is by fiaj^'dxa) + ahJo(fia) =0 or (v. (1) Art. 122) by fi((J^(fxa) — ahJ^t(fx,a) = 0. (8) If now in (5) and (6) /x^, /Uo, fis, &c., are roots of (8), (6) will be the solution of our present problem. 124. It can be shown that Jo(x) = (1) M.r)=0 (2) and a:Jo'(x) + A'/o(.r) = (3) have each an infinite number of real positive roots (v. Riemann, Par. Dif. GL, § 97). The earlier roots of these equations can be computed without serious difficulty from the table for the values of Jo(x) (Table VI., Appendix). The first twelve roots of Jo(x) ==0 and Ji(x) =0 are given in Table IV., Appendix, a table due to Stokes. Large roots of Jo(x) = and of Ji(x) = may be very easily computed from the formulas ^ _ . _ 9r . -050661 _ .053041 .262051 _ x^_ , ^^ .151982 , .015399 .245835 , ^-'■^-' 4.S+1 +(4^ + 1)^ (4. + l)^^"" ^^ given by Stokes in Camb. Phil. Trans., Vol. IX., x'-'^ representing the sth root of Jo(x) = 0, and a-/o(/^,/-) are solutions of V^U=0 and \7'^V=0 if we express Laplace's Equation in terms of Cylindrical Coordinates (v. (1) Art. 123). Hence, if fdS represents the surface integral over any closed surface, we have J(iW,V- VD,U)dS^O by Green's Theorem (v. Art. 92). If we take the cylinder of Art. 123 as our surface, and perform the integrations and simplify the resulting equation, we find CrJ^,(fl^r)Jf^(^x^7•)d^'=— :; [/i^o, Jo(M;«)^/(^/;'0 " A^/''-^o(/iA«)^)'(/^;«)] Hence if jx^. and /Lt, are different roots of j;,(^a) = (), or of ^i(/^''0=*'; or of iJiaJy(fia) — \J^,{/j.'i) = 0, then JrJ,(lx,r)J,(/xp-)dr = 0. (2) EXAMPLE. Obtain (1) Art. 125 directly from Fourier's Equation di^^ r dr ^ 126. We are now able to obtain the developments called for in Art. 123. Let /(/•) = A,J,{^l,r) + A,J,{fi,r) + A,J,{^l,r) + • • • (1) /*!, /X2, /As, &c., being roots of J(t(/xa) =0, or of Ji(fin) =0, or of /xuJi(/jLa) — \Jf)(/jLa) = 0. To determine any coefficient A^ multiply (1) by r Jf^{^fx^r)dr and integrate from zero to a. The first member will become jrf{r)J,{fi,r)dr. Chap. VII.] CYLINDRICAL HARMONIC SERIES. 229 Every term of the second member will vanish by (2) Art. 125 except the term A,fr(J,(fi,r)ydr. fr(J,(fi,r)ydr = - fx(J,(x)ydx = '^ [Wf^,a)y + (^i(/^,«))'] by (10) Art. 122. «^"- -^'- = ,.t(^„(...))' +(■/ ,(.,>■))-] fr'-^'^'^'-'-'^"- (^> The development (1) holds good from r = to r = a (v. Arts. 24, 25, and 88). If fii, fio, fi^, &c., are roots of Jq(/j-<') =0, (2) reduces to If fii, /j,2, fji's, &c., are roots of Jii^fid) =0, (2) reduces to If /Lti, /i2, yu-3, &c., are roots of fiaJi^/Mu) — XJ,^(/xa) =0, (2) reduces to For the important case where /(;■) = 1 ^^ "^ ' ^^ by (9) Art. 122, and (3) reduces to 9 J — - , (7) (4) reduces to J^. = except for k = l when i^f. = and we have Ai= ], 2X (5) reduces to -4„ = ^^^,-j_-^^^^_^ . (8) 230 CYLINDRICAL HARMONICS. [Art. 127. EXAMPLES. 1. Show that in (12) Art. 11 any coefficient A^. has the value given in (3) Art. 126 ; and in the answer to Art. 11, Ex. the value given in (7) Art. 126. 2. Show that if a drumhead be initially distorted so that it has circular symmetry, 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 when the distortion has circular symmetry are proportional to the roots of Jo(a') = ; that the possible nodes for such vibrations are concentric circles whose radii are proportional to the roots of t7o(.r) = 0. 3. 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 temperature is set up. Find the tempera- ture at points on the axis 25 cm., 50 cm., and 75 cm. from the base, and also at a point 25 cm. from the base and 50 cm. from the axis. Ans., 29°.6 ; 47°.6 ; 71°.2 ; 25°.8. 4. An iron cylinder one meter long and twenty centimeters in diameter has its convex surface covered with a so-called non-conducting 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°. Find to the nearest tenth of a degree the temperature of the middle point of the axis, and of the points of the axis twenty centimeters from each end after the temperatures have ceased to change. Given that the conductivity of iron is 0.185 and of cement 0.000162 in C. G. S. units. Find also the temperature of a point on the surface midway between the ends, and of points on the surface twenty centimeters from each end. Find the temperatures of the three points "of the axis, supposing the coating a perfect non-conductor, and again, supposing the coating absent. Neglect the curvature of the coating. Ans., 15°.4 ; 40°.85 ; 72°.8 ; 15°.3 ; 40°.7 ; 72°.5 ; 0°.0 ; 0°.0 ; 1°.3. 127. If instead of considering the cooling of a cylinder as in Art. 123 we have to deal with a cylindrical shell whose curved surfaces are co-axial cylinders, we are obliged to use the Bessel's Functions of the second kind. Let our equations of condition be M 1= when s = , u=f(r) '' z = b, . Then (v. Art. 123) V = sinh {fx,z)^J,(i,,r) - 1^ Al,(/.,r)] u = when r = a, u = " r^c. Chap. VIT.] CYLINDRICAL SHELL. 231 where jx,. is a root of the equation ^o(y^«)-^^A'-o(;^«)=0 (1) will satisfy Laplace's Equation [(1) Art. 123] and all of the equations of condition except the second. is the required solution if /(,.) J^A, [,_^.(^„,) _^^ A-.(^,-)]. (3) The development (3) is easily obtained. Call the parenthesis for the sake of brevity B^^ix^r). Then by the method of Art. 125 we get if we integrate over our cylindrical shell jrB,{ti,r)B,{jiir)dy = (4) if [I,, and /i,^ are roots of (1) ; and by an easy extension of (10) Art. 122 jrlB,{f,,r)Jdv = h{!''W{l^^J - <'\B,;(^.,a)J}. (5) Determining the coefficients in (3) as in Art. 124 and simplifying by the aid of (4) we have (6) EXAMPLE. If a membrane bounded by concentric circles of radius a and radius h, and fastened at the-edges, is initially distorted into a form symmetrical with respect to the centre, and then allowed to vibrate y = ^A, cos {^Ji,ct) [ J-„(/.,r) - ^^^ K^if^.r)^ where A,, is obtained from (6) Art. 127 by replacing c by h. 232 CYLINDRICAL HARMONICS. [Art. 128. 128. If in the cooling of a cylinder u = when ;^ = 0, u = when z = b, and u =/(s) when r = a, the problem is easily solved. If in (2) and (3) Art. 123 fx is replaced by fii we can readily obtain ,v = sin(/A«)Jo(/^n) and z = cos (fiz) t/o(/^*'0 as particular solutions of Laplace's Equation [(1) Art. 123] ; and and is real. Jo(xi) _ 1 + ^ ^ 2M2 "^ 2-. 42. 6^ "*" M = XAsiJ^ where by Art. 31 (7) and (8). Hence is our required solution. 2 r b kirz sm-—— fZ« b /kirriX V^ . k'jrz'^'Vir) u = 2^ A„ sm b /k7rai\ (1) (2) (3) EXAMPLES. 1. If the cylinder is hollow and we have u^O when z = 0, u = when z = b, 11 = when r = c, and u ^=f(z) when r^= a ; then Xa kirz where A^. has the value given in (2) Art. 128, and Ko(xi) = Ko(x{) — Jo(xi) log i = Jo(xi) log aj — ; X\ + h) ^ / kirc i\ -— / kircA\ (i+i+j) + 22 I 2^.4^^ 2^.4^.6^ [v. (4) Art. 120], and is real. 2. A hollow cylinder 6 feet long Avhose inner surface has the radius 3 inches, and whose outer surface has the radius 1 foot, has its bases and outer surface kept at the temperature 0°, and its inner surface at the temperature 100°, until Chap. VII.] TEMPERATURES UNSYMMETRICAL. 233 the permanent state of temperatures is established ; find the temperatures of two points in a plane parallel to the bases and half-way between them, one of which is 6 inches and the other 9 inches from the axis. Ans., 49°.6; 20°.2. 129. If in the problem of Art. 123 the temperatures of the points of the upper base of the cylinder are unsymmetrical so that u ^/(r,<^) when z = b, we have to get particular solutions of Laplace's Equation [(1) Art. 123] for the case where B^u is not equal to zero. We readily find that u =^ sinh (/jLz)[A cos n + B sin ?i']J^(/jL)-) and u = cosh (ijlz)[A cos n(j> + B sin nJrf(r,)J,(fi, ^o.k ■ TT a'^[Ji(/i^,o ')J B,y- = 0, A 2k a jd4Jrf{r,4>) ■" (1 n COS n(f>J„(ti^. r)dr A,k - «W + .:(/x.«)]-^ Cdcf) j rf(r,,,~ = when f = 0.z = when r = a, the solution is ^ = X X ^^^ ^^'''*^ ^^"'' ^°^ '^*^ "^ ^"' " ^^^ "'^)'^«(^'^'') where A^^^, B^^^, A^^,., and B„j, have the values given in Ex. 3. 7. What modifications do the statements made in Ex. 2, Art. 126, need to make them apply to the unsymmetrical case treated in Ex. 6 ? Show that any possible nodal system in Ex. 6 is composed of concentric circles and of radii whose outer extremities are equidistant, v. Eayleigh's Sound, Vol. I., Arts. (202-207). 8. Solve the problem of Art. 127 and of Art. 127, Ex. for the unsym- metrical case. Suggest ion: AJ„(x) + BK„(x) is a solution of Bessel's Equation. 9. Solve the problem of Art. 128 and of Art. 128, Ex. 1, for the case where u=f{z, -{- B sin 7i(f>)Jj,(fji7-i) is a solution of Laplace's Equation, and /'(.:;. cf}) can be developed into a double Fourier's Series [v. (15) Art. 71]. Chap. VII.] EXAMPLES. 235 10. Show that in dealing with a wedge cut from a cylinder by planes passed through the axis, or with a membrane in the form of a circular sector, it may be necessary to use Bessel's Functions of fractional or incommensurable orders. 11. BernouillV s Problem (v. Chapter IX). In considering small transverse vibrations of a uniform, heavy, flexible, inelastic string fastened at one end and initially distorted into some given curve, we have to solve the equation Diij-=(P'{xB^y -\- D^li), subject to the conditions I>^y^=^ when if = 0, y =/(.x) when i! = 0, ?/ ^ when .r ^ a ; the origin being taken at the distance a below the point of suspension and the axis of X taken vertical. Show that y'^Zj ^'^k ^^'^ ^'^'^-^ Ai(/^A-^) 1 k= 1 where B^{x) = l-~-\- ^^ - py|^, + • • ' ='/o(2v':^) and /i^. is a root of the equation ^fix) B,{ix^ x)dx jf(x) J,{2 fji, sl~x)dx 12. As a simple case under Example 10 consider the vibrations of a circular membrane fastened at the perimeter and also along a radius and then initially distorted (v. Rayleigh's Sound, Art. 207). In this case we must modify the formula given in Ex. 6 by dropping out the terms involving cos w^ and by taking •/< = — • The required solution is = X X ^'"'^ ^°^ ^^*^^ ^^^ 2 "^-(^'^■'■^ where /w^, is a root of ^^^ ^^^ = 277 a ^^ ^d<\>^rf{v,^) sin^J-^,(;U,r)cZ7 and ^--^-TT ct}{,J.!{ix,a)J 2 236 CYLINDRICAL HARMONICS. [Akt. 129. For the terms in which m is odd, Jm(x) can be readily obtained from (13) 2 Art. 122, whicli will become a finite sum. For exami)le, (13) Art. 122 gives the values M^) = - \ d [ (l + .^) «i^^ "^ + 1 cos x~j ; &c. 13. The question of the flow of heat in three dimensions involves a problem not unlike the last. Suppose the initial temperatures of all points in a sphere of radius c given, and let the surface be kept at the temperature zero. Then Ave have to solve the equation 7>,« =^;[a(-A.O +^,I'.(sin^D.«) +-^,^l«] (1) ([iv] Art. 1) subject to the conditions 7/^0 Avhen r = c, u =/(>; 6, ) when f = 0. If we assume u — T.B. V where T is a function of t onlj^, B of ;• only, and V of 6 and <^ only, (1) can be broken up into '^+u'a^T=0 (2) m{m. + 1) r+ ^ A(sin OD^ V) + ^ D^ V= (3) (4) Hence T= e" "'"=', V= Y^(fi, ) [v. Art. 102 (2)], and B is still to be found. If in (4) we let x = ar and z = Rsfar it becomes which is satisfied by z=^J^^^(x). (v. Art. 17.) Therefore ^ = "7= '^m + i(«»*)- ya?' Chap. \U.] FLOW OF HEAT IN A SPHERE. 237 f(r, 0,cf>)=^j^^ Cd4, Cf(r, $1 , ,,.)^ L = (D,xr+(n,j/f+(I),^f Y^={F>,xy+{Dpjjy^{D,^z)\ (2) Then if dui is the element of length normal to the surface pi = a, dn^ normal to po = h, and dn^ normal to pg = „ {^D,, r) + D,,(j!^D,, r-) + D,,iJ^D,, r)] , (6) and Laplace's Equation in our curvilinear system is 240 ELLIPSOIDAL HARMONICS. [Art. 132. If it happens that V"/3i = **' ^ == pi will satisfy (7) and we shall have h,hJi,Dj^\=^0. In like manner if V-p. = we have dJj^\=0, and if V-/03 = we have Dp/— y- ) := ; and therefore (7) reduces to hfir-^ F+ hW^V^ hlD^J= (8) when VVi = 0, VV2 = 0, and Ws == 0. 132. If instead of having the value of the Potential Function V given on the surface of a sphere as in our Spherical Harmonic problem, we have it given at all the points on the surface of an oblate spheroid, and are required to find its value at any internal or external point, we can easily get a solution by methods in no essential respect different from those already employed, if only we rightly choose our system of coordinates. If we take an ellipse and an hyperbola having the same foci, and revolve them about the minor axis of the ellipse, we shall get a pair of surfaces which are mutually perpendicular ; a plane through the axis of revolution will cut both the spheroid and the hyperholoid orthogonally. The equations of the three surfaces can be written : — i?,(;r,j,,«,A)=^ + ;^, + |;-l=0 (1) ^(.,2,,,,M)=^ + ^ + i;-l=0 (2) F^(x, y, z, v)=z — vx=^0, (3) where X^> ^^> yu-^, 2 b being the distance between the foci. For all values of A, fi, and v consistent with the inequality above written the surfaces (1), (2), (3) intersect in real points and cut orthogonally. X, fM, and V can be so chosen that the surfaces will intersect in any given point, and therefore can be taken as a set of curvilinear coordinates, and Laplace's Equation can be expressed in terms of them by the aid of Formula [xv] Art. 1. From (1), (2), and (3) we readily get b%l + v') i) 7.2 xV'i'' ^»2(l + v'') (4) Chap. VIII.] SPHEROIDAL COORDINATES. 241 u. A j/i'^ — u,^ av whence D),x = . > D^y = 7 \ 7:^ tt » D^z = - — ; h^l + v'' b^k--b' b\ll + v^ and A? = X^^T^ (^> [v. 130 (2)]. In like manner we get 1 _ A^-/i^ (6) and ^^ = ^^(1 + .-^)-^' ^^> and [xv] Art. 1 becomes which is Laplace's Equation in terms of our SpJieroidal Coordinates \, fi, and v. If now in place of A, ^l, and v we can introduce some function of A, some function of fi, and some function of v which, therefore, will represent the same set of orthogonal surfaces, and if we can choose three functions a, p, and y, which of course are functions of x, y, and z, so that V^a = 0, V^/8 = 0, and W =^ 0, equation (8) must reduce to the simple and sym- metrical form given in [xvi] Art. 1. These functions a, (3, and y are easily found. Equation (8) is V'T— expressed in terms of A, /x, and v. Assume that F is a function of A only ; then D^V=0, and i),F=0, and (8) reduces to A[AN/A--/>2.Ar] = o whence K i> In the same way we get ./^ = -M^, and /3--^^sech-^, (V. Int. Cal. Art. 46, Ex.) dy = 3-1 — :,' and y = ('3 tan"^ v. ' 1 -\- V- Substituting these vahies in (8) and taking (\ = — r^ = I>, and (-3 = 1, (8) reduces at once to or since X == i sec a , fi =^ b sech /3 , and v = tan y , (10) to cos^ a D; V + cosh^ jB D/ T + (cosh'^ yS — cos'-^ a)Z>^' T == (11) which is Laplace's Equation in terms of what we may call Normal Oblate Spheroidal Coordinates. In using (11) it is to be noted that the point whose coordinates are (a, /3, y) is the point of intersection of an oblate spheroid whose semi-axes are b sec a and b tan a, an unparted hyperboloid of revolution whose semi-axes are b sech ^ and b tanh )8, and a plane containing the axis of the system and making the angle y with a fixed plane ; and that if the axis of revolution is the axis of Y and the fixed plane is the plane of XY, the rectangular coordi- nates of (a, (i, y) are x^^b sec a sech /3 cos y, ij^=b tan a tanh (i, z^=b sec a sech j3 sin y (12) [V. (4)]. If now we let a range from to — , fB from -co to 00 , and y from to 27r, we shall be able to represent all points in space ; and if we agree that negative values of /3 shall belong to points below a plane through the origin and perpendicular to the axis of revolution and positive values of /3 to points above that plane, not only shall we have no ambiguity, but also the rectangular coordinates of any point as given in (12) will have their proper signs. Chap. VIII.] SPHEROIDAL COORDINATES. 243 EXAMPLES. 1. If the spheroid is a prolate spheroid, the ellipse and confocal hyperbola must be revolved about the major axis of the ellipse, and the plane must con- tain that axis. In place of equations (1), (2), and (3) of Art. 132 we have, then, f;+^+^-i=o fi' fi~ — Ir fi- — ir z — vij = K'>J/> fi\ \^ -b' ifi-fx' h%i + vy- X-- -f,^' ■'•' A— /x^' "' {X'-b'){lr-fx') where Laplace's Equation becomes P^ A[(x- ^■^)Ar] +p^ />,[(.'- ^=)/), n + ^X?^=^A.[(i+.')Z).F] = o. (1) (l),educesto ^+ ^+ ^^ ^-^_ ^^ D^V =^, (2) where da = — — -, > d(3 = 7:^ — '—-;, > d\ — X' — b'' ' b' — iJi'' l + y'^ a = ctnh~ ^ V > /3 = tanh~ ' ^ ? and y =: tan" ^ v . b b Since X^ b ctnh a, /n = (*> tanh (3, and v = tan y (2) can be reduced to sinh^a Z>,' V-\- cosh^ (3D^-V-\- (sinh-' a + cosh^ ^)i)/ P^= 0. (3) In using (3) it is to be noted that the point (a, ft, y) is the point of inter- section of a prolate spheroid whose semi-axes are b ctnh a and b csch a, a biparted hyperboloid of revolution whose semi-axes are b tanh (3 and b sech /3, and a plane containing the axis of revolution and making the angle y with a fixed plane. 244 ELLIPSOIDAL HARMONICS. [Akt. 133. If the fixed plane is that of (AT) the rectangular coordinates of any point (a, /3, y) are X — 6 ctnh a tanh (3, ij = b csch a sech ^ cos y , z — h csch a sech /3 sin y , and a may range from oo to 0, yS from — oo to oo , and y from to 'Itt. Negative values of /3 are to be taken for points lying to the left of a plane through the origin perpendicular to the axis of revolution. 2. Transform Laplace's Equation in Spherical Coordinates [xiii] Art. 1 to the symmetrical form a- D: V + cosh2 ^ j^'i Y ^ ^osh^ ^ B^ F = 1 (9 where * — ",' yS = logtan-) and y = <^. 3. Transform Laplace's Equation in Cylindrical Coordinates [xiv] Art. 1 to the symmetrical form Bl V^BIV^ e'-B; r = where a = log r , fi=, and y = .t . 133. In each of the cases we have considered, it has been easy to pass from Laplace's Equation in terms of the chosen coordinates representing an orthogonal system of surfaces to the symmetrical form [xvi] Art, 1 ; and it is evident that our new coordinate a is a value of V corresponding to such a distribution that the surfaces obtained by giving particular values to p^ are equipotential surfaces ; that y8 is a value of V corresponding to such a distribution that the surfaces obtained by giving particular values to p., are equipotential surfaces ; and that y is a value of V corresponding to such a distribution that the surfaces obtained by giving particular values to p^ are equipotential svirfaces. a, /3, and y are called by Lame <' thermometric parameters.'''' The condition that these values should exist, for a given system of surfaces, that is, that the distribution described above should be possible, is readily obtained. We shall work it out for a. It is merely the condition that V in Laplace's Equation may be a function of p^ alone. If F is a function of pi alone i>.y="^i>,P., D,v=%D„,, ^'.^•=';^'ap.. Chap. VIII.] THERMOMETRIC PARAMETERS. 245 D,fV = dpi (AjpO'-h dV vPi DJ'V: dp^ dpi Therefore [(D^p^y + (D,,p,y + (I),p,y-] '|^V [D^p, + DJ-p, + D^ '^= (ipi dpi whence (D,p,y + (D^p,y+(D,p,y d^V . (IV dpj; ' dpi where i^i(pi) may be any function of p^ alone. Uur required conditions are then -Jf-^^(pO --r — F,(p.^ —TV — ^3(^3) (1) and when they are fulfilled the original curvilinear coordinates p^, p^, ps, correspond to possible equijjotential or isothermal surfaces, ther mo metric jjarameters a, ^, and y exist, and the reduction of Laplace's Equation to the symmetrical form [xvi] Art. 1 is possible. 134. Returning to our Oblate Spheroid problem of Art. 132 we can proceed as usual to break up our equation (11) Art. 132. Assume that V= L.M.N, where X is a function of a only, J/ of /3 only, and A^of y only. (11) Art. 132 becomes cos^g d^L cosh^ (3 d^3I [cosh^ (S — cos^ a] d^N _ L da^'^ M dfi'^ ' N ^ ~ ^ dV. 1^ cosh2 ^ dm 1 COS^ Lcosh^fi — cos'^ a da^ ' Jfcosh^^S — cos^a 1 d^JY JV dy' ' The first member is independent of y, and the second member is independent of a and /?, and the two members are identically equal. The second member is then independent of a, /?, and y and must be constant; call it n^. We have, then, ^ + .W=0 (1) 246 ELLIPSOIDAL HAKMONICS. [Art. 134. (1) gives us N= A cos ny + i? sin wy. (3) (2) can be written ~T~ "^ ~^ " " ^ ^ ' ^^^ JT' ^ ^ ^^^^'^ ^' d'^L whence cos^a -—^ + ['^^ c'^^" " ~~ "K"^ + 1)]^ = (4) and cosh2 ^ ^+ [m(m + 1) — n^ cosli^ /3]ilf = 0. (5) If we introduce x = tanli ^8 in (5) it becomes (1 - -') "'- 2- f + [»(»' + 1) - I^J ^^= (6) where since x = tanh /? and fi may have any value from — oo to oo , a; may have any value between — 1 and 1. (6) is a familiar equation having for a particular solution M= (1 - xy '^-^1^ = PZ(x) = P^(tanh (S). (7) (V. Arts. 101 and 102) If we introduce in (4) x = tan a it reduces to (l+.,.)g + 2.g+[j^,-,„(»+l)]i = 0. (8) (8) is an unfamiliar equation, but it can be treated as (6) was treated if we take the pains to go back to the beginning and follow the steps of the treat- ment of Legendre's Equation. V This labor can be saved, however, by noting that if we let x = "4 (8) becomes and is identical in form with (6). Hence L = P»(^) and L = (l- f)! '^-^f^ (v. Art. 101), where y = i tan a, are particular solutions of (4). AVe can avoid imaginaries if we use the values L = (-i)"'-"P:,\Q/) and L = P" + " + \1 - y^)! ^^"^"f'^^- (9) ay Chap. VIII.] SPHEROIDAL HARMONICS. 247 Since we assumed V= L. M. N we have V — {A cos ny + B sin 7iy)P,;;(tanh fi) (— /)'« - "P,',;(i tan a) ^ and r= (.4 cos «y + B sin 7^7)P,«(tanh /3) /'" + " + ^ sec" a ^^"^^';»(^^ana) V (10) ^ ^ ^^ '"^ '^^ (f/(aana))« J as particular solutions of (11) Art. 132. If the problem is symmetrical with respect to the axis of the spheroid D-V=^(), 7^2 = and our particular solutions (10) reduce to r= (- 0'«P,„(/ tan a)P^(tanh ^) | and r= /'« + !(),„(*■ tan a) P„,(tanh^). J ^ If, then, V is given on the surface of a spheroid as a function of fi and y, we must express it as a function of tanh ji and y, and shall be obliged to develop it in terms of Spherical Harmonics of tanh y8 and y by the formulas of Chapter VII, using the first equation in (10) for the value of V at an internal point, and the second for the value of V at an external point. If the problem is symmetrical, we must develop in Zonal Harmonics of tanh (i by the formulas of Chapter VI. A convenient form for Q^(i tan a) is obtained from (2) Art. 100 ; it is /dx Hence ^o(* tan a)= — ii ^ ^^^ = — iY| — aj- (13) tan a EXAMPLES. 1. A conductor in the form of an oblate spheroid whose semi-axes are b sec ao and h tan a^ is charged with electricity and is found to be at potential Vq ; find the value of the potential function at any internal or external point. Here Fo= FoPo(tanh /3). Hence at an internal point , and at an external point ^o(aana) _ \2~ V ^ '^«a(*tana,)^"^*'^^^^^-^77r_ >. ^^^ \2 V Since V in (2) involves a only, the equipotential surfaces are all spheroids confocal with the conductor. 248 ELLIPSOIDAL HARMONICS. [Art. 135. 2. The upper half of an oblate spheroid whose semi-axes are h sec a^ and b tan ao is kept at the temperature unity, and the lower half at the tempera- ture zero. Find the permanent temperature at any internal point. 1 , 3Pi(itana) „ ^^ . .,, 7 lP3(itana) ^ a\ \ A71S. u = ^ + 7 TTT^T T Pi(tanh /S) — - • - ^f— ~ P3(tanh p) -\ 2 4Pi(ttanao) ^ 8 2P3(itanao) ^ (v. Art. 93). u may be expressed in terms of x, y, and z without serious difficulty [v. (12) Art. 132]. _1 , 3//_7 1 1 \1hif - \^y{x' + if + ^ - h^) - O^'V] , "~2^4c 8'2'2 hc^-^Wc "^ if 2c ^ lb tan a^ =^ minor axis of spheroid. 135. Let us now find the potential function at an external point due to the attraction of a solid homogeneous oblate spheroid, using the method em- ployed in Arts. 98 and 99. Consider first the potential function due to a shell bounded by the spheroids for which a^= -\- dcj). By (1) Art. 98 we have 4npK^[I)„V,-D„r,-],^^, (1) where p is the density and k the thickness of the shell, Fj the value of the potential function at an internal point, and Vo the value of the potential function at an external point. Let Fi = ^A,J- 0'«P„.(i tan a)P,„(tanh /5) and F. = ^B^J^' + 'Q^il tan a)P,„(tanh /?) [v. (11) Art. 134]. Since Vi and Vo must have the same value when a^= (f> ^- ^™^ P.(/tanc^) ^ '^ ''-J(l+x%F^(xi)f ^'> tan

„Fi = P»„Fi.P»„a. D„V^ = D,V^.D^a (3) Chap. VIIL] ATTBACTION OF A SPHEROID. 249 tan a V, - n = Xi"B.. p.. (tauh « P,„ (,• tan a) f .^^ . tan<|) ^ (Xt )J (/P„,(/tana) /^ dx n d7i = '^ = '-^ = '^ ^' ~ ^' dX = bseca Vtan^a + tanh'-^^S.fZa (4) V. Art. 130 (3), and Art. 132 (5) and (10). 1 [A,a]a sec <^ Vtan^ <^ + tanh^ ^ Hence [ A, F, - Z)„ n]„ _ , = — -^^ Xi^B ^-'f.f'^^ ■- « J * ^secc^N/tan^ + tanh^^^ "'P,„(itan<^) K = [(//^]„ ^ ^ = /; sec and p2 = ^ ^pf^^ sec^ c^(3 tan- c- >/«--> ^'> i^-. Here the first surface is an ellipsoid, the second an unparted hyperboloid, and the third a biparted hyperboloid. Each of the three principal sections of the system consists of confocal conies, and it is well known and is easily shown that the surfaces cut orthogonally. A, /i, and V will be our curvilinear coordinates, and are known as Ellvpsoidal Coordinates. We find without difficulty that m^ ^ b\c' — b^) " c\c^ — l/') ^'^^ (X^-J^^XX^-c^ (f^-PXc'-f^) (P-v^(^-r/^) To avoid ambiguity, we shall suppose that of the nine semi-axes in (1) Vc^ — fi^ is to be taken with the positive sign for a point on the half of the unparted hyperboloid on which z is positive, and with the negative sign for a point on the half on which z is negative ; '^b'^ — v'^ is to be taken with the positive sign for a point on the half of the biparted hyperboloid on which y is positive, and with the negative sign for a point on the half on which y is negative ; i^ is to be taken positive for a point on the half of the biparted hyperboloid on which x is positive, and negative for a point on the half on which X is negative, and that the remaining six are to be always positive. It follows that our Ellipsoidal Coordinates have the disadvantage that to fully fix a point Ave need to know not merely the values of its coordinates A, fx, and V, but the signs of sjc'^ — /a^, and \Jb'^ — iP- as well. 252 ELLIPSOIDAL HARMONICS. [Art. 136. We shall see later, Art. 139, when we come to introduce what we may call the Normal Ellipsoidal Coordinates a, (3, and y that they are free from this disadvantage. It is to be observed that A may range from c to cc, fi from b to c, and v from - b to b. The element of length perpendicular to the Ellipsoid is --l^=V&?s:^-- (^) The element of Ellipsoidal surface is and the element of volume is hi/iJh \/(x:' - b')(x:' - c')(^' - b'')(c' - fi')(b'-v')(c-'-v') The surface integral of any given function of fi and v taken over the ellipsoid is ft c +A(^,.)W - .') V (^, _ t,g _ ;!g_ :^(,. _ ,.) • -^M, (7) where fi(lJ^,v), /^(fJi',^), f3(f^,v) and fi(fJi,v) are the values of the given function on the four quarters of the ellipsoid into which it is divided by the planes of (XY) and (XZ). Laplace's Equation proves reducible to (fi^ - v^)Dl V+ (X:' - v')I)l V + {X' - ix^)D-'^ V=0 (8) Jdv ^(b'-v^){c'-i^'')' (9) Chap. VIII. ] NORMAL ELLIPSOIDAL COORDINATES. 253 a, /3, and y can be expressed as Elliptic Integrals of the first class and are 1 " y = F(^,sm-^f); (10) I mnrl - I =r r; 1 mod - h (A' -a) whence A = — tt^ ( mod ) = '' ( mod - ) snui — a)\ c/ cnaV c/ /^ = :Tf^("^of^(l-^2) j' ^-^^-sny^mod-') (11) b dn/3' (v. Int. Cal. Arts. 179, 192, and 196). 137. If in (8) Art. 136 we assume V= L.M.N where L involves a only, M involves (3 only, and N involves y only, (8) can be written fi- - v' cPL A- - v' (P3f A- - fx' cPN _ L da'^ M dp''^ N dy'~ ^^ (1) is too complicated to be broken up by our usual method. If, however, we let substitute in (1) and make use of the fact that the result must be identically zero, we find that the coefficients are zero for all values of k except A* = and k = 2, and that ao = — ^o = ^o ; and a^ = — &2 = ^2 • Therefore (1) can be broken up into the three equations _=(ao + «,A-)L -^ = («o + (hV')N. 254 ELLIPSOIDAL HARMONICS. [Art. 137. We shall find it convenient to take ^2 ''^s m.(in -\- 1) and a^ as — {U^ + •'-^)i^ whence ' — - [m(m + 1)A=^ - (b' + c')p-]L = dm af^2 + ["K"^ + 1)/^' - (^^' + ''')pW= h - [m(m + l)v' - (b' + c')p]N= 0. (2) If now in (2) we replace a, /3, and y by their values in terms of X, fi, and V, we get (X^ - b^-) (X^ -c')~ + X(X^ -b'- + \'- c^) '^^ dX' d^M ^^^-l^X^,^-c-)— + ^.^^.--b-^Jr^^'-c^)■^^^ d\ [m(m + 1)X- - (b' + c')p^L = dM - [m(m + l)fi' - {P + c')p^3I= (,._^,>)(,2_,.)^+,(,._^,. + ,2_,.)f^^ c?y (3) — \_m{m + 1)1^- — (&2 + t'2)|)]iV^= 0. ^ Whence if L = -E^,(X), it follows that 3I=Ell(fi) and N=Ei;^{v), and that V^E^^(X)E^^{fi)Er.iv) (4) is a solution of Laplace's Equation, (8) Art. 136. The equation (x^- _ i;^) (x^ - cO ^, + a-(a-^ - ^.^ + a-'^ - e=^) -£ dx" [^m{m + l).r-' — {b"" + c>]s; == (5) is known as Lame's Equation, and Et;,(x) as a Lame's Fiinction or an Ellij)- soidal Harmonic. We shall suppose m a positive integer. To get a particular solution of (5) let z = ^a^x^. Substitute in (5) and reduce and we get \_k{k + 1) - m{m + l)]a, - (5^ + c^) i{k + 2)^ -py,^, + h'e^ik + 3) (A- + 4)a,^4 = 0. (6) We have now only to choose a sequence of coefficients satisfying (6), and we may take any two consecutive coefficients arbitrarily. Chap. VIII.] LAMe's FUNCTIONS. 255 (6) which is ordinarily a relation connecting three consecutive coefficients reduces to a relation between two when /.■ = m., when Ic = — 3, and when A' = — 4. If we take «,„ + 2 = 0j '^',h + 4> ^'m + «> &c., will vanish. Let a„^=^l. If m is even the coefficient of ciq in (6) will be zero ; if p has such a value that a_o is zero, a_4, a_6, &c., will be zero, and there will be no terms in the solution involving negative powers of x. If we write the values of a„,_2, «m_4, &c., by the aid of (6) we see that a„,_2 is of the first degree in p, «„,_4 of the second degree in j), &c., and «_2 of the degree tt + 1 iii i>- There are then 77 + 1 values of p which we shall call ^1, ^2 5^*3 5 &c., for which a_^ will vanish, and for which our solutions will be of the form :E^,(x) = a-'« + a,,^ _ 2 X'" - - + (i,,^ _ 4 X'" - ^ ^ \- <^ if m is even. If m is odd, the coefficient of «i in (6) will vanish and we can choose j^ so that a_i shall be zero, and then all coefficients of lower order will vanish. a_i is of the degree — - — in p, and there will be —^ — values 2h, i'25 2h} &c., of 2^ for which ^^^(x) = x"' + '(„,^,x"'-' + a^„_,x"'-' H h ^'i^:-. Following Heine we shall call the solution just obtained Kl,',(x) so that KP(x) = X'" + c(,„ _ 2 X'" - - + r/„, _ , X"' - ^ H (7) terminating with Oq if vi is even, and with ctiX if m is odd. If m is even, there are — +1 of these functions K,>,['(x), K^^-^(x), &c., and there are — ^~— of them if m is odd. The coefficients can be computed by the aid of (6). If in Lame's Equation (5) we let z = r\x- — U- v/e get the equation - [(m + 2) (m - l)x' + 6-- - (^^' + c'>] V = 0. (8) Letting v = ^a^x* we obtain the relation [k(k + 3) - (m + 2){m - l)-]a, - {(F- + c'Xik + 2)^' -7.] + c^(2/c + 5)}a,^2 + ^-V(A- + 3)(A- + 4)«,^, = 0. (9) 256 ELLIPSOIDAL HARMONICS. [Art. 137, Proceeding exactly as before, we find that there are — values q^, ?2> ^S) &c., of j; for which r = ./•"'-' + ^?,„_ 3a,-'"-" H V ^xX if m is even, and — 7, — values for which r — .r'"-' + a„,_^,x'"-^ H \- a^ if m is odd. Calling v\ix-—lr L\n{x) so that ip(^) =.. Vx--^'-[.r'"-' + '',„_, .T'"-" + «,„_5X"'-^ + • • •], (10) terminating with a-^x if m is even and with a^ if ?;i is odd, we have — values of JS'/,;(a;), namely i,?,'(.x), -^„?(a-')) &c., of the form (10) if m is even and — ^— - values if m is odd. By interchanging h and c in (8), (9), and (10) we may show that if Mlix) = s/x- — c- [x"' - ' + a,,^ _ 3 X'" - •■= + a„, _ , x'" -' -\ ] (11) there are — values of ^,^(.r), namely J/,;;'(.r), ^1,'^X^)^ ^^m(^')^ &c., of the form , VI + 1 , . „ . , , (11) if 7)1 is even and — r^ — values if vi is odd. Finally if in Lame's Equation (5) we let z = r\J{x- — lr){x- — c-) we get {X- - h-) (X-- - C-) ^. + 3.r(x'2 - U' + .T- - C-) ^ - \_{vi + 3)(m - 2),/^ - (1/- + 6'^)(/j - 1)]^' = 0. (12) If now we let v — '±o,,x'' we obtain the relation \_k{k + 5) - {in - 2)(m + 3)Jr?, - (//-' + r^)[(;.- + 2)(/.- + 4) + 1 -pyi,^, + /A;^'(A- + 3)(^- + 4)a,,, = 0. (13) Proceeding as before we find that there are — values Si, 5.3, s^, &c., of j^ for which ?> = ji-'"-- + r/„,_4.)""-'' + r/,„_ya:"'-''' H \- n^ if ?« is even, and ^" values for which v = .r"'-- -)- c?,„_^a-'"~^ + • ■ • + '■'i-''' if ^n is odd. Calling rV(^— //-)(.r- — (•-) .X^;(.r) so that X^lx) = ^J{x' - 1>'){J- - (")[.r"-- + a„,_,x'"-' + a„_«.r"'-'^ + • • •] (14) terminating with Qq if 7)i is even and with UiX if ?« is odd, we have — values of E;;,(x), namely -A^,';(a;), ^''^(x), JSr^f(x), &c., of the form (14) if w is even and — - — values if ?m is odd. Chap. VIII.J TABLES OF ELLIPSOIDAL HARMONICS. 257 Summing up our results we see that there are 2?« + l Ellipsoidal Harmonics E^^(x) each o f whic h is a finite sum of the mth degree in x, or in x and \Jx^—U\ or in X and sjx^ — c^, or in x and '^x^ — b^ and >Jx'^ — c\ It was proved by Lame that the 2m + 1 values of j), namely ^^i, ^j.,, jj^, &c., S'l) ^2? ^sj <^c., ri, rj, r^, &g., Si, s.^, s^, &c., were all real, and by Liouville that they were all different. We give tables of the Ellipsoidal Harmonics for m = 0, m = 1, m = 2, and m = 3. The coefficients were obtained by the aid of formulas (6), (9), and (13). Eo(x) E,(x) A(.r) =:0 Mo(x) = N,(x)=0 K,(x)=.r Mi(x) — yjx- — c- iVi(.r) = E,(x) Kp(x) = x'- - ^ [fi' + r- - v/(/r' + c'Y - 3//-V,-] KS^(x) = x' - ^ [//^ + c"' + n/(//-' + c^r - SO-c'] L,(x) =x\/x' — b'' M„(Jc) =xsjx- — c- iY,(x) =s/(a.^-i2)(,r-'_c^) A\(x) iq^(x) : -blx^- '•-) - n/4( ''1 c^)] ^r' + fr)^'-15Z'-< ^/- + cy — 15F-1 - V'(^>-^ + 2cy - -W Lp(x) -. = V1'^ — b%x'- — i(^^ + 2c^ + \/(b'-\-2cy- - 5b- .•^)] 3Ip(x) -- = Vx"^ -cXx-- 1(2^-2 + .^ - sj{2b-' + cy - - 5// .^)] Mp(x) -- = Va^ -c^[x^- 1(2^-^ + ^'^ + V(2^»-^ + c-T- -5^- r')] N,(x) -- =W(: r--b')(x' -'>') 258 ELLIPSOIDAL HARMONICS. [Art. 138. It is to be noted that since in the solution (4) of Laplace's Equation, V= EP(\) El\{fi) El>(v), we have the same m and^j* in each of the three factors, we shall have to deal merely Avith products made up of factors of the same form, for example, A'„f ^ (A) ^„f ^-(m) k:'x^) , l:\x) i,;^(/x) l:;^x^) , &c. ; and that in a solution of the form we shall have for a given m just 2?«, + 1 terms. 138. From the particular solution of Lame's Equation [(5) Art. 137] z = EfJx), we can get by formula (5), Art. 18, the general solution. dx /(lor Making A^O and B = 2m -\-l we get a second form of particular solution of Lame's Equation, z — Fl',(x) where Fj;Jx) = (2m + l)EZ(x) ( ,— ' = (2) We shall call Fp(x) a Lame's Function of the second kind. It is easily seen to approach the value zero as x is indefinitely increased. EXAMPLES. 1. If an ellipsoidal conductor is charged with electricity, and is found to be at potential Vq, show that since Fo= FoA'o(A), F= V,K,(\)K,{p)K,{v) = Vo . at an internal point, and dx V= l\K,{^.)K,(y)^K,(X)J^ sJ(x^-bXx'-c^)lK,{x)f = V, {^{x'-h'^){x^-c')lK,(x)Y-\ r r dx ._ r (ix n _ \c a/ LJ N/(x2-/A)(.r--^-r2) • J N/(.r2-/.^)7^2"^:^J ~ \(^_^ sin-^ -) ' " \c xJ Chap. VIIL] NORMAL ELLIPSOIDAL COORDINATES. 259 whence V^ V, V. (10) Art. 136. 2. Find the value of the potential function at an external point due to the attraction of a solid homogeneous ellipsoid (v. Art. 135). Observe that f /.2 I -.2 o 72 —I and that where 31 is the mass of the ellipsoid. dx Ans. V=M iJ^W^ V/(x-2 — Z*2)(a;2 — c2) , ^ — rA7-(At)A>(^)A>(A) f-=_S= 2n/(A- + c2)2 — 3^*VL " ' ' y \J(x- — //-) (.r^ — c').(iq\x)y ^ ^ ' ^ Y v'(^2-Z-^)(x=^-c2).(A>(x-))0 i 139. If for the sake of brevity we represent - by k, and (1 rj by k' in the formulas (11) Art. 136 we have ^ = ^'^^^^^°^^^^' ^ = ch./3(Ldk'y - = ^'«"yOnodA-) (1) and from these we get without difficulty (v. Int. Cal. Art. 192) v^a"^^^7;^ = ck' en a (mod A-) dn/? ^ '^' \/b- — v-^^^b Q,\iy (mod A-:), VA' — c- ^ -^^ (mod k), sjc- — u- = -- — -^ (mod A-'), V^'" — V- = c dn y (mod /c). dn y8 ^ V / (2) 260 ELLIPSOIDAL HARMONICS. [Art. lit). If Ave let a range from to A', and /3 from to 2/v ', and y from to 4A', where 7i and 7v' are the complete Elliptic Integrals F{k, — \ and i^(A-'. — ) respectively, (a, /3, y) may represent any point in space, and there Avill be no ambiguity in sign (v. Art. 136). We may note that if < /? < A', z is positive ; if K' <. /3 <. -K', z is negative; if < y < A^, x and y are both positive; if Jr{\) c da v/(A2_^2)(;^2_j,2^ UD„V—VD„U = E/;(f.)E^,(v)E,^(fx)E:i(v)(^El^,(X) '^^ - E:!(X) '^^) da /S/(A2 — yLl2)(X^'_i;2) Chap. VIII.] DEVELOPMENT IN LAME S FUNCTIONS. 261 Integrating UD„V — FI>„?7over the whole ellipsoid, and writing the result equal to zero, we have 2A" 4 A' 2A" 4A- Hence unless ES(K)^-Ji;(>.)'^ = 0. (3) But as our ellipsoid may be taken at pleasure, A and a are unrestricted, and if (3) is true it must be trvie identically. If we divide (3) by [Ep(\)J it becomes drE,ux)-\ ,^ ^ E,ia) ;^L:^-I^ ;^ = a constant; and this obviously cannot be true unless w = )ii and 7 ^=p. EXAMPLES. 1. Show that it follows from (2) Art. 140 that A" A- Jd(3 CEP(fji)EP(v)E,'l(lji)E,?(v)(^' - v-)dy = 0. -K -K Suggestio7i : 2K' K' jE„P,(fi)E,^(fi)(fx' -v')d(3 =jEl,](fi)E:i(,x)(fi^ - v'-)dS •IK' + fEi:,(fi)E;!(fi)(f,'- - i,^-)dB. A'' If in the last integral we replace /3 by /3 + 2 A'' it becomes n -A'' V. Arts. 136 and 139 and Int. Cal. Art. 196. 2. Show that 2A" 4 A' A" A- Jdl3f[ES,(fi)Ej:,(v)J(ti' - v')dy = 8jd(3j[Ei:(fi)Ei;,(v)]\fi' - v^ly. 262 ELLIPSOIDAL HARMONICS. [Airr. 141. 141. We can now solve the problem of finding the value of Tat any point in space when it is given at all the points on the surface of the ellipsoid We have first to develop in Ellipsoidal Harmonics a function of yu. and v or rather of a and /3 given at all points on the surface of the ellipsoid in question ; and this is now easily accomplished by our usual method, which leads us to the result m=x A- = 2m + 1 where J,,,^. = " ^^^ (2) Our final solution is m = » A- = jm + 1 p at an internal point; at an external point. Lame has proved rather ingeniously that p^j'[E':!^Xf^)E!:'(^)j(fi' - vyiy can always be found and that it is equal to — multiplied by a rational integral function of the coefficients of ^i^'(a') and of c- and (-) • Of course the labor of obtaining even a few terms of the development of a function that is in the least complicated is enormous. 142. If in Laplace's Equation (8) Art. 136 we let V=Ep(\)U supposing ?7 to be a function of /3 and y only, we get after replacing — '"!, by its value m,(m + 1)A- — (b' + c-> [v. (2) Art. 137] (X' - v'-)Dl U+(\'- fJ?)D'; U + (fjr - v') [m(m + 1)A— (Ir + c')p'] T = U ; (1) Chap. VIIL] CONICAL COORDINATES. 263 and since by hypothesis U is independent of A, the coefficient of X^ in (1) must vanish. Hence Dl U +D';U^ (fx' - v') m(m -]-l)U=0. ' (2) Of course U= UP(fi)EP(v) will satisfy (2). EXAMPLES. 1. Substitute U= EP(fi)EP(v) in (2) Art. 142 and by the aid of (2) Art. 137 show that the equation (2) Art. 142 is satisfied. 2. Obtain (2) Art. 140 directly from (2) Art. 142. .3. Co7iical Coordinates. Consider the system of coordinates defined by the equations fx- jx- — Ir fx- — r- (1) V- v~ — h~ V- — c- diere r^ > /x- > Ir > ir. Show that ^ r^x-v' 2 ''"{y? — lr){v~ — Jr) , 't^iix" — (^^v- — c^) . ^h- ,.2(^2 _ ^2) ' Ih- ^(^2_^2) ' /'3-1- Laplace's Equation is Dl V + Dl V + (fx'- - v')D,.(>^I),. F) = (2) = f-=J^== and fi= t ^'^ J \/(fx' - b') (c' - fx') J \/(f,^ - v') (c^ - z.^') If F= U.B (2) breaks up into where (^^'^)=.n.(m, + l)E, (3) d_/ „dE dr Dl U -f D'^ U + m{m + 1) {fx' -v-)U= 0. (4) (3) gives B = Ar'" + Br-"'-\ (4) gives U=EP(/x)EP(v) (v. Art. 142). So that a solution of (2) is r= Ar"'EP(fx)El](v). But since (2) is Laplace's Equation, ir=^r'«F^(/x, ^), if expressed in Conical Coordinates, must satisfy it, consequently EP(fx)EP(v) must be simply a Spherical Harmonic of the ?/ith degree. 264 ELLIPSOIDAL HARMONICS. [Art. 143. Toroidal Coordinates. 143. Any pair of circles belonging to the orthogonal system obtained and figured in Art. 46 can be represented by the equations 2ax x~ + ?/- + «- sinh a cosh a „ , ., A (1) 2ft// ^ 3- + //- — sin /3 cos /3 if we take 2a instead of 2 as the distance between the points common to the second set of circles. If we rotate the system about the axis of i/ we get a set of spheres and a set of anchor rings which cut orthogonally. These and a set of planes through the axis of revolution will form an orthogonal system of surfaces, and the parameters corresponding to them may be taken as a set of curvilinear coordinates and may be called Toroidal Coordinates. If we take the axis of the system as the axis of Z, the equations of a set of the surfaces may be written 4ft-(a,-- + //-) ^ r-^" + !/ + -^- + 'rj sinh- a cosh- a 2az X- + //■ -\- z- — ; V=0. (2) We cannot proceed further by our usual method, for the assumption that V is a function of a alone, or that V is a function of ft alone, proves to be inadmissible. Indeed, not only are a, ft, and y not thermometric parameters (v. Art. 133), but no thermometric parameters exist, and no possible distribu- tion can make our anchor rings or our spheres a set of equipotential surfaces. We can, however, simplify (2). It can be written i>a( vs/;-) + i)i( FN/;) + —^ D;( FN/;) - i^^aV; + div;) - o. (3) D'^^ r -\- iJ'^s/ )' proves equal to — . j hence if U ^= V^r (3) becomes sinh^ a(DlU+ Z»| U) + J)^ U+\U= 0, (4) for which particular solutions can readily be found by our usual process. (4) can be broken up into the three equations ^'+(/M + i)^V=0 (5) dy- sinh^ a —— — [m(m + 1) + ir sinh- alL =0. (7) da- '- ^ X:= A cos(m -\- 4-)y + -^ sin(?/^ + 4-)y 31= Jj cos nft + Bi sin 7ift. If we introduce into (7) x = ctnh a it becomes a solution of which is L = piix) = (1 - x^y "^"i (^- ^^'^^ 1*^2). It is to be noted that since ctnh a is greater than 1 T, , X .- . cZ"P™('ctnha) P"(ctnh a) = i2 csch" a ,,"*:, — r-^- '"^ ^ (c? ctnh a)" 266 ELLIPSOIDAL HARMONICS. The constant coefficient i- can be rejected and we get U= \_A cos(vi -\-h)y-\-S sin(7?i + h)y](Ai cos nji-\- B^ sin ?i/3)csch" (/"P,„(ctnha) ((^ctnha)" as a particular solution of (4). 1 T. . . 1 N 1 . rf"P„Yctnha) - P, (ctnh a) = csch" a — --^^^^ — -^ .'-' ^ (fZctnha)" I has been called a Toroidal Harmonic. EXAMPLES. 1. Given the value of the potential function at all points on the surface of an anchor ring ; find its value at any point within the ring. Suggestion: If V=f((S, y) when a^ao< the function to be developed is and the development will be in a double Fourier's Series (v. Art. 71). 2. Show that if we let a range from to oc, ^ from — tt to tt, and y from to 27r, each of the double signs on page 264 may be replaced by the minus sign without loss of generality. CHAPTER TX.* HISTORICAL SUMMARY. The method of development in series which has enabled ns in the preceding chapters to solve problems in various branches of mathematical physics, had its origin, as might have been expected, in the theory of the musical vibrations of a stretched string. It was in the year 1753^ that Daniel Bernoulli enunciated the principle of the coexistence of small oscillations, which, in connection with Taylor's and John Bernoulli's theory of the vibrating string, led him to believe that the general solution of this problem could be put in the form of a trigonometric series. This principle also led him and Euler to treat in a similar manner the problems of the vibration of a column of air and of an elastic rod. The problem of the vibration of a heavy string suspended from one end was also treated in the same manner by these mathematicians and deserves special mention here as in it Bessel's functions of the zeroth order appear for the first time.^ In none of these cases, however, was any method given for determining the coefficients of the series. This last remark also applies to the more complicated problems of the vibration of rectangular and circular membranes, which were discussed by Euler ^ in 1764, and in the last of which the general Bessel's functions of integral orders occur. It is in problems connected with astronomy that the first completely successful application of the method here considered occurs. Legendre in a paper published in the Memoires des Savants Strangers for 1785, first introduced the zonal harmonics P^ and applied them to the determination of the attraction of solids of revolution. He was followed by Laplace, who in one of the most remarkable memoirs ever written* determined the potential of a solid differing but little from a sphere by means of the development according to the spherical harmonics y,„. 1 See two articles by Bernoulli and one by Euler in the Memoirs of the Academy of Berlin for this year. 2 See the Transactions of the Academy of St. Petersburg for 1732-33, 1734 and 1781. 3 Transactions of the Academy of St. Petersburg. 4"Th^orie des attractions des sph^roides et de la figure des Plan fetes" Mfemoii'es de I'acadfemie des sciences 1782. This article, although bearing an earlier date than that of Legendre, was really inspired by it. It is here that "Laplace's equation" first appears, occurring, however, only in polar coordinates. * See preface. 268 HISTORICAL SUMMARY. Very closely related to this problem is Gauss's celebrated treatment of the theory of terrestrial magnetism/ which we will for that reason mention here, although it was not published until more than half a century later. This paper is particularly noteworthy as it contains a numerical application of the method on a larger scale than has ever been attempted before or since. After the researches of Legendre and Laplace there was a pause of a quarter of a century until in 1812 Fourier's extensive memoir : Theorie du mouvement de la chaleur dans les corps solides was crowned by the French Academy. Although not printed until the years 1824-26,^ the manuscript of this work was in the meantime accessible to the other French mathematicians presently to be mentioned. The first part of this memoir, which was repro- duced with but few alterations in the Theorie analytique de la chaleur (1822), contains a treatment of the following problems and of practically all of their special cases : (a) The one dimensional flow of heat. (Ij) The two dimensional flow of heat in a rectangle, (c) The three dimensional flow of heat in a rectangular parallelopiped. (d) The flow of heat in a sphere when the temperature depends only on the distance from the centre, (e) The flow of heat in a right circular cylinder when the temperature depends only on the distance from the axis. In these problems not merely the simpler boundary conditions are considered but also the question of radiation into an atmosphere. In special cases of the first three problems just mentioned (when one or more dimensions become infinite) the series degenerate into " Fourier's integrals." More important even than any of these special problems is the great advance which Fourier caused the theory of trigonometric series to make. In a posthumous paper Euler had given the formulae for determining the coefficients,^ but Fourier was the first to assert and to attempt to prove that any function, even though for different values of the argument it is expressed by different analytical formulae, can be developed in such a series. The fact that the real importance of trigonometric series was thus for the first time shown justifies us in associating Fourier's name with them, although, as we have seen, they were known long before his day. Fourier's results were extended by Laplace in 1820* to the general (unsym- metrical) case of the flow of heat in a sphere, and by Poisson* (1821) to the unsymmetrical flow of heat in a cylinder. 1 Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1838. Leipzig, 1839. Reprinted in Gauss's collected works, Vol. V., p. 121. 2 M^moires de racaddmie des sciences for 1819-20 and 1821-22. 3 Lagrange had practically determined these coefficients long before but failed to notice what he had got. 4 Connaissance des Temps pour Tan 1823. 5 Journal de I'ficole Polytechnique, 19*^ Cahier. Although the final forms to which Poisson reduces his results are similar to Fourier's, his methods are very different. ELLIPSOIDAL HARMONICS. • 209 In 1835 Green published a paper ^ in which the method we are considering is employed to determine the potential of a heterogeneous ellipsoid. This paper, in which the analysis is performed at once for space of n dimensions, anticipates much that was subsequently done by others, but has failed to exert an influence proportional to its importance. At about this time Lame began a series of publications which have con- nected his name inseparably with the problem of the permanent state of temperature of an ellipsoid. In the first of these - the equation V'^ F = is transformed to ellipsoidal coordinates and is then broken up into three ordinary differential equations. The rest of the solution, however, is hardly touched upon. Lame's most important work on this subject^ was published in Liouville's Journal in 1839, and in it the complete solution of the problem is given. Lame clearly shows in this paper how he arrived at his solution, by considering first the simpler case of a sphere where, instead of the polar coordinates 6 and , the parameters of two families of confocal cones of the second degree are used as coordinates. This system of curvilinear coordinates, which, Avhen applied to the complete sphere, merely gives the old results of Laplace in a new form, is barely mentioned in Lame's later publications. In the same volume of Liouville's Journal Lame published a second paper in which he applies his results to the special cases of ellipsoids of revolution. These two papers form the starting-point for a series of articles on the same subject by Heine and Liouville. Heine in his doctor dissertation* (1842) determined the potential not merely for the interior of an ellipsoid of revolution when the value of the potential is given on the surface, but also for the exterior of such an ellipsoid and for the shell between two confocal ellipsoids of revolution. Even in the first of these problems, which is equivalent to that of Lame, he simplified Lame's solution materially by showing that the functions used may be reduced to spherical harmonics, while in the other two problems he introduced spherical harmonics of the second kind, which were then new. Shortly afterwards^ Heine and Liouville 1 " On the determination of the exterior and interior attraction of ellipsoids of variable densities." Transactions of the Cambridge Philosophical Society. 2 M^moires des Savants Strangers, Vol. V. Although the volume is dated 1838 this paper (which was reprinted in Liouville's Journal, 1837) must have appeared at least as early as 183-5. 3 "Sur I't^quilibre des Temperatures dans un ellipsoide k trois axes in^gaux." An article by the same author on the two dimensional potential will be found in Vol. I. of this Journal. 4 Reprinted in Crelle's Journal, Vol. 26 (1843). In the same Journal for 1847 F. Neumann discussed the related problem of the magnet- isation of a soft iron ellipsoid of revolution. ^ Heine: Crelle's Journal, Vol. 29, 1845. Liouville: Liouville's Journal, Vol. X., 1845, and Vol. XI., 1846. For a treatment of the problem of the potential of an ellipsoidal shell by means of a development of - in terms of Lamp's functions, see a paper by Heine in Crelle's Journal, Vol. 42, 1851. 270 HISTORICAL SUMMARY. published simultaneously two papers in which they arrived independently of each other at about the same results. In each of these papers attention is called to the fact that the product of two Lame's functions is a spherical harmonic, and this fact is made use of to throw Lame's solution of the problem of the permanent state of temperatures of an ellipsoid into a more elementary form. Besides this the second solution of Lame's equation is introduced for the sake of solving the potential problem for the exterior of the ellipsoid. In thus following up the theory of heat and the related potential problems, we have lost sight of the question of small vibrations, to which during the early part of the century a great deal of attention had been devoted by Poisson, who frequently made use of the method of development in series. In his memoirs^ most of the problems left unfinished by Bernoulli and Euler are thoroughly treated, as well as various slight modifications of them. When, however, he attacked the problem of the vibration of an elastic plate he was unable to make much progress, owing in part to the erroneous form of his boundary conditions. He was, nevertheless, able to solve the problem of the symmetrical vibration of a free circular plate. The complete theory of the vibration of a free circular plate was first given by Kirchhoff.^ Passing now to a new subject, the theory of the equilibrium of an elastic spherical shell, we find a solution by Lame in Liouville's Journal for 1854, and by Sir William Thomson (1862) in the Philosophical Transactions for 1863. Both of these papers consist of an application of the spherical harmonic analysis to this rather complicated problem. Thomson, however, considers besides Lame's problem certain related questions and the form of his analysis is very different from Lame's, being of the same nature as that used in the Appendix B of his Natural Philosophy of which we shall have to speak presently. These investigations form the starting point for a number of recent memoirs among which those of G. H. Darwin on cosmographical questions deserve special mention. Closely related to this last mentioned problem is the theory of the small vibrations of an elastic sphere. While the simplest case of this problem was treated by Poisson in the memoir referred to above, the general solution has been only recently obtained by Jaerisch (1879)^ and Lamb (1882).* The functions involved are the same as those which occur in the problem of the non-stationary flow of heat in a sphere as solved by Laplace. The Appendix B of Thomson and Tait's Natural Philosophy, ^ to which we have already referred, deserves to be regarded as one of the most important 1 See especially the one in the M^moires de I'acad^mie des sciences, Vol. VIII., 1829. 2 Crelle's Journal, Vol. 40, 1850. ^ Crelle's Journal, Vol. 88. * Proc. Lond. Math. Soc. ^ First edition, 1867. This appendix was evidently written as early as 1862, as Thomson refers to it in the memoir quoted above. TOKOIDAL AND CONAL HARMONICS. 271 contributions to the general theory. The way in which spherical harmonics are introduced (as homogeneous functions of the rectangular coordinates) was then new/ and the sokition of the potential problem for a variety of new solids was indicated ; viz., for solids whose boundaries consist of concentric spheres, cones of revolution, and planes. We shall have more to say presently concerning the method employed for the solution of these problems. Although connected only indirectly with the theory we are discussino-, it will be well to mention at this point the method of electrical images which is also due to Sir William Thomson (1845). This method enables us to solve many potential problems for the inverse of any solid when once we have solved it for the solid itself. By means of this method most of the solutions of potential problems obtained by our method may be applied at once with very little modification to systems of curvilinear coordinates derived by inversion from those we have used. It will not be necessary to mention separately problems of this sort, as it is clearly immaterial whether they be solved directly or by means of the method of inversion.^ Eeturning now to the Continent, we find as the next important question taken up the problem of the potential of an anchor ring. The first publication on this subject is a monograph by C. Neumann ^ (1864), but in Riemann's posthumous papers which were not published until 1876, ten years after his death, will be found a short fragment on this subject, which {cf. the last page of Hattendorf's edition of Riemann's lectures : '< Partielle Differentialglei- chungen ") would appear to date back to the winter 1860-61. This fragment is of peculiar interest, as the opening paragraphs clearly show that Riemann had in mind an extended article on the fundamental principles of our subject. We will next mention two papers by Mehler in which the functions known as ^' conal harmonics," which had already been introduced by Thomson in the Appendix B above mentioned, were applied to the solution of two problems in electrostatics. The first of these papers ■* (1868) deals with the solid bounded by two intersecting spheres, while in the second^ (1870) the infinite cone of revolution is treated. Both of these problems are essentially different from those discussed in the '' Appendix B," inasmuch as the infinite series which we usually have degenerate in these cases into definite integrals, just as they do in some simpler cases treated by Fourier. The later of the two papers just quoted also contains valuable information concerning the nature of the 1 The same method was used at about the same time by Clebsch. - A case in point would be the potential problem for the shell between two non-intersectino- eccentric spheres, since these spheres can be inverted into concentric spheres. This problem was treated directly by C. Neumann in a monograph published in Halle in 1862. 3 " Theorie der Elektricitiits- und Warme-Vertheilung in einem Ringe." Halle. ^ 4 Crelle's Journal, Vol. 68, 1868. 5 Jahresbericht des Gymnasiums zu Elbing. 272 HISTORICAL SUMMARY. solution of similar problems for the hyperboloids and paraboloids of revolu- tion. The solutions of these problems are not, however, given. It remains, in order to close the history of this part of the subject, to mention a number of memoirs which although treating entirely new problems are of far less importance than most of those considered up to this point, partly because the solution is not brought to a point where it can be of much immediate use, and partly because most of the methods employed are such as could not fail to present themselves to any one attacking these problems. Of these the first is a paper by Mathieu' on the vibration of an elliptic membrane (1868), in which the functions of the elliptic cylinder occur for the first time. This was followed in the same year by a paper on closely allied subjects by H. Weber,^ in which not merely the case of the complete ellipse is briefly considered, but also that in which the boundary consists of two arcs of confocal ellipses and two arcs of hyperbolas confocal with them. The special case in which the ellipses and hyperbolas become confocal parabolas is also considered, whereby the functions of the parabolic cylinder are for the first time introduced. In Mathieu's " Cours de physique mathematique " (1873) the problem of the non-stationary flow of heat in an ellipsoid is touched upon, and an elaborate though not very satisfactory treatment of the special cases where we have ellipsoids of revolution is given. New functions appear in all of these problems. Of late years C. Baer has supplied a number of missing links in the chain of problems here considered by treating in succession the potential problem for the paraboloid of revolution,^ the parabolic cylinder'' and the general paraboloid.* In the first of these problems Bessel's functions occur, as had already been stated by Mehler, while in the last we find the functions of the elliptic cylinder. For each of the three systems of coordinates employed the same author also touches upon the more general problem of the non-stationary flow of heat, in which new functions occur. Except in the case of the anchor ring we have found so far only such solids treated by our method as are bounded by surfaces of the first or second 1 Liouville's Journal, Vol. XIII. 2 " Ueber die Inteo;ration der partiellen Differentialgleichung — — -f- — — -f- A:% = ." ox- oy- Math. Ann., Vol. I. No physical problem is mentioned in this paper. 3"Ueber das Gleichgewicht und die Bewegung der Warme in einem Rotationspara- boloid." Dissertation, Halle, 1881. * "Die Funktion des parabolischen Cylinders," Gymnasialprogramm Ciistrin, 1883. 5 " Parabolische Coordinaten," Frankfurt, 1888. See also a paper by Greenhill in the Proc. Lond. Math. See, Vol. XIX., 1889 (read Dec. 8, 1887). Also a posthumous paper by Lam6 in Liouville's Journal for 1874, Vol. XIX. CYCLIDIC COORDINATES. 273 degree. Wangerin^ (1875-76) considered in connection with the theory of the potential, more general systems of curvilinear coordinates than had previously been used in physical questions, namely, cyclidic coordinates.- He showed, however, merely how to break up Laplace's equation into three ordinary differential equations.^ An important branch of our theory which we have not yet touched upon dates back to the year 1836, when Sturm published a series of fundamentally important papers in the first two volumes of Liouville's Journal. The physical question which lies at the basis of these papers is the problem of the flow of heat in a heterogeneous bar.* The method here employed depends upon the fact that the functions which occur are characterized by the number of times they vanish in a certain interval. This same idea reappears in Thomson and Tait's Appendix B already referred to, but first finds its full expression in this more general field of the three dimensional potential in an article by Klein : " Ueber Korper welche von confocalen Flachen zweiten Grades begrenzt sind " ^ (1881). Still more recently (1889-90) Klein has in his lectures extended this theory to the treatment of solids bounded by six confocal eyelids, and has indicated how all the potential problems heretofore treated by our method are special cases of this one.** Of late years, especially since the year 1880, the younger English mathe- maticians have done a vast amount of work in the theory we are here considering. Although much of this work is of great value, hardly any of it can be regarded as being a real development of the method ; it is rather an application of it to a great variety of problems. We must therefore content ourselves with giving a mere list of a few of the more important of these papers. Niven: On the Conduction of Heat in Ellipsoids of Revolution. Phil. Trans., 1880. Niven : On the Induction of Electric Currents in Infinite Plates and Spherical Shells. Phil. Trans., 1881. 1 Preisschriften der Jablanowski'schen Gesellschaft, No. XVIII., and Crelle's Journal, Vol. 82. See also, concerning a still further extension, the Berliner Monatsberichten for 1878. 2 Cyclids are a kind of surface of the fourth order (see Salmon's Geom. of three Dimen- sions, p. 527). In his first memoir Wangerin considers only cyclids of revolution. 3 See also a paper by this author in Griinert's Archiv for 1873, where the problem of the equilibrium of elastic solids of revolution is treated. * The similar problem of the vibration of a heterogeneous string under the action of an external force was treated by Maggi (Giornale di Matematiche, 1880). Several special cases are also considered here in detail. 5 Math. Ann., 18. ^ For an exposition of this theory see the treatise : Ueber die Reihenentwickelungen der Potentialtheorie, Leipsic, Teubner, 1894, by the writer of the present chapter. 274 HISTORICAL SUMMARY. Hicks: On Toroidal Functions. Phil. Trans., 1881. Hicks : On the Steady jMotion and Small A^ibrations of a Hollow Vortex. Phil. Trans., 1884, 1885. Lamb: On Ellipsoidal Current Sheets. Phil. Trans., 1887. Chree: The Equations of an Isotropic Elastic Solid in Polar and Cylin- drical Coordinates, their Solution and Application. Camb. Phil. Soc. Trans., XIV., 1889. Hohson: On a Class of Spherical Harmonics of Complex Degree with Applications to Physical Problems. Camb. Phil. Soc. Trans., XIV., 1889. Chree: On some Compound Vibrating Systems. Camb. Phil. Soc. Trans., XV., 1891. Niven: On Ellipsoidal Harmonics. Phil. Trans., 1892. The historical sketch we have just given would naturally require as a supplement some account of the work that has been done on the question of the convergence of the various series which occur. This, however, would carry us too far, and we will content ourselves with mentioning the two fundamental memoirs by Dirichlet in Crelle's Journal, one in 1829 on Fourier's series, and one, which has been criticised to some extent by subse- quent mathematicians, in 1837 on Laplace's spherical harmonic development. Another subject which naturally presents itself here is the theory of the various new functions we have met. Those properties of these functions, however, which the physicist needs have usually been investigated by the physicists themselves in the papers mentioned above ; while any thorough account of the development of the theory of these functions would lead us into the vast region of the modern theory of linear differential equations. We will therefore close by merely giving a list of books which will be found useful by those wishing to continue their study of the subject further. We begin with the books relating directly to physical questions : Fourier: Theorie Analytique de la Chaleur, 1822. Lame : Leqons sur les Eonctions inverses des Transcendantes et les Surfaces isothermes, 1857. Lame: Leqons sur les Coordonnees Curvilignes et leurs diverses Applica- tions, 1859. Mathieu : Cours de Physique Mathematique, 1873. Riemann: Partielle Differentialgleichungen, und deren Anwendung auf physikalische Fragen (edited by Hattendorf), third edition, 1882. F. Neumann: Theorie des Potentials und der Kugelfunktionen (edited by C. Neumann), 1887. Thomson and Tait : Natural Philosophy, second edition, 1879. Bayleigh: Theory of Sound, 1877. Basset: Hydrodynamics, 1888. Love : Theory of Elasticity, 1892. BOOKS OF REFERENCE. 275 Heine : Handbuch der Kugelfuiiktionen (second edition), 1878-81. Ferrers : Spherical Harmonics, 1881. Haentzschel : Reduction der Potentialgleichung auf gewohnliclie Differential- gleichungen, 1893. These last three books would also belong in the following list of books relating to the theory of the various functions we use : Todhunter : The Functions of Laplace, Lame and Bessel, 1875. Lommel: Studien liber die Bessel'schen Punktionen, 1868. F. Neumann : Beitrage zur Theorie der Kugelfunktionen, 1878. And finally concerning the question of convergence : C. Neumann: tjber die nach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden Entwickelungen, 1881. APPENDIX TABLES. Table I., a table of Surface Zonal Harmonics (Legendrians), gives the values of the first seven Harmonics P^ (cos 9), P^ (cos $), ••• P^ (cos 6) for the argument 6 in degrees. It is taken from the Philosophical Magazine for December, 1891, and was computed by Messrs. C. E. Holland, P. E. James, and C. G. Lamb, under the direction of Professor John Perry. Table II., a table of Surface Zonal Harmonics (Legendrians), gives the values of the first seven Harmonics Pi (x), P^{x), •• • P^ (x) for the argument x. It is reduced from the Tables of Legendrian Functions computed under the direction of Dr. J. W. L. Glaisher, and published in the Report of the British Association for the Advancement of Science for the year 1879. Table III., the table of Hyperbolic Functions, gives the values of e-'^, e'~^, sinlix, cosh if, and gdx (Gudermannian of x) for values of x from 0.00 to 1.00; and the values of logsinhx and log cosh x for values of x from 1.00 to 10.0. The values of gd a-, log sinh x, and log cosh x are taken from the Mathematical Tables prepared by Professor J. M. Peirce (Boston: Ginn & Co.). The log sinh a; and log cosh a- for values of x between 0.00 and 1.00 can be obtained from the values given for the Gudermannian of x in the table by the aid of the relations log sinh X. = log tan (gd x) log cosh X = log sec (gd x). Table IV. gives the first twelve roots of Jq (x) = and Ji (x) = each divided by tt. The table is taken from Lord Rayleigh's Sound, Vol. I., page 274, and is due to Professor Stokes, Camb. Phil. Trans., Vol. IX., page 186. Table V. gives the first nine roots of Jq (x) = 0, Ji (x) = 0, -■ • J^ (x) = 0. The table is taken from Rayleigh's Sound, Vol. I., page 274, and is due to Professor J. Bourget, Ann. de I'Ecole Normale, T. III., 1866, page 82. Table VI., the table of Bessel's Functions, gives the values of the Bessel's Functions Jo (^) and J^ (x) for the argument x from ic = to ic = 15. It is taken from Rayleigh's Sound, Vol. I., page 265, and from Lommel's Bessel'sche Functionen. 278 APPENDIX. TABLE I. — Surface Zonal Harmonics. e Pi (cos e) Pa (cos d) PzicosO) Pi{cose) P5(cos^) P6(cose) P7(COS6) 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 .9829 3 .9986 .9959 .9918 .9863 .9795 .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 S .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 .8238 .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 —.3463 26 .8988 .7117 .4670 .2007 -.0489 —.2478 —.3717 27 .8910 .6908 .4319 .1553 -.0964 —.2869 —.3921 28 .8829 .6694 .3964 .1105 -.1415 -.3211 —.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 —.1469 40 .7660 .3802 —.0252 -.3190 —.4197 -.3234 —.1003 41 .7547 .3544 -.0574 —.3416 -.4181 -.2938 —.0534 42 .7431 .3284 -.0887 —.3616 -.4128 -.2611 -.0065 43 .7314 .3023 —.1191 —.3791 —.4038 -.2255 .0398 44 .7193 .2762 —.1485 —.3940 —.3914 -.1878 .0846 45° .7071 .2500 —.1768 —.4062 -.3757 -.1485 .1270 TABLE I. APPENDIX. Surface Zonal Harmonics. 279 6 Pi (cos e) P2(cose) Pa (cos 6*) Pi (cos 6) Ps (cos 6) Pe (cos d) P^ (cos 6) 45° .7071 .2500 -.1768 —.4062 —.3757 —.1485 .1270 46 .6947 .2238 —.2040 -.41.58 —.3568 —.1079 .1666 47 .6820 .1977 —.2300 —.4252 —.33.50 —.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 .0433 — 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 .2949 58 .5299 -.0788 —.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 — .1S41 .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 .2S19 -.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 .0795 -.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 —.2943 79 .1908 —.4454 -.2688 .2443 .2990 —.0989 —.2913 SO .1736 —.4548 —.2474 .2659 .2810 —.1321 -.2835 81 .1564 -.4633 -.2251 .2859 .2606 — . 1635 —.2709 82 .1392 —.4709 —.2020 .3040 .2378 —.1926 — 2536 S3 .1219 —.4777 -.1783 .3203 .2129 —.2193 —.2321 S4 .1045 -.4836 —.1539 .3345 .1861 -.2431 —.2067 85 .0872 -.4886 —.1291 .3468 .1577 -.2638 —.1779 86 .0698 —.4927 -.1038 .3569 .1278 -.2811 — .H60 87 .0523 —.4959 —.0781 .3648 .0969 -.2947 —.1117 88 .0349 -.4982 —.0522 .3704 .0651 —.3045 —.0735 89 .0175 —.4995 —.0262 .3739 .0327 —.3105 —.0381 90° .0000 -.5000 .0000 .3750 .0000 -.3125 .0000 280 TABLE II. APPENDIX. Surface Zoxal Harmoxics. X Pi(.r) Po(a.-) -PsCx) P4(x) A(x) PgW P,(x) 0.00 0.0000 —.5000 0.0000 0.3750 0.0000 —.3125 0.0000 .01 .0100 —.4998 —.0150 .3746 .0187 —.3118 —.0219 .02 .0200 —.4994 —.0300 .3735 .0374 —.3099 —.0436 .03 .0300 —.4986 —.0449 .3716 .0560 —.3066 —.0651 .04 .0400 —.4976 —.0598 .3690 .0744 —.3021 —.0862 .05 .0500 —.4962 —.0747 .3657 .0927 —.2962 —.1069 .06 .0600 —.4946 —.0895 .3616 .1106 —.2891 —.1270 .07 .0700 —.4926 —.1041 .3567 .1283 —.2808 —.1464 .OS .0800 —.4904 —.1187 .3512 .1455 —.2713 —.1651 .09 .0900 —.4878 -.1332 .3449 .1624 —.2606 -.1828 .10 .1000 —.4850 -.1475 .3379 .1788 —.2488 -.1995 .11 .1100 —.4818 —.1617 .3303 .1947 —.2360 —.2151 .12 .1200 -.4784 —.1757 .3219 .2101 —.2220 —.2295 .13 .1300 —.4746 -.1895 .3129 .2248 —.2071 —.2427 .14 .1400 -.4706 —.2031 .3032 .2389 —.1913 —.2545 .15 .1500 —.4662 —.2166 .2928 .2523 —.1746 —.2649 .16 .1600 —.4616 —.2298 .2819 .2650 -.1572 —.2738 .17 .1700 —.4566 —.2427 .2703 .2769 -.1389 —.2812 ; .18 .1800 -.4514 —.2554 .2581 .2880 —.1201 -.2870 .19 .1900 -.4458 —.2679 .2453 .2982 —.1006 —.2911 1 .20 .2000 —.4400 —.2800 .2320 .3075 -.0806 -.2935 .21 .2100 -.4338 —.2918 .2181 .3159 -.0601 —.2943 .22 .2200 —.4274 —.3034 .2037 .3234 —.0394 -.2933 .23 .2300 —.4206 —.3146 .1889 .3299 -.0183 —.2906 .24 .2400 —.4136 —.3254 .1735 .3353 .0029 —.2861 .25 .2500 —.4062 —.3359 .1577 .3397 .0243 -.2799 .26 .2600 -.3986 —.3461 .1415 .3431 .0456 —.2720 .27 .2700 —.3906 —.3558 .1249 .3453 .0669 -.2625 .28 .2800 -.3824 -.3651 .1079 .3465 .0879 -.2512 .29 .2900 -.3738 —.3740 .0906 .3465 .1087 -.2384 .30 .3000 -.3650 —.3825 .0729 .3454 .1292 —.2241 .31 .3100 -.3558 —.3905 .0550 .3431 .1492 -.2082 .32 .3200 —.3464 —.3981 .0369 .3397 .1686 —.1910 .33 .3300 -.3366 —.4052 .0185 .3351 .1873 -.1724 .34 ■ .3400 —.3266 —.4117 —.0000 .3294 .2053 —.1527 .35 .3500 —.3162 —.4178 —.0187 .3225 .2225 —.1318 .36 .3600 —.3056 —.4234 —.0375 .3144 .2388 -.1098 .37 .3700 —.2946 —.4284 —.0564 .3051 .2540 —.0870 .38 .3800 —.2834 —.4328 —.0753 .2948 .2681 —.0635 .39 .3900 -.2718 —.4367 —.0942 .2833 .2810 —.0393 .40 .4000 —.2600 —.4400 —.1130 .2706 .2926 —.0146 .41 .4100 -.2478 —.4427 —.1317 .2569 .3029 .0104 .42 .4200 —.2354 —.4448 —.150+ .2421 .3118 .0356 .43 .4300 —.2226 —.4462 —.1688 .2263 .3191 .0608 .44 .4400 —.2096 —.4470 —.1870 .2095 .3249 .0859 .45 .4500 —.1962 —.4472 —.2050 .1917 .3290 .1106 .46 .4600 —.1826 —.4467 - —.2226 .1730 .3314 .1348 .47 .4700 —.1686 —.4454 —.2399 .1534 .3321 .1584 .48 .4800 —.1544 —.4435 —.2568 .1330 .3310 .1811 .49 .4900 —.1398 —.4409 —.2732 .1118 .3280 .2027 .50 .5000 —.1250 -.4375 -.2891 .0898 .3232 .2231 appe:*dix. TABLE II. — Surface Zonal Harmonics. 281 X Pi (x) P-2 (X) Pz{x) Pi ic) Pb (x) Po(x) P7(X) .50 .5000 —.1250 —.4375 —.2891 .0898 .3232 .2231 .51 .5100 —.1098 —.4334 —.3044 .0673 .3166 .2422 52 .5200 —.0944 —.4285 —.3191 .0441 .3080 .2596 .5.'5 .5300 —.0786 —.4228 —.3332 .0204 .2975 .2753 .54 .5400 -.0626 —.4163 —.3465 —.0037 .2851 .2891 .55 .5500 —.0462 -.4091 —.3590 —.0282 .2708 .3007 .56 .5600 —.0296 —.4010 -.3707 —.0529 .2546 .3102 • 57 .5700 —.0126 —.3920 -.3815 -.0779 .2366 .3172 .58 .5800 .0046 -.3822 -.3914 —.1028 .2168 .3217 .59 .5900 .0222 —.3716 —.4002 -.1278 .1953 .3235 .60 .6000 .0400 -.3600 —.4080 —.1526 .1721 .3226 .61 .6100 .0582 -.3475 —.4146 —.1772 .1473 .3188 .62 .6200 .0766 -.3342 —.4200 —.2014 .1211 .3121 .63 .6300 .0954 —.3199 —.4242 —.2251 .0935 .3023 .64 .6400 .1144 —.3046 —.4270 —.2482 .0646 .2895 .65 .6500 .1338 —.2884 —.4284 —.2705 .0347 .2737 .66 .6600 .1.534 —.2713 -.4284 —.2919 .0038 .2548 .67 .6700 .1734 —.2531 —.4268 —.3122 -.0278 .2329 .68 .6800 .1936 —.2339 —.4236 —.3313 -.0601 .2081 .69 .6900 .2142 —.2137 —.4187 —.3490 —.0926 .1805 .70 .7000 .2350 —.1925 -.4121 —.3652 —.1253 .1502 .71 .7100 .2562 —.1702 —.4036 —.3796 —.1578 .1173 .72 .7200 .2776 —.1469 —.3933 —.3922 —.1899 .0822 .73 .7300 .2994 —.1225 —.3810 —.4026 —.2214 .0450 .74 .7400 .3214 —.0969 —.3666 -.4107 —.2518 .0061 75 .7500 .3438 —.0703 —.3501 —.4164 -.2808 —.0342 '76 .7600 .3664 —.0426 • -.3314 —.4193 —.3081 —.0754 .77 .7700 .3894 —.0137 —.3104 —.4193 —.3333 -.1171 .78 .7800 .4126 .0164 —.2871 —.4162 —.3559 -.1588 .79 .7900 .4362 .0476 -.2613 —.4097 —.3756 —.1999 .80 .8000 .4600 .0800 —.2330 —.3995 —.3918 —.2.397 .81 .8100 .4842 .1136 —.2021 —.3855 —.4041 -.2774 .82 .8200 .5086 .1484 -.1685 —.3674 -.4119 —.3124 .83 .8300 .5334 .1845 —.1321 —.3449 -.4147 -.3437 .84 .8400 .5584 .2218 —.0928 —.3177 —.4120 -.3703 .85 .8500 .5838 .2603 -.0506 —.2857 —.4030 —.3913 .86 .8600 .6094 .3001 —.0053 -.2484 —.3872 —.4055 .87 .8700 .6354 .3413 .0431 —.2056 —.3638 —.4116 .88 .8800 .6616 .3837 .0947 —.1570 —.3322 —.4083 .89 .8900 .6882 .4274 .1496 -.1023 —.2916 —.3942 .90 .9000 .7150 .4725 .2079 —.0411 —.2412 -.3678 .91 .9100 .7422 .5189 .2698 .0268 —.1802 — .3274 .9200 .7696 .5667 .3352 .1017 — .1077- -.2713 .93 .9300 .7974 .6159 .4044 .1842 —.0229 -.1975 .94 .9400 .8254 .6665 .4773 .2744 .0751 -.1040 .95 .9500 .8538 .7184 .5541 .3727 .1875 .0112 ■96 .9600 .8824 .7718 .6349 .4796 .3151 .1506 .97 .9700 .9114 .8267 .7198 .5954 .4590 .3165 .98 .9800 .9406 .8830 .8089 .7204 .6204 .5115 .99 .9900 .9702 .9407 .9022 .8552 .8003 .7384 1.00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 282 APPENDIX. TABLE III. — Hyperbolic Functioxs. X e.r e-x sinhx cosh X gdx 0.00 1.0000 1.0000 0.0000 1.0000 0?0000 .01 1.0100 0.9900 .0100 1.0000 0.5729 .02 1.0202 .9802 .0200 1.0002 1.1458 .03 1.0305 .9704 .0300 1.0004 1.7186 .04 1.0408 .9608 .0400 1.0008 2.2912 .05 1.0513 .9512 .0500 1.0012 2.8636 .06 1.0618 .9418 .0600 1.0018 3.4357 .07 1.0725 .9324 .0701 1.0025 4.0074 .OS 1.0S33 .9231 .0801 1.0032 4.5788 .09 1.0942 .9139 .0901 1.0040 5.1497 .10 1.1052 .9048 .1002 1.0050 5.720 .11 1.1163 .8958 .1102 1.0061 6.290 .12 1.1275 .8869 .1203 1.0072 6.859 .13 1.13SS .8781 .1304 1.0085 7.428 .14 1.1503 .8694 .1405 1.0098 7.995 .15 1.1618 .8607 .1506 1.0113 8.562 .16 1.1735 .8521 .1607 1.0128 9.128 .17 1.1853 .8437 .1708 1.0145 9.694 .IS 1.1972 .8353 .1810 1.0162 10.258 .19 1.2092 .8270 .1911 1.0181 10.821 .20 1.2214 .8187 .2013 1.0201 11.384 .21 1.2337 .8106 .2115 1.0221 11.945 .22 1.2461 .8025 .2218 1.0243 12.505 !23 1.2586 .7945 .2320 1.0266 13.063 .24 1.2712 .7866 .2423 1.0289 13.621 .25 1.2840 .7788 .2526 1.0314 14.177 .26 1.2969 .7711 .2629 1.0340 14.732 .27 1.3100 .7634 .2733 1.0367 15.285 .28 1.3231 .7558 .2837 1.0395 15.837 .29 1.3364 .7483 .2941 1.0423 16.388 .30 1.3499 .7408 .3045 1.0453 16.937 .31 1.3634 .7334 .3150 1.0484 17.484 .32 1.3771 .7261 .3255 1.0516 18.030 32, 1.3910 .7189 .3360 1.0549 18.573 .34 1.4049 .7118 .3466 1.0584 19.116 .35 1.4191 .7047 .3572 1.0619 19.656 .36 1.4333 .6977 .3678 1.0655 20.195 .37 1.4477 .6907 .3785 1.0692 20.732 .38 1.4623 .6839 .3892 1.0731 21.267 .39 1.4770 .6771 .4000 1.0770 21.800 .40 1.4918 .6703 .4108 1.0811 22.331 .41 1.5068 .6636 .4216 1.0852 22.859 .42 1.5220 .6570 .4325 1.0895 23.386 .43 1.5373 .6505 .4434 1.0939 23.911 .44 1.5527 .6440 .4543 1.0984 24.434 .45 1.5 683 .6376 .4653 1.1030 24.955 .46 1.5841 .6313 .4764 1.1077 25.473 .47 1.6000 .6250 .4875 1.1125 25.989 .48 1.6161 .6188 .4986 1.1174 26.503 .49 1.6323 .6126 .5098 1.1225 27.015 0.50 1.6487 0.6065 0.5211 1.1276 27?524 APPENDIX. TABLE III. — Hyperbolic Functiox> 283 X e-r e— sinh X cosh X gdx 0.50 1.6487 0.6065 0.5211 1.1276 27?524 .51 1.6653 .6005 .5324 1.1329 28.031 .52 1.6820 .5945 .5438 1.1383 28.535 .53 1.6989 .5886 .5552 1.1438 29.037 .54 1.7160 .5827 .5666 1.1494 29.537 .55 1.7333 .5770 .5782 1.1551 30.034 .56 1.7507 .5712 .5897 1.1609 30.529 .57 1.76S3 .5655 .6014 1.1669 31.021 .58 1.7860 .5599 .6131 1.1730 31.511 .59 1.8040 .5543 .6248 1.1792 31.998 .60 1.8221 .5488 .6367 1.1855 32.483 .61 1.8404 .5433 .6485 1.1919 32.965 .62 1.8589 .5379 .6605 1.1984 33.444 .63 1.8776 .5326 .6725 1.2051 33.921 .64 1.8965 .5273 .6846 1.2119 34.395 .65 1.9155 .5220 .6967 1.2188 34.867 .66 1.9348 .5169 .7090 1.2258 35.336 .67 1.9542 .5117 .7213 1.2330 35.802 .68 1.9739 .5066 .7336 1.2402 36.265 .69 1.9937 .5016 .7461 1.2476 36.726 .70 2.0138 .4966 .7586 1.2552 37.183 .71 2.0340 .4916 .7712 1.2628 37.638 .72 2.0544 .4867 .7838 1.2706 38.091 .73 2.0751 .4819 .7966 1.2785 38.540 .74 2.0959 .4771 .8094 1.2865 38.987 .75 2.1170 .4724 .8223 1.2947 39.431 .76 2.1383 .4677 .8353 1.3030 39.872 .77 2.1598 .4630 .8484 1.3114 40.310 .78 2.1815 .4584 .8615 • 1.3199 40.746 .79 2.2034 .4538 .8748 1.3286 41.179 .80 2.2255 .4493 .8881 1.3374 41.608 .81 2.2479 .4449 .9015 1.3464 42.035 .82 2.2705 .4404 .9150 1.3555 42.460 .83 2.2933 .4360 .9286 1.3647 42.881 .84 2.3164 .4317 .9423 1.3740 43.299 .85 2.3396 .4274 .9561 1.3835 43.715 .86 2.3632 .4232 .9700 1.3932 44.128 .87 2.3869 .4190 .9840 1.4029 44.537 .88 2.4109 .4148 .9981 1.4128 44.944 .89 2.4351 .4107 1.0122 1.4229 45.348 .90 2.4596 .4066 1.0265 1.4331 45.7.^0 .91 2.4843 .4025 1.0409 1.4434 46.148 .92 2.5093 .3985 1.0554 1.4539 46.544 .93 2.5345 .3946 1.0700 1.4645 46.936 .94 2.5600 .3906 1.0847 1.4753 47.326 .95 2.5857 .3867 1.0995 1.4862 47.713 .96 2.6117 .3829 1.1144 1.4973 48.097 .97 2.6379 .3791 1.1294 1.5085 48.478 .98 2.6645 .3753 1.1446 1.5199 48.857 .99 2.6912 .3716 1.1598 1.5314 49.232 1.00 2.7183 0.3679 1.1752 1.5431 49/505 284 APPENDIX. TABLE III. — Hyperbolic Functions X Lsiuhu; I cosh X X I sinli X I cosh X X I sinh X I cosh X 1.00 0.0701 0.1884 1.50 0.3282 0.3715 2.00 0.5595 0.5754 1.01 .0758 .1917 ].51 .3330 .3754 2.01 .5640 .5796 1.02 .0815 . 1950 1.52 .3378 .3794 2.02 .5685 .5838 1.03 .0871 .1984 1.53 .3426 .3833 2.03 .5730 .5880 1.04 .0927 .2018 1.54 .3474 .3873 2.04 .5775 .5922 1.05 .0982 .2051 1.55 .3521 .3913 2.05 .5820 .5964 1.06 .1038 .2086 1.56 .3569 .3952 2.06 .5865 .6006 1.07 .1093 .2120 1.57 .3616 .3992 2.07 .5910 .6048 l.OS .1148 .2154 ].5S .3663 .4032 2.08 .5955 .6090 1.09 .1203 .2189 1.59 .3711 .4072 2.09 .6000 .6132 1.10 .1257 .2223 1.60 .3758 .4112 2.10 .6044 .6175 1.11 .1311 .2258 1.61 .3805 .4152 2.11 .6089 .6217 1.12 .1365 .2293 1.62 .3852 .4192 2.12 .6134 .6259 1.13 .1419 .2328 1.63 .3899 .4232 2.13 .6178 .6301 1.14 .1472 .2364 1.64 .3946 .4273 2.14 .6223 .6343 1.15 .1525 .2399 1.65 .3992 .4313 2.15 .6268 .6386 1.16 .1578 .2435 1.66 .4039 .4353 2.16 .6312 .6428 1.17 .1631 .2470 1.67 .4086 .4394 2.17 .6357 .6470 1.18 .1684 .2506 1.68 .4132 .4434 2.18 .6401 .6512 1.19 .1736 .2542 1.69 .4179 .4475 2.19 .6446 .6555 1.20 .1788 .2578 1.70 .4225 .4515 2.20 .6491 .6597 1.21 .1840 .2615 1.71 .4272 .4556 2.21 .6535 .6640 1.22 .1892 .2651 1.72 .4318 .4597 2.22 .6580 .6682 1.23 .1944 .2688 1.73 .4364 .4637 2.23 .6624 .6724 1.24 .1995 .2724 1.74 .4411 .4678 2.24 ■ .6668 .6767 1.25 .2046 .2761 1.75 .4457 .4719 2.25 .6713 .6809 1.26 .2098 .2798 1.76 .4503 .4760 2.26 .6757 .6852 1.27 .2148 .2835 1.77 .4549 .4801 2.27 .6802 .6894 1.28 .2199 .2872 1.78 .4595 .4842 2.28 .6846 .6937 1.29 .2250 .2909 1.79 .4641 .4883 2.29 .6890 .6979 1.30 .2300 .2947 l.SO .4687 .4924 2.30 .6935 .7022 1.31 .2351 .2984 1.81 .4733 .4965 2.31 .6979 .7064 1.32 .2401 .3022 1.82 .4778 .5006 2.32 .7023 .7107 1.33 .2451 .3059 1.83 .4824 .5048 2.33 .7067 .7150 1.34 .2501 .3097 1.84 .4870 .5089 2.34 .7112 .7192 1.35 .2551 .3135 1.S5 .4915 .5130 2.35 .7156 .7235 1.36 .2600 .3173 1.86 .4961 .5172 2.36 .7200 .7278 1.37 .2650 .3211 1.87 .5007 .5213 2.37 .7244 .7320 1.38 .2699 .3249 1.88 .5052 .5254 2.38 .7289 .7363 1.39 .2748 .3288 1.89 .5098 .5296 2.38 .7333 .7406 1.40 .2797 .3326 1.90 .5143 .5337 2.40 .7377 .7448 1.41 .2846 .3365 1.91 .5188 .5379 1 2.41 .7421 .7491 1.42 .2895 .3403 1.92 .5234 .5421 2.42 .7465 .7534 1.43 .2944 .3442 1.93 .5279 .5462 2.43 .7509 .7577 1.44- .2993 .3481 1.94 .5324 .5504 2.44 .75.53 .7619 1.45 .3041 .3520 1.95 .5370 .5545 2.45 .7597 .7662 1.46 .3090 .3559 1.96 .5415 .5687 2.46 .7642 .7705 1.47 .3138 .3598 1.97 .5460 .5629 2.47 .7686 .7748 1.48 .3186 .3637 1.98 .5505 .5671 1 2.48 .7730 .7791 1.49 .3234 .3676 1.99 .5550 .5713 2.49 .7774 .7833 1.50 0.3282 0.3715 [ 2.00 0.5595 0.5754 i 2-50 0.7818 0.7876 APPENDIX. TABLE III. — Hyperbolic Functions !85 .T I sinhx I cosh X X I sinh X Z cosh a; X I sinh X I cosh X ' 2.50 0.7S1S 0.7876 2.75 0.8915 0.8951 3.0 1.0008 1.0029 2.51 .7S62 .7919 2.76 .8959 .8994 3.1 1.0444 1.0462 2.52 .7906 .7962 2.77 .9003 .9037 3.2 1.0880 1.0894 2.53 .7950 .8005 2.78 .9046 .9080 3.3 1.1316 1.1327 2.54 .7994 .8048 2.79 .9090 .9123 3.4 1.1751 1.1761 2.55 .8038 .8091 2.80 .9134 .9166 3.5 1.2186 1.2194 j 2.56 .8082 .8134 2.81 .9178 .9209 3.6 1.2621 1.2628 2.57 .8126 .8176 2.82 .9221 .9252 3.7 1.3056 1.3061 2.5S .8169 .8219 2.83 .9265 .9295 3.8 1.3491 1.3495 1 2.59 .8213 .8262 2.84 .9309 .9338 3.9 1.3925 1.3929 2.60 .8257 .8305 2.85 .9353 .9382 4.0 1.4360 1.4363 2.61 .8301 .8348 2.86 .9396 .9425 4.1 1.4795 1.4797 2.62 .8345 .8391 2.87 .9440 .9468 4.2 1.5229 1.5231 2.63 .8389 .8434 2.88 .9484 .9511 4.3 1.5664 1.5665 2.64 .8433 .8477 2.89 .9527 .9554 4.4 1.6098 1.6099 2.65 .8477 .8520 2 90 .9571 .9597 4.5 1.6532 1.6533 1 2.66 .8521 .8563 2.91 .9615 .9641 4.6 1.6967 1.6968 i 2.67 .8564 .8606 2.92 .9658 .9684 4.7 1.7401 1.7402 2.6S .8608 .8649 2.93 .9702 .9727 4.8 1.7836 1.7836 2.69 .8652 .8692 2.94 .9746 .9770 4.9 1.8270 1.8270 2.70 .8696 .8735 2.95 .9789 .9813 5.0 1.8704 1.8705 2.71 .8740 .8778 2.96 .9833 .9856 6.0 2.3047 2.3047 2.72 .8784 .8821 2.97 .9877 .9900 7.0 2.7390 2.7390 2.73 .8827 .8864 2.98 .9920 .9943 8.0 3.1733 3.1733 2.74 .8871 .8907 2.99 .9964 .9986 9.0 3.6076 3.6076 2.75 0.8915 0.8951 3.00 l.OOOS 1.0029 10.0 4.0419 4.0419 286 APPENDIX. TABLE IV. — Roots of Bessel's Functions. -forJo(x) = -forJ"i(x)=0 - for Jo{x) = -for Ji(x)=0 1 2 3 4 s 6 0.7655 1.7571 2.7546 3.7534 4.7527 5.7522 1.2197 2.2330 3.23S3 4.2411 5.2428 6.2439 7 8 9 10 11 12 6.7519 7.7516 8.7514 9.7513 10.7512 11.7511 7.244S 8.2454 9.2459 10.2463 11.2466 12.2469 TABLE v.— Boots OF J„(x) = -.0. 71 = n = l n=2 n — S 71 = 4 71 = 5 1 2.405 3.832 5.135 6.379 7.586 8.780 2 5.520 7.016 8.417 9.760 11.064 12.339 8.654 10.173 11.620 13.017 14.373 15.700 4 11.792 13.323 14.796 16.224 17.616 18.982 5 14.931 16.470 17.960 19.410 20.827 22.220 6 18.071 19.616 21.117 22.583 24.018 25.431 7 21.212 22.760 24.270 25.749 27.200 28.628 8 24.353 25.903 27.421 28.909 30.371 31.813 9 27.494 29.047 30.571 32.050 33.512 34.983 APPENDIX. 287 TABLE VI. — Bessel's Functions. X /o(x) Mx) X Mx) 1 J'i(x) X J,{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 o.s .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 6.1 .1773 —.2559 11.1 —.1528 -.1913 ! 1.2 .6711 .4983 6.2 .2017 —.2329 11.2 —.1330 -.2039 1 1.3 .6201 .5220 6.3 .2238 -.2081 11.3 —.1121 -.2143 1 1.4 .5669 .5419 6.4 .2433 —.1816 11.4 —.0902 —.2225 1 ' 1.5 .5118 .5579 6.5 .2601 -.1538 11.5 —.0677 —.2284 1.6 .4554 .5699 6.6 .2740 —.1250 11.6 —.0446 —.2320 1.7 .3980 .5778 6.7 .2851 —.0953 11.7 —.0213 —.2333 l.S .3400 .5815 6.8 .2931 —.0652 11.8 .0020 —.2323 1.9 .2818 .5812 6.9 .2981 —.0349 11.9 .0250 —.2290 2.0 .2239 .5767 7.0 .3001 -.0047 12.0 .0477 —.2234 1 2.1 .1666 .5683 7.1 .2991 .0252 12.1 .0697 —.2157 ■ 2.2 .1104 .5560 . 7.2 .2951 .0543 12.2 .0908 —.2060 2.3 .0555 .5399 7.3 .2882 .0826 12.3 .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 i 13.1 .2129 —.0489 3.2 —.3202 .2613 8.2 .1222 .2580 1 13.2 .2167 —.0271 ?,.?, —.3443 .2207 8.3 .0960 .2657 1 13.3 .2183 —.0052 3.4 —.3643 .1792 8.4 .0692 .2708 13.4 .2177 .0166 3.5 -.3801 .1374 8.5 .0419 .2731 13.5 .2150 .0380 3.6 —.3918 .0955 S.6 .0146 .2728 13.6 .2101 .0590 3.7 —.3992 .0538 8.7 -.0125 .2697 13.7 .2032 .0791 3.S —.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 1 4.1 —.3887 —.1033 9.1 —.1142 .2324 14.1 .1570 .1488 4.2 —.3766 —.1386 9.2 —.1367 .2174 14.2 .1414 .1626 1 4.3 —.3610 —.1719 9.3 —.1577 .2004 14.3 .1245 .1747 1 4.4 -.3423 -.2028 9.4 —.1768 .1816 14.4 .1065 .1850 4.5 —.3205 -.2311 9.5 —.1939 .1613 14.5 .0875 .1934 i 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 .2069 1 5.0 1 —.1776 —.3276 10.0 —.2459 .0435 15.0 —.0142 .2051 ADVERTISEMENTS LATIN TEXT-BOOKS Allen and Greenough : Latiu Grammar ..... New Caesar (seven books, with vocab., i New Cicero (thirteen orations, with voca New Ovid (illust., with vocab.) 1.50; (w: Sallust's Catiline, (iO cts. ; Cicero de Sen* Allen : New Latin Method Remnants of Early Latin Germania and Agricola of Tacitus . . Collar : Gate to Caesar, 40 cts. ; Gradatim . . Practical Latin Composition .... Collar and Daniell : First Book in Latin .... Bej;niner's Latin Book College Series of Latin Authors: Allen's Annals of Tacitus, Books I.-VL Bennett's Dialogus de Ora.toribus of Tac Greenough's Livy, Books L and II. Greenough and Peck's Livy, Books XXI. Kellogg's Brutus of Cicero Merrill's Catullus Smith's Odes and Epodes of Horace Editions of the text are issued sepan Crowell : Selections from the Latin Poets . . . Crowell and Richardson : Bender's Roman Literature Eaton : Latin Prose : Livy , , Ferguson : Questions on Caesar and Xenophon Fowler : Quintus Curtius Gepp and Haigh : Latin-English Dictionary . . . Ginn & Company : Classical Atlas Greenough: Bucolics and six books of Aeneid (with For other parts of Vergil, see Cata Sight Pamphlets : No. I. Eutropius . Gudeman : Dialogus de Oratoribus of Tacitus . . Halsey : Etymology of Latin and Greek . . . Keep: Essential Uses of the Moods .... Leighton : Latin Lessons, 1.12; First Steps in Lati Post : Latin at Sight Preble and Parker : Handbook of Latin Writing . School Classics : Collar's Aeneid, Book VII., 45cts. ; wi Tetlow's Aeneid, Book VIII. , 45 cts. ; w Shumway : Latin Synonymes Stickney : Cicero de Natura Deorum Terence : Adelphoe, Phormio, Heauton Timorume Tetlow : Inductive Lessons Thacher : Madvig's Latin Grammar Tomlinson : Latin tor Sight Reading White: Latin-Eng. Lexicon, 1.00; Eng.-Latin Li Latin-English and English-Latin Lexicoi White and Waite : Straight Road to Caesar . . . Whiton : Auxilia Vergiliana ; or. First Steps in L Six Weeks' Preparation for Reading Cac Copies sent to Teachers for Examination, loith a view receipt of Introductory Price. The above list is not CiNN & COMPANY, Pub BOSTON, NEW YORK, AND CHICA( JREEK TEXT-BOOKS. INTKOD, PRICE lea of Euripides SI. 00 ek-Eiiglish Word-List .30 11 : Beginner's Greek Composit»ou 90 icuic Orations of Demosthenes 1.00 eu against Tliebes, .§1.00; Anacreontics '.'