LIDKAfi i ^ 1 lip -ftiji ANNEX ALBERT R. MANN LIBRARY New York State Colleges OF Agriculture and Home Economics AT Cornell University _. , Cornell University Library QA 403.B99 Harmonic functions 3 1924 002 952 194 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002952194 HARMONIC FUNCTIONS. MATHEMATICAL MONOGRAPHS EDITED BY Mansfield Merriman and Robert S, Woodward Octavo, Cloth No. 1. History of Modern Mathematics. By David Eugene Smith. $1.25 net. No. 2, Synthetic Projective Geometry. By the Late George Bruce Halsted, SI. 25 net. No. 3. Determinants. By the Late Laenas Gifford Weld. SI. 25 net. No. 4. Hyperbolic Functions. . By the Late James McMahon. S1.25 net. No. 6. Harmonic Functions. By William E. Byerly. $1.25 net. No. 6. Grassmann's Space Analysis. By Edward W. Hyde. $1.25 net. No. 7. ProbabiUty and Theory of Errors. By Robert S. Woodward. $1.25 net. No. 8. Vector Analysis and Quaternions. By the Late Alexander Macfarlane. $1.25 net. No. 9. Differential Equations. By William Woolbey Johnson. $1.25 net. No. 10. The Solution of Equations. By Mansfield Merhiman. $1.25 net. No. 11. Functions of a Complex Variable. By Thomas S. Fibke. $1.25 net. No. 12. The Theory of Relativity. By Robert D. Carmichael. $1.50 net. No. 13. The Theory of Numbers. By Robert X). Carmichael. $1.25 net. No. 14. Algebraic Invariants. By Leonard E. Dickson. $1.50 net. No. 16. Mortality Laws and Statistics. By Robert Henderson. $1.50 net. No. 16. Diophantine Analysis. By Robert D, Carmichael. $1.50 net. No. 17. Ten British Mathematicians. By the Late Alexander Macfahlane. $1.50 net. No. 18. Elliptic Integrals. By Harris Hancock. $1.50 net. No. 19. Empirical Formulas. By Theodore R. Running. $2.00 net. No. 20. Ten British Physicists. By the Late Alexander Macfarlane. $1.50 net. No. 21. The Dynamics of the Airplane. By Kenneth P. Williams. $2.50 net. PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK CHAPMAN & HALL, Limited, LONDON MATHEMATICAL MONOGRAPHS. EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD. No. 5. HARMONIC FUNCTIONS. WILLIAM E. BYERLY, Perkins Professor of Mathematics, Emeritus Harvard University FOURTH EDITIOJST, ENLARGED. 'U. ' ' ' Ml NEW YORK: JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited. 5 Copyright, 1896, BY MANSFIELD MERRIMAN and ROBERT S, WOODWARD UNDER THE TiTLE HIGHER MATHEMATICS, First Edition, September, 189& Second Edition, January, 1898. Third Edition, August, 1900. Fourth Edition, January, 1906* PRESS OF 2/25 BRAUNWORTH & CO. ^ BOOK MANUFACTURERS BROOKLYN, N. Y. EDITORS' PREFACE. The volume called Higher Mathematics, the first edition of which was published in 1896, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume is now discontinued and the chapters are issued in separate form. In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theory of num- bers, the group theory, the calculus of variations, and non- Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this form of publication may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. December, 1905. AUTHOR'S PREFACE. This brief sketch of the Harmonic Functions and their use in Mathematical Physics was written as a chapter of Merriman and Woodward's Higher Mathematics. It was intended to give enough in the way of introduction and illustration to serve as a useful part of the equipment of the general mathematical student, and at the same time to point out to one specially inter- ested in the subject the way to carry on his study and reading toward a broad and detailed knowledge of its more difficult portions. Fourier's Series, Zonal Harmonics, and Bessel's Functions of the order zero are treated at considerable length, with the inten- tion of enabling the reader to use them in actual work in physical problems, and to this end several valuable numerical tables are included in the text. Cambridge, Mass., December, 1905- CONTENTS. Art. I. History and Description Page 7 li. Homogeneous Linear Differential Equations 10 3. Problem in Trigonometric Series 12 4. Problem in Zonal Harmonics 15 ■ 5. Problem in Bessel's Functions 21 6. The Sine Series 26 7. The Cosine Series 30 8. Fourier's Series . ... 32 9. Extension of Fourier's Series 34 10. Dirichlet's Conditions 36 11. Applications of Trigonometric Series 38 12. Properties of Zonal Harmonics 40 13. Problems in Zonal Harmonics 43 14. Additional Forms 4S 15. Development in Terms of Zonal Harmonics 46 16. Formulas for Development . 47 17. Formulas in Zonal Harmonics 50 18. Spherical Harmonics ... 51 19. Bessel's Functions. Properties 52 20. Applications of Bessel's Functions . . 53 21. Development in Terms of Bessel's Functions 55 22. Problems in Bessel's Functions . 58 23. Bessel's Functions of Higher Order 59 24. Lame's Functions 59 Table I. Surface Zonal Harmonics 60 II. Bessel's Functions 62 III. Roots of Bessel's Functions 63 IV. Values of J^{xi) 63 Index 65 HARMONIC FUNCTIONS. Art. 1. History and Description. What is known as the Harmonic Analysis owed its origin and development to the study of concrete problems in various branches of Mathematical Physics, which however all involved the treatment of partial differential equations of the same general form. The use of Trigonometric Series was first suggested by Daniel Bernouilli in 1753 in his researches on the musical vibrations of stretched elastic strings, although Bessel's Func- tions had been already (1732) employed by him and by Euler in dealing with the vibrations of a heavy string suspended from one end; and Zonal and Spherical Harmonics were introduced by Legendre and Laplace in 1782 in dealing with the attrac- tion of soHds of revolution. The analysis was greatly advanced by Fourier in 1812-1824 in his remarkable work on the Conduction of Heat, and im- portant additions have been made by Lam6 (1839) ^nd by a host of modern investigators. The differential equations treated in the problems which have just been enumerated are 9^-«^?> (I) 8 HARMONIC FUNCTIONS, for the transverse vibrations of a musical string ; for small transverse vibrations of a uniform heavy string sus- pended from one end ; d'v d^v d'v ^ ,. which is Laplace's equation ; and for the conduction of heat in a homogeneous solid. Of these Laplace's equation (3), and (4) of which (3) is a special case, are by far the most important, and we shall con- cern ourselves mainly with them in this chapter. As to their interest to engineers and physicists we quote from an article in The Electrician of Jan. 26, 1894, by Professor John Perry: " There is a well-known partial differential equation, which is the same in problems on heat-conduction, motion of fluids, the establishment of .electrostatic or electromagnetic potential, certain motions of viscous fluid, certain kinds of strain and stress, currents in a conductor, vibrations of elastic solids, vibrations of flexible strings or elastic membranes, and innumerable other phenomena. The equation has always to be solved subjeQt to certain boundary or limiting conditions, sometimes as to space and time, sometimes as to space alone, and we know that if we obtain any solution of a particular problem, then that is the true and only solution. Further- more, if a solution, say, of a heat-conduction problem is obtained by any person, that answer is at once applicable to analogous prob- lems in all the other departments of physics. Thus, if Lord Kel- vin draws for us the lines of flow in a simple vortex, he has drawn for us the lines of magnetic force about a circular current; if Lord Rayleigh calculates for us the resistance of the mouth of an organ-pipe, he has also determined the end effect of a bar of iron which is magnetized; when Mr. Oliver Heaviside shows his match- HISTORY AND DESCRIPTION. 9 less skill and familiarity with Bessel's functions in solving electro- magnetic problems, he is solving problems in heat-conductivity or the strains in prismatic shafts. How difficult it is to express exactly the distribution of strain in a twisted square shaft, for example, and yet how easy it is to understand thoroughly when one knows the perfect-fluid analogy! How easy, again, it is to imagine the electric current density everywhere in a conductor when transmitting alter- nating currents when we know Mr. Heaviside's viscous-fluid analogy, or even the heat-conduction analogy! " Much has been written about the correlation of the physical sciences; but when we observe how a young man who has worked almost altogether at heat problems suddenly shows himself ac- quainted with the most difficult investigations in other departments of physics, we may say that the true correlation of the physical sciences lies in the equation of continuity In the Theory of the Potential Function in the Attraction of Gravitation, and in Electrostatics and Electrodynamics,* V'm Laplace's equation (3) is the value of the Potential Func- tion, at any external point {x, y, z), due to any distribution of matter or of electricity; in the theory of the Conduction of Heat in a homogeneous solid f F" is the temperature at any point in the solid after the stationary temperatures have been established, and in the theory of the irrotational flovif of an incompressible fluid % V Is the Velocity Potential Function and (3) is known as the equation of continuity. If vs^e use spherical coordinates, (3) takes the form ira-CrF) , r 4'" V) ,_L_ 3:^-1 _^. ^ ^ ?L dr' ~*~sin6' dd """sin^ tf 90= J ~ ' v5) * See Peirce's Newtonian Potential Function. Boston. f See Fourier's Analytic Tlieory of Heat. London and New York, 1878 ; or Riemann's Partielle Differentiaigleichungen. Brunswick. X See Lamb's Hydrodynamics. London and New York, 1895. 10 HARMONIC FUNCTIONS. and if we use cylindrical coordinates, the form dr' ' r dr ' r' 90' ' d^' = o. (6) In the theory of the Conduction of Heat in a homogene- ous solid,* u in equation (4) is the temperature of any point {x, y, £) of the solid at any time t, and d is a constant deter- mined by experiment and depending on the conductivity and the thermal capacity of the solid. Art. 2. Homogeneous Linear Differential Equations. The general solution of a differential equation is the equa- tion expressing the most general relation between the primi- tive variables which is consistent with the given differential equation and which does not involve differentials or derivatives. A general solution will always contain arbitrary (i.e., undeter- mined) constants or arbitrary functions. A particular solution of a differential equation is a relation between the primitive variables which is consistent with the given differential equation, but which is less general than the general solution, although included in it. Theoretically, every particular solution can be obtained from the general solution by substituting in the general solu- tion particular values for the arbitrary constants or particular functions for the arbitrary functions ; but in practice it is often easy to obtain particular solutions directly from the differential equation when it would be difficult or impossible to obtain the general solution. («) If a problem requiring for its solution the solving of a differential equation is determinate, there must always be given in addition to the differential equation enough outside condi- tions for the determination of all the arbitrary constants or arbitrary functions that enter into the general solution of the equation ; and in dealing with such a problem, if the differen- tial equation can be readily solved the natural method of pro- HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS. 11 cedure is to obtain its general solution, and then to determine the constants or functions by the aid of the given conditions. It often happens, however, that the general solution of the differential equation in question cannot be obtained, and then, since the problem, if determinate, will be solved, if by any means a solution of the equation can be found which will also satisfy the given outside conditions, it is worth while to try to get particular solutions and so to combine them as to form a result which shall satisfy the given conditions without ceasing to satisfy the differential equation. {b) A differential equation is linear when it would be of the first degree if the dependent variable and all its derivatives were regarded as algebraic unknown quantities. If it is linear and contains no term which does not involve the dependent variable or one of its derivatives, it is said to be linear and homogeneous. All the differential equations given in Art. i are linear and homogeneous. (c) If a value of the dependent variable has been found which satisfies a given homogeneous, linear, differential equa- tion, the product formed by multiplying this value by any constant will also be a value of the dependent variable which will satisfy the equation. For if all the terms of the given equation are transposed to the first member, the substitution of the first-named value must reduce that member to zero ; substituting the second value is equivalent to multiplying each term of the result of the first substitution by the same constant factor, which there- fore may be taken out as a factor of the whole first member. The remaining factor being zero, the product is zero and the equation is satisfied. (d) If several values of the dependent variable have been found each of which satisfies the given differential equation, their sum will satisfy the equation ; for if the sum of the values in question is substituted in the equation, each term of the sum 12 HARMONIC FUNCTIONS. will give rise to a set of terms' which must be equal to zero, and therefore the sum of these sets must be zero. {e) It is generally possible to get by some simple device particular solutions of such differential equations as those we have collected in Art. i. The object of this chapter is to find methods of so combining these particular solutions as to satisfy any given conditions which are consistent with the nature of the problem in question. This often requires us to be able to develop any given func- tion of the variables which enter into the expression of these conditions in terms of normal forms suited to the problem with which we happen to be dealing, and suggested by the form of particular solution that we are able to obtain for the differential equation. These normal forms are frequently sines and cosines, but they are often much more complicated functions known as Legendre's Coefficients, or Zonal Harmonics ; Laplace's Coef- ficients, or Spherical Harmonics ; Bessel's Functions, or Cylin- drical Harmonics; Lamp's Functions, or Ellipsoidal Har- monics; etc. Art. 3. Problem in Trigonometric Series. As an illustration let us consider the following problem : A large iron plate n centimeters thick is heated throughout to a uniform temperature of lOO degrees centigrade; its faces are then suddenly cooled to the temperature zero and are kept at that temperature for 5 seconds. What will be the tempera- ture of a point in the middle of the plate at the end of that time? Given a" =0.185 in C.G.S. units. Take the origin of coordinates in one face of the plate and the axis of X perpendicular to that face, and let u be the temperature of any point in the plate t seconds after the cool- ing begins. We shall suppose the flow of heat to be directly across the plate so that at any given time all points in any plane parallel PROBLEM IN TRIGONOMETRIC SERIES. 13 to the faces of the plate will have the same temperature. Then u depends upon a single space-coordinate x ; — = o and — = o, and (4), Art, i, reduces to '^^a^^. (I) -dt ■dx' Obviously, u = ioo° when t = o, (2) u = o when x =^ o, (3) and « = o when x ^^ 7t; (4) and we need to find a solution of (i) which satisfies the con- ditions (2), (3), and (4). We shall begin by getting a particular solution of (i), and we shall use a device which always succeeds when the equa- tion is linear and homogeneous and has constant coefficients. Assume* u = e^'^''\ where fS and / are constants; substi- tute in (i) and divide through by ^P^+V and we get y = a'/f ; and if this condition is satisfied, u = e^'+y' is a solution of (i). u = e^'+'V' is then a solution of (i) no matter what the value of /J. We can modify the form of this solution with advantage. Let /S = iJLi,\ then u = ^-"V'V' is a solution of (i), as is also By (d), Art. 2, ^ = ^-aV«(£l^!l+£l!^ = e—V' cos fAX (5) is a solution, as is also ^ = ^-av«5£ f > = e-"!^" sin MX ; (6) and /^ is entirely arbitrary. * This assumption must be regarded aspurely tentative. It must be tested^ by substituting in the equation, and is justified if it leads to a solution. \ The letter i will be used to represent \^— i. 14 HARMONIC FUNCTIONS. By giving different values to fx we get different particular solutions of (i) ; let us try to so combine them as to satisfy our conditions while continuing to satisfy equation (l). u — e~''i^'" sin jxx is zero when x — o for all values of /* ; it is zero when x = 7t ii /i is a. whole number. If, then, we write u equal to a sum of terms of the form Ae "'""'" sin mx, where m is an integer, we shall have a solution of (i) (see {d), Art. 2) which satisfies (3) and (4). Let this solution be u = A.e-"" sin X -{- A,e-*^" sin 2x -{- A^e-^"" sin 3X-^..., (7) A^, A,, A,, . . . being undetermined constants. When t =0, (7) reduces to u = A^sin X -\- A^ sin 2x -\- A, sin ^x -\- . . . . (8) If now it is possible to develop unity into a series of the form (8) we have only to substitute the coefficients of that series each multiplied by 100 for A^, A,, ^3 ... in (7) to have a solution satisfying (i) and all the equations of condition (2), (3), and (4). We shall prove later (see Art. 6) that I = — sin ;ir + ~ sin 3;ir -f- ~ sin 5^ -j- . . . ^L 3 5 -■ for all values of x between o and 7t, Hence our solution is ^ u = —le-'"' sin x + -e'^""' sin sx + -e-^'" sin 5;t: + . . . (9) To get the answer of the numerical problem we have only to compute the value of u when x = — and / = 5 seconds. As there is no object in going beyond tenths of a degree, four- place tables will more than suffice, and no term of (9) beyond the first will affect the result. Since sin — = l, we have to compute the numerical value of PROBLEM IN ZONAL HARMONICS. 15 = 0.185 and t—t,. log «' = 9.2672 — 10 log 400 = 2.6021 log t — 0.6990 Colog Tt ■=■ 9.5059 — 10 log C^t = 9.9662 — ID colog^" = 9.5982 — 10 log log e = 9.6378 — 10 log log ^"^ = 9.6040 — 10 log u =. 1.7062 log <•■"" = 0.4018 u = 50°.8. If the breadth of the plate had been c centimeters instead of It centimeters it is easy to see that we should have needed the development of unity in a series of the form . . TtX , 2nX , _ . 'i.TlX A^ sm — -\- A, sm -f- A, sin -\- Prob. I. An iron slab 50 centimeters thick is heated to the tem- perature 100 degrees Centigrade throughout. The faces are then sud- denly cooled to zero degrees, and are kept at that temperature for 10 minutes. Find the temperature of a point in the middle of the slab, and of a point 10 centimeters from a face at the end of that time. Assume that 4/ . TTX , I . ZTtX , I . K7tx , \ , I = — sin h - sm - — ■ +- sm f-... from x = o to x = c. 7t\ c 5 ^5 <^ I Ans. 84°.o; 49°.4. Art. 4. Problem in Zonal Harmonics. As a second example let us consider the following problem : Two equal thin hemispherical shells of radius unity placed together to form a spherical surface are separated by a thin layer of air. A charge of statical electricity is placed upon one hemisphere and the other hemisphere is connected with the ground, the first hemisphere is then found to be at poten- tial I, the other hemisphere being of course at potential zero. At what potential is any point in the " field of force" due to the charge? We shall use spherical coordinates and shall let Fbe the potential required. Then Fmust satisfy equation (5), Art. I. 16 HARMONIC FUNCTIONS. But since from the symmetry of the problem V is obviously independent of 0, if we take the diameter perpendicular to the plane separating the two conductors as our polar axis, -^^ is zero, and our equation reduces to r^\rV) I 4'"^^) _. (,) 9r" "f sin "dti Vis given on the surface of our sphere, hence V-f{e) when r = i, (2) where /(<^) = i if o < (9 < -, and /{O) = o if ^ < ^ < tt. Equation (2) and the implied conditions that V is zero at an infinite distance and is nowhere infinite are our conditions. To find particular solutions of (i) we shall use a method which is generally effective. Assume* that F=i?0 where 72 is a function of r but not of 6, and is a function of 6 but not of r. Substitute in (i) and reduce, and we get I rd\rR) ^ I 4"'" ^~de) . (3) R dr" © sin B dO Since the first member of (3) does not contain 6 and the second does not contain r and the two members are identically equal, each must be equal to a constant. Let us call this constant, which is wholly undetermined, mini -\- i) ; then / . J®\ r dHrR) i \ . df) I R dr' ©sin (9 dd \ T j, whence r , , -^ — m(m -\- i)R = o, (4) / d©\ I ^\^sin e-^) ^""^ ihT^ — de — + M'«+i)0 = o. (5) * See the first foot-note on page 175.- PROBLEM IN ZONAL HARMONICS. 17 Equation (4) can be expanded into d'R , dR , , ,r. r-^ + 2r^ - m{m ^\)R = o, and can be solved by elementary methods. Its complete solution is R = Ar" + 5r-»-i. (6) Equation (5) can be simplified by changing the independ- ent variable to x where x = cos 0. It becomes an equation which has been much studied and which is known as Legendre's Equation. We shall restrict m, which is wholly undetermined, to posi- tive whole values, and we can then get particular solutions of (7) by the following device : Assume* that can be expressed as a sum or a series of terms involving whole powers of x multiplied by constant coefficients. Let = Sa^x* and substitute in (7). We get 2[n{n — i)a„x"~'' — n{n -[- i)a„x" -\- m{m -\- i)«„a;"] = o, (8) where the symbol 2 indicates that we are to form all the terms we can by taking successive whole numbers for n. Since (8) must be true no matter what the value of x, the coefficient of any given power of x, as for instance x'', must vanish. Hence {k + 2\k + i)«4+, — k{k + i)«i + m{m. + i)«j = o, m{mA^-C) — k{k-^\) and «.+, = _^_______^^. (g) If now any set of coefficients satisfying the relation (9) be taken, = 'Sa^^ will be a solution of (7). If k-=m, then a^j,^ = o, a^+^ = o, etc. * See the first foot-note on page 175. 18 HARMONIC FUNCTIONS. Since it will answer our purpose if we pick out the simplest set of coefficients that will obey the condition (9), we can take a set including «„. Let us rewrite (9) in the form ^* - (»« _ k){m ^k~ I)' ^^°> We get from (10), beginning with k ■=■ m — 2, m(in — i) "* ' 2 . (2m — l) m{m — \)(m — 2)(»« — 3) '^'"-' ~ 2.i,.(2m- i){2m-i) '^'"' mint — i){m — 2'){m — 3)(»« — 4){»i — 5) '^'"-» ~ 2 . 4 . 6 . (2;« - i){2m - 2,){2m - 5) ^"" ^*'^- If tn is even we see that the set will end with «„; if m is odd, with «,. ^ r mint — i) , = «„ I ;r'« - ,..„ — ^^x 2 ,{2m — i) m{m —i){m — 2){m — 3) m{m - i){m - 2){m - 3) ^^_, _ "l "' 2.4. (2;« — i)(2?« — 3) ■••J' where a„, is entirely arbitrary, is, then, a solution of (7). It is found convenient to take «„, equal to {2m — i){2m — 3) ... I m ! ' and it will be shown later that with this value of «„,,©— i when ;ir = I. © is a function of x and contains no higher powers of x than jt"". It is usual to write it as PJx). We proceed to write out a few values of PJ^x) from the formula p f^) ^ (2^ - 0(2^ - 3) ■ • • I r„ _ m{m- i) ^,„... ""^ ' ni\ L 2. (2m— i) ^(^ - !)(;« - 2){m - 3) ^^,„_. -| "■" 2.4. (2;« — i)(2w« — 3) ■ ■ ■ J ^ > PROBLEM IN ZONAL HARMONICS. 19 We have : Ki2) Plx) = I or F,{cos e) = I, P^{x) = X or P,(cos 0) = cos ff, Pl^) = 4(3^" - I) or ^.(cos ti) = i(3 cos'6/ - i), P^x) = i(s^= - ix) or /'.(cos 6^) = i(5 cos'6' - 3 cos 6), Pkx) = i(3S^' - 30-J^' + 3) or /•^(cos B) = i(3S cos'tf - 30 cos"# + 3), ^.W = i(63^'-70^"+i5^) or P,{C0S d) = 1(63 cos' — 70 cos' 6* + 15 COS ti). We have obtained = P,Xx) as a particular solution of (7), and = P,„(cos ff) as a particular solution of (5). - P,S.^^ or /■((((cos 6^) is a new function, known as a Legendre's Coefficient, or as a Surface Zonal Harmonic, and occurs as a normal form in many important problems. F= r"/',„(cos B) is a particular solution of (i), and r'"/'„(cos 6^) is sometimes called a Solid Zonal Harmonic. V = A,P,{cos B) + A,rP,(cos B) + A/'P,{cos B) + AyPlcosB)+... (13) satisfies (i), is not infinite at any point within the sphere, and reduces to V^A^Plcos d) + A,Pt,zos B) + A,P,{cos B) + A,Plcosff)+... (14) when r := I. A,PXcosff) A,P,(cosB) A^Pjco^B) +d,S£SLa + ... (,5, satisfies (i), is not infinite at any point without the sphere, is equal to zero when r = 00, and reduces to (14) when r = i. If then we can develop /(^) [see eq. (2)] into a series of the form (14), we have only to put the coefficients of this series in place of the A„, A^, A^, ... in (13) to get the value of Ffor a point within the sphere, and in (15) to get the value of Fat a point without the sphere. 30 HARMONIC FUNCTIONS. We shall see later (Art. l6, Prob. 22) that if f{P) = I for o < ^ < — and /(&) = o for — < (S* < ar, /(^) =i + ^/',(cose)-^- \-Plcose) 2 I 4-.V-"o-.^ g 2 +77 -y^'^-^^)-- ('^> Hence our required solution is + — • — r'/',(cos ff)-... (17) ~ 12 2.4 '^ '' ^ '^ at an internal point ; and 2r ' 4 r V = — + - A-PXcose)-l----^Ffcos0) 2r ^ A r ''827-'^ ' I LL.ijJi /-(cose)-... (18) ?t an external point. li r = — and 6=0, (17) reduces to F=-+- H- r4 -^- -1-.-., since P„(i) = i. 2 ' 4 4 8 2 4' ' 12 2.4 4' ' ^ ^ To two decimal places V= 0.68, and the point r = —, = is at potential 0.68. If r = 5 and — — , (18) and Table I, at the end of this chapter, give I ^ I 71.^1 V= 1---— r-o.707i + 5-- — - •-4--O.I768 +... =0.12, 2.545 ''^^82.45* ^ ^ ', and the point ^ ^ 5, '= is at potential o. 12. If the radius of the conductor is a instead of unity, we have only to replace r by - in (17) and (18). PROBLEM IN BESSEL's FUNCTIONS. 21 Prob. 2. One half the surface of a solid sphere 12 inches in di- ameter is kept at the temperature zero and the other half at 100 de- grees centigrade until there is no longer any change of temperature at any point within the sphere. Required the temperature of the center ; of any point in the diametral plane separating the hot and cold hemispheres ; of points 2 inches from the center and in the axis of symmetry ; and of points 3 inches from the center in a di- ameter inclined at an angle of 45° to the axis of symmetry. Ans. 50°; 50°; 73°.9 ; 26°.i ; 77°.! ; 22°.9. Art. 5. Problem in Bessel's Functions. As a last example we shall take the following problem : The base and convex surface of a cylinder 2 feet in diameter and 2 feet high are kept at the temperature zero, and the upper base at 100 degrees centigrade. Find the temperature of a point in the axis one foot from the base, and of a point 6 inches from the axis and one foot from the base, after the permanent state of temperatures has been set up. If we use cylindrical coordinates and take the origin in the base we shall have to solve equation (6), Art. I ; or, represent- ing the temperature by u and observing that from the sym- metry of the problem u is independent of ^''«j + «4_, = 0, whence we obtain '^k--, = — ^'^k (11) as the only relation that need be satisfied by the coefificients in order that R = 2a,,x^ shall be a solution of (10). If k — o, a/i_, = o, «i.4 = o, etc. We can, then, begin with ^ = o as the lowest subscript. From (ii) «4=: PROBLEM IN BESSEL's FUNCTIONS. 33 k' Then a, = - -5, tf^=___, ^. = - ; , etc. Hence ^ = .„[i _ fj + -^^ - -^^ + . . . ], where «„ may be taken at pleasure, is a solution of (lo), pro- vided the series is convergent. Take «„ = i, and then R :^ JJx) where /.(^) = I - '-. + ^, - 2^ 4'. 6" + 2=. 4". 6^ 8" ~ ' ■ • '-'^^ is a solution of (.10). JJ^x) is easily shown to be convergent for all values real or imaginary of x, it is a new and important form, and is called a Bessel's Function of the zero order, or a Cylindrical Har- monic. Equation (10) was obtained from (8) by the substitution of X := fxr \ therefore JAI^^) 2= ^2^4" 2\4'.6=^ is a solution of (8), no matter what the value of )x; and u z= JJ^fxr) sinh {fxz) and u ^ /J^fxr) cosh (/^z) are solutions of (i). u = Jj^fxr) sinh (^^i) satisfies condition (2) whatever the value of fx. In order that it should satisfy condition (3) )x must be so taken that /.(y") = o; (13) that is, n must be a root of the transcendental equation (13). It was shown by Fourier that JS^fji) = O has an infinite num- ber of real positive roots, any one of which can be obtained to any required degree of approximation without serious diffi- culty. Let /^, , yU, , /<3, . . . be these roots ; then u = A.Jlix/) sinh {ix^z) + AJlfx^r) sinh {}x^z) + AJl)x/) sinh {^i^z) + . . . (14) is a Solution of (i) which satisfies (2) and (3). 34 HARMONIC FUNCTIONS. If now we can develop unity into a series of the form I = BJXl^y) + BJljx^r) + BJlixj) + . . . , L sinh (2;u,) sinh (2/ if /{x) = I for o < a: < — , and /(;«:) = o for — < jc < w. ((/) sinh ^ = — -(cosh 7t — i) (cosh TT -|- i) cos X -\ — (cosh 7r — t) cos 2X (cosh « + i) cos 3a: + . . . ; , s o TT" /cos :« cos 2X . cos ^X COS AX \ 32 HARMONIC FUNCTIONS. Art. 8. Fourier's Series. Since a sine series is an odd function of x the development of an odd function of x in such a series must hold good from X ^ — n to X =z n, except perhaps for the value x ^^ o, where it is easily seen that the series is necessarily zero, no matter what the value of the function. In like manner we see that if /{x) is an even function of x its development in a cosine series must be valid from x=- — n\ox=^n. Any function of x can be developed into a Trigonometric series to which it is equal for all values of x between — it and n. Let/(;r) be the given function of x. It can be expressed as the sum of an even function of x and an odd function of x by the following device : /(^) = /(^)+y(-^) ^ A^) -A- ^) (I) identically; but '^ ' is not changed by reversing the sign of x and is therefore an even function of x; and when f(x) — f(— x) we reverse the sign of x, ■'y—L r^A jg aiTected only to the extent of having its sign reversed, and is consequently an odd function of x. Therefore for all values of x between — n and 7t Ax) + A— ^) I z. r z. r I It t ■"^ ' ^ = -i?„ + 0, cos x-\-t>, cos 2x -\- (?3 cos 3;ir + . . . where 2 rM+A-A It J 2 and -^^-^ — --^ = «i sui x -\-a^ sin 2x -{■ a^ ?,m ix -\- . . . where a^^^fM^^^^ n>J 2 sin mx . dx. Fourier's series. 33 b„ and «„ can be simplified a little. b„ = — / -"^ ' ^ -"- ^cos mx. dx 7t *J 2 IT ir = — / f{x) COS mx . dx-\- I J\—x) cos mx .dx\; 0-^ but if we replace ;ir by — *■, we get ff — w / /(— x) COS mx . dx=— I fix) cos mx.dx— I f{x)cos mx.dx, . _„ and we have b,„ — — I f{x) cos mx . dx. In the same way we can reduce the value of a,^ to TT — I /{^^ sin mx.dx. Hence f{x) =— b^-\- b^ CQS X -\- b^ cos 2x -\- b^ cos ^x -\- . .. + «, sin X -\- a^ sin 2x -\- a^ sin 2,x -\- . . . , (2) TT where <5»i = — / f{x) cos »/;? . d';ir, (3) — IT TT and «» = — I f{^) s'" '^■*' • '^■^1 (4) and this development holds for all values of x between — 7t and ;r. The second member of (2) is known as a Fourier's Series. The developments of Arts. 6 and 7 are special cases of development in Fourier's Series. 1 Prob. 6. Show that for all values of x from — tt to ^ 1 I 2 sinh 71 cos X + —COS 2a: cos XXA COS4JC + ..0 2 '5 10 •^ ^ 17 ^ ^ I 34 HARMONIC FUNCTIONS. , 2 sinh a- Pi. 2. ,3- 4- 1I H — sma; sin 2a: + -^ sin 3a: sm 4^ + . . . . TT [_2 5 10 17 J Prob. 7. Show that formula (2), Art. 8, can be written f(x) = - <:„ cos/?„ + c^ cos (x — /?.) + <:, cos {zx - /?,) + c, cos (3* - ^J + . . . y where <:„ = {aj + -J„°)5 and /?„ = tan"' -r^" Prob,. 8. Show that formula (2), Art. 8, can be written f(x) = -c, sin /3„ + c, sin (,r + /^,) + -^^ sin {2X + /?,) 2 + -^s sin (3x + /?,) + . . . where c^ = («„" + ^„'')* and /?„ = tan"'^!-. Art. 9. Extension of Fourier's Series. In developing a function of x into a Trigonometric Series it is often inconvenient to be held within the narrow boundaries X = — X 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 /{x) for all values of X between x =: — c and x = c. Introduce a new variable z = — X, c which is equal to — ;r when x =■ — c, and to n when x ^= c. fix) = /( — z] can be developed in terms of z by Art. 8, (2), (3), and (4). We have /[~zj = - *o -t- *> cos £r + d, cos 2z + <^, cos 30 -j- . . . + a, sin z -\- a^ sin 2z -]- a, sin ^z -\- . . . , (i) where d„ = — / / ^ — zj cos mz . dz, (2) and EXTENSION OF FOURIEr's SERIES, 35 a„ = -^ff\^^) sin ^^ ■ ^ > < ^ /< " \\ X 1 It \: Y /-rs.— - , , / ^ ^X / ^-4 X :i III It \ 7t lY Let us apply this method to the series / = sin ;ir + ^ sin 3;i: -|- ^ sin 5;ir -|- (i) (See (7), Art. 6.) y z= o when ;P = O, — from ;p = O to ;i; =; ;r, and o when x ^ Tt. 4 It must be borne in mind that our curve is periodic, hav- ing the period 27t, and is symmetrical with respect to the origin. The preceding figures represent the first four approxima- 38 HARMONIC FUNCTIONS. tion to this curve. In each figure the curve y = the series, and the approximations in question are drawn in continuous lines, and the preceding approximation and the curve corre- sponding to the term to be added are drawn in dotted lines. Prob. II. Construct successive approximations to the series given in the examples at the end of Art. 6. Prob. 12. Construct successive approximations to the Maclaurin's 3 & X X Series for sinh x, namely x -\ -\ r + ' • • ^ 3-' 5! Art. 11. Applications of Trigonometric Series. («) Three edges of a rectangular plate of tinfoil are kept at potential zero, and the fourth at potential i. At what po- tential is any point in the plate ? Here we have to solve Laplace's Equation (3), Art. I, which, since the problem is two-dimensional, reduces to 9?-+ ^ = °' ^'^ subject to the conditions F = o when x ^^ o, (2) V=o " X ^ a, (3) V=o " y = 0, (4) F = I " 7 = 3. (5) Working as in Art. 3, we readily get sinh §y sin ^x, sinh fty cos ftx, cosh /?y sin ^x, and cosh /?j/ cos ftx as particu- lar values of F satisfying (i), V = sinh — — sin satisfies (i), (2), (3), and (4). psinh ^ sinh^ -| 4 ^ . nx I a_ . ^nx is the required solution, for it reduces to i when y ^ b. See (7), Art. 6. _y = O w. hen X = o, y = o n x^l, dy It t = o, y =^/x ti t — o. APPLICATIONS OF TRIGONOMETRIC SERIES. 39 {p) A harp-string is initially distorted into a given plane curve and then released ; find its motion. The differential equation for the small transverse vibrations of a stretched elastic string is as stated in Art. i. Our conditions if we take one end of the string as origin are (2) (3) (4) (5) Using the method of Art. 3, we easily get as particular solutions of (I) ■y = sin fix sin a fit, y = sin fix cos afit, y = cos fix sin afii, and y = cos fix cos a fit y = sin —J- cos— ^ satisfies (i), (2), (3), and (4), ,— - . mnx mnat k-^ y = ^a„, sin —1— cos — ^— , (o) where «'» = 7 fA^) sin ^7^ • ^^ (?) is our required solution ; for it reduces to/"(j^) when if = 0. See Art. 9. Prob. 13. Three edges of a square sheet of tinfoil are kept at potential zero, and the fourth at potential unity ; at what potential is the centre of the sheet ? Ans. 0.25. Prob. 14. Two opposite edges of a square sheet of tinfoil are kept at potential zero, and the other two at potential unity ; at what potential is the centre of the sheet ? Ans. 0.5. Prob. 15. Two adjacent edges of a square sheet of tinfoil are 40 HARMONIC FUNCTIONS. kept at potential zero, and the other two at potential unity. At what potential is the centre of the sheet ? Ans. 0.5. Prob. 16. Show that if a point whose distance from the end of a harp-string is -th the length of the string is drawn aside by the player's finger to a distance b from its position of equilibrium and then released, the form of the vibrating string at any instant is.given by the equation 2bi^ in— i)7t^^ ^ ' 771=1 / I . mn . viTtx m7Tat\ —5 sin — sin ——r- cos — ;— . \m 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. Prob. 17.* An iron slab 10 centimeters thick is placed between and in contact with two other iron slabs each 10 centimeters thick. The temperature of the middle slab is at first 100 degrees Centigrade throughout, and of the outside slabs zero throughout. The outer faces of the outside slabs are kept at the temperature zero. Re- quired the temperature of a point in the middle of the middle slab fifteen minutes after the slabs have been placed in contact. Given a'' = 0.185 in C.G.S. units. Ans. io°.3. Prob. 18.* Two iron slabs each 20 centimeters thick, one of which is at the temperature zero and the other at 100 degrees Centigrade throughout, are placed together face to face, and their outer faces are kept at the temperature zero. Find the temperature of a point in their common face and of points 10 centimeters from the com- mon face fifteen minutes after the slabs have been put together. Ans. 22°.8; 15°.! ; i7°.2. Art. 12.f Properties of Zonal Harmonics. In Art. 4,3^= Pm{x) was obtained as a particular solution of Legendre's Equation [(7), Art. 4] by the device of assuming that z could be expressed as a sum or a series of terms of the form a„x'^ and then determining the coefficients. We * See Art. 3. \ The student should review Art. 4 before beginning this article. PROPERTIES OF ZONAL HARMONICS. 41 can, however, obtain a particular solution of Legendre's equa- tion by an entirely different method. The potential function for any point {x, jy, s) due to a unit of mass concentrated at a given point (^,, jv,, z^ is F= ^ ^=J ^ , (i) and this must be a particular solution of Laplace's Equation [(3), Art. i], as is easily verified by direct substitution. If we transform (l) to spherical coordinates we get V= , ~ — (2) y r' — 2rr,[cos cos ^x) = i, and P^i-x) = x. Such a table for X = cos ti is given at the end of this chapter. Art. 13. Problems in Zonal Harmonics. In any problem on Potential if Fis independent of so that we can use the form of Laplace's Equation employed in Art. 4, and if the value of J^on the axis of A" is known, and can be expressed as ^a^r" or as ^ ~;^it we can write out our required solution as V = ^a„r^P„, (cos ff) or F = ^ ^ — '- ; for since PJ(l) = I each of these forms reduces to the proper value on the axis ; and as we have seen in Art. 4 each of them satisfies the reduced form of Laplace's Equation. As an example, let us suppose a statical charge of M units of electricity placed on a conductor in the form of a thin circu- lar disk, and let it be required to find the value of the Poten- tial Function at any point in the " field of force " due to the charge. The surface density at a point of the plate at a distance r from its centre is M and all points of the conductor are at potential . See Pierce's 2a Newtonian Potential Function (§ 61). The value of the potential function at a point in the axis ot the plate at the distance x from the plate can be obtained without difficulty by a simple integration, and proves to be r. M ,x' - a' F = — cos-i^— — ,. (i) 2a x A-a ^ ' 44 HARMONIC FUNCTIONS. The second member of (i) is easily developed into a power series. M ,x' -a' cos- 2a x" + a' MVn X , x' x' ^ x' -] ., = — i -5 -\ : — . . . \ n X <" a (2) a L2 a 3fl s« ' 7fl' J ^ v ^ MTa a' , a' a' , 1 -f ^ , , = tLJ-^= + ^-7F'+ ••■J'f^>«- (3) 3^° 5jr° ^x Hence ^, J/r;r y z)\ I I ^' r> / fl\ - 1 ^>,(cos 6/) + . . . ] (4) S «' is our required solution \i r .(cos^) + ...]ifr>«. (5) The series in (4) and (5) are convergent, since they may be obtained from the convergent series (2) and (3) by multiplying the terms by a set of quantities no one of which exceeds one in absolute value. For it will be shown in the next article that P„ (cos 6) always lies between i and — i. Prob. 19. Find the value of the Potential Function due to the attraction of a material circular ring of small cross-section. The value on the axis of the ring can be obtained by a simple M integration, and is , if M is the mass and c the radius of the Vc + r ring. At any point in space, ii r < c c /'„(cos 6)-'- jV,(cos ^) + ^ ^^«(cos 6^) - . . .], and \ir > e ADDITIONAL FORMS. 45 ^=v[j-^'^''°' ^> - 1?''^^''°' ^^ +^.^^'('=°^ ^) - ■ • •]• Art. 14. Additional Forms. (a) We have seen in Art. 12 that P„{x) is the coefficient of s™ in the development of (i — 2xz -\- z')-^ in a power series. (1 - 2X3 + ^') - * = [I — 2{e'' + e- »'■) + s'~\-i = {i - z^iy \\ - ^^»o -*• If we develop (l — ze*^)-^ and (i — ze-^')~^ by the Bi- nomial Theorem their product will give a development for (i — 2XZ -f- z^) - i The coefficient of a'" is easily picked out and reduced, and we get P,„(cos B) = 1.3.5... (2ffi — i) r „ , I .m ^ ^ ^ ^ 2 cos »2(? + 2 — -. r cos (m — 2)d 2.4.6... 2m L I ■ {2m — i) ' 1.3. m(m — I ) , , _ , 1 + 2 7^ Ht '—.cos(m-4)e + ...\ (1) '^ 1 .2.{2m—i){2m — T,) ^ 1-^ I J \ J If m is odd the parenthesis in(i) ends with the term con- taining cos 6 ; if »2 is even, with the term containing cos o, but in the latter case the term in question will not be multiplied by the factor 2, which is common to all the other terms. Since all the coefficients in the second member of (i) are positive, P„,(cos ff) has its maximum value when ~ o, and its value then has already been shown in Art. 12 to be unity. Obviously, then, its minimum value cannot be less than — i. {b) If we integrate the value of P„,{x) given in (11), Art. 4, m times in succession with respect to x, the result will be J ^ C (2fH I^ found to differ from ' -^ ' ' , -ix'' — lY' by terms in- {2in)\ ' volving lower powers of x than the m\h. Hence P^^^-) = ^^^i^' - 'T- (2) 46 HARMONIC FUNCTIONS. c Other forms for PJx), which we give without demon- stration, are Prix) = iy [;ir + I^F^I . cos 0]"'^0. (4) PJ.x)^\f- -=^ -— .. • (f) 'f j/ [j; — 4/;^ - I . cos 0]*"+' (4) and (5) can be verified without difficulty by expanding and integrating. Art. 15. Development in Terms of Zonal Harmonics, Whenever, as in Art. 4, we have the value of the Potential Function given on the surface of a sphere, and this value de- pends only on the distance from the extremity of a diameter, it becomes necessary to develop a function of 6 into a series of the form ^„/>„(cos 9) + AJ>izos ff) + A^Plcos ff) + ...; or, what amounts to the same thing, to develop a function of X into a series of the form A^P^x) + A^Plx) + A^Plx) + . . . . The problem is entirely analogous to that of development in sine-series treated at length in Art. 6, and may be solved by the same method. Assume f{x) = A,Plx) + A,Plx)+A,Plx) + . . . (i) for — I < ;ir < I. Multiply (i) by P„{x)dx and integrate from — I to I. We get 1 _ 1 ff{x)P,lx)dx = "2[A„ fpjx)Plx)dx\ f2) FORMUI AS FOR DEVELOPMENT. 47 We shall show in the next article that / P„,{x)P„{x)dx = o, unless m = n, -1 and that /[/>„(.)] V.- = ^-^. -1 Hence A^ = ^-^^ ff{x)P„,{x)dx. (3) -1 It is important to notice here, as in Art. 6, that the method we have used in obtaining A,^ amounts essentially to deter- mining A„, so that the equation f{x) = A^Plx) + A.P^x) + A^Plx) + . . . + A„P„{x) shall hold good for n -\' i equidistant values of x between — I and I, and taking its limiting value as n is indefinitely in- creased. Art. 16. Formulas for Development. We have seen in Art. 4 that z = P„{x) is a solution of 71 — t — . Legendre's Equation -3- (i — x') -— \-\- m(m -\- \)z = o. (i) Hence -^\_i^ ~ ^'^~^'J + ^(^^+ i)^».(^) = o, (2) Multiply (2) by P„{x) and {3) by P„(x), subtract, transpose, and integrate. We have 1 lm{m + I) - n{fi + i)-]fp„{x)P„{x)dx -1 48 HARMONIC FUNCTIONS. -,Awi[<— •)^-']^' (4) -1 -1 by integration by parts, = o. t Hence J PJ^x)PJ^x)dx = o, (6) -1 unless m ^ n. If in (4) we integrate from x to I instead of from — i to I, we get an important formula. J p.^'')p.^'y'= ^(m+, )-.(«+,) • "> and as a special case, since PX^) = i- _ ^=)^) PJx)dx = p r— V- , (8) unless m = o. 1 To get C\PJ^x)^dx is not particularly difficult. By (2), Art. 14, /I ncTKX — ir a!"!;!: — I)" , , . \-PJ.^Wx =-^;,^^ f ,^^ ■ ^^,. . ^- (9) -1 ^ ' -1 By successive integrations by parts, noting that d'"~' T ar^i^^ — i)"* contains {x' — i)" as a factor li k <. m, and FORMULAS FOR DEVELOPMENT. 49 that — ^^ — 5 — = (2my. we get -1 ^ ■■' -1 1 1 fix" - \Ydx = J{x — iy'{x + lY'dx 1 ^ - fix - i)"~\x + lY'+'dx — 1 ■.\m\ n. ... ^2""+%ml)' 1 Hence /[P,„W]V.y = ^^^ (ii) 1 Prob. 2o. Show that / P^{x)dx = o if w is even and is not zero = (- ^)'^-r-^\- ^•i-'^-y'" ^ if m is odd. 1 \FJx)'Ydx= 1 — . Note that ^ '-" 2m -f- 1 [Pm{x)y is an even function of x. Prob. 2 2. Show that if f{x) = o from .a; = — i to ^ = o, and /{x) = I from X ^ o to x = x, f(pc) = I + 3 p (^) _ 7 Ip (^) + II . L3 p (^) _ _ ^ ^ 24 82 12 2.4 ' JM = 0O Prob. 23. Show that ii^(») = :2 £„J'„,{cos d) where = ^m+l fF{e)P„{cos e) sin ^ dd. 2 50 HARMONIC FUNCTIONS. Prob. 24. Show that CSC ^ = ^[i + 5(^)'a (cos 0) + 9(j;^J^.icos #)+..,]. See (i)j Art. 14. Prob. 25. Show that *" = 1.3.5 r..(2;.+i )B^^ + '^^"^^^ + ^''^ ~ 3)^^^«-.w , ,(2;? + l)(2« — l) „ / s , "1 2.4 J 1 1 Note that fx''F,„{x)dx = -^— f'x" '^'"^^\~ ^y idx, and use the -1 --1 method of integration by parts freely. Prob. 26. Show that if f^is the value of the Potential Function at any point in a field of force, not imbedded in attracting or repel- ling matter; and if V = /{9) when r = a, V = :SA„~F^{cos 6) li r a, 2m -\- I where A^ = ^"^ ^ ^ J 'f{0)P^(cos 0) sin 6^0. Prob. 27. Show that if V — c when r = a ; V =^ c \i r <. a, and F = — if ^ > «. r Art. 17. Formulas in Zonal Harmonics. The following formulas which we give without demonstra- tion may be found useful for reference: dP (x\ -^=(2«-l)/'„.,(^)+(2«-S)/'„_3W+(2«-9)/'„..(;.)+....(l) 1 fp„{x)dx = — 1_[P,_,(^) - P„„(^)]. (3) SPHERICAL HARMONICS. 51 Art. 18. Spherical Harmonics. In problems in Potential where the value of Fis given on the surface of a sphere, but is not independent of the angle (p, we have to solve Laplace's Equation in the form (5), Art. i, and by a treatment analogous to that given in Art. 4 it can be proved that V=r'" cos nd) sin" d—j^^ and V = r'" sin nS sin" ^^^^, where /< = cos d, are particular solutions of (5), Art. i. The factors multiplied by r'" in these values are known as Tesseral Harmonics. They are functions of

) and V=~ Y,„(m, - -d^-iC ^4+2.4=. 6 2.4=.6".8 + --J^3) is called a Bessel's Function of the first order, and is a solution of the equation d'^z' , idz' I I \ , . , which is the result of differentiating (2) with respect to x. A table giving values of /^{x) and JX^) will be found at the end of this chapter. If we write JJi^x) for z in equation (2), then multiply through by xdx and integrate from zero to x, simplifying the resulting equation by integration by parts, we get or, since /.(;^) = -^\ X JxJlx)dx = xJlx). (5) If we write Jlx) for z in equation (2), then multiply through by x'^—^-, and integrate from zero to x, simplifying by inte- gration by parts, we get or X fx(Jlx)ydx = f [(/.W)' + (/.W)']. (6) APPLICATIONS OF BESSEL'S FUNCTIONS. 53 If we replace x by }ix in (2) it becomes (See (8), Art. S). Hence z = jSj^) is a solution of (7). If we substitute in turn in (7) /„(ju,x) and J^f^.x) for z, mul- tiply the first equation by x/J^^.x), the second by xJl^fx^x), subtract the second from the first, simplify by integration by parts, and reduce, we get fxJJjx,x)Jlix,x)dx Hence if //, and ix^ are different roots of Joi/M) — O, or of /.(yua) = O, or of ixa/,[ixd) — X/,(fia) = o, a fxJXl^Kx)J,(fA,x)dx = o. (9) We give without demonstration the following formulas, which are sometimes useful : JX^) = — / COS (x COS "'') + ^/.(^'') = o when r = a, or jja/,{/xa) — a/i/^{jxa) = o. (7) If now in (4) and {$) /^, < M^ < Mz < • • • ^^^ roots of (7), (5) will be the solution of our present problem. It can be shown that /o(^) = o, (8) /M = o. (9) and ^/i{x) — VoW = o (10) have each an infinite number of real positive roots.* The earlier roots of these equations can be obtained without serious difficulty from the table fory„(;ir) and /,(.*) at the end of this chapter. Art. 21. Development in Terms of Bessel's Functions. We shall now obtain the developments tailed for in the last article. Let /{r) = AJXM,r) + AM.I^,^) + AJ,{;i,r) + ... (i) ;u, , /<, , //j , etc., being roots of /„(;"«) = o, or of /,(/(«) = O, or of /Aa/X/ua) — \J,{iAa) = o. To determine any coefificient A,, multiply (i) by rJJ^piiy)dr and integrate from zero to a. The first member will become a Jrf(r)Jlfx^r)dr. Every term of the second member will vanish by (9), Art. 19, except the term a A,fr\J,{,,,r)Jdr. *o by (6), Art. 19. * See Riemann's Partielle Differentialgleichungen, § 97. 56 HARMONIC FUNCTIONS. Hence At = a —, X I rfiAf J.fi,y)dr. (2) The development (i) holds good from r = o to r = « (see Arts. 6 and 1 5). If /<, , /^2 , /<3 , etc., are roots ol JJ^pui) = o, (2) reduces to a If /*!, >W9» /*a> etc., are roots oiJljxci) = o, (2) reduces to If /i, , ;<3 , /<3 , etc., are roots of jxa/^M^) — \/AMa) = O, (2) reduces to For the important case where /"(r) = i a a. ^yt frA^)Jl}^kr)dr= f rj ,{n^r)dr^^^ CxJlx)dx = ^-JSjjl^) (6) by (5), Art. 19; and (3) reduces to 2 (4) reduces to except for ^ = 1 A,=0, [, when fii, = 0, and we have (7) (8) A,= i; (9) (5) reduces to ^ _ 2X , (10) '"'«— (V -i- u:',7^\rc,..^\- Prob. 28. A cylinder of radius one meter and altitude one meter has its upper surface kept at the temperature 100°, and its base and convex surface at the temperature 15°, until the stationary temper- atures are established. Find the temperature at points on the axis 25, 50, and 75 centimeters from the base, and also at a point 25 centimeters from the base and 50 centimeters from the axis. Ans. 29°.6; 47°.6 ; 7i°.2 ; 25°.? DEVELOPMENT IN TERMS OF BESSEL'S FUNCTIONS. 57 Prob. 29. An iron cylinder one meter long and 20 centimeters in diameter has its convex surface covered with a so-called non-con- ducting cement one centimeter thick. One end and the convex surface of the cylinder thus coated are kept at the temperature zero, the other end at the temperature of 100 degrees. Given that the con- ductivity of iron is 0.185 ^^^ °^ cement 0.000162 in C. G. S. units. Find to the nearest tenth of a degree the temperature of the mid- dle point of the axis, and of the points of the axis 20 centimeters from each end after the temperatures have ceased to change. Find also the temperature of a point on the surface midway be- tween the ends, and of points of the surface 20 centimeters from each end. Find the temperatures of the three points of the axis, supposing the coating a perfect non-conductor, and again, suppos- ing the coating absent. Neglect the curvature of the coating. Ans. IS .4 ; 40 .85 ; 72 .8 ; 15 .3 ; 40 .7 ; 72 .5 ; o .0 ; o .0 ; I .3. Prob. 30. If the temperature at any point in an infinitely long cylinder of radius c is initially a function of the distance of the point from the axis, the temperature at any time must satisfy the du 5 /d'u I du\ , K ^ \ ■ •..■II- equation xr = « (^ T — ^J (see Art. i), since it is clearly in- dependent of z and 0. Show that where, if the surface of the cylinder is kept at the temperature zero, ><, , /ij , yUj , . . . are roots of JJ^i^c) = o and Ak is the value given in (3) with c written in place of a ; if the surface of the cylin- der is adiabatic /<,, /^,, A'3. • • • a-i'e roots of J^inc) = o and A^ is ob- tained from (4); and if heat escapes at the surface into air at the tem- perature zero /<,, /i,, /v„...are roots of f^cjXt^c) — 7^/^{mc) = o, and Ak is obtained from (5). Prob. 31. If the cylinder described in problem 29 is very long and is initially at the temperature 100° throughout, and the con- vex surface is kept at the temperature 0°, find the temperature of a point 5 centimeters from the axis 15 minutes after cooling has begun ; first when the cylinder is coated, and second, when the coating is absent. Ans. 97°.2 ; o°.oi. Prob. 32. A circular drumhead of radius a is initially slightly distorted into a given form which is a surface of revolution about the axis of the drum, and is then allowed"to vibrate, and z is the ordinate of any point of the membrane at any time. Assuming that 58 HARMONIC FUNCTIONS. z must satisfy the equation — = ^".(^-^ + ~ ^ l; subject to the con- 9^ ditions z = o when r = a, — = o when t = o, and z = f{r) when or f = o, show that\s; = A^J^piy) cos /+7 2*. 2 \{n + i\n -f- 2) play very much the same part as that played hy J„{x) in the preceding articles. They are known as Bessel's Functions of the ;2th order. In problems concerning hollow cylinders much more complicated functions enter, known as Bessel's Functions of the second kind. For a very brief discussion of these functions the reader is referred to Byerly's Fourier's Series and Spherical Harmonics ; for a much more complete treatment to Gray and Matthews' admirable treatise on Bessel's Functions. Art. 24. Lame's Functions. Complicated problems in Potential and in allied subjects are usually handled by the aid of various forms of curvilinear co- ordinates, and each form has its appropriate Harmonic Func- tions, which are usually extremely complicated. For instance. Lamp's Functions or Ellipsoidal Harmonics are used when solutions of Laplace's Equation in Ellipsoidal coordinates are required ; Toroidal Harmonics when solutions of Laplace's Equation in Toroidal coordinates are needed. For a brief introduction to the theory of these functions see Byerly's Fourier's Series and Spherical Harmonics. 60 HARMONIC FUNCTIONS. Table I. Subface Zonal Hahmonics. e r, (cos 9) Pj (cos 9) Pj (cos fl) P, (cos 9) Ps (cos fl) P. (cos ») P, (cos fl) 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 .9937 .9854 .9758 .9638 .9495 .9329 5 .9963 .9886 .9773 .9623 .9437 .9316 .8961 6 .9945 .9836 .9674 .9459 .9194 .8881 .8533 7 .9935 .9777 .9.557 .9267 .8911 .8476 .7986 S .9903 .9709 .9423 .9048 .8589 .8053 .7448 9 .9877 .9633 .9273 .8803 .8332 .7571 .6831 10 .9848 .9548 .9106 .8533 .7840 .7045 .6164 11 .9816 .9454 .8923 .8238 .7417 .6483 .5461 12 .9781 .9353 .8724 .7920 .6966 .5893 .4733 13 .9744 .9341 .8511 .7582 .6489 .5273 .3940 14 .9703 .9133 .8383 .7224 .5990 .4635 .3319 15 .9659 .8995 .8043 .6847 .5471 .3982 .3454 16 .9613 .8860 .7787 .6454 .4937 .3332 .1699 17 .9568 .8718 .7519 .6046 .4391 .2660 .0961 18 .9511 .8568 .7240 .5634 .3836 .2003 .0389 19 .9455 .8410 .6950 .5193 .3276 .1347 -.0443 20 .9397 .8245 .6649 .4750 .3715 .0719 -.1073 21 .9336 .8074 .6338 .4300 .3156 .0107 -.1662 22 .9373 .7895 .6019 .3845 .1603 -.0481 -.3201 23 .9205 .7710 .5692 .3386 .1057 -.1038 -.2681 24 .9135 .7518 .5357 .3926 .0525 -.1559 -.3095 25 .9063 .7331 .5016 .3465 .0009 -.3053 -.3463 2(} .8988 .7117 .4670 .2007 -.0489 -.2478 -.3717 27 .8910 .6908 .4319 .1553 -.0964 -.2869 -.3921 28 .8829 .6694 .3964 .1105 -.1415 -.3211 -.4053 29 .8746 .6474 .3607 .0665 -.1839 -.3503 -.4114 30 .8660 .6250 .3248 .0334 -.3333 -.3740 -.4101 31 .8573 .6031 .2887 -.0185 -.3595 -.3924 -.4023 32 .8480 .5788 .2527 -.0591 -.3933 -.4052 -.3876 33 .8387 .5551 .3167 -.0983 -.3216 -.4126 -.3670 34 .8390 .5310 .1809 -.1357 -.3473 -.4148 -.3409 35 .8193 .5065 .1454 -.1714 -.3691 -.4115 -.3096 36 .8090 .4818 .1102 -.2053 -.3871 -.4031 —.2738 37 .7986 .4567 .0755 -.2370 -.4011 -.3898 -.3343 38 .7880 .4314 .0413 -.2666 -.4112 -.3719 -.1918 39 .7771 .4059 .0077 -.2940 -.4174 -.3497 -.1469 40 .7660 .3803 -.03.52 -.3190 -.4197 -.3234 -.1003 41 .7547 .3544 -.0574 -.3416 -.4181 -.2938 -.0534 42 .7431 .3384 -.0887 -.3616 -.4128 -.2611 -.0065 43 .7314 .3033 -.1191 -.3791 -.4038 -.3255 .0398 44 .7193 .3763 -.1485 -.3940 -.3914 -.1878 .0846 45° .7071 .2500 -.1768 -.4063 -.3757 -.1485 .1270 TABLES. 61 TaBLB I. SUKFACB ZoNAL HaEMONICS. 6 P, (cos 9) Pj cos 9) Pi (cos 9) P^ (cos 9) Pb (cos 9) Pe (cos 9) P, (cos 9) 46° .7071 .3500 -.1768 -.4063 -.3757 -.1485 .1270 46 .6947 .3338 -.2040 -.4158 -.8568 -.1079 .1666 47 .6820 .1977 -.2800 -.4353 -.3350 -.0645 .2054 48 .6691 .1716 -.3547 -.4370 -.3105 -.0251 .3349 49 .6561 .1456 -.3781 -.4386 -.2836 .0161 .3637 50 .6428 .1198 -.8003 -.4375 -.2545 .0563 .2854 51 .6298 .0941 -.3309 -.4339 -.3335 .0954 .3081 52 .6157 .0686 -.3401 -.4178 -.1910 .1336 .3153 53 .6018 .0483 -.3578 -.4093 -.1571 .1677 .3231 54 .5878 .0182 -.3740 -.3984 -.1323 .3002 .3234 55 .5736 -.0065 -.3886 -.3852 -.0868 .2297 .3191 56 .5592 -.0310 -.4016 -.3698 -.0510 .2559 .8095 57 .5446 -.0551 -.4181 -.3524 -.0150 .3787 .2949 58 .5299 -.07-8 -.4239 -.3331 .0206 .3976 .2752 59 .5150 -.1021 -.4310 -.3119 .0557 .3125 .3511 60 .5000 -.1350 -.4875 -.2891 .0898 .3232 .2231 61 .4848 -.1474 -.4423 -.2647 .1329 .3298 .1916 62 .4695 -.1694 -.4455 -.2390 .1543 .3331 .1571 63 .4540 -.1908 -.4471 -.2131 .1844 .3303 .1203 64 .4384 -.3117 -.4470 -.1841 .2123 .3240 .0818 65 .4226 -.3331 -.4453 -.1552 .3381 .8138 .0433 66 .4067 -.2518 -.4419 -.1256 .3615 .3996 .0081 67 .8907 -.3710 -.4370 -.0955 .3824 .3819 -.0875 68 .3746 -:3896 -.4305 -.0650 .3005 .2605 -.0763 69 .3584 -.3074 -.4325 -.0344 .3158 .2361 -.1135 70 .3420 -.3245 -.4130 -.0088 .8381 .2089 -.1485 71 .3256 -.8410 -.4031 .0267 .3873 .1786 -.1811 72 .3090 -.8568 -.3898 .0568 .8434 .1472 -.3099 73 .2924 -.8718 -.3761 .0864 .3463 .1144 -.3847 74 .3756 -.3860 -.3611 .1158 .3461 .0795 -.2559 75 .2588 -.8995 -.8449 .1434 .3427 .0431 -.2780 76 .2419 -.4112 -.3375 .1705 .3363 .0076 -.2848 77 .2250 -.4241 -.3090 .1964 .8267 -.0284 -.3919 78 .2079 -.4352 -.2894 .3211 .3143 -.0644 -.2943 79 .1908 -.4454 -.2688 .3443 .3990 -.0989 -.2913 80 .1786 -.4548 -.2474 .3659 .3810 -.1821 -.3835 81 .1564 -.4633 -.3351 .3859 .3606 -.1635 -.2709 82 .1393 -.4709 -.2030 .3040 .3378 -.1926 -.2536 83 .1319 —.4777 -.1783 .3208 .2129 -.2198 -.2321 84 .1045 -.4836 -.1589 .3345 .1861 -.3431 -.2067 85 .0872 -.4886 -.1391 .3468 .1577 -.2638 -.1779 86 .0698 -.4927 -.1038 .3569 .1278 -.2811 -.1460 87 .0533 -.4959 -.0781 .8648 .0969 -.2947 -.1117 88 .0849 -.4983 -.0533 .3704 .0651 -.3045 -.0735 89 .0175 -.4995 -.0263 .3789 .0827 -.8105 -.0381 90° .0000 -.5000 .0000 .3750 .0000 -.3125 .0000 63 HARMONIC FUNCTIONS. Table II. Bessel's Functions. X JoW Ji.x) X M") Jiix) X J-o(x) J,ix) 0.0 1.0000 0.0000 5.0 -.1776 -.3376 10.0 -.3459 .0435 0.1 -.9975 .0499 5.1 -.1443 -.3371 10.1 -.8490 .0184 0.3 .9900 .0995 5.3 -.1103 -.8483 10.2 -.3496 .0066 0.3 .9776 .1483 5.8 -.0758 -.8460 10.3 -.3477 -.0313 0.4 .9604 .1960 5.4 -.0413 -.3453 10.4 -.2434 -.0555 0.5 .9385 .3423 5.5 -.0068 -.3414 10.5 -.3366 -.0789 0.6 .9120 .2867 5.6 .0370 -.3343 10.6 -.3876 -.1018 0.7 .8813 .3390 5.7 .0599 -.8341 10.7 — .S«164 -.1834 08 .8463 .3688 5.8 .0917 -.8110 10.8 -.3033 -.1428 0.9 .8075 .4060 5.9 .1320 -.3951 10.9 -.1881 -.1604 1.0 .7653 .4401 6.0 .1506 -.3767 11.0 -.1712 -.1768 1.1 .7196 .4709 61 .1773 -.3559 11.1 -.1528 -.1918 1.3 .6711 .4988 6.3 , .2017 -.2339 11.3 -.1330 -.2039 1.3 .6301 .5330 6.3 .3338 -.2081 11.3 -.1121 -.3143 1.4 .5669 .5419 6.4 .2438 -.1816 11.4 -.0903 -.3885 1.5 .5118 .5579 6.5 .2601 -.1538 11.5 -.0677 -.3284 1.6 .4554 .5699 6.6 .3740 -.1350 11.6 -.0446 -.2820 1.7 .3980 .5778 6.7 .3851 -.0953 11.7 -.0313 -.2333 1.8 .3400 .581p 6.8 .3931 -.0658 11.8 .0020 -.2323 1.9 .3818 .5813 6.9 .2981 -.0349 11.9 .0250 -.2390 •i.O .3339 .5767 - 7.0 .3001 -.0047 13.0 .0477 -.2234 31 .1666 .5683 7.1 .3991 .0252 12.1 .0697 -.2157 3.2 .1104 .5560 7.3 .3951 .0543 12.2 .0908 -.2060 3.3 .0555 .5399 7.3 .3883 .0826 12 8 .1108 -.1943 3.4 .0035 .5202 7.4 ,3786 .1096 13.4 .1296 -.1807 3.5 -.0484 .4971 7.5 .2668 .1352 13.5 .1469 -.1655 8.6 -.0968 .4708 7.6 .2516 .1592 13.6 .1626 -.1487 3.7 -.1434 .4416 7.7 .2346 .1813 12.7 .1766 -.1307 2.8 -.1850 .4097 7.8 .2154 .2014 13 8 .1887 -.1114 3.9 -.2243 .3754 7.9 .1944 .3198 13.9 .1988 -.0918 3.0 -.2601 .3391 8.0 .1717 .2346 13.0 .2069 -.0703 3.1 -.3921 .3009 8.1 .1475 .2476 13 1 .2139 -.0489 3.3 -.3303 .2613 82 .1223 .3580 18.3 .3167 -.0271 3.3 -.8443 .2207 8.3 .0960 .2657 18.8 .3183 -.0058 3.4 -.3643 .1793 8.4 .0692 .2708 13.4 .3177 .0166 3.5 -.3801 .1374 8.5 .0419 .3731 13.5 .3150 .0880 3.6 -.3918 .0955 8.6 .0146 .3728 13.6 .8101 .0590 3.7 -.3993 .0538 8.7 -.0125 .2697 13.7 .3033 .0791 8.8 -.4036 .0128 8.8 -.0892 .2641 13.8 .1948 .0984 3.9 -.4018 -.0373 8.9 -.0653 .3559 18.9 .1836 .1166 4.0 -.3973 -.0660 9.0 -.0903 .2453 14.0 .1711 .1334 4.1 -.3887 -.1033 9.1 -.1143 .2824 14.1 .1570 .1488 4.3 -.8766 -.1386 92 -.1367 .2174 14.3 .1414 .1626 4.3 -.3610 -.1719 9.3 -.1577 .2004 14.3 .1345 .1747 4.4 -.3423 -.3038 9.4 -.1768 .1816 14.4 .1065 .1850 4.5 -.3305 -.3811 9.5 -.1939 .1613 14.5 .0875 .1984 4.6 -.3961 -.2566 9.6 - .2090 .1395 14.6 .0679 .1999 4.7 -.3693 -.3791 9.7 -.8318 .1166 14.7 .0476 .2043 4.8 -.3404 -.3985 9.8 -.2338 .0928 14.8 .0371 .2066 4.9 .-.3097 -.3147 9.9 -.2403 .0684 14.9 .0064 .2069 5.0 -.1776 -.3376 10.0 -.8459 .0435 15.0 -.0143 .2051 TABLES. Table III. — Eoots of Bbssbl's Functions. 63 n x„ for Jo(a;„) = x„ for J\{x^ = IL a;„ for J„{xJ = «„ for Ji(*J = a 1 3 3 4 5 3.4048 5.5301 8.6537 11.7915 14.9309 3 8317 7.0156 10.1785 13.3387 16.4706 6 7 8 9 10 18.0711 31.3116 34.3535 27.4935 30.6346 19.6159 23.7601 25 9037 21J.0468 32.1897 Table IV.— Values of Jo{xi). X Mxi) »> Mxi) X Joixi) 0.0 1.0000 2.0 2 3796 4.0 11.3019 0.1 1.0035 2.1 2.4463 4.1 13.8336 0.3 1.0100 2.3 3.6391 4.2 13.4435 0.3 1.0236 3.3 3.8296 4.3 14.6680 0.4 1.0404 2.4 3.0498 4.4 16.0104 0.5 1.0635 3.5 3.3898 4.5 17.4812 0.6 1.0930 2.6 8.5533 4.6 19.0936 7 1.1268 2.7 3.8417 4.7 30.8585 0.8 1 . 1665 2.8 4 1573 4.8 33.7937 0.9 1.2130 2.9 4.5037 4.9 34.9148 1.0 1.3661 3.0 4.8808 5.0 37.3399 1.1 1.8363 8.1 5.2945 5.1 39.7889 1.2 1.3937 8.2 5.7473 5.3 33.5836 1.3 1.4963 3.3 6.2426 5.8 35.6481 1.4 1.5534 3.4 6.7848 5.4 39.0088 1.5 1.6467 3.5 7.3783 5.5 42.6946 1.6 1.7500 8.6 8.0277 5.6 46.7376 1.7 1.8640 3.7 8.7386 5.7 51.1735 1.8 1.9896 3.8 9.5169 5.8 56.0381 1.9 2.1377 3.9 10.3690 • 5.9 61.3766 INDEX. Bernouilli, Daniel, 7. Bessel's Functions: applications to physical problems, 53-55- development in terms of, 55-56. first used, 7. introductory problem, 21. of the order zero, 23. of higher order, 59. problems, 25, 56-59. properties, 51-53. series for unity, 24, 56. tables, 62-63. Conduction of heat, 7. differential equations for, 8, g, 10, 13, 21, 54. 57- problems, 12-15, 21-zS. 4°, 56, 57- Continuity, equation of, 9. Cosine Series, 30. determination of the coefficients, 30. problems in development, 31. Cylindrical harmonics, 52. Differential equations, 10. arbitrary constants and arbitrary functions, 10. linear, 10. linear and homogeneous, 10. general solution, 10. particular solution, 10. Dirichlet's conditions, 36. Drumhead, vibrations of, 57, 58. Electrical potential problems, 15, 39, 40, 43- Ellipsoidal harmonics, 59. Fourier, 7. Fourier's integral, 35. Fourier's series, 32-36. applications to problems in physicsi 38-40. Dirichlet's conditions of developa- bility, 36. extension of the range, 34-35. graphical representation, 37. problems in development, 33, 34. Harmonic analysis, 7. Harmonics: cylindrical, 12, 21, 25, 51-59, 62- 63- ellipsoidal, 55. spherical, 7, 12, 51. tesseral, 51. toroidal, 59. zonal, 12, 15-21, 40, 50, 6o-6i. Heat V. Conduction of heat, 7 Historical introduction, 7. Introduction, historical and descriptive, 7, 8, 9- Lam^, 7. Lamp's functions, 12, 59. Laplace, 7. INDEX. Laplace's coefficients, 12, 51. Laplace's equation, 17, 41, 43, 51. in cylindrical coordinates, 10, 21. in spherical coordinates, 9, 12. Laplacian, 51. Legendre, 7. Legendre's coefficients, 19. Legendre's equation, 17, 40, 41, 47. Musical strings, 7. differential equation for small vibra- tions, 7. problems, 39, 40. Perry, John, 8. Potential function in attraction: problems, 44, 51. Sine series, 26. determination of the coefficients, 26- 28. examples, 29. for unity, 12, 29. Spherical harmonics, 7, 12, 51. Stationary temperatures: problems, 21, 25, 56, 57, 59. Tesseral harmonics, 51. Toroidal harmonics, 59. Tables, 60-63. Vibrations: of a circular elastic membrane, 57, 58. of a heavy hanging string, 7. of a stretched elastic string, 7, 39, 40. Zonal harmonics: development in terms of, 46—49. first used, 7. introductory problem, 15. problems, 21, 43, 44, 49, 50. properties, 40, 43. short table, 19. special formulas, 50. surface and solid, 19. tables, 60-61. various forms. 45-46. m lUi-AtrauisisimiisStii