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OAT 12zZ ;.AAi TIM :t. 1. T aBLE OF C 0 NT E N TS Pope 1. INTRODUCTI:O ........... ... 1 2. TIE BASIC EQU.TION . . .. 3 2.1 F:oatlonailly Symmetrical Elliptic Ccrd: :* ~ 3 2.2 Sep..ration of the Inve Equation .... .... 4 2.'3 R tuction to a Differential Equation . 2.4 Trirrsformations of the Basic Eqi'ation . 2.5 Co:ur.etion with Mathie's Fur.tions . 6 . SPHERICAL A i CYLINDPIC.L FUJCTIOES .. 7 3.1 Fcw Formulas for Spherical Functions 7 3.2 A iLW Estimates for SphesicL l I'unctions .. 8 3.3 A Few Formulas fo:r Cylinari 'cal Functiorns .. .1 3.4 An 'Etimate for" Cylindrical Functions 11 4. TIHE X FUICT.IO OF TEE FLRET i.D SECO(iT KTlD 13 4.1 DEf.:.ition of thle XFunctions uf the' Fi:.t 'nil Second Ki.nd ................. 1? 4.2 General Qualitec of the Coefficiernt: a, ... 15 4.3 Convergence of the juries LevelD;mvlnt2 of the XFunctions of the Pirst and SeconL Kin" ... ... 16 '4.4 Further Solu.tio;s of the Basic LEqLation anrl! Their Relation to the Xr'uncrtiuns of the F'.rs. and Second Kind . .. 17 4.5 General Relations between the jFu'nrr..f;ins .. 2. 5. TEE ZFUfICTIONS OF TPE FIRST TO FOLU.TH KIID . 21 5.1 DefLa'tion of the ZFun;ctiors of t.he First to Four'zh Kir.. ............. ... 21 5P2 Conernenco of the Series Dev.eloriment of the Z7unijcticns of the First and i'econd Kin .. 22 5.3 General Relations bet.reCr: the ZFiuctions ..... 23 5.4 An,'nmp,;otic Dj)celoper~sts of the ZFunctions 24 55 Fuirt!.er Solutions cf the Basic Eq2otion an. Their Relation to the ZFunctions of tho FIrest nd Second Kind . ... ...... 27 5.6 Lau:rentDevelo]ments fcr 2 e.nd ZFunctione 23 57 Corinecbion between the Z and ZFunctions .... 31 5.8 Wron.ki'e Determin nt .............. 33 59 Other Series Developments of the Solutions of the Enasic Equ.tbion . .... .34 NACA TM No. 1224 Page 6. CALCULATION OF THE COEFFICIEINTS OF THE SERIES DEVELOPMNTS TI T,r3 OF SPBERICAL ALD CYLINDRICAL FUNCTIONS 37 6.1 Continued Fraction Developments . 37 6.2 Method for Numerical Calculation of the Separation Farameter and. the Development Coefficients .... 38 6.3 Power Series for Separation Parameter and Development Coefficients.. ........ .... .39 6.4 Pover Series Developments ... .. .. 44 7. EIGEPFUNCTIONS OF TE BASIC EQUATION ............ 47 7.1 Limitation to v,lI Being Integers; v %i. J 0 47 7.2 Breaking Off of the Series .. .. . .48 7.3 A Few Special Function Values . 50 7.4 Connection between the X and ZFunctions .. 51 7.5 Normaalization eaaL Properties of Orthogonality of the XFunctions of the First Kind .. .. .. 53 7.6 Generalization of F. F. Nevmann's Integral Relation 54 7.7 Zeros of the EigenfPnctions ... 55 7.8 Intesgral Equations for the Eigenfunctions .. 56 &. ASn PTOTICS OF TEE EIGENVALUES ANDD EIGLTiT'UCTIONS .... 57 8.1 Asymptotic Behavior of the Eigenvalues and Eigenfunctions for Large V .. 57 8.2 Asymptotic Behavior of the Eigenvalues for Large Real 7 .. .. . 59 8.3 Asymptotic Behavior of the Eigenfunctions for Large Real 7 . 61 8.4 Asymptotic Behavior of the Eigenvalues for Large Purely Imaginary 7 . .63 8.5 Asymptotic Behavior of the Eigenfunctiorn for Large Purely Imaginary 7 .. ... .. .65 9 EIGENFUNCTIONS OF TEE WAVE EQUATION IN ROTATIONALLY SYMMETRICAL ELLIPTIC COODINATES .. ..... 67 9.1 Larae's Wave Functions of the Prolate Ellipsoid of Revolution .. . . 67 9.2 Lame's Wave Functions of the Oblate Ellipsoid of Revolution . . 69 9.3 Normalization of Lame's Wave Functions for Outside Space Problems .. .... 70 9.4 Development of Lame's Wave Functions in Terms of Spherical and Cylindrical.F*unctions 72 ;iACA TVl Hlo. 1224 Page 10. THE MEliHOD OF iGREET"'E F..2lSN FOP. THE SOLUTIOr Or BOUNDARY VYALTE P;i Problems . . 7: 10.2 Developmexit of the Spnerical Wave and of ;:r. .lane Wave in Terms of Lam1's Ware .".;jc;,n.; .. 76 10.3 Diff'rction of a Scalar Sphlei ',:. !':,ve or Plane Wave on the Ellipsoid of Revolution .. 78 11. TABIES ... .. .. . . .. 80 11.1 C,m'ierts to the Tables . .. .. 10 11.2 Eigonvaluas X (7) and Devolopmont Coeffi m cionts a (7), b ,(7); Represonted. by Broken nor nr Off Powor SSrils in 7 ............ 85 11.3 numerical Magnitude of the Eiornvalu.)sa and the DevcJlom ount Coefficients for a._i'orcnt n,7 and m = 0 . 5 95 11.4 Course of the Curv s X = An(y) or Lor Values of th I x n . .. 102 Digitized by the Internet Archive in 2011 with funding from University of Florida, George A.'Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/lameswavefunctio00unit NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1224 LAME'S WAVE FUNCTIONS OF THE ELLIPSOID OF PEVOLUTION* By J. Meixner 1. INTRODUCTION Lame's wave functions result by separation of the wave equation in elliptic coordinates and by integration of the ordinary differen tial equations thus originating. They are a generalization of Lame's potential functions which originate in the same manner from the potential equation. Lame's wave functions are applied for boundary value problems of the wave equation for regions of space bounded by surfaces of a system of corfocal ellipsoids and hyperboloids. For general elliptic coordinates Lame's wave functions have not been fully calculated so far. Except for a few general properties, not much is known about them. More consideration was given to Lame's wave functions for the case of rotationally oymraetrical elliptic coordinates (called for short, Lame's wave functions of the ellipsoid of revolution). However, even for these functions few results are in existence compared with those for the better known special functions of mathematical physics, such as cylindrical and spherical functions. The first more detailed investigation of Lame's wave functions of the ellipsoid of revolution was made by Niven (reference 1) who with their aid treated a heatconduction problem in the ellipsoid of revolution. However, the numerical values of the coefficients of his series developments in terms of spherical and cylindrical functions as they are given for the lowest indices contain several errors which were taken over into the report by Strutt (reference 2). A more extensive investigation with a greater number of applications was made by Maclaurin (reference 3). M8glich (reference 4), whose mathematical investigation of Lame's wave equation is based on certain linear homogeneous integral equations, obtained results of a *"Die Lameschen Wellenfunktionen des Drehellipsoids." Zentrale fur wissenschaftliches Berichtswesen der Luftfahrtfcrschung des Generalluftzeugmeisters (ZWB) BerlinAdlershof, Forschungsbericht Nr. 1952, Juni 1944. NACA TM No. 1224 more genera~ character. Strutt (reference 2) gives a survey of the state of the theory of Lame's wave functions in 1932; he also demon strates on a large number of examples from acoustics, electrodynamics, optics, wave mechanics, and theory of wave filters, the manifold possibilities of application for these functions. Of the treatises published in the meantime, an investigation by Hanson (reference 5), which contains several new details, should be mentioned, as well as a treatise by Morse (reference 6) on addition theorems, that is, on the development of the plane wave and the spherical wave in terms of Lame's wave functions, furthermore, a number of treatises on the wavemechanical treatment of the ion of the hydrogen molecule referencee 7). Kotani (reference 8) deals with integral equations for Lame's wave functions. In particular, a treatise by Chu and Stratton (reference 9) should be pointed out which settles exhaustively the problem (treated so far only incompletely) of the continuation of the solutions of equation (2.4g) for large and small argument and shows in detail how the entire theory of Mathieu's functions results as a special and boundary case 'rom the general theory of Lame's wave functions. Finally, a treatise by Bouwkamp (reference 10) on the theoretical and numerical treatment of diffraction on a circular aperture is to be mentioned which, for the first time, contains more detailed numerical materialconcerning Lame's wave functions of the ellipsoid of revolution. The main task of the present report on Lame's wave functions of the ellipsoid of revolution will be to compile their most important properties in such a manner that. these functions take on a form which facilitates their application. In this connection an investigation of the solutions of the ordinary homogeneous linear differential equations of the second order, which originate with separation of the wave equation in rotationally symmctrical elliptic coordinates, is of importance; further, it has to be determined what is to be understood in these solutions by functions of the first and second kind, their normalization as well as the description of the behavior ofthese solutions in different domains of the independent variables, in particular, their asymptotic behavior. Here belongs also the indication of a method of numerical calculation of these functions and the presentation of numerical tables. For the purpose of clarity it was necessary to generalize and supplement the existing material in some respects and to simplify some of the calculations and proofs. Therewith the theory of Lane's wave functions of the ellipsoid of revolution as a whole would seem to have reached a development equivalent to the theory of Iathieu's NACA TM No. 1224 +2 *P o 0 P, 0 . 0 0 C" p4 0 0 0 ral O c ca am u  0 a*r '0) Pl 1I ,O LU( 0 0 4 C 0 o. Hrt 0 4P F94 k 0o *H 0 . a) rd *wo H) 0 0 0 4 0 0 rO 0a a O H O) *l j 3 0 OH *rIP. 4 H rl 0 ON 4 p)i ^ H 0 0 SId C H HO Fb P 0 u 9 0 0 Pi A .;) P 0 S crF ri m P H 0 0 ** 0rI0)NX 0 +d Ea .H 4 Cj H *g *X l H m 4 ** HiO d M 02 0P 0'0 R a re 3 ^O *H pa * rl ri to OJ I r o ,) 0 4l 0 0 >A0 >)hO o ar P ,O ; .4 E 54 0 0 r1 0 o 0 0 4p m *ri P U PO, O .)I 9 a u0 rQ ( ,p" 0 Of (U ^ a It zI (C) II 0 rlf II NACA TM No. 1224 2.2 Separation of the Wave Equation Solutions of the wave equation in three dimensions are to be determined. (k = wave number.) Au + k2u = 0 (2.2) of the form u = fl()f2(T)f3(q) (2.3) Then the ordinary differential equations a [ t r l 2 + 1 T . d .  + 1 + ,2 252 21  Yi T12 k2c2c2 Xfl = 0 + k c f2 = 0  k2c2 2 )lf = 0o k2 2 = k2c22 +X7)f2 = 0 2, "3 2f ^p =" T3 are valid for fl, f2, and f3. X and 12 are the separation assumed to be any complex number'= . for a given boundary value problem. parameters. First, they are Tipy can only be determined In particular, g need not (2.4a) (2.5a) (2.4g) (2.5g) (2.6) NACA TM No. 1224 be an integer; this can be recognized, for instance, in the treat ment of an inside space problem in a sector 0 c = qcp of an ellipsoid of revolution. 2.3 Reduction to a Differential Equation The differential equation (2.4g) is designated as the basic equation. (2.5g) is identical vith it;, (2.ha) is transformed into it when 8 is replaced by tit and 1tc2 by vIc'. Therewith the investigation of the differential equations (2.4a), (2.5a), and (2.5g) is reduced to that of the differential equation (2.4g). The baeic domain, hocve:r, is not the same for all cases; it extends from 1 to 1 in the cases (2 .a) and (2.5C), from ] to w in the case (2.g), whereas the basic domain of the differential equation (2.4a) in the transformation to (2.47) will be changed to the domain from 0 to im (or .lse im). It proves, therefore, to be necessary to investigate the differential equation (2.4F) in the entire complex Eplane. 2.4i Transformations of the Basic Equation The basic equation represents a special cas; of the linear homogeneous differential equation of th; second order with four extra essential sing.ilaritiss, t.wo of which are made to join to one essential singulsrity. The. latter is at infinity, the two remaining extra essential singularities are at 1 and 1. The present investigation of th.' basic equation will start with connecting its solutions with the solutions of limiting cases of the basic equation. For k!c2 = O, the basic equation is trans formed into the differential equation o0 the sphrical functions and their associated functions or, as they iill be called here, of the general spherical functions. If one lets the two singu larities at 1 and 1 combine into a single singularity at t = 0, there originates, asic.e from an elementary transformation, the differential equation of the cylindrical functions. This is brought about by the substitution S= 7'1 (2.7) and the abbreviation (2.8) 7 = kc NACA TM No. 1224 if one then performs the limiting process 7,>0. From (2.4g) there originates with fl = (2 72)/2 1 v() (2.9) the differential equation ^2 2)d + 2 3 + .+ 2) d + 2 P. (4 + 1 v 0 o (2.10) In the transition from (2.4g) to (2.4a) is transformed. into itself and 72 need only be replaced by 72 For large distances, that is, r2 = 2 + y2z2+ c2 r 1 r .4 2 S 22 O(2.1la) 2 k 2 + ,f2)lc2_ = kr2 1 + + 0 '+ (2.11g) are valid. Another important limiting case of the basic equation occurs if, of the two singularities of the basic equation located at finite distance, one or both move to infinity. Thenthe differential equation of Laguerre's and Hermite's orthogonal functions, respec tively, is formed. This limitirni case will yield the asL.ymtotics of the eigenvalues and. eigenfunctions for large absolute value of y. 2.5 Connection with Mathieu's Functions Mathieu's functions are, in connection with Lam's wave functions, obtained in two ways. They appear, as is well cnown, in the separa tion of the wave equation in the coordinates of the elliptic cylinder and must, therefore, also appear in the limiting case of Lame's wave functions for the ellipsoid with three axes when one axis becomes NACA TM No. 1224 infinitely long. However, Methieu's differential equation is also obtained, except for an elementary transformation, if p in (2.4g) is set equal to t1/2. This also indicates that it is useful to consider the basic equation not only for J that are integers, but rather for arbitrary' coefficients V and i. The theory of Mathieu's fmlctions is, therefore, a special case of the theory of Lame's vave functions of the ellipsoid of revolution. Although the present report does not yield ner results of Fathieu's functions, it demonstrates hov they fit into a more general picture. 3. SPEPIIC.';L iCmD CYLI'DRIiCA FUi'TIiC. S 3.1 A Few Formulas for Spherical Functions The most important formaulas and theor.ms for spherical and cylindric functions needed belov are compiled and a few estimates for these functions are given, which will be necessary for con siderations on uniform converglence of certain series in terms of such functions. Magnus and Oberhettinrer (.reflernc, 11) is again referred to concerning the notation and add.itional formulas. The general spherical functions sa() and. _,(?) both satisfy the differential equation (1 I) () + v(v + 1) 1 1 ) = C (3.1) "v 1 2 [V and both satisfy the recursion formula (2v + i P() (v )P+1 v + )P1( (3.2) from which by three times repeated. application . 2.pN_2 + + + 2 2  S= (2V + 1)(2V + 3) v+2 + (2v 1)(2v + 3) v (+ [)(v : l)p (1) (3.3) (2v 1)(2v , 1) V2 _ NACA TM No. 1224 v + p4 1 + + + cu (1 + P) rl I rti Sf UA s8 ra HII A m O g ttH ^P 0 4, 0 0 H d4r 0 m 0 (I O 0 O d rI MP (D Ao 0 O rP .a) 4,n (d3 0O OJ *i(D * OJ H v r1 +. HI CC t I (D 0a ^ I rI 5 I :L OJ a.t ''II s^! rLOJ'~ d11 i x INAA Tl o. 1224 ( + . *"I S r r N i Si .cr 4 H H P * > 0 C  *n d d N P r r  0 0 '0 , " 01 C E l o OP " Sd 0 r r S *1 0 C 0 *H a '0 I H , .) , s fv 0 ? >' li + +1t 4 : A 0 > 0) S 0 . 4J j Q (j 0 a o cu  ra * ElT P Pa, S 43O r , 4 0 0 a ,a, C r0 C 0 0 I' B V o .1 L 0 . [ J cl 0 i K P m 0 c o z e a 0 O Nd 3 PP co 0 0 4 P H a R w F i 0 p 0 w I H r t* 0 E p t r'_ 0 3 U D T  'C L c: 4 CI Sro 0 00 20 to ,. ., 9 43, H C r i Ir 5 400 P HG ii p 3 I 0 R 02 1E O 0 31 0. C ' 0' n as r" o in .' ,0  '1 as 4a C, 0 42 I n 0* *  + c. 0 S o \l II  U 0 a + 0') C 3 + P o PI U) tz C' C 4' or (O U a5o r rI 1 1 + ' s 0 C C S0 rit a oo 0 aI .. . cu 0 ou p I' C I >B CO c co * P P m ? r!liJ * * I. E H 0 a s  :. 0 P Uj 3 *rH   I I 4 N :2 (13 =L > I CJ I t C. pr r\ a. +I NACA TM No. 1224 It is valid for 2v < arg z < 21( with the provision that the path of integration for i < arg z2 < 2t and 2it < arg z < it, respec tively, leads past the left of the point u = 0 and is also to be returned there. The estimate has to be made as above. The maximum of (u)r is, if the path of integration is suitably selected, smaller than (Izl2 + )r for positive r's and smaller than br for negative r's, where 8 is a number of the conditions Indicated above, so that (g.)^ (Z + L r> ) V+r z! v <= I z z r for r = 0, 1, 2, (3.8) for r = 1, 2, 3, . respectively, where a(5) and ( are positive, restricted functions independent of r in each closed domain excluding the points I = Tl, . 3,3 A Few Formulas for Cylindrical Functions For the following it is more convenient to introduce not the cylindrical functions themselves but rather the functions (3.9) They both satisfy the differential equation They both satisfy the differential equation 2 v v(v + 1) d + d 2 v = O ( d d2 . (3.10) INCA TM Io. 1224 and the recursion fornxilas V 2 L  2 2V 1 2 2v + 1 f _1 t2 V 2v 1 v2 2 + 1 + (2V 1)(LV + 3) 2(2v 1) _ (2v 1)(2v + 3) v V + 1 +2 V, (3.11) 2V+ vI2 1 2V + Besides, the eimpler recursion formula (v r ) C+1) +V(1) (3.13) is to be noted for the cylindrical obtained by repeated application. series development 00 4 ( ( ) = 'v , 2 2 functions from which (3.12) is Finally, the consistently convergent (3.14) is given with ar t = 0 if t is real and>0. 3.4 An Estimate for Cylindrical Functions For th:" cylindrical functions Z application of (3.13) (Q) one obtains by repeated 2V + t r(p + 1)r v + p + . 12 NACA TM No. 1224 H 0H L ^ I v' +OJ + cu _ 14 *ri .P , r II 0 rr 4o rd +rI 4 I '. I + tr ; o o,  *0 0 ' 0 0 3 H 0  SA + H  So+ S. + 4a P 0H , x I EH ai 0I S 0 l a l* *c .. +O r r H &, 0 1  H 0 o 0 C D O 0 ;4 H S+ ... P r/". ^r 9 0 + + IIO a aI m* 00  54 0 C n + a) Hr 0 ri x m El O E. i d In 0 + T NACA TM No. 1224 is in every closed domain, excluding the points 0 and o, restricted (considered as a function of ); the upper limit dioes not depend on r. If Re v < 1, this relation is valid at least for such r's for which Re (v + r) > 2. A corresponding estimate is valid for negative r's. 4. THE XFUNCTIONS OF THE FIRST A2D SECOND KIND 4.1 Definition of the XFunctions of the First and Second Kind Since for 7 = 0 the basic equation (2.4g) is joined to the differential equation (3.1) of the general spherical functions, it suggests itself to develop the solutions o' the basic equation in terms of spherical functions. One formulates the two at first formal series X" r ()P(l) r(4.) _, r= o 2)r a(l \ irU 2(7)Q ( (2) V Vr V:r r= oo and attempts to determine the coefficients aL (7) and the index v Vr in such a manner that these t.ro series formally satisfy the basic equation and converge. The further problem will be to investigate the convergence properties of the two. series (4.1) and (4.2) in order to determine that, for the two series, one has to deal with analytic functions which, in general, are linearly independent solutions of the basic equation. For the coefficients a (7) the indices V,p and the argument 7 v,r will be omitted where there is no danger of confusion; the same applies to the coefficients to be introduced later for series developments of a similar hind.. The summation index r assumes only even values. The term vith r = 0 in the two series (4.1) and (4.2) is designated as the principal term of the series. In the solutions (4.1) and (4.2) of the basic equation an arbitrary constant factor remains NACA TM No. 1224 undetermined. It may be determined in some way. Then the series (4.1) is denoted as Xfunction of the first kind.and the series (4.2) as Xfunction of the second kind with the argument t and with the indices V, with the parameter y. It will be found that the index v is determined by the separation parameter X; more accurately, there exists a functional relation between X, v, p. and 7 which is expressed by x = (Y) (4.3) V The series (4.1) and (4.2) are now inserted in the basic equation, the differential quotients of the spherical functions are eliminated by means of the differential equation of the spherical functions (3.1), and the factor E2 of the spherical functions is eliminated by application of (3.3). Then there appears an infinite sum of spherical functions with coefficients independent of t which is equal to zero. The disappearance of the individual coefficient is sufficient to this end. This leads to the conditional equations 21r = qrar2 + Prar+2 (r = 0, f2, . 7 with the abbreviation P 2(v + r+ 1)(v + r) 242 1 c = x (v + r l)(V + r) + 72 (vr+ 1)(v 2 1 r (2V + 2r + 3)(2V + 2r 1) (v+ r + + 2)(v+r +.1 (~4. Pr (2v+ 2r + 5)(21 + 2r + 3) S= (v + r i) (v + r 1) (2V + 2r 1)(2V 2r 3) NACA Tb No. 1224 15 4.2 General Oualities of the Coefficients a, The recircion formula for the coefficients a., is int,.r,reted as a difference equation. In order to aroid .:omlic;tionr, the cace of real fractional values of onehalf for v is com:plct.cly e:xcluded. and the case of real integ,rs for v + I an. v " rVe. ctively, is postpone.!. Concerning the behai or of the coefficiets ay,. at infinity, a simple formulation can be obtained ac.or'.inS to iKeuser (reference 12). The equations 1im sup .r1 Ia 2[ roo r (4.!) or '%li cu .r'a : = rl (4.7) are valid. If the behavior of th' coeffici:nts ar ?.t irfinity is given by (4.6) at lert for neftlvee cr pooitivc ri's, they inc.rase too strongly to make a cnvervtenie 0o series (4.1) ..nr (4.2) possible. Therefore, a solution of the difference equ..tion ((4.4) is to be found which shows the behavior (4.7) faor r4m ss c.ll CIa for r.r, Althou.i th..re al.s e:.ists an e::sct sol.it.ion W:ihicl behaves for r co as indizated in (4.7), this solution will in general ehibit for r+r4 th: behhav'or (' .6). 'Onrl;. fcor certain distinct values of the paramieter v (r:e so fir'). the blha.:i ('.7) prevail: for both rm ar r; in:rce, in this manner distinct values of re coordinated to cs:h val~ e 0o v. For y = 0 the c,nr.ittions oare particvJ.rly sialul. There becomes for all r's (v + r )( r) a= (r = 0, .) Th ns can, for a givan v, a.mrue any of the value. (v + r + 1) x (v + r). It is determined by the requirement that the series (4.1) and (4.2) should be reduced to the principal terr' for this case, NACA TM No. 1224 Trhich leads to X = v(v + 1). Now it is further required that under (7) always the value should be understood ~ to v(v + 1) for y 0. The existence of such a distinct Xvalue to each given V, g, and e and its uniqueness will not be proved here; it follows from the method of calculation given in section 6 for the determination of X. From here on, the coefficients a (7) will always represent V,r that solution of the difference equation (4.4) Phich shows the behavior (4.7) for r>t belongs to the value ~ (y), and therefore has the boundary values lim ar 0 (r = 2, t4, 6, .) (4.8) 7=0 Furthermore, the constant factor which is arbitrary in the coeffi cients ar may be determined in a given manner. 4.3 Convergence of the Series Developments of the XFunctions of the First and Second Kind From the estimates (3.6) and (3.8) as well as from the boundary values (4.7) for r>+~o there follows immediately that the series (4.1) and (4.2) in each closed .omacin, which does not include the points = 1l, 0, will converge absolutr'ly and uniformly. One may further conclude that the series (4.1) and (4.2) will converge as well as the exponential series. Since the individual terms of these series are analytic functions in this domain, there follows from the uniform convergence that the.sums of the series themselves will again be analytic functions, the singularities of vhich can lie only at loo, 1.'rtherrore, that the series can be dif ferentiated. termwise, and therewith the fact that the functions rerer:senter, by these series are real solutions of the basic equation. HACA TM No. 1224 4.4 Further Solutions of the Basic Equation and Their Relation to the XFunctions of the First end Second Kind Between the general spherical functions P, P F' v v vI P Q, Q", Q Q' all of which satisfy, the same vl v v v1 v1 differential equation there exist, in general, six linear relations independent of each other. They can be generalized for the Xfurnctions of the first and second. kinr. To this end several relations for the coefficients ap (7) rill be derived. V,r The system of equations (4.1) and the system of equations originating from it by the substitution v>v1 and r>r are identical because of 1I =bI q1' V1,r V,r Vl.r V,r Due to the uniqueness of the solution there follows from it (') % x1)' ((.9) v1 V Furthermore, the constant factor rhich is arbitrary in the ar's can be determined in such a manner that ai (7) = a (.) (4.10) v1,r v,r The system of equations (4.4) end the system of equations originating from it by the substitution s>i become identical if one introduces in the latter instead of the a 's the values V,ri b (7) = L(V + r ) (7) (4.11) v,r r(v r p + 1) P(v + ; 1) V,r NACA TM No. 1224 They are determined. so that bo = ao. Then the equations b" (7) = a" (y) v,r v,r and 0 7(Y) = X1(Y) V V (4.12) (4.13) are valid. After these preparations,at first a relation between X( ( ). V X (1)(;7), and. 3x (y) is derived, since according to (4.1) and (4.12) CO X() (;Y) = v _r=_ r=oo ib (7r)P () V 'r 'v+r is valid. If one expresses in this equation the spherical function P (S) v+r by ( ) V+r( and (4.11)) XV (1 ;) and Q+r () V+r the required relation =Pv r+ l ) +.(1) r(v + i + (under consideration of (4.1), (4.2), 2 epti  7C sin it ) (2)( (4.15) will be found at once. In exactly the same way there result the formulas v1 v (4.16) IIACA TN IJo. 1224 19  co 0 a .4 * I ... f ij 0 ; c/ . r,  ,) i II  4 C ,,, 4o, F i t'' ,', cr 0 ,o "'" Ti 0 > 1 l II r a P c. J . S r  *, ii rc'~i H r C S. o3 P r ,4 1. .P 0 .i ..i ci .l .. XL >  ,1 8 SC .. >  I) = I , I' II I II 1z C. , ri r coi :! ,r1 + + + O > (j d + i. 4,t, ,"  ,D , r c ,  :_ *. K, ?, , R. L II C L'r C C*ri I *. . C *r CT 5 I H 0 C . r >  .; . .) r 0 ' S + , C, C O . C! i ., u"1, T U a' . (r  + 0 ...  1 4^ . Q . ^. ,.<4  ,r r C C S0 i N + ?Z! c, ED . 1:1 >p 0* '. r I I  H o" r H 0 PO N. i i P C2 .l r P r, ' wflc  a C' * H: C 0 '.iP '. C. 0 C N ' 3 r.i rl ) ?2 P ( 0 P P *' '2 H . ,0 ; rF . 2 r 4 *r 4 ? H UH 02 C r < 0 0 .. 0 . ,O * P0 ri ..H / N F=0 E H 9 r/ 5 Q i o ci 0 i *' W rI U0 0 Pi 01 20 NACA TM No. 1224 11 Ci > 4P 4 d H* I0 P ? >0 P + + 0 O O O 0 H o* * 0+ 0) 4) 10 i l H I+0 +) u 41 CC\j ed att HH H Ii eH I < H P O O 1+ p E 'H 5H 5+P Q Id.t en i H 0II H cu 00 + m 0 0 C w a H2 0 * Hpr CH C *S* *CMI 0 0 c 2 *I 2 CH ae I = *>P 0 I r 0) 0 . Pen 0 0) enH "3 i H LA 0 'd HI *H *H 1+ c enP Hd a) r H =L 0 >0 H F ? *rH *H 0 ai * en 0) b I P o , s P *P *r s a d d . I en pe  41 9 rd H a H O1 H 0 2~ . 0 c h 4.? NACA TM No. 1224 These tro general relations which are valid for I >1, are in the ca"e 7 = 1 transformed not to exactly: (h.19) and (4.20); eT1i, namely, represc.ntz in (i .23) end (4.24) an increase of the argument by it, whereby under certain conditions the branch cut may be passed, whereas the argument of 4 in (4.19) and (4.20) is obtained by choosing such a path from E to . that the branch cut extending from c to 1 iill not be passed. 5. THE ZFUIrYCTIOIS OF THE FII'ST TO FOUrTH KITD 5.1 Definition of the ,Functions of the First to Fourth Kind If the tvo e'tra,essential sLmnuilarities of thl basic equa tion (2.1h) are made to join, as indicated in section 2, t]'ere originates, aside from an elementary transforEnation, Beesel's differential equation. Tt therefore suggests itself to attempt a solution of the basic eql.atijon alo by seriess developm.nt2 in terms of cylindrical functions. The functions .a"fined by the series (which are at first formal) series arc jcz Substitution of these serir.s into the basic eqv.tion (2.h) (,t is t to insert it into te transrd frm () (5 o indices 4 and n by means of (3.10) and (3.11), and V r Vr r= oo are defined as Efunctiro.ns of the first and. sconLd. kind. In these sere2 2 '. series ar < n; ar, 2 = O, if arg c = arg r. Substitution of these aeris into the be.sic equation (2 .r) (it is best to insert it into thle trinrsfori'icd form (210) of t. basic equation), elimination of the fist ani Esecond ,ierivativ.s of the indliccs and n by eran of (:.10) and (3.11), and removal of the denomjnal.or 2 by means of (...12) lead.E finally, exactly as in the Xfiunctions of +he first and second, kinrl., to a threeterm recursion system for the bi; (7). It arees vith the v,r recursion system (L.4) for the aP (7), if the 1 there is v,r NACA TM No. 1224 replaced by .. The solution of the recursion system differs from the indices bp (7) defined in (4.11) only by a constant factor; V,r this factor is selected to equal one. Therefore the relation (4.3) formerly found between the index v of the generating functions in (5.1) and (5.2) and the separation parameter X has to be assumed also in this case. As Zfunctions of the third and fourth kind one defines S3 ) (1) ;7) + i 2 (;) (5.3) Z ( y) =Z V (1) ) 182 (5.4) They have the same relation to Hankel's f'.nctions as the Zfunctions of the first and second kind to Bessel's and Neumann's functions. 5.2 Convergence of the Series Developments of the ZFunctions of the First and Second Kind It must now be demonstrated that the series (5.1) and (5.2) converge uniformly in a certain domain. One starts from the estimate (3.17) and from the boundary values (4.7) which are also valid for the b There results v,r lim sup) ri7 < II 1 (55) The convergence is uniform. Thus the series (5.1) and (5.2) con verge uniformly and absolutely in the entire domain { > 1 with the exclusion of the infinitely distant point; they represent NACA TM No. 1224 therefore analytic functions, can be differentiated any number of times termwise with respect to C, and satisfy the basic equation. Only in special cases these series converge also for i1 1. 5.3 General Relations between the ZFumctions The transition to various function branches over the branch cut from c through to 1 is made possible by the general relations. They can be obtained corresponding to the case of the :'fmnctions from the general relations valid for the separate series terms, thus for the cylindrical functions. (Compar IMagnus and Oberhettirner, elsewhere.) They read for  > 7(y that is, 15 > 1 1(l)(eir .yY) = evi Z(l)(t;y) V % (5.6) Zp(2)(eti~ y) = e,(v+l)fi ;4L(2)( t;) V V + 2e7na/2 sin (V +) cot (v (1) ) V (5.7) ZIL(")(e2'ti;Y) = V sin (7 l) V + i e7N/i2 2 t sin (v + & ir sin 7 v 1  e7i/2 e(vl1/2)Ti 2.. t(4)(y 7) sin (v + ) 7i/,12 (v+1/2)i sin 7 +(3) g .() sin (v r+ v sin (7 + 1) (v + In sin (v + 2)it (5.8) (5.9) 24 IACA TM No. 1224 * H H H H S" 0 S00 ca $4 SR)  a c, 0 gg k 0 0 o OJ a r 1 H Ho I + + I +a, a, + o a, N  0 I ,8 n Ig + rd v H +0 H < a, a, H rI, +0 P a 8 1 4z 0 0I~ S*ri +, o a ) a,P IA + N, P  (,0 ri uca PF fl + P a) 0 p' vi 0 C0 > *3 0* i bD F+3 C\C d0 I S  NACA TM No. 1224 This derivation is not accurate since the asymptotic develop ments (5.10) are further dependent upon the condition (i >> Iv + rl, and this condition is not satisfied for all series terms of equa tions (5.1) and (5.2), respectively, since the sum has to be formed over all r's from c to o The fact that the developments (5.12) are valid nevertheless is due to the behavior at infinity of the bar's (compare equation (.7)) according to which tha) series terms with sufficiently large values of r do not contribute noticeably to the Zfunctions. Equation (5.12) is proved as follows. According to general theorems on the asymptotic behavior of the solutions of homogeneous linear differential equations, the coefficients of which are polynomials, one obtains asymptotic series for the solutions by going into the differential equation (2.4g) with a formulation of the form (5.12) and attempts to satisfy it formally. This yields for the present case for the coefficients C(p) the fourterm recursion system (p + )(p ) + (p + p + 7 (p) + +C(p ic 1) [P + l)p + 72 jC(p) + h ( + p)C(p 1) + 472(p + p)(p + p l)C(p 2) = 0 C(l) = C(2) = 0, p = 0, 1, 2, 3, (5.14) from which they can be calculated recursively. This recursion system, however, is satisfied just then when the series (5.13) are substituted for the coefficients C(p). This substitution leads after slight transformation to MACA TM No. 1224 o II rl 54 S ri + 4 + + 5. S4 ra + 4  5  a m + > o > , ,r' 0 . .,I ad 0co al )  P*h q+ H ,1 .o + " 8 , rt 0 + H ) ,rl  , + p. a) a )*H H 'd rO H .H to 03 *rl pP ** 5 0 o *  *rU) C O 0 0 0 I S o H H co 41 0 . rl I 0 03 a) 0) 0 09 4 0 4 P CU' (0 S.r I O a 0 n P .r0 a *a) 3 0 +n> 0 H> Pi 4 0 4 WH 0 3 e 0) m +3 n a a) ri 0) ca 0 A 1 A ) a)rd a a) P D 0 +*H 1 . 4t 1 4  a . + 0 rdH4 o 0) HD O CJ A : +* s < 9 P ... D LP w ,C +' .I ,I 4 CH 0 0) d( 0 H +d '.ri r O r) 0 A 0H O O  P k a a :* ap tiil d t) L B 'rI CD ) C > a.r4I' 0d 4i r0 M +0 pI rd *ra 0 a)H l*H ad ) O r P a fl 0 0 "0 H'I 0VA * +3a 0H.Ht O S0 o0) Pd Cd ,+Q 4.p a) 0)0 00 4a H. H ra) *H 1 4 4 0 S0 i 'd t 0 0a Ms ( 4S3 B3 0 a 4* i m 0 P) 0 On O W *H '4I 0 3 H 0 0 A, PHD .r0 1 1 +0 a) ( P St I ) ) r + Od 0 0 0 (u ri 03 (j w * D E r 4" +D O F*a 9P a) I0 t 0 0 a0 : . a) a) 0d ar pQ T f Ofi O +>  OEai P ,t a 0 cir 0 0 t) l a O g l k 0I O el p rI C C 0 0 0 + Da o2p4 0 0 ,c H Se B (O t QB k u do, m r~~ . H + 4I r + ,S PI l + + a + + ., P1 II IMACA TM No. 1224 5.5 Further Solutions of the Basic Equation and Their Relation to the ZFunctions of the First and Second Kind It is immediately clear that with the Zfunctions Z() and Z p(2)(;7) the functions Zp(1'2 ) nd Zv(l'2)(;y) also VV1 V are solutions of the basic equation. Since there exist only two linearily independent solutions of the basic equation, it must be possible to express all solutions lincarily by two of them. Because of the two relations l() = cos vnn (C) sin vai( ( ) vv v L (5.16) nvl(Q) = cos wvw( ) sin vnn () J there follows irith the aid of equations (4.12) and (4.10) from the definitions (5.1) and (5.2) ZP(1)(L;y) = sin vZW(1)(7) cos vZ4(2) (r;) (5.17) V1 V V Z() () ;y) = cos v7ri(1) ( ;y) sin vwZ' (tC ;) (5.18) vl v v In order to express the functions Z P1l2)(y) y (b 1 7) and 2) ( 7)Y) it rill be practical to mve the asmptotic series; it is sufficient to limit oneself to the first Lerm of the series. Then there becomes Z (3' ) =leva I  i_ b () 0(l r)] (5.19) r=no NACA TM No. 1224 The only difference for the asymptotic series for Z (34 7) is that here al (7) takes the place of b1 (7). There follows v,r v,r immediately that 0o (i;7) ) irb (7) = z(i) y) ira' (7) Sr= ,r r ,r (i = 1, 2) (5.20) By combination of equations (5.17), (5.18), and (5.20) finally, also the solutions Z_!) ();7) of the basic equation can be reduced to the two solutions ZP (1;r). As special cases of equations (5.17) and (5.18) ZN(3q) v) e~ vri ie(3)( ;7) v1 v (5.21) 1 (4) = e i (4) ;7) v1 v should be noted. 5.6 LaurentDevelopments for X and ZFunctions *The X and Zfunctions were introduced wholly independent of each other. Since they all are, however, solutions of the same differential equation, it must be possible to express, for instance, the Zfunctions of the first to fourth kind in general linearily by the Xfunctions of the first and second kind. It will appear that in general the Zfunctions of the first and second kind are not proportional to the Xfunctions of the first and the second kind, respectively. Thus, it is not possible to define simply functions of the first and second kind for the solutions of the basic equation; it must always be added whether one is dealing with X or Zfunctions. NACA TM No. 1224 29 0 + C aj il t O 0 J ' S + O + COO  S*rl >a *i h a P C0 )o )' C) II 0 r C H 4 + * O *f  P P O 0 W P 4' H' O *H 'M 'P I i) .r  8 c a 0 " Pi 0 V a ) c 0 d p a + 0) + C0* 0>+ B 00 P 0(I no NO rt > orI P 0 mp ,i 0 c 0 0 m f co tl i H ) 02 4 z C3 H rI . TI > Cd v ? pa +2O0 ri 4 4 4 Id i *rQ *) T! r r rq , go 0o; o ro P Vfi d i 4 o ID p N i o 0 0 +  O D 4 oM 4 0 O4 O o 9 O H 4 ^^ aa r *0 CD 14 0 I'Da 1Y P 0 O Fi H 40 r O P 0 S cO C O 0 L O r f. 4 O u 0 'I *H *u 0 oI l.  t a a.u1 C > ) 0 or1 *0>C O O P 0 m 0 > s rI 0% m e O CM + 0 H aId M o 0 C3 4 b a0 a Iood ca nd m P L *)i 433 0) 0 0 Orl 9 mo mdH 4) g I >3 O C ro 0 ()>a 0e 4.H C O O Cik 0' ' a .1 (fl 1 rI0H O rd_ C a A S0) 00)00 + 0 rI o'd C a. * cC N, 0ii 4 ; a), o a) 0 43 0L 0 p =L ( a)u 0 Oro o 0 i I r1 I.p a *i r 4 0 mN O > I A m .43 a ) qH 0)O C 0u w00 C S cr4 c 04 W M i 0 4 4 ) 43 02 '4 *ri %) ) A (  )) O) G)/ !4 nin S ) 0A '0g 4'3 +3 4 + rI (3 m) su c. 1a c3 0 o O 8 8N 0* i Hoi ori 11 O o: P .14 s o ( a a) ri 14 +2 ) f rLC!3OO) CO C 0) 0 CO +2 1 H P M CO a Q O 0 I1 0 0 H 0 A 8 1 4.n . J ., C) C. P C OO OO *H m r4 .ri C3 (D SI t r1 r1 p 0(U 0 0 P 0) RC rO 1 H l M 0)024 PlC oirN o a) 0 a+ 4 d t 0) 0 ' C,CJ 0 CO P rI 50 fC rH 1dH *4 pH a * (1) LM 0 t> m +) m 0) E r(D 0) ) a  rd0)iu Id a S9 rd C) o *H ID op d o i r 4i 2o H a M . 30 NACA TM No. 1224 \ tr itr\ + CC ;> It *H g 0 C r+ *H H 0i II' 8a o Cm Cl U) =L P. H 411 + 8 lc 04 i ; j 8 81,xin 0 V0 2 a II B 04' V0 v v ^1 *) Oa F!1 8 cr:or r 8 mII r~c NACA TM No. 1224 5.7 Connection between the X and ZFunctions Due to the equality of the characteristic e:cponents v and v 1, respectively, in equations (5.23) and (5.?4) or equations (5.22) and (5.25), respectively, aX(2) ard IZ(1) on V vI the one hand and 1.(2) and S Z( on the otble only differ every vi v time by a factor independent of 6 Thus one may equate (1 7) = 1 sin (v p)*e (y)()) (E (/) (5.26) v 7 V vi (1)(%+1)ii (2) L: z ( ) =e e () (t(;) (5.27) vi V V The various factors, as sin (v Pi, and co forth, were introduced for convenience. Bctween K l (y) and 7(2)() there V V exists the connection p(2)( 1 ) (V+1)U! p(.) K (7) in (v )e (7) (5.28) v1 = v One further obtains with the aid of equations (4.18) and (5.17) 1)(1) r yi. (1) S( ;) = K (Y) (e coe vn) (C) v v L v " P+V i p 1 (2g) (5.29) V V V V vi z^2^c,;) + sin vn z c;) = eV7 () V(y)c)i2 () 7) (5.30) Tf V, I are integers, these relations are essentially simplified; then the Xfvnctions and theZfunctions of the first and the second kind., respectively, are actually proportional. 32 NACA TM No. 1224 Now the calculation of K (7) is left to be performed. To that end one may select the coefficient of any power of E in equations (6.22) and (525) and carry out the comparison. One obtains P(1) 1 v 22siv . ) S 7 =i pr &) = v,r v,r V,r r= ) (5.31) + + (1 +Sr p(2) 1 (v+P+l)it ")2 s2 is S (Y) e ) r(v + P + s + 1) P irbP (7) vr V 2 + 2 2 x (532) Lira (y) Any even number is to be substituted for s in equations (531) and (5.32). The value of v(i)(7) is independent of the selected TIJ.CA TM No. 1224 special value of s. If one replaces in equation (5.32) p by p and s by s and then multiplies by equation (5.31), r. (7) 4 (7) = e(+l) 7 (5.33) V V is originated. 5.8 Uronski's Detorminarnt Wronski's determinant of the Zfunctions of the firct and second kind are defined by V Z(1) (;) .. ZP(2) (;y) g(2)( ) ()(;7) (5.34) wz v d V v d V From the basic equation (2.4g) their follows in the Imnown way that Wronski's 'etcrminant of any tTo of its solutions is proportional to (~2 l)1. The factor of proportionality is determined by substituting their asymptotic series for the Zfunctions of the first and second kind; it is sufficient to limit oneself to the first terms (5.19). There results m 2 S 11 r= o Wronski's determinant of the Xftmctions of the first and second kind (2V()l) (t) ) (2) tC A Y) (5.36) results from Wg by using equation (5.30). NACA TM No. 1224 (5.37) is originated and therefore W 1 1 7 t2 1 irb ( (Y) v (7) Simplifications result for the important First, one agrees upon omitting the index i zero. No7r there is valid b (7) = a (7) V,r V,r to equation (4.1), because of _n,(1) = 1, X(1)(1;Y) = v; (5.38) special case p = 0. when it has the value and further, according o00 r vr v,r (5.39) Thus, one can also write for 'Wronski's two determinants X (1) ..X (1) (l;7) t 1 (5.40o) 5.9 Other Series Developments of the Solutions of the Basic Equation Niven (reference 1) investigated series developments of the following form (the functions represented by them are called V and Wfuunctions): ( (7) = ( ,2 + 72)/ w1 v ric r(Y) (w) Vr V+r r=>  S.(r) (2) wx NACA TM No. 1224 W4(1)(w;7) = ir a (7)$ (w) (5.42) Vir v+r r=oo The relation of the variable w to E ancd is: 2 = t2 Y2 = (E2 1) (5.43) For the coefficients cr arid dr there results again a threeterm recursion system which can be transformed into equation (4.1L). If the coefficient of the principal term is set equal to ao, the equations Ir + r + y + Cy>r(7) ( M c (7) = a ,(7) Vr r(VMrv + r j +vr =r +r a (7) (5.44) p (0) vde v1 () = all (7) V,r pv + r _+ (V + g + v,r = irtV al (7) (5.45) (o) v,r V are valid. NACA TM No. 1224 The series (5.41) and (5.42) converge uniformly in each closed domain given by 17r < lvw < c; or, expressed in the tplane: 1 < J2 < co. The bounding curve f,2 1 =1 is a lemniscate. Equation (5.42) is, as will be shown later, a limiting case of a general development, which still contains an arbitrary parameter and which yields as a further limiting case the series (5.1) and (5.2) of the Zfunctions. One can immediately give further series developments of V and Wfunctions; to this end one has to replace the functions 'V+r in equations (5.41) and (5.42) by nv+r, or the indices V + r by V r 1, or g by [i, or one has to make two or three of these substitutions simultaneously. One thus obtains a total of eight Vfunctions and eight Wfunctions. Their properties will not be investigated here more closely; it should only be mentioned that all of them also can be expressed linearily by the functions of the first and second k~ d which is done in the simplest way with the aid of the as ::_'c;iic series. Whereas the asymptotic series of the .f.rctions progress with powers of 1 the asjyrattic series of t] V and. Wfunctions one obtains from equation (5.41), and so fort'!, by substitution of the asymptotic series of the cylindrical finc tons, contain powers of (2 72) /, that is, 2 1)12. According to a suggestion by Wilson (reference 7) one can now also set up asymptotic series which progress with powers of ( t l)l. They have compared with the series (5.12) a slight .vant:ge in.: far as a threeterm recursion system results for their coefficients. Correspondingly, for the solutions of the babic equation also developments in terms of cylindrical functions with the argument u t 7 = 7(~ 1) may be given, of the form F1 = ( 1)/2( + 1)1/2 e t ( 7) (5.46) 1 vt V+t where t runs through all integers, the odd as well as the even ones. +t can again be replaced by n and so forth. These developments will, however, not be followed up here. NACA TM No. 1224 6. CALCULATION OF THE COEFFICIENTS OF THE SERIES DEVELOPMENTS IN TERMS OF SPHERICAL AVD CYLINDRICAL FUNCTIONS 6.1 Continued Fraction Developments The solution of the recursion system (4.4) which for r  has the behavior at infinity (4.7), can be represented by the convergent (reference 13) continued fraction ar = 2 r 9pr0r+2 r r/ r+2 I 4r+29r+/ r+2 (6.1) 5r 2r/$r  ^ $r 1j/%.+i (6.1) ar2 1 1 1 The solution which has the behavior (4.7) for r> m can be represented by the convergent continued fraction S Prd r r r/W2 rr 2 r2r4/2r4 (6.2) ar+2 1 1 1 The subnumerators of both continued fractions are in each finite closed domain of y and Xvalues for sufficiently large values of r in the case (6.1), of r in the case (6.2) smaller than onefourth; thus, according to a theorem on uniform convergence of continued fractions, the continued fractions (6.1) and (6.2), respectively, are in each domain of this kind for sufficiently large r's and r's, respectively, uniformly convergent and are therewith regular analytic functions in 7 and X, since the individual approximation fractions are functions of this kind. For not sufficiently large r's and r's, respectively, then follows, that these continued fractions are also analytic functions which, however, need not in every case be regular. A solution of the recursion system (4.4) has now to be found which shows the behavior at infinity (4.7) for r> o as well as for r> . Then the value of ar/ar2 calculated from equation (6.1) must equal the value of this expression calculated NACA TM No. 1224 from equation (6.2). An equationredlts which for given V and g allows calculation of the. separation parameter X as a function of 7. If 7 and X V(V + 1) both are'sufficiently small, the solution k (y) is a regular analytic function of 7 which V assumes for 7 = 0 the value v(v + 1). Thus the X'(7) as well as the ar/ao can be developed in power series in 7 with non vanishing radius of convergence the magnitude of which will not be~ investigated here more closely. *' :6.2 Method' for Numerical Calculation of the Separation  Parameter and the Development Coefficients The representation of the coefficient ar by continued fractions is also for larger values of 7 still particularly suitable for the numerical calculation of the separation parameter X and the are's. Mostly v, &, and .7 are given. Then the values pr, qr, and r + X can be calculated numerically from equation (4.5). One starts from a value for X which is assumed as close as possible to the actual value and calculates for a selected fixed r the expression ar+2/aZr from equation (6.1) as well as from equation (6.2). Then one repeats this calculation with a slightly altered value of X and exaiines whether thereby the agreement of the two values ar+2/ar _is improved. By further variation of X one can finally obtain .,. an agreement of arbitrary accuracy. Therewith one can find the S lue X (7) with any desired accuracy. One more investigation has to be made: whetherthe solution thus found for 7 = 0 goes over continuously into v(v + 1), that is, into X () and not perhaps into X (0); for X (0) Sv+2+2 also is a .solution of the present problem as can be recognized from the fact that equations (6.1) and (6.2) contain the values v and r only in the combination V + r. This question cannot be decided unless one has already a general picture of the functions X (7) as it is given in figure 1 for g = 0, v's that are ntegers and el are integers, and real y S. NACA TM No. 1224 The number of terms of the continued fractions (6.1) and (6.2) to be included in the calculation corresponds to the desired accuracy. For large Irl the partial fractions 7 prqr+2/r r+2 assume the order of magnitude 7 /(16r4); thus the index r' of the last partial fraction to be included will have to be selected at any rate larger than j7j/2. The calculation of the a (y) 's is made by taking the v,r value found for X(7) as a base, and calculating ar+2/ar from equation (6.1) and therefrom a+2/ao, at+f/ao, 6.3 Power Series for Separation Parameter and Development Coefficients For the numerical calculation of the separation parameter and the development coefficients one can for small values of 17l make good use of the power series developments in terms of 7. If one limits oneself in these to the first terms up to the fifth power of 72, inclusive, one obtains, in general,still quite useful approxi mations up to about i72 = 5. Therefore, following, the power series for the N (7) shall be calculated explicitly to 7 0 V 8 inclusive, for the ar/ao to y inclusive. Therewith one more series term is obtained than by Niven (reference l); compared with Niven's cumbersome treatment, the calculation is essentially simplified. For the limiting case 7 = 0 there follows from the recursion system (4.4) (7) + (v + r + 1)(v + r)ar = 0 (6.3) The case where all ar disappear is not of interest since it leads only to identically disappearing solutions of the basic equation. Thus there becomes X (0) = v((v + 1), ar = 0 for r y 0. V NACA TM No. 1224 For y = 0 all 0r with the exception of 0o have nondisappearing limiting values. From the continued fraction (6.1) one can draw the conclusion a = 71 +0 r2r (r = 2, 4, 6, .) and from that further a r = Yr wo (r = 2, 4, 6, .) If one well, there takes the next partial fraction into consideration as results as the next approximation ar 7 ,"'^ 4 P "6 a0 7r J+ 7* +*. + +() (78)j Accordingly,, one obtains ar P2P4 Pr ao 204 r 9rPr2) rr 2/ + o(78) . (r = 2, 4, 6, .) (6.6) 4(92P4 94P6 + + .4 +. . \^2P14 i^4 6 (r = 2,4,6, .. ) (6.7) One now substitutes.. a/ao, from equation (6.6) and a*2/ao from equation (6.7) into the .equation .r= '0 of' the recursion system (4.4) ,and obtains (6.4) (6.5) q2 .14 .+ q , NACA TM No. 1224 ^r2 1 c"P ^ 2 F, q 2p41 R 74 0 + 2 41 V_ This equation permits the calculation of X (y) 10 V as Over (6.0) series in terms of r up to the power 70, inclusive. At first one can see, by having 7 approach 0, that 0o = 0(74). Tharewith, however, 0r also is Iknown for any r with the exception of terms of the second and of higher powers in y2. If ont now inserts c, in'this approximation on the right side of equation (6.8), Do becomes already; correct ji to the third power in If one repeats this procedure with the new values are correct up to 7y, inclusive, there results therewith Y (7) exactly up to 710, inclusive. of this calculation as well as the cflcu.lation of not particularly difficult, therefore the results timnediatuly. In order to make the representation clearer, abbreviations are introduced: 7", inclusive. of the r which finally 'o and The performance the ar's is are giver. the following Or = Dr(1 725) + . where Dr = r(2v + r + 1) 5(= (2 1) (2V + 3)(2v l)(P v + 2r + 3)(2v + 2r 1) 09D P2  D2 D4 D~ ". (6.11) 2q 0 P'4 2 DP,2 P 4 (6.9) (6.10) NACA TM No. 1224 Ai = p22 + P 2 2i + 8 D 2 D2 1 = 0, 1, 2, . Then there becomes 2 2 Sv(v 1) + 2 2v + 2v 2, Sv( + (2 + 3)(2 ( 1) + (ABoB A2 C)78 + [A 3 2AoB, + p2?(22 + 7 10 + 0(712) 2 Y2 12  72, + 746 + 6Ao2 + Al D2 Y4 9' 4 2 ( 2 A 4 y6  AY + Al1 D2 S1 +(6 13 + ) (6.13) P4 Ao + D2) + 4 2 + (710) (6.14) + 5) + 7422 + (6.15) 2 A P Ao 1 82 64 2 A P6 1 0 + 0o(710) 8264 64 2 D 4 L) 2 1 D2 C = 5 D2 (6.12) X" () V P4 . NACA TM No. 1224 a 7 6 6 72 52 + F64 6 + 0(7 1G) aa q 2" a8 q2q q68 + 0(y0) a, D2D4gD8o Y 2 P2 2 1 =7 t 11  2 + 744. 2  5_82 52 . A.l S4) 52 + 0(710) (6.18) 212A, 2 + A 4 P2P4 '2(,_ + 74 (_2 _25.4 + a:6 a r a8 '3 P2P4P6P8 10 ao 7 D.2DDD8 (7 8, D_2 4 6D 8 o(y10) (6.19) (6.20) (6.21) For the case excluded. above where v has fractional values of onehalf, the convergence radii of these series equal zero. It seems therefore probable that the convergence radii ere functions of v which can be infinitely large for special cases, but not in general. (6.16) (6.17) a_2 a. + 4 S A: P AG 6 L D_ D 4 2 2 4 + 6) + o(7 1) P4  (252 D_2 ,6 P2P4P6 6 L NACA TM No. 1224 6.4 Power Series Developments Since occasionally power series developments of the solutions of the basic equation (2.4g) also can be useful, they will be briefly discussed below. One can of course obtain them at once by substituting in the series (4.1) and (4.2) for the Xfunctions of the first and second. kind. the known power series developmnts of the spherical functions in terms of powers of ; one thus obtains power series for the solutions of the basic equation which converge in the circle I0 < 1. The problem of the Laurentseries for 1 < IS < need not be discussed further since they are already calculated in equations (5.22) to (5.25). However, one can obtain these developments directly. Therewith a new method for the calculation of these functions and particularly of X (7) is found. V One starts from the differential equation written in terms of t rather than of . (2.10) which is g+ d[~+ 2 v = (6.22) For the integration one tries the statement v= s=0 (6.23) S ( s +s gVi t Then there results for the gs (the indices p and v as well as the argument y in general are again omitted) the threeterm recursion system (2 1) + 2 ar NACA TM No. 1224 (v + s 4 + 2)(v + a s + l)gs+2 + [(v + 3 + 1)(v ) X g 72g2 = 0 (s = 0, 2, t., .) (6.24) gg *2 There exists a solution with the behavior at frnfinity g > gs2 B2 gs for s)>= and a solution with the behavior at infinity  1 gs2 for s a>. The quotient of the two solutions in independent of s only then when X assumes certain distinct values. As one can see by comparing vith equation (5.23), these are just the values X (7). V From the behavior at infinity of the coefficients gs one can conclude at once that the series (6.23) converges in the domain 1 < Il < . If one substitutes in equation (6.24) for the coefficicnts gs the coefficients calculated already in equation (5.23), there results after elementary transformations ira pl) +rar (2) + r + ;.iaj;; + r v r= v 2 2) L 2:73) + s 1)(2 + r a) v + r.+ 1)(v + r)  Y2(v + s ) = (s = 0, 2 t, .. .) (6.25) These relations can be used, like equation (5.15), for the control of numerically calculated values of the a (7). v,r HACA TM No. 1224 The recursion system (6.24) is, except for the case of v,IL being integers with v k H 0, probably less suitable for the numerical calculation of the X1 () than the continued fractions (6.1) v and (6.2). Ordinary power series with increasing powers of result for the solutions of the basic equation if one sets equal g2 = g4 = = 0 and requires go = 0. Then there results for v the determining equation (v .g) (v 1) = o (6.26) Therefore v has here a meaning different from the one it had so far. The behavior at infinity of the gs for s > is simple: all of them disappear. The behavior at infinity for s  c is given by g 1 or >~. The first case is the standard gs2 s case; the power series converges for }(g < 1. The second case is, for V and beiig integers with v p ( 0, realized for a solution of the basic equation, the Xfunction of the first kind; the power series then converges for all finite . It will be best to make the numerical calculation of the coefficients of these power series which are convergent in the unit circle so that first X (y) will be determined according V to the method given in section'6.1, or, for smaller values of 7, from the series (6,13); the coefficients gs can then be calculated from equation (6.24) for each of the two V values given by equation (6.26). A special but simple problem will then be left: how the two calculated power series are connected with the Xfunctions of the first and second kind. NACA TM No. 1224 7. EIGFNFUNCTIONS OF TEE BASIC EQUATION 7.1 Limitation to v,p Being Integers; v ?= i = 0 The determining factors for the eigonvalues of the separation parameters X and p and, if occasion arises, of the wave coef ficient k, are the domain of space which was taken as a basis and the boundary conditions on its boundary. This treatise is limited to the most important type of eigenvalue problems of this kind; for them the domain of space lies either within an ellipsoid of revolution, or between two confocal ellipsoids of revolution, or outside of an ellipsoid of revolution. The first t :o cases will be called problems of inside space, the last case problem of outside space. The entire domain 1 T= 1, 0 S cp 5 2g becomes then effective for the two coordinates T and cp. Boundary conditions in q and p do not apear then; they are replaced by the require ment that the wave function for n = "l remains finite and that it is single valued, that is, that it has the same value for cp + 2n that it has for cp. The latter requirement leads to p's that are integers, the first one to v's that are integers v > IPI 0. That the Xfunctions of the first kind remain finite at the points T = tl follows directly from the series (4.1) by teling the estimate P ) + \2 112, which is valid for this case, as a basis. Following, n will always be Tlrtten for V and m for p where v and p are real integers; for the present, n m 0 is assumed. Ths case of negative m's, the absolute amount of which is = n, is then obtained at once from equations (4.15), (4.17), and (5.20). The calculation of these special functions was practically settled amongst other thinGs in the lact sections; even though it was assumed there that neither v  i nor v p are integers, almost all results can nevertheless be taken over as simple limiting processes demonstrate. Only a few particularities result, compared with the general case; they will be discussed belov. NACA TM No. 1224 7.2 Breaking Off of the Series If gr, = 0 for a positive r' or pr, = 0 for a negative r', the are's break off to the right or to the left, that is, for q = 0; r for pr, = 0; r' > 0: r' < 0: a = a =a = ,. r' r'+2 r'+4 ar ar 2 r '4 . = 0 = (7.1) . = 0 is valid which follows in the simplest way from the continued fraction developments (6.1) and (6.2), These cases occur when i v is a positive integer or when p + v is not a negative (Sicl) integer. Since it was presumed 0 = m = n the first possibility does not occur, but the second one does always occur, that is, for all admissible m,n. Here again tro cases must be distinguished which are both oriGinated from Pr' = 0: n + m+ 2 = r'> 0 or n+ m+ 1 = r'> 0 In the first case m n is an even number, in the second,an number; ar +2 is the first nonvanishing ar. For the b 's follows from equation (4.12) that they disappear for all r (7.2) odd there r' + 2m. Further, all pm (0) disappear for n + r = m, m + .m 1. n+r The developments of the X and functions of the first kind begin, therefore, for n m = even with i =(l)( ) =am imnpm( ) n n,mn m + am Imn+2(m ,+ Sn,mn+21 m+2 + ()() = (2 im ,m(n ) + bn,mn+2mi.2( ) +* . NACA TM ITo. 1224 and for n m = odd with m(1) m mn+1 m X (1 7) = a i P (g) n n,mn+l Im+1 + am imn1r+3 m () + n,mn+3 m+3 (7.4) ,m(l) =y ( a/2 m kI n= n bnn+1 m 1) + bti 3() '. n ,mn+3 n +3' The series for iZ(1)(;vy) converges for all finite The n corresponding formulas for the fiuctions of the second kind result if the functions +r (t) are replaced by n (Q). The developments for the Xfunctions of the second kind show a special behavior. The spherical functions of the second kind belonging to the vanishing coefficients are, ar_2, become infinitely, large in such a manner that their products have finite limiting values. The coefficients a (7) are defined by Iin a () 0= a Pm (I ) v,r v+r an,r n+r for m = 0, 1, 2, .. and v + r + m> 1, 2, (7.5) Then there becomes m m+n: m an,r ) = lim () r ( + v + m)P(vm)a (y) Yvr for m = 0, 1, 2, and v + r + m  1, 2, . (7.6) NACA TM No. 1224 and the series (4.2) reads r' to xm(2)' ;T( i (n,r( irnr) Mra (7.T) r= oo =r +2 The series (5.41) for Vm(1)(w;y) breaks off only when n m n is an even number for odd values of n m the coefficients cm (7) n,r have the indefinite value o.0 if r  r' with finite limiting value. The series (5.42) for l(1)(v;y), on the other hand, breaks off only when n m is an odd number; for even values of n m the coefficients ed (Y) have the indefinite value o.0 with n,r finite limiting value if r = r'. Similar conditions exist for the other V and Wfunctions. 7.3 A Few Special Function Values From the series (5.1) one obtains when arg (2 l) = for = 0 m(l) im(r/2)m bmn(7)l + for n even n (0) = (7.8) 0 for n m odd m(1 0 for n m even (l) =(7.9) d "\im(7/2)m bmn+i(7)/ ( + ( m) for n m odd Vn,m,,,,,,k 50 NACA TM Io. 1224 The X and gfunctions of the first kind are for the index values n and m considered here either even or odd functions of t or 9, respectively, according to whether n m is an even or odd value. Furthermore, because of em(cos e + i.0) = (i)mQm(co 9) n n X~(2)(0;7) = i x(l)(0; .:1 n (1)  in pm(cos 3 + i.O) there is 2. n for n m even (7.10) ,m(2) adx (0;7) d i (1)ml(07) 2i (or) for n m odd From WTronski's determinant (5.35) follows = f br (b [r=co for n m even (7.12) a.m(1) (o;y) n 6( 7m(2) (0;y) n G2 Srbm (y) n,r r=Co for n m odd Therefrom the Zfunction of the second kind and its derivative with respect to ( for f = 0 can be calculated at once. 7 .4 Connection between the X and ZFunctions If v,p are integers, considerable simplifications occur in the relations (4.15) to (4.20), (4.24), (4.25), (5.6) to (5.9), (5.17), and (5.18). They are so obvious that they need not be discussed further. Equations (5.29) and (5.30) now assume the simple form (7.11) (7.13) 7Zm(l) (0;7) n d(;) ar NACA TM No. 1224 m(i) m(i) m(i)( ) n n =)n (i) n n n. (1 = 1, 2) For the m(i) (7) simpler expressions can be obtained if s n in equations (531) and (5.32) is selected in a suitable manner. The same expressions, however, result in an even simpler way if one substitutes in equation (7.14) and in the derivative of this equation with respect to S, respectively, the special value 0 = 0. m(i) m(i) If one expresses () (O;7) and & (0;y)/di, respectively, according to equations (7.8), (7.9), (7.12), (7.13), using equation (5.13), there originates for n. m = even m(1) 1 1/2 n () = m /2i bmmn(7) n2 )(o0) )r(;+ ) m(2) 1/2m m1 m(1)(O;l) m) n \ m a (7) n,nm P (7.15) and for n m = odd m(1) n (7) = 1 1/2 m(2) 1/2 n b m () n.mn+1 (1) n m(l) (O;)/ Q r( a m) m/ m2 n 2 M ( a1 am ('7) n,nim+l (7.14) S(7.16) mi m1 (O,7 d) r( 5 NACA TM No. 1224 By m(l) (0;O7) and a '(l)(o;y)/d9 the valves of these n n functions are understood which result when E goes towards zero from the positive imaginary half plane. The distinction between mvea a~A odd n m can be avoided if one sets, for instance, P equal zero in calculating the (i)(7) n from equations (5.31) and (5.32); the formulas (7.15) and (7.16), on the other hand, have the advantage of greater simplicity. 7.5 Normalization and Properties of Orthogonality of the XFunctions of the First Kind The eigenvalues of the base equation Xm(y) are always real. n Proof of it is given in the known manner. Equally sJmply it can be sho m that the functions X ()(;7) are orthogonal to each other, that is, I r 1 Xm(1) (;7)xml)C;7) cl = 0 n n (7.17) is valid for n / n'. By inserting the series (4.1) into (7.17) one can also express this property of orthogonality for even differences n n' thus: a ()bm () 2 = 0o n,r n',r+nny') 2n + 2r + 1 For the normalization integral one obtains p1 (I;y) (n + 1) 7) d = n)' f1 r= for n i n' (7.18) 2 m (y)bm (7) (7.19) 2n+2r+l n,r n,r NACA TM No. 1224 7.6 Generalization of F. E. e1umann's Integral Relation In the case m = 0 equation (2.4g) which is the integral F(4,)= 1 1 <1 one obtains a second solution of the basic independent of X(')(;7j) in the form of n (7.20) e ) (y) dt g t n The fact that this integral actually represents a solution of equation (2.4g) is confirmed by substitution. The calculation is reproduced in detail in Bouwkamp (reference 10). For large 5, t and in the denominator of the integrand cancel in first approximation and one can see then at once that J(t) is proportional to the function of the fourth kind. The integral over t can then be evaluated according to equations (8.20) and there originates, because of equation (5.12), J() = i Z (7( ; l) (7;l (;7 n n / 4 iram () n,r (7.21) According to equations (7.8) and (5.33) theZfunctions are now converted to Xfunctions. Because of equation (5.39) there results finally 1 '1 ei (1)(t;y) dt t n. (7.22) Therefrom results for 7 = 0 F. E. Neumann's integral relation between spherical functions of the first and second kind. x(1) (t;) n X(2) (;) 17 ) (7 NACA TM No. 1224 7.7 Zeros of the Eigenfunctions For m > 0 the zeros of the basic equaticn are situated at I = 1, respectively, siace they there have the behavior 2 l)m/2. If one divides the eigenfunctions by this expression, the quotient does not have zeros at t = l1. In order to understand this, one need only enter the basic equation (2.hg) with the expression (t 1 I)m/2 multiplied by a power series in (6 t 1). The zeros of (2 1)"m/2 Xm(1)(;y) are all simple; n if they were not simple, all higher derivatives of the eigenfunctions would have to disappear there also. Since it is, however, a non identically vanishing analytic function, this case can never occur. Further properties of the zerus of the eigenfunctions follow from a simple consideration of continuity: namely, that in a nonsingular point of the basic equation a zero cannot be newly originated for a change of .* and an already existing one cannot vanish. There with the problem of the number of the zeros is essentially reduced to the problem of the number of zeros of L6gendre'3 and their associated polynomials and of Bessel's functions with on index of a fractional value of onehalf. One deals first with the X(l) (;y) with real 9 and 7, n that is, with the eigenfunctions of the prolate ellipsoid of revolution. All zeros are real; for this is valid for 7 = 0. If, namely, for a change of y a complex zero would originate, the conjugatecomplex would originate along with it; but it contradicts the simplicity of the zeros, that a real zero splits into two complex zeros. The number of the zeros in the interval 1 < t < + 1 equals n m, that is, the number of zeros of Pm(g) in this equan interval. The zeros outside of this interval go over into the zeros of Jn+I/2(Y) for 7> O; the asymptotic distribution of the zeros for large t is the same as'the distribution of Jn+1/2(7) for arbitrary 7. For the eigenfunctions of the oblate ellipsoid of revo lution X(m;iy) with real 9 and 7 also n m zeros are n situated in the interval 1 < 9 < 1; but now the remaining zeros Translator's note: missing in the original. NACA TM No. 1224 E 0= c./ cl Oc Cu 0) ,* (D t2 0 r 0 0 C ) 0 0H c PH .' 0) 8 T g 0 t 2a *rI rU O P ( + H 0H o)M 0) C L r4 P S1H 1 0 )u S 0 U .0 r u N H ri 0 0' $4 I 'H + ) 0+ *ri 0 CHO P4 0 f Q4 P O Pr S4 0 p r + 0 4 a c ( *H PI oH r4 r a ,C *r + O O fi, H 0 ,2 M a ) ***I g O O S 0 k 0 2 k) 0 0 riD 0 2 U o 0 0 k N P P 0 0 Z 0 4r rd rP go a) a0 ,s O *id 4 m 0 r P I 0 S* 9 0 1 H H C i * aH ) 0 *P CH 0 05 t Di p Oq4a 0) H S 4* + 2 r 0 0 0 a  0 H mr 0) 0 0 w M H o .m V N  H OH '+1> (> 0O 0 r 5 r0 0 0 *! f5 r m *o 01 0 O H ) H 9  C O a rP C?'I 0 0 H O tan 0 a* r( O OI P k6 CH o, *H P4 3 + l n I I P 0 'H Cd 0) A ? o H o s P ) 0 0) 0 S* *'*( < *rl 0 O 0) *r 0 p oG o o 0 +2 (D O O O1 1 r ,t , 0 0 )nd 0 0 )' ,M D P H (D Kih H ^ ^ t2+D NACA TM No. 1224 The integral equation (7.24) can be generalized. By selecting another path of integration one can, for instance, also express the Zfunction of the second kind (that is, also the Xfunction of the second kind) by an integral over the Xfunction of the first kind; furthermore, equation (7.24) can be generalized to the case of arbitrary v,p,7. However, the respective results shall not be discussed here. Kotani (reference 8) indicated a general principle for obtaining more general integral equations for the X and Zfunctions, respectively. Integral representations for the X andZfunctions have not become known so far. It seems that the integral equations of the type (7.24) or of another kind also can be substituted for them and replace them; thus equation (7.24), for instance, represents a very useful starting point for the investigation of the asymptotics of the Xfunctions of the first kind. The integral equation (7.24) can perhaps also be applied when the values of the Xfunction of the first kind are known only in the interval 1 < t < 1 and are to be calculated for arbitrary real and complex E. (Compare the discussion on the 7asymptotics of the eigenfunctions in the following section.) As M6glich (reference 4) has sho'wm, the integral equation (7.24) can also be used for obtaining developments of the Xfunctions of the first kind in terms of powers of 7. 8. ASYMPTOTICS OF THE EIGENVALUES AND EIGEIFUIECTIONS 8.1 Asymptotic Behavior of the Eigenvalues and Eigenfunctions for Large v The continued fractions (6.1) and (6.2) do not only have the property to yield a development of X (7) in terms of powers of y but in addition one can obtain from them a development in terms of powers of v1. It is more favorable to set up a development in terms of powers of (2v + 1)', because then the odd powers of (2v+ 1) are eliminated because of equation (4.9). The cal culation itself is relatively simple so that the result can be given immediately NACA TM No. 1224 v(v + 1) + 2 + 1 4 162)2 + Y4 V8(2v+ 1)2 + 1 (24 l6P2)Y2 + 46) +] 0 (2V 1) 2(2v + 1)4 2 (8.1) Presumably this series is not convergent but has asymptotic character. In order to form a judgment on the usefulness of the series (8.1) for numerical purposes,, one gives for several cases the numerical value of the remainder term denoted by 0 (2v + 1)6J in comparison with the value of the separation parameter X itself. n 2 4 6 8 xo 0 11.7904 25.2513 47.10958 77.06246 Remainder term 0.134 0.0132 0.00095 0.00017 In a similar way one obtains the following expressions for the development coefficients of the eigenfunctions for large values of v. a = aO a2 a0 (8.2) S + 0 2V + 1)3 8(2v + 1) 2(2V + l)2 2 8(2v + 1) 2(2V 1)2 + o(2v + 1) 3] (8.3) S  + 0 (2 ao 32(2V + 1)2 + 1)3] (8.4) a4 32(2V+ + 0 2 + 1) 3 a0 '32(2v+ 1)2 NACA TM No. 1224 8.2 Asymptotic Behavior of the Eigenvalues for Large Real 7 One limits oneself here to n and m vhich are integers (n > m 0O) and to real large 7 vhich may be assumed to be positive without essential restriction. Thus one obtains the asymptotics of the eigenvalues and eigenfunctions for the wave equation in the coordinates of the prolate ellipsoid of revolution. An approximate picture is obtained if one puts the standard form d2 do2 + of the eigenvalues and eigenfunctions basic equation into tho Liouville  72 cos2 0  2 1 4 sin e q = ( )(1 2)1/4 S= cos 0 and interprets it as interval 0 5 0 I a Schridinger wave equation in the of the potential energy (in suitable units) n2 1 72 cos2 9 4 . sin2 4 sin 6 (8.6) It has for large 7 at 0 = 2 a very narrow minimum and can there very well be approximated by a parable. Then, however, there results just Schridinger's vave equation of the harmonic oscillator, for which eigenvalues and eigenfiunctions are knoim. In order to obtain also higher approximations it suggests itself to attempt a similar formulation as in equations (4.1) and (5.1). One sets F1 =( 2  1)m/2 "2rDT+r ( , S (8.5) (8.7) NACA TM No. 1224 equal, where the Dn are Hermite's orthogonal functions and the functions of the parabolic cylinder, respectively. By substitution of equation (8.7) into equation (2.4g) there results, if one utilizes also the recursion formulas and the differential equation of Hermite's orthogonal functions (see Magnus and Oberhettinger (referencr' 11)), the fiveterm recursion system 4 l r2 + l4k + 87 + r + 1+ 4m2 2(N + r)2 2(N + r) 3r + W4(N + r + 2)(N + r + 1) p+2 + (N + r + 4)(N + r + 3) (T + r + 2)(N + r + l)r = O (r = 0, t2, t4, ) (8.8) The series (8.7) is probably not convergent; it rather represents an asymptotic development in the sense that limits = 0 for all o even r f 0, or, as one concludes from that and from equation (8.8) StL2r 2) = o(71r/2) 190 .+(2 _0(_lr) (8.9) (r = 0, 2, 4, .. ) By a method of successive approximation the r and X can be represented as power series in 71. The calculation is elementary; thus only the result is given. It is NACA TM No. 1224 Xm(y) = (2N + 1)y 1(2 2+ 2N 3 4m2) n 4 (21r + 1) (2 N + 3 8r2) + 1 F8m2(2e + 2N + 1) 64y" 5(4 + 2N3 + + 7N + 3)] + 0("3) (8.10) The connection between N, n, and m is given by counting the zeros. For 7 > w the de7elopment (8.8) is reduced to the principal term with the N real zeros of Eemrite's 1th polynomial, hereas the Xfunction with the indices n and m to be Lapproximated has exactly n m real zeros in the interval 1 < ? < 1. Therefrom follows N = n m. For negative m on3 inserts instead N = n + m. 8.3 Asymptotic Behavior of the Eigenfunctions for Large Real 7 The asymptotic representation of the elgenfunctions results by calculation of the coefficients 5r. They read, aside from the terms of the order y>, NACA TM No. 1224  25N 36) + 271 10ji (N 1) + 2 ) ~2)3n(  1) (N 2)(N 3) r (8.u1) "6 = m o0 12872' 1 2 S204872  u m N! 0 1287 2 ( 6) 8 1 N! S 2204872 (N8). According to the type of derivation, however, the eigenfunctions are approximated by these series only in the interval 1 < t < 1. In order to obtain an asymptotic series also for other t one starts from the integral equation (7.24) and substitutes for the Xfunctions in the integrand the series (8.8). Therewith the asymptotic develop ment of the eigenfunctions for all E is known; in particular, their behavior can be investigated where, besides 7, 6 also is very large. Since now the eigenfunction for all t is asymptotically knownone obtains the solutions of the second kind by calculating the integral in equation (7.24) with the asymptotic series of the eigen function and by means of another appropriate path of integration. . 327 a2 = 1 [i 4y_ + 1(N2 32y = 1+ 327 32y L '00 1 3N+ 27 L 2 NACA TM No. 1224 The zeros of the eigenfunctions located in the interval 1 < 5 < 1 crowd for large 7 more and more around t = 0; in order to understand this, one has only to divide the zeros of Hermite's Nth polynomial by \f and therewith to convert to the Fscale. The domain of validity for equation (8.10) and (8.11) extends over the indicated domain; thus originates, for instance, for m = tl from equation (8.10) the asymptotic representation of the eigenvalues of Mathieu's differential equation found by Ince (reference 14). However, the limits for this domain of validity shall not be submitted to closer investigation here. 8.4 Asymptotic Behavior of the Elgenvalues for Large Purely Imaginary 7 One limits oneself again to n and m that are integers (n = m = 0) and to.purely imaginary y of large absolute value. This procedure yields the asymptotics of the eigenvalues and eigen functions for the coordinates of the oblate ellipsoid of revolution (reference 10) and for the socalled inner equation for the separation of the wave equation of the ion of the hydrogen molecule (reference 7). The method applied in equation (8.2) fails here; yt2 namely would become purely imaginary and the Dn(\ if27) would, for large t, no longer decrease exponentially, but increase exponentially; they would, therefore, be no longer appropriate for the development of the eigenfunctions. The wave mechanical picture of the differential equation (8.5) shows that in the case of purely imaginary 7 two domains with low potential energy are present at 0 = 0 and 6 = 2n, which are separated by a high potential peak with the maximum at e = . One may, therefore, expect beforehand that the eigenvalues will degenerate in first approximation; their splitup is exponentially small in 7 I; it is the larger, the higher the eigenvalue. For each eigenvalue there is an eigenfunction symmetric with respect to e = , that is, a = 0 arnan asymmetric eigenfunction. The mathematical treatment is as follows. A singularity is made to move to infinity. Then one obtains from equation (2.4g), aside from an elementary transformation, the differential equation of 64 NACA TM No. 1224 Laguerrets orthogonal polynomials. This suggests for the solution of equation (2.4g) the formulation of Svartholm (reference 7) (1 e 2m/2 PE > t[ (m) Fl(J) = 1 [ e [ W p(l  (8.12) wherein i7 was set equal to p; again it does not mean an essential restriction if p > 0 is assumed. By substitution of equation (8.12) into the differential equation 2,4g), application of Laguerre's differential equation, and the recursion formulas for Laguerre's polynomials (compare Magnus and Oberhettinger (reference 11)),there originates in the known way for the at a threeterm recursion system. With the abbreviations X = p2 + 2Tp (T2 + 1 m2) + A 2 T = 2N + m + 1 4At = (T + 2t 1)2 m2; Pt = 2t(T 2p + t) (8.13) (8.14) (8.15) the recursion system reads at+l At+ + tl A = (A + Pt)at (t = i, N+l, N+2, . Therefrom follows for A the transcendent equation A A2 A+ = A22 S AO 2 A + P1 A2 +P2 from which A can be obtained as series in terms of powers of Therefrom then.results (8.16) (8.17) 1. NACA TM No. 1224 m(7) = P2 + 2rp (T2 + 1 m2) (T2 + 1 2) 10r2 + 1 2M2(3r2 + 1) + m4] 64p2 [L 533T 512p + lll4T2 + 37 2(23T2+ 25) + 13mj+ o(7 14) 8.5 Asymptotic Behavior of the Eigenfunctions for Large Purely Imaginary 7 For the coefficients of the development (8.12) 0i T+T 1 + +o  1 1 1) +l T+2pJ+ 0171) an0 16p 16p 2 (T 0o 5 L2p a1 i1  =  a2 512 + 1)2 (T + 3) m + 0(171 3) 1)2 2 m + 9 +0( 7 )  1)2 M [(T 3) 2 m2 + oy (8.19) is valid. The significance of T zeros. The principal term and N results again from counting the Sm)[217(1 )] has N zeros which for large jy1 lie all closely to t = 1. The real eigenfunction has again N zeros in the neighborhood of = 1. For odd n m another zero at E = 0 is added. The sum total of the zeros n m equals, therefore, 2N for even n m and 2N + 1 for odd n m; thus (8.18) NACA TM No. 1224 T=n+ 1 =2N + m + for n m =even T = n = 2N + m + 1 for n m = odd is valid. Baber and Hasse' (reference 7) calculated the series (8.18) with the exception of the last two terms; only for the special case N = 0 they give also the last two terms; Bouwkamp (reference 10) calculated the series (8.18) with the exception of the last term for the special case m = 0. The asymptotic series (8.18) can still be used for m = Vt; it then goes over, exactly like equation (8.10), into the asymptotic series for the eigenvalues of Mathieu's functions (reference 14). For large values of ly7 the eigenvalues move closer and closer together in pairs so that the asymptotic series (8.18) for the eigen values of each pair are the same (see equations (8.20)); that is, the difference of the two eigenvalues has a stronger tendency to vanish with increasing (7y than any power of 1/171. (Compare table 11.) The series (8.12) for the eigenfunction is useless in the interval 1 : 5 0. There an approximation must be attempted starting from the point 6 = 1. Since the eigenfunctidns become exponentially small in the neighborhood of 0 = 0, one can build up the eigenfunction in the entire interval 1 < < < 1 by combi nation of the two approximations starting from 1 and 1 and one obtains nm(1)p y) = Constant (1 2)2 co t 4I L2^p( M t=Go Se L [2p(1 ) (8.21) N+t For even n m the positive, for odd n m the negative sign is to be selected; in the one case the eigenfunction is symmetric, in the other antisymmetric with respect to the point g = 0. NACA TM No. 1224 'What was said in section 8.3 is valid for the asymptotic calculation of the eigenfunctions and the functions of the second kind. for any complex g as well as for the limits of the domain of validity in the variables v,g,y of the asymptotic representations. In order to show the use of the asymptotic series for numerical purposes one compares for m = 0 a few eigenvalues with the values resulting from equations (8.10) and (8.18) by giving the value of the remainder term o(y73) and 0(p'4), respectively. n 0 0 0 2 Y2 10 25 100 100 .n(r) 2.305 16.07904 81.02794 45.48967 Remainder term 0.025 0.01616 0.00008 0.01528 9. EIGENFOICTIOI OF TEE WAVE EQUATION IN POTATIONALLY SYMMETRICAL EIPTIC COORDINATES 9.1 Lame's Wave Functions of the Prolate Ellipsoid of Revolution By separation of the wave equation in the coorf.nates of the prolate ellipsoid of revolution one obtains the following solutions of the wave equation u [= (1) k ) + Bg (2) (t;, [CX(1) (i ) + MD (2) (I;YEeiPc + Fe1n (9.1) V v. A, B, C, D, E, F are arbitrary constants, v and p arbitrary real or complex parameters; the significance of 7 is given by equation (2.8), thus 7 is real. The coordinates g and ( = 7A, respectively, n and cp are real as well. Under X41,2) (7y) one V INACA TM No. 1224 understands %(1,2) ( + i X 07). According to the kind of the boundary value problem presented, the arbitrariness concerning the constants and parameters is limited; then such solutions of the wave function have to be determined which remain finite for the entire domain of the eigenfunctions. Following, as before, threedimensional domains only are dealt with which lie inside or outside of an ellipsoid of revolution or between two confocal ellipsoids of revolution. Then the domain of the coordinates T and p is given by i1 I 1, 0 c9 p 2A. The requirement of singlevaluedness and finiteness of the eigen functions then leads t? v = n, p = m, n Iml 0, and D =0. The eigenfunctions are written in the form U(e,1,pik5 = [Bnl)(P ) + m(2)(ti/nlc]ie1 (n = 0, 1, 2, .; m = t+, +2, .., tn) (9.2) The domain of variables in. is denoted by 1 9 2 and g1 t 5 ~2, respectively., For the prolate ellipsoid of revolution there is always 1 For inside space problems 92 = finite, for outside space problems infinite. For inside space problems boundary conditions for S1 and E2 are to be prescribed. This results in two linear homogeneous determining equations for A and B; they can be satisfied only for certain distinct values of 7, that is, for certain eigenfrequencies; in that case they fix the ratio A:B. In case El = 1 a boundary condition can be prescribed only for. t2 > 1; the boundary condition for 5l = 1 is then replaced by the requirement of finiteness of the eigenfunction at the singular point l l= 1; it leads to B = 0. For outside space problems the boundary condition for E2 = is eliminated; the functions (9.2) have for 2 > co for arbitrary A and B an oscillating behavior. One can see that immediately from the asymptotic series (5.12). The boundary condition at 6 = 1 gives the ratio A:B. For l 1 this boundary condition in turn is NACA TM Io. 1224 eliminated and B becomes B = 0. A condition for the frequence does not exist; all wave coefficients are admissible, the spectrum is continuous and extends from k = 0 to k = . 9.2 Lame's Wave Functions of the Oblate Ellipsoid of Revolution The solutions of the wave equation originating by separation of the wave equation in the coordinates of the oblate ellipsoid of revolution are obtained from equation (9.1), by replacing 7 there by tiy. Here also only the tIreedimensional domains characterized in section 9.1 are dealt with and the eigenfunctions can, therefore, be written in the form u (6,,cp;k) = A )(iy) + Bm(;i)(i)ei n n n jn (n = 0, 1, 2, ; m = 0, 1, 2, ,n) (9.3) The domain of variables in T and (p is the same as in the coordinates of the prolate ellipsoid of revolution. The domain of variables in f is again denoted by 1 ( 2. For the oblate ellipsoid of revolution there is 0 = What was said in section 9.1 for l > 1 is valid for inside and outside space problems with l1 > 0. However, whereas there t, = 1 was a singular point of the function of the second kind, here t = 0 is a regular point for all ;functions. Thus, for determination of the eigenvalue problem for Il = 0 in this case, also a boundary condition must be given. The area (t = 0 is a circular disc. If such a circular disc actually exists as a physical object, for instance, a circular screen for problems of diffraction or a circular membrane, the boundary condition on the disc results from the physical problem taken as a basis. If, however, this circular disc has geometrical significance only as singular surface of the coordinate system taken as a basis, for instance, for the determination of the acoustic or electrical natural oscillations inside an oblate ellipsoid of revolution,the eigenfunction together with its derivative must be required to be continuous at this circular disc which leads to B = 0. NACA TM No. 1224 4 h o P 0 0 0 0w 0 o (ri 4 0 H 0 0 0 0 0 14 *4 o a,  0 o Q SH0 40 a, 0 0 0 ~sJ0 4H 0 4 0 +3 4' 00 Srri t 0 S $*4 0 *H 0 3 B r oa 0 Q > O ^?? ed P 00 o N 1p 4 4 o *H C a 00 a rd 0 0 a S a, 4 0 Pd T 8 0 4,0 H4 CU v 3' 1S c I* ** 9 *"< en Cu Cu F v ad u M P 4 x tu 1t SaLA 5" m * 0 ac l r 'd * )0 00 r rI * 0 0 a CO r4 0 o a +' 0 0 40 I rI No O, 0 *H *r4 0 o P (D a, 0 o d rdm 0 Hp a pai HO 0 * m _ 0 P0 rj 0 H *i i^ H a, H a 0a o 0 o r ^^g43 *^ ,C O l 43& *d PE * C, Cl + F< II %. *t !3 NACA TM No. 1224 01 l(Dr () P + 4 2o 4) d 0 0 A 0 4 02 +( aa i 'P() 0 I m t8 to aH H (u P oO I r I B .rl %0 o H ONO 0 0 cI o *S o P4l O 0 0 0 o 0 P I a a 0 *r 0 00 P &i m00 E> 020 0 r H I H cJ 'C II C an IH OM II Co K H I ___ d H e a ,ki UIl GI i IM __ si U0l PI a I p4 a , 01 H W 00 0 ( (0 a O 02 0 r k II o 0 c 3 O 0 H 0 a H 0 +2 P C' 1H *2 C I 013 0) '8 1I1 02 a rl I 01 42 m U ,C ol *P k a I I 0 O 4I 0 +2 SI 0 C 0) 'c C) 0 Tl +2 0 N '?d *H 4 0 4 0 *ri O (D Q) i o n 0 W m 4H .rl cr o P 4 +4 m a l 9 00  41 0 SIC Pa 0 4 Pz 01 r 1 0 4i 0 0) rR 43 A 4 H *H 02 mi i H (U Pd o B, 9 0 P i4 .0 r. a H + C I + II CU F i *r L + II S NACA TM No. 1224 C0 O H 01 * B. I: 4a 0 Co* 0 H 04 a *S * P014 r1 1* 04 Li S m A43 0 0 ad 0 42H 1 0rs 0 0d +3r 04 : em l rd (a 0 S 0 mp 2 +%g a O CO * e o 0 *' r,ro 01 m go o 0 * CO *1 h 0 d M 8r . rt$ * d o m 0 ( 00 $ *4 c $,4 0o NI$ 6 1 $4 + $4 + * S N u cI c/ at + of S +P 00 CI M I P4 4' O H a 4D 10 d0 * o 0 0) 9' ip 0 0 *1 C i (D k  v $4 4 8 U U a r0 N N 31 D ' *1s  'f * v ,o b y NACA TM No. 1224 where r in the argument is given by r2 = x2 + y2 + z2 and must not be confused with the index r which runs through all even numbers. The development (9.9) is given already by Morse (reference 6). One can interpret equation (9.10) as a development of the Zfunctions which contains still an arbitrary parameter r. For n>l there originates, if one divides before by (1 ,2)m/2, the series (5.1); for nf>0 one obtains the series (5.42) for the Wfunctions of the first kind. If one differentiates equation (9.10) with respect to I and sets then n = 0, there results the series (5.41) of the Vfunctions of the first kind. For > O there originates from equation (9.10) the series (4.1) of the Xfunctions of the first kind. At this point one can recognize why the formulations (4.1), (4.2), (5.1), (5.2), (5.41), and (5.42) that is, the series developments considered by Niven (reference l) all had to lead to the same development coefficients ar. Whereas equation (9.9) represents a development in terms of eigenfunctions in polar coordinates which have their origin at the point x = y = z = 0, Lame's wave functions can be developed also 'in terms of eigenfunctions in polar coordinates with the origin x = y = 0, z = c. This development reads, as shown by a simple calculation, jm(l) (C;r) (l ( ) em ()n+tE PM) (9.n1) t=WO If one multiplies by (1 )m/2 and then sets = 1, equation (9.11) is transformed into the development (5.46). For g  one obtains a development in terms of spherical functions multiplied by sin (y7) and cos (y7), respectively; the special case of this development m = 0 is already given by Hanson (reference 5). If one finally develops Lame's wave functions in terms of the eigenfunctions originating by separation of the wave equation in cylindric coordinates, there results, with the aid of equation (7.24), 74 'NACA TM No. 1224 A0 DH  Cu O rd Ol 0 a0 cu 0 O f M *o O fq (a110 r0d >0 0 4 (" C 41 'o *3i a COt) 0 P 0 m o Uo 0 u IOD 0 t oH 4' ao ,o o 4 1d i * r 0q0 0 m we S H O O pr 4d F 20,C  ,DrO 8 H P 0 H tpo a . 19 0 0 0 H "0 P N 8 a H H P P 0 o 4 HO 0O H 0 J P PO D 6 H 0 OG 0 H ) i *rt o 0_0 1 24. P4 44 1 p O 0 V 5 +, 0 AO 1 P 0 0 0 4a O) m 0 0 d 0 qo 0 '60 K "0 1 q4 H +0 f 4P ooo t N o o 9 " go 0 H 0^ 0 0 0oi $4r H oq 0x P 0 o PP S 10 A  d o >g^ m 0 0 *0 M S w) m PFP 0 co H C8 H H Vr Ot ;1 P 0 4 P 0P o 0 >0 H + *P 'O  O 0 4r O C M 43 do o HOP O O rO a, .i o: g i p g ip; E r1^0S1 NACA TM No. 1224 the right side of which is a (recently socalled) Dirac's 5 function. It was introduced first by Sommerfeld (reference 15) and designated by him as prong function. It has a singularity at the source point Q in such a manner that I 6(P,Q) dTp = 1 (10.2) for each domain G which contains the source point Q, whereas the integral has the value zero if the domain G does not contain the source point. One can interpret 6(P,0) as limiting case of a function which has for points of influence P in the neighbor hood of the source point Q a very steep prong whereas it decreases toward the outside very rapidly to zero. The solution of equation (10.1) is for outside space problems uniquely determined only when besides the boundary conditions on the bounding areas which are at a finite distance an additional boundary condition at infinity is required, namely, the outgoing radiation condition (or else the incoming radiation condition) introduced by Sommerfeld (reference 15). According to this condition, u(P,Q) for points P at very large distance from the source point Q should behave like an outgoing (or incoming) wave. One designates this solution because of its special properties as Green's function G(P,Q k) of the wave equation pertaining to the outgoing (or incoming) radiation condition. For physical reasons the case of the incoming radiation condition will not be considered below. All developments of this section are performed for the coordinates of the prolate ellipsoid of revolution; one obtains the corresponding formulas for the coordinates of the oblate ellipsoid of revolution by replacing 7 everywhere by iT. Green's function can be developed in the following way in terms of the eigenfunctions of the continuous spectrum G(P,ojk) = k2 2 m U p,cPp;) u ( nQcPQ7)27 (10.3) Jo k2 K 2 n n, m The integration over i goes from 0 to o, the path of integration deviating at the point i = k in the case of the outgoing radiation NACA TM No. 1224 0' Cu 8rt II it v OJ a 9 * 4 0 0 d 0 a P4S rd 4 Pw 4 I Pi (0 a;n Rm P4 d9 Pa SO ii m ri 4  I m J o 4 4. S a 0 m 11 4! Sm 9 ?4i4 p.( 0 0(D SV ri 3 'n ? 4 ) 0 oan  ) 0 0 rd (k *ri I p p4 m D rd 0 oH s. m 0d a r o? H kto N a> 43a 0 4a i .I D 10 0 of 0 1 . mo k 0 ED sq 1 : (D o a m 0 it N( 0 44 rd r > 0 0 od E *ri Pi4 F H '1 0 r 09* 0 o n ., o r.i 3 H4, iI 0 0 O i a,) 0 ri f 0 0 il 0 P so 0 0 r0e 0 OO P 00 0 1 M (D 0 04 0 m a 00 H r *P O 4 0* 0 0 0 k qA m F I C' + ST 1 Iti pq +U 0 0 11r 0 H 4, 4A2 m r1d Pi1 S0 0o .rl a^l a rd 0 O)  0 0 $4 NACA TM No. 1224 ICD *C o0 0' ^ ^ 4 ^p II S CG "s" 4I 0} P *l 0A r > P1 1$ 4 0 *CH kI F Si 0 i 0l '0 0 Cu 9 8 m* pa I F . a F. .' H ** a II  S n  C \' 11 1 t m 4k I Q 0 *H ri 4 OA a *i m ) LA S0 P 0 0 O ^ 0 A Iat (D (D 'O r' P r a, H 0 (r 0 0+' 0 0 9 0 cr O P 0 Fe'd 0 CiE O5 0 r 8 o p * ':p 1I cr c S I i *s .M of Fd *H ri 5 P + O Fs kO rd 0d 0 0 0 * rl 0 0 r1 doa P, ri t t oP o 0 0r *rl 4 .0 ) 04 hP 0 i~g 0 4I0 0 ox 6  of y r4 *r4 *H $1 k k 0 0 mm P4 PI4 aH mm *rl vi sd cd *O NACA TM No. 1224 F + 4+> CH mo 0 HaO h * 1114 0 *i O o 4' SH rd 4 *H 0 1 4  0.13 rM 0 c r00 .p* 541 H 0 0P ,o 4 ?5I 0.1 *ri 8IM g a II b 0 ,a . 4 0 a) o4 ., a) 'd wH 0 0 m P P .o c O 0 S0 rd Ep c 0 oa c* 50 1 iP 0 0 CO 0 0 *, > m o H + P op 0 0 d4 :3 CH4 0 H540 P o H 0 H 54 o a0 0 0 C. 0 ++i 0 00 P o g ej 0 0, 0 *O  0 0+ O ,O Pi0 4 md 0 O  0 H i a' 0 0 H 8 0 r 'd P W ed cil 3 e 0 O d 'o 0a 00 0 0 0 10 4' 0 10 0 P4 0 od0 0 0 *, ) E i d +D CO 00p S' 0 ca 544 0 4H 0 00 0 0' pI HC . PI ,4 v *rH 04 afl a, NS p4 pE 0 0 + o OU P, &IA 0 I% r1 i+ (n II o' C? 0' 0 54 I cu + 8 m 0 o0 H i II Of l* ALCA TM No. 1224 formed and the contention made that it solves the problem of dif fraction. Actually it represents a wave which comes as spherical vave from the source point Q, satisfies on the diffracting surface the given boundary condition (as does each single term of the sum), and which behaves at infinity like an outgoing spherical wave. If the source point Q, in particular, lies at infinity, Green's functions represent the superposition of a plane wave and of an outgoing spherical wave originating from the diffracting body with an amplitude which, in general, is dependent on direction. Treat ment of the diffraction problem for a source point within a finite region is omitted. One starts immediately from equation (10.8) and contends that the solution of the diffraction problem of a plene wave at the ellipsoid of revolution is given by Sn m() ) () ; n,m n Qi )7) xm(1)() p;7)Xm(l)(n 1;7)eim(OpCP) Im(y) 2 in ir b (7) (10.10) r=o in the case of the boundary condition u = 0 for 5 = fl. Under m(7), one understands therein the factor of normalization (9.8) with AA* + BB = 1. In the case of the boundary condition = 0 for 5 = 5l one has to replace the two Zfunctions with tho argument t1 in equation (10.10) by their derivatives with respect to Cp at the point i'. The first term of the sum in the brackets of equation (10.10) yields,when the sum over n,m is formed,exactly the plane wave (10.8); the second term of the sum gives outgoing spherical waves; furthermore, the wave equation and the boundary (surface) condition are satisfied by each separate term of the sum; the contention is therefore proved. For the diffraction at the infinitely thin wire of finite length, one has to set 1 = 1. Z1m(3)(1;7) then becomes infinitely large n and, in equation (10.10), there remains only the plane wave. Thus an infinitely thin wire does not present an obstacle for a plane wave. NACA TM No. 1224 For the diffraction at the infinitely thin circular disk, 7 is to be replaced in the formulas by i7 aend 1 is to be set equal to zero. (3)(0;iy) has a finite value so that the outgoing spherical waves do not disappear; that is, even an infinitely thin disk represents an essential disturbance for a plane wave striking it. 11. TABLES 11.1 Comnents to the Tables The tables in section 11.2 contain power series developments to y10, inclusive, for the eigenvalue Xm(7) according to n equation (6.13) and to y, inclusive, for the coefficients a (y mn,r and bm (7) according to equations (6.14) to (6.21) and equation (4.11). Furthermore, to 7 inclusive, the coef ficients a la (7), according to equation (7.6), are given nfrie an,o for all those cases where a2/aO, a.4/a0, and a_6/a0 disappear. As far as the values of the coefficients ar/ao and br/bo are not given in the tables, they disappear; then one must use for the Xfunctions of the second kind the series (7.7) and the table for the .c/ao. The region of the n and mvalues in the tables extends from m = 0, 1, 2, .. ., 9 and from n = m, m + 1, .., 9. negative m, which are integers, reference is made to the relation (4.12). ) For The last given digit is, in general, probably certain; only where the following digit after rounding up or off, respectively, is a 5, the last given digit would have to be changed in a few cases by unity. In the cases of the end digits ...5, ..50, .500, and so forth, it is mostly indicated by a line over or under, respectively, the last digit whether the respective decimal fraction had been originated by rounding up or off. N.CA TM No. 1224 For n = 0 the series begin to be useless only at 72 =10; for larger n they can be used up to far larger values of 72. Below, a few of the first eigenvalues for 72 = 10 are given as they follow from the exact numerical calculation and from the poweri series development to 710, inclusive. n 0 2 4 XO(Vo) 2.305040 1.790395 25.25113 n Xn F ) approximation 2.215 11.860 25.25147 Figure 1 gives a survey on the dependence of the lowest eigenvalues on 7. The tables in section 11.3 are taken from the thesis of Bouwkamp (reference 10). They contain the eigenvalues X)(7) for n a number of pairs of values n, 72, and the coefficients am (7) n,r of the pertinent Xfunctions. These latter are fixed. so that 2n + 1 ar ( (ii.I) 2n + 2r + 1 _n,r (11.1) r=c The integral of normalization then (compare equation (7.19)) has the value J X0(i)t; 7)12 2 (11.2) Ln (11.2)n + u1 These tables contain further the values X(1)(l;7) and X(1)(0;7) (1){ n n for even and dX '(0;7) /d for odd n. The signs of the ar are different from those of Bouwkamp since the present series (4.1) and (4.2) contain in the coefficients a factor ir which is missing in reference 10 by Bouwk.mp. 82 NACA TM No. 1224 Since the 72 assume in these tables only negative values, these functions are appropriate for the treatment of problems concerning the oblate ellipsoid of revolution or for the investi gation of the eigenvalues of the ion of the hydrogen molecule, whereas, the tables in section 11.2 where 7 can be positive as well as negative, may be used for problems of the oblate as well as of the prolate ellipsoid of revolution. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM No. 1224 REFERENCES 1. Niven, C.: Philos. Trans. Roy, Soc. (London), vol. 171, 1880, pp. 117151. 2. Strutt, M. J. 0.: Lamesche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 1, 1932, pp. 199323. 3. Maclaurin, R.: Trans. Cambridge Philos. Soc., vol. 17, 1898, pp. 41108. 4. M'glishp F.: Ann. Physik (4), vol. 83, 1927, pp. 609734. 5. Hanson, E. T.: Philos. Trans. Roy. Soc. (London), A, vol. 212, 1933, PP. 223283. 6. Morse, P. M.: Proc. Nst. Acad. Sciences, vol. 21, 1935, pp. 562. 7. Wilson, A. H.: Proc. Roy. Soc. (London), A, vol. 118, 1928, pp. 617635, 635647. Jaffs G.: Z. Physik, vol. 87, 1934, pp. 535544. Baber, W. G., and Hasse, d. R.: Proc. Cambridge Philos. Soc., vol. 31, 1935, pp. 564581. Svartholm, N.: Z. Physik, vol. 111, 1933, pp. 18.6194. 8. Kotani, M.: Proc. Phys.Math. Soc. Japan, III, vol. 15, 1933, PP. 3057. 9. Chu, L., and Stratton, J. A.: Journ. Math. Physics, vol. 20, 1941, pp. 259309. See also Stratton, J. A.: Proc. Nat. Acad. Sciences, vol. 21, 1935, pp. 5156, 316321. 10., Bouwkamp, Ch. J.: Theoretischc en numerieke behandeling van de buJging door en ronde opening. Diss. (Groningen). GroningenBatavia 1941. 11. Magnus, W., and Oberhettinger, F.: Formelrn und Sgtze fur die speziellen Funktionen der mathematischen Physik. Berlin 1943. 12. Kreuser, P.: fber das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen. Diss. (rubingen), BornaLeipzig 1914. 84 NACA TM No. 1224 13. Perron, 0.: Die Lehre von den Kettenbruchen, 2. Aufl., Leipzig und Berlin 1929. 14. Ince, E. L.: Proc. Edinburgh Boyal Soc., vol. 47, 1927, pp. 294301. 15. Sommerfeld, A.: Jahresber. d. Deutschen Math.Ver., vol. 21, 1913. pp. 309353 See also Meixner, J.: Math. Zeitschr., vol. 36, 1933, pp. 677707. NACA TM No. 1224 m m m 11.2 Eignvaluea s (') and Develonment Coefficients a ,,(7), b r(7); Repreenrterl tr, BrokernOff Power Serilea in 7 ABLE i. (r. x 1010 ..5 F, . S1T 1r y I n I t e m = 0, n = 0 + '.:. 2 n = 1 +6000000000 n = *2 jt 78 ir O,1 8 n 3 +5111111illl n 4= +40649 1.506 n = 5 +504273504L n 6 +503030,020 n = 7 4502262.. L3 n : 8 *5017a1,.860 n 9 +.011.005602 m = 1, n = 1 200000000C0,2 n = 2 +428571428 n : 3 +1666666661 n = 4, +9805194.805 n = 5 +4871794872 n = 6 +1.9009090o9 n =7 +4932126697 n = 8 +94 l7368421 n 9 +4 957983193 m = 2, an 2 +148571j29'2 n = i +33i33333333 n = +40?2597o406 n = +54.,897+4352. n : 6 +4 4c.5'L454 n = 7 b6606334u8% n 8 +I 736842105 n = 9 +L769915966 m = 3, n = 3 +U 111111112 n = 1 +27?727?727 n = 5 + 504 27"=,50 n = 6 + 9 39 39 39 39 n =7 4+208144b796 n 8 +4 38596691 n = 9 +=.S0980 922 m = L, n = + 9'0pgO090? n = = +?30769208 n 6 +3090909091 n 7 +3571660633 n = 8 +3894736842 n 9 +11176i.70..9 m = n 5 + 76920769Y2 n 6 +2000000000 n = 7 +2760180995 n 8 +3263157892 n = 9 +3611'45378 m = 6, n = 6 + 666666667v' n = 7 +1764705882 n = 8 +2491228070 n = 9 +29971998880 m = 7, n = 7 + 58823529602 n = 8 +1578947368 n = 9 +2268907563 m = 8, n = 8 + 52631578972 n = 9 +1h485714'29 m = 9, n = 9 + 47619017672 1E1%6 1916  68 .71' 2: S101 ,00918 + 3. 1i0576 * 11298966 + 78741 3L * 581912?L + L 8:65 1 S.62a0LO  4c.714 86  8872'692 + 13647587 * 11902117 4 8894604L + 6698782 + .5174730 + 6099403 + 3320032  194 36 346,  2241h6689  2139871 + "'.84 89. + 3176672 + ..6211 , 300596 . 262083,  9976 306t1  13870427  1920040  877352 + 755560 + 1369050 + 1958920  57793LLL'  9103?:  8 '.9057  20 790 3  530540 + ?l4418  36413297  6274510  4157529  2301599  109649  2bo'00387y:' _ 1499 41l  3460673  2221592  171 l0327"  3332L 31  2822595  12.96627  2535176  938951 k +470') 6ib  6o.0 :1. 476o91_ + 5.9'989 * 116'. * 36,j + 1413 * 628 + 310 +1219048 ,6 + 1644~hO 1182504  1315.03  U166  10150  4008  1807  899 + 36060076 + 127901  309653  10857  39Q01  16771  7911  o052 + 132638v6 + 76421  9'311  51,68  27781  14. 3"  7813 + 5731676 + h ?60  28862  26567  17172  10467 + 2788376 + 26419  8311  12782  10172 * 1484076 + 16310  1336  60o4 + 847376 + 104 35 + 967 + 511776 + 6898 + 33676 Sl8356878 + 25896 i'.1089  '1= 1.2 i5.04 S .3 + 22 4 11  2103478 + a,682  21357  5669  26 + 6 + 7 + 2  5822Y8 + 607 + 6053  53  205 67 22 8  169878  83 + 1759 + 119 32  27  14  578v8  118 + 570 + 129 4 17 4  24.78  80 + 205 4+ 79 + 22  97,8  49 + 80 4 44L  4678  29 + 32 23r8 18  1378 2L290.96T10 + 872.80 +2 12.17  887.76  12.19 + 1 .8,: * 0.94 + 0.15 4 0.03 + 0.01  005.710 10  76.28 + 229.7 ' * 78.30  2;.61 .9 0.08 0.02 4 57.71710  22.60  56.26 + 23.27 1.22  0.58 0.30 0.07 + 17.80710  4.51  18.21 + .55 + 0.53  0.01  0.04 + 5.2 710  0.79  5.,5 + 0.73 + 0.29 + 0.06 + 1.70710  0.07  1.82 + 0,03 + 0.10 + 0.62710 + 0.04 065 0.07 + 0.22710 + 0.0k  0.25 + 0.11710 + 0.,03 + 0.02710 ____ _~I___ NACA TM No. 1224 a (7) m TABE 2. x 1010 Io AS POWER SERIES I 7 m.= 0, = 0 n=1 n=2 n=3 n= 4 n =5 n=6 n= 7 n=8 n=9 =1, n= 1 n=2 n= 3 n=4 n=5 n=6 n= 7 n=8 n=9 m=2,n=2 n= 3 n=4 n=5 n=6 n=7 n=8 n=9 m 3, n=3 n=4 n=5 n=6 n=7 n=8 n=9 m=4,n= 4 n=5 n=6 n=7 n=8 n =9 m= 5, n = 5 n=6 n=7 n=8 n=9 m=6, n=6 n=7 n=8 n=9 m 7, n = 7 n =8 n= 9 m= 8, n= 8 n=9 m=9, n= 9 +111111111172 + 400000000 + 2.44897959 + 176366843 + 137741047 + 112963959 + 95726496 + 83044983 + 73325729 + 65640291 + 13333333372 + 122448980 + 105820106 + 91827365 + 80688542 + 71794872 + 64590542 + 58660583 + 53705693 + 4081632772 + 52910053 + 55096419 + 53792361 + 51282051 + 48442907 + 45624898 + 42964554 + 1763668472 + 27548209 + 32275417 + 34188034 + 34602076 + 34218674 + 33416876 + 918273672 + 16137708 + 20512821 + 23068051 + 24441910 + 25062657 + 537923672 + 10256410 + 13840830 + 16294606 + 17901898 + 341880372 + 6920415 + 9776764 + 11934598 + 2306805672 + 4888382 + 7160759 + 162946172 + 3580380 + 119346072 3527336974 + 3555556 + 302904 + 66996 6 + 21683 + 8738 + 1071 + 2105 + 1177 + 700  355555674  454356  120593  43366  18723  9160  4912  2826  1719  75726074  301482  130098  62411  32716  18420  10990  6877  2344867  151781  87376  50891  30700  19232  12481  9106874  78638  54963  36840  24727  16849  4119274  43185  34735  2590.5  18912  20793 4  25086  22451  18212  111037  15307  14900  667274  9743  41147/ 101900876  151062 + 208818 + 46389 + 18710 + 9467 + 5462 + 3439 + 2306 + 1622 + 6134876  17898 + 16210 + 9870 + 5930 + 3767 + 2524 + 1768 + 1285 + 1222776  1430 + 2386 + 2541 + 2057 + 1579 + 1207 + 931 + 300376 + 165 + 621 + 828 + 812 + 718 + 609 + 91876 + 210 + 242 + 329 + 361 + 353 + 33276 + 131 + 118 + 152 + 178 + 13676 + 71 + 64 + 79 + 6276 + + 37 + 3076 + 25 + 1676 I ~ NACA TM No. 1224 TABLE 3. anL x 101 AND x 1010 AS POWER SERiS a otm ) a (() 1 noa0 m = 0, a 0 n 2 n 3 n34 n n 6 n 7 n 8 n = 9 m 1, n 1 n =2 n 3 n  n 5 n 7 3.8 n =9 * = 2, n 2 n 3 n = n = 5 n 6 n 7 n.7 n 8 n 9 m 3, n = 3 n . n 5 n.6 u 6 n 7 n 8 n 9 m 4, n 6 n =5 n =6 n =7 n 8 n =9 m = 5, n = 5 n =6 n =7 n 8 n =9 a = 6, n 6 n 7 n =8 n 9 S= 7, n 7 n =8 n 9 S= 8, n 8 n 9 S9, n = 9 +190676197" + 4535147 + 2061l31l + 1177274 + 760700 + 531592. + 392250 *+ 30124= + 238563 + 193571 + 90702974 + 687144 + 506546 + 380350 + 295331 + 235350 + 191701 + 159062 + 131010 + 1376297g + 168182 + 163001 + 167662 + 130750 + 115021 + 101209 + 89340 + 33636716 + 54'336 + 63285 + 65375 + 63900 + 60725 + 56853 + o186177 + 21095 + 28018 + 31950 + 33736 + 36112 + I421974 + 9339 + 13693 + 16868 + 18951 + 186871 + 4564 4 722a + 9175 + 91374 + 2110 + 4061 + 4824 + 1354 + 27171 7.69601o6 + 55817 + 3740 + 684 + 189 + 67 + 28 + 13 + 7 + 4  3349076  3740  879  28  111 5 24 13 7  374076  1466  608  277  138  73  42  25  68476  473  277  161 95  p58 37  17076  166  124  86  58  40  527 6  61 58 46 32 1976 28 28 2  876  13  12 376 6 N6 +1 374276 + 24667 + 8970 + 426g + 2355 + 1437 + 940 + 649 + 466 + 346 + 352476 + 2242 * 1421 + 912 + 65' + 170 * 208 + 31L + 261 * 175 + 143 + 1l9 + 11. + 5176 + 79 + + 79 + 72 + 64 + II,6 + 22 + 28 + 32 + 33 + 32 + 376 + 7 + 11 + 13 + 11 * I + 176 + 3 + 6 + 6 + 076 + 1 + 2 + 076 + 0 + 076 NACA TM No. 1224 TABI I4.  x 1010 no(7) AS POWER SERIE IN 7 m= 0, n= 2 n= 3 n=4 n=5 n = 6 n=7 n=8 n=9 a 1, a = 1 n=2 n=3 n=4 n 5 n =6 n 7 n=8 n=9 S= 2,n = 2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 m = 3, n = 3 n=4 n=5 n=6 n=7 n=8 n=9 = 4, n=4 n=5 n=6 n=7 n=8 n=9 m = 5, n = 5 n=6 n=7 n=9 a=6, =6 n 7 n=8 = 9 m = 7, n = 7 n=8 n 9 a=8,n=8 n = 8 n=9 9, n =9  22222222272  171428571  136054422  112233446  95359186  82840237  73202614  65561828 33333333337  666666667  342857143  226757370  168350168  133502861  110453649  94117647  81952286 133333333372  571428571  340136054  235690236  178003814  142011834  117647059  100163902  85714285772  476190476  314253648  228862047  177511793  14379085.  120196685  63492063572  404040404  286077559  216962522  172549020  142050628  50505050572  349650350,  260355030  203921569  165725733  41958042072  307692308  237908497  191222000  35897435972  274509804  218539428  31372549072  247678019  27863777172 + 705467474  1523810  168280  42634  15011  6408  3113  1662 133333333374  63492063 + 9142857 + 841400 + 191852 + 63047 + 25630 + 12009 + 6232  63492063574 + 76190476 + 6310502 + 1342964 + 420316 + 164766 + 75054 + 38087 + 2666666677 + 20614306 + 4178111 + 1260948 + 480568 + 214043 + 106642 + 49471433574 + 9669343 + 2837133 + 1057249 + 462332 + 226858 + 1899335274 + 5419096 + 1993669 + 858611 + 415905 + 94451007 + 3403334 + 1446929 + 693176 + 5414397 4 + 2276636 + 1080271 + 34024467 + 1601021 + 2281456Y7 + 15197176 + 42927  26152  8241  4452  2790  1889  1346 54095238176  6478782  197018 + 16156  7170  5138  3381  2295  1621  8055289076 + 8760622  122277  4268  4295  3369  2449  1778 + 8838009.76  2451288  64139  7860 3572 2475 1829  1295164876  380104  34429  6853 2950 1908  136583176  123724 19430 5180 2347  3701176 52456 11529 3762  12387876  25809 7154  5372076  14022  26503 6 __  ACA TM No. 1224 TABLE 5. x 1010 AND x 1010 AS PWl SI IE IN 7 a 4/ao x 1010 a^/a, x 101 m = 0, n = 4 n =5 n 6 n 7 n =8 n =9 * = 1, n = 3 n = n =5 n 6 n 7 n 8 n 9 a 2, n 2 n 3 n = 14 n = n 6 n=7 n 8 na8 n =9 a 3, n 3 n . n=5 n 6 n 7 n 8 n 9 a 4 n = 4 n =5 n 6 n 7 n 8 n8 n 9 m 5, n 5 n 7 n 8 n 9 n=9 S= 6, n = 6 n=8 n =9 n = 9 S= 8, = 8 n 9 m = 9, 9 + 90702974 + 687144 + 504546_ + 380350 + 295331 + 235350 * 19047619,r4 + 4535147 + 2061431 + 1177274l * 76070o + 531592 + 392250 666666667' * 95238095 + 13605h.2 + 481000o2 + 23545148 + 1369260 + 885992 + 616393 +28571I2867 + 31716032 + 9620010 + 4238186 + 2282100 + 1392273 + 921590 + 634920637 + 17316017 + 7063643 + 3586157 + 2088109 + 1335519 + 288600297k + 11100011 + 5379236 + 3016591 + 1869726 + 166500177' + 7770008 + 1223228 + 2549627 + 108780117 + 57589147 + 3399502 + 7710238r' + 44145503 + 5715647v  673176 + 2610 + 371 + 106 + 39 + 17 + 33862476 + 100968  23492  2592  635  213  86 12698t12776 + 8165608 + 1511520  274072t  259k4  5719  1776  677 * 5925925976 + 8245723  127901  108971  22241  66514  2370 + 2968460176  4144O0O  326912  62911  17587  6162  10853314L76  807274  148289  39919  13555  17 909376  309 39'  80722  26700  59066176  150108  18546  26280576  83016  13519776  160276  1207  852  610  1581076  11212  4830  2557  1524 + 151171676  320667  48h49  14190  6392  3353 3176603276 +10582011  1231668  134546  3622  11063 6705 +142328014276  384800Z  336361  79692  28127  12452  962001076  740001  159385  52235  21792  14o800176  296000  91412  36319  51800176  152353  58111  21476976  89808  13471276 NACA TM No. 1224 m /m TABLE 6. NUMERICAL VALUES OF THE COEFFICIE TS an,r ano APPEARING IN THE SERIES DEVELOPMENTS (7.7) 2 /ao x J10 m = 0, n = 0 +500000000072 66666666674 +6613756676 n = 1 1666666667 +222222222 16825397 c6/ao x o100 m 0, n = 0 +176366876 n= 1 5291005 n = 2 +3703704 n = 3 529101 n = 4 50391 n= 5 7632 m = 1, n = 1 +265957476 n = 2 3703704 n = 3 +1058201 n = 4 + 251953 m = 2, n = 2 +740740776 n= 3 5291005 MACA TM No. 1224 [For btm () TABLE 7. 2 x 10 AS PFOER SERIES IN 7 bm (T) n,o m = 0, there Is br = ar; compare therefore table 2a =1, n 1 n = I* n =2 n =3 n = 7 n = n 7 1=8 n9 m = 2, n n n n n n n n m = 3, n n n n n n n m = n n n n n n m = 5, n n n n n m = 6, n 6 n=7 n=8 n= 9 m = 7, n n n m = 8, n = 8 n 9 a 9, n = 9 +8000000007 2 +k08163265 +264550261 +192837166 +150618612 +123076923 +103806228 + 89620336 + 78768349 +61224489872 +370370370 +?57116621 +193652501 + 153866154 +126874279 +1075144403 + 93089868 +49 382716072 +330578512 +242065627 +188034188 +152219132 +127097931 +10860o48'5 +41322314072 +295857988 +225641026 +179930796 +148280919 +125313283 +35502958672 +266666667 +209919262 +171093368 +143215181 +31111111172 +242214533 +195535278 +162310538 +27861660972 +221606648 +182599356 +24930747972 +204081633 +22675737072 2133333371  1514520  301182  91068  34950  15704  78941  41357  2522 113589037 4  2110372  607123  224681  98147  4821 3  2590S  14900  656560 374  1821369  655319  279902  135080  71* 32  4o0562  40980817  1441703  604588  287353  150013  81244  27186397,  1122807  526811  272002  151296  189213874  878023  449019  247677  13683477*  693939  379959  102089174  555324  78156774 +36808976  59660 + *0526 + 20727 + 11069 + 6158 + 4057 + 2702 + 1885 +18340476  10012 + 11136 + 9149 + 6171 + 4135 + 2844 + 2017 + 8107176 + 1978 + 4651 + 1553 + 357* + 2666 + 1980 + 4129376 + 3851 + 2666 + 2567 + 2193 + 1766 + 21880,6 + 3110 + 1790 + 1601 + 1422 + 1238576 + 2623 + 1285 + 1072 + 741076 + 1939 + 952 + 464* 376 + 1422 + 302576 NACA TM No. 1224 TABLE 8. 4) x 1010 bm (7) nso ABD x 1010 AS PO SfRIES II bm () n,o a = 0, there is br = ar; compare therefore table m = 1, a = 1 n=2 n= 3 n =4 n 5 n = 6 n 7 n 8 n 9 S= 2, = 2 n 3 n =4 n=5 n 6 n 7 n 8 n 9 m = 3, n = 3 n = n = n 9 n=8 a = 4, n = 4 n = 5 n =6 n =7 n =8 n =9 m = 5, n = 5 n = 6 n =7 n=8 n =9 m = 6, n = 6 n = 7 n=8 n =9 m = 7, n= 7 n = 8 n 9 S= 8, n = 8 n = 9 m = 9, n = 9 b4\ x 1010 b6/b x 101 +1360544274 + 4810001 + 2354548 + 1369260 + 885992 + 616393 + 451867 + 344592 + 270999 + 96200107 + 4238186 + 2282100 + 1392273 + 924590 + 652697 + 482428 + 3695144 + 7063643~7 + 3586157 + 2088409 + 1335519 + 913776 + 657857 + 492725 + 53792367 + 3016591 + 1869726 + 1246058 + 877142 + 644333 + 422322874 1 2549627 + 166111i + 1147032 + 828428 + 33995027y + 2172614 + 1474756 + 1049342 + 279336174 + 1868024 + 1311678 + 23350307,' + 1620308 + 19803767 50235176  26177  4104  1021  333  130  58  28  12 26176976  36934  8508  2615  97m  417  198  102 14363376  31198  9154  32k  1362  630  317  8423376  23800  8277  3343  1512  747  5235976  17737  700a  3107  1509  3415976  13231  5770  2761  2319776  9966  4706  1629076 7602  1176976 +9866776 +26909 + 10654 + 5181 + 2874 + 1746 + 1135 + 777 + 554 +6727376 +23'39 +10362 + 5337 + 3056 + 1892 + 1243 + 856 +4687876 +19241 + 9339 + 5094 + 3027 + 1921 + 1284 +3367876 +15565 + 8150 + 4678 + 2881 + 1876 +2h90y76 +12595 + 7017 + 4211 + 2680 +1889276 +10256 + 6016 + 3752 +1465176 + 8423 + 5160 +1158176 + 6981 + 930876 I __ __ _ NACA TM No. 1224 TABLE 9. bm n2L x 1010 AS POWER SERIES IN 7 bm n,o m = 0, there is br = ar; compare therefore table 4] m = 1, n n n n n n n m = 2, n n n n n n m = 3, n n n n n m = 4, n n n n m = 5, n n n m=6, n= 8 n=9 m = 7 n= 9 5714285772 68027211 67340067 63572791 59171598 54901961 50992534 2267573772 33670034 38143674 39447732 39215686 38244400 1122334272 19071837 23668639 26143791 27317428  635727972 11834320 15686272 18211619  394477372  7843137 10926971  261437972  5463486  182116272 +1523810 4 + 252420 + 76741 + 30022 + 13730 + 7005 + 3878 + 42070074 + 191852 + 90068 + 45768 + 25018 + 14542 + 14921874 + 105079 + 64076 + 38917 + 24237 + 630477 + 57668 + 42030 + 29084 + 3020774 + 33024 + 27422 + 159007Y + 19805 + 900274 n = m + 1, b2/bo disappears. [For 3283676 + 4847  2868  2447  1811  1339  1009  815276  610  920  936  816  679  229176  655  476  455  416 76576  374  268  242  29476  199  152  12776  107  6076 _ I__~C I For n = m and NACA TM No. 1224 TABLE 10. X 1010 bm (7) n,o AND na6 x 1010 AS bm (7) n,o POWER SERIES IN 7 For m = 0, there is br = as; compare therefore table 5] b ob x 1010 b4b0o x 1010 m = 1, n = 5 +13742974 156676 n = 6 +168182 371 n = 7 +163007 136 17276 n = 8 +14766~ 59 213 n = 9 +130750 29 203 m = 2, n = 6 + 3363674 37176 n = 7 + 54336 227 6 n = 8 + 63285 127 307 n = 9 + 65375 72 51 m = 3, n = 7 + 1086774 10676 n 8 + 21095 99 6 n = 9 + 28018 72 77 m = 4, n = 8 + 421974 3676 n = 9 + 9339 43 m = 5, n =9 + 186874 1476 For n = m, n = m+ 1, n = m+ 2, n = m+ 3, b4/bo disappears. For n = m, n = m+ n = m+2, n = m + 5, b_/bo disappears. n = m+ 3, n = m+ 4, 