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 cop- 2- 
 
 ANTENNA LABORATORY 
 Technical Report No. 38 
 
 CONFERENCE ROOM 
 
 PROLATE SPHEROIDAL WAVE FUNCTIONS 
 FOR ELECTROMAGNETIC THEORY 
 
 THE LIBRARY OF THE 
 
 MAR 6 1961 
 
 UNIVERSITY OF ILLINOIS 
 
 by 
 
 Walter L. Weeks 
 
 5 June 1959 
 
 Contract No. AF33(616)-6079 
 Project No. 9-(l 3-6278) Task 40572 
 
 Sponsored by: 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 ELECTRICAL ENGINEERING RESEARCH LABORATORY 
 
 ENGINEERING EXPERIMENT STATION 
 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
ANTENNA LABORATORY 
 
 Technical Report No. 38 
 
 PROLATE SPHEROIDAL WAVE FUNCTIONS 
 FOR ELECTROMAGNETIC THEORY 
 
 by 
 
 Walter L» Weeks 
 
 5 June 1959 
 
 Contract AF33(6l6)-6079 
 Project No. 9.(13-6278) Task 40572 
 
 Sponsored by: 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/prolatespheroida38week 
 
, r- c A ENGINEERING LIBRARY 
 
 ABSTRACT 
 
 Some of the results of computations of prolate spheroidal wave func- 
 tions for a specific electromagnetic radiation problem are recorded. A 
 convenient method for machine computations is described. Numerical results 
 in the form of test calculations, graphs , and tables of values are included, 
 
 UNIVERSfTYOfi 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
• li 
 
 ACKNOWLEDGEMENT 
 
 The author wishes to express his deep appreciation to all of the mem- 
 bers of the University of Illinois Digital Computing Laboratory, past and 
 present. Without their unseen work in achieving and maintaining the high 
 level of reliability of ILLIAC, and in the preparation of an excellent and 
 extensive library of service routines, the work described herein could not 
 have been carried out. 
 
CONTENTS 
 
 Abstract i 
 
 Acknowledgement ii 
 
 1. Introduction 1 
 
 2. Determination of Eigenvalues 5 
 
 2 1 - 
 
 3. The Spheroidal Angle Function^ V, / (l - v ) 2 lk 
 
 k-. The Spheroidal Radial Functions., U, . h-6 
 
 5- Summary 56 
 
 References 57 
 
1. INTRODUCTION 
 
 The object of this report is to record some of the results of calcula- 
 tions of prolate spheroidal wave functions which were obtained in the course 
 of the solution of a specific electromagnetic radiation problem. However, 
 a limited amount of peripheral material is included with the hope of in- 
 creasing the utility of the specific results. 
 
 Maxwell's equations in the prolate spheroidal coordinate system (u, 
 
 v, cp ) are partial differential equations which are separable in the usual 
 
 sense only under the condition that the fields have no variation with the 
 
 coordinate <p. With this restriction, the fields are expressible in terms 
 
 of functions, U(u) arid" V(v), which satisfy the following differential equa- 
 ls 
 tions : 
 
 2 
 (U 2 - 1)S+ ((Piu) 2 - k, ) U= (1) 
 
 du 
 
 (1 - v 2 ) ±1 + (^ - (p iv ) 2 ) v = . (2) 
 dv 
 
 The representation for the magnetic field strength can "then take the follow- 
 ing form 
 
 The notation corresponds to that employed by Schelkunoff . That is, 
 in these equations, the values of the coordinate, u, are the reciprocals 
 of the eccentricities of the ellipses which generate the prolate spheroids 
 (l < u < oo ), while the magnitudes of the coordinate, v, correspond to the 
 reciprocals of the eccentricities of the generating hyperbolas (l < v < l). 
 The coordinate, v, is also related to the polar angle, 0, of spherical 
 coordinates through the relation v = -cos 8; the surfaces © = constant 
 are asymptotic cones for the hyperboloidal surfaces, v = constant. The 
 quantity, i, is the semifocal distance (common of course to both the ellipses 
 and the hyperbolas), while k, is the separation constant (eigenvalue) and 
 
2 
 
 the quantity, (3, is defined by 6 = od(|j.€)^. (For a sketch of the coordi- 
 nate system, see reference 2, page 32 or reference h, page 7-) I n most 
 exterior boundary value problems, boundary conditions can be attached to 
 the differential equation, (2), with the result (from Sturm -Liouville theory) 
 that the functions, V v , comprise a complete orthogonal set. The functions 
 to be discussed in this report are those which satisfy Equations (l) and 
 (2) together with the condition V(+ l) =0. 
 
 These functions are cl_osely related to some of those which have been 
 
 3 k 
 
 partially tabulated by Stratton, Morse et al." and Flammer (see Table A). 
 
 The correspondence between the coordinate variables is as follows: u = % , 
 v = -tj. The funet.ii 
 3 and k as follows : 
 
 v = -tj. The functions, V, are related to the functions, S of references 
 
 2 2 
 
 V k = \^ 1 ' Y ) S lk ( 1,e -> m = 1, n =-k) 
 
 // 2,2 
 
 in which a, depends on the normalization. Actually the quantity V / (1-v ) 
 
 is tabulated in this report since this is the quantity which appears in 
 
 the expressions for the fields. The functions, U, are related to some of 
 
 3 k 
 the radial functions tabulated previously ' , as follows s 
 
 U, . = — — — — (notation of reference k) 
 
 / 2 A 
 (u - 1) 
 
 in which b depends on the normalization, or 
 
 U . = — — - < 1Jc > (notation of reference 3) 
 
 k /il 1 | ne nl 
 
 12J /__2 nX 2 ( ik 
 
 1 
 
 (Actually, reference 3 contains no tables of the spheroidal function them- 
 selves but only the coefficients which can be used in the calculation of 
 the functions in a series expansion.) 
 
 It appears that almost every person who has worked with the spheroidal 
 functions has elected to normalize the functions in a different way. There 
 
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is then no standard scheme. The fact that the normalization of the func- 
 tions presented here differs from those used previously therefore requires 
 no apology o The normalization adopted here is convenient for the antenna 
 problem since the integral., 
 
 U ( i=v ) 
 
 which appears frequently in the problems is made equal to unity. 
 
 In electromagnetic problems, the derivatives of the radial functions, 
 dll/du, are needed as often as the functions themselves., Thus, this report 
 includes two linearly independent solutions to Equation (l) together with 
 their derivatives., The solution which represents outward traveling waves 
 is 
 
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 k3 kl k2 
 
 if a time convention e J is adopted to represent sinusoidal time varia- 
 tions. 
 
2. DETERMINATION OF EIGENVALUES 
 
 One difficulty with the calculation of the spheroidal functions is 
 that the eigenvalues depend on the parameter, pi , in such a way that each 
 class (size) of spheroids has a different set of characteristic functions. 
 Thus, for each value of |3i, a new set of eigenvalues must be determined . 
 
 A practical method for the machine calculation of the eigenvalues 
 
 2 
 has been outlined previously . Briefly, the method can be described as 
 
 follows: The spheroidal functions are represented by means of a finite 
 
 number of terms selected from the exact representation 
 
 1 
 
 P 2 °° 1 
 
 V. = (1 - v T S cL P (v) (3) 
 
 k ' _ kn n v ' v ' 
 
 where P is the associated Legendre function (polynomial) of degree one. 
 The eigenvalues are calculated by finding the eigenvalues of a matrix which 
 is generated as follows: The series, (3), or some portion of it, is sub- 
 stituted into the differential Equation, (2), and the properties of the 
 
 2 
 associated Legendre functions are utilized to generate a set of equations 
 
 oo 
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 1 r 1 
 
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 mn 
 
The value of the separation constants, k , which make the determinant equal 
 to zero are the eigenvalues; thus, the problem of eigenvalue determination 
 is equivalent to the problem of finding the eigenvalues of a symmetric 
 matrix. 
 
 Evaluation of the integrals in (5) shows that the elements of the 
 matrix are non-zero only if m = n, or m = n - 2, or m = n + 2. If the 
 normalized associated Legendre functions are employed in (3); the matrix 
 is symmetric and the non-zero elements are determined as follows : 
 
 L^ = k - n(n + 1) - (pi) 2 
 
 L =&£ 
 
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 nm 2n + J 
 
 2 
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 infinite series ((3) and (^)) to be replaced by a finite series of n terms, 
 which in turn means that the determinant in (6) is only of order n. Then, 
 the determination of the n eigenvalues of the associated matrix gives sta- 
 tionary approximations to the eigenvalues of the n characteristic functions 
 most closely represented by the finite number of terms selected from the 
 infinite series (3). It is demonstrated below that only moderate size 
 matrices need be handled in order to simultaneously determine several eigen- 
 values, each good to ten or eleven decimal places. 
 
 The practical question is, "What order of matrices should be handled 
 for accurate and rapid eigenvalue determination?" A good indication of 
 the answer to this question is given by the data in Tables I and II. Table 
 I gives the results of computations of eigenvalues employing different size 
 matrices, for two particular values of p£ (rt/2 and 5«)» The functions hav- 
 ing Pi = n jt/(2 are very useful in the problem of deciding the size of matrix 
 required since, for such values of pi, one of the eigenvalues is known 
 exactly (this follows since if t = pi one solution of (2) consists of sine 
 and cosine functions). In the tables, the exact value which is known 
 should appear as unity since the eigenvalues are scaled by (pi ) . The results 
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10 
 
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11 
 
 the first n non-sero terms, then the lowest n - p eigenvalues are good to 
 about ten decimal places, where p ranges from about three, for (3i of the 
 order of rt/2, to about eight, for pi of the order of 5rt « Thus, if a prob- 
 lem requires the determination of the lowest m eigenvalues, it is only 
 necessary to solve a matrix, of order m/2 + p, where p is less then ten for 
 most cases of practical interest „ (m/2 appears in the foregoing since the 
 odd functions and the even functions can be handled separately. ) 
 
 If because of machine or time limitations, a matrix of order m/2 + p 
 is too large, or if the lowest eigenvalues are "not of interest, it is pos- 
 sible to calculate a group of q intermediate eigenvalues by solving matrices 
 of lower order o That this is so can be seen if the infinite series (3) 
 is replaced by a finite sum as follows ; 
 
 ai h 
 
 \ - a - -r Z a ta < w . (5a: 
 
 n=n 
 o 
 
 In such a case, Equation (k) becomes 
 
 N 
 
 T -a. l = o, 
 
 t^ kn nm 
 
 n=n 
 o 
 
 and the result can be described as the "trimming" of the matrix, as opposed 
 
 to what is usually referred to as "truncation". The process is indicated 
 
 schematically in Figure 1, and is possible because the magnitudes of the 
 
 coefficients, d, , peak at the value d (i.e., the spheroidal functions 
 Kn KJ£ 
 
 most closely resemble the spheroidal functions which have the same number 
 of zeros in the range). 
 
 The results of trial eigenvalue calculations by means of such "trimmed" 
 matrices are summarized in Table II. It appears from these data that if 
 a criterion of ten figures is adopted to specify an acceptable eigenvalue 
 calculation, then the lowest p + 2 and the highest p eigenvalues calculated 
 are unacceptable, where p depends on the magnitude of Bi as indicated above. 
 Thus, the solution for a matrix, whose order is equal to the number, 
 r = I - n + If of non-zero terms in (3 a )j gives q good intermediate eigen- 
 values, where q - r - 2(p + l)„ 
 
12 
 
 
 
 
 Y/ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 K 
 
 > 
 
 
 
 / 
 
 
 
 \ 
 
 
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 y 
 
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 /\ 
 
 
 ^ 
 
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 / / / 
 
 T 
 
 
 
 
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 / 
 
 
 / 
 
 
 /// 
 
 y///z 
 
 // 
 
 original matrix 
 
 "truncated" matrix 
 
 "trimmed" matrix 
 
 FIGURE 1. 
 
 The University of Illinois digital computer (iLLIAC) has two library 
 routines which can he used to find the eigenvalues of a symmetric matrix. 
 One of these (M-l8) can handle matrices of order forty while ^he other 
 (M-20) is capable of handling matrices of order one hundred and twenty 
 eight. Both of these routines were used in the trial calculations. The 
 latter, (M-20), is capable of slightly greater accuracy and was used in 
 the regular production runs. The number of eigenvalues needed was never 
 so large that the "trimming" technique was actually necessary. 
 
 It is worth noting, for the. benefit of those who are not" so fortunate 
 as to have immediate access to a good service routine for finding the eigen- 
 values of a matrix, that when the odd functions and the even functions 
 are handled separately, as they should be, the elements of the matrices 
 which result are zeros except for the main diagonal and the diagonals on 
 either side of the main diagonal. This means that the evaluation of the 
 
13 
 
 associated determinant can be written in terms of a Sturm sequence and 
 a system for the determination of the eigenvalues follows rather naturally 
 from the known algebraic properties of such sequences. (in fact, in view 
 of this, the machine could make the decision as to the orders of matrices 
 which are required*) 
 
 Some eigenvalues for the functions having values of the parameter 
 0i of it/2, jt., 3it/2, 2jt, kit, 5jt, 12 and 16 are included in Table III (eigen- 
 values for Bi = 5^ &) and 12 appear in reference 2). The known eigenvalues 
 were not corrected in the tables, so as to give the user an indication of 
 the accuracy of the calculation. All eigenvalues are scaled by (pi) since 
 this is required in most of the calculations. Figures 2 and 3 give a 
 graphical presentation of the eigenvalue variation as a function of Bi . 
 
14 
 
 
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 H -* L^- O -* CO OJ C— OJ f- OJ b- KN ONVO OJ 
 
 ON-* 4-OKNrl LTNVO OJ 
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 + + + + + + 
 
 
 i-l ON f- LTN KN H 
 
 
 H 
 
 LTNCOCOCO KNONKN-*- KN 
 
 LTNVO VO KN O VO O CO b- 
 
 " C— H VO b-O OJCOltnO 
 
 OJ VO CO OJ -* VO KN-* OJ LTN 
 
 * CO KN LTN-* CO t— O -* VO CO 
 
 KN-*CO-* KN-* CO LTNKNKN-* 
 
 O -* O ON O KNCO VO VO CO OJ 
 
 ' 'COr-IONO hhHOIAOOl 
 
 t~--* ^T LTN O VO LTN t— H L^-VO t~- 
 
 OJCO !>-OnltnKNltnO ONO ltnKN 
 CO OJ b-0JC0-*O b-KNHcOVO 
 
 44KMOOI WOlrlH HOO 
 + + + + + + + + + + + + 
 
 -*OJOCOVO-*OJOCOVO-*OJ 
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16 
 
 -P OJ VO £— LT\ VO VOrlt^t-hrlHfO^OMOrlCMMO 
 
 a ON CO ON ON KN *-n -^-HHVOIr-OJLrNUA^OJVOVOr-ONVOaOOJ 
 O 4 HOOI4 >d VO^aOCO-^ONVOOJLrNKN^ONLfNONrl-^OJh- 
 O H-4C0 IAO rrt ^^VOaOr|COOOVOVOVOCOOJOJI s ~-KNOJONKNO 
 «-^ H IOHO\4 O ojj-4-^04-4-OOJOI>H^O^)^DKM>- CO LfN l>- H 
 rl OlAH IA v_x r HOU)OaiKNKMOfOLrNO\OMAOJ4'CO KMACD H KA 
 CJ VO VO VO I s - KN CO-^I"— V0l<NV0V0KNI>-C0V0H-4-LJ'N-4-rll s ---4-0J-4-Lr\O 
 \ lt\-4- ON ON OJ te LT\irNlT\VOCOOKN(^— HVOOJONVO^IOiro^LrNCOOJCOf- 
 fe jt t— KN K>VO -4" K>OC\IO\HCOrHCOOOOOCOOOOrlON(MOlf\r|iAl^. I s - 
 KN . . . * . 4-O^OOJOt s -^)4J- lA-4 4-^ t-OW^OOJ-ONJ-O^ 
 II lAIOiOl HO II ........................ 
 
 OOOOO (M H OCO t-^)-* KMM H O ONCO l>-VO U) Lf\4-4- fO ai OJ Ol H 
 
 ^ OOOOO **< OJOJOJrHrlr-lrlrlHrlr-IOOOOOOOOOOOOO 
 
 ca + + + + + ca + + + + + + + +. + + +•+ + + +. + ■+ + + + + + + + 
 
 O 00 VO -4- OJ OAt s -ir\hrNHONt s -Lr\K~\ r -IONt^-Lr\KNrlONt s — ltn r-TN rl ON t— lt\ i<N 
 
 ,-| LT\ LTN LT\ LTN LT\ -4" J-J-J-J-KMOrOfOfOaiGiaiaiajrHHrHH 
 
 ^ s-^ O -4- -4 ON CO I 00VO IALAO CM t-^- VO0J-4-COOJKNLf\VO 
 
 Si • -4" CO LT\ C— ON '-n IAH4-OJCOHHOJHOOJ-4VOCDCOOMM 
 
 P -P UNOWO I s - ON q~> OxO O H4 IA4 4 (J\VO-4VO H-4 lAO ^-fO 
 
 B S t-O^0O0\ 11 C— l>-00 -4H^DOJCV100r-IOIO,ON(MaJ(M N~NCO tOv 
 
 "2 O CT\0\O lAON O 0A0>r|00-4-r<~ s ,O--4KN0NOV0 N ~-LrN0JLr\r|V0[ s — O 
 
 "£ CJ LTN CO ON ON ON KN.HVaONCOVOOJD— r1VQOJKNCOOJONLrNCOKNOVQhr\ 
 
 S ^-" -4" KN OJ VO ON OJ -4 OJ LfN-4 ON O I s - CTNCO OJ K> O 1^-4 H I s - H ^O KMA O 
 
 r, -4- ON LT\ H ON CM O^OCO lAGO^O H O lA^O KMA r^VO Lf\0 O l^-ONCO K>VO 
 
 " te VQ f- r- lt\ ON H VO ON t— H H VO I s - -=f" VO KNVO ltn ON ON -4- ltn OJ -4- H -4- KNCO CO 
 
 • • • • • -4" CO KN ON LT\ H CO VO -4" KN OJ OJ CJ KN i/N C— O KN I s - rl VO H t— 
 
 l_4 II H I s - -4- OJ O II ....... , l ............ -. 
 
 H HOOOO LrNKN,HOco^vo^KNOJrlOO\cot s -vou-NLr\_4KNKNOJOJrl 
 
 H ^ OOOOO ^ OJOJOJOJr-irlrlrlrlrlrlrlOOOOOOOOOQOO 
 
 M ca + + + + + cS. ++++■+-=+ + + + + + + + + + + + + + + + + + + 
 
 EH 
 
 O CO VO -4" OJ ON C—LTN KN rl ON I s - LTN KN rl ON r— irNKNHONt— LT\KNrl ON I s - LT\ KN 
 
 H LfNlT\IJALJAU-\^^^^^4KNKNKNKNKNOJOJOJOJOJrlrlrlrl 
 
 •P ON OJ rl ON VO LTNt-I^-LrNHI^-ON-^-ON. ONVOVO^-LTNrHCOON 
 
 a t- KN I s — O KN '—> KNH KMAKNCO OIC004-COHCOVDaiCOHO 
 
 O O H IA t—-=t >d VOOJOJKNLr\OOJOJKNr|CO^-LT\KNLr\ f-JSf VO OJ 
 
 CJ -4- CO -4- OJ -4" nd I s -- 4- s -vOOI^04rlKN4hOHOIr|VOHI s -0\ 
 
 ■^ VO -4" H VO CO O O VO I s — OJ OJ h- VOr!OLr\Lr\riK\KNOVQKNKNOJLr\VQ 
 
 UN(M lAO I s - ^— ' lAMD OliAUNrHK\aiCOOOOKMr\ (T\CO OCO-4 f-CO I s - VO 
 
 OJ COVO-4-ONO I s — LTN VOONUA^LrNONLfNLrNVOHCOCOOVOrrN^t-I^-KNCJ^ 
 
 \ ht^HCO IA fcr 4" H OM"- LfN-4- IA OJ rl H rl rl OJ OJ KN LTN VO CO O OJ UACO rl 
 
 |S' O VQ LTN LfNCO OJ OJLTNOAVOlJNVOON^rlOrl-4-ONVOLrNVOON-4-OJrlOJLr\rl 
 
 ii o^oooJt^oJt-ojco^dvdoJcouAOJ onvo -4* OJ d cd vd lA 
 
 ON CO I s - C— VO VO LTN LTN^F 44lAfAIAW(Mr!rlrlHrlOOO 
 «* OOOOOOOOOOOOOOOOOOOOOOOO 
 
 ca + + + + + + + + +. + + + + + + + + + + + + + + + 
 
 ONt--Lr\KNr!ON|>-Lr\KNH ONt— LrNKNrlONt— LfNKNrl ON I s — LTN KN 
 LTN LT\ LfN LTN LT\-4- -4- -4" -4" -4" KN KN KN KN KN OJ OJOJOJOJ rl H rl rl 
 
 II 
 
 LTN ON I s — CO 
 
 0.1 
 
 
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 OJ 
 
 H 
 
 O 
 
 o 
 
 «^ 
 
 Q 
 
 O 
 
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 O 
 
 ca 
 
 + 
 
 + 
 
 + 
 
 + 
 
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 OJ 
 
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 CJ 
 
 
 
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 ^-^ 
 
 ^ — s 
 
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 fd 
 
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 0) 
 
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 !> 
 
 O 
 
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 <D 
 
 fc! 
 
 ON KNCO O O H 
 
 ^ 
 
 Jf 
 
 VO H CO UNCO CO 
 NN H CO VO fOO 
 
 J? 
 
 II 
 
 HHOOOO 
 
 II 
 
 «=* 
 
 o o o o o o 
 
 "=* 
 
 CO. 
 
 + + + + + + 
 
 CO. 
 
 17 
 
 OJ H IO OJ ON ON VO VO H KMn ON H [>- ON rO 
 
 on ON ON co vo ON!^-rouND--ONONrH ON -* h H 
 
 UNVOrHCOCOHUNONOJIOCO^OVOONOJOJ-* 
 
 ONKNO C— t— CO H UN ON ON O -* KN H VO UN 0J KN t— 
 
 UNC0 ON O -* t— -* KNH H ONVO CO VO UN UN ONVO OJ LT\ 
 
 C— OJ -* -^* KN ON-* ON-* OCO ON b- KN OJ ON OJ KN OJ ON UN 
 
 OOl WCOH rlf-HH 0N-* UN-* H VO ON O OJ-* 0\0\H 
 
 UNON-*ONVOKNOONCO!^-COONrH-* hdhl^OCOcOH 
 
 t— H O UN-* ON ON-* KNCO 00 KN KN ON ON-* UN O H C-- t— KN UN 
 
 H t-KNOWO KN H O OMDcO O\0 H KNVO ON KN D~- rH VO 0JC0 
 
 KNOI O OM^VO lA^I- KNH O ONQ0C0 t-VO lAJ"-* KN KN OJ 0J 
 OJOJOJHHHHHHHHOOOOOOOOOOOO 
 + + + + + + + + + + + + + + + + + + + + + + + 
 
 H OM'-iAtOH 
 
 O CO VO -* 0J O00V0 4- 0J O CO VO -* OJ O CO VO -* 
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 s 
 
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 -P 
 
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 -P 
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 o 
 
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 t— O C--CO IAO (U 
 
 ON t— C-- * H CO >— ' 
 VO KNCO CO C— OJ 
 
 VO UN-* t~- [*- UN OJ 
 
 -* VO OJ t- ONCO H 
 -* H ONVO KN O 
 
 ...... || 
 
 r-\ H O O O O 
 
 O O O O O O ^ 
 
 + + + + + + CO. 
 
 CO UN ONVO CO OJ f-VO KN O LfN -4" OJ H -* t~- 
 
 KN O KN-* -*ONOCJVQltnOJOJOJOON UN-* 
 
 ONOJ OJ -* VO O t— -* ON-* -* ONO H KN KN UN H 
 
 KNO W O LfNOO^- KN4- UALfM^-^ONOLAHOJON 
 
 KNCO -* KN ON UN t~- OJ C~- KN-* ON KN OJ UN t- UN OJ KN UN 
 
 CO CO -* 00 ON C— -* ON-* CO -* OJ -* -* -* ON t— ON OJ KN ON 
 
 ON KN NN 0\ O t~- O ON KN -* O KN OJ t~- ON CO -* ON KN ON 00 LTN 
 
 t— VO O ON UN UN OJ KNH-4- KN l>- D— OJ «N O KN H VO VO KN t— 
 
 VOVO OJ -* O KN H UN-* ON ON UN VO KNVO -* OO C— OJ OJCO O t— 
 
 OJVO HVO OJCO LfNOJ OCO t— C— C— GO ONH KNVO O -* CO -* ON 
 
 1A4" CVI H ON00 VO IA4- (OHO O\C0 t^-VD VO UN-* -* KN (M CJ H 
 OJOJOJOJHHcHHHHiHHOOOOOOOOOOOO 
 + + + + + + + + + + + + + + + + + + + + + + + + 
 
 ON f- LTN KN H 
 
 O CO VO -* OJ O CO VO -* OJ OCOVO-* OJ O CO VO -* 
 
 VO LrNUALfNLOvLfN-*-*-*-*-* KNKNrOKNKAOJ OJ OJ 
 
 CJ OCOVQ4- 
 OIOJHHH 
 
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 a 
 
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 0) 
 
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 > 
 
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 C— LTN-* LTN ON OJ 
 
 ONCO H OJ D-- t- 
 
 Q) 
 
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 « 
 
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 CO t^-ONOJ C— H 
 
 OJ 
 
 n 
 
 tocy HHOO 
 O O O O O O 
 
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 •=y 
 
 O O O O O O 
 
 ^ 
 
 ax 
 
 + + + + + + 
 
 ca 
 
 H -=*■ CO HONOJ O t— O VO -* CO ONLfNO OJVO 
 
 CO CO KNVO ltn ON OJ OJ D— VO OVOrHCOOKNOO 
 
 LfNHCO IOIOOVO O OJ rH D— OJ t— E^- ON ON O OVO 
 
 VOOOUACOCOt-UfAVOOHCvlOO-=i-VO ONCO D— O 
 
 VOOONOJONHCOOt— ONt— O ONUNCO ONO tOCJ OJKN 
 
 u\CJVOUNCJ4-*ONCJO UNt-VO O CJ O UNCO t>— * ON KN 
 
 -* H O CVI E--4-4VO CJ O O lOCACO OMOONOO O UN OJ IO 
 
 co ltn ro H On t— vo ltn-* -* _*-*_* ■ ^j- ltnvo CO ON H -* vo ON OJ 
 
 HKND— tOOOOJVOOJOOOJVOOJOOaivorOHVH IO00 
 
 KNt— HU\OUNOUNOV001C040 h--* HCO LTN IO H ON t— UN 
 ONCO 00 t^ t-VO vo ir\ ia4 4iOfOfOOICJCIHHrlHOOO 
 OOOOOOOOOOOOOOOOOOOOOOOO 
 
 + + + + + + + + + + + + + + + + + + + + + + + + 
 
 H ON 1^ UN hO rH 
 
 OC0V0 4 OJ OCOVD4 OJ OQ0V0 4 CVI OOOVO-* OJ OCQVO* 
 VO UN UN UN UN UN-* ****(OlOIO l IOIOOJCJCICJOJHHH 
 
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 rc\ tr\ o -4- on kn <d 
 
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 VO lAO IA ON J- VO 
 rltAO t^-Hl^ H 
 LTN CM O t— LTN CM 
 || 
 
 rH H H O O O ^ 
 O O O O O O 
 
 + + + + + + ca 
 
 ON KNVO O HCO C— O LTN CO KNVO -4" -=t" O 
 
 h- CM CO VO r-\ H VO CO t- CM ON KN CM t- -3* 00 
 
 LTNVO-d-0\ONO-=i-0>0 rl IAVO 00 O 1^4- rl 
 
 t— KN CM O-HCM O VO 0\-3" 0\-=t- O CM uaVO ON CM 
 
 LfACO O H CM VO KNVO VO t- KNVO HNO\ H OMA4" b- 
 
 ONON-^-VOKNVOLrNrHKNCMcOCM KNKNrHOoOONKNCM 
 
 O VO ltn t— r-\ CO D— ON— t HHIOO\ t— CO CM ONCO H CO CO 
 
 O M OJ lAOMTNM O\C0 CAH4-CO K>, O 0O CO CO O-J-CO-? 
 
 VO CO KN H CM t-LTNUN ONVO 1^- O VO VO ON-=f KMAH ONO IA 
 
 LfNKNH OO\C0 [— b-H>-t— COONrHCM-4-voONCM UN ON CM b- H 
 
 4- ACM H ON 00 b- VO LTN -4" KN CM CM H CD ONCO CC° t>-VO VO LTN Lf\ 
 CMCMCMCMCMHHHHHHHHHrHHOOOOOOOO 
 
 + + + + + + + + + + + + + + + + + + + + + + + + 
 
 cm oco vo 4- 
 
 rH rH 
 
 CM 
 
 OCOVO-4- CMOCOVO-=t-CMOCOVO_3-CMOOOVO-4- (M OCO^O-* 
 CO bhM>- D— vo vovovovo is\u\ is\\s\ir\^- -=*- -3- -3- ^£ KN kn KN 
 
 CD 
 
 •H 
 -P 
 
 o 
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 OWO ON KN LTN-* ^~o 
 
 CM LTNCO KN O CM Ti 
 
 -=t" ONVO ONOO CO <X3 
 
 -4" UN LTN b- O O O 
 
 H t— -=t ON OWO — ' 
 O O VO 00 O CM 
 
 ON ON CM _* CM LTN VO 
 
 O ON-3- O -*^t H 
 VO CM O 00 LTN CM 
 
 . . . «, . . || 
 
 H H H O O O ^ 
 
 O O O O O O ^ 
 
 + + + + + + ca 
 
 CM 
 VO o 
 
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 LTN C— CM t— LTN rH VO t—-=h -4" LTN t~- O KN-4" 
 LrNCMb-rHVOCOHrHLTN. ONVOVOVOKNOLfN 
 KNKNVOVO hCNVO CM CM rH -4" VQ ltn.-* On LTN LTN LT\ 
 D—C0 ONHVOVO CMC0VO O -4~ -4" t-HVO b O CO 
 VO -4" 00 00 LTN b-VO CM -4" -4" OJ b-H-^-COCMONHOO 
 CM b- LTN Lf\00 -4" CM KNb-KNCM-4-OOVOVO ONVO LTN ON LTN 
 O CO CO ON H U~\ O VO N~\ CM CM K"\ LTN ON-* O 00 t— t— ON 
 b-VO ONLTNUAb-KNH KN00VO t— rH CO ON KN ON ON CM CO 
 
 MJ LJ IT— \SJ ON LfA UN L~— IH \ r-1 N N UU VU ^ H OJ UN K N UA UN CM CD 
 b-VO-4-K"NCMCMCMCM KN-4" LfNVOCO O KNLTNCO CM LTN ON -4" CO 
 
 KN CM H. O ONCO t— VO IA-4 KN CJ H O O ONCO t>- l>-VO° lt\ Lf\-4- 
 CMCMCMCMCMHHHrHHHrHHrH,HHOOOOOOOO 
 + + + + + + + + + + + + + + + + + + + + + + + + 
 
 B 
 
 OJ OCO^OJ- 
 H H 
 
 CM ONt--Lr\KNrHONl>-LrNKNHONl>- LTNKN. H ON t— LT\ KN H ON l>- LTN rfN 
 
 C- — C — C — t — 0-VO VOVOVOVO Lr\LrNLr\LfNLrN^-_4-^J--^--d-KNhTNKNK-N 
 
 -p 
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 ^ OOVQ_=t-|>-00 ^-" 
 
 VO KN-3- CO 04- 
 
 K LTN Ox KN f- O VO ^ 
 
 CM -4" CM KN LTN O -4" LTN 
 
 II J-KNWHHO II 
 
 _ oooooo _ 
 
 ^ oooooo ** 
 
 ca + + + + + + ca 
 
 VO W CM t— VO CM ON-=h hhOlrlO l- -• 
 
 LTnO UAb-ONO ON t— CM VO LPvLTNONONrH 
 
 KNKNOCO CM b- b- CM -* b-COVO b-LTNONON 
 
 LTNrH KNVOVOVOCO KNVO OnltnONCM LTnOO t~- H 
 
 H K~\C0VO ONONCOCO KNLr\ONr-fVO-=J- KNb-hTNKN 
 
 I>-C0 CM ONO-4- CM-3- O HVOVO KNltnltnKNO ON H 
 
 LT\t--KN_ = t-cOCO H ONHCO ON -* -=fr ONCO CM H LTNKNCO 
 
 -4* ONVO VOaOCMCNONOLfAr-IOCMVOCM H KNbKNCVI KN 
 
 O ONH t-vOCO-4 OJ4- OCO O lAKN-4 ONbCO CM O rl LA 
 
 COVO KNCMO O\C0c0c0C0OnONHCM-4-V000 H-*C0 CMVOO 
 
 VO -4" KN CM r-\ O C0 , h-VO LTN -4- t<\ CM CM H O ONOO 00 t^-VO VO LTN lA 
 
 CMCMCMCMCVJCMHrHi-lHrHHHHHHOOOOOOOO 
 + + + + + + + + + + + + + + + + + + + + + + + + 
 
 CM O 00 VO -4" CM 
 
 rH rH 
 
 ONL^-LTNrArHONl^-LrNKNH ONb-LTNKNH ONb-LTNKNrHONt— LTnKN 
 tr— b-l>-t^-L>-vovovovovo Lf\Lr\LrvurvirN-=i--=J--=t--=fr-4- hCNKNiOvKN 
 
-p 
 
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 o 
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 3 COiA-*H-*D---OJOJONKNONr--tOOLrN-*KN 
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 1> LTN t— NA^aicOLTNVOLfAOJN^I^-KAaxOXO- 
 
 <i) CO KN O -3- O LT\ ON CO -=1- KN LT\ O -* O- OJ 00 
 
 v-' ONOOI^O O IAI--0AJ- ^4" O HCO h4 
 
 r-|0\OJO^|-UAVOOKNOJLr\VOVOrHKNt— 
 
 VO WOHK>VOHOOCOOMD^4'4in-d--d' 
 
 H K>-4-cOLr\UAO\Lf>LrNO>UALrA0OH0JrHcO 
 
 VO r-H VO OJ CO -* HCO UAKN H ONCO VO -* H 
 
 44-KM<MM01(MrlHHrlddd6d 
 
 ** oooooooooooooooo 
 
 ca + + + + + + + + + + + + + + + + 
 
 CM O00V0 4 CJ O CO VO -* OJ O CO VO -* OJ 
 
 KMoai aiaiaiWHHriHH 
 
 19 
 
 +3 
 Pi 
 
 o 
 
 CD r-l 00 KNVO O OJ KN OJ t--* VO ON H OJ l>- H 
 
 3 '-^ -4 VO -4- LfN VO D--VO 0\HHV0H0JOI>-UA 
 
 Pi Tj VO t^VO o KAVO J-VO iAO\H IAN-4 O 1^- 
 
 °H t3 b- l>- IO fA H f- H iAIAKM3\a3 fOiVO HCO 
 
 -P O H O H OJ t—-* lAlAHHHWCOO LT\ <-\ 
 
 fl v_, vo [— f-_* 0\0 OJ t~- KN t— OACO O OWOVO 
 
 O vOrHr(l>a\Hl^OOHWWh-(MI^Lr\ 
 
 O VO OJh-KNaa\H-*Oa\OJCOHOJOJHKN 
 
 ^— ' H GOOVOLr\VOO!OOJVOLrAVOOOJOJOVO 
 KN ON-* O VO KA O D—-* OJ O ON t— LfN KN O 
 
 h 4KM<NSNGJ(M(MririHHddddd 
 
 H <=»< oooooooooooooooo 
 
 ca + + + + + + + + + + + + + + + + 
 
 a 
 
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 <d tAOJOJOJOJOJHHHHH 
 
 EH 
 
 •P 
 PI 
 O 
 
 o 
 
 ONGOKNOcOHOOK\r-iOOO\VOOKNON 
 
 "-» H t---* ON t— ON-* KM>-H NONHO C*- KN 
 
 t3 LT\OVOI^-ONONOJ-*-*KN r -)OOOiONVOLr\ 
 
 ^ 0AHKNOJc0Oih-r-IHc0Lr\-*0\c000LrN 
 
 O *COV0 lAKN* LtNVO O dVOVO b-KN* Ln 
 
 ^ ON O- H 00 ON O VO t— ONCO VO VO ON O t-VO 
 
 t- KNVO IXNKAOJOJONONOJHOOJOJHO- 
 
 K t— -* KNtAOOD ON KN OJ t— VO LT\-* H t~--* 
 
 LT\ CJ KN t>--* LT\00 LT\VO O t— CO H KN KN O VO 
 
 LAO LTNH b-KNO 0-LT\OJ O ONO-LfNKNO 
 
 **KNNNCvioJOJHHHH6dddd 
 <=* OOOOOOOOOOOOOOOO 
 ca. + + + + + + + + + + + + + + + + 
 
 H ON |>-- lt\ KN H ONb-iA^H On t-- lt\ KN H 
 KNOJOJOJOJOJHHHHH 
 
20 
 
 IOO\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 sno 
 
 80 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 70 
 60 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 
 FIGURE 2. : 
 
 EIGENVALUES OF SPHEROIDAL FUNCTION, 
 
 
 
 40 
 
 \\ 
 
 
 V., AS A F 
 
 JNCTION OF pi (ODD NUMBERED ORDERS) 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 30 
 
 
 
 v^ 
 
 
 
 
 
 
 
 
 
 
 ZQ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 9 
 
 a 
 
 7 
 
 t> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "\ 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 N^ x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 3 
 2 
 
 / 
 
 .9 
 
 ,3 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V? 
 
 9 
 
 
 
 
 
 
 
 
 
 sJ 
 
 ""v^f 
 
 
 
 
 
 
 
 
 v 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 "— 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 .6 
 
 .5 
 
 .4 
 
 V 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~ —. 
 
 .3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 -id 
 
 1 
 
 01 
 
 l 
 
 
 
 
 
 
 
 
 
 
 
 09 
 03 
 .07 
 .(%. 
 .05 
 
 ,09 
 .03 
 
 ,o?. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 & 
 
 IO 
 
 // 
 
 IZ 
 
 13 
 
 e>i 
 
21 
 
 'CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 90 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 93 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 -I 
 
 A 
 
 
 FIGURE 5. eigenvalues of spheroidal function, 
 
 
 
 \\ 
 
 
 V 
 
 AS FUNCTION OF 0| (EVEN NUMBERED ORDERS) 
 
 
 
 so 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 JO 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 U.' 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 !0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 & 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 
 
 
 
 
 
 
 
 
 N6 
 
 <J4 
 
 \fi 
 
 
 
 3 
 
 \ 
 
 
 
 
 \« 
 
 \8 
 
 \tf 
 
 \^ 
 
 
 
 
 
 2 
 
 / 
 
 ,9 
 
 \ 
 
 \ 2 
 
 
 \4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ,8 
 
 ,7 
 ,6 
 .5 
 
 .4 
 .3 
 
 
 
 
 
 
 
 
 
 
 
 
 ~^« 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 OS 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 _S£ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 it 2 
 "2 
 
 S 
 
 /£> 
 
 /2 
 
 /3 
 
 # 
 
22 
 
 1 
 2s2 
 
 3- THE SPHEROIDAL ANGLE FUNCTION, V /(l - v ) 
 
 The machine computation of the functions, V, , Is "based on Equation 
 (3)- From (k) and (5) it is clear that the coefficients, d , satisfy 
 the following three term recursion formula; 
 
 - d 
 
 (n - 2)(n - 1) 
 kn-2 "(2n - l)(2n - 3)~ + kn 
 
 (Pi)' 
 
 n(n + 1) (2n(n + l) - 3) 
 (pi) 2 ' (2n - l)(2n + 3) 
 
 - d 
 
 k,n+2 
 
 ( n + 2)fn;-4- 3) 
 J2n~+ 2)(2n + 5) 
 
 (8) 
 
 The scaled eigenvalues are determined by the process described in Section 
 2. Then, in principle, one can assign a value to the lowest order of co- 
 efficient and proceed to determine the higher orders. However, the form 
 of (8) is not in general suitable for numerical computation, in the upward 
 sense, since the error may -grow with each upward step. To cut down on 
 the error, Equation (8) is rewritten for computation in a downward continued 
 fraction scheme as follows : 
 
 n 
 
 n-2 
 
 Zir2 
 
 2n-3 
 
 n-1) 
 2n^lT 
 
 \ 
 
 n(n+l' 
 (Pi) 2 
 
 2n(n+l)-3) 
 2n-l)(2n+3T 
 
 (n+3)(n+2) n+2 
 
 T2H+37(2n+5) d 
 
 n 
 
 (9) 
 
 From the recursion formula written in this form, it is clear that if n 
 
 is taken large enough, the ratio d _/d may be assumed to be zero with 
 
 negligible error in the calculation of d /d _. In the actual program, 
 
 n n-2 
 
 the straight upward recursion form, (8), was employed to calculate d, , 
 1 < n < k, and the downward continued fraction form, (9), employed for 
 n > ko The coefficients were first generated on the assumption that 
 d . = 1, and then were normalized by dividing each by the quantity 
 
 K.K 
 
 I 
 
 n 
 
 kn 2n + 1 
 
This normalization sets the level of the functions so that the following 
 equation is satisfied; 
 
 - 1 v 2 r 1 
 
 ! -, | --*-_ dv = s d] J / (^) 2 dv ( 10 ) 
 
 /-I C 1 ~ v ) 
 
 n 
 
 since 
 
 , l x 2 a 2n(n + l) 
 (P: ) dv = ~ , -, ~ 
 
 x n 2n + 1 
 
 The machine generation of the associated Legendre polynomials was 
 "based on the straight upward use of the recurrence form 
 
 P Ll (x) = ^ ((2n + 1) x P^ (x) -, (n + l) P^ ( X )), (11 ) 
 
 1 1 2 2 
 
 noting that P =0, and P = (l - x ) = sin 0. A simplified flow chart 
 
 for the calculation of the V v functions appears as Figure 10 of reference 
 
 2. 1 
 
 P P 
 Tables IV to VII are tables of the function, V,/ (l - v ) , for values 
 
 of pi of it/2, 5> 8, and 12. In these tables, the arbitrary constant has 
 been set so that the functions satisfy Equation (10). Figures k to 15 
 are graphs of the lower ten orders of the functions to indicate their be- 
 havior with the parameter j3i . It is clear from the graphs that 
 
 P as k — *■ oo 
 (1-v-) 2 
 
 as has been noted previously by other workers. In fact, if pi. is not too 
 large, the normalized associated Legendre polynomials are very good approxi- 
 mations to the spheroidal angle functions. Figures k to 13 show the angular 
 
2k 
 
 80 30 
 
 (-u- = -cosQ) 
 
 FIGURE k. THE SPHEROIDAL ANGLE FUNCTION 
 OF ORDER ONE FOR DIFFERENT VALUES OF THE 
 PARAMETER pi. THE, COORDINATE IS THE 
 POLAR ANGLE' OF SPHERICAL COORDINATES. © 
 POINTS PLOTTED ARE THOSE OF NORMALISED 
 ASSOCIATED LEGENDRE POLYNOMIAL (pi=0). 
 
25 
 
 JO 
 
 20 
 
 70 
 
 SO 
 
 30 40 50 60 
 
 -*► 9 (^ = -cosG) 
 
 FIGURE 5. THE, SPHEROIDAL ANGLE FUNCTION 
 OF ORDER TWO FOR DIFFERENT VALUES OF TEE 
 PARAMETER pi. THE COORDINATE IS THE 
 POLAR ANGLE OF SPHERICAL COORDINATES. 
 POINTS PLOTTED ARE THOSE OF NORMALIZED 
 ASSOCIATED LEGENDRE POLYNOMIAL (pi=0). 
 
26 
 
 -l.oo 
 
^ 
 
 
 1$ 
 
 /z 
 
 ll 
 IP 
 .1. 
 
 -3 
 ,7 
 .4, 
 .5 
 
 27 
 
 4 2 
 
 t ' 
 
 -,? 
 
 - £ 
 
 -/£ 
 
 
 
 ... .. ,, ,. . — 
 
 / ' /TV 
 
 / i /A V s 
 
 /-\/2 
 
 
 
 \ V \ / 
 
 
 
 
 
 
 
 
 
 \ 1 
 
 
 
 
 \ 
 
 
 
 
 1 1 
 
 
 
 
 \ 
 
 
 
 
 1 1 1 
 
 
 
 
 \ 1 
 
 
 
 
 \ 1 
 
 
 
 
 \ 1 
 
 
 
 
 1 1 1 
 
 
 
 5 5 
 
 jo \ | 1 7 
 
 5 <j 
 
 
 
 v\\\ 
 
 
 FIGURE 7o 
 
 AIMVTTP TPTTft 
 
 TEE SPHERO] 
 
 DAL\ \ \ 1 
 
 YET? \ \ \ 1 
 
 
 FOUR FOR DIFFERENT VAI 
 OF THE PARAMETER SI. 
 COORDINATE IS THE PC 
 ANGLE OF SPHERICAL COC 
 
 ,UES 6 \ \ ' 
 
 THE \ A \ 
 )LAR \ \ \ 
 EDI- ?|\ y 
 
 
 NATES. PC 
 
 THOSE OF I 
 
 ATED LEGI 
 
 ( 
 
 JIMTS PLOTTET 
 IORMALIZED A£ 
 ]NDRE POLYNC& 
 
 Pi=o). 
 
 ) ARE V/ 
 SSOCI- 
 
 1IAL 
 
 iz. 
 
28 
 
 FIGURE 8. THE 
 SPHEROIDAL ANGLE 
 FUNCTION OF ORDER 
 FIVE FOR DIFFERENT 
 VALUES OF THE PARA- 
 METER Pi. THE 
 COORDINATE 9 IS 
 THE POLAR ANGLE 
 OF SPHERICAL COORDI- 
 NATES. POINTS _ 
 PLOTTED ARE THOSE 
 OF NORMALIZED 
 ASSOCIATED LEGENDRE 
 POLYNOMIAL (pl=0] 
 
 -1.0 
 
29 
 
 FIGURE 9. THE SPHEROIDAL ANGLE FUNCTION OF ORDER SIX FOR DIFFERENT 
 VALUES OF THE PARAMETER pi . THE COORDINATE 9 IS THE POLAR ANGLE OF 
 SPHERICAL COORDINATES . POINTS PLOTTED ARE THOSE OF NORMALIZED ASSOCI- 
 ATED LEGENDRE POLYNOMIAL fei=0) . 
 
30 
 
 2.0 
 
 -J.5 
 
 FIGURE 10. THE SPHEROIDAL ANGLE FUNCTION OF ORDER SEVEN FOR DIFFERENT 
 
 VALUES OF THE PARAMETER Bi . THE COORDINATE IS THE POLAR ANGLE OF 
 SPHERICAL COORDINATES. POINTS PLOTTED ARE THOSE OF NORMALIZED ASSOCI- 
 ATED LEGENDRE POLYNOMIAL (pi=0). 
 
31 
 
 -/.£ 
 
 FIGURE 11. THE SPHEROIDAL ANGLE FUNCTION OF ORDER EIGHT FOR DIFFERENT 
 VALUES OF THE PARAMETER Pi . THE COORDINATE © IS THE POLAR ANGLE OF 
 
 SPHERICAL COORDINATE. 
 
52 
 
 -A5 
 
 FIGURE 12. THE SPHEROIDAL ANGLE FUNCTION OF ORDER NINE FOR DIFFERENT 
 VALUES OF THE PARAMETER Pi . THE COORDINATE 9 IS THE POLAR ANGLE OF 
 
 SPHERICAL COORDINATES. 
 
33 
 
 -/.£ 
 
 FIGURE 13- THE SPHEROIDAL ANGLE FUNCTION OF ORDER TEN FOR DIFFERENT 
 VALUES OF THE PARAMETER pi . THE COORDINATE IS THE POLAR ANGLE OF 
 
 SPHERICAL COORDINATES. 
 
lh 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 (i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 II 
 
 -i6 
 
 10 / 
 II / 
 
 .56/ 
 
 
 \ II 
 
 rt)\ 
 •1 \ 
 
 
 o 
 
 / aoo y 
 
 in 
 
 
 
 10 
 
 I 
 
 
 
 10 
 O I" 
 
 
 O 
 1 
 
 Q 
 
 (T> 
 
 00 
 
 nS 
 
 10 
 
 n 
 
 CNJ 
 
 H 
 
 I 
 1 
 
 O 
 
 Ph 
 CQ 
 
 c3 
 
 B 
 
 H 
 B 
 O 
 
 < 
 
 cq 
 < 
 
 S 
 ©I 
 
 I 
 
 H 
 EH 
 O 
 
 Fn 
 
 O 
 
 P 
 H 
 O 
 
 PS 
 
 PL, 
 
 CQ 
 
 H 
 
 H 
 
35 
 
 - 
 < 
 
 '< 
 
 Q 
 M 
 
 C 
 
 EH 
 
 O 
 
 -— 
 
 -r 1 
 
 > 
 O 
 
 § 
 H 
 
 EH 
 
 o 
 
 - 
 H 
 
 o 
 
 - 
 
 LT\ 
 M 
 
56 
 
 Table IV . The spheroidal (angle) function V^(v) fur £! = it/2. The tables are 
 arranged into blocks, each of which contains several orders of the functions, 
 for a given argument. The first number in each block is $1 , presented to two 
 figures. The next number printed (to seven figures) is the argument, v = -cos 9. 
 The function values follow (printed to nine figures) and are arranged so that 
 the next five lines in each block list the ten lower odd orders (i.e., V]_, V*, 
 
 Vjq) and the last five lines give the ten lower even orders 
 
 (i.e., V 2 , V U , V 20 ). (+123 +02 = 12.3). Thus, V 15 (--99619+7) = 
 
 2.096^^115. The arguments are formed from approximately five degree increments 
 on 6. (0 - 5 (5°)90°) 
 
 +16 +01 
 
 +621211384 
 
 +534292^44 
 
 +116372356 
 
 +180638553 
 
 +23487 587 5 
 
 -151930291 
 
 -68350787 1 
 
 -132790025 
 
 -195339236 
 
 -2457 32173 
 
 +16 +01 - 
 +124468499 
 +9842547 46 
 +177 324206 
 +193628510 
 +1347 v 363 48 
 -300455230 
 -121421419 
 -1889814 50 
 -187 663303 
 -108874234 
 
 9961947 +00 
 
 -01 +265468287 +00 
 
 +00 +839542545 +00 
 
 +01 +149093273 +01 
 
 +01 +209644115 +01 
 
 +01 +*55*53031 +01 
 
 +00 -394139535 +00 
 
 +00 -100028983 +01 
 
 +01 -165100918 +01 
 
 +01 -222803150 +01 
 
 +01 -263331270 +01 
 
 9848078 +00 
 
 +00 +515773398 +00 
 
 +00 +142813015 +01 
 
 +01 +196147439 +01 
 
 +01 +174653307 +01 
 
 +01 +799250869 +00 
 
 +00 -747982028 +00 
 
 +01 -161710860 +01 
 
 +01 -198437650 +01 
 
 +01 -156874025 +01 
 
 +01 -487103370 +00 
 
 +16+01 - 
 +187227 923 
 +127969011 
 +156513106 
 +476020129 
 -104921902 
 -442138763 
 -147629090 
 -1407 59^80 
 -753590615 
 +131595844 
 
 96 592 
 +00 
 +01 
 +01 
 +00 
 +01 
 +00 
 +01 
 +01 
 -01 
 +01 
 
 58 +00 
 +736479643 
 +139676388 
 +116340912 
 -329795587 
 -149543606 
 -102539769 
 -162843370 
 -846466196 
 + 713061642 
 +1575527*6 
 
 +16 +01 - 
 +250505878 
 +137637591 
 +699055355 
 -103124951 
 -108599651 
 -573497525 
 -141883720 
 -255069613 
 +127786086 
 +7 35951388 
 
 9396926 +00 
 
 +00 +914607566 
 
 +01 +131291582 
 
 +00 -217123028 
 
 +01 -137374581 
 
 +01 -298562072 
 
 +00 -119823018 
 
 +01 -106486217 
 
 +00 +663227852 
 
 +01 +130726532 
 
 +00 -174210449 
 
 +00 
 
 +01' 
 
 +01 
 +00 
 +01 
 +01 
 +01 
 +00 
 +00 
 +01 
 
 +00 
 +01 
 +00 
 +01 
 +00 
 +01 
 +01 
 +00 
 +01 
 +00 
 
 +16 +01 - 
 +314293383 
 +126364123 
 -339277437 
 -112102022 
 +695193793 
 -691011033 
 -106791039 
 +806375-106 
 +798250453 
 -105854774 
 
 9063078 +00 
 
 +00 +103935708 +01 
 
 +01 +689292558 +00 
 
 +00 -112516612 +01 
 
 +01 -328118098 +00 
 
 +00 +122494834 +01 
 
 +00 -124939019 +01 
 
 +01 -191052733 +00 
 
 +00 +123701544 +01 
 
 +00 -202474395 +00 
 
 +01 -116336515 +01 
 
 +16 +01 
 
 +37 8 4 046 
 
 +966 3296 
 
 -IOO83OI 
 
 +3169985 
 
 +96/5835 
 
 -7911691 
 
 -5195206 
 
 +1136841 
 
 -5919991 
 
 -5454*76 
 
 -86602 
 91 +00 
 08 +00 
 33 +01 
 50 -01 
 39 +00 
 15 +00 
 41 +00 
 38 +01 
 81 +00 
 31 +00 
 
 54 +00 
 +110280398 
 -561331695 
 -964766215 
 +994407007 
 -2203881 17 
 -117491152 
 +613746130 
 +537567920 
 -11318U64 
 +583465327 
 
 +16 +01 -8191 
 
 +442602878 +00 
 +540633432 +00 
 -100391456 +01 
 +9840*0198 +00 
 -30U531559 +00 
 -870562391 +00 
 +848863*44 -01 
 +625851310 +00 
 -102787364 +01 
 +9426 7 3 46 7 +00 
 
 3*0 +00 
 
 +1100 53052 
 -675275851 
 -239909988 
 +701440223 
 -104455826 
 -984697676 
 +102*68573 
 -58 5661855 
 -122386520 
 +769437124 
 
 +16 +01 -7 
 
 +506284771 
 +54*47*669 
 -409706078 
 +6 9827 2929 
 -900003223 
 -926019176 
 +594511656 
 -271539043 
 -772751843 
 +414270912 
 
 6604 
 
 +00 
 
 -01 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 -01 
 
 +00 
 
 44 +00 
 + 10321334-7 
 -9/4537069 
 +825*51050 
 -5794*796* 
 +*64919945 
 -701777916 
 +901354348 
 -990587*67 
 +96525077* 
 -820138825 
 
 +01 
 -01 
 +00 
 +00 
 -01 
 +01 
 +00 
 +00 
 +01 
 +00 
 
 +01 
 +00 
 -01 
 +00 
 +01 
 +00 
 +01 
 +00 
 +00 
 +00 
 
 +01 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
57 
 
 TABLE IV 
 
 +16 +01 - 
 +568777597 
 -376668854 
 +370429033 
 -368059721 
 +366832666 
 -954779013 
 +887782764 
 -881801978 
 +879809099 
 -878867018 
 
 7071068 +00 
 
 +00 +901539339 
 
 +00 -879551853 
 
 +00 +877080512 
 
 +00 -876458813 
 
 +00 +876^60173 
 
 +00 -36002594,0 
 
 +00 +358594459 
 
 +00 -359457117 
 
 +00 +360134804 
 
 +00 -360618528 
 
 +16 +01 -64278 
 +629199478 +00 
 -704870355 +00 
 +859440500 +00 
 -912953764 +00 
 +857 208966 +00 
 -95^98672 +00 
 +90343623:> +00 
 -791041741 +00 
 +586261346 +00 
 -311568374 +0C 
 
 +16 +01 -57 357 
 +686515134 +00 
 -864623540 +00 
 +7847/3039 +00 
 -339413888 +00 
 -263895592 +00 
 -924465400 +00 
 +65574278:? +00 
 -117709523 +00 
 -472602228 +00 
 +84079674 5 +00 
 
 76 +00 
 +717054046 
 -457380149 
 +157939487 
 +159038910 
 -456539592 
 -469254391 
 -313652586 
 +587485223 
 -790529884 
 +898416807 
 
 64 +00 
 +491083610 
 +111740925 
 -649312936 
 +881301753 
 -700486520 
 +333484764 
 -782401551 
 +862078406 
 -P3S20 5001 
 -3727ii8951 
 
 +16 +01 - 
 +7 39576990 
 -83327 9981 
 +226/60833 
 +604253394 
 -82907 5712 
 -863798339 
 +229502414 
 +603756266 
 -829333574 
 +224261879 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 -03 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 +U0 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 -01 
 
 5000000 +00 
 
 +00 +239487813 
 
 +00 +602924659 
 
 +00 -829775785 
 
 +00 +224746300 
 
 +00 +604821758 
 
 +00 +601387315 
 
 +00 -830678407 
 
 +00 +225486071 
 
 +00 +604584742 
 
 +00 -828908432 
 
 +16 +01 -42261 
 +787180333 +00 
 -625536^43 +00 
 -4457 55208 +00 
 +77 5919941 +00 
 +178519681 +00 
 -77 3610561 +00 
 -244177 923 +00 
 +831822242 +00 
 -401571583 -01 
 -817706913 +00 
 
 83 +00 
 -19432237 5 
 +830467681 
 -256441359 
 -741952414 
 + 512358354 
 +770983732 
 -455814097 
 -615204330 
 +666908101 
 +384781446 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 -01 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 (continued) 
 
 
 
 +16 +01 -3420201 +00 
 
 
 +828129472 +00 
 
 -266262988 
 
 +00 
 
 -290627336 +00 
 
 +716053185 
 
 +00 
 
 -809419151 +00 
 
 +525530431 
 
 +00 
 
 +381186337 -02 
 
 -531552392 
 
 +00 
 
 +811026281 +00 
 
 -711457724 
 
 +00 
 
 -656106188 +00 
 
 +822613513 
 
 +00 
 
 -624930398 +00 
 
 +L368134 42 
 
 +00 
 
 +415731646 +00 
 
 -774739677 
 
 +00 
 
 +772147315 +00 
 
 -408816473 
 
 +00 
 
 -145641399 +00 
 
 +632087647 
 
 +00 
 
 +16 +01 -2588191 +00 
 
 
 +861310153 +00 
 
 -482023141 
 
 +00 
 
 +974462500 -01 
 
 + 31587060 7 
 
 +00 
 
 -646458133 +00 
 
 +804989588 
 
 +00 
 
 -748514995 +00 
 
 +491826460 
 
 +00 
 
 -103456002 +00 
 
 -312698294 
 
 +00 
 
 -514791168 +00 
 
 +752090645 
 
 +op 
 
 -802789110 +00 
 
 +640290 545 
 
 +00 
 
 -306587202 +00 
 
 -109422276 
 
 +00 
 
 +496465547 +00 
 
 -75087ki753 
 
 +00 
 
 +804423148 +00 
 
 -642668412 
 
 +00 
 
 +16 +01 -1736482 +00 
 
 
 +885762176 +00 
 
 -649779599 
 
 +00 
 
 +454864743 +00 
 
 -203555136 
 
 +00 
 
 -73^805473 -01 
 
 +342071262 
 
 +00 
 
 -569897^70 +00 
 
 +7^9261706 
 
 +00 
 
 -800861480 +00 
 
 +7/5999522 
 
 +00 
 
 -354384*02 +00 
 
 +571553065 
 
 +00 
 
 -729237548 +00 
 
 +800241135 
 
 +00 
 
 -77515456^ +00 
 
 +6 56738019 
 
 +00 
 
 -459150472 +00 
 
 +206156291 
 
 +00 
 
 +717675199 -01 
 
 -341118iiOO 
 
 +00 
 
 +16 +01 -871557 5 -01 
 
 
 +9007 45487 +00 
 
 -756162948 
 
 +00 
 
 +705144247 +00 
 
 -631577751 
 
 +00 
 
 +538087646 +00 
 
 -427921530 
 
 +00 
 
 +304583563 +00 
 
 -171889407 
 
 +00 
 
 +339047446 -01 
 
 + 105159037 
 
 +00 
 
 -180627002 +00 
 
 +3080 52002 
 
 +00 
 
 -430637266 +00 
 
 +540703271 
 
 +00 
 
 -634536444 +00 
 
 + 709185318 
 
 +00 
 
 -762340841 +00 
 
 +792367147 
 
 +00 
 
 -798339743 +00 
 
 +780069290 
 
 +00 
 
 +16 +01 -3929017 -08 
 
 
 +905793126 +00 
 
 -79260728^ 
 
 +00 
 
 +794940853 +00 
 
 -796226591 
 
 +00 
 
 +796836118 +00 
 
 -7 97164232 
 
 +00 
 
 +7 97O39900 +00 
 
 -797485650 
 
 +00 
 
 +7 97 57 1146 +00 
 
 -797631873 
 
 +00 
 
 -818455288 -08 
 
 +142133155 
 
 -07 
 
 -204419600 -07 
 
 +266942284 
 
 -07 
 
 -329541310 -07 
 
 +392175199 
 
 -07 
 
 -454828096 -07 
 
 +517492558 
 
 -07 
 
 -580164606 -07 
 
 +642841909 
 
 -07 
 
Table V . The spheroidal (angle) function V. (v) for Pi ■ 5. The tables are 
 arranged into blocks, each of which contains severax o -Tiers of the functions, for 
 a given argument. The first number in each block is fa , presented to two figures. 
 The next number printed (to seven figures) is the argument, v « -cos 0. The 
 function values follow (printed to nine figures) and are arranged so that the 
 next four lines in each block list the eight lower odd orders (i.e. V., V,, 
 v 15>) and " the l a8-t four lines give the eight lower even orders (i.e. , 
 
 V2 ' Vl *, V X 5). (+ 123 + 02 = 12.3). Thus, V g (-. 99619^7) - 1.099^2224. 
 
 The arguments are formed from approximately five degree increments on 0. 
 
 (6 - 5°(5°)90°) 
 
 +50 +01 -9961947 +00 
 
 
 +125537088 -01 
 
 +142153469 
 
 +00 
 
 +425647385 +00 
 
 +7 5686 3193 
 
 +00 
 
 +109942224 +01 
 
 +144116003 
 
 +01 
 
 +176902804 +01 
 
 +207009729 
 
 +01 
 
 -541659445 -01 
 
 -271996762 
 
 +00 
 
 -589016956 +00 
 
 -927463132 
 
 +00 
 
 -127119844 +01 
 
 -160765047 
 
 +01 
 
 -192369128 +01 
 
 -22067 7629 
 
 +01 
 
 +50 +01 -9848078 +00 
 
 
 +265232138 -01 
 
 +285897456 
 
 +00 
 
 +809641528 +00 
 
 +133148035 
 
 +01 
 
 +173666962 +01 
 
 + 197230 524 
 
 +01 
 
 +200340 544 +01 
 
 +181899737 
 
 +ol 
 
 -111670901 +U0 
 
 -533049002 
 
 +00 
 
 -108112496 +01 
 
 -155231071 
 
 +01 
 
 -187837101 +01 
 
 -201472056 
 
 +01 
 
 -193779267 +01 
 
 -164979564 
 
 +01 
 
 +50 +01 -9659258 +00 
 
 
 +435097396 -01 
 
 +431829620 
 
 +00 
 
 +111190524 +01 
 
 +158531235 
 
 +01 
 
 +165850399 +01 
 
 +131349740 
 
 +01 
 
 +6 38596179 +00 
 
 -191281398 
 
 +00 
 
 -175692292 +00 
 
 -770787373 
 
 +00 
 
 -13931 i486 +01 
 
 -167 560094 
 
 +01 
 
 -153515636 +01 
 
 -100795662 
 
 +01 
 
 -2298134 13 +00 
 
 +596688569 
 
 +00 
 
 +50 +01 -9396926 +00 
 
 
 +652150632 -01 
 
 +578569453 
 
 +00 
 
 +129648280 +01 
 
 + 145930705 
 
 +01 
 
 +937068339 +00 
 
 -159782331 
 
 -02 
 
 -911688532 +00 
 
 -137167 423 
 
 +01 
 
 -24901296 4 +00 
 
 -970433464 
 
 +00 
 
 -146900092 +01 
 
 -127314275 
 
 +01 
 
 -493974633 +00 
 
 +489646071 
 
 +00 
 
 +121763429 +01 
 
 +135646996 
 
 +01 
 
 +50 +01 -90630 
 +93568590P -01 
 +133570210 +01 
 -620755099 -01 
 -120169075 +01 
 -333657221 +00 
 -129228803 +01 
 +6071**082 +00 
 +937 3158 30 +00 
 
 +50 +01 -8660^ 
 +1306/9567 +00 
 +12159*791 +01 
 -853336319 +00 
 -174291119 +00 
 -430364530 +00 
 -894459643 +00 
 +113006276 +01 
 -424394842 +00 
 
 78 +00 
 +721839799 
 +993586381 
 -102317860 
 -493769103 
 -111*70736 
 -508719600 
 +123353327 
 -42427*006 
 
 54 +00 
 +853767382 
 +323558744 
 -108128957 
 +902290156 
 -118505162 
 +320569768 
 +72820320 6 
 -112913498 
 
 +50 +01 -8191 
 +17,8710472 +00 
 +94409*003 +00 
 -107112275 +01 
 +894159230 +00 
 -537919957 +00 
 -355909942 +00 
 +827 168937 +00 
 -106026204 +01 
 
 +50 +01 -7660 
 +23962767 5 +00 
 +552588250 +00 
 -659170059 +00 
 +831366064 +00 
 -65243^221 +00 
 +204528152 +00 
 -100447*62 -01 
 -265884426 +00 
 
 520 +00 
 
 +962661999 
 -351693263 
 -267568091 
 +82897 7756 
 -116444995 
 +894664884 
 -389137966 
 -289648424 
 
 444 +00 
 
 +103360901 
 -825767632 
 +6686U894 
 -426278939 
 -10417720 2 
 +100014044 
 -993629726 
 +911314520 
 
 +00 
 +00 
 +01 
 +00 
 +01 
 +00 
 +01 
 -01 
 
 +00 
 +00 
 +01 
 +00 
 +01 
 +00 
 +00 
 +01 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 +01 
 +00 
 +00 
 +00 
 
 +01 
 +00 
 +00 
 +00 
 
 +oi 
 
 +01 
 +00 
 +00 
 
TABLE V (continued) 
 
 39 
 
 +50 +01 -70710 
 +314820282 +00 
 +100415876 +00 
 +971323647 -01 
 -178214539 +00 
 -766764485 +00 
 +653271967 +00 
 -7 54347567 +00 
 +7 93553694 +00 
 
 68 +00 
 +105018770 +01 
 -952933515 +00 
 +94002*1822 +00 
 -928322925 +00 
 -817006452 +00 
 +62627 196<i K>0 
 -547184243 +00 
 +504327608 +00 
 
 +50 +01 -642787 
 +404611330 +00 
 -332760496 +00 
 +72863375Z +00 
 -891253680 +00 
 -870318440 +00 
 +876295979 +00 
 -878568184 +00 
 +706818851 +00 
 
 6 +00 
 
 +997444617 
 -70 5178739 
 + 3667 5ii90 5 
 -347898211 
 -5060 5027 7 
 -216238570 
 +417745617 
 -701769188 
 
 +50 +01 -57357 
 +507 7 33016 +00 
 -6605766/9 +00 
 +855937026 +00 
 -488725398 +00 
 -949566P02 +00 
 +815692441 +00 
 -328725245 +U0 
 -32807 9260 +00 
 
 64 +00 
 +865880669 
 -192750713 
 -492240819 
 +853134957 
 -143164028 
 -610720269 
 +876859351 
 -640 527872 
 
 +00 
 +00 
 +00 
 -Od 
 +00 
 -01 
 +00 
 +00 
 
 +U0 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +50 +01 
 
 +620888 54^ 
 
 -810547640 
 
 +421631628 
 
 +481470317 
 
 -9894 42550 
 
 +495512693 
 
 +441975396 
 
 -848465312 
 
 ■5000000 +00 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +50 +01 -42261 
 +7 38557263 +00 
 -746263244 +00 
 -270244147 +00 
 +808262674 +00 
 -975667742 +00 
 +237105336 -01 
 +821922556 +00 
 -162070193 +00 
 
 +655710332 
 +368664128 
 -84776^011 
 
 +349400598 
 +220853344 
 -840882571 
 + 377791861 
 +50 5867604 
 
 33 +00 
 +380136936 
 +740175^65 
 -394623625 
 -675631774 
 +525942409 
 -600996944 
 -502307562 
 +7ii2449634 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +50 +01 -3420201 +00 
 
 
 +853193123 +00 
 
 +6660^1323 
 
 -01 
 
 -4835698 7 +00 
 
 +761363080 
 
 +00 
 
 -749143257 +J0 
 
 +413639417 
 
 +00 
 
 +111996037 +00 
 
 -595561749 
 
 +00 
 
 -897696568 +00 
 
 +715015412 
 
 +00 
 
 -436596420 +00 
 
 -34 5316004 
 
 -01 
 
 +520299801 +00 
 
 -795501729 
 
 +00 
 
 +723379245 +00 
 
 -325782715 
 
 +00 
 
 +50 +01 -2583191 +00 
 
 
 +955901670 + 00 
 
 -246116889 
 
 +00 
 
 -930788697 -01 
 
 +428802831 
 
 +00 
 
 -6924010 55 +00 
 
 +796721565 
 
 +00 
 
 -702592439 +00 
 
 +427122523 
 
 +oo 
 
 -7 5166/170 +00 
 
 +748553690 
 
 +00 
 
 -721231271 +00 
 
 +532930623 
 
 +00 
 
 -199916P45 +00 
 
 -195521966 
 
 +00 
 
 +549733561 +00 
 
 -767452874 
 
 +00 
 
 +50 +01 -17 36 482 +00 
 
 
 +1037 56152 +01 
 
 -513414093 
 
 +00 
 
 +315u66890 +00 
 
 -100056 341 
 
 +00 
 
 -148631684 +00 
 
 +393373977 
 
 +00 
 
 -599930159 +00 
 
 +740609304 
 
 +00 
 
 -342501392 +00 
 
 +617<i31:>66 
 
 +00 
 
 -72610834 4 +00 
 
 +774605953 
 
 +00 
 
 -7^5282479 +00 
 
 +609132009 
 
 +00 
 
 -409*37 :>47 +U0 
 
 +159108341 
 
 +00 
 
 +50 +01 -8715; 
 
 >75 -01 
 
 
 +109021 146 +01 
 
 -693694164 
 
 +00 
 
 +6217 346 7 3 +00 
 
 -571327406 
 
 +00 
 
 +492492424 +00 
 
 -39097 7027 
 
 +00 
 
 +27o65 597 4 +o0 
 
 -142625482 
 
 +00 
 
 -284590534 +00 
 
 +347920542 
 
 +00 
 
 -450241354 +00 
 
 +552185897 
 
 +00 
 
 -6 40235163 +00 
 
 +710387092 
 
 +00 
 
 -759947285 +00 
 
 +78707*999 
 
 +00 
 
 +50 +01 -392901 
 +110840683 +01 
 +7o539o844 +00 
 +77 3034048 +00 
 +785238732 +00 
 -lo020^881 -07 
 -216921827 -07 
 -337619613 -07 
 -460788347 -07 
 
 7 -08 
 
 -757361247 +00 
 -759310031 +00 
 -780670616 +00 
 -788280102 +00 
 +162853378 -07 
 +276734113 -07 
 + 399039186 -07 
 + 522755726 -07 
 
40 
 
 Table VI The spheroidal (angle) function V\(v) for fit = 8. The tables are 
 arranged into blocks , each of which contain* several orders of the functions, 
 for a given argument. The first number in each block is f3i, presented to two 
 figures. The next number printed (to seven figures) is the argument, v = -cos 
 The function values follow (printed to nine figures) and are arranged so that 
 the next five lines in each block list the ten lower odd orders (i.e. V\, V*, 
 
 V-jq) and the last five lines give the ten lower even orders X^-' e 't 
 
 V*, V^, V 20 ). (+123 +02 = 12.3). Thus, V 15 (-. 99619^7) = 2.01867236. 
 
 Trie arguments are formed from approximately five degree increments on 8. 
 (e - 5 8 (5°)90°) 
 
 e. 
 
 +80 +01 -99619 
 +149935262 -02 
 +231502981 +00 
 +971940576 +00 
 +169665152 +U1 
 +2299397 22 +01 
 -939376433 -02 
 -399430516 +00 
 -116003674 +01 
 -186214754 +01 
 -242104281 +01 
 
 47 +00 
 +375040915 
 +588047467 
 +134427508 
 +201867236 
 +252864669 
 -107304021 
 -780825268 
 -152357201 
 
 ' -216486021 
 -262116230 
 
 -01 
 +00 
 +01 
 +01 
 +01 
 +00 
 +00 
 +01 
 +01 
 +01 
 
 +S0 +01 -98^807 
 -J5491 92968 -02 
 +464348522 +00 
 +162709086 +01 
 +206667650 +01 
 +158040468 +01 
 -210176259 -01 
 -77 2339885 +00 
 -182*178345 +01 
 -202959621 +01 
 -133297925 +Q1 
 
 8 +00 
 
 +807766669 
 
 +109123673 
 
 +196172015 
 
 +193397903 
 
 + 104781152 
 
 -222923977 
 
 -138153562 
 
 -204394341 
 
 -178274231 
 
 -733369894 
 
 -01 
 +01 
 +01 
 +01 
 +01 
 +00 
 +01 
 +01 
 +01 
 +00 
 
 +80 +01 -96592 
 +661840422 -02 
 +695656880 +00 
 +174904684 +01 
 +918712228 +00 
 -783929258 +00 
 -374940524 -01 
 -108717660 +01 
 -170888634 +01 
 -504684093 +00 
 +112971933 +01 
 
 58 +00 
 +135825257 
 +142852036 
 + 154525343 
 +622822250 
 -139159998 
 -353846266 
 -165588625 
 -12/449261 
 +377377684 
 +155242128 
 
 +00 
 +01 
 +01 
 -01 
 +01 
 +00 
 +01 
 +01 
 +00 
 +01 
 
 +80 +01 -9396926 +00 
 
 +118366/82 -01 
 +914328870 +00 
 +130128891 +01 
 -654511542 +00 
 -1302807 32 +01 
 -621956304 -01 
 -130486845 +01 
 -902552606 +00 
 +106239073 +01 
 +10 3428690 +CL1 
 
 +208845067 
 +153147252 
 +395671753 
 -132196114 
 -635453979 
 -503898472 
 -152700728 
 +147251610 
 +140420473 
 +158524311 
 
 +00 
 
 +0L 
 
 +00 
 
 +01 
 
 +00 
 
 +00 
 
 +01 
 
 +00 
 
 +01 
 
 +00 
 
 +80 +01 -90630 
 +20627 7011 -01 
 +109803192 +01 
 +459540327 +00 
 -12697 8838 +01 
 +323008238 +00 
 -994890226 -01 
 -138152338 +01 
 +177 178613 +00 
 +113376332 +01 
 -818381610 +00 
 
 78 +00 
 +305555907 
 +136089803 
 -7 47 378824 
 -764206921 
 +114633511 
 -671 125844 
 -102016119 
 +113530875 
 +242472384 
 -124432121 
 
 +00 
 +01 
 +00 
 +00 
 +01 
 +00 
 +01 
 +01 
 +00 
 +01 
 
 +80 +01 -8660254 +00 
 
 +352296*83 
 +121280046 
 -430613707 
 -524095983 
 +llo052523 
 -154704173 
 -127853434 
 +964182236 
 -997660334 
 -870471723 
 
 -01 
 +01 
 +00 
 +00 
 +01 
 +00 
 +01 
 +00 
 -01 
 +00 
 
 +449607402 
 +929295464 
 -116714524 
 +685921209 
 +363214316 
 -844562355 
 -279088007 
 +997309164 
 -1064515/7 
 +244489601 
 
 +00 
 +00 
 +01 
 +00 
 +00 
 +00 
 +00 
 +00 
 +01 
 +00 
 
 +80 +01 -8 
 
 +588-529315 
 
 +121759238 
 
 -981861990 
 
 +653885654 
 
 -840909599 
 
 -23357 5180 
 
 -9802487 38 
 
 +104 926268 
 
 -1031097 95 
 
 +694527995 
 
 191520 +00 
 
 -01 +5798S5923 +00 
 
 +01 +320324716 +00 
 
 +00 -6 58210672 +00 
 
 +00 +992630693 +00 
 
 -01 -103470174 +01 
 
 +00 -100139704 +01 
 
 +00 +4 54087337 +00 
 
 +01 -359824901 -04 
 
 +01 -562717787 +00 
 
 +00 +977716712 +00 
 
 +80 +01 -7 
 
 +957695555 
 
 +107581181 
 
 -948129909 
 
 +976269736 
 
 -100038980 
 
 -340858605 
 
 -515006925 
 
 +439172483 
 
 -567634679 
 
 +744015697 
 
 660444 +00 
 
 -01 +747002647 +00 
 
 +01 -309964135 +00 
 
 +00 +30 5972923 +00 
 
 +00 -129439135 +00 
 
 +01 -119870 542 +00 
 
 +00 -110652910 +01 
 
 +00 +909994418 +00 
 
 +00 -875990500 +00 
 
 +00 +75669634-0 +00 
 
 +00 ^561392002 +00 
 
TABLE J I (continued) 
 
 +80 +01 -7 
 
 +151105150 
 
 +773446344 
 
 -37/800663 
 
 +163284761 
 
 -412185582 
 
 -477 949309 
 
 +30497 5369 
 
 -411917819 
 
 +575395938 
 
 -657 792338 
 
 +80 +01 -6 
 
 +230131406 
 
 +338813489 
 
 +372182034 
 
 -766522631 
 
 +904602236 
 
 -639690030 
 
 +517424250 
 
 -880680996 
 
 +849962738 
 
 -624030253 
 
 0710 
 
 +06 
 
 +00 
 +00 
 +00 
 -01 
 +00 
 -01 
 +00 
 +00 
 +00 
 
 68 +00 
 +91003501* 
 -76567*50* 
 +912697775 
 -944333704 
 +949274112 
 -111714388 
 +905170522 
 -7 9977 3966 
 +716411647 
 -6 57352014 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +01 
 +00 
 +00 
 +00 
 +00 
 
 427876 +00 
 
 +00 +103553411 +01 
 
 +00 -880930841 +00 
 
 +00 +664695313 +00 
 
 +00 -278871992 +00 
 
 +00 -123233073 +00 
 
 +00 -994305904 +00 
 
 +00 +451521039 +00 
 
 +00 +851482042 -01 
 
 +00 -504201 30 5* +00 
 
 +00 +778074586 +00 
 
 +80 +01 
 
 +336915419 
 
 -146403334 
 
 +81992*066 
 
 -694390u34 
 
 +940512663 
 
 -811224299 
 
 +7 91911297 
 
 -624401512 
 
 -60177 2354 
 
 +671029626 
 
 +80 +01 ' - 
 +472394475 
 -557174487 
 +667 983^36 
 +241852433 
 -840413506 
 -966468384 
 +748909078 
 +127959509 
 -824882925 
 +494552537 
 
 57357 
 +00 
 +00 
 +00 
 +00 
 -01 
 +00 
 +00 
 +00 
 -01 
 +00 
 
 64 +00 
 +108130974 
 -605952256 
 -17385490* 
 +760769239 
 -840566161 
 -720803913 
 -209960429 
 +31049719* 
 -774964948 
 +266811915 
 
 500000 
 +00 
 +00 
 +00 
 +00 
 
 +oo 
 
 +00 
 +00 
 +00 
 ! +O0 
 +00 
 
 +00 
 
 +100671245 
 
 -623703461 
 
 -790288130 
 
 +538121472 
 
 +363482493 
 
 -319779209 
 
 -706134797 
 
 +595323557 
 
 +314203261 
 
 -847752138 
 
 +80 +01 -42261 
 +632294226+00 
 -7 62603358 +00 
 +350878393 -Ql 
 +8142760 50 +00 
 -984637121 -6l 
 -107010120+01 
 +396805969 +00 
 +728545862+00 
 -355955385 +00 
 -709248571 +00 
 
 83 +00 
 +788469909 
 +482940540 
 -586561414 
 -534999244 
 +676712873 
 +135832578 
 -739466007 
 -285127816 
 +786863761 
 +157*22246 
 
 +01 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +01 
 -01 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 ;+Q0 
 
 +80 +01 -3420 
 +8056824O1 +00 
 -682570635 +00 
 -595845769 +00 
 +28475724> +00 
 +8024*6092 +00 
 -108431415 +01 
 -117933507 +00 
 +65lo2o757 +00 
 +616965354 +00 
 -346806587 +00 
 
 *01 +00 
 
 +4371^*5968 
 +740130840 
 +*lo763695 
 -683161677 
 -576864050 
 +5334751** 
 -*90*459*5 
 -79133304* 
 -178134611 
 +7*88057*8 
 
 +80 +01 - 
 +975104889 
 -307888013 
 -729579603 
 -610005306 
 +712*39395 
 -979683606 
 -5575^7o08 
 -238190219 
 +625*93714 
 +7 50 5-31231 
 
 +80 +01 - 
 +111894828 
 +138294*65 
 -2607 90 726 
 
 -64 17 06 40 5 
 -733595623 
 -746992656 
 -70-*O4l 764 
 -658212400 
 -323507 34* 
 +179604892 
 
 +80 +01 
 +121 586949 
 +545827221 
 +418708130 
 +2*23o9097 
 -25/47 30 35 
 -405044763 
 --,79466675 
 -645 1614 Id 
 -7 5o734273 
 -77657 7*5* 
 
 +80 +01 - 
 +125011395 
 +705566380 
 +734823457 
 +765172139 
 +778071100 
 -187376777 
 -237397734 
 -349727854 
 -470358131 
 -592662492 
 
 +00 
 +00 
 +00 
 + 00 
 -01 
 +00 
 +00 
 -01 
 +00 
 +00 
 
 91 +00 
 +607 75492* 
 +56*49*734 
 +7 5637 5443 
 +31176544* 
 -444 5**533 
 +7 544*8903 
 +340*1*477 
 -3*9*1*759 
 -78*101880 
 -531*590 46 
 
 17 3648* +00 
 
 +01 -414901663 
 
 +00 +5*3415090 
 
 +00 +466710013 
 
 +00 +75*641050 
 
 +00 +/*7*68474 
 
 +00 +/*1*96866 
 
 +00 +7139*4911 
 
 +o0 +5*3775562 
 
 +00 +/96**3000 
 
 +00 -4*34658*6 
 
 87155 
 +01 " 
 +00 
 +00 
 +00 
 -01 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 39*90 
 +01 
 +00 
 +00 
 +00 
 +00 
 -07 
 -07 
 -07 
 -07 
 -07 
 
 75 -01 
 -72297 3226 
 -481065760 
 -330174244 
 -101364136 
 +155549430 
 +438131739 
 +564143491 
 +70934**16 
 +776504*76 
 +7 5364 767* 
 
 17 -08 
 -336105532 
 -706103849 
 -753538459 
 -772808*07 
 -781845053 
 +*09395483 
 +290599721 
 + 409771690 
 +531354511 
 +6542089*3 
 
 +00 
 +00 
 +00 
 +0o 
 +00 
 
 +oo 
 
 +00 
 +00 
 
 +oo 
 
 +00 
 
 -0* 
 +00 
 +00 
 +00 
 
 +00 
 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 -01 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 -01 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 
 +oo 
 
 -07 
 -07 
 -07 
 -07 
 -07 
 
42 
 
 Table VII- The spheroidal (angle) function V k (v) for ^i = 12. The tables are 
 arranged into blocks, each of which contains several orders of the functions, for 
 a given argument. The first number in each block is fit , presented to two figures. 
 The next number printed (to seven figures) is the argument, v = -cos 9. The 
 function values follow (printed to nine figures) and are arranged so that the next 
 
 six lines in each block list the twelve lower odd orders (i.e. V-^, V*, 
 
 VpO and the last six lines give the twelve lower even orders (i.e., V^ V^, 
 
 V 2i+ ). (+123 +02 = 12.3). Thus, V 2 (-.99619^7) = -.000502628725. 
 
 The arguments are formed from approximately five degree increments on 9. 
 (9 - 5 8 (5°)90°) 
 
 +12 +02 - 
 +507123580 
 +399064670 
 +646094743 
 +151572523 
 +221303702 
 +268234207 
 -502628725 
 -108358956 
 -875763364 
 -170804796 
 -235423606 
 -275708132 
 
 +12 +02 - 
 +170604348 
 +904300418 
 +120146532 
 +211098848 
 +185461326 
 +716591412 
 -133578360 
 -233691332 
 -1536 37196 
 -215142212 
 -163372408 
 -359764679 
 
 +12 +02 - 
 +419022548 
 +16336^317 
 +15657 5348 
 +144967934 
 -351760023 
 -159196834 
 -304026221 
 -390972292 
 -179576337 
 -107011212 
 +793382901 
 +162567442 
 
 9961947 +00 
 
 -04 +278768412 
 
 -01 +236936743 
 
 +00 +109919064 
 
 +01 +188898676 
 
 +01 +248010973 
 
 +01 +281333926 
 
 -05 -118174082 
 
 +00 -424251515 
 
 + 00 -131266647 
 
 +01 -205766366 
 
 +01 -258976540 
 
 +01 -285059977 
 
 984807 
 -0 5 
 -01 
 +01 
 +01 
 +01 
 +00 
 -02 
 +00 
 +01 
 +01 
 +01 
 +00 
 
 8 +00 
 
 +701439148 
 
 +487044677 
 
 +180391225 
 
 +211941 118 
 
 +136410846 
 
 -421505536 
 
 -281932098 
 
 -830906409 
 
 -199561908 
 
 -201873509 
 
 -105499973 
 
 +364017416 
 
 96592 
 -03 
 +00 
 +01 
 +01 
 +00 
 +01 
 -02 
 +00 
 +01 
 +01 
 
 +uo 
 +01 
 
 58 +00 
 +147847098 
 +754213582 
 +184322067 
 +617386633 
 -116277322 
 -153616265 
 -550131545 
 -118633471 
 -171868989 
 -130 54ii425 
 +143476363 
 +133244247 
 
 -QZ 
 +0Q 
 +01 
 +01 
 +01 
 +01 
 -01 
 +00 
 +01 
 +01 
 +01 
 +01 
 
 -02 
 +00 
 +01 
 +01 
 +01 
 
 -oz 
 
 -01 
 +00 
 +01 
 +01 
 +01 
 +00 
 
 -01 
 +00 
 +01 
 +00 
 +01 
 +01 
 
 -01 
 
 +01 
 +01 
 +00 
 +01 
 +01 
 
 + 12 +02 -"P396 
 +100oll942 -J2 
 +271708789 +00 
 + 163538961 +u,l 
 +353924252 - J 1 
 -1 43>60i59 +01 
 -84 8656986 -01 
 -667^24847 -U2 
 -588796194 +00 
 -15467-4208 +01 
 +552537139 +00 
 +132590765 +01 
 -431136047 +00 
 
 926 +00 
 
 +296792292 -ul 
 +102250413 +01 
 +118824387 +01 
 -103004-152 +01 
 -103023254 +01 
 +883553394 +U0 
 -100681050 +U0 
 -143009803 +01 
 -649175087 +00 
 + 1335597ii2 +01 
 +595520836 +00 
 -121372983 +01 
 
 +12 +02 - 
 +23612^7 7 7 
 +4 26 98 4 3U2 
 +1338624 91 
 -108155447 
 -239443^21 
 +124990571 
 -143166908 
 -820945009 
 -831848U4 
 +129480974 
 -363972762 
 -104177876 
 
 +12 +02 
 
 +542802619 
 
 +6321 16052 
 
 +6981165^6 
 
 -107182888 
 
 +109405217 
 
 -38986647 3 
 
 -298354987 
 
 -10546877 
 
 +102133298 
 
 +594704176 
 
 -112834422 
 
 +#00854211 
 
 90630 
 -02 
 +00 
 +01 
 +01 
 +00 
 +01 
 -01 
 +00 
 +00 
 +01 
 +00 
 +01 
 
 86602 
 -02 
 +00 
 +00 
 +01 
 +J1 
 +00 
 -01 
 +01 
 +00 
 +00 
 +01 
 +00 
 
 78 +00 
 +577438418 
 +124602676 
 +123508258 
 -118561184 
 +877299328 
 +617534284 
 -176665181 
 -147802079 
 +573243847 
 +799914618 
 -119315134 
 -686276121 
 
 54 +00 
 +108389559 
 +1346 50840 
 -812772811 
 +749574608 
 +808974979 
 -113965655 
 -296197630 
 -125014629 
 +116822785 
 -705040332 
 -243859787 
 + 104034 542 
 
 -01 
 +01 
 +00 
 +01 
 +00 
 +00 
 +00 
 +01 
 
 +00 
 
 +00 
 +01 
 -01 
 
 +00 
 +01 
 +00 
 
 -01 
 +j0 
 +01 
 +00 
 +01 
 +01 
 +00 
 +00 
 +01 
 

TABLE VII (continued) 
 
 4} 
 
 +12 +02 - 
 +120781193 
 +869342481 
 -109195977 
 -635924527 
 +482389911 
 -889507405 
 -598466/33 
 -122182684 
 +839820864 
 -644778797 
 +2094157 47 
 +355513038 
 
 8191520 +00 
 
 -01 +194351646 +00 
 
 +00 +123120519 +01 
 
 +0Q -108418207 +01 
 
 -01 +104510865 +01 
 
 +00 -806717872 +00 
 
 +00 +317301328 +00 
 
 -01 -469242107 +00 
 
 +01 -727797211 +00 
 
 +00 +7 56950616 +00 
 
 +00 -974728187 +00 
 
 +00 +106622071 +01 
 
 +00 -860957987 +00 
 
 +12 +02 -76604 
 +257927 462 -01 
 +108691628 +01 
 -758805179 +00 
 +87 5126493 +00 
 -85066/389 +00 
 +7 52715407 +00 
 -114435267 +00 
 -122532599 +01 
 +984059452 +00 
 -96 3484151 +00 
 +983021347 +00 
 -100050498 +Jl 
 
 +12 +02 -7 
 +524510836 
 +119555023 
 -9140227 34 
 +7 44453345 
 -546217178 
 +393119471 
 -206617 562 
 -9763127 47 
 +445933300 
 -378228593 
 -210395297 
 +3640371 19 
 
 0710 
 
 -01 
 
 +01 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 -01 
 
 +00 
 
 +00 
 
 44 +00 
 +329095833 
 +841580448 
 -535127326 
 + 492327715 
 -589159600 
 +734787540 
 -692157496 
 -199775940 
 -255010921 
 +252096 549 
 -113124073 
 ••964462118 
 
 68 +00 
 +519734182 
 +223618381 
 +383726684 
 -6 92210665 
 +8253920 55 
 -888063601 
 -933364989 
 +615696619 
 -908414018 
 +938976600 
 -899958083 
 +850428820 
 
 +12 +02 -6 
 
 +100862350 
 
 +109092407 
 
 -4677 46 322 
 
 -251994 330 
 
 +719948010 
 
 -898845561 
 
 -348965601 
 
 -462430481 
 
 -378824511 
 
 +845164366 
 
 -864934597 
 
 +631451541 
 
 42787 
 
 +00 
 
 +01 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 
 +00 
 +00 
 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 
 -01 
 
 +00 
 +00 
 +00 
 
 -01 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 
 +00 
 +00 
 
 6 +00 
 
 +754898431 
 
 -425592171 
 
 +867801537 
 
 -724430828 
 
 +323549633 
 
 +120318020 
 
 -1,12262456 
 
 +864047852 
 
 -557591411 
 
 +358883921 
 
 +471496880 
 
 -779732407 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +01 
 +00 
 +00 
 -02 
 +00 
 +00 
 
 +12 +02 - 
 +182267516 
 +711521679 
 +280080991 
 -843789290 
 +513285100 
 +200906163 
 -5460PO010 
 +185772274 
 -811009997 
 +4 57 26337 3 
 +31387 5230 
 -8192487 82 
 
 +12 +02 - 
 +307823292 
 +113175875 
 +745442170 
 -262651165 
 -682920762 
 +693385679 
 -783409006 
 +69827 o854 
 -4^76^4424 
 -581244230 
 +7 36584807 
 +9028600 30 
 
 +12 +02 - 
 +483532960 
 -496027 97 4 
 +5207 44661 
 +651133177 
 -4192575J1 
 -68226587 1 
 -101/82612 
 +79321957 
 +332848897 
 -6^8038956 
 -475276972 
 +685376461 
 
 +12 +02 - 
 +7 035854 46 
 -824809232 
 -189605288 
 +557028362 
 +702711127 
 -139639163 
 -117763544 
 +388416293 
 +7246571^6 
 +338432673 
 -556832616 
 -702189723 
 
 57357 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 + 00 
 
 64 +00 
 +991338569 
 -805145551 
 +461814932 
 +382455995 
 -849613483 
 +694925527 
 -115848610 
 +560733300 
 +384569131 
 -8 55462514 
 +599987444 
 +566537049 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +01 
 +00 
 +00 
 +00 
 +00 
 
 -01 
 
 5000000 +00 
 
 +00 +114938267 +01 
 
 +00 -684026377 +00 
 
 +00 -401843489 +00 
 
 +00 +777484524 +00 
 
 +00 +426307751 -02 
 
 +00 -780520120 +00 
 
 +00 -947729771 +00 
 
 +00 -110933141 +00 
 
 +00 +797004576 +00 
 
 +00 -109867982 +00 
 
 +00 -743150036 +00 
 
 -01 +6 53468083 +00 
 
 42261! 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +01 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 342020 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +01 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 !3 +00 
 
 +113064666 
 
 -115224838 
 
 -755949182 
 
 -182057057 
 
 +791464092 
 
 +137064317 
 
 -472567 497 
 
 -654151820 
 
 +173917450 
 
 +792134735 
 
 -142768339 
 
 -797332793 
 
 1 +00 
 
 +863591111 
 
 +508657880 
 
 -185581518 
 
 V7674740 53 
 
 -359606141 
 
 +592545939 
 
 + 151324185 
 
 -616034812 
 
 -647715678 
 
 +120110193 
 
 +787732360 
 
 +321253447 
 
 +ul 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
TABLE VII (continued) 
 
 kh 
 
 +12 +02 -25881 
 +945197872 +00 
 -67 435167 5 +00 
 -677495068 +00 
 -385021247 +00 
 +276544 598 +00 
 +761463740 +00 
 -118096219 +01 
 -265960217 +00 
 +279769290 +00 
 +715613183 +00 
 +639532957 +00 
 +20606487 2 -02 
 
 91 +00 
 +361148735 +00 
 +700316244 +00 
 +593806562 +00 
 +732055176 -01 
 -580970100 +00 
 -766336144 +00 
 +684038213 +00 
 -214223273 -01 
 -534373568 +00 
 -760339020 +00 
 -367864725 +00 
 +372330944 +00 
 
 +12 +02 -1 
 
 +116925918 
 
 -113136839 
 
 -428285760 
 
 -689702759 
 
 -737704587 
 
 -488552364 
 
 -970692571 
 
 -703691880 
 
 -481378644 
 
 -151463817 
 
 +304908012 
 
 +675747518 
 
 736482 +00 
 
 +01 -247804019 +00 
 
 +00 +291360801 +00 
 
 +00 +573338034 +00 
 
 +00 +749592802 +00 
 
 +00 +648942710 +00 
 
 +00 +271407780 +00 
 
 +00 +869508964 +00 
 
 +00 +574897491 +00 
 
 +00 +343488378 +00 
 
 +00 -735802629 -01 
 
 +00 -514380016 +00 
 
 +00 -767892038 +00 
 
 +12 +02 -871557 
 +132944123 +01 
 +5104 33923 +00 
 +288187 47 3 +00 
 +125260536 +00 
 -969598685 -01 
 -3327 5927 2 +00 
 -5507 34687 +00 
 -586114227 +00 
 -639606849 +00 
 -733427286 +00 
 -74400357 9 +00 
 -667 361131 +00 
 
 5 -01 
 
 -750464511 
 
 -367586440 
 
 -216599483 
 
 -185582241 
 
 +215713105 
 
 +443620267 
 
 + 5967 54630 
 
 +586868487 
 
 +695937522 
 
 +749654682 
 
 +716391499 
 
 +598120678 
 
 +00 
 +00 
 +00 
 -01 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 
 +12 +02 -39290 
 +138774469 +01 
 +780128561 +00 
 +671644234 +00 
 +725969273 +00 
 +7 53840977 +00 
 +768235277 +00 
 -258660837 -07 
 -306187076 -07 
 -36857 30 39 -07 
 -487100954 -07 
 -607217285 -07 
 *!■ 28 532492 -07 
 
 17,-08 
 -945584848 
 -683902939 
 -701582600 
 -7 42411*15 
 -762090813 
 -772931794 
 +296376286 
 +321232605 
 +4274237 73 
 +547OO7559 
 +667735440 
 +789570520 
 
 +00 
 +00 
 +00 
 +00 
 +00 
 +00 
 -07 
 -07 
 -07 
 -07 
 -07 
 -07 
 
^5 
 
 functions as functions of the polar angle, 9. These graphs are useful in 
 visualizing the distant fields of the various spheroidal modes. On the 
 graphs of the lower seven orders, points have "been plotted to indicate the 
 magnitudes of the corresponding normalized Legendre polynomial (pi = 0). 
 For the higher order functions, the points fall so nearly on the curves 
 for pi = it/ 2 that they were not plotted. Figures ik and 15 show the angu- 
 lar functions for pi = it/ 2 as a function of the coordinate, v. These graphs 
 are useful in visualizing the current distributions of the spheroidal modes 
 along the length of wire-like spheroids. 
 

k6 
 
 k. THE SPHEROIDAL RADIAL FUNCTIONS, U, . 
 
 ' ki 
 
 The computation of one of the two independent solutions to Equation 
 (l) is troublesome. The troublesome one is the solution which grows 
 indefinitely in the neighborhood of u = 1. All known series representa- 
 tions for the function converge very slowly in this latter neighborhood , 
 and hence some alternative scheme is highly desirable. Nonetheless, the 
 functions were calculated in this work by means of a series representation. 
 The reason for this can be understood from a consideration of the follow- 
 ing typical alternative: Given the eigenvalues and some initial conditions, 
 a numerical solution to (l) can be readily obtained by a machine. However, 
 the output of such a calculation consists of many function values of a • 
 function of a given order . Such a type of output is wasteful in many elec- 
 tromagnetic boundary value problems, since the formulations often require 
 many orders of the functions evaluated for only one or two arguments . 
 Thus, a computational system whose output is in the latter form may be 
 more economical, even if the convergence is very slow. Furthermore, the 
 series representation described below provides a convenient way to accom- 
 plish the normalization which is usually desired in electromagnetic radia- 
 tion problems — -namely, one which results in a convenient behavior of the 
 functions of infinite argument. In any case, the series can be used to 
 obtain function values for arguments as close to u = 1 as practicable, 
 and these values employed as initial conditions for a numerical solution 
 over the remainder of the range of interest. Thus, we consider a series 
 representation of the functions. 
 
 Equation (l) bears a strong resemblance to the radial differential 
 equation in spherical coordinates. Consequently, we make the assumption 
 that the prolate spheroidal functions can be expressed in a series of the 
 corresponding functions in the spherical coordinate system, 
 
 T ( X ^ T 
 
 n s 2' n+g 
 
 (where J 1 is the half integral order Bessel function), as follows 
 

vr 
 
 U k= S a kn J n^> (12) 
 
 n=-oo 
 
 If this series is substituted into (l), and the known properties of the 
 Bessel functions employed, it is possible (see reference 2, page 50) to 
 arrive at a recurrence form for the coefficients as follows; 
 
 k,n-2 V(2n - 3)(2n - 1) ,/ k,n ^ (fJ|) 2 - (gn . l)(gp - ?) j 
 
 + a, 
 
 (n + 2)(n + 3) 
 k,n+2 l*( 2n + 3)(2n + 5) 
 
 = (13) 
 
 A glance at the first term in (13) shows that the positive side of 
 
 the recurrence chain breaks off at n = 1 and at n= 2, while from the last 
 
 term it is clear that the negative side of the recurrence chain breaks off 
 
 at n = -2, and n = -3- As a result, the value of a corresponding to one 
 
 of these particular four values of n can be assigned, and on the basis of 
 
 this assignment, a positive (negative) side, odd (even) set of coefficients 
 
 can be determined. Furthermore, if it happens that a = a (or 
 
 a = a ), then the coefficients generated on the positive side from 
 
 a n _ (a n ,_) are the same as those generated on the negative side by a n ^ 
 k,l v k,2' & . & j k,-2 
 
 (a ), since in Equation (13) > n can be changed to - (m + l) without 
 k,-3 
 
 a change in the equation. Thus, suppose for example one of the coefficients, 
 say a , is assigned a value. Then, since Equation (l) is a differential 
 equation of second order, there is still one constant to be assigned. If 
 this second constant is set by putting a = (a assigned), the co- 
 efficients are all determined, and the series gives the spheroidal func- 
 tions of the first kind as follows ; 
 
 00 1 
 
 U kl - \ n V f n^ u >' {lk) 
 
 n=-l,0 
 

48 
 
 where the prime indicates that the sum is on odd (even) values of n only. 
 
 (The even set results from the assignment of a p , with a = 0.) On 
 
 the other hand, if a n _ (a, _) is assigned to fix one of the constants, 
 
 k,-2 k,-3 
 
 while a, _ (a, _ ) is put equal to zero to fix the other, the series is 
 k,2 k,l 
 
 u). (15) 
 
 In practice it is convenient to take advantage of the relationship 
 
 
 °>\ - 
 
 u n „ = 
 
 = 2 a, J 
 
 k2 
 
 kn n 
 
 
 -co 
 
 5 s = (-i)^ N - 
 
 -n v ' n-1 
 
 and so transform the series, (15 )> into a sum over positive values of n, 
 
 QD 
 
 t A 
 
 n=-l,0 
 
 in which the coefficients are identical to those in (l4) , 
 
 The series represented by Equation (lk) converges very nicely, but 
 that of Equation (l6) converges slowly in the neighborhood of u = 1. 
 Furthermore, the scaling of the computation of (l6) so as to remain within 
 
 machine range is a serious problem for small values of |3i since the range 
 
 /v 120 
 
 of the functions N may exceed 10 . Nevertheless, the representation is 
 
 useful for the reasons discussed earlier in this section. A very conven- 
 ient normalization of the functions is obtained by setting the arbitrary 
 constant (a and so on) in such a way that as (3iu — > ao 
 
 U, x = U. . - jU. 
 
 t i(Piu) — -> j k+1 e-^ iu , 
 
 k3 kl ° k2 £3iu-»-ao k XK ' piu->co 
 
 that is, the prolate spheroidal function is made asymptotically equal to 
 
 the corresponding order of spherical function. Since the coefficients 
 
 in the spherical function series for U, -, are exactly those determined from 
 

K 9 
 
 A 
 
 (13); and in view of the asymptotic form indicated above for 1 ; the con- 
 stant is set so that the following equation is satisfied : 
 
 ^ a. j x ' = 1. 
 n=-l,0 
 
 Since the sum is taken to include only odd (even) n, the normalization 
 can he accomplished in the machine by making a positive and changing the 
 
 K.K. 
 
 signs of alternate coefficients in the summation process . 
 
 A simplified flow chart for the machine calculation of these functions 
 appears as Figure 11 of reference 2. As was mentioned in the introduction, 
 the program calculated the derivatives of the functions along with the func- 
 tions (from the derivative series ). All of the required quantites except 
 the eigenvalues are generated by the machine. The coefficients, a , are 
 generated in a manner completely analogous to that described under Equa- 
 tions (8) and (9), i.e., Equation (13) is employed in a straight upward 
 fashion for the range 1 <; n < k, and a downward continued fractions form 
 analogous to (9) employed for the range n > k. The spherical Bessel func- 
 tions are generated using the following recurrence formulas : 
 
 J _ (x) = — — — J (x) - J , (x) (in a downward 
 n+1 v ' x n v ' n-1 v ' x . . , _ 
 
 continued fractions 
 
 representation) 
 
 a 2n + 1 A ^ 
 
 N -, (x) = — — — N (x) - N _ (x) (straight upward) 
 n+1 v x n v ' n-1 v ' v * 
 
 A 
 
 (x) = -f- J n (x) - J n+1 (x) 
 
 A 
 
 & JjL (x) = EJlA J (x) , J (x) 
 dx x ' x n v ' n+1 x ' 
 
 in which 
 
 A . . A 
 
 J^ (x) = N ., (x) = sin x 
 
 () (x) = N_ x (x) = si] 
 
 N^ (x) = -J n (x) = -cos x 
 
50 
 
 dU k2 
 The computation of the functions U and — — utilizes the full capacity 
 
 of the machine; consequently, special checks were built into the program 
 to indicate overflow and underflow, to indicate an insufficient number of 
 terms, and to give a measure of the accuracy. The latter was accomplished 
 by computing and printing the value of the Wronskian along with the values 
 of the function. (it is readily shown that the Wronskian is a constant 
 in this case, in fact equal to |3i.) A facsimile of a typical ILLIAC print 
 out is given in Table II (page 55) of reference 2. 
 
 Tables VIII-X of this report include values of many orders of the 
 functions for pi = 5> 8, and 12, respectively. Each table gives the func- 
 tion values for two values of u, namely u = 1.077 and u = 1.100. 
 

51 
 
 TABLE VIII 
 Prolate Spheroidal (Radial) Functions Which Satisfy Equation (l) 
 
 dU. 
 
 
 
 Pi = 5 
 
 u = I.0770 
 
 order 
 
 TT 
 
 dU kl 
 
 TT 
 
 k 
 
 u kl 
 
 du 
 
 U k2 
 
 1 
 
 . 62324 
 
 4.14536 
 
 .2470 
 
 3 
 
 . 409526 
 
 4.88735 
 
 .8576O 
 
 5 
 
 . 092916 
 
 1.51635 
 
 2.450 
 
 7 
 
 . 008039 
 
 .169277 
 
 - 18.5 
 
 9 
 
 .000376 
 
 . 009800 
 
 - 299 
 
 11 
 
 . 0000113 
 
 . 0003516 
 
 - 81.0 x 10 2 
 
 2 
 
 .560471 
 
 5.35611 
 
 . 52842 
 
 4 
 
 .225716 
 
 3 18500 
 
 - 1.332 
 
 6 
 
 .030017 
 
 •559543 
 
 - 5.95 
 
 8 
 
 . 001852 
 
 .043599 
 
 - 68.9 
 
 10 
 
 . 000068 
 
 .001957 
 
 - 147.0 x 10 2 
 
 12 
 
 . 0000017 
 
 .OOOO57 
 
 - 49.0 x 10 5 
 
 
 
 Pi = 5 
 
 u = 1.1000 
 
 
 u *l 
 
 dU kl 
 du 
 
 U k2 
 
 1 
 
 . 695066 
 
 2.137396 
 
 . 093085 
 
 3 
 
 . 5179119 
 
 4.527658 
 
 .8045937 
 
 5 
 
 .1298707 
 
 I.696II 
 
 2.1789 
 
 7 
 
 . 0124845 
 
 . 218432 
 
 - i4.io 
 
 9 
 
 .000652 
 
 . 014357 
 
 - 201.2 
 
 11 
 
 . 0000218 
 
 .000580 
 
 -4822. 
 
 2 
 
 . 670385 
 
 4.203466 
 
 . 428326 
 
 4 
 
 .300099 
 
 3.27645 
 
 1.26045 
 
 6 
 
 . 044191 
 
 .67414 
 
 - 4.887 
 
 8 
 
 003038 
 
 . 060025 
 
 - 49.35 
 
 10 
 
 .0001253 
 
 .003044 
 
 - 931.7 
 
 12 
 
 . 0000035 
 
 . 0001005 
 
 - 27.54 x 10 
 
 k2 
 du 
 
 6.38O 
 
 1.9745 
 
 13.83 
 
 233. 
 
 r 
 
 55-0 x 10 £ 
 1 
 19.0 x 10 
 
 3.8715 
 
 3.359 
 55.6 
 
 10.8 x 10' 
 31.0 x 10 : 
 13.0 x 10' 
 
 ^k2 
 du 
 
 6.90735 
 2 . 620287 
 
 10.044 
 
 153.8 
 
 324o. 
 
 ] 
 
 10.08 x 10 
 
 4.77219 
 2.89971 
 
 38060 
 
 67.08 
 
 17.3 x 10 
 64.0 x 10 
 

52 
 
 TABLE LX 
 
 Prolate Spheroidal (Radial) Functions Which Satisfy Equation (l) 
 
 Pi = 8 u = 1.0770 
 
 order 
 k 
 
 ' U kl 
 
 " u kl 
 du 
 
 U k2 
 
 1 
 
 . 463061 
 
 -7.42220 
 
 + . 44076 
 
 3 
 
 .746556 
 
 +2.000526 
 
 .035682 
 
 5 
 
 . 5966044 
 
 6.561799 
 
 - .779678 
 
 7 
 
 .18729 
 
 3.1888 
 
 - 1.881 
 
 9 
 
 .026514 
 
 . 6O352 
 
 - 8.265 
 
 11 
 
 .002226 
 
 . O63175 
 
 - 73.6 
 
 13 
 
 . 000127 
 
 . 004301 
 
 - 10.5 x 10 ; 
 
 15 
 
 . 0000053 
 
 . 000208 
 
 - 21 . x 10- 
 
 2 
 
 . 639174 
 
 -2.69708 
 
 + .24741 
 
 1* 
 
 .7395565 
 
 +5 . 466534 
 
 . 392008 
 
 6 
 
 .37689 
 
 5.3138 
 
 - 1.2003 
 
 8 
 
 . O76238 
 
 1.51764 
 
 - 3-542 
 
 10 
 
 . 0081125 
 
 .20759 
 
 - 23.0 
 
 12 
 
 .000555 
 
 . OI728 
 
 -264. 
 
 14 
 
 .0000269 
 
 .OOO983 
 
 - ^5.3 x 10' 
 
 16 
 
 9.74 x 10" T 
 
 4.08 x 10° 5 
 
 - 11.0 x 10 
 
 dU. 
 
 k2 
 
 4 
 
 
 du 
 
 
 10. 
 
 212 
 
 
 10. 
 
 .620 
 
 
 4.83380 
 
 10, 
 n 4 
 
 •69 
 
 
 J.J.H- , 
 15. 
 
 ,1 X 
 
 10 2 
 
 27, 
 
 ,6 x 
 
 ID? 
 
 68, 
 
 ,0 x 
 
 10* 
 
 11, 
 
 .473 
 
 
 7. 
 
 . 91982 
 
 4.304 
 
 
 34.43 
 
 
 398, 
 
 
 
 62. 
 
 ,0 x 
 
 10 2 
 
 •13 
 
 .2 x 
 
 10* 
 
 37. 
 
 .0 x 
 
 10 5 
 
TABLE LX (continued) 
 
 pi = 8 u = 1.100 
 
 53 
 
 dU. 
 
 dU 
 
 order 
 k 
 
 U kl 
 
 kl 
 
 du 
 
 
 U k2 
 
 _ k2 
 du 
 
 1 
 
 .251915 
 
 -10.5244 
 
 
 + . 624434 
 
 5.66977 
 
 3 
 
 ■7495995 
 
 - 1.624533 
 
 
 + . 206136 
 
 10.2257 
 
 5 
 
 .735^9176 
 
 + 5.^82276 
 
 
 - .65^6797 
 
 5.997166 
 
 7 
 
 .265150 
 
 +3.57151 
 
 
 - 1.67399 
 
 7.6233 
 
 9 
 
 . 04264l4 
 
 +8.0334 
 
 
 - 6.195 
 
 70.91 
 
 11 
 
 . 004048 
 
 .096883 
 
 
 - 47.5+ 
 
 83.84 
 
 13 
 15 
 
 17 
 
 .0002601 
 1.22 x 10" 5 
 4.35 x 10" 7 
 
 . 007501 
 4.09 x 10" 
 I.67 X 10" 
 
 4 
 
 ■5 
 
 -594.0 
 
 - 10.7 x 10 5 
 
 - 2.59 x 10 5 
 
 13.63 x 10 5 
 k 
 
 29.8 x 10 
 
 8.47 x 10 6 
 
 2 
 
 . 529^2 
 
 -6.5896 
 
 
 + .48564 
 
 9.O667 
 
 4 
 
 . 836299 
 
 2.95557 
 
 
 - .197499 
 
 8.86799 
 
 6 
 
 . ^99275 
 
 5.29894 
 
 
 - 1.1018 
 
 4.3293 
 
 8 
 
 .115110 
 
 1.86431 
 
 
 - 2.8992 
 
 • 
 
 22 . 544 
 
 10 
 
 .013881 
 
 . 29727 
 
 
 - 15-95 
 
 234.8 
 
 12 
 
 l4 
 
 16 
 18 
 
 .0010716 
 5.385 x 10~ 5 
 
 2.37 x 10 
 
 
 
 7.52 x 10 
 
 . 028296 
 
 .001823 
 
 8.53 x 10" 
 
 3.05 x 10" 
 
 •5 
 •6 
 
 -159.7 
 
 - 2.42 x 10 5 
 
 k 
 
 - 5.08 x 10 
 
 - 1.4l X 10 
 
 3249. 
 
 k 
 6.16 x 10 
 
 1.55 x 10 
 4.93 x 10 7 
 
5* 
 
 TABLE X 
 
 Prolate Spheroidal (Radial) Functions Which Satisfy Equation (l) 
 
 
 
 
 Pi = 12 u 
 
 = 1. 
 
 O77O 
 
 
 order 
 k 
 
 u kl 
 
 
 du 
 
 
 U k2 
 
 dU k2 
 du 
 
 1 
 
 - .429 
 
 
 -15.1 
 
 + 
 
 .455 
 
 - 11.9 
 
 3 
 
 + .030404 
 
 
 -17.436 
 
 + 
 
 . 68389 
 
 +- 2.4862 
 
 5 
 
 + .61802 
 
 
 - 8.08802 
 
 + 
 
 . 46167 
 
 13.376 
 
 7 
 
 + .8694125 
 
 
 + 4.96970 
 
 - 
 
 .306526 
 
 12.0503 
 
 9 
 
 + .1*8697 
 
 
 + 7.2438 
 
 - 
 
 1.201 j 
 
 6.776 
 
 11 
 
 .121*2 
 
 
 + 2.746 
 
 - 
 
 2.93 
 
 32.0 
 
 13 
 
 .01877 
 
 
 .53815 
 
 - 
 
 13-2 
 
 261. 
 
 15 
 
 . 00195 1* 
 
 
 .06804 
 
 - 
 
 99.7 
 
 2.67 x 10 5 
 
 1 
 
 17 
 
 1.51 x 10" 
 
 1* 
 
 . OO615 
 
 - 
 
 1.08 x 10 5 
 
 1 
 
 3.58 x 10 
 
 19 
 
 9.07 x 10* 
 
 •6 
 
 4.21 x 10" 
 
 - 
 
 1.56 x 10 
 
 6.0 x 10 5 
 
 2 
 
 - .23214 
 
 
 -17.793 
 
 + 
 
 . 61022 
 
 - 4.9212 
 
 4 
 
 + .3299966 
 
 
 -14.01197 
 
 + 
 
 . 64341 
 
 + 9.0445 
 
 6 
 
 + .823927 
 
 
 - 1.03704 
 
 + 
 
 .13193 
 
 + 14.399 
 
 8 
 
 .72909 
 
 
 + 7.8242 
 
 - 
 
 .76452 
 
 8.2544 
 
 10 
 
 .2661*1 
 
 
 + 4.9580 
 
 - 
 
 1.7701 
 
 12.10 
 
 12 
 
 .05092 
 
 
 1.296 
 
 - 
 
 5-7^6 
 
 89.4 
 
 14 
 
 . OO631 
 
 1 
 
 .2006 
 
 - 
 
 34.5 
 
 806. 
 
 1 
 
 16 
 
 5 062 x 10" 
 
 -i* 
 
 .0212 
 
 
 515- 
 
 9-5 x 10 
 
 18 
 
 20 
 
 3.81 X 10" 
 2,04 x 10" 
 
 ■5 
 -6 
 
 1.66 x 10 -5 
 
 -4 
 1.00 x 10 
 
 - 
 
 5.96 x 10 5 
 k 
 6.5-. x 10 
 
 1.43 x 10 5 
 2.7 x 10 
 
TABLE X (continued) 
 
 pi = 12 u = 1.100 
 
 55 
 
 du 
 
 order 
 k 
 
 U kl 
 
 
 " u kl 
 du 
 
 U k2 
 
 k2 
 du 
 
 1 
 
 - .654 
 
 
 - 4.08 
 
 + . 0991 
 
 - 17.7 
 
 3 
 
 - .354224 
 
 
 -15.17^9 
 
 + . 62583 
 
 - 7 . O67O 
 
 5 
 
 + .372902 
 
 
 -12.7192 
 
 + .70973 
 
 + .79725 
 
 7 
 
 + .93^839 
 
 
 + .724303 
 
 . OI778I 
 
 + 12.823 
 
 9 
 
 . 65226 
 
 
 + 7-0439 
 
 - 1.0427 
 
 7-1375 
 
 11 
 
 .19583 
 
 
 3.48464 
 
 - 2.3491 
 
 19.^77 
 
 13 
 
 . 03^200 
 
 
 .81498 
 
 - 8.751 
 
 142.3 
 
 15 
 
 . 004071 
 
 -4 
 
 •11955 
 
 - 56.47 
 
 12.89 x 10 2 
 
 4 
 1.52 x 10 
 
 17 
 
 3.574 x 10" 
 
 . 01236 
 
 - 5.32 x 10 
 
 19 
 
 21 
 
 2.43 x 10" 
 1.32 X 10" 
 
 •5 
 
 ■6 
 
 9.62 x 10" 
 5.87 x 10" 5 
 
 - 6.74 x 10 5 
 
 - 1.09 x 10 5 
 
 2.27 x 10 5 
 4.24 x 10 
 
 2 
 
 - 0570093 
 
 
 -IO.8856 
 
 + .385124 
 
 - 13.6954 
 
 4 
 
 - .022515 
 
 
 -15.8977 
 
 + .755952 
 
 + . 789^37 
 
 6 
 
 + .73534 
 
 
 - 6,4388 
 
 + . 44404 
 
 + 12.431 
 
 8 
 
 + .886367 
 
 
 + 5.75711 
 
 .55^568 
 
 + 9.93639 
 
 10 
 
 .38849 
 
 
 + 5.6212 
 
 - 1.5370 
 
 + 8.6492 
 
 12 
 
 .08646 
 
 
 1.8064 
 
 - 4.176 
 
 + 51.54 
 
 i4 
 
 . 01231 
 
 
 .3279 
 
 - 21.03 
 
 414.8 
 
 16 
 
 . 00125 
 
 
 .03997 
 
 -167. 
 
 4.28 x 10 5 
 
 4 
 5.71 x 10 
 
 9.6 x 10 5 
 
 18 
 
 20 
 
 9.59 x 10 = 
 5.80 x 10" 
 
 -5 
 -6 
 
 .00356 
 2.44 x 10 
 
 - 1.83 x 10 5 
 
 - 2.64 x 10 
 
 22 
 
 2.85 x 10' 
 
 -7 
 
 1.34 x 10" 5 
 
 - 4.8 x 10 5 
 
 2.0 x 10 T 
 
56 
 5. SUMMARY 
 
 Some of the results of a computation of prolate spheroidal wave func- 
 tions for a specific electromagnetic radiation problem have "been recorded. 
 A convenient method for machine computation of eigenvalues is described. 
 Numerical results in the form of test calculations, graphs, and tables of 
 eigenvalues for parameter (Bi ) values of it/ 2, it, 3it/2, 2jt, 12, 4jt, 5^, and 
 l6 are included, The method of calculation of the spheroidal angle func- 
 tions is described briefly and tables and graphs of many orders of these 
 functions, for B! of it/2, 5, 8, and 12 are presented. The resemblance of 
 the higher order spheroidal functions to the corresponding orders of normal- 
 ized associated Legendre polynomials is clearly indicated in the graphs. 
 A description of a method of calculation for the spheroidal radial func- 
 tions is presented together with a few numerical results. 
 
 This work demonstrates that, with the aid of a digital computer of 
 moderate speed and capacity, the computation of spheroidal wave functions 
 for electromagnetic theory need not consume excessive time, and hence that 
 detailed solutions of electromagnetic problems in spheroidal coordinate 
 systems can be obtained with the expenditure of only reasonable time and 
 effort. 
 

71 
 
 REFERENCES 
 
 1. Schelkenoff , S. A. Advanced Antenna Theory , Chap. 3; John Wiley and 
 Sons, Inc., New York, 1952. 
 
 2. Weeks, W. L. Dielectric Coated Spheroidal Radiators ., Univ. of Illinois 
 Antenna Laboratory, Tech. Report No. 3k (Contract AF33(6l6)-3220), 
 Sept. 1958. 
 
 3- Stratton, J. A., Morse, P. M. , Chu, L. J., Little, D. C. and Corbato, 
 F. J., Spheroidal Wave Functions . John Wiley and Sons, Inc., New York, 
 1956. 
 
 h. Flammer, C. Spheroidal Wave Functions , Stanford Univ. Press, Stanford, 
 Calif., 1957- 
 
ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 Contract AF3 3(6 16) -31.0 
 
 "Synthesis of Aperture Antennas," Technical Report No. 1, C.T„A„ Johnk 
 October, 1954, ™ 
 
 A Synthesis Method for Broad-band Antenna Impedance Matching Networks," 
 Technical Report No 2, Nicholas Yaru, 1 February 1955 „ 
 
 -The Asymmetrically Excited Spherical Antenna," Technical Report No. 3, 
 Robert C. Hansen, 30 April 1955 „ 
 
 "Analysis of an Airborne Homing System," Technical Report No. 4, Paul E„ 
 Mayes, 1 June 1955, (CONFIDENTIAL) 
 
 "Coupling of Antenna Elements to a Circular Surface Waveguide," Technical 
 Report No, 5, H. 3 E. King and R. H. DuHamel, 30 June 1955 
 
 "Input Impedance of a Spherical Ferrite Antenna wigh a Latitudinal Current," 
 Technical Report No„ 6, W„ L„ Weeks, 20 August 1955 „ 
 
 "Axially Excited Surface Wave Antennas," Technical Report No„ 7, D. E. Royal, 
 10 October 1955 
 
 "Homing Antennas for the F-86F Aircraft (450-2500mc), " Technical Report No. 8 
 P. E Mayes, R F t Hyneman, and R. C, Becker, 20 February 1957 „ (CONFIDENTIAL) 
 
 "Ground Screen Pattern Range," Technical Memorandum No. 1, Roger R Trapp, 
 10 July 1955„ ~~ ~ ~~~ ~~~~ ~~ 
 
 Contract AF33(616)-3220 
 
 "Effective Permeability of Spheroidal Shells," Technical Report No, 9, 
 E„ J„ Scott and R„ H,, DuHamel, 16 April 1956„ 
 
 "An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report 
 No„ 10, D, G„ Berry and J B„ Kreer, 10 May 1956„ 
 
 "'A Technique for Controlling the Radiation from Dielectric Rod Wavequides," 
 Technical Report No. 11, J. W„ Duncan and R H„ DuHamel, 15 July 1956, 
 
 'Directional Characteristics of a U-Shaped Slot Antenna," Technical Report 
 No, 12, Richard C Becker, 30 September 1956, 
 
 "Impedance of Ferrite Loop Antennas," Technical Report No,, 13, V. H„ Rumsey 
 and W„ L. Weeks, 15 October 1956, 
 
"Closely Spaced Transverse Slots in Rectangular Waveguide " Technical 
 Report No, 14, Richard F, Hyneman, 20 December 1956, 
 
 "Distributed Coupling to Surface Wave Antennas,' 5 Technical Report No, 15, 
 Ralph Richard Hodges, Jr,, 5 January 1957, 
 
 "The Characteristic Impedance of the Fin Antenna of Infinite Length," 
 Technical Report No, 1.6 , Robert L., Carrel, 15 January 1957, 
 
 "On the Estimation of Ferrite Loop Antenna Impedance," Technical Report 
 No, 17, Walter L„ Weeks, 10 April 1957. 
 
 "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line 
 Source Antenna," Technical Report No, 18 , Walter L, Weeks, 20 April 1957 , 
 
 "Broadband Logarithmically Periodic Antenna Structures," Technical Report 
 No, 19 , R, H. DuHamel and D. E, Isbell, 1 May 1957, 
 
 "Frequency Independent Antennas," Technical Report No, 20, V. H„ Rumsey, 
 25 October 1957, ~ 
 
 "The Equiangular Spiral Antenna," Technical Report No. 21 , J. D, Dyson, 
 15 September 1957, ~ 
 
 "Experimental Investigation of the Conical Spiral Antenna," Technical 
 Report No, 22, R„ L„ Carrel, 25 May 1957, 
 
 "Coupling Between a Parallel Plate Waveguide and a Surface Waveguide," 
 Technical Report No, 23, E.J, Scott, 10 August 1957, 
 
 "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," 
 Technical Report No, 24, J,,W. Duncan and R.H, DuHamel, August 1957, 
 
 "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
 Cross Section," Technical Report No, 25, Robert L. Carrel, August 1957. 
 
 "Cavity-Backed Slot Antennas," Technical Report No, 26 , R.J. Tector, 
 30 October 1957, 
 
 "Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical 
 Report No, 27, W,L, Weeks, 1 December 1957, ~~ 
 
 "Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," 
 Technical Report No, 28, W.L„ Weeks, 20 December 1957, 
 
 "Measuring the Capacitance per Unit Length of Biconical Structures of 
 
 Arbitrary Cross Section," Technical Report No. 29 , J.D, Dyson, 10 January 1958. 
 
Non-Planar Logarithmically Periodic Antenna Structure, " Technical Report 
 No. 30 , D.W Isbell, 20 February 1958„ 
 
 "Electromagnetic Fields in Rectangular Slots," Technical Report No, 31, 
 N„J„ Kuhn and P.E. Mast, 10 March 1958, 
 
 'The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder," 
 T echnical Report No. 32 , J.W., Duncan, 25 May 1958 „ 
 
 "A Unidirectional Equiangular Spiral Antenna," Technical Report No. 33 , 
 J.D. Dyson, 10 July 1958, 
 
 "Dielectric Coated Spheroidal Radiators," Technical Report No. 34 , 
 W.L. Weeks, 12 September 1958. 
 
 "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report 
 No. 35 , P„E„ Mast, 12 September 1958. ' "" 
 
 Contract AF33 (616)-6079 
 
 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," 
 Technical Report No. 36 , R.H. MacPhie, 30 April 1959. 
 
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