Studies in Radar Cross - Sections -II The Zeros of the Associated Legendre Functions Pmn (A/) of Non-Integral Degree by K. M. Siegel, D. M. Brown, H. E. Hunter, H. A. Alperin and C. W. Quillen Project MX-794 USAF Contract W33 -038-ac-14222 April 1951 2nd Printing, November 1953 Willow Run Research Center Engineering Research Institute University of Michigan UMM-82.

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-82 TABLE OF CONTENTS Section Page Preface iii Abstract iv Nomenclature v Introduction vi I Formulas 1 II Formulas for Q ( ) and Q ( ) When m Is an Integer 4 n n -m III A. The Zeros of P (') When A' Is Close to - 1 and n m Is an Integer 6 -m B. The Zeros of P (.') When A' Is Close to-1 and m n Is Not an Integer 7 IV Application of Macdonald's Formula 9 Appendix 16 References 19 Distribution 20 ii

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 PREFACE For several years the Willow Run Research Center has been interested. in calculating the radar cross-sections of many shapes. Many approximate answers to composite bodies have been obtained in utilizing the methods of physical and geometric optics due to R. C. Spencer (Ref. 1, 2, 3). In the case of a prolate spheroid, answers have been obtained by four different methods: electromagnetic theory, scalar waves, physical optics and geometric optics (Ref. 4). The radar cross-section of an ogive is of particular interest because it is a typical missile shape. Theoretically, it was expected that the radar cross-section of an ogive would approximate that of the ogive's tangent cone. Much analysis has been applied to this problem since 1946. On September 30, 1948, Hansen and Schiff presented an analytical theory for the scattering by a semiinfinite cone (Ref. 5). When put to use, this theory requires numerical solutions to certain functional equations. Carrus and Treuenfels at Massachusetts Institute of Technology set out to compute solutions to these functional equations. Their methods of attack and recorded values appeared in an unpublished Cambridge Research Laboratory report entitled "Tables of Roots and Incomplete Integrals of Associated Legendre Functions of Fractional Orders." These values were then used by the authors and other investigators to obtain numerical solutions to this and other scattering problems. This Carrus and Treuenfels report was recently published (Ref. 6). A review has been made of our scattering research program, and an attempt was made to find out what additional information was available in this field. This investigation showed that both theoretical and computational errors have been made in the published solutions to the cone problem. These computational errors originated in the Carrus and Treuenfels report and were copied by other investigators. This report presents a new method of calculating some of the values of interest. In the process some limits of mathematical interest are obtained. i iii

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN vUMM-82 ABSTRACT This paper derives (by a new method) an equation due to Macdonald for determining the zeros of the associated Legendre functions of order m and non-integral degree n when the argument is close to -1 (Ref. 7). A closed form solution is obtained for the values of Qnm ( ) and Qn-m ( i ) for l close to 1. Certain observations are made concerning errors in a recently published article (Ref. 6). L iv

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM- 82 I L -m n Qm (4 ) n E. 4 F (a, p; y; z) II II (x) II (x) n m EC NOMENCLATURE = the associated Legendre function of the first kind of degree n, order -m and argument. = the associated Legendre function of the second kind of degree n, order -m and argument. = usual polar coordinate angle = cos ( - ) cos e cos 4 = an angle such that 00< 4 < 15~ = Riemann hypergeometric function = Gaussian operator = x! for positive integral x =r (x + 1) where r (x + 1) is the well known G-amma function = a non-integral real number = a real number = approximately equals = the essential contribution of the function as the variable 4 approaches a value close to 1.. differs from 1 by (1 - cos 4). This latter value is at most 0.03407 for = = 15~. i v

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 INTRODUCTION In Part I are stated well-known equations involving the properties of spherical harmonics and the Riemann hypergeometric function. In Part II we use these equations to derive the values of Qnm (,u ) and. Qn-m ( ) when pi is close to 1. In Part III we derive, by a new method, a simple formula due to Macdonald for determining the zeros of either Pn-m ( I) or Pnm ( I') when * is sufficiently small, namely: ~n~~m+k+I (2m + k) 2m /2) n ""' m + k + I I (m m 1) II (k) tan /2) In Part IV we analyze the values obtained in reference 6 and show that some of these values disagree significantly with the exact values. vi

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 PART I FORMULAS For m > 0: m/2 -m n 1 II (m) (1 +- 4 1 + J F ( - n, n + 1; 1 + m; 1 - ) (1) (Ref. 8, p. 404) For m ~ 1, 2, 3, etc.: m ( P (i) = n1c 1 II (- m) (1 + )/ 1 - L F ( - n, n + 1; 1 - m; 1 -) 2 (2) (Ref. 8, p. 586) For m = 0, 1, 2, 3, etc.: m ( II 1 II (n + m) ( n m = 2m II (m) II (n - m) 2 m/2 ~ ) F (m - n, n + m + 1; 1- 4.L) 1 +m; - ) 2 (5) (Ref. 8, p. 586) For m an integer: m ( II (n + m) -m n II (n - m) n (4a) (Ref. 8, p. 205) and m II (n + m) Qn () II (n - m) n (4b) (Ref. 8, p. 196) Combining (3) and (4a) we obtain: -m 1 n ) =2m II (m) (1 - 2 F/ ( 1 - 91 m - n, n + m + 1; 1 + m; 2 (5) 1

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-82 P (- i4) =cos (n + m) it P ( ) - n n 2 sin (n + m)it it Qn ( ) (6) m ( ) = n 2 sin (met) m( (Ref. 8, p. 407) cos mt - II (n + nl) -m ( } (6a) II (, p.-) n) (Ref. 8, p. 230) a b z a (a + 1) b (b + 1) F (a, b; c; z) = 1 + c ( ) + c +' c (c +1) 2 +. 21 (7) (Ref. 9, p. 7) lim F (a, b; c; z) = 1 = E C z 0 F (a, b; c; 1 - 4 ) (8) If a + b - c <0 lim F (a, b; c; ) = fL+ 1 r (c)r (c - a - b) r (c - a) r (c - b) II (c - 1) II (c - a - b - 1) II (c - a - 1) II (c - b - 1) (9) (Ref. 9, p.8) = E C F(a, b; c; i ) For m > 0, from (1) and (8): -m 1 EC P ( ) II () n II (m) (1 - r/ 1 + c (10) For m I 1, 2, 3, etc., from (2) and (8) m/2 EC pm ( ) 1 /II (- ) m For m = 0, 1, 2, 3, etc., from (3) and (8) m 1 II (n + m) ( /2 E P ()2m II () II (n - m) (10a) (11) 2

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ______________ UMM- 82 For m = 0, 1, 2, 3, etc., from (5) and (8): -m 1 ( 2m II (m) (12) it It sin t z = r(z) r(1 - z) II (z - ) II (- z) (13) (Ref. 9, p. 1) 3

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN TU.M-82 PART II FORMULAS FOR Q ( ) AND Q ( ) DERIVED WHEN m IS AN INTEGER, n n.. n IS NOT AN INTEGER AND i IS CLOSE TO 1 In this section, m will be considered an integer. Replacing m by -m in (6) we obtain: _-m.., -m. 2 sin (n- m)t -m P (-i ) = cos (n - m) t P ( ) - (Q ( ) in n i n Simplifying, (14)* -Mn -mn P (- L ) = (- 1) Cm os t P ( L) - n n 2 (- l) sin ) (15) it n Replacing, by - in equation (1) we obtain: (-m 1 1 + - 1 + n - 2 (16) Using equation (9) M/2 ECP -m m/2 II (m - 1) n ) - II (m + n) II (m - n - 1) Combining (15, (17) and (10) we obtain m/2 IT (n-i) n1) - ( 1 - ~~L cos nxit (17) I - I -m ECQ (.) = n II (m + n) II (m - n - 1) 2 (- l) sin n it II (m) \1 + u/, sin ni IK (18) *This equation appears incorrectly in reference 7. 4

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN We observe that the second term is small compared to the first term. Therefore, -m II (m - ) cotm (I/2) n ( )- 2 II (m + n) II (m - n - 1) (- 1)m + 1 sin n n (19) Using the equation (13), equation (19) becomes -m (I ) (- 1 II (m - 1) II (n- 1) II (- n) cotm(/2) n 2 II (m + n) II (m - n - 1) Now using (4b) we have (20) m Q n ()- 1m + +II (m - 1) II (n- 1) II (- n) m -'-)~2 II (n - m) II (: - n - 1) co /2) (21) Noting from (13) that sin (m - n)X IT (m_-_n__ 1) II(n__ m = (_ 1)m + II (m - n - 1) II (n - m) sin (nit) it and sin ni = II (n - 1) II (- n) — ( )m + then Q (, ) n and finally II (m - 1) sin (m - n)i cot (,/2) (22).v 2 sin (nr ) m ( II (m - 1) cotm/2 Q ( "' ) n-f -2 cot (/2) (23) By using equation (4b) -m ),, II (n - m) II (m- 1) cotm( /2) n 2 II (n + m)..... ~~~~~~~~~~ (24) 5

WILLOWR RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 PART III A. THE ZEROS OF P-m (') WHEN 1' IS CLOSE TO - 1 AND m IS AN INTEGER I _ _ ___In L Rewriting equation (14) and replacing -A by i' we obtain (- ) =P ( ) = cos (n - m) P () -- sin (n - m) Q ( n n t (14) (14) We wish to find the values of n such that -m P (A') =O n Upon division, equation (14) becomes n tan (n - m) it =- 2n (25) To find the value of the above expression when [I is close to - 1, (i.e., pi close to 1) and when m is an integer, we may make use of equations (10) and (24), yielding: 1 IT (m) (lI \ m/2 1 - M) 1 + J tan (n - m) it zz --- II (n - m) (26) II (m - 1) cotm(~ /2) - - II (n + m) and 1 t rT II (n + m) tan 2n(/2) 2 n m + k +arctan II (n- m) II (m) II (- 1) A~ I II (n+)tn m$2 (27) where k = 0, 1, 2, 3, etc. 6

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-82 But since the argument of the arctangent is small, we may write n- m + k (28a) and: arctan (x) - x (28b) Using (28a) and (28b), equation (27) now becomes: II (2m + k) tan 2m(/2) II (k) II (m - 1) II (m) (29) Equation (29) is Macdonald's formula as derived here for m an integer. We will now follow references 7 and 8 in deriving Macdonald's formula for non-integral m. B. THE ZEROS OF P ( i') WHEN i' IS CLOSE TO-1 AND m IS NOT AN INTEGER Using equations (6a) and (14) we obtain -m sin m it Pn ( ) tan (n - m) Et II (n - m) p m ( -m) _ cos m P () II (n + m () - cos m P ( ) II (n + m) n n When i is close to 1, by using equations (10) and (10a), equation (30) becomes: sin m i (1 -\ m/2 II (m) k1 +> ta-n (n - m) it %, II (n - m) 1 + 4 m/2 cos m it - mX/2 11 (m) 1 + / II (n + m) II (- m) (31) *It should be pointed out that equation (30) appears incorrectly in reference 8. OurT appears as - j in this reference. 7

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN m-__ 82 I The second term in the denominator is small while large. Thus, the first term is tan (n - m) I sin mn II (n.+ m) II (- m) t 2m ( /2) tan (n11 (m) II (n - m) (32) Using equation (13) then t II (n + m) tan 2m((/2) tan (n - m) it II (m - 1) II (m) II (n - m) (33) and 1 n _m + k + -- IC itII (m + n) tan 2m (/2) arctan II (m- 1) II (m) II (n- m) (34) But the arctangent term is small when ( is small, therefore n m+ k arctan (x) - x (35a) (35b) i and finally II (2m + k) tan 2m(,/2) II (m - 1) II (m) II (k) (36) Thus we have derived Macdonald's formula (without the use of Lagrange's theorem as suggested by references 7 and 8) for all m. It should be pointed out that for integral m m ( ) II (m + n) -m P ( ) =m) n III (n - m) n (4a) and that since II (- - ) has no zeros that the zeros of P ( t1') coII (n - m) n incide with the zeros of P ( T). n 8

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 PART IV APPLICATION OF MACDONALD'S FORMULA Macdonald's formula, n m +k+ II (2m + k) tan 2m ( II (m) II (m - 1) II (k) (/) (29) as we have seen, is approximate. It is of major interest in the back scattering from a cone to find the zeros of Pn ( i'). We will use a cone angle of 300 as the greatest angle to be treated by equation (29). If we analyze the case of axially symmetric back scattering, the angle * will be one-half the cone angle. Since the zeros of the associated Legendre functions for positive and negative integer m's coincide, we may use equation (29) to determine the zeros of either Pnm ( n') or Pnm ( I ). The zeros of p1 ( p') for a 30~ cone are given by (29) as n n^ 1 + k + (k + 2) (k + 1) tan2 (7.50) (38) where k = 0, 1, 2, 3, etc. The first zero would occur when k = 0. Thus by equation (38) we have nl- 1.03466 (39) Reference 6 states this value to be nl - 1.053 (40) The exact value of nl lies between 1.0321 and 1.0316. 1.0321 > nl > 1.0316 (41) 9

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 One of these values was obtained by summing the first 50 terms of the hypergeometric series associated with P 10321 (cos 165~) and comparing the remainder with two geometric progressions. One of these progressions was larger than the remainder, and one was less than the remainder. In this way we found for n = 1.0321 - 0.000045 > F (- 1.0321, 2.0321; 2; 1 - cos 165) > - 0.00027 (42) 2 We also computed the hypergeometric series for n = 1.0316 for 60 terms before summing and found that 1 - cos 1650 + 0.00000369 <F (- 1.0316, 2.0316; 2; - ) < 0.000125 (43) Since we have established a change in sign, we have proven a root exists between n = 1.0521 and n = 1.0316. This is the way in which equation (41) was obtained (for details of this computation see Appendix). Comparing equations (39), (40) and (41) we observe that Macdonald's formula yields a better result for the first zero than the Carrus and Treuenfels report. It should be pointed out that Macdonald's formula cannot be used to find all the zeros, even for angles between 12 i and t. This arises from the fact that in the derivation we replaced arctan x by x; this is only a good approximation when x<< 1. That is, xrII (2m + k) tan 2m (/2) II (m) II (m - 1) II (k) Working with m = 1 as previously, we find that 1 (k + 2) (k + 1) << - it2 (2(44) tan (/2) For the case previously under consideration, i.e.,, = 15~0 (k + 2) (k + 1) << 18.37 10

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 Thus, in this case we are permitted to use Macdonald's formula for at most the first three zeros. When x2 >> 1, the following series may be used for the arctangent: t 1 1 arctan (x) = i- - + 3 - 2 x 5x3 (45) It is also possible by improving upon many of the approximations in the derivation of the Macdonald formula to produce even better results. i 11

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN _______ UMM-82 OBSERVATIONS ON THE ARTICLE, "TABLES OF ROOTS AND INCOMPLETE INTEGRALS OF ASSOCIATED LEGENDRE FUNCTIONS OF FRACTIONAL ORDERS", BY P. A. CARRUS C. G. TREUENFELS (Ref. 6) Tables of Differences of Table 1, page 292, reference 6 Table 1 L. r L Zero No. e = 165 0 e = 170 o e = 175 O 2 - 1 1.030 0.99 1.01 3 - 2 1.067 1.03 1.02 4 - 3 1.075 1.04 1.01 5 - 4 1.079 1.046 1.04 6 - 5 1.081 1.49 1.00 7 - 6 1.082 1.051 1.00 8 - 7 1.085 1.050 1.02 9 - 8 1.085 1.053 1.022 10 - 9 1.087 1.053 1.025 11 - 10 1.087 1.054 1.024 12 - 11 1.088 1.055 1.024 13 - 12 1.089 1.055 1.024 14 - 13 1.088 1.055 1.024 15 - 14 1.089 1.056 1.025 16 - 15 1.089 1.057 1.025 17 - 16 1.090 1.056 1.025 18 - 17 1.089 1.057 1.025 19 - 18 1.090 1.057 1.025 20 - 19 1.090 1.057 1.026 21 - 20 1.090 1.057 1.026 22 - 21 1.090 1.058 1.026 23 - 22 1.090 1.057 1.026 24 - 23 1.090 1.058 1.027 25 - 24 1.090 1.058 1.027 26 - 25 1.090 1.058 1.026 i 12

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN l___ ______________UMM-82 Table 1 (Continued) Zero No. e = 165 o e = 170 o e = 175 o 27 - 26 1.091 1.058 1.027 28 - 27 1.090 1.058 1.027 29 - 28 1.090 1.058 1.027 30 - 29 1.091 1.058 1.027 31 - 30 1.090 1.058 1.027 32 - 31 1.091 1.058 1.028 33 - 32 1.090 1.058 1.027 34 - 33 1.091 1.058 1.028 35 - 34 1.090 1.059 1.027 356 - 35 1.091 1.058 1.028 37 - 36 1.090 1.058 1.027 38 - 37 1.091 1.059 1.028 39 - 38 1.091 1.058 1.028 40 - 39 1.090 1.058 1.027 41 - 40 1.091 1.059 1.029 42 - 41 1.09o 1.058 1.027 43 - 42 1.091 1.059 1.028 44 - 43 1.091 1.058 1.028 45 - 44 1.090 1.059 1.028 46 - 45 1.091 1.058 1.028 47 - 46 1.091 1.059 1.028 48 - 47 1.091 1.058 1.027 49 - 48 1.090 1.059 1.029 50 - 49 1.091 1.058 1.028 15

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 We had previously observed in equation (41) that the exact value of the first zero for 0 = 165~ was lower than is indicated in reference 6. 1.0321 > n > 1.0316 (41) Now it should be pointed out that for 165~ < G < 180~ the value of nl should decrease with increasing 0 (proven in reference 7). As a result, we observe that the first zero for 0 = 1700 cannot be correct, for it is greater than 1.0321 (Ref. 6, Table 1). We also note that one difference is less than an integer; this is impossible. In one case, the successive difference for 0 = constant decreases for increasing k by more than 1 in the last significant figure. This, too, is incorrect as it implies an error other than a mere rounding-off error. References 7 and 8 show that when n is large, the zeros are given by: 1 11 m TI (n + ) -- - (2k + 1) k+i 1 2 4 2 2 Successive differences in n are then given by: n -n it k + 2 k+ i- - Table 2 of Pn-m (cos 8) (46) (47) 0 nk + - nk 165~ 1.0909 1700 1.0588 175~ 1.0286 It should be pointed out that equation (46) should not be used when 0 is very close to it. Although, if it is used, one obtains Table 2. This table predicts most of the differences, obtained from reference 6 and listed in Table 1 of this report, to two decimal places. 14

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN..______________ UMM-82 Thus, if one wants to extend Table 1 of reference 6, he has merely to add to each successive zero the values listed in Table 2 on the preceding page. One may also note that the correct integer values for the associated functions may be obtained by observing the change in sign in the associated Legendre polynomials. An analysis of the associated Legendre polynomials, given in reference 10 for 0 values from 0~ to 90~ in intervals of five degrees, makes it possible (after multiplying through by (- 1)k) to check* all the values listed in Table 1 (Ref. 6) from n = 1 to n = 24 (the extent of the table in reference 10). Thus, any errors that exist in reference 6 should occur to the right of the decimal point, for one could know the exact integer values by examination of the signs of the polynomials. *Example: 1 sin 165~ 1 sin 1650 1 sin 1650 P13 (cos 165~) 6.50140 P14 (cos 165~) n-.86327 Pl5 (Cos 165~0) - 5.23851 Thus, one would expect a zero of Pn (cos 165~) to occur between n = 13 and n = 14, but none between n = 14 and n = 15 for e = 165~. 15

WILLOW, RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-82 APPEDIDX The hypergeometric series t, form Ao + Al + Ax2 +... where,n+1; 2;x) may be written in the - n(n+l) Ao=l, Al= 2-1' ('" A k (i-n-(i+n) k i=l i(i+l) Then: A k-l (k-i-l)(k+n) 1k(k+l) k+ n-+ 1) k(k+l 1- 2 k Let Sk= ji= Aixt 00 Rk= L i=k+l Then: Rk= Ak+l Xk + Ak+Z k+ + -Akl x 1 [l+ X+ Ak X+. ] L kAk+l XAk+l -A hl xk+I = A+l X 1 + Ak X+ Ak+l X+ riAx AX+43 Ak+z 2 2 Ak42 Ak+l1 ] = Alt+ Xk+l Since lim P= 0 and k.-oo lilm A = 1 we have for k-^oo Ak-l k>n < Ak+2 Ak+l Ah+3 Ak 2 < Ak+'+ ~~< 1 Ak+3 --- It follows that 1 1-AK+z Ak+l X< 1 1- x and hence R lies between two specific 16

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ~______________ {~ThUM-82 values which are determined by the sums of the two infinite geometric series. Hence, we show that S - Sk + Rk satisfies the relation a > S > where a and D are positive when n = 1.0316, and a and P are negative for n = 1.0321. On the next page we show a few lines of the form used in the actual computation for the case n = 1.0316. 1 - cos 1650.98296291 2 17

~ ~ ~ 0 (D 0 ( ~ (-n+k) (n+l+k) i(i+l) k -n+k n+l+k xQ (i=k+l) A = II B A x' A x 0 -1.0316 2.0316 -2.09579856 2 -1.04789928 -1.04789928 -1.03004613 1 -0.0o16 3.0316 -0.09579856 6 -0.01596643 0.01673121 0.01616596 2 0.9684 4.0316 3.90420144 12 0.32535012 0.00544350 0.00516999 3 1.9684 5.0316 9.90420144 20 0.49521007 0.00269568 0.00251661 4 2.9684 6.0316 17.90420144 30 0.59680671 0.00160880 0.00147634 H IOD Another procedure used on some of the computation is illustrated below: For n = 1.0316, (n + 1) 1.04789928 2 r r 0 a c 2z C) z 0 PI1 PICt o 50 C) O M) td *^ ~ ~ 0 0 0 ~ k(k+l) 2x. 2 k(k+l) x(l-Pk) Ak xk JAk xk (to be summed after 40 terms) 0 1.0000000000 1.0000000000 1 1.9829629131 -1.0500461289 -1.0500461289 2 3.3276545044 -.01569440537 0.01616596150 3 6.1638271522.3198071017 0.005169989294 4 10.09829629131.4867731350 0.002516611897 5 15.06553086087.5866388668 0.001476342351

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-82 REFERENCES No. Title 1 Spencer, R. C., "Back Scattering from Various Shaped Objects", Air Materiel Commandg Watson Laboratories, Cambridge Field Station, June 1946. 2 Spencer, R. C., "Back Scattering from Various Shaped Surfaces", Air Materiel Command Watson Laboratories, Cambridge Field Station, July 1946. 3 Spencer, R. C., "Reflections from Smooth Curved Surfaces", Massachusetts Institute of Technology Radiation Laboratory Report No. 661, January 1945. 4 Shultz, F. V., "Scattering by a Prolate Spheroid", UMM-42, Willow Run Research Center, University of Michigan, March 1950. 5 Hansen, W. W. and Schiff, L. I., "Theoretical Study of Electromagnetic Waves Scattered from Shaped Metal Surfaces", Quarterly Report No. 4, Microwave Laboratory Department of Physics, Stanford University, September 1948. 6 Carrus, P. A. and Treuenfels, C. G., "Tables of Roots and Incomplete Integrals of Associated Legendre Functions of Fractional Orders", Journal of Mathematics and Physics, Technology Press (M. I. T.), Vol. XXIX, No. 4. 7 Macdonald, H. M., "Zeros of the Spherical Harmonic Pnm ( ) Considered as a Function of n", Proceedings of the London Mathematical Society (1), Vol. XXXI (1900). 8 Hobson, E. W., "Spherical and Ellipsoidal Harmonics", Cambridge University Press, 1931. 9 Magnus and Oberhettinger, "Special Functions of Mathematical Physics", Chelsea Publishing Company, New York, 1949. 10 "Scattering from Spheres", Report No. 4, University of California, Department of Engineering. 19

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-82 DISTRIBUTION Distribution of this report is made in accordance with ANAF-G/M Mailing List No. 14, dated 15 January 1951, to include Part A, Part B and Part C. 20