DASA -1184 UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR, MICHIGAN Annual Report THE EFFECT OF ANISOTROPIC SCATTERING ON RADIATIVE TRANSFER AFSWP STUART W. CIMTRCHILL Professor of Chemical Engineering CHIAO-MIN,' CHIU Associate Professor ofElectrical Engineering JAMES A. IEACOCKJON CE N Assistants in Research' -:. -,.. DEPARTMENT OF THE NAVY, OFFICE OF NAVAL RESEARCH Contract No. Nonr-1224(17) ONR Project No. NR 087-063 March, 1960 Reproduction in whole or in part is permitted for any purpose of the United States Government.

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TABLE OF CONTENTS Page FOREWARD iii LIST OF FIGURES iv NOMENCLATURE v ABSTRACT viii INTRODUCTION 1 PART I - THEORETICAL STUDIES 3 A. The Transport Equation 3 1. The specific intensity 3 2. The distribution function fl(,Qa') 4 3. Boundary conditions and virtual sources 5 4. Green's function and the reciprocity theorem 5 5. Density, current and the integral equations 7 6. Expression of the transport equation in various coordinate 12 systems 7. General discussion of solutions of the transport equation 15 B. Exact Solutions for Isotropic Scattering 17 1. Infinite dispersion 17 2. Semi-infinite dispersion 20 3. A dispersion bounded by parallel planes 29 4. Some applications of exact solutions 34 C. Exact Solutions for Anisotropic Scattering 38 1. General 38 2. Semi-infinite dispersion 39 3. A dispersion bounded by parallel planes 47 4. Discussion of integral formulations 50 PART II - NUMERICAL CALCULATIONS 55 A. Mathematical Formulation for Half-space Problem 55 1. Geometry and coordinate systems 55 2. V-functions 55 3. H-functions 60 4. Intensity of reflected radiation 61 5. Integrated reflection 64 6. The general numerical problem 67

B. Specific Example 71 1. Representation of the phase function 71 2. The t-functions 75 3. The intensity of the reflected radiation 75 4. The integrated reflectance 81 86 SUMMARY AND CONCLUSIONS 86 APPENDIX A - Tables of computed functions 88 Index to tables in Appendix A 89 APPENDIX B - Computer programs 173 Index of contents of Appendix B 174 REFERENCES 226 DISTRIBUTION LIST l~~~~~i

FORWARD This report was typed by Mrs. Marguerite Schaible. Dr. George C. Clark, now with the Continental Oil Company, James R. Street and Larry B. Evans assisted in the early phases of the research. The machine computations were carried out on the IBM-704 at the University of Michigan Computing Center. iii

LIST OF FIGURES PAGE I-1 Representation of geometry for virtual sources 6 I-2 Representation of geometry for Green's function 8 I-3 Representation of geometry used in integrating 10 transport equation I-4 Rectangular coordinates 13 I-5 Cylindrical coordinates 14 I-6 Spherical coordinates 16 I-7 Plane parallel source obliquely incident on a semi 21 infinite dispersion I-8 A point source above a perfect specularly reflecting 36 boundary II-1 Geometry of half-space problem 56 II-2 Coordinate system 57 II-3 Angular distribution of radiation scattered by a 74 spherical particle for several phase functions II-4 Intensity of reflection as function of azimuthal angle, 77 I/Io vs4, parameters of Ai, N=4 and o=0o.4 II-5 Intensity of reflection as function of azimuthal angle, 78 I/Io vs 4, parameters of ol, N=4 and,=0.5 II-6 Intensity of reflection as function of incident angle, 79 I/Io vs,o parameters of 4i, N=4 andl =0~ II-7 Intensity of reflection as function of incident angle, 80 I/Io vs 40o parameters of ~t N=4 andq =1800 II-8 Intensity of reflection as function of emergent angle, 82 I/Io vs p, parameters of )o, N=4 and -=00 II-9 Intensity of reflection as function of emergent angle, 83 I/Io vs. Ai, parameters of, N=4 and ~o=0.4 II-10 Intensity of reflection as function of azimuthal angle, 84 I/Io vs Q, parameters of N, p. =0.4, p=0.5 II-11 Integrated reflectance (albedo of phase), 85 R vs Lo, parameters of N iv

NOMENCLATURE an angular distribution coefficients B(Z) distributed source inside the dispersion Cnm coefficient in spherical harmonics expansion D diameter of the scattering particle E1(Z) defined by eZ dz fl(~',~J)P) fl((~i If (@ )} fl = angular distribution function f(Q) pseudo angular distribution function

R point on boundary of dispersion R position of a source Rm(,) an integral transform of fm (Za) distance from measured along direction of S distance from R measured along direction of 52s

Y spatial angle between directions n and Q g,4) azimuthal and bearing angle representation of 7 nm Dirac delta function) 0 when n f m U)h l- )(+r! angular distribution coefficients for pseudo phase functions a specific direction of incidence on the dispersion s t0 direction of peaked radiation from a source at R nm(iR) components of density function optical thickness of dispersion bounded by parallel planes — (/u) characteristic function for H('I)

ABSTRACT The effect of anisotropic scattering on radiant transfer was investigated theoretically. Methods of obtaining exact solutions of the transport equation were reviewed, classified and generalized. Numerical calculations were carried out for a plane source obliquely incident on a dispersion in a half space. An angular distribution for single scattering containing up to four non-isotropic terms was utilized. The angular distribution of the radiation reflected by the half space and the albedo of the half space were computed.

INTRODUCTION The long range objective of this research is to develop methods for predicting the transmission of thermal radiation through the atmosphere. Theoretical and numerical results for a spherical source of finite volume both above and inside a haze were presented in an earlier report (1). The original objective of the current work was to investigate the influence of stratified layers of haze or clouds on radiant transfer from a spherical source. This objective was shifted during the course of the work as discussed below. The scattering of radiation by dispersed material presents a problem of considerable mathematical difficulty for any physical and geometrical conditions. Exact solutions exist only for highly idealized conditions; primarily for isotropic or Rayleigh scattering; and for a point source and a spherical dispersion or a plane source and parallel plane dispersion. Exact solutions for more complicated conditions are generally so involved as to be impractical for computational purposes. The practical importance of the problem and the analogous problem of neutron scattering by dense material has resulted in the development of a great number of approximate representations and solutions for complex conditions. Examples are the discrete flux representation and the diffusion representation (2). The lack of exact solutions for general conditions makes it difficult to compare and evaluate the various approximations which have been proposed and utilized. As a consequence of a general conference on radiant transfer (3) and subsequent discussions with Dr. S. Chandrasekhar and AFSWP personnel in which this situation was stressed, the immediate objective of the research was shifted to the development of exact solutions for more general conditions.

In particular, attention has been given to the development of exact solutions for non-isotropic scattering. In this report consideration is limited to a plane source oblique to a dispersion bounded by parallel planes or to a dispersion occupying the half space. The theoretical work follows the lead of Chandrasekhar (4)*. In the first part of the report the general mathematical problem is formulated and the various proposed methods of solution are discussed and generalized. In the second part of the report numerical results are presented for the half-space problem for different degrees of anisotropy and for a range of angles of incidence. In the continuation of this research it is planned to extend the theoretical solutions outlined in this report into forms more suitable for numerical computations. Numerical calculations for anisotropic scattering by finite dispersions are planned. The exact results for anisotropic scattering by finite and infinite dispersions will then be used to evaluate the various approximate models which have been proposed. * Hereafter referred to as R. T.

PART I - THEORETICAL STUDIES A. The Transport Equation 1. The Specific Intensity. The distribution of radiation scattered by a dispersion can be best expressed in terms of the specific intensity i(rC,) defined as the energy intercepted per unit time by a unit area at r per unit steradian in the direction Q normal to the area, where r is a vector representing the position coordinates and 2 is a unit vector representing direction. A steady state energy balance for monochromatic energy yields the transport equation: V;( n ( SL) - Q r,f, +% f) I-1 where::Q] = solid angle = unit direction vector r = position vector n = concentration of particles a = interception cross-section of a particle X = albedo for single scattering fi= distribution function for single scattering q = source in the dispersion In terms of the optical distance defined as R 1-2 Equation (I-1) may be expressed in the form t *,,R, t(R, (-a) I-3 -<.~~~~~~~~~~~

The solution of Eq. I-D, in terms of fl, q, and the geometry of dispersion is the objective of this investigation. 2. The Distribution Function f1(n22') The distribution function fl(1,1.) in most cases can be expressed as a function of the cosine of the angle 7 between 2 and a,, hence (f~, (A & ) = f, (costr') I-4 Expanding in terms of Legendre polynomials,' Q.) _ 1 [ I + - a,n f,(cs')]' I-5 where the a's are known as the angular distribution coefficients. Values n of these coefficients have been published for a wide range of values of the index of refraction and the ratio of the circumference of the particle to the wavelength. If n is represented by an azimuth angle ~ and a bearing angle 50 with respect to some orthogonal coordinate axes, and 2' is similarly represented by an Ai' and f', then by the addition theorem of Legendre polynomials: f, (at 4, a, 8X) 4ii-l — - t a[P. (Cosh) F', (CS,,,') + f,(n,,') fn,"): ( wheren "' ae n z-onc6 +ZZere ("-"M~mle p)!Y(. Symbolically, Eq. r-5 may be represented by: where Ynm(a) are normalized spherical harmonics.

3. Boundary Conditions and Virtual Sources. The solution of the transport equation in a region with sources outside the boundary may be reduced to an equivalent problem with no outside sources by considering only once scattered radiations and by replacing the outside sources by virtual sources inside the media. For example, consider the problem where radiation in the direction Qs is incident on the boundary of a dispersion at Rs as shown in Fig. I-1. The incident radiation scattered once by the dispersion may be considered as a virtual source at any point R = Rs + QsS with strength: — ~ - S -- - I-8 v ( (R. = Xea f,(-SL, s) where S is the distance from Rs measured along s'. The transport equation for radiation scattered at least once therefore becomes: f7 ~ SX 1C ROn CS RLn. wf - I-9 with no inward flux at the boundary. The advantages of using virtual sources are: (a) the solution of the inhomogeneous transport equation can in principle be formulated in terms of the familiar Green's function, and (b) the directional nature of the incident radiation is replaced by the more isotropic virtual sources. Hence i(R,a) can be represented more accurately by a finite number of spherical harmonics. Very peaked sources within the dispersion can similarly be replaced by distributed virtual sources. 4. Green's Function and the Reciprocity Theorem. In principle, when the incident radiation is replaced by virtual sources, the fundamental problem is to find the Green's function, i.e., the value of i(R4,) due to a source at R peaked in the direction of %O, which can be represented by a Dirac i-function, (RRo)O 6 (&O).

Dispersion \ i r- Rs Origin Figure I-1 Representation of Geometry for Virtual Sources

Referring to Fig. I-2, if the Green's function G(R,2,Ro0o) can be found, then the solution of any problem simply involves integration of G for the source configuration. However Green's function can be found only in very special cases. The following reciprocity theorem of the Green's function has been proven by Case (5): G(9R3,; Ro.O. - G(Ro,-.; 1I-10 provided that the distribution function of single scattering satisfies: This relationship will be utilized subsequently. 5. Density, Current and the Integral Equations. From Eq. I-9, it can be seen that if the distribution function is represented by a finite number of terms of spherical harmonics such as: ~~~~n:o,o~~~~~ P~-ll then, both qv and i can be represented by a finite series of spherical harmonics, such as: (Q, ) _ Eo Qn,(, t AI-12 and n JOOO~n -13 Since Y (S) is normalized: nm Qnm ('it) i%~ ~~~~~~~~~~~~~~~I-l14

Dispersion with no incident radiat/ion from outside A: source Bj point of observation Origin Figure I-2 Representation of Geometry for Green's Function reometry fr Green's Function

and /Vm (R) = r t (Q. SL) Xl/n (A) A I-15 The functions Qnm(R) and Pnm(R') may be called density functions. Evidently, Qoo (R) is the total strength of the source and poo(R) can be interpreted as the density of radiation. Introducing I-11, I-12, and I-13 into Eq. I-9 yields: h:o - o I-16 + Qnm,R)] Y,.n (,) Let S be the distance measured from the boundary along S as shown in Fig. I-3. Equation I-16 is then reduced to: dS,' E:, Po " I-17 + Qnm(W)],n_ (S') Integrating over the distance SS gives,,'.o. dS e Ynm (. ) ~~0 Ab-~ I-18.X,4 - - E.') Multiplying Eq. I-17 by Ypq(2) and integrating over the volume of the dispersion gives the following integral equations for the density functions: 1d () _ _ ___ R Ld | i [ 1-19 where ~~~~~11~~~~~~~~

Dispersion Rs/ Origin Figure I-3 Representation of Geometry Used in Integrating Transport Equation 10

Equation I-19 can be derived from I-18 by the following change of variables: therefore S. R' drr' =S, dS$ da therefore For isotropic scattering, all the Cnm are equal to zero except COO which equals 1/4Xt. Eq. I-19 then reduces to its more familar form: ( -4some probles f dsc(IR i the, n crrent.,L d )] by In some problems of scattering, the net current, defined by 3J() 1 jIL.(Q C~d L)c I-21 is of interest. Actually the components of J(R) can be seen in the development of i(R,2) to be qi (R). Hence the knowledge of J(R) does not solve the multiple scattering problem completely. However use of J(R), as for example, in the diffusion approximation, sometimes does yield reasonably good results. A divergence relation of J which is widely used in the diffusion approximation for the transport equation is obtained by direct integration of the transport equation: 7 rI (. Q + Q. ). 4Jf (,x.) fc(.ndn i8 I)''' I-3 ll

The result is: 1-22 V7 J() -,(R) + QO('R) 6. Expression of the Transport Equation in Various Coordinate Systems. In practical problems, the geometry of the dispersion necessitates the use of different coordinate systems. The explicit expression 1-23 can be expressed in rectangular, cylindrical, and spherical coordinates as follows: (i) Rectangular coordinates. R is represented by x, y and z, and Q is represented by 0 and O as shown in Fig. I-4. Eq. 1-23 then takes the form: 6in v0 co0 ~ -,- + Sine cosl + + + cos e (,z,,Z,,) e, ) + -24 + dJ ej I a(x,',z, 4, 2') f, (e, s', D, 4) sine 9C' (ii) Cylindrical Coordinates. R is represented by o, and z, and S by S and~ as shown in Fig. I-5. Equation I-23 then takes the form: Cos S(;,"6 +,zA) & S +..-sin6 sina? (e<,,},) _ As5;nx7 F,1 I2-25 -_ i (4',e,6s,=) St |d | i(Cp,z,$, =,)fC~,(6g,a',,)s dol 12

J2 I I I I I I I I I I I I I I Figue ili ectnguar oorInatesy I' I I I I I I I x~~~ Figure 1-4 Rectangular Coordinates

z 8 I_ Z Xso e~y\ Projection of Cy in xy parne Figure I-5 Cylindrical Coordinates

(iii) In Spherical Coordinates. R is represented by R, Q and 9, and Q2 by and A as shown in Fig. I-6. Eq. 1-23 takes the form:,; B; (R. ~.,6, a) cos'i (R.,(9,, ) RR R S6 + s;nS 6sin A,, ha _ _, _ ). ~R c;^6 At a 6 I-26 I 5ir si;n Cose -C;.(R,as,, 4) R S;n _ I ALe,) + 4' ~J d 6'j L(R, 9, 4,5',;') f, (6, a,'s', a') s;n 6'' o o 7. General Discussion of Solutions of the Transport Equation. Exact solutions of the transport equation have been obtained only for very idealized conditions. However these exact solutions serve a very useful function as criteria for evaluation of the accuracy and range of validity of the many approximate solutions that have been developed or proposed for more realistic conditions. From the integral form of the transport equation (Eq. I-19) it is apparent that the order of complexity increases as the distribution function f(n,f') becomes more and more anisotropic. For example, in isotropic scattering, Eq. I-19 reduces to a single integral equation for poo(R). In general if the distribution function contains n terms, there are n(n + 1) coupled integral equations to be solved. For problems involving an infinite medium, Eq. I-19 can in principle be solved by Fourier transforms. For finite dispersions the integral equations are more difficult to solve. In fact, the only existing solutions are for one dimensional problems, specifically, for a half space or a plane slab in which p depends only on one rectangular coordinate. In the next two sections, some of these exact solutions are developed and discussed. 15

I, Figure I-6 Spherical Coordinates 16

B. Exact Solutions for Isotropic Scattering 1. Infinite Dispersion. For isotropic scattering, Eq. I-19 reduces to _IR- R'I f,~ (I 0 = (P)a [1(R 8] -27 The solutions of 1-27 for any distributed source, actual or virtual, can be generated from the Green's function, i.e., from the solution for a unit isotropic source at the center. For this case, gdO(' I) ^ = S(.,~) I-28 For an infinite medium, Eq. I-27 then takes the form: f I+ 4'r _' - 1 The right hand side of Eq. I-29 can be recognized as a convolution integral in three dimensional space. Thus, if the following pair of Fourier transforms are defined: j o ~cR 1-30 r,. (Qu) - (2 ) l(r d k e sh o () I-31 the Fourier transform of Eq. I-29 can be expressed as 5togo: TV~) =-f.,-~ [ n~ d ) 1 k I-32

Hence Jk: (iI-32a The fact the 4o (k) depends only on k indicates the Po (R) depends only on R, which is intuitively correct. The inversion of I-32 yields () ( -oI Equation I-33 considered as a contour integral, has a branch cut due to the multiple value of tan-lk. By the proper choice of a contour about the branch cut, Case (5) obtained: where K sad;sfies the equation I-35 and (- -'+ ( v i) I-36.18 ~ ~ ~ ~ ~~I3

Some characteristic features of this solution, useful as criteria for comparison and evaluation of approximate solutions are given below. p(R) is used to denote poo(0) since no ambiguity is involved here. i. p(R) may be considered as two parts: (a) the asymptotic solution, which is the solution of the homogeneous equation: 07 Us (t) _- K; (lo) = ~ I-37 and (b) the particular solution, ~p(t ) c4 R R I I-38 which does not satisfy the diffusion-type equation, I-37. ii. Near the source, R+O, and the density p(R) is dominated by the particular solution. Far from the source, R-+oo, the asymptotic solution dominates. This is a formal statement of the well known fact that diffusion equation yields a solution which is inaccurate near sources and boundaries. iii. The average density and moment of density are: fIgd~ J R = \,L I-39 _- z 1-40

2. Semi-infinite Dispersion (half-space). For isotropic scattering in a finite medium Eq. I-29 reduces to: e-I~- p'1 1-p(Rd) =frf d W' Z\ -tt' [GJ ( 8(R z I-41 obf. of dietvSlr;On Due to the finite dimensions of the dispersion, direct application of Fourier transforms is impossible. All of the known exact solutions are for the one dimensional problem, i.e., p and q are assumed to be functions of one space coordinate only. By carrying out the integration with respect to the other coordinates, x and y, equation 1-41 is reduced to the familiar inhomogeneous Fredholm equation: dz E E(-z I)[4(Z g())] - 1-42 of d;spers;'n where Various mathematical and physical arguments have been introduced by different investigators for the solution of Eq. I- 42 for purposes of numerical computation. Those arguments can be illustrated by considering the problem of a parallel beam of intensity Lo obliquely incident on a semi-finite dispersion at an angle cos g0 = [0 as shown in Fig. I-7. The equation for the total density (the unscattered beam plus the scattered radiation) is then given by: ) = l e + fdz' E:,(Iz-z'l)(o(Z3) I-453 20

Dispersion //f 10 - Unit intensity FiguLre I-7 Plane Parallel Source Obliquely Incident on a Semi-infinite Dispersion 21

The solution of Eq. I-43 can be used to obtain the specific intensity of the radiation scattered at least once as follows: c.(zke) 4 IZ i-44 (7-'U 42(Z' Z_( I L(Z7A)') de z 1-45 Two arguments used to solve Eq. I-43 will be considered in turn. (a). The physical argument of invariant imbedding*. The numerical solution of Eq. I-43 by iteration is difficult because of slow convergence. As an alternative some gross physical property of the dispersion such as the coefficient of reflection may be defined and the resulting equation solved. The principle possible advantages of such a procedure are: a) the equation may bemore adaptable for numerical calculation than Eq. I-43 and b) the numerical results may be useful in a class of similar problems. The details of this procedure are illustrated in the following paragraphs. Define a reflection coefficient S(k,po) relating the radiation reflected from the half space in the direction -I to the incident radiation, Io, in the direction, o'. Thus 4f(lns,) S(*H O) I-46 This coefficient, S(t,io) can then be used to relate the intensities defined by Eqs. I-44 and I-45 as follows * R. T. 22

0 e 4-' 4, (-A S (.) + I-47 contribution of radiation contribution of radiation not scattered before scattered at least once reaching depth Z before reaching depth Z An interesting mathematical interpretation of the function S(,po0) can be obtained from Eq. 1-45 and I-47 by letting Z = 0: 4 je i 6 (z ) e aclZ 4 iT (,-48 Thus, S(~t,o) is related to the Laplace transform of p(Z) and can be represented as: I-49 An explicit equation for S(k,Yo) can be obtained from Eqs. I-43 through I-48 by the following steps. Differentiate Eq. I-47 with respect to Z and let Z = 0 yielding iI(,-,)- 4~/[ -,* 2-W j' O I (o,AA') S (,,')d,'J I-50 Similarly from Eq. I-44 and I-45, i'Lo,,X) - ( d(o') I1-51 4 i/,,23 25

and iL(~ -9A) = 4-i'Ft I-52 From Eqs. 1-50, I-51 and I-52 then: (14 1 AS(+, sA., -, >( ) d4- 4 I-53 Eqs. I-43 and I-48 can be combined to yield (r'(O) - I[I?1 f f I-54 Eliminating p(0) between Eqs. I-53 and I-54 then gives I-55 I t S. A-) t s a From the reciprocity relations (Eq. I-10), it is easy to show that 5 (A', ) A S(/,') The function S(i,pZ) can be expressed in terms of a single function H(t) defined as:,AA o jf ~A, $ I -56 Combination of Eqs. 1-55 and 56 gives S H (A) F1) I-57 244

Then from Eqs. I-56 and I-57 the following non-linear equations for H(k) is obtained: H (C+) - K t a> z H~ffi X d — I-58 Eq. I-58 is quite adaptable for numerical computation. A solution of equation I-58 is a solution for the entire problem since all of the functions of interest are obtainable from H(i). For example, from Eqs. I-57 and I-49, fto(z) e z 32 t. n A) 4 (,a5.) I-59 Hence p(Z) can be obtained from the Laplace transform. From Eq. I-46, the radiation reflected at boundary of the half-space is distributed as follows: i4t; ) AE/ASI) I-6o (b) Solution by auxiliary equation and the probabilistic interpretation. An ingenius technique to solve Eq. I-44 has been suggested by Ambarzumian (7) in connection with an emissive atmosphere. The method can be extended to the solution of any arbitrary one dimensional source function. If the half-space problem is represented by the inhomogeneous equation for p(Z) with an arbitrary *distributed source)Eq. I-43 can be rewritten as: (z) A 4 (z< > ) I-61

where B(Z) is the distributed source inside the dispersion andA is an operator defined as: ZlP (z) = (Z) E.( "~' O ~7{U )E, (1X ZZI)4~ J1 I-62 Two useful relationships which can readily be derived for this operator are as follows: d nC Ajd fi (Z)}_AtPz(z) + - & _o,, I-63 and If,(zf)Atfz(z)1d z fc()Atf.cz}fZ E I-64 a, II To solve Eq. I-61 an adjoint equation is defined as follows: f (Z,) = f If(Z ~. + eg/A 1-65 Multiplying Eq. I-65 by p(Z) and Eq. I-61 by f(Z,i) and integrating between Z = 0 and oo gives: fLcz) e Kdz =f (z) f (ZA)dz I-66 Combining Eq. I-45 for Z 0 with Eq. I-66 gives Combining Eq. I-45 for Z = 0 with Eq. I-66 gives ( o),f) (z) f (zA) dZ I-67 which provides a solution for i(O,-L) if Eq. I-65 can be solved. 26

To find the solution of I-65 the following steps must be taken. Differentiate Eq. I-65 with respect to Z: c3 F~~~,I I -'Z I C'(z,r)= wn~f'(,))~ + (,X2l eI6 1-68 Divide Eq. I-65 by L and integrate:,~~~~t 1' z / f(z, (= { z AAM " I' tcr~0'1 At A i-69 From Eq. I-65, I-68, and I-69, it can be shown that F(z,/) = A AF(z,A)} where F(z,) = k)+l fA) -,A f(0,) AA) F f~~~~2~~ d(AA~ I-70 One solution is obviously F(Z,) = O Hence 4, =,'z,4)+o-~a^)If >, d7-' I-71 Taking the Laplace transform with respect to pi, yields 1 C ~ ~ ~ ~ N 5 ~~~~~rl,~~~c~~~l) ^,e, C/,,,.) - v(o,,) -- __, ") + ) —-o S,,,),, I C f. % r/ A C_.01/.-I 1-72 where s(',x) = r f(z~) e Ct 27

It can be verified that 1I-73 and that -fl(r,) 2 _; 1,SFIAA I-74 Hence e5ASC, ) (= uA,, f(,,' ) I-75 Substituting this expression for S in Eq. I-74 yields, -(*,A)' I 4 A Cj"/~AA (E,*) - + z/i ) 4(o, a,' I-76 Comparing Eqs. 1-76 and 1-58, obtained by the principle of invariant imbedding shows that f(O,~) _ E(A,) An interesting interpretation of H(i) and f(Z,) is given by Ueno( 8 ). If 4 at any depth is considered to be a stochastic variable depending on Z, then f(Z,O)d1. is the probability of finding M between j and i + dCi at depth Z in terms of f(O,i). Equation I-71 for the probability density f(Z,p) can be derived from the Chapman-Kolmogorov equation for the transition probability of a Markov chain ( 8). 28

Although f(O,k) is mathematically the same as H(p), it has a wider physical interpretation, in the sense that the problem of any source distribution inside the dispersion can be solved (see Eqs. 1-61 and I-67). In the particular case of a parallel plane beam incident in the direction ~o B (z) t,.e I-77 and the distribution of the reflected radiation is 4i (o, 1) 4sT r (zI) dZ WI, I f a) Clo/do H ()AA 4 Ad H S (God} ~) =A AA - + A which is the same as Eq. i-60. 3. A Dispersion Bounded by Parallel Planes. The extention of the technique discussed in preceeding sections for the problem of a dispersion bounded by parallel planes at Z = a and Z = b is straightforward. The equation to be treated now is to )ra = Ita {a(I+ )}z b+ 1 -78 where the operator ~ab is defined as /%~ _. abf = f(z,) E,(Lz-z')4z_ I-79 29.

It can be shown that dZ ab A.l,fj )} + (a) E(z-,) f(b)E, (b-Z) ~~41 ~Z ~~~bLJ~~ i-80 Therefore a function f(Z,.) satisfying f(z,)= Aaof6Cz)} + I-81 is defined so that f /(z)e cz -C BCz)7(z, )Jz a 1-82 The left side of Eq. I-82 yields the distribution of radiation emerging from the surface Z = a. Similarly, the radiation emerging from the surface Z = b is b - 6b-Z)/ b I z~z) ez d z 1-83 Using arguments similar to those used in the derivation of Eq. 1-71, it is found that Eq. I-76 is satisfied if f(Z,.) satisfies the following equation:' f (z, A) -,,,,a 50- ~ f~ b *,) X f~v~Z')g *1I-84 30

Eq. I-84 of course could be interpreted as the probability distribution function f(Z,A) in terms of the unknown boundary values f(a,0I) and f(b,j). To solve Eq. I-84 numerically, the following transformations are defined,~b - Cz-&)/A,5 (.,,) =- j f (Z,A.) e z I-85 and b -(b-Z VA'T 4(',~ - f(z,)eJ d -86. (b-4 Z, ) e It can then be shown mathematically that S (AA,') - 5 C,AA) -87 T (#, S ) lr T (>, >~ ) -87 T C~t T(CAA)AA 1-88 Ji z -a )_AA Sf" (Z,) e ( z - f (b,,,) e _ A(, + I-89 + (,S(,,,) and f(z,,,,) e dJ Z - (b,,A,,) -.(:C,, ),,(:e I-go 31

Applying the transforms I-85 and I-86 to Eq. I-84 Yields,,' - ]4 )AA fE(/ IA+ Ir 0/)^ - jb,) -;.. f J4 " I — 91 and, ( ) + i (-cr —,yc' 1I-92 -f(.b,, Oe) e +, V Now, from Eq. 1-79 and 1-81, it can be seen that: f(A(&,-" + 040 ( ) J'-93 and f (bAA) - e + i I. b, /Ic) f3 acyl, r~~cA Ir' I-94 Therefore, [ - $] 7,',), =- f(,) )f(4),,') - f(b, )f(a,5') 1-95 32

and',,o, - f(b,) cB, 4 ) I-96 Hence, the following pair of coupled integral equations are obtained 1~~~(a,,&A~~) 4f (% f' ( sa,d) - (b,) fl b,,u') 9 f-, -,,-97 and -f'( AA) f(b,,d) - f( ) - e + J C-AA I-98 Both f(a,k), f(b,9i) are functions of (b-a) and. Chandrasekhar* derived Eqs. I-97 and I-98 by the principle of invariant imbedding. In his notation f(a,) = X(A) and f(b,.) =Y(k) The effect of the finite boundary in multiple scattering can be inferred by comparing f(a,k) with H(k) for various optical thicknesses, (b-a). Considerable effort was expended to develop an efficient program for the numerical solutions of the coupled integral equations I-97 and 1-98. Completion of this work was however temporarily deferred in order to complete the investigation of the effect of anisotropic scattering in a half space. * R. T.

4. Some Applications of Exact Solutions. The solution for a point source in an infinite medium given by Eq. I-33 can be used to find the solutions for any distributed sources by simple integration. Some simple examples are given below: i. For an infinitely long line source of unit intensity per unit length coinciding with the z-axis, the density at any point R (in cylindrical coordinates is f (R ii For an i pa a I-99 ii. For an infinite plane source of unit intensity per unit area at the plane z = o, the density at any depth Z is given by: v(z) A rf 0( Jg zZ)+ P o - 2I 1~a 1~ a ) r7I-100 Izi iii. For a spherical shell source of optical radius A and zero thickness and unit intensity per unit area, the density at any point R inside or outside the source is PO2R) f (00 A 2A R,),2 IR+ A lg-Aj 0 I-101 34

Another class of problems that can be expressed in terms of poo is the problem involving a perfect, specularly reflecting boundary. Consider a unit point source placed at a distance Zo from a perfect, specularly reflecting boundary as shown in Fig. I-8. It is evident that the density at any point (X,Y,Z) can be obtained from that due to original source and an image located at -Z Thus, O0 (X, {'4 z =(Z Z.)) + X2 X~ % (g Z IF ) -102 The complete solution for a parallel plane dispersion can be adapted to the problem of a finite spherical dispersion with radial symmetric sources inside. For example, for a spherical dispersion of radius A with internal symmetrical sources depending on R only, Eq. I-41 can be expressed as: i( f dv' e4 Rl (t) + (R'Vl. o*;J s A I-103 Since p depends on the values of R only, the integration can be carried out partially, yielding: )( E ( RBR) J' I- 1-104 35

Point source Zo ~x,y, Z I Perfect specularly reflecting boundary I -Ao I Image Figure I-8 A Point Source Above a Perfect Specularly Reflecting Boundary

with the condition, B(-R) = B(R). Eq. I-104 is of the same mathematical form as Eq. 1-78 except that the source and density are modified by the factor R. Unfortunately, the distribution of emergent radiation depends on the knowledge of p(R), instead of the integral transform of p(R), hence numerical integration is necessary. 37

C. Exact Solutions for Anisotropic Scattering 1. General In the case of anisotropic scattering, the distribution function for a single scattering is given by 4 hi 2 ani,.'TE(cos') I-5 where ( is the angle between the directions "a and Q. If Q and n' are represented by an azimuth angle Q (cos @ = A) and a bearing angle ~ with respect to any fixed axis, Eq. I- 6 takes the form: WWM I-105 The integral formulation of the transport equation then involves the solution of the eet of coupled integral equations given by Eq. I-19. No successful attempt has been made to solve these equations in the general case. For one dimensional. cases involving a dispersion in half-infinite space or between parallel planes, the principle of invariant imbedding discussed in Section B can be applied as discussed by Chandrasekhar*. The result of such a treatment, is a set of coupled integral equations, whose solution can be applied * R. T. 38

in a whole class of problems comparable to H(i) in section B. In principle, these coupled equations can be solved numerically. In part II, numerical results are given for a single case. Unfortunately, as the number of terms in Eq. 1-105 is increased, i.e., as the distribution of single scattered radiation becomes more peaked, the numerical solution becomes prohibitively difficult even with a modern high-speed computer. It appears possible to reduce the solution of the set of coupled integral equations to the solution of a single integral equation, which is very desirable in terms of numerical computation. A great deal of effort has been expended to complete such a formulation but thus far success has not been obtained. In the following sections Chandrasekhar's results are re-derived and some possible simplifications in the numerical calculation procedure are discussed. 2. Semi-infinite Dispersion (half-space) For an infinite half space with a source function depending only on Z (see Fig. I-7), the transport equation takes the form, n w.fJ ('L(#-4) P~) P ))m~ 2C. 1-106 Letting (Z ) m (Zaps) cm 1-107 39

and oEq. I-106 can be decoupled into an infinite set of equations: n >, A,IA jim(Z ~A)' a(Z ( I) 4 #4~m c1r -110 + 2 bL; P( \ ) 7(AW' )',(z,M'),' where The boundary conditions for the solution of Eq. I-110 are m(Z,) +4*' O ) Z o I-112 n (Z -,- ) -O, Z - The solution of Eq. I-109 for m >N is trivial. For the set of Eq. I-110, the component densities f Q(A') ti (Z,') d,' I= ) I-113 may be defined.

Thus, z +(z EE),'Z ) I-114 The solution of Eq. I-114, subject to the boundary conditions 1-112, is evidently: (, )-,' )/ fg I(z-t' + z 1AL u F A)f ) e P(jZ') cIZ' I-115 and. -) (~,-A) iz + 14 r.0 I-116,(Z Z) p d I dT A (-/A)I p..(_(z-z)/.,, a,/A.'~- ~ e f z) S Z Substituting into Eq. 1-113, yields the integral equation for pm(z): k fz) j 2 ) z I -117 n 0w t4 41

where [z' z - d> e~z z lz',- (z Pa) P-d S)d,z o 8,,,(Z P (-)-118 and the kernel function Nnk(x) is defined as: l Pc cn(-) d(-,) dXo 42

From Eq. 1-115, I-116 and 1-107, it is evident the radiation scattered at least once can be represented by Y iz,,~) C E 6 (gz,. ) <t M I-120 where z i(z, ) F (, ) d Z I-121 and z,(z -,,,- ~.,, P.(')dz v b:n. m it _(Z-2)>it 1-122 The scattered radiation at Z 0 O is given by: Z oo, -,,, 4) — Z: U " i.,,-s) I-125 where mnt p /.A m I-124 k= t k which is related to the Laplace transform of Pk(z), just as in Eq. I-48. The radiant flux density leaving the dispersion at the plane Z = is: E(o) — d, ~.,ud(. (-,) = / P (-A,)( z, P(Z')dz C)~~~~~~~~~~

For a parallel plane beam of intensity Io incident at Z = O in the direction (4Oy9= 0) z f o: e) I-125 The solution more adaptable to numerical calculation then Eq. I-117 can be obtained by using the auxiliary equation: z-,, Ai (z-z - 7 /z fA,A ez, -. f 1I-126 In terms of fn(Z,.), 00 -PP~~I~e e; e Zd'/ dc, ^ ^pa t Z/ u X n(z)d I 127 A differential equation for fn(Z[) can be obtained by a process similar to that used to derive Eq. I-71. Briefly,(a)differentiating Eq. I-126 yields C\a f (Z A (,) P() AZ/A 4- -(. j (Z,,) f.~ + J5k(Z>)lu rdZ +r 1<P pi'o' -g -129 44

(b) Multiplying Eq. 1-126 by PS, CO and integrating gives f'.,7zt, ) f P)(Q) Poy) -(S~s z/ I-130 (c) Multiplying Eq. I-130 by Ws f (o,) summing over S = m to N, and subtracting from Eq. I-129 yields:'t: (z,,~ +~ = div' noS tl -'~;*,- z,',, - f,(o,,., JAA) h=~~~~~~~ ~o 0 I~~1-131 (d) Since Eq. I-131 must be satisfied for all n, it must be satisfied by in Vn5 0 ( n= V+I,.) ", t) ) 4'5

(e) Let the Fourier transform of fm(z, ) be * e dZ -R(,. I-133 o Therefore ( } + /X~ ) EAA^y)- 2S AA) ) Rn LA X) PSMs I-134 (f) Introducing;R (As; #) Ph(/4 ) =,U) I-135 n - produces - )R"(1 AA= {( ) I- 136xtfR( ( ) P() -136 (g) From Eq. 1-126 it can be shown that (of~, A4 +88k___ (____ -d' v I-137 and 6Uc8' R U xf(, A * I-138 Thus (R) s ~') 8 (AA' AA) i (~, O}2;s(~18 A') I-139 S rb' and I A,~*) A* $(Z)$(osI-140 AA+,AA

(h) Substitution in Eq. I-140 yields a system of integral equations: (O,,a)f P,,()+ fk(o,)c, )f(, foe a = o,e, I, -", Al i-141 The solution of this set of equations is discussed in Section II. 3. A dispersion bounded by parallel planes The extension of the formulation in the previous section parallel plane dispersion defined by the space between D= 0 and Z = 2Z is straightforward. Therefore the procedure will be outlined and the details omitted. a) The equation to be solved is (See Eq. I-117) jcz) Z) r I f A&z') M j(z-7) dZ I-142 b) The auxiliary equation is A-) (zA f(Z) t z) Ir -143 c) The differential equations for fn(Z,p.) are 47

ZIAz 4f( A) 4= e t( )JL Z.ftio) (X J + d) By a procedure similar to that used to derive Eq. I-141 and also Eqs. 1-97 and 98, it can be shown that fm'(o,) and fmC(ZL~) are solutions of the following (I P +, -.o-, P,(,,) (o ) (o,,'). (,,)f, -5 [.e t, ('I-146 Thise a equations havcan of course e also been derived by Chandrasear* using the principle of ilvariance. In his notation: * R. T. 48

tb (OJ n) ) ( a oh)-147 As 7;) E. (1-1 r t/A As 2-*o, Eq. I-145 reduces to Eq. 1-141. 49

4. Discussion of Integral Formulations. The "exact solutions" discussed above are not explicit solutions but are expressed in terms of certain functions which must be evaluated numerically. For example, for the half-infinite dispersion, the set of inhomogeneous linear integral equations, 1-117, is transformed to the somewhat simpler set of inhomogeneous, non-linear integral equations, such as Eq. 1-141. The numerical labor involved in solving such a set increases drastically as the number of terms involved in the distribution function fl(glAt) is increased and is prohibitive for a very peaked distribution involving say ten or more terms. However, it seems possible that the numerical procedure can be reduced to the solution of a single integral equation. In the following paragraphs, such a possibility is discussed. To simplify the mathematical notations, only the case m = 0 is considered, thereby reducing Eq. 1-145 to #kttz>; P/(z) oF &x) 4 ~),_ ~l V I-149 where Z is now considered.a complex variable. The mathematical problem is to find se solution of Eq. I-149 such that Q(Z)is bounded in the interval 0o Z I.Evidently, k(VZ is analytic except in the interval-/4ZzO. Using the recurrence relations for Legendre polynomials, i+ Z)a PI (Z),) +4 k PJ,(z) I-150 it is seen that: (rk+l,, A ) I) (2)+ 0,(bz) + 2; k), 1 -)50-151.50

Eq. I-151 may be considered as a set of N simultaneous homogeneous equation for N + 1 functions 4k(Z) involving unknown numerical coefficients such as cr)k(-x)dy. It follows, thereforethat each 4Pk(Z) is proportional to anN x N determinant whose elements are linear functions of Z. Therefore 1S U k() K(z) I-152 where Vk(Z) are polynomials of degreeNor less, and K(Z) is arbitrary in so far as Eq. I-151 is concerned except that it is bounded for o04Zkand may be "normalized" such that whbn i H + Introducing Eq. I-152 into Eq. I-149 gives: RZ) Z), K.P(2j) Z U (Z) Z+ I-153 Eq. I-153 can be re-written as r) tF< - Z, ('r 4 ( C) 4 &') Ud aI + Since the factor'(Z) P(ZP) -,)(Z) is divisible by Z + X, the second integration is Eq. I-154 is a polynomial of degree N - 1 or less. Equation I-154 can therefore be represented by 51

KLZ) Z (l. Jo l -_ t <+J where the term in the bracket is a polynomial of degree N or less. From Eq. I-155 it can be concluded that I~tc)' =~ | id-, i S e) (X I-156 and U;D(Z)_ K)(Z'4; j t.r(@) t (4(Z)1kE4)-LY%)Pk(Zgi7 I 1-57 The proof of Eq. I-156 and 1-157 can be stated as follows: a) For each k the right hand side of Eq. I-155 is analytic except at zeros of P() b) But the left hand side is independent of k, hence must be analytic everywhere, except possibly at infinity c) Since (b) is true for k=N, the expression must be constant. d) From the normalizing condition that K(Z) = 1 as Z-PO+ this constant must be 1, hence Eq. I-155 and I-156 are established. Thus, if an explicit expression can be found for Ewo(- ) U()U(-7) without actually solving Eq. I-149, then K(Z) can be obtained from the solution of the single integral Eq. I-156, and the polynomials U%)can be obtained in terms of the moments K(Z). Hence,the problem of solving a set of complex integral equations such at I-149 can be reduced to the solution of a single integral, Eq. I-156 and some algebraic equations.

To show the feasibility of such a reduction, consider the case of N = 1. Eq. I-151 can then be written as zLf(Z) [-LcOo A0o]i A, (z)- Z Al I-158 where Ak ( ( ) pR( ) K( ) X * -l159 It is evident therefore that Uo(,.. I-,, AoI-16 and l gr- ~4.7 I-161 Thus, Wo +,,'/ - towtA + So A / - A I-162 Now,from Eq. I-155,4 it can easily be shown that atot'z+ o Ato, AloI-13 _,[ O A107-163 Hence Eq. I-162 is reduced to WO(X)Wo( X)Y),f() t( t)= t ALwo c,)(I WO Hence K(Z) can be solved from Eq. I-156. The explicit forms of Uo(X) and U1(X) can now be obtained by determining A,, and Ao~. To accomplish this, the moments of K(Z) are first taken: 53

From Eq. 1-159, 1-160, I-161 A0 = to.- A O I-166 and A =6 f 1-iA1 I-167 Solution of the above yields, A00 = ~ —~r? I-168 1 4 and AID,0. <l'[- 2 zI-169 Relations Eq. I-169 and I-168 were given and verified by Chandrasekhar but the generalization to any arbitrary terms of distribution function has not been mentioned. Considerable time has been spent in extension of the above and results are expectedi;n the continuation of the research.

PART II - NUMERICAL CALCULATIONS A. Mathematical Formulation for Half-space Problems 1. Geometry and Coordinate System. This study is concerned with the irradiation of an infinite halfspace by a uniform parallel flux as indicated in Figure II-1. The Z = 0 plane represents the plane interface between region (2) containing the source and region (1) consisting of a uniform dispersion of multiple scattering paro o o N ticles, characterized by the constants mo. ol, co2, --- N a. The objective of this study is to determine the intensity of the reflected radiation (the diffuse reflectance) and the integrated reflectance (the phase albedo) as a function of the dispersion characteristics. The coordinate system used in the analysis is shown in Figure II-2. An incident flux of intensity I~ strikes the interface at an angle Q0 with respect to the normal (Z axis). The direction of I, the reflected intensity leaving the dispersion is defined by the angle @, measured from the normal, and the azimuthal angle ~, measured from Y axis. For simplicity the coordinate system is oriented so that o = O., i.e., Io lies in the YZ plane. 2. y/ Functions. The following integral equations for the two basic functions S and were derived in section I for the slab problem, i.e., for a plane parallel media of finite thickness:.k-m9I I t,c -0 (1 )i,4uA+ A CA'A (1-147) 55A

#Z -X region (2) Figure II Geomety of Halfr gi!on (I) Figure II-1 Geometry of Half-space Problem

Jo 80 den ~~~~~I xy plane = media interface IO = direction of incident flux. - I = direct/on of reflected flux under consideration a - cos-'. 8 =cos-/i Figure II-2 Coordinate System

1, Pd f ) - 8 4 () -f tX G/ k ih 1 L r (Pin _.U tb(S, ) (, J-A (I-148) where, X- cos5, - ioptical thickness of media (number of mean free paths for scattering) f (~a) = associated Lengindre polynominals N number of terms expansion of phase function, L)j - coefficients of phase function expansion (a- W)! For half-space problems, the thickness,:, equals infinity and equations (I-147) and (I-148) reduce to: on W ( (P], a,AA (II-1) w,.e.e ^,,,,,,,..,; m o,*-,,r and co;CS; (') ~. (II-82) 5i8

For any value of N, equation II-1 yields a system of (N + 1) (2 + 1) simultaneous integral equations which define the functions t (/). As an example, with the notation F,/ ()/.)- _ (O () Y I (), the system of integral equation has been expanded as follows for N = 2: i A.+ F - f, F.i + — dA' (II-3a) 3/,~r) ie r f' C ~- F, fs o *'/ f(II-3a) ( _F. F + f 3,, (II-3b) Z {sw,) +', (N) _ (t-s ) t / t [ - |., d)' " ~4(II-3d) - = 3,. )~ + z- [ F-3e) + F; (,CZ. _, -+ - J [ F - F I da' (II-3f) It is such systems of integral equations that have to be solved to obtain tabular values of the 5 functions at various /s. 59

3. H Functions. By definitionChandrasekhars H function is*: H (r, - I I*A H( I + is, f(l where, t4 ('). some characteristic function. For the isotropic case: 47 (, ) _ a constant Then H() He () H 6,,( ) AA)T d"' (I-5) which is identical to the integral equation for o()obtained from Equation II-1 for the case of N = C, + tA- + 1 to i t. (/AJ )A s C' (II-6) Thus, for isotropic scattering,- Values for the H functions have been tabulated.** * R. T., p. 105. ** R. T., p. 125.

4. Intensity of Reflected Radiation. The intensity of radiation reflected by a plane parallel atmosphere in the direction k,q7) may be expressed in terms of a scattering function.** -4(, ) - ~TT x S (t cp, IA)A (II-7) where, incident flux intensity I(,A, 0) _ intensity of flux reflected in direction s,'). A,.= optical thickness of atmosphere S - scattering function The scattering function is*** where,?, a ao): (, - Hi) (- " 4 (. - (II-8) )9 kAf~) ) ** R. T., page 161 *** R. T., page 180 and 177 61

For the half-space problem, so that equation II-8 reduces to, ~S (I~,= A (2 z c~ ) >1 (-sr F' (,AA,.A.) (11-9) where, F4 (,qU, ) r(.) The diffuse reflected intensity may then be written as, Io& ft~e_ Z (JZC$ ) #) (- ) FFS V,) (II-10) For N = 1, =4 tCM-4) F (. o) - F. (,,) + 2f, co (II-ll) For N = 2, I CAA,) _ +'l,4 rIFC, i. (,-") 4.-(++,A,,) O.. 0) F,,,) FC,,,, + 4-~c(~) *ri(,A+) + +(F f, (,)co + F-(F,(AA,,oi9Cs 4 (II-12) For N = 3, + 2(F.) (A ) -,+ F,A.-) + f.,,,jcosfo + ) 4 (f,( +,) - F(,A.ae))co5 z + (Fu. Y9cos3ciJ (I1-15) 62~~~~~~

For N = 4, 1.(S4..) 4 -1T Cs3+~),F,, ( Ao.), - F'O(,.).)F F(;(gA. +Z (F:...) t- F2, +(: e.) - FA C,,., ) cos + (FL (,l ) FA,,. j+ f o.) Go s 2+ + 2F ( CAAAe) F4 }(.))CoS3 + Z(F:.,4))(os41j (II-14) For the isotropic case (N = 0), the equation for the intensity of reflected radiation is* o (O,,). t ie9 It is seen that for this latter case the intensity is independent of the azimuthal angle qP. * R. T., page 124. 63

5. Integrated Reflection. The integrated reflectance, or the albedo of the half-space is defined as the ratio of reflected to incident power passing through a unit area of the interface. The flux, I, was defined as the radiant energy passing through a unit area normal to the direction of propagation per unit time. Therefore, the incident power striking a unit area of the interface is Io x 4. Similarly, the total reflected power passing out of a unit area of the interface is lJf I(,.)xJ dg. The total albedo is then expressed as, Je' I'e Itf, ) OV dA do R - -', (II-16) Replacing I(t 4 ) by II-10 and noting that Li j F(&4m) co5"s c I() o I mWI:o lT (Ad) j~ e =o (II-17) yields IR =,Lo z 5 dA (II-18) But by equation (II-1),fiAG1L X ( A1f Fe (,') - + 77 dk' (II-19) 64

Comparing equations (II-18) and (II-19), it is seen that, go, A H(I- R) (11-20) and hence that R = (II-20a) This relationship is valid for all N 1 and Fio)O. Equation II-20a provides a simple method for calculating the albedo, once the. ([I) function has been established. It eliminates the necessity of performing the two integrations to find f |T, ) t l and the first momentj iA(%,.)JJOXL. This result has apparently not been derived previously. Equation II-20a degenerates at 0o = 0 for any N, and at any Po for N = 0 (isotropic scattering). At o = 0, the reflectance may be calculated from equation II-18. z O I;(-,)'I; (a ) o 2 z (-.1) o P() o (II-21) in 0( = 0 t Since Pl(o) = 0 for godd, the following system is obtained: For N = 0, R(- -o)'-[ o () a (II-22) For N = 1, R(p O) r do'G o V /6 (II-23) 65

For N = 2, = ):- L-[:g ()] d~ ~ ( 1-24) For N = 3, R(A.:o) -=' iz r fto. ~[a +:>.oA1c (II-25) For N = 4, R (A: a) - f[(: L:) - 3 i"'!'A) + a' (gt' ()j IS, (11-26) For the isotropic case of N = 0, albedo may be determined from the H functions, as shown below. From equation II-18, for N = 0, By equations II-6 and II-5, R o H(AI) f' tfI H,.),,U. d,,A (11-27) From the general properties of H functions, it is known that * 8s+ fP- [)(dj A +14', (II-28) For the case of' i), it is seen that |Jt(~ HT ) AcJ r'coY ~J.)'[A- J. l -I [l A]2 (II-29) * R. T., page 107, equation 13. 66

and Y] H(8~,[~;;,-s.) ~ -~ol~,J I~;- (II-30) Equation II-30 may be used to calculate the total reflectance (albedo) for isotropic scattering with any hw and any particle albedo W. 6. The General Numerical Problem. As shown in the previous section,the intensity of the reflected radiation I (o/1>j > ) for a particular phase function for single scattering can be expressed in terms of sets of simultaneous, non-linear, Fredholm-type, integral equations. The numerical problem then is to solve these sets in order to obtain values for the intensity I in terms of its parameters. The most straight forward method of obtaining numerical solutions of integral equations is simple iteration using the original equations. For example, in the case of a single integral equation of the type where f(x) is the function to be found g(x) and +(x) are known functions of x and % is some known constant, the method of simple iteration involves (1) choosing an initial fo(x), (2) evaluating the right side of equation II-31 using some mechanical quadrature method - thus evaluating a new f,(x), (3) testing to see if fo(x) = f,(x),and

(4) if they are not equal everywhere, repeating the process using f, (x) in place of fo(x). This procedure is then repeated until fn+l(X) = fn(X) within acceptable limits. This procedure is only advisable where the computations are to be done on a high speed computer of the order of the IBM 704. The rate of convergence of the iteration procedure depends very strongly on the value of the parameter s in equation II-31. A slight variation of this iteration method was successfully used to compute solutions to many sets of simultaneous'-functions where the albedo for single scattering was less than 1.0. For values of the albedo equal to 1.0, the convergence of the iteration procedure was so slow that the method breaks down for practical purposes. The problem of convergence is more complicated than a simple numerical test. The trial functions fo(x), fl(x), —- are necessarily tabular since the integrals are to be evaluated numerically, hence convergence of the tabular functions fo(x), fl(x),..... to some fn(x) assures that some tabular function, f (x) has been found which satisfies equation II-31. However, this does not guarantee that fn(x) = f(x). Convergence in this latter sense has not been shown for the /-functions computed in this report. Numerically, though, it was found that as the mesh for the mechanical integration was refined, the convergent values of the tabular (4-functions did not change. Thus, without proof, it is probably true that the convergent values of the tabular SL-functions are a good approximation of the true qPfunctions. In order to speed the convergence of the tabular functions, the method of exponential extrapolation (9) was used. This is a method whereby given three previous iterates, a fourth can be computed by a simple arithmetic formula, thus saving the computation of an iteration. The method's theoretical basis is that

if three consecutive iterates yield a straight line when plotted on semilog paper against iteration number, then the iterates value after an infinity of iterations can be calculated straight off. Extrapolation methods cannot be applied to the process of iterating tabular functions indescriminately. If every value of the function at O:- xi- 1.0 were to be extrapolated, the iteration procedure might be unstable. A satisfactory compromise is an extrapolation of the function at every xi in proportion to the amount the function was extrapolated at xi = 1.0. Also, in order to be sure that the extrapolated function will not upset the iteration procedure, extrapolation is to be applied only when the three iterates used to construct it do approximate a straight line. Simple iteration is then used until the functions have again smoothed out and extrapolation can be used, or until convergence is reached and the process is terminated. As Chandrasekhar points out*, when only a single equation is involved, a rearrangement of equation II-31 is far better for computational purposes. If the notation is changed to agree with his, the single equation or H-function is rewritten FCt) L /IAl /A Equation II-32 was used for computing single integral equations. Once the various t(-functions have been computed for a given phase function, it is a straight forward matter to evaluate the intensity of the reflected radiation and the integrated reflectance. These follow immediately from the formulas derived above and summarized below: * R. T., p. 123.

Function Equation No. I/Io for N = 1,2,3,4 II-11,12,13 and 14 I/I for N = 0 II-15 o R forA#0, N = 1,2,3,4 II-20 R for/'=0, N = 1,2,3,4 II-23,24,25, and 26 R for any N = 0 II-30 70

B. Specific Example 1. Representation of the Phase Function. As stated previously, when the phase function for single scattering is fixed, the.-functions can be written down, and the emergent intensity can be calculated. The construction of a phase function for scattering by a spherical particle depends on the wave length A of the radiation, the diameter of the scattering particle D and its index of refraction m. This problem is fully described by Chu and Churchill (10). In summary, the phase function f(Q) or in Chandrasekharts notation*4i p(cosQ) can be written eQ I ~l. ~Ft@) - tar f 4 E an P (ca m a) (II-33) where f(G) is the fraction of randomly polarized radiation scattered by a spherical particle into a unit solid angle in the direction Q, measured from the incident direction. an(a,m) angular distribution coefficients P (cosQ) are Legendre polynomials a = CD/% Therefore, theoretically, no matter how non-isotropic the scattering is, the half-plane problem can be solved. Practically, howevers the number of terms in f(Q) must be kept to a minimum or the numerical problem of solving the sets of simultaneous *-functions becomes excessive. For an example, consider the half-plane problem where radiation of wavelength X= 0.55 microns is being scattered by spherical particles of diameter D = 0.4 microns and index of refraction m = 1.44. The angular distribution coefficients an of Equation II-33 for these conditions are found to be,** * R. T., p. 149. ** unpublished values. 71

a0 = 1.00000 a4 = 0.33726 a7 = 0.00167 al = 1.93878 a5 = 0.08261 a8 = 0.00015 a2 = 1.75235 a6 = 0.01391 a9 = 0.00001 a= 0.85757 If all the an's were to be included in the phase function, it would be necessary to solve 10 sets of simultaneous 0-functions of order 10, 9,...3, 2,1. This is a formidable calculation. The approach taken for this example was to construct pseudo phase functions of reasonable numbers of terms from the actual phase function. Or, if f(B{ ~ ~ (.)&* 1 + 9 co@)te ) 4 *. + X (c )| (II-34) is to be the pseudo phase function which is to replace equation II-33, conditions must be put down in order to determine the t new angular distribution coefficients Y i........ Moments of the two phase functions were equated to fix the I s: JoC c) co o h S it,,- (II-35) The resulting phase functions for the various numbers of terms are: N- 1, $F(B&) 4nl ++34 i5, PI(c (11-36) s= a, f(e)= 4-r /?34 8 i4 Z z 24153 P,(. ~)] (II-37),'9)~ 44lI+ A O8o(036 I(b') + P. (co94 a + (11-38) + i.3i37 gP(& ) 72

So-r ~s~(u~,e) rs41PzP(ev/ - - (II-39) + o.5?4%9 I (co) O.S14 0.4I a These pseudo phase functions f(G) and the actual f(g) are plotted in Figure II-3 vs the angle 0. It is noted that equating the moments enables the pseudo phase functions to approximate the actual phase function fairly well in the forward direction for N = 2,3, or 4. However, only for N = 4 does the back scattering agree very well. The representation of an actual phase function for single scattering by a small number of terms using the method of equating moments can be expected to be meaningful only if the number of terms in the actual phase function is small. This can be seen in Figure II-3, where for a = 2.2, four terms were required before a good degree of approximation was obtained. For fewer than four terms, the pseudo angular distribution functions did not agree well with the actual distribution in the back-scattering region from 90~ to 1800. This is because the conditions set down to determine the constants pertain only to the forward direction. Other conditions might be chosen in order to balance the agreement between the forward and backward directions, however, conditions that would accomplish this and keep the integral of the phase function normalized would be very difficult to use. It is to be noted that representation of the actual phase function with four terms reduced the number of equations to be solved from 55 to 15. The phase functions of equations II-33,34,36,37,38 and 39 are for non-absorbing particles (wo = 1.0). For partially absorbing particles with complex indices of refraction, the angular distribution coefficients must be recomputed for every value of wo. As an approximation the phase functions were assumed to be wof(o) where f(Q) is the phase function of a non-absorbing particle. 73

.5 NV=O.4 o N =2 0 N =3 X N -4 actual (G 2.2, m /.44) 2 f (9) -. 1 _ _ _ __ _ _ _ _ _ o 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Figure II-5 Angular Distribution of Radiation Scattered by a Spherical Particle for Several Phase Functions

2. The Ji-Functions. For the pseudo phase functions of N = 2,3, and 4, appropriate -ffunctions were computed for HO= 0.9, 0.6 and 0.3. For N = 1, Chandrasekhar has written the solution for intensity in terms of H(1)-functions, and therefore in this case H(i)-functions were computed using equation II-32 for the same three WO s. These values are tabulated in Appendix A. For isotropic scattering, Chandrasekhar has tabulated the relevant H-functions. For N = 2,3 and 4 the (/hfunctions were computed at intervals of 0.05 except for Q = m = N, where the interval was 0.01. Additionally, $L-functions, for N = 2,3, and 4 were computed for an interval of 0.01. The number of significant figures obtained in the calculations are unknown because the error introduced by using finite integration processes in non-linear integral equations is almost impossible to assess. However, if the error of using Simpson's rule is used as an estimate, the tabular functions are probably good to four decimal places. There is no error due to truncation of the iteration process before convergenceJbecause,for the mcSused in these calculations, the tabular functions converged to as many significant figures as were carried with computation. 3. The Intensity of the Reflected Radiation. The normalized intensity of the reflected radiation, I/Io(10/A; /,)O ) was computed for the four pseudo phase functions at o = 0.9. A base value of'o = 0 was used. Values for/g= 0.2(0.2)1.0 were chosen and for every/ig, values ofl = 0.0(0.1)1.0 and for every <>lvalues of < = 0(10)1800 were calculated. These values are tabulated in Appendix A. Values for NA = 4, which was shown to represent the phase function for a = 2.2 and m = 1.44 reasonably well, are presented graphically below. * R. T., p. 145. **I R. T., p. 125.

In general, the reflected intensity, I, is a function of the three directional-angles cos-Tc, cos-, and 0 (see Figure II-2). The function I/Ifo(~}) may be thought of as a 3-dimensional surface in 4-dimensional space. A graphic picture of this function may be obtained by 2-dimensional crosssectional plots of the I/Io surface. Figure II-4 represents a cross sectional plot of I/I versus 0 ato= 0.4, with parameters off. It is seen that the flux reflected in a direction normal to the media interface (f = 1.0) is of the same magnitude of all azimuthal angles. -1 As the angle Q = cost increases, the intensity of reflection becomes disproportionately greater in the forward direction. Thus, for/1A 1, the reflected intensity is a maximum at 0 = 0 and decreases to a minimum at the side (,u 1200). A second smaller maximum is observed at 0 = 1800~. In Figure II-5-I/Io is plotted versus 0 for parameters of/f, at/ = 0.5. As might be expected, the intensity of reflection is independent of 0 when the incident irradiation strikes normal to the media interface (at/0= 1). For nonnormal incidence,/e8 f 1, the intensities of reflections again show the tendency to peak in the forward and rear directions. It may also be seen that the maximum intensity which occurs in the 0 = 0 direction, increases as/?o decreases and goes through a maximum at about ro = 0.4. This effect is more readily observed in Figure II-6, in which I/Io is plotted versus /V0 for ~ = 0 withiA as a parameter. It is seen that for small values ofj, including/4L = 0.5, the intensities go through a definite maximum. The intensity close to the normal direction (* +l) is less and the maximum is less evident. At /L= 0.8, the intensity actually shows a steady increase from 0 at/,0 to a high of 0.074 at/, = 1, with no maximum in between. A similar plot for = 1800 is shown in Figure II-7.

9010 00~20 ~ —-6 \180e -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' *;.02 - ~ 0 8 — 1046- 06/./2/14 /1G 18 0 e ~~~~ --,8 N 4 ~ 0. 4 Figure II-4 Intensity of Reflection as Function of Azimuthal Angle, I/Io vs 0, parameters of p., N=4 and ~o=0.4

900 /20 60 180 1~ 1 I I I ~ - I. ~;02 —-04.06-i.10-712.14.6.16 co 2r=2 i,=3 0.5 Figure IT-5 Intensity of Reflection as Function of Azimuthal Angle, I/I, vs 0, parameters of tl- -TAL n-na.,-n C

.30.28 0.9 N =4 26 i00.24.22.20 ___ /IO.16.14.1X'.06.02 ~.1.2.3.4.5.6.7.8.9 1.0 Figure II-6 Intensity of Reflection as Function of Incident Angle, I/Io vs tio parameters of U, N=4 and0=0o 79

o089= Spwu t='r Jo s a'@m~rd'~0T SA OI/I TITUV;uPpFTuI JO uoTlcun, ss UOT.TaIJaT t go X0.TsuaGTI L-II a@n.TL 0'1 6' 8' L_' 9' __ _ _' I' 00 I 0' 00". 20" 90 OI/I /' O" 60' J~~~~~~~~60=0

The variation of the reflected intensity on the forward direction as a function of emergent angle (Q) is shown on Figure 11-8. For/Jof 0.8 and less, it is seen that the reflected intensity is greatest at large emergent angles (L-e 0) and decreases asj approaches unity. This is not true for normal or near normal irradiation. As /c approaches unity, it is seen that the maximum reflected intensity is at low emergent angles ( $' 1.0) and is at minimum for large emergent angles (A- 0.0). The effect of the phase function is indicated in Figure II-9 in which the variation of I/I with and 0 is contrasted for N = 4 and N = 0, and in Figure II-10 in which the variation in I/Io with # is compared for N = 0,1,2,3, and 4. 4. The Integrated Reflectance. Values of the integrated reflectance are given in Appendix A, Tables A27 and A28. Figure II-11 shows that the pseudo phase functions for N = 1,2,3,4 all yield an integrated reflectance considerably less than the value for the isotropic case, N = 0. The variation of the reflectance with the angle of incidence is also relatively insensitive to the number of terms in the phase function. Figure II-lI also indicates that as the direction of the incident radiation moves toward normal incidence, the difference between the non-isotropic representations and the isotropic one becomes greater. At /A= 0 the reduction is approximately 104, while at/c'l.0, the reductions is approximately 45%. For all N, the integrated reflectance is a non-increasing function of / 81

.4 A'o -0.2. =0.9,44 N =4 ~60e.40 ~c,=0.4 Fi ure II-8 Intensity cf Reflection s Function of Emergert Angle, /I vs 1, par meters of,.S6 ]i=4 a-n~ O~.'.32 L 0.6..28 24.20.12 8.o.04.04__.1.2.3.4.5.6.7.8.9 1.0 82

uTO= or pus tl=N'pjo sIamLacGMe d' r SA 0I/I aTiuv u a mJOam: o uoptounO se uoTfoalTJGa Jo sua 6-II aan WaftO/I //D'O N —. — lo Alt 10 = V

90 /20 60 80/ —— 02_ -04 -. 06 -/O -.10. 4.16.16 =4 Nx —2 =0.8 Figure II-10 Intensity of Reflection as Function of Azimuthal Angle, I/Io vs D, parameters of N,,uo=0.4, p.=0.5

wO - 0.9.9.8 w 0 Lr I, at o N Z.68.I C3. vs p., parameters of N

SUMMARY The solution of the transport equation in terms of integral equations is described in general terms. The solution of these equations, by the physical principle of invariance and the use of auxiliary operators, and the probabilistic interpretation are discussed. The formal solutions for a semi-infinite or finite parallel plane dispersion with an obliquely incident parallel-plane source are derived by using auxiliary operators. The possibility of simplifying the numerical evaluation of the formal solutions is also discussed. Numerical computations were carried out for a parallel-plane source obliquely incident on a semi-infinite dispersion. The albedo of the half space and the intensity of the reflected radiation were computed for several phase functions for single scattering, a complete range of albedos for single scattering, and a complete range of angles of incidence for the source. One-, two-, three-, four- and five- term representations were used for the phase function for single scattering. The results maoy be interpreted in two ways. The coefficients of the phase functions may be considered to be completely arbitrary. The one-, two-, three-, four- and five- term phase functions are then interpreted as five real distributions for single scattering. Comparison of the reflected intensity and albedo of the half space for one phase function with another then indicates the effect of an arbitrary degree of anisotropy. The coefficients for the phase functions were actually chosen to obtain the best possible two-, three-, four- and five- term representations for the actual angular distribution for itD/k = 2.2 and m = 1.44. The five term phase function 86

provides a good representation of this distribution hence the reflected intensity and albedo obtained for the five-term phase function can be interpreted as a good approximation for a = 2.2 and m = 1.44. Comparison of the results with previous results for isotropic scattering reveals that the albedo is reduced considerably and the distribution of the reflected intensity is changed critically for the non-isotropic scattering. Thus the assumption of isotropic scattering is not valid for the computation of these quantities. 87

APPENDIX A Tables of Computed Functions The computed values of the intermediate functions H and ~, and the final functions of I/Io and R are tabulated on the following pages. In most cases the values were printed directly from computer output cards or magnetic tape and hence are in the corresponding format. The printed values are in the floating point format of the E conversion of FORTRAN in which the number is followed by the code E + mn. The number mn indicates the number of places the decimal is to be shifted, to the left for E-mn, to the right for E + mn. The succeeding values of the function are printed from left to right across the page. The range and interval of the independent variable are indicated in the headings of Tables A-1 through A-25 but are printed with the function in Tables A-26 through A-28. As an example, the last six values in Table A-1 are.95 0.40311659.96 0.36174924.97 0.31431157.98 0.25747042.99 0.18264862 1.00 0.00000000 88

Index to Tables in Appendix A Table No. Page A-1 H and V -functions, N = 1, wo = 0.9 90 A-2 H and qY -functions, N = 1, wo = o.6 91 A-53 H and yv -functions, N = 1, w = 0.3 92 A-4 V -functions, N = 2, w = O 93 A-5 p -functions, N = 2, = 0.6 95 A-6 V -functions, N = 2, co = 0.3 97 A-7 W-functions, N = 3, wo = 0.9 99 A-8 (W-functions, N = 3, wo = o.6 101 A-9 9)-functions, N = 3, C = 0.3 103 A-10 q -functions, N = 4, 0 =.9 105 A-11 y -functions, N = 4, wo = 0.6 108 A-12 p -functions, N.= 4, w = 0.3 111 A-13 zero order y -functions, N = 2, 2 = 0.9, 0.01 intervals 114 A-14 zero order -functions, N = 2, 0 = 0.6, 0.01 intervals 116 A-15 zero order -functions, N = 2, w0= 0.3, 0.01 intervals 118 A-16 zero order -functions, N = 23, w= 0., 0.01 intervals 120 A-17 zero order p-functions, N = 31, o = 0.6, O.01 intervals 122 A-17 zero order.-functions, N = 3, o = 0.6, 0.01 intervals 124 A-19 zero order i -functions, N = 34, = 0.3, 0.01 intervals 126 A-20 zero order y -functions, N = 4, cO = 0.69 0.01 intervals 129 A-21 zero order 9 -functions, N = 4, co = 0.1, 0.01 intervals 132 A-22 normalized reflected intensity I/T for N = 1, c = 0.9 135 A-23 normalized reflected intensity I/I for N = 2, o = 0.9 14 A-24 normalized reflected intensity I/Io for N = 3, 0o = 0.9 153 A-25 normalized reflected intensity I/Io for N = 4, o = 0.9 162 A-26 normalized reflected intensity I/Io for N =, = 0.9 171 o 0 o 172 A-27 total reflectance, N = 1,2,3 and 4, o =0.9 173 A-28 total reflectance, N = 0, o = 0.9 89

Table A-1 N = 1 = 0.9 R(W) = 0.0(o.o01)1.o 1.00000OOE 00 1.0283370E 00 1.0507700E 00 1.0711854E 00 1.0903159E 00 1.1085109E 00 1.1259733E 00 1.1428348E 00 1.1591876E 00 1.1750995E 00 1.1906225E 00 1.2057974E 00 1.2206572E 00 1.2352290E 00 1.2495356E 00 1.2635961E 00 1.2774270E 00 1.2910428E 00 1.3044559E 00 1.3176773E 00 1.3307169E 00 1.3435835E 00 1.3562848E 00 1.3688281E 00 1.3812199E 00 1.3934660E 00 1.4055718E 00 1.4175425E 00 1.4293824E 00 1.4410960E 00 1.4526871E 00 1.4641595E 00 1.4755165E 00 1.4867615E 00 1.4978974E 00 1.5089272E 00 1.5198534E 00 1.5306786E 00 1.5414052E 00 1.5520355E 00 1.5625716E 00 1.5730157E 00 1.5833696E 00 1.5936352E 00 1,6038143E 00 1.6139086E 00 1*6239198E 00 1.6338494E 00 1.6436989E 00 1.6534697E 00 1.6631633E 00 1.6727810E 00 1.6823241E 00 1.6917937E 00 1.7011912E 00 1.7105176E 00 1.7197742E 00 1.7289619E 00 1.7380819E 00 1.7471350E 00 1.7561225E 00 1.7650451E 00 1.7739039E 00 1.7826997E 00 1.7914334E 00 1.8001059E 00 1.8087180E 00 1.8172705E 00 1.8257642E 00 1.8341998E 00 1.8425782E 00 1.8509001E 00 1.8591661E 00 1.8673769E 00 1.8755333E 00 1.8836360E O0 1.89168t55 00 1.8996825E 00 1.9076276E O0 1.9155214E 00 1.9233646E 00 1.9311577E 00 1.9389013E 00 1.9465959E 00 1.9542421E 00 1*9618404E 00 1.9693914E 00 1.9768956E 00 1.9843534E 00 1.9917655E 00 1.9991322E 00 2,0064541E 00 2.013.7316E 00 2.0209651E 00 2.0281552E 00 2.0353024E 00 2.0424069E 00 2.0494693E 00 2.0564900E 00 2.0634694:E 00 260704078E 00 = 0.0(o.o01)1.o 10OOOOOOOE 00 1.0179011E 00 1*0305114E 00 1.0411276E 00 1.0504357E 00 1.0587575E 00 1,0662804E 00 1.0731270E 00 1.0793837E 00 1.0851142E 00 1.0903679E 00 1.0951835E 00 10995924E 00 1.1036205E 00 1*1072894E 00 1.1106173E 00 1.1136199E 00 1.1163105E 00 1.1187007E 00 1.1208009E 00 1.1226196E 00 1*1241649E 00 1*12544~35E 00 1.1264616E 00 1.1272245E 00 1.1277368E 00 1.1280029E 00 1.1280263E 00 1.1278102E 00 1.1273575E 00 1.1266706E 00 1.1257516E 00 1.1246021E 00 1.1232236E 00 1.1216174E 00 1.1197842E 00 1.1177247E 00 1.1154393E 00 1.1129280E 00 1.1101909E 00 1.1072276E 00 1.1040375E 00 1.1006199E 00 1.0969738E 00 1.0930980E 00 1.0889910E 00 1.0846512E 00 1.0800767E 00 1.0752654E 00 1.0702149E 00 1.0649227E 00 1.0593859E 00 1.0536012E 00 1.0475655E 00 1.0412748E 00 1,0347254E 00 1l0279127E 00 1.0208321E 00 1,0134787E 00 1.0058469E 00 9.9793087E-01 9.8972436E-01 9.8122057E-01 9.7241215E-01 9.6329120E-01 9.5384926E-01 9.4407706E-01 9.3396472E-01 9.2350144E-01 991267556E-01 9.0147442E-01 8, 8988421E-01 8.7788990E-01 8,6547506E-01 8.5262167E-01 8.3930992E-01 8.'2551799E-01 8.1122172E-01 7.9639427E-01 7.8100575E-01 7.6502256E-01 7.4840696E-01 7.3111615E-01 7.1310130E-01 6.9430638E-01 6.7466648E-01 6.5410583E-01 6.3253494E-01 6.0984693E-01 5.8591235E-01 5.6057186E-01 5.3362552E-01 5.0481660E-01 4.7380579E-01 4*4012826E-01 4.0311659E-01 3.6174924E-01 3.1431157E-01 2.5747042E-01 1.8264862,E-01. 0. 9o

Table A-2 N1 =0.6 = 0.0(o.ol)1.o 1.OOOOOOOE 00 1.0176695E 00 1.0312603E 00 1.0433998E 00 1..0545938E 00 1.0650865E 00 1.0750216E 00 1.0844936E 00 1.0935694E 00 1.1022987E 00 1.1107201E 00 1.1188644E 00 1.1267567E 00 1.1344179E 00 1.1418656E 00 1.1491151E 00 1.1561793E 00 1.1630700E 00 1.1697972E 00 1.1763700E 00 1.1827966E 00 1.1890842E 00 1l. 952395E 00 1*2012687E 00 1.2071772E 00 1.2129702E 00 1.218.6523E 00 1.2242281E 00 1.2297014E 00 1l2350762E 00 1.2403559E 00 1.2455439E 00 1.2506432E 00 1.2556569E 00 1.2605875E 00 1.2654379E 00 1.2702103E 00 1.2749072E 00 1.279'5308E 00 1.2840832E 00 1.2885664E 00 1.2929822E 00 1.2973326E 00 1.3016192E 00 1.3058438E 00 113100079E 00 1.3141130E 00 1.3181607E 00 1.3221522E 00 1.3260891E 00 1.3299725E 00 1.3338038E 00 1.3375841E 00 1.3413145E 00 1. 3449963E 00 1.3486305E 00 1.3522181E 00 1.3557601E 00 1.3592575E 00 1.3627113E 00 1.3661224E 00 1,3694915E 00.1,3728197E 00 1.3761077E 00 1.13793563E 00 1-3825664E 00 1.3857385E 00 1.3888736E 00 1.3919723E 00 1.3950352E 00 1.3980631E 00 1.4010566E 00 1.4040164E 00 1.4069430E 00 1.4098371E 00 1.4126993E 00 1.4155300E 00 1.4183299E 00 I.4210996E 00'1.4238394E 00 1.4265501E 00 1.4292319E 00 1.4318855E 00 1.4345113E 00 1.4371098E 00 1.4396814E 00 1.4422266E 00 1.4447457E 00 1.4472393E 00 1*4497077E 00 1 45.21514E 00 1.4545707E 00 1.4569660E 00 1.4593376E 00 1.4616860E 00 1 4640116E 00 1 4663146E 00 1 4685953E 00 1.4708543E 00 1,4730916E 00 1.4753078E 00 qt(A~&L~~) A = 0.0(0.01)1.0 1.O0000000E 00 1.0114239E 00.1.0192629E 00 1.0257310E 00 1.0312916E 00 1.0361627E 00 1.0404715E 00 1.0443011E 00 1.0477101E 00Q 1.0507419E 00 1.0534298E 00 1.0557999E 00 1.0578735E 00 1.05966'79E 00 1.06119-17E 00 1.0624749E 00 1.0635100E 00 1.0643119E 00 1.064.8881E 00 1.0652453E 00 1.0653892'E 00 1.0653248E 00 1.0650565E 00 1.0645881E 00 1.0639228E 00 1.0630636E 00 1.0620129E 00 1.0607729E 00 1.05934'53E 00 1.0577317E 00 1.0559332E 00 1.0539510E 00 1.0517857E 00 1.0494.379E 00 1.0469078E 00 1,0441958E 00 1.0413015E 00 1.0382249E 00 1.0349654E 00 1.0315224E 00 1.0278952E 00 1.0240829E 00 1.0200841E 00 1.0158977E 00 1.0115221E 00'1.0069558E 00 1.0021967E 00 9.9724299E-01 9.9209229E-01 9.8674222E-01 9,8.119016E-01 9.7543324E-01 9.6946839E-01 9.6329225E-01 9.5690131E-01 9.5029171E-01 9,4345934E-01 9.3639976E-01 9.2910830E-01 9.2157983E-01 9.1380892E-01 9.057-8973E-01 8.9751602E-01 8.8898104E-01 8.8017753E-01 8.7109770E-01 8.6173321E-01 -8.5207492E-01 8.4211311E-OI 8*.3183717E-01 8.2123560E-01'8.1029590E'-01 7.9900451E-01 7.8734657E-01 7.7530581E-01 7.6286436E-01 7.5000260E-01 7.3669872E-01 7.2292857E-01 7.0866526E-01' 6.9387860E-01 6.7853462E-01 6.6259486E-01' 6.4601545E-01 6.2874605E-01 6.1072834E-01 5.9189425E-01 5.7216338E-01 5.5143972E-01 5.2960700E-01 5.0652205E-01 4.8200524E-01 4.5582584E-01 4.2767891E-01 3.9714659E-01 3.6362882E-01 3.2620693E-01 2.8333859E-01 2.3202474E-01 1,6454576E-01 0. 91

Table A-3 N = 1 w=o - 0.3 ) =.0(o.o01)1.o 1.OOOOOOOE 00 1.0084846E 00 1.0148796E 00 1.0205148E 00 1.0256506E 00 1.0304140E 00 1.0348804E 00 1.0390996E 00 1.0431076E 00 1.0469308E 00 1.0505902E 00 1.0541025E 00 1.0574814E 00 1.0607383E 00 1.0638830E 00 1.U669239E UOU 1U0698681t UO 1oU727223E O0 1,0754919E 00 1.0781821E 00 1,0807973E 00 1.0833418E 00 10858192E 00 1,0882328E 00 1.0905859E 00 1.0928813E 00 1.0951215E 00 1.0973091E OQ 1.0994464E 00 1.10153.52E 00 1.1035778E 00 1.1055759E 00 1.1075312E 00 1.1094453E 00 1.1113198E 00 1.1131561E 00 1.1149555E 00 11167194E 00 1.1184489E 00 1.1201451E 00 1.1218092E 00 1,1234421E 00 1.1250449E 00 1.1266185E 00 1.1281639E 00 1.1296817E 00 1.1311729E 00 1.1326382E 00 1.1340784E 00 1.1354942E 00 1.1368862E 00 1.1382552E 00 1.1396016E 00 1.1409263E 00 1.1422296E 00 1.1435122E 00 1.1447746E 00 1.1460173E 00 1.1472408E 00 1.1484456E 00 1.1496322E 00 1.1508010E 00 1.1519523E 00 1.1530868E 00 1,1542046E 00 1.1553062E 00 1.1563921E 00 1.1574625E 00 1.1585178E 00 1.1595583E 00 1.1605844E 00 1 1615964E 00 1.1625946E 00 1,1635793E 00 1.1645507E 00 1.1655091E 00 1.1664550E 00 1.1673883E 00 1.1683096E 00 1o1692189E 00 1.1701-165E 00 1.1710027E 00 1.1718777E 00 1.1727418E 00 1 1735950E 00 1.1744377E 00 1.1752700E 00 1.1760921E 00 1,1769044E 00 1.1777068E 00 1.1784996E 00 1.1792830E 00 1.18005.71E 00 1.1808222E 00 1.1815784E 00 1.1823258E 00 11830646E 00 1.1837950E 00 1l1845171'E 00 1.1852310E 00 1.1859369E 00 = 0.0(0.01)1.0 1.OOOOOOOE 00 1.0054860E 00 1.0091214E 00 1.0120200E 00 1.0144158E 00 1.0164192E 00 1.0180944t'00 1.0194832E 0( 1.0206153E 00 1.0215123E 00 1.0221910E 00 1.0226642E 00 1.0'229426E 00 1.0230345E 00 1.0229472E 00.O0226863E CO0 1,0222569E 00 1.0216631E 00 1.0209084E 00 1.0199958E 00 1.0189278E 00 1.0177066E 00 1.0163339E 00 1.0148112E 00 1.0131398E 00 1.0113207E 00 1.0093545E 00 1.0072420E 00 1.0049834E 00 1.0025790E 00 1.0000288E 00 9,9733281E-01 9.9449072E-01 9.9150215E-01 9.8836658E-01 9*8508-341E-01 9.8165187E-01 9.7807101E-01 9.7433985E-01 9. 7045724E-l01 9.6642190E-01 9.6223243E-01 9.5788730E-01 9.5338484E-01 9.4872324E-01 9.4390059E-01 9.3891475E-01 9*3376354E-01 9.2844452E-01 9.2295514E-01 9.1729267E-01 9.1145420E-01 9.0543659E-01 8.9923654E-01 8.9285053E-01 8.8627478E-01 8.7950531E-01 8.7253783E-01 8,6536779E-01 8.5799034E-01 *8.5040027E-01 8.4259207E-01 8.3455980E-01 8.2629712E-01 8.1779722E-01 810905282E-01 8.0005608E-01 7.9079852E-01 7.8127107E-01 7.7146386E-01 7.6136626E-01 7.5096670E-01 7.4025258E-01 7.2921020E-01 7.1782455E-01 7.0607916E-01 6.9395589E-01 6.8143471E-Q1 6.6849337E-01 6.5510714E-01 6.4124826E-01 6,2688549E-01 6.*1198351E-01 5.9650199E-01 5.8039473E-01 5.6360816E-01 5.4607977E-01 5.2773577E-01 5,0848801E-01 4. 8822973E-01 4.6682951E-01 4.4412246E-01 4,1989673E-01 3.9387233E-01 3.6566554E-01 3.341Z528E-L- 3.U002U07(0t-01 2,6069588E-01 2,1343414E-01 1.5132783E-01 0. 92

Table A-4 N= 2 o, = 0.9 2(II) i = 0.0(0.05)1.0 1.OO0000OE 00 1.1264548E 00 1.2082793E 00 1,2733841E 00 1.3267149E 00 1.3706533E 00 1.4066158E 00 1.4355569E 00. 1.4581745E 00 1.4750093E 00 1.48649.82E 00 1.4930054E 00 1.4948419E 00 1.4'922786E'00 1.4855542E 00 1.4748826E 00 1.4604564E 00 1.4424508E 00 1,4210262E 00 1.3963301E 00 1.368'4989E 00 0. 1,9984339E-02 4..3576432E-02 7.0055772E-02 9.9026692E-02 1.3020 245E-01 1.6335203E-01 1.9827907E-01 2.3481141E-01 2.7279513E-01 3.1209080E-0.1 3.5257086E-01 3.9411789E-01 4.3662315E-01 4.7998556E-01 5.2411085E-01 5.6891079E-01 6.1430273E-01 6,6020897E-01 7.0655639E-01 7.532760:9E-01 -5. 0000000-01-5.39.26057E-01-5.4950182E-01-5.4498410E-01-5.2818701E-01 -5 0022963E-01-4.6171979E-01-4. 1302330E-01-3.5437488E-01-2 8593139E-01 -2.0780059E-01-1.2005774E-01-2.2755796E-02 8.4067984E-02 2.0038761E-01 3.2618539E-01 4.6144972E-01 6.0617367E-01 7.6035370E-01 9.2398894E-01 1.0970805E 00 (/') _ _ = o.o0(0o.o05)1.o0 1.OOOOOOOE 00 1.0549172E 00 1.0797455E 00 1.0913002E 00 1.0929237E 00 1.0863083E 00 1.0724916E 00 1.0521740E.00 1.0258484E 00 9.9385951E-01 9.5643032E-01 9.1366905E-01 8.6555901E-01 8.1192983E-01 7.523987E-l01 6.8625672E-01 6.1223723E-01 5.2798660E-01 4.2860129E-01 3.0083761E-01 O. 0. 1.1558236E-01 2.4109793E-01 3.7259167E-01 35.0734775E-01 6.4303775E-01 7.7748515E-01 9.0855154E-01 1.0340590E 00 1.1517210E 00 1.2590677E 00 1.3533529E 00 1.4314261'E 00 1.4895430E 00 1.5230623E 00 1.5259229E 00 1.4896481E 00 1.4011854E 00 1.2371883E 00 9.4233382E-01 0. 93

Table A-4 cont. N =2 D =0. 9 z 2 ~~u GII) 0.0(oIi= o.o(o.oi)i.o 3.0000000E 00 3.0475735E 00 3.0794794E 00 3.1053199E 00 3.1270935E 00 3.1457361E 00 3.1617919E 00 3.1756161E 00 3.1874578E 00 3*1975014E 00 3.2058879E 00 3.2127289E 00 3.2181141E 00 3.22.21176E 00 3.2248011E 00 3.2262169E 00 3.2264097E 00 3.2254181E 00 3.2232758E 00 3.2200123E 00 3.2156535E 00 3.2102228E 00 3.2037407E 00 3.1962259E 00 3.1876950E 00 3.1781632E 00 3.1676442E 00 3.1561507E 00 3.1436939E 00 3.1302845E 00 3.1159320E 00 3.1006453E 00 3.0844328E 00 3.0673018E 00 3.0492596E 00 3.0303126E 00 3.0104669E 00 2.9897283E 00 2.9681020E 00 2.9455931E 00 2.9222061E 00 2.8979454E 00 2.8728152E 00 2.8468193E 00?.8199613E 00 2.7922446E 00 2.7636726E 00 2.7342482E 00 2.7039743E 00 2.6728538E 00 2.6408890E 00 2.6080826E 00 2.5744369E 00 2.5399540E 00 2.5046362E 00 2.4684853E 00 2.4315033E 00 2.3936921E 00 2.3550534E 00 2.3155887E 00 2.2752998E 00 2.2341881E 00 2.1922551E 00 2.1495021E 00 2.1059306E 00 2.0615416E 00 2.0163366E 00 1.9703166E 00 1.9234827E 00 1.8758361E 00 1.8273777E 00 1.7781086E 00 1.7280296E 00 1.6771418E 00 1.6254460E 00 1.5729430E 00 1.5196337E 00 1.4655189E 00 1.4105993E 00 1.35.48756E 00 1.2983485E 00 1.2410188E 00 1.1828871-E 00 1.1239540E 00 1.0642201E 00 1.0036860E 00 9.4235234E-01 8.8021959E-01 8.1728835E-01 7.5355908E-01 6.8903230E-01 6*2370852E-01 5.5758819E-01 4.9067175E-01 4.2295964E-01 3.5445231E-01 2*8515016E-01 2.1505359E-01 1.4416301E-01 7.2478739E-02 94

Table A-5 N =2 uoo =0.6 qJ (L) A = 0.0(0.05)1.0 1.0000000E 00 1.0708033E 00 1*1103139E 00 1.1383042E 00 1.1585776E 00 1.1730250E 00 1.1827946E 00 1.1886671E 00 1.1912129E 00 1.1908693E 00 1.1879841E 00 1*1828408E 00 1.1766754E 00 1.1666876E 00 1.1560483E 00 1.1439057E 00 1.1303891E 00 1.1156123E 00 1.0996762E 00 1.0826706E 00 1.0646760E 00 0. 3.5077586E-02 7.3647319E-02 1.1469450E-01 11.5767226E-01 2.0220175E-01 2*.4799405E-01 2.9481741E-01 3.4248012E-01 3.9082025E-01 4.3969893E-01 4.8899573E-01 5.3860523E-01 5.8843448E-01 6 3840099E-01 6.8843119E-01 7.3845915E-01 7.8842554E-01 8.3827678E-01 8.8796434E-01 9.3744407E-01 -5.0000OOOE-0 1-5 222 50938-E-1- 5. 2349768E-01-5.1261586E-01-4. 9151173E-01 -4. 6090080E-01-4.2115359E-01-3. 7248068E-01-3. 1500827E-01-2.4881412E-01 -1.7394655E-01-9.0435241E-02 1.7022863E-03 1.0245657E-01 2.1182354E-01 3.2980285E-01 4.5639659E-01 5.9160855E-01 7.3544361E-01 8.8790736E-01 1.0490058E 00,) = o.o(o.05)1.o 1.OOOOOO00000E 00 1.0341964E 00 1.0474086E 00 1.0511089E 00 1.0476032E 00 1.0380346E UO 1.U230753L 0U0 1.UU31475E O0 9.7851:411E-U1 9.4931794E-01 9. 1559740E-01 8.7728540E-01 8.3419392E-01 7.8597973E-01 7.3207976E-01 6.7158951E-01 6.0301880E-01- 5.2373904E-01 4. 2846592E-01 3.0329179E-01 0. 0. 1.2848249E-01 2.6356301E-01 4.0186038E-01 5.4100739E-01 6.7895759E-01 8.1378.607E-'01 9.4358952E-01 1.0664158E 00 1.1802000E 00 1.2826931E 00 1.3713744E 00 1.4433274E 00 1.4950578E 00 1.5222001E 00 1.5190103E 00 1.4773993E 00 1'.3848311E 00 1.2187519E 00 9.2543422E-01 0. 95

Table A-5 cont. N = 2 0o =0.6 ~ 2 W~~~~~~~~~~~ o.o(o.ol)l.o 3.0000000E 00 3.0306444E 00 3.0506034E 00 3.0663151E 00 3.0791244E 00 3.0896620E 00 3.0982945E 00 3.1052610E 00 3.1107293E 00 3*1148230E 00 3.1176369E 00 3.1192456E 00 3.1197093E 00 3.1190'773E 00 3.1173908E 00 3.1146847E 00 3.1109885E G0 3.1063279E 0O0 3.1007251E 00 3.0941997E 00 3.0867686E 00 3.0784473E 00 3.0692493E 00 3.0591868E 00 3.0482706E 00 3.0365107E 00 3.0239160E 00 3.0104947E 00 2.9962542E 00 2.9812014E 00 2.9653424E 00 2.9486830E 00. 2.9312285E 00 2.9129838E 00 2.8939534E 00 2.8741417E 00 2.8535523E 0O0 2.8321890E 00 2.8100553E 00 2.?7871541E 00 2,7634886E 00 2,7390614E 00 2.7138753E 00 2.6879326E 00 2.6612357E 00 2.6337867E 00 2.6055878E 00 2.5766406E 00 2,5469473E 00 2*5165093E 00 2.4853285E 00 2,4534063E 00 2.4207442E 00 2.3873435E 00 2.3532057E 00 2.3183319E 00 2.2827234E Q0 2.2463813E 00 2.2093066E 00 2.1715005E 00 2.1329638E 00 2.0936976E 00 2,0537027E 00 2.0129800E 00 1.971530'4E O0 1*9293545E 00 1.8864532E 00 1.8428272E 00 1.7984771E 00 1.7534037E 00 1.7076076E 00 1.6610893E 00 1.6138494E 00 1.5658886E 00 1,5172074E 00 1.4678062E 00 1.4176857E 00 1.3668462E 00 1.3152882E 00 1.2630122E 0 1.2100186E 00 1.1563079E 00 1.10(18 8 04E 00 1.0467365E 00 9.9087662E-01 9.3430107E-01 8.7701021E-01 8.1900438E-01 7.6028394E-01 7.0084915E-0 1 6.40700'32E-01 5.7983778E-01 5.1826180E-01 4.5597265E-01 3.929705'8E-01 3.2925587E-01 2*6482876E-01 1.9968950E-01 1.3383834E-01 6.7275454E-02.

Table A-6 N = 2 = 0.3 o(F),u= o0.0(0.05)1.0 1.0000000E 00 1.0318822E 00 1.0480444E 00 1.0586778E 00 1.Q0657710E 00 1.0703009E 0.0 1*07'28504E 00 1.0738072E 00 1.0734480E 00 1.0719797E 00 1.0695627E 00 1.06632.44E 00 1.0623683E 00 1.0577799E 00 1.0526306E 00 1.'0469809E 00 1.0408826E 00' 1.0343802E 00 1.0275123E 00 1.0203129E 00 1.0128116E 00 0. 4.3615760E-02.8.9167032E-02 1.35985.18E-01 18372721E-01 2.3216694E'-01 2,8113971E-01 3.3051907E-01 3.8020454E-01 4.3011438E-01 4.8'018'088E-01 5.3034718E-01 5.8056501E-01 6,3079295E-01 6.80995'24E-01 7.3114077E-01 7.8120227E-01 8.3115578E-01 8.8098007E-01 9-o3065630E-01 9.8016766E-01 ~-5.0000000E-01-5.0841818E-01-5.0249.240E-01-4.8706768E-01-4.62944.85E-0 1 -4.3045863E-01-3.8977536E'-01-3.4098544E-0.1-2.8414043E-01-2 1927027E-0 1 -1 i4639225E-01-6.5515941E-02 2.3353891E-02 1.2021554E-01 2. 2506920E-01 3.3791632E-01 4.5875909E-01 5.8760015E-01 7.2444241E-01 8.6928887E-01 1.0221425E 00 / (P) s = o.o(o.05)1.0 OOOO00000E 00.10156574E 00 1.0195660E 00 1.0175564E 00 1.0107857E 00 9.9979613E-01 9.8487247E-01 9.6615511E-01 9.4368426E-01 9. 1741637E-01 8.8722601E-01 8.5289618E-01 8. 1409613E-01 7,7034147E-01 7.2092344E-01 6.6477952E-01 6.0023677E-01 5.2443612E-01 4.3175968E-01 3.076752.7E-01 0. 0. 1.3972299E-01 2.8230300E-01 4.2532108E-01 5.6694506E-01 7.0551.473E-01 8.3941516E-01 9.6700'746E-01 1.0865749E 00 1.1962693E 00 1.2940481E 00 1.3775920E 00 1.4441890E 00 1.4905585E 00 1.5125688E 00 1.5047450E 00 1.4593299E 00 1.3642388E 00 1.197630.6E 00 9.0726796E-01 0. 07.

Table A-6 cont. N = 2'O =0.3 z o ct~(W) g. 0.0(0.01)1.0 3.0000000E 00 3.0147511E 00 3.0238456E 00 3.0305186E 0 3.0354477E 00 3.0389505E 00 3.0412117E 00 3.0423514E 00 390424539E 00 3.0415813E 00 3.0397807E 00 3.0370896E 00 3.0335380E 00 3.0291502E 00 3.0239469E 00 3.0179451E 00 3.0111596E 00 3.0036030E 00 2.9952862E 00 2.9862188E 00 2.9764092E 00 2.9658649E 00 2.9545924E 00 2o9425978E 00 2.9298864E 00 2.9164629E 00 2.9023317E 00 2.8874967E 00 2,8719616E 00 2.8557296E 00 2.8388036E 00 2.8211866E 00 2,8028811E 00 2.7838892E 00 2.7642133E 00 2.7438554E 00 2.7228173E 00 2.7011007E 00 2.6787073E 00 2,6556386E 00 2.6318959E 00 2.6074807E 00 2.5823940E 00 2.556637VE 00 2.5302113E 00 2."5031172E 00 2.4753560E 00 2.4469287E 00 2.4178359E 00 2.3880786E 00 2.3576575E 00 2.3265733E 00 2.2948267E 00 2.2624184E 00 2.2293490E 00 2.1956190E 00 2.1612292E 00 2.1261798E 00 2.0904715E 00 2.0541048E 00 2.0170800E 00 1.9793977E 00 1.9410583E 00 1.9020621E 00 1.8624095E 00 1.8221010E 00 1,.7811368E 00 1.7395172E O0 1.6972427E 00 1.6543134E 00' 1.6107298E 00 1.5664921E 00 1.5216005E 00 1,4760552E 00 1.4298567E 00 1 3835E 1350050 00 1.3355005 87343 0 1.2385337E 00 1. 1890'7185E 00 1.1389579E 00 10881921E (0 1.0367747E 00 9.8470581E-01 9,3198559E-01 8.7861420E-01 8.2459183E-01 7.6991860E-01 7.1459469E-01 6,5862022E-01 6,0'199532E-01 5.4472015E-01 4,8679485E-01 4.2821950E-01 3,6899424E-01 3.0911921E-01 2.4859449E-011 1.8742022E-01 1.2559650E-01 6,3123392E-02 98

Table, A-7 N = 3 (o = 9 ) =0.0(0.05)1.0 1.OO00000E 00 1.1090948E 00 1.1784873E 00 1.2325616E 00 1.2759064E 00 1.3108692E 00 1.3389520E 00 1.3612517E 00 1.3786396E 00 1.3918488E 00 1.4015206E 00 1.4082323E 00 1.4125140E 00 1.4148596E 00 1.4157351E 00 1.4155833E 00 1.4148283E 00 1.4138783E 00 1.4131279E 00 1.4129598E 00 1.4137464E 00 0. 2.1563794E-02 4.69356'88E-02 7.5427417E-02 1.0665153E-01 1.4032027E-01 1.7619636E-01 2.1407374E-01 2.5376803E-01 2.9511120E-01 3.3794818E-01 3.8213462E-01 4.2753528E-01 4.7402284E-01 5.2147696E-01 5.6978353E-01 6.1883409E-01 6.6852530E-01. 7.1875851E-01 7.6943936E-01 8.2047750E-0 1 -5.0000000E-01-5.2949727E-01-5.3279612E-01-5.2244912E-01-5.0082937E-01,46911364E-01-4o.2802757E-01-3.7808274E-01-3 *1967439E-01-2.5312873E-01 -1.7872825E-01-9.6.726561E-02-7.3574306E-03 8.9159335E-02 1.9261395E-01 3.0280400E-01 4.1953248E-01 5.4260649E-01 6.7183617E-01 8.0703399E-01 9.4801418E-01 0. - 5.4683519E-02-1.1291829E-01-1.7111009E-01-2.2652079E-O1 -2.7668791E-01-3.1927947E-01-3. 5203687E-01-3.7274702E-01-3. 7922696E-01 _3.6931474E-01-3. 4086349E-01-2.9173760E-01-2. 1980995E-0:-1.2296005E-01' 9.2734861E-04 1.5396334E-01 3.3825561E-01 5.5590906E-01 8.0902628E-01 1.0997079E 00 (= 0.0(0o.o05)1.0.r. 1.0000000E 00 1.0833873E 00 1.1258603E 00. 1.1516815E 00 1.1653839E 00 1.1691686E 00 1.1643265E 00 1.1516879E 00 1.1318053E 00 1.1050358E O00 1.0715789E 00 1.0314857E 0:0 9.8464812E-01 9.3076187E-01 8.6925316E-017.9913609E-01 7.1872346E-01 6.2496949E-01 5.-1166148E-01 3.6229973E-01 0. 0.. 1.02'31153E-01 2.1479731E-01 3.3299377E-01 4.5402256E-01 5.7552296E-01 6.9536554E-01 8.:1152112E-01 9.2197800E-01 1.0246739E 00 1.1174270E 00 1.1978541E 00 1,2632600E 00 1.3.104766E 00Q 1.3356055E 00 1.3335758E 00 1.2 973001E 00 1.2158406E 00 1.0695433E 00 8.1154013E- 01 0.. -1.5000000E 00-1.5630100E 00-1.5346645E 00-1,4513514E 00-1.3203204E 00 -1.1460468E 00-9.3243051E-O1-6.8352874E-01-4.0390873E-01-9.888 6697E-02 2.2525211E-01 5.6099938E-01 8.9928755E-01 1.2290027E 00 1.5362262E OQ 1.8029595E 00 2.0047706E 00 2.1058496E 00 2.0463701E 00 1.6963576E 00 O. 99N

Table A-7 cont. N 3 aJ = 0.9 z O Y () =0.0(0.05)1.0 2=a 2 3.0000000E 00 3.0726222E 00 3.0837607E 00 3.0629796E' 00 3.0163701E 00 2.9471084E 00 2.8572460E 00 2.7482872E 00 2.6214317E 00 2.4776934E 00 2.3179659E 00 2.1430603E 00 1.9537298E 00 1.7506855E 00 1.5346067E 00 1.3061488E 00 1.0659484E 00 8.1462760E-01 5.5279668E-01 2.8105654E-01 0, 113 0. 6.8097395E-01 1.3803135E 00 2.0769213E 00 2.7544169E 00 3.3980141E 00 3.9937092E 00 4.5279500E 00 4.9874697E 00 5.3591934E 00 5.6301806E 00 5.7875888E 00 5.8186484E 00 5.7106450E 00 5.4509072E 00 5.0267981E 00 4.4257069E 00 3.6350453E 00 2..6422426E 00 1.4347427E 00 0. = 0.o(o.oi)i.o 1.5000000E 01 1.5110094E 01 1.5176933E 01 l.5225381E 01 1.5260625E 01 1.5285092E 01 1.5300189E 01 1.5306837E 01 1.5305678E 01 1.5297191E 01 1.5281742E 01 1.5259623E 01 1.5231074E 01 1.5196292E 01 1.5155448E 01 1.5108688E 01 1.5056142E 01 1.4997923E 01 1.4934138E 01 1.4864882E 01 1.4790244E 01 1.4710306E 01 1.4625148E 01 1.4534845E 01 1.4439469E 01 1.4339090E 01 1.4233778E 01 1.4123599E 01 1.4008620E 01 1.3888907E 01 1.3764527E 01 1.3635545E 01 1.3502027E 01 1.3364040E 01 1.3221652E 01 1.3074930E 01 1.2923945E 01 1.2768765E 01 1.2609463E 01 1.2446111E 01 1.2278785E 01 1*210755'9E 01 1.1932511E 01 1.1753720E 01 1.1571269E 01 1.1385241E 01 1.1195720E 01 1.10'02795E 01 1.0806554E 01 1.0607091E 01 1.0404501E 01 1.0198881E 01 9.9903309E 00 9.7789543E 00 9.5648575E 00 9.3481488E 00 9.1289423E 00 8.9073527E 00 8.6835006E 00 8.4575088E 00 8.2295047E 00 7.9996198E 00 7.7679902E 00 7.5347562E 00 7.3000627E 00 7.0640606E 00 6.8269052E 00 6.5887579E 00 6,3497868E 00 6.1101650E 00 5,.8700736E 00 5.6297008E 00 5.3892426E 00 5,1489028E 00 4.9088954E 00 4.6694434E 00 4,4307806E 00 4.1931524E 00 3.9568165E 00 3.7220452E 00 3.4891253E 00 3.2583606E 00 3.0300745E 00 2.804610.6E 00 2.5823371E 00 2.3636488E 00 2.1489723E 00 1l9387703E 00 1,.7335479E 00 1.5338608E 00 1.3403259E 00 1.1'536350E 00 9.7457412E-01 8,0405068E-01 6.431335 3E-01 4.9311577E-01 3.5561940E-01 2.3278839E-01 1.2770019E-01 4.5498747E-02 100

Table A-8 N.= 3 co o.o o6 0 (,) Ii = 0.0(0.05)1.0 ho 1.0000000EE 00 1.0615362E 00 1.0949317E 00 1.1174879E 00 1.1327569E 00 1.1426112E 00 1.1482622E 00 1.1505899E 00 1'1502812E 00 1.1479001E 00 1.1439257E 00 1.1387759E 00 1.1328228E 00 1.1264029E 00 1.1198242E 00 1.1133721E 00 1.1073128E 00 1.1018972E 00 1.0973624E 00 1.Q0939345E 00 1.0918298E 00 0. 3.5581290E-02 7.4788772E-02 1.1667202E-01 1.6069660E-01 2.0648122E-01 2.5372840E-01 3.0219505E-01 3.5167682E-01 4.0'199860E-01 4.5300822E-01 5 0457206E-01 5.5657170E-01 6.0890151E-01 6.6146663E-01 7.1418146E-01 7,6696838E-01 8.1975666E-01 8.7248170E-01 9.2508416E-01 9. 7750943E-01 -5.0000000E-0- 69E-9E-01-5. 14086331E-01-5. 0003.157E-0 1-4. 76304.10E-01 -4.4365391E- 1-4 0252532E-01- 53 22000E-01-2.9596.384E'01 -2.30939.08E-01 -1.5830132E-01-7.8189282E-02 9.2692935E-03 1.0395397E-01 2.0574916E-01 3.1454259E-01 4.3022413E-01 5.5268516E-01 6.8181792E-01 8.1751525E-01 9,5967025E-01 0. -6.3367311E-02-1.2877538E-01-1.9293524E-0-.1 25328528E-01 — 3.0750225E-01-3.5337704E-01-3.8876494E-01-4. 1156120E-0 1-4.1 968744E-01 -4.1108345E-01-3.8370197E-01-3.3550524E-01-2.6446258E-01-1.6854871E-01 -4.5742542E-02 1.0597378E-01 2.8861550E-01 5.0419584E-0.1 7.5472631E-01 1.0422171E 00 (g)t (1,= 0.0(0.05)1.0 1.00OOOOOOE 00 1.0504607E 00 1.0727726E 00 1*0835190E 00 1.0858254E 00 1.0811799E 00 1.0704241E 00 1.0540681E 00 1,0324162E 00 1.0056239E 00 9.7372026E-01 9.3661003E-01 8.9405798E-01 8.4565129E-01 7.9072750E-01 7.2823845E-01 6,5647690E-01 5.7245905E-01 4.7023242E-01 3.3423610E-01 0. 0. 1.2194987E-01 2..5084645E-01 3.8277185E-01 5.1517922E-01 6.4596297E-01 7.7320448E-01 8.9505306E-01 1.0096484E 00 1.1150552E 00 1.2091943E 00 1.2897604E 00 1.3541083E 00 1.3990868E 00 1.4207719E 00 1.4140059E 00 1.3715180E 00 1.2820105E 00 1.1250748E 00 8.5185369E-01 0. -1.5000000E 00-1.5322047E 00-1.4906456E 00-1.3998143E 00-1.2648534E 00 -1.0892702E 00-8.7642132E-01-6.3UO1386E-')1-3.5436054E-01-5.4569604E-02 2.6326344E-01 5.9182065E-01 9.2224030E-01 1,2436442E 00 1.5423959E 00 1.8008499E 00 1.9950379E 00 2.0898035E 00 2,0263570E 00 1.6768151E 00 O. 101

Table A-8 cont. N = 3 =0.6 cLXQ(.) IL = 0.0(0.05)1.0 3.0000000E 00 3.0450343E 00 3.0440150E 00 3,0168959E 00 2.9677760E 00 2.8987708E 00 2.8112314E 00 2.7061370E 00 2.5842608E 00 2.4462506E 00 2.2926728E 00 2.1240387E 00 1.9408210E 00 1.7434639E 00 1.5323909E 00 1.3080093E 00 1.0707142E 00 8.2089100E-01 5.5891744E-01 2.8516507E-01 O,. 2=3 0. 7.0425296E-01 1.4172309E 00 2.1207588E 00 2.8000478E 00 3.4414534E 00 4.0318910E 00 4.5585974E 00 5.0090114E 00 3.3707073E 00 5.6313534E 00 5.7786866E 00 5.8004941E 00 5.6846014E 00 5.4188638E 00 4.9911599E 00 4.3893863E 00 3.6014552E 00 2.6152901E 00 1.4188250E 00 0. = o.o(o.oi)i.o A_3'1.5000000E 01 1.5071744E 01 1.5113130E 01 1.5140770E 01 1..5158135E 01 1.5166847E 01 1.5167850E 01 1.5161759E 01 1.5149006E 01 1.5129915E 01 1.5104731E 01 1.5073653E 01 1.5036846E 01 1.4994446E 01 1.4946572E 01 1.4893326E 01 1.4834800E 01 1.4771079E 01 1.47U2238L 01 1.4628350E 01 1.4549482E 01 1.4465697E 01 1.4377057E 01 1.4283621E 01 1.4185450E 01 1.4082599E 01 1.3975125E 01 1.3863087E 01 1,3746539E 01 1.3625539E 01 1.3500145E 01 1.3370414E 01 1.3236405E 01 1.3098178E 01 1.2955793E 01 1.2809313E 01 1.2658800E 01 1.2504318E 01 1.2345935E 01 1.2183716E 01 1.2017732E 01 1.*1848052E 01 1.1674750E 01 1,1497901E 01 1.1317580E 01 1.1133868E 01 1.0946843E 01 1.0756590E 01 1.0563194E 01 1.0366743E 01 1.0167328E 01 9.9650404E 00 9.7599775E 00 9.5522375E 00 9.3419223E 00 9.1291366E 00 8.9139887E 00 8.6965898E 00 8.4770555E 00 8.2555041E 00 8.0320581E 00 7.8068444E 00 7.5799941E 00 7.3516425E 00 7.1219295E 00 6.8910007E 00 6.6590062E 00 6.4261022E 00 6.1924505E 00 5.9582192E 00 5.7235828E 00 5o4887239E 00 5.2538318E 00 5,0191040E 00 4.7847477E 00 4.5509786E 00 4.3180234E 00 4.0861197E 00 3.8555172E 00 3.6264799E 00 3.3992858E 00 3.1742293E 00 2.9516241E 00 2.7318033E 00 2.5151241E 00'.2.3019696E 00 2.0927536E 00 1.8879251E 00 1.6879743E 00 1.4934407E 00 1.3049231E 00 1.1230934E 00 9.4871482E-01 7.8266877E-01 6.2599360E-01 4.7994581E-01 3.4610169E-01 2.2654523E-01 1.2426844E-01 4o4273596E-02 2.9604660E-09 102

Table A-9 N = 3 5o = 0.3 = 0.o(o.05)1.0 1.0OOOOOOOE 00 1.0279870E 00 1.0417084E 00 1.0501423E 00 1.0551783E 00 1.0578003E 00 1.0586318E 00 1.0581136E 00. 1.0565805E 00 1.0542995E 00 1.0514917E 00 1.0483453E 00 1l0450246E 00 1.0416750E 00 1.0384279.E 00 1.0354031E 00 1.0327114E 00 1*0304561E 00 1.0287340E 00 1.0276372E 00 1.0272530E 00 b-x 0. 4.3704636E-CI2 8.9433695E-02 1.3654454E-01 1.8469159E-01 2.3364073E-01 2.8321857E-01 3.332901'7E-01 3,8374732E-01 4.3450121E-01 4.8547776E-01 5.3661440E-01 5.8785762E-01 6.3916129E-'01 6.9048535E-01 7.4179468E-01 7.9305838E-01 8.4424911E-01 8.9534245E-01 9.4631662E-01 9.9715205E-01 -5.000000OE-Q01-5.0591574E-01-4.9834439E-01-4.8154094E-01-4.5627105E-01 -4.2288787E-01-3.8159384E-01-3.3252279E-01-2.7577319E-01-2.1142375E-01 -1.3954161E-01-6.0186862E-02 2.6584796E-02 1.2072043E-01 2.2216879E-01 3.3087977E-01 4.4680381E-01 5.6989182E-01 7.0009477E-01 8.3736377E-01 9.8164984E-01 0. -6.9596310E-02-1.393311 7E-01-20.652990E-01-2.6894571E-0 1 — 3.2447081E-01-3.,7106291E-01-4.067159.4E-01-4.2944581E-01-4.3728258E-01 -4. 28.26585E -01-4.0044180E-01-3. 5186133E-01-2. 8057880E-0 1-1. 8465108E-01 — 6.2,136980E-02 8.8903261E-02 2.7040826E-01 4.8431579E-01 7.3256292E-01 1!0170862E 00 = 0.0(0.05)1.0 1OO0000000E 00 1.0228527E 00 1.0304611E 00 1.03'12290E 00 1.0267123E 00 1.0176152E 00 1.0042999E 00 9.8694564E-01 9.6561066E-01 9.4025666E-01 9.1075399E-01 8.76'87367E-01 8.3826513E-01 7.9441503E-01 7.4457349E-01 6-.8761898E-01 6.2 178989E-01 5.4408242E-01 4.4860741E-01 3.2016313E-0.1 0.. 0. 1.3719613E-01 2.7744155E-01 4,1800193E-01' 5.5694676E-01 6.9258432E-01 8.23.30189E-01 9.;4748524E-01 1,0634608E 00 1.1694408E 00 1.2634616E 0'0 1.3433038E 00 1.4063804E 00 1.4495686E 00 1,4689389E 00 1.4592842E 00 1,4132197E 00 1.3192216E 00 1.1564160E 00 8.7475048E-01 0. -1.5000000 E 00-1.5045376E 00-1,4521012E 00-1.3552852E 00-1.2172655E 00 -1. 0407106E 00-8.2854022E-01-5.8419780E-01-3.1181291E-01-1.6345929E-02 2.9624516E-01 6.1878567E-01 9.4257526E-01 1,2569271E 00 1.5484403E 00 1.7997694E 00 1.9873447E 00 2.0765758E 00 2.0095716E 00 1.6602579E 00 0. 103

Table A-9 cont. N 3 coo =.3 y,~j([l) W = 0.0(0.05)1.0 3.0000000E 00 3.0182081E 00 3.0059396E 00 2.9732356E 00 2.9221513E 00 2.8537368E 00 2.7686557E 00 2.6673841E 00 2.5502946E 00 2.4176967E 00 2.2698597E 00 2.1070251E 00 1.9294156E 00 1.7372395E 00 1.5306952E 00 1.3099730E 00 1.0752572E 00 8.2672763E-01 5.6456007E-01 2.8892749E-01 O. ~-3 0. 7.2673199E-01 1.4522674E 00 2.1617366E 00 2.8420739E 00 3.4808214E 00 4.0658173E 00 4.5850684E 00 5.0266872E 00 5.3788565E 00 5.6298077E 00 5.7678069E 00 5.7811466E 00 5.6581385E 00 5.3871093E 00 4.9563974E 00 4.3543505E 00 3.5693240E 00 2.5896789E 00 1.4037814E 00 u. _~.. t(PL) IL= o0.0(0.01)1.0 1.5000000E 01 1.5034319E 01 l.5051201E 01 1.5058970E 01 1.5059368E 01 1.5053210E 01 11.5040971E 01 1.5022963E 01 1.4999407E 01 1.4970468E 01 1,4936275E 01 1.4896935E 01 1.4852535E 01 1.4803151E 01 1.4748851E 01 1.4689696E 01 1.4625740E 01 1.4557037E 01 1.4483635E 01 1.4405582E 01 1.4322925E 01 1.4235708E 01 1.4143977E 01 1.4047776E 01 1.3947150E 01 1.3842144E 01 1.3732805E 01 1.3619178E 01 1.3501310E 01 1.3379251E 01 1.3253048E 01 1.3122752E 01 1.2988415E 01 1.2850089E 01 1.2707829E 01 1.2561690E 01 1.2411731E 01 1.2258010E 01 1,2100587E 01 1.1939525E 01 1.1774888E 011,1606742E 01 11435156E 01 1,1260198E 01 1.1081942E 01 1.0900463E 01 1.0715835E 01 1.0528139E 01 1.0337455E 01 1.0143868E 01 9.9474635E 00 9.7483310E 00 9.5465618E 00 9,3422509E 00 9.1354960E 00 8.9263976E 00 8.7150599E 00 8.5015898E 00 8.2860990E 00 8.0687008E 00 7.8495142E 00 7.6286613E 00 7.4062687E 00 7..1824673E 00 6.9573923E 00 6.7311844E 00 6.5039890E 00 6.2759570E 00 6.0472454E 00 5.8180169E 00 5.5884407E 00 5.3586934E OQ 5.1289589E 00 4.8994286E 00 4.6703033E 00 4.4417925E 00' 4.2 141159E 00 3.9875044E 00 3.7622003E 00 3.5384597E 00 3.3165526E 00 3.0967652E 00 2.8794020E 00 2.664*7867E 00 2.4532663E 00 2.2452131E 00 2.0o410293E 00 1.8411515E 00 1.6460561E 00 1.4562678E 00 1.2723690E 00 1.0950130E 00 9.2494259E-01 7.6301515E-01 6.1024113E-01 4.6784347E-01 3.3735662E-01 2.2080958E-01 1.2111605E-0 1 4.3148311E-02 104

Table A-10 N =4 c =0.9 o o G (=) W - 0.0:(0.05)1.0 1.0000000E 00 1.1231375E 00 1.2011527E 00 1.2621302E.00 1.3111667E 0.0 1.3507940E 00 1.3825815E 00 1.4076407E 00 1.4268309-E 0 1.4408587E 00 1.4503309E 00 1.4557858E 00 1.4577126E 00 1.4565637E 00 1.4527639E 00 1.4467162E 00 1.4388062E 00 1429405900 1.412940588758E 00 1.4075672E 00 1.39582406E 00 0. 2.0563221E-02 4.4864428E-02 7.2161933E-02 1.0204653E-01 1.3422083E-01 1.6844522E-01 2.0451667E-01 2.4225824E-01 2.8151313E-01 3.2214102E-01 3.6401548E-01 4.0702218E-01 4.5105764E-01 4.960280.7E-01 5.4184857E-01 5.8844246E-01 6.35-74072E-01 6.836-8143E-01 7.3220-937E-01 7.8127570E-01 -5.000000E-01-5.3730446E-01-5.4542412E-01-5s3875390E-01-5. 1988894E-01 -4.9006003E-01-4o4998626E-01-4. 0014534E-01-3. 4088472E-01-2. 7247504E-01 -1.9513876E-01-1. 0906682E-01-1.4428792E-02 8.8620570E-02 1.9993693E-01 3.1938346E-01 4.4682857E-01 5.8214462E-01 7.2520657E-01 8.7589140E-01 1.0340773E 00 2=3 0. -5.1218991E-02-1.0613981E-0 1-1. 6114920E-0 1-2. 1356086E-0 1 OM2.6D99324E-01-3.0121441E-01-3.3208107E-01-3.5150897E-01-3.5745656E-01 -3.4791528E-01-3.2090338E-01-2.7446173E-O 1-2.0665111E-01-1.1555004E-01 7.4653141E-04 1.4412846E-0.1 3.1647113E-01 5.1963614E-01 7.5547177E -0i 1.0258134E 00 3.7500000E-01 3.8566769E-01 3.6381742E-01 3.2095630E-01 2.6041761E-01 1.8521776E-01 9.8688968E-02 4.6748942E-03-9.2391152E-02-1.8744974E-01 -2. 7479629E-1-3 480 7040-01-43.480 24983-0-423646080- 01-44. 0990171E-01 -3.4998929E-01-2.3.420986E-01-5.2192007E-02 2.0710914E-01 5.5541197E-01 1.0051067E 00 105

Table A-10 cont. N = 4 w = 0.9 uQ(lL) Op = 0.0(0.05)1.0 1.0000000E 00 1.0660856E 00 1.0969229E 00 1.1127863E 00 1.1177157E 00 1.1137791E 00 1.1022606E 00 1.0840363E 00 1.0597277E 00 1.0297689E 00 9.9443585E-01 9.5385085E-01 9.0796691E-01 8.5652919E-01 7.989999.1E-01 7.3441728E-01 6.6111334E-01 5.7608058E-01 4.7324515E-01 3.3672204E-01 0. 0. 1.1055477E —01 2.3148482E-01 3.5846934E-01 4.8861459E-01 6.1948975E-01 7*4886406E-01 8,7458310E-01 9,9448779E-01 1.1063452E 00 1.20'77760E 00 1.2961664E 00 1.3685487E 00 1.4214239E 00 1.4504806E 00 1.4501086E 00 1.4124720E 00 1.3254959E 00 1.1675265E 00 8.8705009E-01 OoI. 3 -1.5000000E 00-1.5424643E 00-1.5037935E 00-1.4142518E 00-1.2802932E 00 -1.- 1061340E 00-8.9557884E-01-6.52.63162E-01-3.8179269E-01-8.8263856E-02 2.2185859E-01 5.4135728E-01 8.6158992E-01 1.1720455E 00 1.4596488E 00 1.7075866E 00 1.8931323E 00 1.9830647E 00 1.9219289E 00 1.5890967E 00 O0 i.. 4 0. -3.1976508E-01-6.4288525E-01-9.4611319E-01-1.2110006E 00 -1.4213503E 00-1.5628486E 00-1 —6231679E 00-1,5922668E 00-1.,4628221E 00 -1,2307968E 00-8.9617580E-01-4.6394336E-01 5.4566420E-02 6.3969711E-01 1.2603775E 00 1 8687875E 00 2*3898644E 00 2.6961001E 00 2.5210654E O0 0, t (~)' = 0o.o0(0.05)1.0,: 2 3.0000000E 00 3.1292553E 00 3.1731762E 00 3.1780499E 00 3.1524838E 00 3.1006201E 00 3.0249185E 00 2.9270388E 00 2.8082043E 00 2.6693773E 00 2.5113538E 00 2.3348183E 00 2.1403781E 00 1.9285847E 00 1.6999490E 00 1.4549517E 00 1.1940500E 00 9.1768358E-01 6.2627806E-01 3.2024827E-01 0. 0. 6.3542046E-01 1.2937987E 00 1.9514777E 00 2.5914965E 00 3.1991142E 00 3.7606856E 00 4.2632151E 00 4,6941396E 00 5.0412079E 00 5.2924081E 00 5.4359222E 00 5.4600950E 00 5.3534133E 00 5.1044908E 00 4.7020572E 00 4,13494.98E 00 3.3921081E 00 2.4625680E 00 1.3354592E 00 0. — 7.5000000E 00-7.5630679E 00-7.1848102E 00-6.4859326E 00-5.5028053E 00 -4.2697867E 00-2.8257914E 00-1.2162734E 00 5.0599800E-01 2.2805867E 00 4.0392108E 00 5.7056454E 00 7.1956666E 00 8.4170175E 00 9.2693866E 00 9.6443942E 00 9.4255830E 00 8.4884129E 00 6.7002548E 00 3.9203897E 00 O.... l06

Table A-10 cont., N=4 =o0.9 41J (,uG)O = 0.0(0.05)1.0 A-3 1.5000000E 01 1.5079197E 01 1.4960619E 01 1.4701783E 01 1.4316316E 01 1.3813214E 01 1.3200306E 01 1.2485396E 01 1.1676781E 01 1.0783558E 01 9.8158625E 00 8.7851034E 00 7.7042320E 00 6.5880981E 00 5'.4539491E 00 4.3221872E 0.0 3.2176045E 00 2.1716087E 00 1.2268595E 00 4.49697885-01 0. 0. 5.0906641E 00 1.0135060E 01 1.4989715E 01 1.9527815E 012.3631557E 01 2.7191491E 01 3.0107403E 01 3.2289864E 01 3.3662233E 01 3.4163091E 01 3.3749219E 01 3.2399284E 01 3.0118550E 01 2.6945165E 01 2.2958998E. 01 1.8295000E 01 1.3165498E 01 7.9032649E 00 3.0686793E 00 O..... -......:-~ ~. i...;..-~-~-.. ---— ~~ ~ ~. — ~.. 4~( CL~~~),~ ~= o.o(o.oi)l.o'_~~~__~~.._.__. _ __............._ _ L..- -.... J. 4 1.0500000E 02 1.0523350E 02 1.05332-35E 02 1.0535637E 02 1.0531810E 02 1.0522339E 02 1.0507571E 02 1.0487,735E 02 1.0462996E 02 1.0433482E 02 1.0399299E 02 1.0360535E 02 1.0317267E 02 1.0269568E 02 1.0'217503E 02 1.0161137E 02 1.0100530E 02 1.0035743rE 02 9.9668372E 01 9.8938719E 01 9.8169088E 01 9.7360095E 01 9.6512374E 01 9.5626574E 01 9.4703356E 01 9.3743401E 01 9.2747402E 01 9.1716084E 01 9.0650177E 01 8.9550446E 01 8.8417665E 01 8.7252640E 01 8.6056193E 01 8.4829170E 01 8.3572441E 01 8.2286903E 01 8.0973470E 01 7.9633082E 01 7.8266706E 01 7.6875333E 01 7.5459971E 01 7.4021664E 01 7.2561468E 01 7.1080474E 01 6.957979-4E 01 6.8060569E 01 6.6523958E 01 6.4971145E 01 6.3403348E 01 6.182180.1E 01 6.0227769E 01 5.8622539E 01 5.7007425E 01 5,5383766E 01 5.3752925E 01 5.2116293E 01 5.0475286E 01 4.8831344E 01 4.7185934E 01 4.5540547E 01 4.3896699E 01 4.2255937E 01 4.0619828E 01 3.8989965E 01 3.7367969E 01 3.5755487E 01 3.4154190E 01 3.2565773E 01 3.0991962E 01 2.9434502E -01 2.7895170E 01 2.6375766E 01 2.4878114E 01 2.3404066E 01 2.1955501E 01 2.0534321E 01 1.9142455E 01 1.7781857E 01 1.64545Q8E 01 1.5162415E 01 1.3907609E 01 1.2692148E 01 1.1518117E 01 1.0387624E 01 9.3028058E 00 8.2658223E 00 7.2788612E 00 6.3441350E 00 5.4638830E 00 4.6403692E 00 3.8758841E 00 3.1727444E 00O 2.5332920E 00 1.9598947E 00 1.4549463E 00 1.0208668E 00 6.6010150E-01 3.7512201E-01 1.6842555E-01 4.2535154E-02 107

Table A-11 N - 4 o = o.6 o 10(jl) ~~g= 0.0(0.05)1.0 1.0000000E 00 1.0692143E 00 1.1067794E 00 1.1325780E 00 1.1505182E 00 1.1625927E 00 1.1700493E 00 1.1737667E 00 1.1744113E 00 1.1725159E 00 1.1685224E 00 1.1628077E 00 1.1557008E 00 1.1474936E 00 1.1384493E 00 1.1288078E 00 1.1187902E 00 1.1086021E 00 1.0984363E 00 1.0884745E 00 1.0788895E 00 Q21 0. 3.5304442E-02 7.4202704E-02 1.1566126E-01 1.5911768E-01 2.0418087E-01 2.5055293E-01 2.9799642E-01 3.4631713E-01 3.9535373E-01 4.4497094E-01 4.9505485E-01 5.4550938E-01 5.9625365E- 01 6*4721993E-01 6.9835200E-01 7.4960387E-01 8.0093862E-01 8.5232753E-01 9.0374935E-01 9.5518960E-01 422 -5.000000OE-01-5.2145649E-01-5.2128.149E-01-5.0920190E-01-4. 8693530E-01 -4.5526531E-01-4.1463011E-01-3.6530794E-01-3.0749284E-01-2.4133066E-01 -1.6693811E-(1-8.4413638E-02 6*1561307E-03 1.0469228E-01 2.1112096E-01 3.2537167E-01 4.4737604E-01 5.7706709E-01 7.1437855E-01 8.5924460E-01 1.0115995E 00 0. -6.1533396E-02- 1.2520941E-0 1-18767529E-01-2, 4638844E-01 - 2.9907284E-01-3.4358242E-01-3.7784702.E-01-3.9984594E-01-4.0759352E- 01 -3.9913043E-01-3.7251810E-01-3.2583519E-01-2.5717505E-0 1-1.6464397E-01 -4.6359924E- U2 9.95483(72t-U2 2'.749421ZE-U1 4.8167303E-01 "7.2158361E-O1 9.9650764E-01 3.750OOOO0000E-01 3.7837865E-01 3.5429228E-01 3.1067138E-01 2.5020998E-01 1.7562806E-01 9.0089557E-02-2.6714892E-03-9.8301778E-02-1.9180646E-01 — 2.7753789E-01-3.4918947E-01-3.99792'51E-01-4.2171365E-01-4.0665348E-01 -3.4564629E-01-2.2905924E-01-4.6591848E-02 2. 1272466E-01 5.6052716E-01 1.0091187E 00 108

Table-e A-ll cont. N=4 m =0.6.~Y'(p) ~ Ir=~0.0(0.05)1.0 1.0000000E 00 1.0406589E 00 1.0568878E 00 1.0625380E 00 1.0603958E 00 1.0518577E 00 1.0377580E 00 1.0186336E 00 9,9482989E-01 9.6654742E-01 9.3385922E-01 8.9670923E-01 8.5489376E-01 8.0802172E-01 7.5544091E-01 6.9610068E-01 6.2827925E-01 5.4897235E-01 4.5222849E-01 3.2265135E-01 0. 0. 1.2599670E-01 2.5898491E-01 3.9524721E-01 5.-3226238E-01 6.6789815E-01 8.0018340E-01 9.2719773E-01 1.0469965E 00 1.1575452E 00 1.2566495E 00 1.3418686E 00 1.4103969E 00 1.4588882E 00 1.4831750E 00 1.4777818E 00 1.4349987E 00 1.3428633E 00 1.17.98070E 00 8.9429897E-01 0. -1!.5000000E 00-1.5195042E 00-1.4718061E 00-1.3773160E 00-1.2406775E 00 -1, 0652514E 00-8 5435108E-01-6.1166523E-01-3.4147774E-01-4.8836414E-02 2.6027212E-01 5.7870440E-01 8.9785913E-01 1.2072316E 00 1.4937122E 00 1.7403988E 00 1.9243966E 00 2.0121.822E 00 1.9477268E 00 1*6090535E 00 0. 0. -3.3997692E-01-6.. 7704034E-01-99009480E-01-1.2617434E 00 -1.4764787E 00-1.6205002E 00-1.6818884E 00-1,6509384E 00-1.5206099E 00 -1.2871026E 00-9.5059536E-01-5.1622217E-01 4.5823706E-03 5.9212114E-01 1.2153607E 00 1.8266317E 00 2.3512073E 00 2.6622803E 00 2.4951399E 00 O. = 0.0(0.05)1.0 3.0000000E 00 3.0800062E 00 3*0984347E 00 3.0863596E 00 3.0495268E 00 2.9906720E 00 2.9114031E 00 2.8127935E 00 2.6956266E 00 2.5605133E 00 2.4079546E 00 2.2383784E 00 2.0521616E 00 1.8496449E 00 1.6311423E 00 1l3969480E 00 1.1473408E 00 8.8258795E-01 6.0294739E-01 3.0866989E-01 0. 0. 6.7711553E-01 1.3660195E O0 2.0466010E 00 2.7036235E 00 3.3234456E 00 3.8932029E 00 4'.4004870E 00 4.8331892E 00 5.1794153E 00 5.4274341E 00 5.5656456E 00 5.5825595E 00 5.4667807E 00 5.2069988E 00 4,7919806E 00 4.2105648E 00 3.4516577E 00 2.5042298E 00 1.3573136E 00 0. -7.5000000E 00-7.4884062E 00-7.0855614E 00-6.3755072E 00-5.3887236E 00 -4.1570790E 00-2,7181739E 00-1.1166371.E 00 5.9537184E-01 2.3579386E 00' 4.1032807E 00 5.7556872E 00 7.2314834E 00 8,4390117E 00 9e2786165E 00 9.6426380E 00 9.4154058E 00 8.4732361E 00 6,6844280E 00 3.9092636E 00 0. 109

Table A-11 cont. N =4 woX =0.6 tI~(3,) p = 0.0(0.05)1.0 f-3 1.5000000E 01 1.5033412E 01 1.4897735E 01 1.4631797E,01 1.4245394E 01 1.3745516E 01 1.3138686E 01 1.2431741E 01 1.1632205E 01 1,0748527E 01 9.7902884E 00 8.7684176E 00 7.6954561E 00 6.5859140E 00 5*4567736E 00 4.3282586E 00 3.2250862E 00 2.1787241E 00 1.2320810E 00 4.5206403E-01 0. 0. 5.1380029E 00.1.0205951E 01 i15069221E 01 1.9605462E 01 2.3699906E 01 2.7245443E 01 3.0143802E 01 3.2307220E 01 3.3660516E 01 3.4143557E 01 3.3714241E 01 3.2352179E 01 3,0063396E 01 2.6886581E 01 2.2901889E 01 1.8244236'E 01 1.3125485E 01 7,8773134E 00 3.0579076E 00 00(0.01)1.0 =o.o(o.oi)i.o 1.0500000E 02 1.0514807E 02 1.0519236E 02 1.0517281E 02 1.0509779E 02 1.0497'124E 02 1.0479551E 02 1.0457216E 02 ].0430236E 02 1.0398703E 02 1.0362693E 02 1.0322273E 02 1*0277504E 02 1.0228444E 02 1!.175146E 02 1.0117665E 02 1.0056053E,02 9.9903641E 01 9,9206523E 01 9.8469722E 01 9.7693806E 01 9.6879344E 01 9.6026936E 01 9.5137190E 01 9.4210736E. 01 9.3248227E 01 9.2250329E 01 9.1217741E 01 9.0151169E 01 8.9051356E 01 8.7919055E 01 8.6755053E 01 8.5560155E 01 8.4335184E 01 8.3080999E 01 8.1798472E'01 8.0488510E 01. 7.9152032E 01 7.7789991E 01 7,6403359E 01 7.4993136E 01 7.3560348E 01 7.2106037E 01 7.0631281E 01 6.9137177E 01 6.7624850E 01 6.6095445E 01 6.4550138E 01 6.2990127E 01 6,1416635E 01 5-9830911E 01 5.8234231E 01 5.6627893E 01 5.5013223E 01 5.3391569E 01 5.1764309E 01 5.0132842E 01 4*8498597E 01 4.6863022E 01 4,.5227599E 01 4.3593826E 01 4.1963235E 01 4.0337378E 01 3.8717834E 01 3.7106208E 01 3.5504130E 01 3.3913257E 01 3.233526.7E 01 3.0771870E 01 2.9224'797E 01 2.7695805E 01 2,6186679E 01 2.4699227E 01 2.3235284E 01 2.1796710E 01 2.0385390E 01 1.9003237E 01 1.7652186E 0,1 1.6334200E 01 l.5051268E 01 1.3805403E 01 1.2598644E 01 1.1433056E 01 1.0310729E 01 9.233780'9E 00 8.2043514E 00 7.2,246088E 00 6.2967459E 00 5.4229811E 00 4.6055586E 00 3.8467482E 00 3.1488455E 00 2.51417.17E 00 1.9450731E 00 1.4439222E 00 1.0131170E'00 6.5508112E-01 3.7226382E-01 1.6713994E-01 4.2209908E-02 0. 110

Table A-12 N 4 o =.3 5G(L) = o0.0(0.05)1.0 1.0000000E 00 1.0312699E 00 1.0466201E 00 1.0563116E O0 1.0623933E.00 1.0658960E 00 1.0674516E 00 1.0674937E 00 1.0663423E 00 1.0642456E 00 10.0614040E 00 1.0579836E 00 1.0541258E 00 1,049952'9E 00 1*0455730E 00 1.0410824E 00 1*0365685E 00 1.0321110E 00 1.0277838E 00 1.023'6553E 00 1.0197.899E 00 0. 4.3684486E-02 8.9363239E-0.2 1.3635051E-01 1.8429074E-01 2.3294893-E-O1 2.8215478E-01 3.3177890E-01 3.8172028E-01 4.3189874E-01 4.8225012E-01 5.3272288E-01 5.8327574E-01 6.3387586E-01 6.8449759E-01 7.3512133E-01 7.8573276E-01 8.3632216E-01 8.8688394E-01 9.374161 1E-01 9 8792002E-01 -5.OOOOOOO E-01-5.0796942E-01 —5.0153390E-01-4. 8557560E-01-4. 6092932E-0.1 -4.2 796218E-0 1-3.8687234E-01-3.3778170E-O 1-2. 8077320E-01-2.1590814E-0 1 -1.4323523E-01-6.2795550E-02 2.537.4570E-02 1.2124174E-01 2.2477437E-01 3.3594204E-01 4.5471494E-01 5.8106369E-01 7o.1495900E-01 8.5637162E-01 1.0052722E 00 0. -.6,8834528E-02-1.3784539E-01-2.043'1774E-01-2.6601423E-01'-3.2085107E-01-3.6681747E-01-4 ~ 0194415E-0 1-4.2428821E-01-4.3192491E-O 1 ~4'.'42294292E-01-3.9544129E-01-3.4752-751E-01-2. 7731627E-01-1. 8292849E- 1,.6.2490736E-02 8.5865252E-02 2.6400287E-01 4.7378095E-01 7.1705386E-01 9.9567174E-01 3.7500000E-01 3.7170235E-01 3.4563093E-01 3.0120211E-01 2.4054875E-01 1.6615623E-01 8.1067580E-02-1. 1056205E-02-1..0591323E-01-1.9855063E-01O -2.8335958E-01-3,5407265E-01-4. 0376272E-01-4,2484167E-01-4. 0905980E- 1 -3.4750609E-01-2.3060776E-01-4.8129948E-02 2. 1082468E-01 5.5781573E-01'1.0050630E 00 111

Table A-12 cont. N = 4 a = 0.3 4j(I) 1 = 0.0(0.05)1.0 1.0000000E 00 1.0185428E 00 1.0236357E 00 1.0223162E 00 1.0159800E 00 1.0052919E 00 9.9061333E-01 9.7213811E-01 9.4994463Ew01 9.2401589E-01 8.9424235E-01 8.6041212E-01 8.2.218754E-01 7.7906279E-01 7.3028908E-01 6.7473842E-01 6.1063443E-01 5.3495072E-01 4.4177406E-01 3.1591829E-01 0. 0. 1.3876398E-01 2.8060039E -01 4.2289364E-01 5.6372736E-01 7.0139655E-01 8.3426328E-01 9.6068056E-01 1.0789358E 00 1.1871960E 00 1.2834450E 00 1.3654019E 00 1.4304066E 00 1.4752480E 00 1.4958858E 00 1.4869687E 00 1.4409124E 00 1.3458922E 00 1.1805111E 00 8.9351598E-.01 O. -1.5000000E 00-1.4985969E 00-1.4433802E 00-1.3449207E 00-1.206'1621E 00 -1.0297113E 00-8.847417E-01-5.7589107E-01-3. 0608254E-01-1.3984227E-02 2.9449417E-01 6.1225791E-01 9o3C72796E-01 1.2393930E 00 1.5251011E 00 1..7708592E 00 1.9536083E 00 2.0395356E 00 1.9720660E 00 1.6279427E 00 0. 0. -3.5721563E-01-7.0505962E-O1 —1.0250412E 00-1. 3009471E 0'0 -1.5179980E 00-1.662905.3E 00-1.7241392E 00-1.6923054E 00-1.5606169E.00 -1o.3254810E 00-9.8724264E-01-5.5116106E-01-2,8750957E-02 5.6026410E-01 1,1848936E 00 1.7976098E 00 2.3239852E 00 2.6378181E 00 2.4758304E 00 0. = 0.0(0.05)1.0 i:Z 3*0000000E 00 3.0346469E 00 3.0311856E 00 3,0052263E 00 2.9596315E 00 2.8957497E 00 2.8143651E 00 2.7159965E 00 2.6010193E 00 2.4697239E 00 2.3223468E 00 2.1590888E 00 1.9801260E 00 1,7856168E 00 1.5757068E 00 13505317E 00 1.1102199E 00 8.5489438E-01 5o846732,3E-01 2.9967127E-01 0. 0. 7.1438271E-01 1.4290990E 00 2.1282229E 00 2.7984468E 00 3.4273074E 00 4.0027480E 00 4.5)29468E 00 4.9462362E 00 5o.2910569E 00 5.5359319E 00 5.6694502E 00 5.6802554E 00 5.5570382E 00 5..2885320E 00 4.8635079E 00 4,2707729E 00 3.4991674E 00 2o5375635E 00 1l3748641E 00 0. -7.5000000E 00-7.4176278E 00-6.9926188E 00-6.2726992E 00-5..2827923E 00 -4.0525092E 00-2.6182994E 00-l.0U240847E 00 6.7849664E-U1 Z.4299833E 00 4.1630368E 00 5.8024132E 00 7.2649474E 00 8.4595438E 00 9.2871695E 00 9.6408515E 00 9.4056746E OQ 8.4587795E 00 6.6693609E 00 3.8986685E 00 0~ 112

Table A-12 cont. N-4 4 = ~=' = 0.0(0.05)1.0 1.5000000E 01 1.4988282E 01 1.4836065E 01 1.4563429E 01 1.4176338E 01 1.36797 97E 01 1.3079043E 01 1.2379968E 01 1.1589343E 01 1.0714992E 01 9.7659596E 00 8.7527127E 00 7.6874006E 00 6.5842087E 00 5.4598679.E 00 4.3344338E 00 3.2325571E 00 2.1857685E 00 1.2372246E 00 4+.5438696E-01 0. 0. 5.1845459E 00 l.0275205E 01 1.5146425E 01 1.9680371E 01 2.3765305E 01 2.7296440E 01 3.0177410E 01 3.2322086E 01 3.3656708E 01 3.4122389E 01 3.3678105E 01 3.2304379E 01 3.0007959E 01 2.6828051E 01 2.2845075E 01 1.8193895E 01 1.3085903E 01 7.8516930E 00 3.0472911E 00 0. ) = 0o.o0(o.o01)1.o 1.0500000E 02 1,0506324E 02 1.0505359E 02 1.0499107E 02 1.0487987E 02 1.0472202E 02 1.0451876E 02 1.0427093E 02 1,0397919E 02 1,.0364410E 02 1i0326615E 02 1.0284580E 02 1.0238347E 02 10187960E 02 1.0133463E 02 1.0074898E 02 1.0012311E 02 9.9457476E 01 9.8752558E.01 9.8008852E 1O 9.7226868E 01 9.6407143E 01 9.5550232E 01 9.4656708E 01 9.3727175E 01 9.2762251E 01 9.1762576E 01 9.0728825E 01 8.9661684E 01 8.8561869E 01 8.7430115E 01 8.6267189E 01 8.5073875E 01 8.3850985E 01 8.2599354E 01 8.1319844E 01 8.0013338E 01 7.8680747E 01 7o7323006E 01 7.5941074E 01 7.4535933E 01 7.3108597E 01 7.1660095E 01 7.0191488E 01 6.8703864E 01 6.7198328E 01 6.5676019E 01 6.4138094E 01 6.2585738E 01 6.1020161E 01 5.9442601E 01 5o7854316E 01 5.6256592E 01 5..4650742E 01 5.3038102E 01 5.1420032E 01 4.9797921E 01 4.8173180E 01 4.6547248E 01 4.4921587E. 01 4.3297684E 01 4.1677056E 01 4.0061240E 01 3.8451800E 01 3.6850325E 01 3.5258432E 01 3.3677760E 01 3.2109975E 01 3.0556768E 01 2.9019856E 01 2.7500980E 01 2.6001908E 01 2.4524432E 01 2.3070369E 01 2.1641564E 01 2..0239886E 01 1.8867228E 01 1.7525510E 01 1.6216676E 01 1.4942698E 01 1.3705570E 01 1.2507314E 01 1.1349977E 01 1.0235629E 01 9.1663696E 00 8.1443200E 00 7.1716290E 00 6.2504696E 00 5.3830414E 00 4.5715681E 00 3.8182995E 00 3.1255113E 00 2.4955037E 00 1.9306027E 00 1.4331595E 00 1.0055514E 00 6.5018016E-01 3.6947370E-01 1.6588500E-01 4.1892429E-02 O..

Table A-13 N =2 ao =0.9 = 0.0(0.01)1.0 J- o 1.0000000E 00 1.0357323E 00 1.0621984E 00 1.0854243E 00 1.1065419E 00 1.1260912E 00 1.1443864E 00 1.1616322E 00 1.1779717E 00 1.1935110E 00 1.2083312E 00 1.2224966E 00 1.2360595E 00 1.2490627E 00 1.2615424E 00 1.2735297E 00 1.2850509E 00 1.2961294E 00 1,3067856E 00 13170375E 00 1.3269014E 00 1.3363918E 00 1.3455219E 00 1.3543039E 00 1.3627484E 00 1.3708658E 00 1.3786653E 00 1.3861555E 00 1.3933445E 00 1.4002397E 00 1.4068480E 00 1.4131762E 00 1.4192302E 00 1.4250161E 00 1.4305393E 00 1.4358048E 00 1.4408178E 00 1.4455830E 00 1.4501048E 00 1.4543874E 00 1.4584351E 00 1.4622516E 00 1.4658408E 00 1.4692062E 00 1.4723514E 00 1.4752796E 00 1.4779941E 00 1.4804980E 00 1.4827943E 00 1.4848859E 00 1.4867754E 00 1l4884659E 00 1.4899596E 00 1.4912594E 00 1.4923676E 00 1.4932867E 00 1.4940190E 00 1.4945667E 00 1.4949321E 00 1.4951174E 00 1.4951245E 00 1.4949557E 00 1.4946128E 00 1.4940978E 00 1.4934127E 00 1.4925593E 00 1.4915395E 00 1.4903549E 00 1l4890074E 00 1.4874986E 00 1.4858302E 00 1.4840040E 00 1.4820213E 00 1.4798838E 00 1l4775931E 00 1.4751506E 00 1.4725580E 00 1.4698166E 00 1.4669277E 00 1*4638928E 00 1.4607133E 00 1,4573907E 00 1.4539261E 00 1,4503208E 00 1.4465762E 00 1.4426934E 00 1.4386739E 00 1.4345188E 00 1.4302291E 00 1.4258062E 00 1.4212510E 00 1.4165653E 00 1.4117495E 00 1.4068051E 00 1.4017328E 00 1.3965340E 00 1.3912099E 00 1.3857612E 00 1,3801892E 00 1.3744946E 00 1.3686785E 00 0. 3.6267447E-03 7.4622310E-03 1.1474294E-02 1.5646375E-02 1.9967274E-02 2.4428536E-02 2.9023324E-02 3.3745929E-02 3,8591402E-02 4.3555384E-02 4.8633964E-02 5.3823586E-02 5.9121004E-02 6.4523189E-02 7.0027319E-02 7.5630792E-02 8.1331062E-02 8.7125782E-02 9.3012731E-02 9.8989741E-02 1.0505477E-01 1.1120584E-01 1,1744107E-01 1.2375860E-01 1.3015671E-01 1.3663364E-01 1.4318781E-01 1.4981754E-01 1.5652131E-01 1.6329758E-01 1.7014491E-01 1.7706182E-01 1.8404690E-01 1.9109884E-01 1.9821626E-01 2.0539783E-01 2.1264230E-01 2.1994843E-01 2.2731497E-01 2.3474070E-01 2.4222449E-01 2.4976515E-01 2.5736154E-01 2.6501261E-01 Z.(12L717E-U1 2.804(7425-01 2.8828270E-U1 2.9614154E-01 3.0404973E-01 3.1200630E-01 3.2001025E-01 3.2806061E-01 3.3615642E-01 3o4429677E-01 3.5248068E-01 3.6070727E-01 3.6897571E-01 3.7728501E-01 3,8563434E-01 3.9402288E-01 4.0244973E-01 4.1091414E-01 4,1941524E-01 4.27.95222E-01 4.3652428E-01 4.4513068E-01 4.5377057E-01 4.6244325E-01 4.7114792E-01 4.7988390E-01 4.8865036E-01 4.9744667E-01 5.0627211E-01 5.1512592E-01 5.ZU 40UU46L-U1 D.3291D599t-01 5.41850Ub2-UE 5508113-U1 5508113.599691E-01 5.6880682E-01 5.7784042E-01 58689703E-01 5.8689703E-01 5.9597619E-01 6.0507715E-01 6.1419937E —U01 6.2334213.E-1 6.3Z5U4'UL-u1 6o41687U0E-01 6.5088810E-0i 6.6010748E-01 6.6934443E-01 6.'7859848E-01 6.8786909E-01 6.9715582E-01 70645802E-01 577503E-01 75 1 0 6 5 1 E - 1 72510651E-01 73445182E-01 7.4381055E-01 7.5318215E-01,,~~~~~~~~~~~~~1~

Table A-13 cont. -5. 0000000-E-0 1-5. 13670036-01-5. 2238525E-0 1-5. 29189-56E-0 1-5. 3466246E-0 1 -5. 3908194E-0 1-5.426 1020E-01-5.4535291E-0 1-5. 4738407E-0 1-5.4875820E-01 -5.4951698E-0 1-5.4969317E-01-5.49 31320E-01-5.4839867E-01-5 4696760 E-0 1 -5 *45035 16E-0 1-5 * 4261432E- 0 1-5. 39716 14E-01-5. 3635038E-0 1-5. 3252546E-01 -5.2824870E-01-5.2352679E-01-5.1836552E-0 1-5. 1277015E-01-50674522E-0 1 -5. 0029504E-0-4. 9342 1-4.7842909342335E-0 14 8613371E-4. 7842909E- 14 7031240E-0 1.-4.6178612E-01-4. 5285283E-01-4.4351448E-01-4. 3377325E-01-4. 2363090E-01 -4. 1308905E-01-4.021493 /E-01-3. 9081328E-0 1-3.,7908200E-0 1-3.,6695694E-01 -3.5443902E-01-3 4152953E-01-3.2822922-0 1-3. 1453918E-0 1-30046023E-01 -2. 8599306E-01-2.7113853E-01-2.5589721E-O 1-2. 4026981E-O 1-2.2425692E-01 -2. 0785906E-0 1-1. 910 7678E-01-1. 73 91053E-O 1-1. 5636077E-0 1-1. 842 789E-0 1 -1. 2011234E-01-1.0141445E-01-8.2334580E-02-6. 2872957E-02-4.3030009E-02 -2.2805876E-02-2.2009644E-03 1.8784618E-02 4.0150589E-02 6.1896766E-02 8.4022968E-02 1.0652892E-01 1.2941461E-01 1.5267978E-01 1. fb32440E-01 2.0034820E-01 2.2475114E-01 2.4953311E-01 2.7469397E-01 3.0023368E-01 3.2615-212E-01 3.5244923E-01 3.7912495E-01 4.0617920E-01 4.3361192E-01 4.6142313E-01 4.8961266E-01 5.1818060E-01 5,4712683E-01 5.7645137E-01 6.06154i8E-01 6.3623524E-01 6.6669444E-01 6.9753205E-01 7.2874780E-01 7.6034177E-01 7.9231387E-01 8.2466-427E-01 8.5739298E-01 8. 049987E-01 9.2398499E-01 9.5784838E-01 9.9209006E-01 1.0267100E 00 1.0617083E 00 1.0970850E 00 115

Table A-14 O = ouN = 0. 6 = 0.0(0.01)1.0 1.OO00000OE 00 1.0213865E 00 1.0364067E 00 1.0491485E 00 1.0603981E 00 1.0705358E 00 1.0797855E 00 1.0882943E 00 1,0961670E 00 1.1034-812E 00 1.1102977E 00 1.1166644E 00 1.1226210E 00 1.1282003E 00 1.1334302E 00 1.1383346E 00 1.1429343E 00 1.'1472476E O0 1.1512906E 00 1.1550777E 00 1.1586219E 00 1.1619350E 00 1.1650274E 00 1.1679090E 00 1.1705887E 00 1.1730746E 00 1.1753745E 00 1.1774952E 00 1.1794434E 00 1.1812252E, 00 1.1828464E'00 1.1843122E 00 1.1856278E 00 1.1867979E 00 1.1878270E 00 1.1887194E 00 1.1894791E 00 1.1901099E 00 1.1906155E 00 1.1909993E 00 1.1912646E 00 1.1914147E 00 1.1914524E 00 1.1913806E 00 1.1912023E 00 1.1909198E 00 1.1905359E 00 1.1900530E 00 1.1894733E 00 1.1887992E 00 1.1880327E 00 1.1871761E 00 1.1862313E 00 1,1852003E 00 1.1840849E 00 1.1828870E 00 1.1816083E 00 1.1802505E 00 1.1788152E 00 1.1773041E 00 1.1757187E 00 1.1740604E 00 1.1723307E 00 1.1705311E 00 1.1686629E 00 1.1667273E 00 1.1647258E 00 1.1626595E 00 1.1605297E 00 1.1583375E 00 1.1560841E 00 1.i537706E 00 1.1513981E 00 1.1489676E 00 1.1464802E 00 1.1439370E 00 1.1413388E 00 1.1386866E 00 1.1359813E 00 1.1332240E 00 1.1304153E 00 1.1275563E 00 1.1246478E 00 1.1216905E 00 1.1186853E 00 1.1156330E 00 1.1125344E 00 1.1093901E 00 1.1062010E 00 1.1029677E 00 1.0996909E 00 1.0963714E 00 1.0930098E 00 1.0896067E 00 1.0861629E 00 1.0826789E O0 1.0791553E 00 1.0755928E O0 1.0719919L 00 1.0683532E 00 1.0646773E 00 0. 6.6268133E-03 1.3480913E-02 2,0519478E-02 2.7719833E-02 3.5066393E-02 4.2547245E- 02 5.0152740E-02 5.7874762E-02 6.5706301E-02 7.3641185E-02 8.1673911E-02 8.9799494E-02 9.8013396E-02 1.0631145E-01 1 4689( 8E-01 1 2314481 t-01 1- 3167319E-0U 1. 4027175L-01 1 4893755E-0 1 1.5766778E-01 1.6645979E-01 1.7531106E-01 1.8421921E-01 1.9318194E-01 2.0219710E-01 2.1126258E-01 2.2037639E-01 2.2953663E-01 2.3874145E-01 2.4798909E-01 2.5727784E-01 2.6660607E-01 2.7597220E-01 2.3537471E-01 2.9481212E-01 3.0428301E-01 3.1378600E-01 3.2331978E-01 3.3288303E-01 3,4247453E-01 3o.5209305E-01 3.6173742E-01 3.,7140653E-01 3. 8109920E-01 3.9081443E-01 4.0055113E-01 4.1030831E-01 4.2008498E-01 4.2988017E-01 4.3969295E-01 4.4952241E-01 4.5936767E-01 4.6922786E-01 4*7910214E-01 4.8898969E-01 4.9888971E-01 5.0880144E-01 5.1872409E- 01 5.2865693E-01 5.3859923E-01 5.4855031E-01 5.5850945E-01 5.6847597E-01 5.7844926E-01 5.8842862E-01 5.9841345E-01 6.0840315E-01 6.1839707E-01 6.2839468E-01 6.3839538E-01 6.4839859E-01 6.5840379E-01 6. 6841043E-01 6.7841798E-01 6.8842593E-01 6.9843377E-01 7.0844101E-01 7.1844717E-01 7.2845177E-01 7.3845435E-01 7.4845444E-01 7.5845163E-01 7.6844545E-01 7.7843549E-01 7.8842131E-01 7.9840253E-01 8.0837871E-01 8.1834949E-01 8.2831446E-01 8.3827324L-01 8.4822547E-01 8.5817078E-01 8.6810881E-01 8.7803919E-01 8.8796160E-01 8.9787568E-01 9.0778112E-01 9.1767757E-01 9.2756472E-01 9.*3744224E-01 116

Table A-14 cont. -5000000OE-0O1-5~0858457E-01-511374788E-01-51753.95E-0-1-5.2035336E-01 -5.2237858E-01-5.2372558E-01-5.2446619E-01-5.2465064E-01-5.2431581E-0O -5.2348979E-01-5.2219457E-01-5.2044777E-01-5.1826376E-01-5.1565439E-01 -5.1262966E-01-5.0919795E-01-5.0536549E-1-501 14145E-01-4.652820'E-01 -4.9153141E-01-4.86152 1E-01-4 8040317E-01-4. 742,7851E-01-4.6778404E-0 1 -4.6092230E-01-4.5369552E-01-4.4610572E-01-4.3815473E-01-4.2984415E-01 -4.2117544E-01-4.12174994E-01-4.0276884E-01-3.9303322E-01 -3.8294405E-01 -3.7250224E-01-3.6170857E-01-3.5056380E-01-3.3906858E-01-3.2722352E-01.3.1502917E-01-3.o0248603E-01-2.8959456E-01-2.7635517E-01-2.,6276824E-01 -2.488341 1E-01-2.3455310E-01-2.1992548E-O1 —2.0495152E-01-1.8963143E-O1 -1. 7396544E-01-1.o5795373E-01-1.4159647E-01-1.*2489382E-01-1.-0784590E-O1 9. 0452845E-02-7.2714763E-02-5.4631749E-O2-3.6203882E-02-11.7431242E-02 1.6861157E-03 2.1148132E-02 4.0954756E-02 6.1105960E-02 8.1601709E-02 1.0244197E-01 1.2362673E-01 1.4515598E-01 1.6702973E-O1 1.8924795E-01 2*1181066E-01 2.3471784E-01 2.5796954E-01 2.8156575E-01 3o0550651E-O1 3.2979179E-01 3.5442167E-01 3.7939616E-Q1 4.0471527E-01 4.3037903E-O1 4.5638751E-01 4.8274072E-01 5.0943868E-01 5.3648144E-01 5.6386907E-01 5.9160157E-O01 6.1967901E-O01 6.4810140EO1 6.7686881E-0O1 7.0598129E-O1 7.3543886E-01 7.6524159E-01 7.9538952E-01 8.2588269E-01 8.5,672115E-01 8.8790494E-01 9.1943415E-01 9.5130879E-01 9.8352891E-01 1.0160946E 00 10490058E 00 117

'Table A-15 N = 2 w =0.3 t1,(W) W _0.0(0.01)1.0 1.0000000E 00 1.0100409E 00 1.0168528E 00 1.0225054E 00 1.0274024E 00 1.0317399E 00 1.0356340E 00 1.0391613E 00 1.0423761E 00 1.0453192E 00 1.0480222E 00 1.0505101E 00 1.0528037E 00 1.0549200E 00 1.0568737E 00 1.0586771E 00 1.0603411E 00 1.0618751E 00 1.0632876E 00 1.0645858E 00 1.0657765E 00 1.0668656E 00 1.0678587E 00 1.0687606E 00 1.0695758E 00 1.0703086E 00 1.0709627E 00 1.0715415E 00 1.0720486E 00 1.0724867E 00 1.0728588E 00 1.0731676E 00 1.0734154E 00 1.0736046E 00 1.0737374E 00 1.0738159E 00 1.0738419E 00 1.0738172E 00 1.0737437E 00 1.0736230E 00 1.0734565E 00 1.0732459E 00 1.0729924E 00 1.0726974E 00 1.0723622E 00 1.0719880E 00 1.0715759E 00 1.0711271E 00 1,0706426E 00 1.0701234E 00 1.0695705E 00 1.0689849E 00 1*0683673E 00 1.0677187E 00 1.0670399E 00 1.0663317E 00 1.0655948E 00 1.0648300E 00 1.0640380E 00 1.0632195E 00 1.0623750E 00 1.0615053E 00 1.0606110E 00 1,0596'926E 00 1.0587508E 00 1.0577860E 00 1.0567987E 00 1.0557896E 00 1.0547592E 00 1.0537078E 00 1.0526359E 00 1.0515441E 00 1.0504327E 00 1.0493022E 00 1.0481530E 00 1.0469855E 00 1.0458000E 00 1.0445970E 00 1.0433768E 00 1*.3421398E 00 1.0408863E 00 1.0396166E 00 1.0383311E 00 1.0370302E 00 1.0357140E 00 1.0343829E 00 1.0330372E 00 1.0316772E 00 1.0303032E 00 1.0289154E 00 1.0275141E 00 1.0260995E 00 1.0246719E 00 1.0232316E 00 1.0217787E 00 1.0203136E 00 1.0188363E 00 1.0173473E 00 1.0158466E 00 1.0143345E 00 1.0128111E 00 0. 8.4952139E-03 1.7126655E-02 2.5865541E-02 3.4696583E-02 4.3609291E-02 5.2595694E-02 6.1649376E-02 7.0764982E-02 7.9937929E-02 8.9164218E-02 9.8440310E-02 1.0776304E-01 1.1712955E-01 1.2653723E-01 1.3598370E-01 1.4546676E-01 1.5498438E-01 1,6453467E-01 1.7411586E-01 1.8372631E-01 1.9336447E-01 2.0302888E-01 2.1271818E-01 2.2243107E-01 2.3216632E-01 2.4192277E-01 2.5169930E-01 2.6149488E-01 2.7130851E-01 2.8113923E-01 2.9098613E-01 3.0084833E-01 3.1072501E-01 3.20615388E-01 3.30518-66E-01 3.4043413E-01 3.5036107E-01 3.6029883E-01 3.7024674E-01 3.8020417E-01 3.9017054E-01 4.0014526E-01 4.1012778E-01 4.2011754E-01 4.3011404E-01 4.4011676E-01 4.5012524E-01 4.6013899E-01 4.7015758E-01 4.8018055E-01 4.9020748E-01 5.0023797E-01 5.1027162E-01 5.2030804E-01 5,3034686E-01 5.4038772E-01 5.5043027E-01 5.6047416E-01 5.7051907E-01 5.8056468E-01 5.9061068E-01 6.0065677E-01 6.1070264E-0-1 6.2074802E-01 6.3079263E-01 6.4083620E-01 6.5087847E-01 6.6091917E-01 6.7095808E-01 6.8099493E-01 6.9102950E-01 7.0106156E-01 7,1109088E-01 7.2111726E-0I (.3114046L-01 7.4116U3UtL-01 7.511765 7E-01 7.6118906E-01 7.7119759E-01 7.8120198E-01 7.9120204E-01 8.0119759E-01 8.1118846E-01 8.2117449E-01 8.3115549E-01 8.4113133E-01 8.5110183E-01 8.6106684E-01 8.7102-622E-01 8.8097981E-01 8.9092747E-01 9,0086907E-01 9.1080445E-01 9.2073350E-01 9.3065608E-01 9.4057204E-01 9.5048130E-01 9.6038369E-01 9.7027912E-01 9.8016746E-01 118

Table A-15 cont. -5.O000000E-01-5.0402133E-01-5. 0617766E-01-5. 0750191E-O-5.0819348E-01 -5.0834809E-01-5 ~ 0802137E-01-50724942E-01-5.0605736E-01-5,0446353E-01 -5.0248184E-01-5.0012310E-01-4.9739593E-01-4.9430729E-01-4.9086293E-01 -4.8706761E-01-4.8292533E-01-4.7843949E-01-4.7361300E-01-4.6844835E-01 -4.6294770E-01-4.5711291E-01-4.5094564E-01-4.4444732E-01-4.3761922E-01 -4.3046244E-01-4.2297799E-01-4.1516672E-01-4.0702944E-01-3.9856682E-01 -3.8977949E-01-3.0q66800E-01-3.7123284E-01-3.6147447E-01-3.5139328E-01 -3.4098962E-01-3.3026382E-01-3.1921616E-01-3.0784692E-01-2.9615630E-01 -2.8414453E-01-2.7181179E-01-2.5915824E-01-2.4618405E-01-2.3288933E-01 -2. 1927421E-O1-20533881E-O1-20533881-.9108320E-01-1i.7650749E-01-1.6161172E-01 -1.4639599E-01-1.3086035E-01-1.1500482E-01-9.8829480E-02-8.2334343E-02 -6.5519449E-02-4. 8384821E-02-3. 0930484E-02-1.3156451E-02 4.9372622E-0'3 -2.3350654E-02 4.2083710E-02 6.1136429E-02 8.0508816E-02 1.0020087E-01 1.2021259E-01 1.4054399E-01 1.6119507E-01 1.8216585E-01 2.0345635E-01 2.2506656E-0.1 2.4699650E-01 2.6924621E-01 2.9181568E-01 3.1470495E-01 3.37914U1E-01 3.614429UE-O1 3.8529164E-01 4.0946025E-01 4. 3394873E-01 4.5875713E-01 4.8388547E-01 5.0933375E-01 5.3510200E-01 5.6119028E-01 5.8759857E-01 6.1432691E-01 6.4137532E-0-1 6.6874382E-01 6.9643245E-01 7.2444122E-01 7.5277016E-01 7.8141930E-01 8.1038864E-01 8.3967823E-01Q 8.6928807E-01 8.9921824E-01 9.2946869E-01 9.6003947E-01 9.9093063E-01 1.0221422E 00 119

Table A-16 N = 3 o = o.9 R50 = o.o(o.oi)i.o 1.0000000E 00 1.0309100E 00 1.0538031E 00 1.0738438E 00 1.0920096E 00 1.1087697E 00 1.1243983E 00 1.1390746E 00 1.1529256E 00 1.1660455E 00 1.1785070E 00 1.1903685E 00 1.2016773E 00 1.2124735E 00 1.2227904E 00 1.2326572E 00 1.2420994E 00 1.2511390E 00 1.2597964E 00 1.2680895E 00 1.2760345E 00 1.2836459E 00 1.2909377E 00 1.2979230E 00 1.3046127E 00 1.3110180E 00 1.3171492E 00 1.3230158E 00 1.3286265E 00 1.3339910E 00 1.3391172E 00 1.3440123E 00 1.3486840E 00 1.3531401E 00 1.3573865E 00 1.3614307E 00 1.3652785E 00 1.3689367E 00 1.3724110E 00 1.3757072E 00 1.3788309E 00 1.3817878E 00 1.3845834E 00 1.3872220E 00 1.3897098E 00 1.3920513E 00 1.3942515E 00 1.3963147E 00 1.3982464E 00 1.4000509E 00 1.4017329E 00 1.4032960E 00 1.4047455E 00 1.4060853E 00 1.4073198E 00 1.4084528E 00 1.4094886E 00 1.'4104314E 00 1.4112853E 00 1.412054+3E 00 1.4127414E 00 1.4133518E 00 1.4138879E 00 1.4143544E 00 1.4147541E 00 1.4150933E 00 1.4153721E 00 1.4155966E 00 1.4157691.E 00 1.4158935E 00 1.4159730E 00 1.4160121E 00 1.4160127E 00 1.4159795E 00 1.4159161E 00 1.4158242E 00 1.4157091E 00 1.4155732E 00 1*4154188E 00 1.4152505E 00 1.4150703E 00 1.4148846E 00 1.4146930E 00 1*4144995E 00 1.4143084E 00 1.4141207E 00 1.4139428E 00 1.4137767E 00 1.4136228E 00 1.4134860E 00 1.4133689E 00 1.4132777E 00 1.4132120E 00 1.4131752E 00 1.4131692E 00 1.4131979E 00 1.4132684E 00 1.4133795E 00 1.4135331E 00 1.4137324E 00 1.4139809E 00 0. 3.9317971E-03 8.0764093E-03 1.2403781E-02 1.6898378E-02 2.1549609E-02 2.6349446E-02 3.1291503E-02 3.6370222E-02 4.1580897E-02 4.6919324E-02 5.2381722E-02 5.7964693E-02 6.3664867E-02 6.9479477E-02 7.5405672E-02 8.1440845E-02 8.7582562E-02 9.3828341E-02 1.0017597E-01 1.0662330E-01 1.1316835E-01 1.1980900E-01 1.2654314E-01 1.3336914E-01 1.4028511E-01 1.4728926E-01 1.5437990E-01 1.6155555E-01 1.6881411E-01 1.7615410E-01 1.8357423E-01 1.9107289E-01 1.9864828E-01 2.0629941E-01 2.1402443E-01 2.2i82224E-01 2.2969115E-01 2.3762993E-01 2.4563724E-01 2.5371179E-01 2.6185223E-01 2.7005724E-01 2.7832588E-01 2.8665652E-01 2.9504810E-01 3.0349941E-01 3.1200943E-01 3.2057669E-01 3.2920012E-01 3.3787855E-01 3.4661117E-01 3.5539656E-01 3.6423365E-01 3. 7312144E-01 3.8205888E-01 3.9104495E-01 4.0007850E-01 4.0915844E-01 4.1828378E-01 4.2745382E-01 4.3666701E-01 4.4592305E-01 4.5522047E-01 4.6455878E.-01 4.7393580E-01 4.8335221E-01 4.9280593E-01 5.0229661E-01 5.1182325E-01 5.2138506E-01 5.3098062E-01 5.4060981E-01 5.5027111E-01 5.5996362E-01 5.6968719E-01 5.7944013E-01 5.8922195E-01 5.9903237E-01 6.0886979E-01 6.1873410E-01 6.2862296E-01 6.3853729E-01 6.4847.578E-01 6.5843726E-01 6.6842187E-01 6.7842731E-01 6.8845326E-01 6.9850032E-01 7.0856684E-01 7.1865230E-01 7.2875440E-01 7.2875440 E-1 7.3887401E-01 7,4901035E-01 7.5916305E-01 7.6933103E-01 7.7951152E-01 7.8970589E-01 7.9991387E-01 8.1013466E-01 8.2036728E-01 120

Table A-16 cont. -5.OOO0OOE-01-5.1101307E- 01-5. 1 7l7-ZE2-01-5.2270734E-01-5.2650202E-01 -5.2933830E-01-5.3135860E-01-5.3265644E-01-5.3329802E-01-5.3333250E-01 -5.3279791E.-01-5.3172450E-01-5. 3013695E-01-5. 2805588E-01-5.2549853E-O1 -5.2247983E-01-5.1901270E-01-5.15 10842E-01-5. 1077712E- 01-5.0602770E-01 -5.0086823E-01-4.9530599E-01-4.8934765E-01-4.8299944E-01-4.7626685E-O1 -4.6915514E-01-4.6166919E-01-4.5381353E-01-4.4559235E-01-4.3700982E-0! -4 * 2806968E-01-4. 1877 548E-01-4. 0913062E-Q 1-3. 9913849E-01-3. 8880207E-01 -3.7812445E-01-3*6710845E-01-3.5575688E-01-3.4407245E-01-3.3205768E-01 -3.1971509E-01-3.0704712E-01-2.9405621E-01-2.8074449E-01-2.6711436E-01 -2.5316796E-01-2.3890745E-01-2.*2433489E-01-2. 0945241E-01-1.9426197E-01 -1. 7876565E-01-1.6296522E-01-1.4686278E-01-1.3046011E-01-1.1375917E-01 -9.6761705E-02-7.9469544E-02-6.1884577E-02-4.4008491E-02-2.5843117E-02 -7.3900246E-03 1.1348859E-02 3.0371988E-02 4.,9677585E-02 6.9264045E-02 8.9129511E-02 1.0927259E-01 1.2969148E-01 1.5038467E-01 1.7135046E-01 1.9258728E-01 2.1409344E-01 2.3586746E-01 2.5790772E-01 2.8021266E-01 3.0278068E-01 3.2561021E-01 3.4869974E-01 3.7204770E-01 3.9565256E-01 4.1951281E-01 4.4362679E-01 4.6799310E-01 4,9261025E-01 5.1747658E-01 5.4259066E-01 5.6795100E-01 5.9355599E-01 6.1940430E-01 6.4549426E-01 6.7182449E-01 6.9839337E-01 7.2519960E-01 7,.5224146E-01 7.7951765E-01 8.0702668E-01 8.3476691E-01 8.6273694E-01 8.9093535E-01 9.1936070E-0'1 9.4801137E-01 h=3 0. -1.0362426E-02-2.1089169E-02-3.2083447E-02-4.3288315E-02 -5.4660707E-02-6.6164724E-02-7.7768866E-02-8.9444429E-02-1.0116486E-01 -1. 1290513E-01-1.2464139E-01-1. 3635077E-01-1. 4801104E-0 1-1. 5960078E-01 -1.7109896E-01-1.8248503E-01-1.9373884E-01-2.0484047E-01-2.1577037E-01 -2.2650922E-01-2.3703795E-01-2.4733759E-01-2.5738930E-01-2.6717460E-01 -2.7667499E-01-2.8587212E-01-2.9474776E-01-3.0328386E-01-3.1146221E-01 -3.1926488E-01-3.2667409E-01-3.3367201E-01-3.4024078E-01-3.4636288E-01 -3. 5202052E-0 1-3.5719625E-0 1-3 3. 61872660341E-01-3. 6603152E-01-3.6965617E-01 -3.7272897E-01-3.7523247E-01-3.7714929E-01-3. 7846231E-01-3.7915403E-01 -3.7920 728 E-0 1-3.7860479E-0 1-3. 7732942E-O 01-3. 7536386E-01-3.7269099E-01 -3.6929362E-01-3.6515478E-01-3.6025716E-01-3.5458377E-01- 3.4811746E-01 -3.4084125E-01-3.3273805E-01-3.2379078E-01-3.1398240E-01-3.0329591E-01 -2.9171446E-01-2.7922079E-01-2.6579818E-01-2. 5142950E-0.1-2.3609794E-01 -2*1978611E-01-2* 0247764E-01-1.8415521E-01-1.6480209E-01-1.4440132E-01 -1. 2293598E-01-1. 0038902E-01-7. 6743807E-02-5.1983228E- 02-2. 6090396E-02 9.5136836E-04 2.9159199E-02 5.8549834E-02 8,9140037E-02 1.2094680E-01 1.5398683E-01 1.8827745E-01 2.2383498E-01 2.6067634E-01 2,9881859E-01 3 * 3827828E-01 3.7907262E-01 4.2121838E-01 4.6473188E-01 5.0963037E-01 5.5593045E-01 6.0364945E-01 6.5280367E-01 7.0341000E-01 7.5548513E-01 8.0904589E-01 8.6410975E-01 9.2069293E-01 9,7881205E-01 1.0384839E 00 1.0997255E 00 121

Table A-17 N = 3 o =o0.6 o o Y (R) 11-=0.0(0.01)1.0 =o.o(o.o 1.0000000E 00 1.0186197E 00 1.0317246E 00 1.0428128E 00 1.0525622E 00 1.0613024E 00 1.0692288E 00 1.0764709E 00 1.0831212E 00 1.0892494E 00 1.0949100E 00 1.1001469E 00 1.1049965E 00 1.1094893E 00 1.1136514E 00 1.1175056E 00 1.1210719E 00 1*1243680E 00 1.1274098E 00 1.1302117E 00 1.1327868E 00 1.1351468E 00 1.1373030E 00 1.1392653E 00 1.1410434E 00 1.1426460E 00 1.1440813E 00 1.1453572E 00 1.1464809E 00 1.1474595E 00 1..1482994E 00 1.1490068E 00 1. 1495879E 00 1.1500482E 00 1.150.3932E 00 1.1506281E 00 1.1507579E 00 1.1507874E 00 1.1507213E 00 1.1505640E 00 1.1503198E 00 1.1499930E 00 1.1495877E 00 1.1491076E 00 1.1485567E 00 1.1479386E 00 1.1472570E 00 1.1465153E 00 1.1457170E 00 1.1448654E 00 1.1439637E 00 1.1430151E 00 1.1420228E 00 1.1409897E 00 1.1399188E 00 1.1388131E 00 1.1376753E 00 1.1365083E 00 1.1353148E 00 1.1340974E 00 1.1328589E 00 1.1316017E 00 1.1303286E 00 1.1290418E 00 1.1277440E 00 1.1264375E 00 1.1251247E 00 1.1238080E 00 1.1224896E 00 1.1211719E 00 1.1198571E 00 1.1185474E 00 1*1172450E 00 1.1159520E 00 1.1146707E 00 1.1134030E 00 1.1121510E 00 1.1109169E 00 1.1097026E 00 1.1085101E 00 1.1073415E 00 1.1061986E 00 1.1050834E 00 1.1039979E 00 1,1029439E 00 11019233E 00 1.1009379E 00 1.0999897E 00 1.0990804E 00 1.0982118E 00 1.0973857E 00 1.0966039E 00 1.0958682E 00 1.0951803E 00 1.0945419E 00 1,0939548E 00 1.0934206E 00 10929410E 00 1.0925177E 00 1.0921524E 00 1*0918467E O0 O, 6b.7275U077-03 1,3678675E-U2 2.081563UE-0Z 2.811802TE-02 3.5571682E-02 4.3165659E-02 5.0890986E-02 5.8740046E-02 6.6706204E-02 741783572E-UO2 6.2966646L-02 9o1251205E-U2 9.9632217E-02 10810578E-0 1 1. 1666809E-01 1253403154487E-01 142.3404487E5283E-01 14285283E-01 5173646E-01 lob6Obv9Zt 0u1 1,6b~/1952E-0I Id -UI 1*f^93tU I 1,- 9.I49 3 - 0 1 2.0647743E-01 2.1581825E-01 2.2521532E-01 2.3466663E-01 2.4417026E-01 Z. 5372435E-01 2 63327U8-01 7674E- 2 7297674 E- 2. 8267163E-01 29241014E-01 3.0219069E-01 3.1201175E-01 3.2187184E-01 3.3'176953E-01 3.4170341E-01 3.5167215E-01 3.6167441E-001 37817744370892E-01 3,8l77443E-Ol 3.9186973E-01 4.0199364E-01 4.1214501E-01 4. 2232271E-01 4. 3252565E-01 4-.4275278E-01 4.53UU3U4E-01 4.6327541E-01 4. 7356892E- 1 4. 8388259E-01 4.9421549E-01 5 0456668E-01 5.149352 6E-01 5. 2 0 36E-01 5.3572110E-01 5.4613665E-01 5.5656618E-01 5.6700889E-01 5.7746398E-01 5.8793068E-01 5.9840823E-01 6.0889590E-01 6.1939295E-01 6.2989868E-01 6.4041238E-01 6.5093337E-01 6.6146100E-01 6.7199458E-01 6.8253348E-01 6,9307707E-01 7.0362473E-01 7.1417585E-01 7.2472984E-01 7. 3528609E-01 7.4584405E-01 7. 56403.16E-01 7.6696284E-01 7.7752256E-01 7.8808 1 7 8E-01 7. 9863998E-01 8.0919664E-01 8.1975126E-01 8.3030332E-01 8.4085236E-01 8.5139788E-01 8.6193941E-01 8.7247647E-50 8.8300862E-01 8.8300862E-01 89353540E-01 9.040'5637E'-01 9.1457111E-01 9.2507915E-01 9.3558011E-01 9.4607354E-01 9.5655905E-01 9.6703625E-01 9. 777504 696E- 01 122

Table A-17 cont. -5.0000000E-01-5.0698981E-01-5.1099163E-01-5.1376064E-01-5.1563688E-01 -5.1678481E-01-5.1730101E-01-5.1724390E-01-5.1667317E-01-5.1560698E-01 -5 * 1407586E-01-5 * 1 210005E-0-5. 0969596E-01-5. 0687718E-01-5.0365512E-01 -5.0003950E-01-4.9603869E-01-4.9165999E-01-4.869098lE-01-4.8179384E-01 -4.7631716E-01-4. 7048432E-0 1-4.6429945E-01-4.5776632E-01-4.5088835E-01 -4.4366871E-01-4. 3611031E-01-4.2821587E-01-4. 1998790E-01-4.1142877E-01 -4.0254070E-01-3.9332576E-01-3.8378594E-01-3.7392310E-01-3.6373904E-01 -3. 5323544E-01-3.t4241393E-01-3.3127606E-01-3. 1982335E-01-3. 0805722E-01 -2.9597908E-01-2.8359026E-01-2.7089209E-01-2.5788582E-01-2.445721lE-01 -2.3095393E-01-21703070E-01-2. 0280413E-01-1.8827538E-01-1.7344552E-01 -1. 5831567E-01- 1. 4288687E-0 1-1. 27 160 17E-0 1-1. 1113662E-0 1-9.48 17228E-02 -7.8203001E-02-6.1294930E-02-4.4093996E-02-2.6601174E-02-8.8174229E-03 9.2563117E-03 2.7619075E-02 4.6269929E-02 6.5207958E-02 8.4432228E-02 1.0394,182E-01 1.2373582E-1 14381334E-01 1.6443813347346E-01 1.8486417346530E-01 2.0573795E-01 2.2694052E-01 2.4842213E-01 2.7018191E-01 2.9221897E-01 3.1453241E-01 3.3712139E-01 3.5998503E-01 3.8312244E-01 4.0653275E-01 4.3021512E-01 4.5416868E-01 4.7839255E-01 5.0288586E-01 5.2764778E-01 5.5267743E-01 5.7797395E-01 6.0353647E-01 6.2936417E-01 6.5545616E-01 6.8181159E-01 7.0842961E-01 7.3530936E-01 7.6244999E-01 7.8985063E-01 8.1751043E-01 8.4542856E-01 8.7360414E-01 9.0203632E-01 9.3072425E-01 9.5966709E-01 - 3 0. -1.2255591E-02-2.4786538E-02-3.7510679E-02-5.0378310E-02 -6.3351180E-02-7.6396968E-02-8.9486984E-02-1.0259498E-01-1.1569647E-01 -1.2876827E-01-1.4178820E-01-1. 6758771E-01-1.8032641E-01 -1.9293130E-01-2.0538305E-01-2.1766266E-01-2.2975140E-01-2.4163079E-01 -2.5328256E-01-2.6468864E-01-2.7583'112E-01-2.8669224E-01-2.9725437E-01 -3.0750000E-01-3. 1741173E-01-32697227E-01-3. 3616439E-01-34497096E-01 -3.5337492E-01-3.6135927E-01-3.6890708E-01-3. 7600149E-01-3.8262565E-01 — 3. 8876280E-0 1-3.9-439619E-0 1-3. 9950915E-O 1-4. 0408500E-0 1-4.'08 10714E-01 -4.1155898E-01-4.1442396E-01-4. 1668555E-01-4.1832726E-01-4. 1933260E-01 -4.1968513E-01-4. 1936840E-01-4. 1836602E-01-4. 1666158E-01-4. 1423871E-01 -4.1108106E-0 1-4.0717228E-01-4. 0249605E-01-3. 9703605E-0 1-3. 9077598E-01 -3.8369956E-01-3.7579051E-01-3.6703256E-01-3.5740947E-01-3.4690498E-01 -3. 3550287E-01-3.2318692E-0 1-3. 0994088E-01-2. 9574858E-0 1-2. 8059380E-01 -2.6446035E-01-2.4733204E-01-2.2919269E-01-2.1002614E-01-1.8981621E-01 -1.6854673E-01-1.4620155E-01-1.2276454E-01-9.8219514E-02-7.2550354E-02 -4.5740924E-02-1.7775080E-02 1.1363297E-02 4. 1690345E-02 7.3222172E-02 1.0597490E-01 1.3996463E-01 1.7520752E-01 2.1171962E-01 2.'951707E-01 2.8861596E-01 3.2903239E-01 3.7078248E-01 4.1388228E-01 4.5834792E-01 5.0419548E-01 5.5144102E-01 6.0010067E-0i 6.5019046E-01 7.0172653E-01 7.5472494E-01 8.0920173E-01 8.6517302E-01 9.2265488E-01 9.8166335E-01 1.0422145E 00

Table A-18 N = 3 o = 0.3 L = o0.0(0.01)1.0 100OU000E 00 1.0088117t 00 1.0148200E 00 1.0197954E 00 1.0240865E 00 1.0278639E 00 1.0312297E 00 1.0342518E 00 1.0369789E 00 1.0394478E 00 1*0416874E 00 1.0437211E 00 1.0455681E 00 1.0472446E 00 1.0487645E 00 1.0501399E 00 1.0513814E 00 1.0524983E 00 1.0534991E 00 1.0543911E 00 1,0551813E 00 1.0558759E 00 1.0564805E 00 1.0570005E 00 1.0574406E 00 1.0578054E 00 1*0580991E 00 1.0583256E 00 1,0584886E 00 1.0585916E 00 1.0586378E 00 1.0586303E 00 1.0585721E 00 1.0584659E 00 1.0583144E 00 1.0581200E 00 1.0578853E 00 1.0576124E 00 1.0573037E 00 1.0569612E 00 1.0565870E 00 1,0561830E 00 1.0557511E 00 1.0552931E 00 1.0548109E 00 1.0543060E 00 1.0537801E 00 1.0532348E 00 1.0526717E 00 1.052092"3E 00 1.U-O149BOL U0 1.05080U2E 00 1.05U2(04E 00 1.0496398E UU 1.0489997E O0 1.0483515E 00 1.0476963E 00 1.0470355E 00 1.0463701E 00 1.0457014E 00 1.0450305E 00 1.0443584E 00 1.0436863E 00 1.0430153E 00 1.0423464E 00 1.0416806E 00 1.0410189E 00 1*0403$23E 00 1.0397119E 00 1.390685E 00 1.0384331E 00 1.0378066E 00 1..0371899E 00 1.0365840E 00 1.0359897E 00 1.0354079E 00 1.0348394E 00 1.0342851E 00 1.0337459E O0 1.0332225E 00 1.U327157E Ou 1.0322265E UO 1.0317555E 00 1.0313035E 00 1.0308714E 00 1.0304598E 00 1.0300696E 00 1.0297015E 00 1.0293563E 00 1.0290346E 00 1.0287372E 00 1.0284649E 00 1.0282182E 00 1.0279980E 00 1.0278050E 00 1.0276398E 00 1.,0275030E 00 1.0273955E 00 1.0273179E 00 1.0272708E 00 1.0272550E 00 0. 8.5120984E-03 1.7158501E-02 2.5913695E-02 3.4763789E-02 4.3699073E-02 5.2712044E-02 6.1796581E-02 7.0947515E-02 8.0160372E-02 8.9431216E-02 9.8756530E-02 1.0813315E-01 1.1755819E-01 1.2702902E-01 1.3654319E-01 1.4609847E-01 1.5569275E-01 1.6532409E-01 1.74990665-01 1.8469073E-01 1.9442269E-01 2.0418501E-01 2,1397626E-01 2.23795065-01 2.3364012E-01 2.4351020E-01 2.5340413E-01 2.6332079E-01 2.7325911E-01 2.8321807E-01 2.9319670E-01 3.0319406E-01 3.1320926E-01 3.2324143E-01 3.3328975E-01 3.4335344E-01 3.5343171E-01 3.6352386E-01 3.73629165-01 3.8374694E-01 3.9387654E-01 4.0401733E-01 4.1416870E-01 4.2433006E-01 4.3450084E-01 4.4468050E-01 4.5486849E-01 4.6506431E-01 4.7526744E-01 4.8547742E-01 4.9569377E-01 5.0591604E-01 5.1614380E-01 5.2637661E-01 5.3661406E-01 5.4685575E-01 5.5710129E-01 5.6735030E-01 5.7760242E-01 5.8785728E-01 5.9811454E-01 6.0837386E-01 6.1863492E-01 6.28897395-01 6.3916096E-01 6.4942533E-01 6.5969020E-01 6.6995529E-01 6.8022032E-01 6.9048501E-01 7.0074910E-01 7.1101232E-01 7.2127444E-01 7.31535185-01 7.4179434E-01 7.5205166E-01 7.6230690E-01 7'7255986E-01 7.8281031E-01 7.9305804E-01 8.0330284E-01 8.1354451E-01 8.2378285E-01 8.3401765E-01 8.4424875E-01 8.5447593E-01 8.6469904E-01 8.7491788E-01 8. 8513229E-01 8.9534209E-01 9.0554711E-01 9.1574719E-01 9.2594218E-01 9.36131-92E-01 9.4631625E-01 9.5649502E-01 9.6666809E-01 9.7683532E-01 9.8699656E-01 9.9715167E-01 124

Table A-18 cont. -5 OOOOOOOE-01-5. 0328649E-01-5. 0492511E- 015. 0579955E-0 1-5. 0608186E-01 -5.0585520E-01-5.0516832E-01-5.0405313E-01-5.0253202E-01-5.0062157E-01 -4. 9833444E-01-4.9568061E-01-4. 9266816E-01-4. 8930370E-0 1-4.8559276E-01 -4.8154004E-01-4.7714952E-01-4.7242469E-01-4.6736855E-01-4.6198378E-01 -4.5627275E-01-4.5023754E-01-4.4388008E-01-4.3720207E-01-4.3020508E-01 -4.2289051E-01-4.1525971E-01-4.0731385E-01-3.9905408E-01-3.9048143E-01 -3.8159688E-01-3.7240136E-01-3.6289570E-01-3.5308075E-01-3.4295727E-01 -3.3252600E-01-3.2178764E-0-3. 10 74287E-01-2. 9939234E-01-2.8773666E-01 -2.7577645E-01-2.6351226E-01- 2.5094469E-01-2.3807426E-01-2.2490153E-01 -2.1142699E-0- 1.9765117E-01- 1.8357455E-01-1.6919764E-01-1. 5452089E-01 -1. 3954479E- 01-1. 2426981E-01-1.0869638E-01-9. 2824968E-02-7.6656011E-02 -6.0189951E-02-4.3427222E-02-2.6368255E-02-9.0134737E-03 8.6366947E-03 2.6581840E-02 4.4821538E-02 6.3355374E-02 8.2182944E-02 1.0130383E-01 1.2071763E-0 1 1.4042394E-0 1 1.6042234E-01 l.8071246E-01 2.01-29387E-01 2.2216620E-01 2.4332901E-01 2.6478193-E-01 2. 8652457E-01 3.0855653E-01 3.3087738E-01 3.5348677E-01 3.7638430E-01 3.9956956E-01 4.2304215E-01 4.4680170E-01 4.7084782E-01 4.9518009E-01 5.1979814E-01 5.4470158E-01 5.6989000E-01 5.9536304E-01 6.2112026E-01 6.4716132E-01 6.7348580E-01 7.0009331E-01 7.2698347E-01 7.5415589E-01 7.8161015E-01 8.0934589E-01 8.3736269E-01 8.6566020E-01 8.9423801E-01 9.2309569E-01 9.5223290E-01 9.8164922E-01 0. -1.3723616E-02-2.7589801E-02-4. 1544771E-02-5.5553476E-02 -6 o9587526E-02-8. 3621993E-02-9. 7634085E-02-1.1160245E-0 1-1.o5 50675E-01 -1. 3932743E-01-1 5304555E-01-1.6664261E-01I-1.8010051E-O 1-1.9340149E-01 -2.0652801E-01-2.1946278E-01-2.3218870E-01-2.4468882E-01-2.56946~6E-01 -2.66894461E-01-2.8066703E-01-2.9209714E- 1-3. 0321855E-0 1-3.1401495E-01 -3.2447009E-01-3.3456779E-01-3.4429193E-01-3o5362642E-01-3.6255523E-01 -3. 7106236E-01-3o7913186E-01-7913186E-01-3 8674780-01-39389429E-0 1-40055546E-01 -4.0671547E-01-4. 1235851E-01-4* 1746879E-0 1-4. 2203053E-0 1-4. 2602796E-01 -4.2944538E-01-4.3226703E-01-4.o3447723E-01-44.3606027E-01-4.3700047E-01 -4.3728217E-01-4.3688971E-01-4..3580744E-01-4.3401973E-01-4.3151094E-01 -4,2826545E-01-42426766E-1-4 242 6766E-01-4. 1950.196E-01-4. 1395274E-01-4.0760443E-01 -4 0044144E-01-3 *9244817E-01-3. 8360906E-0 1-3. 7390854E-0 1-3.6333105E-01 -3.5186102E-01-3.3948291E-01-3.2618115E-01-3.1194021E-01.-2.9674453E-01 -'2.8057858E-01-2.6342681E-01-2.4527370E-01-2.2610372E-01:'-2.0590133E-01 -1. 8465100E-0 1-1. 6233720E-0 1-1. 3894445E-0 1-1.1445718E-0 1-8.885988.8E-02 -6 2137074E-02-3.4273207E-02-5.2527866E-03 2.4939710E-02 5.6319780E-02 8.8902929E-02 1.2270466E-01 1.5774051E-01 1.9402594E-01 2,3157648E-01 2.7040761E-01 3.1053485E-01 3.5197370E-01 3.9473992E-01 4.3884816E-01 4.8431478E-01 5 3115496E-01 5.7938424E-01 6.2901806E-01 6.8007196E-01 7.3256143E-01 7.8650192E-01 8.,4190896E-01 8.9879801E-01 9o5718457E-01 1.0170842E 00 125

Table A-19 N = 4 m 0=o0.9 = 0.0(0.01)1.0 a _o 1.0000000E 00 1.0351210E 00 1.0609466E 00 1.0835065E 00 1.1039350E 00 1.1227742E 00 1.1403401E 00 1.1568386E 00 1.1724145E 00 1.1871749E 00 1.2012024E 00 1.2145625E 00 1.2273085E 00 1.2394846E 00 1.2511282E 00 1.2622715E 00 1.2729421E 00 1.2831645E 00 1.2929602E 00 1.3023486E 00 1.3113471E 00 1.3199715E 00 1.3282360E 00 1.3361541E 00 1.3437378E 00 1.3509985E 00 1.3579468E 00 1.3645924E 00 1.3709448E 00 1.3770125E 00 1.3828038E 00 1.3883267E 00 1.3935885E 00 1.3985963E 00 1.4033569E 00 1.4078770E 00 1.4121625E 00 1.4162198E 00 1.4200543E 00 1.4236721E 00 1.4270783E 00 1.4302781E 00 1.4332768E 00 1,4360792E 00 1.4386903E 00 1.4411145E 00 1.4433569E 00 1.4454216E QO 1.4473130E 00 1.4490354E 00 1.4505932E 00 1.4519902E 00 1.4532308E 00 1.4543187E 00 1*4552580E 00 1l4560522E 00 1.4567054E 00 1.4572215E 00 1.4576036E 00 1.4578559E 00 1.4579813E 00 1.4579839E 00 1.4578670E 00 1,4576337E 00 1.4572881E 00 1.4568330E 00 1.4562720E 00 1.4556081E 00 1,4548447E 00 1.4539849E 00 1.4530322E 00 1.451'9894E 00 1.4508602E 00 1,.4496468E 00 1.4483534E 00 1.4469823E 00 1.4455365E 00 1.4440192E 00 1,4424336E 00 1*4407825E 00 1.4390689E 00 1.4372958E 00 1.4354659E 0(0 1.4335821E 00 1.4316475E 00 1.4296649E 00 1.4276368E 00 1.4255666E 00 1.4234573E 00 1.4213108E 00 1.4191305E 00 1.4169191E 00 1.4146796E 00 1.4124142E 00 1.4101261E 00 1.4078179E 00 1.4054924E 00 1.4031524E 00 1.4008001E 00 1.3984388E 00 1o3960710E 00 0. 3.7299700E-03 7.6756319E-03 1.1803954E-02 1.6097946E-02 2.0546103E-02 2,5139728E-02 2.9871807E-02 3.4736450E-02 3.9728569E-02 4.4843655E-02 5.0077686E-02 5.5426998E-02 6.0888190E-02 6.6458182E-02 7.2134022E-02 7.7912986E-02 8.3792490E-02 8.9770085E-02 9.5843443E-02 1.0201033E-01 1.0826862E-01 1.1461627E-01 1,2105130E-01 1.2757186E-01 1.3417604E-01 1.4086213E-01 1.4762841E-01 1,5447319E-01 161394YO90E-01 1.6839192E-01 1.7546275E-01 1.8260596E-01 118982000E-01 1.9710357E-01 2.044551.7E-01 21187358E-01 2.1935740E-01 2.2690546E-01 2*3451636E-01 2.4218899E-01 2.4992214E-01 2.5771461E-01 2.6556527E-01 2.7347299E-01 2.8143675E-01 2.8945531E-01 2.9752774E-01 3.0565303E-01 3.1383014E-01 3.2205798E-01 3.3033578E-01 3. 3866242E-01 3.4703706E-01 3.5545871E-01 3.6392661E-01 3.7243972E-01 3.8099722E-01 3.8959836E-01 3.9824215E-01 4.0692804E-01 4.1565500E-01 4.2442228E-01 4.3322932E-01 4.4207503E-01 4.5095893E-01 4.5988016E-01 4.o6883815E-01 4o7783213E-01 4,8686153E-01 4.9592549E-01 5.05023521 5.0502352E-01 5.1415473E-01 5.2131890E-01 5.3251503E-01 5.4174269E-01 5.5100138E-01 5.6029043E-01 5.6960927E-01 5.7895727E-01 5.8833399E-01 5.9773884E-01 6.0717143E-01 6.1663116E-01 6-.2611754E-01 6.3563006E-01 6.4516841E-01 6.5473191E-01 6.6432002E-01 6.7393264E-01 6.8356917E-01 6.9322915E-01 7.0291212E-01 7.1261789E-01 7.2234594E-01 7.3209587E-01 7.4186729E-01 7.5165980E-01 7.6147331E-01 7.7130722E-01 7.81 16122E-01 -5.0000000E-01-5.1330171E-01-5.2163479E-0 1-5.2804594E-01-5.3311655E-01 -5.3712615E-01-5.4023815E-01-5.4255939E-01-5.4416494E-01-5.4511035E-01 5. 4543827E-01-5.4518242E-01-5.4437015E-O 1-5.4302401E-O 1-5.4 1 16291E-O 1 -5.3880294E-01-5.3595794E-01-5.3263996-01-5.28853E01-5.246260001 126

Table A-19 cont. -5S. 1994767E-01-5. 1483200E-01-5.0928571E-01-5.O0331491E-01-4.9692515E.-01 -4.90 12153E-01-4.8290873E-01-4.7529106E-01-4.6727257E-1-4. 5885698E-01 -4.5004777E-01-4.4084822E-01-4.3 126138E-01-4.2129017E-01-4. 1093730E-01 -4.0020539E-01-3.89096809689E-01-3.7761415E-01-3.6575938E-01-3.5353477E-01 -3.4U94233E-U1-3.2 7984U6E-01-3. 1466.184E-U1-3.0097749E 01-2.869328 1E-01 -2. 7252946E-01-2. 5776916E-O 1-2.4265346E-0 1-2. 2718393E-0 1-2. 1 136209E-01 — 1l.9518947E-01-1.7866744E-011- 1.6179744E-01-l.4458087E- 01-1.2701907E-01 -1.0911332E-01-9.0864998E-02-7.2275354E-02-5.3345616E-02-3. 4077053E-02 -1.4470824E-02 5.4718167E-03 2.-5749682E-02 4.6361690E-02 6.7306553E-02 8.8583229E-02 1.1019055E-01 1.3212743E-01 1.5439275E-01 1.7698544E-01 1.9990439E-01 2.2314851E0-1- 2.4671673E-01 2.7060807E-01 2.9482137E-01 3.1935563E-01 3.4420984E-01 3.6938296E-01 3.9487392E-01 4.2068171E-01 4.4680535E-01 4.7324379E-01 4.9999605E-01 5.2706111E-01 5.5443799E-01 5.8212566E-01 6.1012317E-01 6.3842949E-01 6.6704360E-01 6.9596461E-01 7.2519149E-01 7.5472323E-01 7.8455890E-01 8.1469751 8.1469750E-0 8.4513804E-01 8.7587957E-01 9.0692114E-01 9.3826169E-01 9.6990037E-01 1.0018361E 00 1.0340680E UO =-3 0. -9.6467454E-03-1. 9673049E-2-2.9976510E-02-4.0497797E-02 -5. 1192579E-02-6.2024238E-02-7.2960835E-02-8.3973565E-02-9.5035863E-02 -1. 0612283E-01-1.1721085E-01-1. 2827735E-01-1. 3930056E-01-11. 5025943E-01 -1. 6113344E-01-1.7 190259E-01-1. 8254727E-01-1.9304826E-01-2. 0338665E-01 -2. 1354377E-01-2.2350125E-01-2.3324091E-01-2.4274477E-01-2.5199504E-01 -2.6097406E-01-2.6966437E-01-2.7804861E-01-2.8610951E-01-2.9383000E-01'-3.Uii31U5L-U1 —.U08181(4h —I1 —3.14?[92b -UOi-3.20968 0-3L-I-3.263386L-01 -3.3205770E-01-3.3692389E-01-3.4131593E-01-3.4521752E-01-3.4861223E-01 -.5i148 8U_ —3.ID 3 Ul2Eo-I5.btb 310E-1 13.556;c7840U — 1-J3. 5416U/1-U1 -3.5743011E-01-3. 5682448E-01-3.5558330E-01-3. 5369069E-01-3.5113080E-01 -3.4788775E-01 —3. 3439458 - 1-3 928934E-01-3. 3390250E-01-3.2776962E-01 -3.2087518E-01-3.1320344E-01-3.0473887E-01-2.9546597E-01-2..8536916E-01 -2.7443307E-01-2.6264213E-01-2.4998094E-01-2.3643420E-01-2.2198636E-01 -2* 06622.21E-O 1-1. 90 32637E-01-1. 7308360E-01-1. 5487862E-01-1. 356962 1E-01 -1.I15521U9E-01-9.4338121E -02-7.2132058E-2'-4.8887906E 02-2.4590370E-02 7.7552534E-04 2.7224855E-02 5.4772688E-02 8.3434080E-02 1.1322403E-01 1.4415749E-01 1.7624945E-01 2.0951481E-01 2.4396848E-01 2.7962539E-01 3.1650040E-01 3.5460831E-01 3.9396403E-01 4.3458240E-01 4.7647809E-01 5.1966590E-01 5.6416060E-01 6.0997695E-01 6.571a953E-01 7.0563315E-01 7.5550242E-01 8.0675200E-01 8.5939658E-01 9.1345057E-01 9.6892873E-01 1.0258456E 00 3..7500000E-01 3.8239097E-01 3.8561020E-01 3.8695289E-01 3.8685853E-01 3.8553918E-01 3.8312014E-01 3.7968501E-01 3.7529452E-01 3.6999577E-01 3.6382733E-01 3.5682218E-01 3.49-00960E-01 3.4041649E-01 3.3106809E-01 3.2098871E-01 3.1020208E-01 2.9873172E-01 2.8660116E-01 2.7383.417E-0t 2.6045484E-01 2.4648778E-01 2.3195814E-01 2.1689173E-01 2.0131506E-01 1.8525542E-01 1.6874091E-01 1.5180043E-01 1.3446378E-01 1.16761'64E-01 9.8725628E 02 8.0388278E-02 6.1783081E-02 4.2944510E-02 2.3907996E-02 4. 7099558E-03-1.4612183E-02-3.40 19990E-02-5. 3474044E-02-7. 2933859E-02 -9 * 2357967E- 2 -1.1170385E-0 1-1 3U992797E-U 1- 1 ~ 4998573E-U 1-1. 6883150 E-O 1 -1. 8741862E-01-2.0569934E-01-2.2362492E-01-2. 4114554E-01-2.5821033E-01 -2. 7476737E-01-2.9076357E-01-3.0614503E-01-3. 2085647E-01-3. 3484186E-01 -3.4804385E-01-3.6040424E -01-3.7186357E-01-3.8236149E-01-3.9183637E-01 -4.0022582E-01-4.0746602E-01-4.1349239E-01-4.1823901E-0 1-4.2163914E-01 -4.2362475E-01-4.2412696E-01-4.2307558E-01-4.2039956E-01-4.1602659E-01 127

Table A-19 cont. -4. 0 9 8 8 349E-01-40189575E —-O11-3.-91-98-811E-01-'-348.0 8 3 8 4E - 0 1-3. 6 6 1 0 5 5 6 E- 0 1 -3.4997440E-01-3.3161084E-01-3.1093385E-01-2.8786177E-0 1- 26231139E-01 -2. 3419889E-0O1-2.034388 E-01-1. 6994544E-0 1-1.3363103E-0 1-9.4407579E-02 -5.2185341E-,02-6.8740738E-03 4.1618113E-02 9.3383443E-02 1.4851616E-01 2.0711073E-01 2.6926329E-01 3.s3507057E-01 4.0463095E-01 4.7804317E-01 5.5540776E-01 6.3682567E-01 7.2239970E-01 8.1223265E-01 9.0642952E-01 1*0050956E 00 128

Table A-20 N=4 o =o.6 o o = 0.0(0.01)1.0 h~o 1.OOOOOOOE 00 1.0211032E 00 1.0358219E 00 1.0482449E 00 1.0591596E 00 1.0689472E 00 1.0778325E 00 1.0859637E 00 1.0934462E 00 1.1003589E 00 1.1067629E 00 1.1127075E 00 1.1182330E 00 1.1233730E 00 1.1281563E 00 1.1326077E 00 1.1367487E 00 1.1405984E 00 1.1441739E 00 1.1474903E 00 1.1505614E 00 1.1533998E 00 1.1560167E 00 1.1584229E 00 1.1606279E 00 1.1626408E 00 1.1644700E 00 1.1661233E 00 1.1676081E 00 1.1689312E 00 1.1700991E 00 1.1711181E 00 1.1719939E 00 1.1727321E 00 1.1733380E 00 1.1738165E 00 1.1741725E 00 1.1744106E 00 1.1745351E 00 1.1745503E 00 1.1744603E 00 1.1742689E 00 1.1739799E 00 1.1735970E 00 1.1731237E 00 1.1725633E 00 1.1719192E 00 1.1711946E 00 1.1703925E 00 1.1695159E 00 1.1685677E 00 1.1675509E 00 1.1664681E 00 1.1653221E 00 1.1641154E 00 1.1628506E 00 1.1615303E 00 1.1601567E 00 1.1587325E 00 1.1572598E 00 1.1557409E 00 1.1541781E 00 1.1525736E 00 1.1509294E 00 1.1492478E 00 1.1475307E 00 1.1457802E 00 1.1439982E 00 1.14'21868E 00 1.1403478E 00 1.1384831E 00 1.1365946E 00 1.1346841E 00 1.1327534E 00 1.1308042E 00 1.1288383E 00 1.1268574E 00 1.1248632E 00 1.1228574E 00 1.1208415E 00 1.1188173E 00 1.1167864E 00 1.1147502E 00 1..1127104E 00 1.1106685E 00 1.1086260E 00 1.1065845E 00 1.1045453E 00 1.1025101E 00 1.1004802E 00 1.0984571E 00 1.0964422E 00 1.0944369E 00 1.0924425E 00 1.0904606E 00 1.0884925E 00 1.0865394E 00 1.0846028E 00 1.0826840E 00 1.,)807843E 00 1.0789050E 00 Q12 0. 6.6629212E-03 1.3557999E-02 2.0642138E-02 2.7892464E-02 3.5293213E-02 4.2832305E-02 5.0499931E-02 5.8287818E —02 6.6188809E-02 7.4196589E-02 8.2305510E-02 9.0510459E-02 9.8806766E-02 1.0719013E-01 1.1565658E-01 1.2420241E-01 1.3282415E-01 1.4151855E-01 1.5028255E-01 1.5911326E-01 1.6800795E-01 1.7696400E-01 1.8597897E-01 1.9505048E-01 2.0417629E-01 2.1335428E-01 2.2258237E-01 2.3185861E-01 2.4118110E-01 2.5054806E-01 2.5995774E-01 2.6940847E-01 2.7889864E-01 2.8842673E-01 2.9799123E-01 3.0759071E-01 3.1722381E-01 3.2688918E-01 3.3658554E-01 3.4631164E-01 3.5606630E-01 3.6584837E-01 3.7565672E-01 3.8549027E-01 3.9534799E-01 4.0522887E-01 4.1513194E-01 4.2505626E-01 4.3500092E-01 4.4496504E-01 4.5494777E-01 4.6494830E-01 4.7496583E-01 4.8499960E-01 4.9504886E-01 5.0511290E-01 5.1519102E-01 5.2528257E-01 5.3538690E-01 5.4550338E-01 5.5563140E-01 5.6577039E-01 5.7591981E-01 5.8607908E-01 5.9624771E-01 6.0642518E-01 6.1661102E-01 6.2680475E-01 6.3700593E-01 6.4721413E-01 6.5742893E-01 6.6764994E-01 6.7787676E-011 6.8810905E-01 6.9834642E-01 7.0858858E-01 7.1883518E-01 7.2908590E-01 7.3934047E-01 7.4959860E-O1 7.5986001E-01 7.7012447E-01 7.8039172E-01 7.9066153E-01 8.0093369E-01 8.1120801E-01 8.2148428E-01 8.3176231E-01 8.4204196E-01 8.5232304E-01 8.6260543E-01 8.7288896E-01 8.8317354E-01 8.9345903E-01 9.0374535E-01 9.1403238E-01 9.2432004E-01 9.3460826E-01 9.4489698E-01 9.55 18614E-01 -5.OOOOOOOE-01-5 0838739E-01-5 1334594E-01-5.1692623E-01-5.1952291E-01 -5.2132599E-01-5.2244647E-01-5. 2295688E-01-5.2290801E-01-5.2233737E-01 -5.2 127359E-01-5 * 1973925E-01-5. 1775253E-01-5. 1532836E-0 1-5. 1247917E-Ol -5.0921546E-01-5.0554621E- 01-5.0147915E-01-4.9702103E-01-4.9217775E-01 129

Table A-20 cont. -4.8695455E-01-4.8135604E-01-4.7538639E-01-4.6904932E-01-4.6234822E-01 -4.5528615E-01-4.4786588E-01-4.4008998E-01-4.3196079E-01-4.2348049E-01 -4.1465107E-01-4.0547441E-01-3.9595224E-01-3.8608617E-01-3.7587775E-01 -3.6532839E-0O1-3.5443944E-01-3.4321218E-01-3.3164783E-01-3.1974753E-01 -3.75 1239E- 1-2 9494344E - 1-2. 8204169E-01-2. 68 80810E-01-2.5524360E-01 -2.4134907E-01-2.2712539E-01-2.1257336E-01-1.9769382E-01-1.82487i2E-Q1 -1.6695524E-01-1.5109772E-01-1.3491568E-01-1.1840982E-01-1.o0158083E-01 -8.4429379E-02-6.6956145E-02-4.9161770E-02-3. 1046896E-02-1.2612152E-02 6.1418493E-03 2.5214485E-02 4.4605159E-02 6.4313276E-02 8.4338243E-02 1.0467947E-01 1.2533640E-01 1.4630843E-01 1'.6759502E-01 1.8919560E-01 2.1110961E-01 2.3333648E-01 2.5587567E-01 2.7872664E-01 3.0188884E-01 3.2536170E-01 3.4914471E-01 3.7323732E-01 3.9763899E-01 4.2234917E-01 4.4736736E-01 4.7269300E-01 4.9832557E-01 5.2426452E -01 5. 5050935E-01 5.7705952E-01 6.0391449E-01 6.3107373E-01 6.5853673E-01 6.8630297E-01 7.1437188E-01 7.4274297E-01 7.7141572E-01 8.0o38956E-01 8.2966403E-01 8.5923853E-01 8.8911260E-01 9.1928568E-01 9.4975723E-01 9.8052676E-01 1.0115937E 00 0. -1.1865774E-02-2.4023837E-02-3. 6384589E-02-4.8894998E-02 - 6. 15 14901E-02-7.4210773E-02-8.6953148E-02-9. 9715284E-02-1.1247239E-01 -1.2 520115E-0 1-1 3787934E-01- 1 5048567E-01-1. 6299955E -01-1. 7540 1 01E-01 -1.8767058E-01-1.9978921E-01-2.1173820E-01-2.2349921E-01-2.3505413E-O1 -2.4'638511E-01-2 5747453E-01-2. 6830493E-01-2. 7885906E-0 1-2. 8911977E-01 -2.9907010E-01-3.0869319E-01-3.1797232E-01-3.2689082E-01-3.3543219E-01 -3.4357996E-01-3.513177bE-01-3. 5862933E-01-3.6549844E-01-3.7190892E-01 -3. 7784471E-01-3.8328975E-01-3 ~ 8822808E-01-3.9264377E-0'1-3 9652094E-01 -3. 9984377E-01-4U,259645L-U1-4 U4(76326E-U1 -4.o632848E-01-4.0727643E-01 -4.0759149E-01-4.0725807E-01-4.0626058E-01-4.0458350E-01-4.02.21131E-01 -3.9912856E-01-3.9531978E-01-3.9076956E-01-3.8546250E-01-3.7938323E-01 -3.7251641E-01-3.6484673E-01-3.5635888E-01-3.4703758E-01-3.368676QE-01 -3.2583369E-01-3. 1392066E-01-3 * 0111330E-01-2.8739645E-01-2.7275497E-01 -2.5717372E-01-2.4063759E-01-2.2313150E-01-2.0464037E-01-1.8514914E-01 -1.6464276E-01-1 4310623E-01-1. 2052456E-01-9. 6882716E-02-7.2165765E-02 -4.6358747E-02-1.9446-717E-02 8.5852351E-03 3.7752030E-02 6.8068543E-02 9.9549648E-02 1.3221020E-01 1.6606508E-01 2.0112906E-01 2.3741700E-01 2.7494366E-01 3.1372387E-01 3.5377237E-01 3.9510390E-01 4.3773324E-01 4.8167509E-01 5.2694415E-01 5.7355515E-01 6.2152274E-01 6.7086163E-01 7.2158647E-01 7.7371186E-01 8.2725247E-01 8.8222291E-01 9.3863777E-01 9.9651167E-01 3.7500000E-01 3.7965412E-01 3.8128416E-01 3.8142790E-01 3. 8038037E-01 3.7828406E-01 3.7522351E-01 3.7125564E-01 3.6642242E-01 3.6075713E-01 3.5428766E-01 3.4703862E-01 3.3903251E-01 3.3029062E-01 3.2083358E-01 3. 1068170E-01 2.9985535E-01 2.8837513E-01 2.7626203E-01 2.6353756E-01 2.5022384E-01 2.3634370E-01 2.2192071E-01 2.0697926E-01 1.9154456E-01 1.7564270E-01 1.5930071E-01 1.4254650E-01 1.2540895E-01 1.0791789E-01 9.0104123E-02 7.1999452E-02 5.3636668E-02 3. 5049570E-02 1.6272973E-02 -2.6573150E-03-2.1704432E-02-4.0830495E-02-5.9996609E-02-7;9162809E-02 -9.8288109E-02-1. 1733048E-01-1.3624685E-01-1. 5499308E-01-1. 7352399E-01 -1.9179338E-01-2.0975392E-01-2.2735731E-01-2.4455417E-01-2.6129404E-01 -2.7752543E-01-2 9319570E-01-3. 0825134E-01-3. 2263 752E-0 1-3. 3629862E-01 — 3.4917774E-01-3.6121708E-01-3.7235764E-01-3.8253949E-01-3.9170147E-01 -3.9978158E-01-4.0671650E-01-4.1244208E-0 1-4. 1689286E-01-4. 2000256E-01 -4.2170362E-01-4.2192764E-01-4.2060490E-01-4.1766486E-01-4.1303567E-01 150

Table A-20 cont. — 4. 0664467E-01-3.9841787E-01-3.8828049E-0 1-3.7615634E-01-3.6196856E-01 -3.4563884E-01-3.2 708818E-O 1-3.0623611E-O 1-28300152E-1- 7 830017752E-01-2.573 0 177E-01 -2.2905362E-01-1.9817226E-01-1.6457237E-01-1.2816699E-01-8.8868673E-02 -4.6588288E-02-1.2362261E-03 4.7278786E-02 9.9048331E-02 15416599E-01 2.1272555E-01 2.7482251E-01 3.4055292E-01 4.1001444E-01 4.8330515E-01 5.6052486E-01 6.4177381E-01 7.2715406E-01 8.1676769E-01 9.1071895E-01 1.0091123E 00 131

Table A-21 N=4 -oo= ~-3 tQ (). = o0.0(0.01)1.0 1.0000000E 00 1.0099380E 00 1.0166368E 00 1.0221664E 00 1.0269311E 00 1.0311277E 00 1.0'34'8730E 00 1.0382440E 00 1.0412959E 00 1.0440698E 00 1.0465978E 00 1.0489057E 00 1.0510146E 00 1.0529420E 00 1.0547031E 00 1.0563108E 00 1.0577762E 00 1.0591094E 00 1.0603191E 00 1.0614131E 00 1.0623986E 00 1.0632820E 00 1.0640690E 00 1.0647650E 00 1.0653750E 00 1.0659034E 00 1.0663545E 00 1.0667322E 00 1.0670401E 00 1.0672815E 00 1.0674598E 00 1.0675778E 00 1.0676384E 00 1.0676444E 00 1,0675981E 00 1.0675020E 00 1.0673584E 00 1.0671694E 00 1.0669371E 00 1.0666635E 00 1.0663504E 00 1.0659996E 00 1.0656129E 00 1.0651920E 00 1.0647383E 00 1.0642534E 00 1.0637389E 00 1.0631961E 00 1.0626263E 00 1.0620310E 00 1.0614114E 00 1.0607687E 00 1.0601041E 00 1.0594188L 00 1'.0587139E 00 1.0579905E 00 1.0572496E 00 1.0564924E 00 1.0557197E 00 1.0549326E 00 1.0541321E 00 1.0533189E 00 1.0524942E 00 1.0516586E 00 1.0508131E 00 1.0499586E 00 1.0490958E 00 1.0482256E 00 1.0473487E 00 1.0464659E 00 1.0455780E 00 1.0446858E 00 1.0437898E 00 1.0428908E 00 1.0419896E 00 1.0410869E 00 1.0401832E 00 1.0392792E 00 1.0383757E 00 1.0374732E 00 1.U365(724E 00 1U0356738t O0 1.034715T 00 1l0338859E 00 1.0347781E 0338859E 0329978E 00 1.0321143E 00 1.0312361E 00 1.0303636E 00 1.0294975E 00 1.0286383E 00 1.027.7865E 00 1.0269427E 00 1.0261074E 00 1,0252810E 00 1.0244642E 00 1.0236575E 00 1.0228613E 00 1.0220761E 00 1.0213025E 00 1.0205409E 00 1.0197918E 00 #:LI UO 8.5032601E-03 1.7145780E-02 2.5898591.E-02 3.4746222E-02 4.3678014E-02 5.2685834E-02 6.1763113E-02 7.0904353E-02 8.0104831E-02 8.9.360427E-02 9.8667476E-02 10802270E-01 1.1742313E-01 1.268660 7E-01 1 3634904E-01 1.4586974E-01 1 4586974E-01 1.5542608E-0 1 6501607E-0 1 1. 7463790E-0 1 1..8428984E-01 1.9397030E-01 2.0367775E-001 2.1341079E-01 2,.316806E- 01 2.3294831E-01 2.4275033E-01 2.5257299E-01 2.6241522E-01. 2,7227597E-01 2.8215429E-01 2.9204926E-01 3.0195998E-01 3,1188562E-01 3.2182538E-01 3.3177848E-01 3.4174422E-01 3.5172187E-01 3,6171079E-01 3.7171033E-01 3.8171989E-01 3.9173888E-01 4.0176674E-01 4. 1180295 E-O1 4.2184698E-01 4.3189836E-01 4.4195662E-01 4.5202130E-01 4.6209199E-01 4.7216827E-01 4.8224974E-01 4.9233603E-01 5.0242679E-G1 5.1252167E-01 5.2262034E-01 5.3272249E-01 5.4282780E-01 5. 5293602E-01 5.6304685E-01 5.73'16003E-01 5.8327532E-01 5.9339249E-01 6.0351129E-01 6.1363152E-01 6.2375297E-01 6.3387545E-01 6.4399878E-01 6.5412278E-01 6. 6424728E-01 6.7437212E-01 6.8449718E-01 6.9462231E-01 7. 0474735E-01 7.1487222E-0 1 7.2499678E-01 7.3512093E-01 7.4524458E-01 7.5536764E-01 7.6549000E-01 7.7561161E-01 [.8573239E-01 7.9585227E-01 8.0597121E-01 8'.1608914E-01 8.2620603E-01 8.3632184E-01 8.4643652E-01 8.5655007E-01 8.6666245E-01 8.7677367E-01 8.8688368E-01 8.9699250E-01 9.0710014E-01 9.1720657E-01 9.2731185E-01 9.3741596E-01 9.4751892E-01 9.5762078E-01 9.6772155E-01 9.7782128E-01 9.8791999E-01 aa P -5.0000000E-01-5.0393842E-015.0015E-1-5 0600815E- 1-5. 0724231E-0 1-5.0784072E-01 - -5.0789941E-01-5.0747434E-01-5.0660196E-01-5.0530764E-01-5.0361005E-01 -5. 0152338E-01-4.9905871E-O 1-4.96 22494E-01-4.9302932E-01-4.8947784E-01 4.8557556E-01-4.8132672E-O1-4.7673500 —4.7180355E-01-4.6653513E132

Table A-21 cont. -4.6093215E-01-4.5499675E-01-4.4873083E-01-4.4213606E-01-4.3521398E-01 -4.2796596E-01-4.2039322E-01~-4.1249690E-O1-4.0427805E-01-3.9573758E-01 -3.8687639E-01-3.7769528E-01-3.6819498E-01-3.5837620E-O 1-3.4823959E-01 -3.3778576E-01-3.270'1527,E -01-3.1592866E-O1-3.0452645E-01-2.9280c911E-01 -2. 8077711E-0 1-2. 6843088E-O 1-2. 55 77085E-0 1-2.427974 1E-01-2. 2951095E-01 -2. 1591184E-01-2.0200044E-01-1.8777709E-01-1.7324213E-01-1. 5839588E-01 -1.4323868E-01-1.2777081E-01-1. 1199259E-01-9.5904307E-02-7. 9506251E-02 -6.279'8709E.-02-4.5781957E-02-2.845627.2E-02-1.0821922E-02 7.! 208244E-03 2.5371711E-02 4.3930472E-02 6.2796850E-02.8.1970595E-02 1.0145145E-01 1.2123917E-01 1.4133350E-01 1.6173420E-01 1.8244102E-01 2.0345373E-01 2.2477208E-01 2.4639581E-01 2.6832472E-01 2.9055854E-01 3.1309705E-01 3.3593999E-01 3.5908713E-01- 3.8253827E-01 4.0629312E-01 4,3035147E-01 4.5471311E-01 4.7937776E- 01 5.0434521E-01 5.2961521E-01 5.5518757E-01 5.8106200E-01 6.0723830E-01 6.3371621E-01 6.6049552E-01 6.8757598E-01 7.1495737E-01 7.4263944E-01 7.7062199E-01 7.9890473E-01 8.2748748E-01 8.5636996E-01 8.8555198E-01 9.1503327E-01 9.4481360E-01 9.7489274E-01 1.0052705E 00 3=3 0. -1.3558661E-02-2.7269788E-02-4. 1074653E-02-5. 4936049E-02 -6.8824376E-02-8.2713962E-02-9.6581540E-02-1.1040545E-01-1. 2416520E-01 -1. 3784114E-01-1.5 141427E-01-1.6487817876E-01-1 9133439E-01 -2.0431566E-01-2.1710541E-01-2.2968672E-01-2.420428 E-01-2.5415710E-01 -2.6601310E-01-2.7759449E-01-2.8888501E-01-2.9986854E-01-3.1052900E-01 -o3.2085043E-01-303081690E-O1-13.4041257E-01-3.4962165E-01-3.5842839E-01 -3.6681711E-01-3.7477216E-01-3.8227791E-0.1-3. 8931883E-01-3.9587935E-01 -4.0194398E-01-4.0749724E-01-4.1252370E-01-4.1700794E-01-4.2093454E-01 -4.2428117E-U 1-4. 2705346 -01-4.2921508E-O 1-4.3075773E-1-4. 3166613E-O 1 -4.3192499E-01-4.3151908E-01-4.3043315E-01-4.2865198E-01-4.2616036E-01 -4.2294311E-0 1-4.1898504E-01-4.1427101E-01-4. 0878584E-01-4.0251440E-01 -3. 9544158E-01-3.8755224E-01-3.7883129E-01-3.6926363E-01-3.5883420E-01 -3. 4752789E-0 1- 3.3532968E-01-3.2222448E-01-3.0819728E-01-2.9323304E-01 -2.7731672E-01-2.6043332E-01-2.4256784E-01-2.2370528E-01-2.0383066E-01 -1. 8292897E-01-1. 6098527E-01-1. 3798461E-0 1-1. 1391200E-0 1-8. 8752511E-02 -6,2491210E-02-3.5113156E-02-6.6034374E-03 2.3052872E-02 5.3870669E-02 8.5864851E-02 1,1905031E-01 1.5344196E-01 1.8905464E-01 2.2590324E-01 2.6400262E-01 3.0336764E-01 3.4401315E-01 3.8595397E-01 4.2920498E-01 4.7378097E-01 5,1969675E-01 5.6696718E-01 6.1560703E-01 6.6563112E-01 7.1705426E-01 7.6989119E-0 1 8.2415674E-01 8.7986565E-01 9.3703270E-01 9.9567268E-01 3.'7500000E-01 3.7706536E-01 3.7724760E-01 3.7631932E-01 3.7442957E-01 3.7165118E-01 3.6802834E-01 3.6359197E-01 3.5836603E-01 3.5237066E-01 3.4562390E-01 3.3814268E-01 3.2994344E-01 3.2104255E-01 3..145661E-01 3.0120263E-01 2.9029814E-01 2.7876140E-01 2.6661135E-01 2*5386775E-01 2.4055123E-01 2.2668326E-01 2.1228625E-01 1.'9738357E-01 1.8199950E-01 1.6615931E-01 1.4988929E-01 1.3321667E-01 1.1616970E-01 9.8777671E-02 8.1070862E-02 6.3080585E-02 4,.4839181E-02 2.6380024E-02 7.7375299E-03 -1.1052874E-02-2.9954693E-02-4.8930395E-02-6.7941423E-02-8.6948143E-02 -1.0590990E-01-1.2478496E-01-1.4353059E-01-1.6210295E-01-1.8045716E-01.41.9854732E-01-2.1632643E-01-2.3374650E-01-2. 5075843E-0 1-2.6731 2 10 E-01 -2.83 35631E-O 1-2. 9883874E-01-3. 1370623E-01-3. 2790425E-0 1-3.4137753E-01 -3.5406948E-01-3.6592267E-01-3.7687843E-01-3.8687719E-01-3.9585814E-01 -4 0375969E-01-4 105 1883E-0 1-4. 1607184E-0 1-4. 2035 365-0 1-4. 2329835E-01 -4. 248 3880E-0 1-4. 249 0700E-0 1-'4. 2343368E-0 1-4. 2034869E-0 1-4. 15 58064E-01 133

Table A-21 cont. -4.0905732E-01-4.0070518E-01-3.9044990E-01-3.7821578E-01-3..6392643E-01 -3.4750403E-O1-3.288.700O7E-01-3.O0794459E-C)1-2.84647OOE-0O1-2. 5889518E-0 1 -2.3060643E-01-1.9969648E-01-1.6608056E-01-1i2967228E-01-9.0384752E-02 -4.8129429E-02-2.8172923E-03 4.5642329E-02 9.7340415E-02 1.5236992E-01 2.U1824.2EL-U1 2,7279756E-01 3.3838594E-01 4.0768619E-01 4.8079570E-01 5.5781360E-01 6.3883949E-01 7.2397469E-01 8.1332058E-01 9.0698072E-01 1.0050589E 00 134

Table A-22 N=1 = 0.900 000 = 1.561 120 2 o.78o 560 90 o(lo) 8o Pio= 0.2. = 0.0 0 22977836E-00 0 2276.5961E-0 0 O 22 13-6772E-O0 0 2.21109387E-00 0.1 97 15 0 202 E-00 0 1799-6044E-00' 0 16004685tE-00 0o 1380t450E-00 0*114532:84E-00 0o90315343E-01 s06-609784.1E-01 0 42616177E-0i 0 62058383.2 E-0 1 0 6-70 2.3958E-03 -0 O 1'6519535 E-0 1 -0 -30463181E-O 1 -0o40737035E-01 -0 47028928E-01'.-0 491-4768.6. E-01 lO= 0.2, = 0.1 0 17082.206E-00 0.16.928 91E-00 0.o 16470 826E-00 0 ~ 15.7:24007E-00 0 t14710426E-00 0.0 13460882E-00 0.12013.339E-00- 0 10411.781E-00 0 87048-7 14E-0 1 0 6-94.44723 E-0 1 0 5 1.840734E-01 0 o 3 47 7 1.632 E-0 1 O 18756056E-01 0 6.42806289E-02 -0 a 82148235E-02 -0 18350627E-01 -0,25818813E-01 -0 a30 392466E-01 -00 31932615E-01 Io — 0.2,u 0.2 0 134613,77E-00 0 13342449E-00 0 12.989279-E-00 0 124125 977 E- 00 01136:29927E-00 0.O10665 048E-00 0,a95472792E-01 0 831058t E- 1 0 699253.31E-01 0 56331810E-01 0 42738.291E-01 0 295I57803E-01 017190830E-01 0o60131367E-02 -0s36356521Et-02 -0 11462358E- 1 -0O17:229173E-O01 -0 *20760874E-01 -0 21950153E-01 o=_ 0.2 u = 0.3 0.11040562E-00 10945076E-G 0 0- iO66-152100E-o 0.10198511E-o0 0,95701153E-01 0,87954269E —018, 0.78979845E-0 1. 0,69050565E0-01 0 58468126E-01 O 4-47554067E-01 00 36640009 E- 1 0 260 57 5 69E-01 0,16128290E-01 0.71538670E-02 -0,59301837E-03 -0*68769769E-02 -0 11507077E-01 -0. 14342633E-0 1 -0 1.529 7490E-0 1 11o= 0.2 -=L o.4 0 92626598E-0 1 0 9 1844616E-0'.. 8 9 5 2243 C 0 — 1...... 857 3 05 9 9 E-01 0O80584335E-01 0,74240005E-01 0, 66890379E-01.a58758771E-0t 0,50092255 E-O 1 0 41154161E-01 -0 32216065E-01 0 235495502-01 O 15417942E-01 0o80683172E-02 0.17239863E-02 -O ~ 3 42 2 2 773E-0 2 - 072141087E-02 -0,95362948E-02 -0 10318277E-01 135

Table A-22 cont. o0= 0.2 1 =0.5 078636611E-01 0, 77991951E-01 O 76077557E-01 0 * 72951597E-01 0 6870905.2E-01 10 63741784E-01 C 9 750t7 4 4 E7 16.205 E-O 1 043571590E-01-1 0.36203085E-01 0 288345811 E-0 0 2.1689964E-01 0 14986323E-01 0,89273413E-02' 36971 10E-02 -0 54542653E-03 -0,367138.66E-02 -CC, 5585709E-02 -0,62304417E- 02 [Lo= 0.2, = o.6 0. 6-7001288E-O1 0, 66472 695E2-01 0 6.4929'2'9'77-ol -0 -i o, 62339828E-0 1 0,58661129E-01 O'.54572579E-01 0 49604481E-0i1 0 441077'91E-01 0*38249522E-01 0*32207674E-01 C0s26165826E-01. *20307557E-01 O1t4810867E-01 0 98427702E-02 0*55542i86E-02 0.20755196E-02 -0 648762'926E-03 -0 *20573481E-02 -0 25859413E-02 __ o= 0.2 0.7 0.56819745E-01 0,563953002-O1 0.55134860E-01 0O5307&723E-01 0O50283425E-01 0 46839839E-01 0,42850596E-01 O0 38436907E-0 1 0,33732881E-01 02 18.88 1446E-01 0,24030011-G01.O 19325984 —01 0*14912296E-01 0,10923053E-01 0,74794664E-02 o 46861682E-02 0 26280 3 14E-02 912E-02 0136 7 2 E -02 0 94314557E-03 0= 0.2 0 = 0.8 O *473.84442E-01 0 47060262E-0 1 0.46097574E-01 0 44525629E-01 0O42392 188 E-01 0e 39762075E-01 0 3 6 7 15 2.06E-01 o0 33344157E-0 1 o 2.9751357E-01 0e 26045970E-01 0 22340583E-0 1 0. 1747783E-0 1 0, 1.5376734E-L 0 O1232986.4E-01 0. 96997520aE-02 0O75663111E-02 0*59943651E-02 0.50316.772E-02 0,4707497 7E-02 %= 0.2 = 0.9 0 3 7 7 98-8 9 E- 0 1 0,37582241E-01 O 3694'095 8 E-0.1 G 35893825E-0 1 0 3447265GE-0 1 0 32720.6-40E-01 0 30691003E-01 0 28445419E-01 0,26052117E-01 023'5b83817E-01 0' 21115518E-01 0,18722216E-01 0,.16476631E-01 01 14446995E-01 0 12694976E-01 O 11273810E-01 0 * 10226677.-O01 0,95853933E-02 0 93694450E-02 Co= 0.2 = 1.0 0*214141195E-01 0 o21-4L4 195E-01 00214114195E-01 0,21414195E-01 0*21-414195E-01 0*21414195E-01 0,214195E-01 0221414195E-01 o121441495E-01 0,6214141951-01 Ct21414195E-0 *21414195E-01 2 414 95 — 0,21414195E-01 0*21414195E-01 0*21414195E-01 %b= 0.4 = 0.0 0 i23774242E-00 0,23565271E-OG C022944709E-00 0,2193141OE-00 0 20 5 5663E-00 O0 18860754E-00 0 16 8 969 9 E- 0 0 147 23672 E-00 0 12407701E-O0 0 10019155E-00 0,76306091-01 0 53146379E-O 1 0 31416113E-01 0 o 11'775553E-01 -0 51785356E-02 -018931005E-0 i -0.29063995E-01 -0,35269623E-01 -037359329E-01_=

Table A-22 cont. it.= o.4 = 0.1 O 21188684E-00 0 21006400OE-00 0O 2 20465 0 87 E - 0 19581192E-00 O, i1831i572LE-0J0 O, 165902676E- 00 0 15.189442E-0O0 O 1293'923C-C QO11273714E-00 0,919019957-01 O- 7106684.4-01 0 Q86476 -O-01 0 31909 72E- 01 0 1477722 3E- 1 -0 117.32747E-04 -0o( 12007932 E-0 1 -O 20846O 83 -0 1 -0C 626 0015E - 1 -0 e 28082856E-01 Po= 0.4 -L =0.2 O.18525320E-00 0 18368923E-00 0.17944:86E 4O0 e 17146119E-00 016116867E-00 Q 614848001E-C00 6.0337806E-00 O 1175i754E-00 Qr i 0 0 1:8 4 5 1 E90 0 9 0810018451E-00 08230.8322. - 6..4432131-1 O47099101EO. 0.3083,58863 - i 16t36633E-0 C E344 972E-02 -0 6844556E-02 -0,14428218E-01 -019072590E-01 - Q 2C365535-0i pC= 0.4 ~ = 0.3 O 01624966'6E-00 0 16115128E-00 0 l.5 7 15603-00 0. 1350632 29 E-00 0,14177829E-00 0.13086305E-00 J011821823E-00 O.10422803E-00 0 89317532E-01 0,73939791E-01, 0,58562052E-01 0,4365155 7E-01 0.29661336E-1 0 1 i70 1652E-01 061012901E-02 -0*27527113E-02 -092764300E-02 -0 *13271705E-01 -0 14617084E-01 %IJO.= o.4 OQ 14284554E-00 0 14168864E-00 O0, 13825313E-00 0 1326-43 36E-0 O1 i2502981E-O0 O* 11564379E-o0 0 i 1047705UE-00 O 92740332E-0 1 0679918797E-01 0 *66695478E-01 0.53472158E-01 0.40650622H-0i 0 28620446E-0i 0 17747161E-01 O 83611441E-02 0 74758792E-03 -0486217557E-02 -0. 82976962.E-02 -0,94545873E-02 = 0o.4 [L =0.-5 O 12546442E-00 O 12447536E-00 0 121538244E-00 0. 11674231E-00 0 11023328E-00 0 10220893E-00 0 *92913073E-0iG 0 82628159E-01 0 716t66693E-O 1 0 60361729E-01 0, 49056765E -01 0 a 38095298E-01 0,27810386E-01 0*185 14529E-01 0 *i10490176E -01 0.39811464 E-02 -0,8!47A787.81E-n3.rsI7 6 -O _ 3751904E-02 -0. 409607E - P= 0.4 = o.6 0 01096 b' 98 E-O 0.10 883 83E'-00 0. 1 5372E-00.. 60.5 10230891E-00 0.96819304E-01 0.90051702E-01 0.82211733E-01 O. 73537611E-01 0 4.292894E-0 1 0.5475848 0 E-01 0 452240 65 E-01i 0 35979 349 E-0 1 0.27305228E3-01 0 s19465259E-_01 012697:655E-01 0. 72080503E-02 0.31632408E-02 0, 68612617E-03 -0. 14802727?E-03 Io= 0.4 1 = 0.7 0,94838243E-01 0.941D3218E-01 0 92118958E-01 0,.88797275E-01 0.84289093E-01 0.78731393E-01 0e7-2293043E-01 0&65169667E-01 0O57577709E-01 0,49747843E-01 0.41917977E-01 0,34326018E-01 O. 27202644E-01 0 20 764294E-0 1 O 15 2G6593 E-01 0. 10698412E-01 0.737677272E-02 0, 53424679E-02 0 46574430E-02 1537

Table A-22 cont. L c0-.4 = 0.8 0O80301893E-01 0 79769 002E-O 1 0 478186521>L-O1 O 75602532E-G1 0 72095548E-0 1 0 *67772129E-0 1 0 627-6363-9E-0 1 -0 572 2.22258E-0 i 0O51316357E-01 0 645225386E-01 O 39134-413E-01 0. 33228 13E-0.1 Oe27687132E-0 1 02267864O-O1 0.1835522 1E0 - 01, 14848237E-0 1 0 12264249E-0 1 0 10681767E-01 0 10 148876E-0 1 - 0oo.4 = o.9 0 64836498E-o 1 0o 64476058E-01 0 63405690E-0 1 0 61657914E-01 0 59285838E-0 1 0*56361536E-01 0.52973860E-01 0.49225744E-01 Oe45231073E-01 0 e41111221E'-01 0,36991370E-01 0 32996699E-01 0 29248583E-0 1 0 e 25 860908E-01 0O 22936605E-01 0 * 205 645 2 9 E- 01 0 o18816753E-01 O., 1'7746384E-01 0 17385944E-01 o= 0.4, = 1.0 0 37343307E-01 0 37343307E-01 0 e 37343307E-01 0 37343307E-0 1 0*37343307E-01 0*37343307E-01 0 37343307E-01 0 37343307E-01 0e37343307E-01 0*37343307E-01 0*37343307E-01 0 37343307E-01 0O37343307E-01 0.,37343307E-01 0 37343307E-01 %-b= 0o6 l, = 0.0 0O 22.998939E-0.0 0,22810596E-00 0*22251290E-00 0,2.13380 7ct?-00 0*20098523E-00 o0 185704-72E-00 16800292E-00 0 * 1484i769L -00 0O12754412E-00 0 10601644E'00 0 84488770E-01 0 o.3615.201E-O 0 4-402 9974E-0 1 0 26328-175E-01 0 147-6.9E-01 -06 1 3472749E-02 -010480018E-01 -o0 *16073073E-01 -0,17956502-0E-1 )J= 0.6 0 = o.1 0*21936238E-00 0 21.760213E-00 0 * 21.237484E-00 0* 20363938E-00 O 1922.5506E-00 0 17797387E-00 0 16142976E-00 0 143t2538E-00 O 12361692'E-00 0 O 10349713E'00 O, 3377341E-01 0 38-6.8-8.83 E -O O*45564508E-01 0.2902038-8E-01 0*14739201E-01 0.31 5 48337 E-02 -0 53 805873 E-02 -0 1060,7864E-0 1 -0 12368117E-0 o- 0.6, = 0.2 0,20100386E-00 0 19941808E-0 0 0,19470893'E-00 O 18701946-00 O 17658339E-00 O 16371773E-00 0 * 14881344E-O0 0 13232337E-00 0Q 11474856E-00 O 96623022E-01 0 78497478E-01 _C 3092267 0E-0i 0 44432601E-O 1 0 29528310E-0 1 0 166,62655E-01 0O62265588E-02 -Oi —14628877.E-02 -0 61720444E-02 -0 77578241E-02 Clo 0.6 1 = 0-3 0, O18242 054E-00 O. 18100587E-00 0. 1768048-4E-00. 0 16994512E-00 0 16063'511E-00 0 14915771E-00 0,13586164E-00 0 12 15091tE-00 0O10547248E-00 0.89302754E-01 0-73133022EL-01 Q35745459.8L-0i 0.42743865E-01 0 29447800-001 0 1797C395'-01 0 86603905L-E0.18006615E-02 -0 24003593E-02 -0 38150282 —02 138

Table A-22 cont. p o= 0.6 - 0.4 O 164497485E-nO O.16324624 E-00 0O15953057c-00 O 15346336E-00 0,i14522895E.-0 O, 01357755E-00 0*12331760C-00 O, i 1030641E-00 0.96439341E-0 I 0, 82137720E-01 0 67836097E-01 i 53969023'6-0I 0*40957641o-0 1 -0 2919788E-01 0 19046482E-0i O. 1)o61275E-ui 0.47448605 2-02 0 1 0291885E-02 -03 222C4162Z-03 Ib= ~o.6 4 = 0.5'O 1 473 1 4 82 E-OO 0 6 1462208s0E-uo Q 14297198E-00 Os 13766707E-00 0O 1j3046727E-00 0, 12159132E-00 0o11130893E-00. 99932520E- 01 0,87807754E-01 0, 75303041E-0O1 03627.98326'L-01.0 50673560E-0I 0O39297150E-01 0,29014760E-01 0,2138814E-1: C 12939007E-ui 0*7634099'2-02 O 43852776E-02 3,32912567E-02 Pt= 0.6 4 = 0.6 0 13-070 358 E_-C00 0 12976381-E-C O. *.12697307E-C30 O. 12241.615E-00 OL 162315-00 O. 10860705E-00 O, 99774472E-0C1 0. 900v02120E-01 O679586931E-0Oi 0G.68'45365E-0 i 0C580C3798IE-01 0.47688609E-0 i 0,37916258E-01 0,2905367itE-01 O21439223E-01, 15274579E-01 0,10717656E-01 0, 79269141E-02 Co$ 9871489E-02 0= 0.6 i7 = 0.7 0a114345605-00 0,11356197E-00 0.11123489-00 010743507E-00 0102.27797E-00 0,95920287E-01 0 8855187E-01 0 80406456E-0 0,71721692E-01 0,62764777E-01 0*53807862E-01 045123099E-01 03-6974369E-01 0O 29609267E-01 0,23251577E-01 0 18094478E-0i 0 14294662E-0o1 0* 1196758.6E-01 0 11183958E —01 = 0.6 o = 0.8 0,97690304E-01 0 97072792E-01 Q 95239014E-01 O0 92244692E-01 0 88.1-80804E-0 1 0 * 83170831E-01 0 77366999E-01 0 * 7094.56.5 1E-O 1 0.64101902E-01 0 *57043692E-01 0 49985480E-01 0 43141731E-01 O * 36720385E-0 1 0,309 16552E-0 0 578 E - 0 1 0 2506578 2 1 8 4269.1 E- 0 1 0, 18848368E-01.O 17014591E-01 0 16397077E-01 %b= 0.6, =0.9 0, 794-56624E-01 0.79034304E-01 O 7778018 E-01 0 1 O 757.323 58 E- 0 1 0*72953057E-01 0'69526726E-01 0,65557474E-01 0"61165904E-01 0, 5648 5 4.50 F - r5!h58326-0! 0I CA46831203F-0! 042150748_. -01 0 37759177E-0 1 0 s 33789925 E-01 0 3 363594E-01 O 27584293E-01 0*25536470E-01 0 24282346E-01 0*23-8600285-01 )o= 0.6 1 = 1.0 0,46583565E-01 0 46583565E-01 0,46583565E-0i 0,46583565 E-O 0,46583565E-O 1 0 *465 83565E-01 0e46583565E-01 0 46583565 t-01,0.46583563 E-O 1 0.46583563E-O1 0 4658355655-01 0, 46583565E-01 046583565E-O 1 0 46583565E-01 0.46583565E-01 0.46583565E-01 0 46583565 E-O 1 046583365E-01 40., 465835 65._- 01 139

Table A-22 cont. =o.8 = 0.0 0o.2039384E-0 0 O020249499E-O0 0.198207'31E-00 Os 19120607E-00 O.18170400E-OC 0 16998982E-00 O. 15641947E-00 0 14140526E-00 0 12540340E-00 0 10890010E-00 0 92396796L-01 0 76394'936E-01 0*61380729E-01 0 * 47810374E-0 1 0 * 36096197 9 -0 C 26594132E-0 1 0e19592891E-01 O0 15305205E-01O 013861353E-01 4g= 0.8 1 = 0.1'020152573E-00 0.20012633E-00 O.19597063E-00 0.18918492E-00 -O. 17997537E-o0 0 * 16862180E-00 O 15t546920E-00 0. 14091719E-00 0 1 2540793E-00 0 10941267E-00 0 93417404E-01 0O 77908146E O1 0 63356137E-0 1 0 50203533E-01 0 38349966E-01 0 e 29640414E-01 O 22854700E-01 0 e 18699007E-01 O 17299602E-01 0o= 0.8 ~ = 0.2 0 18953776E-00 0. 18824105E-00 0 18439030E-00 0 0 17810 252E-00 0.16956875E-00 0 15904830E-00 0.14686082E-00 0. 13337663 E-00 0.11900543E-00 O 10418383E-00 089362333E-01 0_74991131E-01 0*61506937E-01 0 49319460E-01 0638799008E-01 0*30265244E-01 O,23977461E-0 1 0, 20126709E-01 0818829991E-01 go= 0.8,L =.3 0.17543860E -O0 0.17425551E-00 0.17074219 E00 0.16500540E- 00 0e15721943E-00 0. 14762088E-00 GO 13650137E-00 0 12419878E-00 01t1108690E-00 0 97564151E-01 0.84041395E-01 0 70929522E-0 1 0O58626930E-01 0.47507424E-01 0 379C8864E-01 0 3012.2902E- 1 0 24386108E-01 0 20872792E-01 0e 19689704E-01 p0= 0.8 u = O.4 0-16060378E-00' 0*15953800E-00 015637304E-00 O 15120506E-0O O 14419109>E00 0. 13554426E-00 0. 12552'727E-00 0 11444451E-O00 O I0263271E-00 0 *90450769E-01 0 78268826E-O1 O. 66457024E-,0,2 55374262>E-01 0.45357280E-0 0 36710440E-01 0. 29696474E-0 1 0,24528496E-01 0 21363534E-01 0.20297752E-01 C) = o.8 ~ = o.5 0.14550292E-00 0*14455672E-0C-0 o14174683E-00 o. 013715867E-00 0*13093161E-00 0 12325487E-00 O 11436172- 00 0 10452235E-00,94035745tE-01 0.832205186>E-01 0.72405291E-01 0, 61918679E-0 1 6.~20T93iE'-01 0.43186156E-0o1 Oe35509422E-01 0.29282368E-01 0 24694195E-01 0 2 1 84 3 -0 0 20938106E-01 no- 0.8 4 = 0.6 0,13025374E-0. 0, 12943039E-00 0.12695 535E- 00 0.12299292E-00 O 0 11757440E-00O 0 e 11089444-00. C 10315600E-00 0. 94594203E-01 GOe5469203E-O i 0.76058256E-0O1 0 666647308E-01 0. 57522309>-01 0,48960514E-0i 0 41222070E-01 0,34542105E-01 0 29123589E-01 0. 231 157E-01 0,226661 21E-0i1 0 21862769>-01 140

Table A-22 cont. go= 0.8, = 0.7 O. 1t475 341 E-00 0 O11i405922E-00 O 11199776E-00 0 o 10863166E-00 0 10406320E-00 0 984-31186E-01 0. 91906746E-0 1 90 84681 t18E-0 1 0O76994643E-0 1 0 669060078E-01 0o61125514E-01 0.o53432038E-01 0,46.213412E-0i 0 39068971E-01 0. 34056954E-01 0 0 29488491E-01 0O26122390E-01 0o 24060S29E-01 0 23366744E-01 lo= 0.8 = 0.8 0.98-629453E-01 098077164E-01 0 9 6 430 7 7 5 E -O 0 93759021E-001 0690 124374E-01 0 o85643569E-01 0 oi0452757E-01 O.74709652E-01 0.68 588760E-01 0 62276060E-01 0U 55963361E-0 1 0. 49842468 E-0 1 0 44099365E-0i 0O 38908551E-01 034427746E-01 0O30793099E-01 0O28115045E-C01 0 2647'4957 E-01 0*25922667E -'0 oLr= 0.8 g =. 0.9 8080.1 789E-01 o 80420902E-01 o.79289818E-01 0. 77442 901E-o 0.7493-6269E-01 0 o 7 1846-08c8E-01 O 682662.48-01 0l C 64.3055.22E-0 0 60084255E-01 0 55730708E-01 0*51377161E-01 0.47155893E-01 0 43195168 E-0-1 0 3391528E-01 036525145E-01 0. 340185-14E-01 0.32171598E-01 0.31040512E-01 0 o 30659626E-01 no= 0. 8 - 1.0 0, 49433742E-01 0. 49433742E-01 Co 49433-7-42E-0 1 0 4943-3742E-O 1 0*4943374.2E-01 0 49433742E-01 0 o 49433742 E-0 1 0. 49433742E-01 O *49433742E-O 0O.49433742E-01 0 49433.742E-01 0 o 4943374.2E-0 1 0o49433742E-01 0 o49433742E-01 0*49433742E-01 0 49433742E-0 1 0O49433742E-01 0 49433742E-01 0 49433B742 E-01 go= 1.0 = 0.0 0 109:46152E-00 0 10946152E-00 0 # 10946152E-00 0 109 461.52E-00 O 10946152E-00 0O. 10946 152E-00 0 10946152E-00 0 1094615.2 E-00 10946152E-00 10946152E-00 0.10946152E-00 0.10946152E-00 0C10946152E-00 O.10946152E-00 0I 10946152E-00 0o 10946152E-00 0 10946152E-00 0.&10946t152E-00 O.10946152E-00 0 10 946152 E-0 0 Co= 1.0 o = o.1 0 11'149361E-00 0 * 11149361E-00 0 11149361E-00 0.1114936 1E-00 0 11149361E-00 0'1114:9361E-00 0 l4 1493.61E-00 O.111493-61E-00 O. 11149361E-00 O 11149361E-00 O0 11149361E- -00 0 111493-6 1E-00 OI 1 1 149361E-00 0*11149361E-00 01 11149361E-00 0O11493161E-00 0 11 49361E-.00. 0 -. 111-49361 E-00 - O * 111 493631-00 - Cio= 1.0 u = 0.2 0. 10707097 E-00 0. 10707097E-00 0 170707097E-00 0. 107 07097 E-o00 0 10707097E-00 0 10707097E-00 G. 107u7097E-00 0. 10707097E-00 0 10707097E-00 0. 10707097E-00 O 10707097E-00 0 10707097E-00O 0 10707097E-00 0. 10707097E-00 0 10707097E-00 O 10707097E-00 0.10707097E-00 0.10707097E -00 0.107G7097E-0 0' 141

Table A-22 cont. 0o= 1.0 ~ = 0.3 0 1006857E-00 0.10068157E-00. 00100 68157E-000 O01006815O7 E-00 0*10068157E-00 0. 10068 157E-00 0( 10068157E-00 Os 10068157E-00 0#10068157E-00 0s 10068 157E-00 0 1006815 7E-00 OS 10068157E-00 0 10068157E-00 0 10068 157E-00 0, 10068157E-00 O 100 68157E-00 O 10068157E-00. O, 100681 57-E-QO 0 10068157E-00 o=.0o,- = 0.4 0 93358269E-0 1 0 93358 269 E-O 1 09 3358269E-0 0 ~ 358269 E-O 1 0. 93358269E-01 0 *93358269E-01 0 * 933582369E-01 0, 93358269 E-01 0 933 5-8 26 9E-O 1 0 ~ 9 3335 8269 E-O 1 0 93 35826-9 E-O 1 O 9 33582.69 E-O 1 0o93358269E-01 0*93358269E-01 0 93358269c9-01 0 * 933582.69E-01 0 *93358269E-01 0 6 93358269E-0 1 0 o 93358269E- 1 11o= 1.0 L =- 0.5 085587588E-01 0 88558758:8E-0 1 0 O 85587588E-01 0 8 55.8758 8 E- 1 0 o 85587588E-01 0 855875-88E-01 O 85587588E-01 0O8558758;E-01 0 *855.875888E-0 1 0 885587 588E-0 1 0 85 587588E-01 0 8558775,8:8 E-01 0. 8 5587 588 E-O 1 0 855 8 7 588 E-01 0 o 8 5587588E-01 0 8.5.5.87 5. 8-8 E-O 1 O,85:587588E-O1 0,8558758.8E-01 0o855.875.88E-01 Lo= 1.0 4 = 0.6 6 77639274 E-0 1 0 77639274E-01 0 o 77639274E-0 1 O 77.639274E-0 1 O 776392'74E-01 7 76.3 9 2 7 4 E - 1 077663929274E-01 7763274639274E- 1 0 77'639 274E-0 1 o 7 7763 9 2 74E -0 1 0 o 777639274 E -O 0 1 0 77639274E-0O 0 77639274E-0 1 0, 77.639274E-01 0 *776392.74E-01 0 77639274E-01 O 77639274E-01 0 7763927.4E-0 1 0.'77639274E-01 Io= 1.0, - 0.7 0O69674293E-01 0 69674293E-0 t 0 6.967,4293 E-01 0. 696742.93 E-O 1 069674293E-01 0 &9674293E- O1 0 69674293E-01 0, 9674293 E-0 1 0 69674293E-O 1 0 o 69674293E-01 0 o 6967429.3E-0 1 0 i 69674293E-O 1 O 69A674293E-0 1 O-0.* 696 74 29 3 E-O 1 0 69674293E-01 - 0o696.74293E-01 0 69674293E-01 0 69.674293E-01 0. 6967429.3E-01 go= 1.0. = 0.8 0*61792 177E-01 0 61792177E-01 0.6179:2177E-01 0 o 61792t77E-01 0 *617921 77E-0'1. 0 61792177E'-01" 0 61792177E-0 1 0O61792177E-01 0 61792177E-01 0 o61792177E-01 0 61792177E-0i 0O61792177E-01 0 61792177E-01 0o61792177E-01 0,61792177E-01 0O61792177E-01 0 e6172177i-01 0o61792177E-01 g.o= 1.0 ~ = 0.9 0 e 54055209E-01 55209E-0 1 545405520901 0 55529E-01 0 54055209E-01 0 54055209E-0 1 0 e54055209E-01 0 o54055209E-01 0 * 54055209E- 01 0.54055209E-01 0. 5401 05455209E-01 4055209-1 4055209E-01 0.54055209E-01 0 o54055209E-01 0o54055209E-01 0,54055209E-O1 0.54055209E-01 0.54055209E-01 0,54055209E-01

Table A-22 cont. Lo= 1.0 =1.0 O 46502034E-0 1 0 o46502034E-01 0 *46502034E-01 0, 46502034E-0 i 0 46502034E-01 0 4.6502034E-01 0 46'502'034E-01 0s4635 02034E -0 1 0.46502034E-01 0o46502034E-O 1 046502034E-0i1 0. s.4'.6 5 02.034 E-i 1 0 46502034E-01 0 o46502034E-01 046502034E-01 O 46502034E-0 1.0*46502034E-01 O 46502034E-01 0 46502034E-01 143

Table A-23 N=2'oIO = 0.900 000 1 = 1.561 120 (0 = 2.017 380 7805 560 - o.336 230 2 = 0.84 057;UIo(s.0) 8 = 0 (10)180 90= 0.2 p= 0.0 O i40Z2248. SZE-O U,39'Z4UZ63E-QU: U * B6 bb6 -)UfEU 1'O 0U'ict52896E-ou 0826383564E-00 0 620 227899E-00I 0 14077363E-00 0 84988758E-01 O,39717443E-0.1 0 83573649E -0Z -Us (743634Z3t-UZ -u -U bUb0652t-02 0 49997269E-0 2 0 27731417E-01 0, 55817953E-01 0 6 84361674E-01 0 # 1086949-92E-00 0 1 24980026E-0- 0 1307074t-U.U. o0= 0.2 P= 0.1 0 2 925595UE-UU U0,e2855t996].UUE, u.za zlb t E'00 U O U -00 0 19473699E-00 0 6 15129668E-00.Os 10 795848E-00 0. 68746681E-01,U 0,i530i7 U 5 9 S 1- U' 3&s -U0i U 46 0S3.bSt =-02' U "04 I 5796E-02 0 14588.103E-0 1 0 3 1262028E-01 0,51685273E-01 0o72356073E-01 08993890E-0 1 0 I 10 b 38Eb - Us 1 58 18t6E'0U 0I = 02 = 0.2 UO21Z U 20/7E-'U 0UoiOUY81 I- -U I u 2U u5u6E00 r176O9 UlE-Ou' 0O,14838789E-00 0. 1606 992E-00 083886058E-01 54850973E-01 0,3ia50 728E-0i.O +6iU6Euoa-U u 81314i0~83E-02 O 88094985E-02 0 16577926E-01 0 29540 2 55E-01 0 45 2 6 5 8 5 5 E-0 1 0561109421E-01 U, 7455 2 77 6 E-0 1 0 8 3 755E' 0 i 8667 535E-01 ~o = 0.2 0 = 0.3 0 i 17 1+4U26E-030 O 16775 006E-00 05 1 56192 45E-00 O, i 13825871 E-00 0,11578088E-00 0 *91020827E-01 0*66411086E-01 0, 44278307E-01 0.2'6582 l09E- i 0 O 1470288E-Ci 0U.92856359E-02 O U1021'0908E-0 i 0. 16607881E0 1 2699559-01 9 0 39477947E-O 1 O 51997036E-O 1 U 4U5 b ( L.S49-0 0 I9-i U V 7i 31 E-i' I

Table A-23 cont. o, = 0.2 - = 0.4 0 1342 1 1 88E-0 0 - u u 1312 i 2 5E-uO U 6 I227 2 E-00 0 829347 E-0 0690911769E-01 0 671789566E-01 O.52821022E-01 0*358 16424E-01 O62228'367E-1 a "0.Uii3334i54E-0Ui 09424i80iE-02- 010459226E-01 0,15751279E-01 0 24133623E-01 0O34117629E-01: Oi44086807E-01 U 0.52 50U40 -5OL- " 0e -O81055E 0 "0 0.... 16U 0724403E-0-1 %, =6.2 p= 0.5 U l4U8'UY 7I UU 1U. 7i6i-00 0Iltu U Y 624E-01 0V.q:419 670 O E- V 1I 0,70870280E-01 0,56227736E-01 0*41731162E-01 028776871E-01 "Oe 1iSBI-E 3 J -ui'..u.......OE1 i' 01- 0 1'" 04u' U:b0694E0- 2....-i I00iu778iu05Eni-0 ~zi" O 14395395E-0 1 O 2 i085553E-01 0 28989457E-01 0 36849768E-01 0.7,'3471t$1U i l U Ut 47i6-(0)b(U1 -O U'+94-1(+745 t0 EO i.. o = 0.2 = 0.6 u.. e'678i.4001-8 E0 Vi e( 71'+.....it"U 0e72IZ i +.E3-02 1 EI''aI+I ~505 3 E-01'... 0 5 4092961E-0 1 0 43170638E-01 0 a 32376777E-01 0, 22759926E-0 1 U,5~1'9711E'01 0.10309259E-"0i.0.83703207E'02 0-9309 6530E-'.2 0.12712332E-01 0 17890515E-01 0 23965285 E-01 0 29985238E-01 0I35 c.-4687E-Q1.t I 840'8' 0';.9E 1,' o,::.3e99-c o0 = 0.2 11'= 0.7 0 &57261761E-01 0,56002937E1 0 56007- 1 52359769E-01 0 a 467 1-66-7 5 E-0 1 a 039-66'44177E-0 1' 0e 31V 3 31.876E-0 i O 2I4+3v125 BE-0' 0 175 i518899EL-01 0 12210239E-01 0 88102981E-02 0 751744:14E-02 0 1 82758912E-02 1078-.-t8..t'1 - Gl' i.456/U3 E",O i. i Z 2.E 1..u - 25 I 20t559E*o' U O 26'9648 16 E-O 1 0 e 29388759E-0'1 0 & 30 23 7019 E- 1 I0 = 0.2. = 0.8 0,38236197E-01 0; 37'432823E.-01 0 0351078696E-01 0 7 31507085E-'01 Oi 7 U7 O t-u. tJ. U-U.U I1(SLobI [ u~ Ue(L.IbUltUJ. ~ZiLj UI 0 95139732E-02 0 7357 9 3 66E-02 0 65-492714E-02 O, 70517629E-02 0 086'751 136E-02 0# 11101839E-01 0 a 1392925.8E-0 U. 0.t167It714 E-0i i 0 190,64491E-01 0 a20619167E-01 0 a 21 163.165 E'0 1 L = 0.2,u -= o0.9 0.21213567E-0 1 0 20820583E-01 O,19682901E-01 O0 17919529E-01..O, 15713.423E-01 13290315E -1 0 1089 2-/17-0.1 - ",8/'2 i8t0-O2 0 *70631182E-02 0 596129-54E-02 0.55095173E-02 0O 56921839E-02 0 6419. 19E-O 17.3E.- 1U E,.i3-46E-0 i 0. 11275630 E-0 1 0 *12009676E-01 0.1 12266737E-0 1 o=0 - 0. I = 1.0 0&46234217E-02 0,46234217E-02 0O46234-Z17E-02 0,46234217E-02 O, 46-2342 i 7 E-O 2 0 6 462 34.217h-02 0 1 46 2342i F'0 1 7 E.-0 0!~ 46 2 34-2 i 7 E-O 2 0,46234 1E-02 34217E-02 0462427E-02 046234217 E-02 0.462321 7E-0 2 0,46234217E-0 2 0,46234217E-02 0146234217E-02 145

Table A-23 cont. o. - 0.4 = 0.0 U 37 76U1(- tt-00 0 3685 t3E-00- 0 34247753E-00 30188727E-00 0625087063E-00 0 19443128E-00 013795931E-00 0 86624146E-01 044809Y52E-u i U. 155981u5E-0ui 0U5411 0514E-03 -0 55t073i43E-03 010518166E-01 0,30596109E-01 0.55619478E-01 0.81152753E-01.'. i2lU w6 ~L 5t-Uo' 1 1 i Oeu N It27 i52E-00.0. 4 =.1 - u6324iw506E-00 0,3 i 70. i 3.E-0 05 062951.6254E-00 0 6.16455E-00 0.21850403E-00 0e17142966E-00 0 12451372E-00 0 82134185E-01 U U479' u 60 u 3E'0ip3~t 0.027'-3460790V i' 1i41 3354E'1A 0 25103496E-0 1 04436303'40E-01 0666198731E-01 0. 88980997E"01 05iWU;345t-UU U i 1o 2 Z5824-U: 0 e i2 1579464'E-00....... =,o.4: 0.2 0 O 18182354E00 0,1435.7913E-00 O 10564204E-00 0 71632851E-01. U04453673 tE-: 1 4' U, 6266b8,31 i-0 0i 84,63O0E-01 0209i 845,49tE-01 0,31502560E-O 1 0.48267249E-01 0e68235261E-01 O0 88173617E-01 0.~ 10 500862 — u 0 u i iuu-0 I1 0e t 1 2 048iE-0 O:o = 0.4 0 =0.3 U. L93 4(i UU U,.~10 U UO'yv0,+40~ 5t177381 32E-00 O,z14939E-00 0 l1uz85,428.8-00' 0a88184518E-01 0 $i12 i0943F-01.21139QF''0' 0'0 0. 1 0' 8 8' w 00,40 112692E-0 1 0, 26621459E-01 0 21525942E-01 0 246s - 95E-01 0 *34666109E-0 1 0, 49740947E-01 0 67318942E-01 0 +8416 84726 E-0 1 1 USYzY!4butzut'Ul s' u 5:1,s1X u -u- I OA10 %_, Vi — ~ |- 0.04.4 0 17667473E00' 0 a 17267 154E;00. t 16109443 E-0 O o 143 901 1E-400 0,12087385E-00 0a 96505647'E-01 0e72615885E-01 0 5 1612450E-01 0 35504702 E0'1 0 e 25677037E-01 0 * 227'38890E-0 1 0' 26468711 E-0 I O 35858199E-01 0 O4.9250875E-01 0 6:4557812E-01 0O 79523963E-0 1 0 920125;89E-01 01-0027304E-00' 0O103'15938E-00 Lo = 0.4.i = 0.5 U1'392973:6 E-0UC u. e136i7/bI -U-E U e'I2Ib:6bU'E-00 U U I. 14-3:5"sYE-aU-. 0 95950222E-01 0 *77153841E-01 0 58:865240E-01 0.42991292E-01 o 3 1 1 1 58'9 E-0 1. o 0 aZ43 I 8' 7 I E-0 1 U' 23 55 -795 EU-U ib i.Z-i 6':-Oz0 I O.3565735 6E-0 1 0, 7318359E-01 0 60'39 3682E-O 1 O, 73048139E-01 0 e 833-546 U5 1lE-01 -' 9U4669'52 E-1O U 92'8'8Ib.604'E-01 o = 0 4 o.6 0.74071974E-01 O 60181877E-01 0,46794028E-O01 0,35362927E-01 02 a70900Ui0 E-0 1 0227841 2q.E-0i 271 0 &i9 9 —02E0 0 1026853.602E-01 0,34354228E-0 1 0 44189578E-01 0.55013096E-01 0 65380953E-01 u eY7304-13OE -0 i. 0 9547'595E-0 - 0! 8 i 043-45 E- 01 146

Table A-23 cont. J._ =o0.4 = 0-7 U0773 (722i E-0i 0675746442E-0i 01710450i0E-01 0663822807E-01 0O54922882Em0 1 0,45380616E-01 0 363000007E-0.1 0s 28721927E-01 U eZDSUuZ 9t-U 1 * i20u 9Et-jU 0 2203433iE0 i OU 258&3t.653E-0 iI 0 32079094E-0 1 0.39954316E-0 1 0148456070E-01 0 656511976E-0 1. 063i 1294L-U I 6 "0'4-5Z [t8'tE -0 1 u 6e 54uE-0i l = 0.. 4, = 0.8 O*SZUOO84uL3t-O i UI ioU5tLiU u044 386553E-01 43 745607E-t1I 0.3837086 0 E32675933E-0 1 O32675933E-1 0 2 7 60 5 1 EO1 023094828E-01 "0 204041Z9E-O i 0 196'38tU4E-01 U *2U9141 74E-0li 0.24099419E-0 i 1 0,28832666E-01 0,34563947E-01 0*4062090'9E-01 0,46289324E-01 0 5u0898749E-01 U 0 5 3902 978U' E-0 0 54945637 E-0I, 0o = 0.4 = 0.9 O.63045 7226E-01 i.030022073E-0i 0*28774934E-01.0 26:883'-66E-01 0 24604400E-0 1 0 *22250751E-01 0.20153859E-01 0, 18621499E-01 p0 i 789 195E-0 i 9u O i8138755Et0i Oi 19378003E-01 0, 21534183E-01 0,24411917E-01 0#27724806E-01 0*31128126E-01 034258741E-01 0i 3 8777467E-01 u, 38408i3E-01' 0 oo387347-01 - o = 0.4 =.i t 72>3073E'01 0 i 167230-7_oU 3E__-'0 - i u0 r.16723073E-0 i 01672 073f -0 0O16723073E-01 016723073E-01 0 16723073E-0 1 0 16723073E-01 0 16723073E-01' 01672-373E-01 " 0V1-6723073'E-01 0167 2073 E-0 016723073E-01 0.16723073E-01 0*16723073E-01 0 16723073E-01 0OUi IZlZ3U-.-ul u b7I/-0 /t-0i 0O16 i2.30 3E-0i = 0.6 0.0 0.30773314E-UU. 309.3E -U0,UU476U279834U3'1. OZZ +(66b07(E-0L0 0&20711548E-00 0&16214746E-00 0 11699194E-00 0,75709970E-01 0-, 417 747998 E-u 1 1 756.8 739 E-U I U44037763t- Z U 2-1 b 66E-02 09463549 0 E-02 0 23911-627 E-01 0 42372434E-01 0 4 614234-46E-0 1 Ue((t9b("5I t -U I u'O's Zui1 U*S1LOU5?btOfuJ gLO = 0.6 = 0.1,OeY27812'46E-00U 027!8i669 0 0- U48UE-00 - Z 4bb6SEU00 0622506829E-00 0' 18940365E-00 0 15003922E-00 0 c Ii079220E-00 0 & 75318147E-01 U 04b / i'0;'3~tm-U I U Z-ti252956t-0i 01i..i-i U I''1I -=0 0 i830725734E-0i 0 s27104769E-0 1 0 42452736E-01 0 61187077E-0O1 0 e801 17427E-01 0.9 062'05530E-U ~1 U" 10'6b' +466b E -U U 0I. is1U0'2763E0'0 l.o = 0.6. = 0.2' UZ3b674464b+-'' O Z i42 I4E- 0 * 2 0i i 8 zE-00 0 i t 92i3 i6 t E-O0 0,16227888E5-00 0.12951191E-00 0.97130331E-01 0.68279777E-01 U- e455-9 34E-O 1. l 30 92 777E-0i t 0 25 1 i0961E-0i 1.. 2792 5 95 8E-0 1 O 38136996E-0 1 0 53671546E-01 0 71895856E-01 O 0 89955715 E-O 1 010514063E-00 0,11522669E-O0 0,1i875799E-00 147

Table A-23 cont. "o = 0.6 0i-19659774( -0 U1E-00' 017i95622i7-00 0 16000108E-00 0613564825E-00 0 1 0910571E-00 0 83161380E-01 0i60465531E-01 o 04322565E-u 1 32956286E-0i 0 30313546E-01' 0i35033653E-01 0*45982462E-01 0.61309414E-01 0""86O6851E-O1 08 95605309E-01 U 01 9U 887 E-( t-UU "'O1i89wi43E-U0 U 0 122239925-0.......,u 0_.6 = o.4 0 0,159575 90E-UU U,1560L73B2 -UU 0 i1459648E-00 0 1u I39U97E-00 0.11110796E-00 0 690272816E-01 0 70191041E01 0 # 53044390E-01'O * "Ub63U14t-0'US U 6-5+i1f6 6t E-UI U06,+..4154+'5b IE-( ui I U' (4U'8U4U2t-U1 0 51531341E-01 066284367E-01 0* 8251.96.44E-01 0. 980 71431E-01 O 0 11089562E-00 O 11932 139E00 0 1.222562ZE"O0 0to = 0.6 0.5 -O.12622361 -00 U'O61z526Zb6t-OUU U*11 r5 (5E-tUU UOiU83LtE-0UU 0689173575E-0 12 0 6735 15'282E-0 1 058707178E-01 0 46490364E-01 0 38288612E-01 0 # 35038 142E-01 0) 370 76318-LU I1 0 * 44'"10Z6 1U 0- 1 0 55216515E-0O1 0 69027774E-01 0e83825572E-01 0 97786254E-0 1 — 1-0-I46E-O O 4.6 00 1 I 1 1)6 1E I'0 t 1 1 94230uE-UU iz = 0.6 o = 0.6 96 75 +5 40 E-Q i 01 9479i96E-0 0 6 89 1 52556E-0 08 550i1 0 E-0u 1 670077328E-01 0 59072139E-01 0.48954100E-01 0641050363E-01 "0~'36k+336 6(LU- " U63578-300UUttUi 0,63~i 0454E-0i 0'6b7i9090E-0i 0,57241225E-01 0669725862E-01 0,.82773942E-01 0O94903836E-01 Ua I U +-(L4(LO-0OUU.u I...1 I i 4~-00!t Oa 0iTi3326r75-E0'K =.6 = O.7 0&71387821E-01 0 70090321E' 1 0&66376944.E-01 0 060762765E-01 0,54034653E-0 1: 047153009E-01 0 41131432E-01 0 36908858E-01 0.35229871E-01 0 36547734E-01 0,40962101E-01 0 48199147E-01,5 isb36 / Z7L-U I ubb./Ibtll-UI.U a(L-L22235E-LU0I U 0 935.0 7t~-E-0i 0 97396761E-01 0 10259941E-00 0610439842E-00. ~, = 0.6 o. 0,50598820E-01 0 649891669E-01 0,47883868E-01 0,44901805E-01 0,414426(8E-01' 03810i046E-uUI'U 06!568zEL-UI U" j.,Z99UZ-0i-u 0, 34841277E-0 1 0 37412350E-01 0O42026337E-01 0 48450880E-01 06 56225388E-0 1 0 0'64 70U04.6EE-u1 U *(3 1 393UE- u t i ( O 8073 54-'4zE-0 1 o0,86765644E-0 1 0Q 90640181E-01 0. 91975944E-01 UOl = 0.6 = 0.9 0,35730203E-0 1 0 35520086E-01 0 34941771E-0 1 0, 34144279E-0 1 "033353232E-0i O1U.3283971.t-0i O3282/24E-01 0: 33-77264'94E-0l O 35548291E-O01 0 38423697E-01 0 42310998E-01 O 47046429E-01.... U5354.btb-01 -".'5-78730u0u "0u 0.66'3i86r4.bE-01 0.b67861it5u-o~ 0 71537995E-O01 O 73873314E-01 0 74675092E-01

Table A-23 cont.,o = O.6 1 = 1.0 0 39608456E -0 1 O 39608456E-01 0 39608.456E-01 0 *39608456E-01 0 39608456,E-0 u 960845b6E-u0 0 3960 456 E-0 1. 0 39608456E-0i 0 39608456E-0 1 0 39608456E'01 0 a 3960845-6E-01 0 e 39608456E,- 01 u5968L5UI UoE-i *I u3t8456E-0U- 0.3 608-4ti+56E-0i... 0.39-6 8456E-0i 3960Ol 0&456E-01 0 39608456E-01 0 39608456E-01,O = 0.8 1 = 0.0 S i i95724 -'00 U10 69i'26i4-0 0eiW7O94M620E-00'0 1594800E-00 O 13486991E-00 0 10739811E-000 o*79532210E-01 o O 53653348E-O 1 0&31't87I9E- i LU 0 U 154488ib6E-0 t..05i83978 19E-02 0 1i2664565 rME-02' 0 3 4739830E-02 0 96195359E-O2 0 18 3419 54E-01 0 & 2 77433 0 7 E-0 1 u O36u3 504E=0.t i ui 16 U686tE-0 1 Ob 4Q 60 7?721 E-0 1.. ~_ = 0.8; = 0.1 0 Osi7j8831iE-00 u I0 i7499:-664E-00...16387888E-0 0 i0 I....'......0. 0 12484718E-00 0 10076495E-00 0.76627372E-01 0, 54625379E-01 U.66157u -0 1 i ~ E i U /:: 7 L,3 1 i'"' 1-tr 1 0 Q.17196616. —0 I' C - 1*ut69772 t. - Ol 1 0O20712354E-01 0288-81997E-O1 039180406E-01 0,49743585E-01 u 6-58 t9303Ili t-0 648655 t6E0 1 0 6700 1 169E-0Ui Lo = 0.8: = 0.2 U i 521:9.4 479Et0tJ uS I 249Eit 1 -0u up~'+c.i.d.... " E-00'i u a 20o v 34E-0 0 7E 0 1.08031 8 5 E-00 0. 88304700E-01.0. 6884.6526E-01 0 5 1564 340 E-0 1 0.38055893'-01 0.29431?45oE-0i....01 926i970 8'5E'01 u 0'O,:f05 E _-0 1. 0 34700'453E-0 1 0, 44407355E-01 0 55717032E-01 0 0 66-885614E-01 0 76257964E-0 1 0 8247 66-8E-01 Art.084652656E-01 "o = 0.8 = 0.3 uoi27 24E00 ti2475.2i9 E-M00 0O 1 727420E-00 0... Oi t46E-0U 0 * 91481937E-0 1 0 76055303E-01 0.61174756E-01 O0' 48451507E-0 1 u 3 92 z 7'3Z E-LO u1 O 34 i 36846E - O 0 4 3.26 78 8 E-O0i 0 3 70 35 E-iO 1 0s46924176E-01 0,57735111E-01 0 69648784E-01 0 e 81069738E-01 o =0o.8 0 1 040168UE-U- u OU 0U2U2U73 E-U'u u. 96 Z 77 O yY30 5 -0 - 0 1'4: 9 i2 l4+t-O 1 0 76741719E-01 0 65351-865E-01 0, 54728 102E-01 0 s46189655E-01 0.4088259E.0u iC 0 3 709E"'0i 0.'i82834.-0 3+ is PM4 6 3E-0 iaC 0 57665331E-O 1 0 69127893E-01 0 & 81241817E-01 0 e 925 78647E-01 O. 1U 9 7/-9-0tUU U. i 0u78U595E-00 O. 0 i0-989i 27E-00 *. -L 1Uo = o0.O8 = 0.5 A 0.64515904E-01 0 56807477E-01 050022269E-01 0 45197677E-0 1 O0.431579ZZE-01 0' *444- 7 Z U11 1 U*i'' 0, 637(UE-0i O U5825" 1E-O 0167020569E-01 0.78660071E-01 0,90558812E-01 0,10147067E-00 O.110232 7E-0o 0 11 590154-U0 Q 11 t8621b6-E-U 149

Table A-23 cont. 0.6 ~ o = 0.8 CL = o.6 0O67465U95 fE1 i U,66bt5.26226.-u1 0,63845158E-0i U n98690 t4u -0 0155256905E-01 0*50813797E-01 O,47382439E-01 0 45732097E-01.46455036E-01 u 09&8ji35 0i U.5630351:18E-01 I O' 6460Ii73E-0 0 74967184E-0 1 0 86276063E-01 0 97519188E-01 0 s10764726E-00 O e il!'68 5I/D'3 E-ULU l " Um' t~(UU- UI ZZ -b39A+D I UU0 o = 0.8 = 0.7 0 5 553933 03 E;O 1 0. 548924v59E101.. 0 3497. Z.t8LiIUI O1~_-3 3 Z t-i-i 049413271E-01 0,47753432E-08 0.47116727E-01 0,48020983E-01 o' U t45 1 1t- 1 U*( i t:)-'t'U / t -'i 6Z. 1-3'2Ui J. UI I fi+ UUU t-0U 0 81348652E'0 1 0 91761147E-01 0 10 185962E-00 0 11080676E-00 O " i178322 4E-m.O' 1' z L 2 351619E t' U *i385(.6E-00....tU )1 = 0.8 - o.8 _o -0 0.8 0,4767891.2E-01 0 48:211294E-01 049684.727E-01 0&5238+6476E-01 0856905132~E-0u1 PM, - 6-i U+064E'0 0 s85775 567E-0 1 0 94608784E-01 0. 10297328E-00.0,11026157E-00 0. 1i 5' ZI U 16 tE0' 00 1 1i 9 50:6"4'~. E O 0 ot i 2 07 t O. vt EO _ = 0.8 ~ = 0,9 0 0Q51672085E0 1 0 53545681E-01 0 56144062-01 1 059554122E-01 0-Q 6380744+8ElO I 4 0 16V86DO'foE- i- 0 t+996/+E0 i I O 80807289E-0t'U1 0 8 7 214116 -01 0 s 9348 8 572E01 0 & 99 2741.67 E-0 1 1 0 10421862 E-00 076077834E-01 0 076077834E01" 0am 76077834E-01 0.76077834E-01 U, 1 6 0 7 7.' 0 1 Q 60 ~ ~ — r- i T 8t0 1-U i I= 1.0 = 0.0 0 -O e+9958'rE-u'2 0U 9.49-2587E-t' u.. 99'499587E02'"0,99499587E-02 0, 99499587E-0 2 o0 99499587E-02 0A 99499587E-02 P0 99499587E-02 0, 99499587'E-0 2 0 e 99499587E-02 0 7 99499587E-02 0 e 99499587E-02 0 i 9949958 7E20-O' 0 i 994Y'95.8E-0 2 99'499587t- 0'go = 1.0,.= 0.1 0'15970413E-0 1' 0 1'0J3VE-1' V.'0 15'9 04' I- V s I " 15I 9704+ E- i O1570413E-01 15970413E-01 0 7 015970433E-01 O,,l2Yo4E'-OlU' * 1-"' i -vAAo+-.9 (UA1.-L a1 Ai.'+i3E-0i Wll'...''i3E-0iO. 15970413E-0 1 O 15970413E-01 0 159-70413E-01 0. 15970413E-01 O970 4i9 E0i2.... 0#149(0+iL oi.........0 9f i o39E-05 150

Table A-23 cont. Cuo = 1. - 0.2 0,23if1i09E-0i u0i~f~u'~~~ 823ii09E-0i 0,-23117'1095-0iIV, i.E-o i 0*23t17109E-01 0 a23117109E-01 023117109-01 023117109E-01 2 3 7 1 0 9 E- 1 0,23117109E-01 0 23117109E-01 0&23117109E-01 0,23117109E-01'0823-ii7i09L-0 1,23ii7i9E-0U1..0231i7109E-0i o = 1.0 1 = 0.3 0 063i737401E-01 063173740tLE-01- O 3i73173740iE-0i - 013i737+01Em0i 0O31737401E-01 0 31737401E-01 0*31737401E-01 0301737401E-01. 63i731740iE-0i 0N3i 7374 iE'0i0 03i73740iE-0 0. 3i737401E-01 0,31737401E-01 0, 31737401E-01 0,31737401E-01 0 31737401E-01 0,3i-.~74'i E-i:L. -0,i -7'01-0. _ 3' 0_31731- I0f1' ITEM = 1.0 0 =0.4 o'- 0 84t 80'7b684-Li....... 0, 4i876 U1E0i; UO~+~i 8078+E=0i' 0,I4i80'64E- iv O0 41807684E-0 1 0 41807684E-01 0,41807684E-1 04E-01 0 41807684E-01 U04.i t0764E-0I Us 4i8013~.E+l4 i- U, 4'+i8756U- t84. E- i 04i-8 vI68-E-0 0 641807684E-01 0.41807684E-01 0*41807684E-01 0 41807684E-0 1 Oa4z18 6u4 ue4u 764E' 0 u1 uO1 i068 - 4 O41U 0 476 E0i =, = 1.0 =.5 0o 0 5 3 2601T_6E-01: 5 326 o06 tE-tr 0 5 5:326 010 6 E-AV1 i A5 32 i 601V 0 E-V 1 0,53260106E-0 1 0 53260106E-01 0 53260106E-01 0 53260106E-0 1 0 532601 06E-0 1 00. 53260 106E-0 O 5 360 106E-1 0 v 5. 32601 0 E-O - 0. 5 3260106E-0 1 0 53260106E-01 0.53260106E-01 0 53260106E-0 1......0 i532460uiot-0..06326iE"0 "5326006E-0.. j = 1.0 F = 0.6 0,660i4094E-0 1 0 660 t094E-01 0 6601494E-01 O6-60 1409.4E- 1 0 6:6014094E-01 066014094E6-01 0,66014094E-01 0. 660 14094E-01 66 iO-U94-E- u i'0 06" 66 i 4: t - - 0, i u, 66uJ.,+u,+r'-010....0 - o 1 u.,-,. 7- r -.94E1 0 66014094E-0 1 0O 660 14094E-01 0,66014094E-01 0, 660 140 94E-O 1 D *bb_.....,,_ V....,........... u*'0,66 i409'4E-0 Ji 0 i4-094- - i u.66 i 409-'E-0 - CL = 1. 0,u-~ = 0.7 Po = 1.0 O 79986770E-01 0,799866-770E-01 0.79986770E01 7O79986770E-0 1 0,79986770E-01' 0'79986770E -01 07998677OL-U01 0,79986770E-0! 0,79986770E0 1 0, 799867:70E"01 0 79986770E-01 0 679986770E-01 0'*s 9986 (Ot.-Oi U.'"b" (I O. U I U& 7(9bb(: utr-"- i U' U e i8 C u_ E-0 it O0 79986770E-01 0 79986770E-01 0 7998-6770E-01 iz~ = 1.0, = o.8 0e95097291E-0 1 0 95097291E-h01 0 95097291E-01 0 95097291E-0Ol 0.95097291E-01 0o 95097291E-01 0*95097291E-01 0.95097291E-01l 0O09 i- 5. 0 *'9.59 ( 2c2 i E-i 065'29i -0ui'"10 io 0.95097291E-01 0,95097291E-01 0695097291E-01 151

Table A-23 cant. ~o = 1.0 A = 0.9 0,11126883E00 11126883E-00 0 11126883E t116883E-00 0 111 26883E-00 0,11126883E-m0O OlllZ'6b83t-uu0 U011126683E-u C 0,.i12i6883E-00.11126883E-00 0 1.1126883E-00 0, 11 126883E-00 0, 111268 83 E-00 0 1112I 2.6.863E-0'o 0U,*1i268E bUI-ibtt-UU83E ii-u 1i126883E-00 0 11126883E-00 O. 11126883E-00 0 #11126883E-00 W. = 1.0 10 0,12.842935E-00 0. 12842935E-00 0 12842935E-0 0 a 128:42935E-00 U' 0i28.42935E-00 0, i28.429'35E-00 -01i.842935E-N00 Oei. 2:-429'SCE. -00 0 12842935E-00 O 0 12842935E-00 OC 1:2,842935 E-00 0, 12l842935 E-00 o 6 z 14Y t -U vU,. 1CO<Z 7 33r' D - 5 Y 5 — U V 14 CV 7 r~-u * 1:~ O,12842935E-00 0 12842935E-00 06128;42935E-A00 152

Table A-24 N=3 = 0. 900goo 000 = 1.872 320 ~ 0 (D0 = L.~U( ~)u 0.244 770 2 - 1.187 530 3 = 1.244 770 1 M1 0.936 160 m2 = 0.197 922 )3 0.103 731 2 =.49 480 a -2 0.010 373 3= o.001 729 3 ioe~~ (%)q = 0(10)180 i0 - 0.2 p - 0.0 0O45368283E-00 0 o 43807219E-00 0*39404-523E-00 0 32936433E-00 0 e 25495642 E-0O0 O 18243177E-00 0 12153520E-00 0.'78'158334E-01 0O53386951E-01 0 43782199 9E-01 0. 4277221 E-01 0 42:75 0446E-01 O e37291420E0 1 0, 22907946E-01 -O.11849933E-03 -0 280 132:96E-01 -0 e54776364E-01 -Oo74054220E-01 -705220-01 - 81069600E-01.1 = 0.2 i,= 0.1 O o 3-2-310843E-0O0 0. 6 312-57340E-00 O O 28'28509"9E-00 O 2 3914906E-OO 0.18879208E-00 0 *13954970E-00 0.97922510E-01 0O67809697E-01 0 49878484E84 1 Oe41768110E-01 0 68 OE-39043193E-01 0 e 36618789 E-0 1 0.30263790E-01 0o17779858E-01 -O 44890134E-03 -0o 21662610E-0 1 -0 416067'35E-01 -O 558'22347E-01 -0 *60971916E-01 Po = 0.2 p = 0.2 O 23873679E-00 O 23121554E-OO 0 e 20 999907E-00 O-..,i78812 14 E00 0 14288681E-00 0 *10775996E-00 0 78042248E-01 0 56465007E-0 1 0O 43435660E-0 1 0 6371963'79 -01 0 a 3452281-8E-01 0 31760-560E-0 1 O 25913411E-01 0 e 15491503E-01 0 * 89883006E-03 -O 15'73O691 E-O 1 -0 31187:531E-0 1 -O 42136922E-01 -0.46092447E-01 o0.2 = =o.3 0 o18036910E-00......... ^.0e I7'8764OE'-00- O 15938654E-00 -. O. 1 3663007E-00 0. 11043726E-00 0 84850993E-01 0,63220138E-01 0o47500614E-O 1 O37 934 230E-01 0.33174859E-01 0., 308138-1E-01 0 028153!36E-01 O 23013768E-0 1 0 1436 393E-0 1 O, 25865298E-02 -O 10627798E-0 1 -0 22804112E-01 -O 31388270E-01 -0) 34482644E- 01 153

Table A-24 cont. -o = 0.2 o = 0.4 0 13680454E'-00 0,13280411 -00 0!12152632E-00 0 10496897E-00 0.85930459E-01 0.67356368E-01 0 51672976E-01 0 40275554E-01 0,33297054E-01 0.29707914E-01 0O27710484E-01 0*25315133E-01 O020939502E-01 0) 1386961 1E-01 0,44618160E-02- -O 59631953E-02 -015 498754E-01 _ -022193404E-01 -019404E-1-024602127E-01 go = 0.2 = 0.5 10 281354E-00 0 99968 197E-01 091949449E-01 0 80184785E-0. 0666670717E-01 0* 53502717E-01 0O42'396601E-01 0O34322713E-01 -_. 29343379E-01. 0 26689810E-01 0 t 25051431E-01 0. 022995219E-01 19399814EE-01 0i 13787369E-01 0 64656098E-02 -0 15547860E-02 -0o88409694E-02 -0 393 6 597E -01 -0u 15 766757E-01 4o = 0.2 [ = 0.6 0,75706389E-01 0,73777594E-01 0,68343438E-01 0*60375227E-01 0*51229326E-01 0 42'325024E-01 0 34817208E-01 0 29'347347E-0 1 0 *25934882E-01 0 24033595E-01 0Q22733863E-01 0. 21053508E-0 1 0 18237396E-01 O 13984884E-01 0, 85448021E-02 0 26542044E-02 -0 26602590E-02 - 323048E-02 -.6 3623048E, 02 -O 76895443E-02 4u = 0.2 = 0.7 0 5 4024254E-01 0,528 24665 E-O 1 0 49445417E-01 O 44491415E-01 0*38805992E-01 0.33268809E'-01 O, 28590962E-01. 0 25159883E-01 O, 22973098 E-0 1 0' 21676056E-01 0(20691703E-01 0 t19405265E-0 1 0O17353601E-O1 0a 14367855E-01 0.10631367E-01 0,66380458E-02 0 306345'78E-02 0 5 8455655E-03 -0 30 234490E-03 go = 0.2 4 = 0.8 O*36998073E01 0, 36364902E-01 0 34580670E-01 0 3'1962813E-01 0,28953140E-01 0,26011026E-01 0123505366E-01 0 a2163'3'277E-0 1 0 20386200E-01 0 19571368E-01 0 18881863E-01 0 a 17995656E,-01 0 16676452 E-01 0. 14849006E-01 O 1 2628630E-O01 a0 10297170E-0 1 0,82321968E-02 0,680'88347E-02 0.63009930E-02 4 =0.2 4 =.g9 0, 24345 0 66E-O 1 0 * 24 2001 9E'O1 0,23484389E-01 0'22546903 E-0 1 0, 2 1 458 7-2 0 E-0 1 0. 20 376679 E -01 0 1 9 4 2 6 2 5 6 E-01 ) 18673865E-01 0 1811 5676E-01 0. 1768-5591E-01 0 17279979E-01 0. 16792259E-0 1 0,16147835E-01 0.15329849E-01 0*14388747E-01 0*13433002E-01 O0 12603481E-01 0O12038320E-01 0O11837747E-01 40 =0.2 = 1.0 0*159929'33E-01 15992933E-01 O- 15992933E-01 0 15992933E-Q. 0,15-992933E-01 0 15992933E-01' 0 15992933E-01 0*15992933E-01 O 15992933E-0 1 0 15992933E-0 1 0.15992933E-0 1 0 15992933E-0 1 - O i 159929-33E-01 015992933E-01 0 15992933E-01 0 15992933E-0 1 0 15992933E-01 0 15992933E-01 0 15992933E-01 154

Table A-24 cont. ~o = 0.4 = 0.0 n,4n 8787E-00._, 1 84943-oL-0.0 35448632E-00 0 O2995o 1 En 0 1 -0n 0423610000E- 0 0O1740237OE00 0 0o12151565E-0o 0 83607951 EO 1 0.61 3261.05E-01.0 51910774E-01 0 49965322E-.01 0 O.491921.42E-01 0 * 44237626E-O 1 0 *32152642E-01 0 1309367'3E-01 -0 O 989.4'1697E-02 -031916397E-01 -Q 147770847E-01 -O 535394-14E-01 %O =o.4 = Q.1 0O 33943268E-00.0 32901987Ei00 0 299-64009E-.00 0 0 256433 14E- OO 0 20 662 238 E-00 0 1t57'8 5 945E-00 0 1 1:65 28-60 E-00 0 86432208E-O 1 0 681'74420E-01 0 59373888E-01 0 55619'437E-O1 Ol 51891264E-01 0 4405'5539E-O 1 0 30031428E-01 Oe 10 327719E-01 -O' 12176298E-O I -0 33122575E-01 -047972741E-O 1 -0 53339474E-01 =oL =~4 =0.2 o 0O 27'360909E-00 0 26560 824E-00 04 24305264E-00 0 209g'3794-00O 0 17186092E-00 0 1347'I'273E-00 0 10334595E-OO 0 8055t109E-01 0 66594110E-O 1 0 5-94 15 829E-0 1 0e 55420969E-01 O 5.0630266E-0 1 0 e 41879004E-01 4 I277397223E-01 0 s 892'36325 E-02 -0 * 119263 89E-0 1 -O 30997507E-01 -! 0 44386807E-01 -O z49204 253 E-O 1 %O = 0.4 =.3 O.21 80 8926E-00 0 21 204639-00 O1 9:50 3 283 33E-0E-0 0 170096 5 8 E-O-O O'141i:5232'3 E-00 0O 113772'17E-00 0 9046471'3E-01 0- 735940 20E-0 1 O 63184940E-01 0e5747-6711E-01 0 53567414E-01 0 483'38'157E-01 O 3'9403414E-01 0 25830841EO1 0 8:4394260E-02 -O 10399040E-01 -O' 2739484 0 E-O 1 -0 e. 392'33 08 3.E-O 1 -0 43 4770 22 E-O 1 - 04 = 0.4 0O 1471'86710E —0 0 0 1674'2779E-00 015494'O43 -00 015E-O 136.69034E-00 0o11:585248E-00 095715'985E-O1 0'78900793E-01 0 66770153E-01 0 59189104E-01 0 5468:6584E-01 0.50971076E-01 0 45656 42E-01 0e36997394E-01 0 24441104E-01 0 88368289E02 - O 77421253E02 -O 4 22 5 20 3 3'3 E-0 1 -0 3274 15 75E-1 -0 3:6 39 3872 E-0 1 o0 = o.4 = 0.5 0 e 13367216E-00 0. 130 54816E-00 121'7'192E00 1 O10897905 E-00 0 94429918E-01 0 80441'923E01 0 688191389E-01 0 # 604-32258E-01 5505:4048E-0.1 0 51500057E-01 0O48026115E-01 0 -428783 95 E-01 0 34844399 E-0 1 0 e 23655 049E-01 0 10128033E-0 1 - 0 3987911 8E-02 -0. *16,42766 0E-0 1 - -0,24973401 0Er1 - 0. 2801 t73:58E-01 ~1 =.o4 CL =.6 O. 10253562E-00 0. 10047877E-00 0. 94708197E-01 0 8'63188 17E-O 1 0 768'13821E-0 1 0. 67713802E-01 0.O6016735!1E-01 0. 54668'479E-0 1 0.33066903E-01 0e23542024E-01 0e12336604E-01 0.85317783E-03 -0. 91490338E-O 2 -O 15972309E-01 -O 18394759E-0 1 155

Table A-24 cont. ~o = 0.4 = 0.7 0*77779005E-01 0,76563413E-01 0*73156984E-01 0O68215098E-01 0,62629305E-01 0 57284587E-01 0o52820006E-01 0 49459380E-0 1 0 46960039E-01 0 44695607E-01 0o41852126E-0i1 0 37685484E-01 0o31770407E-01 0 24172325E-01 0o 5493007E-01 0 o67737126E-02 -0.72262599E-03 — 0 57963568E-02 -0. 75909968E-02 ~o = o.4 0 = o.8 0ij 58.9 9171E-01 0 58347651E-01 0. 56689941 E-,01 0 54282700E-01 0O51550303E-01 0 48900823E-01 0 46605656E-01 0 o44717908E-01 0a430543613_E1-O 041248492E-01 0,38862234E-01 0&35527535E-01 0*31079429E-0 1 025643519E-01 0o19651728E-01 0 13778587E-0 1 0., 88109875E-02 0. 0,54821230 E-02 0.43102159E-02 C1o =0o.4 1.9 0 45643652E-O 1 0 45454584E-01 0 44920149E-01 0 441 27876E- 1 0 43190937E-0 1 0 42209583E-0 0* 41235051E-01 Os40247573E-01 0O 39156402E-01 0. 37823664E-0;1 036107031E-001 0,33910473E-01 0 31229447E-01 0. 28177468E-01 0. 24985 342E-01 0 219 70958E-01 19485136E-01 O 17845315E-01 0 o 17272 349E- 0 1 o = 0.4 1.0 O0 34.438033 E-0 1 0 344'38033E-01 0 34.43803-3 E- 01 0 3-4438033 E-O 1 0 34438033E-0 1 0 344 38 033E-01 0 34'4'38033 E-O 1 0 a 3`4438033 E-0 1 0. 3 4438033E-O 01 0 3443'8033E-01 0,34438033E-01 0O3 4438033E-0 1 0 3 4438033E-0 1 0 o 34438 033 E-01 0 * 34438 033 E-0 1 O 3 44 3 80 33 E-0 1 O 34438033E-01 0 34438033E-01 0 34438033E-01 [ =0.6.=oo 0 e 30607177E-00 0 s 29684712E-00 0 o 27078564E-00 0 s 23235591E-00..0. 18786816E-00 0 14406784E-00 0.106 8571E00 000 79279782 8 -0 1 O 62763333E-01 0 55354332E-01 0 o 53512567E-01 O 53008288E-0 01 0 50180127E-01 0 42924944E-0 1 0o31185053E-01 0 * 16:82425.4E-0 1 0 29542454E-02 -070770133E-02 -0 o 10734439E-0 1 0 = o.6 = 0.1 ~L = 0.6 0.27082954E-00 0 26334578E'-00 0*24222537E-00 0*21114828E-00 0 17528531E-00 0 14011216E-00 0 11019488E-00 O, 88249414E-01 0o74701933E-01 0 67844087F-01 0O64520568E-01 0o6'149772E-0. 0 54795889E-01 0 44004536E-01 0 29184753E-01 0. 1 2447707E-0 1 -0 30377941E-02 -0 o 13981517E-01 -0 17930949E-01 =. = 0.6 0.2 0O22711916E-00 0 22133278E-300 020503031E-00 01*8112569E- 00 0. 15368798E-00O O 12697507E-00 0 10445 162E-0C)0 0 88042044E-0Q 1 0,77804647E-01 0 72100785E-01 0,68201588E-01 0o63160524E'-0 0o54712188E-01 0.41954652E-01 0. 25634405E-01 0 o79626136E-02 -0 79807776E-02 -0 19086915E-01 -0O 23068633E-01 156

Table A-24 cont. [o = 0.6, = 0.3 0 -1T8757769-E-0O 0O 18325926E-0O 0617111684E-00 0 15338463E-00 O 133159t6E-00 0. 1 1363298E-00 0 o97323132E-01 0 O 85496378E-0l 0177940298E-01 0 73136123E-01 068781964E-01 -0O62521664E-01 0o52680162E-01 0 38805579E-01 0.21870958E-01 0#40807542E-02 -0.11666198E-01 -0 22512843E-01 -0.26381252E-01 0 p = o.6, = 0.4 0,15380343E-00 0 15071816E-00 0&14206229E-00 0 12947822E-00 0.11522073E-00 010 157070.'-00 0 o 90 251026E-.-01 082002719E-01 0 76-436295E-01 0.72199038E-01 0O67413381E-01 0 60267907E-0 1 0 49600354E-0 1 0 a 353130'34E-01 0 18504904E-01 0 12797653E-02 -0 4 13723552E-0 1 -0. 23958466E-0 1 -O 27592141E-01 po = 0.6 =.5 0O12596361E-00 0612388590E-00 0.11807025E-00 0. 10965338E-00 06 0017626E-00 0 91157445E-01 0,83673538E-01 0 78065073E-01 0 73'852248E-01 0 69888201E-01 0 64712245E-01 0O57009879E- 01 0,46055125E-01 0. 320 13024E-01 0. 16015121E-01 -0 a 20186446E-04 -0. 13783355E-01 -0 23088548E-01 -0 26378159E-01 = 0.6 = 0.6 o 0 10378624E-00 0 10250010E-00 0O98907051E-01 0 93724035E-01 O0 87905192E-01 O 82353685E-01 0, 77643590E-01 0, 73834186E-01 0*70434580E-01 0 66535474E-01 0o61078306E-01 0 53192439E-0 1 0.42509689E-01 0 29:368185E-01 0*14844157E-01 0 59390450E-03 -0, 11462462E-0 1 -0 19541781E-01 -0 o 22385974E-01 o = 0o.6 ~ = 0.7 0,86788718E-01 0 s86085626E-01 0 8412076'7E-01 0. 81279375E-01 0 78057950E-01 0o74892329E-01 0o71996319E-01 0 69262417E-0 1 0 66259889E-01 0 62338597E01 0*56816768E-01 0 49205680E-01 0 39411057E-01 0 27853790E-01 0 15471049.E-01 0 35883670E-02 -0 *63137469E-02 -0 o 12887166E-01 -0 15190794E-01 Cro = 0.6 =.8 0 74244262E-01 0 73921351E-01 0 73009303E-01 0 71654907E-01 0&70032861E-01 0,68265089E-01 0 a66349035E- 01 064121131E-01 0o61272639E-01 0o 57420329E-01 0O52218149E-01 0 s 45482864E-01 0 s 37300047E-0 1 0 28079212E-01 018537854E1 18537854E-01 96110249E-02 0o23017159E-02 -0 24971256E-02 -0 41699051E-02 ~,o = 0.6 = 0.9 0.64725620E-01 0,64'587937E-01 0.64180980E-01 0.63516905E-01 0,62598039E-01 0 61397722E-O0 0.59846444E-01 0,57831194E-01 0.55212515E-01 0,51858637E-01 0o47690532E-01 0,42727374E-01l 0.37120175E-01 0131162722E-0 ] 0.25273447E-01 0 19948406E-01 0,15692382E-01 0,12940945E-01 0o11989035E- 01 157

Table A-24 cont. = 0.6 4 = 1.0 0 0e45703425E-01 0 a45703425E-01. 0*45703L425E-01 0 45703425E-0i1 O 45703425E-01 0 45703425E-01 0,45703425E-01 0 457(03425E-0 1 0,45703425E-01 045703425-01 45703425l,)04570342501 4573425E-01 0O45703425E-01 0 45703425E-01 0,45703425E-01 0 457034255-01 0,45703425E-01 0 45703425E5-0 045703425E-01 4I = 0.8 = 0o.o n, 17853039E-o00 0. 1720968-00 0 161 9 742 1E-00 014384256E-00 0, 12268070E-00 0. 10158377E-00 0.83234154E-01 0 69389296E-0 1 o 0,60616057E-01 0 563233'72E-01 0*55068154E-01 0 * 55040947E-01 0 545:95789E-01 0 52683334E-01 0 49078465E-01 0,44353777E-01 0,39621814E-01 0 36132380E-01 0o34849136E-01 %Lo = 0.8 4 = 0.1 0. 16635718E-00 0 1 62993931E-00 0_ C 1 5328 104E-00 0 * 13902988E-00 0. 2250505E-00 0 s 10617-132E-00 0 92099084E-01 0 * 81544578E-0 1 0. 74750593E-01 0 *71009645F-01 0 68962775E-01 0 i 67046694E-01 0,63965570E-0'1 0 59060790E-01 0 52482858E-01 0.45125496E-0 1 O0.38347714E4-01 O. 3'3567'3'395-01. O 31 843542E-01: = 0o.8 I = 0.2 0 0, 14799229E-00 0a 14545961E-00 0 13832268E-00 0 1 2785125 E-0 O.1158125'5E-00 0&10404-410E-00 0,94021463E-01 0O 86533111E-01 0 81544804E01 0 72854711 0 75274528571-01 0. 7527452E 7'1982624E-01 0 66'705810E-01 0 59396023 5514521E-01 505145215-01 0 41188683E-01 0 329 2 8 787:E-0 1 0 27235339E-01 0 3 25 203972E-0 1, = o.8 0 = 0.3 0e131457765-00 0 12967538E-00 0e12466772E-00 0 11736342E-00 0 #10903531E-00 0 10096673E-00 0*94120135E-01 0 88898184E-01 0a850629-53E-01 O 81836086E-01 0 o78153492E-01 0 73006050E-01 O 65773674E-0 1 0 * 56460930E-01 0 45767926E-01 O 34973657E-0 1 0 25658938E-01 0, 19339065E-01 0 17100958E-01 4 = o.8 0 =o.4 011787834E-00 0 11669530F-00' ) 13379885-00 0,10856540E-00 OslO310061E-00 0 97801644E-01 0 93211312E-01.0 89435816E-0 1 0,86108726E-01 0,82496983E-01 0*77724468E-01 0 71055068E-01 0O62158859E-01 0,51287038E-01 0,39303456E5-01 027557174E-01 0, 7621975E-01 0&10964245E-01 0 86204313E-02 =o.8 4 = 0.5 0.10723340O E-00 0O 10649882E-00 (0 10444044E-00 0 10144631E-00 0 98017277E-01 0 94594840E-01 0 91399439E-0)1 0 88330959E-01 0 84966735E-01 0 806654-48e-0o1 0, 74755043E-01 0' 66757546E-0 1 0,56590480E-01 0,446873755-01 0.31998392E-01 0 19861654E-01 0 97689296E-0 2 0 30 772 867E-02 0 73358803E-03 158

Table A-24 cont. -o = 0.8 =.6 0,98992349E-01 0 o98561803E-01 0 97345740E-01 0 95539878E-01 0.93377148E-01 0 91020121E-01 0 88465382E-01 0 885494842E-01 0O81696854E-01 0 76560440E-01 0 6962420OE-01 0e60643821E-01 O 49733397E-01 0' 37438951E-0 I 0Q 24717141E-0l1 0 128 14701 E-01 O.30689564E-02 -0 m 33294989E-02 -0 5 5598716E-02 O ~=.8, = 0.7 09'2227306E-01 0 91967326E-01 0 91212752E-01 -0O90022696E-01 O 88445363E-01 0o 86462528E-01 0O 83945458E-01 0 80643129E-0 1 0 76215243E-01 0 *7030.9822E-01 0 62671081E-01 0 6 53252304E-0 1 0.42303488E-01 0.30406614E-01 0 18441996E-01 0 74848596E-02 -0 13516832E-02 -0, 70971l 498E-02 -0 909044t80E-02,1 = 0.8 =.8 O, 85427412E-01 0 08522231 7-01 0 8'4602941E-01 O 835505()8F-01 0 82015219E-01 0 79898 307E-01 077043275E-01 0, 7'32-458 1 7E-0 1 0.68287288F-01 0.61989485F-01 0. 54281180E-01 0, 45262 79 F-01 0 35241439E —01 0 24753523E-01 0 a14513864E-01 06 53456913E-02 -O 1 9 2 9125 1 E-0 2 _ 0660 98756E-02 -0 8225.'4694E-02 - = 0.8, = 0.9 0 75535590E-01 0 75307196E-01 0 74610675E-01 0 73411712E-0 1 0 71652999E-01 0 69256680E-01 0 66131961E-01 0, 621 89835E-01 057365 140E-01 O, 51643690E-01 0 i 45 0 89 845 E-01 0 e 378 6 83.66.E-0 1 0O30254151E-01 0 22624947E-01 0*15435056E-01 0 *91718583E-02 0,43008061E-02 0&12074979E-02 0 14670896E-03 Lu-o. =.8 1.0 0O39296651E-01 0 e 39296651E-01 0O 39296651E-01 01 3929665 1E-0 O 0O 392-96651E-01 0. 39296651E-01 0, 39296651E-01 0 e 39296651E-0 1 0O39 296651E-01 966 392966551E-01 39 1 3 2966651E5 1 E0 1 0,39296651E-01 0 39296651E-01 0 39296 651E-01 0 3 92 96651E-01 O039296651E-01 0 39296651E-01 0 39296651E-01 ~o = 1*.0 0.0 0 0 56458156E-01 0 56458 156E-01 0156458156E-01 0 a 56458156-E-01 0 56;4581'56E-01 0. 56458 15-6Er O 0 56458 156E-01 0* 56458156E-0 1 0 56458156E-01 0.56458156E-01 056458156E-01 0 * 56458156 E-0 1 0*56458156E-01 0 56458 156E-01 0.5645 8"15'6E-01 0, 56458156.E-01 0O56458156E-01 0 56458156E-01 0 56458156E-01. -. = 1.0 ~ = 0.1 0.71049959E-0-1 0.71049959E-Oi1 0 i 71049959'E01 i 0 71049959E-01 0.71049959E-01 0.71049959E-01 0.71049959E-01 0,71049959E-01 0O71049959E-01 0.71049959E-01 0,71049959E-01 0.71049959E-01 0.71049959E-01 0,71049959E-01 0e71049959E-01 0.71049959E-01 0,71049959E-01 0.71049959E-0 1 0,71049959E-01 159

Table A-24 cont. 1o = 1.0 = 0.2 0 79964668 E-0 1 0 79964668E-01 0 e 79964668E-01 0 s 79'964668E-0 1 0,79964668E-O 1 0 79964668E-01 0 e 79964668E-03' 0 # 79964668E-0 1 0 7996.4668E-01 0 * 79964668E-0 1 0 79964668E-01 0 6 79964668E-0 I. 0 e79'964668E-O 1 0 79964668E-01 0 79964668E-01 0 e 79964668E-0.1 0 79964668E-001 0 79964668E-01 C e 79964668E-01 %o = 1.0 = 0. 0 e84973890E-01 0 84973890E-01 O O 84973890E-01 0, 849738 9 E-O 1 0 * 8497'3890E-0 1 0 84973890E-01 0 e 8:4973890E-0i 0. 8497'3890C) E-0 1 Oe84973890E-0 1 O 84973890E-01 0 84973890E-0 1, 0,84973890E-01 0 84973'890E-01 0 84973890E-01 0 84973'8905E-01 O 84973890E-0 1 0 o 84973 89 0 E-O 1 O e 849'73 890E-01 0,s8'497S89 0 E-0 1 po = 1.0 u = 0.4 0.86095085E-01 0 86O95085E-01 O, 86095085E-01 0 8 86095085E-01 0*86095085E-01 0*86095085EO 1 0 86095085 E-01 0 86095085E-01. O86095085E-01 0e86095085E-01'0 86095085E-01 0O8609'5085E-01 O,86095085E-01 0 8609'5085E-01 0 8609508'5E-01 0,8'6095085E-01 Oi86095085E-01 86095085E1 0 860950 1 086095085 E-01 1 = 1.0 0.5 0*83208209E-01 0.83208209E-01 0.83208209E-01 0.83"208209E-01 0, 83208209E-01 0.83208209E-01 0832081 83208209E-01 0'208209E-01 0.83208209E-.01 083208209-01 0 83208209E-01 0.83208209E-01.. 0 8'3208209E-01 0.83208209"-01 0.83208209E-01 0e83208209E-01 0.83208209E-01 0. i83208209E-01 -. 0. 83208209E-01 L1o = 1.0 o.6.,76'1723'76 E-0 1 0 76 172376.E-01 0,7617'2376E-01 0,76'1'723765E-001 0 76172376E-01 0 76172376E-01 0 76172376E-01 0 761723'76 E- 0 1 0 76172376E-01 0.7'6172'376E-01 0*76172376E-01 0 761723'76E-01 076172376E-01 0 7617237617276E-01 0 76172376E-01 0 e 761723'76E-0 1 0 7617276E-01 0*76172376E-01 0*76172376E-01 Po = 1.0 = 07 0.64851527E-01 0C 64851527E-01 0O64851527E-01 0 r64851527E-01 0 e64851527E-01 0 648515 27E-7E1 6485152E01 0 e 64851'527E-01 0O64851527E-01 0 e648515277E-O 1 00648 51527E-01 0*64851527E-01 0 64851527E-01 0 64851527E-01 0 64851527E-0 1 0 64851527E-01 0 64851527E-01 0 6485'1527E-01 1o = 1.0, = 0.8 0O49120814E-O1 0 e49120814E-01 0e49120814E-01 0 e49120814E-01 0.49120814E-01 0 49120814E-01 0e49120814E-01 0 49120814E-0 1 0a49120814E-9120814E1 49120814F-01 0,4912081401 01 49120814E-01 0e49120814E-01 0 49120814E-01 0O49120814E-O1 0 49120814E-01l 0a49120814E-01 n 491208140-01 0.49120814E-01 160

Table A-24 cont. A- =1.0 ~ = o.g 0628867568E-01 0 288675681,-01 C028867568E-01 0.28867568E-01 0,28867568E-0 1 0 *28867568E-01 0 o 28867568E-'01 0 28867568E-01 O28867568E-01 O 28867568E-01 0,28867568E-01 0,28867568E-01 O 2886'7568E-01 0O 28867568E-01 0 28867568E-01 0 X"28867568E-01 0 &28867568E01 028867568E-0-1 2886756F-01 0 28 867568E-01 Cu - 1.0 IU,= 1.0 0 0*39905872E-02 0. 39905872E-02 0C 39905872E-02 0 39905872E-02 0 39905872E-02 0,39905872E-02 0 *39905872E-02 01 9905872E-02 0 39905872E-02 0 3'9905872E-02 0C39905872E-02 0 39905872E-02 0,39905872E-02 0 39905872E-02 O 39905872E-02 0 39905872E-02 0,39905872E-02 0 39905872E-02 0 39905872E-02 161

Table A-25 N=4 0.900 000 = 1.695 000 = 1.741 600 o 1 3 = 0.535 480 04 = 0.487 630 = 0.847 500 = 0.290 267 = 0.044 623 = 0.024 382 2 222 -0004 462 2 2 = 0.072 567 2D = -.4 462 = 0.001 355 = 0.000 744 = 0.000 097 4 = 0.000 012 io (po)= 0(10)180 0 0,0 = 0.2 ~ = 0.0 0.46150572E-00 0.44383705E-0 0.39485060E-00 0.32523838E -00 0.24881817E-00 0.17822653E-00 0.12152027E-00 0.80892910E-01 0.53757687E-01 O.35437176E-01 0.22080608E-01 0.12462653E-01 0.79569260E-02 0.10903481E -01 0.22360296E-01 0.40487910E -01 0.60459551E-01 0.76004286E -01 0.81866801E-01 ~o = 0.2 0.1 0.32574496E-00oo 0.31406613E-00 0.28164012E-00 0.23541155E-00 0.18435964E-00oo 0.13671062E-00 0.97747184E-01 0.6899490o7E-01 0.48940241E-01 0.34775090E-01 0.24264051E-01 0.16859812E-01 0.13618840E-01 0.16080258E-01 0.24774104E-01 0.38187283E -01 0. 52769685E-01 0.64036117E-01 0. 68270849E -01 t= =0.2 ~ = 0.2 0.23838691E-00 o 0.23022976E-00 0.20756523E -00 0.17519839E-00 0.13933720E-00 0.10566349E-00 0.77825839E-01 0.56894236E-01 0.41877146E-01 0.30940403E-01 0.22691897E-01 0. 16922486E-01 0.14498988E-01 0.16537328E-01 0.23336657ET-01 0.33663898E-01 0.44797247Eo-01o 0.53358082E-01 0.56568828E-01 -o = 0.2 4 = 0.3 0.17829240E-00 0.17248568E-00 0.15633865E-00 0.13323558E-00 0.10754574E -00 0.83264869E -01 0.62960116E -01 0.47398692E -01 0.35926288E-01 0.27333590E-01 0.20755237E-01 0.16168904E-01 0.37842988E-01 0.44417685E-01 0.46879296E-01 162

Table A-25 cont. o = 0.2, =.4 0.13390994E -00 0.12980322E-00 0.11836997E-00 0.10196733E -00 0.83637784E-01 0.66164509E-01 0.51342013E-01 0.39726130E-01 0.30906215E-01 0.24111994E-01 0.18833068E-01 0.15148639E-01 0.13624718E-01 0.14857362E-01 0.18917495E-01 0.25002369E-01 0.31498865E-01 0.36463540E-01 0.38320000E-01 i = 0.2 = 0.5 0 99823245E-01 0.96995367E-01 0.89107903E-01 0.77745587E-01 0.64955352E-01 0.52614252E-01 0.41943839E-01 0.33347665E-01 0.26598169E-01 0.21243899E-01 0.17018394E-01 0.14047440E-01 0.12761777E-01 0.13573531E-01 0.16494392E-01 0.20905096E-01 0.25614952E-01 0.29210907E-01 0.30554689E-01 z = 0.2, = 0.6 0.73178476E-01 0.71323112E-01 0.66132747E-01 0.58607031E-01 0.50040338E -01 0.41627141E-o01 0.34159996E-01 0.27930101E-01 0.22845340E-01 0.18683176E-01 0.15340162E2-01 0.12951025E-01 0.11821355E-01 0.12214158E-01 0.14105913E-01 0.17046252E-01 0.20208417E-01 0.22627380E-01 0.23531739E-01, = 0.2, = 0.7 0 0.52302193E-01 0.51176536E-01 0.48011915E-01 0.43374615E-01 0.38002256E-01 0.32585077E-01 0.27598714E-01 0.23248980E -01 0.19536137E-01 0.16391882E-01 0.13811955E-01 0.11914373E-01 0.10893862E-01 0.10897210E-01 0.11886133E-01 0.13563593E-01 0.15411786E-01 0.16838006E-01 0.17372838E-o01'-~ = 0.2 o = 0.8 O.36075779E-o1 O.35476067E-01 0. 33775503E-01 0.31238485E-01 0.28214351E-01 0.25040305E-01 0.21966632E-01 0.19130666E-01 0.16582476E-01 0.14341458E-01 0.12449925E-01 0.10993446E-01 0.10076009E -01 0.97622076E-02 0.10017014E-01 0. 1677145E-01 0.11474905E -01 0.12111484E -01 0.12353116E -01 C. = 0.2 = 0.9 0 0.23516694E-01 0.23268607E-01 0.22554006E-01 0.21453815E-01 0.20079336E-01 0.18546783E-01 0.16956827E-01 0.15385699E-01 0.13888659E-01 0.12510790E-01 0.11296962E-01 0.10293902E-01 O.95418993E-02 0.90597238E-02 0.88306608E-02 o.87981601E.-02 0.88759061E-02 0.89708157E-02 0.90113845E-02 [o = 0.2 1.0 0.10884311E-01 0.10884311E-01 0.10884311E-01 0.10884311E-01 0 10884311E-01 0. 10884311E-01 0.10884311E-01 0.10884311E-01 0.10884311E -01 0.10884311E -01 0.10884311E -01 0.10884311E -01 0.10884311E-01 0.10884311E-01 0.10884311E -01 0.10884311E -01 0.10884311E-01 0.10884311E-01 0.10884311E-01 163

Table A-25 cont. = 0.4 ~ = 0.0 0 O.40393784E-OO 0.38954035E-00 0.349531t32E-00 0.29239372E-00 0.22913735E-00 0.16992181E-00 0.12137248E-00 0.85531487E-01 0.60648981E —01 0.43237060E-01 0.30326054E -01 0.20881742E -01 0.15829509E-01 0.16842000E-01 0.24641194E-01 0.37769500E -01 0.52515664E-01 0.64083164E-01 0.68458736E-01 O o = 0.41 O.33518295E -00 0.32418277E -00.29357738E -00 0.24974269E -00 0.20093669E -00 0.15475739E-00 0.11614801E -00 0.86659601E-01 O.6510o6937E-01 0. 49151167E -01 0.36979960E-01 0.28288160E -01 0.24157844E-01 0.25999340E-01 0.34175166E -01 0.47072534E-01 0.61156635E-01 0.72045518E -0 0.76138115E-01 Lo = 0.4 = 0 0.26781988E-00 0.25960644E-00 0.23673995E-00 0.20393468E-00 0.16727556E-00 0.13232902E -00 0.10268402E-00 0. 79452260E-01 0.61812431E -01 0.48223989E -p 0. 37666135E -01 0.30297277E-01 0.27249435E-01 0.29714724E -01 0.37834990E -01 0.50004736E -01 0.62997729E-01 0.72927079E-01 0. 76640000E-01 =~o = 0.4 = 0.3 0.21195832E-00 0.20593513E-00 0.18915081E-00 0.16501555E-00 0.13791585E-00 0.11184227E-00 0.89348530E-01 0.71226932E-01 0.56948781E-01 0.45574459E-01 0.36642811E-01 0.30590598E-01 0.28505640E-01 0.31358745E -01 0.39126246E-01 0.50280146E-01 0.61955796E-01 0.70786631E-01 0.74073659E-01 o=' = 0.44 0.16628754E-00 0.16198696E-00 0.14998358E-00 0.13265751E-00 0.11306141E-00 0.93961311E-01 0.77123931E -01 0.63119326E-01 0.51662473E-01 0.42268793E-01 0.34863135E-01 0.30021120E-01 0.28715258E-01 0.31693102E-01 0.38808380E-01 0.48665842E-01 0.58805312E-01 0.66402469E-01 0.69218375E-01 = = 4 0.5 0.12920019E -00 0.12624246E-00 0.11796410E -00 0.10593962E-00 0.92185234E-01 0.78527127E -01 0.66142945E-01 0.55453480E-01 O.46371425E-01 0.38744569E-01 0.32742576E-01 0.28970937E-01 0.28236717E-01 0.31071073E-01 0.37246632E-01 o.45548802E-01 0.53956010E -01 0.60200901E-01 0.62506460E -01 = 0.4 = o.6 o 0.99316329E-01 0.97382919E-01 0.91946156E-01 0.83968212E-01 0.74683022E-01 0.65216368E-01 0.56317759E-0 1 0.48309917E-01 0.41253320E-01 0.35217211E-01 0.30496593E-01 0.27642247E-01 0.27265962E-01 0.29701065E-01 0.34682167E-01 0.41216158E-01 O.47741902E-01 0.52550644E-01 0.54319395E-01 164

Table A-25 cont. L= -0.4 4 = 0.7 0 0.75427777E -01 0.74247693E-ol0 0.70903724E -01 0.65917006E -01 O.59961777E-01 0.53668310E-01 0.47488426E-01 0.41677625E-01 0.36387585E-01 0.31805383E-01 0.28249945E-01 0.26155782E-01 0.25930735E-01 0.27742153E-Olq 0.31331638E-01 0.35958978E-ol01 0.40528065E-01 0.43871173E-01 0.45096640E-01 = 0.4 =.8 0.56350411E-01 0.55696048E-01 0.53819547E-01 0.50953317E-01 0.47406625E-01 o.43487590E-01 0.39451794E-01 0.35497908E-01 0.31804737E-01 0.28580370E-01 0.26085043E-01 0.24600486E-01 0.24345457E-01 0.25367691E —01 0.27461430E-01 0.30157026E-01 0.32804044E-01 0.34731703E-01 0.35436515E-01 o = 0.4 - 0.9 0 0.40630243E-01 0.40324689E-01 0.39434654E-01 0.38034175E-01 0.36230002E-01 0.34144906E-01 0.31906211E-01 0.29643014E-01 0.27489693E-01 0.25588419E-01 0.24082364E-01 0.23095179E-01 0.22699636E-01 0.22885957E-01 0.23543587E-01 0.24469060E-01 0.25403008E -01 0.26089564E -01 0.26341330E-01 = 0.4 = 1.0 0.2286185E -01 0.2286153E -01 0.22861853E -01 0.228618E -01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861855E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-O1 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 0.22861853E-01 ~o = 0.6 = 0.0 0.29794430E -00 0.28854416E-00 0.26231311E-00 0.22451674E-00 0.18204410E -00 0.14135842E-00 0.10684469E-00 0.80121429E-01 0.60451309E-01 0.45922755E-Olq 0.34784498E-01 0.26318053E-01 0.20911026E-01 0o.19406253E-01 0.22150379E-01 0.28290127E-01 0.35715850E-01 0.41710927E-01 0.44003505E-01 Lo =0 o.6 = 0.1 0.26240110E-00 0.25500984E-00 0.23435629E-00 0.20449648E-o00 0.17071313E-00 0.13792779E-00 0.10944937E-00 0.86500473E-01 0.68605757E-01 0.54553289E-01 0.43415356E-01 0.35136607E-01 0. 30044281E-ol 0.30185079E-01 0.34478097E -01 O.42160070E-01 0.50851045E-01 0.57659906E-01 0.60231292E -01 =0 0 o.6 O = 0.2 0.21953503E -00 0.21396933E-00 0.19839824E-00 0.17582109E-00 0.i5012101E-00 0.12488142E-00 0.10247999E-00 0.83790304E-l01 0.68536019E-01 0. 56049528E-01 0. 46020486E-01 0.38853078E-01 O.35464064E-01 0.36642473E-01l 0.42317741E -01 0.51138756E -01 0.60625252E -01 0.67882140E-01 0.70595215E -01 165

Table 25 cont. on = 0.6 = 0.3 0.18136378E -00 0 0.17728753E-00 0.16586252E -00 0.14922111E -00 0.13010902E-00 0.11103969E-00 0.93669849E-01 0.78632323E-01 0.65841256E-01 0.55067152E-01 0.46449851E-01 O. 40679947E-01 0.38729269E-01 0.41264302E-01 0.48038121E-01 0.57590402E -01 0.67456155E-01 0.74853525E-01 0.77595314E-01 ~o = 0.6 = 0.4 0.14897449E-00 0.14607438E -00 0.13791923E-00 0.12595232E-00 0.11202453E-00 0.97824553E-01 0.84476636E-o1 0.72464877E-01 0.61679982E-Olq O.52825816E-01 0.45744891E-01 0.41463371E-01 O.40898945E-01 0.44515600E-01 0.52023251E-01 0.61824238E-01 0.71612854E-01 0.78825966E -01 0.81479092E-01 0.6 = 0.5 0.12202354E-00 0.12002303E -00 0.11436789E-00 0.10597424E-00 O.96017124E-01 0.85576867E-01 0.75404294E-01 0.65902145E-01 0.57307715E-01 0.49955250E-01 0.44450301E-01 0.41626639E-01 0.42264618E-01 0.46677270E-01 0.54355877E-01 0.63865857E-01 0. 73091800E-01 0.79784749E-01 0.82229348E -01 =0.6 =.6 0.99728698E-01 0.98392282E-01 0.94584758E-01 0.88841249E -01 0.81855020E -01 0.74283995E-01 0.66633058E-01 0. 59264114E-01 0.52514810E-01 0.46842945E -01 O.42890489E-01 0.41393381E -01 0.42939006E-01 0.47657511E-01 0.54985240E-Olq 0.63630072E-01 0.71798214E-01 0.77637251E-01 0.79755712E-01 = 0.6 = 0.7 0.81129516E-01 0.80265951E-01 0.77780370E-01 0.73955078E-01 0.69168201E-01 0.63807677E -01 0.58225938E-01 0.52756424E-01 O.47774702E-01 o.43754819E-01 0.41263819E-01 0.40861117E-01 0.42916072E-01 o.47405694E-01 0.53780899E-01 0.60978620E-01 O.67609039E-01 0.72280169E -01 0.73963497E -01 o:; 0.6 = 0.8 0.65158364E-01 0.64626706E-01 0.63080497E-01 0.60655170E-01 O.57546707E-01 0. 53985840E-01 0. 50227398E-01 O. 46558890E-ol O.43316957E-01 0.40889087E-01 0.39678314E-01 0.40022712E-01 0.42084428E-Olq 0.45744599E-01 0.50549749E-01 0.55745640E-01.6-01 0075 70.64797779E -01 go =.6 =0.9 o.50637898E-01 0.50360165E-01 0.49548384E-01 0.48265271E-l01 O.46611000E-01 0.44719151E-01 0.42753620E-01 0.40905271E-01 0.39384376E-01 0.38403743E-01 0.38149479E-01 0o.38741366E-01 O.40191282E-01 O.42372970E-01 0.45017087E -01 0. 47740426E-01 O.50108552E-01 0.51719958E-01 0.52291178E-01

Table A-25 cont. o = 0.6 =1.0 0.36431806E -01 0.36431806E-01 0.36431806E-01 0.36431806E-01 0.36431806E-01 0.36431806E-01 0.36431806E-01 0.36431806E-01 0.36431806E-01 0.364318o6E-01 0.364318o6E-01 0.36431806E-01 0.364318o6E-ol 0.364318o6E- 0.3643189 006E-01 0.36431806E-01 0.36431806E-01 0.36431806E-01 = 0.8 0 = 0.0 0.17233661E-00 0.16828933E-00 0.15688222E-00 0.14009661E-00 0.12057687E-00 0.10090490E-00 0.82994100E-0o 0.67799231E-01 0.55399019E-01 o.45354380E-01 0.37146404E-01 0.30491420E-01 0.25413966E-01 0.22087649E-01 0.20566601E -01 0.20574922E-01 0.21480714E -01 0.22479289E -01 0.22899096E-01 =-o.8 =0.1 0.1617i145E-00 0.15852761E-00 0.14952919E-00 0.13620393E-00 0.12052733E-00 0.10441575E-00 0.89284824E-01 0. 75862535E-01 O.64288091E-01 0.54405295E-01 0.46083661E-01 0.39406527E-01 0.34645217E-01 5217E2168E-01.2052168-01 0.31598814E-01 0.32812226E-01 0.34815809E-01 0.36578331E-01 0.37271372E-01 o =0.8 =0.2 0.14430311E-00 0.14190426E-00 0.13510201E-00 0.12495394E-00 0.11285 740E -00 0.1oo0016122E -00 0.87866528E -01 0.76522663E-01 0.66329905E-01 0.57365832E-01 0.49799698E-01 0.43973786E-01 0.40304036E-01 0.39048830E -01 0.40068056E-01 0.42708578E-01 0.45899622E-01 0.48445937E-01 0.49412462E-01 = o.8 = 0.3 0.12768770E-00 0.12590981E-00 0.12084408E-00 0.11320817E-00 0.10394922E-00 0.93986174E-01 0.84023392E-01 0.74505371E-01 0.65710129E-01 0.57909308E-01 0.51489833E-01 0.46951744E-01 0.44761523E-01 0.45118794E-01 0.47749031E-01 0.51837667E-01 0.56167986E-01 0.59438384E-01 0.60652751E-01 40 = 0.8 0.4 0.11270082E-00 0.11139210E-00 0.10763909E-00 0.10190663E-00 0.94813250E-01 0.86975180E-01 0.78903589E-01 0.70995817E -01 0.63609476E-01 0.57160740E-01 0.52170087E-01 0.49200971E-01 0.48690914E-01 0.50735381E -01 0.54922859E-01 0.60314053E-01 0.65608089E-01 0.69463408E-01 0. 70873030E-01 40 =0.8 4 = 0.5 0.99220436E-01 O.98256894E-01 0.95473240E-01 0.91159467E-01 0.85712107E-01 0. 79551984E-01 0. 73077743E-01 o.66674617E -01 0.60765285E-01 O.55844366E -01 0.52487408E-01 0. 51234794E-01 0.52433489E-01 0.56051518E-01 0.61559863E-01 0.67949773E-01 0.73911310E-01 0.78137403E-01 0. 79664318E -01 167

Table 25 cont. =o.6 % = o.8 O _ 0.6 o.86877821E:-01 0.86168942E-01 0.84107330E-01 0.80873561E -01 0.76728944E-01 0.71981121E-01 0.66969865E -01 0.62078520E -01 0.57755943E-01 0.54518782E-01 0.52904418E-01 0.53363617E-01 O.56112571E-01 0.60992800E-01 0.67399664El-01 0.74327521E-01 o.805.43427E-01 0.84854731E-01 q o.8639704E-l01 IL =.8 =0.7 0.75337762E-0 1 0.74829021E-01 0.73344627E -01 0 0.71005233E -01 0.67999371E-01 0.64574254E-01 0.61034307E-01 0.57745440E-01 0.55133157E-01 0.53657342E-01 0.53750562E-01 0.55720851E-01 0.59639242E-01 0.65248290E -01 0.71932031E-01 0.78775617E-0 1 0.84716637Eo01.88758530E-01 0.90191525E-01 = 0.8 - =.8 0.64540421E -01 0.64214033E-01 0.63265192E -01 0.61784093E-01 O.59919056E-01 0.57874885E-01 0.55910721E-01 0.54333560E-01 0.53480756E-01 0.53684685E-01 0.55217295E-01 0.58220518E-01 0.62638243E-01 0.68171844E-01 0.74280833E-01 0.80240956E-01 0.85256337E-01 0.88604629E-01 0.89781123E-01 o =0.8 =0.9 0.55271707E-01 0.55142932E-01 0.54777632E-01 0.54238316E-01 0.53627099E-01 0.53081743E-01 0.52768318E-01 0.52868739E-01 0.53561584E —01 0.54995877E-01 0.57259839E-01 0.60349552E-01 0.64145035E-01 0.68402033E-01 0.72766110E-01 0.76811191E-01 0.80098503E-01 0.82245768E-01 0.82292353E-01 40 = 0.8 = 1.0 0.57967425E-01 0.57967425E-01 0.57967425E-01 0.57967425E-01 0.57967425E -01 0.57967425E-01 0.57967425E-01 0.57967425E-01 0.57967425E-01 0.57967425E -01 0.57967425E-01 0.57967425E-01 0. 57967425E-01 0. 57967425 7967425E-01 0. 5 7967425E-0. 57967425E-01 0.57967425E-01 0.57967425E-01 0.57967425E -01 = 1.0 - =0.0 0.42936990E -01 0.429369go E-o 0.2936990E-01 4293699 0E-01.42936990E-01 0.42936990E-01 0.42936990E-01 0.42936990E-01 0. 42936990E-01 0. 42936990E-01 0.42936990E-01 0.42936990E-01 0. 42936990E -01 0.42936990E-01 0.42936990E-01 0.42936990E-01 0.42936990E-01 O.42936990E-01 O. 42936990E -01 0.42936990E-01 = 1.0 p = 0.1 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 0.50919585E-01 168

Table A-25 cont. 4 = 1.0 ~ =0.2 0.54421556E-01 0.54421556E-01 0.54421556E-0i 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0.54421556E-01 0. 54421556E-01 0.54421556E-01 0.54421556E-01 %L = 1.0 0.3 0. 56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0. 56145927E-01 0. 56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 0.56145927E-01 = 1.0 = 0.4 0 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0.57154635E-01 0. 57154635E-01 0.57154635E-01 = 1.0 =0.5 0.58375879E-01 0. 58375879E-01 0.58375879E-01 0.58375879E-01 0. 58375879E-01 0. 58375879E-01 0.58375879E-01 0. 58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 0.58375879E-01 o0 = 1.0. =0.6 0.60719676E-01 0.60719676E-01 0.60719676E-01 0.60719676E-01 0.60719676E -01 0.60719676E -01 0.60719676E-01 0.60719676E -01 0.60719676E-01 0.60719676E-0 1 0.60719676E-01 0.60719676E-01 0.60719676E-0 1 0.60719676E-01 0.60719676E-01 0.60719676E-01 0. 60719676E -0 0. 60719676E -01. 60719676E -01 60719676E -01. = 1.0. = 0.7 0.65103649E -01.65165103649E-0o1 o'65103649E-01 0.65103649E-01 0.65103649E-o1 o.65103649E-o1 0.65103649E-01 0.651053649E-01 0.65103649E-01 0.65103649E-01 0.65103649E-0o 0.65103649E-01 0.65103649E-o1 0.65103649E-01 0.65103649E-01 0.6510.65103649E3649E-01 65103649E-01 65103649E.65103649E-o1 0.65103649E-o1 o.65103649E-01 p = 1.0 p = 0.8.725928-01 0.7259281-01 0.724592 -01 0.7245928-01 O.72459281E-01 0.72459281E-01 0.72459281E-01 0.72459281E-01 0.72459281E -01 0.72459281E -01 0.72459281E -01 0.72459281E -01 O.72459281E -01 0.72459281E-01 0.72459281E-01 0.72459281E-01 O.72459281E-01 C.72459281E-01 C.72459281E-01 C.72459281E-01 O.72459281E-01 0.72459281E-01 0.72459281E-01 169

Table A-25 cont. =1.0 =0.9 c o = 1.0 R = 0R9 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 0.83732596E-01 so = 1.0 =1.0 0.99883246E-01 0.99883246E -01 0.99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 O. 99883246E-01 0.99883246E-01 0.99883246E-01 0. 99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 0.99883246E-01 170

Table A-26 Normalized Reflected Intensity, I/Io, for Isotropic Scattering (N = 0) 0 = 0.4 Lo = 0.9 A EH(Z), I/Io o 1.0 0.1059 0.1 1.1722 0.0993 0.2 1.2914 0.0916 0.3 1.3914 0.0842 0.4 1.4785 0.0783 0.5 1.5560 0.0732 0.6 1.6259 o.o688 0.7 1.6893 0.0650 0.8 1.7474 0.0617 o.9 1.8008 0.0587 1.0 1.8501 0.0560 171

Table A-27 Total Reflectances o0 = 0.9 N =l N =2 N= 3 N = 4 o K a ~(NO) R ( R tGo)R R 0 0.0.6262 0.0.6499 0.0.6198 0.0.6403 0.1.0445.5553.o436.5642.0469.5307.0449.5514 0.2.0994.5030.0990.5049.1067.4667.1020.4898 0.3.1628.4574.1634.4555.1762.4127.1684.4385 rro 0.4.2335.4164.2348.4130.2538.3656.2423.3944 0.5.3106.3788.3121. 3758.3379.3241.3221 ~3557 0.6 3936.3441.3941.3431.4275.2874.4070.3216 0.7.4818.3118.4800.3143.5215.2550.4960.2914 0.8.5747.2816.5689.2889.6188.2265.5884.2644 0.9.6720.2533.6602.2664.7188.2014.6837.2404 1.0.7733.2267.7533.2467.8205.1795.7813.2187

Table A-28 Total Reflectances for Isotropic Scattering (N=O) Lo0 R 0.6837 0.1.6292 0.2.5915 0.3.5599 0.4.5324 0.5.5078 0.6.4857 0.7 A4657 o.8.4473 0.9.4387 1.0.4148 173

APPENDIX B Computer Programs The FORTRAN programs used with the IBM-704 computer are presented in the following pages with brief explanatory notes. 174

TABLE OF CONTENTS I. I and H-function Programs Page 175 Introduction 176 Program for 2 simultaneous integral equations 183 Program for 3 simultaneous integral equations 191 Program for 4 simultaneous integral equations 198 Program for 5 simultaneous integral equations 203 Program for H-function Gauss Quadrature, 1 parameter characteristic function 211 Program for H-function Simpson's Rule, arbitrary characteristic function II. Normalized Reflected Intensities, I/I0 217 Introduction 218 Program for N = 1 220 Program for N = 2 222 Program for N = 3 224 Program for N = 4 175

I. ~ and H-function. For N = 4, the computational problem consists of solving sets of 5, 4, 3, 2 and 1 simultaneous integral equations. For N = 3, the sets involved are 4, 3, 2 and 1. For N = 2, they are 3, 2 and 1. Hence general programs were written to handle 5, 4, 3 and 2 simultaneous integral equations without regard to the phase function the set comes from. These programs have the same type of skeleton regardless of the number of equations and various subroutines which were written to handle these parts of the program that would be unique for a given phase function. These are found under the section "Subroutines". As stated the -function and H-function programs follow the same general outline: a. read in constants peculiar to the situation, i.e., interval length for integration, coefficients in the phase functions, etc. b. calculate known functions peculair to the situation i.e., the various Legendre polynomials. c. calculate the first guess for the 4 -functions d. go through several iterations testing each time if the functions are changing uniformly enough to use exponential extrapolation or convergence is reached e. finally use the extrapolation f. go back to step (d) and repeat until convergence is reached For the ( -functions, integration was carried out by Simpson's rule for either 20, 50 or 100 intervals or in steps of 0.05, 0.02 or 0.01. For the 50 or 100 intervals a provision was placed between b and c to read in converged results for 20 intervals and use these as first guesses for the smaller intervals calculations. For the H-functions both Simpson's rule and 48 point Gauss Quadrature were used for the integrations. For +-functions where m ~ 0 it is best to define new functions which are of the same form as 4-functions whose m = O and compute these. This keeps the absolute values of the tabular functions down and reduces the variations. 176

For 2 simultaneous integral equations. Constants: Z3 = integration interval Z4 = phase function coefficients Z5 = phase function coefficients ZO = set extrapolation value-to use if calculated one becomes too large Z07 = test number to see if functions are changing to slowly to apply extrapolation number Z8 = test to see if functions at 1.0 are changing approximately linearly Z9 = exit test for iteration convergence, applied at f(1.0) KO = iteration counter between extrapolation K1 = interval length K2 = auxiliary read constant K3 = extrapolation counter K4 = total iteration counter K5 = convergence counter test. K6 = iteration number test-for exit before convergence K7 = secondary iteration counter K8 = auxiliary read interval K9 = test to enter Calc 3 subroutine J5 = set to zero Functions: X,Y = the iterated functions E _ function to convert X, Y back to (/-functions Subroutines: Calc 4 - subroutine to compute transformation functions Simp A - iteration subroutine Calc 3 - subroutine to calc f -functions from iterated functions 177

FORTRAN PROGRAM FOR 2 SIMULTANEOUS INTEGRAL EQUATIONS. DIMENSION X(404),Y(404),E(101) 41 READ INPUT TAPE 7,42,Z3,Z4,Z5,ZO,Z07,Z8,Z9,D25,KO, K1,K2,K3,K4,K5, 1 K6,K7,K8,J5,K9 42 FORMAT (E9.2,1P2E14.7/5E9.2/115) WRITE OUTPUT TAPE 6,43,Z3,Z4,Z5 43 FORMAT (1H1,E9.2,1P2E14.7) CALL CALC4 (Z3,K1,E) WRITE OUTPUT TAPE 6,250,(E(I),I=1,101,K1) 250 FORMAT (1HO,7E17.8/(1H,7E17.8)) IF(K2)55,55,46 46 READ INPUT TAPE 7,47,(X(I),I=1,101,K8) READ INPUT TAPE 7,47,(Y(I),I=1,101,K8) 47 FORMAT(lP5E14.7) CALL WRI (101,K8,X,Y) IF (K2-1) 2,2,49 49 CALL EXTEND (X) CALL EXTEND (Y) CALL WRI (101,K1,X,Y) GO TO 2 55 Zl=O. DO 1 I1=1,101,K1 X( I1 )=1. Y(I1) =Z1 1 Z1=Z 1+Z3 02'CALL SIMPA (Z3,X,Y,EK1,Z4,Z5) D24=X(( 101 )-D50 D29=Y( 101 )-D51 D50=X( 1.01 ) D51=Y( 101 ) IF(KO-1 1)102,15, 102 102 IF (ABSF (D24)-Z9) 103,103,8 103 IF(ABSF(D29)-Z9) 104104,104-8 104 K5=K5+1 GO TO 15 8 IF (J5). 107,107,228 107 CALL TEST(J1,D22,D23,D24,D25,Z07)' CALL TEST (J2, D27, D28,D29,D30,Z07) 14 J4=J1+J2 IF (J4-2) 124,19, 124 124 CALL -CALC1(J1,D21,D22,D23,D24,D25) CALL CALC1(J2,D26,D27D28, D29,D30) 15 WRITE' OUTPUT TAPE 6,131,K4,D22,D27,D25,D30,,D24,D29 131 FORMAT (1HO,I3,6E17.8) CALL-'WRI (101,K1,X,Y) D21=D22 133 D23=D24 D26=D27 D28=D29 KO=KO+1 K4=K4+1 IF (K4-K6) 144,144,23-2 144 IF(KS-3) 145,1 45,232 178

145 IF( ABSF( D25 )-Z8 ) 146,146,2 146 IF (ABSF(D30)-Z8) 36,36,2 36 K2=-1 DO 9 I=l1,101,K1 I4=I 1+101 X(I4)=X(I1) 09 Y(I4)=Y(I1) 23 IF(K2)152,24,30 152 I1=202 K2=0 GO TO 25 24 I1=303 K2= 1 25 CALL SIMPA- (Z3,X,Y,E,K1,Z4,Z5) DO 29 I=1,101,K1 I4=I+I 1 X( I4)=X( I) 29 Y(14)=Y(I) GO TO 23 30 K3=K3+1 191 FORMAT(1HO, I5) CALL CALC2 (202,303,404, X,Z13,ZO) CALL CALC2 (202,303,4 404,Y,Z14,ZO ) 34 IO=Kl+1 DO 31 I5=I0,101,K1 i2=I5+101 3= I5+202 X( I5)=X(I2)-(X( I3)-X:(I2) )*Z13 31 Y( I5)=Y( I2)-(Y( I3)-Y((I2 ) )*Z14 KO=1 D25=1. J5=O0 J2=0 J1=0 GO TO 2 19 J5= 1 228 J5=J5+1 K4K4+ 1 IF(J5-K7) 2,36,36.232 WRITE OUTPUT TAPE 6,191,K4 CALL WRI (10l1,KlX,Y) IF (K5-3) 240,240,241 241 IF (K9) 240,240,242 242 CALL CALC3 (X,Y,Z3,K1,E) CALL WRI (101i,K,XY) 240 PUNCH 239(X(I),I=I1,101,Kl) PUNCH 239, (Y(I), I=1,1O1,K1) 239 FORMAT(1P5E14.7) D 30=0 ~ GO TO 41 179

SUBROUTINE WRI (NUNl,A,B) DIMENSION A(lu4)9,B(1L4) WRITE OUTPUT TAPE 6,1,(A(I),I=lNOq,Nl) WRITE OUTPUT TAPE 6,1,(-3 (I) I =1,NON1) 1 FORMAT ( 1HO,7E1.-78/( 1H,97E17.8) ) RETURN SUBROUTINE SIMPA (Z3,X,YEK1,Z4,Z5) DIMENSION X(101),Y(101),E(101),A(10) ZI=O. 74 I3=K1+1 75 DO 3 I5=I3,101,K1 76 ZI=Z1+Z3 77 Z2=O. DO 1 I=1,10 1 A(I)=O. Z1l0=Z4*X(15) 82 I7=-1 83 DO 4 18=13,101,K1 84 Z2=Z2+Z3 Z14=(Z4*X(I5)*XI() 8)-Z5*Y()Y I5 )*Y(I8) )*E( 18)/(1.+Z2/Z1) IF (I8-101) 2,5,2 2 IF(I7) 797,6 7 A(1)=A(1)+Z14 A 2 )A (2)+Z14*Z2 95 GO TO 4 6 A(6)=A(6)+Z14 A(7)=A.(7 )+Z1 4*Z2 4!7=-17 5 X( 15)=1.+(Z1O+4*A(1)+2'*A(6)+Z14)*Z3/6~ 3Y ( 15 ) =Z1-(4.*A (2)+2.*A(7)+Z 14*Z2)*Z3/6., RETURN 180

SPECIAL SUBROUTINES FOR 2 SIMULTANEOUS 2 TERM SUBROUTINE CALC3 (X,Y,Z3,KlE) DIMENSION X(1O1),Y(1L1),E(101) Zl=O. DO 1 I=ll1O1,K1 D=SQRTFE(I) )....................... X( I )=D*X( I) Y I )=3.*D*Y( I) 0 1 Z 1=Z1+Z3 RET IJU R N SUBROUTINE CALC4(Z3K —1E) DIMENSION E(101) DO 53 I1=1,101,K1 E ( I )=1. -Z1*Z1 53 Z1=Z1+Z3 RETURN SPECIAL SUBROUTINES FOR 2 SIMULTANEOUS 1 TERM SUBROUTINE CALC4(Z3,K1,E) DIMENSION E( 1G1 Z 1=0. DO 5.3 I1=1,101,Kl...E....... _._ I..!._!.-_..1,....................................._....... 53 Z1=Z1+Z3 RETURN 181

SPECIAL SUBROUTINES FOR 2 SIMULTANEOUS 3 TERM SUBROUTINE CALC3 (XY,Z3,K1,E) DIMENSION X( 101),Y(101) E(101 ) 21=0. DO 1 I=1,101,K1 D=3.*( 1.-Zl*Z1) X(I)=D-*X(I) Y( I )=5.*D*Y( I) -- ---------- - - - -'~ - - - - 01 ZI=Z1+Z3 X(101)=0. Y(101)=0o RETURN SUBROUTINE CALC4(Z3,K1,E) DIMENSION E(101) ZI=O. DO 53 I1=1,101,K1.......E_(..1)=.(1.,_,Z l*Zl!)* (1.-Zl*Z1) 53 Z1=Z1+Z3 E(101)=0o RETURN 182,

SPECIAL SUBROUTINES FOR 2 SIMULTANEOUS 4 TERM SUBROUTINE CALC3 (XY*Z39K1lE) ~~~~~~~~~~~~~~~~~~~~~~~~~~~.....................DiM E ~'..''""~-)-;'-'i( —--- 0 DIMENSION X(101);Y(101),E(1O1) Zl=O, z 1= 0. _ _ _ _ _ _ _ _ _ _ _ _ _ __ __ DO 1 I=19101,pKl D=SQRTF(E(I)) Y( I )=105*D*Y( I) X(I)=15.*D*X(I) 01 Z1=Z1+Z3 RETURN _ _ _ SUBROUTINE CALC4(Z39K19E) __ DIMENSION E(101) Zi=O~o D0 53 I 11=11,TD=1I-Zi*Z1 E(I1) D*D*D 53 Zi=ZI+Z3 E(101)=0o RETURN 183'

For 3 simultaneous integral equations. Constants: Z3 = integration interval Z4 = phase function coefficients Z5 = phase function coefficients ZO = set extrapolation value-to use if calculated one becomes too large Z07 = test number to see if functions are changing to slowly to apply extrapolation number Z8 = test to see if functions at 1.0 are changing approximately linearly Z9 = exit test for iteration convergence, applied at f(1.0) KO = iteration counter between extrapolation K1 = interval length K2 = auxiliary read constant K3 = extrapolation counter K4 = total iteration counter K5 = convergence counter test K6 = iteration number test-for exit before convergence K7 = secondary iteration counter K8 = auxiliary read interval N1 = test to enter Calc 3 subroutine J5 = set to zero Functions: X, Y, C = the iterated functions E = P1(I) H = transformation function Subroutines: Calc 4 = subroutine to compute transformation functions Simp 2 = iteration subroutine Calc 3 = subroutine to calculate If -functions from iterated functions 184

FORTRAN PROGRAM FOR 3 SIMULTANEOUS INTEGRAL EQUATIONS DIMENSION X(404),Y(404),C(404),E(101),H(1011) 41 READ INPUT TAPE 7,42,Z3,Z4,Z5,Z6,ZOZ07,Z8,Z9,D25, 1KO,K1,K2,K3,K4,K5- K6,K7K8,J5, N1 42 FORMAT(E9.2, 1P3E14.7/5E9.2/1115) 43 WRITE OUTPUT TAPE 6-44,Z3,Z4,, Z5Z6, ZZ07,Z8, Z9, D25, 1KOK1,K2,K3K4, K5,K6,K 9K7,K8,J5,N1 44 FORMAT(1Hl1E9.2,1P3E14.7/1H,-5E9.2/1H,1115) CALL CALC4.(Z3,KK1, EH) WRITE OUTPUT TAPE 6,250,(E(I),I=1 101,K 1),(H(I),I1,101,KK1) 250 FORMAT (1HO,7E17.8/(1H,7E17.8)) 45 IF(K2)55,55946 46 READ INPUT TAPE 7,47,(X(I),=1,101,K8) READ INPUT TAPE 7,47,(Y(I)oI=1,101,K8) READ INPUT TAPE 7,47,(C(I)I:1,101,K8)_ 47 FORMAT(1PSE14.7) CALL WRI(lOl1K8XYY, C) IF (K2-1) 292,49 49 CALL EXTEND (X) CALL EXTEND (Y) CALL.EXTEND (C) CALL WRI(1lO1,K1,XY, C) GO TO 2....... -'5 -5~...'I": -'O-.~................................................ 55 Z1=0. 56 DO 1 I1=1,101,K1 57 X(I1)=1. 58 Y(I1)=Z1 C(I1)=E(Il) 1 Z1=Z1+Z3_ 2 CALL SIMP2(Z3-X,Y,C,K1,E,Z4,Z5,Z6, H) D24=X(101)-D50 D29=Y(101)-D51 D34=C( 101)-D52 D050=X(101)........_ D51=Y( 101) ) _ _ _ _ D52=C ( 101) 101 IF(KO-1) 102,152,102 102 IF(ABSF(D24)-Z9)1 03,1'03,8 103 IF(ABSF(D29)-Z9)104,1048,8..104 IF- (-ABSF( D'34 - -Z9) 10 5; i05 —,"8 105 KS5KS+1.106 GO TO 15 8 IF (J5) 107,1.07,228 107 CALL TEST(Jl,D22,D23,D24,D25,Z07) CALL TEST ( J2,D27,D28,D29,D30,Z07)' CALL TEST( J3,D32,D33,D34,D35,Z07 )14 J4+J1+J2+J3 123 IF(J4-3)124,19,124-........... 124 CALL CALC1(J1, D2, D22, D23D21, 2 4D25) CALL CALC 1 (J2, D26, D27,D28,D29t"D30) CALL CALC1(J3,D31,D32,D33,D34+,D35) 15 WRITE OUTPUT TAPE 6,1-3i','Kz+, D22,,D27,D32,D25,D30D35,D24,D29,D34+ 1-31 FORMAT( 1HO, I3,3E17.8/6E17.8) 185

CALL WRI (101,K1,X,Y,C ) D21=D22... 133 D23=D24 136 D31=D32 134 D26=D27 135 D28=D29 137 D33=D34 141 KO0KO+1 142 K4=K4+1 143 IF(K4-K6)144,144,232 144 I F(K5-3) 1459145,232 145 IF(ABSF(D25)-Z8) 146,146,2 146 IF(ABSF(D30)-Z8)147,147,2 147 IF(ABSF(D35)-Z8)36,362. 36 K2=- 1 148 DO 9 I1=1,1O1,K1 149 I4=I+101 150 X(I4)=X(I1) 151 Y(14)=Y(I1) 9 C(14):C(1 ) 23 IF(K2)152*24930 152 11=202 153~~~~~~~~~~~~~~~~~~~~~~` K2 -. ~ ~` -`~ -...~'~~-.-~~~ —-. —-. —~.-.-..-.^ —-—' —-.. ——. —-_-_ 153 K2=O 154 GO TO 25 24 11=303 155 K2=1 25 CALL -SIMP2(Z3,X,Y,C,K-,E,Z4,Z5,Z6,H) DO 29 I=I,1'01,K1 I,4(+I 1 X-(.I4):=X( I) Y(I4)=Y( I) 29 C(I14)=C(I) 189 GO TO 23 30 K3=K3+1........ 191 FORMAT(1H.OI5) CA LL CALC2 (202,3u3,404, X,Z13,ZO ) CALL CALC2 (202, 303,404,qYZ14, ZO) CALL CALC2 ( 202,.303,404, C,Z 15,Z0 _ ) 34 IO=KI+1 205 DO 31 I5=I0,101,K1 207 I2=15+101 208 13=15+202 209 Z10.=(X(I3)-X(I2))*Z13 210 Z11=(Y(I3)-Y(12))*Z14 211 Z12=(C( I 3)-C(I 2) )*Z15 212 X(I5)=X(I2)-Z11 G _ _ 213 Y(I5)=Y(12)-Z11 3 1 C II 5)=C(I2)-Z12 218 KO=1 219 D25=1. 220 J5=0 22 1 3 3= 0 _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 222 J2=0 223 Jl=O 224 GO TO 2.....1 9.. 3 5 =11?_. _ _ _...-_............... _ _ _ 228 J535+1 229 K4=K4+1:86

— 2 iF(JS — K7) 2,3 6,36 232 WRITE OUTPUT TAPE 6,19.1,K4 CALL WRI ( 1O1,K1XY, C) IF (K5-3) 240,240,241 241 IF (N1)240,240,242 242 CALL CALC3(XYqC'Z3qK1) CALL WRI (101,K1,XqY,C) 240 PUNCH 239, (X(I)( I=1,1011, K1) PUNCH 239, (Y(I), I-'1,11, K1l) PUNCH 239, (C(I), I=lUl,K1) _ 239 FORMAT (1P5E14.7) D30=0. D35=0. 247 GO TO 41

SUBROUTINE WRI(NO,N1,A,B,C) DIMEN SI ON A(104),B(104 ),C(104) WRITE OUTPUT TAPE 6,1,(C(I),.I-,NO,NN1) WRITE OUTPUT TAPE 6,1,(A(I.),I=1,NO,qN1) WRITE OUTPUT TAPE 6,1,(B(I ),I=1,N0,N1) 1 FORMAT (1HO,7E17.8/(1H,7E17.8)) RETURN SUBROUTINE SIM4P2 (Z3,X,Y,CK1,E,Z4,Z5,Z6,H) bIMENSION X(101),Y(101),C(101),E 101 01) A( 1) H(1 Zl=Oo 74 I 3=Kl+l 75 DO 3 I5=I3,101,K1 76 Zl=Zl+Z3 77 Z2=0. DO 1 I=1,10 1 A(I)=O. Z10Z4*X( I5 )-Z6*C( I5)*.5

SPECIAL SUBROUTINES FOR 3 SIMULTANEOUS 4 TERM SUBROUTINE CALC3(X,Y,C,Z3, K 1 ) DI-MENSION X( 101 ),Y( 11),C(101) ZI=O. DO 1 I=1,lOl,K1 D=3.* ( i.-Z*Zl) X(I) D*X ( I ) Y( I )=5.*D*Y( I) C( I )=5.*D*C( I) 01 Zl=ZI+Z3 X (101) -=0 Y(101)=0. C(101)=0. RETURN SUBROUTINE CALC4(Z3,K1,E,H) DIMENSION E(11O), H(101) ZI=O. DO 53 I1=l,101-O,K1 E I1 ). 5 (7.*Z 1Z1-1. ) H( I1)=(1.-Z1*Z1 )*(1.-Z1*Z1) 53 Z1=Z1+Z3 H(la 10.1 ) =0. RETURN 189

SPECIAL SUBROUTI NES FOR 3 SIMULTANEOUS 3 TERM SUBROUTINE CALC3(X,Y,C,Z3,K1) DIMENSION X(101),Y101),C(101) Z1=0, DO 1 I=1,101,K1l D=SQRTF (1.-Z1*Z1') X( I )=D*X(I') Y( I )=3*D*Y( I) C(I)3*D*C( I ) 01 Zl=Z1+Z3 X ( 101 ) =0. Y ('101.) =0. C( 101 )=0. RETURN' SUBROUTINE CALC4(Z3,K1,EHi) DIMENSION E(101),H(101) z1=0. DO 53 I1=l,101,K1 E(I1 ).5*(5.*Z*Zl —1. ) H( I ) = 1.-Z1*Z1 53 Z1iZ 1+Z3 H(101)=0. RETURN 190

For 4 simultaneous integral equations. Constants: Z3 = integration interval Z4 = phase function coefficients -7.c; - -n'hq. fl'nt+.-inn nrnefficients

FORTRAN PROGRAM FOR 4 SIMULTANEOUS INTEGRAL EQUATIONS, 40 DIMENSION X(404),Y(404),C(404),D(404),E(101),F(101),H(101)

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255 Z10=(X(I3)-X(I2) )*Z14 256. Z11=(Y(I3)-Y( I2) )*Z1 5 257 Z12=(C(I3)-C(I22) )*Z16 258 Z13=(D(I3)-D(I2))*Z17 2,59 X(I5)=X(I2)-Z10 260 Y(I5)=Y(.I2)-Z11 261 C(I5)=C( I2)-Z12 31 D(I5)=D(I2)-Z13 267 KO=1 268 D25= 1. 269 J6=0 270 J4=0 271 J3=0 272- J2=0 273 J1=0

SUBROUTINE WRI(NO,N1,A,B,C,D) DIMENSION A.(1010,B(104),C(104),D(104) WRITE OUTPUT TAPE 6,1,(A(I),I=l,NO,N1) WRITE OUTPUT TAPE 6;1,(B(I),'I=1,NON1) WRITE OUTPUT TAPE 6,1,(C(I),I=,NON1) WRITE OUTPUT TAPE 6q1q(D(I)qI.=1qN0qNl) 1 FORMAT (1HO,7E17.8/(1H 97E17.8)) RETURN SUBROUTINE SIMP3(Z3,XY,C,D,K1,EFZ4, Z5, Z6,Z7,H) DIMENSIONX( 101) Y( 101),C(lul) 1D( 101 E( 101),F( 101) A( 10) H( 101) Z1=O, 67 I3=K1+1 68 DO 3 I5=I3,10'1,K1. 69 ZI=Z1+Z3 70 Z2=0, DO 1 I=1,10O 1 A( I )=0, Z10O=Z4*X(I5)-Z6*C(I5)*,5 Z12=-Z1O*.5 74 I7=-1 75 DO 4 I8=I3,101,K1 76 Z2=Z2+Z3 Z14=(Z4*X(I55*X(I8)-Z5*Y (15)*YI18)+Z6*CI15)*CI18-Z7*D(I 5)*D(18) i1 H(I8)/ (1.+Z2/Z1) IF (I8-101) 2-)5,2 2 IF(17) 7,7,6 7 A(1)=A(1)+Z14 A(2)=A(2)+Z14*Z2 A(3)=A(3)+Z14*E(I8) A(4)=A(4)+Z14*F(18) 85 GO TO 4 6 A(6)=A(6)+Z14 A(7)=A(7)+Z14*Z2 A(8)=A(8)+Z14*E(18) A(9)=A(9)+Z14*F(I 8) 4 I7=- I7 5 x(15)=1.+CZ1O4~A1+4.* ~)+14)+2.*()Z*3/6.-_ ___ Y(I5)=Zi-(4.*A(2)+2.*A(7)+Z1 4*Z2)*Z3/6. C(15)=E( I5)+Z2+4.*A3)+2.*A( 8)+Z14*E(I8) )*Z3/6. 3 D(15)=F(15)-(4.*A(4)+2.*A(9)+Z14*F(18))*Z3/6. RETURN 196

SPECIAL SUBROUTINES FOR 4 SIMULTANEOUS 3 TERM SUBROUTINE CALC4 (Z3,K1,E,F,H) DIMENSION E(101),F(101),H(101). Zl=O,. DO 1 I=,1O01,K1 E(I)=.5* ( 5.*Zl*Z-. ) F(I)=.5*Z1*(7.*Zl*Z-3. ) H( I )=1.-Zl*Zl 1 Z1=Z1+Z3 H(101) =0. RETURN SUBROUTINE CALC3 (X,Y-C,D,Z3,K1,H) DIMENSION X(101),Y(101'),C(101),D(1 01),H( 101) DO 1 I=1,101,K1 F=SQRTF(H(I)) X( I )=F*X( I) Y( I )=3.*F*Y(I) C(I )=3.*F*C( I) D(I )5.*F*D( I ) 1 Zl=Z.1+Z3 RETURN 197

SPECIAL SUBROUT'INES FOR 4'SIMULTANEOUS 4 TERM SUBROUTINE CALC4 (Z3,K1,E,FH) DIMENSION E(101),F( 101),H(1 ) z1=0, DO 1 I=1,1Ol1,K1 E( I )=. 5*(3.*Z1*Zl-1.) F( I ) =. 5*Z1*(5.*Z1*Z1-3. ) H(I)=1. 1 Z1=Z1+Z3 RETURN 198

For 5 simultaneous integral equations. Constants: Z3 = integration interval Z4,Z5,Z6,Z7,Z18 = phase function coefficients ZO = set extrapolation value-to use if calculated one becomes too large Z07 = test number to see if functions are changing to slowly to apply extrapolation number Z8 = test to see if functions at 1.0 are changing approximately linearly Z9 = exit test for iteration convergence, applied at f(1.0) KO = iteration counter between extrapolation K1 = interval length K2 = auxiliary read constant K3 = extrapolation counter K4 = total.-iteration counter K5'= convergence counter test K6 = iteration number test-for exit before convergence K7 = secondary iteration counter K8 = auxiliary read interval J6 = set to zero Functions: X, Y, C, D, P = the /-functions E,F,G = Legendre polynomials; P1(Q), P2(y), P3(Q) Subroutines: Extend - to interpolate a 20 interval function to a 100 interval function Wri - write the t -functions out Simp 4 - subroutine where the computation of the n + 1 functions from the n- functions takes place Test - test to see if the rate of convergence has become so slow as to erase all significant numbers in extrapolation formulas. Calc 1 - subroutine for testing smoothness of functions at x = 1.0 Calc 2 - calculate extrapolation multiplier 199

FORTRAN PROGRAM FOR 5 SIMULTAEOUS INTEGRAL EQUATIONS DIMENSION X(404), Y(404), C(404), D(404), P(404), E(101), F(101), 1G(101) 41 READ INPUT TAPE 7,42, Z4,Z5,Z6,Z7,Z18,ZOZ07,Z3,Z8,Z9,D25,KO,'K1,K2 1,K3,'K4,K5,K6,K7,K8,J6 42 FORMAT (lP5E14.7/6E9.2/101 15) WRITE OUTPUT TAPE 6,44,Z4,Z5,Z6,Z7,Z18,ZO,Z07,Z3,Z8,Z9,D25, 1KO, K 1,K2,K3,K4,K5,K6, K7, K8,.J6 44 FORMAT(lHl.,1PSE14.7/lH,6E9.2/1H,10I5) IF(K2)58,58,46 46. READ INPUT TAPE 7,47, (X(I), I=1,101,K8) READ INPUT TAPE 7,47, (Y(I), I=1,1O1,K8) READ INPUT TAPE 7,47, (C(I), I=1,101,K8) READ INPUT TAPE 7,47, (D(I), I=l,lOl,K8) READ INPUT TAPE 7,47, (P(I), I=1,101,K9K) 47 FORMAT(1P5E14.7) CALL WRI (101,K8,X,Y,C,D,P) IF (K2-1) 48,48,49 49 CALL EXTEND(X) CALL EXTEND(Y) CALL EXTEND(C) CALL EXTEND(D) CALL EXTEND(P) CALL WRI(101,K1,X,Y,C,D,P) 48 Z=O. DO 400 I1=1,101,K1 E(I )=.5*(3.*ZI*Z-1- 1.) F( I )=.5*Z1*(5.*Z.1*Zl-3.) G(I1)=.12.5*(35..*Zl**4.-30.*Z*Zl+3. ) 400 Z1=Z1+Z3 GO TO 2 58 Zl=0. DO1 I1=1,101,K1, X ( I 1 )=. Y (I1) =Z1 C (I1)=.5*( 3.*Zl*Z1-1. ) D( I I ) =.5*Z1* ( 5.*Zl*Zl1-3. ) P ( I1 )=.125*( 35.*Z;**4.-30.*Zl*Zl+3, ) E( I1 )=C( I 1 F(I1) =D( I1) G(I1)=P(I1) 1 Z1=Z1+Z3 2 CALL SIMP4(Z3tX,Y,CD,P,K1, E,F,GZ4,Z5,Z6,Z7, 18) D24=X (101)-D50 D29=Y (101 )-D51 D34=C( 101 ) -D:52 D39=D( 101)-D53 D44=P (101 )-D54 D50=X (101) D51=Y ( 101) D52=C ( 101 ) D53=D 101) D54=P( 101) 200

- t.-K -I )!1 ~i5,1 18/~ 118 IF(ABSF(D24)-Z9) 1 19, 119,8 119 I FABSF(D29)-Z9)12, 1 20, 8 120 IF(ABSF(D34.)-Z 9)121,121,8 121 IF(ABSF(D39)-Z9)123,123,8 123 IF(ABSF(D44)-Z9)122,122,8 122 K5=K5+1 GOT015 8 IF (J6) 124,124,279 124 CALL TEST(J1,D22,D23,D24,D25,Z07) CALL TEST(J2,D27,D28,D29,D30,Z07) CALL TEST (J3,D32 D33,D33,D34,D35,Z07) CALL TEST ( J4,D37,D38,D39,D40,Z07 ) CALL TEST ( J7,D42,D43,D44,D.45,Z07) 16 J5=J 1++J3+J 4+J7 IF'( J5-5)146,19,146 146 CALL CALC1(J1,D21,D22,D23,D24, D25 ) CALL CALC1 (J2,D26, D27,D28,D29, D30) CALL CALC1(J33,D31 D32,D33,D34,D35 ) CALL CA LC (J4,D36,D37,D38,D39,D40 ) CALL CALC1(J7, D41, D42,D43,D44, D45 ) 1-5 WRITE OUTPUT TAPE 6,155, K4,D22,D27,D32,D37,D42,9D25,D30, D35, 1D40,D45,D24,D29,D34,D39,D44 155 FORMAT (1HO:,I3,5E17.8/5E17.8/5E17.8) D21 = D 22 D23=D24 D26=D27 D28=D29 D31=D32 D33=D34 D36=D37 D38=D39 D41=D42 D43=D44 168 KO=KO+1 169 K4=K4+1 170 IF(K4-K6) 171,171,283 171 IF (K5-3) 172,172,283

DO 29 I=l,lOl,Kl I4=I+I11 X(I4)-X( I) Y(I4)=Y(I) C (I4)C( I ) D(I4)=D(I) 29 P(I4)=P(I) GO-TO 23 30 K3=K3+1 231 FORMAT (1HO,I5) CALL C'ALC2 (202,.303,404, X,Z14,ZO ) CALL CALC2 (202,303,404,Y,Z15,ZO) CALL CALC2 (202,303,404,C, Z16,ZO) CALL CALC2 (202,303404.,D,Z17,ZO ) CALL CALC2 (202,303,404,P,Z20,ZO 250 I OK1+l 251 DO 31 I5=I0,101,K1 I2=I5+101 I3=I 5+202 Z10= (X ( I'3 )-X( 2 ) )*Z14 Z11=(Y( I3)-Y(I2) )*Z15 Z12=(C(I3)-C( I2) )*Z16 Z1.3'= ('D'( I3 )-D( I2) )*Z17 Z19= ( P(I3) - P(I2)) * Z20 X(I5)=X( 1I2)-Z1O Y( I5)=Y( 12)-Zll C( I5)=C( 12)-Z12 D (.'I 5 ) =D ( I 2 ) -Z 13 31 P('I5)=P(I2)-Z19 267 K0=1 D25=1. J6-0 J4=0 J3=0 J2=0 J1=O J7=0 274 GOT02 19 J6=1 279 J6=J6+1 K4=K4+1 IF(J6-K7)2, 36 36 283 WRITE OUTPUT TAPE 6, 231,K4 CALL WRI (101 Kl,X,Y,C',D,P-) PUNCH 289, (X(I), I=1,101,K1) PUNCH 289, (Y(I), I=1.,1Oi,K1) PUNCH 289, (C(I), I ll,101,K1) PUNCH 289, (D(I),. I=l,101,K1) PUNCH 289, (P(I), I-l,101,K1) 28.9 FORMAT ( 1P5E14 7) D30=O. D35=0. D40=O. D45=0. GOT041 202

SUBROUTINE WRI(NO,N1,A,B,C,D,E) DIMENSION A(104),B(104),C(104),D(104),E (104) WRITE OUTPUT TAPE 6,1,(A(I)t,I=l,NO,N1) WRITE OUTPUT TAPE 6,1,(B(I)tI =N,N,N1) WRITE OUTPUT TAPE 6,1,(C( I),I=1,NO,N1) WRITE OUTPUT TAPE 6,1., ( D(I), I=,NO,N1 ) WRITE OUTPUT TAPE 6,1,('E(I)JI=,1,N,N1') 1 FORMAT (1HO,7E17.8/(1H,7E17.8)) RETURN SUBROUTINE SIMP4(Z3sX,Y,C,DP,K1,E, F,G,Z4,Z5.Z6,Z7,Z18) DIMENSION X( 101 ),Y( 10l),C( lUl),D( 0l1 ),P( 101),E ( ll) ),F( 101),G( 101) 1,A(10) I 3=K1+1 D03 I 5=I3,101,K1 Z1=Z1+Z3 Z2=0. DO 1 I=1,10 1 A(I)=0. Z 1O=Z4*X ( I5 )-Z6*C( 5 )*. 5+Z18*P ( 5 )*. 375Z12=-ZO*. 5 Z 19=.375*Z10 I 7=-1 D04. I8=I3,101,K1 Z2=Z2+Z3 Z14=(Z4*X( 15 )*X( I18)-Z5*Y( I5 )*Y( I8)+Z6*C ( I)*C( I8) 1-Z7*D( I5)*D( I8 )+Z18*P(I5)*P(I8) ) / (1.+Z2/Z1) IF (I8-101) 2,592 2 IF(I7) 7,7,6 7 A(1)=A(1)+Z14 A(2)=A(2 )+Z14*Z2 A(3)=A(3)+Z14*E(18) A(4)=A(4)+Z14*F(I8) A('5)=A(5)+Z14*G(I8) GOT04 6 A(6)=A(6)+Z14 A(7) =A(7)+Z14*Z2 A(8 )=A(8 )+Zl4*E( 18) A(9)=A(9)+Z14*F (I8) A(10)=A(10O)+Zl4*G( I8 4 I7-I7 5 X( 15 ) =1.+ (Z1lO+4.*A( 1 )+2.*A(6)+Z14)*Z3/6. Y ( I5 )=Z-(4.*A ( 2)+2.*A(7)+Z14*Z2 )*Z3/6. C(I5 ) E(I5 )+(Z12+4.*A(3)+2.*A(8)+Z14*E(I8))*Z3/6. D( I5)=F( I5)- (4.*A(4)+2.*A(9)+14*FU I8) )*Z3/6. 3 P( I5)=G(I5)+(Z19+4.*A(5)+2.*A( 10)+Z14*G(I8))*Z3/6. RETURN 203

H-function program - Gauss Quadrature 1 parameter characteristic function Constants: N = number of Gauss numbers W = parameter for characteristic function ZO = set extrapolation value-to use if calculated one becomes too large Z7 = test number to see if functions are changing too slowly to apply extrapolation Z8 = test to see if functions at 1.0 are changing approximately linearly Z9 = exit test for iteration convergence, applied at f(l.0) KO = iteration counter between extrapolation K2 = auxiliary read constant K3 = extrapolation counter K4 = total iteration counter K5 = convergence counter test K6 = iteration number test-for exit before convergence K7 = secondary iteration counter K8 = auxiliary read interval Functions: Z = abscissa values for Gauss numbers G = Gauss numbers Y = characteristic function X = iterated function at Gauss abscissa values H = intermediate function even intervals Subroutines: Chara = calculates characteristic functions and integrates it for term in integral equation Gaush = iteration subroutine Tnter = interpolation routine for even intervals Calc 3 = calculates related' -function if required 204

FORTRAN PROGRAM FOR H FUNCTION-GAUSS QUADRATURE-ONE PARAMETER CHARACTERI DIMENSION Z(100),G(100), Y(100),X(400),H(101) 1 READ INPUT TAPE 7_2,N,W.ZO.,Z.7,Z8Z9.,D5,KO,K2,K3, K4,K5,K6, K7,K8 02 FORMAT (I5,E17o8/5E9.2/8I5) READ INPUT TAPE 7q39(Z(I),oI1l,-N) _ __ __ READ INPUT TAPE 7,3,(G(I),I=1,N) 03 FORMAT (4E17.8) WRITE OUTPUT TAPE 6,4,N,W 04 FORMAT (l1H1,I5,ElE7, 8) CALL CHARA (Z,G-Y,N-W,SO) N2 N +N N3=N+N2 N4=N+N3 I-......... (<2 -—,3'% "3 IF (K2) 3434,33 33 READ INPUT TAPE 7,25,(X(I),I=1,N) GO TO 12 34 DO 5 I=1,N 05 X(I.)=1. 12 CALL GAUSH (Z,GXYSO,N) D4=X(N)-D6 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ D6=X(N) IF (KO-1) 7,7,6 06 IF (ABSF(D4)-Z9) 8,8,9 08 K5=K5+1 GO TO 7 09 CALL TEST (J1,D2,D3,D4,D5,Z7) IF (J1) 10,10,28 10 D2=D4/D3 D5=D2-D1 07 WRITE OUTPUT TAPE 6'11,K4'D4,D2,D5 WRITE OUT PUT TAPE 6,32,(X(I),I=1, N) 11 FORMAT (IHO,I3,3E17.8) D1=D2 D3=D4 KO=KO+l K4=K4+1 IF (K4-K6) 14,29',29 14 IF (K5-5) 15151,30 15 IF (ABSF(D5)-Z8) 13_12,12 13 K2=-1 DO 16 I=1,N J=I+N 16 X(J)=X(I) 18 IF (K2) 20,22,19 20 I 1=N2 K2=0 GO TO 21 22 I11N.3 K2=1 21 CALL GAUSH (ZGX,YS0,N) 32 FORMAT (1HO,7E17.8/(IH.,7E17.8)).DO 17 I=1,N0 205

17 X(J)=X(I) GO TO 18 19 K3=K3+1 CALL CALC2 (N2,N3,N4,X,D7,ZO) DO 23 I=1,N I2=N+I I3=N2+I 23 X(I.)=X(I2)-(X(I3)-X (I2) )*D7 KO=1 D5=1. Jl=O GO TO 12 28 J1=J1+1 K4=K4+1 IF (Ji -K7) 12,13,13 29 WRITE OUTPUT TAPE 6,26,K4 WRITE OUTPUT TAPE 6,27,(X(I),I=1, N). P'UNCH 25,(X(I),I=1,lN) 25 FORMAT (1P5E14.7) 26 FORMAT (1HOI5) 27 FORMAT ('1HO,5E20.8/(1H, 5E208)) GO TO 1 30 CALL TNTER (Z,G,X,Y,H,N,SO) CALL CALC3(HX) N=101 GO TO 29 206

SUBROUTINE GAUSH (Z,G,X,Y,SO,N) DIMENSION Z(100)G(100),G,X( 100)',Y(100) DO 1 I=1,N S=O. DO 2 J=1,N F=X(J)*Y(J)/(1.+Z( I)/Z(J)) 02 S=S+F*G(J) 1 X(I)=1./(SO+.5*S) RETURN SUBROUTINE TNTER (ZG,X,Y,H,N,S0) DIMENSION Z(100),G ( 1 0) ),H ( 1 1 ),Y(100),X(101) U=O. H(1 )=1. DO 1 I=2,101 U=U+.01 S=0. DO 2 J=1,N F=X(J)*Y(J)/( 1.+U/Z(J)) 02 S=S.+F*G(J) 1 H( I )=1./(SO+. 5*) WRITE OUTPUT TAPE 6,3,(H(I),I=1,101) 03 FORMAT (1HO,5E20.8/(1H,5E20.8)) R-ETURN 207

SPECIAL SUBROUTINES FOR H-FUNCTION 1 TERM SUBROUTINE CALC3 (HX) DIMENSION H(101),X(101) z1=O. DO 1 I=1,101 D=SQRTF( 1.-Z 1*Z 1 ) X( I )=D*H( I) 01 Z1l=Z 1+.01 X(101)=O. RETURN SUBROUTINE CHARA (ZG,Y,NW,SO0) DIMENSION Z(100),G(100'),Y(100) DO 1 I=1,N 1 Y(I)= (1.-Z(I)*Z(I))*W WRITE OUTPUT TAPE 6,3,(Y(I),Iz,N)......^..3..FORMA'T'-TI H'E- -,-TT-IH.... —T'E ~7E T-I 50=0. DO 5 I'=1,N 05 SO=SO+Y(I)*G(I) _ SO=SQRTF( 1.i.-S'O.)-___ __ —_ WRITE OUTPUT TAPE 6,8,SO............... —FOR"MA —- H —E17.8 7 RETURN 208

SPECIAL SUBROUTINES FOR H-FUNCTION 2 TERM SUBROUTINE CALC3 (H X) DIMENSION H( 101 ),Xi( 101) Z1=O. DO 1 I=1,101 X( I )=3.*( 1.-Zl*Z1)*H( I) 01 Zl=Zl+.01 X( 1.01)-=0. RETURN SUBROUTINE CHARA (Z,G,Y,N,W,SO) DIMENSION Z( 00),G(100),Y(100) DO 1 I=1,N 01 Y( I) =W*(1.-Z( I)*Z(I))*( 1.-Z( I )*Z i) ) WRITE OUTPUT TAPE 6,3,(Y(I) I=1,N) 3 FORMAT (1HO,7E17.8/(1H,7E17.8) ) - SO=O. DO 5 I=1 N 05 SO=SO+Y(I)*G(I) SO=SQRTF ( 1.-SO ) WRITE OUTPUT TAPE 6,8,SO0 08 FORMAT (1HO,E17.8) RETURN 209

SPECIAL SUBROUTINES FOR H-FUNCTION'3 TERM SUBROUTINE CALC3 (HX) DIMENSION H(101),X(101) Z1=0O, DO 1 I=1,101 D= SQRTF(1-Zi).....-........-............... X(I)=15.*D*D*D*H(I) X.(101)=0. RETURN.SUBROUTINE CHARA (ZG;YN,-S 0) DIMENSION Z(100),G(100),Y(100) DO 1 I=1,N D=1.-Z( I )*Z(I) 1 Y(I)=W*D*D*D WRITE OUTPUT TAPE 6,3,(Y(I),I=1,N).3 FORMAT-(1HO7E178/(1H, 97E17.8) ) SO=0. DO 5 I.=1,-N......... 05 SO=SO+Y(I)*G(I) SO=SQRTF ( I.-SO) WRITE OUTPUT TAPE 6,8,S0 08 FORMAT (1HOE-178) RETURN 210

SPECIAL SUBROUTINES FOR H-FUNCTION 4 TERM SUBROUTINE CALC3 (HtX) DIMENSION H(101),X(1O1) Z1=0O. DO 1 I=1101O X( I )=105.*( l-Z 1*Z1 )*( 1.-Z*Z1 )*H(I ) 01 Z1=Zl1+.01. X (101) =0. RETURN SUBROUTINE CHARA (Z,G,Y,NW, S0) DIMENSION Z(100),G(100),Y(100) DO 1 I=19N D=1.-Z( I )*Z(I) 1 Y(I)' i=W*D*D*D*D WRITE OUTPUT TAPE 6,3q(Y(I),I=1t N)_ _ _ _ _ _ 3 FORMAT (1H0O,7E178/(1H,7E17.8)) SO=O. DO 5 I=1,N _05 50SO+Y(I)*G(I) SO-SQRTFC1.-SO) WRITE OUTPUT TAPE 68,SO950 08 FORMAT (1HOE17.8). RETURN 211

H-function program - Simpson's rule arbitrary characteristic function The first section calculates the characteristic function. The iteration program runs from statement 2 to 137. What follows 137calculates moments with n arbitrary. The characteristic function is of the form fwo where and 212

FORTRAN PROGRAM FOR H FUNCTION-SIMPSON,S RULE-ARBITRARY CHARACTERISTIC 201 DIMENSION X(404), Y(101), Z(35),D(35) 202 READINPUTTAPE7,203,K1,Jl,Xl,ZO,(Z ( I ), I1,J1) 203 FORMAT (2I5,2E17.8/(E17 8) ) WRITE OUTPUT TAPE 6,155,K1,Jl,X1,ZO,((Z(I),I=l,J1) 155 FORMAT(1Hl,2I5,2E17.8/(4E17.8)) 204 XO.=O 205 DO 224 I1=1,101,K1 206 X 3 =- 1 207 X5-=1. 208 X6=XO*(ZO-1.) 209 (8= 1. 210 X9=XO 211 X4=ZO-Z ( 1 )*XO*XO* (ZO-1 ) 212 X2=0. 213 DO 222 JO=2,J1 214 X2=X2+1. 215 X7=((Z(JO-1)-2.*X2-1. )*XO*X6-X2*X5)/(X2+*.) 216 X10=((2*X2+1. ) *XO*X9-X2*X8 ) / ( X2+1. ) 217 X3=-X3 218 X4=X4+Z (JO)*X7*X 1O*X3 219 X5=X6 220 X6=X7 221 X8 X9 222 X9=X10 223 Y(I1)=X4 224 XO=XO+X1 225 WRITE OUTPUT TAPE 6, 9, (Y(I) I=1,101,K1) IF (ZO-1.) 226,238,238 226 I7=1 227 Cl=Y(1) 228 13=K 1+ 229 I4-101-K1 230 DO 235 I1=I3,I4,K1 231 I7=-I7 232 IF (I7) 236,233,233 233 C1=C1+2.*Y(Il) 234 GOT0235 236 Cl-C1+4.*Y(I1) 235 CO=CO 237 C1=C1+Y (101) CI=SQRTF( 1.-Xl*C1/3. ) GO TO 239 238 Cl=O. 239 WRITE OUTPUT TAPE 6, 240,C1 240 FORMAT (1H,E17.8) 241 J1=0 2 READ INPUT TAPE 7, 3, Z3,ZO,Z7, Z8, Z9,D5, 1 K0.K2,K3K4,K5,K6 K7, K8 3 FORMAT (,6E9.2/81I'5) 4 WRITE OUTPUT TAPE 6,156,Z3,Z0,Z7,Z8,Z9,D5, 1 KO,K2,K3,K4,K5,K6,K7,-K8 156 FORMAT (1HO,6E9.2/1H,8I5) 213

5 IF(K2)11,ll,6 6 READ INPUT TAPE 7,7,(X(I), I=1,101,K8) 7 FORMAT(1P5E14.7) 8 WRITE OUTPUT TAPE 6,9,(X(I),I=1,101,K8) 9 FORMAT(1H0,7E17.8/(1H,7E17.8)) IF (K2-1) 26,26*152 152 DO 153 I1=1,96,5 I2=I 1+5 A1=(X(I2)-X(Ii))/5. I=I2-1 DO 153 I3=I1,I I4=I 3+1 153 X( 14)=X( I3)+A1 WRITE OUTPUT TAPE 6,9,(X(I),I=1,101) GO TO 26 11 DO 12 I=1,101,K1 12 X( I)=1. 26 CALL SIMPO(Z3,X,Y,K1,C1) D4=X( 101 )-D6 D6=X ( 101) 48 IF(KO-1)61*61,49 49 IF(ABSF(D4)-Z9)50,50,52 50 K5=K5+1 51 GO TO 61 52 CALL TEST(J1,D2",D3,D4,D5,Z7) IF (Jl) 59,59,126 59 D2=D4/D3 60 D5=D2-D1 61 WRITE OUTPUT TAPE 6,62,K4,D4,D2,D5 ~62 FORMA T ( 1 HO, I 3, 3 E 1 7 8 ) 63 D1=D2 64 D3=D4 66 KO=KO+1 67 K4=K4+1 138 IF(K4-K6)68,1299129 68 IF ( K5-5 ) 69, 69, 140 69 IF(ABSF(D5)-Z8)70,26,26 70 K2=-1 71 DO. 73 I1=1,101,-K1 72 I4=I1+101 73 X(I4)=X(I1) 74 IF(K2)75,78,103 75 I 1=202 76 K2=0 77 GO TO 80 78 I1=303 79 K2= 1 80 CALL SIMPO(Z3,X,Y,K1,C1) DO 98 I=1,101,-K1 I4= I+I -1 98 X(I4)=X(I) 102 GO TO 74 103 K3=K3+1 105 FORMAT (1HO, 15 ) CALL CALC2(202,303,404,X,D7,ZO) I3=Kl+l 112 DO 116 I5=I3,101,K1 113 I2=I 5+101 214

114 I1=I5+202 115 D8=(X(I1)-X(I2))*D7 116 X(I5 )=X(I2)-D8 120 K0= 1 121 D5=1. 122 J1=0 123 GO TO 26 126 Jl=J1+1 127 K4=K4+1 128 IF(J1-K7)26,70,70 129 WRITE OUTPUT TAPE 6,105,K4 117 WRITE OUTPUT TAPE 6,9,(X(I),I=1,101,K1) PUNCH 136, (X(I), I=1,101,K1) 136 FORMAT (P5E14..7) 137 GOTO 202 140 READ INPUT TAPE 7,141,I1 141 *FORMAT (I5) WRITE OUTPUT TAPE 6,151,I1 151 FORMAT (1H0,I5) Zl=O. Z1=0. KO=-1 D(1 )=1. DO 142 I2=2,Il 142 D(I2)=0. I2=101-K1 I3=Kl+l DO 146 JO=I3,I2,K1 Z1=Zl+Z3 Z2=1. Y(1) =X (,JO) DO. 143 I4=2,I1 Z2=Z2*Z1 143 Y(I4)=Z2*X(JO) IF.(KO) 147,144,144 144 DO'145 I4=1, IL1 145 D(I4)=D( I4)+2.*Y( I4) GO TO 146 147 DO 148 I4=1,I1 148 D(I4)=D(I4).+4.*Y(I 4) 146 KO=-KO DO 149 I4=l,I1 149 D(I4)=(D(I4)+X(101))*Z3/3. -WRITE OUTPUT TAPE 6,150, (D'(I2),I2=2, I1),D('1) 150 FORMAT ('1HO,4E17.8) PUNCH 154,(D(I2),I2=2,I1),D(1) 154 FORMAT (4E17.8) GO TO 129 215

SUBROUTINE SIMPO(Z3,X,Y,K1,C1) DIMENSION X(101.),Y(101) Zl=O. I3=K1+1 DO 1 I5=I3,101,K1 Z1=Z1+Z3 Z2=0. A2=O. I7-17-l DO 2 I8=I3,101,K1 Z2=Z2+Z3 D7=X(I8)*Y(I8)/(1.+Z1/Z2) IF (I8-101) 3,1,3 3 IF (I7) 5,5,6 5 A1=A1+D7 GO TO 2 6 A2=A2+D7 2 I7=-I7 1 X( I5 )=1./(C1+Z3*(4.*A1+2.*A2+D7)/6. ) RETURN 216

SUBROUTINE TEST (J,B1,B2,B3B4,A1) IF (J) 3,3,2 3 IF (ABSF(B2-B3)-A1) 1,1,2 1 "J=1 B1=0. B4=0. B3=0. 2 RETURN SUBROUTINE CALC1 (J,B1,B2,B3,B4,B5) IF(J) 1,1,2 1 B2=B4/B3 B5=B2-B1 2 RETURN SUBROUTINE CAL'C2 (N2, N3,N4,A, D..D1 ) DIMENSION A(404) D=A(N2)-2.*A(N3)+A(N4) IF (D) 1,2,1 1 D=(A(N3)-A(N2 ))/D IF (D) 3,3,2 3 IF (D1+D) 2,4.4 2 D=-D1 4 KEI URN 217

II. Normalized Reflected Intensities. The computations of I/Io as functions of i,+, and 0o for N = 1,2,3,4 is based on equations II-11 through II-14 respectively. The general flow scheme for all four programs are the same: a) Read in computation specifications, IU = ji between available y-functions NUX = number of >o'S for which (I/I )'s are to be computed NU = number of It's for which (I/Io0s are to be computed NP = number of ~'s for which (I/Io)'s are to be computed b) Read in the values of,o Ai, and 4 for which (I/IO)'s are to be computed c) Read in physical constants of the scattering media, Wo, N,.... d) Read in tabular values of k (L) functions e) Compute the reflection intensities according to equations II-11 through II-14 f) Print out results of I/Io for desired's,p's, and do's. It should be noted that the N = 1 case was treated slightly differently than the others. For N = 1, values of H(I) were read in place of G (t) and ~(l). Values for these two f-functions were then computed from the H-function according to relations developed by Chandrasekhar*. * R. T., p. 140. 218

FORTRAN PROGRAM FOR DIFFUSE REFLECTION FOR CASE N=l1 N = 1 PI = 3.14159265 DIMENSION AZZ(101),AZ1 (101),All( 101 ),UX( 101),U( 101 ), IP(181 ), x P(181),BRAD(181),FN (181 ), JUX(101),LU(101) 55 READ INPUT TAPE 7,3,IU,NUX, NU,NP I = 180 /(NP-1) DO 96 M= 1, NP IP(M) = (M-1) * I FIP = IP(M) P (M.) = FIP / 57.295780 96 FN(M) = 2.0 *COS(P(M)) READ INPUT TAPE 7,3, (JUX(I),I=1,NUX) READ INPUT TAPE 7,3, (LU(I),I=1,NU) READ INPUT TAPL 7,4,WZZ,X,'ALPZ,ALP P1 DIMENSION H(101), H1(101) READ INPUT TAPE 7,5, (H(I),I=1,101) WZI1 = X*WZZ Wll = WZ1/2.0 Q= 2.0*(1*0-WZZ)/(20- (WZZ*ALPZ) ) C = W11*ALP1*Q DO 95 I=1,101,IU DOPE = I-1 FMU = DOPE / 100.0 AZZ( I ) H (I)*( 1.0-(C*FMU) ) 95 AZl(I) = H(I)*Q*FMU READ INPUT TAPE 7,5,(A11(I),I=1,101) WRITE' OUTPUT TAPE 6,6,NIU,NUX,NU,.NP WRITE OUTPUT TAPE 6,7,WWZZtWZ1,W11 WRITE OUTPUT TAPE 6,11,(AZZ(I),I=1,101, IU) WRITE OUTPUT TAPE 6,11,(AZ 1 ( I ),I =,101,I U WRITE OUTPUT TAPE 6,11,(A11(I),I=1,101) DO 99 I = 1,NUX J = JUX(I) + 1 FJUX = JUX(I) UX(I) = FJUX / 100.0 WRITE OUTPUT TAPE'6,8,UX(I) DO 98 K = 1, NU L = LU(K) + 1 FLU = LU(K) U(K) = FLU / 1.00.0 WRITE OUTPUT TAPE 6,9,U(K) TERMA = WZZ*AZZ(L)*AZZ(J)-WZ1*AZ1(L)*AZ1(J) TERMB = W11*A11(L)*A11(J) FACTOR = UX(I) /(o..O*Pl*('UX (I)+U(K))) DO 97 M = 1,NP 97 BRAD.(M) = (TERMA+TLRMB*FN (M)) ) FACTOR 98 WR.ITE OUTPUT TAPE 6,10, (IP(M), BRAD(M), M=1,NP) 99 CONTINUE 3 FORMAT (717) 219

4 FORMAT (4'E17.8) 5 FORMAT (1P5E14.7) 6 FORMAT(1H1//42H I2,419) 7 FORMAT (1HO/(32H 3E20.8)) 8 FORMAT(1HO/E67.8) 9 FORMAT(1HO/E18.8) 10 FORMAT(1H I11,E18.8,I11,E18.8,IllE18.8, I11,E18.8) 11 FORMAT (iHO 5E20.8/(1H 5E20.8)) GO TO 55 220

FORTRAN PROGRAM FOR DIFFUSE REFLECTION FOR CASE N=2' N = 2 Pi = 3.14159265 DIMENSION AZZ(101), AZl(101), AZ2(101), A11(lO1), A12(101), X A22(101), UX(101), U(1.01), IP(181)', P(181), BRAD(181), X FNB(181), FNC(181),JUX(lO11),LU(101) 55 READ INPUT TAPE 7,3,.IU,NUX,NU',NP I = 180 /(NP-1) DO 96 M= 1, NP IP(M) = (M-1) * I FIP = IP(M) P(M) = FIP / 57.295780 FNB(M) = 2.0 * COSF(P(M)) 96 FNC(M) = 2.0 * COSF(2.0*P(M)) READ INPUT TAPE 7,3, (JUX(I),I=1,NUX) READ INPUT TAPE 7,3, (LU(I),I=1,NU) READ INPUT TAPE 7,4,WZZ,WZ1,WZ2,Wll,W12,W22 READ INPUT TAPE 7,5, (AZZ(I),I=.l,101,IU) READ INPUT TAPE 7,5, (AZ1'(I),I=1,1Ol,IU) READ INPUT TAPE 7,5., (AZ2(I),I:l,1Oi,IU) READ INPUT TAPE 7,5, (All(I),I=l,lOl,IU) READ INPUT TAPE 7,5, (A12(I),I=1,lO1,IU) READ INPUT TAPE 7,5, (A22(I),I=,lOl1)' WRITE OUTPUT TAPE 6,6,N,IU,NUX,NU,NP WRITE OUTPUT TAPE 6,7,WZZ,WZ1,WZ2,Wll,W12,W22 WRITE OUTPUT TAPE 6,.11,(AZZ(I),I=l,101,IU) WRITE OUTPUT TAPE 6,11,(AZ1(I),I=l,lOl,IU) WRITE OUTPUT TAPE 6,1l,(AZ2(I),I=1,101O,IU) WRITE OUTPUT TAPE 6,11,(Al'l(I),I=l, 0lIU) WRITE OUTPUT TAPE 6,11,(A12(I),I=l,lOl,IU) WRITE OUTPUT TAPE 6,11,(A22(I),I=1,10.1) DO 99 I = 1,NUX J = JUX(I) + 1 FJUX = JUX(I) UX(I) = FJUX / 100.0 WRITE OUTPUT TAPE 6,8,UX(I)'DO 98 K = 1, NU L = LU(K) + 1 FLU = LU(K) U(K) = FLU / 100.0 WRITE OUTPUT TAPE 6,9,U(K) TERMA = WZZ*AZZ(L)*AZZ(J)-NZ1*AZ1(L)*AZI(J)+WZ2*AZ2(L)*AZ2(J). TERMB = W11*A11(L)*A11(J)-W12*A12(L)*Al(J) TERMC = W22*A22(L)*A22(J) FACTOR = UX(I) /(4.0*PI*(UX(I)+U(K))) DO 97 M = 1,NP 97 BRAD(M) = (TERMA + TERMB * FNB(M) + TERMC * FNC(M)) * FACTOR 98 WRITE OUTPUT TAPE 6,10, (IP(M), BRAD(M), M=19NP) 99 CONTINUE 3 FORMAT (717) 221

4 FORMAT (4E17.3) 5 FORt. AT (125E'4.7) 6 FORAT ( 1HI//42H I 2,419) 7 FORMAT(1HO/(39H 2E20.8) ) 8 FORMAT ( 1HO/E67.8 ) 9 FORMAT ( 1HO/E18.8 ) 10 FORMAT( 1H I 11,E18.8, I11,E18.8, I 11,E18.8, I11,E18*8) 11 FORMAT (1HO 5E20.8/(1H 5E20.S)) GO TO 55 222

FORTRAN PROGRAM FOR DIFFUSE REFLECTION FOR CASE N=3 N 3 PI = 3.14159265 DIMENSION AZZ(101), AZ1(101), AZ2(101), A11(101), A12(101), X A22(101), UX(lUl), U(lOl), IP(181),o P(181)., BRAD(181), X FNB(181), FNC(181),JUX(101)*LU(101) DIMENSION AZ3(10'1),A13(101),A23(101),A33(101),FND(181) 55 READ INPUT TAPE 7,3,IU,NUX,NUNP I = 180 /(NP-1) DO 96 M: 1, NP IP(M) = (M-1) * I FIP = IP(M) P(M) = FIP / 57.295780 FNB(M) = 2.0*COS(P(M)) FNC(M) = 2.0*COS(2.0*P(M)) 96 FND(M) = 2.0*COS(3.0*P(M)) READ INPUT TAPE 7,3, (JUX(I),I=1,NUX) READ INPUT TAPE 7,3, (LU(I),I=1;NU) READ INPUT TAP' 7,4,WZZ,WZ1,WZ2,WZ3,Wll,W12,W13,WZ2,W23,W33 READ INPUT TAPE 7,5, (AZZ(I),I=l,101,IU) READ lINPUT TAPE 7,D, (AZ2l(I)tl=,I=lllIU) READ INPUT TAPE 7.,5, (AZ2(I),I=1,101,IU) READ INPUT TAPE 7,5, (AZ3(I), I=1,101, IU) READ INPUT TAPE 7,5, (A11(I),I=1,101,IU) READ INPUT TAPE 7,5, (A12(I), 1-,10iIU) READ INPUT TAPE 7,5,(A13(I),I=1,101,IU) READ INPUT TAPE 7,,(AZZ(I ),I =1,U1,IU) READ INPUT TAPE 7,5,(A23(I),I=,101, IU) READ INPUT TAPE 7,5,(A33(I),I=1,101) WRITE OUTPUT TAPE 6,6,N,IU,NUX,NU,NP WRITE OUTPUT TAPE 6,7,WZZ,WZ1,WZ2,WZ3, W11,W12, W16,W22,W23,W33 WRITE OUTPUT TAPE 6,11,(AZZ(I),I=1,0l1,IU) WRITE OUTPUT TAPE 6,11,(AZ1(I),I=1, 1,IU) WRITE OUTPUT TAPE 6,11,(AZ2UI),I=lol,0,IU) WRITE OUTPUT TAPE 6,11,(AZ3(I),I=1,101O,IU) WRITE OUTPUT TAPE 6,11,(All(I),I=1,101,IU) WRITE OUTPUT TAPE 6,'11,(A12(I),I=,io01,IU) WRITE OUTPUT TAPE 6,11,(A13(I)J,I1,101,IU) WR ITE OUTPUT TAPE 6, I, ( AZ2 (I )=1,110i,IU) WRITE OUTPUT TAPE 6,11,(A23(I),I=1,101,IU) WRITE OUTPUT TAPE 6,11,(A33(I'i).,I=1,101) DO 99 I = 1,NUX J = JUX( ) + 1 FJUX = JUX(I) UX(I) = FJUX / 1UU-.O WRITE OUTPUT TAPE 6,8,UX(I) DO 98 K = 1, NU L = LU(K) + 1 FLU = LU(K) U(K) = FLU / 100.0 223

WRITE OUTPUT TAPE 6,9,U(K) TERMA = WZZ*AZZ(L)*AZZ(J )-WZ1*AZ1(L)*AZ1(J)+WZ2*AZ2(L)*AZ2(J)X WZ3*AZ3(L)*AZ3(J) TERMB = W11*A11(L)*A11(J)-W12*Al2(L)*A12(J)+W13*A13(L)*A13(J) TERMC = W22*A22 ( L)*A22 (J )-W23*A23( L )*A23 (J ) TERMD = W33*A33 (L)*A33(J) FACTOR = UX(I) /(4.0*PI*(UX(I)+U(K))) DO 97 M = 1,NP 97 BRAD (M) = ( TERMA+TERMB*FNB (M) +TE.RMC*FNC ( M ) +TERMD*FND ( M) ) *FACTOR 98 WRITE OUTPUT TAPE 6,10, (IP(M), BRAD(M), M=1,NP) 99 CONT INUE 3 FORMAT (717) 4 FORMAT (4E17.8) 5 FORMAT (1P5E14.7) 6 FORMAT( 1H//42H I2,4I9) 7 FORMAT(1HO/(39H 2E20.8)) 8 FORMAT(1HO/E67.8) 9 FORMAT( 1HO/E18.8 ) 10 FORMAT ( 1H Ill,E18.8, I ll,E18.8,Ill,E188, I 11,El8. 8) 1.1 FORMAT (1HO 5E20.8/(lH 5E20.8)') G.O TO 55 224

FORTRAN PROGRAM FOR DIFFUSE REFLECTION FOR CASE N=4 N =4 PI - 3.14159265 DIMENSION AZZ(101), AZ1(101), AZ2(101'), All(101), A12(101), X A22(101), UX(101), U(10 1), IP(181), P(181), BRAD(181), X FNB(181), FNC(181),JUX(101),LU(101) DIMENSION AZ3(101),A13(101),A23(101),A33(101)',FND(181) DIMENSION AZ4(101),A14(101),A24(101),A34(101),A44(101),FNE(181) 55 READ INPUT TAPE 7,3,IU,NUX,NU,NP I = 180 /(NP-1) DO 96 M= 1, NP IP(M) = (M-1) * I FIP = IP(M) P(M) = FIP / 57.295780 FNB(M) = 2.0*COS(P(M)) FNC(M) = 2.0*COS(2.O*P(M)) FND(M) = 2.0*COS(3.0*P(M)) 96 FNE(M) = 2.0*COS(4.0*P(M)) READ INPUT TAPE 7,3, (JUX(I),I=1,NUX) READ INPUT TAPE 7,3, (LU(I),I=1,NU) READ INPUT TAPE 7,4, WZZ,WzlWZZ,WZ3,WZ4,W1,W12,W'13,W14,W22, X W23,W24,W33,W34,W44 READ INPUT TAPE 7,5, (AZZ(I),I=1,lO,101IU) READ INPUT TAPE 7,5, (AZ1(I),I=1,O1,IU) READ INPUT TAPE 7,5, (AZ2(I),I=1,10,IU) READ INPUT TAPE 7,5,(AZ3(I),I=1,101,IU) READ INPUT TAPE 7,5,(AZ4(I),I=l,1Ol,IU) READ INPUT TAPE 7,5, (All(I),I=I,O11,IU) READ INPUT TAPE 7,5, (A12(I),I1,101,,IU) READ INPUT TAPE 7,5,(A13(I),I=1,101IU) READ INPUT TAPE 7,5,(A14(I),I=1,101,IU) READ INPUT TAPE 7,5,(A22(I),I=1,101,IU) READ INPUT TAPE 7,5,(A23(I),I=1,101, IU) READ INPUT TAPE 7,5,(A24(I)','I=1,101,IU) READ INPUT TAPE 7,5,(A33(I),I=1,101,.IU) READ INPUT TAPE 7,5,(A34(I),I=1,101,IU.) READ INPUT TAPE 7,5,(A44(I).,I=1,101) WRITE OUTPUT TAPE 6,6,N,IU,NUX,NU,NP WRITE OUTPUT TAPE 6,7,WZZWZ1,WZ2,WZ3,WZ4,W11,W12,W13,Wl4,W22, X W23,W24,W33,W34,W44 WRITE OUTPUT TAPE 6,11,(AZZ(I),I=1,101,IU) WRITE OUTPUT TAPE 6,11,(AZ,1(I),I=1,101,IU) WRITE OUTPUT TAPE 6,.11,(AZ2(I),I=11lO1,IU) WRITE OUTPUT TAPE 6,11,(AZ3(I),I=1,1OL.,IU) WRITE OUTPUT TAPE 6,11,(AZ4(I),I=1,10'1,IU) WRITE OUTPUT TAPE 6,11,(Al(I),I=l,l101,IU) WRITE OUTPUT TAPE 6,11,(A12(I),I=l,101,IU) WRITE OUTPUT TAPE 6,11,(A13(I),I=1,10l,IU) WRITE OUTPUT TAPE 6,11,(A14(I),I=1,1Ol,IU) WRITE OUTPUT TAPE 6,11,(A22(I),I=1,101,IU) 225

WRITE OUTPUT TAPE 6,11',(A23(I),I=1 101,IU) WRITE OUTPUT TAPE 6,11,(A24(I)',I=1,101,IU) WRITE OUTPUT TAPE 6,11,(A33(I),I=1,lO1,IU) WRITE OUTPUT TAPE 6,11,(A34(IC),I=1,101, IU) WK I TE OUTPUT TAPE b, 1, ( A44 ( 1), =l,101 ) DO 99 I = 1,NUX J = JUX(I) + 1 FJUX = JUX(I) UX(I) = FJUX / 100.0 WRITE OUTPUT TAPE 6,8,UX(I) DO 98 K = 1, NU L = LU(K) + 1 FLU = LU(K) U(K) = FLU / 100.0 WRITE OUTPUT TAPE 6,9,U(K) TERMA = WZZ*AZZ ( L ) *AZZ ( J )-WZ1*AZ1 ( L)*AZ1 (J )+WZ2*AZ2 ( L) *AZ2 (J) X -WZ3*AZ3(L)*AZ3J )+WZ4*AZ4 L).*AZ4 (J) TERMB = Wll*All( L)*All (J)-W12*A12(L)*A12(J)+W13*A13(L)*A13( J)X W14*A14(L)*A14 (J ) TERMC = W22*A22(L)*A22(J)-W23*A23(L)*A23(J)+W24*A24(L)*A24(J) TERMD = W33*A3 3 ( L)*A33(J)-W34*A34(L)*A34 ( J ) TERME = W44*A44(L)*A44(J) FACTOR = UX(I) /(4.0*PI*(UX(I)+U(K))) DO 97 M = 1,NP 97 BRAD(M)= (TERMA+TERMB*FNB ( M ) +TERMC*FNC ( M) +TERMD*FND M ) + X TERME*FNE(M)) * FACTOR 98 WRITE OUTPUT TAPE 6,10, ( IP(M), BRAD(M), M=1,NP) 99 CONTINUE 3 FORMAT (7I7) 4 FORMAT (4E17.8) 5 FORMAT (1P5E14.7) 6 FORMAT( 1H1///42H 12,4I9) 7 FORMAT (C1HO/(32H 3E20.8)) 8 FORMAT(1HO/E67.8) 9 FORMAT (HO/E18.8)10 FORMAT (1H I i, E18.8, I 11,E18.8, I 11,E18.8, I1, E18.8) 11 FORMAT (1HO 5E20.8/(1H 5E20.-8)) GO TO 55 226

REFERENCES 1. S. W. Churchill, J. H. Chin, G. C. Clark, B. K. Larkin and J. A. Leacock "The Transmission of Thermal Radiation through Real Atmospheres" - Rept. No. AFSWP-1035, ASTIA No. The University of Michigan, April 1957. 2. C. M. Chu and S. W. Churchill, "Multiple Scattering by Randomly Distributed Obstacles - Methods of Solution", IRE Trans. on Antennas and Propagation, AP-4, 142 (1956). 3. Symposium on "Atmospheric Transmission of Thermal Radiation" held at AFSWP Headquarters, Washington, D. C., April 16, 1957. 4. S. Chandrasekhar, "Radiative Transfer", Oxford at the Clarendon Press, 1950. 5. K. M. Case, F. de Hoffman and G. Placzek, "Introduction to the Theory of Neutron Diffusion", Vol. I, U. S. Government Printing Office, Washington 25, D. C. 6. R. Bellman and R. Kalaba, Proc. Nat. Acad. Sci. U.S.A., 42, 629 (1956). 7. V. Kourganoff, "Basic Methods in Transfer Problems", Oxford Press, London (1952). 8. S. Ueno, J. Math. and Mech. 7, 628 (1958). 9. D. R. Hartree, Numerical Analysis Oxford, Clarendon Press, 1958, p. 30. 10. C. M. Chu and S. W. Churchill, Representation of the Angular Distribution of Radiation Scattered by a Spherical Particle, J. Opt. Soc. Am. 45, 958 (1955). 227

DISTRIBUTION LIST Addressee Army No. of Cys. Deputy Chief of Staff for Military Operations, DA, Washington 25, D. C. 1 ATTN: Director of SW and R Chief of Research and Development, DA, Washington 25, D.C., ATTN: 1 Atomics Division Chief of Ordnance, DA, Washington 25, D.C., ATTN: ORDTN 1 Chief Signal Officer, DA, Combat Dev. and Ops. Div., Washington 25, D.C. ATTN: SIGCO-4 The Surgeon General, DA, Washington 25, D.C. ATTN: MEDNE 1 Chief Chemical Officer, DA, Washington 25, D.C. 2 Chief of Engineers, DA, Washington 25, D.C. ATTN: ENGNB 3 The Quartermaster General, DA, Washington 25, D.C. ATTN: R and D Division 1 Chief of Transportation, DA, Washington 25, D.C. ATTN: Military Planning 1 and Intelligence Division Commanding General, U.S. Continental Army Command, Ft.Monroe, Va. 4 President, U.S. Army Artillery Board, Ft. Sill, Oklahoma 1 President, U.S. Army Infantry Board; Ft. Benning, Georgia 1 President, U.S. Army Air Defense Board, Ft. Bliss, Texas 1 Commandant, Command and General Staff College, Ft. Leavenworth, Kansas, 1 ATTN: Archives Commanding General, Army Medical Service School, Brooke Army Medical 1 Center, Ft. Sam Houston, Texas Director, Special Weapons Development, Hq CONARC, Ft. Bliss, Texas 1 ATTN: Capt Chester I. Peterson Commandant, Walter Reed Army Institute of Research, Walter Reed Army 1 Medical Center, Washington 25, D. C. ATTN: Dept of Biophysics Commandant, Chemical Corps School, Chemical Corps Training Command, 1 Ft. McClellan, Alabama Commanding General, U.S. Army Chemical Corps, Research and Development 2 Command, Washington 25, D. C. Commanding General, The Engineer Center, Ft. Belvoir, Virginia 3 ATTN: Ass't Commandant, Engineer School 228

Addressee Army No. of Cys. Commanding General, Aberdeen Proving Ground, Aberdeen Proving Ground, 2 Md. ATTN: Director, Ballistics Research Laboratory Commanding Officer, Engineer Research and Development Laboratory, Ft. 1 Belvoir, Va., ATTN: Chief, Tech Support Branch Commanding Officer, Picatinny Arsenal, Dover, N.J. ATTN: ORDBB-TK 1 Commanding Officer, Frankford Arsenal, Bridge and Tacony St. Philadelphia 1 Pa. Commanding Officer, Army Medical Research Laboratory, Ft. Knox, Ky. 1 Commanding Officer, Chemical Warefare Lab, Army Chemical Center, Md. 2 ATTN: Tech Library Commanding Officer, Transportation Research Command, Ft.Eustis, Va, ATTN: 1 Tech Info Div. Commanding Officer, U.S. Army Signal R and D Lab, Ft. Monmouth,NJ. ATTN: 1 Technical Documents, Evans Area Director, Waterways Experiment Station, P.O.Box 631, Vicksburg, Miss. 1 ATTN: Library Director, Operations Research Office, The Johns Hopkins University, 1 6935 Arlington Rd., Bethesda 14, Maryland Commanding General, Quartermaster R and D Command, Quartermaster R and D 2 Center Natick, Mass. ATTN: CBR Liaison Officer Navy Chief of Naval Operations, DN, Washington 25, D.C. ATTN: OP-75 2 Chief of Naval Operations, DN, Washington 25, D.C. ATTN: OP-03EG 1 Chief, Bureau of Medicine and Surgery, DN, Washington 25, D.C. ATTN: 1 Special Weapons Defense Division Chief, Bureau of Naval Weapons, DN, Washington 25, D.C. 3 Chief, Bureau of Ships, DN, Washington 25, D.C. ATTN: Code 423 2 Chief, Bureau of Supplies and Accounts, DN, Washington 25, D.C. 1 Chief, Bureau of Yards and Docks, DN, Washington 25, D.C. ATTN: D-440 1 Chief of Naval Research, DN, Washington 25, D.C. ATTN: Code 811 1 229

Addressee Navy No. of Cys. Commander-in-Chief, U.S. Pacific Fleet, FPO, San Francisco, Calif. 1 Commander-in-Chief, U.S. Atlantic Fleet, U.S. Naval Base, Norfolk 11, Va. 1 Commandant of the Marine Corps, DN, Washington 25, D.C. ATTN: Code AO3H 4 President, U.S. Naval War College, Newport, R.I. 1 Superintendent, U.S. Naval Postgraduate School, Monterey, California 1 Commanding Officer, U.S. Naval Schools Command, U.S. Naval Station, 2 Treasure Island, San Francisco, Calif. Commanding Officer, Nuclear Weapons Training Center, Atlantic, Naval 2 Base, Norfolk 11, Va., ATTN: Nuclear Warefare Dept. Commanding Officer, Nuclear Weapons Tng Center, Pacific, Naval Station, 2 North Island, San Diego 35, Calif. Commanding Officer, Air Development Squadron 5, VX-5, China Iake, Calif. 1 Commanding Officer, U.S. Naval Damage Control Training Center, Naval Base, 1 Philadelphia 12, Pa. ATTN: ABC Defense Course Commander, U.S. Naval Ordnance Laboratory, White Oak, Silver Spring, Md. ATTN: EE Division 1 R Division 1 Commander, U.S. Naval Ordnance Test Station, China Lake, Calif. 1 Commanding Officer, U.S. Naval Medical Research Institute, National 1 Naval Medical Center, Bethesda 14, Md. Commanding Officer, U.S. Naval Air Development Center, Johnsville, Pa. 1 ATTN: Dr. J.D. Hardy, Aviation Acceleration Lab. Director, U.S. Naval Research Laboratory, Washington 25, D.C. 1 ATTN:; Code 2029 Commander, New York Naval Shipyard, Brooklyn 1, N.Y., ATTN: Director, 1 The Material Laboratory Commanding Officer and Director, U.S. Naval Electronics Lab, San Diego, Cal. 1 Commanding Officer and Director, U.S. Naval Radiological Defense Lab, San 3 Francisco, 24, Calif. ATTN: Tech Info Div Commanding Officer, U.S. Naval Development Center, Johnsville, Pa. 1 Officer-in-charge, U.S. Naval Supply Research and Development Facility, 1 Naval Supply Center, Bayonne, N.J. Commanding Officer, Naval Medical Fld Research Lab, Camp Lejune, N.C. 1 230

Addressee Navy No. of Cys. Commander-in-Chief, Pacific, Fleet Post Office, San Francisco, Calif. 1 Air Force Assistant for Atomic Energy, Hq USAF, Washington 25, D.C. 1 Deputy Chief of Staff, Operations, Hq USAF, Washington 25, D.C. ATTN: 1 Operations Analysis Deputy Chief of Staff, Plans and Programs, Hq USAF, Washington 25, D.C. 1 ATTN: War Plans Division Director of Research and Development, DCS/D, Hq USAF, Washington 25, D.C. 1 ATTN: Guidance and Weapons Division Air Force Intelligence Center, Hq USAF, ACS/I CAFCIN-3Vl) Washington 25, D.C. 2 The Surgeon General, Hq USAF, Washington 25, D.C. ATTN: Bio Defense 1 Branch, Prev Med Division Commander-in-Chief, Strategic Air Command, OFFutt AFB, Neb. ATTN: OAWS 1 Commander, Tactical Air Command, Langley AFB, Va. ATTN: Doc Sec Branch Ass 1 Commander, Air Defense Command, Ent AFB, Colorado, ATTN: Assistant 1 for Atomic Energy, ADLDC Commander, Air Material Command, Wright-Patterson AFB, Ohio 2 Commander, Air Research and Development Command, Andrews AFB, Washington 3 25, D.C. ATTN: RDRWA Commander, Air Proving Ground Command, Eglin AFB, Fla, ATTN: PGTRIL 1 Director, Air University Library, Maxwell AFB, Alabama 2 Commandant School of Aviation Medicine, USAF Aerospace Medical Center 2 (ATC) Brooks AFB, Tex. ATTN: Col Gerrith L. Hekhnis Commander, Wright Air Development Center, Wright-Patterson AFB,Ohio 1 ATTN: WCOSI Commander, AF Cambridge Research Center, L.G. Hanscom Fld, Bedford, Mass. 1 ATTN: CRQST-2 Commander, AF Special Weapons Center, Kirtland AFB, N.M. ATTN: Tech Info 3 Office Commander, 3415th Tech Training Wing, Lowry AFB, Colorado, ATTN: Dept of 1 Weapons Training 231

Other DOD Activities Addressee No. of cys. Director, Weapons Systems Evaluation Group, OSD, Room 1E880, The 1 Pentagon, Washington 25, D.C. U. S. Documents Officer, Office of the United States National Military 1 Representative - SHAPE, APO 55, New York, N.Y. Director of Defense Research and Engineering, Washington 25, D.C., 1 ATTN: Tech Library Commandant, Armed Forces Staff College, Norfolk 11, Va. ATTN: Library 1 Commander, Field Command, DASA, Sandia Base, Albuquerque, New Mexico 21 Chief, Defense Atomic Support Agency, Washington 25, D.C. 8 Commander, ASTIA, Arlington Hall Station, Arlington 12, Va. ATTN: TIPDR 15 Others Dr. Hermann E. Pearse, University of Rochester, Atomic Energy Project 1 P. O. Box 287, Station 3, Rochester 20, New York ATTN: Tech Report Control Unit Sandia Corporation, Sandia Base, Albuquerque, New Mexico, ATTN: Classified 1 Document Division Los Alamos Scientific Lab, P. O. Box 1663, Loa Alamos, N.M. ATTN: Report 1 Librarian Medical College of Virginia, 12th and Broad Sts, Box 222, Richmond 19, Va. 1 ATTN: Security Officer (For Dr. William T. Ham) Director, Lincoln Lab, Massachusetts Institute of Technology, P.O. Box 73, 1 Lexington 73, Mass, ATTN: Publications (For Prof. G.C.Williams) Los Alamos Scientific lab, P.O.Box 1663, Los Alamos, N.M. ATTN: Report 1 Librarian (For Dr.Alvin C. Graves) Prof. Hoyt C. Hottel, Lincoln Lab,,Massachusetts Institute of Technology 1 P.O. Box 73, Lexington 73, Mass. 232