ENGN UMR1290 THE UfiPV'ERSF TY OF MICHIGAN ENGINEERING LiBRARY -- MOTION OF' A CHARGED PARTICUE IN A SPATIAlLY PERIODIC MAGNETIC FIELD - Timothy P. Coffey

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Motion of a Charged Particle in a Spatially Periodic Magnetic Field* Timothy P. Coffey+ Dept.. of Physics, University of Michigan Degenerate perturbation theory is employed to discuss the motion of a charged particle in a consbtan-t magnetic field on which is super". imposed a weak, transverse, ispatiallyi periodic magnetic field. A first order solution of, the equations of motion is presented. It is shown that the secular motion is -periodici.1in time. The significance of this result with respect to. the, "(tability of protons in the inner Van Allen belt is discussed. 1. INTRODUCTION In the preceding paperI (henceforth cited as I) we have presented a new formulation of classical perturbation theory. There we illustrated the non-degenerate form of this 1

theory by discussing the van der Pol equation. The- van derPol equation has, of course, been adequately discussed by many authors using a variety of techniques. In the present paper, however, we shall discuss a problem which has not been adequately treated in previous publications. Here we shall employ degenerate perturbation theory to discuss the interaction between a charged particle and a constant magnetic field on which is superimposed a weak, transverse, spatially periodic magnetic field. This interaction has played an important role in recent discussions of the stability of protons in the inner Van Allen belt. For example, Dragt2 and Wentzel3 have argued that a resonant interaction between the charged particle and a periodic magnetic field would cause a breakdown of the adiabatic invariance of the particle's orbital magnetic moment. They further argue that such a breakdown of the adiabatic invariance of the magnetic moment would destroy the magnetic trapping effect. This reasoning has led them to assert that a periodic disturbance (produced, for example, by a hydromagnetic wave) on the geomagnetic field is responsible for the removal of protons, which would otherwise be trapped, from the inner Van Allen belt. Of crucial importance to their assertions is the assumption that a weak periodic disturbance can cause a large change in the orbital magnetic moment. 2

In what, follows, we shall obtain a complete first order solution of the equations of motion in the case where the periodic field is a sinusoid. We shall find that the secular changes produced by such a field are of bounded variation. In particular, the "average" magnetic moment is a periodic function of time. The relative fluctuation in the "average" magnetic moment depends upon the ratio of the particle's cyclotron radius to the wavelength of the periodic disturbance: The fluctuation is large when the ratio is small and small when the ratio is large. The essential point is that a resonant interaction between the particle and the periodic field is not sufficient to cause large changes in the magnetic moment. This is contrary to the assumptions of Dragt and Wentzel. Aside from its application to the question of stability of protons in the inner Van Allen belt, the example which we shall discuss is an interesting mathematical exercise. It illustrates quite nicely many of the phenomena which are characteristic of non-linear oscillatory systems. For example, the ideas of secular growth, stability and instability, and synchronous and non-synchronous behavior arise in a very natural way. The example also illustrates that a non-linear resonance is considerably more complicated than a linear resonance. Our program is as follows: In section 2. we derive; Hamilton's equations of motion which describe the 3

interaction between the particle and the field. In section 3. we introduce the appropriate perturbation theory and obtain the differential equations. which describe the secular motion. In section 4. we perform a phase plane analysis in order to characterize the secular motion. In section 5. we obtain an explicit solution of the differential equations which describe the secular motion. In section 6. we discuss the behavior of the secular motion under resonance conditions. The final section summarizes the main conclusions of the paper. 2. THE EQUATIONS OF MOTION In the Cartesian reference frame x, y, z the magnetic field is taken to have the form B= [B1sin kz, 0, Bj. (2.1) This field can be described by the vector potential A -BoY, (Bl/k)cos kz, o]. (2.2) The non-relativistic Hamiltonian H which describes- the system is 2 H = (1/2m){p- (e/c)A, (2.3) = (1/2m){ x + mOy] + py - (mwl/k)cos kz + (ihm~C~~2 E! Z 4

where 0 = eBo/me, (1 = eBl/mc (2.4) In order to prepare the system for perturbation theory we introduce the new canonical momenta J, pr, PZ and their conjugate coordinates n, r, and Z as follows: x = r - (2J/mo )l/2cos, P = P (2.5) Y = -(l/m()o)pr + (2J/m(o)l/2sin'p = (2mwoJ)l/2cos 4, z = Z, P= PZ The quantities r, p/mwo, and Z are the Cartesian coordinates of the guiding center. In the unperturbed state the particle gyrates about this center with angular velocity m in a circle of radius (2J/mo )1/2. We shall measure the 0 0 time in units of the rotation period. To do this we introduce a new independent variable t = moto (2.6) It is straightforward to show that the Hamiltonian h which is appropriate to the new variables is 5

h = J + (1/2m)P2 - (/k)(2mJ)/2co cos kZ + (e 2mo/2k2)cos2 kZ, (2.7) where E = (l/%o = 31/Bo. (2.8) Hamilton's equations of motion are found from Eq.(2.7) to be J = -(C/2k)(2mo0J)l/2 sin(p*+ kZ) + sin('- kZ) > (2.9a) P' = -(c/2)(2mJ )1/2 s in (+ kZ) - sin (- kZ ) (2.9b) + ( 2mOo/k)cos kZ sin kZ, kZ' = kPz/m0o, (2.9c) =1- (e/2k)(mc%0/2J)l/2[cos(*.+ kZ) + cos(4- kZ2),(2.9d) where, for example, J dJ/dt. (2.10) The system of differential equations (2.9) is in the standard form to which the perturbation theory of I is

applicable. The parameter of smallness is e = B1/Bo. We see from equations (2.9) that the sum angle "-I+ kZ contributes only small amplitude, rapid fluctuations to the motion. However, the difference angle4/4- kZ can give rise to secular motion when'I - kZ' = O(~). The system (2.9) must, therefore, be treated by degenerate perturbation theory. We shall carry out this treatment in the next section. 3. PERTURBATION THEORY In this section we shall perform first order perturbation theory according to the formalism presented in I. Our object is to separate the rapidly fluctuating motion from the secular motion. In order to do this we introduce new variables U, V, K, and 0 as follows: kZ = kU + CD1(U,V,K,$) + 0(e ), (3.la) P V + eE1(U,V,K,0) + o0(2), (3.b) J = K + eF1(U,V,K,$) + 0(es2 (3.lc) lr= 0 + eGC(U,V,K,0) + 0(e2), (3.ld) where D1, El, F1, and G1 are required to be periodic functions of 0 and of kU with period 21T. The variables U, V,K, and $ are to contain the secular motion and D1, E1, F1,

and G1 are to contain the rapidly fluctuating motion. In order to guarantee that U,V,K, and / represent the secular motion we require that kU' = (kV/mwo) + Eal(U,V,K,$) + 0(E2),~ (352a) V' = Ebl(U,V,K,$) + 0(E2), (3.2b) K' = EA1(U,V,K,0) + O(E2), (3.2c) 0' = 1 + EB1(U,V,K,$) + 0(E2). (3.2d) The functions al, bl, Al, and B1 are to contain only those combinations of U and 0 which can give rise to secular motion. The precise manner in which this choice is made is fully described in I. If we substitute the ansatz (3.1) and (3.2) into equations (2.9) and retain only terms through first order in E, then we obtain the following set of equations: Al + 8 F1 = -(1/2k)(2mw0K)1/2 sin($ + kU) (3.3a) + sin(- kU), bl + 0 E1 =-(1/2)(2mtwK)l/2 K sin( + kU) (3,3b) - s.in( - kU), 8

al + O D1 = (k/mo0)El, (3.3c) B1 + 0 G1 = -(1/2k)(mxo/2K)l/2 cos($ + kU) (3.3d) + cos($ - kU)1, where the operator o = (kV/mwo)a /akU + a /a$. (3.4) The difference angle 9 = - kU can give rise to secular behavior when it is s-lowly varying. In order that U, V, K, and $ shall contain all of the secular motion we muat absorb the 9 dependence into the functions Al, al, bl, and B1. We, therefore, choose A1 -(1/2k) (2moK) l/2 in 0, ( 3. 5a) B1= -(1/2k)(mw /2K) 1/2cos 9, (3.5b) al- O. (3,50) a1 = O, (3.5c) b = (1/2) (2moK) 1/2 sin e. (3.5d) With this choice of A1, B1, al, and bl we find that

D1 = (k/2 2)(2K/mw o) /2sin(0 + kU), (3.6a) E1 = (1/2)(2moK)l/2cos($ + kU), (3.6b) F = 1/2 (3.6c)ooK/ F1 = (1/2k2) (2moK)/2cos( + kco), (3.6c) G1 = -(1/2k,2)(mo/2K)1/2sin($ + kU), (3.6d) where <J2 = 1 + (kV/mo). (3.7) When our choice for A1, BI, al, and bl is substituted into equations (3.2) we find that K' = -(c/2k)(2mo K) /2sin $, (3.8a) = (E/2)(2mwoK)1/2sin 0, (3.8b) @' = 1 - (kV/mco) - (E/2k)(mo /2K) 1/2cos G, (3.8c) where 0' = $' - kU'. It follows from equations (3.8a,b) that K + (V/k) = I, (3.9) where I is a constant. The system of equations (3.8), 10

therefore, reduces. to two equations relating the two variables K and Q. In the next section we shall use these equations to obtain some general information concerning K and O without producing an explicit solution. 4. PHASE PLANE ANALYSISWe begin this section by introducing the new variables a and b which are defined by the equations. a = R cos a, b = R sin 0, (4.1) where R = k(2K/mwo)l/2 (4.2) The quantity R measures the ratio of the cyclotron radius to the fundamental period of the disturbance. The equations of motion for a and b are found from equations. (3.8) to be a' = -(1 - C + R2/2)b, (4.3a) b' = (1 - C + R2/2)a - E/2, (4.3b) where C = k2I/mw (4.c) is a constant. These equations give rise to the differential form l - C + (R2/2)bdb +f[1 - C + (R2/2)a - (E/2)3da = O. (4.4) This is an exact differential whose integral is

R4 + 4(1 - C)R2 - 4ea = M, (4.5) where M is a constant. Eq. (4.5) expresses the conservas tion of energy through first order. It follows from the Hamiltonian (2.7) that, to first order in E~ M = (8k2h/mwo) - 4C2. (4.6) The important aspects of the motion can be illustrated. by plotting Eq. (4.5) in the a-b plane. Before doing this it is useful to examine the points where a' and b' are simultaneously zero. These are the points of equilibrium and are usually termed singular points. It follows from equations (4.3) that the singular points are to be found from the equations b = 0, (4.7a) a3 + 2(1 - C)a - s = 0. (4.7b) If e is sufficiently small the cubic equation; will have three real roots; we shall assume this to be the case. These roots, which we shall call al, a2, a3, are approximately as follows: a1 = 2 -2(1 - C)/3 / (3/4)1/2 - /6] + O( 0, (4.8a) 12

a2 =2 L-2(1 - C)/31 - L(3/4/1/2 _ 6/6] + o(2)<o, (4.8b) a3 = 2 2(1 - C)/ ( ) + (O( )<o, (4.8c) where = -(e/2) [-2(1 - C)/3]. (4.9) The nature of these singular pointsi can be determined by examining the behavior of the motion in their vicinity. In order to do thias we let a = ai + 9 b=91, (4.10) where ai is one of the singular points and S and 87 represent small displacements from this s-ingular point. Upon substituting equations (4.10) into equations. (4.3) andretaining terms: through first order in ~ and? we find that g = -1 - C + (a2/2)]7, (4.11a) "P= [1 - + (3ai/2). (4.11b) We aeek solutions in the form S =;oea I ='h - Pe. (4.12) 13

These solutions are valid if A = + -i/ - C + (a/2)[ - + (3a/2}. (4.13) When the values of ai as expressed by equations (4.8) are substituted into equations (4.13) we find that we. can classify the singular points as to their stability. This classification is given in Table 1. Singular Points Nature of Singular Point al complex Center (stable) a2 real Saddle point (unstable) a3 complex Center (s table) Table I. Classification of singular points Table I. shows that the trajectories must close about the point a3 and they must also close about the point a1. Furthermore, it follows from Eq. (4.5) that, for large values of R, the trajectories have the form R4 = constant (4.14) and are, therefore, circles centered about the origin. The; manner in which these requirements are satisfied is shown in Figure 1. There several trajectories are plotted for a particular physical situation. The parameters such as the 14

Figure 1. Trajectories of a 50 Mev proton in the a-b plane. k = 6.28x10-8 cm-1, o = 370 rad/sec, s =.01. 15

energy, background field strength, etc., have been assigned values which are appropriate to a proton which is moving in the inner Van Allen belt at a distance of two earth radii. However,the value of ~ which was used in Figure 1. was chosen to be about ten times larger than what- one would expect at two earth radii.4 This larger value of s was used to facilitate the plotting of the trajectories. The. trajectories in Figure 1. consist of a family of closed curves. This means that K, and consequently V, is a periodic function of time; we shall obtain the period in the next section. the trajectories can be divided into two groups: those which are centered about the point a3 and those which are centered about the point al1.'he motion correspond. ing to the first group is non-synchronous since the difference angle, 4 increases without bound. The motion corresponding to the second group isa synchronous since, for these trajectories, the angle @ oscillates between well defined limits. The- synchronous and non-synchronous regions are separated by the trajectory, called a separatrix, which passes through the unstable point a2. The largest fluctuations in Koccuron trajectories which pass close to the separatrix. As one moves away from the separatrix into the non-synchronous regions the trajectories rapidly become circles centered about the point a3. As: one moves away from the separatrix into the synchronous region the fluctuations in K and O become smaller until, at the point al, they vanish. The separatrix, therefore, determines the range of values of K and e for which maximum 16

resonance occurs. In the following sections we shall obtain the time dependence of K and we shall estimate the total fluctuation in K under resonance conditions. 5. TIlVE DEPENDENCE OF THE MOTION In order to find the explicit time dependence of K and an expression for the period T we introduce a new variable S = R2= (2k2/m%)K. (5.1) It follows from equations (3.8) that S' = -Rsain. (5.2) The right hand side of Eq. (5.2) can be expressed as a function of S alone by making use of equations (4.5) and (4.6). A straightforward calculation gives S' = -(1/4){-S4 - 8IS3 - (8N + 16L2)S2 (5.3;) - 16(2IN -2)S - 16N2] 1/2 where L = 1 - C, = 2 2Ch/I. (5.4) If we denote the roots of the quartic in the square bracket in (5.3) 17

by S1, S2, S3, and S4, then Eq. (5.3) becomes 1/2 -= -(1/4)[(s - s)(S - S2)(S - s3)( - )/25 This is a first order differential equation for S(T) which is solvable in terms of elliptic functions. The solutions depend upon the nature of the roots S1, S2, S3, and S4; we must distinguish the case of two real roots from the case of four real roots. Case of two Real Roots It should be clear frHm Figure 1. that two real roots corresponds to motion in the synchronous region. We shall arrange the real roots S1 and S2 such that S1 S2. The complex roots S3 and S4 may be written as S3 m~ in, S4 m - in (5.6) In this case Eq. (5.5) has the solution5 S1B + S2A + (S2A- S1B)cnL t -to)/4g] (57) S( ) -.., A + B + (A - B))c n [(- )/4'g where A =(S1-m) +n, B2 = (S2 -m) + n, (5.8) 18

and g = (AB)-1/2 (5.9) The function cn(x) is a Jacobi elliptic function. The modulus e of the elliptic function is given by IC2 (S1 - 2)2 - (A - B)2 (5.10) = ~...........~....... (5.10) 4AB The constant Xt is chosen to satisfy the initial conditions. The function cn(x) is periodic in x with period 4K, K being the complete elliptic integral of the first kind. It follows, that S(V), and hence K(z), is a periodic function of Z with period T given by -1/2 T = 16/1(AB), (5.11) where the modulus ic of the complete elliptic integral is given by Eq. (5.10) Case of four Real Roots Four real roots corresponds to motion in the non-synchronous region. We consider first the case where S1> S 1 S2> S3>S4. In this case the solution of Eq. (5.5) is S2(S1 - S3) - (S1- S1 - S2)sn2[(t -')/4g],(5.12) S -S (1 - S2)sn[( )/4g ] 19

where g = 2(S1 - S3)(S2 - S4 (5-13) The! modulus kC of the Jacobi elliptic sine function is given by I2 (S1 S2)(S3 - S4) (5.14) (S1 - S3)(S2 - S4) The function sn2(x) is periodic in x with period 2 K, K being the complete elliptic integral of the first kind. It follows that S(t), and hence K(r), is a periodic function of t with period -1/2 T = 32K [(S1 - S3)(S2 - s41 (5.15) In the remaining case where S1 2 S2 _S3S S? S4 we find that S4(S1- S3) + Sl(S3 - S )sn2 (-T-)/4g,(516) S(T) =35 -1 3 + (S3 - S4)s[(t - t)/4g] where g is given by Eq. (5.13) and where the modulus KC of theJacobi elliptic sine function is given by Eq. (5.14). It follows that S(T) is a periodic function of " with period given by Eq. (5.15), We have now. de~termined the function S(z) in the 20

synchronous and non-synchronous regions. A knowledge of the function S(t) immediately determines K(Z), V(t), and cos. @(T) These latter functions allow us to find the explicit time dependence of O(T) and kU(T). We shall not attempt to do this since the resulting expressions would add little to our understanding of the motion. The s~ignificant point is that K(t) and V(t) are strictly periodic functions of Z. In the next section we shall estimate the maximum fluctuation in K(t) 6. BEHAVIOR UNDER RESONANCE CONDITIONS Exact resonance occurs when the particle traverses a single period of the sinusoidal field in one cyclotron period. This exact resonance is nearly fulfilled when kV/mco = 1. (6.1) If we substitute condition (6.1) into Eq. (2.7) and neglect the first order term we find that k2K/m0o = (k2h/mwo) - (1/2). (6.2) Equations (4.5), (4.6), (641), and (6.2) allow! us to- find a value of the constant C which corresponds to near resonance. The appropriate value of C is found to be C = d + 1/2, (6.3) 21

where d = kZh/m(oo. With this value of C the quartic in Eq. (5.3) becomes S4 + (4 - 8d)S3 + (6 - 24d + 24d2)S2 (6.4) + (4 - 24d + 48d2 - 32d3 - 16e2)S + 1 - 8d + 24d2 - 32d3 + 16d4. The roots of this quartic are approximately as follows. S1,2 = (2d - 1) + 2(2d - 1)1/4 1/2 + O(E), (65) 1l,2 1) s + o(-), (6.5) S3,4 = (2d - 1) + i2(2d -1) /4l/2 + O(E). (6.6) Since two roots are complex it follows- that the trajectory corresponding to condition (6. 3) lies in the synchronous region. Now, by definition, d = k2h/mo = k2H/mw2o (6.7) where H is the energy. Thus 2d - 1 = 2/m)[H - (m2 /2k2/m) (6.8) = (2k2/mO) t(mv2 /2) + (p2 /2m) (m)o2/2k2)+ O(e) where v is the initial transverse velocity and Pz is the 22

initial longitudinal momentum. According to Eq. (6.1) pz~ = mo /k + 0(s). (6.9) 0 0 Upon substituting Eq. (6.9) into Eq. (6.8) we find that 2d - 1 = k vf /o/ + 0(E). (6.10) We now: define the relative fluctuationAK in K as follows~: 2(max - Kmin) 2(S1 - S2) (Kmax + Kmin) (S1 + S2) Upon making use of equations (6.5), (6.10), and (6.11) we find that A K = 4(kv, /o0-3e/ 2 + 0(s). (6.12) If we denote the initial value of K by Ko, then vo and Ko are related by the expression im2v/2 = Koo + O(s). (6.13) It follows that kv/0'o = k(2Ko/mo0)l/2 + 0(s) (6.14) 23

where Ro is 2 7v times the ratio of the initial cyclotron radius to the wavelength of the disturbance. Upon substituting Eq. (6.14) into Eq. (6.12) we find that K = 4Ro3/2 /2 +0(). (6.15) Thus the relative fluctuation in K under resonance conditions depends not only on C but also on the ratio of the cyclotron radius. to the wavelength of the periodic disturbance. This dependence of the relative change in K upon Ro is not surprising. An increase of the wavelength of the disturbance requires a corresponding increase in the longitudinal particle velocity in order to achieve resonance. Associated with the increase in the longitudinal velocity is a decrease in the transverse velocity and hence a decrease in absolute value of K. This decrease in absolute value contributes to the increase in the relative change. The ideas developed above are best illustrated through an example. We first observe from Eq. (6.15) that relative fluctuations of order unity will occur when R0 (16E) (6.16) Now consider the trajectories plotted in Figure 1. These trajectories correspond to a 50 Mev proton moving in a disturbance whose wavelength is 108 cm (1000 km) and to a field strength ratio s =.01. It follows from Eq. (6.16) that, for s =.01, relative fluctuations of order unity 24

will occur when Ro = (.16)1/3 =.543 (6.17) lies in the resonance region. However, it is clear from Figure 1. that Ro =.543 lies well outside the resonance region. Thus K(t) for a 50 Mev proton moving in a periodic disturbance whose wavelength is 1000km undergoes only relatively small fluctuations. Let us now increase the wavelength of the disturbance to 1.5x108 cm (1500 km) and hold the other parameters fixed. The trajectories in the a-b plane for this situation are plotted in Figure 2. Inspection of Figure 2. reveals that Ro =.543 lies in the resonance region. Thus, as is evident from Figure 2., K(t) for a 50 Mev proton moving in a periodic disturbance whose wavelength is 1500 km can undergo relative fluctuations of order unity. Here we have graphic evidence of the influence of Ro on the relative fluctuation of K(t). We shall conclude this section with a few remarks relating our results to previously published work. The average orbital magnetic moment It(V) is related to K(z) by the equation L(T) = eK(T) /mc. (6.18) Therefore, what has been said above about K(r) also holds 25

-1.0 I.0 a -1.0 Figure 2. Trajectories of a 50 Mev proton in the a-b plane. k = 4.19x10-8 cm-1, Co = 370 rad/sec, ~ =.01. 26

for I(I). This means that~ vt(t) is a periodic function of;. This result was not obtained in either the work of Dragt or that of Wentzel. Furthermore,the relative fluctuation in tL(t) (that is: the fluctuation measured with respect to the mean value of 11) depends upon the ratio of the cyclotron radius to the wavelength of the periodic disturbance. This. fact was not appreciated by Dragt or Wentzel. For example, Dragt 8 concludes that, when wo = 370rad/sec and C 4.01, the orbital magnetic moment of a 160 Mev proton will be significantly effected by a disturbance having a wavelength of 1000 km. However, we have shown in Figure 1. that, under the some conditions, the magnetic moment of a 50 Mev proton is only slightly effected by a 1000 km wave. The resonant cyclotron radius of a 160 Mev proton is- greater than that of a 50 Iiev proton. It follows, therefore, from Eq. (6.15) that the effect of a 1000 km wave on a 160 Mev proton will be smaller than its- effect on a 50 Mev proton. Thus, we are forced to conclude that a 1000 km wave will not appreciably effect the orbital magnetic moment of a 160 Mev proton, under the conditions stated above. It is apparent from the above discussion that the interaction between a charged particle and a weak, spatially periodic magnetic field is considerably different from that. envisioned by Dragt and Wentzel. This means that their explanations of the origin of the distribution of protons" in the inner Van Allen belt can not be considered satisfactory. It does not mean that their work is without merit. 27

The work of Dragt is especially noteworthy since he makes substantial progress in accounting for the proton distribution in the inner Van Allen belt via heuristic arguments. What is evident from all of this is that the origin of the distribution of protons in the inner Van Allen belt must be carefully re-examined in view of the true nature, of the interaction between the protons and spatially periodic magnetic fields. Until this is done the origin of the proton distribution in the inner Van Allen belt must remain an open question. 7. CONCLUSION We have shown that the perturbation theory presented in I yields a complete first order solution for the motion of a charged particle in a constant magnetic field on which is superimposed a weak, spatially periodic magnetic field. The significant result from a physical viewpoint is the periodic behavior of the secular motion. It is hoped that we have succeeded in showing that this periodic behavior of the secular motion gives rise to serious objections to previous work concerning the stability of protons in the inner Van Allen belt. 28

Footnotes Supported in part by the National Science Foundation. +Present address: E. G. and G. Inc. Boston, Mass. 1. T. P. Coffey and G. W. Ford, J. Math. Phys. —. 2. A. J. Dragt, J. Geophys. Research, 66, (1961). 3. D. Wentzel, J. Geophys. Research, 66, (1961) 4. Dragt, op cit., p. 1647. 5. P. Fo Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), Eq. (259.00), p. 133. 6. Byrd and Friedman, op cit., Eq. (256.00), p. 120. 7. Byrd and Friedman, op cit., Eq. (252.00), p. 103. 8. Dragt, op cit., p. 1646. 29

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