[45 ]
ON THE SCATTERING OF FAST NEUTRONS BY PROTONS*
BY A. PAIS
Communicated by P. A. M.
DIRAC
Received 18 June 1945
1. Introduction. Recent experiments by Amaldi arid his collaborators on the scattering of high-energy neutrons (of 10—15MeV.) by protons (2) have disclosed a considerable anisotropy in the angular distribution of the scattered particles. Theoretical
discussions of this problem show an interesting feature in that the results depend
sensitively on the basic assumptions involved with regard to the charge dependence
of the neutron-proton interaction. This can be seen in particular from calculations by
Rarita and Schwinger(3) and by Ferretti(4). The former authors started from the
assumption of a distance dependence of this interaction represented by a square well,
while the angular and spin dependence included terms of the axial dipole type. If the
charge dependence was further assumed to be of the ' symmetrical' type, they found
a value for the anisotropy in strong disagreement with experiment, whereas the total
cross-section agreed with the measured value; a 'neutral' theory, on the other hand,
yielded agreement as regards anisotropy, but a total cross-section too large by a factor
of the order of 1-5. Ferretti investigated the scattering on Bethe's neutral meson
theory (5) and found satisfactory agreement with regard to both angular distribution
and total cross-section. It should be stressed that all calculations mentioned were
performed in the approximation in which only the contributions of the S- and
P-waves are considered.
From the point of view of the 'mixed' theory of Moller and Rosenfeld(6) the scattering has been treated by Ferretti (7) and Hulthen(8); these authors have put it beyond
doubt that Amaldi's results are irreconcilable with the symmetrical form of this theory!.
* The present paper was completed at the end of 1943, but its publication was then withheld
on account of the unacceptable restrictions imposed by the Germans on the choice of the language
to be used in scientific work in Holland.
The variational method here expounded by Pais may be regarded, from a systematic point of
view, as a special case of a more general one, which was developed independently by Hulth6n(l)
some time later; this circumstance does not, in my opinion, render its publication superfluous.
In a recent work, which will probably be published soon, I propose a modification of the nonstatic interaction operator of the symmetrical mixed theory, as a result of which no first order
perturbation of the energy of the deuteron ground state exists. The use of this modified operator
would thus cause a corresponding change of certain constants appearing in the numerical calculations of §4 below. In fact, the values taken over from Hulthe'n's work should be replaced by
those adopted by Ferretti; but Hultheii's S-phases would, moreover, have to be recomputed.
For this last reason, and since the whole numerical work of this paper can have only illustrative
significance, it has not been thought worth while to change anything in the text (except for a
few additional remarks).
L. ROSBNFELD
Utrecht, 19 May 1945
f This, of course, does not affect the general argument of Moller and Rosenfeld concerning the
inadmissibility of a spin-orbit coupling term in the potential of 'static' nuclear interaction.
46
A. P A I S
Moreover, it was pointed out by Hulthen that, in the energy region concerned, the
S- and P-approximation is too rough, and that higher phases must be taken into
account. It would therefore be of interest to find a suitable quick way of determining
these phases. It is the aim of this note to describe such a method for computing P- and
higher phases which holds generally in all cases in which the average interaction
potential between the scattered particle and the scattering centre is small compared
with the influence of the centrifugal force.
I shall apply the general considerations to two cases in which the radial interaction
is given by Yukawa's potential, namely, the symmetrical mixed theory just mentioned,
and the neutral mixed theory, according to which the nuclear forces are described by
a particular combination of neutral scalar and pseudovector mesonfields(see (6), p. 34).
The latter case is only of formal interest, however. Indeed, it appears from as yet
unpublished investigations by Prof. L. Rosenfeld that the neutral mixed theory cannot
be maintained as a possible description of nuclear phenomena, essentially because it
does not yield a quadrupole moment of the deuteron.
2. Method of determining the phases. If the radial wave function r-^V^r), corresponding to a value I of the azimuthal quantum number, satisfies an equation of the
type
x
Yl = r*Jl+i(kr) is a suitable solution if V = 0 (J is a Bessel function of the first kind).
For such cases in which the influence of V is small compared with that of the term
1(1 + 1) r~2, we now try to describe this 'perturbation' by making the following assumption for Y,:
(2)
Y, = riJ[+i+Al(kr),
where A, is to be determined by variational methods. The occurrence of V is thus marked
by a 'shift of order' of the Bessel function. This assumption is quite appropriate to
our problem, as (2) becomes asymptotically
Thus, if Aj is known, the phase 7)l of the wave is determined by
T)l = - \n\.
(3)
It will be clear beforehand that (2) will not apply to iS-waves, as in this case the condition ' V small compared with the centrifugal force' does not apply. Generally we
should expect (2) to hold the better the larger I is.
We fix \ by
r
Inserting (2), we get(9)
provided that 21 + 2A; + 1 > 0, and that the integral in the second term of (4) converges.
Under these conditions (4) holds for any potential Vt(r).
On the scattering of fast neutrons by protons
47
We now apply (4) to the case of the mixed theories. Comparing the 'non-static
interaction'* with the 'static' neutron-proton potential it can be seen(7,8) that, as
far as its role as scattering field is concerned, the first interaction may be neglected for
energies which are of interest at presentf. The radial wave function describing the
motion relative to the centre of gravity of the neutron-proton system therefore satisfies
an equation of the type (1) in which M now denotes the mass of the nucleons (the
mass difference of the proton and neutron is neglected) and
¥• =
\ME0lh2.
Here Eo is the kinetic energy of the incoming neutron. V also depends on the relative
position of the spins. We have
00
^
where s = 1(0) denotes triplet (singlet) states, Mm is the meson rest mass; the spin
dependence is involved in the coefficient bls whose numerical value depends on the
magnitude of the nuclear binding constants. Hence (4) becomes (10)
•\
J = 0, a = —,
(6)
where Q is a Legendre function of the second kind:
i
( n + l ) ( » + 2) 1
2.(2n + 3)
(n+l)(n + 2)(n + 3)(n + 4) 1
2.4.(2» + 3)(2» + 5) x*
"1
J
According to (6), A,g->0 for a->0 as well as for a-^-ooJ, for all l> 0.
3. The angular distribution. The differential cross-section of the process is
d
where
®
=
15*5 S Bn cos- 6dQ. = p S f$n cos" 6dQ,
C= § ,
(7)
y?n = |j>, 5 n = 4 ' + 3 ^ .
^ is the angle of deviation in the centre of gravity system. Asn is the same function of
the singlet phases as A% is of the triplet phases. Considering the approximation in
which the v up to I = 5 are taken into account, we have (some of the numerical coefficients have suitably been rounded ofiF)
Ao = 4sin2v/0// + 25sin 2 i/ 2 -5a 02 // + 46sin 2 ^ 4 + 7a 04 -17a 24 ,
Ax = 6a01 / - 15a12 / - 21a03 + 52aM / + 20a14 - 71a34 / + 41a05 - 103a25 + 140a45,
A2 = 36 sin2 ^ / - 150 sin21/2 + 15a02 / + 440 sin2 TJ3 - 63a13 j - 910 sin2 T/4 - 67aM
+ 220a24 / + 2000 sin2 7)5 + 124a15 - 430a35,
* Cf. (6), equation (116).
•f The influence of the non-static potential on the proton-neutron scattering has been discussed
by Hulth6n(8). In view of the above-mentioned work by Rosenfeld, this effect needs reconsideration, but the general conclusion about its order of magnitude will not be affected.
t For x^l we have(ll) Qn(a:)~log {l/(a;2- 1)}.
48
A. PAIS
A3 = 45a12 / + 35a,)3 - 245a23 / - 202a14 + 825a34 / - 190a05 + 7900^ - 2040a45,
At = 225sm 2 7 2 /-1470sm 2 %+105a 13 /+5600sin 2 T/ 4 +79a 04 -700a 24 /
- 18600 sin2 v/5 - 580a15 + 2740o35,
A5 = 262a 23 /+236a 14 -2000a 34 /+173a 05 -1880a 25
A6 = 1225sin 2 i/ 3 /-10-630sin 2 i/ 4 + 590a
A7 = 1380a34/+1380o25-13-430a45,
A8 = 6200 sin2 7ji f - 66-500 sin2 rj5 + 3040a35,
A9 = 6810a45,
A10 = 30-000 sin2 y5,
aik = 4 sin2 t}i sin2 7jk + sin 2i]i sin 2v/fc.
with*
The / signs separate the terms which have to be added to the foregoing in pushing the
approximation one step further.
We now define the ' angular anisotropy' A as the ratio of the intensities of the
scattered particles at 6 = n and 6 = \n:
A i = S ( - l ) - 5 n / 5 0 = S(-l)»/?»(8)
By using the well-known formula which expresses && in terms of the zonal harmonics
6), we get, remembering that
p (o)-(
iv
{2n)l
and that au = 4 sin2 7^
'(2m+l)(2n+l) (0^ + 3 0
A=
:
If A< 1 we have 'forward scattering', while A> 1 corresponds to 'backward scattering'.
I t is a characteristic feature of nuclear forces of the symmetrical type that they lead,
in any approximation for the energies of interest at present, to A > 1, as can be seen
in the following way. First consider the S-P approximation. Then, if, for ' triplet' as
well as for ' singlet' scattering, we have
ao^esin2^,
(9)
there will be backward scattering, as (9) is a sufficient condition for A> 1. For small
energies (9) becomes, remembering that i)1 < 0,
sin9/0(cosv/0-|v/1|sini/())> - f | ^ | ,
which is satisfied for the rj0 values concerned (even if ij0 = \n, we would still have
* In the case of attractive as well as repulsive ordinary forces, all aik > 0. Since (for Eg 4= 0)
cos 7/i cos 7/t< 1, the sign of aik is the same as that of sin 27]t sin 2t]k. We thus have in the case
of exchange forces (for both attraction or repulsion in iS-states) aik > 0 if both i and k are either
even or odd, aik < 0 in the other cases.
On the scattering of fast neutrons by protons
49
— I Vi I > — 11 Vi {)• To get an insight into the case of large energies, we note that (9)
can be written as
(9 a)
esin2^ 1 <sin 2 i; 1 ,
sin2%
|
k
^
I t will be seen in the next section that TJ1 > — \it, while 5/0 lies in the first quadrant.
Thus (9a) is satisfied. However, for large energies the #-P-approximation is insufficient,
but we know* that the neglect of rjl (I ^ 2) involves an underestimate of A. Thus there
is backward scattering for the energies concerned.
The situation is more complicated in the case of a neutral theory (yielding attractive
forces) with 7i>0. Again consider the S-P approximation; a sufficient condition for
forward scattering is a01 > 6 sin2 T\X which, for small energies, gives
sin 1/O(cos 7j0 + i/i sin i)0) > f ^
This is satisfied, because for these energies VolVi^ 1- ^ o r large energies this relation
cannot be employed. Instead, we get again (9 a) as a sufficient condition for backward
scattering, which can be fulfilled if e is sufficiently small. But if (9a) is only satisfied
for either triplet or singlet phases, A may still be greater than 1 for a suitable ratio
between the singlet and triplet phases. Thus, for large energies a neutral theory may,
in the approximation considered, lead to backward scattering. Of course this approximation is not sufficient in the energy region considered, and the example of a neutral
theory, which we will discuss in § 4, will indeed illustrate the fact that, although A is
slightly larger than 1, if we restrict ourselves to the inclusion of ?/(P) only, it becomes
less than 1 if higher phases are taken into account, in accordance with the circumstance
that the neglect of r\ (I ^ 2) leads to an overestimate of A. Still, the result just obtained
on the #-P-approximation deserves some interest in itself, as it casts additional doubt
on the applicability of this approximation on which many theoretical discussions have
been based.
4. Numerical evaluations, (a) Symmetrical mixed theory. I have computed the
phases for the neutron energy and meson mass chosen by Ferretti (Eo = 13-5 MeV.,
M^ = 177me, a = 0-762) and by Hulthen (Eo = 14-5MeV., J 4 , = 200me, a = 0-64).
Ferretti takes for bla (see (5)) the values that result from (6), equations (101)-(105),
namely,
(10)
{y
6W = - ( l - 2 ( - 1)') 1-69,
whereas Hulthen has corrected the nuclear binding constants occurring in the static
interaction for the influence of the non-static potentialf. This changes the coefficient
0-107 in (10) to about half its value, and leads therefore to a decrease of the triplet
phases. Furthermore, this author replaces the coefficient 1-69 by 1-50 which again
results in a decrease in the rfa. In order to discuss the reliability of the method of phase
* This can be seen by estimating the contribution of 7/ (l>2) by means of the Born-approximation, cf. §4 and (8).
"f See, however, footnote * on p. 45.
PSP42, I
4
50
A. PAIS
calculation developed here, we have, in calculating the phases for the respective energies,
used the same 6-values as the mentioned authors, arriving at the following results:
Eo = 13-5 MeV., Mm = Yllm,
I
7}i (triplet)
0
1
2
3
4
5
1-52
-0-08
0-071
-0-007
0-007
-0-0008
(-0-062)
(0-055)
(-0-0054)
(0-0054)
(-0-0006)
Tji (singlet)
0-9
-0-36
0-042
-0-035
0-0039
-0-0039
(-0-33)
(0-038)
(-0-031)
(0-0035)
(-0-0035)
Eo =14-5 MeV., Mm = 200me
Tji (triplet)
7li (singlet)
1-28
-0055
0-042
-0-0037
0-0030
- 0-0003
0-83
-0-29
0-030
-0024
0-0022
-0-0018
As was mentioned above, the present method does not apply to I = 0. The values of the
$-phases, given here, have been taken over from the papers cited. For EQ = 13-5 MeV.,
Mn = 177me, we have indicated between brackets the T/'S obtained with the 6-values
corresponding to those of Hulthen. Apart from the causes previously mentioned, the
larger values of the phases found for 13-5MeV., compared with those at 14-5MeV.,
are essentially due to the larger range of the nuclear forces employed in the former case.
An appropriate check is provided by the comparison of the 1 P-phase, found here,
with that computed by Hulthen by numerical integration. This author has obtained
for rj{xP), from which originates the main contribution to the anisotropy, a value of
— O28x with which our value of —0-29 is in very good agreement. ij(3P) has been
calculated by Hulthen by expanding TJ in a series in b and including the term in b2.
His value of — 0-0552 practically coincides with the present result.
In calculating the 7}(l^2) which are small compared with TJ(P), (6) can be simplified.
Since A <| I and since Qn(x) varies, for given x and for intervals of n concerned, very
slowly with n, we may write
/,
/
i
that is, using (3),
(11)
which is the result that is obtained by the Born approximation. Using (11) instead
of (6) gives for tj^D) an error of 4 %. The agreement which was found with regard to the
P-phases justifies the application of (11), which was also used by Hulthen in this case,
to the higher phases. I t may be remarked that, if higher energy neutron sources
become available, and it becomes of interest to compute the phases for the corresponding energy regions, we shall still have in (6) a sufficiently accurate formula for
obtaining the y's, while the applicability of (11) might become questionable.
Ferretti has,found that T/( 1 P) = — 0-45, which, in absolute value, is larger by a factor
1-25 than our result. However, this author has computed r/(P) by essentially applying
the Born approximation to P-waves*, and this method generally leads to an overestimate of the absolute value of the phases of states in which the neutron-proton force
is repulsive, (b < 0), as can be seen by comparing (6) with (ll)t* Cf. (7) equation (6).
t Qi+b(x) decreases with increasing A. For positive b the situation is therefore the reverse.
Cf. also (8), pp. 10 and 15.
On the scattering of fast neutrons by protons
51
We can now compute the anisotropy defined by (8); A is directly comparable with
the experimental results of Amaldi, according to which A ~ 0-50 for Eo = 13-5MeV.
The following values are obtained for A and for the total cross-section 0 :
Highest I
included
0
1
2
3
4
5
Eo = 14-5 MeV ., Mm = 20O?ne
Eo= 13-5 MeV ., Mm = 177me
A
24
<B in 10~ cm.
1
1-51
2-06
2-43
2-52
2-58
0-71
0-81
0-82
0-82
0-82
0-82
2
A
1
1-47
1-95
215
219
2-21
$ in lO"24 cm. 2
0-60
0-65
0-66
0-66
0-66
0-66
While, as regards O, it is obviously sufficient to go only as far as I = 1, the convergence
of A is, in accordance with Hulthen's result, much slower. By means of extrapolation
one can obtain an estimate of the exact value of A, from which it is inferred that the
.D-, F-, ^-approximations involve an error of about 15-20, 6 and 3 % respectively.
It is gratifying to note the very good agreement of O, as computed with Hulthen's
parameters, with the experimental values of 0-70 and 0-66 x 10~24 cm.2 for average
neutron energies of 14 and 15MeV. respectively (12).
(b) Neutral mixed theory. Here the static interaction operator is of the form
As the values of Gx and G2 are fixed by means of experimental data concerning the
3
S- and ^-states of the deuteron, 60 s, and thus the phases for ^-states, are the same in
the neutral as in the symmetrical theory (at any rate if we disregard any eventual
influence of the non-static potential). Indicating the 6's of the symmetrical and the
neutral theory by 6 s and bN respectively, we have
bftl = i [ l - 2 ( - l)'+i]6ft, bfr0 = - [ 1 - 2 ( - 1)']6£.
(12)
Using the same two sets of parameters as in the symmetrical case (cf. footnote * on p. 45)
the following results have been obtained:
Eo = 13-5 MeV -,
'
0
1
2
3
4
5
Vi.i
1-52
0-29
0071
0021
0-007
0-0024
Mm = mm.
V',0
0-90
0-17
0-042
0-012
0-0039
0-0013
Eo = 14-5 MeV ., Mm = 200m,.
V'.i
1-28
0-232
0-042
0011
0003
0-00009
Vt.o
0-83
0-144
0-030
0-008
0-002
0-0006
The S, D, G, ... phases are the same as in the symmetrical theory (cf. (12)). The phases
for I = 3,5,... are obtained from the corresponding ones in the symmetrical case by
remembering that these ?/'s are proportional to b (cf. (11)), and thus are directly found
by means of (12); rj{l) has again been calculated with the help of (6). All the T/'S are
52
A. PAIS
positive, which is characteristic of (attractive) Wigner forces. Contrary to the symmetrical theory, where, due to the properties of T*1* . T<2), rj is, in the triplet (singlet)
case, larger (smaller) than ?} in the singlet (triplet) case if I is even (odd), here all the
triplet v's are larger than the singlet i/'s.
For A andO we get the following results:
Highest I .
included
0
1
2
3
4
Eo = 13-5 MeV., Mm = Him,
Eo = 14-5 MeV., Mm = 200me
2
A
O in 10-" cm.
A
<J> in- 10~21 cm. 2
1
1-04
0-76
0-87
0-86
0-71
0-87
1
1
0-98
1
0-99
0-60
0-70
0-70
0-70
0-70
•0-89
0-89
0-89
A varies in a more complicated way with the approximation used than in the symmetrical case and, for Eo = 13-5 MeV., exhibits the feature, to the possibility of which
we already drew attention in § 3, of being greater than 1 in the $-P-approximation.
The marked quantitative difference of A(S — P) here and in the neutral theories of
Rarita and Schwinger and of Bethe*findsits origin essentially in the difference in the
angular and radial dependence of the various interaction potentials.
5. Conclusion. The experimental evidence is in favour of forward scattering, and,
as was mentioned in § 1, is in good quantitative agreement with the neutral theories of
Rarita and Schwinger and of Bethe (especially with the former one) if the $-P-approximation is used. In these theories we have to deal with other types of radial and angular
dependence of the static nuclear forces than in the mixed theories. Therefore, we may
not conclude from the above that, in these cases, the relative contributions of the v's
for I Js 2 will be as large as in the mixed cases, but anyhow they will be of considerable
importance; the neglect of higher than P-phases is therefore hot justified. Even apart
from the general objections which can be raised against any neutral theory of nuclear
forces, it remains, therefore, to be seen whether the experiments are quantitatively
reconcilable with any of the current theories at all.
I am much indebted to Prof. L. Rosenfeld for valuable discussions.
APPENDIX
The energy loss of fast neutrons in travelling through a hydrogenic substance depends
on the anisotropy in the following way. Let w(x) dx be the probability that the energy
of the neutron in one collision is reduced to x times its original value. It is easily seen
that
- l)ndx,
w(x)dx = y
n=0
* A survey of the theoretical results is given in (8), Table 3.
On the scattering of fast neutrons by protons
For the average value x of x we find
,
53
a=
and, for the average of log a;,
D
2
T>
"
-1
-"2n+i
The following table gives the values of a and log a; on the symmetrical and the neutral
mixed theory, using Ferretti's values of the various parameters.
Symmetrical
Neutral
Highest I
included
a
log*
a
log a;
0
1
2
3
0
- 0-046
- 0-073
-0-07 7
-1
-M
-119
-1-23
0
017
0-23
0-23
-1
-0-80
-Q-72
-0-72
The /S-approximation yields the well-known results of Fermi (Bic. Sci. 7 (1936), 13).
The stationary energy distribution of neutrons emitted by a point source and travelling through an infinite block of' paraffin' has been discussed by Ageno (II Nuovo dm.
1 (1943), 41), assuming the differential cross-section to be proportional to l + 6cos0.
In view of the results obtained here, it might be of some interest to consider this distribution for the general case of (7). The general theory seems quite difficult, however,
as the knowledge of the dependence of the Bn on the energy of the incoming neutrons
is required. If we assume the Bn to be energy-independent (which for high energies
seems justified, as remarked by Ageno), the integro-differential equation for the
stationary distribution N(E) of the neutrons can be solved to any approximation I.
Let N®(E) be the expression for N(E) if the phases up to the Ith are included, let Eo
be the energy of the neutrons emitted by the source and let x = E/Eo. We find
= Z®(x)N<*>\E),
Z®{x) = 2 7/ia;fci+1.
with
i=0
The kt are the 21 +1 roots of
22!
i=0
a (?)
^
1
= 0,
'
T
l
2l
B
S
n=
3= 0
whereas the t)i are obtained from the following set of 21 + 1 linear equations
i+S
= 0 (n = 0,!,...,»).
We have Z^x) = 1; in this case we get Fermi's formula for isotropic scattering. If
we put BJB0 = b, Ageno's formula, loc. cit. equation (8), for the case dO ~ 1 + b cos d
is easily found from our result.
54
A. PAIS
REFERENCES
(1) HOT/THEN, L. Fysiogr. Sallsk. Lund Forhandl. 14 (1944), no. 21.
(2) AMALDI, E., BOCCIARELLI, D., FERRETTI, B. and TRABACCHI, G.
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
Naturwissenschaften,
48 (1942), 39.
RARITA, W., SCHWINGER, J . and NYE, H. A. Phys..Rev. 59 (1941), 201.
RARITA, W. and SCHWINGER, J . Phys. Rev. 59 (1941), 436.
FERRETTI, B. Ric. Sci. (1941), 843, 993.
BETHE, H. A. Phys. Rev. 57 (1940), 260, 390.
MOLLER, C. and ROSENFELD, L. Det Kgl. Danske Vid. Selsk. Mat. Fys. Medd. 17 (1940), 8.
FERRETTI, B. II Nuovo Cim. 1 (1943), 25.
HUM-HEN, L. Ark. Mat. Astr. och Fys. 29A (1943), no. 33; 30A (1943), no. 9.
WATSON, G. N. Theory of Bessel functions (Cambridge 1922), p. 403.
See reference (9), p . 389.
Cf. WHITTAKER, E. T. and WATSON, G. N. Modern analysis, 4th ed. (Cambridge 1935),
pp. 299 (Ex. 18) and 317.
SALANT, E. and RAMSEY, N. Phys. Rev. 57 (1940), 1075.
AGENO, M., AMALDI, E., BOCCIAREIXI, D. and TRABACCHI, G. II NUOVO Cim. 1 (1943), 253.
INSTITUUT VOOR THEORETISCHE NATUTTRKTJNDE
RIJKS-UNIVERSITEIT, UTRECHT
ON MATHISSON'S VARIATIONAL EQUATION OF
RELATIVISTIC DYNAMICS
B Y S. SHANMUGADHASAN
Communicated by P. A. M. DIRAC
Received 17 April 1945
Mathisson(4) has derived the variational equation of relativistic dynamics for a noncomplete system (with the necessary modifications when the energy-tensor becomes
infinite on the world line) to be of the form
dxdJfi + mWdxdfldJ/1+...]ds
= 0,
where E,a is an arbitrary vector field vanishing, together with all its derivatives, at the
limits sx and s2. In this equation the infinite sequence of the m tensors are characteristic
of the physical system while the X's (calculated from the distribution of the energytensor Taf over the surface of the world tube) are characteristic of the forces. The
variational equation is said to be solved when the system of relations between the m's,
the X's and the path L, compatible with the variations allowed to the £'s, is found.
Mathisson(5) has given the method of solution and has applied it to the solution of a
specialized equation. The present paper consists of an investigation of the solution of
the general variational equation using the methods given in Mathisson's papers. The
solution discussed can be applied to the general theory of spinning particles.